OLIGOPOLISTICMARKETS WITH PRICE- SETTING FIRMSt
The Existence of Equilibrium with Price-Setting Firms
By ERIC MASKIN*
Ever since Joseph Bertrand (1883), econ- worth model when market demand as a func- omists have been interested in static models tion of price is continuous, downward slop- of oligopoly where firms set prices. Francis ing, and equal to zero for a sufficiently high Edgeworth's 1925 critique of Bertrand recog- price. In their study of Bertrand-Edgeworth nized, however, that, except in the case of competition in large economies, Beth Allen constant marginal costs, there are serious and Martin Hellwig (1983) extended this re- equilibrium existence problems when firms sult to demand curves that do not necessarily produce a homogeneous good. In particular, intersect the horizontal axis and need not Edgeworth proposed a modification of slope downward. Bertrand's model in which firms have zero There have been several treatments of cost marginal cost up to some fixed capacity. He functions more general than the Bertrand- showed that, unless demand is highly elastic, Edgeworth variety. R. Gertner (1985) estab- price equilibrium may fail to exist. lished the existence of symmetric equilibrium Mixed strategies provide one way of avoid- in a model where firms are identical, have ing this nonexistence problem, as various convex or concave costs, and choose output authors have noted. Martin Beckmann levels at the same time as prices. Also in a (1965), for instance, explicitly calculated model of identical firms, H. Dixon (1984) mixed strategy equilibria in a symmetric proved existence when firms have convex example of the Bertrand-Edgeworth model. costs and produce to order, that is, produce However, a general treatment of mixed after other firms' prices are realized. strategies has suffered from the fact that the In this paper, I present some existence standard equilibrium existence lemmas (see, results that do not require symmetry and for example, K. Fan, 1952, and I. Glicks- permit fairly general cost functions. These berg, 1952), require continuous payoff func- findings pertain both to the simultaneous tions, whereas, in price-setting oligopoly, choice of price and production level and to payoffs are inherently discontinuous the the formulation where a firm's output is set firm charging the lowest price captures the only after it knows others' prices. Existence whole market. in the former case is proved by direct appli- Recently, Partha Dasgupta and I (1986) cation of the Dasgupta-Maskin/Simon theo- and Leo Simon (1984) developed several ex- rems, as are the existence results mentioned istence theorems for discontinuous games. above. The latter case, however, requires Dasgupta and I used one of the theorems to some additional argument. I give a sketch of establish the general existence of mixed this argument below; the details and more strategy equilibrium in the Bertrand-Edge- general results can be found in Dixon and myself (1986).
1. The Model tDiscussant: Richard Schmalensee, Massachusetts In- stitute of Technology. For simplicity, I shall consider only two *Department of Economics, Harvard University, firms; all results generalize immediately to Cambridge, MA 02138. I thank the Sloan Foundation and the NSF for research support. I am indebted to any finite number. Firms produce the same Martin Hellwig for helpful comments on an earlier good, and firm i, i = 1, 2, has total cost func- version of this paper. tion ci(xi), where xi is the firm's output
382 VOL. 76 NO. 2 OLIGOPOLISTIC MARKETS WITH PRICE-SETTING FIRMS 383 level. We assume share is a nondecreasing function of its supply and that a positive supply implies a (1) ci(x) is continuous and nondecreasing; positive share. Two examples of Gi's satisfy- ci(O) = ?. ing (4) are Firms face an industry demand curve F: R+ Gi(p,s, S2) -* R +, where (2) F is continuous with F(O) = K for some F(p), if s1+s2>0 { Sl+S2 K > 0; pF(p) - ci(F(p) is maximized at > 0, and, if there are multiple Pi - if s1+s2=O maximizers, Pi is the largest. Let p= 2F(p), max pi. and Gi(p, sI, s2) = a1F(p), where a1 + Firm i may have a constraint Ki on its a2 =1. production level. In view of (2), we may The third line of (3) requires that the firm assume, without loss of generality, that Ki charging the higher price get less than full < K. market demand. Moreover, if the two firms Firm i's strategy consists of choosing a are charging approximately the same price, price pi E [0, pI and supply si E [0, K1. Given the demand facing the high-price firm is the firms' strategies the demand facing firm i approximately full market demand at that is price minus the supply of the other firm. Two examples of Hi's satisfying (5) are (3) di(pj,S1,P2,52) Hi(PI, P2' sj) = max{F(pi)- sJ,O} and
