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)—holds for the Cournot ﬁrm. As we will see from an analysis of the particular form that the marginal revenue term takes for the Cournot ﬁrm, price is above the perfectly competitive level (above marginal cost) but below the level in a perfect cartel that maximizes ﬁrms’ joint proﬁts. In order for Equation 15.2 to equal 0, price must exceed marginal cost by the magnitude of the ‘‘wedge’’ term P 0(Q)qi. If the Cournot ﬁrm produces another unit on top of its existing production of qi units, then, because demand is downward sloping, the additional unit causes market price to decrease by P 0(Q), leading to a loss of revenue of P 0(Q)qi (the wedge term) from ﬁrm i’s existing production. To compare the Cournot outcome with the perfect cartel outcome, note that the objec- tive for the cartel is to maximize joint proﬁt: n 1 j X ¼ pj ¼ P Q ð Þ n n qj $ 1 j X ¼ 1 j X ¼ Cjð : qjÞ Taking the ﬁrst-order condition of Equation 15.3 with respect to qi gives @ @qi n 1 j X ¼ pj! qj C 0i ð qiÞ $ ¼ 0: MR MC This ﬁrst-order condition is similar to Equation 15.2 except that the wedge term, \|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} \|ﬄﬄ{zﬄﬄ qj ¼ P 0 Q, Q ð Þ (15:3) (15:4) (15:5) is larger in magnitude with a perfect cartel than with Cournot ﬁrms. In maximizing joint proﬁts, the cartel
. The solution for the Nash equilibrium follows Example 8.6 closely. Proﬁts for the two Cournot ﬁrms are a a P P Q ð Q ð p1 ¼ p2 ¼ cq1 ¼ ð cq2 ¼ ð q1 $ Þ q2 $ Þ Using the ﬁrst-order conditions to solve for the best-response functions, we obtain q2 $ 2 q1 ¼ Solving Equations 15.8 simultaneously yields the Nash equilibrium c q1, Þ q2: c Þ q1 $ q1 $ q2 $ q2 $ q1 $ 2 q2 "1 ¼ q"2 ¼ a c : $ 3 (15:7) (15:8) (15:9) Thus, total output is Q" (2/3)(a implies an equilibrium price of P" þ functions (Equations 15.7) implies p"1 ¼ G" 2=9 a Þð p"2 ¼ ð p"1 ¼ $ ¼ 2. Þ ¼ $ ¼ c c). Substituting total output into the inverse demand curve 2c)/3. Substituting price and outputs into the proﬁt (a 2, so total market proﬁt equals c p"2 ¼ ð Þ a Þð 1=9 $ Perfect cartel. The objective function for a perfect cartel involves joint proﬁts q1 $ a q1 þ ð Þ The two ﬁrst-order conditions for maximizing Equation 15.10 with respect to q1 and q2 are the same: a p2 ¼ ð p1 þ q1 $ q2 $ q2 $ q2: Þ (15:10) $ $ c c @ @q1 ð p1 þ p2Þ ¼ @ @q2 ð p1 þ p2Þ ¼ a 2q1 $ 2q2 $ c $ ¼ 0: (15:11) The ﬁrst-order conditions do not pin down market shares for ﬁrms in a perfect cartel because they produce identical products at constant marginal cost. But Equation 15.11 does pin down total output: q"1 þ. Substituting total output into inverse demand implies that c Q" q"2 �
curves for ﬁrm 1 increase until point M is reached, which is the monopoly outcome for ﬁrm 1. q2 a − c BR1(q2 π1 = 100 π1 = 200 BR2(q1) a − c q1 538 Part 6: Market Power Figure 15.2 displays ﬁrms’ isoproﬁt curves. An isoproﬁt curve for ﬁrm 1 is the locus of quantity pairs providing it with the same proﬁt level. To compute the isoproﬁt curve associated with a proﬁt level of (say) 100, we start by setting Equation 15.7 equal to 100: q1 ¼ Þ Then we solve for q2 to facilitate graphing the isoproﬁt: a p1 ¼ ð q2 $ q1 $ $ c 100: (15:12) q2 ¼ a c q1 $ 100 q1 : (15:13) $ $ Several example isoproﬁts for ﬁrm 1 are shown in the ﬁgure. As proﬁt increases from 100 to 200 to yet higher levels, the associated isoproﬁts shrink down to the monopoly point, which is the highest isoproﬁt on the diagram. To understand why the individual isoproﬁts are shaped like frowns, refer back to Equation 15.13. As ql approaches 0, the last term ( 100 /q1) dominates, causing the left side of the frown to turn down. As ql increases, the ql term in Equation 15.13 begins to dominate, causing the right side of the frown to turn down. $ $ Figure 15.3 shows how to use best-response diagrams to quickly tell how changes in such underlying parameters as the demand intercept a or marginal cost c would affect the equilibrium. Figure 15.3a depicts an increase in both ﬁrms’ marginal cost c. The best responses shift inward, resulting in a new equilibrium that involves lower output for both. Although ﬁrms have the same marginal cost in this example, one can imagine a model in which ﬁrms have different marginal cost parameters and so can be varied independently. Figure 15.3b depicts an increase in just ﬁrm
1’s marginal cost; only ﬁrm 1’s best response shifts. The new equilibrium involves lower output for ﬁrm 1 and higher output for ﬁrm 2. Although ﬁrm 2’s best response does not shift, it still increases its output as it anticipates a reduction in ﬁrm 1’s output and best responds to this anticipated output reduction. QUERY: Explain why ﬁrm 1’s individual isoproﬁts reach a peak on its best-response function in Figure 15.2. What would ﬁrm 2’s isoproﬁts look like in Figure 15.2? How would you represent an increase in demand intercept a in Figure 15.3? FIGURE 15.315Shifting Cournot Best Responses Firms’ initial best responses are drawn as solid lines, resulting in a Nash equilibrium at point E 0. Panel (a) depicts an increase in both ﬁrms’ marginal costs, shifting their best responses—now given by the dashed lines—inward. The new intersection point, and thus the new equilibrium, is point E 00. Panel (b) depicts an increase in just ﬁrm 1’s marginal cost. q2 q2 BR1(q2) BR1(q2) E ′ E″ E ′ E″ BR2(q1) q1 BR2(q1) q1 (a) Increase in both firms’ marginal costs (b) Increase in firm 1’s marginal cost Chapter 15: Imperfect Competition 539 to n ¼1 Varying the number of Cournot ﬁrms The Cournot model is particularly useful for policy analysis because it can represent the whole range of outcomes from perfect competition to perfect cartel/monopoly (i.e., the whole range of points between C and M in Figure 15.1) by varying the number of ﬁrms 1. For simplicity, consider the case of identical ﬁrms, which here n from n ¼ means the n ﬁrms sharing the same cost function C(qi). In equilibrium, ﬁrms will produce the same share of total output: qi ¼ Q/n into Equation 15.12, the wedge term becomes P 0(Q)Q/n. The wedge term disappears as n grows large;
Q a cqi ¼ ð i $ $ Q qi $ Þ qi: c Þ (15:14pi @qi ¼ a 2qi $ Q $ i $ c $ ¼ 0, (15:15) 1, 2,..., n. which holds for all i ¼ The key to solving the system of n equations for the n equilibrium quantities is to recognize that the Nash equilibrium involves equal quantities because ﬁrms are symmetric. Symmetry implies that (15:16) (15:17) (15:18) Q" $ q"i ¼ Substituting Equation 15.16 into 15.15 yields 2q"i $ ð n i ¼ Q" $ $ a nq"i $ q"i ¼ ð n $ 1 q"i : Þ 1 q"i $ Þ c $ ¼ 0, a or q"i ¼ ð : = c Þ Þ Total market output is n ð $ þ 1 and market price is P" Q" nq": (15:19) þ Substituting for q"i, Q", and P" into the ﬁrm’s proﬁt Equation 15.14, we have that total proﬁt for all ﬁrms is þ P" np"15:20) 540 Part 6: Market Power ¼ Setting n 1 in Equations 15.18–15.20 gives the monopoly outcome, which gives the same price, total output, and proﬁt as in the perfect cartel case computed in Example 15.1. Letting n grow without bound in Equations 15.18–15.20 gives the perfectly competitive outcome, the same outcome computed in Example 15.1 for the Bertrand case. QUERY: We used the trick of imposing symmetry after taking the ﬁrst-order condition for ﬁrm i’s quantity choice. It might seem simpler to impose symmetry before taking the ﬁrst-order condition. Why would this be a mistake? How would the incorrect expressions for quantity, price, and proﬁt compare with the correct ones here? Prices or quantities? Moving from price competition in the Bertrand model to quantity competition in the Cournot model changes the market outcome dramatically. This change is surprising on ﬁrst thought. After all, the
monopoly outcome from Chapter 14 is the same whether we assume the monopolist sets price or quantity. Further thought suggests why price and quantity are such different strategic variables. Starting from equal prices, a small reduction in one ﬁrm’s price allows it to steal all the market demand from its competitors. This sharp beneﬁt from undercutting makes price competition extremely ‘‘tough.’’ Quantity competition is ‘‘softer.’’ Starting from equal quantities, a small increase in one ﬁrm’s quantity has only a marginal effect on the revenue that other ﬁrms receive from their existing output. Firms have less of an incentive to outproduce each other with quantity competition than to undercut each other with price competition. An advantage of the Cournot model is its realistic implication that the industry grows more competitive as the number n of ﬁrms entering the market increases from monopoly to perfect competition. In the Bertrand model there is a discontinuous jump from monopoly to perfect competition if just two ﬁrms enter, and additional entry beyond two has no additional effect on the market outcome. An apparent disadvantage of the Cournot model is that ﬁrms in real-world markets tend to set prices rather than quantities, contrary to the Cournot assumption that ﬁrms choose quantities. For example, grocers advertise prices for orange juice, say, $3.00 a container, in newpaper circulars rather than the number of containers it stocks. As we will see in the next section, the Cournot model applies even to the orange juice market if we reinterpret quantity to be the ﬁrm’s capacity, deﬁned as the most the ﬁrm can sell given the capital it has in place and other available inputs in the short run. Capacity Constraints For the Bertrand model to generate the Bertrand paradox (the result that two ﬁrms essentially behave as perfect competitors), ﬁrms must have unlimited capacities. Starting from equal prices, if a ﬁrm lowers its price the slightest amount, then its demand essentially doubles. The ﬁrm can satisfy this increased demand because it has no capacity constraints, giving ﬁrms a big incentive to undercut. If the undercutting ﬁrm could not serve all the demand at its lower price because of capacity constraints, that would leave some residual demand for
