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ay introduce additional barriers to entry. Search costs may prevent consumers from ﬁnding new entrants with lower Chapter 15: Imperfect Competition 563 prices and/or higher quality than existing ﬁrms. Product differentiation may raise entry barriers because of strong brand loyalty. Existing ﬁrms may bolster brand loyalty through expensive advertising campaigns, and softening this brand loyalty may require entrants to conduct similarly expensive advertising campaigns. Existing ﬁrms may take other strategic measures to deter entry, such as committing to a high capacity or output level, engaging in predatory pricing, or other measures discussed in previous sections. % Long-run equilibrium Consider the following game-theoretic model of entry in the long run. A large number of symmetric ﬁrms are potential entrants into a market. Firms make their entry decisions simultaneously. Entry requires the expenditure of sunk cost K. Let n be the number of ﬁrms that decide to enter. In the next stage, the n ﬁrms engage in some form of competition over a sequence of periods during which they earn the present discounted value of some constant proﬁt stream. To simplify, we will usually collapse the sequence of periods of competition into a single period. Let g (n) be the proﬁt earned by an individual ﬁrm in this competition subgame [not including the sunk cost, so g (n) is a gross proﬁt]. Presumably, the more ﬁrms in the market, the more competitive the market is and the less an individual ﬁrm earns, so g 0(n) < 0. We will look for a subgame-perfect equilibrium in pure strategies.16 This will be the number of ﬁrms, n", satisfying two conditions. First, the n" entering ﬁrms earn enough to cover their entry cost: g (n") K. Otherwise, at least one of them would have preferred to have deviated to not entering. Second, an additional ﬁrm cannot earn enough to cover its K. Otherwise, a ﬁrm that remained out of the market could have entry cost: g (n" proﬁtably deviated by entering. Given that g 0(n) < 0, we can put these two conditions together and say that n" is the greatest integer satisfying g (n") K. 1) þ % # This condition is reminiscent of the zero-proﬁt condition for long-run equilibrium under perfect competition. The slight nuance here is that active ﬁrms are allowed to earn positive proﬁts. Especially if K is large relative to the size of the market, there may only be a few long-run entrants (thus, the market looks like a canonical oligopoly) earning well above what they need to cover their sunk costs, yet an additional ﬁrm does not enter because its entry would depress individual proﬁt enough that the entrant could not cover its large sunk cost. Is the long-run equilibrium efﬁcient? Does the oligopoly involve too few or too many ﬁrms relative to what a benevolent social planner would choose for the market? Suppose the social planner can choose the number of ﬁrms (restricting entry through licenses and promoting entry through subsidizing the entry cost) but cannot regulate prices or other competitive conduct of the ﬁrms once in the market. The social planner would choose n to maximize CS n ð Þ þ ng n ð Þ $ nK; (15:73) where CS(n) is equilibrium consumer surplus in an oligopoly with n ﬁrms, ng(n) is total equilibrium proﬁt (gross of sunk entry costs) across all ﬁrms, and nK is the total expenditure on sunk entry costs. Let n"" be the social planner’s optimum. In general, the long-run equilibrium number of ﬁrms, n", may be greater or less than the social optimum, n"", depending on two offsetting effects: the appropriability effect and the business-stealing effect. 16A symmetric mixed-strategy equilibrium also exists in which sometimes more and sometimes fewer ﬁrms enter than can cover their sunk costs. There are multiple pure-strategy equilibria depending on the identity of the n" entrants, but n" is uniquely identiﬁed. 564 Part 6: Market Power • The social planner takes account of the beneﬁt of increased consumer surplus from lower prices, but ﬁrms do not appropriate consumer surplus and so do not take into account this beneﬁt. This appropriability effect would lead a social planner to choose more entry than in the long-run equilibrium: n"" > n". • Working in the opposite direction is that entry causes the proﬁts of existing ﬁrms to decrease, as indicated by the derivative g 0(n) < 0. Entry increases the competitiveness of the market, destroying some of ﬁrms’ proﬁts. In addition, the entrant ‘‘steals’’ some market share from existing ﬁrms—hence the term business-stealing effect. The marginal ﬁrm does not take other ﬁrms’ loss in proﬁts when making its entry decision, whereas the social planner would. The business-stealing effect biases long-run equilibrium toward more entry than a social planner would choose: n"" < n". Depending on the functional forms for demand and costs, the appropriability effect dominates in some cases, and there is less entry in long-run equilibrium than is efﬁcient. In other cases, the business-stealing dominates, and there is more entry in long-run equilibrium than is efﬁcient, as in Example 15.11. EXAMPLE 15.11 Cournot in the Long Run Long-run equilibrium. Return to Example 15.3 of a Cournot oligopoly. We will determine the long-run equilibrium number of ﬁrms in the market. Let K be the sunk cost a ﬁrm must pay to enter the market in an initial entry stage. Suppose there is one period of Cournot competition after entry. To further simplify the calculations, assume that a 0. Substituting these values back into Example 15.3, we have that an individual ﬁrm’s gross proﬁt is 1 and 15:74) The long-run equilibrium number of ﬁrms is the greatest integer n" satisfying g (n") Ignoring integer problems, n" satisﬁes K. % n" ¼ 1 Kp $ 1: (15:75) Social planner’s problem. We ﬁrst compute the individual terms in the social planner’s objective function (Equation 15.73). Consumer surplus equals the area of the shaded triangle in Figure 15.9, which, using the formula for the area of a triangle, is ﬃﬃﬃﬃ CS ð n Þ ¼ 1 2 Q ð n Þ½ a P n ð $ Þ15:76) here the last equality comes from substituting for price and quantity from Equations 15.18 and 15.19. Total proﬁts for all ﬁrms (gross of sunk costs) equal the area of the shaded rectangle: ng 15:77) Substituting from Equations 15.76 and 15.77 into the social planner’s objective function (Equation 15.73) gives þ After removing positive constants, the ﬁrst-order condition with respect to n is nK, (15:78) (15:79) Chapter 15: Imperfect Competition 565 FIGURE 15.915Proﬁt and Consumer Surplus in Example 15.11 Equilibrium for n ﬁrms drawn for the demand and cost assumptions in Example 15.11. Consumer surplus, CS(n), is the area of the shaded triangle. Total proﬁts ng(n) for all ﬁrms (gross of sunk costs) is the area of the shaded rectangle. Price 1 P(n) c = 0 CS(n) ng(n) Demand Q(n) 1 Quantity implying that Ignoring integer problems, this is the optimal number of ﬁrms for a social planner. n"" 1 K 1=3 $ 1: ¼ (15:80) Comparison. If K < 1 (a condition required for there to be any entry), then n"" < n", and so there is more entry in long-run equilibrium than a social planner would choose. To take a particular numerical example, let K 1.15, implying that the market would be 2.16 and n"" a duopoly in long-run equilibrium, but a social planner would have preferred a monopoly. 0.1. Then n" ¼ ¼ ¼ QUERY: If the social planner could set both the number of ﬁrms and the price in this example, what choices would he or she make? How would these compare to long-run equilibrium? Feedback effect We found that certain factors decreased the stringency of competition and increased ﬁrms’ proﬁts (e.g., quantity rather than price competition, product differentiation, search costs, discount factors sufﬁcient to sustain collusion). A feedback effect is that the more proﬁtable the market is for a given number of ﬁrms, the more ﬁrms will enter the market, making the market more competitive and less proﬁtable than it would be if the number of ﬁrms were ﬁxed. To take an extreme example, compare the Bertrand and Cournot games. Taking as given that the market involves two identical producers, we would say that the Bertrand 566 Part 6: Market Power game is much more competitive and less proﬁtable than the Cournot game. This conclusion would be reversed if ﬁrms facing a sunk entry cost were allowed to make rational entry decisions. Only one ﬁrm would choose to enter the Bertrand market. A second ﬁrm would drive gross proﬁt to zero, and so its entry cost would not be covered. The long-run equilibrium outcome would involve a monopolist and thus the highest prices and proﬁts possible, exactly the opposite of our conclusions when the number of ﬁrms was ﬁxed! On the other hand, the Cournot market may have space for several entrants driving prices and proﬁts below their monopoly levels in the Bertrand market. The moderating effect of entry should lead economists to be careful when drawing conclusions about oligopoly outcomes. Product differentiation, search costs, collusion, and other factors may reduce competition and increase proﬁts in the short run, but they may also lead to increased entry and competition in the long run and thus have ambiguous effects overall on prices and proﬁts. Perhaps the only truly robust conclusions about prices and proﬁts in the long run involve sunk costs. Greater sunk costs constrain entry even in the long run, so we can conﬁdently say that prices and proﬁts will tend to be higher in industries requiring higher sunk costs (as a percentage of sales) to enter.17 Innovation At the end of the previous chapter, we asked which market structure—monopoly or perfect competition—leads to more innovation in new products and cost-reducing processes. If monopoly is more innovative, will the long-run beneﬁts of innovation offset the short-run deadweight loss of monopoly? The same questions can be asked in the context of oligopoly. Do concentrated market structures, with few ﬁrms perhaps charging high prices,
provide better incentives for innovation? Which is more innovative, a large or a small ﬁrm? An established ﬁrm or an entrant? Answers to these questions can help inform policy toward mergers, entry regulation, and small-ﬁrm subsidies. As we will see with the aid of some simple models, there is no deﬁnite answer as to what level of concentration is best for long-run total welfare. We will derive some general trade-offs, but quantifying these trade-offs to determine whether a particular market would be more innovative if it were concentrated or unconcentrated will depend on the nature of competition for innovation, the nature of competition for consumers, and the speciﬁcation of demand and cost functions. The same can be said for determining what ﬁrm size or age is most innovative. The models we introduce here are of product innovations, the invention of a product (e.g., plasma televisions) that did not exist before. Another class of innovations is that of process innovations, which reduce the cost of producing existing products—for example, the use of robot technology in automobile manufacture. Monopoly on innovation Begin by supposing that only a single ﬁrm, call it ﬁrm 1, has the capacity to innovate. For example, a pharmaceutical manufacturer may have an idea for a malaria vaccine that no other ﬁrm is aware of. How much would the ﬁrm be willing to complete research and development for the vaccine and to test it with large-scale clinical trials? How does this willingness to spend (which we will take as a measure of the innovativeness of the ﬁrm) depend on concentration of ﬁrms in the market? 17For more on robust conclusions regarding industry structure and competitiveness, see J. Sutton, Sunk Costs and Market Structure (Cambridge, MA: MIT Press, 1991). Chapter 15: Imperfect Competition 567 Suppose ﬁrst that there is currently no other vaccine available for malaria. If ﬁrm 1 successfully develops the vaccine, then it will be a monopolist. Letting GM be the monopoly proﬁt, ﬁrm 1 would be willing to spend as much as GM to develop the vaccine. Next, to examine the case of a less concentrated market, suppose that another ﬁrm (ﬁrm 2) already has a vaccine on the market for which ﬁrm 1’s would be a therapeutic substitute. If ﬁrm 1 also develops its vaccine, the ﬁrms compete as duopolists. Let pD be the duop0, but pD > 0 in other oly proﬁt. In a Bertrand model with identical products, pD ¼ models—for example, models involving quantity competition or collusion. Firm 1 would be willing to spend as much as pD to develop the vaccine in this case. Comparing the two cases, because GM > pD, it follows that ﬁrm 1 would be willing to spend more (and, by this measure, would be more innovative) in a more concentrated market. The general principle here can be labeled a dissipation effect: Competition dissipates some of the proﬁt from innovation and thus reduces incentives to innovate. The dissipation effect is part of the rationale behind the patent system. A patent grants monopoly rights to an inventor, intentionally restricting competition to ensure higher proﬁts and greater innovation incentives. Another comparison that can be made is to see which ﬁrm, 1 or 2, has more of an incentive to innovate given that it has a monopoly on the initial idea. Firm 1 is initially out of the market and must develop the new vaccine to enter. Firm 2 is already in the malaria market with its ﬁrst vaccine but can consider developing a second one as well, which we will continue to assume is a perfect substitute. As shown in the previous paragraph, ﬁrm 1 would be willing to pay up to pD for the innovation. Firm 2 would not be willing to pay anything because it is currently a monopolist in the malaria vaccine market and would continue as a monopolist whether or not it developed the second medicine. (Crucial to this conclusion is that the ﬁrm with the initial idea can decline to develop it but still not worry that the other ﬁrm will take the idea; we will change this assumption in the next subsection.) Therefore, the potential competitor (ﬁrm 1) is more innovative by our measure than the existing monopolist (ﬁrm 2). The general principle here has been labeled a replacement effect: Firms gain less incremental proﬁt and thus have less incentive to innovate if the new product replaces an existing product already making proﬁt than if the ﬁrm is a new entrant in the market. The replacement effect can explain turnover in certain industries where old ﬁrms become increasingly conservative and are eventually displaced by innovative and quickly growing startups, as Microsoft displaced IBM as the dominant company in the computer industry and as Google now threatens to replace Microsoft. Competition for innovation New ﬁrms are not always more innovative than existing ﬁrms. The dissipation effect may counteract the replacement effect, leading old ﬁrms to be more innovative. To see this trade-off requires yet another variant of the model. Suppose now that more than one ﬁrm has an initial idea for a possible innovation and that they compete to see which can develop the idea into a viable product. For example, the idea for a new malaria vaccine may have occurred to scientists in two ﬁrms’ laboratories at about the same time, and the ﬁrms may engage in a race to see who can produce a viable vaccine from this initial idea. Continue to assume that ﬁrm 2 already has a malaria vaccine on the market and that this new vaccine would be a perfect substitute for it. The difference between the models in this and the previous section is that if ﬁrm 2 does not win the race to develop the idea, then the idea does not simply fall by the wayside but rather is developed by the competitor, ﬁrm 1. Firm 2 has an incentive to win the innovation competition to prevent ﬁrm 1 from becoming a competitor. Formally, if ﬁrm 1 wins the innovation competition, then it enters the market and is a competitor with ﬁrm 2, 568 Part 6: Market Power earning duopoly proﬁt pD. As we have repeatedly seen, this is the maximum that ﬁrm 1 would pay for the innovation. Firm 2’s proﬁt is GM if it wins the competition for the innovation but pD if it loses and ﬁrm 1 wins. Firm 2 would pay up to the difference, GM $ pD, for the innovation. If GM > 2pD—that is, if industry proﬁt under a monopoly is greater than under a duopoly, which it is when some of the monopoly proﬁt is dissipated by duopoly competition—then GM $ pD > pD, and ﬁrm 2 will have more incentive to innovate than ﬁrm 1. This model explains the puzzling phenomenon of dominant ﬁrms ﬁling for ‘‘sleeping patents’’: patents that are never implemented. Dominant ﬁrms have a substantial incentive—as we have seen, possibly greater than entrants’—to ﬁle for patents to prevent entry and preserve their dominant position. Whereas the replacement effect may lead to turnover in the market and innovation by new ﬁrms, the dissipation effect may help preserve the position of dominant ﬁrms and retard the pace of innovation. SUMMARY Many markets fall between the polar extremes of perfect competition and monopoly. In such imperfectly competitive markets, determining market price and quantity is complicated because equilibrium involves strategic interaction among the ﬁrms. In this chapter, we used the tools of game theory developed in Chapter 8 to study strategic interaction in oligopoly markets. We ﬁrst analyzed oligopoly ﬁrms’ short-run choices such as prices and quantities and then went on to analyze ﬁrms’ longer-run decisions such as product location, innovation, entry, and the deterrence of entry. We found that in modeling assumptions may lead to big changes in equilibrium outcomes. Therefore, predicting behavior in oligopoly markets may be difﬁcult based on theory alone and may require knowledge of particular industries and careful empirical analysis. Still, some general principles did emerge from our theoretical analysis that aid in understanding oligopoly markets. seemingly small changes • One of the most basic oligopoly models, the Bertrand model involves two identical ﬁrms that set prices simultaneously. The equilibrium resulted in the Bertrand paradox: Even though the oligopoly is the most concentrated possible, ﬁrms behave as perfect competitors, pricing at marginal cost and earning zero proﬁt. • The Bertrand paradox is not the inevitable outcome in an oligopoly but can be escaped by changing assumptions underlying the Bertrand model—for example, allowing for quantity competition, differentiated products, search costs, capacity constraints, or repeated play leading to collusion. • As in the Prisoners’ Dilemma, ﬁrms could proﬁt by coordinating on a less competitive outcome, but this outcome will be unstable unless ﬁrms can explicitly collude by forming a legal cartel or tacitly collude in a repeated game. • For tacit collusion to sustain supercompetitive proﬁts, ﬁrms must be patient enough that the loss from a price war in future periods to punish undercutting exceeds the beneﬁt from undercutting in the current period. • Whereas a nonstrategic monopolist prefers ﬂexibility to respond to changing market conditions, a strategic oligopolist may prefer to commit to a single choice. The ﬁrm can commit to the choice if it involves a sunk cost that cannot be recovered if the choice is later reversed. • A ﬁrst mover can gain an advantage by committing to a different action from what it would choose in the Nash equilibrium of the simultaneous game. To deter entry, the ﬁrst mover should commit to reducing the entrant’s proﬁts using an aggressive ‘‘top dog’’ strategy (high output or low price). If it does not deter entry, the ﬁrst mover should commit to a strategy leading its rival to compete less aggressively. This is sometimes a ‘‘top dog’’ and sometimes a ‘‘puppy dog’’ strategy, depending on the slope of ﬁrms’ best responses. • Holding the number of ﬁrms in an oligopoly constant in the short run, the introduction of a factor that softens competition (e.g., product dif
ferentiation, search costs, collusion) will increase ﬁrms’ proﬁt, but an offsetting effect in the long run is that entry—which tends to reduce oligopoly proﬁt—will be more attractive. • Innovation may be even more important than low prices for total welfare in the long run. Determining which oligopoly structure is the most innovative is difﬁcult because offsetting effects (dissipation and replacement) are involved. Chapter 15: Imperfect Competition 569 PROBLEMS 15.1 Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by Q 150 P. $ ¼ a. Calculate the proﬁt-maximizing price–quantity combination for this monopolist. Also calculate the monopolist’s proﬁt. b. Suppose instead that there are two ﬁrms in the market facing the demand and cost conditions just described for their identical products. Firms choose quantities simultaneously as in the Cournot model. Compute the outputs in the Nash equilibrium. Also compute market output, price, and ﬁrm proﬁts. c. Suppose the two ﬁrms choose prices simultaneously as in the Bertrand model. Compute the prices in the Nash equilibrium. Also compute ﬁrm output and proﬁt as well as market output. d. Graph the demand curve and indicate where the market price–quantity combinations from parts (a)–(c) appear on the curve. 15.2 Suppose that ﬁrms’ marginal and average costs are constant and equal to c and that inverse market demand is given by P where a, b > 0. a ¼ $ bQ, a. Calculate the proﬁt-maximizing price–quantity combination for a monopolist. Also calculate the monopolist’s proﬁt. b. Calculate the Nash equilibrium quantities for Cournot duopolists, which choose quantities for their identical products simultaneously. Also compute market output, market price, and ﬁrm and industry proﬁts. c. Calculate the Nash equilibrium prices for Bertrand duopolists, which choose prices for their identical products simultane- ously. Also compute ﬁrm and market output as well as ﬁrm and industry proﬁts. d. Suppose now that there are n identical ﬁrms in a Cournot model. Compute the Nash equilibrium quantities as functions of n. Also compute market output, market price, and ﬁrm and industry proﬁts. e. Show that the monopoly outcome from part (a) can be reproduced in part (d) by setting n 1, that the Cournot duopoly 2 in part (d), and that letting n approach inﬁnity yields ¼ outcome from part (b) can be reproduced in part (d) by setting n the same market price, output, and industry proﬁt as in part (c). ¼ ¼ $ 15.3 Let ci be the constant marginal and average cost for ﬁrm i (so that ﬁrms may have different marginal costs). Suppose demand is given by P Q. 1 a. Calculate the Nash equilibrium quantities assuming there are two ﬁrms in a Cournot market. Also compute market output, market price, ﬁrm proﬁts, industry proﬁts, consumer surplus, and total welfare. b. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in ﬁrm 1’s cost would change the equilibrium. Draw a representative isoproﬁt for ﬁrm 1. 15.4 Suppose that ﬁrms 1 and 2 operate under conditions of constant average and marginal cost but that ﬁrm 1’s marginal cost is c1 ¼ 20P. a. Suppose ﬁrms practice Bertrand competition, that is, setting prices for their identical products simultaneously. Compute the Nash equilibrium prices. (To avoid technical problems in this question, assume that if ﬁrms charge equal prices, then the low-cost ﬁrm makes all the sales.) 10 and ﬁrm 2’s is c2 ¼ 8. Market demand is Q 500 $ ¼ b. Compute ﬁrm output, ﬁrm proﬁt, and market output. c. Is total welfare maximized in the Nash equilibrium? If not, suggest an outcome that would maximize total welfare, and compute the deadweight loss in the Nash equilibrium compared with your outcome. 15.5 Consider the following Bertrand game involving two ﬁrms producing differentiated products. Firms have no costs of production. Firm 1’s demand is $ where b > 0. A symmetric equation holds for ﬁrm 2’s demand. q1 ¼ 1 p1 þ bp2, 570 Part 6: Market Power a. Solve for the Nash equilibrium of the simultaneous price-choice game. b. Compute the ﬁrms’ outputs and proﬁts. c. Represent the equilibrium on a best-response function diagram. Show how an increase in b would change the equilibrium. Draw a representative isoproﬁt curve for ﬁrm 1. 15.6 Recall Example 15.6, which covers tacit collusion. Suppose (as in the example) that a medical device is produced at constant average and marginal cost of $10 and that the demand for the device is given by 5,000 Q ¼ $ 100P: The market meets each period for an inﬁnite number of periods. The discount factor is d. a. Suppose that n ﬁrms engage in Bertrand competition each period. Suppose it takes two periods to discover a deviation because it takes two periods to observe rivals’ prices. Compute the discount factor needed to sustain collusion in a subgame-perfect equilibrium using grim strategies. b. Now restore the assumption that, as in Example 15.7, deviations are detected after just one period. Next, assume that n is not given but rather is determined by the number of ﬁrms that choose to enter the market in an initial stage in which entrants must sink a one-time cost K to participate in the market. Find an upper bound on n. Hint: Two conditions are involved. 15.7 Assume as in Problem 15.1 that two ﬁrms with no production costs, facing demand Q 150 P, choose quantities q1 and q2. ¼ $ a. Compute the subgame-perfect equilibrium of the Stackelberg version of the game in which ﬁrm 1 chooses q1 ﬁrst and then ﬁrm 2 chooses q2. b. Now add an entry stage after ﬁrm 1 chooses q1. In this stage, ﬁrm 2 decides whether to enter. If it enters, then it must sink cost K2, after which it is allowed to choose q2. Compute the threshold value of K2 above which ﬁrm 1 prefers to deter ﬁrm 2’s entry. c. Represent the Cournot, Stackelberg, and entry-deterrence outcomes on a best-response function diagram. 15.8 Recall the Hotelling model of competition on a linear beach from Example 15.5. Suppose for simplicity that ice cream stands can locate only at the two ends of the line segment (zoning prohibits commercial development in the middle of the beach). This question asks you to analyze an entry-deterring strategy involving product proliferation. a. Consider the subgame in which ﬁrm A has two ice cream stands, one at each end of the beach, and B locates along with A at the right endpoint. What is the Nash equilibrium of this subgame? Hint: Bertrand competition ensues at the right endpoint. b. If B must sink an entry cost KB, would it choose to enter given that ﬁrm A is in both ends of the market and remains there after entry? c. Is A’s product proliferation strategy credible? Or would A exit the right end of the market after B enters? To answer these questions, compare A’s proﬁts for the case in which it has a stand on the left side and both it and B have stands on the right to the case in which A has one stand on the left end and B has one stand on the right end (so B’s entry has driven A out of the right side of the market). Analytical Problems 15.9 Herﬁndahl index of market concentration One way of measuring market concentration is through the use of the Herﬁndahl index, which is deﬁned as qi/Q is ﬁrm i’s market share. The higher is H, the more concentrated the industry is said to be. Intuitively, more where st ¼ concentrated markets are thought to be less competitive because dominant ﬁrms in concentrated markets face little competitive pressure. We will assess the validity of this intuition using several models. H ¼ n s2 i , 1 i X ¼ Chapter 15: Imperfect Competition 571 a. If you have not already done so, answer Problem 15.2d by computing the Nash equilibrium of this n-ﬁrm Cournot game. Also compute market output, market price, consumer surplus, industry proﬁt, and total welfare. Compute the Herﬁndahl index for this equilibrium. b. Suppose two of the n ﬁrms merge, leaving the market with n 1 ﬁrms. Recalculate the Nash equilibrium and the rest of the items requested in part (a). How does the merger affect price, output, proﬁt, consumer surplus, total welfare, and the Herﬁndahl index? $ c. Put the model used in parts (a) and (b) aside and turn to a different setup: that of Problem 15.3, where Cournot duopolists face different marginal costs. Use your answer to Problem 15.3a to compute equilibrium ﬁrm outputs, market output, price, consumer surplus, industry proﬁt, and total welfare, substituting the particular cost parameters c1 ¼ 1/4. Also compute the Herﬁndahl index. c2 ¼ d. Repeat your calculations in part (c) while assuming that ﬁrm 1’s marginal cost c1 falls to 0 but c2 stays at 1/4. How does the cost change affect price, output, proﬁt, consumer surplus, total welfare, and the Herﬁndahl index? e. Given your results from parts (a)–(d), can we draw any general conclusions about the relationship between market concen- tration on the one hand and price, proﬁt, or total welfare on the other? 15.10 Inverse elasticity rule Use the ﬁrst-order condition (Equation 15.2) for a Cournot ﬁrm to show that the usual inverse elasticity rule from Chapter 11 holds under Cournot competition (where the elasticity is associated with an individual ﬁrm’s residual demand, the demand left after all rivals sell their output on the market). Manipulate Equation 15.2 in a different way to obtain an equivalent version of the inverse elasticity rule: P MC $ P si eQ, P , ¼ $ where si ¼ elasticity rule with that for a monopolist from the previous chapter. qi/Q is ﬁrm i’s market share and eQ, P is the elasticity of market demand. Compare this version of the inverse 15.11 Competition on a circle Hotelling’s model of competition on a linear beach is used widely in many applications, but one application that is difﬁcult to study in the model is free entry. Free entry is easiest to study in a model with symmetric ﬁrms, but more than two ﬁrms on a line cannot be symmetric because those located nearest the endpoints will have only one neighbori
ng rival, whereas those located nearer the middle will have two. To avoid this problem, Steven Salop introduced competition on a circle.18 As in the Hotelling model, demanders are located at each point, and each demands one unit of the good. A consumer’s surplus equals v (the value of consuming the good) minus the price paid for the good as well as the cost of having to travel to buy from the ﬁrm. Let this travel cost be td, where t is a parameter measuring how burdensome travel is and d is the distance traveled (note that we are here assuming a linear rather than a quadratic travel-cost function, in contrast to Example 15.5). Initially, we take as given that there are n ﬁrms in the market and that each has the same cost function Ci ¼ cqi, where K is the sunk cost required to enter the market [this will come into play in part (e) of the question, where we consider free entry] and c is the constant marginal cost of production. For simplicity, assume that the circumference of the circle equals 1 and that the n ﬁrms are located evenly around the circle at intervals of 1/n. The n ﬁrms choose prices pi simultaneously. þ K a. Each ﬁrm i is free to choose its own price (pi) but is constrained by the price charged by its nearest neighbor to either side. Let p" be the price these ﬁrms set in a symmetric equilibrium. Explain why the extent of any ﬁrm’s market on either side (x) is given by the equation tx p þ ¼ p" þ t[(1/n) x]. $ b. Given the pricing decision analyzed in part (a), ﬁrm i sells qi ¼ proﬁt-maximizing price for this ﬁrm as a function of p", c, t, and n. c. Noting that in a symmetric equilibrium all ﬁrms’ prices will be equal to p", show that pi ¼ d. Show that a ﬁrm’s proﬁts are t/n2 K in equilibrium. e. What will the number of ﬁrms n" be in long-run equilibrium in which ﬁrms can freely choose to enter? intuitively. p" $ þ ¼ c 2x because it has a market on both sides. Calculate the t/n. Explain this result 18See S. Salop, ‘‘Monopolistic Competition with Outside Goods,’’ Bell Journal of Economics (Spring 1979): 141–56. 572 Part 6: Market Power f. Calculate the socially optimal level of differentiation in this model, deﬁned as the number of ﬁrms (and products) that minimizes the sum of production costs plus demander travel costs. Show that this number is precisely half the number calculated in part (e). Hence this model illustrates the possibility of overdifferentiation. 15.12 Signaling with entry accommodation This question will explore signaling when entry deterrence is impossible; thus, the signaling ﬁrm accommodates its rival’s entry. Assume deterrence is impossible because the two ﬁrms do not pay a sunk cost to enter or remain in the market. The setup of the model will follow Example 15.4, so the calculations there will aid the solution of this problem. In particular, ﬁrm i’s demand is given by ai $ where ai is product i’s attribute (say, quality). Production is costless. Firm 1’s attribute can be one of two values: either a1 ¼ in which case we say ﬁrm 1 is the low type, or a1 ¼ across periods for simplicity. 1, 2, in which case we say it is the high type. Assume there is no discounting qi ¼ pi þ , pj 2 a. Compute the Nash equilibrium of the game of complete information in which ﬁrm 1 is the high type and ﬁrm 2 knows that ﬁrm 1 is the high type. b. Compute the Nash equilibrium of the game in which ﬁrm 1 is the low type and ﬁrm 2 knows that ﬁrm 1 is the low type. c. Solve for the Bayesian–Nash equilibrium of the game of incomplete information in which ﬁrm 1 can be either type with equal probability. Firm 1 knows its type, but ﬁrm 2 only knows the probabilities. Because we did not spend time this chapter on Bayesian games, you may want to consult Chapter 8 (especially Example 8.7). d. Which of ﬁrm 1’s types gains from incomplete information? Which type would prefer complete information (and thus would have an incentive to signal its type if possible)? Does ﬁrm 2 earn more proﬁt on average under complete information or under incomplete information? e. Consider a signaling variant of the model chat has two periods. Firms 1 and 2 choose prices in the ﬁrst period, when ﬁrm 2 has incomplete information about ﬁrm 1’s type. Firm 2 observes ﬁrm 1’s price in this period and uses the information to update its beliefs about ﬁrm 1’s type. Then ﬁrms engage in another period of price competition. Show that there is a separating equilibrium in which each type of ﬁrm 1 charges the same prices as computed in part (d). You may assume that, if ﬁrm 1 chooses an out-of-equilibrium price in the ﬁrst period, then ﬁrm 2 believes that ﬁrm 1 is the low type with probability 1. Hint: To prove the existence of a separating equilibrium, show that the loss to the low type from trying to pool in the ﬁrst period exceeds the second-period gain from having convinced ﬁrm 2 that it is the high type. Use your answers from parts (a)–(d) where possible to aid in your solution. SUGGESTIONS FOR FURTHER READING Carlton, D. W., and J. M. Perloff. Modern Industrial Organization, 4th ed. Boston: Addison-Wesley, 2005. Classic undergraduate text on industrial organization that covers theoretical and empirical issues. Kwoka, J. E., Jr., and L. J. White. The Antitrust Revolution, 4th ed. New York: Oxford University Press, 2004. Summarizes economic arguments on both sides of a score of important recent antitrust cases. Demonstrates the policy relevance of the theory developed in this chapter. J. Richards, and G. Norman. Pepall, L., D. Industrial Organization: Contemporary Theory and Practice, 2nd ed. Cincinnati, OH: Thomson South-Western, 2002. An undergraduate textbook providing a simple but thorough treatment of oligopoly theory. Uses the Hotelling model in a variety of additional applications including advertising. Sutton, J. Sunk Costs and Market Structure. Cambridge, MA: MIT Press, 1991. Argues that the robust predictions of oligopoly theory regard the size and nature of sunk costs. Second half provides detailed case studies of competition in various manufacturing industries. Tirole, J. The Theory of Industrial Organization. Cambridge, MA: MIT Press, 1988. A comprehensive survey of the topics discussed in this chapter and a host of others. Standard text used in graduate courses, but selected sections are accessible to advanced undergraduates. STRATEGIC SUBSTITUTES AND COMPLEMENTS EXTENSIONS We saw in the chapter that one can often understand the nature of strategic interaction in a market simply from the slope of ﬁrms’ best-response functions. For example, we argued that a ﬁrst mover that wished to accept rather than deter entry should commit to a strategy that leads its rival to behave less aggressively. What sort of strategy this is depends on the slope of ﬁrms’ best responses. If best responses slope downward, as in a Cournot model, then the ﬁrst mover should play a ‘‘top dog’’ strategy and produce a large quantity, leading its rival to reduce its production. If best responses slope upward, as in a Bertrand model with price competition for differentiated products, then the ﬁrst mover should play a ‘‘puppy dog’’ strategy and charge a high price, leading its rival to increase its price as well. More generally, we have seen repeatedly that best-response function diagrams are often helpful in understanding the nature of Nash equilibrium, how the Nash equilibrium changes with parameters of the model, how incomplete information might affect the game, and so forth. Simply knowing the slope of the best-response function is often all one needs to draw a usable best-response function diagram. By analogy to similar deﬁnitions from consumer and producer theory, game theorists deﬁne ﬁrms’ actions to be strategic substitutes if an increase in the level of the action (e.g., output, price, investment) by one ﬁrm is met by a decrease in that action by its rival. On the other hand, actions are strategic complements if an increase in an action by one ﬁrm is met by an increase in that action by its rival. E15.1 Nash equilibrium To make these ideas precise, suppose that ﬁrm 1’s proﬁt, p1(a1, a2), is a function of its action a1 and its rival’s (ﬁrm 2’s) action a2. (Here we have moved from subscripts to superscripts for indicating the ﬁrm to which the proﬁts belong to make room for subscripts that will denote partial derivatives.) Firm 2’s proﬁt function is denoted similarly. A Nash equilibrium is a proﬁle of actions for each ﬁrm, , such that each ﬁrm’s equilibrium action is a best response to the other’s. Let BR1(a2) be ﬁrm 1’s best-response function, and let BR2(a1) be ﬁrm 2’s; then a Nash equilibrium is given by a"1 ¼ and a"2 ¼ BR1ð E15.2 Best-response functions in more detail The ﬁrst-order condition for ﬁrm 1’s action choice is a"1, a"2Þ ð BR2ð . a"1Þ a"2Þ p1 a1, a2Þ ¼ 1ð 0, (i) where subscripts for p represent partial derivatives with respect to its various arguments. A unique maximum, and thus a unique best response, is guaranteed if we assume that the proﬁt function is concave: p1 11ð a1, a2Þ < 0: (ii) Given a rival’s action a2, the solution to Equation i for a maximum is ﬁrm 1’s best-response function: a1 ¼ BR1ð Since the best response is unique, BR1(a2) is indeed a function rather than a correspondence (see Chapter 8 for more on correspondences). : a2Þ (iii) The strategic relationship between actions is determined by the slope of the best-response functions. If best responses < 0], are downward sloping [i.e., if BR01ð a1Þ then a1 and a2 are strategic substitutes. If best responses are > 0], then upward sloping [i.e., if BR01ð a2Þ a1 and a2 are strategic complements. a2Þ > 0 and BR02ð < 0 and BR02ð a1Þ E15.3 Inferences from the proﬁt function We just saw that a direct route for determining whether actions are strategic substitutes or complements is ﬁrst to solve explicitly for best-response functions and then to differentiate them. In some applications, however, it is difﬁcult or impossible to ﬁnd an explicit solution to Equation i. We can still determine wheth
er actions are strategic substitutes or complements by drawing inferences directly from the proﬁt function. Substituting Equation iii into the ﬁrst-order condition of Equation i gives p1 BR1ð 1ð a2Þ , a2Þ ¼ 0: (iv) Totally differentiating Equation iv with respect to a2 yields, after dropping the arguments of the functions for brevity, p1 11BR01 þ Rearranging Equation v gives the derivative of response function: p1 12 ¼ 0: BR01 ¼ $ p1 12 p1 11 : (v) the best- (vi) In view of the second-order condition (Equation ii), the denominator of Equation vi is negative. Thus, the sign of BR01 is 574 Part 6: Market Power the same as the sign of the numerator, p1 12 > 0 implies BR01 > 0 and p1 12 < 0 implies BR01 < 0. The strategic relationship between the actions can be inferred directly from the cross-partial derivative of the proﬁt function. 12. That is, p1 E15.4 Cournot model In the Cournot model, proﬁts are given as a function of the two ﬁrms’ quantities: p1 q1, q2Þ ¼ ð The ﬁrst-order condition is q1P q1, q2Þ $ ð C : q1Þ ð (vii) p1 1 ¼ P q1P 0 q1 þ ð q2Þ þ q1 þ ð as we have already seen (Equation 15.2). The derivative of Equation viii with respect to q2 is, after dropping functions’ arguments for brevity, q2Þ $ , q1Þ ð (viii) C 0 (ix) p1 12 ¼ q1P 00 P 0: 0 and so p1 þ Because P 0 < 0, the sign of p1 12 will depend on the sign of P00—that is, the curvature of demand. With linear demand, P00 12 is clearly negative. Quantities are strategic substitutes in the Cournot model with linear demand. Figure 15.2 illustrates this general principle. This ﬁgure is drawn for an example involving linear demand, and indeed the best responses are downward sloping. ¼ More generally, quantities are strategic substitutes in the Cournot model unless the demand curve is ‘‘very’’ convex (i.e., unless P00 is positive and large enough to offset the last term in Equation ix). For a more detailed discussion see Bulow, Geanakoplous, and Klemperer (1985). E15.5 Bertrand model with differentiated products In the Bertrand model with differentiated products, demand can be written as q1 ¼ D1 p1, p2Þ : ð (x) See Equation 15.24 for a related expression. Using this notation, proﬁt can be written as p1 ð C ¼ ¼ D1 ð C q1Þ ð p1, p2Þ $ p1q1 $ p1D1 : p1, p2ÞÞ ð The ﬁrst-order condition with respect to p1 is D1 p1D1 p1, p2Þ p1, p2Þ þ 1ð ð D1 D1 : C 0 p1, p2Þ p1, p2ÞÞ 1ð ð ð The cross-partial derivative is, after dropping functions’ arguments for brevity, p1 1 ¼ (xii) (xi) $ p1 12 ¼ p1D1 12 þ D1 2 $ C 0D1 12 $ C 00D1 2D1 1: (xiii) Interpreting this mass of symbols is no easy task. In the 0) and linear special case of constant marginal cost (C00 ¼ 0 12 ¼ D1 ð , the sign of p1 Þ 12 is given by the sign of D1 demand 2 (i.e., how a ﬁrm’s demand is affected by changes in the rival’s price). In the usual case when the two goods are themselves substitutes, we have D1 2 > 0 and so p1 12 > 0. That is, prices are strategic complements. The terminology here can seem contradictory, so the result bears repeating: If the goods that the ﬁrms sell are substitutes, then the variables the ﬁrms choose— prices—are strategic complements. Firms in such a duopoly would either raise or lower prices together (see Tirole, 1988). We saw an example of this in Figure 15.4. The ﬁgure was drawn for the case of linear demand and constant marginal cost, and we saw that best responses are upward sloping. E15.6 Entry accommodation in a sequential game Consider a sequential game in which ﬁrm 1 chooses a1 and then ﬁrm 2 chooses a2. Suppose ﬁrm 1 ﬁnds it more proﬁtable to accommodate than to deter ﬁrm 2’s entry. Because ﬁrm 2 moves after ﬁrm 1, we can substitute ﬁrm 2’s best response into ﬁrm 1’s proﬁt function to obtain a1, BR2ð ð Firm 1’s ﬁrst-order condition is p1 : a1ÞÞ p1 1 þ p1 2BR02 S 0: ¼ (xiv) (xv) \|ﬄﬄ{zﬄﬄ} By contrast, the ﬁrst-order condition from the simultaneous game (see Equation i) is simply p1 0. The ﬁrst-order conditions from the sequential and simultaneous games differ in the term S. This term captures the strategic effect of moving ﬁrst—that is, whether the ﬁrst mover would choose a higher or lower action in the sequential game than in the simultaneous game. 1 ¼ The sign of S is determined by the signs of the two factors in S. We will argue in the next paragraph that these two factors will typically have the same sign (both positive or both negative), implying that S > 0 and hence that the ﬁrst mover will typically distort its action upward in the sequential game compared with the simultaneous game. This result conﬁrms the ﬁndings from several of the examples in the text. In Figure 15.6, we see that the Stackelberg quantity is higher than the Cournot quantity. In Figure 15.7, we see that the price leader distorts its price upward in the sequential game compared with the simultaneous one. Section E15.3 showed that the sign of BR02 is the same as the sign of p2 12. If there is some symmetry to the market, then the sign of p2 12 will be the same as the sign of p1 2 and p1 12 will have the same sign. For example, consider the case of Cournot competition. By Equation 15.1, ﬁrm 1’s proﬁt is q1 þ ð 12. Typically, p1 p1 ¼ q1 $ : q1Þ q2Þ (xvi) C P ð Therefore, p1 2 ¼ P 0 q1 þ ð q2Þ q1: (xvii) Because demand is downward sloping, it follows that p1 Differentiating Equation xvii with respect to q1 yields 2 < 0. p1 12 ¼ P 0 þ q1P 00: (xviii) 0) This expression is also negative if demand is linear (so P 00 or if demand is not too convex (so the last term in Equation xviii does not swamp the term P 0). ¼ E15.7 Extension to general investments The model from the previous section can be extended to general investments—that is, beyond a mere commitment to a quantity or price. Let K1 be this general investment—(say) advertising, investment in lower-cost manufacturing, or product positioning—sunk at the outset of the game. The two ﬁrms then choose their product-market actions a1 and a2 (representing prices or quantities) simultaneously in the second period. Firms’ proﬁts in this extended model are, respectively, p1 a1, a2, K1Þ ð and p2 : a1, a2Þ ð (xix) The analysis is simpliﬁed by assuming that ﬁrm 2’s proﬁt is not directly a function of K1, although ﬁrm 2’s proﬁt will indirectly depend on K1 in equilibrium because equilibrium actions will depend on K1. Let a"1ð be ﬁrms’ K1Þ actions in a subgame-perfect equilibrium: and a"2ð K1Þ a"1ð a"2ð a"2ð a"1ð Because ﬁrm 2’s proﬁt function does not depend directly on K1 in Equation xix, neither does its best response in Equation xx. K1Þ ¼ K1Þ ¼ BR1ð BR2ð K1Þ K1ÞÞ , , K1Þ : (xx) The analysis here draws on Fudenberg and Tirole (1984) and Tirole (1988). Substituting from Equation xx into Equation xix, the ﬁrms’ Nash equilibrium proﬁts in the subgame following ﬁrm 1’s choice of K1 are K1Þ K1Þ Fold the game back to ﬁrm 1’s ﬁrst-period choice of K1. Because ﬁrm 1 wants to accommodate entry, it chooses K1 to "(K1). Totally differentiating p1 maximize p1 "(K1), the ﬁrstorder condition is K1Þ ¼ ð K1Þ ¼ ð , a"2ð , a"2ð K1Þ K1ÞÞ , , K1Þ : a"1ð a"1ð p1 p2 p1 p2 (xxi) ð ð " " dp1 " dK1 ¼ ¼ p1 1 p1 2 da"1 dK1 þ da"2 dK1 þ S p1 2 da"2 dK1 þ @p1 @K1 @p1 @K1 : (xxii) The second equality in Equation xxii holds by the envelope \|ﬄﬄﬄ{zﬄﬄﬄ} theorem. (The envelope theorem just says that p1 da"1=dK1 1 & disappears because a1 is chosen optimally in the second period, so p1 0 by the ﬁrst-order condition for a1.) The ﬁrst of the remaining two terms in Equation xxii, S, is the strategic effect of an increase in K1 on ﬁrm 1’s proﬁt through ﬁrm 2’s 1 ¼ Chapter 15: Imperfect Competition 575 action. If ﬁrm 1 cannot make an observable commitment to K1, then S disappears from Equation xxii and only the last term, the direct effect of K1 on ﬁrm 1’s proﬁt, will be present. The sign of S determines whether ﬁrm 1 strategically overor underinvests in K1 when it can make a strategic commitment. We have the following steps: sign S ð Þ ¼ ¼ da"2 dK1 1BR02 sign p2 1 " sign p2 " # da"1 dK1 # sign ¼ dp2 " dK1 BR02 : # (xxiii) " The ﬁrst line of Equation xxiii holds if there is some symme2 equals the sign of p2 try to the market, so that the sign of p1 1. The second line follows from differentiating a"2ð in EquaK1Þ tion xx. The third line follows by totally differentiating p2 " in Equation xxi: p2 2 da"2 dK1 dp2 " dK1 ¼ ¼ p2 1 p2 1 da"1 dK1 þ da"1 dK1 , (xxiv) where the second equality again follows from the envelope theorem. By Equation xxiii, the sign of the strategic effect S is determined by the sign of two factors. The ﬁrst factor, dp2 "/dK1, indicates the effect of K1 on ﬁrm 2’s equilibrium proﬁt in the subgame. If dp2 "/dK1 < 0, then an increase in K1 harms ﬁrm 2, and we say that investment makes ﬁrm 1 ‘‘tough.’’ If dp2 "/ dK1 > 0, then an increase in K1 beneﬁts ﬁrm 2, and we say that investment makes ﬁrm 1 ‘‘soft.’’ The second factor, BR02, is the slope of ﬁrm 2’s best response, which depends on whether actions a1 and a2 are strategic substitutes or complements. Each of the two terms in S can have one of two signs for a total of four possible combinations, displayed in Table 15.1. If investment makes ﬁrm 1 ‘‘tough,’’ then the strategic effect S leads ﬁrm 1 to reduce K1 if actions are strategic complements or to increase K1 if actions are strategic substitutes. The opposite is true if investment makes ﬁrm 1 ‘‘soft.’’ For example, actions could be prices in a Bertrand model with differentiated products and thus would be strategic complements. Investment K1 could be advertising that steals market share from ﬁrm 2. Table 15.1 indicates that, when K1 is observable, ﬁrm 1 should strategically underinvest to induce less aggressive price competition from ﬁrm 2. E15.8 Most-favored customer program The preceding analysis applies even if K1 is not a continuous investment variable but instead a 0–1 choice. For example, consider the decision by ﬁrm 1 of whether to start a most-favored customer program (studied in Cooper, 1986). A most-favored customer program rebates the price difference (sometimes in addition
to a premium) to past customers if the ﬁrm lowers its 576 Part 6: Market Power TABLE 15.1 STRATEGIC EFFECT WHEN ACCOMMODATING ENTRY ‘‘Tough’’ (dp2 "/dK1 < 0) ‘‘Soft’’ (dp2 "/dK1 > 0) Firm 1’s Investment Actions Strategic Complements (BR 0 > 0) Strategic Substitutes (BR 0 < 0) Underinvest ( Overinvest ( ) Overinvest ( þ Underinvest ( ) $ ) $ ) þ price in the future. Such a program makes ﬁrm 1 ‘‘soft’’ by reducing its incentive to cut price. If ﬁrms compete in strategic complements (say, in a Bertrand model with differentiated products), then Table 15.1 says that ﬁrm 1 should ‘‘overinvest’’ in the most-favored customer program, meaning that it should be more willing to implement the program if doing so is observable to its rival. The strategic effect leads to less aggressive price competition and thus to higher prices and proﬁts. One’s ﬁrst thought might have been that such a mostfavored customer program should be beneﬁcial to consumers and lead to lower prices because the clause promises payments back to them. As we can see from this example, strategic considerations sometimes prove one’s initial intuition wrong, suggesting that caution is warranted when examining strategic situations. E15.9 Trade policy The analysis in Section E15.7 applies even if K1 is not a choice by ﬁrm 1 itself. For example, researchers in international trade sometimes take K1 to be a government’s policy choice on behalf of its domestic ﬁrms. Brander and Spencer (1985) studied a model of international trade in which exporting ﬁrms from country 1 engage in Cournot competition with domestic ﬁrms in country 2. The actions (quantities) are strategic substitutes. The authors ask whether the government of country 1 would want to implement an export subsidy program, a decision that plays the role of K1 in their model. An export subsidy makes exporting ﬁrms ‘‘tough’’ because it effectively lowers their marginal costs, increasing their exports to country 2 and reducing market price there. According to Table 15.1, the government of country 1 should overinvest in the it is observable to subsidy policy, adopting the policy if domestic ﬁrms in country 2 but not otherwise. The model explains why countries unilaterally adopt export subsidies and other trade interventions when free trade would be globally efﬁcient (at least in this simple model). Our analysis can be used to show that Brander and Spencer’s rationalization of export subsidies may not hold up under alternative assumptions about competition. If exporting ﬁrms and domestic ﬁrms were to compete in strategic complements (say, Bertrand competition in differentiated products rather than Cournot competition), then an export subsidy would be a bad idea according to Table 15.1. Country 1 should then underinvest in the export subsidy (i.e., not adopt it) to avoid overly aggressive price competition. E15.10 Entry deterrence Continue with the model from Section E15.7, but now suppose that ﬁrm 1 prefers to deter rather than accommodate entry. Firm 1’s objective is then to choose K1 to reduce ﬁrm 2’s proﬁt p2 " to zero. Whether ﬁrm 1 should distort K1 upward or downward to accomplish this depends only on the sign of dp2 "/dK1—that is, on whether investment makes ﬁrm 1 ‘‘tough’’ or ‘‘soft’’—and not on whether actions are strategic substitutes or complements. If investment makes ﬁrm 1 ‘‘tough,’’ it should overinvest to deter entry relative to the case in which it cannot observably commit to investment. On the other hand, it should underinvest to deter entry. if investment makes ﬁrm 1 ‘‘soft,’’ For example, if K1 is an investment in marginal cost reduction, this likely makes ﬁrm 1 ‘‘tough’’ and so it should overinvest to deter entry. If K1 is an advertisement that increases demand for the whole product category more than its own brand (advertisements for a particular battery brand involving an unstoppable, battery-powered bunny may increase sales of all battery brands if consumers have difﬁculty remembering exactly which battery was in the bunny), then this will likely make ﬁrm 1 ‘‘soft,’’ so it should underinvest to deter entry. References Brander, J. A., and B. J. Spencer. ‘‘Export Subsidies and International Market Share Rivalry.’’ Journal of International Economics 18 (February 1985): 83–100. Bulow, J., G. Geanakoplous, and P. Klemperer. ‘‘Multimarket Oligopoly: Strategic Substitutes and Complements.’’ Journal of Political Economy (June 1985): 488–511. Cooper, T. ‘‘Most-Favored-Customer Pricing and Tacit Collusion.’’ Rand Journal of Economics 17 (Autumn 1986): 377–88. Fudenberg, D., and J. Tirole. ‘‘The Fat Cat Effect, the Puppy Dog Ploy, and the Lean and Hungry Look.’’ American Economic Review, Papers and Proceedings 74 (May 1984): 361–68. Tirole, J. The Theory of Industrial Organization. Cambridge, MA: MIT Press, 1988, chap. 8. This page intentionally left blank Pricing in Input Markets P A R T SEVEN Chapter 16 Labor Markets Chapter 17 Capital and Time Our study of input demand in Chapter 11 was quite general in that it can be applied to any factor of production. In Chapters 16 and 17 we take up several issues speciﬁcally related to pricing in the labor and capital markets. Chapter 16 focuses mainly on labor supply. Most of our analysis deals with various aspects of individual labor supply. In successive sections we look at the supply of hours of work, decisions related to the accumulation of human capital, and modeling the job search process. For each of these topics, we show how the decisions of individuals affect labor market equilibria. The ﬁnal sections of Chapter 16 take up some aspects of imperfect competition in labor markets. In Chapter 17 we examine the market for capital. The central purpose of the chapter is to emphasize the connection between capital and the allocation of resources over time. Some care is also taken to integrate the theory of capital into the models of ﬁrms’ behavior we developed in Part 4. A brief appendix to Chapter 17 presents some useful mathematical results about interest rates. In The Principles of Political Economy and Taxation, Ricardo wrote: The produce of the earth . . . is divided among three classes of the community, namely, the proprietor of the land, the owner of the stock of capital necessary for its cultivation, and the laborers by whose industry it is cultivated. To determine the laws which regulate this distribution is the principal problem in Political Economy.* The purpose of Part 7 is to illustrate how the study of these ‘‘laws’’ has advanced since Ricardo’s time. *D. Ricardo, The Principles of Political Economy and Taxation (1817; reprinted, London: J. M. Dent and Son, 1965), p. 1. 579 This page intentionally left blank C H A P T E R SIXTEEN Labor Markets In this chapter we examine some aspects of input pricing that are related particularly to the labor market. Because we have already discussed questions about the demand for labor (or any other input) in some detail in Chapter 11, here we will be concerned primarily with analyzing the supply of labor. We start by looking at a simple model of utility maximization that explains individuals’ supply of work hours to the labor market. Subsequent sections then take up various generalizations of this model. Allocation Of Time In Part 2 we studied the way in which an individual chooses to allocate a ﬁxed amount of income among a variety of available goods. Individuals must make similar choices in deciding how they will spend their time. The number of hours in a day (or in a year) is absolutely ﬁxed, and time must be used as it ‘‘passes by.’’ Given this ﬁxed amount of time, any individual must decide how many hours to work; how many hours to spend consuming a wide variety of goods, ranging from cars and television sets to operas; how many hours to devote to self-maintenance; and how many hours to sleep. By examining how individuals choose to divide their time among these activities, economists are able to understand the labor supply decision. Simple two-good model For simplicity we start by assuming there are only two uses to which an individual may devote his or her time—either engaging in market work at a real wage rate of w per hour or not working. We shall refer to nonwork time as ‘‘leisure,’’ but this word is not meant to carry any connotation of idleness. Time not spent in market work can be devoted to work in the home, to self-improvement, or to consumption (it takes time to use a television set or a bowling ball).1 All of those activities contribute to an individual’s well-being, and time will be allocated to them in what might be assumed to be a utility-maximizing way. More speciﬁcally, assume that an individual’s utility during a typical day depends on consumption during that period (c) and on hours of leisure enjoyed (h): Notice that in writing this utility function we have used two ‘‘composite’’ goods, consumption and leisure. Of course, utility is actually derived by devoting real income and utility c, h U ð : Þ ¼ (16:1) 1Perhaps the ﬁrst formal theoretical treatment of the allocation of time was given by G. S. Becker in ‘‘A Theory of the Allocation of Time,’’ Economic Journal 75 (September 1965): 493–517. 581 582 Part 7: Pricing in Input Markets time to the consumption of a wide variety of goods and services.2 In seeking to maximize utility, an individual is bound by two constraints. The ﬁrst of these concerns the time that is available. If we let l represent hours of work, then l h 24: (16:2) þ ¼ That is, the day’s time must be allocated either to work or to leisure (nonwork). A second constraint records the fact that an individual can purchase consumption items only by working (later in this chapter we will allow for the availability of nonlabor income). If the real hourly market wage rate the individual can earn is given by w, then the income constraint is given by Combining the two constraints, we have wl: c ¼ c 24 w ð h Þ % ¼ or (16:3) (16:4) c wh 24w: (16:5) þ ¼ This combined constraint
has an important interpretation. Any person has a ‘‘full income’’ given by 24w. That is, an individual who worked all the time would have this much command over real consumption goods each day. Individuals may spend their full income either by working (for real income and consumption) or by not working and thereby enjoying leisure. Equation 16.5 shows that the opportunity cost of consuming leisure is w per hour; it is equal to earnings forgone by not working. Utility maximization The individual’s problem, then, is to maximize utility subject to the full income constraint. Given the Lagrangian expression 24w ð the ﬁrst-order conditions for a maximum are Þ þ c, h ¼ U k ð + c wh , Þ % % @+ @c ¼ @+ @h ¼ @U @c % @U @h % 0, k ¼ wk 0: ¼ Dividing the two lines in Equation 16.7, we obtain @U=@h @U=@c ¼ w ¼ MRS : h for c Þ ð Hence we have derived the following principle. (16:6) (16:7) (16:8 Utility-maximizing labor supply decision. To maximize utility given the real wage w, the individual should choose to work that number of hours for which the marginal rate of substitution of leisure for consumption is equal to w. 2The production of goods in the home has received considerable attention, especially since household time allocation diaries have become available. For a survey, see R. Granau, ‘‘The Theory of Home Production: The Past Ten Years’’ in J. T. Addison, Ed. Recent Developments in Labor Economics. (Cheltenham, UK: Elgar Reference Collection, 2007), vol. 1, pp 235–43. Chapter 16: Labor Markets 583 Of course, the result derived in Equation 16.8 is only a necessary condition for a maximum. As in Chapter 4, this tangency will be a true maximum provided the MRS of leisure for consumption is diminishing. Income and substitution effects of a change in w A change in the real wage rate (w) can be analyzed in a manner identical to that used in Chapter 5. When w increases, the ‘‘price’’ of leisure becomes higher: a person must give up more in lost wages for each hour of leisure consumed. As a result, the substitution effect of an increase in w on the hours of leisure will be negative. As leisure becomes more expensive, there is reason to consume less of it. However, the income effect will be positive—because leisure is a normal good, the higher income resulting from a higher w will increase the demand for leisure. Thus, the income and substitution effects work in opposite directions. It is impossible to predict on a priori grounds whether an increase in w will increase or decrease the demand for leisure time. Because leisure and work are mutually exclusive ways to spend one’s time, it is also impossible to predict what will happen to the number of hours worked. The substitution effect tends to increase hours worked when w increases, whereas the income effect—because it increases the demand for leisure time—tends to decrease the number of hours worked. Which of these two effects is the stronger is an important empirical question.3 A graphical analysis The two possible reactions to a change in w are illustrated in Figure 16.1. In both graphs, the initial wage is w0, and the initial optimal choices of c and h are given by the point c0, h0. FIGURE 16.1 Income and Substitution Effects of a Change in the Real Wage Rate w Because the individual is a supplier of labor, the income and substitution effects of an increase in the real wage rate (w) work in opposite directions in their effects on the hours of leisure demanded (or on hours of work). In (a) the substitution effect (movement to point S) outweighs the income effect, and a higher wage causes hours of leisure to decrease to h1. Therefore, hours of work increase. In (b) the income effect is stronger than the substitution effect, and h increases to h1. In this case, hours of work decrease. Consumption Consumption c = w0(24 − h) c = w1(24 − h) c1 c0 S S c1 c0 c = w1(24 − h) c = w0(24 − h) U1 U0 h1 h0 Leisure h0 h1 (a) (b) U1 U0 Leisure 3If the family is taken to be the relevant decision unit, then even more complex questions arise about the income and substitution effects that changes in the wages of one family member will have on the labor force behavior of other family members. 584 Part 7: Pricing in Input Markets When the wage rate increases to w1, the optimal combination moves to point c1, h1. This movement can be considered the result of two effects. The substitution effect is represented by the movement of the optimal point from c0, h0 to S and the income effect by the movement from S to c1, h1. In the two panels of Figure 16.1, these two effects combine to produce different results. In panel (a) the substitution effect of an increase in w outweighs the income effect, and the individual demands less leisure (h1 < h0). Another way of saying this is that the individual will work longer hours when w increases. In panel (b) of Figure 16.1 the situation is reversed. The income effect of an increase in w more than offsets the substitution effect, and the demand for leisure increases (h1 > h0). The individual works shorter hours when w increases. In the cases examined in Chapter 5 this would have been considered an unusual result—when the ‘‘price’’ of leisure increases, the individual demands more of it. For the case of normal consumption goods, the income and substitution effects work in the same direction. Only for ‘‘inferior’’ goods do they differ in sign. In the case of leisure and labor, however, the income and substitution effects always work in opposite directions. An increase in w makes an individual better-off because he or she is a supplier of labor. In the case of a consumption good, individuals are made worse-off when a price increases because they are consumers of that good. We can summarize this analysis as follows Income and substitution effects of a change in the real wage. When the real wage rate increases, a utility-maximizing individual may increase or decrease hours worked. The substitution effect will tend to increase hours worked as the individual substitutes earnings for leisure, which is now relatively more costly. On the other hand, the income effect will tend to reduce hours worked as the individual uses his or her increased purchasing power to buy more leisure hours. We now turn to examine a mathematical development of these responses that provides additional insights into the labor supply decision. A Mathematical Analysis Of Labor Supply To derive a mathematical statement of labor supply decisions, it is helpful ﬁrst to amend the budget constraint slightly to allow for the presence of nonlabor income. To do so, we rewrite Equation 16.3 as c wl n, (16:9) þ where n is real nonlabor income and may include such items as dividend and interest income, receipt of government transfer beneﬁts, or simply gifts from other persons. Indeed, n could stand for lump-sum taxes paid by this individual, in which case its value would be negative. ¼ Maximization of utility subject to this new budget constraint would yield results virtually identical to those we have already derived. That is, the necessary condition for a maximum described in Equation 16.8 would continue to hold as long as the value of n is unaffected by the labor-leisure choices being made; that is, so long as n is a lump-sum receipt or loss of income,4 the only effect of introducing nonlabor income into the 4In many situations, however, n itself may depend on labor supply decisions. For example, the value of welfare or unemployment beneﬁts a person can receive depends on his or her earnings, as does the amount of income taxes paid. In such cases the slope of the individual’s budget constraint will no longer be reﬂected by the real wage but must instead reﬂect the net return to additional work after taking increased taxes and reductions in transfer payments into account. For some examples, see the problems at the end of this chapter. Chapter 16: Labor Markets 585 analysis is to shift the budget constraints in Figure 16.1 outward or inward in a parallel manner without affecting the trade-off rate between earnings and leisure. This discussion suggests that we can write the individual’s labor supply function as l(w, n) to indicate that the number of hours worked will depend both on the real wage rate and on the amount of real nonlabor income received. On the assumption that leisure is a normal good, @l / @n will be negative; that is, an increase in n will increase the demand for leisure and (because there are only 24 hours in the day) reduce l. Before studying wage effects on labor supply (@l / @w), we will ﬁnd it helpful to consider the dual problem to the individual’s primary utility-maximization problem. Dual statement of the problem As we showed in Chapter 5, related to the individual’s primal problem of utility maximization given a budget constraint is the dual problem of minimizing the expenditures necessary to attain a given utility level. In the present context, this problem can be phrased as choosing values for consumption (c) and leisure time (h l ) such that the amount of spending, 24 % ¼ E c wl n, (16:10) % U(c, h)] is as small as possible. As in required to attain a given utility level [say, U0 ¼ Chapter 5, solving this minimization problem will yield exactly the same solution as solving the utility-maximization problem. % ¼ Now we can apply the envelope theorem to the minimum value for these extra expenditures calculated in the dual problem. Speciﬁcally, a small change in the real wage will change the minimum expenditures required by @E @w ¼ % l: (16:11) Intuitively, each $1 increase in w reduces the required value of E by $l, because that is the extent to which labor earnings are increased by the wage change. This result is similar to Shephard’s lemma in the theory of production (see Chapter 11); here the result shows that a labor supply function can be calculated from the expenditure function by partial differentiation. Because utility is held constant in the dual expenditure minimization appro
ach, this function should be interpreted as a ‘‘compensated’’ (constant utility) labor supply function, which we will denote by lc(w, U ) to avoid confusing it with the uncompensated labor supply function l(w, n) introduced earlier. Slutsky equation of labor supply Now we can use these concepts to derive a Slutsky-type equation that reﬂects the substitution and income effects that result from changes in the real wage. We begin by recognizing that the expenditures being minimized in the dual problem of Equation 16.11 play the role of nonlabor income in the primal utility-maximization problem. Hence, by deﬁnition, for the utility-maximizing choice we have w, E ½ Partial differentiation of both sides of Equation 16.12 with respect to w yields w, U ð w, n ð w, U Þ’ ¼ Þ ¼ : Þ ð l l lc @lc @w ¼ @l @w þ @l @E ( @E @w , and by using the envelope relation from Equation 16.11 for @E/@w we obtain @lc @w ¼ @l @w % l @l @E ¼ @l @w % l @l @n : (16:12) (16:13) (16:14) 586 Part 7: Pricing in Input Markets Introducing a slightly different notation for the compensated labor supply function, (16:15) (16:16) and then rearranging terms gives the ﬁnal Slutsky equation for labor supply: @lc @w ¼ @l @w , U U0 ¼ ! ! ! ! @l @w ¼ @l @w l @l @n : þ U U0 ! ! ! ! ¼ In words (as we have previously shown), the change in labor supplied in response to a change in the real wage can be disaggregated into the sum of a substitution effect in which utility is held constant and an income effect that is analytically equivalent to an appropriate change in nonlabor income. Because the substitution effect is positive (a higher wage increases the amount of work chosen when utility is held constant) and the term @l / @n is negative, this derivation shows that the substitution and income effects work in opposite directions. The mathematical development supports the earlier conclusions from our graphical analysis and suggests at least the theoretical possibility that labor supply might respond negatively to increases in the real wage. The mathematical development also suggests that the importance of negative income effects may be greater the greater is the amount of labor itself being supplied. EXAMPLE 16.1 Labor Supply Functions Individual labor supply functions can be constructed from underlying utility functions in much the same way that we constructed demand functions in Part 2. Here we will begin with a fairly extended treatment of a simple Cobb–Douglas case and then provide a shorter summary of labor supply with CES utility. 1. Cobb–Douglas utility. Suppose that an individual’s utility function for consumption, c, and leisure, h, is given by U c, h ð Þ ¼ cahb, (16:17) and assume for simplicity that a income constraint that shows how consumption can be ﬁnanced, þ ¼ b 1. This person is constrained by two equations: (1) an where n is nonlabor income; and (2) a total time constraint wl c ¼ þ n, h l þ ¼ 1, (16:18) (16:19) where we have arbitrarily set the available time to be 1. By combining the ﬁnancial and time constraints into a ‘‘full income’’ constraint, we can arrive at the following Lagrangian expression for this utility-maximization problem: % First-order conditions for a maximum are Þ þ þ ¼ U c, h ð k w ð n + wh c % Þ ¼ cahb w k ð þ n % þ wh c : Þ % (16:20) @+ @c ¼ @+ @h ¼ @+ @k ¼ ac% bhb bcah% a 0, k ¼ kw 0, ¼ % % w n wh c % ¼ 0: % þ (16:21) Chapter 16: Labor Markets 587 Dividing the ﬁrst of these by the second yields % Substitution into the full income constraint then yields the familiar results ah bc ¼ ah or wh c: (16:2216:23) In words, this person spends a ﬁxed fraction, a, of his or her full consumption and the complementary fraction, b % The labor supply function for this person is then given by n) on a, on leisure (which costs w per unit). income (w ¼ þ 1 l w % bn w : (16:24) 2. Properties of the Cobb–Douglas labor supply function. This labor supply function shares many of the properties exhibited by consumer demand functions derived from Cobb– Douglas utility. For example, if n b proportion of his or her time to working, no matter what the wage rate. Income and substitution effects of a change in w are precisely offsetting in this case, just as they are with cross-price effects in Cobb–Douglas demand functions. 0—this person always devotes 1 0 then @l / @w ¼ ¼ % On the other hand, if n > 0, then @l / @w > 0. When there is positive nonlabor income, this person spends bn of it on leisure. But leisure ‘‘costs’’ w per hour, so an increase in the wage means that fewer hours of leisure can be bought. Hence, an increase in w increases labor supply. Finally, observe that @l / @n < 0. An increase in nonlabor income allows this person to buy more leisure, so labor supply decreases. One interpretation of this result is that transfer programs (such as welfare beneﬁts or unemployment compensation) reduce labor supply. Another interpretation is that lump-sum taxation increases labor supply. But actual tax and transfer programs are seldom lump sum—usually they affect net wage rates as well. Hence any precise prediction requires a detailed look at how such programs affect the budget constraint. 3. CES labor supply. In the Extensions to Chapter 4 we derived the general form for demand functions generated from a CES (constant elasticity of substitution) utility function. We can apply that derivation directly here to study CES labor supply. Speciﬁcally, if utility is given by U c, h ð Þ ¼ cd d þ hd d , then budget share equations are given by sc ¼ sh ¼ c w þ wh w þ n ¼ 1 n ¼ 1 1 , wj þ 1 j w% þ , where k d/(d ¼ % and 1). Solving explicitly for leisure demand gives h ¼ w n þ w1 j % w þ l w w1 % w þ n j : % w1 % (16:25) (16:26) (16:27) (16:28) 588 Part 7: Pricing in Input Markets It is perhaps easiest to explore the properties of this function by taking some examples. If 0.5 and k 1, the labor supply function is d ¼ ¼ % l m, n ð Þ ¼ w2 w n % w2 ¼ þ 1 1 n=w2 1=w % þ : (16:29) ¼ 0, then clearly @l / @w > 0; because of the relatively high degree of substitutability If n between consumption and leisure in this utility function, the substitution effect of a higher wage outweighs the income effect. On the other hand, if d 0.5, then the labor supply function is 1 and k ¼ % ¼ l w, n ð Þ ¼ w0:5 w n % w0:5 ¼ 1 n=w0:5 w0:5 % 1 : (16:30) Now (when n utility function, the income effect outweighs the substitution effect in labor supply.5 þ 0) @l / @w < 0; because there is a smaller degree of substitutability in the ¼ þ FIGURE 16.2 Construction of the Market Supply Curve for Labor QUERY: Why does the effect of nonlabor income in the CES case depend on the consumption/leisure substitutability in the utility function? Market Supply Curve For Labor We can plot a curve for market supply of labor based on individual labor supply decisions. At each possible wage rate we add together the quantity of labor offered by each individual to arrive at a market total. One particularly interesting aspect of this procedure is that, as the wage rate increases, more individuals may be induced to enter the labor force. Figure 16.2 illustrates this possibility for the simple case of two people. For a real As the real wage increases, there are two reasons why the supply of labor may increase. First, higher real wages may cause each person in the market to work more hours. Second, higher wages may induce more individuals (for example, individual 2) to enter the labor market. Real wage S2 Real wage S1 S Real wage w3 w2 w1 Hours Hours Total labor supply (a) Individual 1 (b) Individual 2 (c) The market 5In the Cobb–Douglas case (d 0, k ¼ ¼ 0), the constant-share result (for n 0) is given by l(w, n) (w % ¼ n)/2w 0.5 % ¼ n/2w. ¼ Chapter 16: Labor Markets 589 wage below w1, neither individual chooses to work. Consequently, the market supply curve of labor (Figure 16.2c) shows that no labor is supplied at real wages below w1. A wage in excess of w1 causes individual 1 to enter the labor market. However, as long as wages fall short of w2, individual 2 will not work. Only at a wage rate above w2 will both individuals participate in the labor market. In general, the possibility of the entry of new workers makes the market supply of labor somewhat more responsive to wagerate increases than would be the case if the number of workers were assumed to be ﬁxed. The most important example of higher real wage rates inducing increased labor force participation is the labor force behavior of married women in the United States in the post–World War II period. Since 1950 the percentage of working married women has increased from 32 percent to over 65 percent; economists attribute this, at least in part, to the increasing wages that women are able to earn. Labor Market Equilibrium Equilibrium in the labor market is established through the interaction of individuals’ labor supply decisions with ﬁrms’ decisions about how much labor to hire. That process is illustrated by the familiar supply–demand diagram in Figure 16.3. At a real wage rate of w), the quantity of labor demanded by ﬁrms is precisely matched by the quantity supplied by individuals. A real wage higher than w) would create a disequilibrium in which the quantity of labor supplied is greater than the quantity demanded. There would be some involuntary unemployment at such a wage, and this may create pressure for the real wage to decrease. Similarly, a real wage lower than w) would result in disequilibrium behavior because ﬁrms would want to hire more workers than are available. In the scramble to hire workers, ﬁrms may bid up real wages to restore equilibrium. A real wage of w) creates an equilibrium in the labor market with an employment level of l ). Real wage w * S D l * Quantity of labor FIGURE 16.3 Equilibrium in the Labor Market 590 Part 7: Pricing in Input Markets Possible reasons for disequilibria in the labor market are a major topic in macroeconomics, especially in relationship to the business cycle. Perceived failures of the market to adjust to chang
ing equilibria have been blamed on ‘‘sticky’’ real wages, inaccurate expectations by workers or ﬁrms about the price level, the impact of government unemployment insurance programs, labor market regulations and minimum wages, and intertemporal work decisions by workers. We will encounter a few of these applications later in this chapter and in Chapters 17 and 19. Equilibrium models of the labor market can also be used to study a number of questions about taxation and regulatory policy. For example, the partial equilibrium tax incidence modeling illustrated in Chapter 12 can be readily adapted to the study of employment taxation. One interesting possibility that arises in the study of labor markets is that a given policy intervention may shift both demand and supply functions—a possibility we examine in Example 16.2. EXAMPLE 16.2 Mandated Beneﬁts A number of recent laws have mandated that employers provide special beneﬁts to their workers such as health insurance, paid time off, or minimum severance packages. The effect of such mandates on equilibrium in the labor market depends importantly on how the beneﬁts are valued by workers. Suppose that, prior to implementation of a mandate, the supply and demand for labor are given by a c lS ¼ lD ¼ þ % bw, dw: (16:31) Setting lS ¼ lD yields an equilibrium wage of c b a d : w) ¼ % þ Now suppose that the government mandates that all ﬁrms provide a particular beneﬁt to their workers and that this beneﬁt costs t per unit of labor hired. Therefore, unit labor costs t. Suppose also that the new beneﬁt has a monetary value to workers of k per increase to w unit of labor supplied—hence the net return from employment increases to w k. Equilibrium in the labor market then requires that (16:32) þ þ A bit of manipulation of this expression shows that the net wage is given by )) ¼ a d % bk b c b % þ þ þ dt d ¼ w) % dt d : bk b þ þ (16:33) (16:34) ¼ 0), then the mandate is just like a tax If workers derive no value from the mandated beneﬁt (k on employment: employees pay a share of the tax given by the ratio d / (b d), and the equilibrium quantity of labor hired decreases. Qualitatively similar results will occur so long as k < t. On the other hand, if workers value the beneﬁt at precisely its cost (k t), then the new t) and the equilibrium level of wage decreases precisely by the amount of this cost (w)) ¼ employment does not change. Finally, if workers value the beneﬁt at more than it costs the ﬁrm to provide it (k > t—a situation where one might wonder why the beneﬁt was not already then the equilibrium wage will decrease by more than the beneﬁt costs and provided), equilibrium employment will increase. w) % ¼ þ QUERY: How would you graph this analysis? Would its conclusions depend on using linear supply and demand functions? Chapter 16: Labor Markets 591 Wage variation The labor market equilibrium illustrated in Figure 16.3 implies that there is a single marketclearing wage established by the supply decisions of households and the demands of ﬁrms. The most cursory examination of labor markets would suggest that this view is far too simplistic. Even in a single geographical area wages vary signiﬁcantly among workers, perhaps by a multiple of 10, or even 50. Of course, such variation probably has some sort of supply–demand explanation, but possible reasons for the differentials are obscured by thinking of wages as being determined in a single market. In this section we look at three major causes of wage differences: (1) human capital; (2) compensating wage differentials; and (3) job search uncertainty. In the ﬁnal sections of the chapter we look at a fourth set of causes—imperfect competition in the labor market. Human capital Workers vary signiﬁcantly in the skills and other attributes they bring to a job. Because ﬁrms pay wages commensurate with the values of workers’ productivities, such differences can clearly lead to large differences in wages. By drawing a direct analogy to the ‘‘physical’’ capital used by ﬁrms, economists6 refer to such differences as differences in ‘‘human capital.’’ Such capital can be accumulated in many ways by workers. Elementary and secondary education often provides the foundation for human capital—the basic skills learned in school make most other learning possible. Formal education after high school can also provide a variety of jobrelated skills. College and university courses offer many general skills, and professional schools provide speciﬁc skills for entry into speciﬁc occupations. Other types of formal education may also enhance human capital, often by providing training in speciﬁc tasks. Of course, elementary and secondary education is compulsory in many countries, but postsecondary education is often voluntary, and thus attendance may be more amenable to economic analysis. In particular, the general methods to study a ﬁrm’s investment in physical capital (see Chapter 17) have been widely applied to the study of individuals’ investments in human capital. Workers may also acquire skills on the job. As they gain experience their productivity will increase, and presumably, they will be paid more. Skills accumulated on the job may sometimes be transferable to other possible employment. Acquiring such skills is similar to acquiring formal education and hence is termed general human capital. In other cases, the skills acquired on a job may be quite speciﬁc to a particular job or employer. These skills are termed speciﬁc human capital. As Example 16.3 shows, the economic consequences of these two types of investment in human capital can be quite different. EXAMPLE 16.3 General and Speciﬁc Human Capital Suppose that a ﬁrm and a worker are entering into a two-period employment relationship. In the ﬁrst period the ﬁrm must decide on how much to pay the worker (w1) and how much to invest in general (g) and speciﬁc (s) human capital for this worker. Suppose that the value of the worker’s marginal product is v1 in the ﬁrst period. In the second period, the value of the worker’s marginal product is given by: v2ð v1 þ where v g and vs represent the increase in human capital as a result of the ﬁrm’s investments in period one. Assume also that both investments are proﬁtable in that vg > pss > pgg and v s (16:35) Þ ¼ Þ þ s ð g, s v g vs g g Þ ð ð Þ s ð Þ 6Widespread use of the term human capital is generally attributed to the American economist T. W. Schultz. An important pioneering work in the ﬁeld is G. Becker, Human Capital: A Theoretical and Empirical Analysis with Special Reference to Education (New York: National Bureau of Economic Research, 1964). 592 Part 7: Pricing in Input Markets (where pg and ps are the per-unit prices of providing the different types of skills). Proﬁts7 for the ﬁrm are given by p1 ¼ p2 ¼ p ¼ v1 % v1 þ p1 þ w1 % v g g ð p2 ¼ pgg pss % vs w2 s Þ % Þ þ ð v g g 2v1 þ ð (16:36) pgg vs s ð pss w1 % w2 þ % Þ % w2, and the ﬁrm wishes to maximize two-period proﬁts. Þ % where w2 is the second-period wage paid to the worker. In this contractual situation, the worker wishes to maximize w1 þ Competition in the labor market will play an important role in the contract chosen in this situation because the worker can always choose to work somewhere else. If he or she is paid the marginal product in this alternative employment, alternative wages must be w1 ¼ v1 and increase the worker’s w2 ¼ alternative wage rate, but investments in speciﬁc human capital do not because, by deﬁnition, such skills are useless on other jobs. If the ﬁrm sets wages equal to these alternatives, proﬁts are given by investments in general human capital . Note that v1 þ v g g ð Þ and the ﬁrm’s optimal choice is to set g investment in general human capital, its proﬁt-maximizing decision is to refrain from such investing. ¼ v s p ¼ pss Þ % pgg s ð 0. Intuitively, if the ﬁrm cannot earn any return on its (16:37) % From the worker’s point of view, however, this decision would be nonoptimal. He or she would command a higher wage with such added human capital. Hence, the worker may opt to pay for his or her own general human capital accumulation by taking a reduction in ﬁrst-period wages. Total wages are then given by w ¼ w2 ¼ w1 þ and the ﬁrst-order condition for an optimal g for the worker is @v g(g) / @g pg. Note that this is the same optimality condition that would prevail if the ﬁrm could capture all the gains from its investment in general human capital. Note also that the worker could not opt for this optimal contract if legal restrictions (such as a minimum wage law) prevented him or her from paying for the human capital investment with lower ﬁrst-period wages. 2w1 þ (16:38) pgg, Þ % ¼ g ð v g The ﬁrm’s ﬁrst-order condition for a proﬁt-maximizing choice of s immediately follows from Equation 16.37—@vs(s)/@s ps. Once the ﬁrm makes this investment, however, it must decide how, if at all, the increase in the value of the marginal product is to be shared with the worker. This is ultimately a bargaining problem. The worker can threaten to leave the ﬁrm unless he or she gets a share of the increased marginal product. On the other hand, the ﬁrm can threaten to invest little in speciﬁc human capital unless the worker promises to stay around. A number of outcomes seem plausible depending on the success of the bargaining strategies employed by the two parties. ¼ QUERY: Suppose that the ﬁrm offered to provide a share of the increased marginal product given by avs(s) with the worker (where 0 1). How would this affect the ﬁrm’s investment in s? How might this sharing affect wage bargaining in future periods? * * a One ﬁnal type of investment in human capital might be mentioned—investments in health. Such investments can occur in many ways. Individuals can purchase preventive care to guard against illness, they may take other actions (such as exercise) with the same goal, or they may purchase medical care to restore health if they have contracted an illness. All of these actions are
intended to augment a person’s ‘‘health capital’’ (which is one component of human capital). There is ample evidence that such capital pays off in 7For simplicity we do not discount future proﬁts here. Chapter 16: Labor Markets 593 terms of increased productivity; indeed, ﬁrms themselves may wish to invest in such capital for the reasons outlined in Example 16.3. All components of human capital have certain characteristics that differentiate them from the types of physical capital also used in the production process. First, acquisition of human capital is often a time-consuming process. Attending school, enrolling in a jobtraining program, or even daily exercise can take many hours, and these hours will usually have signiﬁcant opportunity costs for individuals. Hence, human capital acquisition is often studied as part of the same time allocation process that we looked at earlier in this chapter. Second, human capital, once obtained, cannot be sold. Unlike the owner of a piece of machinery, the owner of human capital may only rent out that capital to others—the owner cannot sell the capital outright. Hence, human capital is perhaps the most illiquid way in which one can hold assets. Finally, human capital depreciates in an unusual way. Workers may indeed lose skills as they get older or if they are unemployed for a long time. However, the death of a worker constitutes an abrupt loss of all human capital. That, together with their illiquidity, makes human capital investments rather risky. Compensating wage differentials Differences in working conditions are another reason why wages may differ among workers. In general one might expect that jobs with pleasant surroundings would pay less (for a given set of skills) and jobs that are dirty or dangerous would pay more. In this section we look at how such ‘‘compensating wage differentials’’ might arise in competitive labor markets. Consider ﬁrst a ﬁrm’s willingness to provide good working conditions. Suppose that the ﬁrm’s output is a function of the labor it hires (l) and the amenities it provides to its workers (A). Hence q f(l, A). We assume that amenities themselves are productive ( fA > 0) and exhibit diminishing marginal productivity ( fAA < 0). The ﬁrm’s proﬁts are ¼ l, A p ð Þ ¼ pf l, A ð Þ % wl % pAA (16:39) where p, w, and pA are, respectively, the price of the ﬁrm’s output, the wage rate paid, and the price of amenities. For a ﬁxed wage, the ﬁrm can choose proﬁt-maximizing levels for its two inputs, l) and A). The resulting equilibrium will have differing amenity levels among ﬁrms because these amenities will have different productivities in different applications (happy workers may be important for retail sales, but not for managing oil reﬁneries). In this case, however, wage levels will be determined independent of amenities. Consider now the possibility that wage levels can change in response to amenities provided on the job. Speciﬁcally, assume that the wage actually paid by a ﬁrm is given by w A)), where k represents the implicit price of a unit of amenity—an implicit price that will be determined in the marketplace (as we shall show). Given this possibility, the ﬁrm’s proﬁts are given by w0 % k(A % ¼ l, A p ð Þ ¼ pf ð l, A w0 % Þ % ½ A k ð % l A)Þ’ % pAA and the ﬁrst-order condition for a proﬁt-maximizing choice of amenities is @p @A ¼ pfA þ kl pA ¼ 0 or pfA ¼ pA % % kl: (16:40) (16:41) Hence, the ﬁrm will have an upward sloping ‘‘supply curve’’ for amenities in which a higher level of k causes the ﬁrm to choose to provide more amenities to its workers (a fact derived from the assumed diminishing marginal productivity of amenities). A worker’s valuation of amenities on the job is derived from his or her utility function, U(w, A). The worker will choose among employment opportunities in a way that maxiA)). As in other models of mizes utility subject to the budget constraint w k(A w0 % ¼ % 594 Part 7: Pricing in Input Markets utility maximization, the ﬁrst-order conditions for this constrained maximum problem can be manipulated to yield: MRS UA Uw ¼ k: ¼ (16:42) That is, the worker will choose a job that offers a combination of wages and amenities for which his or her MRS is precisely equal to the (implicit) price of amenities. Therefore, the utility-maximizing process will generate a downward sloping ‘‘demand curve’’ for amenities (as a function of k). An equilibrium value of k can be determined in the marketplace by the interaction of the aggregate supply curve for ﬁrms and the aggregate demand curve for workers. Given this value of k, actual levels of amenities will differ among ﬁrms according to the speciﬁcs of their production functions. Individuals will also take note of the implicit price of amenities in sorting themselves among jobs. Those with strong preferences for amenities will opt for jobs that provide them, but they will also accept lower wages in the process. Inferring the extent to which compensating such wage differentials explains wage variation in the real world is complicated by the many other factors that affect wages. Most importantly, linking amenities to wage differentials across individuals must also account for possible differences in human capital among these workers. The simple observation that some unpleasant jobs do not seem to pay very well is not necessarily evidence against the theory of compensating wage differentials. The presence or absence of such differentials can be determined only by comparing workers with the same levels of human capital. Job search Wage differences can also arise from differences in the success that workers have in ﬁnding good job matches. The primary difﬁculty is that the job search process involves uncertainty. Workers new to the labor force may have little idea of how to ﬁnd work. Workers who have been laid off from jobs face special problems, in part because they lose the returns to the speciﬁc human capital they have accumulated unless they are able to ﬁnd another job that uses these skills. Therefore, in this section we will look brieﬂy at the ways economists have tried to model the job search process. Suppose that the job search process proceeds as follows. An individual samples the available jobs one at a time by calling a potential employer or perhaps obtaining an interview. The individual does not know what wage will be offered by the employer until he or she makes the contact (the ‘‘wage’’ offered might also include the value of various fringe beneﬁts or amenities on the job). Before making the contact, the job seeker does know that the labor market reﬂects a probability distribution of potential wages. This probability density function (see Chapter 2) of potential wages is given by f(w). The job seeker spends an amount c on each employer contact. One way to approach the job seeker strategy is to argue that he or she should choose the number of employer contacts (n) for which the marginal beneﬁt of further searching (and thereby possibly ﬁnding a higher wage) is equal to the marginal cost of the additional contact. Because search encounters diminish returns,8 such an optimal n) will generally exist, although its value will depend on the precise shape of the wage distribution. Therefore, individuals with differing views of the distribution of potential wages may adopt differing search intensities and may ultimately settle for differing wage rates. n 8The probability that a job seeker will encounter a speciﬁc high wage, say, w0, for the ﬁrst time on the nth employer contact is is the cumulative distribution of wages showing the probability that wages are less than given by or equal to a given level; see Chapter 2). Hence the expected maximum wage after n contacts is wn is a fairly simple matter to show that wn wdw. It Þ w0Þ’ ð (where F max diminishes as n increases. max ¼ w0Þ w Þ w ð 1f % F ½ wn 1f w Þ max % Chapter 16: Labor Markets 595 Setting the optimal search intensity on a priori grounds may not be the best strategy in this situation. If a job seeker encountered an especially attractive job on, say, the third employer contact, it would make little sense for him or her to continue looking. An alternative strategy would be to set a ‘‘reservation wage’’ and take the ﬁrst job that offered this wage. An optimal reservation wage (wr) would be set so that the expected gain from one more employer contact should be equal to the cost of that contact. That is, wr should be chosen so that c ¼ 1 ð wr w ð % f wrÞ w ð dw: Þ (16:43) Equation 16.43 makes clear that an increase in c will cause this person to reduce his or her reservation wage. Hence people with high search costs may end the job search process with low wages. Alternatively, people with low search costs (perhaps because the search is subsidized by unemployment beneﬁts) will opt for higher reservation wages and possibly higher future wages, although at the cost of a longer search. Examining issues related to job search calls into question the deﬁnition of equilibrium in the labor market. Figure 16.3 implies that labor markets will function smoothly, settling at an equilibrium wage at which the quantity of labor supplied equals the quantity demanded. In a dynamic context, however, it is clear that labor markets experience considerable ﬂows into and out of employment and that there may be signiﬁcant frictions involved in this process. Economists have developed a number of models that explore what ‘‘equilibrium’’ might look like in a market with search unemployment, but we will not pursue these here.9 Monopsony In The Labor Market In many situations ﬁrms are not price-takers for the inputs they buy. That is, the supply curve for, say, labor faced by the ﬁrm is not inﬁnitely elastic at the prevailing wage rate. It often may be necessary for the ﬁrm to offer a wage above that currently prevailing if it is to attract more employees. In order to study such situations, it is most convenient to examine the polar case of monopsony (a single b
uyer) in the labor market. If there is only one buyer in the labor market, then this ﬁrm faces the entire market supply curve. To increase its hiring of labor by one more unit, it must move to a higher point on this supply curve. This will involve paying not only a higher wage to the ‘‘marginal worker’’ but also additional wages to those workers already employed. Therefore, the marginal expense associated with hiring the extra unit of labor (MEl) exceeds its wage rate. We can show this result mathematically as follows. The total cost of labor to the ﬁrm is wl. Hence the change in those costs brought about by hiring an additional worker is MEl ¼ @wl @l ¼ l @w @l : w þ (16:44) In the competitive case, @w/@l 0 and the marginal expense of hiring one more worker is simply the market wage, w. However, if the ﬁrm faces a positively sloped labor supply curve, then @w/ @l > 0 and the marginal expense exceeds the wage. These ideas are summarized in the following deﬁnition. ¼ 9For a pioneering example, see P. Diamond, ‘‘Wage Determination and Efﬁciency in Search Equilibrium,’’ Review of Economic Studies XLIX (1982): 217–27. 596 Part 7: Pricing in Input Markets Marginal input expense. The marginal expense (ME) associated with any input is the increase in total costs of the input that results from hiring one more unit. If the ﬁrm faces an upward-sloping supply curve for the input, the marginal expense will exceed the market price of the input. A proﬁt-maximizing ﬁrm will hire any input up to the point at which its marginal revenue product is just equal to its marginal expense. This result is a generalization of our previous discussion of marginalist choices to cover the case of monopsony power in the labor market. As before, any departure from such choices will result in lower proﬁts for the ﬁrm. If, for example, MRPl > MEl , then the ﬁrm should hire more workers because such an action would increase revenues more than costs. Alternatively, if MRPl < MEl , then employment should be reduced because that would lower costs more rapidly than revenues. Graphical analysis The monopsonist’s choice of labor input is illustrated in Figure 16.4. The ﬁrm’s demand curve for labor (D) is drawn negatively sloped, as we have shown it must be.10 Here also FIGURE 16.4 Pricing in a Monopsonistic Labor Market If a ﬁrm faces a positively sloped supply curve for labor (S), it will base its decisions on the marginal expense of additional hiring (MEl). Because S is positively sloped, the MEl curve lies above S. The curve S can be thought of as an ‘‘average cost of labor curve,’’ and the MEl curve is marginal to S. At l1 the equilibrium condition MEl ¼ Notice that the monopsonist buys less labor than would be bought if the labor market were perfectly competitive (l)). MRPl holds, and this quantity will be hired at a market wage rate w1. Wage w * w1 D S MEl S D l1 l * Labor input per period 10Figure 16.4 is intended only as a pedagogic device and cannot be rigorously defended. In particular, the curve labeled D, although it is supposed to represent the ‘‘demand’’ (or marginal revenue product) curve for labor, has no precise meaning for the monopsonist buyer of labor, because we cannot construct this curve by confronting the ﬁrm with a ﬁxed wage rate. Instead, the ﬁrm views the entire supply curve, S, and uses the auxiliary curve MEl to choose the most favorable point on S. In a strict sense, there is no such thing as the monopsonist’s demand curve. This is analogous to the case of a monopoly, for which we could not speak of a monopolist’s ‘‘supply curve.’’ Chapter 16: Labor Markets 597 the MEl curve associated with the labor supply curve (S) is constructed in much the same way that the marginal revenue curve associated with a demand curve can be constructed. Because S is positively sloped, the MEl curve lies everywhere above S. The proﬁt-maximizing level of labor input for the monopsonist is given by l1, for at this level of input the proﬁt-maximizing condition holds. At l1 the wage rate in the market is given by w1. Notice that the quantity of labor demanded falls short of that which would be hired in a perfectly competitive labor market (l )). The ﬁrm has restricted input demand by virtue of its monopsonistic position in the market. The formal similarities between this analysis and that of monopoly presented in Chapter 14 should be clear. In particular, the ‘‘demand curve’’ for a monopsonist consists of a single point given by l1, w1. The monopsonist has chosen this point as the most desirable of all points on the supply curve, S. A different point will not be chosen unless some external change (such as a shift in the demand for the ﬁrm’s output or a change in technology) affects labor’s marginal revenue product.11 EXAMPLE 16.4 Monopsonistic Hiring To illustrate these concepts in a simple context, suppose a coal mine’s workers can dig two tons of coal per hour and coal sells for $10 per ton. Therefore, the marginal revenue product of a coal miner is $20 per hour. If the coal mine is the only hirer of miners in a local area and faces a labor supply curve of the form then this ﬁrm must recognize that its hiring decisions affect wages. Expressing the total wage bill as a function of l, 50w, l ¼ (16:45) l2 50 , wl ¼ (16:46) permits the mine operator (perhaps only implicitly) to calculate the marginal expense associated with hiring miners: MEl ¼ @wl @l ¼ l 25 : (16:47) Equating this to miners’ marginal revenue product of $20 implies that the mine operator should hire 500 workers per hour. At this level of employment the wage will be $10 per hour—only half the value of the workers’ marginal revenue product. If the mine operator had been forced by market competition to pay $20 per hour regardless of the number of miners hired, then market 1,000 rather than the 500 hired under equilibrium would have been established with l monopsonistic conditions. ¼ QUERY: Suppose the price of coal increases to $15 per ton. How would this affect the monopsonist’s hiring and the wages of coal miners? Would the miners beneﬁt fully from the increase in their MRP? 11A monopsony may also practice price discrimination in all of the ways described for a monopoly in Chapter 14. For a detailed discussion of the comparative statics analysis of factor demand in the monopoly and monopsony cases, see W. E. Diewert, ‘‘Duality Approaches to Microeconomic Theory,’’ in K. J. Arrow and M. D. Intriligator, Eds., Handbook of Mathematical Economics (Amsterdam: North-Holland, 1982), vol. 2, pp. 584–90. 598 Part 7: Pricing in Input Markets Labor Unions Workers may at times ﬁnd it advantageous to join together in a labor union to pursue goals that can more effectively be accomplished by a group. If association with a union were wholly voluntary, we could assume that every union member derives a positive beneﬁt from belonging. Compulsory membership (the ‘‘closed shop’’), however, is often used to maintain the viability of the union organization. If all workers were left on their own to decide on membership, their rational decision might be not to join the union, thereby avoiding dues and other restrictions. However, they would beneﬁt from the higher wages and better working conditions that have been won by the union. What appears to be rational from each individual worker’s point of view may prove to be irrational from a group’s point of view, because the union is undermined by ‘‘free riders.’’ Therefore, compulsory membership may be a necessary means of maintaining the union as an effective bargaining agent. Unions’ goals A good starting place for our analysis of union behavior is to describe union goals. A ﬁrst assumption we might make is that the goals of a union are in some sense an adequate representation of the goals of its members. This assumption avoids the problem of union leadership and disregards the personal aspirations of those leaders, which may be in conﬂict with rank-and-ﬁle goals. Therefore, union leaders are assumed to be conduits for expressing the desires of the membership.12 In some respects, unions can be analyzed in the same way as monopoly ﬁrms. The union faces a demand curve for labor; because it is the sole source of supply, it can choose at which point on this curve it will operate. The point actually chosen by the union will obviously depend on what particular goals it has decided to pursue. Three possible choices are illustrated in Figure 16.5. For example, the union may choose to offer that quantity of labor that maximizes the total wage bill (w Æ l ). If this is the case, it will offer that quantity for which the ‘‘marginal revenue’’ from labor demand is equal to 0. This quantity is given by l1 in Figure 16.5, and the wage rate associated with this quantity is w1. Therefore, the point E1 is the preferred wage-quantity combination. Notice that at wage rate w1 there may be an excess supply of labor, and the union must somehow allocate available jobs to those workers who want them. Another possible goal the union may pursue would be to choose the quantity of labor that would maximize the total economic rent (that is, wages less opportunity costs) obtained by those members who are employed. This would necessitate choosing that quantity of labor for which the additional total wages obtained by having one more employed union member (the marginal revenue) are equal to the extra cost of luring that member into the market. Therefore, the union should choose that quantity, l2, at which the marginal revenue curve crosses the supply curve.13 The wage rate associated with this quantity is w2, and the desired wage-quantity combination is labeled E2 in the diagram. With the wage w2, many individuals who desire to work at the prevailing wage are left unemployed. Perhaps the union may ‘‘tax’’ the large economic rent earned by those who do work to transfer income to those who don’t. 12Much recent analysis, however, revolves around whether ‘‘potential’’ union members have some voi
ce in setting union goals and how union goals may affect the desires of workers with differing amounts of seniority on the job. 13Mathematically, the union’s goal is to choose l so as to maximize wl – (area under S), where S is the compensated supply curve for labor and reﬂects workers’ opportunity costs in terms of forgone leisure. FIGURE 16.5 Three Possible Points on the Labor Demand Curve That a Monopolistic Union Might Choose Chapter 16: Labor Markets 599 A union has a monopoly in the supply of labor, so it may choose its most preferred point on the demand curve for labor. Three such points are shown in the ﬁgure. At point E1, total labor payments (w Æ l ) are maximized; at E2, the economic rent that workers receive is maximized; and at E3, the total amount of labor services supplied is maximized. Real wage MR D w2 w1 w3 E2 E1 E3 S D l 2 l 1 l 3 Quantity of labor per period A third possibility would be for the union to aim for maximum employment of its members. This would involve choosing the point w3, l3, which is precisely the point that would result if the market were organized in a perfectly competitive way. No employment greater than l3 could be achieved, because the quantity of labor that union members supply would be reduced for wages less than w3. EXAMPLE 16.5 Modeling a Union In Example 16.4 we examined a monopsonistic hirer of coal miners who faced a supply curve given by 50w: l ¼ (16:48) To study the possibilities for unionization to combat this monopsonist, assume (contrary to Example 16.4) that the monopsonist has a downward-sloping marginal revenue product for labor curve of the form MRP 70 % ¼ 0:1l: (16:49) It is easy to show that, in the absence of an effective union, the monopsonist in this situation will choose the same wage-hiring combination it did in Example 16.4; 500 workers will be hired at a wage of $10. If the union can establish control over labor supply to the mine owner, then several other options become possible. The union could press for the competitive solution, for example. A contract of l 11.66 would equate supply and demand. Alternatively, the union could act as a monopolist facing the demand curve given by Equation 16.49. It could calculate the marginal increment yielded by supplying additional workers as 583, w ¼ ¼ d l ð ( MRP dl Þ 70 % ¼ 0:2l: (16:50) 600 Part 7: Pricing in Input Markets The intersection between this ‘‘marginal revenue’’ curve and the labor supply curve (which indicates the ‘‘marginal opportunity cost’’ of workers’ labor supply decisions) yields maximum rent to the unions’ workers: or l 50 ¼ 70 % 0:2l 3,500 11l: ¼ (16:51) (16:52) Therefore, such a calculation would suggest a contract of l 318 and a wage (MRP) of $38.20. The fact that both the competitive and union monopoly supply contracts differ signiﬁcantly from the monopsonist’s preferred contract indicates that the ultimate outcome here is likely to be determined through bilateral bargaining. Notice also that the wage differs signiﬁcantly depending on which side has market power. ¼ QUERY: Which, if any, of the three wage contracts described in this example might represent a Nash equilibrium? EXAMPLE 16.6 A Union Bargaining Model Game theory can be used to gain insights into the economics of unions. As a simple illustration, suppose a union and a ﬁrm engage in a two-stage game. In the ﬁrst stage, the union sets the wage rate its workers will accept. Given this wage, the ﬁrm then chooses its employment level. This two-stage game can be solved by backward induction. Given the wage w speciﬁed by the union, the ﬁrm’s second-stage problem is to maximize l ð where R is the total revenue function of the ﬁrm expressed as a function of employment. The ﬁrst-order condition for a maximum here (assuming that the wage is ﬁxed) is the familiar (16:53) Þ % wl ¼ R p R0 l ð Þ ¼ w: Assuming l ) solves Equation 16.54, the union’s goal is to choose w to maximize utility w, l ð w w, l)ð ½ U U Þ’ , Þ ¼ and the ﬁrst-order condition for a maximum is or U1 þ U2l 0 0 ¼ (16:54) (16:55) (16:56) l 0: U1=U2 ¼ % In words, the union should choose w so that its MRS is equal to the absolute value of the slope of the ﬁrm’s labor demand function. The w), l ) combination resulting from this game is clearly a Nash equilibrium. (16:57) Efﬁciency of the labor contract. The labor contract w), l ) is Pareto inefﬁcient. To see this, notice that Equation 16.57 implies that small movements along the ﬁrm’s labor demand curve (l) leave the union equally well-off. But the envelope theorem implies that a decrease in w must increase proﬁts to the ﬁrm. Hence there must exist a contract w p, l p (where w p < w) and l p > l )) with which both the ﬁrm and union are better off. Chapter 16: Labor Markets 601 The inefﬁciency of the labor contract in this two-stage game is similar to the inefﬁciency of some of the repeated Nash equilibria we studied in Chapter 15. This suggests that, with repeated rounds of contract negotiations, trigger strategies might be developed that form a subgame-perfect equilibrium and maintain Pareto-superior outcomes. For a simple example, see Problem 16.10. QUERY: Suppose the ﬁrm’s total revenue function differed depending on whether the economy was in an expansion or a recession. What kinds of labor contracts might be Pareto optimal? SUMMARY In this chapter we examined some models that focus on pricing in the labor market. Because labor demand was already treated as being derived from the proﬁt-maximization hypothesis in Chapter 11, most of the new material here focused on labor supply. Our primary ﬁndings were as follows. • A utility-maximizing individual will choose to supply an amount of labor at which his or her marginal rate of substitution of leisure for consumption is equal to the real wage rate. • An increase in the real wage creates substitution and income effects that work in opposite directions in affecting the quantity of labor supplied. This result can be summarized by a Slutsky-type equation much like the one already derived in consumer theory. • A competitive labor market will establish an equilibrium real wage at which the quantity of labor supplied by individuals is equal to the quantity demanded by ﬁrms. PROBLEMS • Wages may vary among workers for a number of reasons. Workers may have invested in different levels of skills and therefore have different productivities. Jobs may differ in their characteristics, thereby creating compensating wage differentials. And individuals may experience differing degrees of job-ﬁnding success. Economists have developed models that address all of these features of the labor market. • Monopsony power by ﬁrms on the demand side of the labor market will reduce both the quantity of labor hired and the real wage. As in the monopoly case, there will also be a welfare loss. • Labor unions can be treated analytically as monopoly suppliers of labor. The nature of labor market equilibrium in the presence of unions will depend importantly on the goals the union chooses to pursue. 16.1 Suppose there are 8,000 hours in a year (actually there are 8,760) and that an individual has a potential market wage of $5 per hour. a. What is the individual’s full income? If he or she chooses to devote 75 percent of this income to leisure, how many hours will be worked? b. Suppose a rich uncle dies and leaves the individual an annual income of $4,000 per year. If he or she continues to devote 75 percent of full income to leisure, how many hours will be worked? c. How would your answer to part (b) change if the market wage were $10 per hour instead of $5 per hour? d. Graph the individual’s supply of labor curve implied by parts (b) and (c). 16.2 As we saw in this chapter, the elements of labor supply theory can also be derived from an expenditure-minimization approach. cah1–a. Then the Suppose a person’s utility function for consumption and leisure takes the Cobb–Douglas form U(c, h) expenditure-minimization problem is ¼ minimize c w 24 ð % h Þ % s.t. U c, h ð Þ ¼ cah1 % a U: ¼ 602 Part 7: Pricing in Input Markets a. Use this approach to derive the expenditure function for this problem. b. Use the envelope theorem to derive the compensated demand functions for consumption and leisure. c. Derive the compensated labor supply function. Show that @l c / @w > 0. d. Compare the compensated labor supply function from part (c) to the uncompensated labor supply function in Example 16.2 0). Use the Slutsky equation to show why income and substitution effects of a change in the real wage are precisely (with n offsetting in the uncompensated Cobb–Douglas labor supply function. ¼ 16.3 A welfare program for low-income people offers a family a basic grant of $6,000 per year. This grant is reduced by $0.75 for each $1 of other income the family has. a. How much in welfare beneﬁts does the family receive if it has no other income? If the head of the family earns $2,000 per year? How about $4,000 per year? b. At what level of earnings does the welfare grant become zero? c. Assume the head of this family can earn $4 per hour and that the family has no other income. What is the annual budget constraint for this family if it does not participate in the welfare program? That is, how are consumption (c) and hours of leisure (h) related? d. What is the budget constraint if the family opts to participate in the welfare program? (Remember, the welfare grant can only be positive.) e. Graph your results from parts (c) and (d). f. Suppose the government changes the rules of the welfare program to permit families to keep 50 percent of what they earn. How would this change your answers to parts (d) and (e)? g. Using your results from part (f), can you predict whether the head of this family will work more or less under the new rules described in part (f )? 16.4 Suppose demand for labor is given by and supply is given by l 50w ¼ % 450 þ 100w, l ¼ where l represents the number of people employed and w is the real wage rate per hour. a.
