Split (1)

train · 2.77k rows

text stringlengths 644 10k
rice effects include income effects, they need not be symmetric. That is, @xi/@pj does not necessarily equal @xj/@pi. PROBLEMS • Focusing only on the substitution effects from price changes eliminates this ambiguity because substitution effects are symmetric; that is, @xc j =@pi. Now two goods are deﬁned as net (or Hicksian) substitutes if @xc i =@pj < 0. Hicks’ ‘‘second law of demand’’ shows that net substitutes are more prevalent. i =@pj > 0 and net complements if @xc i =@pj ¼ @xc • If a group of goods has prices that always move in unison, then expenditures on these goods can be treated as a ‘‘composite commodity’’ whose ‘‘price’’ is given by the size of the proportional change in the composite goods’ prices. • An alternative way to develop the theory of choice among market goods is to focus on the ways in which market goods are used in household production to yield utility-providing attributes. This may provide additional insights into relationships among goods. 6.1 Heidi receives utility from two goods, goat’s milk (m) and strudel (s), according to the utility function Þ ¼ a. Show that increases in the price of goat’s milk will not affect the quantity of strudel Heidi buys; that is, show that @s/@pm ¼ b. Show also that @m/@ps ¼ c. Use the Slutsky equation and the symmetry of net substitution effects to prove that the income effects involved with the 0. 0. & U m, s ð m s: derivatives in parts (a) and (b) are identical. d. Prove part (c) explicitly using the Marshallian demand functions for m and s. 6.2 Hard Times Burt buys only rotgut whiskey and jelly donuts to sustain him. For Burt, rotgut whiskey is an inferior good that exhibits Giffen’s paradox, although rotgut whiskey and jelly donuts are Hicksian substitutes in the customary sense. Develop an intuitive explanation to suggest why an increase in the price of rotgut whiskey must cause fewer jelly donuts to be bought. That is, the goods must also be gross complements. Chapter 6: Demand Relationships among Goods 201 6.3 Donald, a frugal graduate student, consumes only coffee (c) and buttered toast (bt). He buys these items at the university cafeteria and always uses two pats of butter for each piece of toast. Donald spends exactly half of his meager stipend on coffee and the other half on buttered toast. a. In this problem, buttered toast can be treated as a composite commodity. What is its price in terms of the prices of butter (pb) and toast (pt)? b. Explain why @c/@pbt ¼ c. Is it also true here that @c/@pb and @c/@pt are equal to 0? 0. 6.4 Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by utility = b p, t & & where each letter stands for miles traveled by a speciﬁc mode. Suppose that the ratio of the price of train travel to that of bus travel (pt/pb) never changes. a. How might one deﬁne a composite commodity for ground transportation? b. Phrase Sarah’s optimization problem as one of choosing between ground ( g) and air ( p) transportation. c. What are Sarah’s demand functions for g and p? d. Once Sarah decides how much to spend on g, how will she allocate those expenditures between b and t? 6.5 Suppose that an individual consumes three goods, x1, x2, and x3, and that x2 and x3 are similar commodities (i.e., cheap kp3, where k < 1—that is, the goods’ prices have a constant relationship to one and expensive restaurant meals) with p2 ¼ another. a. Show that x2 and x3 can be treated as a composite commodity. b. Suppose both x2 and x3 are subject to a transaction cost of t per unit (for some examples, see Problem 6.6). How will this transaction cost affect the price of x2 relative to that of x3? How will this effect vary with the value of t? c. Can you predict how an income-compensated increase in t will affect expenditures on the composite commodity x2 and x3? Does the composite commodity theorem strictly apply to this case? d. How will an income-compensated increase in t affect how total spending on the composite commodity is allocated between x2 and x3? 6.6 Apply the results of Problem 6.5 to explain the following observations: a. It is difﬁcult to ﬁnd high-quality apples to buy in Washington State or good fresh oranges in Florida. b. People with signiﬁcant babysitting expenses are more likely to have meals out at expensive (rather than cheap) restaurants than are those without such expenses. c. Individuals with a high value of time are more likely to ﬂy the Concorde than those with a lower value of time. d. Individuals are more likely to search for bargains for expensive items than for cheap ones. Note: Observations (b) and (d) form the bases for perhaps the only two murder mysteries in which an economist solves the crime; see Marshall Jevons, Murder at the Margin and The Fatal Equilibrium. 6.7 In general, uncompensated cross-price effects are not equal. That is, @xi @pj 6¼ @xj @pi : Use the Slutsky equation to show that these effects are equal if the individual spends a constant fraction of income on each good regardless of relative prices. (This is a generalization of Problem 6.1.) 202 Part 2: Choice and Demand 6.8 Example 6.3 computes the demand functions implied by the three-good CES utility function U x, y. Use the demand function for x in Equation 6.32 to determine whether x and y or x and z are gross substitutes or gross complements. b. How would you determine whether x and y or x and z are net substitutes or net complements? Analytical Problems 6.9 Consumer surplus with many goods In Chapter 5, we showed how the welfare costs of changes in a single price can be measured using expenditure functions and compensated demand curves. This problem asks you to generalize this to price changes in two (or many) goods. a. Suppose that an individual consumes n goods and that the prices of two of those goods (say, p1 and p2) increase. How would you use the expenditure function to measure the compensating variation (CV) for this person of such a price increase? b. A way to show these welfare costs graphically would be to use the compensated demand curves for goods x1 and x2 by assuming that one price increased before the other. Illustrate this approach. c. In your answer to part (b), would it matter in which order you considered the price changes? Explain. d. In general, would you think that the CV for a price increase of these two goods would be greater if the goods were net sub- stitutes or net complements? Or would the relationship between the goods have no bearing on the welfare costs? 6.10 Separable utility A utility function is called separable if it can be written as U1ð x where Ui0 > 0, Ui00 < 0, and U1, U2 need not be the same function. x, y ð Þ ¼ U Þ þ U2ð y , Þ a. What does separability assume about the cross-partial derivative Uxy? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to conclude deﬁnitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb–Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter. 6.11 Graphing complements Graphing complements is complicated because a complementary relationship between goods (under Hicks’ deﬁnition) cannot occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more) goods. In his review of complementarity, Samuelson provides a way of illustrating the concept with a two-dimensional indifference curve diagram (see the Suggested Readings). To examine this construction, assume there are three goods that a consumer might choose. The quantities of these are denoted by x1, x2, and x3. Now proceed as follows. a. Draw an indifference curve for x2 and x3, holding the quantity of x1 constant at x0 1. This indifference curve will have the customary convex shape. b. Now draw a second (higher) indifference curve for x2, x3, holding x1 constant at x0 h. For this new indifference curve, show the amount of extra x2 that would compensate this person for the loss of x1; call this amount j. Similarly, show that amount of extra x3 that would compensate for the loss of x1 and call this amount k. c. Suppose now that an individual is given both amounts j and k, thereby permitting him or her to move to an even higher x2, x3 indifference curve. Show this move on your graph, and draw this new indifference curve. d. Samuelson now suggests the following deﬁnitions: • If the new indifference curve corresponds to the indifference curve when x1 ¼ • If the new indifference curve provides more utility than when x1 ¼ x0 1 % x0 1 % 2h, goods 2 and 3 are complements. 2h, goods 2 and 3 are independent. 1 % Chapter 6: Demand Relationships among Goods 203 • If the new indifference curve provides less utility than when x1 ¼ graphical deﬁnitions are symmetric. x0 1 % 2h, goods 2 and 3 are substitutes. Show that these e. Discuss how these graphical deﬁnitions correspond to Hicks’ more mathematical deﬁnitions given in the text. f. Looking at your ﬁnal graph, do you think that this approach fully explains the types of relationships that might exist between x2 and x3? 6.12 Shipping the good apples out Details of the analysis suggested in Problems 6.5 and 6.6 were originally worked out by Borcherding and Silberberg (see the Suggested Readings) based on a supposition ﬁrst proposed by Alchian and Allen. These authors look at how a transaction charge affects the relative demand for two closely substitutable items. Assume that goods x2 and x3 are close substitutes and are subject to a transaction charge of t per unit. Suppose also that good 2 is the more expensive of the two goods (i.e., ‘‘good apples’’ as opposed to ‘‘cooking apples’’). Hence the transaction charge lowers the r
elative price of the more expensive good =@t > 0 [i.e., (p2 þ (where we use compensated demand functions to eliminate pesky income effects). Borcherding and Silberberg show this result will probably hold using the following steps. t)/(p3 + t) decreases as t increases]. This will increase the relative demand for the expensive good if @ 2=xc xc 3Þ ð a. Use the derivative of a quotient rule to expand @ b. Use your result from part (a) together with the fact that, in this problem, @xc =@t. 2=xc xc 3Þ ð i =@t that the derivative we seek can be written as @xc i =@p2 þ ¼ @xc i =@p3 for i = 2, 3, to show @ xc 2=xc 3Þ @t ð ¼ xc 2 xc 3 s22 x2 þ # s23 x2 % s32 x3 % s33 x3 , $ @xc i =@pj. where sij ¼ c. Rewrite the result from part (b) in terms of compensated price elasticities: ec ij ¼ @xc i @pj & pj xc i : d. Use Hicks’ third law (Equation 6.26) to show that the term in brackets in parts (b) and (c) can now be written as [(e22 – e23)(1/p2 – 1/p3) + (e21 – e31)/p3]. e. Develop an intuitive argument about why the expression in part (d) is likely to be positive under the conditions of this problem. Hints: Why is the ﬁrst product in the brackets positive? Why is the second term in brackets likely to be small? f. Return to Problem 6.6 and provide more complete explanations for these various ﬁndings. SUGGESTIONS FOR FURTHER READING Borcherding, T. E., and E. Silberberg. ‘‘Shipping the Good Apples Out—The Alchian-Allen Theorem Reconsidered.’’ Journal of Political Economy (February 1978): 131–38. Good discussion of the relationships among three goods in demand theory. See also Problems 6.5 and 6.6. Hicks, J. R. Value and Capital, 2nd ed. Oxford, UK: Oxford University Press, 1946. See Chapters I–III and related appendices. Proof of the composite commodity theorem. Also has one of the ﬁrst treatments of net substitutes and complements. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Explores the consequences of the symmetry of compensated cross-price effects for various aspects of demand theory. Rosen, S. ‘‘Hedonic Prices and Implicit Markets.’’ Journal of Political Economy (January/February 1974): 34–55. Nice graphical and mathematical treatment of the attribute approach to consumer theory and of the concept of ‘‘markets’’ for attributes. Samuelson, P. A. ‘‘Complementarity—An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory.’’ Journal of Economic Literature (December 1977): 1255–89. Reviews a number of deﬁnitions of complementarity and shows the connections among them. Contains an intuitive, graphical discussion and a detailed mathematical appendix. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Good discussion of expenditure functions and the use of indirect utility functions to illustrate the composite commodity theorem and other results. EXTENSIONS SIMPLIFYING DEMAND AND TWO-STAGE BUDGETING In Chapter 6 we saw that the theory of utility maximization in its full generality imposes rather few restrictions on what might happen. Other than the fact that net cross-substitution effects are symmetric, practically any type of relationship among goods is consistent with the underlying theory. This situation poses problems for economists who wish to study consumption behavior in the real world—theory just does not provide much guidance when there are many thousands of goods potentially available for study. There are two general ways in which simpliﬁcations are made. The ﬁrst uses the composite commodity theorem from Chapter 6 to aggregate goods into categories within which relative prices move together. For situations where economists are speciﬁcally interested in changes in relative prices within a category of spending (such as changes in the relative prices of various forms of energy), however, this process will not do. An alternative is to assume that consumers engage in a two-stage process in their consumption decisions. First they allocate income to various broad groupings of goods (e.g., food, clothing) and then, given these expenditure constraints, they maximize utility within each of the subcategories of goods using only information about those goods’ relative prices. In that way, decisions can be studied in a simpliﬁed setting by looking only at one category at a time. This process is called two-stage budgeting. In these Extensions, we ﬁrst look at the general theory of twostage budgeting and then turn to examine some empirical examples. E6.1 Theory of two-stage budgeting The issue that arises in two-stage budgeting can be stated succinctly: Does there exist a partition of goods into m nonoverlapping groups (denoted by r 1, m) and a separate budget ¼ (lr) devoted to each category such that the demand functions for the goods within any one category depend only on the prices of goods within the category and on the category’s budget allocation? That is, can we partition goods so that demand is given by xið p1, . . . , pn, I xi rð 2 for r 1, m? That it might be possible to do this is suggested by comparing the following two-stage maximization problem, Þ ¼ ¼ 2 pi r, IrÞ (i) ð p1, . . . , pn, I1, . . . , ImÞ max x1, . . . , xnÞ x1, ..., xn ð U " V , ¼ and s.t. r i X 2 pixi ) Ir, r ¼ 1, m # (ii) max I1, ..., Im V , s.t. M 1 r X ¼ I, Ir¼ to the utility-maximization problem we have been studying, n s.t. max xi U x1, . . . , xnÞ ð pixi ) I: (iii) 1 i X ¼ Without any further restrictions, these two maximization processes will yield the same result; that is, Equation ii is just a more complicated way of stating Equation iii. Thus, some restrictions have to be placed on the utility function to ensure that the demand functions that result from solving the twostage process will be of the form speciﬁed in Equation i. Intuitively, it seems that such a categorization of goods should work providing that changes in the price of a good in one category do not affect the allocation of spending for goods in any category other than its own. In Problem 6.9 we showed a case where this is true for an ‘‘additively separable’’ utility function. Unfortunately, this proves to be a special case. The more general mathematical restrictions that must be placed on the utility function to justify two-stage budgeting have been derived (see Blackorby, Primont, and Russell, 1978), but these are not especially intuitive. Of course, economists who wish to study decentralized decisions by consumers (or, perhaps more importantly, by ﬁrms that operate many divisions) must do something to simplify matters. Now we look at a few applied examples. E6.2 Relation to the composition commodity theorem Unfortunately, neither of theoretical the approaches to demand simpliﬁcation is completely satisfying. The composite commodity theorem requires that the relative prices for goods within one group remain constant over time, an assumption that has been rejected during many different historical periods. two available Chapter 6: Demand Relationships among Goods 205 On the other hand, the kind of separability and two-stage budgeting indicated by the utility function in Equation i also requires strong assumptions about how changes in prices for a good in one group affect spending on goods in any other group. These assumptions appear to be rejected by the data (see Diewert and Wales, 1995). Economists have tried to devise even more elaborate, hybrid methods of aggregation among goods. For example, Lewbel (1996) shows how the composite commodity theorem might be generalized to cases where within-group relative prices exhibit considerable variability. He uses this generalization for aggregating U.S. consumer expenditures into six large groups (i.e., food, clothing, household operation, medical care, transportation, and recreation). Using these aggregates, he concludes that his procedure is much more accurate than assuming two-stage budgeting among these expenditure categories. E6.3 Homothetic functions and energy demand One way to simplify the study of demand when there are many commodities is to assume that utility for certain subcategories of goods is homothetic and may be separated from the demand for other commodities. This procedure was followed by Jorgenson, Slesnick, and Stoker (1997) in their study of energy demand by U.S. consumers. By assuming that demand functions for speciﬁc types of energy are proportional to total spending on energy, the authors were able to concentrate their empirical study on the topic that is of most interest to them: estimating the price elasticities of demand for various types of energy. They conclude that most types of energy (i.e., electricity, natural gas, gasoline) have fairly elastic demand functions. Demand appears to be most responsive to price for electricity. References Blackorby, Charles, Daniel Primont, and R. Robert Russell. Duality, Separability and Functional Structure: Theory and Economic Applications. New York: North Holland, 1978. Diewert, W. Erwin, and Terrence J. Wales. ‘‘Flexible Functional Forms and Tests of Homogeneous Separability.’’ Journal of Econometrics (June 1995): 259–302. Jorgenson, Dale W., Daniel T. Slesnick, and Thomas M. Stoker. ‘‘Two-Stage Budgeting and Consumer Demand for Energy.’’ In Dale W. Jorgenson, Ed., Welfare, vol. 1: Aggregate Consumer Behavior, pp. 475–510. Cambridge, MA: MIT Press, 1997. Lewbel, Arthur. ‘‘Aggregation without Separability: A Standardized Composite Commodity Theorem.’’ American Economic Review (June 1996): 524–43. This page intentionally left blank Uncertainty and Strategy P A R T THREE Chapter 7 Uncertainty Chapter 8 Game Theory This part extends the analysis of individual choice to more complicated settings. In Chapter 7 we look at individual behavior in uncertain situations. A decision is no longer associated with a single outcome but a number of more or less likely ones. We describe why people generally dislike the risk involv
ed in such situations. We seek to understand the steps they take to mitigate risk, including buying insurance, acquiring more information, and preserving options. Chapter 8 looks at decisions made in strategic situations in which a person’s well-being depends not just on his or her own actions but also on the actions of others and vice versa. This leads to a certain circularity in analyzing strategic decisions, which we will resolve using the tools of game theory. The equilibrium notions we develop in studying such situations are widely used throughout economics. Although this part can be regarded as the natural extension of the analysis of consumer choice from Part 2 to more complicated settings, it applies to a much broader set of decision-makers, including ﬁrms, other organizations, even whole countries. For example, game theory will provide the framework to study imperfect competition among few ﬁrms in Chapter 15. 207 This page intentionally left blank C H A P T E R SEVEN Uncertainty In this chapter we explore some of the basic elements of the theory of individual behavior in uncertain situations. We discuss why individuals do not like risk and the various methods (buying insurance, acquiring more information, and preserving options) they may adopt to reduce it. More generally, the chapter is intended to provide a brief introduction to issues raised by the possibility that information may be imperfect when individuals make utility-maximizing decisions. The Extensions section provides a detailed application of the concepts in this chapter to the portfolio problem, a central problem in ﬁnancial economics. Whether a well-informed person can take advantage of a poorly informed person in a market transaction (asymmetric information) is a question put off until Chapter 18. Mathematical Statistics Many of the formal tools for modeling uncertainty in economic situations were originally developed in the ﬁeld of mathematical statistics. Some of these tools were reviewed in Chapter 2, and in this chapter we will make a great deal of use of the concepts introduced there. Speciﬁcally, four statistical ideas will recur throughout this chapter. • Random variable: A random variable is a variable that records, in numerical form, the possible outcomes from some random event.1 • Probability density function (PDF): A function f (x) that shows the probabilities associated with the possible outcomes from a random variable. • Expected value of a random variable: The outcome of a random variable that will occur ‘‘on average.’’ The expected value is denoted by E(x). If x is a discrete random n variable with n outcomes, then E i x variable, then E ð . If x is a continuous random • Variance and standard deviation of a random variable: These concepts measure the dispersion of a random variable about its expected value. In the discrete case, x Var ð þ1 %1 ½ R n xi % 1 ½ i ¼ dx. The standard deviation is the square root of the variance. x Þ ; in the continuous case, Var r2 x ¼ x E Þ’ ð Þ ¼ x % 1 xif ¼ x ð dx. 2f ð P þ1 %1 xiÞ xiÞ r2 P 2f xf Þ’ E ð ð Þ R As we shall see, all these concepts will come into play when we begin looking at the decision-making process of a person faced with a number of uncertain outcomes that can be conceptually represented by a random variable. 1When it is necessary to distinguish between random variables and nonrandom variables, we will use the notation ~x to denote the fact that the variable x is random in that it takes on a number of potential randomly determined outcomes. Often, however, it will not be necessary to make the distinction because randomness will be clear from the context of the problem. 209 210 Part 3: Uncertainty and Strategy Fair Gambles and The Expected Utility Hypothesis A ‘‘fair’’ gamble is a speciﬁed set of prizes and associated probabilities that has an expected value of zero. For example, if you ﬂip a coin with a friend for a dollar, the expected value of this gamble is zero because E x ð 0:5 $1 0:5 $1 0, (7.1) Þ þ where wins are recorded with a plus sign and losses with a minus sign. Similarly, a game that promised to pay you $10 if a coin came up heads but would cost you only $1 if it came up tails would be ‘‘unfair’’ because Þ ¼ Þ ¼ ðþ ð% E x ð Þ ¼ 0:5 ðþ $10 Þ þ 0:5 ð% $1 Þ ¼ $4:50: (7.2) This game can easily be converted into a fair game, however, simply by charging you an entry fee of $4.50 for the right to play. It has long been recognized that most people would prefer not to take fair gambles.2 Although people may wager a few dollars on a coin ﬂip for entertainment purposes, $1 million or they would generally balk at playing a similar game whose outcome was $1 million. One of the ﬁrst mathematicians to study the reasons for this unwillingness % to engage in fair bets was Daniel Bernoulli in the eighteenth century.3 His examination of the famous St. Petersburg paradox provided the starting point for virtually all studies of the behavior of individuals in uncertain situations. þ St. Petersburg paradox In the St. Petersburg paradox, the following gamble is proposed: A coin is ﬂipped until a head appears. If a head ﬁrst appears on the nth ﬂip, the player is paid $2n. This gamble has an inﬁnite number of outcomes (a coin might be ﬂipped from now until doomsday and never come up a head, although the likelihood of this is small), but the ﬁrst few can easily be written down. If xi represents the prize awarded when the ﬁrst head appears on the ith trial, then x1 ¼ $2, x2 ¼ The probability of getting a head for the ﬁrst time on the ith trial is ity of getting (i Equation 7.3 are i; it is the probabil1) tails and then a head. Hence the probabilities of the prizes given in $8, . . . , xn ¼ $4, x3 ¼ (7.3) 1 2Þ % ð $2n: p1 ¼ Therefore, the expected value of the gamble is inﬁnite: , p2 ¼ , p3 ¼ , . . . , pn ¼ 1 2 1 4 1 8 1 2n : pixi ¼ 1 1 þ 1 2i 1=2i ( ( ( ¼1 . (7.4) (7.5) 2The gambles discussed here are assumed to yield no utility in their play other than the prizes; hence the observation that many individuals gamble at ‘‘unfair’’ odds is not necessarily a refutation of this statement. Rather, such individuals can reasonably be assumed to be deriving some utility from the circumstances associated with the play of the game. Therefore, it is possible to differentiate the consumption aspect of gambling from the pure risk aspect. 3The paradox is named after the city where Bernoulli’s original manuscript was published. The article has been reprinted as D. Bernoulli, ‘‘Exposition of a New Theory on the Measurement of Risk,’’ Econometrica 22 (January 1954): 23–36. Chapter 7: Uncertainty 211 Some introspection, however, should convince anyone that no player would pay very much (much less than inﬁnity) to take this bet. If we charged $1 billion to play the game, we would surely have no takers, despite the fact that $1 billion is still considerably less than the expected value of the game. This then is the paradox: Bernoulli’s gamble is in some sense not worth its (inﬁnite) expected dollar value. Expected utility Bernoulli’s solution to this paradox was to argue that individuals do not care directly about the dollar prizes of a gamble; rather, they respond to the utility these dollars provide. If we assume that the marginal utility of wealth decreases as wealth increases, the St. Petersburg gamble may converge to a ﬁnite expected utility value even though its expected monetary value is inﬁnite. Because the gamble only provides a ﬁnite expected utility, individuals would only be willing to pay a ﬁnite amount to play it. Example 7.1 looks at some issues related to Bernoulli’s solution. EXAMPLE 7.1 Bernoulli’s Solution to the Paradox and Its Shortcomings Suppose, as did Bernoulli, that the utility of each prize in the St. Petersburg paradox is given by xiÞ ¼ ð This logarithmic utility function exhibits diminishing marginal utility (i.e., U0 > 0 but U 00 < 0), and the expected utility value of this game converges to a ﬁnite number: : xiÞ ln ð (7.6) U expected utility ¼ ¼ piU xiÞ ð 1 2i 2i ln ð : Þ (7.7 Some manipulation of this expression yields4 the result that the expected utility from this gamble is 1.39. Therefore, an individual with this type of utility function might be willing to invest resources that otherwise yield up to 1.39 units of utility (a certain wealth of approximately $4 provides this utility) in purchasing the right to play this game. Thus, assuming that the large prizes promised by the St. Petersburg paradox encounter diminishing marginal utility permitted Bernoulli to offer a solution to the paradox. Unbounded utility. Unfortunately, Bernoulli’s solution to the St. Petersburg paradox does not completely solve the problem. As long as there is no upper bound to the utility function, the paradox can be regenerated by redeﬁning the gamble’s prizes. For example, with the logarithmic e2i , in which case utility function, prizes can be set as xi ¼ U (7.8) e2 ln 2i i xiÞ ¼ ð ½ ’ ¼ and the expected utility from the gamble would again be inﬁnite. Of course, the prizes in this redeﬁned gamble are large. For example, if a head ﬁrst appears on the ﬁfth ﬂip, a person would 4Proof: expected utility i 2i ( ¼ 1 1 i X ¼ ln 2 ¼ ln 2 i 2i . 1 1 i X ¼ But the value of this ﬁnal inﬁnite series can be shown to be 2.0. Hence expected utility 2 ln 2 1.39. ¼ ¼ 212 Part 3: Uncertainty and Strategy ¼ $79 trillion, although the probability of winning this would be only 1/25 win e25 0.031. The idea that people would pay a great deal (say, trillions of dollars) to play games with small probabilities of such large prizes seems, to many observers, to be unlikely. Hence in many respects the St. Petersburg game remains a paradox. ¼ QUERY: Here are two alternative solutions to the St. Petersburg paradox. For each, calculate the expected value of the original game. 1. Suppose individuals assume that any probability less than 0.01 is in fact zero. 2. Suppose that the utility from the St. Petersburg prizes is given by U
xiÞ ¼ ð ! xi 1,000,000 1,000,000, if xi ) if xi > 1,000,000: The von Neumann–Morgenstern Theorem Among many contributions relevant to Part 3 of our text, in their book The Theory of Games and Economic Behavior, John von Neumann and Oscar Morgenstern developed a mathematical foundation for Bernoulli’s solution to the St. Petersburg paradox.5 In particular, they laid out basic axioms of rationality and showed that any person who is rational in this way would make choices under uncertainty as though he or she had a utility function over money U(x) and maximized the expected value of U(x) (rather than the expected value of the monetary payoff x itself). Although most of these axioms seem eminently reasonable at ﬁrst glance, many important questions about their tenability have been raised.6 We will not pursue these questions here, however. The von Neumann–Morgenstern utility index To begin, suppose that there are n possible prizes that an individual might win by participating in a lottery. Let these prizes be denoted by x1, x2,…, xn, and assume that these have been arranged in order of ascending desirability. Therefore, x1 is the least preferred prize for the individual and xn is the most preferred prize. Now assign arbitrary utility numbers to these two extreme prizes. For example, it is convenient to assign 0, 1, U U x1Þ ¼ ð xnÞ ¼ ð but any other pair of numbers would do equally well.7 Using these two values of utility, the point of the von Neumann–Morgenstern theorem is to show that a reasonable way exists to assign speciﬁc utility numbers to the other prizes available. Suppose that we choose any other prize, say, xi. Consider the following experiment. Ask the individual to state the probability, say, pi, at which he or she would be indifferent between xi with (7.9) 5J. von Neumann and O. Morgenstern, The Theory of Games and Economic Behavior (Princeton, NJ: Princeton University Press, 1944). The axioms of rationality in uncertain situations are discussed in the book’s appendix. 6For a discussion of some of the issues raised in the debate over the von Neumann–Morgenstern axioms, especially the assumption of independence, see C. Gollier, The Economics of Risk and Time (Cambridge, MA: MIT Press, 2001), chap. 1. 7Technically, a von Neumann–Morgenstern utility index is unique only up to a choice of scale and origin—that is, only up to a ‘‘linear transformation.’’ This requirement is more stringent than the requirement that a utility function be unique up to a monotonic transformation. Chapter 7: Uncertainty 213 % certainty, and a gamble offering prizes of xn with probability pi and x1 with probability pi). It seems reasonable (although this is the most problematic assumption in the (1 von Neumann–Morgenstern approach) that such a probability will exist: The individual will always be indifferent between a gamble and a sure thing, provided that a high enough probability of winning the best prize is offered. It also seems likely that pi will be higher the more desirable xi is; the better xi is, the better the chance of winning xn must be to get the individual to gamble. Therefore, the probability pi measures how desirable the prize xi is. In fact, the von Neumann–Morgenstern technique deﬁnes the utility of xi as the expected utility of the gamble that the individual considers equally desirable to xi: U xiÞ ¼ piÞ ð Because of our choice of scale in Equation 7.9, we have xnÞ þ ð ð piU % U 1 : x1Þ ð U xiÞ ¼ ð pi ( 1 1 0 piÞ ( % þ ð ¼ pi: (7.10) (7.11) By judiciously choosing the utility numbers to be assigned to the best and worst prizes, we have been able to devise a scale under which the utility index attached to any other prize is simply the probability of winning the top prize in a gamble the individual regards as equivalent to the prize in question. This choice of utility indices is arbitrary. Any other two numbers could have been used to construct this utility scale, but our initial choice (Equation 7.9) is a particularly convenient one. Expected utility maximization In line with the choice of scale and origin represented by Equation 7.9, suppose that a utility index pi has been assigned to every prize xi. Notice in particular that p1 ¼ 1, and that the other utility indices range between these extremes. Using these utility indices, we can show that a ‘‘rational’’ individual will choose among gambles based on their expected ‘‘utilities’’ (i.e., based on the expected value of these von Neumann–Morgenstern utility index numbers). 0, pn ¼ As an example, consider two gambles. Gamble A offers x2 with probability a and x3 a). Gamble B offers x4 with probability b and x5 with probability b). We want to show that this person will choose gamble A if and only if the with probability (1 (1 expected utility of gamble A exceeds that of gamble B. Now for the gambles: % % expected utility of A expected utility of B aU bU ¼ ¼ 1 x2Þ þ ð ð x4Þ , x3Þ ð : x5Þ ð (7.12) Substituting the utility index numbers (i.e., p2 is the ‘‘utility’’ of x2, and so forth) gives expected utility of A expected utility of B 1 ap2 þ ð 1 bp4 þ ð a b p3, Þ p5: Þ % % ¼ ¼ (7.13) We wish to show that the individual will prefer gamble A to gamble B if and only if 1 b 1 a % ap2 þ ð p3 > bp4 þ ð Þ To show this, recall the deﬁnitions of the utility index. The individual is indifferent between x2 and a gamble promising x1 with probability (1 p2) and xn with probability p2. We can use this fact to substitute gambles involving only x1 and xn for all utilities in Equation 7.13 (even though the individual is indifferent between these, the assumption that this substitution can be made implicitly assumes that people can see through complex lottery combinations). After a bit of messy algebra, we can conclude that gamble A is p5: Þ (7.14) % % 214 Part 3: Uncertainty and Strategy a)p3, and gamble B is equivalent to a gamble promising xn with probability ap2 + (1 % equivalent to a gamble promising xn with probability bp4 + (1 b)p5. The individual will presumably prefer the gamble with the higher probability of winning the best prize. Consequently, he or she will choose gamble A if and only if ap2 þ ð p3 > bp4 þ ð Þ But this is precisely what we wanted to show. Consequently, we have proved that an individual will choose the gamble that provides the highest level of expected (von Neumann– Morgenstern) utility. We now make considerable use of this result, which can be summarized as follows. p5: b Þ (7.15 Expected utility maximization. If individuals obey the von Neumann–Morgenstern axioms of behavior in uncertain situations, they will act as though they choose the option that maximizes the expected value of their von Neumann–Morgenstern utility. Risk Aversion Economists have found that people tend to avoid risky situations, even if the situation amounts to a fair gamble. For example, few people would choose to take a $10,000 bet on the outcome of a coin ﬂip, even though the average payoff is 0. The reason is that the gamble’s money prizes do not completely reﬂect the utility provided by the prizes. The utility that people obtain from an increase in prize money may increase less rapidly than the dollar value of these prizes. A gamble that is fair in money terms may be unfair in utility terms and thus would be rejected. In more technical terms, extra money may provide people with decreasing marginal utility. A simple example can help explain why. An increase in income from, say, $40,000 to $50,000 may substantially increase a person’s well-being, ensuring he or she does not have to go without essentials such as food and housing. A further increase from $50,000 to $60,000 allows for an even more comfortable lifestyle, perhaps providing tastier food and a bigger house, but the improvement will likely not be as great as the initial one. Starting from a wealth of $50,000, the individual would be reluctant to take a $10,000 bet on a coin ﬂip. The 50 percent chance of the increased comforts that he or she could have with $60,000 does not compensate for the 50 percent chance that he or she will end up with $40,000 and perhaps have to forgo some essentials. These effects are only magniﬁed with riskier gambles, that is, gambles having even more variable outcomes.8 The person with initial wealth of $50,000 would likely be even more reluctant to take a $20,000 bet on a coin ﬂip because he or she would face the prospect of ending up with only $30,000 if the ﬂip turned out badly, severely cutting into life’s essentials. The equal chance of ending up with $70,000 is not adequate compensation. On the other hand, a bet of only $1 on a coin ﬂip is relatively inconsequential. Although the person may still decline the bet, he or she would not try hard to do so because his or her ultimate wealth hardly varies with the outcome of the coin toss. Risk aversion and fair bets This argument is illustrated in Figure 7.1. Here W0 represents an individual’s current wealth and U(W ) is a von Neumann–Morgenstern utility index (we will call this a utility 8Often the statistical concepts of variance and standard deviation are used to measure. We will do so at several places later in this chapter. Chapter 7: Uncertainty 215 FIGURE 7.1 Utility of Wealth from Two Fair Bets of Differing Variability If the utility-of-wealth function is concave (i.e., exhibits a diminishing marginal utility of wealth), then this person will refuse fair bets. A 50–50 chance of winning or losing h dollars, for example, yields less expected utility [EU(A)] than does refusing the bet. The reason for this is that winning h dollars means less to this individual than does losing h dollars. Utility U(W0) EU(A) = U(CEA) EU(B) U(W) W0 – 2h W0 + h W0 W0 + h W0 + 2h Wealth (W) CEA function from now on) that reﬂects how he or she feels about various levels of wealth.9 In the ﬁgure, U(W ) is drawn as a concave function of W to reﬂect the assumption of a diminishing marginal utility. Now suppose this person is offered two fair gambles: gamble A, whi
ch is a 50–50 chance of winning or losing $h, and gamble B, which is a 50–50 chance of winning or losing $2h. The utility of current wealth is U(W0), which is also the expected value of current wealth because it is certain. The expected utility if he or she participates in gamble A is given by EU(A): EU A ð Þ ¼ 1 2 U W0 þ ð h Þ þ 1 2 U W0 % ð h , Þ and the expected utility of gamble B is given by EU(B): EU B ð Þ ¼ 1 2 U W0 þ ð 2h Þ þ 1 2 U W0 % ð . 2h Þ (7.16) (7.17) Equation 7.16 shows that the expected utility from gamble A is halfway between the utility h and the utility from favorable outcome W0 + h. from the unfavorable outcome W0 % Likewise, the expected utility from gamble B is halfway between the utilities from unfavorable and favorable outcomes, but payoffs in these outcomes vary more than with gamble A. 9Technically, U(W ) is an indirect utility function because it is the consumption allowed by wealth that provides direct utility. In Chapter 17 we will take up the relationship between consumption-based utility functions and their implied indirect utility of wealth functions. 216 Part 3: Uncertainty and Strategy It is geometrically clear from the ﬁgure that10 U > EU W0Þ ð A Þ ð Therefore, this person will prefer to keep his or her current wealth rather than taking either fair gamble. If forced to choose a gamble, the person would prefer the smaller one (A) to the large one (B). The reason for this is that winning a fair bet adds to enjoyment less than losing hurts. (7.18) > EU B Þ ð : Risk aversion and insurance As a matter of fact, this person might be willing to pay some amount to avoid participating in any gamble at all. Notice that a certain wealth of CEA provides the same expected utility as does participating in gamble A. CEA is referred to as the certainty equivalent of gamble A. The individual would be willing to pay up to W0 % CEA to avoid participating in the gamble. This explains why people buy insurance. They are giving up a small, certain amount (the insurance premium) to avoid the risky outcome they are being insured against. The premium a person pays for automobile collision insurance, for example, provides a policy that agrees to repair his or her car should an accident occur. The widespread use of insurance would seem to imply that aversion to risk is prevalent. In fact, the person in Figure 7.1 would pay even more to avoid taking the larger gamble, B. As an exercise, try to identify the certainty equivalent CEB of gamble B and the amount the person would pay to avoid gamble B on the ﬁgure. The analysis in this section can be summarized by the following deﬁnition Risk aversion. An individual who always refuses fair bets is said to be risk averse. If individuals exhibit a diminishing marginal utility of wealth, they will be risk averse. As a consequence, they will be willing to pay something to avoid taking fair bets. EXAMPLE 7.2 Willingness to Pay for Insurance To illustrate the connection between risk aversion and insurance, consider a person with a current wealth of $100,000 who faces the prospect of a 25 percent chance of losing his or her $20,000 automobile through theft during the next year. Suppose also that this person’s von Neumann–Morgenstern utility function is logarithmic; that is, U(W ) ln (W ). If this person faces next year without insurance, expected utility will be ¼ no insurance EU ð 0:75U 100,000 ð 0:75 ln 100,000 11:45714: 0:25U 80,000 Þ ð 0:25 ln 80,000 Þ þ þ Þ ¼ ¼ ¼ (7.19) In this situation, a fair insurance premium would be $5,000 (25 percent of $20,000, assuming that the insurance company has only claim costs and that administrative costs are $0). 10Technically this result is a direct consequence of Jensen’s inequality in mathematical statistics. The inequality states that if x is a random variable and f(x) is a strictly concave function of that variable, then E[ f (x)] < f [E(x)]. In the utility context, this means that if utility is concave in a random variable measuring wealth (i.e., if U 0(W) > 0 and U 00(W) < 0), then the expected utility of wealth will be less than the utility associated with the expected value of W. With gamble A, for example, EU(A) < U(W0) because, as a fair gamble, A provides expected wealth W0. Chapter 7: Uncertainty 217 Consequently, if this person completely insures the car, his or her wealth will be $95,000 regardless of whether the car is stolen. In this case then, EU fair insurance ð U 95,000 Þ ð 95,000 ln Þ ð 11:46163: Þ ¼ ¼ ¼ (7.20) This person is made better off by purchasing fair insurance. Indeed, he or she would be willing to pay more than the fair premium for insurance. We can determine the maximum insurance premium (x) by setting EU maximum-premium insurance ð 100,000 U ð ln 100,000 ð 11:45714 Solving this equation for x yields or 100,000 x % ¼ e11:45714; 5,426: x ¼ (7.21) (7.22) (7.23) This person would be willing to pay up to $426 in administrative costs to an insurance company (in addition to the $5,000 premium to cover the expected value of the loss). Even when these costs are paid, this person is as well off as he or she would be when facing the world uninsured. QUERY: Suppose utility had been linear in wealth. Would this person be willing to pay anything more than the actuarially fair amount for insurance? How about the case where utility is a convex function of wealth? Measuring Risk Aversion In the study of economic choices in risky situations, it is sometimes convenient to have a quantitative measure of how averse to risk a person is. The most commonly used measure of risk aversion was initially developed by J. W. Pratt in the 1960s.11 This risk aversion measure, r (W ), is deﬁned as r W U 00 W ð U 0ð W Because the distinguishing feature of risk-averse individuals is a diminishing marginal utility of wealth [U 00(W) < 0], Pratt’s measure is positive in such cases. The measure is invariant with respect to linear transformations of the utility function, and therefore not affected by which particular von Neumann–Morgenstern ordering is used. Þ ¼ % (7.24) Þ Þ ð : Risk aversion and insurance premiums A useful feature of the Pratt measure of risk aversion is that it is proportional to the amount an individual will pay for insurance against taking a fair bet. Suppose the winnings from such a fair bet are denoted by the random variable h (which takes on both 11J . W. Pratt, ‘‘Risk Aversion in the Small and in the Large,’’ Econometrica (January/April 1964): 122–36. 218 Part 3: Uncertainty and Strategy 0. Now let p be the size of positive and negative values). Because the bet is fair, E(h) the insurance premium that would make the individual exactly indifferent between taking the fair bet h and paying p with certainty to avoid the gamble7.25) where W is the individual’s current wealth. We now expand both sides of Equation 7.25 using Taylor’s series.12 Because p is a ﬁxed amount, a linear approximation to the right side of the equation will sufﬁce: U W ð W ð For the left side, we need a quadratic approximation to allow for the variability in the gamble, h: higher-order terms: p Þ ¼ (7.26) pU hU 0 W ð Þ þ h2 2 U 00 W ð Þ þ higher-order terms (7.27 h2 ð 2 Þ U 00 W ð Þ þ higher-order terms. (7.28) If we recall that E(h) represent E(h2)/2, we can equate Equations 7.26 and 7.28 as ¼ 0 and then drop the higher-order terms and use the constant k to or U W ð Þ % pU 0 W ð Þ ﬃ U W ð Þ % kU 00 W ð Þ (7.29) kU 00 W ð U 0ð W ð p : Þ ¼ kr W ﬃ % Þ Þ That is, the amount that a risk-averse individual is willing to pay to avoid a fair bet is approximately proportional to Pratt’s risk aversion measure.13 Because insurance premiums paid are observable in the real world, these are often used to estimate individuals’ risk aversion coefﬁcients or to compare such coefﬁcients among groups of individuals. Therefore, it is possible to use market information to learn a bit about attitudes toward risky situations. (7.30) Risk aversion and wealth An important question is whether risk aversion increases or decreases with wealth. Intuitively, one might think that the willingness to pay to avoid a given fair bet would decrease as wealth increases because decreasing marginal utility would make potential losses less serious for high-wealth individuals. This intuitive answer is not necessarily correct, however, because decreasing marginal utility also makes the gains from winning gambles less attractive. Thus, the net result is indeterminate; it all depends on the precise shape of the utility function. Indeed, if utility is quadratic in wealth, W U ð Þ ¼ a þ bW þ cW2, (7.31) 12Taylor’s series provides a way of approximating any differentiable function around some point. If f (x) has derivatives of all f (x) + hf 0(x) + (h2/2)f 00(x) + higher-order terms. The point-slope formula in algebra is orders, it can be shown that f (x + h) a simple example of Taylor’s series. 13In this case, the factor of proportionality is also proportional to the variance of h because Var(h) an illustration where this equation ﬁts exactly, see Example 7.3. E[h — E(h)]2 E(h2). For ¼ ¼ ¼ where b > 0 and c < 0, then Pratt’s risk aversion measure is r W ð Þ ¼ % U 00 W ð U 0ð W Þ Þ 2c 2cW % þ , ¼ b which, contrary to intuition, increases as wealth increases. On the other hand, if utility is logarithmic in wealth, then we have U W ð Þ ¼ ln , W ð Þ U 00 W ð U 0ð W which does indeed decrease as wealth increases The exponential utility function U W ð Þ ¼ % AW e% exp AW Þ ð% ¼ % Chapter 7: Uncertainty 219 (7.32) (7.33) (7.34) (7.35) (where A is a positive constant) exhibits constant absolute risk aversion over all ranges of wealth because now r W ð Þ ¼ % U 00 W ð U 0ð W Þ Þ ¼ AW A2e% Ae% AW ¼ A: (7.36) This feature of the exponential utility function14 can be used to provide some numeri- cal estimates of the willingness to pay to avoid gambles, as the next example shows. EXAMPLE 7.3 Constant Risk Aversion Suppose an individual whose initial wealth is W0 and whose utility function exhibits cons
tant absolute risk aversion is facing a 50–50 chance of winning or losing $1,000. How much ( f ) would he or she pay to avoid the risk? To ﬁnd this value, we set the utility of W0 % f equal to the expected utility from the gamble: f % ½% exp W0 % A ð A W0 þ ð W0 % A ð Because the factor AW0) is contained in all the terms in Equation 7.37, this may be divided out, thereby showing that (for the exponential utility function) the willingness to pay to avoid a given gamble is independent of initial wealth. The remaining terms 1,000 Þ’ 1; 000 Þ’ Þ’ ¼ % % 0:5 exp 0:5 exp ½% ½% (7.37) exp( % % : exp Af ð Þ ¼ 0:5 exp ð% 1,000A 0:5 exp ð Þ þ 1,000A Þ 0.0001, then f (7.38) can now be used to solve for f for various values of A. If A 49.9; a person with this degree of risk aversion would pay approximately $50 to avoid a fair bet of $1,000. Alternatively, if A 147.8 to avoid the gamble. Because intuition suggests that these values are not unreasonable, values of the risk aversion parameter A in these ranges are sometimes used for empirical investigations. 0.0003, this more risk-averse person would pay f ¼ ¼ ¼ ¼ Normally distributed risk. The constant risk aversion utility function can be combined with the assumption that a person faces a random shock to his or her wealth that follows a Normal distribution (see Chapter 2) to arrive at a particularly simple result. Speciﬁcally, if a person’s 14Because the exponential utility function exhibits constant (absolute) risk aversion, it is sometimes abbreviated by the term CARA utility. 220 Part 3: Uncertainty and Strategy risky wealth follows a Normal distribution with mean m and variance s2, then the probability density function for wealth is given by f m)/s]. If this [(W AW, then expected utility from his person has a utility function for wealth given by U(W) or her risky wealth is z2=2, where z e% e% Þ ﬃﬃﬃﬃﬃ ¼ % W ð 2pp Þ ¼ ð 1= % ¼ U E ½ W ð Þ’ ¼ 1 ð %1 U W ð f Þ W ð Þ dW 1 2pp ¼ ð e% AWe%½ð W % =r l Þ % 2=2 dW: ’ (7.39) ﬃﬃﬃﬃﬃ Perhaps surprisingly, this integration is not too difﬁcult to accomplish, although it does take patience. Performing this integration and taking a variety of monotonic transformations of the resulting expression yields the ﬁnal result that U E ½ W ð Þ’ ﬃ l % Ar2 2 : (7.40) Hence expected utility is a linear function of the two parameters of the wealth probability density function, and the individual’s risk aversion parameter (A) determines the size of the negative effect of variability on expected utility. For example, suppose a person has invested his or her funds so that wealth has an expected value of $100,000 but a standard deviation (s) of $10,000. Therefore, with the Normal distribution, he or she might expect wealth to decrease below $83,500 about 5 percent of the time and increase above $116,500 a similar fraction of the time. With these parameters, expected utility is given by E[U(W )] ¼ 0.0001 95,000. Hence this person receives the same utility from his or her risky wealth as would be obtained from a certain wealth of $95,000. A more risk-averse person might have A 0.0003, and in this case the certainty equivalent of his or her wealth would be $85,000. 4, expected utility is given by 100,000 (A/2)(10,000)2. If A % (104)2 100,000 ¼ 10% 10% 0.5 ¼ ¼ % ¼ ( ( 4 QUERY: Suppose this person had two ways to invest his or her wealth: Allocation 1, m1 ¼ 107,000 and s1 ¼ 2,000. How would this person’s attitude toward risk affect his or her choice between these allocations?15 10,000; Allocation 2, m2 ¼ 102,000 and s2 ¼ Relative risk aversion It seems unlikely that the willingness to pay to avoid a given gamble is independent of a person’s wealth. A more appealing assumption may be that such willingness to pay is inversely proportional to wealth and that the expression rr W ð Þ ¼ Wr W ð Þ ¼ % W U 00 W ð U 0ð W Þ Þ (7.41) might be approximately constant. Following the terminology proposed by J. W. Pratt,16 the rr (W ) function deﬁned in Equation 7.41 is a measure of relative risk aversion. The power utility function W, R U ð Þ ¼ ! WR=R if R < 1, R ln W if R 0 ¼ 0 6¼ (7.42) 15This numerical example (roughly) approximates historical data on real returns of stocks and bonds, respectively, although the calculations are illustrative only. 16Pratt, ‘‘Risk Aversion.’’ Chapter 7: Uncertainty 221 exhibits diminishing absolute risk aversion, W r ð Þ ¼ % U 00 W ð U 0ð W Þ Þ R ð 2 WR 1 % Þ % 1 WR % 1 R , % W ¼ ¼ % but constant relative risk aversion:17 rr W ð Þ ¼ Wr W ð Þ ¼ 1 % R: (7.43) (7.44) Empirical evidence is generally consistent with values of R in the range of 1. Hence individuals seem to be somewhat more risk averse than is implied by the logarithmic utility function, although in many applications that function provides a reasonable approximation. It is useful to note that the constant relative risk aversion utility function in Equation 7.42 has the same form as the general CES utility function we ﬁrst described in Chapter 3. This provides some geometric intuition about the nature of risk aversion that we will explore later in this chapter. 3 to % % EXAMPLE 7.4 Constant Relative Risk Aversion An individual whose behavior is characterized by a constant relative risk aversion utility function will be concerned about proportional gains or loss of wealth. Therefore, we can ask what fraction of initial wealth ( f ) such a person would be willing to give up to avoid a fair gamble of, say, 10 percent of initial wealth. First, we assume R 0, so the logarithmic utility function is appropriate. Setting the utility of this individual’s certain remaining wealth equal to the expected utility of the 10 percent gamble yields ¼ ln 1 ½ð % f W0’ ¼ Þ 0:5 ln 1:1W0Þ þ ð 0:5 ln 0:9W0Þ ð : (7.45) Because each term contains ln W0, initial wealth can be eliminated from this expression: ln 1 ð % f Þ ¼ 0:5 1:1 ln ð ½ Þ þ ln 0:9 ð Þ’ ¼ ln ð 0:99 0:5:99 0:5 Þ ¼ 0:995 hence and 0:005: f ¼ (7.46) Thus, this person will sacriﬁce up to 0.5 percent of wealth to avoid the 10 percent gamble. A similar calculation can be used for the case R 2 to yield ¼ % 0:015: f ¼ (7.47) Hence this more risk-averse person would be willing to give up 1.5 percent of his or her initial wealth to avoid a 10 percent gamble. QUERY: With the constant relative risk aversion function, how does this person’s willingness to pay to avoid a given absolute gamble (say, of 1,000) depend on his or her initial wealth? 17Some authors write the utility function in Equation 7.42 as U(W) R. In this case, a is the relative risk aversion measure. The constant relative risk aversion function is sometimes abbreviated as CRRA utility. a) and seek to measure a a/(1 W1 ¼ % ¼ % 1 % 222 Part 3: Uncertainty and Strategy Methods for Reducing Uncertainty and Risk We have seen that risk-averse people will avoid gambles and other risky situations if possible. Often it is impossible to avoid risk entirely. Walking across the street involves some risk of harm. Burying one’s wealth in the backyard is not a perfectly safe investment strategy because there is still some risk of theft (to say nothing of inﬂation). Our analysis thus far implies that people would be willing to pay something to at least reduce these risks if they cannot be avoided entirely. In the next four sections, we will study each of four different methods that individuals can take to mitigate the problem of risk and uncertainty: insurance, diversiﬁcation, ﬂexibility, and information. Insurance We have already discussed one such strategy: buying insurance. Risk-averse people would pay a premium to have the insurance company cover the risk of loss. Each year, people in the United States spend more than half a trillion dollars on insurance of all types. Most commonly, they buy coverage for their own life, for their home and cars, and for their health care costs. But insurance can be bought (perhaps at a high price) for practically any risk imaginable, ranging from earthquake insurance for a house along a fault line to special coverage for a surgeon against a hand injury. A risk-averse person would always want to buy fair insurance to cover any risk he or she faces. No insurance company could afford to stay in business if it offered fair insurance (in the sense that the premium exactly equals the expected payout for claims). Besides covering claims, insurance companies must also maintain records, collect premiums, investigate fraud, and perhaps return a proﬁt to shareholders. Hence an insurance customer can always expect to pay more than an actuarially fair premium. If people are sufﬁciently risk averse, they will even buy unfair insurance, as shown in Example 7.2; the more risk averse they are, the higher the premium they would be willing to pay. Several factors make insurance difﬁcult or impossible to provide. Large-scale disasters such as hurricanes and wars may result in such large losses that the insurance company would go bankrupt before it could pay all the claims. Rare and unpredictable events (e.g., war, nuclear power plant accidents) offer reliable track record for insurance companies to establish premiums. Two other reasons for absence of insurance coverage relate to the informational disadvantage the company may have relative to the customer. In some cases, the individual may know more about the likelihood that they will suffer a loss than the insurance company. Only the ‘‘worst’’ customers (those who expect larger or more likely losses) may end up buying an insurance policy. This adverse selection problem may unravel the whole insurance market unless the company can ﬁnd a way to control who buys (through some sort of screening or compulsion). Another problem is that having insurance may make customers less willing to take steps to avoid losses, for example, driving more recklessly with auto insurance or eating fatty foods and smoking with health insurance. This so-called moral hazard problem again may impair the insurance market unless the insurance company can ﬁnd a way t
o cheaply monitor customer behavior. We will discuss the adverse selection and moral hazard problems in more detail in Chapter 18, and discuss ways the insurance company can combat these problems, which besides the above strategies include offering only partial insurance and requiring the payment of deductibles and copayments. Chapter 7: Uncertainty 223 Diversiﬁcation A second way for risk-averse individuals to reduce risk is by diversifying. This is the economic principle behind the adage, ‘‘Don’t put all your eggs in one basket.’’ By suitably spreading risk around, it may be possible to reduce the variability of an outcome without lowering the expected payoff. The most familiar setting in which diversiﬁcation comes up is in investing. Investors are routinely advised to ‘‘diversify your portfolio.’’ To understand the wisdom behind this advice, consider an example in which a person has wealth W to invest. This money can be invested in two independent risky assets, 1 and 2, which have equal expected values (the mean returns are m1 ¼ 2). A person whose undiversiﬁed portfolio, UP, includes just one of the assets (putting all his or her m2 and would face ‘‘eggs’’ in that ‘‘basket’’) would earn an expected return of mUP ¼ a variance of r2 m2) and equal variances (the variances are r2 m1 ¼ 1 ¼ r2 r2 r2 2. Suppose instead the individual chooses a diversiﬁed portfolio, DP. Let a1 be the fraca1 in the second. We will see that the person tion invested in the ﬁrst asset and a2 ¼ can do better than the undiversiﬁed portfolio in the sense of getting a lower variance without changing the expected return. The expected return on the diversiﬁed portfolio does not depend on the allocation across assets and is the same as for either asset alone: % 1 UP ¼ 1 ¼ a1Þ To see this, refer back to the rules for computed expected values from Chapter 2. The variance will depend on the allocation between the two assets: a1l1 þ ð lDP ¼ l1 ¼ l2 ¼ (7.48) l2: % 1 r2 DP ¼ 1r2 a2 1 þ ð 1 a1Þ % 2r2 2 ¼ ð 1 2a1 þ 2a2 1Þ r2 1: % (7.49) This calculation again can be understood by reviewing the section on variances in Chapter 2. There you will be able to review the two ‘‘facts’’ used in this calculation: First, the variance of a constant times a random variable is that constant squared times the variance of a random variable; second, the variance of independent random variables, because their covariance is 0, equals the sum of the variances. Choosing a1 to minimize Equation 7.49 yields a1 ¼ r2 2 . Therefore, the 1 optimal portfolio spreads wealth equally between the two assets, maintaining the same expected return as an undiversiﬁed portfolio but reducing variance by half. Diversiﬁcation works here because the assets’ returns are independent. When one return is low, there is a chance the other will be high, and vice versa. Thus, the extreme returns are balanced out at least some of the time, reducing the overall variance. Diversiﬁcation will work in this way as long as there is not perfect correlation in the asset returns so that they are not effectively the same asset. The less correlated the assets are, the better diversiﬁcation will work to reduce the variance of the overall portfolio. 2 and r2 DP ¼ 1 The example, constructed to highlight the beneﬁts of diversiﬁcation as simply as possible, has the artiﬁcial element that asset returns were assumed to be equal. Diversiﬁcation was a ‘‘free lunch’’ in that the variance of the portfolio could be reduced without reducing the expected return compared with an undiversiﬁed portfolio. If the expected return from one of the assets (say, asset 1) is higher than the other, then diversiﬁcation into the other asset would no longer be a ‘‘free lunch’’ but would result in a lower expected return. Still, the beneﬁts from risk reduction can be great enough that a risk-averse investor would be willing to put some share of wealth into the asset with the lower expected return. A practical example of this idea is related to advice one would give to an employee of a ﬁrm with a stock purchase plan. Even if the plan allows employees to buy shares of the company’s stock at a generous discount compared with the market, the employee may still be 224 Part 3: Uncertainty and Strategy advised not to invest all savings in that stock because otherwise the employee’s entire savings, to say nothing of his or her salary and perhaps even house value (to the extent house values depend on the strength of businesses in the local economy), is tied to the fortunes of a single company, generating a tremendous amount of risk. The Extensions provide a much more general analysis of the problem of choosing the optimal portfolio. However, the principle of diversiﬁcation applies to a much broader range of situations than ﬁnancial markets. For example, students who are uncertain about where their interests lie or about what skills will be useful on the job market are well advised to register for a diverse set of classes rather than exclusively technical or artistic ones. Flexibility Diversiﬁcation is a useful method to reduce risk for a person who can divide up a decision by allocating small amounts of a larger sum among a number of different choices. In some situations, a decision cannot be divided; it is all or nothing. For example, in shopping for a car, a consumer cannot combine the attributes that he or she likes from one model (say, fuel efﬁciency) with those of another (say, horsepower or power windows) by buying half of each; cars are sold as a unit. With all-or-nothing decisions, the decisionmaker can obtain some of the beneﬁt of diversiﬁcation by making ﬂexible decisions. Flexibility allows the person to adjust the initial decision, depending on how the future unfolds. The more uncertain the future, the more valuable this ﬂexibility. Flexibility keeps the decision-maker from being tied to one course of action and instead provides a number of options. The decision-maker can choose the best option to suit later circumstances. A good example of the value of ﬂexibility comes from considering the fuels on which cars are designed to run. Until now, most cars were limited in how much biofuel (such as ethanol made from crops) could be combined with petroleum products (such as gasoline or diesel) in the fuel mix. A purchaser of such a car would have difﬁculties if governments passed new regulations increasing the ratio of ethanol in car fuels or banning petroleum products entirely. New cars have been designed that can burn ethanol exclusively, but such cars are not useful if current conditions continue to prevail because most ﬁlling stations do not sell fuel with high concentrations of ethanol. A third type of car has internal components that can handle a variety of types of fuel, both petroleumbased and ethanol, and any proportions of the two. Such cars are expensive to build because of the specialized components involved, but a consumer might pay the additional expense anyway because the car would be useful whether or not biofuels become more important over the life of the car.18 Types of options The ability of ‘‘ﬂexible-fuel’’ cars to be able to burn any mix of petroleum-based fuels and biofuels is valuable because it provides the owner with more options relative to a car that can run on only one type of fuel. Readers are probably familiar with the notion that options are valuable from another context where the term is frequently used—ﬁnancial markets— where one hears about stock options and other forms of options contracts. There is a close connection between the option implicit in the ﬂexible-fuel cars and these option contracts that we will investigate in more detail. Before discussing the similarities between the options arising in different contexts, we introduce some terms to distinguish them. 18While the current generation of ﬂexible-fuel cars involve state-of-the-art technology, the ﬁrst such car, produced back in 1908, was Henry Ford’s Model-T, one of the top-selling cars of all time. The availability of cheap gasoline may have swung the market toward competitors’ single-fuel cars, spelling the demise of the Model-T. For more on the history of this model, see L. Brooke, Ford Model T: The Car That Put the World on Wheels (Minneapolis: Motorbooks, 2008). Chapter 7: Uncertainty 225 Financial option contract. A ﬁnancial option contract offers the right, but not the obligation, to buy or sell an asset (such as a share of stock) during some future period at a certain price Real option. A real option is an option arising in a setting outside of ﬁnancial markets. The ﬂexible-fuel car can be viewed as an ordinary car combined with an additional real option to burn biofuels if those become more important in the future. Financial option contracts come in a variety of forms, some of which can be complex. There are also many different types of real options, and they arise in many different settings, sometimes making it difﬁcult to determine exactly what sort of option is embedded in the situation. Still, all options share three fundamental attributes. First, they specify the underlying transaction, whether it is a stock to be traded or a car or fuel to be purchased. Second, they specify a period over which the option may be exercised. A stock option may specify a period of 1 year, for example. The option embedded in a ﬂexible-fuel car preserves the owner’s option during the operating life of the car. The longer the period over which the option extends, the more valuable it is because the more uncertainty that can be resolved during this period. Third, the option contract speciﬁes a price. A stock option might sell for a price of $70. If this option is later traded on an exchange, its price might vary from moment to moment as the markets move. Real options do not tend to have explicit prices, but sometimes implicit prices can be calculated. For example, if a ﬂexible-fuel car costs $5,000 more than an otherwise equivalent car that burns one type of
fuel, then this $5,000 could be viewed as the option price. Model of real options Let x embody all the uncertainty in the economic environment. In the case of the ﬂexible-fuel car, x might reﬂect the price of fossil fuels relative to biofuels or the stringency of government regulation of fossil fuels. In terms of the section on statistics in Chapter 2, x is a random variable (sometimes referred to as the ‘‘state of the world’’) that can take on possibly many different values. The individual has some number, I 1, … , n, of choices currently available. Let Ai (x) be the payoffs provided by choice i, where the argument (x) allows each choice to provide a different pattern of returns depending on how the future turns out. ¼ Figure 7.2a illustrates the case of two choices. The ﬁrst choice provides a decreasing payoff as x increases, indicated by the downward slope of A1. This might correspond to ownership of a car that runs only on fossil fuels; as biofuels become more important than fossil fuels, the value of a car burning only fossil fuels decreases. The second choice provides an increasing payoff, perhaps corresponding to ownership of a car that runs only on biofuels. Figure 7.2b translates the payoffs into (von Neumann–Morgenstern) utilities that the person obtains from the payoffs by graphing U(Ai) rather than Ai. The bend introduced in moving from payoffs to utilities reﬂects the diminishing marginal utility from higher payoffs for a risk-averse person. If the person does not have the ﬂexibility provided by a real option, he or she must make the choice before observing how the state x turns out. The individual should choose the single alternative that is best on average. His or her expected utility from this choice is max U E f ½ A1Þ AnÞ’g : ð (7.50) 226 Part 3: Uncertainty and Strategy FIGURE 7.2 The Nature of a Real Option Panel (a) shows the payoffs and panel (b) shows the utilities provided by two alternatives across states of the world (x). If the decision has to be made upfront, the individual chooses the single curve having the highest expected utility. If the real option to make either decision can be preserved until later, the individual can obtain the expected utility of the upper envelope of the curves, shown in bold. Payoff Utility A2 U(A2) U(A1) State x x′ A1 State x x′ (a) Payoffs from alternatives (b) Utilities from alternatives Figure 7.2 does not provide enough information to judge which expected utility is higher because we do not know the likelihoods of the different x’s, but if the x’s are about equally likely, then it looks as though the individual would choose the second alternative, providing higher utility over a larger range. The individual’s expected utility from this choice is E[U(A2)]. On the other hand, if the real option can be preserved to make a choice that responds to which state of the world x has occurred, the person will be better off. In the car application, the real option could correspond to buying a ﬂexible-fuel car, which does not lock the buyer into one fuel but allows the choice of whatever fuel turns out to be most common or inexpensive over the life of the car. In Figure 7.2, rather than choosing a single alternative, the person would choose the ﬁrst option if x < x 0 and the second option if x > x 0. The utility provided by this strategy is given by the bold curve, which is the ‘‘upper envelope’’ of the curves for the individual options. With a general number (n) of choices, expected utility from this upper envelope of individual options is E max f ½ U , . . . , U A1Þ ð A1Þ’g : ð (7.51) The expected utility in Equation 7.51 is higher than in 7.50. This may not be obvious at ﬁrst glance because it seems that simply swapping the order of the expectations and ‘‘max’’ operators should not make a difference. But indeed it does. Whereas Equation 7.50 is the expected utility associated with the best single utility curve, Equation 7.51 is the expected utility associated with the upper envelope of all the utility curves.19 19The result can be proved formally using Jensen’s inequality, introduced in footnote 10. The footnote discusses the implications of Jensen’s inequality for concave functions: E[ f (x)] f [E(x)]. Jensen’s inequality has the reverse implication for convex functions: E[ f (x)] f [E(x)]. In other words, for convex functions, the result is greater if the expectations operator is applied outside of the function than if the order of the two is reversed. In the options context, the ‘‘max’’ operator has the properties of a convex function. This can be seen from Figure 7.2b, where taking the upper envelope ‘‘convexiﬁes’’ the individual curves, turning them into more of a V-shape. + ) Chapter 7: Uncertainty 227 FIGURE 7.3 More Options Cannot Make the Individual Decision-Maker Worse Off The addition of a third alternative to the two drawn in Figure 7.2 is valuable in (a) because it shifts the upper envelope (shown in bold) of utilities up. The new alternative is worthless in (b) because it does not shift the upper envelope, but the individual is not worse off for having it. Utility Utility U(A3) U(A2) U(A1) State x U(A2) U(A3) U(A1) State x (a) Additional valuable option (b) Additional worthless option More options are better (generally) Adding more options can never harm an individual decision-maker (as long as he or she is not charged for them) because the extra options can always be ignored. This is the essence of options: They give the holder the right—but not the obligation—to choose them. Figure 7.3 illustrates this point, showing the effect of adding a third option to the two drawn in Figure 7.2. In the ﬁrst panel, the person strictly beneﬁts from the third option because there are some states of the world (the highest values of x in the ﬁgure) for which it is better than any other alternative, shifting the upper envelope of utilities (the bold curve) up. The third option is worthless in the second panel. Although the third option is not the worst option for many states of the world, it is never the best and so does not improve the upper envelope of utilities relative to Figure 7.2. Still, the addition of the third option is not harmful. This insight may no longer hold in a strategic setting with multiple decision-makers. In a strategic setting, economic actors may beneﬁt from having some of their options cut off. This may allow a player to commit to a narrower course of action that he or she would not have chosen otherwise, and this commitment may affect the actions of other parties, possibly to the beneﬁt of the party making the commitment. A famous illustration of this point is provided in one of the earliest treatises on military strategy, by Sun Tzu, a Chinese general writing in 400 BC. It seems crazy for an army to destroy all means of retreat, burning bridges behind itself and sinking its own ships, among other measures. Yet this is what Sun Tzu advocated as a military tactic. If the second army observes that the ﬁrst cannot retreat and will ﬁght to the death, it may retreat itself before engaging the ﬁrst. We will analyze such strategic issues more formally in the next chapter on game theory. Computing option value We can push the analysis further to derive a mathematical expression for the value of a real option. Let F be the fee that has to be paid for the ability to choose the best 228 Part 3: Uncertainty and Strategy alternative after x has been realized instead of before. The individual would be willing to pay the fee as long as E max f U ½ x A1ð Anð ð ½ Þ % F Þ’g + max U E ½ f x A1ð ð ÞÞ Anð ð : ÞÞ’g (7.52) The right side is the expected utility from making the choice beforehand, repeated from Equation 7.50. The left side allows for the choice to be made after x has occurred, a beneﬁt, but subtracts the fee for option from every payoff. The fee is naturally assumed to be paid up front, and thus reduces wealth by F whichever option is chosen later. The real option’s value is the highest F for which Equation 7.52 is still satisﬁed, which of course is the F for which the condition holds with equality. EXAMPLE 7.5 Value of a Flexible-Fuel Car 1 % x be the payoff from a fossil-fuel–only car and A2(x) Let’s work out the option value provided by a ﬂexible-fuel car in a numerical example. Let A1(x) x be the payoff from a biofuel¼ only car. The state of the world, x, reﬂects the relative importance of biofuels compared with fossil fuels over the car’s lifespan. Assume x is a random variable that is uniformly distributed between 0 and 1 (the simplest continuous random variable to work with here). The statistics section in Chapter 2 provides some detail on the uniform distribution, showing that the 1 in the special case when the uniform random probability density function (PDF) is f (x) variable ranges between 0 and 1. ¼ ¼ Risk neutrality. To make the calculations as easy as possible to start, suppose ﬁrst that the car buyer is risk neutral, obtaining a utility level equal to the payoff level. Suppose the buyer is forced to choose a biofuel car. This provides an expected utility of E A2’ ¼ ½ 1 0 ð x A2ð f Þ x ð Þ dx ¼ x dx x2 7.53) x & & & & where the integral simpliﬁes because f (x) 1. Similar calculations show that the expected utility from buying a fossil-fuel car is also 1/2. Therefore, if only single-fuel cars are available, the person is indifferent between them, obtaining expected utility 1/2 from either. ¼ Now suppose that a ﬂexible-fuel car is available, which allows the buyer to obtain either A1(x) or A2(x), whichever is higher under the latter circumstances. The buyer’s expected utility from this car is max E ½ A1, A2Þ’ ¼ ð ¼ 1 max ð % x, x f x ð Þ Þ dx ¼ 1 x dx x2 dx x Þ % þ x dx 1 1 2 ð (7.54) The second line in Equation 7.54 follows from the fact that the two integrals in the preceding expression are symmetric. Because the buyer’s utility exactly equals the payoffs, we can compute the option value of the ﬂexible-fuel car directly by taking
the difference between the expected payoffs in Equations 7.53 and 7.54, which equals 1/4. This is the maximum premium the person would pay for the ﬂexible-fuel car over a single-fuel car. Scaling payoffs to more realistic levels by multiplying by, say, $10,000, the price premium (and the option value) of the ﬂexible-fuel car would be $2,500. This calculation demonstrates the general insight that options are a way of dealing with uncertainty that have value even for risk-neutral individuals. The next part of the example investigates whether risk aversion makes options more or less valuable. Chapter 7: Uncertainty 229 Risk aversion. Now suppose the buyer is risk averse, having von Neumann–Morgenstern utility function U xp . The buyer’s expected utility from a biofuel car is x ð Þ ¼ ﬃﬃﬃ A2Þ A2ð Þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p dx f x ð Þ ¼ 1 2 dx 7.55) which is the same as from a fossil-fuel car, as similar calculations show. Therefore, a single-fuel car of whatever type provides an expected utility of 2/3. The expected utility from a ﬂexible-fuel car that costs F more than a single-fuel car is E max ½ f U A1ð x ð Þ % F , U Þ A2ð ð x Þ % F Þ’g ¼ ¼ ¼ ¼ max p 1 ð % x % F , p dx x ð Þ 1 x F f Þ % ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p x F % ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dx 2 ¼ 1 p 1 ð 2 x F dx % ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 0 ð 1 2 0 ð 2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F x dx % p 1 % ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F 1 % 1 2 du 7.56) FIGURE 7.47Graphical Method for Computing the Premium for a Flexible-Fuel Car To ﬁnd the maximum premium F that the risk-averse buyer would be willing to pay for the ﬂexible-fuel car, we plot the expected utility from a single-fuel car from Equation 7.55 and from the ﬂexible-fuel car from Equation 7.56 and see the value of F where the curves cross. Expected utility 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Single fuel Flexible fuel 0.0 0.00 0.10 0.20 0.30 0.40 F 0.50 230 Part 3: Uncertainty and Strategy The calculations involved in Equation 7.56 are somewhat involved and thus require some discussion. The second line relies on the symmetry of the two integrals appearing there, which allows us to collapse them into two times the value of one of them, and we chose the simpler of the two for these purposes. The third line uses the change of variables u F to simplify the integral. (See Equation 2.135 in Chapter 2 for another example of the change-of-variables trick and further discussion.) % ¼ x To ﬁnd the maximum premium the buyer would pay for a ﬂexible-fuel car, we can set Equations 7.55 and 7.56 equal and solve for F. Unfortunately the resulting equation is too complicated to be solved analytically. One simple approach is to graph the last line of Equation 7.56 for a range of values of F and eyeball where the graph hits the required value of 2/3 from Equation 7.55. This is done in Figure 7.4, where we see that this value of F is slightly less than 0.3 (0.294 to be more precise). Therefore, the risk-averse buyer is willing to pay a premium of 0.294 for the ﬂexible-fuel car, which is also the option value of this type of car. Scaling up by $10,000 for more realistic monetary values, the price premium would be $2,940. This is $440 more than the risk-neutral buyer was willing to pay. Thus, the option value is greater in this case for the risk-averse buyer. QUERY: Does risk aversion always increase option value? If so, explain why. If not, modify the example with different shapes to the payoff functions to provide an example where the riskneutral buyer would pay more. Option value of delay Society seems to frown on procrastinators. ‘‘Do not put off to tomorrow what you can do today’’ is a familiar maxim. Yet the existence of real options suggests a possible value in procrastination. There may be a value in delaying big decisions—such as the purchase of a car—that are not easily reversed later. Delaying these big decisions allows the decisionmaker to preserve option value and gather more information about the future. To the outside observer, who may not understand all the uncertainties involved in the situation, it may appear that the decision-maker is too inert, failing to make what looks to be the right decision at the time. In fact, delaying may be exactly the right choice to make in the face of uncertainty. Choosing one course of action rules out other courses later. Delay preserves options. If circumstances continue to be favorable or become even more so, the action can still be taken later. But if the future changes and the action is unsuitable, the decision-maker may have saved a lot of trouble by not making it. The value of delay can be seen by returning to the car application. Suppose for the sake of this example that only single-fuel cars (of either type, fossil fuel or biofuel) are available on the market; ﬂexible-fuel cars have not yet been invented. Even if circumstances start to favor the biofuel car, with the number of ﬁlling stations appearing to tip toward offering biofuels, the buyer may want to hold off buying a car until he or she is more sure. This may be true even if the buyer is forgoing considerable consumer surplus from the use of a new car during the period of delay. The problem is that if biofuels do not end up taking over the market, the buyer may be left with a car that is hard to fuel up and hard to trade in for a car burning the other fuel type. The buyer would be willing to experience delay costs up to F to preserve ﬂexibility. The value of delay hinges on the irreversibility of the underlying decision. If in the car example the buyer manufacturer could recover close to the purchase price by selling it on the used-car market, there would be no reason to delay purchasing. But it is well known that the value of a new car decreases precipitously once it is driven off the car lot (we will discuss reasons for this including the ‘‘lemons effect’’ in Chapter 18); therefore, it may not be so easy to reverse the purchase of a car. Chapter 7: Uncertainty 231 Implications for cost–beneﬁt analysis To an outside observer, delay may seem like a symptom of irrationality or ignorance. Why is the decision-maker overlooking an opportunity to take a beneﬁcial action? The chapter has now provided several reasons why a rational decision-maker might not want to pursue an action even though the expected beneﬁts from the action outweigh the expected costs. First, a risk-averse individual might avoid a gamble even if it provided a positive expected monetary payoff (because of the decreasing marginal utility from money). And option value provides a further reason for the action not to be undertaken: The decision-maker might be delaying until he or she has more certainty about the potential results of the decision. Many of us have come across the cost–beneﬁt rule, which says that an action should be taken if anticipated costs are less than beneﬁts. This is generally a sensible rule, providing the correct course of action in simple settings without uncertainty. One must be more careful in applying the rule in settings involving uncertainty. The correct decision rule is more complicated because it should account for risk preferences (by converting payoffs into utilities) and for the option value of delay, if present. Failure to apply the simple cost–beneﬁt rule in settings with uncertainty may indicate sophistication rather than irrationality.20 Information The fourth method of reducing the uncertainty involved in a situation is to acquire better information about the likely outcome that will arise. We have already considered a version of this in the previous section, where we considered the strategy of preserving options while delaying a decision until better information is received. Delay involved some costs, which can be thought of as a sort of ‘‘purchase price’’ for the information acquired. Here, we will be more direct in considering information as a good that can be purchased directly and analyze in greater detail why and how much individuals are willing to pay for it. Information as a good By now it should be clear to the reader that information is a valuable economic resource. We have seen an example already: A buyer can make a better decision about which type of car to buy if he or she has better information about the sort of fuels that will be readily available during the life of the car. But the examples do not end there. Shoppers who know where to buy high-quality goods cheaply can make their budgets stretch further than those who do not; doctors can provide better medical care if they are up to date on the latest scientiﬁc research. The study of information economics has become one of the major areas in current research. Several challenges are involved. Unlike the consumer goods we have been studying thus far, information is difﬁcult to quantify. Even if it could be quantiﬁed, information has some technical properties that make it an unusual sort of good. Most information is durable and retains value after it has been used. Unlike a hot dog, which is consumed only once, knowledge of a special sale can be used not only by the person who 20Economists are puzzled by consumers’ reluctance to install efﬁcient appliances even though the savings on energy bills are likely to defray the appliances’ purchase price before long. An explanation from behavioral economics is that consumers are too ignorant to perform the cost–beneﬁt calculations or are too impatient to wait for the energy savings to accumulate. K. Hassett and G. Metcalf, in ‘‘Energy Conservation Investment: Do Consumers Discount the Future Correctly?’’ Energy Policy (June 1993): 710–16, suggest that consumer inertia may be rational delay in the face of ﬂuctuating energy prices. See Problem 7.9 for a related numerical example. 232 Part 3: Uncertainty and Strategy discovers it but also by anyone else with whom the information is shared. The friends then may gain from this information even though they do not have to spend anything to obtain it. Indeed, in a special case of this situation, information has the characteristi
c of a pure public good (see Chapter 19). That is, the information is both nonrival, in that others may use it at zero cost, and nonexclusive, in that no individual can prevent others from using the information. The classic example of these properties is a new scientiﬁc discovery. When some prehistoric people invented the wheel, others could use it without detracting from the value of the discovery, and everyone who saw the wheel could copy it freely. Information is also difﬁcult to sell because the act of describing the good that is being offered to a potential consumer gives it away to them. These technical properties of information imply that market mechanisms may often operate imperfectly in allocating resources to information provision and acquisition. After all, why invest in the production of information when one can just acquire it from others at no cost? Therefore, standard models of supply and demand may be of relatively limited use in understanding such activities. At a minimum, models have to be developed that accurately reﬂect the properties being assumed about the informational environment. Throughout the latter portions of this book, we will describe some of the situations in which such models are called for. Here, however, we will pay relatively little attention to supply–demand equilibria and will instead focus on an example that illustrates the value of information in helping individuals make choices under uncertainty. Quantifying the value of information We already have all the tools needed to quantify the value of information from the section on option values. Suppose again that the individual is uncertain about what the state of the world (x) will be in the future. He or she needs to make one of n choices today (this allows us to put aside the option value of delay and other issues we have already studied). As before, Ai(x) represents the payoffs provided by choice i. Now reinterpret F as the fee charged to be told the exact value that x will take on in the future (perhaps this is the salary of the economist hired to make such forecasts). The same calculations from the option section can be used here to show that the maximum such F is again the value for which Equation 7.52 holds with equality. Just as this was the value of the real option in that section, here it is the value of information. The value of information would be lower than this F if the forecast of future conditions were imperfect rather than perfect as assumed here. Other factors affecting an individual’s value of information include the extent of uncertainty before acquiring the information, the number of options he or she can choose between, and his or her risk preferences. The more uncertainty resolved by the new information, the more valuable it is, of course. If the individual does not have much scope to respond to the information because of having only a limited range of choices to make, the information will not be valuable. The degree of risk aversion has ambiguous effects on the value of information (answering the Query in Example 7.5 will provide you with some idea why). The State-Preference Approach to Choice Under Uncertainty Although our analysis in this chapter has offered insights on a number of issues, it seems rather different from the approach we took in other chapters. The basic model of utility maximization subject to a budget constraint seems to have been lost. To make further progress in the study of behavior under uncertainty, we will develop some new Chapter 7: Uncertainty 233 techniques that will permit us to bring the discussion of such behavior back into the standard choice-theoretic framework. States of the world and contingent commodities We start by pushing a bit further on an idea already mentioned, thinking about an uncertain future in term of states of the world. We cannot predict exactly what will happen, say, tomorrow, but we assume that it is possible to categorize all the possible things that might happen into a ﬁxed number of well-deﬁned states. For example, we might make the crude approximation of saying that the world will be in only one of two possible states tomorrow: It will be either ‘‘good times’’ or ‘‘bad times.’’ One could make a much ﬁner gradation of states of the world (involving even millions of possible states), but most of the essentials of the theory can be developed using only two states. A conceptual idea that can be developed concurrently with the notion of states of the world is that of contingent commodities. These are goods delivered only if a particular state of the world occurs. As an example, ‘‘$1 in good times’’ is a contingent commodity that promises the individual $1 in good times but nothing should tomorrow turn out to be bad times. It is even possible, by stretching one’s intuitive ability somewhat, to conceive of being able to purchase this commodity: I might be able to buy from someone the promise of $1 if tomorrow turns out to be good times. Because tomorrow could be bad, this good will probably sell for less than $1. If someone were also willing to sell me the contingent commodity ‘‘$1 in bad times,’’ then I could assure myself of having $1 tomorrow by buying the two contingent commodities ‘‘$1 in good times’’ and ‘‘$1 in bad times.’’ Utility analysis Examining utility-maximizing choices among contingent commodities proceeds formally in much the same way we analyzed choices previously. The principal difference is that, after the fact, a person will have obtained only one contingent good (depending on whether it turns out to be good or bad times). Before the uncertainty is resolved, however, the individual has two contingent goods from which to choose and will probably buy some of each because he or she does not know which state will occur. We denote these two contingent goods by Wg (wealth in good times) and Wb (wealth in bad times). Assuming that utility is independent of which state occurs21 and that this individual believes that good times will occur with probability p, the expected utility associated with these two contingent goods is WgÞ þ ð ð This is the magnitude this individual seeks to maximize given his or her initial wealth, W. Wg, WbÞ ¼ ð : WbÞ (7.57) p Þ pU % U V 1 ð Prices of contingent commodities Assuming that this person can purchase $1 of wealth in good times for pg and $1 of wealth in bad times for pb, his or her budget constraint is then pgWg þ The price ratio pg /pb shows how this person can trade dollars of wealth in good times for 0.20, the sacriﬁce of $1 of wealth dollars in bad times. If, for example, pg ¼ 0.80 and pb ¼ pbWb: (7.58) W ¼ 21This assumption is untenable in circumstances where utility of wealth depends on the state of the world. For example, the utility provided by a given level of wealth may differ depending on whether an individual is ‘‘sick’’ or ‘‘healthy.’’ We will not pursue such complications here, however. For most of our analysis, utility is assumed to be concave in wealth: U 0(W) > 0, U 00(W) < 0. 234 Part 3: Uncertainty and Strategy in good times would permit this person to buy contingent claims yielding $4 of wealth should times turn out to be bad. Whether such a trade would improve utility will, of course, depend on the speciﬁcs of the situation. But looking at problems involving uncertainty as situations in which various contingent claims are traded is the key insight offered by the state-preference model. Fair markets for contingent goods If markets for contingent wealth claims are well developed and there is general agreement about the likelihood of good times (p), then prices for these claims will be actuarially fair—that is, they will equal the underlying probabilities: pg ¼ pb ¼ Hence the price ratio pg /pb will simply reﬂect the odds in favor of good times: pg pb ¼ p, 1 p: % p p 1 : (7.59) (7.60) In our previous example, if pg ¼ 4. In this case the odds in favor of good times would be stated as ‘‘4 to 1.’’ Fair markets for contingent claims (such as insurance markets) will also reﬂect these odds. An analogy is provided by the ‘‘odds’’ quoted in horse races. These odds are ‘‘fair’’ when they reﬂect the true probabilities that various horses will win. 0.2, then p/(1 p) p) (1 % ¼ % ¼ ¼ p % 0.8 and pb ¼ Risk aversion We are now in a position to show how risk aversion is manifested in the state-preference model. Speciﬁcally, we can show that, if contingent claims markets are fair, then a utilitymaximizing individual will opt for a situation in which Wg ¼ Wb; that is, he or she will arrange matters so that the wealth ultimately obtained is the same no matter what state occurs. As in previous chapters, maximization of utility subject to a budget constraint requires that this individual set the MRS of Wg for Wb equal to the ratio of these ‘‘goods’’ prices: MRS @V=@Wg @V=@Wb ¼ ¼ pU 0 p WgÞ ð U 0ð Þ WbÞ pg pb ¼ 1 ð : (7.61) % In view of the assumption that markets for contingent claims are fair (Equation 7.60), this ﬁrst-order condition reduces to or22 U 0 WgÞ ð U 0ð WbÞ 1 ¼ Wg ¼ Wb: (7.62) Hence this individual, when faced with fair markets in contingent claims on wealth, will be risk averse and will choose to ensure that he or she has the same level of wealth regardless of which state occurs. 22This step requires that utility be state independent and that U 0(W) > 0. Chapter 7: Uncertainty 235 FIGURE 7.5 Risk Aversions in the State-Preference Model The line I represents the individual’s budget constraint for contingent wealth claims: W = pgWg + pbWb. If the market for contingent claims is actuarially fair [pg /pb = p/(1 p)], then utility maximization will occur on the certainty line where Wg = Wb = W,. If prices are not actuarially fair, the budget constraint may resemble I 0, and utility maximization will occur at a point where Wg > Wb. % Wb W* Certainty line I I′ U1 Wg W* A graphic analysis Figure 7.5 illustrates risk aversion with a graph. This individual’s budget constraint (I) is shown to be tangent t
o the U1 indifference curve where Wg ¼ Wb—a point on the ‘‘certainty line’’ where wealth (W,) is independent of which state of the world occurs. At W, the slope of the indifference curve [p /(1 p)] is precisely equal to the price ratio pg /pb. If the market for contingent wealth claims were not fair, utility maximization might not occur on the certainty line. Suppose, for example, that p /(1 4 but that pg /pb ¼ 2 because ensuring wealth in bad times proves costly. In this case the budget constraint would resemble line I0 in Figure 7.5, and utility maximization would occur below the certainty line.23 In this case this individual would gamble a bit by opting for Wg > Wb because claims on Wb are relatively costly. Example 7.6 shows the usefulness of this approach in evaluating some of the alternatives that might be available. p) ¼ % % EXAMPLE 7.6 Insurance in the State-Preference Model We can illustrate the state-preference approach by recasting the auto insurance illustration from Example 7.2 as a problem involving the two contingent commodities ‘‘wealth with no theft’’ (Wg) and ‘‘wealth with a theft’’ (Wb). If, as before, we assume logarithmic utility and that the probability of a theft (i.e., 1 p) is 0.25, then % 23Because (as Equation 7.61 shows) the MRS on the certainty line is always p /(1 must occur below the line. % p), tangencies with a ﬂatter slope than this 236 Part 3: Uncertainty and Strategy expected utility 0:75U WgÞ þ ð 0:75 ln Wg þ 0:25U WbÞ ð 0:25 ln Wb: ¼ ¼ (7.63) If the individual takes no action, then utility is determined by the initial wealth endowment, W,g ¼ 100,000 and W,b ¼ 80,000, so expected utility 0:75 ln 100,000 11:45714: þ ¼ ¼ 0:25 ln 80,000 (7.64) To study trades away from these initial endowments, we write the budget constraint in terms of the prices of the contingent commodities, pg and pb: pgW,g þ Assuming that these prices equal the probabilities of the two states ( pg ¼ constraint can be written pbW,b ¼ pgWg þ pbWb: (7.65) 0.75, pb ¼ 0.25), this 0:75 100,000 ð 0:25 80,000 ð 95,000 0:75Wg þ 0:25Wb; Þ þ Þ ¼ that is, the expected value of wealth is $95,000, and this person can allocate this amount between Wb Wg and Wb. Now maximization of utility with respect to this budget constraint yields Wg ¼ 95,000. Consequently, the individual will move to the certainty line and receive an expected ¼ utility of ¼ (7.66) expected utility ln 95,000 ¼ ¼ 11:46163; (7.67) a clear improvement over doing nothing. To obtain this improvement, this person must be able to transfer $5,000 of wealth in good times (no theft) into $15,000 of extra wealth in bad times (theft). A fair insurance contract would allow this because it would cost $5,000 but return $20,000 should a theft occur (but nothing should no theft occur). Notice here that the wealth changes promised by insurance—dWb/dWg ¼ 3—exactly equal the negative p/(1 of the odds ratio ¼ % % 15,000/ 3. 0.75/0.25 5,000 ¼ % ¼ % p) % % A policy with a deductible provision. A number of other insurance contracts might be utility improving in this situation, although not all of them would lead to choices that lie on the certainty line. For example, a policy that cost $5,200 and returned $20,000 in case of a theft would permit this person to reach the certainty line with Wg ¼ 94,800 and expected utility ln 94,800 ¼ ¼ (7.68) Wb ¼ 11:45953; which also exceeds the utility obtainable from the initial endowment. A policy that costs $4,900 and requires the individual to incur the ﬁrst $1,000 of a loss from theft would yield Wg ¼ Wb ¼ then 100,000 80,000 4,900 4,900 95,100, ¼ 19,000 % % þ 94,100; ¼ expected utility 0:75 ln 95,100 11:46004: þ ¼ ¼ 0:25 ln 94,100 (7.69) (7.70) Although this policy does not permit this person to reach the certainty line, it is utility improving. Insurance need not be complete to offer the promise of higher utility. QUERY: What is the maximum amount an individual would be willing to pay for an insurance policy under which he or she had to absorb the ﬁrst $1,000 of loss? Chapter 7: Uncertainty 237 Risk aversion and risk premiums The state-preference model is also especially useful for analyzing the relationship between risk aversion and individuals’ willingness to pay for risk. Consider two people, each of whom starts with a certain wealth, W,. Each person seeks to maximize an expected utility function of the form V Wg, WbÞ ¼ ð p WR g R þ ð 1 WR b R : p Þ % (7.71) Here the utility function exhibits constant relative risk aversion (see Example 7.4). Notice also that the function closely resembles the CES utility function we examined in Chapter 3 and elsewhere. The parameter R determines both the degree of risk aversion and the degree of curvature of indifference curves implied by the function. A risk-averse individual will have a large negative value for R and have sharply curved indifference curves, such as U1 shown in Figure 7.6. A person with more tolerance for risk will have a higher value of R and ﬂatter indifference curves (such as U2).24 Suppose now these individuals are faced with the prospect of losing h dollars of wealth in bad times. Such a risk would be acceptable to individual 2 if wealth in good times were to increase from W, to W2. For the risk-averse individual 1, however, wealth would have FIGURE 7.6 Risk Aversion and Risk Premiums Indifference curve U1 represents the preferences of a risk-averse person, whereas the person with preferences represented by U2 is willing to assume more risk. When faced with the risk of losing h in bad times, person 2 will require compensation of W2 % larger amount given by W1 % W, in good times, whereas person 1 will require a W,. Wb W* W* − h Certainty line U1 U2 Wg W* W1W2 24Tangency of U1 and U2 at W, is ensured because the MRS along the certainty line is given by p /(1 value of R. % p) regardless of the 238 Part 3: Uncertainty and Strategy to increase to W1 to make the risk acceptable. Therefore, the difference between W1 and W2 indicates the effect of risk aversion on willingness to assume risk. Some of the problems in this chapter make use of this graphic device for showing the connection between preferences (as reﬂected by the utility function in Equation 7.71) and behavior in risky situations. Asymmetry of Information One obvious implication of the study of information acquisition is that the level of information that an individual buys will depend on the per-unit price of information messages. Unlike the market price for most goods (which we usually assume to be the same for everyone), there are many reasons to believe that information costs may differ signiﬁcantly among individuals. Some individuals may possess speciﬁc skills relevant to information acquisition (e.g., they may be trained mechanics), whereas others may not possess such skills. Some individuals may have other types of experience that yield valuable information, whereas others may lack that experience. For example, the seller of a product will usually know more about its limitations than will a buyer because the seller will know precisely how the good was made and where possible problems might arise. Similarly, large-scale repeat buyers of a good may have greater access to information about it than would ﬁrst-time buyers. Finally, some individuals may have invested in some types of information services (e.g., by having a computer link to a brokerage ﬁrm or by subscribing to Consumer Reports) that make the marginal cost of obtaining additional information lower than for someone without such an investment. All these factors suggest that the level of information will sometimes differ among the participants in market transactions. Of course, in many instances, information costs may be low and such differences may be minor. Most people can appraise the quality of fresh vegetables fairly well just by looking at them, for example. But when information costs are high and variable across individuals, we would expect them to ﬁnd it advantageous to acquire different amounts of information. We will postpone a detailed study of such situations until Chapter 18. SUMMARY The goal of this chapter was to provide some basic material for the study of individual behavior in uncertain situations. The key concepts covered are listed as follows. lute risk aversion (CARA) function and the constant relative risk aversion (CRRA) function. Neither is completely satisfactory on theoretical grounds. • The most common way to model behavior under uncertainty is to assume that individuals seek to maximize the expected utility of their actions. • Individuals who exhibit a diminishing marginal utility of wealth are risk averse. That is, they generally refuse fair bets. • Risk-averse individuals will wish to insure themselves completely against uncertain events if insurance premiums are actuarially fair. They may be willing to pay more than actuarially fair premiums to avoid taking risks. • Two utility functions have been extensively used in the study of behavior under uncertainty: the constant abso- • Methods for reducing the risk involved in a situation include transferring risk to those who can bear it more effectively through insurance, spreading risk across several activities through diversiﬁcation, preserving options for dealing with the various outcomes that arise, and acquiring information to determine which outcomes are more likely. • One of the most extensively studied issues in the economics of uncertainty is the ‘‘portfolio problem,’’ which asks how an investor will split his or her wealth among available assets. A simple version of the problem is used to illustrate the value of diversiﬁcation in the text; the Extensions provide a detailed analysis. Chapter 7: Uncertainty 239 • Information is valuable because it permits individuals to make better decisions in uncertain situations. Information can be most valuable when individuals have some ﬂexibility in their decision making. • The state-preference approach allows decision making under unc
ertainty to be approached in a familiar choice-theoretic framework. PROBLEMS 7.1 George is seen to place an even-money $100,000 bet on the Bulls to win the NBA Finals. If George has a logarithmic utilityof-wealth function and if his current wealth is $1,000,000, what must he believe is the minimum probability that the Bulls will win? 7.2 Show that if an individual’s utility-of-wealth function is convex then he or she will prefer fair gambles to income certainty and may even be willing to accept somewhat unfair gambles. Do you believe this sort of risk-taking behavior is common? What factors might tend to limit its occurrence? 7.3 An individual purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all the eggs carried on any one trip will be broken during the trip. The individual considers two strategies: (1) take all 12 eggs in one trip; or (2) take two trips with 6 eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on average, 6 eggs will remain unbroken after the trip home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. Which strategy will be preferable? c. Could utility be improved further by taking more than two trips? How would this possibility be affected if additional trips were costly? 7.4 Suppose there is a 50–50 chance that a risk-averse individual with a current wealth of $20,000 will contract a debilitating disease and suffer a loss of $10,000. a. Calculate the cost of actuarially fair insurance in this situation and use a utility-of-wealth graph (such as shown in Figure 7.1) to show that the individual will prefer fair insurance against this loss to accepting the gamble uninsured. b. Suppose two types of insurance policies were available: (1) a fair policy covering the complete loss; and (2) a fair policy covering only half of any loss incurred. Calculate the cost of the second type of policy and show that the individual will generally regard it as inferior to the ﬁrst. 7.5 Ms. Fogg is planning an around-the-world trip on which she plans to spend $10,000. The utility from the trip is a function of how much she actually spends on it (Y), given by U Y ð Þ ¼ ln Y: a. If there is a 25 percent probability that Ms. Fogg will lose $1,000 of her cash on the trip, what is the trip’s expected utility? b. Suppose that Ms. Fogg can buy insurance against losing the $1,000 (say, by purchasing traveler’s checks) at an ‘‘actuarially fair’’ premium of $250. Show that her expected utility is higher if she purchases this insurance than if she faces the chance of losing the $1,000 without insurance. c. What is the maximum amount that Ms. Fogg would be willing to pay to insure her $1,000? 240 Part 3: Uncertainty and Strategy 7.6 In deciding to park in an illegal place, any individual knows that the probability of getting a ticket is p and that the ﬁne for receiving the ticket is f. Suppose that all individuals are risk averse (i.e., U 00(W) < 0, where W is the individual’s wealth). Will a proportional increase in the probability of being caught or a proportional increase in the ﬁne be a more effective deterrent to illegal parking? Hint: Use the Taylor series approximation U(W f ) % ¼ U(W) % f U 0(W) þ ( f 2/2)U 00(W ). 7.7 A farmer believes there is a 50–50 chance that the next growing season will be abnormally rainy. His expected utility function has the form expected utility 1 2 ¼ ln Y NR þ 1 2 ln Y R, where YNR and YR represent the farmer’s income in the states of ‘‘normal rain’’ and ‘‘rainy,’’ respectively. a. Suppose the farmer must choose between two crops that promise the following income prospects: Crop Wheat Corn YNR YR $28,000 $19,000 $10,000 $15,000 Which of the crops will he plant? b. Suppose the farmer can plant half his ﬁeld with each crop. Would he choose to do so? Explain your result. c. What mix of wheat and corn would provide maximum expected utility to this farmer? d. Would wheat crop insurance—which is available to farmers who grow only wheat and which costs $4,000 and pays off $8,000 in the event of a rainy growing season—cause this farmer to change what he plants? 7.8 In Equation 7.30 we showed that the amount an individual is willing to pay to avoid a fair gamble (h) is given by 0.5E(h2)r(W ), where r(W ) is the measure of absolute risk aversion at this person’s initial level of wealth. In this p problem we look at the size of this payment as a function of the size of the risk faced and this person’s level of wealth. ¼ a. Consider a fair gamble (v) of winning or losing $1. For this gamble, what is E(v2)? b. Now consider varying the gamble in part (a) by multiplying each prize by a positive constant k. Let h value of E(h2)? kv. What is the ¼ c. Suppose this person has a logarithmic utility function U(W) d. Compute the risk premium ( p) for k ¼ 0.5, 1, and 2 and for W ¼ ln W. What is a general expression for r(W )? 10 and 100. What do you conclude by comparing the six ¼ values? 7.9 Return to Example 7.5, in which we computed the value of the real option provided by a ﬂexible-fuel car. Continue to assume that the payoff from a fossil-fuel–burning car is A1(x) x. Now assume that the payoff from the biofuel car is higher, A2(x) 2x. As before, x is a random variable uniformly distributed between 0 and 1, capturing the relative availability of biofuels versus fossil fuels on the market over the future lifespan of the car. % ¼ ¼ 1 a. Assume the buyer is risk neutral with von Neumann–Morgenstern utility function U (x) ¼ a ﬂexible-fuel car that allows the buyer to reproduce the payoff from either single-fuel car. x. Compute the option value of b. Repeat the option value calculation for a risk-averse buyer with utility function U c. Compare your answers with Example 7.5. Discuss how the increase in the value of the biofuel car affects the option value x ð Þ ¼ xp . ﬃﬃﬃ provided by the ﬂexible-fuel car. Analytical Problems Chapter 7: Uncertainty 241 7.10 HARA Utility The CARA and CRRA utility functions are both members of a more general class of utility functions called harmonic absolute g, where the various parameters risk aversion (HARA) functions. The general form for this function is U (W ) obey the following restrictions: y(m + W/g) ½ð 1, W=g > 0, =g Þ % g ’ > 0. The reasons for the ﬁrst two restrictions are obvious; the third is required so that U 0 > 0. a. Calculate r (W ) for this function. Show that the reciprocal of this expression is linear in W. This is the origin of the term harmonic in the function’s name. b. Show that when m note 17). 0 and y [(1 ¼ % ¼ g)/g] g % 1, this function reduces to the CRRA function given in Chapter 7 (see foot- c. Use your result from part (a) to show that if g ﬁ d. Let the constant found in part (c) be represented by A. Show that the implied form for the utility function in this case is , then r (W ) is a constant for this function. 1 the CARA function given in Equation 7.35. e. Finally, show that a quadratic utility function can be generated from the HARA function simply by setting g f. Despite the seeming generality of the HARA function, it still exhibits several limitations for the study of behavior in uncer- ¼ % 1. tain situations. Describe some of these shortcomings. 7.11 Prospect theory Two pioneers of the ﬁeld of behavioral economics, Daniel Kahneman and Amos Tversky (winners of the Nobel Prize in economics in 2002), conducted an experiment in which they presented different groups of subjects with one of the following two scenarios: • • Scenario 1: In addition to $1,000 up front, the subject must choose between two gambles. Gamble A offers an even chance of winning $1,000 or nothing. Gamble B provides $500 with certainty. Scenario 2: In addition to $2,000 given up front, the subject must choose between two gambles. Gamble C offers an even chance of losing $1,000 or nothing. Gamble D results in the loss of $500 with certainty. a. Suppose Standard Stan makes choices under uncertainty according to expected utility theory. If Stan is risk neutral, what choice would he make in each scenario? b. What choice would Stan make if he is risk averse? c. Kahneman and Tversky found 16 percent of subjects chose A in the ﬁrst scenario and 68 percent chose C in the second scenario. Based on your preceding answers, explain why these ﬁndings are hard to reconcile with expected utility theory. d. Kahneman and Tversky proposed an alternative to expected utility theory, called prospect theory, to explain the experimental results. The theory is that people’s current income level functions as an ‘‘anchor point’’ for them. They are risk averse over gains beyond this point but sensitive to small losses below this point. This sensitivity to small losses is the opposite of risk aversion: A risk-averse person suffers disproportionately more from a large than a small loss. (1) Prospect Pete makes choices under uncertainty according to prospect theory. What choices would he make in Kahne- man and Tversky’s experiment? Explain. (2) Draw a schematic diagram of a utility curve over money for Prospect Pete in the ﬁrst scenario. Draw a utility curve for him in the second scenario. Can the same curve sufﬁce for both scenarios, or must it shift? How do Pete’s utility curves differ from the ones we are used to drawing for people like Standard Stan? 7.12 More on the CRRA function For the CRRA utility function (Equation 7.42), we showed that the degree of risk aversion is measured by 1 we showed that the elasticity of substitution for the same function is given by 1/(1 each other. Using this result, discuss the following questions. % R. In Chapter 3 R). Hence the measures are reciprocals of % a. Why is risk aversion related to an individual’s willingness to substitute wealth between states of the world? What phenom- enon is being captured by both concepts? 242 Part 3: Uncertainty and Strategy b. How would you interpret th
e polar cases R c. A rise in the price of contingent claims in ‘‘bad’’ times (pb) will induce substitution and income effects into the demands for Wg and Wb. If the individual has a ﬁxed budget to devote to these two goods, how will choices among them be affected? Why might Wg rise or fall depending on the degree of risk aversion exhibited by the individual? in both the risk-aversion and substitution frameworks? 1 and R ¼ %1 ¼ d. Suppose that empirical data suggest an individual requires an average return of 0.5 percent before being tempted to invest in an investment that has a 50–50 chance of gaining or losing 5 percent. That is, this person gets the same utility from W0 as from an even bet on 1.055 W0 and 0.955 W0. (1) What value of R is consistent with this behavior? (2) How much average return would this person require to accept a 50–50 chance of gaining or losing 10 percent? Note: This part requires solving nonlinear equations, so approximate solutions will sufﬁce. The comparison of the risk– reward trade-off illustrates what is called the equity premium puzzle in that risky investments seem actually to earn much more than is consistent with the degree of risk aversion suggested by other data. See N. R. Kocherlakota, ‘‘The Equity Premium: It’s Still a Puzzle,’’ Journal of Economic Literature (March 1996): 42–71. 7.13 Graphing risky investments Investment in risky assets can be examined in the state-preference framework by assuming that W, dollars invested in an asset with a certain return r will yield W,(1 + r) in both states of the world, whereas investment in a risky asset will yield W,(1 + rg) in good times and W,(1 + rb) in bad times (where rg > r > rb). a. Graph the outcomes from the two investments. b. Show how a ‘‘mixed portfolio’’ containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals’ attitudes toward risk will determine the mix of risk-free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual’s utility takes the constant relative risk aversion form (Equation 7.42), explain why this person will not change the fraction of risky assets held as his or her wealth increases.25 7.14 The portfolio problem with a Normally distributed risky asset In Example 7.3 we showed that a person with a CARA utility function who faces a Normally distributed risk will have expected utility of the form E W is its variance. Use this fact to solve for the optimal portfolio allocation for a person with a CARA utility function who must invest k of his or her wealth in a Normally distributed risky asset whose expected return is mr and variance in return is r2 r (your answer should depend on A). Explain your results intuitively. W, where mW is the expected value of wealth and r2 r2 Þ lW % ð W ð Þ’ ¼ A=2 U ½ SUGGESTIONS FOR FURTHER READING Arrow, K. J. ‘‘The Role of Securities in the Optimal Allocation of Risk Bearing.’’ Review of Economic Studies 31 (1963): 91–96. Introduces the state-preference concept and interprets securities as claims on contingent commodities. _____. ‘‘Uncertainty and the Welfare Economics of Medical Care.’’ American Economic Review 53 (1963): 941–73. Excellent discussion of the welfare implications of insurance. Has a clear, concise, mathematical appendix. Should be read in conjunction with Pauly’s article on moral hazard (see Chapter 18). Bernoulli, D. ‘‘Exposition of a New Theory on the Measurement of Risk.’’ Econometrica 22 (1954): 23–36. Reprint of the classic analysis of the St. Petersburg paradox. Dixit, A. K., and R. S. Pindyck. Investment under Uncertainty. Princeton, NJ: Princeton University Press, 1994. Focuses mainly on the investment decision by ﬁrms but has good coverage of option concepts. Friedman, M., and L. J. Savage. Choice.’’ Journal of Political Economy 56 (1948): 279–304. ‘‘The Utility Analysis of Analyzes why individuals may both gamble and buy insurance. Very readable. Gollier, Christian. The Economics of Risk and Time. Cambridge, MA: MIT Press, 2001. Contains a complete treatment of many of the issues discussed in this chapter. Especially good on the relationship between allocation under uncertainty and allocation over time. 25This problem is based on J. E. Stiglitz, ‘‘The Effects of Income, Wealth, and Capital Gains Taxation in Risk Taking,’’ Quarterly Journal of Economics (May 1969): 263–83. Chapter 7: Uncertainty 243 Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green. Microeconomic Theory. New York: Oxford University Press, 1995, chap. 6. Provides a good summary of the foundations of expected utility theory. Also examines the ‘‘state independence’’ assumption in detail and shows that some notions of risk aversion carry over into cases of state dependence. Pratt, J. W. ‘‘Risk Aversion in the Small and in the Large.’’ Econometrica 32 (1964): 122–36. Theoretical development of risk-aversion measures. Fairly technical treatment but readable. Rothschild, M., and J. E. Stiglitz. ‘‘Increasing Risk: 1. A Deﬁnition.’’ Journal of Economic Theory 2 (1970): 225–43. Develops an economic deﬁnition of what it means for one gamble to be ‘‘riskier’’ than another. A sequel article in the Journal of Economic Theory provides economic illustrations. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Chapter 13 provides a nice introduction to the relationship between statistical concepts and expected utility maximization. Also shows in detail the integration mentioned in Example 7.3. EXTENSIONS THE PORTFOLIO PROBLEM One of the classic problems in the theory of behavior under uncertainty is the issue of how much of his or her wealth a risk-averse investor should invest in a risky asset. Intuitively, it seems that the fraction invested in risky assets should be smaller for more risk-averse investors, and one goal of our analysis in these Extensions will be to show that formally. We will then see how to generalize the model to consider portfolios with many such assets, ﬁnally working up to the Capital Asset Pricing model, a staple of ﬁnancial economics courses. E7.1 Basic model with one risky asset To get started, assume that an investor has a certain amount of wealth, W0, to invest in one of two assets. The ﬁrst asset yields a certain return of rf , whereas the second asset’s return is a random variable, r. If we let the amount invested in the risky asset be denoted by k, then this person’s wealth at the end of one period will be rf Þ þ W0 % 1 W0ð rf Þ þ 1 k Þð : rf Þ % (i) % rf ) > 0, and this will imply k Notice three things about this end-of-period wealth. First, W is a random variable because its value depends on r. Second, k can be either positive or negative here depending on whether this person buys the risky asset or sells it short. As we shall see, how0. ever, in the usual case E(r Finally, notice also that Equation i allows for a solution in which k > W0. In this case, this investor would leverage his or her investment in the risky asset by borrowing at the risk-free rate rf. If we let U(W ) represent this investor’s utility function, then the von Neumann–Morgenstern theorem states that he or she will choose k to maximize E[U(W)]. The ﬁrst-order condition for such a maximum is 1 W0ð ð rf ÞÞ’ k ð @E @E % + U r ½ Þ’ ½ U W ð @k ¼ (ii) E U 0 r ½ ¼ ( ð % In calculating this ﬁrst-order condition, we can differentiate through the expected value operator, E. See Chapter 2 for a discussion of differentiating integrals (of which an expected value operator is an example). Equation ii involves the expected value of the product of marginal utility and the term r rf is positive or negative will depend on how well the risky assets rf. Both of these terms are random. Whether r % % þ rf Þ þ @k rf Þ’ ¼ 0: perform over the next period. But the return on this risky asset will also affect this investor’s end-of-period wealth and thus will affect his or her marginal utility. If the investment does well, W will be large and marginal utility will be relatively low (because of diminishing marginal utility). If the investment does poorly, wealth will be relatively low and marginal utility will be relatively high. Hence in the expected rf value calculation in Equation ii, negative outcomes for r will be weighted more heavily than positive outcomes to take the utility consequences of these outcomes into account. If the expected value in Equation ii were positive, a person could increase his or her expected utility by investing more in the risky asset. If the expected value were negative, he or she could increase expected utility by reducing the amount of the risky asset held. Only when the ﬁrst-order condition holds will this person have an optimal portfolio. % % Two other conclusions can be drawn from Equation ii. rf) > 0, an investor will choose positive First, as long as E(r amounts of the risky asset. To see why, notice that meeting Equation ii will require that fairly large values of U 0 be attached to situations where r rf turns out to be negative. That can only happen if the investor owns positive amounts of the risky asset so that end-of-period wealth is low in such situations. % A second conclusion from Equation ii is that investors who are more risk averse will hold smaller amounts of the risky asset. Again, the reason relates to the shape of the U 0 function. For risk-averse investors, marginal utility rises rapidly as wealth falls. Hence they need relatively little exposure to potential negative outcomes from holding the risky asset to satisfy Equation ii. E7.2 CARA utility To make further progress on the portfolio problem requires that we make some speciﬁc assumptions about the investor’s utility function. Suppose it is given by the CARA form: AW ). Then the marginal utility function is U(W) exp( given by U 0(W) AW ); substituting for end-of-period A exp( we
