BERTRAND COMPETITION AND COURNOT OUTCOMES: FURTHER RESULTS

Transcription

1 BERTRAND COMPETITION AND COURNOT OUTCOMES: FURTHER RESULTS Ncolas Boccard * & Xaver Wauthy ** SEPTEMBER 1999 ABSTRACT In ths note, we extend the classcal result of Kreps & Schenkman [1983] to an olgopolstc settng where capacty s modelled as an mperfect commtment devce. To ths end, we retan the cost structure put froward n Dxt [1980],.e. we allow frms to produce beyond nstalled capactes but n ths case they have to ncur an addtonal unt cost. When the unt cost of producng beyond capactys large enoough, Cournot outcomes always obtan n the unque subgame perfect equlbrum whereas when t s low, there s a contnuum of equlbra. Each of them nvolve an dentcal aggregate capacty whch converges to the compettve one as the extraz unt cost tends to zero. Keywords : Prce Competton, Capacty Commtment JEL Classfcaton : D43, F13, L13 * CORE & Unversté de Lège. Emal : boccard@core.ucl.ac.be ** CORE and CEREC, Facultés Unverstares Sant Lous. xwauthy@fusl.ac.be

2 1) INTRODUCTI ON Accordng to the Bertrand paradox ''two s enough for compettve outcomes''. Ths result however s well-known to rely on the hypothess of constant margnal cost and for ths reason lacks any generalty (ths crtque of Bertrand [1883] dates at least back to Edgeworth [1925]). Stll, ths paradox s often contrasted wth the Cournot outcomes and reconclng these two approaches has been the am of many papers. The most spectacular result n ths feld s to be found n Kreps & Schenkman [1983] (hereafter KS). Two frms nvest n capactes and then compete n prces wth constant margnal cost up to capacty and an abrtrarly large margnal cost above capacty; nvestng nto lmted capacty has a strategc value because t amounts to commt not to be aggressve n the prcng game. KS reconclate the Cournot and Bertrand approaches by showng that the Cournot outcome s the unque Subgame Perfect Equlbrum of ther game. Ths result also has been much crtczed. In partcular because of the partcular ratonng rule, the effcent one, retaned for the analyss. Stll, the fact that capacty commtment relaxes prce competton and drves equlbrum outcomes towards cournotan ones s much less controversal. An open queston remans: to whch extent s the "rgd capacty" assumpton central for ths result to hold? In other words, would smoothly ncreasng margnal cost (or weaker forms of quanttatve constrants) lead to smlar outcomes? Ths queston s nteresqtng from a purely theoretcal pont of vew but t also belongs to the class of generalzatons that seem necessary to pursue robust emprcal studes on olgopolstc markets. As argued for nstance by Trole [1988] (chap. 5, p.244), "the Bertrand and Cournot models should not be vewed as two rval models gvng contradctory predctons of the outcome of competton n a gven market. (After all, frms almost always compete n prces.) Rather, they are meant to depct markets wth dfferent cost structures." It seems far however to say that a rgorous lnk between the shape of margnal cost and equlbrum prce s stll mssng under strateg prce competton. The man goal of the present paper conssts precsely n showng how the whole range of prces, from Bertrand towards Cournot ones, can be sustaned as equlbrum outcomes n olgopolstc ndustres, dependng on the shape of margnal costs. Prce competton under decreasng returns to scale, or weaker forms of capacty constrants, has been recently studed n the lterature. Klemperer and Meyer [1989] have sdhed lght on ths ssue usng the concept of supply -functon equlbrum. In ther model however, the prce s not fully strategc n the sense that they are ntemately related to supply. Dastdar [1995], [1997] consders a prce Bertrand competton wth contnuously 2