What will be the equilibrium levels for w and l in this market? b. Suppose the government wishes to increase the equilibrium wage to $4 per hour by offering a subsidy to employers for each person hired. How much will this subsidy have to be? What will the new equilibrium level of employment be? How much total subsidy will be paid? c. Suppose instead that the government declared a minimum wage of $4 per hour. How much labor would be demanded at this price? How much unemployment would there be? d. Graph your results. 16.5 Carl the clothier owns a large garment factory on an isolated island. Carl’s factory is the only source of employment for most of the islanders, and thus Carl acts as a monopsonist. The supply curve for garment workers is given by where l is the number of workers hired and w is their hourly wage. Assume also that Carl’s labor demand (marginal revenue product) curve is given by 80w, l ¼ a. How many workers will Carl hire to maximize his proﬁts, and what wage will he pay? b. Assume now that the government implements a minimum wage law covering all garment workers. How many workers will Carl now hire, and how much unemployment will there be if the minimum wage is set at $4 per hour? 400 l ¼ % 40MRPl: Chapter 16: Labor Markets 603 c. Graph your results. d. How does a minimum wage imposed under monopsony differ in results as compared with a minimum wage imposed under perfect competition? (Assume the minimum wage is above the market-determined wage.) 16.6 The Ajax Coal Company is the only hirer of labor in its area. It can hire any number of female workers or male workers it wishes. The supply curve for women is given by and for men by lf ¼ 100wf lm ¼ 9w2 m, where wf and wm are the hourly wage rates paid to female and male workers, respectively. Assume that Ajax sells its coal in a perfectly competitive market at $5 per ton and that each worker hired (both men and women) can mine 2 tons per hour. If the ﬁrm wishes to maximize proﬁts, how many female and male workers should be hired, and what will the wage rates be for these two groups? How much will Ajax earn in proﬁts per hour on its mine machinery? How will that result compare to one in which Ajax was constrained (say, by market forces) to pay all workers the same wage based on the value of their marginal products? 16.7 Universal Fur is located in Clyde, Bafﬁn Island, and sells high-quality fur bow ties throughout the world at a price of $5 each. The production function for fur bow ties (q) is given by where x is the quantity of pelts used each week. Pelts are supplied only by Dan’s Trading Post, which obtains them by hiring Eskimo trappers at a rate of $10 per day. Dan’s weekly production function for pelts is given by 240x q ¼ % 2x2, where l represents the number of days of Eskimo time used each week. ﬃﬃ lp , x ¼ a. For a quasi-competitive case in which both Universal Fur and Dan’s Trading Post act as price-takers for pelts, what will be the equilibrium price (px) and how many pelts will be traded? b. Suppose Dan acts as a monopolist, while Universal Fur continues to be a price-taker. What equilibrium will emerge in the pelt market? c. Suppose Universal Fur acts as a monopsonist but Dan acts as a price-taker. What will the equilibrium be? d. Graph your results, and discuss the type of equilibrium that is likely to emerge in the bilateral monopoly bargaining between Universal Fur and Dan. 16.8 Following in the spirit of the labor market game described in Example 16.6, suppose the ﬁrm’s total revenue function is given by and the union’s utility is simply a function of the total wage bill: 10l R ¼ % l2 w, l U ð Þ ¼ wl: a. What is the Nash equilibrium wage contract in the two-stage game described in Example 16.6? b. Show that the alternative wage contract w0 l 0 c. Under what conditions would the contract described in part (b) be sustainable as a subgame-perfect equilibrium? 4 is Pareto superior to the contract identiﬁed in part (a). ¼ ¼ 604 Part 7: Pricing in Input Markets Analytical Problems 16.9 Compensating wage differentials for risk An individual receives utility from daily income (y), given by U y ð Þ ¼ 100y 1 2 % y2: The only source of income is earnings. Hence y wl, where w is the hourly wage and l is hours worked per day. The individual knows of a job that pays $5 per hour for a certain 8-hour day. What wage must be offered for a construction job where hours of work are random—with a mean of 8 hours and a standard deviation of 6 hours—to get the individual to accept this more ‘‘risky’’ job? Hint: This problem makes use of the statistical identity ¼ E x2 ð Þ ¼ Var x E x2 ð : Þ þ 16.10 Family labor supply A family with two adult members seeks to maximize a utility function of the form U c, h1, h2Þ , ð where c is family consumption and h1 and h2 are hours of leisure of each family member. Choices are constrained by h2Þ þ where w1 and w2 are the wages of each family member and n is nonlabor income. h1Þ þ w2ð w1ð 24 24 % ¼ % c n, a. Without attempting a mathematical presentation, use the notions of substitution and income effects to discuss the likely signs of the cross-substitution effects @h1 / @w2 and @h2 / @w1. b. Suppose that one family member (say, individual 1) can work in the home, thereby converting leisure hours into consump- tion according to the function c1 ¼ where f 0 > 0 and f 00 < 0. How might this additional option affect the optimal division of work among family members? , h1Þ ð f 16.11 A few results from demand theory The theory developed in this chapter treats labor supply as the mirror image of the demand for leisure. Hence, the entire body of demand theory developed in Part 2 of the text becomes relevant to the study of labor supply as well. Here are three examples. a. Roy’s identity. In the Extensions to Chapter 5 we showed how demand functions can be derived from indirect utility functions by using Roy’s identity. Use a similar approach to show that the labor supply function associated with the utilitymaximization problem described in Equation 16.20 can be derived from the indirect utility function by l w, n ð Þ ¼ @V @V =@w w, n Þ ð =@n w, n Þ ð : Illustrate this result for the Cobb–Douglas case described in Example 16.1. b. Substitutes and complements. A change in the real wage will affect not only labor supply, but also the demand for speciﬁc items in the preferred consumption bundle. Develop a Slutsky-type equation for the cross-price effect of a change in w on a particular consumption item and then use it to discuss whether leisure and the item are (net or gross) substitutes or complements. Provide an example of each type of relationship. c. Labor supply and marginal expense. Use a derivation similar to that used to calculate marginal revenue for a given demand curve to show that MEl ¼ w(1 þ 1 / el, w). Chapter 16: Labor Markets 605 ¼ þ 16.12 Intertemporal labor supply It is relatively easy to extend the single-period model of labor supply presented in Chapter 16 to many periods. Here we look at a simple example. Suppose that an individual makes his or her labor supply and consumption decisions over two periods.14 Assume that this person begins period 1 with initial wealth W0 and that he or she has 1 unit of time to devote to work or leisure in each period. Therefore, the two-period budget constraint is given by W0 ¼ c2, where the w’s are the real wage rates prevailing in each period. Here we treat w2 as uncertain, so utility in period 2 will also be uncertain. If we assume utility is additive across the two periods, we have E[U(c1, h1, c2, h2)] c1 þ U(c1, h1) E[U(c2, h2)]. w2(1 w1(1 h2) h1) % % % % a. Show that the ﬁrst-order conditions for utility maximization in period 1 are the same as those shown in Chapter 16; in par- ¼ ticular, show MRS(c1 for h1) c1. (Note that because w2 is a random variable, V is also random.) w1. Explain how changes in W0 will affect the actual choices of c1 and h1. b. Explain why the indirect utility function for the second period can be written as V(w2, W)), where W) ¼ c. Use the envelope theorem to show that optimal choice of W) requires that the Lagrange multipliers for the wealth constraint in the two periods obey the condition l1 ¼ E(l2) (where l1 is the Lagrange multiplier for the original problem and l2 is the implied Lagrange multiplier for the period 2 utility-maximization problem). That is, the expected marginal utility of wealth should be the same in the two periods. Explain this result intuitively. W0 þ w1(1 h1) % % d. Although the comparative statics of this model will depend on the speciﬁc form of the utility function, discuss in general terms how a governmental policy that added k dollars to all period 2 wages might be expected to affect choices in both periods. SUGGESTIONS FOR FURTHER READING Ashenfelter, O. C., and D. Card. Handbook of Labor Economics, 3. Amsterdam: North Holland, 1999. Contains a variety of high-level essays on many labor market topics. Survey articles on labor supply and demand in volumes 1 and 2 (1986) are also highly recommended. Becker, G. ‘‘A Theory of the Allocation of Time.’’ Economic Journal (September 1965): 493–517. One of the most inﬂuential papers in microeconomics. Becker’s observations on both labor supply and consumption decisions were revolutionary. Binger, B. R., and E. Hoffman. Microeconomics with Calculus, 2nd ed. Reading, MA: Addison-Wesley, 1998. Chapter 17 has a thorough discussion of the labor supply model, including some applications to household labor supply. Hamermesh, D. S. Labor Demand. Princeton, NJ: Princeton University Press, 1993. The author offers a complete coverage of both theoretical and empirical issues. The book also has nice coverage of dynamic issues in labor demand theory. Silberberg, E., and W. Suen. The Structure of Economics: A Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill, 2001. Provides a nice discussion of the dual approach to labor supply theory. 14Here we assume that the individual does not disco
unt utility in the second period and that the real interest rate between the two periods is zero. Discounting in a multiperiod context is taken up in Chapter 17. The discussion in that chapter also generalizes the approach to studying changes in the Lagrange multiplier over time shown in part (c). This page intentionally left blank Capital and Time In this chapter we provide an introduction to the theory of capital. In many ways that theory resembles our previous analysis of input pricing in general—the principles of proﬁt-maximizing input choice do not change. But capital theory adds an important time dimension to economic decision making; our goal here is to explore that extra dimension. We begin with a broad characterization of the capital accumulation process and the notion of the rate of return. Then we turn to more speciﬁc models of economic behavior over time. Capital and the Rate of Return When we speak of the capital stock of an economy, we mean the sum total of machines, buildings, and other reproducible resources in existence at some point in time. These assets represent some part of an economy’s past output that was not consumed but was instead set aside to be used for production in the future. All societies, from the most primitive to the most complex, engage in capital accumulation. Hunters in a primitive society taking time off from hunting to make arrows, individuals in a modern society using part of their incomes to buy houses, or governments taxing citizens in order to purchase dams and post ofﬁce buildings are all engaging in essentially the same sort of activity: Some portion of current output is being set aside for use in producing output in future periods. As we saw in the previous chapter, this is also true for human capital— individuals invest time and money in improving their skills so that they can earn more in the future. Present ‘‘sacriﬁce’’ for future gain is the essential aspect of all capital accumulation. Rate of return The process of capital accumulation is pictured schematically in Figure 17.1. In both panels of the ﬁgure, society is initially consuming level c0 and has been doing so for some time. At time t1 a decision is made to withhold some output (amount s) from current consumption for one period. Starting in period t2, this withheld consumption is in some way put to use producing future consumption. An important concept connected with this process is the rate of return, which is earned on that consumption that is put aside. In panel (a), for example, all of the withheld consumption is used to produce additional output only in period t2. Consumption is increased by amount x in period t2 and then returns to the long-run level c0. Society has saved in one year in order to splurge in the next year. The (one-period) rate of return from this activity is deﬁned as follows. 607 608 Part 7: Pricing in Input Markets FIGURE 17.1 Two Views of Capital Accumulation In (a), society withdraws some current consumption (s) to gorge itself (with x extra consumption) in the next period. The one-period rate of return would be measured by x/s 1. The society in (b) takes a more long-term view and uses s to increase its consumption perpetually by y. The perpetual rate of return would be given by y/s. ! Consumption Consumption c0 x s c0 y s t1 t2 t3 Time (a) One-period return t1 t2 t3 (b) Perpetual return Time Single-period rate of return. The single-period rate of return (r1) on an investment is the extra consumption provided in period 2 as a fraction of the consumption forgone in period 1. That is, r1 : (17:1) If x > s (if more consumption comes out of this process than went into it), we would say that the one-period rate of return to capital accumulation is positive. For example, if withholding 100 units from current consumption permitted society to consume an extra 110 units next year, then the one-period rate of return would be 110 100 ! 1 ¼ 0:10 or 10 percent. In panel (b) of Figure 17.1, society takes a more long-term view in its capital accumulation. Again, an amount s is set aside at time t1. Now, however, this set-aside consumption is used to increase the consumption level for all periods in the future. If the permanent level of consumption is increased to c0 þ y, we deﬁne the perpetual rate of return as follows Perpetual rate of return. The perpetual rate of return (r consumption expressed as a fraction of the initial consumption forgone. That is, 1 ) is the permanent increment to future y s : r 1 ¼ (17:2) If capital accumulation succeeds in raising c0 permanently, then r will be positive. For example, suppose that society set aside 100 units of output in period t1 to be devoted to capital accumulation. If this capital would permit output to be increased by 10 units for every period in the future (starting at time period t2), the perpetual rate of return would be 10 percent. 1 Chapter 17: Capital and Time 609 When economists speak of the rate of return to capital accumulation, they have in mind something between these two extremes. Somewhat loosely we shall speak of the rate of return as being a measure of the terms at which consumption today may be turned into consumption tomorrow (this will be made more explicit soon). A natural question to ask is how the economy’s rate of return is determined. Again, the equilibrium arises from the supply and demand for present and future goods. In the next section we present a simple two-period model in which this supply–demand interaction is demonstrated. Determining the Rate of Return In this section we will describe how operation of supply and demand in the market for ‘‘future’’ goods establishes an equilibrium rate of return. We begin by analyzing the connection between the rate of return and the ‘‘price’’ of future goods. Then we show how individuals and ﬁrms are likely to react to this price. Finally, these actions are brought together (as we have done for the analysis of other markets) to demonstrate the determination of an equilibrium price of future goods and to examine some of the characteristics of that solution. Rate of return and price of future goods For most of the analysis in this chapter, we assume there are only two periods to be considered: the current period (denoted by the subscript 0) and the next period (subscript 1). We will use r to denote the (one-period) rate of return between these two periods. Hence as deﬁned in the previous section, Dc1 Dc0 ! 1, r ¼ (17:3) where the D notation indicates the change in consumption during the two periods. Note that throughout this discussion we are using the absolute values of the changes in consumption as in Equations 17.1 and 17.2. Rewriting Equation 17.3 yields Dc1 Dc0 ¼ r, 1 þ or Dc0 Dc1 ¼ 1 1 þ : r (17:4) (17:5) The term on the left of Equation 17.5 records how much c0 must be forgone if c1 is to be increased by 1 unit; that is, the expression represents the relative ‘‘price’’ of 1 unit of c1 in terms of c0. Thus we have deﬁned the price of future goods. Price of future goods. The relative price of future goods ( p1) is the quantity of present goods that must be forgone to increase future consumption by 1 unit. That is, p1 ¼ Dc0 Dc1 ¼ 1 1 þ : r (17:6) We now proceed to develop a demand–supply analysis of the determination of p1. By so doing we also will have developed a theory of the determination of r, the rate of return in this simple model. 1This price is identical to the discount factor introduced in connection with repeated games in Chapter 8. 610 Part 7: Pricing in Input Markets Demand for future goods The theory of the demand for future goods is one further application of the utilitymaximization model developed in Part 2 of this book. Here the individual’s utility depends on present and future consumption [i.e., utility U(c0, c1)], and he or she must decide how much current wealth (W ) to allocate to these two goods.2 Wealth not spent on current consumption can be invested at the rate of return r to obtain consumption next period. As before, p1 reﬂects the present cost of future consumption, and the individual’s budget constraint is given by ¼ ¼ W p1c1: c 0 þ This constraint is illustrated in Figure 17.2. If the individual chooses to spend all of his or her wealth on c0, then total current consumption will be W with no consumption occurr). That ring in period 2. Alternatively, if c0 ¼ þ r) in is, if all wealth is invested at the rate of return r, current wealth will grow to W(1 period 2.3 0, then c1 will be given by W/p1 ¼ W(1 (17:7) þ FIGURE 17.2 Individual’s Intertemporal Utility Maximization When faced with the intertemporal budget constraint W by choosing to consume c$0 currently and c$1 in the next period. A decrease in p1 (an increase in the rate of return, r) will cause c1 to increase, but the effect on c0 is indeterminate because substitution and income effects operate in opposite directions (assuming that both c0 and c1 are normal goods). p1c1, the individual will maximize utility c0 þ ¼ Future consumption (c1) W/p1 W = c0 + p1c1 c1* U2 U1 U0 c0* W Current consumption (c0) 2For an analysis of the case where the individual has income in both periods, see Problem 17.1. 3This observation yields an alternative interpretation of the intertemporal budget constraint, which can be written in terms of the rate of return as This illustrates that it is the ‘‘present value’’ of c1 that enters into the individual’s current budget constraint. The concept of present value is discussed in more detail later in this chapter. W c0 þ 1 ¼ c1 þ : r Chapter 17: Capital and Time 611 Utility maximization Imposing the individual’s indifference curve map for c0 and c1 onto the budget constraint in Figure 17.2 illustrates utility maximization. Here utility is maximized at the point c$0, c$1. The individual consumes c$0 currently and chooses to save W c$0 to consume next period. This future consumption can be found from the budget constraint as ! or p1c$1 ¼ W ! c$0 c$1 ¼ W ð ! p1 c$0Þ W 1 c$0Þð
! þ r : Þ ¼ ð (17:8) (17:9) (17:10) In words, wealth that is not currently consumed ð return, r, and will grow to yield c$1 in the next period. W c$0Þ ! is invested at the rate of EXAMPLE 17.1 Intertemporal Impatience Individuals’ utility-maximizing choices over time will obviously depend on how they feel about the relative merits of consuming currently or waiting to consume in the future. One way of reﬂecting the possibility that people exhibit some impatience in their choices is to assume that the utility from future consumption is implicitly discounted in the individual’s mind. For example, we might assume that the utility function for consumption, U(c), is the same in both periods but that period 1’s utility is discounted in the individual’s mind by d) (where d > 0). If the intertemporal utility function is also a ‘‘rate of time preference’’ of 1/(1 separable (for more discussion of this concept, see the Extensions to Chapter 6), we can write with U 0 > 0, U 00 < 0 þ ð Þ c0, c1Þ ¼ ð Maximization of this function subject to the intertemporal budget constraint c0Þ þ ð : c1Þ ð þ U U U d 1 1 yields the following Lagrangian expression: W c0 þ 1 ¼ c1 þ r k W ! and the ﬁrst-order conditions for a maximum are c0, c1Þ þ ð ¼ U + c0 ! 1 ! c1 þ , r " @+ @c0 ¼ @+ @c1 ¼ @+ @k ¼ U 0 c0Þ ! , k ¼ c1Þ ! ð c1 c0 ! 1 0: r ¼ þ k þ 1 0, r ¼ Dividing the ﬁrst and second of these and rearranging terms gives4 U 0 c0Þ : c1Þ ð (17:11) (17:12) (17:13) (17:14) (17:15) Because the utility function for consumption is assumed to be the same in two periods, we can d, that c0 > c1 if d > r [to obtain U 0(c0) < U 0(c1) requires c0 > c1], conclude that c0 ¼ c1 if r ¼ 4Equation 17.15 is sometimes called the ‘‘Euler equation’’ for intertemporal utility maximization. As we show, once a speciﬁc utility function is deﬁned, the equation indicates how consumption changes over time. 612 Part 7: Pricing in Input Markets and that c0 < c1 for r > d. Therefore, whether this individual’s consumption increases or decreases from period 0 to period 1 will depend on exactly how impatient he or she is. Although a consumer may have a preference for present goods (d > 0), he or she may still consume more in the future than in the present if the rate of return received on savings is high enough. Consumption smoothing. Because utility functions exhibit diminishing marginal utility of consumption, individuals will seek to equalize their consumption across periods. The extent of such smoothing will depend on the curvature of the utility function. Suppose, for example, that an individual’s utility function takes the CES form U c Þ ¼ ð # c R=R if R if R ln c ð Þ 0 and R 0: 1, ’ 6¼ ¼ (17:16) Suppose also that this person’s rate of time preference is d be written as ¼ 0. In this case Equation 17.15 can or c1 1 c0 ¼ ð r Þ þ 1= ð 1 ! R Þ: (17:17) ¼ 0, this person will equalize consumption no matter what his or her utility function is. But If r a positive interest rate will encourage unequal consumption because in that case future goods are relatively cheaper. The degree to which a positive interest rate will encourage consumption inequality is determined by the value of R (which is sometimes referred to as the ‘‘coefﬁcient of r and so, with a ﬂuctuation aversion’’ in this context). For example, if R 5 percent interest rate, consumption in period 1 will be 5 percent higher than in period 0. On the other hand, if this person is more averse to consumption ﬂuctuations, then R might take a value such as 3. In this case (with a 5 percent interest rate), 0 then c1/c0 ¼ ¼ þ 1 ! c1 1 c0 ¼ ð r Þ þ 0:25 1:05 ¼ ð 0:25 Þ ¼ 1:012: (17:18) That is, consumption in period 1 will be only about 1 percent higher than in period 0. The real interest rate has a substantially smaller effect in encouraging this person to depart from an equalized consumption pattern when he or she is averse to ﬂuctuations. QUERY: Empirical data show that per capita consumption has increased at an annual rate of approximately 2 percent in the U.S. economy over the past 50 years. What real interest rate would be needed to make this increase utility maximizing (again assuming that d 0)? Note: We will return to the relationship between consumption smoothing and the real interest rate in Example 17.2. Problem 17.12 shows how intertemporal discount rates that follow a hyperbolic pattern can be used to explain why people may sometimes make decisions that seem ‘‘shortsighted.’’ ¼ Effects of changes in r A comparative statics analysis of the equilibrium illustrated in Figure 17.2 is straightforward. If p1 decreases (that is, if r increases), then both income and substitution effects will cause more c1 to be demanded—except in the unlikely event that c1 is an inferior good. Hence the demand curve for c1 will be downward sloping. An increase in r effectively lowers the price of c1, and consumption of that good thereby increases. This demand curve is labeled D in Figure 17.3. Before leaving our discussion of individuals’ intertemporal decisions, we should point out that the analysis does not permit an unambiguous statement to be made about the sign of @c0/@p1. In Figure 17.2, substitution and income effects work in opposite directions, and thus no deﬁnite prediction is possible. A decrease in p1 will cause the individual to substitute c1 for c0 in his or her consumption plans. But the decrease in p1 increases FIGURE 17.3 Determination of the Equilibrium Price of Future Goods Chapter 17: Capital and Time 613 The point p$1, c$1 represents an equilibrium in the market for future goods. The equilibrium price of future goods determines the rate of return via Equation 17.16. Price (p1) p1* D s s D c1* Future consumption (c1) the real value of wealth, and this income effect causes both c0 and c1 to increase. Phrased somewhat differently, the model illustrated in Figure 17.2 does not permit a deﬁnite prediction about how changes in the rate of return affect current-period wealth accumulation (saving). A higher r produces substitution effects that favor more saving and income effects that favor less. Ultimately, then, the direction of the effect is an empirical question. Supply of future goods In one sense the analysis of the supply of future goods is quite simple. We can argue that an increase in the relative price of future goods ( p1) will induce ﬁrms to produce more of them, because the yield from doing so is now greater. This reaction is reﬂected in the positively sloped supply curve S in Figure 17.3. It might be expected that, as in our previous perfectly competitive analysis, this supply curve reﬂects the increasing marginal costs (or diminishing returns) ﬁrms experience when attempting to turn present goods into future ones through capital accumulation. Unfortunately, by delving deeper into the nature of capital accumulation, one runs into complications that have occupied economists for hundreds of years.5 Basically, all of these derive from problems in developing a tractable model of the capital accumulation process. For our model of individual behavior this problem did not arise, because we could assume that the ‘‘market’’ quoted a rate of return to individuals so they could adapt their behavior to it. We shall also follow this route when describing ﬁrms’ investment decisions later in the chapter. But to develop an adequate model of capital accumulation by ﬁrms, we must describe precisely how c0 is ‘‘turned into’’ c1, and doing so would take us too far aﬁeld into the intricacies of capital theory. Instead, we will be content to draw the supply curve in Figure 17.3 with a positive slope on the presumption that such a 5For a discussion of some of this debate, see M. Blaug, Economic Theory in Retrospect, rev. ed. (Homewood, IL: Richard D. Irwin, 1978), chap. 12. 614 Part 7: Pricing in Input Markets shape is intuitively reasonable. Much of the subsequent analysis in this chapter may serve to convince you that this is indeed the case. Equilibrium price of future goods Equilibrium in the market shown in Figure 17.3 is at p$1, c$1. At that point, individuals’ supply and demand for future goods are in balance, and the required amount of current goods will be put into capital accumulation to produce c$1 in the future.6 There are a number of reasons to expect that p1 will be less than 1; that is, it will cost less than the sacriﬁce of one current good to ‘‘buy’’ one good in the future. As we showed in Example 17.1, it might be argued that individuals require some reward for waiting. Everyday adages (‘‘a bird in the hand is worth two in the bush,’’ ‘‘live for today’’) and more substantial realities (the uncertainty of the future and the ﬁniteness of life) suggest that individuals are generally impatient in their consumption decisions. Hence, capital accumulation such as that shown in Figure 17.3 will take place only if the current sacriﬁce is in some way worthwhile. There are also supply reasons for believing p1 will be less than 1. All of these involve the idea that capital accumulation is ‘‘productive’’: Sacriﬁcing one good today will yield more than one good in the future. Some simple examples of the productivity of capital investment are provided by such pastoral activities as the growing of trees or the aging of wine and cheese. Tree nursery owners and vineyard and dairy operators ‘‘abstain’’ from selling their wares in the belief that time will make them more valuable in the future. Although it is obvious that capital accumulation in a modern industrial society is more complex than growing trees (consider building a steel mill or an electric power system), economists believe the two processes have certain similarities. In both cases, investing current goods makes the production process longer and more complex, thereby increasing the contribution of other resources used in production. The equilibrium rate of return is determined in the Figure 17.3 shows how the equilibrium price of future goods market for those goods. Because p
resent and future consumption consists of the same homogeneous good, this will also determine the equilibrium rate of return according to the relationship p$1Þ ð p$1 ¼ 1 1 þ r$ or r$ 1 p$1 : ! p$1 ¼ (17:19) 0:95, then r$ Because p$1 will be less than 1, this equilibrium rate of return will be positive. For exam0:05, and we would say that the rate of return is ple, if p$1 ¼ ‘‘5 percent.’’ By withholding 1 unit of consumption in year 0, an individual would be able to purchase 1.05 units of consumption in period 1. Hence the equilibrium rate of return shows the terms on which goods can be reallocated over time for both individuals and ﬁrms. 0:05=0:95 ¼ ( Rate of return, real interest rates, and nominal interest rates The concept of the rate of return that we have been analyzing here is sometimes used synonymously with the related concept of the ‘‘real’’ interest rate. In this context, both are taken to refer to the real return that is available from capital accumulation. This 6This is a much simpliﬁed form of an analysis originally presented by I. Fisher, The Rate of Interest (New York: Macmillan, 1907). Chapter 17: Capital and Time 615 concept must be differentiated from the nominal interest rate actually available in ﬁnancial markets. Speciﬁcally, if overall prices are expected to increase by ˙pe between two periods (that is, ˙pe ¼ 0:10 for a 10 percent inﬂation rate), then we would expect the nominal interest rate (i) to be given by the equation 1 i 1 1 r Þð þ ˙peÞ , þ ¼ ð þ (17:20) because a would-be lender would expect to be compensated for both the opportunity cost of not investing in real capital (r) and for the general increase in prices . Expansion of Equation 17.17 yields ˙peÞ ð 1 i 1 r ˙pe þ r˙pe; þ ˙pe is small, we have the simpler approximation þ ¼ þ (17:21) and assuming r ) i r ˙pe: (17:22) þ If the real rate of return is 4 percent (0.04) and the expected rate of inﬂation is 10 percent (0.10), then the nominal interest rate would be approximately 14 percent (0.14). Therefore, the difference between observed nominal interest rates and real interest rates may be substantial in inﬂationary environments. ¼ EXAMPLE 17.2 Determination of the Real Interest Rate A simple model of real interest rate determination can be developed by assuming that consumption grows at some exogenous rate, g. For example, suppose that the only consumption good is perishable fruit and that this fruit comes from trees that are growing at the rate g. More realistically, g might be determined by macroeconomic forces, such as the rate of technical change in the Solow growth model (see the Extensions to Chapter 9). No matter how the growth rate is determined, the real interest rate must adjust so that consumers are willing to accept this rate of growth in consumption. Optimal consumption. The typical consumer wants his or her consumption pattern to maximize the utility received from this consumption over time. That is, the goal is to maximize utility 1 ð 0 ¼ e! dtU t c ð ð ÞÞ dt, (17:23) where d is the rate of pure time preference. At each instant of time, this person earns a wage w and earns interest r on his or her capital stock k. Hence this person’s capital evolves according to the equation dk dt ¼ w rk c ! þ (17:24) and is bound by the endpoint constraints k(0) Hamiltonian for this dynamic optimization problem (see Chapter 2) yields 0 and k( ) 1 ¼ ¼ 0. Setting up the augmented e! dtU rk k c Þ þ ! dk dt : Therefore, the ‘‘maximum principle’’ requires: Hc ¼ Hk ¼ e! dtU 0 rk þ k c Þ ! ð dk 0 dt ¼ 0; ¼ or rk ¼ ! dk dt : (17:25) (17:26) 616 Part 7: Pricing in Input Markets Solving the differential equation implied by the second of these conditions yields the conclusion that l rt, and substituting this into the ﬁrst of the conditions shows that e! ¼ U 0 c Þ ¼ ð Hence consistent with our results in Example 17.1, marginal utility should increase or decrease over time depending on the relationship between the rate of time preference and the real rate of 1, Equation 17.27 interest. When utility takes the CES form of U(c) gives the explicit solution: c R/R and U0(c) (17:27) cR ¼ ¼ eð ! r d ! t: Þ , % ex. Thus, if r > d, then consumption should increase over time, but the extent this increase should be affected by how willing this person is to tolerate unequal (17:28) ! ! t c ð Þ ¼ exp ¼ # t r 1 d R where exp{x} of consumption. Real interest rate determination. The only ‘‘price’’ in this simple economy is the real interest rate. This rate must adjust so that consumers will accept the rate of growth of consumption that is being determined exogenously. Hence it must be the case that g ¼ d R r 1 ! ! or r d 1 þ ð g: R Þ ! ¼ (17:29) ¼ 0, then the real rate of interest will equal the rate of time preference. With a positive If g growth rate of consumption, the real interest rate must exceed the rate of time preference to encourage people to accept consumption growth. Real interest rate paradox. Equation 17.29 provides the basis for what is sometimes termed interest rate paradox.’’ Over time, real consumption grows at about 1.6 percent the ‘‘real 3. Hence per year in the U.S. economy, and other evidence suggests that R is around even when the rate of time preference is zero, the real interest rate should be at least r 0 þ (1 0.048 (that is, about 5 percent). But empirical evidence shows that the real, risk-free rate in the United States over the past 75 years has been only about 2 percent—far lower than it should be. Either there is something wrong with this model, or people are more ﬂexible in their consumption decisions than is believed. 2) Æ 0.016 2 or ! ! ¼ ¼ þ QUERY: How should the results of this example be augmented to allow for the possibility that g may be subject to random ﬂuctuations? (See also Problem 17.9.) The Firm’s Demand For Capital Firms rent machines in accordance with the same principles of proﬁt maximization we derived in Chapter 11. Speciﬁcally, in a perfectly competitive market, the ﬁrm will choose to hire that number of machines for which the marginal revenue product is precisely equal to their market rental rate. In this section we ﬁrst investigate the determinants of this market rental rate, and implicitly assume all machines are rented from other ﬁrms. Later in the section we will see that this analysis is little changed when ﬁrms actually own the machines they use. Determinants of market rental rates Consider a ﬁrm in the business of renting machines to other ﬁrms. Suppose the ﬁrm owns a machine (say, a car or a backhoe) that has a current market price of p. How much Chapter 17: Capital and Time 617 will the ﬁrm charge its clients for the use of the machine? The owner of the machine faces two kinds of costs: depreciation on the machine and the opportunity cost of having its funds tied up in a machine rather than in an investment earning the current available rate of return. If it is assumed that depreciation costs per period are a constant percentage (d) of the machine’s market price and that the real interest rate is given by r, then the total costs to the machine owner for one period are given by pr pd p ð If we assume that the machine rental market is perfectly competitive, then no long-run proﬁts can be earned by renting machines. The workings of the market will ensure that the rental rate per period for the machine (v) is exactly equal to the costs of the machine owner. Hence we have the basic result that (17:30 The competitive rental rate is the sum of forgone interest and depreciation costs the machine’s owner must pay. For example, suppose the real interest rate is 5 percent (i.e., 0.05) and the physical depreciation rate is 15 percent (0.15). Suppose also that the current market price of the machine is $10,000. Then, in this simple model, the machine would have an annual rental rate of $2,000 [ 0.15)] per year; $500 of this $10,000 would represent the opportunity cost of the funds invested in the machine, and the remaining $1,500 would reﬂect the physical costs of deterioration. (0.05 þ ¼ * (17:31) Nondepreciating machines In the hypothetical case of a machine that does not depreciate (d can be written as ¼ v P ¼ r: 0), Equation 17.31 (17:32) In equilibrium an inﬁnitely long-lived (nondepreciating) machine is equivalent to a perpetual bond (see the Appendix to this chapter) and hence must ‘‘yield’’ the market rate of return. The rental rate as a percentage of the machine’s price must be equal to r. If v/p > r, then everyone would rush out to buy machines, because renting out machines would yield more than rates of return elsewhere. Similarly, if v/p < r, then no one would be in the business of renting out machines, because more could be made on alternative investments. Ownership of machines Our analysis so far has assumed that ﬁrms rent all of the machines they use. Although such rental does take place in the real world (for example, many ﬁrms are in the business of leasing airplanes, trucks, freight cars, and computers to other ﬁrms), it is more common for ﬁrms to own the machines they use. A ﬁrm will buy a machine and use it in combination with the labor it hires to produce output. The ownership of machines makes the analysis of the demand for capital somewhat more complex than that of the demand for labor. However, by recognizing the important distinction between a stock and a ﬂow, we can show that these two demands are quite similar. A ﬁrm uses capital services to produce output. These services are a ﬂow magnitude. It is the number of machine-hours that is relevant to the productive process (just as it is labor-hours), not the number of machines per se. Often, however, the assumption is made that the ﬂow of capital services is proportional to the stock of machines (100 machines, if fully employed for 1 hour, can deliver 100 machine-hours of service); therefore, these two different concepts are often used synonymously. If during a period a ﬁrm 618 Part 7: Pricing in Input Markets desires a certain number o
f machine-hours, this is usually taken to mean that the ﬁrm desires a certain number of machines. The ﬁrm’s demand for capital services is also a demand for capital.7 A proﬁt-maximizing ﬁrm in perfect competition will choose its level of inputs so that the marginal revenue product from an extra unit of any input is equal to its cost. This result also holds for the demand for machine-hours. The cost of capital services is given by the rental rate (v) in Equation 17.31. This cost is borne by the ﬁrm whether it rents the machine in the open market or owns the machine itself. In the former case it is an explicit cost, whereas in the latter case the ﬁrm is essentially in two businesses: (1) producing output, and (2) owning machines and renting them to itself. In this second role the ﬁrm’s decisions would be the same as any other machine rental ﬁrm because it incurs the same costs. The fact of ownership, to a ﬁrst approximation, is irrelevant to the determination of cost. Hence our prior analysis of capital demand applies to the owners by case as well Demand for capital. A proﬁt-maximizing ﬁrm that faces a perfectly competitive rental market for capital will hire additional capital input up to the point at which its marginal revenue product (MRPk) is equal to the market rental rate, v. Under perfect competition, the rental rate will reﬂect both depreciation costs and opportunity costs of alternative investments. Thus we have MRPk 17:33) Theory of investment If a ﬁrm obeys the proﬁt-maximizing rule of Equation 17.33 and ﬁnds that it desires more capital services than can be provided by its currently existing stock of machinery, then it has two choices. First, it may hire the additional machines that it needs in the rental market. This would be formally identical to its decision to hire additional labor. Second, the ﬁrm can buy new machinery to meet its needs. This second alternative is the one most often chosen; we call the purchase of new equipment by the ﬁrm investment. Investment demand is an important component of ‘‘aggregate demand’’ in macroeconomic theory. It is often assumed this demand for plant and equipment (i.e., machines) is inversely related to the real rate of interest, or what we have called the ‘‘rate of return.’’ Using the analysis developed in this part of the text, we can demonstrate the links in this argument. A decrease in the real interest rate (r) will, ceteris paribus, decrease the rental rate on capital (Equation 17.31). Because forgone interest represents an implicit cost for the owner of a machine, a decrease in r in effect reduces the price (i.e., the rental rate) of capital inputs. This decrease in v implies that capital has become a relatively less expensive input; this will prompt ﬁrms to increase their capital usage. Present Discounted Value Approach to Investment Decisions When a ﬁrm buys a machine, it is in effect buying a stream of net revenues in future periods. To decide whether to purchase the machine, the ﬁrm must compute the present 7Firms’ decisions on how intensively to use a given capital stock during a period are often analyzed as part of the study of business cycles. Chapter 17: Capital and Time 619 discounted value of this stream.8 Only by doing so will the ﬁrm have taken adequate account of the effects of forgone interest. This provides an alternative approach to explaining the investment decision. Consider a ﬁrm in the process of deciding whether to buy a particular machine. The machine is expected to last n years and will give its owner a stream of monetary returns (i.e., marginal revenue products) in each of the n years. Let the return in year i be represented by Ri. If r is the present real interest rate and if this rate is expected to prevail for the next n years, then the present discounted value (PDV) of the net revenue ﬂow from the machine to its owner is given by PDV R1 R2 Rn r þ Þ 1 ð n : (17:34) þ Þ This present discounted value represents the total value of the stream of payments provided by the machine—once adequate account is taken of the fact that these payments occur in different years. If the PDV of this stream of payments exceeds the price ( p) of the machine, then the ﬁrm, and other similar ﬁrms, should make the purchase. Even when the effects of the interest payments the ﬁrm could have earned on its funds had it not purchased the machine are taken into account, the machine promises to return more than its prevailing price. On the other hand, if p > PDV, the ﬁrm would be better off to invest its funds in some alternative that promises a rate of return of r. When account is taken of forgone interest, the machine does not pay for itself. Thus, in a competitive market, the only equilibrium that can prevail is that in which the price of a machine is equal to the present discounted value of the net revenues from the machine. Only in this situation will there be neither an excess demand for machines nor an excess supply of machines. Hence, market equilibrium requires that p PDV ¼ R1 R2 Rn r þ Þ 1 ð n : (17:35) þ We shall now use this condition to show two situations in which the present discounted value criterion of investment yields the same equilibrium conditions described earlier in the chapter. Þ Simple case Assume ﬁrst that machines are inﬁnitely long lived and that the marginal revenue product (Ri) is the same in every year. This uniform return also will equal the rental rate for machines (v), because that is what another ﬁrm would pay for the machine’s use during any period. With these simplifying assumptions, we may write the present discounted value from machine ownership as PDV 17:36 8See the Appendix to this chapter for an extended discussion of present discounted value. 620 Part 7: Pricing in Input Markets But in equilibrium p PDV, so ¼ or , (17:37) (17:38) as was already shown in Equation 17.32. For this case, the present discounted value criterion gives results identical to those outlined in the previous section. General case Equation 17.31 can also be derived for the more general case in which the rental rate on machines is not constant over time and in which there is some depreciation. This analysis is most easily carried out by using continuous time. Suppose that the rental rate for a new machine at anytime s is given by v(s). Assume also that the machine depreciates exponentially at the rate of d.9 Therefore, the net rental rate (and the marginal revenue product) of a machine decreases over time as the machine gets older. In year s, the net rental rate of an old machine bought in a previous year (t) would be v s d e! Þ ð t s ! ð Þ, (17:39) ! because s t is the number of years over which the machine has been decaying. For example, suppose that a machine is bought new in 2005. Its net rental rate in 2010 then would be 5d to account the rental rate earned by new machines in 2010 [v(2010)] discounted by the e! for the amount of depreciation that has taken place over the ﬁve years of the machine’s life. If the ﬁrm is considering buying the machine when it is new in year t, it should discount all of these net rental amounts back to that date. Therefore, the present value of the net rental in year s discounted back to year t is (if r is the interest rate) s ð r e! t s ! ð Þv d e! t s ! ð Þ eð r d þ Þ v e!ð r d þ s Þ (17:40) Þ Þ because, again, (s t) years elapse from when the machine is bought until the net rental is received. Therefore, the present discounted value of a machine bought in year t is the sum (integral) of these present values. This sum should be taken from year t (when the machine is bought) over all years into the future: ! ¼ s ð PDV t ð Þ ¼ 1 ð t eð !ð r d s ds: Þ þ (17:41) Since in equilibrium the price of the machine at year t [ p(t)] will be equal to this present value, we have the following fundamental equation: p t ð Þ ¼ 1 ð t eð !ð r d þ s ds: Þ (17:42) 9In this view of depreciation, machines are assumed to ‘‘evaporate’’ at a ﬁxed rate per unit of time. This model of decay is in many ways identical to the assumptions of radioactive decay made in physics. There are other possible forms that physical depreciation might take; this is just one that is mathematically tractable. It is important to keep the concept of physical depreciation (depreciation that affects a machine’s productivity) distinct from accounting depreciation. The latter concept is important only in that the method of accounting depreciation chosen may affect the rate of taxation on the proﬁts from a machine. From an economic point of view, however, the cost of a machine is a sunk cost: any choice on how to ‘‘write off’’ this cost is to some extent arbitrary. Chapter 17: Capital and Time 621 This rather formidable equation is simply a more complex version of Equation 17.35 and can be used to derive Equation 17.31. First rewrite the equation as p t ð Þ ¼ eð !ð r d þ s ds: Þ (17:43) Now differentiate with respect to t, using the rule for taking the derivative of a product: dp t Þ ð dt ¼ ð r r t d þ Þ d eð !ð r d þ s ds Þ eð !ð r d t Þ þ Hence dp t ð dt Þ : (17:44) (17:45) dp(t)/dt This is precisely the result shown earlier in Equation 17.31 except that the term has been added. The economic explanation for the presence of this added term is that it represents the capital gains accruing to the owner of the machine. If the machine’s price can be expected to increase, for example, the owner may accept somewhat less than d)p for its rental.10 On the other hand, if the price of the machine is expected to (r decrease [dp(t)/dt < 0], the owner will require more in rent than is speciﬁed in Equation 17.31. If the price of the machine is expected to remain constant over time, then dp(t)/dt 0 and the equations are identical. This analysis shows there is a deﬁnite relationship between the price of a machine at anytime, the stream of future proﬁts the machine promises, and the current rental rate for the machine. þ ! ¼ EXAMPLE 17.3 Cutting Down a Tree As an example of the PDV
criterion, consider the case of a forester who must decide when to cut down a growing tree. Suppose the value of the tree at any time, t, is given by f (t) where f 0ð and that l dollars were invested initially as payments to workers t ð who planted the tree. Assume also that the (continuous) market interest rate is given by r. When the tree is planted, the present discounted value of the tree owner’s proﬁts is given by > 0, f 00ð t < 0 Þ Þ Þ PDV t ð Þ ¼ e! rtf t ð Þ ! l, (17:46) which is simply the difference between (the present value of) revenues and present costs. The forester’s decision, then, consists of choosing the harvest date t to maximize this value. As always, this value may be found by differentiation: or, dividing both sides by e! rt, dPDV dt t ð Þ ¼ e! rtf 0 t ð Þ ! re! rtf ! rf t ð Þ ¼ 0: (17:47) (17:48) 10For example, rental houses in suburbs with rapidly appreciating house prices will usually rent for less than the landlord’s actual costs because the landlord also gains from price appreciation. 622 Part 7: Pricing in Input Markets Therefore, : r ¼ Þ Þ t f 0ð t f ð Two features of this optimal condition are worth noting. First, observe that the cost of the initial labor input drops out upon differentiation. This cost is (even in a literal sense) a ‘‘sunk’’ cost that is irrelevant to the proﬁt-maximizing decision. Second, Equation 17.49 can be interpreted as saying the tree should be harvested when the rate of interest is equal to the proportional rate of growth of the tree. This result makes intuitive sense. If the tree is growing more rapidly than the prevailing interest rate, then its owner should leave his or her funds invested in the tree, because the tree provides the best return available. On the other hand, if the tree is growing less rapidly than the prevailing interest rate, then the tree should be cut and the funds obtained from its sale should be invested elsewhere at the rate r. (17:49) Equation 17.49 is only a necessary condition for a maximum. By differentiating Equation 17.48 again, it is easy to see that it is also required that, at the chosen value of t, f 00 t ð Þ ! rf 0 t ð Þ < 0 (17:50) if the ﬁrst-order conditions are to represent a true maximum. Because we assumed f 0(t) > 0 < 0 (the growth slows over time), it is clear that this (the tree is always growing) and f 00 condition holds. t ð Þ A numerical illustration. Suppose trees grow according to the equation This equation always exhibits a positive growth rate [ f 0(t) > 0] and, because f t ð Þ ¼ exp 0:4 f tp : g ﬃﬃ t f 0ð t f ð Þ Þ 0:2 tp , ¼ (17:51) (17:52) the tree’s proportional growth rate diminishes over time. If the real interest rate were, say, 0.04, ﬃﬃ then we could solve for the optimal harvesting age as or so 0:04 :2 tp ¼ ﬃﬃ (17:53) 0:2 0:4 ¼ 5, ¼ tp ﬃﬃ t$ 25: (17:54) ¼ Up to 25 years of age, the volume of wood in the tree is increasing at a rate in excess of 4 percent per year, so the optimal decision is to permit the tree to stand. But for t > 25, the annual growth rate decreases below 4 percent, and thus the forester can ﬁnd better investments—perhaps planting new trees. A change in the interest rate. If the real interest rate increases to 5 percent, then Equation 17.53 becomes 0:05 r ¼ ¼ 0:2 tp , ﬃﬃ (17:55) Chapter 17: Capital and Time 623 2 16: ¼ (17:56) and the optimal harvest age would be t$ ¼ 0:2 0:05 & ’ The higher real interest rate discourages investment in trees by prompting the forester to choose an earlier harvest date.11 QUERY: Suppose all prices (including those of trees) were increasing at 10 percent per year. How would this change the optimal harvesting results in this problem? Natural Resource Pricing Pricing of natural resources has been a concern of economists at least since the time of Thomas Malthus. A primary issue has been whether the market system can achieve a desirable allocation of such resources given their ultimately ﬁnite and exhaustible nature. In this section we look at a simple model of resource pricing to illustrate some of the insights that economic analysis can provide. Proﬁt-maximizing pricing and output Suppose that a ﬁrm owns a ﬁnite stock of a particular resource. Let the stock of the resource at any time be denoted by x(t) and current production from this stock by q(t). Hence the stock of this resource evolves according to the differential equation dx t ð Þ dt ¼ ˙17:57) ð x and x( 0 Þ ¼ where we use the dot notation to denote a time derivative. The stock of this resource is constrained by x 0. Extraction of this resource exhibits constant average and marginal cost for changes in output levels, but this cost may change over time. Hence the ﬁrm’s total costs at any point in time are C(t) c(t)q(t). The ﬁrm’s goal then is to maximize the present discounted value of proﬁts subject to the constraint given in Equation 17.57. If we let p(t) be the price of the resource at time t, then the present value of future proﬁts is given by ) , e! rt dt, (17:58) where r is the real interest rate (assumed to be constant throughout our analysis). Setting up the augmented Hamiltonian for this dynamic optimization problem yields ! Þ, ¼ ½ rt k t q ð ½! þ Þ, þ x t ð Þ dk dt : (17:59) The maximum principle applied to this dynamic problem has two ﬁrst-order conditions for a maximum: rt e! c t ð Þ, k ! ¼ 0, Hq ¼ ½ t p ð dk Hx ¼ dt ¼ Þ ! 0: (17:60) 11For further tree-related economics, see Problems 17.4 and 17.11. 624 Part 7: Pricing in Input Markets The second of these conditions implies that the ‘‘shadow price’’ of the resource stock should remain constant over time. Because producing a unit of the resource reduces the stock by precisely 1 unit no matter when it is produced, any time path along which this shadow price changed would be nonoptimal. If we now solve the ﬁrst-order condition for l and differentiate with respect to time, we get (using the fact that dl/dt 0) ¼ t dk ð Þ dt ¼ 0 ˙k ˙p rt ˙c e! Þ r p rt: e! c Þ ! rt and rearranging terms provides an equation that explains how the price ¼ ð ! ¼ ! ð (17:61) Dividing by e! of the resource must change over time: (17:62) ˙p r p ˙c: c Þ þ ð ¼ ! Notice that the price change has two components. The second component shows that price changes must follow any changes in marginal extraction costs. The ﬁrst shows that, even if extraction costs do not change, there will be an upward trend in prices that reﬂects the scarcity value of the resource. The ﬁrm will have an incentive to delay some resource production only if so refraining will yield a return equivalent to the real interest rate. Otherwise it is better for the ﬁrm to sell all its resource assets and invest the funds elsewhere. This result, ﬁrst noted12 by Harold Hotelling in the early 1930s, can be further simpliﬁed by assuming that marginal extraction costs are always zero. In this case, Equation 17.62 reduces to the simple differential equation whose solution is rp, ˙p ¼ p0ert: p ¼ (17:63) (17:64) That is, prices increase exponentially at the real rate of interest. More generally, suppose that marginal costs also follow an exponential trend given by where g may be either positive or negative. In this case, the solution to the differential Equation 17.62 is t c ð Þ ¼ c 0egt, (17:65) p c0Þ This makes it even clearer that the resource price is inﬂuenced by two trends: an increasing scarcity rent that reﬂects the asset value of the resource, and the trend in marginal extraction costs. p0 ! (17:66) Þ ¼ ð t ð þ ert c0egt: EXAMPLE 17.4 Can Resource Prices Decrease? Although Hotelling’s original observation suggests that natural resource prices should increase at the real rate of interest, Equation 17.66 makes clear that this conclusion is not unambiguous. If marginal extraction costs decrease because of technical advances (i.e., if g is negative), then it is possible that the resource price will decrease. The conditions that would lead to decreasing resource prices can be made more explicit by calculating the ﬁrst and second time derivatives of price in Equation 17.66: 12H. Hotelling, ‘‘The Economics of Exhaustible Resources,’’ Journal of Political Economy (April 1931): 137–75. Chapter 17: Capital and Time 625 ert c0Þ þ gc0egt, (17:67) r dp dt ¼ d2p dt2 ¼ ð p0 ! r2 c0Þ Because the second derivative is always positive, we need only examine the sign of the ﬁrst derivative at t 0 to conclude when prices decrease. At this initial date, p0 ! þ ð g2c0egt > 0: ert ¼ c0Þ þ Hence prices will decrease (at least initially), providing p0 ! ð r dp dt ¼ g ! r > c0 : p0 ! c0 gc0: (17:68) (17:69) Clearly this condition cannot be met if marginal extraction costs are increasing over time (g > 0). But if costs are decreasing, a period of decreasing real price is possible. For example, if r 0.05 and g 0.02, then prices would decrease provided initial scarcity rents were less than 40 percent of extraction costs. Although prices must eventually increase, a fairly abundant resource that experienced signiﬁcant decreases in extraction costs could have a relatively long period of decreasing prices. This seems to have been the case for crude oil, for example. ¼ ! ¼ QUERY: Is the ﬁrm studied in this section a price-taker? How would the analysis differ if the ﬁrm were a monopolist? (See also Problem 17.10.) Generalizing the model The description of natural resource pricing given here provides only a brief glimpse of this important topic.13 Some additional issues that have been considered by economists include social optimality, substitution, and renewable resources. Social optimality. Are the price trends described in Equation 17.66 economically efﬁcient? That is, do they maximize consumer surplus in addition to maximizing the ﬁrm’s proﬁts? Our previous discussion of optimal consumption over time suggests that the marginal utility of consumption should change in certain prescribed ways if the consumer is to remain on his or her optimal path. Because individuals will consume any resource up to the point at which its price is p
roportional to marginal utility, it seems plausible that the price trends calculated here might be consistent with optimal consumption. But a more complete analysis would need to introduce the consumer’s rate of time preference and his or her willingness to substitute for an increasingly high-priced resource, so there is no clear-cut answer. Rather, the optimality of the path indicated by Equation 17.66 will depend on the speciﬁcs of the situation. Substitution. A related issue is how substitute resources should be integrated into this analysis. A relatively simple answer is provided by considering how the initial price ( p0) should be chosen in Equation 17.66. If that price is such that the initial price–quantity combination is a market equilibrium, then—assuming all other ﬁnite resource prices follow a similar time trend—relative resource prices will not change and (with certain utility functions) the price–quantity time paths for all of them may constitute an equilibrium. An alternative approach would be to assume that a perfect substitute for the resource will be developed at some date in the future. If this new resource is available in perfectly elastic 13For a sampling of dynamic optimization models applied to natural resource issues, see J. M. Conrad and C. W. Clark, Natural Resource Economics: Notes and Problems (Cambridge: Cambridge University Press, 2004). 626 Part 7: Pricing in Input Markets supply, then its availability would put a cap on the price or the original resource; this also would have implications for p0 (see Problem 17.7). But all of these solutions to modeling substitutability are special cases. To model the situation more generally requires a dynamic general equilibrium model capable of capturing interactions in many markets. Renewable resources. A ﬁnal complication that might be added to the model of resource pricing presented here is the possibility that the resource in question is not ﬁnite: it can be renewed through natural or economic actions. This would be the case for timber or ﬁshing grounds, where various types of renewal activities are possible. The formal consideration of renewable resources requires a modiﬁcation of the differential equation deﬁning changes in the resource stock, which no longer takes the simple form given in Equation 17.57. Speciﬁcation of proﬁt-maximizing price trajectories in such cases can become quite complicated. SUMMARY In this chapter we examined several aspects of the theory of capital, with particular emphasis on integrating it with the theory of resource allocation over time. Some of the results were as follows. • The rate of return (or real interest rate) is an important element in the overall costs associated with capital ownership. It is an important determinant of the market rental rate on capital, v. • Capital accumulation represents the sacriﬁce of present for future consumption. The rate of return measures the terms at which this trade can be accomplished. • The rate of return is established through mechanisms much like those that establish any equilibrium price. The equilibrium rate of return will be positive, reﬂecting not only individuals’ relative preferences for present over future goods but also the positive physical productivity of capital accumulation. • Future returns on capital investments must be discounted at the prevailing real interest rate. Use of such present value notions provides an alternative way to approach studying the ﬁrm’s investment decisions. • Individual wealth accumulation, natural resource pricing, and other dynamic problems can be studied using the techniques of optimal control theory. Often such models will yield competitive-type results. PROBLEMS 17.1 An individual has a ﬁxed wealth (W) to allocate between consumption in two periods (c1 and c2). The individual’s utility function is given by and the budget constraint is where r is the one-period interest rate. U c1, c2Þ , ð W c1 þ 1 ¼ c2 þ , r a. Show that, in order to maximize utility given this budget constraint, the individual should choose c1 and c2 such that the b. Show that @c2/@r 0 but that the sign of @c1/@r is ambiguous. If @c1/@r is negative, what can you conclude about the price MRS (of c1 for c2) is equal to 1 r. þ elasticity of demand for c2? - c. How would your conclusions from part (b) be amended if the individual received income in each period (y1 and y2) such that the budget constraint is given by y1 ! c1 þ c2 r ¼ y2 ! 1 þ 0? Chapter 17: Capital and Time 627 17.2 Assume that an individual expects to work for 40 years and then retire with a life expectancy of an additional 20 years. Suppose also that the individual’s earnings increase at a rate of 3 percent per year and that the interest rate is also 3 percent (the overall price level is constant in this problem). What (constant) fraction of income must the individual save in each working year to be able to ﬁnance a level of retirement income equal to 60 percent of earnings in the year just prior to retirement? 17.3 As scotch whiskey ages, its value increases. One dollar of scotch at year 0 is worth V If the interest rate is 5 percent, after how many years should a person sell scotch in order to maximize the PDV of this sale? 0:15t 2 f Þ ¼ exp tp t ð ! g dollars at time t. ﬃﬃ 17.4 As in Example 17.3, suppose trees are produced by applying 1 unit of labor at time 0. The value of the wood contained in a tree is given at any time t by f (t). If the market wage rate is w and the real interest rate is r, what is the PDV of this production process, and how should t be chosen to maximize this PDV? a. If the optimal value of t is denoted by t$, show that the ‘‘no pure proﬁt’’ condition of perfect competition will necessitate that Can you explain the meaning of this expression? w ¼ e! rtf t$ ð : Þ b. A tree sold before t$ will not be cut down immediately. Rather, it still will make sense for the new owner to let the tree continue to mature until t$. Show that the price of a u-year-old tree will be weru and that this price will exceed the value of the wood in the tree [ f (u)] for every value of u except u t$ (when these two values are equal). c. Suppose a landowner has a ‘‘balanced’’ woodlot with one tree of ‘‘each’’ age from 0 to t$. What is the value of this woodlot? ¼ Hint: It is the sum of the values of all trees in the lot. d. If the value of the woodlot is V, show that the instantaneous interest on V (that is, r Æ V) is equal to the ‘‘proﬁts’’ earned at each instant by the landowner, where by proﬁts we mean the difference between the revenue obtained from selling a fully matured tree [ f (t$)] and the cost of planting a new one (w). This result shows there is no pure proﬁt in borrowing to buy a woodlot, because one would have to pay in interest at each instant exactly what would be earned from cutting a fully matured tree. 17.5 This problem focuses on the interaction of the corporate proﬁts tax with ﬁrms’ investment decisions. a. Suppose (contrary to fact) that proﬁts were deﬁned for tax purposes as what we have called pure economic proﬁts. How would a tax on such proﬁts affect investment decisions? b. In fact, proﬁts are deﬁned for tax purposes as ! where depreciation is determined by governmental and industry guidelines that seek to allocate a machine’s costs over its ‘‘useful’’ lifetime. If depreciation were equal to actual physical deterioration and if a ﬁrm were in long-run competitive equilibrium, how would a tax on p0 affect the ﬁrm’s choice of capital inputs? ! ¼ p0 pq wl depreciation, c. Given the conditions of part (b), describe how capital usage would be affected by adoption of ‘‘accelerated depreciation’’ policies, which specify depreciation rates in excess of physical deterioration early in a machine’s life but much lower depreciation rates as the machine ages. d. Under the conditions of part (c), how might a decrease in the corporate proﬁts tax affect capital usage? 17.6 A high-pressure life insurance salesman was heard to make the following argument: ‘‘At your age a $100,000 whole life policy is a much better buy than a similar term policy. Under a whole life policy you’ll have to pay $2,000 per year for the ﬁrst four years but nothing more for the rest of your life. A term policy will cost you $400 per year, essentially forever. If you live 35 years, you’ll pay only $8,000 for the whole life policy, but $14,000 ( $400 Æ 35) for the term policy. Surely, the whole life is a better deal.’’ Assuming the salesman’s life expectancy assumption is correct, how would you evaluate this argument? Speciﬁcally, ¼ calculate the present discounted value of the premium costs of the two policies assuming the interest rate is 10 percent. 628 Part 7: Pricing in Input Markets 17.7 Suppose that a perfect substitute for crude oil will be discovered in 15 years and that the price of this substitute will be the equivalent of an oil price of $125 per barrel. Suppose the current marginal extraction cost for oil is $7 per barrel. Assume also that the real interest rate is 5 percent and that real extraction costs decrease at a rate of 2 percent annually. If crude oil prices follow the path described in Equation 17.66, what should the current price of crude oil be? Does your answer shed any light on actual pricing in the crude oil market? Analytical Problems 17.8 Capital gains taxation Suppose an individual has W dollars to allocate between consumption this period (c0) and consumption next period (c1) and that the interest rate is given by r. a. Graph the individual’s initial equilibrium and indicate the total value of current-period savings (W b. Suppose that, after the individual makes his or her savings decision (by purchasing one-period bonds), the interest rate decreases to r 0. How will this alter the individual’s budget constraint? Show the new utility-maximizing position. Discuss how the individual’s improved position can be interpreted as resulting from a ‘‘capital gain’’ on his or her initial bond
purchases. c0). ! c. Suppose the tax authorities wish to impose an ‘‘income’’ tax based on the value of capital gains. If all such gains are valued in terms of c0 as they are ‘‘accrued,’’ show how those gains should be measured. Call this value G1. d. Suppose instead that capital gains are measured as they are ‘‘realized’’—that is, capital gains are deﬁned to include only that portion of bonds that is cashed in to buy additional c0. Show how these realized gains can be measured. Call this amount G2. e. Develop a measure of the true increase in utility that results from the decrease in r, measured in terms of c0. Call this ‘‘true’’ capital gain G3. Show that G3 < G2 < G1. What do you conclude about a tax policy that taxes only realized gains? Note: This problem is adapted from J. Whalley, ‘‘Capital Gains Taxation and Interest Rate Changes,’’ National Tax Journal (March 1979): 87–91. 17.9 Precautionary saving and prudence The Query to Example 17.2 asks how uncertainty about the future might affect a person’s savings decisions. In this problem we explore this question more fully. All of our analysis is based on the simple two-period model in Example 17.1. a. To simplify matters, assume that r d in Equation 17.15. If consumption is certain, this implies that u 0(c0) ¼ c p c1. But suppose that consumption in period 1 will be subject to a zero-mean random shock, so that c1 ¼ 1 þ u 0(c1) or c0 ¼ x, where c p 1 is planned period-1 consumption and x is a random variable with an expected value of 0. Describe why, in this context, utility maximization requires u 0(c0) E[u 0(c1)]. ¼ b. Use Jensen’s inequality (see Chapters 2 and 7) to show that this person will opt for c p ¼ 1 > c0 if and only if u 0 is convex—that is, if and only if u000 > 0. c. Kimball14 suggests using the term ‘‘prudence’’ to describe a person whose utility function is characterized by u 000 > 0. Describe why the results from part (b) show that such a deﬁnition is consistent with everyday usage. d. In Example 17.2 we showed that real interest rates in the U.S. economy seem too low to reconcile actual consumption growth rates with evidence on individuals’ willingness to experience consumption ﬂuctuations. If consumption growth rates were uncertain, would this explain or exacerbate the paradox? 17.10 Monopoly and natural resource prices Suppose that a ﬁrm is the sole owner of a stock of a natural resource. a. How should the analysis of the maximization of the discounted proﬁts from selling this resource (Equation 17.58) be modi- ﬁed to take this fact into account? b. Suppose that the demand for the resource in question had a constant elasticity form q(t) change the price dynamics shown in Equation 17.62? a[p(t)]b. How would this ¼ c. How would the answer to Problem 17.7 be changed if the entire crude oil supply were owned by a single ﬁrm? 14M. S. Kimball, ‘‘Precautionary Savings in the Small and in the Large,’’ Econometrica (January 1990): 53–73. Chapter 17: Capital and Time 629 17.11 Renewable timber economics The calculations in Problem 17.4 assume there is no difference between the decisions to cut a single tree and to manage a woodlot. But managing a woodlot also involves replanting, which should be explicitly modeled. To do so, assume a lot owner is considering planting a single tree at a cost w, harvesting the tree at t$, planting another, and so forth forever. The discounted stream of proﬁts from this activity is then V w ¼ ! þ rt e r2t e rnt e. Show that the total value of this planned harvesting activity is given by b. Find the value of t that maximizes V. Show that this value solves the equation V ¼ t f Þ ! ð rt e! ! w 1 ! w: t$ ð c. Interpret the results of part (b): How do they reﬂect optimal usage of the ‘‘input’’ time? Why is the value of t$ speciﬁed in t$ ð t$ ð Þ ¼ Þ þ rV rf f 0 : Þ part (b) different from that in Example 17.2? d. Suppose tree growth (measured in constant dollars) follows the logistic function f t ð Þ ¼ 50= 1 ð þ e10 ! 0:1t : Þ What is the maximum value of the timber available from this tree? e. If tree growth is characterized by the equation given in part (d), what is the optimal rotation period if r Does this period produce a ‘‘maximum sustainable’’ yield? f. How would the optimal period change if r decreased to 0.04? 0.05 and w 0? ¼ ¼ Note: The equation derived in part (b) is known in forestry economics as Faustmann’s equation. 17.12 Hyperbolic discounting The notion that people might be ‘‘shortsighted’’ was formalized by David Laibson in ‘‘Golden Eggs and Hyperbolic Discounting’’ (Quarterly Journal of Economics, May 1997, pp. 443–77). In this paper the author hypothesizes that individuals maximize an intertemporal utility function of the form where 0 < b < 1 and 0 < d < 1. The particular time pattern of these discount factors leads to the possibility of shortsightedness. utility U ctÞ dsU ct ð , sÞ þ þ a. Laibson suggests hypothetical values of b 0.99. Show that, for these values, the factors by which future consumption is discounted follow a general hyperbolic pattern. That is, show that the factors decrease signiﬁcantly for period t 1 and then follow a steady geometric rate of decrease for subsequent periods. 0.6 and d ¼ ¼ b. Describe intuitively why this pattern of discount rates might lead to shortsighted behavior. c. More formally, calculate the MRS between ct 2 at time 1. Explain why, with a constant real interest rate, this would imply ‘‘dynamically inconsistent’’ choices over time. 1 and ct t Speciﬁcally, how would the relationship between optimal ct 2 at time t. Compare this to the MRS between ct 2 differ from these two perspectives? þ d. Laibson explains that the pattern described in part (c) will lead ‘‘early selves’’ to ﬁnd ways to constrain ‘‘future selves’’ and 1 and ct 1 and ct þ þ þ þ þ þ so achieve full utility maximization. Explain why such constraints are necessary. e. Describe a few of the ways in which people seek to constrain their future choices in the real world. 630 Part 7: Pricing in Input Markets SUGGESTIONS FOR FURTHER READING Blaug, M. Economic Theory in Retrospect, rev. ed. Homewood, IL: Richard D. Irwin, 1978, chap. 12. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Good review of Austrian capital theory and of attempts to conceptualize the capital accumulation process. Conrad, J. M., and C. W. Clark. Natural Resource Economics: Notes and Problems. Cambridge: Cambridge University Press, 2004. Provides several illustrations of how optimal control theory can be applied to problems in natural resource pricing. Dixit, A. K. Optimization in Economic Theory, 2nd ed. New York: Oxford University Press, 1990. Extended treatment of optimal control theory in a fairly easyto-follow format. Dorfman, R. ‘‘An Economic Interpretation of Optimal Control Theory.’’ American Economic Review 59 (December 1969): 817–31. Uses the approach of this chapter to examine optimal capital accumulation. Excellent intuitive introduction. Hotelling, H. Journal of Political Economy 39 (April 1931): 137–75. ‘‘The Economics of Exhaustible Resources.’’ Fundamental work on allocation of natural resources. Analyzes both competitive and monopoly cases. Chapter 20 offers extensive coverage of issues in deﬁning equilibrium over time. The discussion of ‘‘overlapping generations’’ models is especially useful. Ramsey, F. P. ‘‘A Mathematical Theory of Saving.’’ Economic Journal 38 (December 1928): 542–59. One of the ﬁrst uses of the calculus of variations to solve economic problems. Solow, R. M. Capital Theory and the Rate of Return. Amsterdam: North-Holland, 1964. Lectures on the nature of capital. Very readable. Sydsaeter, K., A. Strom, and P. Berck. Economists’ Mathematical Manual, 3rd ed. Berlin: Springer-Verlag, 2000. Chapter 27 provides a variety of formulas that are valuable for ﬁnance and growth theory. APPENDIX TO The Mathematics of Compound Interest The purpose of this appendix is to gather some simple results concerning the mathematics of compound interest. These results have applications in a wide variety of economic problems that range from macroeconomic policy to the optimal way of raising Christmas trees. We assume there is a current prevailing market interest rate of i per period—say, of one year. This interest rate is assumed to be both certain and constant over all future periods.1 If $1 is invested at this rate i and if the interest is then compounded (i.e., future interest is paid on post interest earned), then: at the end of one period, $1 will be at the end of two periods, $1 will be Þ and at the end of n periods, $1 will be þ ð $1 3 1 i $1 3 1 ð þ 2; i Þ Similarly, $N grows like $1 3 1 ð þ n: i Þ $N 3 1 ð þ n: i Þ Present Discounted Value The present value of $1 payable one period from now is þ This is simply the amount an individual would be willing to pay now for the promise of $1 at the end of one period. Similarly, the present value of $1 payable n periods from now is $1 1 : i $1 1 ð þ i Þ n , 1The assumption of a constant i is obviously unrealistic. Because problems introduced by considering an interest rate that varies from period to period greatly complicate the notation without adding a commensurate degree of conceptual knowledge, such an analysis is not undertaken here. In many cases the generalization to a varying interest rate is merely a trivial application of the notion that any multiperiod interest rate can be regarded as resulting from compounding several single-period rates. If we let rij be the interest rate prevailing between periods i and j (where i < j), then 1 rij ¼ ð 1 þ ri, i þ 1Þ þ ð 1 þ ri þ 1, i þ þ 2Þ þ ) ) ) þ ð 1 rj : 1, jÞ ! þ 632 Part 7: Pricing in Input Markets and the present value of $N payable n periods from now is Þ The present discounted value of a stream of payments N0, N1, N2, … , Nn (where the þ $N i n : 1 ð subscripts indicate the period in which the payment is to be made) is PDV N0 þ ¼ N1 1 ð þ N2 Nn i þ
Þ 1 ð n : (17A:1) The PDV is the amount an individual would be willing to pay in return for a promise to receive the stream N0, N1, N2, … , Nn. It represents the amount that would have to be invested now if one wished to duplicate the payment stream. Annuities and perpetuities An annuity is a promise to pay $N in each period for n periods, starting next period. The PDV of such a contract is PDV : Let d = 1/(1 + i); then, PDV ¼ ¼ ¼ þ ) ) ) þ d2 d2 N d þ ð Nd 1 ð Nd d þ dn d þ 1 1 ! ! : ’ & dn Þ þ ) ) ) þ dn 1 ! Þ (17A:2) (17A:3) Observe that Therefore, for an annuity of inﬁnite duration, lim n !1 dn 0: ¼ PDV of infinite annuity lim n !1 ¼ PDV ¼ Nd & 1 1 d ’ ! ; (17A:4) by the deﬁnition of d, Nd & ’ This case of an inﬁnite-period annuity is sometimes called a perpetuity or a consol. The formula simply says that the amount that must be invested if one is to obtain $N per period forever is simply $N/i, because this amount of money would earn $N in interest each period (i Æ $N/i $N). þ & & ’ (17A:5) ¼ The special case of a bond An n-period bond is a promise to pay $N each period, starting next period, for n periods. It also promises to return the principal (face) value of the bond at the end of n periods. Chapter 17: Capital and Time 633 If the principal value of the bond is $P (usually $1,000 in the U.S. bond market), then the present discounted value of such a promise is PDV 17A:6) Again, let d = 1/(1 i); then, þ PDV Nd þ ¼ Nd2 dn: (17A:7) Equation 17A.7 can be looked at in another way. Suppose we knew the price (say, B) at which the bond is currently trading. Then we could ask what value of i gives the bond a PDV equal to B. To ﬁnd this i we set PDV B ¼ ¼ Nd þ Nd2 dn: (17A:8) Because B, N, and P are known, we can solve this equation for d and hence for i.2 The i that solves the equation is called the yield on the bond and is the best measure of the return actually available from the bond. The yield of a bond represents the return available both from direct interest payments and from any price differential between the initial price (B) and the maturity price (P). Notice that, as i increases, PDV decreases. This is a precise way of formulating the well-known concept that bond prices (PDVs) and interest rates (yields) are inversely correlated. Continuous Time Thus far our approach has dealt with discrete time—the analysis has been divided into periods. Often it is more convenient to deal with continuous time. In such a case the interest on an investment is compounded ‘‘instantaneously’’ and growth over time is ‘‘smooth.’’ This facilitates the analysis of maximization problems because exponential functions are more easily differentiated. Many ﬁnancial intermediaries (for example, savings banks) have adopted (nearly) continuous interest formulas in recent years. Suppose that i is given as the (nominal) interest rate per year but that half this nominal rate is compounded every six months. Then, at the end of one year, the investment of $1 would have grown to $17A:9) Observe that this is superior to investing for one year at the simple rate i, because interest has been paid on interest; that is17A:10) 2Because this equation is an nth-degree polynomial, there are in reality n solutions (roots). Only one of these solutions is the relevant one reported in bond tables or on calculators. The other solutions are either imaginary or unreasonable. In the present example there is only one real solution. 634 Part 7: Pricing in Input Markets TABLE 17A.1 EFFECTIVE ANNUAL INTEREST RATES FOR SELECTED CONTINUOUSLY COMPOUNDED RATES Continuously Compounded Rate 3.0% Effective Annual Rate 3.05% 4.0 5.0 5.5 6.0 6.5 7.0 8.0 9.0 10.0 4.08 5.13 5.65 6.18 6.72 7.25 8.33 9.42 10.52 Consider the limit of this process: for the nominal rate of i per period, consider the amount that would be realized if i were in fact ‘‘compounded n times during the period.’’ Letting n ﬁ , we have 1 lim n !17A:11) This limit exists and is simply ei, where e is the base of natural logarithms (the value of e is approximately 2.72). It is important to note that ei > (1 i)—it is much better to have continuous compounding over the period than to have simple interest. þ We can ask what continuous rate r yields the same amount at the end of one period as the simple rate i. We are looking for the value of r that solves the equation er 1 : i ¼ ð þ Þ Hence r 1 ln ð þ : i Þ ¼ (17A:12) (17A:13) Using this formula, it is a simple matter to translate from discrete interest rates into continuous ones. If i is measured as a decimal yearly rate, then r is a yearly continuous rate. Table 17A.1 shows the effective annual interest rate (i) associated with selected interest rates (r) that are continuously compounded.3 Tables similar to 17A.1 often appear in the windows of savings banks advertising the ‘‘true’’ yields on their accounts. Continuous growth One dollar invested at a continuous interest rate of r will become V $1 erT (17A:14) ) after T years. This growth formula is a convenient one to work with. For example, it is easy to show that the instantaneous relative rate of change in V is, as would be expected, simply given by r: ¼ 3To compute the ﬁgures in Table 17A.1, interest rates are used in decimal rather than percent form (that is, a 5 percent interest rate is recorded as 0.05 for use in Equation 17A.12). Chapter 17: Capital and Time 635 relative rate of change dV= dt ¼ V ¼ rert ert ¼ r: (17A:15) Continuous interest rates also are convenient for calculating present discounted values. Suppose we wished to calculate the PDV of $1 to be paid T years from now. This would be given by4 $1 erT ¼ $1 3 e! rT : (17A:16) The logic of this calculation is exactly the same as that used in the discrete time analysis of this appendix: future dollars are worth less than present dollars. Payment streams One interesting application of continuous discounting occurs in calculating the PDV of $1 per period paid in small installments at each instant of time from today (time 0) until time T. Because there would be an inﬁnite number of payments, the mathematical tool of integration must be used to compute this result: PDV ¼ T ð 0 e! rt dt: (17A:17) What this expression means is that we are adding all the discounted dollars over the time period 0 to T. The value of this deﬁnite integral is given by PDV ! ! ¼ ¼ T rt e! r 0 ) ) rT e! ) ) r þ As T approaches inﬁnity, this value becomes PDV 1 r , ¼ 1 r : (17A:18) (17A:19) as was the case for the inﬁnitely long annuity considered in the discrete case. Continuous discounting is particularly convenient for calculating the PDV of an arbitrary stream of payments over time. Suppose that f(t) records the number of dollars to be paid during period t. Then the PDV of the payment at time t is ð and the PDV of the entire stream from the present time (year 0) until year T is given by Þ e! rtf t , (17A:20) PDV ¼ T ð 0 e! rt dt: f t ð Þ (17A:21) Often, economic agents may seek to maximize an expression such as that given in Equation 17A.21. Use of continuous time makes the analysis of such choices straightforward because standard calculus methods of maximization can be used. 4In physics this formula occurs as an example of ‘‘radioactive decay.’’ If 1 unit of a substance decays continuously at the rate d dT units will remain. This amount never exactly reaches zero no matter how large T is. Depreciation can then, after T periods, e! be treated the same way in capital theory. 636 Part 7: Pricing in Input Markets Duration The use of continuous time can also clarify a number of otherwise rather difﬁcult ﬁnancial concepts. For example, suppose we wished to know how long, on average, it takes for an individual to receive a payment from a given payment stream, f (t). The present value of the stream is given by V ¼ T ð 0 e! rt dt: f t ð Þ Differentiation of this value by the discount factor, e! r, yields @V @e! r ¼ T ð 0 tf t ð r e! Þ t ð 1 Þ dt, ! and the elasticity of this change is given by @V @e! r ) e ¼ T 0 tf r e! V ¼ Ð rt dt e! t Þ ð V : (17A:22) (17A:23) (17A:24) Hence the elasticity of the present value of this payment stream with respect to the annual discount factor (which is similar to, say, the elasticity of bond prices with respect to changes in interest rates) is given by the ratio of the present value of a time-weighted stream of payments to an unweighted stream. Conceptually, then, this elasticity represents the average time an individual must wait to receive the typical payment. In the ﬁnancial press this concept is termed the duration of the payment stream. This is an important measure of the volatility of the present value of such a stream with respect to interest rate changes.5 5As an example, a duration of 8 years would mean that the mean length of time that the individual must wait for the typical payment is 8 years. It also means that the elasticity of the value of this stream with respect to the discount factor is 8.0. Because the elasticity of the discount factor itself with respect to the interest rate is simply r, the elasticity of the value of the stream with respect to this interest rate is 0.05, for example, then the elasticity of the present value of this stream with 8r. If r 0.40. respect to r is ! ! ¼ ! This page intentionally left blank Market Failure P A R T EIGHT Chapter 18 Asymmetric Information Chapter 19 Externalities and Public Goods In this part we look more closely at some of the reasons why markets may perform poorly in allocating resources. We will also examine some of the ways in which such market failures might be mitigated. Chapter 18 focuses on situations where some market participants are better informed than others. In such cases of asymmetric information, establishing efﬁcient contracts between these parties can be quite complicated and may involve a variety of strategic choices. We will see that in many situations the ﬁrst-best, fully informed solution is not attainable. Therefore, second-best solutions that may involve so
me efﬁciency losses must be considered. Externalities are the principal topic of Chapter 19. The ﬁrst part of the chapter is concerned with situations in which the actions of one economic actor directly affect the well-being of another actor. We show that, unless these costs or beneﬁts can be internalized into the decision process, resources will be misallocated. In the second part of the chapter we turn to a particular type of externality, that posed by ‘‘public goods’’: goods that are both nonexclusive and nonrival. We show that markets will often underallocate resources to such goods, so other ways of ﬁnancing (such as compulsory taxation) should be considered. Chapter 19 concludes with an examination of how voting may affect this process. 639 This page intentionally left blank C H A P T E R EIGHTEEN Asymmetric Information Markets may not be fully efﬁcient when one side has information that the other side does not (asymmetric information). Contracts with more complex terms than simple per-unit prices may be used to help solve problems raised by such asymmetric information. The two important classes of asymmetric information problems studied in this chapter include moral hazard problems, in which one party’s actions during the term of the contract are unobservable to the other, and adverse selection problems, in which a party obtains asymmetric information about market conditions before signing the contract. Carefully designed contracts may reduce such problems by providing incentives to reveal one’s information and take appropriate actions. But these contracts seldom eliminate the inefﬁciencies entirely. Surprisingly, unbridled competition may worsen private information problems, although a carefully designed auction can harness competitive forces to the auctioneer’s advantage. Complex Contracts as a Response to Asymmetric Information So far, the transactions we have studied have involved simple contracts. We assumed that ﬁrms bought inputs from suppliers at constant per-unit prices and likewise sold output to consumers at constant per-unit prices. Many real-world transactions involve much more complicated contracts. Rather than an hourly wage, a corporate executive’s compensation usually involves complex features such as the granting of stock, stock options, and bonuses. Insurance policies may cap the insurer’s liability and may require the customer to bear costs in the form of deductibles and copayments. In this chapter, we will show that such complex contracts may arise as a way for transacting parties to deal with the problem of asymmetric information. Asymmetric information Transactions can involve a considerable amount of uncertainty. The value of a snow shovel will depend on how much snow falls during the winter season. The value of a hybrid car will depend on how much gasoline prices increase in the future. Uncertainty need not lead to inefﬁciency when both sides of a transaction have the same limited knowledge concerning the future, but it can lead to inefﬁciency when one side has better information. The side with better information is said to have private information or, equivalently, asymmetric information. There are several sources of asymmetric information. Parties will often have ‘‘inside information’’ concerning themselves that the other side does not have. Consider the case 641 642 Part 8: Market Failure the burden of of health insurance. A customer seeking insurance will often have private information about his or her own health status and family medical history that the insurance company does not. Consumers in good health may not bother to purchase health insurance at the prevailing rates. A consumer in poor health would have higher demand for insurance, large anticipated medical expenses to the insurer. wishing to shift A medical examination may help the insurer learn about a customer’s health status, but examinations are costly and may not reveal all of the customer’s private health information. The customer will be reluctant to report family medical history and genetic disease honestly if the insurer might use this information to deny coverage or increase premiums. Other sources of asymmetric information arise when what is being bought is an agent’s service. The buyer may not always be able to monitor how hard and well the agent is working. The agent may have better information about the requirements of the project because of his or her expertise, which is the reason the agent was hired in the ﬁrst place. For example, a repairer called to ﬁx a kitchen appliance will know more about the true severity of the appliance’s mechanical problems than does the homeowner. Asymmetric information can lead to inefﬁciencies. Insurance companies may offer less insurance and charge higher premiums than if they could observe the health of potential clients and could require customers to obey strict health regimens. The whole market may unravel as consumers who expect their health expenditures to be lower than the average insured consumer’s withdraw from the market in successive stages, leaving only the few worst health risks as consumers. With appliance repair, the repairer may pad his or her bill by replacing parts that still function and may take longer than needed—a waste of resources. The value of contracts Contractual provisions can be added in order to circumvent some of these inefﬁciencies. An insurance company can offer lower health insurance premiums to customers who submit to medical exams or who are willing to bear the cost of some fraction of their own medical services. Lower-risk consumers may be more willing than high-risk consumers to submit to medical exams and to bear a fraction of their medical expenses. A homeowner may buy a service contract that stipulates a ﬁxed fee for keeping the appliance running rather than a payment for each service call and part needed in the event the appliance breaks down. Although contracts may help reduce the inefﬁciencies associated with asymmetric information, rarely do they eliminate the inefﬁciencies altogether. In the health insurance example, having some consumers undertake a medical exam requires the expenditure of real resources. Requiring low-risk consumers to bear some of their own medical expenditures means that they are not fully insured, which is a social loss to the extent that a risk-neutral insurance company would be a more efﬁcient risk bearer than a risk-averse consumer. A ﬁxed-fee contract to maintain an appliance may lead the repairer to supply too little effort, overlooking potential problems in the hope that nothing breaks until after the service contract expires (and so then the problems become the homeowner’s). Principal-Agent Model Models of asymmetric information can quickly become quite complicated, and thus, before considering a full-blown market model with many suppliers and demanders, we will devote much of our analysis to a simpler model—called a principal-agent model—in which there is only one party on each side of the market. The party who proposes the Chapter 18: Asymmetric Information 643 contract is called the principal. The party who decides whether or not to accept the contract and then performs under the terms of the contract (if accepted) is called the agent. The agent is typically the party with the asymmetric information. We will use ‘‘she’’ for the principal and ‘‘he’’ for the agent to facilitate the exposition. Two leading models Two models of asymmetric information are studied most often. In a ﬁrst model, the agent’s actions taken during the term of the contract affect the principal, but the principal does not observe these actions directly. The principal may observe outcomes that are correlated with the agent’s actions but not the actions themselves. This ﬁrst model is called a hidden-action model. For historical reasons stemming from the insurance context, the hidden-action model is also called a moral hazard model. In a second model, the agent has private information about the state of the world before signing the contract with the principal. The agent’s private information is called his type, consistent with our terminology from games of private information studied in Chapter 8. The second model is thus called a hidden-type model. For historical reasons stemming from its application in the insurance context, which we discuss later, the hidden-type model is also called an adverse selection model. As indicated by Table 18.1, the hidden-type and hidden-action models cover a wide variety of applications. Note that the same party might be a principal in one setting and an agent in another. For example, a company’s CEO is the principal in dealings with the company’s employees but is the agent of the ﬁrm’s shareholders. We will study several of the applications from Table 18.1 in detail throughout the remainder of this chapter. First, second, and third best In a full-information environment, the principal could propose a contract to the agent that maximizes their joint surplus and captures all of this surplus for herself, leaving the agent with just enough surplus to make him indifferent between signing the contract or not. This outcome is called the ﬁrst best, and the contract implementing this outcome is called the ﬁrst-best contract. The ﬁrst best is a theoretical benchmark that is unlikely to be achieved in practice because the principal is rarely fully informed. The outcome that maximizes the principal’s surplus subject to the constraint that the principal is less well informed than the agent is called the second best, and the contract that implements this TABLE 18.1 APPLICATIONS OF THE PRINCIPAL-AGENT MODEL Principal Shareholders Manager Agent Manager Employee Homeowner Appliance repairer Agent’s Private Information Hidden Type Managerial skill Job skill Skill, severity of appliance malfunction Hidden Action Effort, executive decisions Effort Effort, unnecessary repairs Student Monopoly Tutor Customer Subjec
t knowledge Preparation, patience Value for good Care to avoid breakage Health insurer Insurance purchaser Preexisting condition Parent Child Moral ﬁber Risky activity Delinquency 644 Part 8: Market Failure outcome is called the second-best contract. Adding further constraints to the principal’s problem besides the informational constraint—for example, restricting contracts to some simple form such as constant per-unit prices—leads to the third best, the fourth best, and so on, depending on how many constraints are added. Since this chapter is in the part of the book that examines market failures, we will be interested in determining how important a market failure is asymmetric information. Comparing the ﬁrst to the second best will allow us to quantify the reduction in total welfare due to asymmetric information. Also illuminating is a comparison of the second and third best. This comparison will indicate how surpluses are affected when moving from simple contracts in the third best to potentially quite sophisticated contracts in the second best. Of course, the principal’s surplus cannot decrease when she has access to a wider range of contracts with which to maximize her surplus. However, total welfare—the sum of the principal’s and agent’s surplus in a principal-agent model—may decrease. Figure 18.1 suggests why. In the example in panel (a) of the ﬁgure, the complex contract increases the total welfare ‘‘pie’’ that is divided between the principal and the agent. The principal likes the complex contract because it allows her to obtain a roughly constant share of a bigger pie. In panel (b), the principal likes the complex contract even though the total welfare pie is smaller with it FIGURE 18.1 The Contracting ‘‘Pie’’ The total welfare is the area of the circle (‘‘pie’’); the principal’s surplus is the area of the shaded region. In panel (a), the complex contract increases total welfare and the principal’s surplus along with it because she obtains a constant share. In panel (b), the principal offers the complex contract—even though this reduces total welfare—because the complex contract allows her to appropriate a larger share. Simple, third-best contract Complex, second-best contract (a) Complex contract increases parties’ joint surplus Simple, third-best contract Complex, second-best contract (b) Complex contract increases principal’s share of surplus Chapter 18: Asymmetric Information 645 than with the simple contract. The complex contract allows her to appropriate a larger slice at the expense of reducing the pie’s total size. The different cases in panels (a) and (b) will come up in the applications analyzed in subsequent sections. Hidden Actions The ﬁrst of the two important models of asymmetric information is the hidden-action model, also sometimes called the moral hazard model in insurance and other contexts. The principal would like the agent to take an action that maximizes their joint surplus (and given that the principal makes the contract offer, she would like to appropriate most of the surplus for herself). In the application to the owner-manager relationship that we will study, the owner would like the manager whom she hires to show up during business hours and work diligently. In the application to the accident insurance, the insurance company would like the insured individual to avoid accidents. The agent’s actions may be unobservable to the principal. Observing the action may require the principal to monitor the agent at all times, and such monitoring may be prohibitively expensive. If the agent’s action is unobservable, then he will prefer to shirk, choosing an action to suit himself rather than the principal. In the owner-manager application, shirking might mean showing up late for work and slacking off while on the job; in the insurance example, shirking might mean taking more risk than the insurance company would like. Although contracts cannot prevent shirking directly by tying the agent’s compensation to his action—because his action is unobservable—contracts can mitigate shirking by tying compensation to observable outcomes. In the owner-manager application, the relevant observable outcome might be the ﬁrm’s proﬁt. The owner may be able to induce the manager to work hard by tying the manager’s pay to the ﬁrm’s proﬁt, which depends on the manager’s effort. The insurance company may be able to induce the individual to take care by having him bear some of the cost of any accident. Often, the principal is more concerned with the observable outcome than with the agent’s unobservable action anyway, so it seems the principal should do just as well by conditioning the contract on outcomes as on actions. The problem is that the outcome may depend in part on random factors outside of the agent’s control. In the owner-manager application, the ﬁrm’s proﬁt may depend on consumer demand, which may depend on unpredictable economic conditions. In the insurance application, whether an accident occurs depends in part on the care taken by the individual but also on a host of other factors, including other people’s actions and acts of nature. Tying the agent’s compensation to partially random outcomes exposes him to risk. If the agent is risk averse, then this exposure causes disutility and requires the payment of a risk premium before he will accept the contract (see Chapter 7). In many applications, the principal is less risk averse and thus is a more efﬁcient risk bearer than the agent. In the owner-manager application, the owner might be one of many shareholders who each hold only a small share of the ﬁrm in a diversiﬁed portfolio. In the insurance application, the company may insure a large number of agents, whose accidents are uncorrelated, and thus face little aggregate risk. If there were no issue of incentives, then the agent’s compensation should be independent of risky outcomes, completely insuring him against risk and shifting the risk to the efﬁcient bearer: the principal. The second-best contract strikes the optimal balance between incentives and insurance, but it does not provide as strong incentives or as full insurance as the ﬁrst-best contract. In the following sections, we will study two speciﬁc applications of the hidden-action model. First, we will study employment contracts signed between a ﬁrm’s owners and a manager who runs the ﬁrm on behalf of the owners. Second, we will study contracts offered by an insurance company to insure an individual against accident risk. 646 Part 8: Market Failure Owner-Manager Relationship Modern corporations may be owned by millions of dispersed shareholders who each own a small percentage of the corporation’s stock. The shareholders—who may have little expertise in the line of business and who may own too little of the ﬁrm individually to devote much attention to it—delegate the operation of the ﬁrm to a managerial team consisting of the chief executive ofﬁcer (CEO) and other ofﬁcers. We will simplify the setting and suppose that the ﬁrm has one representative owner and one manager. The owner, who plays the role of the principal in the model, offers a contract to the manager, who plays the role of the agent. The manager decides whether to accept the employment contract and, if so, how much effort e 0 to exert. An increase in e increases the ﬁrm’s gross proﬁt (not including payments to the manager) but is personally costly to the manager.1 ! Assume the ﬁrm’s gross proﬁt pg takes the following simple form: (18:1) e pg ¼ e: þ Gross proﬁt is increasing in the manager’s effort e and also depends on a random variable e, which represents demand, cost, and other economic factors outside of the manager’s control. Assume that e is normally distributed with mean 0 and variance s2. The manager’s personal disutility (or cost) of undertaking effort c(e) is increasing [c 0(e) > 0] and convex [c 00(e) > 0]. Let s be the salary—which may depend on effort and/or gross proﬁt, depending on what the owner can observe—offered as part of the contract between the owner and manager. Because the owner represents individual shareholders who each own a small share of the ﬁrm as part of a diversiﬁed portfolio, we will assume that she is risk neutral. Letting net proﬁt pn equal gross proﬁt minus payments to the manager, pn ¼ the risk-neutral owner wants to maximize the expected value of her net proﬁt: pg $ s, (18:2) E pnÞ $ To introduce a trade-off between incentives and risk, we will assume the manager is risk averse; in particular, we assume the manager has a utility function with respect to salary whose constant absolute risk aversion parameter is A > 0. We can use the results from Example 7.3 to show that his expected utility is Þ ¼ þ $ ð ð (18:3 Var 18:4) We will examine the optimal salary contract that induces the manager to take appropriate effort e under different informational assumptions. We will study the ﬁrst-best contract, when the owner can observe e perfectly, and then the second-best contract, when there is asymmetric information about e. First best (full-information case) With full information, it is relatively easy to design an optimal salary contract. The owner can pay the manager a ﬁxed salary s’ if he exerts the ﬁrst-best level of effort e’ (which we will compute shortly) and nothing otherwise. The manager’s expected utility from the contract can be found by substituting the expected value [E(s’) s’] and variance 0] of the ﬁxed salary as well as the effort e’ into Equation 18.4. For the [Var(s’) ¼ ¼ 1Besides effort, (e) could represent distasteful decisions such as ﬁring unproductive workers. Chapter 18: Asymmetric Information 647 manager to accept the contract, this expected utility must exceed what he would obtain from his next-best job offer: U E ð Þ ¼ s’ $ e’Þ ! c ð 0, (18:5) where we have assumed for simplicity that he obtains 0 from his next-best job offer. In principal-agent models, a condition like Equation 18.5 is called a participation constraint,
ensuring the agent’s participation in the contract. The owner optimally pays the lowest salary satisfying Equation 18.5: s’ ¼ owner’s net proﬁt then is c(e’). The e’Þ c pnÞ ¼ ð ð which is maximized for e’ satisfying the ﬁrst-order condition s’Þ ¼ e’ $ e’ $ E E ð , e’Þ ¼ ð At an optimum, the marginal cost of effort, c0(e’), equals the marginal beneﬁt, 1. c0 1: (18:6) (18:7) Second best (hidden-action case) If the owner can observe the manager’s effort, then she can implement the ﬁrst best by simply ordering the manager to exert the ﬁrst-best effort level. If she cannot observe effort, the contract cannot be conditioned on e. However, she can still induce the manager to exert some effort if the manager’s salary depends on the ﬁrm’s gross proﬁt. The manager is given performance pay: the more the ﬁrm earns, the more the manager is paid. Suppose the owner offers a salary to the manager that is linear in gross proﬁt: pgÞ ¼ bpg, þ a ð s (18:8) where a is the ﬁxed component of salary and b measures the slope, sometimes called the power, of the incentive scheme. If b 0, then the salary is constant and, as we saw, provides no effort incentives. As b increases toward 1, the incentive scheme provides increasingly powerful incentives. The ﬁxed component a can be thought of as the manager’s base salary and b as the incentive pay in the form of stocks, stock options, and performance bonuses. ¼ The owner-manager relationship can be viewed as a three-stage game. In the ﬁrst stage, the owner sets the salary, which amounts to choosing a and b. In the second stage, the manager decides whether or not to accept the contract. In the third stage, the manager decides how much effort to exert conditional on accepting the contract. We will solve for the subgame-perfect equilibrium of this game by using backward induction, starting with the manager’s choice of e in the last stage and taking as given that the manbpg and accepted it. Substituting from Equation 18.8 ager was offered salary scheme a into Equation 18.4, the manager’s expected utility from the linear salary is þ E a ð bpgÞ $ þ A 2 Var a ð þ bpgÞ $ c ð : e Þ (18:9) Reviewing a few facts about expectations and variances of a random variable will help us simplify Equation 18.9. First note that E a ð see Equation 2.179. Furthermore, bpgÞ ¼ þ E a ð þ be þ be Þ ¼ a be bE e ð þ þ Þ ¼ a þ be; (18:10) Var a ð þ bpgÞ ¼ a Var ð þ be þ be Þ ¼ b2 Var e Þ ¼ ð b2r2; (18:11) 648 Part 8: Market Failure see Equation 2.186. Therefore, Equation 18.9 reduces to manager’s expected utility be a þ $ ¼ Ab2r2 2 $ e c ð : Þ (18:12) The ﬁrst-order condition for the e maximizing the manager’s expected utility yields c0 e ð b: Þ ¼ Because c(e) is convex, the marginal cost of effort c 0(e) is increasing in e. Hence, as shown in Figure 18.2, the higher is the power b of the incentive scheme, the more effort e the manager exerts. The manager’s effort depends only on the slope, b, and not on the ﬁxed part, a, of his incentive scheme. (18:13) Now fold the game back to the manager’s second-stage choice of whether to accept the contract. The manager accepts the contract if his expected utility in Equation 18.12 is non-negative or, upon rearranging, if a e c ð Þ þ ! Ab2r2 2 $ be: (18:14) The ﬁxed part of the salary, a, must be high enough for the manager to accept the contract. Next, fold the game back to the owner’s ﬁrst-stage choice of the parameters a and b of the salary scheme. The owner’s objective is to maximize her expected surplus, which (upon substituting from Equation 18.10 into 18.3) is ¼ subject to two constraints. The ﬁrst constraint (Equation 18.14) is that the manager must accept the contract in the second stage. As mentioned in the previous section, this is called a $ owner’s surplus e 1 ð a, b Þ $ (18:15) FIGURE 18.2 Manager’s Effort Responds to Increased Incentives Because the manager’s marginal cost of effort, c 0(e), slopes upward, an increase in the power of the incentive scheme from b1 to b2 induces the manager to increase his effort from e1 to e2. c′(e) b2 b1 e e1 e2 Chapter 18: Asymmetric Information 649 participation constraint. Although Equation 18.14 is written as an inequality, it is clear that the owner will keep lowering a until the condition holds with equality, since a does not affect the manager’s effort and since the owner does not want to pay the manager any more than necessary to induce him to accept the contract. The second constraint (Equation 18.13) is that the manager will choose e to suit himself rather than the owner, who cannot observe e. This is called the incentive compatibility constraint. Substituting the constraints into Equation 18.15 allows us to express the owner’s surplus as a function only of the manager’s effort: e e c ð $ Þ $ Ar2 c 0 ½ 2 e ð Þ 2 ) . The second-best effort e’’ satisﬁes the ﬁrst-order condition c0 e’’Þ ¼ ð 1 1 Ar2c 00ð e’’Þ : þ (18:16) (18:17) The right-hand side of Equation 18.17 is also equal to the power b’’ of the incentive scheme in the second best, since c 0(e’’) b’’ by Equation 18.13. 1. The second-best effort is less than 1 and thus is less than the ﬁrst-best effort e’ ¼ The presence of asymmetric information leads to lower equilibrium effort. If the owner cannot specify e in a contract, then she can induce effort only by tying the manager’s pay to ﬁrm proﬁt; however, doing so introduces variation into his pay for which the riskaverse manager must be paid a risk premium. This risk premium (the third term in Equation 18.16) adds to the owner’s cost of inducing effort. ¼ If effort incentives were not an issue, then the risk-neutral owner would be better-off bearing all risk herself and insuring the risk-averse manager against any ﬂuctuations in proﬁt by offering a constant salary, as we saw in the ﬁrst-best problem. Yet if effort is unobservable then a constant salary will not provide any incentive to exert effort. The second-best contract trades off the owner’s desire to induce high effort (which would come from setting b close to 1) against her desire to insure the risk-averse manager against variations in his salary (which would come from setting b close to 0). Hence the resulting value of b’’ falls somewhere between 0 and 1. In short, the fundamental trade-off in the owner-manager relationship is between incentives and insurance. The more risk averse is the manager (i.e., the higher is A), the more important is insurance relative to incentives. The owner insures the manager by reducing the dependence of his salary on ﬂuctuating proﬁt, reducing b’’ and therefore e’’. For the same reason, the more that proﬁt varies owing to factors outside of the manager’s control (i.e., the higher is s2), the lower is b’’ and e’’.2 EXAMPLE 18.1 Owner-Manager Relationship As a numerical example of some of these ideas, suppose the manager’s cost of effort has the simple form c(e) e2/2 and suppose s2 1. ¼ ¼ First best. The ﬁrst-best level of effort satisﬁes c 0(e’) that the manager exerts ﬁrst-best effort e’ ¼ 1. A ﬁrst-best contract speciﬁes 1 in return for a ﬁxed salary of 1/2, which leaves e’ ¼ ¼ 2A study has conﬁrmed that CEOs and other top executives receive more powerful incentives if they work for ﬁrms with less volatile stock prices. See R. Aggarwal and A. Samwick, ‘‘The Other Side of the Trade-off: The Impact of Risk on Executive Compensation,’’ Journal of Political Economy 107 (1999): 65–105. 650 Part 8: Market Failure the manager indifferent between accepting the contract and pursuing his next-best available job (which we have assumed provides him with utility 0). The owner’s net proﬁt equals 1/2. Second best. The second-best contract depends on the degree of the manager’s risk aversion 1.3 Then, by Equation 18.17, the second-best level of measured by A. Suppose ﬁrst that A effort is e’’ ¼ 1/2 as well. To compute the ﬁxed part a’’ of the manager’s salary, recall that Equation 18.14 holds as an equality in the second best and substitute the variables computed so far, yielding a’’ ¼ 0. The manager receives no ﬁxed pay but does receive incentive pay equal to 50 cents for every dollar of gross proﬁt. Substituting the variables computed into Equation 18.15, we see that the owner’s expected net proﬁt is 1/4. 1/2, and b’’ ¼ ¼ ¼ 2, so that the manager is more risk averse. The second-best effort 1/3, and b’’ decreases to 1/3 as well. The ﬁxed part of the manager’s salary 1/18. The owner’s expected net proﬁt decreases to 1/6. Now suppose A decreases to e’’ ¼ increases to a’’ ¼ Empirical evidence. In an inﬂuential study of performance pay, Jensen and Murphy estimated that b 0.003 for top executives in a sample of large U.S. ﬁrms, which is orders of magnitude smaller than the values of b’’ we just computed.4 The fact that real-world incentive schemes are less sensitive to performance than theory would indicate is a puzzle for future research to unravel. ¼ QUERY: How would the analysis change if the owners did not perfectly observe gross proﬁt but instead depended on the manager for a self-report? Could this explain the puzzle that top executives’ incentives are unexpectedly low-powered? Comparison to standard model of the ﬁrm It is natural to ask how the results with hidden information about the manager’s action compare to the standard model of a perfectly competitive market with no asymmetric information. First, the presence of hidden information raises a possibility of shirking and inefﬁciency that is completely absent in the standard model. The manager does not exert as much effort as he would if effort were observable. Even if the owner does as well as she can in the presence of asymmetric information to provide incentives for effort, she must balance the beneﬁts of incentives against the cost of exposing the manager to too much risk. Second, although the manager can be regarded as an input like any other (capital, labor, materials, and so forth) in the standard model, he becomes a unique sort of input when his actions are hidden information. It is not
