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A price-setting game with a nonatomic fringe
Economics Letters 69 (2000) 63–69 www.elsevier.com / locate / econbase
A price-setting game with a nonatomic fringe ´ * Attila Tasnadi U ´ ter ´ 8, Department of Mathematics, Budapest University of Economic Sciences and Public Administration, Fovam H-1093 Budapest, Hungary Received 15 November 1999; accepted 10 March 2000
Abstract This paper extends the Bertrand–Edgeworth price-setting game with finitely many firms to a game with infinitely many firms. Taking a market with one significant firm and a nonatomic fringe, we present a microfoundation of dominant-firm price leadership. 2000 Elsevier Science S.A. All rights reserved. Keywords: Bertrand–Edgeworth; Dominant firm; Price leadership JEL classification: D43; L13
1. Introduction In the following we shall consider a homogenous good market with one significant firm and a nonatomic fringe containing many infinitesimal firms. Our model may be regarded as an extension of the Bertrand–Edgeworth game in which there are finitely many firms to a game with infinitely many firms. Mixed measure theoretic models have been considered, for instance, by Gabszewicz and Mertens (1971), Shitovitz (1973), and Okuno et al. (1980) in a general equilibrium framework. Sadanand and Sadanand (1996) used in their analysis a partial equilibrium model containing a dominant firm and a nonatomic competitive fringe in order to investigate the timing of quantity-setting oligopoly games. Our model may be considered as the price-setting counterpart of their model. In our analysis we will assume that the large firm is the exogenously specified first mover while the small firms follow simultaneously. We will show that our model gives a game theoretic foundation of dominant-firm price leadership. In the dominant-firm price leadership model introduced by Forchheimer (see Scherer and Ross, 1990) there is one large firm and many small firms. Furthermore, the *Tel.: 136-1-387-0834; fax: 136-1-387-0834. ´ E-mail address: [email protected] (A. Tasnadi). 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 00 )00264-0
2000 Elsevier Science S.A. All rights reserved.
64
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
large firm is able to set the price on the market and the firms in the competitive fringe act as price takers. Therefore, the large producer sets a price by maximizing profit subject to its residual demand curve. The large firm’s residual demand curve can be obtained as the horizontal difference of the demand curve and the aggregate supply curve of the competitive fringe. The problem is that this model is not based on the firms’ individual profit maximizing behavior since it does not explain the large firm’s price-setting behavior nor why small firms act as price takers. A game theoretic foundation of price leadership on a duopolistic market was given by Deneckere and Kovenock (1992). They extended the capacity constrained Bertrand–Edgeworth duopoly game to a two-stage game with the timing of price decision in stage one. While they assumed constant average costs up to some capacity levels, in our analysis we shall assume strictly convex cost functions.
2. The model We denote the set of producers by V. Let us denote by vd [ V the dominant firm and by Vc [V \hvdj the competitive fringe. There is a s -algebra ! given above the set of producers such that hvdj [ !. We suppose that there is a finite measure m given above the set of producers such that vd is its only m -atom with mshvdjd 5 1 and the restriction of m to Vc is nonatomic. Let us denote by P[f0, p]g the set of possible prices. We denote by ss p, vd the supply of producer v [ V at price level p. Assumption 2.1. We assume that s is a bounded and ! measurable function for all p [ P. Furthermore, we assume that s is differentiable for all v with respect to p, that s is integrable for all p [ P with respect to m, and that ss0, vd 5 0, ≠ss p, vd / ≠p . 