Bertrand–Edgeworth equilibria with unobservable output, uncoordinated consumers and large number of firms

Economics Letters 63 (1999) 207–211

Bertrand–Edgeworth equilibria with unobservable output, uncoordinated consumers and large number of firms Prabal Roy Chowdhury* ITDD, CSDILE, School of International Studies ( SIS), Jawaharlal Nehru University ( JNU), New Delhi-110067, India Received 25 September 1998; accepted 13 January 1999

Abstract We examine a Bertrand–Edgeworth model of price competition where the output levels of the firms are unobservable and consumers are uncoordinated. We show that if the number of firms is large enough then any price, greater than the competitive one, can be sustained as a pure strategy Nash equilibrium.  1999 Elsevier Science S.A. All rights reserved. Keywords: Pure strategy Bertrand equilibria; Coordination failure JEL classification: D43; D41; L13

1. Introduction We examine a Bertrand–Edgeworth model of price competition where the firms simultaneously decide on both their price and output levels. Moreover, the firms are free to supply less than the quantity demanded. In such models it is generally argued that the Edgeworth (1897) paradox holds, i.e. equilibrium in pure strategies do not exist.1 In this paper we seek to develop some conditions which ensure the existence of such pure strategy equilibria.2 The point of departure of our approach lies in the way consumer behaviour is modelled. We assume that consumers cannot observe the output level of the different firms. Moreover, since firms have an incentive to overstate their output levels (so as to attract demand), output announcements by firms lack credibility. Thus consumers base their decisions on price announcements only. Furthermore, consumers are uncoordinated, and, given the price level, randomly decide which one of the firms (charging this price) to approach. Let us consider a situation where there are n firms, all of whom charge the same price p. We do not model consumer behaviour formally, but, given a large number of consumers, it seems reasonable to *Tel.: 10091-11-6107676; fax: 10091-11-6165886. E-mail address: [email protected] (P.R. Chowdhury) 1 See either Dixon (1987), theorem 1, or Friedman (1988), theorem 1, for a formal statement of this result. 2 An alternative approach is to look for equilibria in mixed strategies (often using the fixed point theorems for discontinuous games developed by Dasgupta and Maskin (1986a, 1986b)). This is the approach taken, among others, by Allen and Hellwig (1986), Maskin (1986) and Vives (1986). 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00040-3

 1999 Elsevier Science S.A. All rights reserved.

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assume that the total demand d( p) will be distributed among the n firms in some given ratio, say equally. Hence the demand facing each of these firms would be d( p) /n. Note that as the number of firms increases, the demand going to any one firm tends to zero. It is this property which drives our result. For technical reasons we allow the prices to vary over a grid, where the grid can be as small as we like. We then argue that if the number of firms is large enough, then any price larger than the competitive one can be sustained as a Nash equilibrium. The paper closest to our own is by Simon (1984), who shows that in a general equilibrium framework with strategic consumers, price setting equilibrium in pure strategies can be sustained. In contrast, we assume that consumers are uncoordinated and less than perfectly informed, rather than strategic. In a couple of papers, Dixon (1987, 1990) also demonstrates the existence of pure strategy equilibria in Bertrand–Edgeworth models of price competition. However, while Dixon (1987) examines epsilon-Nash rather than exact Nash equilibria, Dixon (1990) is driven by an assumption on the firm side (that they avoid turning customers away), rather than on consumers. Finally, Dastidar (1995) uses a Bertrand–Chamberlin framework, where firms supply all demand, to demonstrate that equilibria in pure strategies exist.

2. The model There are N identical firms, all producing the same homogeneous good, where we assume that N is as large as we require. The demand function is q 5 d( p), where d:[0, `) → [0, `), and the common cost function of all the firms is C(q) 5

Ha0, 1 c(q),

if q . 0, otherwise,

(1)

where a . 0. Throughout we maintain the following assumptions on the demand and the cost functions. • A.1: The demand function d( p) is bounded. • A.2: The variable cost c(q) is once continuously differentiable, increasing and strictly convex. Moreover, c(0) 5 0.3 We assume that prices vary over a grid.4 This assumption is not too unrealistic given that there is a smallest possible unit beyond which money is not divisible. From a purely technical point of view this assumption allows us to avoid some open set problems associated with this game. Define the set of feasible prices P 5 h p0 , p1 , . . . j, where p0 5 0, and pi 5 pi21 1 b, ;i [ h1, 2, . . . j. Thus firm i’s strategy consists of a choosing both a price pi [ P and an output qi [ [0, `). All firms move simultaneously. 3

Since we have a fixed cost of production a, c(0) 5 0 is not a restriction. Other papers that use this assumption include Ray Chaudhuri (1996). Dasgupta and Maskin (1986a) discuss the sensitivity of equilibrium outcomes to the size of the grid. 4

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We then specify the residual demand function. Consider a situation where there are n firms, all of whom charge the price p. Let the number of firms charging a price strictly less than p be m, and let the price and the output vector of these firms be Pm 5 h p1 , . . . , pm j, and Q m 5 hq1 , . . . , qm j respectively. Then the residual demand function is defined as follows: R(n, p, Pm , Q m ) 5 f(n)D( p, Pm , Q m ).