enough to pay a ﬁxed unit price for this input as a ﬁrm would the rental rate for capital or the market price for materials. How productive the manager is depends on how his compensation is structured. The same can be said for any sort of labor input: workers may shirk on the job unless monitored or given incentives not to shirk. Moral Hazard in Insurance Another important context in which hidden actions lead to inefﬁciencies is the market for insurance. Individuals can take a variety of actions that inﬂuence the probability that a risky event will occur. Car owners can install alarms to deter theft; consumers can eat healthier foods to prevent illness. In these activities, utility-maximizing individuals will 3To make the calculations easier, we have scaled A up from its more realistic values in Chapter 7 and have rescaled several other parameters as well. 4M. Jensen and K. Murphy, ‘‘Performance Pay and Top-Management Incentives,’’ Journal of Political Economy 98 (1990): 5–64. Chapter 18: Asymmetric Information 651 pursue risk reduction up to the point at which marginal gains from additional precautions are equal to the marginal cost of these precautions. In the presence of insurance coverage, however, this calculation may change. If a person is fully insured against losses, then he or she will have a reduced incentive to undertake costly precautions, which may increase the likelihood of a loss occurring. In the automobile insurance case, for example, a person who has a policy that covers theft may not bother to install a car alarm. This behavioral response to insurance coverage is termed moral hazard Moral hazard. The effect of insurance coverage on an individual’s precautions, which may change the likelihood or size of losses. The use of the term ‘‘moral’’ to describe this response is perhaps unfortunate. There is nothing particularly ‘‘immoral’’ about the behavior being described, since individuals are simply responding to the incentives they face. In some applications, this response might even be desirable. For example, people with medical insurance may be encouraged to seek early treatment because the insurance reduces their out-of-pocket cost of medical care. But, because insurance providers may ﬁnd it costly to measure and evaluate such responses, moral hazard may have important implications for the allocation of resources. To examine these, we need a model of utility-maximizing behavior by insured individuals. Mathematical model Suppose a risk-averse individual faces the possibility of incurring a loss (l) that will reduce his initial wealth (W0). The probability of loss is p. An individual can reduce p by spending more on preventive measures (e).5 Let U(W) be the individual’s utility given wealth W. An insurance company (here playing the role of principal) offers an insurance contract involving a payment x to the individual if a loss occurs. The premium for this coverage is p. If the individual takes the coverage, then his wealth in state 1 (no loss) and state 2 (loss) are W1 ¼ W2 ¼ W0 $ W0 $ e e $ $ p p and l $ þ x, and his expected utility is ð The risk-neutral insurance company’s objective is to maximize expected proﬁt: $ 1 ð U p Þ W1Þ þ pU : W2Þ ð expected insurance profit p $ ¼ px: (18:18) (18:19) (18:20) First-best insurance contract In the ﬁrst-best case, the insurance company can perfectly monitor the agent’s precautionary effort e. It sets e and the other terms of the insurance contract (x and p) to maximize its expected proﬁt subject to the participation constraint that the individual accepts the contract: 1 ð p U Þ W1Þ þ ð $ pU W2Þ ! ð !U, (18:21) 5For consistency, we use the same variable e as we did for managerial effort. In this context, since e is subtracted from the individual’s wealth, e should be thought of as either a direct expenditure or the monetary equivalent of the disutility of effort. 652 Part 8: Market Failure where !U is the highest utility the individual can attain in the absence of insurance. It is clear that the insurance company will increase the premium until the participation constraint holds with equality. Thus, the ﬁrst-best insurance contract is the solution to a maximization problem subject to an equality constraint, which we can use Lagrange methods to solve. The associated Lagrangian is + p px 1 k ½ð $ U p Þ W1Þ þ ð þ pU W2Þ $ ð $ ¼ !U : ) The ﬁrst-order conditions are @+ @p ¼ 0 ¼ 1 1 k ½ð $ U 0 p Þ W0 $ ð e $ $ p Þ þ pU 0 W0 $ ð e l x , Þ) þ $ @+ @x ¼ $ @+ @e ¼ $ p þ @p @e 0 0 ¼ ¼ kpU 0 W0 $ p @e ½ ! þ U ð W0 $ ð W0 $ pU 0 W0 $ ð p e W0 $ ) þ $ : (18:22) (18:23) (18:24) (18:25) These conditions may seem complicated, but they have simple implications. Equations 18.23 and 18.24 together imply " W0 $ ð which in turn implies x ance. Substituting for l from Equation 18.26 into Equation 18.25 and noting x l. This is the familiar result that the ﬁrst best involves full insurl, we have W0 $ ð p e W0 $ ð Þ þ , x (18:26) pU p @e l $ 1: ¼ ¼ (18:27) At an optimum, the marginal social beneﬁt of precaution (the reduction in the probability of a loss multiplied by the amount of the loss) equals the marginal social cost of precaution (which here is just 1). In sum, the ﬁrst-best insurance contract provides the individual with full insurance but requires him to choose the socially efﬁcient level of precaution. Second-best insurance contract To obtain the ﬁrst best, the insurance company would need to monitor the insured individual to ensure that the person was constantly taking the ﬁrst-best level of precaution, e’. In the case of insurance for automobile accidents, the company would have to make sure that the driver never exceeds a certain speed, always keeps alert, and never drives while talking on his cell phone, for example. Even if a black-box recorder could be installed to constantly track the car’s speed, it would still be impossible to monitor the driver’s alertness. Similarly, for health insurance, it would be impossible to watch everything the insured party eats to make sure he doesn’t eat anything unhealthy. Assume for simplicity that the insurance company cannot monitor precaution e at all, so that e cannot be speciﬁed by the contract directly. This second-best problem is similar to the ﬁrst-best except that a new constraint must to be added: an incentive compatibility constraint specifying that the agent is free to choose the level of precaution that suits him and maximizes his expected utility, 1 ð p U Þ W1Þ þ ð $ pU : W2Þ ð (18:28) Chapter 18: Asymmetric Information 653 Unlike the ﬁrst best, the second-best contract will typically not involve full insurance. W2. But then the Under full insurance, x ¼ insured party’s expected utility from Equation 18.28 is l and (as Equation 18.18 shows) W1 ¼ W1Þ ¼ ð which is maximized by choosing the lowest level of precaution possible, e W0 $ ð p Þ $ U U e , (18:29) 0. To induce the agent to take precaution, the company should provide him only partial insurance. Exposing the individual to some risk induces him to take at least some precaution. The company will seek to offer just the right level of partial insurance: not too much insurance (else the agent’s precaution drops too low) and not too little insurance (else the agent would not be willing to pay much in premiums). The principal faces the same trade-off in this insurance example as in the owner-manager relationship studied previously: incentives versus insurance. ¼ The solution for the optimal second-best contract is quite complicated, given the general functional forms for utility that we are using.6 Example 18.2 provides some further practice on the moral hazard problem with speciﬁc functional forms. EXAMPLE 18.2 Insurance and Precaution against Car Theft In Example 7.2 we examined the decision by a driver endowed with $100,000 of wealth to purchase insurance against the theft of a $20,000 car. Here we reexamine the market for theft insurance when he can also take the precaution of installing a car alarm that costs $1,750 and that reduces the probability of theft from 0.25 to 0.15. No insurance. In the absence of insurance, the individual can decide either not to install the alarm, in which case (as we saw from Example 7.2) his expected utility is 11.45714, or to install the alarm, in which case his expected utility is 0:85 ln 100,000 ð $ 1,750 Þ þ 0:15 ln 100,000 ð $ 1,750 $ 20,000 Þ ¼ 11:46113: (18:30) He prefers to install the device. First best. The ﬁrst-best contract maximizes the insurance company’s proﬁt given that it requires the individual to install an alarm and can costlessly verify whether the individual has complied. The ﬁrst-best contract provides full insurance, paying the full $20,000 if the car is stolen. The highest premium p that the company can charge leaves the individual indifferent between accepting the full-insurance contract and going without insurance: Solving for p yields ln 100,000 ð $ 1,750 p $ Þ ¼ 11:46113: 98,250 p $ ¼ e11:46113, (18:31) (18:32) implying that p 3,298. (Note that the e in Equation 18.32 is the number 2.7818 … , not the individual’s precaution.) The company’s proﬁt equals the premium minus the expected payout: 3,298 20,000) $298. (0.15 ¼ $ * ¼ Second best. If the company cannot monitor whether the individual has installed an alarm, then it has two choices. It can induce him to install the alarm by offering only partial insurance, or it can disregard the alarm and provide him with full insurance. 6For more analysis see S. Shavell, ‘‘On Moral Hazard and Insurance,’’ Quarterly Journal of Economics (November 1979): 541–62. 654 Part 8: Market Failure If the company offers full insurance, then the individual will certainly save the $1,750 by not installing the alarm. The highest premium that the company can charge him solves 100; 000 ln ð 11:46113, Þ ¼ $ p implying that p 5,048. The company’s proﬁt is then 5,048 (0.25 20,000) $48. ¼ On the other hand, the company can induce the individual to install the alarm if it reduces the p
ayment after theft from the full $20,000 down to $3,374 and lowers the premium to $602. (These second-best contractual terms were computed by the authors using numerical methods; we will forgo the complicated computations and just take these terms as given.) Let’s check that the individual would indeed want to install the alarm. His expected utility if he accepts the contract and installs the alarm is $ * ¼ (18:33) 0:85 ln 100,000 ð 0:15 ln 1,750 $ 100,000 ð 602 Þ $ 1,750 $ 3,374 the same as if he accepts the contract and does not install the alarm: 20,000 602 $ þ $ þ 11:46113, (18:34) Þ ¼ 0:75 ln 100,000 602 Þ $ ð 0:25 ln 100,000 $ ð (18:35) $ also the same as he obtains if he goes without insurance. So he weakly prefers to accept the contract and install the alarm. The insurance company’s proﬁt is 602 $96. Thus, partial insurance is more proﬁtable than full insurance when the company cannot observe precaution. 11:46113, 20,000 3,374) 3,374 (0.15 602 Þ ¼ * þ $ þ ¼ QUERY: What is the most that the insurance company would be willing to spend in order to monitor whether the individual has installed an alarm? Competitive insurance market So far in this chapter we have studied insurance using the same principal-agent framework as we used to study the owner-manager relationship. In particular, we have assumed that a monopoly insurance company (principal) makes a take-it-or-leave-it offer to the individual (agent). This is a different perspective than in Chapter 7, where we implicitly assumed that insurance is offered at fair rates—that is, at a premium that just covers the insurer’s expected payouts for losses. Fair insurance would arise in a perfectly competitive insurance market. With competitive insurers, the ﬁrst best maximizes the insurance customer’s expected utility given that the contract can specify his precaution level. The second best maximizes the customer’s expected utility under the constraint that his precaution level must be induced by having the contract offer only partial insurance. Our conclusions about the moral hazard problem remain essentially unchanged when moving from a monopoly insurer to perfect competition. The ﬁrst best still involves full insurance and a precaution level satisfying Equation 18.27. The second best still involves partial insurance and a moderate level of precaution. The main difference is in the distribution of surplus: insurance companies no longer earn positive proﬁts, since the extra surplus now accrues to the individual. EXAMPLE 18.3 Competitive Theft Insurance Return to Example 18.2, but now assume that car theft insurance is sold by perfectly competitive companies rather than by a monopolist. First best. If companies can costlessly verify whether or not the individual has installed an alarm, then the ﬁrst-best contract requires him to install the alarm and fully insures him for a Chapter 18: Asymmetric Information 655 premium of 3,000. This is a fair insurance premium because it equals the expected payout for a loss: 3,000 20,000. Firms earn zero proﬁt at this fair premium. The individual’s expected utility increases to 11.46426 from the 11.46113 of Example 18.2. 0.15 * ¼ Second best. Suppose now that insurance companies cannot observe whether the individual has installed an alarm. The second-best contract is similar to that computed in Example 18.2 except that the $96 earned by the monopoly insurer is effectively transferred back to the customer in the form of a reduced premium charged by competing insurers. The equilibrium premium is p 506 and the payment for loss is x 3,374. ¼ ¼ QUERY: Which case—monopoly or perfect competition—best describes the typical insurance market? Which types of insurance (car, health, life, disability) and which countries do you think have more competitive markets? Hidden Types Next we turn to the other leading variant of principal-agent model: the model of hidden types. Whereas in the hidden-action model the agent has private information about a choice he has made, in the hidden-type model he has private information about an innate characteristic he cannot choose. For example, a student’s type may be his innate intelligence as opposed to an action such as the effort he expends in studying for an exam. At ﬁrst glance, it is not clear why there should be a fundamental economic difference between hidden types and hidden actions that requires us to construct a whole new model (and devote a whole new section to it). The fundamental economic difference is this: In a hidden-type model, the agent has private information before signing a contract with the principal; in a hidden-action model, the agent obtains private information afterward. Having private information before signing the contract changes the game between the principal and the agent. In the hidden-action model, the principal shares symmetric information with the agent at the contracting stage and so can design a contract that extracts all of the agent’s surplus. In the hidden-type model, the agent’s private information at the time of contracting puts him in a better position. There is no way for the principal to extract all the surplus from all types of agents. A contract that extracts all the surplus from the ‘‘high’’ types (those who beneﬁt more from a given contract) would provide the ‘‘low’’ types with negative surplus, and they would refuse to sign it. The principal will try to extract as much surplus as possible from agents through clever contract design. She will even be willing to shrink the size of the contracting pie, sacriﬁcing some joint surplus in order to obtain a larger share for herself [as in panel (b) of Figure 18.1]. To extract as much surplus as possible from each type while ensuring that low types are not ‘‘scared off,’’ the principal will offer a contract in the form of a cleverly designed menu that includes options targeted to each agent type. The menu of options will be more proﬁtable for the principal than a contract with a single option, but the principal will still not be able to extract all the surplus from all agent types. Since the agent’s type is hidden, he cannot be forced to select the option targeted at his type but is free to select any of the options, and this ability will ensure that the high types always end up with positive surplus. To make these ideas more concrete, we will study two applications of the hidden-type model that are important in economics. First we will study the optimal nonlinear pricing problem, and then we will study private information in insurance. 656 Part 8: Market Failure Nonlinear Pricing In the ﬁrst application of the hidden-type model, we consider a monopolist (the principal) who sells to a consumer (the agent) with private information about his own valuation for the good. Rather than allowing the consumer to purchase any amount he wants at a constant price per unit, the monopolist offers the consumer a nonlinear price schedule. The nonlinear price schedule is a menu of different-sized bundles at different prices, from which the consumer makes his selection. In such schedules, the larger bundle generally sells for a higher total price but a lower per-unit price than a smaller bundle. Our approach builds on the analysis of second-degree price discrimination in Chapter 14. Here we analyze general nonlinear pricing schedules, the most general form of second-degree price discrimination. (In the earlier chapter, we limited our attention to a simpler form of second-degree price discrimination involving two-part tariffs.) The linear, two-part, and general nonlinear pricing schedules are plotted in Figure 18.3. The ﬁgure graphs the total tariff—the total cost to the consumer of buying q units—for the three different schedules. Basic and intermediate economics courses focus on the case of a constant per-unit price, which is called a linear pricing schedule. The linear pricing schedule is graphed as a straight line that intersects the origin (because nothing needs to be paid if no units are purchased). The two-part tariff is also a straight line, but its intercept—reﬂecting the ﬁxed fee—is above the origin. The darkest curve is a general nonlinear pricing schedule. Examples of nonlinear pricing schedules include a coffee shop selling three different sizes—say, a small (8-ounce) cup for $1.50, a medium (12-ounce) cup for $1.80, and a large (16-ounce) cup for $2.00. Although larger cups cost more in total, they cost less per FIGURE 18.3 Shapes of Various Pricing Schedules The graph shows the shape of three different pricing schedules. Thicker curves are more complicated pricing schedules and so represent more sophisticated forms of second-degree price discrimination. Total tariff Linear Two-part Nonlinear 0 q Chapter 18: Asymmetric Information 657 ounce (18.75 cents per ounce for the small, 15 for the medium, and 12.5 for the large). The consumer does not have the choice of buying as much coffee as he wants at a given per-ounce price; instead he must pick one of these three menu options, each specifying a particular bundled quantity. In other examples, the ‘‘q’’ that is bundled in a menu item is the quality of a single unit of the product rather than the quantity or number of units. For example, an airline ticket involves a single unit (i.e., a single ﬂight) whose quality varies depending on the class of the ticket, which ranges from ﬁrst class, with fancy drinks and meals and plush seats offering plenty of leg room, to coach class, with peanuts for meals and small seats having little leg room. Mathematical model To understand the economic principles involved in nonlinear pricing, consider a formal model in which a single consumer obtains surplus U uv q ð Þ $ T ¼ (18:36) from consuming a bundle of q units of a good for which he pays a total tariff of T. The ﬁrst term in the consumer’s utility function, yv(q), reﬂects the consumer’s beneﬁt from consumption. Assume v0(q) > 0 and v 00 < 0, implying that the consumer prefers q Þ more
of the good to less but that the marginal beneﬁt of more units is decreasing. The consumer’s type is given by y, which can be high (yH) with probability b and low (yL) with probability 1 b. The high type enjoys consuming the good more than the low type: 0 < yL < yH. The total tariff T paid by the consumer for the bundle is subtracted from his beneﬁt to compute his net surplus. $ ð For simplicity, we are assuming that there is a single consumer in the market. The analysis would likewise apply to markets with many consumers, a proportion b of which are high types and 1 b of which are low types. The only complication in extending the model to many consumers is that we would need to assume that consumers cannot divide bundles into smaller packages for resale among themselves. (Of course, such repackaging would be impossible for a single unit of the good involving a bundle of quality; and reselling may be impossible even for quantity bundles if the costs of reselling are prohibitive.) $ Suppose the monopolist has a constant marginal and average cost c of producing a unit of the good. Then the monopolist’s proﬁt from selling a bundle of q units for a total tariff of T is P T ¼ $ cq: (18:37) First-best nonlinear pricing In the ﬁrst-best case, the monopolist can observe the consumer’s type y before offering him a contract. The monopolist chooses the contract terms q and T to maximize her proﬁt subject to Equation 18.37 and subject to a participation constraint that the consumer accepts the contract. Setting the consumer’s utility to 0 if he rejects the contract, the participation constraint may be written as uv q ð Þ $ T ! 0: (18:38) The monopolist will choose the highest value of T satisfying the participation constraint: T yv (q). Substituting this value of T into the monopolist’s proﬁt function yields ¼ uv q ð Þ $ cq: (18:39) 658 Part 8: Market Failure Taking the ﬁrst-order condition and rearranging provides a condition for the ﬁrst-best quantity: c: uv 0 q Þ ¼ ð This equation is easily interpreted. In the ﬁrst best, the marginal social beneﬁt of increased quantity on the left-hand side [the consumer’s marginal private beneﬁt, yv 0(q)] equals the marginal social cost on the right-hand side [the monopolist’s marginal cost, c]. yH, yL. The tariffs are set so and that offered to the low type as to extract all the type’s surplus. The ﬁrst best for the monopolist is identical to what we termed ﬁrst-degree price discrimination in Chapter 14. The ﬁrst-best quantity offered to the high type ð satisﬁes the equation for y satisﬁes Equation 18.40 for y q’L Þ ð q’HÞ (18:40) ¼ ¼ It is instructive to derive the monopolist’s ﬁrst best in a different way, using methods similar to those used to solve the consumer’s utility maximization problem in Chapter 4. The contract (q, T ) can be thought of as a bundle of two different ‘‘goods’’ over which the monopolist has preferences. The monopolist regards T as a good (more money is better than less) and q as a bad (higher quantity requires higher production costs). Her indifference curve (actually an isoproﬁt curve) over (q, T ) combinations is a straight line with slope c. To see this, note that the slope of the monopolist’s indifference curve is her marginal rate of substitution: MRS @P=@q @P=@T ¼ $ ð$ c Þ 1 ¼ c. ¼ $ (18:41) The monopolist’s indifference curves are drawn as dashed lines in Figure 18.4. Because q is a bad for the monopolist, her indifference curves are higher as one moves toward the upper left. FIGURE 18.4 First-Best Nonlinear Pricing The consumer’s indifference curves over the bundle of contractual terms are drawn as solid lines (the thicker one for the high type and thinner for the low type); the monopolist’s isoproﬁts are drawn as dashed lines. Point A is the ﬁrst-best contract option offered to the high type, and point B is that offered to the low type Chapter 18: Asymmetric Information 659 H) and the low type’s (labeled U 0 Figure 18.4 also draws indifference curves for the two consumer types: the high type’s (labeled U 0 L). Because T is a bad for consumers, higher indifference curves for both types of consumer are reached as one moves toward the lower right. The U 0 H indifference curve for the high type is special because it intersects the origin, implying that the high type gets the same surplus as if he didn’t sign the contract at all. The ﬁrst-best contract offered by the monopolist to the high type is point A, at which the highest indifference curve for the monopolist still intersects the high type’s U 0 H indifference curve and thus still provides the high type with non-negative surplus. This is a point of tangency between the contracting parties’ indifference curves—that is, a point at which the indifference curves have the same slope. The monopolist’s indifference curves have slope c everywhere, as we saw in Equation 18.41. The slope of type y’s indifference curve is the marginal rate of substitution: MRS @U=@q @U=@T ¼ $ ¼ uv 0 q ð Þ 1 ¼ $ uv 0 q ð : Þ (18:42) Equating the slopes gives the same condition for the ﬁrst best as we found in Equation 18.40 (marginal social beneﬁt equals marginal social cost of an additional unit). The same arguments imply that point B is the ﬁrst-best contract offered to the low type, and we can again verify that Equation 18.40 is satisﬁed there. To summarize, the ﬁrst-best contract offered to each type speciﬁes a quantity (q’H or q’L , respectively) that maximizes social surplus given the type of consumer and a tariff (T’H or T’L , respectively) that allows the monopolist to extract all of the type’s surplus. Second-best nonlinear pricing Now suppose that the monopolist does not observe the consumer’s type when offering him a contract but knows only the distribution (y yL with probability 1 b). As Figure 18.5 shows, the ﬁrst-best contract would no longer ‘‘work’’ because the high type obtains more utility (moving from the indifference curve labeled U 0 H) by choosing the bundle targeted to the low type (B) rather than the bundle targeted to him (A). In other words, choosing A is no longer incentive compatible for the high type. To keep the high type from choosing B, the monopolist must reduce the high type’s tariff, offering C instead of A. $ H to the one labeled U 2 yH with probability b and y ¼ ¼ The substantial reduction in the high type’s tariff (indicated by the downwardpointing arrow) puts a big dent in the monopolist’s expected proﬁt. The monopolist can do better than offering the menu of contracts (B, C): she can distort the low type’s bundle in order to make it less attractive to the high type. Then the high type’s tariff need not be reduced as much to keep him from choosing the wrong bundle. Figure 18.6 shows how this new contract would work. The monopolist reduces the quantity in the low type’s bundle (while reducing the tariff so that the low type stays on his U 0 L indifference curve and thus continues to accept the contract), offering bundle D rather than B. The high type obtains less utility from D than B, as D reaches only his U 1 H indifference curve and is short of his U 2 H indifference curve. To keep the high type from choosing D, the monopolist need only lower the high type’s tariff by the amount given by the vertical distance between A and E rather than all the way down to C. Relative to (B, C), the second-best menu of contracts (D, E) trades off a distortion in the low type’s quantity (moving from the ﬁrst-best quantity in B to the lower quantity in D and destroying some social surplus in the process) against an increase in the tariff that can be extracted from the high type in moving from C to E. An attentive student might wonder why the monopolist would want to make this trade-off. After all, the monopolist must reduce the low type’s tariff in moving from B to D or else the low 660 Part 8: Market Failure FIGURE 18.5 First Best Not Incentive Compatible The ﬁrst-best contract, involving points A and B, is not incentive compatible if the consumer has private information about his type. The high type can reach a higher indifference curve by choosing the bundle (B) that is targeted at the low type. To keep him from choosing B, the monopolist must reduce the high type’s tariff by replacing bundle A with C. T 0 0 U H Reduction in tariff type would refuse to accept the contract. How can we be sure that this reduction in the low type’s tariff doesn’t more than offset any increase in the high type’s tariff? The reason is that a reduction in quantity harms the high type more than it does the low type. As Equation 18.42 shows, the consumer’s marginal rate of substitution between contractual terms (quantity and tariff) depends on his type y and is higher for the high type. Since the high type values quantity more than does the low type, the high type would pay more to avoid the decrease in quantity in moving from B to D than would the low type. Further insight can be gained from an algebraic characterization of the second best. The second-best contract is a menu that targets bundle (qH, TH) at the high type and (qL, TL) at the low type. The contract maximizes the monopolist’s expected proﬁt, subject to four constraints: b ð T H $ cqHÞ þ ð 1 $ b Þð T L $ , cqLÞ uLv qLÞ $ ð uHv qHÞ $ ð T L ! qLÞ $ ð T H ! qHÞ $ ð 0, T L ! T H ! 0, uLv qHÞ $ ð uHv qLÞ $ ð T H, T L: uLv uHv (18:43) (18:44) (18:45) (18:46) (18:47) Chapter 18: Asymmetric Information 661 FIGURE 18.6 Second-Best Nonlinear Pricing The second-best contract is indicated by the circled points D and E. Relative to the incentive-compatible contract found in Figure 18.5 (points B and C), the second-best contract distorts the low type’s quantity (indicated by the move from B to D) in order to make the low type’s bundle less attractive to the high type. This allows the principal to charge tariff to the high type (indicated by the move from C to E). *qL qL ** qH q The ﬁrst two are participation constraints for the low and high type of consumer, ensuring that th
ey accept the contract rather than forgoing the monopolist’s good. The last two are incentive compatibility constraints, ensuring that each type chooses the bundle targeted to him rather than the other type’s bundle. As suggested by the graphical analysis in Figure 18.6, only two of these constraints play a role in the solution. The most important constraint was to keep the high type from choosing the low type’s bundle; this is Equation 18.47 (incentive compatibility constraint for the high type). The other relevant constraint was to keep the low type on his U 0 L indifference curve to prevent him from rejecting the contract; this is Equation 18.44 (participation constraint for the low type). Hence, Equations 18.44 and 18.47 hold with equality in the second best. The other two constraints can be ignored, as can be seen in Figure 18.6. The high than if he type’s second-best bundle E puts him on a higher indifference curve , so the high type’s participation constraint (Equation 18.45) can rejects the contract be safely ignored. The low type would be on a lower indifference curve if he chose the high type’s bundle (E) rather than his own (D), so the low type’s incentive compatibility constraint (Equation 18.46) can also be safely ignored. U 0 HÞ ð U 1 HÞ ð Treating Equations 18.44 and 18.47 as equalities and using them to solve for TL and TH yields T L ¼ uLv qLÞ ð (18:48) 662 Part 8: Market Failure and T H ¼ ¼ qLÞ) þ ð qLÞ) þ ð By substituting these expressions for TL and TH into the monopolist’s objective function (Equation 18.39), we convert a complicated maximization problem with four inequality constraints into the simpler unconstrained problem of choosing qL and qH to maximize qHÞ $ ð qHÞ $ ð uH½ uH½ T L uLv : qLÞ ð (18:49) v v v v v v b f uH½ qHÞ $ ð cqHgþ ð The low type’s quantity satisﬁes the ﬁrst-order condition with respect to qL, which (upon considerable rearranging) yields qLÞ) þ ð qLÞ $ ð qLÞ $ ð cqL) (18:50) uLv uLv $ Þ½ b 1 : uLv 0 q’’L Þ ¼ ð c þ uH $ b ð 1 uLÞ v 0 b $ q’’L Þ ð : (18:51) The last term is clearly positive, and thus the equation implies that uLv 0 > c, whereas uLv 0 c in the ﬁrst best. Since v(q) is concave, we see that the second-best quantity is lower than the ﬁrst best, verifying the insight from our graphical analysis that the low type’s quantity is distorted downward in the second best to extract surplus from the high type. q’L Þ ¼ ð q’’L Þ ð The high type’s quantity satisﬁes the ﬁrst-order condition from the maximization of Equation 18.43 with respect to qH; upon rearranging, this yields uHv 0 q’’H Þ ¼ ð This condition is identical to the ﬁrst best, implying that there is no distortion of the high type’s quantity in the second best. There is no reason to distort the high type’s quantity because there is no higher type from whom to extract surplus. The result that the highest type is offered an efﬁcient contract is often referred to as ‘‘no distortion at the top.’’ (18:52) c: Returning to the low type’s quantity, how much the monopolist distorts this quantity downward depends on the probabilities of the two consumer types or—equivalently, in a model with many consumers—on the relative proportions of the two types. If there are many low types (b is low), then the monopolist would not be willing to distort the low type’s quantity very much, because the loss from this distortion would be substantial and there would be few high types from whom additional surplus could be extracted. The more high types (the higher is b), the more the monopolist is willing to distort the low type’s quantity downward. Indeed, if there are enough high types, the monopolist may decide not to serve the low types at all and just offer one bundle that would be purchased by the high types. This would allow the monopolist to squeeze all the surplus from the high types because they would have no other option. EXAMPLE 18.4 Monopoly Coffee Shop 15). Assume v The college has a single coffee shop whose marginal cost is 5 cents per ounce of coffee. The representative customer is equally likely to be a coffee hound (high type with yH ¼ 20) or a regular Joe (low type with yL ¼ First best. Substituting the functional form v quantities [yv 0(q) The tariff extracts all of each type’s surplus [T T’H ¼ ﬃﬃﬃ 2 qp into the condition for ﬁrst-best Þ ¼ (y/c)2. Therefore, q’L ¼ 16. ¼ yv (q)], here implying that T’L ¼ 90 and q ð c] and rearranging, we have q 160. The shop’s expected proﬁt is 9 and q’H ¼ qp . q ð Þ ¼ ¼ ¼ ﬃﬃﬃ 2 Chapter 18: Asymmetric Information 663 1 2 ð T’H $ cq’HÞ þ 1 2 ð T’L $ cq’L Þ ¼ 62:5 (18:53) cents per customer. The ﬁrst best can be implemented by having the owner sell a 9-ounce cup for 90 cents to the low type and a 16-ounce cup for $1.60 to the high type. (Somehow the barista can discern the customer’s type just by looking at him as he walks in the door.) Incentive compatibility when types are hidden. The ﬁrst best is not incentive compatible if the barista cannot observe the customer’s type. The high type obtains no surplus from the 16ounce cup sold at $1.60. If he instead paid 90 cents for the 9-ounce cup, he would obtain a surplus of yHv (9) 30 cents. Keeping the same cup sizes as in the ﬁrst best, the price for the large cup would have to be reduced by 30 cents (to $1.30) in order to keep the high type from buying the small cup. The shop’s expected proﬁt from this incentive compatible menu is 90 ¼ $ 1 2 ð 130 5 + $ 16 Þ þ 1 2 ð 90 9 5 + $ Þ ¼ 47:5: (18:54) Second best. The shop can do even better by reducing the size of the small cup to make it less attractive to high demanders. The size of the small cup in the second best satisﬁes Equation 18.51, which, for the functional forms in this example, implies that or, rearranging, uLq$ 1=2 L ¼ c uH $ þ ð 1=2 q$ L uLÞ q’’L ¼ 2uL $ c $ 2 uH ¼ $ % 2 + 15 $ 5 2 20 ¼ % 4: The highest price that can be charged without losing the low-type customers is (18:55) (18:56) uLv T’’L ¼ q’’L Þ ¼ ð The large cup is the same size as in the ﬁrst best: 16 ounces. It can be sold for no more than $1.40 or else the coffee hound would buy the 4-ounce cup instead. Although the total tariff for the large cup is higher at $1.40 than for the small cup at 60 cents, the unit price is lower (8.75 cents versus 15 cents per ounce). Hence the large cup sells at a quantity discount. Þ ¼ ﬃﬃﬃ (18:57) 2 Þð 60: 15 ð 4p The shop’s expected proﬁt is 1 2 ð 140 5 + $ 16 Þ þ 1 2 ð 60 5 4 + $ Þ ¼ 50 (18:58) cents per consumer. Reducing the size of the small cup from 9 to 4 ounces allows the shop to recapture some of the proﬁt lost when the customer’s type cannot be observed. QUERY: In the ﬁrst-best menu, the price per ounce is the same (10 cents) for both the low and high type’s cup. Can you explain why it is still appropriate to consider this a nonlinear pricing scheme? Adverse Selection in Insurance For the second application of the hidden-type model, we will return to the insurance market in which an individual with state-independent preferences and initial income W0 faces the prospect of loss l. Assume the individual can be one of two types: a high-risk type with probability of loss pH or a low-risk type with probability pL, where pH > pL. We will ﬁrst assume the insurance company is a monopolist; later we will study the case 664 Part 8: Market Failure of competitive insurers. The presence of hidden risk types in an insurance market is said to lead to adverse selection. Insurance tends to attract more risky than safe consumers (the ‘‘selection’’ in adverse selection) because it is more valuable to risky types, yet risky types are more expensive to serve (the ‘‘adverse’’ in adverse selection). Adverse selection. The problem facing insurers that risky types are both more likely to accept an insurance policy and more expensive to serve. As we will see, if the insurance company is clever then it can mitigate the adverse selection problem by offering a menu of contracts. The policy targeted to the safe type offers only partial insurance so that it is less attractive to the high-risk type. First best In the ﬁrst best, the insurer can observe the individual’s risk type and offer a different policy to each. Our previous analysis of insurance makes it clear that the ﬁrst best involves full insurance for each type, so the insurance payment x in case of a loss equals the full amount of the loss l. Different premiums are charged to each type and are set to extract all of the surplus that each type obtains from the insurance. The solution is shown in Figure 18.7 (the construction of this ﬁgure is discussed further in Chapter 7). Without insurance, each type ﬁnds himself at point E. Point A (resp., B) is the ﬁrstbest policy offered to the high-risk (resp., low-risk) type. Points A and B lie on the certainty line because both are fully insured. Since the premiums extract each type’s surplus from insurance, both types are on their indifference curves through the no-insurance point E. The high type’s premium is higher, so A is further down the certainty line toward the origin than is B.7 Second best If the monopoly insurer cannot observe the agent’s type, then the ﬁrst-best contracts will not be incentive compatible: the high-risk type would claim to be low risk and take full insurance coverage at the lower premium. As in the nonlinear pricing problem, the second best will involve a menu of contracts. Other principles from the nonlinear pricing problem also carry over here. The high type continues to receive the ﬁrst-best quantity (here, full insurance)—there is no distortion at the top. The low type’s quantity is distorted downward from the ﬁrst best, so he receives only partial insurance. Again we see that, with hidden types, the principal is willing to sacriﬁce some social surplus in order to extract some of the surplus the agent would otherwise derive from his private information. 7Mathematically, A appears further down the certainty line than B in Figure 18.7 because the high type’s indifference curve thr
ough E is ﬂatter than the low type’s. To see this, note that expected utility equals (1 pU(W2) and so the MRS is given by p)U(W1) dW1 dW2 ¼ 1 ð $ U 0ð p $ Þ W2Þ pU 0ð $ W1Þ : þ At a given (W1, W2) combination on the graph, the marginal rates of substitution differ only because the underlying probabilities of loss differ. Since 1 pH $ pH 1 < pL $ pL , it follows that the high-risk type’s indifference curve will be ﬂatter. This proof follows the analysis presented in M. Rothschild and J. Stiglitz, ‘‘Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information,’’ Quarterly Journal of Economics (November 1976): 629–50. Chapter 18: Asymmetric Information 665 FIGURE 18.7 First Best for a Monopoly Insurer In the ﬁrst best, the monopoly insurer offers policy A to the high-risk type and B to the low-risk type. Both types are fully insured. The premiums are sufﬁciently high to keep each type on his indifference curve through the no-insurance point (E). W2 0 U H Certainty line 0 U L B A E 0 W1 Figure 18.8 depicts the second best. If the insurer tried to offer a menu containing the ﬁrst-best contracts A and B, then the high-risk type would choose B rather than A. To maintain incentive compatibility, the insurer distorts the low type’s policy from B along its indifference curve U 0 L down to D. The low type is only partially insured, and this allows the insurer to extract more surplus from the high type. The high type continues to be fully insured, but the increase in his premium shifts his policy down the certainty line to C. EXAMPLE 18.5 Insuring the Little Red Corvette The analysis of automobile insurance in Example 18.2 (which is based on Example 7.2) can be recast as an adverse selection problem. Suppose that the probability of theft depends not on the act of installing an antitheft device but rather on the color of the car. Because thieves prefer red to gray cars, the probability of theft is higher for red cars (pH ¼ 0.15). First best. The monopoly insurer can observe the car color and offer different policies for different colors. Both colors are fully insured for the $20,000 loss of the car. The premium is the maximum amount that each type would be willing to pay in lieu of going without insurance; as computed in Example 7.2, is $5,426 for the high type (red cars). Similar calculations show that a gray-car owner’s expected utility if he is not insured is 11.4795, and the maximum premium he would be willing to pay for full insurance is $3,287. Although the insurer pays more claims for red cars, the higher associated premium more than compensates, and thus the expected proﬁt from a policy sold for a red car is 5,426 $426 versus 3,287 0.25) than for gray cars (pL ¼ $287 for a gray car. 0.25 Æ 20,000 this amount 0.15 Æ 20,000 $ ¼ $ ¼ 666 Part 8: Market Failure FIGURE 18.8 Second Best for a Monopoly Insurer Second-best insurance policies are represented by the circled points: C for the high-risk type and D for the low-risk type. W2 Certainty line W1 Second best. Suppose the insurer does not observe the color of the customer’s car and knows only that 10 percent of all cars are red and the rest are gray. The second-best menu of insurance policies—consisting of a premium/insurance coverage bundle ( pH, xH) targeted for high-risk, red cars and ( pL, xL) for low-risk, gray cars—is indicated by the circled points in Figure 18.8. Red cars are fully insured: xH ¼ 20,000. To solve for the rest of the contractual parameters, observe that xH, pH, and pL can be found as the solution to the maximization of expected insurer proﬁt 0:1 pH $ ð 0:25 + 20,000 0:9 pL $ 0:15xLÞ ð Þ þ (18:59) subject to a participation constraint for the low type, 0:85 ln pLÞ þ and to an incentive compatibility constraint for the high type, 100,000 ð 100,000 ð pL $ 0:15 ln $ $ 20,000 xLÞ ! þ 11:4795, (18:60) þ $ ln 0:75 ln 20,000 pL $ pLÞ $ pHÞ ! 100,000 ð 100,000 ð $ 0:25 ln 100,000 ð Participation and incentive compatibility constraints for the other types can be ignored, just as in the nonlinear pricing problem. This maximization problem is too difﬁcult to solve by hand, but it can be solved numerically using popular spreadsheet programs or other mathematical software. The second-best values that result are x’’H ¼ 20,000, p’’H ¼ QUERY: Look at the spreadsheet associated with this example on the website for this textbook. Play around with different probabilities of the two car colors. What happens when red cars become sufﬁciently common? (Even if you cannot access the spreadsheet, you should be able to guess the answer.) 11,556, and p’’L ¼ 4,154, x’’L ¼ : xLÞ (18:61) 1,971. þ Chapter 18: Asymmetric Information 667 Competitive insurance market Assume now that insurance is provided not by a monopoly but rather by a perfectly competitive market, resulting in fair insurance. Figure 18.9 depicts the equilibrium in which insurers can observe each individual’s risk type. Lines EF and EG are drawn with slopes pL)/pL pH)/pH, respectively, and show the market opportunities for each person to trade and W1 for W2 by purchasing fair insurance.8 The low-risk type is sold policy F, and the high-risk type is sold policy G. Each type receives full insurance at a fair premium. (1 (1 $ $ $ $ However, the outcome in Figure 18.9 is unstable if insurers cannot observe risk types. The high type would claim to be low risk and take contract F. But then insurers that offered F would earn negative expected proﬁt: at F, insurers break even serving only the low-risk types, so adding individuals with a higher probability of loss would push the company below the break-even point. FIGURE 18.9 Competitive Insurance Equilibrium with Perfect Information With perfect information, the competitive insurance market results in full insurance at fair premiums for each type. The high type is offered policy G; the low type, policy F. W2 Certainty line F E G 0 W1 8To derive these slopes, called odds ratios, note that fair insurance requires the premium to satisfy p and W2 yields ¼ px. Substituting into W1 W1 ¼ W2 ¼ Hence a $1 increase in the insurance payment (x) reduces W1 by p and increases W2 by 1 W0 $ x l ¼ þ px W0 $ W0 $ W0 $ : l p. $ 668 Part 8: Market Failure The competitive equilibrium with unobservable types is shown in Figure 18.10. The equilibrium is similar to the second best for a monopoly insurer. A set of policies is offered that separates the types. The high-risk type is fully insured at point G, the same policy as he was offered in the ﬁrst best. The low-risk type is offered policy J, which features partial insurance. The low type would be willing to pay more for fuller insurance, preferring a policy such as K. Because K is below line EF, an insurer would earn positive proﬁt from selling such a policy to low-risk types only. The problem is that K would also attract high-risk types, leading to insurer losses. Hence insurance is rationed to the low-risk type. With hidden types, the competitive equilibrium must involve a set of separating contracts; it cannot involve a single policy that pools both types. This can be shown with the aid of Figure 18.11. To be accepted by both types and allow the insurer to at least break even, the pooling contract would have to be a point (such as M) within triangle EFG. But M cannot be a ﬁnal equilibrium because at M there exist further trading opportunities. To see this, note that—as indicated in the ﬁgure and discussed earlier in the chapter—the indifference curve for the high type (UH) is ﬂatter than that for the low type (UL). Consequently, there are insurance policies such as N that are unattractive to high-risk types, attractive to low-risk types, and proﬁtable to insurers (because such policies lie below EF ). FIGURE 18.10 Competitive Insurance Equilibrium with Hidden Types With hidden types, the high-risk type continues to be offered ﬁrst-best policy G but the low-risk type is rationed, receiving only partial insurance at J in order to keep the high-risk type from pooling. W2 Certainty line F K G J E UH UL 0 W1 FIGURE 18.11 Impossibility of a Competitive Pooling Equilibrium Chapter 18: Asymmetric Information 669 Pooling contract M cannot be an equilibrium because there exist insurance policies such as N that are proﬁtable to insurers and are attractive to low-risk types but not to high-risk types. W2 0 F M G N Certainty line E UL UH W1 Assuming that no barriers prevent insurers from offering new contracts, policies such as N will be offered and will ‘‘skim the cream’’ of low-risk individuals from any pooling equilibrium. Insurers that continue to offer M are left with the ‘‘adversely selected’’ individuals, whose risk is so high that insurers cannot expect to earn any proﬁt by serving them. EXAMPLE 18.6 Competitive Insurance for the Little Red Corvette Recall the automobile insurance analysis in Example 18.5, but now assume that insurance is provided by a competitive market rather than a monopolist. Under full information, the competitive equilibrium involves full insurance for both types at a fair premium of (0.25)(20,000) $5,000 for high-risk, red cars and (0.15)(20,000) $3,000 for low-risk, gray cars. ¼ If insurers cannot observe car colors, then in equilibrium the coverage for the two types will still be separated into two policies. The policy targeted for red cars is the same as under full information. The policy targeted for gray cars involves a fair premium ¼ and an insurance level that does not give red-car owners an incentive to deviate by pooling on the gray-car policy: pL ¼ 0:15xL (18:62) 0:75 ln 100,000 ð pLÞ þ $ 0:25 ln 100,000 ð pL $ $ 20,000 þ Equations 18.62 and 18.63 can be solved numerically, yielding pL ¼ QUERY: How much more would gray-car owners be willing to pay for full insurance? Would an insurer proﬁt from selling full insurance at this higher premium if it sold only to owners of gray cars? Why then do the companies ration insurance to gray cars by insuring them partially? ln xLÞ ¼ 9
5,000 ð 453 and xL ¼ : Þ 3,020. (18:63) 670 Part 8: Market Failure Market Signaling In all the models studied so far, the uninformed principal moved ﬁrst—making a contract offer to the agent, who had private information. If the information structure is reversed and the informed player moves ﬁrst, then the analysis becomes much more complicated, putting us in the world of signaling games studied in Chapter 8. When the signaler is a principal who is offering a contract to an agent, the signaling games become complicated because the strategy space of contractual terms is virtually limitless. Compare the simpler strategy space of Spence’s education signaling game in Chapter 8, where the worker chose one of just two actions: to obtain an education or not. We do not have space to delve too deeply into complex signaling games here nor to repeat Chapter 8’s discussion of simpler signaling games. We will be content to gain some insights from a few simple applications. Signaling in competitive insurance markets In a competitive insurance market with adverse selection (i.e., hidden risk types), we saw that the low-risk type receives only partial insurance in equilibrium. He would beneﬁt from report of his type, perhaps hiring an independent auditor to certify that type so the reporting would be credible. The low-risk type would be willing to pay the difference between his equilibrium and his ﬁrst-best surplus in order to issue such a credible signal. It is important that there be some trustworthy auditor or other way to verify the authenticity of such reports, because a high-risk individual would now have an even greater incentive to make false reports. The high-risk type may even be willing to pay a large bribe to the auditor for a false report. EXAMPLE 18.7 Certifying Car Color Return to the competitive market for automobile insurance from Example 18.6. Let R be the most that the owner of a gray car would be willing to pay to have his car color (and thus his type) certiﬁed and reported to the market. He would then be fully insured at a fair premium of $3,000, earning surplus ln(100,000 R). In the absence of such a certiﬁed report, his expected surplus is 3,000 $ $ 0:85 ln 100,000 ð 11:4803: ¼ $ Solving for R in the equation 453 Þ þ 0:15 ln 100,000 ð $ 453 $ 20,000 3,020 Þ þ (18:64) ln(100,000 453 R 11:4803 (18:65) Þ ¼ 207. Thus the low-risk type would be willing to pay up to $207 to have a credible $ $ yields R report of his type issued to the market. ¼ The owner of the red car would pay a bribe as high as $2,000—the difference between his fair premium with full information ($5,000) and the fair premium charged to an individual known to be of low risk ($3,000). Therefore, the authenticity of the report is a matter of great importance. QUERY: How would the equilibrium change if reports are not entirely credible (i.e., if there is some chance the high-risk individual can successfully send a false report about his type)? What incentives would an auditor have to maintain his or her reputation for making honest reports? Chapter 18: Asymmetric Information 671 Market for lemons Markets for used goods raise an interesting possibility for signaling. Cars are a leading example: having driven the car over a long period of time, the seller has much better information about its reliability and performance than a buyer, who can take only a short test drive. Yet even the mere act of offering the car for sale can be taken as a signal of car quality by the market. The signal is not positive: the quality of the good must be below the threshold that would have induced the seller to keep it. As George Akerlof showed in the article for which he won the Nobel Prize in economics, the market may unravel in equilibrium so that only the lowest-quality goods, the ‘‘lemons,’’ are sold.9 To gain more insight into this result, consider the used-car market. Suppose there is a continuum of qualities from low-quality lemons to high-quality gems and that only the owner of a car knows its type. Because buyers cannot differentiate between lemons and gems, all used cars will sell for the same price, which is a function of the average car quality. A car’s owner will choose to keep it if the car is at the upper end of the quality spectrum (since a good car is worth more than the prevailing market price) but will sell the car if it is at the low end (since these are worth less than the market price). This reduction in average quality of cars offered for sale will reduce market price, leading would-be sellers of the highest-quality remaining cars to withdraw from the market. The market continues to unravel until only the worst-quality lemons are offered for sale. The lemons problem leads the market for used cars to be much less efﬁcient than it would be under the standard competitive model in which quality is known. (Indeed, in the standard model the issue of quality does not arise, because all goods are typically assumed to be of the same quality.) Whole segments of the market disappear—along with the gains from trade in these segments—because higher-quality items are no longer traded. In the extreme, the market can simply break down with nothing (or perhaps just a few of the worst items) being sold. The lemons problem can be mitigated by trustworthy used-car dealers, by development of car-buying expertise by the general public, by sellers providing proof that their cars are trouble-free, and by sellers offering money-back guarantees. But anyone who has ever shopped for a used car knows that the problem of potential lemons is a real one. EXAMPLE 18.8 Used-Car Market Suppose the quality q of used cars is uniformly distributed between 0 and 20,000. Sellers value their cars at q. Buyers (equal in number to the sellers) place a higher value on cars, q b, so there are gains to be made from trade in the used-car market. Under full information about quality, all used cars would be sold. But this does not occur when sellers have private information about quality and buyers know only the distribution. Let p be the market price. Sellers offer their cars for sale if and only if q p. The quality of a car offered for sale is thus uniformly distributed between 0 and p, implying that expected quality is , þ p ð 0 1 q p $ % dq ¼ p 2 (18:66) 9G. A. Akerlof, ‘‘The Market for ‘Lemons’: Quality Uncertainty and the Market Mechanism,’’ Quarterly Journal of Economics (August 1970): 488–500. 672 Part 8: Market Failure (see Chapter 2 for background on the uniform distribution). Hence, a buyer’s expected net surplus is 18:67) There may be multiple equilibria, but the one with the most sales involves the highest value of p for which Equation 18.67 is non-negative: b 2b. Only a fraction 2b/20,000 of the cars are sold. As b decreases, the market for used cars dries up. 0, implying that p’ ¼ p/2 ¼ $ QUERY: What would the equilibrium look like in the full-information case? Auctions The monopolist has difﬁculty extracting surplus from the agent in the nonlinear pricing problem because high-demand consumers could guarantee themselves a certain surplus by choosing the low demanders’ bundle. A seller can often do better if several consumers compete against each other for her scarce supplies in an auction. Competition among consumers in an auction can help the seller solve the hidden-type problem, because highvalue consumers are then pushed to bid high so they don’t lose the good to another bidder. In the setting of an auction, the principal’s ‘‘offer’’ is no longer a simple contract or menu of contracts as in the nonlinear pricing problem; instead, her offer is the format of the auction itself. Different formats might lead to substantially different outcomes and more or less revenue for the seller, so there is good reason for sellers to think carefully about how to design the auction. There is also good reason for buyers to think carefully about what bidding strategies to use. Auctions have received a great deal of attention in the economics literature ever since William Vickery’s seminal work, for which he won the Nobel Prize in economics.10 Auctions continue to grow in signiﬁcance as a market mechanism and are used for selling such goods as airwave spectrum, Treasury bills, foreclosed houses, and collectibles on the Internet auction site eBay. There are a host of different auction formats. Auctions can involve sealed bids or open outcries. Sealed-bid auctions can be ﬁrst price (the highest bidder wins the object and must pay the amount bid) or second price (the highest bidder still wins but need only pay the next-highest bid). Open-outcry auctions can be either ascending, as in the socalled English auction where buyers yell out successively higher bids until no one is willing to top the last, or descending, as in the so-called Dutch auction where the auctioneer starts with a high price and progressively lowers it until one of the participants stops the auction by accepting the price at that point. The seller can decide whether or not to set a ‘‘reserve clause,’’ which requires bids to be over a certain threshold else the object will not be sold. Even more exotic auction formats are possible. In an ‘‘all-pay’’ auction, for example, bidders pay their bids even if they lose. A powerful and somewhat surprising result due to Vickery is that, in simple settings (risk-neutral bidders who each know their valuation for the good perfectly, no collusion, etc.), many of the different auction formats listed here (and more besides) provide the monopolist with the same expected revenue in equilibrium. To see why this result is 10W. Vickery, ‘‘Counterspeculation, Auctions, and Competitive Sealed Tenders,’’ Journal of Finance (March 1961): 8–37. Chapter 18: Asymmetric Information 673 surprising, we will analyze two auction formats in turn—a ﬁrst-price and a second-price sealed-bid auction—supposing that a single object is to be sold. In the ﬁrst-price sealed-bid auction, all bidders simultaneously submit s
ecret bids. The auctioneer unseals the bids and awards the object to the highest bidder, who pays his or her bid. In equilibrium, it is a weakly dominated strategy to submit a bid b greater than or equal to the buyer’s valuation v Weakly dominated strategy. A strategy is weakly dominated if there is another strategy that does at least as well against all rivals’ strategies and strictly better against at least one. A buyer receives no surplus if he bids b v no matter what his rivals bid: if the buyer ¼ loses, he gets no surplus; if he wins, he must pay his entire surplus back to the seller and again gets no surplus. By bidding less than his valuation, there is a chance that others’ valuations (and consequent bids) are low enough that the bidder wins the object and derives a positive surplus. Bidding more than his valuation is even worse than just bidding his valuation. There is good reason to think that players avoid weakly dominated strategies, meaning here that bids will be below buyers’ valuations. In a second-price sealed-bid auction, the highest bidder pays the next-highest bid rather than his own. This auction format has a special property in equilibrium. All bidding strategies are weakly dominated by the strategy of bidding exactly one’s valuation. Vickery’s analysis of second-price auctions and of the property that they induce bidders to reveal their valuations has led them to be called Vickery auctions. We will prove that, in this kind of auction, bidding something other than one’s true valuation is weakly dominated by bidding one’s valuation. Let v be a buyer’s valuation and b his bid. If the two variables are not equal, then there are two cases to consider: ~ either b < v or b > v. Consider the ﬁrst case (b < v). Let b be the highest rival bid. If ~ b > v, then the buyer loses whether his bid is b or v, so there is a tie between the strat~ b < b, then the buyer wins the object whether his bid is b or v and his payment egies. If ~ is the same (the second-highest bid, b) in either case, so again we have a tie. We no longer ~ have a tie if b lies between b and v. If the buyer bids b, then he loses the object and obtains no surplus. If he bids v, then he wins the object and obtains a net surplus of ~ b > 0, so bidding v is strictly better than bidding b < v in this case. Similar logic v shows that bidding v weakly dominates bidding b > v. $ The reason that bidding one’s valuation is weakly dominant is that the winner’s bid does not affect the amount he has to pay, for that depends on someone else’s (the second-highest bidder’s) bid. But bidding one’s valuation ensures the buyer wins the object when he should. With an understanding of equilibrium bidding in second-price auctions, we can compare ﬁrst- and second-price sealed-bid auctions. Each format has plusses and minuses with regard to the revenue the seller earns. On the one hand, bidders shade their bids below their valuations in the ﬁrst-price auction but not in the second-price auction, a ‘‘plus’’ for second-price auctions. On the other hand, the winning bidder pays the highest bid in the ﬁrst-price auction but only the second-highest bid in the secondprice auction, a ‘‘minus’’ for second-price auctions. The surprising result proved by Vickery is that these plusses and minuses balance perfectly, so that both auction types provide the seller with the same expected revenue. Rather than working through a general proof of this revenue equivalence result, we will show in Example 18.9 that it holds in a particular case. 674 Part 8: Market Failure EXAMPLE 18.9 Art Auction Suppose two buyers (1 and 2) bid for a painting in a ﬁrst-price sealed-bid auction. Buyer i’s valuation, vi, is a random variable that is uniformly distributed between 0 and 1 and is independent of the other buyer’s valuation. Buyers’ valuations are private information. We will look for a symmetric equilibrium in which buyers bid a constant fraction of their valuations, bi ¼ Symmetric equilibrium. Given that buyer 1 knows his own type v1 and knows buyer 2’s equilibrium strategy b2 ¼ kv2, buyer 1 best responds by choosing the bid b1 maximizing his expected surplus kvi. The remaining step is to solve for the equilibrium value of k. þ Pr(1 wins auction)(v1 $ b1 > b2Þð Pr v1 $ ð b1 > kv2Þð Pr v1 $ ð v2 < b1=k Pr v1 $ ð Þð b1 k ð v1 $ ¼ ¼ ¼ : b1Þ b1) b1Þ b1Þ b1Þ ¼ Pr(1 loses auction)(0) (18:68) We have ignored the possibility of equal bids, because they would only occur in equilibrium if buyers had equal valuations yet the probability is zero that two independent and continuous random variables equal each other. The only tricky step in Equation 18.68 is the last one. The discussion of cumulative distribution functions in Chapter 2 shows that the probability Pr(v2 < x) can be written as Pr v2 < x ð Þ ¼ x ð $1 f v2Þ ð dv2, (18:69) where f is the probability density function. But for a random variable uniformly distributed between 0 and 1 we have x ð 0 f v2Þ ð dv2 ¼ x ð 0 1 ð Þ dv2 ¼ x, (18:70) so Pr(v2 < b1/k) b1/k. ¼ Taking the ﬁrst-order condition of Equation 18.68 with respect to b1 and rearranging yields 1/2, implying that buyers shade their valuations down by half in forming v1/2. Hence k’ ¼ b1 ¼ their bids. Order statistics. Before computing the seller’s expected revenue from the auction, we will introduce the notion of an order statistic. If n independent draws are made from the same distribution and if they are arranged from smallest to largest, then the kth lowest draw is called the kth-order statistic, denoted X(k). For example, with n random variables, the nth-order statistic X(n) is the largest of the n draws; the (n 1) is the second largest; and so on. Order statistics are so useful that statisticians have done a lot of work to characterize their properties. For instance, statisticians have computed that if n draws are taken from a uniform distribution between 0 and 1, then the expected value of the kth-order statistic is 1)th-order statistic X(n $ $ This formula may be found in many standard statistical references. E X ð k ð ÞÞ ¼ n k þ : 1 (18:71) Chapter 18: Asymmetric Information 675 Expected revenue. The expected revenue from the ﬁrst-price auction equals E max ð b1, b2ÞÞ ¼ ð 1 2 E max ð v1, v2ÞÞ ð : (18:72) But max(v1, v2) is the largest-order statistic from two draws of a uniform random variable between 0 and 1, the expected value of which is 2/3 (according to Equation 18.71). Therefore, the expected revenue from the auction equals (1/2)(2/3) 1/3. ¼ Second-price auction. Suppose that the seller decides to use a second-price auction to sell the painting. In equilibrium, buyers bid their true valuations: bi ¼ vi. The seller’s expected revenue is E(min(b1, b2)) because the winning bidder pays an amount equal to the loser’s bid. But min(b1, b2) min(v1, v2), and the latter is the ﬁrst-order statistic for two draws from a random variable uniformly distributed between 0 and 1 whose expected value is 1/3 (according to Equation 18.71). This is the same expected revenue generated by the ﬁrst-price auction. ¼ QUERY: In the ﬁrst-price auction, could the seller try to boost bids up toward buyers’ valuations by specifying a reservation price r such that no sale is made if the highest bid does not exceed r? What are the trade-offs involved for the seller from such a reservation price? Would a reservation price help boost revenue in a second-price auction? In more complicated economic environments, the many different auction formats do not necessarily yield the same revenue. One complication that is frequently considered is supposing that the good has the same value to all bidders but that they do not know exactly what that value is: each bidder has only an imprecise estimate of what his or her valuation might be. For example, bidders for oil tracts may have each conducted their own surveys of the likelihood that there is oil below the surface. All bidders’ surveys taken together may give a clear picture of the likelihood of oil, but each one separately may give only a rough idea. For another example, the value of a work of art depends in part on its resale value (unless the bidder plans on keeping it in the family forever), which in turn depends on others’ valuations; each bidder knows his or her own valuation but perhaps not others’. An auction conducted in such an economic environment is called a common values auction. The most interesting issue that arises in a common values setting is the so-called winner’s curse. The winning bidder realizes that every other bidder probably thought the good was worth less, meaning that he or she probably overestimated the value of the good. The winner’s curse sometimes leads inexperienced bidders to regret having won the auction. Sophisticated bidders take account of the winner’s curse by shading down their bids below their (imprecise) estimates of the value of the good, so they never regret having won the auction in equilibrium. Analysis of the common values setting is rather complicated, and the different auction formats previously listed no longer yield equivalent revenue. Roughly speaking, auctions that incorporate other bidders’ information in the price paid tend to provide the seller with more revenue. For example, a second-price auction tends to be better than a ﬁrstprice auction because the price paid in a second-price auction depends on what other bidders think the object is worth. If other bidders thought the object was not worth much, then the second-highest bid will be low and the price paid by the winning bidder will be low, precluding the winner’s curse. 676 Part 8: Market Failure SUMMARY In this chapter we have provided a survey of some issues that arise in modeling markets with asymmetric information. Asymmetric information can lead to market inefﬁciencies relative to the ﬁrst-best benchmark, which assumes perfect information. Cleverly designed contracts can often help recover some of this lost surplus. We examined some of the following
speciﬁc issues. • Asymmetric information is often studied using a principal-agent model in which a principal offers a contract to an agent who has private information. The two main variants of the principal-agent model are the models of hidden actions and of hidden types. • • In a hidden-action model (called a moral hazard model in an insurance context), the principal tries to induce the agent to take appropriate actions by tying the agent’s payments to observable outcomes. Doing so exposes the agent to random ﬂuctuations in these outcomes, which is costly for a risk-averse agent. In a hidden-type model (called an adverse selection model in an insurance context), the principal cannot extract all the surplus from high types because they can always gain positive surplus by pretending to be a low type. In an effort to extract the most surplus possible, the principal offers a menu of contracts from which PROBLEMS different types of agent can select. The principal distorts the quantity in the contract targeted to low types in order to make this contract less attractive to high types, thus extracting more surplus in the contract targeted to the high types. • Most of the insights gained from the basic form of the is a principal-agent model, monopolist, carry over to the case of competing principals. The main change is that agents obtain more surplus. in which the principal • The lemons problem arises when sellers have private information about the quality of their goods. Sellers whose goods are higher than average quality may refrain from selling at the market price, which reﬂects the average quality of goods sold on the market. The market may collapse, with goods of only the lowest quality being offered for sale. • The principal can extract more surplus from agents if several of them are pitted against each other in an auction setting. In a simple economic environment, a variety of common auction formats generate the same revenue for the seller. Differences in auction format may generate different levels of revenue in more complicated settings. 18.1 A personal-injury lawyer works as an agent for his injured plaintiff. The expected award from the trial (taking into account the plaintiff ’s probability of prevailing and the damage award if she prevails) is l, where l is the lawyer’s effort. Effort costs the lawyer l 2/2. a. What is the lawyer’s effort, his surplus, and the plaintiff ’s surplus in equilibrium when the lawyer obtains the customary 1/3 contingency fee (i.e., the lawyer gets 1/3 of the award if the plaintiff prevails)? b. Repeat part (a) for a general contingency fee of c. c. What is the optimal contingency fee from the plaintiff ’s perspective? Compute the associated surpluses for the lawyer and plaintiff. d. What would be the optimal contingency fee from the plaintiff ’s perspective if she could ‘‘sell’’ the case to her lawyer [i.e., if she could ask him for an up-front payment in return for a speciﬁed contingency fee, possibly higher than in part (c)]? Compute the up-front payment (assuming that the plaintiff makes the offer to the lawyer) and the associated surpluses for the lawyer and plaintiff. Do they do better in this part than in part (c)? Why do you think selling cases in this way is outlawed in many countries? 18.2 Solve for the optimal linear price per ounce of coffee that the coffee shop would charge in Example 18.4. How does the shop’s proﬁt compare to when it uses nonlinear prices? Hint: Your ﬁrst step should be to compute each type’s demand at a linear price p. Chapter 18: Asymmetric Information 677 18.3 Return to the nonlinear pricing problem facing the monopoly coffee shop in Example 18.4, but now suppose the proportion of high demanders increases to 2/3 and the proportion of low demanders decreases to 1/3. What is the optimal menu in the second-best situation? How does the menu compare to the one in Example 18.4? 18.4 Suppose there is a 50–50 chance that an individual with logarithmic utility from wealth and with a current wealth of $20,000 will suffer a loss of $10,000 from a car accident. Insurance is competitively provided at actuarially fair rates. a. Compute the outcome if the individual buys full insurance. b. Compute the outcome if the individual buys only partial insurance covering half the loss. Show that the outcome in part (a) is preferred. c. Now suppose that individuals who buy the partial rather than the full insurance policy take more care when driving, reducing the damage from loss from $10,000 to $7,000. What would be the actuarially fair price of the partial policy? Does the individual now prefer the full or the partial policy? 18.5 Suppose that left-handed people are more prone to injury than right-handed people. Lefties have an 80 percent chance of suffering an injury leading to a $1,000 loss (in terms of medical expenses and the monetary equivalent of pain and suffering) but righties have only a 20 percent chance of suffering such an injury. The population contains equal numbers of lefties and righties. Individuals all have logarithmic utility-of-wealth functions and initial wealth of $10,000. Insurance is provided by a monopoly company. a. Compute the ﬁrst best for the monopoly insurer (i.e., supposing it can observe the individual’s dominant hand). b. Take as given that, in the second best, the monopolist prefers not to serve righties at all and targets only lefties. Knowing this, compute the second-best menu of policies for the monopoly insurer. c. Use a spreadsheet program (such as the one on the website associated with Example 18.5) or other mathematical software to solve numerically the constrained optimization problem for the second best. Make sure to add constraints bounding the insurance payments for righties: 0 xR is binding and so righties are not served in the second best. 1,000. Establish that the constraint 0 xR , , , 18.6 Consider the same setup as in Problem 18.5, but assume that insurance is offered by competitive insurers. a. Ignore the issue of whether consumers’ insurance decisions are rational for now and simply assume that the equal numbers of lefties and righties both purchase full insurance whatever the price. If insurance companies cannot distinguish between consumer types and thus offer a single full-insurance contract, what would the actuarially fair premium for this contract be? b. Which types will buy insurance at the premium calculated in (a)? c. Given your results from part (b), will the insurance premiums be correctly computed? 18.7 Suppose 100 cars will be offered on the used-car market. Let 50 of them be good cars, each worth $10,000 to a buyer, and let 50 be lemons, each worth only $2,000. a. Compute a buyer’s maximum willingness to pay for a car if he or she cannot observe the car’s quality. b. Suppose that there are enough buyers relative to sellers that competition among them leads cars to be sold at their maxi- mum willingness to pay. What would the market equilibrium be if sellers value good cars at $8,000? At $6,000? 18.8 Consider the following simple model of a common values auction. Two buyers each obtain a private signal about the value of an object. The signal can be either high (H ) or low (L) with equal probability. If both obtain signal H, the object is worth 1; otherwise, it is worth 0. a. What is the expected value of the object to a buyer who sees signal L? To a buyer who sees signal H? b. Suppose buyers bid their expected value computed in part (a). Show that they earn negative proﬁt conditional on observing signal H—an example of the winner’s curse. 678 Part 8: Market Failure Analytical Problems 18.9 Doctor-patient relationship Consider the principal-agent relationship between a patient and doctor. Suppose that the patient’s utility function is given by UP (m, x), where m denotes medical care (whose quantity is determined by the doctor) and x denotes other consumption goods. The patient faces budget constraint Ic ¼ x, where pm is the relative price of medical care. The doctor’s utility function is given by Ud (Id) Up—that is, the doctor derives utility from income but, being altruistic, also derives utility from the patient’s well-being. Moreover, the additive speciﬁcation implies that the doctor is a perfect altruist in the sense that his or her utility increases one-for-one with the patient’s. The doctor’s income comes from the patient’s medical expenditures: Id ¼ pmm. Show that, in this situation, the doctor will generally choose a level of m that is higher than a fully informed patient would choose. pmm þ þ 18.10 Diagrams with three types Suppose the agent can be one of three types rather than just two as in the chapter. a. Return to the monopolist’s problem of computing the optimal nonlinear price. Represent the ﬁrst best in a schematic dia- gram by modifying Figure 18.4. Do the same for the second best by modifying Figure 18.6. b. Return to the monopolist’s problem of designing optimal insurance policies. Represent the ﬁrst best in a schematic diagram by modifying Figure 18.7. Do the same for the second best by modifying Figure 18.8. 18.11 Increasing competition in an auction A painting is auctioned to n bidders, each with a private value for the painting that is uniformly distributed between 0 and 1. a. Compute the equilibrium bidding strategy in a ﬁrst-price sealed-bid auction. Compute the seller’s expected revenue in this auction. Hint: Use the formula for the expected value of the kth-order statistic for uniform distributions in Equation 18.71. b. Compute the equilibrium bidding strategy in a second-price sealed-bid auction. Compute the seller’s expected revenue in this auction using the hint from part (a). c. Do the two auction formats exhibit revenue equivalence? d. For each auction format, how do bidders’ strategies and the seller’s revenue change with an increase in the number of bidders? eiÞ ¼ ð 18.12 Team effort Increasing the size of a team that creates a joint product may dull incentives, as thi
s problem will illustrate.11 Suppose n partners together produce a revenue of R en; here ei is partner i’s effort, which costs him c e2 i =2 to exert. e1 þ + + + þ ¼ a. Compute the equilibrium effort and surplus (revenue minus effort cost) if each partner receives an equal share of the revenue. b. Compute the equilibrium effort and average surplus if only one partner gets a share. Is it better to concentrate the share or to disperse it? c. Return to part (a) and take the derivative of surplus per partner with respect to n. Is surplus per partner increasing or decreasing in n? What is the limit as n increases? d. Some commentators say that ESOPs (employee stock ownership plans, whereby part of the ﬁrm’s shares are distributed among all its workers) are beneﬁcial because they provide incentives for employees to work hard. What does your answer to part (c) say about the incentive properties of ESOPs for modern corporations, which may have thousands of workers? 11The classic reference on the hidden-action problem with multiple agents is B. Holmstro¨m, ‘‘Moral Hazard in Teams,’’ Bell Journal of Economics (Autumn 1982): 324–40. Chapter 18: Asymmetric Information 679 SUGGESTIONS FOR FURTHER READING Bolton, P., and M. Dewatripont. Contract Theory. Cambridge, MA: MIT Press, 2005. Comprehensive graduate textbook treating all topics in this chapter and many other topics in contract theory. Krishna, V. Auction Theory. San Diego: Academic Press, 2002. Advanced text on auction theory. Lucking-Reiley, D. ‘‘Using Field Experiments to Test Equivalence between Auction Formats: Magic on the Internet.’’ American Economic Review (December 1999): 1063–80. Tests the revenue equivalence theorem by selling Magic playing cards over the Internet using various auction formats. Milgrom, P. ‘‘Auctions and Bidding: A Primer.’’ Journal of Economic Perspectives (Summer 1989): 3–22. Intuitive discussion of methods used and research questions explored in the ﬁeld of auction theory. Rothschild, M., and J. Stiglitz. ‘‘Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information.’’ Quarterly Journal of Economics (November 1976): 629–50. Presents a nice graphic treatment of the adverse selection problem. Contains ingenious illustrations of various possibilities for separating equilibria. Salanie´, B. The Economics of Contracts: A Primer. Cambridge, MA: MIT Press, 1997. A concise treatment of contract theory at a deeper level than this chapter. Shavell, S. Economic Analysis of Accident Law. Cambridge, MA: Harvard University Press, 1987. Classic reference analyzing the effect of different laws on the level of precaution undertaken by victims and injurers. Discusses how the availability of insurance affects parties’ behavior. EXTENSIONS NONLINEAR PRICING WITH A CONTINUUM OF TYPES In this extension, we will expand the analysis of nonlinear pricing to allow for a continuum of consumer types rather than just two. The extension will be especially valuable for students who are interested in seeing new applications of optimal control techniques introduced in Chapter 2 to applications beyond dynamic choice problems. Be warned that the mathematics used here is some of the most complicated in the book. For those not interested in practicing optimal control, the main point to take away from this extension is ‘‘reassurance’’: we can rest assured that the conclusions we have drawn from the simple two-type model in this chapter indeed hold in more general settings. Besides drawing on Chapter 2, the extension draws on Section 2.3.3 of Bolton and Dewatripont (2005). E18.1 Remaining questions about hiddentype models We analyzed the simplest possible hidden-type model in Chapter 18. The agent’s type could be one of only two possible values. In the nonlinear pricing application, for example, the agent was a consumer who could have high or low demand. In the application to adverse selection in insurance, the agent was an individual who could have a high or low probability of an accident. We derived a number of insights from the analysis, including that the low type’s contract was distorted downward relative to the ﬁrst best, although the high type’s contract was not. The latter insight was summarized as ‘‘no distortion at the top.’’ The analysis left a number of open questions. How general are the ﬁrst-order conditions characterizing the second-best contract? Does ‘‘no distortion at the top’’ mean that only the highest type’s contract is efﬁcient, or that all but the very lowest type’s are, or something in between? Does the monopolist want to serve all types, or will the lowest types be left off the menu? We cannot tell by analyzing a two-type model, but we can answer these questions by extending the analysis to a continuous distribution of types. As mentioned previously, the other motivation for this extension is to show the power of the optimal control methods introduced in Chapter 2 for solving problems beyond dynamic choice problems. E18.2 Nonlinear pricing model For concreteness, we will focus our analysis on the nonlinear pricing problem for a monopolist. The monopolist offers a menu of bundles, one for each type y, where a bundle is a speciﬁcation of a quantity q(y) and a total tariff for this quantity T (y). The consumer has private information about his type, but the monopolist knows only the distribution from which y is drawn. Let j(y) be the associated probability density function and F(y) the cumulative distribution function. Suppose all types fall in the interval between yL at the low end and yH at the high end. (Review the section on probability and statistics from Chapter 2 for these and other concepts used in this extension.) As before, the consumer’s utility function is U (y) ¼ T (y). The monopolist’s proﬁt from serving type y yv(q(y)) is G(y) cq(y), where c is the constant marginal and T (y) average cost of production. $ ¼ $ E18.3 First best The ﬁrst best is easy to solve for. Each type is offered the socially optimal quantity, which satisﬁes the condition yv 0(q) c. Each type is charged the tariff that extracts all of his surplus T(y) yv(q(y)). The monopolist earns proﬁt yv(q(y)) clearly all of the social surplus. ¼ cq(y), which is ¼ $ E18.4 Second best The monopolist’s second-best pricing scheme is the menu of bundles q(y) and T (y) that maximizes its expected proﬁt, uH ð uL P u u Þ ð u Þ ð du ¼ uH ð uL T u ð ½ Þ $ u cq ð u ð u Þ Þ) du, (i) subject to participation and incentive compatibility constraints for the consumer. As we have seen, the participation constraint is a concern only for the lowest type that the monopolist serves. Then all types will participate as long as yL does. The relevant participation constraint is thus uLv q ð ð uLÞÞ $ T uLÞ ! ð 0: (ii) That all types participate in the contract does not require the monopolist to serve them with a positive quantity. The monopolist may choose to offer the null contract (zero quantity and tariff ) to a range of types. By reducing some types down to the null contract, the monopolist can extract even more surplus from higher types. Incentive compatibility requires additional discussion. Incentive compatibility requires that type prefer its bundle to ~ u any other type’s, say, q In other words, . ~ Þ ð uv u. Taking the ﬁrstu T ¼ Þ ð order condition with respect to ~ ~ u u Þ ¼ ð ð ~ u ð is maximized at and T ~ u ~ u yields ~ u ÞÞ $ iii) uv 0 for ~ u T 0 ÞÞ ¼ u; q0 0 Þ that is, uv 0 q ð u ÞÞ ð q0 u ð T 0 u Þ ¼ ð 0: Þ $ Equation iv is both necessary and sufﬁcient for incentive compatibility under a set of conditions that hold in many examples but are a bit too technical to discuss here. (iv) E18.5 Rewriting the problem There are too many derivatives in Equation iv for us to apply the optimal control methods from Chapter 2. The analogous equation in Chapter 2 (Equation 2.148) has only one derivative. To obtain a workable incentive compatibility constraint, observe that ÞÞ þ , ÞÞ uv 0 q ð ð u ÞÞ v) where the second line follows from Equation iv. Now we have expressed the incentive compatibility constraint in a form with only one derivative, as required. Since the differential equation U 0(y) v(q(y)) involves the derivative of U(y) rather than of T(y), we can make the substitution T(y) U(y) everywhere in the maximization problem to put it in terms of q(y) and U(y) rather than q(y) and T(y). yv(q(y)) ¼ $ ¼ The reformulated problem is to maximize uH ð uL uv q ð u ÞÞ $ ð U u ð ½ Þ $ u cq Þ) ð u u ð Þ du (vi) subject to the participation constraint (inequality ii) and the incentive compatibility constraint U 0(y) v (q(y)). By Equation 2.150, the Hamiltonian associated with the optimal control problem is ¼ U H uv ¼ ½ þ u cq ð Þ $ u U k0 Þ ð u u q ÞÞ $ ð ð ð u u q v k ÞÞ þ ð ð Þ ð To see how this Hamiltonian is constructed, y is here playing the role played by t in Chapter 2, q(y) is playing the role of control variable c (t), U(y) is playing the role of the state variable x(t), u u Þ) ð u : Þ ð (vii) Þ uv q Þ $ ð is playing the role of f , and U 0(y) of differential equation ÞÞ $ cq ð v (q(y)) is playing the role (viii) Þ) ¼ Chapter 18: Asymmetric Information 681 E18.6 Optimal control solution Analogous to the conditions @H/@c 0 from Equation 2.151, here the conditions for the optimal control solution are 0 and @H/@x ¼ ¼ @H @q ¼ ½ uv 0 q ð u ÞÞ $ , @H @U ¼ $ u ð u Þ þ k0 u ð Þ ¼ 0: (x) (xi) To cast these conditions in a more useful form, we shall eliminate the Lagrange multiplier. The second equation implies l0(y) j(y). By the fundamental theorem of calculus (discussed in Chapter 2), ¼ uHÞ $ k ð k u Þ ¼ ð uH ð u uH ds k0 s ð Þ ds u s ð Þ (xii) ¼ ð u U 1 uHÞ $ U ð u , U Þ ð $ where F(yH) 1 because F is a cumulative distribution function, which equals 1 when evaluated at the greatest possible value of the random variable. Therefore uHÞ þ U k ð u 1xiii) ¼ since l(yH) 0 [there are no types above yH from whom to extract surplus, so the value from distorting type yH’s contract
as measured by l(yH) is 0]. Substituting into Equation x and rearranging yields uv 0 q ð u ÞÞ : ÞÞ (xiv) ¼ This equation tells us a lot about the second best. Because F(yH) 1, for the highest type the equation reduces to yHv 0(q(yH)) c, the ﬁrst-best condition. We again have ‘‘no distortion at the top’’ for the high type, but all other types face some downward distortion in q(y). To see this, note that yv 0(q(y)) > c for these other types, implying that q(y) is less than the ﬁrst best for all y < yH. ¼ E18.7 Uniform example Suppose y is uniformly distributed between 0 and 1 and that y. Equation xiv q v ð Þ ¼ implies that ﬃﬃﬃ qp . Then j(y) 1 and F(y) ¼ ¼ 2 dx t ð Þ dt ix) q(u) = 2u $ c 1 % $ 2 : (xv) 682 Part 8: Market Failure FIGURE E18.1 Nonlinear Pricing Schedule for Continuum of Types The graph is based on calculations for uniformly distributed types. Larger bundles receive per-unit price discount. T 16 14 12 10 8 6 4 2 0 Slope of chord ! average price per unit 0 20 40 60 80 100 q It is apparent from Equation xv that only types above 1/2 are served. By leaving types below 1/2 unserved, the monopolist can extract more surplus from the higher-value consumers whom it does serve. To compute the tariff, observe that T u Þ ¼ ð u ð 1=2 u T 0 s ð Þ ds sv 0 q ð s ð ÞÞ q0 s Þ ð ds ð 1=2 4u2 ¼ ¼ 1 , $ 2c (xvi) where the ﬁrst equality holds by the fundamental theorem of calculus, the second by Equation iv, and the third by Equation xv. Figure E18.1 graphs the resulting nonlinear pricing schedule. Each point on the schedule is a bundle targeted at a particular type. The implied per-unit price can be found by looking at the slope of the chord from the origin to the graph. It is clear that this chord is decreasing as q increases, implying that the per-unit price is decreasing, which in turn implies that the schedule involves quantity discounts for large purchases. Reference Bolton, P., and M. Dewatripont. Contract Theory. Cambridge, MA: MIT Press, 2005. This page intentionally left blank C H A P T E R NINETEEN Externalities and Public Goods In Chapter 13 we looked brieﬂy at a few problems that may interfere with the allocational efﬁciency of perfectly competitive markets. Here we will examine two of those problems—externalities and public goods—in more detail. This examination has two purposes. First, we wish to show clearly why the existence of externalities and public goods may distort the allocation of resources. In so doing it will be possible to illustrate some additional features of the type of information provided by competitive prices and some of the circumstances that may diminish the usefulness of that information. Our second reason for looking more closely at externalities and public goods is to suggest ways in which the allocational problems they pose might be mitigated. We will see that, at least in some cases, the efﬁciency of competitive market outcomes may be more robust than might have been anticipated. Deﬁning Externalities Externalities occur because economic actors have effects on third parties that are not reﬂected in market transactions. Chemical makers spewing toxic fumes on their neighbors, jet planes waking up people, and motorists littering the highway are, from an economic point of view, all engaging in the same sort of activity: they are having a direct effect on the well-being of others that is outside market channels. Such activities might be contrasted to the direct effects of markets. When I choose to purchase a loaf of bread, for example, I (perhaps imperceptibly) increase the price of bread generally, and that may affect the well-being of other bread buyers. But such effects, because they are reﬂected in market prices, are not externalities and do not affect the market’s ability to allocate resources efﬁciently.1 Rather, the increase in the price of bread that results from my increased purchase is an accurate reﬂection of societal preferences, and the price increase helps ensure that the right mix of products is produced. That is not the case for toxic chemical discharges, jet noise, or litter. In these cases, market prices (of chemicals, air travel, or disposable containers) may not accurately reﬂect actual social costs because they may take no account of the damage being done to third parties. Information being conveyed by market prices is fundamentally inaccurate, leading to a misallocation of resources. As a summary, then, we have developed the following deﬁnition. 1Sometimes effects of one economic agent on another that take place through the market system are termed pecuniary externalities to differentiate such effects from the technological externalities we are discussing. Here the use of the term externalities will refer only to the latter type, because these are the only type with consequences for the efﬁciency of resource allocation by competitive markets. 685 686 Part 8: Market Failure Externality. An externality occurs whenever the activities of one economic actor affect the activities of another in ways that are not reﬂected in market transactions. Before analyzing in detail why failing to take externalities into account can lead to a misallocation of resources, we will examine a few examples that should clarify the nature of the problem. Interﬁrm externalities To illustrate the externality issue in its simplest form, we consider two ﬁrms: one producing good x and the other producing good y. The production of good x is said to have an external effect on the production of y if the output of y depends not only on the inputs chosen by the y-entrepreneur but also on the level at which the production of x is carried on. Notationally, the production function for good y can be written as y f k, l; x ð , Þ ¼ (19:1) where x appears to the right of the semicolon to show that it is an effect on production over which the y-entrepreneur has no control.2 As an example, suppose the two ﬁrms are located on a river, with ﬁrm y being downstream from x. Suppose ﬁrm x pollutes the river in its productive process. Then the output of ﬁrm y may depend not only on the level of inputs it uses itself but also on the amount of pollutants ﬂowing past its factory. The level of pollutants, in turn, is determined by the output of ﬁrm x. In the production function shown by Equation 19.1, the output of ﬁrm x would have a negative marginal physical productivity @y/@x < 0. Increases in x output would cause less y to be produced. In the next section we return to analyze this case more fully, since it is representative of most simple types of externalities. Beneﬁcial externalities The relationship between two ﬁrms may be beneﬁcial. Most examples of such positive externalities are rather bucolic in nature. Perhaps the most famous, proposed by J. Meade, involves two ﬁrms, one producing honey (raising bees) and the other producing apples.3 Because the bees feed on apple blossoms, an increase in apple production will improve productivity in the honey industry. The beneﬁcial effects of having well-fed bees are a positive externality to the beekeeper. In the notation of Equation 19.1, @y/@x would now be positive. In the usual perfectly competitive case, the productive activities of one ﬁrm have no direct effect on those of other ﬁrms: @y/@x 0. ¼ Externalities in utility Externalities also can occur if the activities of an economic actor directly affect an individual’s utility. Most common examples of environmental externalities are of this type. From an economic perspective it makes little difference whether such effects are created by ﬁrms (in the form, say, of toxic chemicals or jet noise) or by other individuals (litter or, perhaps, the noise from a loud radio). In all such cases the amount of such activities would enter directly into the individual’s utility function in much the same way as ﬁrm x’s output entered into ﬁrm y’s production function in Equation 19.1. As in the case of ﬁrms, such externalities may sometimes be beneﬁcial (you may actually like the song being played on your neighbor’s radio). So, again, a situation of zero externalities can be 2We will ﬁnd it necessary to redeﬁne the assumption of ‘‘no control’’ considerably as the analysis of this chapter proceeds. 3J. Meade, ‘‘External Economies and Diseconomies in a Competitive Situation,’’ Economic Journal 62 (March 1952): 54–67. Chapter 19: Externalities and Public Goods 687 regarded as the middle ground in which other agents’ activities have no direct effect on individuals’ utilities. One special type of utility externality that is relevant to the analysis of social choices arises when one individual’s utility depends directly on the utility of someone else. If, for example, Smith cares about Jones’s welfare, then we could write his or her utility function (US) as utility USð , x1, . . . , xn; UJÞ ¼ (19:2) where x1, . . . , xn are the goods that Smith consumes and UJ is Jones’s utility. If Smith is altruistic and wants Jones to be well off (as might happen if Jones were a close relative), @US /@UJ would be positive. If, on the other hand, Smith were envious of Jones, then it might be the case that @US /@UJ would be negative; that is, improvements in Jones’s utility make Smith worse off. The middle ground between altruism and envy would occur if Smith were indifferent to Jones’s welfare (@US /@UJ ¼ 0), and that is what we have usually assumed throughout this book (for a brief discussion, see the Extensions to Chapter 3). Public goods externalities Goods that are ‘‘public’’ or ‘‘collective’’ in nature will be the focus of our analysis in the second half of this chapter. The deﬁning characteristic of these goods is nonexclusion; that is, once the goods are produced (either by the government or by some private entity), they provide beneﬁts to an entire group—perhaps to everyone. It is technically impossible to restrict these beneﬁts to the speciﬁc group of individuals who pay for them, so the beneﬁts are available to all. As we mentioned in Chapter
13, national defense provides the traditional example. Once a defense system is established, all individuals in society are protected by it whether they wish to be or not and whether they pay for it or not. Choosing the right level of output for such a good can be a tricky process, because market signals will be inaccurate. Externalities and Allocative Inefﬁciency Externalities lead to inefﬁcient allocations of resources because market prices do not accurately reﬂect the additional costs imposed on or beneﬁts provided to third parties. To illustrate these inefﬁciencies requires a general equilibrium model, because inefﬁcient allocations in one market throw into doubt the efﬁciency of market-determined outcomes everywhere. Here we choose a very simple and, in some ways, rather odd general equilibrium model that allows us to make these points in a compact way. Speciﬁcally, we assume there is only one person in our simple economy and that his or her utility depends on the quantities of x and y consumed. Consumption levels of these two goods are denoted by xc and yc, so ð This person has initial stocks of x and y (denoted by x$ and y$) and can either consume these directly or use them as intermediary goods in production. To simplify matters, we assume that good x is produced using only good y, according to the production function ¼ utility U : xc, ycÞ ð where subscript o refers to outputs and i to inputs. To illustrate externalities, we assume that the output of good y depends not only on how much x is used as an input in the production process but also on the x production level itself. Hence this would model a xo ¼ f , yiÞ (19:3) (19:4) 688 Part 8: Market Failure situation, say, where y is downriver from ﬁrm x and must cope with the pollution created by production of x output. The production function for y is given by , xi, xoÞ ð where g1 > 0 (more x input produces more y output), but g2 < 0 (additional x output reduces y output because of the externality involved). yo ¼ (19:5) g The quantities of each good in this economy are constrained by the initial stocks avail- able and by the additional production that takes place: xc þ yc þ xi ¼ yi ¼ xo þ yo þ x$, y$: (19:6) (19:7) Finding the efﬁcient allocation The economic problem for this society, then, is to maximize utility subject to the four constraints represented by Equations 19.4–19.7. To solve this problem we must introduce four Lagrange multipliers. The Lagrangian expression for this maximization problem is + xc, ycÞ þ U k2½ ð yc þ xi ’ xc þ k3ð and the six ﬁrst-order conditions for a maximum are yiÞ ’ x$ Þ þ f k1½ ð xo ’ xo( þ k4ð ¼ þ xi, xoÞ ’ g ð y$ yo ’ yi ’ yo( , Þ (19:8) i 0, ½ ( ii k3 ¼ 0, k4 ¼ k3 ¼ 0, k4 ¼ k2 g2 ’ 0: k4 ¼ Eliminating the ls from these equations is a straightforward process. Taking the ratio of Equations i and ii yields the familiar result @+=@xc ¼ @+=@yc ¼ @+=@xi ¼ @+=@yi ¼ @+=@xo ¼ ’ @+=@yo ¼ ’ U1 þ U2 þ k2 g1 þ k1 fy þ k1 þ k2 ’ k3 ¼ (19:9) ( ½ iii ( vi iv 0, 0, v ½ ½ ( ½ ( ½ ( MRS U1 U2 ¼ k3 k4 : ¼ But Equations iii and vi also imply MRS k3 k4 ¼ k2 g1 k2 ¼ ¼ g1: (19:10) (19:11) Hence optimality in y production requires that the individual’s MRS in consumption equal the marginal productivity of x in the production of y. This conclusion repeats the result from Chapter 13, where we showed that efﬁcient output choice requires that dy/dx in consumption be equal to dy/dx in production. To achieve efﬁciency in x production, we must also consider the externality that this production poses to y. Combining Equations iv–vi gives k2 g2 ’ k1 þ k4 k1 ’ k4 þ k2 g2 k4 ¼ MRS ¼ ¼ k3 k4 ¼ 1 fy ’ g2: (19:12) Chapter 19: Externalities and Public Goods 689 Intuitively, this equation requires that the individual’s MRS must also equal dy/dx obtained through x production. The ﬁrst term in the expression, 1/fy, represents the reciprocal of the marginal productivity of y in x production—this is the ﬁrst component of dy/dx as it relates to x production. The second term, g2, represents the negative impact that added x production has on y output—this is the second component of dy/dx as it relates to x production. This ﬁnal term occurs because of the need to consider the externality from x production. If g2 were zero, then Equations 19.11 and 19.12 would represent essentially the same condition for efﬁcient production, which would apply to both x and y. With the externality, however, determining an efﬁcient level of x production is more complex. Inefﬁciency of the competitive allocation Reliance on competitive pricing in this simple model will result in an inefﬁcient allocation of resources. With equilibrium prices px and py, a utility-maximizing individual would opt for MRS ¼ px=py (19:13) and the proﬁt-maximizing producer of good y would choose x input according to px ¼ pyg1: (19:14) Hence the efﬁciency condition (Equation 19.11) would be satisﬁed. But the producer of good x would choose y input so that py ¼ px fy or px py ¼ 1 fy : (19:15) That is, the producer of x would disregard the externality that its production poses for y and so the other efﬁciency condition (Equation 19.12) would not be met. This failure results in an overproduction of x relative to the efﬁcient level. To see this, note that the marginal product of y in producing x ( fy) is smaller under the market allocation represented by Equation 19.15 than under the optimal allocation represented by Equation 19.12. More y is used to produce x in the market allocation (and hence more x is produced) than is optimal. Example 19.1 provides a quantitative example of this nonoptimality in a partial equilibrium context. EXAMPLE 19.1 Production Externalities As a partial equilibrium illustration of the losses from failure to consider production externalities, suppose two newsprint producers are located along a river. The upstream ﬁrm (x) has a production function of the form 2,000l1=2 x , x ¼ (19:16) where lx is the number of workers hired per day and x is newsprint output in feet. The downstream ﬁrm (y) has a similar production function, but its output may be affected by the chemicals ﬁrm x pours into the river: y ¼ ( 2,000l1=2 2,000l1=2 y y x ð ’ a x0Þ for x > x0, x0, for x ) (19:17) where x0 represents the river’s natural capacity for neutralizing pollutants. If a 0, then x’s production process has no effect on ﬁrm y, but if a < 0, an increase in x above x0 causes y’s output to decrease. ¼ 690 Part 8: Market Failure Assuming newsprint sells for $1 per foot and workers earn $50 per day, ﬁrm x will maximize proﬁts by setting this wage equal to labor’s marginal revenue product: 50 p * ¼ @x @lx ¼ 1,000l’ x 1=2 : (19:18) The solution then is lx ¼ workers. Each ﬁrm will produce 40,000 feet of newsprint. 400. If a ¼ 0 (there are no externalities), ﬁrm y will also hire 400 Effects of an externality. When ﬁrm x does have a negative externality (a < 0), its proﬁtmaximizing hiring decision is not affected—it will still hire lx ¼ 40,000. ¼ 0.1 But for ﬁrm y, labor’s marginal product will be lower because of this externality. If a and x0 ¼ 38,000, for example, then proﬁt maximization will require 400 and produce x ¼ ’ 50 p * ¼ @y @ly ¼ 1=2 1,000l’ y 1=2 468l’ y : ¼ ¼ 1,000l’ y 1=2 2,000 ð ’ Þ x ð 0:1 38,000 0:1 ’ Þ ’ (19:19) Solving this equation for ly shows that ﬁrm y now hires only 87 workers because of this lowered productivity. Output of ﬁrm y will now be y 87 2,000 ð 1=2 Þ 2,000 ð ’ Þ 0:1 ¼ ¼ 8,723: (19:20) Because of the externality (a externality (a 0). ¼ 0.1), newsprint output will be lower than without the ¼ ’ Inefﬁciency. We can demonstrate that decentralized proﬁt maximization is inefﬁcient in this situation by imagining that ﬁrms x and y merge and that the manager must decide how to allocate the combined workforce. If one worker is transferred from ﬁrm x to ﬁrm y, then x output becomes for ﬁrm y, 1=2 2,000 399 Þ ð 39,950; x ¼ ¼ 2,000 88 ð 8,796: Þ y ¼ ¼ 1=2 0:1 1,950 ð ’ Þ (19:21) (19:22) Total output has increased by 23 feet of newsprint with no change in total labor input. The market-based allocation was inefﬁcient because ﬁrm x did not take into account the negative effect of its hiring decisions on ﬁrm y. Marginal productivity. This can be illustrated in another way by computing the true social marginal productivity of labor input to ﬁrm x. If that ﬁrm were to hire one more worker, its own output would increase to 2,000 x ¼ 1=2 401 Þ ð ¼ 40,050: (19:23) As proﬁt maximization requires, the (private) marginal value product of the 401st worker is equal to the wage. But increasing x’s output now also has an effect on ﬁrm y—its output decreases by about 21 units. Hence the social marginal revenue product of labor to ﬁrm x $21). That is why the manager of a merged ﬁrm would actually amounts to only $29 ($50 ﬁnd it proﬁtable to shift some workers from ﬁrm x to ﬁrm y. ’ QUERY: Suppose a How would such an externality affect the allocation of labor? ¼ þ 0.1. What would that imply about the relationship between the ﬁrms? Chapter 19: Externalities and Public Goods 691 Solutions to the Externality Problem Incentive-based solutions to the allocational harm of externalities start from the basic observation that output of the externality-producing activity is too high under a marketdetermined equilibrium. Perhaps the ﬁrst economist to provide a complete analysis of this distortion was A. C. Pigou, who in the 1920s suggested that the most direct solution would simply be to tax the externality-creating entity.4 All incentive-based solutions to the externality problem stem from this basic insight.5 A graphic analysis Figure 19.1 provides the traditional illustration of an externality together with Pigou’s taxation solution. The competitive supply curve for good x also represents that good’s private marginal costs of production (MC). When the demand for x is given by DD, the market equilibrium will occur at x1. The external costs involved in x production create a divergence between private marginal costs (MC) and over
all social marginal costs (MC 0)—the vertical distance between the two curves represents the costs that x FIGURE 19.1 Graphic Analysis of an Externality The demand curve for good x is given by DD. The supply curve for x represents the private marginal costs (MC) involved in x production. If x production imposes external costs on third parties, social marginal costs (MC 0) will exceed MC by the extent of these costs. Market equilibrium occurs at x1 and, at this output level, social marginal costs exceed what consumers pay for good x. A tax of amount t that reﬂects the costs of the externalities would achieve the efﬁcient output of x—given by output level x2. Price, costs D MC′ S = MC p2 p1 t D x2 x1 Output of x per period 4A. C. Pigou, The Economics of Welfare (London: MacMillan, 1920). Pigou also recognized the importance of subsidizing goods that yield positive externalities. 5We do not discuss purely regulatory solutions here, although the study of such solutions forms an important part of most courses in environmental economics. See W. J. Baumol and W. E. Oates, The Theory of Environmental Policy, 2nd ed. (Cambridge: Cambridge University Press, 2005) and the Extensions to this chapter. 692 Part 8: Market Failure production poses for third parties (in our examples, only on ﬁrm y). Notice that the per-unit costs of these externalities need not be constant, independent of x output. In the ﬁgure, for example, the size of these external costs increases as x output expands (i.e., MC 0 and MC become further apart). At the market-determined output level x1, the comprehensive social marginal cost exceeds the market price p1, thereby indicating that the production of x has been pushed ‘‘too far.’’ It is clear from the ﬁgure that the optimal output level is x2, at which the market price p2 paid for the good now reﬂects all costs. As is the case for any tax, imposition of a Pigovian tax would create a vertical wedge between the demand and supply curves for good x. In Figure 19.1 this optimal tax is shown as t. Imposition of this tax serves to reduce output to x2, the social optimum. Tax collections equal the precise amount of external harm that x production causes. These collections might be used to compensate ﬁrm y for these costs, but this is not crucial to the analysis. Notice here that the tax must be set at the level of harm prevailing at the optimum (i.e., at x2), not at the level of harm at the original market equilibrium (x1). This point is also made in the next example and more completely in the next section by returning to our simple general equilibrium model. EXAMPLE 19.2 A Pigovian Tax on Newsprint The inefﬁciency in Example 19.1 arises because the upstream newsprint producer (ﬁrm x) takes no account of the effect that its production has on ﬁrm y. A suitably chosen tax on ﬁrm x can cause it to reduce its hiring to a level at which the externality vanishes. Because the river can absorb the pollutants generated with an output of x 38,000, we might consider imposing a tax (t) on the ﬁrm’s output that encourages it to reduce output to this level. Because output will be 38,000 if lx ¼ 361, we can calculate t from the labor demand condition: ¼ or 1 ð t 1 MPL ¼ ð Þ ’ ’ t 1,000 Þ 361 Þ ð ’ 0:5 50, ¼ 0:05: t ¼ (19:24) (19:25) Such a 5 percent tax would effectively reduce the price ﬁrm x receives for its newsprint to $0.95 and provide it with an incentive to reduce its hiring by 39 workers. Now, because the river can handle all the pollutants that x produces, there is no externality in the production function of ﬁrm y. It will hire 400 workers and produce 40,000 feet of newsprint per day. Observe that total newsprint output is now 78,000, a signiﬁcantly higher ﬁgure than would be produced in the untaxed situation. The taxation solution provides a considerable improvement in the efﬁciency of resource allocation. QUERY: The tax rate proposed here (0.05) seems rather small given the signiﬁcant output gains obtained relative to the situation in Example 19.1. Can you explain why? Would a merged ﬁrm opt for x 38,000 even without a tax? ¼ Taxation in the general equilibrium model pyg2. That is, The optimal Pigovian tax in our general equilibrium model is to set t the per-unit tax on good x should reﬂect the marginal harm that x does in reducing y output, valued at the market price of good y. Notice again that this tax must be based on the value of this externality at the optimal solution; because g2 will generally be a function of the level of x output, a tax based on some other output level would be inappropriate. ¼ ’ Chapter 19: Externalities and Public Goods 693 With the optimal tax, ﬁrm x now faces a net price for its output of px ’ y input according to t and will choose Hence the resulting allocation of resources will achieve py ¼ ð px ’ t Þ fy: MRS px py ¼ 1 fy þ t py ¼ 1 fy ’ ¼ g2, (19:26) (19:27) which is precisely what is required for optimality (compare to the efﬁciency condition, Equation 19.12). The Pigovian taxation solution can be generalized in a variety of ways that provide insights about the conduct of policy toward externalities. For example, in an economy with many x-producers, the tax would convey information about the marginal impact that output from any one of these would have on y output. Hence the tax scheme mitigates the need for regulatory attention to the speciﬁcs of any particular ﬁrm. It does require that regulators have enough information to set taxes appropriately—that is, they must know ﬁrm y’s production function. Pollution rights An innovation that would mitigate the informational requirements involved with Pigovian taxation is the creation of a market for ‘‘pollution rights.’’ Suppose, for example, that ﬁrm x must purchase from ﬁrm y rights to pollute the river they share. In this case, x’s decision to purchase these rights is identical to its decision to choose its output level, because it cannot produce without them. The net revenue x receives per unit is given by px ’ r, where r is the payment the ﬁrm must make for each unit it produces. Firm y must decide how many rights to sell to ﬁrm x. Because it will be paid r for each right, it must ‘‘choose’’ x output to maximize its proﬁts: xi, x0Þ þ ð the ﬁrst-order condition for a maximum is py ¼ pyg rx0; @py @x0 ¼ py g2 þ r ¼ 0 or r py g2: ¼ ’ (19:28) (19:29) Equation 19.29 makes clear that the equilibrium solution to pricing in the pollution rights market will be identical to the Pigovian tax equilibrium. From the point of view of ﬁrm x, it makes no difference whether a tax of amount t is paid to the government or a royalty r of the same amount is paid to ﬁrm y. So long as t r (a condition ensured by Equation 19.29), the same efﬁcient equilibrium will result. ¼ The Coase theorem In a famous 1960 paper, Ronald Coase showed that the key feature of the pollution rights equilibrium is that these rights be well deﬁned and tradable with zero transaction costs.6 The initial assignment of rights is irrelevant because subsequent trading will always yield the same efﬁcient equilibrium. In our example we initially assigned the rights to ﬁrm y, allowing that ﬁrm to trade them away to ﬁrm x for a per-unit fee r. If the rights had been assigned to ﬁrm x instead, that ﬁrm still would have to impute some cost to using these rights themselves rather than selling them to ﬁrm y. This calculation, in combination with ﬁrm y’s decision about how many such rights to buy, will again yield an efﬁcient result. 6R. Coase, ‘‘The Problem of Social Cost,’’ Journal of Law and Economics 3 (October 1960): 1–44. 694 Part 8: Market Failure To illustrate the Coase result, assume that ﬁrm x is given xT rights to produce (and to pollute). It can choose to use some of these to support its own production (x0), or it may sell some to ﬁrm y (an amount given by xT x0). Gross proﬁts for x are given by ’ x0 þ rxT px ’ r f Þ ð yiÞ þ ¼ ð rxT (19:30) px ¼ and for y by pxx0 þ xT r ð x0Þ ¼ ð px ’ r Þ ’ py ¼ py g xi, x0Þ ’ ð r xT ð : x0Þ ’ (19:31) Clearly, proﬁt maximization in this situation will lead to precisely the same solution as in the case where ﬁrm y was assigned the rights. Because the overall total number of rights (xT) is a constant, the ﬁrst-order conditions for a maximum will be exactly the same in the two cases. This independence of initial rights assignment is usually referred to as the Coase theorem. Although the results of the Coase theorem may seem counterintuitive (how can the level of pollution be independent of who initially owns the rights?), it is in reality nothing more than the assertion that, in the absence of impediments to making bargains, all mutually beneﬁcial transactions will be completed. When transaction costs are high or when information is asymmetric, initial rights assignments will matter because the sorts of trading implied by the Coase theorem may not occur. Therefore, it is the limitations of the Coase theorem that provide the most interesting opportunities for further analysis. This analysis has been especially far reaching in the ﬁeld of law and economics,7 where the theorem has been applied to such topics as tort liability laws, contract law, and product safety legislation (see Problem 19.4). Attributes of Public Goods We now turn our attention to a related set of problems about the relationship between competitive markets and the allocation of resources: those raised by the existence of public goods. We begin by providing a precise deﬁnition of this concept and then examine why such goods pose allocational problems. We then brieﬂy discuss theoretical ways in which such problems might be mitigated before turning to examine how actual decisions on public goods are made through voting. The most common deﬁnitions of public goods stress two attributes of such goods: nonexclusivity and nonrivalness. We now describe these attributes in detail. Nonexclusivity The ﬁrst property that distinguishes public goods concerns whether individuals may be excluded f
rom the beneﬁts of consuming the good. For most private goods such exclusion is indeed possible: I can easily be excluded from consuming a hamburger if I don’t pay for it. In some cases, however, such exclusion is either very costly or impossible. National defense is the standard example. Once a defense system is established, everyone in a country beneﬁts from it whether they pay for it or not. Similar comments apply, on a more local level, to goods such as mosquito control or a program to inoculate against disease. In these cases, once the programs are implemented, no one in the community can be excluded from those beneﬁts whether he or she pays for them or not. Hence we can divide goods into two categories according to the following deﬁnition. 7The classic text is R. A. Posner, Economic Analysis of Law, 4th ed. (Boston: Little, Brown, 1992). A more mathematical approach is T. J. Miceli, Economics of the Law (New York: Oxford University Press, 1997). Chapter 19: Externalities and Public Goods 695 Exclusive goods. A good is exclusive if it is relatively easy to exclude individuals from beneﬁting from the good once it is produced. A good is nonexclusive if it is impossible (or costly) to exclude individuals from beneﬁting from the good. Nonrivalry A second property that characterizes public goods is nonrivalry. A nonrival good is one for which additional units can be consumed at zero social marginal cost. For most goods, of course, consumption of additional amounts involves some marginal costs of production. Consumption of one more hot dog requires that various resources be devoted to its production. However, for certain goods this is not the case. Consider, for example, having one more automobile cross a highway bridge during an off-peak period. Because the bridge is already in place, having one more vehicle cross requires no additional resource use and does not reduce consumption elsewhere. Similarly, having one more viewer tune in to a television channel involves no additional cost, even though this action would result in additional consumption taking place. Therefore, we have developed the following deﬁnition Nonrival goods. A good is nonrival if consumption of additional units of the good involves zero social marginal costs of production. Typology of public goods The concepts of nonexclusion and nonrivalry are in some ways related. Many nonexclusive goods are also nonrival. National defense and mosquito control are two examples of goods for which exclusion is not possible and additional consumption takes place at zero marginal cost. Many other instances might be suggested. The concepts, however, are not identical: some goods may possess one property but not the other. For example, it is impossible (or at least very costly) to exclude some ﬁshing boats from ocean ﬁsheries, yet the arrival of another boat clearly imposes social costs in the form of a reduced catch for all concerned. Similarly, use of a bridge during off-peak hours may be nonrival, but it is possible to exclude potential users by erecting toll booths. Table 19.1 presents a cross-classiﬁcation of goods by their possibilities for exclusion and their rivalry. Several examples of goods that ﬁt into each of the categories are provided. Many of the examples, other than those in the upper left corner of the table (exclusive and rival private goods), are often produced by governments. That is especially the case for nonexclusive goods because, as we shall see, it is difﬁcult to develop ways of paying for such goods other than through compulsory taxation. Nonrival goods often are privately produced (there are, after all, private bridges, swimming pools, and highways that consumers must pay to use) as long as nonpayers can be excluded from consuming them.8 Still, we will use the following stringent deﬁnition, which requires both conditions. 8Nonrival goods that permit imposition of an exclusion mechanism are sometimes referred to as club goods, because provision of such goods might be organized along the lines of private clubs. Such clubs might then charge a ‘‘membership’’ fee and permit unlimited use by members. The optimal size of a club is determined by the economies of scale present in the production process for the club good. For an analysis, see R. Cornes and T. Sandler, The Theory of Externalities, Public Goods, and Club Goods (Cambridge: Cambridge University Press, 1986). 696 Part 8: Market Failure TABLE 19.1 EXAMPLES SHOWING THE TYPOLOGY OF PUBLIC AND PRIVATE GOODS Rival Yes No Yes Hot dogs, automobiles, houses Exclusive No Fishing grounds, public grazing land, clean air Bridges, swimming pools, satellite television transmission (scrambled) National defense, mosquito control, justice Public good. A good is a (pure) public good if, once produced, no one can be excluded from beneﬁting from its availability and if the good is nonrival—the marginal cost of an additional consumer is zero. Public Goods and Resource Allocation To illustrate the allocational problems created by public goods, we again employ a simple general equilibrium model. In this model there are only two individuals—a single-person economy would not experience problems from public goods because he or she would incorporate all of the goods’ beneﬁts into consumption decisions. We denote these two individuals by A and B. There are also only two goods in this economy. Good y is an ordinary private good, and each person begins with an allocation of this good given by y A $ and y B $, respectively. Each person may choose to consume some of his or her y directly or to devote some portion of it to the production of a single public good, x. The amounts contributed are given by y A s , and the public good is produced according to the production function s and y B Resulting utilities for these two people in this society are given by and U A x; yA $ ð ’ yA s Þ (19:32) (19:33) (19:34) U B x; yB ð $ yB s Þ ’ Notice here that the level of public good production, x, enters identically into each person’s utility function. This is the way in which the nonexclusivity and nonrivalry characteristics of such goods are captured mathematically. Nonexclusivity is reﬂected by the fact that each person’s consumption of x is the same and independent of what he or she contributes individually to its production. Nonrivalry is shown by the fact that the consumption of x by each person is identical to the total amount of x produced. Consumption of x beneﬁts by A does not diminish what B can consume. These two characteristics of good x constitute the barriers to efﬁcient production under most decentralized decision schemes, including competitive markets. The necessary conditions for efﬁcient resource allocation in this problem consist of s ) that maximize, say, A’s choosing the levels of public goods subscriptions (y A utility for any given level of B’s utility. The Lagrangian expression for this problem is s and y B + U A x19:35) Chapter 19: Externalities and Public Goods 697 where K is a constant level of B’s utility. The ﬁrst-order conditions for a maximum are @+ @yA ’ kU B 1 f 0 0, ¼ @+ @yB s ¼ U A 1 f 0 kU B 2 þ ’ kU B 1 f 0 0: ¼ A comparison of these two equations yields the immediate result that kU B 2 ¼ U A 2 : (19:36) (19:37) (19:38) As might have been expected here, optimality requires that the marginal utility of y consumption for A and B be equal except for the constant of proportionality, l. This equation may now be combined with either Equation 19.36 or 19.37 to derive the optimality condition for producing the public good x. Using Equation 19.36, for example, gives or, more simply, U A 1 U A 2 þ kU B 1 kU B 2 ¼ 1 f 0 MRSA MRSB þ 1 f 0 : ¼ (19:39) (19:40) The intuition behind this condition, which was ﬁrst articulated by P. A. Samuelson,9 is that it is an adaptation of the efﬁciency conditions described in Chapter 13 to the case of public goods. For such goods, the MRS in consumption must reﬂect the amount of y that all consumers would be willing to give up to get one more x, because everyone will obtain the beneﬁts of the extra x output. Hence it is the sum of each individual’s MRS that should be equated to dy/dx in production (here given by 1/f 0). Failure of a competitive market Production of goods x and y in competitive markets will fail to achieve this allocational goal. With perfectly competitive prices px and py, each individual will equate his or her MRS to the price ratio px /py. A producer of good x would also set 1/f 0 to be equal to px /py, as would be required for proﬁt maximization. This behavior would not achieve the optimality condition expressed in Equation 19.40. The price ratio px /py would be ‘‘too low’’ in that it would provide too little incentive to produce good x. In the private market, a consumer takes no account of how his or her spending on the public good beneﬁts others, so that consumer will devote too few resources to such production. The allocational failure in this situation can be ascribed to the way in which private markets sum individual demands. For any given quantity, the market demand curve reports the marginal valuation of a good. If one more unit were produced, it could then be consumed by someone who would value it at this market price. For public goods, the value of producing one more unit is in fact the sum of each consumer’s valuation of that extra output, because all consumers will beneﬁt from it. In this case, then, individual demand curves should be added vertically (as shown in Figure 19.2) rather than horizontally (as they are in competitive markets). The resulting price on such a public good 9P. A. Samuelson, ‘‘The Pure Theory of Public Expenditure,’’ Review of Economics and Statistics (November 1954): 387–89. 698 Part 8: Market Failure FIGURE 19.2 Derivation of the Demand for a Public Good For a public good, the price individuals are willing to pay for one more unit (their ‘‘marginal valuations’’) is equal to the sum of what each individual would p