the higher-priced ﬁrm and would decrease the incentive to undercut. Consider a two-stage game in which ﬁrms build capacity in the ﬁrst stage and ﬁrms choose prices p1 and p2 in the second stage.4 Firms cannot sell more in the second stage 4The model is due to D. Kreps and J. Scheinkman, Outcomes,’’ Bell Journal of Economics (Autumn 1983): 326–37. ‘‘Quantity Precommitment and Bertrand Competition Yield Cournot Chapter 15: Imperfect Competition 541 than the capacity built in the ﬁrst stage. If the cost of building capacity is sufﬁciently high, it turns out that the subgame-perfect equilibrium of this sequential game leads to the same outcome as the Nash equilibrium of the Cournot model. To see this result, we will analyze the game using backward induction. Consider the second-stage pricing game supposing the ﬁrms have already built capacities q1 and q2 in the ﬁrst stage. Let p be the price that would prevail when production is at capacity for both ﬁrms. A situation in which p1 ¼ p2 < p (15:21) is not a Nash equilibrium. At this price, total quantity demanded exceeds total capacity; therefore, ﬁrm 1 could increase its proﬁts by raising price slightly and continuing to sell q1. Similarly, p1 ¼ p2 > p (15:22) is not a Nash equilibrium because now total sales fall short of capacity. At least one ﬁrm (say, ﬁrm 1) is selling less than its capacity. By cutting price slightly, ﬁrm 1 can increase its proﬁts by selling up to its capacity, q1. Hence the Nash equilibrium of this secondstage game is for ﬁrms to choose the price at which quantity demanded exactly equals the total capacity built in the ﬁrst stage:5 p: p1 ¼ p2 ¼ Anticipating that the price will be set such that ﬁrms sell all their capacity, the ﬁrststage capacity choice game is essentially the same as the Cournot game. Therefore, the equilibrium quantities, price, and proﬁts will be the same as in the Cournot game. Thus
, even in markets (such as orange juice sold in grocery stores) where it looks like ﬁrms are setting prices, the Cournot model may prove more realistic than it ﬁrst seems. (15:23) Product Differentiation Another way to avoid the Bertrand paradox is to replace the assumption that the ﬁrms’ products are identical with the assumption that ﬁrms produce differentiated products. Many (if not most) real-world markets exhibit product differentiation. For example, toothpaste brands vary somewhat from supplier to supplier—differing in ﬂavor, ﬂuoride content, whitening agents, endorsement from the American Dental Association, and so forth. Even if suppliers’ product attributes are similar, suppliers may still be differentiated in another dimension: physical location. Because demanders will be closer to some suppliers than to others, they may prefer nearby sellers because buying from them involves less travel time. Meaning of ‘‘the market’’ The possibility of product differentiation introduces some fuzziness into what we mean by the market for a good. With identical products, demanders were assumed to be indifferent about which ﬁrm’s output they bought; hence they shop at the lowest-price ﬁrm, leading to the law of one price. The law of one price no longer holds if demanders strictly 5For completeness, it should be noted that there is no pure-strategy Nash equilibrium of the second-stage game with unequal p2). The low-price ﬁrm would have an incentive to increase its price and/or the high-price ﬁrm would have an inprices (p1 6¼ centive to lower its price. For large capacities, there may be a complicated mixed-strategy Nash equilibrium, but this can be ruled out by supposing the cost of building capacity is sufﬁciently high. 542 Part 6: Market Power prefer one supplier to another at equal prices. Are green-gel and white-paste toothpastes in the same market or in two different ones? Is a pizza parlor at the outskirts of town in the same market as one in the middle of town? With differentiated products, we will take the market to be a group of closely related products that are more substitutable among each other (as measured by cross-price elasticities) than with goods outside the group. We will be somewhat loose with this
deﬁnition, avoiding precise thresholds for how high the cross-price elasticity must be between goods within the group (and how low with outside goods). Arguments about which goods should be included in a product group often dominate antitrust proceedings, and we will try to avoid this contention here. ¼ Bertrand competition with differentiated products Return to the Bertrand model but now suppose there are n ﬁrms that simultaneously choose prices pi (i 1,..., n) for their differentiated products. Product i has its own speciﬁc attributes ai, possibly reﬂecting special options, quality, brand advertising, or location. A product may be endowed with the attribute (orange juice is by deﬁnition made from oranges and cranberry juice from cranberries), or the attribute may be the result of the ﬁrm’s choice and spending level (the orange juice supplier can spend more and make its juice from fresh oranges rather than from frozen concentrate). The various attributes serve to differentiate the products. Firm i’s demand is qið i is a list of all other ﬁrms’ prices besides i’s, and A i, ai, A $, iÞ $ pi, P where P attributes besides i’s. Firm i’s total cost is $ (15:24) i is a list of all other ﬁrms’ $ and proﬁt is thus Cið qi, aiÞ (15:25) pi ¼ piqi $ : qi, aiÞ Cið With differentiated products, the proﬁt function (Equation 15.26) is differentiable, so we do not need to solve for the Nash equilibrium on a case-by-case basis as we did in the Bertrand model with identical products. We can solve for the Nash equilibrium as in the Cournot model, solving for best-response functions by taking each ﬁrm’s ﬁrst-order condition (here with respect to price rather than quantity). The ﬁrst-order condition from Equation 15.26 with respect to pi is (15:26) @pi @pi ¼ qi þ pi @qi @pi @Ci @qi & @qi @pi $ 0: ¼ A B (15:27) \|ﬄ�
��ﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} \|ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} The ﬁrst two terms (labeled A) on the right side of Equation 15.27 are a sort of marginal revenue—not the usual marginal revenue from an increase in quantity, but rather the marginal revenue from an increase in price. The increase in price increases revenue on existing sales of qi units, but we must also consider the negative effect of the reduction in sales (@qi/@pi multiplied by the price pi) that would have been earned on these sales. The last term, labeled B, is the cost savings associated with the reduced sales that accompany an increased price. The Nash equilibrium can be found by simultaneously solving the system of ﬁrst-order 1,..., n. If the attributes ai are also choice conditions in Equation 15.27 for all i ¼ Chapter 15: Imperfect Competition 543 variables (rather than just endowments), there will be another set of ﬁrst-order conditions to consider. For ﬁrm i, the ﬁrst-order condition with respect to ai has the form @pi @ai ¼ pi @qi @ai $ @Ci @ai $ @Ci @qi & @qi @ai ¼ 0: (15:28) The simultaneous solution of these ﬁrst-order conditions can be complex, and they yield few deﬁnitive conclusions about the nature of market equilibrium. Some insights from particular cases will be developed in the next two examples. EXAMPLE 15.4 Toothpaste as a Differentiated Product Suppose that two ﬁrms produce toothpaste, one a green gel and the other a white paste. To simplify the calculations, suppose that production is costless. Demand for product i is qi ¼ ai $ The positive coefﬁcient on pj, the other good’s price, indicates that the goods are gross substitutes. Firm i’s demand is increasing in the attribute ai, which we will take to be demanders’ inherent preference for the variety in question; we will suppose that this is an endowment rather than a choice variable for the