0 for all v. The supply function ss ? , vd of firm v [ V can be obtained from its cost function cs ? , vd. Suppose that cs0, vd 5 0 for all v [ V. Then in the opposite direction we can reconstruct the cost functions from the supply functions since ss ? , vd is invertible by Assumption 2.1. Hence, we have ≠c / ≠qs q, vd 5 p if and only if ss p, vd 5 q. The supply of producers A [ ! is given at price level p by Ss p, Ad[eA ss p, vd dmsvd. Assumption 2.1 assures that function Ss p, Ad is differentiable with respect to p and that ≠S / ≠ps p, Ad 5 eA ≠s / ≠ps p, vd dmsvd. Hence, it follows that ≠S / ≠ps p, Ad . 0 for any set A with positive measure, which means that Ss p, Ad is strictly increasing in p. We will model dominant-firm price leadership by a price-setting game. The price actions of the producers’ are given by a measurable function p: V → P that we will call from now on a price profile. Let us denote by 3 the set of price profiles. Lemma 2.2. For any price profile p function fsvd[sspsvd, vd is measurable and integrable with respect to m. Proof. We pick an arbitrary set A [ ! of producers and an arbitrary price profile p [ 3. We will construct an increasing sequence s fnd n[N of type V → P measurable functions that converges n n n n n n pointwise to f. Let p i 5 ip] / 2 , where i 5 0, 1, . . . ,2 ; let A i 5hv [ Au p i 21 # psvd , p i j, where n n n n n n i 5 1, . . . ,2 2 1; and let A i 5hv [ Au p i21 # psvd # p i j, where i 5 2 . Define the functions fn 5 o 2i 51
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
65
n
1A ni ss p in21 , ?d and gn 5 o 2i 51 1A in ss p ni , ?d, where 1A denotes the characteristic function of set A. Obviously, n n fn # f # gn , eA fnsvd dmsvd 5 o i251 Ss p ni21 , A ni d, and eA gnsvd dmsvd 5 o i251 Ss p ni , A ni d for any n [ N. For any e . 0 there exists a d . 0 such that usspsvd, vd 2 ss p9, vdu , e, if upsvd 2 p9u , d,
because ss ? , vd is continuous for any v [ V. We can choose a sufficiently large number n [ N so that n n upsvd 2 p i u , d for some i 5 0, 1, . . . ,2 2 1. Hence, 2n
fsvd 2 fnsvd 5 sspsvd, vd 2
O 1 svdss p
i51
A ni
n i21
,vd , e.
Therefore, it follows that the sequence s fnd n [N converges increasingly pointwise to function f and thus f is measurable. Now, we can conclude by Lebesgue’s monotone convergence theorem that eA fsvd dmsvd exists and that 2n
E fsvd dmsvd 5lim E f svd dmsvd 5lim O Ss p n
n→`
A
n →`
A
n i 21
n
, A i d.
i51
Furthermore, eA fsvd dmsvd has to be finite since f # gn and eA gnsvd dmsvd is finite for any n [ N. h The supply of producers A [ ! at price profile p [ 3 can be defined by Sˆ sp,Ad[eA sspsvd, vd dmsvd because of Lemma 2.2. The consumers side is given by the demand function D. Assumption 2.3. We assume that the demand curve is a continuously differentiable, decreasing 1 function and intersects both axis. Formally D [ # , D9 , 0, Ds]pd 5 0 and Ds pd . 0 for all p [f0, ]pd. We denote by Bsp, vd the set of those producers setting prices below producer v [ V, and by Csp, vd the set of those producers in the fringe setting the same price as producer v [ V. Formally, Bsp, vd[hv 9 [ V upsv 9d , psvdj and Csp, vd: 5hv 9 [ Vc upsv 9d 5 psvdj. Assuming efficient rationing of consumers, we define the demand served by the firms in the following manner:
Dsp, vd[ sspsvd, vd, 0,
H
J
Dspsvdd 2 Sˆ sp, Bsp, vdd sspsvd, vdmin 1, ]]]]]]] , Sˆ sp, Csp, vdd
if Sˆ sp, Csp, vdd . 0 and Dspsvdd $ Sˆ sp, Bsp, vdd; if Sˆ sp, Csp, vdd 5 0 and Dspsvdd $ Sˆ sp, Bsp, vdd; if Dspsvdd , Sˆ sp, Bsp, vdd;
(1)
for any firm v [ Vc and 1 Dsp, vdd[minhsspsvdd, vdd, sDspsvddd 2 Sˆ sp, Bsp, vdd < Csp, vdddd j
(2)
for the dominant firm vd . The definitions (1) and (2) assume that the dominant firm serves the consumers at a given price level after the competitive fringe has already sold its supply. However, this
66
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