ay. Hence, for public goods, the demand curve must be derived by a vertical summation rather than the horizontal summation used in the case of private goods. Price D1 + D2 + D3 = D 3 2 1 3 2 D D3 D2 D1 Quantity per period demand curve will then reﬂect, for any level of output, how much an extra unit of output would be valued by all consumers. But the usual market demand curve will not properly reﬂect this full marginal valuation. Inefﬁciency of a Nash equilibrium An alternative approach to the production of public goods in competitive markets might rely on individuals’ voluntary contributions. Unfortunately, this also will yield inefﬁcient results. Consider the situation of person A, who is thinking about contributing sA of his or her initial y endowment to public goods production. The utility maximization problem for A is then choose sA to maximize U A[ f sB), y A $ sA þ ð sA]. ’ The ﬁrst-order condition for a maximum is ’ or U A 1 U A 2 ¼ MRSA 1 f 0 : ¼ (19:41) (19:42) Because a similar logic will apply to person B, the efﬁciency condition of Equation 19.40 will once more fail to be satisﬁed. Again the problem is that each person considers only his or her beneﬁt from investing in the public good, taking no account of the beneﬁts provided to others. With many consumers, this direct beneﬁt may be very small indeed. (For example, how much do one person’s taxes contribute to national defense in the United States?) In this case, any one person may opt for sA ¼ 0 and become a pure ‘‘free rider,’’ hoping to beneﬁt from the expenditures of others. If every person adopts this strategy, then no resources will be subscribed to public goods. Example 19.3 illustrates the free-rider problem in a situation that may be all too familiar. Chapter 19: Externalities and Public Goods 699 EXAMPLE 19.3 Purchasing a Public Good: The Roommates’ Dilemma To illustrate the nature of the public good problem numerically, suppose two roommates with identical preferences derive utility from the number of music compact disks (CDs, denoted by x) in their shared music collection and on the number of granola bars ( y) eaten. The speciﬁc utility function for i 1, 2 is given by ¼ Uið x, yiÞ ¼ x1=2y1=2 i : (19:43) ¼ x1 þ Utility for each roommate depends on the total number of CDs (x x2) in their collection but only on the number of granola bars eaten by the individual. Hence in this problem a CD is a public good and a granola bar is a private good. (We could justify the classiﬁcation of CDs as a public good by assuming that the purchaser of the CD cannot exclude his or her roommate from borrowing and playing it on their shared sound system. Playing the CD once does not diminish its value when played again, so there is nonrivalry in CD consumption.) Assume each roommate has $300 to spend and that px ¼ Nash equilibrium. We ﬁrst consider the outcome if the roommates make their consumption decisions independently without coming to a more or less formal agreement about how many CDs to buy. Roommate 1’s decision depends on how many CDs roommate 2 buys and vice versa. We are in a strategic situation for which we need the tools of game theory from Chapter 8 to analyze. We will look for the Nash equilibrium, in which both roommates are playing a best response. $10 and py ¼ $1. roommate 2. Roommate 1 maximizes utility To ﬁnd roommate 1’s best response, take as given the number x2 of CDs purchased by x1 þ x2Þ 10x1 þ ¼ subject to the budget constraint 1=2y1=2 i (19:44) 300 y1, ð leading to the Lagrangian The ﬁrst-order conditions with respect to roommate 1 choice variables are + x1 þ ¼ ð x2Þ 1=2y1=2 i þ k 300 ð ’ 10x1 ’ : y1Þ @+ @x1 ¼ @+ @y1 ¼ 1 2 ð 1 2 ð x1 þ x1 þ x2Þ 1=2 y1=2 ’ i ’ 10k 1=2 1=2 y’ i x2Þ k ’ ¼ 0 ¼ 0: (19:45) (19:46) Solving Equations 19.46 in the usual way gives y1 ¼ x1 þ ð which, when substituted into 1’s budget constraint and rearranged, gives the best-response function , x2Þ (19:47) 10 x1 ¼ 15 ’ x2 2 : (19:48) Because the problem is symmetric, roommate 2’s best-response function will have the same form: x2 ¼ 15 ’ x1 2 : (19:49) These best-response functions reﬂect a free-rider problem in that the more CDs one roommate is expected to purchase, the fewer CDs the other wants to buy. Solving Equations 19.48 and 19.49 simultaneously gives x$1 ¼ y$2 ¼ into Equation 19.47 gives y$1 ¼ 200. Nash equilibrium utilities are U $1 ¼ x$2 ¼ U $2 + 63:2. 10, and substituting this 700 Part 8: Market Failure Efﬁcient allocation. We saw that the efﬁcient level of a public good can be calculated by setting the sum of each person’s MRS equal to the good’s price ratio. In this example, the MRS for roommate i is MRSi ¼ @Ui=@x @Ui=@yi ¼ yi x : Hence the condition for efﬁciency is MRS1 þ MRS2 ¼ y1 x þ y2 x ¼ px py ¼ 10 1 : Consequently, which can be substituted into the combined budget constraint y1 þ y2 ¼ 10x, 600 10x y1 þ y2 þ ¼ (19:50) (19:51) (19:52) (19:53) ¼ 67:1. U $$2 + 30 and y$$1 þ to obtain x$$ 300 (double stars distinguish efﬁcient values from the Nash y$$2 ¼ equilibrium ones with single stars). Assuming each roommate eats half (150) of the granola bars, the resulting utilities are U $$1 ¼ Comparison. In the Nash equilibrium, too little of the public good (CDs) is purchased. The most efﬁcient outcome has them purchasing ﬁve more CDs than they would on their own. It might be possible for them to come to a formal or informal agreement to buy more CDs, perhaps putting money in a pool and purchasing them together; the utility of both could simultaneously be increased this way. In the absence of such an agreement, the roommates face a similar dilemma as the players in the Prisoners’ Dilemma: the Nash equilibrium (both ﬁnk) is Pareto dominated by another outcome (their utility is higher if both are silent). QUERY: Solve the problem for three roommates. In what sense has the public good problem become worse with more players? How would an increase in the number of roommates affect their ability to enforce a cooperative agreement to buy more CDs? Lindahl Pricing of Public Goods An important conceptual solution to the public goods problem was ﬁrst suggested by the Swedish economist Erik Lindahl10 in the 1920s. Lindahl’s basic insight was that individuals might voluntarily consent to be taxed for beneﬁcial public goods if they knew that others were also being taxed. Speciﬁcally, Lindahl assumed that each individual would be presented by the government with the proportion of a public good’s cost he or she would be expected to pay and then reply (honestly) with the level of public good output he or she would prefer. In the notation of our simple general equilibrium model, individual A would be quoted a speciﬁc percentage (aA) and then asked the level of public goods that he or she would want given the knowledge that this fraction of total cost would have to be paid. To answer that question (truthfully), this person would choose that overall level of public goods output, x, that maximizes utility ¼ U A[x, y A $ 1 aAf ’ x)]. ð ’ (19:54) 10Excerpts from Lindahl’s writings are contained in R. A. Musgrave and A. T. Peacock, Eds., Classics in the Theory of Public Finance (London: Macmillan, 1958). Chapter 19: Externalities and Public Goods 701 The ﬁrst-order condition for this utility-maximizing choice of x is given by U A 1 ’ aU B 2 1 f 0! " ¼ 0 or MRSA aA f 0 : ¼ (19:55) Individual B, presented with a similar choice, would opt for a level of public goods satisfying : (19:56) aB f 0 An equilibrium would then occur where aA MRSB ¼ aB 1—that is, where the level of public goods expenditure favored by the two individuals precisely generates enough in tax contributions to pay for it. For in that case ¼ þ MRSA MRSB þ aA aB 19:57) and this equilibrium would be efﬁcient (see Equation 19.40). Hence, at least on a conceptual level, the Lindahl approach solves the public good problem. Presenting each person with the equilibrium tax share ‘‘price’’ will lead him or her to opt for the efﬁcient level of public goods production. EXAMPLE 19.4 Lindahl Solution for the Roommates y$$2 ¼ Lindahl pricing provides a conceptual solution to the roommates’ problem of buying CDs in Example 19.3. If ‘‘the government’’ (or perhaps social convention) suggests that each roommate will pay half of CD purchases, then each would face an effective price of CDs of $5. Since the utility functions for the roommates imply that half of each person’s total income of $300 will be spent on CDs, it follows that each will be willing to spend $150 on such music and will, if each is honest, report that he or she would like to have 15 CDs. Hence the solution will be x$$ 30 and y$$1 ¼ 150. This is indeed the efﬁcient solution calculated in Example 19.3. This solution works if the government knows enough about the roommates’ preferences that it can set the payment shares in advance and stick to them. Knowing that the roommates have symmetric preferences in this example, it could set equal payment shares a1 ¼ a2 ¼ 1=2 , and rest assured that both will honestly report the same demands for the public good, x$$ 30. If, however, the government does not know their preferences, it would have to tweak the payment shares based on their reports to make sure the reported demands end up being equal as required for the Lindahl solution to be ‘‘in equilibrium.’’ Anticipating the effect of their reports on their payment shares, the roommates would have an incentive to underreport demand. In fact, this underreporting would lead to the same outcome as in the Nash equilibrium from Example 19.3. ¼ ¼ QUERY: Although the 50–50 sharing in this example might arise from social custom, in fact the optimality of such a split is a special feature of this problem. What is it about this problem that leads to such a Lindahl outcome? Under what conditions would Lindahl prices result in other than a 50–50 sharing? Shortcomings of the Lindahl solution Unfortunately, Lindahl’s solution is only a conceptual one. We have already seen in our examinatio
n of the Nash equilibrium for public goods production and in our roommates’ example that the incentive to be a free rider in the public goods case is very strong. This fact makes it difﬁcult to envision how the information necessary to compute equilibrium 702 Part 8: Market Failure Lindahl shares might be obtained. Because individuals know their tax shares will be based on their reported demands for public goods, they have a clear incentive to understate their true preferences—in so doing they hope that the ‘‘other guy’’ will pay. Hence, simply asking people about their demands for public goods should not be expected to reveal their true demands. We will discuss more sophisticated mechanisms for eliciting honest demand reports at the end of the chapter. Local public goods Some economists believe that demand revelation for public goods may be more tractable at the local level.11 Because there are many communities in which individuals might reside, they can indicate their preferences for public goods (i.e., for their willingness to pay Lindahl tax shares) by choosing where to live. If a particular tax burden is not utility maximizing then people can, in principle, ‘‘vote with their feet’’ and move to a community that does provide optimality. Hence, with perfect information, zero costs of mobility, and enough communities, the Lindahl solution may be implemented at the local level. Similar arguments apply to other types of organizations (such as private clubs) that provide public goods to their members; given a sufﬁciently wide spectrum of club offerings, an efﬁcient equilibrium might result. Of course, the assumptions that underlie the purported efﬁciency of such choices by individuals are quite strict. Even minor relaxation of these assumptions may yield inefﬁcient results owing to the fragile nature of the way in which the demand for public goods is revealed. EXAMPLE 19.5 The Relationship between Environmental Externalities and Public Goods Production In recent years, economists have begun to study the relationship between the two issues we have been discussing in this chapter: externalities and public goods. The basic insight from this examination is that one must take a general equilibrium view of these problems in order to identify solutions that are efﬁcient overall. Here we illustrate this point by returning to the computable general equilibrium model ﬁrms described in Chapter 13 (see Example 13.4). To simplify matters we will now assume that this economy includes only a single representative person whose utility function is given by utility U ð ¼ x, y, l, g, c Þ ¼ x0:5y0:3l 0:2g0:1c0:2, (19:58) where we have added terms for the utility provided by public goods ( g), which are initially ﬁnanced by a tax on labor, and by clean air (c). Production of the public good requires capital k0.5 l 0.5; there is an externality in the and labor input according to the production function g 0.2y. The production production of good y, so that the quantity of clean air is given by c functions for goods x and y remain as described in Example 13.4, as do the endowments of k and l . Hence our goal is to allocate resources in such a way that utility is maximized. 10 ¼ ’ ¼ Base case: Optimal public goods production with no Pigovian tax. If no attempt is level of public goods made to control the externality in this problem, then the optimal 2.93 and this is ﬁnanced by a tax rate of 0.25 on labor. Output of good production requires g y in this case is 29.7, and the quantity of clean air is given by c 4.06. Overall utility in this situation is U 19.34. This is the highest utility that can be obtained in this situation without regulating the externality. 5.94 10 ¼ ¼ ’ ¼ ¼ 11The classic reference is C. M. Tiebout, ‘‘A Pure Theory of Local Expenditures,’’ Journal of Political Economy (October 1956): 416–24. Chapter 19: Externalities and Public Goods 703 A Pigovian tax. As suggested by Figure 19.1, a unit tax on the production of good y may improve matters in this situation. With a tax rate of 0.1, for example, output of good y is 4.52), and the revenue generated is used to expand public reduced to y 27.4 (c 19.38. By carefully specifying how the goods production to g revenue generated by the Pigovian tax is used, a general equilibrium model permits a more complete statement of welfare effects. 10 3.77. Utility is increased to U ¼ ¼ 5.48 ’ ¼ ¼ ¼ The ‘‘double dividend’’ of environmental taxes. The solution just described is not optimal, however. Production of public goods is actually too high in this case, since the revenues from environmental taxes are also used to pay for public goods. In fact, simulations show that optimality can be achieved by reducing the labor tax to 0.20 and public goods production to g 19.43. This result is sometimes referred to as the ‘‘double dividend’’ of environmental taxation: not only do these taxes reduce externalities relative to the untaxed situation (now c 4.40), but also the extra governmental revenue made available thereby may permit the reduction of other distorting taxes. 3.31. With these changes, utility expands even further to U 5.60 10 ¼ ¼ ¼ ¼ ’ QUERY: Why does the quantity of clean air decrease slightly when the labor tax is reduced relative to the situation where it is maintained at 0.25? More generally, describe whether environmental taxes would be expected always to generate a double dividend. Voting and Resource Allocation Voting is used as a social decision process in many institutions. In some instances, individuals vote directly on policy questions. That is the case in some New England town meetings, many statewide referenda (for example, California’s Proposition 13 in 1977), and for many of the national policies adopted in Switzerland. Direct voting also characterizes the social decision procedure used for many smaller groups and clubs such as farmers’ cooperatives, university faculties, or the local Rotary Club. In other cases, however, societies have found it more convenient to use a representative form of government, in which individuals vote directly only for political representatives, who are then charged with making decisions on policy questions. For our study of public choice theory, we will begin with an analysis of direct voting. This is an important subject not only because such a procedure applies to many cases but also because elected representatives often engage in direct voting (in Congress, for example), and the theory we will illustrate applies to those instances as well. Majority rule Because so many elections are conducted on a majority rule basis, we often tend to regard that procedure as a natural and, perhaps, optimal one for making social choices. But even a cursory examination indicates that there is nothing particularly sacred about a rule requiring that a policy obtain 50 percent of the vote to be adopted. In the U.S. Constitution, for example, two thirds of the states must adopt an amendment before it becomes law. And 60 percent of the U.S. Senate must vote to limit debate on controversial issues. Indeed, in some institutions (Quaker meetings, for example), unanimity may be required for social decisions. Our discussion of the Lindahl equilibrium concept suggests there may exist a distribution of tax shares that would obtain unanimous support in voting for public goods. But arriving at such unanimous agreements is usually thwarted by emergence of the free-rider problem. Examining in detail the forces that lead societies to move 704 Part 8: Market Failure TABLE 19.2 PREFERENCES THAT PRODUCE THE PARADOX OF VOTING Choices: A—Low Spending B—Medium Spending C—High Spending Preferences Smith A B C Jones B C A Fudd C A B away from unanimity and to choose some other determining fraction would take us too far aﬁeld here. We instead will assume throughout our discussion of voting that decisions will be made by majority rule. Readers may wish to ponder for themselves what kinds of situations might call for a decisive proportion of other than 50 percent. The paradox of voting In the 1780s, the French social theorist M. de Condorcet observed an important peculiarity of majority rule voting systems—they may not arrive at an equilibrium but instead may cycle among alternative options. Condorcet’s paradox is illustrated for a simple case in Table 19.2. Suppose there are three voters (Smith, Jones, and Fudd) choosing among three policy options. For our subsequent analysis we will assume the policy options represent three levels of spending (A low, B medium, or C high) on a particular public good, but Condorcet’s paradox would arise even if the options being considered did not have this type of ordering associated with them. Preferences of Smith, Jones, and Fudd among the three policy options are indicated in Table 19.2. These preferences give rise to Condorcet’s paradox. Consider a vote between options A and B. Here option A would win, because it is favored by Smith and Fudd and opposed only by Jones. In a vote between options A and C, option C would win, again by 2 votes to 1. But in a vote of C versus B, B would win and we would be back where we started. Social choices would endlessly cycle among the three alternatives. In subsequent votes, any choice initially decided upon could be defeated by an alternative, and no equilibrium would ever be reached. In this situation, the option ﬁnally chosen will depend on such seemingly nongermane issues as when the balloting stops or how items are ordered on an agenda—rather than being derived in some rational way from the preferences of voters. Single-peaked preferences and the median voter theorem Condorcet’s voting paradox arises because there is a degree of irreconcilability in the preferences of voters. Therefore, one might ask whether restrictions on the types of preferences allowed could yield situations where equilibrium voting outcomes are more likely. A fundamental result about this probability was discovered by Duncan Black i
n 1948.12 Black showed that equilibrium voting outcomes always occur in cases where the issue being voted upon is one-dimensional (such as how much to spend on a public good) and where voters’ preferences are ‘‘single peaked.’’ To understand what the notion of single peaked means, consider again Condorcet’s paradox. In Figure 19.3 we illustrate the 12D. Black, ‘‘On the Rationale of Group Decision Making,’’ Journal of Political Economy (February 1948): 23–34. Chapter 19: Externalities and Public Goods 705 FIGURE 19.3 Single-Peaked Preferences and the Median Voter Theorem This ﬁgure illustrates the preferences in Table 19.2. Smith’s and Jones’s preferences are single peaked, but Fudd’s have two local peaks and these yield the voting paradox. If Fudd’s preferences had instead been single peaked (the dashed line), then option B would have been chosen as the preferred choice of the median voter (Jones). Utility Fudd Fudd (alternate) Jones Smith A B C Quantity of public good preferences that gave rise to the paradox by assigning hypothetical utility levels to options A, B, and C that are consistent with the preferences recorded in Table 19.2. For Smith and Jones, preferences are single peaked: as levels of public goods expenditures increase, there is only one local utility-maximizing choice (A for Smith, B for Jones). Fudd’s preferences, on the other hand, have two local maxima (A and C). It is these preferences that produced the cyclical voting pattern. If instead Fudd had the preferences represented by the dashed line in Figure 19.3 (where now C is the only local utility maximum), then there would be no paradox. In this case, option B would be chosen because that option would defeat both A and C by votes of 2 to 1. Here B is the preferred choice of the ‘‘median’’ voter (Jones), whose preferences are ‘‘between’’ the preferences of Smith and the revised preferences of Fudd. Black’s result is quite general and applies to any number of voters. If choices are unidimensional13 and if preferences are single peaked, then majority rule will result in the selection of the project that is most favored by the median voter. Hence, that voter’s preferences will determine what public choices are made. This result is a key starting point for many models of the political process. In such models, the median voter’s preferences dictate policy choices—either because that voter determines which policy gets a majority of votes in a direct election or because the median voter will dictate choices in competitive elections in which candidates must adopt policies that appeal to this voter. A Simple Political Model To illustrate how the median voter theorem is applied in political models, suppose a community is characterized by a large number (n) of voters each with an income given by yi. 13The result can be generalized a bit to deal with multidimensional policies if individuals can be characterized in their support for such policies along a single dimension. 706 Part 8: Market Failure The utility of each voter depends on his or her consumption of a private good (ci) and of a public good ( g) according to the additive utility function utility of person i U i ¼ ci þ f g , Þ ð ¼ (19:59) where fg > 0 and fgg < 0. Each voter must pay income taxes to ﬁnance g. Taxes are proportional to income and are imposed at a rate t. Therefore, each person’s budget constraint is given by ci ¼ ð yi: Þ ’ 1 t The government is also bound by a budget constraint: n where y A denotes average income for all voters. g ¼ tny A, tyi ¼ 1 X (19:60) (19:61) Given these constraints, the utility of person i can be written as a function of his or her choice of g only: g U ið 19:62) Utility maximization for person i shows that his or her preferred level of expenditures on the public good satisﬁes dU i dg ¼ ’ yi ny A þ g f gð Þ ¼ 0 or g 1 f ’ g ¼ yi ny A ! : " (19:63) This shows that desired spending on g is inversely related to income. Because (in this model) the beneﬁts of g are independent of income but taxes increase with income, highincome voters can expect to have smaller net gains (or even losses) from public spending than can low-income voters. The median voter equilibrium If g is determined here through majority rule, its level will be chosen to be that level favored by the ‘‘median voter.’’ In this case, voters’ preferences align exactly with incomes, so g will be set at that level preferred by the voter with median income (y m). Any other level for g would not get 50 percent of the vote. Hence, equilibrium g is given by g$ ¼ 1 f ’ g y m ny 19:64) In general, the distribution of income is skewed to the right in practically every political jurisdiction in the world. With such an income distribution, ym < y A, and the difference between the two measures becomes larger the more skewed is the income distribution. Hence Equation 19.64 suggests that, ceteris paribus, the more unequal is the income distribution in a democracy, the higher will be tax rates and the greater will be spending on public goods. Similarly, laws that extend the vote to increasingly poor segments of the population can also be expected to increase such spending. Optimality of the median voter result Although the median voter theorem permits a number of interesting positive predictions about the outcome of voting, the normative signiﬁcance of these results is more difﬁcult Chapter 19: Externalities and Public Goods 707 to pinpoint. In this example, it is clear that the result does not replicate the Lindahl voluntary equilibrium—high-income voters would not voluntarily agree to the taxes imposed.14 The result also does not necessarily correspond to any simple criterion for social welfare. For example, under a ‘‘utilitarian’’ social welfare criterion, g would be chosen so as to maximize the sum of utilities: n n SW ’ $ yi y A þ f g ð ¼ Þ & ny A g nf g : Þ ð þ ’ (19:65) The optimal choice for g is then found by differentiation: dSW dg ¼ ’ 1 nf g ¼ þ 0, or g19:66) which shows that a utilitarian choice would opt for the level of g favored by the voter with average income. That output of g would be smaller than that favored by the median voter because y m < y A. In Example 19.6 we take this analysis a bit further by showing how it might apply to governmental transfer policy. EXAMPLE 19.6 Voting for Redistributive Taxation Suppose voters were considering adoption of a lump-sum transfer to be paid to every person and ﬁnanced through proportional taxation. If we denote the per-person transfer by b, then each individual’s utility is now given by and the government budget constraint is U i ¼ ci þ b nb ¼ tny A or b ty A: ¼ (19:67) (19:68) ¼ For a voter whose income is greater than average, utility would be maximized by choosing b 0, because such a voter would pay more in taxes than he or she would receive from the transfer. Any voter with less than average income will gain from the transfer no matter what the y A. tax rate is. Hence such voters (including the decisive median voter) will opt for t That is, they would vote to fully equalize incomes through the tax system. Of course, such a tax scheme is unrealistic—primarily because a 100 percent tax rate would undoubtedly create negative work incentives that reduce average income. 1 and b ¼ ¼ To capture such incentive effects, assume15 that each person’s income has two components, one responsive to tax rates [ yi(t)] and one not responsive (ni). Assume also that the average value of ni is 0 but that its distribution is skewed to the right, so nm < 0. Now utility is given by t yið U i ¼ ð 1 ni( þ (19:69) Þ þ Þ ½ ’ b: t 14Although they might if the beneﬁts of g were also proportional to income. 15What follows represents a much simpliﬁed version of a model ﬁrst developed by T. Romer in ‘‘Individual Welfare, Majority Voting, and the Properties of a Linear Income Tax,’’ Journal of Public Economics (December 1978): 163–68. 708 Part 8: Market Failure Assuming that each person ﬁrst optimizes over those variables (such as labor supply) that affect yi (t), the ﬁrst-order condition16 for a maximum in his or her political decisions about t and b then becomes (using the government budget constraint in Equation 19.68) dU i dt ¼ ’ ni þ dy A dt ¼ t 0: Hence for voter i the optimal redistributive tax rate is given by ti ¼ ni dy A=dt : (19:70) (19:71) Assuming political competition under majority rule voting will opt for that policy favored by the median voter, the equilibrium rate of taxation will be nm dy A=dt (19:72) ¼ t$ : Because both nm and dy A/dt are negative, this rate of taxation will be positive. The optimal tax will be greater the farther nm is from its average value (i.e., the more unequally income is distributed). Similarly, the larger are distortionary effects from the tax, the smaller the optimal tax. This model then poses some rather strong testable hypotheses about redistribution in the real world. QUERY: Would progressive taxation be more likely to raise or lower t$ in this model? Voting Mechanisms The problems involved in majority rule voting arise in part because such voting is simply not informative enough to provide accurate appraisals of how people value public goods. This situation is in some ways similar to some of the models of asymmetric information examined in the previous chapter. Here voters are more informed than is the government about the value they place on various tax-spending packages. Resource allocation would be improved if mechanisms could be developed that encourage people to be more accurate in what they reveal about these values. In this section we examine two such mechanisms. Both are based on the basic insight from Vickrey second-price auctions (see Chapter 18) that incorporating information about other bidders’ valuations into decisionmakers’ calculations can yield a greater likelihood of revealing truthful valuations. The Groves mechanism In a 1973 paper, T. Groves proposed a way to incorporate the Vickrey insight into a method for en
couraging people to reveal their demands for a public good.17 To illustrate this mechanism, suppose that there are n individuals in a group and each has a private (and unobservable) net valuation vi for a proposed taxation–expenditure project. In seeking information about these valuations, the government states that, should the project be undertaken, each person will receive a transfer given by ti ¼ i j X 6¼ vj e (19:73) 16Equation 19.70 can be derived from 19.69 through differentiation and by recognizing that dyi /dt tion of individual optimization. 17T. Groves, ‘‘Incentives in Teams,’’ Econometrica (July 1973): 617–31. ¼ 0 because of the assump- Chapter 19: Externalities and Public Goods 709 vj represents the valuation reported by person j and the summation is taken over where all individuals other than person i. If the project is not undertaken, then no transfers are made. Given this setup, the problem for voter i is to choose his or her reported net valuation e so as to maximize utility, which is given by utility vi þ ¼ ti ¼ vi þ vj: (19:74) Since the project will be undertaken only if project to be undertaken only if it increases utility (i.e., vi þ utility-maximizing strategy is to set vi ¼ each person to be truthful in his or her reporting of valuations for the project. e i j X 6¼ vi and since each person will wish the vj > 0), it follows that a i j 6¼ vi. Hence, the Groves mechanism encourages e P P 1 ¼ e e n i The Clarke mechanism A similar mechanism was proposed by E. Clarke, also in the early 1970s.18 This mechanism also envisions asking individuals about their net valuations for some public project, but it focuses mainly on ‘‘pivotal voters’’—those whose reported valuations can change the overall evaluation from negative to positive or vice versa. For all other voters, there are no special transfers, on the presumption that reporting a nonpivotal valuation will not change either the decision or the (zero) payment, so he or she might as well report truthfully. For voters reporting pivotal valuations, however, the Clarke mechanism incorporates a Pigovian-like tax (or transfer) to encourage truth telling. To see how this works, suppose that the net valuations reported by all other voters are negvj < 0 , but that a truthful statement of the valuation by person i would ative Þ ð . Here, as for the Groves mechanism, a make the project acceptable Þ P transfer of ti ¼ vj (which in this case would be negative—i.e., a tax) would ene courage this pivotal voter to report ~vi ¼ vi. Similarly, if all other individuals reported e i ~vj > 0 but inclusion of person i’s evaluation of valuations favorable to a project e j Þ ð 6¼ the project would make it unfavorable, then a transfer of ti ¼ vj (which in this i j 6¼ vi also. Overall, then, case is positive) would encourage this pivotal voter to choose vi ¼ P the Clarke mechanism is also truth revealing. Notice that in this case the transfers play much the same role that Pigovian taxes did in our examination of externalities. If other voters view a project as unfavorable, then voter i must compensate them for accepting it. On the other hand, if other voters ﬁnd the project acceptable, then voter i must be sufﬁciently against the project that he or she cannot be ‘‘bribed’’ by other voters into accepting it. vi þ ð vj > 0 i j 6¼ P P P i 6¼ j 6¼ e e i j Generalizations The voter mechanisms we have been describing are sometimes called VCG mechanisms after the three pioneering economists in this area of research (Vickrey, Clarke, and Groves). These mechanisms can be generalized to include multiple governmental projects, alternative concepts of voter equilibrium, or an inﬁnite number of voters. One assumption behind the mechanisms that does not seem amenable to generalization is the quasi-linear utility functions that we have been using throughout. Whether this assumption provides a good approximation for modeling political decision making remains an open question, however. 18E. Clarke, ‘‘Multipart Pricing for Public Goods,’’ Public Choice (Fall 1971): 19–33. 710 Part 8: Market Failure SUMMARY In this chapter we have examined market failures that arise from externality (or spillover) effects involved in the consumption or production of certain types of goods. In some cases it may be possible to design mechanisms to cope with these externalities in a market setting, but important limits are involved in such solutions. Some speciﬁc issues we examined were as follows. • Externalities may cause a misallocation of resources because of a divergence between private and social marginal cost. Traditional solutions to this divergence include mergers among the affected parties and adoption of suitable (Pigovian) taxes or subsidies. • If transaction costs are small, then private bargaining among the parties affected by an externality may bring social and private costs into line. The proof that resources will be efﬁciently allocated under such circumstances is sometimes called the Coase theorem. • Public goods provide beneﬁts to individuals on a nonexclusive basis—no one can be prevented from consuming such goods. Such goods are also usually PROBLEMS nonrival in that the marginal cost of serving another user is zero. • Private markets will tend to underallocate resources to public goods because no single buyer can appropriate all of the beneﬁts that such goods provide. • A Lindahl optimal tax-sharing scheme can result in an efﬁcient allocation of resources to the production of public goods. However, computing these tax shares requires substantial information that individuals have incentives to hide. • Majority rule voting does not necessarily lead to an efﬁcient allocation of resources to public goods. The median voter theorem provides a useful way of modeling the actual outcomes from majority rule in certain situations. • Several truth-revealing voting mechanisms have been developed. Whether these are robust to the special assumptions made or capable of practical application remain unresolved questions. 19.1 A ﬁrm in a perfectly competitive industry has patented a new process for making widgets. The new process lowers the ﬁrm’s average cost, meaning that this ﬁrm alone (although still a price taker) can earn real economic proﬁts in the long run. a. If the market price is $20 per widget and the ﬁrm’s marginal cost is given by MC production for the ﬁrm, how many widgets will the ﬁrm produce? 0.4q, where q is the daily widget ¼ b. Suppose a government study has found that the ﬁrm’s new process is polluting the air and estimates the social marginal cost of 0.5q. If the market price is still $20, what is the socially optimal level of production widget production by this ﬁrm to be SMC for the ﬁrm? What should be the rate of a government-imposed excise tax to bring about this optimal level of production? ¼ c. Graph your results. 19.2 On the island of Pago Pago there are 2 lakes and 20 anglers. Each angler can ﬁsh on either lake and keep the average catch on his particular lake. On Lake x, the total number of ﬁsh caught is given by where lx is the number of people ﬁshing on the lake. For Lake y, the relationship is Fy 5ly: ¼ Fx 10lx ’ ¼ 1 2 l2 x, a. Under this organization of society, what will be the total number of ﬁsh caught? b. The chief of Pago Pago, having once read an economics book, believes it is possible to increase the total number of ﬁsh caught by restricting the number of people allowed to ﬁsh on Lake x. What number should be allowed to ﬁsh on Lake x in order to maximize the total catch of ﬁsh? What is the number of ﬁsh caught in this situation? c. Being opposed to coercion, the chief decides to require a ﬁshing license for Lake x. If the licensing procedure is to bring about the optimal allocation of labor, what should the cost of a license be (in terms of ﬁsh)? d. Explain how this example sheds light on the connection between property rights and externalities. Chapter 19: Externalities and Public Goods 711 19.3 Suppose the oil industry in Utopia is perfectly competitive and that all ﬁrms draw oil from a single (and practically inexhaustible) pool. Assume that each competitor believes that it can sell all the oil it can produce at a stable world price of $10 per barrel and that the cost of operating a well for one year is $1,000. Total output per year (Q) of the oil ﬁeld is a function of the number of wells (n) operating in the ﬁeld. In particular, and the amount of oil produced by each well (q) is given by 500n Q ¼ ’ n2, Q n ¼ q ¼ 500 n: ’ (19:75) a. Describe the equilibrium output and the equilibrium number of wells in this perfectly competitive case. Is there a diver- gence between private and social marginal cost in the industry? b. Suppose now that the government nationalizes the oil ﬁeld. How many oil wells should it operate? What will total output be? What will the output per well be? c. As an alternative to nationalization, the Utopian government is considering an annual license fee per well to discourage overdrilling. How large should this license fee be if it is to prompt the industry to drill the optimal number of wells? 19.4 There is considerable legal controversy about product safety. Two extreme positions might be termed caveat emptor (let the buyer beware) and caveat vendor (let the seller beware). Under the former scheme producers would have no responsibility for the safety of their products: Buyers would absorb all losses. Under the latter scheme this liability assignment would be reversed: Firms would be completely responsible under law for losses incurred from unsafe products. Using simple supply and demand analysis, discuss how the assignment of such liability might affect the allocation of resources. Would safer products be produced if ﬁrms were strictly liable under law? How do possible information asymmetries affect your results? 19.5 Suppose a monopoly produces a harmful externality. Use the concept of consumer surplus in a partial equilibrium diagram to a
nalyze whether an optimal tax on the polluter would necessarily be a welfare improvement. 19.6 Suppose there are only two individuals in society. Person A’s demand curve for mosquito control is given by for person B, the demand curve for mosquito control is given by qn ¼ 100 p; ’ qb ¼ 200 p: ’ a. Suppose mosquito control is a pure public good; that is, once it is produced, everyone beneﬁts from it. What would be the optimal level of this activity if it could be produced at a constant marginal cost of $120 per unit? b. If mosquito control were left to the private market, how much might be produced? Does your answer depend on what each person assumes the other will do? c. If the government were to produce the optimal amount of mosquito control, how much will this cost? How should the tax bill for this amount be allocated between the individuals if they are to share it in proportion to beneﬁts received from mosquito control? 19.7 Suppose the production possibility frontier for an economy that produces one public good (y) and one private good (x) is given by 100y2 x2 þ ¼ 5,000: 712 Part 8: Market Failure This economy is populated by 100 identical individuals, each with a utility function of the form utility = xiyp , where xi is the individual’s share of private good production ( everyone beneﬁts equally from its level of production. ¼ x/100). Notice that the public good is nonexclusive and that ﬃﬃﬃﬃﬃﬃ a. If the market for x and y were perfectly competitive, what levels of those goods would be produced? What would the typical individual’s utility be in this situation? b. What are the optimal production levels for x and y? What would the typical individual’s utility level be? How should consumption of good x be taxed to achieve this result? Hint: The numbers in this problem do not come out evenly, and some approximations should sufﬁce. Analytical Problems 19.8 More on Lindahl equilibrium The analysis of public goods in this chapter exclusively used a model with only two individuals. The results are readily generalized to n persons—a generalization pursued in this problem. a. With n persons in an economy, what is the condition for efﬁcient production of a public good? Explain how the character- istics of the public good are reﬂected in these conditions. b. What is the Nash equilibrium in the provision of this public good to n persons? Explain why this equilibrium is inefﬁcient. Also explain why the underprovision of this public good is more severe than in the two-person cases studied in the chapter. c. How is the Lindahl solution generalized to n persons? Is the existence of a Lindahl equilibrium guaranteed in this more complex model? 19.9 Taxing pollution Suppose that there are n ﬁrms each producing the same good but with differing production functions. Output for each of these ﬁrms depends only on labor input, so the functions take the form qi ¼ fi (li). In its production activities each ﬁrm also produces some pollution, the amount of which is determined by a ﬁrm-speciﬁc function of labor input of the form gi (li). a. Suppose that the government wishes to place a cap of amount K on total pollution. What is the efﬁcient allocation of labor among ﬁrms? b. Will a uniform Pigovian tax on the output of each ﬁrm achieve the efﬁcient allocation described in part (a)? c. Suppose that, instead of taxing output, the Pigovian tax is applied to each unit of pollution. How should this tax be set? Will the tax yield the efﬁcient allocation described in part (a)? d. What are the implications of the problem for adopting pollution control strategies? (For more on this topic see the Exten- sions to this chapter.) 19.10 Vote trading Suppose there are three individuals in society trying to rank three social states (A, B, and C ). For each of the methods of social choice indicated, develop an example to show how the resulting social ranking of A, B, and C will be intransitive (as in the paradox of voting) or indeterminate. a. Majority rule without vote trading. b. Majority rule with vote trading. c. Point voting where each voter can give 1, 2, or 3 points to each alternative and the alternative with the highest point total is selected. 19.11 Public choice of unemployment beneﬁts Suppose individuals face a probability of u that they will be unemployed next year. If they are unemployed they will receive unemployment beneﬁts of b, whereas if they are employed they receive w (1 t), where t is the tax used to ﬁnance unemployment beneﬁts. Unemployment beneﬁts are constrained by the government budget constraint ub tw (1 u). ’ ¼ ’ Chapter 19: Externalities and Public Goods 713 a. Suppose the individual’s utility function is given by yiÞ d is the degree of constant relative risk aversion. What would be the utility-maximizing choices for b and t? ¼ ð U d=d, b. How would the utility-maximizing choices for b and t respond to changes in the probability of unemployment, u? c. How would b and t change in response to changes in the risk aversion parameter d? where 1 ’ 19.12 Probabilistic voting Probabilistic voting is a way of modeling the voting process that introduces continuity into individuals’ voting decisions. In this way, calculus-type derivations become possible. To take an especially simple form of this approach, suppose there are n voters and two candidates (labeled A and B) for elective ofﬁce. Each candidate proposes a platform that promises a net gain or loss to i and uB each voter. These platforms are denoted by uA 1, . . . , n. The probability that a given voter will vote for uB candidate A is given by pA , where f 0 > 0 > f 00. The probability that the voter will vote for candidate B is i Þ( pA pB i . i ¼ a. How should each candidate chose his or her platform so as to maximize the probability of winning the election subject to i , where i uA i Þ ’ Uið Uið i ¼ ’ ¼ 1 ½ f the constraint i uA i ¼ i uB i ¼ 0? (Do these constraints seem to apply to actual political candidates?) b. Will there exist a Nash equilibrium in platform strategies for the two candidates? c. Will the platform adopted by the candidates be socially optimal in the sense of maximizing a utilitarian social welfare? P P [Social welfare is given by SW i Uið uiÞ .] ¼ P SUGGESTIONS FOR FURTHER READING Alchian, A., and H. Demsetz. ‘‘Production, Information Costs, and Economic Organization.’’ American Economic Review 62 (December 1972): 777–95. Uses externality arguments to develop a theory of economic organizations. Barzel, Y. Economic Analysis of Property Rights. Cambridge: Cambridge University Press, 1989. Provides a graphical analysis of several economic questions that are illuminated through use of the property rights paradigm. Black, D. ‘‘On the Rationale of Group Decision Making.’’ Journal of Political Economy (February 1948): 23–34. Reprinted in K. J. Arrow and T. Scitovsky, Eds., Readings in Welfare Economics. Homewood, IL: Richard D. Irwin, 1969. Early development of the median voter theorem. Buchanan, J. M., and G. Tullock. The Calculus of Consent. Ann Arbor: University of Michigan Press, 1962. Classic analysis of the properties of various voting schemes. Cheung, S. N. S. ‘‘The Fable of the Bees: An Economic Investigation.’’ Journal of Law and Economics 16 (April 1973): 11–33. Empirical study of how the famous bee-orchard owner externality is handled by private markets in the state of Washington. Coase, R. H. ‘‘The Market for Goods and the Market for Ideas.’’ American Economic Review 64 (May 1974): 384– 91. Speculative article about notions of externalities and regulation in the ‘‘marketplace of ideas.’’ ———. ‘‘The Problem of Social Cost.’’ Journal of Law and Economics 3 (October 1960): 1–44. Classic article on externalities. Many fascinating historical legal cases. Cornes, R., and T. Sandler. The Theory of Externalities, Public Goods, and Club Goods. Cambridge: Cambridge University Press, 1986. Good theoretical analysis of many of the issues raised in this chapter. Good discussions of the connections between returns to scale, excludability, and club goods. Demsetz, H. ‘‘Toward a Theory of Property Rights.’’ American Economic Review, Papers and Proceedings 57 (May 1967): 347–59. Brief development of a plausible theory of how societies come to deﬁne property rights. Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. New York: Oxford University Press, 1995. Chapter 11 covers much of the same ground as this chapter does, though at a somewhat more abstract level. Olson, M. The Logic of Collective Action. Cambridge, MA: Harvard University Press, 1965. Analyzes the effects of individual incentives on the willingness to undertake collective action. Many fascinating examples. Persson, T., and G. Tabellini. Political Economics: Explaining Economic Policy. Cambridge, MA: MIT Press, 2000. A complete summary of recent models of political choices. Covers voting models and issues of institutional frameworks. Posner, R. A. Economic Analysis of Law, 5th ed. Boston:Little, Brown, 1998. In many respects the ‘‘bible’’ of the law and economics movement. Posner’s arguments are not always economically correct but are unfailingly interesting and provocative. Samuelson, P. A. ‘‘The Pure Theory of Public Expenditures.’’ Review of Economics and Statistics 36 (November 1954): 387–89. Classic statement of the efﬁciency conditions for public goods production. EXTENSIONS POLLUTION ABATEMENT Although our discussion of externalities focused on how Pigovian taxes can make goods’ markets operate more efﬁciently, similar results also apply to the study of the technology of pollution abatement. In these Extensions we brieﬂy review this alternative approach. We assume there are only two ﬁrms, A and B, and that their output levels (qA and qB , respectively) are ﬁxed throughout our discussion. It is an inescapable scientiﬁc principle that production of physical goods (as opposed to services) must obey the conservation of matter. Hence production of qA and qB is certain to involve some e
mission by-products, eA and eB. The physical amounts of these emissions (or at least their harmful components) can be abated using inputs zA and zB (which cost p per unit). The resulting levels of emissions are given by f A qA, zAÞ ¼ ð eA and f B qB, zBÞ ¼ ð eB, (i) where, for each ﬁrm’s abatement function, f1 > 0 and f2 < 0. E19.1 Optimal abatement If a regulatory agency has decided that e$ represents the maximum allowable level of emissions from these ﬁrms, then this level would be achieved at minimal cost by solving the Lagrangian expression + pzA þ First-order conditions for a minimum are pzB $ : Þ p þ Hence we have kf A 2 ¼ 0 and p + kf B 2 ¼ 0: (ii) (iii) (iv) k p=f A p=f B 2 : ¼ ’ 2 ¼ ’ This equation makes the rather obvious point that costminimizing abatement is achieved when the marginal cost of abatement (universally referred to as MAC in the environmental literature) is the same for each ﬁrm. A uniform standard that required equal emissions from each ﬁrm would not be likely to achieve that efﬁcient result—considerable cost savings might be attainable under equalization of MACs as compared to such uniform regulation. E19.2 Emission taxes The optimal solution described in Equation iv can be achieved by imposing an emission tax (t) equal to l on each ﬁrm (presumably this tax would be set at a level that reﬂects the marginal harm that a unit of emissions causes). With this tax, tf i(qi, zi), which does each ﬁrm seeks to minimize pzi þ indeed yield the efﬁcient solution (v) t p=f A p=f B 2 : ¼ ’ 2 ¼ ’ Notice that, as in the analysis of Chapter 19, one beneﬁt of the taxation solution is that the regulatory authority need not know the details of the ﬁrms’ abatement functions. Rather, the ﬁrms themselves make use of their own private information in determining abatement strategies. If these functions differ signiﬁcantly among ﬁrms, then it would be expected that emissions reductions would also differ. Emission taxes in the United Kingdom Hanley, Shogren, and White (1997) review a variety of emission taxation schemes that have been implemented in the United Kingdom. They show that marginal costs of pollution abatement vary signiﬁcantly (perhaps as much as thirtyfold) among ﬁrms. Hence, relative to uniform regulation, the cost savings from taxation schemes can be quite large. For example, the authors review a series of studies of the Tees estuary that report annual cost savings in the range of £10 million (1976 pounds). The authors also discuss some of the taxes complications that arise in setting efﬁcient efﬂuent when emission streams do not have a uniform mix of pollutants or when pollutants may accumulate to dangerous levels over time. E19.3 Tradable permits As we illustrated in Chapter 19, many of the results achievable through Pigovian taxation can also be achieved through a tradable permit system. In this case, the regulatory agency would set the number of permits (s$) equal to e$ and allocate these permits in some way among ﬁrms (sA þ s$). Each ﬁrm may then buy or sell any number of permits desired but must also ensure that its emissions are equal to the number of permits it holds. If the market price of permits is given by ps, then each ﬁrm’s problem is again to minimize pzi þ ei ’ which yields an identical solution to that derived in Equations iv and v with ps ¼ l. Hence the tradable permit solution sB ¼ , siÞ psð (vi) ¼ t Chapter 19: Externalities and Public Goods 715 equivalence may vanish once the dynamics of innovation in pollution abatement technology are considered. Of course, both procedures offer incentives to adopt new technologies: If a new process can achieve a given emission reduction at a lower MAC, it will be adopted under either scheme. Yet in a detailed analysis of dynamics under the two approaches, Milliman and Prince (1989) argue that taxation is better. Their reasoning is that the taxation approach encourages a more rapid diffusion of new abatement technology because incremental proﬁts attainable from adoption are greater than with permits. Such rapid diffusion may also encourage environmental agencies to adopt more stringent emission targets because these targets will now more readily meet cost–beneﬁt tests. References Hanley, N., J. F. Shogren, and B. White. Environmental Ecoin Theory and Practice. New York: Oxford nomics University Press, 1997. Milliman, S. R., and R. Prince. ‘‘Firm Incentive to Promote Technological Change in Pollution Control.’’ Journal of Environmental Economics and Management (November 1989): 247–65. Schmalensee, R., P. L. Joskow, A. D. Ellerman, J. P. Montero, and E. M. Bailey. ‘‘An Interim Evaluation of the Sulfur Dioxide Trading Program.’’ Journal of Economic Perspectives (Summer 1998): 53–68. would be expected to yield the same sort of cost savings as do taxation schemes. SO2 trading The U.S. Clean Air Act of 1990 established the ﬁrst large-scale program of tradable emission permits. These focused on sulfur dioxide emissions with the goal of reducing acid rain arising from power-plant burning of coal. Schmalensee et al. (1998) review early experiences under this program. They conclude that it is indeed possible to establish large and well-functioning markets in emission permits. More than ﬁve million (one-ton) emission permits changed hands in the most recent year examined—at prices that averaged about $150 per permit. The authors also show that ﬁrms using the permit system employed a wide variety of compliance strategies. This suggests that the ﬂexibility inherent in the permit system led to considerable cost savings. One interesting aspect of this review of SO2 permit trading is the authors’ speculations about why the permit prices were only about half what had been expected. They attribute a large part of the explanation to an initial ‘‘overinvestment’’ in emission cleaning technology by power companies in the mistaken belief that permit prices, once the system was implemented, would be in the $300–$400 range. With such large ﬁxed-cost investments, the marginal cost of removing a ton of SO2 may have been as low as $65/ton, thereby exerting a significant downward force on permit prices. E19.4 Innovation Although taxes and tradable permits appear to be mathematically equivalent in the models we have been describing, this This page intentionally left blank Brief Answers to Queries The following brief answers to the queries that accompany each example in the text may help students test their understanding of the concepts being presented. 2.4 For different constants, each production possibility frontier is a successively larger quarter ellipse centered at the origin. CHAPTER 1 1.1 If price depends on quantity, differentiation of p(q) q would be more complicated. This would lead to the concept of marginal revenue—a topic we encounter in many places in this book. ! 1.2 The reduced form in Equation 1.16 shows that @p"/@a ¼ 1/225. So, if a increases by 450, p" should increase by 2— which is what a direct solution shows. 1.3 200p If all labor is devoted to x production, then x ¼ 180p 14.1 with full employment and x 13.4 with ﬃﬃﬃﬃﬃﬃﬃﬃ unemployment. Hence the efﬁciency cost of unemployment is 0.7 units of x. Similar calculations show that the efﬁciency cost in terms of good y is about 1.5 units of that good. With reductions in both goods, one would need to know the relative price of x in terms of y in order to aggregate the losses. ¼ ¼ ﬃﬃﬃﬃﬃﬃﬃﬃ ¼ CHAPTER 2 2.1 The ﬁrst-order condition for a maximum is @p/@l 50/ 25, p" 250. 0, l" lp 10 $ ¼ ¼ ¼ ¼ ﬃﬃ 2.2 No, only the exponential function (or a function that approximates it over a range) has constant elasticity. 2.3 Putting all the terms over a common denominator gives y 165 3p ¼ ¼ 55 p . Hence, @y @p ¼ $ 55 p2. 1, ¼ ¼ 2. For y 10, the ‘‘circle’’ is a single point. 0 because x1 would always be set at b for opti- 2.5 These would be concentric circles centered at x1 ¼ x2 ¼ 2.6 @y"/@b mality, and the term (x1 $ 2.7 With x1 þ x2 & x1 þ 2.8 A circular ﬁeld encloses maximal area for minimum perimeter. Proof requires a limit argument. x2 ¼ 3, the unconstrained optimum is attainable. b) would vanish. 0.5, x2 ¼ 2, x1 ¼ 1.5. Now y" 9.5. For ¼ 2.9 The local maximum is also a global maximum here. The constancy of the second derivative implies the slope of the function decreases at a constant rate. 2.10 This function resembles an inverted cone that has only one highest point. 2.11 A linear constraint would be represented by a plane in these three-dimensional ﬁgures. Such a plane would have a unique tangency to the surfaces in both Figures 2.4(a) and 2.4(c). For an unconstrained maximum, however, the plane would be horizontal, so only Figure 2.4(a) would have a maximum. 2.12 Such a transformation would not preserve homogeneity. However it would not affect the trade-off between the x’s: for any constant k, x2 / x1. f1 / f2 ¼ $ $ 717 718 Brief Answers to Queries 2.13 Total variable costs of this expansion would be 110 0:2q dq 0:1q2 ¼ 110 100 ¼ 100 ð 1;210 1;000 $ 210: ¼ # # # 100(1,500) from total costs when q This could also be calculated by subtracting total costs when q 110(1,710). ¼ Fixed costs would cancel out in this subtraction. ¼ 2.14 As we show in Chapter 17, a higher value for d will cause wine to be consumed earlier. A lower value for g will make the consumer less willing to experience consumption ﬂuctuations. 2.15 If g(x) is concave, then values of this function will increase less rapidly than does x itself. Hence E[ g (x)] < g[E(x)]. In Chapter 7 this is used to explain why a person with a diminishing marginal utility of wealth will be risk averse. 2 ¼ ¼ (b (b 6, r2 a)2/12 2.16 Using the results from Example 2.15 for the uniform disa)/2 tribution gives mx ¼ x ¼ $ ¼ 120.5 12, and sx ¼ 3.464. In this case, 57.7 percent ( 3.464/12) of the distribution is within one standard ! ¼ deviation of the mean. This is less than the comparable ﬁgure for the Normal distribution because the uniform distribution is not bunched around t
he mean. However, unlike the Normal, the entire uniform distribution is within two standard deviations of the mean because that distribution does not have long tails. $ CHAPTER 3 3.1 The derivation here holds utility constant to create an implicit relationship between y and x. Changes in x also implicitly change y because of this relationship (Equation 3.11). 3.2 The MRS is not changed by such a doubling in Examples 1 and 3. In Example 2, the MRS would be changed (1 because (1 2x)/(1 x)/(1 2y). y) þ þ 6¼ þ þ 3.3 For homothetic functions, the MRS is the same for every point along a positively sloped ray through the origin. 3.4 The indifference curves here are ‘‘horizontally parallel.’’ That is, for any given level of y, the MRS is the same no matter what the value of x is. One implication of this (as we shall see in Chapter 4) is that the effect of additional income on purchases of good y is zero—after a point all extra income is channeled into the good with constant marginal utility (good x). CHAPTER 4 4.1 Constant shares imply @x/@py ¼ 0. Notice py does not enter into Equation 4.23; px does not enter into 4.24. 0 and @y/@px ¼ 4.2 Budget shares are not affected by income, but they may be affected by changes in relative prices. This is the case for all homothetic functions. 4.3 Since a doubling of all prices and nominal income does not change the budget constraint, it will not change utility-maximizing choices. Indirect utility is homogeneous of degree zero in all prices and nominal income. ! ! 1 2 ¼ 30.5 3, E(1,3,2) 4.4 In the Cobb-Douglas case, with py ¼ ¼ 2 6.93, so this person should have his or ! her income reduced by a lump-sum 1.07 to compensate for the fall in prices. In the ﬁxed proportions case, the original consumption bundle now costs 7, so the com1.0. Notice that with ﬁxed proportions pensation is the consumption bundle does not change, but with the Cobb-Douglas, the new choice is x 1.15 because this person takes advantage of the reduction in the price of y. 3.46, y $ ¼ ¼ CHAPTER 5 5.1 The shares equations computed from Equations 5.5 or 5.7 show that this individual always spends all of his or her income regardless of px, py, and I. That is, the shares sum to one. ¼ ¼ ¼ 5.2 If x 0.5I/px then I 100 and px ¼ In Equation 5.11, x 0.5(100/1) ¼ 2.0, the Cobb-Douglas predicts x ¼ 16.67. The CES is more responsive to price. x 1 imply that x 50. 50 also. If px rises to 25. The CES implies ¼ 5.3 Since proportional changes in px and py do not induce substitution effects, holding U constant implies that 100/6 ¼ ¼ x and y will not change. That should be true for all compensated demand functions. 5.4 A larger exponent for, say, x in the Cobb-Douglas function will increase the share of income devoted to that good and increase the relative importance of the income effect in the Slutsky decomposition. This is easiest to see using the Slutsky equation in elasticity form (Example 5.5). 5.5 1 Consider the Cobb-Douglas case for which ex;px ¼ $ regardless of budget shares. The Slutsky equation in elasticity terms shows that, because the income effect sx, the compensated price here is $ elasticity is ec . This occurs sxÞ because proportional changes in x demand will be larger when the share devoted to that good is smaller because they are starting from a smaller base. sxex,I ¼ $ x;px ¼ ¼ $ sx ¼ $ð sx(1) ex;px þ $ 1 5.6 Typically it is assumed that demand goes to zero at some ﬁnite price when calculating total consumer surplus. The speciﬁc assumption made does not affect calculations of changes in consumer surplus. CHAPTER 6 6.1 Since @x/@py includes both income and substitution effects, this derivative could be 0 if the effects offset each other. The conclusion that @x/@py ¼ 0 implies the goods must be used in ﬁxed proportions would hold only if the income effect of this price change were 0. 6.2 Asymmetry can occur with homothetic preferences since, although substitution effects are symmetric, income effects may differ in size. 6.3 Since the relationships between py, pz, and ph never change, the maximization problem will always be solved the same way. Brief Answers to Queries 719 ﬂip is over $1 million. The value of the game in this case is 19 1,000,000/219 $20.91. þ ¼ 7.2 With linear utility, the individual would care only about expected dollar values and would be indifferent about buying actuarially fair insurance. When utility U is a convex function of wealth (U > 0, U 00 > 0), the individual prefers to gamble and will buy insurance only if it costs less than is actuarially justiﬁed. 7.3 If A 10$ 4: ¼ CE CE ð ð 0:5 107;000 102;000; $ 0:5 102;000 101;800: $ ! ! 4 10$ 104 ! ð 2 Þ 4 10$ 106 4 ! ! #1 #2 Þ ¼ ¼ Þ ¼ ¼ So the riskier allocation is preferred. On the other hand, if A 4 then the less risky allocation is preferred. 10$ 3 ¼ ! 7.4 Willingness to pay is a declining function of wealth 0 the person will pay 50 to (Equation 7.43). With R ¼ 10,000 but only 5 if W0 ¼ avoid a 1,000 bet if W0 ¼ 2 he or she will pay 149 to avoid a 100,000. With R ¼ 100,000. 10,000 but only 15 if W0 ¼ 1,000 bet if W0 ¼ 7.5 Option value may be low for a risk-averse person if one of the choices is relatively safe. Reworking the example 1=2 shows that the option value is 0.125 with A1ð x for the risk-neutral person but only about 0.11 for the risk-averse one. Þ ¼ 7.6 The actuarially fair price for such a policy is 0.25 19,000 would pay (x) solves the equation ! 4,750. The maximum amount the individual ¼ 11:45714 ¼ 0:75 ln ð 0:25 ln ð þ 100;000 $ 99;000 x Þ x $ : Þ Solving this yields an approximate value of x $5,120. This person would be willing to pay up to $370 in administrative costs for the deductible policy. ¼ CHAPTER 7 CHAPTER 8 7.1 In case 1, the probability of seven heads is less than 0.01. Hence the value of the original game is $6. In case 2, the prize for obtaining the ﬁrst head on the twentieth 8.1 Best responses are not unique, so the game has no dominant strategies. The extensive form looks like Figure 8.1 with different payoffs. 720 Brief Answers to Queries 8.2 No dominant strategies. (Paper, scissors) isn’t a Nash equilibrium because player 1 would deviate to rock. 8.3 If the wife plays mixed strategy (1/9, 8/9) and the husband plays (4/5, 1/5), then his expected payoff is 4/9. If she plays (1, 0) and he plays (4/5, 1/5), his expected payoff is 4/5. If he plays (4/5, 1/5), her best response is to play ballet. $ cH ¼ 8.10 In equilibrium, type H obtains an expected payoff of j"w cH. This exceeds the payoff of 0 from cL $ deviating to NE. Type L pools with type H on E with probability e". But de"/d Pr(H) w)/p. Since this expression is positive, type L must increase its probability of playing E to offset an increase in Pr(H) and still keep player 2 indifferent between J and NJ. (p $ ¼ 8.4 Players earn 2/3 in the mixed-strategy Nash equilibrium. This is less than the payoff even in the less desirable of the two pure-strategy Nash equilibria. Symmetry might favor the mixed-strategy Nash equilibrium. CHAPTER 9 9.1 Now, with k 11: ¼ 8.5 The Nash equilibrium would involve higher quantities for both if their beneﬁts increased. If herder 2’s beneﬁt decreased, his or her quantity would fall and the other’s would rise. 8.6 Yes. Letting p be the probability that player 1 is type 6, player 2’s expected payoff from choosing L is 2p. t This is at least as high as 2’s expected payoff of 4(1 p) 2/3. from choosing R if p $ ¼ & 8.7 Moving from incomplete to full information increases herder 1’s output and decreases the rival’s if 1 is the high type. The opposite is true if 1 is the low type. The high type prefers full information and would like to somehow signal its type; the low type prefers incomplete information and would like to conceal its type. q MPl APl ¼ ¼ ¼ 72;600l2 145;200l 72;600l $ 1;331l3; 3;993l2; $ 1;331l2: $ In this case, APl reaches its maximal value at l rather than at l 30. 27.3 ¼ ¼ ¼ l < 20. 9.2 fl Since k and l enter f symmetrically, if k l then fk ¼ fll. Hence, the numerator of Equation 9.21 and fkk ¼ will be negative if fkl > fll. Combining Equations 9.24 l) shows this holds for and 9.25 (and remembering k k ¼ 9.3 0; 4 isoquant contains the points k The q ¼ 4. It is therefore fairly 0, l k sharply convex. It seems possible that an L-shaped isoquant might be approximated for particular coefﬁcients of the linear and radical terms. 1; and k ¼ 1, l 4, l ¼ ¼ ¼ ¼ ¼ ¼ 8.8 Obtaining an education informs the ﬁrm about the worker’s ability and thus may increase the high-skill worker’s salary. The separating equilibrium would not exist if the low-skill worker could get an education more cheaply than the high-skill one. 8.9 The proposed pooling outcome cannot be an equilibrium if the ﬁrm’s posterior beliefs equal its priors after unexpectedly seeing an uneducated worker. Then its beliefs would be the same whether or not it encountered an educated worker, it would have the same best response, and workers would deviate from E. If the ﬁrm has pessimistic posteriors following NE, then the outcome is an equilibrium because the ﬁrm’s best response to NE would be NJ, inducing both types of worker to pool on E. 9.4 Because the composite technical change factor is y ¼ 0.3 implies that technical aj improvements in labor will be weighted more highly in determining the overall result. a)e, a value of a (1 ¼ $ þ CHAPTER 10 ¼ ¼ 16, l 8/5, k 0.5, k/l ¼ 0.5, then r 2, then r 96. 10.1 If s ¼ and C If s C Notice that changes in s also change the scale of the production function, so the total cost ﬁgures cannot be compared directly. ¼ 1,080. 120, and 128/5, 1, k/l 60, k ¼ $ 2, l ¼ ¼ ¼ ¼ ¼ ¼ 10.2 w1 s). If The expression for unit costs is (v1 $ $ $ v. For s > 0 s the function is increasingly convex, showing that large increases in w can be offset by small decreases in v. þ 0 then this function is linear in w s)1/(1 þ ¼ s 10.3 The elasticities are given by the exponents in the cost functions and are unaffected by technical change as modeled here. ¼ ¼1 . With
w 10.4 In this case s 4v, cost minimization could use the inputs in any combination (for q constant) without changing costs. A rise in w would cause the ﬁrm to switch to using only capital and would not affect total costs. This shows that the impact on costs of an increase in the price of a single input depends importantly on the degree of substitution. 10.5 Because capital costs are ﬁxed in the short run, they do not affect short-run marginal costs (in mathematical terms, the derivative of a constant is zero). Capital costs do, however, affect short-run average costs. In Figure 10.9 an increase in v would shift MC, AC, and all of the SATC curves upward, but would leave the SMC curves unaffected. CHAPTER 11 11.1 If MC P ¼ 7.50, R 5, proﬁt maximization requires q 25. Now ¼ 62.50. 187.50, C 125, and p ¼ ¼ ¼ 11.2 Factors other than p can be incorporated into the constant term a. These would shift D and MR but would not affect the elasticity calculations. ¼ 11.3 When w rises to 15, supply shifts inward to q 8P/5. When k increases to 100, supply shifts outward to q 25P/6. A change in v would not affect short-run marginal cost or the shutdown decision. ¼ ¼ 11.4 A change in v has no effect on SMC but it does affect ﬁxed costs. A change in w would affect SMC and shortrun supply. 11.5 A rise in wages for all ﬁrms would shift the market supply curve upward, raising the product price. Because Brief Answers to Queries 721 total output must fall given a negatively sloped demand curve, each ﬁrm must produce less. Again, both substitution and output effects would then be negative. CHAPTER 12 12.1 The ability to sum incomes in this linear case would require that each person have the same coefﬁcient for income. Because each person faces the same price, aggregation requires only adding the price coefﬁcients. 12.2 A value for b other than 0.5 would mean that the exponent of price would not be 1.0. The higher b is, the more price elastic is short-run supply. 12.3 Following steps similar to those used to derive Equation 12.30 yields eP;b eQ;b ¼ $ $ eS;P eQ;P 0.5, so eP, b ¼ $ eQ,w ¼ $ Here eQ, b ¼ ¼ 0.227. Multiplication by 0.20 (since wages rose 20 percent) predicts a price rise of 4.5 percent, very close to the number in the example. 0.5)/2.2 $ ( 12.4 The short-run supply curve is given by Qs ¼ þ 750, and the short-term equilibrium price is $643. Each ﬁrm earns approximately $2,960 in proﬁts in the short run. 0.5P 12.5 Total and average costs for Equation 12.55 exceed those for Equation 12.42 for q > 15.9. Marginal costs for Equation 12.55 always exceed those for Equation 12.42. Optimal output is lower with Equation 12.55 than with Equation 12.42 because marginal costs increase more than average costs. 12.6 Losses from a given restriction in quantity will be greater when supply and/or demand is less elastic. The actor with the least elastic response will bear the greater share of the loss. 12.7 An increase in t unambiguously increases deadweight loss. Because increases in t reduce quantity, however, total tax revenues are subject to countervailing effects. Indeed, if t/(P 1/eQ,P then dtQ/dt < 0. t) þ & $ 722 Brief Answers to Queries CHAPTER 13 13.1 An increase in labor input will shift the ﬁrst frontier out uniformly. In the second case, such an increase will shift the y-intercept out farther than the x-intercept because good y uses labor intensively. 13.2 In all three scenarios the total value of output is 200w, composed half of wages and half of proﬁts. With the shift in supply, consumers still devote 100w to each good. Purchases of x are twice those of y because y costs twice as much. With the shift in demand, the consumer spends 20w on good x and 180w on good y. But good y now costs three times what x costs, so consumers buy only three times as much y as they do x. 13.3 All efﬁcient allocations require the ratio of x to y to be relatively high for A and low for B. Hence, when good x is allocated evenly, A must get less than half the amount of y available and B must get more than half. Because efﬁciency requires 2yA=xA ¼ 0:5yB=xB and the symmetry xA=yA for of the utility functions requires yB=xB ¼ 0:5yB. 2yA, xB ¼ equal utility, we can conclude xA ¼ 333.3, yB ¼ 333.3, xB ¼ 666.7, yA ¼ Utility for both parties is about 496. So xA ¼ 666.7. 13.4 The consumers here also spend some of their total income on leisure. For person 1, say, total income with the equilibrium prices is 40 11.4. The CobbDouglas exponents imply that this person will spend half of this on good x. Hence, total spending on that good will be 5.7, which is also equal to the quantity of x bought (15.7) multiplied by this good’s equilibrium price (0.363). 0.136 0.248 24 þ ¼ ! ! 13.5 No—such redistribution could not make both better-off owing to the excess burden of the tax. CHAPTER 14 14.1 The increase in ﬁxed costs would not alter the output decisions because it would not affect marginal costs. It would, however, raise AC by 5 and reduce proﬁts to 12,500. With the new C function, MC would rise to 0.15Q. In this case, Q" ¼ 14.2 For the linear case, an increase in a would increase price by a/2. A shift in the price intercept has an effect 22,000, and p 400, P" 10,000. 80, C ¼ ¼ ¼ similar to an increase in marginal cost in this case. In the constant elasticity case, the term a does not enter into the calculation of price. For a given elasticity of demand, the gap between price and marginal cost is the same no matter what a is. 14.3 With e 1.5, the ratio of monopoly to competitive consumer surplus is 0.58 (Equation 14.19). Proﬁts represent 19 percent of competitive consumer surplus (Equation 14.21). ¼ $ ¼ 0, P 14.4 100. Total proﬁts are given by the trianIf Q gular area between the demand curve and the MC curve, 33,333. So p less ﬁxed costs. This area is 0.5(100)(666) 23,333. 33,333 10,000 ¼ ¼ ¼ $ ¼ 14.5 One must be careful when summing the demand functions. For P > 12, there is no demand in market 2, so the monopoly solution in that case yields proﬁts of 81. For P < 12, market demand is Q Q/3. In this case the monopoly price would be 11. Proﬁts would be (11 75, so it is still not worthwhile ¼ to serve market 2. Proﬁts are maximized when P 3P or P 15. 15 48 16 6) $ $ ¼ $ ¼ ! ¼ CHAPTER 15 15.1 Members of a perfect cartel produce less than their best responses, so cartels may be unstable. 15.2 A point on ﬁrm 1’s best response must involve a tangency between 1’s isoproﬁt and a horizontal line of height q2. This isoproﬁt reaches a peak at this point. Firm 2’s isoproﬁts look something like right parentheses that peak on 2’s best-response curve. An increase in demand intercept would shift out both best responses, resulting in higher quantities in equilibrium. 15.3 The ﬁrst-order condition is the mathematical representation of the optimal choice. Imposing symmetry before taking a ﬁrst-order condition is like allowing ﬁrm i to choose the others’ outputs as well as its own. Making this mistake would lead to the monopoly rather than the Cournot outcome in this example. 15.4 An increase in the demand intercepts would shift out both best responses, leading to an increase in equilibrium prices. Brief Answers to Queries 723 15.5 Locating in the same spot leads to marginal cost pricing as in the Bertrand model with homogeneous products. Locating at opposite ends of the beach results in the softest price competition and the highest prices. 16.2 The conclusion does not depend on linearity. So long as the demand and supply curves are conventionally shaped, the curves will be shifted vertically by the parameters t and k. 15.6 It is reasonable to suppose that competing gas stations monitor each other’s prices and could respond to a price change within the day, so one day would be a reasonable period length. A year would be a reasonable period for producers of small cartons of milk for school lunches, because the contracts might be renegotiated each new school year. 15.7 Reverting to the stage-game Nash equilibrium is a less harsh punishment in a Cournot model (ﬁrms earn positive proﬁt) than a Bertrand model (ﬁrms earn zero proﬁt). 15.8 Firms might race to be the ﬁrst to market, investing in research and development and capacity before sufﬁcient demand has materialized. In this way, they may compete away all the proﬁts from being ﬁrst, a possible explanation for the puncturing of the dot-com bubble. Investors may even have overestimated the advantages of being ﬁrst in the affected industries. 15.9 In most industries, price can be changed quickly—perhaps instantly—whereas quantity may be more difﬁcult to adjust, requiring the installation of more capacity. Thus, price is more difﬁcult to commit to. Among other ways, ﬁrms can commit to prices by mentioning price in their national advertising campaigns, by offering price guarantees, and by maintaining a long-run reputation for not discounting list price. 15.10 Entry reduces market shares and lower prices from tougher competition, so one ﬁrm may earn enough proﬁt to cover its ﬁxed cost where two ﬁrms would not. 15.11 The social planner would have one ﬁrm charge marginal cost prices. This would eliminate any deadweight loss from pricing and also economize on ﬁxed costs. CHAPTER 16 16.1 Nonlabor income permits the individual to ‘‘buy’’ leisure but the amount of such purchases depends on labor-leisure substitutability. 16.3 With this sharing, Equation 16.37 becomes p $ a)vs(s) pss and proﬁt maximization requires that @vs/@s a). Hence the ﬁrm will invest less in speciﬁc human capital. In future bargaining, workers might be willing to accept a lower a in exchange for the ﬁrm’s paying some of the costs of general human capital. pgg p s/(1 $ $ $ ¼ (1 ¼ 16.4 Now MRS $30 per hour. In this case, the monopsony will hire 750 workers, and wages will be $15 per hour. As before, the wages remains at only half the MRP. ¼ 16.5 The monopsonist wants to be on its demand for labor curve; the union (presumably) wants to be on the labor supply curve of its membe
rs. Only the supply-demand equilibrium (l 11.67) satisﬁes both these ¼ curves. Whether this is indeed a Nash equilibrium depends, among other things, on whether the union deﬁnes its payoffs as being accurately reﬂected by the labor supply curve. 583, w ¼ 16.6 If the ﬁrm is risk neutral, workers risk averse, optimal contracts might have lower wages in exchange for more-stable income. CHAPTER 17 17.1 Using Equation 17.17 yields c1/c0 ¼ Hence 1 ¼ ¼ R ¼ (1.02)1 $ 0.082. r þ 3 then r R. If R ¼ $ 1.02 (1 ¼ 0 then r R). r)1/(1 $ 0.02; if þ ¼ 17.2 If g is uncertain, the future marginal utility of consumption will be a random variable. If U 0(c) is convex, its expected value with uncertain growth will be greater than its value when growth is at its expected value. The effect is similar to what would occur with a lower growth rate. Equation 17.29 shows that the risk-free interest rate must fall to accommodate such a lower g. 17.3 With an inﬂation rate of 10 percent, the nominal value of the tree would rise at an additional 10 percent per 724 Brief Answers to Queries year. But such revenues would have to be discounted by an identical amount to calculate real proﬁts so the optimal harvesting age would not change. 17.4 For a monopolist, an equation similar to Equation 17.62 would hold with marginal revenue replacing price. With a constant elasticity demand curve, price would have the same growth rate under monopoly as under perfect competition. CHAPTER 18 18.1 The manager would have an incentive to overstate gross proﬁts unless some discipline were imposed by an audit. If audits are costly, the efﬁcient arrangement might involve few audits with harsh punishments for false reports. If harsh punishments are impossible, the power of the manager’s incentives might have to be reduced. 18.2 The insurer would be willing to pay the difference between its ﬁrst- and second-best proﬁts, 298 $202. 96 $ ¼ 18.3 Insurance markets are generally thought to be fairly competitive, except where regulation has limited entry. It is hard to say which segment is most competitive. The fact that the individuals purchase car insurance whereas ﬁrms purchase health insurance on behalf of their employees ‘‘in bulk’’ may affect the nature of competition. 18.4 A linear price would allow the consumer to buy whatever number of ounces desired at the 10 cents per ounce price. Here the consumer is restricted to two cup sizes: 4 or 16 ounces. 18.5 The insurance company decides to offer just one policy targeted to red cars and ignores gray cars. 18.6 Gray-car owners obtain utility of 11.48033 in the competitive equilibrium under asymmetric information. They would obtain the same utility under full insurance with a premium of $3,207. The difference between this and the equilibrium premium ($453) is $2,754. Any premium between $3,000 and $3,207 would allow an insurance company to break even from its sales just for gray cars. The problem is that red-car owners would deviate to the policy, causing the company to make negative proﬁt. 18.7 If the reports are fairly credible, then gray cars may still be able to get as full insurance with reporting as without, but not as full as with 100 percent credibility. Auditors have short-run incentives to take bribes to issue ‘‘gray’’ reports. In the long run, dishonesty will reduce the fees the auditor can charge. He or she would like to maintain high fees by establishing a reputation for honest reporting (which would be ruined if ever discovered to be dishonest). 18.8 If there are fewer sellers than buyers, then all the cars b. will sell. A car of quality q will sell at a price of q If there are fewer buyers than sellers, then all buyers will purchase a car but some cars will be left unsold (a random selection of them). The equilibrium price will equal the car’s quality: q. þ 18.9 Yes, reservation prices can often help. The trade-offs involved in increasing the reservation price are, on the one hand, that buyers are encouraged to increase their bids, but, on the other hand, that the probability the object goes unsold increases. In a second-price auction, buyers bid their valuations without a reservation price, and a reservation price would not induce them to bid above their valuations. CHAPTER 19 19.1 Production of x would have a beneﬁcial impact on y so labor would be underallocated to x by competitive markets. 19.2 The tax is relatively small because of the nature of the externality that vanishes with only a relatively minor reduction in x output. A merged ﬁrm would also ﬁnd x 38,000 to be a proﬁt-maximizing choice. ¼ 19.3 With two roommates, 2/3 of the efﬁcient level of the public good is supplied in equilibrium. With three roommates, in equilibrium each supplies 7.5 for a total of 22.5, only half the efﬁcient level (45 units total). 19.4 The roommates have identical preferences here and therefore identical marginal rates of substitution. If each pays half the price of the public good then the sum of their MRSs will be precisely the ratio of the price of the public good to the price of the private good, as required in Equation 19.40. With differing MRSs, the sharing might depart from 50 50 to ensure efﬁciency $ 19.5 Reduction of the labor tax increases after-tax income and the demand for good y. With a ﬁxed Pigovian tax, pollution rises. More generally, the likelihood of a dou- Brief Answers to Queries 725 ble dividend depends on the precise demand relationship in people’s utility functions between clean air and the other items being taxed (here, labor). 19.6 Progressive taxation should raise t" because the median voter can gain more revenue from high-income tax payers without incurring high tax costs. This page intentionally left blank Solutions to Odd-Numbered Problems Only very brief solutions to most of the odd-numbered problems in the text are given here. Complete solutions to all of the problems are contained in the Solutions Manual, which is available to instructors upon request. CHAPTER 2 2.1 a. 8x, 6y b. 8, 12 c. 8xdx d. dy/dx e. x f. dy/dx g. U ¼ ¼ 6ydy 1, U þ ¼ # ¼ ¼ # 16 contour line is an ellipse. 4x/3y. (4)(1) 2/3. (3)(4) 16. þ ¼ 2.3 Both approaches yield x y 0.5. ¼ ¼ 2.5 a. The ﬁrst-order condition for a maximum is 40 0, so t$ b. Substitution yields f (t$) 40/g. ¼ ¼ ¼ 800/g. So @f(t$)/@g ¼ # c. This follows because @f/@g d. @f/@g 0.5(40/g)2 0.5g(40/g)2 40(40/g) þ ¼ # 800/g2. 0.5(t$)2. ¼ # 0.8, so each 0.1 increase ¼ # ¼ # in g reduces maximum height by 0.08. gt # þ 5. With k 2.7 a. First-order conditions require f1 ¼ x2 ¼ ¼ 4, x1 ¼ # b. With k 4. c. x1 ¼ 5. Because marginal 15, x2 ¼ 20, x1 ¼ d. With k value of x1 is constant, every addition to k beyond 5 adds only to that variable. ¼ 0, x2 ¼ ¼ 10, x1 ¼ 1. 1. Hence, f2 ¼ 5. 2.9 Since fii < 0, the condition for concavity implies that the matrix of second-order partials is negative deﬁnite. Hence the quadratic form involving [ f1, f2] will be negative as required for quasi-concavity. The converse is not true, as shown by the Cobb-Douglas function with a b > 1. þ f 00 d(d 2.11 1)xd a. b. Since f11, f22 < 0 and f12, f21 ¼ c. This preserves quasi-concavity but not concavity. obviously holds. 2 < 0. ¼ # 0, Equation 2.98 # 2.13 a. From Equation 2.85, a function in one variable is concave if f 00(x) < 0. Using the quadratic Taylor to approximate f (x) near a point a Þð # :5f 00 a ð Þ þ Þð because f 00 and # Þð Þð ( # Þ þ Þ b. From Equation 2.98, a function in 2 variables is f 2 12 > 0 and we also know that 1 þ 0. This is the third term of a, concave if f11f22 # due to the concavity of the function, 0.5( f11dx2 f22dx2 2f12dx1dx2 þ 2) the quadratic Taylor expansion where dx dy y þ ( ¼ f1(a,b)(x b). Which shows that any concave function must lie on or below its tangent plane. b. Thus, we have f (x, y) a) x ¼ # f (a,b) f2(a,b)(y # # þ # ( 2.15 a. Use Var(x) (E(x))2). b. Let y (x c. First part is trivial. Let E[(x # ¼ E(x))2] E(x2 2xE(x) # þ ¼ ¼ mx)2 and apply Markov’s inequality to y. xi=n E l nl=n # Var X d. Var(X) ð k ¼ Var(X) e. If r2 k ¼ Þ ¼ ¼ 2k Þ ¼ ¼ # 0.5. In this case Var(X) nr2=n2 (2k2 ¼ r2=n. P 1)s2 which is minimized for 0.5s2. If, say, k 0.7, 0.58s2 so it is not changed all that much. ¼ rr2 2, the weighted average is minimized if r). 1 ¼ r/(l þ ¼ ¼ X ð ¼ þ 727 728 Solutions to Odd-Numbered Problems CHAPTER 3 3.1 a. No b. Yes c. Yes d. No e. Yes 3.3 The shape of the marginal utility function is not necessarily an indicator of convexity of indifference curves. min(h, 2b, m, 0.5r). 3.5 a. U(h, b, m, r) ¼ b. A fully condimented hot dog c. $1.60 d. $2.10—an increase of 31 percent. e. Price would increase only to $1.725—an increase of 7.8 percent. f. Raise prices so that a fully condimented hot dog rises in price to $2.60. This would be equivalent to a lump-sum reduction in purchasing power. 3.7 a. Indifference curve is linear—MRS b. a c. Just knowing the MRS at a known point can iden- 2, b 1/3. ¼ ¼ ¼ 1. tify the ratio of the Cobb-Douglas exponents. 3.9 a.–c. See detailed solutions. 3.11 It ¼ depend on y or vice versa. 3.1(b) is a counterexample. MUx/MUy Æ MUx doesn’t follows, since MRS 3.13 a. MRS fx / fy ¼ ¼ fxx ¼ b. fxy ¼ reduces to # y. 0, so the condition for quasi-concavity 1/y2 < 0. c. An indifference curve is given by y d. Marginal utility of x is constant, marginal utility of y diminishes. As income rises, consumers will eventually choose only added x. exp(k x). # ¼ e. y could be a particular good, whereas x could be ‘‘everything else.’’ 3.15 a. U$ ¼ ¼ b. Because the reference bundle has y a. Hence, b(U$) aba(1 ¼ b) # U$. 0, it is not possible to attain any speciﬁed utility level by replicating this bundle. ¼ c. a is given by the length of a vector in the direction of the reference bundle from the initial endowment to the target indifference curve. See detailed solutions. d. This follows directly from the convexity of indiffer- ence curves. See detailed solutions. CHAPTER 4 4.1 a. t b. t 4.3 a. c b. c ¼ ¼ ¼ ¼ 5 and s ¼ 5/2 and s 2. ¼ 4. Costs $2 so needs e
xtra $1. 10, b 4, b ¼ ¼ 3, and U 1, and U 127. 79. ¼ ¼ 4.5 b. g c. Utility d. E ¼ ¼ I/(pg þ m ¼ ¼ m(2pg þ pv/2); v v ¼ pv). pv). I/(2pg þ pv). ¼ I/(2pg þ 4.7 a. See detailed solutions. b. Requires expenditure of 12. c. Subsidy is 5/9 per unit. Total cost of subsidy is 5. d. Expenditures to reach U 2 are 9.71. To reach ¼ 3 requires 4.86 more. A subsidy on good x U must be 0.74 per unit and costs 8.29. ¼ e. With ﬁxed proportions the lump sum and single good subsidy would cost the same. pxU/a. If px/py > a/b then ¼ a/b then E pxU/a ¼ ¼ pyU/b. 4.9 If px/py < a/b then E pyU/b. If px/py ¼ E ¼ 4.11 a. Set MRS b. Set d c. Use pxx/pyy ¼ 0. ¼ px/py. (px/py)d/(d # 1). ¼ 4.13 a. See detailed solutions. b. Multiplying prices and income by 2 does not change V. c. Obviously @V/@I > 0. d. @V/@px , @V/@py < 0. e. Just exchange I and V. f. Multiplying the prices by 2 doubles E. g. Just take partials. h. Show @E/@px > 0, @2E/@p2 x < 0. Solutions to Odd-Numbered Problems 729 CHAPTER 5 CHAPTER 6 5.1 a. U b. x x 3 8 y. x þ I/px if px ( 0 if px > 3 8 py. 3 8 py. ¼ ¼ ¼ d. Changes in py don’t affect demand until reverse the inequality. Just two points (or vertical lines). e. 5.3 a. It is obvious since px/py doesn’t change. b. No good is inferior. 5.5 a. x I px # 2px , y I px þ 2py . ¼ ¼ b. V ð pxÞ þ 4pxpy Hence, changes in py do not affect x, but changes in px do affect y. I 2 and so E 4pxpyV px. ¼ ¼ c. The compensated demand function for x depends on py, whereas the uncompensated function did not. ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p # 5.7 a. Use the Slutsky equation in elasticity form. Because sheh, I ¼ b. Compensated price elasticity is zero for both goods, there are no substitution effects, eh, ph ¼ 0 # ¼ # 0:5. 0:5 # 0 which are consumed in ﬁxed proportions. 2=3 so eh,ph ¼ # c. Now sh ¼ d. For a ham and cheese sandwich (sw), esw; psw ¼ # 1 epsw, ph ¼ ð# esw, ph ¼ Þ ) esw, psw ) ¼ # 2=3. 0:5. 0:5 5.9 a. pxI@x=@I I2 @sx @I ¼ the result. pxx # . Multiplication by I sx ¼ I2 pxx gives b.-d. All of these proceed as in part (a). e. Use Slutsky equation—see detailed solutions. 5.11 a. Just follow the approaches used in the two-good cases in the text (see detailed solutions). 5.13 a. ln E px, py, U ð a0 þ þ 2 g22ð Þ ¼ a2 log py þ a1 ln px þ 2 g11ð g12 ln px ln py þ ln pyÞ þ 1 1 2 ln pxÞ Ub0pb1 x pb2 y . b. Doubling all prices adds ln2 to the log of the expenditure function, thereby doubling it (with U held constant). a1 þ þ g12 ln py g11 ln px þ p b2 y . Ub0b1p b1 # c. sx ¼ 1 x 6.1 a. Convert this to a Cobb-Douglas with a Result follows from prior examples. b ¼ ¼ 0.5. b. Also follows from Cobb-Douglas. c. Set @m/@ps ¼ d. Use the Cobb-Douglas representation. substitution effects. @s/@pm and cancel the symmetric they 6.3 a. pbt ¼ b. Since pc and I are constant, c 2pb þ pt. constant. I/2pc is also ¼ c. Yes—since changes in pb or pt affect only pbt . 6.5 p3(kx2 þ p3x3 ¼ a. p2x2 þ x3). t)/(p3 þ (p2 þ b. Relative price ¼ Approaches p2/p3 < 1 as t ﬁ 0. Approaches 1 as t ﬁ . 1 So, an increase in t raises the relative price of x2. c. Does not strictly apply since changes in t change t). relative prices. d. May reduce spending on x2—the effect on x3 is uncertain. 6.7 Show xi Æ @xj /@I ¼ substitution effects. xj Æ @xi /@I and use symmetry of net 1, 6.9 a. CV ¼ E p01, p02, !p3, : : : , !pn, !U ð E # Þ p1, p2, !p3, : : : , !pn, !U ð . Þ b. See graphs in detailed solutions—note that change in one price shifts compensated demand curve in the other market. c. Symmetry of cross-price effects implies that order is irrelevant. d. Smaller for complements than for substitutes. 6.11 See graphs in detailed solutions or in Samuelson reference. 2 CHAPTER 7 7.1 P 0.525. ¼ 7.3 a. One trip: expected value Two trip: expected value 0.25 Æ 12 6. ¼ 0.5 Æ 0 þ 0.25 Æ 0 0.5 Æ 12 0.5 Æ 6 þ 6. ¼ þ ¼ ¼ 730 Solutions to Odd-Numbered Problems b. Two-trip strategy is preferred because of smaller CHAPTER 8 variance. c. Adding trips reduces variance, but at a diminishing rate. So desirability depends on the trips’ cost. 7.5 a. E(U) b. E(U) c. $260. ¼ ¼ 0.75 ln(10,000) ln(9,750) 0.25 ln(9,000) 9.1840. 9.1850—insurance is preferable. ¼ þ ¼ 7.7 a. Plant corn. b. Yes, a mixed crop should be chosen. Diversiﬁcation increases variance, but takes advantage of wheat’s high yield. c. 44 percent wheat, 56 percent corn. d. The farmer would only plant wheat. 7.9 a. E A2½ + ¼ 1 and E max A1, A2Þ ð + ¼ ½ 7=6 implying the option value is 1=6. 2p =3 0:94. The graph of E max b. E U 2 ’ + ¼ ½ U reaches this same value F A2 # , U F ﬃﬃﬃ ð ð Þ 0:24. for an option value of F ÞÞ+ ’ c. Making one choice more attractive reduces option A2Þ ð A1 # ð ½ value. 7.11 a. Risk-neutral Stan indifferent among A-D. b. Risk-averse Stan should choose safe option in each scenario (B in 1 and D in 2). c. Most subjects chose C in Scenario 2, but a risk- averse person should choose D. d. (1) Depends, but could make same choices as most experimental subjects. (2) See detailed answers for graph. Curve has to shift because of kink at anchor point. Pete’s curves are convex below anchor and concave above, while Stan’s are concave everywhere. 7.13 a. See graph in detailed answers. b. Mixed portfolios lie on a segment between the risky and riskless assets. c. Risk aversion is indicated by sharper bend to indifference curves. A person with L-shaped indifference curves (inﬁnitely risk averse) would hold no risky asset. d. A CRRA investor has homothetic indifference curves. 8.1 a. (C, F). b. Each player randomizes over the two actions with equal probability. c. Players each earn 4 in the pure-strategy equilibrium. Players 1 and 2 earn 6 and 7, respectively, in the mixed-strategy equilibrium. d. The extensive form is similar to Figures 18.1 and 18.2 but has three branches from each node rather than two. 8.3 a. The extensive form is similar to Figure 8.9. b. (Do not veer, veer) and (veer, do not veer). c. Players randomize with equal probabilities over the two actions. d. Teen 2 has four contingent strategies: always veer, never veer, do the same as Teen 1, and do the opposite of Teen 1. e. The ﬁrst is (do not veer, always veer), the second is (do not veer, do the opposite), and the third is (veer, never veer). f. (Do not veer, do the opposite) is a subgame-perfect equilibrium. 8.5 a. If all play blond, then one would prefer to deviate to brunette to obtain a positive payoff. If all play brunette, then one would prefer to deviate to blond for payoff a rather than b. b. Playing brunette provides a certain payoff of b and blond provides a payoff of a with probability 1 (the probability no other player approaches (1 # the blond). Equating the two payoffs yields p$ (b/a)1/(n # p)n # 1). ¼ # 1 c. The probability the blond is approached by at least one male equals 1 minus the probability no males 1). This (b/a)n/(n p$)n approach her: 1 # # expression is decreasing in n because n/(n 1) is decreasing in n and b/a is a fraction. (1 # # ¼ # 1 3.5 2.5 8.7 l2/4 for a. The best-response function is lLC ¼ þ l2/4 for the low-cost type of player 1, lHC ¼ þ !l1/4 for player 2, the high-cost type, and l2 ¼ þ where !l1 is the average for player 1. Solving these 3:5, and l$2 ¼ 4:5, l$HC ¼ equations yields l$LC ¼ c. The low-cost type of player 1 earns 20.25 in the Bayesian-Nash equilibrium and 20.55 in the fullinformation game, so it would prefer to signal its 4. 3 Solutions to Odd-Numbered Problems 731 þ c. ta bf (k, l); f (tk, tl) ¼ @f (tk, tl)/@t Æ t/f (k, l) 1 this is just a At t (a b. d., e. Apply the deﬁnitions using the derivatives from b)ta þ ¼ þ þ ¼ b. part (a). 9.7 a. b0 ¼ 0. 1 2 b1 l=k k=l b. MPk ¼ . c. In general, s is not constant. If b2 ¼ b3 ¼ ﬃﬃﬃﬃﬃﬃ p ﬃﬃﬃﬃﬃﬃ p 0, s s ¼1 b2 þ 1. If b1 ¼ ; MPL ¼ . b3 þ 1 2 b1 ¼ 0, 9.9 a. If f (tk, tl ) l ) then eq,t ¼ tf (k, t/f (tk, tl ). If t ﬁ 1 then f (k, l )/f (k, l ) ¼ @f (tk, tl )/@t Æ 1. ¼ b. Apply Euler’s theorem and use part (a): f (k, l) fll. 2(1 fkk þ c. eq,t ¼ d. The production function has an upper bound of q). Hence q < 0.5 implies eq,t > 1 and # q > 0.5 implies eq,t < 1. ¼ 1. q ¼ 9.11 a. Apply Euler’s theorem to each fi. 2, k2fkk þ 1)f (k, l). 2klfkl þ b. With n ¼ 1, this implies fkl > 0. If k > 1, it is even If k clearer that fkl must be positive. For k < 1, the case is not so clear. l2fll ¼ k(k ¼ # c. Implies that fij > 0 is more common for k d. ( k(k 1). ai)2 1. ¼ # ai ¼ # 10.1 a. By deﬁnition. C(q1, 0) is the cost of producing just good 1 in one ﬁrm. b. By C 0, q2Þ q2 ð C q1, q2Þ q < assumption, ð . Multiplying respectively by q1 and q2 and q1, q2Þ q < C and C q1, 0 Þ ð q1 ð summing gives the economies-of-scope condition. 10.3 a. C q v=5 ð ¼ w=10 Þ þ 50, SC . AC 10v b. For q v=5 MC þ ¼ wq=10, SAC w=10. 10v=q þ ¼ ¼ þ ¼ w=10. .5. For q c. AC ( w=10, SMC MC .3. SMC ¼ ¼ ¼ ¼ 50, SAC 10=q :3, þ ¼ ( type if it could. Similar calculations show that the high-cost player would like to hide its type. 0; proposer offers r$ 8.9 a. Responder accepts any r b. Same as in a. c. (1) Responder accepts any r . (2) Proposer offers exactly r$ Þ (3) In Dictator Game, proposer still offers r$ so less even split than Ultimatum Game. a= 1 ð a= 1 ð 2a . Þ 2a þ þ , ¼ , 0. ¼ 0, ¼ , 8.11 a. The condition for cooperation to be sustainable 1, so one pewith one period of punishment is d , riod of punishment is not enough. Two periods of punishment are enough as long as d2 0, or d 0.62. , # þ d 1 2d11/(1 value of the payoffs from cooperating, 2/(1 d10)/(1 exceed that from deviating, 3 1 b. The required condition is that the present discounted d), d) 0. þ Using numerical or graphical methods, this condition can be shown to be d 0.50025, not much stricter than the condition for cooperation with inﬁnitely many periods of punishment (d d). Simplifying, 2d # # , # d11 1/2). d(1 # # # þ , , CHAPTER 9 9.1 a. k b. k c. k 8 and l ¼ 10 and l ¼ 9, l ¼ hours). ¼ 8. ¼ 5. ¼ 6.5, k d. The isoquant is linear between solutions (a) and (b). ¼ ¼ ¼ 100, l 33, l ¼ ¼ 40, l 100, C 132, C 10,000. 8,250. ¼ ¼ 160, C 10, k ¼ 10, k ¼ 12.13, k 9.3 a. q b. q c. q d. Car
la’s ability to inﬂuence the decision depends on whether she can impose any costs on the bar if she is unhappy serving the additional tables. Such ability depends on whether Carla is a draw for Cheers’ customers. 10,000. ¼ ¼ ¼ 9.5 Let A bkalb 1 > 0, # a. 1 for simplicity. ¼ aka 1lb > 0, fl ¼ fk ¼ # 2lb < 0, ka a a 1 fkk ¼ # # ð Þ kalb 2 < 0, b b 1 fll ¼ # ð # Þ 1lb abka 1 > 0: fkl ¼ flk ¼ # # fk Æ k/q a, eq,l ¼ b. eq,k ¼ ¼ fl Æ l/q 10.5 a. First, show SC 125 q2 2=100. Set up ¼ Lagrangian for cost minimization L q1 # q2), yielding q1 ¼ q2 1=25 .25 q2. l(q SC ¼ þ þ þ # b. ¼ 9.5, and l ¼ ¼ 5.75 (fractions of P P CHAPTER 10 732 Solutions to Odd-Numbered Problems b. SC 125 q2=125, SMC ¼ 2q=125 , SAC ¼ þ q=125. ¼ 125=q SMC(100) $3.20. þ ¼ $1.60, SMC(125) $2.00, SMC(200) ¼ ¼ c. Distribution across plants irrelevant in long run. C 2q; AC MC d. Distribute output evenly across plants. ¼ ¼ ¼ 2. 10.7 a. Let B and l b. q v1=2 þ ¼ @C=@w ¼ 1 1. 1 k# ¼ l# þ w1=2. Then k Bw# 1=2q. ¼ @C=@v ¼ ¼ Bv# 1=2q (w/b)1–s]1/(1–s). ¼ [(v/a)/(w/b)]s so wl/vk q1/g[(v/a)1–s þ qa–ab–b vawb. b/a. 10.9 a. C ¼ b. C ¼ c. wl/vk (v/w)s–1 (b/a)s. l/k d. Labor’s relative share is an increasing function of b/a. If s > 1, labor’s share moves in the same direction as v/w. If s < 1, labor’s relative share moves in the opposite direction to v/w. This accords with intuition on how substitutability should affect shares. ¼ ¼ 10.11 @ ln Ci=@ ln wj # a. sij ¼ @ ln Cj=@ ln wi # b. sij ¼ c. See detailed solutions. @ ln Cj =@ ln wj ¼ @ ln Ci=@ ln wi ¼ exc i , wj # exc j , wi# exc exc j , wj . i , wi. CHAPTER 11 11.1 a. q b. p c. q 50. 200. 5P # ¼ ¼ ¼ 50. wq2/4. 11.3 a. C ¼ b. p(P, w) c. q d. ¼ l(P, w) ¼ 2P/w. P 2/w. P 2/w2. ¼ 11.5 a. Diminishing returns is needed to ensure that a proﬁt-maximizing output choice exists. (w v)q2/100, P (P, v, w) 20, P v) þ 50P/(w ¼ 13,500. þ ¼ 25P2/(w 6,000. þ v). ¼ ¼ b. C(v, w, q) c. q d. q ¼ @P/@P 30, P ¼ ¼ ¼ 11.7 a., b. q a bP, P q/b aq)/b, ¼ ¼ a/b, and the mr curve has double the a/b, R # (q2 þ 2q/b ¼ mr slope of the demand curve, so d ¼ # # Pq ¼ mr # q/b. ¼ # 1/e) P(1 c. mr d. It follows since e ¼ þ 1/b). P(1 þ @q/@P Æ P/q. ¼ ¼ 11.9 b. Diminishing returns is needed to ensure increasing marginal cost. c. s determines how ﬁrms adapt to disparate input prices. d. q @P=@P 1 # KPg= Þ v1 Þ. ð The size of s does not affect the supply elasticity, but greater substitutability implies that increases in one input price will shift the supply curve less. w1 þ Þ # # Þð r ð r 1 # g 1 # g= e. See detailed solutions. 11.11 a. Follow the indicated steps. By analogy to part c of Problem 11.10, @q$=@v @k=@P. b. As argued in the text, @l=@w ¼ # 0. By similar argu0, implying the last term of the ( ments, @k=@v displayed equation in part a is positive. ( c. First, differentiate the deﬁnitional relation with respect to w. Second, differentiate the relation with respect to v, and use this expression to substitute result for @k$=@w substitute @ls=@k$. @l$=@w. Finally, the d. The increase in long vs. short-run costs from a wage increase w 0 < w 00 can be compared by combining three facts: • C(v, w 0, q) • C(v, w 00, q) • SC(v, w 0, q, k 0) for k 0 SC(v, w 00, q, k 00) for k 00 ¼ ¼ SC(v, w 00, q, k 0). kc(v, w 0, q) kc(v, w 00, q) SC(v, w 00, q, k 00) ¼ ¼ ¼ ( 11.13 a. See detailed answers for proof. b. The formula for cross-price elasticity of input demand weighs both terms by the share of the other input. The effect of a change in the price of the other input will depend primarily on the importance of this other input. c. Using Shephard’s lemma and an implication of Euler’s Theorem (Cww ¼ # AKL: ALL ¼ # vKCwvC wLCwCv ¼ # Sk SL vCwv=w) shows 1=16 and xs 11.15 If the assets are separate, the equilibrium investments a2=16, yielding joint surplus are xs G ¼ 3=16 . If GM acquires both assets, equilibrium ð þ a2=4, yielding joint 0 and xb investments are xb surplus a2=4. The latter joint surplus is higher if 3p . a2 Þ ﬃﬃﬃ CHAPTER 12 12.1 a. q b. Q c. P Pp 10 1,000 ﬃﬃﬃ 25; Q ¼ ¼ ¼ 20. # Pp # 3,000. ﬃﬃﬃ ¼ 2,000. # ¼ 10,000P. 5.99. 12.3 6. a. P ¼ 60,100 b. q ¼ 6.01, P c. P ¼ 600. d. eq,p ¼ # a 0 P 6. ¼ b 0 Q 359,800 – 59,950P. ¼ 6.002; P c 0 P ¼ d 0 eq,p ¼ # 5.998. ¼ 0.6; eq,p ¼ # 3,597. 50, Q 72, Q 12.5 20, P a. n b. n 24, P c. The increase for the makers 10, and w 14, and w $5,368. The linear approximation for the supply curve yields approximately the same result. 1,000, q 1,728, q 200. 288 12.7 a. P ¼ b. P ¼ c. D PS ¼ d. D rents 11, Q 12, Q ¼ ¼ 750. 750. ¼ 500, and r ¼ 1,000, and r 1. ¼ 2. 12.9 a. Long-run equilibrium requires P a a MC k=q AC bq ¼ þ þ k=b P ¼ 2 q ¼ b. Want A p BP # Hence n ﬃﬃﬃﬃﬃﬃﬃ supply A A # # ¼ ¼ ¼ a þ kbp demand. ﬃﬃﬃﬃﬃ ¼ kbp B . 2 a þ ð Þ kbp B a þ ð k=bp ﬃﬃﬃﬃﬃ . Þ MC. AC ¼ 2bq Hence ¼ þ ¼ nq ¼ n k=b ¼ p ﬃﬃﬃﬃﬃﬃﬃ c. A has a positive effect on n. That makes sense since A reﬂects the ‘‘size’’ of the market. If a > 0, the effect of B on n is clearly negative. ﬃﬃﬃﬃﬃ ﬃﬃﬃﬃ d. Fixed costs (k) have a negative effect on n. Higher marginal costs raise price and therefore reduce the number of ﬁrms. 12.11 a. Use the deadweight loss formula from Problem 12.10: n n Solutions to Odd-Numbered Problems 733 n T @+=@T ¼ Thus ti ¼ # # 1 i ¼ P eS # tipixi ¼ eDÞ 0 =eSeD ¼ k ð b. The above formula suggests 1=eDÞ 1=eS # k ð that higher taxes should be applied to goods with more inelastic supply and demand. c. This result was obtained under a set of very restric- tive assumptions. CHAPTER 13 900; 9x2 900; x 10, ¼ ¼ 9 on the production possibility frontier, c. ¼ þ ¼ 20. 2x, x2 2(2x)2 13.1 b. If y y ¼ If x y ¼ If x Hence RPT is approximately 2 20:24. 819=2 11 on the frontier, y ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0.25. p ¼ ¼ ¼ ¼ 779=2 ¼ Dy/Dx p # ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ # ¼ ( # 19:74. 0.50)/ ¼ ¼ C C ¼ ¼ þ þ Cloth. 100. 150. 13.3 Let F Food, C a. Labor constraint: F b. Land constraint: 2F c. Outer frontier satisﬁes both constraints. d. Frontier is concave because it must satisfy both constraints. Since the RPT 1 for the labor constraint and 2 for the land constraint, the production possibility frontier of part (c) exhibits an increasing RPT; hence it is concave. e. Constraints intersect at F 50, C ¼ ¼ 1 so PF /PC ¼ 2. F < 50, dC/dF dC/dF 5 ¼ # 2 so PF /PC ¼ ¼ # 4, then PF / PC ¼ f. If for consumers dC/dF 1.1, consumers will g. If PF / PC ¼ choose F 50 since both price lines are ¼ ‘‘tangent’’ to production possibility frontier at its kink. h. 0.8F ¼ # 1.9 or PF / PC ¼ 50, C ¼ þ 125, F since capital constraint is nowhere binding. 100. Capital constraint: C ¼ 111.1. This results in the same PPF 0.9C 0, C ¼ ¼ 0, F 5 4. ¼ ¼ 50. For 1. For F > 50, ¼ 13.5 a. The contract curve is a straight line. Only equilib- rium price ratio is PH/PC ¼ 4/3. b. Initial equilibrium on the contract curve. c. Not on the contract curve—equilibrium is between DW + ¼ 1 i ¼ P @+=@ki ¼ tipixi k T tiÞ þ # ð 1 i " # ¼ P eDeS= 2tipixi # eDÞ+ eS # ð :5 ½ kpixi ¼ 0 40H, 80C and 48H, 96C. d. Smith takes everything; Jones starves. 734 Solutions to Odd-Numbered Problems 13.7 a. px ¼ 26.2, y b. px ¼ 30.2, y c. Raises price of labor and relative price of x. 0.374, py ¼ 22.3. ¼ 0.284, py ¼ 18.5. ¼ 0.238, pk ¼ 0.338, pk ¼ 0.124, pl ¼ 0.162, pl ¼ 0.264, x 0.217, x ¼ ¼ 13.9 Computer simulations show that increasing returns to scale is still compatible with a concave production possibility frontier provided the input intensities of the two goods are suitably different. 13.11 a. Doubling prices leaves excess demands unchanged. b. Since, by Walras’ law, p1ED1 ¼ 0. The excess demand in market 1 can be calculated ex3p2 p1p2þ plicitly as: =p2 2p1p3Þ 1. This is also homogeneous of degree 0 in the prices. c. p2/p1 ¼ 0 and ED1 ¼ 6p2p3 þ 3, p3/p1 ¼ ED1 ¼ ð 3 þ 2 # 2p2 5. CHAPTER 14 14.1 24, P a. Q ¼ b. MC P c. Consumer surplus sumer surplus loss ¼ 5 and Q 288. ¼ ¼ ¼ ¼ 29, and p 576. ¼ 48. 1,152. Under monopoly, con576, deadweight ¼ ¼ 288, proﬁts ¼ 14.3 a. Q b. Q c. Q 14.5 a. P b. A ¼ ¼ ¼ ¼ ¼ 25, P 20, P 40, P ¼ ¼ ¼ 35, and p 50, and p 30, and p 625. 800. 800. ¼ ¼ ¼ 15, Q 3, P 5, C 15, Q ¼ ¼ ¼ ¼ 65, and p ¼ 6.05, and p 10. 12.25. ¼ 14.7 a. Under competition: P Under monopoly: P 10, Q ¼ 16, Q 500, CS ¼ 200, CS ¼ 400. 2,500. ¼ ¼ b. See graph in detailed solutions. c. Loss of 2,100, of which 800 is transferred to monopoly proﬁts, 400 is a loss from increased costs (not relevant in usual analysis), and 900 is a deadweight loss. ¼ 14.9 First-order conditions for a maximum imply X C(X)/C 0(X)—that is, X is chosen independently of Q. ¼ 14.11 a. @U/@Q b. P # Q(@P/@Q) @C/@Q 0, @U/@X 0, ¼ @C/@Q 0. ¼ # @C/@X þ @P/@X Æ Q @C/@X 0. ¼ # ¼ c. Using the hint, parts (a) and (b) imply @SW/@Q # Q(@P/@Q) > 0. @U/@X d. @SW/@X # ¼ @P/@X Æ Q, where the derivatives are calculated at the monopolist’s proﬁt-maximizing choices. It is generally not possible to sign this expression. # ¼ CHAPTER 15 15.1 a. Pm b. Pc c. Pb 75, Pm Qm ¼ qc 50, pc i ¼ 0, Qb ¼ 2,500. i ¼ 150, pb 0. 5,625. ¼ i ¼ ¼ ¼ ¼ 15.3 a. Equilibrium quantities are qc (1 i ¼ c2)/3, P c ¼ pc 1 þ ¼ CSc. Further, Qc (2 c1 # ¼ # c2)2/9, Pc pc 2c1 þ (1 i ¼ Pc c2)2/18, and Wc c1 # b. The diagram looks like Figure 15.2. A reduction in ﬁrm 1’s cost would shift its best response out, increasing its equilibrium output and reducing 2’s. 2ci þ # c1 þ (1 þ 2, CSc pc ¼ cj)/3. c2)/3, (2 # ¼ þ # 1/(2 (1 b). # 2b)/(2 15.5 a. p$i ¼ b); p$i ¼ b. q$i ¼ c. The diagram would look like Figure 15.4. An increase in b would shift out both best responses and result in higher equilibrium prices for both. b)2. 1/(2 # # # 75=2. 15.7 a. q$1 ¼ 75, q$2 ¼ b. If ﬁrm 1 accommodates 2’s entry, it earns 2,812.5. 2’s deter To produce 1 needs K2p . Firm 1’s proﬁt from operating !q1 ¼ 2 150 alone in the market and producing this output is , which exceeds 2,812.5 if 150 Þ ð K2 , ﬃﬃﬃﬃﬃ 2 # 120.6. ﬃﬃﬃﬃﬃ K2p 2 entry, K2p ﬃﬃﬃﬃﬃ to # Þð c)/(n þ nc)/(n (a # (a þ 1)]2, CS$ 1)b. Further, Q$ 1), P$ (n2/b) Æ [(a n(a ¼ np$i ¼ þ c)/(n ¼ þ c)2/b]. Because ﬁrms are symmet- 15.9 a. q$i ¼ P$ ¼ (n þ [n /(n þ ric, si ¼ ¼ b. We can obtain a rough idea of the effect of merger by se
eing how the variables in part (a) change with a reduction in n. Per-ﬁrm output, price, industry ¼ 1)] Æ [(a 1/ n, implying H c)/(n # þ (n/b)[(a # 1)]2, and W$ ¼ # n(1/n)2 1)b, c)/ 1/n. # ¼ Solutions to Odd-Numbered Problems 735 c h for I < 8,000. That is: for h < or 14,000 6,000 hours, welfare grant creates a kink in the budget constraint at 6,000 hours of leisure. ¼ þ 16.5 a. For MEl ¼ and l ¼ 200/80 $2.50. ¼ MRPl, l/40 10 l/40 so 2l/40 ¼ 200. Get w from supply curve: w # ¼ l/80 10 ¼ ¼ b. For Carl, the marginal expense of labor now equals $4.00. Setting this equal the minimum wage—wm ¼ 240. to the MRP yields l c. Under perfect competition, a minimum wage means higher wages but fewer workers employed. Under monopsony, a minimum wage may result in higher wages and more workers employed. ¼ 16.7 a. Since q 1,200x ¼ ¼ 240x # @TR/@x ¼ 10x2. MRP 2x2, total revenue is 5q 1,200 20x. ¼ # lp . Total cost ¼ @C/@x ﬃﬃ MC 600. # Production of pelts x 10x2. Marginal cost tition, price of pelts 20x; x MC ¼ ¼ 30, px ¼ b. From Dan’s perspective, demand for pelts ¼ ¼ ¼ ¼ 1,200 20x, R px Æ x 1,200x ¼ ¼ 1,200 ¼ 20x. Yields x ¼ ¼ # ginal revenue: @R/@x marginal cost # # 20, px ¼ c. From UF’s perspective, supply of pelts ¼ 20x2 and MEx ¼ ¼ 1,200 MRPx ¼ 40x ¼ # 20, px ¼ 400. ¼ 40x. So MEx ¼ ¼ ¼ a solution of x px, total cost pxx 20x ¼ wl ¼ ¼ ¼ 20x. Under compepx ¼ 20x, MRP MRP ¼ 20x2. Mar40x set equal to 800. MC ¼ @C/@x 20x with 16.9 U E U E ½ ½ yjob1Þ+ ¼ ð yjob2Þ+ ¼ ð ¼ 40 100 ) E wh U ð ½ 800w # # Þ+ ¼ 0:5 ) ½ 0:5 1,600 ) E 100wh ½ 36w2 3; 200: wh + Þ 800w ¼ 0:5 # 64w2 2 ð + ¼ þ 50w2: # 16.11 a. @V/@w ¼ b. @xi/@w c. MEl ¼ l(1 ¼ (@V/@w)/(@V/@n). h) ¼ # ll(w, n), @V/@n l, l(w, n) ¼ @xi/@w\|U w ¼ @wl/@l ¼ constant þ ¼ l@w/@l ¼ þ l[@xi/@n]. w[1 þ 1/(el,w)]. index increase. Total 1/4 into the answers for proﬁt, and the Herﬁndahl output, consumer surplus, and welfare decrease. c2 ¼ 1/4, Q$ 1/8, and W$ 1/2, P$ 1/4. Also, H 1/2, P$ 1/2. c. Substituting c1 ¼ 15.3, we have q$i ¼ 1/8, CS$ ¼ d. Substituting c1 ¼ for 15.3, we have q$1 ¼ 5/12, P$ P$ ¼ 107/288. Also, H ¼ ¼ 0 and c2 ¼ 5/12, q$2 ¼ 29/144, CS$ ¼ 29/49. ¼ ¼ 1/4 into the answers 7/12, 49/288, and W$ 2/12, Q$ ¼ ¼ ¼ ¼ e. Comparing part (a) with (b) suggests that increases in the Herﬁndahl index are associated with lower welfare. The opposite is evidenced in the comparison of part (c) to (d). ¼ 15.11 a. This is the indifference condition for a consumer located distance x from ﬁrm i. b. The proﬁt-maximizing price is p c. Setting p answer. ¼ t/n)/2. p$ and solving for p$ gives the speciﬁed (p$ þ þ ¼ c d. Substituting p p$ c t/n into the proﬁt func- ¼ tion gives the speciﬁed answer. K þ ¼ ¼ # 0 and solving for n yields n$ . transportation costs equal e. Setting t/n2 t=K the number of f. Total p ﬃﬃﬃﬃﬃﬃﬃﬃ half-segments between ﬁrms, 2n, times the transportation costs of consumers on the half segment, 1=2n t=8n2. Total ﬁxed cost equal nF. The 0 number of ﬁrms minimizing the sum of the two is R n$$ tx dx t=K 1=2 ¼ ¼ . ¼ ð Þ CHAPTER 16 p ﬃﬃﬃﬃﬃﬃﬃﬃ 16.1 a. Full income b. c. d. Supply is asymptotic to 2,000 hours as w rises. 1,400 hours. 1,700 hours. 2,000 hours. 40,000; l ¼ ¼ ¼ ¼ l l 16.3 a. Grant If I I I b. Grant ¼ ¼ ¼ ¼ ¼ 8,000. ¼ # 0.75(I). Grant Grant Grant 6,000 0 2,000 4,000 0 when 6,000 6,000. 4,500. 3,000. 0.75I ¼ ¼ ¼ # 0, I ¼ 6,000/0.75 CHAPTER 17 ¼ c. Assume there are 8,000 hours in the year. Full 4 Income d. Full Income ¼ - 8,000 32,000 c þ ¼ 4h. ¼ 32,000 32,000 38,000 ¼ ¼ ¼ þ þ # grant 6,000 24,000 0.75 Æ 4(8,000 4h c 3h h) # ¼ þ # þ 17.1 b. Income and substitution effects work in opposite directions. If @c1/@r < 0, then c2 is price elastic. c. Budget constraint passes through y1, y2, and rotates through this point as r changes. Income effect depends on whether y1 > c1 or y1 < c1 initially. 736 Solutions to Odd-Numbered Problems 17.3 25 years 17.5 a. Not at all. b. Tax would be on opportunity cost of capital. c. Taxes are paid later, so cost of capital is reduced. d. If tax rates decline, the beneﬁt of accelerated depre- ciation is reduced. 17.7 Using equation 17.66, we get e.75(p0 # p(15) ¼ e.75p0 # p(15) ¼ e.75p0 # 125 ¼ 63.6. p0 ¼ c0) þ e.75c0 þ 7(e.75 e# þ 0.3 c0e# 0.3 c0e# 0.3) 17.9 a. Maximizes expected utility. b. If marginal utility is convex, applying Jensen’s function implies E[U 0(c1)] > U 0(c0). So must increase next period’s inequality to that U 0[E(c1)] consumption to yield equality. ¼ c. Part (b) shows that this person will save more when next period’s consumption is random. d. Prompting added precautionary savings would require an even higher r, exacerbating the paradox. x þ # þ x) x2 … for x < 1. 17.11 a. Use x/(1 ¼ b. See detailed solutions for derivative. c. The increased output from a higher t must be balanced against (1) the delay in getting the ﬁrst yield, and (2) the opportunity cost of a delay in all future rotations. f (t) is asymptotic to 50 as t ﬁ . 1 d. e. t$ f. t$ 100. 104.1. ¼ ¼ CHAPTER 18 18.1 l$ a. tiff’s is (2/3)l$ l$ ¼ is c (1 c). ¼ b. # 1/3. The lawyer’s surplus is 1/18 and the plain- 2/9. ¼ c. The lawyer’s surplus is c2/2 and the plaintiffs c$ ¼ c. The optimal contingency fee for the plaintiff is 1/2. Her surplus is 1/4 and the lawyer’s is 1/8. d. With a 100% contingency fee, the lawyer chooses 1 and earns a surplus of 1/2, which the plain- l$ tiff can extract up front by selling the case to him. ¼ uLv 150. 18.3 The low type’s second-best quantity satisﬁes Equation 18.51 at the new parameter values: q$$L ¼ 1. The tariff is T $$L ¼ 30. The high type’s quantity is the 1 Þ ¼ ð 16. The tariff just satissame as in the ﬁrst best: q$$H ¼ ﬁes incentive compatibility: T $$H ¼ 18.5 a. With no insurance, a lefty’s expected utility is 9.1261 and a righty’s is 9.1893. The monopolist fully insures both at a premium that reduces each 208. to his no-insurance utility: pL ¼ 808 and pH ¼ b. Lefties receive the same policy as in part (a). c. Lefties are fully insured. The second-best values of the other policy terms (pL, pR, and xR) maximize (0.8)(1,000)]/2 the insurer’s expected proﬁt [pL # 0.2xR]/2 subject to the righty’s participaþ tion and lefty’s incentive compatibility constraints. A spreadsheet calculation shows that the solution is 808, p$$R ¼ approximately p$$L ¼ 0, and x$$R ¼ [pR # 0. (1/2)(2,000) 18.7 a. (1/2)(10,000) b. If sellers value cars at $8,000, only lemons will be sold at a market price of $2,000. If sellers value cars at $6,000, all cars will be sold at a market price of $6,000. $6,000. þ ¼ 18.9 The optimum of the fully informed patient satisﬁes (@Up/@m)/(@Up/@x) pm. The doctor’s pm@Up/@x 0. optimum satisﬁes pmU 0d þ Rearranging, MRS < pm, implying that the doctor chooses more medical care. pm or MRS @Up/@m ¼ ¼ # ¼ n i # Q b1) 18.11 a. Bidder 1 maximizes Pr(b1 > max(b2, . . . , bn))(v1 # b1), 2Pr(vi < b1/k) assuming rivals which equals (v1# ¼ use linear bidding strategies, which in turn equals 1. Maximizing with respect to b1 b1)(b1/k)n (v1 # # 1)/n. Expected revenue is E(v(n)) v1(n yields b1 ¼ (n 1), using the for1)/n. This equals (n # # mula for the expected value of the maximum order statistic v(n). b. Buyers bid bi ¼ (n 1). 1)/(n # c. Yes. d. Bids converge to valuations in the ﬁrst-price auction but don’t change in the second-price auction. Expected revenue approaches 1. vi. Expected revenue is E(v(n 1)/(n 1)) þ þ ¼ # CHAPTER 19 b. y 19.1 a. P b. P 19.3 a. n 20 and q 20, q 50. 40, MC ¼ ¼ ¼ ¼ 16, and tax 4. ¼ ¼ 400. The externality arises because one well’s ¼ drilling affects all wells’ output. 200. b. n ¼ c. Fee ¼ 2,000/well. 19.5 The tax will improve matters only if the output restriction required by the externality exceeds the output restriction brought about by the monopoly. 19.7 a. Roughly speaking, individuals would free-ride on each other under perfect competition, producing 0. More rigorously, in y ’ MRSi, the Nash equilibrium, each sets RPT ¼ .704, and utility yielding x$i ¼ .704. ¼ 0 and obtaining utility 7:04, x$ 70.4, y$ ¼ ’ ¼ Solutions to Odd-Numbered Problems 737 5, x 50, x/100 0.5, and utility ¼ ¼ ¼ 2:5p . ¼ 19.9 a. Want g0i to be the same for all ﬁrms. b. A uniform tax will not achieve the result ﬃﬃﬃﬃﬃﬃ in part (a). # ¼ (p c. In general, optimal pollution tax is t w/f 0) Æ 1/g 0, which will vary from ﬁrm to ﬁrm. However, if ﬁrms have simple linear production functions given ali, then a uniform tax can achieve efﬁby qi ¼ ciency even if gi differs among ﬁrms. In this case the optimal tax is t w)/a, where l is the l(a value of the Lagrange multiplier in the social optimum described in part (a). ¼ # d. It is more efﬁcient to tax pollution than to tax output. 19.11 a. Choose b and t so that y is the same in each state. Requires t U. ¼ b. b always equals (1 c. No. Because this person is risk averse, he or she will t)w and t U. ¼ # always opt for equal income in each state. This page intentionally left blank Glossary of Frequently Used Terms Some of the terms that are used frequently in this book are deﬁned below. The reader may wish to use the index to ﬁnd those sections of the text that give more complete descriptions of these concepts. Coase Theorem Result attributable to R Coase: if bargaining costs are zero, an efﬁcient allocation of resources can be attained in the presence of externalities through reliance on bargaining among the parties involved. A Adverse Selection The problem facing insurers that risky types are both more likely to accept an insurance policy and more expensive to serve. Asymmetric Information A situation in which an agent on one side of a transaction has information that the agent on the other side does not have. Average Cost Total cost per unit of output: AC =q Bayesian-Nash Equilibrium A strategy proﬁle in a twoplayer simultaneous-move game in which player 1 has private information. This generalizes the Nash equilibrium concept to allow for player 2’s beliefs about player 1’s type. Bertrand Paradox The Nash equilibrium in a simultaneous-move pricing game is competitive pricing even when there are only two
