Economics Letters 68 (2000) 279±285 www.elsevier.com/locate/econbase

Bertrand competition and Cournot outcomes: further results

Nicolas Boccardab, , Xavier Wauthy *

a ,34Voie du Roman Pays, 1348 LLN, Brussels, Belgium bCEREC, Facultes Universitaires Saint-Louis, Blvd du Jardin Botanique 43, B-1000 Brussels, Belgium Received 27 September 1999; accepted 8 February 2000

Abstract

The model of Kreps and Scheinkman where ®rms choose capacities and then compete in price is extended to . Further, capacity is an imperfect commitment device: ®rms can produce beyond capacities at an additional unit cost u. When u is larger than the Cournot price, the Cournot obtains in the unique perfect equilibrium. When u decreases from the Cournot price towards zero, the whole range of prices, from Cournot to Bertrand, is obtained in equilibrium.  2000 Elsevier Science S.A. All rights reserved.

Keywords: Price competition; Capacity commitment

JEL classi®cation: D43; F13; L13

1. Introduction

According to the Bertrand paradox, `two is enough for competitive outcomes.' This result, however, is well known to rely on the hypothesis of constant and for this reason lacks any generality [this critique of Bertrand (1883) dates at least back to Edgeworth (1925)]. Still, this paradox is often contrasted with the Cournot outcomes and reconciling these two approaches has been the aim of many papers, the most spectacular being Kreps and Scheinkman (1983) (hereafter KS). Two ®rms invest in capacities and then compete in prices with constant marginal cost up to capacity and an arbitrarily large marginal cost above capacity. Investing in limited capacity has a strategic value because it amounts to committing not to be aggressive in the pricing game. KS reconcile the Cournot and Bertrand approaches by showing that the Cournot outcome is the unique subgame perfect equilibrium of their game. This result has also been much criticised. In particular because of the particular rationing rule, the ef®cient one, retained for the analysis. Still, the

*Corresponding author. Tel.: 132-2-211-7949; fax: 132-2-211-7943. E-mail address: [email protected] (X. Wauthy)

0165-1765/00/$ ± see front matter  2000 Elsevier Science S.A. All rights reserved. PII: S0165-1765(00)00256-1 280 N. Boccard, X. Wauthy / Economics Letters 68 (2000) 279 ±285 fact that capacity commitment relaxes price competition and drives equilibrium outcomes towards Cournotian ones is much less controversial. An open question remains: to what extent is the `rigid capacity' assumption central for this result to hold? In other words, would smoothly increasing marginal cost (or weaker forms of quantitative constraints) lead to similar outcomes? This question is interesting from a theoretical point of view, but it also belongs to the class of generalisations that seem necessary to pursue robust empirical studies on oligopolistic markets. As argued, for instance by Tirole (1988) (Chapter 5, p. 244), ``the Bertrand and Cournot models should not be viewed as two rival models giving contradictory predictions of the outcome of competition in a given market. (After all, ®rms almost always compete in prices.) Rather, they are meant to depict markets with different cost structures.'' It seems fair, however, to say that a rigorous link between the shape of marginal cost and equilibrium price is still missing under strategic price competition. The main goal of the present paper consists precisely in showing how the whole range of prices, from Bertrand towards Cournot, can be sustained as equilibrium outcomes in oligopolistic industries, depending on the shape of marginal costs. In this letter we extend the KS result to the case of imperfect commitment and many ®rms. Firms commit to capacities in a ®rst stage and then compete in price in the second stage. In the second stage, they may produce beyond installed capacity, but have to incur for this an extra unit cost, denoted by u. The subgame perfect equilibria of this game exhibit the following features. Capacity has its full commitment value whenever producing beyond capacity entails an additional unit cost at least equal to the corresponding Cournot price. The lower the value of the additional unit cost, the closer the price to the competitive benchmark.

