In classical competitive equilibrium, the rms in a market maximize prots taking price as given, each rm produces the quantity at which price equals ma

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1 Relative Prots and Competitive Equilibrium Cheng-Zhong Qin and Charles Stuart Department of Economics University of California at Santa Barbara Santa Barbara, California August 30, 2001 Abstract In competitive equilibrium, rms take prices as given and set quantities that equate prices and marginal costs. The price-taking that underlies competitive equilibrium is traditionally justied by an assumption that the number of rms is innite. We show that, when a nite number of rms compete in a market by maximizing relative prots and equilibrium is symmetric, resource allocation is identical to that in competitive equilibrium. Without symmetry when rms maximize relative prots, \quasi-competitive" equilibria can arise in which price is the market average of marginal costs. Keywords: oligopoly, competitive equilibrium, Nash equilibrium, price-taking, relative prot. 0

2 In classical competitive equilibrium, the rms in a market maximize prots taking price as given, each rm produces the quantity at which price equals marginal cost, and price is determined by the intersection of demand and supply. Competitive equilibrium has a central position in the theory of markets owing largely to the rst welfare theorem, which formalizes Adam Smith's idea that a system of competitive markets lead to economic eciency. The price-taking behavior by rms that underlies competitive equilibrium is traditionally justied by the \competitive" assumption that the number of rms is innite. 1 of any single rm has no eect on market price. Under this assumption, the production decision We show that allocations identical to those in classical competitive equilibrium may also obtain in oligopolies with nite numbers of rms that maximize relative prots. 2 (Relative prot is the dierence between own prot and a weighted average of the prots of the other rms in the market, or between own prot and the expected prot of other rms.) One view of the result is that we obtain competitive allocations by substituting a dierent notion of competition. Specically,weweaken the notion of competition by assuming only a nite number of rms, but strengthen it by assuming that a rm benets, through increased relative prot, from reductions in prots of other rms. Because an increase in a rm's output reduces the market price and 1 See e.g. Novshek and Sonnenschein (1980). 2 Related results are in Shubik and Levitan (1980), Jones (1980), Donaldson and Neary (1984), and Lundgren (1996). Shubik and Levitan consider the case of symmetric duopoly with linear demand in which each rm maximizes the minimum relative prot the opponent can induce. They show that the outcome \is the ecient point or competitive solution." Donaldson and Neary, generalizing Jones, provide a result similar to our theorem 1 except that they use a slightly more restricted notion of relative prot (see below). Lundgren studies how to implement relative prot maximization as a device for preventing collusion. His appendix A contains a result similar to our theorem 1 and his appendix B contains an example of duopoly that is a special case of our theorem 3. 1

3 thereby also reduces the revenues but not the costs of other rms, maximization of relative prots induces greater \competitive" pressure downward on price than does maximization of absolute prots. The assumption that rms maximize relative prots is not standard but may be justied under some circumstances. 3 Two cases may be of particular importance. First, managers of rms may have incentives to maximize relative prots when, as is common practice, stockholders judge and reward a manager by comparing the prot obtained by the manager with prots obtained by managers of other rms. 4 Second, relative outcomes determine tness in evolutionary environments, so evolutionary pressures may induce a tendency to maximize relative prots. 5 We take relative prot maximization as given and characterize market equilibrium. After describing an oligopoly with a nite number of rms that maximize relative prots (section I), we show that, when a nite number of rms maximize relative prot (as opposed to own or absolute prot), then any symmetric market (Nash) equilibrium is a competitive equilibrium and vice verse (section II). In section III, we show how the results generalize to settings in which rms have dierent cost functions and equilibria may be asymmetric. In section IV, we generalize further to consider dierentiated-good oligopoly, which is equivalent to a system of monopolistic markets with linked demands. In these latter cases, equilibria need not be competitive but are close to competitive in a sense made precise below. Thus even though each rm in Nash equilibrium sees the market price as dependent on the rm's quantity, the resulting allocation is close to or the same as in a perfectly competitive market in which each rm takes price as given. Because the assumption that rms have no (or little) 3 Smith (1759) and Veblen (1899) stressed the importance of relative outcomes. 4 This justication for relative prot maximization follows Shubik (1971), who points out that \In Wall Street, for example, much of industry analysis is based upon comparisons between rms." 5 Schaer (1989) shows that, if the number of rms is nite, rms that maximize relative prots survive better than rms that maximize absolute prots. 2