; their intersection (E ) is the Nash equilibrium. Isoproﬁt curves for ﬁrm 1 increase moving out along ﬁrm 1’s best-response function. p2 p2* a2 + c 2 BR1(p2) E 0 a1 + c 2 p1* BR2(p1) π1 = 200 π1 = 100 p1 The smile turns up as p1 approaches 0 because the denominator of 100/p1 approaches 0. The smile turns up as p1 grows large because then the second term on the right side of Equation 15.35 grows large. Isoproﬁt curves for ﬁrm 1 increase as one moves away from the origin along its best-response function. QUERY: How would a change in the demand intercepts be represented on the diagram? EXAMPLE 15.5 Hotelling’s Beach A simple model in which identical products are differentiated because of the location of their suppliers (spatial differentiation) was provided by H. Hotelling in the 1920s.6 As shown in Figure 15.5, two ice cream stands, labeled A and B, are located along a beach of length L. The stands make identical ice cream cones, which for simplicity are assumed to be costless to produce. Let a and b represent the ﬁrms’ locations on the beach. (We will take the locations of the ice cream stands as given; in a later example we will revisit ﬁrms’ equilibrium location choices.) Assume that demanders are located uniformly along the beach, one at each unit of length. Carrying ice cream a distance d back to one’s beach umbrella costs td 2 because ice cream melts more the higher the temperature t and the further one must walk.7 Consistent with the Bertrand assumption, ﬁrms choose prices pA and pB simultaneously. Determining demands. Let x be the location of the consumer who is indifferent between buying from the two ice cream stands. The following condition must be satisﬁed by x: pA þ t x ð $ 2 a Þ pB þ t b ð $ ¼ 2: x Þ (15:36) 6H. Hotelling, ‘‘Stability in Competition,’’ Economic Journal 39 (1929): 41–57. 7The assumption of quadratic ‘
‘transportation costs’’ turns out to simplify later work, when we compute ﬁrms’ equilibrium locations in the model. Chapter 15: Imperfect Competition 545 FIGURE 15.515Hotelling’s Beach Ice cream stands A and B are located at points a and b along a beach of length L. The consumer who is indifferent between buying from the two stands is located at x. Consumers to the left of x buy from A and to the right buy from B. A’s demand B’s demand 0 a x b L The left side of Equation 15.36 is the generalized cost of buying from A (including the price paid and the cost of transporting the ice cream the distance x a). Similarly, the right side is the generalized cost of buying from B. Solving Equation 15.36 for x yields $ b a þ 2 þ x ¼ pB $ b 2t $ ð pA a Þ : (15:37) If prices are equal, the indifferent consumer is located midway between a and b. If A’s price is less than B’s, then x shifts toward endpoint L. (This is the case shown in Figure 15.5.) Because all demanders between 0 and x buy from A and because there is one consumer per unit distance, it follows that A’s demand equals x: qAð pA, pB, a The remaining L $ x consumers constitute B ’s demand: pB $ b 2t $ ð pA a Þ : (15:38) qBð pB, pA, b pA $ b 2t $ ð pB a Þ : (15:39) Solving for Nash equilibrium. The Nash equilibrium is found in the same way as in Example 15.4 except that, for demands, we use Equations 15.38 and 15.39 in place of Equation 15.29. Skipping the details of the calculations, the Nash equilibrium prices are p"A ¼ p" 2L Þð 4L Þð $ $ 15:40) These prices will depend on the precise location of the two stands and will differ from each other. For example, if we assume that the beach is L 70 $2:90. These price yards, and t differences arise only from the locational aspects of this problem—the cones themselves are identical and costless to produce. Because A
is somewhat more favorably located than B, it can charge a higher price for its cones without losing too much business to B. Using Equation 15.38 shows that $0.001 (one tenth of a penny), then p"A ¼ 100 yards long, a ¼ $3:10 and p"B ¼ 40 yards, b ¼ ¼ ¼ 3:10 110 2 þ x ¼ 2 ð Þð $ 0:001 2:90 110 Þ Þð 52, * (15:41) 546 Part 6: Market Power so stand A sells 52 cones, whereas B sells only 48 despite its lower price. At point x, the consumer is indifferent between walking the 12 yards to A and paying $3.10 or walking 18 yards to B and paying $2.90. The equilibrium is inefﬁcient in that a consumer slightly to the right of x would incur a shorter walk by patronizing A but still chooses B because of A’s power to set higher prices. Equilibrium proﬁts are p"A ¼ p"B ¼ t b 18 ð t b 18 ð a a 2L Þð 4L Þð þ $ a a þ $ $ $ 2, 2: b b Þ Þ (15:42) Somewhat surprisingly, the ice cream stands beneﬁt from faster melting, as measured here by 100, a the transportation cost t. For example, if we take L $0.001 as in ¼ $140 (rounding to the nearest dollar). If $160 and p"B ¼ the previous paragraph, then p"A ¼ transportation costs doubled to t $320 and ¼ p"B ¼ The transportation/melting cost is the only source of differentiation in the model. If t 0, then we can see from Equation 15.40 that prices equal 0 (which is marginal cost given that production is costless) and from Equation 15.42 that proﬁts equal 0—in other words, the Bertrand paradox results. ¼ $0.002, then proﬁts would double to p"A ¼ 70, and t $280. 40, b ¼ ¼ ¼ QUERY: What happens to prices and proﬁts if ice cream stands locate in the same spot? If they locate at the opposite ends of the beach? Consumer search
pt, and the demander would have to pay the cost s to ﬁnd this other price. Less extreme equilibria are found in models where consumers have different search costs.9 For example, suppose one group of consumers can search for free and another group has to pay s per search. In equilibrium, there will be some price dispersion across stores. One set of stores serves the low–search-cost demanders (and the lucky high– search-cost consumers who happen to stumble on a bargain). These bargain stores sell at marginal cost. The other stores serve the high–search-cost demanders at a price that makes these demanders indifferent between buying immediately and taking a chance that the next price search will uncover a bargain store. Tacit Collusion In Chapter 8, we showed that players may be able to earn higher payoffs in the subgameperfect equilibrium of an inﬁnitely repeated game than from simply repeating the Nash equilibrium from the single-period game indeﬁnitely. For example, we saw that, if players are patient enough, they can cooperate on playing silent in the inﬁnitely repeated version of the Prisoners’ Dilemma rather than ﬁnking on each other each period. From the perspective of oligopoly theory, the issue is whether ﬁrms must endure the Bertrand paradox (marginal cost pricing and zero proﬁts) in each period of a repeated game or whether they might instead achieve more proﬁtable outcomes through tacit collusion. A distinction should be drawn between tacit collusion and the formation of an explicit cartel. An explicit cartel involves legal agreements enforced with external sanctions if the agreements (e.g., to sustain high prices or low outputs) are violated. Tacit collusion can only be enforced through punishments internal to the market—that is, only those that can be generated within a subgame-perfect equilibrium of a repeated game. Antitrust laws generally forbid the formation of explicit cartels, so tacit collusion is usually the only way for ﬁrms to raise prices above the static level. Finitely repeated game Taking the Bertrand game to be the stage game, Selten’s theorem from Chapter 8 tells us that repeating the stage game any ﬁnite number of times T does not change the outcome. The only subgame-perfect equilibrium of the ﬁnitely repeated Bertrand game is to repeat the stage-game Nash equilibrium—marginal
cost pricing—in each of the T periods. The game unravels through backward induction. In any subgame starting in period T, the unique Nash equilibrium will be played regardless of what happened before. Because the outcome in period T 1 does not affect the outcome in the next period, it is as though period T 1 is the last period, and the unique Nash equilibrium must be played then, too. Applying backward induction, the game unravels in this manner all the way back to the ﬁrst period. $ $ Inﬁnitely repeated game If the stage game is repeated inﬁnitely many periods, however, the folk theorem applies. The folk theorem indicates that any feasible and individually rational payoff can be sustained each period in an inﬁnitely repeated game as long as the discount factor, d, is close enough to unity. Recall that the discount factor is the value in the present period of one 9The following model is due to S. Salop and J. Stiglitz, ‘‘Bargains and Ripoffs: A Model of Monopolistically Competitive Price Dispersion,’’ Review of Economic Studies 44 (1977): 493–510. 548 Part 6: Market Power dollar earned one period in the future—a measure, roughly speaking, of how patient players are. Because the monopoly outcome (with proﬁts divided among the ﬁrms) is a feasible and individually rational outcome, the folk theorem implies that the monopoly outcome must be sustainable in a subgame-perfect equilibrium for d close enough to 1. Let’s investigate the threshold value of d needed. First suppose there are two ﬁrms competing in a Bertrand game each period. Let GM denote the monopoly proﬁt and PM the monopoly price in the stage game. The ﬁrms may collude tacitly to sustain the monopoly price—with each ﬁrm earning an equal share of the monopoly proﬁt—by using the grim trigger strategy of continuing to collude as long as no ﬁrm has undercut PM in the past but reverting to the stage-game Nash equilibrium of marginal cost pricing every period from then on if any ﬁrm deviates by undercutting. Successful tacit collusion provides the proﬁt stream V collude PM ¼ 2 þ d & PM 1 d þ PM ¼ ¼ 2 ð PM 2 2 þ