3 ncreasng margnal costs and homogeneous products. Magg [1996] ntroduces mperfect commtment n capacty games usng the set up developed by Dxt [1980] n a market for dfferentated products. In hs model, margnal cost s modelled as a stepwse, dscontnuous functon, beng constant up to the capacty level where t umps up to a hgher level. Magg obtan a "cournotan-lke" outcome n hs unque subgame perfect equlbrum. Stll, both Magg's and Dastdar's result does not compare at all wth that of KS and more generally wth the standard lterature on capacty-constraned prcng games because they totally rule out any form of ratonng. Whether frms are allowed to raton consumers or not s central when studyng prce competton under decreasng returns to scale. From a techncal pont of vew, t s wellknown that ratonng tends to destroy payoffs' concavty, and therefore pure strategy equlbra. Under weak forms of capacty constrants (such as those retaned by Dastdar and Magg) producng beyond capactes s always feasble but not necessarly proftable. We are nclned to beleve that frms should be allowed not to meet full demand whenever ths last strategy s proftable. As a consequence, we wll allow for consumers' ratonng. In the next secton, we extend the KS result to the case of mperfect commtment and many frms. To ths end we retan the cost framework put forward by Dxt [1980] and Magg [1996]. The chef mert of the cost structure proposed by these authors s that t allows us to parametrze the commtment value of capactes by the heght of the upward ump n margnal cost. Frms commt to capactes n a frst stage and then compete n prce n the second stage. In the second stage, they may produce beyond nstalled capacty but have to ncur for ths an extra unt cost, denoted by θ. The subgame perfect equlbra of ths game exhbt the followng features. If the margnal cost θ of producng beyond capacty s larger than Cournot prce, then Cournot equlbrum always obtan n the unque subgame perfect equlbrum. If θ s less than the cournot prce, there s a contnuum of subgame perfect equlbra but the prce on the equlbrum path s always θ whch mples that the aggregate quantty converges toward the Bertrand-compettve soluton at the lmt θ = 0. In other words, capacty has ts full commtment value whenever producng beyond capacty entals an addtonal unt cost at least equal to the correspondng Cournot prce. The lower the value of the addtonal unt cost, the closer the prce to the compettve benchmark. 3

4 2) BERT RAND- EDGEWORTH COMPETITION AND I MPERFECT COMMI T MENT The game tree we are consderng s dentcal to that of KS. Some n 2 frms choose some costly capactes and then compete n prce n the market for an homogeneous product. Ratonng, f any, s organzed accordng to the effcent ratonng rule. Under the effcent ratonng rule, low prcng frms get served frst and tes are broken evenly. The frm exhbtng the hghest prce s left wth a resdual demand (f any) whch s smply defned as a functon of the other frms' aggregate capacty. The cost structure n the prcng game s borrowed from Dxt [1980]. The margnal cost up to the capacty level s w.l.o.g. zero whle t s some postve θ beyond (ths ump typcally measures the legal wage gap for overtme work). Thus, n our capacty-prce game tree, frms nvest n capactes x at cost c(x ) and then compete n prces for the demand 0 f q x functon D(.) wth the same cost structure mc( q)= for all n. θ f q > x Note that Dxt [1980] uses ths set-up under quantty competton whereas Magg [1996] retans t for analyzng prce competton under product dfferentaton. Note also, that the orgnal KS model corresponds to the partcular case where θ s nfnte; observe also that any value larger than the zero-demand market clearng prce D -1 (0) would trvally yeld ther result. As a benchmark we consder the basc model of Cournot olgopolstc competton among n frms havng the same convex cost functon c(.). The aggregate consumer demand s D(p), ts nverse P(x). Let x x be the total quantty produced by frm 's opponents. A nl producton s clearly optmal f x - > D(0), otherwse frm 's proft s xpx ( + x ) cx ( ). We assume throughout that xp(x+z) s concave n x for all z. Therefore, the proft maxmzng quantty r c (x - ) s unque and decreasng 1 n x -. We denote r(x - ) the best reply wth zero producton cost, t wll play a central role n the study of prce competton. The symmetrc Cournot-Nash equlbrum s the soluton x < D(0)/n of x = r ( n 1 ) x. For nstance, wth constant margnal cost c and lnear demand P(z) = 1 c ( ) z, we get rc ()= z 1 c z 2 and x = 1 c + 1 n. 1 Formally r c (z) s the soluton of P( x + z) + xp ( x + z ) c ( x ) = 0. Let f( x, z) xp( x+ z) c( x) and 2 2 f r dfferentate the equaton to obtan c f = 2 x z x z r c Px ( + z ) + xpx ( + z ) = [ ] z 2Px 10 ;. ( + z ) + xpx ( + z ) ( cx ) 4