(2)

We assume that the residual demand function satisfies the following assumption. A.3 • (i) f(n) is strictly decreasing in n, and lim n →` f(n) 5 0. • (ii) D( p, Pm , 0 m ) 5 d( p), where 0 m is the m-dimensional null vector. • (iii) D( p, Pm , Q m ) is decreasing in Q m .5 Observe that assumption A.3(i) follows from our basic premise that output is unobservable and that consumers are uncoordinated. The other two assumptions simply reflect the fact that R(n, p, Pm , Q m ) is a residual demand function. In fact these assumptions are general enough to include many different rationing rules, including the parallel and the proportional rationing rules as special cases. Let p ( p, q) denote the profit of a firm charging a price p and selling an amount q. Next define S(n, p, Pm , Q m ) 5 minhc 9 21 ( p), R(n, p, Pm , Q m )j.6

(3)

We can now define the supply function of a firm s(n, p, Pm , Q m ) 5

H

S(n, p, Pm , Q m ), 0,

if p ( p, S(n, p, Pm , Q m )) $ 0, otherwise.

(4)

Thus we follow Edgeworth (1897) in assuming that firms supply at most till the output level where price equals marginal cost, rather than Chamberlin (1933), who assumes that firms meet the whole of the demand coming to them. We then define the competitive price p * as the minimum price such that a single firm, supplying a positive amount, makes a profit of zero. We assume that the game is non-trivial in the sense that such a p * always exists. Definition. Consider any p [ P such that p $ p * . We define n( p) to be the largest possible number of firms that can charge p, supply a strictly positive amount and make non-negative profits. Thus n( p) is the largest possible integer such that

p ( p, S(n( p), p, Pm , 0 m )) $ 0 . p ( p, S(n( p) 1 1, p, Pm , 0 m )).

(5)

1 Of course, if o n f(n) 5 1, then f(n) 5 ]. Also note that instead of assuming that the f(n) and the D( p, Pm , Q m ) functions are n the same for all firms, we can instead assume that they are of the form fi (n), and Di ( p, Pm , Q m ). However, if an appropriately modified version of assumption A.3 holds, then our result goes through. 6 21 Note that the only role played by the assumption of a strictly convex cost function is to ensure that c 9 ( p) is defined. It is straightforward to relax this assumption but only at the cost of additional notational complexity. 5

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If there is some p $ p * such that, p ( p, S(n, p, Pm , 0 m )) , 0, ;n $ 1, then we define n( p) 5 0. It is easy to see that for all p such an n( p) exists and is unique. This follows since we can use assumption A.3(i) and the fact that d( p) is bounded to observe that, • (i) p ( p, S(n, p, Pm , 0 m )) is decreasing in n, and • (ii) lim n →` p ( p, S(n, p, Pm , 0 m )) 5 2 a , 0. ˆ Definition. Let Pn( pˆ ) be the n(pˆ )-dimensional p-vector, and let Q n( pˆ ) be the n(pˆ )-dimensional s(n(pˆ ), ˆ Pm , 0 m )-vector. We then define n( p, pˆ ), where p . p, ˆ as the largest possible integer such that p,

p ( p, S(n( p, pˆ ), p, Pn( pˆ ) , Q n( pˆ ) )) $ 0 . p ( p, S(n( p, pˆ ) 1 1, p, Pn( pˆ ) , Q n( pˆ ) )).

(6)