alth, we have exp A exp ½% 1 W0ð A ð 1 AW0ð rf Þ þ þ exp rf Þ’ k r ð % Ak rf ÞÞ’ r % ð : rf Þ’ ¼ ½% þ That is, the marginal utility function can be separated into a random part and a nonrandom part (both initial wealth and ½% (iii) the risk-free rate are nonrandom). Hence the optimality condition from Equation ii can be written as ½ ½ r E 0: % % % U 0 (iv) r ( ð ½% ð% rf Þ’ ¼ rf Þ’ ¼ A exp exp E rf Þ’ þ rf ÞÞ ( ð 1 AW0ð r Ak % ð Now we can divide by the exponential function of initial leaving an optimality condition that involves only wealth, terms in k, A, and r rf . Solving this condition for the optimal level of k can in general be difﬁcult (but see Problem 7.14). Regardless of the speciﬁc solution, however, Equation iv shows that this optimal investment amount will be a constant regardless of the level of initial wealth. Hence the CARA function implies that the fraction of wealth that an investor holds in risky assets should decrease as wealth increases—a conclusion that seems precisely contrary to empirical data, which tend to show the fraction of wealth held in risky assets increasing with wealth. If we instead assumed utility took the CRRA rather than the CARA form, we could show (with some patience) that all individuals with the same risk tolerance will hold the same fraction of wealth in risky assets, regardless of their absolute levels of wealth. Although this conclusion is slightly more in accord with the facts than is the conclusion from the CARA function, it still falls short of explaining why the fraction of wealth held in risky assets tends to increase with wealth. E7.3 Portfolios of many risky assets Additional insight can be gained if the model is generalized to allow for many risky assets. Let the return on each of n risky assets be the random variable ri (i 1,…, n). The expected values and variances of these assets’ returns are denoted by E(ri) i , respectively. An investor who invests a portion of his or her wealth in a portfolio of these assets will obtain a random return (rp) given by mi and Var(riÞ ¼ r2 ¼ ¼ rp ¼ n 1 i X ¼ airi, (v) where ai ( 1 ai ¼ asset i and where return on this portfolio will be + n i ¼ 0) is the fraction of the risky portfolio held in 1. In this situation, the expected P E rpÞ ¼ ð lp ¼ aili: n 1 i X ¼ (vi) If the returns of each asset are independent, then the variance of the portfolio’s return will be Var(rpÞ ¼ r2 p ¼ i r2 a2 i : n 1 i X ¼ (vii) If the returns are not independent, Equation vii would have to be modiﬁed to take covariances among the returns into account. Using this general notation, we now proceed to look at some aspects of this portfolio allocation problem. Chapter 7: Uncertainty 245 E7.4 Optimal portfolios With many risky assets, the optimal portfolio problem can be divided into two steps. The ﬁrst step is to consider portfolios of just the risky assets. The second step is to add in the riskless one. To solve for the optimal portfolio of just the risky assets, one can proceed as in the text, where in the section on diversiﬁcation we looked at the optimal investment weights across just two risky assets. Here, we will choose a general set of asset weightings (the ai) to minimize the variance (or standard deviation) of the portfolio for each potential expected return. The solution to this problem yields an ‘‘efﬁciency frontier’’ for risky asset portfolios such as that represented by the line EE in Figure E7.1. Portfolios that lie below this frontier are inferior to those on the frontier because they offer lower expected returns for any degree of risk. Portfolio returns above the frontier are unattainable. Sharpe (1970) discusses the mathematics associated with constructing the EE frontier. Now add a risk-free asset with expected return mf and sf ¼ 0, shown as point R in Figure E7.1. Optimal portfolios will now consist of mixtures of this asset with risky ones. All such portfolios will lie along the line RP in the ﬁgure, because this shows the maximum return attainable for each value of s for various portfolio allocations. These allocations will contain only one speciﬁc set of risky assets: the set represented by point M. In equilibrium this will be the ‘‘market portfolio’’ consisting of all capital assets held in proportion to their market valuations. This market portfolio will provide an expected return of mM and a standard deviation of that return of sM. The equation for the line RP that represents any mixed portfolio is given by the linear equation lp ¼ lf þ lf lM % rM rp: ( (viii) This shows that the market line RP permits individual investors to ‘‘purchase’’ returns in excess of the risk-free return (mM % mf) by taking on proportionally more risk (sP /sM). For choices on RP to the left of the market point M, sP /sM < 1 and mf < mP < mM. High-risk points to the right of M—which can be obtained by borrowing to produce a leveraged portfolio—will have sP/sM > 1 and will promise an expected return in excess of what is provided by the market portfolio ( mP > mM ). Tobin (1958) was one of the ﬁrst economists to recognize the role that risk-free assets play in identifying the market portfolio and in setting the terms on which investors can obtain returns above risk-free levels. E7.5 Individual choices Figure E7.2 illustrates the portfolio choices of various investors facing the options offered by the line RP. This ﬁgure illustrates the type of portfolio choice model previously described in this chapter. Individuals with low tolerance for risk (I ) will opt for portfolios that are heavily weighted toward the riskfree asset. Investors willing to assume a modest degree of risk 246 Part 3: Uncertainty and Strategy FIGURE E7.1 Efﬁcient Portfolios The frontier EE represents optimal mixtures of risky assets that minimize the standard deviation of the portfolio, sP, for each expected return, mP. A risk-free asset with return mf offers investors the opportunity to hold mixed portfolios along RP that mix this risk-free asset with the market portfolio, M FIGURE E7.2 Investor Behavior and Risk Aversion Given the market options RP, investors can choose how much risk they wish to assume. Very risk-averse investors (UI) will hold mainly risk-free assets, whereas risk takers (UIII) will opt for leveraged portfolios. UIII P UII UI M P R f P (II ) will opt for portfolios close to the market portfolio. Highrisk investors (III ) may opt for leveraged portfolios. Notice that all investors face the same ‘‘price’’ of risk (mM % mf) with their expected returns being determined by how much relative risk (sP/sM) they are willing to incur. Notice also that the risk associated with an investor’s portfolio depends only on the fraction of the portfolio invested in the market portfolio (a) 2 because r2 a a and so Þ the investor’s choice of portfolio is equivalent to his or her choice of risk. 0. Hence sP/sM ¼ 1 M þ ð a2r2 P ¼ % ( Mutual funds The notion of portfolio efﬁciency has been widely applied to the study of mutual funds. In general, mutual funds are a good answer to small investors’ diversiﬁcation needs. Because such funds pool the funds of many individuals, they are able to achieve economies of scale in transactions and management costs. This permits fund owners to share in the fortunes of a much wider variety of equities than would be possible if each acted alone. But mutual fund managers have incentives of their own; therefore, the portfolios they hold may not always be perfect representations of the risk attitudes of their clients. For example, Scharfstein and Stein (1990) developed a model that shows why mutual fund managers have incentives to ‘‘follow the herd’’ in their investment picks. Other studies, such as the classic investigation by Jensen (1968), ﬁnd that mutual fund managers are seldom able to attain extra returns large enough to offset the expenses they charge investors. In recent years this has led many mutual fund buyers to favor ‘‘index’’ funds that seek simply to duplicate the market average (as represented, say, by the Standard and Poor’s 500 stock index). Such funds have low expenses and therefore permit investors to achieve diversiﬁcation at minimal cost. E7.6 Capital asset pricing model Although the analysis of E7.5 shows how a portfolio that mixes a risk-free asset with the market portfolio will be priced, it does not describe the risk–return trade-off for a single asset. Because (assuming transactions are costless) an investor can always avoid risk unrelated to the overall market by choosing to diversify with a ‘‘market portfolio,’’ such ‘‘unsystematic’’ risk will not warrant any excess return. An asset will, however, earn an excess return to the extent that it contributes to overall market risk. An asset that does not yield such extra returns would not be held in the market portfolio, so it would not be held at all. This is the fundamental insight of the capital asset pricing model (CAPM). To examine these results formally, consider a portfolio that combines a small amount (a) of an asset with a random return of x with the market portfolio (which has a random return of M ). The return on this portfolio (z) would be given by z ax 1 þ ð % a M: Þ ¼ (ix) Chapter 7: Uncertainty 247 The expected return is lz ¼ alx þ ð 1 a lM Þ % with variance r2 z ¼ a2r2 1 x þ ð a Þ % 2r2 M þ 2a 1 ð % a rx,M, Þ (x) (xi) where sx,M is the covariance between the return on x and the return on the market. But our previous analysis shows lz ¼ lf þ ð lM % lf Þ ( rz rM : (xii) Setting Equation x equal to xii and differentiating with respect to a yields @lz @a ¼ lx % lM ¼ lf lM % rM @rz @a : (xiii) By calculating @rz=@a from Equation xi and taking the limit as a approaches zero, we get lM % rM rx,M % rM lM ¼ lx % (xiv) r2 M lf , ( ’ or, rearranging terms, lx ¼ lf þ ð lM % lf Þ ( rx;M r2 M : (xv) Again, risk has a reward of mM % mf, but now the quantity of risk is measured by rx,M=r2 M. This ratio of the covariance between the return x and the market to the variance of the market
return is referred to as the beta coefﬁcient for the asset. Estimated beta coefﬁcients for ﬁnancial assets are reported in many publications. Studies of the CAPM This version of the CAPM carries strong implications about the determinants of any asset’s expected rate of return. Because of this simplicity, the model has been subject to a large number of empirical tests. In general these ﬁnd that the model’s measure of systemic risk (beta) is indeed correlated with expected returns, whereas simpler measures of risk (e.g., the standard deviation of past returns) are not. Perhaps the most inﬂuential early empirical test that reached such a conclusion was that of Fama and MacBeth (1973). But the CAPM itself explains only a small fraction of differences in the returns of various assets. And contrary to the CAPM, a number of authors have found that many other economic factors signiﬁcantly affect expected returns. Indeed, a prominent challenge to the CAPM comes from one of its original founders—see Fama and French (1992). References Fama, E. F., and K. R. French. ‘‘The Cross Section of Expected Stock Returns.’’ Journal of Finance 47 (1992): 427–66. 248 Part 3: Uncertainty and Strategy Fama, E. F., and J. MacBeth. ‘‘Risk Return and Equilibrium.’’ Sharpe, W. F. Portfolio Theory and Capital Markets. New Journal of Political Economy 8 (1973): 607–36. York: McGraw-Hill, 1970. Jensen, M. ‘‘The Performance of Mutual Funds in the Period 1945–1964.’’ Journal of Finance (May 1968): 386–416. Scharfstein, D. S., and J. Stein. ‘‘Herd Behavior and Investment.’’ American Economic Review (June 1990): 465–89. Tobin, J. ‘‘Liquidity Preference as Behavior towards Risk.’’ Review of Economic Studies (February 1958): 65–86. This page intentionally left blank C H A P T E R EIGHT Game Theory This chapter provides an introduction to noncooperative game theory, a tool used to understand the strategic interactions among two or more agents. The range of applications of game theory has been growing constantly, including all areas of economics (from labor economics to macroeconomics) and other ﬁelds such as political science and biology. Game theory is particularly useful in understanding the interaction between ﬁrms in an oligopoly, so the concepts learned here will be used extensively in Chapter 15. We begin with the central concept of Nash equilibrium and study its application in simple games. We then go on to study reﬁnements of Nash equilibrium that are used in games with more complicated timing and information structures. Basic Concepts Thus far in Part 3 of this text, we have studied individual decisions made in isolation. In this chapter we study decision making in a more complicated, strategic setting. In a strategic setting, a person may no longer have an obvious choice that is best for him or her. What is best for one decision-maker may depend on what the other is doing and vice versa. For example, consider the strategic interaction between drivers and the police. Whether drivers prefer to speed may depend on whether the police set up speed traps. Whether the police ﬁnd speed traps valuable depends on how much drivers speed. This confusing circularity would seem to make it difﬁcult to make much headway in analyzing strategic behavior. In fact, the tools of game theory will allow us to push the analysis nearly as far, for example, as our analysis of consumer utility maximization in Chapter 4. There are two major tasks involved when using game theory to analyze an economic situation. The ﬁrst is to distill the situation into a simple game. Because the analysis involved in strategic settings quickly grows more complicated than in simple decision problems, it is important to simplify the setting as much as possible by retaining only a few essential elements. There is a certain art to distilling games from situations that is hard to teach. The examples in the text and problems in this chapter can serve as models that may help in approaching new situations. The second task is to ‘‘solve’’ the given game, which results in a prediction about what will happen. To solve a game, one takes an equilibrium concept (e.g., Nash equilibrium) and runs through the calculations required to apply it to the given game. Much of the chapter will be devoted to learning the most widely used equilibrium concepts and to practicing the calculations necessary to apply them to particular games. A game is an abstract model of a strategic situation. Even the most basic games have three essential elements: players, strategies, and payoffs. In complicated settings, it is sometimes also necessary to specify additional elements such as the sequence of moves 251 252 Part 3: Uncertainty and Strategy and the information that players have when they move (who knows what when) to describe the game fully. Players Each decision-maker in a game is called a player. These players may be individuals (as in poker games), ﬁrms (as in markets with few ﬁrms), or entire nations (as in military conﬂicts). A player is characterized as having the ability to choose from among a set of possible actions. Usually the number of players is ﬁxed throughout the ‘‘play’’ of the game. Games are sometimes characterized by the number of players involved (two-player, three-player, or n-player games). As does much of the economic literature, this chapter often focuses on two-player games because this is the simplest strategic setting. We will label the players with numbers; thus, in a two-player game we will have players 1 and 2. In an n-player game we will have players 1, 2,…, n, with the generic player labeled i. Strategies Each course of action open to a player during the game is called a strategy. Depending on the game being examined, a strategy may be a simple action (drive over the speed limit or not) or a complex plan of action that may be contingent on earlier play in the game (say, speeding only if the driver has observed speed traps less than a quarter of the time in past drives). Many aspects of game theory can be illustrated in games in which players choose between just two possible actions. Let S1 denote the set of strategies open to player 1, S2 the set open to player 2, and S1 be a particular strategy chosen by S2 the particular strategy chosen by player 2, Si for player i. A strategy proﬁle will refer to a listing of particular strategies (more generally) Si the set open to player i. Let s1 2 player 1 from the set of possibilities, s2 2 and si 2 chosen by each of a group of players. Payoffs The ﬁnal return to each player at the conclusion of a game is called a payoff. Payoffs are measured in levels of utility obtained by the players. For simplicity, monetary payoffs (say, proﬁts for ﬁrms) are often used. More generally, payoffs can incorporate nonmonetary factors such as prestige, emotion, risk preferences, and so forth. In a two-player game, u1(s1, s2) denotes player 1’s payoff given that he or she chooses s1 and the other player chooses s2 and similarly u2(s2, s1) denotes player 2’s payoff.1 The fact that player 1’s payoff may depend on player 2’s strategy (and vice versa) is where the strategic interdependence shows up. In an n-player game, we can write the payoff of a i), which depends on player i’s own strategy si and the proﬁle generic player i as ui(si, s s 1, … , sn) of the strategies of all players other than i. (s1, … , si 1, si ! i ¼ ! ! þ Prisoners’ Dilemma The Prisoners’ Dilemma, introduced by A. W. Tucker in the 1940s, is one of the most famous games studied in game theory and will serve here as a nice example to illustrate all the notation just introduced. The title stems from the following situation. Two suspects are arrested for a crime. The district attorney has little evidence in the case and is eager to extract a confession. She separates the suspects and tells each: ‘‘If you ﬁnk on your companion but your companion doesn’t ﬁnk on you, I can promise you a reduced 1Technically, these are the von Neumann–Morgenstern utility functions from the previous chapter. Chapter 8: Game Theory 253 (one-year) sentence, whereas your companion will get four years. If you both ﬁnk on each other, you will each get a three-year sentence.’’ Each suspect also knows that if neither of them ﬁnks then the lack of evidence will result in being tried for a lesser crime for which the punishment is a two-year sentence. Boiled down to its essence, the Prisoners’ Dilemma has two strategic players: the suspects, labeled 1 and 2. (There is also a district attorney, but because her actions have already been fully speciﬁed, there is no reason to complicate the game and include her in the speciﬁcation.) Each player has two possible strategies open to him: ﬁnk or remain {ﬁnk, silent}. To avoid negative silent. Therefore, we write their strategy sets as S1 ¼ numbers we will specify payoffs as the years of freedom over the next four years. For example, if suspect 1 ﬁnks and suspect 2 does not, suspect 1 will enjoy three years of freedom and suspect 2 none, that is, u1(ﬁnk, silent) 3 and u2(silent, ﬁnk) S2 ¼ 0. ¼ ¼ Normal form The Prisoners’ Dilemma (and games like it) can be summarized by the matrix shown in Figure 8.1, called the normal form of the game. Each of the four boxes represents a different combination of strategies and shows the players’ payoffs for that combination. The usual convention is to have player 1’s strategies in the row headings and player 2’s in the column headings and to list the payoffs in order of player 1, then player 2 in each box. Thinking strategically about the Prisoners’ Dilemma Although we have not discussed how to solve games yet, it is worth thinking about what we might predict will happen in the Prisoners’ Dilemma. Studying Figure 8.1, on ﬁrst thought one might predict that both will be silent. This gives the most total years of freedom for both (four) compared with any other outcome. Thinking a bit deeper, this may not be the best prediction in the game. Imagine ourselves
in player 1’s position for a moment. We do not know what player 2 will do yet because we have not solved out the game, so let’s investigate each possibility. Suppose player 2 chose to ﬁnk. By ﬁnking ourselves we would earn one year of freedom versus none if we remained silent, so ﬁnking is better for us. Suppose player 2 chose to remain silent. Finking is still better for us than remaining silent because we get three rather than two years of freedom. Regardless of what the other player does, ﬁnking is better for us than being silent because it results in an extra year of freedom. Because players are symmetric, the same reasoning holds if we FIGURE 8.1 Normal Form for the Prisoners’ Dilemma Suspect 2 Fink Silent Fink u1 ! 1, u2 ! 1 u1 ! 3, u2 ! 0 1 t c e p s u S Silent u1 ! 0, u2 ! 3 u1 ! 2, u2 ! 2 254 Part 3: Uncertainty and Strategy imagine ourselves in player 2’s position. Therefore, the best prediction in the Prisoners’ Dilemma is that both will ﬁnk. When we formally introduce the main solution concept— Nash equilibrium—we will indeed ﬁnd that both ﬁnking is a Nash equilibrium. The prediction has a paradoxical property: By both ﬁnking, the suspects only enjoy one year of freedom, but if they were both silent they would both do better, enjoying two years of freedom. The paradox should not be taken to imply that players are stupid or that our prediction is wrong. Rather, it reveals a central insight from game theory that pitting players against each other in strategic situations sometimes leads to outcomes that are inefﬁcient for the players.2 The suspects might try to avoid the extra prison time by coming to an agreement beforehand to remain silent, perhaps reinforced by threats to retaliate afterward if one or the other ﬁnks. Introducing agreements and threats leads to a game that differs from the basic Prisoners’ Dilemma, a game that should be analyzed on its own terms using the tools we will develop shortly. Solving the Prisoners’ Dilemma was easy because there were only two players and two strategies and because the strategic calculations involved were fairly straightforward. It would be useful to have a systematic way of solving this as well as more complicated games. Nash equilibrium provides us with such a systematic solution. Nash Equilibrium In the economic theory of markets, the concept of equilibrium is developed to indicate a situation in which both suppliers and demanders are content with the market outcome. Given the equilibrium price and quantity, no market participant has an incentive to change his or her behavior. In the strategic setting of game theory, we will adopt a related notion of equilibrium, formalized by John Nash in the 1950s, called Nash equilibrium.3 Nash equilibrium involves strategic choices that, once made, provide no incentives for the players to alter their behavior further. A Nash equilibrium is a strategy for each player that is the best choice for each player given the others’ equilibrium strategies. The next several sections provide a formal deﬁnition of Nash equilibrium, apply the concept to the Prisoners’ Dilemma, and then demonstrate a shortcut (involving underlining payoffs) for picking Nash equilibria out of the normal form. As at other points in the chapter, the reader who wants to avoid wading through a lot of math can skip over the notation and deﬁnitions and jump right to the applications without losing too much of the basic insight behind game theory. A formal deﬁnition Nash equilibrium can be deﬁned simply in terms of best responses. In an n-player game, strategy si is a best response to rivals’ strategies s i if player i cannot obtain a strictly higher payoff with any other possible strategy, s0i 2 ! Si, given that rivals are playing s i Best response. si is a best response for player i to rivals’ strategies s s0i 2 s0i, s ! for all iÞ & uið uið si, s iÞ ! ! Si. i , denoted si 2 BRi(s i), if ! (8:1) 2When we say the outcome is inefﬁcient, we are focusing just on the suspects’ utilities; if the focus were shifted to society at large, then both ﬁnking might be a good outcome for the criminal justice system—presumably the motivation behind the district attorney’s offer. 3John Nash, ‘‘Equilibrium Points in n-Person Games,’’ Proceedings of the National Academy of Sciences 36 (1950): 48–49. Nash is the principal ﬁgure in the 2001 ﬁlm A Beautiful Mind (see Problem 8.5 for a game-theory example from the ﬁlm) and co-winner of the 1994 Nobel Prize in economics. Chapter 8: Game Theory 255 A technicality embedded in the deﬁnition is that there may be a set of best responses i). rather than a unique one; that is why we used the set inclusion notation si 2 There may be a tie for the best response, in which case the set BRi(s i) will contain more than one element. If there is not a tie, then there will be a single best response si and we can simply write si ¼ We can now deﬁne a Nash equilibrium in an n-player game as follows. BRi(s BRi(s i). ! ! ! Nash equilibrium. A Nash equilibrium is a strategy proﬁle player i s’i 2 such that, for each i. That is, is a best response to the other players’ equilibrium strategies s’ ! 1, 2, … , n, s’i i). ¼ BRi (s’ ! s’1, s’2, . . . , s’n " ! These deﬁnitions involve a lot of notation. The notation is a bit simpler in a two-player is a Nash equilibrium if s’1 and s’2 are mutual best game. In a two-player game, responses against each other: s’1, s’2 ! s’1, s’2Þ & " u1ð u1ð s1, s’2Þ for all s1 2 S1 and u2ð s’1, s’2Þ & u2ð s2, s’1Þ for all S2: s2 2 (8:2) (8:3) A Nash equilibrium is stable in that, even if all players revealed their strategies to each other, no player would have an incentive to deviate from his or her equilibrium strategy and choose something else. Nonequilibrium strategies are not stable in this way. If an outcome is not a Nash equilibrium, then at least one player must beneﬁt from deviating. Hyper-rational players could be expected to solve the inference problem and deduce that all would play a Nash equilibrium (especially if there is a unique Nash equilibrium). Even if players are not hyper-rational, over the long run we can expect their play to converge to a Nash equilibrium as they abandon strategies that are not mutual best responses. Besides this stability property, another reason Nash equilibrium is used so widely in economics is that it is guaranteed to exist for all games we will study (allowing for mixed strategies, to be deﬁned below; Nash equilibria in pure strategies do not have to exist). The mathematics behind this existence result are discussed at length in the Extensions to this chapter. Nash equilibrium has some drawbacks. There may be multiple Nash equilibria, making it hard to come up with a unique prediction. Also, the deﬁnition of Nash equilibrium leaves unclear how a player can choose a best-response strategy before knowing how rivals will play. Nash equilibrium in the Prisoners’ Dilemma Let’s apply the concepts of best response and Nash equilibrium to the example of the Prisoners’ Dilemma. Our educated guess was that both players will end up ﬁnking. We will show that both ﬁnking is a Nash equilibrium of the game. To do this, we need to show that ﬁnking is a best response to the other players’ ﬁnking. Refer to the payoff matrix in Figure 8.1. If player 2 ﬁnks, we are in the ﬁrst column of the matrix. If player 1 also ﬁnks, his payoff is 1; if he is silent, his payoff is 0. Because he earns the most from ﬁnking given player 2 ﬁnks, ﬁnking is player 1’s best response to player 2’s ﬁnking. Because players are symmetric, the same logic implies that player 2’s ﬁnking is a best response to player 1’s ﬁnking. Therefore, both ﬁnking is indeed a Nash equilibrium. We can show more: that both ﬁnking is the only Nash equilibrium. To do so, we need to rule out the other three outcomes. Consider the outcome in which player 1 ﬁnks and player 2 is silent, abbreviated (ﬁnk, silent), the upper right corner of the 256 Part 3: Uncertainty and Strategy matrix. This is not a Nash equilibrium. Given that player 1 ﬁnks, as we have already said, player 2’s best response is to ﬁnk, not to be silent. Symmetrically, the outcome in which player 1 is silent and player 2 ﬁnks in the lower left corner of the matrix is not a Nash equilibrium. That leaves the outcome in which both are silent. Given that player 2 is silent, we focus our attention on the second column of the matrix: The two rows in that column show that player 1’s payoff is 2 from being silent and 3 from ﬁnking. Therefore, silent is not a best response to ﬁnk; thus, both being silent cannot be a Nash equilibrium. To rule out a Nash equilibrium, it is enough to ﬁnd just one player who is not playing a best response and thus would want to deviate to some other strategy. Considering the outcome (ﬁnk, silent), although player 1 would not deviate from this outcome (he earns 3, which is the most possible), player 2 would prefer to deviate from silent to ﬁnk. Symmetrically, considering the outcome (silent, ﬁnk), although player 2 does not want to deviate, player 1 prefers to deviate from silent to ﬁnk, so this is not a Nash equilibrium. Considering the outcome (silent, silent), both players prefer to deviate to another strategy, more than enough to rule out this outcome as a Nash equilibrium. Underlining best-response payoffs A quick way to ﬁnd the Nash equilibria of a game is to underline best-response payoffs in the matrix. The underlining procedure is demonstrated for the Prisoners’ Dilemma in Figure 8.2. The ﬁrst step is to underline the payoffs corresponding to player 1’s best 1 in responses. Player 1’s best response is to ﬁnk if player 2 ﬁnks, so we underline u1 ¼ the upper left box, and to ﬁnk if player 2 is silent, so we underline u1 ¼ 3 in the upper left box. Next, we move to underlining the payoffs corresponding to player 2’s best 1 in responses. Player 2’s best response is to ﬁnk if player 1 ﬁnks, so we underline u2 ¼ the upper left box, and to ﬁnk if player 1 is si
lent, so we underline u2 ¼ 3 in the lower left box. Now that the best-response payoffs have been underlined, we look for boxes in which every player’s payoff is underlined. These boxes correspond to Nash equilibria. (There may be additional Nash equilibria involving mixed strategies, deﬁned later in the chapter.) In Figure 8.2, only in the upper left box are both payoffs underlined, verifying that (ﬁnk, ﬁnk)—and none of the other outcomes—is a Nash equilibrium. FIGURE 8.2 Underlining Procedure in the Prisoners’ Dilemma Suspect 2 Fink Silent Fink u1 ! 1, u2 ! 1 u1 ! 3, u2 ! 0 1 t c e p s u S Silent u1 ! 0, u2 ! 3 u1 ! 2, u2 ! 2 Chapter 8: Game Theory 257 Dominant strategies (Fink, ﬁnk) is a Nash equilibrium in the Prisoners’ Dilemma because ﬁnking is a best response to the other player’s ﬁnking. We can say more: Finking is the best response to all the other player’s strategies, ﬁnk and silent. (This can be seen, among other ways, from the underlining procedure shown in Figure 8.2: All player 1’s payoffs are underlined in the row in which he plays ﬁnk, and all player 2’s payoffs are underlined in the column in which he plays ﬁnk.) A strategy that is a best response to any strategy the other players might choose is called a dominant strategy. Players do not always have dominant strategies, but when they do there is strong reason to believe they will play that way. Complicated strategic considerations do not matter when a player has a dominant strategy because what is best for that player is independent of what others are doing. Dominant strategy. A dominant strategy is a strategy s’i for player i that is a best response to all strategy proﬁles of other players. That is, s’i 2 s BRið for all s iÞ i. ! ! Note the difference between a Nash equilibrium strategy and a dominant strategy. A strategy that is part of a Nash equilibrium need only be a best response to one strategy proﬁle of other players—namely, their equilibrium strategies. A dominant strategy must be a best response not just to the Nash equilibrium strategies of other players but to all the strategies of those players. If all players in a game have a dominant strategy, then we say the game has a dominant strategy equilibrium. As well as being the Nash equilibrium of the Prisoners’ Dilemma, (ﬁnk, ﬁnk) is a dominant strategy equilibrium. It is generally true for all games that a dominant strategy equilibrium, if it exists, is also a Nash equilibrium and is the unique such equilibrium. Battle of the Sexes The famous Battle of the Sexes game is another example that illustrates the concepts of best response and Nash equilibrium. The story goes that a wife (player 1) and husband (player 2) would like to meet each other for an evening out. They can go either to the ballet or to a boxing match. Both prefer to spend time together than apart. Conditional on being together, the wife prefers to go to the ballet and the husband to the boxing match. The normal form of the game is presented in Figure 8.3. For brevity we dispense with the Player 2 (Husband) Boxing Ballet Ballet 2, 1 0 ( Boxing 0, 0 1 FIGURE 8.3 Normal Form for the Battle of the Sexes 258 Part 3: Uncertainty and Strategy u1 and u2 labels on the payoffs and simply re-emphasize the convention that the ﬁrst payoff is player 1’s and the second is player 2’s. We will examine the four boxes in Figure 8.3 and determine which are Nash equilibria and which are not. Start with the outcome in which both players choose ballet, written (ballet, ballet), the upper left corner of the payoff matrix. Given that the husband plays ballet, the wife’s best response is to play ballet (this gives her her highest BR1(ballet). [We do not need the payoff in the matrix of 2). Using notation, ballet BR1(ballet)’’ because the husband has only fancy set-inclusion symbol as in ‘‘ballet one best response to the wife’s choosing ballet.] Given that the wife plays ballet, the husband’s best response is to play ballet. If he deviated to boxing, then he would earn 0 rather than 1 because they would end up not coordinating. Using notation, ballet ¼ BR2(ballet). Thus, (ballet, ballet) is indeed a Nash equilibrium. Symmetrically, (boxing, boxing) is a Nash equilibrium. ¼ 2 Consider the outcome (ballet, boxing) in the upper left corner of the matrix. Given the husband chooses boxing, the wife earns 0 from choosing ballet but 1 from choosing boxing; therefore, ballet is not a best response for the wife to the husband’s choosing boxing. In notation, ballet = BR1(boxing). Hence (ballet, boxing) cannot be a Nash 2 equilibrium. [The husband’s strategy of boxing is not a best response to the wife’s playing ballet either; thus, both players would prefer to deviate from (ballet, boxing), although we only need to ﬁnd one player who would want to deviate to rule out an outcome as a Nash equilibrium.] Symmetrically, (boxing, ballet) is not a Nash equilibrium either. The Battle of the Sexes is an example of a game with more than one Nash equilibrium (in fact, it has three—a third in mixed strategies, as we will see). It is hard to say which of the two we have found thus far is more plausible because they are symmetric. Therefore, it is difﬁcult to make a ﬁrm prediction in this game. The Battle of the Sexes is also an example of a game with no dominant strategies. A player prefers to play ballet if the other plays ballet and boxing if the other plays boxing. Figure 8.4 applies the underlining procedure, used to ﬁnd Nash equilibria quickly, to the Battle of the Sexes. The procedure veriﬁes that the two outcomes in which the players succeed in coordinating are Nash equilibria and the two outcomes in which they do not coordinate are not. Examples 8.1 and 8.2 provide additional practice in ﬁnding Nash equilibria in more complicated settings (a game that has many ties for best responses in Example 8.1 and a game that has three strategies for each player in Example 8.2). FIGURE 8.4 Underlining Procedure in the Battle of the Sexes Player 2 (Husband) Boxing Ballet Ballet 2, 1 0 ( Boxing 0, 0 1, 2 Chapter 8: Game Theory 259 EXAMPLE 8.1 The Prisoners’ Dilemma Redux In this variation on the Prisoners’ Dilemma, a suspect is convicted and receives a sentence of four years if he is ﬁnked on and goes free if not. The district attorney does not reward ﬁnking. Figure 8.5 presents the normal form for the game before and after applying the procedure for underlining best responses. Payoffs are again restated in terms of years of freedom. FIGURE 8.58The Prisoners’ Dilemma Redux (a) Normal form Suspect 2 Fink Silent Fink 0, 0 1, 0 Silent 0, 1 1b) Underlining procedure Suspect 2 Fink Silent Fink 0, 0 1, 0 Silent 0, 1 1 Ties for best responses are rife. For example, given player 2 ﬁnks, player 1’s payoff is 0 whether he ﬁnks or is silent. Thus, there is a tie for player 1’s best response to player 2’s ﬁnking. This is an example of the set of best responses containing more than one element: BR1 (ﬁnk) {ﬁnk, silent}. ¼ The underlining procedure shows that there is a Nash equilibrium in each of the four boxes. Given that suspects receive no personal reward or penalty for ﬁnking, they are both indifferent between ﬁnking and being silent; thus, any outcome can be a Nash equilibrium. QUERY: Does any player have a dominant strategy? EXAMPLE 8.2 Rock, Paper, Scissors Rock, Paper, Scissors is a children’s game in which the two players simultaneously display one of three hand symbols. Figure 8.6 presents the normal form. The zero payoffs along the diagonal show that if players adopt the same strategy then no payments are made. In other cases, the payoffs indicate a $1 payment from loser to winner under the usual hierarchy (rock breaks scissors, scissors cut paper, paper covers rock). As anyone who has played this game knows, and as the underlining procedure reveals, none of the nine boxes represents a Nash equilibrium. Any strategy pair is unstable because it offers 260 Part 3: Uncertainty and Strategy at least one of the players an incentive to deviate. For example, (scissors, scissors) provides an incentive for either player 1 or 2 to choose rock; (paper, rock) provides an incentive for player 2 to choose scissors. FIGURE 8.68Rock, Paper, Scissors (a) Normal form Player 2 Rock Paper Scissors Rock 0, 0 −1, 1 1, −1 1 r e y a l P Paper 1, −1 0, 0 −1, 1 Scissors −1, 1 1, −1 0, 0 (b) Underlining procedure Player 2 Rock Paper Scissors Rock 0, 0 −1, 1 1, −1 1 r e y a l P Paper 1, −1 0, 0 −1, 1 Scissors −1, 1 1, −1 0, 0 The game does have a Nash equilibrium—not any of the nine boxes in the ﬁgure but in mixed strategies, deﬁned in the next section. QUERY: Does any player have a dominant strategy? Why is (paper, scissors) not a Nash equilibrium? Mixed Strategies Players’ strategies can be more complicated than simply choosing an action with certainty. In this section we study mixed strategies, which have the player randomly select from several possible actions. By contrast, the strategies considered in the examples thus far have a player choose one action or another with certainty; these are called pure strategies. For example, in the Battle of the Sexes, we have considered the pure strategies of choosing either ballet or boxing for sure. A possible mixed strategy in this game would be Chapter 8: Game Theory 261 to ﬂip a coin and then attend the ballet if and only if the coin comes up heads, yielding a 50–50 chance of showing up at either event. Although at ﬁrst glance it may seem bizarre to have players ﬂipping coins to determine how they will play, there are good reasons for studying mixed strategies. First, some games (such as Rock, Paper, Scissors) have no Nash equilibria in pure strategies. As we will see in the section on existence, such games will always have a Nash equilibrium in mixed strategies; therefore, allowing for mixed strategies will enable us to make predictions in such games where it was impossible to do so otherwise. Second, strategies involving randomization are natural and fam
iliar in certain settings. Students are familiar with the setting of class exams. Class time is usually too limited for the professor to examine students on every topic taught in class, but it may be sufﬁcient to test students on a subset of topics to induce them to study all the material. If students knew which topics were on the test, then they might be inclined to study only those and not the others; therefore, the professor must choose the topics at random to get the students to study everything. Random strategies are also familiar in sports (the same soccer player sometimes shoots to the right of the net and sometimes to the left on penalty kicks) and in card games (the poker player sometimes folds and sometimes bluffs with a similarly poor hand at different times).4 rm i , . . . , rm r1 , where rm i Formal deﬁnitions To be more formal, suppose that player i has a set of M possible actions Ai ¼ i , . . . , aM i , . . . , am a1 , where the subscript refers to the player and the superscript to the i g f different choices. A mixed strategy is a probability distribution over the M actions, i , . . . , rM is a number between 0 and 1 that indicates the si ¼ ð i Þ probability of player i playing action am i . The probabilities in si must sum to unity: rM r1 In the Battle of the Sexes, for example, both players have the same two actions of ballet and boxing, so we can write A1 ¼ {ballet, boxing}. We can write a mixed strategy as a pair of probabilities (s, 1 s), where s is the probability that the player chooses ballet. The probabilities must sum to unity, and so, with two actions, once the probability of one action is speciﬁed, the probability of the other is determined. Mixed strategy (1/3, 2/3) means that the player plays ballet with probability 1/3 and boxing with probability 2/3; (1/2, 1/2) means that the player is equally likely to play ballet or boxing; (1, 0) means that the player chooses ballet with certainty; and (0, 1) means that the player chooses boxing with certainty. A2 ¼ i ¼ ! 1. In our terminology, a mixed strategy is a general category that includes the special case of a pure strategy. A pure strategy is the special case in which only one action is played with positive probability. Mixed strategies that involve two or more actions being played with positive probability are called strictly mixed strategies. Returning to the examples from the previous paragraph of mixed strategies in the Battle of the Sexes, all four strategies (1/3, 2/3), (1/2, 1/2), (1, 0), and (0, 1) are mixed strategies. The ﬁrst two are strictly mixed, and the second two are pure strategies. With this notation for actions and mixed strategies behind us, we do not need new deﬁnitions for best response, Nash equilibrium, and dominant strategy. The deﬁnitions introduced when si was taken to be a pure strategy also apply to the case in which si is taken to be a mixed strategy. The only change is that the payoff function ui(si, s i), rather ! 4A third reason is that mixed strategies can be ‘‘puriﬁed’’ by specifying a more complicated game in which one or the other action is better for the player for privately known reasons and where that action is played with certainty. For example, a history professor might decide to ask an exam question about World War I because, unbeknownst to the students, she recently read an interesting journal article about it. See John Harsanyi, ‘‘Games with Randomly Disturbed Payoffs: A New Rationale for MixedStrategy Equilibrium Points,’’ International Journal of Game Theory 2 (1973): 1–23. Harsanyi was a co-winner (along with Nash) of the 1994 Nobel Prize in economics. 262 Part 3: Uncertainty and Strategy than being a certain payoff, must be reinterpreted as the expected value of a random payoff, with probabilities given by the strategies si and s i. Example 8.3 provides some practice in computing expected payoffs in the Battle of the Sexes. ! EXAMPLE 8.3 Expected Payoffs in the Battle of the Sexes Let’s compute players’ expected payoffs if the wife chooses the mixed strategy (1/9, 8/9) and the husband (4/5, 1/5) in the Battle of the Sexes. The wife’s expected payoff is U1 , U1ð # $ # $ 16 : 45 U1ð Þ þ ballet, ballet Þ þ boxing, ballet # $ U1ð ballet, boxing Þ boxing, boxing U1ð 8:4) To understand Equation 8.4, it is helpful to review the concept of expected value from Chapter 2. The expected value of a random variable equals the sum over all outcomes of the probability of the outcome multiplied by the value of the random variable in that outcome. In the Battle of the Sexes, there are four outcomes, corresponding to the four boxes in Figure 8.3. Because players randomize independently, the probability of reaching a particular box equals the product of the probabilities that each player plays the strategy leading to that box. Thus, for example, the probability (boxing, ballet)—that is, the wife plays boxing and the husband plays ballet—equals (8/9) (4/5). The probabilities of the four outcomes are multiplied by the value of the relevant random variable (in this case, players 1’s payoff) in each outcome. ) Next we compute the wife’s expected payoff if she plays the pure strategy of going to ballet [the same as the mixed strategy (1, 0)] and the husband continues to play the mixed strategy (4/5, 1/5). Now there are only two relevant outcomes, given by the two boxes in the row in which the wife plays ballet. The probabilities of the two outcomes are given by the probabilities in the husband’s mixed strategy. Therefore, ballet, ballet U 1ð ballet, boxing U 1ð Þ U 1 ballet # $ Finally, we will compute the general expression for the wife’s expected payoff when she plays h): If the wife plays ballet with w) and the husband plays (h, 1 (8:5 : mixed strategy (w, 1 probability w and the husband with probability h, then ! ! 1 5 # $ Þ þ 8 5 w, 1 U 1ðð , w Þ h, 1 ð ! h ! ÞÞ ¼ ð 1 Þð h ! ballet, boxing U 1ð Þ Þ h w w Þ þ ð ballet, ballet U 1ð Þ Þð boxing, ballet w U 1ð h 1 Þ Þ Þð ! þ ð U 1ð h 1 w 1 Þ ! Þð ! þ ð h w 1 w 2 Þð Þ þ ð Þð ¼ ð ! Þð h 1 1 w 1 ! Þð ! þ ð Þ Þð 3hw: w h 1 þ ¼ ! ! Þð h 0 boxing, boxing 1 Þ þ ð Þ w ! Þð h 0 Þð Þ (8:6) QUERY: What is the husband’s expected payoff in each case? Show that his expected payoff is 2 3hw in the general case. Given the husband plays the mixed strategy (4/5, 1/5), what strategy provides the wife with the highest payoff? 2w 2h ! ! þ Chapter 8: Game Theory 263 Computing mixed-strategy equilibria Computing Nash equilibria of a game when strictly mixed strategies are involved is a bit more complicated than when pure strategies are involved. Before wading in, we can save a lot of work by asking whether the game even has a Nash equilibrium in strictly mixed strategies. If it does not, having found all the pure-strategy Nash equilibria, then one has ﬁnished analyzing the game. The key to guessing whether a game has a Nash equilibrium in strictly mixed strategies is the surprising result that almost all games have an odd number of Nash equilibria.5 Let’s apply this insight to some of the examples considered thus far. We found an odd number (one) of pure-strategy Nash equilibria in the Prisoners’ Dilemma, suggesting we need not look further for one in strictly mixed strategies. In the Battle of the Sexes, we found an even number (two) of pure-strategy Nash equilibria, suggesting the existence of a third one in strictly mixed strategies. Example 8.2—Rock, Paper, Scissors—has no purestrategy Nash equilibria. To arrive at an odd number of Nash equilibria, we would expect to ﬁnd one Nash equilibrium in strictly mixed strategies. EXAMPLE 8.4 Mixed-Strategy Nash Equilibrium in the Battle of the Sexes ! A general mixed strategy for the wife in the Battle of the Sexes is (w, 1 – w) and for the husband is (h, 1 h), where w and h are the probabilities of playing ballet for the wife and husband, respectively. We will compute values of w and h that make up Nash equilibria. Both players have a continuum of possible strategies between 0 and 1. Therefore, we cannot write these strategies in the rows and columns of a matrix and underline best-response payoffs to ﬁnd the Nash equilibria. Instead, we will use graphical methods to solve for the Nash equilibria. Given players’ general mixed strategies, we saw in Example 8.3 that the wife’s expected payoff is w, 1 U 1ðð w , Þ h, 1 ð ! ! h ÞÞ ¼ h 1 ! ! w þ 3hw: (8:7) As Equation 8.7 shows, the wife’s best response depends on h. If h < 1/3, she wants to set w as 1. When low as possible: w h 1/3, her expected payoff equals 2/3 regardless of what w she chooses. In this case there is a tie for the best response, including any w from 0 to 1. 0. If h > 1/3, her best response is to set w as high as possible: w ¼ ¼ ¼ In Example 8.3, we stated that the husband’s expected payoff is h, 1 U 2ðð h , Þ w, 1 ð ! ! w ÞÞ ¼ 2h 2 ! ! 2w þ 3hw: (8:8) When w < 2/3, his expected payoff is maximized by h is maximized by h expected payoff of 2/3 regardless. 1; and when w ¼ ¼ 0; when w > 2/3, his expected payoff 2/3, he is indifferent among all values of h, obtaining an ¼ The best responses are graphed in Figure 8.7. The Nash equilibria are given by the intersection points between the best responses. At these intersection points, both players are best responding to each other, which is what is required for the outcome to be a Nash equilibrium. There are three Nash equilibria. The points E1 and E2 are the pure-strategy Nash equilibria we found before, with E1 corresponding to the pure-strategy Nash equilibrium in which both play boxing and E2 to that in which both play ballet. Point E3 is the strictly mixed-strategy Nash equilibrium, which can be spelled out as ‘‘the wife plays ballet with probability 2/3 and boxing with probability 1/3 and the husband plays ballet with probability 1/3 and boxing with probability 2/3.’’ More succinctly, having deﬁned w and h, we may write the equilibrium as ‘‘w’ 2/3 and h’ 1/3.’’ ¼ ¼ 5John Harsanyi, ‘‘Oddness of the Number of Equilibrium Points
: A New Proof,’’ International Journal of Game Theory 2 (1973): 235–50. Games in which there are ties between payoffs may have an even or inﬁnite number of Nash equilibria. Example 8.1, the Prisoners’ Dilemma Redux, has several payoff ties. The game has four pure-strategy Nash equilibria and an inﬁnite number of different mixed-strategy equilibria. 264 Part 3: Uncertainty and Strategy FIGURE 8.78Nash Equilibria in Mixed Strategies in the Battle of the Sexes Ballet is chosen by the wife with probability w and by the husband with probability h. Players’ best responses are graphed on the same set of axes. The three intersection points E1, E2, and E3 are Nash equilibria. The Nash equilibrium in strictly mixed strategies, E3, is w’ 2/3 and h’ 1/3. ¼ ¼ h 1 2/3 1/3 E1 0 E2 Husband’s best response, BR2 E3 Wife’s best response, BR1 w 1/3 2/3 1 QUERY: What is a player’s expected payoff in the Nash equilibrium in strictly mixed strategies? How does this payoff compare with those in the pure-strategy Nash equilibria? What arguments might be offered that one or another of the three Nash equilibria might be the best prediction in this game? Example 8.4 runs through the lengthy calculations involved in ﬁnding all the Nash equilibria of the Battle of the Sexes, those in pure strategies and those in strictly mixed strategies. A shortcut to ﬁnding the Nash equilibrium in strictly mixed strategies is based on the insight that a player will be willing to randomize between two actions in equilibrium only if he or she gets the same expected payoff from playing either action or, in other words, is indifferent between the two actions in equilibrium. Otherwise, one of the two actions would provide a higher expected payoff, and the player would prefer to play that action with certainty. Suppose the husband is playing mixed strategy (h, 1 probability h and boxing with probability 1 ballet is ! h), that is, playing ballet with h. The wife’s expected payoff from playing ! U 1ð Þ ¼ ð Her expected payoff from playing boxing is ballet, (h, 1 h) ! h 2 Þð 1 0 h Þð ! Þ ¼ Þ þ ð 2h: (8:9) boxing, (h, 1 h) U 1ð h 0 Þ þ ð 1 Þð For the wife to be indifferent between ballet and boxing in equilibrium, Equations 8.9 and 8.10 must be equal: 2h 1/3. Similar calculations based on the husband’s indifference between playing ballet and boxing in equilibrium show that the h, implying h’ Þ ¼ ð Þ ¼ ¼ ¼ ! ! ! ! Þð 1 h 1 1 h: (8:10) Chapter 8: Game Theory 265 wife’s probability of playing ballet in the strictly mixed strategy Nash equilibrium is w’ 2/3. (Work through these calculations as an exercise.) ¼ Notice that the wife’s indifference condition does not ‘‘pin down’’ her equilibrium mixed strategy. The wife’s indifference condition cannot pin down her own equilibrium mixed strategy because, given that she is indifferent between the two actions in equilibrium, her overall expected payoff is the same no matter what probability distribution she plays over the two actions. Rather, the wife’s indifference condition pins down the other player’s—the husband’s—mixed strategy. There is a unique probability distribution he can use to play ballet and boxing that makes her indifferent between the two actions and thus makes her willing to randomize. Given any probability of his playing ballet and boxing other than (1/3, 2/3), it would not be a stable outcome for her to randomize. Thus, two principles should be kept in mind when seeking Nash equilibria in strictly mixed strategies. One is that a player randomizes over only those actions among which he or she is indifferent, given other players’ equilibrium mixed strategies. The second is that one player’s indifference condition pins down the other player’s mixed strategy. Existence Of Equilibrium One of the reasons Nash equilibrium is so widely used is that a Nash equilibrium is guaranteed to exist in a wide class of games. This is not true for some other equilibrium concepts. Consider the dominant strategy equilibrium concept. The Prisoners’ Dilemma has a dominant strategy equilibrium (both suspects ﬁnk), but most games do not. Indeed, there are many games—including, for example, the Battle of the Sexes—in which no player has a dominant strategy, let alone all the players. In such games, we cannot make predictions using dominant strategy equilibrium, but we can using Nash equilibrium. The Extensions section at the end of this chapter will provide the technical details behind John Nash’s proof of the existence of his equilibrium in all ﬁnite games (games with a ﬁnite number of players and a ﬁnite number of actions). The existence theorem does not guarantee the existence of a pure-strategy Nash equilibrium. We already saw an example: Rock, Paper, Scissors in Example 8.2. However, if a ﬁnite game does not have a purestrategy Nash equilibrium, the theorem guarantees that it will have a mixed-strategy Nash equilibrium. The proof of Nash’s theorem is similar to the proof in Chapter 13 of the existence of prices leading to a general competitive equilibrium. The Extensions section includes an existence theorem for games with a continuum of actions, as studied in the next section. Continuum Of Actions Most of the insight from economic situations can often be gained by distilling the situation down to a few or even two actions, as with all the games studied thus far. Other times, additional insight can be gained by allowing a continuum of actions. To be clear, we have already encountered a continuum of strategies—in our discussion of mixed strategies—but still the probability distributions in mixed strategies were over a ﬁnite number of actions. In this section we focus on continuum of actions. Some settings are more realistically modeled via a continuous range of actions. In Chapter 15, for example, we will study competition between strategic ﬁrms. In one model (Bertrand), ﬁrms set prices; in another (Cournot), ﬁrms set quantities. It is natural to allow ﬁrms to choose any non-negative price or quantity rather than artiﬁcially restricting them to just two prices (say, $2 or $5) or two quantities (say, 100 or 1,000 units). Continuous actions have several other advantages. The familiar methods from calculus can often be used to solve for Nash equilibria. It is also possible to analyze how the equilibrium 266 Part 3: Uncertainty and Strategy actions vary with changes in underlying parameters. With the Cournot model, for example, we might want to know how equilibrium quantities change with a small increase in a ﬁrm’s marginal costs or a demand parameter. Tragedy of the Commons Example 8.5 illustrates how to solve for the Nash equilibrium when the game (in this case, the Tragedy of the Commons) involves a continuum of actions. The ﬁrst step is to write down the payoff for each player as a function of all players’ actions. The next step is to compute the ﬁrst-order condition associated with each player’s payoff maximum. This will give an equation that can be rearranged into the best response of each player as a function of all other players’ actions. There will be one equation for each player. With n players, the system of n equations for the n unknown equilibrium actions can be solved simultaneously by either algebraic or graphical methods. EXAMPLE 8.5 Tragedy of the Commons The term Tragedy of the Commons has come to signify environmental problems of overuse that arise when scarce resources are treated as common property.6 A game-theoretic illustration of this issue can be developed by assuming that two herders decide how many sheep to graze on the village commons. The problem is that the commons is small and can rapidly succumb to overgrazing. To add some mathematical structure to the problem, let qi be the number of sheep that 1, 2 grazes on the commons, and suppose that the per-sheep value of grazing on the herder i commons (in terms of wool and sheep-milk cheese) is ¼ v q1, q2Þ ¼ ð 120 q1 þ ! ð : q2Þ (8:11) This function implies that the value of grazing a given number of sheep is lower the more sheep are around competing for grass. We cannot use a matrix to represent the normal form of this game of continuous actions. Instead, the normal form is simply a listing of the herders’ payoff functions u1ð u2ð q1, q2Þ ¼ q1, q2Þ ¼ q1v q2v q1, q2Þ ¼ ð q1, q2Þ ¼ ð 120 120 q1ð q2ð q1 ! q1 ! q2Þ , : q2Þ ! ! To ﬁnd the Nash equilibrium, we solve herder 1’s value-maximization problem: max q1 f q1ð 120 q1 ! : q2Þg ! The ﬁrst-order condition for a maximum is or, rearranging, 120 2q1 ! q2 ¼ 0 ! q1 ¼ 60 ! q2 2 ¼ BR1ð : q2Þ (8:12) (8:13) (8:14) (8:15) Similar steps show that herder 2’s best response is q1 2 ¼ q2 ¼ 60 ! : q1Þ that satisﬁes Equations 8.15 and 8.16 The Nash equilibrium is given by the pair simultaneously. Taking an algebraic approach to the simultaneous solution, Equation 8.16 can ! be substituted into Equation 8.15, which yields BR2ð q’1, q’2 (8:16) " 6This term was popularized by G. Hardin, ‘‘The Tragedy of the Commons,’’ Science 162 (1968): 1243–48. Chapter 8: Game Theory 267 60 1 2 q1 ¼ on rearranging, this implies q’1 ¼ 40 as well. Thus, each herder will graze 40 sheep on the common. Each earns a payoff of 1,600, as can be seen by substituting q’1 ¼ q’2 ¼ Equations 8.15 and 8.16 can also be solved simultaneously using graphical methods. Figure 8.8 plots the two best responses on a graph with player 1’s action on the horizontal axis and ! ! 40. Substituting q’1 ¼ 40 into the payoff function in Equation 8.13. q1 2 40 into Equation 8.17 implies q’2 ¼ (8:17) 60 % & ; FIGURE 8.88Best-Response Diagram for the Tragedy of the Commons The intersection, E1, between the two herders’ best responses is the Nash equilibrium. An increase in the per-sheep value of grazing in the Tragedy of the Commons shifts out herder 1’s best response, resulting in a Nash equilibrium E2 in which herder 1 grazes more sheep (and herder 2, fewer sheep) than in the original Nash equilibrium. BR1(q2) q2 120 60 40 E1 E2 0 40 60 BR2(q1) 120 q1 player 2’s on the verti
cal axis. These best responses are simply lines and thus are easy to graph in this example. (To be consistent with the axis labels, the inverse of Equation 8.15 is actually what is graphed.) The two best responses intersect at the Nash equilibrium E1. The graphical method is useful for showing how the Nash equilibrium shifts with changes in the parameters of the problem. Suppose the per-sheep value of grazing increases for the ﬁrst herder while the second remains as in Equation 8.11, perhaps because the ﬁrst herder starts raising merino sheep with more valuable wool. This change would shift the best response out for herder 1 while leaving herder 2’s the same. The new intersection point (E2 in Figure 8.8), which is the new Nash equilibrium, involves more sheep for 1 and fewer for 2. The Nash equilibrium is not the best use of the commons. In the original problem, both herders’ per-sheep value of grazing is given by Equation 8.11. If both grazed only 30 sheep, then each would earn a payoff of 1,800, as can be seen by substituting q1 ¼ 30 into Equation 8.13. Indeed, the ‘‘joint payoff maximization’’ problem q2Þg max q1 þ q1, q2fð 30 or, more generally, by any q1 and q2 that sum to 60. is solved by q1 ¼ QUERY: How would the Nash equilibrium shift if both herders’ beneﬁts increased by the same amount? What about a decrease in (only) herder 2’s beneﬁt from grazing? max q1, q2fð q2 ¼ q1, q2Þg ¼ ð q2 ¼ q1 ! q1 þ v q2Þ q2Þð (8:18) 120 ! 268 Part 3: Uncertainty and Strategy As Example 8.5 shows, graphical methods are particularly convenient for quickly determining how the equilibrium shifts with changes in the underlying parameters. The example shifted the beneﬁt of grazing to one of herders. This exercise nicely illustrates the nature of strategic interaction. Herder 2’s payoff function has not changed (only herder 1’s has), yet his equilibrium action changes. The second herder observes the ﬁrst’s higher beneﬁt, anticipates that the ﬁrst will increase the number of sheep he grazes, and reduces his own grazing in response. The Tragedy of the Commons shares with the Prisoners’ Dilemma the feature that the Nash equilibrium is less efﬁcient for all players than some other outcome. In the Prisoners’ Dilemma, both ﬁnk in equilibrium when it would be more efﬁcient for both to be silent. In the Tragedy of the Commons, the herders graze more sheep in equilibrium than is efﬁcient. This insight may explain why ocean ﬁshing grounds and other common resources can end up being overused even to the point of exhaustion if their use is left unregulated. More detail on such problems—involving what we will call negative externalities—is provided in Chapter 19. Sequential Games In some games, the order of moves matters. For example, in a bicycle race with a staggered start, it may help to go last and thus know the time to beat. On the other hand, competition to establish a new high-deﬁnition video format may be won by the ﬁrst ﬁrm to market its technology, thereby capturing an installed base of consumers. Sequential games differ from the simultaneous games we have considered thus far in that a player who moves later in the game can observe how others have played up to that moment. The player can use this information to form more sophisticated strategies than simply choosing an action; the player’s strategy can be a contingent plan with the action played depending on what the other players have done. To illustrate the new concepts raised by sequential games—and, in particular, to make a stark contrast between sequential and simultaneous games—we take a simultaneous game we have discussed already, the Battle of the Sexes, and turn it into a sequential game. Sequential Battle of the Sexes Consider the Battle of the Sexes game analyzed previously with all the same actions and payoffs, but now change the timing of moves. Rather than the wife and husband making a simultaneous choice, the wife moves ﬁrst, choosing ballet or boxing; the husband observes this choice (say, the wife calls him from her chosen location), and then the husband makes his choice. The wife’s possible strategies have not changed: She can choose the simple actions ballet or boxing (or perhaps a mixed strategy involving both actions, although this will not be a relevant consideration in the sequential game). The husband’s set of possible strategies has expanded. For each of the wife’s two actions, he can choose one of two actions; therefore, he has four possible strategies, which are listed in Table 8.1. TABLE 8.1 HUSBAND’S CONTINGENT STRATEGIES Contingent Strategy Always go to the ballet Follow his wife Do the opposite Always go to boxing Written in Conditional Format (ballet \| ballet, ballet \| boxing) (ballet \| ballet, boxing \| boxing) (boxing \| ballet, ballet \| boxing) (boxing \| ballet, boxing \| boxing) Chapter 8: Game Theory 269 The vertical bar in the husband’s strategies means ‘‘conditional on’’ and thus, for example, ‘‘boxing \| ballet’’ should be read as ‘‘the husband chooses boxing conditional on the wife’s choosing ballet.’’ Given that the husband has four pure strategies rather than just two, the normal form (given in Figure 8.9) must now be expanded to eight boxes. Roughly speaking, the normal form is twice as complicated as that for the simultaneous version of the game in Figure 8.2. This motivates a new way to represent games, called the extensive form, which is especially convenient for sequential games. Extensive form The extensive form of a game shows the order of moves as branches of a tree rather than collapsing everything down into a matrix. The extensive form for the sequential Battle of the Sexes is shown in Figure 8.10a. The action proceeds from left to right. Each node (shown as a dot on the tree) represents a decision point for the player indicated there. The ﬁrst move belongs to the wife. After any action she might take, the husband gets to move. Payoffs are listed at the end of the tree in the same order (player 1’s, player 2’s) as in the normal form. Contrast Figure 8.10a with Figure 8.10b, which shows the extensive form for the simultaneous version of the game. It is hard to harmonize an extensive form, in which moves happen in progression, with a simultaneous game, in which everything happens at the same time. The trick is to pick one of the two players to occupy the role of the second mover but then highlight that he or she is not really the second mover by connecting his or her decision nodes in the same information set, the dotted oval around the nodes. The dotted oval in Figure 8.10b indicates that the husband does not know his wife’s move when he chooses his action. It does not matter which player is picked for ﬁrst and second mover in a simultaneous game; we picked the husband in the ﬁgure to make the extensive form in Figure 8.10b look as much like Figure 8.10a as possible. The similarity between the two extensive forms illustrates the point that that form does not grow in complexity for sequential games the way the normal form does. We FIGURE 8.9 Normal Form for the Sequential Battle of the Sexes Husband (Ballet \| Ballet Ballet \| Boxing) (Ballet \| Ballet Boxing \| Boxing) (Boxing \| Ballet Ballet \| Boxing) (Boxing \| Ballet Boxing \| Boxing) Ballet 2, 1 2, 1 0, 0 0, 0 e f i W Boxing 0, 0 1, 2 0, 0 1, 2 270 Part 3: Uncertainty and Strategy FIGURE 8.10 Extensive Form for the Battle of the Sexes In the sequential version (a), the husband moves second, after observing his wife’s move. In the simultaneous version (b), he does not know her choice when he moves, so his decision nodes must be connected in one information set. Ballet 1 Boxing 2 2 Ballet Boxing Ballet Boxing 2, 1 0, 0 0, 0 1, 2 Ballet 1 Boxing 2 2 Ballet Boxing Ballet Boxing 2, 1 0, 0 0, 0 1, 2 (a) Sequential version (b) Simultaneous version next will draw on both normal and extensive forms in our analysis of the sequential Battle of the Sexes. Nash equilibria To solve for the Nash equilibria, return to the normal form in Figure 8.9. Applying the method of underlining best-response payoffs—being careful to underline both payoffs in cases of ties for the best response—reveals three pure-strategy Nash equilibria: 1. wife plays ballet, husband plays (ballet \| ballet, ballet \| boxing); 2. wife plays ballet, husband plays (ballet \| ballet, boxing \| boxing); 3. wife plays boxing, husband plays (boxing \| ballet, boxing \| boxing). As with the simultaneous version of the Battle of the Sexes, here again we have multiple equilibria. Yet now game theory offers a good way to select among the equilibria. Consider the third Nash equilibrium. The husband’s strategy (boxing \| ballet, boxing \| boxing) involves the implicit threat that he will choose boxing even if his wife chooses ballet. This threat is sufﬁcient to deter her from choosing ballet. Given that she chooses boxing in equilibrium, his strategy earns him 2, which is the best he can do in any outcome. Thus, the outcome is a Nash equilibrium. But the husband’s threat is not credible—that is, it is an empty threat. If the wife really were to choose ballet ﬁrst, then he would give up a payoff of 1 by choosing boxing rather than ballet. It is clear why he would want to threaten to choose boxing, but it is not clear that such a threat should be Chapter 8: Game Theory 271 believed. Similarly, the husband’s strategy (ballet \| ballet, ballet \| boxing) in the ﬁrst Nash equilibrium also involves an empty threat: that he will choose ballet if his wife chooses boxing. (This is an odd threat to make because he does not gain from making it, but it is an empty threat nonetheless.) Another way to understand empty versus credible threats is by using the concept of the equilibrium path, the connected path through the extensive form implied by equilibrium strategies. In Figure 8.11, which reproduces the extensive form of the sequential Battle of the Sexes from Figure 8.10, a dotted line is used to identify the equilibrium path for the third of the listed Nash eq
uilibria. The third outcome is a Nash equilibrium because the strategies are rational along the equilibrium path. However, following the wife’s choosing ballet—an event that is off the equilibrium path—the husband’s strategy is irrational. The concept of subgame-perfect equilibrium in the next section will rule out irrational play both on and off the equilibrium path. Subgame-perfect equilibrium Game theory offers a formal way of selecting the reasonable Nash equilibria in sequential games using the concept of subgame-perfect equilibrium. Subgame-perfect equilibrium is a reﬁnement that rules out empty threats by requiring strategies to be rational even for contingencies that do not arise in equilibrium. Before deﬁning subgame-perfect equilibrium formally, we need a few preliminary deﬁnitions. A subgame is a part of the extensive form beginning with a decision node and including everything that branches out to the right of it. A proper subgame is a subgame FIGURE 8.11 Equilibrium Path In the third of the Nash equilibria listed for the sequential Battle of the Sexes, the wife plays boxing and the husband plays (boxing \| ballet, boxing \| boxing), tracing out the branches indicated with thick lines (both solid and dashed). The dashed line is the equilibrium path; the rest of the tree is referred to as being ‘‘off the equilibrium path.’’ Ballet 2 Ballet Boxing 1 Boxing Ballet 2 Boxing 2, 1 0, 0 0, 0 1, 2 272 Part 3: Uncertainty and Strategy that starts at a decision node not connected to another in an information set. Conceptually, this means that the player who moves ﬁrst in a proper subgame knows the actions played by others that have led up to that point. It is easier to see what a proper subgame is than to deﬁne it in words. Figure 8.12 shows the extensive forms from the simultaneous and sequential versions of the Battle of the Sexes with boxes drawn around the proper subgames in each. The sequential version (a) has three proper subgames: the game itself and two lower subgames starting with decision nodes where the husband gets to move. The simultaneous version (b) has only one decision node—the topmost node—not connected to another in an information set. Hence this verion has only one subgame: the whole game itself Subgame-perfect equilibrium. A subgame-perfect equilibrium is a strategy proﬁle s’1, s’2, . . . , s’n that is a Nash equilibrium on every proper subgame. ! " A subgame-perfect equilibrium is always a Nash equilibrium. This is true because the whole game is a proper subgame of itself; thus, a subgame-perfect equilibrium must be a Nash equilibrium for the whole game. In the simultaneous version of the Battle of the Sexes, there is nothing more to say because there are no subgames other than the whole game itself. In the sequential version, subgame-perfect equilibrium has more bite. Strategies must not only form a Nash equilibrium on the whole game itself; they must also form Nash FIGURE 8.12 Proper Subgames in the Battle of the Sexes The sequential version in (a) has three proper subgames, labeled A, B, and C. The simultaneous version in (b) has only one proper subgame: the whole game itself, labeled D. A B 2 Ballet 2, 1 D Ballet 2 Ballet Boxing Ballet Boxing 1 Boxing C Ballet 2 Boxing 0, 0 0, 0 1, 2 1 Boxing Ballet 2 Boxing 2, 1 0, 0 0, 0 1, 2 (a) Sequential (b) Simultaneous Chapter 8: Game Theory 273 equilibria on the two proper subgames starting with the decision points at which the husband moves. These subgames are simple decision problems, so it is easy to compute the corresponding Nash equilibria. For subgame B, beginning with the husband’s decision node following his wife’s choosing ballet, he has a simple decision between ballet (which earns him a payoff of 1) and boxing (which earns him a payoff of 0). The Nash equilibrium in this simple decision subgame is for the husband to choose ballet. For the other subgame, C, he has a simple decision between ballet, which earns him 0, and boxing, which earns him 2. The Nash equilibrium in this simple decision subgame is for him to choose boxing. Therefore, the husband has only one strategy that can be part of a subgame-perfect equilibrium: (ballet \| ballet, boxing \| boxing). Any other strategy has him playing something that is not a Nash equilibrium for some proper subgame. Returning to the three enumerated Nash equilibria, only the second is subgame perfect; the ﬁrst and the third are not. For example, the third equilibrium, in which the husband always goes to boxing, is ruled out as a subgame-perfect equilibrium because the husband’s strategy (boxing \| boxing) is not a Nash equilibrium in proper subgame B. Thus, subgame-perfect equilibrium rules out the empty threat (of always going to boxing) that we were uncomfortable with earlier. More generally, subgame-perfect equilibrium rules out any sort of empty threat in a sequential game. In effect, Nash equilibrium requires behavior to be rational only on the equilibrium path. Players can choose potentially irrational actions on other parts of the extensive form. In particular, one player can threaten to damage both to scare the other from choosing certain actions. Subgame-perfect equilibrium requires rational behavior both on and off the equilibrium path. Threats to play irrationally—that is, threats to choose something other than one’s best response—are ruled out as being empty. Backward induction Our approach to solving for the equilibrium in the sequential Battle of the Sexes was to ﬁnd all the Nash equilibria using the normal form and then to seek among those for the subgame-perfect equilibrium. A shortcut for ﬁnding the subgame-perfect equilibrium directly is to use backward induction, the process of solving for equilibrium by working backward from the end of the game to the beginning. Backward induction works as follows. Identify all the subgames at the bottom of the extensive form. Find the Nash equilibria on these subgames. Replace the (potentially complicated) subgames with the actions and payoffs resulting from Nash equilibrium play on these subgames. Then move up to the next level of subgames and repeat the procedure. Figure 8.13 illustrates the use of backward induction in the sequential Battle of the Sexes. First, we compute the Nash equilibria of the bottom-most subgames at the husband’s decision nodes. In the subgame following his wife’s choosing ballet, he would choose ballet, giving payoffs 2 for her and 1 for him. In the subgame following his wife’s choosing boxing, he would choose boxing, giving payoffs 1 for her and 2 for him. Next, substitute the husband’s equilibrium strategies for the subgames themselves. The resulting game is a simple decision problem for the wife (drawn in the lower panel of the ﬁgure): a choice between ballet, which would give her a payoff of 2, and boxing, which would give her a payoff of 1. The Nash equilibrium of this game is for her to choose the action with the higher payoff, ballet. In sum, backward induction allows us to jump straight to the subgame-perfect equilibrium in which the wife chooses ballet and the husband chooses (ballet \| ballet, boxing \| boxing), bypassing the other Nash equilibria. Backward induction is particularly useful in games that feature many rounds of sequential play. As rounds are added, it quickly becomes too hard to solve for all the Nash 274 Part 3: Uncertainty and Strategy FIGURE 8.13 Applying Backward Induction The last subgames (where player 2 moves) are replaced by the Nash equilibria on these subgames. The simple game that results at right can be solved for player 1’s equilibrium action. Ballet 2 Ballet Boxing 1 Boxing Ballet 2 Boxing 2, 1 0, 0 0, 0 1, 2 Ballet 1 Boxing 2 plays ballet \| ballet payoﬀ 2, 1 2 plays boxing \| boxing payoﬀ 1, 2 equilibria and then to sort through which are subgame-perfect. With backward induction, an additional round is simply accommodated by adding another iteration of the procedure. Repeated Games In the games examined thus far, each player makes one choice and the game ends. In many real-world settings, players play the same game over and over again. For example, the players in the Prisoners’ Dilemma may anticipate committing future crimes and thus playing future Prisoners’ Dilemmas together. Gasoline stations located across the street from each other, when they set their prices each morning, effectively play a new pricing game every day. The simple constituent game (e.g., the Prisoners’ Dilemma or the gasoline-pricing game) that is played repeatedly is called the stage game. As we saw with the Prisoners’ Dilemma, the equilibrium in one play of the stage game may be worse for all players than some other, more cooperative, outcome. Repeated play of the stage game opens up the possibility of cooperation in equilibrium. Players can adopt trigger strategies, whereby they continue to cooperate as long as all have cooperated up to that point but revert to playing the Nash equilibrium if anyone deviates from cooperation. We will investigate the conditions under which trigger strategies work to increase players’ payoffs. As is standard in game theory, we will focus on subgame-perfect equilibria of the repeated games. Finitely repeated games For many stage games, repeating them a known, ﬁnite number of times does not increase the possibility for cooperation. To see this point concretely, suppose the Prisoners’ Chapter 8: Game Theory 275 Dilemma were played repeatedly for T periods. Use backward induction to solve for the subgame-perfect equilibrium. The lowest subgame is the Prisoners’ Dilemma stage game played in period T. Regardless of what happened before, the Nash equilibrium on this subgame is for both to ﬁnk. Folding the game back to period T 1, trigger strategies that 1 are ruled out. Although a condition period T play on what happens in period T player might like to promise to play cooperatively in period T and thus reward the other for playing cooperatively in period T 1, we have just seen th
at nothing that happens in 1 affects what happens subsequently because players both ﬁnk in period T regardperiod T 1 were the last, and the Nash equilibrium of this subgame is less. It is as though period T again for both to ﬁnk. Working backward in this way, we see that players will ﬁnk each period; that is, players will simply repeat the Nash equilibrium of the stage game T times. ! ! ! ! ! Reinhard Selten, winner of the Nobel Prize in economics for his contributions to game theory, showed that this logic is general: For any stage game with a unique Nash equilibrium, the unique subgame-perfect equilibrium of the ﬁnitely repeated game involves playing the Nash equilibrium every period.7 If the stage game has multiple Nash equilibria, it may be possible to achieve some cooperation in a ﬁnitely repeated game. Players can use trigger strategies, sustaining cooperation in early periods on an outcome that is not an equilibrium of the stage game, by threatening to play in later periods the Nash equilibrium that yields a worse outcome for the player who deviates from cooperation.8 Rather than delving into the details of ﬁnitely repeated games, we will instead turn to inﬁnitely repeated games, which greatly expand the possibility of cooperation. Inﬁnitely repeated games With ﬁnitely repeated games, the folk theorem applies only if the stage game has multiple equilibria. If, like the Prisoners’ Dilemma, the stage game has only one Nash equilibrium, then Selten’s result tells us that the ﬁnitely repeated game has only one subgame-perfect equilibrium: repeating the stage-game Nash equilibrium each period. Backward induction starting from the last period T unravels any other outcomes. With inﬁnitely repeated games, however, there is no deﬁnite ending period T from which to start backward induction. Outcomes involving cooperation do not necessarily end up unraveling. Under some conditions the opposite may be the case, with essentially anything being possible in equilibrium of the inﬁnitely repeated game. This result is sometimes called the folk theorem because it was part of the ‘‘folk wisdom’’ of game theory before anyone bothered to prove it formally. One difﬁculty with inﬁnitely repeated games involves adding up payoffs across periods. An inﬁnite stream of low payoffs sums to inﬁnity just as an inﬁnite stream of high payoffs. How can the two streams be ranked? We will circumvent this problem with the aid of discounting. Let d be the discount factor (discussed in the Chapter 17 Appendix) measuring how much a payoff unit is worth if received one period in the future rather than today. In Chapter 17 we show that d is inversely related to the interest rate.9 If the interest rate is high, then a person would much rather receive payment today than next period because investing 7R. Selten, ‘‘A Simple Model of Imperfect Competition, Where 4 Are Few and 6 Are Many,’’ International Journal of Game Theory 2 (1973): 141–201. 8J. P. Benoit and V. Krishna, ‘‘Finitely Repeated Games,’’ Econometrica 53 (1985): 890–940. 9Beware of the subtle difference between the formulas for the present value of an annuity stream used here and in Chapter 17 Appendix. There the payments came at the end of the period rather than at the beginning as assumed here. So here the present value of $1 payment per period from now on is $1 $1 d ( þ þ $1 ( d2 d3 $1 ( þ þ ::: ¼ 1 $1 ! : d 276 Part 3: Uncertainty and Strategy today’s payment would provide a return of principal plus a large interest payment next period. Besides the interest rate, d can also incorporate uncertainty about whether the game continues in future periods. The higher the probability that the game ends after the current period, the lower the expected return from stage games that might not actually be played. Factoring in a probability that the repeated game ends after each period makes the setting of an inﬁnitely repeated game more believable. The crucial issue with an inﬁnitely repeated game is not that it goes on forever but that its end is indeterminate. Interpreted in this way, there is a sense in which inﬁnitely repeated games are more realistic than ﬁnitely repeated games with large T. Suppose we expect two neighboring gasoline stations to play a pricing game each day until electric cars replace gasoline-powered ones. It is unlikely the gasoline stations would know that electric cars were coming in exactly T 2,000 days. More realistically, the gasoline stations will be uncertain about the end of gasoline-powered cars; thus, the end of their pricing game is indeterminate. ¼ Players can try to sustain cooperation using trigger strategies. Trigger strategies have them continuing to cooperate as long as no one has deviated; deviation triggers some sort of punishment. The key question in determining whether trigger strategies ‘‘work’’ is whether the punishment can be severe enough to deter the deviation in the ﬁrst place. Suppose both players use the following speciﬁc trigger strategy in the Prisoners’ Dilemma: Continue being silent if no one has deviated; ﬁnk forever afterward if anyone has deviated to ﬁnk in the past. To show that this trigger strategy forms a subgameperfect equilibrium, we need to check that a player cannot gain from deviating. Along the equilibrium path, both players are silent every period; this provides each with a payoff of 2 every period for a present discounted value of V eq ! 2d 2d2 d2 þ d þ þ þ þ 2d3 d3 þ ( ( ( þ ( ( (Þ : d (8:19) A player who deviates by ﬁnking earns 3 in that period, but then both players ﬁnk every period from then on—each earning 1 per period for a total presented discounted payoff of V dev 3 3 ¼ ¼ þ ð d þ 1 Þð Þð d2 d2 d3 1 Þð Þ þ The trigger strategies form a subgame-perfect equilibrium if V eq ! ¼ 1 d (8:20) V dev, implying that & 2 d 3 : (8:21 and rearranging, we obtain d After multiplying through by 1 1/2. In other words, players will ﬁnd continued cooperative play desirable provided they do not discount future gains from such cooperation too highly. If d < 1/2, then no cooperation is possible in the inﬁnitely repeated Prisoners’ Dilemma; the only subgame-perfect equilibrium involves ﬁnking every period. ! & The trigger strategy we considered has players revert to the stage-game Nash equilibrium of ﬁnking each period forever. This strategy, which involves the harshest possible punishment for deviation, is called the grim strategy. Less harsh punishments include the so-called tit-for-tat strategy, which involves only one round of punishment for cheating. Because the grim strategy involves the harshest punishment possible, it elicits cooperation for the largest range of cases Chapter 8: Game Theory 277 (the lowest value of d) of any strategy. Harsh punishments work well because, if players succeed in cooperating, they never experience the losses from the punishment in equilibrium.10 The discount factor d is crucial in determining whether trigger strategies can sustain cooperation in the Prisoners’ Dilemma or, indeed, in any stage game. As d approaches 1, grim-strategy punishments become inﬁnitely harsh because they involve an unending stream of undiscounted losses. Inﬁnite punishments can be used to sustain a wide range of possible outcomes. This is the logic behind the folk theorem for inﬁnitely repeated games. Take any stage-game payoff for a player between Nash equilibrium one and the highest one that appears anywhere in the payoff matrix. Let V be the present discounted value of the inﬁnite stream of this payoff. The folk theorem says that the player can earn V in some subgame-perfect equilibrium for d close enough to 1.11 Incomplete Information In the games studied thus far, players knew everything there was to know about the setup of the game, including each others’ strategy sets and payoffs. Matters become more complicated, and potentially more interesting, if some players have information about the game that others do not. Poker would be different if all hands were played face up. The fun of playing poker comes from knowing what is in your hand but not others’. Incomplete information arises in many other real-world contexts besides parlor games. A sports team may try to hide the injury of a star player from future opponents to prevent them from exploiting this weakness. Firms’ production technologies may be trade secrets, and thus ﬁrms may not know whether they face efﬁcient or weak competitors. This section (and the next two) will introduce the tools needed to analyze games of incomplete information. The analysis integrates the material on game theory developed thus far in this chapter with the material on uncertainty and information from the previous chapter. Games of incomplete information can quickly become complicated. Players who lack full information about the game will try to use what they do know to make inferences about what they do not. The inference process can be involved. In poker, for example, knowing what is in your hand can tell you something about what is in others’. A player who holds two aces knows that others are less likely to hold aces because two of the four aces are not available. Information on others’ hands can also come from the size of their bets or from their facial expressions (of course, a big bet may be a bluff and a facial expression may be faked). Probability theory provides a formula, called Bayes’ rule, for making inferences about hidden information. We will encounter Bayes’ rule in a later section. The relevance of Bayes’ rule in games of incomplete information has led them to be called Bayesian games. To limit the complexity of the analysis, we will focus on the simplest possible setting throughout. We will focus on two-player games in which one of the players (player 1) has private information and the other (player 2) does not. The analysis of games of incomplete information is divided into two sections. The next section begins with the simple case in which the players move simultaneously. Th
e subsequent section then 10Nobel Prize–winning economist Gary Becker introduced a related point, the maximal punishment principle for crime. The principle says that even minor crimes should receive draconian punishments, which can deter crime with minimal expenditure on policing. The punishments are costless to society because no crimes are committed in equilibrum, so punishments never have to be carried out. See G. Becker, ‘‘Crime and Punishment: An Economic Approach,’’ Journal of Political Economy 76 (1968): 169–217. Less harsh punishments may be suitable in settings involving uncertainty. For example, citizens may not be certain about the penal code; police may not be certain they have arrested the guilty party. 11A more powerful version of the folk theorem was proved by D. Fudenberg and E. Maskin (‘‘The Folk Theorem in Repeated Games with Discounting or with Incomplete Information,’’ Econometrica 54 (1986) 533–56). Payoffs below even the Nash equilibrium ones can be generated by some subgame-perfect equilibrium, payoffs all the way down to players’ minmax level (the lowest level a player can be reduced to by all other players working against him or her). 278 Part 3: Uncertainty and Strategy analyzes games in which the informed player 1 moves ﬁrst. Such games, called signaling games, are more complicated than simultaneous games because player 1’s action may signal something about his or her private information to the uninformed player 2. We will introduce Bayes’ rule at that point to help analyze player 2’s inference about player 1’s hidden information based on observations of player 1’s action. Simultaneous Bayesian Games In this section we study a two-player, simultaneous-move game in which player 1 has private information but player 2 does not. (We will use ‘‘he’’ for player 1 and ‘‘she’’ for player 2 to facilitate the exposition.) We begin by studying how to model private information. Player types and beliefs John Harsanyi, who received the Nobel Prize in economics for his work on games with incomplete information, provided a simple way to model private information by introducing player characteristics or types.12 Player 1 can be one of a number of possible such types, denoted t. Player 1 knows his own type. Player 2 is uncertain about t and must decide on her strategy based on beliefs about t. Formally, the game begins at an initial node, called a chance node, at which a particular value tk is randomly drawn for player 1’s type t from a set of possible types T {t1, …, tk , … , tK}. Let Pr(tk) be the probability of drawing the particular type tk. Player 1 sees which type is drawn. Player 2 does not see the draw and only knows the probabilities, using them to form her beliefs about player 1’s type. Thus, the probability that player 2 places on player 1’s being of type tk is Pr(tk). ¼ Because player 1 observes his type t before moving, his strategy can be conditioned on t. Conditioning on this information may be a big beneﬁt to a player. In poker, for example, the stronger a player’s hand, the more likely the player is to win the pot and the more aggressively the player may want to bid. Let s1(t) be player 1’s strategy contingent on his type. Because player 2 does not observe t, her strategy is simply the unconditional one, s2. As with games of complete information, players’ payoffs depend on strategies. In Bayesian games, payoffs may also depend on types. Therefore, we write player 1’s payoff as u1(s1(t), s2, t) and player 2’s as u2(s2, s1(t), t). Note that t appears in two places in player 2’s payoff function. Player 1’s type may have a direct effect on player 2’s payoffs. Player 1’s type also has an indirect effect through its effect on player 1’s strategy s1(t), which in turn affects player 2’s payoffs. Because player 2’s payoffs depend on t in these two ways, her beliefs about t will be crucial in the calculation of her optimal strategy. Figure 8.14 provides a simple example of a simultaneous Bayesian game. Each player chooses one of two actions. All payoffs are known except for player 1’s payoff when 1 chooses U and 2 chooses L. Player 1’s payoff in outcome (U, L) is identiﬁed as his type, t. There are two possible values for player 1’s type, t 0, each occurring with equal probability. Player 1 knows his type before moving. Player 2’s beliefs are that each type has probability 1/2. The extensive form is drawn in Figure 8.15. 6 and t ¼ ¼ Bayesian–Nash equilibrium Extending Nash equilibrium to Bayesian games requires two small matters of interpretation. First, recall that player 1 may play a different action for each of his types. Equilibrium requires that player 1’s strategy be a best response for each and every one of his types. Second, recall that player 2 is uncertain about player 1’s type. Equilibrium requires 12J. Harsanyi, ‘‘Games with Incomplete Information Played by Bayesian Players,’’ Management Science 14 (1967–68): 159–82, 320–34, 486–502. Chapter 8: Game Theory 279 FIGURE 8.14 6 with probability 1/2 and t t ¼ 0 with probability 1/2. ¼ Simple Game of Incomplete Information Player 2 L R U t, 2 0, 0 D 2, 0 2, 4 1 r e y a l P FIGURE 8.15 Extensive Form for Simple Game of Incomplete Information This ﬁgure translates Figure 8.14 into an extensive-form game. The initial chance node is indicated by an open circle. Player 2’s decision nodes are in the same information set because she does not observe player 1’s type or action before moving. t = 6 Pr = 1/2 t = 0 Pr = 1/, 2 0, 0 2, 0 2, 4 0, 2 0, 0 2, 0 2, 4 280 Part 3: Uncertainty and Strategy that player 2’s strategy maximize an expected payoff, where the expectation is taken with respect to her beliefs about player 1’s type. We encountered expected payoffs in our discussion of mixed strategies. The calculations involved in computing the best response to the pure strategies of different types of rivals in a game of incomplete information are similar to the calculations involved in computing the best response to a rival’s mixed strategy in a game of complete information. Interpreted in this way, Nash equilibrium in the setting of a Bayesian game is called Bayesian–Nash equilibrium. Next we provide a formal deﬁnition of the concept for reference. Given that the notation is fairly dense, it may be easier to ﬁrst skip to Examples 8.6 and 8.7, which provide a blueprint on how to solve for equilibria in Bayesian games you might come across Bayesian–Nash equilibrium. In a two-player, simultaneous-move game in which player 1 has private information, a Bayesian–Nash equilibrium is a strategy proﬁle is a best response to s’2 for each type t t such that s’1ð T of player 1, , s’2Þ Þ s’1ð t ð Þ 2 , s’2, t U 1ð t s’1ð Þ U 1ð s01, s’2, t Þ Þ & for all s01 2 S1, (8:22) and such that s’2 is a best response to s’1ð t U 2ð , tkÞ & tkÞ Pr ð s’2, s’1ð tkÞ given player 2’s beliefs Pr(tk) about player 1’s types: Þ Pr tkÞ ð U 2ð s02, s’1ð tkÞ , tkÞ for all s02 2 S2: (8:23) T tk2 X T tk2 X Because the difference between Nash equilibrium and Bayesian–Nash equilibrium is only a matter of interpretation, all our previous results for Nash equilibrium (including the existence proof ) apply to Bayesian–Nash equilibrium as well. EXAMPLE 8.6 Bayesian–Nash Equilibrium of Game in Figure 8.15 To solve for the Bayesian–Nash equilibrium of the game in Figure 8.15, ﬁrst solve for the informed player’s (player 1’s) best responses for each of his types. If player 1 is of type t 0, then he would choose D rather than U because he earns 0 by playing U and 2 by playing D regardless of what player 2 does. If player 1 is of type t 6, then his best response is U to player 2’s playing L and D to her playing R. This leaves only two possible candidates for an equilibrium in pure strategies: ¼ ¼ 1 plays 1 plays t j t U ð D ð ¼ 6, D 6, D t j t ¼ 0 0 Þ and 2 plays L; and 2 plays R: ¼ 0), The ﬁrst candidate cannot be an equilibrium because, given that player 1 plays (U \|t player 2 earns an expected payoff of 1 from playing L. Player 2 would gain by deviating to R, earning an expected payoff of 2. 6, D\|t ¼ ¼ ¼ Þ j j The second candidate is a Bayesian–Nash equilibrium. Given that player 2 plays R, player 1’s best response is to play D, providing a payoff of 2 rather than 0 regardless of his type. Given that both types of player 1 play D, player 2’s best response is to play R, providing a payoff of 4 rather than 0. QUERY: If the probability that player 1 is of type t be a Bayesian–Nash equilibrium? If so, compute the threshold probability. ¼ 6 is high enough, can the ﬁrst candidate Chapter 8: Game Theory 281 EXAMPLE 8.7 Tragedy of the Commons as a Bayesian Game For an example of a Bayesian game with continuous actions, consider the Tragedy of the Commons in Example 8.5 but now suppose that herder 1 has private information regarding his value of grazing per sheep: where herder 1’s type is t ¼ type) with probability 1/3. Herder 2’s value remains the same as in Equation 8.11. 130 (the ‘‘high’’ type) with probability 2/3 and t ¼ q1, q2, t v1ð t q1 þ ! ð , q2Þ Þ ¼ (8:24) 100 (the ‘‘low’’ To solve for the Bayesian–Nash equilibrium, we ﬁrst solve for the informed player’s (herder 1’s) best responses for each of his types. For any type t and rival’s strategy q2, herder 1’s valuemaximization problem is (8:25) (8:26) (8:27) max q1 f q1v1ð q1, q2, t Þg ¼ max q1 f t q1ð q1 ! : q2Þg ! The ﬁrst-order condition for a maximum is t 2q1 ! ! Rearranging and then substituting the values t 0: q2 ¼ 130 and t ¼ and q1L ¼ q2 2 100, we obtain ¼ q2 2 q1H ¼ where q1H is the quantity for the ‘‘high’’ type of herder 1 (i.e., the t ‘‘low’’ type (the t 100 type). 65 50 ! ! , Next we solve for herder 2’s best response. Herder 2’s expected payoff is ¼ 100 type) and q1L for the ¼ 2 3 ½ q2ð 120 q1H ! q2Þ+ þ ! 1 3 ½ 120 q2ð q1L ! q2Þ+ ¼ q2ð ! 120 q1 ! , q2Þ ! (8:28) where q1H þ Rearranging the ﬁrst-order condition from the maximization of Equation 8.28 with respect to q2 gives q1 ¼ (8:29) q1L: 2 3 1 3 q2 ¼ 60 ! q1 2 : (8:30) Substituting
for q1H and q1L from Equation 8.27 into Equation 8.29 and then substituting the resulting expression for q1 into Equation 8.30 yields q2 ¼ 30 þ q2 4 , (8:31) 40. Substituting q’2 ¼ 40 back into Equation 8.27 implies q’1H ¼ implying that q’2 ¼ 30: q’1L ¼ Figure 8.16 depicts the Bayesian–Nash equilibrium graphically. Herder 2 imagines playing against an average type of herder 1, whose average best response is given by the thick dashed line. The intersection of this best response and herder 2’s at point B determines herder 2’s equilibrium quantity, q’2 ¼ 40 is given 40. The best response of the low (resp. high) type of herder 1 to q’2 ¼ by point A (resp. point C). For comparison, the full-information Nash equilibria are drawn when herder 1 is known to be the low type (point A0) or the high type (point C 0). 45 and QUERY: Suppose herder 1 is the high type. How does the number of sheep each herder grazes change as the game moves from incomplete to full information (moving from point C 0 to C)? What if herder 1 is the low type? Which type prefers full information and thus would like to signal its type? Which type prefers incomplete information and thus would like to hide its type? We will study the possibility player 1 can signal his type in the next section. 282 Part 3: Uncertainty and Strategy FIGURE 8.168Equilibrium of the Bayesian Tragedy of the Commons Best responses for herder 2 and both types of herder 1 are drawn as thick solid lines; the expected best response as perceived by 2 is drawn as the thick dashed line. The Bayesian–Nash equilibrium of the incomplete-information game is given by points A and C; Nash equilibria of the corresponding fullinformation games are given by points A0 and C 0. q2 High type’s best response Low type’s best response 40 A′ A CB C′ 0 30 40 45 2’s best response q1 Signaling Games In this section we move from simultaneous-move games of private information to sequential games in which the informed player, player 1, takes an action that is observable to player 2 before player 2 moves. Player 1’s action provides information, a signal, that player 2 can use to update her beliefs about player 1’s type, perhaps altering the way player 2 would play in the absence of such information. In poker, for example, player 2 may take a big raise by player 1 as a signal that he has a good hand, perhaps leading player 2 to fold. A ﬁrm considering whether to enter a market may take the incumbent ﬁrm’s low price as a signal that the incumbent is a low-cost producer and thus a tough competitor, perhaps keeping the entrant out of the market. A prestigious college degree may signal that a job applicant is highly skilled. The analysis of signaling games is more complicated than simultaneous games because we need to model how player 2 processes the information in player 1’s signal and then updates her beliefs about player 1’s type. To ﬁx ideas, we will focus on a concrete application: a version of Michael Spence’s model of job-market signaling, for which he won the Nobel Prize in economics.13 13M. Spence, ‘‘Job-Market Signaling,’’ Quarterly Journal of Economics 87 (1973): 355–74. Chapter 8: Game Theory 283 ¼ ¼ Job-market signaling H) or low-skilled Player 1 is a worker who can be one of two types, high-skilled (t (t L). Player 2 is a ﬁrm that considers hiring the applicant. A low-skilled worker is completely unproductive and generates no revenue for the ﬁrm; a high-skilled worker generates revenue p. If the applicant is hired, the ﬁrm must pay the worker w (think of this wage as being ﬁxed by government regulation). Assume p > w > 0. Therefore, the ﬁrm wishes to hire the applicant if and only if he or she is high-skilled. But the ﬁrm cannot observe the applicant’s skill; it can observe only the applicant’s prior education. Let cH be the high type’s cost of obtaining an education and cL the low type’s cost. Assume cH < cL, implying that education requires less effort for the high-skilled applicant than the low-skilled one. We make the extreme assumption that education does not increase the worker’s productivity directly. The applicant may still decide to obtain an education because of its value as a signal of ability to future employers. Figure 8.17 shows the extensive form. Player 1 observes his or her type at the start; player 2 observes only player 1’s action (education signal) before moving. Let Pr(H) and Pr(L) be player 2’s beliefs before observing player 1’s education signal that player 1 is high- or low-skilled. These are called player 1’s prior beliefs. Observing player 1’s action will lead player 2 to revise his or her beliefs to form what are called posterior beliefs. For FIGURE 8.17 Job-Market Signaling Player 1 (worker) observes his or her own type. Then player 1 chooses to become educated (E) or not (NE ). After observing player 1’s action, player 2 (ﬁrm) decides to make him or her a job offer (J ) or not (NJ ). The nodes in player 2’s information sets are labeled n1, … , n4 for reference. E 1 Pr(H) NE E NE Pr(L) 1 2 n1 2 n2 2 n3 2 n4 J NJ J NJ J NJ J NJ w − cH, π − w −cH, 0 w − cL, −w −cL, 0 w, π − w 0, 0 w, −w 0, 0 284 Part 3: Uncertainty and Strategy example, the probability that the worker is high-skilled is conditional on the worker’s having obtained an education, Pr(H\|E), and conditional on no education, Pr(H\|NE). Player 2’s posterior beliefs are used to compute his or her best response to player 1’s education decision. Suppose player 2 sees player 1 choose E. Then player 2’s expected payoff from playing J is w Pr H E p w, (8:32) Pr H ð E j p w E L Pr ð Þð ! Þ þ Þð! where the left side of this equation follows from the fact that because L and H are the Pr(H\|E). Player 2’s payoff from playing NJ is 0. To determine only types, Pr(L\|E) the best response to E, player 2 compares the expected payoff in Equation 8.32 to 0. Player 2’s best response is J if and only if Pr(H\|E) > w/p The question remains of how to compute posterior beliefs such as Pr(H\|E). Rational players use a statistical formula, called Bayes’ rule, to revise their prior beliefs to form posterior beliefs based on the observation of a signal. Bayes’ rule Bayes’ rule gives the following formula for computing player 2’s posterior belief Pr(H\|E)14: Pr E H ð j Þ ¼ Pr E ð H j Þ Similarly, Pr(H\|E) is given by E Pr ð j H Pr ð H H Pr Þ ð L E Pr j ð Þ Þ þ : L Pr ð Þ Þ (8:33) Pr H ð j NE Þ ¼ Pr NE H H NE Pr j ð Þ H Pr Þ þ ð Pr H ð Þ NE Pr ð L Pr L j Two sorts of probabilities appear on the left side of Equations 8.33 and 8.34: Þ Þ Þ ð ð j : (8:34) • • the prior beliefs Pr(H) and Pr(L); the conditional probabilities Pr(E\|H), Pr(NE\|L), and so forth. The prior beliefs are given in the speciﬁcation of the game by the probabilities of the different branches from the initial chance node. The conditional probabilities Pr(E\|H), Pr(NE\|L), and so forth are given by player 1’s equilibrium strategy. For example, Pr(E\|H) is the probability that player 1 plays E if he or she is of type H; Pr(NE\|L) is the probability that player 1 plays NE if he or she is of type L; and so forth. As the schematic diagram in Figure 8.18 summarizes, Bayes’ rule can be thought of as a ‘‘black box’’ that takes prior beliefs and strategies as inputs and gives as outputs the beliefs we must know to solve for an equilibrium of the game: player 2’s posterior beliefs. 14Equation 8.33 can be derived from the deﬁnition of conditional probability in footnote 25 of Chapter 2. (Equation 8.34 can be derived similarly.) By deﬁnition, Reversing the order of the two events in the conditional probability yields Pr H ð E j Þ ¼ Pr ð H and E E Pr ð Þ Þ : or, after rearranging, Pr H E ð j Þ ¼ Pr ð Þ H and E H Pr ð H j E ð Þ Þ Pr : H ð Þ Pr H and E ð Þ ¼ Pr Substituting the preceding equation into the ﬁrst displayed equation of this footnote gives the numerator of Equation 8.33. The denominator follows because the events of player 1’s being of type H or L are mutually exclusive and jointly exhaustive, so Pr E ð Pr ð Pr ð Þ ¼ ¼ E and H Þ þ H Pr E ð H j Þ Þ þ Pr E and L ð L E Pr j ð Þ Þ Pr ð L : Þ Chapter 8: Game Theory 285 FIGURE 8.18 Bayes’ Rule as a Black Box Bayes’ rule is a formula for computing player 2’s posterior beliefs from other pieces of information in the game. Inputs Player 2’s prior beliefs Player 1’s strategy Bayes’ rule Output Player 2’s posterior beliefs When player 1 plays a pure strategy, Bayes’ rule often gives a simple result. Suppose, 0 or, in other words, that player 1 obtains for example, that Pr(E\|H) an education if and only if he or she is high-skilled. Then Equation 8.33 implies 1 and Pr(E\|L) ¼ ¼ H Pr Pr ð Þ þ H 0 Þ ( Pr ( 1: Pr L ð Þ ¼ (8:35) That is, player 2 believes that player 1 must be high-skilled if it sees player 1 choose E. 1—that is, suppose player 1 On the other hand, suppose that Pr(E\|H) obtains an education regardless of his or her type. Then Equation 8.33 implies Pr(E\|L) ¼ ¼ Pr Pr ð Þ þ H 1 Þ ( Pr ( Pr L Þ ð ¼ H Pr ð , Þ (8:36) Pr(L) because Pr(H) 1. That is, seeing player 1 play E provides no information about player 1’s type, so player 2’s posterior belief is the same as his or her prior one. More generally, q, then Bayes’ rule implies that if player 2 plays the mixed strategy Pr(E\|H) p and Pr(E\|L) ¼ þ ¼ ¼ H Pr ð j E Þ ¼ p Pr p Pr H ð H ð Þ q Pr Þ þ : L ð Þ (8:37) Perfect Bayesian equilibrium With games of complete information, we moved from Nash equilibrium to the reﬁnement of subgame-perfect equilibrium to rule out noncredible threats in sequential games. For the same reason, with games of incomplete information we move from Bayesian-Nash equilibrium to the reﬁnement of perfect Bayesian equilibrium Perfect Bayesian equilibrium. A perfect Bayesian equilibrium consists of a strategy proﬁle and a set of beliefs such that • • at each information set, the strategy of the player moving there maximizes his or her expected payoff, where the expectation is taken with respect to his or her beliefs; and at each information set, where possible, the beliefs of t
he player moving there are formed using Bayes’ rule (based on prior beliefs and other players’ strategies). 286 Part 3: Uncertainty and Strategy The requirement that players play rationally at each information set is similar to the requirement from subgame-perfect equilibrium that play on every subgame form a Nash equilibrium. The requirement that players use Bayes’ rule to update beliefs ensures that players incorporate the information from observing others’ play in a rational way. The remaining wrinkle in the deﬁnition of perfect Bayesian equilibrium is that Bayes’ rule need only be used ‘‘where possible.’’ Bayes’ rule is useless following a completely unexpected event—in the context of a signaling model, an action that is not played in equilibrium by any type of player 1. For example, if neither H nor L type chooses E in the job-market signaling game, then the denominators of Equations 8.33 and 8.34 equal zero and the fraction is undeﬁned. If Bayes’ rule gives an undeﬁned answer, then perfect Bayesian equilibrium puts no restrictions on player 2’s posterior beliefs and thus we can assume any beliefs we like. As we saw with games of complete information, signaling games may have multiple equilibria. The freedom to specify any beliefs when Bayes’ rule gives an undeﬁned answer may support additional perfect Bayesian equilibria. A systematic analysis of multiple equilibria starts by dividing the equilibria into three classes—separating, pooling, and hybrid. Then we look for perfect Bayesian equilibria within each class. In a separating equilibrium, each type of player 1 chooses a different action. Therefore, player 2 learns player 1’s type with certainty after observing player 1’s action. The posterior beliefs that come from Bayes’ rule are all zeros and ones. In a pooling equilibrium, different types of player 1 choose the same action. Observing player 1’s action provides player 2 with no information about player 1’s type. Pooling equilibria arise when one of player 1’s types chooses an action that would otherwise be suboptimal to hide his or her private information. In a hybrid equilibrium, one type of player 1 plays a strictly mixed strategy; it is called a hybrid equilibrium because the mixed strategy sometimes results in the types being separated and sometimes pooled. Player 2 learns a little about player 1’s type (Bayes’ rule reﬁnes player 2’s beliefs a bit) but does not learn player 1’s type with certainty. Player 2 may respond to the uncertainty by playing a mixed strategy itself. The next three examples solve for the three different classes of equilibrium in the job-market signaling game. EXAMPLE 8.8 Separating Equilibrium in the Job-Market Signaling Game Pr(L\|NE) A good guess for a separating equilibrium is that the high-skilled worker signals his or her type by getting an education and the low-skilled worker does not. Given these strategies, player 2’s beliefs must 0 according to Bayes’ rule. Conditional on Pr(L\|E) be Pr(H\|E) these beliefs, if player 2 observes that player 1 obtains an education, then player 2 knows it must be at node n1 rather than n2 in Figure 8.17. Its best response is to offer a job (J), given the payoff of w > 0. If player 2 observes that player 1 does not obtain an eduation, then player 2 knows it p must be at node n4 rather than n3, and its best response is not to offer a job (NJ) because 0 > 1 and Pr(H\|NE) w. ¼ ¼ ¼ ¼ ! The last step is to go back and check that player 1 would not want to deviate from the separating strategy (E\|H, NE\|L) given that player 2 plays (J\|E, NJ\|NE). Type H of player 1 earns w cH by obtaining an education in equilibrium. If type H deviates and does not obtain an education, then he or she earns 0 because player 2 believes that player 1 is type L and does not offer a job. For type H cH > 0. Next, turn to type L of player 1. Type L earns 0 not to prefer to deviate, it must be that w by not obtaining an education in equilibrium. If type L deviates and obtains an education, then he cL because player 2 believes that player 1 is type H and offers a job. For type L not or she earns w cL < 0. Putting these conditions together, there is separating to prefer to deviate, we must have w equilibrium in which the worker obtains an education if and only if he or she is high-skilled and in which the ﬁrm offers a job only to applicants with an education if and only if cH < w < cL. ! ! ! ! ! Another possible separating equilibrium is for player 1 to obtain an education if and only if he or she is low-skilled. This is a bizarre outcome—because we expect education to be a signal of high rather than low skill—and fortunately we can rule it out as a perfect Bayesian equilibrium. Chapter 8: Game Theory 287 Player 2’s best response would be to offer a job if and only if player 1 did not obtain an education. Type L would earn cL from playing E and w from playing NE, so it would deviate to NE. ! QUERY: Why does the worker sometimes obtain an education even though it does not raise his or her skill level? Would the separating equilibrium exist if a low-skilled worker could obtain an education more easily than a high-skilled one? EXAMPLE 8.9 Pooling Equilibria in the Job-Market Signaling Game Let’s investigate a possible pooling equilibrium in which both types of player 1 choose E. For player 1 not to deviate from choosing E, player 2’s strategy must be to offer a job if and only if the worker is educated—that is, ( J\|E, NJ\|NE). If player 2 does not offer jobs to educated workers, then player 1 might as well save the cost of obtaining an education and choose NE. If player 2 offers jobs to uneducated workers, then player 1 will again choose NE because he or she saves the cost of obtaining an education and still earns the wage from the job offer. Next, we investigate when ( J\|E, NJ\|NE) is a best response for player 2. Player 2’s posterior beliefs after seeing E are the same as his or her prior beliefs in this pooling equilibrium. Player 2’s expected payoff from choosing J is Pr L ð w Þ Pr Pr H ð H ð p Þð p Þ w w: Pr H ð E p w Pr E L j ð w j Þð ! Þ þ Þ þ Þð! Þð! ! ! Þ ¼ ¼ For J to be a best response to E, Equation 8.38 must exceed player 2’s zero payoff from choosing NJ, which on rearranging implies that Pr(H) w/p. Player 2’s posterior beliefs at nodes n3 and n4 are not pinned down by Bayes’ rule because NE is never played in equilibrium and so seeing player 1 play NE is a completely unexpected event. Perfect Bayesian equilibrium allows us to specify any probability distribution we like for the posterior beliefs Pr(H\|NE) at node n3 and Pr(L\|NE) at node n4. Player 2’s payoff from choosing NJ is 0. For NJ to be a best response to NE, 0 must exceed player 2’s expected payoff from playing J: (8:38) & 0 > Pr NE p w Pr Þ þ where the right side follows because Pr(H\|NE) ! Þð j H ð w NE L j Þð! ð Pr(L\|NE) Pr H ð NE p Þ Þ ¼ j ! 1. Rearranging yields Pr(H\|NE) w, w/p. In sum, for there to be a pooling equilibrium in which both types of player 1 obtain an education, we need Pr(H\|NE) Pr(H). The ﬁrm has to be optimistic about the proportion of skilled workers in the population—Pr(H) must be sufﬁciently high—and pessimistic about the skill level of uneducated workers—Pr(H\|NE) must be sufﬁciently low. In this equilibrium, type L pools with type H to prevent player 2 from learning anything about the worker’s skill from the education signal. w/p ¼ þ , , , (8:39) The other possibility for a pooling equilibrium is for both types of player 1 to choose NE. There are a number of such equilibria depending on what is assumed about player 2’s posterior beliefs out of equilibrium (i.e., player 2’s beliefs after he or she observes player 1 choosing E). Perfect Bayesian equilibrium does not place any restrictions on these posterior beliefs. Problem 8.12 asks you to search for various of these equilibria and introduces a further reﬁnement of perfect Bayesian equilibrium (the intuitive criterion) that helps rule out unreasonable out-ofequilibrium beliefs and thus implausible equilibria. QUERY: Return to the pooling outcome in which both types of player 1 obtain an education. Consider player 2’s posterior beliefs following the unexpected event that a worker shows up with no education. Perfect Bayesian equilibrium leaves us free to assume anything we want about these posterior beliefs. Suppose we assume that the ﬁrm obtains no information from the ‘‘no education’’ signal and so maintains its prior beliefs. Is the proposed pooling outcome an equilibrium? What if we assume that the ﬁrm takes ‘‘no education’’ as a bad signal of skill, believing that player 1’s type is L for certain? 288 Part 3: Uncertainty and Strategy EXAMPLE 8.10 Hybrid Equilibria in the Job-Market Signaling Game One possible hybrid equilibrium is for type H always to obtain an education and for type L to randomize, sometimes pretending to be a high type by obtaining an education. Type L e. Player 2’s strategy is to randomizes between playing E and NE with probabilities e and 1 offer a job to an educated applicant with probability j and not to offer a job to an uneducated applicant. ! We need to solve for the equilibrium values of the mixed strategies e’ and j’ and the posterior beliefs Pr(H\|E) and Pr(H\|NE) that are consistent with perfect Bayesian equilibrium. The posterior beliefs are computed using Bayes’ rule: Pr H ð Þ ePr Þ þ Þ ¼ Pr H ð L Þ ð ¼ Pr H ð Pr E H ð j 0. Pr H ð 1 e ½ Þ ! Þ þ Pr H ð Þ+ (8:40) and Pr(H\|NE) ¼ For type L of player 1 to be willing to play a strictly mixed strategy, he or she must get the same expected payoff from playing E—which equals jw cL, given player 2’s mixed strategy—as from playing NE—which equals 0 given that player 2 does not offer a job to uneducated applicants. Hence jw ¼ Player 2 will play a strictly mixed strategy (conditional on observing E) only if he or she gets cL ¼ the same expected payoff from playing J, which equals 0 or, solving for j, j’ cL/w. ! ! Pr E H ð j p Þð ! w Þ þ Pr E L j ð w Þð! Pr E H ð j