5 We are now n a poston to solve the two-stage game wth mperfect capacty commtment and prce competton. The partcular shape of margnal costs affects the analyss of prce subgames. Recall that the Edgeworth's argument conssts n showng that upwards prce devaton may be proftable when other frms are lkely to raton consumers. Ths requres frst that the demand addressed to them exceeds ther aggregate capacty and second that they are not wllng to meet demand beyond capacty. In the KS model, the second condton s always satsfed snce the cost of producng beyond capacty s prohbtve whereas ths s no longer the case n our model: a frm s wllng to meet any level of demand, beyond ts nstalled capacty provded the prce s above θ. It s only for prces below that level that the Edgeworth's argument apples, otherwse the standard Bertrand analyss apples. Fgure 1 below helps to understand the nature of prce competton. Consder the case of two fms. A frm wll perform ratonng f ts prce s less than θ, whch makes t unproftable to sell beyond capacty, and f the demand t faces s larger than ts capacty, thus the relevant threshold for ratonng s mn θ, 1. In the regon where the two frms { } name prce above ther respectve treshold, a standard Bertrand competton apples. p 2 k frm 1 ratons Bertrand competton mn{ θ,1 k2} frms 1 & 2 raton frm 2 ratons p 1 mn{ θ,1 k1} Fgure 1 As appears from nspecton of Fgure 1, the analyss developed by KS for the prcng games under capacty constrants apples here n a truncated part of the strategy space. Lemma 1 characterzes equlbra n all possble prce subgames. 5

6 Lemma 1 Whenever x D( θ ), p θ s the unque pure strategy equlbrum. Whenever x D( θ ), f the largest capacty x 1 s less than r(x -1 ), the equlbrum s the pure strategy P(x 1 + x -1 ), otherwse t s a mxed strategy equlbrum. Consderng an olgopoly nstead of a duopoly requres a dfferent orderng of the arguments used by KS n ther lemmas 2 to 5 but no real novelty s ntroduced. Stll, the proof s somewhat techncal and has been relegated to the appendx. We present here the lne of reasonng. We start by showng that ether the equlbrum s the pure strategy θ or that there are more than two frms playng the lower bound p of all equlbrum dstrbutons. Then we show that those frms are constraned at p so that ther equlbrum payoff s Π * = px. 2 Next, we show that f the sum of capactes s lesser than D(θ) then the equlbrum has to be θ because any frm s able to sell all of ts capacty at ths prce. In the remanng cases the equlbrum s n mxed strateges and we can ntroduce p * P( x ) < θ. We then show that frms who play the upper bound p of all equlbrum dstrbutons satsfy ) or ). ( ) ) rx ( )< x and p = P r( x )+ x : those frms have a large capacty and p does not depend on t but on what other frms dd choose. ) p = p * must hold n whch case p * s the equlbrum. Lastly, usng a complex formula borrowed from KS, we show that frms playng p n equlbrum have the same capacty and also the largest one. Relyng on Lemma 1, we may state our theorem whch extends the result of KS to olgopoly and mperfect commtment. For n symmetrc frms and effcent ratonng, the Cournot outcome emerge as the subgame perfect equlbrum outcome of the capactyprcng game as soon as the ex-post margnal cost θ s larger than the Cournot prce. Theorem 1 If θ > Pnx ( ), the symmetrc Cournot-Nash nvestment x followed by P(n x ) s the unque subgame perfect equlbrum of Γ. If θ < Pnx ( ), there s a contnuum of SPE who nevertheless satsfy D( θ ) =. It converges toward the Bertrand soluton. x n 2 In the KS settng, there are only two frms thus all frms derve ths payoff whch eases the rest of the proof. 6