{F( pi), if Pi< Pj Hi(P1, P2, Sj) = max F(pj)p -IF(pi)j. = G( p, S1, S2), if P1=P2=P
Hi(Pl,P2,s1j), if Pi>P, The former rule is called parallel rationing (see Richard Levitan and Martin Shubik, where Gi is a function such that 1972), whereas the latter is proportional ra- tioning (see Edgeworth). (4) GI(p, SI S2)+ G2(P 1, S2) =Fp); In the case where a firm sets production at if s > 0, G >0; if si?sj, Gi?Gj; the same time as price (production in ad- and min{ Gi(p, sl, S2), Si } is continuous vance), firm i's payoff is in si, (6) Pixi-Ci (si), and Hi(pI, P2, sj) is a function such that where
(5) Hi ( P 1 P2, Sj) < F( pi); Hi( P, P, Sj) (7) xi = min{ si, di(p1, sl, P2' SM} = F( p) - s; and Hi is continuous. whereas in the case where it produces to order, the firm's payoff is The first line of (3) simply posits that the firm charging the lower price attracts the (8) pixi-ci(xi). entire market demand. The second line stipulates that, if firms 11. Equilibriumin DiscontinuousGames charge the same demand, they split the market in some way that depends on their Let us present a special case of the main supplies. Condition (4) tells us that a firm's existence theorems in Dasgupta's and my 384 AEA PAPERS AND PROCEEDINGS MA Y1986 article, and in Simon. For i = 1,2, let Ai be a continuous at (p, si, P2' S2), and so (9) holds convex, compact subset of R2. The set Ai is automatically. If Pi and s, are both positive, player i's strategy space. Let Ui:A1 x A2 > then dl(p1, sl, P2' S2) is lower semicontinu- R, the payoff function for player i, be (a) ous from the left. That is, at points of dis- bounded and (b) continuous except at points continuity (where Pi = P2), firm l's demand (Pl,'Si, P2' 52) e Al X A2 where Pi = P2. As- jumps downward for a sequence { pl } of sume, furthermore, that Ui is weakly lower prices converging to p1 from below. Hence, semicontinuous in (pi, si). That is, for any (9) holds for any convergent sequence {(1p, (Pl,sl) there exists a sequence {(pn,sn)} sn )} where pn converges to Pi from below. converging to (P', s') such that no two pj 's Finally, observe that discontinuities simply and no two sn 's are the same and such that, entail a shift in demand from one firm to the for any (P2, S2), other. Thus, although U1 and U2 are discon- tinuous, their sum is not. I conclude that the (g) liim n1( ,5,n2s5 proposition applies. (P1 ,sl) -~(pi,si) THEOREM 1: Given (1)-(5), a mixed strat- Un(pP, )2)S? 51' P2' egy equilibrium exists in the case of produc- tion in advance (where firm i's profit is given and similarly for player 2. Finally, suppose by (6)). that L_1JJUis upper semicontinuous. I turn next to production to order. Here PROPOSITION: Given the stated assump- we encounter a difficulty in the application tions, a mixed strategy equilibriumexists. That of the proposition; namely, the sum of prof- is, there exist probability measures (4, ,*2) its need no longer be continuous nor even such that upper semicontinuous. Although discontinui- ties still involve a shift in demand between firms, they now also entail a shift in produc- fUJ(P1, 51, P2, S2)dp4(p1, s1)x d* ( P2, S2) tion. Thus an increase in production by a less efficient firm at the expense of a more effi- f U1(Pi, 51' P2' s2) dp1(p1, s1) cient one may induce a fall in total profit. To deal with this difficulty, I modify the strategy spaces and payoff functions some- xdp*(p2,S2) for all p on Al; what to restore upper semicontinuity. This will enable us to conclude that an equi- librium for the modified payoff functions s2) x fU2(p1' Sl' P2' dp4 dp* exists. I then argue that the strategies for this equilibrium remain in equilibrium for the ?fU2(p1,s11P2,S2)dp4 x dM2, original payoff functions. I first strengthen the assumptions about cost functions. In addition to (1), we require for all12 on A2. (10) ci ( x ) is strictly convex. 