p Þ ! w, Þ ¼ (8:41) as from playing NJ, which equals 0. Setting Equation 8.41 equal to 0, substituting for Pr(H\|E) from Equation 8.40, and then solving for e gives w e’ ¼ p ð w ½ ! 1 ! Pr Þ Pr H ð H ð Þ Þ+ : (8:42) QUERY: To complete our analysis: In this equilibrium, type H of player 1 cannot prefer to deviate from E. Is this true? If so, can you show it? How does the probability of type L trying to ‘‘pool’’ with the high type by obtaining an education vary with player 2’s prior belief that player 1 is the high type? Experimental Games Experimental economics is a recent branch of research that explores how well economic theory matches the behavior of experimental subjects in laboratory settings. The methods are similar to those used in experimental psychology—often conducted on campus using undergraduates as subjects—although experiments in economics tend to involve incentives in the form of explicit monetary payments paid to subjects. The importance of experimental economics was highlighted in 2002, when Vernon Smith received the Nobel Prize in economics for his pioneering work in the ﬁeld. An important area in this ﬁeld is the use of experimental methods to test game theory. Experiments with the Prisoners’ Dilemma There have been hundreds of tests of whether players ﬁnk in the Prisoners’ Dilemma as predicted by Nash equilibrium or whether they play the cooperative outcome of Silent. In one experiment, subjects played the game 20 times with each player being matched with a different, anonymous opponent to avoid repeated-game effects. Play converged to the Nash equilibrium as subjects gained experience with the game. Players played the cooperative Chapter 8: Game Theory 289 action 43 percent of the time in the ﬁrst ﬁve rounds, falling to only 20 percent of the time in the last ﬁve rounds.15 As is typical with experiments, subjects’ behavior tended to be noisy. Although 80 percent of the decisions were consistent with Nash equilibrium play by the end of the experiment, 20 percent of them still were anomalous. Even when experimental play is roughly consistent with the predictions of theory, it is rarely entirely consistent. Experiments with the Ultimatum Game Experimental economics has also tested to see whether subgame-perfect equilibrium is a good predictor of behavior in sequential games. In one widely studied sequential game, the Ultimatum Game, the experimenter provides a pot of money to two players. The ﬁrst mover (Proposer) proposes a split of this pot to the second mover. The second mover (Responder) then decides whether to accept the offer, in which case players are given the amount of money indicated, or reject the offer, in which case both players get nothing. In the subgame-perfect equilibrium, the Proposer offers a minimal share of the pot, and this is accepted by the Responder. One can see this by applying backward induction: The Responder should accept any positive division no matter how small; knowing this, the Proposer should offer the Responder only a minimal share. In experiments, the division tends to be much more even than in the subgame-perfect equilibrium.16 The most common offer is a 50–50 split. Responders tend to reject offers giving them less than 30 percent of the pot. This result is observed even when the pot is as high as $100, so that rejecting a 30 percent offer means turning down $30. Some economists have suggested that the money players receive may not be a true measure of their payoffs. They may care about other factors such as fairness and thus obtain a beneﬁt from a more equal division of the pot. Even if a Proposer does not care directly about fairness, the fear that the Responder may care about fairness and thus might reject an uneven offer out of spite may lead the Proposer to propose an even split. The departure of experimental behavior from the predictions of game theory was too systematic in the Ultimatum Game to be attributed to noisy play, leading some game theorists to rethink the theory and add an explicit consideration for fairness.17 Experiments with the Dictator Game To test whether players care directly about fairness or act out of fear of the other player’s spite, researchers experimented with a related game, the Dictator Game. In the Dictator Game, the Proposer chooses a split of the pot, and this split is implemented without input from the Responder. Proposers tend to offer a less-even split than in the Ultimatum Game but still offer the Responder some of the pot, suggesting that Proposers have some residual concern for fairness. The details of the experimental design are crucial, however, as one ingenious experiment showed.18 The experiment was designed so that the experimenter would never learn which Proposers had made which offers. With this element of anonymity, Proposers almost never gave an equal split to Responders and indeed took the whole pot for themselves two thirds of the time. Proposers seem to care more about appearing fair to the experimenter than truly being fair. 15R. Cooper, D. V. DeJong, R. Forsythe, and T. W. Ross, ‘‘Cooperation Without Reputation: Experimental Evidence from Prisoner’s Dilemma Games,’’ Games and Economic Behavior (February 1996): 187–218. 16For a review of Ultimatum Game experiments and a textbook treatment of experimental economics more generally, see D. D. Davis and C. A. Holt, Experimental Economics (Princeton, NJ: Princeton University Press, 1993). 17See, for example, E. Fehr and K.M. Schmidt, ‘‘A Theory of Fairness, Competition, and Cooperation,’’ Quarterly Journal of Economics (August 1999): 817–868. 18E. Hoffman, K. McCabe, K. Shachat, and V. Smith, ‘‘Preferences, Property Rights, and Anonymity in Bargaining Games,’’ Games and Economic Behavior (November 1994): 346–80. 290 Part 3: Uncertainty and Strategy Evolutionary Games And Learning The frontier of game-theory research regards whether and how players come to play a Nash equilibrium. Hyper-rational players may deduce each others’ strategies and instantly settle on the Nash equilibrium. How can they instantly coordinate on a single outcome when there are multiple Nash equilibria? What outcome would real-world players, for whom hyper-rational deductions may be too complex, settle on? Game theorists have tried to model the dynamic process by which an equilibrium emerges over the long run from the play of a large population of agents who meet others at random and play a pairwise game. Game theorists analyze whether play converges to Nash equilibrium or some other outcome, which Nash equilibrium (if any) is converged to if there are multiple equilibria, and how long such convergence takes. Two models, which make varying assumptions about the level of players’ rationality, have been most widely studied: an evolutionary model and a learning model. In the evolutionary model, players do not make rational decisions; instead, they play the way they are genetically programmed. The more successful a player’s strategy in the population, the more ﬁt is the player and the more likely will the player survive to pass his or her genes on to future generations and thus the more likely the strategy spreads in the population. Evolutionary models were initially developed by John Maynard Smith and other biologists to explain the evolution of such animal behavior as how hard a lion ﬁghts to win a mate or an ant ﬁghts to defend its colony. Although it may be more of a stretch to apply evolutionary models to humans, evolutionary models provide a convenient way of analyzing population dynamics and may have some direct bearing on how social conventions are passed down, perhaps through culture. In a learning model, players are again matched at random with others from a large population. Players use their experiences of payoffs from past play to teach them how others are playing and how they themselves can best respond. Players usually are assumed to have a degree of rationality in that they can choose a static best response given their beliefs, may do some experimenting, and will update their beliefs according to some reasonable rule. Players are not fully rational in that they do not distort their strategies to affect others’ learning and thus future play. Game theorists have investigated whether more- or less-sophisticated learning strategies converge more or less quickly to a Nash equilibrium. Current research seeks to integrate theory with experimental study, trying to identify the speciﬁc algorithms that real-world subjects use when they learn to play games. SUMMARY This chapter provided a structured way to think about strategic situations. We focused on the most important solution concept used in game theory, Nash equilibrium. We then progressed to several more reﬁned solution concepts that are in standard use in game theory in more complicated settings (with sequential moves and incomplete information). Some of the principal results are as follows. • All games have the same basic components: players, strategies, payoffs, and an information structure. • Games can be written down in normal form (providing a payoff matrix or payoff functions) or extensive form (providing a game tree). • Strategies can be simple actions, more complicated plans contingent on others’ actions, or even probability distributions over simple actions (mixed strategies). • A Nash equilibrium is a set of strategies, one for each player, that are mutual best responses. In other words, a player’s strategy in a Nash equilibrium is optimal given that all others play their equilibrium strategies. • A Nash equilibrium always exists in ﬁnite games (in mixed if not pure strategies). Chapter 8: Game Theory 291 • Subgame-perfect equilibrium is a reﬁnement of Nash equilibrium that helps to rule out equilibria in sequential games involving noncredible threats. • Repeating a stage game a large number of times introduces the possibility of using punishment strategies to attain higher payoffs than if the stage game is played on
ce. If players are sufﬁciently patient in an inﬁnitely repeated game, then a folk theorem holds implying that essentially any payoffs are possible in the repeated game. • In games of private information, one player knows more about his or her ‘‘type’’ than another. Players maximize their expected payoffs given knowledge of their own type and beliefs about the others’. • In a perfect Bayesian equilibrium of a signaling game, the second mover uses Bayes’ rule to update his or her beliefs about the ﬁrst mover’s type after observing the ﬁrst mover’s action. • The frontier of game-theory research combines theory with experiments to determine whether players who may not be hyper-rational come to play a Nash equilibrium, which particular equilibrium (if there are more than one), and what path leads to the equilibrium. PROBLEMS 8.1 Consider the following game: Player 2 E F D A 7, 6 5, 8 0, 8 7, 6 1, 1 C 0, 0 1, 1 4, 4 a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the ﬁrst two actions. c. Compute players’ expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game. 8.2 The mixed-strategy Nash equilibrium in the Battle of the Sexes in Figure 8.3 may depend on the numerical values for the payoffs. To generalize this solution, assume that the payoff matrix for the game is given by Player 2 (Husband) Ballet Boxing Ballet K, 1 0 ( Boxing 0, 0 1, K where K & 1. Show how the mixed-strategy Nash equilibrium depends on the value of K. 292 Part 3: Uncertainty and Strategy 8.3 The game of Chicken is played by two macho teens who speed toward each other on a single-lane road. The ﬁrst to veer off is branded the chicken, whereas the one who does not veer gains peer-group esteem. Of course, if neither veers, both die in the resulting crash. Payoffs to the Chicken game are provided in the following table. Teen 2 Veer Does not veer Veer 2, 2 1, 3 1 n e e T Does not veer 3, 1 0, 0 a. Draw the extensive form. b. Find the pure-strategy Nash equilibrium or equilibria. c. Compute the mixed-strategy Nash equilibrium. As part of your answer, draw the best-response function diagram for the mixed strategies. d. Suppose the game is played sequentially, with teen 1 moving ﬁrst and committing to this action by throwing away the steering wheel. What are teen 2’s contingent strategies? Write down the normal and extensive forms for the sequential version of the game. e. Using the normal form for the sequential version of the game, solve for the Nash equilibria. f. Identify the proper subgames in the extensive form for the sequential version of the game. Use backward induction to solve for the subgame-perfect equilibrium. Explain why the other Nash equilibria of the sequential game are ‘‘unreasonable.’’ 8.4 Two neighboring homeowners, i average beneﬁt per hour is ¼ 1, 2, simultaneously choose how many hours li to spend maintaining a beautiful lawn. The li þ and the (opportunity) cost per hour for each is 4. Homeowner i’s average beneﬁt is increasing in the hours neighbor j spends on his own lawn because the appearance of one’s property depends in part on the beauty of the surrounding neighborhood. 10 ! , lj 2 a. Compute the Nash equilibrium. b. Graph the best-response functions and indicate the Nash equilibrium on the graph. c. On the graph, show how the equilibrium would change if the intercept of one of the neighbor’s average beneﬁt functions fell from 10 to some smaller number. 8.5 The Academy Award–winning movie A Beautiful Mind about the life of John Nash dramatizes Nash’s scholarly contribution in a single scene: His equilibrium concept dawns on him while in a bar bantering with his fellow male graduate students. They notice several women, one blond and the rest brunette, and agree that the blond is more desirable than the brunettes. The Nash character views the situation as a game among the male graduate students, along the following lines. Suppose there are n males who simultaneously approach either the blond or one of the brunettes. If male i alone approaches the blond, then he is successful in getting a date with her and earns payoff a. If one or more other males approach the blond along with i, the competition causes them all to lose her, and i (as well as the others who approached her) earns a payoff of zero. On the other hand, male i earns a payoff of b > 0 from approaching a brunette because there are more brunettes than males; therefore, i is certain to get a date with a brunette. The desirability of the blond implies a > b. a. Argue that this game does not have a symmetric pure-strategy Nash equilibrium. b. Solve for the symmetric mixed-strategy equilibrium. That is, letting p be the probability that a male approaches the blond, ﬁnd p’. Chapter 8: Game Theory 293 c. Show that the more males there are, the less likely it is in the equilibrium from part (b) that the blond is approached by at least one of them. Note: This paradoxical result was noted by S. Anderson and M. Engers in ‘‘Participation Games: Market Entry, Coordination, and the Beautiful Blond,’’ Journal of Economic Behavior & Organization 63 (2007): 120–37. 8.6 The following game is a version of the Prisoners’ Dilemma, but the payoffs are slightly different than in Figure 8.1. Suspect 2 Fink Silent Fink 0, 0 3, −1 Silent −1, 3 1. Verify that the Nash equilibrium is the usual one for the Prisoners’ Dilemma and that both players have dominant strat- egies. b. Suppose the stage game is repeated inﬁnitely many times. Compute the discount factor required for their suspects to be able to cooperate on silent each period. Outline the trigger strategies you are considering for them. 8.7 Return to the game with two neighbors in Problem 8.5. Continue to suppose that player i’s average beneﬁt per hour of work on landscaping is li þ Continue to suppose that player 2’s opportunity cost of an hour of landscaping work is 4. Suppose that player 1’s opportunity cost is either 3 or 5 with equal probability and that this cost is player 1’s private information. 10 ! : lj 2 a. Solve for the Bayesian–Nash equilibrium. b. Indicate the Bayesian–Nash equilibrium on a best-response function diagram. c. Which type of player 1 would like to send a truthful signal to player 2 if it could? Which type would like to hide his or her private information? 8.8 In Blind Texan Poker, player 2 draws a card from a standard deck and places it against her forehead without looking at it but so player 1 can see it. Player 1 moves ﬁrst, deciding whether to stay or fold. If player 1 folds, he must pay player 2 $50. If player 1 stays, the action goes to player 2. Player 2 can fold or call. If player 2 folds, she must pay player 1 $50. If player 2 calls, the card is examined. If it is a low card (2–8), player 2 pays player 1 $100. If it is a high card (9, 10, jack, queen, king, or ace), player 1 pays player 2 $100. a. Draw the extensive form for the game. b. Solve for the hybrid equilibrium. c. Compute the players’ expected payoffs. Analytical Problems 8.9 Fairness in the Ultimatum Game Consider a simple version of the Ultimatum Game discussed in the text. The ﬁrst mover proposes a division of $1. Let r be the 1=2. Then the other player share received by the other player in this proposal (so the ﬁrst mover keeps 1 moves, responding by accepting or rejecting the proposal. If the responder accepts the proposal, the players are paid their r), where 0 , ! , r 294 Part 3: Uncertainty and Strategy shares; if the responder rejects it, both players receive nothing. Assume that if the responder is indifferent between accepting or rejecting a proposal, he or she accepts it. a. Suppose that players only care about monetary payoffs. Verify that the outcome mentioned in the text in fact occurs in the unique subgame-perfect equilibrium of the Ultimatum Game. b. Compare the outcome in the Ultimatum Game with the outcome in the Dictator Game (also discussed in the text), in which the proposer’s surplus division is implemented regardless of whether the second mover accepts or rejects (so it is not much of a strategic game!). c. Now suppose that players care about fairness as well as money. Following the article by Fehr and Schmidt cited in the text, suppose these preferences are represented by the utility function x1 ! where x1 is player 1’s payoff and x2 is player 2’s (a symmetric function holds for player 2). The ﬁrst term reﬂects the usual desire for more money. The second term reﬂects the desire for fairness, that the players’ payoffs not be too unequal. The parameter a measures how intense the preference for fairness is relative to the desire for more money. Assume a < 1=2. x1, x2Þ ¼ x1 ! U1ð x2j a j , 1. Solve for the responder’s equilibrium strategy in the Ultimatum Game. 2. Taking into account how the second mover will respond, solve for the proposer’s equilibrium strategy r’ in the Ultima- tum Game. (Hint: r’ will be a corner solution, which depends on the value of a.) 3. Continuing with the fairness preferences, compare the outcome in the Ultimatum Game with that in the Dictator Game. Find cases that match the experimental results described in the text, in particular in which the split of the pot of money is more even in the Ultimatum Game than in the Dictator Game. Is there a limit to how even the split can be in the Ultimatum Game? 8.10 Rotten Kid Theorem In A Treatise on the Family (Cambridge, MA: Harvard University Press, 1981), Nobel laureate Gary Becker proposes his famous Rotten Kid Theorem as a sequential game between the potentially rotten child (player 1) and the child’s parent (player 2). The child moves ﬁrst, choosing an action r that affects his own income ! 1ð and the income of the parent r ! 2ð ! 02ð < 0 r r . Later, the parent moves, leaving a monetary bequest L to the child. The child cares only for his own utility, Þ Þ½ aU1, where a > 0 reﬂects the parent’s altruism toward the
child. Prove that, ! 2 ! ! 1 þ U1ð , but the parent maximizes U2ð L Þ in a subgame-perfect equilibrium, the child will opt for the value of r that maximizes ! 1 þ ! 2 even though he has no altruistic intentions. Hint: Apply backward induction to the parent’s problem ﬁrst, which will give a ﬁrst-order condition that implicitly determines L’; although an explicit solution for L’ cannot be found, the derivative of L’ with respect to r—required in the child’s ﬁrst-stage optimization problem—can be found using the implicit function rule. ! 01ð r > 0 Þ þ Þ½ L Þ + + 8.11 Alternatives to Grim Strategy Suppose that the Prisoners’ Dilemma stage game (see Figure 8.1) is repeated for inﬁnitely many periods. a. Can players support the cooperative outcome by using tit-for-tat strategies, punishing deviation in a past period by reverting to the stage-game Nash equilibrium for just one period and then returning to cooperation? Are two periods of punishment enough? b. Suppose players use strategies that punish deviation from cooperation by reverting to the stage-game Nash equilibrium for 10 periods before returning to cooperation. Compute the threshold discount factor above which cooperation is possible on the outcome that maximizes the joint payoffs. 8.12 Reﬁnements of perfect Bayesian equilibrium Recall the job-market signaling game in Example 8.9. a. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an educa- tion (NE) and where the ﬁrm offers an uneducated worker a job. Be sure to specify beliefs as well as strategies. b. Find the conditions under which there is a pooling equilibrium where both types of worker choose not to obtain an education (NE) and where the ﬁrm does not offer an uneducated worker a job. What is the lowest posterior belief that the worker is low-skilled conditional on obtaining an education consistent with this pooling equilibrium? Why is it more natural to think that a low-skilled worker would never deviate to E and thus an educated worker must be high-skilled? Cho and Kreps’s intuitive criterion is one of a series of complicated reﬁnements of perfect Bayesian equilibrium that rule out equilibria based on unreasonable posterior beliefs as identiﬁed in this part; see I. K. Cho and D. M. Kreps, ‘‘Signalling Games and Stable Equilibria,’’ Quarterly Journal of Economics 102 (1987): 179–221. Chapter 8: Game Theory 295 SUGGESTIONS FOR FURTHER READING Fudenberg, D., and J. Tirole. Game Theory. Cambridge, MA: MIT Press, 1991. Rasmusen, E. Games and Information, 4th ed. Malden, MA: Blackwell, 2007. A comprehensive survey of game theory at the graduate-student level, although selected sections are accessible to advanced undergraduates. Holt, C. A. Markets, Games, & Strategic Behavior. Boston: Pearson, 2007. An undergraduate games. text with emphasis on experimental An advanced undergraduate text with many real-world applications. Watson, Joel. Strategy: An Introduction to Game Theory. New York: Norton, 2002. An undergraduate text that balances rigor with simple exam2 games). Emphasis on bargaining and contractples (often 2 ) ing examples. EXTENSIONS EXISTENCE OF NASH EQUILIBRIUM This section will sketch John Nash’s original proof that all ﬁnite games have at least one Nash equilibrium (in mixed if not in pure strategies). We will provide some of the details of the proof here; the original proof is in Nash (1950), and a clear textbook presentation of the full proof is provided in Fudenberg and Tirole (1991). The section concludes by mentioning a related existence theorem for games with continuous actions. Nash’s proof is similar to the proof of the existence of a general competitive equilibrium in Chapter 13. Both proofs rely on a ﬁxed point theorem. The proof of the existence of Nash equilibrium requires a slightly more powerful theorem. Instead of Brouwer’s ﬁxed point theorem, which applies to functions, Nash’s proof relies on Kakutani’s ﬁxed point theorem, which applies to correspondences—more general mappings than functions. E8.1 Correspondences versus functions A function maps each point in a ﬁrst set to a single point in a second set. A correspondence maps a single point in the ﬁrst set to possibly many points in the second set. Figure E8.1 illustrates the difference. An example of a correspondence that we have already seen is the best response, BRi(s–i). The best response need not map other players’ strategies si into a single strategy that is a best response for player i. There may be ties among several best responses. As shown in Figure 8.4, in the Battle of the Sexes, the husband’s best response to the wife’s playing the mixed strategy of going to ballet with probability 2/3 and boxing with probability 1/3 (or just w 2/3 for short) is not just a single point but the whole interval of possible mixed strategies. Both the husband’s and the wife’s best in this ﬁgure are correspondences, not functions. responses ¼ The reason Nash needed a ﬁxed point theorem involving correspondences is precisely than just because his proof works with players’ best responses to prove existence. functions rather FIGURE E8.1 Comparision of Functions and Correspondences The function graphed in (a) looks like a familiar curve. Each value of x is mapped into a single value of y. With the correspondence graphed in (b), each value of x may be mapped into many values of y. Thus, correspondences can have bulges as shown by the shaded regions in (b). y y (a) Function x (b) Correspondence x Chapter 8: Game Theory 297 FIGURE E8.2 Kakutani’s Conditions on Correspondences The correspondence in (a) is not convex because the dashed vertical segment between A and B is not inside the correspondence. The correspondence in (b) is not upper semicontinuous because there is a path (C) inside the correspondence leading to a point (D) that, as indicated by the open circle, is not inside the correspondence. Both (a) and (b) fail to have ﬁxed points. f(x) 1 f(x) 1 45° A B 45° C D (a) Correspondence that is not convex x 1 x (b) Correspondence that is not upper semicontinuous E8.2 Kakutani’s ﬁxed point theorem Here is the statement of Kakutani’s ﬁxed point theorem: Any convex, upper-semicontinuous correspondence [ f(x)] from a closed, bounded, convex set into itself has at least one ﬁxed point (x’) such that x’ f(x’). 2 Comparing the statement of Kakutani’s ﬁxed point theorem with Brouwer’s in Chapter 13, they are similar except for the substitution of ‘‘correspondence’’ for ‘‘function’’ and for the conditions on the correspondence. Brouwer’s theorem requires the function to be continuous; Kakutani’s theorem requires the correspondence to be convex and upper semicontinuous. These properties, which are related to continuity, are less familiar and worth spending a moment to understand. Figure E8.2 provides examples of correspondences violating (a) convexity and (b) upper semicontinuity. The ﬁgure shows why the two properties are needed to guarantee a ﬁxed point. Without both properties, the correspondence can ‘‘jump’’ across the 45! line and thus fail to have a ﬁxed point—that is, a point for which x f(x). ¼ E8.3 Nash’s proof We use R(s) to denote the correspondence that underlies Nash’s existence proof. This correspondence takes any proﬁle (s1, s2, … , sn) (possibly mixed) and of players’ strategies s maps it into another mixed strategy proﬁle, the proﬁle of best responses: ¼ A ﬁxed point of the correspondence is a strategy for which s’ R(s’); this is a Nash equilibrium because each player’s strategy is a best response to others’ strategies. 2 The proof checks that all the conditions involved in Kakutani’s ﬁxed point theorem are satisﬁed by the best-response correspondence R(s). First, we need to show that the set of mixed-strategy proﬁles is closed, bounded, and convex. Because a strategy proﬁle is just a list of individual strategies, the set of strategy proﬁles will be closed, bounded, and convex if each player’s strategy set Si has these properties individually. As Figure E8.3 shows for the case of two and three actions, the set of mixed strategies over actions has a simple shape.1 The set is closed (contains its boundary), bounded (does not go off to inﬁnity in any direction), and convex (the segment between any two points in the set is also in the set). We then need to check that the best-response correspondence R(s) is convex. Individual best responses cannot look like Figure E8.2a because if any two mixed strategies such as A and B are best responses to others’ strategies, then mixed strategies between them must also be best responses. For example, in the Battle of the Sexes, if (1/3, 2/3) and (2/3, 1/3) are best responses for the husband against his wife’s playing (2/3, 1/3) (where, in each pair, the ﬁrst number is the probability of playing ballet and the second of playing boxing), then mixed strategies between the two such as (1/2, 1/2) must also be best responses for him. Figure 8.4 showed that in fact all s R ð BR1ð s Þ ¼ ð ; BR2ð s 1Þ ! ! s , . . . , BRnð : nÞÞ ! 2Þ (i) 1Mathematicians study them so frequently that they have a special name for such a set: a simplex. 298 Part 3: Uncertainty and Strategy FIGURE E8.3 Set of Mixed Strategies for an Individual Player 1’s set of possible mixed strategies over two actions is given by the diagonal line segment in (a). The set for three actions is given by the shaded triangle on the three-dimensional graph in (b). 2 p1 1 3 p1 1 0 1 2 p1 0 (a) Two actions 1 p1 1 1 1 p1 (b) Three actions possible mixed strategies for the husband are best responses to the wife’s playing (2/3, 1/3). Finally, we need to check that R(s) is upper semicontinuous. Individual best responses cannot look like in Figure E8.2b. They cannot have holes like point D punched out of them because payoff functions ui(si, s–i) are continuous. Recall that payoffs, when written as functions of mixed strategies, are actually expected values with probabilities given by the strateg
ies si and s–i. As Equation 2.176 showed, expected values are linear functions of the underlying probabilities. Linear functions are, of course, continuous. E8.4 Games with continuous actions Nash’s existence theorem applies to ﬁnite games—that is, games with a ﬁnite number of players and actions per player. Nash’s theorem does not apply to games that feature continuous actions, such as the Tragedy of the Commons in Example 8.5. Is a Nash equilibrium guaranteed to exist for these games, too? Glicksberg (1952) proved that the answer is ‘‘yes’’ as long as payoff functions are continuous. References Fudenberg, D., and J. Tirole. Game Theory. Cambridge, MA: MIT Press, 1991, sec. 1.3. Glicksberg, I. L. ‘‘A Further Generalization of the Kakutani Fixed Point Theorem with Application to Nash Equilibrium Points.’’ Proceedings of the National Academy of Sciences 38 (1952): 170–74. Nash, John ‘‘Equilibrium Points in n-Person Games.’’ Proceedings of the National Academy of Sciences 36 (1950): 48–49. This page intentionally left blank Production and Supply P A R T FOUR Chapter 9 Production Functions Chapter 10 Cost Functions Chapter 11 Proﬁt Maximization In this part we examine the production and supply of economic goods. Institutions that coordinate the transformation of inputs into outputs are called ﬁrms. They may be large institutions (such as Google, Sony, or the U.S. Department of Defense) or small ones (such as ‘‘Mom and Pop’’ stores or self-employed individuals). Although they may pursue different goals (Google may seek maximum proﬁts, whereas an Israeli kibbutz may try to make members of the kibbutz as well off as possible), all ﬁrms must make certain basic choices in the production process. The purpose of Part 4 is to develop some tools for analyzing those choices. In Chapter 9 we examine ways of modeling the physical relationship between inputs and outputs. We introduce the concept of a production function, a useful abstraction from the complexities of real-world production processes. Two measurable aspects of the production function are stressed: its returns to scale (i.e., how output expands when all inputs are increased) and its elasticity of substitution (i.e., how easily one input may be replaced by another while maintaining the same level of output). We also brieﬂy describe how technical improvements are reﬂected in production functions. The production function concept is then used in Chapter 10 to discuss costs of production. We assume that all ﬁrms seek to produce their output at the lowest possible cost, an assumption that permits the development of cost functions for the ﬁrm. Chapter 10 also focuses on how costs may differ between the short run and the long run. In Chapter 11 we investigate the ﬁrm’s supply decision. To do so, we assume that the ﬁrm’s manager will make input and output choices to maximize proﬁts. The chapter concludes with the fundamental model of supply behavior by proﬁt-maximizing ﬁrms that we will use in many subsequent chapters. 301 This page intentionally left blank C H A P T E R NINE Production Functions The principal activity of any ﬁrm is to turn inputs into outputs. Because economists are interested in the choices the ﬁrm makes in accomplishing this goal, but wish to avoid discussing many of the engineering intricacies involved, they have chosen to construct an abstract model of production. In this model the relationship between inputs and outputs is formalized by a production function of the form q f k, l, m, . . . ð , Þ ¼ (9:1) where q represents the ﬁrm’s output of a particular good during a period,1 k represents the machine (i.e., capital) usage during the period, l represents hours of labor input, m represents raw materials used,2 and the notation indicates the possibility of other variables affecting the production process. Equation 9.1 is assumed to provide, for any conceivable set of inputs, the engineer’s solution to the problem of how best to combine those inputs to get output. Marginal Productivity In this section we look at the change in output brought about by a change in one of the productive inputs. For the purposes of this examination (and indeed for most of the purposes of this book), it will be more convenient to use a simpliﬁed production function deﬁned as follows Production function. The ﬁrm’s production function for a particular good, q, q f k, l ð , Þ ¼ (9:2) shows the maximum amount of the good that can be produced using alternative combinations of capital (k) and labor (l). Of course, most of our analysis will hold for any two inputs to the production process we might wish to examine. The terms capital and labor are used only for convenience. Similarly, it would be a simple matter to generalize our discussion to cases involving 1Here we use a lowercase q to represent one ﬁrm’s output. We reserve the uppercase Q to represent total output in a market. Generally, we assume that a ﬁrm produces only one output. Issues that arise in multiproduct ﬁrms are discussed in a few footnotes and problems. 2In empirical work, raw material inputs often are disregarded, and output, q, is measured in terms of ‘‘value added.’’ 303 304 Part 4: Production and Supply more than two inputs; occasionally, we will do so. For the most part, however, limiting the discussion to two inputs will be helpful because we can show these inputs on twodimensional graphs. Marginal physical product To study variation in a single input, we deﬁne marginal physical product as follows Marginal physical product. The marginal physical product of an input is the additional output that can be produced by using one more unit of that input while holding all other inputs constant. Mathematically, marginal physical product of capital marginal physical product of labor MPk ¼ MPl ¼ ¼ ¼ @q @k ¼ @q @l ¼ f k, f l: (9:3) Notice that the mathematical deﬁnitions of marginal product use partial derivatives, thereby properly reﬂecting the fact that all other input usage is held constant while the input of interest is being varied. For example, consider a farmer hiring one more laborer to harvest the crop but holding all other inputs constant. The extra output this laborer produces is that farmhand’s marginal physical product, measured in physical quantities, such as bushels of wheat, crates of oranges, or heads of lettuce. We might observe, for example, that 50 workers on a farm are able to produce 100 bushels of wheat per year, whereas 51 workers, with the same land and equipment, can produce 102 bushels. The marginal physical product of the 51st worker is then 2 bushels per year. Diminishing marginal productivity We might expect that the marginal physical product of an input depends on how much of that input is used. Labor, for example, cannot be added indeﬁnitely to a given ﬁeld (while keeping the amount of equipment, fertilizer, and so forth ﬁxed) without eventually exhibiting some deterioration in its productivity. Mathematically, the assumption of diminishing marginal physical productivity is an assumption about the second-order partial derivatives of the production function: @MPk @k ¼ @MPl @l ¼ @ 2f @k2 ¼ @ 2 f @l2 ¼ fkk ¼ fll ¼ f11 < 0, f22 < 0: (9:4) The assumption of diminishing marginal productivity was originally proposed by the nineteenth-century economist Thomas Malthus, who worried that rapid increases in population would result in lower labor productivity. His gloomy predictions for the future of humanity led economics to be called the ‘‘dismal science.’’ But the mathematics of the production function suggests that such gloom may be misplaced. Changes in the marginal productivity of labor over time depend not only on how labor input is growing but also on changes in other inputs, such as capital. That is, we must also be concerned with @MPl/@k flk. In most cases, flk > 0, thus, declining labor productivity as both l and k increase is not a foregone conclusion. Indeed, it appears that labor productivity has risen signiﬁcantly since Malthus’ time, primarily because increases in capital inputs (along with technical improvements) have offset the impact of decreasing marginal productivity alone. ¼ Chapter 9: Production Functions 305 Average physical productivity In common usage, the term labor productivity often means average productivity. When it is said that a certain industry has experienced productivity increases, this is taken to mean that output per unit of labor input has increased. Although the concept of average productivity is not nearly as important in theoretical economic discussions as marginal productivity is, it receives a great deal of attention in empirical discussions. Because average productivity is easily measured (say, as so many bushels of wheat per labor-hour input), it is often used as a measure of efﬁciency. We deﬁne the average product of labor (APl) to be APl ¼ output labor input ¼ q l ¼ f ð k, l l Þ : (9:5) Notice that APl also depends on the level of capital used. This observation will prove to be important when we examine the measurement of technical change at the end of this chapter. EXAMPLE 9.1 A Two-Input Production Function Suppose the production function for ﬂyswatters during a particular period can be represented by q f k, l ð ¼ Þ ¼ 600k2l2 k3l3: $ (9:6) To construct the marginal and average productivity functions of labor (l) for this function, we must assume a particular value for the other input, capital (k). Suppose k 10. Then the production function is given by ¼ 60,000l 2 q ¼ $ 1,000l 3: Marginal product. The marginal productivity function (when k 10) is given by ¼ MPl ¼ @q @l ¼ 120,000l 3,000l2, $ (9:7) (9:8) which diminishes as l increases, eventually becoming negative. This implies that q reaches a maximum value. Setting MPl equal to 0, yields or 120,000l 3,000l2 0 ¼ $ 40l l2 ¼ (9:9) (9:10) ¼ as the point at which q reaches its maximum value. Labor input beyond 40 units per period actually reduces total output. For example, when l 32 million ﬂyswatters, wher
eas when l 50, production of ﬂyswatters amounts to only 25 million. 40, Equation 9.7 shows that q ¼ ¼ l 40 (9:11) ¼ ¼ Average product. To ﬁnd the average productivity of labor in ﬂyswatter production, we divide q by l, still holding k 10: APl ¼ q l ¼ 60,000l $ 1,000l2: (9:12) 306 Part 4: Production and Supply Again, this is an inverted parabola that reaches its maximum value when @APl 60,000 2,000l 0, (9:13) ¼ which occurs when l 30. At this value for labor input, Equation 9.12 shows that APl ¼ ¼ 900,000, and Equation 9.8 shows that MPl is also 900,000. When APl is at a maximum, average and marginal productivities of labor are equal.3 @l ¼ $ Notice the relationship between total output and average productivity that is illustrated by this example. Even though total production of ﬂyswatters is greater with 40 workers (32 million) than with 30 workers (27 million), output per worker is higher in the second case. With 40 workers, each worker produces 800,000 ﬂyswatters per period, whereas with 30 workers each worker produces 900,000. Because capital input (ﬂyswatter presses) is held constant in this deﬁnition of productivity, the diminishing marginal productivity of labor eventually results in a declining level of output per worker. QUERY: How would an increase in k from 10 to 11 affect the MPl and APl functions here? Explain your results intuitively. Isoquant Maps And The Rate Of Technical Substitution To illustrate possible substitution of one input for another in a production function, we use its isoquant map. Again, we study a production function of the form q f (k, l), with the understanding that ‘‘capital’’ and ‘‘labor’’ are simply convenient examples of any two inputs that might happen to be of interest. An isoquant (from iso, meaning ‘‘equal’’) records those combinations of k and l that are able to produce a given quantity of output. For example, all those combinations of k and l that fall on the curve labeled ‘‘q 10’’ in Figure 9.1 are capable of producing 10 units of output per period. This isoquant then records the fact that there are many alternative ways of producing 10 units of output. One way might be represented by point A: We would use lA and kA to produce 10 units of output. Alternatively, we might prefer to use relatively less capital and more labor and therefore would choose a point such as B. Hence we may deﬁne an isoquant as follows Isoquant. An isoquant shows those combinations of k and l that can produce a given level of output (say, q0). Mathematically, an isoquant records the set of k and l that satisﬁes f k, l ð Þ ¼ q0: (9:14) As was the case for indifference curves, there are inﬁnitely many isoquants in the k–l plane. Each isoquant represents a different level of output. Isoquants record successively higher levels of output as we move in a northeasterly direction. Presumably, using more 3This result is general. Because at a maximum l Æ MPl ¼ q or MPl ¼ APl. @APl @l ¼ l % MPl $ l2 q , Chapter 9: Production Functions 307 FIGURE 9.1 An Isoquant Map Isoquants record the alternative combinations of inputs that can be used to produce a given level of output. The slope of these curves shows the rate at which l can be substituted for k while keeping output constant. The negative of this slope is called the (marginal) rate of technical substitution (RTS). In the ﬁgure, the RTS is positive and diminishing for increasing inputs of labor. k per period kA A kB lA B lB q = 30 q = 20 q = 10 l per period ¼ 20 and of each of the inputs will permit output to increase. Two other isoquants (for q q 30) are shown in Figure 9.1. You will notice the similarity between an isoquant map and the individual’s indifference curve map discussed in Part 2. They are indeed similar concepts because both represent ‘‘contour’’ maps of a particular function. For isoquants, however, the labeling of the curves is measurable—an output of 10 units per period has a quantiﬁable meaning. Therefore, economists are more interested in studying the shape of production functions than in examining the exact shape of utility functions. ¼ The marginal rate of technical substitution (RTS) The slope of an isoquant shows how one input can be traded for another while holding output constant. Examining the slope provides information about the technical possibility of substituting labor for capital. A formal deﬁnition follows Marginal rate of technical substitution. The marginal rate of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant. In mathematical terms, RTS l for k ð Þ ¼ $ dk dl : q0 q ! ¼ ! ! ! (9:15) In this deﬁnition, the notation is intended as a reminder that output is to be held constant as l is substituted for k. The particular value of this trade-off rate will depend not only on the level of output but also on the quantities of capital and labor being used. Its value depends on the point on the isoquant map at which the slope is to be measured. 308 Part 4: Production and Supply RTS and marginal productivities To examine the shape of production function isoquants, it is useful to prove the following result: The RTS (of l for k) is equal to the ratio of the marginal physical productivity of labor (MPl) to the marginal physical productivity of capital (MPk). Imagine using Equation 9.14 to graph the q0 isoquant. We would substitute a sequence of increasing values of l and see how k would have to adjust to keep output constant at q0. The graph of the isoquant is really the graph of the implicit function k(l) satisfying ð Just as we did in the section on implicit functions in Chapter 2 (see in particular Equation 2.22), we can use the chain rule to differentiate Equation 9.16, giving Þ q0 9:16) fk dk dl þ fl ¼ 0 ¼ MPk dk dl þ MPl, (9:17) where the initial 0 appears because q0 is being held constant; therefore, the derivative of the left side of Equation 9.16 with respect to l equals 0. Rearranging Equation 9.17 gives RTS l for k ð Þ ¼ $ dk dl MPl MPk q0 ¼ : (9:18) q ¼ Hence the RTS is given by the ratio of the inputs’ marginal productivities. ! ! ! ! Equation 9.18 shows that those isoquants that we actually observe must be negatively sloped. Because both MPl and MPk will be non-negative (no ﬁrm would choose to use a costly input that reduced output), the RTS also will be positive (or perhaps zero). Because the slope of an isoquant is the negative of the RTS, any ﬁrm we observe will not be operating on the positively sloped portion of an isoquant. Although it is mathematically possible to devise production functions whose isoquants have positive slopes at some points, it would not make economic sense for a ﬁrm to opt for such input choices. Reasons for a diminishing RTS The isoquants in Figure 9.1 are drawn not only with a negative slope (as they should be) but also as convex curves. Along any one of the curves, the RTS is diminishing. For high ratios of k to l, the RTS is a large positive number, indicating that a great deal of capital can be given up if one more unit of labor becomes available. On the other hand, when a lot of labor is already being used, the RTS is low, signifying that only a small amount of capital can be traded for an additional unit of labor if output is to be held constant. This assumption would seem to have some relationship to the assumption of diminishing marginal productivity. A hasty use of Equation 9.18 might lead one to conclude that an increase in l accompanied by a decrease in k would result in a decrease in MPl, an increase in MPk, and, therefore, a decrease in the RTS. The problem with this quick ‘‘proof ’’ is that the marginal productivity of an input depends on the level of both inputs—changes in l affect MPk and vice versa. It is not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone. To see why this is so mathematically, assume that q f(k, l) and that fk and fl are positive (i.e., the marginal productivities are positive). Assume also that fkk < 0 and fll < 0 (that the marginal productivities are diminishing). To show that isoquants are convex, we would like to show that d(RTS)/dl < 0. Because RTS fl=fkÞ dl fl /fk, we have (9:19) dRTS dl ¼ ¼ ¼ d ð : Chapter 9: Production Functions 309 Because fl and fk are functions of both k and l, we must be careful in taking the derivative of this expression: dRTS dl ¼ fkð fll þ f lk % dk=dl flð 2 Þ $ fkÞ ð fkl þ fkk % dk=dl Þ : (9:20) Using the fact that dk/dl we have fl /fk along an isoquant and Young’s theorem ( fkl ¼ ¼ $ flk), f 2 k fll $ dRTS dl ¼ 2fk fl fkl þ fkÞ ð Because we have assumed fk > 0, the denominator of this function is positive. Hence the whole fraction will be negative if the numerator is negative. Because fll and fkk are both assumed to be negative, the numerator deﬁnitely will be negative if fkl is positive. If we can assume this, we have shown that dRTS/dl < 0 (that the isoquants are convex).4 (9:21) : 3 f 2 l fkk Importance of cross-productivity effects flk should be positive. Intuitively, it seems reasonable that the cross-partial derivative fkl ¼ If workers had more capital, they would have higher marginal productivities. Although this is probably the most prevalent case, it does not necessarily have to be so. Some production functions have fkl < 0, at least for a range of input values. When we assume a diminishing RTS (as we will throughout most of our discussion), we are therefore making a stronger assumption than simply diminishing marginal productivities for each input— speciﬁcally, we are assuming that marginal productivities diminish ‘‘rapidly enough’’ to compensate for any possible negative cross-productivity effects. Of course, as we shall see later, with three or more inputs, things become even more complicated. EXAMPLE 9.2 A Diminishing RTS In Example 9.1, the production function for ﬂyswatters was given by ¼ General marginal productivity functions for this production function are Þ ¼ $ q f
k, l ð 600k2l2 k3l3: MPl ¼ MPk ¼ fl ¼ fk ¼ @q @l ¼ @q @k ¼ 1,200k2l 3k3l2, $ 1,200kl2 3k2l3: $ (9:22) (9:23) Notice that each of these depends on the values of both inputs. Simple factoring shows that these marginal productivities will be positive for values of k and l for which kl < 400. Because and fll ¼ 1,200k2 6k3l $ fkk ¼ 1,200l2 6kl3, $ (9:24) it is clear that this function exhibits diminishing marginal productivities for sufﬁciently large values of k and l. Indeed, again by factoring each expression, it is easy to show that fll, fkk < 0 if 4As we pointed out in Chapter 2, functions for which the numerator in Equation 9.21 is negative are called (strictly) quasiconcave functions. 310 Part 4: Production and Supply kl > 200. However, even within the range 200 < kl < 400 where the marginal productivity relations for this function behave ‘‘normally,’’ this production function may not necessarily have a diminishing RTS. Cross-differentiation of either of the marginal productivity functions (Equation 9.23) yields fkl ¼ flk ¼ 2,400kl $ 9k2l2, (9:25) which is positive only for kl < 266. Therefore, the numerator of Equation 9.21 will deﬁnitely be negative for 200 < kl < 266, but for larger-scale ﬂyswatter factories the case is not so clear because fkl is negative. When fkl is negative, increases in labor input reduce the marginal productivity of capital. Hence the intuitive argument that the assumption of diminishing marginal productivities yields an unambiguous prediction about what will happen to the RTS ( fl/fk) as l increases and k decreases is incorrect. ¼ It all depends on the relative effects on marginal productivities of diminishing marginal productivities (which tend to reduce fl and increase fk) and the contrary effects of cross-marginal productivities (which tend to increase fl and reduce fk). Still, for this ﬂyswatter case, it is true that the RTS is diminishing throughout the range of k and l where marginal productivities are positive. For cases where 266 < kl < 400, the diminishing marginal productivities exhibited by the function are sufﬁcient to overcome the inﬂuence of a negative value for fkl on the convexity of isoquants. QUERY: For cases where k l, what can be said about the marginal productivities of this production function? How would this simplify the numerator for Equation 9.21? How does this permit you to more easily evaluate this expression for some larger values of k and l? ¼ Returns to Scale We now proceed to characterize production functions. A ﬁrst question that might be asked about them is how output responds to increases in all inputs together. For example, suppose that all inputs were doubled: Would output double or would the relationship not be so simple? This is a question of the returns to scale exhibited by the production function that has been of interest to economists ever since Adam Smith intensively studied the production of pins. Smith identiﬁed two forces that came into operation when the conceptual experiment of doubling all inputs was performed. First, a doubling of scale permits a greater division of labor and specialization of function. Hence there is some presumption that efﬁciency might increase—production might more than double. Second, doubling of the inputs also entails some loss in efﬁciency because managerial overseeing may become more difﬁcult given the larger scale of the ﬁrm. Which of these two tendencies will have a greater effect is an important empirical question. These concepts can be deﬁned technically as followsk, l) and if all inputs are multiplied Returns to scale. If the production function is given by q by the same positive constant t (where t > 1), then we classify the returns to scale of the production function by ¼ Effect on Output Returns to Scale f (tk, tl ) tf (k, l ) ¼ f (tk, tl ) < tf (k, l ) f (tk, tl ) > tf (k, l ) tq tq tq ¼ ¼ ¼ Constant Decreasing Increasing Chapter 9: Production Functions 311 In intuitive terms, if a proportionate increase in inputs increases output by the same proportion, the production function exhibits constant returns to scale. If output increases less than proportionately, the function exhibits diminishing returns to scale. And if output increases more than proportionately, there are increasing returns to scale. As we shall see, it is theoretically possible for a function to exhibit constant returns to scale for some levels of input usage and increasing or decreasing returns for other levels.5 Often, however, economists refer to returns to scale of a production function with the implicit understanding that only a fairly narrow range of variation in input usage and the related level of output is being considered. Constant returns to scale There are economic reasons why a ﬁrm’s production function might exhibit constant returns to scale. If the ﬁrm operates many identical plants, it may increase or decrease production simply by varying the number of them in current operation. That is, the ﬁrm can double output by doubling the number of plants it operates, and that will require it to employ precisely twice as many inputs. Empirical studies of production functions often ﬁnd that returns to scale are roughly constant for the ﬁrms studied (at least around for outputs close to the ﬁrms’ established operating levels—the ﬁrms may exhibit increasing returns to scale as they expand to their established size). For all these reasons, the constant returns-to-scale case seems worth examining in somewhat more detail. When a production function exhibits constant returns to scale, it meets the deﬁnition of ‘‘homogeneity’’ that we introduced in Chapter 2. That is, the production is homogeneous of degree 1 in its inputs because f (tk, tl) = t1f (k, l) = tq. (9:26) In Chapter 2 we showed that, if a function is homogeneous of degree k, its derivatives are homogeneous of degree k 1. In this context this implies that the marginal productivity functions derived from a constant returns-to-scale production function are homogeneous of degree 0. That is, $ MPk ¼ MPl ¼ @f @f k, l ð Þ @k ¼ k, l ð Þ @l ¼ @f @f tk, tl ð @k tk, tl ð @l Þ , Þ for any t > 0. In particular, we can let t 1/ l in Equations 9.27 and get ¼ MPk ¼ MPl ¼ @f @f k=l, 1 ð @k k=l, 1 ð @l Þ , Þ : (9:27) (9:28) That is, the marginal productivity of any input depends only on the ratio of capital to labor input, not on the absolute levels of these inputs. This fact is especially important, for example, in explaining differences in productivity among industries or across countries. 5A local measure of returns to scale is provided by the scale elasticity, deﬁned as eq, t ¼ @f tk, tl ð @t Þ t tk, tl , Þ % f ð where this expression is to be evaluated at t level of input usage. For some examples using this concept, see Problem 9.9. ¼ 1. This parameter can, in principle, take on different values depending on the 312 Part 4: Production and Supply Homothetic production functions One consequence of Equations 9.28 is that the RTS ( MPl/MPk) for any constant returns-to-scale production function will depend only on the ratio of the inputs, not on their absolute levels. That is, such a function will be homothetic (see Chapter 2)—its isoquants will be radial expansions of one another. This situation is shown in Figure 9.2. Along any ray through the origin (where the ratio k/l does not change), the slopes of successively higher isoquants are identical. This property of the isoquant map will be useful to us on several occasions. ¼ A simple numerical example may provide some intuition about this result. Suppose a large bread order (consisting of, say, 200 loaves) can be ﬁlled in one day by three bakers working with three ovens or by two bakers working with four ovens. Therefore, the RTS of ovens for bakers is one for one—one extra oven can be substituted for one baker. If this production process exhibits constant returns to scale, two large bread orders (totaling 400 loaves) can be ﬁlled in one day, either by six bakers with six ovens or by four bakers with eight ovens. In the latter case, two ovens are substituted for two bakers, so again the RTS is one for one. In constant returns-to-scale cases, expanding the level of production does not alter trade-offs among inputs; thus, production functions are homothetic. A production function can have a homothetic indifference curve map even if it does not exhibit constant returns to scale. As we showed in Chapter 2, this property of homotheticity is retained by any monotonic transformation of a homogeneous function. Hence increasing or decreasing returns to scale can be incorporated into a constant returns-toscale function through an appropriate transformation. Perhaps the most common such transformation is exponential. Thus, if f(k, l) is a constant returns-to-scale production FIGURE 9.2 Isoquant Map for a Constant Returns-toScale Production Function Because a constant returns-to-scale production function is homothetic, the RTS depends only on the ratio of k to l, not on the scale of production. Consequently, along any ray through the origin (a ray of constant k/l), the RTS will be the same on all isoquants. An additional feature is that the isoquant labels increase proportionately with the inputs. k per period per period Chapter 9: Production Functions 313 function, we can let F(k, l) = [ f (k, l)]g, (9:29) where g is any positive exponent. If g > 1, then F(tk, tl ) = [ f (tk, tl )]g = [tf (k, l )]g = tg[ f (k, l )]g = tgF(k, l ) > tF(k, l ) (9:30) ¼ for any t > 1. Hence this transformed production function exhibits increasing returns to scale. The exponent g captures the degree of the increasing returns to scale. A doubling of 2 but an eight-fold increase if inputs would lead to a four-fold increase in output if g 3. An identical proof shows that the function F exhibits decreasing returns to scale g for g < 1. Because this function remains homothetic through all such transformations, we have shown that there are important cases where the issue of ret
urns to scale can be separated from issues involving the shape of an isoquant. In these cases, changes in the returns to scale will just change the labels on the isoquants rather than their shapes. In the next section, we will look at how shapes of isoquants can be described. ¼ The n-input case The deﬁnition of returns to scale can be easily generalized to a production function with n inputs. If that production function is given by and if all inputs are multiplied by t > 1, we have q f x1, x2, . . . , xnÞ ð ¼ f (tx1, tx2, . . . , txn) = t kf (x1, x2, . . . , xn) = t kq (9:31) (9:32) 1, the production function exhibits constant returns to scale. for some constant k. If k Decreasing and increasing returns to scale correspond to the cases k < 1 and k > 1, respectively. ¼ The crucial part of this mathematical deﬁnition is the requirement that all inputs be increased by the same proportion, t. In many real-world production processes, this provision may make little economic sense. For example, a ﬁrm may have only one ‘‘boss,’’ and that number would not necessarily be doubled even if all other inputs were. Or the output of a farm may depend on the fertility of the soil. It may not be literally possible to double the acres planted while maintaining fertility because the new land may not be as good as that already under cultivation. Hence some inputs may have to be ﬁxed (or at least imperfectly variable) for most practical purposes. In such cases, some degree of diminishing productivity (a result of increasing employment of variable inputs) seems likely, although this cannot properly be called ‘‘diminishing returns to scale’’ because of the presence of inputs that are held ﬁxed. The Elasticity Of Substitution Another important characteristic of the production function is how ‘‘easy’’ it is to substitute one input for another. This is a question about the shape of a single isoquant rather than about the whole isoquant map. Along one isoquant, the rate of technical substitution will decrease as the capital–labor ratio decreases (i.e., as k/l decreases); now we wish to deﬁne some parameter that measures this degree of responsiveness. If the RTS does not change at all for changes in k/l, we might say that substitution is easy because the ratio of the marginal productivities of the two inputs does not change as the input mix changes. Alternatively, if the RTS changes rapidly for small changes in k/l, we would say that 314 Part 4: Production and Supply substitution is difﬁcult because minor variations in the input mix will have a substantial effect on the inputs’ relative productivities. A scale-free measure of this responsiveness is provided by the elasticity of substitution, a concept we encountered informally in our discussion of CES utility functions. Here we will work on providing a more formal deﬁnition. For discrete changes, the elasticity of substitution is given by percent D k=l percent DRTS ¼ ð Þ r ¼ D k=l ð k=l Þ 4 DRTS RTS ¼ k=l D Þ ð DRTS % : RTS k=l ð Þ (9:33) More often, we will be interested in considering small changes; therefore, a modiﬁcation of Equation 9.33 will be of more interest: k=l d Þ ð d RTS % RTS k=l ¼ k=l d ln Þ ð d ln RTS : r ¼ (9:34) The logarithmic expression follows from mathematical derivations along the lines of Example 2.2 from Chapter 2. All these equations can be collected in the following formal deﬁnition Elasticity of substitution. For the production function q f (k, l), the elasticity of substitution (s) measures the proportionate change in k/l relative to the proportionate change in the RTS along an isoquant. That is, ¼ percent D percent DRTS ¼ k=l ð Þ r ¼ k=l d Þ ð d RTS % RTS k=l ¼ k=l d ln ð d ln RTS ¼ Þ k=l d ln ð Þ fl =fkÞ d ln ð : (9:35) Because along an isoquant k/l and RTS move in the same direction, the value of s is always positive. Graphically, this concept is illustrated in Figure 9.3 as a movement from point A to point B on an isoquant. In this movement, both the RTS and the ratio k/l will change; we are interested in the relative magnitude of these changes. If s is high, then the RTS will not change much relative to k/l and the isoquant will be close to linear. On the other hand, a low value of s implies a rather sharply curved isoquant; the RTS will change by a substantial amount as k/l changes. In general, it is possible that the elasticity of substitution will vary as one moves along an isoquant and as the scale of production changes. Often, however, it is convenient to assume that s is constant along an isoquant. If the production function is also homothetic, then—because all the isoquants are merely radial blowups—s will be the same along all isoquants. We will encounter such functions later in this chapter and in many of the end of chapter problems.6 The n-input case Generalizing the elasticity of substitution to the many-input case raises several complications. One approach is to adopt a deﬁnition analogous to Equation 9.35; that is, to deﬁne 6The elasticity of substitution can be phrased directly in terms of the production function and its derivatives in the constant returns-to-scale case as But this form is cumbersome. Hence usually the logarithmic deﬁnition in Equation 9.35 is easiest to apply. For a compact summary, see P. Berck and K. Sydsaeter, Economist’s Mathematical Manual (Berlin, Germany: Springer-Verlag, 1999), chap. 5. r ¼ fl fk, l fk % f % Chapter 9: Production Functions 315 FIGURE 9.3 Graphic Description of the Elasticity of Substitution In moving from point A to point B on the q q0 isoquant, both the capital–labor ratio (k/l) and the RTS will change. The elasticity of substitution (s) is deﬁned to be the ratio of these proportional changes; it is a measure of how curved the isoquant is. ¼ k per period A RTSA RTSB (k /l ) A (k /l ) B B q = q0 l per period the elasticity of substitution between two inputs to be the proportionate change in the ratio of the two inputs to the proportionate change in the RTS between them while holding output constant.7 To make this deﬁnition complete, it is necessary to require that all inputs other than the two being examined be held constant. However, this latter requirement (which is not relevant when there are only two inputs) restricts the value of this potential deﬁnition. In real-world production processes, it is likely that any change in the ratio of two inputs will also be accompanied by changes in the levels of other inputs. Some of these other inputs may be complementary with the ones being changed, whereas others may be substitutes, and to hold them constant creates a rather artiﬁcial restriction. For this reason, an alternative deﬁnition of the elasticity of substitution that permits such complementarity and substitutability in the ﬁrm’s cost function is generally used in the n-good case. Because this concept is usually measured using cost functions, we will describe it in the next chapter. 7That is, the elasticity of substitution between input i and input j might be deﬁned as rij ¼ xi=xjÞ @ ln ð @ ln fj=fiÞ ð for movements along f (x1, x2, . . ., xn) all inputs other than i and j be held constant when considering movements along the q0 isoquant. q0. Notice that the use of partial derivatives in this deﬁnition effectively requires that ¼ 316 Part 4: Production and Supply Four Simple Production Functions In this section we illustrate four simple production functions, each characterized by a different elasticity of substitution. These are shown only for the case of two inputs, but generalization to many inputs is easily accomplished (see the Extensions for this chapter). Case 1: Linear (r 5 ¥) Suppose that the production function is given by Þ ¼ It is easy to show that this production function exhibits constant returns to scale: For any t > 1, ¼ þ q f k, l ð ak bl: (9:36) f tk, tl ð Þ ¼ atk btl ak t ð þ bl ¼ þ Þ ¼ k, l tf ð : Þ (9:37) All isoquants for this production function are parallel straight lines with slope b/a. Such an isoquant map is pictured in Figure 9.4a. Because the RTS is constant along any straight-line isoquant, the denominator in the deﬁnition of s (Equation 9.35) is equal to 0 and hence s is inﬁnite. Although this linear production function is a useful example, it is rarely encountered in practice because few production processes are characterized by such ease of substitution. Indeed, in this case, capital and labor can be thought of as perfect substitutes for each other. An industry characterized by such a production function could use only capital or only labor, depending on these inputs’ prices. It is hard to envision such a production process: Every machine needs someone to press its buttons, and every laborer requires some capital equipment, however modest. $ Case 2: Fixed proportions (r 5 0) Production functions characterized by s 0 have L-shaped isoquants as depicted in ¼ Figure 9.4b. At the corner of an L-shaped isoquant, a negligible increase in k/l causes an inﬁnite increase in RTS because the isoquant changes suddenly from horizontal to vertical there. Substituting 0 for the change in k/l in the numerator of the formula for s in Equation 9.33 and inﬁnity for the change in RTS in the denominator implies s 0. A ﬁrm would always operate at the corner of an isoquant. Operating anywhere else is inefﬁcient because the same output could be produced with fewer inputs by moving along the isoquant toward the corner. ¼ As drawn in Figure 9.4, the corners of the isoquants all lie along the same ray from the origin. This illustrates the important special case of a ﬁxed-proportions production function. Because the ﬁrm always operates at the corner of some isoquant, and all isoquants line up along the same ray, it must be the case that the ﬁrm uses inputs in the ﬁxed proportions given by the slope of this ray regardless of how much it produces.8 The inputs are perfect complements in that, starting from the ﬁxed proportion, an increase in one input is useless unless the other is increased as w
ell. The mathematical form of the ﬁxed-proportions production function is given by q , a, b > 0, Þ where the operator ‘‘min’’ means that q is given by the smaller of the two values in parentheses. For example, suppose that ak < bl; then q ak, and we would say that capital is the binding constraint in this production process. The employment of more labor would ak, bl ð (9:38) min ¼ ¼ 8Production functions with s along a nonlinear curve from the origin rather than lining up along a ray. ¼ 0 need not be ﬁxed proportions. The other possibility is that the corners of the isoquants lie Chapter 9: Production Functions 317 FIGURE 9.4 Isoquant Maps for Simple Production Functions with Various Values for s Three possible values for the elasticity of substitution are illustrated in these ﬁgures. In (a), capital and labor are perfect substitutes. In this case, the RTS will not change as the capital–labor ratio changes. In (b), the ﬁxed–proportions case, no substitution is possible. The capital–labor ratio is ﬁxed at b/a. A case of limited substitutability is illustrated in (c). k per period k per period σ = ∞ Slope = __−β α __q3 α q1 q2 q3 l per period __q3 β (b) (a) k per period σ = 0 q3 q2 q1 l per period σ = 1 q3 q2 q1 l per period (c) not increase output, and hence the marginal product of labor is zero; additional labor is superﬂuous in this case. Similarly, if ak > bl, then labor is the binding constraint on output, and additional capital is superﬂuous. When ak bl, both inputs are fully utilized. b/a, and production takes place at a vertex on the isoquant When this happens, k/l map. If both inputs are costly, this is the only cost-minimizing place to operate. The locus of all such vertices is a straight line through the origin with a slope given by b/a.9 ¼ ¼ 9With the form reﬂected by Equation 9.38, the ﬁxed-proportions production function exhibits constant returns to scale because Þ ¼ for any t > 1. As before, increasing or decreasing returns can be easily incorporated into the functions by using a nonlinear transformation of this functional form—such as [ f (k, l)]g, where g may be greater than or less than 1. Þ ¼ Þ ¼ Þ % f tk, tl ð atk, btl min ð t min ð ak, bl tf k, l ð 318 Part 4: Production and Supply The ﬁxed-proportions production function has a wide range of applications. Many machines, for example, require a certain number of people to run them, but any excess labor is superﬂuous. Consider combining capital (a lawn mower) and labor to mow a lawn. It will always take one person to run the mower, and either input without the other is not able to produce any output at all. It may be that many machines are of this type and require a ﬁxed complement of workers per machine.10 Case 3: Cobb–Douglas (r 5 1) 1, called a Cobb–Douglas production function,11 The production function for which s provides a middle ground between the two polar cases previously discussed. Isoquants for the Cobb–Douglas case have the ‘‘normal’’ convex shape and are shown in Figure 9.4c. The mathematical form of the Cobb–Douglas production function is given by ¼ q = f (k, l ) = Akalb, (9:39) where A, a, and b are all positive constants. The Cobb–Douglas function can exhibit any degree of returns to scale, depending on the values of a and b. Suppose all inputs were increased by a factor of t. Then tk, tl f ð Þ ¼ ¼ A tk ð b ta þ a tl ð Þ k, l ð Ata bkalb þ ¼ b Þ : Þ (9:40) b ¼ þ 1, the Cobb–Douglas function exhibits constant returns to scale Hence if a b > 1, then the function exhibits because output also increases by a factor of t. If a b < 1 corresponds to the decreasing returnsincreasing returns to scale, whereas a to-scale case. It is a simple matter to show that the elasticity of substitution is 1 for the Cobb–Douglas function.12 This fact has led researchers to use the constant returnsto-scale version of the function for a general description of aggregate production relationships in many countries. þ þ The Cobb–Douglas function has also proved to be useful in many applications because it is linear in logarithms: ln q ¼ ln A þ a ln k þ b ln l: (9:41) The constant a is then the elasticity of output with respect to capital input, and b is the elasticity of output with respect to labor input.13 These constants can sometimes be 10The lawn mower example points up another possibility, however. Presumably there is some leeway in choosing what size of lawn mower to buy. Hence before the actual purchase, the capital–labor ratio in lawn mowing can be considered variable: Any device, from a pair of clippers to a gang mower, might be chosen. Once the mower is purchased, however, the capital–labor ratio becomes ﬁxed. 11Named after C. W. Cobb and P. H. Douglas. See P. H. Douglas, The Theory of Wages (New York: Macmillan Co., 1934), pp. 132–35. 12For the Cobb–Douglas function, RTS fl fk ¼ ¼ 1 bAkalb $ 1lb ¼ aAka $ b a k l or Hence 13See Problem 9.5. ln RTS ln ð ¼ b=a Þ þ ln : k=l ð Þ @ ln k=l @ ln RTS ¼ 1: r ¼ Chapter 9: Production Functions 319 estimated from actual data, and such estimates may be used to measure returns to scale (by examining the sum a b) and for other purposes. þ Case 4: CES production function A functional form that incorporates all three previous cases and allows s to take on other values as well is the constant elasticity of substitution (CES) production function ﬁrst introduced by Arrow et al. in 1961.14 This function is given by q f k, l kq g=q lq (9:42) ð ) ¼ 1, r ( 0, and g > 0. This function closely resembles the CES utility function disfor r cussed in Chapter 3, although now we have added the exponent g/r to permit explicit introduction of returns-to-scale factors. For g > 1 the function exhibits increasing returns to scale, whereas for g < 1 it exhibits decreasing returns. Þ ¼ ½ þ 6¼ Direct application of the deﬁnition of s to this function15 gives the important result that r ¼ 1 1 $ : q (9:43) Hence the linear, ﬁxed-proportions, and Cobb–Douglas cases correspond to r r Cobb–Douglas cases requires a limit argument. 1, 0, respectively. Proof of this result for the ﬁxed-proportions and , and r ¼ $1 ¼ $ ¼ Often the CES function is used with a distributional weight, a (0 the relative signiﬁcance of the inputs: a ) ) 1), to indicate q f k, l ð ¼ akq 1 lq =q: (9:44) With constant returns to scale and r form ¼ 0, this function converges to the Cobb–Douglas q f k, l ð ¼ Þ ¼ kal1 $ a: EXAMPLE 9.3 A Generalized Leontief Production Function Suppose that the production function for a good is given by q f k lp : k % ﬃﬃﬃﬃﬃﬃﬃ (9:45) (9:46) 14K. J. Arrow, H. B. Chenery, B. S. Minhas, and R. M. Solow, ‘‘Capital–Labor Substitution and Economic Efﬁciency,’’ Review of Economics and Statistics (August 1961): 225–50. 15For the CES function we have RTS fl fk ¼ ¼ g=q ð g=q ð Þ % Þ % q g qð g qð $ q $ =g Þ =g Þ % % 1 qlq $ qkq $ Applying the deﬁnition of the elasticity of substitution then yields @ ln k=l ð @ ln RTS ¼ Þ 1 r ¼ 1 $ : q Notice in this computation that the factor r cancels out of the marginal productivity functions, thereby ensuring that these marginal productivities are positive even when r is negative (as it is in many cases). This explains why r appears in two different places in the deﬁnition of the CES function. 320 Part 4: Production and Supply This function is a special case of a class of functions named for the Russian-American economist Wassily Leontief.16 The function clearly exhibits constant returns to scale because (9:47) (9:48) Marginal productivities for the Leontief function are ﬃﬃﬃﬃ f tk, tl ð Þ ¼ tk tl þ þ 2t klp tf k, l ð : Þ ¼ 1 1 fk ¼ fl ¼ þ ð 0:5, k=l k=l $ Þ 0:5: Þ þ ð Hence marginal productivities are positive and diminishing. As would be expected (because this function exhibits constant returns to scale), the RTS here depends only on the ratio of the two inputs RTS fl fk ¼ ¼ 1 1 k=l k=l 0:5 0:9:49) This RTS diminishes as k/l falls, so the isoquants have the usual convex shape. There are two ways you might calculate the elasticity of substitution for this production function. First, you might notice that in this special case the function can be factored as q k l þ þ ¼ klp 2 kp ¼ ð lp 2 Þ þ k0:5 ¼ ð l 0:5 2, Þ þ (9:50) which makes clear that this function has a CES form with r elasticity of substitution here is s r) ﬃﬃﬃﬃ 1/(1 2. ﬃﬃﬃ ﬃﬃ 0.5 and g ¼ ¼ 1. Hence the Of course, in most cases it is not possible to do such a simple factorization. A more exhaustive approach is to apply the deﬁnition of the elasticity of substitution given in footnote 6 of this chapter: ¼ $ ¼ k=l 0:5 ( $ Þ r ¼ f fk fl fk l ¼ % 1 ½ þ ð k=l k=l 2 þ ð k=l 0:5 ð 0:5 Þ 0:5 Þ 0:5 Þ 1 (½ 0:5= q % ð þ ð klp 0:5 k=l 0:5 $ Þ k=l ð ﬃﬃﬃﬃﬃ $ Þ Þ (9:51) 2: 1 þ ¼ þ ð þ Notice that in this calculation the input ratio (k/l) drops out, leaving a simple result. In other applications, one might doubt that such a fortuitous result would occur and hence doubt that the elasticity of substitution is constant along an isoquant (see Problem 9.7). But here the result 2 is intuitively reasonable because that value represents a compromise between the that s ) and its elasticity of substitution for this production function’s linear part (q Cobb–Douglas part (q 2k0.5l 0.5, s 0:5 ¼ ¼1 l, s 1). þ ¼ ¼ k ¼ ¼ QUERY: What can you learn about this production function by graphing the q Why does this function generalize the ﬁxed-proportions case? ¼ 4 isoquant? Technical Progress Methods of production improve over time, and it is important to be able to capture these improvements with the production function concept. A simpliﬁed view of such progress is provided by Figure 9.5. Initially, isoquant q0 records those combinations of capital and labor that can be used to produce an output level of q0. Following the development of superior production techniques, this isoquant shifts to q00. Now the same level of output 16Leontief was a pioneer in the development of input–output analysis. In input–output analysis, production is assumed to take place with a ﬁxed-proportions technology. T
he Leontief production function generalizes the ﬁxed-proportions case. For more details see the discussion of Leontief production functions in the Extensions to this chapter. Chapter 9: Production Functions 321 FIGURE 9.5 Technical Progress Technical progress shifts the q0 isoquant toward the origin. The new q0 isoquant, q00, shows that a given level of output can now be produced with less input. For example, with k1 units of capital it now only takes l1 units of labor to produce q0, whereas before the technical advance it took l2 units of labor. k per period k2 k1 q0 q′0 l1 l2 l per period can be produced with fewer inputs. One way to measure this improvement is by noting that with a level of capital input of, say, k1, it previously took l2 units of labor to produce q0, whereas now it takes only l1. Output per worker has risen from q0/l2 to q0/l1. But one must be careful in this type of calculation. An increase in capital input to k2 would also have permitted a reduction in labor input to l1 along the original q0 isoquant. In this case, output per worker would also increase, although there would have been no true technical progress. Use of the production function concept can help to differentiate between these two concepts and therefore allow economists to obtain an accurate estimate of the rate of technical change. Measuring technical progress The ﬁrst observation to be made about technical progress is that historically the rate of growth of output over time has exceeded the growth rate that can be attributed to the growth in conventionally deﬁned inputs. Suppose that we let q A ¼ t ð k, l f Þ be the production function for some good (or perhaps for society’s output as a whole). The term A(t) in the function represents all the inﬂuences that go into determining q other than k (machine-hours) and l (labor-hours). Changes in A over time represent technical progress. For this reason, A is shown as a function of time. Presumably dA/dt > 0; particular levels of input of labor and capital become more productive over time. (9:52) Þ ð 322 Part 4: Production and Supply Growth accounting Differentiating Equation 9.52 with respect to time gives df k, l ð dt Þ % dq dt ¼ ¼ dA dt % dA dt % A f k, l f ð @f @k % dk dt þ @f @l % dl dt : & % Þ Dividing by q gives or dq=dt dA=dt q ¼ A þ @f =@k k, l f dk dt þ % @f =@l k, l f dl dt % ð Þ ð Þ dq=dt dA=dt q ¼ A þ @f @k % k k, l f ð % Þ dk=dt k þ @f @l % l k, l f ð % Þ dl=dt l : (9:53) (9:54) (9:55) Now for any variable x, (dx/dt)/x is the proportional rate of growth of x per unit of time. We shall denote this by Gx.17 Hence Equation 9.55 can be written in terms of growth rates as Gq ¼ GA þ @f @k % k k, l f ð Þ Gk þ % @f @l % l k, l f ð % Þ Gt: (9:56) But and @f @k % k k, l f ð ¼ Þ @q @k % k q ¼ @f @l % l k, l f ð ¼ Þ @q @l % l q ¼ elasticity of output with respect to capital eq, k ¼ (9:57) elasticity of output with respect to labor eq, l: ¼ (9:58) Therefore, our growth equation ﬁnally becomes eq, kGk þ This shows that the rate of growth in output can be broken down into the sum of two components: growth attributed to changes in inputs (k and l) and other ‘‘residual’’ growth (i.e., changes in A) that represents technical progress. GA þ Gq ¼ eq, lGl: (9:59) Equation 9.59 provides a way of estimating the relative importance of technical progress (GA) in determining the growth of output. For example, in a pioneering study of the entire U.S. economy between the years 1909 and 1949, R. M. Solow recorded the following values for the terms in the equation18: Gq ¼ Gl ¼ Gk ¼ eq, l ¼ eq, k ¼ 2:75 percent per year, 1:00 percent per year, 1:75 percent per year, 0:65, 0:35: (9:60) 17Two useful features of this deﬁnition are: (1) Gx Æ y ¼ Gy. sum of each one’s growth rate; and (2) Gx/y ¼ 18R. M. Solow, ‘‘Technical Progress and the Aggregate Production Function,’’ Review of Economics and Statistics 39 (August 1957): 312–20. Gy—that is, the growth rate of a product of two variables is the Gx þ Gx $ Chapter 9: Production Functions 323 0:35 1:75 ð Þ (9:61) Consequently, GA ¼ ¼ ¼ ¼ eq, lGl $ Gq $ 0:65 2:75 1:00 ð $ 0:65 2:75 1:50: $ $ eq, kGk Þ $ 0:60 The conclusion Solow reached then was that technology advanced at a rate of 1.5 percent per year from 1909 to 1949. More than half of the growth in real output could be attributed to technical change rather than to growth in the physical quantities of the factors of production. More recent evidence has tended to conﬁrm Solow’s conclusions about the relative importance of technical change. Considerable uncertainty remains, however, about the precise causes of such change. EXAMPLE 9.4 Technical Progress in the Cobb–Douglas Production Function The Cobb–Douglas production function provides an especially easy avenue for illustrating technical progress. Assuming constant returns to scale, such a production function with technical progress might be represented by q t A ð f Þ k, l ð ¼ Þ ¼ t A ð kal1 Þ a: $ (9:62) If we also assume that technical progress occurs at a constant exponential (y), then we can write A(t) Aeyt, and the production function becomes ¼ A particularly easy way to study the properties of this type of function over time is to use ‘‘logarithmic differentiation’’: a: Aeutkal1 $ q ¼ (9:63) @ ln q @t ¼ @ ln q @q u a % þ ¼ @q @t ¼ % @ ln k @q=@t q ¼ Gq ¼ ln A @ ½ þ ut þ a ln k @t 1 þ ð $ a Þ ln l ( 1 @t þ ð @ ln l Þ % @t ¼ a $ u aGk þ ð 1 þ $ a Gl: Þ (9:64) ¼ Thus, this derivation just repeats Equation 9.59 for the Cobb–Douglas case. Here the technical change factor is explicitly modeled, and the output elasticities are given by the values of the exponents in the Cobb–Douglas. The importance of technical progress can be illustrated numerically with this function. 4. Then, at 0.5, and that a ﬁrm uses an input mix of k l Suppose A t 10, y ¼ 0, output is 40( 0.03, a 10 Æ 40.5 Æ 40.5). After 20 years (t ¼ 20), the production function becomes ¼ ¼ ¼ k0:5l 0:5 10 1:82 % ð 10e0:03:20k0:5l 0:5 ¼ ¼ In year 20, the original input mix now yields q q ¼ and l would increase from 10 (q/l 40/4) to 18.2 ( case would have been true technical progress. 72.8 in year 0, but it would have taken a lot more inputs. For example, with k 72.8. Of course, one could also have produced 13.25 4, output is indeed 72.8 but much more capital is used. Output per unit of labor input 72.8/4) in either circumstance, but only the ﬁrst ¼ ¼ ¼ ¼ ¼ Þ 18:2k0:5l 0:5: ¼ (9:65) ¼ ¼ q Input-augmenting technical progress. It is tempting to attribute the increase in the average productivity of labor in this example to, say, improved worker skills, but that would be misleading in the Cobb–Douglas case. One might just as well have said that output per unit of 324 Part 4: Production and Supply capital increased from 10 to 18.2 over the 20 years and attribute this increase to improved machinery. A plausible approach to modeling improvements in labor and capital separately is to assume that the production function is a Þ where j represents the annual rate of improvement in capital input and e represents the annual rate of improvement in labor input. But because of the exponential nature of the Cobb–Douglas function, this would be indistinguishable from our original example: eetl ð 1 $ Þ (9:66) eutk A ð a, ¼ q q ¼ au Ae½ 1 $ a Þ e ( tkal1 þð a $ Aeutkal1 $ a, ¼ (9:67) aj where y (1 – a)e. Hence to study technical progress in individual inputs, it is necessary either to adopt a more complex way of measuring inputs that allows for improving quality or (what amounts to the same thing) to use a multi-input production function. ¼ þ QUERY: Actual studies of production using the Cobb–Douglas tend to ﬁnd a 0.3. Use this ﬁnding together with Equation 9.67 to discuss the relative importance of improving capital and labor quality to the overall rate of technical progress. * SUMMARY In this chapter we illustrated the ways in which economists conceptualize the production process of turning inputs into outputs. The fundamental tool is the production function, form—assumes that output per which—in its simplest period (q) is a simple function of capital and labor inputs during that period, q f (k, l). Using this starting point, we developed several basic results for the theory of production. ¼ • If all but one of the inputs are held constant, a relationship between the single-variable input and output can be derived. From this relationship, one can derive the marginal physical productivity (MP) of the input as the change in output resulting from a one-unit increase in the use of the input. The marginal physical productivity of an input is assumed to decrease as use of the input increases. • The entire production function can be illustrated by its isoquant map. The (negative of the) slope of an isoquant is termed the marginal rate of technical substitution (RTS) because it shows how one input can be substituted for another while holding output constant. The RTS is the ratio of the marginal physical productivities of the two inputs. • Isoquants are usually assumed to be convex—they obey the assumption of a diminishing RTS. This assumption cannot be derived exclusively from the assumption of diminishing marginal physical productivities. One must also be concerned with the effect of changes in one input on the marginal productivity of other inputs. • The returns to scale exhibited by a production function record how output responds to proportionate increases in all inputs. If output increases proportionately with input use, there are constant returns to scale. If there are greater than proportionate increases in output, there are increasing returns to scale, whereas if there are less than proportionate increases in output, there are decreasing returns to scale. • The elasticity of substitution (s) provides a measure of how easy it is to substitute one input for another in production. A high s implies nearly linear isoquants, whereas a low s implies that isoquants are nearly L-shaped. • Technical progress shifts the entir
e production function and its related isoquant map. Technical improveimproved, more ments may arise from the use of productive inputs or from better methods of economic organization. Chapter 9: Production Functions 325 PROBLEMS 9.1 Power Goat Lawn Company uses two sizes of mowers to cut lawns. The smaller mowers have a 22-inch deck. The larger ones combine two of the 22-inch decks in a single mower. For each size of mower, Power Goat has a different production function, given by the rows of the following table. Output per Hour (square feet) Capital Input (# of 2200 mowers) Labor Input Small mowers Large mowers 5000 8000 1 2 1 1 40,000 square feet isoquant for the ﬁrst production function. How much k and l would be used if these a. Graph the q ¼ factors were combined without waste? b. Answer part (a) for the second function. c. How much k and l would be used without waste if half of the 40,000-square-foot lawn were cut by the method of the ﬁrst production function and half by the method of the second? How much k and l would be used if one fourth of the lawn were cut by the ﬁrst method and three fourths by the second? What does it mean to speak of fractions of k and l? d. Based on your observations in part (c), draw a q ¼ 40,000 isoquant for the combined production functions. 9.2 Suppose the production function for widgets is given by kl q ¼ $ 0:8k2 $ 0:2l2, where q represents the annual quantity of widgets produced, k represents annual capital input, and l represents annual labor input. a. Suppose k 10; graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point? ¼ b. Again assuming that k c. Suppose capital inputs were increased to k d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale? 10, graph the MPl curve. At what level of labor input does MPl ¼ 20. How would your answers to parts (a) and (b) change? 0? ¼ ¼ 9.3 Sam Malone is considering renovating the bar stools at Cheers. The production function for new bar stools is given by 0:1k0:2l 0:8, q ¼ where q is the number of bar stools produced during the renovation week, k represents the number of hours of bar stool lathes used during the week, and l represents the number of worker hours employed during the period. Sam would like to provide 10 new bar stools, and he has allocated a budget of $10,000 for the project. a. Sam reasons that because bar stool lathes and skilled bar stool workers both cost the same amount ($50 per hour), he might as well hire these two inputs in equal amounts. If Sam proceeds in this way, how much of each input will he hire and how much will the renovation project cost? b. Norm (who knows something about bar stools) argues that once again Sam has forgotten his microeconomics. He asserts that Sam should choose quantities of inputs so that their marginal (not average) productivities are equal. If Sam opts for this plan instead, how much of each input will he hire and how much will the renovation project cost? c. On hearing that Norm’s plan will save money, Cliff argues that Sam should put the savings into more bar stools to provide seating for more of his USPS colleagues. How many more bar stools can Sam get for his budget if he follows Cliff ’s plan? d. Carla worries that Cliff ’s suggestion will just mean more work for her in delivering food to bar patrons. How might she convince Sam to stick to his original 10-bar stool plan? 326 Part 4: Production and Supply 9.4 Suppose that the production of crayons (q) is conducted at two locations and uses only labor as an input. The production function in location 1 is given by q1 ¼ a. If a single ﬁrm produces crayons in both locations, then it will obviously want to get as large an output as possible given the labor input it uses. How should it allocate labor between the locations to do so? Explain precisely the relationship between l1 and l2. 1 and in location 2 by q2 ¼ 50l 0:5 2 . 10l 0:5 b. Assuming that the ﬁrm operates in the efﬁcient manner described in part (a), how does total output (q) depend on the total amount of labor hired (l)? 9.5 As we have seen in many places, the general Cobb–Douglas production function for two inputs is given by q k, l f ð ¼ Þ ¼ Akalb, where 0 < a < 1 and 0 < b < 1. For this production function: a. Show that fk > 0, f1 > 0, fkk < 0, fll < 0, and fkl ¼ flk > 0. b. Show that eq, k ¼ a and eq, l ¼ c. In footnote 5, we deﬁned the scale elasticity as b. eq, t ¼ @f tk, tl ð @t Þ % f t tk, tl ð Þ , where the expression is to be evaluated at t the scale elasticity and the returns to scale of the production function agree (for more on this concept see Problem 9.9). 1. Show that, for this Cobb–Douglas function, eq, t ¼ b. Hence in this case ¼ þ a d. Show that this function is quasi-concave. e. Show that the function is concave for a þ b ) 1 but not concave for a b > 1. þ 9.6 Suppose we are given the constant returns-to-scale CES production function q kq ¼ ½ 1=q: lq ( þ a. Show that MPk ¼ b. Show that RTS ¼ c. Determine the output elasticities for k and l; and show that their sum equals 1. d. Prove that (q/k)1–r and MPl ¼ (k/l )1–r; use this to show that s (q/l )1–r. 1/(1 r). $ ¼ and hence that q l ¼ r @q @l # $ ln q l ’ ( r ln ¼ @q @l # $ : Note: The latter equality is useful in empirical work because we may approximate @q/@l by the competitively determined wage rate. Hence s can be estimated from a regression of ln(q/l) on ln w. 9.7 Consider a generalization of the production function in Example 9.3: q b0 þ ¼ b1 klp b2k þ þ b3l, ﬃﬃﬃﬃ Chapter 9: Production Functions 327 where 0 bi ) ) 1, i ¼ 0, . . . , 3: a. If this function is to exhibit constant returns to scale, what restrictions should be placed on the parameters b0, . . . , b3? b. Show that, in the constant returns-to-scale case, this function exhibits diminishing marginal productivities and that the marginal productivity functions are homogeneous of degree 0. c. Calculate s in this case. Although s is not in general constant, for what values of the b’s does s 0, 1, or ? 1 ¼ 9.8 Show that Euler’s theorem implies that, for a constant returns-to-scale production function [q f (k, l )], ¼ q k fk % þ l: fl % ¼ Use this result to show that, for such a production function, if MPl > APl then MPk must be negative. What does this imply about where production must take place? Can a ﬁrm ever produce at a point where APl is increasing? Analytical Problems 9.9 Local returns to scale A local measure of the returns to scale incorporated in a production function is given by the scale elasticity eq, t ¼ evaluated at t a. Show that if the production function exhibits constant returns to scale, then eq, t ¼ b. We can deﬁne the output elasticities of the inputs k and l as ¼ 1. 1. @f (tk, tl )/@t Æ t/q @f @f eq, k ¼ eq, l ¼ eq, l. k, l ð @k k, l ð @l Þ % Þ % k q l q , : Show that eq, t ¼ eq, k þ c. A function that exhibits variable scale elasticity is Show that, for this function, eq,t > 1 for q < 0.5 and that eq, t < 1 for q > 0.5. d. Explain your results from part (c) intuitively. Hint: Does q have an upper bound for this production function? q 1 þ ¼ ð k$ 1 1l$ 1: $ Þ 9.10 Returns to scale and substitution Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases. a. In footnote 6 we pointed out that, in the constant returns-to-scale case, the elasticity of substitution for a two-input production function is given by r ¼ f : fk fl fkl % Suppose now that we deﬁne the homothetic production function F as F k, l ð Þ ¼ ½ g, f k, l ð Þ( where f (k, l) is a constant returns-to-scale production function and g is a positive exponent. Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function f. b. Show how this result can be applied to both the Cobb–Douglas and CES production functions. i xifi ¼ kf , and P 328 Part 4: Production and Supply 9.11 More on Euler’s theorem Suppose that a production function f(x1, x2, . . ., xn) is homogeneous of degree k. Euler’s theorem shows that this fact can be used to show that the partial derivatives of f are homogeneous of degree k – 1. a. Prove that 1 ¼ b. In the case of n n i P ¼ P n 1 xixj fij ¼ j ¼ 2 and k ¼ 1 k ð k 1, what kind of restrictions does the result of part (a) impose on the second-order partial f . Þ $ derivative f12? How do your conclusions change when k > 1 or k < 1? c. How would the results of part (b) be generalized to a production function with any number of inputs? d. What are the implications of this problem for the parameters of the multivariable Cobb–Douglas production function f (x1, x2, . . . , xn) ¼ Q 1x ai i n i ¼ for ai + 0? SUGGESTIONS FOR FURTHER READING Clark, J. M. ‘‘Diminishing Returns.’’ In Encyclopaedia of the Social Sciences, vol. 5. New York: Crowell-Collier and Macmillan, 1931, pp. 144–46. Lucid discussion of the historical development of the diminishing returns concept. Douglas, P. H. ‘‘Are There Laws of Production?’’ American Economic Review 38 (March 1948): 1–41. A nice methodological analysis of the uses and misuses of production functions. Ferguson, C. E. The Neoclassical Theory of Production and Distribution. New York: Cambridge University Press, 1969. A thorough discussion of production function theory (as of 1970). Good use of three-dimensional graphs. Fuss, M., and D. McFadden. Production Economics: A Dual Approach to Theory and Application. Amsterdam: NorthHolland, 1980. An approach with a heavy emphasis on the use of duality. Mas-Collell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Chapter 5 provides a sophisticated, if somewhat spare, review of production theory. The use of the proﬁt function
(see Chapter 11) is sophisticated and illuminating. Shephard, R. W. Theory of Cost and Production Functions. Princeton, NJ: Princeton University Press, 1978. Extended analysis of the dual relationship between production and cost functions. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Thorough analysis of the duality between production functions and cost curves. Provides a proof that the elasticity of substitution can be derived as shown in footnote 6 of this chapter. Stigler, G. J. ‘‘The Division of Labor Is Limited by the Extent of the Market.’’ Journal of Political Economy 59 (June 1951): 185–93. Careful tracing of the evolution of Smith’s ideas about economies of scale. MANY-INPUT PRODUCTION FUNCTIONS EXTENSIONS Most of the production functions illustrated in Chapter 9 can be easily generalized to many-input cases. Here we show this for the Cobb–Douglas and CES cases and then examine two ﬂexible forms that such production functions might take. In all these examples, the a’s are non-negative parameters and the n inputs are represented by x1, . . ., xn. 1. Because this parameter is so constrained in and sij, ¼ the Cobb–Douglas function, the function is generally not used in econometric analyses of microeconomic data on ﬁrms. However, the function has a variety of general uses in macroeconomics, as the next example illustrates. E9.1 Cobb–Douglas The many-input Cobb–Douglas production function is given by q ¼ xai i : n 1 i Y ¼ a. This function exhibits constant returns to scale if n 1 i X ¼ ai ¼ 1: (i) (ii) b. In the constant-returns-to-scale Cobb–Douglas function, ai is the elasticity of q with respect to input xi. Because 0 ai < 1, each input exhibits diminishing marginal productivity. c. Any degree of increasing returns to scale can be incorpo- ) rated into this function, depending on e ¼ n ai: 1 i X ¼ (iii) d. The elasticity of substitution between any two inputs in this production function is 1. This can be shown by using the deﬁnition given in footnote 7 of this chapter: rij ¼ xi=xjÞ @ ln ð fj=fiÞ @ ln ð : Here Hence fj fi ¼ aj aix j aixai i 1 $ 1 $ j xai i aj j ¼ i x aj ai % xi xj : i 6¼ Q j 6¼ Q ln fj fi# $ ln ¼ aj ai# $ ln þ xi xj# $ The Solow growth model The many-input Cobb–Douglas production function is a primary feature of many models of economic growth. For example, Solow’s (1956) pioneering model of equilibrium growth can be most easily derived using a two-input constantreturns-to-scale Cobb–Douglas function of the form Akal1 $ a, q ¼ (iv) where A is a technical change factor that can be represented by exponential growth of the form Dividing both sides of Equation iv by l yields eat: A ¼ where eat ^ ka, ^q ¼ q^ ¼ ^ q=l and k ¼ k=l (v) (vi) Solow shows that economies will evolve toward an equilibrium value of ^k (the capital–labor ratio). Hence cross-country differences in growth rates can be accounted for only by differences in the technical change factor, a. Two features of Equation vi argue for including more inputs in the Solow model. First, the equation as it stands is incapable of explaining the large differences in per capita output (^q) that are observed around the world. Assuming 0.3, say (a ﬁgure consistent with many empirical studies), a it would take cross-country differences in k/l of as much as 4,000,000 to 1 to explain the 100-to-1 differences in per capita income observed—a clearly unreasonable magnitude. By introducing additional inputs, such as human capital, these differences become more explainable. ¼ A second shortcoming of the simple Cobb–Douglas formulation of the Solow model is that it offers no explanation of the technical change parameter, a—its value is determined ‘‘exogenously.’’ By adding additional factors, it becomes easier to understand how the parameter a may respond to economic 330 Part 4: Production and Supply incentives. This is the key insight of literature on ‘‘endogenous’’ growth theory (for a summary, see Romer, 1996). Then the ﬁnal production function might Douglas form: take a Cobb– E9.2 CES The many-input constant elasticity of substitution (CES) production function is given by q ¼ h X e=q aixq i i , q 1: ) (vii) a. By substituting txi for each output, it is easy to show that this function exhibits constant returns to scale for e 1. For e > 1, the function exhibits increasing returns to scale. b. The production function exhibits diminishing marginal ¼ productivities for each input because r 1. ) c. As in the two-input case, the elasticity of substitution here is given by r ¼ 1 1 $ , q (viii) and this elasticity applies to substitution between any two of the inputs. Checking the Cobb–Douglas in the Soviet Union One way in which the multi-input CES function is used is to determine whether the estimated substitution parameter (r) is consistent with the value implied by the Cobb– 1). For example, in a study of ﬁve Douglas (r major industries in the former Soviet Union, E. Bairam (1991) ﬁnds that the Cobb–Douglas provides a relatively good explanation of changes in output in most major manufacturing sectors. Only for food processing does a lower value for s seem appropriate. 0, s ¼ ¼ The next three examples illustrate ﬂexible-form production functions that may approximate any general function of n inputs. In the Chapter 10 extensions, we examine the cost function analogs to some of these functions, which are more widely used than the production functions themselves. E9.3 Nested production functions In some applications, Cobb–Douglas and CES production functions are combined into a ‘‘nested’’ single function. To accomplish this, the original n primary inputs are categorized into, say, m general classes of inputs. The speciﬁc inputs in each of these categories are then aggregated into a single composite input, and the ﬁnal production function is a function of these m composites. For example, assume there are three primary inputs, x1, x2, x3. Suppose, however, that x1 and x2 are relatively closely related in their use by ﬁrms (e.g., capital and energy), whereas the third input (labor) is relatively distinct. Then one might want to use a CES aggregator function to construct a composite input for capital services of the form x4 ¼ ½ gxq 1 þ (1 1=q: g)xq 2 ( $ (ixx) This structure allows the elasticity of substitution between x1 and x2 to take on any value [s r)] but constrains the elasticity of substitution between x3 and x4 to be one. A variety of other options are available depending on how precisely the embedded functions are speciﬁed. 1/(1 $ ¼ The dynamics of capital/energy substitutability Nested production functions have been widely used in studies that seek to measure the precise nature of the substitutability between capital and energy inputs. For example, Atkeson and Kehoe (1999) use a model rather close to the one speciﬁed in Equations ix and x to try to reconcile two facts about the way in which energy prices affect the economy: (1) Over time, use of energy in production seems rather unresponsive to price (at least in the short run); and (2) across countries, energy prices seem to have a large inﬂuence over how much energy is used. By using a capital service equation of the form given in Equation ix with a low 2.3)—along with a Cobb– degree of substitutability (r Douglas production function that combines labor with capital services—they are able to replicate the facts about energy prices fairly well. They conclude, however, that this model implies a much more negative effect of higher energy prices on economic growth than seems actually to have been the case. Hence they ultimately opt for a more complex way of modeling production that stresses differences in energy use among capital investments made at different dates. ¼ $ E9.4 Generalized Leontief aij xixjp , ﬃﬃﬃﬃﬃﬃﬃ aji. where aij ¼ a. The function considered in Problem 9.7 is a simple case of this function for the case n 3, the function ¼ would have linear terms in the three inputs along with three radical terms representing all possible cross-products of the inputs. 2. For n ¼ b. The function exhibits constant returns to scale, as can be shown by using txi. Increasing returns to scale can be incorporated into the function by using the transformation qe, q0 ¼ e > 1: c. Because each input appears both linearly and under the radical, the function exhibits diminishing marginal productivities to all inputs. d. The restriction aij ¼ aji is used to ensure symmetry of the second-order partial derivatives. E9.5 Translog n ln q a0 þ ¼ 1 i X ¼ ai ln xi þ 0: aij ln xi ln xj, aij ¼ aji: a. Note that the Cobb–Douglas function is a special case of aij ¼ b. As for the Cobb–Douglas, this function may assume any this function where a0 ¼ 0 for all i, j. degree of returns to scale. If n 1 i X ¼ ai ¼ 1 and n 1 j X ¼ aij ¼ 0 for all i, then this function exhibits constant returns to scale. The proof requires some care in dealing with the double summation. c. Again, the condition aij ¼ aji is required to ensure equality of the cross-partial derivatives. Immigration Because the translog production function incorporates a large number of substitution possibilities among various inputs, it has been widely used to study the ways in which newly arrived workers may substitute for existing workers. Of particular interest is the way in which the skill level of immigrants may lead to differing reactions in the demand for skilled and Chapter 9: Production Functions 331 unskilled workers in the domestic economy. Studies of the United States and many other countries (e.g., Canada, Germany, and France) have suggested that the overall size of such effects is modest, especially given relatively small immigration ﬂows. But there is some evidence that unskilled immigrant workers may act as substitutes for unskilled domestic workers but as complements to skilled domestic workers. Hence increased immigration ﬂows may exacerbate trends toward increasing wage differ
entials. For a summary, see Borjas (1994). References Atkeson, Andrew, and Patrick J. Kehoe. ‘‘Models of Energy Use: Putty-Putty versus Putty-Clay.’’ American Economic Review (September 1999): 1028–43. Bairam, Erkin. ‘‘Elasticity of Substitution, Technical Progress and Returns to Scale in Branches of Soviet Industry: A New CES Production Function Approach.’’ Journal of Applied Economics (January–March 1991): 91–96. Borjas, G. J. ‘‘The Economics of Immigration.’’ Journal of Economic Literature (December 1994): 1667–717. Romer, David. Advanced Macroeconomics. New York: McGraw- Hill, 1996. Solow, R. M. ‘‘A Contribution to the Theory of Economic (February Growth.’’ Quarterly Journal of Economics 1956): 65–94. This page intentionally left blank C H A P T E R TEN Cost Functions In this chapter we illustrate the costs that a ﬁrm incurs when it produces output. In Chapter 11, we will pursue this topic further by showing how ﬁrms make proﬁt-maximizing input and output decisions. Deﬁnitions of Costs Before we can discuss the theory of costs, some difﬁculties about the proper deﬁnition of ‘‘costs’’ must be cleared up. Speciﬁcally, we must distinguish between (1) accounting cost and (2) economic cost. The accountant’s view of cost stresses out-of-pocket expenses, historical costs, depreciation, and other bookkeeping entries. The economist’s deﬁnition of cost (which in obvious ways draws on the fundamental opportunity-cost notion) is that the cost of any input is given by the size of the payment necessary to keep the resource in its present employment. Alternatively, the economic cost of using an input is what that input would be paid in its next best use. One way to distinguish between these two views is to consider how the costs of various inputs (labor, capital, and entrepreneurial services) are deﬁned under each system. Labor costs Economists and accountants regard labor costs in much the same way. To accountants, expenditures on labor are current expenses and hence costs of production. For economists, labor is an explicit cost. Labor services (labor-hours) are contracted at some hourly wage rate (w), and it is usually assumed that this is also what the labor services would earn in their best alternative employment. The hourly wage, of course, includes costs of fringe beneﬁts provided to employees. Capital costs In the case of capital services (machine-hours), the two concepts of cost differ. In calculating capital costs, accountants use the historical price of the particular machine under investigation and apply some more-or-less arbitrary depreciation rule to determine how much of that machine’s original price to charge to current costs. Economists regard the historical price of a machine as a ‘‘sunk cost,’’ which is irrelevant to output decisions. They instead regard the implicit cost of the machine to be what someone else would be willing to pay for its use. Thus, the cost of one machine-hour is the rental rate for that machine in its best alternative use. By continuing to use the machine itself, the ﬁrm is 333 334 Part 4: Production and Supply implicitly forgoing what someone else would be willing to pay to use it. This rental rate for one machine-hour will be denoted by v.1 Suppose a company buys a computer for $2,000. An accountant applying a ‘‘straightline’’ depreciation method over ﬁve years would regard the computer as having a cost of $400 a year. An economist would look at the market value of the computer. The availability of much faster computers in subsequent years can cause the second-hand price of the original computer to decrease precipitously. If the second-hand price decreases all the way to, for example, $200 after the ﬁrst year, the economic cost will be related to this $200; the original $2,000 price will no longer be relevant. (All these yearly costs can easily be converted into computer-hour costs, of course.) The distinction between accounting and economic costs of capital largely disappears if the company rents it at a price of v each period rather than purchasing. Then v reﬂects a current company expenditure that shows up directly as an accounting cost; it also reﬂects the market value of one period’s use of the capital and thus is an opportunity/economic cost. Costs of entrepreneurial services The owner of a ﬁrm is a residual claimant who is entitled to whatever extra revenues or losses are left after paying other input costs. To an accountant, these would be called proﬁts (which might be either positive or negative). Economists, however, ask whether owners (or entrepreneurs) also encounter opportunity costs by working at a particular ﬁrm or devoting some of their funds to its operation. If so, these services should be considered an input, and some cost should be imputed to them. For example, suppose a highly skilled computer programmer starts a software ﬁrm with the idea of keeping any (accounting) proﬁts that might be generated. The programmer’s time is clearly an input to the ﬁrm, and a cost should be attributed to it. Perhaps the wage that the programmer might command if he or she worked for someone else could be used for that purpose. Hence some part of the accounting proﬁts generated by the ﬁrm would be categorized as entrepreneurial costs by economists. Economic proﬁts would be smaller than accounting proﬁts and might be negative if the programmer’s opportunity costs exceeded the accounting proﬁts being earned by the business. Similar arguments apply to the capital that an entrepreneur provides to the ﬁrm. Economic costs In this book, not surprisingly, we use economists’ deﬁnition of cost Economic cost. The economic cost of any input is the payment required to keep that input in its present employment. Equivalently, the economic cost of an input is the remuneration the input would receive in its best alternative employment. Use of this deﬁnition is not meant to imply that accountants’ concepts are irrelevant to economic behavior. Indeed, accounting procedures are integrally important to any manager’s decision-making process because they can greatly affect the rate of taxation to be applied against proﬁts. Accounting data are also readily available, whereas data on economic costs must often be developed separately. Economists’ deﬁnitions, however, do have 1Sometimes the symbol r is chosen to represent the rental rate on capital. Because this variable is often confused with the related but distinct concept of the market interest rate, an alternative symbol was chosen here. The exact relationship between v and the interest rate is examined in Chapter 17. Chapter 10: Cost Functions 335 the desirable features of being broadly applicable to all ﬁrms and of forming a conceptually consistent system. Therefore, they are best suited for a general theoretical analysis. Simplifying assumptions As a start, we will make two simpliﬁcations about the inputs a ﬁrm uses. First, we assume that there are only two inputs: homogeneous labor (l, measured in labor-hours) and homogeneous capital (k, measured in machine-hours). Entrepreneurial costs are included in capital costs. That is, we assume that the primary opportunity costs faced by a ﬁrm’s owner are those associated with the capital that the owner provides. Second, we assume that inputs are hired in perfectly competitive markets. Firms can buy (or sell) all the labor or capital services they want at the prevailing rental rates (w and v). In graphic terms, the supply curve for these resources is horizontal at the prevailing factor prices. Both w and v are treated as ‘‘parameters’’ in the ﬁrm’s decisions; there is nothing the ﬁrm can do to affect them. These conditions will be relaxed in later chapters (notably Chapter 16), but for the moment the price-taker assumption is a convenient and useful one to make. Therefore, with these simpliﬁcations, total cost C for the ﬁrm during the period is given by where, as before, l and k represent input usage during the period. total cost C ¼ ¼ wl þ vk (10:1) Relationship between proﬁt maximization and cost minimization Let’s look ahead to the next chapter on proﬁt maximization and compare the analysis here with the analysis in that chapter. We will deﬁne economic proﬁts (p) as the difference between the ﬁrm’s total revenues (R) and its total costs (C). Suppose the ﬁrm takes the market price ( p) for its total output (q) as given and that its production function is q f (k, l). Then its proﬁt can be written ¼ p R C pq wl vk pf k # wl # vk: (10:2) Equation 10.2 shows that the economic proﬁts obtained by this ﬁrm are a function of the amount of capital and labor employed. If, as we will assume in many places in this book, this ﬁrm seeks maximum proﬁts, then we might study its behavior by examining how k and l are chosen to maximize Equation 10.2. This would, in turn, lead to a theory of supply and to a theory of the ‘‘derived demand’’ for capital and labor inputs. In the next chapter we will take up those subjects in detail. Here, however, we wish to develop a theory of costs that is somewhat more general, applying not only to ﬁrms that are price-takers on their output markets (perfect competitors) but also to those whose output choice affects the market price (monopolies and oligopolies). The more general theory will even apply to nonproﬁts (as long as they are interested in operating efﬁciently). The other advantage of looking at cost minimization separately from proﬁt maximization is that it is simpler to analyze this small ‘‘piece’’ in isolation and only later add the insights obtained into the overall ‘‘puzzle’’ of the ﬁrm’s operations. The conditions derived for cost-minimizing input choices in this chapter will emerge again as a ‘‘by-product’’ of the analysis of the maximization of proﬁts as speciﬁed in Equation 10.2. Hence we begin the study of costs by ﬁnessing, for the moment, a discussion of output choice. That is, we assume that for some reason the ﬁrm has decided to produce a particular output level (say
, q0). The ﬁrm will of course earn some revenue R from this output choice, but we will ignore revenue for now. We will focus solely on the question of how the ﬁrm can produce q0 at minimal cost. 336 Part 4: Production and Supply Cost-Minimizing Input Choices Mathematically, this is a constrained minimization problem. But before proceeding with a rigorous solution, it is useful to state the result to be derived with an intuitive argument. To minimize the cost of producing a given level of output, a ﬁrm should choose that point on the q0 isoquant at which the rate of technical substitution of l for k is equal to the ratio w/v: It should equate the rate at which k can be traded for l in production to the rate at which they can be traded in the marketplace. Suppose that this were not true. 10, In particular, suppose that the ﬁrm were producing output level q0 using k ¼ and assume that the RTS were 2 at this point. Assume also that w $1, and hence that w/v 1 (which is unequal to 2). At this input combination, the cost of producing q0 is $20. It is easy to show this is not the minimal input cost. For example, q0 11; we can give up two units of k and keep can also be produced using k output constant at q0 by adding one unit of l. But at this input combination, the cost of producing q0 is $19, and hence the initial input combination was not optimal. A contradiction similar to this one can be demonstrated whenever the RTS and the ratio of the input costs differ. ¼ $1, v 8 and l 10, l ¼ ¼ ¼ ¼ ¼ Mathematical analysis Mathematically, we seek to minimize total costs given q Lagrangian f(k, l ) ¼ ¼ q0. Setting up the , Þ’ the ﬁrst-order conditions for a constrained minimum are q0 # k ½ k, l wl vk ¼ þ þ ð + f @f @l ¼ @f @k ¼ 0, 0, @+ @l ¼ @+ @k ¼ @+ @k ¼ w # k k v # q0 # k, l f ð Þ ¼ 0, or, dividing the ﬁrst two equations, w v ¼ @f =@l @f =@k ¼ RTS of l for k Þ : ð (10:3) (10:4) (10:5) This says that the cost-minimizing ﬁrm should equate the RTS for the two inputs to the ratio of their prices. Further interpretations These ﬁrst-order conditions for minimal costs can be manipulated in several different ways to yield interesting results. For example, cross-multiplying Equation 10.5 gives f k v ¼ f l w : (10:6) That is, for costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs. If increasing one input promised to increase output by a greater amount per dollar spent than did another input, costs would not be minimal—the ﬁrm should hire more of the input that promises a bigger ‘‘bang per buck’’ and less of the more costly (in terms of productivity) input. Any input that cannot meet the common beneﬁt–cost ratio deﬁned in Equation 10.6 should not be hired at all. Chapter 10: Cost Functions 337 Equation 10.6 can, of course, also be derived from Equation 10.4, but it is more in- structive to derive its inverse: (10:7) This equation reports the extra cost of obtaining an extra unit of output by hiring either added labor or added capital input. Because of cost minimization, this marginal cost is the same no matter which input is hired. This common marginal cost is also measured by the Lagrange multiplier from the cost-minimization problem. As is the case for all constrained optimization problems, here the Lagrange multiplier shows how much in extra costs would be incurred by increasing the output constraint slightly. Because marginal cost plays an important role in a ﬁrm’s supply decisions, we will return to this feature of cost minimization frequently. Graphical analysis Cost minimization is shown graphically in Figure 10.1. Given the output isoquant q0, we wish to ﬁnd the least costly point on the isoquant. Lines showing equal cost are parallel straight lines with slopes w/v. Three lines of equal total cost are shown in Figure 10.1; C1 < C2 < C3. It is clear from the ﬁgure that the minimum total cost for producing q0 is given by C1, where the total cost curve is just tangent to the isoquant. The associated inputs are l c and k c, where the superscripts emphasize that these input levels are a solution to a cost-minimization problem. This combination will be a true minimum if the isoquant is convex (if the RTS diminishes for decreases in k/l ). The mathematical and graphic analyses arrive at the same conclusion, as follows. # FIGURE 10.1 Minimization of Costs Given q q0 ¼ A ﬁrm is assumed to choose k and l to minimize total costs. The condition for this minimization is that the rate at which k and l can be traded technically (while keeping q q0) should be equal to the rate at which these inputs can be traded in the market. In other words, the RTS (of l for k) should be set equal to the price ratio w/v. This tangency is shown in the ﬁgure; costs are minimized at C1 by choosing inputs k c and l c. ¼ k per period C1 C2 kc lc C3 q0 l per period 338 Part 4: Production and Supply Cost minimization. To minimize the cost of any given level of output (q0), the ﬁrm should produce at that point on the q0 isoquant for which the RTS (of l for k) is equal to the ratio of the inputs’ rental prices (w/v). Contingent demand for inputs Figure 10.1 exhibits the formal similarity between the ﬁrm’s cost-minimization problem and the individual’s expenditure-minimization problem studied in Chapter 4 (see Figure 4.6). In both problems, the economic actor seeks to achieve his or her target (output or utility) at minimal cost. In Chapter 5 we showed how this process is used to construct a theory of compensated demand for a good. In the present case, cost minimization leads to a demand for capital and labor input that is contingent on the level of output being produced. Therefore, this is not the complete story of a ﬁrm’s demand for the inputs it uses because it does not address the issue of output choice. But studying the contingent demand for inputs provides an important building block for analyzing the ﬁrm’s overall demand for inputs, and we will take up this topic in more detail later in this chapter. The ﬁrm’s expansion path A ﬁrm can follow the cost-minimization process for each level of output: For each q, it ﬁnds the input choice that minimizes the cost of producing it. If input costs (w and v) remain constant for all amounts the ﬁrm may demand, we can easily trace this locus of cost-minimizing choices. This procedure is shown in Figure 10.2. The curve 0E records the cost-minimizing tangencies for successively higher levels of output. For example, the minimum cost for producing output level q1 is given by C1, and inputs k1 and l1 are used. Other tangencies in the ﬁgure can be interpreted in a similar way. The locus of these FIGURE 10.2 The Firm’s Expansion Path The ﬁrm’s expansion path is the locus of cost-minimizing tangencies. Assuming ﬁxed input prices, the curve shows how inputs increase as output increases. k per period E k1 q3 q2 q1 C1 C2 C3 0 l1 l per period Chapter 10: Cost Functions 339 tangencies is called the ﬁrm’s expansion path because it records how input expands as output expands while holding the prices of the inputs constant. As Figure 10.2 shows, the expansion path need not be a straight line. The use of some inputs may increase faster than others as output expands. Which inputs expand more rapidly will depend on the shape of the production isoquants. Because cost minimization requires that the RTS always be set equal to the ratio w/v, and because the w/v ratio is assumed to be constant, the shape of the expansion path will be determined by where a particular RTS occurs on successively higher isoquants. If the production function exhibits constant returns to scale (or, more generally, if it is homothetic), then the expansion path will be a straight line because in that case the RTS depends only on the ratio of k to l. That ratio would be constant along such a linear expansion path. It would seem reasonable to assume that the expansion path will be positively sloped; that is, successively higher output levels will require more of both inputs. This need not be the case, however, as Figure 10.3 illustrates. Increases of output beyond q2 cause the quantity of labor used to decrease. In this range, labor would be said to be an inferior input. The occurrence of inferior inputs is then a theoretical possibility that may happen, even when isoquants have their usual convex shape. Much theoretical discussion has centered on the analysis of factor inferiority. Whether inferiority is likely to occur in real-world production functions is a difﬁcult empirical question to answer. It seems unlikely that such comprehensive magnitudes as ‘‘capital’’ and ‘‘labor’’ could be inferior, but a ﬁner classiﬁcation of inputs may bring inferiority to light. For example, the use of shovels may decrease as production of building foundations (and the use of backhoes) increases. In this book we shall not be particularly concerned with the analytical issues raised by this possibility, although complications raised by inferior inputs will be mentioned in a few places. FIGURE 10.3 Input Inferiority With this particular set of isoquants, labor is an inferior input because less l is chosen as output expands beyond q2. k per period E q4 q3 q2 q1 0 l per period 340 Part 4: Production and Supply EXAMPLE 10.1 Cost Minimization The cost-minimization process can be readily illustrated with two of the production functions we encountered in the last chapter. 1. Cobb–Douglas: q f (k, l ) kal b. For this case, the relevant Lagrangian expression for minimizing the cost of producing, say, q0 is ¼ ¼ q0 # ð and the ﬁrst-order conditions for a minimum are wl vk þ þ ¼ k + kal b , Þ @+ @k ¼ @+ @l ¼ @+ @k ¼ kaka 1l b # v # 0, ¼ w kbkal b # 1 0, ¼ kal b 0: ¼ # q0 # Dividing the second of these by the ﬁrst yields bkalb aka w v ¼ 1 # 1lb ¼ # b a ( k l , (10:8) (10:9) (10:10) which again shows that costs are minimized when the ratio of the inputs’ prices is equal to the RTS. Because the Cobb–Douglas function is homothetic, the RTS depends only on t