7 Proof If x D( θ ) x, the equlbrum of the prcng game played after ( x ) n s the pure strategy θ and frm 's payoff n Γ s θx. If D( θ) x < x r( x ), the equlbrum s the pure strategy P( x + x) and frm s pad f( x, x) = xp( x + x) c( x). Ths functon s concave wth a maxmum at rc ( x ) < r( x ). If x > r( x ), the equlbrum s n mxed strateges and frm earns g( x, x ) = R( x ) c( x ). ( )= ( ) Notce that P( x + x) =θ when x = D( θ ) x and g x,( r x ) f x,( r x ). Hence the frst perod payoff as a functon of x s contnuous. Moreover at x = D( θ ) x, f = θ+ xpx ( + x ) cx ( ) < θ and the slope of g s steeper than that of f as the second x perod payoff becomes constant. The payoff functon s thus concave n x for any x - ; ts average over the equlbrum dstrbutons of the others frms s concave too, meanng that the best reply of frm s always a pure strategy. Because ths apples for all frms, the equlbrum s n pure strateges and satsfes x = max D( θ ) x, r ( x ) for all. { } c If D( θ) x > rc ( x ) for some x then D( θ ) = x + x. From ths we deduce that D( θ ) = x + x must hold for all other frms, so that x = rc ( x ) > D( θ ) x s mpossble. Therefore the canddate equlbra are all vectors ( x ) n satsfyng D( θ ) = x and D( θ ) > rc ( x ) + x for all. The symmetrc equlbrum D ( θ )/ n exsts n f t s larger than the Cournot canddate x.e., f θ Pnx ( ) the Cournot prce. Solvng for D( θ ) = rc ( y) + y yelds a value y * that crcumvents the range of asymmetrc equlbra; they are gven by the constrant n, x y * n addton to D( θ ) = x, thus ths set s a smplex. Now f θ >P( nx), the equlbrum s unque. The case for n = 2 s KS, thus consder n > 2 and let m x 1 2. If x 1 = r c (m+x 2 ) and x 2 = r c (m+x 1 ), then x 1 and x 2 are solutons of z = h( z) rc( m+ rc( m+ z) ). But snce 0 > r c () z > 1 (cf. footnote 5) t must be the case that hz ( ) = r m+ r( m+ z) r ( m z) ( ) + <1, thus h has a unque fxed pont so that x 1 = x 2. c c c By repetton of the argument to all pars, we conclude that the equlbrum s unque and symmetrc: t s the Cournot quantty x. n Fgure 2 below llustrates our fndngs for n = 2, a lnear demand D(p) = 1 p and zero margnal cost. The Cournot quantty 1/3 s found at the ntersecton of the two best reply functons (dashed lnes on Fgure 2). If the sum of quanttes x 1 + x 2 s less than D(θ) = 1 θ then frms are not able to avod the tradtonnal Bertrand competton, t s only for large aggregate capacttes that the prce equlbrum result n Cournot payoffs. Thus for θ > 1/3 (recall that the KS hypothess was θ > 1), very low capacty choces push frms toward 7

8 the lne D(θ) = x 1 + x 2, then for larger capactes the Cournot competton apples and leads to the symmetrc equlbrum choce of 1/3. For a θ' smaller than 1/3, the area where Bertrand competton apples ncorporates the prevous equlbrum meanng that frms are nduce to buld more capacty because the ferce prce competton yeld too small margns. There s now a contnüm of equlbra where frms share the market but not too asymmetrcally as the Cournot best reples provde lower bounds on one's equlbrum capacty. x 2 1 θ' 1 θ 1/2 1 Fgure 2 x 1 4) FI NAL REMARKS Many researches have amed at reconclng Cournotan outcomes wth the explct prce mechansm nvolved n the Bertrand model. These researches have been successful to the extent that they have been able to combne the two features of olgopolstc ndustres whch are lmted scales of producton or ncreasng margnal costs and prce settng behavour. The man challenge n ths respect conssts n dealng wth the ssue of quanttatve constrants (non-constant margnal cost) at the prce competton stage whch tends to make t unproftable for the frms to meet full demand. Ths n turn generates ratonng possbltes whch are at the heart of the Edgeworth's crtque. The ssue of ratonng n prcng games s best understood by studyng closely the allocaton process of a non compettve market wth prce-settng frms. Three stages are needed to correctly descrbe ths process. In the frst stage frms name prces and consumers address demand to frms. In the second one, frms decde on ther sales and possbly raton consumers. In the thrd stage, ratoned consumers possbly report ther demand to nonratonng frms who may or may not accept them. In the case of homogeneous products and 8