assumption is not necessary, but we impose it only for the purely technical reason of avoiding the need to have a competitive fringe setting their prices arbitrarily close to, but below, the dominant firm’s price. This simplification has already been applied by Deneckere and Kovenock (1992) in their analysis. Now, we are ready to define the profit function of firm v [ V as psp, vd[psvdDsp, vd 2 csDsp, vd, vd. Our next assumption concerns the timing of the game in that we suppose that firm vd is the exogenously given first mover. Assumption 2.4. We suppose that firm vd set its price first and that the firms in the fringe set their prices simultaneously already in the knowledge of the price set by the dominant firm. For any price profile p and any firm v [ V we denote by p 2 v the restriction of p to the set V \hvj, that is p 2 v contains the price decisions of firm v ’s rivals. We will also write spsvd, p 2 vd for p. Definition 2.5. Let the dominant firms’s action be pd [ P. We call the price profile p* with p*svdd 5 pd a Nash equilibrium of stage two, if
* vd, vd pssp*svd, p *2 vd, vd $ pss p, p 2 for any firm v [ Vc and for any price p [ P. The following behavior is called the dominant-firm price-setting behavior: firm vd is maximizing profit subject to its residual demand function D dr s pd[sDs pd 2 Ss p, Vcdd 1 . The existence of a price p *d [ P maximizing the residual profit function p dr s pd[ pD dr s pd 2 csD dr s pd, vdd on set P is guaranteed by the continuity of the residual profit function. We want to show that in a subgame perfect Nash equilibrium of the price-setting game the dominant firm and all other firms set price p *d . Proposition 2.6. The extensive game kV, 3, p l has a unique subgame perfect Nash equilibrium under Assumptions 2.1, 2.3, and 2.4. The equilibrium price profile is given by p*svd 5 p *d for all v in V. Proof. Let the dominant-firm’s action be any pd [ P. We have to distinguish between three cases: (i) Ss pd , Vcd , Ds pdd 2 ss pd , vdd, (ii) Ds pdd 2 ss pd , vdd # Ss pd , Vcd # Ds pdd, and (iii) Ds pdd , Ss pd , Vcd. Let pl be the price level for which Ss p, Vcd 5 Ds pd 2 ss p, vdd holds and let pu be the price level for which Ds pd 5 Ss p, Vcd. Such prices pl and pu exist uniquely because of Assumptions 2.1 and 2.3. In case of (i) we have pl . pd since Ss ? , Vcd is a strictly increasing continuous function, and D is a decreasing continuous function. Firm v [ Vc will not set its price below pl , because at price pl it can sell its entire supply independently from its rivals’ actions and its profit function by Assumption 2.1 increases on the interval f0, plg. Therefore, it follows that at price pd the dominant firm will sell ss pd , vdd amount of product. In case of (ii) any firm v [ Vc will not set its price below pd because at price pd a firm in the fringe can sell its entire supply. Let us denote by p the price profile for that psvd 5 pd for all v [ V. Furthermore, suppose that there is a Nash equilibrium price profile p9 of the subgame so that p9svdd 5 pd , p9svd $ pd for all v [ Vc and p9 . p above a set with positive measure. Let A [ Vc > ! be the set of those producers in the fringe for which p9svd . pd . Denote by U the set of those
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
67
producers in A that cannot sell their entire supply, i.e. U [hv [ AuDsp9, vd , ssp9svd, vdj. Let A p [hv [ V up9svd # pj, F[h p [ PuSˆ sp9, A pd # Ds pdj, and ps [sup F. Of course, ps $ pd because Sˆ sp9, A pd 5 0 for any p , pd . It can be verified that if ps 5 pd , then any firm v [ A will not sell anything at all. Thus, p9 cannot be a Nash equilibrium of the subgame. Therefore, in what follows we suppose that ps . pd . Since A p increases as p increases it follows that either F 5f0, psg or F 5f0, psd. First, suppose that F 5f0, psg. Then ps ,p] because otherwise Sˆ sp9, A ]pd 5 Sˆ sp9, Vd # Dsp]d would follow, which is in contradiction to Sˆ sp9, Vd . Ss pd , Vd $ Ds pdd $ Dsp]d because of the properties of Sˆ and D. V \U 5hvdj