he ratio of the two inputs. If the ratio of input costs does not change, the ﬁrms will use the same input ratio no matter how much it produces—that is, the expansion path will be a straight line through the origin. b ¼ 0.5, w As a numerical example, suppose a 3, and that the ﬁrm wishes ¼ to produce q0 ¼ 40. The ﬁrst-order condition for a minimum requires that k 4l. Inserting that into the production function (the ﬁnal requirement in Equation 10.9), we have q0 ¼ k0.5l0.5 40 80, and ¼ total costs are given by vk 480. That this is a true cost minimum 12(20) is suggested by looking at a few other input combinations that also are capable of producing 40 units of output: 2l. Thus, the cost-minimizing input combination is l 20 and k 3(80) 12, v wl 40, l 10, l 160, l ¼ ¼ ¼ 40, C 160, C 10, C ¼ ¼ ¼ 600, 2,220, 600: (10:11) Any other input combination able to produce 40 units of output will also cost more than 480. Cost minimization is also suggested by considering marginal productivities. At the optimal point MPk ¼ MPl ¼ f k ¼ f l ¼ 0:5k# 0:5k0:5l # 0:5l 0:5 0:5 ¼ 0:5 0:5 20=80 ð 80=20 ð 0:5 Þ 0:5 Þ 0:25, ¼ 1:0; (10:12) ¼ hence at the margin, labor is four times as productive as capital, and this extra productivity precisely compensates for the higher unit price of labor input. ¼ 2. CES: q f (k, l ) (kr ¼ þ ¼ l r)g/r. Again we set up the Lagrangian expression + vk wl k ½ kq q0 # ð þ l q g=q Þ , ’ þ þ ¼ and the ﬁrst-order conditions for a minimum are @+ @k ¼ @+ @l ¼ @+ @g ¼ v g=q k ð # kq Þð l q ð Þ þ g q # ==q k # ð kq q0 # ð þ kq Þð l q ð Þ þ g # =: 0, 0, ¼ ¼ (10:13) (10:14) Chapter 10: Cost Functions 341 Dividing the ﬁrst two of these equations causes a lot of this mass of symbols to drop out, leaving =r , or 10:15) ¼ 1). With the Cobb where s 1/(1 – r) is the elasticity of substitution. Because the CES function is also homothetic, the cost-minimizing input ratio is independent of the absolute level of production. Douglas result (when The result in Equation 10.15 is a simple generalization of the Cobb Douglas, the cost-minimizing capital–labor ratio changes directly s in proportion to changes in the ratio of wages to capital rental rates. In cases with greater substitutability (s > 1), changes in the ratio of wages to rental rates cause a greater than proportional increase in the cost-minimizing capital–labor ratio. With less substitutability (s < 1), the response is proportionally smaller. # ¼ # QUERY: In the Cobb–Douglas numerical example with w/v minimizing input ratio for producing 40 units of output was k/l value change for s would total costs be? 4, we found that the cost¼ 4. How would this 80/20 ¼ 0.5? What actual input combinations would be used? What 2 or s ¼ ¼ ¼ Cost Functions We are now in a position to examine the ﬁrm’s overall cost structure. To do so, it will be convenient to use the expansion path solutions to derive the total cost function Total cost function. The total cost function shows that, for any set of input costs and for any output level, the minimum total cost incurred by the ﬁrm is C C v, w, q ð : Þ ¼ (10:16) Figure 10.2 makes clear that total costs increase as output, q, increases. We will begin by analyzing this relationship between total cost and output while holding input prices ﬁxed. Then we will consider how a change in an input price shifts the expansion path and its related cost functions. Average and marginal cost functions Although the total cost function provides complete information about the output–cost relationship, it is often convenient to analyze costs on a per-unit of output basis because that approach corresponds more closely to the analysis of demand, which focused on the price per unit of a commodity. Two different unit cost measures are widely used in economics: (1) average cost, which is the cost per unit of output; and (2) marginal cost, which is the cost of one more unit of output Average and marginal cost functions. The average cost function (AC) is found by computing total costs per unit of output: average cost AC v, w, q ð Þ ¼ ¼ C v, w, q ð q Þ : (10:17) 342 Part 4: Production and Supply The marginal cost function (MC ) is found by computing the change in total costs for a change in output produced: marginal cost MC v, w, q ð Þ ¼ ¼ @C v, w, q ð @q Þ : (10:18) Notice that in these deﬁnitions, average and marginal costs depend both on the level of output being produced and on the prices of inputs. In many places throughout this book, we will graph simple two-dimensional relationships between costs and output. As the definitions make clear, all such graphs are drawn on the assumption that the prices of inputs remain constant and that technology does not change. If input prices change or if technology advances, cost curves generally will shift to new positions. Later in this chapter, we will explore the likely direction and size of such shifts when we study the entire cost function in detail. Graphical analysis of total costs Figures 10.4a and 10.5a illustrate two possible shapes for the relationship between total cost and the level of the ﬁrm’s output. In Figure 10.4a, total cost is simply proportional to output. Such a situation would arise if the underlying production function exhibits constant returns to scale. In that case, suppose k1 units of capital input and l1 units of labor input are required to produce one unit of output. Then C q ð ¼ 1 Þ ¼ v k1 þ w l1: (10:19) To produce m units of output requires mk1 units of capital and ml1 units of labor because of the constant returns-to-scale assumption.2 Hence wml1 ¼ , 1 Þ ¼ vmk1 þ q C m ð ( vk1 þ ð Þ ¼ ¼ wl1Þ (10:20) q ð m m ¼ C and the proportionality between output and cost is established. The situation in Figure 10.5a is more complicated. There it is assumed that initially the total cost curve is concave; although initially costs increase rapidly for increases in output, that rate of increase slows as output expands into the midrange of output. Beyond this middle range, however, the total cost curve becomes convex, and costs begin to increase progressively more rapidly. One possible reason for such a shape for the total cost curve is that there is some third factor of production (say, the services of an entrepreneur) that is ﬁxed as capital and labor usage expands. In this case, the initial concave section of the curve might be explained by the increasingly optimal usage of the entrepreneur’s services—he or she needs a moderate level of production to use his or her skills fully. Beyond the point of inﬂection, however, the entrepreneur becomes overworked in attempting to coordinate production, and diminishing returns set in. Hence total costs increase rapidly. A variety of other explanations have been offered for the cubic-type total cost curve in Figure 10.5a, but we will not examine them here. Ultimately, the shape of the total cost curve is an empirical question that can be determined only by examining realworld data. In the Extensions to this chapter, we illustrate some of the literature on cost functions. 2The input combination (ml1, mk1) minimizes the cost of producing m units of output because the ratio of the inputs is still k1/l1 and the RTS for a constant returns-to-scale production function depends only on that ratio. FIGURE 10.4 Total, Average, and Marginal Cost Curves for the Constant Returns-to-Scale Case Chapter 10: Cost Functions 343 In (a) total costs are proportional to output level. Average and marginal costs, as shown in (b), are equal and constant for all output levels. Total costs (a) Average and marginal costs (b) C Output per period AC = MC Output per period Graphical analysis of average and marginal costs Information from the total cost curves can be used to construct the average and marginal cost curves shown in Figures 10.4b and 10.5b. For the constant returns-to-scale case (Figure 10.4), this is simple. Because total costs are proportional to output, average and marginal costs are constant and equal for all levels of output.3 These costs are shown by the horizontal line AC MC in Figure 10.4b. For the cubic total cost curve case (Figure 10.5), computation of the average and marginal cost curves requires some geometric intuition. As the deﬁnition in Equation 10.18 makes clear, marginal cost is simply the slope of the total cost curve. Hence because of ¼ 3Mathematically, because C ¼ aq (where a is the cost of one unit of output), AC C q ¼ a ¼ ¼ @C @q ¼ MC: 344 Part 4: Production and Supply FIGURE 10.5 Total, Average, and Marginal Cost Curves for the Cubic Total Cost Curve Case If the total cost curve has the cubic shape shown in (a), average and marginal cost curves will be U-shaped. In (b) the marginal cost curve passes through the low point of the average cost curve at output level q). Total costs (a) Average and marginal costs C Output per period MC AC (b) q* Output per period the assumed shape of the curve, the MC curve is U-shaped, with MC falling over the concave portion of the total cost curve and rising beyond the point of inﬂection. Because the slope is always positive, however, MC is always greater than 0. Average costs (AC) start out being equal to marginal cost for the ‘‘ﬁrst’’ unit of output.4 As output expands, however, AC exceeds MC because AC reﬂects both the marginal cost of the last unit produced 4Technically, AC MC at q ¼ ¼ 0. This can be shown by L’Hoˆpital’s rule, which states that if f (a) g (a) ¼ ¼ 0, then In this case, C 0 at q ¼ ¼ 0, and thus or which was to be shown. lim lim 0ð Þ AC lim 0 q ! lim 0 q ! ¼ C q ¼ lim 0 q ! @C=@q 1 ¼ MC lim 0 q ! AC ¼ MC at q 0, ¼ Chapter 10: Cost Functions 345 and the higher marginal costs of the previously produced units. As long as AC > MC, average costs must be decreasing. Because the lower costs of the newly produced units are below average cost, they continue to pull average costs downward. Marginal costs increase, however, and eventually (at q)) equal average cost. Beyond this point, MC > AC, and average costs will increase b
ecause they are pulled upward by increasingly higher marginal costs. Consequently, we have shown that the AC curve also has a Ushape and that it reaches a low point at q), where AC and MC intersect.5 In empirical studies of cost functions, there is considerable interest in this point of minimum average cost. It reﬂects the minimum efﬁcient scale (MES) for the particular production process being examined. The point is also theoretically important because of the role it plays in perfectly competitive price determination in the long run (see Chapter 12). Cost Functions and Shifts in Cost Curves The cost curves illustrated in Figures 10.4 and 10.5 show the relationship between costs and quantity produced on the assumption that all other factors are held constant. Speciﬁcally, construction of the curves assumes that input prices and the level of technology do not change.6 If these factors do change, the cost curves will shift. In this section, we delve further into the mathematics of cost functions as a way of studying these shifts. We begin with some examples. EXAMPLE 10.2 Some Illustrative Cost Functions In this example we calculate the cost functions associated with three different production functions. Later we will use these examples to illustrate some of the general properties of cost functions. ¼ 1. Fixed Proportions: q f (k, l ) min(ak, bl ). The calculation of cost functions from their underlying production functions is one of the more frustrating tasks for economics students. Thus, let’s start with a simple example. What we wish to do is show how total costs depend on input costs and on quantity produced. In the ﬁxed-proportions case, we know bl. Hence that production will occur at a vertex of the L-shaped isoquants where q total costs are ak ¼ ¼ ¼ C q a # $ This is indeed the sort of function we want because it states total costs as a function of v, w, and q only together with some parameters of the underlying production function. q b ! " v, w, q ð v a þ q ! (10:21) w b Þ ¼ wl vk þ ¼ ¼ þ " w v : 5Mathematically, we can ﬁnd the minimum AC by setting its derivative equal to 0: or @AC @q ¼ @ C=q Þ ð @q ¼ q ( ð @C=@q q2 C 1 ( Þ # q ( MC q2 C # ¼ 0, ¼ MC q ( C # ¼ 0 or MC C=q AC: ¼ ¼ ¼ AC when AC is minimized. Thus, MC 6For multiproduct ﬁrms, an additional complication must be considered. For such ﬁrms it is possible that the costs associated with producing one output (say, q1) are also affected by the amount of some other output being produced (q2). In this case the ﬁrm is said to exhibit ‘‘economies of scope,’’ and the total cost function will be of the form C(v, w, q1, q2). Hence q2 must also be held constant in constructing the q1 cost curves. Presumably increases in q2 shift the q1 cost curves downward. 346 Part 4: Production and Supply Because of the constant returns-to-scale nature of this production function, it takes the special form C v, w, q ð Þ ¼ q C : v, w, 1 Þ ð (10:22) That is, total costs are given by output times the cost of producing one unit. Increases in input prices clearly increase total costs with this function, and technical improvements that take the form of increasing the parameters a and b reduce costs. 2. Cobb–Douglas: q f (k, l ) kalb. This is our ﬁrst example of burdensome computation, but we can clarify the process by recognizing that the ﬁnal goal is to use the results of cost minimization to replace the inputs in the production function with costs. From Example 10.1 we know that cost minimization requires that ¼ ¼ and so : k ¼ (10:23) (10:24) Substitution into the production function permits a solution for labor input in terms of q, v, and w as kal 10:25) or lc v, w, q ð Þ ¼ q1=a þ a= ð a þ b Þ w# a= ð b b a ! " a b Þva= ð a b Þ: þ (10:26) þ A similar set of manipulations gives kc v, w, q ð Þ ¼ b= b a q1= wb= ð Now we are ready to derive total costs as a b Þv# b= ð a þ b Þ: (10:27) þ C v, w, q ð a/(a b)a# vk c wl c þ ¼ Þ ¼ q1= ð a þ ÞBva= b ð a b Þw b= ð a þ b Þ, (10:28) þ þ þ ¼ þ b/(a (a b)b# b)—a constant that involves only the parameters a and b. where B Although this derivation was a bit messy, several interesting aspects of this Cobb–Douglas cost function are readily apparent. First, whether the function is a convex, linear, or concave b < 1), function of output depends on whether there are decreasing returns to scale (a þ b > 1). Second, an constant returns to scale (a increase in any input price increases costs, with the extent of the increase being determined by the relative importance of the input as reﬂected by the size of its exponent in the production function. Finally, the cost function is homogeneous of degree 1 in the input prices—a general feature of all cost functions, as we shall show shortly. 1), or increasing returns to scale (a ¼ þ þ b 3. CES: q f (k, l ) l r)g/r. For this case, your authors will mercifully spare you the algebra. To derive the total cost function, we use the cost-minimization condition speciﬁed in Equation 10.15, solve for each input individually, and eventually get (kr ¼ ¼ þ C v, w, q ð Þ ¼ ¼ vk þ q1=g wl v1 # ð ¼ r q1=g vq= ð w1 # = Þ þ þ r Þ, wq= ð q # 1 Þ q # =q 1 Þ ð Þ (10:29) where the elasticity of substitution is given by s 1/(1 – r). Once again the shape of the total cost is determined by the scale parameter (g) for this production function, and the cost ¼ Chapter 10: Cost Functions 347 function increases in both of the input prices. The function is also homogeneous of degree 1 in those prices. One limiting feature of this form of the CES function is that the inputs are given equal weights—hence their prices are equally important in the cost function. This feature of the CES is easily generalized, however (see Problem 10.9). QUERY: How are the various substitution possibilities inherent in the CES function reﬂected in the CES cost function in Equation 10.29? Properties of cost functions These examples illustrate some properties of total cost functions that are general. 1. Homogeneity. The total cost functions in Example 10.2 are all homogeneous of degree 1 in the input prices. That is, a doubling of input prices will precisely double the cost of producing any given output level (you might check this out for yourself). This is a property of all proper cost functions. When all input prices double (or are increased by any uniform proportion), the ratio of any two input prices will not change. Because cost minimization requires that the ratio of input prices be set equal to the RTS along a given isoquant, the cost-minimizing input combination also will not change. Hence the ﬁrm will buy exactly the same set of inputs and pay precisely twice as much for them. One implication of this result is that a pure, uniform inﬂation in all input costs will not change a ﬁrm’s input decisions. Its cost curves will shift upward in precise correspondence to the rate of inﬂation. 2. Total cost functions are nondecreasing in q, v, and w. This property seems obvious, but it is worth dwelling on it a bit. Because cost functions are derived from a costminimization process, any decrease in costs from an increase in one of the function’s arguments would lead to a contradiction. For example, if an increase in output from q1 to q2 caused total costs to decrease, it must be the case that the ﬁrm was not minimizing costs in the ﬁrst place. It should have produced q2 and thrown away an output of q2 # q1, thereby producing q1 at a lower cost. Similarly, if an increase in the price of an input ever reduced total cost, the ﬁrm could not have been minimizing its costs in the ﬁrst place. To see this, suppose the ﬁrm was using the input combination (l1, k1) and that w increases. Clearly that will increase the cost of the initial input combination. But if changes in input choices caused total costs to decrease, that must imply that there was a lower-cost input mix than (l1, k1) initially. Hence we have a contradiction, and this property of cost functions is established.7 3. Total cost functions are concave in input prices. It is probably easiest to illustrate this property with a graph. Figure 10.6 shows total costs for various values of an input price, say, w, holding q and v constant. Suppose that initially input prices w0 and v0 prevail 7A formal proof could also be based on the envelope theorem as applied to constrained minimization problems. Consider the Lagrangian in Equation 10.3. As was pointed out in Chapter 2, we can calculate the change in the objective in such an expression (here, total cost) with respect to a change in a variable by differentiating the Lagrangian. Performing this differentiation yields @C @q ¼ @C @v ¼ @C @w ¼ @+ @q ¼ @+ @v ¼ @+ @w ¼ MC k ð¼ Þ * 0, k c l c 0, * 0: * Not only do these envelope results prove this property of cost functions, but they also are useful in their own right, as we will show later in this chapter. 348 Part 4: Production and Supply FIGURE 10.6 Cost Functions Are Concave in Input Prices With input prices w 0 and v 0, total costs of producing q0 are C (v 0, w 0, q0). If the ﬁrm does not change its input mix, costs of producing q0 would follow the straight line CPSEUDO. With input substitution, actual costs C (v 0, w, q0) will fall below this line, and hence the cost function is concave in w. Costs C(v′, w′, q0) CPSEUDO C(v′, w, q0) w′ w and that total output q0 is produced at total cost C(v0, w0, q0) using cost-minimizing inputs l 0 and k 0. If the ﬁrm did not change its input mix in response to changes in wages, then its total cost curve would be linear as reﬂected by the line CPSEUDO(v 0, w, q0) v 0k 0 wl 0 in the ﬁgure. But a cost-minimizing ﬁrm probably would change the input mix it þ uses to produce q0 when wages change, and these actual costs C(v 0, w, q0) would fall below the ‘‘pseudo’’ costs. Hence the total cost function must have the concave shape shown in Figure 10.6. One implication of this ﬁnding is that costs will be lower when a ﬁrm faces input prices that ﬂuctuate around a given level than when they rem
ain constant at that level. With ﬂuctuating input prices, the ﬁrm can adapt its input mix to take advantage of such ﬂuctuations by using a lot of, say, labor when its price is low and economizing on that input when its price is high. ¼ 4. Average and marginal costs. Some, but not all, of these properties of total cost functions carry over to their related average and marginal cost functions. Homogeneity is one property that carries over directly. Because C(tv, tw, q) tC(v, w, q), we have ¼ AC ð tv, tw, q Þ ¼ C tv, tw, q ð q Þ tC v, w, q ð q Þ ¼ tAC ð ¼ v, w, q Þ (10:30) and8 MC tv, tw, q ð Þ ¼ @C ð tv, tw, q Þ @q ¼ t@C v, w, q Þ ð @q tMC ð ¼ v, w, q : Þ (10:31) 8This result does not violate the theorem that the derivative of a function that is homogeneous of degree k is homogeneous of 1 because we are differentiating with respect to q and total costs are homogeneous with respect to input prices only. degree k # Chapter 10: Cost Functions 349 The effects of changes in q, v, and w on average and marginal costs are sometimes ambiguous, however. We have already shown that average and marginal cost curves may have negatively sloped segments, so neither AC nor MC is nondecreasing in q. Because total costs must not decrease when an input price increases, it is clear that average cost is increasing in w and v. But the case of marginal cost is more complex. The main complication arises because of the possibility of input inferiority. In that (admittedly rare) case, an increase in an inferior input’s price will actually cause marginal cost to decrease. Although the proof of this is relatively straightforward,9 an intuitive explanation for it is elusive. Still, in most cases, it seems clear that the increase in the price of an input will increase marginal cost as well. Input substitution A change in the price of an input will cause the ﬁrm to alter its input mix. Hence a full study of how cost curves shift when input prices change must also include an examination of substitution among inputs. The previous chapter provided a concept measuring how substitutable inputs are—the elasticity of substitution. Here we will modify the deﬁnition, using some results from cost minimization, so that it is expressed only in terms of readily observable variables. The modiﬁed deﬁnition will turn out to be more useful for empirical work. Recall the formula for the elasticity of substitution from Chapter 9, repeated here: d k=l ð Þ d RTS ( RTS k=l ¼ d ln k=l Þ ð d ln RTS : r ¼ (10:32) But the cost-minimization principle says that RTS stituting gives a new version of the elasticity of substitution:10 of l for k ð Þ ¼ w=v at an optimum. Sub- s ¼ d k=l Þ ð d w=v Þ ð w=v k=l ¼ ( d ln k=l Þ ð d ln w=v Þ ð , (10:33) distinguished by changing the label from s to s. The elasticities differ in two respects. Whereas s applies to any point on any isoquant, s applies only to a single point on a single isoquant (the equilibrium point where there is a tangency between the isoquant and an equal total cost line). Although this would seem to be a drawback of s, the big advantage of focusing on the equilibrium point is that s then involves only easily observable variables: input amounts and prices. By contrast, s involves the RTS, the slope of an isoquant. Knowledge of the RTS would require detailed knowledge of the production process that even the ﬁrm’s engineers may not have, let alone an outside observer. In the two-input case, s must be non-negative; an increase in w/v will be met by an increase in k/l (or, in the limiting ﬁxed-proportions case, k/l will stay constant). Large values of s indicate that ﬁrms change their input proportions signiﬁcantly in response to changes in relative input prices, whereas low values indicate that changes in input prices have relatively little effect. 9The proof follows the envelope theorem results presented in footnote 7. Because the MC function can be derived by differentiation from the Lagrangian for cost minimization, we can use Young’s theorem to show @MC @v ¼ @ ð @+=@q Þ @v ¼ @ 2L @v@q ¼ @ 2+ @q@v ¼ @k @q : Hence, if capital is a normal input, an increase in v will raise MC whereas, if capital is inferior, an increase in v will actually reduce MC. 10This deﬁnition is usually attributed to R. G. D. Allen, who developed it in an alternative form in his Mathematical Analysis for Economists (New York: St. Martin’s Press, 1938), pp. 504–9. 350 Part 4: Production and Supply Substitution with many inputs Instead of just the two inputs k and l, now suppose there are many inputs to the production process (x1, x2, … , xn) that can be hired at competitive rental rates (w1, w2, … , wn). Then the elasticity of substitution between any two inputs (sij) is deﬁned as follows Elasticity of substitution. The elasticity of substitution between inputs xi and xj is given by sij ¼ @ @ xi=xjÞ ð wj=wiÞ ð wj=wi xi=xj ¼ @ ln @ ln ( xi=xjÞ ð wj=wiÞ ð , (10:34) where output and all other input prices are held constant. A subtle point that did not arise in the two-input case regards what is assumed about the ﬁrm’s usage of the other inputs besides i and j. Should we perform the thought experiment of holding them ﬁxed as are other input prices and output? Or should we take into account the adjustment of these other inputs to achieve cost minimization? The latter assumption has proved to be more useful in economic analysis; therefore, that is the one we will take to be embodied in Equation 10.34.11 For example, a major topic in the theory of ﬁrms’ input choices is to describe the relationship between capital and energy inputs. The deﬁnition in Equation 10.34 would permit a researcher to study how the ratio of energy to capital input changes when relative energy prices increase while permitting the ﬁrm to make any adjustments to labor input (whose price has not changed) that would be required for cost minimization. Hence this would give a realistic picture of how ﬁrms behave with regard to whether energy and capital are more like substitutes or complements. Later in this chapter we will look at this deﬁnition in a bit more detail because it is widely used in empirical studies of production. Quantitative size of shifts in cost curves We have already shown that increases in an input price will raise total, average, and (except in the inferior input case) marginal costs. We are now in a position to judge the extent of such increases. First, and most obviously, the increase in costs will be inﬂuenced importantly by the relative signiﬁcance of the input in the production process. If an input constitutes a large fraction of total costs, an increase in its price will raise costs signiﬁcantly. An increase in the wage rate would sharply increase home-builders’ costs because labor is a major input in construction. On the other hand, a price increase for a relatively minor input will have a small cost impact. An increase in nail prices will not raise home costs much. A less obvious determinant of the extent of cost increases is input substitutability. If ﬁrms can easily substitute another input for the one that has increased in price, there may be little increase in costs. Increases in copper prices in the late 1960s, for example, had little impact on electric utilities’ costs of distributing electricity because they found they could easily substitute aluminum for copper cables. Alternatively, if the ﬁrm ﬁnds it difﬁcult or impossible to substitute for the input that has become more costly, then costs may increase rapidly. The cost of gold jewelry, along with the price of gold, rose rapidly during the early 1970s because there was simply no substitute for the raw input. 11This deﬁnition is attributed to the Japanese economist M. Morishima, and these elasticities are sometimes referred to as Morishima elasticities. In this version, the elasticity of substitution for substitute inputs is positive. Some authors reverse the order of subscripts in the denominator of Equation 10.31, and in this usage the elasticity of substitution for substitute inputs is negative. Chapter 10: Cost Functions 351 It is possible to give a precise mathematical statement of the quantitative sizes of all these effects by using the elasticity of substitution. To do so, however, would risk further cluttering the book with symbols.12 For our purposes, it is sufﬁcient to rely on the previous intuitive discussion. This should serve as a reminder that changes in the price of an input will have the effect of shifting ﬁrms’ cost curves, with the size of the shift depending on the relative importance of the input and on the substitution possibilities that are available. Technical change Technical improvements allow the ﬁrm to produce a given output with fewer inputs. Such improvements obviously shift total costs downward (if input prices stay constant). Although the actual way in which technical change affects the mathematical form of the total cost curve can be complex, there are cases where one may draw simple conclusions. Suppose, for example, that the production function exhibits constant returns to scale and that technical change enters that function as described in Chapter 9 [i.e., q A(t)f(k, l ) where A(0) 1]. In this case, total costs in the initial period are given by ¼ ¼ qC0ð Because the same inputs that produced one unit of output in period 0 are also the costminimizing way of producing A(t) units of output in period t, we know that v, w, 1 Þ v, w, q C0ð (10:35) Þ ¼ : v, w, 1 C0ð v, w, A t Ctð ð ÞÞ ¼ Þ ¼ Therefore, we can compute the total cost function in period t as v, w, q C0ð t A ð Þ Hence total costs decrease over time at the rate of technical change.13 v, w, 1 qC0ð t A ð qCtð v, w, q v, w, 1 Ctð Ctð v, w, 1 Þ : (10:36) Þ : (10:37) Note that in this case technical change is ‘‘neutral’’ in that it does not affect the ﬁrm’s input choices (as long as input prices stay constant). This neutrality result might not hold in cases where technical progress takes a more complex form or w
here there are variable returns to scale. Even in these more complex cases, however, technical improvements will cause total costs to decrease. EXAMPLE 10.3 Shifting the Cobb–Douglas Cost Function In Example 10.2 we computed the Cobb–Douglas cost function as C v, w, q ð Þ ¼ q1= a ð ÞBva= b ð þ a þ Þw b= b a ð b Þ, þ (10:38) 12For a complete statement, see C. Ferguson, Neoclassical Theory of Production and Distribution (Cambridge, UK: Cambridge University Press, 1969), pp. 154–60. 13To see that the indicated rates of change are the same, note ﬁrst that the rate of change of technical progress is while the rate of change in total cost is using Equation 10.34. r t ð Þ ¼ A0 t ð t A ð Þ Þ ; @Ct @t ( 1 Ct ¼ t C0A0ð 2 t A Þ ð 1 Ct ¼ Þ ( t A0ð 352 Part 4: Production and Supply where B (a assume that a ¼ b)a# þ b ¼ ¼ a/(a b)b# þ b/(a b). As in the numerical illustration in Example 10.1, let’s þ 0.5, in which case the total cost function is greatly simpliﬁed: Þ ¼ This function will yield a total cost curve relating total costs and output if we specify particular values for the input prices. If, as before, we assume v 12, then the relationship is 3 and w C v, w, q ð 2qv0:5w0:5: (10:39) 3, 12, q C ð Þ ¼ 2q 12q, ¼ (10:40) ¼ 36p ¼ and, as in Example 10.1, it costs 480 to produce 40 units of output. Here average and marginal costs are easily computed as ﬃﬃﬃﬃﬃ 12, AC MC ¼ ¼ C q ¼ @C @q ¼ 12: (10:41) As expected, average and marginal costs are constant and equal to each other for this constant returns-to-scale production function. Changes in input prices. If either input price were to change, all these costs would change also. For example, if wages were to increase to 27 (an easy number with which to work), costs would become C ð 3, 27, q AC MC Þ ¼ ¼ ¼ 2q 18, 18: 81p ﬃﬃﬃﬃﬃ 18q, ¼ (10:42) Notice that an increase in wages of 125 percent increased costs by only 50 percent here, both because labor represents only 50 percent of all costs and because the change in input prices encouraged the ﬁrm to substitute capital for labor. The total cost function, because it is derived from the cost-minimization assumption, accomplishes this substitution ‘‘behind the scenes’’— reporting only the ﬁnal impact on total costs. Technical progress. Let’s look now at the impact that technical progress can have on costs. Speciﬁcally, assume that the Cobb–Douglas production function is q t A ð k0:5l 0:5 Þ ¼ ¼ e:03tk0:5l 0:5: (10:43) That is, we assume that technical change takes an exponential form and that the rate of technical change is 3 percent per year. Using the results of the previous section (Equation 10.37) yields v, w, q Ctð Þ ¼ v, w, q C0ð t A ð Þ Þ ¼ 2qv0:5w 0:5e# :03t: (10:44) if input prices remain the same, then total costs decrease at the rate of technical Thus, improvement—that is, at 3 percent per year. After, say, 20 years, costs will be (with v 36p :60 e# ( 12q 0:55 ( ð Þ ¼ ¼ 6:6q, C20ð 3, 12, q Þ ¼ AC20 ¼ MC20 ¼ 2q 6:6, 6:6: ﬃﬃﬃﬃﬃ 3, w 12) ¼ ¼ (10:45) Consequently, costs will have decreased by nearly 50 percent as a result of the technical change. This would, for example, more than have offset the wage increase illustrated previously. QUERY: In this example, what are the elasticities of total costs with respect to changes in input costs? Is the size of these elasticities affected by technical change? Chapter 10: Cost Functions 353 Contingent demand for inputs and Shephard’s lemma As we described earlier, the process of cost minimization creates an implicit demand for inputs. Because that process holds quantity produced constant, this demand for inputs will also be ‘‘contingent’’ on the quantity being produced. This relationship is fully reﬂected in the ﬁrm’s total cost function and, perhaps surprisingly, contingent demand functions for all the ﬁrm’s inputs can be easily derived from that function. The process involves what has come to be called Shephard’s lemma,14 which states that the contingent demand function for any input is given by the partial derivative of the total cost function with respect to that input’s price. Because Shephard’s lemma is widely used in many areas of economic research, we will provide a relatively detailed examination of it. The intuition behind Shephard’s lemma is straightforward. Suppose that the price of labor (w) were to increase slightly. How would this affect total costs? If nothing it seems that costs would increase by approximately the amount of else changed, labor (l) that the ﬁrm was currently hiring. Roughly speaking then, @C/@w l, and that is what Shephard’s lemma claims. Figure 10.6 makes roughly the same point graphically. Along the ‘‘pseudo’’ cost function all inputs are held constant; therefore, an increase in the wage increases costs in direct proportion to the amount of labor used. Because the true cost function is tangent to the pseudo-function at the current wage, its slope (i.e., its partial derivative) also will show the current amount of labor input demanded. ¼ Technically, Shephard’s lemma is one result of the envelope theorem that was ﬁrst discussed in Chapter 2. There we showed that the change in the optimal value in a constrained optimization problem with respect to one of the parameters of the problem can be found by differentiating the Lagrangian for that optimization problem with respect to this changing parameter. In the cost-minimization case, the Lagrangian is + vk wl k q ½ # k, l f ð Þ’ þ þ ¼ and the envelope theorem applied to either input is @C @C ð ð v, w, q @v v, w, q @w Þ Þ ¼ ¼ @+ @+ ð ð v, w, q, k @v v, w, q, k @w Þ Þ ¼ ¼ kc v, w, q ð Þ , lc v, w, q ð Þ , (10:46) (10:47) where the notation is intended to make clear that the resulting demand functions for capital and labor input depend on v, w, and q. Because quantity produced enters these functions, input demand is indeed contingent on that variable. This feature of the demand functions is also reﬂected by the ‘‘c’’ in the notation.15 Hence the demand relations in Equation 10.47 do not represent a complete picture of input demand because they still depend on a variable that is under the ﬁrm’s control. In the next chapter, we will complete the study of input demand by showing how the assumption of proﬁt maximization allows us to effectively replace q in the input demand relationships with the market price of the ﬁrm’s output, p. 14Named for R. W. Shephard, who highlighted the important relationship between cost functions and input demand functions in his Cost and Production Functions (Princeton, NJ: Princeton University Press, 1970). 15The notation mirrors that used for compensated demand curves in Chapter 5 (which were derived from the expenditure function). In that case, such demand functions were contingent on the utility target assumed. 354 Part 4: Production and Supply EXAMPLE 10.4 Contingent Input Demand Functions In this example, we will show how the total cost functions derived in Example 10.2 can be used to derive contingent demand functions for the inputs capital and labor. 1. Fixed Proportions: C(v, w, q) functions are simple: q(v/a þ ¼ w/b). For this cost function, contingent demand k c l c v, w, q ð Þ ¼ v, w, q ð Þ ¼ @C @C v, w, q ð @v v, w, q ð @10:48) To produce any particular output with a ﬁxed proportions production function at minimal cost, the ﬁrm must produce at the vertex of its isoquants no matter what the inputs’ prices are. Hence the demand for inputs depends only on the level of output, and v and w do not enter the contingent input demand functions. Input prices may, however, affect total input demands in the ﬁxed proportions case because they may affect how much the ﬁrm decides to sell. 2. Cobb–Douglas: C(v, w, q) ¼ ier but also more instructive: q1/(a þ b) Bva/(a þ b)wb/(a þ b). In this case, the derivation is mess- q1= a ð b ÞBv# þ b= a ð b Þw b, b= a ð b ÞB w v # $ ÞBv a= b a q1= þ ð a ð b Þw# a= ð þ (10:49) a þ b Þ k c v, w, q ð Þ ¼ ¼ l c v, w, q ð Þ ¼ b ( @C @v ¼ a a þ @C @w ¼ b a a b ( þ q1= ð a þ a b ( b þ q1= ð ¼ a b ( þ a= ð # a þ b Þ: a b ÞB þ w v # $ Consequently, the contingent demands for inputs depend on both inputs’ prices. If we assume a 2), these reduce to 0.5 (so :5 0:5 0:5 ¼ ¼ ¼ l c k c 0:5 0:5 Þ ¼ Þ ¼ 3, w ¼ 0:5 (10:50) 12, and q v, w, q ð v, w # $ 40, Equations 10.50 yield the result we obtained previously: With v that the ﬁrm should choose the input combination k 20 to minimize the cost of producing 40 units of output. If the wage were to increase to, say, 27, the ﬁrm would choose the input combination k 40/3 to produce 40 units of output. Total costs would increase from 480 to 520, but the ability of the ﬁrm to substitute capital for the now more expensive labor does save considerably. For example, the initial input combination would now cost 780. 3. CES: C(v, w, q) w 1–s)1/(1–s). The importance of input substitution is shown even more clearly with the contingent demand functions derived from the CES function. For that function, q1/g(v 1–s 120, l 80, w, q ð Þ ¼ ¼ l c v, w, q ð Þ ¼ ¼ @C @v ¼ 1 q1=g v1 ð @C @w ¼ 1 q1=g v1 ð 1 r ( # r # þ 1 w1 r q1=g v1 # ð r= Þ q1= ( w1 r # þ v1 ð r= Þ þ r Þv# þ r Þw# r: 1 ð # w1 # r 1 # r Þ ð r= Þ 1 ð # r r v# Þ r, (10:51) w1 1 ð r # r # r= Þ Þ 1 ð # r r w# Þ Chapter 10: Cost Functions 355 These functions collapse when s with either more (s 2) or less (s middle ground. If we assume constant returns to scale (g 40, then contingent demands for the inputs when s q 1 (the Cobb–Douglas case), but we can study examples 0.5) substitutability and use Cobb–Douglas as the 12, and 1) and v ¼ 2 are 3, 12, 40 ð 3, 12, 40 ð Þ ¼ 40 40 3# ð 3# ð 1 1 þ 1 12# 1 12# 2 2 # Þ # Þ ¼ 3# 2 ( 12# ¼ 2 25:6, 1:6: (10:52) (10:53) þ That is, the level of capital input is 16 times the amount of labor input. With less substitutability (s 0.5), contingent input demands are , 12, 40 ð 3, 12, 40 ð Þ ¼ 40 40 30:5 ð 30:5 ð þ 120:5 120:5 1 1 Þ ( 0:5 3# 0:5 12# ¼ 120, 60: ( Þ ¼ Þ ¼ þ Thus, in this case, capital input is only twice as large as labor input. Although these va
rious cases cannot be compared directly because different values for s scale output differently, we can, as an example, look at the consequence of a increase in w to 27 in the lowsubstitutability case. With w 53.3. In this case, the 27, the ﬁrm will choose k cost savings from substitution can be calculated by comparing total costs when using the initial input combination ( 1,980) to total costs with the optimal combination ( 1,919). Hence moving to the optimal input combination reduces total costs by only about 3 percent. In the Cobb–Douglas case, cost savings are over 20 percent. ¼ 27 Æ 53.3 3 Æ 160 3 Æ 120 27 Æ 60 160 QUERY: How would total costs change if w increased from 12 to 27 and the production function took the simple linear form q 4l? What light does this result shed on the other cases in this example? þ ¼ k Shephard’s Lemma and the Elasticity of Substitution One especially nice feature of Shephard’s lemma is that it can be used to show how to derive information about input substitution directly from the total cost function through differentiation. Using the deﬁnition in Equation 10.34 yields Ci=CjÞ ð wj=wiÞ ð xi=xjÞ ð wj=wiÞ ð @ ln @ ln @ ln @ ln sij ¼ (10:54) ¼ , where Ci and Cj are the partial derivatives of the total cost function with respect to the input prices. Once the total cost function is known (perhaps through econometric estimation), information about substitutability among inputs can thus be readily obtained from it. In the Extensions to this chapter, we describe some of the results that have been obtained in this way. Problems 10.11 and 10.12 provide some additional details about ways in which substitutability among inputs can be measured. Short-Run, Long-Run Distinction It is traditional in economics to make a distinction between the ‘‘short run’’ and the ‘‘long run.’’ Although no precise temporal deﬁnition can be provided for these terms, the general purpose of the distinction is to differentiate between a short period during which 356 Part 4: Production and Supply economic actors have only limited ﬂexibility in their actions and a longer period that provides greater freedom. One area of study in which this distinction is important is in the theory of the ﬁrm and its costs because economists are interested in examining supply reactions over differing time intervals. In the remainder of this chapter, we will examine the implications of such differential response. To illustrate why short-run and long-run reactions might differ, assume that capital input is held ﬁxed at a level of k1 and that (in the short run) the ﬁrm is free to vary only its labor input.16 Implicitly, we are assuming that alterations in the level of capital input are inﬁnitely costly in the short run. As a result of this assumption, the short-run production function is where this notation explicitly shows that capital inputs may not vary. Of course, the level of output still may be changed if the ﬁrm alters its use of labor. q f k1, l ð , Þ ¼ (10:55) Short-run total costs Total cost for the ﬁrm continues to be deﬁned as C vk wl (10:56) ¼ SC þ for our short-run analysis, but now capital input is ﬁxed at k1. To denote this fact, we will write vk1 þ where the S indicates that we are analyzing short-run costs with the level of capital input ﬁxed. Throughout our analysis, we will use this method to indicate short-run costs. Usually we will not denote the level of capital input explicitly, but it is understood that this input is ﬁxed. The cost concepts introduced earlier—C, AC, MC—are in fact long-run concepts because, in their deﬁnitions, all inputs were allowed to vary freely. Their longrun nature is indicated by the absence of a leading S. (10:57) wl, ¼ Fixed and variable costs The two types of input costs in Equation 10.57 are given special names. The term vk1 is referred to as (short-run) ﬁxed costs; because k1 is constant, these costs will not change in the short run. The term wl is referred to as (short-run) variable costs—labor input can indeed be varied in the short run. Hence we have the following deﬁnitions Short-run ﬁxed and variable costs. Short-run ﬁxed costs are costs associated with inputs that cannot be varied in the short run. Short-run variable costs are costs of those inputs that can be varied to change the ﬁrm’s output level. The importance of this distinction is to differentiate between variable costs that the ﬁrm can avoid by producing nothing in the short run and costs that are ﬁxed and must be paid regardless of the output level chosen (even zero). Nonoptimality of short-run costs It is important to understand that total short-run costs are not the minimal costs for producing the various output levels. Because we are holding capital ﬁxed in the short run, 16Of course, this approach is for illustrative purposes only. In many actual situations, labor input may be less ﬂexible in the short run than is capital input. Chapter 10: Cost Functions 357 the ﬁrm does not have the ﬂexibility of input choice that we assumed when we discussed cost minimization earlier in this chapter. Rather, to vary its output level in the short run, the ﬁrm will be forced to use ‘‘nonoptimal’’ input combinations. The RTS will not necessarily be equal to the ratio of the input prices. This is shown in Figure 10.7. In the short run, the ﬁrm is constrained to use k1 units of capital. To produce output level q0, it will use l0 units of labor. Similarly, it will use l1 units of labor to produce q1 and l2 units to produce q2. The total costs of these input combinations are given by SC0, SC1, and SC2, respectively. Only for the input combination k1, l1 is output being produced at minimal cost. Only at that point is the RTS equal to the ratio of the input prices. From Figure 10.7, it is clear that q0 is being produced with ‘‘too much’’ capital in this short-run situation. Cost minimization should suggest a southeasterly movement along the q0 isoquant, indicating a substitution of labor for capital in production. Similarly, q2 is being produced with ‘‘too little’’ capital, and costs could be reduced by substituting capital for labor. Neither of these substitutions is possible in the short run. Over a longer period, however, the ﬁrm will be able to change its level of capital input and will adjust its input usage to the cost-minimizing combinations. We have already discussed this ﬂexible case earlier in this chapter and shall return to it to illustrate the connection between long-run and short-run cost curves. FIGURE 10.7 ‘‘Nonoptimal’’ Input Choices Must Be Made in the Short Run Because capital input is ﬁxed at k, in the short run the ﬁrm cannot bring its RTS into equality with the ratio of input prices. Given the input prices, q0 should be produced with more labor and less capital than it will be in the short run, whereas q2 should be produced with more capital and less labor than it will be. k per period k1 SC2 SC0 SC1 = C q2 q1 q0 l 0 l1 l2 l per period 358 Part 4: Production and Supply Short-run marginal and average costs Frequently, it is more useful to analyze short-run costs on a per-unit of output basis rather than on a total basis. The two most important per-unit concepts that can be derived from the short-run total cost function are the short-run average total cost function (SAC) and the short-run marginal cost function (SMC). These concepts are deﬁned as total costs total output ¼ change in total costs SC q , SAC ¼ SMC ¼ change in output ¼ @SC @q , (10:58) where again these are deﬁned for a speciﬁed level of capital input. These deﬁnitions for average and marginal costs are identical to those developed previously for the long-run, fully ﬂexible case, and the derivation of cost curves from the total cost function proceeds in exactly the same way. Because the short-run total cost curve has the same general type of cubic shape as did the total cost curve in Figure 10.5, these short-run average and marginal cost curves will also be U-shaped. Relationship between short-run and long-run cost curves It is easy to demonstrate the relationship between the short-run costs and the fully ﬂexible long-run costs that were derived previously in this chapter. Figure 10.8 shows this relationship for both the constant returns-to-scale and cubic total cost curve cases. Short-run total costs for three levels of capital input are shown, although of course it would be possible to show many more such short-run curves. The ﬁgures show that long-run total costs (C) are always less than short-run total costs, except at that output level for which the assumed ﬁxed capital input is appropriate to long-run cost minimization. For example, as in Figure 10.7, with capital input of k1 the ﬁrm can obtain full cost minimization when q1 is produced. Hence short-run and long-run total costs are equal at this point. For output levels other than q1, however, SC > C, as was the case in Figure 10.7. Technically, the long-run total cost curves in Figure 10.8 are said to be an ‘‘envelope’’ of their respective short-run curves. These short-run total cost curves can be represented parametrically by short-run total cost SC v, w, q, k ð , Þ ¼ (10:59) and the family of short-run total cost curves is generated by allowing k to vary while holding v and w constant. The long-run total cost curve C must obey the short-run relationship in Equation 10.59 and the further condition that k be cost minimizing for any level of output. A ﬁrst-order condition for this minimization is that @SC(v, w, q, k) @k 0: ¼ (10:60) Solving Equations 10.59 and 10.60 simultaneously then generates the long-run total cost function. Although this is a different approach to deriving the total cost function, it should give precisely the same results derived earlier in this chapter—as the next example illustrates. Chapter 10: Cost Functions 359 FIGURE 10.8 Two Possible Shapes for Long-Run Total Cost Curves By considering all possible levels of capital input, the long-run total cost curve (C ) can be trace
d. In (a), the underlying production function exhibits constant returns to scale: In the long run, although not in the short run, total costs are proportional to output. In (b), the long-run total cost curve has a cubic shape, as do the short-run curves. Diminishing returns set in more sharply for the short-run curves, however, because of the assumed ﬁxed level of capital input. Total costs SC(k1) SC(k2) C SC(k0) q0 q1 q2 Output per period (a) Constant returns to scale Total costs C SC(k2) SC(k1) SC(k0) q0 q1 q2 Output per period (b) Cubic total cost curve case 360 Part 4: Production and Supply EXAMPLE 10.5 Envelope Relations and Cobb–Douglas Cost Functions Again we start with the Cobb–Douglas production function q input constant at k1. Thus, in the short run, ¼ kalb, but now we hold capital and total costs are given by 1 l b ka q ¼ or l ¼ q1=bk# 1 a=b , SC v, w, q, k1Þ ¼ ð vk1 þ wl vk1 þ ¼ wq1=bk# 1 a=b : (10:61) (10:62) Notice that the ﬁxed level of capital enters into this short-run total cost function in two ways: (1) k1 determines ﬁxed costs; and (2) k1 also in part determines variable costs because it determines how much of the variable input (labor) is required to produce various levels of output. To derive long-run costs, we require that k be chosen to minimize total costs: @SC v, w, q, k Þ ð @k v þ ¼ a # b ( wq1=bk#ð a þ b)=b 0: ¼ (10:63) Although the algebra is messy, this equation can be solved for k and substituted into Equation 10.62 to return us to the Cobb–Douglas cost function: C v, w, q ð Þ ¼ Bq1= ð a þ Þva= b ð a þ Þw b= b a ð b Þ: þ (10:64) Numerical example. If we again let a function is b ¼ ¼ 0.5, v ¼ 3, and w ¼ 12, then the short-run cost SC 3, 12, q; k1Þ ¼ ð 3k1 þ 12q2k# 1 1 : (10:65) In Example 10.1 we found that the cost-minimizing level of capital input for q 40 was k Equation 10.65 shows that short-run total costs for producing 40 units of output with k1 ¼ ¼ 80. ¼ 80 is SC 3, 12, q, 80 ð Þ ¼ 3:80 12 q2 ( þ 240 þ ¼ 240 ¼ 240 3q2 20 þ 1 80 ¼ ( 480, (10:66) which is just what we found before. We can also use Equation 10.65 to show how costs differ in 40, short-run the short and long run. Table 10.1 shows that, for output levels other than q costs are larger than long-run costs and that this difference is proportionally larger the farther one gets from the output level for which k 80 is optimal. ¼ ¼ ¼ 240 3q2/20 TABLE 10.1 DIFFERENCE BETWEEN SHORT-RUN AND LONG-RUN TOTAL COST, k 80 q 10 20 30 40 50 60 70 80 SC ¼ C 12q ¼ 120 240 360 480 600 720 840 960 þ 255 300 375 480 615 780 975 1,200 Chapter 10: Cost Functions 361 TABLE 10.2 UNIT COSTS IN THE LONG RUN AND THE SHORT RUN, k 80 ¼ AC 12 12 12 12 12 12 12 12 MC 12 12 12 12 12 12 12 12 SAC 25.5 15.0 12.5 12.0 12.3 13.0 13.9 15.0 SMC 3 6 9 12 15 18 21 24 q 10 20 30 40 50 60 70 80 It is also instructive to study differences between the long-run and short-run per-unit costs 12. We can compute the short-run equivalents (when MC in this situation. Here AC k 80) as ¼ ¼ ¼ SAC SMC ¼ ¼ SC q ¼ @SC @q ¼ 240 q þ 6q 20 : 3q 20 , (10:67) Both of these short-run unit costs are equal to 12 when q 40. However, as Table 10.2 shows, short-run unit costs can differ signiﬁcantly from this ﬁgure, depending on the output level that the ﬁrm produces. Notice in particular that short-run marginal cost increases rapidly as output 40 because of diminishing returns to the variable input (labor). This expands beyond q conclusion plays an important role in the theory of short-run price determination. ¼ ¼ QUERY: Explain why an increase in w will increase both short-run average cost and short-run marginal cost in this illustration, but an increase in v affects only short-run average cost. Graphs of per-unit cost curves The envelope total cost curve relationships exhibited in Figure 10.8 can be used to show geometric connections between short-run and long-run average and marginal cost curves. These are presented in Figure 10.9 for the cubic total cost curve case. In the ﬁgure, shortrun and long-run average costs are equal at that output for which the (ﬁxed) capital input is appropriate. At q1, for example, SAC(k1) AC because k1 is used in producing q1 at ¼ minimal costs. For movements away from q1, short-run average costs exceed long-run average costs, thus reﬂecting the cost-minimizing nature of the long-run total cost curve. Because the minimum point of the long-run average cost curve (AC) plays a major role in the theory of long-run price determination, it is important to note the various curves that pass through this point in Figure 10.9. First, as is always true for average and marginal cost curves, the MC curve passes through the low point of the AC curve. At q1, long-run average and marginal costs are equal. Associated with q1 is a certain level of capital input (say, k1); the short-run average cost curve for this level of capital input is tangent to the AC curve at its minimum point. The SAC curve also reaches its minimum at output level q1. For movements away from q1, the AC curve is much ﬂatter than the SAC curve, and this reﬂects the greater ﬂexibility open to ﬁrms in the long run. Short-run costs increase rapidly because capital inputs are ﬁxed. In the long run, such inputs are 362 Part 4: Production and Supply FIGURE 10.9 Average and Marginal Cost Curves for the Cubic Cost Curve Case This set of curves is derived from the total cost curves shown in Figure 10.8. The AC and MC curves have the usual U-shapes, as do the short-run curves. At q1, long-run average costs are minimized. The conﬁguration of curves at this minimum point is important. Costs MC SMC(k2) SAC(k2) AC SAC(k1) SMC(k1) SAC(k0) SMC(k0) q0 q1 q2 Output per period not ﬁxed, and diminishing marginal productivities do not occur so abruptly. Finally, because the SAC curve reaches its minimum at q1, the short-run marginal cost curve (SMC) also passes through this point. Therefore, the minimum point of the AC curve brings together the four most important per-unit costs: At this point, AC ¼ MC ¼ SAC ¼ SMC: (10:68) For this reason, as we shall show in Chapter 12, the output level q1 is an important equilibrium point for a competitive ﬁrm in the long run. SUMMARY In this chapter we examined the relationship between the level of output a ﬁrm produces and the input costs associated with that level of production. The resulting cost curves should generally be familiar to you because they are widely used in most courses in introductory economics. Here we have shown how such curves reﬂect the ﬁrm’s underlying production function and the ﬁrm’s desire to minimize costs. By developing cost curves from these basic foundations, we were able to illustrate a number of important ﬁndings. • A ﬁrm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the rate of technical substitution (RTS) is equal to the ratio of the inputs’ rental prices. • Repeated application of this minimization procedure yields the ﬁrm’s expansion path. Because the expansion path shows how input usage expands with the level of output, it also shows the relationship between output level and total cost. That relationship is summarized by the total cost function, C(v, w, q), which shows production costs as a function of output levels and input prices. Chapter 10: Cost Functions 363 • • Input demand functions can be derived from the ﬁrm’s total cost function through partial differentiation. These input demand functions will depend on the quantity of output that the ﬁrm chooses to produce and are therefore called ‘‘contingent’’ demand functions. In the short run, the ﬁrm may not be able to vary some inputs. It can then alter its level of production only by changing its employment of variable inputs. In so doing, it may have to use nonoptimal, higher-cost input combinations than it would choose if it were possible to vary all inputs. • The ﬁrm’s average cost (AC C/q) and marginal cost @C/@q) functions can be derived directly from (MC the total cost function. If the total cost curve has a general cubic shape, then the AC and MC curves will be U-shaped. ¼ ¼ • All cost curves are drawn on the assumption that the input prices are held constant. When input prices change, cost curves will shift to new positions. The extent of the shifts will be determined by the overall importance of the input whose price has changed and by the ease with which the ﬁrm may substitute one input for another. Technical progress will also shift cost curves. PROBLEMS 10.1 Suppose that a ﬁrm produces two different outputs, the quantities of which are represented by q1 and q2. In general, the ﬁrm’s C(0, q2) > C(q1, q2) for all total costs can be represented by C(q1, q2). This function exhibits economies of scope if C(q1, 0) output levels of either good. þ a. Explain in words why this mathematical formulation implies that costs will be lower in this multiproduct ﬁrm than in two single-product ﬁrms producing each good separately. b. If the two outputs are actually the same good, we can deﬁne total output as q C/q) decreases as q increases. Show that this ﬁrm also enjoys economies of scope under the deﬁnition provided here. q1 þ ¼ q2. Suppose that in this case average cost ( ¼ 10.2 Professor Smith and Professor Jones are going to produce a new introductory textbook. As true scientists, they have laid out the production function for the book as where q hours spent working by Jones. ¼ the number of pages in the ﬁnished book, S S1=2J 1=2, q ¼ the number of working hours spent by Smith, and J the number of ¼ ¼ After having spent 900 hours preparing the ﬁrst draft, time which he valued at $3 per working hour, Smith has to move on to other things and cannot contribute any more to the book. Jones, whose labor is valued at $12 per working hour, will revise Smith’s draft to complete the book. a. How many hours will Jones have to spend to produce a ﬁnished book of 150 pages? Of 300 pages? Of 450 pages? b. What is the margi
nal cost of the 150th page of the ﬁnished book? Of the 300th page? Of the 450th page? 10.3 Suppose that a ﬁrm’s ﬁxed proportion production function is given by q min 5k, 10l ð : Þ ¼ a. Calculate the ﬁrm’s long-run total, average, and marginal cost functions. b. Suppose that k is ﬁxed at 10 in the short run. Calculate the ﬁrm’s short-run total, average, and marginal cost functions. c. Suppose v 3. Calculate this ﬁrm’s long-run and short-run average and marginal cost curves. 1 and w ¼ ¼ 10.4 A ﬁrm producing hockey sticks has a production function given by klp : 2 q ¼ ﬃﬃﬃﬃ 364 Part 4: Production and Supply ¼ In the short run, the ﬁrm’s amount of capital equipment is ﬁxed at k for l is w $4. 100. The rental rate for k is v $1, and the wage rate ¼ ¼ a. Calculate the ﬁrm’s short-run total cost curve. Calculate the short-run average cost curve. b. What is the ﬁrm’s short-run marginal cost function? What are the SC, SAC, and SMC for the ﬁrm if it produces 25 hockey sticks? Fifty hockey sticks? One hundred hockey sticks? Two hundred hockey sticks? c. Graph the SAC and the SMC curves for the ﬁrm. Indicate the points found in part (b). d. Where does the SMC curve intersect the SAC curve? Explain why the SMC curve will always intersect the SAC curve at its lowest point. Suppose now that capital used for producing hockey sticks is ﬁxed at k in the short run. e. Calculate the ﬁrm’s total costs as a function of q, w, v, and k. f. Given q, w, and v, how should the capital stock be chosen to minimize total cost? g. Use your results from part (f) to calculate the long-run total cost of hockey stick production. h. For w $4, v short-run curves computed in part (e) by examining values of k of 100, 200, and 400. ¼ ¼ $1, graph the long-run total cost curve for hockey stick production. Show that this is an envelope for the 10.5 An enterprising entrepreneur purchases two factories to produce widgets. Each factory produces identical products, and each has a production function given by q ¼ kili , 1, 2: i ¼ ﬃﬃﬃﬃﬃﬃ p The factories differ, however, in the amount of capital equipment each has. In particular, factory 1 has k1 ¼ factory 2 has k2 ¼ a. If the entrepreneur wishes to minimize short-run total costs of widget production, how should output be allocated between 100. Rental rates for k and l are given by w 25, whereas $1. ¼ ¼ v the two factories? b. Given that output is optimally allocated between the two factories, calculate the short-run total, average, and marginal cost curves. What is the marginal cost of the 100th widget? The 125th widget? The 200th widget? c. How should the entrepreneur allocate widget production between the two factories in the long run? Calculate the long-run total, average, and marginal cost curves for widget production. d. How would your answer to part (c) change if both factories exhibited diminishing returns to scale? 10.6 Suppose the total-cost function for a ﬁrm is given by a. Use Shephard’s lemma to compute the (constant output) demand functions for inputs l and k. b. Use your results from part (a) to calculate the underlying production function for q. qw2=3v1=3: C ¼ 10.7 Suppose the total-cost function for a ﬁrm is given by C q v ð þ ¼ 2 vwp w : Þ þ a. Use Shephard’s lemma to compute the (constant output) demand function for each input, k and l. b. Use the results from part (a) to compute the underlying production function for q. c. You can check the result by using results from Example 10.2 to show that the CES cost function with s ﬃﬃﬃﬃﬃﬃ generates this total-cost function. 0.5, r 1 ¼ # ¼ 10.8 In a famous article [J. Viner, ‘‘Cost Curves and Supply Curves,’’ Zeitschrift fur Nationalokonomie 3 (September 1931): 23–46], Viner criticized his draftsman who could not draw a family of SAC curves whose points of tangency with the U-shaped AC curve were also the minimum points on each SAC curve. The draftsman protested that such a drawing was impossible to construct. Whom would you support in this debate? Chapter 10: Cost Functions 365 Analytical Problems 10.9 Generalizing the CES cost function The CES production function can be generalized to permit weighting of the inputs. In the two-input case, this function is q f k, l ð ¼ Þ ¼ ½ð ak Þ q g=q: bl q Þ ’ þ ð a. What is the total-cost function for a ﬁrm with this production function? Hint: You can, of course, work this out from scratch; easier perhaps is to use the results from Example 10.2 and reason that the price for a unit of capital input in this production function is v/a and for a unit of labor input is w/b. 1, it can be shown that this production function converges to the Cobb–Douglas form q b. If g 1 and a ¼ b þ ¼ kal b as ¼ r ﬁ 0. What is the total cost function for this particular version of the CES function? c. The relative labor cost share for a two-input production function is given by wl/vk. Show that this share is constant for the Cobb–Douglas function in part (b). How is the relative labor share affected by the parameters a and b? d. Calculate the relative labor cost share for the general CES function introduced above. How is that share affected by changes in w/v? How is the direction of this effect determined by the elasticity of substitution, s? How is it affected by the sizes of the parameters a and b? 10.10 Input demand elasticities The own-price elasticities of contingent input demand for labor and capital are deﬁned as elc , w ¼ @l c @w ( w lc , ekc, v ¼ @kc @v ( v kc : a. Calculate elc, w and ekc, v for each of the cost functions shown in Example 10.2. b. Show that, in general, elc, w þ 0. c. Show that the cross-price derivatives of contingent demand functions are equal—that is, show that @lc/@v this fact to show that slelc, v ¼ total cost (vk/C). ekc, v ¼ @kc/@w. Use skekc, w where sl, sk are, respectively, the share of labor in total cost (wl/C) and of capital in ¼ d. Use the results from parts (b) and (c) to show that slel c,w þ e. Interpret these various elasticity relationships in words and discuss their overall relevance to a general theory of input skekc, w ¼ 0. demand. 10.11 The elasticity of substitution and input demand elasticities The deﬁnition of the (Morishima) elasticity of substitution sij in Equation 10.54 can be recast in terms of input demand elasticities. This illustrates the basic asymmetry in the deﬁnition. a. Show that if only wj changes, sij ¼ b. Show that if only wi changes, sji ¼ c. Show that if the production function takes the general CES form q elasticities are the same: sij ¼ 1/(1 n xq i s. This is the only case in which the Morishima deﬁnition is symmetric. ex c i ,wj # ex c j ,wi # r) 0, then all of the Morishima 1=q for r j ,wj : i ,wi : ex c ex c # ¼ ¼ 6¼ ’ & P 10.12 The Allen elasticity of substitution Many empirical studies of costs report an alternative deﬁnition of the elasticity of substitution between inputs. This alternative deﬁnition was ﬁrst proposed by R. G. D. Allen in the 1930s and further clariﬁed by H. Uzawa in the 1960s. This deﬁnition builds directly on the production function-based elasticity of substitution deﬁned in footnote 6 of Chapter 9: Aij ¼ CijC/CiCj, where the subscripts indicate partial differentiation with respect to various input prices. Clearly, the Allen deﬁnition is symmetric. a. Show that Aij ¼ b. Show that the elasticity of si with respect to the price of input j is related to the Allen elasticity by esi, pj ¼ c. Show that, with only two inputs, Akl ¼ 1 for the Cobb–Douglas case and Akl ¼ d. Read Blackorby and Russell (1989: ‘‘Will the Real Elasticity of Substitution Please Stand Up?’’) to see why the Morishima i ,wj =sj, where sj is the share of input j in total cost. s for the CES case. Aij # : 1 Þ sjð ex c deﬁnition is preferred for most purposes. 366 Part 4: Production and Supply SUGGESTIONS FOR FURTHER READING Allen, R. G. D. Mathematical Analysis for Economists. New York: St. Martin’s Press, 1938, various pages—see index. Complete (though dated) mathematical analysis of substitution possibilities and cost functions. Notation somewhat difﬁcult. Blackorby, C., and R. R. Russell. ‘‘Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/ Uzawa and Morishima Elasticities).’’ American Economic Review (September 1989): 882–88. A nice clariﬁcation of the proper way to measure substitutability among many inputs in production. Argues that the Allen/ Uzawa deﬁnition is largely useless and that the Morishima deﬁnition is by far the best. Ferguson, C. E. The Neoclassical Theory of Production and Distribution. Cambridge: Cambridge University Press, 1969, Chap. 6. Nice development of cost curves; especially strong on graphic analysis. Fuss, M., and D. McFadden. Production Economics: A Dual Approach to Theory and Applications. Amsterdam: NorthHolland, 1978. Difﬁcult and quite complete treatment of the dual relationship between production and cost functions. Some discussion of empirical issues. Knight, H. H. ‘‘Cost of Production and Price over Long and Short Periods.’’ Journal of Political Economics 29 (April 1921): 304–35. Classic treatment of the short-run, long-run distinction. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Chapters 7–9 have a great deal of material on cost functions. Especially recommended are the authors’ discussions of ‘‘reciprocity effects’’ and their treatment of the short-run-long, run distinction as an application of the Le Chatelier principle from physics. Sydsaeter, K., A. Strom, and P. Berck. Economists’ Mathematical Manual, 3rd ed. Berlin: Springer-Verlag, 2000. Chapter 25 provides a succinct summary of the mathematical concepts in this chapter. A nice summary of many input cost functions, but beware of typos. THE TRANSLOG COST FUNCTION EXTENSIONS ¼ The two cost functions studied in Chapter 10 (the Cobb– Douglas and the CES) are very restrictive in the substitution possibilities they permit. The Cobb–Douglas implicitly assumes that s 1 between
any two inputs. The CES permits s to take any value, but it requires that the elasticity of substitution be the same between any two inputs. Because empirical economists would prefer to let the data show what the actual substitution possibilities among inputs are, they have tried to ﬁnd more ﬂexible functional forms. One especially popular such form is the translog cost function, ﬁrst made popular by Fuss and McFadden (1978). In this extension we will look at this function. E10.1 The translog with two inputs In Example 10.2, we calculated the Cobb–Douglas cost b) multi function in the two-input case as C(v, w, q) b). If we take the natural logarithm of this we b)wb/(a þ Bq1/(a þ va/(a þ ¼ + have b ln q Þ ¼ ln B ln C v, w, q ð a 1 That is, the log of total costs is linear in the logs of output and the input prices. The translog function generalizes this by permitting second-order terms in input prices: Þ’ ln v ln w: þ Þ’ Þ’ þ ½ þ ½ a= b= (i) ln C v, w, q ð Þ ¼ ln q a0 þ þ a3(ln v)2 + a4ð a5 ln v ln w, þ a1 ln v + a2 ln w ln w)2 þ (ii) where this function implicitly assumes constant returns to scale (because the coefﬁcient of ln q is 1.0)—although that need not be the case. Some of the properties of this function are: • • • 0. a2 ¼ a5 ¼ a4 ¼ For the function to be homogeneous of degree 1 in 1 and input prices, it must be the case that a1 þ a3 þ a4 þ This function includes the Cobb–Douglas as the special case a3 ¼ 0. Hence the function can be used to test statistically whether the Cobb–Douglas is appropriate. Input shares for the translog function are especially easy (@ ln C)/(@ ln wi). to compute using the result that si ¼ In the two-input case, this yields a5 ¼ • @ ln C @ ln v ¼ @ ln C @ ln w ¼ a1 þ a2 þ sk ¼ sl ¼ 2a3 ln v a5 ln w, þ 2a4 ln w a5 ln v: þ (iii) 0) these In the Cobb–Douglas case (a3 ¼ shares are constant, but with the general translog function they are not. a5 ¼ a4 ¼ • Calculating the elasticity of substitution in the translog case proceeds by using the result given in Problem 10.11 that skl ¼ calculation is el c,w. Making this straightforward (provided one keeps track of how to use logarithms): ek c,w # (iv) ek c,w ¼ @ ln Cv @ ln w ¼ @ ln C ¼ h # @ ln C @ ln v @ ln C v ( # @ ln w $ ln @ ln C @ ln v # @2 ln C @v@w ¼ $ i ln v þ @ ln w @ ln sk @sk a5 sk : sl þ 0 sl # ¼ þ ( Observe that, in the Cobb–Douglas case (a5 ¼ 0), the contingent price elasticity of demand for k with respect to the wage has a simple form: ek c,w ¼ sl. A 2a4=sl similar set of manipulations yields el c,w ¼ # sk þ sk. Bringing in the Cobb–Douglas case, el c,w ¼ # and, these two elasticities together yields skl ¼ ¼ ¼ el c,w a5 sk # ek c,w # sk þ sl þ sla5 # 1 sksl þ 2a4 sl 2ska4 : (v) 1, as Again, in the Cobb–Douglas case we have skl ¼ should have been expected. The Allen elasticity of substitution (see Problem 10.12) for the translog function is Akl ¼ a5 /sk sl. This function can also be used to calculate that the (contingent) crossa5=sk, price elasticity of demand is ek c,w ¼ slAk l ¼ sl þ as was shown previously. Here again, Ak l ¼ 1 in the Cobb–Douglas case. In general, however, the Allen and Morishima deﬁnitions will differ even with just two inputs. þ 1 368 Part 4: Production and Supply E10.2 The many-input translog cost function Most empirical studies include more than two inputs. The translog cost function is especially easy to generalize to these situations. If we assume there are n inputs, each with a price of wi (i 1, … , n) then this function is ¼ C w1, . . . , wn, q ð Þ ¼ ln q a0 þ þ n n ai ln wi n 1 i X ¼ (vi) 0:5 aij ln wi ln wj where we have once again assumed constant returns to scale. This function requires aij ¼ j appears twice in the ﬁnal double sum (which explains the presence of the 0.5 in the expression). For this function to be homogeneous of degree 1 in the input prices, it must be the case that 0. Two useful properties 1 ai ¼ of this function are: P aji, so each term for which i 1 aij ¼ 1 and P 6¼ n i n i ¼ ¼ • Input shares take the linear form n aij ln wj: (vii) si ¼ ai þ 1 j X ¼ Again, this shows why the translog is usually estimated in a share form. Sometimes a term in ln q is also added to the share equations to allow for scale effects on the shares (see Sydsæter, Strøm, and Berck, 2000). The elasticity of substitution between any two inputs in the translog function is given by sjaij # sisj sij ¼ (viii) siajj þ 1 : • Hence substitutability can again be judged directly from the parameters estimated for the translog function. E10.3 Some applications The translog cost function has become the main choice for empirical studies of production. Two factors account for this popularity. First, the function allows a fairly complete characterization of substitution patterns among inputs—it does not require that the data ﬁt any prespeciﬁed pattern. Second, the function’s format incorporates input prices in a ﬂexible way so that one can be reasonably sure that he or she has controlled for such prices in regression analysis. When such control is assured, measures of other aspects of the cost function (such as its returns to scale) will be more reliable. One example of using the translog function to study input substitution is the study by Westbrook and Buckley (1990) of the responses that shippers made to changing relative prices the of moving goods that resulted from deregulation of railroad and trucking industries in the United States. The authors look speciﬁcally at the shipping of fruits and vegetables from the western states to Chicago and New York. They ﬁnd relatively high substitution elasticities among shipping options and so conclude that deregulation had signiﬁcant welfare beneﬁts. Doucouliagos and Hone (2000) provide a similar analysis of deregulation of dairy prices in Australia. They show that changes in the price of raw milk caused dairy processing ﬁrms to undertake signiﬁcant changes in input usage. They also show that the industry adopted signiﬁcant new technologies in response to the price change. An interesting study that uses the translog primarily to judge returns to scale is Latzko’s (1999) analysis of the U.S. mutual fund industry. He ﬁnds that the elasticity of total costs with respect to the total assets managed by the fund is less than 1 for all but the largest funds (those with more than $4 billion in assets). Hence the author concludes that money management exhibits substantial returns to scale. A number of other studies that use the translog to estimate economies of scale focus on municipal services. For example, Garcia and Thomas (2001) look at water supply systems in local French communities. They conclude that there are signiﬁcant operating economies of scale in such systems and that some merging of systems would make sense. Yatchew (2000) reaches a similar conclusion about electricity distribution in small communities in Ontario, Canada. He ﬁnds that there are economies of scale for electricity distribution systems serving up to about 20,000 customers. Again, some efﬁciencies might be obtained from merging systems that are much smaller than this size. References Doucouliagos, H., and P. Hone. ‘‘Deregulation and Subequilibrium in the Australian Dairy Processing Industry.’’ Economic Record (June 2000): 152–62. Fuss, M., and D. McFadden, Eds. Production Economics: A Dual Approach to Theory and Applications. Amsterdam: North Holland, 1978. Garcia, S., and A. Thomas. ‘‘The Structure of Municipal Water Supply Costs: Application to a Panel of French Local Communities.’’ Journal of Productivity Analysis (July 2001): 5–29. Latzko, D. ‘‘Economies of Scale in Mutual Fund Administra- tion.’’ Journal of Financial Research (Fall 1999): 331–39. Sydsæter, K., A. Strøm, and P. Berck. Economists’ Mathemati- cal Manual, 3rd ed. Berlin: Springer-Verlag, 2000. Westbrook, M. D., and P. A. Buckley. ‘‘Flexible Functional Forms and Regularity: Assessing the Competitive Relationship between Truck and Rail Transportation.’’ Review of Economics and Statistics (November 1990): 623–30. Yatchew, A. ‘‘Scale Economies in Electricity Distribution: A Semiparametric Analysis.’’ Journal of Applied Econometrics (March/April 2000): 187–210. This page intentionally left blank C H A P T E R ELEVEN Profit Maximization In Chapter 10 we examined the way in which ﬁrms minimize costs for any level of output they choose. In this chapter we focus on how the level of output is chosen by proﬁtmaximizing ﬁrms. Before investigating that decision, however, it is appropriate to discuss brieﬂy the nature of ﬁrms and the ways in which their choices should be analyzed. The Nature and Behavior of Firms In this chapter, we delve deeper into the analysis of decisions made by suppliers in the market. The analysis of the supply/ﬁrm side of the market raises questions that did not come up in our previous analysis of the demand/consumer side. Whereas consumers are easy to identify as single individuals, ﬁrms come in all shapes and sizes, ranging from a corner ‘‘mom and pop’’ grocery store to a vast modern corporation, supplying hundreds of different products produced in factories operating across the globe. Economists have long puzzled over what determines the size of ﬁrms, how their management is structured, what sort of ﬁnancial instruments should be used to fund needed investment, and so forth. The issues involved turn out to be rather deep and philosophical. To make progress in this chapter, we will continue to analyze the standard ‘‘neoclassical’’ model of the ﬁrm, which brushes most of these deeper issues aside. We will provide only a hint of the deeper issues involved, returning to a fuller discussion in the Extensions to this chapter. Simple model of a ﬁrm Throughout Part 4, we have been examining a simple model of the ﬁrm without being explicit about the assumptions involved. It is worth being a bit more explicit here. The ﬁrm has a technology given by the production function, say f (k, l). The ﬁrm is run by an entrepreneur who makes all
the decisions and receives all the proﬁts and losses from the ﬁrm’s operations. The combination of these elements—production technology, entrepreneur, and inputs used (labor l, capital k, and others)—together constitutes what we will call the ‘‘ﬁrm.’’ The entrepreneur acts in his or her own self-interest, typically leading to decisions that maximize the ﬁrm’s proﬁts, as we will see. Complicating factors Before pushing ahead further with the analysis of the simple model of the ﬁrm, which will occupy most of this chapter, we will hint at some complicating factors. In the simple model just described, a single party—the entrepreneur—makes all the decisions and receives all the returns from the ﬁrm’s operations. With most large corporations, decisions and returns are separated among many parties. Shareholders are really the owners 371 372 Part 4: Production and Supply of the corporation, receiving returns in the form of dividends and stock returns. But shareholders do not run the ﬁrm; the average shareholder may own hundreds of different ﬁrms’ stock through mutual funds and other holdings and could not possibly have the time or expertise to run all these ﬁrms. The ﬁrm is run on shareholders’ behalf usually by the chief executive ofﬁcer (CEO) and his or her management team. The CEO does not make all the decisions but delegates most to managers at one of any number of levels in a complicated hierarchy. The fact that ﬁrms are often not run by the owner leads to another complication. Whereas the shareholders may like proﬁts to be maximized, the manager may act in his or her own interest rather than the interests of the shareholders. The manager may prefer the prestige from expanding the business empire beyond what makes economic sense, may seek to acquire expensive perks, and may shy away from proﬁtable but uncomfortable actions such as ﬁring redundant workers. Different mechanisms may help align the manager’s interests with those of the shareholder. Managerial compensation in the form of stock and stock options may provide incentives for proﬁt maximization as might the threat of ﬁring if a poorly performing ﬁrm goes bankrupt or is taken over by a corporate raider. But there is no telling that such mechanisms will work perfectly. Even a concept as simple as the size of the ﬁrm is open to question. The simple deﬁnition of the ﬁrm includes all the inputs it uses to produce its output, for example, all the machines and factories involved. If part of this production process is outsourced to another ﬁrm using its machines and factories, then several ﬁrms rather than one are responsible for supply. A classic example is provided by the automaker, General Motors (GM).1 Initially GM purchased the car bodies from another ﬁrm, Fisher Body, who designed and made these to order; GM was only responsible for ﬁnal assembly of the body with the other auto parts. After experiencing a sequence of supply disruptions over several decades, GM decided to acquire Fisher Body in 1926. Overnight, much more of the production—the construction of the body and ﬁnal assembly—was concentrated in a single ﬁrm. What then should we say about the size of a ﬁrm in the auto-making business? Is the combination of GM and Fisher Body after the acquisition or the smaller GM beforehand a better deﬁnition of the ‘‘ﬁrm’’ in this case? Should we expect the acquisition of Fisher Body to make any real economic difference to the auto market, say, reducing input supply disruptions, or is it a mere name change? These are deep questions we will touch on in the Extensions to this chapter. For now, we will take the size and nature of the ﬁrm as given, speciﬁed by the production function f(k, l). Relationship to consumer theory Part 2 of this book was devoted to understanding the decisions of consumers on the demand side of the market; this Part 4 is devoted to understanding ﬁrms on the supply side. As we have already seen, there are many common elements between the two analyses, and much of the same mathematical methods can be used in both. There are two essential differences that merit all the additional space devoted to the study of ﬁrms. First, as just discussed, ﬁrms are not individuals but can be much more complicated organizations. We will mostly ‘‘ﬁnesse’’ this difference by assuming that the ﬁrm is represented by the entrepreneur as an individual decision-maker, dealing with the complications in more detail in the Extensions. 1GM’s acquisition of Fisher Body has been extensively analyzed by economists. See, for example, B. Klein, ‘‘Vertical Integration as Organization Ownership: the Fisher-Body–General Motors Relationship Revisited,’’ Journal of Law, Economics and Organization (Spring 1988): 199–213. Chapter 11: Prof it Maximization 373 Another difference between ﬁrms and consumers is that we can be more concrete about the ﬁrm’s objectives than a consumer’s. With consumers, there is ‘‘no accounting for taste.’’ There is no telling why one consumer likes hot dogs more than hamburgers and another consumer the opposite. By contrast, it is usually assumed that ﬁrms do not have an inherent preference regarding the production of hot dogs or hamburgers; the natural assumption is that it produces the product (or makes any number of other decisions) earning the most proﬁt. There are certainly a number of caveats with the proﬁt-maximization assumption, but if we are willing to make it, we can push the analysis farther than we did with consumer theory. Proﬁt Maximization Most models of supply assume that the ﬁrm and its manager pursue the goal of achieving the largest economic proﬁts possible. The following deﬁnition embodies this assumption and also reminds the reader of the deﬁnition of economic proﬁts Proﬁt-maximizing ﬁrm. The ﬁrm chooses both its inputs and its outputs with the sole goal of maximizing economic proﬁts, the difference between its total revenues and its total economic costs. This assumption—that ﬁrms seek maximum economic proﬁts—has a long history in economic literature. It has much to recommend it. It is plausible because ﬁrm owners may indeed seek to make their asset as valuable as possible and because competitive markets may punish ﬁrms that do not maximize proﬁts. This assumption comes with caveats. We already noted in the previous section that if the manager is not the owner of the ﬁrm, he or she may act in a self-interested way and not try to maximize owner wealth. Even if the manager is also the owner, he or she may have other concerns besides wealth, say, reducing pollution at a power plant or curing illness in developing countries in a pharmaceutical lab. We will put such other objectives aside for now, not because they are unrealistic but rather because it is hard to say exactly which of the broad set of additional goals are most important to people and how much they matter relative to wealth. The social goals may be addressed more efﬁciently by maximizing the ﬁrm’s proﬁt and then letting the owners use their greater wealth to fund other goals directly through taxes or charitable contributions. In any event, a rich set of theoretical results explaining actual ﬁrms’ decisions can be derived using the proﬁt-maximization assumption; thus, we will push ahead with it for most of the rest of the chapter. Proﬁt maximization and marginalism If ﬁrms are strict proﬁt maximizers, they will make decisions in a ‘‘marginal’’ way. The entrepreneur will perform the conceptual experiment of adjusting those variables that can be controlled until it is impossible to increase proﬁts further. This involves, say, looking at the incremental, or ‘‘marginal,’’ proﬁt obtainable from producing one more unit of output, or at the additional proﬁt available from hiring one more laborer. As long as this incremental proﬁt is positive, the extra output will be produced or the extra laborer will be hired. When the incremental proﬁt of an activity becomes zero, the entrepreneur has pushed that activity far enough, and it would not be proﬁtable to go further. In this chapter, we will explore the consequences of this assumption by using increasingly sophisticated mathematics. 374 Part 4: Production and Supply Output choice First we examine a topic that should be familiar: what output level a ﬁrm will produce to obtain maximum proﬁts. A ﬁrm sells some level of output, q, at a market price of p per unit. Total revenues (R) are given by q p ð where we have allowed for the possibility that the selling price the ﬁrm receives might be affected by how much it sells. In the production of q, certain economic costs are incurred and, as in Chapter 10, we will denote these by C (q). q Þ ¼ ð (11:1) Þ $ q, R The difference between revenues and costs is called economic proﬁts (p). We will recap this deﬁnition here for reference Economic proﬁt. A ﬁrm’s economic proﬁts are the difference between its revenues and costs: economic profits 11:2) Because both revenues and costs depend on the quantity produced, economic proﬁts will also. The necessary condition for choosing the value of q that maximizes proﬁts is found by setting the derivative of Equation 11.2 with respect to q equal to 0:2 dp dq ¼ p 0 q ð Þ ¼ dR dq % dC dq ¼ 0, so the ﬁrst-order condition for a maximum is that dR dq ¼ dC dq : (11:3) (11:4) In the previous chapter, the derivative dC/dq was deﬁned to be marginal cost, MC. The other derivative, dR/dq, can be deﬁned analogously as follows Marginal revenue. Marginal revenue is the change in total revenue R resulting from a change in output q: marginal revenue MR ¼ dR dq : ¼ (11:5) With the deﬁnitions of MR and MC in hand, we can see that Equation 11.4 is a mathematical statement of the ‘‘marginal revenue equals marginal cost’’ rule usually studied in introductory economics courses. The rule is important enough to be highlighted as an optimization principle Proﬁt maximization. To maximize economic proﬁts, the ﬁrm should choose output q& at which marginal revenue is equal to marginal cost. Th
at is, MR q& ð Þ ¼ MC q& ð : Þ (11:6) 2Notice that this is an unconstrained maximization problem; the constraints in the problem are implicit in the revenue and cost functions. Speciﬁcally, the demand curve facing the ﬁrm determines the revenue function, and the ﬁrm’s production function (together with input prices) determines its costs. Chapter 11: Prof it Maximization 375 Second-order conditions Equation 11.4 or 11.5 is only a necessary condition for a proﬁt maximum. For sufﬁciency, it is also required that d2p dq2 q dp 0 ð dq Þ ¼ q q, q& q ¼ (11:7) or that ‘‘marginal’’ proﬁt must decrease at the optimal level of output, q&. For q less than q&, proﬁt must increase [p 0(q) > 0]; for q greater than q&, proﬁt must decrease [p 0(q) < 0]. Only if this condition holds has a true maximum been achieved. Clearly the condition holds if marginal revenue decreases (or remains constant) in q and marginal cost increases in q. Graphical analysis These relationships are illustrated in Figure 11.1, where the top panel depicts typical cost and revenue functions. For low levels of output, costs exceed revenues; thus, economic proﬁts are negative. In the middle ranges of output, revenues exceed costs; this means that proﬁts are positive. Finally, at high levels of output, costs rise sharply and again exceed revenues. The vertical distance between the revenue and cost curves (i.e., proﬁts) is shown in Figure 11.1b. Here proﬁts reach a maximum at q&. At this level of output it is also true that the slope of the revenue curve (marginal revenue) is equal to the slope of the cost curve (marginal cost). It is clear from the ﬁgure that the sufﬁcient conditions for a maximum are also satisﬁed at this point because proﬁts are increasing to the left of q& and decreasing to the right of q&. Therefore, output level q& is a true proﬁt maximum. This is not so for output level q&&. Although marginal revenue is equal to marginal cost at this output, proﬁts are in fact at a local minimum there. Marginal Revenue Marginal revenue is simple to compute when a ﬁrm can sell all it wishes without having any effect on market price. The extra revenue obtained from selling one more unit is just this market price. A ﬁrm may not always be able to sell all it wants at the prevailing market price, however. If it faces a downward-sloping demand curve for its product, then more output can be sold only by reducing the good’s price. In this case the revenue obtained from selling one more unit will be less than the price of that unit because to get consumers to take the extra unit, the price of all other units must be lowered. This result can be easily demonstrated. As before, total revenue (R) is the product of the quantity sold (q) times the price at which it is sold (p), which may also depend on q. Using the product rule to compute the derivative, marginal revenue is MR q Þ ¼ ð dR dq ¼ p d ½ q ð dq Þ $ q ( p q $ þ ¼ dp dq : (11:8) Notice that the marginal revenue is a function of output. In general, MR will be different for different levels of q. From Equation 11.8 it is easy to see that if price does not 0), marginal revenue will be equal to price. In change as quantity increases (dp/dq ¼ 376 Part 4: Production and Supply FIGURE 11.1 Marginal Revenue Must Equal Marginal Cost for Proﬁt Maximization Proﬁts, deﬁned as revenues (R) minus costs (C), reach a maximum when the slope of the revenue function (marginal revenue) is equal to the slope of the cost function (marginal cost). This equality is only a necessary condition for a maximum, as may be seen by comparing points q& (a true maximum) and q&& (a local minimum), points at which marginal revenue equals marginal cost. C R q ** q * Output per period q * Output per period Revenues, costs (a) Proﬁts 0 Losses (b) this case we say that the ﬁrm is a price-taker because its output decisions do not inﬂuence the price it receives. On the other hand, if price decreases as quantity increases (dp/dq < 0), marginal revenue will be less than price. A proﬁt-maximizing manager must know how increases in output will affect the price received before making an optimal output decision. If increases in q cause market price to decrease, this must be taken into account. Chapter 11: Prof it Maximization 377 EXAMPLE 11.1 Marginal Revenue from a Linear Demand Function Suppose a shop selling sub sandwichs (also called grinders, torpedoes, or, in Philadelphia, hoagies) faces a linear demand curve for its daily output over period (q) of the form Solving for the price the shop receives, we have 100 q ¼ % 10p: q % 10 þ p ¼ 10, and total revenues (as a function of q) are given by The sub ﬁrm’s marginal revenue function is pq R ¼ ¼ q2 % 10 þ 10q: MR dR dq ¼ q % 5 þ ¼ 10, (11:9) (11:10) (11:11) (11:12) and in this case MR < p for all values of q. If, for example, the ﬁrm produces 40 subs per day, Equation 11.10 shows that it will receive a price of $6 per sandwich. But at this level of output Equation 11.12 shows that MR is only $2. If the ﬁrm produces 40 subs per day, then total revenue will be $240 ( 40), whereas if it produced 39 subs, then total revenue would be 39) because price will increase slightly when less is produced. Hence the $6.1 $238 ( marginal revenue from the 40th sub sold is considerably less than its price. Indeed, for q 50, marginal revenue is zero (total revenues are a maximum at $250 50), and any further expansion in daily sub output will result in a reduction in total revenue to the ﬁrm. $6 $5 * * ¼ * ¼ ¼ ¼ To determine the proﬁt-maximizing level of sub output, we must know the ﬁrm’s marginal costs. If subs can be produced at a constant average and marginal cost of $4, then Equation MC at a daily output of 30 subs. With this level of output, each sub will 11.12 shows that MR $4) Æ 30]. Although price exceeds average and marginal sell for $7, and proﬁts are $90 [ cost here by a substantial margin, it would not be in the ﬁrm’s interest to expand output. With q % $4.00) Æ 35]. Marginal revenue, not price, is the primary determinant of proﬁt-maximizing behavior. 35, for example, price will decrease to $6.50 and proﬁts will decrease to $87.50 [ ($6.50 ($7 % ¼ ¼ ¼ ¼ QUERY: How would an increase in the marginal cost of sub production to $5 affect the output decision of this ﬁrm? How would it affect the ﬁrm’s proﬁts? Marginal revenue and elasticity The concept of marginal revenue is directly related to the elasticity of the demand curve facing the ﬁrm. Remember that the elasticity of demand (eq, p) is deﬁned as the percentage change in quantity demanded that results from a 1 percent change in price: eq, p ¼ dq=q dp=p ¼ dq dp $ p q : Now, this deﬁnition can be combined with Equation 11.8 to give MR p þ ¼ q dp dq ¼ $ q p : dp dq eq , p : # (11:13) As long as the demand curve facing the ﬁrm is negatively sloped, then eq, p < 0 and marginal revenue will be less than price, as we have already shown. If demand is elastic 378 Part 4: Production and Supply TABLE 11.1 RELATIONSHIP BETWEEN ELASTICITY AND MARGINAL REVENUE 1 eq, p < % eq, p ¼ % 1 eq, p > % 1 MR > 0 0 MR ¼ MR < 0 % (eq, p < 1), then marginal revenue will be positive. If demand is elastic, the sale of one more unit will not affect price ‘‘very much,’’ and hence more revenue will be yielded by the sale. In fact, if demand facing the ﬁrm is inﬁnitely elastic (eq , p ¼ %1 ), marginal revenue will equal price. The ﬁrm is, in this case, a price-taker. However, if demand is inelastic (eq, p > 1), marginal revenue will be negative. Increases in q can be obtained only through ‘‘large’’ decreases in market price, and these decreases will cause total revenue to decrease. % The relationship between marginal revenue and elasticity is summarized by Table 11.1. Price–marginal cost markup If we assume the ﬁrm wishes to maximize proﬁts, this analysis can be extended to illustrate the connection between price and marginal cost. Setting MR MC in Equation 11.13 yields ¼ or, after rearranging, MC ¼ p 1 " þ 1 eq, p # p MC % p 1 eq, p ¼ 1 eq, pj j : ¼ % (11:14) where the last equality holds if demand is downward sloping and thus eq, p < 0. This formula for the percentage ‘‘markup’’ of price over marginal cost is sometimes called the Lerner index after the economist Abba Lerner, who ﬁrst proposed it in the 1930s. The markup depends in a speciﬁc way on the elasticity of demand facing the ﬁrm. First, notice that this demand must be elastic (eq, p < 1) for this formula to make any sense. If demand were inelastic, the ratio in Equation 11.14 would be greater than 1, which is impossible if a positive MC is subtracted from a positive p in the numerator. This simply reﬂects that, when demand is inelastic, marginal revenue is negative and cannot be equated to a positive marginal cost. It is important to stress that it is the demand facing the ﬁrm that must be elastic. This may be consistent with an inelastic market demand for the product in question if the ﬁrm faces competition from other ﬁrms producing the same good. % Equation 11.14 implies that the percentage markup over marginal cost will be higher 1. If the demand facing the ﬁrm is inﬁnitely elastic (perhaps because , and there is no MC). On the other hand, with an elasticity of demand of, say, eq, p ¼ % 2, the closer eq , p is to there are many other ﬁrms producing the same good), then eq, p ¼ %1 markup (p the markup over marginal cost will be 50 percent of price; that is, ( p MC)/p 1/2. % ¼ % ¼ Marginal revenue curve Any demand curve has a marginal revenue curve associated with it. If, as we sometimes assume, the ﬁrm must sell all its output at one price, it is convenient to think of the demand curve facing the ﬁrm as an average revenue curve. That is, the demand curve shows the revenue per unit (in other words, the price) yielded by alternative output choices. The marginal revenue curve, on the other hand, shows the extra revenue FIGURE 11.2 Market Demand Curve and Associated Marginal Revenue Curve Chapter 11: Prof it Maximization 379 Because the
demand curve is negatively sloped, the marginal revenue curve will fall below the demand (‘‘average revenue’’) curve. For output levels beyond q1, MR is negative. At q1, total revenues (p1 Æ q1) are a maximum; beyond this point, additional increases in q cause total revenues to decrease because of the concomitant decreases in price. Price p1 0 D (average revenue) q1 Quantity per period MR provided by the last unit sold. In the usual case of a downward-sloping demand curve, the marginal revenue curve will lie below the demand curve because, according to Equation 11.8, MR < p. In Figure 11.2 we have drawn such a curve together with the demand curve from which it was derived. Notice that for output levels greater than q1, marginal revenue is negative. As output increases from 0 to q1, total revenues (p Æ q) increase. However, at q1 total revenues (p1 Æ q1) are as large as possible; beyond this output level, price decreases proportionately faster than output increases. In Part 2 we talked in detail about the possibility of a demand curve’s shifting because of changes in income, prices of other goods, or preferences. Whenever a demand curve does shift, its associated marginal revenue curve shifts with it. This should be obvious because a marginal revenue curve cannot be calculated without referring to a speciﬁc demand curve. EXAMPLE 11.2 The Constant Elasticity Case In Chapter 5 we showed that a demand function of the form has a constant price elasticity of demand equal to function for this function, ﬁrst solve for p: b. To compute the marginal revenue % apb q ¼ (11:15) 1=b p ¼ 1 a " # q1=b ¼ k q1=b, (11:16) 380 Part 4: Production and Supply where k ¼ (1/a)1/b. Hence and pq R ¼ ¼ kqð 1 þ =b b Þ MR ¼ dR=dq 1 þ b ¼ b kq1=b b p: 1 þ b ¼ (11:17) ¼ 0.5p. For a more elastic case, suppose b For this particular function, MR is proportional to price. If, for example, eq, p ¼ 2, then 0.9p. The MR curve MR ¼ p; , then MR approaches the demand curve as demand becomes more elastic. Again, if b that is, in the case of inﬁnitely elastic demand, the ﬁrm is a price-taker. For inelastic demand, on the other hand, MR is negative (and proﬁt maximization would be impossible). 10; then MR ¼ %1 ¼ % ¼ % ¼ b QUERY: Suppose demand depended on other factors in addition to p. How would this change the analysis of this example? How would a change in one of these other factors shift the demand curve and its marginal revenue curve? Short-Run Supply by a Price-Taking Firm We are now ready to study the supply decision of a proﬁt-maximizing ﬁrm. In this chapter we will examine only the case in which the ﬁrm is a price-taker. In Part 6 we will look at other cases in considerably more detail. Also, we will focus only on supply decisions in the short run here. Long-run questions concern entry and exit by ﬁrms and are the primary focus of the next chapter. Therefore, the ﬁrm’s set of short-run cost curves is the appropriate model for our analysis. ¼ Proﬁt-maximizing decision Figure 11.3 shows the ﬁrm’s short-run decision. The market price3 is given by P&. Therefore, the demand curve facing the ﬁrm is a horizontal line through P&. This line is labeled MR as a reminder that an extra unit can always be sold by this price-taking ﬁrm P& without affecting the price it receives. Output level q& provides maximum proﬁts because at q& price is equal to short-run marginal cost. The fact that proﬁts are positive can be seen by noting that price at q& exceeds average costs. The ﬁrm earns a proﬁt on each unit sold. If price were below average cost (as is the case for P&&&), the ﬁrm would have a loss on each unit sold. If price and average cost were equal, proﬁts would be zero. Notice that at q& the marginal cost curve has a positive slope. This is required if proﬁts are to be a MC on a negatively sloped section of the marginal cost curve, then true maximum. If P this would not be a point of maximum proﬁts because increasing output would yield more in revenues (price times the amount produced) than this production would cost (marginal cost would decrease if the MC curve has a negative slope). Consequently, proﬁt maximization requires both that P MC and that marginal cost increase at this point.4 ¼ ¼ 3We will usually use an uppercase italic P to denote market price here and in later chapters. When notation is complex, however, we will sometimes revert to using a lowercase p. 4Mathematically: because proﬁt maximization requires (the ﬁrst-order condition) p q ð Þ ¼ Pq C , q Þ ð % and (the second-order condition) p 0 q Þ ¼ ð P % MC q Þ ¼ ð 0 q Þ Hence, it is required that MC 0(q) > 0; marginal cost must be increasing. Þ ¼ % MC 0 q ð p 00 ð < 0: Chapter 11: Prof it Maximization 381 FIGURE 11.3 Short-Run Supply Curve for a Price-Taking Firm In the short run, a price-taking ﬁrm will produce the level of output for which SMC example, the ﬁrm will produce q&. The SMC curve also shows what will be produced at other prices. For prices below SAVC, however, the ﬁrm will choose to produce no output. The heavy lines in the ﬁgure represent the ﬁrm’s short-run supply curve. P. At P&, for ¼ Market price P ** P * = MR P * Ps 0 SMC SAC SAVC q * q * q ** Quantity per period The ﬁrm’s short-run supply curve The positively sloped portion of the short-run marginal cost curve is the short-run supply curve for this price-taking ﬁrm. That curve shows how much the ﬁrm will produce for every possible market price. For example, as Figure 11.3 shows, at a higher price of P&& the ﬁrm will produce q&& because it is in its interest to incur the higher marginal costs entailed by q&&. With a price of P&&&, on the other hand, the ﬁrm opts to produce less (q&&&) because only a lower output level will result in lower marginal costs to meet this lower price. By considering all possible prices the ﬁrm might face, we can see by the marginal cost curve how much output the ﬁrm should supply at each price. The shutdown decision. For low prices we must be careful about this conclusion. Should market price fall below Ps (the ‘‘shutdown price’’), the proﬁt-maximizing decision would be to produce nothing. As Figure 11.3 shows, prices less than Ps do not cover average variable costs. There will be a loss on each unit produced in addition to the loss of all ﬁxed costs. By shutting down production, the ﬁrm must still pay ﬁxed costs but avoids the losses incurred on each unit produced. Because, in the short run, the ﬁrm cannot close down and avoid all costs, its best decision is to produce no output. On the other hand, a price only slightly above Ps means the ﬁrm should produce some output. Although proﬁts may be negative (which they will be if price falls below short-run average total costs, the case at P&&&), the proﬁt-maximizing decision is to continue production as long as variable costs are covered. Fixed costs must be paid in any case, and any price that covers variable costs will provide revenue as an offset to 382 Part 4: Production and Supply the ﬁxed costs.5 Hence we have a complete description of this ﬁrm’s supply decisions in response to alternative prices for its output. These are summarized in the following deﬁnition Short-run supply curve. The ﬁrm’s short-run supply curve shows how much it will produce at various possible output prices. For a proﬁt-maximizing ﬁrm that takes the price of its output as given, this curve consists of the positively sloped segment of the ﬁrm’s short-run marginal cost above the point of minimum average variable cost. For prices below this level, the ﬁrm’s proﬁtmaximizing decision is to shut down and produce no output. Of course, any factor that shifts the ﬁrm’s short-run marginal cost curve (such as changes in input prices or changes in the level of ﬁxed inputs used) will also shift the short-run supply curve. In Chapter 12 we will make extensive use of this type of analysis to study the operations of perfectly competitive markets. EXAMPLE 11.3 Short-Run Supply In Example 10.5 we calculated the short-run total-cost production function as function for the Cobb–Douglas SC v, w, q, k1Þ ¼ ð vk1 þ wq1=bk% 1 a=b , (11:18) where k1 is the level of capital input that is held constant in the short run.6 Short-run marginal cost is easily computed as SMC v, w, q, k1Þ ¼ ð @SC @q ¼ w b qð 1 % b Þ a=b =bk% 1 : (11:19) Notice that short-run marginal cost increases in output for all values of q. Short-run proﬁt maximization for a price-taking ﬁrm requires that output be chosen so that market price (P) is equal to short-run marginal cost: SMC w b ¼ qð 1 % b Þ a=b =bk% 1 P, ¼ and we can solve for quantity supplied as b= ð 1 % b Þ % ka= 1 1 % b Þ ð P b= ð 1 % b Þ: q ¼ w b " # (11:20) (11:21) This supply function provides a number of insights that should be familiar from earlier economics courses: (1) The supply curve is positively sloped—increases in P cause the ﬁrm to 5Some algebra may clarify matters. We know that total costs equal the sum of ﬁxed and variable costs, and that proﬁts are given by p If q ¼ 0, then variable costs and revenues are 0, and thus The ﬁrm will produce something only if p > SC ¼ SFC þ SVC, R SC ¼ % P q $ % ¼ SFC % SVC: p ¼ % SFC: % SFC. But that means that p q > SVC or p > SVC=q: $ 6Because capital input is held constant, the short-run cost function exhibits increasing marginal cost and will therefore yield a unique proﬁt-maximizing output level. If we had used a constant returns-to-scale production function in the long run, there would have been no such unique output level. We discuss this point later in this chapter and in Chapter 12. Chapter 11: Prof it Maximization 383 produce more because it is willing to incur a higher marginal cost;7 (2) the supply curve is shifted to the left by increases in the wage rate, w—that is, for any given output price, less is supplied with a higher wage; (3) the supply curve is shifted outward by increases in capital input, k1—with more capital in the short run, the ﬁrm incurs a given level of short-run marginal cost at a higher
output level; and (4) the rental rate of capital, v, is irrelevant to short-run supply decisions because it is only a component of ﬁxed costs. Numerical example. We can pursue once more the numerical example from Example 10.5, where a 80. For these speciﬁc parameters, the supply function is 0.5, v 3, w b ¼ ¼ ¼ ¼ p1 $ ¼ 40 P w ¼ 40P 12 ¼ 10P 3 : $ (11:22) 12, and k1 ¼ w k1Þ 0:5 $ ð % 1 1 % q ¼ $ That this computation is correct can be checked by comparing the quantity supplied at various prices with the computation of short-run marginal cost in Table 10.2. For example, if P 12, 40 will be supplied, and Table 10.2 shows that this then the supply function predicts that q 24, an output level of 80 would will agree with the P be supplied, and again Table 10.2 shows that when q 6) would cause less to be produced (q ¼ SMC rule. If price were to double to P 24. A lower price (say, P ¼ 80, SMC 20). ¼ ¼ ¼ ¼ ¼ Before adopting Equation 11.22 as the supply curve in this situation, we should also check 0 SMC rule? From Equation 11.18 we know that short-run variable costs are the ﬁrm’s shutdown decision. Is there a price where it would be more proﬁtable to produce q than to follow the P given by ¼ ¼ and so SVC ¼ wq 1=bk% 1 a=b SVC q ¼ wqð 1 % b Þ a=b =bk% 1 : (11:23) (11:24) ¼ ¼ A comparison of Equation 11.24 with Equation 11.19 shows that SVC/q < SMC for all values of q provided that b < 1. Thus, in this problem there is no price low enough such that, by following the P SMC rule, the ﬁrm would lose more than if it produced nothing. ¼ In our numerical example, consider the case P 10. Total revenue would be R 3. With such a low price, the ﬁrm would opt 255 for q ¼ 225. Although the situation is dismal (see Table 10.1). Hence proﬁts would be p ¼ for the ﬁrm, it is better than opting for q 0. If it produces nothing, it avoids all variable (labor) costs but still loses 240 in ﬁxed costs of capital. By producing 10 units of output, its revenues 15) and contribute 15 to offset slightly the loss of cover variable costs (R ﬁxed costs. 30, and total short-run costs would be SC ¼ % SVC SC 30 15 % % % ¼ ¼ ¼ ¼ ¼ R QUERY: How would you graph the short-run supply curve in Equation 11.22? How would the curve be shifted if w rose to 15? How would it be shifted if capital input increased to k1 ¼ 100? How would the short-run supply curve be shifted if v fell to 2? Would any of these changes alter the ﬁrm’s determination to avoid shutting down in the short run? Proﬁt Functions Additional insights into the proﬁt-maximization process for a price-taking ﬁrm8 can be obtained by looking at the proﬁt function. This function shows the ﬁrm’s (maximized) proﬁts as depending only on the prices that the ﬁrm faces. To understand the logic of its construction, remember that economic proﬁts are deﬁned as 7In fact, the short-run elasticity of supply can be read directly from Equation 11.21 as b/(1 8Much of the analysis here would also apply to a ﬁrm that had some market power over the price it received for its product, but we will delay a discussion of that possibility until Part 5. b). % 384 Part 4: Production and Supply p Pq C P f k, l vk wl: (11:25) ¼ % Þ % ð ¼ Only the variables k and l [and also q f (k, l)] are under the ﬁrm’s control in this expression. The ﬁrm chooses levels of these inputs to maximize proﬁts, treating the three prices P, v, and w as ﬁxed parameters in its decision. Looked at in this way, the ﬁrm’s maximum proﬁts ultimately depend only on these three exogenous prices (together with the form of the production function). We summarize this dependence by the proﬁt function Proﬁt function. The ﬁrm’s proﬁt function shows its maximal proﬁts as a function of the prices that the ﬁrm faces: P, v, w P ð Þ ¼ max k, l k, l p ð Þ ¼ Pf max k, l ½ k, l ð Þ % vk wl : ( % (11:26) In this deﬁnition we use an upper case P to indicate that the value given by the function is the maximum proﬁts obtainable given the prices. This function implicitly incorporates the form of the ﬁrm’s production function—a process we will illustrate in Example 11.4. The proﬁt function can refer to either long-run or short-run proﬁt maximization, but in the latter case we would need also to specify the levels of any inputs that are ﬁxed in the short run. Properties of the proﬁt function As for the other optimized functions we have already looked at, the proﬁt function has a number of properties that are useful for economic analysis. 1. Homogeneity. A doubling of all the prices in the proﬁt function will precisely double proﬁts—that is, the proﬁt function is homogeneous of degree 1 in all prices. We have already shown that marginal costs are homogeneous of degree 1 in input prices; hence a doubling of input prices and a doubling of the market price of a ﬁrm’s output will not change the proﬁt-maximizing quantity it decides to produce. However, because both revenues and costs have doubled, proﬁts will double. This shows that with pure inﬂation (where all prices rise together) ﬁrms will not change their production plans, and the levels of their proﬁts will just keep up with that inﬂation. 2. Proﬁt functions are nondecreasing in output price, P. This result seems obvious—a ﬁrm could always respond to an increase in the price of its output by not changing its input or output plans. Given the deﬁnition of proﬁts, they must increase. Hence if the ﬁrm changes its plans, it must be doing so to make even more proﬁts. If proﬁts were to decrease, the ﬁrm would not be maximizing proﬁts. 3. Proﬁt functions are nonincreasing in input prices, v, and w. Again, this feature of the proﬁt function seems obvious. A proof is similar to that used above in our discussion of output prices. 4. Proﬁt functions are convex in output prices. This important feature of proﬁt functions says that the proﬁts obtainable by averaging those available from two different output prices will be at least as large as those obtainable from the average9 of the two prices. Mathematically, P P1, v, w ð Þ þ 2 P2, v, w P ð P Þ + P1 þ 2 " P2 , v, w : # (11:27) 9Although we only discuss a simple averaging of prices here, it is clear that with convexity a condition similar to Equation 11.27 holds for any weighted average price P 1. P2 where 0 1 t t tP1 þ ð ¼ % Þ , , Chapter 11: Prof it Maximization 385 The intuitive reason for this convexity is that, when ﬁrms can freely adapt their decisions to two different prices, better results are possible than when they can make only one set P2)/2 of choices in response to the single average price. More formally, let P3 ¼ and let qi, ki, li represent the proﬁt-maximizing output and input choices for these various prices. Then (P1 þ P ð P3, v, w Þ - P3q3 % vk3 % wl3 ¼ , - P1q3 % P1q1 % vk3 % 2 vk1 % 2 wl3 wl1 P P1, v, w ð P Þ þ 2 þ P2q3 % P2q2 % þ P2, v, w ð Þ wl3 wl2 vk3 % 2 vk2 % 2 , (11:28) which proves Equation 11.27. The key step is Equation 11.28. Because (q1, k1, l1) is the proﬁt-maximizing combination of output and inputs when the market price is P1, it must generate as much proﬁt as any other choice, including (q3, k3, l3). By similar reasoning, the proﬁt from (q2, k2, l2) is at least as much as that from (q3, k3, l3) when the market price is P2. The convexity of the proﬁt function has many applications to topics such as price stabilization. Envelope results Because the proﬁt function reﬂects an underlying process of unconstrained maximization, we may also apply the envelope theorem to see how proﬁts respond to changes in output and input prices. This application of the theorem yields a variety of useful results. Speciﬁcally, using the deﬁnition of proﬁts shows that @P ð P, v, w @P Þ q ð ¼ P, v, w , Þ @P ð P, v, w @v Þ k P, v, w ð , Þ ¼ % @P ð P, v, w @w Þ l ð ¼ % P, v, w : Þ (11:29) (11:30) (11:31) Again, these equations make intuitive sense: A small change in output price will increase proﬁts in proportion to how much the ﬁrm is producing, whereas a small increase in the price of an input will reduce proﬁts in proportion to the amount of that input being used. The ﬁrst of these equations says that the ﬁrm’s supply function can be calculated from its proﬁt function by partial differentiation with respect to the output price.10 The second and third equations show that input demand functions11 can also be derived from the proﬁt functions. Because the proﬁt function itself is homogeneous of degree 1, all the functions described in Equations 11.29–11.31 are homogeneous of degree 0. That is, a doubling of both output and input prices will not change the input levels that the ﬁrm chooses, nor will this change the ﬁrm’s proﬁt-maximizing output level. All these ﬁndings also have short-run analogs, as will be shown later with a speciﬁc example. 10This relationship is sometimes referred to as ‘‘Hotelling’s lemma’’—after the economist Harold Hotelling, who discovered it in the 1930s. 11Unlike the input demand functions derived in Chapter 10, these input demand functions are not conditional on output levels. Rather, the ﬁrm’s proﬁt-maximizing output decision has already been taken into account in the functions. Therefore, this demand concept is more general than the one we introduced in Chapter 10, and we will have much more to say about it in the next section. 386 Part 4: Production and Supply Producer surplus in the short run In Chapter 5 we discussed the concept of ‘‘consumer surplus’’ and showed how areas below the demand curve can be used to measure the welfare costs to consumers of price changes. We also showed how such changes in welfare could be captured in the individual’s expenditure function. The process of measuring the welfare effects of price changes for ﬁrms is similar in short-run analysis, and this is the topic we pursue here. However, as we show in the next chapter, measuring the welfare impact of price changes for producers in the long run requires a different approach because most such long-term effects are felt not by ﬁrms themselves but rather by their input suppliers. In