2. Bertrand±Edgeworth competition and imperfect commitment

The game tree is identical to KS except for the oligopoly setting: n $ 2 ®rms choose costly capacities and then compete in price in the market for an homogeneous product. Rationing, if any, is organised according to the ef®cient rationing rule. Under this rule, low pricing ®rms are served ®rst and ties are broken evenly. The ®rm exhibiting the highest price is left with a residual demand (if any) which is simply de®ned as a function of the other ®rms' aggregate capacity. The cost structure in the pricing game is borrowed from Dixit (1980). The marginal cost up to the capacity level is w.l.o.g. zero while it is some positive u beyond (this jump typically measures the legal wage gap for overtime work). Thus, in our capacity-price game tree, ®rm i invests in capacity xiiat cost c(x ), then all ®rms compete in prices for the demand function D(.) with the same production cost structure

0, if q # x , c9(q) 5 i for all i # n. i H u,ifq . xi, The original KS model corresponds to an in®nite u; still, any value larger than the zero-demand market clearing price D 21(0) would trivially yield their result. As a benchmark we consider the basic model of Cournot oligopolistic competition among n ®rms having the same convex cost function c(.). The aggregate consumer demand is D( p), its inverse P(x).

Let x2ij;o ±ijx be the total quantity produced by ®rm i's opponents. A nil production is clearly optimal if x2iii. D(0), otherwise ®rm i's pro®t is xP(x 1 x2ii) 2 c(x ). We assume that xP(x 1 z)is N. Boccard, X. Wauthy / Economics Letters 68 (2000) 279 ±285 281

concave in x for all z. Therefore, the pro®t maximising quantity rc(x2i ) is the unique solution of P(x 1 z) 1 xP~(x 1 z) 2 c~(x) 5 0 and satis®es

≠r P~(x 1 z) 1 xPÈ(x 1 z) ]c 52]]]]]]]]] [ [21;0]. ≠z 2P~(x 1 z) 1 xPÈ(x 1 z) 2 cÈ(x)

We denote r(x2i ) the best reply with zero production cost; it plays a central role in the price competition. The symmetric Cournot± is the solution xÅ , D(0)/n of x 5 rc((n 2 1)x). With constant marginal cost c and demand P(z) 5 1 2 z, we obtain

1 2 c 2 z r (z) 5 ]]] c 2 and

1 2 c xÅ 5 ]]. n 1 1

We are now in a position to solve the two-stage game with imperfect capacity commitment and price competition. The particular shape of marginal costs affects the analysis of price . An upward price deviation may be pro®table when other ®rms are likely to ration consumers. This requires ®rst that the demand addressed to them exceeds their aggregate capacity and second that they are not willing to meet demand beyond capacity, i.e. the price is below u. Otherwise, the standard Bertrand analysis applies as shown in Lemma 1.

Lemma 1. The price equilibrium is the pure u if oiix # D(u ), the pure strategy P(x1 1 x21 ) if oiix . D(u ) and x1 # r(x21 ), a mixed strategy equilibrium otherwise.

Proof. W.l.o.g. ®rm 1 has the largest capacity in the subgame following the play of (x ) . Let p and ii#ni] pÅ be the lower and upper bounds of ®rm i's equilibrium distribution F . Let H ; arg min p , i ii] #ni] p ; min p , HÅ ; arg max pÅÅand p ; max p Å. The cardinal of a set E is denoted by [E. ]]i#ni i#ni i#ni Existence of an equilibrium is guaranteed by Theorem 5 of Dasgupta and Maskin (1986). By lowering its price a ®rm always bene®ts from an increase in demand (this property is not in¯uenced by our rationing rule); its payoff is therefore left lower semi-continuous (l.s.c.) in its price, thus weakly l.s.c. The sum of payoffs is u.s.c. because discontinuous shifts in demand occur only when two ®rms or more derive the same pro®t.

Claim 1. [H] . 1 or the equilibrium is the pure strategy u.

If [H 5 1, some ®rm k enjoys demand D( pki)on[p;min[¤ [Hip ] and two cases occur: ] ]]] (i) p ,u. If ®rm k is constrained at p its revenue is the strictly increasing function px,a ]] kk contradiction to p being in the support of the equilibrium distribution F . Thus, D(p) , x and ]]kk because pD( p ) is a non-constant function, it must be the case that p is the price kk ] P(r(0)). Since no other price (irrespective of what may play the other ®rms) can yield the 282 N. Boccard, X. Wauthy / Economics Letters 68 (2000) 279 ±285

monopoly payoff, ®rm k must be playing the pure strategy P(r(0)), but then any other ®rm i undercuts it, contradicting the optimality of its own equilibrium strategy. (ii) p $u. We are contemplating the classical Bertrand price competition, the outcome of which is ] pricing at the marginal cost u.