⁄ E, then p9svd [ ⁄ F and therefore Sˆ sp9, A p9svdd 5 Sˆ sp9, Bsp9, Clearly, vd is contained in both sets. If v [ ˆ vdd 1 Ssp9, Csp9, vdd . Dsp9svdd. Thus, v [ U and E . V \U. If v [ U, then regarding the definition of D (Eq. (1)) we obtain Sˆ sp9, A p9svdd . Dsp9svdd, which implies that p9svd [ ⁄ F. Therefore, v [ ⁄ E and V \U 5 E [ !. Let V [hv [ V up9svd 5 psj [ !. Set V has positive measure since Sˆ sp9, V \Ud # Ds psd , Sˆ sp9, A psd 5 Sˆ sp9, V
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
68
dp r ≠s ≠c ≠s ]]d s pd 5 p ] s p, vdd 1 ss p, vdd 2 ] sss p, vdd, vdd ] s p, vdd 5 ss p, vdd . 0 dp ≠p ≠q ≠p the residual profit function p rds pd is strictly increasing on the interval s0, pld. Therefore, pd $ pl , which means that the dominant firm set its price in the region corresponding to case (ii). Thus, the dominant firm will set its price to p *d in order to maximize its profits and the firms in the fringe will also set their price to p *d . This completes the proof of the proposition. h In order to present a complete model of dominant-firm price leadership we should resolve Assumption 2.4. Following Deneckere and Kovenock (1992) we shall investigate the outcome of the simultaneous-move price-setting game. Thereafter we have to compare the firm’s profits in the simultaneous-move game to the profits in the extensive game. In case of the simultaneous-move game there is a lack of equilibrium in pure strategies because the large firm will have an incentive to slightly undercut price p *d and therefore it will trigger a price war. The competitive price will not be an equilibrium either since the large firm will prefer price p *d to it. Hence, one has to consider equilibrium in mixed strategies. The crucial point is that we cannot even guarantee the existence of a mixed strategy equilibrium. The main existence theorems for games with discontinuous payoffs given by Dasgupta and Maskin (1986), Simon (1987) and the recent one by Reny (1999) can be applied only in the case of finitely many firms.
3. Concluding remarks The price-setting game with a continuum of firms, as presented in Section 2, may also be formulated for the case of more than one large firm, but the equilibrium behavior of the model would be quite different. The random rationing rule for the Bertrand–Edgeworth game with a continuum of firms may be specified by the following residual demand function:
1
D rs pd[Ds pd 1 2
E
sspsvd, vd ]]] dmsvd Dspsvdd
hv 9[ V upsv 9d,pj
2
1
.
Our proposition remains valid in the case of random rationing because it can be shown that in the relevant price region of the large firm the small firms will follow the price set by the large firm. Therefore, the large firm has to maximize the following residual demand function: D rds pd 5 Ds pds1 2 Scs pd /Ds pdd 5 Ds pd 2 Scs pd.
References Dasgupta, P., Maskin, E., 1986. The existence of equilibria in discontinuous games, I: Theory. Review of Economic Studies 53, 1–26. Deneckere, R., Kovenock, D., 1992. Price leadership. Review of Economic Studies 59, 143–162.
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Gabszewicz, J., Mertens, J.F., 1971. An equivalence theorem for the core of an economy whose atoms are not ‘too’ big. Econometrica 39, 713–722. Okuno, M., Postlewaite, A., Roberts, J., 1980. Oligopoly and competition in large markets. American Economic Review 70, 22–31. Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056. Sadanand, A., Sadanand, V., 1996. Firm scale and the endogenous timing of entry: a choice between commitment and flexibility. Journal of Economic Theory 70, 516–530. Scherer, F.M., Ross, D., 1990. Industrial Market Structure and Economic Performance. Houghton Mifflin Company, Boston. Shitovitz, B., 1973. Oligopoly in markets with a continuum of traders. Econometrica 41, 467–502. Simon, L., 1987. Games with discontinuous payoffs. Review of Economic Studies 54, 569–597.