d2 þ 1 1 d # d2 PM & 2 þ & & & þ & & &Þ : (15:43) d " þ þ $ d2 " # factors Refer to Chapter 8 for a discussion of adding up a series of discount 1. We need to check that a ﬁrm has no incentive to deviate. By undercutting the collusive price PM slightly, a ﬁrm can obtain essentially all the monopoly proﬁt for itself in the current period. This deviation would trigger the grim strategy punishment of marginal cost pricing in the second and all future periods, so all ﬁrms would earn zero proﬁt from there on. Hence the stream of proﬁts from deviating is V deviate þ & & & GM. For this deviation not to be proﬁtable we must have V collude % ¼ V deviate or, on substituting, PM 2 " 1 1 " # $ % d # PM: (15:44) Rearranging Equation 15.44, the condition reduces to d 1/2. To prevent deviation, ﬁrms must value the future enough that the threat of losing proﬁts by reverting to the one-period Nash equilibrium outweighs the beneﬁt of undercutting and taking the whole monopoly proﬁt in the present period. % EXAMPLE 15.6 Tacit Collusion in a Bertrand Model Bertrand duopoly. Suppose only two ﬁrms produce a certain medical device used in surgery. The medical device is produced at constant average and marginal cost of $10, and the demand for the device is given by 5,000 Q ¼ $ 100P: (15:45) If the Bertrand game is played in a single period, then each ﬁrm will charge $10 and a total of 4,000 devices will be sold. Because the monopoly price in this market is $30, ﬁrms have a clear incentive to consider collusive strategies. At the monopoly price, total proﬁts each period are $40,000, and each ﬁrm’s share of total proﬁts is $20,000. According to Equation 15.44, collusion at the monopoly price is sustainable if 20,000 1 1 " $ % d # 40,000 (15:46) or if d % 1/2, as
we saw. Chapter 15: Imperfect Competition 549 % Is the condition d 1/2 likely to be met in this market? That depends on what factors we consider in computing d, including the interest rate and possible uncertainty about whether the game will continue. Leave aside uncertainty for a moment and consider only the interest rate. If the period length is one year, then it might be reasonable to assume an annual interest rate of 10%. As shown in the Appendix to Chapter 17, d r 10%, then 0.91. This value of d clearly exceeds the threshold of 1/2 needed to sustain collusion. For d d to be less than the 1/2 threshold for collusion, we must incorporate uncertainty into the discount factor. There must be a signiﬁcant chance that the market will not continue into the next period—perhaps because a new surgical procedure is developed that renders the medical device obsolete. r); therefore, if r ¼ ¼ 1/(1 þ ¼ ¼ We focused on the best possible collusive outcome: the monopoly price of $30. Would collusion be easier to sustain at a lower price, say $20? No. At a price of $20, total proﬁts each period are $30,000, and each ﬁrm’s share is $15,000. Substituting into Equation 15.44, collusion can be sustained if 15,000 1 1 " $ % d # 30,000, (15:47) % 1/2. Whatever collusive proﬁt the ﬁrms try to sustain will cancel out from again implying d both sides of Equation 15.44, leaving the condition d 1/2. Therefore, we get a discrete jump in ﬁrms’ ability to collude as they become more patient—that is, as d increases from 0 to 1.10 For d below 1/2, no collusion is possible. For d above 1/2, any price between marginal cost and the monopoly price can be sustained as a collusive outcome. In the face of this multiplicity of subgame-perfect equilibria, economists often focus on the one that is most proﬁtable for the ﬁrms, but the formal theory as to why ﬁrms would play one or another of the equilibria is still unsettled. % Bertrand oligopoly. Now suppose n ﬁrms produce the medical device.
The monopoly proﬁt continues to be $40,000, but each ﬁrm’s share is now only $40,000/n. By undercutting the monopoly price slightly, a ﬁrm can still obtain the whole monopoly proﬁt for itself regardless of how many other ﬁrms there are. Replacing the collusive proﬁt of $20,000 in Equation 15.46 with $40,000/n, we have that the n ﬁrms can successfully collude on the monopoly price if or 40,000 n 1 1 " $ % d # 40,000, d 1 $ % 1 n : (15:48) (15:49) Taking the ‘‘reasonable’’ discount factor of d 0.91 used previously, collusion is possible when 11 or fewer ﬁrms are in the market and impossible with 12 or more. With 12 or more ﬁrms, the only subgame-perfect equilibrium involves marginal cost pricing and zero proﬁts. ¼ Equation 15.49 shows that tacit collusion is easier the more patient are ﬁrms (as we saw before) and the fewer of them there are. One rationale used by antitrust authorities to challenge certain mergers is that a merger may reduce n to a level such that Equation 15.49 begins to be satisﬁed and collusion becomes possible, resulting in higher prices and lower total welfare. QUERY: A period can be interpreted as the length of time it takes for ﬁrms to recognize and respond to undercutting by a rival. What would be the relevant period for competing gasoline stations in a small town? In what industries would a year be a reasonable period? 10The discrete jump in ﬁrms’ ability to collude is a feature of the Bertrand model; the ability to collude increases continuously with d in the Cournot model of Example 15.7. 550 Part 6: Market Power EXAMPLE 15.7 Tacit Collusion in a Cournot Model Suppose that there are again two ﬁrms producing medical devices but that each period they now engage in quantity (Cournot) rather than price (Bertrand) competition. We will again investigate the conditions under which ﬁrms can collude on the monopoly outcome. To generate the monopoly outcome in a period, ﬁrms
need to produce 1,000 each; this leads to a price of $30, total proﬁts of $40,000, and ﬁrm proﬁts of $20,000. The present discounted value of the stream of these collusive proﬁts is V collude ¼ 20,000 1 " 1 $ d : # (15:50) Computing the present discounted value of the stream of proﬁts from deviating is somewhat complicated. The optimal deviation is not as simple as producing the whole monopoly output oneself and having the other ﬁrm produce nothing. The other ﬁrm’s 1,000 units would be provided to the market. The optimal deviation (by ﬁrm 1, say) would be to best respond to ﬁrm 2’s output of 1,000. To compute this best response, ﬁrst note that if demand is given by Equation 15.45, then inverse demand is given by 50 P ¼ $ Q 100 : (15:51) Firm 1’s proﬁt is p1 ¼ Pq1 $ cq1 ¼ q1 40 $ q2 q1 þ 100 $ % : (15:52) Taking the ﬁrst-order condition with respect to q1 and solving for q1 yields the best-response function q1 ¼ 2,000 q2 2 : $ (15:53) Firm 1’s optimal deviation when ﬁrm 2 produces 1,000 units is to increase its output from 1,000 to 1,500. Substituting these quantities into Equation 15.52 implies that ﬁrm 1 earns $22,500 in the period in which it deviates. How much ﬁrm 1 earns in the second and later periods following a deviation depends on the trigger strategies ﬁrms use to punish deviation. Assume that ﬁrms use the grim strategy of reverting to the Nash equilibrium of the stage game—in this case, the Nash equilibrium of the Cournot game—every period from then on. In the Nash equilibrium of the Cournot game, each ﬁrm best responds to the other in accordance with the best-response function in Equation 15.53 (switching subscripts in the case of ﬁrm 2). Solving these best-response equations simultaneously 4,000=3 and