Claim 2. ;i [H, P * 5px . ] ii] • p ,u. If ®rm i [H deviates to p2 (this is shorthand for p 2 ´ where ´ is a small positive real ]]]] number) its demand may jump upward; in order for p to be in the support of an equilibrium ] distribution it must be the case that this does not happen, thus the payoff is continuous at p. If the 22 2 ] demand at p is D(p ) it must be the case that p , P(r(0)), otherwise ®rm i would deviate ]] ] downward, thus D(p2) . r(0), which is an upper bound on capacity investment as it is the optimal ] quantity with zero cost for a monopoly (in a subgame perfect equilibrium we can eliminate strictly dominated strategies in the ®rst stage). Therefore, ®rm i is constrained at p and we get P * 5px . ]]ii • p $u. We saw that u is the unique possible price equilibrium. The claim is even valid for all ®rms ] since sales beyond capacity neither generate losses nor bene®ts.

Claim 3. If x1 1 x21 # D(u ), u is the unique price equilibrium.

The only case we need to consider is p ,u. If ®rm i plays u 2 .p then the other ®rms that are less ]] expensive receive full demand, but serve only their capacities so that ®rm i receives more than 22 D(u ) 2 x2ii$ x , thus P i(u ,F2iii) 5u x . P *, a contradiction. From now on we study the case where p* ; P(x1 1 x21 ) ,u.

ÅÅAB Å Å A Å B Claim 4. H 5 H < H , where j [ H if r(x2jj) , x and j [ H if pÅ 5 p*.

Let Cjj( p ) ; p jminhx j,maxh0,D( p j) 2 x2jjj be the payoff to ®rm j when it names a price pj . maxi[¤ HiÅ hpÅÅj.Ifp j# max i±jihp j then ®rm j gets at least the payoff C jj( p ), thus this function must be maximal at pÅ to sustain this price as a member of the support of an equilibrium strategy. Firm j cannot be fully served at pÅ 1 for otherwise it would deviate upward, thus it will only sell units with zero marginal cost. Two cases can occur. If Cjj( p )5p j(D( p j) 2 x2j ) in the neighbourhood of pÅ;we study the alternate formulation of pro®ts yP(y 1 x2j ). The arg max is r(x2j ), thus pÅ 5 P(r(x2j ) 1 x2j ) and since ®rm j is not constrained at pÅ, it must be true that r(x2jj) , x . Besides the equilibrium payoff in that case is P j* 5 R(x2j ), where R(x) ; r(x)P(r(x) 1 x). Using the envelope theorem we obtain ~~ 2 R(x) 5 r(x)P(r(x) 1 x) , 0. If, on the other hand, Cjj( p ) 5 xp jjat pÅ then the upper price is pÅ 5 P(xj 1 x2jj) and we have x # r(x2j ).

Å B ± 5 Claim 5. If H , the equilibrium is p* ; P(x1 1 x21 ).

Observe that p**guarantees the revenue pxi to any ®rm i. Indeed, if all other ®rms are less expensive, they are served ®rst but the residual demand addressed to ®rm i is precisely its capacity. If HÅ B ± 5, the equilibrium must be the pure strategy p* for all ®rms.

Claim 6. If HÅÅÅBA5 5, then H 5 H 5 h1j, the large capacity ®rm. N. Boccard, X. Wauthy / Economics Letters 68 (2000) 279 ±285 283

ÅÅA ⇒ ~ Let j [ H 5 H .Ifx1 . xj then x21 , x2j r(x21 ) 1 x21 , r(x2j ) 1 x2j as r(z) .21. Hence, ®rm