A price-setting game with a nonatomic fringe ´ * Attila Tasnadi U ´ ter ´ 8, Department of Mathematics, Budapest University of Economic Sciences and Public Administration, Fovam H-1093 Budapest, Hungary Received 15 November 1999; accepted 10 March 2000
Abstract This paper extends the Bertrand–Edgeworth price-setting game with finitely many firms to a game with infinitely many firms. Taking a market with one significant firm and a nonatomic fringe, we present a microfoundation of dominant-firm price leadership. 2000 Elsevier Science S.A. All rights reserved. Keywords: Bertrand–Edgeworth; Dominant firm; Price leadership JEL classification: D43; L13
1. Introduction In the following we shall consider a homogenous good market with one significant firm and a nonatomic fringe containing many infinitesimal firms. Our model may be regarded as an extension of the Bertrand–Edgeworth game in which there are finitely many firms to a game with infinitely many firms. Mixed measure theoretic models have been considered, for instance, by Gabszewicz and Mertens (1971), Shitovitz (1973), and Okuno et al. (1980) in a general equilibrium framework. Sadanand and Sadanand (1996) used in their analysis a partial equilibrium model containing a dominant firm and a nonatomic competitive fringe in order to investigate the timing of quantity-setting oligopoly games. Our model may be considered as the price-setting counterpart of their model. In our analysis we will assume that the large firm is the exogenously specified first mover while the small firms follow simultaneously. We will show that our model gives a game theoretic foundation of dominant-firm price leadership. In the dominant-firm price leadership model introduced by Forchheimer (see Scherer and Ross, 1990) there is one large firm and many small firms. Furthermore, the *Tel.: 136-1-387-0834; fax: 136-1-387-0834. ´ E-mail address: [email protected] (A. Tasnadi). 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 00 )00264-0
2000 Elsevier Science S.A. All rights reserved.
64
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
large firm is able to set the price on the market and the firms in the competitive fringe act as price takers. Therefore, the large producer sets a price by maximizing profit subject to its residual demand curve. The large firm’s residual demand curve can be obtained as the horizontal difference of the demand curve and the aggregate supply curve of the competitive fringe. The problem is that this model is not based on the firms’ individual profit maximizing behavior since it does not explain the large firm’s price-setting behavior nor why small firms act as price takers. A game theoretic foundation of price leadership on a duopolistic market was given by Deneckere and Kovenock (1992). They extended the capacity constrained Bertrand–Edgeworth duopoly game to a two-stage game with the timing of price decision in stage one. While they assumed constant average costs up to some capacity levels, in our analysis we shall assume strictly convex cost functions.
2. The model We denote the set of producers by V. Let us denote by vd [ V the dominant firm and by Vc [V \hvdj the competitive fringe. There is a s -algebra ! given above the set of producers such that hvdj [ !. We suppose that there is a finite measure m given above the set of producers such that vd is its only m -atom with mshvdjd 5 1 and the restriction of m to Vc is nonatomic. Let us denote by P[f0, p]g the set of possible prices. We denote by ss p, vd the supply of producer v [ V at price level p. Assumption 2.1. We assume that s is a bounded and ! measurable function for all p [ P. Furthermore, we assume that s is differentiable for all v with respect to p, that s is integrable for all p [ P with respect to m, and that ss0, vd 5 0, ≠ss p, vd / ≠p . 