4 latitude for altering the prices at which they sell is strong as a description of many actual markets, results here broaden theoretical support for competitive outcomes. I. A SYMMETRIC OLIGOPOLY SETTING: ABSOLUTE AND RELATIVE PROFITS To abstract from issues of competition on the demand side, we consider a single market in which consumers are price takers whose behavior is represented by a non-negative aggregate inverse demand function P (Q): The market contains a nite number n 2 of identical rms that produce and sell a homogeneous commodity. Let q i denote the quantity sold by rm i and let (q 1 q 2 ::: q n ) denote the vector of quantities set by the n rms. The total quantity sold by all rms is Q = Pi q i: Dene Q ;i = Q ; q i : Assume that each rm has the same non-negative cost function C(q). The absolute prot of rm i =1 ::: n is then a i (q i Q ;i )=P(q i + Q ;i )q i ; C(q i ): To dene relative prot in a symmetric setting, let be an n n non-negative matrix with elements! ij that satisfy! ii = 0 and Pj! ij = 1 for i = 1 ::: n: For given the relative prot of rm i =1 ::: n is then r i (q i Q ;i )= a i (q i Q ;i ) ;X j! ij a j(q j Q ;j ): In this formulation,! ij is the weight placed on the absolute prot of rm j as an object of comparison in rm i's relative prot. The condition that row sums of equal one says that each rm compares its absolute prot with the absolute prot of, on average, one other rm. (The condition is dropped in section III when we generalize to allow rms not to be identical.) An important case with row sums equal to one occurs when! ij =1=(n ; 1) for all j 6= i and all i (equal weighting). Each rm's relative prot is then the dierence between 3

5 own absolute prot and the average absolute prot of the other rms in the market. This case may describe a number of real-world stories. One is that managers seek \status" specied as the dierence between own prot and the market-average prot. 6 A second is that each manager is evaluated and rewarded by stockholders on the basis of the dierence between the prot obtained by the manager and the average prot obtained by the other managers in the market. A third is that each manager receives a reward ex post that increases with the dierence between own prot and the prot of one other rm, but each operates ex ante under uncertainty and places probability 1=(1 ; n) on each other rm's prot being the object of comparison. 7 that! ij The opposite extreme from equal weighting is \all-on-one" weighting, meaning = 1 for each i and some j 6= i: Under all-on-one weighting, each rm i maximizes the dierence between i's absolute prot and the absolute prot of a single other rm. This can represent the maximand of managers who compete for status (relative prot) against the manager of only one other rm, or who are rewarded by equity-holders on the basis of a prot comparison with only one other rm. (Each rm need not have the same comparison rm.) Cases between the extremes of equal weighting and all-on-one weighting are also covered by the formulation here. A competitive equilibrium is a pair (p (q 1 q 2 ::: q n)) of a price and a vector 6 The assumption that!ii = 0 rules out! ii = 1 but is otherwise without loss of generality, which implies that maximization of the dierence between own prot and the market-average prot is equivalent to maximization of the dierence between P own prot and the average prot of all other rms. Namely, if 0 <! ii < 1 then a ;! i i ij a = (1 ;! j ii)( a ; Pj6=i!0 i ij a ) where j! 0 ij =! ij =(1 ;! ii ) for all j 6= i and maximization of a i ;Pi! ij a j is equivalent to maximization of a i ;Pj6=i!0 ij a j. 7 It is not necessary to restrict the interpretation of the third case to ex post comparisons with only one other rm. To illustrate, suppose there are three rms, with rm i's manager believing that i's prot will be compared ex post against each other single rm's prot with probability x and against the average prot of the other two rms with probability 1 ; 2x.! ij = x +(1; 2x)(1=2) = 1=2 fori 6= j. In this case, 4