If there is some p . pˆ such that p ( p, S(n, p, Pn( pˆ ) , Q n( pˆ ) )) , 0, ;n $ 1, then we define n( p, pˆ ) 5 0. We are now in a position to state and prove the main result of this paper. Proposition 1. Consider any pˆ [ P such that pˆ $ p * . Then the price pˆ can be sustained as a Nash equilibrium using the following strategies: ˆ and supply s(n(pˆ ), p, ˆ Pm , 0 m ). 1. There are exactly n(pˆ ) firms all of whom charge the price p, ˆ * 2. Consider any p [ P, such that p . p $ p . Then there are at least n( p) 1 1 firms all of whom charge p and have an output of zero. ˆ Then there are at least n( p, pˆ ) 1 1 firms all of whom charge 3. Consider any p [ P, such that p . p. p and have an output of zero. Proof. From the definition of n( p) it follows that none of the firms charging a price different from pˆ can charge pˆ and make a gain. Similarly, none of the firms charging pˆ can deviate to any price less ˆ For prices greater than p, ˆ it is sufficient to recall the definition of n( p, pˆ ) and the fact that the than p. residual demand function is decreasing in Q m . h ˆ there are just enough firms charging pˆ The idea is quite simple. For any given equilibrium price p, to ensure that these firms break even. For any other price p, there are too many firms who charge this price. Coordination failure among consumers ensures that none of these firms have a sufficient demand. This precludes deviation to this price p by other firms, thus sustaining pˆ as an equilibrium price. Thus contrary to the standard intuition, even a very large number of firms does not drive prices to the competitive level. We then observe some properties of the equilibrium outcomes under Proposition 1. First, notice 21 that, if c 9 (pˆ ) , f(n(pˆ ))d(pˆ ), then the equilibrium outcome would involve an excess demand. In a model where firms are free to supply less than the quantity demanded and where the cost function is ˆ discontinuous at zero, this, of course, is only to be expected. Second, it is possible that for some p, ˆ n(p ) 5 0. Then the equilibrium involves zero transactions and there is complete market failure. Third, Proposition 1 implies that even with decreasing average costs, the equilibrium price pˆ may be greater

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than the so called contestable price p * (see Baumol et al. (1982)). Thus in the presence of coordination failure, decreasing average costs, identical technologies and a large number of firms (a proxy for free entry) is not enough to guarantee the contestable price. Finally, note that we restrict ˆ the supply is zero. If, however, attention to equilibrium outcomes where at any price greater than p, 21 ˆ firms c 9 (pˆ ) , f(n(pˆ ))d(pˆ ), then there may be other equilibria where, for some price greater than p, supply a positive amount. Clearly Proposition 1 is driven by our specification of the residual demand function. The result may break down if either of the two underlying premises, that of unobservable output and uncoordinated consumers, fail to hold. Assume, to begin with, that demand is observable, and consider a situation where there are n firms all of them charging the price p. Suppose only one of these firms, say firm 1, supplies a positive amount. In this case all the demand would go to firm 1 since the consumers can observe that firm 1 is the only firm that can meet their demand. Now consider Proposition 1. With a demand function like this, for b small, it would be profitable for firms charging a price of pˆ to to increase its price, so that Proposition 1 breaks down. (This is essentially the Edgeworth (1897) paradox.) Next consider the case where the consumers can coordinate their actions. Suppose at a given price the consumers form a deterministic ranking over the firms charging this price, say firm 1, firm 2 etc, and then approach these firms en masse, following the given ranking. Thus all the consumers first approach firm 1, then, if there is any demand left, they approach firm 2, etc. It is clear that Proposition 1 breaks down in this case also and for the same reason. In conclusion, the idea that coordination problems can lead to market failure is not new. What this paper demonstrates is that such problems may ensure the existence of pure strategy equilibria in Bertrand–Edgeworth models. Furthermore, existence seems to require very mild restrictions on the demand and the cost functions.

References Allen, B., Hellwig, M., 1986. Bertrand–Edgeworth oligopoly in large markets. Review of Economic Studies 53, 175–204. Baumol, W., Panzar, J., Willig, R., 1982. Contestable Markets and the Theory of Industrial Structure, Harcourt Brace Jovanovich, New York. Chamberlin, E.H., 1933. The Theory of Monopolistic Competition, Harvard University Press, Cambridge. Dasgupta, P., Maskin, E., 1986a. The existence of equilibrium in discontinuous economic games, I: Theory. Review of Economic Studies 53, 1–26. Dasgupta, P., Maskin, E., 1986b. The existence of equilibrium in discontinuous economic games, II: Applications. Review of Economic Studies 53, 27–41. Dastidar, K.G., 1995. On the existence of pure strategy Bertrand equilibrium. Economic Theory 5, 19–32. Dixon, H., 1987. Approximate Bertrand equilibria in a replicated industry. Review of Economic Studies 54, 47–62. Dixon, H., 1990. Bertrand–Edgeworth equilibria when firms avoid turning customers away. Journal of Industrial Economics 39, 131–146. Edgeworth, F., 1897. La teoria pura del monopolio. Giornale Degli Economisti 40, 13–31. Friedman, J.W., 1988. On the strategic importance of prices versus quantities. Rand Journal of Economics 19, 607–622. Maskin, E., 1986. The existence of equilibrium with price setting firms. American Economic Review 76, 382–386. Ray Chaudhuri, P., 1996. Contestability as a Bertrand equilibrium. Economics Letters 50, 237–242. Simon, L., 1984. Bertrand, the Cournot paradigm and the theory of perfect competition. Review of Economic Studies 51, 209–230. Vives, X., 1986. Rationing rules and Bertrand–Edgeworth equilibria in large markets. Economics Letters 18, 140–153.