0 for all v. The supply function ss ? , vd of firm v [ V can be obtained from its cost function cs ? , vd. Suppose that cs0, vd 5 0 for all v [ V. Then in the opposite direction we can reconstruct the cost functions from the supply functions since ss ? , vd is invertible by Assumption 2.1. Hence, we have ≠c / ≠qs q, vd 5 p if and only if ss p, vd 5 q. The supply of producers A [ ! is given at price level p by Ss p, Ad[eA ss p, vd dmsvd. Assumption 2.1 assures that function Ss p, Ad is differentiable with respect to p and that ≠S / ≠ps p, Ad 5 eA ≠s / ≠ps p, vd dmsvd. Hence, it follows that ≠S / ≠ps p, Ad . 0 for any set A with positive measure, which means that Ss p, Ad is strictly increasing in p. We will model dominant-firm price leadership by a price-setting game. The price actions of the producers’ are given by a measurable function p: V → P that we will call from now on a price profile. Let us denote by 3 the set of price profiles. Lemma 2.2. For any price profile p function fsvd[sspsvd, vd is measurable and integrable with respect to m. Proof. We pick an arbitrary set A [ ! of producers and an arbitrary price profile p [ 3. We will construct an increasing sequence s fnd n[N of type V → P measurable functions that converges n n n n n n pointwise to f. Let p i 5 ip] / 2 , where i 5 0, 1, . . . ,2 ; let A i 5hv [ Au p i 21 # psvd , p i j, where n n n n n n i 5 1, . . . ,2 2 1; and let A i 5hv [ Au p i21 # psvd # p i j, where i 5 2 . Define the functions fn 5 o 2i 51
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
65
n
1A ni ss p in21 , ?d and gn 5 o 2i 51 1A in ss p ni , ?d, where 1A denotes the characteristic function of set A. Obviously, n n fn # f # gn , eA fnsvd dmsvd 5 o i251 Ss p ni21 , A ni d, and eA gnsvd dmsvd 5 o i251 Ss p ni , A ni d for any n [ N. For any e . 0 there exists a d . 0 such that usspsvd, vd 2 ss p9, vdu , e, if upsvd 2 p9u , d,
because ss ? , vd is continuous for any v [ V. We can choose a sufficiently large number n [ N so that n n upsvd 2 p i u , d for some i 5 0, 1, . . . ,2 2 1. Hence, 2n
fsvd 2 fnsvd 5 sspsvd, vd 2
O 1 svdss p
i51
A ni
n i21
,vd , e.
Therefore, it follows that the sequence s fnd n [N converges increasingly pointwise to function f and thus f is measurable. Now, we can conclude by Lebesgue’s monotone convergence theorem that eA fsvd dmsvd exists and that 2n
E fsvd dmsvd 5lim E f svd dmsvd 5lim O Ss p n
n→`
A
n →`
A
n i 21
n
, A i d.
i51
Furthermore, eA fsvd dmsvd has to be finite since f # gn and eA gnsvd dmsvd is finite for any n [ N. h The supply of producers A [ ! at price profile p [ 3 can be defined by Sˆ sp,Ad[eA sspsvd, vd dmsvd because of Lemma 2.2. The consumers side is given by the demand function D. Assumption 2.3. We assume that the demand curve is a continuously differentiable, decreasing 1 function and intersects both axis. Formally D [ # , D9 , 0, Ds]pd 5 0 and Ds pd . 0 for all p [f0, ]pd. We denote by Bsp, vd the set of those producers setting prices below producer v [ V, and by Csp, vd the set of those producers in the fringe setting the same price as producer v [ V. Formally, Bsp, vd[hv 9 [ V upsv 9d , psvdj and Csp, vd: 5hv 9 [ Vc upsv 9d 5 psvdj. Assuming efficient rationing of consumers, we define the demand served by the firms in the following manner:
Dsp, vd[ sspsvd, vd, 0,
H
J
Dspsvdd 2 Sˆ sp, Bsp, vdd sspsvd, vdmin 1, ]]]]]]] , Sˆ sp, Csp, vdd
if Sˆ sp, Csp, vdd . 0 and Dspsvdd $ Sˆ sp, Bsp, vdd; if Sˆ sp, Csp, vdd 5 0 and Dspsvdd $ Sˆ sp, Bsp, vdd; if Dspsvdd , Sˆ sp, Bsp, vdd;
(1)
for any firm v [ Vc and 1 Dsp, vdd[minhsspsvdd, vdd, sDspsvddd 2 Sˆ sp, Bsp, vdd < Csp, vdddd j
(2)
for the dominant firm vd . The definitions (1) and (2) assume that the dominant firm serves the consumers at a given price level after the competitive fringe has already sold its supply. However, this
66
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