that proﬁts are implies that p"1 ¼ p"2 ¼ $17,778. Firm 1’s present discounted value of the stream of proﬁts from deviation is the Nash equilibrium outputs are q"1 ¼ q"2 ¼ V deviate ¼ ¼ ¼ 22,500 22,500 $22,500 þ 17,778 d 17,778 þ þ ð & 17,778 & $17,778 þ d 1 Þð d d þ & þ : 1 d # $ " d2 d2 17,778 þ d3 & þ & & & þ & & &Þ We have V collude V deviate if % $20,000 or, after some algebra, if d " 0.53. % 1 $ 1 % d # $22,500 þ $17,778 " d 1 d # $ Unlike with the Bertrand stage game, with the Cournot stage game there is a possibility of some collusion for discount factors below 0.53. However, the outcome would have to involve higher outputs and lower proﬁts than monopoly. (15:54) (15:55) Chapter 15: Imperfect Competition 551 QUERY: The beneﬁt to deviating is lower with the Cournot stage game than with the Bertrand stage game because the Cournot ﬁrm cannot steal all the monopoly proﬁt with a small deviation. Why then is a more stringent condition (d 0.5) needed to collude on the monopoly outcome in the Cournot duopoly compared with the Bertrand duopoly? 0.53 rather than d % % Longer-Run Decisions: Investment, Entry, And Exit The chapter has so far focused on the most basic short-run decisions regarding what price or quantity to set. The scope for strategic interaction expands when we introduce longer-run decisions. Take the case of the market for cars. Longer-run decisions include whether to update the basic design of the car, a process that might take up to two years to complete. Longer-run decisions may also include investing in robotics to lower production costs, moving manufacturing plants closer to consumers and cheap inputs, engaging in a new advertising campaign, and entering or exiting certain product lines (say, ceasing the production of station wagons or starting production of hybrid cars). In making such decisions, an oligopolist must consider how rivals will respond to it. Will competition with
existing rivals become tougher or milder? Will the decision lead to the exit of current rivals or encourage new ones to enter? Is it better to be the ﬁrst to make such a decision or to wait until after rivals move? Flexibility versus commitment Crucial to our analysis of longer-run decisions such as investment, entry, and exit is how easy it is to reverse a decision once it has been made. On ﬁrst thought, it might seem that it is better for a ﬁrm to be able to easily reverse decisions because this would give the ﬁrm more ﬂexibility in responding to changing circumstances. For example, a car manufacturer might be more willing to invest in developing a hybrid-electric car if it could easily change the design back to a standard gasoline-powered one should the price of gasoline (and the demand for hybrid cars along with it) decrease unexpectedly. Absent strategic considerations—and so for the case of a monopolist—a ﬁrm would always value ﬂexibility and reversibility. The ‘‘option value’’ provided by ﬂexibility is discussed in further detail in Chapter 7. Surprisingly, the strategic considerations that arise in an oligopoly setting may lead a ﬁrm to prefer its decision be irreversible. What the ﬁrm loses in terms of ﬂexibility may be offset by the value of being able to commit to the decision. We will see a number of instances of the value of commitment in the next several sections. If a ﬁrm can commit to an action before others move, the ﬁrm may gain a ﬁrst-mover advantage. A ﬁrm may use its ﬁrst-mover advantage to stake out a claim to a market by making a commitment to serve it and in the process limit the kinds of actions its rivals ﬁnd proﬁtable. Commitment is essential for a ﬁrst-mover advantage. If the ﬁrst mover could secretly reverse its decision, then its rival would anticipate the reversal and the ﬁrms would be back in the game with no ﬁrst-mover advantage. We already encountered a simple example of the value of commitment in the Battle of the Sexes game from Chapter 8. In the simultaneous version of the model, there were three Nash equilibria. In one pure
-strategy equilibrium, the wife obtains her highest payoff by attending her favorite event with her husband, but she obtains lower payoffs in the other two equilibria (a pure-strategy equilibrium in which she attends her less favored 552 Part 6: Market Power event and a mixed-strategy equilibrium giving her the lowest payoff of all three). In the sequential version of the game, if a player were given the choice between being the ﬁrst mover and having the ability to commit to attending an event or being the second mover and having the ﬂexibility to be able to meet up with the ﬁrst wherever he or she showed up, a player would always choose the ability to commit. The ﬁrst mover can guarantee his or her preferred outcome as the unique subgame-perfect equilibrium by committing to attend his or her favorite event. Sunk costs Expenditures on irreversible investments are called sunk costs Sunk cost. A sunk cost is an expenditure on an investment that cannot be reversed and has no resale value. Sunk costs include expenditures on unique types of equipment (e.g., a newsprint-making machine) or job-speciﬁc training for workers (developing the skills to use the newsprint machine). There is sometimes confusion between sunk costs and what we have called ﬁxed costs. They are similar in that they do not vary with the ﬁrm’s output level in a production period and are incurred even if no output is produced in that period. But instead of being incurred periodically, as are many ﬁxed costs (heat for the factory, salaries for secretaries and other administrators), sunk costs are incurred only once in connection with a single investment.11 Some ﬁxed costs may be avoided over a sufﬁciently long run—say, by reselling the plant and equipment involved—but sunk costs can never be recovered because the investments involved cannot be moved to a different use. When the ﬁrm makes a sunk investment, it has committed itself to that investment, and this may have important consequences for its strategic behavior. First-mover advantage in the Stackelberg model The simplest setting to illustrate the ﬁrst-mover advantage is in the Stackelberg model, named after the economist who ﬁrst analyzed it.12 The model is similar to a duopoly version of the Cournot
model except that—rather than simultaneously choosing the quantities of their identical outputs—ﬁrms move sequentially, with ﬁrm 1 (the leader) choosing its output ﬁrst and then ﬁrm 2 (the follower) choosing after observing ﬁrm 1’s output. We use backward induction to solve for the subgame-perfect equilibrium of this sequential game. Begin with the follower’s output choice. Firm 2 chooses the output q2 that maximizes its own proﬁt, taking ﬁrm 1’s output as given. In other words, ﬁrm 2 best responds to ﬁrm 1’s output. This results in the same best-response function for ﬁrm 2 as we computed in the Cournot game from the ﬁrst-order condition (Equation 15.2). Label this best-response function BR2(q1). Turn then to the leader’s output choice. Firm 1 recognizes that it can inﬂuence the follower’s action because the follower best responds to 1’s observed output. Substituting BR2(q1) into the proﬁt function for ﬁrm 1 given by Equation 15.1, we have q1 þ ð BR2ð p1 ¼ q1 $ q1ÞÞ : q1Þ (15:56) C1ð P 11Mathematically, the notion of sunk costs can be integrated into the per-period total cost function as þ where S is the per-period amortization of sunk costs (e.g., the interest paid for funds used to ﬁnance capital investments), Ft is Ft; but if the production the per-period ﬁxed costs, c is marginal cost, and qt is per-period output. If qt ¼ period is long enough, then some or all of Ft may also be avoidable. No portion of S is avoidable, however. 12H. von Stackelberg, The Theory of the Market Economy, trans. A. T. Peacock (New York: Oxford University Press, 1952). 0, then Ct ¼ þ ¼ S Ct (qt) S Ft þ cqt, Chapter 15: Imperfect Competition 553 The ﬁrst-order condition with respect to q1 is @p1 @
q1 ¼ P Q ð Þ þ P 0 Q q1 þ Þ ð P 0 Q ð Þ BR 20 q1 q1Þ ð C i0 qiÞ ¼ ð $ 0: (15:57) S \|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} This is the same ﬁrst-order condition computed in the Cournot model (see Equation 15.2) except for the addition of the term S, which accounts for the strategic effect of ﬁrm 1’s output on ﬁrm 2’s. The strategic effect S will lead ﬁrm 1 to produce more than it would have in a Cournot model. By overproducing, ﬁrm 1 leads ﬁrm 2 to reduce q2 by ; the fall in ﬁrm 2’s output increases market price, thus increasing the the amount BR02ð revenue that ﬁrm 1 earns on its existing sales. We know that q2 decreases with an increase in ql because best-response functions under quantity competition are generally downward sloping; see Figure 15.2 for an illustration. q1Þ The strategic effect would be absent if the leader’s output choice were unobservable to the follower or if the leader could reverse its output choice in secret. The leader must be able to commit to an observable output choice or else ﬁrms are back in the Cournot game. It is easy to see that the leader prefers the Stackelberg game to the Cournot game. The leader could always reproduce the outcome from the Cournot game by choosing its Cournot output in the Stackelberg game. The leader can do even better by producing more than its Cournot output, thereby taking advantage of the strategic effect S. EXAMPLE 15.8 Stackelberg Springs Recall the two natural-spring owners from Example 15.1. Now, rather than having them choose outputs simultaneously as in the Cournot game, assume that they choose outputs sequentially as in the Stackelberg game, with ﬁrm 1 being the