6 of quantities that satises p = P (Q )andp q i ; C(q i ) p q i ; C(q i ) for any q i 0 and for i =1 ::: n: We refer to such a quantity vector (q1 q 2 ::: q n) as a competitive allocation. A symmetric competitive equilibrium is a competitive equilibrium with q 1 = q 2 = ::: = qn, with the common quantity denoted q below. A Nash equilibrium under relative prot maximization is a quantity vector (q1 q 2 ::: q n) that satises r i (q i Q ;i) r (q i Q ;i) for any q i 0 and for i =1 ::: n: We sometimes also refer to such a quantity vector as a Nash allocation. A symmetric Nash equilibrium is a Nash equilibrium with q 1 = q 2 = ::: = qn, with the common quantity denoted q below. II. SYMMETRIC EQUILIBRIA A simple example suggests why the allocation in Nash equilibrium when all rms maximize relative prots is the same as the competitive allocation. 8 Consider a duopoly with dierentiable demand and cost functions (primes denote derivatives) where relative prot equals own prot minus opponent prot (! 12 =! 21 = 1 and! 11 =! 22 = 0). In equilibrium when both rms sell positive quantities, the rate of change in rm 1's relative prot given a small change in its quantity is P 0 (q 1 + q2)q 1 + fp (q 1 + q2) ; C 0 (q1)g ;P 0 (q 1 + q2)q 2: In Nash equilibrium, this sum of four terms equals zero. In competitive equilibrium, price equals marginal cost, so the two terms in braces in the middle of the expression equal zero. This suggests that the Nash and competitive allocations are the same if the two outer terms sum to zero, which is clearly the case in symmetric equilibrium. An interpretation is that price-taking in competitive equilibrium means eectively that a rm's behavior is not inuenced on net by consideration of price changes caused by changes in the quantity the rm sells. In Nash equilibrium when both rms 8 The example is similar to the case studied by Shubik and Levitan (1980). 5

7 maximize relative prots, a rm takes account of the loss in own revenue due to the price reduction caused by an increase in the quantity the rm sells. This adds the term P 0 q 1 to rm 1's rst-order condition. On the other hand, a rm maximizing relative prot also takes account of the loss in the other rm's revenue and hence prot caused by the induced price reduction. This adds the term ;P 0 q 2 to rm 1's rst-order condition. Under symmetry, q 1 = q2 so the two additional terms net. 9 This logic generalizes to any nite number of rms under weaker assumptions on demand and cost functions. The following two theorems establish that a price and a symmetric allocation are a competitive equilibrium if and only if the allocation is a symmetric Nash equilibrium of the game in which each rm's payo is its relative prot. Proofs are in an appendix: 10 Theorem 1 Suppose P is continuous, C is convex, and is non-negative with! ii = 0 and Pj! ij =1 for i =1 ::: n: Then every symmetric Nash allocation of the game in which each rm's payo is its relative prot is a symmetric competitive allocation. Although theorem 1 does not rule out equilibria in which q = 0 mild conditions ensure that q > 0: One such condition is that there exists q > 0 such that p(q) >C(q)=q that is, there is a quantity thatwould yield positive prot if only one rm were in the market and set that quantity. To see this, let (q q ::: q ) be a symmetric Nash equilibrium. Then P (q + Q ;i)q ; C(q) ;X! ij [P (q + Q ;i)q ; C(q )] 0 (1) j 9 It is in this sense that the decrease in competitiveness caused by going from an innite (in a traditional competitive model) to a nite number of rms is just balanced by the increase in competitiveness caused by going from absolute prot maximization to relative prot maximization. 10 Donaldson and Neary's (1984) theorem 1 diers from ours in that thay consider only equal weighting (! ij =1=(n ; 1) for all j 6= i and all i) and make stronger assumptions on cost functions to ensure that a unique Nash equilibrium exists. 6