assumption is not necessary, but we impose it only for the purely technical reason of avoiding the need to have a competitive fringe setting their prices arbitrarily close to, but below, the dominant firm’s price. This simplification has already been applied by Deneckere and Kovenock (1992) in their analysis. Now, we are ready to define the profit function of firm v [ V as psp, vd[psvdDsp, vd 2 csDsp, vd, vd. Our next assumption concerns the timing of the game in that we suppose that firm vd is the exogenously given first mover. Assumption 2.4. We suppose that firm vd set its price first and that the firms in the fringe set their prices simultaneously already in the knowledge of the price set by the dominant firm. For any price profile p and any firm v [ V we denote by p 2 v the restriction of p to the set V \hvj, that is p 2 v contains the price decisions of firm v ’s rivals. We will also write spsvd, p 2 vd for p. Definition 2.5. Let the dominant firms’s action be pd [ P. We call the price profile p* with p*svdd 5 pd a Nash equilibrium of stage two, if
* vd, vd pssp*svd, p *2 vd, vd $ pss p, p 2 for any firm v [ Vc and for any price p [ P. The following behavior is called the dominant-firm price-setting behavior: firm vd is maximizing profit subject to its residual demand function D dr s pd[sDs pd 2 Ss p, Vcdd 1 . The existence of a price p *d [ P maximizing the residual profit function p dr s pd[ pD dr s pd 2 csD dr s pd, vdd on set P is guaranteed by the continuity of the residual profit function. We want to show that in a subgame perfect Nash equilibrium of the price-setting game the dominant firm and all other firms set price p *d . Proposition 2.6. The extensive game kV, 3, p l has a unique subgame perfect Nash equilibrium under Assumptions 2.1, 2.3, and 2.4. The equilibrium price profile is given by p*svd 5 p *d for all v in V. Proof. Let the dominant-firm’s action be any pd [ P. We have to distinguish between three cases: (i) Ss pd , Vcd , Ds pdd 2 ss pd , vdd, (ii) Ds pdd 2 ss pd , vdd # Ss pd , Vcd # Ds pdd, and (iii) Ds pdd , Ss pd , Vcd. Let pl be the price level for which Ss p, Vcd 5 Ds pd 2 ss p, vdd holds and let pu be the price level for which Ds pd 5 Ss p, Vcd. Such prices pl and pu exist uniquely because of Assumptions 2.1 and 2.3. In case of (i) we have pl . pd since Ss ? , Vcd is a strictly increasing continuous function, and D is a decreasing continuous function. Firm v [ Vc will not set its price below pl , because at price pl it can sell its entire supply independently from its rivals’ actions and its profit function by Assumption 2.1 increases on the interval f0, plg. Therefore, it follows that at price pd the dominant firm will sell ss pd , vdd amount of product. In case of (ii) any firm v [ Vc will not set its price below pd because at price pd a firm in the fringe can sell its entire supply. Let us denote by p the price profile for that psvd 5 pd for all v [ V. Furthermore, suppose that there is a Nash equilibrium price profile p9 of the subgame so that p9svdd 5 pd , p9svd $ pd for all v [ Vc and p9 . p above a set with positive measure. Let A [ Vc > ! be the set of those producers in the fringe for which p9svd . pd . Denote by U the set of those
´ / Economics Letters 69 (2000) 63 – 69 A. Tasnadi
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producers in A that cannot sell their entire supply, i.e. U [hv [ AuDsp9, vd , ssp9svd, vdj. Let A p [hv [ V up9svd # pj, F[h p [ PuSˆ sp9, A pd # Ds pdj, and ps [sup F. Of course, ps $ pd because Sˆ sp9, A pd 5 0 for any p , pd . It can be verified that if ps 5 pd , then any firm v [ A will not sell anything at all. Thus, p9 cannot be a Nash equilibrium of the subgame. Therefore, in what follows we suppose that ps . pd . Since A p increases as p increases it follows that either F 5f0, psg or F 5f0, psd. First, suppose that F 5f0, psg. Then ps ,p] because otherwise Sˆ sp9, A ]pd 5 Sˆ sp9, Vd # Dsp]d would follow, which is in contradiction to Sˆ sp9, Vd . Ss pd , Vd $ Ds pdd $ Dsp]d because of the properties of Sˆ and D. V \U 5hvdj