general, it is this long-run approach that will prove more useful for our subsequent study of the welfare impacts of price changes. Because the proﬁt function is nondecreasing in output prices, we know that if P2 > P1 then P ð P2, . . . P P1, . . . ð Þ , Þ + and it would be natural to measure the welfare gain to the ﬁrm from the price change as welfare gain P P2, . . . ð Þ % P P1, . . . ð Þ : ¼ (11:32) Figure 11.4 shows how this value can be measured graphically as the area bounded by the two prices and above the short-run supply curve. Intuitively, the supply curve shows the minimum price that the ﬁrm will accept for producing its output. Hence when market If price increases from P1 to P2, then the increase in the ﬁrm’s proﬁts is given by area P2ABP1. At a price of P1, the ﬁrm earns short-run producer surplus given by area PsCBP1. This measures the increase in short-run proﬁts for the ﬁrm when it produces q1 rather than shutting down when price is Ps or below. Market price P2 P1 Ps SMC A B C q1 q2 q FIGURE 11.4 Changes in Short-Run Producer Surplus Measure Firm Proﬁts Chapter 11: Prof it Maximization 387 price increases from P1 to P2, the ﬁrm is able to sell its prior output level (q1) at a higher q1) for which, at the margin, it likewise price and also opts to sell additional output (q2 % earns added proﬁts on all but the ﬁnal unit. Hence the total gain in the ﬁrm’s proﬁts is given by area P2 ABP1. Mathematically, we can make use of the envelope results from the previous section to derive welfare gain P P2, . . . ð Þ % P P1, . . . ð Þ ¼ ¼ P2 ð P1 @P @P dP ¼ P2 ð P1 dP: P q ð Þ (11:33) Thus, the geometric and mathematical measures of the welfare change agree. Using this approach, we can also measure how much the ﬁrm values the right to produce at the prevailing market price relative to a situation where it would produce no output. If we denote the short-run shutdown price as Ps (which may or may not be a price of zero), then the extra proﬁts available from facing a price of P1 are deﬁned to be producer surplus: producer surplus P ð ¼ P1, . . . P Ps, . . . ð Þ ¼ Þ % P1 ð Ps dP: q P ð Þ (11:34) This is shown as area P1BCPs deﬁnition. in Figure 11.4. Hence we have the following formal Producer surplus. Producer surplus is the extra return that producers earn by making transactions at the market price over and above what they would earn if nothing were produced. It is illustrated by the size of the area below the market price and above the supply curve. In this deﬁnition, we have made no distinction between the short run and the long run, although our development thus far has involved only short-run analysis. In the next chapter, we will see that the same deﬁnition can serve dual duty by describing producer surplus in the long run, so using this generic deﬁnition works for both concepts. Of course, as we will show, the meaning of long-run producer surplus is different from what we have studied here. One more aspect of short-run producer surplus should be pointed out. Because the vk1; that is, ﬁrm produces no output at its shutdown price, we know that II(PS,…) proﬁts at the shutdown price are solely made up of losses of all ﬁxed costs. Therefore, ¼ % producer surplus P1, . . . P ð P1, . . . P ð P PS, . . . ð Þ % Þ vk1Þ ¼ Þ % ð% ¼ ¼ P P1, . . . ð Þ þ vk1: (11:35) That is, producer surplus is given by current proﬁts being earned plus short-run ﬁxed costs. Further manipulation shows that magnitude can also be expressed as producer surplus PS, . . . P Þ % ð wl1 þ vk1 % In words, a ﬁrm’s short-run producer surplus is given by the extent to which its revenues exceed its variable costs—this is, indeed, what the ﬁrm gains by producing in the short run rather than shutting down and producing nothing. P1, . . . P ð P1q1 % Þ vk1 ¼ P1q1 % (11:36) ¼ ¼ wl1: 388 Part 4: Production and Supply EXAMPLE 11.4 A Short-Run Proﬁt Function These various uses of the proﬁt function can be illustrated with the Cobb–Douglas production kalb and because we treat capital as ﬁxed at k1 in the function we have been using. Because q short run, it follows that proﬁts are ¼ % To ﬁnd the proﬁt function we use the ﬁrst-order conditions for a maximum to eliminate l from this expression: ¼ p Pka 1 l b vk1 % w l: (11:37) @p @l ¼ so bPka bPka 1 " 1= b 1 % ð Þ : # (11:38) (11:39) We can simplify the process of substituting this back into the proﬁt equation by letting A : Making use of this shortcut, we have w ¼ ð bPka 1 Þ ’ P ð P, v, w, k1Þ ¼ Pka 1 Ab= ð wA1= b 1 Þ % % Þ Pka 1 1 " wb= b ð % vk1 % A 1 w % ÞP1= 1 ð % # b 1 % wA1= ð b % 1 % b bb= ð b 1 Þ % ¼ ¼ Þka= ð 1 1 % b Þ vk1: % b ð 1 Þ % vk1 (11:40) Though admittedly messy, this solution is what was promised—the ﬁrm’s maximal proﬁts are expressed as a function of only the prices it faces and its technology. Notice that the ﬁrm’s ﬁxed costs (vk1) enter this expression in a simple linear way. The prices the ﬁrm faces determine the extent to which revenues exceed variable costs; then ﬁxed costs are subtracted to obtain the ﬁnal proﬁt number. Because it is always wise to check that one’s algebra is correct, let’s try out the numerical 12, and k1 ¼ 80, we know that at a 20. Hence 0. The ﬁrm will just break even at a 0.5, v 12 the ﬁrm will produce 40 units of output and use labor input of l example we have been using. With a price of P proﬁts will be p price of P % 12. Using the proﬁt function yields 12 Æ 20 12 Æ 40 3 Æ 80 3, v, w, k1Þ ¼ P ð P 12, 3, 12, 80 ð Þ ¼ 0:25 $ 1 12% 122 $ 80 3 $ % $ 80 ¼ 0: (11:41) Thus, at a price of 12, the ﬁrm earns 240 in proﬁts on its variable costs, and these are precisely offset by ﬁxed costs in arriving at the ﬁnal total. With a higher price for its output, the ﬁrm earns positive proﬁts. If the price falls below 12, however, the ﬁrm incurs short-run losses.12 Hotelling’s lemma. We can use the proﬁt function in Equation 11.40 together with the envelope theorem to derive this ﬁrm’s short-run supply function: P, v, w, k1Þ ¼ q ð b= ð b % 1 Þ @P @P ¼ w b " # ka= 1 ð b 1 % P b= ð 1 % b Þ, Þ (11:42) which is precisely the short-run supply function that we calculated in Example 11.3 (see Equation 11.21). 12In Table 10.2 we showed that if q 40, then SAC ¼ ¼ 12. Hence zero proﬁts are also indicated by P 12 ¼ ¼ SAC. Chapter 11: Prof it Maximization 389 Producer surplus. We can also use the supply function to calculate the ﬁrm’s short-run producer surplus. To do so, we again return to our numerical example: a 12, 3, w and k1 ¼ 10P/3 and the shutdown price is zero. Hence at a price of P ¼ 80. With these parameters, the short-run supply relationship is q 12, producer surplus is 0.5, v ¼ ¼ ¼ ¼ b ¼ producer surplus 12 ¼ ð 0 10P 3 dP ¼ 10P 2 6 12 0 ¼ ! ! ! ! 240: (11:43) This precisely equals short-run proﬁts at a price of 12 (p vk1 240). If price were to rise to (say) 15, then producer surplus would increase to 375, 0) plus short-run ﬁxed costs ( ¼ which would still consist of 240 in ﬁxed costs plus total proﬁts at the higher price (P 3 Æ 80 ¼ ¼ ¼ 135). ¼ QUERY: How is the amount of short-run producer surplus here affected by changes in the rental rate for capital, v? How is it affected by changes in the wage, w? Proﬁt Maximization and Input Demand Thus far, we have treated the ﬁrm’s decision problem as one of choosing a proﬁt-maximizing level of output. But our discussion throughout has made clear that the ﬁrm’s outin fact, determined by the inputs it chooses to use, a relationship that is put is, summarized by the production function q f(k, l). Consequently, the ﬁrm’s economic proﬁts can also be expressed as a function of only the inputs it uses: ¼ p k, l ð Þ ¼ Pq C q ð % Þ ¼ P f k, l ð vk wl : Þ þ Þ % ð (11:44) Viewed in this way, the proﬁt-maximizing ﬁrm’s decision problem becomes one of choosing the appropriate levels of capital and labor input.13 The ﬁrst-order conditions for a maximum are @p @k ¼ @p @l ¼ P P @f @k % @f @l % 0, v ¼ 0: w ¼ (11:45) (11:46) These conditions make the intuitively appealing point that a proﬁt-maximizing ﬁrm should hire any input up to the point at which the input’s marginal contribution to revenue is equal to the marginal cost of hiring the input. Because the ﬁrm is assumed to be a price-taker in its hiring, the marginal cost of hiring any input is equal to its market price. The input’s marginal contribution to revenue is given by the extra output it produces (the marginal product) times that good’s market price. This demand concept is given a special name as follows Marginal revenue product. The marginal revenue product is the extra revenue a ﬁrm receives when it uses one more unit of an input. In the price-taking14 case, MRPl ¼ Pfl and MRPk ¼ Pfk. 13Throughout our discussion in this section, we assume that the ﬁrm is a price-taker; thus, the prices of its output and its inputs can be treated as ﬁxed parameters. Results can be generalized fairly easily in the case where prices depend on quantity. 14If the ﬁrm is not a price-taker in the output market, then this deﬁnition is generalized by using marginal revenue in place of price. That is, MRPl ¼ MR Æ MPl. A similiar derivation holds for capital input. @R/@q Æ @q/@l @R/@l ¼ ¼ 390 Part 4: Production and Supply Hence proﬁt maximization requires that the ﬁrm hire each input up to the point at which its marginal revenue product is equal to its market price. Notice also that the proﬁt-maximizing Equations 11.45 and 11.46 also imply cost minimization because RTS w/v. fl/fk ¼ ¼ Second-order conditions Because the proﬁt function in Equation 11.44 depends on two variables, k and l, the secondorder conditions for a proﬁt maximum are somewhat more complex than in the singlevariable case we examined earlier. In Chapter 2 we showed that, to ensure a true maximum, the proﬁt function must be concave. That is, p kk ¼ f kk < 0, p ll ¼ f ll < 0, (11:47) and f 2 kl > 0: p2 kl ¼ (11:48) p kkp ll % f kk f ll % Therefore, concavity of the proﬁt relationship amounts to requiring that the production function itself be concave. N
otice that diminishing marginal productivity for each input is not sufﬁcient to ensure increasing marginal costs. Expanding output usually requires the ﬁrm to use more capital and more labor. Thus, we must also ensure that increases in capital input do not raise the marginal productivity of labor (and thereby reduce marginal cost) by a large enough amount to reverse the effect of diminishing marginal productivity of labor itself. Therefore, Equation 11.47 requires that such cross-productivity effects be relatively small—that they be dominated by diminishing marginal productivities of the inputs. If these conditions are satisﬁed, then marginal costs will increase at the proﬁt-maximizing choices for k and l, and the ﬁrst-order conditions will represent a local maximum. Input demand functions In principle, the ﬁrst-order conditions for hiring inputs in a proﬁt-maximizing way can be manipulated to yield input demand functions that show how hiring depends on the prices that the ﬁrm faces. We will denote these demand functions by capital demand labor demand ¼ k l , P, v, w Þ ð : P, v, w Þ (11:49) ð ¼ Notice that, contrary to the input demand concepts discussed in Chapter 10, these demand functions are ‘‘unconditional’’—that is, they implicitly permit the ﬁrm to adjust its output to changing prices. Hence these demand functions provide a more complete picture of how prices affect input demand than did the contingent demand functions introduced in Chapter 10. We have already shown that these input demand functions can also be derived from the proﬁt function through differentiation; in Example 11.5, we show that process explicitly. First, however, we will explore how changes in the price of an input might be expected to affect the demand for it. To simplify matters, we look only at labor demand, but the analysis of the demand for any other input would be the same. In general, we conclude that the direction of this effect is unambiguous in all cases—that is, @l/@w 0 no matter how many inputs there are. To develop some intuition for this result, we begin with some simple cases. , Single-input case One reason for expecting @l/@w to be negative is based on the presumption that the marginal physical product of labor decreases as the quantity of labor employed increases. A Chapter 11: Prof it Maximization 391 P Æ MPl: decrease in w means that more labor must be hired to bring about the equality w A decrease in w must be met by a decrease in MPl (because P is ﬁxed as required by the ceteris paribus assumption), and this can be brought about by increasing l. That this argument is strictly correct for the case of one input can be shown as follows. With one input, Equation 11.44 is the sole ﬁrst-order condition for proﬁt maximization, rewritten here in a slightly different form: ¼ Pfl % w F l, w, P ð ¼ Þ ¼ 0: (11:50) where F is just a shorthand we will use to refer to the left side of Equation 11.50. If w changes, the optimal value of l must adjust so that this condition continues to hold, which deﬁnes l as an implicit function of w. Applying the rule for ﬁnding the derivative of an implicit function in Chapter 2 (Equation 2.23 in particular) gives dl dw ¼ % @F=@w @F=@l ¼ w Pfll , 0, (11:51) where the ﬁnal inequality holds because the marginal productivity of labor is assumed to be diminishing ( fll , 0). Hence we have shown that, at least in the single-input case, a ceteris paribus increase in the wage will cause less labor to be hired. Two-input case For the case of two (or more) inputs, the story is more complex. The assumption of a diminishing marginal physical product of labor can be misleading here. If w falls, there will not only be a change in l but also a change in k as a new cost-minimizing combination of inputs is chosen. When k changes, the entire fl function changes (labor now has a different amount of capital to work with), and the simple argument used previously cannot be made. First we will use a graphic approach to suggest why, even in the two-input case, @l/@w must be negative. A more precise, mathematical analysis is presented in the next section. Substitution effect In some ways, analyzing the two-input case is similar to the analysis of the individual’s response to a change in the price of a good that was presented in Chapter 5. When w falls, we can decompose the total effect on the quantity of l hired into two components. The ﬁrst of these components is called the substitution effect. If q is held constant at q1, then there will be a tendency to substitute l for k in the production process. This effect is illustrated in Figure 11.5a. Because the condition for minimizing the cost of producing q1 w/v, a fall in w will necessitate a movement from input combination requires that RTS A to combination B. And because the isoquants exhibit a diminishing RTS, it is clear from the diagram that this substitution effect must be negative. A decrease in w will cause an increase in labor hired if output is held constant. ¼ Output effect It is not correct, however, to hold output constant. It is when we consider a change in q (the output effect) that the analogy to the individual’s utility-maximization problem breaks down. Consumers have budget constraints, but ﬁrms do not. Firms produce as much as the available demand allows. To investigate what happens to the quantity of output produced, we must investigate the ﬁrm’s proﬁt-maximizing output decision. A change in w, because it changes relative input costs, will shift the ﬁrm’s expansion path. Consequently, all the ﬁrm’s cost curves will be shifted, and probably some output level 392 Part 4: Production and Supply FIGURE 11.5 The Substitution and Output Effects of a Decrease in the Price of a Factor When the price of labor falls, two analytically different effects come into play. One of these, the substitution effect, would cause more labor to be purchased if output were held constant. This is shown as a movement from point A to point B in (a). At point B, the cost-minimizing condition (RTS w/v) is satisﬁed for the new, lower w. This change in w/v will also shift the ﬁrm’s expansion path and its marginal cost curve. A normal situation might be for the MC curve to shift downward in response to a decrease in w as shown in (b). With this new curve (MC 0) a higher level of output (q2) will be chosen. Consequently, the hiring of labor will increase (to l2), also from this output effect. ¼ k per period Price k1 k2 A C B P q2 q1 MC MC′ l1 l2 l per period q1 q2 Output per period (a) The isoquant map (b) The output decision other than q1 will be chosen. Figure 11.5b shows what might be considered the ‘‘normal’’ case. There, the fall in w causes MC to shift downward to MC 0. Consequently, the proﬁtmaximizing level of output rises from q1 to q2. The proﬁt-maximizing condition (P ¼ MC) is now satisﬁed at a higher level of output. Returning to Figure 11.5a, this increase in output will cause even more l to be demanded as long as l is not an inferior input (see below). The result of both the substitution and output effects will be to move the input choice to point C on the ﬁrm’s isoquant map. Both effects work to increase the quantity of labor hired in response to a decrease in the real wage. The analysis provided in Figure 11.5 assumed that the market price (or marginal revenue, if this does not equal price) of the good being produced remained constant. This would be an appropriate assumption if only one ﬁrm in an industry experienced a fall in unit labor costs. However, if the decline were industry wide, then a slightly different analysis would be required. In that case, all ﬁrms’ marginal cost curves would shift outward, and hence the industry supply curve (which as we will see in the next chapter is the sum of ﬁrm’s individual supply curves) would shift also. Assuming that output demand is downward sloping, this will lead to a decline in product price. Output for the industry and for the typical ﬁrm will still increase and (as before) more labor will be hired, but the precise cause of the output effect is different (see Problem 11.11). Cross-price effects We have shown that, at least in simple cases, @l/@w is unambiguously negative; substitution and output effects cause more labor to be hired when the wage rate falls. From Figure 11.5 it should be clear that no deﬁnite statement can be made about how capital usage responds to the wage change. That is, the sign of @k/@w is indeterminate. In the simple two-input case, a fall in the wage will cause a substitution away from capital; that Chapter 11: Prof it Maximization 393 is, less capital will be used to produce a given output level. However, the output effect will cause more capital to be demanded as part of the ﬁrm’s increased production plan. Thus, substitution and output effects in this case work in opposite directions, and no deﬁnite conclusion about the sign of @k/@w is possible. A summary of substitution and output effects The results of this discussion can be summarized by the following principle Substitution and output effects in input demand. When the price of an input falls, two effects cause the quantity demanded of that input to rise: 1. the substitution effect causes any given output level to be produced using more of the input; and 2. the fall in costs causes more of the good to be sold, thereby creating an additional output effect that increases demand for the input. Conversely, when the price of an input rises, both substitution and output effects cause the quantity demanded of the input to decline. We now provide a more precise development of these concepts using a mathematical approach to the analysis. A mathematical development Our mathematical development of the substitution and output effects that arise from the change in an input price follows the method we used to study the effect of price changes in consumer theory. The ﬁnal result is a Slutsky-style equation that resembles the one we derived in Chapter 5. How
ever, the ambiguity stemming from Giffen’s paradox in the theory of consumption demand does not occur here. We start with a reminder that we have two concepts of demand for any input (say, labor): (1) the conditional demand for labor, denoted by lc(v, w, q); and (2) the unconditional demand for labor, which is denoted by l(P, v, w). At the proﬁt-maximizing choice for labor input, these two concepts agree about the amount of labor hired. The two concepts also agree on the level of output produced (which is a function of all the prices): P, v, w l ð l c v, w, q ð ð Þ ¼ P, v, w ÞÞ : (11:52) Differentiation of this expression with respect to the wage (and holding the other prices constant) yields @l P, v, w ð @w Þ ¼ @l c v, w, q ð @w Þ þ @l c v, w, q ð @q Þ $ @q P, v, w ð @w Þ : (11:53) Thus, the effect of a change in the wage on the demand for labor is the sum of two components: a substitution effect in which output is held constant; and an output effect in which the wage change has its effect through changing the quantity of output that the ﬁrm opts to produce. The ﬁrst of these effects is clearly negative—because the production function is quasi-concave (i.e., it has convex isoquants), the output-contingent demand for labor must be negatively sloped. Figure 11.5b provides an intuitive illustration of why the output effect in Equation 11.53 is negative, but it can hardly be called a proof. The particular complicating factor is the possibility that the input under consideration (here, labor) may be inferior. Perhaps oddly, inferior inputs also have negative output effects, but for rather 394 Part 4: Production and Supply arcane reasons that are best relegated to a footnote.15 The bottom line, however, is that Giffen’s paradox cannot occur in the theory of the ﬁrm’s demand for inputs: Input demand functions are unambiguously downward sloping. In this case, the theory of proﬁt maximization imposes more restrictions on what might happen than does the theory of utility maximization. In Example 11.5 we show how decomposing input demand into its substitution and output components can yield useful insights into how changes in input prices affect ﬁrms. EXAMPLE 11.5 Decomposing Input Demand into Substitution and Output Components To study input demand we need to start with a production function that has two features: (1) The function must permit capital–labor substitution (because substitution is an important part of the story); and (2) the production function must exhibit increasing marginal costs (so that the second-order conditions for proﬁt maximization are satisﬁed). One function that satisﬁes these conditions is a three-input Cobb–Douglas function when one of the inputs is held k0.25l0.25g0.5, where k and l are the familiar capital and labor ﬁxed. Thus, let q inputs and g is a third input (size of the factory) that is held ﬁxed at g 16 (square meters?) for ¼ 4k0.25l0.25. We assume that all our analysis. Therefore, the short-run production function is q the factory can be rented at a cost of r per square meter per period. To study the demand for (say) labor input, we need both the total cost function and the proﬁt function implied by this production function. Mercifully, your author has computed these functions for you as f (k, l, g) ¼ ¼ ¼ and C v, w, r, q ð Þ ¼ q2v 0:5w 0:5 8 16r þ P P, v, w, r ð Þ ¼ 2P 2v% 0:5w% 0:5 16r: % (11:54) (11:55) As expected, the costs of the ﬁxed input ( g) enter as a constant in these equations, and these costs will play little role in our analysis. Envelope results. Labor-demand relationships can be derived from both of these functions through differentiation: and l c v, w, r, q ð Þ ¼ @C @w ¼ q 2v 0:5w% 16 0:5 l P, v, w, r ð Þ ¼ @P @w ¼ P 2v% 0:5w% 1:5: (11:56) (11:57) These functions already suggest that a change in the wage has a larger effect on total labor demand than it does on contingent labor demand because the exponent of w is more negative in the total demand equation. That is, the output effect must also play a role here. To see that directly, we turn to some numbers. 15In words, an increase in the price of an inferior reduces marginal cost and thereby increases output. But when output increases, less of the inferior input is hired. Hence the end result is a decrease in quantity demanded in response to an increase in price. A formal proof makes extensive use of envelope relationships. The output effect equals @l c @q $ @q @w ¼ @l c @q $ @ 2P @w @P ¼ @l c @q $ % @l @P 2 @lc @q " # @q @P ¼ % $ 2 @lc @q " # @ 2P @P 2 , $ ¼ % # where the ﬁrst step holds by Equation 11.52, the second by Equation 11.29, the third by Young’s theorem and Equation 11.31, the fourth by Equation 11.52, and the last by Equation 11.29. But the convexity of the proﬁt function in output prices implies the last factor is positive, so the whole expression is clearly negative. " Chapter 11: Prof it Maximization 395 Numerical example. Let’s start again with the assumed values that we have been using in several previous examples: v 60. Let’s ﬁrst calculate what output the ﬁrm 3, w will choose in this situation. To do so, we need its supply function: 12, and P ¼ ¼ ¼ q P, v, w, r ð Þ ¼ @P @P ¼ 4Pv% 0:5w% 0:5: (11:58) ¼ ¼ With this function and the prices we have chosen, the ﬁrm’s proﬁt-maximizing output level is 40. With these prices and an output level of 40, both of the demand functions (surprise) q 50. Because the RTS here is given by k/l, we also know that predict that the ﬁrm will hire l k/l w/v; therefore, at these prices k ¼ Suppose now that the wage rate rises to w 27 but that the other prices remain unchanged. The ﬁrm’s supply function (Equation 11.58) shows that it will now produce q 26.67. The rise in the wage shifts the ﬁrm’s marginal cost curve upward, and with a constant output price, this causes the ﬁrm to produce less. To produce this output, either of the labor-demand functions can be used to show that the ﬁrm will hire l 133.3 because of the large reduction in output. 14.8. Hiring of capital will also fall to k 200. ¼ ¼ ¼ ¼ ¼ We can decompose the fall in labor hiring from l 40 even though the wage rose, Equation 11.56 shows that it would have used l 14.8 into substitution and output effects by using the contingent demand function. If the ﬁrm had continued to produce q 33.33. Capital input would have increased to k 300. Because we are holding output constant at its initial level of q 40, these changes represent the ﬁrm’s substitution effects in response to the higher wage. 50 to l ¼ ¼ ¼ ¼ ¼ ¼ The decline in output needed to restore proﬁt maximization causes the ﬁrm to cut back on its output. In doing so it substantially reduces its use of both inputs. Notice in particular that, in this example, the rise in the wage not only caused labor usage to decline sharply but also caused capital usage to fall because of the large output effect. QUERY: How would the calculations in this problem be affected if all ﬁrms had experienced the rise in wages? Would the decline in labor (and capital) demand be greater or smaller than found here? SUMMARY In this chapter we studied the supply decision of a proﬁtmaximizing ﬁrm. Our general goal was to show how such a ﬁrm responds to price signals from the marketplace. In addressing that question, we developed a number of analytical results. • To maximize proﬁts, the ﬁrm should choose to produce that output level for which marginal revenue (the revenue from selling one more unit) is equal to marginal cost (the cost of producing one more unit). • If a ﬁrm is a price-taker, then its output decisions do not affect the price of its output; thus, marginal revenue is given by this price. If the ﬁrm faces a downwardsloping demand for its output, however, then it can sell more only at a lower price. In this case marginal revenue will be less than price and may even be negative. • Marginal revenue and the price elasticity of demand are related by the formula MR P 1 " where P is the market price of the ﬁrm’s output and eq, p is the price elasticity of demand for its product. , # ¼ þ 1 eq, p • The supply curve for a price-taking, proﬁt-maximizing ﬁrm is given by the positively sloped portion of its marginal cost curve above the point of minimum average variable cost (AVC). If price falls below minimum AVC, the ﬁrm’s proﬁt-maximizing choice is to shut down and produce nothing. • The ﬁrm’s reactions to changes in the various prices it faces can be studied through use of its proﬁt function, P(P, v, w). That function shows the maximum proﬁts 396 Part 4: Production and Supply that the ﬁrm can achieve given the price for its output, the prices of its input, and its production technology. The proﬁt function yields particularly useful envelope results. Differentiation with respect to market price yields the supply function, whereas differentiation with respect to any input price yields (the negative of ) the demand function for that input. • Short-run changes in market price result in changes to the ﬁrm’s short-run proﬁtability. These can be measured graphically by changes in the size of producer surplus. The proﬁt function can also be used to calculate changes in producer surplus. • Proﬁt maximization provides a theory of the ﬁrm’s derived demand for inputs. The ﬁrm will hire any input up to the point at which its marginal revenue product to its per-unit market price. Increases in the price of an input will induce substitution and output effects that cause the ﬁrm to reduce hiring of that input. is just equal PROBLEMS 11.1 John’s Lawn Mowing Service is a small business that acts as a price-taker (i.e., MR mowing is $20 per acre. John’s costs are given by ¼ P ). The prevailing market price of lawn the number of acres John chooses to cut a day. total cost 0:1q 2 ¼ 10q 50, þ þ where q ¼ a. How many acres should John choose to cut to maximize proﬁt? b. Calculate John’s maximum daily proﬁt. c. Graph these results, and label John’s supply curve. 11.2 Universal Widget produces high-quality widge
ts at its plant in Gulch, Nevada, for sale throughout the world. The cost function for total widget production (q) is given by total cost 0.25q2. ¼ 2PA) and Lapland (where the Widgets are demanded only in Australia (where the demand curve is given by qA ¼ demand curve is given by qL ¼ qL. If Universal Widget can control the quantities supplied to each market, how many should it sell in each location to maximize total proﬁts? What price will be charged in each location? 4PL); thus, total demand equals q qA þ 100 100 % ¼ % 11.3 The production function for a ﬁrm in the business of calculator assembly is given by lp , 2 q ¼ where q denotes ﬁnished calculator output and l denotes hours of labor input. The ﬁrm is a price-taker both for calculators (which sell for P) and for workers (which can be hired at a wage rate of w per hour). ﬃﬃ a. What is the total cost function for this ﬁrm? b. What is the proﬁt function for this ﬁrm? c. What is the supply function for assembled calculators [q(P, w)]? d. What is this ﬁrm’s demand for labor function [l(P, w)]? e. Describe intuitively why these functions have the form they do. 11.4 The market for high-quality caviar is dependent on the weather. If the weather is good, there are many fancy parties and caviar sells for $30 per pound. In bad weather it sells for only $20 per pound. Caviar produced one week will not keep until the next week. A small caviar producer has a cost function given by Chapter 11: Prof it Maximization 397 0.5q2 C ¼ 5q þ þ 100, where q is weekly caviar production. Production decisions must be made before the weather (and the price of caviar) is known, but it is known that good weather and bad weather each occur with a probability of 0.5. a. How much caviar should this ﬁrm produce if it wishes to maximize the expected value of its proﬁts? b. Suppose the owner of this ﬁrm has a utility function of the form utility pp , ¼ where p is weekly proﬁts. What is the expected utility associated with the output strategy deﬁned in part (a)? ﬃﬃﬃ c. Can this ﬁrm owner obtain a higher utility of proﬁts by producing some output other than that speciﬁed in parts (a) and (b)? Explain. d. Suppose this ﬁrm could predict next week’s price but could not inﬂuence that price. What strategy would maximize expected proﬁts in this case? What would expected proﬁts be? 11.5 The Acme Heavy Equipment School teaches students how to drive construction machinery. The number of students that the 10 min(k, l )r, where k is the number of backhoes the ﬁrm rents per week, l is the school can educate per week is given by q number of instructors hired each week, and g is a parameter indicating the returns to scale in this production function. ¼ a. Explain why development of a proﬁt-maximizing model here requires 0 < g < 1. b. Supposing g c. If v 500, and P d. If the price students are willing to pay rises to P e. Graph Acme’s supply curve for student slots, and show that the increase in proﬁts calculated in part (d) can be plotted on 600, how many students will Acme serve and what are its proﬁts? 0.5, calculate the ﬁrm’s total cost function and proﬁt function. 900, how much will proﬁts change? ¼ 1000, w ¼ ¼ ¼ ¼ that graph. 11.6 Would a lump-sum proﬁts tax affect the proﬁt-maximizing quantity of output? How about a proportional tax on proﬁts? How about a tax assessed on each unit of output? How about a tax on labor input? 11.7 This problem concerns the relationship between demand and marginal revenue curves for a few functional forms. a. Show that, for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. b. Show that, for any linear demand curve, the vertical distance between the demand and marginal revenue curves is where b (< 0) is the slope of the demand curve. 1/b Æ q, % c. Show that, for a constant elasticity demand curve of the form q aPb, the vertical distance between the demand and marginal revenue curves is a constant ratio of the height of the demand curve, with this constant depending on the price elasticity of demand. ¼ d. Show that, for any downward-sloping demand curve, the vertical distance between the demand and marginal revenue curves at any point can be found by using a linear approximation to the demand curve at that point and applying the procedure described in part (b). e. Graph the results of parts (a)–(d) of this problem. 11.8 How would you expect an increase in output price, P, to affect the demand for capital and labor inputs? a. Explain graphically why, if neither input is inferior, it seems clear that a rise in P must not reduce the demand for either factor. b. Show that the graphical presumption from part (a) is demonstrated by the input demand functions that can be derived in the Cobb–Douglas case. c. Use the proﬁt function to show how the presence of inferior inputs would lead to ambiguity in the effect of P on input demand. 398 Part 4: Production and Supply Analytical Problems 11.9 A CES proﬁt function With a CES production function of the form q s)(g g)(v1 P(P, v, w) % s)g/(1 % KP1/(1 w1 % % % s ¼ þ lq kq ¼ ð 1), where s þ g=q a whole lot of algebra is needed to compute the proﬁt function as Þ ¼ r) and K is a constant. 1/(1 % a. If you are a glutton for punishment (or if your instructor is), prove that the proﬁt function takes this form. Perhaps the easiest way to do so is to start from the CES cost function in Example 10.2. b. Explain why this proﬁt function provides a reasonable representation of a ﬁrm’s behavior only for 0 < g < 1. c. Explain the role of the elasticity of substitution (s) in this proﬁt function. d. What is the supply function in this case? How does s determine the extent to which that function shifts when input prices change? e. Derive the input demand functions in this case. How are these functions affected by the size of s? 11.10 Some envelope results Young’s theorem can be used in combination with the envelope results in this chapter to derive some useful results. a. Show that @l(P, v, w)/@v @k(P, v, w)/@w. Interpret this result using substitution and output effects. b. Use the result from part (a) to show how a unit tax on labor would be expected to affect capital input. c. Show that @q/@w d. Use the result from part (c) to discuss how a unit tax on labor input would affect quantity supplied. @l/@P. Interpret this result. ¼ % ¼ 11.11 Le Chaˆ telier’s Principle Because ﬁrms have greater ﬂexibility in the long run, their reactions to price changes may be greater in the long run than in the short run. Paul Samuelson was perhaps the ﬁrst economist to recognize that such reactions were analogous to a principle from physical chemistry termed the Le Chaˆtelier’s Principle. The basic idea of the principle is that any disturbance to an equilibrium (such as that caused by a price change) will not only have a direct effect but may also set off feedback effects that enhance the response. In this problem we look at a few examples. Consider a price-taking ﬁrm that chooses its inputs to maximize a proﬁt vk. This maximization process will yield optimal solutions of the general function of the form P(P, v, w) form q&(P, v, w), l&(P, v, w), and k&(P, v, w). If we constrain capital input to be ﬁxed at k in the short run, this ﬁrm’s short-run responses can be represented by qs Pf (k, l) and ls wl ¼ % % a. Using the deﬁnitional relation q&(P, v, w) P, w, k Þ ð P, w, k ð qs(P, w, k&(P, v, w)), show that . Þ ¼ @q& @P ¼ @q s @P þ 2 : % @k& @P " # @k& @v Do this in three steps. First, differentiate the deﬁnitional relation with respect to P using the chain rule. Next, differentiate the deﬁnitional relation with respect to v (again using the chain rule), and use the result to substitute for @q s=@k in the initial derivative. Finally, substitute a result analogous to part (c) of Problem 11.10 to give the displayed equation. b. Use the result from part (a) to argue that @q&=@P @q s=@P. This establishes Le Chaˆtelier’s Principle for supply: Long-run supply responses are larger than (constrained) short-run supply responses. + c. Using similar methods as in parts (a) and (b), prove that Le Chaˆtelier’s Principle applies to the effect of the wage on labor @l s=@w, demand. That is, starting from the deﬁnitional relation l &(P, v, w) implying that long-run labor demand falls more when wage goes up than short-run labor demand (note that both of these derivatives are negative). l s(P, w, k&(P, v, w)), show that @l&=@w ¼ , d. Develop your own analysis of the difference between the short- and long-run responses of the ﬁrm’s cost function [C (v, w, q)] to a change in the wage (w). 11.12 More on the derived demand with two inputs The demand for any input depends ultimately on the demand for the goods that input produces. This can be shown most explicitly by deriving an entire industry’s demand for inputs. To do so, we assume that an industry produces a homogeneous D(P), where good, Q, under constant returns to scale using only capital and labor. The demand function for Q is given by Q P is the market price of the good being produced. Because of the constant returns-to-scale assumption, P AC. MC Throughout this problem let C(v, w, 1) be the ﬁrm’s unit cost function. ¼ ¼ ¼ Chapter 11: Prof it Maximization 399 a. Explain why the total industry demands for capital and labor are given by K b. Show that ¼ QCv and L QCw . ¼ @K @v ¼ QCvv þ D 0C 2 v and @L @w ¼ QCww þ D 0C2 w: c. Prove that Cvv ¼ d. Use the results from parts (b) and (c) together with the elasticity of substitution deﬁned as s Cww ¼ Cvw Cvw: and w % v v % w CCvw/CvCw to show that ¼ @K @v ¼ wL Q $ rK vC þ D 0K 2 Q2 and @L @w ¼ vK Q $ rL wC þ D 0L2 Q2 : e. Convert the derivatives in part (d) into elasticities to show that eL, w ¼ % eK, v ¼ % where eQ , P is the price elasticity of demand for the product being produced. sK eQ , P sLr and þ sK r þ sLeQ , P, f. Discuss the importance of the results in part (e) using the notions of substitution and out
put effects from Chapter 11. Note: The notion that the elasticity of the derived demand for an input depends on the price elasticity of demand for the output being produced was ﬁrst suggested by Alfred Marshall. The proof given here follows that in D. Hamermesh, Labor Demand (Princeton, NJ: Princeton University Press, 1993). 11.13 Cross-price effects in input demand With two inputs, cross-price effects on input demand can be easily calculated using the procedure outlined in Problem 11.12. a. Use steps (b), (d), and (e) from Problem 11.12 to show that eK, w ¼ sLð r eQ , PÞ þ and eL, v ¼ sK ð r þ : eQ , PÞ b. Describe intuitively why input shares appear somewhat differently in the demand elasticities in part (e) of Problem 11.12 than they do in part (a) of this problem. c. The expression computed in part (a) can be easily generalized to the many-input case as exi, wj ¼ , where Aij is the Allen elasticity of substitution deﬁned in Problem 10.12. For reasons described in Problems 10.11 and 10.12, this approach to input demand in the multi-input case is generally inferior to using Morishima elasticities. One oddity might be mentioned, however. For the case i sL (ALL þ eQ , P), and if we jumped to the conclusion that ALL ¼ s in the two-input case, then this would contradict the result from Problem 11.12. You can resolve this parasK/sL) Æ s and so dox by using the deﬁnitions from Problem 10.12 to show that, with two inputs, ALL ¼ ( % there is no disagreement. j this expression seems to say that eL , w ¼ sK/sL) Æ AKL ¼ Aij þ eQ, PÞ ( % sjð ¼ 11.14 Proﬁt functions and technical change Suppose that a ﬁrm’s production function exhibits technical improvements over time and that the form of the function is q f (k, l, t). In this case, we can measure the proportional rate of technical change as ¼ @ ln q @t ¼ ft f (compare this with the treatment in Chapter 9). Show that this rate of change can also be measured using the proﬁt function as @ ln q @t ¼ P, v, w, t P ð Pq Þ @ ln P @t : $ That is, rather than using the production function directly, technical change can be measured by knowing the share of proﬁts in total revenue and the proportionate change in proﬁts over time (holding all prices constant). This approach to measuring technical change may be preferable when data on actual input levels do not exist. 400 Part 4: Production and Supply 11.15 Property rights theory of the ﬁrm This problem has you work through some of the calculations associated with the numerical example in the Extensions. Refer to the Extensions for a discussion of the theory in the case of Fisher Body and General Motors (GM), who we imagine are deciding between remaining as separate ﬁrms or having GM acquire Fisher Body and thus become one (larger) ﬁrm. xF, xGÞ ¼ Let the total surplus that the units generate together be S G , where xF and xG are the investments ð undertaken by the managers of the two units before negotiating, and where a unit of investment costs $1. The parameter a measures the importance of GM’s manager’s investment. Show that, according to the property rights model worked out in the Extensions, it is efﬁcient for GM to acquire Fisher Body if and only if GM’s manager’s investment is important enough, in particular, if a > x1/2 F þ ax1/2 3p . ﬃﬃﬃ SUGGESTIONS FOR FURTHER READING Hart, O. Firms, Contracts, and Financial Structure. Oxford, UK: Oxford University Press, 1995. Samuelson, P. A. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press, 1947. Discusses the philosophical issues addressed by alternative theories of the ﬁrm. Derives further results for the property rights theory discussed in the Extensions. Hicks, J. R. Value and Capital, 2nd ed. Oxford, UK: Oxford University Press, 1947. The Appendix looks in detail at the notion of factor complementarity. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Provides an elegant introduction to the theory of production using vector and matrix notation. This allows for an arbitrary number of inputs and outputs. Early development of the proﬁt function idea together with a nice discussion of the consequences of constant returns to scale for market equilibrium. Pages 36–46 have extensive applications of Le Chaˆtelier’s Principle (see Problem 11.11). Sydsaeter, K., A. Strom, and P. Berck. Economists’ Mathematical Manual, 3rd ed. Berlin: Springer-Verlag, 2000. Chapter 25 offers formulas for a number of proﬁt and factor demand functions. Varian, H. R. Microeconomic Analysis, 3rd ed. New York: W. W. Norton, 1992. Includes an entire chapter on the proﬁt function. Varian offers a novel approach for comparing short- and long-run responses using Le Chaˆtelier’s Principle. BOUNDARIES OF THE FIRM EXTENSIONS Chapter 11 provided fairly straightforward answers to the questions of what determines the boundaries of a ﬁrm and its objectives. The ﬁrm is identiﬁed by the production function f (k, l ) it uses to produce its output, and the ﬁrm makes its input and output decisions to maximize proﬁt. Ronald Coase, winner of the Nobel Prize in economics in 1991, was the ﬁrst to point out (back in the 1930s) that the nature of the ﬁrm is a bit more subtle than that. The ﬁrm is one way to organize the economic transactions necessary for output to be produced and sold, transactions including the purchase of inputs, ﬁnancing of investment, advertising, management, and so forth. But these transactions could also be conducted in other ways: Parties could sign long-term contracts or even just trade on a spot market; see Coase (1937). There is a sense in which ﬁrms and spot markets are not just different ways of organizing transactions but polar opposites. Moving a transaction within a ﬁrm is tantamount to insulating the transaction from short-term market forces, eliminating price signals, by placing it inside a more durable institution. This presents a puzzle. Economists are supposed to love markets—why are they then so willing to take the existence of ﬁrms for granted? On the other hand, if ﬁrms are so great, why is there not just one huge ﬁrm that controls the whole economy, removing all transactions from the market? Clearly, a theory is needed to explain why there are ﬁrms of intermediate sizes, and why these sizes vary across different industries and even across different ﬁrms in the same industry. To make the ideas in the Extensions concrete, we will couch the discussion in terms of the classic case of Fisher Body and General Motors (GM) mentioned at the beginning of Chapter 11. Recall that Fisher Body was the main supplier of auto bodies to GM, which GM would assemble with other auto parts into a car that it then sold to consumers. At ﬁrst the ﬁrms operated separately, but GM acquired Fisher Body in 1926 after a series of supply disruptions. We will narrow the broad question of where ﬁrm boundaries should be set down to the question of whether it made economic sense for GM and Fisher Body to merge into a single ﬁrm. E11.1 Common features of alternative theories A considerable amount of theoretical and empirical research continues to be directed toward the fundamental question of the nature of the ﬁrm, but it is fair to say that it has not pro- vided a ‘‘ﬁnal answer.’’ Reﬂecting this uncertainty, the Extensions present two different theories that have been proposed as alternatives to the neoclassical model studied in Chapter 11. The ﬁrst is the property rights theory associated with the work of Sanford Grossman, Oliver Hart, and John Moore. The second is the transactions cost theory associated with the work of Oliver Williamson, co-winner of the Nobel Prize in economics in 2009.1 The theories share some features. Both acknowledge that if all markets looked like the supply–demand model encountered in principles courses—where a large number of suppliers and buyers trade a commodity anonymously—that would be the most efﬁcient way to organize transactions, leaving no role for ﬁrms. However, it is unrealistic to assume that all markets look that way. Three factors often present—uncertainty, complexity, and specialization—lead markets to look more like negotiations among a few market participants. We can see how these three factors would have operated in the GM–Fisher Body example. The presence of uncertainty and complexity would have made it difﬁcult for GM to sign contracts years in advance for auto bodies. Such contracts would have to specify how the auto bodies should be designed, but successful design depends on the vagaries of consumer taste, which are difﬁcult to predict (after all, large tail ﬁns were popular at one point in history) and hard to specify in writing. The best way to cope with uncertainty and complexity may be for GM to negotiate the purchase of auto bodies at the time they are needed for assembly rather than years in advance at the signing of a long-term contract. The third factor, specialization, leads to obvious advantages. Auto bodies that are tailored to GM’s styling and other technical requirements would be more valuable than ‘‘generic’’ ones. But specialization has the drawback of limiting GM to a small set of suppliers rather than buying auto bodies as it would an input on a competitive commodity market. Markets exhibiting these three factors—uncertainty, complexity, and specialization—will not involve the sale of perfect long-term contracts in a competitive equilibrium with large numbers of suppliers and demanders. Rather, they will often involve few parties, perhaps just two, negotiating often not far 1Seminal articles on the property rights theory are Grossman and Hart (1986) and Hart and Moore (1990). See Williamson (1979) for a comprehensive treatment of the transactions cost theory. Gibbons (2005) provides a good summary of these and other alternatives to the neoclassical model. 402 Part 4: Production and Supply in advance of when the input is required. This makes the alternative theories of the ﬁrm in
teresting. If the alternative theories merely compared ﬁrms to perfectly competitive markets, markets would always end up ‘‘winning’’ in the comparison. Instead, ﬁrms are compared to negotiated sales, a more subtle comparison without an obvious ‘‘winner.’’ We will explore the subtle comparisons offered by the two different theories next. E11.2 Property rights theory To make the analysis of this alternative theory as stark as possible, suppose that there are just two owner-managers, one who runs Fisher Body and one who runs GM. Let S(xF, xG) be the total surplus generated by the transaction between Fisher Body and GM, the sum of both ﬁrm’s proﬁts (Fisher Body from its sale of auto bodies to GM and GM from its sale of cars to consumers). Instead of being a function of capital and labor or input and output prices, we now put those factors aside and just write surplus as a function of two new variables: the investments made by Fisher Body (xF) and GM (xG). The surplus function subtracts all production costs (just as the producer surplus concept from Chapter 11 did) but does not subtract the cost of the investments xF and xG. The investments are sunk before negotiations between them over the transfer of the auto bodies. The investments include, for example, any effort made by Fisher Body’s manager to improve the precision of its metal-cutting dies and to reﬁne the shapes to GM’s speciﬁcations, as well as the effort expended by GM’s manager in designing and marketing the car and tailoring its assembly process to use the bodies. Both result in a better car model that can be sold at a higher price and that generates more proﬁt (not including the investment effort). For simplicity, assume one unit of investment costs a manager $1, implying that investment level xF costs Fisher Body’s manager xF dollars and that the marginal cost of investment for both parties is 1. Before computing the equilibrium investment levels under various ownership structures, as a benchmark we will compute the efﬁcient investment levels. The efﬁcient levels maximize total surplus minus investment costs, S xF, xGÞ % ð xF % xG: (i) The ﬁrst-order conditions for maximization of this objective are @S @xF ¼ @S @xG ¼ 1: (ii) The efﬁcient investment levels equalize the total marginal beneﬁt with the marginal cost. Next, let’s compute equilibrium investment levels under various ownership structures. Assume the investments are too complicated to specify in a contract before they are undertaken. So too is the speciﬁcation of the auto bodies themselves. Instead, starting with the case in which Fisher Body and GM are separate ﬁrms, they must bargain over the terms of trade of the auto bodies (prices, quantities, nature of the product) when they are needed. There is a large body of literature on how to model bargaining (we will touch on this a bit more in Chapter 13 when we introduce Edgeworth boxes and contract curves). To make the analysis as simple as possible, we will not solve for all the terms of the bargain but will just assume that they come to an agreement to split any gains from the transaction equally.2 Because cars cannot be produced without auto bodies, no surplus is generated if parties do not consummate a deal. Therefore, the gain from bargaining is the whole surplus, S(xF, xG). The investment expenditures are not part of the negotiation because they were sunk =2 in before. Fisher Body and GM each end up with S equilibrium from bargaining. xF, xGÞ ð To solve for equilibrium investments, subtract Fisher Body’s cost of investment from its share of the bargaining gains, yielding the objective function 1 2 S xF, xGÞ % ð Taking the ﬁrst-order condition with respect to xF and rearranging yields the condition (iii) xF: 1 2 @S @xF" ¼ 1: (iv) # The left side of Equation iv is the marginal beneﬁt to Fisher Body from additional investment: Fisher Body receives its bargaining share, half, of the surplus. The right side is the marginal cost, which is 1 because investment xF is measured in dollar terms. As usual, the optimal choice (here investment) equalizes marginal beneﬁt and marginal cost. A similar condition characterizes GM’s investment decision: 1 2 @S @xG" ¼ 1: (v) # In sum, if Fisher Body and GM are separate ﬁrms, investments are given by Equations iv and v. If instead GM acquires Fisher Body so they become one ﬁrm, the manager of the auto body subsidiary is now in a worse bargaining position. He or she can no longer extract half of the bargaining surplus by threatening not to use Fisher Body’s assets to produce bodies for GM; the assets are all under GM’s control. To make the point as clear as possible, assume that Fisher Body’s manager obtains no bargaining surplus; GM obtains all of it. Without the prospect of a return, the manager will not undertake any investment, implying xF ¼ 0. On the other hand, because GM’s manager now obtains the whole surplus S(xF, xG), the objective function determining his or her investment is now 1 2 S xF, xGÞ % ð xG: (vi) 2This is a special case of so-called Nash bargaining, an inﬂuential bargaining theory developed by the same John Nash behind Nash equilibrium. yielding ﬁrst-order condition @S @xG ¼ 1: (vii) When both parties were in separate ﬁrms, each had less than efﬁcient investment incentives (compare the ﬁrst-order conditions in the efﬁcient outcome in Equation ii with Equations iv and v) because they only obtain half the bargaining surplus. Combining the two units under GM’s ownership further dilutes Fisher Body’s investment incentives, reducing its 0, but boosts GM’s, so investment all the way down to xF ¼ that GM’s ﬁrst-order condition resembles the efﬁcient one. Intuitively, asset ownership gives parties more bargaining power, and this bargaining power in turn protects the party from having the returns from their investment appropriated by the other party in bargaining.3 Of course there is only so much bargaining power to go around. A shift of assets from one party to another will increase one’s bargaining power at the expense of the other’s. Therefore, a trade-off is involved in merging two units into one; the merger only makes economic sense under certain conditions. If GM’s investment is much more important for surplus, then it will be efﬁcient to allocate ownership over all the assets to GM. If both units’ investments are roughly equally important, then maintaining both parties’ bargaining power by apportioning some of the assets to each might be a good idea. If Fisher Body’s investment is the most important, then having Fisher Body acquire GM may produce the most efﬁcient structure. More speciﬁc recommendations would depend on functional forms, as will be illustrated in the following numerical example. E11.3 Numerical example For a simple numerical example of the property rights theory, x1/2 let S G . The ﬁrst-order condition for the ð efﬁcient level of Fisher Body’s investment is xF, xGÞ ¼ x1/2 F þ 1 1/2 x% F ¼ 2 1/4. Likewise, x&G ¼ implying x&F ¼ ing the investment costs is 1/2. 1, 1/4. Total surplus subtract- If Fisher Body and GM remain separate ﬁrms, half the surplus from each party’s investment is ‘‘held up’’ by the other party. Fisher Body’s ﬁrst-order condition is Chapter 11: Prof it Maximization 403 the integrated ﬁrm bargaining surplus. The manager of obtains all the bargaining surplus and invests at the efﬁcient level, x&G ¼ 1/4. Overall, total surplus subtracting investment costs is 1/4. Combining the ﬁrms decreases Fisher Body’s investment and increases GM’s, but the net effect is to make them jointly worse off; therefore, the ﬁrms should remain separate. If GM’s investment were more important than Fisher Body’s, ax1/2 merging them could be efﬁcient. Let S G , where a allows the impact of GM’s investment on surplus to vary. One of the problems at the end of this chapter asks you to show that having GM’s manager own all assets is more efﬁcient than keeping the ﬁrms separate for high enough a, in particular, a > xF, xGÞ ¼ ð x1/2 F þ 3p . ﬃﬃﬃ E11.4 Transaction cost theory Next, turn to the second alternative theory of the ﬁrm—the it shares transaction cost theory. As discussed previously, many common elements with the property rights theory, but there are subtle differences. With the property rights theory, the main beneﬁt of restructuring the ﬁrm was to get the right incentives for investments made before bargaining. With the transaction cost theory, the main beneﬁt is to reduce haggling costs at the time of bargaining. Let hF be a costly action undertaken by Fisher Body at the time of bargaining that increases its bargaining power at the expense of GM. We loosely interpret this action as ‘‘haggling,’’ but more concretely it could be a costly signal such as was seen in the Spence education signaling game in Chapter 8, or it could represent bargaining delay or an input supplier strike. GM can take a similar haggling action, hG. Rather than ﬁxing the bargaining shares at 1/2 each, we now assume a(hF, hG) is the share accruing to Fisher Body and 1 a(hF, hG) is the share accruing to GM, where a is between 0 and 1 and is increasing in hF and decreasing in hG. For simplicity, assume that the marginal cost for one unit of the haggling action is $1, implying a haggling level of hF costs Fisher Body hF dollars and of hG costs GM hG dollars. To abstract from some of the bargaining issues in the previous theory, assume that investments are made at the time of bargaining rather than beforehand, so that in principle they can be set at the efﬁcient levels x&F and x&G satisfying Equation ii. % 1/2 x% F ¼ 1 4 1/16. Likewise, xG ¼ 1: 1/16. Thus, parties are implying xF ¼ underinvesting relative to the efﬁcient outcome. Total surplus subtracting investment costs is only 3/8. If GM acquires Fisher Body, the manager of the auto body 0) because he or she obtains no unit does not invest (xF ¼ 3The appropriation of the returns from one party’s investment by the other party in bargaining is called the hold-up
problem, referring to the colorful image of a bandit holding up a citizen at gunpoint. Nothing illegal is happening here; hold up is just a feature of bargaining. hG ¼ The efﬁcient outcome is for investments to be set at x&F and x&G and for parties not to undertake any haggling actions: hF ¼ 0. Haggling does not generate any more total surplus but rather reallocates it from one party to another. If Fisher Body and GM are separate ﬁrms, they will undertake some of these actions, much like the prisoners were led to ﬁnk on each other in equilibrium of the Prisoners’ Dilemma in Chapter 8 when it would have been better for the two of them to remain silent. Fisher Body’s objective function determining its equilibrium level of haggling is a hF, hGÞ½ x&F, x&GÞ % ð ð x&G( % x&F % (viii) hF, S 404 Part 4: Production and Supply where it is assumed the parties naturally would agree on the investments maximizing their joint surplus. Fisher Body’s ﬁrst-order condition is, after rearranging, @a @xF ½ x&F, x&GÞ % S ð x&F % x&G( ¼ 1: Similarly, GM will have ﬁrst-order condition @a @xG ½ S x&F, x&GÞ % ð x&F % x&G( ¼ 1: (ix) (x) The main point to take away from these somewhat complicated conditions is that both parties will engage in some wasteful haggling if they remain separate. If instead GM acquires Fisher Body and they become one ﬁrm, assume this enables GM to authorize what investment levels should be undertaken without having to resort to bar0, a savgaining. This rules out haggling; therefore, hF ¼ ings with this organizational structure. In many accounts of the transactions cost theory, that is the end of the story. Combining separate units together in the same ﬁrm reduces haggling, and thus ﬁrms are always more efﬁcient than markets when haggling costs are signiﬁcant. The trouble with stopping there with the model is that there is no trade-off associated with ﬁrms: In theory, one large ﬁrm should operate the entire economy, which is certainly an unrealistic outcome. hG ¼ One way to generate a trade-off is to assume that there is drawback to having one party (here GM) make a unilateral decision. One natural drawback is that GM may not choose the efﬁcient investment levels, either because it lacks valuable information to which the manager of the auto body unit is privy or because the manager of the merged ﬁrm makes the investment for his or her own beneﬁt rather than to maximize joint surplus. Letting ~xF and ~xG be the investment levels authorized by the manager of the merged ﬁrm, total surplus as a result of the merger is S ~xF, ~xGÞ % ð ~xF % ~xG, (xi) distortion relative to the haggling costs, which in turn depends on functional forms. E11.5 Classic empirical studies Early empirical studies of these alternative theories of the ﬁrm were not designed to distinguish between these speciﬁc theories (or additional alternatives). The focus was instead on seeing whether the conditions pushing input markets away from perfect competition toward negotiated sales—uncertainty, complexity, and specialization leading to few bargaining parties—could help explain the decision to have a transaction occur within the boundaries of a ﬁrm rather than having it occur between separate parties. Monteverde and Teece (1982) surveyed engineers at U.S. auto manufacturers about more than 100 parts assembled together to make cars, asking them how much engineering effort was required to design the part and whether the part was specialized to a single manufacturer. The authors found that these variables had a signiﬁcant positive effect on the decision of the manufacturer to produce the part in house rather than purchasing from a separate supplier. Masten (1984) found similar results in the aerospace industry. Anderson and Schmittlein (1984) found that proxies for complexity and specialization could help explain why some electronic components were sold by sales representatives employed by the manufacturers themselves and some by independent operators. References Anderson, E., and D. C. Schmittlein. ‘‘Integration of the Sales Force: An Empirical Examination.’’ Rand Journal of Economics (Autumn 1984): 385–95. Coase, R. H. ‘‘The Nature of the Firm.’’ Economica (Novem- ber 1937): 386–405. Gibbons, R. ‘‘Four Formal(izable) Theories of the Firm?’’ Journal of Economic Behavior and Organization (October 2005): 200–45. Hart, O. Firms, Contracts, and Financial Structure. Oxford, compared with total surplus when the ﬁrms remain separate, UK: Oxford University Press, 1995. S x&F, x&GÞ % ð x&F % x&G % hF % hG: (xii) The trade-offs involved in different ﬁrm structures are apparent from a comparison of these equations: Giving GM the unilateral authority to make the investment decision avoids any haggling costs but may result in inefﬁcient investment levels. Whether it is more efﬁcient to keep the ﬁrms separate or to merge the two units together and have one manager control them depends on the signiﬁcance of the investment Masten, S. E. ‘‘The Organization of Production: Evidence from the Aerospace Industry.’’ Journal of Law and Economics (October 1984): 403–17. Monteverde, K., and D. J. Teece. ‘‘Supplier Switching Costs and Vertical Integration in the Automobile Industry.’’ Bell Journal of Economics (Spring 1982): 206–13. Williamson, O. ‘‘Transaction Cost Economics: The Governance of Contractual Relations.’’ Journal of Law and Economics (October 1979): 233–61. This page intentionally left blank Competitive Markets P A R T FIVE Chapter 12 The Partial Equilibrium Competitive Model Chapter 13 General Equilibrium and Welfare In Parts 2 and 4 we developed models to explain the demand for goods by utility-maximizing individuals and the supply of goods by proﬁt-maximizing ﬁrms. In the next two parts we will bring together these strands of analysis to discuss how prices are determined in the marketplace. The discussion in this part concerns competitive markets. The principal characteristic of such markets is that ﬁrms behave as price-takers. That is, ﬁrms are assumed to respond to market prices, but they believe they have no control over these prices. The primary reason for such a belief is that competitive markets are characterized by many suppliers; therefore, the decisions of any one of them indeed has little effect on prices. In Part 6 we will relax this assumption by looking at markets with only a few suppliers (perhaps only one). For these cases, the assumption of pricetaking behavior is untenable; thus, the likelihood that ﬁrms’ actions can affect prices must be taken into account. Chapter 12 develops the familiar partial equilibrium model of price determination in competitive markets. The principal result is the Marshallian ‘‘cross’’ diagram of supply and demand that we ﬁrst discussed in Chapter 1. This model illustrates a ‘‘partial’’ equilibrium view of price determination because it focuses on only a single market. In the concluding sections of the chapter we show some of the ways in which such models are applied. A speciﬁc focus is on illustrating how the competitive model can be used to judge the welfare consequences for market participants of changes in market equilibria. Although the partial equilibrium competitive model is useful for studying a single market in detail, it is inappropriate for examining relationships among markets. To capture such cross-market effects requires the development of ‘‘general’’ equilibrium models—a topic we take up in Chapter 13. There we show how an entire economy can be viewed as a system of interconnected competitive markets that determine all prices simultaneously. We also examine how welfare consequences of various economic questions can be studied in this model. 407 This page intentionally left blank C H A P T E R TWELVE The Partial Equilibrium Competitive Model In this chapter we describe the familiar model of price determination under perfect competition that was originally developed by Alfred Marshall in the late nineteenth century. That is, we provide a fairly complete analysis of the supply–demand mechanism as it applies to a single market. This is perhaps the most widely used model for the study of price determination. Market Demand In Part 2 we showed how to construct individual demand functions that illustrate changes in the quantity of a good that a utility-maximizing individual chooses as the market price and other factors change. With only two goods (x and y) we concluded that an individual’s (Marshallian) demand function can be summarized as quantity of x demanded x(px, py, I). ¼ (12:1) Now we wish to show how these demand functions can be added up to reﬂect the 1, n) to represent demand of all individuals in a marketplace. Using a subscript i (i each person’s demand function for good x, we can deﬁne the total demand in the market as ¼ market demand for X n ¼ 1 i X ¼ xið : px, py, IiÞ (12:2) Notice three things about this summation. First, we assume that everyone in this marketplace faces the same prices for both goods. That is, px and py enter Equation 12.2 without person-speciﬁc subscripts. On the other hand, each person’s income enters into his or her own speciﬁc demand function. Market demand depends not only on the total income of all market participants but also on how that income is distributed among consumers. Finally, observe that we have used an uppercase X to refer to market demand—a notation we will soon modify. The market demand curve Equation 12.2 makes clear that the total quantity of a good demanded depends not only on its own price but also on the prices of other goods and on the income of each person. To construct the market demand curve for good X, we allow px to vary while holding py and the income of each person constant. Figure 12.1 shows this construction for the case where there are only two consumers in the market. For each potential price of x, 409 410 Part 5: Competitive Markets FIGURE 12.1 Construction of a Market Demand Curve from Individual Demand Curves A market demand curve is the
‘‘horizontal sum’’ of each individual’s demand curve. At each price the quantity demanded in the market is the sum of the amounts each individual demands. For example, at p$x the demand in the market is x$1 þ x$2 ¼ X$. px px px px* x1 x1 x1* x2 x2* x2 X* X X (a) Individual 1 (b) Individual 2 (c) Market demand the point on the market demand curve for X is found by adding up the quantities demanded by each person. For example, at a price of p$x, person 1 demands x$1 and person 2 demands x$2. The total quantity demanded in this two-person market is the x$2). Therefore, the point p$x, X$ is one point on sum of these two amounts (X$ the market demand curve for X. Other points on the curve are derived in a similar way. Thus, the market demand curve is a ‘‘horizontal sum’’ of each individual’s demand curve.1 x$1 þ ¼ Shifts in the market demand curve The market demand curve summarizes the ceteris paribus relationship between X and px. It is important to keep in mind that the curve is in reality a two-dimensional representation of a many-variable function. Changes in px result in movements along this curve, but changes in any of the other determinants of the demand for X cause the curve to shift to a new position. A general increase in incomes would, for example, cause the demand curve to shift outward (assuming X is a normal good) because each individual would choose to buy more X at every price. Similarly, an increase in py would shift the demand curve to X outward if individuals regarded X and Y as substitutes, but it would shift the demand curve for X inward if the goods were regarded as complements. Accounting for all such shifts may sometimes require returning to examine the individual demand functions that constitute the market relationship, especially when examining situations in which the distribution of income changes and thereby raises some incomes while reducing others. To keep matters straight, economists usually reserve the term change in quantity demanded for a movement along a ﬁxed demand curve in response to a change in px. Alternatively, any shift in the position of the demand curve is referred to as a change in demand. 1Compensated market demand curves can be constructed in exactly the same way by summing each individual’s compensated demand. Such a compensated market demand curve would hold each person’s utility constant. Chapter 12: The Partial Equilibrium Competitive Model 411 EXAMPLE 12.1 Shifts in Market Demand These ideas can be illustrated with a simple set of linear demand functions. Suppose individual 1’s demand for oranges (x, measured in dozens per year) is given by2 x1 ¼ 10 2px þ 0:1I1 þ & 0:5py, where px ¼ I1 ¼ py ¼ price of oranges (dollars per dozen), individual 1’s income (in thousands of dollars), price of grapefruit (a gross substitute for oranges—dollars per dozen). Individual 2’s demand for oranges is given by x2 ¼ Hence the market demand function is 17 px þ 0:05I2 þ & 0:5py: X px, py, I1, I2Þ ¼ ð x1 þ x2 ¼ 27 3px þ 0:1I1 þ 0:05I2 þ & py: (12:3) (12:4) (12:5) Here the coefﬁcient for the price of oranges represents the sum of the two individuals’ coefﬁcients, as does the coefﬁcient for grapefruit prices. This reﬂects the assumption that orange and grapefruit markets are characterized by the law of one price. Because the individuals have differing coefﬁcients for income, however, the demand function depends on each person’s income. To graph Equation 12.5 as a market demand curve, we must assume values for I1, I2, and py (because the demand curve reﬂects only the two-dimensional relationship between x and px). If I1 ¼ 4, then the market demand curve is given by 20, and py ¼ 40, I2 ¼ X 27 3px þ 4 & þ 1 þ 4 ¼ ¼ 36 & 3px, (12:6) which is a simple linear demand curve. If the price of grapefruit were to increase to py ¼ the curve would, assuming incomes remain unchanged, shift outward to 6, then þ whereas an income tax that took 10 (thousand dollars) from individual 1 and transferred it to individual 2 would shift the demand curve inward to þ ¼ & & ¼ X 27 3px þ 4 1 6 38 3px, (12:7) X 27 3px þ 3 & þ ¼ 1:5 4 þ ¼ 35:5 3px & (12:8) because individual 1 has a larger marginal effect of income changes on orange purchases. All these changes shift the demand curve in a parallel way because, in this linear case, none of them affects either individual’s coefﬁcient for px. In all cases, an increase in px of 0.10 (ten cents) would cause X to decrease by 0.30 (dozen per year). QUERY: For this linear case, when would it be possible to express market demand as a linear I2)? Alternatively, suppose the individuals had differing coefﬁcients function of total income (I1 þ for py. Would that change the analysis in any fundamental way? Generalizations Although our construction concerns only two goods and two individuals, it is easily generalized. Suppose there are n goods (denoted by xi, i 1, n. Assume also that there are m individuals in society. Then the jth individual’s demand for 1, n) with prices pi, i ¼ ¼ 2This linear form is used to illustrate some issues in aggregation. It is difﬁcult to defend this form theoretically, however. For example, it is not homogeneous of degree 0 in all prices and income. 412 Part 5: Competitive Markets the ith good will depend on all prices and on Ij, the income of this person. This can be denoted by where i 1, n and j ¼ ¼ xi, j ¼ xi, jð , p1, . . . , pn, IjÞ (12:9) 1, m. Using these individual demand functions, market demand concepts are provided by the following deﬁnition Market demand. The market demand function for a particular good (Xi) is the sum of each individual’s demand for that good: Xið p1, . . . , pn, I1, . . . , ImÞ ¼ m 1 j X ¼ xi, jð : p1, . . . , pn, IjÞ (12:10) The market demand curve for Xi is constructed from the demand function by varying pi while holding all other determinants of Xi constant. Assuming that each individual’s demand curve is downward sloping, this market demand curve will also be downward sloping. Of course, this deﬁnition is just a generalization of our previous discussion, but three features warrant repetition. First, the functional representation of Equation 12.10 makes clear that the demand for Xi depends not only on pi but also on the prices of all other goods. Therefore, a change in one of those other prices would be expected to shift the demand curve to a new position. Second, the functional notation indicates that the demand for Xi depends on the entire distribution of individuals’ incomes. Although in many economic discussions it is customary to refer to the effect of changes in aggregate total purchasing power on the demand for a good, this approach may be a misleading simpliﬁcation because the actual effect of such a change on total demand will depend on precisely how the income changes are distributed among individuals. Finally, although they are obscured somewhat by the notation we have been using, the role of changes in preferences should be mentioned. We have constructed individuals’ demand functions with the assumption that preferences (as represented by indifference curve maps) remain ﬁxed. If preferences were to change, so would individual and market demand functions. Hence market demand curves can clearly be shifted by changes in preferences. In many economic analyses, however, it is assumed that these changes occur so slowly that they may be implicitly held constant without misrepresenting the situation. A simpliﬁed notation Often in this book we look at only one market. To simplify the notation, in these cases we use QD to refer to the quantity of the particular good demanded in this market and P to denote its market price. As always, when we draw a demand curve in the Q–P plane, the ceteris paribus assumption is in effect. If any of the factors mentioned in the previous section (e.g., other prices, individuals’ incomes, or preferences) should change, the Q–P demand curve will shift, and we should keep that possibility in mind. When we turn to consider relationships among two or more goods, however, we will return to the notation we have been using up until now (i.e., denoting goods by x and y or by xi). Chapter 12: The Partial Equilibrium Competitive Model 413 Elasticity of market demand When we use this notation for market demand, we will also use a compact notation for the price elasticity of the market demand function: price elasticity of market demand eQ, P ¼ ¼ @QDð P, P 0, I @P P QD , Þ ’ (12:11) where the notation is intended as a reminder that the demand for Q depends on many factors other than its own price, such as the prices of other goods (P 0) and the incomes of all potential demanders (I). These other factors are held constant when computing the own-price elasticity of market demand. As in Chapter 5, this elasticity measures the proportionate response in quantity demanded to a 1 percent change in a good’s price. Market demand is also characterized by whether demand is elastic (eQ, P < 1) or inelastic (0 > eQ, P > 1). Many of the other concepts examined in Chapter 5, such as the crossprice elasticity of demand or the income elasticity of demand, also carry over directly into the market context:3 & & cross-price elasticity of market demand income elasticity of market demand ¼ ¼ @QDð @QDð P, P 0, I @P 0 P, P 0, I @I P 0 QD I QD , : Þ Þ ’ ’ (12:12) Given these conventions about market demand, we now turn to an extended examination of supply and market equilibrium in the perfectly competitive model. Timing of the Supply Response In the analysis of competitive pricing, it is important to decide the length of time to be allowed for a supply response to changing demand conditions. The establishment of equilibrium prices will be different if we are talking about a short period during which most inputs are ﬁxed than if we are envisioning a long-run process in which it is possible for new ﬁrms to enter an industry. For this reason, it has been traditional in economics to discuss pricing in three different time per
iods: (1) very short run, (2) short run, and (3) long run. Although it is not possible to give these terms an exact chronological deﬁnition, the essential distinction being made concerns the nature of the supply response that is assumed to be possible. In the very short run, there is no supply response: The quantity supplied is ﬁxed and does not respond to changes in demand. In the short run, existing ﬁrms may change the quantity they are supplying, but no new ﬁrms can enter the industry. In the long run, new ﬁrms may enter an industry, thereby producing a ﬂexible supply response. In this chapter we will discuss each of these possibilities. Pricing in the Very Short Run In the very short run, or the market period, there is no supply response. The goods are already ‘‘in’’ the marketplace and must be sold for whatever the market will bear. In this situation, price acts only as a device for rationing demand. Price will adjust to clear the market of the quantity that must be sold during the period. Although the market price 3In many applications, market demand is modeled in per capita terms and treated as referring to the ‘‘typical person.’’ In such applications it is also common to use many of the relationships among elasticities discussed in Chapter 5. Whether such aggregation across individuals is appropriate is discussed brieﬂy in the Extensions to this chapter. 414 Part 5: Competitive Markets FIGURE 12.2 Pricing in the Very Short Run When quantity is ﬁxed in the very short run, price acts only as a device to ration demand. With quantity ﬁxed at Q$, price P1 will prevail in the marketplace if D is the market demand curve; at this price, individuals are willing to consume exactly that quantity available. If demand should shift upward to D 0, the equilibrium market price would increase to P2. Price D P2 P1 D′ S D′ D S Q * Quantity per period may act as a signal to producers in future periods, it does not perform such a function in the current period because current-period output is ﬁxed. Figure 12.2 depicts this situation. Market demand is represented by the curve D. Supply is ﬁxed at Q$, and the price that clears the market is P1. At P1, individuals are willing to take all that is offered in the market. Sellers want to dispose of Q$ without regard to price (suppose that the good in question is perishable and will be worthless if it is not sold in the very short run). Hence P1, Q$ is an equilibrium price–quantity combination. If demand should shift to D 0, then the equilibrium price would increase to P2 but Q$ would stay ﬁxed because no supply response is possible. The supply curve in this situation is a vertical straight line at output Q$. The analysis of the very short run is not particularly useful for many markets. Such a theory may adequately represent some situations in which goods are perishable or must be sold on a given day, as is the case in auctions. Indeed, the study of auctions provides a number of insights about the informational problems involved in arriving at equilibrium prices, which we take up in Chapter 18. But auctions are unusual in that supply is ﬁxed. The far more usual case involves some degree of supply response to changing demand. It is presumed that an increase in price will bring additional quantity into the market. In the remainder of this chapter, we will examine this process. Before beginning our analysis, we should note that increases in quantity supplied need not come only from increased production. In a world in which some goods are durable (i.e., last longer than a single period), current owners of these goods may supply them in increasing amounts to the market as price increases. For example, even though the supply of Rembrandts is ﬁxed, we would not want to draw the market supply curve for these paintings as a vertical line, such as that shown in Figure 12.2. As the price of Rembrandts increases, individuals and museums will become increasingly willing to part with them. From a market point of view, therefore, the supply curve for Rembrandts will have an upward slope, even though no new production takes place. A similar analysis would Chapter 12: The Partial Equilibrium Competitive Model 415 follow for many types of durable goods, such as antiques, used cars, vintage baseball cards, or corporate shares, all of which are in nominally ‘‘ﬁxed’’ supply. Because we are more interested in examining how demand and production are related, we will not be especially concerned with such cases here. Short-Run Price Determination In short-run analysis, the number of ﬁrms in an industry is ﬁxed. These ﬁrms are able to adjust the quantity they produce in response to changing conditions. They will do this by altering levels of usage for those inputs that can be varied in the short run, and we shall investigate this supply decision here. Before beginning the analysis, we should perhaps state explicitly the assumptions of this perfectly competitive model Perfect competition. A perfectly competitive market is one that obeys the following assumptions. 1. There are a large number of ﬁrms, each producing the same homogeneous product. 2. Each ﬁrm attempts to maximize proﬁts. 3. Each ﬁrm is a price-taker: It assumes that its actions have no effect on market price. 4. Prices are assumed to be known by all market participants—information is perfect. 5. Transactions are costless: Buyers and sellers incur no costs in making exchanges (for more on this and the previous assumption, see Chapter 18). Throughout our discussion we continue to assume that the market is characterized by a large number of demanders, each of whom operates as a price-taker in his or her consumption decisions. Short-run market supply curve In Chapter 11 we showed how to construct the short-run supply curve for a single proﬁtmaximizing ﬁrm. To construct a market supply curve, we start by recognizing that the quantity of output supplied to the entire market in the short run is the sum of the quantities supplied by each ﬁrm. Because each ﬁrm uses the same market price to determine how much to produce, the total amount supplied to the market by all ﬁrms will obviously depend on price. This relationship between price and quantity supplied is called a shortrun market supply curve. Figure 12.3 illustrates the construction of the curve. For simplicity assume there are only two ﬁrms, A and B. The short-run supply (i.e., marginal cost) curves for ﬁrms A and B are shown in Figures 12.3a and 12.3b. The market supply curve shown in Figure 12.3c is the horizontal sum of these two curves. For example, at a price of P1, ﬁrm A is willing to supply qA 1 . Therefore, at this price the total supply in the market is given by Q1, which is equal to qA qB 1 . The other points on the curve are constructed in an identical way. Because each ﬁrm’s supply curve has a positive slope, the market supply curve will also have a positive slope. The positive slope reﬂects the fact that short-run marginal costs increase as ﬁrms attempt to increase their outputs. 1 and ﬁrm B is willing to supply qB 1 þ Short-run market supply More generally, if we let qi(P, v, w) represent the short-run supply function for each of the n ﬁrms in the industry, we can deﬁne the short-run market supply function as follows. 416 Part 5: Competitive Markets FIGURE 12.3 Short-Run Market Supply Curve The supply (marginal cost) curves of two ﬁrms are shown in (a) and (b). The market supply curve (c) is the horizontal sum of these curves. For example, at P1 ﬁrm A supplies qA 1 , and total market supply is given by Q1 ¼ 1 , ﬁrm B supplies qB qA 1 þ qB 1 . P P1 P P SA SB S 1q A q A 1q B q B (a) Firm A (b) Firm B (c) The market Q1 Total output per period Short-run market supply function. The short-run market supply function shows total quantity supplied by each ﬁrm to a market: P, v, w QSð Þ ¼ n 1 i X ¼ P, v, w qið : Þ (12:13) Notice that the ﬁrms in the industry are assumed to face the same market price and the same prices for inputs.4 The short-run market supply curve shows the two-dimensional relationship between Q and P, holding v and w (and each ﬁrm’s underlying technology) constant. The notation makes clear that if v, w, or technology were to change, the supply curve would shift to a new location. Short-run supply elasticity One way of summarizing the responsiveness of the output of ﬁrms in an industry to higher prices is by the short-run supply elasticity. This measure shows how proportional changes in market price are met by changes in total output. Consistent with the elasticity concepts developed in Chapter 5, this is deﬁned as follows Short-run elasticity of supply (es, P). eS, P ¼ percentage change in Q supplied percentage change in P @QS @P ’ P QS : ¼ (12:14) 4Several assumptions that are implicit in writing Equation 12.13 should be highlighted. First, the only one output price (P) enters the supply function—implicitly ﬁrms are assumed to produce only a single output. The supply function for multiproduct ﬁrms would also depend on the prices of the other goods these ﬁrms might produce. Second, the notation implies that input prices (v and w) can be held constant in examining ﬁrms’ reactions to changes in the price of their output. That is, ﬁrms are assumed to be price-takers for inputs—their hiring decisions do not affect these input prices. Finally, the notation implicitly assumes the absence of externalities—the production activities of any one ﬁrm do not affect the production possibilities for other ﬁrms. Models that relax these assumptions will be examined at many places later in this book. Chapter 12: The Partial Equilibrium Competitive Model 417 Because quantity supplied is an increasing function of price (@QS=@P > 0s), the supply elasticity is positive. High values for eS, P imply that small increases in market price lead to a relatively large supply response by ﬁrms because marginal costs do not increase steeply and input price interaction effects are small. Alternatively, a lo
w value for eS, P implies that it takes relatively large changes in price to induce ﬁrms to change their output levels because marginal costs increase rapidly. Notice that, as for all elasticity notions, computation of eS, P requires that input prices and technology be held constant. To make sense as a market response, the concept also requires that all ﬁrms face the same price for their output. If ﬁrms sold their output at different prices, we would need to deﬁne a supply elasticity for each ﬁrm. EXAMPLE 12.2 A Short-Run Supply Function In Example 11.3 we calculated the general short-run supply function for any single ﬁrm with a two-input Cobb–Douglas production function as b= 1 & b Þ ð ka= ð 1 1 & b Þ P b= ð b 1 & Þ: (12:15) qið 3, w P, v, w & w b Þ ¼ ! " 12, and k1 ¼ ¼ If we let a function b ¼ ¼ 0.5, v ¼ 80, then this yields the simple, single-ﬁrm supply P, v, w 12 qið 10P 3 : (12:16) ¼ Now assume that there are 100 identical such ﬁrms and that each ﬁrm faces the same market prices for both its output and its input hiring. Given these assumptions, the short-run market supply function is given by Þ ¼ P, v, w QSð 12 Þ ¼ ¼ qi ¼ 100 1 i X ¼ 1 i X ¼ 100 10P 1,000P 3 : (12:17) 3 ¼ Thus, at a price of (say) P supplying 40 units. We can compute the short-run elasticity of supply in this situation as 12, total market supply will be 4,000, with each of the 100 ﬁrms ¼ eS, P ¼ @QSð P, v, w @P P QS ¼ 1; 000 3 Þ ’ P ’ 1; 000P=3 ¼ 1; (12:18) this might have been expected, given the unitary exponent of P in the supply function. Effect of an increase in w. If all the ﬁrms in this marketplace experienced an increase in the wage they must pay for their labor input, then the short-run supply curve would shift to a new position. To calculate the shift, we must return to the single ﬁrm’s supply function (Equation 12.15) and now use a new wage, say, w 15. If none of the other parameters of the problem have changed (the ﬁrm’s production function and the level of capital input it has in the short run), the supply function becomes ¼ and the market supply function is P, v, w qið 15 Þ ¼ ¼ 8P 3 P, v, w QSð 15 ¼ Þ ¼ 8P 3 ¼ 800P 3 : 100 1 i X ¼ (12:19) (12:20) ¼ 12, now this industry will supply only QS ¼ 3,200, with each ﬁrm 32. In other words, the supply curve has shifted upward because of the increase in 1. Thus, at a price of P producing qi ¼ the wage. Notice, however, that the price elasticity of supply has not changed—it remains eS, P ¼ QUERY: How would the results of this example change by assuming different values for the weight of labor in the production function (i.e., for a and b)? 418 Part 5: Competitive Markets Equilibrium price determination We can now combine demand and supply curves to demonstrate the establishment of equilibrium prices in the market. Figure 12.4 shows this process. Looking ﬁrst at Figure 12.4b, we see the market demand curve D (ignore D 0 for the moment) and the short-run supply curve S. The two curves intersect at a price of P1 and a quantity of Q1. This price– quantity combination represents an equilibrium between the demands of individuals and the costs of ﬁrms. The equilibrium price P1 serves two important functions. First, this price acts as a signal to producers by providing them with information about how much should be produced: To maximize proﬁts, ﬁrms will produce that output level for which marginal costs are equal to P1. In the aggregate, production will be Q1. A second function of the price is to ration demand. Given the market price P1, utility-maximizing individuals will decide how much of their limited incomes to devote to buying the particular good. At a price of P1, total quantity demanded will be Q1, and this is precisely the amount that will be produced. Hence we deﬁne equilibrium price as follows Equilibrium price. An equilibrium price is one at which quantity demanded is equal to quantity supplied. At such a price, neither demanders nor suppliers have an incentive to alter their economic decisions. Mathematically, an equilibrium price P$ solves the equation or, more compactly, P$, P 0, I QDð P$, v, w QSð Þ Þ ¼ P$ QDð Þ ¼ P$ QSð : Þ (12:21) (12:22) FIGURE 12.4 Interactions of Many Individuals and Firms Determine Market Price in the Short Run Market demand curves and market supply curves are each the horizontal sum of numerous components. These market curves are shown in (b). Once price is determined in the market, each ﬁrm and each individual treat this price as a ﬁxed parameter in their decisions. Although individual ﬁrms and persons are important in determining price, their interaction as a whole is the sole determinant of price. This is illustrated by a shift in an individual’s demand curve to d 0. If only one individual reacts in this way, market price will not be affected. However, if everyone exhibits an increased demand, market demand will shift to D 0; in the short run, price will increase to P2. Price Price SMC SAC D′ D P 2 P 1 Price S d′ d D′ D d′ d q 1 q 2 Output per period Q 1 Q 2 Total output per period q 1 q 2 q 1′ Quantity demanded per period (a) A typical firm (b) The market (c) A typical individual Chapter 12: The Partial Equilibrium Competitive Model 419 The deﬁnition given in Equation 12.22 makes clear that an equilibrium price depends on the values of many exogenous factors, such as incomes or prices of other goods and of ﬁrms’ inputs. As we will see in the next section, changes in any of these factors will likely result in a change in the equilibrium price required to equate quantity supplied to quantity demanded. The implications of the equilibrium price (P1) for a typical ﬁrm and a typical individual are shown in Figures 12.4a and 12.4c, respectively. For the typical ﬁrm the price P1 will cause an output level of q1 to be produced. The ﬁrm earns a small proﬁt at this particular price because short-run average total costs are covered. The demand curve d (ignore d 0 for the moment) for a typical individual is shown in Figure 12.4c. At a price of P1, this individual demands q1. By adding up the quantities that each individual demands at P1 and the quantities that each ﬁrm supplies, we can see that the market is in equilibrium. The market supply and demand curves provide a convenient way of making such a summation. Market reaction to a shift in demand The three panels in Figure 12.4 can be used to show two important facts about short-run market equilibrium: the individual’s ‘‘impotence’’ in the market and the nature of short-run supply response. First, suppose that a single individual’s demand curve were to shift outward to d 0, as shown in Figure 12.4c. Because the competitive model assumes there are many demanders, this shift will have practically no effect on the market demand curve. Consequently, market price will be unaffected by the shift to d 0, that is, price will remain at P1. Of course, at this price, the person for whom the demand curve has shifted will consume slightly more (q 01), as shown in Figure 12.4c. But this amount is a tiny part of the market. If many individuals experience outward shifts in their demand curves, the entire market demand curve may shift. Figure 12.4b shows the new demand curve D 0. The new equilibrium point will be at P2, Q2; at this point, supply–demand balance is re-established. Price has increased from P1 to P2 in response to the demand shift. Notice also that the quantity traded in the market has increased from Q1 to Q2. The increase in price has served two functions. First, as in our previous analysis of the very short run, it has acted to ration demand. Whereas at P1 a typical individual demanded q 01, at P2 only q 02 is demanded. The increase in price has also acted as a signal to the typical ﬁrm to increase production. In Figure 12.4a, the ﬁrm’s proﬁt-maximizing output level has increased from q1 to q2 in response to the price increase. That is what we mean by a short-run supply response: An increase in market price acts as an inducement to increase production. Firms are willing to increase production (and to incur higher marginal costs) because the price has increased. If market price had not been permitted to increase (suppose that government price controls were in effect), then ﬁrms would not have increased their outputs. At P1 there would now be an excess (unﬁlled) demand for the good in question. If market price is allowed to increase, a supply–demand equilibrium can be re-established so that what ﬁrms produce is again equal to what individuals demand at the prevailing market price. Notice also that, at the new price P2, the typical ﬁrm has increased its proﬁts. This increasing proﬁtability in the short run will be important to our discussion of long-run pricing later in this chapter. Shifts in Supply and Demand Curves: a Graphical Analysis In previous chapters we established many reasons why either a demand curve or a supply curve might shift. These reasons are brieﬂy summarized in Table 12.1. Although most of these merit little additional explanation, it is important to note that a change in the 420 Part 5: Competitive Markets TABLE 12.1 REASONS FOR SHIFTS IN DEMAND OR SUPPLY CURVES Demand Curves Shift Because Supply Curves Shift Because Incomes change Input prices change Prices of substitutes or complements change Technology changes Preferences change Number of producers changes number of ﬁrms will shift the short-run market supply curve (because the sum in Equation 12.13 will be over a different number of ﬁrms). This observation allows us to tie together short-run and long-run analysis. It seems likely that the types of changes described in Table 12.1 are constantly occurring in real-world markets. When either a supply curve or a demand curve does shift, equilibrium price and quantity will change. In this section we investigate graphically the relative magnitudes of such changes. In the next section we show the results mathematically. Shifts in supply curves: Importance of the shape of the demand curve Consider
ﬁrst a shift inward in the short-run supply curve for a good. As in Example 12.2, such a shift might have resulted from an increase in the prices of inputs used by ﬁrms to produce the good. Whatever the cause of the shift, it is important to recognize that the effect of the shift on the equilibrium level of P and Q will depend on the shape of the demand curve for the product. Figure 12.5 illustrates two possible situations. The demand curve in Figure 12.5a is relatively price elastic; that is, a change in price substantially affects quantity demanded. For this case, a shift in the supply curve from S to S 0 will cause equilibrium price to increase only moderately (from P to P 0), whereas quantity decreases sharply (from Q to Q 0). Rather than being ‘‘passed on’’ in higher prices, the In (a) the shift upward in the supply curve causes price to increase only slightly while quantity decreases sharply. This results from the elastic shape of the demand curve. In (b) the demand curve is inelastic; price increases substantially, with only a slight decrease in quantity. Price D P′ P S′ S S′ S Price D P′ P S′ S Q′ Q Q per period Q′ Q Q per period (a) Elastic demand (b) Inelastic demand FIGURE 12.5 Effect of a Shift in the Short-Run Supply Curve Depends on the Shape of the Demand Curve Chapter 12: The Partial Equilibrium Competitive Model 421 increase in the ﬁrms’ input costs is met primarily by a decrease in quantity (a movement down each ﬁrm’s marginal cost curve) and only a slight increase in price. This situation is reversed when the market demand curve is inelastic. In Figure 12.5b a shift in the supply curve causes equilibrium price to increase substantially while quantity is little changed. The reason for this is that individuals do not reduce their demands much if prices increase. Consequently, the shift upward in the supply curve is almost entirely passed on to demanders in the form of higher prices. Shifts in demand curves: Importance of the shape of the supply curve Similarly, a shift in a market demand curve will have different implications for P and Q, depending on the shape of the short-run supply curve. Two illustrations are shown in Figure 12.6. In Figure 12.6a the supply curve for the good in question is inelastic. In this situation, a shift outward in the market demand curve will cause price to increase substantially. On the other hand, the quantity traded increases only slightly. Intuitively, what has happened is that the increase in demand (and in Q) has caused ﬁrms to move up their steeply sloped marginal cost curves. The concomitant large increase in price serves to ration demand. Figure 12.6b shows a relatively elastic short-run supply curve. Such a curve would occur for an industry in which marginal costs do not increase steeply in response to output increases. For this case, an increase in demand produces a substantial increase in Q. However, because of the nature of the supply curve, this increase is not met by great cost increases. Consequently, price increases only moderately. These examples again demonstrate Marshall’s observation that demand and supply simultaneously determine price and quantity. Recall his analogy from Chapter 1: Just as it is impossible to say which blade of a scissors does the cutting, so too is it impossible to attribute price solely to demand or to supply characteristics. Rather, the effect of FIGURE 12.6 Effect of a Shift in the Demand Curve Depends on the Shape of the Short-Run Supply Curve In (a), supply is inelastic; a shift in demand causes price to increase greatly, with only a small concomitant increase in quantity. In (b), on the other hand, supply is elastic; price increases only slightly in response to a demand shift. D′ D Price P′ P S Price D′ D S P′ P D′ S D Q Q′ Q per period Q Q′ (a) Inelastic supply (b) Elastic supply S D′ D Q per period 422 Part 5: Competitive Markets shifts in either a demand curve or a supply curve will depend on the shapes of both curves. Mathematical Model of Market Equilibrium A general mathematical model of the supply–demand process can further illuminate the comparative statics of changing equilibrium prices and quantities. Suppose that the demand function is represented by D , Þ QD ¼ P, a ð where a is a parameter that allows us to shift the demand curve. It might represent consumer income, prices of other goods (this would permit the tying together of supply and demand in several related markets), or changing preferences. In general we expect that @D=@P Da may have any sign, depending precisely on what the parameter a means. Using this same procedure, we can write the supply relationship as DP < 0, but @D=@a (12:23) ¼ ¼ P, b , Þ ð where b is a parameter that shifts the supply curve and might include such factors as input prices, technical changes, or (for a multiproduct ﬁrm) prices of other potential outputs. Here @S=@P Sb may have any sign. The model is closed by requiring that, in equilibrium,5 SP > 0, but @S=@b QS ¼ (12:24) ¼ ¼ S QD ¼ QS: (12:25) To analyze the effect of a small change in one of the exogenous parameters (a or b) on market equilibrium requires a bit of calculus.6 Suppose we are interested in the impact of a shift in demand (a) while keeping the supply function ﬁxed (i.e., holding b constant). Differentiation of the demand and supply functions yields: dQD da ¼ dQS da ¼ Þ P, a dD ð da P, b Þ ð da ¼ dS ¼ SP dP da : DP dP da þ Da (12:26) Notice that the only effect on supply here occurs through the impact of market price— the exogenous factors in the supply function are held constant. Maintenance of market equilibrium for this shift in demand requires that dQD da ¼ dQS da : (12:27) 5The model could be further modiﬁed to show how the equilibrium quantity supplied is to be allocated among the ﬁrms in the industry. If, for example, the industry is composed of n identical ﬁrms, then the output of any one of them would be given by Q n : q ¼ In the short run with n ﬁxed this would add little to our analysis. In the long run, however, n must also be determined by the model as we show later in this chapter. 6This type of analysis is usually called comparative statics analysis because we are comparing two equilibrium positions but are not especially concerned with the ‘‘dynamics’’ of how the market moves from one equilibrium to the other. Chapter 12: The Partial Equilibrium Competitive Model 423 Hence we can solve for the change in equilibrium price as or, after a bit of algebra, DP dP da þ Da SP ¼ dP da (12:28) : DP (12:29) dP da ¼ Da SP & Because the denominator of this expression is positive, the overall sign of dP=da will depend only on the sign of Da—that is, on how the change of the exogenous factor a affects demand. For example, if a represents consumer income, we would expect Da to be positive and thus dP=da would be positive. That is, an increase in income would be expected to increase equilibrium price. On the other hand, if a represented the price of a (gross) complement, we would expect Da to be negative and dP=da would also be negative. An increase in the price of a complementary good would be expected to reduce P. It would be a simple matter to repeat the steps in Equations 12.27–12.29 to derive a similar expression for how a shift in supply (b) would affect the equilibrium price. An elasticity interpretation Further algebraic manipulation of Equation 12.29 yields a more useful comparative statics result. Multiplying both sides of that equation by a/P gives eP,a ¼ dP da ’ a P ¼ Dað SP & a=Q DPÞ ’ DP ’ Da SP & Þ P=Q ¼ a P eQ,a eS,P & eQ,P : (12:30) ¼ ð Because all the elasticities in this equation may be available from empirical studies, this equation can be a convenient way to make rough estimates of the effects of various events on equilibrium prices. As an example, suppose again that a represents consumer income and that there is interest in predicting how an increase in income affects the equilibrium price of, say, automobiles. Suppose empirical data suggest that eQ, I ¼ 3.0 and 1.0. eQ, P ¼ & Substituting these ﬁgures into Equation 12.30 yields 1.2 (these ﬁgures are from Table 12.3; see Extensions) and assume that eS, P ¼ eQ, a ¼ eP,a ¼ ¼ eQ,a eS,P & 3:0 2:2 ¼ eQ,P ¼ 1:36: 3:0 1:0 1:2 Þ & ð& (12:31) Therefore, the empirical elasticity estimates suggest that each 1 percent increase in consumer incomes results in a 1.36 percent increase in the equilibrium price of automobiles. Estimates of other kinds of shifts in supply or demand can be similarly modeled by using the type of calculus-based approach provided in Equations 12.26–12.29. EXAMPLE 12.3 Equilibria with Constant Elasticity Functions An even more complete analysis of supply–demand equilibrium can be provided if we use speciﬁc functional forms. Constant elasticity functions are especially useful for this purpose. Suppose the demand for automobiles is given by P, I QDð Þ ¼ 0:1P& 1:2 I3; (12:32) 424 Part 5: Competitive Markets here price (P) is measured in dollars, as is real family income (I). The supply function for automobiles is P, w QSð Þ ¼ 6,400Pw& 0:5, (12:33) where w is the hourly wage of automobile workers. Notice that the elasticities assumed here 1). If the values for the 3.0, and eS, P ¼ are those used previously in the text (eQ, P ¼ & ‘‘exogenous’’ variables I and w are $20,000 and $25, respectively, then demand–supply equilibrium requires 1.2, eQ, I ¼ QD ¼ ¼ 0:1P& Qs ¼ 1:2I3 ¼ ð 6,400Pw& 8 3 1011 0:5 P& Þ 1,280P 1:2 ¼ or or P2:2 8 3 1011 ¼ ð =1,280 Þ ¼ 6:25 3 108 P$ Q$ ¼ ¼ 9,957, 1;280 P$ ’ ¼ 12,745,000: (12:34) (12:35) Hence the initial equilibrium in the automobile market has a price of nearly $10,000 with approximately 13 million cars being sold. A shift in demand. A 10 percent increase in real family income, all other factors remaining constant, would shift the demand function to QD ¼ ð 1:06 3 1012 (12:36) P& Þ 1:2 and, proceeding as before, or P 2:2 1:06 3 1012 ¼ ð =1,280 Þ ¼ 8:32 3 108 P$ Q$ ¼ ¼ 11,339, 14,514,000: (12:37) (12:38)
As we predicted earlier, the 10 percent increase in real income made car prices increase by nearly 14 percent. In the process, quantity sold increased by approximately 1.77 million automobiles. A shift in supply. An exogenous shift in automobile supply as a result, say, of changing auto workers’ wages would also affect market equilibrium. If wages were to increase from $25 to $30 per hour, the supply function would shift to 30 ð returning to our original demand function (with I 6,400P Qsð P, w Þ ¼ or P2:2 8 3 1011 ¼ ð 0:5 & Þ ¼ 1,168P; $20,000) then yields ¼ =1,168 Þ 6:85 3 108 ¼ P$ Q$ ¼ ¼ 10,381, 12,125,000: (12:39) (12:40) (12:41) Therefore, the 20 percent increase in wages led to a 4.3 percent increase in auto prices and to a decrease in sales of more than 600,000 units. Changing equilibria in many types of markets can be approximated by using this general approach together with empirical estimates of the relevant elasticities. QUERY: Do the results of changing auto workers’ wages agree with what might have been predicted using an equation similar to Equation 12.30? Chapter 12: The Partial Equilibrium Competitive Model 425 Long-Run Analysis We saw in Chapter 10 that, in the long run, a ﬁrm may adapt all its inputs to ﬁt market conditions. For long-run analysis, we should use the ﬁrm’s long-run cost curves. A proﬁtmaximizing ﬁrm that is a price-taker will produce the output level for which price is equal to long-run marginal cost (MC). However, we must consider a second and ultimately more important inﬂuence on price in the long run: the entry of entirely new ﬁrms into the industry or the exit of existing ﬁrms from that industry. In mathematical terms, we must allow the number of ﬁrms, n, to vary in response to economic incentives. The perfectly competitive model assumes that there are no special costs of entering or exiting from an industry. Consequently, new ﬁrms will be lured into any market in which (economic) proﬁts are positive. Similarly, ﬁrms will leave any industry in which proﬁts are negative. The entry of new ﬁrms will cause the short-run industry supply curve to shift outward because there are now more ﬁrms producing than there were previously. Such a shift will cause market price (and industry proﬁts) to decrease. The process will continue until no ﬁrm contemplating entry would be able to earn a proﬁt in the industry.7 At that point, entry will cease and the industry will have an equilibrium number of ﬁrms. A similar argument can be made for the case in which some of the ﬁrms are suffering short-run losses. Some ﬁrms will choose to leave the industry, and this will cause the supply curve to shift to the left. Market price will increase, thus restoring proﬁtability to those ﬁrms remaining in the industry. ¼ MC (which is required for proﬁt maximization) and P Equilibrium conditions To begin with we will assume that all the ﬁrms in an industry have identical cost functions; that is, no ﬁrm controls any special resources or technologies.8 Because all ﬁrms are identical, the equilibrium long-run position requires that each ﬁrm earn exactly zero economic proﬁts. In graphic terms, the long-run equilibrium price must settle at the low point of each ﬁrm’s long-run average total cost curve. Only at this point do the two equilibrium conditions P AC (which is required for zero proﬁt) hold. It is important to emphasize, however, that these two equilibrium conditions have rather different origins. Proﬁt maximization is a goal of ﬁrms. Therefore, the P MC rule derives from the behavioral assumptions we have made about ﬁrms and is similar to the output decision rule used in the short run. The zero-proﬁt condition is not a goal for ﬁrms; ﬁrms obviously would prefer to have large, positive proﬁts. The long-run operation of the market, however, forces all ﬁrms to accept a level of zero economic proﬁts (P AC) because of the willingness of ﬁrms to enter and to leave an industry in response to the possibility of making supranormal returns. Although the ﬁrms in a perfectly competitive industry may earn either positive or negative proﬁts in the short run, in the long run only a level of zero proﬁts will prevail. Hence we can summarize this analysis by the following deﬁnition Long-run competitive equilibrium. A perfectly competitive market is in long-run equilibrium if there are no incentives for proﬁt-maximizing ﬁrms to enter or to leave the market. This will occur when (a) the number of ﬁrms is such that P AC and (b) each ﬁrm operates at the low point of its long-run average cost curve. MC ¼ ¼ 7Remember that we are using the economists’ deﬁnition of proﬁts here. These proﬁts represent a return to the owner of a business in excess of that which is strictly necessary to stay in the business. 8If ﬁrms have different costs, then low-cost ﬁrms can earn positive long-run proﬁts, and such extra proﬁts will be reﬂected in the price of the resource that accounts for the ﬁrm’s low costs. In this sense the assumption of identical costs is not restrictive because an active market for the ﬁrm’s inputs will ensure that average costs (which include opportunity costs) are the same for all ﬁrms. See also the discussion of Ricardian rent later in this chapter. 426 Part 5: Competitive Markets Long-Run Equilibrium: Constant Cost Case To discuss long-run pricing in detail, we must make an assumption about how the entry of new ﬁrms into an industry affects the prices of ﬁrms’ inputs. The simplest assumption we might make is that entry has no effect on the prices of those inputs—perhaps because the industry is a relatively small hirer in its various input markets. Under this assumption, no matter how many ﬁrms enter (or leave) this market, each ﬁrm will retain the same set of cost curves with which it started. This assumption of constant input prices may not be tenable in many important cases, which we will look at in the next section. For the moment, however, we wish to examine the equilibrium conditions for a constant cost industry. Initial equilibrium Figure 12.7 demonstrates long-run equilibrium in this situation. For the market as a whole (Figure 12.7b), the demand curve is given by D and the short-run supply curve by SS. Therefore, the short-run equilibrium price is P1. The typical ﬁrm (Figure 12.7a) will produce output level q1 because, at this level of output, price is equal to short-run marginal cost (SMC). In addition, with a market price of P1, output level q1 is also a long-run equilibrium position for the ﬁrm. The ﬁrm is maximizing proﬁts because price is equal to long-run marginal costs (MC). Figure 12.7a also implies our second long-run equilibrium property: Price is equal to long-run average costs (AC). Consequently, economic proﬁts are zero, and there is no incentive for ﬁrms either to enter or to leave the industry. Therefore, the market depicted in Figure 12.7 is in both short-run and long-run equilibrium. FIGURE 12.7 Long-Run Equilibrium for a Perfectly Competitive Industry: Constant Cost Case An increase in demand from D to D 0 will cause price to increase from P1 to P2 in the short run. This higher price will create proﬁts in the industry, and new ﬁrms will be drawn into the market. If it is assumed that the entry of these new ﬁrms has no effect on the cost curves of the ﬁrms in the industry, then new ﬁrms will continue to enter until price is pushed back down to P1. At this price, economic proﬁts are zero. Therefore, the long-run supply curve (LS) will be a horizontal line at P1. Along LS, output is increased by increasing the number of ﬁrms, each producing q1. Price P 2 P 1 Price D′ D SMC MC AC SS SS′ SS SS′ LS D′ D q1 q2 Quantity per period Q1 Q2 Q3 Total quantity per period (a) A typical firm (b) Total market Chapter 12: The Partial Equilibrium Competitive Model 427 Firms are in equilibrium because they are maximizing proﬁts, and the number of ﬁrms is stable because economic proﬁts are zero. This equilibrium will tend to persist until either supply or demand conditions change. Responses to an increase in demand Suppose now that the market demand curve in Figure 12.7b shifts outward to D 0. If SS is the relevant short-run supply curve for the industry, then in the short run, price will increase to P2. The typical ﬁrm, in the short run, will choose to produce q2 and will earn proﬁts on this level of output. In the long run, these proﬁts will attract new ﬁrms into the market. Because of the constant cost assumption, this entry of new ﬁrms will have no effect on input prices. New ﬁrms will continue to enter the market until price is forced down to the level at which there are again no pure economic proﬁts. Therefore, the entry of new ﬁrms will shift the short-run supply curve to SS0, where the equilibrium price (P1) is re-established. At this new long-run equilibrium, the price–quantity combination P1, Q3 will prevail in the market. The typical ﬁrm will again produce at output level q1, although now there will be more ﬁrms than in the initial situation. Inﬁnitely elastic supply We have shown that the long-run supply curve for the constant cost industry will be a horizontal straight line at price P1. This curve is labeled LS in Figure 12.7b. No matter what happens to demand, the twin equilibrium conditions of zero long-run proﬁts (because free entry is assumed) and proﬁt maximization will ensure that no price other than P1 can prevail in the long run.9 For this reason, P1 might be regarded as the ‘‘normal’’ price for this commodity. If the constant cost assumption is abandoned, however, the long-run supply curve need not have this inﬁnitely elastic shape, as we show in the next section. EXAMPLE 12.4 Inﬁnitely Elastic Long-Run Supply Handmade bicycle frames are produced by a number of identically sized ﬁrms. Total (long-run) monthly costs for a typical ﬁrm are given by C q ð Þ ¼ q3 & 20q2 þ 100q þ 8,000; (12:42) where q is the number of frames produced per month. Demand for handmade bicycle frames is given by QD ¼ 2,500
3P, & (12:43) where QD is the quantity demanded per month and P is the price per frame. To determine the long-run equilibrium in this market, we must ﬁnd the low point of the typical ﬁrm’s average cost curve. Because AC C q ð Þ q ¼ ¼ q2 & 20q þ 100 8,000 q þ (12:44) 9These equilibrium conditions also point out what seems to be, somewhat imprecisely, an ‘‘efﬁcient’’ aspect of the long-run equilibrium in perfectly competitive markets: The good under investigation will be produced at minimum average cost. We will have much more to say about efﬁciency in the next chapter. 428 Part 5: Competitive Markets and MC @C q Þ ð @q ¼ ¼ 3q2 40q & þ 100 (12:45) and because we know this minimum occurs where AC q2 & 20q þ 100 8,000 þ q ¼ or ¼ 3q2 MC, we can solve for this output level: 40q 100 þ þ 2q2 20q & 8,000 q , ¼ (12:46) which has a convenient solution of q 20. With a monthly output of 20 frames, each producer has a long-run average and marginal cost of $500. This is the long-run equilibrium price of bicycle frames (handmade frames cost a bundle, as any cyclist can attest). With P $500, Equation 12.43 shows QD ¼ 1,000. Therefore, the equilibrium number of ﬁrms is 50. When each of these 50 ﬁrms produces 20 frames per month, supply will precisely balance what is demanded at a price of $500. ¼ ¼ If demand in this problem were to increase to QD ¼ 3,000 3P, & (12:47) then we would expect long-run output and the number of frames to increase. Assuming that entry into the frame market is free and that such entry does not alter costs for the typical bicycle maker, the long-run equilibrium price will remain at $500 and a total of 1,500 frames per month will be demanded. That will require 75 frame makers, so 25 new ﬁrms will enter the market in response to the increase in demand. QUERY: Presumably, the entry of frame makers in the long run is motivated by the short-run proﬁtability of the industry in response to the increase in demand. Suppose each ﬁrm’s shortrun costs were given by SC 20,000. Show that short-run proﬁts are zero þ when the industry is in long-term equilibrium. What are the industry’s short-run proﬁts as a result of the increase in demand when the number of ﬁrms stays at 50? 1,500q 50q2 ¼ & Shape of the Long-Run Supply Curve Contrary to the short-run situation, long-run analysis has little to do with the shape of the (long-run) marginal cost curve. Rather, the zero-proﬁt condition centers attention on the low point of the long-run average cost curve as the factor most relevant to long-run price determination. In the constant cost case, the position of this low point does not change as new ﬁrms enter the industry. Consequently, if input prices do not change, then only one price can prevail in the long run regardless of how demand shifts—the long-run supply curve is horizontal at this price. Once the constant cost assumption is abandoned, this need not be the case. If the entry of new ﬁrms causes average costs to rise, the long-run supply curve will have an upward slope. On the other hand, if entry causes average costs to decline, it is even possible for the long-run supply curve to be negatively sloped. We shall now discuss these possibilities. Increasing cost industry The entry of new ﬁrms into an industry may cause the average costs of all ﬁrms to increase for several reasons. New and existing ﬁrms may compete for scarce inputs, thus driving up their prices. New ﬁrms may impose ‘‘external costs’’ on existing ﬁrms (and on themselves) in the form of air or water pollution. They may increase the demand for Chapter 12: The Partial Equilibrium Competitive Model 429 FIGURE 12.8 An Increasing Cost Industry Has a Positively Sloped LongRun Supply Curve Initially the market is in equilibrium at P1, Q1. An increase in demand (to D0) causes price to increase to P2 in the short run, and the typical ﬁrm produces q2 at a proﬁt. This proﬁt attracts new ﬁrms into the industry. The entry of these new ﬁrms causes costs for a typical ﬁrm to increase to the levels shown in (b). With this new set of curves, equilibrium is re-established in the market at P3, Q3. By considering many possible demand shifts and connecting all the resulting equilibrium points, the long-run supply curve (LS) is traced out. Price Price SMC MC P 3 AC P 2 P 1 SMC MC AC Price D P 2 P 3 P 1 D′ SS SS′ LS D′ D q 1 Output q 2 per period q 3 Output per period Q 1 Q 2 Q 3 Output per period (a) Typical firm before entry (b) Typical firm after entry (c) The market tax-ﬁnanced services (e.g., police forces, sewage treatment plants), and the required taxes may show up as increased costs for all ﬁrms. Figure 12.8 demonstrates two market equilibria in such an increasing cost industry. The initial equilibrium price is P1. At this price the typical ﬁrm produces q1, and total industry output is Q1. Suppose now that the demand curve for the industry shifts outward to D 0. In the short run, price will rise to P2 because this is where D 0 and the industry’s short-run supply curve (SS) intersect. At this price the typical ﬁrm will produce q2 and will earn a substantial proﬁt. This proﬁt then attracts new entrants into the market and shifts the short-run supply curve outward. Suppose that this entry of new ﬁrms causes the cost curves of all ﬁrms to increase. The new ﬁrms may compete for scarce inputs, thereby driving up the prices of these inputs. A typical ﬁrm’s new (higher) set of cost curves is shown in Figure 12.8b. The new long-run AC), and at this price Q3 is equilibrium price for the industry is P3 (here P3 ¼ demanded. We now have two points (P1, Q1 and P3, Q3) on the long-run supply curve. All other points on the curve can be found in an analogous way by considering all possible shifts in the demand curve. These shifts will trace out the long-run supply curve LS. Here LS has a positive slope because of the increasing cost nature of the industry. Observe that the LS curve is ﬂatter (more elastic) than the short-run supply curves. This indicates the greater ﬂexibility in supply response that is possible in the long run. Still, the curve is upward sloping, so price increases with increasing demand. This situation is probably common; we will have more to say about it in later sections. MC ¼ Decreasing cost industry Not all industries exhibit constant or increasing costs. In some cases, the entry of new ﬁrms may reduce the costs of ﬁrms in an industry. For example, the entry of new ﬁrms may provide a larger pool of trained labor from which to draw than was previously available, thus reducing the costs associated with the hiring of new workers. Similarly, the entry of new ﬁrms may provide a ‘‘critical mass’’ of industrialization, which permits the development of more efﬁcient transportation and communications networks. Whatever 430 Part 5: Competitive Markets FIGURE 12.9 A Decreasing Cost Industry Has a Negatively Sloped Long-Run Supply Curve In (c), the market is in equilibrium at P1, Q1. An increase in demand to D0 causes price to increase to P2 in the short run, and the typical ﬁrm produces q2 at a proﬁt. This proﬁt attracts new ﬁrms to the industry. If the entry of these new ﬁrms causes costs for the typical ﬁrm to decrease, a set of new cost curves might look like those in (b). With this new set of curves, market equilibrium is re-established at P3, Q3. By connecting such points of equilibrium, a negatively sloped long-run supply curve (LS) is traced out. Price P2 P1 AC SMC MC Price Price D′ SS SMC MC AC P3 D LS P2 P1 P3 q1 q2 Output per period q3 Output per period D Q1 Q2 (a) Typical firm before entry (b) Typical firm after entry (c) The market SS′ LS D′ Output Q3 per period the exact reason for the cost reductions, the ﬁnal result is illustrated in the three panels of Figure 12.9. The initial market equilibrium is shown by the price–quantity combination P1, Q1 in Figure 12.9c. At this price the typical ﬁrm produces q1 and earns exactly zero in economic proﬁts. Now suppose that market demand shifts outward to D 0. In the short run, price will increase to P2 and the typical ﬁrm will produce q2. At this price level, positive proﬁts are being earned. These proﬁts cause new entrants to come into the market. If this entry causes costs to decline, a new set of cost curves for the typical ﬁrm might resemble those shown in Figure 12.9b. Now the new equilibrium price is P3; at this price, Q3 is demanded. By considering all possible shifts in demand, the long-run supply curve, LS, can be traced out. This curve has a negative slope because of the decreasing cost nature of the industry. Therefore, as output expands, price falls. This possibility has been used as the justiﬁcation for protective tariffs to shield new industries from foreign competition. It is assumed (only occasionally correctly) that the protection of the ‘‘infant industry’’ will permit it to grow and ultimately to compete at lower world prices. Classiﬁcation of long-run supply curves Thus, we have shown that the long-run supply curve for a perfectly competitive industry may assume a variety of shapes. The principal determinant of the shape is the way in which the entry of ﬁrms into the industry affects all ﬁrms’ costs. The following deﬁnitions cover the various possibilities Constant, increasing, and decreasing cost industries. An industry supply curve exhibits one of three shapes. Constant cost: Entry does not affect input costs; the long-run supply curve is horizontal at the longrun equilibrium price. Increasing cost: Entry increases input costs; the long-run supply curve is positively sloped. Decreasing cost: Entry reduces input costs; the long-run supply curve is negatively sloped. Chapter 12: The Partial Equilibrium Competitive Model 431 Now we show how the shape of the long-run supply curve can be further quantiﬁed. Long-Run Elasticity of Supply The long-run supply curve for an industry incorporates information on internal ﬁrm adjustments to changing prices and changes in the number of ﬁrms and input costs in res
