Title: Bertrand-Edgeworth Competition, Demand Unertainty, and Asymmetri Outomes * Authors: Stanley S. Reynolds Bart J. Wilson Department of Eonomis Eonomi Siene Laboratory College of Business & Publi Admin. University of Arizona University of Arizona Tuson, Arizona Tuson, Arizona Running Title: Unertain Demand & Asymmetri Outomes Corresponding Author: Stanley S. Reynolds Department of Eonomis MClelland Hall University of Arizona Tuson, Arizona 8572 phone: (520) 62-625 FAX: (520) 62-8450 email: reynolds@bpa.arizona.edu * We are grateful for omments from anonymous referees and an assoiate editor and from seminar partiipants at the University of Arizona, U.S. Department of Justie, Center for Eonomi Researh and Graduate Eduation at Charles University, Juan Carlos III Universidad de Madrid, Wissenshaftszentrum in Berlin, and the Eonometri Soiety 998 Winter Meetings. This paper is a substantially revised version of a hapter of Wilson s 997 University of Arizona Ph.D. thesis. Wilson aknowledges support from a National Siene Foundation dissertation grant. Reynolds did muh of his work while visiting CERGE at Charles University in Prague and aknowledges support from that institution.
Abstrat: We analyze investment and priing inentives in a symmetri Bertrand- Edgeworth framework with unertain demand. Firms hoose prodution apaities before observing demand. Pries are hosen after demand is observed. If the extent of demand variation exeeds a threshold level then a symmetri equilibrium in pure strategies for apaities does not exist. A smaller firm has no inentive (ex ante) to expand its apaity beause apaity expansion would redue its expeted revenue in the event that demand is lower than expeted. Output pries are predited to have positive variane when demand is low and zero variane when demand is high. 2
. Introdution In this paper we explore how demand variability and relatively inflexible prodution apaity interat to influene priing and investment inentives in an oligopoly setting. Our point of departure is the Kreps and Sheinkman [7] two-stage model of apaity investment and priing. Kreps and Sheinkman (hereafter, KS) show that in a game in whih duopolists hoose apaities in stage one and pries in stage two, the equilibrium involves prodution levels and apaities that orrespond to Cournot output levels and pries that are set to lear the market. Their analysis provides a sort of vindiation of the Cournot analysis by showing that a simple reformulation exists in whih firms atually hoose their pries, and yet the final outome repliates the outome of the basi Cournot analysis. We introdue demand unertainty into the KS formulation. At the time when firms hoose their apaities in stage one, the firms are unertain about the level of demand. After installing their apaities, the firms observe the realization of demand and hoose pries in stage two. The timing of deisions and aess to demand information that we speify are onsistent with the view that apaity hoie is a long-run deision made before all market onditions are known and that prie hoie is a short-run deision made after observing urrent market onditions. We show that a symmetri equilibrium in pure strategies for apaity hoies fails to exist if the degree of variability in demand exeeds a threshold level, in spite of the fat that firms payoffs are symmetri. Asymmetri equilibria are shown to exist for partiular assumptions on the demand funtion and the distribution of demand shoks. The apaities in suh an asymmetri equilibrium are not equal to Cournot outputs for the orresponding 3
game of output hoie with demand unertainty. Hene, this paper provides another instane in whih the KS vindiation of Cournot is not robust. The absene of symmetri equilibria is based on a differene in expeted revenue between a large (high apaity) firm and a smaller (lower apaity) firm, when demand is low. If demand is low, then the marginal revenue assoiated with extra apaity for a large firm is zero; the large firm s expeted revenue is ompletely determined by the smaller firm s apaity. However, if demand is low then the marginal revenue assoiated with extra apaity for a smaller firm may be negative. Additional apaity for the smaller firm lowers the distribution of subgame equilibrium pries in a way that may redue expeted revenue for the smaller firm. This differene in marginal revenue between a large firm and a smaller firm yields a disontinuity in the apaity reation funtion at symmetri apaity pairs. This result reveals an error in a paper that analyzes ollusion among firms faing flutuating demand. Staiger and Wolak [4] formulate a model in whih our two-stage game is infinitely repeated. Defetions from ollusion are punished by reversion to an equilibrium of the two-stage game. Staiger and Wolak identify a symmetri equilibrium in apaity hoies for the Nash reversion game; however, their proof fails to reognize the lak of differentiability of the expeted revenue funtion at symmetri apaity pairs. The ollusive outome for the repeated game an not be supported as an equilibrium by the symmetri pure strategies in apaity hoies that they identify. It is possible to support a ollusive outome that is qualitatively similar to the one analyzed by Staiger and Wolak in [4] through the use of symmetri optimal punishment strategies of the type onsidered by Abreu []. 4