8 for any q 0andall i: Because Pj! ij =1 (1) can be rewritten: P (q + Q ;i)(q ; q ) C(q) ; C(q ): Then q = 0 implies P (q)q C(q) for any q > 0 which contradicts the additional condition. Theorem 2 Suppose P is downward sloping and is non-negative with! ii = 0 and Pj! ij = 1 for i = 1 ::: n. Then every symmetric competitive allocation is a symmetric Nash allocation of the game in which each rm's payo is its relative prot. III. GENERALIZATION TO ASYMMETRIC SETTINGS The theorems of the preceding section assume that all rms have the same cost function and that equilibria are symmetric. The symmetry condition is strong, particularly if cost functions are permitted to dier across rms. To study potentially asymmetric cases with dierent cost functions, let rm i have non-negative, convex cost function C i remove the restriction on that Pn j=1! ij = 1 for i = 1 ::: n and maintain the assumptions that! ii = 0 and! ij 0 for i j =1 ::: n: A generalization of the theorems of the previous section is possible without symmetry although the result is weaker. We say that an equilibrium is quasicompetitive if it has a market price that equals the average of marginal costs across rms with positive output. Quasi-competitive equilibrium entails substantial competition and is conceptually close to competitive equilibrium. Quasi-competitive equilibrium reduces to competitive equilibrium under symmetry when rms have the same marginal costs. Quasi-competitive equilibria can arise without symmetry: Theorem 3 Suppose P and C i are dierentiable and is non-negative with! ii =0 for i = 1 ::: n. If (q 1 q 2 ::: q n) is any Nash equilibrium of the game in which each 7

9 rm's payo is its relative X prot, then the associated X equilibrium price satises P (Q )= Ci(q 0 i )=n + (! j ; 1)P 0 (Q )q j =n (2) i2n j2n where N = fj 2 N j q j > 0g, n is the number of rms in N and! j = Pi2N! ij for j 2 N : Theorem 3 takes account of the possibility that some rms may have zero outputs in equilibrium. A polar case is that when n = 1 (2) implies that P (Q ) is monopoly price, because! ii = 0 then implies! j = 0 for the rm with positive output. More generally, let denote the submatrix of that corresponds to the rows and columns in N which is the set of rms with positive equilibrium outputs. The jth column sum of is then! j and the theorem implies that equilibrium is quasi-competitive if all such column sums equal one. 11 If column sums of do not equal one and demand slopes down, then the second sum on the right-hand side of (2) need not equal zero. If column sums tend to exceed one for rms with relatively large outputs so rms with large outputs are disproportionately the objects of comparison, for instance, then the second sum is negative and equilibrium price is smaller than the average of marginal costs. If column sums tend to be less than one for rms with relatively large outputs, on the other hand, then the second sum is positive and equilibrium price is larger than the average of marginal costs. It is straightforward to understand this pattern and equation (2) generally. As noted, a marginal increase in rm i's quantity has a \direct" positive eect on i's relative prot equal to the market price, a \direct" negative eect equal to i's marginal cost, and two \indirect" eects that arise because the market price falls. The rst indirect eect is that a small increase in q i reduces the market price and hence the absolute 11 Lundgren (1996, appendix B) contains an example of duopoly in which column sums equal one and equilibrium is quasi-competitive. 8

10 prot of each other rm j by P 0 (Q )q j in equilibrium, which increases i's relative prot by Pj2N! ij P 0 (Q )q j. This is a benet to i: The market average of such indirect benets across all rms is Pi2N Pj2N! ij P 0 (Q )q j =n = P 0 (Q ) Pj2N! j q j =n : The second indirect eect is that a small increase in q i reduces the market price and hence lowers i's revenue and relative prot by P 0 (Q )q i. This is a cost to i. The market average of such indirect costs across all rms is Pi2N P 0 (Q )q i =n = P 0 (Q ) Pj2N q j =n : The second sum in (2) is precisely the market-average indirect benet of quantity increases minus the market-average indirect cost. When this term is zero in equilibrium, price changes induced by quantity changes have no net incentive eect on the total equilibrium quantity and hence price. In this case, (2), says that direct eects of quantity increases on revenue and cost characterize equilibrium so equilibrium price equals the market average of marginal costs. Alternatively, if the marketaverage indirect benet of quantity increases exceeds the market-average indirect cost in equilibrium, then average incentives to produce output are great and equilibrium price is below the market average of marginal costs. Finally, if the market-average indirect benet of quantity changes is less than the market-average indirect cost in equilibrium, then average incentives to produce output are low and equilibrium price is above the market average of marginal costs. IV. DIFFERENTIATED-GOOD OLIGOPOLY Instead of restricting attention to oligopoly in the market for a single, homogeneous good, we now characterize equilibrium for dierentiated-good oligopoly. As is standard when n 2 rms sell dierentiated goods, consumers are assumed to be price takers whose behavior is represented by n non-negative aggregate inverse demand functions. 12 Formally, P i (q i q ;i ) now denotes the inverse demand faced by 12 see e.g. Gabszewicz and Vial (1971), Singh and Vives (1984), Klemperer and Meyer (1986). 9