ponse to proﬁt opportunities. All these supply responses are summarized in the following elasticity concept Long-run elasticity of supply. The long-run elasticity of supply (eLS,P) records the proportionate change in long-run industry output in response to a proportionate change in product price. Mathematically, eLS, P ¼ percentage change in Q percentage change in P ¼ @QLS @P ’ P QLS : (12:48) The value of this elasticity may be positive or negative depending on whether the industry exhibits increasing or decreasing costs. As we have seen, eLS,P is inﬁnite in the constant cost case because industry expansions or contractions can occur without having any effect on product prices. Empirical estimates It is obviously important to have good empirical estimates of long-run supply elasticities. These indicate whether production can be expanded with only a slight increase in relative price (i.e., supply is price elastic) or whether expansions in output can occur only if relative prices increase sharply (i.e., supply is price inelastic). Such information can be used to assess the likely effect of shifts in demand on long-run prices and to evaluate alternative policy proposals intended to increase supply. Table 12.2 presents several long-run supply elasticity estimates. These relate primarily (although not exclusively) to natural resources because economists have devoted considerable attention to the implications of increasing demand for the prices of such resources. As the table makes clear, these estimates vary widely depending on the spatial and geological properties of the particular resources involved. All the estimates, however, suggest that supply does respond positively to price. Comparative Statics Analysis of Long-Run Equilibrium Earlier in this chapter we showed how to develop a simple comparative statics analysis of changing short-run equilibria in competitive markets. By using estimates of the long-run elasticities of demand and supply, exactly the same sort of analysis can be conducted for the long run as well. For example, the hypothetical auto market model in Example 12.3 might serve equally well for long-run analysis, although some differences in interpretation might be required. Indeed, in applied models of supply and demand it is often not clear whether the author intends his or her results to reﬂect the short run or the long run, and some care must be taken to understand how the issue of entry is being handled. 432 Part 5: Competitive Markets TABLE 12.2 SELECTED ESTIMATES OF LONG-RUN SUPPLY ELASTICITIES Agricultural acreage Corn Cotton Wheat Aluminum Chromium Coal (eastern reserves) Natural gas (U.S. reserves) Oil (U.S. reserves) Urban housing Density Quality 0.18 0.67 0.93 Nearly inﬁnite 0–3.0 15.0–30.0 0.20 0.76 5.3 3.8 SOURCES: Agricultural acreage—M. Nerlove, ‘‘Estimates of the Elasticities of Supply of Selected Agricultural Commodities,’’ Journal of Farm Economics 38 (May 1956): 496–509. Aluminum and chromium—estimated from U.S. Department of Interior, Critical Materials Commodity Action Analysis (Washington, DC: U.S. Government Printing Ofﬁce, 1975). Coal—estimated from M. B. Zimmerman, ‘‘The Supply of Coal in the Long Run: The Case of Eastern Deep Coal,’’ MIT Energy Laboratory Report No. MITEL 75–021 (September 1975). Natural gas—based on estimate for oil (see text) and J. D. Khazzoom, ‘‘The FPC Staff’s Econometric Model of Natural Gas Supply in the United States,’’ The Bell Journal of Economics and Management Science (Spring 1971): 103–17. Oil—E. W. Erickson, S. W. Millsaps, and R. M. Spann, ‘‘Oil Supply and Tax Incentives,’’ Brookings Papers on Economic Activity 2 (1974): 449–78. Urban housing—B. A. Smith, ‘‘The Supply of Urban Housing,’’ Journal of Political Economy 40 (August 1976): 389–405. Industry structure One aspect of the changing long-run equilibria in a perfectly competitive market that is obscured by using a simple supply–demand analysis is how the number of ﬁrms varies as market equilibria change. Because—as we will see in Part 6—the functioning of markets may in some cases be affected by the number of ﬁrms, and because there may be direct public policy interest in entry and exit from an industry, some additional analysis is required. In this section we will examine in detail determinants of the number of ﬁrms in the constant cost case. Brief reference will also be made to the increasing cost case, and some of the problems for this chapter examine that case in more detail. Shifts in demand Because the long-run supply curve for a constant cost industry is inﬁnitely elastic, analyzing shifts in market demand is particularly easy. If the initial equilibrium industry output is Q0 and if q$ represents the output level for which the typical ﬁrm’s long-run average cost is minimized, then the initial equilibrium number of ﬁrms (n0) is given by n0 ¼ Q0 q$ . (12:49) A shift in demand that changes equilibrium output to Q1 will, in the long run, change the equilibrium number of ﬁrms to n1 ¼ Q1 q$ , (12:50) Chapter 12: The Partial Equilibrium Competitive Model 433 and the change in the number of ﬁrms is given by n1 & n0 ¼ Q0 : Q1 & q$ (12:51) That is, the change in the equilibrium number of ﬁrms is completely determined by the extent of the demand shift and by the optimal output level for the typical ﬁrm. Changes in input costs Even in the simple constant cost industry case, analyzing the effect of an increase in an input price (and hence an upward shift in the inﬁnitely elastic long-run supply curve) is relatively complicated. First, to calculate the decrease in industry output, it is necessary to know both the extent to which minimum average cost is increased by the input price increase and how such an increase in the long-run equilibrium price affects total quantity demanded. Knowledge of the typical ﬁrm’s average cost function and of the price elasticity of demand permits such a calculation to be made in a straightforward way. But an increase in an input price may also change the minimum average cost output level for the typical ﬁrm. Such a possibility is illustrated in Figure 12.10. Both the average and marginal costs have been shifted upward by the input price increase, but because average cost has shifted up by a relatively greater extent than the marginal cost, the typical ﬁrm’s optimal output level has increased from q$0 to q$1. If the relative sizes of the shifts in cost curves were reversed, however, the typical ﬁrm’s optimal output An increase in the price of an input will shift average and marginal cost curves upward. The precise effect of these shifts on the typical ﬁrm’s optimal output level (q$) will depend on the relative magnitudes of the shifts. Average and marginal costs MC1 MC0 AC1 AC0 q0* *q1 Output per period FIGURE 12.10 An Increase in an Input Price May Change Long-Run Equilibrium Output for the Typical Firm 434 Part 5: Competitive Markets level would have decreased.10 Taking account of this change in optimal scale, Equation 12.51 becomes n1 & n0 ¼ Q1 q$1 & Q0 q$0 , (12:52) and a number of possibilities arise. If q$1 ( q$0, the decrease in quantity brought about by the increase in market price will deﬁnitely cause the number of ﬁrms to decrease. However, if q$1 < q$0, then the result will be indeterminate. Industry output will decrease, but optimal ﬁrm size also will decrease, thus the ultimate effect on the number of ﬁrms depends on the relative magnitude of these changes. A decrease in the number of ﬁrms still seems the most likely outcome when an input price increase causes industry output to decrease, but an increase in n is at least a theoretical possibility. EXAMPLE 12.5 Increasing Input Costs and Industry Structure An increase in costs for bicycle frame makers will alter the equilibrium described in Example 12.4, but the precise effect on market structure will depend on how costs increase. The effects of an increase in ﬁxed costs are fairly clear: The long-run equilibrium price will increase and the size of the typical ﬁrm will also increase. This latter effect occurs because an increase in ﬁxed costs increases AC but not MC. To ensure that the equilibrium condition for AC MC holds, output (and MC) must also increase. For example, if an increase in shop rents causes the typical frame maker’s costs to increase to ¼ C(q) q3 20q2 100q 11,616, (12:53) ¼ þ & it is an easy matter to show that MC 22. Therefore, the increase in rent has AC when q increased the efﬁcient scale of bicycle frame operations by 2 bicycle frames per month. At q 22, the long-run average cost and the marginal cost are both 672, and that will be the longrun equilibrium price for frames. At this price ¼ ¼ ¼ þ ¼ so there will be room in the market now for only 22 ( 22) ﬁrms. The increase in ﬁxed costs resulted not only in an increase in price but also in a signiﬁcant reduction in the number of frame makers (from 50 to 22). 484 ¼ ) & QD ¼ 2,500 3P 484, (12:54) Increases in other types of input costs may, however, have more complex effects. Although a complete analysis would require an examination of frame makers’ production functions and their related input choices, we can provide a simple illustration by assuming that an increase in some variable input prices causes the typical ﬁrm’s total cost function to become C(q) q3 & ¼ 8q2 þ 100q þ 4,950. (12:55) 10A mathematical proof proceeds as follows. Optimal output q is deﬁned such that $ Differentiating both sides of this expression by (say) v yields AC v, w, q$ ð Þ ¼ MC ð v, w, q$ : Þ but @AC=@q$ ¼ 0 because average costs are minimized. Manipulating terms, we obtain @AC @v þ @AC @q$ ’ @q$ @v ¼ @MC @v þ @MC @q$ ’ @q$ @v ; @q$ @v ¼ @MC @q$! 1 & " ’ ! @AC @v & @MC @v : " Because @MC=@q > 0 at the minimum AC, it follows that @q$=@v will be positive or negative depending on the sizes of the relative shifts in the AC and MC curves. Chapter 12: The Partial Equilibrium Competitive Model 435 Now MC AC 3q2 16q & q2 8q & þ ¼ ¼ 100 þ 100 þ and 4,950 q : Settin
g MC AC yields ¼ 2q2 8q & ¼ 4,950 q , (12:56) (12:57) which has a solution of q signiﬁcantly reduced the optimal size for frame shops. With q AC 15. Therefore, this particular change in the total cost function has 15, Equations 12.56 show 535, and with this new long-run equilibrium price we have MC ¼ ¼ ¼ ¼ QD ¼ These 895 frames will, in equilibrium, be produced by about 60 ﬁrms (895 59.67— problems do not always work out evenly!). Even though the increase in costs results in a higher price, the equilibrium number of frame makers expands from 50 to 60 because the optimal size of each shop is now smaller. (12:58) 2,500 895. 3P 15 ¼ ) ¼ & QUERY: How do the total, marginal, and average functions derived from Equation 12.55 differ from those in Example 12.4? Are costs always greater (for all levels of q) for the former cost curve? Why is long-run equilibrium price higher with the former curves? (See footnote 10 for a formal discussion.) Producer Surplus in the Long Run In Chapter 11 we described the concept of short-run producer surplus, which represents the return to a ﬁrm’s owners in excess of what would be earned if output were zero. We showed that this consisted of the sum of short-run proﬁts plus short-run ﬁxed costs. In long-run equilibrium, proﬁts are zero and there are no ﬁxed costs; therefore, all such short-run surplus is eliminated. Owners of ﬁrms are indifferent about whether they are in a particular market because they could earn identical returns on their investments elsewhere. Suppliers of ﬁrms’ inputs may not be indifferent about the level of production in a particular industry, however. In the constant cost case, of course, input prices are assumed to be independent of the level of production on the presumption that inputs can earn the same amount in alternative occupations. But in the increasing cost case, entry will bid up some input prices and suppliers of these inputs will be made better off. Consideration of these price effects leads to the following alternative notion of producer surplus Producer surplus. Producer surplus is the extra return that producers make by making transactions at the market price over and above what they would earn if nothing were produced. It is illustrated by the size of the area below the market price and above the supply curve. Although this is the same deﬁnition we introduced in Chapter 11, the context is now different. Now the ‘‘extra returns that producers make’’ should be interpreted as meaning ‘‘the higher prices that productive inputs receive.’’ For short-run producer surplus, the gainers from market transactions are ﬁrms that are able to cover ﬁxed costs and possibly 436 Part 5: Competitive Markets earn proﬁts over their variable costs. For long-run producer surplus, we must penetrate back into the chain of production to identify who the ultimate gainers from market transactions are. It is perhaps surprising that long-run producer surplus can be shown graphically in much the same way as short-run producer surplus. The former is given by the area above the long-run supply curve and below equilibrium market price. In the constant cost case, long-run supply is inﬁnitely elastic, and this area will be zero, showing that returns to inputs are independent of the level of production. With increasing costs, however, longrun supply will be positively sloped and input prices will be bid up as industry output expands. Because this notion of long-run producer surplus is widely used in applied analysis (as we show later in this chapter), we will provide a formal development. Ricardian rent Long-run producer surplus can be most easily illustrated with a situation ﬁrst described by David Ricardo in the early part of the nineteenth century.11 Assume there are many parcels of land on which a particular crop might be grown. These range from fertile land (low costs of production) to poor, dry land (high costs). The long-run supply curve for the crop is constructed as follows. At low prices only the best land is used. As output increases, higher-cost plots of land are brought into production because higher prices make it proﬁtable to use this land. The long-run supply curve is positively sloped because of the increasing costs associated with using less fertile land. Market equilibrium in this situation is illustrated in Figure 12.11. At an equilibrium price of P$, owners of both the low-cost and the medium-cost ﬁrms earn (long-run) profits. The ‘‘marginal ﬁrm’’ earns exactly zero economic proﬁts. Firms with even higher costs stay out of the market because they would incur losses at a price of P$. Proﬁts earned by the intramarginal ﬁrms can persist in the long run, however, because they reﬂect a return to a unique resource—low-cost land. Free entry cannot erode these proﬁts even over the long term. The sum of these long-run proﬁts constitutes long-run producer surplus, as given by area P$EB in Figure 12.11d. Equivalence of these areas can be shown by recognizing that each point in the supply curve in Figure 12.11d represents minimum average cost for some ﬁrm. For each such ﬁrm, P AC represents proﬁts per unit of output. Total long-run proﬁts can then be computed by summing over all units of output.12 & 11See David Ricardo, The Principles of Political Economy and Taxation (1817; reprinted London: J. M. Dent and Son, 1965), chap. 2 and chap. 32. 12More formally, suppose that ﬁrms are indexed by i (i In the long-run equilibrium, Q$ also the inverse of the supply function (competitive price as a function of quantity supplied) is given by P ACi and P$ the indexing of ﬁrms, price is determined by the highest cost ﬁrm in the market: P Now, in long-run equilibrium, proﬁts for ﬁrm i are given by 1,…, n) from lowest to highest cost and that each ﬁrm produces q$. n$q$ (where n$ is the equilibrium number of ﬁrms and Q$ is total industry output). Suppose P (Q). Because of P (n$q$). P (Q$) P (iq$) ¼ ¼ ¼ ¼ ¼ ¼ ¼ and total proﬁts are given by which is the shaded area in Figure 12.11d. pi ¼ ð P$ & ACiÞ q$, p ¼ n$ 0 ð n$ ¼ 0 ð pi di n$ P$ 0 ð n$ ¼ ð ACiÞ & q$ di P$q$ di & n$ 0 ð ACiq$ di ¼ ¼ P$n$q$ & 0 ð Q$ iq$ P ð q$ di Þ P$Q$ & 0 ð P Q Þ ð dQ, Chapter 12: The Partial Equilibrium Competitive Model 437 FIGURE 12.11 Ricardian Rent Owners of low-cost and medium-cost land can earn long-run proﬁts. Long-run producers’ surplus represents the sum of all these rents—area P$EB in (d). Usually Ricardian rents will be capitalized into input prices. Price P * Price P * MC AC Price P * MC AC (a) Low-cost firm q * Quantity per period q * (b) Medium-cost firm Quantity per period MC AC Price P * B E S D q * (c) Marginal firm Quantity per period (d) The market Q * Quantity per period Capitalization of rents The long-run proﬁts for the low-cost ﬁrms in Figure 12.11 will often be reﬂected in prices for the unique resources owned by those ﬁrms. In Ricardo’s initial analysis, for example, one might expect fertile land to sell for more than an untillable rock pile. Because such prices will reﬂect the present value of all future proﬁts, these proﬁts are said to be ‘‘capitalized’’ into inputs’ prices. Examples of capitalization include such disparate phenomena as the higher prices of nice houses with convenient access for commuters, the high value of rock and sport stars’ contracts, and the lower value of land near toxic waste sites. Notice that in all these cases it is market demand that determines rents—these rents are not traditional input costs that indicate forgone opportunities. Input supply and long-run producer surplus It is the scarcity of low-cost inputs that creates the possibility of Ricardian rent. If lowcost farmland were available at inﬁnitely elastic supply, there would be no such rent. More generally, any input that is ‘‘scarce’’ (in the sense that it has a positively sloped supply curve to a particular industry) will obtain rents in the form of earning a higher return than would be obtained if industry output were zero. In such cases, increases in output 438 Part 5: Competitive Markets not only raise ﬁrms’ costs (and thereby the price for which the output will sell) but also generate factor rents for inputs. The sum of all such rents is again measured by the area above the long-run supply curve and below equilibrium price. Changes in the size of this area of long-run producer surplus indicate changing rents earned by inputs to the industry. Notice that, although long-run producer surplus is measured using the market supply curve, it is inputs to the industry that receive this surplus. Empirical measurements of changes in long-run producer surplus are widely used in applied welfare analysis to indicate how suppliers of various inputs fare as conditions change. The ﬁnal sections of this chapter illustrate several such analyses. Economic Efﬁciency and Welfare Analysis Long-run competitive equilibria may have the desirable property of allocating resources ‘‘efﬁciently.’’ Although we will have far more to say about this concept in a general equilibrium context in Chapter 13, here we can offer a partial equilibrium description of why the result might hold. Remember from Chapter 5 that the area below a demand curve and above market price represents consumer surplus—the extra utility consumers receive from choosing to purchase a good voluntarily rather than being forced to do without it. Similarly, as we saw in the previous section, producer surplus is measured as the area below market price and above the long-run supply curve, which represents the extra return that productive inputs receive rather than having no transactions in the good. Overall then, the area between the demand curve and the supply curve represents the sum of consumer and producer surplus: It measures the total additional value obtained by market participants by being able to make market transactions in this good. It seems clear that this total area is maximized at the competitive market equilibrium. A graphic proof Figure 12.12 shows a si
mpliﬁed proof. Given the demand curve (D) and the long-run supply curve (S), the sum of consumer and producer surplus is given by distance AB for the ﬁrst unit produced. Total surplus continues to increase as additional output is produced—up to the competitive equilibrium level, Q$. This level of production will be achieved when price is at the competitive level, P$. Total consumer surplus is represented by the light shaded area in the ﬁgure, and total producer surplus is noted by the darker shaded area. Clearly, for output levels less than Q$ (say, Q1), total surplus would be reduced. One sign of this misallocation is that, at Q1, demanders would value an additional unit of output at P1, whereas average and marginal costs would be given by P2. Because P1 > P2, total welfare would clearly increase by producing one more unit of output. A transaction that involved trading this extra unit at any price between P1 and P2 would be mutually beneﬁcial: Both parties would gain. The total welfare loss that occurs at output level Q1 is given by area FEG. The distribution of surplus at output level Q1 will depend on the precise (nonequilibrium) price that prevails in the market. At a price of P1, consumer surplus would be reduced substantially to area AFP1, whereas producers might gain because producer surplus is now P1 FGB. At a low price such as P2 the situation would be reversed, with producers being much worse off than they were initially. Hence the distribution of the welfare losses from producing less than Q$ will depend on the price at which transactions are FIGURE 12.12 Competitive Equilibrium and Consumer/Producer Surplus Chapter 12: The Partial Equilibrium Competitive Model 439 At the competitive equilibrium (Q$), the sum of consumer surplus (shaded lighter gray) and producer surplus (shaded darker) is maximized. For an output level Q1 < Q$, there is a deadweight loss of consumer and producer surplus that is given by area FEG. Price A P1 P * P2 B F G E S D 0 Q1 Q * Quantity per period conducted. However, the size of the total loss is given by FEG, regardless of the price settled upon.13 A mathematical proof Mathematically, we choose Q to maximize consumer surplus producer surplus [U(Q) PQ] PQ & dQ dQ 3 5 12.59 ð Þ where U(Q) is the utility function of the representative consumer and P(Q) is the longAC run supply relation. In long-run equilibria along the long-run supply curve, P(Q) MC. Maximization of Equation 12.59 with respect to Q yields ¼ U 0(Q) P(Q) AC MC, ¼ ¼ so maximization occurs where the marginal value of Q to the representative consumer is equal to market price. But this is precisely the competitive supply–demand equilibrium because the demand curve represents consumers’ marginal valuations, whereas the supply curve reﬂects marginal (and, in long-term equilibrium, average) cost. ¼ ¼ (12:60) 13Increases in output beyond Q$ also clearly reduce welfare. 440 Part 5: Competitive Markets Applied welfare analysis The conclusion that the competitive equilibrium maximizes the sum of consumer and producer surplus mirrors a series of more general economic efﬁciency ‘‘theorems’’ we will examine in Chapter 13. Describing the major caveats that attach to these theorems is best delayed until that more extended discussion. Here we are more interested in showing how the competitive model is used to examine the consequences of changing economic conditions on the welfare of market participants. Usually such welfare changes are measured by looking at changes in consumer and producer surplus. In the ﬁnal sections of this chapter, we look at two examples. EXAMPLE 12.6 Welfare Loss Computations Use of consumer and producer surplus notions makes possible the explicit calculation of welfare losses from restrictions on voluntary transactions. In the case of linear demand and supply curves, this computation is especially simple because the areas of loss are frequently triangular. For example, if demand is given by QD ¼ 10 & P (12:61) and supply by P 2, QS ¼ then market equilibrium occurs at the point P$ 3 would create a gap between what demanders are willing to pay (PD ¼ 7) and what suppliers require (PS ¼ 5). The welfare loss from restricting transactions is given by a triangle with a base of 2 ( 5) and a height of 1 (the difference between Q$ and Q). Hence the welfare loss is $1 if P is measured in dollars per unit and Q is measured in units. More generally, the loss will be measured in the units in which P Æ Q is measured. 4. Restriction of output to Q Q ¼ PD & PS ¼ þ ¼ 6, Q$ (12:62) 10 Computations with constant elasticity curves. More realistic results can usually be obtained by using constant elasticity demand and supply curves based on econometric studies. In Example 12.3 we examined such a model of the U.S. automobile market. We can simplify that example a bit by assuming that P is measured in thousands of dollars and Q in millions of automobiles and that demand is given by and supply by 200P& 1:2 QD ¼ QS ¼ 1.3P: (12:63) (12:64) Equilibrium in the market is given by P$ 12.8. Suppose now that government policy restricts automobile sales to 11 (million) to control emissions of pollutants. An approximation to the direct welfare loss from such a policy can be found by the triangular method used earlier. 9.87, Q$ ¼ ¼ ¼ With Q 8.46. Hence the welfare loss 11, we have PD ¼ 2:38. Here the ‘‘triangle’’ is given by 0.5(PD & units are those of P times Q: billions of dollars. Therefore, the approximate14 value of the welfare loss is $2.4 billion, which might be weighed against the expected gain from emissions control. 11.1 and PS ¼ 11:1 0:5 & ð (11/200)–0.83 Q PS)(Q$ & 11/1.3 8:46 ¼ Þ ¼ ¼ 12:8 Þ ’ ð Þ ¼ 11 & 14A more precise estimate of the loss can be obtained by integrating PD & 12.8. With exponential demand and supply curves, this integration is often easy. In the present case, the technique yields an estimated welfare loss of 2.28, showing that the triangular approximation is not too bad even for relatively large price changes. Hence we will primarily use such approximations in later analysis. PS over the range Q 11 to Q ¼ ¼ Chapter 12: The Partial Equilibrium Competitive Model 441 Distribution of loss. In the automobile case, the welfare loss is shared about equally by consumers given by 0.5(PD & Q P$) & 8.46) Æ (12.8 1.27. Because the price elasticity of demand is somewhat greater (in absolute value) than the price elasticity of supply, consumers incur less than half the loss and producers somewhat more than half. With a more price elastic demand curve, consumers would incur a smaller share of the loss. and producers. An approximation for (Q$ & 11) ¼ losses 1:11 and for producers by 0.5(9.87 consumers’ ’ & 12:8 9:87 11:1 Þ ¼ Þ ¼ 0:5 11 & & Þð is ð QUERY: How does the size of the total welfare loss from a quantity restriction depend on the elasticities of supply and demand? What determines how the loss will be shared? Price Controls and Shortages Sometimes governments may seek to control prices at below equilibrium levels. Although adoption of such policies may be based on noble motives, the controls deter long-run supply responses and create welfare losses for both consumers and producers. A simple analysis of this possibility is provided by Figure 12.13. Initially the market is in long-run equilibrium at P1, Q1 (point E). An increase in demand from D to D 0 would cause the price to rise to P2 in the short run and encourage entry by new ﬁrms. Assuming this market is characterized by increasing costs (as reﬂected by the positively sloped long-run supply curve LS), price would decrease somewhat as a result of this entry, ultimately settling at P3. If these price changes were regarded as undesirable, then the government could, in FIGURE 12.13 Price Controls and Shortages A shift in demand from D to D0 would increase price to P2 in the short run. Entry over the long run would yield a ﬁnal equilibrium of P3, Q3. Controlling the price at P1 would prevent these actions and yield a shortage of Q4 & Q1. Relative to the uncontrolled situation, the price control yields a transfer from producers to consumers (area P3CEP1) and a deadweight loss of forgone transactions given by the two areas AE0C and CE0E. Price P 2 P 3 P 1 A C E SS LS E′ D D′ Q 1 Q 3 Q 4 Quantity per period 442 Part 5: Competitive Markets Q1. principle, prevent them by imposing a legally enforceable ceiling price of P1. This would cause ﬁrms to continue to supply their previous output (Q1); but, because at P1 demanders now want to purchase Q4, there will be a shortage given by Q4 & Welfare evaluation The welfare consequences of this price-control policy can be evaluated by comparing consumer and producer surplus measures prevailing under this policy with those that would have prevailed in the absence of controls. First, the buyers of Q1 gain the consumer surplus given by area P3CEP1 because they can buy this good at a lower price than would exist in an uncontrolled market. This gain reﬂects a pure transfer from producers out of the amount of producer surplus that would exist without controls. What current consumers have gained from the lower price, producers have lost. Although this transfer does not represent a loss of overall welfare, it does clearly affect the relative well-being of the market participants. Second, the area AE 0C represents the value of additional consumer surplus that would have been attained without controls. Similarly, the area CE0E reﬂects additional producer surplus available in the uncontrolled situation. Together, these two areas (i.e., area AE 0E) represent the total value of mutually beneﬁcial transactions that are prevented by the government policy of controlling price. This is, therefore, a measure of the pure welfare costs of that policy. Disequilibrium behavior The welfare analysis depicted in Figure 12.13 also suggests some of the types of behavior that might be expected as a result of the price-control policy. Assuming that observed mar
ket outcomes are generated by Q(P1Þ ¼ min ½ QD(P1), QS(P1)], (12:65) suppliers will be content with this outcome, but demanders will not because they will be forced to accept a situation of excess demand. They have an incentive to signal their dissatisfaction to suppliers through increasing price offers. Such offers may not only tempt existing suppliers to make illegal transactions at higher than allowed prices but may also encourage new entrants to make such transactions. It is this kind of activity that leads to the prevalence of black markets in most instances of price control. Modeling the resulting transactions is difﬁcult for two reasons. First, these may involve non–price-taking behavior because the price of each transaction must be individually negotiated rather than set by ‘‘the market.’’ Second, nonequilibrium transactions will often involve imperfect information. Any pair of market participants will usually not know what other transactors are doing, although such actions may affect their welfare by changing the options available. Some progress has been made in modeling such disequilibrium behavior using game theory techniques (see Chapter 18). However, other than the obvious prediction that transactions will occur at prices above the price ceiling, no general results have been obtained. The types of black-market transactions undertaken will depend on the speciﬁc institutional details of the situation. Tax Incidence Analysis The partial equilibrium model of competitive markets has also been widely used to study the impact of taxes. Although, as we will point out, these applications are necessarily limited by their inability to analyze tax effects that spread through many markets, they do provide important insights on a number of issues. Chapter 12: The Partial Equilibrium Competitive Model 443 A mathematical model of tax incidence The effect of a per-unit tax can be most easily studied using the mathematical model of supply and demand that was introduced previously. Now, however, we need to make a distinction between the price paid by demanders (PD) and the price received by suppliers (PS) because a per-unit tax (t) introduces a ‘‘wedge’’ between these two magnitudes: PS ¼ If we let the demand and supply functions for this taxed good be given by D(PD) and S(PS), respectively, then equilibrium requires that PD & (12:66) t. Differentiation with respect to the tax rate, t, yields: D PDÞ ¼ ð S PSÞ ¼ ð S PD & ð t : Þ DP dPD dt ¼ SP dPD dt & SP: Rearranging terms then produces the ﬁnal result that dPD dt ¼ SP SP & DP ¼ eS eS & , eD (12:67) (12:68) (12:69) where eS and eD represent the price elasticities of supply and demand and the ﬁnal equation is derived by multiplying both numerator and denominator by P/Q. A similar set of manipulations for the change in supply price gives Because eD + 0 and eS ( dPS dt ¼ eD eS & : eD 0, these calculations provide the obvious results dPD dt ( dPS dt + 0, 0: (12:70) (12:71) (12:72) If eD ¼ 1 and the per-unit tax is com0 (demand is perfectly inelastic), then dPD/dt 1 and the tax is pletely paid by demanders. Alternatively, if eD ¼ &1 wholly paid by producers. More generally, dividing Equation 12.70 by Equation 12.69 yields ¼ , then dPS/dt ¼ & dPS=dt dPD=dt ¼ & eD es , & (12:73) which shows that the actor with the less elastic responses (in absolute value) will experience most of the price change occasioned by the tax. A welfare analysis Figure 12.14 permits a simpliﬁed welfare analysis of the tax incidence issue. Imposition of the unit tax, t, creates a vertical wedge between the supply and demand curves, and the quantity traded declines to Q$$. Demanders incur a loss of consumer surplus given by area PDFEP$, of which PDFHP$ is transferred to the government as a portion of total tax revenues. The balance of total tax revenues (P$HGPS) is paid by producers, who incur a total loss of producer surplus given by area P$EGPS. Notice that the reduction in combined consumer and producer surplus exceeds total tax revenues collected by area FEG. 444 Part 5: Competitive Markets FIGURE 12.14 Tax Incidence Analysis Imposition of a speciﬁc tax of amount t per unit creates a ‘‘wedge’’ between the price consumers pay (PD) and what suppliers receive (PS). The extent to which consumers or producers pay the tax depends on the price elasticities of demand and supply. Price PD P * PS ** Q * Output per period This area represents a ‘‘deadweight’’ loss that arises because some mutually beneﬁcial transactions are discouraged by the tax. In general, the sizes of all the various areas illustrated in Figure 12.14 will be affected by the price elasticities involved. To determine the ﬁnal incidence of the producers’ share of the tax would require an explicit analysis of input markets—the burden of the tax would be reﬂected in reduced rents for those inputs characterized by relatively inelastic supply. More generally, a complete analysis of the incidence question requires a general equilibrium model that can treat many markets simultaneously. We discuss such models in the next chapter. Deadweight loss and elasticity All non–lump-sum taxes involve deadweight losses because they alter the behavior of economic actors. The size of such losses will depend in a rather complex way on the elasticities of demand and supply in the market. A linear approximation to the size of this deadweight loss triangle for a small tax, t, is given by: DW 0:5t ¼ & dQ dt ’ t 0:5t2 dQ dt : ¼ & (12:74) Here the negative sign is needed because dQ/dt < 0, and we wish our deadweight loss ﬁgure to be positive. Now, by deﬁnition, the price elasticity of demand at the initial equilibrium (P0, Q0) is eD ¼ dQ dP ’ P0 Q0 ¼ dQ=dt dP=dt ’ P0 Q0 or dQ dt ¼ eD dP dt ’ Q0 P0 : (12:75) Chapter 12: The Partial Equilibrium Competitive Model 445 Thus, we can combine Equations 12.74, 12.75, and 12.69 to get a ﬁnal expression for the deadweight loss of this tax: DW 0:5t2 ¼ & eDeS eS & eD ’ Q0 P0 ¼ & 0:5 t P0! " 2 eDeS eS & eD P0Q0: (12:76) Clearly, deadweight losses are zero in cases in which either eD or eS is zero because then the tax does not alter the quantity of the good traded. More generally, deadweight losses are smaller in situations where eD or eS is small. In principle, Equation 12.76 can be used to evaluate the deadweight losses accompanying any complex tax system. This information might provide some insights on how a tax system could be designed to minimize the overall ‘‘excess burden’’ involved in collecting a needed amount of tax revenues (see Problems 12.9 and 12.10). Notice also that DW is proportional to the square of the tax rate—marginal excess burden increases with the tax rate. Transaction costs Although we have developed this discussion in terms of tax incidence theory, models incorporating a wedge between buyers’ and sellers’ prices have a number of other applications in economics. Perhaps the most important of these involve costs associated with making market transactions. In some cases these costs may be explicit. Most real estate transactions, for example, take place through a third-party broker, who charges a fee for the service of bringing buyer and seller together. Similar explicit transaction fees occur in the trading of stocks and bonds, boats and airplanes, and practically everything that is sold at auction. In all these instances, buyers and sellers are willing to pay an explicit fee to an agent or broker who facilitates the transaction. In other cases, transaction costs may be largely implicit. Individuals trying to purchase a used car, for example, will spend considerable time and effort reading classiﬁed advertisements and examining vehicles, and these activities amount to an implicit cost of making the transaction. EXAMPLE 12.7 The Excess Burden of a Tax In Example 12.6 we examined the loss of consumer and producer surplus that would occur if automobile sales were cut from their equilibrium level of 12.8 (million) to 11 (million). An auto tax of $2,640 (i.e., 2.64 thousand dollars) would accomplish this reduction because it would introduce exactly the wedge between demand and supply price that was calculated previously. Because we have assumed eD ¼ & 1.0 in Example 12.6 and because initial spending on automobiles is approximately $126 (billion), Equation 12.76 predicts that the excess burden from the auto tax would be 1.2 and eS ¼ DW 0:5 ¼ 2:64 9:87 2 1:2 2:2 ! " ¼ 126 2:46: (12:77) ! " This loss of 2.46 billion dollars is approximately the same as the loss from emissions control calculated in Example 12.6. It might be contrasted to total tax collections, which in this case amount to $29 billion ($2,640 per automobile times 11 million automobiles in the post-tax equilibrium). Here, the deadweight loss equals approximately 8 percent of total tax revenues collected. Marginal burden. An incremental increase in the auto tax would be relatively more costly in terms of excess burden. Suppose the government decided to round the auto tax upward to a ﬂat $3,000 per car. In this case, car sales would drop to approximately 10.7 (million). Tax collections would amount to $32.1 billion, an increase of $3.1 billion over what was computed previously. 446 Part 5: Competitive Markets Equation 12.76 can be used to show that deadweight losses now amount to $3.17 billion—an increase of $0.71 billion above the losses experienced with the lower tax. At the margin, additional deadweight losses amount to approximately 23 percent (0.72/3.1) of additional revenues collected. Hence marginal and average excess burden computations may differ signiﬁcantly. QUERY: Can you explain intuitively why the marginal burden of a tax exceeds its average burden? Under what conditions would the marginal excess burden of a tax exceed additional tax revenues collected? To the extent that transaction costs are on a per-unit basis (as they are in the real estate, securities, and auction examples), our previous taxation example applies exactly. From the poi
nt of view of the buyers and sellers, it makes little difference whether t represents a per-unit tax or a per-unit transaction fee because the analysis of the fee’s effect on the market will be the same. That is, the fee will be shared between buyers and sellers depending on the speciﬁc elasticities involved. Trading volume will be lower than in the absence of such fees.15 A somewhat different analysis would hold, however, if transaction costs were a lump-sum amount per transaction. In that case, individuals would seek to reduce the number of transactions made, but the existence of the charge would not affect the supply–demand equilibrium itself. For example, the cost of driving to the supermarket is mainly a lump-sum transaction cost on shopping for groceries. The existence of such a charge may not signiﬁcantly affect the price of food items or the amount of food consumed (unless it tempts people to grow their own), but the charge will cause individuals to shop less frequently, to buy larger quantities on each trip, and to hold larger inventories of food in their homes than would be the case in the absence of such a cost. Effects on the attributes of transactions More generally, taxes or transaction costs may affect some attributes of transactions more than others. In our formal model, we assumed that such costs were based only on the physical quantity of goods sold. Therefore, the desire of suppliers and demanders to minimize costs led them to reduce quantity traded. When transactions involve several dimensions (such as quality, risk, or timing), taxes or transaction costs may affect some or all of these dimensions—depending on the precise basis on which the costs are assessed. For example, a tax on quantity may cause ﬁrms to upgrade product quality, or informationbased transaction costs may encourage ﬁrms to produce less risky, standardized commodities. Similarly, a per-transaction cost (travel costs of getting to the store) may cause individuals to make fewer but larger transactions (and to hold larger inventories). The possibilities for these various substitutions will obviously depend on the particular circumstances of the transaction. We will examine several examples of cost-induced changes in attributes of transactions in later chapters.16 15This analysis does not consider possible beneﬁts obtained from brokers. To the extent that these services are valuable to the parties in the transaction, demand and supply curves will shift outward to reﬂect this value. Hence trading volume may expand with the availability of services that facilitate transactions, although the costs of such services will continue to create a wedge between sellers’ and buyers’ prices. 16For the classic treatment of this topic, see Y. Barzel, ‘‘An Alternative Approach to the Analysis of Taxation,’’ Journal of Political Economy (December 1976): 1177–97. Chapter 12: The Partial Equilibrium Competitive Model 447 • If shifts in long-run equilibrium affect input prices, this will also affect the welfare of input suppliers. Such welfare changes can be measured by changes in long-run producer surplus. • The twin concepts of consumer and producer surplus provide useful ways of measuring the welfare impact on market participants of various economic changes. Changes in consumer surplus represent the monetary value of changes in consumer utility. Changes in producer in the monetary returns that inputs receive. represent changes surplus • The competitive model can be used to study the impact of various economic policies. For example, it can be used to illustrate the transfers and welfare losses associated with price controls. • The competitive model can also be applied to study taxation. The model illustrates both tax incidence (i.e., who bears the actual burden of a tax) and the welfare losses associated with taxation (the excess burden). Similar conclusions can be derived by using the competitive model to study transaction costs. SUMMARY In this chapter we developed a detailed model of how the equilibrium price is determined in a single competitive market. This model is basically the one ﬁrst fully articulated by Alfred Marshall in the latter part of the nineteenth century. It remains the single most important component of all of microeconomics. Some of the properties of this model we examined may be listed as follows. • Short-run equilibrium prices are determined by the interaction of what demanders are willing to pay (demand) and what existing ﬁrms are willing to produce (supply). Both demanders and suppliers act as price-takers in making their respective decisions. • In the long run, the number of ﬁrms may vary in response to proﬁt opportunities. free entry is assumed, then ﬁrms will earn zero economic proﬁts over the long run. Therefore, because ﬁrms also maximize proﬁts, the long-run equilibrium condition is P MC AC. ¼ If ¼ • The shape of the long-run supply curve depends on how the entry of new ﬁrms affects input prices. If entry has no impact on input prices, the long-run supply curve will be horizontal (inﬁnitely elastic). If entry increases input prices, the long-run supply curve will have a positive slope. PROBLEMS 12.1 Suppose there are 100 identical ﬁrms in a perfectly competitive industry. Each ﬁrm has a short-run total cost function of the form C q ð Þ ¼ 1 300 q3 þ 0:2q2 4q þ þ 10: a. Calculate the ﬁrm’s short-run supply curve with q as a function of market price (P). b. On the assumption that there are no interaction effects among costs of the ﬁrms in the industry, calculate the short-run industry supply curve. c. Suppose market demand is given by Q 200P ¼ & þ 8,000. What will be the short-run equilibrium price–quantity combination? 12.2 Suppose there are 1,000 identical ﬁrms producing diamonds. Let the total cost function for each ﬁrm be given by where q is the ﬁrm’s output level and w is the wage rate of diamond cutters. C(q, w) q2 ¼ þ wq, a. If w 10, what will be the ﬁrm’s (short-run) supply curve? What is the industry’s supply curve? How many diamonds will ¼ be produced at a price of 20 each? How many more diamonds would be produced at a price of 21? b. Suppose the wages of diamond cutters depend on the total quantity of diamonds produced, and suppose the form of this relationship is given by here Q represents total industry output, which is 1,000 times the output of the typical ﬁrm. 0.002Q ; w ¼ 448 Part 5: Competitive Markets In this situation, show that the ﬁrm’s marginal cost (and short-run supply) curve depends on Q. What is the industry supply curve? How much will be produced at a price of 20? How much more will be produced at a price of 21? What do you conclude about the shape of the short-run supply curve? 12.3 A perfectly competitive market has 1,000 ﬁrms. In the very short run, each of the ﬁrms has a ﬁxed supply of 100 units. The market demand is given by 160,000 Q ¼ & 10,000P. a. Calculate the equilibrium price in the very short run. b. Calculate the demand schedule facing any one ﬁrm in this industry. c. Calculate what the equilibrium price would be if one of the sellers decided to sell nothing or if one seller decided to sell 200 units. d. At the original equilibrium point, calculate the elasticity of the industry demand curve and the elasticity of the demand curve facing any one seller. Suppose now that, in the short run, each ﬁrm has a supply curve that shows the quantity the ﬁrm will supply (qi) as a function of market price. The speciﬁc form of this supply curve is given by Using this short-run supply response, supply revised answers to (a)–(d). qi ¼ & 200 þ 50P. 12.4 A perfectly competitive industry has a large number of potential entrants. Each ﬁrm has an identical cost structure such that long-run average cost is minimized at an output of 20 units (qi ¼ 20). The minimum average cost is $10 per unit. Total market demand is given by 1,500 Q ¼ & 50P. a. What is the industry’s long-run supply schedule? b. What is the long-run equilibrium price (P$)? The total industry output (Q$)? The output of each ﬁrm (q$)? The number of ﬁrms? The proﬁts of each ﬁrm? c. The short-run total cost function associated with each ﬁrm’s long-run equilibrium output is given by & Calculate the short-run average and marginal cost function. At what output level does short-run average cost reach a minimum? ¼ þ C(q) 0.5q2 10q 200. d. Calculate the short-run supply function for each ﬁrm and the industry short-run supply function. e. Suppose now that the market demand function shifts upward to Q 2,000 ¼ part (b) for the very short run when ﬁrms cannot change their outputs. & 50P. Using this new demand curve, answer f. In the short run, use the industry short-run supply function to recalculate the answers to (b). g. What is the new long-run equilibrium for the industry? 12.5 Suppose that the demand for stilts is given by 1,500 Q ¼ & 50P and that the long-run total operating costs of each stilt-making ﬁrm in a competitive industry are given by C(q) ¼ 0.5q2 10q. & Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is given by where w is the annual wage paid. QS ¼ 0.25w, Chapter 12: The Partial Equilibrium Competitive Model 449 Suppose also that each stilt-making ﬁrm requires one (and only one) entrepreneur (hence the quantity of entrepreneurs hired is equal to the number of ﬁrms). Long-run total costs for each ﬁrm are then given by C(q, w) 0.5q2 ¼ 10q w. þ & a. What is the long-run equilibrium quantity of stilts produced? How many stilts are produced by each ﬁrm? What is the long-run equilibrium price of stilts? How many ﬁrms will there be? How many entrepreneurs will be hired, and what is their wage? b. Suppose that the demand for stilts shifts outward to How would you now answer the questions posed in part (a)? c. Because stilt-making entrepreneurs are the cause of the upward-sloping long-run supply curve in this problem, they will receive all rents generated as industry output expands. Calculate the inc
rease in rents between parts (a) and (b). Show that this value is identical to the change in long-run producer surplus as measured along the stilt supply curve. 2,428 Q ¼ & 50P. 12.6 The handmade snuffbox industry is composed of 100 identical ﬁrms, each having short-run total costs given by and short-run marginal costs given by where q is the output of snuffboxes per day. STC ¼ 0.5q2 10q 5 þ þ SMC q þ ¼ 10, a. What is the short-run supply curve for each snuffbox maker? What is the short-run supply curve for the market as a whole? b. Suppose the demand for total snuffbox production is given by 1,100 Q ¼ & 50P. What will be the equilibrium in this marketplace? What will each ﬁrm’s total short-run proﬁts be? c. Graph the market equilibrium and compute total short-run producer surplus in this case. d. Show that the total producer surplus you calculated in part (c) is equal to total industry proﬁts plus industry short-run ﬁxed costs. e. Suppose the government imposed a $3 tax on snuffboxes. How would this tax change the market equilibrium? f. How would the burden of this tax be shared between snuffbox buyers and sellers? g. Calculate the total loss of producer surplus as a result of the taxation of snuffboxes. Show that this loss equals the change in total short-run proﬁts in the snuffbox industry. Why do ﬁxed costs not enter into this computation of the change in short-run producer surplus? 12.7 The perfectly competitive videotape-copying industry is composed of many ﬁrms that can copy ﬁve tapes per day at an average cost of $10 per tape. Each ﬁrm must also pay a royalty to ﬁlm studios, and the per-ﬁlm royalty rate (r) is an increasing function of total industry output (Q): Demand is given by 0.002Q. r ¼ 1,050 Q ¼ & 50P. a. Assuming the industry is in long-run equilibrium, what will be the equilibrium price and quantity of copied tapes? How many tape ﬁrms will there be? What will the per-ﬁlm royalty rate be? b. Suppose that demand for copied tapes increases to 1,600 Q ¼ & 50P. 450 Part 5: Competitive Markets In this case, what is the long-run equilibrium price and quantity for copied tapes? How many tape ﬁrms are there? What is the per-ﬁlm royalty rate? c. Graph these long-run equilibria in the tape market, and calculate the increase in producer surplus between the situations described in parts (a) and (b). d. Show that the increase in producer surplus is precisely equal to the increase in royalties paid as Q expands incrementally from its level in part (b) to its level in part (c). e. Suppose that the government institutes a $5.50 per-ﬁlm tax on the ﬁlm-copying industry. Assuming that the demand for copied ﬁlms is that given in part (a), how will this tax affect the market equilibrium? f. How will the burden of this tax be allocated between consumers and producers? What will be the loss of consumer and producer surplus? g. Show that the loss of producer surplus as a result of this tax is borne completely by the ﬁlm studios. Explain your result intuitively. 12.8 The domestic demand for portable radios is given by & where price (P) is measured in dollars and quantity (Q) is measured in thousands of radios per year. The domestic supply curve for radios is given by ¼ Q 5,000 100P, 150P. Q ¼ a. What is the domestic equilibrium in the portable radio market? b. Suppose portable radios can be imported at a world price of $10 per radio. If trade were unencumbered, what would the new market equilibrium be? How many portable radios would be imported? c. If domestic portable radio producers succeeded in having a $5 tariff implemented, how would this change the market equilibrium? How much would be collected in tariff revenues? How much consumer surplus would be transferred to domestic producers? What would the deadweight loss from the tariff be? d. How would your results from part (c) be changed if the government reached an agreement with foreign suppliers to ‘‘voluntarily’’ limit the portable radios they export to 1,250,000 per year? Explain how this differs from the case of a tariff. 12.9 Suppose that the market demand for a product is given by QD ¼ given by C(q) bq2. aq k ¼ þ þ BP. Suppose also that the typical ﬁrm’s cost function is A & a. Compute the long-run equilibrium output and price for the typical ﬁrm in this market. b. Calculate the equilibrium number of ﬁrms in this market as a function of all the parameters in this problem. c. Describe how changes in the demand parameters A and B affect the equilibrium number of ﬁrms in this market. Explain your results intuitively. d. Describe how the parameters of the typical ﬁrm’s cost function affect the long-run equilibrium number of ﬁrms in this example. Explain your results intuitively. Analytical Problems 12.10 Ad valorem taxes Throughout this chapter’s analysis of taxes we have used per-unit taxes—that is, a tax of a ﬁxed amount for each unit traded in the market. A similar analysis would hold for ad valorem taxes—that is, taxes on the value of the transaction (or, what amounts to the same thing, proportional taxes on price). Given an ad valorem tax rate of t (t 0.05 for a 5 percent tax), the gap between the price demanders pay and what suppliers receive is given by PD ¼ t)PS. (1 ¼ þ Chapter 12: The Partial Equilibrium Competitive Model 451 a. Show that for an ad valorem tax d ln PD dt ¼ eS eS & eD and d ln PS dt ¼ eD eS & : eD b. Show that the excess burden of a small tax is c. Compare these results with those derived in this chapter for a unit tax. DW 0:5 ¼ & eDeS eS & eD t2P0Q0: 12.11 The Ramsey formula for optimal taxation The development of optimal tax policy has been a major topic in public ﬁnance for centuries.17 Probably the most famous result in the theory of optimal taxation is due to the English economist Frank Ramsey, who conceptualized the problem as how to structure a tax system that would collect a given amount of revenues with the minimal deadweight loss.18 Speciﬁcally, suppose there are n goods (xi with prices pi) to be taxed with a sequence of ad valorem taxes (see Problem 12.10) whose rates n are given by ti (i 1 tipixi. Ramsey’s problem is for a ﬁxed T to choose i ¼ tax rates that will minimize total deadweight loss DW a. Use the Lagrange multiplier method to show that the solution to Ramsey’s problem requires ti ¼ b. Interpret the Ramsey result intuitively. c. Describe some shortcomings of the Ramsey approach to optimal taxation. 1, n). Therefore, total tax revenue is given by T 1 DW ¼ is the Lagrange multiplier for the tax constraint. 1/eD), where l l(1/eS & ¼ . tiÞ ð P P ¼ ¼ n i 12.12 Cobweb models One way to generate disequilibrium prices in a simple model of supply and demand is to incorporate a lag into producer’s supply response. To examine this possibility, assume that quantity demanded in period t depends on price in that period QD but that quantity supplied depends on the previous period’s price—perhaps because farmers refer to that price bPtÞ a ð & in planting a crop t ¼ c QS ð t ¼ dPt . 1Þ & þ a. What is the equilibrium price in this model (P$ ¼ b. If P0 represents an initial price for this good to which suppliers respond, what will the value of P1 be? c. By repeated substitution, develop a formula for any arbitrary Pt as a function of P0 and t. d. Use your results from part (a) to restate the value of Pt as a function of P0, P$, and t. e. Under what conditions will Pt converge to P$ as t ﬁ f. Graph your results for the case a Pt–1) for all periods, t. Pt ¼ 1 1, d 4, b 2, c ? ¼ ¼ ¼ 1, and P0 ¼ ¼ cobweb model. 0. Use your graph to discuss the origin of the term SUGGESTIONS FOR FURTHER READING Arnott, R. ‘‘Time for Revision on Rent Control?’’ Journal of Economic Perspectives (Winter 1995): 99–120. Provides an assessment of actual ‘‘soft’’ rent-control policies and provides a rationale for them. deMelo, J., and D. G. Tarr. ‘‘The Welfare Costs of U.S. Quotas in Textiles, Steel, and Autos.’’ Review of Economics and Statistics (August 1990): 489–97. A nice study of the quota question in a general equilibrium context. Finds that the quotas studied have the same quantitative effects as a tariff rate of about 20 percent. Knight, F. H. Risk, Uncertainty and Proﬁt. Boston: Houghton Mifﬂin, 1921, chaps. 5 and 6. Classic treatment of the role of economic events in motivating industry behavior in the long run. 17The seventeenth-century French ﬁnance minister Jean-Baptiste Colbert captured the essence of the problem with his memorable statement that ‘‘the art of taxation consists in so plucking the goose as to obtain the largest possible amount of feathers with the smallest amount of hissing.’’ 18See F. Ramsey, ‘‘A Contribution to the Theory of Taxation,’’ Economic Journal (March 1927): 47–61. 452 Part 5: Competitive Markets Marshall, A. Principles of Economics, 8th ed. New York: Crowell-Collier and Macmillan, 1920, book 5, chaps. 1, 2, and 3. Salanie, B. The Economics of Taxation. Cambridge, MA: MIT Press, 2003. Classic development of the supply–demand mechanism. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995, chap. 10. Provides a compact analysis at a high level of theoretical precision. There is a good discussion of situations where competitive markets may not reach an equilibrium. Reynolds, L. G. ‘‘Cut-Throat Competition.’’ American Economic Review 30 (December 1940): 736–47. Critique of the notion that there can be ‘‘too much’’ competition in an industry. Robinson, J. ‘‘What Is Perfect Competition?’’ Quarterly Journal of Economics 49 (1934): 104–20. Critical discussion of the perfectly competitive assumptions. This provides a compact study of many issues in taxation. Describes a few simple models of incidence and develops some general equilibrium models of taxation. Stigler, G. J. plated.’’ Journal of Political Economy 65 (1957): 1–17. ‘‘Perfect Competition, Historically Contem- Fascinating discussion of the historical development of the competitive model. Varian, H. R. Microeconomic
Analysis, 3rd ed. New York: W. W. Norton, 1992, chap. 13. Terse but instructive coverage of many of the topics in this chapter. The importance of entry is stressed, although the precise nature of the long-run supply curve is a bit obscure. DEMAND AGGREGATION AND ESTIMATION EXTENSIONS In Chapters 4–6 we showed that the assumption of utility maximization implies individual demand functions: several properties for • • • • the functions are continuous; the functions are homogeneous of degree 0 in all prices and income; income-compensated substitution effects are negative; and cross-price substitution effects are symmetric. In this extension we will examine the extent to which these properties would be expected to hold for aggregated market demand functions and what, if any, restrictions should be placed on such functions. In addition, we illustrate some other issues that arise in estimating these aggregate functions and some results from such estimates. E12.1 Continuity The continuity of individual demand functions clearly implies the continuity of market demand functions. But there are situations in which market demand functions may be continuous, whereas individual functions are not. Consider the case where goods—such as an automobile—must be bought in large, discrete units. Here individual demand is discontinuous, but the aggregated demands of many people are (nearly) continuous. E12.2 Homogeneity and income aggregation Because each individual’s demand function is homogeneous of degree 0 in all prices and income, market demand functions are also homogeneous of degree 0 in all prices and individual incomes. However, market demand functions are not necessarily homogeneous of degree 0 in all prices and total income. To see when demand might depend just on total income, suppose individual i’s demand for X is given by xi ¼ ai(P Þ þ b(P)yi, 1, n, i ¼ (i) where P is the vector of all market prices, ai(P) is a set of individual-speciﬁc price effects, and b(P) is a marginal propensity-to-spend function that is the same across all individuals (although the value of this parameter may depend on market prices). In this case the market demand functions will depend on P and on total income: n yi: (ii) g ¼ 1 i X ¼ This shows that market demand reﬂects the behavior of a single ‘‘typical’’ consumer. Gorman (1959) shows that this is the most general form of demand function that can represent such a typical consumer. E12.3 Cross-equation constraints Suppose a typical individual buys k items and that expenditures on each are given by pjxj ¼ k 1 i X ¼ aijpi þ bjy, j ¼ 1, k: (iii) If expenditures on these k items exhaust total income, that is, k 1 j X ¼ pjxj ¼ y, then summing over all goods shows that and that k 1 j X ¼ aij ¼ 0 for all i k 1 j X ¼ 1 bj ¼ (iv) (v) (vi) for each person. This implies that researchers are generally not able to estimate expenditure functions for k goods independently. Rather, some account must be taken of relationships between the expenditure functions for different goods. E12.4 Econometric practice The degree to which these theoretical concerns are reﬂected in the actual practices of econometricians varies widely. At the least sophisticated level, an equation similar to Equation iii might be estimated directly using ordinary least squares (OLS) with little attention to the ways in which the assumptions might be violated. Various elasticities could be calculated 454 Part 5: Competitive Markets TABLE 12.3 REPRESENTATIVE PRICE AND INCOME ELASTICITIES OF DEMAND Price Elasticity Income Elasticity Food Medical services Housing Rental Owner occupied Electricity Automobiles Gasoline Beer Wine Marijuana Cigarettes Abortions Transatlantic air travel Imports Money 0.21 & 0.18 & 0.18 1.20 & & 1.14 & 1.20 & 0.55 & 0.26 & 0.88 & 1.50 & 0.35 & 0.81 & 1.30 & 0.58 & 0.40 & 0.28 þ 0.22 þ 1.00 þ 1.20 þ 0.61 þ 3.00 þ 1.60 þ 0.38 þ 0.97 þ 0.00 0.50 þ 0.79 þ 1.40 þ 2.73 þ 1.00 þ Note: Price elasticity refers to interest rate elasticity. SOURCES: Food: H. Wold and L. Jureen, Demand Analysis (New York: John Wiley & Sons, 1953): 203. Medical services: income elasticity from R. Andersen and L. Benham, ‘‘Factors Affecting the Relationship between Family Income and Medical Care Consumption,’’ in Herbert Klarman, Ed., Empirical Studies in Health Economics (Baltimore: Johns Hopkins University Press, 1970); price elasticity from W. C. Manning et al., ‘‘Health Insurance and the Demand for Medical Care: Evidence from a Randomized Experiment,’’ American Economic Review (June 1987): 251–77. Housing: income elasticities from F. de Leeuw, ‘‘The Demand for Housing,’’ Review for Economics and Statistics (February1971); price elasticities from H. S. Houthakker and L. D. Taylor, Consumer Demand in the United States (Cambridge, MA: Harvard University Press, 1970): 166–67. Electricity: R. F. Halvorsen, ‘‘Residential Demand for Electricity,’’ unpublished Ph.D. dissertation, Harvard University, December 1972. Automobiles: Gregory C. Chow, Demand for Automobiles in the United States (Amsterdam: North Holland, 1957). Gasoline: C. Dahl, ‘‘Gasoline Demand Survey,’’ Energy Journal 7 (1986): 67–82. Beer and wine: J. A. Johnson, E. H. Oksanen, M. R. Veall, and D. Fritz, ‘‘Short-Run and LongRun Elasticities for Canadian Consumption of Alcoholic Beverages,’’ Review of Economics and Statistics (February 1992): 64–74. Marijuana: T. C. Misket and F. Vakil, ‘‘Some Estimate of Price and Expenditure Elasticities among UCLA Students,’’ Review of Economics and Statistics (November 1972): 474–75. Cigarettes: F. Chalemaker, ‘‘Rational Addictive Behavior and Cigarette Smoking,’’ Journal of Political Economy (August 1991): 722–42. Abortions: M. H. Medoff, ‘‘An Economic Analysis of the Demand for Abortions,’’ Economic Inquiry (April 1988): 253–59. Transatlantic air travel: J. M. Cigliano, ‘‘Price and Income Elasticities for Airline Travel,’’ Business Economics (September 1980): 17–21. Imports: M. D. Chinn, ‘‘Beware of Econometricians Bearing Estimates,’’ Journal of Policy Analysis and Management (Fall 1991): 546–67. Money: D. L. Hoffman and R. H. Rasche, ‘‘Long-Run Income and Interest Elasticities of Money Demand in the United States,’’ Review of Economics and Statistics (November 1991): 665–74. directly from this equation—although, because of the linear form used, these would not be constant for changes in pi or y. A constant elasticity formulation of Equation iii would be ln pjxjÞ ¼ ð k 1 i X ¼ aij ln piÞ þ ð bj ln y, j 1, k, ¼ (vii) where price and income elasticities would be given directly by 1, exj,pj ¼ exj,pi ¼ exj,y ¼ aj,j & ai,j ð i bj’ , j Þ 6¼ (viii) Notice here, however, that no speciﬁc attention is paid to biases introduced by the use of aggregate income or by the disregard of possible cross-equation restrictions such as those in Equations v and vi. Further restrictions are also implied the demand functions by the homogeneity of each of , although this restriction too is often 1 1 aij þ ð Þ disregarded in the development of simple econometric P estimates. bj ¼ & k i ¼ More sophisticated studies of aggregated demand equations seek to remedy these problems by explicitly considering potential income distribution effects and by estimating entire systems of demand equations. Theil (1971, 1975) provides a good introduction to some of the procedures used. Econometric results Table 12.3 reports a number of economic estimates of representative price and income elasticities drawn from a variety of Chapter 12: The Partial Equilibrium Competitive Model 455 sources. The original sources for these estimates should be consulted to determine the extent to which the authors have been attentive to the theoretical restrictions outlined prethese estimates accord fairly well with viously. Overall, intuition—the demand for transatlantic air travel is more price elastic than is the demand for medical care, for example. Perhaps somewhat surprising are the high price and income elasticities for owner-occupied housing because ‘‘shelter’’ is often regarded in everyday discussion as a necessity. The high estimated income elasticity of demand for automobiles probably conﬂates the measurement of both quantity and quality demanded. But it does suggest why the automobile industry is so sensitive to the business cycle. References Gorman, W. M. ‘‘Separable Utility and Aggregation.’’ Econo- metrica (November 1959): 469–81. Theil, H. Principles of Econometrics. New York: John Wiley & Sons, 1971, pp. 326–46. ———. Theory and Measurement of Consumer Demand, vol. 1. Amsterdam: North Holland, 1975, chaps. 5 and 6. This page intentionally left blank CHAPTER THIRTEEN General Equilibrium and Welfare The partial equilibrium models of perfect competition that were introduced in Chapter 12 are clearly inadequate for describing all the effects that occur when changes in one market have repercussions in other markets. Therefore, they are also inadequate for making general welfare statements about how well market economies perform. Instead, what is needed is an economic model that permits us to view many markets simultaneously. In this chapter we will develop a few simple versions of such models. The Extensions to the chapter show how general equilibrium models are applied to the real world. Perfectly Competitive Price System The model we will develop in this chapter is primarily an elaboration of the supply– demand mechanism presented in Chapter 12. Here we will assume that all markets are of the type described in that chapter and refer to such a set of markets as a perfectly competitive price system. The assumption is that there is some large number of homogeneous goods in this simple economy. Included in this list of goods are not only consumption items but also factors of production. Each of these goods has an equilibrium price, established by the action of supply and demand.1 At this set of prices, every market is cleared in the sense that suppliers are willing to supply the quantity that is demand
ed and consumers will demand the quantity that is supplied. We also assume that there are no transaction or transportation charges and that both individuals and ﬁrms have perfect knowledge of prevailing market prices. The law of one price Because we assume zero transaction cost and perfect information, each good obeys the law of one price: A homogeneous good trades at the same price no matter who buys it or which ﬁrm sells it. If one good traded at two different prices, demanders would rush to buy the good where it was cheaper, and ﬁrms would try to sell all their output where the good was more expensive. These actions in themselves would tend to equalize the price of the good. In the perfectly competitive market, each good must have only one price. This is why we may speak unambiguously of the price of a good. 1One aspect of this market interaction should be made clear from the outset. The perfectly competitive market determines only relative (not absolute) prices. In this chapter, we speak only of relative prices. It makes no difference whether the prices of apples and oranges are $.10 and $.20, respectively, or $10 and $20. The important point in either case is that two apples can be exchanged for one orange in the market. The absolute level of prices is determined mainly by monetary factors—a topic usually covered in macroeconomics. 457 458 Part 5: Competitive Markets Behavioral assumptions The perfectly competitive model assumes that people and ﬁrms react to prices in speciﬁc ways. 1. There are assumed to be a large number of people buying any one good. Each person takes all prices as given and adjusts his or her behavior to maximize utility, given the prices and his or her budget constraint. People may also be suppliers of productive services (e.g., labor), and in such decisions they also regard prices as given.2 2. There are assumed to be a large number of ﬁrms producing each good, and each ﬁrm produces only a small share of the output of any one good. In making input and output choices, ﬁrms are assumed to operate to maximize proﬁts. The ﬁrms treat all prices as given when making these proﬁt-maximizing decisions. These various assumptions should be familiar because we have been making them throughout this book. Our purpose here is to show how an entire economic system operates when all markets work in this way. A Graphical Model of General Equilibrium with Two Goods We begin our analysis with a graphical model of general equilibrium involving only two goods, which we will call x and y. This model will prove useful because it incorporates many of the features of far more complex general equilibrium representations of the economy. General equilibrium demand Ultimately, demand patterns in an economy are determined by individuals’ preferences. For our simple model we will assume that all individuals have identical preferences, which can be represented by an indifference curve map3 deﬁned over quantities of the two goods, x and y. The beneﬁt of this approach for our purposes is that this indifference curve map (which is identical to the ones used in Chapters 3–6) shows how individuals rank consumption bundles containing both goods. These rankings are precisely what we mean by ‘‘demand’’ in a general equilibrium context. Of course, we cannot illustrate which bundles of commodities will be chosen until we know the budget constraints that demanders face. Because incomes are generated as individuals supply labor, capital, and other resources to the production process, we must delay any detailed illustration until we have examined the forces of production and supply in our model. General equilibrium supply Developing a notion of general equilibrium supply in this two-good model is a somewhat more complex process than describing the demand side of the market because we have two goods simultaneously. Our not thus far illustrated production and supply of 2Hence, unlike our partial equilibrium models, incomes are endogenously determined in general equilibrium models. 3There are some technical problems in using a single indifference curve map to represent the preferences of an entire community of individuals. In this case the marginal rate of substitution (i.e., the slope of the community indifference curve) will depend on how the available goods are distributed among individuals: The increase in total y required to compensate for a one-unit reduction in x will depend on which speciﬁc individual(s) the x is taken from. Although we will not discuss this issue in detail here, it has been widely examined in the international trade literature. Chapter 13: General Equilibrium and Welfare 459 approach is to use the familiar production possibility curve (see Chapter 1) for this purpose. By detailing the way in which this curve is constructed, we can illustrate, in a simple context, the ways in which markets for outputs and inputs are related. Edgeworth box diagram for production Construction of the production possibility curve for two outputs (x and y) begins with the assumption that there are ﬁxed amounts of capital and labor inputs that must be allocated to the production of the two goods. The possible allocations of these inputs can be illustrated with an Edgeworth box diagram with dimensions given by the total amounts of capital and labor available. In Figure 13.1, the length of the box represents total labor-hours, and the height of the box represents total capital-hours. The lower left corner of the box represents the ‘‘origin’’ for measuring capital and labor devoted to production of good x. The upper right corner of the box represents the origin for resources devoted to y. Using these conventions, any point in the box can be regarded as a fully employed allocation of the available resources between goods x and y. Point A, for example, represents an allocation in which the indicated number of labor hours are devoted to x production together with a speciﬁed number of hours of capital. Production of good y uses whatever labor and capital are ‘‘left over.’’ Point A in Figure 13.1, for example, also shows the exact amount of labor and capital used in the production of good y. Any other point in the box has a similar interpretation. Thus, the Edgeworth box shows every possible way the existing capital and labor might be used to produce x and y. FIGURE 13.1 Construction of an Edgeworth Box Diagram for Production The dimensions of this diagram are given by the total quantities of labor and capital available. Quantities of these resources devoted to x production are measured from origin Ox; quantities devoted to y are measured from Oy. Any point in the box represents a fully employed allocation of the available resources to the two goods. Labor for x Labor in y production Labor for y O y Capital in y production Capital in x production O x Labor in x production A Total labor 460 Part 5: Competitive Markets Efﬁcient allocations Many of the allocations shown in Figure 13.1 are technically inefﬁcient in that it is possible to produce both more x and more y by shifting capital and labor around a bit. In our model we assume that competitive markets will not exhibit such inefﬁcient input choices (for reasons we will explore in more detail later in the chapter). Hence we wish to discover the efﬁcient allocations in Figure 13.1 because these illustrate the production outcomes in this model. To do so, we introduce isoquant maps for good x (using Ox as the origin) and good y (using Oy as the origin), as shown in Figure 13.2. In this ﬁgure it is clear that the arbitrarily chosen allocation A is inefﬁcient. By reallocating capital and labor, one can produce both more x than x2 and more y than y2. The efﬁcient allocations in Figure 13.2 are those such as P1, P2, P3, and P4, where the isoquants are tangent to one another. At any other points in the box diagram, the two goods’ isoquants will intersect, and we can show inefﬁciency as we did for point A. At the points of tangency, however, this kind of unambiguous improvement cannot be made. In going from P2 to P3, for example, more x is being produced, but at the cost of less y being produced; therefore, P3 is not ‘‘more efﬁcient’’ than P2—both of the points are efﬁcient. Tangency of the isoquants for good x and good y implies that their slopes are equal. That is, the RTS of capital for labor is equal in x and y production. Later we will show how competitive input markets will lead ﬁrms to make such efﬁcient input choices. Therefore, the curve joining Ox and Oy that includes all these points of tangency shows all the efﬁcient allocations of capital and labor. Points off this curve are inefﬁcient in that unambiguous increases in output can be obtained by reshufﬂing inputs between the two goods. Points on the curve OxOy are all efﬁcient allocations, however, because more x can be produced only by cutting back on y production and vice versa. FIGURE 13.2 Edgeworth Box Diagram of Efﬁciency in Production This diagram adds production isoquants for x and y to Figure 13.1. It then shows technically efﬁcient ways to allocate the ﬁxed amounts of k and l between the production of the two outputs. The line joining Ox and Oy is the locus of these efﬁcient points. Along this line, the RTS (of l for k) in the production of good x is equal to the RTS in the production of y. O y y1 P3 y2 P2 x1 Total l Total k y4 y3 P1 O x P4 x3 A x4 x2 Chapter 13: General Equilibrium and Welfare 461 FIGURE 13.3 Production Possibility Frontier The production possibility frontier shows the alternative combinations of x and y that can be efﬁciently produced by a ﬁrm with ﬁxed resources. The curve can be derived from Figure 13.2 by varying inputs between the production of x and y while maintaining the conditions for efﬁciency. The negative of the slope of the production possibility curve is called the rate of product transformation (RPT). Quantity of y O x y4 y3 y2 y1 P1 P2 A P3 P4 x1 x2 x3 x4 O y Quantity of x Production possib
ility frontier The efﬁciency locus in Figure 13.2 shows the maximum output of y that can be produced for any preassigned output of x. We can use this information to construct a production possibility frontier, which shows the alternative outputs of x and y that can be produced with the ﬁxed capital and labor inputs. In Figure 13.3 the OxOy locus has been taken from Figure 13.2 and transferred onto a graph with x and y outputs on the axes. At Ox, for example, no resources are devoted to x production; consequently, y output is as large as is possible with the existing resources. Similarly, at Oy, the output of x is as large as possible. The other points on the production possibility frontier (say, P1, P2, P3, and P4) are derived from the efﬁciency locus in an identical way. Hence we have derived the following deﬁnition Production possibility frontier. The production possibility frontier shows the alternative combinations of two outputs that can be produced with ﬁxed quantities of inputs if those inputs are employed efﬁciently. Rate of product transformation The slope of the production possibility frontier shows how x output can be substituted for y output when total resources are held constant. For example, for points near Ox on 462 Part 5: Competitive Markets 1/4; this the production possibility frontier, the slope is a small negative number—say, implies that, by reducing y output by 1 unit, x output could be increased by 4. Near Oy, on the other hand, the slope is a large negative number (say, 5), implying that y output must be reduced by 5 units to permit the production of one more x. The slope of the production possibility frontier clearly shows the possibilities that exist for trading y for x in production. The negative of this slope is called the rate of product transformation (RPT). ! ! Rate of product transformation. The rate of product transformation (RPT) between two outputs is the negative of the slope of the production possibility frontier for those outputs. Mathematically, RPT of x for y ð Þ ¼ !½ slope of production possibility frontier dy dx ð along OxOyÞ , ¼ ! & (13:1) The RPT records how x can be technically traded for y while continuing to keep the available productive inputs efﬁciently employed. Shape of the production possibility frontier The production possibility frontier illustrated in Figure 13.3 exhibits an increasing RPT. For output levels near Ox, relatively little y must be sacriﬁced to obtain one more x (–dy/dx is small). Near Oy, on the other hand, additional x may be obtained only by substantial reductions in y output (–dy/dx is large). In this section we will show why this concave shape might be expected to characterize most production situations. A ﬁrst step in that analysis is to recognize that RPT is equal to the ratio of the marginal cost of x (MCx) to the marginal cost of y (MCy). Intuitively, this result is obvious. Suppose, for example, that x and y are produced only with labor. If it takes two labor hours to produce one more x, we might say that MCx is equal to 2. Similarly, if it takes only one labor hour to produce an extra y, then MCy is equal to 1. But in this situation it is clear that the RPT is 2: two y must be forgone to provide enough labor so that x may be increased by one unit. Hence the RPT is equal to the ratio of the marginal costs of the two goods. More formally, suppose that the costs (say, in terms of the ‘‘disutility’’ experienced by factor suppliers) of any output combination are denoted by C(x, y). Along the production possibility frontier, C(x, y) will be constant because the inputs are in ﬁxed supply. If we call this constant level of costs C, we can write C 0. It is this implicit function that underlies the production possibility frontier. Applying the results from Chapter 2 for such a function yields: x, y Þ ! ¼ C ð RPT dy dx jC ¼ x Cx Cy ¼ ! MCx MCy : (13:2) To demonstrate reasons why the RPT might be expected to increase for clockwise movements along the production possibility frontier, we can proceed by showing why the ratio of MCx to MCy should increase as x output expands and y output contracts. We ﬁrst present two relatively simple arguments that apply only to special cases; then we turn to a more sophisticated general argument. Diminishing returns The most common rationale offered for the concave shape of the production possibility frontier is the assumption that both goods are produced under conditions of diminishing returns. Hence increasing the output of good x will raise its marginal cost, whereas Chapter 13: General Equilibrium and Welfare 463 decreasing the output of y will reduce its marginal cost. Equation 13.2 then shows that the RPT will increase for movements along the production possibility frontier from Ox to Oy. A problem with this explanation, of course, is that it applies only to cases in which both goods exhibit diminishing returns to scale, and that assumption is at variance with the theoretical reasons for preferring the assumption of constant or even increasing returns to scale as mentioned elsewhere in this book. Specialized inputs If some inputs were ‘‘more suited’’ for x production than for y production (and vice versa), the concave shape of the production frontier also could be explained. In that case, increases in x output would require drawing progressively less suitable inputs into the production of that good. Therefore, marginal costs of x would increase. Marginal costs for y, on the other hand, would decrease because smaller output levels for y would permit the use of only those inputs most suited for y production. Such an argument might apply, for example, to a farmer with a variety of types of land under cultivation in different crops. In trying to increase the production of any one crop, the farmer would be forced to grow it on increasingly unsuitable parcels of land. Although this type of specialized input assumption has considerable importance in explaining a variety of real-world phenomena, it is nonetheless at variance with our general assumption of homogeneous factors of production. Hence it cannot serve as a fundamental explanation for concavity. Differing factor intensities Even if inputs are homogeneous and production functions exhibit constant returns to scale, the production possibility frontier will be concave if goods x and y use inputs in different proportions.4 In the production box diagram of Figure 13.2, for example, good x is capital intensive relative to good y. That is, at every point along the OxOy contract curve, the ratio of k to l in x production exceeds the ratio of k to l in y production: The bowed curve OxOy is always above the main diagonal of the Edgeworth box. If, on the other hand, good y had been relatively capital intensive, the OxOy contract curve would have been bowed downward below the diagonal. Although a formal proof that unequal factor intensities result in a concave production possibility frontier will not be presented here, it is possible to suggest intuitively why that occurs. Consider any two points on the frontier OxOy in Figure 13.3—say, P1 (with coordinates x1, y4) and P3 (with coordinates x3, y2). One way of producing an output combination ‘‘between’’ P1 and P3 would be to produce the combination x1 þ 2 x3 , y4 þ 2 y2 : Because of the constant returns-to-scale assumption, that combination would be feasible and would fully use both factors of production. The combination would lie at the midpoint of a straight-line chord joining points P1 and P3. Although such a point is feasible, it is not efﬁcient, as can be seen by examining points P1 and P3 in the box diagram of Figure 13.2. Because of the bowed nature of the contract curve, production at a point midway between P1 and P3 would be off the contract curve: Producing at a point such as P2 would provide more of both goods. Therefore, the production possibility frontier in Figure 13.3 must ‘‘bulge out’’ beyond the straight line P1P3. Because such a proof could be constructed for any two points on OxOy, we have shown that the frontier is concave; that is, the RPT increases as the output of good X increases. When production is reallocated in a northeast 4If, in addition to homogeneous factors and constant returns to scale, each good also used k and l in the same proportions under optimal allocations, then the production possibility frontier would be a straight line. 464 Part 5: Competitive Markets direction along the OxOy contract curve (in Figure 13.3), the capital–labor ratio decreases in the production of both x and y. Because good x is capital intensive, this change increases MCx. On the other hand, because good y is labor intensive, MCy decreases. Hence the relative marginal cost of x (as represented by the RPT) increases. Opportunity cost and supply The production possibility curve demonstrates that there are many possible efﬁcient combinations of the two goods and that producing more of one good necessitates cutting back on the production of some other good. This is precisely what economists mean by the term opportunity cost. The cost of producing more x can be most readily measured by the reduction in y output that this entails. Therefore, the cost of one more unit of x is best measured as the RPT (of x for y) at the prevailing point on the production possibility frontier. The fact that this cost increases as more x is produced represents the formulation of supply in a general equilibrium context. EXAMPLE 13.1 Concavity of the Production Possibility Frontier In this example we look at two characteristics of production functions that may cause the production possibility frontier to be concave. Diminishing returns. Suppose that the production of both x and y depends only on labor input and that the production functions for these goods are lxÞ ¼ ð lyÞ ¼ f ð Hence production of each of these goods exhibits diminishing returns to scale. If total labor supply is limited by x ¼ y ¼ (13:3) f l 0:5 x , l 0:5 y : then simpl
e substitution shows that the production possibility frontier is given by lx þ ly ¼ 100, x2 y2 þ ¼ 100 for x, y 0: ( (13:4) (13:5) In this case, the frontier is a quarter-circle and is concave. The RPT can now be computed directly from the equation for the production possibility frontier (written in implicit form as f x, y 100 0): x2 y2 ð Þ ¼ þ ! ¼ RPT dy dx ¼ !ð! fx fyÞ ¼ 2x 2y ¼ x y , ¼ ! (13:6) and this slope increases as x output increases. A numerical illustration of concavity starts by noting that the points (10, 0) and (0, 10) both lie on the frontier. A straight line joining these two points would also include the point (5, 5), but that point lies below the frontier. If equal 50p , which yields amounts of labor are devoted to both goods, then production is x more of both goods than the midpoint. ¼ ¼ y ﬃﬃﬃﬃﬃ Factor intensity. To show how differing factor intensities yield a concave production possibility frontier, suppose that the two goods are produced under constant returns to scale but with different Cobb–Douglas production functions: x y f g ¼ ¼ k, l ð k, l ð Þ ¼ Þ ¼ k0:5 x l 0:5 x , k0:25 l 0:75 y y : (13:7) Chapter 13: General Equilibrium and Welfare 465 Suppose also that total capital and labor are constrained by It is easy to show that kx þ ky ¼ 100, lx þ ly ¼ 100: RTSx ¼ kx lx ¼ jx, RTSy ¼ 3ky ly ¼ 3jy, (13:8) (13:9) RTSy or ki/li. Being located on the production possibility frontier requires RTSx ¼ where ki ¼ 3ky. That is, no matter how total resources are allocated to production, being on the kx ¼ production possibility frontier requires that x be the capital-intensive good (because, in some sense, capital is more productive in x production than in y production). The capital–labor ratios in the production of the two goods are also constrained by the available resources: ky ly ¼ kx lx þ kx þ lx þ ly)—that is, a is the share of total labor devoted to x production. Using the 3ky, we can ﬁnd the input ratios of the two goods in terms of the overall 100 100 ¼ ajx þ ð ky lx þ jy ¼ Þ (13:10) ly ¼ ly þ ! 1, a 1 lx/(lx þ where a ¼ condition that kx ¼ allocation of labor: jy ¼ 1 , 2a jx ¼ 1 : 2a (13:11) 1 þ 3 þ Now we are in a position to phrase the production possibility frontier in terms of the share of labor devoted to x production: j0:5 x lx ¼ ¼ j0:5 x a 100 ð Þ ¼ 100a " 3 1 2a # þ 0:5 , j0:25 y ly ¼ ¼ j0:25 y a 1 ð ! Þð 100 Þ ¼ 1 100 ð ! a Þ x y (13:12) 0:25 : 1 1 " þ 2a # We could push this algebra even further to eliminate a from these two equations to get an explicit functional form for the production possibility frontier that involves only x and y, but we 0 (x production gets no can show concavity with what we already have. First, notice that if a labor or capital inputs), then x 0. Hence a 0.39, say, then linear production possibility frontier would include the point (50, 50). But if a ¼ 1, we have x 100. With a 100, y 0, y ¼ ¼ ¼ ¼ ¼ ¼ 0:5 3 x y ¼ 100a 1 " 2a # þ 1 100 ð ! a ¼ 1 Þ " 3 1:78 0:5 50:6, ¼ # 0:25 39 " 0:25 61 ¼ 1 1:78 ¼ ¼ 1 2a (13:13) 52:8, # which shows that the actual frontier is bowed outward beyond a linear frontier. It is worth repeating that both of the goods in this example are produced under constant returns to scale and that the two inputs are fully homogeneous. It is only the differing input intensities involved in the production of the two goods that yields the concave production possibility frontier. þ " # QUERY: How would an increase in the total amount of labor available shift the production possibility frontiers in these examples? Determination of equilibrium prices Given these notions of demand and supply in our simple two-good economy, we can the now illustrate how equilibrium prices are determined. Figure 13.4 shows PP, 466 Part 5: Competitive Markets FIGURE 13.4 Determination of Equilibrium Prices With a price ratio given by px/py, ﬁrms will produce x1, y1; society’s budget constraint will be given by line C. With this budget constraint, individuals demand x01 and y01; that is, there is an excess demand for good x and an excess supply of good y. The workings of the market will move these prices toward their equilibrium levels p)x, p)y . At those prices, society’s budget constraint will be given by line C), and supply and demand will be in equilibrium. The combination x), y) of goods will be chosen. Quantity of y C P C* y1 y* y 1′ −px * Slope = ____ * py E −px Slope = ____ py C U3 U2 x1 x* P x1′ C* U1 Quantity of x production possibility frontier for the economy, and the set of indifference curves represents individuals’ preferences for these goods. First, consider the price ratio px/py. At this price ratio, ﬁrms will choose to produce the output combination x1, y1. Proﬁtmaximizing ﬁrms will choose the more proﬁtable point on PP. At x1, y1 the ratio of the two goods’ prices (px/py) is equal to the ratio of the goods’ marginal costs (the RPT); thus, proﬁts are maximized there. On the other hand, given this budget constraint (line C),5 individuals will demand x01, y01. Consequently, with these prices, there is an excess demand for good x (individuals demand more than is being produced) but an excess supply of good y. The workings of the marketplace will cause px to increase and py to decrease. The price ratio px/py will increase; the price line will take on a steeper slope. Firms will respond to these price changes by moving clockwise along the production possibility frontier; that is, they will increase their production of good x and decrease their production of good y. Similarly, individuals will respond to the changing prices by substituting y for x in their consumption choices. These actions of both ﬁrms and individuals serve to eliminate the excess demand for x and the excess supply of y as market prices change. 5It is important to recognize why the budget constraint has this location. Because px and py are given, the value of total producpy Æ y1. This is the value of ‘‘GDP’’ in the simple economy pictured in Figure 13.4. It is also, therefore, the total tion is px Æ x1 þ income accruing to people in society. Society’s budget constraint therefore passes through x1, y1 and has a slope of –px/py. This is precisely the budget constraint labeled C in the ﬁgure. Chapter 13: General Equilibrium and Welfare 467 Equilibrium is reached at x), y) with a price ratio of p)x=p)y. With this price ratio,6 supply and demand are equilibrated for both good x and good y. Given px and py, ﬁrms will produce x) and y) in maximizing their proﬁts. Similarly, with a budget constraint given by C ), individuals will demand x) and y). The operation of the price system has cleared the markets for both x and y simultaneously. Therefore, this ﬁgure provides a ‘‘general equilibrium’’ view of the supply–demand process for two markets working together. For this reason we will make considerable use of this ﬁgure in our subsequent analysis. Comparative Statics Analysis As in our partial equilibrium analysis, the equilibrium price ratio p)x=p)y illustrated in Figure 13.4 will tend to persist until either preferences or production technologies change. This competitively determined price ratio reﬂects these two basic economic forces. If preferences were to shift, say, toward good x, then px/py would increase and a new equilibrium would be established by a clockwise move along the production possibility curve. More x and less y would be produced to meet these changed preferences. Similarly, technical progress in the production of good x would shift the production possibility curve outward, as illustrated in Figure 13.5. This would tend to decrease the relative price of x and increase the quantity of x consumed (assuming x is a normal good). In the ﬁgure the quantity of y FIGURE 13.5 Effects of Technical Progress in x Production Technical advances that lower marginal costs of x production will shift the production possibility frontier. This will generally create income and substitution effects that cause the quantity of x produced to increase (assuming x is a normal good). Effects on the production of y are ambiguous because income and substitution effects work in opposite directions. Quantity of y y1 y0 E1 E0 U1 U0 x0 x1 Quantity of x 6Notice again that competitive markets determine only equilibrium relative prices. Determination of the absolute price level requires the introduction of money into this barter model. 468 Part 5: Competitive Markets consumed also increases as a result of the income effect arising from the technical advance; however, a slightly different drawing of the ﬁgure could have reversed that result if the substitution effect had been dominant. Example 13.2 looks at a few such effects. EXAMPLE 13.2 Comparative Statics in a General Equilibrium Model To explore how general equilibrium models work, let’s start with a simple example based on the production possibility frontier in Example 13.1. In that example we assumed that production of both goods was characterized by decreasing returns x and also that total labor available was given by lx þ 100. The resulting production possibility frontier was given by ly ¼ x2 x/y. To complete this model we assume that the typical individual’s 100, and RPT ¼ utility function is given by U(x, y) x0.5y0.5, so the demand functions for the two goods are and y l0:5 x l0:5 y y2 ¼ ¼ þ ¼ ¼ x y x px, py, I ð ¼ Þ ¼ y px, py, I ð ¼ Þ ¼ 0:5I px 0:5I py , : (13:14) Base-case equilibrium. Proﬁt maximization by ﬁrms requires that px/py ¼ x/y, and utility-maximizing demand requires that px/py ¼ ¼ x/y ¼ shows that RPT y/x. Thus, equilibrium requires that y. Inserting this result into the equation for the production possibility frontier MCx/MCy ¼ y/x, or x ¼ x) y) ¼ ¼ 50p ﬃﬃﬃﬃﬃ ¼ 7:07 and px py ¼ 1: (13:15) This is the equilibrium for our base case with this model. that The budget constraint. The budget constraint faces individuals is not especially transparent in this illustration; therefore, it may be useful to discuss it explicitly. To bring some degree of abs
olute pricing into the model, let’s consider all prices in terms of the wage rate, w. Because total labor supply is 100, it follows that total labor income is 100w. However, because of the diminishing returns assumed for production, each ﬁrm also earns proﬁts. For ﬁrm x, say, 50p . Therefore, the the total cost function is C(w, x) proﬁts for ﬁrm x are px ¼ 50w. A similar computation shows that proﬁts for ﬁrm y are also given by 50w. Because general equilibrium models must obey the national income identity, we assume that consumers are also shareholders in the two ﬁrms and treat these proﬁts also as part of their spendable incomes. Hence total consumer income is wx2, so px ¼ MCx ¼ wx2 (px – wx)x ¼ ¼ wlx ¼ ¼ ¼ (px – ACx)x 2wx 2w ﬃﬃﬃﬃﬃ ¼ total income ¼ ¼ labor income + profits 50w 100w ð Þ ¼ þ 2 200w: This income will just permit consumers to spend 100w on each good by buying price of 2w 50p , so the model is internally consistent. (13:16) 50p units at a ﬃﬃﬃﬃﬃ A shift in supply. There are only two ways in which this base-case equilibrium can be disturbed: (1) by changes in ‘‘supply’’—that is, by changes in the underlying technology of this economy; or (2) by changes in ‘‘demand’’—that is, by changes in preferences. Let’s ﬁrst consider changes in technology. Suppose that there is technical improvement in x production so that the production function is x 100, and RPT . Now the production possibility frontier is given by x2/4 x/4y. Proceeding as before to ﬁnd the equilibrium in this model: 2l 0:5 x y2 þ ¼ ¼ ﬃﬃﬃﬃﬃ ¼ Chapter 13: General Equilibrium and Welfare 469 px py ¼ px py ¼ x 4y ð y x ð supply , Þ demand , Þ so x2 ¼ 4y2 and the equilibrium is 2 50p , x) ¼ y) ¼ 50p and ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ px py ¼ 1 2 : (13:17) (13:18) Technical improvements in x production have caused its relative price to decrease and the consumption of this good to increase. As in many examples with Cobb–Douglas utility, the income and substitution effects of this price decrease on y demand are precisely offsetting. Technical improvements clearly make consumers better off, however. Whereas utility was previously given 0:5 50p by U x, y ð Þ 0:5 50p Þ ð ﬃﬃﬃﬃﬃ 50p 2 ¼ ð 10. Technical change has increased consumer welfare substantially. 7:07, now it has increased to U A shift in demand. If consumer preferences were to switch to favor good y as U(x, y) x0.1y0.9, then demand functions would be given by x equilibrium would require px/py ¼ to arrive at an overall equilibrium, we have ¼ 0.9I/py, and demand 0.1I/px and y y/9x. Returning to the original production possibility frontier x0:5y0:5 50p Þ ¼ 2p ¼ x, y ð x0:5y0:5 ¼ ¼ Þ ¼ ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ ¼ ¼ ¼ ﬃﬃﬃ * px py ¼ px py ¼ x y ð y 9x ð supply , Þ demand , Þ so 9x2 ¼ y2 and the equilibrium is given by 10p , x) ¼ y) ¼ 3 10p and ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃ px py ¼ 1 3 (13:19) (13:20) Hence the decrease in demand for x has signiﬁcantly reduced its relative price. Observe that in this case, however, we cannot make a welfare comparison to the previous cases because the utility function has changed. QUERY: What are the budget constraints in these two alternative scenarios? How is income distributed between wages and proﬁts in each case? Explain the differences intuitively. General Equilibrium Modeling and Factor Prices This simple general equilibrium model reinforces Marshall’s observations about the importance of both supply and demand forces in the price determination process. By providing an explicit connection between the markets for all goods, the general equilibrium model makes it possible to examine more complex questions about market relationships than is possible by looking at only one market at a time. General equilibrium modeling also permits an examination of the connections between goods and factor markets; we can illustrate that with an important historical case. 470 Part 5: Competitive Markets The Corn Laws debate High tariffs on grain imports were imposed by the British government following the Napoleonic wars. Debate over the effects of these Corn Laws dominated the analytical efforts of economists between the years 1829 and 1845. A principal focus of the debate concerned the effect that elimination of the tariffs would have on factor prices—a question that continues to have relevance today, as we will see. The production possibility frontier in Figure 13.6 shows those combinations of grain (x) and manufactured goods (y) that could be produced by British factors of production. Assuming (somewhat contrary to actuality) that the Corn Laws completely prevented trade, market equilibrium would be at E with the domestic price ratio given by p)x=p)y. Removal of the tariffs would reduce this price ratio to p0x=p0y. Given that new ratio, Britain would produce combination A and consume combination B. Grain imports would amount to xB – xA, and these would be ﬁnanced by export of manufactured goods equal to yA – yB. Overall utility for the typical British consumer would be increased by the opening of trade. Therefore, use of the production possibility diagram demonstrates the implications that relaxing the tariffs would have for the production of both goods. Trade and factor prices By referring to the Edgeworth production box diagram (Figure 13.2) that lies behind the production possibility frontier (Figure 13.3), it is also possible to analyze the effect of FIGURE 13.6 Analysis of the Corn Laws Debate Reduction of tariff barriers on grain would cause production to be reallocated from point E to point A; consumption would be reallocated from E to B. If grain production is relatively capital intensive, the relative price of capital would decrease as a result of these reallocations. Output of manufactured goods (y) P yA yE yB * * Slope = −px /py A Slope = −p′x /p′y E B U2 U1 xA xE xBP Output of grain (x) Chapter 13: General Equilibrium and Welfare 471 tariff reductions on factor prices. The movement from point E to point A in Figure 13.6 is similar to a movement from P3 to P1 in Figure 13.2, where production of x is decreased and production of y is increased. This ﬁgure also records the reallocation of capital and labor made necessary by such a move. If we assume that grain production is relatively capital intensive, then the movement from P3 to P1 causes the ratio of k to l to increase in both industries.7 This in turn will cause the relative price of capital to decrease (and the relative price of labor to increase). Hence we conclude that repeal of the Corn Laws would be harmful to capital owners (i.e., landlords) and helpful to laborers. It is not surprising that landed interests fought repeal of the laws. Political support for trade policies The possibility that trade policies may affect the relative incomes of various factors of production continues to exert a major inﬂuence on political debates about such policies. In the United States, for example, exports tend to be intensive in their use of skilled labor, whereas imports tend to be intensive in unskilled labor input. By analogy to our discussion of the Corn Laws, it might thus be expected that further movements toward free trade policies would result in increasing relative wages for skilled workers and in decreasing relative wages for unskilled workers. Therefore, it is not surprising that unions representing skilled workers (the machinists or aircraft workers) tend to favor free trade, whereas unions of unskilled workers (those in textiles, shoes, and related businesses) tend to oppose it.8 A Mathematical Model of Exchange Although the previous graphical model of general equilibrium with two goods is fairly instructive, it cannot reﬂect all the features of general equilibrium modeling with an arbitrary number of goods and productive inputs. In the remainder of this chapter we will illustrate how such a more general model can be constructed, and we will look at some of the insights that such a model can provide. For most of our presentation we will look only at a model of exchange—quantities of various goods already exist and are merely traded among individuals. In such a model there is no production. Later in the chapter we will look brieﬂy at how production can be incorporated into the general model we have constructed. Vector notation Most general equilibrium modeling is conducted using vector notation. This provides great ﬂexibility in specifying an arbitrary number of goods or individuals in the models. Consequently, this seems to be a good place to offer a brief introduction to such notation. A vector is simply an ordered array of variables (which each may take on speciﬁc values). Here we will usually adopt the convention that the vectors we use are column vectors. Hence we will write an n 1 column vector as: + 7In the Corn Laws debate, attention centered on the factors of land and labor. 8The ﬁnding that the opening of trade will raise the relative price of the abundant factor is called the Stolper–Samuelson theorem after the economists who rigorously proved it in the 1950s. 472 Part 5: Competitive Markets 2 x ¼ x1 x2 : : : xn 13:21) 1 column + where each xi is a variable that can take on any value. If x and y are two n vectors, then the (vector) sum of them is deﬁned as x1 x2 : : : xn y1 y2 : : : yn x1 þ x2 þ : : : xn þ y1 y2 3 7 7 7 7 7 7 5 yn 6 6 6 6 6 6 4 : (13:22) Notice that this sum only is deﬁned if the two vectors are of equal length. In fact, checking the length of vectors is one good way of deciding whether one has written a meaningful vector equation. The (dot) product of two vectors is deﬁned as the sum of the component-by-component product of the elements in the two vectors. That is: xy x1y1 þ x2y2 þ * * * þ ¼ xnyn: (13:23) Notice again that this operation is only deﬁned if the vectors are of the same length. With these few concepts we are now ready to illustrate the general equilibrium model of exchange. Utility, initial endowments, and budget constraints In our model of exchange there are assumed to be n goods and m individuals. Each individual gai
ns utility from the vector of goods he or she consumes ui(xi) where i 1. . . m. Individuals also possess initial endowments of the goods given by xi. Individuals are free to exchange their initial endowments with other individuals or to keep some or all the endowment for themselves. In their trading individuals are assumed to be price-takers—that is, they face a price vector (p) that speciﬁes the market price for each of the n goods. Each individual seeks to maximize utility and is bound by a budget constraint that requires that the total amount spent on consumption equals the total value of his or her endowment: ¼ pxi ¼ pxi: (13:24) Although this budget constraint has a simple form, it may be worth contemplating it for a minute. The right side of Equation 13.24 is the market value of this individual’s endowment (sometimes referred to as his or her full income). He or she could ‘‘afford’’ to consume this endowment (and only this endowment) if he or she wished to be self-sufﬁcient. But the endowment can also be spent on some other consumption bundle (which, presumably, provides more utility). Because consuming items in one’s own endowment has an opportunity cost, the terms on the left of Equation 13.24 consider the costs of all items that enter into the ﬁnal consumption bundle, including endowment goods that are retained. Demand functions and homogeneity The utility maximization problem outlined in the previous section is identical to the one we studied in detail in Part 2 of this book. As we showed in Chapter 4, one outcome of Chapter 13: General Equilibrium and Welfare 473 this process is a set of n individual demand functions (one for each good) in which quantities demanded depend on all prices and income. Here we can denote these in vector form as xi . These demand functions are continuous, and, as we showed in ChapÞ ter 4, they are homogeneous of degree 0 in all prices and income. This latter property can be indicated in vector notation by p, pxi ð xi tp, tpxi ð Þ ¼ xi p, pxi ð Þ (13:25) for any t > 0. This property will be useful because it will permit us to adopt a convenient normalization scheme for prices, which, because it does not alter relative prices, leaves quantities demanded unchanged. Equilibrium and Walras’ law Equilibrium in this simple model of exchange requires that the total quantities of each good demanded be equal to the total endowment of each good available (remember, there is no production in this model). Because the model used is similar to the one originally developed by Leon Walras,9 this equilibrium concept is customarily attributed to him Walrasian equilibrium. Walrasian equilibrium is an allocation of resources and an associated price vector, p), such that m 1 i X ¼ xi p), p)xi ð Þ ¼ m xi, 1 i X ¼ (13:26) where the summation is taken over the m individuals in this exchange economy. The n equations in Equation 13.26 state that in equilibrium demand equals supply in each market. This is the multimarket analog of the single market equilibria examined in the previous chapter. Because there are n prices to be determined, a simple counting of equations and unknowns might suggest that the existence of such a set of prices is guaranteed by the simultaneous equation solution procedures studied in elementary algebra. Such a supposition would be incorrect for two reasons. First, the algebraic theorem about simultaneous equation systems applies only to linear equations. Nothing suggests that the demand equations in this problem will be linear—in fact, most examples of demand equations we encountered in Part 2 were deﬁnitely nonlinear. A second problem with Equation 13.26 is that the equations are not independent of one another—they are related by what is known as Walras’ law. Because each individual in this exchange economy is bound by a budget constraint of the form given in Equation 13.24, we can sum over all individuals to obtain m m pxi ¼ 1 i X ¼ 1 i X ¼ pxi or m 1 i X ¼ xi p ð ! xi Þ ¼ 0: (13:27) In words, Walras’ law states that the value of all quantities demanded must equal the value of all endowments. This result holds for any set of prices, not just for equilibrium 9The concept is named for the nineteenth century French/Swiss economist Leon Walras, who pioneered the development of general equilibrium models. Models of the type discussed in this chapter are often referred to as models of Walrasian equilibrium, primarily because of the price-taking assumptions inherent in them. 474 Part 5: Competitive Markets prices.10 The general lesson is that the logic of individual budget constraints necessarily creates a relationship among the prices in any economy. It is this connection that helps to ensure that a demand–supply equilibrium exists, as we now show. Existence of equilibrium in the exchange model The question of whether all markets can reach equilibrium together has fascinated economists for nearly 200 years. Although intuitive evidence from the real world suggests that this must indeed be possible (market prices do not tend to ﬂuctuate wildly from one day to the next), proving the result mathematically proved to be rather difﬁcult. Walras himself thought he had a good proof that relied on evidence from the market to adjust prices toward equilibrium. The price would increase for any good for which demand exceeded supply and decrease when supply exceeded demand. Walras believed that if this process continued long enough, a full set of equilibrium prices would eventually be found. Unfortunately, the pure mathematics of Walras’ solution were difﬁcult to state, and ultimately there was no guarantee that a solution would be found. But Walras’ idea of adjusting prices toward equilibrium using market forces provided a starting point for the modern proofs, which were largely developed during the 1950s. A key aspect of the modern proofs of the existence of equilibrium prices is the choice of a good normalization rule. Homogeneity of demand functions makes it possible to use any absolute scale for prices, providing that relative prices are unaffected by this choice. Such an especially convenient scale is to normalize prices so that they sum to one. Consider an arbitrary set of n non-negative prices p1, p2 . . . pn. We can normalize11 these to form a new set of prices p0i ¼ pi n : pk (13:28) 1 k ¼ P These new prices will have the properties that maintained: n 1 k ¼ P pk pk ¼ 1 and that relative price ratios are (13:29) p0k ¼ pi pj : p0i p0j ¼ pi= pj P $P Because this sort of mathematical process can always be done, we will assume, without loss of generality, that the price vectors we use (p) have all been normalized in this way. Therefore, proving the existence of equilibrium prices in our model of exchange amounts to showing that there will always exist a price vector p) that achieves equilibrium in all markets. That is, m 1 i X ¼ xi p), p)xi ð Þ ¼ xi or xi p), p)xi ð Þ ! m 1 i X ¼ xi ¼ 0 or p) z ð Þ ¼ 0, (13:30) where we use z(p) as a shorthand way of recording the ‘‘excess demands’’ for goods at a particular set of prices. In equilibrium, excess demand is zero in all markets.12 10Walras’ law holds trivially for equilibrium prices as multiplication of Equation 13.26 by p shows. 11This is possible only if at least one of the prices is nonzero. Throughout our discussion we will assume that not all equilibrium prices can be zero. 12Goods that are in excess supply at equilibrium will have a zero price. We will not be concerned with such ‘‘free goods’’ here. FIGURE 13.7 A Graphical Illustration of Brouwer’s Fixed Point Theorem Chapter 13: General Equilibrium and Welfare 475 Because any continuous function must cross the 45! line somewhere in the unit square, this function must have a point for which f (x)) x). This point is called a ﬁxed point. ¼ f (x) 1 f (x) Fixed point f (x) 45° 0 x 1 x Now consider the following way of implementing Walras’ idea that goods in excess demand should have their prices increased, whereas those in excess supply should have their prices reduced.13 Starting from any arbitrary set of prices, p0, we deﬁne a new set, p1, as f ð k z ð , p0Þ p1 ¼ p0Þ ¼ p0 þ where k is a positive constant. This function will be continuous (because demand functions are continuous), and it will map one set of normalized prices into another (because of our assumption that all prices are normalized). Hence it will meet the conditions of the Brouwer’s ﬁxed point theorem, which states that any continuous function from a closed compact set onto itself (in the present case, from the ‘‘unit simplex’’ onto itself) f (x). The theorem is illustrated for a single dimenwill have a ‘‘ﬁxed point’’ such that x sion in Figure 13.7. There, no matter what shape the function f(x) takes, as long as it is continuous, it must somewhere cross the 45! line and at that point x (13:31) f(x). ¼ If we let p) represent the ﬁxed point identiﬁed by Brouwer’s theorem for Equation ¼ 13.31, we have: p) f p) ð ¼ Þ ¼ p) p) k z ð Þ : þ (13:32) ¼ Hence at this point z(p)) 0; thus, p) is an equilibrium price vector. The proof that Walras sought is easily accomplished using an important mathematical result developed a few years after his death. The elegance of the proof may obscure the fact that it uses a number of assumptions about economic behavior such as: (1) price-taking by all parties; (2) homogeneity of demand functions; (3) continuity of demand functions; and (4) presence of budget constraints and Walras’ law. All these play important roles in showing that a system of simple markets can indeed achieve a multimarket equilibrium. 13What follows is an extremely simpliﬁed version of the proof of the existence of equilibrium prices. In particular, problems of free goods and appropriate normalizations have been largely assumed away. For a mathematically correct proof, see, for example, G. Debreu, Theory of Value (New York: John Wiley & Sons, 1959). 476 Part 5: Competitive Markets First theorem of welfare economics Given that