leader and ﬁrm 2 the follower. Firm 2’s output. We will solve for the subgame-perfect equilibrium using backward induction, starting with ﬁrm 2’s output choice. We already found ﬁrm 2’s best-response function in Equation 15.8, repeated here: q2 ¼ a $ c : q1 $ 2 (15:58) Firm 1’s output. Now fold the game back to solve for ﬁrm 1’s output choice. Substituting ﬁrm 2’s best response from Equation 15.58 into ﬁrm 1’s proﬁt function from Equation 15.56 yields a q1 $ $ p1 ¼ h $ a $ c q1 $ 2 q1 ¼ c i $ % 1 a 2 ð q1 $ c Þ q1: $ Taking the ﬁrst-order condition, @p1 @q1 ¼ 1 2 ð a 2q1 $ c Þ ¼ $ 0, (15:59) (15:60) $ $ 1=8 To provide a numerical example, suppose a =2. Substituting q"1 back into ﬁrm 2’s best-response function gives c Þ a and solving gives q"1 ¼ ð 1=16 =4. Proﬁts are p"1 ¼ ð q"2 ¼ ð a c a Þð Þ 30, 120 and c $900. Firm 1 produces twice as much and earns twice as much as ﬁrm 2. p"1 ¼ Recall from the simultaneous Cournot game in Example 15.1 that, for these numerical values, total market output was 80 and total industry proﬁt was 3,200, implying that each of the two $1,600. Therefore, when ﬁrm 1 is the ﬁrms produced 80/2 2. c Þ 0. Then q"1 ¼ 2 and p"2 ¼ ð ¼ 40 units and earned $3,200/2 $1,800, and p"2 ¼ 60, q"2 ¼ $ ¼ c Þ $ Þð a ¼ ¼ 554 Part 6: Market Power ﬁrst mover in a sequential game, it produces
(60 1,600)/1,600 12.5% more than in the simultaneous game. 40)/40 $ ¼ 50% more and earns (1,800 ¼ $ Graphing the Stackelberg outcome. Figure 15.6 illustrates the Stackelberg equilibrium on a best-response function diagram. The leader realizes that the follower will always best respond, so the resulting outcome will always be on the follower’s best-response function. The leader effectively picks the point on the follower’s best-response function that maximizes the leader’s proﬁt. The highest isoproﬁt (highest in terms of proﬁt level, but recall from Figure 15.2 that higher proﬁt levels are reached as one moves down toward the horizontal axis) is reached at the point S of tangency between ﬁrm 1’s isoproﬁt and ﬁrm 2’s best-response function. This is the Stackelberg equilibrium. Compared with the Cournot equilibrium at point C, the Stackelberg equilibrium involves higher output and proﬁt for ﬁrm 1. Firm 1’s proﬁt is higher because, by committing to the high output level, ﬁrm 2 is forced to respond by reducing its output. Commitment is required for the outcome to stray from ﬁrm 1’s best-response function, as happens at point S. If ﬁrm 1 could secretly reduce q1 (perhaps because q1 is actual capacity that can be secretly reduced by reselling capital equipment for close to its purchase price to a manufacturer of another product that uses similar capital equipment), then it would move back to its best response, ﬁrm 2 would best respond to this lower quantity, and so on, following the dotted arrows from S back to C. FIGURE 15.615Stackelberg Game Best-response functions from the Cournot game are drawn as thick lines. Frown-shaped curves are ﬁrm 1’s isoproﬁts. Point C is the Nash equilibrium of the Cournot game (invoking simultaneous output choices). The Stackelberg equilibrium is point S, the point at which the highest isoproﬁt for ﬁrm 1 is reached on ﬁrm 2’s best-response function
. At S, ﬁrm 1’s isoproﬁt is tangent to ﬁrm 2’s best-response function. If ﬁrm 1 cannot commit to its output, then the outcome function unravels, following the dotted line from S back to C. q2 BR1(q2) C S BR2(q1) q1 Chapter 15: Imperfect Competition 555 QUERY: What would be the outcome if the identity of the ﬁrst mover were not given and instead ﬁrms had to compete to be the ﬁrst? How would ﬁrms vie for this position? Do these considerations help explain overinvestment in Internet ﬁrms and telecommunications during the ‘‘dot-com bubble?’’ Contrast with price leadership In the Stackelberg game, the leader uses what has been called a ‘‘top dog’’ strategy,13 aggressively overproducing to force the follower to scale back its production. The leader earns more than in the associated simultaneous game (Cournot), whereas the follower earns less. Although it is generally true that the leader prefers the sequential game to the simultaneous game (the leader can do at least as well, and generally better, by playing its Nash equilibrium strategy from the simultaneous game), it is not generally true that the leader harms the follower by behaving as a ‘‘top dog.’’ Sometimes the leader beneﬁts by behaving as a ‘‘puppy dog,’’ as illustrated in Example 15.9. EXAMPLE 15.9 Price-Leadership Game Return to Example 15.4, in which two ﬁrms chose price for differentiated toothpaste brands simultaneously. So that the following calculations do not become too tedious, we make the simplifying assumptions that a1 ¼ 0. Substituting these parameters back into Example 15.4 shows that equilibrium prices are 2/3 0.444 for each 0.667 and proﬁts are 4/9 ﬁrm. a2 ¼ 1 and c ¼ * * Now consider the game in which ﬁrm 1 chooses price before ﬁrm 2.14 We will solve for the subgame-perfect equilibrium using backward induction, starting with ﬁrm 2’s move. Firm 2’s best response to its
game in which ﬁrm 1 moves ﬁrst. At L, ﬁrm 1’s isoproﬁt is tangent to ﬁrm 2’s best response. p2 L BR2(p1) B BR1(p2) p1 The leader needs a moderate price increase (from 0.667 to 0.714) to induce the follower to increase its price slightly (from 0.667 to 0.679), so the leader’s proﬁts do not increase as much as the follower’s. QUERY: What choice variable realistically is easier to commit to, prices or quantities? What business strategies do ﬁrms use to increase their commitment to their list prices? We say that the ﬁrst mover is playing a ‘‘puppy dog’’ strategy in Example 15.9 because it increases its price relative to the simultaneous-move game; when translated into outputs, this means that the ﬁrst mover ends up producing less than in the simultaneousmove game. It is as though the ﬁrst mover strikes a less aggressive posture in the market and so leads its rival to compete less aggressively. A comparison of Figures 15.6 and 15.7 suggests the crucial difference between the games that leads the ﬁrst mover to play a ‘‘top dog’’ strategy in the quantity game and a ‘‘puppy dog’’ strategy in the price game: The best-response functions have different slopes. The goal is to induce the follower to compete less aggressively. The slopes of the best-response functions determine whether the leader can best do that by playing aggressively itself or by softening its strategy. The ﬁrst mover plays a ‘‘top dog’’ strategy in the sequential quantity game or indeed any game in which best responses slope down. When best responses slope down, playing more aggressively induces a rival to respond by competing less aggressively. Conversely, the ﬁrst mover plays a ‘‘puppy dog’’ strategy in the price game or any game in which best responses slope up. When best responses slope up, playing less aggressively induces a rival to respond by competing less aggressively. Chapter 15: Imperfect Competition 557 Therefore, knowing the slope of ﬁr
ms’ best responses provides considerable insight into the sort of strategies ﬁrms will choose if they have commitment power. The Extensions at the end of this chapter provide further technical details, including shortcuts for determining the slope of a ﬁrm’s best-response function just by looking at its proﬁt function. Strategic Entry Deterrence We saw that, by committing to an action, a ﬁrst mover may be able to manipulate the second mover into being a less aggressive competitor. In this section we will see that the ﬁrst mover may be able to prevent the entry of the second mover entirely, leaving the ﬁrst mover as the sole ﬁrm in the market. In this case, the ﬁrm may not behave as an unconstrained monopolist because it may have distorted its actions to fend off the rival’s entry. In deciding whether to deter the second mover’s entry, the ﬁrst mover must weigh the costs and beneﬁts relative to accommodating entry—that is, allowing entry to happen. Accommodating entry does not mean behaving nonstrategically. The ﬁrst mover would move off its best-response function to manipulate the second mover into being less competitive, as described in the previous section. The cost of deterring entry is that the ﬁrst mover would have to move off its best-response function even further than it would if it accommodates entry. The beneﬁt is that it operates alone in the market and has market demand to itself. Deterring entry is relatively easy for the ﬁrst mover if the second mover must pay a substantial sunk cost to enter the market. EXAMPLE 15.10 Deterring Entry of a Natural Spring Recall Example 15.8, where two natural-spring owners choose outputs sequentially. We now add an entry stage: In particular, after observing ﬁrm 1’s initial quantity choice, ﬁrm 2 decides whether to enter the market. Entry requires the expenditure of sunk cost K2, after which ﬁrm 2 can choose output. Market demand and cost are as in Example 15.8. To simplify the 0 [implying that inverse calculations, we will take the speciﬁc numerical values a demand is P(Q) Q, and that production is cost