11 rm i = 1 ::: n where q ;i is the vector of quantities of all rms other than rm i: Given the relative prot of rm i is r i (q i q ;i )=P i (q i q ;i )q i ; C i (q i ) ;X j! ij [P j (q i q ;i )q j ; C j (q j )]: This generalizes the specication in previous sections in that i's output and the output of each other rm j 6= i may have any degree of substitutability previously, perfect substitutability was assumed. We prove in the appendix: Theorem 4 Suppose P i and C i are dierentiable and is non-negative with! ii =0 for i = 1 ::: n. If q = (q 1 q 2 ::: q n) is any Nash equilibrium of the game in which each rm's payo is its relative prot, then the associated equilibrium prices P i (q ) X! q j (3) for i 2 N satisfy X X X P i (q )= Ci(q 0 i )+ i2n i2n j2n where N = fj 2 N j q j > j! (q j(q ) ij i2n j If all goods are perfect substitutes so P i (q i q ;i ) = P (q i + Q ;i ) for all i, then theorem 4 reduces to theorem 3. Without perfect substitutability, however, neither within-condition cancellation nor cross-condition cancellation is possible so the result diers from theorem 3. If all goods are substitutes and own-price eects dominate cross-price eects (or if all goods are complements), for instance, then the marketaverage equilibrium price exceeds the market-average marginal cost. An implication of theorem 4 is that relative prot maximization does not suce to ensure that equilibrium is close to quasi-competitive even if all column sums of are close to one. Namely, dierentiated-good oligopoly can be interpreted as a system of monopolistic markets in which each market's demand depends on the quantities sold in all markets. Suppose the monopolists maximize relative prots dened as dierences between own prots and weighted sums of prots of other monopolists. 10

12 From theorem 4, the average equilibrium price is then close to the average marginal cost so equilibrium is close to quasi-competitive if all goods are close substitutes and all column sums of are close to one. Equilibrium need not be close to quasicompetitive, however, if the goods sold by the dierent monopolists are not close substitutes. CONCLUDING REMARK The analysis here shows how relative prot maximization results in greater competitive pressure downward on price than does absolute prot maximization. Specically, relative prot maximization can lead to an allocation identical to or close to the competitive allocation even if the number of rms is nite. Given the eciency properties of competitive equilibrium, this suggests that resources may be allocated (approximately) eciently in market systems even if the number of competing rms is small. Such a suggestion is consistent with Harberger's (1954) empirical estimate that the eciency costs from prices in excess of competitive prices in manufacturing are on the order of tenths of a percent of GNP in the U.S., and his conclusion that \in the big picture of our manufacturing economy, we need not apologize for treating it as competitive, for in fact it is awfully close to being so." 13 If concern with relative prots leads to greater economic eciency than does concern with absolute prots, then one might expand Adam Smith's idea that selfinterest in the marketplace can be good for society. Concern with relative prots means that rms or their managers gain from reductions in the prots of competitors and may take actions in part because these actions reduce competitors' prots. Such actions are sometimes termed spiteful. Thus in a marketplace with a nite number of rms, self-interest and even spitefulness may sometimes be good for society. 13 Laboratory-market experiments commonly nd that outcomes are close to purely competitive even with small numbers of players see Holt (1995) for a survey of the experimental evidence. 11