ﬁrms. s0i; s si; s Uið iÞ % $ s BRið si is a best response for player i to Best Response rivals’ strategies, s_i, denoted by ; if Uið si 2 iÞ $ C Ceteris Paribus Assumption The assumption that all other relevant factors are held constant when examining the inﬂuence of one particular variable in an economic model. Reﬂected in mathematical terms by the use of partial differentiation. for all s0i 2 iÞ $ si: Function showing relaCompensated Demand Function tionship between the price of a good and the quantity consumed while holding real income (or utility) constant. Denoted by xc(px, py, U). Compensated Price Elasticity The price elasticity of the compensated demand function xc exc; px ¼ Compensating Variation (CV) The compensation required to restore a person’s original utility level when prices change. @xc=@px & px; py; U ð . That is, px=xc. Þ Compensating Wage Differentials Differences in real wages that arise when the characteristics of occupations cause workers in their supply decisions to prefer one job over another. Complements (Gross) Two goods such that if the price of one rises, the quantity consumed of the other will fall. Goods x and y are gross complements if @x=@py < 0. See also Substitutes (Gross). Complements (Net) Two goods such that if the price of one rises, the quantity consumed of the other will fall, holding real income (utility) constant. Goods x and y are net complements if @x=@pyjU ¼ U < 0: – Such compensated cross-price effects are symmetric, that is, @x=@pyjU – U ¼ See also Substitutes (Net). Also called Hicksian substitutes and complements. @y=@pxjU U : – ¼ ¼ 739 740 Glossary of Frequently Used Terms Composite Commodity A group of goods whose prices all move together—the relative prices of goods in the group do not change. Such goods can be treated as a single commodity in many applications. Concave Function A function that lies everywhere below its tangent plane. Constant Cost Industry An industry in which expansion of output and entry by new ﬁrms has no effect on the cost curves of individual ﬁrms. Constant Returns to Scale See Returns to Scale. Consumer Surplus The area below the Marshallian demand curve and above market price. Shows what an individual would pay for the right to make voluntary transactions at this price. Changes in consumer surplus can be used to measure the welfare effects of price changes. Contingent Input Demand See Input Demand Functions. Contour Line The set of points along which a function has a constant value. Useful for graphing threedimensional functions in two dimensions. Individuals’ indifference curve maps and ﬁrms’ production isoquant maps are examples. Contract Curve The set of all the efﬁcient allocations of goods among those individuals in an exchange economy. Each of these allocations has the property that no one individual can be made better off without making someone else worse off. Cost Function See Total Cost Function. Cournot Equilibrium Equilibrium in duopoly quantitysetting game. A similar concept applies to an n-person game. Cross-price Elasticity of Demand ; ex; py ¼ px; py; I function x Þ ð For the demand py=x. @x=@py & D Deadweight Loss A loss of mutually beneﬁcial transactions. Losses in consumer and producer surplus that are not transferred to another economic agent. Decreasing Cost Industry An industry in which expansion of output generates cost-reducing externalities that cause the cost curves of those ﬁrms in the industry to shift downward. Decreasing Returns to Scale See Returns to Scale. Demand Curve A graph showing the ceteris paribus relationship between the price of a good and the quantity of that good purchased. A two-dimensional representation of the demand function x . Þ This is referred to as ‘‘Marshallian’’ demand to differentiate it from the compensated (Hicksian) demand concept. px; py; I ¼ x ð Diminishing Marginal Productivity Product. See Marginal Physical Diminishing Marginal Rate of Substitution Rate of Substitution. See Marginal Discount Factor The degree to which a payoff next period is discounted in making this period’s decisions; denoted by d in the text. If r is the single-period interest rate, then usually d = 1/(1 þ Discrimination, Price Occurs whenever a buyer or seller is able to use its market power effectively to separate markets and to follow a different price policy in each market. See also Price Discrimination. r). Dominant Strategy A strategy, s(i , for player i that is a best response to the all-strategy proﬁle of other players. Duality The relationship between any constrained maximization problem and its related ‘‘dual’’ constrained minimization problem. E Economic Cost The opportunity cost of using a particular good or resource. Economic Efﬁciency Exists when resources are allocated so that no activity can be increased without cutting back on some other activity. See also ParetoEfﬁcient Allocation. Edgeworth Box Diagram A graphic device used to demonstrate economic efﬁciency. Most frequently used to illustrate the contract curve in an exchange economy, but also useful in the theory of production. & x=y. @y=@x Elasticity A unit-free measure of the proportional effect of one variable on another. If y = f(x), then ey; x ¼ Entry Conditions Characteristics of an industry that determine the ease with which a new ﬁrm may begin production. Under perfect competition, entry is assumed to be costless, whereas in a monopolistic industry there are signiﬁcant barriers to entry. Envelope Theorem A mathematical result: the change in the maximum value of a function brought about by a change in a parameter of the function can be found by partially differentiating the function with respect to the parameter (when all other variables take on their optimal values). Glossary of Frequently Used Terms 741 Equilibrium A situation in which no actors have an incentive to change their behavior. At an equilibrium price, the quantity demanded by individuals is exactly equal to that which is supplied by all ﬁrms. Fixed Costs Costs that do not change as the level of output changes in the short run. Fixed costs are in many respects irrelevant to the theory of short-run price determination. See also Variable Costs. Equivalent Variation The added cost of attaining the new utility level when prices change. Euler’s Theorem A mathematical theorem: if f is homogeneous of degree k, then x1; . . . ; xnÞ ð f1x1 þ f2x2 þ & & & þ fnxn ¼ kf x1; . . . ; xnÞ ð : Exchange Economy An economy in which the supply of goods is ﬁxed (that is, no production takes place). The available goods, however, may be reallocated among individuals in the economy. Expansion Path The locus of those cost-minimizing input combinations that a ﬁrm will choose to produce various levels of output (when the prices of inputs are held constant). Expected Utility The average utility expected from a risky situation. If there are n outcomes, x1; . . . ; xn with probabilities p1; . . . ; pn pi ¼ utility is given by , then the expected 1 Þ ð P p2U E U ð Þ ¼ p1U x1Þ þ ð x2Þ þ & & & þ ð pnU : xnÞ ð Expenditure Function A function derived from the individual’s dual expenditure minimization problem. Shows the minimum expenditure necessary to achieve a given utility level: expenditures E(px, py, U). ¼ Externality An effect of one economic agent on another that is not taken into account by normal market behavior. F Financial Option Contract A contract offering the right, but not the obligation, to buy or sell an asset during some future period at a certain price. First-Mover Advantage The advantage that may be gained by the player who moves ﬁrst in a game. First-Order Conditions Mathematical conditions that must necessarily hold if a function is to take on its maximum or minimum value. Usually show that any activity should be increased to the point at which marginal beneﬁts equal marginal costs. First Theorem of Welfare Economics Every Walrasian Equilibrium is Pareto Optimal. G General Equilibrium Model A model of an economy that portrays the operation of many markets simultaneously. Giffen’s Paradox A situation in which the increase in a good’s price leads individuals to consume more of the good. Arises because the good in question is inferior and because the income effect induced by the price change is stronger than the substitution effect. H Hidden Action An action taken by one party to a contract that cannot be directly observed by the other party. Hidden Type A characteristic of one party to a contract that cannot be observed by the other party prior to agreeing to the contract. Homogeneous Function A function, f (x1, x2, . . . , xn), is homogeneous of degree k if mkf f ð ð mx1; mx2; . . . ; mxnÞ ¼ : x1; x2; . . . ; xnÞ Homothetic Function A function that can be represented as a monotonic transformation of a homogeneous function. The slopes of the contour lines for such a function depend only on the ratios of the variables that enter the function, not on their absolute levels. I Income and Substitution Effects Two analytically different effects that come into play when an individual is faced with a changed price for some good. Income effects arise because a change in the price of a good will affect an individual’s purchasing power. Even if purchasing power is held constant, however, substitution effects will cause individuals to reallocate their expectations. Substitution effects are reﬂected in movements along an indifference curve, whereas income effects entail a movement to a different indifference curve. See also Slutsky Equation. I=x. For the demand function Income Elasticity of Demand @x=@I ; ex; I ¼ x Þ px; py; I & ð Increasing Cost Industry An industry in which the expansion of output creates cost-increasing externalities, which cause the cost curves of those ﬁrms in the industry to shift upward. Increasing Returns to Scale See Returns to Scale. 742 Glossary of Frequently Used Terms Indifference Curve Map A contour map of an individual’s utility function showing those altern
ative bundles of goods from which the individual derives equal levels of welfare. Indirect Utility Function A representative of utility as a function of all prices and income. Individual Demand Curve The ceteris paribus relationship between the quantity of a good an individual chooses to consume and the good’s price. A two-dimensional representation of x x (px , py , I) for one person. ¼ Inferior Good A good that is bought in smaller quantities as an individual’s income rises. Inferior Input A factor of production that is used in smaller amounts as a ﬁrm’s output expands. Input Demand Functions These functions show how input demand for a proﬁt-maximizing ﬁrm is based on input prices and on the demand for output. The input demand function for labor, for example, can be written as l output. Contingent input demand functions [l c(v, w, q)] are derived from cost minimization and do not necessarily reﬂect proﬁt-maximizing output choices. l (P, v, w), where P is the market price of the ﬁrm’s ¼ Isoquant Map A contour map of the ﬁrm’s production function. The contours show the alternative combinations of productive inputs that can be used to produce a given level of output. K First-order conditions for an Kuhn-Tucker Conditions optimization problem in which inequality constraints are present. These are generalizations of the ﬁrst-order conditions for optimization with equality constraints. Marginal Input Expense The increase in total costs that results from hiring one more unit of an input. Marginal Physical Product (MP ) The additional output that can be produced by one more unit of a particular input while holding all other inputs constant. It is usually assumed that an input’s marginal productivity diminishes as additional units of the input are put into use while holding other inputs ﬁxed. If q @q=@l. f(k, l), MPl ¼ ¼ Marginal Rate of Substitution (MRS ) The rate at which an individual is willing to trade one good for another while remaining equally well off. The MRS is the absolute value of the slope of an indifference curve. MRS dy=dx ¼ $ U . – jU ¼ Marginal Revenue (MR ) The additional revenue obtained by a ﬁrm when it is able to sell one more unit of output. MR q=@q @p p 1 ð þ 1=eq; pÞ . ¼ ¼ & Marginal Revenue Product (MRP ) The extra revenue that accrues to a ﬁrm when it sells the output that is produced by one more unit of some input. In the case of labor, for example, MRPl = MR Æ MPl. Marginal Utility (MU ) The extra utility that an individual receives by consuming one more unit of a particular good. Market Demand The sum of the quantities of a good demanded by all individuals in a market. Will depend on the price of the good, prices of other goods, each consumer’s preferences, and on each consumer’s income. Market Period A very short period over which quantity supplied is ﬁxed and not responsive to changes in market price. L Limit Pricing Choice of low-price strategies to deter entry. Mixed Strategy A strategy in which a player chooses which pure strategy to play probabilistically. Lindahl Equilibrium A hypothetical solution to the public goods problem: the tax share that each individual pays plays the same role as an equilibrium market price in a competitive allocation. Long Run See Short Run–Long Run Distinction. Lump Sum Principle The demonstration that general purchasing power taxes or transfers are more efﬁcient than taxes or subsidies on individual goods. M Marginal Cost (MC ) The additional cost incurred by producing one more unit of output: MC @C=@q. ¼ Monopoly An industry in which there is only a single seller of the good in question. Monopsony An industry in which there is only a single buyer of the good in question. Moral Hazard The effect of insurance coverage on individuals’ decisions to undertake activities that may change the likelihood or sizes of losses. N Nash Equilibrium A strategy proﬁle such that, for each player i, si is a best response to the other i. players’ equilibrium strategies s( $ s(1; s(2; . . . ; s(nÞ ð Normal Good A good for which quantity demanded increases (or stays constant) as an individual’s income increases. Normative Analysis Economic analysis that takes a position on how economic actors or markets should operate. O Oligopoly An industry in which there are only a few sellers of the good in question. Opportunity Cost Doctrine The simple, though farreaching, observation that the true cost of any action can be measured by the value of the best alternative that must be forgone when the action is taken. Output and Substitution Effects Come into play when a change in the price of an input that a ﬁrm uses causes the ﬁrm to change the quantities of inputs it will demand. The substitution effect would occur even if output were held constant, and it is reﬂected by movements along an isoquant. Output effects, on the other hand, occur when output levels change and the ﬁrm moves to a new isoquant. P Paradox of Voting ity rule voting may not yield a determinate outcome but may instead cycle among alternatives. Illustrates the possibility that major- Pareto Efﬁcient Allocation An allocation of resources in which no one individual can be made better off without making someone else worse off. Partial Equilibrium Model A model of a single market that ignores repercussions in other markets. Perfect Competition The most widely used economic model: there are assumed to be a large number of buyers and sellers for any good, and each agent is a price taker. See also Price Taker. Positive Analysis Economic analysis that seeks to explain and predict actual economic events. Present Discounted Value (PDV ) The current value of a sum of money that is payable sometime in the future. Takes into account the effect of interest payments. Selling identical goods at different Price Discrimination prices. Requires sellers to have the ability to prevent resale. There are three types: ﬁrst degree—selling each unit at a different price to the individual willing to pay the most for it (‘‘perfect price discrimination’’); second degree— adopting price schedules that give buyers an incentive to Glossary of Frequently Used Terms 743 separate themselves into differing price categories; third degree—charging different prices in separated markets. For the demand function Price Elasticity of Demand x px=x. @x=@px & ; ex; px ¼ Þ px; py; I ð Price Taker An economic agent that makes decisions on the assumption that these decisions will have no effect on prevailing market prices. Principal-Agent Relationship The hiring of one person (the agent) by another person (the principal) to make economic decisions. Prisoners’ Dilemma Originally studied in the theory of games but has widespread applicability. The crux of the dilemma is that each individual, faced with the uncertainty of how others will behave, may be led to adopt a course of action that proves to be detrimental for all those individuals making the same decision. A strong coalition might have led to a solution preferred by everyone in the group. Producer Surplus 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. Production Function A conceptual mathematical function that records the relationship between a ﬁrm’s inputs and its outputs. If output is a function of capital and labor only, this would be denoted by q f (k, l). ¼ Production Possibility Frontier The locus of all the alternative quantities of several outputs that can be produced with ﬁxed amounts of productive inputs. Proﬁt Function The relationship between a ﬁrm’s maximum proﬁts (P) and the output and input prices it faces: P ¼ P*(P, v, w). Proﬁts The difference between the total revenue a ﬁrm receives and its total economic costs of production. Economic proﬁts equal zero under perfect competition in the long run. Monopoly proﬁts may be positive, however. Property Rights the rights of owners. Legal speciﬁcation of ownership and Public Good A good that once produced is available to all on a nonexclusive basis. Many public goods are also nonrival—additional individuals may beneﬁt from the good at zero marginal costs. 744 Glossary of Frequently Used Terms Q Quasi-concave Function A function for which the set of all points for which f (X) > k is convex. R Rate of Product Transformation (RPT ) The rate at which one output can be traded for another in the productive process while holding the total quantities of inputs constant. The RPT is the absolute value of the slope of the production possibility frontier. Rate of Return The rate at which present goods can be transformed into future goods. For example, a oneperiod rate of return of 10 percent implies that forgoing 1 unit of output this period will yield 1.10 units of output next period. Rate of Technical Substitution (RTS ) The rate at which one input may be traded off against another in the productive process while holding output constant. The RTS is the absolute value of the slope of an isoquant. RTS dk dl ¼ $ : q0 q ¼ ! ! ! ! Real Option An option arising in a setting outside of ﬁnancial markets. Rent Payments to a factor of production that are in excess of that amount necessary to keep it in its current employment. Rental Rate The cost of hiring one machine for one hour. Denoted by v in the text. Rent-Seeking Activities Economic agents engage in rent-seeking activities when they utilize the political process to generate economic rents that would not ordinarily occur in market transactions. Returns to Scale A way of classifying production functions that records how output responds to proportional increases in all inputs. If a proportional increase in all inputs causes output to increase by a smaller proportion, the production function is said to exhibit decreasing returns to scale. If output increases by a greater proportion than the inputs, the production function exhibi
ts increasing returns. Constant returns to scale is the middle ground where both inputs and outputs increase by the same proportions. Mathematically, if f increasing returns, k = 1 constant returns, and k < 1 decreasing returns. ; k > 1 implies mk; ml mkf Þ ¼ k; l ð Þ ð concave [that is, U 0 risk aversion is measured by r Relative risk aversion is measured by > 0; U 00 ]. Absolute W ð =U 0 U 00 W ð Þ . Þ W rr ð Þ ¼ $ WU 00 W ð U 0ð W Þ Þ : S Second-Order Conditions Mathematical conditions required to ensure that points for which ﬁrst-order conditions are satisﬁed are indeed true maximum or true minimum points. These conditions are satisﬁed by functions that obey certain convexity assumptions. Second Theorem of Welfare Economics Any Pareto optimal allocation can be attained as a Walrasian equilibrium by suitable transfers of initial endowments. Shephard’s Lemma Application of the envelope theorem, which shows that a consumer’s compensated demand functions and a ﬁrm’s (constant output) input demand functions can be derived from partial differentiation of expenditure functions or total cost functions, respectively. Shifting of a Tax Market response to the imposition of a tax that causes the incidence of the tax to be on some economic agent other than the one who actually pays the tax. Short Run, Long Run Distinction A conceptual distinction made in the theory of production that differentiates between a period of time over which some inputs are regarded as being ﬁxed and a longer period in which all inputs can be varied by the producer. Signaling Actions taken by individuals in markets characterized by hidden types in an effort to identify their true type. Slutsky Equation A mathematical representation of the substitution and income effects of a price change on utility-maximizing choices: @x=@px ¼ : @x=@pxjU Þ Social Welfare Function A hypothetical device that records societal views about equity among individuals. @x=@I ð – U $ X ¼ Subgame-Perfect Equilibrium A strategy proﬁle s(1; s(2; . . . ; s(nÞ ð every proper subgame. that constitutes a Nash equilibrium for Risk Aversion Unwillingness to accept fair bets. Arises when an individual’s utility of wealth function is Substitutes (Gross) Two goods such that if the price of one increases, more of the other good will be Glossary of Frequently Used Terms 745 demanded. That is x and y are gross substitutes if @x=@py > 0. See also Complements; Slutsky Equation. Substitutes (Net) Two goods such that if the price of one increases, more of the other good will be demanded if utility is held constant. That is, x and y are net substitutes if U Utility Function A mathematical conceptualization of the way in which individuals rank alternative bundles of commodities. If there are only two goods, x and y, utility is denoted by @x=@pyjU ¼ U > 0: – utility ¼ U(x, y). Net substitutability is symmetric in that @x=@pyjU @y=@pxjU ¼ See also Complements; Slutsky Equation. – U ¼ ¼ U : – See Income and Substitution Substitution Effects Effects; Output and Substitution Effects; Slutsky Equation. Sunk Cost An expenditure on an investment that cannot be reversed and has no resale value. For a proﬁt-maximizing ﬁrm, a func- Supply Function tion that shows quantity supplied (q) as a function of output price (P) and input prices (v, w): q(P, v, w). q ¼ Increases in production prompted by Supply Response changing demand conditions and market prices. Usually a distinction is made between short-run and longrun supply responses. T Tacit Collusion Choice of cooperative (monopoly) strategies without explicit collusion. Total Cost Function The relationship between (minimized) total costs, output, and input prices C(v, w, q). C ¼ V Variable Costs Costs that change in response to changes in the level of output being produced by a ﬁrm. This is in contrast to ﬁxed costs, which do not change. von Neumann–Morgenstern Utility A ranking of outcomes in uncertain situations such that individuals choose among these outcomes on the basis of their expected utility values. W Wage The cost of hiring one worker for one hour. Denoted by w in the text. Walrasian Equilibrium An allocation of resources and an associated price vector such that quantity demanded equals quantity supplied in all markets at these prices (assuming all parties act as price-takers). Walrasian Price Adjustment The assumption that markets are cleared through price adjustments in response to excess demand or supply. Z Zero-Sum Game A game in which winnings for one player are losses for the other player. This page intentionally left blank Index Author names are in italics; glossary terms are in boldface. A AC. See Average cost function (AC) Addiction, 112–113 Adverse selection, 222, 663–669 competitive insurance market and, 665–669 ﬁrst-best contract, 663 second-best contract, 663–665 Agents asymmetric information and, 642 deﬁned, 643 principal-agent model, 642–645, 720 Aggregation Cournot, 166–167 Engel, 166 of goods, 194, 204–205 AIDS (almost ideal demand system), 142–143, 184 Aizcorbe, Ana M., 182 Alcoa, entry deterrence by, 557–558 Aleskerov, Fuad, 112 Allocation of time, 581–584 graphical analysis, 583–584 income and substitution effects of change in real wage rate, 583 two-good model, 581–582 utility maximization, 582–583 Almost ideal demand system (AIDS), 142–143, 184 Altruism, 113, 117–118, 687 Anderson, E., 404 Annuities, 632 Antiderivatives calculating, 59–60 deﬁned, 58 Antitrust laws Alcoa, 558 explicit cartels and, 547 Standard Oil Company, 561 Appropriability effect, 563–564 Aquinas, St. Thomas, 10 Assumptions of nonsatiation, 120 testing, 4 See also Ceteris paribus assumption Asymmetric information, 238, 641–676 adverse selection in insurance, 663–669 auctions, 672–675 complex contracts as response to, 641–642 gross deﬁnitions, 190–191 hidden actions, 645 hidden types, 655–656 market signaling, 670–672 moral hazard in insurance, 650–655 nonlinear pricing, 656–663, 680–682 owner-manager relationship, 646–650 principal-agent model, 642–645 Atkeson, Andrew, 330 Attributes model, 198–199 Attributes of goods. See Household production models Auctions, 672–675 Automobiles ﬂexibility in fuel usage, 224–225, 228–230 tied sales, 529 used-car market, signaling in, 671–672 Average cost (AC), 341–342 deﬁned, 341 graphical analysis of, 343–345 properties of, 348–349 Average physical productivity, 305–306 Average revenue curve, 378–379 Axioms of rational choice, 89–90 B Backward induction, 273–274 Bairam, E., 330 Barriers to entry, 501–503 creation of, 502–503 legal, 502 oligopolies and, 562–563 technical, 501–502 Battle of the Sexes backward induction in, 273–274 expected payoffs in, 261 extensive form for, 269–270 formal deﬁnitions, 260 mixed strategies in, 262–263 Nash equilibrium in, 257–260, 270–271 subgame-perfect equilibrium, 271–273 Bayesian games, 277–282 Bayesian-Nash equilibrium, 278–282, 285–288 deﬁned, 280 games of incomplete information, 280 Tragedy of the Commons, 281–282 Bayes’ rule, 277, 284–285 Becker, Gary, 113, 277 Behrman, Jere R., 141 Beliefs of players, 283–285 posterior, 283–285 prior, 283–285 Beneﬁt-cost ratio, 41–42 Beneﬁts, mandated, 590 Bentham, Jeremy, 90 Bernat, G. A., 496 Bernoulli, Daniel, 210–212 Bertrand, J., 533 Bertrand game, 265, 531–534, 540 Cournot game versus, 540 differentiated products, 542–546, 574 feedback effect, 565–566 Nash equilibrium of, 533–534 natural-spring duopoly in, 536–537 tacit collusion in, 548–549 Bertrand paradox, 534 Best response 747 748 Index Cournot model, 537 deﬁned, 254 imperfect competition, 573 payoffs in, 255–257 Tragedy of the Commons, 267 Beta coefﬁcients, 247 Binomial distribution, 69 expected values of, 71–72 variances and standard deviations for, 73 Black, Duncan, 704–705 Blackorby, Charles, 204 Bolton, P., 680 Bonds, 632–633 Borjas, G. J., 331 Brander, J. A., 576 Brouwer’s theorem, 297, 475 Brown, D. K., 495 Buckley, P. A., 368 Budget constraints attributes model, 198–199 mathematical model of exchange, 472, 483–484 in two-good case, 119 Budget shares, 126–128, 141–143 almost ideal demand system, 142–143 CES utility, 142 linear expenditure system, 141–142 variability of, 141 Burniaux, J. M., 496 Business-stealing effect, 563–564 C Calculus, fundamental theorem of, 61 Capacity constraints, 540–541 Capital, 607–626 accumulation of, 607–608 capitalization of rents, 437 costs, 333–334 demand for, 616–618 energy substitutability and, 330 natural resource pricing, 623–626 present discounted value approach, 618–623 rate of return, 609–616 time and, 631–636 Capital asset pricing model (CAPM), 247 CARA (constant absolute risk aversion) function, 219, 244–245 Cardinal properties, 57–58 Cartels, 531–532 antitrust laws and, 547 natural-spring duopoly, 536–537 CDF (cumulative distribution function), 71 Central limit theorem, 70 CEOs (chief executive ofﬁcers), 372 Certainty equivalent, 216 CES utility, 104–105, 319–320, 330 budget shares and, 126–128, 142 cost functions, 346–347 demand elasticities and, 168–169 labor supply, 587–588 Ceteris paribus assumption, 5–6 partial derivatives and, 27 in utility-maximizing choices, 90–91 CGE models. See Computable general equilibrium (CGE) models Chain rule, 25, 30–32 Chance nodes, 278 Change in demand, 410 Change of variable, 59 Changes in income, 147–148 Chief executive ofﬁcers (CEOs), 372 China, changing demands for food in, 183–184 Choice, 112–113 individual, portfolio problem, 245–247 rational, axioms of, 89–90 special preferences, 112–113 See also State-preference model; Utility Clarke, E., 709 Clarke mechanism, 709 Classiﬁcation of long-run supply curves, 430–431 Closed shops, 598 CO2 reduction strategies, 496 Coase, Ronald, 401, 513, 693 Coase theorem, 693–694 Cobb-Douglas production function, 318–319 cost functions, 346 envelope relations and, 360–361 shifting, 351–352 Solow growth model, 329–330 technical progress in, 323–324 Cobb-Douglas utility, 102–103, 183 corner solutions, 125–128 labor supply and, 586–587 Commitment versus ﬂexibility,
551–552 Comparative statistics analysis, 422 changes in input costs, 433–435 in general equilibrium model, 467–469 industry structure, 432 shifts in demand, 432–433 Compensated cross-price elasticity of demand, 165 Compensated demand curves, 155–159 compensating variation and, 170 deﬁned, 157 relationship between compensated/ uncompensated curves, 158–159 relationship to uncompensated curves, 160–163 Shephard’s lemma, 157–158 See also Competitive insurance market; Imperfect competition Competitive insurance market adverse selection and, 665–669 equilibrium with hidden types, 668 equilibrium with perfect information, 667 moral hazard and, 654–655 signaling in, 670 See also Insurance Competitive price system, 457–458 behavioral assumptions, 458 law of one price, 457 Complements, 189–191 asymmetry of gross deﬁnitions, 190–191 gross, 190 imperfect competition, 573–576 net, 191–192 perfect, 103–104 Completeness and preferences, 89 Composite commodities, 193–196 generalizations and limitations, 194–196 housing costs as, 195–196 theorem, 194 two-stage budgeting and, 204–205 Compound interest, mathematics of, 631–636 Computable general equilibrium (CGE) models, 485–489, 495 economic insights from, 487–489 solving, 486 structure of, 486 Computers and empirical analysis, 18 Concave functions, 51, 53–54, 83–84 Concavity of production possibility frontier, 464–465 quasi-concave functions, 53–55 Condorcet, M. de, 704 Consols (perpetuities), 624–625, 632 Constant absolute risk aversion (CARA) function, 219, 244–245 Constant cost industry, 426–428 deﬁned, 430 inﬁnitely elastic supply, 427–428 initial equilibrium, 426–427 responses to increase in demand, 427 Constant elasticity, 379–380 Constant elasticity of substitution (CES) function. See CES utility Constant relative risk aversion, 220 Constant relative risk aversion (CRRA) Compensated demand functions, 157, function, 221 159–160 Compensated own-price elasticity of demand, 165 Compensating variation (CV), 170 Compensating wage differentials, 591–594 Competition allocative inefﬁciency and, 689–690 failure of competitive market, 697–698 for innovation, 567–568 perfect, 415, 426, 720 Constant returns to scale, 311 Constant risk aversion, 219–220 Constrained maximization, 39–45, 84–85 duality, 42–45 envelope theorem in, 45–46 ﬁrst-order conditions and, 40 formal problem, 39–40 Lagrange multiplier method, 39, 41–42 optimal fences and, 43–45 second-order conditions and, 52–53 Index 749 Consumer price index (CPI), 181–184 Consumer search, 546–547 Consumer surplus, 169–174 consumer welfare and expenditure function, 169–170 deﬁned, 173 overview, 170–172 using compensated demand curve to show CV, 170 welfare changes and Marshallian demand curve, 172–174 Consumer theory, relationship of ﬁrm to, 372–373 Consumption convexity and balance in, 96–99 of goods, utility from, 91 See also Indifference curves Contingent commodities fair markets for, 234 prices of, 233–234 states of world and, 233 Contingent input demand, 354–355 cost-minimizing input choices, 338 Shephard’s lemma and, 353–355 Continuity partial equilibrium competitive model, 453 preferences and, 89–90 Continuous actions, games with, 298 Continuous random variables, 67–68 Continuous time, 633–636 continuous growth, 634 duration, 636 payment streams, 635 Continuum of actions, 265–268 Contour lines, 34, 112 Contract curves, 463–464, 477–481 Contracts, 641–642 asymmetric information, 641–642 ﬁrst-best, 643, 651–652, 660, 666 second-best, 643–644, 652–654, 666–667 value of, 642 Controlled experiments, 6 Convex functions, 83–84 Convex indifference curves, 95–96, 97, 100–101 Convexity, 96–99 Corn Laws debate, 470–471 Correspondences, functions versus, 296–297 Cost-beneﬁt analysis, 231 Cost curves per-unit, 361–362 shifts in, 345–355 See also Cost functions Cost functions, 333–363 average and marginal, 341–342, 343–345 cost-minimizing input choices, 336–341 deﬁnitions of costs, 333–335 graphical analysis of total costs, 342–343 homogeneity, 347 input prices and, 347–348 Shephard’s lemma and elasticity of substitution, 355 shifts in cost curves and, 345–355 short-run, long-run distinction, 355–362 translog, 367–368 uncompensated, 158–159 See also Compensated demand curves Demand elasticities, 163–169 Cost industry decreasing, 429–430 increasing, 428–429 Cost minimization illustration of process, 340–341 principle of, 338 relationship between proﬁt maximization and, 335 Cournot, Antoine, 166, 534 Cournot aggregation, 166–167 Cournot equilibrium, 554 Cournot game, 265–266, 534–540 feedback effect, 565–566 imperfect competition, 574 long-run equilibrium and, 564–565 Nash equilibrium of Cournot game, 535–538 natural-spring duopoly, 536–537 prices versus quantities, 540 tacit collusion in, 550–551 varying number of ﬁrms and, 539–540 Covariance, 74–76 CPI (consumer price index), 181–184 Cross-partial derivatives, 50 Cross-price effects asymmetry in, 190–191 net substitutes and complements, 192 proﬁt maximization and input demand, 392–393 Slutsky decomposition, 188–189 Cross-price elasticity of demand, 163 Cross-productivity effects, 309–310 CRRA (constant relative risk aversion) function, 221 Cumulative distribution function (CDF), 71 CV (compensating variation), 170, 716 D Deadweight loss, 444–445 Deaton, Angus, 143 Decrease in price, graphical analysis of, 149 Decreasing cost industry, 430 Decreasing returns to scale, 312–313 Deﬁnite integrals deﬁned, 60 differentiating, 62–63 Delay, option value of, 230 Demand. See Supply and demand Demand aggregation and estimation, 453–455 Demand curves deﬁned, 152 demand functions and, 154–155 importance of shape of supply curve, 421–422 importance to supply curves, 420–421 individual, 152–155 shifts in, 154, 421 compensated price elasticities, 165 Marshallian, 163–164 price elasticity and total spending, 164 price elasticity of demand, 164 relationships among, 165–169 Demand functions, 145–147 demand curves and, 154–155 indirect utility function, 128 mathematical model of exchange, 472–473 Demand relationships among goods, 187–200 attributes of goods, 197–200 composite commodities, 193–196 home production, 197–200 implicit prices, 197–200 net substitutes and complements, 191–192 overview, 187 simplifying demand and two-stage budgeting, 204–205 substitutability with many goods, 193 substitutes and complements, 189–191 two-good case, 187–189 Derivatives cross-partial, 50 deﬁned, 22 homogeneity and, 56 partial, 26–30 rules for ﬁnding, 24–25 second, 23–24 value of at point, 22–23 Deterring entry. See Entry deterrence/ accommodation Dewatripont, M., 680 Diamond, Peter, 546 Dictator game, 289 Diewert, W. Erwin, 205 Differentiated products. See Product differentiation Diminishing marginal productivity. See Marginal physical product (MP) Diminishing marginal rate of substitution. See Marginal rate of substitution (MRS) Diminishing returns, 462–463 Diminishing RTS. See Rate of technical substitution (RTS) Direct approach, 4 Discount factor, 275–277, 547–550 Discrete random variables, 67–68 Discrimination, price. See Price discrimination Disequilibrium behavior, 442 Dissipation effect, 567 Diversiﬁcation, 223–224 Dominant strategies deﬁned, 257 Nash equilibrium, 257, 265 Doucouliagos, H., 368 Dual expenditure-minimization problem, 132 750 Index Duality, 42–45 Dufﬁeld, James A., 112 Durability of goods, 512 Dutch MIMIC model, 495 Dynamic optimization, 63–66 maximum principle, 64–66 optimal control problem, 63–64 Dynamic views of monopoly, 523 E Economic costs, 334–335 deﬁned, 334 Economic efﬁciency concept of, 17 welfare analysis and, 438–441 Economic goods, in utility functions, 92 Economic models, 3–18 ceteris paribus assumption, 5–6 economic theory of value, 9–17 modern developments in, 17–18 optimization assumptions, 7–8 positive-normative distinction, 8–9 structure of economic models, 6–7 theoretical models, 3 veriﬁcation of, 4–5 Economic proﬁts, 374 Edgeworth, Francis Y., 17 Edgeworth box diagram, 459, 460 Efﬁciency allocative inefﬁciency, 687–690 concept of, 17 efﬁcient allocations, 460–461, 688–689 Pareto efﬁcient allocation, 476 welfare analysis and, 438–441 Elasticity general deﬁnition of, 28–29 interpretation in mathematical model of market equilibrium, 423–424 marginal revenue and, 377–378 of substitution, 313–315, 355 of supply, 431 Elasticity of demand compensated cross-price, 165 compensated own-price, 165 cross-price, 163 monopolies and, 509–510 price, 163–164 Elasticity of substitution, 104, 350 deﬁned, 314 graphic description of, 314–315 See also CES utility Empirical analysis computers and, 18 importance of, 5 Empirical estimates, 431 Endogenous variables, 6–7 Energy capital and, 330 homothetic functions and, 205 Engel, Ernst, 141 Engel aggregation, 166 Engel’s law, 141 Entrepreneurial service costs, 334 Entry conditions. See Entry deterrence/ accommodation Entry deterrence/accommodation barriers to entry, 501–503 entry-deterrence model, 559–560 imperfect competition, 562–566, 576 in sequential game, 574–575 strategic entry deterrence, 557–559 Envelope theorem, 35–39 Cobb–Douglas cost functions and, 360–361 in constrained maximization problems, 45–46 direct, time-consuming approach, 36–37 envelope shortcut, 37 many-variable case, 37–39 proﬁt function, 385 Shephard’s lemma and, 353 speciﬁc example of, 35–36 Environmental externalities, 702–703 Equilibrium Bayesian-Nash, 278–282, 285–288 computable, 485–489, 495 existence of, 265 median voter, 706 separating, 286–287, 561 subgame-perfect, 271–273, 721 Walrasian, 473, 484–485 See also General equilibrium; Nash equilibrium; Partial equilibrium model Equilibrium path, 271 Equilibrium point, 11 Equilibrium price deﬁned, 418 determination of, 418–419, 465–467 of future goods, 613, 614 supply-demand equilibrium, 12 Equilibrium rate of return, 614 Euler’s theorem, 56, 193 Evolutionary games and learning, 290 Exact price indices, 183–184 Exchange, mathematical model of, 471–482 demand functions and homogeneity, 472–473 equilibrium and Walras’ law, 4
73–474 existence of equilibrium in exchange model, 474–475 ﬁrst theorem of welfare economics, 476–478 second theorem of welfare economics, 478–481 social welfare functions, 481–482 utility, initial endowments, and budget constraints, 472 vector notation, 471–472 Exchange economy, 479–481 Exchange value, labor theory of, 10 Exclusive goods, 695 Exogenous variables, 6–7 Expansion path, 338–340 Expected utility, 210–214 Expected value, 70–72, 209 Expenditure functions, 169–170 deﬁned, 132–133 properties of, 134–135 substitution bias and, 182 Expenditure minimization, 131–134 Experimental games, 288–289 Dictator game, 289 Prisoners’ Dilemma, 288–289 Ultimatum game, 289 Exponential distribution, 70 expected values of random variables, 72 variances and standard deviations, 73 Extensive form games of incomplete information, 279 of sequential games, 269–270 Externalities, 685–710 allocative inefﬁciency and, 687–690 deﬁned, 686, 718 deﬁning, 685–687 graphic analysis of, 691 solutions to externality problem, 691–694 F Factor intensities, 463–464 Factor prices, 470–471 Fair bets, 214–216 Fair gambles, 210–211 Fair markets for contingent goods, 234 Fama, E. F., 247 Farmland reserve pricing, 529 Feedback effect, 565–566 Feenstra, Robert C., 183 Financial option contracts, 225 Finitely repeated games, 274–275, 547 Firms, 371–373 complicating factors, 371–372 expansion path, cost-minimizing input choices, 338–341 in oligopoly setting, 562–566 proﬁt maximization, 401–404 relationship to consumer theory, 372–373 simple model of, 371 First-best contracts, 643, 660 adverse selection and, 663 monopoly insurers, 666 moral hazard and, 651–652 nonlinear pricing, 680 principal-agent model, 643–645 First-best nonlinear pricing, 657–659 First-degree price discrimination, 514–515 First-mover advantage, 552–555 First-order conditions, 123–124 Lagrange multiplier method, 40 for maximum, 23, 33–34, 120–121 First theorem of welfare economics, 476–478 deﬁned, 477 Edgeworth box diagram, 478 Fisher Body, 372, 401–404 Index 751 General Motors (GM), 372, 401–404 Giffen, Robert, 152 Giffen’s paradox, 151–152 Glicksberg, I. L., 298 GM (General Motors), 372, 401–404 Goods changes in price of, 149–153 demand relationships among, 187–200, 204–205 durability of, 512 exclusive, 695 fair markets for contingent, 234 future, 609–610, 614 inferior, 147–148, 150–151 information as, 231–232 nonrival, 695 normal, 147–148 See also Demand relationships among goods; Public goods Gorman, W. M., 453 Gould, Brain W., 184 Government procurement, 496 Graaﬂund, J. J., 495 Grim strategy, 276 Gross complements, 188, 190 Gross deﬁnitions, asymmetry of, 190–191 Grossman, Michael, 113 Grossman, Sanford, 401 Gross substitutes, 188, 190 Groves, T., 708 Groves mechanism, 708–709 Growth accounting, 322–324 Gruber, Jonathan, 113 H Habits and addiction, 113 Hanley, N., 714 Hanson, K., 496 Harsanyi, John, 278 Hart, Oliver, 401 Hausman, Jerry, 182–183 Hayashi, Fumio, 141 Hessian matrix, 83–84 Heterogeneous demand, 512–513 Hicks, John, 192–194 Hicksian demand curves, 155–159 income aggregation and, 453 mathematical model of exchange, 472–473 proﬁt functions, 384 Homogeneous functions, 55–58 derivatives and, 56 Euler’s theorem, 56 homothetic functions, 56–58 Homothetic functions, 56–58, 205, 312–313 Homothetic preferences, 105 Hone, P., 368 Hotelling, Harold, 385, 544 Hotelling’s beach model, 544–546 Hotelling’s lemma, 385 Household production models, 197–200 corner solutions, 199–200 illustrating budget constraints, 198–199 linear attributes model, 198 overview, 197–198 Housing costs, as composite commodity, 195–196 Human capital, 591 Hybrid equilibria, 286, 288 I Immigration, 331 Imperfect competition, 531–568 Bertrand model, 533–534 capacity constraints, 540–541 Cournot model, 534–540 entry of ﬁrms, 562–566 innovation, 566–568 longer-run decisions, 551–557 pricing and output, 531–532 product differentiation, 541–547 signaling, 559–562 strategic entry deterrence, 557–559 strategic substitutes and complements, 573–576 tacit collusion, 547–551 Implicit (shadow) prices, 197–200 Implicit functions, 32–33 Income, changes in, 147–148 Income aggregation, 453 Income effects, 145–177 relationship between compensated and uncompensated, 158–159 Shephard’s lemma, 157–158 consumer surplus, 169–174 demand concepts and evaluation of price indices, 181–184 Hicksian demand functions. See Demand demand curves and functions, 145–147, Fixed costs short-run, 356 sunk costs versus, 552 Fixed point, 297 Fixed-proportions production function, 316–318, 345–346 Fixed supply, allocating, 65–67 Flexibility, 224–231 commitment versus, 551–552 computing option value, 227–230 implications for cost–beneﬁt analysis, 231 model of real options, 225–227 number of options, 227 option value of delay, 230 types of options, 224–225 Folk theorem for inﬁnitely repeated games, 275–277, 547–548 Foundations of Economic Analysis (Samuelson), 17 Friedman, Milton, 4 Fudenberg, D., 296, 575 Full-information case, 646–647 Functional form and elasticity, 28–29 Fundamental theorem of calculus, 61 Fuss, M., 367 Future goods, 609–610, 614 G Game theory, 251–291 basic concepts, 251–252 continuum of actions, 265–268 evolutionary games and learning, 290 existence of equilibrium, 265, 296–298 experimental games, 288–289 incomplete information, 277–278 mixed strategies, 260–265 Nash equilibrium, 254–260 payoffs, 252 players, 252 Prisoners’ Dilemma, 252–254 repeated games, 274–277 sequential games, 268–274 signaling games, 282–288 simultaneous Bayesian games, 278–282 strategies, 252 Garcia, S., 368 Gaussian (Normal) distribution, 70, 72–74 Gelauff, G. M. M., 495 General equilibrium, 457–489 comparative statistics analysis, 467–469 mathematical model of exchange, 471–482 mathematical model of production and functions Hicksian substitutes and complements, exchange, 482–485 191–192 modeling and factor prices, 469–471 perfectly competitive price system, 457–458 with two goods, 458–467 General equilibrium model, 14, 469–471, 692–693 computable, 485–489 simple, 487–488 welfare and, 495–496 Hicks’ second law of demand, 193 Hidden actions, 643, 645, 647–650 Hidden types, 643, 655–656, 668, 680 Hoffmann, S., 496 Hold-up problem, 403 Homogeneity of demand, 146–147, 165–166 expenditure functions, 134 153–159 demand elasticities, 163–169 income changes, 147–148 price changes, 149–153, 160–163 real wage rate changes, 583 two-good case, 187–188 See also Substitution effects Income elasticity of demand, 163 Incomplete-information games, 277–280 Increasing cost industry, 430 Increasing returns to scale, 311–313 752 Index Independent variables, 32 Indifference curve maps, 94–95, 102 Indifference curves convexity of, 95–96, 100–101 deﬁned, 93 maps, 94–95, 102 mathematics of, 99–101 and transitivity, 95 two-good case, 187 utility maximization in attributes model, 199–200 Indirect approach, 4 Indirect utility function, 128–129 Individual demand curves, 153–155 Industry structure, 432 Inequality constraints, 46–48 complementary slackness, 47–48 slack variables, 46–47 solution using Lagrange multipliers, 47 two-variable example, 46 Inferior goods, 147–148, 150–151 Inferior inputs, 339 Inﬁnitely elastic long-run supply, 427–428 Integration, 58–60 antiderivatives, 58–60 deﬁnite integrals, 60 differentiating deﬁnite integral, 62–63 fundamental theorem of calculus, 61 by parts, 59 Interest rates, 614–616 Interﬁrm externalities, 686 Inverse elasticity rule, 504 Investments, 552, 618–623 diversiﬁcation, 223–224 portfolio problem, 244–247 theory of, 618 Isoquant maps, 306–310 constant returns-to-scale production function, 312 elasticity of substitution, 315 importance of cross-productivity effects, 309–310 input inferiority, 339 rate of technical substitution, 307–309 simple production functions, 317 technical progress, 321 Isoquants, deﬁned, 306 Inﬁnitely repeated games, 275–277, See also Isoquant maps; Rate of technical 547–551 Information, 231–232 in economic models, 18 as good, 231–232 quantifying value of, 232 See also Asymmetric information Initial endowments, 472 Innovation, 566–568 competition for, 567–568 monopoly on, 566–567 Input costs changes in, 433–435 industry structure and, 434–435 Input demand decomposing into substitution and output components, 394–395 proﬁt maximization and, 388–395 Input demand functions, 390 Inputs contingent demand for, and Shephard’s lemma, 353–355 substitution, 349, 350 supply, and long-run producer surplus, 437–438 See also Cost minimization; Labor markets Insurance adverse selection, 222, 663–669 asymmetric information, 642 competitive theft, 654–655 moral hazard, 650–655 precaution against car theft, 653–654 premiums, 217–218 risk aversion and, 216–217 in state-preference model, 235–236 willingness to pay for, 216–217 See also Competitive insurance market substitution (RTS) J Jackman, Patrick C., 182 Jensen, M., 247 Jensen’s inequality, 216, 225 Job-market signaling, 283–284 hybrid equilibrium in, 288 pooling equilibrium in, 287 separating equilibrium in, 286–287 Jorgenson, Dale W., 205 K Kakutani’s ﬁxed point theorem, 297 Kehoe, Patrick J., 142, 330 Kehoe, Timothy J., 142 Koszegi, Botond, 113 Kuhn-Tucker conditions, 48 Kwoka, J. E., 529 L Labor costs, 333 mandated beneﬁts, 590 productivity, 304–305 Labor markets, 581–601 allocation of time, 581–584 equilibrium in, 589–590 labor unions, 598–601 market supply curve for labor, 588–589 mathematical analysis of labor supply, 584–588 monopsony in labor market, 595–597 wage variation, 591–595 Labor supply, 584–588 dual statement of problem, 585 Slutsky equation of labor supply, 585–588 Labor theory of exchange value, 10 Labor unions, 598–601 bargaining model, 600–601 modeling, 599–600 Lagrangian multiplier as beneﬁt–cost ratio, 41–42 interpreting, 41 method for, 39 in n-good utility maximization, 124 solution using, 47 Lancaster, K.J., 198 Latzko, D., 368 Law of one price, 457 Leading principal minors, 83 Learning games, 290 Legal barriers to entry, 502 Lemons, market for, 671–67
2 Leontief, Wassily, 320 Leontief production functions, 319–320, 330 Lerner, Abba, 378 Lerner index, 378 LES (linear expenditure system), 141–142 Lewbel, Arthur, 205 Lightning calculations, 117 Limitations and composite commodities, 197–200 Lindahl, Erik, 700 Lindahl equilibrium, 700–703 local public goods, 702–703 shortcomings of, 701–702 Linear attributes model, 198 Linear expenditure system (LES), 141–142 Linear pricing, 656 Linear production function, 316 Local public goods, 702–703 Locay, L., 529 Long run. See Short-run, long-run distinction Long-run analysis elasticity of supply, 431 long-run equilibrium, 425–428, 431–435 overview, 425 producer surplus in, 435–438 shape of supply curve, 428–431 Long-run competitive equilibrium, 425 Long-run cost curves, 358–361 Long-run elasticity of supply, 431 Long-run equilibrium comparative statistics analysis of, 431–435 conditions for, 425 constant cost case, 426–428 Cournot model, 564–565 in oligopoly, 563–565 Long-run producer surplus, 438–441 Long-run supply curves, 427–428 Lump sum principle, 129–131 M MacBeth, J., 247 Majority rule, 703–704 Malthus, Thomas, 304 Many-good case, 106 Marginal beneﬁt, 41 Marginal costs (MC), 341–342, 343–345, 348–349 deﬁned, 341 graphical analysis of, 343–344 pricing, 519–520 Marginal expense (ME), 596 Marginalism, 10–11, 373 Marginal physical product (MP), 304 Marginal productivity, 303–306 average physical productivity, 305–306 diminishing, 305–306 marginal physical product, 304 rate of technical substitution, 308 Marginal rate of substitution (MRS) deﬁned, 93 indifference curves, 99–100 with many goods, 106 Marginal revenue (MR), 375–380 curves, 378–380 deﬁned, 374 and elasticity, 377–378 from linear demand functions, 377 price–marginal cost markup, 378 Marginal revenue product (MRP), 389 Marginal utility (MU), 99–100, 124, 215–218, 244 Market basket index, 181–182 Market demand, 409–413 deﬁned, 412 elasticity of market demand, 413 generalizations, 411–412 market demand curve, 409–410 shifts in, 411 shifts in market demand curve, 410–411 simpliﬁed notation, 412 Market period, 413 Markets meaning of, 541–542 reaction to shift in demand, 419 rental rates, 616–617 separation, third-degree price discrimination through, 515–517 tools for studying, 18 Market supply curve, 415–416, 588–589 Marshall, Alfred, 11, 409 Marshallian demand, 163–164, 172–174, 182–183 Marshallian substitutes and complements, 190 Marshallian supply-demand synthesis, 11–14 Masten, S. E., 404 Matrix algebra constrained maxima, 84–85 quasi-concavity, 85 Maximal punishment principle for crime, 277 Maximization, 84 of one variable, 21–25 of several variables, 33–35 Maximum principle, 64–66 MC. See Marginal costs (MC) McFadden, D., 367 ME (marginal expense), 596 Meade, J., 686 Median voter theorem, 705–708 median voter equilibrium, 706 optimality of median voter result, 706–708 overview, 704–705 MES (minimum efﬁcient scale), 345 Mexico, NAFTA and, 142, 495 Microsoft, 567 Milliman, S. R., 715 Minimization of costs, 335, 338, 340–341 of expenditures, 131–134 Minimum efﬁcient scale (MES), 345 Mixed strategies, 260–265 computing mixed-strategy equilibria, 263–265 formal deﬁnitions, 261–262 Modern economics, founding of, 10 Monjardet, Bernard, 112 Monopolies, 501–524 allocational effects of, 508 barriers to entry, 501–503 coffee shop example, 662–663 deﬁned, 501 distributional effects of, 508 dynamic views of, 523 on innovation, 566–567 linear two-part tariffs, 528–529 natural, 501 price determination for, 503 price discrimination, 513–519 product quality and durability, 510–513 proﬁt maximization and output choice, 503–507 regulation of, 519–523 resource allocation and, 507–510 simple demand curves, 507 welfare losses and elasticity, 509–510 Monopoly output, 503–504, 506 Monopoly rents, 505 Monopsonies, 595–597 Monotonic transformations, 56–58 Monteverde, K., 404 Moore, John, 401 Moral hazard, 222, 650–655 competitive insurance market, 654–655 deﬁned, 651 ﬁrst-best insurance contract, 651–652 mathematical model, 651 second-best insurance contract, 652–654 Morgenstern, Oscar, 212 Morishima, M., 350 Morishima elasticities, 350 Most-favored customer program, 575–576 MP (marginal physical product), 304 MR. See Marginal revenue (MR) Index 753 MRP (marginal revenue product), 389 MRS. See Marginal rate of substitution (MRS) MU (marginal utility), 99–100, 124, 215–218, 244 Muellbauer, John, 143 Murphy, Kevin M., 113 Multivariable Calculus, 26–35 calculating partial derivatives, 26–27 chain rule with many variables, 30–32 elasticity, 28–29 ﬁrst-order conditions for maximum, 33–34 implicit functions, 32–33 partial derivatives, 26–30 second-order conditions, 34–35 Young’s theorem, 30 Mutual funds, 247 N NAFTA (North American Free Trade Agreement), 142, 495 Nash, John, 254, 296 Nash bargaining, 402 Nash equilibrium, 254–260 in Battle of the Sexes, 257–260 of Bertrand game, 533–534 of Cournot game, 535–538 deﬁned, 255 dominant strategies, 257 existence of, 296–298 formal deﬁnition, 254–255 imperfect competition, 573 inefﬁciency of, 698–700 in Prisoners’ Dilemma, 254–255 in sequential games, 270–271 underlining best-response payoffs, 255–257 Natural monopolies, 501, 519–520 Natural resource pricing, 623–626 decrease in prices, 624–625 proﬁt-maximizing pricing and output, 623–625 renewable resources, 626 social optimality, 625 substitution, 625–626 Natural-spring duopoly, 536–539 deterring entry, 557–558 Stackelberg model, 553–555 Natural-spring oligopoly, 539–540 Negative deﬁnite, 83–84 Negative externalities, 268 Nested production functions, 330 Net complements, 191–192, 716 Net substitutes, 191–192, 721 New goods bias, 182–183 n-good case, 122–128 corner solutions, 124–128 ﬁrst-order conditions, 123 implications of ﬁrst-order conditions, 123–124 interpreting Lagrange multiplier, 124 Nicoletti, G., 496 754 Index n-input case elasticity of substitution, 314–315 returns to scale, 313 Nominal interest rates, 614–616 Nondepreciating machines, 617 Nonexclusive goods, 694–695 Nonhomothetic preferences, 105 Nonlinear pricing, 656–663 with continuum of types, 680–682 ﬁrst-best case, 657–659 mathematical model, 657 second-best case, 659–663 Nonoptimality of short-run costs, 356–367 Nonrival goods, 695 Nonuniqueness of utility measures, 90 Normal (Gaussian) distribution, 70, 72–74 Normal form for Battle of the Sexes, 257 for Prisoners’ Dilemma, 252 Normal goods, 147–148 Normative analysis, 8–9 North American Free Trade Agreement (NAFTA), 142, 495 O Oczkowski, E., 142 Oi, Walter, 518 Oligopolies, 531–568 Bertrand model, 533–534 capacity constraints, 540–541 Cournot model, 534–540 deﬁned, 531 entry of ﬁrms, 562–566 innovation, 566–568 longer-run decisions, 551–557 pricing and output, 531–532 product differentiation, 541–547 signaling, 559–562 strategic entry deterrence, 557–559 strategic substitutes and complements, 573–576 tacit collusion, 547–551 See also Cournot game Oliviera-Martins, J., 496 Opportunity cost doctrine, 15, 464–465 Optimal control problem, 63–64 Optimality of median voter result, 706–708 Optimization assumptions, 7–8 dynamic, 63–66 Ordinal properties, 57–58 Output choice, 374 Output effects principle of, 393 proﬁt maximization and input demand, 391–392, 393 Outputs imperfect competition, 531–532 monopolies and, 503–504, 506 proﬁt-maximizing, for natural resources, 623–625 Owner-manager relationship, 646–650 comparison to standard model of ﬁrm, 650 full-information case, 646–647 hidden-action case, 647–650 Ownership of machines, 617–618 Political support for trade policies, 471 Pollution abatement of, 714–715 CO2 reduction strategies, 496 emission taxes in the United Kingdom, P Paradox of voting, 704 Pareto, Vilfredo, 17, 476 Pareto efﬁcient allocation, 476 Pareto superiority, 528–529 Partial derivatives calculating, 26–27 ceteris paribus assumption and, 27 deﬁned, 26 second-order, 29 units of measurement and, 27–28 Partial equilibrium model, 14, 409–447 comparative statistics analysis, 431–435 demand aggregation and estimation, 453–455 economic efﬁciency and welfare analysis, 438–441 long-run analysis, 425 long-run elasticity of supply, 431 long-run equilibrium, 426–428 market demand, 409–413 mathematical model of market equilibrium, 422–424 price controls and shortages, 441–442 pricing in very short run, 413–415 producer surplus in long run, 435–438 shape of long-run supply curve, 428–431 shifts in supply and demand curves, 419–422 short-run price determination, 415–419 tax incidence analysis, 442–446 timing of supply response, 413 Payoffs, 252 in Battle of the Sexes, 261 in best response, 255–257 in Rock, Paper, Scissors game, 259–260 PDF. See Probability density function (PDF) PDV. See Present discounted value (PDV) Perfect Bayesian equilibrium, 285 Perfect competition, 457–458 behavioral assumptions, 458 deﬁned, 415 law of one price, 457 long-run equilibrium, 426 Perfect complements, 103–104 Perfect price discrimination, 514–515 Perfect substitutes, 103 Perpetual rate of return, 608 Perpetuities (consols), 624–625, 632 Per-unit cost curves, 361–362 Philip, N. E., 142 Pigou, A. C., 691 Pigovian taxes, 691–693, 703 Players, 252, 278, 283–285 Point-slope formula, 33 714 pollution rights, 693 Pontryagin, L. S., 64 Pooling equilibrium, 286, 561 in competitive insurance market, 668 in job-market signaling game, 287 Portfolio problem, 244–247 CARA utility, 244–245 individual choices, 245–247 many risky assets, 245 mutual funds, 247 one risky asset, 244 optimal portfolios, 245 studies of CAPM, 247 Positive analysis, 9 Positive deﬁnite, 83–84 Positive-normative distinction, 8–9 Posterior beliefs, 283–285 Pratt, J. W., 217, 220 Pratt’s risk aversion measure, 217–219 Predatory pricing, 561–562 Predictions, testing, 4–5 Preferences, 89–107 axioms of rational choice, 89–90 many-good case, 106 mathematics of indifference curves, 99–101 overview, 89 trades and substitution, 92–99 utility, 90–92 utility functions for speciﬁc, 102–105 Present discounted value (PDV), 631–633 annuities and
perpetuities, 632 bonds, 632–633 investment decisions, 618–623 Price controls and shortages, 441–442 disequilibrium behavior, 442 welfare evaluation, 442 Price discrimination, 513–517 deﬁned, 513 perfect, 514–515 second-degree, 517–519 third-degree, 515–517 Price dispersion, 546–547 Price elasticity, 163, 164 Price–marginal cost markup, 378 Prices of contingent commodities, 233–234 of future goods, 609, 614 imperfect competition, 531–532 implicit, 197–200 law of one, 457 predatory, 561–562 response to changes in, 160–163 in short-run analysis, 415–419 versus value, 9–10 Index 755 in very short run, 413–415 welfare effects of, 172–174 See also Bertrand game; Consumer surplus; Equilibrium price; Expenditure functions; Natural resource pricing; Nonlinear pricing; Price discrimination Price schedules, 517–519 Price takers, 376, 380–384 Primont, Daniel, 204 Prince, R., 715 Principal-agent relationship, 642–645 Principles of Economics (Marshall), 11 Prior beliefs, 283–285 Prisoners’ Dilemma, 252–254 experiments with, 288–289 ﬁnitely repeated games, 274–275 inﬁnitely repeated games, 276 Nash equilibrium in, 255 normal form, 252 thinking strategically about, 252–254 variation of, 259 Private information. See Asymmetric information Probability density function (PDF) deﬁned, 68, 209 examples of, 68–70 random variables, 67 Producer surplus deﬁned, 387, 435 in long run, 435–438 in short run, 386–389 Product differentiation, 541–547 Bertrand competition with, 542–546 Bertrand model, 574 consumer search and price dispersion, 546–547 Hotelling’s beach model, 544–546 meaning of ‘‘market,’’ 541–542 toothpaste as a differentiated product, 543–544 Production and exchange, mathematical model of, 482–485 budget constraints and Walras’ law, 483–484 Walrasian equilibrium, 484 Welfare economics in Walrasian model with production, 484–485 Production externalities, 689–690 Production functions, 302–324, 329–331 CES, 319–320, 330 Cobb–Douglas, 318–319, 329–330 deﬁned, 303 elasticity of substitution, 313–315 ﬁxed proportions, 316–318 generalized Leontief, 330 isoquant maps and rate of technical substitution, 306–310 linear, 316 marginal productivity, 303–306 nested, 330 returns to scale, 310–313 technical progress, 320–324 translog, 331 two-input, 305–306 Quantifying value of information, 232 Quantitative size of shifts in cost curves, Production possibility frontier, 14–17, 350–351 461–462 concavity of, 464–465 deﬁned, 461 implicit functions and, 32–33 Proﬁt functions, 384–389 envelope results, 385 properties of, 384–385 short-run, 386–389 Proﬁt maximization, 371–396 boundaries of ﬁrm, 401–404 cost minimization and, 335 decisions, 380–381 ﬁnding derivatives and, 25 functions of variable, 49 graphical analysis, 375 input demand and, 388–395 marginalism and, 373 marginal revenue and, 375–380 by monopolies, 503–507 nature and behavior of ﬁrms, 371–373 optimization assumptions and, 7–8 output choice and, 374 overview, 4 principle of, 374 proﬁt functions, 384–389 second-order conditions and, 375 short-run supply by price-taking ﬁrm, 380–384 testing assumptions of, 4 testing predictions of, 4–5 Proﬁts, 374 monopolies, 504–505 See also Proﬁt functions; Proﬁt maximization Proper subgames, 271–273 Properties of expenditure functions, 134–135 Property rights, 402–403 Public goods attributes of, 694–696 deﬁned, 696 derivation of the demand for, 698 environmental externalities and production of, 702–703 externalities, 687 Lindahl pricing of, 700–703 resource allocation and, 696–700 Roommates’ dilemma, 699–701 simple political model, 705–708 voting and resource allocation, 703–705 voting mechanisms, 708–709 Puppy dog strategy, 555–556, 558–559, 573 Pure inﬂation, 146 Pure strategies, 259–260 Q Quality choice models and, 112–113 of products, 511–512 Quasi-concave function, 53–55 concave functions and, 53–54 convex indifference curves, 100 Quasi-concavity, 85 R Random variables continuous, 67–68 deﬁned, 209 discrete, 67–68 expected value of, 209 and probability density functions, 67 variance and standard deviation of, 209 Rate of product transformation (RPT), 461–462 Rate of return demand for future goods, 610 effects of changes in, 612–613 equilibrium, 614 interest rates, 614–616 overview, 607–609 price of future goods and, 609 regulation of, 521–522 supply of future goods, 613–614 utility maximization, 611–612 Rate of technical substitution (RTS) deﬁned, 307 diminishing, 308–309 importance of cross-productivity effects, 309–310 marginal productivities and, 308 reasons for diminishing, 308–309 Rational choice, axioms of, 89–90 Real interest rates, 614–616 Real option theory, 225–227 Reinsdorf, Marshall B., 183 Relative risk aversion, 220–221 Renewable resources, 626 Rent capitalization of, 437 monopoly, 505 Ricardian, 436–437 Rental rates, 616–617 Repeated games, 274–277 ﬁnitely, 274–275, 547 inﬁnitely, 275–277, 547–551 Replacement effect, 567 Resource allocation monopoly and, 507–510 public goods and, 696–700 Returns to scale, 310–313 constant, 311 deﬁned, 310 homothetic production functions, 312–313 n-input case, 313 Revealed preference theory, 174–176 graphical approach, 175 negativity of substitution effect, 175–176 756 Index Ricardian rent, 436–437 Ricardo, David, 10, 436 Risk aversion, 214–217 constant, 219–220 constant relative, 221 deﬁned, 216 fair bets and, 214–216 insurance and, 216–217 measuring, 217–221 state-preference approach to choice, 234–235 See also Uncertainty Risk premiums, 237–238 Robinson, S., 496 Rock, Paper, Scissors game, 259–260 Rockefeller, John D., 561 Rodriguez, A., 529 Roy’s identity, 182–183 RPT (rate of product transformation), 461–462, 720–721 RTS. See Rate of technical substitution (RTS) Russell, R. Robert, 204 S SAC (short-run average total cost function), 358, 361–362 St. Petersburg paradox, 210–212 Samuelson, Paul, 17, 174 Scarf, Herbert, 486 Scharfstein, D. S., 247 Schmalensee, R., 715 Schmittlein, D. C., 404 Schumpeter, J. A., 523 Second-best contracts adverse selection, 663–665 monopoly insurer, 666–667 moral hazard, 652–654 nonlinear pricing, 680–681 principal-agent model, 643–644 Second-best nonlinear pricing, 659–663 Second-degree price discrimination, 517–519 Second derivatives, 23–24 Second-order conditions, 23, 375, 390 concave and convex functions, 51, 83–84 constrained maxima, 84–85 curvature and, 48–55 for maximum, 84, 121–122 quasi-concavity, 85 several variables, 34–35 Second-order partial derivatives, 29–30 Second-party preferences, 113 Second theorem of welfare economics, 478–481 Selﬁshness, 117–118 Selten, Reinhard, 275 Separating equilibrium, 286–287, 561 Sequential Battle of the Sexes game, 268–269 Sequential games, 268–274 backward induction, 273–274 Battle of the Sexes, 268–269 extensive form, 269–270 Nash equilibria, 270–271 subgame-perfect equilibrium, 271–273 Shadow (implicit) prices, 197–200 Sharpe, W. F., 245 Shephard, R. W., 157 Shephard’s lemma, 157–158 contingent demand for inputs and, 353–355 deﬁned, 721 elasticity of substitution and, 355 net substitutes and complements, 192 Shogren, J. F., 714 Short run, long run distinction, 355–362 ﬁxed and variable costs, 356 graphs of per-unit cost curves, 361–362 nonoptimality of, 356–367 relationship between long-run cost curves and, 358–361 short-run marginal and average costs, 358 total costs, 356 Short-run analysis, 355–362 ﬁxed and variable costs, 356 graphs of per-unit cost curves, 361–362 nonoptimality of, 356–367 price determination, 415–419 producer surplus in, 388–395 relationship between long-run cost curves and, 358–361 short-run marginal and average costs, 358 total costs, 356 Short-run average total cost function (SAC), 358, 361–362 Short-run ﬁxed costs, 356 Short-run marginal cost function (SMC), 358, 361–362 Short-run market supply function, 416–417 Short-run supply curve, 381–384, 416 Short-run supply elasticity, 416 Short-run variable costs, 356 Shutdown decision, 381–384 Signaling, 559–562, 670–672 in competitive insurance markets, 670 entry-deterrence model, 559–560 market for lemons, 671–672 pooling equilibrium, 561 predatory pricing, 561–562 separating equilibrium, 561 Signaling games, 278, 282–288 Bayes’ rule, 284–285 job-market signaling, 283–284 perfect Bayesian equilibrium, 285–288 Simplexes, 297 Simultaneous games, 278–282 Bayesian–Nash equilibrium, 278–282 player types and beliefs, 278 sequential games versus, 268–272 Single-input case, 390–391 Single-peaked preferences, 704–705 Single-period rate of return, 608 Single variable calculus, 21–25 derivatives, 22 ﬁrst-order condition for maximum, 23 rules for ﬁnding derivatives, 24–25 second derivatives, 23–24 second-order conditions and curvature, 23, 48–49 value of derivative at point, 22–23 Slesnick, Daniel T., 205 Slutsky, Eugen, 161 Slutsky equation, 161–163 for cross-price effects, 188–189 of labor supply, 585–588 two-good case, 187–188 SMC (short-run marginal cost function), 358, 361–362 Smith, Adam, 10, 118, 310, 476 Smith, John Maynard, 290 Smith, R. B. W., 529 Smith, Vernon, 288 Social optimality, 625 Social welfare function, 481–482 Solow, R. M., 322–323, 329 Solow growth model, 329–330 Specialized inputs, 463 Special preferences, 112–113 habits and addiction, 113 quality, 112–113 second-party preferences, 113 threshold effects, 112 Spence, Michael, 282 Spencer, B. J., 576 Stackelberg, H. von, 552 Stackelberg model, 552–555 Stage games, 274–276 Standard deviation, 72–74 State-preference model, 232–238 contingent commodities, 233 fair markets for contingent goods, 234 graphic analysis of, 235–236 insurance in, 235–236 prices of contingent commodities, 233–234 risk aversion in, 234–235, 237–238 states of world and contingent commodities, 233 utility analysis, 233 States of the world, 233 Stein, J., 247 Stigler, George J., 113 Stock options, 224–225 Stocks, 61–62 Stoker, Thomas M., 205 Stone–Geary utility function, 142 Strategic entry deterrence, 557–559 Strategies, 252 dominant, 257, 265, 717 grim, 276 mixed, 260–265, 719 portfolio problem, 244–247 in Prisone
rs’ Dilemma, 252–254 puppy dog and top dog, 555–556, 558–559, 573 pure, 259–260 trigger, 274–276 Strictly mixed strategies, 260 Subgame-perfect equilibrium, 271–273 Subramanian, S., 496 Substitutes, 189–191 asymmetry of gross deﬁnitions, 190–191 elasticity of, Shephard’s lemma and, 355 gross, 190 imperfect competition, 573–576 with many goods, 193 of natural resources, 625–626 net, 191–192 perfect, 103 See also Trades and substitution Substitution bias expenditure functions and, 182 market basket index, 182 Substitution effects, 149–151, 161, 392 consumer surplus, 169–174 demand concepts and evaluation of price indices, 181–184 demand curves and functions, 153–159 demand elasticities, 163–169 demand functions, 145–147 impact on demand elasticities, 167–169 negativity of, 175–176 net substitutes and complements, 192 price changes, 149–153, 160–163 principle of, 393 proﬁt maximization and input demand, 391, 393 real wage rate changes, 583 revealed preference and, 174–176 two-good case, 187–188 See also Income effects Sunk costs, 552 Sun Tzu, 225 Supply and demand, 112–113 equilibrium, 12–13, 458–459 shifts in, 432–433 special preferences, 112–113 synthesis, 11–14 Supply curve importance of shape of demand curve, 420–421 importance to demand curves, 421–422 long-run, 428–431 monopoly, 506–507 reasons for shifts in, 420 Supply elasticity, 416–417 elasticity of, 431 Supply function, 382–383, 385, 388, 395, 415–417 Supply response, 413 Swan, Peter, 512 Swan’s independence assumption, 512 T Tacit collusion, 547–551 in Bertrand model, 548–549 in Cournot model, 550–551 in ﬁnitely repeated games, 547 in inﬁnitely repeated games, 547–551 Tariffs, two-part, 518–519, 528–529 Taxation environmental, 714 excess burden of, 445–446, 488–489 in general equilibrium model, 495, 692–693 Pigovian, 691–693, 703 voting for redistributive, 707–708 Tax incidence analysis, 442–446 deadweight loss and elasticity, 444–445 effects on attributes of transactions, 446 mathematical model of tax incidence, 443 transaction costs, 445–446 welfare analysis, 443–444 Taylor’s series, 80, 218 Technical barriers to entry, 501–502 Technical progress, 320–324 in Cobb–Douglas production function, 323–324 effects on production, 467 growth accounting, 322–324 measuring, 321 Teece, D. J., 404 Testing assumptions, 4 predictions, 4–5 Theil, H., 454 Theoretical models, 3 Theory of Games and Economic Behavior, The (von Neumann and Morgenstern), 212 Third-best outcome, 643–645 Third-degree price discrimination, 515–517 Thomas, A., 368 Threshold effects, 112 Tied sales, 529 Time allocation of, 581–584 capital and, 631–636 inconsistency, 512–513 Timing of supply response, 413 Tirole, J., 296, 575 Tobin, J., 245 Top dog strategy, 555–556, 558–559, 573 Total cost function, 341 Trade general equilibrium models, 495 imperfect competition, 576 political support for, 471 prices, 470–471 Trades and substitution, 92–99 convexity, 95–99 indifference curve map, 94–95 marginal rate of substitution, 92–94 transitivity, 95 Tragedy of the Commons, 266–268, 281–282 Transaction costs, 403–404, 445–446 Transitivity Index 757 indifference curves and, 95 preferences and, 89 Translog cost function applications of, 368 many-input, 368 with two inputs, 367 Translog production function, 331 Trigger strategies, 274–276 Tucker, A. W., 252 Two-good model allocation of time, 581–582 demand relationships among goods, 187–189 Two-good utility maximization, 119–122 budget constraint, 119 corner solutions, 122–129 ﬁrst-order conditions for maximum, 120–121 second-order conditions for maximum, 121–122 Two-input case, 391 Two-part pricing, 656 Two-part tariffs, 518–519, 528–529 Two-stage budgeting homothetic functions and energy demand, 205 relation to composition commodity theorem, 204–205 theory of, 204 Two-tier pricing systems, 520–521 Typology of public goods, 695–696 U Ultimatum game, 289 Uncertainty, 209–239 asymmetry of information, 238 diversiﬁcation, 223–224 in economic models, 18 expected utility hypothesis, 210–212 fair gambles, 210–211 ﬂexibility, 224–231 information as a good, 231–232 insurance, 222 mathematical statistics, 209 measuring risk aversion, 217–221 methods for reducing risk and, 222 portfolio problem, 244–247 risk aversion, 214–217 state-preference approach to choice under, 232–238 von Neumann–Morgenstern theorem, 212–214 Uncompensated demand curves, 158–159 Uniform distribution, 69–70, 72–73 Used-car market, signaling in, 671–672 Utility, 90–92 arguments of functions, 91–92 ceteris paribus assumption, 90–91 from consumption of goods, 91 deﬁned, 92 economic goods, 92 externalities in, 686–687 758 Index functions for speciﬁc preferences, 102–105 mathematical model of exchange, 472 maximization, 582–583, 611–612 nonuniqueness of measures, 90 See also CES utility; Cobb-Douglas utility; Indifference curves; Preferences Utility maximization, 117–136 altruism and selﬁshness, 117–118 in attributes model, 198–199 budget shares and, 141–143 expenditure minimization, 131–134 graphical analysis of two-good case, 119–122 indirect utility function, 128–129 individual’s intertemporal, 610–612 initial survey, 118 labor supply, 582 and lightning calculations, 117 lump sum principle, 129–131 n-good case, 122–128 properties of expenditure functions, 134–135 See also Demand relationships among exogenous, 6–7 independent, 32 independent, implicit functions and, 32 random, 67–68, 209 Variance, 72–74, 209 Vector notation, 471–472 Vedenov, Dmitry V., 112 Veriﬁcation of economic models, 4–5 importance of empirical analysis, 5 proﬁt-maximization model, 4 testing assumptions, 4 testing predictions, 4–5 Vickery, William, 672 Villarreal, Hector J., 184 von Neumann, John, 212 von Neumann-Morgenstern theorem, 212–214 Walrasian price adjustment, 473, 484–485 Walras’ law equilibrium and, 473–474 mathematical model of production and exchange, 483–484 Water-diamond paradox, 10, 14 Weakly dominated strategy, 673 Wealth and risk aversion measurement, 218–220 Wealth of Nations, The (Smith), 10 Welfare analysis, 443–444 applied analysis, 440–441 economic efﬁciency and, 438–441 economics, 17 effects of price changes, 172–174 evaluation, price controls and shortages, expected utility maximization, 442 213–214 utility index, 212–213 goods; Income effects; Substitution effects von Neumann-Morgenstern utility, V Value early economic thoughts on, 9–10 economic theory of, 9–17 labor theory of exchange, 10 of options, 227–230 Value and Capital (Hicks), 193–194 Value in exchange concept, 10 Value in use concept, 10 Variable costs, 356 Variables change of variable, 59 endogenous, 6–7 212–214, 216, 252 Voting, 708–709 Clarke mechanism, 709 Groves mechanism, 708–709 mechanisms, generalizations of, 709 median voter theorem, 705–708 resource allocation and, 703–705 W Wages compensating differentials, 591–594 variation in, 591–595 Wales, Terrence J., 205 Walras, Leon, 14, 473–474 ﬁrst theorem of welfare economics, 476–478 general equilibrium and, 495–496 general equilibrium models and, 495–496 loss computations, 440–441 monopolies and, 509–510 second theorem of welfare economics, 478–481 Westbrook, M. D., 368 Wetzstein, Michael E., 112 White, B., 714 Williamson, Oliver, 401 Y Yatchew, A., 368 Young’s theorem, 30
tan, and Somalia, has an economic dimension. If you are going to be part of solving those problems, you need to be able to understand them. Economics is crucial. • It is hard to overstate the importance of economics to good citizenship. You need to be able to vote intelligently on budgets, regulations, and laws in general. When the U.S. government came close to a standstill at the end of 2012 due to the “fiscal cliff,” what were the issues? Did you know? • A basic understanding of economics makes you a well-rounded thinker. When you read articles about economic issues, you will understand and be able to evaluate the writer’s argument. When you hear classmates, co-workers, or political candidates talking about economics, you will be able to distinguish between common sense and nonsense. You will find new ways of thinking about current events and about personal and business decisions, as well as current events and politics. The study of economics does not dictate the answers, but it can illuminate the different choices. 1.2 \| Microeconomics and Macroeconomics By the end of this section, you will be able to: • Describe microeconomics • Describe macroeconomics • Contrast monetary policy and fiscal policy Economics is concerned with the well-being of all people, including those with jobs and those without jobs, as well as those with high incomes and those with low incomes. Economics acknowledges that production of useful goods and services can create problems of environmental pollution. It explores the question of how investing in education helps to develop workers’ skills. It probes questions like how to tell when big businesses or big labor unions are operating in a way that benefits society as a whole and when they are operating in a way that benefits their owners or members at the expense of others. It looks at how government spending, taxes, and regulations affect decisions about production and consumption. It should be clear by now that economics covers considerable ground. We can divide that ground into two parts: Microeconomics focuses on the actions of individual agents within the economy, like households, workers, and businesses. Macroeconomics looks at the economy as a whole. It focuses on broad issues such as growth of production, the number of unemployed people, the inflationary increase in prices, government deficits, and levels of exports and imports. Microeconomics and macroeconomics are not separate subjects, but rather complementary perspectives on the overall subject of the economy. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 1 \| Welcome to Economics! 15 To understand why both microeconomic and macroeconomic perspectives are useful, consider the problem of studying a biological ecosystem like a lake. One person who sets out to study the lake might focus on specific topics: certain kinds of algae or plant life; the characteristics of particular fish or snails; or the trees surrounding the lake. Another person might take an overall view and instead consider the lake's ecosystem from top to bottom; what eats what, how the system stays in a rough balance, and what environmental stresses affect this balance. Both approaches are useful, and both examine the same lake, but the viewpoints are different. In a similar way, both microeconomics and macroeconomics study the same economy, but each has a different viewpoint. Whether you are scrutinizing lakes or economics, the micro and the macro insights should blend with each other. In studying a lake, the micro insights about particular plants and animals help to understand the overall food chain, while the macro insights about the overall food chain help to explain the environment in which individual plants and animals live. In economics, the micro decisions of individual businesses are influenced by whether the macroeconomy is healthy. For example, firms will be more likely to hire workers if the overall economy is growing. In turn, macroeconomy's performance ultimately depends on the microeconomic decisions that individual households and businesses make. Microeconomics What determines how households and individuals spend their budgets? What combination of goods and services will best fit their needs and wants, given the budget they have to spend? How do people decide whether to work, and if so, whether to work full time or part time? How do people decide how much to save for the future, or whether they should borrow to spend beyond their current means? What determines the products, and how many of each, a firm will produce and sell? What determines the prices a firm will charge? What determines how a firm will produce its products? What determines how many workers it will hire? How will a firm finance its business? When will a firm decide to expand, downsize, or even close? In the microeconomics part of this book, we will learn about the theory of consumer behavior, the theory of the firm, how markets for labor and other resources work, and how markets sometimes fail to work properly. Macroeconomics What determines the level of economic activity in a society? In other words, what determines how many goods and services a nation actually produces? What determines how many jobs are available in an economy? What determines a nation’s standard of living? What causes the economy to speed up or slow down? What causes firms to hire more workers or to lay them off? Finally, what causes the economy to grow over the long term? We can determine an economy's macroeconomic health by examining a number of goals: growth in the standard of living, low unemployment, and low inflation, to name the most important. How can we use government macroeconomic policy to pursue these goals? A nation's central bank conducts monetary policy, which involves policies that affect bank lending, interest rates, and financial capital markets. For the United States, this is the Federal Reserve. A nation's legislative body determines fiscal policy, which involves government spending and taxes. For the United States, this is the Congress and the executive branch, which originates the federal budget. These are the government's main tools. Americans tend to expect that government can fix whatever economic problems we encounter, but to what extent is that expectation realistic? These are just some of the issues that we will explore in the macroeconomic chapters of this book. 1.3 \| How Economists Use Theories and Models to Understand Economic Issues By the end of this section, you will be able to: Interpret a circular flow diagram • • Explain the importance of economic theories and models • Describe goods and services markets and labor markets 16 Chapter 1 \| Welcome to Economics! Figure 1.5 John Maynard Keynes One of the most influential economists in modern times was John Maynard Keynes. (Credit: Wikimedia Commons) John Maynard Keynes (1883–1946), one of the greatest economists of the twentieth century, pointed out that economics is not just a subject area but also a way of thinking. Keynes (Figure 1.5) famously wrote in the introduction to a fellow economist’s book: “[Economics] is a method rather than a doctrine, an apparatus of the mind, a technique of thinking, which helps its possessor to draw correct conclusions.” In other words, economics teaches you how to think, not what to think. Watch this video (http://openstax.org/l/Keynes) about John Maynard Keynes and his influence on economics. Economists see the world through a different lens than anthropologists, biologists, classicists, or practitioners of any other discipline. They analyze issues and problems using economic theories that are based on particular assumptions about human behavior. These assumptions tend to be different than the assumptions an anthropologist or psychologist might use. A theory is a simplified representation of how two or more variables interact with each other. The purpose of a theory is to take a complex, real-world issue and simplify it down to its essentials. If done well, this enables the analyst to understand the issue and any problems around it. A good theory is simple enough to understand, while complex enough to capture the key features of the object or situation you are studying. Sometimes economists use the term model instead of theory. Strictly speaking, a theory is a more abstract representation, while a model is a more applied or empirical representation. We use models to test theories, but for this course we will use the terms interchangeably. For example, an architect who is planning a major office building will often build a physical model that sits on a tabletop to show how the entire city block will look after the new building is constructed. Companies often build models of their new products, which are more rough and unfinished than the final product, but can still demonstrate how the new product will work. A good model to start with in economics is the circular flow diagram (Figure 1.6). It pictures the economy as consisting of two groups—households and firms—that interact in two markets: the goods and services market in which firms sell and households buy and the labor market in which households sell labor to business firms or other employees. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 1 \| Welcome to Economics! 17 Figure 1.6 The Circular Flow Diagram The circular flow diagram shows how households and firms interact in the goods and services market, and in the labor market. The direction of the arrows shows that in the goods and services market, households receive goods and services and pay firms for them. In the labor market, households provide labor and receive payment from firms through wages, salaries, and benefits. Firms produce and sell goods and services to households in the market for goods and services (or product market). Arrow “A” indicates this. Households pay for goods and services, which beco
mes the revenues to firms. Arrow “B” indicates this. Arrows A and B represent the two sides of the product market. Where do households obtain the income to buy goods and services? They provide the labor and other resources (e.g. land, capital, raw materials) firms need to produce goods and services in the market for inputs (or factors of production). Arrow “C” indicates this. In return, firms pay for the inputs (or resources) they use in the form of wages and other factor payments. Arrow “D” indicates this. Arrows “C” and “D” represent the two sides of the factor market. Of course, in the real world, there are many different markets for goods and services and markets for many different types of labor. The circular flow diagram simplifies this to make the picture easier to grasp. In the diagram, firms produce goods and services, which they sell to households in return for revenues. The outer circle shows this, and represents the two sides of the product market (for example, the market for goods and services) in which households demand and firms supply. Households sell their labor as workers to firms in return for wages, salaries, and benefits. The inner circle shows this and represents the two sides of the labor market in which households supply and firms demand. This version of the circular flow model is stripped down to the essentials, but it has enough features to explain how the product and labor markets work in the economy. We could easily add details to this basic model if we wanted to introduce more real-world elements, like financial markets, governments, and interactions with the rest of the globe (imports and exports). Economists carry a set of theories in their heads like a carpenter carries around a toolkit. When they see an economic issue or problem, they go through the theories they know to see if they can find one that fits. Then they use the theory to derive insights about the issue or problem. Economists express theories as diagrams, graphs, or even as mathematical equations. (Do not worry. In this course, we will mostly use graphs.) Economists do not figure out the answer to the problem first and then draw the graph to illustrate. Rather, they use the graph of the theory to help them figure out the answer. Although at the introductory level, you can sometimes figure out the right answer without applying a model, if you keep studying economics, before too long you will run into issues and problems that you will need to graph to solve. We explain both micro and macroeconomics in terms of theories and models. The most well-known theories are probably those of supply and demand, but you will learn a number of others. 18 Chapter 1 \| Welcome to Economics! 1.4 \| How To Organize Economies: An Overview of Economic Systems By the end of this section, you will be able to: • Contrast traditional economies, command economies, and market economies • Explain gross domestic product (GDP) • Assess the importance and effects of globalization Think about what a complex system a modern economy is. It includes all production of goods and services, all buying and selling, all employment. The economic life of every individual is interrelated, at least to a small extent, with the economic lives of thousands or even millions of other individuals. Who organizes and coordinates this system? Who insures that, for example, the number of televisions a society provides is the same as the amount it needs and wants? Who insures that the right number of employees work in the electronics industry? Who insures that televisions are produced in the best way possible? How does it all get done? There are at least three ways that societies organize an economy. The first is the traditional economy, which is the oldest economic system and is used in parts of Asia, Africa, and South America. Traditional economies organize their economic affairs the way they have always done (i.e., tradition). Occupations stay in the family. Most families are farmers who grow the crops using traditional methods. What you produce is what you consume. Because tradition drives the way of life, there is little economic progress or development. Figure 1.7 A Command Economy Ancient Egypt was an example of a command economy. (Credit: Jay Bergesen/ Flickr Creative Commons) Command economies are very different. In a command economy, economic effort is devoted to goals passed down from a ruler or ruling class. Ancient Egypt was a good example: a large part of economic life was devoted to building pyramids, like those in Figure 1.7, for the pharaohs. Medieval manor life is another example: the lord provided the land for growing crops and protection in the event of war. In return, vassals provided labor and soldiers to do the lord’s bidding. In the last century, communism emphasized command economies. In a command economy, the government decides what goods and services will be produced and what prices it will charge for them. The government decides what methods of production to use and sets wages for workers. The government provides many necessities like healthcare and education for free. Currently, Cuba and North Korea have command economies. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 1 \| Welcome to Economics! 19 Figure 1.8 A Market Economy Nothing says “market” more than The New York Stock Exchange. (Credit: Erik Drost/ Flickr Creative Commons) Although command economies have a very centralized structure for economic decisions, market economies have a very decentralized structure. A market is an institution that brings together buyers and sellers of goods or services, who may be either individuals or businesses. The New York Stock Exchange (Figure 1.8) is a prime example of a market which brings buyers and sellers together. In a market economy, decision-making is decentralized. Market economies are based on private enterprise: the private individuals or groups of private individuals own and operate the means of production (resources and businesses). Businesses supply goods and services based on demand. (In a command economy, by contrast, the government owns resources and businesses.) Supply of goods and services depends on what the demands. A person’s income is based on his or her ability to convert resources (especially labor) into something that society values. The more society values the person’s output, the higher the income (think Lady Gaga or LeBron James). In this scenario, market forces, not governments, determine economic decisions. Most economies in the real world are mixed. They combine elements of command and market (and even traditional) systems. The U.S. economy is positioned toward the market-oriented end of the spectrum. Many countries in Europe and Latin America, while primarily market-oriented, have a greater degree of government involvement in economic decisions than the U.S. economy. China and Russia, while over the past several decades have moved more in the direction of having a market-oriented system, remain closer to the command economy end of the spectrum. The Heritage Foundation provides information about how free and thus market-oriented different countries' are, as the following Clear It Up feature discusses. For a similar ranking, but one that defines freedom more broadly, see the Cato Foundation's Human Freedom Index (https://openstax.org/l/cato) . What countries are considered economically free? Who is in control of economic decisions? Are people free to do what they want and to work where they want? Are businesses free to produce when they want and what they choose, and to hire and fire as they wish? Are banks free to choose who will receive loans, or does the government control these kinds of choices? Each year, researchers at the Heritage Foundation and the Wall Street Journal look at 50 different categories of economic freedom for countries around the world. They give each nation a score based on the extent of economic freedom in each category. The 2016 Heritage Foundation’s Index of Economic Freedom report ranked 178 countries around the world: Table 1.1 lists some examples of the most free and the least free countries. Several additional countries were not ranked because of extreme instability that made judgments about economic freedom impossible. These countries include Afghanistan, Iraq, Libya, Syria, Somalia, and Yemen. The assigned rankings are inevitably based on estimates, yet even these rough measures can be useful for discerning trends. In 2015, 101 of the 178 included countries shifted toward greater economic freedom, although 77 of the countries shifted toward less economic freedom. In recent decades, the overall trend has been a higher level of economic freedom around the world. 20 Chapter 1 \| Welcome to Economics! Most Economic Freedom Least Economic Freedom 1. Hong Kong 2. Singapore 3. New Zealand 4. Switzerland 5. Australia 6. Canada 7. Chile 8. Ireland 9. Estonia 10. United Kingdom 11. United States 12. Denmark 167. Timor-Leste 168. Democratic Republic of Congo 169. Argentina 170. Equatorial Guinea 171. Iran 172. Republic of Congo 173. Eritrea 174. Turkmenistan 175. Zimbabwe 176. Venezuela 177. Cuba 178. North Korea Table 1.1 Economic Freedoms, 2016 (Source: The Heritage Foundation, 2016 Index of Economic Freedom, Country Rankings, http://www.heritage.org/index/ranking) Regulations: The Rules of the Game Markets and government regulations are always entangled. There is no such thing as an absolutely free market. Regulations always define the “rules of the game” in the economy. Economies that are primarily market-oriented have fewer regulations—ideally just enough to maintain an even playing field for participants. At a minimum, these laws govern matters like safeguarding private property against theft, protecting people from violence, enforcing legal contracts, preventing fraud, and collecting taxes. Conversely, even the most command-oriented economies operate using m
arkets. How else would buying and selling occur? The government heavily regulates decisions of what to produce and prices to charge. Heavily regulated economies often have underground economies (or black markets), which are markets where the buyers and sellers make transactions without the government’s approval. The question of how to organize economic institutions is typically not a black-or-white choice between all market or all government, but instead involves a balancing act over the appropriate combination of market freedom and government rules. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 1 \| Welcome to Economics! 21 Figure 1.9 Globalization Cargo ships are one mode of transportation for shipping goods in the global economy. (Credit: Raul Valdez/Flickr Creative Commons) The Rise of Globalization Recent decades have seen a trend toward globalization, which is the expanding cultural, political, and economic connections between people around the world. One measure of this is the increased buying and selling of goods, services, and assets across national borders—in other words, international trade and financial capital flows. Globalization has occurred for a number of reasons. Improvements in shipping, as illustrated by the container ship in Figure 1.9, and air cargo have driven down transportation costs. Innovations in computing and telecommunications have made it easier and cheaper to manage long-distance economic connections of production and sales. Many valuable products and services in the modern economy can take the form of information—for example: computer software; financial advice; travel planning; music, books and movies; and blueprints for designing a building. These products and many others can be transported over telephones and computer networks at ever-lower costs. Finally, international agreements and treaties between countries have encouraged greater trade. Table 1.2 presents one measure of globalization. It shows the percentage of domestic economic production that was exported for a selection of countries from 2010 to 2015, according to an entity known as The World Bank. Exports are the goods and services that one produces domestically and sells abroad. Imports are the goods and services that one produces abroad and then sells domestically. Gross domestic product (GDP) measures the size of total production in an economy. Thus, the ratio of exports divided by GDP measures what share of a country’s total economic production is sold in other countries. Country 2010 2011 2012 2013 2014 2015 Higher Income Countries United States Belgium Canada France Middle Income Countries Brazil Mexico South Korea 12.4 76.2 29.1 26.0 10.9 29.9 49.4 13.6 81.4 30.7 27.8 11.9 31.2 55.7 13.6 82.2 30.0 28.1 12.6 32.6 56.3 13.5 82.8 30.1 28.3 12.6 31.7 53.9 13.5 84.0 31.7 29.0 11.2 32.3 50.3 12.6 84.4 31.5 30.0 13.0 35.3 45.9 Table 1.2 The Extent of Globalization (exports/GDP) (Source: http://databank.worldbank.org/data/) 22 Chapter 1 \| Welcome to Economics! Country 2010 2011 2012 2013 2014 2015 Lower Income Countries Chad China India Nigeria 36.8 29.4 22.0 25.3 38.9 28.5 23.9 31.3 36.9 27.3 24.0 31.4 32.2 26.4 24.8 18.0 34.2 23.9 22.9 18.4 29.8 22.4 - - Table 1.2 The Extent of Globalization (exports/GDP) (Source: http://databank.worldbank.org/data/) In recent decades, the export/GDP ratio has generally risen, both worldwide and for the U.S. economy. Interestingly, the share of U.S. exports in proportion to the U.S. economy is well below the global average, in part because large economies like the United States can contain more of the division of labor inside their national borders. However, smaller economies like Belgium, Korea, and Canada need to trade across their borders with other countries to take full advantage of division of labor, specialization, and economies of scale. In this sense, the enormous U.S. economy is less affected by globalization than most other countries. Table 1.2 indicates that many medium and low income countries around the world, like Mexico and China, have also experienced a surge of globalization in recent decades. If an astronaut in orbit could put on special glasses that make all economic transactions visible as brightly colored lines and look down at Earth, the astronaut would see the planet covered with connections. Despite the rise in globalization over the last few decades, in recent years we've seen significant pushback against globalization from people across the world concerned about loss of jobs, loss of political sovereignty, and increased economic inequality. Prominent examples of this pushback include the 2016 vote in Great Britain to exit the European Union (i.e. Brexit), and the election of Donald J. Trump for President of the United States. Hopefully, you now have an idea about economics. Before you move to any other chapter of study, be sure to read the very important appendix to this chapter called The Use of Mathematics in Principles of Economics. It is essential that you learn more about how to read and use models in economics. Decisions ... Decisions in the Social Media Age The world we live in today provides nearly instant access to a wealth of information. Consider that as recently as the late 1970s, the Farmer’s Almanac, along with the Weather Bureau of the U.S. Department of Agriculture, were the primary sources American farmers used to determine when to plant and harvest their crops. Today, farmers are more likely to access, online, weather forecasts from the National Oceanic and Atmospheric Administration or watch the Weather Channel. After all, knowing the upcoming forecast could drive when to harvest crops. Consequently, knowing the upcoming weather could change the amount of crop harvested. Some relatively new information forums, such as Facebook, are rapidly changing how information is distributed; hence, influencing decision making. In 2014, the Pew Research Center reported that 71% of online adults use Facebook. This social media forum posts topics ranging from the National Basketball Association, to celebrity singers and performers, to farmers. Information helps us make decisions as simple as what to wear today to how many reporters the media should send to cover a crash. Each of these decisions is an economic decision. After all, resources are scarce. If the media send ten reporters to cover an accident, they are not available to cover other stories or complete other tasks. Information provides the necessary knowledge to make the best possible decisions on how to utilize scarce resources. Welcome to the world of economics! This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 1 \| Welcome to Economics! 23 KEY TERMS circular flow diagram a diagram that views the economy as consisting of households and firms interacting in a goods and services market and a labor market command economy an economy where economic decisions are passed down from government authority and where the government owns the resources division of labor the way in which different workers divide required tasks to produce a good or service economics the study of how humans make choices under conditions of scarcity economies of scale when the average cost of producing each individual unit declines as total output increases exports products (goods and services) made domestically and sold abroad fiscal policy economic policies that involve government spending and taxes globalization the trend in which buying and selling in markets have increasingly crossed national borders goods and services market a market in which firms are sellers of what they produce and households are buyers gross domestic product (GDP) measure of the size of total production in an economy imports products (goods and services) made abroad and then sold domestically labor market the market in which households sell their labor as workers to business firms or other employers macroeconomics the branch of economics that focuses on broad issues such as growth, unemployment, inflation, and trade balance market interaction between potential buyers and sellers; a combination of demand and supply market economy an economy where economic decisions are decentralized, private individuals own resources, and businesses supply goods and services based on demand microeconomics the branch of economics that focuses on actions of particular agents within the economy, like households, workers, and business firms model see theory monetary policy policy that involves altering the level of interest rates, the availability of credit in the economy, and the extent of borrowing private enterprise system where private individuals or groups of private individuals own and operate the means of production (resources and businesses) scarcity when human wants for goods and services exceed the available supply specialization when workers or firms focus on particular tasks for which they are well-suited within the overall production process theory a representation of an object or situation that is simplified while including enough of the key features to help us understand the object or situation traditional economy typically an agricultural economy where things are done the same as they have always been done underground economy a market where the buyers and sellers make transactions in violation of one or more 24 Chapter 1 \| Welcome to Economics! government regulations KEY CONCEPTS AND SUMMARY 1.1 What Is Economics, and Why Is It Important? Economics seeks to solve the problem of scarcity, which is when human wants for goods and services exceed the available supply. A modern economy displays a division of labor, in which people earn income by specializing in what they produce and then use that income to purchase the products they need or want. The division of labor allows individuals and firms to specialize and to produce more for several reasons: a) It allows the agents to focus
on areas of advantage due to natural factors and skill levels; b) It encourages the agents to learn and invent; c) It allows agents to take advantage of economies of scale. Division and specialization of labor only work when individuals can purchase what they do not produce in markets. Learning about economics helps you understand the major problems facing the world today, prepares you to be a good citizen, and helps you become a well-rounded thinker. 1.2 Microeconomics and Macroeconomics Microeconomics and macroeconomics are two different perspectives on the economy. The microeconomic perspective focuses on parts of the economy: individuals, firms, and industries. The macroeconomic perspective looks at the economy as a whole, focusing on goals like growth in the standard of living, unemployment, and inflation. Macroeconomics has two types of policies for pursuing these goals: monetary policy and fiscal policy. 1.3 How Economists Use Theories and Models to Understand Economic Issues Economists analyze problems differently than do other disciplinary experts. The main tools economists use are economic theories or models. A theory is not an illustration of the answer to a problem. Rather, a theory is a tool for determining the answer. 1.4 How To Organize Economies: An Overview of Economic Systems We can organize societies as traditional, command, or market-oriented economies. Most societies are a mix. The last few decades have seen globalization evolve as a result of growth in commercial and financial networks that cross national borders, making businesses and workers from different economies increasingly interdependent. SELF-CHECK QUESTIONS 1. What is scarcity? Can you think of two causes of scarcity? 2. Residents of the town of Smithfield like to consume hams, but each ham requires 10 people to produce it and takes a month. If the town has a total of 100 people, what is the maximum amount of ham the residents can consume in a month? 3. A consultant works for $200 per hour. She likes to eat vegetables, but is not very good at growing them. Why does it make more economic sense for her to spend her time at the consulting job and shop for her vegetables? 4. A computer systems engineer could paint his house, but it makes more sense for him to hire a painter to do it. Explain why. 5. What would be another example of a “system” in the real world that could serve as a metaphor for micro and macroeconomics? 6. Suppose we extend the circular flow model to add imports and exports. Copy the circular flow diagram onto a sheet of paper and then add a foreign country as a third agent. Draw a rough sketch of the flows of imports, exports, and the payments for each on your diagram. 7. What is an example of a problem in the world today, not mentioned in the chapter, that has an economic dimension? This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 1 \| Welcome to Economics! 25 8. The chapter defines private enterprise as a characteristic of market-oriented economies. What would public enterprise be? Hint: It is a characteristic of command economies. 9. Why might Belgium, France, Italy, and Sweden have a higher export to GDP ratio than the United States? REVIEW QUESTIONS 10. Give the three reasons that explain why the division of increases an economy’s level of production. labor 11. What are three reasons to study economics? How did 15. economics? John Maynard Keynes define 16. Are households primarily buyers or sellers in the goods and services market? In the labor market? 12. What is the difference between microeconomics and macroeconomics? 17. Are firms primarily buyers or sellers in the goods and services market? In the labor market? 13. What are examples of agents? individual economic 18. What are the three ways that societies can organize themselves economically? What 14. macroeconomics? are the three main goals of 19. What is globalization? How do you think it might have affected the economy over the past decade? CRITICAL THINKING QUESTIONS 20. Suppose you have a team of two workers: one is a baker and one is a chef. Explain why the kitchen can produce more meals in a given period of time if each worker specializes in what they do best than if each worker tries to do everything from appetizer to dessert. 24. Macroeconomics is an aggregate of what happens at the microeconomic level. Would it be possible for what happens at the macro level to differ from how economic agents would react to some stimulus at the micro level? Hint: Think about the behavior of crowds. 21. Why would division of labor without trade not work? 25. Why is it unfair or meaningless to criticize a theory as “unrealistic?” 22. Can you think of any examples of free goods, that is, goods or services that are not scarce? 23. A balanced federal budget and a balance of trade are secondary goals of macroeconomics, while growth in the standard of living (for example) is a primary goal. Why do you think that is so? 26. Suppose, as an economist, you are asked to analyze an issue unlike anything you have ever done before. Also, suppose you do not have a specific model for analyzing that issue. What should you do? Hint: What would a carpenter do in a similar situation? 27. Why do you think that most modern countries’ economies are a mix of command and market types? 28. Can you think of ways that globalization has helped you economically? Can you think of ways that it has not? 26 Chapter 1 \| Welcome to Economics! This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 27 2 \| Choice in a World of Scarcity Figure 2.1 Choices and Tradeoffs In general, the higher the degree, the higher the salary, so why aren’t more people pursuing higher degrees? The short answer: choices and tradeoffs. (Credit: modification of work by “Jim, the Photographer”/Flickr Creative Commons) Choices ... To What Degree? In 2015, the median income for workers who hold master's degrees varies from males to females. The average of the two is $2,951 weekly. Multiply this average by 52 weeks, and you get an average salary of $153,452. Compare that to the median weekly earnings for a full-time worker over 25 with no higher than a bachelor’s degree: $1,224 weekly and $63,648 a year. What about those with no higher than a high school diploma in 2015? They earn just $664 weekly and $34,528 over 12 months. In other words, says the Bureau of Labor Statistics (BLS), earning a bachelor’s degree boosted salaries 54% over what you would have earned if you had stopped your education after high school. A master’s degree yields a salary almost double that of a high school diploma. Given these statistics, we might expect many people to choose to go to college and at least earn a bachelor’s degree. Assuming that people want to improve their material well-being, it seems like they would make those choices that provide them with the greatest opportunity to consume goods and services. As it turns out, the analysis is not nearly as simple as this. In fact, in 2014, the BLS reported that while almost 88% of the population in the United States had a high school diploma, only 33.6% of 25–65 year olds had bachelor’s degrees, and only 7.4% of 25–65 year olds in 2014 had earned a master’s. This brings us to the subject of this chapter: why people make the choices they make and how economists explain those choices. 28 Chapter 2 \| Choice in a World of Scarcity Introduction to Choice in a World of Scarcity In this chapter, you will learn about: • How Individuals Make Choices Based on Their Budget Constraint • The Production Possibilities Frontier and Social Choices • Confronting Objections to the Economic Approach You will learn quickly when you examine the relationship between economics and scarcity that choices involve tradeoffs. Every choice has a cost. In 1968, the Rolling Stones recorded “You Can’t Always Get What You Want.” Economists chuckled, because they had been singing a similar tune for decades. English economist Lionel Robbins (1898–1984), in his Essay on the Nature and Significance of Economic Science in 1932, described not always getting what you want in this way: The time at our disposal is limited. There are only twenty-four hours in the day. We have to choose between the different uses to which they may be put. ... Everywhere we turn, if we choose one thing we must relinquish others which, in different circumstances, we would wish not to have relinquished. Scarcity of means to satisfy given ends is an almost ubiquitous condition of human nature. Because people live in a world of scarcity, they cannot have all the time, money, possessions, and experiences they wish. Neither can society. This chapter will continue our discussion of scarcity and the economic way of thinking by first introducing three critical concepts: opportunity cost, marginal decision making, and diminishing returns. Later, it will consider whether the economic way of thinking accurately describes either how we make choices and how we should make them. 2.1 \| How Individuals Make Choices Based on Their Budget Constraint By the end of this section, you will be able to: • Calculate and graph budget constraints • Explain opportunity sets and opportunity costs • Evaluate the law of diminishing marginal utility • Explain how marginal analysis and utility influence choices Consider the typical consumer’s budget problem. Consumers have a limited amount of income to spend on the things they need and want. Suppose Alphonso has $10 in spending money each week that he can allocate between bus tickets for getting to work and the burgers that he eats for lunch. Burgers cost $2 each, and bus tickets are 50 cents each. We can see Alphonso's budget problem in Figure 2.2. Figure 2.2 The Budget Constraint: Alphonso’s Consumption Choice Opportunity Frontier Each point on the budget constraint represents a combination of burgers and bus tickets whose total cost
adds up to Alphonso’s budget of $10. The relative price of burgers and bus tickets determines the slope of the budget constraint. All along the budget set, giving up one burger means gaining four bus tickets. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 29 The vertical axis in the figure shows burger purchases and the horizontal axis shows bus ticket purchases. If Alphonso spends all his money on burgers, he can afford five per week. ($10 per week/$2 per burger = 5 burgers per week.) However, if he does this, he will not be able to afford any bus tickets. Point A in the figure shows the choice (zero bus tickets and five burgers). Alternatively, if Alphonso spends all his money on bus tickets, he can afford 20 per week. ($10 per week/$0.50 per bus ticket = 20 bus tickets per week.) Then, however, he will not be able to afford any burgers. Point F shows this alternative choice (20 bus tickets and zero burgers). If we connect all the points between A and F, we get Alphonso's budget constraint. This indicates all the combination of burgers and bus tickets Alphonso can afford, given the price of the two goods and his budget amount. If Alphonso is like most people, he will choose some combination that includes both bus tickets and burgers. That is, he will choose some combination on the budget constraint that is between points A and F. Every point on (or inside) the constraint shows a combination of burgers and bus tickets that Alphonso can afford. Any point outside the constraint is not affordable, because it would cost more money than Alphonso has in his budget. The budget constraint clearly shows the tradeoff Alphonso faces in choosing between burgers and bus tickets. Suppose he is currently at point D, where he can afford 12 bus tickets and two burgers. What would it cost Alphonso for one more burger? It would be natural to answer $2, but that’s not the way economists think. Instead they ask, how many bus tickets would Alphonso have to give up to get one more burger, while staying within his budget? Since bus tickets cost 50 cents, Alphonso would have to give up four to afford one more burger. That is the true cost to Alphonso. The Concept of Opportunity Cost Economists use the term opportunity cost to indicate what one must give up to obtain what he or she desires. The idea behind opportunity cost is that the cost of one item is the lost opportunity to do or consume something else. In short, opportunity cost is the value of the next best alternative. For Alphonso, the opportunity cost of a burger is the four bus tickets he would have to give up. He would decide whether or not to choose the burger depending on whether the value of the burger exceeds the value of the forgone alternative—in this case, bus tickets. Since people must choose, they inevitably face tradeoffs in which they have to give up things they desire to obtain other things they desire more. View this website (http://openstaxcollege.org/l/linestanding) for an example of opportunity cost—paying someone else to wait in line for you. A fundamental principle of economics is that every choice has an opportunity cost. If you sleep through your economics class, the opportunity cost is the learning you miss from not attending class. If you spend your income on video games, you cannot spend it on movies. If you choose to marry one person, you give up the opportunity to marry anyone else. In short, opportunity cost is all around us and part of human existence. The following Work It Out feature shows a step-by-step analysis of a budget constraint calculation. Read through it to understand another important concept—slope—that we further explain in the appendix The Use of Mathematics in Principles of Economics. 30 Chapter 2 \| Choice in a World of Scarcity Understanding Budget Constraints Budget constraints are easy to understand if you apply a little math. The appendix The Use of Mathematics in Principles of Economics explains all the math you are likely to need in this book. Therefore, if math is not your strength, you might want to take a look at the appendix. Step 1: The equation for any budget constraint is: Budget = P1 × Q1 + P2 × Q2 where P and Q are the price and quantity of items purchased (which we assume here to be two items) and Budget is the amount of income one has to spend. Step 2. Apply the budget constraint equation to the scenario. In Alphonso’s case, this works out to be: Budget = P1 × Q1 + P2 × Q2 $10 budget = $2 per burger × quantity of burgers + $0.50 per bus ticket × quantity of bus tickets $10 = $2 × Qburgers + $0.50 × Qbus tickets Step 3. Using a little algebra, we can turn this into the familiar equation of a line: y = b + mx For Alphonso, this is: Step 4. Simplify the equation. Begin by multiplying both sides of the equation by 2: $10 = $2 × Qburgers + $0.50 × Qbus tickets 2 × 10 = 2 × 2 × Qburgers + 2 × 0.5 × Qbus tickets 20 = 4 × Qburgers + 1 × Qbus tickets Step 5. Subtract one bus ticket from both sides: 20 – Qbus tickets = 4 × Qburgers Divide each side by 4 to yield the answer: 5 – 0.25 × Qbus tickets = Qburgers or Qburgers = 5 – 0.25 × Qbus tickets Step 6. Notice that this equation fits the budget constraint in Figure 2.2. The vertical intercept is 5 and the slope is –0.25, just as the equation says. If you plug 20 bus tickets into the equation, you get 0 burgers. If you plug other numbers of bus tickets into the equation, you get the results (see Table 2.1), which are the points on Alphonso’s budget constraint. Point Quantity of Burgers (at $2) Quantity of Bus Tickets (at 50 cents) A B C D 5 4 3 2 Table 2.1 0 4 8 12 This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 31 Point Quantity of Burgers (at $2) Quantity of Bus Tickets (at 50 cents) E F 1 0 Table 2.1 16 20 Step 7. Notice that the slope of a budget constraint always shows the opportunity cost of the good which is on the horizontal axis. For Alphonso, the slope is −0.25, indicating that for every bus ticket he buys, he must give up 1/4 burger. To phrase it differently, for every four tickets he buys, Alphonso must give up 1 burger. There are two important observations here. First, the algebraic sign of the slope is negative, which means that the only way to get more of one good is to give up some of the other. Second, we define the slope as the price of bus tickets (whatever is on the horizontal axis in the graph) divided by the price of burgers (whatever is on the vertical axis), in this case $0.50/$2 = 0.25. If you want to determine the opportunity cost quickly, just divide the two prices. Identifying Opportunity Cost In many cases, it is reasonable to refer to the opportunity cost as the price. If your cousin buys a new bicycle for $300, then $300 measures the amount of “other consumption” that he has forsaken. For practical purposes, there may be no special need to identify the specific alternative product or products that he could have bought with that $300, but sometimes the price as measured in dollars may not accurately capture the true opportunity cost. This problem can loom especially large when costs of time are involved. For example, consider a boss who decides that all employees will attend a two-day retreat to “build team spirit.” The out-of-pocket monetary cost of the event may involve hiring an outside consulting firm to run the retreat, as well as room and board for all participants. However, an opportunity cost exists as well: during the two days of the retreat, none of the employees are doing any other work. Attending college is another case where the opportunity cost exceeds the monetary cost. The out-of-pocket costs of attending college include tuition, books, room and board, and other expenses. However, in addition, during the hours that you are attending class and studying, it is impossible to work at a paying job. Thus, college imposes both an out-of-pocket cost and an opportunity cost of lost earnings. What is the opportunity cost associated with increased airport security measures? After the terrorist plane hijackings on September 11, 2001, many steps were proposed to improve air travel safety. For example, the federal government could provide armed “sky marshals” who would travel inconspicuously with the rest of the passengers. The cost of having a sky marshal on every flight would be roughly $3 billion per year. Retrofitting all U.S. planes with reinforced cockpit doors to make it harder for terrorists to take over the plane would have a price tag of $450 million. Buying more sophisticated security equipment for airports, like three-dimensional baggage scanners and cameras linked to face recognition software, could cost another $2 billion. However, the single biggest cost of greater airline security does not involve spending money. It is the opportunity cost of additional waiting time at the airport. According to the United States Department of Transportation (DOT), there were 895.5 million systemwide (domestic and international) scheduled service passengers in 2015. Since the 9/11 hijackings, security screening has become more intensive, and consequently, the procedure takes longer than in the past. Say that, on average, each air passenger spends 32 Chapter 2 \| Choice in a World of Scarcity an extra 30 minutes in the airport per trip. Economists commonly place a value on time to convert an opportunity cost in time into a monetary figure. Because many air travelers are relatively high-paid business people, conservative estimates set the average price of time for air travelers at $20 per hour. By these backof-the-envelope calculations, the opportunity cost of delays in airports could be as much as 800 million × 0.5 hours × $20/hour, or $8 billion per year. Clearly, the opportunity costs of waiting time can be just as important as costs that involve direct spending. In some cases, realizing the opportu