less]. To further simplify, we will abstract from ﬁrm 1’s entry decision and assume that it has already sunk any cost needed to enter before the start of the game. We will look for conditions under which ﬁrm 1 prefers to deter rather than accommodate ﬁrm 2’s entry. 120 and c 120 ¼ $ ¼ ¼ Accommodating entry. Start by computing ﬁrm 1’s proﬁt if it accommodates ﬁrm 2’s entry, denoted pacc 1. This has already been done in Example 15.8, in which there was no issue of deterring qacc =2 ﬁrm 2’s entry. There we found ﬁrm 1’s equilibrium output to be 1 and its proﬁt to Þ ¼ a be 0, we have 120 and c ð qacc 1 ¼ a ð pacc 1. Substituting the speciﬁc numerical values a 2=8 $ Þ ¼ 60 and pacc 1 ¼ ð 2=8 Þ 1,800. 120 ¼ ¼ $ ¼ $ 0 c c Deterring entry. Next, compute ﬁrm 1’s proﬁt if it deters ﬁrm 2’s entry, denoted pdet deter entry, ﬁrm 1 needs to produce an amount qdet responds to qdet 15.58 that ﬁrm 2’s best-response function is 1. To 1 high enough that, even if ﬁrm 2 best 1, it cannot earn enough proﬁt to cover its sunk cost K2. We know from Equation Substituting for q2 in ﬁrm 2’s proﬁt function (Equation 15.7) and simplifying gives q2 ¼ 120 q1 : $ 2 120 $ 2 2 qdet 1 # p2 ¼ " K2: $ (15:63) (15:64) 558 Part 6: Market Power Setting ﬁrm 2’s proﬁt in Equation 15.64 equal to 0 and solving yields $ is the ﬁrm 1 output needed to keep ﬁrm 2 out of the market. At this output level, ﬁrm 1’s ﬃ�
�ﬃﬃﬃ qdet 1 ¼ 120 K2p ; 2 (15:65) qdet 1 proﬁt is which we found by substituting qdet Equation 15.7. We also set q2 ¼ alone in the market. pdet 1 ¼ 1, a K2p 120 2 K2p 2, $ (15:66) * ﬃﬃﬃﬃﬃ 0 into ﬁrm 1’s proﬁt function from 0 because, if ﬁrm 1 is successful in deterring entry, it operates ) ﬃﬃﬃﬃﬃ 120, and c ¼ ¼ Comparison. The ﬁnal step is to juxtapose p acc ﬁrm 1 prefers deterring to accommodating entry. To simplify the algebra, let x pdet 1 ¼ to ﬁnd the condition under which K2p. Then 2 and p det pacc 1 ¼ if 1 1 ﬃﬃﬃﬃﬃ (15:67) Applying the quadratic formula yields x2 $ 120x 1,800 0: ¼ þ 120 + x ¼ p 2 7,200 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : (15:68) Taking the smaller root (because we will be looking for a minimum threshold), we have x (rounding to the nearest decimal). Substituting x yields 17.6 K2p and solving for K2 17.6 into x ¼ ¼ ¼ 2 ﬃﬃﬃﬃﬃ K2 ¼ 2 x 2 $ % 2 17:6 2 # ¼ " 77: * (15:69) 1 ¼ 77, then entry is so cheap for ﬁrm 2 that ﬁrm 1 would have to increase its output all the If K2 ¼ way to qdet 102 in order to deter entry. This is a signiﬁcant distortion above what it would produce when accommodating entry: qacc 60. If K2 < 77, then the output distortion needed to deter entry wastes so much proﬁt that ﬁrm 1 prefers to accommodate
entry. If K2 > 77, output need not be distorted as much to deter entry; thus, ﬁrm 1 prefers to deter entry. 1 ¼ K. QUERY: Suppose the ﬁrst mover must pay the same entry cost as the second, K1 ¼ Suppose further that K is high enough that the ﬁrst mover prefers to deter rather than accommodate the second mover’s entry. Would this sunk cost not be high enough to keep the ﬁrst mover out of the market, too? Why or why not? K2 ¼ A real-world example of overproduction (or overcapacity) to deter entry is provided by the 1945 antitrust case against Alcoa, a U.S. aluminum manufacturer. A U.S. federal court ruled that Alcoa maintained much higher capacity than was needed to serve the market as a strategy to deter rivals’ entry, and it held that Alcoa was in violation of antitrust laws. To recap what we have learned in the last two sections: with quantity competition, the ﬁrst mover plays a ‘‘top dog’’ strategy regardless of whether it deters or accommodates the second mover’s entry. True, the entry-deterring strategy is more aggressive than the entry-accommodating one, but this difference is one of degree rather than kind. However, with price competition (as in Example 15.9), the ﬁrst mover’s entry-deterring strategy would differ in kind from its entry-accommodating strategy. It would play a ‘‘puppy dog’’ Chapter 15: Imperfect Competition 559 strategy if it wished to accommodate entry because this is how it manipulates the second mover into playing less aggressively. It plays a ‘‘top dog’’ strategy of lowering its price relative to the simultaneous game if it wants to deter entry. Two general principles emerge. • Entry deterrence is always accomplished by a ‘‘top dog’’ strategy whether competition is in quantities or prices, or (more generally) whether best-response functions slope down or up. The ﬁrst mover simply wants to create an inhospitable environment for the second mover. If ﬁrm 1 wants to accommodate entry, whether it should play a ‘‘pupp
y dog’’ or ‘‘top dog’’ strategy depends on the nature of competition—in particular, on the slope of the best-response functions. • Signaling The preceding sections have shown that the ﬁrst mover’s ability to commit may afford it a big strategic advantage. In this section we will analyze another possible ﬁrst-mover advantage: the ability to signal. If the second mover has incomplete information about market conditions (e.g., costs, demand), then it may try to learn about these conditions by observing how the ﬁrst mover behaves. The ﬁrst mover may try to distort its actions to manipulate what the second learns. The analysis in this section is closely tied to the material on signaling games in Chapter 8, and the reader may want to review that material before proceeding with this section. The ability to signal may be a plausible beneﬁt of being a ﬁrst mover in some settings in which the beneﬁt we studied earlier—commitment—is implausible. For example, in industries where the capital equipment is readily adapted to manufacture other products, costs are not very ‘‘sunk’’; thus, capacity commitments may not be especially credible. The ﬁrst mover can reduce its capacity with little loss. For another example, the price– leadership game involved a commitment to price. It is hard to see what sunk costs are involved in setting a price and thus what commitment value it has.15 Yet even in the absence of commitment value, prices may have strategic, signaling value. Entry-deterrence model Consider the incomplete information game in Figure 15.8. The game involves a ﬁrst mover (ﬁrm 1) and a second mover (ﬁrm 2) that choose prices for their differentiated products. Firm 1 has private information about its marginal cost, which can take on one of two values: high with probability Pr(H) or low with probability Pr(L) Pr(H). In period 1, ﬁrm 1 serves the market alone. At the end of the period, ﬁrm 2 observes ﬁrm 1’s price and decides whether to enter the market. If it enters, it sinks an entry cost K2 and learns the true level of ﬁrm 1’s costs; then
ﬁrms compete as duopolists in the second period, choosing prices for differentiated products as in Example 15.4 or 15.5. (We do not need to be speciﬁc about the exact form of demands.) If ﬁrm 2 does not enter, it obtains a payoff of zero, and ﬁrm 1 again operates alone in the market. Assume there is no discounting between periods. $ ¼ 1 Firm 2 draws inferences about ﬁrm 1’s cost from the price that ﬁrm 1 charges in the ﬁrst period. Firm 2 earns more if it competes against the high-cost type because the 15The Query in Example 15.9 asks you to consider reasons why a ﬁrm may be able to commit to a price. The ﬁrm may gain commitment power by using contracts (e.g., long-term supply contracts with customers or a most-favored customer clause, which ensures that if the ﬁrm lowers price in the future to other customers, then the favored customer gets a rebate on the price difference). The ﬁrm may advertise a price through an expensive national advertising campaign. The ﬁrm may have established a valuable reputation as charging ‘‘everyday low prices.’’ 560 Part 6: Market Power FIGURE 15.8 Signaling for Entry Deterrence Firm 1 signals its private information about its cost (high H or low L) through the price it sets in the ﬁrst period. Firm 2 observes ﬁrm 1’s price and then decides whether to enter. If ﬁrm 2 enters, the ﬁrms compete as duopolists; otherwise, ﬁrm 1 operates alone on the market again in the second period. Firm 2 earns positive proﬁt if and only if it enters against a high cost rival. Pr(H) Pr(L) 1 1 H p1 L p1 L p1 2 2 2 E NE E NE E NE M1 H + D1 H H, D2 2M1 H, 0 M1 H − R + D1 H H, D2 H − R, 0 2M1 M1 L + D1 L L, D2 2M1 L, 0 i be the duopoly proﬁt (not including entry costs) for ﬁrm i