1 obtains R(x211) # P * by naming P(r(x21 ) 1 x21 ).P(r(x2j ) 1 x2j ) 5 pÅÅ. We now prove that xR1 (m 1 x1 ) , xRjj(mÅÅ1 x ), where m 5 x212j. Let Q(z) ; zR(m ÅÅÅÅ1 z) 5 zr(m 1 z)P(m 1 z 1 r(m 1 z)) so that Q~ (z) 5 (r(mÅÅÅ1 z) 2 z)P(m 1 z 1 r(m 1 z)) obtains by the envelope theorem. If r(m Å1 x ) , x , then ~ jj Q ,0: we are done. Otherwise, r(mÅÅ1 xjj) . x implies that r(m 1 x) 5 x is solved for x* greater than xj since r(.) is decreasing. By the same token, x**. xjjimplies that the solution y to r(mÅ 1 x) 5 x has to be greater than x*. Finally, r(mÅ 1 y**) 5 x , x implies y , x . This is crucial as the positiveness ~ j 11 of Q on [xj;x**] will be offset by its negativeness on the large interval [x ;x1] as we now show:

x**x1 x y* ~~~~ 21 ] Q(x1 ) 2 Q(xjjjjj) 5EQ 1EQ #EQ 1EQ 5 Q(y**) 2 Q(x ) 5 yR(r (x )) 2 xR(m 1 x ) xxjjxx** 21 ]]] 5 yxP***jj(x 1 r (x j)) 2 xR j(m 1 x j) 5 x j(yP(x j1m 1 y ) 2 R(m 1 x j)) , 0 by de®nition of R(.).

xxjj P j* 5 R(x2j ) , ]]R(x21111) # P **, P 5p [F211(p )(D(p 1) 2 x21 ) 1 (12 F211(p ))D(p 1)]. xx11 ]]] ]]

If ®rm j plays p2 its demand is F (p )(D(p ) 2 x ) 1 (12 F (p ))D(p ), which is larger than ]]]]]1 212j 1121 212j 11 the demand of ®rm 1 because there is more weight on the monopolistic term, therefore Pj(p1,F2j ) . Å A ] P j*, the desired contradiction. We have shown that 1 [ H . Since pÅ 5 P(r(x2j ) 1 x2j ) holds true for any j [ HÅÅAA, members of H must have the same (largest) capacity. j

Relying on Lemma 1, we may state our theorem which extends the result of KS to oligopoly and imperfect commitment. For n symmetric ®rms and ef®cient rationing, the Cournot outcome emerges as the subgame perfect equilibrium outcome of the capacity-pricing game as soon as the ex-post marginal cost u is larger than the Cournot price.

Theorem 1.

• If u $ P(nxÅÅÅ), the symmetric Cournot±Nash investment x followed by the price P(nx ) is the unique subgame perfect equilibrium of G. • If u , P(nxÅ ), there is a continuum of SPE of G who nevertheless satisfy D(u ) 5 oi#nix . They converge toward the Bertrand solution as u tends to zero.

Proof. If xi # D(u ) 2 x2i, the equilibrium of the pricing game played after (xii)#n is the pure strategy u and ®rm i's payoff in G is uxi.IfD(u ) 2 x2ii, x # r(x2i ), the equilibrium is the pure strategy P(x2ii1 x ) and ®rm i is paid f(x2ii,x ) 5 xP i(x2ii1 x ) 2 c(x i). This function is concave with a maximum at rc(x2i ) , r(x2ii). If x . r(x2i ), the equilibrium is in mixed strategies and ®rm i earns g(x2ii,x ) 5 R(x2ii) 2 c(x ). Notice that P(x 1 x ) 5u when x 5 D(u ) 2 x and g(x ,r(x )) 5 f(x ,r(x )). Hence the ®rst 2ii i 2i 2i 2i 2i 2i ~ period payoff as a function of xiiis continuous. Moreover, at x 5 D(u ) 2 x2ii, ≠f/≠x 5u 1 xP i(x2i 1 ~ xii) 2 c(x ) ,u and the slope of g is steeper than that of f as the second period payoff becomes 284 N. Boccard, X. Wauthy / Economics Letters 68 (2000) 279 ±285

constant. The payoff function is thus concave in xi for any x2i; its average over the equilibrium distributions of the others ®rms is also concave, meaning that the best reply of ®rm i is always a pure strategy. Because this applies for all ®rms, the equilibrium is in pure strategies and satis®es xi 5 maxhD(u ) 2 x2ic,r (x2i )j for all i. If D(u ) 2 x2jc. r (x2jj) for some x then D(u ) 5 x2jj1 x . From this we deduce that D(u ) 5 x2ii1 x must hold for all other ®rms, so that xic5 r (x2i ) . D(u ) 2 x2i is impossible. Hence the candidate equilibria are all vectors (xii)#ni satisfying D(u ) 5 o #nix and D(u ) . r c(x2j ) 1 x2j for all j. The D(u )/n exists if it is larger than the Cournot candidate xÅÅ, i.e. if u # P(nx ), the