nity cost can alter behavior. Imagine, for example, that you spend $8 on lunch every day at work. You may know perfectly well that bringing a lunch from home would cost only $3 a day, so the opportunity cost of buying lunch at the restaurant is $5 each day (that is, the $8 buying lunch costs minus the $3 your lunch from home would cost). Five dollars each day does not seem to be that much. However, if you project what that adds up to in a year—250 days a year × $5 per day equals $1,250, the cost, perhaps, of a decent vacation. If you describe the opportunity cost as “a nice vacation” instead of “$5 a day,” you might make different choices. Marginal Decision-Making and Diminishing Marginal Utility The budget constraint framework helps to emphasize that most choices in the real world are not about getting all of one thing or all of another; that is, they are not about choosing either the point at one end of the budget constraint or else the point all the way at the other end. Instead, most choices involve marginal analysis, which means examining the benefits and costs of choosing a little more or a little less of a good. People naturally compare costs and benefits, but often we look at total costs and total benefits, when the optimal choice necessitates comparing how costs and benefits change from one option to another. You might think of marginal analysis as “change analysis.” Marginal analysis is used throughout economics. We now turn to the notion of utility. People desire goods and services for the satisfaction or utility those goods and services provide. Utility, as we will see in the chapter on Consumer Choices, is subjective but that does not make it less real. Economists typically assume that the more of some good one consumes (for example, slices of pizza), the more utility one obtains. At the same time, the utility a person receives from consuming the first unit of a good is typically more than the utility received from consuming the fifth or the tenth unit of that same good. When Alphonso chooses between burgers and bus tickets, for example, the first few bus rides that he chooses might provide him with a great deal of utility—perhaps they help him get to a job interview or a doctor’s appointment. However, later bus rides might provide much less utility—they may only serve to kill time on a rainy day. Similarly, the first burger that Alphonso chooses to buy may be on a day when he missed breakfast and is ravenously hungry. However, if Alphonso has a burger every single day, the last few burgers may taste pretty boring. The general pattern that consumption of the first few units of any good tends to bring a higher level of utility to a person than consumption of later units is a common pattern. Economists refer to this pattern as the law of diminishing marginal utility, which means that as a person receives more of a good, the additional (or marginal) utility from each additional unit of the good declines. In other words, the first slice of pizza brings more satisfaction than the sixth. The law of diminishing marginal utility explains why people and societies rarely make all-or-nothing choices. You would not say, “My favorite food is ice cream, so I will eat nothing but ice cream from now on.” Instead, even if you get a very high level of utility from your favorite food, if you ate it exclusively, the additional or marginal utility from those last few servings would not be very high. Similarly, most workers do not say: “I enjoy leisure, so I’ll never work.” Instead, workers recognize that even though some leisure is very nice, a combination of all leisure and no income is not so attractive. The budget constraint framework suggests that when people make choices in a world of scarcity, they will use marginal analysis and think about whether they would prefer a little more or a little less. A rational consumer would only purchase additional units of some product as long as the marginal utility exceeds the opportunity cost. Suppose Alphonso moves down his budget constraint from Point A to Point B to Point C and further. As he consumes more bus tickets, the marginal utility of bus tickets will diminish, while the opportunity cost, that is, the marginal utility of foregone burgers, will increase. Eventually, the opportunity cost will exceed the marginal utility of an additional bus ticket. If Alphonso is rational, he won’t purchase more bus tickets once the marginal utility just equals the opportunity cost. While we can’t (yet) say exactly how many bus tickets Alphonso will buy, that number is unlikely to be the most he can afford, 20. Sunk Costs In the budget constraint framework, all decisions involve what will happen next: that is, what quantities of goods will This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 33 you consume, how many hours will you work, or how much will you save. These decisions do not look back to past choices. Thus, the budget constraint framework assumes that sunk costs, which are costs that were incurred in the past and cannot be recovered, should not affect the current decision. Consider the case of Selena, who pays $8 to see a movie, but after watching the film for 30 minutes, she knows that it is truly terrible. Should she stay and watch the rest of the movie because she paid for the ticket, or should she leave? The money she spent is a sunk cost, and unless the theater manager is sympathetic, Selena will not get a refund. However, staying in the movie still means paying an opportunity cost in time. Her choice is whether to spend the next 90 minutes suffering through a cinematic disaster or to do something—anything—else. The lesson of sunk costs is to forget about the money and time that is irretrievably gone and instead to focus on the marginal costs and benefits of current and future options. For people and firms alike, dealing with sunk costs can be frustrating. It often means admitting an earlier error in judgment. Many firms, for example, find it hard to give up on a new product that is doing poorly because they spent so much money in creating and launching the product. However, the lesson of sunk costs is to ignore them and make decisions based on what will happen in the future. From a Model with Two Goods to One of Many Goods The budget constraint diagram containing just two goods, like most models used in this book, is not realistic. After all, in a modern economy people choose from thousands of goods. However, thinking about a model with many goods is a straightforward extension of what we discussed here. Instead of drawing just one budget constraint, showing the tradeoff between two goods, you can draw multiple budget constraints, showing the possible tradeoffs between many different pairs of goods. In more advanced classes in economics, you would use mathematical equations that include many possible goods and services that can be purchased, together with their quantities and prices, and show how the total spending on all goods and services is limited to the overall budget available. The graph with two goods that we presented here clearly illustrates that every choice has an opportunity cost, which is the point that does carry over to the real world. 2.2 \| The Production Possibilities Frontier and Social Choices By the end of this section, you will be able to: Interpret production possibilities frontier graphs • • Contrast a budget constraint and a production possibilities frontier • Explain the relationship between a production possibilities frontier and the law of diminishing returns • Contrast productive efficiency and allocative efficiency • Define comparative advantage Just as individuals cannot have everything they want and must instead make choices, society as a whole cannot have everything it might want, either. This section of the chapter will explain the constraints society faces, using a model called the production possibilities frontier (PPF). There are more similarities than differences between individual choice and social choice. As you read this section, focus on the similarities. Because society has limited resources (e.g., labor, land, capital, raw materials) at any point in time, there is a limit to the quantities of goods and services it can produce. Suppose a society desires two products, healthcare and education. The production possibilities frontier in Figure 2.3 illustrates this situation. 34 Chapter 2 \| Choice in a World of Scarcity Figure 2.3 A Healthcare vs. Education Production Possibilities Frontier This production possibilities frontier shows a tradeoff between devoting social resources to healthcare and devoting them to education. At A all resources go to healthcare and at B, most go to healthcare. At D most resources go to education, and at F, all go to education. Figure 2.3 shows healthcare on the vertical axis and education on the horizontal axis. If the society were to allocate all of its resources to healthcare, it could produce at point A. However, it would not have any resources to produce education. If it were to allocate all of its resources to education, it could produce at point F. Alternatively, the society could choose to produce any combination of healthcare and education on the production possibilities frontier. In effect, the production possibilities frontier plays the same role for society as the budget constraint plays for Alphonso. Society can choose any combination of the two goods on or inside the PPF. However, it does not have enough resources to produce outside the PPF. Most importantly, the production possibilities frontier clearly shows the tradeoff between healthcare and education. Suppose society has chosen to operate at point B, and it is considering producing more education. Because the PPF is downward sloping from left to right, the only way society can obtain more education is by giving up some healthcare. That is the tradeoff society fac
es. Suppose it considers moving from point B to point C. What would the opportunity cost be for the additional education? The opportunity cost would be the healthcare society has to forgo. Just as with Alphonso’s budget constraint, the slope of the production possibilities frontier shows the opportunity cost. By now you might be saying, “Hey, this PPF is sounding like the budget constraint.” If so, read the following Clear It Up feature. What’s the difference between a budget constraint and a PPF? There are two major differences between a budget constraint and a production possibilities frontier. The first is the fact that the budget constraint is a straight line. This is because its slope is given by the relative prices of the two goods, which from the point of view of an individual consumer, are fixed, so the slope doesn't change. In contrast, the PPF has a curved shape because of the law of the diminishing returns. Thus, the slope is different at various points on the PPF. The second major difference is the absence of specific numbers on the axes of the PPF. There are no specific numbers because we do not know the exact amount of resources this imaginary economy has, nor do we know how many resources it takes to produce healthcare and how many resources it takes to produce education. If this were a real world example, that data would be available. Whether or not we have specific numbers, conceptually we can measure the opportunity cost of additional education as society moves from point B to point C on the PPF. We measure the additional education by the horizontal distance between B and C. The foregone healthcare is given by the vertical distance between B and C. The slope of the PPF between B and C is (approximately) the vertical distance (the “rise”) over the horizontal distance (the “run”). This is the opportunity cost of the additional education. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 35 The Shape of the PPF and the Law of Diminishing Returns The budget constraints that we presented earlier in this chapter, showing individual choices about what quantities of goods to consume, were all straight lines. The reason for these straight lines was that the relative prices of the two goods in the consumption budget constraint determined the slope of the budget constraint. However, we drew the production possibilities frontier for healthcare and education as a curved line. Why does the PPF have a different shape? To understand why the PPF is curved, start by considering point A at the top left-hand side of the PPF. At point A, all available resources are devoted to healthcare and none are left for education. This situation would be extreme and even ridiculous. For example, children are seeing a doctor every day, whether they are sick or not, but not attending school. People are having cosmetic surgery on every part of their bodies, but no high school or college education exists. Now imagine that some of these resources are diverted from healthcare to education, so that the economy is at point B instead of point A. Diverting some resources away from A to B causes relatively little reduction in health because the last few marginal dollars going into healthcare services are not producing much additional gain in health. However, putting those marginal dollars into education, which is completely without resources at point A, can produce relatively large gains. For this reason, the shape of the PPF from A to B is relatively flat, representing a relatively small drop-off in health and a relatively large gain in education. Now consider the other end, at the lower right, of the production possibilities frontier. Imagine that society starts at choice D, which is devoting nearly all resources to education and very few to healthcare, and moves to point F, which is devoting all spending to education and none to healthcare. For the sake of concreteness, you can imagine that in the movement from D to F, the last few doctors must become high school science teachers, the last few nurses must become school librarians rather than dispensers of vaccinations, and the last few emergency rooms are turned into kindergartens. The gains to education from adding these last few resources to education are very small. However, the opportunity cost lost to health will be fairly large, and thus the slope of the PPF between D and F is steep, showing a large drop in health for only a small gain in education. The lesson is not that society is likely to make an extreme choice like devoting no resources to education at point A or no resources to health at point F. Instead, the lesson is that the gains from committing additional marginal resources to education depend on how much is already being spent. If on the one hand, very few resources are currently committed to education, then an increase in resources used can bring relatively large gains. On the other hand, if a large number of resources are already committed to education, then committing additional resources will bring relatively smaller gains. This pattern is common enough that economists have given it a name: the law of diminishing returns, which holds that as additional increments of resources are added to a certain purpose, the marginal benefit from those additional increments will decline. (The law of diminishing marginal utility that we introduced in the last section is a more specific case of the law of diminishing returns.) When government spends a certain amount more on reducing crime, for example, the original gains in reducing crime could be relatively large. However, additional increases typically cause relatively smaller reductions in crime, and paying for enough police and security to reduce crime to nothing at all would be tremendously expensive. The curvature of the production possibilities frontier shows that as we add more resources to education, moving from left to right along the horizontal axis, the original gains are fairly large, but gradually diminish. Thus, the slope of the PPF is relatively flat. By contrast, as we add more resources to healthcare, moving from bottom to top on the vertical axis, the original gains are fairly large, but again gradually diminish. Thus, the slope of the PPF is relatively steep. In this way, the law of diminishing returns produces the outward-bending shape of the production possibilities frontier. Productive Efficiency and Allocative Efficiency The study of economics does not presume to tell a society what choice it should make along its production possibilities frontier. In a market-oriented economy with a democratic government, the choice will involve a mixture of decisions by individuals, firms, and government. However, economics can point out that some choices are unambiguously better than others. This observation is based on the concept of efficiency. In everyday usage, efficiency refers to lack of waste. An inefficient machine operates at high cost, while an efficient machine operates at lower cost, because it is not wasting energy or materials. An inefficient organization operates with long delays and high costs, while an efficient organization meets schedules, is focused, and performs within budget. The production possibilities frontier can illustrate two kinds of efficiency: productive efficiency and allocative efficiency. Figure 2.4 illustrates these ideas using a production possibilities frontier between healthcare and 36 education. Chapter 2 \| Choice in a World of Scarcity Figure 2.4 Productive and Allocative Efficiency Productive efficiency means it is impossible to produce more of one good without decreasing the quantity that is produced of another good. Thus, all choices along a given PPF like B, C, and D display productive efficiency, but R does not. Allocative efficiency means that the particular mix of goods being produced—that is, the specific choice along the production possibilities frontier—represents the allocation that society most desires. Productive efficiency means that, given the available inputs and technology, it is impossible to produce more of one good without decreasing the quantity that is produced of another good. All choices on the PPF in Figure 2.4, including A, B, C, D, and F, display productive efficiency. As a firm moves from any one of these choices to any other, either healthcare increases and education decreases or vice versa. However, any choice inside the production possibilities frontier is productively inefficient and wasteful because it is possible to produce more of one good, the other good, or some combination of both goods. For example, point R is productively inefficient because it is possible at choice C to have more of both goods: education on the horizontal axis is higher at point C than point R (E2 is greater than E1), and healthcare on the vertical axis is also higher at point C than point R (H2 is great than H1). We can show the particular mix of goods and services produced—that is, the specific combination of selected healthcare and education along the production possibilities frontier—as a ray (line) from the origin to a specific point on the PPF. Output mixes that had more healthcare (and less education) would have a steeper ray, while those with more education (and less healthcare) would have a flatter ray. Allocative efficiency means that the particular combination of goods and services on the production possibility curve that a society produces represents the combination that society most desires. How to determine what a society desires can be a controversial question, and is usually a discussion in political science, sociology, and philosophy classes as well as in economics. At its most basic, allocative efficiency means producers supply the quantity of each product that consumers demand. Only one of the productively efficient choices will be the allocatively efficient choice for society as
a whole. Why Society Must Choose In Welcome to Economics! we learned that every society faces the problem of scarcity, where limited resources conflict with unlimited needs and wants. The production possibilities curve illustrates the choices involved in this dilemma. Every economy faces two situations in which it may be able to expand consumption of all goods. In the first case, a society may discover that it has been using its resources inefficiently, in which case by improving efficiency and producing on the production possibilities frontier, it can have more of all goods (or at least more of some and less of none). In the second case, as resources grow over a period of years (e.g., more labor and more capital), the economy grows. As it does, the production possibilities frontier for a society will tend to shift outward and society will be able to afford more of all goods. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 37 However, improvements in productive efficiency take time to discover and implement, and economic growth happens only gradually. Thus, a society must choose between tradeoffs in the present. For government, this process often involves trying to identify where additional spending could do the most good and where reductions in spending would do the least harm. At the individual and firm level, the market economy coordinates a process in which firms seek to produce goods and services in the quantity, quality, and price that people want. However, for both the government and the market economy in the short term, increases in production of one good typically mean offsetting decreases somewhere else in the economy. The PPF and Comparative Advantage While every society must choose how much of each good or service it should produce, it does not need to produce every single good it consumes. Often how much of a good a country decides to produce depends on how expensive it is to produce it versus buying it from a different country. As we saw earlier, the curvature of a country’s PPF gives us information about the tradeoff between devoting resources to producing one good versus another. In particular, its slope gives the opportunity cost of producing one more unit of the good in the x-axis in terms of the other good (in the y-axis). Countries tend to have different opportunity costs of producing a specific good, either because of different climates, geography, technology, or skills. Suppose two countries, the US and Brazil, need to decide how much they will produce of two crops: sugar cane and wheat. Due to its climatic conditions, Brazil can produce quite a bit of sugar cane per acre but not much wheat. Conversely, the U.S. can produce large amounts of wheat per acre, but not much sugar cane. Clearly, Brazil has a lower opportunity cost of producing sugar cane (in terms of wheat) than the U.S. The reverse is also true: the U.S. has a lower opportunity cost of producing wheat than Brazil. We illustrate this by the PPFs of the two countries in Figure 2.5. Figure 2.5 Production Possibility Frontier for the U.S. and Brazil The U.S. PPF is flatter than the Brazil PPF implying that the opportunity cost of wheat in terms of sugar cane is lower in the U.S. than in Brazil. Conversely, the opportunity cost of sugar cane is lower in Brazil. The U.S. has comparative advantage in wheat and Brazil has comparative advantage in sugar cane. When a country can produce a good at a lower opportunity cost than another country, we say that this country has a comparative advantage in that good. Comparative advantage is not the same as absolute advantage, which is when a country can produce more of a good. Comparative advantage is not the same as absolute advantage, which is when a country can produce more of a good. In our example, Brazil has an absolute advantage in sugar cane and the U.S. has an absolute advantage in wheat. One can easily see this with a simple observation of the extreme production points in the PPFs of the two countries. If Brazil devoted all of its resources to producing wheat, it would be producing at point A. If however it had devoted all of its resources to producing sugar cane instead, it would be producing a much larger 38 Chapter 2 \| Choice in a World of Scarcity amount than the U.S., at point B. The slope of the PPF gives the opportunity cost of producing an additional unit of wheat. While the slope is not constant throughout the PPFs, it is quite apparent that the PPF in Brazil is much steeper than in the U.S., and therefore the opportunity cost of wheat generally higher in Brazil. In the chapter on International Trade you will learn that countries’ differences in comparative advantage determine which goods they will choose to produce and trade. When countries engage in trade, they specialize in the production of the goods in which they have comparative advantage, and trade part of that production for goods in which they do not have comparative advantage. With trade, manufacturers produce goods where the opportunity cost is lowest, so total production increases, benefiting both trading parties. 2.3 \| Confronting Objections to the Economic Approach By the end of this section, you will be able to: • Analyze arguments against economic approaches to decision-making • • Contrast normative statements and positive statements Interpret a tradeoff diagram It is one thing to understand the economic approach to decision-making and another thing to feel comfortable applying it. The sources of discomfort typically fall into two categories: that people do not act in the way that fits the economic way of thinking, and that even if people did act that way, they should try not to. Let’s consider these arguments in turn. First Objection: People, Firms, and Society Do Not Act Like This The economic approach to decision-making seems to require more information than most individuals possess and more careful decision-making than most individuals actually display. After all, do you or any of your friends draw a budget constraint and mutter to yourself about maximizing utility before you head to the shopping mall? Do members of the U.S. Congress contemplate production possibilities frontiers before they vote on the annual budget? The messy ways in which people and societies operate somehow doesn’t look much like neat budget constraints or smoothly curving production possibilities frontiers. However, the economics approach can be a useful way to analyze and understand the tradeoffs of economic decisions. To appreciate this point, imagine for a moment that you are playing basketball, dribbling to the right, and throwing a bounce-pass to the left to a teammate who is running toward the basket. A physicist or engineer could work out the correct speed and trajectory for the pass, given the different movements involved and the weight and bounciness of the ball. However, when you are playing basketball, you do not perform any of these calculations. You just pass the ball, and if you are a good player, you will do so with high accuracy. Someone might argue: “The scientist’s formula of the bounce-pass requires a far greater knowledge of physics and far more specific information about speeds of movement and weights than the basketball player actually has, so it must be an unrealistic description of how basketball passes actually occur.” This reaction would be wrongheaded. The fact that a good player can throw the ball accurately because of practice and skill, without making a physics calculation, does not mean that the physics calculation is wrong. Similarly, from an economic point of view, someone who shops for groceries every week has a great deal of practice with how to purchase the combination of goods that will provide that person with utility, even if the shopper does not phrase decisions in terms of a budget constraint. Government institutions may work imperfectly and slowly, but in general, a democratic form of government feels pressure from voters and social institutions to make the choices that are most widely preferred by people in that society. Thus, when thinking about the economic actions of groups of people, firms, and society, it is reasonable, as a first approximation, to analyze them with the tools of economic analysis. For more on this, read about behavioral economics in the chapter on Consumer Choices. Second Objection: People, Firms, and Society Should Not Act This Way The economics approach portrays people as self-interested. For some critics of this approach, even if self-interest is an accurate description of how people behave, these behaviors are not moral. Instead, the critics argue that people should be taught to care more deeply about others. Economists offer several answers to these concerns. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 39 First, economics is not a form of moral instruction. Rather, it seeks to describe economic behavior as it actually exists. Philosophers draw a distinction between positive statements, which describe the world as it is, and normative statements, which describe how the world should be. Positive statements are factual. They may be true or false, but we can test them, at least in principle. Normative statements are subjective questions of opinion. We cannot test them since we cannot prove opinions to be true or false. They just are opinions based on one's values. For example, an economist could analyze a proposed subway system in a certain city. If the expected benefits exceed the costs, he concludes that the project is worthy—an example of positive analysis. Another economist argues for extended unemployment compensation during the Great Depression because a rich country like the United States should take care of its less fortunate citizens—an example of normative analysis. Even if the line between positive and normat
ive statements is not always crystal clear, economic analysis does try to remain rooted in the study of the actual people who inhabit the actual economy. Fortunately however, the assumption that individuals are purely self-interested is a simplification about human nature. In fact, we need to look no further than to Adam Smith, the very father of modern economics to find evidence of this. The opening sentence of his book, The Theory of Moral Sentiments, puts it very clearly: “How selfish soever man may be supposed, there are evidently some principles in his nature, which interest him in the fortune of others, and render their happiness necessary to him, though he derives nothing from it except the pleasure of seeing it.” Clearly, individuals are both self-interested and altruistic. Second, we can label self-interested behavior and profit-seeking with other names, such as personal choice and freedom. The ability to make personal choices about buying, working, and saving is an important personal freedom. Some people may choose high-pressure, high-paying jobs so that they can earn and spend considerable amounts of money on themselves. Others may allocate large portions of their earnings to charity or spend it on their friends and family. Others may devote themselves to a career that can require much time, energy, and expertise but does not offer high financial rewards, like being an elementary school teacher or a social worker. Still others may choose a job that does consume much of their time or provide a high level of income, but still leaves time for family, friends, and contemplation. Some people may prefer to work for a large company; others might want to start their own business. People’s freedom to make their own economic choices has a moral value worth respecting. Is a diagram by any other name the same? When you study economics, you may feel buried under an avalanche of diagrams. Your goal should be to recognize the common underlying logic and pattern of the diagrams, not to memorize each one. This chapter uses only one basic diagram, although we present labels. The consumption budget constraint and the production possibilities frontier for society, as a whole, are the same basic diagram. Figure 2.6 shows an individual budget constraint and a production possibilities frontier for two goods, Good 1 and Good 2. The tradeoff diagram always illustrates three basic themes: scarcity, tradeoffs, and economic efficiency. it with different sets of The first theme is scarcity. It is not feasible to have unlimited amounts of both goods. Even if the budget constraint or a PPF shifts, scarcity remains—just at a different level. The second theme is tradeoffs. As depicted in the budget constraint or the production possibilities frontier, it is necessary to forgo some of one good to gain more of the other good. The details of this tradeoff vary. In a budget constraint we determine, the tradeoff is determined by the relative prices of the goods: that is, the relative price of two goods in the consumption choice budget constraint. These tradeoffs appear as a straight line. However, a curved line represents the tradeoffs in many production possibilities frontiers because the law of diminishing returns holds that as we add resources to an area, the marginal gains tend to diminish. Regardless of the specific shape, tradeoffs remain. The third theme is economic efficiency, or getting the most benefit from scarce resources. All choices on the production possibilities frontier show productive efficiency because in such cases, there is no way to increase the quantity of one good without decreasing the quantity of the other. Similarly, when an individual makes a choice along a budget constraint, there is no way to increase the quantity of one good without decreasing the quantity of the other. The choice on a production possibilities set that is socially preferred, or the choice on an 40 Chapter 2 \| Choice in a World of Scarcity individual’s budget constraint that is personally preferred, will display allocative efficiency. The basic budget constraint/production possibilities frontier diagram will recur throughout this book. Some examples include using these tradeoff diagrams to analyze trade, environmental protection and economic output, equality of incomes and economic output, and the macroeconomic tradeoff between consumption and investment. Do not allow the different labels to confuse you. The budget constraint/production possibilities frontier diagram is always just a tool for thinking carefully about scarcity, tradeoffs, and efficiency in a particular situation. Figure 2.6 The Tradeoff Diagram Both the individual opportunity set (or budget constraint) and the social production possibilities frontier show the constraints under which individual consumers and society as a whole operate. Both diagrams show the tradeoff in choosing more of one good at the cost of less of the other. Third, self-interested behavior can lead to positive social results. For example, when people work hard to make a living, they create economic output. Consumers who are looking for the best deals will encourage businesses to offer goods and services that meet their needs. Adam Smith, writing in The Wealth of Nations, named this property the invisible hand. In describing how consumers and producers interact in a market economy, Smith wrote: Every individual…generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it. By preferring the support of domestic to that of foreign industry, he intends only his own security; and by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain. And he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention…By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it. The metaphor of the invisible hand suggests the remarkable possibility that broader social good can emerge from selfish individual actions. Fourth, even people who focus on their own self-interest in the economic part of their life often set aside their own narrow self-interest in other parts of life. For example, you might focus on your own self-interest when asking your employer for a raise or negotiating to buy a car. Then you might turn around and focus on other people when you volunteer to read stories at the local library, help a friend move to a new apartment, or donate money to a charity. Selfinterest is a reasonable starting point for analyzing many economic decisions, without needing to imply that people never do anything that is not in their own immediate self-interest. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 41 Choices ... To What Degree? What have we learned? We know that scarcity impacts all the choices we make. An economist might argue that people do not obtain a bachelor’s or master’s degree because they do not have the resources to make those choices or because their incomes are too low and/or the price of these degrees is too high. A bachelor’s or a master’s degree may not be available in their opportunity set. The price of these degrees may be too high not only because the actual price, college tuition (and perhaps room and board), is too high. An economist might also say that for many people, the full opportunity cost of a bachelor’s or a master’s degree is too high. For these people, they are unwilling or unable to make the tradeoff of forfeiting years of working, and earning an income, to earn a degree. Finally, the statistics we introduced at the start of the chapter reveal information about intertemporal choices. An economist might say that people choose not to obtain a college degree because they may have to borrow money to attend college, and the interest they have to pay on that loan in the future will affect their decisions today. Also, it could be that some people have a preference for current consumption over future consumption, so they choose to work now at a lower salary and consume now, rather than postponing that consumption until after they graduate college. 42 Chapter 2 \| Choice in a World of Scarcity KEY TERMS allocative efficiency when the mix of goods produced represents the mix that society most desires budget constraint all possible consumption combinations of goods that someone can afford, given the prices of goods, when all income is spent; the boundary of the opportunity set comparative advantage when a country can produce a good at a lower cost in terms of other goods; or, when a country has a lower opportunity cost of production invisible hand Adam Smith's concept that individuals' self-interested behavior can lead to positive social outcomes law of diminishing marginal utility as we consume more of a good or service, the utility we get from additional units of the good or service tends to become smaller than what we received from earlier units law of diminishing returns as we add additional increments of resources to producing a good or service, the marginal benefit from those additional increments will decline marginal analysis examination of decisions on the margin, meaning a little more or a little less from the status quo normative statement statement which describes how the world should be opportunity cost measures cost by what we give up/forfeit in exchange; opportunity cost measures the value of the forgone alternative opportunity set all possible combinations of consumption that someone can afford given the prices of goods and the individual’s income positive statement statement which describes the world as it is production possibilities frontier (PPF) a diagram that shows the productively efficient combinations of two products that an economy can produce given the resources it has available.
productive efficiency when it is impossible to produce more of one good (or service) without decreasing the quantity produced of another good (or service) sunk costs costs that we make in the past that we cannot recover utility satisfaction, usefulness, or value one obtains from consuming goods and services KEY CONCEPTS AND SUMMARY 2.1 How Individuals Make Choices Based on Their Budget Constraint Economists see the real world as one of scarcity: that is, a world in which people’s desires exceed what is possible. As a result, economic behavior involves tradeoffs in which individuals, firms, and society must forgo something that they desire to obtain things that they desire more. Individuals face the tradeoff of what quantities of goods and services to consume. The budget constraint, which is the frontier of the opportunity set, illustrates the range of available choices. The relative price of the choices determines the slope of the budget constraint. Choices beyond the budget constraint are not affordable. Opportunity cost measures cost by what we forgo in exchange. Sometimes we can measure opportunity cost in money, but it is often useful to consider time as well, or to measure it in terms of the actual resources that we must forfeit. Most economic decisions and tradeoffs are not all-or-nothing. Instead, they involve marginal analysis, which means they are about decisions on the margin, involving a little more or a little less. The law of diminishing marginal utility points out that as a person receives more of something—whether it is a specific good or another resource—the This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 2 \| Choice in a World of Scarcity 43 additional marginal gains tend to become smaller. Because sunk costs occurred in the past and cannot be recovered, they should be disregarded in making current decisions. 2.2 The Production Possibilities Frontier and Social Choices A production possibilities frontier defines the set of choices society faces for the combinations of goods and services it can produce given the resources available. The shape of the PPF is typically curved outward, rather than straight. Choices outside the PPF are unattainable and choices inside the PPF are wasteful. Over time, a growing economy will tend to shift the PPF outwards. The law of diminishing returns holds that as increments of additional resources are devoted to producing something, the marginal increase in output will become increasingly smaller. All choices along a production possibilities frontier display productive efficiency; that is, it is impossible to use society’s resources to produce more of one good without decreasing production of the other good. The specific choice along a production possibilities frontier that reflects the mix of goods society prefers is the choice with allocative efficiency. The curvature of the PPF is likely to differ by country, which results in different countries having comparative advantage in different goods. Total production can increase if countries specialize in the goods in which they have comparative advantage and trade some of their production for the remaining goods. 2.3 Confronting Objections to the Economic Approach The economic way of thinking provides a useful approach to understanding human behavior. Economists make the careful distinction between positive statements, which describe the world as it is, and normative statements, which describe how the world should be. Even when economics analyzes the gains and losses from various events or policies, and thus draws normative conclusions about how the world should be, the analysis of economics is rooted in a positive analysis of how people, firms, and governments actually behave, not how they should behave. SELF-CHECK QUESTIONS 1. Suppose Alphonso’s town raised the price of bus tickets to $1 per trip (while the price of burgers stayed at $2 and his budget remained $10 per week.) Draw Alphonso’s new budget constraint. What happens to the opportunity cost of bus tickets? 2. Return to the example in Figure 2.4. Suppose there is an improvement in medical technology that enables more healthcare with the same amount of resources. How would this affect the production possibilities curve and, in particular, how would it affect the opportunity cost of education? 3. Could a nation be producing in a way that is allocatively efficient, but productively inefficient? 4. What are the similarities between a consumer’s budget constraint and society’s production possibilities frontier, not just graphically but analytically? Individuals may not act in the rational, calculating way described by the economic model of decision making, 5. measuring utility and costs at the margin, but can you make a case that they behave approximately that way? 6. Would an op-ed piece in a newspaper urging the adoption of a particular economic policy be a positive or normative statement? 7. Would a research study on the effects of soft drink consumption on children’s cognitive development be a positive or normative statement? REVIEW QUESTIONS 8. Explain why scarcity leads to tradeoffs. 10. What is comparative advantage? 9. Explain why individuals make choices that are directly on the budget constraint, rather than inside the budget constraint or outside it. 11. What does a production possibilities frontier illustrate? 44 Chapter 2 \| Choice in a World of Scarcity 12. Why is a production possibilities frontier typically drawn as a curve, rather than a straight line? 16. What is the difference between a positive and a normative statement? 13. Explain why societies cannot make a choice above their production possibilities frontier and should not make a choice below it. Is 17. the economic model of decision-making intended as a literal description of how individuals, firms, and the governments actually make decisions? 14. What are diminishing marginal returns? 15. What efficiency? is productive efficiency? Allocative CRITICAL THINKING QUESTIONS 19. Suppose Alphonso’s town raises the price of bus tickets from $0.50 to $1 and the price of burgers rises from $2 to $4. Why is the opportunity cost of bus tickets unchanged? Suppose Alphonso’s weekly spending money increases from $10 to $20. How is his budget constraint affected from all three changes? Explain. 20. During the Second World War, Germany’s factories were decimated. It also suffered many human casualties, both soldiers and civilians. How did the war affect Germany’s production possibilities curve? 18. What are four responses to the claim that people should not behave in the way described in this chapter? 21. It is clear that productive inefficiency is a waste since resources are used in a way that produces less goods and services than a nation is capable of. Why is allocative inefficiency also wasteful? 22. What assumptions about the economy must be true for the invisible hand to work? To what extent are those assumptions valid in the real world? 23. Do economists have any particular expertise at making normative arguments? In other words, they have expertise at making positive statements (i.e., what will happen) about some economic policy, for example, but do they have special expertise to judge whether or not the policy should be undertaken? PROBLEMS information to answer Use this the following 4 questions: Marie has a weekly budget of $24, which she likes to spend on magazines and pies. 26. Draw Marie’s budget constraint with pies on the horizontal axis and magazines on the vertical axis. What is the slope of the budget constraint? 24. If the price of a magazine is $4 each, what is the maximum number of magazines she could buy in a week? If the price of a pie is $12, what is the maximum 25. number of pies she could buy in a week? 27. What is Marie’s opportunity cost of purchasing a pie? This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 45 3 \| Demand and Supply Figure 3.1 Farmer’s Market Organic vegetables and fruits that are grown and sold within a specific geographical region should, in theory, cost less than conventional produce because the transportation costs are less. That is not, however, usually the case. (Credit: Modification of work by Natalie Maynor/Flickr Creative Commons) Why Can We Not Get Enough of Organic? Organic food is increasingly popular, not just in the United States, but worldwide. At one time, consumers had to go to specialty stores or farmers' markets to find organic produce. Now it is available in most grocery stores. In short, organic is part of the mainstream. Ever wonder why organic food costs more than conventional food? Why, say, does an organic Fuji apple cost $1.99 a pound, while its conventional counterpart costs $1.49 a pound? The same price relationship is true for just about every organic product on the market. If many organic foods are locally grown, would they not take less time to get to market and therefore be cheaper? What are the forces that keep those prices from coming down? Turns out those forces have quite a bit to do with this chapter’s topic: demand and supply. Introduction to Demand and Supply In this chapter, you will learn about: • Demand, Supply, and Equilibrium in Markets for Goods and Services • Shifts in Demand and Supply for Goods and Services • Changes in Equilibrium Price and Quantity: The Four-Step Process 46 Chapter 3 \| Demand and Supply • Price Ceilings and Price Floors An auction bidder pays thousands of dollars for a dress Whitney Houston wore. A collector spends a small fortune for a few drawings by John Lennon. People usually react to purchases like these in two ways: their jaw drops because they think these are high prices to pay for such goods or they think these are rare, desirable items and the amount paid seems right. Visit this website (http://openstaxcollege.org/l/celebauction) to read a list of bizarre items that have been purchased f
or their ties to celebrities. These examples represent an interesting facet of demand and supply. When economists talk about prices, they are less interested in making judgments than in gaining a practical understanding of what determines prices and why prices change. Consider a price most of us contend with weekly: that of a gallon of gas. Why was the average price of gasoline in the United States $3.71 per gallon in June 2014? Why did the price for gasoline fall sharply to $1.96 per gallon by January 2016? To explain these price movements, economists focus on the determinants of what gasoline buyers are willing to pay and what gasoline sellers are willing to accept. As it turns out, the price of gasoline in June of any given year is nearly always higher than the price in January of that same year. Over recent decades, gasoline prices in midsummer have averaged about 10 cents per gallon more than their midwinter low. The likely reason is that people drive more in the summer, and are also willing to pay more for gas, but that does not explain how steeply gas prices fell. Other factors were at work during those 18 months, such as increases in supply and decreases in the demand for crude oil. This chapter introduces the economic model of demand and supply—one of the most powerful models in all of economics. The discussion here begins by examining how demand and supply determine the price and the quantity sold in markets for goods and services, and how changes in demand and supply lead to changes in prices and quantities. 3.1 \| Demand, Supply, and Equilibrium in Markets for Goods and Services By the end of this section, you will be able to: Identify a demand curve and a supply curve • Explain demand, quantity demanded, and the law of demand • • Explain supply, quantity supplied, and the law of supply • Explain equilibrium, equilibrium price, and equilibrium quantity First let’s first focus on what economists mean by demand, what they mean by supply, and then how demand and supply interact in a market. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 47 Demand for Goods and Services Economists use the term demand to refer to the amount of some good or service consumers are willing and able to purchase at each price. Demand is fundamentally based on needs and wants—if you have no need or want for something, you won't buy it. While a consumer may be able to differentiate between a need and a want, but from an economist’s perspective they are the same thing. Demand is also based on ability to pay. If you cannot pay for it, you have no effective demand. By this definition, a homeless person probably has no effective demand for shelter. What a buyer pays for a unit of the specific good or service is called price. The total number of units that consumers would purchase at that price is called the quantity demanded. A rise in price of a good or service almost always decreases the quantity demanded of that good or service. Conversely, a fall in price will increase the quantity demanded. When the price of a gallon of gasoline increases, for example, people look for ways to reduce their consumption by combining several errands, commuting by carpool or mass transit, or taking weekend or vacation trips closer to home. Economists call this inverse relationship between price and quantity demanded the law of demand. The law of demand assumes that all other variables that affect demand (which we explain in the next module) are held constant. We can show an example from the market for gasoline in a table or a graph. Economist call a table that shows the quantity demanded at each price, such as Table 3.1, a demand schedule. In this case we measure price in dollars per gallon of gasoline. We measure the quantity demanded in millions of gallons over some time period (for example, per day or per year) and over some geographic area (like a state or a country). A demand curve shows the relationship between price and quantity demanded on a graph like Figure 3.2, with quantity on the horizontal axis and the price per gallon on the vertical axis. (Note that this is an exception to the normal rule in mathematics that the independent variable (x) goes on the horizontal axis and the dependent variable (y) goes on the vertical. Economics is not math.) Table 3.1 shows the demand schedule and the graph in Figure 3.2 shows the demand curve. These are two ways to describe the same relationship between price and quantity demanded. Price (per gallon) Quantity Demanded (millions of gallons) $1.00 $1.20 $1.40 $1.60 $1.80 $2.00 $2.20 800 700 600 550 500 460 420 Table 3.1 Price and Quantity Demanded of Gasoline 48 Chapter 3 \| Demand and Supply Figure 3.2 A Demand Curve for Gasoline The demand schedule shows that as price rises, quantity demanded decreases, and vice versa. We graph these points, and the line connecting them is the demand curve (D). The downward slope of the demand curve again illustrates the law of demand—the inverse relationship between prices and quantity demanded. Demand curves will appear somewhat different for each product. They may appear relatively steep or flat, or they may be straight or curved. Nearly all demand curves share the fundamental similarity that they slope down from left to right. Demand curves embody the law of demand: As the price increases, the quantity demanded decreases, and conversely, as the price decreases, the quantity demanded increases. Confused about these different types of demand? Read the next Clear It Up feature. Is demand the same as quantity demanded? In economic terminology, demand is not the same as quantity demanded. When economists talk about demand, they mean the relationship between a range of prices and the quantities demanded at those prices, as illustrated by a demand curve or a demand schedule. When economists talk about quantity demanded, they mean only a certain point on the demand curve, or one quantity on the demand schedule. In short, demand refers to the curve and quantity demanded refers to the (specific) point on the curve. Supply of Goods and Services When economists talk about supply, they mean the amount of some good or service a producer is willing to supply at each price. Price is what the producer receives for selling one unit of a good or service. A rise in price almost always leads to an increase in the quantity supplied of that good or service, while a fall in price will decrease the quantity supplied. When the price of gasoline rises, for example, it encourages profit-seeking firms to take several actions: expand exploration for oil reserves; drill for more oil; invest in more pipelines and oil tankers to bring the oil to plants for refining into gasoline; build new oil refineries; purchase additional pipelines and trucks to ship the gasoline to gas stations; and open more gas stations or keep existing gas stations open longer hours. Economists call this positive relationship between price and quantity supplied—that a higher price leads to a higher quantity supplied and a lower price leads to a lower quantity supplied—the law of supply. The law of supply assumes that all other variables that affect supply (to be explained in the next module) are held constant. Still unsure about the different types of supply? See the following Clear It Up feature. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 49 Is supply the same as quantity supplied? In economic terminology, supply is not the same as quantity supplied. When economists refer to supply, they mean the relationship between a range of prices and the quantities supplied at those prices, a relationship that we can illustrate with a supply curve or a supply schedule. When economists refer to quantity supplied, they mean only a certain point on the supply curve, or one quantity on the supply schedule. In short, supply refers to the curve and quantity supplied refers to the (specific) point on the curve. Figure 3.3 illustrates the law of supply, again using the market for gasoline as an example. Like demand, we can illustrate supply using a table or a graph. A supply schedule is a table, like Table 3.2, that shows the quantity supplied at a range of different prices. Again, we measure price in dollars per gallon of gasoline and we measure quantity supplied in millions of gallons. A supply curve is a graphic illustration of the relationship between price, shown on the vertical axis, and quantity, shown on the horizontal axis. The supply schedule and the supply curve are just two different ways of showing the same information. Notice that the horizontal and vertical axes on the graph for the supply curve are the same as for the demand curve. Figure 3.3 A Supply Curve for Gasoline The supply schedule is the table that shows quantity supplied of gasoline at each price. As price rises, quantity supplied also increases, and vice versa. The supply curve (S) is created by graphing the points from the supply schedule and then connecting them. The upward slope of the supply curve illustrates the law of supply—that a higher price leads to a higher quantity supplied, and vice versa. Price (per gallon) Quantity Supplied (millions of gallons) $1.00 $1.20 $1.40 $1.60 $1.80 $2.00 500 550 600 640 680 700 Table 3.2 Price and Supply of Gasoline 50 Chapter 3 \| Demand and Supply Price (per gallon) Quantity Supplied (millions of gallons) $2.20 720 Table 3.2 Price and Supply of Gasoline The shape of supply curves will vary somewhat according to the product: steeper, flatter, straighter, or curved. Nearly all supply curves, however, share a basic similarity: they slope up from left to right and illustrate the law of supply: as the price rises, say, from $1.00 per gallon to $2.20 per gallon, the quantity supplied increases from 500 gallons to 720 gallons. Conversely, as the price falls, the quantity supplied decreases. Equilibrium
—Where Demand and Supply Intersect Because the graphs for demand and supply curves both have price on the vertical axis and quantity on the horizontal axis, the demand curve and supply curve for a particular good or service can appear on the same graph. Together, demand and supply determine the price and the quantity that will be bought and sold in a market. Figure 3.4 illustrates the interaction of demand and supply in the market for gasoline. The demand curve (D) is identical to Figure 3.2. The supply curve (S) is identical to Figure 3.3. Table 3.3 contains the same information in tabular form. Figure 3.4 Demand and Supply for Gasoline The demand curve (D) and the supply curve (S) intersect at the equilibrium point E, with a price of $1.40 and a quantity of 600. The equilibrium is the only price where quantity demanded is equal to quantity supplied. At a price above equilibrium like $1.80, quantity supplied exceeds the quantity demanded, so there is excess supply. At a price below equilibrium such as $1.20, quantity demanded exceeds quantity supplied, so there is excess demand. Price (per gallon) Quantity demanded (millions of gallons) Quantity supplied (millions of gallons) $1.00 $1.20 $1.40 $1.60 $1.80 800 700 600 550 500 500 550 600 640 680 Table 3.3 Price, Quantity Demanded, and Quantity Supplied This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 51 Price (per gallon) Quantity demanded (millions of gallons) Quantity supplied (millions of gallons) $2.00 $2.20 460 420 700 720 Table 3.3 Price, Quantity Demanded, and Quantity Supplied Remember this: When two lines on a diagram cross, this intersection usually means something. The point where the supply curve (S) and the demand curve (D) cross, designated by point E in Figure 3.4, is called the equilibrium. The equilibrium price is the only price where the plans of consumers and the plans of producers agree—that is, where the amount of the product consumers want to buy (quantity demanded) is equal to the amount producers want to sell (quantity supplied). Economists call this common quantity the equilibrium quantity. At any other price, the quantity demanded does not equal the quantity supplied, so the market is not in equilibrium at that price. In Figure 3.4, the equilibrium price is $1.40 per gallon of gasoline and the equilibrium quantity is 600 million gallons. If you had only the demand and supply schedules, and not the graph, you could find the equilibrium by looking for the price level on the tables where the quantity demanded and the quantity supplied are equal. The word “equilibrium” means “balance.” If a market is at its equilibrium price and quantity, then it has no reason to move away from that point. However, if a market is not at equilibrium, then economic pressures arise to move the market toward the equilibrium price and the equilibrium quantity. Imagine, for example, that the price of a gallon of gasoline was above the equilibrium price—that is, instead of $1.40 per gallon, the price is $1.80 per gallon. The dashed horizontal line at the price of $1.80 in Figure 3.4 illustrates this above equilibrium price. At this higher price, the quantity demanded drops from 600 to 500. This decline in quantity reflects how consumers react to the higher price by finding ways to use less gasoline. Moreover, at this higher price of $1.80, the quantity of gasoline supplied rises from the 600 to 680, as the higher price makes it more profitable for gasoline producers to expand their output. Now, consider how quantity demanded and quantity supplied are related at this above-equilibrium price. Quantity demanded has fallen to 500 gallons, while quantity supplied has risen to 680 gallons. In fact, at any above-equilibrium price, the quantity supplied exceeds the quantity demanded. We call this an excess supply or a surplus. With a surplus, gasoline accumulates at gas stations, in tanker trucks, in pipelines, and at oil refineries. This accumulation puts pressure on gasoline sellers. If a surplus remains unsold, those firms involved in making and selling gasoline are not receiving enough cash to pay their workers and to cover their expenses. In this situation, some producers and sellers will want to cut prices, because it is better to sell at a lower price than not to sell at all. Once some sellers start cutting prices, others will follow to avoid losing sales. These price reductions in turn will stimulate a higher quantity demanded. Therefore, if the price is above the equilibrium level, incentives built into the structure of demand and supply will create pressures for the price to fall toward the equilibrium. Now suppose that the price is below its equilibrium level at $1.20 per gallon, as the dashed horizontal line at this price in Figure 3.4 shows. At this lower price, the quantity demanded increases from 600 to 700 as drivers take longer trips, spend more minutes warming up the car in the driveway in wintertime, stop sharing rides to work, and buy larger cars that get fewer miles to the gallon. However, the below-equilibrium price reduces gasoline producers’ incentives to produce and sell gasoline, and the quantity supplied falls from 600 to 550. When the price is below equilibrium, there is excess demand, or a shortage—that is, at the given price the quantity demanded, which has been stimulated by the lower price, now exceeds the quantity supplied, which had been depressed by the lower price. In this situation, eager gasoline buyers mob the gas stations, only to find many stations running short of fuel. Oil companies and gas stations recognize that they have an opportunity to make higher profits by selling what gasoline they have at a higher price. As a result, the price rises toward the equilibrium level. Read Demand, Supply, and Efficiency for more discussion on the importance of the demand and supply model. 3.2 \| Shifts in Demand and Supply for Goods and 52 Chapter 3 \| Demand and Supply Services By the end of this section, you will be able to: Identify factors that affect demand • • Graph demand curves and demand shifts Identify factors that affect supply • • Graph supply curves and supply shifts The previous module explored how price affects the quantity demanded and the quantity supplied. The result was the demand curve and the supply curve. Price, however, is not the only factor that influences demand, nor is it the only thing that influences supply. For example, how is demand for vegetarian food affected if, say, health concerns cause more consumers to avoid eating meat? How is the supply of diamonds affected if diamond producers discover several new diamond mines? What are the major factors, in addition to the price, that influence demand or supply? Visit this website (http://openstaxcollege.org/l/toothfish) to read a brief note on how marketing strategies can influence supply and demand of products. What Factors Affect Demand? We defined demand as the amount of some product a consumer is willing and able to purchase at each price. That suggests at least two factors in addition to price that affect demand. Willingness to purchase suggests a desire, based on what economists call tastes and preferences. If you neither need nor want something, you will not buy it. Ability to purchase suggests that income is important. Professors are usually able to afford better housing and transportation than students, because they have more income. Prices of related goods can affect demand also. If you need a new car, the price of a Honda may affect your demand for a Ford. Finally, the size or composition of the population can affect demand. The more children a family has, the greater their demand for clothing. The more driving-age children a family has, the greater their demand for car insurance, and the less for diapers and baby formula. These factors matter for both individual and market demand as a whole. Exactly how do these various factors affect demand, and how do we show the effects graphically? To answer those questions, we need the ceteris paribus assumption. The Ceteris Paribus Assumption A demand curve or a supply curve is a relationship between two, and only two, variables: quantity on the horizontal axis and price on the vertical axis. The assumption behind a demand curve or a supply curve is that no relevant economic factors, other than the product’s price, are changing. Economists call this assumption ceteris paribus, a Latin phrase meaning “other things being equal.” Any given demand or supply curve is based on the ceteris paribus assumption that all else is held equal. A demand curve or a supply curve is a relationship between two, and only two, variables when all other variables are kept constant. If all else is not held equal, then the laws of supply and demand will not necessarily hold, as the following Clear It Up feature shows. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 53 When does ceteris paribus apply? We typically apply ceteris paribus when we observe how changes in price affect demand or supply, but we can apply ceteris paribus more generally. In the real world, demand and supply depend on more factors than just price. For example, a consumer’s demand depends on income and a producer’s supply depends on the cost of producing the product. How can we analyze the effect on demand or supply if multiple factors are changing at the same time—say price rises and income falls? The answer is that we examine the changes one at a time, assuming the other factors are held constant. For example, we can say that an increase in the price reduces the amount consumers will buy (assuming income, and anything else that affects demand, is unchanged). Additionally, a decrease in income reduces the amount consumers can afford to buy (assuming price, and anything else that affects demand, is unchanged). This is what the ceteris paribus assu
mption really means. In this particular case, after we analyze each factor separately, we can combine the results. The amount consumers buy falls for two reasons: first because of the higher price and second because of the lower income. How Does Income Affect Demand? Let’s use income as an example of how factors other than price affect demand. Figure 3.5 shows the initial demand for automobiles as D0. At point Q, for example, if the price is $20,000 per car, the quantity of cars demanded is 18 million. D0 also shows how the quantity of cars demanded would change as a result of a higher or lower price. For example, if the price of a car rose to $22,000, the quantity demanded would decrease to 17 million, at point R. The original demand curve D0, like every demand curve, is based on the ceteris paribus assumption that no other economically relevant factors change. Now imagine that the economy expands in a way that raises the incomes of many people, making cars more affordable. How will this affect demand? How can we show this graphically? Return to Figure 3.5. The price of cars is still $20,000, but with higher incomes, the quantity demanded has now increased to 20 million cars, shown at point S. As a result of the higher income levels, the demand curve shifts to the right to the new demand curve D1, indicating an increase in demand. Table 3.4 shows clearly that this increased demand would occur at every price, not just the original one. Figure 3.5 Shifts in Demand: A Car Example Increased demand means that at every given price, the quantity demanded is higher, so that the demand curve shifts to the right from D0 to D1. Decreased demand means that at every given price, the quantity demanded is lower, so that the demand curve shifts to the left from D0 to D2. 54 Chapter 3 \| Demand and Supply Price Decrease to D2 Original Quantity Demanded D0 Increase to D1 $16,000 17.6 million 22.0 million $18,000 16.0 million 20.0 million $20,000 14.4 million 18.0 million $22,000 13.6 million 17.0 million $24,000 13.2 million 16.5 million $26,000 12.8 million 16.0 million Table 3.4 Price and Demand Shifts: A Car Example 24.0 million 22.0 million 20.0 million 19.0 million 18.5 million 18.0 million Now, imagine that the economy slows down so that many people lose their jobs or work fewer hours, reducing their incomes. In this case, the decrease in income would lead to a lower quantity of cars demanded at every given price, and the original demand curve D0 would shift left to D2. The shift from D0 to D2 represents such a decrease in demand: At any given price level, the quantity demanded is now lower. In this example, a price of $20,000 means 18 million cars sold along the original demand curve, but only 14.4 million sold after demand fell. When a demand curve shifts, it does not mean that the quantity demanded by every individual buyer changes by the same amount. In this example, not everyone would have higher or lower income and not everyone would buy or not buy an additional car. Instead, a shift in a demand curve captures a pattern for the market as a whole. In the previous section, we argued that higher income causes greater demand at every price. This is true for most goods and services. For some—luxury cars, vacations in Europe, and fine jewelry—the effect of a rise in income can be especially pronounced. A product whose demand rises when income rises, and vice versa, is called a normal good. A few exceptions to this pattern do exist. As incomes rise, many people will buy fewer generic brand groceries and more name brand groceries. They are less likely to buy used cars and more likely to buy new cars. They will be less likely to rent an apartment and more likely to own a home. A product whose demand falls when income rises, and vice versa, is called an inferior good. In other words, when income increases, the demand curve shifts to the left. Other Factors That Shift Demand Curves Income is not the only factor that causes a shift in demand. Other factors that change demand include tastes and preferences, the composition or size of the population, the prices of related goods, and even expectations. A change in any one of the underlying factors that determine what quantity people are willing to buy at a given price will cause a shift in demand. Graphically, the new demand curve lies either to the right (an increase) or to the left (a decrease) of the original demand curve. Let’s look at these factors. Changing Tastes or Preferences From 1980 to 2014, the per-person consumption of chicken by Americans rose from 48 pounds per year to 85 pounds per year, and consumption of beef fell from 77 pounds per year to 54 pounds per year, according to the U.S. Department of Agriculture (USDA). Changes like these are largely due to movements in taste, which change the quantity of a good demanded at every price: that is, they shift the demand curve for that good, rightward for chicken and leftward for beef. Changes in the Composition of the Population The proportion of elderly citizens in the United States population is rising. It rose from 9.8% in 1970 to 12.6% in 2000, and will be a projected (by the U.S. Census Bureau) 20% of the population by 2030. A society with relatively more children, like the United States in the 1960s, will have greater demand for goods and services like tricycles and day care facilities. A society with relatively more elderly persons, as the United States is projected to have by 2030, has a higher demand for nursing homes and hearing aids. Similarly, changes in the size of the population can affect the demand for housing and many other goods. Each of these changes in demand will be shown as a shift in the demand curve. Changes in the prices of related goods such as substitutes or complements also can affect the demand for a product. A This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 55 substitute is a good or service that we can use in place of another good or service. As electronic books, like this one, become more available, you would expect to see a decrease in demand for traditional printed books. A lower price for a substitute decreases demand for the other product. For example, in recent years as the price of tablet computers has fallen, the quantity demanded has increased (because of the law of demand). Since people are purchasing tablets, there has been a decrease in demand for laptops, which we can show graphically as a leftward shift in the demand curve for laptops. A higher price for a substitute good has the reverse effect. Other goods are complements for each other, meaning we often use the goods together, because consumption of one good tends to enhance consumption of the other. Examples include breakfast cereal and milk; notebooks and pens or pencils, golf balls and golf clubs; gasoline and sport utility vehicles; and the five-way combination of bacon, lettuce, tomato, mayonnaise, and bread. If the price of golf clubs rises, since the quantity demanded of golf clubs falls (because of the law of demand), demand for a complement good like golf balls decreases, too. Similarly, a higher price for skis would shift the demand curve for a complement good like ski resort trips to the left, while a lower price for a complement has the reverse effect. Changes in Expectations about Future Prices or Other Factors that Affect Demand While it is clear that the price of a good affects the quantity demanded, it is also true that expectations about the future price (or expectations about tastes and preferences, income, and so on) can affect demand. For example, if people hear that a hurricane is coming, they may rush to the store to buy flashlight batteries and bottled water. If people learn that the price of a good like coffee is likely to rise in the future, they may head for the store to stock up on coffee now. We show these changes in demand as shifts in the curve. Therefore, a shift in demand happens when a change in some economic factor (other than price) causes a different quantity to be demanded at every price. The following Work It Out feature shows how this happens. Shift in Demand A shift in demand means that at any price (and at every price), the quantity demanded will be different than it was before. Following is an example of a shift in demand due to an income increase. Step 1. Draw the graph of a demand curve for a normal good like pizza. Pick a price (like P0). Identify the corresponding Q0. See an example in Figure 3.6. Figure 3.6 Demand Curve We can use the demand curve to identify how much consumers would buy at any given price. Step 2. Suppose income increases. As a result of the change, are consumers going to buy more or less pizza? The answer is more. Draw a dotted horizontal line from the chosen price, through the original quantity demanded, to the new point with the new Q1. Draw a dotted vertical line down to the horizontal axis and label the new Q1. Figure 3.7 provides an example. 56 Chapter 3 \| Demand and Supply Figure 3.7 Demand Curve with Income Increase With an increase in income, consumers will purchase larger quantities, pushing demand to the right. Step 3. Now, shift the curve through the new point. You will see that an increase in income causes an upward (or rightward) shift in the demand curve, so that at any price the quantities demanded will be higher, as Figure 3.8 illustrates. Figure 3.8 Demand Curve Shifted Right With an increase in income, consumers will purchase larger quantities, pushing demand to the right, and causing the demand curve to shift right. Summing Up Factors That Change Demand Figure 3.9 summarizes six factors that can shift demand curves. The direction of the arrows indicates whether the demand curve shifts represent an increase in demand or a decrease in demand. Notice that a change in the price of the good or service itself is not listed among the factors that can shift a demand curve. A change in t
he price of a good or service causes a movement along a specific demand curve, and it typically leads to some change in the quantity demanded, but it does not shift the demand curve. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 57 Figure 3.9 Factors That Shift Demand Curves (a) A list of factors that can cause an increase in demand from D0 to D1. (b) The same factors, if their direction is reversed, can cause a decrease in demand from D0 to D1. When a demand curve shifts, it will then intersect with a given supply curve at a different equilibrium price and quantity. We are, however, getting ahead of our story. Before discussing how changes in demand can affect equilibrium price and quantity, we first need to discuss shifts in supply curves. How Production Costs Affect Supply A supply curve shows how quantity supplied will change as the price rises and falls, assuming ceteris paribus so that no other economically relevant factors are changing. If other factors relevant to supply do change, then the entire supply curve will shift. Just as we described a shift in demand as a change in the quantity demanded at every price, a shift in supply means a change in the quantity supplied at every price. In thinking about the factors that affect supply, remember what motivates firms: profits, which are the difference between revenues and costs. A firm produces goods and services using combinations of labor, materials, and machinery, or what we call inputs or factors of production. If a firm faces lower costs of production, while the prices for the good or service the firm produces remain unchanged, a firm’s profits go up. When a firm’s profits increase, it is more motivated to produce output, since the more it produces the more profit it will earn. When costs of production fall, a firm will tend to supply a larger quantity at any given price for its output. We can show this by the supply curve shifting to the right. Take, for example, a messenger company that delivers packages around a city. The company may find that buying gasoline is one of its main costs. If the price of gasoline falls, then the company will find it can deliver messages more cheaply than before. Since lower costs correspond to higher profits, the messenger company may now supply more of its services at any given price. For example, given the lower gasoline prices, the company can now serve a greater area, and increase its supply. Conversely, if a firm faces higher costs of production, then it will earn lower profits at any given selling price for its products. As a result, a higher cost of production typically causes a firm to supply a smaller quantity at any given price. In this case, the supply curve shifts to the left. Consider the supply for cars, shown by curve S0 in Figure 3.10. Point J indicates that if the price is $20,000, the quantity supplied will be 18 million cars. If the price rises to $22,000 per car, ceteris paribus, the quantity supplied will rise to 20 million cars, as point K on the S0 curve shows. We can show the same information in table form, as in Table 3.5. 58 Chapter 3 \| Demand and Supply Figure 3.10 Shifts in Supply: A Car Example Decreased supply means that at every given price, the quantity supplied is lower, so that the supply curve shifts to the left, from S0 to S1. Increased supply means that at every given price, the quantity supplied is higher, so that the supply curve shifts to the right, from S0 to S2. Price Decrease to S1 Original Quantity Supplied S0 Increase to S2 $16,000 10.5 million $18,000 13.5 million $20,000 16.5 million $22,000 18.5 million $24,000 19.5 million $26,000 20.5 million 12.0 million 15.0 million 18.0 million 20.0 million 21.0 million 22.0 million Table 3.5 Price and Shifts in Supply: A Car Example 13.2 million 16.5 million 19.8 million 22.0 million 23.1 million 24.2 million Now, imagine that the price of steel, an important ingredient in manufacturing cars, rises, so that producing a car has become more expensive. At any given price for selling cars, car manufacturers will react by supplying a lower quantity. We can show this graphically as a leftward shift of supply, from S0 to S1, which indicates that at any given price, the quantity supplied decreases. In this example, at a price of $20,000, the quantity supplied decreases from 18 million on the original supply curve (S0) to 16.5 million on the supply curve S1, which is labeled as point L. Conversely, if the price of steel decreases, producing a car becomes less expensive. At any given price for selling cars, car manufacturers can now expect to earn higher profits, so they will supply a higher quantity. The shift of supply to the right, from S0 to S2, means that at all prices, the quantity supplied has increased. In this example, at a price of $20,000, the quantity supplied increases from 18 million on the original supply curve (S0) to 19.8 million on the supply curve S2, which is labeled M. Other Factors That Affect Supply In the example above, we saw that changes in the prices of inputs in the production process will affect the cost of production and thus the supply. Several other things affect the cost of production, too, such as changes in weather or other natural conditions, new technologies for production, and some government policies. Changes in weather and climate will affect the cost of production for many agricultural products. For example, in This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 59 2014 the Manchurian Plain in Northeastern China, which produces most of the country's wheat, corn, and soybeans, experienced its most severe drought in 50 years. A drought decreases the supply of agricultural products, which means that at any given price, a lower quantity will be supplied. Conversely, especially good weather would shift the supply curve to the right. When a firm discovers a new technology that allows the firm to produce at a lower cost, the supply curve will shift to the right, as well. For instance, in the 1960s a major scientific effort nicknamed the Green Revolution focused on breeding improved seeds for basic crops like wheat and rice. By the early 1990s, more than two-thirds of the wheat and rice in low-income countries around the world used these Green Revolution seeds—and the harvest was twice as high per acre. A technological improvement that reduces costs of production will shift supply to the right, so that a greater quantity will be produced at any given price. Government policies can affect the cost of production and the supply curve through taxes, regulations, and subsidies. For example, the U.S. government imposes a tax on alcoholic beverages that collects about $8 billion per year from producers. Businesses treat taxes as costs. Higher costs decrease supply for the reasons we discussed above. Other examples of policy that can affect cost are the wide array of government regulations that require firms to spend money to provide a cleaner environment or a safer workplace. Complying with regulations increases costs. A government subsidy, on the other hand, is the opposite of a tax. A subsidy occurs when the government pays a firm directly or reduces the firm’s taxes if the firm carries out certain actions. From the firm’s perspective, taxes or regulations are an additional cost of production that shifts supply to the left, leading the firm to produce a lower quantity at every given price. Government subsidies reduce the cost of production and increase supply at every given price, shifting supply to the right. The following Work It Out feature shows how this shift happens. Shift in Supply We know that a supply curve shows the minimum price a firm will accept to produce a given quantity of output. What happens to the supply curve when the cost of production goes up? Following is an example of a shift in supply due to a production cost increase. Step 1. Draw a graph of a supply curve for pizza. Pick a quantity (like Q0). If you draw a vertical line up from Q0 to the supply curve, you will see the price the firm chooses. Figure 3.11 provides an example. Figure 3.11 Supply Curve You can use a supply curve to show the minimum price a firm will accept to produce a given quantity of output. Step 2. Why did the firm choose that price and not some other? One way to think about this is that the price is composed of two parts. The first part is the cost of producing pizzas at the margin; in this case, the cost of producing the pizza, including cost of ingredients (e.g., dough, sauce, cheese, and pepperoni), the cost of the pizza oven, the shop rent, and the workers' wages. The second part is the firm’s desired profit, which is determined, among other factors, by the profit margins in that particular business. If you add these two parts together, you get the price the firm wishes to charge. The quantity Q0 and associated price P0 give you one point on the firm’s supply curve, as Figure 3.12 illustrates. 60 Chapter 3 \| Demand and Supply Figure 3.12 Setting Prices The cost of production and the desired profit equal the price a firm will set for a product. Step 3. Now, suppose that the cost of production increases. Perhaps cheese has become more expensive by $0.75 per pizza. If that is true, the firm will want to raise its price by the amount of the increase in cost ($0.75). Draw this point on the supply curve directly above the initial point on the curve, but $0.75 higher, as Figure 3.13 shows. Figure 3.13 Increasing Costs Leads to Increasing Price Because the cost of production and the desired profit equal the price a firm will set for a product, if the cost of production increases, the price for the product will also need to increase. Step 4. Shift the supply curve through this point. You will see that an increase in cost causes an upward (or a leftward) shift of the supply curve so that at any price, the
quantities supplied will be smaller, as Figure 3.14 illustrates. Figure 3.14 Supply Curve Shifts When the cost of production increases, the supply curve shifts upwardly to a new price level. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 61 Summing Up Factors That Change Supply Changes in the cost of inputs, natural disasters, new technologies, and the impact of government decisions all affect the cost of production. In turn, these factors affect how much firms are willing to supply at any given price. Figure 3.15 summarizes factors that change the supply of goods and services. Notice that a change in the price of the product itself is not among the factors that shift the supply curve. Although a change in price of a good or service typically causes a change in quantity supplied or a movement along the supply curve for that specific good or service, it does not cause the supply curve itself to shift. Figure 3.15 Factors That Shift Supply Curves (a) A list of factors that can cause an increase in supply from S0 to S1. (b) The same factors, if their direction is reversed, can cause a decrease in supply from S0 to S1. Because demand and supply curves appear on a two-dimensional diagram with only price and quantity on the axes, an unwary visitor to the land of economics might be fooled into believing that economics is about only four topics: demand, supply, price, and quantity. However, demand and supply are really “umbrella” concepts: demand covers all the factors that affect demand, and supply covers all the factors that affect supply. We include factors other than price that affect demand and supply are included by using shifts in the demand or the supply curve. In this way, the two-dimensional demand and supply model becomes a powerful tool for analyzing a wide range of economic circumstances. 3.3 \| Changes in Equilibrium Price and Quantity: The Four-Step Process By the end of this section, you will be able to: Identify equilibrium price and quantity through the four-step process • • Graph equilibrium price and quantity • Contrast shifts of demand or supply and movements along a demand or supply curve • Graph demand and supply curves, including equilibrium price and quantity, based on real-world examples Let’s begin this discussion with a single economic event. It might be an event that affects demand, like a change in income, population, tastes, prices of substitutes or complements, or expectations about future prices. It might be an event that affects supply, like a change in natural conditions, input prices, or technology, or government policies that affect production. How does this economic event affect equilibrium price and quantity? We will analyze this question using a four-step process. Step 1. Draw a demand and supply model before the economic change took place. To establish the model requires four standard pieces of information: The law of demand, which tells us the slope of the demand curve; the law of supply, which gives us the slope of the supply curve; the shift variables for demand; and the shift variables for supply. From this model, find the initial equilibrium values for price and quantity. Step 2. Decide whether the economic change you are analyzing affects demand or supply. In other words, does the 62 Chapter 3 \| Demand and Supply event refer to something in the list of demand factors or supply factors? Step 3. Decide whether the effect on demand or supply causes the curve to shift to the right or to the left, and sketch the new demand or supply curve on the diagram. In other words, does the event increase or decrease the amount consumers want to buy or producers want to sell? Step 4. Identify the new equilibrium and then compare the original equilibrium price and quantity to the new equilibrium price and quantity. Let’s consider one example that involves a shift in supply and one that involves a shift in demand. Then we will consider an example where both supply and demand shift. Good Weather for Salmon Fishing Supposed that during the summer of 2015, weather conditions were excellent for commercial salmon fishing off the California coast. Heavy rains meant higher than normal levels of water in the rivers, which helps the salmon to breed. Slightly cooler ocean temperatures stimulated the growth of plankton, the microscopic organisms at the bottom of the ocean food chain, providing everything in the ocean with a hearty food supply. The ocean stayed calm during fishing season, so commercial fishing operations did not lose many days to bad weather. How did these climate conditions affect the quantity and price of salmon? Figure 3.16 illustrates the four-step approach, which we explain below, to work through this problem. Table 3.6 also provides the information to work the problem. Figure 3.16 Good Weather for Salmon Fishing: The Four-Step Process Unusually good weather leads to changes in the price and quantity of salmon. Price per Pound Quantity Supplied in 2014 Quantity Supplied in 2015 Quantity Demanded $2.00 $2.25 $2.50 $2.75 $3.00 $3.25 80 120 160 200 230 250 Table 3.6 Salmon Fishing 400 480 550 600 640 670 840 680 550 450 350 250 This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 63 Price per Pound Quantity Supplied in 2014 Quantity Supplied in 2015 Quantity Demanded $3.50 270 700 200 Table 3.6 Salmon Fishing Step 1. Draw a demand and supply model to illustrate the market for salmon in the year before the good weather conditions began. The demand curve D0 and the supply curve S0 show that the original equilibrium price is $3.25 per pound and the original equilibrium quantity is 250,000 fish. (This price per pound is what commercial buyers pay at the fishing docks. What consumers pay at the grocery is higher.) Step 2. Did the economic event affect supply or demand? Good weather is an example of a natural condition that affects supply. Step 3. Was the effect on supply an increase or a decrease? Good weather is a change in natural conditions that increases the quantity supplied at any given price. The supply curve shifts to the right, moving from the original supply curve S0 to the new supply curve S1, which Figure 3.16 and Table 3.6 show. Step 4. Compare the new equilibrium price and quantity to the original equilibrium. At the new equilibrium E1, the equilibrium price falls from $3.25 to $2.50, but the equilibrium quantity increases from 250,000 to 550,000 salmon. Notice that the equilibrium quantity demanded increased, even though the demand curve did not move. In short, good weather conditions increased supply of the California commercial salmon. The result was a higher equilibrium quantity of salmon bought and sold in the market at a lower price. Newspapers and the Internet According to the Pew Research Center for People and the Press, increasingly more people, especially younger people, are obtaining their news from online and digital sources. The majority of U.S. adults now own smartphones or tablets, and most of those Americans say they use them in part to access the news. From 2004 to 2012, the share of Americans who reported obtaining their news from digital sources increased from 24% to 39%. How has this affected consumption of print news media, and radio and television news? Figure 3.17 and the text below illustrates using the four-step analysis to answer this question. Figure 3.17 The Print News Market: A Four-Step Analysis A change in tastes from print news sources to digital sources results in a leftward shift in demand for the former. The result is a decrease in both equilibrium price and quantity. Step 1. Develop a demand and supply model to think about what the market looked like before the event. The demand curve D0 and the supply curve S0 show the original relationships. In this case, we perform the analysis without 64 Chapter 3 \| Demand and Supply specific numbers on the price and quantity axis. Step 2. Did the described change affect supply or demand? A change in tastes, from traditional news sources (print, radio, and television) to digital sources, caused a change in demand for the former. Step 3. Was the effect on demand positive or negative? A shift to digital news sources will tend to mean a lower quantity demanded of traditional news sources at every given price, causing the demand curve for print and other traditional news sources to shift to the left, from D0 to D1. Step 4. Compare the new equilibrium price and quantity to the original equilibrium price. The new equilibrium (E1) occurs at a lower quantity and a lower price than the original equilibrium (E0). The decline in print news reading predates 2004. Print newspaper circulation peaked in 1973 and has declined since then due to competition from television and radio news. In 1991, 55% of Americans indicated they received their news from print sources, while only 29% did so in 2012. Radio news has followed a similar path in recent decades, with the share of Americans obtaining their news from radio declining from 54% in 1991 to 33% in 2012. Television news has held its own over the last 15 years, with a market share staying in the mid to upper fifties. What does this suggest for the future, given that two-thirds of Americans under 30 years old say they do not obtain their news from television at all? The Interconnections and Speed of Adjustment in Real Markets In the real world, many factors that affect demand and supply can change all at once. For example, the demand for cars might increase because of rising incomes and population, and it might decrease because of rising gasoline prices (a complementary good). Likewise, the supply of cars might increase because of innovative new technologies that reduce the cost of car production, and it might decrease as a result of new government regulations requiring the installation of costly pollution-control technology. Moreover, rising incomes a
nd population or changes in gasoline prices will affect many markets, not just cars. How can an economist sort out all these interconnected events? The answer lies in the ceteris paribus assumption. Look at how each economic event affects each market, one event at a time, holding all else constant. Then combine the analyses to see the net effect. A Combined Example The U.S. Postal Service is facing difficult challenges. Compensation for postal workers tends to increase most years due to cost-of-living increases. At the same time, increasingly more people are using email, text, and other digital message forms such as Facebook and Twitter to communicate with friends and others. What does this suggest about the continued viability of the Postal Service? Figure 3.18 and the text below illustrate this using the four-step analysis to answer this question. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 65 Figure 3.18 Higher Compensation for Postal Workers: A Four-Step Analysis (a) Higher labor compensation causes a leftward shift in the supply curve, a decrease in the equilibrium quantity, and an increase in the equilibrium price. (b) A change in tastes away from Postal Services causes a leftward shift in the demand curve, a decrease in the equilibrium quantity, and a decrease in the equilibrium price. Since this problem involves two disturbances, we need two four-step analyses, the first to analyze the effects of higher compensation for postal workers, the second to analyze the effects of many people switching from “snail mail” to email and other digital messages. Figure 3.18 (a) shows the shift in supply discussed in the following steps. Step 1. Draw a demand and supply model to illustrate what the market for the U.S. Postal Service looked like before this scenario starts. The demand curve D0 and the supply curve S0 show the original relationships. Step 2. Did the described change affect supply or demand? Labor compensation is a cost of production. A change in production costs caused a change in supply for the Postal Service. Step 3. Was the effect on supply positive or negative? Higher labor compensation leads to a lower quantity supplied of postal services at every given price, causing the supply curve for postal services to shift to the left, from S0 to S1. Step 4. Compare the new equilibrium price and quantity to the original equilibrium price. The new equilibrium (E1) occurs at a lower quantity and a higher price than the original equilibrium (E0). Figure 3.18 (b) shows the shift in demand in the following steps. Step 1. Draw a demand and supply model to illustrate what the market for U.S. Postal Services looked like before this scenario starts. The demand curve D0 and the supply curve S0 show the original relationships. Note that this diagram is independent from the diagram in panel (a). Step 2. Did the change described affect supply or demand? A change in tastes away from snail mail toward digital messages will cause a change in demand for the Postal Service. Step 3. Was the effect on demand positive or negative? A change in tastes away from snailmail toward digital messages causes lower quantity demanded of postal services at every given price, causing the demand curve for postal services to shift to the left, from D0 to D1. Step 4. Compare the new equilibrium price and quantity to the original equilibrium price. The new equilibrium (E2) occurs at a lower quantity and a lower price than the original equilibrium (E0). The final step in a scenario where both supply and demand shift is to combine the two individual analyses to determine what happens to the equilibrium quantity and price. Graphically, we superimpose the previous two diagrams one on top of the other, as in Figure 3.19. 66 Chapter 3 \| Demand and Supply Figure 3.19 Combined Effect of Decreased Demand and Decreased Supply Supply and demand shifts cause changes in equilibrium price and quantity. Following are the results: Effect on Quantity: The effect of higher labor compensation on Postal Services because it raises the cost of production is to decrease the equilibrium quantity. The effect of a change in tastes away from snail mail is to decrease the equilibrium quantity. Since both shifts are to the left, the overall impact is a decrease in the equilibrium quantity of Postal Services (Q3). This is easy to see graphically, since Q3 is to the left of Q0. Effect on Price: The overall effect on price is more complicated. The effect of higher labor compensation on Postal Services, because it raises the cost of production, is to increase the equilibrium price. The effect of a change in tastes away from snail mail is to decrease the equilibrium price. Since the two effects are in opposite directions, unless we know the magnitudes of the two effects, the overall effect is unclear. This is not unusual. When both curves shift, typically we can determine the overall effect on price or on quantity, but not on both. In this case, we determined the overall effect on the equilibrium quantity, but not on the equilibrium price. In other cases, it might be the opposite. The next Clear It Up feature focuses on the difference between shifts of supply or demand and movements along a curve. What is the difference between shifts of demand or supply versus movements along a demand or supply curve? One common mistake in applying the demand and supply framework is to confuse the shift of a demand or a supply curve with movement along a demand or supply curve. As an example, consider a problem that asks whether a drought will increase or decrease the equilibrium quantity and equilibrium price of wheat. Lee, a student in an introductory economics class, might reason: “Well, it is clear that a drought reduces supply, so I will shift back the supply curve, as in the shift from the original supply curve S0 to S1 on the diagram (Shift 1). The equilibrium moves from E0 to E1, the equilibrium quantity is lower and the equilibrium price is higher. Then, a higher price makes farmers more likely to supply the good, so the supply curve shifts right, as shows the shift from S1 to S2, shows on the diagram (Shift 2), so that the equilibrium now moves from E1 to E2. The higher price, however, also reduces demand and so causes demand to shift back, like the shift from the original demand curve, D0 to D1 on the diagram (labeled This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 67 Shift 3), and the equilibrium moves from E2 to E3.” Figure 3.20 Shifts of Demand or Supply versus Movements along a Demand or Supply Curve A shift in one curve never causes a shift in the other curve. Rather, a shift in one curve causes a movement along the second curve. At about this point, Lee suspects that this answer is headed down the wrong path. Think about what might be wrong with Lee’s logic, and then read the answer that follows. Answer: Lee’s first step is correct: that is, a drought shifts back the supply curve of wheat and leads to a prediction of a lower equilibrium quantity and a higher equilibrium price. This corresponds to a movement along the original demand curve (D0), from E0 to E1. The rest of Lee’s argument is wrong, because it mixes up shifts in supply with quantity supplied, and shifts in demand with quantity demanded. A higher or lower price never shifts the supply curve, as suggested by the shift in supply from S1 to S2. Instead, a price change leads to a movement along a given supply curve. Similarly, a higher or lower price never shifts a demand curve, as suggested in the shift from D0 to D1. Instead, a price change leads to a movement along a given demand curve. Remember, a change in the price of a good never causes the demand or supply curve for that good to shift. Think carefully about the timeline of events: What happens first, what happens next? What is cause, what is effect? If you keep the order right, you are more likely to get the analysis correct. In the four-step analysis of how economic events affect equilibrium price and quantity, the movement from the old to the new equilibrium seems immediate. As a practical matter, however, prices and quantities often do not zoom straight to equilibrium. More realistically, when an economic event causes demand or supply to shift, prices and quantities set off in the general direction of equilibrium. Even as they are moving toward one new equilibrium, a subsequent change in demand or supply often pushes prices toward another equilibrium. 3.4 \| Price Ceilings and Price Floors By the end of this section, you will be able to: • Explain price controls, price ceilings, and price floors • Analyze demand and supply as a social adjustment mechanism To this point in the chapter, we have been assuming that markets are free, that is, they operate with no government intervention. In this section, we will explore the outcomes, both anticipated and otherwise, when government does intervene in a market either to prevent the price of some good or service from rising “too high” or to prevent the price of some good or service from falling “too low”. 68 Chapter 3 \| Demand and Supply Economists believe there are a small number of fundamental principles that explain how economic agents respond in different situations. Two of these principles, which we have already introduced, are the laws of demand and supply. Governments can pass laws affecting market outcomes, but no law can negate these economic principles. Rather, the principles will become apparent in sometimes unexpected ways, which may undermine the intent of the government policy. This is one of the major conclusions of this section. Controversy sometimes surrounds the prices and quantities established by demand and supply, especially for products that are considered necessities. In some cases, discontent over prices turns into public pressure on politicians, who may then pass legi
slation to prevent a certain price from climbing “too high” or falling “too low.” The demand and supply model shows how people and firms will react to the incentives that these laws provide to control prices, in ways that will often lead to undesirable consequences. Alternative policy tools can often achieve the desired goals of price control laws, while avoiding at least some of their costs and tradeoffs. Price Ceilings Laws that government enact to regulate prices are called price controls. Price controls come in two flavors. A price ceiling keeps a price from rising above a certain level (the “ceiling”), while a price floor keeps a price from falling below a given level (the “floor”). This section uses the demand and supply framework to analyze price ceilings. The next section discusses price floors. A price ceiling is a legal maximum price that one pays for some good or service. A government imposes price ceilings in order to keep the price of some necessary good or service affordable. For example, in 2005 during Hurricane Katrina, the price of bottled water increased above $5 per gallon. As a result, many people called for price controls on bottled water to prevent the price from rising so high. In this particular case, the government did not impose a price ceiling, but there are other examples of where price ceilings did occur. In many markets for goods and services, demanders outnumber suppliers. Consumers, who are also potential voters, sometimes unite behind a political proposal to hold down a certain price. In some cities, such as Albany, renters have pressed political leaders to pass rent control laws, a price ceiling that usually works by stating that landlords can raise rents by only a certain maximum percentage each year. Some of the best examples of rent control occur in urban areas such as New York, Washington D.C., or San Francisco. Rent control becomes a politically hot topic when rents begin to rise rapidly. Everyone needs an affordable place to live. Perhaps a change in tastes makes a certain suburb or town a more popular place to live. Perhaps locally-based businesses expand, bringing higher incomes and more people into the area. Such changes can cause a change in the demand for rental housing, as Figure 3.21 illustrates. The original equilibrium (E0) lies at the intersection of supply curve S0 and demand curve D0, corresponding to an equilibrium price of $500 and an equilibrium quantity of 15,000 units of rental housing. The effect of greater income or a change in tastes is to shift the demand curve for rental housing to the right, as the data in Table 3.7 shows and the shift from D0 to D1 on the graph. In this market, at the new equilibrium E1, the price of a rental unit would rise to $600 and the equilibrium quantity would increase to 17,000 units. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 69 Figure 3.21 A Price Ceiling Example—Rent Control The original intersection of demand and supply occurs at E0. If demand shifts from D0 to D1, the new equilibrium would be at E1—unless a price ceiling prevents the price from rising. If the price is not permitted to rise, the quantity supplied remains at 15,000. However, after the change in demand, the quantity demanded rises to 19,000, resulting in a shortage. Price Original Quantity Supplied Original Quantity Demanded New Quantity Demanded $400 12,000 $500 15,000 $600 17,000 $700 19,000 $800 20,000 Table 3.7 Rent Control 18,000 15,000 13,000 11,000 10,000 23,000 19,000 17,000 15,000 14,000 Suppose that a city government passes a rent control law to keep the price at the original equilibrium of $500 for a typical apartment. In Figure 3.21, the horizontal line at the price of $500 shows the legally fixed maximum price set by the rent control law. However, the underlying forces that shifted the demand curve to the right are still there. At that price ($500), the quantity supplied remains at the same 15,000 rental units, but the quantity demanded is 19,000 rental units. In other words, the quantity demanded exceeds the quantity supplied, so there is a shortage of rental housing. One of the ironies of price ceilings is that while the price ceiling was intended to help renters, there are actually fewer apartments rented out under the price ceiling (15,000 rental units) than would be the case at the market rent of $600 (17,000 rental units). Price ceilings do not simply benefit renters at the expense of landlords. Rather, some renters (or potential renters) lose their housing as landlords convert apartments to co-ops and condos. Even when the housing remains in the rental market, landlords tend to spend less on maintenance and on essentials like heating, cooling, hot water, and lighting. The first rule of economics is you do not get something for nothing—everything has an opportunity cost. Thus, if renters obtain “cheaper” housing than the market requires, they tend to also end up with lower quality housing. Price ceilings are enacted in an attempt to keep prices low for those who need the product. However, when the market price is not allowed to rise to the equilibrium level, quantity demanded exceeds quantity supplied, and thus a shortage occurs. Those who manage to purchase the product at the lower price given by the price ceiling will benefit, but sellers of the product will suffer, along with those who are not able to purchase the product at all. Quality is also likely to deteriorate. Price Floors A price floor is the lowest price that one can legally pay for some good or service. Perhaps the best-known example 70 Chapter 3 \| Demand and Supply of a price floor is the minimum wage, which is based on the view that someone working full time should be able to afford a basic standard of living. The federal minimum wage in 2016 was $7.25 per hour, although some states and localities have a higher minimum wage. The federal minimum wage yields an annual income for a single person of $15,080, which is slightly higher than the Federal poverty line of $11,880. As the cost of living rises over time, the Congress periodically raises the federal minimum wage. Price floors are sometimes called “price supports,” because they support a price by preventing it from falling below a certain level. Around the world, many countries have passed laws to create agricultural price supports. Farm prices and thus farm incomes fluctuate, sometimes widely. Even if, on average, farm incomes are adequate, some years they can be quite low. The purpose of price supports is to prevent these swings. The most common way price supports work is that the government enters the market and buys up the product, adding to demand to keep prices higher than they otherwise would be. According to the Common Agricultural Policy reform passed in 2013, the European Union (EU) will spend about 60 billion euros per year, or 67 billion dollars per year (with the November 2016 exchange rate), or roughly 38% of the EU budget, on price supports for Europe’s farmers from 2014 to 2020. Figure 3.22 illustrates the effects of a government program that assures a price above the equilibrium by focusing on the market for wheat in Europe. In the absence of government intervention, the price would adjust so that the quantity supplied would equal the quantity demanded at the equilibrium point E0, with price P0 and quantity Q0. However, policies to keep prices high for farmers keeps the price above what would have been the market equilibrium level—the price Pf shown by the dashed horizontal line in the diagram. The result is a quantity supplied in excess of the quantity demanded (Qd). When quantity supplied exceeds quantity demanded, a surplus exists. Economists estimate that the high-income areas of the world, including the United States, Europe, and Japan, spend roughly $1 billion per day in supporting their farmers. If the government is willing to purchase the excess supply (or to provide payments for others to purchase it), then farmers will benefit from the price floor, but taxpayers and consumers of food will pay the costs. Agricultural economists and policy makers have offered numerous proposals for reducing farm subsidies. In many countries, however, political support for subsidies for farmers remains strong. This is either because the population views this as supporting the traditional rural way of life or because of industry's lobbying power of the agro-business. Figure 3.22 European Wheat Prices: A Price Floor Example The intersection of demand (D) and supply (S) would be at the equilibrium point E0. However, a price floor set at Pf holds the price above E0 and prevents it from falling. The result of the price floor is that the quantity supplied Qs exceeds the quantity demanded Qd. There is excess supply, also called a surplus. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 71 3.5 \| Demand, Supply, and Efficiency By the end of this section, you will be able to: • Contrast consumer surplus, producer surplus, and social surplus • Explain why price floors and price ceilings can be inefficient • Analyze demand and supply as a social adjustment mechanism The familiar demand and supply diagram holds within it the concept of economic efficiency. One typical way that economists define efficiency is when it is impossible to improve the situation of one party without imposing a cost on another. Conversely, if a situation is inefficient, it becomes possible to benefit at least one party without imposing costs on others. Efficiency in the demand and supply model has the same basic meaning: The economy is getting as much benefit as possible from its scarce resources and all the possible gains from trade have been achieved. In other words, the optimal amount of each good and service is produced and consumed. Consumer Surplus, Producer Surplus, Social Surplus Consider a market for tablet compu
ters, as Figure 3.23 shows. The equilibrium price is $80 and the equilibrium quantity is 28 million. To see the benefits to consumers, look at the segment of the demand curve above the equilibrium point and to the left. This portion of the demand curve shows that at least some demanders would have been willing to pay more than $80 for a tablet. For example, point J shows that if the price were $90, 20 million tablets would be sold. Those consumers who would have been willing to pay $90 for a tablet based on the utility they expect to receive from it, but who were able to pay the equilibrium price of $80, clearly received a benefit beyond what they had to pay. Remember, the demand curve traces consumers’ willingness to pay for different quantities. The amount that individuals would have been willing to pay, minus the amount that they actually paid, is called consumer surplus. Consumer surplus is the area labeled F—that is, the area above the market price and below the demand curve. Figure 3.23 Consumer and Producer Surplus The somewhat triangular area labeled by F shows the area of consumer surplus, which shows that the equilibrium price in the market was less than what many of the consumers were willing to pay. Point J on the demand curve shows that, even at the price of $90, consumers would have been willing to purchase a quantity of 20 million. The somewhat triangular area labeled by G shows the area of producer surplus, which shows that the equilibrium price received in the market was more than what many of the producers were willing to accept for their products. For example, point K on the supply curve shows that at a price of $45, firms would have been willing to supply a quantity of 14 million. The supply curve shows the quantity that firms are willing to supply at each price. For example, point K in Figure 72 Chapter 3 \| Demand and Supply 3.23 illustrates that, at $45, firms would still have been willing to supply a quantity of 14 million. Those producers who would have been willing to supply the tablets at $45, but who were instead able to charge the equilibrium price of $80, clearly received an extra benefit beyond what they required to supply the product. The amount that a seller is paid for a good minus the seller’s actual cost is called producer surplus. In Figure 3.23, producer surplus is the area labeled G—that is, the area between the market price and the segment of the supply curve below the equilibrium. The sum of consumer surplus and producer surplus is social surplus, also referred to as economic surplus or total surplus. In Figure 3.23 we show social surplus as the area F + G. Social surplus is larger at equilibrium quantity and price than it would be at any other quantity. This demonstrates the economic efficiency of the market equilibrium. In addition, at the efficient level of output, it is impossible to produce greater consumer surplus without reducing producer surplus, and it is impossible to produce greater producer surplus without reducing consumer surplus. Inefficiency of Price Floors and Price Ceilings The imposition of a price floor or a price ceiling will prevent a market from adjusting to its equilibrium price and quantity, and thus will create an inefficient outcome. However, there is an additional twist here. Along with creating inefficiency, price floors and ceilings will also transfer some consumer surplus to producers, or some producer surplus to consumers. Imagine that several firms develop a promising but expensive new drug for treating back pain. If this therapy is left to the market, the equilibrium price will be $600 per month and 20,000 people will use the drug, as shown in Figure 3.24 (a). The original level of consumer surplus is T + U and producer surplus is V + W + X. However, the government decides to impose a price ceiling of $400 to make the drug more affordable. At this price ceiling, firms in the market now produce only 15,000. As a result, two changes occur. First, an inefficient outcome occurs and the total surplus of society is reduced. The loss in social surplus that occurs when the economy produces at an inefficient quantity is called deadweight loss. In a very real sense, it is like money thrown away that benefits no one. In Figure 3.24 (a), the deadweight loss is the area U + W. When deadweight loss exists, it is possible for both consumer and producer surplus to be higher, in this case because the price control is blocking some suppliers and demanders from transactions they would both be willing to make. A second change from the price ceiling is that some of the producer surplus is transferred to consumers. After the price ceiling is imposed, the new consumer surplus is T + V, while the new producer surplus is X. In other words, the price ceiling transfers the area of surplus (V) from producers to consumers. Note that the gain to consumers is less than the loss to producers, which is just another way of seeing the deadweight loss. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 73 Figure 3.24 Efficiency and Price Floors and Ceilings (a) The original equilibrium price is $600 with a quantity of 20,000. Consumer surplus is T + U, and producer surplus is V + W + X. A price ceiling is imposed at $400, so firms in the market now produce only a quantity of 15,000. As a result, the new consumer surplus is T + V, while the new producer surplus is X. (b) The original equilibrium is $8 at a quantity of 1,800. Consumer surplus is G + H + J, and producer surplus is I + K. A price floor is imposed at $12, which means that quantity demanded falls to 1,400. As a result, the new consumer surplus is G, and the new producer surplus is H + I. Figure 3.24 (b) shows a price floor example using a string of struggling movie theaters, all in the same city. The current equilibrium is $8 per movie ticket, with 1,800 people attending movies. The original consumer surplus is G + H + J, and producer surplus is I + K. The city government is worried that movie theaters will go out of business, reducing the entertainment options available to citizens, so it decides to impose a price floor of $12 per ticket. As a result, the quantity demanded of movie tickets falls to 1,400. The new consumer surplus is G, and the new producer surplus is H + I. In effect, the price floor causes the area H to be transferred from consumer to producer surplus, but also causes a deadweight loss of J + K. This analysis shows that a price ceiling, like a law establishing rent controls, will transfer some producer surplus to consumers—which helps to explain why consumers often favor them. Conversely, a price floor like a guarantee that farmers will receive a certain price for their crops will transfer some consumer surplus to producers, which explains why producers often favor them. However, both price floors and price ceilings block some transactions that buyers and sellers would have been willing to make, and creates deadweight loss. Removing such barriers, so that prices and quantities can adjust to their equilibrium level, will increase the economy’s social surplus. Demand and Supply as a Social Adjustment Mechanism The demand and supply model emphasizes that prices are not set only by demand or only by supply, but by the interaction between the two. In 1890, the famous economist Alfred Marshall wrote that asking whether supply or demand determined a price was like arguing “whether it is the upper or the under blade of a pair of scissors that cuts a piece of paper.” The answer is that both blades of the demand and supply scissors are always involved. The adjustments of equilibrium price and quantity in a market-oriented economy often occur without much government direction or oversight. If the coffee crop in Brazil suffers a terrible frost, then the supply curve of coffee shifts to the left and the price of coffee rises. Some people—call them the coffee addicts—continue to drink coffee and pay the higher price. Others switch to tea or soft drinks. No government commission is needed to figure out how to adjust coffee prices, which companies will be allowed to process the remaining supply, which supermarkets in which cities will get how much coffee to sell, or which consumers will ultimately be allowed to drink the brew. Such adjustments in response to price changes happen all the time in a market economy, often so smoothly and rapidly that we barely notice them. Think for a moment of all the seasonal foods that are available and inexpensive at certain times of the year, like fresh corn in midsummer, but more expensive at other times of the year. People alter their diets and restaurants alter their menus in response to these fluctuations in prices without fuss or fanfare. For both the U.S. economy and the world 74 Chapter 3 \| Demand and Supply economy as a whole, markets—that is, demand and supply—are the primary social mechanism for answering the basic questions about what is produced, how it is produced, and for whom it is produced. Why Can We Not Get Enough of Organic? Organic food is grown without synthetic pesticides, chemical fertilizers or genetically modified seeds. In recent decades, the demand for organic products has increased dramatically. The Organic Trade Association reported sales increased from $1 billion in 1990 to $35.1 billion in 2013, more than 90% of which were sales of food products. Why, then, are organic foods more expensive than their conventional counterparts? The answer is a clear application of the theories of supply and demand. As people have learned more about the harmful effects of chemical fertilizers, growth hormones, pesticides and the like from large-scale factory farming, our tastes and preferences for safer, organic foods have increased. This change in tastes has been reinforced by increases in income, which allow people to purchase pricier products, and has made organic foods more mainstream. This has led t