high-cost type’s price will be higher, and as we saw in Examples 15.4 and 15.5, the higher the rival’s price for a differentiated product, the higher the ﬁrm’s own demand and proﬁt. Let D t {1, 2} if ﬁrm 1 is of type t 2, so that ﬁrm 2 earns more than its entry cost if it faces the high-cost type but not if it faces the lowcost type. Otherwise, the information in ﬁrm 1’s signal would be useless because ﬁrm 2 would always enter or always stay out regardless of ﬁrm 1’s type. 2 2 < K2 < D H {L, H }. To make the model interesting, we will suppose D L 2 To simplify the model, we will suppose that the low-cost type only has one relevant 1. The high-cost type can 1, or it 1. Presumably, the optimal monopoly price 1 be ﬁrm 1’s monopoly proﬁt if it is 1 if it 1 if it is the low type). Let R be the high type’s loss relative to the 1 rather than its optimal in the ﬁrst period, then it earns action in the ﬁrst period—namely, setting its monopoly price p L choose one of two prices: can set the monopoly price associated with its type, pH can choose the same price as the low type, p L is increasing in marginal cost; thus, p L 1 < p H of type t is the high type and p L optimal monopoly proﬁt in the ﬁrst period if it charges p L monopoly price p H MH 1. Let M t {L, H } (the proﬁt if it is alone and charges its optimal monopoly price pH 1. Thus, if the high type charges p H 1 R. 1 in that period, but if it charges p L 1, it earns MH 2 1 $ Chapter 15: Imperfect Competition 561 Separating equilibrium We will look for two kinds of perfect Bayesian equilibria: separating and pooling. In a separating equilibrium, the different types of the ﬁrst mover must choose different actions. Here, there is only one such possibility for ﬁrm 1: The low-cost type chooses p L 1 and
the high-cost type chooses pH 1. Firm 2 learns ﬁrm 1’s type from these actions perfectly and stays out on seeing p1 1 and enters on seeing p H 1. It remains to check whether the high-cost type would prefer to deviate to p L 1. In equilibrium, the high type earns a total proﬁt of MH 1 in the ﬁrst period because it charges its optimal monopoly price, and D H in the second because ﬁrm 2 enters and the ﬁrms compete as duopolists. If the 1 high type were to deviate to p L R in the ﬁrst period, the loss R coming from charging a price other than its ﬁrst-period optimum, but ﬁrm 2 would think it is the low type and would not enter. Hence ﬁrm 1 would earn MH 1 in the second period, for a total of 2MH R across periods. For deviation to be unproﬁtable we must have 1, then it would earn MH 1 : MH 1 $ 1 þ D H 1 $ MH 1 þ DH 1 % 2MH 1 $ R (15:70) or (after rearranging) R MH DH 1 : (15:71) 1 $ That is, the high-type’s loss from distorting its price from its monopoly optimum in the ﬁrst period exceeds its gain from deterring ﬁrm 2’s entry in the second period. % If the condition in Equation 15.71 does not hold, there still may be a separating equilibrium in an expanded game in which the low type can charge other prices besides pL 1. The high type could distort its price downward below pL 1, increasing the ﬁrst-period loss the high type would suffer from pooling with the low type to such an extent that the high type would rather charge pH 1 even if this results in ﬁrm 2’s entry. Pooling equilibrium If the condition in Equation 15.71 does not hold, then the high type would prefer to pool with the low type if pooling deters entry. Pooling deters entry if ﬁrm 2’s prior belief that ﬁrm 1 is the high type, Pr(H)—which is equal to its posterior belief in a pooling equilibrium—is low enough that ﬁrm 2’s
expected payoff from entering, H Pr ð is less than its payoff of zero from staying out of the market. 1 2 þ ½ H Pr ð 2 $ K2, $ Þ- D H Þ D L (15:72) Predatory pricing The incomplete-information model of entry deterrence has been used to explain why a rational ﬁrm might want to engage in predatory pricing, the practice of charging an artiﬁcially low price to prevent potential rivals from entering or to force existing rivals to exit. The predatory ﬁrm sacriﬁces proﬁts in the short run to gain a monopoly position in future periods. Predatory pricing is prohibited by antitrust laws. In the most famous antitrust case, dating back to 1911, John D. Rockefeller—owner of the Standard Oil Company that controlled a substantial majority of reﬁned oil in the United States—was accused of attempting to monopolize the oil market by cutting prices dramatically to drive rivals out and then raising prices after rivals had exited the market or sold out to Standard Oil. Predatory pricing remains a controversial antitrust issue because of the difﬁculty in distinguishing between predatory conduct, which authorities would like to prevent, and competitive conduct, which authorities would like to promote. In addition, economists initially had 562 Part 6: Market Power trouble developing game-theoretic models in which predatory pricing is rational and credible. Suitably interpreted, predatory pricing may emerge as a rational strategy in the incomplete-information model of entry deterrence. Predatory pricing can show up in a separating equilibrium—in particular, in the expanded model where the low-cost type can separate only by reducing price below its monopoly optimum. Total welfare is actually higher in this separating equilibrium than it would be in its full-information counterpart. Firm 2’s entry decision is the same in both outcomes, but the low-cost type’s price may be lower (to signal its type) in the predatory outcome. Predatory pricing can also show up in a pooling equilibrium. In this case it is the high-cost type that charges an artiﬁcially low price, pricing below its ﬁrst-period optimum to keep ﬁrm 2 out of the market. Whether social welfare is lower in the pooling equilibrium than in a full-information setting is unclear. In the ﬁrst period, price is lower (and total welfare presumably higher) in the pooling
equilibrium than in a fullinformation setting. On the other hand, deterring ﬁrm 2’s entry results in higher secondperiod prices and lower welfare. Weighing the ﬁrst-period gain against the second-period loss would require detailed knowledge of demand curves, discount factors, and so forth. The incomplete-information model of entry deterrence is not the only model of predatory pricing that economists have developed. Another model involves frictions in the market for ﬁnancial capital that stem perhaps from informational problems (between borrowers and lenders) of the sort we will discuss in Chapter 18. With limits on borrowing, ﬁrms may only have limited resources to ‘‘make a go’’ in a market. A larger ﬁrm may force ﬁnancially strapped rivals to endure losses until their resources are exhausted and they are forced to exit the market. How Many Firms Enter? To this point, we have taken the number of ﬁrms in the market as given, often assuming that there are at most two ﬁrms (as in Examples 15.1, 15.3, and 15.10). We did allow for a general number of ﬁrms, n, in some of our analysis (as in Examples 15.3 and 15.7) but were silent about how this number n was determined. In this section, we provide a gametheoretic analysis of the number of ﬁrms by introducing a ﬁrst stage in which a large number of potential entrants can each choose whether to enter. We will abstract from ﬁrst-mover advantages, entry deterrence, and other strategic considerations by assuming that ﬁrms make their entry decisions simultaneously. Strategic considerations are interesting and important, but we have already developed some insights into strategic considerations from the previous sections and—by abstracting from them—we can simplify the analysis here. Barriers to entry For the market to be oligopolistic with a ﬁnite number of ﬁrms rather than perfectly competitive with an inﬁnite number of inﬁnitesimal ﬁrms, some factors, called barriers to entry, must eventually make entry unattractive or impossible. We discussed many of these factors at length in the previous chapter on monopoly. If a sunk cost is required to enter the market, then—even if ﬁrms can
freely choose whether to enter—only a limited number of ﬁrms will choose to enter in equilibrium because competition among more than that number would drive proﬁts below the level needed to recoup the sunk entry cost. Government intervention in the form of patents or licensing requirements may prevent ﬁrms from entering even if it would be proﬁtable for them to do so. Some of the new concepts discussed in this chapter may 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’ Dile
mma, 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 differentiation, 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 pro�
��t-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 pro�
. 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 proﬁ
le 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 whether 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 K1Þ�
�,, 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 approach, 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
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 changing 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 buyer) 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 voice 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,
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 discount 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

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