Cournot price. Solving for D(u ) 5 rc(y) 1 y yields a value y* that circumvents the range of asymmetric equilibria; they are given by the set of constraints ;i # n, xi $ y* in addition to D(u ) 5 oi#nix . Now if u . P(nxÅÅ), the equilibrium is unique. Let m ; x2122 (zero if n 5 2). If x1 5 rc(mÅÅ1 x22) and x 5 rc(m 1 x112), then x and x are solutions of z 5 h(z) ; rcc(m ÅÅ1 r (m 1 z)). But ~~~ since we previously showed that 0 . rccc(z) .21 it must be the case that h(z) 5 r (m 1 r (mÅ 1 ~ z))rc(mÅ 1 z) , 1, thus h has a unique ®xed point so that x125 x . By repetition of the argument to all pairs, we conclude that the equilibrium is unique and symmetric: it is the Cournot quantity xÅ. j

For a linear demand D( p) 5 1 2 p and zero marginal cost the Cournot quantity is 1/3. If the sum of quantities x121 x is less than D(u ) 5 1 2u then ®rms are not able to avoid . It is only for large aggregate capacities that the price equilibrium result in Cournot payoffs. Thus for u . 1/3, very low capacity choices lead ®rms to invest more (x121 x reach D(u )), then for larger capacities the applies and leads to the symmetric equilibrium choice of 1/3. For u , 1/3, the area where Bertrand competition applies incorporates the previous equilibrium meaning that ®rms are induced to build more capacity because the ®erce price competition yields too small margins. There is now a continuum of equilibria where ®rms share the market, but not too asymmetrically, as the Cournot best replies provide lower bounds on one's equilibrium capacity

(condition xi $ y* above).

3. Conclusion

Many researches have aimed at reconciling Cournotian outcomes with the explicit price mechanism involved in the Bertrand model. These researches have been successful to the extent that they have been able to combine the two features of oligopolistic industries which are limited scales of production (increasing marginal costs) and price setting behaviour. The main challenge in this respect consists in dealing with the issue of quantitative constraints (non-constant marginal cost) at the price competition stage which tends to make it unpro®table for the ®rms to meet full demand. This in turn generates rationing possibilities which are at the heart of Edgeworth's critique. The issue of rationing in pricing games is best understood by studying closely the allocation process of a non-competitive market with price-setting ®rms. Three stages are needed to correctly describe this process. In the ®rst stage, ®rms name prices and consumers address demand to ®rms. In the second stage, ®rms decide on their sales and possibly ration consumers. In the third stage, rationed consumers possibly report their demand to non-rationing ®rms who may or may not accept them. In the case of homogeneous products and perfect display of prices, the low pricing ®rm receives all the demand at the end of the ®rst stage. Under constant returns to scale, this ®rm is willing to meet any N. Boccard, X. Wauthy / Economics Letters 68 (2000) 279 ±285 285 demand level so that it always chooses to serve all consumers in the second stage; the third stage is then irrelevant. With decreasing returns to scale, it may not be optimal to meet full demand in the ®rst stage. Our paper takes this possibility into account, contrarily to a recent stream of literature where rationing is ruled by assumption (cf., e.g., Dastidar, 1995, 1997; Maggi, 1996). In our model the marginal cost structure varies from the Bertrand one to the Kreps and Scheinkman one through a single parameter measuring the commitment value of capacities. Within this framework we obtain the whole range of prices, from Bertrand to Cournot ones, as subgame perfect equilibrium outcomes.

Acknowledgements

N.B. gratefully acknowledges ®nancial support from CommunauteÁ FrancËaise de Belgique, DRS, ARC No. 98/03.

References

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