o an increased demand for organic foods. Graphically, the demand curve has shifted right, and we have moved up the supply curve as producers have responded to the higher prices by supplying a greater quantity. In addition to the movement along the supply curve, we have also had an increase in the number of farmers converting to organic farming over time. This is represented by a shift to the right of the supply curve. Since both demand and supply have shifted to the right, the resulting equilibrium quantity of organic foods is definitely higher, but the price will only fall when the increase in supply is larger than the increase in demand. We may need more time before we see lower prices in organic foods. Since the production costs of these foods may remain higher than conventional farming, because organic fertilizers and pest management techniques are more expensive, they may never fully catch up with the lower prices of non-organic foods. As a final, specific example: The Environmental Working Group’s “Dirty Dozen” list of fruits and vegetables, which test high for pesticide residue even after washing, was released in April 2013. The inclusion of strawberries on the list has led to an increase in demand for organic strawberries, resulting in both a higher equilibrium price and quantity of sales. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 75 KEY TERMS ceteris paribus other things being equal complements other goods that are often used together so that consumption of one good tends to enhance consumption of the consumer surplus the extra benefit consumers receive from buying a good or service, measured by what the individuals would have been willing to pay minus the amount that they actually paid deadweight loss the loss in social surplus that occurs when a market produces an inefficient quantity demand the relationship between price and the quantity demanded of a certain good or service demand curve a graphic representation of the relationship between price and quantity demanded of a certain good or service, with quantity on the horizontal axis and the price on the vertical axis demand schedule a table that shows a range of prices for a certain good or service and the quantity demanded at each price economic surplus see social surplus equilibrium the situation where quantity demanded is equal to the quantity supplied; the combination of price and quantity where there is no economic pressure from surpluses or shortages that would cause price or quantity to change equilibrium price the price where quantity demanded is equal to quantity supplied equilibrium quantity the quantity at which quantity demanded and quantity supplied are equal for a certain price level excess demand at the existing price, the quantity demanded exceeds the quantity supplied; also called a shortage excess supply at the existing price, quantity supplied exceeds the quantity demanded; also called a surplus factors of production the resources such as labor, materials, and machinery that are used to produce goods and services; also called inputs inferior good a good in which the quantity demanded falls as income rises, and in which quantity demanded rises and income falls inputs the resources such as labor, materials, and machinery that are used to produce goods and services; also called factors of production law of demand the common relationship that a higher price leads to a lower quantity demanded of a certain good or service and a lower price leads to a higher quantity demanded, while all other variables are held constant law of supply the common relationship that a higher price leads to a greater quantity supplied and a lower price leads to a lower quantity supplied, while all other variables are held constant normal good a good in which the quantity demanded rises as income rises, and in which quantity demanded falls as income falls price what a buyer pays for a unit of the specific good or service price ceiling a legal maximum price price control government laws to regulate prices instead of letting market forces determine prices 76 Chapter 3 \| Demand and Supply price floor a legal minimum price producer surplus the extra benefit producers receive from selling a good or service, measured by the price the producer actually received minus the price the producer would have been willing to accept quantity demanded the total number of units of a good or service consumers are willing to purchase at a given price quantity supplied the total number of units of a good or service producers are willing to sell at a given price shift in demand at every price when a change in some economic factor (other than price) causes a different quantity to be demanded shift in supply every price when a change in some economic factor (other than price) causes a different quantity to be supplied at shortage at the existing price, the quantity demanded exceeds the quantity supplied; also called excess demand social surplus the sum of consumer surplus and producer surplus substitute other a good that can replace another to some extent, so that greater consumption of one good can mean less of the supply the relationship between price and the quantity supplied of a certain good or service supply curve a line that shows the relationship between price and quantity supplied on a graph, with quantity supplied on the horizontal axis and price on the vertical axis supply schedule a table that shows a range of prices for a good or service and the quantity supplied at each price surplus at the existing price, quantity supplied exceeds the quantity demanded; also called excess supply total surplus see social surplus KEY CONCEPTS AND SUMMARY 3.1 Demand, Supply, and Equilibrium in Markets for Goods and Services A demand schedule is a table that shows the quantity demanded at different prices in the market. A demand curve shows the relationship between quantity demanded and price in a given market on a graph. The law of demand states that a higher price typically leads to a lower quantity demanded. A supply schedule is a table that shows the quantity supplied at different prices in the market. A supply curve shows the relationship between quantity supplied and price on a graph. The law of supply says that a higher price typically leads to a higher quantity supplied. The equilibrium price and equilibrium quantity occur where the supply and demand curves cross. The equilibrium occurs where the quantity demanded is equal to the quantity supplied. If the price is below the equilibrium level, then the quantity demanded will exceed the quantity supplied. Excess demand or a shortage will exist. If the price is above the equilibrium level, then the quantity supplied will exceed the quantity demanded. Excess supply or a surplus will exist. In either case, economic pressures will push the price toward the equilibrium level. 3.2 Shifts in Demand and Supply for Goods and Services Economists often use the ceteris paribus or “other things being equal” assumption: while examining the economic impact of one event, all other factors remain unchanged for analysis purposes. Factors that can shift the demand curve for goods and services, causing a different quantity to be demanded at any given price, include changes in tastes, population, income, prices of substitute or complement goods, and expectations about future conditions and prices. Factors that can shift the supply curve for goods and services, causing a different quantity to be supplied at any given price, include input prices, natural conditions, changes in technology, and government taxes, regulations, or subsidies. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 77 3.3 Changes in Equilibrium Price and Quantity: The Four-Step Process When using the supply and demand framework to think about how an event will affect the equilibrium price and quantity, proceed through four steps: (1) sketch a supply and demand diagram to think about what the market looked like before the event; (2) decide whether the event will affect supply or demand; (3) decide whether the effect on supply or demand is negative or positive, and draw the appropriate shifted supply or demand curve; (4) compare the new equilibrium price and quantity to the original ones. 3.4 Price Ceilings and Price Floors Price ceilings prevent a price from rising above a certain level. When a price ceiling is set below the equilibrium price, quantity demanded will exceed quantity supplied, and excess demand or shortages will result. Price floors prevent a price from falling below a certain level. When a price floor is set above the equilibrium price, quantity supplied will exceed quantity demanded, and excess supply or surpluses will result. Price floors and price ceilings often lead to unintended consequences. 3.5 Demand, Supply, and Efficiency Consumer surplus is the gap between the price that consumers are willing to pay, based on their preferences, and the market equilibrium price. Producer surplus is the gap between the price for which producers are willing to sell a product, based on their costs, and the market equilibrium price. Social surplus is the sum of consumer surplus and producer surplus. Total surplus is larger at the equilibrium quantity and price than it will be at any other quantity and price. Deadweight loss is loss in total surplus that occurs when the economy produces at an inefficient quantity. SELF-CHECK QUESTIONS 1. Review Figure 3.4. Suppose the price of gasoline is $1.60 per gallon. Is the quantity demanded higher or lower than at the equilibrium price of $1.40 per gallon? What about the quantity supplied? Is there a shortage or a surplus in the market? If so, how much? 2. Why do economists use the ceteris paribus assumption? In an analysis of the market for paint, an economist discovers the facts listed below. State whether each of
these 3. changes will affect supply or demand, and in what direction. a. There have recently been some important cost-saving inventions in the technology for making paint. b. Paint is lasting longer, so that property owners need not repaint as often. c. Because of severe hailstorms, many people need to repaint now. d. The hailstorms damaged several factories that make paint, forcing them to close down for several months. 4. Many changes are affecting the market for oil. Predict how each of the following events will affect the equilibrium price and quantity in the market for oil. In each case, state how the event will affect the supply and demand diagram. Create a sketch of the diagram if necessary. a. Cars are becoming more fuel efficient, and therefore get more miles to the gallon. b. The winter is exceptionally cold. c. A major discovery of new oil is made off the coast of Norway. d. The economies of some major oil-using nations, like Japan, slow down. e. A war in the Middle East disrupts oil-pumping schedules. f. Landlords install additional insulation in buildings. g. The price of solar energy falls dramatically. h. Chemical companies invent a new, popular kind of plastic made from oil. 5. Let’s think about the market for air travel. From August 2014 to January 2015, the price of jet fuel increased roughly 47%. Using the four-step analysis, how do you think this fuel price increase affected the equilibrium price and quantity of air travel? 78 Chapter 3 \| Demand and Supply 6. A tariff is a tax on imported goods. Suppose the U.S. government cuts the tariff on imported flat screen televisions. Using the four-step analysis, how do you think the tariff reduction will affect the equilibrium price and quantity of flat screen TVs? 7. What is the effect of a price ceiling on the quantity demanded of the product? What is the effect of a price ceiling on the quantity supplied? Why exactly does a price ceiling cause a shortage? 8. Does a price ceiling change the equilibrium price? 9. What would be the impact of imposing a price floor below the equilibrium price? 10. Does a price ceiling increase or decrease the number of transactions in a market? Why? What about a price floor? 11. If a price floor benefits producers, why does a price floor reduce social surplus? REVIEW QUESTIONS 12. What determines the level of prices in a market? 13. What does a downward-sloping demand curve mean about how buyers in a market will react to a higher price? 14. Will demand curves have the same exact shape in all markets? If not, how will they differ? 15. Will supply curves have the same shape in all markets? If not, how will they differ? is 16. What the relationship between quantity demanded and quantity supplied at equilibrium? What is the relationship when there is a shortage? What is the relationship when there is a surplus? 17. How can you locate the equilibrium point on a demand and supply graph? 18. If the price is above the equilibrium level, would you predict a surplus or a shortage? If the price is below the equilibrium level, would you predict a surplus or a shortage? Why? 19. When the price is above the equilibrium, explain to how market equilibrium. Do the same when the price is below the equilibrium. the market price forces move 20. What is the difference between the demand and the quantity demanded of a product, say milk? Explain in words and show the difference on a graph with a demand curve for milk. 21. What is the difference between the supply and the quantity supplied of a product, say milk? Explain in words and show the difference on a graph with the supply curve for milk. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 22. When analyzing a market, how do economists deal with the problem that many factors that affect the market are changing at the same time? 23. Name some factors that can cause a shift in the demand curve in markets for goods and services. 24. Name some factors that can cause a shift in the supply curve in markets for goods and services. 25. How does one analyze a market where both demand and supply shift? 26. What causes a movement along the demand curve? What causes a movement along the supply curve? 27. Does a price ceiling attempt to make a price higher or lower? 28. How does a price ceiling set below the equilibrium level affect quantity demanded and quantity supplied? 29. Does a price floor attempt to make a price higher or lower? 30. How does a price floor set above the equilibrium level affect quantity demanded and quantity supplied? 31. What is consumer surplus? How is it illustrated on a demand and supply diagram? 32. What is producer surplus? How is it illustrated on a demand and supply diagram? 33. What is total surplus? How is it illustrated on a demand and supply diagram? 34. What is the relationship between total surplus and economic efficiency? 35. What is deadweight loss? Chapter 3 \| Demand and Supply 79 44. Use the four-step process to analyze the impact of a reduction in tariffs on imports of iPods on the equilibrium price and quantity of Sony Walkman-type products. 45. Suppose both of these events took place at the same time. Combine your analyses of the impacts of the iPod and the tariff reduction to determine the likely impact on the equilibrium price and quantity of Sony Walkmantype products. Show your answer graphically. 46. Most government policy decisions have winners and losers. What are the effects of raising the minimum wage? It is more complex than simply producers lose and workers gain. Who are the winners and who are the losers, and what exactly do they win and lose? To what extent does the policy change achieve its goals? 47. Agricultural price supports result in governments holding large inventories of agricultural products. Why do you think the government cannot simply give the products away to poor people? 48. Can you propose a policy that would induce the market to supply more rental housing units? 49. What term would an economist use to describe what happens when a shopper gets a “good deal” on a product? 50. Explain why voluntary transactions improve social welfare. 51. Why would a free market never operate at a quantity greater than the equilibrium quantity? Hint: What would be required for a transaction to occur at that quantity? CRITICAL THINKING QUESTIONS 36. Review Figure 3.4. Suppose the government decided that, since gasoline is a necessity, its price should be legally capped at $1.30 per gallon. What do you anticipate would be the outcome in the gasoline market? 37. Explain why the following statement is false: “In the goods market, no buyer would be willing to pay more than the equilibrium price.” 38. Explain why the following statement is false: “In the goods market, no seller would be willing to sell for less than the equilibrium price.” 39. Consider the demand for hamburgers. If the price of a substitute good (for example, hot dogs) increases and the price of a complement good (for example, hamburger buns) increases, can you tell for sure what will happen to the demand for hamburgers? Why or why not? Illustrate your answer with a graph. 40. How do you suppose the demographics of an aging population of “Baby Boomers” in the United States will affect the demand for milk? Justify your answer. 41. We know that a change in the price of a product causes a movement along the demand curve. Suppose consumers believe that prices will be rising in the future. How will that affect demand for the product in the present? Can you show this graphically? 42. Suppose there is a soda tax to curb obesity. What should a reduction in the soda tax do to the supply of sodas and to the equilibrium price and quantity? Can you show this graphically? Hint: Assume that the soda tax is collected from the sellers. 43. Use the four-step process to analyze the impact of the advent of the iPod (or other portable digital music players) on the equilibrium price and quantity of the Sony Walkman (or other portable audio cassette players). PROBLEMS 52. Review Figure 3.4 again. Suppose the price of gasoline is $1.00. Will the quantity demanded be lower or higher than at the equilibrium price of $1.40 per gallon? Will the quantity supplied be lower or higher? Is there a shortage or a surplus in the market? If so, of how much? 80 Chapter 3 \| Demand and Supply 55. Table 3.9 illustrates the market's demand and supply for cheddar cheese. Graph the data and find the equilibrium. Next, create a table showing the change in quantity demanded or quantity supplied, and a graph of the new equilibrium, in each of the following situations: a. The price of milk, a key input for cheese production, rises, so that the supply decreases by 80 pounds at every price. b. A new study says that eating cheese is good for your health, so that demand increases by 20% at every price. Price per Pound Qd Qs $3.00 $3.20 $3.40 $3.60 $3.80 $4.00 Table 3.9 750 700 650 620 600 590 540 600 650 700 720 730 53. Table 3.8 shows information on the demand and supply for bicycles, where the quantities of bicycles are measured in thousands. Price Qd Qs $120 $150 $180 $210 $240 Table 3.8 50 40 32 28 24 36 40 48 56 70 a. What is the quantity demanded and the quantity supplied at a price of $210? b. At what price is the quantity supplied equal to 48,000? c. Graph the demand and supply curve for bicycles. How can you determine the equilibrium price and quantity from the graph? How can you determine the equilibrium price and quantity from the table? What are the equilibrium price and equilibrium quantity? If the price was $120, what would the quantities demanded and supplied be? Would a shortage or surplus exist? If so, how large would the shortage or surplus be? d. 54. The computer market in recent years has seen many more computers sell at much lower prices. What shift in demand or supply is most likely to explain this outcome? Sketch a demand and supply diagram and explain your reasoning for each. a.
A rise in demand b. A fall in demand c. A rise in supply d. A fall in supply This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 3 \| Demand and Supply 81 56. Table 3.10 shows the supply and demand for movie tickets in a city. Graph demand and supply and identify the equilibrium. Then calculate in a table and graph the effect of the following two changes. a. Three new nightclubs open. They offer decent bands and have no cover charge, but make their money by selling food and drink. As a result, demand for movie tickets falls by six units at every price. b. The city eliminates a tax that it placed on all local entertainment businesses. The result is that the quantity supplied of movies at any given price increases by 10%. Price per Pound Qd Qs $5.00 $6.00 $7.00 $8.00 $9.00 Table 3.10 26 24 22 21 20 16 18 20 21 22 57. A low-income country decides to set a price ceiling on bread so it can make sure that bread is affordable to the poor.Table 3.11 provides the conditions of demand and supply. What are the equilibrium price and equilibrium quantity before the price ceiling? What will the excess demand or the shortage (that is, quantity demanded minus quantity supplied) be if the price ceiling is set at $2.40? At $2.00? At $3.60? Price Qd Qs $1.60 $2.00 $2.40 $2.80 $3.20 $3.60 $4.00 Table 3.11 9,000 8,500 8,000 7,500 7,000 6,500 6,000 5,000 5,500 6,400 7,500 9,000 11,000 15,000 82 Chapter 3 \| Demand and Supply This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 83 4 \| Labor and Financial Markets Figure 4.1 People often think of demand and supply in relation to goods, but labor markets, such as the nursing profession, can also apply to this analysis. (Credit: modification of work by "Fotos GOVBA"/Flickr Creative Commons) Baby Boomers Come of Age The Census Bureau reports that as of 2013, 20% of the U.S. population was over 60 years old, which means that almost 63 million people are reaching an age when they will need increased medical care. The baby boomer population, the group born between 1946 and 1964, is comprised of approximately 74 million people who have just reached retirement age. As this population grows older, they will be faced with common healthcare issues such as heart conditions, arthritis, and Alzheimer’s that may require hospitalization, long-term, or at-home nursing care. Aging baby boomers and advances in life-saving and lifeextending technologies will increase the demand for healthcare and nursing. Additionally, the Affordable Care Act, which expands access to healthcare for millions of Americans, has further increase the demand, although with the election of Donald J. Trump, this increase may not be sustained. According to the Bureau of Labor Statistics, registered nursing jobs are expected to increase by 16% between 2014 and 2024. The median annual wage of $67,490 (in 2015) is also expected to increase. The BLS forecasts that 439,000 new nurses will be in demand by 2022. These data tell us, as economists, that the market for healthcare professionals, and nurses in particular, will face several challenges. Our study of supply and demand will help us to analyze what might happen in the 84 Chapter 4 \| Labor and Financial Markets labor market for nursing and other healthcare professionals, as we will discuss in the second half of this case at the end of the chapter. Introduction to Labor and Financial Markets In this chapter, you will learn about: • Demand and Supply at Work in Labor Markets • Demand and Supply in Financial Markets • The Market System as an Efficient Mechanism for Information The theories of supply and demand do not apply just to markets for goods. They apply to any market, even markets for things we may not think of as goods and services like labor and financial services. Labor markets are markets for employees or jobs. Financial services markets are markets for saving or borrowing. When we think about demand and supply curves in goods and services markets, it is easy to picture the demanders and suppliers: businesses produce the products and households buy them. Who are the demanders and suppliers in labor and financial service markets? In labor markets job seekers (individuals) are the suppliers of labor, while firms and other employers who hire labor are the demanders for labor. In financial markets, any individual or firm who saves contributes to the supply of money, and any who borrows (person, firm, or government) contributes to the demand for money. As a college student, you most likely participate in both labor and financial markets. Employment is a fact of life for most college students: According to the National Center for Educational Statistics, in 2013 40% of full-time college students and 76% of part-time college students were employed. Most college students are also heavily involved in financial markets, primarily as borrowers. Among full-time students, about half take out a loan to help finance their education each year, and those loans average about $6,000 per year. Many students also borrow for other expenses, like purchasing a car. As this chapter will illustrate, we can analyze labor markets and financial markets with the same tools we use to analyze demand and supply in the goods markets. 4.1 \| Demand and Supply at Work in Labor Markets By the end of this section, you will be able to: • Predict shifts in the demand and supply curves of the labor market • Explain the impact of new technology on the demand and supply curves of the labor market • Explain price floors in the labor market such as minimum wage or a living wage Markets for labor have demand and supply curves, just like markets for goods. The law of demand applies in labor markets this way: A higher salary or wage—that is, a higher price in the labor market—leads to a decrease in the quantity of labor demanded by employers, while a lower salary or wage leads to an increase in the quantity of labor demanded. The law of supply functions in labor markets, too: A higher price for labor leads to a higher quantity of labor supplied; a lower price leads to a lower quantity supplied. Equilibrium in the Labor Market In 2015, about 35,000 registered nurses worked in the Minneapolis-St. Paul-Bloomington, Minnesota-Wisconsin metropolitan area, according to the BLS. They worked for a variety of employers: hospitals, doctors’ offices, schools, health clinics, and nursing homes. Figure 4.2 illustrates how demand and supply determine equilibrium in this labor market. The demand and supply schedules in Table 4.1 list the quantity supplied and quantity demanded of nurses at different salaries. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 85 Figure 4.2 Labor Market Example: Demand and Supply for Nurses in Minneapolis-St. Paul-Bloomington The demand curve (D) of those employers who want to hire nurses intersects with the supply curve (S) of those who are qualified and willing to work as nurses at the equilibrium point (E). The equilibrium salary is $70,000 and the equilibrium quantity is 34,000 nurses. At an above-equilibrium salary of $75,000, quantity supplied increases to 38,000, but the quantity of nurses demanded at the higher pay declines to 33,000. At this above-equilibrium salary, an excess supply or surplus of nurses would exist. At a below-equilibrium salary of $60,000, quantity supplied declines to 27,000, while the quantity demanded at the lower wage increases to 40,000 nurses. At this belowequilibrium salary, excess demand or a shortage exists. Annual Salary Quantity Demanded Quantity Supplied $55,000 $60,000 $65,000 $70,000 $75,000 $80,000 45,000 40,000 37,000 34,000 33,000 32,000 20,000 27,000 31,000 34,000 38,000 41,000 Table 4.1 Demand and Supply of Nurses in Minneapolis-St. Paul-Bloomington The horizontal axis shows the quantity of nurses hired. In this example we measure labor by number of workers, but another common way to measure the quantity of labor is by the number of hours worked. The vertical axis shows the price for nurses’ labor—that is, how much they are paid. In the real world, this “price” would be total labor compensation: salary plus benefits. It is not obvious, but benefits are a significant part (as high as 30 percent) of labor compensation. In this example we measure the price of labor by salary on an annual basis, although in other cases we could measure the price of labor by monthly or weekly pay, or even the wage paid per hour. As the salary for nurses rises, the quantity demanded will fall. Some hospitals and nursing homes may reduce the number of nurses they hire, or they may lay off some of their existing nurses, rather than pay them higher salaries. Employers who face higher nurses’ salaries may also try to replace some nursing functions by investing in physical equipment, like computer monitoring and diagnostic systems to monitor patients, or by using lower-paid health care aides to reduce the number of nurses they need. As the salary for nurses rises, the quantity supplied will rise. If nurses’ salaries in Minneapolis-St. Paul-Bloomington are higher than in other cities, more nurses will move to Minneapolis-St. Paul-Bloomington to find jobs, more people will be willing to train as nurses, and those currently trained as nurses will be more likely to pursue nursing as a fulltime job. In other words, there will be more nurses looking for jobs in the area. 86 Chapter 4 \| Labor and Financial Markets At equilibrium, the quantity supplied and the quantity demanded are equal. Thus, every employer who wants to hire a nurse at this equilibrium wage can find a willing worker, and every nurse who wants to work at this equilibrium salary can find a job. In Figure 4.2, the supply curve (S) and demand curve (D) intersect at the equilibrium point (E). The equilibrium quantity of nurses in the Minneapolis-St. Paul-Bloomin
gton area is 34,000, and the equilibrium salary is $70,000 per year. This example simplifies the nursing market by focusing on the “average” nurse. In reality, of course, the market for nurses actually comprises many smaller markets, like markets for nurses with varying degrees of experience and credentials. Many markets contain closely related products that differ in quality. For instance, even a simple product like gasoline comes in regular, premium, and super-premium, each with a different price. Even in such cases, discussing the average price of gasoline, like the average salary for nurses, can still be useful because it reflects what is happening in most of the submarkets. When the price of labor is not at the equilibrium, economic incentives tend to move salaries toward the equilibrium. For example, if salaries for nurses in Minneapolis-St. Paul-Bloomington were above the equilibrium at $75,000 per year, then 38,000 people want to work as nurses, but employers want to hire only 33,000 nurses. At that aboveequilibrium salary, excess supply or a surplus results. In a situation of excess supply in the labor market, with many applicants for every job opening, employers will have an incentive to offer lower wages than they otherwise would have. Nurses’ salary will move down toward equilibrium. In contrast, if the salary is below the equilibrium at, say, $60,000 per year, then a situation of excess demand or a shortage arises. In this case, employers encouraged by the relatively lower wage want to hire 40,000 nurses, but only 27,000 individuals want to work as nurses at that salary in Minneapolis-St. Paul-Bloomington. In response to the shortage, some employers will offer higher pay to attract the nurses. Other employers will have to match the higher pay to keep their own employees. The higher salaries will encourage more nurses to train or work in Minneapolis-St. Paul-Bloomington. Again, price and quantity in the labor market will move toward equilibrium. Shifts in Labor Demand The demand curve for labor shows the quantity of labor employers wish to hire at any given salary or wage rate, under the ceteris paribus assumption. A change in the wage or salary will result in a change in the quantity demanded of labor. If the wage rate increases, employers will want to hire fewer employees. The quantity of labor demanded will decrease, and there will be a movement upward along the demand curve. If the wages and salaries decrease, employers are more likely to hire a greater number of workers. The quantity of labor demanded will increase, resulting in a downward movement along the demand curve. Shifts in the demand curve for labor occur for many reasons. One key reason is that the demand for labor is based on the demand for the good or service that is produced. For example, the more new automobiles consumers demand, the greater the number of workers automakers will need to hire. Therefore the demand for labor is called a “derived demand.” Here are some examples of derived demand for labor: • The demand for chefs is dependent on the demand for restaurant meals. • The demand for pharmacists is dependent on the demand for prescription drugs. • The demand for attorneys is dependent on the demand for legal services. As the demand for the goods and services increases, the demand for labor will increase, or shift to the right, to meet employers’ production requirements. As the demand for the goods and services decreases, the demand for labor will decrease, or shift to the left. Table 4.2 shows that in addition to the derived demand for labor, demand can also increase or decrease (shift) in response to several factors. Factors Demand for Output Results When the demand for the good produced (output) increases, both the output price and profitability increase. As a result, producers demand more labor to ramp up production. Table 4.2 Factors That Can Shift Demand This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 87 Factors Education and Training Technology Results A well-trained and educated workforce causes an increase in the demand for that labor by employers. Increased levels of productivity within the workforce will cause the demand for labor to shift to the right. If the workforce is not well-trained or educated, employers will not hire from within that labor pool, since they will need to spend a significant amount of time and money training that workforce. Demand for such will shift to the left. Technology changes can act as either substitutes for or complements to labor. When technology acts as a substitute, it replaces the need for the number of workers an employer needs to hire. For example, word processing decreased the number of typists needed in the workplace. This shifted the demand curve for typists left. An increase in the availability of certain technologies may increase the demand for labor. Technology that acts as a complement to labor will increase the demand for certain types of labor, resulting in a rightward shift of the demand curve. For example, the increased use of word processing and other software has increased the demand for information technology professionals who can resolve software and hardware issues related to a firm’s network. More and better technology will increase demand for skilled workers who know how to use technology to enhance workplace productivity. Those workers who do not adapt to changes in technology will experience a decrease in demand. Number of Companies Government Regulations An increase in the number of companies producing a given product will increase the demand for labor resulting in a shift to the right. A decrease in the number of companies producing a given product will decrease the demand for labor resulting in a shift to the left. Complying with government regulations can increase or decrease the demand for labor at any given wage. In the healthcare industry, government rules may require that nurses be hired to carry out certain medical procedures. This will increase the demand for nurses. Less-trained healthcare workers would be prohibited from carrying out these procedures, and the demand for these workers will shift to the left. Price and Availability of Other Inputs Labor is not the only input into the production process. For example, a salesperson at a call center needs a telephone and a computer terminal to enter data and record sales. If prices of other inputs fall, production will become more profitable and suppliers will demand more labor to increase production. This will cause a rightward shift in the demand curve for labor. The opposite is also true. Higher prices for other inputs lower demand for labor. Table 4.2 Factors That Can Shift Demand Click here (http://openstaxcollege.org/l/Futurework) to read more about “Trends and Challenges for Work in the 21st Century.” 88 Chapter 4 \| Labor and Financial Markets Shifts in Labor Supply The supply of labor is upward-sloping and adheres to the law of supply: The higher the price, the greater the quantity supplied and the lower the price, the less quantity supplied. The supply curve models the tradeoff between supplying labor into the market or using time in leisure activities at every given price level. The higher the wage, the more labor is willing to work and forego leisure activities. Table 4.3 lists some of the factors that will cause the supply to increase or decrease. Factors Number of Workers Required Education Government Policies Results An increased number of workers will cause the supply curve to shift to the right. An increased number of workers can be due to several factors, such as immigration, increasing population, an aging population, and changing demographics. Policies that encourage immigration will increase the supply of labor, and vice versa. Population grows when birth rates exceed death rates. This eventually increases supply of labor when the former reach working age. An aging and therefore retiring population will decrease the supply of labor. Another example of changing demographics is more women working outside of the home, which increases the supply of labor. The more required education, the lower the supply. There is a lower supply of PhD mathematicians than of high school mathematics teachers; there is a lower supply of cardiologists than of primary care physicians; and there is a lower supply of physicians than of nurses. Government policies can also affect the supply of labor for jobs. Alternatively, the government may support rules that set high qualifications for certain jobs: academic training, certificates or licenses, or experience. When these qualifications are made tougher, the number of qualified workers will decrease at any given wage. On the other hand, the government may also subsidize training or even reduce the required level of qualifications. For example, government might offer subsidies for nursing schools or nursing students. Such provisions would shift the supply curve of nurses to the right. In addition, government policies that change the relative desirability of working versus not working also affect the labor supply. These include unemployment benefits, maternity leave, child care benefits, and welfare policy. For example, child care benefits may increase the labor supply of working mothers. Long term unemployment benefits may discourage job searching for unemployed workers. All these policies must therefore be carefully designed to minimize any negative labor supply effects. Table 4.3 Factors that Can Shift Supply This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 89 A change in salary will lead to a movement along labor demand or labor supply curves, but it will not shift those curves. However, other events like those we have outlined here will cause either the demand or the supply of labor to shift, and thus will move th
e labor market to a new equilibrium salary and quantity. Technology and Wage Inequality: The Four-Step Process Economic events can change the equilibrium salary (or wage) and quantity of labor. Consider how the wave of new information technologies, like computer and telecommunications networks, has affected low-skill and high-skill workers in the U.S. economy. From the perspective of employers who demand labor, these new technologies are often a substitute for low-skill laborers like file clerks who used to keep file cabinets full of paper records of transactions. However, the same new technologies are a complement to high-skill workers like managers, who benefit from the technological advances by having the ability to monitor more information, communicate more easily, and juggle a wider array of responsibilities. How will the new technologies affect the wages of high-skill and low-skill workers? For this question, the four-step process of analyzing how shifts in supply or demand affect a market (introduced in Demand and Supply) works in this way: Step 1. What did the markets for low-skill labor and high-skill labor look like before the arrival of the new technologies? In Figure 4.3 (a) and Figure 4.3 (b), S0 is the original supply curve for labor and D0 is the original demand curve for labor in each market. In each graph, the original point of equilibrium, E0, occurs at the price W0 and the quantity Q0. Figure 4.3 Technology and Wages: Applying Demand and Supply (a) The demand for low-skill labor shifts to the left when technology can do the job previously done by these workers. (b) New technologies can also increase the demand for high-skill labor in fields such as information technology and network administration. Step 2. Does the new technology affect the supply of labor from households or the demand for labor from firms? The technology change described here affects demand for labor by firms that hire workers. Step 3. Will the new technology increase or decrease demand? Based on the description earlier, as the substitute for low-skill labor becomes available, demand for low-skill labor will shift to the left, from D0 to D1. As the technology complement for high-skill labor becomes cheaper, demand for high-skill labor will shift to the right, from D0 to D1. Step 4. The new equilibrium for low-skill labor, shown as point E1 with price W1 and quantity Q1, has a lower wage and quantity hired than the original equilibrium, E0. The new equilibrium for high-skill labor, shown as point E1 with price W1 and quantity Q1, has a higher wage and quantity hired than the original equilibrium (E0). Thus, the demand and supply model predicts that the new computer and communications technologies will raise the pay of high-skill workers but reduce the pay of low-skill workers. From the 1970s to the mid-2000s, the wage gap widened between high-skill and low-skill labor. According to the National Center for Education Statistics, in 1980, for example, a college graduate earned about 30% more than a high school graduate with comparable job experience, but by 2014, a college graduate earned about 66% more than an otherwise comparable high school graduate. Many economists believe that the trend toward greater wage inequality across the U.S. economy is due to improvements in 90 technology. Chapter 4 \| Labor and Financial Markets this website (http://openstaxcollege.org/l/oldtechjobs) to read about Visit relevance in today’s workforce. ten tech skills that have lost Price Floors in the Labor Market: Living Wages and Minimum Wages In contrast to goods and services markets, price ceilings are rare in labor markets, because rules that prevent people from earning income are not politically popular. There is one exception: boards of trustees or stockholders, as an example, propose limits on the high incomes of top business executives. The labor market, however, presents some prominent examples of price floors, which are an attempt to increase the wages of low-paid workers. The U.S. government sets a minimum wage, a price floor that makes it illegal for an employer to pay employees less than a certain hourly rate. In mid-2009, the U.S. minimum wage was raised to $7.25 per hour. Local political movements in a number of U.S. cities have pushed for a higher minimum wage, which they call a living wage. Promoters of living wage laws maintain that the minimum wage is too low to ensure a reasonable standard of living. They base this conclusion on the calculation that, if you work 40 hours a week at a minimum wage of $7.25 per hour for 50 weeks a year, your annual income is $14,500, which is less than the official U.S. government definition of what it means for a family to be in poverty. (A family with two adults earning minimum wage and two young children will find it more cost efficient for one parent to provide childcare while the other works for income. Thus the family income would be $14,500, which is significantly lower than the federal poverty line for a family of four, which was $24,250 in 2015.) Supporters of the living wage argue that full-time workers should be assured a high enough wage so that they can afford the essentials of life: food, clothing, shelter, and healthcare. Since Baltimore passed the first living wage law in 1994, several dozen cities enacted similar laws in the late 1990s and the 2000s. The living wage ordinances do not apply to all employers, but they have specified that all employees of the city or employees of firms that the city hires be paid at least a certain wage that is usually a few dollars per hour above the U.S. minimum wage. Figure 4.4 illustrates the situation of a city considering a living wage law. For simplicity, we assume that there is no federal minimum wage. The wage appears on the vertical axis, because the wage is the price in the labor market. Before the passage of the living wage law, the equilibrium wage is $10 per hour and the city hires 1,200 workers at this wage. However, a group of concerned citizens persuades the city council to enact a living wage law requiring employers to pay no less than $12 per hour. In response to the higher wage, 1,600 workers look for jobs with the city. At this higher wage, the city, as an employer, is willing to hire only 700 workers. At the price floor, the quantity supplied exceeds the quantity demanded, and a surplus of labor exists in this market. For workers who continue to have a job at a higher salary, life has improved. For those who were willing to work at the old wage rate but lost their jobs with the wage increase, life has not improved. Table 4.4 shows the differences in supply and demand at different wages. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 91 Figure 4.4 A Living Wage: Example of a Price Floor The original equilibrium in this labor market is a wage of $10/ hour and a quantity of 1,200 workers, shown at point E. Imposing a wage floor at $12/hour leads to an excess supply of labor. At that wage, the quantity of labor supplied is 1,600 and the quantity of labor demanded is only 700. Wage Quantity Labor Demanded Quantity Labor Supplied $8/hr $9/hr $10/hr $11/hr $12/hr $13/hr $14/hr 1,900 1,500 1,200 900 700 500 400 500 900 1,200 1,400 1,600 1,800 1,900 Table 4.4 Living Wage: Example of a Price Floor The Minimum Wage as an Example of a Price Floor The U.S. minimum wage is a price floor that is set either very close to the equilibrium wage or even slightly below it. About 1% of American workers are actually paid the minimum wage. In other words, the vast majority of the U.S. labor force has its wages determined in the labor market, not as a result of the government price floor. However, for workers with low skills and little experience, like those without a high school diploma or teenagers, the minimum wage is quite important. In many cities, the federal minimum wage is apparently below the market price for unskilled labor, because employers offer more than the minimum wage to checkout clerks and other low-skill workers without any government prodding. Economists have attempted to estimate how much the minimum wage reduces the quantity demanded of low-skill labor. A typical result of such studies is that a 10% increase in the minimum wage would decrease the hiring of unskilled workers by 1 to 2%, which seems a relatively small reduction. In fact, some studies have even found no effect of a higher minimum wage on employment at certain times and places—although these studies are controversial. Let’s suppose that the minimum wage lies just slightly below the equilibrium wage level. Wages could fluctuate according to market forces above this price floor, but they would not be allowed to move beneath the floor. In this situation, the price floor minimum wage is nonbinding —that is, the price floor is not determining the market outcome. Even if the minimum wage moves just a little higher, it will still have no effect on the quantity of 92 Chapter 4 \| Labor and Financial Markets employment in the economy, as long as it remains below the equilibrium wage. Even if the government increases minimum wage by enough so that it rises slightly above the equilibrium wage and becomes binding, there will be only a small excess supply gap between the quantity demanded and quantity supplied. These insights help to explain why U.S. minimum wage laws have historically had only a small impact on employment. Since the minimum wage has typically been set close to the equilibrium wage for low-skill labor and sometimes even below it, it has not had a large effect in creating an excess supply of labor. However, if the minimum wage increased dramatically—say, if it doubled to match the living wages that some U.S. cities have considered—then its impact on reducing the quantity demanded of employment would be far greater. As of 2017, many U.S. states are set to increase their minimum wage to $15 per h
our. We will see what happens. The following Clear It Up feature describes in greater detail some of the arguments for and against changes to minimum wage. What’s the harm in raising the minimum wage? Because of the law of demand, a higher required wage will reduce the amount of low-skill employment either in terms of employees or in terms of work hours. Although there is controversy over the numbers, let’s say for the sake of the argument that a 10% rise in the minimum wage will reduce the employment of low-skill workers by 2%. Does this outcome mean that raising the minimum wage by 10% is bad public policy? Not necessarily. If 98% of those receiving the minimum wage have a pay increase of 10%, but 2% of those receiving the minimum wage lose their jobs, are the gains for society as a whole greater than the losses? The answer is not clear, because job losses, even for a small group, may cause more pain than modest income gains for others. For one thing, we need to consider which minimum wage workers are losing their jobs. If the 2% of minimum wage workers who lose their jobs are struggling to support families, that is one thing. If those who lose their job are high school students picking up spending money over summer vacation, that is something else. Another complexity is that many minimum wage workers do not work full-time for an entire year. Imagine a minimum wage worker who holds different part-time jobs for a few months at a time, with bouts of unemployment in between. The worker in this situation receives the 10% raise in the minimum wage when working, but also ends up working 2% fewer hours during the year because the higher minimum wage reduces how much employers want people to work. Overall, this worker’s income would rise because the 10% pay raise would more than offset the 2% fewer hours worked. Of course, these arguments do not prove that raising the minimum wage is necessarily a good idea either. There may well be other, better public policy options for helping low-wage workers. (The Poverty and Economic Inequality chapter discusses some possibilities.) The lesson from this maze of minimum wage arguments is that complex social problems rarely have simple answers. Even those who agree on how a proposed economic policy affects quantity demanded and quantity supplied may still disagree on whether the policy is a good idea. 4.2 \| Demand and Supply in Financial Markets By the end of this section, you will be able to: Identify the demanders and suppliers in a financial market • • Explain how interest rates can affect supply and demand • Analyze the economic effects of U.S. debt in terms of domestic financial markets • Explain the role of price ceilings and usury laws in the U.S. United States' households, institutions, and domestic businesses saved almost $1.3 trillion in 2015. Where did that savings go and how was it used? Some of the savings ended up in banks, which in turn loaned the money to individuals or businesses that wanted to borrow money. Some was invested in private companies or loaned to This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 93 government agencies that wanted to borrow money to raise funds for purposes like building roads or mass transit. Some firms reinvested their savings in their own businesses. In this section, we will determine how the demand and supply model links those who wish to supply financial capital (i.e., savings) with those who demand financial capital (i.e., borrowing). Those who save money (or make financial investments, which is the same thing), whether individuals or businesses, are on the supply side of the financial market. Those who borrow money are on the demand side of the financial market. For a more detailed treatment of the different kinds of financial investments like bank accounts, stocks and bonds, see the Financial Markets chapter. Who Demands and Who Supplies in Financial Markets? In any market, the price is what suppliers receive and what demanders pay. In financial markets, those who supply financial capital through saving expect to receive a rate of return, while those who demand financial capital by receiving funds expect to pay a rate of return. This rate of return can come in a variety of forms, depending on the type of investment. The simplest example of a rate of return is the interest rate. For example, when you supply money into a savings account at a bank, you receive interest on your deposit. The interest the bank pays you as a percent of your deposits is the interest rate. Similarly, if you demand a loan to buy a car or a computer, you will need to pay interest on the money you borrow. Let’s consider the market for borrowing money with credit cards. In 2015, almost 200 million Americans were cardholders. Credit cards allow you to borrow money from the card's issuer, and pay back the borrowed amount plus interest, although most allow you a period of time in which you can repay the loan without paying interest. A typical credit card interest rate ranges from 12% to 18% per year. In May 2016, Americans had about $943 billion outstanding in credit card debts. About half of U.S. families with credit cards report that they almost always pay the full balance on time, but one-quarter of U.S. families with credit cards say that they “hardly ever” pay off the card in full. In fact, in 2014, 56% of consumers carried an unpaid balance in the last 12 months. Let’s say that, on average, the annual interest rate for credit card borrowing is 15% per year. Thus, Americans pay tens of billions of dollars every year in interest on their credit cards—plus basic fees for the credit card or fees for late payments. Figure 4.5 illustrates demand and supply in the financial market for credit cards. The horizontal axis of the financial market shows the quantity of money loaned or borrowed in this market. The vertical or price axis shows the rate of return, which in the case of credit card borrowing we can measure with an interest rate. Table 4.5 shows the quantity of financial capital that consumers demand at various interest rates and the quantity that credit card firms (often banks) are willing to supply. 94 Chapter 4 \| Labor and Financial Markets Figure 4.5 Demand and Supply for Borrowing Money with Credit Cards In this market for credit card borrowing, the demand curve (D) for borrowing financial capital intersects the supply curve (S) for lending financial capital at equilibrium E. At the equilibrium, the interest rate (the “price” in this market) is 15% and the quantity of financial capital loaned and borrowed is $600 billion. The equilibrium price is where the quantity demanded and the quantity supplied are equal. At an above-equilibrium interest rate like 21%, the quantity of financial capital supplied would increase to $750 billion, but the quantity demanded would decrease to $480 billion. At a below-equilibrium interest rate like 13%, the quantity of financial capital demanded would increase to $700 billion, but the quantity of financial capital supplied would decrease to $510 billion. Interest Rate (%) Quantity of Financial Capital Demanded (Borrowing) ($ billions) Quantity of Financial Capital Supplied (Lending) ($ billions) 11 13 15 17 19 21 $800 $700 $600 $550 $500 $480 $420 $510 $600 $660 $720 $750 Table 4.5 Demand and Supply for Borrowing Money with Credit Cards The laws of demand and supply continue to apply in the financial markets. According to the law of demand, a higher rate of return (that is, a higher price) will decrease the quantity demanded. As the interest rate rises, consumers will reduce the quantity that they borrow. According to the law of supply, a higher price increases the quantity supplied. Consequently, as the interest rate paid on credit card borrowing rises, more firms will be eager to issue credit cards and to encourage customers to use them. Conversely, if the interest rate on credit cards falls, the quantity of financial capital supplied in the credit card market will decrease and the quantity demanded will fall. Equilibrium in Financial Markets In the financial market for credit cards in Figure 4.5, the supply curve (S) and the demand curve (D) cross at the equilibrium point (E). The equilibrium occurs at an interest rate of 15%, where the quantity of funds demanded and the quantity supplied are equal at an equilibrium quantity of $600 billion. If the interest rate (remember, this measures the “price” in the financial market) is above the equilibrium level, then an excess supply, or a surplus, of financial capital will arise in this market. For example, at an interest rate of 21%, the quantity of funds supplied increases to $750 billion, while the quantity demanded decreases to $480 billion. At this This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 95 above-equilibrium interest rate, firms are eager to supply loans to credit card borrowers, but relatively few people or businesses wish to borrow. As a result, some credit card firms will lower the interest rates (or other fees) they charge to attract more business. This strategy will push the interest rate down toward the equilibrium level. If the interest rate is below the equilibrium, then excess demand or a shortage of funds occurs in this market. At an interest rate of 13%, the quantity of funds credit card borrowers demand increases to $700 billion, but the quantity credit card firms are willing to supply is only $510 billion. In this situation, credit card firms will perceive that they are overloaded with eager borrowers and conclude that they have an opportunity to raise interest rates or fees. The interest rate will face economic pressures to creep up toward the equilibrium level. The FRED database publishes some two dozen measures of interest rates, including interest rates on credit cards, the FRED website automobile loans
, personal (https://openstax.org/l/FRED_stlouis) . loans, mortgage loans, and more. You can find these at Shifts in Demand and Supply in Financial Markets Those who supply financial capital face two broad decisions: how much to save, and how to divide up their savings among different forms of financial investments. We will discuss each of these in turn. Participants in financial markets must decide when they prefer to consume goods: now or in the future. Economists call this intertemporal decision making because it involves decisions across time. Unlike a decision about what to buy from the grocery store, people make investment or savings decisions across a period of time, sometimes a long period. Most workers save for retirement because their income in the present is greater than their needs, while the opposite will be true once they retire. Thus, they save today and supply financial markets. If their income increases, they save more. If their perceived situation in the future changes, they change the amount of their saving. For example, there is some evidence that Social Security, the program that workers pay into in order to qualify for government checks after retirement, has tended to reduce the quantity of financial capital that workers save. If this is true, Social Security has shifted the supply of financial capital at any interest rate to the left. By contrast, many college students need money today when their income is low (or nonexistent) to pay their college expenses. As a result, they borrow today and demand from financial markets. Once they graduate and become employed, they will pay back the loans. Individuals borrow money to purchase homes or cars. A business seeks financial investment so that it has the funds to build a factory or invest in a research and development project that will not pay off for five years, ten years, or even more. Thus, when consumers and businesses have greater confidence that they will be able to repay in the future, the quantity demanded of financial capital at any given interest rate will shift to the right. For example, in the technology boom of the late 1990s, many businesses became extremely confident that investments in new technology would have a high rate of return, and their demand for financial capital shifted to the right. Conversely, during the 2008 and 2009 Great Recession, their demand for financial capital at any given interest rate shifted to the left. To this point, we have been looking at saving in total. Now let us consider what affects saving in different types of financial investments. In deciding between different forms of financial investments, suppliers of financial capital will have to consider the rates of return and the risks involved. Rate of return is a positive attribute of investments, but risk is a negative. If Investment A becomes more risky, or the return diminishes, then savers will shift their funds to Investment B—and the supply curve of financial capital for Investment A will shift back to the left while the supply curve of capital for Investment B shifts to the right. The United States as a Global Borrower In the global economy, trillions of dollars of financial investment cross national borders every year. In the early 2000s, financial investors from foreign countries were investing several hundred billion dollars per year more in the U.S. economy than U.S. financial investors were investing abroad. The following Work It Out deals with one of the macroeconomic concerns for the U.S. economy in recent years. 96 Chapter 4 \| Labor and Financial Markets The Effect of Growing U.S. Debt Imagine that foreign investors viewed the U.S. economy as a less desirable place to put their money because of fears about the growth of the U.S. public debt. Using the four-step process for analyzing how changes in supply and demand affect equilibrium outcomes, how would increased U.S. public debt affect the equilibrium price and quantity for capital in U.S. financial markets? Step 1. Draw a diagram showing demand and supply for financial capital that represents the original scenario in which foreign investors are pouring money into the U.S. economy. Figure 4.6 shows a demand curve, D, and a supply curve, S, where the supply of capital includes the funds arriving from foreign investors. The original equilibrium E0 occurs at interest rate R0 and quantity of financial investment Q0. Figure 4.6 The United States as a Global Borrower Before U.S. Debt Uncertainty The graph shows the demand for financial capital from and supply of financial capital into the U.S. financial markets by the foreign sector before the increase in uncertainty regarding U.S. public debt. The original equilibrium (E0) occurs at an equilibrium rate of return (R0) and the equilibrium quantity is at Q0. Step 2. Will the diminished confidence in the U.S. economy as a place to invest affect demand or supply of financial capital? Yes, it will affect supply. Many foreign investors look to the U.S. financial markets to store their money in safe financial vehicles with low risk and stable returns. Diminished confidence means U.S. financial assets will be seen as more risky. Step 3. Will supply increase or decrease? When the enthusiasm of foreign investors’ for investing their money in the U.S. economy diminishes, the supply of financial capital shifts to the left. Figure 4.7 shows the supply curve shift from S0 to S1. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 97 Figure 4.7 The United States as a Global Borrower Before and After U.S. Debt Uncertainty The graph shows the demand for financial capital and supply of financial capital into the U.S. financial markets by the foreign sector before and after the increase in uncertainty regarding U.S. public debt. The original equilibrium (E0) occurs at an equilibrium rate of return (R0) and the equilibrium quantity is at Q0. Step 4. Thus, foreign investors’ diminished enthusiasm leads to a new equilibrium, E1, which occurs at the higher interest rate, R1, and the lower quantity of financial investment, Q1. In short, U.S. borrowers will have to pay more interest on their borrowing. The economy has experienced an enormous inflow of foreign capital. According to the U.S. Bureau of Economic Analysis, by the third quarter of 2015, U.S. investors had accumulated $23.3 trillion of foreign assets, but foreign investors owned a total of $30.6 trillion of U.S. assets. If foreign investors were to pull their money out of the U.S. economy and invest elsewhere in the world, the result could be a significantly lower quantity of financial investment in the United States, available only at a higher interest rate. This reduced inflow of foreign financial investment could impose hardship on U.S. consumers and firms interested in borrowing. In a modern, developed economy, financial capital often moves invisibly through electronic transfers between one bank account and another. Yet we can analyze these flows of funds with the same tools of demand and supply as markets for goods or labor. Price Ceilings in Financial Markets: Usury Laws As we noted earlier, about 200 million Americans own credit cards, and their interest payments and fees total tens of billions of dollars each year. It is little wonder that political pressures sometimes arise for setting limits on the interest rates or fees that credit card companies charge. The firms that issue credit cards, including banks, oil companies, phone companies, and retail stores, respond that the higher interest rates are necessary to cover the losses created by those who borrow on their credit cards and who do not repay on time or at all. These companies also point out that cardholders can avoid paying interest if they pay their bills on time. Consider the credit card market as Figure 4.8 illustrators. In this financial market, the vertical axis shows the interest rate (which is the price in the financial market). Demanders in the credit card market are households and businesses. Suppliers are the companies that issue credit cards. This figure does not use specific numbers, which would be hypothetical in any case, but instead focuses on the underlying economic relationships. Imagine a law imposes a price ceiling that holds the interest rate charged on credit cards at the rate Rc, which lies below the interest rate R0 that would otherwise have prevailed in the market. The horizontal dashed line at interest rate Rc in Figure 4.8 shows the price ceiling. The demand and supply model predicts that at the lower price ceiling interest rate, the quantity demanded of credit card debt will increase from its original level of Q0 to Qd; however, the quantity supplied of credit card debt will decrease from the original Q0 to Qs. At the price ceiling (Rc), quantity demanded will exceed quantity supplied. Consequently, a number of people who want to have credit cards and are willing to pay the prevailing interest rate will find that companies are unwilling to issue cards to them. The result will be a credit shortage. 98 Chapter 4 \| Labor and Financial Markets Figure 4.8 Credit Card Interest Rates: Another Price Ceiling Example The original intersection of demand D and supply S occurs at equilibrium E0. However, a price ceiling is set at the interest rate Rc, below the equilibrium interest rate R0, and so the interest rate cannot adjust upward to the equilibrium. At the price ceiling, the quantity demanded, Qd, exceeds the quantity supplied, Qs. There is excess demand, also called a shortage. Many states do have usury laws, which impose an upper limit on the interest rate that lenders can charge. However, in many cases these upper limits are well above the market interest rate. For example, if the interest rate is not allowed to rise above 30% per year, it can still fluctuate below that level according to market forces. A price ceiling that is set at a relatively hig
h level is nonbinding, and it will have no practical effect unless the equilibrium price soars high enough to exceed the price ceiling. 4.3 \| The Market System as an Efficient Mechanism for Information By the end of this section, you will be able to: • Apply demand and supply models to analyze prices and quantities • Explain the effects of price controls on the equilibrium of prices and quantities Prices exist in markets for goods and services, for labor, and for financial capital. In all of these markets, prices serve as a remarkable social mechanism for collecting, combining, and transmitting information that is relevant to the market—namely, the relationship between demand and supply—and then serving as messengers to convey that information to buyers and sellers. In a market-oriented economy, no government agency or guiding intelligence oversees the set of responses and interconnections that result from a change in price. Instead, each consumer reacts according to that person’s preferences and budget set, and each profit-seeking producer reacts to the impact on its expected profits. The following Clear It Up feature examines the demand and supply models. Why are demand and supply curves important? The demand and supply model is the second fundamental diagram for this course. (The opportunity set model that we introduced in the Choice in a World of Scarcity chapter was the first.) Just as it would be foolish to try to learn the arithmetic of long division by memorizing every possible combination of numbers that can be divided by each other, it would be foolish to try to memorize every specific example of demand and supply in this chapter, this textbook, or this course. Demand and supply is not primarily a list of examples. It is a This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 99 model to analyze prices and quantities. Even though demand and supply diagrams have many labels, they are fundamentally the same in their logic. Your goal should be to understand the underlying model so you can use it to analyze any market. Figure 4.9 displays a generic demand and supply curve. The horizontal axis shows the different measures of quantity: a quantity of a good or service, or a quantity of labor for a given job, or a quantity of financial capital. The vertical axis shows a measure of price: the price of a good or service, the wage in the labor market, or the rate of return (like the interest rate) in the financial market. The demand and supply model can explain the existing levels of prices, wages, and rates of return. To carry out such an analysis, think about the quantity that will be demanded at each price and the quantity that will be supplied at each price—that is, think about the shape of the demand and supply curves—and how these forces will combine to produce equilibrium. We can also use demand and supply to explain how economic events will cause changes in prices, wages, and rates of return. There are only four possibilities: the change in any single event may cause the demand curve to shift right or to shift left, or it may cause the supply curve to shift right or to shift left. The key to analyzing the effect of an economic event on equilibrium prices and quantities is to determine which of these four possibilities occurred. The way to do this correctly is to think back to the list of factors that shift the demand and supply curves. Note that if more than one variable is changing at the same time, the overall impact will depend on the degree of the shifts. When there are multiple variables, economists isolate each change and analyze it independently. Figure 4.9 Demand and Supply Curves The figure displays a generic demand and supply curve. The horizontal axis shows the different measures of quantity: a quantity of a good or service, a quantity of labor for a given job, or a quantity of financial capital. The vertical axis shows a measure of price: the price of a good or service, the wage in the labor market, or the rate of return (like the interest rate) in the financial market. We can use the demand and supply curves explain how economic events will cause changes in prices, wages, and rates of return. An increase in the price of some product signals consumers that there is a shortage; therefore, they may want to economize on buying this product. For example, if you are thinking about taking a plane trip to Hawaii, but the ticket turns out to be expensive during the week you intend to go, you might consider other weeks when the ticket might be cheaper. The price could be high because you were planning to travel during a holiday when demand for traveling is high. Maybe the cost of an input like jet fuel increased or the airline has raised the price temporarily to see how many people are willing to pay it. Perhaps all of these factors are present at the same time. You do not need to analyze the market and break down the price change into its underlying factors. You just have to look at the ticket price and decide whether and when to fly. In the same way, price changes provide useful information to producers. Imagine the situation of a farmer who grows 100 Chapter 4 \| Labor and Financial Markets oats and learns that the price of oats has risen. The higher price could be due to an increase in demand caused by a new scientific study proclaiming that eating oats is especially healthful. Perhaps the price of a substitute grain, like corn, has risen, and people have responded by buying more oats. The oat farmer does not need to know the details. The farmer only needs to know that the price of oats has risen and that it will be profitable to expand production as a result. The actions of individual consumers and producers as they react to prices overlap and interlock in markets for goods, labor, and financial capital. A change in any single market is transmitted through these multiple interconnections to other markets. The vision of the role of flexible prices helping markets to reach equilibrium and linking different markets together helps to explain why price controls can be so counterproductive. Price controls are government laws that serve to regulate prices rather than allow the various markets to determine prices. There is an old proverb: “Don’t kill the messenger.” In ancient times, messengers carried information between distant cities and kingdoms. When they brought bad news, there was an emotional impulse to kill the messenger. However, killing the messenger did not kill the bad news. Moreover, killing the messenger had an undesirable side effect: Other messengers would refuse to bring news to that city or kingdom, depriving its citizens of vital information. Those who seek price controls are trying to kill the messenger—or at least to stifle an unwelcome message that prices are bringing about the equilibrium level of price and quantity. However, price controls do nothing to affect the underlying forces of demand and supply, and this can have serious repercussions. During China’s “Great Leap Forward” in the late 1950s, the government kept food prices artificially low, with the result that 30 to 40 million people died of starvation because the low prices depressed farm production. This was communist party leader Mao Zedong's social and economic campaign to rapidly transform the country from an agrarian economy to a socialist society through rapid industrialization and collectivization. Changes in demand and supply will continue to reveal themselves through consumers’ and producers’ behavior. Immobilizing the price messenger through price controls will deprive everyone in the economy of critical information. Without this information, it becomes difficult for everyone—buyers and sellers alike—to react in a flexible and appropriate manner as changes occur throughout the economy. Baby Boomers Come of Age The theory of supply and demand can explain what happens in the labor markets and suggests that the demand for nurses will increase as healthcare needs of baby boomers increase, as Figure 4.10 shows. The impact of that increase will result in an average salary higher than the $67,490 earned in 2015 referenced in the first part of this case. The new equilibrium (E1) will be at the new equilibrium price (Pe1).Equilibrium quantity will also increase from Qe0 to Qe1. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 101 Figure 4.10 Impact of Increasing Demand for Nurses 2014-2024 In 2014, the median salary for nurses was $67,490. As demand for services increases, the demand curve shifts to the right (from D0 to D1) and the equilibrium quantity of nurses increases from Qe0 to Qe1. The equilibrium salary increases from Pe0 to Pe1. Suppose that as the demand for nurses increases, the supply shrinks due to an increasing number of nurses entering retirement and increases in the tuition of nursing degrees. The leftward shift of the supply curve in Figure 4.11 captures the impact of a decreasing supply of nurses. The shifts in the two curves result in higher salaries for nurses, but the overall impact in the quantity of nurses is uncertain, as it depends on the relative shifts of supply and demand. Figure 4.11 Impact of Decreasing Supply of Nurses between 2014 and 2024 The increase in demand for nurses shown in Figure 4.10 leads to both higher prices and higher quantities demanded. As nurses retire from the work force, the supply of nurses decreases, causing a leftward shift in the supply curve and higher salaries for nurses at Pe2. The net effect on the equilibrium quantity of nurses is uncertain, which in this representation is less than Qe1, but more than the initial Qe0. 102 Chapter 4 \| Labor and Financial Markets While we do not know if the number of nurses will increase or decrease relative to their initial employment, we know they will have higher salaries. This OpenStax book
is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 103 KEY TERMS interest rate the “price” of borrowing in the financial market; a rate of return on an investment minimum wage a price floor that makes it illegal for an employer to pay employees less than a certain hourly rate usury laws laws that impose an upper limit on the interest rate that lenders can charge KEY CONCEPTS AND SUMMARY 4.1 Demand and Supply at Work in Labor Markets In the labor market, households are on the supply side of the market and firms are on the demand side. In the market for financial capital, households and firms can be on either side of the market: they are suppliers of financial capital when they save or make financial investments, and demanders of financial capital when they borrow or receive financial investments. In the demand and supply analysis of labor markets, we can measure the price by the annual salary or hourly wage received. We can measure the quantity of labor various ways, like number of workers or the number of hours worked. Factors that can shift the demand curve for labor include: a change in the quantity demanded of the product that the labor produces; a change in the production process that uses more or less labor; and a change in government policy that affects the quantity of labor that firms wish to hire at a given wage. Demand can also increase or decrease (shift) in response to: workers’ level of education and training, technology, the number of companies, and availability and price of other inputs. The main factors that can shift the supply curve for labor are: how desirable a job appears to workers relative to the alternatives, government policy that either restricts or encourages the quantity of workers trained for the job, the number of workers in the economy, and required education. 4.2 Demand and Supply in Financial Markets In the demand and supply analysis of financial markets, the “price” is the rate of return or the interest rate received. We measure the quantity by the money that flows from those who supply financial capital to those who demand it. Two factors can shift the supply of financial capital to a certain investment: if people want to alter their existing levels of consumption, and if the riskiness or return on one investment changes relative to other investments. Factors that can shift demand for capital include business confidence and consumer confidence in the future—since financial investments received in the present are typically repaid in the future. 4.3 The Market System as an Efficient Mechanism for Information The market price system provides a highly efficient mechanism for disseminating information about relative scarcities of goods, services, labor, and financial capital. Market participants do not need to know why prices have changed, only that the changes require them to revisit previous decisions they made about supply and demand. Price controls hide information about the true scarcity of products and thereby cause misallocation of resources. SELF-CHECK QUESTIONS 1. 2. In the labor market, what causes a movement along the demand curve? What causes a shift in the demand curve? In the labor market, what causes a movement along the supply curve? What causes a shift in the supply curve? 3. Why is a living wage considered a price floor? Does imposing a living wage have the same outcome as a minimum wage? In the financial market, what causes a movement along the demand curve? What causes a shift in the demand 4. curve? 5. In the financial market, what causes a movement along the supply curve? What causes a shift in the supply curve? 104 Chapter 4 \| Labor and Financial Markets If a usury law limits interest rates to no more than 35%, what would the likely impact be on the amount of loans 6. made and interest rates paid? 7. Which of the following changes in the financial market will lead to a decline in interest rates: a. a rise in demand b. a fall in demand c. a rise in supply d. a fall in supply 8. Which of the following changes in the financial market will lead to an increase in the quantity of loans made and received: a. a rise in demand b. a fall in demand c. a rise in supply d. a fall in supply 9. Identify the most accurate statement. A price floor will have the largest effect if it is set: a. b. c. d. substantially above the equilibrium price slightly above the equilibrium price slightly below the equilibrium price substantially below the equilibrium price Sketch all four of these possibilities on a demand and supply diagram to illustrate your answer. 10. A price ceiling will have the largest effect: substantially below the equilibrium price slightly below the equilibrium price substantially above the equilibrium price slightly above the equilibrium price a. b. c. d. Sketch all four of these possibilities on a demand and supply diagram to illustrate your answer. 11. Select the correct answer. A price floor will usually shift: a. demand b. supply c. both d. neither Illustrate your answer with a diagram. 12. Select the correct answer. A price ceiling will usually shift: a. demand b. supply c. both d. neither REVIEW QUESTIONS 13. What is the “price” commonly called in the labor market? 15. Name some factors that can cause a shift in the demand curve in labor markets. 14. Are households demanders or suppliers in the goods market? Are firms demanders or suppliers in the goods market? What about the labor market and the financial market? 16. Name some factors that can cause a shift in the supply curve in labor markets. 17. How do economists define equilibrium in financial markets? This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 4 \| Labor and Financial Markets 105 18. What would be a sign of a shortage in financial markets? 19. Would usury laws help or hinder resolution of a shortage in financial markets? 20. Whether the product market or the labor market, what happens to the equilibrium price and quantity for each of the four possibilities: increase in demand, decrease in demand, increase in supply, and decrease in supply. CRITICAL THINKING QUESTIONS 21. Other than the demand for labor, what would be another example of a “derived demand?” 22. Suppose that a 5% increase in the minimum wage causes a 5% reduction in employment. How would this affect employers and how would it affect workers? In your opinion, would this be a good policy? 23. Under what circumstances would a minimum wage be a nonbinding price floor? Under what circumstances would a living wage be a binding price floor? 24. Suppose the U.S. economy began to grow more rapidly than other countries in the world. What would be the likely impact on U.S. financial markets as part of the global economy? 25. If the government imposed a federal interest rate ceiling of 20% on all loans, who would gain and who would lose? 26. Why are the factors that shift the demand for a product different from the factors that shift the demand for labor? Why are the factors that shift the supply of a product different from those that shift the supply of labor? PROBLEMS 27. During a discussion several years ago on building a pipeline to Alaska to carry natural gas, the U.S. Senate passed a bill stipulating that there should be a guaranteed minimum price for the natural gas that would flow through the pipeline. The thinking behind the bill was that if private firms had a guaranteed price for their natural gas, they would be more willing to drill for gas and to pay to build the pipeline. a. Using the demand and supply framework, predict the effects of this price floor on the price, quantity demanded, and quantity supplied. b. With the enactment of this price floor for natural gas, what are some of the likely unintended consequences in the market? c. Suggest some policies other than the price floor that the government can pursue if it wishes to encourage drilling for natural gas and for a new pipeline in Alaska. 28. Identify each of the following as involving either demand or supply. Draw a circular flow diagram and label the flows A through F. (Some choices can be on both sides of the goods market.) 29. Predict how each of the following events will raise or lower the equilibrium wage and quantity of oil workers in Texas. In each case, sketch a demand and supply diagram to illustrate your answer. a. Households in the labor market b. Firms in the goods market c. Firms in the financial market d. Households in the goods market e. Firms in the labor market f. Households in the financial market a. The price of oil rises. b. New oil-drilling equipment is invented that is cheap and requires few workers to run. c. Several major companies that do not drill oil open factories in Texas, offering many well-paid jobs outside the oil industry. d. Government imposes costly new regulations to make oil-drilling a safer job. 106 Chapter 4 \| Labor and Financial Markets 30. Predict how each of the following economic changes will affect the equilibrium price and quantity in the financial market for home loans. Sketch a demand and supply diagram to support your answers. a. The number of people at the most common ages for home-buying increases. b. People gain confidence that the economy is 32. Imagine that to preserve the traditional way of life in small fishing villages, a government decides to impose a price floor that will guarantee all fishermen a certain price for their catch. a. Using the demand and supply framework, the effects on the price, quantity predict demanded, and quantity supplied. b. With the enactment of this price floor for fish, likely unintended are what the consequences in the market? some of c. Suggest some policies other than the price floor to make it possible for small fishing villages to continue. 33. What happens to the price and the quantity bought and sold in the cocoa market if countries producing cocoa experience a drought and a new study is released
demonstrating the health benefits of cocoa? Illustrate your answer with a demand and supply graph. growing and that their jobs are secure. c. Banks that have made home loans find that a larger number of people than they expected are not repaying those loans. d. Because of a threat of a war, people become uncertain about their economic future. e. The overall diminishes. level of saving in the economy f. The federal government changes its bank regulations in a way that makes it cheaper and easier for banks to make home loans. 31. Table 4.6 shows the amount of savings and borrowing in a market for loans to purchase homes, measured in millions of dollars, at various interest rates. What is the equilibrium interest rate and quantity in the capital financial market? How can you tell? Now, imagine that because of a shift in the perceptions of foreign investors, the supply curve shifts so that there will be $10 million less supplied at every interest rate. Calculate the new equilibrium interest rate and quantity, and explain why the direction of the interest rate shift makes intuitive sense. Interest Rate Qs Qd 5% 6% 7% 8% 9% 10% Table 4.6 130 135 140 145 150 155 170 150 140 135 125 110 This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 107 5 \| Elasticity Figure 5.1 Netflix On-Demand Media Netflix, Inc. is an American provider of on-demand Internet streaming media to many countries around the world, including the United States, and of flat rate DVD-by-mail in the United States. (Credit: modification of work by Traci Lawson/Flickr Creative Commons) That Will Be How Much? Imagine going to your favorite coffee shop and having the waiter inform you the pricing has changed. Instead of $3 for a cup of coffee, you will now be charged $2 for coffee, $1 for creamer, and $1 for your choice of sweetener. If you pay your usual $3 for a cup of coffee, you must choose between creamer and sweetener. If you want both, you now face an extra charge of $1. Sound absurd? Well, that is similar to the situation Netflix customers found themselves in—they faced a 60% price hike to retain the same service in 2011. In early 2011, Netflix consumers paid about $10 a month for a package consisting of streaming video and DVD rentals. In July 2011, the company announced a packaging change. Customers wishing to retain both streaming video and DVD rental would be charged $15.98 per month, a price increase of about 60%. In 2014, Netflix also raised its streaming video subscription price from $7.99 to $8.99 per month for new U.S. customers. The company also changed its policy of 4K streaming content from $9.00 to $12.00 per month that year. 108 Chapter 5 \| Elasticity How would customers of the 18-year-old firm react? Would they abandon Netflix? Would the ease of access to other venues make a difference in how consumers responded to the Netflix price change? We will explore the answers to those questions in this chapter, which focuses on the change in quantity with respect to a change in price, a concept economists call elasticity. Introduction to Elasticity In this chapter, you will learn about: • Price Elasticity of Demand and Price Elasticity of Supply • Polar Cases of Elasticity and Constant Elasticity • Elasticity and Pricing • Elasticity in Areas Other Than Price Anyone who has studied economics knows the law of demand: a higher price will lead to a lower quantity demanded. What you may not know is how much lower the quantity demanded will be. Similarly, the law of supply states that a higher price will lead to a higher quantity supplied. The question is: How much higher? This chapter will explain how to answer these questions and why they are critically important in the real world. To find answers to these questions, we need to understand the concept of elasticity. Elasticity is an economics concept that measures responsiveness of one variable to changes in another variable. Suppose you drop two items from a second-floor balcony. The first item is a tennis ball. The second item is a brick. Which will bounce higher? Obviously, the tennis ball. We would say that the tennis ball has greater elasticity. Consider an economic example. Cigarette taxes are an example of a “sin tax,” a tax on something that is bad for you, like alcohol. Governments tax cigarettes at the state and national levels. State taxes range from a low of 17 cents per pack in Missouri to $4.35 per pack in New York. The average state cigarette tax is $1.69 per pack. The 2014 federal tax rate on cigarettes was $1.01 per pack, but in 2015 the Obama Administration proposed raising the federal tax nearly a dollar to $1.95 per pack. The key question is: How much would cigarette purchases decline? Taxes on cigarettes serve two purposes: to raise tax revenue for government and to discourage cigarette consumption. However, if a higher cigarette tax discourages consumption considerably, meaning a greatly reduced quantity of cigarette sales, then the cigarette tax on each pack will not raise much revenue for the government. Alternatively, a higher cigarette tax that does not discourage consumption by much will actually raise more tax revenue for the government. Thus, when a government agency tries to calculate the effects of altering its cigarette tax, it must analyze how much the tax affects the quantity of cigarettes consumed. This issue reaches beyond governments and taxes. Every firm faces a similar issue. When a firm considers raising the sales price, it must consider how much a price increase will reduce the quantity demanded of what it sells. Conversely, when a firm puts its products on sale, it must expect (or hope) that the lower price will lead to a significantly higher quantity demanded. 5.1 \| Price Elasticity of Demand and Price Elasticity of Supply By the end of this section, you will be able to: • Calculate the price elasticity of demand • Calculate the price elasticity of supply Both the demand and supply curve show the relationship between price and the number of units demanded or supplied. Price elasticity is the ratio between the percentage change in the quantity demanded (Qd) or supplied (Qs) and the corresponding percent change in price. The price elasticity of demand is the percentage change in the quantity demanded of a good or service divided by the percentage change in the price. The price elasticity of supply is the percentage change in quantity supplied divided by the percentage change in price. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 109 We can usefully divide elasticities into three broad categories: elastic, inelastic, and unitary. An elastic demand or elastic supply is one in which the elasticity is greater than one, indicating a high responsiveness to changes in price. Elasticities that are less than one indicate low responsiveness to price changes and correspond to inelastic demand or inelastic supply. Unitary elasticities indicate proportional responsiveness of either demand or supply, as Table 5.1 summarizes. If . . . Then . . . And It Is Called . . . % change in quantity > % change in price % change in quantity = % change in price % change in quantity < % change in price % change in quantity % change in price > 1 % change in quantity % change in price = 1 % change in quantity % change in price < 1 Elastic Unitary Inelastic Table 5.1 Elastic, Inelastic, and Unitary: Three Cases of Elasticity Before we delve into the details of elasticity, enjoy this article (http://openstaxcollege.org/l/Super_Bowl) on elasticity and ticket prices at the Super Bowl. To calculate elasticity along a demand or supply curve economists use the average percent change in both quantity and price. This is called the Midpoint Method for Elasticity, and is represented in the following equations: % change in quantity = % change in price = ⎞ ⎠/2 ⎛ Q2 – Q1 ⎝Q2 + Q1 P2 – P1 ⎝P2 + P1 ⎛ ⎞ ⎠/2 × 100 × 100 The advantage of the Midpoint Method is that one obtains the same elasticity between two price points whether there is a price increase or decrease. This is because the formula uses the same base (average quantity and average price) for both cases. Calculating Price Elasticity of Demand Let’s calculate the elasticity between points A and B and between points G and H as Figure 5.2 shows. 110 Chapter 5 \| Elasticity Figure 5.2 Calculating the Price Elasticity of Demand We calculate the price elasticity of demand as the percentage change in quantity divided by the percentage change in price. First, apply the formula to calculate the elasticity as price decreases from $70 at point B to $60 at point A: % change in quantity = 3,000 – 2,800 (3,000 + 2,800)/2 × 100 × 100 = 200 2,900 = 6.9 % change in price = 60 – 70 (60 + 70)/2 × 100 × 100 = –10 65 = –15.4 Price Elasticity of Demand = 6.9% –15.4% = 0.45 Therefore, the elasticity of demand between these two points is 6.9% –15.4% which is 0.45, an amount smaller than one, showing that the demand is inelastic in this interval. Price elasticities of demand are always negative since price and quantity demanded always move in opposite directions (on the demand curve). By convention, we always talk about elasticities as positive numbers. Mathematically, we take the absolute value of the result. We will ignore this detail from now on, while remembering to interpret elasticities as positive numbers. This means that, along the demand curve between point B and A, if the price changes by 1%, the quantity demanded will change by 0.45%. A change in the price will result in a smaller percentage change in the quantity demanded. For example, a 10% increase in the price will result in only a 4.5% decrease in quantity demanded. A 10% decrease in the price will result in only a 4.5% increase in the quantity demanded. Price elasticities of demand are negative numbers indicating that the demand curve is downward sloping, but we read them as absolute values. Th
e following Work It Out feature will walk you through calculating the price elasticity of demand. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 111 Finding the Price Elasticity of Demand Calculate the price elasticity of demand using the data in Figure 5.2 for an increase in price from G to H. Has the elasticity increased or decreased? Step 1. We know that: Price Elasticity of Demand = % change in quantity % change in price Step 2. From the Midpoint Formula we know that: % change in quantity = % change in price = ⎛ Q2 – Q1 ⎝Q2 + Q1)/2 P2 – P1 ⎝P2 + P1)/2 ⎛ × 100 × 100 Step 3. So we can use the values provided in the figure in each equation: % change in quantity = 1,600 – 1,800 ⎝1,600 + 1,800)/2 ⎛ × 100 × 100 = –200 1,700 = –11.76 % change in price = 130 – 120 (130 + 120)/2 × 100 × 100 = 10 125 = 8.0 Step 4. Then, we can use those values to determine the price elasticity of demand: Price Elasticity of Demand = % change in quantity % change in price = –11.76 8 = 1.47 Therefore, the elasticity of demand from G to is H 1.47. The magnitude of the elasticity has increased (in absolute value) as we moved up along the demand curve from points A to B. Recall that the elasticity between these two points was 0.45. Demand was inelastic between points A and B and elastic between points G and H. This shows us that price elasticity of demand changes at different points along a straight-line demand curve. Calculating the Price Elasticity of Supply Assume that an apartment rents for $650 per month and at that price the landlord rents 10,000 units are rented as Figure 5.3 shows. When the price increases to $700 per month, the landlord supplies 13,000 units into the market. By what percentage does apartment supply increase? What is the price sensitivity? 112 Chapter 5 \| Elasticity Figure 5.3 Price Elasticity of Supply We calculate the price elasticity of supply as the percentage change in quantity divided by the percentage change in price. Using the Midpoint Method, % change in quantity = 13,000 – 10,000 (13,000 + 10,000)/2 × 100 = 3,000 11,500 = 26.1 × 100 % change in price = $700 – $650 ⎝$700 + $650)/2 ⎛ × 100 × 100 = 50 675 = 7.4 Price Elasticity of Supply = 26.1% 7.4% = 3.53 Again, as with the elasticity of demand, the elasticity of supply is not followed by any units. Elasticity is a ratio of one percentage change to another percentage change—nothing more—and we read it as an absolute value. In this case, a 1% rise in price causes an increase in quantity supplied of 3.5%. The greater than one elasticity of supply means that the percentage change in quantity supplied will be greater than a one percent price change. If you're starting to wonder if the concept of slope fits into this calculation, read the following Clear It Up box. Is the elasticity the slope? It is a common mistake to confuse the slope of either the supply or demand curve with its elasticity. The slope is the rate of change in units along the curve, or the rise/run (change in y over the change in x). For example, in Figure 5.2, at each point shown on the demand curve, price drops by $10 and the number of units demanded increases by 200 compared to the point to its left. The slope is –10/200 along the entire demand curve and does not change. The price elasticity, however, changes along the curve. Elasticity between points A and B was 0.45 and increased to 1.47 between points G and H. Elasticity is the percentage change, which is a different calculation from the slope and has a different meaning. When we are at the upper end of a demand curve, where price is high and the quantity demanded is low, a small change in the quantity demanded, even in, say, one unit, is pretty big in percentage terms. A change This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 113 in price of, say, a dollar, is going to be much less important in percentage terms than it would have been at the bottom of the demand curve. Likewise, at the bottom of the demand curve, that one unit change when the quantity demanded is high will be small as a percentage. Thus, at one end of the demand curve, where we have a large percentage change in quantity demanded over a small percentage change in price, the elasticity value would be high, or demand would be relatively elastic. Even with the same change in the price and the same change in the quantity demanded, at the other end of the demand curve the quantity is much higher, and the price is much lower, so the percentage change in quantity demanded is smaller and the percentage change in price is much higher. That means at the bottom of the curve we'd have a small numerator over a large denominator, so the elasticity measure would be much lower, or inelastic. As we move along the demand curve, the values for quantity and price go up or down, depending on which way we are moving, so the percentages for, say, a $1 difference in price or a one unit difference in quantity, will change as well, which means the ratios of those percentages and hence the elasticity will change. 5.2 \| Polar Cases of Elasticity and Constant Elasticity By the end of this section, you will be able to: • Differentiate between infinite and zero elasticity • Analyze graphs in order to classify elasticity as constant unitary, infinite, or zero There are two extreme cases of elasticity: when elasticity equals zero and when it is infinite. A third case is that of constant unitary elasticity. We will describe each case. Infinite elasticity or perfect elasticity refers to the extreme case where either the quantity demanded (Qd) or supplied (Qs) changes by an infinite amount in response to any change in price at all. In both cases, the supply and the demand curve are horizontal as Figure 5.4 shows. While perfectly elastic supply curves are for the most part unrealistic, goods with readily available inputs and whose production can easily expand will feature highly elastic supply curves. Examples include pizza, bread, books, and pencils. Similarly, perfectly elastic demand is an extreme example. However, luxury goods, items that take a large share of individuals’ income, and goods with many substitutes are likely to have highly elastic demand curves. Examples of such goods are Caribbean cruises and sports vehicles. Figure 5.4 Infinite Elasticity The horizontal lines show that an infinite quantity will be demanded or supplied at a specific price. This illustrates the cases of a perfectly (or infinitely) elastic demand curve and supply curve. The quantity supplied or demanded is extremely responsive to price changes, moving from zero for prices close to P to infinite when prices reach P. Zero elasticity or perfect inelasticity, as Figure 5.5 depicts, refers to the extreme case in which a percentage change in price, no matter how large, results in zero change in quantity. While a perfectly inelastic supply is an extreme example, goods with limited supply of inputs are likely to feature highly inelastic supply curves. Examples include diamond rings or housing in prime locations such as apartments facing Central Park in New York City. Similarly, 114 Chapter 5 \| Elasticity while perfectly inelastic demand is an extreme case, necessities with no close substitutes are likely to have highly inelastic demand curves. This is the case of life-saving drugs and gasoline. Figure 5.5 Zero Elasticity The vertical supply curve and vertical demand curve show that there will be zero percentage change in quantity (a) demanded or (b) supplied, regardless of the price. Constant unitary elasticity, in either a supply or demand curve, occurs when a price change of one percent results in a quantity change of one percent. Figure 5.6 shows a demand curve with constant unit elasticity. Constant unitary elasticity, in either a supply or demand curve, occurs when a price change of one percent results in a quantity change of one percent. Figure 5.6 shows a demand curve with constant unit elasticity. Using the midpoint method, you can calculate that between points A and B on the demand curve, the price changes by 28.6% and quantity demanded also changes by 28.6%. Hence, the elasticity equals 1. Between points B and C, price again changes by 28.6% as does quantity, while between points C and D the corresponding percentage changes are 22.2% for both price and quantity. In each case, then, the percentage change in price equals the percentage change in quantity, and consequently elasticity equals 1. Notice that in absolute value, the declines in price, as you step down the demand curve, are not identical. Instead, the price falls by $2.00 from A to B, by a smaller amount of $1.50 from B to C, and by a still smaller amount of $0.90 from C to D. As a result, a demand curve with constant unitary elasticity moves from a steeper slope on the left and a flatter slope on the right—and a curved shape overall. Notice that in absolute value, the declines in price, as you step down the demand curve, are not identical. Instead, the price falls by $23 from A to B, by a smaller amount of $1.50 from B to C, and by a still smaller amount of $.90 from C to D. As a result, a demand curve with constant unitary elasticity has a steeper slope on the left and a flatter slope on the right—and a curved shape overall. Figure 5.6 A Constant Unitary Elasticity Demand Curve A demand curve with constant unitary elasticity will be a curved line. Notice how price and quantity demanded change by an identical percentage amount between each pair of points on the demand curve. Unlike the demand curve with unitary elasticity, the supply curve with unitary elasticity is represented by a straight This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 115 line, and that line goes through the origin. In each pair of points on the supply curve there is an equal difference in quantity of 30. However, in percentag
e value, using the midpoint method, the steps are decreasing as one moves from left to right, from 28.6% to 22.2% to 18.2%, because the quantity points in each percentage calculation are getting increasingly larger, which expands the denominator in the elasticity calculation of the percentage change in quantity. Consider the price changes moving up the supply curve in Figure 5.7. From points D to E to F and to G on the supply curve, each step of $1.50 is the same in absolute value. However, if we measure the price changes in percentage change terms, using the midpoint method, they are also decreasing, from 28.6% to 22.2% to 18.2%, because the original price points in each percentage calculation are getting increasingly larger in value, increasing the denominator in the calculation of the percentage change in price. Along the constant unitary elasticity supply curve, the percentage quantity increases on the horizontal axis exactly match the percentage price increases on the vertical axis—so this supply curve has a constant unitary elasticity at all points. Figure 5.7 A Constant Unitary Elasticity Supply Curve A constant unitary elasticity supply curve is a straight line reaching up from the origin. Between each pair of points, the percentage increase in quantity supplied is the same as the percentage increase in price. 5.3 \| Elasticity and Pricing By the end of this section, you will be able to: • Analyze how price elasticities impact revenue • Evaluate how elasticity can cause shifts in demand and supply • Predict how the long-run and short-run impacts of elasticity affect equilibrium • Explain how the elasticity of demand and supply determine the incidence of a tax on buyers and sellers Studying elasticities is useful for a number of reasons, pricing being most important. Let’s explore how elasticity relates to revenue and pricing, both in the long and short run. First, let’s look at the elasticities of some common goods and services. Table 5.2 shows a selection of demand elasticities for different goods and services drawn from a variety of different studies by economists, listed in order of increasing elasticity. 116 Housing Goods and Services Elasticity of Price Chapter 5 \| Elasticity Transatlantic air travel (economy class) Rail transit (rush hour) Electricity Taxi cabs Gasoline Transatlantic air travel (first class) Wine Beef Transatlantic air travel (business class) Kitchen and household appliances Cable TV (basic rural) Chicken Soft drinks Beer New vehicle Rail transit (off-peak) Computer Cable TV (basic urban) Cable TV (premium) Restaurant meals 0.12 0.12 0.15 0.20 0.22 0.35 0.40 0.55 0.59 0.62 0.63 0.69 0.64 0.70 0.80 0.87 1.00 1.44 1.51 1.77 2.27 Table 5.2 Some Selected Elasticities of Demand Note that demand for necessities such as housing and electricity is inelastic, while items that are not necessities such as restaurant meals are more price-sensitive. If the price of a restaurant meal increases by 10%, the quantity demanded will decrease by 22.7%. A 10% increase in the price of housing will cause only a slight decrease of 1.2% in the quantity of housing demanded. Read this article (http://openstaxcollege.org/l/Movietickets) for an example of price elasticity that may have affected you. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 117 Does Raising Price Bring in More Revenue? Imagine that a band on tour is playing in an indoor arena with 15,000 seats. To keep this example simple, assume that the band keeps all the money from ticket sales. Assume further that the band pays the costs for its appearance, but that these costs, like travel, and setting up the stage, are the same regardless of how many people are in the audience. Finally, assume that all the tickets have the same price. (The same insights apply if ticket prices are more expensive for some seats than for others, but the calculations become more complicated.) The band knows that it faces a downward-sloping demand curve; that is, if the band raises the ticket price and, it will sell fewer seats. How should the band set the ticket price to generate the most total revenue, which in this example, because costs are fixed, will also mean the highest profits for the band? Should the band sell more tickets at a lower price or fewer tickets at a higher price? The key concept in thinking about collecting the most revenue is the price elasticity of demand. Total revenue is price times the quantity of tickets sold. Imagine that the band starts off thinking about a certain price, which will result in the sale of a certain quantity of tickets. The three possibilities are in Table 5.3. If demand is elastic at that price level, then the band should cut the price, because the percentage drop in price will result in an even larger percentage increase in the quantity sold—thus raising total revenue. However, if demand is inelastic at that original quantity level, then the band should raise the ticket price, because a certain percentage increase in price will result in a smaller percentage decrease in the quantity sold—and total revenue will rise. If demand has a unitary elasticity at that quantity, then an equal percentage change in quantity will offset a moderate percentage change in the price—so the band will earn the same revenue whether it (moderately) increases or decreases the ticket price. If Demand Is . . . Elastic Then . . . Therefore . . . % change in Qd > % change in P A given % rise in P will be more than offset by a larger % fall in Q so that total revenue (P × Q) falls. Unitary % change in Qd = % change in P A given % rise in P will be exactly offset by an equal % fall in Q so that total revenue (P × Q) is unchanged. Inelastic % change in Qd < % change in P A given % rise in P will cause a smaller % fall in Q so that total revenue (P × Q) rises. Table 5.3 Will the Band Earn More Revenue by Changing Ticket Prices? What if the band keeps cutting price, because demand is elastic, until it reaches a level where it sells all 15,000 seats in the available arena? If demand remains elastic at that quantity, the band might try to move to a bigger arena, so that it could slash ticket prices further and see a larger percentage increase in the quantity of tickets sold. However, if the 15,000-seat arena is all that is available or if a larger arena would add substantially to costs, then this option may not work. Conversely, a few bands are so famous, or have such fanatical followings, that demand for tickets may be inelastic right up to the point where the arena is full. These bands can, if they wish, keep raising the ticket price. Ironically, 118 Chapter 5 \| Elasticity some of the most popular bands could make more revenue by setting prices so high that the arena is not full—but those who buy the tickets would have to pay very high prices. However, bands sometimes choose to sell tickets for less than the absolute maximum they might be able to charge, often in the hope that fans will feel happier and spend more on recordings, T-shirts, and other paraphernalia. Can Businesses Pass Costs on to Consumers? Most businesses face a day-to-day struggle to figure out ways to produce at a lower cost, as one pathway to their goal of earning higher profits. However, in some cases, the price of a key input over which the firm has no control may rise. For example, many chemical companies use petroleum as a key input, but they have no control over the world market price for crude oil. Coffee shops use coffee as a key input, but they have no control over the world market price of coffee. If the cost of a key input rises, can the firm pass those higher costs along to consumers in the form of higher prices? Conversely, if new and less expensive ways of producing are invented, can the firm keep the benefits in the form of higher profits, or will the market pressure them to pass the gains along to consumers in the form of lower prices? The price elasticity of demand plays a key role in answering these questions. Imagine that as a consumer of legal pharmaceutical products, you read a newspaper story that a technological breakthrough in the production of aspirin has occurred, so that every aspirin factory can now produce aspirin more cheaply. What does this discovery mean to you? Figure 5.8 illustrates two possibilities. In Figure 5.8 (a), the demand curve is highly inelastic. In this case, a technological breakthrough that shifts supply to the right, from S0 to S1, so that the equilibrium shifts from E0 to E1, creates a substantially lower price for the product with relatively little impact on the quantity sold. In Figure 5.8 (b), the demand curve is highly elastic. In this case, the technological breakthrough leads to a much greater quantity sold in the market at very close to the original price. Consumers benefit more, in general, when the demand curve is more inelastic because the shift in the supply results in a much lower price for consumers. Figure 5.8 Passing along Cost Savings to Consumers Cost-saving gains cause supply to shift out to the right from S0 to S1; that is, at any given price, firms will be willing to supply a greater quantity. If demand is inelastic, as in (a), the result of this cost-saving technological improvement will be substantially lower prices. If demand is elastic, as in (b), the result will be only slightly lower prices. Consumers benefit in either case, from a greater quantity at a lower price, but the benefit is greater when demand is inelastic, as in (a). Aspirin producers may find themselves in a nasty bind here. The situation in Figure 5.8, with extremely inelastic demand, means that a new invention may cause the price to drop dramatically while quantity changes little. As a result, the new production technology can lead to a drop in the revenue that firms earn from aspirin sales. However, if strong competition exists between aspirin producer, each producer may have little choice bu