the forces of supply and demand can establish equilibrium prices in the general equilibrium model of exchange we have developed, it is natural to ask what are the welfare consequences of this ﬁnding. Adam Smith14 hypothesized that market forces provide an ‘‘invisible hand’’ that leads each market participant to ‘‘promote an end [social welfare] which was no part of his intention.’’ Modern welfare economics seeks to understand the extent to which Smith was correct. Perhaps the most important welfare result that can be derived from the exchange model is that the resulting Walrasian equilibrium is ‘‘efﬁcient’’ in the sense that it is not possible to devise some alternative allocation of resources in which at least some people are better off and no one is worse off. This deﬁnition of efﬁciency was originally developed by Italian economist Vilfredo Pareto in the early 1900s. Understanding the deﬁnition is easiest if we consider what an ‘‘inefﬁcient’’ allocation might be. The total quantities of goods included in initial endowments would be allocated inefﬁciently if it were possible, by shifting goods around among individuals, to make at least one person better off (i.e., receive a higher utility) and no one worse off. Clearly, if individuals’ preferences are to count, such a situation would be undesirable. Hence we have a formal deﬁnition Pareto efﬁcient allocation. An allocation of the available goods in an exchange economy is efﬁcient if it is not possible to devise an alternative allocation in which at least one person is better off and no one is worse off. A proof that all Walrasian equilibria are Pareto efﬁcient proceeds indirectly. Suppose that p) generates a Walrasian equilibrium in which the quantity of goods consumed by each person is denoted by )xk . Now assume that there is some alternative Þ ¼ allocation of the available goods 0xk such that, for at least one person, say, 1 . . . m person i, it is that case that 0xi is preferred to )xi. For this person, it must be the case that )0 xi > p) )xi (13:33) because otherwise this person would have bought the preferred bundle in the ﬁrst place. If all other individuals are to be equally well off under this new proposed allocation, it must be the case for them that p)0 xk p) )xk k 1 . . . m, k i: (13:34) ¼ If the new bundle were less expensive, such individuals could not have been minimizing expenditures at p). Finally, to be feasible, the new allocation must obey the quantity constraints 6¼ ¼ m m 1 i X ¼ Multiplying Equation 13.35 by p)yields 0xi ¼ xi: 1 i X ¼ m p)0 xi m 1 i X ¼ p) xi, ¼ 1 i X ¼ 14Adam Smith, The Wealth of Nations (New York: Modern Library, 1937) p. 423. (13:35) (13:36) Chapter 13: General Equilibrium and Welfare 477 but Equations 13.33 and 13.34 together with Walras’ law applied to the original equilibrium imply that m 1 i X ¼ m m p)0 xi > p) )xi 1 i X ¼ p) xi: ¼ 1 i X ¼ (13:37) Hence we have a contradiction and must conclude that no such alternative allocation can exist. Therefore, we can summarize our analysis with the following deﬁnition First theorem of welfare economics. Every Walrasian equilibrium is Pareto efﬁcient. The signiﬁcance of this ‘‘theorem’’ should not be overstated. The theorem does not say that every Walrasian equilibrium is in some sense socially desirable. Walrasian equilibria can, for example, exhibit vast inequalities among individuals arising in part from inequalities in their initial endowments (see the discussion in the next section). The theorem also assumes price-taking behavior and full information about prices—assumptions that need not hold in other models. Finally, the theorem does not consider possible effects of one individual’s consumption on another. In the presence of such externalities even a perfect competitive price system may not yield Pareto optimal results (see Chapter 19). Still, the theorem does show that Smith’s ‘‘invisible hand’’ conjecture has some validity. The simple markets in this exchange world can ﬁnd equilibrium prices, and at those equilibrium prices the resulting allocation of resources will be efﬁcient in the Pareto sense. Developing this proof is one of the key achievements of welfare economics. x xA and yB A graphic illustration of the ﬁrst theorem In Figure 13.8 we again use the Edgeworth box diagram, this time to illustrate an exchange economy. In this economy there are only two goods (x and y) and two individuals (A and B). The total dimensions of the Edgeworth box are determined by the total quantities of the two goods available (x and y). Goods allocated to individual A are recorded using 0A as an origin. Individual B gets those quantities of the two goods that are ‘‘left over’’ and can be measured using 0B as an origin. Individual A’s indifference curve map is drawn in the usual way, whereas individual B’s map is drawn from the perspective of 0B. Point E in the Edgeworth box represents the initial endowments of these two individuals. Individual A starts with xA and yA. Individual B starts with xB A for person A and U 2 ¼ The initial endowments provide a utility level of U 2 A (point B). Or we could increase person A’s utility to U 3 B for person B. These levels are clearly inefﬁcient in the Pareto sense. For example, we could, by reallocating the available goods,15 increase person B’s utility to U 3 B while holding person A’s utility constant at U 2 A while keeping person B on the U 2 B indifference curve (point A). Allocations A and B are Pareto efﬁcient, however, because at these allocations it is not possible to make either person better off without making the other worse off. There are many other efﬁcient allocations in the Edgeworth box diagram. These are identiﬁed by the tangencies of the two individuals’ indifference curves. The set of all such efﬁcient points is shown by the line joining OA to OB. This line is sometimes called the ‘‘contract curve’’ because it represents all the Paretoefﬁcient contracts that might be reached by these two individuals. Notice, however, that (assuming that no individual would voluntarily opt for a contract that made him or her yA: ¼ ! ! y 15This point UB xB, yB could in principle be subject to the constraint UA xA, yA ð Þ ð U 2 A. See Example 13.3. Þ ¼ found by solving the following constrained optimization problem: Maximize 478 Part 5: Competitive Markets FIGURE 13.8 The First Theorem of Welfare Economics With initial endowments at point E, individuals trade along the price line PP until they reach point E). This equilibrium is Pareto efﬁcient worse off ) only contracts between points B and A are viable with initial endowments given by point E. The line PP in Figure 13.8 shows the competitively established price ratio that is guaranteed by our earlier existence proof. The line passes through the initial endowments (E) and shows the terms at which these two individuals can trade away from these initial positions. Notice that such trading is beneﬁcial to both parties—that is, it allows them to get a higher utility level than is provided by their initial endowments. Such trading will continue until all such mutual beneﬁcial trades have been completed. That will occur at allocation E) on the contract curve. Because the individuals’ indifference curves are tangent at this point, no further trading would yield gains to both parties. Therefore, the competitive allocation E) meets the Pareto criterion for efﬁciency, as we showed mathematically earlier. Second theorem of welfare economics The ﬁrst theorem of welfare economics shows that a Walrasian equilibrium is Pareto efﬁcient, but the social welfare consequences of this result are limited because of the role played by initial endowments in the demonstration. The location of the Walrasian equilibrium at E) in Figure 13.8 was signiﬁcantly inﬂuenced by the designation of E as the starting point for trading. Points on the contract curve outside the range of AB are not attainable through voluntary transactions, even though these may in fact be more socially desirable than E) (perhaps because utilities are more equal). The second theorem of welfare economics addresses this issue. It states that for any Pareto optimal allocation of resources there exists a set of initial endowments and a related price vector such that this allocation is also a Walrasian equilibrium. Phrased another way, any Pareto optimal allocation of resources can also be a Walrasian equilibrium, providing that initial endowments are adjusted accordingly. Chapter 13: General Equilibrium and Welfare 479 FIGURE 13.9 The Second Theorem of Welfare Economics If allocation Q) is regarded as socially optimal, this allocation can be supported by any initial endowments on the price line P0P0. To move from E to, say, Q would require transfers of initial endowments. P' E* A Q* E O B Q P graphical proof of the second theorem should sufﬁce. Figure 13.9 repeats the key aspects of the exchange economy pictures in Figure 13.8. Given the initial endowments at point E, all voluntary Walrasian equilibrium must lie between points A and B on the contract curve. Suppose, however, that these allocations were thought to be undesirable— perhaps because they involve too much inequality of utility. Assume that the Pareto optimal allocation Q) is believed to be socially preferable, but it is not attainable from the initial endowments at point E. The second theorem states that one can draw a price line through Q) that is tangent to both individuals’ respective indifference curves. This line is denoted by P 0P 0 in Figure 13.9. Because the slope of this line shows potential trades these individuals are willing to make, any point on the line can serve as an initial endowment from which trades lead to Q). One such point is denoted by Q. If a benevolent government wished to ensure that Q) would emerge as a Walrasian equilibrium, it would have to transfer initial endowments of the goods from E to Q (making person A better off and person B worse off in the process). E
XAMPLE 13.3 A Two-Person Exchange Economy To illustrate these various principles, consider a simple two-person, economy. Suppose that total quantities of the goods are ﬁxed at x y utility takes the Cobb–Douglas form: ¼ two-good exchange 1,000. Person A’s ¼ UAð and person B’s preferences are given by: xA, yAÞ ¼ A y1=3 x2=3 A , UBð xB, yBÞ ¼ B y2=3 x1=3 B : (13:38) (13:39) 480 Part 5: Competitive Markets Notice that person A has a relative preference for good x and person B has a relative preference for good y. Hence you might expect that the Pareto-efﬁcient allocations in this model would have the property that person A would consume relatively more x and person B would consume relatively more y. To ﬁnd these allocations explicitly, we need to ﬁnd a way of dividing the available goods in such a way that the utility of person A is maximized for any preassigned utility level for person B. Setting up the Lagrangian expression for this problem, we have: UAð Substituting for the explicit utility functions assumed here yields xA, yAÞ ¼ xA, yAÞ þ UBð 1,000 k ½ Lð xA, 1,000 ! yAÞ ! : U B& ! xA, yAÞ ¼ and the ﬁrst-order conditions for a maximum are A þ 1,000 Lð ½ð k ! A y1=3 x2=3 1=3 xAÞ 1,000 ð ! 2=3 yAÞ , U B& ! @ L @xA ¼ @ L @yA ¼ 2 3 1 3 1=3 2=3 yA xA" # xA yA" # k 3 ! ! " 2k 3 " 1,000 1,000 yA xA ! ! 1,000 1,000 # xA yA # ! ! 2=3 0; ¼ 1=3 0: ¼ Moving the terms in l to the right and dividing the top equation by the bottom gives or yA 2 xA" # 1 2 ¼ " 1,000 1,000 yA xA ! ! # xA 1,000 ! 4yA xA ¼ 1,000 ! : yA (13:40) (13:41) (13:42) (13:43) This equation allows us to identify all the Pareto optimal allocations in this exchange economy. For example, if we were to arbitrarily choose xA ¼ 500, Equation 13.43 would become 4yA 1,000 yA ¼ ! 1 so xB ¼ yA ¼ 200, yB ¼ 800: (13:44) This allocation is relatively favorable to person B. At this point on the contract curve UA ¼ 5002/32001/3 683. Notice that although the available quantity of x is divided evenly (by assumption), most of good y goes to person B as efﬁciency requires. 369, UB ¼ 5001/38002/3 ¼ ¼ Equilibrium price ratio. To calculate the equilibrium price ratio at this point on the contract curve, we need to know the two individuals’ marginal rates of substitution. For person A, and for person B MRS @UA=@xA @UA=@yA ¼ 2 yA xA ¼ 2 200 500 ¼ ¼ 0:8 MRS @UB=@xB @UB=@yB ¼ ¼ 0:5 yA xA ¼ 0:5 800 500 ¼ 0:8: (13:45) (13:46) 0.8. Hence the marginal rates of substitution are indeed equal (as they should be), and they imply a price ratio of px/py ¼ Initial endowments. Because this equilibrium price ratio will permit these individuals to trade 8 units of y for each 10 units of x, it is a simple matter to devise initial endowments optimum. Consider, consistent with endowment the 1, the value of person A’s initial 680. If px ¼ 350, yA ¼ xA ¼ endowment is 600. If he or she spends two thirds of this amount on good x, it is possible to Pareto 650, yB ¼ for 0.8, py ¼ this 320; xB ¼ example, Chapter 13: General Equilibrium and Welfare 481 ¼ purchase 500 units of good x and 200 units of good y. This would increase utility from UA ¼ 3502/3 3201/3 340 to 369. Similarly, the value of person B’s endowment is 1,200. If he or she spends one third of this on good x, 500 units can be bought. With the remaining two thirds of the value of the endowment being spent on good y, 800 units can be bought. In the process, B’s utility increases from 670 to 683. Thus, trading from the proposed initial endowment to the contract curve is indeed mutually beneﬁcial (as shown in Figure 13.8). QUERY: Why did starting with the assumption that good x would be divided equally on the contract curve result in a situation favoring person B throughout this problem? What point on the contract curve would provide equal utility to persons A and B? What would the price ratio of the two goods be at this point? Social welfare functions Figure 13.9 shows that there are many Pareto-efﬁcient allocations of the available goods in an exchange economy. We are assured by the second theorem of welfare economics that any of these can be supported by a Walrasian system of competitively determined prices, providing that initial endowments are adjusted accordingly. A major question for welfare economics is how (if at all) to develop criteria for choosing among all these allocations. In this section we look brieﬂy at one strand of this large topic—the study of social welfare functions. Simply put, a social welfare function is a hypothetical scheme for ranking potential allocations of resources based on the utility they provide to individuals. In mathematical terms: ½ ¼ SW x1 U1ð Social Welfare x2 , U2ð Þ The ‘‘social planner’s’’ goal then is to choose allocations of goods among the m individuals in the economy in a way that maximizes SW. Of course, this exercise is a purely conceptual one—in reality there are no clearly articulated social welfare functions in any economy, and there are serious doubts about whether such a function could ever arise from some type of democratic process.16 Still, assuming the existence of such a function can help to illuminate many of the thorniest problems in welfare economics. . . . , Umð (13:47) : Þ& xm , Þ A ﬁrst observation that might be made about the social welfare function in Equation 13.47 is that any welfare maximum must also be Pareto efﬁcient. If we assume that every individual’s utility is to ‘‘count,’’ it seems clear that any allocation that permits further Pareto improvements (that make one person better off and no one else worse off) cannot be a welfare maximum. Hence achieving a welfare maximum is a problem in choosing among Pareto-efﬁcient allocations and their related Walrasian price systems. We can make further progress in examining the idea of social welfare maximization by considering the precise functional form that SW might take. Speciﬁcally, if we assume utility is measurable, using the CES form can be particularly instructive: SW U1, U2, ð . . . , UmÞ : R , (13:48) Because we have used this functional form many times before in this book, its properties should by now be familiar. Speciﬁcally, if R 1, the function becomes: ¼ U1 þ . . . , UmÞ ¼ U1, U2, ð U2 þ (13:49) Um: SW . . . þ 16The ‘‘impossibility’’ of developing a social welfare function from the underlying preferences of people in society was ﬁrst studied by K. Arrow in Social Choice and Individual Values, 2nd ed. (New York: Wiley, 1963). There is a large body of literature stemming from Arrow’s initial discovery. 482 Part 5: Competitive Markets Thus, utility is a simple sum of the utility of every person in the economy. Such a social welfare function is sometimes called a utilitarian function. With such a function, social welfare is judged by the aggregate sum of utility (or perhaps even income) with no regard for how utility (income) is distributed among the members of society. At the other extreme, consider the case R . In this case, social welfare has a ‘‘ﬁxed proportions’’ character and (as we have seen in many other applications), . . . , UmÞ ¼ . . . , Um& U1, U2, U1, U2, Min SW ð ½ : (13:50) ¼ !1 Therefore, this function focuses on the worse-off person in any allocation and chooses that allocation for which this person has the highest utility. Such a social welfare function is called a maximin function. It was made popular by the philosopher John Rawls, who argued that if individuals did not know which position they would ultimately have in society (i.e., they operate under a ‘‘veil of ignorance’’), they would opt for this sort of social welfare function to guard against being the worse-off person.17 Our analysis in Chapter 7 suggests that people may not be this risk averse in choosing social arrangements. However, Rawls’ focus on the bottom of the utility distribution is probably a good antidote to thinking about social welfare in purely utilitarian terms. It is possible to explore many other potential functional forms for a hypothetical welfare function. Problem 13.14 looks at some connections between social welfare functions and the income distribution, for example. But such illustrations largely miss a crucial point if they focus only on an exchange economy. Because the quantities of goods in such an economy are ﬁxed, issues related to production incentives do not arise when evaluating social welfare alternatives. In actuality, however, any attempt to redistribute income (or utility) through taxes and transfers will necessarily affect production incentives and therefore affect the size of the Edgeworth box. Therefore, assessing social welfare will involve studying the trade-off between achieving distributional goals and maintaining levels of production. To examine such possibilities we must introduce production into our general equilibrium framework. A Mathematical Model of Production and Exchange Adding production to the model of exchange developed in the previous section is a relatively simple process. First, the notion of a ‘‘good’’ needs to be expanded to include factors of production. Therefore, we will assume that our list of n goods now includes inputs whose prices also will be determined within the general equilibrium model. Some inputs for one ﬁrm in a general equilibrium model are produced by other ﬁrms. Some of these goods may also be consumed by individuals (cars are used by both ﬁrms and ﬁnal consumers), and some of these may be used only as intermediate goods (steel sheets are used only to make cars and are not bought by consumers). Other inputs may be part of individuals’ initial endowments. Most importantly, this is the way labor supply is treated in general equilibrium models. Individuals are endowed with a certain number of potential labor hours. They may sell these to ﬁrms by taking jobs at competitively determined wages, or they may choose to consume the hours themselves in the form of ‘‘leisure,’’ In making such choices we continue to assume that individuals maximize utility.18 We will assume that the
re are r ﬁrms involved in production. Each of these ﬁrms is bound by a production function that describes the physical constraints on the ways the 17J. Rawls, A Theory of Justice (Cambridge, MA: Harvard University Press, 1971). 18A detailed study of labor supply theory is presented in Chapter 16. Chapter 13: General Equilibrium and Welfare 483 ﬁrm can turn inputs into outputs. By convention, outputs of the ﬁrm take a positive sign, whereas inputs take a negative sign. Using this convention, each ﬁrm’s production plan can 1 . . . r), which contains both positive and be described by an n negative entries. The only vectors that the ﬁrm may consider are those that are feasible given the current state of technology. Sometimes it is convenient to assume each ﬁrm produces only one output. But that is not necessary for a more general treatment of production. 1 column vector, y j( j + ¼ Firms are assumed to maximize proﬁts. Production functions are assumed to be sufﬁciently convex to ensure a unique proﬁt maximum for any set of output and input prices. This rules out both increasing returns to scale technologies and constant returns because neither yields a unique maxima. Many general equilibrium models can handle such possibilities, but there is no need to introduce such complexities here. Given these assumptions, the proﬁts for any ﬁrm can be written as: py j if pjð p Þ ¼ < 0: p 0 if pjð Þ pjð y j (13:51) and Þ ( ¼ p 0 Hence this model has a ‘‘long run’’ orientation in which ﬁrms that lose money (at a particular price conﬁguration) hire no inputs and produce no output. Notice how the convention that outputs have a positive sign and inputs a negative sign makes it possible to phrase proﬁts in a compact way.19 Budget constraints and Walras’ law In an exchange model, individuals’ purchasing power is determined by the values of their initial endowments. Once ﬁrms are introduced, we must also consider the income stream that may ﬂow from ownership of these ﬁrms. To do so, we adopt the simplifying assumption that each individual owns a predeﬁned share, si ð of all ﬁrms. That is, each person owns an ‘‘index fund’’ that can claim a proportionate share of all ﬁrms’ proﬁts. We can now rewrite each individual’s budget constraint (from Equation 13.24) as: of the proﬁts si ¼ where 1 i ¼ P 1 Þ m pxi si ¼ r 1 j X ¼ pyj pxi þ i ¼ 1 . . . m: (13:52) Of course, if all ﬁrms were in long-run equilibrium in perfectly competitive industries, all proﬁts would be zero and the budget constraint in Equation 13.52 would revert to that in Equation 13.24. But allowing for long-term proﬁts does not greatly complicate our model; therefore, we might as well consider the possibility. As in the exchange model, the existence of these m budget constraints implies a constraint of the prices that are possible—a generalization of Walras’ law. Summing the budget constraints in Equation 13.52 over all individuals yields: and letting x Walras’ law xi m p xi yj xi, (13:53) 1 i X ¼ xi provides a simple statement of P p px ð Þ ¼ P px: py p ð Þ þ (13:54) 19As we saw in Chapter 11, proﬁt functions are homogeneous of degree 1 in all prices. Hence both output supply functions and input demand functions are homogeneous of degree 0 in all prices because they are derivatives of the proﬁt function. 484 Part 5: Competitive Markets Notice again that Walras’ law holds for any set of prices because it is based on individuals’ budget constraints. Walrasian equilibrium As before, we deﬁne a Walrasian equilibrium price vector (p)) as a set of prices at which demand equals supply in all markets simultaneously. In mathematical terms this means that: x p) y p) x: (13:55) ð Þ þ Þ ¼ ð Initial endowments continue to play an important role in this equilibrium. For example, it is individuals’ endowments of potential labor time that provide the most important input for ﬁrms’ production processes. Therefore, determination of equilibrium wage rates is a major output of general equilibrium models operating under Walrasian conditions. Examining changes in wage rates that result from changes in exogenous inﬂuences is perhaps the most important practical use of such models. As in the study of an exchange economy, it is possible to use some form of ﬁxed point theorem20 to show that there exists a set of equilibrium prices that satisfy the n equations in Equation 13.55. Because of the constraint of Walras’ law, such an equilibrium price vector will be unique only up to a scalar multiple—that is, any absolute price level that preserves relative prices can also achieve equilibrium in all markets. Technically, excess demand functions 13:56) are homogeneous of degree 0 in prices; therefore, any price vector for which z(p)) 0 0 and t > 0. Frequently it is convenient to norwill also have the property that z(t p)) malize prices so that they sum to one. But many other normalization rules can also be used. In macroeconomic versions of general equilibrium models it is usually the case that the absolute level of prices is determined by monetary factors. ¼ ¼ Welfare economics in the Walrasian model with production Adding production to the model of an exchange economy greatly expands the number of feasible allocations of resources. One way to visualize this is shown in Figure 13.10. There PP represents that production possibility frontier for a two-good economy with a ﬁxed endowment of primary factors of production. Any point on this frontier is feasible. Consider one such allocation, say, allocation A. If this economy were to produce xA and yA, we could use these amounts for the dimensions of the Edgeworth exchange box shown inside the frontier. Any point within this box would also be a feasible allocation of the available goods between the two people whose preferences are shown. Clearly a similar argument could be made for any other point on the production possibility frontier. Despite these complications, the ﬁrst theorem of welfare economics continues to hold in a general equilibrium model with production. At a Walrasian equilibrium there are no further market opportunities (either by producing something else or by reallocating the available goods among individuals) that would make some one individual (or group of individuals) better off without making other individuals worse off. Adam Smith’s ‘‘invisible hand’’ continues to exert its logic to ensure that all such mutually beneﬁcial opportunities are exploited (in part because transaction costs are assumed to be zero). 20For some illustrative proofs, see K. J. Arrow and F. H. Hahn, General Competitive Analysis (San Francisco: Holden-Day, 1971) chap. 5. FIGURE 13.10 Production Increases the Number of Feasible Allocations Chapter 13: General Equilibrium and Welfare 485 Any point on the production possibility frontier PP can serve as the dimensions of an Edgeworth exchange box. Quantity of y P y A A P x A Quantity of x Again, the general social welfare implications of the ﬁrst theorem of welfare economics are far from clear. There is, of course, a second theorem, which shows that practically any Walrasian equilibrium can be supported by suitable changes in initial endowments. One also could hypothesize a social welfare function to choose among these. But most such exercises are rather uninformative about actual policy issues. More interesting is the use of the Walrasian mechanism to judge the hypothetical impact of various tax and transfer policies that seek to achieve speciﬁc social welfare criteria. In this case (as we shall see) the fact that Walrasian models stress interconnections among markets, especially among product and input markets, can yield important and often surprising results. In the next section we look at a few of these. Computable General Equilibrium Models Two advances have spurred the rapid development of general equilibrium models in recent years. First, the theory of general equilibrium itself has been expanded to include many features of real-world markets such as imperfect competition, environmental externalities, and complex tax systems. Models that involve uncertainty and that have a dynamic structure also have been devised, most importantly in the ﬁeld of macroeconomics. A second related trend has been the rapid development of computer power and the associated software for solving general equilibrium models. This has made it possible to 486 Part 5: Competitive Markets study models with virtually any number of goods and types of households. In this section we will brieﬂy explore some conceptual aspects of these models.21 The Extensions to the chapter describe a few important applications. Structure of general equilibrium models Speciﬁcation of any general equilibrium model begins by deﬁning the number of goods to be included in the model. These ‘‘goods’’ include not only consumption goods but also intermediate goods that are used in the production of other goods (e.g., capital equipment), productive inputs such as labor or natural resources, and goods that are to be produced by the government (public goods). The goal of the model is then to solve for equilibrium prices for all these goods and to study how these prices change when conditions change. Some of the goods in a general equilibrium model are produced by ﬁrms. The technology of this production must be speciﬁed by production functions. The most common such speciﬁcation is to use the types of CES production functions that we studied in Chapters 9 and 10 because these can yield some important insights about the ways in which inputs are substituted in the face of changing prices. In general, ﬁrms are assumed to maximize their proﬁts given their production functions and given the input and output prices they face. Demand is speciﬁed in general equilibrium models by deﬁning utility functions for various types of households. Utility is treated as a function both of goods that are consumed and of inputs that are not supplied to the marketplace (e.g., available labor that is not
supplied to the market is consumed as leisure). Households are assumed to maximize utility. Their incomes are determined by the amounts of inputs they ‘‘sell’’ in the market and by the net result of any taxes they pay or transfers they receive. Finally, a full general equilibrium model must specify how the government operates. If there are taxes in the model, how those taxes are to be spent on transfers or on public goods (which provide utility to consumers) must be modeled. If government borrowing is allowed, the bond market must be explicitly modeled. In short, the model must fully specify the ﬂow of both sources and uses of income that characterize the economy being modeled. Solving general equilibrium models Once technology (supply) and preferences (demand) have been speciﬁed, a general equilibrium model must be solved for equilibrium prices and quantities. The proof earlier in this chapter shows that such a model will generally have such a solution, but actually ﬁnding that solution can sometimes be difﬁcult—especially when the number of goods and households is large. General equilibrium models are usually solved on computers via modiﬁcations of an algorithm originally developed by Herbert Scarf in the 1970s.22 This algorithm (or more modern versions of it) searches for market equilibria by mimicking the way markets work. That is, an initial solution is speciﬁed and then prices are raised in markets with excess demand and lowered in markets with excess supply until an equilibrium is found in which all excess demands are zero. Sometimes multiple equilibria will occur, but usually economic models have sufﬁcient curvature in the underlying production and utility functions that the equilibrium found by the Scarf algorithm will be unique. 21For more detail on the issues discussed here, see W. Nicholson and F. Westhoff, ‘‘General Equilibrium Models: Improving the Microeconomics Classroom,’’ Journal of Economic Education (Summer 2009): 297–314. 22Herbert Scarf with Terje Hansen, On the Computation of Economic Equilibria (New Haven, CT: Yale University Press, 1973). Chapter 13: General Equilibrium and Welfare 487 Economic insights from general equilibrium models General equilibrium models provide a number of insights about how economies operate that cannot be obtained from the types of partial equilibrium models studied in Chapter 12. Some of the most important of these are: • All prices are endogenous in economic models. The exogenous elements of models are preferences and productive technologies. • All ﬁrms and productive inputs are owned by households. All income ultimately accrues to households. • Any model with a government sector is incomplete if it does not specify how tax receipts are used. • The ‘‘bottom line’’ in any policy evaluation is the utility of households. Firms and governments are only intermediaries in getting to this ﬁnal accounting. • All taxes distort economic decisions along some dimension. The welfare costs of such distortions must always be weighed against the beneﬁts of such taxes (in terms of public good production or equity-enhancing transfers). Some of these insights are illustrated in the next two examples. In later chapters we will return to general equilibrium modeling whenever such a perspective seems necessary to gain a more complete understanding of the topic being covered. EXAMPLE 13.4 A Simple General Equilibrium Model Let’s look at a simple general equilibrium model with only two households, two consumer goods (x and y), and two inputs (capital k and labor l). Each household has an ‘‘endowment’’ of capital and labor that it can choose to retain or sell in the market. These endowments are denoted by k1, l1 and k2, l2, respectively. Households obtain utility from the amounts of the consumer goods they purchase and from the amount of labor they do not sell into the market (i.e., leisure li ! li). The households have simple Cobb–Douglas utility functions: ¼ U 1 ¼ 1 y0:3 x0:5 1 ð l1 ! 0:2, l1Þ U 2 ¼ 2 y0:4 x0:4 2 ð l2 ! 0:2: l2Þ (13:57) Hence household 1 has a relatively greater preference for good x than does household 2. Notice that capital does not enter into these utility functions directly. Consequently, each household will provide its entire endowment of capital to the marketplace. Households will retain some labor, however, because leisure provides utility directly. Production of goods x and y is characterized by simple Cobb–Douglas technologies: k0:2 x l 0:8 x , x ¼ k0:8 y l 0:2 y : y ¼ (13:58) Thus, in this example, production of x is relatively labor intensive, whereas production of y is relatively capital intensive. To complete this model we must specify initial endowments of capital and labor. Here we assume that k1 ¼ 40, l1 ¼ 24 and k2 ¼ 10, l2 ¼ 24: (13:59) Although the households have equal labor endowments (i.e., 24 ‘‘hours’’), household 1 has signiﬁcantly more capital than does household 2. Base-case simulation. Equations 13.57–13.59 specify our complete general equilibrium model in the absence of a government. A solution to this model will consist of four equilibrium prices (for x, y, k, and l ) at which households maximize utility and ﬁrms maximize proﬁts.23 23Because ﬁrms’ production functions are characterized by constant returns to scale, in equilibrium each earns zero proﬁts; therefore, there is no need to specify ﬁrm ownership in this model. 488 Part 5: Competitive Markets Because any general equilibrium model can compute only relative prices, we are free to impose a price-normalization scheme. Here we assume that the prices will always sum to unity. That is, Solving24 for these prices yields px þ py þ pk þ pl ¼ 1: px ¼ 0:363, py ¼ 0:253, pk ¼ 0:136, pl ¼ 0:248: (13:60) (13:61) At these prices, total production of x is 23.7 and production of y is 25.1. The utility-maximizing choices for household 1 are x1 ¼ y1 ¼ for household 2, these choices are 15:7, 8:1; l1 ! l1 ¼ 24 ! 14:8 9:2, U 1 ¼ ¼ 13:5; (13:62) 8:1, 11:6; y2 ¼ x2 ¼ l2 ¼ Observe that household 1 consumes quite a bit of good x but provides less in labor supply than does household 2. This reﬂects the greater capital endowment of household 1 in this base-case simulation. We will return to this base case in several later simulations. 5:9, U 2 ¼ l2 ! (13:63) 8:75: 18:1 24 ¼ ! QUERY: How would you show that each household obeys its budget constraint in this simulation? Does the budgetary allocation of each household exhibit the budget shares that are implied by the form of its utility function? EXAMPLE 13.5 The Excess Burden of a Tax In Chapter 12 we showed that taxation may impose an excess burden in addition to the tax revenues collected because of the incentive effects of the tax. With a general equilibrium model we can show much more about this effect. Speciﬁcally, assume that the government in the economy of Example 13.4 imposes an ad valorem tax of 0.4 on good x. This introduces a wedge between what demanders pay for this good x (px) and what suppliers receive for the good (p0x ¼ (1 – t)px ¼ 0.6px). To complete the model we must specify what happens to the revenues generated by this tax. For simplicity we assume that these revenues are rebated to the households in a 50–50 split. In all other respects the economy remains as described in Example 13.4. Solving for the new equilibrium prices in this model yields px = 0.472, py = 0.218, pk = 0.121, pl = 0.188. (13:64) At these prices, total production of x is 17.9, and total production of y is 28.8. Hence the allocation of resources has shifted signiﬁcantly toward y production. Even though the relative px/py ¼ 2.17) has increased signiﬁcantly price of x experienced by consumers ( from its value (of 1.43) in Example 13.4, the price ratio experienced by ﬁrms (0.6px/py ¼ 1.30) has decreased somewhat from this prior value. Therefore, one might expect, based on a partial equilibrium analysis, that consumers would demand less of good x and likewise that ﬁrms would similarly produce less of that good. Partial equilibrium analysis would not, however, allow us to predict the increased production of y (which comes about because the relative price 0.472/0.218 ¼ ¼ 24The computer program used to ﬁnd these solutions is accessible at www.amherst.edu/ CompEquApplet.html. - fwesthoff/compequ/FixedPoints Chapter 13: General Equilibrium and Welfare 489 of y has decreased for consumers but has increased for ﬁrms) nor the reduction in relative input prices (because there is less being produced overall). A more complete picture of all these effects can be obtained by looking at the ﬁnal equilibrium positions of the two households. The posttax allocation for household 1 is for household 2, x1 ¼ 11:6, y1 ¼ 15:2, l1 ! l1 ¼ 11:8, U1 ¼ 12:7; x2 ¼ 6.3, y2 ¼ 13.6, l2 ! l2 ¼ 7.9, U 2 ¼ 8.96. (13:65) (13:66) Hence imposition of the tax has made household 1 considerably worse off: utility decreases from 13.5 to 12.7. Household 2 is made slightly better off by this tax and transfer scheme, primarily because it receives a relatively large share of the tax proceeds that come mainly from household 1. Although total utility has decreased (as predicted by the simple partial equilibrium analysis of excess burden), general equilibrium analysis gives a more complete picture of the distributional consequences of the tax. Notice also that the total amount of labor supplied decreases as a result of the tax: total leisure increases from 15.1 (hours) to 19.7. Therefore, imposition of a tax on good x has had a relatively substantial labor supply effect that is completely invisible in a partial equilibrium model. QUERY: Would it be possible to make both households better off (relative to Example 13.4) in this taxation scenario by changing how the tax revenues are redistributed? SUMMARY This chapter has provided a general exploration of Adam Smith’s conjectures about the efﬁciency properties of competitive markets. We began with a description of how to model many competitive markets simultaneously a
nd then used that model to make a few statements about welfare. Some highlights of this chapter are listed here. • Preferences and production technologies provide the building blocks on which all general equilibrium models are based. One particularly simple version of such a model uses individual preferences for two goods together with a concave production possibility frontier for those two goods. • Competitive markets can establish equilibrium prices by making marginal adjustments in prices in response to information about the demand and supply for individual goods. Walras’ law ties markets together so that such a solution is assured (in most cases). • General equilibrium models can usually be solved by using computer algorithms. The resulting solutions yield many insights about the economy that are not obtainable from partial equilibrium analysis of single markets. • Competitive prices will result in a Pareto-efﬁcient allocation of resources. This is the ﬁrst theorem of welfare economics. • Factors that interfere with competitive markets’ abilities to achieve efﬁciency include (1) market power, (2) externalities, (3) existence of public goods, and (4) imperfect information. We explore all these issues in detail in later chapters. • Competitive markets need not yield equitable distributions of resources, especially when initial endowments are highly skewed. In theory, any desired distribution can be attained through competitive markets accompanied by appropriate transfers of initial endowments (the second theorem of welfare economics). But there are many practical problems in implementing such transfers. 490 Part 5: Competitive Markets PROBLEMS 13.1 Suppose the production possibility frontier for guns (x) and butter (y) is given by x 2 2y 2 þ ¼ 900. a. Graph this frontier. b. If individuals always prefer consumption bundles in which y c. At the point described in part (b), what will be the RPT and hence what price ratio will cause production to take place at 2x, how much x and y will be produced? ¼ that point? (This slope should be approximated by considering small changes in x and y around the optimal point.) d. Show your solution on the ﬁgure from part (a). 13.2 Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (x) or chicken soup (y). Smith’s utility function is given by whereas Jones’ is given by x 0:3y 0:7, U S ¼ U J ¼ The individuals do not care whether they produce x or y, and the production function for each good is given by x0:5y0:5: where l is the total labor devoted to production of each good. a. What must the price ratio, px /py, be? b. Given this price ratio, how much x and y will Smith and Jones demand? Hint: Set the wage equal to 1 here. c. How should labor be allocated between x and y to satisfy the demand calculated in part (b)? 2l and y x ¼ 3l, ¼ 13.3 Consider an economy with just one technique available for the production of each good. Good Food Cloth Labor per unit output Land per unit output 1 2 1 1 a. Suppose land is unlimited but labor equals 100. Write and sketch the production possibility frontier. b. Suppose labor is unlimited but land equals 150. Write and sketch the production possibility frontier. c. Suppose labor equals 100 and land equals 150. Write and sketch the production possibility frontier. Hint: What are the intercepts of the production possibility frontier? When is land fully employed? Labor? Both? d. Explain why the production possibility frontier of part (c) is concave. e. Sketch the relative price of food as a function of its output in part (c). f. If consumers insist on trading 4 units of food for 5 units of cloth, what is the relative price of food? Why? g. Explain why production is exactly the same at a price ratio of pF /pC ¼ h. Suppose that capital is also required for producing food and clothing and that capital requirements per unit of food and per unit of clothing are 0.8 and 0.9, respectively. There are 100 units of capital available. What is the production possibility curve in this case? Answer part (e) for this case. 1.1 as at pF /pC ¼ 1.9. 13.4 Suppose that Robinson Crusoe produces and consumes ﬁsh (F) and coconuts (C). Assume that, during a certain period, he has decided to work 200 hours and is indifferent as to whether he spends this time ﬁshing or gathering coconuts. Robinson’s production for ﬁsh is given by Chapter 13: General Equilibrium and Welfare 491 and for coconuts by F ¼ lF p ﬃﬃﬃﬃ C ¼ lC , where lF and lC are the number of hours spent ﬁshing or gathering coconuts. Consequently, p ﬃﬃﬃﬃ lC þ Robinson Crusoe’s utility for ﬁsh and coconuts is given by 200: lF ¼ a. If Robinson cannot trade with the rest of the world, how will he choose to allocate his labor? What will the optimal levels utility = . Cp F * ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ of F and C be? What will his utility be? What will be the RPT (of ﬁsh for coconuts)? b. Suppose now that trade is opened and Robinson can trade ﬁsh and coconuts at a price ratio of pF /pC ¼ 2/1. If Robinson continues to produce the quantities of F and C from part (a), what will he choose to consume once given the opportunity to trade? What will his new level of utility be? c. How would your answer to part (b) change if Robinson adjusts his production to take advantage of the world prices? d. Graph your results for parts (a), (b), and (c). 13.5 Smith and Jones are stranded on a desert island. Each has in his possession some slices of ham (H) and cheese (C). Smith is a choosy eater and will eat ham and cheese only in the ﬁxed proportions of 2 slices of cheese to 1 slice of ham. His utility function is given by US ¼ slices of ham and 200 slices of cheese. Jones is more ﬂexible in his dietary tastes and has a utility function given by UJ ¼ 3C. Total endowments are 100 min(H, C/2). 4H þ a. Draw the Edgeworth box diagram that represents the possibilities for exchange in this situation. What is the only exchange ratio that can prevail in any equilibrium? b. Suppose Smith initially had 40H and 80C. What would the equilibrium position be? c. Suppose Smith initially had 60H and 80C. What would the equilibrium position be? d. Suppose Smith (much the stronger of the two) decides not to play by the rules of the game. Then what could the ﬁnal equilibrium position be? 13.6 In the country of Ruritania there are two regions, A and B. Two goods (x and y) are produced in both regions. Production functions for region A are given by q here lx and ly are the quantities of labor devoted to x and y production, respectively. Total labor available in region A is 100 units; that is, xA ¼ yA ¼ p lx , ; ﬃﬃﬃﬃ ly ﬃﬃﬃﬃ Using a similar notation for region B, production functions are given by lx + ly ¼ 100. There are also 100 units of labor available in region B: 1 xB ¼ 2 1 p yB ¼ 2 lx , ﬃﬃﬃﬃ ly ﬃﬃﬃﬃ : q lx + ly ¼ 100. 492 Part 5: Competitive Markets a. Calculate the production possibility curves for regions A and B. b. What condition must hold if production in Ruritania is to be allocated efﬁciently between regions A and B (assuming labor cannot move from one region to the other)? c. Calculate the production possibility curve for Ruritania (again assuming labor is immobile between regions). How much total y can Ruritania produce if total x output is 12? Hint: A graphical analysis may be of some help here. 13.7 Use the computer algorithm discussed in footnote 24 to examine the consequences of the following changes to the model in Example 13.4. For each change, describe the ﬁnal results of the modeling and offer some intuition about why the results worked as they did. a. Change the preferences of household 1 to U 1 ¼ b. Reverse the production functions in Equation 13.58 so that x becomes the capital-intensive good. c. Increase the importance of leisure in each household’s utility function. 1 (l1 ! 1 y0:2 x0:6 l1)0:2: Analytical Problems 13.8 Tax equivalence theorem Use the computer algorithm discussed in footnote 24 to show that a uniform ad valorem tax of both goods yields the same equilibrium as does a uniform tax on both inputs that collects the same revenue. Note: This tax equivalence theorem from the theory of public ﬁnance shows that taxation may be done on either the output or input sides of the economy with identical results. 13.9 Returns to scale and the production possibility frontier The purpose of this problem is to examine the relationships among returns to scale, factor intensity, and the shape of the production possibility frontier. Suppose there are ﬁxed supplies of capital and labor to be allocated between the production of good x and good y. The production functions for x and y are given (respectively) by where the parameters a, b, g, and d will take on different values throughout this problem. Using either intuition, a computer, or a formal mathematical approach, derive the production possibility frontier for x and y x ¼ kal b and y kgl d, ¼ in the following cases. a. a b. a c. a d. a e. a f/2, g 1/2, g g d ¼ 0.6, g 0.7, g ¼ ¼ ¼ ¼ ¼ ¼ 1/2. 1/3, d d ¼ 2/3. 0.2, d 0.6, d 2/3. ¼ 2/3. 1.0. 0.8. ¼ ¼ Do increasing returns to scale always lead to a convex production possibility frontier? Explain. 13.10 The trade theorems The construction of the production possibility curve shown in Figures 13.2 and 13.3 can be used to illustrate three important ‘‘theorems’’ in international trade theory. To get started, notice in Figure 13.2 that the efﬁciency line Ox,Oy is bowed above the main diagonal of the Edgeworth box. This shows that the production of good x is always ‘‘capital intensive’’ relative to the x > k production of good y. That is, when production is efﬁcient, y no matter how much of the goods are produced. Demonstration of the trade theorems assumes that the price ratio, p px/py, is determined in international markets—the % & ¼ domestic economy must adjust to this ratio (in trade jargon, the country under examination is assumed to be ‘‘a small country in a large world’’). % & k l l a. Factor price e
qualization theorem: Use Figure 13.4 to show how the international price ratio, p, determines the point in the Edgeworth box at which domestic production will take place. Show how this determines the factor price ratio, w/v. If production functions are the same throughout the world, what will this imply about relative factor prices throughout the world? Chapter 13: General Equilibrium and Welfare 493 b. Stolper–Samuelson theorem: An increase in p will cause the production to move clockwise along the production possibility frontier—x production will increase and y production will decrease. Use the Edgeworth box diagram to show that such a move will decrease k/l in the production of both goods. Explain why this will cause w/v to decrease. What are the implications of this for the opening of trade relations (which typically increases the price of the good produced intensively with a country’s most abundant input). c. Rybczynski theorem: Suppose again that p is set by external markets and does not change. Show that an increase in k will increase the output of x (the capital-intensive good) and reduce the output of y (the labor-intensive good). 13.11 An example of Walras’ law Suppose there are only three goods (x1, x2, x3) in an economy and that the excess demand functions for x2 and x3 are given by ED2 ¼ ! ED3 ¼ ! 3p2 p1 þ 4p2 p1 ! 2p3 p1 ! 2p3 p1 ! 1, 2: a. Show that these functions are homogeneous of degree 0 in p1, p2, and p3. b. Use Walras’ law to show that, if ED2 ¼ 0, then ED1 must also be 0. Can you also use Walras’ law to calculate ED1? c. Solve this system of equations for the equilibrium relative prices p2 /p1 and p3 /p1. What is the equilibrium value for p3 /p2? ED3 ¼ ¼ 13.12 Productive efﬁciency with calculus In Example 13.3 we showed how a Pareto efﬁciency exchange equilibrium can be described as the solution to a constrained maximum problem. In this problem we provide a similar illustration for an economy involving production. Suppose that there is only one person in a two-good economy and that his or her utility function is given by U(x, y). Suppose also that this economy’s production possibility frontier can be written in implicit form as T(x, y) 0. a. What is the constrained optimization problem that this economy will seek to solve if it wishes to make the best use of its available resources? b. What are the ﬁrst-order conditions for a maximum in this situation? c. How would the efﬁcient situation described in part (b) be brought about by a perfectly competitive system in which this individual maximizes utility and the ﬁrms underlying the production possibility frontier maximize proﬁts. d. Under what situations might the ﬁrst-order conditions described in part (b) not yield a utility maximum? 13.13 Initial endowments, equilibrium prices, and the ﬁrst theorem of welfare economics In Example 13.3 we computed an efﬁcient allocation of the available goods and then found the price ratio consistent with this allocation. That then allowed us to ﬁnd initial endowments that would support this equilibrium. In that way the example demonstrates the second theorem of welfare economics. We can use the same approach to illustrate the ﬁrst theorem. Assume again that the utility functions for persons A and B are those given in the example. a. For each individual, show how his or her demand for x and y depends on the relative prices of these two goods and on the 1 and let p represent the price of x (relative initial endowment that each person has. To simplify the notation here, set py ¼ to that of y). Hence the value of, say, A’s initial endowment can be written as pxA þ b. Use the equilibrium conditions that total quantity demanded of goods x and y must equal the total quantities of these two goods available (assumed to be 1,000 units each) to solve for the equilibrium price ratio as a function of the initial endowments of the goods held by each person (remember that initial endowments must also total 1,000 for each good). yA. c. For the case xA ¼ d. Describe in general terms how changes in the initial endowments would affect the resulting equilibrium prices in this 500, compute the resulting market equilibrium and show that it is Pareto efﬁcient. yA ¼ model. Illustrate your conclusions with a few numerical examples. 13.14 Social welfare functions and income taxation The relationship between social welfare functions and the optimal distribution of individual tax burdens is a complex one in welfare economics. In this problem, we look at a few elements of this theory. Throughout we assume that there are m individuals in the economy and that each individual is characterized by a skill level, ai, which indicates his or her ability to earn income. Without loss of generality suppose also that individuals are ordered by increasing ability. Pretax income itself is 494 Part 5: Competitive Markets I(ai, ci). Suppose also that determined by skill level and effort, ci, which may or may not be sensitive to taxation. That is, Ii ¼ 0: Finally, the government wishes to choose a schedule of , w0 > 0, w00 < 0, w the utility cost of effort is given by w 0 ð income taxes and transfers, T(I), which maximizes social welfare subject to a government budget constraint satisfying m c Þ ð Þ ¼ T IiÞ ¼ ð R (where R is the amount needed to ﬁnance public goods). 1 i ¼ P a. Suppose that each individual’s income is unaffected by effort and that each person’s utility is given by ui ¼ ui[Ii – T(Ii) – c(c)]. Show that maximization of a CES social welfare function requires perfect equality of income no matter what the precise form of that function. (Note: for some individuals T(Ii) may be negative.) b. Suppose now that individuals’ incomes are affected by effort. Show that the results of part (a) still hold if the government based income taxation on ai rather than on Ii. c. In general show that if income taxation is based on observed income, this will affect the level of effort individuals under- take. d. Characterization of the optimal tax structure when income is affected by effort is difﬁcult and often counterintuitive. Diamond25 shows that the optimal marginal rate schedule may be U-shaped, with the highest rates for both low- and highincome people. He shows that the optimal top rate marginal rate is given by 1 1 eL,wÞð kiÞ ð þ ! 1 eL,wÞð 1 2eL,w þ ð ! þ where ki(0 1) is the top income person’s relative weight in the social welfare function and eL,w is the elasticity of labor supply with respect to the after-tax wage rate. Try a few simulations of possible values for these two parameters, and describe what the top marginal rate should be. Give an intuitive discussion of these results. ImaxÞ ¼ ki , kiÞ T 0 , ð , SUGGESTIONS FOR FURTHER READING Arrow, K. J., and F. H. Hahn. General Competitive Analysis. Amsterdam: North-Holland, 1978, chaps. 1, 2, and 4. Harberger, A. ‘‘The Incidence of the Corporate Income Tax.’’ Journal of Political Economy (January/February 1962): 215–40. Sophisticated mathematical treatment of general equilibrium analysis. Each chapter has a good literary introduction. Nice use of a two-sector general equilibrium model to examine the ﬁnal burden of a tax on capital. Debreu, G. Theory of Value. New York: John Wiley & Sons, 1959. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford, UK: Oxford University Press, 1995. Basic reference; difﬁcult mathematics. Does have a good introductory chapter on the mathematical tools used. Debreu, G. ‘‘Existence of Competitive Equilibrium.’’ In K. J. Arrow and M. D. Intriligator, Eds., Handbook of Mathematical Economics, vol. 2. Amsterdam: North-Holland, 1982, pp. 697–743. Fairly difﬁcult survey of existence proofs based on ﬁxed point theorems. Contains a comprehensive set of references. Ginsburgh, V., and M. Keyzer. The Structure of Applied General Equilibrium Models. Cambridge, MA: MIT Press, 1997. Detailed discussions of the problems in implementing computable general equilibrium models. Some useful references to the empirical literature. Part Four is devoted to general equilibrium analysis. Chapters 17 (existence) and 18 (connections to game theory) are especially useful. Chapters 19 and 20 pursue several of the topics in the Extensions to this chapter. Salanie, B. Microeconomic Models of Market Failure. Cambridge, MA: MIT Press, 2000. Nice summary of the theorems of welfare economics along with detailed analyses of externalities, public goods, and imperfect competition. Sen, A. K. Collective Choice and Social Welfare. San Francisco: Holden-Day, 1970, chaps. 1 and 2. Basic reference on social choice theory. Early chapters have a good discussion of the meaning and limitations of the Pareto efﬁciency concept. 25P. Diamond ‘‘Optimal income taxation: An example with a U-shaped pattern of optimal marginal tax rates’’ American Economic Review, March 1998, pages 83–93 COMPUTABLE GENERAL EQUILIBRIUM MODELS EXTENSIONS As discussed brieﬂy in Chapter 13, recent improvements in computer technology have made it feasible to develop computable general equilibrium (CGE) models of considerable detail. These may involve literally hundreds of industries and individuals, each with somewhat different technologies or preferences. The general methodology used with these models is to assume various forms for production and utility functions, and then choose particular parameters of those functions based on empirical evidence. Numerical general equilibrium solutions are then generated by the models and compared with real-world data. After ‘‘calibrating’’ the models to reﬂect reality, various policy elements in the models are varied as a way of providing general equilibrium estimates of those policy changes. In this extension we brieﬂy review a few of these types of applications. the overall impact of E13.1 Trade models One of the ﬁrst uses for applied general equilibrium models was to the study of the impact of trade barriers. Because much of the debate over the effects of such barriers (or
of their reduction) focuses on impacts on real wages, such general equilibrium models are especially appropriate for the task. Two unusual features tend to characterize such models. First, because the models often have an explicit focus on domestic versus foreign production of speciﬁc goods, it is necessary to introduce a large degree of product differentiation into individuals’ utility functions. That is, ‘‘U.S. textiles’’ are treated as being different from ‘‘Mexican textiles’’ even though, in most trade theories, textiles might be treated as homogeneous goods. Modelers have found they must allow for only limited substitutability among such goods if their models are to replicate actual trade patterns. A second feature of CGE models of trade is the interest in incorporating increasing returns-to-scale technologies into their production sectors. This permits the models to capture one of the primary advantages of trade for smaller economies. Unfortunately, introduction of the increasing returns-to-scale assumption also requires that the models depart from perfectly competitive, price-taking assumptions. Often some type of markup pricing, together with Cournot-type imperfect competition (see Chapter 15), is used for this purpose. North American free trade Some of the most extensive CGE modeling efforts have been devoted to analyzing the impact of the North American Free Trade Agreement (NAFTA). Virtually all these models ﬁnd that the agreement offered welfare gains to all the countries involved. Gains for Mexico accrued primarily because of reduced U.S. trade barriers on Mexican textiles and steel. Gains to Canada came primarily from an increased ability to beneﬁt from economies of scale in certain key industries. Brown (1992) surveys a number of CGE models of North American free trade and concludes that gains on the order of 2–3 percent of GDP might be experienced by both countries. For the United States, gains from NAFTA might be considerably smaller; but even in this case, signiﬁcant welfare gains were found to be associated with the increased competitiveness of domestic markets. E13.2 Tax and transfer models A second major use of CGE models is to evaluate potential changes in a nation’s tax and transfer policies. For these applications, considerable care must be taken in modeling the factor supply side of the models. For example, at the margin, the effects of rates of income taxation (either positive or negative) can have important labor supply effects that only a general tax/ equilibrium approach can model properly. Similarly, transfer policy can also affect savings and investment decisions, and for these too it may be necessary to adopt more detailed modeling procedures (e.g., differentiating individuals by age to examine effects of retirement programs). The Dutch MIMIC model Probably the most elaborate tax/transfer CGE model is that developed by the Dutch Central Planning Bureau—the Micro Macro Model to Analyze the Institutional Context (MIMIC). This model puts emphasis on social welfare programs and on some of the problems they seek to ameliorate (most notably unemployment, which is missing from many other CGE models). Gelauff and Graaﬂund (1994) summarize the main features of the MIMIC model. They also use it to analyze such policy proposals as the 1990s tax reform in the Netherlands and potential changes to the generous unemployment and disability beneﬁts in that country. 496 Part 5: Competitive Markets E13.3 Environmental models CGE models are also appropriate for understanding the ways in which environmental policies may affect the economy. In such applications, the production of pollutants is considered as a major side effect of the other economic activities in the model. By specifying environmental goals in terms of a given reduction in these pollutants, it is possible to use these models to study the economic costs of various strategies for achieving these goals. One advantage of the CGE approach is to provide some evidence on the impact of environmental policies on income distribution—a topic largely omitted from more narrow, industry-based modeling efforts. Assessing CO2 reduction strategies Concern over the possibility that CO2 emissions in various energy-using activities may be contributing to global warming has led to a number of plans for reducing these emissions. Because the repercussions of such reductions may be widespread and varied, CGE modeling is one of the preferred assessment methods. Perhaps the most elaborate such model is that developed by the Organisation for Economic Co-operation and Development (OECD)—the General Equilibrium Environmental (GREEN) model. The basic structure of this model is described by Burniaux, Nicoletti, and Oliviera-Martins (1992). The model has been used to simulate various policy options that might be adopted by European nations to reduce CO2 emissions, such as institution of a carbon tax or increasingly stringent emissions regulations for automobiles and power plants. In general, these simulations suggest that economic costs of these policies would be relatively modest given the level of restrictions currently anticipated. But most of the policies would have adverse distributional effects that may require further attention through government transfer policy. E13.4 Regional and urban models A ﬁnal way in which CGE models can be used is to examine economic issues that have important spatial dimensions. Construction of such models requires careful attention to issues of transportation costs for goods and moving costs associated with labor mobility because particular interest is focused on where transactions occur. Incorporation of these costs into CGE models is in many ways equivalent to adding extra levels of product differentiation because these affect the relative prices of otherwise homogeneous goods. Calculation of equilibria in regional markets can be especially sensitive to how transport costs are speciﬁed. Changing government procurement CGE regional models have been widely used to examine the local impact of major changes in government spending policies. For example, Hoffmann, Robinson, and Subramanian (1996) use a CGE model to evaluate the regional impact of reduced defense expenditures on the California economy. They ﬁnd that the size of the effects depends importantly on the assumed costs of migration for skilled workers. A similar ﬁnding is reported by Bernat and Hanson (1995), who examine possible reductions in U.S. price-support payments to farms. Although such reductions would offer overall efﬁciency gains to the economy, they could have signiﬁcant negative impacts on rural areas. References Bernat, G. A., and K. Hanson. ‘‘Regional Impacts of Farm Programs: A Top-Down CGE Analysis.’’ Review of Regional Studies (Winter 1995): 331–50. Brown, D. K. ‘‘The Impact of North American Free Trade Area: Applied General Equilibrium Models.’’ In N. Lus-tig, B. P. Bosworth, and R. Z. Lawrence, Eds., North American Free Trade: Assessing the Impact. Washington, DC: Brookings Institution, 1992, pp. 26–68. Burniaux, J. M., G. Nicoletti, and J. Oliviera-Martins. ‘‘GREEN: A Global Model for Quantifying the Costs of Policies to Curb CO2 Emissions.’’ OECD Economic Studies (Winter 1992): 49–92. Gelauff, G. M. M., and J. J. Graaﬂund. Modeling Welfare State Reform. Amsterdam: North Holland, 1994. Hoffmann, S., S. Robinson, and S. Subramanian. ‘‘The Role of Defense Cuts in the California Recession: Computable General Equilibrium Models and Interstate Fair Mobility.’’ Journal of Regional Science (November 1996): 571–95. This page intentionally left blank Market Power P A R T SIX Chapter 14 Monopoly Chapter 15 Imperfect Competition In this part we examine the consequences of relaxing the assumption that ﬁrms are price-takers. When ﬁrms have some power to set prices, they will no longer treat them as ﬁxed parameters in their decisions but will instead treat price setting as one part of the proﬁt-maximization process. Usually this will mean prices no longer accurately reﬂect marginal costs and the efﬁciency theorems that apply to competitive markets no longer hold. Chapter 14 looks at the relatively simple case where there is only a single monopoly supplier of a good. This supplier can choose to operate at any point on the demand curve for its product that it ﬁnds most proﬁtable. Its activities are constrained only by this demand curve, not by the behavior of rival producers. As we shall see, this offers the ﬁrm a number of avenues for increasing proﬁts, such as using novel pricing schemes or adapting the characteristics of its product. Although such decisions will indeed provide more proﬁts for the monopoly, in general they will also result in welfare losses for consumers (relative to perfect competition). In Chapter 15 we consider markets with few producers. Models of such markets are considerably more complicated than are markets of monopoly (or of perfect competition, for that matter) because the demand curve faced by any one ﬁrm will depend in an important way on what its rivals choose to do. Studying the possibilities will usually require game-theoretic ideas to capture accurately the strategic possibilities involved. Hence you should review the basic game theory material in Chapter 8 before plunging into Chapter 15, whose general conclusion is that outcomes in markets with few ﬁrms will depend crucially on the details of how the ‘‘game’’ is played. In many cases the same sort of inefﬁciencies that occur in monopoly markets appear in imperfectly competitive markets as well. 499 This page intentionally left blank C H A P T E R FOURTEEN Monopoly A monopoly is a single ﬁrm that serves an entire market. This single ﬁrm faces the market demand curve for its output. Using its knowledge of this demand curve, the monopoly makes a decision on how much to produce. Unlike the perfectly competitive ﬁrm’s output decision (which has no effect on market price), the monopoly’s o
utput decision will, in fact, determine the good’s price. In this sense, monopoly markets and markets characterized by perfect competition are polar-opposite cases Monopoly. A monopoly is a single supplier to a market. This ﬁrm may choose to produce at any point on the market demand curve. At times it is more convenient to treat monopolies as having the power to set prices. Technically, a monopoly can choose that point on the market demand curve at which it prefers to operate. It may choose either market price or quantity, but not both. In this chapter we will usually assume that monopolies choose the quantity of output that maximizes proﬁts and then settle for the market price that the chosen output level yields. It would be a simple matter to rephrase the discussion in terms of price setting, and in some places we shall do so. Barriers to Entry The reason a monopoly exists is that other ﬁrms ﬁnd it unproﬁtable or impossible to enter the market. Therefore, barriers to entry are the source of all monopoly power. If other ﬁrms could enter a market, then the ﬁrm would, by deﬁnition, no longer be a monopoly. There are two general types of barriers to entry: technical barriers and legal barriers. Technical barriers to entry A primary technical barrier is that the production of the good in question may exhibit decreasing marginal (and average) costs over a wide range of output levels. The technology of production is such that relatively large-scale ﬁrms are low-cost producers. In this situation (which is sometimes referred to as natural monopoly), one ﬁrm may ﬁnd it proﬁtable to drive others out of the industry by cutting prices. Similarly, once a monopoly has been established, entry will be difﬁcult because any new ﬁrm must produce at relatively low levels of output and therefore at relatively high average costs. It is important to stress that the range of declining costs need only be ‘‘large’’ relative to the market in questhe tion. Declining costs on some absolute scale are not necessary. For example, 501 502 Part 6: Market Power production and delivery of concrete does not exhibit declining marginal costs over a broad range of output when compared with the total U.S. market. However, in any particular small town, declining marginal costs may permit a monopoly to be established. The high costs of transportation in this industry tend to isolate one market from another. Another technical basis of monopoly is special knowledge of a low-cost productive technique. The monopoly has an incentive to keep its technology secret; but unless this technology is protected by a patent (see next paragraph), this may be extremely difﬁcult. Ownership of unique resources—such as mineral deposits or land locations, or the possession of unique managerial talents—may also be a lasting basis for maintaining a monopoly. Legal barriers to entry Many pure monopolies are created as a matter of law rather than as a matter of economic conditions. One important example of a government-granted monopoly position is the legal protection of a product by a patent or copyright. Prescription drugs, computer chips, and Disney animated movies are examples of proﬁtable products that are shielded (for a time) from direct competition by potential imitators. Because the basic technology for these products is uniquely assigned to one ﬁrm, a monopoly position is established. The defense made of such a governmentally granted monopoly is that the patent and copyright system makes innovation more proﬁtable and therefore acts as an incentive. Whether the beneﬁts of such innovative behavior exceed the costs of having monopolies is an open question that has been much debated by economists. A second example of a legally created monopoly is the awarding of an exclusive franchise to serve a market. These franchises are awarded in cases of public utility (gas and electric) service, communications services, the post ofﬁce, some television and radio station markets, and a variety of other situations. Usually the restriction of entry is combined with a regulatory cap on the price the franchised monopolist is allowed to charge. The argument usually put forward in favor of creating these franchised monopolies is that the industry in question is a natural monopoly: average cost is diminishing over a broad range of output levels, and minimum average cost can be achieved only by organizing the industry as a monopoly. The public utility and communications industries are often considered good examples. Certainly, that does appear to be the case for local electricity and telephone service where a given network probably exhibits declining average cost up to the point of universal coverage. But recent deregulation in telephone services and electricity generation show that, even for these industries, the natural monopoly rationale may not be all-inclusive. In other cases, franchises may be based largely on political rationales. This seems to be true for the postal service in the United States and for a number of nationalized industries (airlines, radio and television, banking) in other countries. Creation of barriers to entry Although some barriers to entry may be independent of the monopolist’s own activities, other barriers may result directly from those activities. For example, ﬁrms may develop unique products or technologies and take extraordinary steps to keep these from being copied by competitors. Or ﬁrms may buy up unique resources to prevent potential entry. The De Beers cartel, for example, controls a large fraction of the world’s diamond mines. Finally, a would-be monopolist may enlist government aid in devising barriers to entry. It may lobby for legislation that restricts new entrants to ‘‘maintain an orderly market’’ or for health and safety regulations that raise potential entrants’ costs. Because the monopolist has both special knowledge of its business and signiﬁcant incentives to pursue these goals, it may have considerable success in creating such barriers to entry. FIGURE 14.1 Proﬁt Maximization and Price Determination for a Monopoly Chapter 14: Monopoly 503 The attempt by a monopolist to erect barriers to entry may involve real resource costs. Maintaining secrecy, buying unique resources, and engaging in political lobbying are all costly activities. A full analysis of monopoly should involve not only questions of cost minimization and output choice (as under perfect competition) but also an analysis of proﬁt-maximizing creation of entry barriers. However, we will not provide a detailed investigation of such questions here.1 Instead, we will take the presence of a single supplier on the market, and this single ﬁrm’s cost function, as given. Proﬁt Maximization and Output Choice To maximize proﬁts, a monopoly will choose to produce that output level for which marginal revenue is equal to marginal cost. Because the monopoly, in contrast to a perfectly competitive ﬁrm, faces a negatively sloped market demand curve, marginal revenue will be less than the market price. To sell an additional unit, the monopoly must lower its price on all units to be sold if it is to generate the extra demand necessary to absorb this marginal unit. The proﬁt-maximizing output level for a ﬁrm is then the level Q! in Figure 14.1. At that level, marginal revenue is equal to marginal costs, and proﬁts are maximized. A proﬁt-maximizing monopolist produces that quantity for which marginal revenue is equal to marginal cost. In the diagram this quantity is given by Q!, which will yield a price of P! in the market. Monopoly proﬁts can be read as the rectangle of P!EAC. Price, costs D P* C MC E A AC D MR Q* Output per period 1For a simple treatment, see R. A. Posner, ‘‘The Social Costs of Monopoly and Regulation,’’ Journal of Political Economy 83 (August 1975): 807–27. 504 Part 6: Market Power Given the monopoly’s decision to produce Q!, the demand curve D indicates that a market price of P! will prevail. This is the price that demanders as a group are willing to pay for the output of the monopoly. In the market, an equilibrium price–quantity combination of P!, Q! will be observed. Assuming P! > AC, this output level will be proﬁtable, and the monopolist will have no incentive to alter output levels unless demand or cost conditions change. Hence we have the following principle Monopolist’s output. A monopolist will choose to produce that output for which marginal revenue equals marginal cost. Because the monopolist faces a downward-sloping demand curve, market price will exceed marginal revenue and the ﬁrm’s marginal cost at this output level. The inverse elasticity rule, again In Chapter 11 we showed that the assumption of proﬁt maximization implies that the gap between a price of a ﬁrm’s output and its marginal cost is inversely related to the price elasticity of the demand curve faced by the ﬁrm. Applying Equation 11.14 to the case of monopoly yields P MC " P 1 eQ, P , ¼ " (14:1) where now we use the elasticity of demand for the entire market (eQ, P) because the monopoly is the sole supplier of the good in question. This observation leads to two general conclusions about monopoly pricing. First, a monopoly will choose to operate only in regions in which the market demand curve is elastic (eQ, P < 1). If demand were inelastic, then marginal revenue would be negative and thus could not be equated to marginal cost (which presumably is positive). Equation 14.1 also shows that eQ, P > 1 implies an (implausible) negative marginal cost. " " ¼ A second implication of Equation 14.1 is that the ﬁrm’s ‘‘markup’’ over marginal cost (measured as a fraction of price) depends inversely on the elasticity of market demand. For example, if eQ, P ¼ " 2MC, whereas if eQ, P ¼ 2, then Equation 14.1 shows that P 1.11MC. Notice also that if the elasticity of demand were constant along 10, then P " ¼ the entire demand curve, the proportional markup over marginal cost would remain unchanged in response to changes in
input costs. Therefore, market price moves proportionally to marginal cost: Increases in marginal cost will prompt the monopoly to increase its price proportionally, and decreases in marginal cost will cause the monopoly to reduce its price proportionally. Even if elasticity is not constant along the demand curve, it seems clear from Figure 14.1 that increases in marginal cost will increase price (although not necessarily in the same proportion). As long as the demand curve facing the monopoly is downward sloping, upward shifts in MC will prompt the monopoly to reduce output and thereby obtain a higher price.2 We will examine all these relationships mathematically in Examples 14.1 and 14.2. Monopoly proﬁts Total proﬁts earned by the monopolist can be read directly from Figure 14.1. These are shown by the rectangle P!EAC and again represent the proﬁt per unit (price minus average cost) times the number of units sold. These proﬁts will be positive if market price exceeds average total cost. If P! < AC, however, then the monopolist can operate only at a long-term loss and will decline to serve the market. 2The comparative statics of a shift in the demand curve facing the monopolist are not so clear, however, and no unequivocal prediction about price can be made. For an analysis of this issue, see the discussion that follows and Problem 14.4. Chapter 14: Monopoly 505 Because (by assumption) no entry is possible into a monopoly market, the monopolist’s positive proﬁts can exist even in the long run. For this reason, some authors refer to the proﬁts that a monopoly earns in the long run as monopoly rents. These proﬁts can be regarded as a return to that factor that forms the basis of the monopoly (e.g., a patent, a favorable location, or a dynamic entrepreneur); hence another possible owner might be willing to pay that amount in rent for the right to the monopoly. The potential for proﬁts is the reason why some ﬁrms pay other ﬁrms for the right to use a patent and why concessioners at sporting events (and on some highways) are willing to pay for the right to the concession. To the extent that monopoly rights are given away at less than their true market value (as in radio and television licensing), the wealth of the recipients of those rights is increased. Although a monopoly may earn positive proﬁts in the long run,3 the size of such proﬁts will depend on the relationship between the monopolist’s average costs and the demand for its product. Figure 14.2 illustrates two situations in which the demand, marginal revenue, and marginal cost curves are rather similar. As Equation 14.1 suggests, the price-marginal cost markup is about the same in these two cases. But average costs in Figure 14.2a are considerably lower than in Figure 14.2b. Although the proﬁt-maximizing decisions are similar in the two cases, the level of proﬁts ends up being different. In Figure 14.2a the monopolist’s price (P!) exceeds the average cost of producing Q! (labeled AC!) by a large extent, and sigAC! and the monopoly earns niﬁcant proﬁts are obtained. In Figure 14.2b, however, P! zero economic proﬁts, the largest amount possible in this case. Hence large proﬁts from a monopoly are not inevitable, and the actual extent of economic proﬁts may not always be a good guide to the signiﬁcance of monopolistic inﬂuences in a market. ¼ FIGURE 14.2 Monopoly Proﬁts Depend on the Relationship between the Demand and Average Cost Curves Both monopolies in this ﬁgure are equally ‘‘strong’’ if by this we mean they produce similar divergences between market price and marginal cost. However, because of the location of the demand and average cost curves, it turns out that the monopoly in (a) earns high proﬁts, whereas that in (b) earns no proﬁts. Consequently, the size of proﬁts is not a measure of the strength of a monopoly. Price D P* C* Price D P* = AC* D MC AC MR MC AC D MR Q* Quantity per period Q* Quantity per period (a) Monopoly with large proﬁts (b) Zero-proﬁt monopoly 3As in the competitive case, the proﬁt-maximizing monopolist would be willing to produce at a loss in the short run as long as market price exceeds average variable cost. 506 Part 6: Market Power There is no monopoly supply curve In the theory of perfectly competitive markets presented in Part 4, it was possible to speak of an industry supply curve. We constructed the long-run supply curve by allowing the market demand curve to shift and observing the supply curve that was traced out by the series of equilibrium price–quantity combinations. This type of construction is not possible for monopolistic markets. With a ﬁxed market demand curve, the supply ‘‘curve’’ for a monopoly will be only one point—namely, that price–quantity combination for which MR MC. If the demand curve should shift, then the marginal revenue curve would also shift, and a new proﬁt-maximizing output would be chosen. However, connecting the resulting series of equilibrium points on the market demand curves would have little meaning. This locus might have a strange shape, depending on how the market demand curve’s elasticity (and its associated MR curve) changes as the curve is shifted. In this sense the monopoly ﬁrm has no well-deﬁned ‘‘supply curve.’’ Each demand curve is a unique proﬁt-maximizing opportunity for a monopolist. ¼ EXAMPLE 14.1 Calculating Monopoly Output Suppose the market for Olympic-quality Frisbees (Q, measured in Frisbees bought per year) has a linear demand curve of the form or 2,000 Q ¼ " 20P 100 P ¼ " Q 20 , and let the costs of a monopoly Frisbee producer be given by C Q ð Þ ¼ 0:05Q2 10,000: þ (14:2) (14:3) (14:4) To maximize proﬁts, this producer chooses that output level for which MR lem we must phrase both MR and MC as functions of Q alone. Toward this end, write total revenue as MC. To solve this prob- ¼ Consequently, and Q P ’ ¼ 100Q Q2 20 : " MR 100 ¼ Q 10 ¼ " MC ¼ 0:1Q Q! ¼ 500, P! 75: ¼ At the monopoly’s preferred output level, Þ ¼ C Q ð AC 2 500 Þ ð 0:05 22,500 þ 45: ¼ 500 ¼ 10,000 ¼ 22,500, Using this information, we can calculate proﬁts as p P! ¼ ð AC Þ ’ " Q! 75 ¼ ð 45 Þ ’ " 500 ¼ 15,000: (14:5) (14:6) (14:7) (14:8) (14:9) Observe that at this equilibrium there is a large markup between price (75) and marginal cost 50). Yet as long as entry barriers prevent a new ﬁrm from producing Olympic(MC quality Frisbees, this gap and positive economic proﬁts can persist indeﬁnitely. 0.1Q ¼ ¼ QUERY: How would an increase in ﬁxed costs from 10,000 to 12,500 affect the monopoly’s output plans? How would proﬁts be affected? Suppose total costs shifted to C(Q) 10,000. How would the equilibrium change? 0.075Q2 ¼ þ Chapter 14: Monopoly 507 EXAMPLE 14.2 Monopoly with Simple Demand Curves We can derive a few simple facts about monopoly pricing in cases where the demand curve facing the monopoly takes a simple algebraic form and the ﬁrm has constant marginal costs (i.e., C(Q) cQ and MC c). ¼ ¼ Linear demand. Suppose that the inverse demand function facing the monopoly is of the linear form P aQ a 2bQ. Hence proﬁt maximization requires that bQ. In this case, PQ bQ2 and MR dPQ/dQ " " ¼ " ¼ ¼ ¼ a MR a " ¼ 2bQ ¼ MC ¼ c or Q c : a " 2b ¼ (14:10) Inserting this solution for the proﬁt-maximizing output level back into the inverse demand function yields a direct relationship between price and marginal cost: c " 2 ¼ An interesting implication is that, in this linear case, dP/dc of any increase in marginal cost will show up in the market price of the monopoly product.4 þ 2 1/2. That is, only half of the amount (14:11) bQ : Constant elasticity demand. If the demand curve facing the monopoly takes the constant elasticity form Q 1/e), and thus proﬁt maximization requires aP e (where e is the price elasticity of demand), then we know MR P( or " Because it must be the case that e < 1 for proﬁt maximization, price will clearly exceed marginal cost, and this gap will be larger the closer e is to e) and so any given increase in marginal cost will increase price by more than this amount. Of course, as we have already pointed out, the proportional increase in marginal cost and price will be the same. That is, eP, c ¼ QUERY: The demand function in both cases is shifted by the parameter a. Discuss the effects of such a shift for both linear and constant elasticity demand. Explain your results intuitively. 1. Notice also that dP/dc dP=dc e/(1 c=P ¼ ¼ " þ 1: ’ (14:12) Monopoly and Resource Allocation In Chapter 13 we brieﬂy mentioned why the presence of monopoly distorts the allocation MR < P, of resources. Because the monopoly produces a level of output for which MC the market price of its good no longer conveys accurate information about production costs. Hence consumers’ decisions will no longer reﬂect true opportunity costs of production, and resources will be misallocated. In this section we explore this misallocation in some detail in a partial equilibrium context. ¼ Basis of comparison To evaluate the allocational effect of a monopoly, we need a precisely deﬁned basis of comparison. A particularly useful comparison is provided by a perfectly competitive industry. It is convenient to think of a monopoly as arising from the ‘‘capture’’ of such a competitive industry and to treat the individual ﬁrms that constituted the competitive industry as now 4Notice that when c curve. ¼ 0, we have P ¼ a/2. That is, price should be halfway between zero and the price intercept of the demand 508 Part 6: Market Power being single plants in the monopolist’s empire. A prototype case would be John D. Rockefeller’s purchase of most of the U.S. petroleum reﬁneries in the late nineteenth century and his decision to operate them as part of the Standard Oil trust. We can then compare the performance of this monopoly with the performance of the previously competitive industry to arrive at a statement about the welfare consequences of monopoly. A graphical analysis Figure 14.3 provides a graphical analysis of the welfare effects of monopoly. If thi
s market were competitive, output would be Qc—that is, production would occur where price is equal to long-run average and marginal cost. Under a simple single-price monopoly, output would be Qm because this is the level of production for which marginal revenue is equal to marginal cost. The restriction in output from Qc to Qm represents the misallocation brought about through monopolization. The total value of resources released by this output restriction is shown in Figure 14.3 as area FEQcQm. Essentially, the monopoly closes down some of the plants that were operating in the competitive case. These transferred inputs can be productively used elsewhere; thus, area FEQcQm is not a social loss. FIGURE 14.3 Allocational and Distributional Effects of Monopoly Monopolization of this previously competitive market would cause output to be reduced from Qc to Qm. Productive inputs worth FEQcQm are reallocated to the production of other goods. Consumer surplus equal to PmBCPc is transferred into monopoly proﬁts. Deadweight loss is given by BEF. Price A Pm Pc G MC Transfer from consumers to firm B E Deadweight loss C F D Value of transferred inputs MR Qm Qc Quantity per period Chapter 14: Monopoly 509 The restriction in output from Qc to Qm involves a total loss in consumer surplus of PmBEPc. Part of this loss, PmBCPc , is transferred to the monopoly as increased proﬁt. Another part of the consumers’ loss, BEC, is not transferred to anyone but is a pure deadweight loss in the market. A second source of deadweight loss is given by area CEF. This is an area of lost producer surplus that does not get transferred to another source.5 The total deadweight loss from both sources is area BEF, sometimes called the deadweight loss triangle because of its roughly triangular shape. The gain PmBCPc in monopoly proﬁt from an increased price more than compensates for its loss of producer surplus CEF from the output reduction so that, overall, the monopolist ﬁnds reducing output from Qc to Qm to be proﬁtable. To illustrate the nature of this deadweight loss, consider Example 14.1, in which we calculated an equilibrium price of $75 and a marginal cost of $50. This gap between price and marginal cost is an indication of the efﬁciency-improving trades that are forgone through monopolization. Undoubtedly, there is a would-be buyer who is willing to pay, say, $60 for an Olympic Frisbee but not $75. A price of $60 would more than cover all the resource costs involved in Frisbee production, but the presence of the monopoly prevents such a mutually beneﬁcial transaction between Frisbee users and the providers of Frisbee-making resources. For this reason, the monopoly equilibrium is not Pareto optimal—an alternative allocation of resources would make all parties better off. Economists have made many attempts to estimate the overall cost of these deadweight losses in actual monopoly situations. Most of these estimates are rather small when viewed in the context of the whole economy.6 Allocational losses are larger, however, for some narrowly deﬁned industries. EXAMPLE 14.3 Welfare Losses and Elasticity The allocational effects of monopoly can be characterized fairly completely in the case of constant marginal costs and a constant price elasticity demand curve. To do so, assume again that constant marginal (and average) costs for a monopolist are given by c and that the demand curve has a constant elasticity form of ¼ where e is the price elasticity of demand (e < market will be Q P e, (14:13) 1). We know the competitive price in this " and the monopoly price is given by Pc ¼ c Pm ¼ 1 c 1=e þ : The consumer surplus associated with any price (P0) can be computed as (14:14) (14:15) CS ¼ ¼ ¼ 1 P0 ð 1 P Q ð dP Þ P edP P0 ð P e e 1 1 þ 1 P0 $ 14:16) 5More precisely, region CEF represents lost producer surplus (equivalently, lost proﬁt) if output were reduced holding prices constant at Pc. To understand how to measure producer surplus on a graph, review the section on producer surplus in Chapter 11, especially Figure 11.4. 6 The classic study is A. Harberger, Harberger estimates that such losses constitute about 0.1 percent of gross national product. ‘‘Monopoly and Resource Allocation,’’ American Economic Review (May 1954): 77–87. 510 Part 6: Market Power Hence under perfect competition we have and, under monopoly, CSc ! CSm ¼ " c 1= Taking the ratio of these two surplus measures yields CSm CSc ¼ 1 1=14:17) (14:18) (14:19) ¼ " If e 2, for example, then this ratio is 1/2: consumer surplus under monopoly is half what it is under perfect competition. For more elastic cases this ﬁgure decreases a bit (because output restrictions under monopoly are more signiﬁcant). For elasticities closer to 1, the ratio increases. " Proﬁts. The transfer from consumer surplus into monopoly proﬁts can also be computed fairly easily in this case. Monopoly proﬁts are given by cQm ¼ pm ¼ ¼ PmQm " c=e 1===e " e c Qm " c 1=e 1 e þ " 1 e : ’ Dividing this expression by Equation 14.17 yields pm CSc =14:20) (14:21) ¼ " For e competition is transferred into monopoly proﬁts. Therefore, monopoly in this case is also a fourth of competition. 2 this ratio is 1/4. Hence one fourth of the consumer surplus enjoyed under perfect loss from the level of consumer surplus under perfect the deadweight QUERY: Suppose e How much is transferred into monopoly proﬁts? Why do these results differ from the case e 1.5. What fraction of consumer surplus is lost through monopolization? 2? ¼ " ¼ " Monopoly, Product Quality, and Durability The market power enjoyed by a monopoly may be exercised along dimensions other than the market price of its product. If the monopoly has some leeway in the type, quality, or diversity of the goods it produces, then it would not be surprising for the ﬁrm’s decisions to differ from those that might prevail under a competitive organization of the market. Whether a monopoly will produce higher-quality or lower-quality goods than would be produced under competition is unclear, however. It all depends on the ﬁrm’s costs and the nature of consumer demand. Chapter 14: Monopoly 511 A formal treatment of quality Suppose consumers’ willingness to pay for quality (X) is given by the inverse demand function P(Q, X), where @P @Q < 0 and @P @X > 0: If the costs of producing Q and X are given by C(Q, X), the monopoly will choose Q and X to maximize " The ﬁrst-order conditions for a maximum are ¼ p P Q, X ð Q Þ C Q, X ð Þ : @p @Q ¼ Q, X P ð Þ þ Q @P @Q " CQ ¼ 0, @p @X ¼ Q @P @X " CX ¼ 0: (14:22) (14:23) (14:24) The ﬁrst of these equations repeats the usual rule that marginal revenue equals marginal cost for output decisions. The second equation states that, when Q is appropriately set, the monopoly should choose that level of quality for which the marginal revenue attainable from increasing the quality of its output by one unit is equal to the marginal cost of making such an increase. As might have been expected, the assumption of proﬁt maximization requires the monopolist to proceed to the margin of proﬁtability along all the dimensions it can. Notice, in particular, that the marginal demander’s valuation of quality per unit is multiplied by the monopolist’s output level when determining the proﬁt-maximizing choice. The level of product quality chosen under competitive conditions will also be the one that maximizes net social welfare: SW Q! ¼ 0 ð P Q, X ð Þ dQ C ð " Q, X , Þ (14:25) where Q! is the output level determined through the competitive process of marginal cost pricing, given X. Differentiation of Equation 14.25 with respect to X yields the ﬁrst-order condition for a maximum: @SW @X ¼ Q! 0 ð Q, X PXð Þ dQ CX ¼ " 0: (14:26) The monopolist’s choice of quality in Equation 14.24 targets the marginal consumer. The monopolist cares about the marginal consumer’s valuation of quality because increasing the attractiveness of the product to the marginal consumer is how it increases sales. The perfectly competitive market ends up providing a quality level in Equation 14.26, maximizing total consumer surplus (the total after subtracting the cost of providing that quality level), which is the same as the quality level that maximizes consumer surplus for the average consumer.7 Therefore, even if a monopoly and a perfectly competitive industry choose the same output level, they might opt for differing quality levels because each is 7The average marginal valuation (AV) of product quality is given by AV Q! ¼ 0 ð Q, X PX ð Þ dQ=Q: Hence Q 14.24. ’ AV ¼ Cx is the quality rule adopted to maximize net welfare under perfect competition. Compare this with Equation 512 Part 6: Market Power concerned with a different margin in its decision making. Only by knowing the speciﬁcs of the problem is it possible to predict the direction of these differences. For an example, see Problem 14.9; more detail on the theory of product quality and monopoly is provided in Problem 14.11. The durability of goods Much of the research on the effect of monopolization on quality has focused on durable goods. These are goods such as automobiles, houses, or refrigerators that provide services to their owners over several periods rather than being completely consumed soon after they are bought. The element of time that enters into the theory of durable goods leads to many interesting problems and paradoxes. Initial interest in the topic started with the question of whether monopolies would produce goods that lasted as long as similar goods produced under perfect competition. The intuitive notion that monopolies would ‘‘underproduce’’ durability (just as they choose an output below the competitive level) was soon shown to be incorrect by the Australian economist Peter Swan in the early 1970s.8 Swan’s insight was to view the demand for durable goods as the demand for a ﬂow of services (i.e., automobile transportation) over several periods. He argued that both a monopoly and a competitive market would seek to minimize the cost o
f providing this ﬂow to consumers. The monopoly would, of course, choose an output level that restricted the ﬂow of services to maximize proﬁts, but—assuming constant returns to scale in production—there is no reason that durability per se would be affected by market structure. This result is sometimes referred to as Swan’s independence assumption. Output decisions can be treated independently from decisions about product durability. Subsequent research on the Swan result has focused on showing how it can be undermined by different assumptions about the nature of a particular durable good or by relaxing the implicit assumption that all demanders are the same. For example, the result depends critically on how durable goods deteriorate. The simplest type of deterioration is illustrated by a durable good, such as a lightbulb, that provides a constant stream of services until it becomes worthless. With this type of good, Equations 14.24 and 14.26 are identical, so Swan’s independence result holds. Even when goods deteriorate smoothly, the independence result continues to hold if a constant ﬂow of services can be maintained by simply replacing what has been used—this requires that new goods and old goods be perfect substitutes and inﬁnitely divisible. Outdoor house paint may, more or less, meet this requirement. On the other hand, most goods clearly do not. It is just not possible to replace a run-down refrigerator with, say, half of a new one. Once such more complex forms of deterioration are considered, Swan’s result may not hold because we can no longer fall back on the notion of providing a given ﬂow of services at minimal cost over time. In these more complex cases, however, it is not always the case that a monopoly will produce less durability than will a competitive market—it all depends on the nature of the demand for durability. Time inconsistency and heterogeneous demand Focusing on the service ﬂow from durable goods provides important insights on durability, but it does leave an important question unanswered—when should the monopoly produce the actual durable goods needed to provide the desired service ﬂow? Suppose, for example, that a lightbulb monopoly decides that its proﬁt-maximizing output decision is to supply the services provided by 1 million 60-watt bulbs. If the ﬁrm decides to 8P. L. Swan, ‘‘Durability of Consumption Goods,’’ American Economic Review (December 1970): 884–94. Chapter 14: Monopoly 513 produce 1 million bulbs in the ﬁrst period, what is it to do in the second period (say, before any of the original bulbs burn out)? Because the monopoly chooses a point on the service demand curve where P > MC, it has a clear incentive to produce more bulbs in the second period by cutting price a bit. But consumers can anticipate this, so they may reduce their ﬁrst-period demand, waiting for a bargain. Hence the monopoly’s proﬁtmaximizing plan will unravel. Ronald Coase was the ﬁrst economist to note this ‘‘time inconsistency’’ that arises when a monopoly produces a durable good.9 Coase argued that its presence would severely undercut potential monopoly power—in the limit, competitive pricing is the only outcome that can prevail in the durable goods case. Only if the monopoly can succeed in making a credible commitment not to produce more in the second period can it succeed in its plan to achieve monopoly proﬁts on the service ﬂow from durable goods. Recent modeling of the durable goods question has examined how a monopolist’s choices are affected in situations where there are different types of demanders.10 In such cases, questions about the optimal choice of durability and about credible commitments become even more complicated. The monopolist must not only settle on an optimal scheme for each category of buyers, it must also ensure that the scheme intended for (say) type-1 demanders is not also attractive to type-2 demanders. Studying these sorts of models would take us too far aﬁeld, but some illustrations of how such ‘‘incentive compatibility constraints’’ work are provided in the Extensions to this chapter and in Chapter 18. Price Discrimination In some circumstances a monopoly may be able to increase proﬁts by departing from a single-price policy for its output. The possibility of selling identical goods at different prices is called price discrimination.11 Price discrimination. A monopoly engages in price discrimination if it is able to sell otherwise identical units of output at different prices. Examples of price discrimination include senior citizen discounts for restaurant meals (which could instead be viewed as a price premium for younger customers), coffee sold at a lower price per ounce when bought in larger cup sizes, and different (net) tuition charged to different college students after subtracting their more or less generous ﬁnancial aid awards. A ‘‘nonexample’’ of price discrimination might be higher auto insurance premiums charged to younger drivers. It might be clearer to think of the insurance policies sold to younger and older drivers as being different products to the extent that younger drivers are riskier and result in many more claims having to be paid. Whether a price discrimination strategy is feasible depends crucially on the inability of buyers of the good to practice arbitrage. In the absence of transactions or information costs, the ‘‘law of one price’’ implies that a homogeneous good must sell everywhere for the same price. Consequently, price discrimination schemes are doomed to failure because demanders who can buy from the monopoly at lower prices will be more attractive sources 9R. Coase, ‘‘Durability and Monopoly,’’ Journal of Law and Economics (April 1972): 143–49. 10For a summary, see M. Waldman, ‘‘Durable Goods Theory for Real World Markets,’’ Journal of Economic Perspectives (Winter 2003): 131–54. 11A monopoly may also be able to sell differentiated products at differential price–cost margins. Here, however, we treat price discrimination only for a monopoly that produces a single homogeneous product. Price discrimination is an issue in other imperfectly competitive markets besides monopoly but is easiest to study in the simple case of a single ﬁrm. 514 Part 6: Market Power of the good—for those who must pay high prices—than is the monopoly itself. Proﬁtseeking middlemen will destroy any discriminatory pricing scheme. However, when resale is costly or can be prevented entirely, then price discrimination becomes possible. First-degree or perfect price discrimination If each buyer can be separately identiﬁed by a monopolist, then it may be possible to charge each the maximum price he or she would willingly pay for the good. This strategy of perfect (or ﬁrst-degree) price discrimination would then extract all available consumer surplus, leaving demanders as a group indifferent between buying the monopolist’s good or doing without it. The strategy is illustrated in Figure 14.4. The ﬁgure assumes that buyers are arranged in descending order of willingness to pay. The ﬁrst buyer is willing to pay up to P1 for Q1 units of output; therefore, the monopolist charges P1 and obtains total revenues of P1Q1, as indicated by the lightly shaded rectangle. A second buyer is willing to pay up to P2 Q1) for Q2 " from this buyer. Notice that this strategy cannot succeed unless the second buyer is unable to resell the output he or she buys at P2 to the ﬁrst buyer (who pays P1 > P2). Q1 units of output; therefore, the monopolist obtains total revenue of P2(Q2 " The monopolist will proceed in this way up to the marginal buyer, the last buyer who is willing to pay at least the good’s marginal cost (labeled MC in Figure 14.4). Hence total quantity produced will be Q!. Total revenues collected will be given by the area DEQ!0. All consumer surplus has been extracted by the monopolist, and there is no deadweight loss in this situation. (Compare Figures 14.3 and 14.4.) Therefore, the allocation of resources under perfect price discrimination is efﬁcient, although it does entail a large transfer from consumer surplus into monopoly proﬁts. FIGURE 14.4 Perfect Price Discrimination Under perfect price discrimination, the monopoly charges a different price to each buyer. It sells Q1 units at P1, Q2 " will be DEQ!0. Q1 units at P2, and so forth. In this case the ﬁrm will produce Q!, and total revenues Price D P1 P2 MC E D 0 Q1 Q2 Q* Quantity per period Chapter 14: Monopoly 515 EXAMPLE 14.4 First-Degree Price Discrimination Consider again the Frisbee monopolist in Example 14.1. Because there are relatively few highquality Frisbees sold, the monopolist may ﬁnd it possible to discriminate perfectly among a few world-class ﬂippers. In this case, it will choose to produce that quantity for which the marginal buyer pays exactly the marginal cost of a Frisbee: 100 P ¼ " Q 20 ¼ MC ¼ 0:1Q: Hence and, at the margin, price and marginal cost are given by Q! ¼ 666 ¼ Now we can compute total revenues by integration: ¼ P MC 66:6: Q! P Q Þ ð 0 ð 55,511: R ¼ ¼ dQ ¼ 100Q ! Q2 40 " 666 Q ¼ " Q ¼ 0 Total costs are total proﬁts are given by C Q ð Þ ¼ 0:05Q2 R p ¼ " þ C 10,000 ¼ 32,178; 23,333, ¼ (14:27) (14:28) (14:29) (14:30) (14:31) which represents a substantial increase over the single-price policy examined in Example 14.1 (which yielded 15,000). QUERY: What is the maximum price any Frisbee buyer pays in this case? Use this to obtain a geometric deﬁnition of proﬁts. Third-degree price discrimination through market separation First-degree price discrimination poses a considerable information burden for the monopoly—it must know the demand function for each potential buyer. A less stringent requirement would be to assume the monopoly can separate its buyers into relatively few identiﬁable markets (such as ‘‘rural–urban,’’ ‘‘domestic–foreign,’’ or ‘‘prime-time–offprime’’) and pursue a separate monopoly pricing policy in each market. Knowledge of the price elasticities of demand in these markets is sufﬁcient to
pursue such a policy. The monopoly then sets a price in each market according to the inverse elasticity rule. Assuming that marginal cost is the same in all markets, the result is a pricing policy in which 1 ei Pi 1 ! þ Pj 1 ! þ ¼ " 1 ej " (14:32) or þ þ where Pi and Pj are the prices charged in markets i and j, which have price elasticities of demand given by ei and ej. An immediate consequence of this pricing policy is that the (14:33) , Pi Pj ¼ 1 ð 1 ð 1=ejÞ 1=eiÞ 516 Part 6: Market Power FIGURE 14.5 Separated Markets Raise the Possibility of Third-Degree Price Discrimination If two markets are separate, then a monopolist can maximize proﬁts by selling its product at different prices in the two markets. This would entail choosing that output for which MC MR in each of the markets. The diagram shows that the market with a less elastic demand curve will be charged the higher price by the price discriminator. ¼ Price P1 P2 MC D1 MR1 D2 MR2 Quantity in market 1 Q1 0 Q2 Quantity in market 2 proﬁt-maximizing price will be higher in markets in which demand is less elastic. If, for example, ei ¼ " 4/3—prices will be one third higher in market i, the less elastic market. 3, then Equation 14.33 shows that Pi/Pj ¼ 2 and ej ¼ " Figure 14.5 illustrates this result for two markets that the monopoly can serve at constant marginal cost (MC). Demand is less elastic in market 1 than in market 2; thus, the gap between price and marginal revenue is larger in the former market. Proﬁt maximization requires that the ﬁrm produce Q!1 in market 1 and Q!2 in market 2, resulting in a higher price in the less elastic market. As long as arbitrage between the two markets can be prevented, this price difference can persist. The two-price discriminatory policy is clearly more proﬁtable for the monopoly than a single-price policy would be because the ﬁrm can always opt for the latter policy should market conditions warrant. The welfare consequences of third-degree price discrimination are, in principle, ambiguous. Relative to a single-price policy, the discriminating policy requires raising the price in the less elastic market and reducing it in the more elastic one. Hence the changes have an offsetting effect on total allocational losses. A more complete analysis suggests the intuitively plausible conclusion that the multiple-price policy will be allocationally superior to a single-price policy only in situations in which total output is increased through discrimination. Example 14.5 illustrates a simple case of linear demand curves in which a single-price policy does result in greater allocational losses.12 12For a detailed discussion, see R. Schmalensee, ‘‘Output and Welfare Implications of Monopolistic Third-Degree Price Discrimination,’’ American Economic Review (March 1981): 242–47. See also Problem 14.13. Chapter 14: Monopoly 517 EXAMPLE 14.5 Third-Degree Price Discrimination Suppose that a monopoly producer of widgets has a constant marginal cost of c products in two separated markets whose inverse demand functions are ¼ 6 and sells its 24 Q1 P1 ¼ and P2 ¼ Notice that consumers in market 1 are more eager to buy than are consumers in market 2 in the sense that the former are willing to pay more for any given quantity. Using the results for linear demand curves from Example 14.2 shows that the proﬁt-maximizing price–quantity combinations in these two markets are: 0:5Q2: (14:34) 12 " " 24 6 þ 2 ¼ P!1 ¼ 99. We can With this pricing strategy, proﬁts are p " compute the deadweight losses in the two markets by recognizing that the competitive output (with P 6) in market 1 is 18 and in market 2 is 12: 15, Q!1 ¼ (15 9, Q!2 ¼ 6) Æ 6 81 9, P!2 ¼ 6) Æ 9 (14:35) MC 18 (9 ¼ ¼ þ þ ¼ " 6: 12 6 þ 2 ¼ ¼ ¼ DW2 6 DW ¼ ¼ ¼ DW1 þ 0:5 P!1 " ð 40:5 9 þ ¼ 18 Þð " 49:5: 9 Þ þ 0:5 P!2 " ð 6 12 Þð 6 Þ " (14:36) A single-price policy. In this case, constraining the monopoly to charge a single price would reduce welfare. Under a single-price policy, the monopoly would simply cease serving market 2 because it can maximize proﬁts by charging a price of 15, and at that price no widgets will be bought in market 2 (because the maximum willingness to pay is 12). Therefore, total deadweight loss in this situation is increased from its level in Equation 14.36 because total potential consumer surplus in market 2 is now lost: DW1 þ DW2 ¼ (14:37) 12 ð 76:5: 40:5 40:5 DW Þ ¼ 0:5 36 12 ¼ þ " þ ¼ " Þð 6 0 This illustrates a situation where third-degree price discrimination is welfare improving over a single-price policy—when the discriminatory policy permits ‘‘smaller’’ markets to be served. Whether such a situation is common is an important policy question (e.g., consider the case of U.S. pharmaceutical manufacturers charging higher prices at home than abroad). QUERY: Suppose these markets were no longer separated. How would you construct the market demand in this situation? Would the monopolist’s proﬁt-maximizing single price still be 15? Second-Degree Price Discrimination Through Price Schedules The examples of price discrimination examined in the previous section require the monopoly to separate demanders into a number of categories and then choose a proﬁtmaximizing price for each such category. An alternative approach would be for the monopoly to choose a (possibly rather complex) price schedule that provides incentives for demanders to separate themselves depending on how much they wish to buy. Such schemes include quantity discounts, minimum purchase requirements or ‘‘cover’’ charges, and tie-in sales. These plans would be adopted by a monopoly if they yielded greater proﬁts than would a single-price policy, after accounting for any possible costs of 518 Part 6: Market Power implementing the price schedule. Because the schedules will result in demanders paying different prices for identical goods, this form of (second-degree) price discrimination is feasible only when there are no arbitrage possibilities. Here we look at one simple case. The Extensions to this chapter and portions of Chapter 18 look at other aspects of second-degree price discrimination. Two-part tariffs One form of pricing schedule that has been extensively studied is a linear two-part tariff, under which demanders must pay a ﬁxed fee for the right to consume a good and a uniform price for each unit consumed. The prototype case, ﬁrst studied by Walter Oi, is an amusement park (perhaps Disneyland) that sets a basic entry fee coupled with a stated marginal price for each amusement used.13 Mathematically, this scheme can be represented by the tariff any demander must pay to purchase q units of a good: T q ð Þ ¼ a þ pq, (14:38) where a is the ﬁxed fee and p is the marginal price to be paid. The monopolist’s goal then is to choose a and p to maximize proﬁts, given the demand for this product. Because the average price paid by any demander is given by T q ¼ a q þ p, p ¼ (14:39) this tariff is feasible only when those who pay low average prices (those for whom q is large) cannot resell the good to those who must pay high average prices (those for whom q is small). One approach described by Oi for establishing the parameters of this linear tariff would be for the ﬁrm to set the marginal price, p, equal to MC and then set a to extract the maximum consumer surplus from a given set of buyers. One might imagine buyers being arrayed according to willingness to pay. The choice of p MC would then maximize consumer surplus for this group, and a could be set equal to the surplus enjoyed by the least eager buyer. He or she would then be indifferent about buying the good, but all other buyers would experience net gains from the purchase. ¼ ¼ This feasible tariff might not be the most proﬁtable, however. Consider the effects on proﬁts of a small increase in p above MC. This would result in no net change in the profits earned from the least willing buyer. Quantity demanded would drop slightly at the margin where p MC, and some of what had previously been consumer surplus (and therefore part of the ﬁxed fee, a) would be converted into variable proﬁts because now p > MC. For all other demanders, proﬁts would be increased by the price increase. Although each will pay a bit less in ﬁxed charges, proﬁts per unit bought will increase to a greater extent.14 In some cases it is possible to make an explicit calculation of the optimal two-part tariff. Example 14.6 provides an illustration. More generally, however, optimal schedules will depend on a variety of contingencies. Some of the possibilities are examined in the Extensions to this chapter. 13W. Y. Oi, ‘‘A Disneyland Dilemma: Two-Part Tariffs for a Mickey Mouse Monopoly,’’ Quarterly Journal of Economics (February 1971): 77–90. Interestingly, the Disney empire once used a two-part tariff but abandoned it because the costs of administering the payment schemes for individual rides became too high. Like other amusement parks, Disney moved to a single-admissions price policy (which still provided them with ample opportunities for price discrimination, especially with the multiple parks at Disney World). 14This follows because qi(MC) > q1(MC), where qi(MC) is the quantity demanded when p MC for all except the least willing buyer (person 1). Hence the gain in proﬁts from an increase in price above MC, Dpqi(MC), exceeds the loss in proﬁts from a smaller ﬁxed fee, Dpq1(MC). ¼ Chapter 14: Monopoly 519 EXAMPLE 14.6 Two-Part Tariffs To illustrate the mathematics of introduced in Example 14.5 but now assume that they apply to two speciﬁc demanders: q1 ¼ q2 ¼ p1, 2p2, two-part tariffs, 24 24 " " where now the p’s refer to the marginal prices faced by these two buyers.15 let’s return to the demand equations (14:40) ¼ MC p2 ¼ An Oi tariff. Implementing the two-part tariff suggested by Oi would require the monopolist to set p1 ¼ 12. With this marginal price, 6. Hence in this case, q1 ¼ 6) Æ 12]. 0.5 Æ (12 demander 2 (the less eager of the two) obtains consumer surplus of 36 [ That is th
e maximal entry fee that might be charged without causing this person to leave the 6q. If the market. Consequently, the two-part tariff in this case would be T(q) monopolist opted for this pricing scheme, its proﬁts would be AC R 30 18 and q2 ¼ T ð 72 (14:41) q2Þ 36 ¼ " ¼ " þ ¼ C p q1Þ þ 6 ’ þ T q2Þ " ð 6 30 ’ " q1 þ ð 72: ¼ ¼ ¼ These fall short of those obtained in Example 14.5. The optimal tariff. The optimal two-part tariff in this situation can be computed by noting that total proﬁts with such a tariff are p q2). Here the entry fee, a, must þ equal the consumer surplus obtained by person 2. Inserting the speciﬁc parameters of this problem yields MC)(q1 þ 2a (p " ¼ 12 p 0:5 24 18p ¼ ¼ ð ¼ 2q2ð ’ 2p Þð " p2: " p p " 12 Þð 6 q2Þ q1 þ 3p 48 " Þð Þ (14:42) 2p)(12 9. 9 and a 0.5(24 Hence maximum proﬁts are obtained when p " " 6, and the 15 and q2 ¼ 9q. With this tariff, q1 ¼ Therefore, the optimal tariff is T(q) 6) Æ (15 monopolist’s proﬁts are 81 [ 6)]. The monopolist might opt for this 2(9) pricing scheme if it were under political pressure to have a uniform pricing policy and to agree not to price demander 2 ‘‘out of the market.’’ The two-part tariff permits a degree of differential pricing but appears ‘‘fair’’ because all buyers face the same schedule. ¼ þ 9 (9 þ " 9:75 p) ¼ ¼ ¼ þ ¼ p1 ¼ 9:60, p2 ¼ ð Þ QUERY: Suppose a monopolist could choose a different entry fee for each demander. What pricing policy would be followed? Regulation Of Monopoly The regulation of natural monopolies is an important subject in applied economic analysis. The utility, communications, and transportation industries are highly regulated in most countries, and devising regulatory procedures that induce these industries to operate in a desirable way is an important practical problem. Here we will examine a few aspects of the regulation of monopolies that relate to pricing policies. Marginal cost pricing and the natural monopoly dilemma Many economists believe it is important for the prices charged by regulated monopolies to reﬂect marginal costs of production accurately. In this way the deadweight loss may be 15The theory of utility maximization that underlies these demand curves is that the quantity demanded is determined by the marginal price paid, whereas the entry fee a determines whether q 0 might instead be optimal. ¼ 520 Part 6: Market Power FIGURE 14.6 Price Regulation for a Decreasing Cost Monopoly Because natural monopolies exhibit decreasing average costs, marginal costs decrease below average costs. Consequently, enforcing a policy of marginal cost pricing will entail operating at a loss. A price of PR, for example, will achieve the goal of marginal cost pricing but will necessitate an operating loss of GFEPR. Price D PA C G PR A B MR QA F E AC MC D QR Quantity per period minimized. The principal problem raised by an enforced policy of marginal cost pricing is that it will require natural monopolies to operate at a loss. Natural monopolies, by deﬁnition, exhibit decreasing average costs over a broad range of output levels. The cost curves for such a ﬁrm might look like those shown in Figure 14.6. In the absence of regulation, the monopoly would produce output level QA and receive a price of PA for its product. Proﬁts in this situation are given by the rectangle PAABC. A regulatory agency might instead set a price of PR for the monopoly. At this price, QR is demanded, and the marginal cost of producing this output level is also PR. Consequently, marginal cost pricing has been achieved. Unfortunately, because of the negative slope of the ﬁrm’s average cost curve, the price PR ( marginal cost) decreases below average costs. With this regulated price, the monopoly must operate at a loss of GFEPR. Because no ﬁrm can operate indeﬁnitely at a loss, this poses a dilemma for the regulatory agency: Either it must abandon its goal of marginal cost pricing, or the government must subsidize the monopoly forever. ¼ Two-tier pricing systems One way out of the marginal cost pricing dilemma is the implementation of a multiprice system. Under such a system the monopoly is permitted to charge some users a high price while maintaining a low price for marginal users. In this way the demanders paying the high price in effect subsidize the losses of the low-price customers. Such a pricing scheme is shown in Figure 14.7. Here the regulatory commission has decided that some users will pay a relatively high price, P1. At this price, Q1 is demanded. Other users Chapter 14: Monopoly 521 FIGURE 14.7 Two-Tier Pricing Schedule By charging a high price (P1) to some users and a low price (P2) to others, it may be possible for a regulatory commission to (1) enforce marginal cost pricing and (2) create a situation where the proﬁts from one class of user (P1DBA) subsidize the losses of the other class (BFEC). Price P1 A P2 D B C Q1 F E D Q2 AC MC Quantity per period (presumably those who would not buy the good at the P1 price) are offered a lower price, Q1. Consequently, a total outP2. This lower price generates additional demand of Q2 " put of Q2 is produced at an average cost of A. With this pricing system, the proﬁts on the sales to high-price demanders (given by the rectangle P1DBA) balance the losses incurred on the low-priced sales (BFEC). Furthermore, for the ‘‘marginal user,’’ the marginal cost pricing rule is being followed: It is the ‘‘intramarginal’’ user who subsidizes the ﬁrm so it does not operate at a loss. Although in practice it may not be so simple to establish pricing schemes that maintain marginal cost pricing and cover operating costs, many regulatory commissions do use price schedules that intentionally discriminate against some users (e.g., businesses) to the advantage of others (consumers). Rate of return regulation Another approach followed in many regulatory situations is to permit the monopoly to charge a price above marginal cost that is sufﬁcient to earn a ‘‘fair’’ rate of return on investment. Much analytical effort is then devoted to deﬁning the ‘‘fair’’ rate concept and to developing ways in which it might be measured. From an economic point of view, some of the most interesting questions about this procedure concern how the regulatory activity affects the ﬁrm’s input choices. If, for example, the rate of return allowed to ﬁrms exceeds what owners might obtain on investment under competitive circumstances, there will be an incentive to use relatively more capital input than would truly minimize costs. And if regulators delay in making rate decisions, this may give ﬁrms cost-minimizing 522 Part 6: Market Power incentives that would not otherwise exist. We will now brieﬂy examine a formal model of such possibilities.16 A formal model Suppose a regulated utility has a production function of the form This ﬁrm’s actual rate of return on capital is then deﬁned as q k, l f ð : Þ ¼ pf k, l Þ " ð k wl , s ¼ (14:43) (14:44) where p is the price of the ﬁrm’s output (which depends on q) and w is the wage rate for labor input. If s is constrained by regulation to be equal to (say) s, then the ﬁrm’s problem is to maximize proﬁts ¼ subject to this regulatory constraint. The Lagrangian for this problem is Þ " " p pf k, l ð wl vk + pf k, l ð Þ " ¼ wl vk k wl ½ þ sk pf : k, l ð Þ) " þ " (14:45) (14:46) Notice that if l maximizing ﬁrm. If l ¼ 0, regulation is ineffective and the monopoly behaves like any proﬁt- 1, Equation 14.46 reduces to ¼ + s v k, Þ " which, assuming s > v (which it must be if the ﬁrm is not to earn less than the prevailing rate of return on capital elsewhere), means this monopoly will hire inﬁnite amounts of capital—an implausible result. Hence 0 < l < 1. The ﬁrst-order conditions for a maximum are ¼ ð (14:47) @+ @l ¼ @+ @k ¼ @+ @k ¼ pfl " pfk " wl sk " þ pf " 0, pf1Þ ¼ " 0, pfkÞ ¼ k, l ð Þ ¼ 0: (14:48) The ﬁrst of these conditions implies that the regulated monopoly will hire additional labor input up to the point at which pfl ¼ w—a result that holds for any proﬁt-maximizing ﬁrm. For capital input, however, the second condition implies that or 1 ð " pfk ¼ k Þ v " ks pfk ¼ v 1 ks " " Because s > v and l < 1, Equation 14.50 implies pfk < v: Þ : (14:49) (14:50) (14:51) 16This model is based on H. Averch and L. L. Johnson, Economic Review (December 1962): 1052–69. ‘‘Behavior of the Firm under Regulatory Constraint,’’ American Chapter 14: Monopoly 523 The ﬁrm will hire more capital (and achieve a lower marginal productivity of capital) than it would under unregulated conditions. Therefore, ‘‘overcapitalization’’ may be a regulatory-induced misallocation of resources for some utilities. Although we shall not do so here, it is possible to examine other regulatory questions using this general analytical framework. Dynamic Views Of Monopoly The static view that monopolistic practices distort the allocation of resources provides the principal economic rationale for favoring antimonopoly policies. Not all economists believe that the static analysis should be deﬁnitive, however. Some authors, most notably J. A. Schumpeter, have stressed the beneﬁcial role that monopoly proﬁts can play in the process of economic development.17 These authors place considerable emphasis on innovation and the ability of particular types of ﬁrms to achieve technical advances. In this context the proﬁts that monopolistic ﬁrms earn provide funds that can be invested in research and development. Whereas perfectly competitive ﬁrms must be content with a normal return on invested capital, monopolies have ‘‘surplus’’ funds with which to undertake the risky process of research. More important, perhaps, the possibility of attaining a monopolistic position—or the desire to maintain such a position—provides an important incentive to keep one step ahead of potential competitors. Innovations in new products and cost-saving production techniques may be integrally related to the possibility of monopolization. Without such a monopolistic position, the full beneﬁts of inno