111. Equilibriumin Oligopoly For i = 1, 2 firm i 's supply function is Let us take Ai=[O,p]x[O,Ki] in our oligopoly model. In the case of production in si(p)=arg max [px-c1(x)]. advance, we can readily verify that firms' x e [0, K] payoffs satisfy the hypotheses of the above proposition. From (2), profits are clearly From (1) and (10), si(p) is well-defined, bounded. Because F, the Hi's and the ci's nondecreasing, and continuous. Notice that are continuous, profits are continuous except by assuming that costs are convex, we where Pl = P2- If Pi = 0 or s, = 0, then U1 is can conclude that si = si(p) maximizes VOL. 76 NO. 2 OLIGOPOLISTIC MARKETS WITH PRICE-SETTING FIRMS 385 min{di, si}p - ci(min{di, si}), for any di. continuous at p, and if p >0, Ui is lower Accordingly, in the modified model, let firm semicontinuous from the left. Hence, apply- i's strategy space be Ai = [0, p]. For any ing the above proposition (and simply ignor- price p, let Vj(p) = pxi - ci(xi) and W1((p) ing the extra dimension of the strategy space), = pyi - ci(yi), where xi = min{si(p), F(p)} I conclude that the modified game has a and yi = min{ si(p), max{ F(p)- sj(p), O}}. mixed strategy equilibrium (al, j2) Then, I claim that (alDj2) also represents a mixed strategy equilibrium of the original (11) W1(p) (12) W2(p) > max{ W11(p)+ V2(p), W2(p) + V11(p)}. (17) fU1(P1, S1(PD,P2,s2(P2))di2 If {(pl, pn)} is a sequence converging to ? Ul(Pl,sl(Pl),P2,s2(P2))dji2 (p, p) then for all P1. (14) lim sup E Ui(pnI 1(Pp), Pn, sP)) Now suppose that for some p? and s? E < max{ W1(p) + V2(p), W2(p) + V1(p), [0, K1], f ULJ( s1(P1), P2' s2(P2)) dA2 EUi(P, S1(P) PIS2(P))) Define < fu1(PO1,S1P2,s2(P2))d ci2. (15) Ut(P1, P2) Then, in view of (17), U1(p , s?, p?, s2(pO)) > U1(P, Si( Po), Pi, S2( Pi0)) and M2(P1) > 0. Thus there exists P slightly less than sIUi(Pl, S1( PJ) P2, S2( P2)), 'f P1 * P2 P0 such that tZi(p), if Pl=P2=P f U1(P, s1(P1), P2' s2(P2)) cAL2 Combining (13)-(15), it can be deduced that XUi is upper semicontinuous. If p =0, Ui is > fi U1(P1,s1(PD),P2,s2(P2)) ciA2, 386 A EA PAPERS AND PROCEEDINGS MA Y 1986 a contradiction of (17). I conclude that Equilibrium in Discontinuous Economic Games, 1 and 2," Review of Economic Studies, forthcoming 1986. f U1( p1, s1( Pr1)IP2, 52( P2)) dAii2 Dixon, H., "Existence of Mixed Strategy Equilibria in a Price-Setting Oligopoly with Convex Costs," Economic Letters, 1984, U(PI S1i, P2, s2(P2)) dfi2 16, 205-12. _ and Maskin, E., "The Existence of for all Pi and sl. Equilibrium with Price-Setting Firms," I have demonstrated mimeo., Harvard University, 1986. Edgeworth, F., Papers Relating to Political THEOREM 2: Given (1)-(5) and (10), a Economy, London: Macmillan, 1925. mixed strategy equilibriumexists in the case of Fan, K., "Fixed Point and Minimax Theo- production to order (where firm i 's profit is rems in Locally Convex Topological Lin- given by (8)). ear Spaces," Proceedings of the National Academy of Sciences, 1952, 38, 121-26. REFERENCES Gertner,R., "Simultaneous Move Price-Quan- tity Games and Non-Market Clearing Allen, B. and Hellwig, M., " Bertrand-Edge- Equilibrium," mimeo., MIT, 1985. worth Oligopoly in Large Markets," Glicksberg,I., "A Further Generalization of mimeo., University of Bonn, 1983. the Kakutani Fixed Point Theorem with Beckmann, M., " Edgeworth-Bertrand Duop- Application to Nash Equilibrium Points," oly Revisited," in R. Henn, ed. Operations Proceedings of the American Mathematical Research-Verfahren, Vol. III, Meisenheim: Society, 1952 38, 170-74. Sonderdruck, Verlag, Anton Hein, 1965, Levitan, R. and Shubik, M., "Price Duopoly 55-68. and Capacity Constraints," International Bertrand, J., "Review of Cournot's 'Rech- Economic Review, February 1972, 13, ercher sur la theorie mathematique de la 111-22. richesse'," Journal des Savants, 1883, Simon, L., "Games with Discontinuous Pay- 499-508. offs, 1: Theory," mimeo., University of Dasgupta, P. and Maskin, E., "Existence of California-Berkeley, 1984.