13 APPENDIX Proof of Theorem 1: Let (q q ::: q ) be a symmetric Nash allocation of the game in which each rm's payo is its relative prot. Then for any i and q 0 P (q + Q ;i)q ; C(q) ; Pj! ij[p (q + Q ;i)q ; C(q )] P (q + Q ;i)q ; C(q ) ; Pj! ij[p (q + Q ;i)q ; C(q )] or equivalently P (q + Q ;i )(q ; q ) C(q) ; C(q ): (4) Suppose rst that q > 0: Because C i is convex on < + theorem 23.1 of Rockafellar (1970, pp ) implies that both the right derivative C 0 (q + ) and the left derivative C 0 (q ; ) exist at q and that C 0 (q ; ) C 0 (q + ): From (4), when q>q and when q<q. From (5), (6), and the continuity ofp, P (q + Q ;i) C(q) ; C(q ) q ; q (5) P (q + Q ;i) C(q) ; C(q ) q ; q (6) C 0 (q ; ) P (q + Q ;i) C 0 (q + ) which from theorem 23.2 of Rockafellar (1970, pp. 216) implies P (q + Q ;i)(q ; q ) C(q) ; C(q ) or P (q + Q ;i)q ; C(q) P (q + Q ;i)q ; C(q ): This says that (p (q q ::: q )) with p = P (Q ) is a symmetric competitive equilibrium. 12

14 Now suppose that q = 0: Again, theorem 23.1 of Rockafellar (1970) implies that the right derivative ofc exists at 0 and that C 0 (0 + )q C(q) ; C(0) (7) for any q 0: Because P is continuous, (5) implies P (0) C 0 (0 + ): (8) Together, (7) and (8) yield P (0)q C(q) ; C(0) so q =0maximizes i's absolute prot at price p = P (0): Thus (p (0 0 ::: 0)) is a symmetric competitive equilibrium. 2 Proof of Theorem 2: Let (q q ::: q ) be a symmetric competitive allocation. For any i and q 0 P (q + Q ;i)q ; C(q) ; Pj! ij[p (q + Q ;i)q ; C(q )] = P (q + Q ;i)q ; C(q) ; P (q + Q ;i)q + C(q ) (9) = P (q + Q ;i)(q ; q ) ; [C(q) ; C(q )]: Because q is the competitive quantity for rm i P (q + Q ;i)(q ; q ) C(q) ; C(q ): (10) Combining (9) and (10), P (q + Q ;i)q ; C(q) ; Pj! ij[p (q + Q ;i)q ; C(q )] [P (q + Q ;i) ; P (q + Q ;i)](q ; q ) 0 13

15 where the last inequality uses the fact that P is downward sloping. Because P (q + Q ;i)q ; C(q i ) ; Pj! ij[p (q + Q ;j)q ; C(q j )] = P (q + Q ;i)q ; C(q ) ; [P (q + Q ;i)q ; C(q )] =0 it follows that r i (q Q ;i) r i (q i Q ;i) for any q 0 and all i: Thus (q q ::::q ) is a symmetric Nash allocation of the game in which each rm's payo is its relative prot. 2 equivalent to Proof of Theorem 3: Note rst that r i (q i Q ;i) r i (q i Q ;i) for q i 0 is P (Q )(q i ; q i )+ ; P (q i + Q ;i) ; P (Q ) q i ; Pj! ij; P (qi + Q ;i) ; P (Q ) q j C i (q i ) ; C i (q i ) for q i 0: Thus if q i > 0 then X P (Q )+P 0 (Q )q i ; j2n! ij P 0 (Q )q j = C 0 i(q i ): (11) Summing both sides of (11) X over i 2 N and rearranging X terms: P (Q )= Ci(q 0 i )=n + P 0 (Q ) (! j q j ; q j )=n : 2 i2n j2n Proof of Theorem 4: First, r i (q i q ;i) r i (q i q ;i) for q i 0 is equivalent to P i (q )(q i ; q i )+ ; P i (q i q ;i) ; P i (q ) q i ; Pj! ij; Pj (q i q ;i) ; P j (q ) q j for q i 0: Thus if q i C i (q i ) ; C i (q i ) > 0 then P i (q i(q i q i ; j! (q ) ij q j = i(q 0 i ): (12) j2n i Summing both sides of (12) over i 2 N and rearranging terms: X X X P i (q )= Ci(q 0 i )+ i2n i2n j2n j! (q j(q ) ij i2n j 14! q j : 2

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