⁄ E, then p9svd [ ⁄ F and therefore Sˆ sp9, A p9svdd 5 Sˆ sp9, Bsp9, Clearly, vd is contained in both sets. If v [ ˆ vdd 1 Ssp9, Csp9, vdd . Dsp9svdd. Thus, v [ U and E . V \U. If v [ U, then regarding the definition of D (Eq. (1)) we obtain Sˆ sp9, A p9svdd . Dsp9svdd, which implies that p9svd [ ⁄ F. Therefore, v [ ⁄ E and V \U 5 E [ !. Let V [hv [ V up9svd 5 psj [ !. Set V has positive measure since Sˆ sp9, V \Ud # Ds psd , Sˆ sp9, A psd 5 Sˆ sp9, V
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dp r ≠s ≠c ≠s ]]d s pd 5 p ] s p, vdd 1 ss p, vdd 2 ] sss p, vdd, vdd ] s p, vdd 5 ss p, vdd . 0 dp ≠p ≠q ≠p the residual profit function p rds pd is strictly increasing on the interval s0, pld. Therefore, pd $ pl , which means that the dominant firm set its price in the region corresponding to case (ii). Thus, the dominant firm will set its price to p *d in order to maximize its profits and the firms in the fringe will also set their price to p *d . This completes the proof of the proposition. h In order to present a complete model of dominant-firm price leadership we should resolve Assumption 2.4. Following Deneckere and Kovenock (1992) we shall investigate the outcome of the simultaneous-move price-setting game. Thereafter we have to compare the firm’s profits in the simultaneous-move game to the profits in the extensive game. In case of the simultaneous-move game there is a lack of equilibrium in pure strategies because the large firm will have an incentive to slightly undercut price p *d and therefore it will trigger a price war. The competitive price will not be an equilibrium either since the large firm will prefer price p *d to it. Hence, one has to consider equilibrium in mixed strategies. The crucial point is that we cannot even guarantee the existence of a mixed strategy equilibrium. The main existence theorems for games with discontinuous payoffs given by Dasgupta and Maskin (1986), Simon (1987) and the recent one by Reny (1999) can be applied only in the case of finitely many firms.
3. Concluding remarks The price-setting game with a continuum of firms, as presented in Section 2, may also be formulated for the case of more than one large firm, but the equilibrium behavior of the model would be quite different. The random rationing rule for the Bertrand–Edgeworth game with a continuum of firms may be specified by the following residual demand function:
1
D rs pd[Ds pd 1 2
E
sspsvd, vd ]]] dmsvd Dspsvdd
hv 9[ V upsv 9d,pj
2
1
.
Our proposition remains valid in the case of random rationing because it can be shown that in the relevant price region of the large firm the small firms will follow the price set by the large firm. Therefore, the large firm has to maximize the following residual demand function: D rds pd 5 Ds pds1 2 Scs pd /Ds pdd 5 Ds pd 2 Scs pd.
References Dasgupta, P., Maskin, E., 1986. The existence of equilibria in discontinuous games, I: Theory. Review of Economic Studies 53, 1–26. Deneckere, R., Kovenock, D., 1992. Price leadership. Review of Economic Studies 59, 143–162.
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Gabszewicz, J., Mertens, J.F., 1971. An equivalence theorem for the core of an economy whose atoms are not ‘too’ big. Econometrica 39, 713–722. Okuno, M., Postlewaite, A., Roberts, J., 1980. Oligopoly and competition in large markets. American Economic Review 70, 22–31. Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056. Sadanand, A., Sadanand, V., 1996. Firm scale and the endogenous timing of entry: a choice between commitment and flexibility. Journal of Economic Theory 70, 516–530. Scherer, F.M., Ross, D., 1990. Industrial Market Structure and Economic Performance. Houghton Mifflin Company, Boston. Shitovitz, B., 1973. Oligopoly in markets with a continuum of traders. Econometrica 41, 467–502. Simon, L., 1987. Games with discontinuous payoffs. Review of Economic Studies 54, 569–597.