t to search for and implement any breakthrough that allows it to reduce production costs. After all, if one firm decides not to implement such a cost-saving technology, other firms that do can drive them out of business. Since demand for food is generally inelastic, farmers may often face the situation in Figure 5.8 (a). That is, a surge in This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 119 production leads to a severe drop in price that can actually decrease the total revenue that farmers receive. Conversely, poor weather or other conditions that cause a terrible year for farm production can sharply raise prices so that the total revenue that the farmer receives increases. The Clear It Up box discusses how these issues relate to coffee. How do coffee prices fluctuate? Coffee is an international crop. The top five coffee-exporting nations are Brazil, Vietnam, Colombia, Indonesia, and Ethiopia. In these nations and others, 20 million families depend on selling coffee beans as their main source of income. These families are exposed to enormous risk, because the world price of coffee bounces up and down. For example, in 1993, the world price of coffee was about 50 cents per pound. In 1995 it was four times as high, at $2 per pound. By 1997 it had fallen by half to $1.00 per pound. In 1998 it leaped back up to $2 per pound. By 2001 it had fallen back to 46 cents a pound. By early 2011 it rose to about $2.31 per pound. By the end of 2012, the price had fallen back to about $1.31 per pound. The reason for these price fluctuations lies in a combination of inelastic demand and shifts in supply. The elasticity of coffee demand is only about 0.3; that is, a 10% rise in the price of coffee leads to a decline of about 3% in the quantity of coffee consumed. When a major frost hit the Brazilian coffee crop in 1994, coffee supply shifted to the left with an inelastic demand curve, leading to much higher prices. Conversely, when Vietnam entered the world coffee market as a major producer in the late 1990s, the supply curve shifted out to the right. With a highly inelastic demand curve, coffee prices fell dramatically. Figure 5.8 (a) illustrates this situation. Elasticity also reveals whether firms can pass higher costs that they incur on to consumers. Addictive substances, for which demand is inelastic, are products for which producers can pass higher costs on to consumers. For example, the demand for cigarettes is relatively inelastic among regular smokers who are somewhat addicted. Economic research suggests that increasing cigarette prices by 10% leads to about a 3% reduction in the quantity of cigarettes that adults smoke, so the elasticity of demand for cigarettes is 0.3. If society increases taxes on companies that produce cigarettes, the result will be, as in Figure 5.9 (a), that the supply curve shifts from S0 to S1. However, as the equilibrium moves from E0 to E1, governments mainly pass along these taxes to consumers in the form of higher prices. These higher taxes on cigarettes will raise tax revenue for the government, but they will not much affect the quantity of smoking. If the goal is to reduce the quantity of cigarettes demanded, we must achieve it by shifting this inelastic demand back to the left, perhaps with public programs to discourage cigarette use or to help people to quit. For example, anti-smoking advertising campaigns have shown some ability to reduce smoking. However, if cigarette demand were more elastic, as in Figure 5.9 (b), then an increase in taxes that shifts supply from S0 to S1 and equilibrium from E0 to E1 would reduce the quantity of cigarettes smoked substantially. Youth smoking seems to be more elastic than adult smoking—that is, the quantity of youth smoking will fall by a greater percentage than the quantity of adult smoking in response to a given percentage increase in price. 120 Chapter 5 \| Elasticity Figure 5.9 Passing along Higher Costs to Consumers Higher costs, like a higher tax on cigarette companies for the example we gave in the text, lead supply to shift to the left. This shift is identical in (a) and (b). However, in (a), where demand is inelastic, companies largely can pass the cost increase along to consumers in the form of higher prices, without much of a decline in equilibrium quantity. In (b), demand is elastic, so the shift in supply results primarily in a lower equilibrium quantity. Consumers suffer in either case, but in (a), they suffer from paying a higher price for the same quantity, while in (b), they suffer from buying a lower quantity (and presumably needing to shift their consumption elsewhere). Elasticity and Tax Incidence The example of cigarette taxes demonstrated that because demand is inelastic, taxes are not effective at reducing the equilibrium quantity of smoking, and they mainly pass along to consumers in the form of higher prices. The analysis, or manner, of how a tax burden is divided between consumers and producers is called tax incidence. Typically, the tax incidence, or burden, falls both on the consumers and producers of the taxed good. However, if one wants to predict which group will bear most of the burden, all one needs to do is examine the elasticity of demand and supply. In the tobacco example, the tax burden falls on the most inelastic side of the market. If demand is more inelastic than supply, consumers bear most of the tax burden, and if supply is more inelastic than demand, sellers bear most of the tax burden. The intuition for this is simple. When the demand is inelastic, consumers are not very responsive to price changes, and the quantity demanded reduces only modestly when the tax is introduced. In the case of smoking, the demand is inelastic because consumers are addicted to the product. The government can then pass the tax burden along to consumers in the form of higher prices, without much of a decline in the equilibrium quantity. Similarly, when a government introduces a tax in a market with an inelastic supply, such as, for example, beachfront hotels, and sellers have no alternative than to accept lower prices for their business, taxes do not greatly affect the equilibrium quantity. The tax burden now passes on to the sellers. If the supply was elastic and sellers had the possibility of reorganizing their businesses to avoid supplying the taxed good, the tax burden on the sellers would be much smaller. The tax would result in a much lower quantity sold instead of lower prices received. Figure 5.10 illustrates this relationship between the tax incidence and elasticity of demand and supply. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 121 Figure 5.10 Elasticity and Tax Incidence An excise tax introduces a wedge between the price paid by consumers (Pc) and the price received by producers (Pp). The vertical distance between Pc and Pp is the amount of the tax per unit. Pe is the equilibrium price prior to introduction of the tax. (a) When the demand is more elastic than supply, the tax incidence on consumers Pc – Pe is lower than the tax incidence on producers Pe – Pp. (b) When the supply is more elastic than demand, the tax incidence on consumers Pc – Pe is larger than the tax incidence on producers Pe – Pp. The more elastic the demand and supply curves, the lower the tax revenue. In Figure 5.10 (a), the supply is inelastic and the demand is elastic, such as in the example of beachfront hotels. While consumers may have other vacation choices, sellers can’t easily move their businesses. By introducing a tax, the government essentially creates a wedge between the price paid by consumers Pc and the price received by producers Pp. In other words, of the total price paid by consumers, part is retained by the sellers and part is paid to the government in the form of a tax. The distance between Pc and Pp is the tax rate. The new market price is Pc, but sellers receive only Pp per unit sold, as they pay Pc-Pp to the government. Since we can view a tax as raising the costs of production, this could also be represented by a leftward shift of the supply curve, where the new supply curve would intercept the demand at the new quantity Qt. For simplicity, Figure 5.10 omits the shift in the supply curve. The tax revenue is given by the shaded area, which we obtain by multiplying the tax per unit by the total quantity sold Qt. The tax incidence on the consumers is given by the difference between the price paid Pc and the initial equilibrium price Pe. The tax incidence on the sellers is given by the difference between the initial equilibrium price Pe and the price they receive after the tax is introduced Pp. In Figure 5.10 (a), the tax burden falls disproportionately on the sellers, and a larger proportion of the tax revenue (the shaded area) is due to the resulting lower price received by the sellers than by the resulting higher prices paid by the buyers. Figure 5.10 (b) describes the example of the tobacco excise tax where the supply is more elastic than demand. The tax incidence now falls disproportionately on consumers, as shown by the large difference between the price they pay, Pc, and the initial equilibrium price, Pe. Sellers receive a lower price than before the tax, but this difference is much smaller than the change in consumers’ price. From this analysis one can also predict whether a tax is likely to create a large revenue or not. The more elastic the demand curve, the more likely that consumers will reduce quantity instead of paying higher prices. The more elastic the supply curve, the more likely that sellers will reduce the quantity sold, instead of taking lower prices. In a market where both the demand and supply are very elastic, the imposition of an excise tax generates low revenue. Some believe that excise taxes hurt mainly the specific industries they target. For example, the medical device excise tax, in effect
since 2013, has been controversial for it can delay industry profitability and therefore hamper startups and medical innovation. However, whether the tax burden falls mostly on the medical device industry or on the patients depends simply on the elasticity of demand and supply. Long-Run vs. Short-Run Impact Elasticities are often lower in the short run than in the long run. On the demand side of the market, it can sometimes be difficult to change Qd in the short run, but easier in the long run. Consumption of energy is a clear example. In the short run, it is not easy for a person to make substantial changes in energy consumption. Maybe you can carpool to work sometimes or adjust your home thermostat by a few degrees if the cost of energy rises, but that is about all. 122 Chapter 5 \| Elasticity However, in the long run you can purchase a car that gets more miles to the gallon, choose a job that is closer to where you live, buy more energy-efficient home appliances, or install more insulation in your home. As a result, the elasticity of demand for energy is somewhat inelastic in the short run, but much more elastic in the long run. Figure 5.11 is an example, based roughly on historical experience, for the responsiveness of Qd to price changes. In 1973, the price of crude oil was $12 per barrel and total consumption in the U.S. economy was 17 million barrels per day. That year, the nations who were members of the Organization of Petroleum Exporting Countries (OPEC) cut off oil exports to the United States for six months because the Arab members of OPEC disagreed with the U.S. support for Israel. OPEC did not bring exports back to their earlier levels until 1975—a policy that we can interpret as a shift of the supply curve to the left in the U.S. petroleum market. Figure 5.11 (a) and Figure 5.11 (b) show the same original equilibrium point and the same identical shift of a supply curve to the left from S0 to S1. Figure 5.11 How a Shift in Supply Can Affect Price or Quantity The intersection (E0) between demand curve D and supply curve S0 is the same in both (a) and (b). The shift of supply to the left from S0 to S1 is identical in both (a) and (b). The new equilibrium (E1) has a higher price and a lower quantity than the original equilibrium (E0) in both (a) and (b). However, the shape of the demand curve D is different in (a) and (b), being more elastic in (b) than in (a). As a result, the shift in supply can result either in a new equilibrium with a much higher price and an only slightly smaller quantity, as in (a), with more inelastic demand, or in a new equilibrium with only a small increase in price and a relatively larger reduction in quantity, as in (b), with more elastic demand. Figure 5.11 (a) shows inelastic demand for oil in the short run similar to that which existed for the United States in 1973. In Figure 5.11 (a), the new equilibrium (E1) occurs at a price of $25 per barrel, roughly double the price before the OPEC shock, and an equilibrium quantity of 16 million barrels per day. Figure 5.11 (b) shows what the outcome would have been if the U.S. demand for oil had been more elastic, a result more likely over the long term. This alternative equilibrium (E1) would have resulted in a smaller price increase to $14 per barrel and larger reduction in equilibrium quantity to 13 million barrels per day. In 1983, for example, U.S. petroleum consumption was 15.3 million barrels a day, which was lower than in 1973 or 1975. U.S. petroleum consumption was down even though the U.S. economy was about one-fourth larger in 1983 than it had been in 1973. The primary reason for the lower quantity was that higher energy prices spurred conservation efforts, and after a decade of home insulation, more fuel-efficient cars, more efficient appliances and machinery, and other fuel-conserving choices, the demand curve for energy had become more elastic. On the supply side of markets, producers of goods and services typically find it easier to expand production in the long term of several years rather than in the short run of a few months. After all, in the short run it can be costly or difficult to build a new factory, hire many new workers, or open new stores. However, over a few years, all of these are possible. In most markets for goods and services, prices bounce up and down more than quantities in the short run, but quantities often move more than prices in the long run. The underlying reason for this pattern is that supply and demand are often inelastic in the short run, so that shifts in either demand or supply can cause a relatively greater This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 123 change in prices. However, since supply and demand are more elastic in the long run, the long-run movements in prices are more muted, while quantity adjusts more easily in the long run. 5.4 \| Elasticity in Areas Other Than Price By the end of this section, you will be able to: • Calculate the income elasticity of demand and the cross-price elasticity of demand • Calculate the elasticity in labor and financial capital markets through an understanding of the elasticity of labor supply and the elasticity of savings • Apply concepts of price elasticity to real-world situations The basic idea of elasticity—how a percentage change in one variable causes a percentage change in another variable—does not just apply to the responsiveness quantity supplied and quantity demanded to changes in the price of a product. Recall that quantity demanded (Qd) depends on income, tastes and preferences, the prices of related goods, and so on, as well as price. Similarly, quantity supplied (Qs) depends on factors such as the cost of production, as well as price. We can measure elasticity for any determinant of quantity supplied and quantity demanded, not just the price. Income Elasticity of Demand The income elasticity of demand is the percentage change in quantity demanded divided by the percentage change in income. Income elasticity of demand = % change in quantity demanded % change in income For most products, most of the time, the income elasticity of demand is positive: that is, a rise in income will cause an increase in the quantity demanded. This pattern is common enough that we refer to these goods as normal goods. However, for a few goods, an increase in income means that one might purchase less of the good. For example, those with a higher income might buy fewer hamburgers, because they are buying more steak instead, or those with a higher income might buy less cheap wine and more imported beer. When the income elasticity of demand is negative, we call the good an inferior good. We introduced the concepts of normal and inferior goods in Demand and Supply. A higher level of income causes a demand curve to shift to the right for a normal good, which means that the income elasticity of demand is positive. How far the demand shifts depends on the income elasticity of demand. A higher income elasticity means a larger shift. However, for an inferior good, that is, when the income elasticity of demand is negative, a higher level of income would cause the demand curve for that good to shift to the left. Again, how much it shifts depends on how large the (negative) income elasticity is. Cross-Price Elasticity of Demand A change in the price of one good can shift the quantity demanded for another good. If the two goods are complements, like bread and peanut butter, then a drop in the price of one good will lead to an increase in the quantity demanded of the other good. However, if the two goods are substitutes, like plane tickets and train tickets, then a drop in the price of one good will cause people to substitute toward that good, and to reduce consumption of the other good. Cheaper plane tickets lead to fewer train tickets, and vice versa. The cross-price elasticity of demand puts some meat on the bones of these ideas. The term “cross-price” refers to the idea that the price of one good is affecting the quantity demanded of a different good. Specifically, the cross-price elasticity of demand is the percentage change in the quantity of good A that is demanded as a result of a percentage change in the price of good B. Cross-price elasticity of demand = % change in Qd of good A % change in price of good B Substitute goods have positive cross-price elasticities of demand: if good A is a substitute for good B, like coffee and tea, then a higher price for B will mean a greater quantity consumed of A. Complement goods have negative crossprice elasticities: if good A is a complement for good B, like coffee and sugar, then a higher price for B will mean a 124 Chapter 5 \| Elasticity lower quantity consumed of A. Elasticity in Labor and Financial Capital Markets The concept of elasticity applies to any market, not just markets for goods and services. In the labor market, for example, the wage elasticity of labor supply—that is, the percentage change in hours worked divided by the percentage change in wages—will reflect the shape of the labor supply curve. Specifically: Elasticity of labor supply = % change in quantity of labor supplied % change in wage The wage elasticity of labor supply for teenage workers is generally fairly elastic: that is, a certain percentage change in wages will lead to a larger percentage change in the quantity of hours worked. Conversely, the wage elasticity of labor supply for adult workers in their thirties and forties is fairly inelastic. When wages move up or down by a certain percentage amount, the quantity of hours that adults in their prime earning years are willing to supply changes but by a lesser percentage amount. In markets for financial capital, the elasticity of savings—that is, the percentage change in the quantity of savings divided by the percentage change in interest rates—will describe the shape of the supply curve for financial capital. That is: Elasticity of
savings = % change in quantity of financial s vings % change in interest rate Sometimes laws are proposed that seek to increase the quantity of savings by offering tax breaks so that the return on savings is higher. Such a policy will have a comparatively large impact on increasing the quantity saved if the supply curve for financial capital is elastic, because then a given percentage increase in the return to savings will cause a higher percentage increase in the quantity of savings. However, if the supply curve for financial capital is highly inelastic, then a percentage increase in the return to savings will cause only a small increase in the quantity of savings. The evidence on the supply curve of financial capital is controversial but, at least in the short run, the elasticity of savings with respect to the interest rate appears fairly inelastic. Expanding the Concept of Elasticity The elasticity concept does not even need to relate to a typical supply or demand curve at all. For example, imagine that you are studying whether the Internal Revenue Service should spend more money on auditing tax returns. We can frame the question in terms of the elasticity of tax collections with respect to spending on tax enforcement; that is, what is the percentage change in tax collections derived from a given percentage change in spending on tax enforcement? With all of the elasticity concepts that we have just described, some of which are in Table 5.4, the possibility of confusion arises. When you hear the phrases “elasticity of demand” or “elasticity of supply,” they refer to the elasticity with respect to price. Sometimes, either to be extremely clear or because economists are discussing a wide variety of elasticities, we will call the elasticity of demand or the demand elasticity the price elasticity of demand or the “elasticity of demand with respect to price.” Similarly, economists sometimes use the term elasticity of supply or the supply elasticity, to avoid any possibility of confusion, the price elasticity of supply or “the elasticity of supply with respect to price.” However, in whatever context, the idea of elasticity always refers to percentage change in one variable, almost always a price or money variable, and how it causes a percentage change in another variable, typically a quantity variable of some kind. Income elasticity of demand = % change in Qd % change in income Cross-price elasticity of demand = % change in Qd of good A % change in price of good B Table 5.4 Formulas for Calculating Elasticity This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 125 Wage elasticity of labor supply = % change in quantity of labor supplied % change in wage Wage elasticity of labor demand = % change in quantity of labor demanded % change in wage Interest rate elasticity of savings = % change in quantity of savings % change in interest rate Interest rate elasticity of borrowing = % change in quantity of borrowing % change in interest rate Table 5.4 Formulas for Calculating Elasticity That Will Be How Much? How did the 60% price increase in 2011 end up for Netflix? It has been a very bumpy ride. Before the price increase, there were about 24.6 million U.S. subscribers. After the price increase, 810,000 infuriated U.S. consumers canceled their Netflix subscriptions, dropping the total number of subscribers to 23.79 million. Fast forward to June 2013, when there were 36 million streaming Netflix subscribers in the United States. This was an increase of 11.4 million subscribers since the price increase—an average per quarter growth of about 1.6 million. This growth is less than the 2 million per quarter increases Netflix experienced in the fourth quarter of 2010 and the first quarter of 2011. During the first year after the price increase, the firm’s stock price (a measure of future expectations for the firm) fell from about $33.60 per share per share to just under $7.80. By the end of 2016, however, the stock price was at $123 per share. Today, Netflix has more than 86 million subscribers million subscribers in fifty countries. What happened? Obviously, Netflix company officials understood the law of demand. Company officials reported, when announcing the price increase, in the loss of about 600,000 existing subscribers. Using the elasticity of demand formula, it is easy to see company officials expected an inelastic response: this could result = –600,000/[(24 million + 24.6 million)/2] $6/[($10 + $16)/2] = –600,000/24.3 million $6/$13 = –0.025 0.46 = –0.05 In addition, Netflix officials had anticipated the price increase would have little impact on attracting new customers. Netflix anticipated adding up to 1.29 million new subscribers in the third quarter of 2011. It is true this was slower growth than the firm had experienced—about 2 million per quarter. Why was the estimate of customers leaving so far off? In the more than two decades since Netflix had been founded, there was an increase in the number of close, but not perfect, substitutes. Consumers now had choices ranging from Vudu, Amazon Prime, Hulu, and Redbox, to retail stores. Jaime Weinman reported in Maclean’s that Redbox kiosks are “a five-minute drive for less from 68 percent of Americans, and it seems that many people still find a five-minute drive more convenient than loading up a movie online.” It seems that in 2012, many consumers still preferred a physical DVD disk over streaming video. What missteps did the Netflix management make? In addition to misjudging the elasticity of demand, by failing 126 Chapter 5 \| Elasticity to account for close substitutes, it seems they may have also misjudged customers’ preferences and tastes. Yet, as the population increases, the preference for streaming video may overtake physical DVD disks. Netflix, the source of numerous late night talk show laughs and jabs in 2011, may yet have the last laugh. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity KEY TERMS 127 constant unitary elasticity when a given percent price change in price leads to an equal percentage change in quantity demanded or supplied cross-price elasticity of demand the percentage change in the quantity of good A that is demanded as a result of a percentage change in good B elastic demand when the elasticity of demand is greater than one, indicating a high responsiveness of quantity demanded or supplied to changes in price elastic supply when the elasticity of either supply is greater than one, indicating a high responsiveness of quantity demanded or supplied to changes in price elasticity an economics concept that measures responsiveness of one variable to changes in another variable elasticity of savings the percentage change in the quantity of savings divided by the percentage change in interest rates inelastic demand when the elasticity of demand is less than one, indicating that a 1 percent increase in price paid by the consumer leads to less than a 1 percent change in purchases (and vice versa); this indicates a low responsiveness by consumers to price changes inelastic supply when the elasticity of supply is less than one, indicating that a 1 percent increase in price paid to the firm will result in a less than 1 percent increase in production by the firm; this indicates a low responsiveness of the firm to price increases (and vice versa if prices drop) infinite elasticity the extremely elastic situation of demand or supply where quantity changes by an infinite amount in response to any change in price; horizontal in appearance perfect elasticity see infinite elasticity perfect inelasticity see zero elasticity price elasticity the relationship between the percent change in price resulting in a corresponding percentage change in the quantity demanded or supplied price elasticity of demand percentage change in the quantity demanded of a good or service divided the percentage change in price price elasticity of supply percentage change in the quantity supplied divided by the percentage change in price tax incidence manner in which the tax burden is divided between buyers and sellers unitary elasticity when the calculated elasticity is equal to one indicating that a change in the price of the good or service results in a proportional change in the quantity demanded or supplied wage elasticity of labor supply the percentage change in hours worked divided by the percentage change in wages zero inelasticity the highly inelastic case of demand or supply in which a percentage change in price, no matter how large, results in zero change in the quantity; vertical in appearance 128 Chapter 5 \| Elasticity KEY CONCEPTS AND SUMMARY 5.1 Price Elasticity of Demand and Price Elasticity of Supply Price elasticity measures the responsiveness of the quantity demanded or supplied of a good to a change in its price. We compute it as the percentage change in quantity demanded (or supplied) divided by the percentage change in price. We can describe elasticity as elastic (or very responsive), unit elastic, or inelastic (not very responsive). Elastic demand or supply curves indicate that quantity demanded or supplied respond to price changes in a greater than proportional manner. An inelastic demand or supply curve is one where a given percentage change in price will cause a smaller percentage change in quantity demanded or supplied. A unitary elasticity means that a given percentage change in price leads to an equal percentage change in quantity demanded or supplied. 5.2 Polar Cases of Elasticity and Constant Elasticity Infinite or perfect elasticity refers to the extreme case where either the quantity demanded or supplied changes by an infinite amount in response to any change in price at all. Zero elasticity refers to the extreme case in which a percentage change in price, no matter how large, results in zero change in quantity. Constant unitary elasticity in either a su
pply or demand curve refers to a situation where a price change of one percent results in a quantity change of one percent. 5.3 Elasticity and Pricing In the market for goods and services, quantity supplied and quantity demanded are often relatively slow to react to changes in price in the short run, but react more substantially in the long run. As a result, demand and supply often (but not always) tend to be relatively inelastic in the short run and relatively elastic in the long run. A tax incidence depends on the relative price elasticity of supply and demand. When supply is more elastic than demand, buyers bear most of the tax burden, and when demand is more elastic than supply, producers bear most of the cost of the tax. Tax revenue is larger the more inelastic the demand and supply are. 5.4 Elasticity in Areas Other Than Price Elasticity is a general term, that reflects responsiveness. It refers to the change of one variable divided by the percentage change of a related variable that we can apply to many economic connections. For instance, the income elasticity of demand is the percentage change in quantity demanded divided by the percentage change in income. The cross-price elasticity of demand is the percentage change in the quantity demanded of a good divided by the percentage change in the price of another good. Elasticity applies in labor markets and financial capital markets just as it does in markets for goods and services. The wage elasticity of labor supply is the percentage change in the quantity of hours supplied divided by the percentage change in the wage. The elasticity of savings with respect to interest rates is the percentage change in the quantity of savings divided by the percentage change in interest rates. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 5 \| Elasticity 129 SELF-CHECK QUESTIONS 1. From the data in Table 5.5 about demand for smart phones, calculate the price elasticity of demand from: point B to point C, point D to point E, and point G to point H. Classify the elasticity at each point as elastic, inelastic, or unit elastic. Points Table 5.5 60 70 80 90 100 110 120 130 3,000 2,800 2,600 2,400 2,200 2,000 1,800 1,600 2. From the data in Table 5.6 about supply of alarm clocks, calculate the price elasticity of supply from: point J to point K, point L to point M, and point N to point P. Classify the elasticity at each point as elastic, inelastic, or unit elastic. Point Price Quantity Supplied J K L M N P Table 5.6 $8 $9 $10 $11 $12 $13 50 70 80 88 95 100 3. Why is the demand curve with constant unitary elasticity concave? 4. Why is the supply curve with constant unitary elasticity a straight line? 5. The federal government decides to require that automobile manufacturers install new anti-pollution equipment that costs $2,000 per car. Under what conditions can carmakers pass almost all of this cost along to car buyers? Under what conditions can carmakers pass very little of this cost along to car buyers? 130 Chapter 5 \| Elasticity 6. Suppose you are in charge of sales at a pharmaceutical company, and your firm has a new drug that causes bald men to grow hair. Assume that the company wants to earn as much revenue as possible from this drug. If the elasticity of demand for your company’s product at the current price is 1.4, would you advise the company to raise the price, lower the price, or to keep the price the same? What if the elasticity were 0.6? What if it were 1? Explain your answer. 7. What would the gasoline price elasticity of supply mean to UPS or FedEx? 8. The average annual income rises from $25,000 to $38,000, and the quantity of bread consumed in a year by the average person falls from 30 loaves to 22 loaves. What is the income elasticity of bread consumption? Is bread a normal or an inferior good? 9. Suppose the cross-price elasticity of apples with respect to the price of oranges is 0.4, and the price of oranges falls by 3%. What will happen to the demand for apples? REVIEW QUESTIONS 10. What is the formula for calculating elasticity? 11. What is the price elasticity of demand? Can you explain it in your own words? 12. What is the price elasticity of supply? Can you explain it in your own words? 13. Describe the general appearance of a demand or a supply curve with zero elasticity. 14. Describe the general appearance of a demand or a supply curve with infinite elasticity. If demand is elastic, will shifts in supply have a 15. larger effect on equilibrium quantity or on price? If demand is inelastic, will shifts in supply have a 16. larger effect on equilibrium price or on quantity? If supply is elastic, will shifts in demand have a 17. larger effect on equilibrium quantity or on price? CRITICAL THINKING QUESTIONS elasticity 25. Transatlantic air travel in business class has an estimated 0.62, while transatlantic air travel in economy class has an estimated price elasticity of 0.12. Why do you think this is the case? demand of of 26. What is the relationship between price elasticity and position on the demand curve? For example, as you move up the demand curve to higher prices and lower quantities, what happens to the measured elasticity? How would you explain that? This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 If supply is inelastic, will shifts in demand have a 18. larger effect on equilibrium price or on quantity? 19. Would you usually expect elasticity of demand or supply to be higher in the short run or in the long run? Why? 20. Under which circumstances does the tax burden fall entirely on consumers? 21. What is the formula for the income elasticity of demand? 22. What is the formula for the cross-price elasticity of demand? 23. What is the formula for the wage elasticity of labor supply? 24. What is the formula for elasticity of savings with respect to interest rates? 27. Can you think of an industry (or product) with near infinite elasticity of supply in the short term? That is, what is an industry that could increase Qs almost without limit in response to an increase in the price? 28. Would you expect supply to play a more significant role in determining the price of a basic necessity like food or a luxury like perfume? Explain. Hint: Think about how the price elasticity of demand will differ between necessities and luxuries. Chapter 5 \| Elasticity 131 29. A city has built a bridge over a river and it decides to charge a toll to everyone who crosses. For one year, the city charges a variety of different tolls and records information on how many drivers cross the bridge. The city thus gathers information about elasticity of demand. If the city wishes to raise as much revenue as possible from the tolls, where will the city decide to charge a toll: in the inelastic portion of the demand curve, the elastic portion of the demand curve, or the unit elastic portion? Explain. 30. In a market where the supply curve is perfectly inelastic, how does an excise tax affect the price paid by consumers and the quantity bought and sold? PROBLEMS 31. Economists define normal goods as having a positive income elasticity. We can divide normal goods into two types: Those whose income elasticity is less than one and those whose income elasticity is greater than one. Think about products that would fall into each category. Can you come up with a name for each category? 32. Suppose you could buy shoes one at a time, rather the cross-price than in pairs. What do you predict elasticity for left shoes and right shoes would be? 33. The equation for a demand curve is P = 48 – 3Q. What is the elasticity in moving from a quantity of 5 to a quantity of 6? 38. Say that a certain stadium for professional football has 70,000 seats. What is the shape of the supply curve for tickets to football games at that stadium? Explain. 34. The equation for a demand curve is P = 2/Q. What is the elasticity of demand as price falls from 5 to 4? What is the elasticity of demand as the price falls from 9 to 8? Would you expect these answers to be the same? 35. The equation for a supply curve is 4P = Q. What is the elasticity of supply as price rises from 3 to 4? What is the elasticity of supply as the price rises from 7 to 8? Would you expect these answers to be the same? 36. The equation for a supply curve is P = 3Q – 8. What is the elasticity in moving from a price of 4 to a price of 7? 37. The supply of paintings by Leonardo Da Vinci, who painted the Mona Lisa and The Last Supper and died in 1519, is highly inelastic. Sketch a supply and demand diagram, paying attention to the appropriate elasticities, to illustrate that demand for these paintings will determine the price. 39. When someone’s kidneys fail, the person needs to have medical treatment with a dialysis machine (unless or until they receive a kidney transplant) or they will die. Sketch a supply and demand diagram, paying attention to the appropriate elasticities, to illustrate that the supply of such dialysis machines will primarily determine the price. 40. Assume that the supply of low-skilled workers is fairly elastic, but the employers’ demand for such workers is fairly inelastic. If the policy goal is to expand employment for low-skilled workers, is it better to focus on policy tools to shift the supply of unskilled labor or on tools to shift the demand for unskilled labor? What if the policy goal is to raise wages for this group? Explain your answers with supply and demand diagrams. 132 Chapter 5 \| Elasticity This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 133 6 \| Consumer Choices Figure 6.1 Investment Choices We generally view higher education as a good investment, if one can afford it, regardless of the state of the economy. (Credit: modification of work by Jason Bache/Flickr Creative Commons) "Eeny, Meeny, Miney, Moe"—Making Choices The 2008–2009 Great Recession touched families around the globe. I
n too many countries, workers found themselves out of a job. In developed countries, unemployment compensation provided a safety net, but families still saw a marked decrease in disposable income and had to make tough spending decisions. Of course, non-essential, discretionary spending was the first to go. Even so, there was one particular category that saw a universal increase in spending world-wide during that time—an 18% uptick in the United States, specifically. You might guess that consumers began eating more meals at home, increasing grocery store spending; however, the Bureau of Labor Statistics’ Consumer Expenditure Survey, which tracks U.S. food spending over time, showed “real total food spending by U.S. households declined five percent between 2006 and 2009.” So, it was not groceries. What product would people around the world demand more of during tough economic times, and more importantly, why? (Find out at chapter’s end.) That question leads us to this chapter’s topic—analyzing how consumers make choices. For most consumers, using “eeny, meeny, miney, moe” is not how they make decisions. Their decision-making processes have been educated far beyond a children’s rhyme. Introduction to Consumer Choices In this chapter, you will learn about: 134 Chapter 6 \| Consumer Choices • Consumption Choices • How Changes in Income and Prices Affect Consumption Choices • How Consumer Choices Might Not Always be Rational Microeconomics seeks to understand the behavior of individual economic agents such as individuals and businesses. Economists believe that we can analyze individuals’ decisions, such as what goods and services to buy, as choices we make within certain budget constraints. Generally, consumers are trying to get the most for their limited budget. In economic terms they are trying to maximize total utility, or satisfaction, given their budget constraint. Everyone has their own personal tastes and preferences. The French say: Chacun à son goût, or “Each to his own taste.” An old Latin saying states, De gustibus non est disputandum or “There’s no disputing about taste.” If people base their decisions on their own tastes and personal preferences, however, then how can economists hope to analyze the choices consumers make? An economic explanation for why people make different choices begins with accepting the proverbial wisdom that tastes are a matter of personal preference. However, economists also believe that the choices people make are influenced by their incomes, by the prices of goods and services they consume, and by factors like where they live. This chapter introduces the economic theory of how consumers make choices about what goods and services to buy with their limited income. The analysis in this chapter will build on the budget constraint that we introduced in the Choice in a World of Scarcity chapter. This chapter will also illustrate how economic theory provides a tool to systematically look at the full range of possible consumption choices to predict how consumption responds to changes in prices or incomes. After reading this chapter, consult the appendix Indifference Curves to learn more about representing utility and choice through indifference curves. 6.1 \| Consumption Choices By the end of this section, you will be able to: • Calculate total utility • Propose decisions that maximize utility • Explain marginal utility and the significance of diminishing marginal utility Information on the consumption choices of Americans is available from the Consumer Expenditure Survey carried out by the U.S. Bureau of Labor Statistics. Table 6.1 shows spending patterns for the average U.S. household. The first row shows income and, after taxes and personal savings are subtracted, it shows that, in 2015, the average U.S. household spent $48,109 on consumption. The table then breaks down consumption into various categories. The average U.S. household spent roughly one-third of its consumption on shelter and other housing expenses, another one-third on food and vehicle expenses, and the rest on a variety of items, as shown. These patterns will vary for specific households by differing levels of family income, by geography, and by preferences. Average Household Income before Taxes Average Annual Expenditures Food at home Food away from home Housing Apparel and services $62,481 $48.109 $3,264 $2,505 $16,557 $1,700 Table 6.1 U.S. Consumption Choices in 2015 (Source: http://www.bls.gov/cex/csxann13.pdf) This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 135 Transportation Healthcare Entertainment Education Personal insurance and pensions All else: alcohol, tobacco, reading, personal care, cash contributions, miscellaneous $7,677 $3,157 $2,504 $1,074 $5,357 $3,356 Table 6.1 U.S. Consumption Choices in 2015 (Source: http://www.bls.gov/cex/csxann13.pdf) Total Utility and Diminishing Marginal Utility To understand how a household will make its choices, economists look at what consumers can afford, as shown in a budget constraint (or budget line), and the total utility or satisfaction derived from those choices. In a budget constraint line, the quantity of one good is on the horizontal axis and the quantity of the other good on the vertical axis. The budget constraint line shows the various combinations of two goods that are affordable given consumer income. Consider José's situation, shown in Figure 6.2. José likes to collect T-shirts and watch movies. In Figure 6.2 we show the quantity of T-shirts on the horizontal axis while we show the quantity of movies on the vertical axis. If José had unlimited income or goods were free, then he could consume without limit. However, José, like all of us, faces a budget constraint. José has a total of $56 to spend. The price of T-shirts is $14 and the price of movies is $7. Notice that the vertical intercept of the budget constraint line is at eight movies and zero T-shirts ($56/$7=8). The horizontal intercept of the budget constraint is four, where José spends of all of his money on T-shirts and no movies ($56/14=4). The slope of the budget constraint line is rise/run or –8/4=–2. The specific choices along the budget constraint line show the combinations of affordable T-shirts and movies. Figure 6.2 A Choice between Consumption Goods José has income of $56. Movies cost $7 and T-shirts cost $14. The points on the budget constraint line show the combinations of affordable movies and T-shirts. José wishes to choose the combination that will provide him with the greatest utility, which is the term economists use to describe a person’s level of satisfaction or happiness with his or her choices. Let’s begin with an assumption, which we will discuss in more detail later, that José can measure his own utility with something called utils. (It is important to note that you cannot make comparisons between the utils of individuals. If one person gets 20 utils from a cup of coffee and another gets 10 utils, this does not mean than the first person gets more enjoyment from the coffee than the other or that they enjoy the coffee twice as much. The reason why is that utils are subjective to an individual. The way one person measures utils is not the same as the way someone else does.) Table 6.2 shows how José’s utility is connected with his T-shirt or movie consumption. The first column 136 Chapter 6 \| Consumer Choices of the table shows the quantity of T-shirts consumed. The second column shows the total utility, or total amount of satisfaction, that José receives from consuming that number of T-shirts. The most common pattern of total utility, in this example, is that consuming additional goods leads to greater total utility, but at a decreasing rate. The third column shows marginal utility, which is the additional utility provided by one additional unit of consumption. This equation for marginal utility is: MU = change in total utility change in quantity Notice that marginal utility diminishes as additional units are consumed, which means that each subsequent unit of a good consumed provides less additional utility. For example, the first T-shirt José picks is his favorite and it gives him an addition of 22 utils. The fourth T-shirt is just something to wear when all his other clothes are in the wash and yields only 18 additional utils. This is an example of the law of diminishing marginal utility, which holds that the additional utility decreases with each unit added. Diminishing marginal utility is another example of the more general law of diminishing returns we learned earlier in the chapter on Choice in a World of Scarcity. The rest of Table 6.2 shows the quantity of movies that José attends, and his total and marginal utility from seeing each movie. Total utility follows the expected pattern: it increases as the number of movies that José watches rises. Marginal utility also follows the expected pattern: each additional movie brings a smaller gain in utility than the previous one. The first movie José attends is the one he wanted to see the most, and thus provides him with the highest level of utility or satisfaction. The fifth movie he attends is just to kill time. Notice that total utility is also the sum of the marginal utilities. Read the next Work It Out feature for instructions on how to calculate total utility. T-Shirts (Quantity) Total Utility Marginal Utility Movies (Quantity) Total Utility Marginal Utility 1 2 3 4 5 6 7 8 22 43 63 81 97 111 123 133 22 21 20 18 16 14 12 10 1 2 3 4 5 6 7 8 16 31 45 58 70 81 91 100 16 15 14 13 12 11 10 9 Table 6.2 Total and Marginal Utility Table 6.3 looks at each point on the budget constraint in Figure 6.2, and adds up José’s total utility for five possible combinations of T-shirts and movies. Point T-Shirts Movies Total Utility 81 + 0 = 81 63 + 31 = 94 43 + 58 = 101 22 + 81 = 103 Table 6.3 Finding the Choice with the Highest Utility This OpenStax book is available for free at h
ttp://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 137 Point T-Shirts Movies Total Utility T 0 8 0 + 100 = 100 Table 6.3 Finding the Choice with the Highest Utility Calculating Total Utility Let’s look at how José makes his decision in more detail. Step 1. Observe that, at point Q (for example), José consumes three T-shirts and two movies. Step 2. Look at Table 6.2. You can see from the fourth row/second column that three T-shirts are worth 63 utils. Similarly, the second row/fifth column shows that two movies are worth 31 utils. Step 3. From this information, you can calculate that point Q has a total utility of 94 (63 + 31). Step 4. You can repeat the same calculations for each point on Table 6.3, in which the total utility numbers are shown in the last column. For José, the highest total utility for all possible combinations of goods occurs at point S, with a total utility of 103 from consuming one T-shirt and six movies. Choosing with Marginal Utility Most people approach their utility-maximizing combination of choices in a step-by-step way. This approach is based on looking at the tradeoffs, measured in terms of marginal utility, of consuming less of one good and more of another. For example, say that José starts off thinking about spending all his money on T-shirts and choosing point P, which corresponds to four T-shirts and no movies, as Figure 6.2 illustrates. José chooses this starting point randomly as he has to start somewhere. Then he considers giving up the last T-shirt, the one that provides him the least marginal utility, and using the money he saves to buy two movies instead. Table 6.4 tracks the step-by-step series of decisions José needs to make (Key: T-shirts are $14, movies are $7, and income is $56). The following Work It Out feature explains how marginal utility can effect decision making. Try Choice 1: P Choice 2: Q Which Has 4 Tshirts and 0 movies 3 Tshirts and 2 movies Total Utility Marginal Gain and Loss of Utility, Compared with Previous Choice Conclusion 81 from 4 T-shirts + 0 from 0 movies = 81 – – 63 from 3 T-shirts + 31 from 0 movies = 94 Loss of 18 from 1 less T-shirt, but gain of 31 from 2 more movies, for a net utility gain of 13 Q is preferred over P Table 6.4 A Step-by-Step Approach to Maximizing Utility 138 Chapter 6 \| Consumer Choices Try Choice 3: R Choice 4: S Choice 5: T Which Has 2 Tshirts and 4 movies 1 T-shirt and 6 movies 0 Tshirts and 8 movies Total Utility Marginal Gain and Loss of Utility, Compared with Previous Choice Conclusion 43 from 2 T-shirts + 58 from 4 movies = 101 Loss of 20 from 1 less T-shirt, but gain of 27 from two more movies for a net utility gain of 7 R is preferred over Q 22 from 1 T-shirt + 81 from 6 movies = 103 Loss of 21 from 1 less T-shirt, but gain of 23 from two more movies, for a net utility gain of 2 0 from 0 T-shirts + 100 from 8 movies = 100 Loss of 22 from 1 less T-shirt, but gain of 19 from two more movies, for a net utility loss of 3 S is preferred over R S is preferred over T Table 6.4 A Step-by-Step Approach to Maximizing Utility Decision Making by Comparing Marginal Utility José could use the following thought process (if he thought in utils) to make his decision regarding how many T-shirts and movies to purchase: Step 1. From Table 6.2, José can see that the marginal utility of the fourth T-shirt is 18. If José gives up the fourth T-shirt, then he loses 18 utils. Step 2. Giving up the fourth T-shirt, however, frees up $14 (the price of a T-shirt), allowing José to buy the first two movies (at $7 each). Step 3. José knows that the marginal utility of the first movie is 16 and the marginal utility of the second movie is 15. Thus, if José moves from point P to point Q, he gives up 18 utils (from the T-shirt), but gains 31 utils (from the movies). Step 4. Gaining 31 utils and losing 18 utils is a net gain of 13. This is just another way of saying that the total utility at Q (94 according to the last column in Table 6.3) is 13 more than the total utility at P (81). Step 5. Thus, for José, it makes sense to give up the fourth T-shirt in order to buy two movies. José clearly prefers point Q to point P. Now repeat this step-by-step process of decision making with marginal utilities. José thinks about giving up the third T-shirt and surrendering a marginal utility of 20, in exchange for purchasing two more movies that promise a combined marginal utility of 27. José prefers point R to point Q. What if José thinks about going beyond R to point S? Giving up the second T-shirt means a marginal utility loss of 21, and the marginal utility gain from the fifth and sixth movies would combine to make a marginal utility gain of 23, so José prefers point S to R. However, if José seeks to go beyond point S to point T, he finds that the loss of marginal utility from giving up the first T-shirt is 22, while the marginal utility gain from the last two movies is only a total of 19. If José were to choose point T, his utility would fall to 100. Through these stages of thinking about marginal tradeoffs, José again concludes that S, with one T-shirt and six movies, is the choice that will provide him with the highest level of total utility. This step-by-step approach will reach the same conclusion regardless of José’s starting point. We can develop a more systematic way of using this approach by focusing on satisfaction per dollar. If an item costing $5 yields 10 utils, then it’s worth 2 utils per dollar spent. Marginal utility per dollar is the amount of additional utility José receives divided by the product's price. Table 6.5 shows the marginal utility per dollar for José's T shirts This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 139 and movies. marginal utility per dollar = marginal utility price If José wants to maximize the utility he gets from his limited budget, he will always purchase the item with the greatest marginal utility per dollar of expenditure (assuming he can afford it with his remaining budget). José starts with no purchases. If he purchases a T-shirt, the marginal utility per dollar spent will be 1.6. If he purchases a movie, the marginal utility per dollar spent will be 2.3. Therefore, José’s first purchase will be the movie. Why? Because it gives him the highest marginal utility per dollar and is affordable. Next, José will purchase another movie. Why? Because the marginal utility of the next movie (2.14) is greater than the marginal utility of the next T-shirt (1.6). Note that when José has no T- shirts, the next one is the first one. José will continue to purchase the next good with the highest marginal utility per dollar until he exhausts his budget. He will continue purchasing movies because they give him a greater "bang for the buck" until the sixth movie which gives the same marginal utility per dollar as the first T-shirt purchase. José has just enough budget to purchase both. So in total, José will purchase six movies and one T-shirt. Quantity of TShirts Total Utility Marginal Utility 1 2 3 4 5 6 7 22 43 63 81 97 111 123 22 21 20 18 16 14 12 Marginal Utility per Dollar 22/$14=1.6 21/$14=1.5 20/$14=1.4 18/$14=1.3 16/$14=1.1 14/$14=1 12/$14=1.2 Quantity of Movies Total Utility Marginal Utility 1 2 3 4 5 6 7 16 31 45 58 70 81 91 16 15 14 13 12 11 10 Marginal Utility per Dollar 16/$7=2.3 15/$7=2.14 14/$7=2 13/$7=1.9 12/$7=1.7 11/$7=1.6 10/$7=1.4 Table 6.5 Marginal Utility per Dollar A Rule for Maximizing Utility This process of decision making suggests a rule to follow when maximizing utility. Since the price of T-shirts is twice as high as the price of movies, to maximize utility the last T-shirt that José chose needs to provide exactly twice the marginal utility (MU) of the last movie. If the last T-shirt provides less than twice the marginal utility of the last movie, then the T-shirt is providing less “bang for the buck” (i.e., marginal utility per dollar spent) than José would receive from spending the same money on movies. If this is so, José should trade the T-shirt for more movies to increase his total utility. If the last T-shirt provides more than twice the marginal utility of the last movie, then the T-shirt is providing more “bang for the buck” or marginal utility per dollar, than if the money were spent on movies. As a result, José should buy more T-shirts. Notice that at José’s optimal choice of point S, the marginal utility from the first T-shirt, of 22 is exactly twice the marginal utility of the sixth movie, which is 11. At this choice, the marginal utility per dollar is the same for both goods. This is a tell-tale signal that José has found the point with highest total utility. We can write this argument as a general rule: If you always choose the item with the greatest marginal utility per dollar spent, when your budget is exhausted, the utility maximizing choice should occur where the marginal utility per dollar spent is the same for both goods. A sensible economizer will pay twice as much for something only if, in the marginal comparison, the item confers MU1 P1 = MU2 P2 140 Chapter 6 \| Consumer Choices twice as much utility. Notice that the formula for the table above is: = 11 22 $14 $7 1.6 = 1.6 The following Work It Out feature provides step by step guidance for this concept of utility-maximizing choices. Maximizing Utility The general rule, MU1 P1 = MU2 P2 , means that the last dollar spent on each good provides exactly the same marginal utility. This is the case at point S. So: Step 1. If we traded a dollar more of movies for a dollar more of T-shirts, the marginal utility gained from Tshirts would exactly offset the marginal utility lost from fewer movies. In other words, the net gain would be zero. Step 2. Products, however, usually cost more than a dollar, so we cannot trade a dollar’s worth of movies. The best we can do is trade two movies for another T-shirt, since in this example T-shirts cost twice what a movie does. Step
3. If we trade two movies for one T-shirt, we would end up at point R (two T-shirts and four movies). Step 4. Choice 4 in Table 6.4 shows that if we move to point R, we would gain 21 utils from one more T-shirt, but lose 23 utils from two fewer movies, so we would end up with less total utility at point R. In short, the general rule shows us the utility-maximizing choice, which is called the consumer equilibrium. There is another equivalent way to think about this. We can also express the general rule as the ratio of the prices of the two goods should be equal to the ratio of the marginal utilities. When we divide the price of good 1 by the price of good 2, at the utility-maximizing point this will equal the marginal utility of good 1 divided by the marginal utility of good 2. P1 P2 = MU1 MU2 Along the budget constraint, the total price of the two goods remains the same, so the ratio of the prices does not change. However, the marginal utility of the two goods changes with the quantities consumed. At the optimal choice of one T-shirt and six movies, point S, the ratio of marginal utility to price for T-shirts (22:14) matches the ratio of marginal utility to price for movies (of 11:7). Measuring Utility with Numbers This discussion of utility began with an assumption that it is possible to place numerical values on utility, an assumption that may seem questionable. You can buy a thermometer for measuring temperature at the hardware store, but what store sells an “utilimometer” for measuring utility? While measuring utility with numbers is a convenient assumption to clarify the explanation, the key assumption is not that an outside party can measure utility but only that individuals can decide which of two alternatives they prefer. To understand this point, think back to the step-by-step process of finding the choice with highest total utility by comparing the marginal utility you gain and lose from different choices along the budget constraint. As José compares each choice along his budget constraint to the previous choice, what matters is not the specific numbers that he places on his utility—or whether he uses any numbers at all—but only that he personally can identify which choices he prefers. In this way, the step-by-step process of choosing the highest level of utility resembles rather closely how many people make consumption decisions. We think about what will make us the happiest. We think about what things cost. We think about buying a little more of one item and giving up a little of something else. We choose what provides us This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 141 with the greatest level of satisfaction. The vocabulary of comparing the points along a budget constraint and total and marginal utility is just a set of tools for discussing this everyday process in a clear and specific manner. It is welcome news that specific utility numbers are not central to the argument, since a good utilimometer is hard to find. Do not worry—while we cannot measure utils, by the end of the next module, we will have transformed our analysis into something we can measure—demand. 6.2 \| How Changes in Income and Prices Affect Consumption Choices By the end of this section, you will be able to: • Explain how income, prices, and preferences affect consumer choices • Contrast the substitution effect and the income effect • Utilize concepts of demand to analyze consumer choices • Apply utility-maximizing choices to governments and businesses Just as we can use utility and marginal utility to discuss making consumer choices along a budget constraint, we can also use these ideas to think about how consumer choices change when the budget constraint shifts in response to changes in income or price. Because we can use the budget constraint framework to analyze how quantities demanded change because of price movements, the budget constraint model can illustrate the underlying logic behind demand curves. How Changes in Income Affect Consumer Choices Let’s begin with a concrete example illustrating how changes in income level affect consumer choices. Figure 6.3 shows a budget constraint that represents Kimberly’s choice between concert tickets at $50 each and getting away overnight to a bed-and-breakfast for $200 per night. Kimberly has $1,000 per year to spend between these two choices. After thinking about her total utility and marginal utility and applying the decision rule that the ratio of the marginal utilities to the prices should be equal between the two products, Kimberly chooses point M, with eight concerts and three overnight getaways as her utility-maximizing choice. Figure 6.3 How a Change in Income Affects Consumption Choices The utility-maximizing choice on the original budget constraint is M. The dashed horizontal and vertical lines extending through point M allow you to see at a glance whether the quantity consumed of goods on the new budget constraint is higher or lower than on the original budget constraint. On the new budget constraint, Kimberly will make a choice like N if both goods are normal goods. If overnight stays is an inferior good, Kimberly will make a choice like P. If concert tickets are an inferior good, Kimberly will make a choice like Q. Now, assume that the income Kimberly has to spend on these two items rises to $2,000 per year, causing her budget constraint to shift out to the right. How does this rise in income alter her utility-maximizing choice? Kimberly will 142 Chapter 6 \| Consumer Choices again consider the utility and marginal utility that she receives from concert tickets and overnight getaways and seek her utility-maximizing choice on the new budget line, but how will her new choice relate to her original choice? We can replace the possible choices along the new budget constraint into three groups, which the dashed horizontal and vertical lines that pass through the original choice M in the figure divide. All choices on the upper left of the new budget constraint that are to the left of the vertical dashed line, like choice P with two overnight stays and 32 concert tickets, involve less of the good on the horizontal axis but much more of the good on the vertical axis. All choices to the right of the vertical dashed line and above the horizontal dashed line—like choice N with five overnight getaways and 20 concert tickets—have more consumption of both goods. Finally, all choices that are to the right of the vertical dashed line but below the horizontal dashed line, like choice Q with four concerts and nine overnight getaways, involve less of the good on the vertical axis but much more of the good on the horizontal axis. All of these choices are theoretically possible, depending on Kimberly’s personal preferences as expressed through the total and marginal utility she would receive from consuming these two goods. When income rises, the most common reaction is to purchase more of both goods, like choice N, which is to the upper right relative to Kimberly’s original choice M, although exactly how much more of each good will vary according to personal taste. Conversely, when income falls, the most typical reaction is to purchase less of both goods. As we defined in the chapter on Demand and Supply and again in the chapter on Elasticity, we call goods and services normal goods when a rise in income leads to a rise in the quantity consumed of that good and a fall in income leads to a fall in quantity consumed. However, depending on Kimberly’s preferences, a rise in income could cause consumption of one good to increase while consumption of the other good declines. A choice like P means that a rise in income caused her quantity consumed of overnight stays to decline, while a choice like Q would mean that a rise in income caused her quantity of concerts to decline. Goods where demand declines as income rises (or conversely, where the demand rises as income falls) are called “inferior goods.” An inferior good occurs when people trim back on a good as income rises, because they can now afford the more expensive choices that they prefer. For example, a higher-income household might eat fewer hamburgers or be less likely to buy a used car, and instead eat more steak and buy a new car. How Price Changes Affect Consumer Choices For analyzing the possible effect of a change in price on consumption, let’s again use a concrete example. Figure 6.4 represents Sergei's consumer choice, who chooses between purchasing baseball bats and cameras. A price increase for baseball bats would have no effect on the ability to purchase cameras, but it would reduce the number of bats Sergei could afford to buy. Thus a price increase for baseball bats, the good on the horizontal axis, causes the budget constraint to rotate inward, as if on a hinge, from the vertical axis. As in the previous section, the point labeled M represents the originally preferred point on the original budget constraint, which Sergei has chosen after contemplating his total utility and marginal utility and the tradeoffs involved along the budget constraint. In this example, the units along the horizontal and vertical axes are not numbered, so the discussion must focus on whether Sergei will consume more or less of certain goods, not on numerical amounts. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 143 Figure 6.4 How a Change in Price Affects Consumption Choices The original utility-maximizing choice is M. When the price rises, the budget constraint rotates clockwise. The dashed lines make it possible to see at a glance whether the new consumption choice involves less of both goods, or less of one good and more of the other. The new possible choices would be fewer baseball bats and more cameras, like point H, or less of both goods, as at point J. Choice K would mean that the higher price of bats led to exactly the
same quantity of bat consumption, but fewer cameras. Theoretically possible, but unlikely in the real world, we rule out choices like L because they would mean that a higher price for baseball bats means a greater consumption of baseball bats. After the price increase, Sergei will make a choice along the new budget constraint. Again, we can divide his choices into three segments by the dashed vertical and horizontal lines. In the upper left portion of the new budget constraint, at a choice like H, Sergei consumes more cameras and fewer bats. In the central portion of the new budget constraint, at a choice like J, he consumes less of both goods. At the right-hand end, at a choice like L, he consumes more bats but fewer cameras. The typical response to higher prices is that a person chooses to consume less of the product with the higher price. This occurs for two reasons, and both effects can occur simultaneously. The substitution effect occurs when a price changes and consumers have an incentive to consume less of the good with a relatively higher price and more of the good with a relatively lower price. The income effect is that a higher price means, in effect, the buying power of income has been reduced (even though actual income has not changed), which leads to buying less of the good (when the good is normal). In this example, the higher price for baseball bats would cause Sergei to buy fewer bats for both reasons. Exactly how much will a higher price for bats cause Sergei's bat consumption to fall? Figure 6.4 suggests a range of possibilities. Sergei might react to a higher price for baseball bats by purchasing the same quantity of bats, but cutting his camera consumption. This choice is the point K on the new budget constraint, straight below the original choice M. Alternatively, Sergei might react by dramatically reducing his bat purchases and instead buy more cameras. The key is that it would be imprudent to assume that a change in baseball bats will only or primarily affect the good's price whose price is changed, while the quantity consumed of other goods remains the same. Since Sergei purchases all his products out of the same budget, a change in the price of one good can also have a range of effects, either positive or negative, on the quantity consumed of other goods. In short, a higher price typically causes reduced consumption of the good in question, but it can affect the consumption of other goods as well. Read this article (http://openstaxcollege.org/l/vending) about machines. the potential of variable prices in vending 144 Chapter 6 \| Consumer Choices The Foundations of Demand Curves Changes in the price of a good lead the budget constraint to rotate. A rotation in the budget constraint means that when individuals are seeking their highest utility, the quantity that is demanded of that good will change. In this way, the logical foundations of demand curves—which show a connection between prices and quantity demanded—are based on the underlying idea of individuals seeking utility. Figure 6.5 (a) shows a budget constraint with a choice between housing and “everything else.” (Putting “everything else” on the vertical axis can be a useful approach in some cases, especially when the focus of the analysis is on one particular good.) We label the preferred choice on the original budget constraint that provides the highest possible utility M0. The other three budget constraints represent successively higher prices for housing of P1, P2, and P3. As the budget constraint rotates in, and in, and in again, we label the utility-maximizing choices M1, M2, and M3, and the quantity demanded of housing falls from Q0 to Q1 to Q2 to Q3. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 145 Figure 6.5 The Foundations of a Demand Curve: An Example of Housing (a) As the price increases from P0 to P1 to P2 to P3, the budget constraint on the upper part of the diagram rotates clockwise. The utility-maximizing choice changes from M0 to M1 to M2 to M3. As a result, the quantity demanded of housing shifts from Q0 to Q1 to Q2 to Q3, ceteris paribus. (b) The demand curve graphs each combination of the price of housing and the quantity of housing demanded, ceteris paribus. The quantities of housing are the same at the points on both (a) and (b). Thus, the original price of housing (P0) and the original quantity of housing (Q0) appear on the demand curve as point E0. The higher price of housing (P1) and the corresponding lower quantity demanded of housing (Q1) appear on the demand curve as point E1. Thus, as the price of housing rises, the budget constraint rotates clockwise and the quantity consumed of housing falls, ceteris paribus (meaning, with all other things being the same). We graph this relationship—the price of housing rising from P0 to P1 to P2 to P3, while the quantity of housing demanded falls from Q0 to Q1 to Q2 to Q3—on the demand curve in Figure 6.5 (b). The vertical dashed lines stretching between the top and bottom of Figure 6.5 show that the quantity of housing demanded at each point is the same in both (a) and (b). We ultimately determine the shape of a demand curve by the underlying choices about maximizing utility subject to a budget constraint. While economists may not be able to measure “utils,” they can certainly measure price and quantity demanded. Applications in Government and Business The budget constraint framework for making utility-maximizing choices offers a reminder that people can react to a change in price or income in a range of different ways. For example, in the winter months of 2005, costs for heating homes increased significantly in many parts of the country as prices for natural gas and electricity soared, due in large part to the disruption caused by Hurricanes Katrina and Rita. Some people reacted by reducing the quantity demanded of energy; for example, by turning down the thermostats in their homes by a few degrees and wearing a heavier sweater inside. Even so, many home heating bills rose, so people adjusted their consumption in other ways, too. As you learned in the chapter on Elasticity, the short run demand for home heating is generally inelastic. Each 146 Chapter 6 \| Consumer Choices household cut back on what it valued least on the margin. For some it might have been some dinners out, or a vacation, or postponing buying a new refrigerator or a new car. Sharply higher energy prices can have effects beyond the energy market, leading to a widespread reduction in purchasing throughout the rest of the economy. A similar issue arises when the government imposes taxes on certain products, such as on gasoline, cigarettes, and alcohol. Say that a tax on alcohol leads to a higher price at the liquor store. The higher price of alcohol causes the budget constraint to pivot left, and alcoholic beverage consumption is likely to decrease. However, people may also react to the higher price of alcoholic beverages by cutting back on other purchases. For example, they might cut back on snacks at restaurants like chicken wings and nachos. It would be unwise to assume that the liquor industry is the only one affected by the tax on alcoholic beverages. Read the next Clear It Up to learn about how who controls the household income influences buying decisions. The Unifying Power of the Utility-Maximizing Budget Set Framework An interaction between prices, budget constraints, and personal preferences determine household choices. The flexible and powerful terminology of utility-maximizing gives economists a vocabulary for bringing these elements together. Not even economists believe that people walk around mumbling about their marginal utilities before they walk into a shopping mall, accept a job, or make a deposit in a savings account. However, economists do believe that individuals seek their own satisfaction or utility and that people often decide to try a little less of one thing and a little more of another. If we accept these assumptions, then the idea of utility-maximizing households facing budget constraints becomes highly plausible. Does who controls household income make a difference? In the mid-1970s, the United Kingdom made an interesting policy change in its “child allowance” policy. This program provides a fixed amount of money per child to every family, regardless of family income. Traditionally, the child allowance had been distributed to families by withholding less in taxes from the paycheck of the family wage earner—typically the father in this time period. The new policy instead provided the child allowance as a cash payment to the mother. As a result of this change, households have the same level of income and face the same prices in the market, but the money is more likely to be in the mother's purse than in the father's wallet. Should this change in policy alter household consumption patterns? Basic models of consumption decisions, of the sort that we examined in this chapter, assume that it does not matter whether the mother or the father receives the money, because both parents seek to maximize the family's utility as a whole. In effect, this model assumes that everyone in the family has the same preferences. In reality, the share of that the father or mother controls does affect what the household consumes. When the mother controls a larger share of family income a number of studies, in the United Kingdom and in a wide variety of other countries, have found that the family tends to spend more on restaurant meals, child care, and women’s clothing, and less on alcohol and tobacco. As the mother controls a larger share of household resources, children’s health improves, too. These findings suggest that when providing assistance to poor families, in high-income countries and low-income countries alike, the monetary amount of assistance is not all that matters: it also matters which family member actually receives th
e money. The budget constraint framework serves as a constant reminder to think about the full range of effects that can arise from changes in income or price, not just effects on the one product that might seem most immediately affected. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 147 6.3 \| Behavioral Economics: An Alternative Framework for Consumer Choice By the end of this section, you will be able to: • Evaluate the reasons for making intertemporal choices • • Analyze why people in America tend to save such a small percentage of their income Interpret an intertemporal budget constraint As we know, people sometimes make decisions that seem “irrational” and not in their own best interest. People’s decisions can seem inconsistent from one day to the next and they even deliberately ignore ways to save money or time. The traditional economic models assume rationality, which means that people take all available information and make consistent and informed decisions that are in their best interest. (In fact, economics professors often delight in pointing out so-called “irrational behavior” each semester to their new students, and present economics as a way to become more rational.) However, a new group of economists, known as behavioral economists, argue that the traditional method omits something important: people’s state of mind. For example, one can think differently about money if one is feeling revenge, optimism, or loss. These are not necessarily irrational states of mind, but part of a range of emotions that can affect anyone on a given day. In addition, actions under these conditions are predictable, if one better understands the underlying environment. Behavioral economics seeks to enrich our understanding of decision-making by integrating the insights of psychology into economics. It does this by investigating how given dollar amounts can mean different things to individuals depending on the situation. This can lead to decisions that appear outwardly inconsistent, or irrational, to the outside observer. The way the mind works, according to this view, may seem inconsistent to traditional economists but is actually far more complex than an unemotional cost-benefit adding machine. For example, a traditional economist would say that if you lost a $10 bill today, and also received an extra $10 in your paycheck, you should feel perfectly neutral. After all, –$10 + $10 = $0. You are the same financially as you were before. However, behavioral economists have conducted research that shows many people will feel some negative emotion, such as anger or frustration, after those two things happen. We tend to focus more on the loss than the gain. We call this loss aversion, where a $1 loss pains us 2.25 times more than a $1 gain helps us, according to the economists Daniel Kahneman and Amos Tversky in a famous 1979 article in the journal Econometrica. This insight has implications for investing, as people tend to “overplay” the stock market by reacting more to losses than to gains. This behavior looks irrational to traditional economists, but is consistent once we understand better how the mind works, these economists argue. Traditional economists also assume human beings have complete self control, but, for instance, people will buy cigarettes by the pack instead of the carton even though the carton saves them money, to keep usage down. They purchase locks for their refrigerators and overpay on taxes to force themselves to save. In other words, we protect ourselves from our worst temptations but pay a price to do so. One way behavioral economists are responding to this is by establishing ways for people to keep themselves free of these temptations. This includes what we call “nudges” toward more rational behavior rather than mandatory regulations from government. For example, up to 20 percent of new employees do not enroll in retirement savings plans immediately, because of procrastination or feeling overwhelmed by the different choices. Some companies are now moving to a new system, where employees are automatically enrolled unless they “opt out.” Almost no-one opts out in this program and employees begin saving at the early years, which are most critical for retirement. Another area that seems illogical is the idea of mental accounting, or putting dollars in different mental categories where they take different values. Economists typically consider dollars to be fungible, or having equal value to the individual, regardless of the situation. You might, for instance, think of the $25 you found in the street differently from the $25 you earned from three hours working in a fast food restaurant. You might treat the street money as “mad money” with little rational regard to getting the best value. This is in one sense strange, since it is still equivalent to three hours of hard work in the restaurant. Yet the “easy come-easy go” mentality replaces the rational economizer because of the situation, or context, in which you attained the money. 148 Chapter 6 \| Consumer Choices In another example of mental accounting that seems inconsistent to a traditional economist, a person could carry a credit card debt of $1,000 that has a 15% yearly interest cost, and simultaneously have a $2,000 savings account that pays only 2% per year. That means she pays $150 a year to the credit card company, while collecting only $40 annually in bank interest, so she loses $130 a year. That doesn’t seem wise. The “rational” decision would be to pay off the debt, since a $1,000 savings account with $0 in debt is the equivalent net worth, and she would now net $20 per year. Curiously, it is not uncommon for people to ignore this advice, since they will treat a loss to their savings account as higher than the benefit of paying off their credit card. They do not treat the dollars as fungible so it looks irrational to traditional economists. Which view is right, the behavioral economists’ or the traditional view? Both have their advantages, but behavioral economists have at least identified trying to describe and explain behavior that economists have historically dismissed as irrational. If most of us are engaged in some “irrational behavior,” perhaps there are deeper underlying reasons for this behavior in the first place. "Eeny, Meeny, Miney, Moe"—Making Choices In what category did consumers worldwide increase their spending during the Great Recession? Higher education. According to the United Nations Educational, Scientific, and Cultural Organization (UNESCO), enrollment in colleges and universities rose one-third in China and almost two-thirds in Saudi Arabia, nearly doubled in Pakistan, tripled in Uganda, and surged by three million—18 percent—in the United States. Why were consumers willing to spend on education during lean times? Both individuals and countries view higher education as the way to prosperity. Many feel that increased earnings are a significant benefit of attending college. U.S. Bureau of Labor Statistics data from May 2012 supports this view, as Figure 6.6 shows. They show a positive correlation between earnings and education. The data also indicate that unemployment rates fall with higher levels of education and training. Why spend the money to go to college during recession? Because if you are unemployed (or underemployed, working fewer hours than you would like), the opportunity cost of your time is low. If you’re unemployed, you don’t have to give up work hours and income by going to college. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 149 Figure 6.6 The Impact of Education on Earnings and Unemployment Rates, 2012 Those with the highest degrees in 2012 had substantially lower unemployment rates; whereas, those with the least formal education suffered from the highest unemployment rates. The national median average weekly income was $815, and the nation unemployment average in 2012 was 6.8%. (Source: U.S. Bureau of Labor Statistics, May 22, 2013) 150 Chapter 6 \| Consumer Choices KEY TERMS behavioral economics a branch of economics that seeks to enrich the understanding of decision-making by integrating the insights of psychology and by investigating how given dollar amounts can mean different things to individuals depending on the situation budget constraint (or budget line) consumer’s limited income shows the possible combinations of two goods that are affordable given a consumer equilibrium point on the budget line where the consumer gets the most satisfaction; this occurs when the ratio of the prices of goods is equal to the ratio of the marginal utilities. diminishing marginal utility the common pattern that each marginal unit of a good consumed provides less of an addition to utility than the previous unit fungible the idea that units of a good, such as dollars, ounces of gold, or barrels of oil are capable of mutual substitution with each other and carry equal value to the individual income effect a higher price means that, in effect, the buying power of income has been reduced, even though actual income has not changed; always happens simultaneously with a substitution effect marginal utility the additional utility provided by one additional unit of consumption marginal utility per dollar the additional satisfaction gained from purchasing a good given the price of the product; MU/Price substitution effect when a price changes, consumers have an incentive to consume less of the good with a relatively higher price and more of the good with a relatively lower price; always happens simultaneously with an income effect total utility satisfaction derived from consumer choices KEY CONCEPTS AND SUMMARY 6.1 Consumption Choices Economic analysis of household behavior is based on the assumption that people seek the highest level of utility or satisfaction. Individuals are the only judge of their own
utility. In general, greater consumption of a good brings higher total utility. However, the additional utility people receive from each unit of greater consumption tends to decline in a pattern of diminishing marginal utility. We can find the utility-maximizing choice on a consumption budget constraint in several ways. You can add up total utility of each choice on the budget line and choose the highest total. You can select a starting point at random and compare the marginal utility gains and losses of moving to neighboring points—and thus eventually seek out the preferred choice. Alternatively, you can compare the ratio of the marginal utility to price of good 1 with the marginal utility to price of good 2 and apply the rule that at the optimal choice, the two ratios should be equal: MU1 P1 = MU2 P2 6.2 How Changes in Income and Prices Affect Consumption Choices The budget constraint framework suggest that when income or price changes, a range of responses are possible. When income rises, households will demand a higher quantity of normal goods, but a lower quantity of inferior goods. When the price of a good rises, households will typically demand less of that good—but whether they will demand a much lower quantity or only a slightly lower quantity will depend on personal preferences. Also, a higher price for one good can lead to more or less demand of the other good. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 151 6.3 Behavioral Economics: An Alternative Framework for Consumer Choice People regularly make decisions that seem less than rational, decisions that contradict traditional consumer theory. This is because traditional theory ignores people’s state of mind or feelings, which can influence behavior. For example, people tend to value a dollar lost more than a dollar gained, even though the amounts are the same. Similarly, many people over withhold on their taxes, essentially giving the government a free loan until they file their tax returns, so that they are more likely to get money back than have to pay money on their taxes. SELF-CHECK QUESTIONS 1. Jeremy is deeply in love with Jasmine. Jasmine lives where cell phone coverage is poor, so he can either call her on the land-line phone for five cents per minute or he can drive to see her, at a round-trip cost of $2 in gasoline money. He has a total of $10 per week to spend on staying in touch. To make his preferred choice, Jeremy uses a handy utilimometer that measures his total utility from personal visits and from phone minutes. Using the values in Table 6.6, figure out the points on Jeremy’s consumption choice budget constraint (it may be helpful to do a sketch) and identify his utility-maximizing point. Round Trips Total Utility Phone Minutes Total Utility 10 Table 6.6 0 80 150 210 260 300 330 200 180 160 140 0 20 40 60 80 100 120 140 160 180 200 0 200 380 540 680 800 900 980 1040 1080 1100 2. Take Jeremy’s total utility information in Exercise 6.1, and use the marginal utility approach to confirm the choice of phone minutes and round trips that maximize Jeremy’s utility. 3. Explain all the reasons why a decrease in a product's price would lead to an increase in purchases. 4. As a college student you work at a part-time job, but your parents also send you a monthly “allowance.” Suppose one month your parents forgot to send the check. Show graphically how your budget constraint is affected. Assuming you only buy normal goods, what would happen to your purchases of goods? REVIEW QUESTIONS 5. Who determines how much utility an individual will receive from consuming a good? 6. Would you expect total utility to rise or fall with additional consumption of a good? Why? 152 Chapter 6 \| Consumer Choices 11. As a general rule, is it safe to assume that a change in the price of a good will always have its most significant impact on the quantity demanded of that good, rather than on the quantity demanded of other goods? Explain. 12. Why does a change in income cause a parallel shift in the budget constraint? 15. Income effects depend on the income elasticity of demand for each good that you buy. If one of the goods you buy has a negative income elasticity, that is, it is an inferior good, what must be true of the income elasticity of the other good you buy? 7. Would you expect marginal utility to rise or fall with additional consumption of a good? Why? Is it possible for total utility to increase while 8. marginal utility diminishes? Explain. 9. If people do not have a complete mental picture of total utility for every level of consumption, how can they find their utility-maximizing consumption choice? 10. What is the rule relating the ratio of marginal utility to prices of two goods at the optimal choice? Explain why, if this rule does not hold, the choice cannot be utility-maximizing. CRITICAL THINKING QUESTIONS 13. Think back to a purchase that you made recently. How would you describe your thinking before you made that purchase? 14. The rules of politics are not always the same as the rules of economics. In discussions of setting budgets for government agencies, there is a strategy called “closing the Washington Monument.” When an agency faces the unwelcome prospect of a budget cut, it may decide to close a high-visibility attraction enjoyed by many people (like the Washington Monument). Explain in terms of diminishing marginal utility why the Washington Monument strategy is so misleading. Hint: If you are really trying to make the best of a budget cut, should you cut the items in your budget with the highest marginal utility or the Washington Monument strategy cut the items with the highest marginal utility or the lowest marginal utility? the lowest marginal utility? Does This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 6 \| Consumer Choices 153 17. If a 10% decrease in the price of one product that you buy causes an 8% increase in quantity demanded of that product, will another 10% decrease in the price cause another 8% increase (no more and no less) in quantity demanded? PROBLEMS 16. Praxilla, who lived in ancient Greece, derives utility from reading poems and from eating cucumbers. Praxilla gets 30 units of marginal utility from her first poem, 27 units of marginal utility from her second poem, 24 units of marginal utility from her third poem, and so on, with marginal utility declining by three units for each additional poem. Praxilla gets six units of marginal utility for each of her first three cucumbers consumed, five units of marginal utility for each of her next three cucumbers consumed, four units of marginal utility for each of the following three cucumbers consumed, and so on, with marginal utility declining by one for every three cucumbers consumed. A poem costs three bronze coins but a cucumber costs only one bronze coin. Praxilla has 18 bronze coins. Sketch Praxilla’s budget set between poems and cucumbers, placing poems on the vertical axis and cucumbers on the horizontal axis. Start off with the choice of zero poems and 18 cucumbers, and calculate the changes in marginal utility of moving along the budget line to the next choice of one poem and 15 cucumbers. Using this step-bystep process based on marginal utility, create a table and identify Praxilla’s utility-maximizing choice. Compare the marginal utility of the two goods and the relative prices at the optimal choice to see if the expected relationship holds. Hint: Label the table columns: 1) Choice, 2) Marginal Gain from More Poems, 3) Marginal Loss from Fewer Cucumbers, 4) Overall Gain or Loss, 5) Is the previous choice optimal? Label the table rows: 1) 0 Poems and 18 Cucumbers, 2) 1 Poem and 15 Cucumbers, 3) 2 Poems and 12 Cucumbers, 4) 3 Poems and 9 Cucumbers, 5) 4 Poems and 6 Cucumbers, 6) 5 Poems and 3 Cucumbers, 7) 6 Poems and 0 Cucumbers. 154 Chapter 6 \| Consumer Choices This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 7 \| Production, Costs, and Industry Structure 155 7 \| Production, Costs, and Industry Structure Figure 7.1 Amazon is an American international electronic commerce company that sells books, among many other things, shipping them directly to the consumer. Until recently there were no brick and mortar Amazon stores. (Credit: modification of work by William Christiansen/Flickr Creative Commons) Amazon In less than two decades, Amazon.com has transformed the way consumers sell, buy, and even read. Prior to Amazon, independent bookstores with limited inventories in small retail locations primarily sold books. There were exceptions, of course. Borders and Barnes & Noble offered larger stores in urban areas. In the last decade, however, independent bookstores have mostly disappeared, Borders has gone out of business, and Barnes & Noble is struggling. Online delivery and purchase of books has overtaken the more traditional business models. How has Amazon changed the book selling industry? How has it managed to crush its competition? A major reason for the giant retailer’s success is its production model and cost structure, which has enabled Amazon to undercut the competitors' prices even when factoring in the cost of shipping. Read on to see how firms great (like Amazon) and small (like your corner deli) determine what to sell, at what output, and price. Introduction to Production, Costs, and Industry Structure In this chapter, you will learn about: • Explicit and Implicit Costs, and Accounting and Economic Profit • Production in the Short Run 156 Chapter 7 \| Production, Costs, and Industry Structure • Costs in the Short Run • Production in the Long Run • Costs in the Long Run This chapter is the first of four chapters that explores the theory of the firm. This theory explains how firms behave. What does that mean? Let’s define what we mean by the firm. A firm (or producer or business) combines inputs of labor, capital, land, and
raw or finished component materials to produce outputs. If the firm is successful, the outputs are more valuable than the inputs. This activity of production goes beyond manufacturing (i.e., making things). It includes any process or service that creates value, including transportation, distribution, wholesale and retail sales. Production involves a number of important decisions that define a firm's behavior. These decisions include, but are not limited to: • What product or products should the firm produce? • How should the firm produce the products (i.e., what production process should the firm use)? • How much output should the firm produce? • What price should the firm charge for its products? • How much labor should the firm employ? The answers to these questions depend on the production and cost conditions facing each firm. That is the subject of this chapter. The answers also depend on the market structure for the product(s) in question. Market structure is a multidimensional concept that involves how competitive the industry is. We define it by questions such as these: • How much market power does each firm in the industry possess? • How similar is each firm’s product to the products of other firms in the industry? • How difficult is it for new firms to enter the industry? • Do firms compete on the basis of price, advertising, or other product differences? Figure 7.2 illustrates the range of different market structures, which we will explore in Perfect Competition, Monopoly, and Monopolistic Competition and Oligopoly. Figure 7.2 The Spectrum of Competition Firms face different competitive situations. At one extreme—perfect competition—many firms are all trying to sell identical products. At the other extreme—monopoly—only one firm is selling the product, and this firm faces no competition. Monopolistic competition and oligopoly fall between the extremes of perfect competition and monopoly. Monopolistic competition is a situation with many firms selling similar, but not identical products. Oligopoly is a situation with few firms that sell identical or similar products. Let's examine how firms determine their costs and desired profit levels. Then we will discuss the origins of cost, both in the short and long run. Private enterprise, which can be private individual or group business ownership, characterizes the U.S. economy. In the U.S. system, we have the option to organize private businesses as sole proprietorships (one owner), partners (more than one owner), and corporations (legal entitles separate from the owners. When people think of businesses, often corporate giants like Wal-Mart, Microsoft, or General Motors come to mind. However, firms come in all sizes, as Table 7.1 shows. The vast majority of American firms have fewer than 20 employees. As of 2010, the U.S. Census Bureau counted 5.7 million firms with employees in the U.S. economy. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 7 \| Production, Costs, and Industry Structure 157 Slightly less than half of all the workers in private firms are at the 17,000 large firms, meaning they employ more than 500 workers. Another 35% of workers in the U.S. economy are at firms with fewer than 100 workers. These small-scale businesses include everything from dentists and lawyers to businesses that mow lawns or clean houses. Table 7.1 does not include a separate category for the millions of small “non-employer” businesses where a single owner or a few partners are not officially paid wages or a salary, but simply receive whatever they can earn. Number of Employees Firms (% of total firms) Number of Paid Employees (% of total employment) Total 0–9 10–19 20–99 5,734,538 112.0 million 4,543,315 (79.2%) 12.3 million (11.0%) 617,089 (10.8%) 8.3 million (7.4%) 475,125 (8.3%) 18.6 million (16.6%) 100–499 81,773 (1.4%) 15.9 million (14.2%) 500 or more 17,236 (0.30%) 50.9 million (49.8%) Table 7.1 Range in Size of U.S. Firms (Source: U.S. Census, 2010 www.census.gov) 7.1 \| Explicit and Implicit Costs, and Accounting and Economic Profit By the end of this section, you will be able to: • Explain the difference between explicit costs and implicit costs • Understand the relationship between cost and revenue Each business, regardless of size or complexity, tries to earn a profit: Profi = Total Revenue – Total Cost Total revenue is the income the firm generates from selling its products. We calculate it by multiplying the price of the product times the quantity of output sold: Total Revenue = Price × Quantity We will see in the following chapters that revenue is a function of the demand for the firm’s products. Total cost is what the firm pays for producing and selling its products. Recall that production involves the firm converting inputs to outputs. Each of those inputs has a cost to the firm. The sum of all those costs is total cost. We will learn in this chapter that short run costs are different from long run costs. We can distinguish between two types of cost: explicit and implicit. Explicit costs are out-of-pocket costs, that is, actual payments. Wages that a firm pays its employees or rent that a firm pays for its office are explicit costs. Implicit costs are more subtle, but just as important. They represent the opportunity cost of using resources that the firm already owns. Often for small businesses, they are resources that the owners contribute. For example, working in the business while not earning a formal salary, or using the ground floor of a home as a retail store are both implicit costs. Implicit costs also include the depreciation of goods, materials, and equipment that are necessary for a company to operate. (See the Work It Out feature for an extended example.) These two definitions of cost are important for distinguishing between two conceptions of profit, accounting profit, and economic profit. Accounting profit is a cash concept. It means total revenue minus explicit costs—the difference between dollars brought in and dollars paid out. Economic profit is total revenue minus total cost, including both explicit and implicit costs. The difference is important because even though a business pays income taxes based on 158 Chapter 7 \| Production, Costs, and Industry Structure its accounting profit, whether or not it is economically successful depends on its economic profit. Calculating Implicit Costs Consider the following example. Fred currently works for a corporate law firm. He is considering opening his own legal practice, where he expects to earn $200,000 per year once he establishes himself. To run his own firm, he would need an office and a law clerk. He has found the perfect office, which rents for $50,000 per year. He could hire a law clerk for $35,000 per year. If these figures are accurate, would Fred’s legal practice be profitable? Step 1. First you have to calculate the costs. You can take what you know about explicit costs and total them: Office ental : Law clerk's salary : Total explicit costs : $50,000 +$35,000 ____________ $85,000 Step 2. Subtracting the explicit costs from the revenue gives you the accounting profit. Revenues : Explicit costs : Accounting profi : $200,000 –$85,000 ____________ $115,000 However, these calculations consider only the explicit costs. To open his own practice, Fred would have to quit his current job, where he is earning an annual salary of $125,000. This would be an implicit cost of opening his own firm. Step 3. You need to subtract both the explicit and implicit costs to determine the true economic profit: Economic profi = total revenues – explicit costs – implicit costs = $200,000 – $85,000 – $125,000 = –$10,000 per year Fred would be losing $10,000 per year. That does not mean he would not want to open his own business, but it does mean he would be earning $10,000 less than if he worked for the corporate firm. Implicit costs can include other things as well. Maybe Fred values his leisure time, and starting his own firm would require him to put in more hours than at the corporate firm. In this case, the lost leisure would also be an implicit cost that would subtract from economic profits. Now that we have an idea about the different types of costs, let’s look at cost structures. A firm’s cost structure in the long run may be different from that in the short run. We turn to that distinction in the next few sections. 7.2 \| Production in the Short Run By the end of this section, you will be able to: • Understand the concept of a production function • Differentiate between the different types of inputs or factors in a production function • Differentiate between fixed and variable inputs • Differentiate between production in the short run and in the long run • Differentiate between total and marginal product • Understand the concept of diminishing marginal productivity In this chapter, we want to explore the relationship between the quantity of output a firm produces, and the cost of This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 7 \| Production, Costs, and Industry Structure 159 producing that output. We mentioned that the cost of the product depends on how many inputs are required to produce the product and what those inputs cost. We can answer the former question by looking at the firm’s production function. Figure 7.3 The production process for pizza includes inputs such as ingredients, the efforts of the pizza maker, and tools and materials for cooking and serving. (Credit: Haldean Brown/Flickr Creative Commons) Production is the process (or processes) a firm uses to transform inputs (e.g. labor, capital, raw materials) into outputs, i.e. the goods or services the firm wishes to sell. Consider pizza making. The pizzaiolo (pizza maker) takes flour, water, and yeast to make dough. Similarly, the pizzaiolo may take tomatoes, spices, and water to make pizza sauce. The cook rolls out the dough, brushes on the pizza sauce, and adds
cheese and other toppings. The pizzaiolo uses a peel—the shovel-like wooden tool-- to put the pizza into the oven to cook. Once baked, the pizza goes into a box (if it’s for takeout) and the customer pays for the good. What are the inputs (or factors of production) in the production process for this pizza? Economists divide factors of production into several categories: • Natural Resources (Land and Raw Materials) - The ingredients for the pizza are raw materials. These include the flour, yeast, and water for the dough, the tomatoes, herbs, and water for the sauce, the cheese, and the toppings. If the pizza place uses a wood-burning oven, we would include the wood as a raw material. If the establishment heats the oven with natural gas, we would count this as a raw material. Don’t forget electricity for lights. If, instead of pizza, we were looking at an agricultural product, like wheat, we would include the land the farmer used for crops here. • Labor – When we talk about production, labor means human effort, both physical and mental. The pizzaiolo was the primary example of labor here. He or she needs to be strong enough to roll out the dough and to insert and retrieve the pizza from the oven, but he or she also needs to know how to make the pizza, how long it cooks in the oven and a myriad of other aspects of pizza-making. The business may also have one or more people to work the counter, take orders, and receive payment. • Capital – When economists uses the term capital, they do not mean financial capital (money); rather, they mean physical capital, the machines, equipment, and buildings that one uses to produce the product. In the case of pizza, the capital includes the peel, the oven, the building, and any other necessary equipment (for example, tables and chairs). • Technology – Technology refers to the process or processes for producing the product. How does the pizzaiolo combine ingredients to make pizza? How hot should the oven be? How long should the pizza cook? What is the best oven to use? Gas or wood burning? Should the restaurant make its own dough, sauce, cheese, toppings, or should it buy them? • Entrepreneurship – Production involves many decisions and much knowledge, even for something as simple as pizza. Who makes those decisions? Ultimately, it is the entrepreneur, the person who creates the business, 160 Chapter 7 \| Production, Costs, and Industry Structure whose idea it is to combine the inputs to produce the outputs. The cost of producing pizza (or any output) depends on the amount of labor capital, raw materials, and other inputs required and the price of each input to the entrepreneur. Let’s explore these ideas in more detail. We can summarize the ideas so far in terms of a production function, a mathematical expression or equation that explains the engineering relationship between inputs and outputs: Q = f ⎡ ⎣NR, L, K, t, E⎤ ⎦ The production function gives the answer to the question, how much output can the firm produce given different amounts of inputs? Production functions are specific to the product. Different products have different production functions. The amount of labor a farmer uses to produce a bushel of wheat is likely different than that required to produce an automobile. Firms in the same industry may have somewhat different production functions, since each firm may produce a little differently. One pizza restaurant may make its own dough and sauce, while another may buy those pre-made. A sit-down pizza restaurant probably uses more labor (to handle table service) than a purely take-out restaurant. We can describe inputs as either fixed or variable. Fixed inputs are those that can’t easily be increased or decreased in a short period of time. In the pizza example, the building is a fixed input. Once the entrepreneur signs the lease, he or she is stuck in the building until the lease expires. Fixed inputs define the firm’s maximum output capacity. This is analogous to the potential real GDP shown by society’s production possibilities curve, i.e. the maximum quantities of outputs a society can produce at a given time with its available resources. Variable inputs are those that can easily be increased or decreased in a short period of time. The pizzaiolo can order more ingredients with a phone call, so ingredients would be variable inputs. The owner could hire a new person to work the counter pretty quickly as well. Economists often use a short-hand form for the production function: Q = f ⎡ ⎣L, K ⎤ ⎦, where L represents all the variable inputs, and K represents all the fixed inputs. Economists differentiate between short and long run production. The short run is the period of time during which at least some factors of production are fixed. During the period of the pizza restaurant lease, the pizza restaurant is operating in the short run, because it is limited to using the current building—the owner can’t choose a larger or smaller building. The long run is the period of time during which all factors are variable. Once the lease expires for the pizza restaurant, the shop owner can move to a larger or smaller place. Let’s explore production in the short run using a specific example: tree cutting (for lumber) with a two-person crosscut saw. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 7 \| Production, Costs, and Industry Structure 161 Figure 7.4 Production in the short run may be explored through the example of lumberjacks using a two-person saw. (Credit: Wknight94/Wikimedia Commons) Since by definition capital is fixed in the short run, our production function becomes − Q = f [L, K ] or Q = f [L] This equation simply indicates that since capital is fixed, the amount of output (e.g. trees cut down per day) depends only on the amount of labor employed (e.g. number of lumberjacks working). We can express this production function numerically as Table 7.2 below shows. # Lumberjacks # Trees (TP) MP 1 4 4 2 10 6 3 12 2 4 13 1 5 13 0 Table 7.2 Short Run Production Function for Trees Note that we have introduced some new language. We also call Output (Q) Total Product (TP), which means the amount of output produced with a given amount of labor and a fixed amount of capital. In this example, one lumberjack using a two-person saw can cut down four trees in an hour. Two lumberjacks using a two-person saw can cut down ten trees in an hour. We should also introduce a critical concept: marginal product. Marginal product is the additional output of one more worker. Mathematically, Marginal Product is the change in total product divided by the change in labor: MP = ΔTP / ΔL . In the table above, since 0 workers produce 0 trees, the marginal product of the first worker is four trees per day, but the marginal product of the second worker is six trees per day. Why might that be the case? It’s because of the nature of the capital the workers are using. A two-person saw works much better with two persons than with one. Suppose we add a third lumberjack to the story. What will that person’s marginal product be? What will that person contribute to the team? Perhaps he or she can oil the saw's teeth to keep it sawing smoothly or he or she could 162 Chapter 7 \| Production, Costs, and Industry Structure bring water to the two people sawing. What you see in the table is a critically important conclusion about production in the short run: It may be that as we add workers, the marginal product increases at first, but sooner or later additional workers will have decreasing marginal product. In fact, there may eventually be no effect or a negative effect on output. This is called the Law of Diminishing Marginal Product and it’s a characteristic of production in the short run. Diminishing marginal productivity is very similar to the concept of diminishing marginal utility that we learned about in the chapter on consumer choice. Both concepts are examples of the more general concept of diminishing marginal returns. Why does diminishing marginal productivity occur? It’s because of fixed capital. We will see this more clearly when we discuss production in the long run. We can show these concepts graphically as Figure 7.5 and Figure 7.6 illustrate. Figure 7.5 graphically shows the data from Table 7.2. Figure 7.6 shows the more general cases of total product and marginal product curves. This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 7 \| Production, Costs, and Industry Structure 163 Figure 7.5 164 Chapter 7 \| Production, Costs, and Industry Structure Figure 7.6 7.3 \| Costs in the Short Run By the end of this section, you will be able to: • Understand the relationship between production and costs • Understand that every factor of production has a corresponding factor price • Analyze short-run costs in terms of total cost, fixed cost, variable cost, marginal cost, and average cost • Calculate average profit • Evaluate patterns of costs to determine potential profit We’ve explained that a firm’s total costs depend on the quantities of inputs the firm uses to produce its output and the cost of those inputs to the firm. The firm’s production function tells us how much output the firm will produce with given amounts of inputs. However, if we think about that backwards, it tells us how many inputs the firm needs to produce a given quantity of output, which is the first thing we need to determine total cost. Let’s move to the second This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 7 \| Production, Costs, and Industry Structure 165 factor we need to determine. For every factor of production (or input), there is an associated factor payment. Factor payments are what the firm pays for the use of the factors of production. From the firm’s perspective, factor payments are costs. From the owner of each factor’s perspective, factor payments are income. Factor payments include: • Raw materials prices for raw m
aterials • Rent for land or buildings • Wages and salaries for labor • Interest and dividends for the use of financial capital (loans and equity investments) • Profit for entrepreneurship. Profit is the residual, what’s left over from revenues after the firm pays all the other costs. While it may seem odd to treat profit as a “cost”, it is what entrepreneurs earn for taking the risk of starting a business. You can see this correspondence between factors of production and factor payments in the inside loop of the circular flow diagram in Figure 1.6. We now have all the information necessary to determine a firm’s costs. A cost function is a mathematical expression or equation that shows the cost of producing different levels of output. Q Cost 1 $32.50 2 $44 3 $52 4 $90 Table 7.3 Cost Function for Producing Widgets What we observe is that the cost increases as the firm produces higher quantities of output. This is pretty intuitive, since producing more output requires greater quantities of inputs, which cost more dollars to acquire. What is the origin of these cost figures? They come from the production function and the factor payments. The discussion of costs in the short run above, Costs in the Short Run, was based on the following production function, which is similar to Table 7.3 except for "widgets" instead of trees. Workers (L) Widgets (Q) 1 0.2 2 0.4 3 0.8 3.25 1 4.4 2 5.2 3 6 3.5 7 3.8 8 3.95 9 4 Table 7.4 We can use the information from the production function to determine production costs. What we need to know is how many workers are required to produce any quantity of output. If we flip the order of the rows, we “invert” the production function so it shows L = f ⎛ ⎠ . ⎝Q⎞ Widgets (Q) Workers (L) 0.2 1 0.4 2 0.8 3 1 3.25 2 4.4 3 5.2 3.5 6 3.8 7 3.95 8 4 9 Table 7.5 Now focus on the whole number quantities of output. We’ll eliminate the fractions from the table: Widgets (Q) 1 2 3 4 Table 7.6 166 Chapter 7 \| Production, Costs, and Industry Structure Workers (L) 3.25 4.4 5.2 9 Table 7.6 Suppose widget workers receive $10 per hour. Multiplying the Workers row by $10 (and eliminating the blanks) gives us the cost of producing different levels of output. Widgets (Q) Workers (L) × Wage Rate per hour = Cost Table 7.7 1.00 3.25 $10 2.00 4.4 $10 3.00 5.2 $10 4.00 9 $10 $32.50 $44.00 $52.00 $90.00 This is same cost function with which we began! (shown in Table 7.3) Now that we have the basic idea of the cost origins and how they are related to production, let’s drill down into the details. Average and Marginal Costs The cost of producing a firm’s output depends on how much labor and physical capital the firm uses. A list of the costs involved in producing cars will look very different from the costs involved in producing computer software or haircuts or fast-food meals. We can measure costs in a variety of ways. Each way provides its own insight into costs. Sometimes firms need to look at their cost per unit of output, not just their total cost. There are two ways to measure per unit costs. The most intuitive way is average cost. Average cost is the cost on average of producing a given quantity. We define average cost as total cost divided by the quantity of output produced. AC = TC / Q If producing two widgets costs a total of $44, the average cost per widget is $44 / 2 = $22 per widget. The other way of measuring cost per unit is marginal cost. If average cost is the cost of the average unit of output produced, marginal cost is the cost of each individual unit produced. More formally, marginal cost is the cost of producing one more unit of output. Mathematically, marginal cost is the change in total cost divided by the change in output: MC = ΔTC / ΔQ . If the cost of the first widget is $32.50 and the cost of two widgets is $44, the marginal cost of the second widget is $44 − $32.50 = $11.50. We can see the Widget Cost table redrawn below with average and marginal cost added. Q Total Cost Average Cost Marginal Cost 1 $32.50 $32.50 $32.50 2 $44.00 $22.00 $11.50 3 $52.00 $17.33 $8.00 4 $90.00 $22.50 $38.00 Table 7.8 Extended Cost Function for Producing Widgets Note that the marginal cost of the first unit of output is always the same as total cost. Fixed and Variable Costs We can decompose costs into fixed and variable costs. Fixed costs are the costs of the fixed inputs (e.g. capital). Because fixed inputs do not change in the short run, fixed costs are expenditures that do not change regardless of the level of production. Whether you produce a great deal or a little, the fixed costs are the same. One example is the rent This OpenStax book is available for free at http://cnx.org/content/col12170/1.7 Chapter 7 \| Production, Costs, and Industry Structure 167 on a factory or a retail space. Once you sign the lease, the rent is the same regardless of how much you produce, at least until the lease expires. Fixed costs can take many other forms: for example, the cost of machinery or equipment to produce the product, research and development costs to develop new products, even an expense like advertising to popularize a brand name. The amount of fixed costs varies according to the specific line of business: for instance, manufacturing computer chips requires an expensive factory, but a local moving and hauling business can get by with almost no fixed costs at all if it rents trucks by the day when needed. Variable costs are the costs of the variable inputs (e.g. labor). The only way to increase or decrease output is by increasing or decreasing the variable inputs. Therefore, variable costs increase or decrease with output. We treat labor as a variable cost, since producing a greater quantity of a good or service typically requires more workers or more work hours. Variable costs would also include raw materials. Total costs are the sum of fixed plus variable costs. Let's look at another example. Consider the barber shop called “The Clip Joint” in Figure 7.7. The data for output and costs are in Table 7.9. The fixed costs of operating the barber shop, including the space and equipment, are $160 per day. The variable costs are the costs of hiring barbers, which in our example is $80 per barber each day. The first two columns of the table show the quantity of haircuts the barbershop can produce as it hires additional barbers. The third column shows the fixed costs, which do not change regardless of the level of production. The fourth column shows the variable costs at each level of output. We calculate these by taking the amount of labor hired and multiplying by the wage. For example, two barbers cost: 2 × $80 = $160. Adding together the fixed costs in the third column and the variable costs in the fourth column produces the total costs in the fifth column. For example, with two barbers the total cost is: $160 + $160 = $320. Labor Quantity Fixed Cost Variable Cost Total Cost 1 2 3 4 5 6 7 16 40 60 72 80 84 82 $160 $160 $160 $160 $160 $160 $160 Table 7.9 Output and Total Costs $80 $160 $240 $320 $400 $480 $560 $240 $320 $400 $480 $560 $640 $720 Figure 7.7 How Output Affects Total Costs At zero production, the fixed costs of $160 are still present. As production increases, variable costs are added to fixed costs, and the total cost is the sum of the two. At zero production, the fixed costs of $160 are still present. As production increases, we add variable costs to fixed 168 Chapter 7 \| Production, Costs, and Industry Structure costs, and the total cost is the sum of the two. Figure 7.7 graphically shows the relationship between the quantity of output produced and the cost of producing that output. We always show the fixed costs as the vertical intercept of the total cost curve; that is, they are the costs incurred when output is zero so there are no variable costs. You can see from the graph that once production starts, total costs and variable costs rise. While variable costs may initially increase at a decreasing rate, at some point they begin increasing at an increasing rate. This is caused by diminishing marginal productivity which we discussed earlier in the Production in the Short Run section of this chapter, which is easiest to see with an example. As the number of barbers increases from zero to one in the table, output increases from 0 to 16 for a marginal gain (or marginal product) of 16. As the number rises from one to two barbers, output increases from 16 to 40, a marginal gain of 24. From that point on, though, the marginal product diminishes as we add each additional barber. For example, as the number of barbers rises from two to three, the marginal product is only 20; and as the number rises from three to four, the marginal product is only 12. To understand the reason behind this pattern, consider that a one-man barber shop is a very busy operation. The single barber needs to do everything: say hello to people entering, answer the phone, cut hair, sweep, and run the cash register. A second barber reduces the level of disruption from jumping back and forth between these tasks, and allows a greater division of labor and specialization. The result can be increasing marginal productivity. However, as the shop adds other barbers, the advantage of each additional barber is less, since the specialization of labor can only go so far. The addition of a sixth or seventh or eighth barber just to greet people at the door will have less impact than the second one did. This is the pattern of diminishing marginal productivity. As a result, the total costs of production will begin to rise more rapidly as output increases. At some point, you may even see negative returns as the additional barbers begin bumping elbows and getting in each other’s way. In this case, the addition of still more barbers would actually cause output to decrease, as the last row of Table 7.9 shows. This pattern of diminishing marginal productivity is common in production. As another example, consider the problem of irrigating a crop on a farmer’s field

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