vation could not be obtained by the innovating ﬁrm. Schumpeter stresses the point that the monopolization of a market may make it less costly for a ﬁrm to plan its activities. Being the only source of supply for a product eliminates many of the contingencies that a ﬁrm in a competitive market must face. For example, a monopoly may not have to spend as much on selling expenses (e.g., advertising, brand identiﬁcation, and visiting retailers) as would be the case in a more competitive industry. Similarly, a monopoly may know more about the speciﬁc demand curve for its product and may more readily adapt to changing demand conditions. Of course, whether any of these purported beneﬁts of monopolies outweigh their allocational and distributional disadvantages is an empirical question. Issues of innovation and cost savings cannot be answered by recourse to a priori arguments; detailed investigation of real-world markets is a necessity. SUMMARY In this chapter we have examined models of markets in which there is only a single monopoly supplier. Unlike the competitive case investigated in Part 4, monopoly ﬁrms do not exhibit price-taking behavior. Instead, the monopolist can choose the price–quantity combination on the market demand curve that is most proﬁtable. A number of consequences then follow from this market power. • The most proﬁtable level of output for the monopolist is the one for which marginal revenue is equal to marginal cost. At this output level, price will exceed marginal cost. The proﬁtability of the monopolist will depend on the relationship between price and average cost. • Relative to perfect competition, monopoly involves a loss of consumer surplus for demanders. Some of this is transferred into monopoly proﬁts, whereas some of the loss in consumer supply represents a deadweight loss of overall economic welfare. 17See, for example, J. A. Schumpeter, Capitalism, Socialism and Democracy, 3rd ed. (New York: Harper & Row, 1950), especially chap. 8. 524 Part 6: Market Power • Monopolists may opt for different levels of quality than would perfectly competitive ﬁrms. Durable goods monopolists may be constrained by markets for used goods. • A monopoly may be able to increase its proﬁts further through price discrimination—that is, charging different prices to different categories of buyers. The ability the monopoly to practice price discrimination of depends on its ability to prevent arbitrage among buyers. • Governments often choose to regulate natural monopolies (ﬁrms with diminishing average costs over a broad range of output levels). The type of regulatory mechanisms adopted can affect the behavior of the regulated ﬁrm. PROBLEMS 14.1 A monopolist can produce at constant average and marginal costs of AC given by Q 53 P. ¼ " MC ¼ ¼ 5. The ﬁrm faces a market demand curve a. Calculate the proﬁt-maximizing price–quantity combination for the monopolist. Also calculate the monopolist’s proﬁts. b. What output level would be produced by this industry under perfect competition (where price c. Calculate the consumer surplus obtained by consumers in case (b). Show that this exceeds the sum of the monopolist’s proﬁts and the consumer surplus received in case (a). What is the value of the ‘‘deadweight loss’’ from monopolization? marginal cost)? ¼ 14.2 A monopolist faces a market demand curve given by a. If the monopolist can produce at constant average and marginal costs of AC Q 70 p: " ¼ MC 6, what output level will the monop- ¼ ¼ olist choose to maximize proﬁts? What is the price at this output level? What are the monopolist’s proﬁts? b. Assume instead that the monopolist has a cost structure where total costs are described by C Q ð Þ ¼ 0:25Q2 5Q þ " 300: With the monopolist facing the same market demand and marginal revenue, what price–quantity combination will be chosen now to maximize proﬁts? What will proﬁts be? c. Assume now that a third cost structure explains the monopolist’s position, with total costs given by C Q ð Þ ¼ 0:0133Q3 5Q þ " 250: Again, calculate the monopolist’s price–quantity combination that maximizes proﬁts. What will proﬁt be? Hint: Set MC MR as usual and use the quadratic formula to solve the second-order equation for Q. ¼ d. Graph the market demand curve, the MR curve, and the three marginal cost curves from parts (a), (b), and (c). Notice that the monopolist’s proﬁt-making ability is constrained by (1) the market demand curve (along with its associated MR curve) and (2) the cost structure underlying production. 14.3 A single ﬁrm monopolizes the entire market for widgets and can produce at constant average and marginal costs of Originally, the ﬁrm faces a market demand curve given by AC MC 10: ¼ ¼ 60 Q ¼ " P: a. Calculate the proﬁt-maximizing price–quantity combination for the ﬁrm. What are the ﬁrm’s proﬁts? b. Now assume that the market demand curve shifts outward (becoming steeper) and is given by 45 Q ¼ " 0:5P: Chapter 14: Monopoly 525 What is the ﬁrm’s proﬁt-maximizing price–quantity combination now? What are the ﬁrm’s proﬁts? c. Instead of the assumptions of part (b), assume that the market demand curve shifts outward (becoming ﬂatter) and is given by What is the ﬁrm’s proﬁt-maximizing price–quantity combination now? What are the ﬁrm’s proﬁts? d. Graph the three different situations of parts (a), (b), and (c). Using your results, explain why there is no real supply curve for a monopoly. 100 Q ¼ " 2P: 14.4 Suppose the market for Hula Hoops is monopolized by a single ﬁrm. a. Draw the initial equilibrium for such a market. b. Now suppose the demand for Hula Hoops shifts outward slightly. Show that, in general (contrary to the competitive case), it will not be possible to predict the effect of this shift in demand on the market price of Hula Hoops. c. Consider three possible ways in which the price elasticity of demand might change as the demand curve shifts: It might increase, it might decrease, or it might stay the same. Consider also that marginal costs for the monopolist might be increasing, decreasing, or constant in the range where MR MC. Consequently, there are nine different combinations of ¼ types of demand shifts and marginal cost slope conﬁgurations. Analyze each of these to determine for which it is possible to make a deﬁnite prediction about the effect of the shift in demand on the price of Hula Hoops. 14.5 Suppose a monopoly market has a demand function in which quantity demanded depends not only on market price (P) but also on the amount of advertising the ﬁrm does (A, measured in dollars). The speciﬁc form of this function is The monopolistic ﬁrm’s cost function is given by Q 20 ¼ ð P 1 Þð þ " 0:1A " 0:01A2 : Þ a. Suppose there is no advertising (A yield? What will be the monopoly’s proﬁts? ¼ C ¼ 10Q 15 A: þ þ 0). What output will the proﬁt-maximizing ﬁrm choose? What market price will this b. Now let the ﬁrm also choose its optimal level of advertising expenditure. In this situation, what output level will be chosen? What price will this yield? What will the level of advertising be? What are the ﬁrm’s proﬁts in this case? Hint: This can be worked out most easily by assuming the monopoly chooses the proﬁt-maximizing price rather than quantity. 14.6 Suppose a monopoly can produce any level of output it wishes at a constant marginal (and average) cost of $5 per unit. Assume the monopoly sells its goods in two different markets separated by some distance. The demand curve in the ﬁrst market is given by and the demand curve in the second market is given by Q1 ¼ 55 " P1, Q2 ¼ 70 " 2P2: a. If the monopolist can maintain the separation between the two markets, what level of output should be produced in each market, and what price will prevail in each market? What are total proﬁts in this situation? b. How would your answer change if it costs demanders only $5 to transport goods between the two markets? What would be the monopolist’s new proﬁt level in this situation? c. How would your answer change if transportation costs were zero and then the ﬁrm was forced to follow a single-price policy? d. Now assume the two different markets 1 and 2 are just two individual consumers. Suppose the ﬁrm could adopt a linear two-part tariff under which marginal prices charged to the two consumers must be equal but their lump-sum entry fees might vary. What pricing policy should the ﬁrm follow? 526 Part 6: Market Power 14.7 Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of $10 per unit. Monopolized marginal costs increase to $12 per unit because $2 per unit must be paid to lobbyists to retain the widget producers’ favored position. Suppose the market demand for widgets is given by QD ¼ 1,000 50P: " a. Calculate the perfectly competitive and monopoly outputs and prices. b. Calculate the total loss of consumer surplus from monopolization of widget production. c. Graph your results and explain how they differ from the usual analysis. 14.8 Suppose the government wishes to combat the undesirable allocational effects of a monopoly through the use of a subsidy. a. Why would a lump-sum subsidy not achieve the government’s goal? b. Use a graphical proof to show how a per-unit-of-output subsidy might achieve the government’s goal. c. Suppose the government wants its subsidy to maximize the difference between the total value of the good to consumers and the good’s total cost. Show that, to achieve this goal, the government should set where t is the per-unit subsidy and P is the competitive price. Explain your result intuitively. t P ¼ " 1 eQ, P , 14.9 Suppose a monopolist produces alkaline batteries that may have various useful lifetimes (X). Suppose also that consumers’ (inverse) demand depends on batteries’ lifetimes and quantity (Q) purchased according to the function where g 0 < 0. That is, consumers care only about the product of quantity times lifetime: They are willing to pay equally for many short-lived batteries or few long-lived ones. Assume also that ba
ttery costs are given by P Q where C 0(X) > 0. Show that, in this case, the monopoly will opt for the same level of X as does a competitive industry even though levels of output and prices may differ. Explain your result. Hint: Treat XQ as a composite commodity. C Q, X ð C X ð Q, Þ Analytical Problems 14.10 Taxation of a monopoly good The taxation of monopoly can sometimes produce results different from those that arise in the competitive case. This problem looks at some of those cases. Most of these can be analyzed by using the inverse elasticity rule (Equation 14.1). a. Consider ﬁrst an ad valorem tax on the price of a monopoly’s good. This tax reduces the net price received by the t)—where t is the proportional tax rate. Show that, with a linear demand curve and constant monopoly from P to P(1 marginal cost, the imposition of such a tax causes price to increase by less than the full extent of the tax. " b. Suppose that the demand curve in part (a) were a constant elasticity curve. Show that the price would now increase by pre- cisely the full extent of the tax. Explain the difference between these two cases. c. Describe a case where the imposition of an ad valorem tax on a monopoly would cause the price to increase by more than the tax. d. A speciﬁc tax is a ﬁxed amount per unit of output. If the tax rate is t per unit, total tax collections are tQ. Show that the imposition of a speciﬁc tax on a monopoly will reduce output more (and increase price more) than will the imposition of an ad valorem tax that collects the same tax revenue. 14.11 More on the welfare analysis of quality choice An alternative way to study the welfare properties of a monopolist’s choices is to assume the existence of a utility function for the customers of the monopoly of the form utility U(Q, X), where Q is quantity consumed and X is the quality associated with that quantity. A social planner’s problem then would be to choose Q and X to maximize social welfare as represented by SW C(Q, X). U(Q, X) ¼ ¼ " Chapter 14: Monopoly 527 a. What are the ﬁrst-order conditions for a welfare maximum? b. The monopolist’s goal is to choose the Q and X that maximize p tions for this maximization? P(Q, X) Æ Q " ¼ C(Q, X). What are the ﬁrst-order condi- c. Use your results from parts (a) and (b) to show that, at the monopolist’s preferred choices, @SW/@Q > 0. That is, as we have already shown, prove that social welfare would be improved if more were produced. Hint: Assume that @U/@Q P. d. Show that, at the monopolist’s preferred choices, the sign of @SW/@X is ambiguous—that is, it cannot be determined (on the sole basis of the general theory of monopoly) whether the monopolist produces either too much or too little quality. ¼ 14.12 The welfare effects of third-degree price discrimination In an important 1985 article,18 Hal Varian shows how to assess third-degree price discrimination using only properties of the indirect utility function (see Chapter 3). This problem provides a simpliﬁed version of his approach. Suppose that a single good is sold in two separated markets. Quantities in the two markets are designated by q1, q2 with prices p1, p2. Consumers of the good are assumed to be characterized by an indirect utility function that takes a quasi-linear form: V( p1, p2, I) I. þ Income is assumed to have an exogenous component (!I), and the monopoly earns proﬁts of p q2), where c is marginal and average cost (which is assumed to be constant). v( p1, p2) c(q1 þ ¼ p2q2 " p1q1 þ ¼ a. Given this setup, let’s ﬁrst show some facts about this kind of indirect utility function. (1) Use Roy’s identity (see the Extensions to Chapter 5) to show that the Marshallian demand functions for the two goods in this problem are given by qi ( p1, p2, I) @v/@pi. ¼ " (2) Show that the function v ( p1, p2) is convex in the prices. (3) Because social welfare (SW) can be measured by the indirect utility function of the consumers, show that the welfare Dp. How does this expression compare with the notion (intro- impact of any change in prices is given by DSW Dv duced in Chapter 12) that any change in welfare is the sum of changes in consumer and producer surplus? b. Suppose now that we wish to compare the welfare associated with a single-price policy for these two markets, p1 ¼ with the welfare associated with different prices in the two markets, p1 ¼ change in social welfare from adopting a two-price policy is given by DSW order Taylor expansion for the function v around p!1, p!2 together with Roy’s identity and the fact that v is convex. p, p!2. Show that an upper bound to the p!1 and p2 ¼ . Hint: Use a ﬁrstq2Þ q!1 þ c p * ð Þð p2 ¼ q!2 " q1 " þ " ¼ c. Show why the results of part (b) imply that, for social welfare to increase from the adoption of the two-price policy, total quantity demanded must increase. two-price policy is given by DSW d. Use an approach similar to that taken in part (b) to show that a lower bound to the change in social welfare from adopting a p!2 " e. Notice that the approach taken here never uses the fact that the price–quantity combinations studied are proﬁt maximizing for the monopolist. Can you think of situations (other than third-degree price discrimination) where the analysis here might apply? Note: Varian shows that the bounds for welfare changes can be tightened a bit in the price discrimination case by using proﬁt maximization. . Can you interpret this lower bound condition? q1Þ þ ð q!1 " Þð p!1 " q!2 " q2Þ c Þð + ð c SUGGESTIONS FOR FURTHER READING Posner, R. A. Regulation.’’ 807–27. ‘‘The Social Costs of Monopoly and Journal of Political Economy 83 (1975): An analysis of the probability that monopolies will spend resources on the creation of barriers to entry and thus have higher costs than perfectly competitive ﬁrms. Schumpeter, J. A. Capitalism, Socialism and Democracy, 3rd ed. New York: Harper & Row, 1950. Classic defense of the role of the entrepreneur and economic proﬁts in the economic growth process. Spence, M. ‘‘Monopoly, Quality, and Regulation.’’ Bell Journal of Economics (April 1975): 417–29. Develops the approach to product quality used in this text and provides a detailed analysis of the effects of monopoly. Stigler, G. J. ‘‘The Theory of Economic Regulation.’’ Bell Journal of Economics and Management Science 2 (Spring 1971): 3. Early development of the ‘‘capture’’ hypothesis of regulatory behavior—that the industry captures the agency supposed to regulate it and uses that agency to enforce entry barriers and further enhance proﬁts. Tirole, J. The Theory of Industrial Organization. Cambridge, MA: MIT Press, 1989, chaps. 1–3. A complete analysis of the theory of monopoly pricing and product choice. Varian, H. R. Microeconomic Analysis, 3rd ed. New York: W. W. Norton, 1992, chap. 14. Provides a succinct analysis of the role of incentive compatibility constraints in second-degree price discrimination. 18H. R. Varian, ‘‘Price Discrimination and Social Welfare,’’ American Economic Review (September 1985): 870–75. EXTENSIONS OPTIMAL LINEAR TWO-PART TARIFFS In Chapter 14 we examined a simple illustration of ways in which a monopoly may increase proﬁts by practicing seconddegree price discrimination—that is, by establishing price (or ‘‘outlay’’) schedules that prompt buyers to separate themselves into distinct market segments. Here we pursue the topic of linear tariff schedules a bit further. Nonlinear pricing schedules are discussed in Chapter 18. E14.1 Structure of the problem To examine issues related to price schedules in a simple context for each demander, we deﬁne the ‘‘valuation function’’ as (i) við q q pið q si, þ Þ ¼ Þ ’ where pi(q) is the inverse demand function for individual i and si is consumer surplus. Hence vi represents the total value to individual i of undertaking transactions of amount q, which includes total spending on the good plus the value of consumer surplus obtained. Here we will assume (a) there are only two demanders1 (or homogeneous groups of demanders) and (b) person 1 has stronger preferences for this good than person 2 in the sense that > v2ð q v1ð q (ii) Þ Þ for all values of q. The monopolist is assumed to have constant marginal costs (denoted by c) and chooses a tariff (revenue) schedule, T(q), that maximizes proﬁts given by c ð p T T ¼ q1 þ ð q1Þ þ ð q2Þ " where qi represents the quantity chosen by person i. In selecting a price schedule that successfully distinguishes among consumers, the monopolist faces two constraints. To ensure that the low-demand person (2) is served, it is necessary that , q2Þ (iii) 0: T (iv) v2ð q2Þ " q2Þ + ð That is, person 2 must derive a net beneﬁt from her optimal choice, q2. Person 1, the high-demand individual, must also obtain a net gain from his chosen consumption level (q1) and must prefer this choice to the output choice made by person 2: q1Þ + v1ð If the monopolist does not recognize this ‘‘incentive compatibility’’ constraint, it may ﬁnd that person 1 opts for the q2Þ " q1Þ " : q2Þ ð v1ð (v) T T ð 1Generalizations to many demanders are nontrivial. For a discussion, see Wilson (1993, chaps. 2–5). portion of the price schedule intended for person 2, thereby destroying the goal of obtaining self-selected market separation. Given this general structure, we can proceed to illustrate a number of interesting features of the monopolist’s problem. E14.2 Pareto superiority Permitting the monopolist to depart from a simple singleprice scheme offers the possibility of adopting ‘‘Pareto superior’’ tariff schedules under which all parties to the transaction are made better off. For example, suppose the monopolist’s proﬁt-maximizing price is pM. At this price, person 2 consumes qM 2 and receives a net value from this consumption of v2ð qM 2 Þ " pMqM 2 : A tariff schedule for which T q ð Þ ¼ pMq pq þ a % qM for q 2 , * for q > qM 2 , (vi) (vii) where a > 0 and c < p < pM, may yield increased proﬁts for the monopolist as well as increased welfare for person 1. Speciﬁcally, consi
der values of a and p such that a pqM 1 ¼ þ pMqM 1 or a pM " ¼ ð p qM 1 , Þ (viii) where qM represents consumption of person 1 under a single1 price policy. In this case, a and p are set so that person 1 can still afford to buy qM 1 under the new price schedule. Because p < pM, however, he will opt for q!1 > qM 1 . Because person 1 could have bought qM 1 but chose q!1 instead, he must be better off under the new schedule. The monopoly’s proﬁts are now given by p a pq1 þ þ ¼ pMqM 2 " q1 þ c ð qM 2 Þ and a p " pM ¼ pq1 þ q1 " c ð where pM is the monopoly’s single-price proﬁts qM 1 þ ð pM " . Substitution for a from Equation viii shows qM 2 Þ) 1 " ½¼ ð þ qM , 1 Þ pMqM c 3 Þ p pM ¼ ð p " c q1 " Þð " qM 1 Þ > 0: (xi) (ix) (x) Chapter 14: Monopoly 529 Hence this new price schedule also provides more proﬁts to the monopoly, some of which might be shared with person 2. The price schedule is Pareto superior to a single monopoly price. The notion that multipart schedules may be Pareto superior has been used not only in the study of price discrimination but also in the design of optimal tax schemes and auction mechanisms (see Willig, 1978). Pricing a farmland reserve The potential Pareto superiority of complex tariff schedules is used by R. B. W. Smith (1995) to estimate a least-cost method for the U.S. government to ﬁnance a conservation reserve program for farmland. The speciﬁc plan the author studies would maintain a 34-million-acre reserve out of production in any given year. He calculates that use of carefully constructed (nonlinear) tariff schedules for such a program might cost only $1 billion annually E14.3 Tied sales Sometimes a monopoly will market two goods together. This situation poses a number of possibilities for discriminatory pricing schemes. Consider, for example, laser printers that are sold with toner cartridges or electronic game players sold with patented additional games. Here the pricing situation is similar to that examined in Chapter 14—usually consumers buy only one unit of the basic product (the printer or camera) and thereby pay the ‘‘entry’’ fee. Then they consume a variable number of tied products (toner and ﬁlm). Because our analysis in Chapter 14 suggests that the monopoly will choose a price for its tied product that exceeds marginal cost, there will be a welfare loss relative to a situation in which the tied good is produced competitively. Perhaps for this reason, tied sales are prohibited by law in some cases. Prohibition may not necessarily increase welfare, however, if the monopoly declines to serve low-demand consumers in the absence of such a practice (Oi, 1971). Automobiles and wine One way in which tied sales can be accomplished is through creation of a multiplicity of quality variants that appeal to different classes of buyers. Automobile companies have been especially ingenious at devising quality variants of their basic models (e.g., the Honda Accord comes in DX, LX, EX, and SX conﬁgurations) that act as tied goods in separating buyers into various market niches. A 1992 study by J. E. Kwoka examines one speciﬁc U.S. manufacturer (Chrysler) and shows how market segmentation is achieved through quality variation. The author calculates that signiﬁcant transfer from consumer surplus to ﬁrms occurs as a result of such segmentation. Generally, this sort of price discrimination in a tied good will be infeasible if that good is also produced under competitive conditions. In such a case the tied good will sell for marginal cost, and the only possibility for discriminatory behavior open to the monopolist is in the pricing of its basic good (i.e., by varying ‘‘entry fees’’ among demanders). In some special cases, however, choosing to pay the entry fee will confer monopoly power in the tied good on the monopolist even though it is otherwise reduced under competitive conditions. For example, Locay and Rodriguez (1992) examine the case of restaurants’ pricing of wine. Here group decirestaurant may confer sions monopoly power to the restaurant owner in the ability to practice wine price discrimination among buyers with strong grape preferences. Because the owner is constrained by the need to attract groups of customers to the restaurant, the power to price discriminate is less than under the pure monopoly scenario. to patronize a particular References Kwoka, J. E. ‘‘Market Segmentation by Price-Quality Schedules: Some Evidence from Automobiles.’’ Journal of Business (October 1992): 615–28. Locay, L., and A. Rodriguez. ‘‘Price Discrimination in Competitive Markets.’’ Journal of Political Economy (October 1992): 954–68. Oi, W. Y. ‘‘A Disneyland Dilemma: Two-Part Tariffs on a Mickey Mouse Monopoly.’’ Quarterly Journal of Economics (February 1971): 77–90. Smith, R. B. W. ‘‘The Conservation Reserve Program as a Least Cost Land Retirement Mechanism.’’ American Journal of Agricultural Economics (February 1995): 93–105. Willig, R. ‘‘Pareto Superior Non-Linear Outlay Schedules.’’ Bell Journal of Economics (January 1978): 56–69. Wilson, W. Nonlinear Pricing. Oxford: Oxford University Press, 1993. This page intentionally left blank C H A P T E R FIFTEEN Imperfect Competition This chapter discusses oligopoly markets, falling between the extremes of perfect competition and monopoly Oligopoly. A market with relatively few ﬁrms but more than one. Oligopolies raise the possibility of strategic interaction among ﬁrms. To analyze this strategic interaction rigorously, we will apply the concepts from game theory that were introduced in Chapter 8. Our game-theoretic analysis will show that small changes in details concerning the variables ﬁrms choose, the timing of their moves, or their information about market conditions or rival actions can have a dramatic effect on market outcomes. The ﬁrst half of the chapter deals with short-term decisions such as pricing and output, and the second half covers longer-term decisions such as investment, advertising, and entry. Short-Run Decisions: Pricing And Output It is difﬁcult to predict exactly the possible outcomes for price and output when there are few ﬁrms; prices depend on how aggressively ﬁrms compete, which in turn depends on which strategic variables ﬁrms choose, how much information ﬁrms have about rivals, and how often ﬁrms interact with each other in the market. For example, consider the Bertrand game studied in the next section. The game involves two identical ﬁrms choosing prices simultaneously for their identical products in their one meeting in the market. The Bertrand game has a Nash equilibrium at point C in Figure 15.1. Even though there may be only two ﬁrms in the market, in this equilibrium they behave as though they were perfectly competitive, setting price equal to marginal cost and earning zero proﬁt. We will discuss whether the Bertrand game is a realistic depiction of actual ﬁrm behavior, but an analysis of the model shows that it is possible to think up rigorous game-theoretic models in which one extreme—the competitive outcome—can emerge in concentrated markets with few ﬁrms. At the other extreme, as indicated by point M in Figure 15.1, ﬁrms as a group may act as a cartel, recognizing that they can affect price and coordinate their decisions. Indeed, they may be able to act as a perfect cartel and achieve the highest possible proﬁts— namely, the proﬁt a monopoly would earn in the market. One way to maintain a cartel is to bind ﬁrms with explicit pricing rules. Such explicit pricing rules are often prohibited by antitrust law. But ﬁrms need not resort to explicit pricing rules if they interact on the market repeatedly; they can collude tacitly. High collusive prices can be maintained with 531 532 Part 6: Market Power FIGURE 15.1 Pricing and Output under Imperfect Competition Market equilibrium under imperfect competition can occur at many points on the demand curve. In the ﬁgure, which assumes that marginal costs are constant over all output ranges, the equilibrium of the Bertrand game occurs at point C, also corresponding to the perfectly competitive outcome. The perfect cartel outcome occurs at point M, also corresponding to the monopoly outcome. Many solutions may occur between points M and C, depending on the speciﬁc assumptions made about how ﬁrms compete. For example, the equilibrium of the Cournot game might occur at a point such as A. The deadweight loss given by the shaded triangle increases as one moves from point C to M. Price PM PA PC M 1 2 A 3 C MC D MR QM QA QC Quantity the tacit threat of a price war if any ﬁrm undercuts. We will analyze this game formally and discuss the difﬁculty of maintaining collusion. The Bertrand and cartel models determine the outer limits between which actual prices in an imperfectly competitive market are set (one such intermediate price is represented by point A in Figure 15.1). This band of outcomes may be wide, and given the plethora of available models there may be a model for nearly every point within the band. For example, in a later section we will show how the Cournot model, in which ﬁrms set quantities rather than prices as in the Bertrand model, leads to an outcome (such as point A) somewhere between C and M in Figure 15.1. It is important to know where the industry is on the line between points C and M because total welfare (as measured by the sum of consumer surplus and ﬁrms’ proﬁts; see Chapter 12) depends on the location of this point. At point C, total welfare is as high as possible; at point A, total welfare is lower by the area of the shaded triangle 3. In Chapter 12, this shortfall in total welfare relative to the highest possible level was called deadweight loss. At point M, deadweight loss is even greater and is given by the area of shaded regions 1, 2, and 3. The closer the imperfectly competitive outcome to C and the farther from M, the higher is total welfare and the better off society will be.1 1Because this section deals with short-run decision variables (price and quantity), t
he discussion of total welfare in this paragraph focuses on short-run considerations. As discussed in a later section, an imperfectly competitive market may produce considerably more deadweight loss than a perfectly competitive one in the short run yet provide more innovation incentives, leading to lower production costs and new products and perhaps higher total welfare in the long run. The patent system intentionally impairs competition by granting a monopoly right to improve innovation incentives. Chapter 15: Imperfect Competition 533 Bertrand Model The Bertrand model is named after the economist who ﬁrst proposed it.2 The model is a game involving two identical ﬁrms, labeled 1 and 2, producing identical products at a constant marginal cost (and constant average cost) c. The ﬁrms choose prices p1 and p2 simultaneously in a single period of competition. Because ﬁrms’ products are perfect subp2. Let stitutes, all sales go to the ﬁrm with the lowest price. Sales are split evenly if p1 ¼ D( p) be market demand. We will look for the Nash equilibrium. The game has a continuum of actions, as does Example 8.5 (the Tragedy of the Commons) in Chapter 8. Unlike Example 8.5, we cannot use calculus to derive best-response functions because the proﬁt functions are not differentiable here. Starting from equal prices, if one ﬁrm lowers its price by the smallest amount, then its sales and proﬁt would essentially double. We will proceed by ﬁrst guessing what the Nash equilibrium is and then spending some time to verify that our guess was in fact correct. Nash equilibrium of the Bertrand game c. That is, The only pure-strategy Nash equilibrium of the Bertrand game is p"1 ¼ the Nash equilibrium involves both ﬁrms charging marginal cost. In saying that this is the only Nash equilibrium, we are making two statements that need to be veriﬁed: This outcome is a Nash equilibrium, and there is no other Nash equilibrium. p"2 ¼ To verify that this outcome is a Nash equilibrium, we need to show that both ﬁrms are playing a best response to each other—or, in other words, that neither ﬁrm has an incentive to deviate to some other strategy. In equilibrium, ﬁrms charge a price equal to marginal cost, which in turn is equal to average cost. But a price equal to average cost means ﬁrms earn zero proﬁt in equilibrium. Can a ﬁrm earn more than the zero it earns in equilibrium by deviating to some other price? No. If it deviates to a higher price, then it will make no sales and therefore no proﬁt, not strictly more than in equilibrium. If it deviates to a lower price, then it will make sales but will be earning a negative margin on each unit sold because price would be below marginal cost. Thus, the ﬁrm would earn negative proﬁt, less than in equilibrium. Because there is no possible proﬁtable deviation for the ﬁrm, we have succeeded in verifying that both ﬁrms’ charging marginal cost is a Nash equilibrium. p2. The same conclusions would be reached taking 2 to be the low-price ﬁrm. It is clear that marginal cost pricing is the only pure-strategy Nash equilibrium. If prices exceeded marginal cost, the high-price ﬁrm would gain by undercutting the other c p"2 ¼ slightly and capturing all the market demand. More formally, to verify that p"1 ¼ is the only Nash equilibrium, we will go one by one through an exhaustive list of cases for various values of p1, p2, and c, verifying that none besides p1 ¼ c is a Nash equilibrium. To reduce the number of cases, assume ﬁrm 1 is the low-price ﬁrm—that is, p l # p1. Case (i) cannot c on every unit it sells, and be a Nash equilibrium. Firm 1 earns a negative margin pl $ because it makes positive sales, it must earn negative proﬁt. It could earn higher proﬁt by deviating to a higher price. For example, ﬁrm 1 could guarantee itself zero proﬁt by deviating to p1 ¼ Case (ii) cannot be a Nash equilibrium either. At best, ﬁrm 2 gets only half of market p2) and at worst gets no demand (if p1 < p2). Firm 2 could capture all demand (if p1 ¼ the market demand by undercutting ﬁrm 1’s price by a tiny amount e. This e could be There are three exhaustive cases: (i) c > p1, (ii) c < p1, and (iii) c p2 ¼ ¼ c. 2J. Bertrand, ‘‘The´orie Mathematique de la Richess Sociale,’’ Journal de Savants (1883): 499–508. 534 Part 6: Market Power p2 chosen small enough that market price and total market proﬁt are hardly affected. If p1 ¼ before the deviation, the deviation would essentially double ﬁrm 2’s proﬁt. If pl < p2 before the deviation, the deviation would result in ﬁrm 2 moving from zero to positive proﬁt. In either case, ﬁrm 2’s deviation would be proﬁtable. c, which we saw is a Nash equilibrium. p2 ¼ Case (iii) includes the subcase of p1 ¼ p1 < p2. This subcase cannot be a p2 is c The only remaining subcase in which p1 # Nash equilibrium: Firm 1 earns zero proﬁt here but could earn positive proﬁt by deviating to a price slightly above c but still below p2. ¼ c. % p"n ¼ p"2 ¼ & & & ¼ Although the analysis focused on the game with two ﬁrms, it is clear that the same 2. The Nash equilibrium of the n-ﬁrm outcome would arise for any number of ﬁrms n Bertrand game is p"1 ¼ Bertrand paradox The Nash equilibrium of the Bertrand model is the same as the perfectly competitive outcome. Price is set to marginal cost, and ﬁrms earn zero proﬁt. This result—that the Nash equilibrium in the Bertrand model is the same as in perfect competition even though there may be only two ﬁrms in the market—is called the Bertrand paradox. It is paradoxical that competition between as few as two ﬁrms would be so tough. The Bertrand paradox is a general result in the sense that we did not specify the marginal cost c or the demand curve; therefore, the result holds for any c and any downward-sloping demand curve. In another sense, the Bertrand paradox is not general; it can be undone by changing various of the model’s other assumptions. Each of the next several sections will present a different model generated by changing a different one of the Bertrand assumptions. In the next section, for example, we will assume that ﬁrms choose quantity rather than price, leading to what is called the Cournot game. We will see that ﬁrms do not end up charging marginal cost and earning zero proﬁt in the Cournot game. In subsequent sections, we will show that the Bertrand paradox can also be avoided if still other assumptions are changed: if ﬁrms face capacity constraints rather than being able to produce an unlimited amount at cost c, if products are slightly differentiated rather than being perfect substitutes, or if ﬁrms engage in repeated interaction rather than one round of competition. Cournot Model The Cournot model, named after the economist who proposed it,3 is similar to the Bertrand model except that ﬁrms are assumed to simultaneously choose quantities rather than prices. As we will see, this simple change in strategic variable will lead to a big change in implications. Price will be above marginal cost, and ﬁrms will earn positive proﬁt in the Nash equilibrium of the Cournot game. It is somewhat surprising (but nonetheless an important point to keep in mind) that this simple change in choice variable matters in the strategic setting of an oligopoly when it did not matter with a monopoly: The monopolist obtained the same proﬁt-maximizing outcome whether it chose prices or quantities. We will start with a general version of the Cournot game with n ﬁrms indexed by 1, . . . , n. Each ﬁrm chooses its output qi of an identical product simultaneously. qn, i The outputs are combined into a total industry output Q ¼ q1 þ q2 þ & & & þ ¼ 3A. Cournot, Researches into the Mathematical Principles of the Theory of Wealth, trans. N. T. Bacon (New York: Macmillan, 1897). Although the Cournot model appears after Bertrand’s in this chapter, Cournot’s work, originally published in 1838, predates Bertrand’s. Cournot’s work is one of the ﬁrst formal analyses of strategic behavior in oligopolies, and his solution concept anticipated Nash equilibrium. Chapter 15: Imperfect Competition 535 resulting in market price P(Q). Observe that P(Q) is the inverse demand curve corresponding to the market demand curve Q D(P). Assume market demand is down¼ ward sloping and so inverse demand is, too; that is, P 0(Q) < 0. Firm i’s proﬁt equals its total revenue, P(Q)qi, minus its total cost, Ci(qi): : qiÞ qi $ Þ pi ¼ (15:1) Cið Q ð P Nash equilibrium of the Cournot game Unlike the Bertrand game, the proﬁt function (15.1) in the Cournot game is differentiable; hence we can proceed to solve for the Nash equilibrium of this game just as we did in Example 8.5, the Tragedy of the Commons. That is, we ﬁnd each ﬁrm i’s best response by taking the ﬁrst-order condition of the objective function (15.1) with respect to qi: @pi @qi ¼ P Q ð Þ þ P 0 qi Q ð Þ C 0i ð qiÞ $ ¼ 0: MR MC (15:2) ¼ Equation 15.2 must hold for all i \|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} 1, . . . , n in the Nash equilibrium. \|ﬄﬄ{zﬄﬄ} According to Equation 15.2, the familiar condition for proﬁt maximization from Chapter 11—marginal revenue (MR) equals marginal cost (MC)—holds for the Cournot ﬁrm. As we will see from an analysis of the particular form that the marginal revenue term takes for the Cournot ﬁrm, price is above the perfectly competitive level (above marginal cost) but below the level in a perfect cartel that maximizes ﬁrms’ joint proﬁts. In order for Equation 15.2 to equal 0, price must exceed marginal cost by the magnitude of the ‘‘wedge’’ term P 0(Q)qi. If the Cournot ﬁrm produces another unit on top of its existing production of qi units, then, because demand is downward sloping, the additional unit causes market price to decrease by P 0(Q), leading to a loss of revenue of P 0(Q)qi (the wedge term) from ﬁrm i’s existing production. To compare the Cournot outcome with the perfect cartel outcome, note that the objec- tive for the cartel is to maximize joint proﬁt: n 1 j X ¼ pj ¼ P Q ð Þ n n qj $ 1 j X ¼ 1 j X ¼ Cjð : qjÞ Taking the ﬁrst-order c
ondition of Equation 15.3 with respect to qi gives @ @qi n 1 j X ¼ pj ! qj C 0i ð qiÞ $ ¼ 0: MR MC This ﬁrst-order condition is similar to Equation 15.2 except that the wedge term, \|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} \|ﬄﬄ{zﬄﬄ qj ¼ P 0 Q, Q ð Þ (15:3) (15:4) (15:5) is larger in magnitude with a perfect cartel than with Cournot ﬁrms. In maximizing joint proﬁts, the cartel accounts for the fact that an additional unit of ﬁrm i’s output, by reducing market price, reduces the revenue earned on all ﬁrms’ existing output. Hence P 0(Q) is multiplied by total cartel output Q in Equation 15.5. The Cournot ﬁrm accounts for the reduction in revenue only from its own existing output qi. Hence Cournot ﬁrms will end up overproducing relative to the joint proﬁt-maximizing outcome. That is, the extra production in the Cournot outcome relative to a perfect cartel will end up in lower joint 536 Part 6: Market Power proﬁt for the ﬁrms. What ﬁrms would regard as overproduction is good for society because it means that the Cournot outcome (point A, referring back to Figure 15.1) will involve more total welfare than the perfect cartel outcome (point M in Figure 15.1). EXAMPLE 15.1 Natural-Spring Duopoly As a numerical example of some of these ideas, we will consider a case with just two ﬁrms and simple demand and cost functions. Following Cournot’s nineteenth-century example of two natural springs, we assume that each spring owner has a large supply of (possibly healthful) water and faces the problem of how much to provide the market. A ﬁrm’s cost of pumping and bottling qi liters is Ci(qi) cqi, implying that marginal costs are a constant c per liter. Inverse ¼ demand for spring water is Q ð where a is the demand intercept (measuring the strength of spring water demand) and Q q2 is total spring water output. We will now examine various models of how this market might operate. q1 þ (15:6) Þ ¼ Q, $ ¼ P a Bertrand model. In the Nash equilibrium of the Bertrand game, the two ﬁrms set price equal to marginal cost. Hence market price is P" c, total output is Q" c, ﬁrm proﬁt is ¼ 0, and total proﬁt for all ﬁrms is G" 0. For the Bertrand quantity to be positive we p"i ¼ must have a > c, which we will assume throughout the problem. ¼ ¼ $ a Cournot model. The solution for the Nash equilibrium follows Example 8.6 closely. Proﬁts for the two Cournot ﬁrms are a a P P Q ð Q ð p1 ¼ p2 ¼ cq1 ¼ ð cq2 ¼ ð q1 $ Þ q2 $ Þ Using the ﬁrst-order conditions to solve for the best-response functions, we obtain q2 $ 2 q1 ¼ Solving Equations 15.8 simultaneously yields the Nash equilibrium c q1, Þ q2: c Þ q1 $ q1 $ q2 $ q2 $ q1 $ 2 q2 "1 ¼ q"2 ¼ a c : $ 3 (15:7) (15:8) (15:9) Thus, total output is Q" (2/3)(a implies an equilibrium price of P" þ functions (Equations 15.7) implies p"1 ¼ G" 2=9 a Þð p"2 ¼ ð p"1 ¼ $ ¼ 2. Þ ¼ $ ¼ c c). Substituting total output into the inverse demand curve 2c)/3. Substituting price and outputs into the proﬁt (a 2, so total market proﬁt equals c p"2 ¼ ð Þ a Þð 1=9 $ Perfect cartel. The objective function for a perfect cartel involves joint proﬁts q1 $ a q1 þ ð Þ The two ﬁrst-order conditions for maximizing Equation 15.10 with respect to q1 and q2 are the same: a p2 ¼ ð p1 þ q1 $ q2 $ q2 $ q2: Þ (15:10) $ $ c c @ @q1 ð p1 þ p2Þ ¼ @ @q2 ð p1 þ p2Þ ¼ a 2q1 $ 2q2 $ c $ ¼ 0: (15:11) The ﬁrst-order conditions do not pin down market shares for ﬁrms in a perfect cartel because they produce identical products at constant marginal cost. But Equation 15.11 does pin down total output: q"1 þ . Substituting total output into inverse demand implies that c Q" q"2 ¼ $ Þ the cartel price is P" c). Substituting price and quantities into Equation 15.10 implies a total cartel proﬁt of G" 1=2 a Þð (1/2)(a þ (1/4)(a c)2. ¼ ð ¼ ¼ $ Chapter 15: Imperfect Competition 537 Comparison. Moving from the Bertrand model to the Cournot model to a perfect cartel, because a > c we can show that quantity Q" decreases from a c). It can also be shown that price P" and industry proﬁt G" increase. For example, if a 0 (implying Q and that production is costless), then market quantity is 120 that inverse demand is P(Q) with Bertrand competition, 80 with Cournot competition, and 60 with a perfect cartel. Price increases from 0 to 40 to 60 across the cases, and industry proﬁt increases from 0 to 3,200 to 3,600. $ 120 and c c) to (1 / 2)(a c to (2 / 3)(a 120 $ $ $ ¼ ¼ ¼ QUERY: In a perfect cartel, do ﬁrms play a best response to each other’s quantities? If not, in which direction would they like to change their outputs? What does this say about the stability of cartels? EXAMPLE 15.2 Cournot Best-Response Diagrams Continuing with the natural-spring duopoly from Example 15.1, it is instructive to solve for the Nash equilibrium using graphical methods. We will graph the best-response functions given in the intersection between the best responses is the Nash equilibrium. As Equation 15.8; background, you may want to review a similar diagram (Figure 8.8) for the Tragedy of the Commons. The linear best-response functions are most easily graphed by plotting their intercepts, as =3, shown in Figure 15.2. The best-response functions intersect at the point q"1 ¼ c Þ which was the Nash equilibrium of the Cournot game computed using algebraic methods in Example 15.1. a q"2 ¼ ð $ FIGURE 15.215Best-Response Diagram for Cournot Duopoly Firms’ best responses are drawn as thick lines; their intersection (E ) is the Nash equilibrium of the Cournot game. Isoproﬁt curves for ﬁrm 1 increase until point M is reached, which is the monopoly outcome for ﬁrm 1. q2 a − c BR1(q2 π1 = 100 π1 = 200 BR2(q1) a − c q1 538 Part 6: Market Power Figure 15.2 displays ﬁrms’ isoproﬁt curves. An isoproﬁt curve for ﬁrm 1 is the locus of quantity pairs providing it with the same proﬁt level. To compute the isoproﬁt curve associated with a proﬁt level of (say) 100, we start by setting Equation 15.7 equal to 100: q1 ¼ Þ Then we solve for q2 to facilitate graphing the isoproﬁt: a p1 ¼ ð q2 $ q1 $ $ c 100: (15:12) q2 ¼ a c q1 $ 100 q1 : (15:13) $ $ Several example isoproﬁts for ﬁrm 1 are shown in the ﬁgure. As proﬁt increases from 100 to 200 to yet higher levels, the associated isoproﬁts shrink down to the monopoly point, which is the highest isoproﬁt on the diagram. To understand why the individual isoproﬁts are shaped like frowns, refer back to Equation 15.13. As ql approaches 0, the last term ( 100 /q1) dominates, causing the left side of the frown to turn down. As ql increases, the ql term in Equation 15.13 begins to dominate, causing the right side of the frown to turn down. $ $ Figure 15.3 shows how to use best-response diagrams to quickly tell how changes in such underlying parameters as the demand intercept a or marginal cost c would affect the equilibrium. Figure 15.3a depicts an increase in both ﬁrms’ marginal cost c. The best responses shift inward, resulting in a new equilibrium that involves lower output for both. Although ﬁrms have the same marginal cost in this example, one can imagine a model in which ﬁrms have different marginal cost parameters and so can be varied independently. Figure 15.3b depicts an increase in just ﬁrm 1’s marginal cost; only ﬁrm 1’s best response shifts. The new equilibrium involves lower output for ﬁrm 1 and higher output for ﬁrm 2. Although ﬁrm 2’s best response does not shift, it still increases its output as it anticipates a reduction in ﬁrm 1’s output and best responds to this anticipated output reduction. QUERY: Explain why ﬁrm 1’s individual isoproﬁts reach a peak on its best-response function in Figure 15.2. What would ﬁrm 2’s isoproﬁts look like in Figure 15.2? How would you represent an increase in demand intercept a in Figure 15.3? FIGURE 15.315Shifting Cournot Best Responses Firms’ initial best responses are drawn as solid lines, resulting in a Nash equilibrium at point E 0. Panel (a) depicts an increase in both ﬁrms’ marginal costs, shifting their best responses—now given by the dashed lines—inward. The new intersection point, and thus the new equilibrium, is point E 00. Panel (b) depicts an increase in just ﬁrm 1’s marginal cost. q2 q2 BR1(q2) BR1(q2) E ′ E″ E ′ E″ BR2(q1) q1 BR2(q1) q1 (a) Increase in both firms’ marginal costs (b) Increase in firm 1’s marginal cost Chapter 15: Imperfect Competition 539 to n ¼1 Varying the number of Cournot ﬁrms The Cournot model is particularly useful for policy analysis because it can represent the whole range of outcomes from perfect competition to perfect cartel/monopoly (i.e., the whole range of points between C and M in Figure 15.1) by varying the number of ﬁrms 1. For simplicity, consider the case of identical ﬁrms, which here n from n ¼ means the n ﬁrms sharing the same cost function C(qi). In equilibrium, ﬁrms will produce the same share of total output: qi ¼ Q/n into Equation 15.12, the wedge term becomes P 0(Q)Q/n. The wedge term disappears as n grows large; ﬁrms become inﬁnitesimally small. An inﬁnitesimally small ﬁrm effectively becomes a pricetaker because it produces so little that any decrease in market price from an increase in output hardly affects its revenue. Price approaches marginal cost and the market outcome approaches the perfectly competitive one. As n decreases to 1, the wedge term approaches that in Equation 15.5, implying the Cournot outcome approaches that of a perfect cartel. As the Cournot ﬁrm’s market share grows, it internalizes the revenue loss from a decrease in market price to a greater extent. Q/n. Substituting qi ¼ EXAMPLE 15.3 Natural-Spring Oligopoly Return to the natural springs in Example 15.1, but now consider a variable number n of ﬁrms rather than just two. The proﬁt of one of them, ﬁrm i, is a qi ¼ ð pi ¼ $ Þ $ It is convenient to express total output as Q qi þ qi is the output of all Q ﬁrms except for i. Taking the ﬁrst-order condition of Equation 15.14 with respect to qi, we recognize that ﬁrm i takes Q i as a given and thus treats it as a constant in the differentiation, qi $ $ i, where
Q a cqi ¼ ð i $ $ Q qi $ Þ qi: c Þ (15:14pi @qi ¼ a 2qi $ Q $ i $ c $ ¼ 0, (15:15) 1, 2, . . . , n. which holds for all i ¼ The key to solving the system of n equations for the n equilibrium quantities is to recognize that the Nash equilibrium involves equal quantities because ﬁrms are symmetric. Symmetry implies that (15:16) (15:17) (15:18) Q" $ q"i ¼ Substituting Equation 15.16 into 15.15 yields 2q"i $ ð n i ¼ Q" $ $ a nq"i $ q"i ¼ ð n $ 1 q"i : Þ 1 q"i $ Þ c $ ¼ 0, a or q"i ¼ ð : = c Þ Þ Total market output is n ð $ þ 1 and market price is P" Q" nq": (15:19) þ Substituting for q"i , Q", and P" into the ﬁrm’s proﬁt Equation 15.14, we have that total proﬁt for all ﬁrms is þ P" np"15:20) 540 Part 6: Market Power ¼ Setting n 1 in Equations 15.18–15.20 gives the monopoly outcome, which gives the same price, total output, and proﬁt as in the perfect cartel case computed in Example 15.1. Letting n grow without bound in Equations 15.18–15.20 gives the perfectly competitive outcome, the same outcome computed in Example 15.1 for the Bertrand case. QUERY: We used the trick of imposing symmetry after taking the ﬁrst-order condition for ﬁrm i’s quantity choice. It might seem simpler to impose symmetry before taking the ﬁrst-order condition. Why would this be a mistake? How would the incorrect expressions for quantity, price, and proﬁt compare with the correct ones here? Prices or quantities? Moving from price competition in the Bertrand model to quantity competition in the Cournot model changes the market outcome dramatically. This change is surprising on ﬁrst thought. After all, the monopoly outcome from Chapter 14 is the same whether we assume the monopolist sets price or quantity. Further thought suggests why price and quantity are such different strategic variables. Starting from equal prices, a small reduction in one ﬁrm’s price allows it to steal all the market demand from its competitors. This sharp beneﬁt from undercutting makes price competition extremely ‘‘tough.’’ Quantity competition is ‘‘softer.’’ Starting from equal quantities, a small increase in one ﬁrm’s quantity has only a marginal effect on the revenue that other ﬁrms receive from their existing output. Firms have less of an incentive to outproduce each other with quantity competition than to undercut each other with price competition. An advantage of the Cournot model is its realistic implication that the industry grows more competitive as the number n of ﬁrms entering the market increases from monopoly to perfect competition. In the Bertrand model there is a discontinuous jump from monopoly to perfect competition if just two ﬁrms enter, and additional entry beyond two has no additional effect on the market outcome. An apparent disadvantage of the Cournot model is that ﬁrms in real-world markets tend to set prices rather than quantities, contrary to the Cournot assumption that ﬁrms choose quantities. For example, grocers advertise prices for orange juice, say, $3.00 a container, in newpaper circulars rather than the number of containers it stocks. As we will see in the next section, the Cournot model applies even to the orange juice market if we reinterpret quantity to be the ﬁrm’s capacity, deﬁned as the most the ﬁrm can sell given the capital it has in place and other available inputs in the short run. Capacity Constraints For the Bertrand model to generate the Bertrand paradox (the result that two ﬁrms essentially behave as perfect competitors), ﬁrms must have unlimited capacities. Starting from equal prices, if a ﬁrm lowers its price the slightest amount, then its demand essentially doubles. The ﬁrm can satisfy this increased demand because it has no capacity constraints, giving ﬁrms a big incentive to undercut. If the undercutting ﬁrm could not serve all the demand at its lower price because of capacity constraints, that would leave some residual demand for the higher-priced ﬁrm and would decrease the incentive to undercut. Consider a two-stage game in which ﬁrms build capacity in the ﬁrst stage and ﬁrms choose prices p1 and p2 in the second stage.4 Firms cannot sell more in the second stage 4The model is due to D. Kreps and J. Scheinkman, Outcomes,’’ Bell Journal of Economics (Autumn 1983): 326–37. ‘‘Quantity Precommitment and Bertrand Competition Yield Cournot Chapter 15: Imperfect Competition 541 than the capacity built in the ﬁrst stage. If the cost of building capacity is sufﬁciently high, it turns out that the subgame-perfect equilibrium of this sequential game leads to the same outcome as the Nash equilibrium of the Cournot model. To see this result, we will analyze the game using backward induction. Consider the second-stage pricing game supposing the ﬁrms have already built capacities q1 and q2 in the ﬁrst stage. Let p be the price that would prevail when production is at capacity for both ﬁrms. A situation in which p1 ¼ p2 < p (15:21) is not a Nash equilibrium. At this price, total quantity demanded exceeds total capacity; therefore, ﬁrm 1 could increase its proﬁts by raising price slightly and continuing to sell q1. Similarly, p1 ¼ p2 > p (15:22) is not a Nash equilibrium because now total sales fall short of capacity. At least one ﬁrm (say, ﬁrm 1) is selling less than its capacity. By cutting price slightly, ﬁrm 1 can increase its proﬁts by selling up to its capacity, q1. Hence the Nash equilibrium of this secondstage game is for ﬁrms to choose the price at which quantity demanded exactly equals the total capacity built in the ﬁrst stage:5 p: p1 ¼ p2 ¼ Anticipating that the price will be set such that ﬁrms sell all their capacity, the ﬁrststage capacity choice game is essentially the same as the Cournot game. Therefore, the equilibrium quantities, price, and proﬁts will be the same as in the Cournot game. Thus, even in markets (such as orange juice sold in grocery stores) where it looks like ﬁrms are setting prices, the Cournot model may prove more realistic than it ﬁrst seems. (15:23) Product Differentiation Another way to avoid the Bertrand paradox is to replace the assumption that the ﬁrms’ products are identical with the assumption that ﬁrms produce differentiated products. Many (if not most) real-world markets exhibit product differentiation. For example, toothpaste brands vary somewhat from supplier to supplier—differing in ﬂavor, ﬂuoride content, whitening agents, endorsement from the American Dental Association, and so forth. Even if suppliers’ product attributes are similar, suppliers may still be differentiated in another dimension: physical location. Because demanders will be closer to some suppliers than to others, they may prefer nearby sellers because buying from them involves less travel time. Meaning of ‘‘the market’’ The possibility of product differentiation introduces some fuzziness into what we mean by the market for a good. With identical products, demanders were assumed to be indifferent about which ﬁrm’s output they bought; hence they shop at the lowest-price ﬁrm, leading to the law of one price. The law of one price no longer holds if demanders strictly 5For completeness, it should be noted that there is no pure-strategy Nash equilibrium of the second-stage game with unequal p2). The low-price ﬁrm would have an incentive to increase its price and/or the high-price ﬁrm would have an inprices (p1 6¼ centive to lower its price. For large capacities, there may be a complicated mixed-strategy Nash equilibrium, but this can be ruled out by supposing the cost of building capacity is sufﬁciently high. 542 Part 6: Market Power prefer one supplier to another at equal prices. Are green-gel and white-paste toothpastes in the same market or in two different ones? Is a pizza parlor at the outskirts of town in the same market as one in the middle of town? With differentiated products, we will take the market to be a group of closely related products that are more substitutable among each other (as measured by cross-price elasticities) than with goods outside the group. We will be somewhat loose with this deﬁnition, avoiding precise thresholds for how high the cross-price elasticity must be between goods within the group (and how low with outside goods). Arguments about which goods should be included in a product group often dominate antitrust proceedings, and we will try to avoid this contention here. ¼ Bertrand competition with differentiated products Return to the Bertrand model but now suppose there are n ﬁrms that simultaneously choose prices pi (i 1, . . . , n) for their differentiated products. Product i has its own speciﬁc attributes ai, possibly reﬂecting special options, quality, brand advertising, or location. A product may be endowed with the attribute (orange juice is by deﬁnition made from oranges and cranberry juice from cranberries), or the attribute may be the result of the ﬁrm’s choice and spending level (the orange juice supplier can spend more and make its juice from fresh oranges rather than from frozen concentrate). The various attributes serve to differentiate the products. Firm i’s demand is qið i is a list of all other ﬁrms’ prices besides i’s, and A i, ai, A $ , iÞ $ pi, P where P attributes besides i’s. Firm i’s total cost is $ (15:24) i is a list of all other ﬁrms’ $ and proﬁt is thus Cið qi, aiÞ (15:25) pi ¼ piqi $ : qi, aiÞ Cið With differentiated products, the proﬁt function (Equation 15.26) is differentiable, so we do not need to solve for the Nash equilibrium on a case-by-case basis as we did in the Bertrand model with identical products. We can solve for the Nash equilibrium as in the Cournot model, solving for best-response functions by taking each ﬁrm’s ﬁrst-order condition (here with respect to price rather than quantity). The ﬁrst-order condition from Equation 15.26 with respect to pi is (15:26) @pi @pi ¼ qi þ pi @qi @pi @Ci @qi & @qi @pi $ 0: ¼ A B (15:27) \|ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} \|ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} The ﬁrst two terms (labeled A) on the right side of Equation 15.27 are a sort of marginal revenue—not
the usual marginal revenue from an increase in quantity, but rather the marginal revenue from an increase in price. The increase in price increases revenue on existing sales of qi units, but we must also consider the negative effect of the reduction in sales (@qi/@pi multiplied by the price pi) that would have been earned on these sales. The last term, labeled B, is the cost savings associated with the reduced sales that accompany an increased price. The Nash equilibrium can be found by simultaneously solving the system of ﬁrst-order 1, . . . , n. If the attributes ai are also choice conditions in Equation 15.27 for all i ¼ Chapter 15: Imperfect Competition 543 variables (rather than just endowments), there will be another set of ﬁrst-order conditions to consider. For ﬁrm i, the ﬁrst-order condition with respect to ai has the form @pi @ai ¼ pi @qi @ai $ @Ci @ai $ @Ci @qi & @qi @ai ¼ 0: (15:28) The simultaneous solution of these ﬁrst-order conditions can be complex, and they yield few deﬁnitive conclusions about the nature of market equilibrium. Some insights from particular cases will be developed in the next two examples. EXAMPLE 15.4 Toothpaste as a Differentiated Product Suppose that two ﬁrms produce toothpaste, one a green gel and the other a white paste. To simplify the calculations, suppose that production is costless. Demand for product i is qi ¼ ai $ The positive coefﬁcient on pj, the other good’s price, indicates that the goods are gross substitutes. Firm i’s demand is increasing in the attribute ai, which we will take to be demanders’ inherent preference for the variety in question; we will suppose that this is an endowment rather than a choice variable for the ﬁrm (and so will abstract from the role of advertising to promote preferences for a variety). pi þ (15:29) : pj 2 Algebraic solution. Firm i’s proﬁt is pi ai $ $ 0 because i’s production is costless. The ﬁrst-order condition for proﬁt maximization piqi $ qiÞ ¼ pi ¼ pi þ (15:30) Cið % , pj 2 where Ci(qi) with respect to pi is ¼ @pi @pi ¼ ai $ 2pi þ pj 2 ¼ 0: Solving for pi gives the following best-response functions for i 1, 2: a1 þ $ Solving Equations 15.32 simultaneously gives the Nash equilibrium prices p1 ¼ p2 ¼ % % , : 1 2 p2 2 1 2 ¼ p1 a2 þ 2 $ The associated proﬁts are p"i ¼ 8 15 ai þ 2 15 aj: (15:31) (15:32) (15:33) ai þ Firm i’s equilibrium price is not only increasing in its own attribute, ai, but also in the other product’s attribute, aj. An increase in aj causes ﬁrm j to increase its price, which increases ﬁrm i’s demand and thus the price i charges. p"i ¼ (15:34) " # aj : 2 8 15 2 15 Graphical solution. We could also have solved for equilibrium prices graphically, as in Figure 15.4. The best responses in Equation 15.32 are upward sloping. They intersect at the Nash equilibrium, point E. The isoproﬁt curves for ﬁrm 1 are smile-shaped. To see this, take the expression for ﬁrm 1’s proﬁt in Equation 15.30, set it equal to a certain proﬁt level (say, 100), and solve for p2 to facilitate graphing it on the best-response diagram. We have 100 p1 þ p2 ¼ p1 $ (15:35) a1: 544 Part 6: Market Power FIGURE 15.415Best Responses for Bertrand Model with Differentiated Products Firm’ best responses are drawn as thick lines; their intersection (E ) is the Nash equilibrium. Isoproﬁt curves for ﬁrm 1 increase moving out along ﬁrm 1’s best-response function. p2 p2* a2 + c 2 BR1(p2) E 0 a1 + c 2 p1* BR2(p1) π1 = 200 π1 = 100 p1 The smile turns up as p1 approaches 0 because the denominator of 100/p1 approaches 0. The smile turns up as p1 grows large because then the second term on the right side of Equation 15.35 grows large. Isoproﬁt curves for ﬁrm 1 increase as one moves away from the origin along its best-response function. QUERY: How would a change in the demand intercepts be represented on the diagram? EXAMPLE 15.5 Hotelling’s Beach A simple model in which identical products are differentiated because of the location of their suppliers (spatial differentiation) was provided by H. Hotelling in the 1920s.6 As shown in Figure 15.5, two ice cream stands, labeled A and B, are located along a beach of length L. The stands make identical ice cream cones, which for simplicity are assumed to be costless to produce. Let a and b represent the ﬁrms’ locations on the beach. (We will take the locations of the ice cream stands as given; in a later example we will revisit ﬁrms’ equilibrium location choices.) Assume that demanders are located uniformly along the beach, one at each unit of length. Carrying ice cream a distance d back to one’s beach umbrella costs td 2 because ice cream melts more the higher the temperature t and the further one must walk.7 Consistent with the Bertrand assumption, ﬁrms choose prices pA and pB simultaneously. Determining demands. Let x be the location of the consumer who is indifferent between buying from the two ice cream stands. The following condition must be satisﬁed by x: pA þ t x ð $ 2 a Þ pB þ t b ð $ ¼ 2: x Þ (15:36) 6H. Hotelling, ‘‘Stability in Competition,’’ Economic Journal 39 (1929): 41–57. 7The assumption of quadratic ‘‘transportation costs’’ turns out to simplify later work, when we compute ﬁrms’ equilibrium locations in the model. Chapter 15: Imperfect Competition 545 FIGURE 15.515Hotelling’s Beach Ice cream stands A and B are located at points a and b along a beach of length L. The consumer who is indifferent between buying from the two stands is located at x. Consumers to the left of x buy from A and to the right buy from B. A’s demand B’s demand 0 a x b L The left side of Equation 15.36 is the generalized cost of buying from A (including the price paid and the cost of transporting the ice cream the distance x a). Similarly, the right side is the generalized cost of buying from B. Solving Equation 15.36 for x yields $ b a þ 2 þ x ¼ pB $ b 2t $ ð pA a Þ : (15:37) If prices are equal, the indifferent consumer is located midway between a and b. If A’s price is less than B’s, then x shifts toward endpoint L. (This is the case shown in Figure 15.5.) Because all demanders between 0 and x buy from A and because there is one consumer per unit distance, it follows that A’s demand equals x: qAð pA, pB, a The remaining L $ x consumers constitute B ’s demand: pB $ b 2t $ ð pA a Þ : (15:38) qBð pB, pA, b pA $ b 2t $ ð pB a Þ : (15:39) Solving for Nash equilibrium. The Nash equilibrium is found in the same way as in Example 15.4 except that, for demands, we use Equations 15.38 and 15.39 in place of Equation 15.29. Skipping the details of the calculations, the Nash equilibrium prices are p"A ¼ p" 2L Þð 4L Þð $ $ 15:40) These prices will depend on the precise location of the two stands and will differ from each other. For example, if we assume that the beach is L 70 $2:90. These price yards, and t differences arise only from the locational aspects of this problem—the cones themselves are identical and costless to produce. Because A is somewhat more favorably located than B, it can charge a higher price for its cones without losing too much business to B. Using Equation 15.38 shows that $0.001 (one tenth of a penny), then p"A ¼ 100 yards long, a ¼ $3:10 and p"B ¼ 40 yards, b ¼ ¼ ¼ 3:10 110 2 þ x ¼ 2 ð Þð $ 0:001 2:90 110 Þ Þð 52, * (15:41) 546 Part 6: Market Power so stand A sells 52 cones, whereas B sells only 48 despite its lower price. At point x, the consumer is indifferent between walking the 12 yards to A and paying $3.10 or walking 18 yards to B and paying $2.90. The equilibrium is inefﬁcient in that a consumer slightly to the right of x would incur a shorter walk by patronizing A but still chooses B because of A’s power to set higher prices. Equilibrium proﬁts are p"A ¼ p"B ¼ t b 18 ð t b 18 ð a a 2L Þð 4L Þð þ $ a a þ $ $ $ 2, 2: b b Þ Þ (15:42) Somewhat surprisingly, the ice cream stands beneﬁt from faster melting, as measured here by 100, a the transportation cost t. For example, if we take L $0.001 as in ¼ $140 (rounding to the nearest dollar). If $160 and p"B ¼ the previous paragraph, then p"A ¼ transportation costs doubled to t $320 and ¼ p"B ¼ The transportation/melting cost is the only source of differentiation in the model. If t 0, then we can see from Equation 15.40 that prices equal 0 (which is marginal cost given that production is costless) and from Equation 15.42 that proﬁts equal 0—in other words, the Bertrand paradox results. ¼ $0.002, then proﬁts would double to p"A ¼ 70, and t $280. 40, b ¼ ¼ ¼ QUERY: What happens to prices and proﬁts if ice cream stands locate in the same spot? If they locate at the opposite ends of the beach? Consumer search and price dispersion Hotelling’s model analyzed in Example 15.5 suggests the possibility that competitors may have some ability to charge prices above marginal cost and earn positive proﬁts even if the physical characteristics of the goods they sell are identical. Firms’ various locations— closer to some demanders and farther from others—may lead to spatial differentiation. The Internet makes the physical location of stores less relevant to consumers, especially if shipping charges are independent of distance (or are not assessed). Even in this setting, ﬁrms can avoid the Bertrand paradox if we drop the assumption that demanders know every ﬁrm’s price in the market. Instead we will assume that demanders face a small cost s, called a search cost, to visit the store (or click to its website) to ﬁnd its price. Peter Diamond, winner of the Nobel Prize in economics in 2010, developed a model in which demanders search by picking one of the n stores at random and learning its price. Demanders know the equilibrium distribution of prices but not which store is charging which price. Demanders get their ﬁrst price search for free but then must pay s for additional searches. They need at most one unit of the good, and they all have the same gross surplus v for the one unit.8 Not only do stores manage to avoid the Bertrand paradox in this model, they obtain the polar opposite outco
me: All charge the monopoly price v, which extracts all consumer surplus! This outcome holds no matter how small the search cost s is—as long as s is positive (say, a penny). It is easy to see that all stores charging v is an equilibrium. If all charge the same price v, then demanders may as well buy from the ﬁrst store they search because additional searches are costly and do not end up revealing a lower price. It can also be seen that this is the only equilibrium. Consider any outcome in which at least one store charges less than v, and consider the lowest-price store (label it i) in this outcome. 8P. Diamond, ‘‘A Model of Price Adjustment,’’ Journal of Economic Theory 3 (1971): 156–68. Chapter 15: Imperfect Competition 547 Store i could raise its price pi by as much as s and still make all the sales it did before. The lowest price a demander could expect to pay elsewhere is no less than pt, and the demander would have to pay the cost s to ﬁnd this other price. Less extreme equilibria are found in models where consumers have different search costs.9 For example, suppose one group of consumers can search for free and another group has to pay s per search. In equilibrium, there will be some price dispersion across stores. One set of stores serves the low–search-cost demanders (and the lucky high– search-cost consumers who happen to stumble on a bargain). These bargain stores sell at marginal cost. The other stores serve the high–search-cost demanders at a price that makes these demanders indifferent between buying immediately and taking a chance that the next price search will uncover a bargain store. Tacit Collusion In Chapter 8, we showed that players may be able to earn higher payoffs in the subgameperfect equilibrium of an inﬁnitely repeated game than from simply repeating the Nash equilibrium from the single-period game indeﬁnitely. For example, we saw that, if players are patient enough, they can cooperate on playing silent in the inﬁnitely repeated version of the Prisoners’ Dilemma rather than ﬁnking on each other each period. From the perspective of oligopoly theory, the issue is whether ﬁrms must endure the Bertrand paradox (marginal cost pricing and zero proﬁts) in each period of a repeated game or whether they might instead achieve more proﬁtable outcomes through tacit collusion. A distinction should be drawn between tacit collusion and the formation of an explicit cartel. An explicit cartel involves legal agreements enforced with external sanctions if the agreements (e.g., to sustain high prices or low outputs) are violated. Tacit collusion can only be enforced through punishments internal to the market—that is, only those that can be generated within a subgame-perfect equilibrium of a repeated game. Antitrust laws generally forbid the formation of explicit cartels, so tacit collusion is usually the only way for ﬁrms to raise prices above the static level. Finitely repeated game Taking the Bertrand game to be the stage game, Selten’s theorem from Chapter 8 tells us that repeating the stage game any ﬁnite number of times T does not change the outcome. The only subgame-perfect equilibrium of the ﬁnitely repeated Bertrand game is to repeat the stage-game Nash equilibrium—marginal cost pricing—in each of the T periods. The game unravels through backward induction. In any subgame starting in period T, the unique Nash equilibrium will be played regardless of what happened before. Because the outcome in period T 1 does not affect the outcome in the next period, it is as though period T 1 is the last period, and the unique Nash equilibrium must be played then, too. Applying backward induction, the game unravels in this manner all the way back to the ﬁrst period. $ $ Inﬁnitely repeated game If the stage game is repeated inﬁnitely many periods, however, the folk theorem applies. The folk theorem indicates that any feasible and individually rational payoff can be sustained each period in an inﬁnitely repeated game as long as the discount factor, d, is close enough to unity. Recall that the discount factor is the value in the present period of one 9The following model is due to S. Salop and J. Stiglitz, ‘‘Bargains and Ripoffs: A Model of Monopolistically Competitive Price Dispersion,’’ Review of Economic Studies 44 (1977): 493–510. 548 Part 6: Market Power dollar earned one period in the future—a measure, roughly speaking, of how patient players are. Because the monopoly outcome (with proﬁts divided among the ﬁrms) is a feasible and individually rational outcome, the folk theorem implies that the monopoly outcome must be sustainable in a subgame-perfect equilibrium for d close enough to 1. Let’s investigate the threshold value of d needed. First suppose there are two ﬁrms competing in a Bertrand game each period. Let GM denote the monopoly proﬁt and PM the monopoly price in the stage game. The ﬁrms may collude tacitly to sustain the monopoly price—with each ﬁrm earning an equal share of the monopoly proﬁt—by using the grim trigger strategy of continuing to collude as long as no ﬁrm has undercut PM in the past but reverting to the stage-game Nash equilibrium of marginal cost pricing every period from then on if any ﬁrm deviates by undercutting. Successful tacit collusion provides the proﬁt stream V collude PM ¼ 2 þ d & PM 1 d þ PM ¼ ¼ 2 ð PM 2 2 þ d2 þ 1 1 d # d2 PM & 2 þ & & & þ & & &Þ : (15:43) d " þ þ $ d2 " # factors Refer to Chapter 8 for a discussion of adding up a series of discount 1 . We need to check that a ﬁrm has no incentive to deviate. By undercutting the collusive price PM slightly, a ﬁrm can obtain essentially all the monopoly proﬁt for itself in the current period. This deviation would trigger the grim strategy punishment of marginal cost pricing in the second and all future periods, so all ﬁrms would earn zero proﬁt from there on. Hence the stream of proﬁts from deviating is V deviate þ & & & GM. For this deviation not to be proﬁtable we must have V collude % ¼ V deviate or, on substituting, PM 2 " 1 1 " # $ % d # PM: (15:44) Rearranging Equation 15.44, the condition reduces to d 1/2. To prevent deviation, ﬁrms must value the future enough that the threat of losing proﬁts by reverting to the one-period Nash equilibrium outweighs the beneﬁt of undercutting and taking the whole monopoly proﬁt in the present period. % EXAMPLE 15.6 Tacit Collusion in a Bertrand Model Bertrand duopoly. Suppose only two ﬁrms produce a certain medical device used in surgery. The medical device is produced at constant average and marginal cost of $10, and the demand for the device is given by 5,000 Q ¼ $ 100P: (15:45) If the Bertrand game is played in a single period, then each ﬁrm will charge $10 and a total of 4,000 devices will be sold. Because the monopoly price in this market is $30, ﬁrms have a clear incentive to consider collusive strategies. At the monopoly price, total proﬁts each period are $40,000, and each ﬁrm’s share of total proﬁts is $20,000. According to Equation 15.44, collusion at the monopoly price is sustainable if 20,000 1 1 " $ % d # 40,000 (15:46) or if d % 1/2, as we saw. Chapter 15: Imperfect Competition 549 % Is the condition d 1/2 likely to be met in this market? That depends on what factors we consider in computing d, including the interest rate and possible uncertainty about whether the game will continue. Leave aside uncertainty for a moment and consider only the interest rate. If the period length is one year, then it might be reasonable to assume an annual interest rate of 10%. As shown in the Appendix to Chapter 17, d r 10%, then 0.91. This value of d clearly exceeds the threshold of 1/2 needed to sustain collusion. For d d to be less than the 1/2 threshold for collusion, we must incorporate uncertainty into the discount factor. There must be a signiﬁcant chance that the market will not continue into the next period—perhaps because a new surgical procedure is developed that renders the medical device obsolete. r); therefore, if r ¼ ¼ 1/(1 þ ¼ ¼ We focused on the best possible collusive outcome: the monopoly price of $30. Would collusion be easier to sustain at a lower price, say $20? No. At a price of $20, total proﬁts each period are $30,000, and each ﬁrm’s share is $15,000. Substituting into Equation 15.44, collusion can be sustained if 15,000 1 1 " $ % d # 30,000, (15:47) % 1/2. Whatever collusive proﬁt the ﬁrms try to sustain will cancel out from again implying d both sides of Equation 15.44, leaving the condition d 1/2. Therefore, we get a discrete jump in ﬁrms’ ability to collude as they become more patient—that is, as d increases from 0 to 1.10 For d below 1/2, no collusion is possible. For d above 1/2, any price between marginal cost and the monopoly price can be sustained as a collusive outcome. In the face of this multiplicity of subgame-perfect equilibria, economists often focus on the one that is most proﬁtable for the ﬁrms, but the formal theory as to why ﬁrms would play one or another of the equilibria is still unsettled. % Bertrand oligopoly. Now suppose n ﬁrms produce the medical device. The monopoly proﬁt continues to be $40,000, but each ﬁrm’s share is now only $40,000/n. By undercutting the monopoly price slightly, a ﬁrm can still obtain the whole monopoly proﬁt for itself regardless of how many other ﬁrms there are. Replacing the collusive proﬁt of $20,000 in Equation 15.46 with $40,000/n, we have that the n ﬁrms can successfully collude on the monopoly price if or 40,000 n 1 1 " $ % d # 40,000, d 1 $ % 1 n : (15:48) (15:49) Taking the ‘‘reasonable’’ discount factor of d 0.91 used previously, collusion is possible when 11 or fewer ﬁrms are in the market and impossible with 12 or more. With 12 or more ﬁrms, the only subgame-perfect equilibrium involves marginal cost pricing and zero proﬁts. ¼ Equation 15.49 shows that tacit collusion is easier the more patient are ﬁrms (as we saw before) and the fewer of them there are. One rationale used by antitrust authorities to
challenge certain mergers is that a merger may reduce n to a level such that Equation 15.49 begins to be satisﬁed and collusion becomes possible, resulting in higher prices and lower total welfare. QUERY: A period can be interpreted as the length of time it takes for ﬁrms to recognize and respond to undercutting by a rival. What would be the relevant period for competing gasoline stations in a small town? In what industries would a year be a reasonable period? 10The discrete jump in ﬁrms’ ability to collude is a feature of the Bertrand model; the ability to collude increases continuously with d in the Cournot model of Example 15.7. 550 Part 6: Market Power EXAMPLE 15.7 Tacit Collusion in a Cournot Model Suppose that there are again two ﬁrms producing medical devices but that each period they now engage in quantity (Cournot) rather than price (Bertrand) competition. We will again investigate the conditions under which ﬁrms can collude on the monopoly outcome. To generate the monopoly outcome in a period, ﬁrms need to produce 1,000 each; this leads to a price of $30, total proﬁts of $40,000, and ﬁrm proﬁts of $20,000. The present discounted value of the stream of these collusive proﬁts is V collude ¼ 20,000 1 " 1 $ d : # (15:50) Computing the present discounted value of the stream of proﬁts from deviating is somewhat complicated. The optimal deviation is not as simple as producing the whole monopoly output oneself and having the other ﬁrm produce nothing. The other ﬁrm’s 1,000 units would be provided to the market. The optimal deviation (by ﬁrm 1, say) would be to best respond to ﬁrm 2’s output of 1,000. To compute this best response, ﬁrst note that if demand is given by Equation 15.45, then inverse demand is given by 50 P ¼ $ Q 100 : (15:51) Firm 1’s proﬁt is p1 ¼ Pq1 $ cq1 ¼ q1 40 $ q2 q1 þ 100 $ % : (15:52) Taking the ﬁrst-order condition with respect to q1 and solving for q1 yields the best-response function q1 ¼ 2,000 q2 2 : $ (15:53) Firm 1’s optimal deviation when ﬁrm 2 produces 1,000 units is to increase its output from 1,000 to 1,500. Substituting these quantities into Equation 15.52 implies that ﬁrm 1 earns $22,500 in the period in which it deviates. How much ﬁrm 1 earns in the second and later periods following a deviation depends on the trigger strategies ﬁrms use to punish deviation. Assume that ﬁrms use the grim strategy of reverting to the Nash equilibrium of the stage game—in this case, the Nash equilibrium of the Cournot game—every period from then on. In the Nash equilibrium of the Cournot game, each ﬁrm best responds to the other in accordance with the best-response function in Equation 15.53 (switching subscripts in the case of ﬁrm 2). Solving these best-response equations simultaneously 4,000=3 and that proﬁts are implies that p"1 ¼ p"2 ¼ $17,778. Firm 1’s present discounted value of the stream of proﬁts from deviation is the Nash equilibrium outputs are q"1 ¼ q"2 ¼ V deviate ¼ ¼ ¼ 22,500 22,500 $22,500 þ 17,778 d 17,778 þ þ ð & 17,778 & $17,778 þ d 1 Þð d d þ & þ : 1 d # $ " d2 d2 17,778 þ d3 & þ & & & þ & & &Þ We have V collude V deviate if % $20,000 or, after some algebra, if d " 0.53. % 1 $ 1 % d # $22,500 þ $17,778 " d 1 d # $ Unlike with the Bertrand stage game, with the Cournot stage game there is a possibility of some collusion for discount factors below 0.53. However, the outcome would have to involve higher outputs and lower proﬁts than monopoly. (15:54) (15:55) Chapter 15: Imperfect Competition 551 QUERY: The beneﬁt to deviating is lower with the Cournot stage game than with the Bertrand stage game because the Cournot ﬁrm cannot steal all the monopoly proﬁt with a small deviation. Why then is a more stringent condition (d 0.5) needed to collude on the monopoly outcome in the Cournot duopoly compared with the Bertrand duopoly? 0.53 rather than d % % Longer-Run Decisions: Investment, Entry, And Exit The chapter has so far focused on the most basic short-run decisions regarding what price or quantity to set. The scope for strategic interaction expands when we introduce longer-run decisions. Take the case of the market for cars. Longer-run decisions include whether to update the basic design of the car, a process that might take up to two years to complete. Longer-run decisions may also include investing in robotics to lower production costs, moving manufacturing plants closer to consumers and cheap inputs, engaging in a new advertising campaign, and entering or exiting certain product lines (say, ceasing the production of station wagons or starting production of hybrid cars). In making such decisions, an oligopolist must consider how rivals will respond to it. Will competition with existing rivals become tougher or milder? Will the decision lead to the exit of current rivals or encourage new ones to enter? Is it better to be the ﬁrst to make such a decision or to wait until after rivals move? Flexibility versus commitment Crucial to our analysis of longer-run decisions such as investment, entry, and exit is how easy it is to reverse a decision once it has been made. On ﬁrst thought, it might seem that it is better for a ﬁrm to be able to easily reverse decisions because this would give the ﬁrm more ﬂexibility in responding to changing circumstances. For example, a car manufacturer might be more willing to invest in developing a hybrid-electric car if it could easily change the design back to a standard gasoline-powered one should the price of gasoline (and the demand for hybrid cars along with it) decrease unexpectedly. Absent strategic considerations—and so for the case of a monopolist—a ﬁrm would always value ﬂexibility and reversibility. The ‘‘option value’’ provided by ﬂexibility is discussed in further detail in Chapter 7. Surprisingly, the strategic considerations that arise in an oligopoly setting may lead a ﬁrm to prefer its decision be irreversible. What the ﬁrm loses in terms of ﬂexibility may be offset by the value of being able to commit to the decision. We will see a number of instances of the value of commitment in the next several sections. If a ﬁrm can commit to an action before others move, the ﬁrm may gain a ﬁrst-mover advantage. A ﬁrm may use its ﬁrst-mover advantage to stake out a claim to a market by making a commitment to serve it and in the process limit the kinds of actions its rivals ﬁnd proﬁtable. Commitment is essential for a ﬁrst-mover advantage. If the ﬁrst mover could secretly reverse its decision, then its rival would anticipate the reversal and the ﬁrms would be back in the game with no ﬁrst-mover advantage. We already encountered a simple example of the value of commitment in the Battle of the Sexes game from Chapter 8. In the simultaneous version of the model, there were three Nash equilibria. In one pure-strategy equilibrium, the wife obtains her highest payoff by attending her favorite event with her husband, but she obtains lower payoffs in the other two equilibria (a pure-strategy equilibrium in which she attends her less favored 552 Part 6: Market Power event and a mixed-strategy equilibrium giving her the lowest payoff of all three). In the sequential version of the game, if a player were given the choice between being the ﬁrst mover and having the ability to commit to attending an event or being the second mover and having the ﬂexibility to be able to meet up with the ﬁrst wherever he or she showed up, a player would always choose the ability to commit. The ﬁrst mover can guarantee his or her preferred outcome as the unique subgame-perfect equilibrium by committing to attend his or her favorite event. Sunk costs Expenditures on irreversible investments are called sunk costs Sunk cost. A sunk cost is an expenditure on an investment that cannot be reversed and has no resale value. Sunk costs include expenditures on unique types of equipment (e.g., a newsprint-making machine) or job-speciﬁc training for workers (developing the skills to use the newsprint machine). There is sometimes confusion between sunk costs and what we have called ﬁxed costs. They are similar in that they do not vary with the ﬁrm’s output level in a production period and are incurred even if no output is produced in that period. But instead of being incurred periodically, as are many ﬁxed costs (heat for the factory, salaries for secretaries and other administrators), sunk costs are incurred only once in connection with a single investment.11 Some ﬁxed costs may be avoided over a sufﬁciently long run—say, by reselling the plant and equipment involved—but sunk costs can never be recovered because the investments involved cannot be moved to a different use. When the ﬁrm makes a sunk investment, it has committed itself to that investment, and this may have important consequences for its strategic behavior. First-mover advantage in the Stackelberg model The simplest setting to illustrate the ﬁrst-mover advantage is in the Stackelberg model, named after the economist who ﬁrst analyzed it.12 The model is similar to a duopoly version of the Cournot model except that—rather than simultaneously choosing the quantities of their identical outputs—ﬁrms move sequentially, with ﬁrm 1 (the leader) choosing its output ﬁrst and then ﬁrm 2 (the follower) choosing after observing ﬁrm 1’s output. We use backward induction to solve for the subgame-perfect equilibrium of this sequential game. Begin with the follower’s output choice. Firm 2 chooses the output q2 that maximizes its own proﬁt, taking ﬁrm 1’s output as given. In other words, ﬁrm 2 best responds to ﬁrm 1’s output. This results in the same best-response function for ﬁrm 2 as we computed in the Cournot game from the ﬁrst-order condition (Equation 15.2). Label this best-response function BR2(q1). Turn then to the leader’s output choice. Firm 1 recognizes that it can inﬂuence the follower’s action because the follower best responds to 1’s observed output. Substituting BR2(q1) into the proﬁt function for ﬁrm 1 given by Equation 15.1, we h
ave q1 þ ð BR2ð p1 ¼ q1 $ q1ÞÞ : q1Þ (15:56) C1ð P 11Mathematically, the notion of sunk costs can be integrated into the per-period total cost function as þ where S is the per-period amortization of sunk costs (e.g., the interest paid for funds used to ﬁnance capital investments), Ft is Ft; but if the production the per-period ﬁxed costs, c is marginal cost, and qt is per-period output. If qt ¼ period is long enough, then some or all of Ft may also be avoidable. No portion of S is avoidable, however. 12H. von Stackelberg, The Theory of the Market Economy, trans. A. T. Peacock (New York: Oxford University Press, 1952). 0, then Ct ¼ þ ¼ S Ct (qt) S Ft þ cqt, Chapter 15: Imperfect Competition 553 The ﬁrst-order condition with respect to q1 is @p1 @q1 ¼ P Q ð Þ þ P 0 Q q1 þ Þ ð P 0 Q ð Þ BR 20 q1 q1Þ ð C i0 qiÞ ¼ ð $ 0: (15:57) S \|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} This is the same ﬁrst-order condition computed in the Cournot model (see Equation 15.2) except for the addition of the term S, which accounts for the strategic effect of ﬁrm 1’s output on ﬁrm 2’s. The strategic effect S will lead ﬁrm 1 to produce more than it would have in a Cournot model. By overproducing, ﬁrm 1 leads ﬁrm 2 to reduce q2 by ; the fall in ﬁrm 2’s output increases market price, thus increasing the the amount BR02ð revenue that ﬁrm 1 earns on its existing sales. We know that q2 decreases with an increase in ql because best-response functions under quantity competition are generally downward sloping; see Figure 15.2 for an illustration. q1Þ The strategic effect would be absent if the leader’s output choice were unobservable to the follower or if the leader could reverse its output choice in secret. The leader must be able to commit to an observable output choice or else ﬁrms are back in the Cournot game. It is easy to see that the leader prefers the Stackelberg game to the Cournot game. The leader could always reproduce the outcome from the Cournot game by choosing its Cournot output in the Stackelberg game. The leader can do even better by producing more than its Cournot output, thereby taking advantage of the strategic effect S. EXAMPLE 15.8 Stackelberg Springs Recall the two natural-spring owners from Example 15.1. Now, rather than having them choose outputs simultaneously as in the Cournot game, assume that they choose outputs sequentially as in the Stackelberg game, with ﬁrm 1 being the leader and ﬁrm 2 the follower. Firm 2’s output. We will solve for the subgame-perfect equilibrium using backward induction, starting with ﬁrm 2’s output choice. We already found ﬁrm 2’s best-response function in Equation 15.8, repeated here: q2 ¼ a $ c : q1 $ 2 (15:58) Firm 1’s output. Now fold the game back to solve for ﬁrm 1’s output choice. Substituting ﬁrm 2’s best response from Equation 15.58 into ﬁrm 1’s proﬁt function from Equation 15.56 yields a q1 $ $ p1 ¼ h $ a $ c q1 $ 2 q1 ¼ c i $ % 1 a 2 ð q1 $ c Þ q1: $ Taking the ﬁrst-order condition, @p1 @q1 ¼ 1 2 ð a 2q1 $ c Þ ¼ $ 0, (15:59) (15:60) $ $ 1=8 To provide a numerical example, suppose a =2. Substituting q"1 back into ﬁrm 2’s best-response function gives c Þ a and solving gives q"1 ¼ ð 1=16 =4. Proﬁts are p"1 ¼ ð q"2 ¼ ð a c a Þð Þ 30, 120 and c $900. Firm 1 produces twice as much and earns twice as much as ﬁrm 2. p"1 ¼ Recall from the simultaneous Cournot game in Example 15.1 that, for these numerical values, total market output was 80 and total industry proﬁt was 3,200, implying that each of the two $1,600. Therefore, when ﬁrm 1 is the ﬁrms produced 80/2 2. c Þ 0. Then q"1 ¼ 2 and p"2 ¼ ð ¼ 40 units and earned $3,200/2 $1,800, and p"2 ¼ 60, q"2 ¼ $ ¼ c Þ $ Þð a ¼ ¼ 554 Part 6: Market Power ﬁrst mover in a sequential game, it produces (60 1,600)/1,600 12.5% more than in the simultaneous game. 40)/40 $ ¼ 50% more and earns (1,800 ¼ $ Graphing the Stackelberg outcome. Figure 15.6 illustrates the Stackelberg equilibrium on a best-response function diagram. The leader realizes that the follower will always best respond, so the resulting outcome will always be on the follower’s best-response function. The leader effectively picks the point on the follower’s best-response function that maximizes the leader’s proﬁt. The highest isoproﬁt (highest in terms of proﬁt level, but recall from Figure 15.2 that higher proﬁt levels are reached as one moves down toward the horizontal axis) is reached at the point S of tangency between ﬁrm 1’s isoproﬁt and ﬁrm 2’s best-response function. This is the Stackelberg equilibrium. Compared with the Cournot equilibrium at point C, the Stackelberg equilibrium involves higher output and proﬁt for ﬁrm 1. Firm 1’s proﬁt is higher because, by committing to the high output level, ﬁrm 2 is forced to respond by reducing its output. Commitment is required for the outcome to stray from ﬁrm 1’s best-response function, as happens at point S. If ﬁrm 1 could secretly reduce q1 (perhaps because q1 is actual capacity that can be secretly reduced by reselling capital equipment for close to its purchase price to a manufacturer of another product that uses similar capital equipment), then it would move back to its best response, ﬁrm 2 would best respond to this lower quantity, and so on, following the dotted arrows from S back to C. FIGURE 15.615Stackelberg Game Best-response functions from the Cournot game are drawn as thick lines. Frown-shaped curves are ﬁrm 1’s isoproﬁts. Point C is the Nash equilibrium of the Cournot game (invoking simultaneous output choices). The Stackelberg equilibrium is point S, the point at which the highest isoproﬁt for ﬁrm 1 is reached on ﬁrm 2’s best-response function. At S, ﬁrm 1’s isoproﬁt is tangent to ﬁrm 2’s best-response function. If ﬁrm 1 cannot commit to its output, then the outcome function unravels, following the dotted line from S back to C. q2 BR1(q2) C S BR2(q1) q1 Chapter 15: Imperfect Competition 555 QUERY: What would be the outcome if the identity of the ﬁrst mover were not given and instead ﬁrms had to compete to be the ﬁrst? How would ﬁrms vie for this position? Do these considerations help explain overinvestment in Internet ﬁrms and telecommunications during the ‘‘dot-com bubble?’’ Contrast with price leadership In the Stackelberg game, the leader uses what has been called a ‘‘top dog’’ strategy,13 aggressively overproducing to force the follower to scale back its production. The leader earns more than in the associated simultaneous game (Cournot), whereas the follower earns less. Although it is generally true that the leader prefers the sequential game to the simultaneous game (the leader can do at least as well, and generally better, by playing its Nash equilibrium strategy from the simultaneous game), it is not generally true that the leader harms the follower by behaving as a ‘‘top dog.’’ Sometimes the leader beneﬁts by behaving as a ‘‘puppy dog,’’ as illustrated in Example 15.9. EXAMPLE 15.9 Price-Leadership Game Return to Example 15.4, in which two ﬁrms chose price for differentiated toothpaste brands simultaneously. So that the following calculations do not become too tedious, we make the simplifying assumptions that a1 ¼ 0. Substituting these parameters back into Example 15.4 shows that equilibrium prices are 2/3 0.444 for each 0.667 and proﬁts are 4/9 ﬁrm. a2 ¼ 1 and c ¼ * * Now consider the game in which ﬁrm 1 chooses price before ﬁrm 2.14 We will solve for the subgame-perfect equilibrium using backward induction, starting with ﬁrm 2’s move. Firm 2’s best response to its rival’s choice p1 is the same as computed in Example 15.4—which, on substituting a2 ¼ 0 into Equation 15.32, is 1 and c ¼ p2 ¼ 1 2 þ p1 4 : (15:61) Fold the game back to ﬁrm 1’s move. Substituting ﬁrm 2’s best response into ﬁrm 1’s proﬁt function from Equation 15.30 gives (15:62) p1 ¼ p1 1 & p1 þ $ 1 2 1 2 þ p1 4 p1 8 ð ¼ 10 : 7p1Þ $ " Taking the ﬁrst-order condition and solving for the equilibrium price, we obtain p"1 * Substituting into Equation 15.61 gives p"2 * p"2 * simultaneous one, but now the follower earns even more than the leader. 0:714. 0:446 and 0:460. Both ﬁrms’ prices and proﬁts are higher in this sequential game than in the 0:679. Equilibrium proﬁts are p"1 * # ’ As illustrated in the best-response function diagram in Figure 15.7, ﬁrm 1 commits to a high price to induce ﬁrm 2 to raise its price also, essentially ‘‘softening’’ the competition between them. 13‘‘Top dog,’’ ‘‘puppy dog,’’ and other colorful labels for strategies are due to D. Fudenberg and J. Tirole, ‘‘The Fat Cat Effect, the Puppy Dog Ploy, and the Lean and Hungry Look,’’ American Economic Review Papers and Proceedings 74 (1984): 361–68. 14Sometimes this game is called the Stackelberg price game, although technically the original Stackelberg game involved quantity competition. 556 Part 6: Market Power FIGURE 15.715Price-Leadership Game Thick lines are best-response functions from the game in which ﬁrms choose prices for differentiated products. U-shaped curves are ﬁrm 1’s isoproﬁts. Point B is the Nash equilibrium of the simultaneous game, and L is the subgame-perfect equilibrium of the sequential game in which ﬁrm 1 moves ﬁrst. At L, ﬁrm 1’s isoproﬁt is tangent to ﬁrm 2’s best response. p2 L BR2(p1) B BR1(p2) p1 The leader needs a moderate price increase (from 0.667 to 0.714) to induce the follower to increase its price slightly (from 0.667 to 0.679), so the leader’s proﬁts do not increase as much as the follower’s. QUERY: What choice variable realistically is easier to commit to, prices or quantities? What business strategies do ﬁrms use to increase their commitment to their list prices? We say that the ﬁrst mover is playing a ‘‘puppy dog’’ strategy in Example 15.9 because it increases its price relative to the simultaneous-move game; when translated into outputs, this means that the ﬁrst mover ends up producing less than in the simultaneousmove game. It is as though the ﬁrst mover strikes a less aggressive posture in the market and so leads its rival to compete less aggressively. A com
parison of Figures 15.6 and 15.7 suggests the crucial difference between the games that leads the ﬁrst mover to play a ‘‘top dog’’ strategy in the quantity game and a ‘‘puppy dog’’ strategy in the price game: The best-response functions have different slopes. The goal is to induce the follower to compete less aggressively. The slopes of the best-response functions determine whether the leader can best do that by playing aggressively itself or by softening its strategy. The ﬁrst mover plays a ‘‘top dog’’ strategy in the sequential quantity game or indeed any game in which best responses slope down. When best responses slope down, playing more aggressively induces a rival to respond by competing less aggressively. Conversely, the ﬁrst mover plays a ‘‘puppy dog’’ strategy in the price game or any game in which best responses slope up. When best responses slope up, playing less aggressively induces a rival to respond by competing less aggressively. Chapter 15: Imperfect Competition 557 Therefore, knowing the slope of ﬁrms’ best responses provides considerable insight into the sort of strategies ﬁrms will choose if they have commitment power. The Extensions at the end of this chapter provide further technical details, including shortcuts for determining the slope of a ﬁrm’s best-response function just by looking at its proﬁt function. Strategic Entry Deterrence We saw that, by committing to an action, a ﬁrst mover may be able to manipulate the second mover into being a less aggressive competitor. In this section we will see that the ﬁrst mover may be able to prevent the entry of the second mover entirely, leaving the ﬁrst mover as the sole ﬁrm in the market. In this case, the ﬁrm may not behave as an unconstrained monopolist because it may have distorted its actions to fend off the rival’s entry. In deciding whether to deter the second mover’s entry, the ﬁrst mover must weigh the costs and beneﬁts relative to accommodating entry—that is, allowing entry to happen. Accommodating entry does not mean behaving nonstrategically. The ﬁrst mover would move off its best-response function to manipulate the second mover into being less competitive, as described in the previous section. The cost of deterring entry is that the ﬁrst mover would have to move off its best-response function even further than it would if it accommodates entry. The beneﬁt is that it operates alone in the market and has market demand to itself. Deterring entry is relatively easy for the ﬁrst mover if the second mover must pay a substantial sunk cost to enter the market. EXAMPLE 15.10 Deterring Entry of a Natural Spring Recall Example 15.8, where two natural-spring owners choose outputs sequentially. We now add an entry stage: In particular, after observing ﬁrm 1’s initial quantity choice, ﬁrm 2 decides whether to enter the market. Entry requires the expenditure of sunk cost K2, after which ﬁrm 2 can choose output. Market demand and cost are as in Example 15.8. To simplify the 0 [implying that inverse calculations, we will take the speciﬁc numerical values a demand is P(Q) Q, and that production is costless]. To further simplify, we will abstract from ﬁrm 1’s entry decision and assume that it has already sunk any cost needed to enter before the start of the game. We will look for conditions under which ﬁrm 1 prefers to deter rather than accommodate ﬁrm 2’s entry. 120 and c 120 ¼ $ ¼ ¼ Accommodating entry. Start by computing ﬁrm 1’s proﬁt if it accommodates ﬁrm 2’s entry, denoted pacc 1 . This has already been done in Example 15.8, in which there was no issue of deterring qacc =2 ﬁrm 2’s entry. There we found ﬁrm 1’s equilibrium output to be 1 and its proﬁt to Þ ¼ a be 0, we have 120 and c ð qacc 1 ¼ a ð pacc 1 . Substituting the speciﬁc numerical values a 2=8 $ Þ ¼ 60 and pacc 1 ¼ ð 2=8 Þ 1,800. 120 ¼ ¼ $ ¼ $ 0 c c Deterring entry. Next, compute ﬁrm 1’s proﬁt if it deters ﬁrm 2’s entry, denoted pdet deter entry, ﬁrm 1 needs to produce an amount qdet responds to qdet 15.58 that ﬁrm 2’s best-response function is 1 . To 1 high enough that, even if ﬁrm 2 best 1 , it cannot earn enough proﬁt to cover its sunk cost K2. We know from Equation Substituting for q2 in ﬁrm 2’s proﬁt function (Equation 15.7) and simplifying gives q2 ¼ 120 q1 : $ 2 120 $ 2 2 qdet 1 # p2 ¼ " K2: $ (15:63) (15:64) 558 Part 6: Market Power Setting ﬁrm 2’s proﬁt in Equation 15.64 equal to 0 and solving yields $ is the ﬁrm 1 output needed to keep ﬁrm 2 out of the market. At this output level, ﬁrm 1’s ﬃﬃﬃﬃﬃ qdet 1 ¼ 120 K2p ; 2 (15:65) qdet 1 proﬁt is which we found by substituting qdet Equation 15.7. We also set q2 ¼ alone in the market. pdet 1 ¼ 1 , a K2p 120 2 K2p 2 , $ (15:66) * ﬃﬃﬃﬃﬃ 0 into ﬁrm 1’s proﬁt function from 0 because, if ﬁrm 1 is successful in deterring entry, it operates ) ﬃﬃﬃﬃﬃ 120, and c ¼ ¼ Comparison. The ﬁnal step is to juxtapose p acc ﬁrm 1 prefers deterring to accommodating entry. To simplify the algebra, let x pdet 1 ¼ to ﬁnd the condition under which K2p . Then 2 and p det pacc 1 ¼ if 1 1 ﬃﬃﬃﬃﬃ (15:67) Applying the quadratic formula yields x2 $ 120x 1,800 0: ¼ þ 120 + x ¼ p 2 7,200 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : (15:68) Taking the smaller root (because we will be looking for a minimum threshold), we have x (rounding to the nearest decimal). Substituting x yields 17.6 K2p and solving for K2 17.6 into x ¼ ¼ ¼ 2 ﬃﬃﬃﬃﬃ K2 ¼ 2 x 2 $ % 2 17:6 2 # ¼ " 77: * (15:69) 1 ¼ 77, then entry is so cheap for ﬁrm 2 that ﬁrm 1 would have to increase its output all the If K2 ¼ way to qdet 102 in order to deter entry. This is a signiﬁcant distortion above what it would produce when accommodating entry: qacc 60. If K2 < 77, then the output distortion needed to deter entry wastes so much proﬁt that ﬁrm 1 prefers to accommodate entry. If K2 > 77, output need not be distorted as much to deter entry; thus, ﬁrm 1 prefers to deter entry. 1 ¼ K. QUERY: Suppose the ﬁrst mover must pay the same entry cost as the second, K1 ¼ Suppose further that K is high enough that the ﬁrst mover prefers to deter rather than accommodate the second mover’s entry. Would this sunk cost not be high enough to keep the ﬁrst mover out of the market, too? Why or why not? K2 ¼ A real-world example of overproduction (or overcapacity) to deter entry is provided by the 1945 antitrust case against Alcoa, a U.S. aluminum manufacturer. A U.S. federal court ruled that Alcoa maintained much higher capacity than was needed to serve the market as a strategy to deter rivals’ entry, and it held that Alcoa was in violation of antitrust laws. To recap what we have learned in the last two sections: with quantity competition, the ﬁrst mover plays a ‘‘top dog’’ strategy regardless of whether it deters or accommodates the second mover’s entry. True, the entry-deterring strategy is more aggressive than the entry-accommodating one, but this difference is one of degree rather than kind. However, with price competition (as in Example 15.9), the ﬁrst mover’s entry-deterring strategy would differ in kind from its entry-accommodating strategy. It would play a ‘‘puppy dog’’ Chapter 15: Imperfect Competition 559 strategy if it wished to accommodate entry because this is how it manipulates the second mover into playing less aggressively. It plays a ‘‘top dog’’ strategy of lowering its price relative to the simultaneous game if it wants to deter entry. Two general principles emerge. • Entry deterrence is always accomplished by a ‘‘top dog’’ strategy whether competition is in quantities or prices, or (more generally) whether best-response functions slope down or up. The ﬁrst mover simply wants to create an inhospitable environment for the second mover. If ﬁrm 1 wants to accommodate entry, whether it should play a ‘‘puppy dog’’ or ‘‘top dog’’ strategy depends on the nature of competition—in particular, on the slope of the best-response functions. • Signaling The preceding sections have shown that the ﬁrst mover’s ability to commit may afford it a big strategic advantage. In this section we will analyze another possible ﬁrst-mover advantage: the ability to signal. If the second mover has incomplete information about market conditions (e.g., costs, demand), then it may try to learn about these conditions by observing how the ﬁrst mover behaves. The ﬁrst mover may try to distort its actions to manipulate what the second learns. The analysis in this section is closely tied to the material on signaling games in Chapter 8, and the reader may want to review that material before proceeding with this section. The ability to signal may be a plausible beneﬁt of being a ﬁrst mover in some settings in which the beneﬁt we studied earlier—commitment—is implausible. For example, in industries where the capital equipment is readily adapted to manufacture other products, costs are not very ‘‘sunk’’; thus, capacity commitments may not be especially credible. The ﬁrst mover can reduce its capacity with little loss. For another example, the price– leadership game involved a commitment to price. It is hard to see what sunk costs are involved in setting a price and thus what commitment value it has.15 Yet even in the absence of commitment value, prices may have strategic, signaling value. Entry-deterrence model Consider the incomplete information game in Figure 15.8. The game involves a ﬁrst mover (ﬁrm 1) and a second mover (ﬁrm 2) that choose prices for their differentiated products. Firm 1 has private information about its marginal cost, which can take on one of two values: high with probability Pr(H) or low with probability Pr(L) Pr(H). In period 1, ﬁrm 1 serves the market alone. At the end of the period, ﬁrm 2 observes ﬁrm 1’s price and decides whether to enter the market. If it enters, it sinks an entry cost K2 and learns the true level of ﬁrm 1’s costs; then ﬁrms compete as duopolists in the second period, choosing prices for differentiated products as in Example 15.4 or 15.5. (We do not need to be speciﬁc about the exact form of demands.) If ﬁrm 2 does not enter, it obtains a p
ayoff of zero, and ﬁrm 1 again operates alone in the market. Assume there is no discounting between periods. $ ¼ 1 Firm 2 draws inferences about ﬁrm 1’s cost from the price that ﬁrm 1 charges in the ﬁrst period. Firm 2 earns more if it competes against the high-cost type because the 15The Query in Example 15.9 asks you to consider reasons why a ﬁrm may be able to commit to a price. The ﬁrm may gain commitment power by using contracts (e.g., long-term supply contracts with customers or a most-favored customer clause, which ensures that if the ﬁrm lowers price in the future to other customers, then the favored customer gets a rebate on the price difference). The ﬁrm may advertise a price through an expensive national advertising campaign. The ﬁrm may have established a valuable reputation as charging ‘‘everyday low prices.’’ 560 Part 6: Market Power FIGURE 15.8 Signaling for Entry Deterrence Firm 1 signals its private information about its cost (high H or low L) through the price it sets in the ﬁrst period. Firm 2 observes ﬁrm 1’s price and then decides whether to enter. If ﬁrm 2 enters, the ﬁrms compete as duopolists; otherwise, ﬁrm 1 operates alone on the market again in the second period. Firm 2 earns positive proﬁt if and only if it enters against a high cost rival. Pr(H) Pr(L) 1 1 H p1 L p1 L p1 2 2 2 E NE E NE E NE M1 H + D1 H H, D2 2M1 H, 0 M1 H − R + D1 H H, D2 H − R, 0 2M1 M1 L + D1 L L, D2 2M1 L, 0 i be the duopoly proﬁt (not including entry costs) for ﬁrm i high-cost type’s price will be higher, and as we saw in Examples 15.4 and 15.5, the higher the rival’s price for a differentiated product, the higher the ﬁrm’s own demand and proﬁt. Let D t {1, 2} if ﬁrm 1 is of type t 2 , so that ﬁrm 2 earns more than its entry cost if it faces the high-cost type but not if it faces the lowcost type. Otherwise, the information in ﬁrm 1’s signal would be useless because ﬁrm 2 would always enter or always stay out regardless of ﬁrm 1’s type. 2 2 < K2 < D H {L, H }. To make the model interesting, we will suppose D L 2 To simplify the model, we will suppose that the low-cost type only has one relevant 1 . The high-cost type can 1 , or it 1 . Presumably, the optimal monopoly price 1 be ﬁrm 1’s monopoly proﬁt if it is 1 if it 1 if it is the low type). Let R be the high type’s loss relative to the 1 rather than its optimal in the ﬁrst period, then it earns action in the ﬁrst period—namely, setting its monopoly price p L choose one of two prices: can set the monopoly price associated with its type, pH can choose the same price as the low type, p L is increasing in marginal cost; thus, p L 1 < p H of type t is the high type and p L optimal monopoly proﬁt in the ﬁrst period if it charges p L monopoly price p H MH 1 . Let M t {L, H } (the proﬁt if it is alone and charges its optimal monopoly price pH 1 . Thus, if the high type charges p H 1 R. 1 in that period, but if it charges p L 1 , it earns MH 2 1 $ Chapter 15: Imperfect Competition 561 Separating equilibrium We will look for two kinds of perfect Bayesian equilibria: separating and pooling. In a separating equilibrium, the different types of the ﬁrst mover must choose different actions. Here, there is only one such possibility for ﬁrm 1: The low-cost type chooses p L 1 and the high-cost type chooses pH 1 . Firm 2 learns ﬁrm 1’s type from these actions perfectly and stays out on seeing p1 1 and enters on seeing p H 1 . It remains to check whether the high-cost type would prefer to deviate to p L 1 . In equilibrium, the high type earns a total proﬁt of MH 1 in the ﬁrst period because it charges its optimal monopoly price, and D H in the second because ﬁrm 2 enters and the ﬁrms compete as duopolists. If the 1 high type were to deviate to p L R in the ﬁrst period, the loss R coming from charging a price other than its ﬁrst-period optimum, but ﬁrm 2 would think it is the low type and would not enter. Hence ﬁrm 1 would earn MH 1 in the second period, for a total of 2MH R across periods. For deviation to be unproﬁtable we must have 1 , then it would earn MH 1 : MH 1 $ 1 þ D H 1 $ MH 1 þ DH 1 % 2MH 1 $ R (15:70) or (after rearranging) R MH DH 1 : (15:71) 1 $ That is, the high-type’s loss from distorting its price from its monopoly optimum in the ﬁrst period exceeds its gain from deterring ﬁrm 2’s entry in the second period. % If the condition in Equation 15.71 does not hold, there still may be a separating equilibrium in an expanded game in which the low type can charge other prices besides pL 1. The high type could distort its price downward below pL 1, increasing the ﬁrst-period loss the high type would suffer from pooling with the low type to such an extent that the high type would rather charge pH 1 even if this results in ﬁrm 2’s entry. Pooling equilibrium If the condition in Equation 15.71 does not hold, then the high type would prefer to pool with the low type if pooling deters entry. Pooling deters entry if ﬁrm 2’s prior belief that ﬁrm 1 is the high type, Pr(H)—which is equal to its posterior belief in a pooling equilibrium—is low enough that ﬁrm 2’s expected payoff from entering, H Pr ð is less than its payoff of zero from staying out of the market. 1 2 þ ½ H Pr ð 2 $ K2, $ Þ- D H Þ D L (15:72) Predatory pricing The incomplete-information model of entry deterrence has been used to explain why a rational ﬁrm might want to engage in predatory pricing, the practice of charging an artiﬁcially low price to prevent potential rivals from entering or to force existing rivals to exit. The predatory ﬁrm sacriﬁces proﬁts in the short run to gain a monopoly position in future periods. Predatory pricing is prohibited by antitrust laws. In the most famous antitrust case, dating back to 1911, John D. Rockefeller—owner of the Standard Oil Company that controlled a substantial majority of reﬁned oil in the United States—was accused of attempting to monopolize the oil market by cutting prices dramatically to drive rivals out and then raising prices after rivals had exited the market or sold out to Standard Oil. Predatory pricing remains a controversial antitrust issue because of the difﬁculty in distinguishing between predatory conduct, which authorities would like to prevent, and competitive conduct, which authorities would like to promote. In addition, economists initially had 562 Part 6: Market Power trouble developing game-theoretic models in which predatory pricing is rational and credible. Suitably interpreted, predatory pricing may emerge as a rational strategy in the incomplete-information model of entry deterrence. Predatory pricing can show up in a separating equilibrium—in particular, in the expanded model where the low-cost type can separate only by reducing price below its monopoly optimum. Total welfare is actually higher in this separating equilibrium than it would be in its full-information counterpart. Firm 2’s entry decision is the same in both outcomes, but the low-cost type’s price may be lower (to signal its type) in the predatory outcome. Predatory pricing can also show up in a pooling equilibrium. In this case it is the high-cost type that charges an artiﬁcially low price, pricing below its ﬁrst-period optimum to keep ﬁrm 2 out of the market. Whether social welfare is lower in the pooling equilibrium than in a full-information setting is unclear. In the ﬁrst period, price is lower (and total welfare presumably higher) in the pooling equilibrium than in a fullinformation setting. On the other hand, deterring ﬁrm 2’s entry results in higher secondperiod prices and lower welfare. Weighing the ﬁrst-period gain against the second-period loss would require detailed knowledge of demand curves, discount factors, and so forth. The incomplete-information model of entry deterrence is not the only model of predatory pricing that economists have developed. Another model involves frictions in the market for ﬁnancial capital that stem perhaps from informational problems (between borrowers and lenders) of the sort we will discuss in Chapter 18. With limits on borrowing, ﬁrms may only have limited resources to ‘‘make a go’’ in a market. A larger ﬁrm may force ﬁnancially strapped rivals to endure losses until their resources are exhausted and they are forced to exit the market. How Many Firms Enter? To this point, we have taken the number of ﬁrms in the market as given, often assuming that there are at most two ﬁrms (as in Examples 15.1, 15.3, and 15.10). We did allow for a general number of ﬁrms, n, in some of our analysis (as in Examples 15.3 and 15.7) but were silent about how this number n was determined. In this section, we provide a gametheoretic analysis of the number of ﬁrms by introducing a ﬁrst stage in which a large number of potential entrants can each choose whether to enter. We will abstract from ﬁrst-mover advantages, entry deterrence, and other strategic considerations by assuming that ﬁrms make their entry decisions simultaneously. Strategic considerations are interesting and important, but we have already developed some insights into strategic considerations from the previous sections and—by abstracting from them—we can simplify the analysis here. Barriers to entry For the market to be oligopolistic with a ﬁnite number of ﬁrms rather than perfectly competitive with an inﬁnite number of inﬁnitesimal ﬁrms, some factors, called barriers to entry, must eventually make entry unattractive or impossible. We discussed many of these factors at length in the previous chapter on monopoly. If a sunk cost is required to enter the market, then—even if ﬁrms can freely choose whether to enter—only a limited number of ﬁrms will choose to enter in equilibrium because competition among more than that number would drive proﬁts below the level needed to recoup the sunk entry cost. Government intervention in the form of patents or licensing requirements may prevent ﬁrms from entering even if it would be proﬁtable for them to do so. Some of the new concepts discussed in this chapter m

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.