9 perfect dsplay of prces, the low prcng frm receves all the demand at the end of the frst stage. Under constant returns to scale, ths frm s wllng to meet any demand level so that t always choose to serve all consumers n the second stage; the thrd stage s then rrelevant. Wth decreasng returns to scale, thngs are qute dfferent because t may not be optmal to meet full demand n the frst stage. Curously enough, Edgeworth's classcal approach has been abrdged n several recent papers. Ther authors consder decreasng returns to scale; hence they have to deal wth the fact that frms are not always wllng to meet demand. Yet, t s generally asserted that "frms meet demand" wthout a word of explanaton. There s nether reference to an external mechansm that forces frms to meet demand nor reference to some reputaton effect n a larger game that would make ths restrcton tenable. Ratonng s thus ruled out by assumpton. Replaced n our three stages process, we can nterpret the equlbra of these prce competton games as Nash equlbra whch are not subgame perfect. In those equlbra ndeed, each frm names a prce and threaten ts opponents to take the non optmal decson to serve all of ts clentele n the subsequent stage. Stll the threat s never carred out on the equlbrum path. Kuhn [1994], Magg [1996], Bulow, Geanakoplos & Klemperer [1985], Vves [1990], and Dastdar [1995], [1997] are promnent examples 3 where such an odd vson of prce competton s endorsed. 4 Yet what an economst probably has n mnd when ntroducng quanttatve constrants nto prcng models s not that frms commt to ncur losses on hgh levels of sales but rather commt not to call for such large sales, precsely because ths would ental losses. Obvously, when a frm names a prce p, t does not threaten the other frms to make losses by later sellng unts havng a margnal cost larger than p. A frm has thus two basc optons when settng ts prce: t ether undercuts opponents to receve a large demand and possbly serves only a fracton of t or t charges a hgh prce n order to beneft from ratonng spllovers. Allowng for ratonng drves us back to the analyss ntated by Edgeworth [1925] and popularzed by KS. In the settng of these last authors, the quanttatve restrcton may seem too effectve snce t s physcally mpossble to produce beyond capactes at the prcng stage. In the present paper we have thus consdered the mperfect commtment of Dxt [1980] wthn an olgopolstc framework. Our framework thus exhbts as ts two 3 The frst two papers state the hypothess mplctly, the next two use a footnote but wthout ustfcaton; only Dastdar makes an effort by referrng to Dxon [1990] (see the precedng footnote). 4 What s lost exactly of the competton process when ratonng s forbdden s largely an open queston. In a companon paper however we show that when ratonng s forbdden wthn the KS framerwork, many SPE outcomes, ncludng the collusve ones can be sustaned. 9

10 polar cases the Bertrand and the KS's cost structure. We have then generalzed of the fndngs of KS n ths framework to show that, dependng on the value of the commtment, the whole range of prces between Bertrand and Cournot prces can be sustaned as part of subgame perfect equlbra. 10

11 Proof of Lemma 1 ( ) If the large capacty x 1 s less than r x 1, the equlbrum s the pure strategy Px + x, otherwse t s a mxed strategy equlbrum. ( ) 1 1 Proof W.l.o.g. frm 1 has the largest capacty n the subgame followng the play of ( x ) n. Let p and p be the lower and upper bounds of frm 's equlbrum dstrbuton F. Let H argmn p, p mn p, H argmax p and p max p. n n n Exstence of an equlbrum s guaranteed by theorem 5 of Dasgupta & Maskn [1986]. By lowerng ts prce a frm always benefts from an ncrease n demand (ths property s not nfluenced by our ratonng rule), ts payoff s therefore left lower semcontnuous (l.s.c.) n ts prce, thus weakly l.s.c.. The sum of payoffs s u.s.c. because dscontnuous shfts n demand occur only when two frms or more derve the same proft. n Clam 1 # H > 1 or the equlbrum s θ. If # H = 1, some frm k enoys demand D(p k ) on p; mn p # H. ) p <θ. If frm k s constraned at p ts revenue s the strctly ncreasng functon p k x k, a contradcton to p beng n the support of the equlbrum dstrbuton F k. Thus, D( p) < x k and because p k D(p k ) s a non-constant functon, t must be the case that p s the monopoly prce P(r(0)). Snce no other prce (rrespectve of what may play the other frms) can yeld the monopoly payoff, frm k must be playng the pure strategy P(r(0)) but then any other frm undercuts t, contradctng the optmalty of ts own equlbrum strategy. ) p θ. We are contemplatng the classcal Bertrand prce competton whose outcome s prcng at the margnal cost θ. * Clam 2 H, Π = px ) p <θ. If frm H devates to p - (ths s a shorthand for p ε where ε s a small postve real number) ts demand may ump upward; n order for p to be n the support of an equlbrum dstrbuton t must be the case that ths does not happen, thus the payoff s contnuous at p. If the demand at p - s D(p - ) t must be the case that p - < P(r(0)) for 11

12 otherwse frm would devate downward, thus D( p - ) > r(0) whch s an upper bound on capacty nvestment 5 as t s the optmal quantty wth zero cost for a monopoly. Therefore frm s constraned at p and we get Π * = px. ) p θ. We saw that θ s the unque possble prce equlbrum. The clam s even vald for all frms snce sales beyond capacty nether generate losses nor benefts. Clam 3 If x1+ x 1 D( θ ), θ s the unque prce equlbrum. The only case we need to consder s p <θ. If frm plays θ > p then the other frms that are less expensve receves full demand but serve only ther capactes so that frm receves more than D( θ) x x thus Π ( θ, F )= θ x > Π * a contradcton. From now on we study the case where p * P( x + x )< A B 1 1 θ. Clam 4 H = H H where H f r( x )< x and H f p = p A { { }} Let Ψ ( p ) p.mn x,max 0, D( p ) x be the payoff to frm when t names a prce p > max{ }. If p max{ p} then frm gets at least the payoff Ψ ( p ), thus ths H p functon must be maxmal at p to sustan ths prce as a member of the support of an equlbrum strategy. Frm cannot be fully served at p + for otherwse t would devate upward thus t wll only sell unts wth zero margnal cost. Two cases can occur. If Ψ ( p ) = p D( p ) x n a neghbourhood of p; we study the alternate formulaton of profts ( ) x B ( ) yp( y + ). The argmax s r( x ) so that p = P r( x ) + x and snce frm s not constraned at p, t must be true that r( x ) < x. Furthermore the equlbrum payoff n that * case s Π = Rx ( ) where R( x) r( x) P ( r( x) + x ). Usng the envelope theorem we obtan Rx ( ) = rxprx ( ) ( ( ) + x)< 0. If on the other hand, Ψ ( p ) = x p at p - then the upper prce s ( ) p = P x + x and we have x r( x ). * ( ) Clam 5 If H B, the equlbrum s p * P x + x In a Subgame Perfect Equlbrum we can elmnate strctly domnated strateges n the frst stage. 12

13 Observe that p * guarantees the revenue p * x to any frm. Indeed, f all other frms are less expensve, they are served frst but the resdual demand addressed to frm s precsely ts capacty. If H B, the equlbrum must be the pure strategy p * for all frms. Clam 6 If H B = then H = H A ={} 1.e., the large capacty frm Let H = H A. If x 1 > x then x 1 < x r( x 1) + x 1 < r( x ) + x as rz ( ) > 1 (cf. footnote 5). Hence, frm 1 obtans a payoff R( x 1) Π 1 * by namng P( r( x 1) + x 1 ) > Prx ( ( ) + x )= p. We now prove that x1r( m + x1)< xr( m+ x) where m = x 1 (zero f n = 2). Let us defne Θ( z) zr( m+ z) = zr( m+ z) P( m+ z+ r( m+ z) ). The envelope theorem mples Θ ( z) = ( r( m+ z) z) P( m+ z+ r( m+ z) ). If rm ( + x) < x then Θ < 0 and we are done. Otherwse rm ( + x) > x mples that rm ( + x) = xs solved for a x * greater than x snce r(.) s decreasng. By the same token, x * > x mples that the soluton y * to rm ( + x) = x has to be greater than x * *. Fnally, rm ( + y) = x < x 1 mples y * < x 1. Ths s crucal because the postveness of Θ * on [ x ; x ] wll be offset by ts negatveness on the * large nterval [ x ; x 1 ] as the followng development shows. * * * x x x y * * x x x x Θ( ) Θ( ) Θ 1 x x = + Θ Θ + Θ = Θ( y ) Θ( x ) 1 = ( ) + = ( + ) + * * = x ( y P( x + m+ y ) R( m+ x )) < 0 by defnton of R(.). * -1 * -1 y R r ( x ) x R( m x ) y x P x r ( x ) x R( m x ) So far we have proved that Π * x Rx Π Π x Rx x = ( ) < ( 1) 1 * < 1 * whch s equal to 1 x1 p F p D p x F p D p 1[ 1 ( 1 ) ( ( 1 ) 1)+ ( 1 1 ( 1 ) ) ( 1 ) ]. If frm plays p t obtans demand 1 F 1 ( p )( D( p ) x )+( F p ) D p ( ) ( ) whch s larger than the demand of frm because there s more weght on the monopolstc demand term, therefore Π p F Π 1, * ( )> the desred contradcton. We have thus shown that 1 H A. Snce p = P r( x ) + x holds true for any H A, members of H A must have the same (largest) capacty. * ( ) The two cases that occur n clam 4 now make sense: f a frm names prces larger than other frms ( H A ), t must be the one wth the greatest capacty and furthermore,the excess must be large enough. 13

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