Another result from our analysis is that the variane of pries may be higher for a low demand realization than for a high demand realization. We find that the type of priing subgame equilibrium, pure or mixed, depends on the realization of demand. For high demand realizations, the priing equilibrium is in pure strategies with the firms utilizing all of their apaity. However, for low demand realizations, the subgame priing equilibrium may be in mixed strategies, with total output less than apaity. The equilibrium in the KS model always involves firms produing at full apaity. We indiate the impliations of our results for empirial work in the last setion of the paper. 2. The Model and Preliminary Results We analyze a two-stage duopoly model of apaity investment and priing in a homogeneous produt market. In stage one firms simultaneously invest in prodution apaity. The firms are unertain about the size of the market for their output when they make investment deisions. Before stage two, the unertainty about the size of the market is resolved. The firms engage in prie ompetition in stage two, having observed the level of market demand and investments made in stage one. The amount of apaity hosen by firm one is x; the amount of apaity hosen by firm two is y. Let a represent the level of market demand. In stage one the firms do not know a but they do know that a will be the realization of a random variable with a partiular distribution and with support ontained in [ aa, ]. In stage two firms hoose their pries after observing prodution apaities and the level of demand. A subgame in stage two is defined by the triple, ( x, ya, ). Our assumptions on the demand funtion follow KS, with the added feature of a demand level variable. Let the market inverse demand funtion be Pqa (, ) (prie as a 5
funtion of quantity q, given demand level a) and let the orresponding ordinary demand funtion be Dpa (, ) (quantity demanded as a funtion of prie p, given a). ASSUMPTION. The funtion P( q, a) is stritly positive for all q ( 0, q( a)), where q ( a) min{ q 0: P( q, a) = 0}. P( q, a) is twie ontinuously differentiable, stritly dereasing, and onave in q for q ( 0, q( a)). For q q( a), P( q, a) = 0. For ( qa, ) suh that P( q, a) is positive, P( q, a) is stritly inreasing in a. We must speify how ustomers are rationed when the firm offering the low prie has insuffiient apaity to serve all ustomers who wish to buy. We follow the speifiation used by Kreps and Sheinkman [7] and Osborne and Pithik [9], [0]. ASSUMPTION 2. Demand is rationed aording to the effiient-rationing rule. 2 There are no osts of prodution, apart from apaity investment osts. A onstant marginal ost of prodution for output up to a firm s apaity an be introdued by subtrating this marginal ost from the inverse demand funtion. ASSUMPTION 3. The marginal ost of apaity is onstant and equal to for eah firm, where, 0< < E[ P( 0, a)]. 3 While we do not model firms as hoosing their output levels diretly, it will prove to be useful to have results from Cournot games of output hoie at our disposal. Consider firm i s optimal output hoie when its rival selets output qj q ( a ). bq (, a) j arg max 0 q q( a) q i j { q ( P( q + q, a)} i i j 6
bq (, a) is the best response funtion in a Cournot output hoie game if i s rival puts q j on j the market, when prodution is ostless and when the demand level is a. In Lemma One KS show that the following results follow from Assumptions and 2: (i) b is noninreasing in q j, is ontinuously differentiable and stritly dereasing in q j over the range where it is positive (given a), bq ( j, a) (ii) b ( qj, a) = q j, with strit inequality for ( qj, a) suh that b( qj, a) > 0, (iii) there is a unique Nash equilibrium for eah demand level a, suh that eah firm hooses output qa $( ), satisfying qa $( ) = bqa ($( ), a). Define the funtion, R( q, a) b( q, a) P( b( q, a) + q, a). This funtion indiates the j j j j revenue a firm earns by hoosing its best output response to rival output q j when the level of demand is a. We onsider a seond Cournot output hoie game in whih outputs are hosen before the demand level realization and prodution has onstant marginal ost,. The equilibrium of this game serves as a benhmark against whih results from the two-stage game may be ompared. Consider firm i s optimal output hoie when its rival selets output qj q ( a ). b ( q ) j arg max 0 qi q( a) q j { EqPq [ ( + q, a) q]} i i j i Under assumptions 3 the objetive funtion is twie ontinuously differentiable and stritly onave in q i. Thus, there is a unique best response for eah q j ( 0, q( a)). The ondition < E[ P( 0, a)] from assumption 3 implies that b ( 0) > 0. The properties of b are analogous to those of b, as speified in the preeding paragraph. The reation funtion 7
b is noninreasing in q j and is ontinuously differentiable with 0 > b ( q ) > over the range where it is positive. Therefore, there is a unique Nash equilibrium for this seond output hoie game suh that eah firm hooses output q ~, satisfying q ~ ( ~ = b q ). We now return to our analysis of the two-stage model of investment and priing. A priing subgame is defined by the triple, (x,y,a). The nature of subgame equilibria an be desribed by dividing the apaity spae into three regions, as depited in Figure One: A( a) = {( x, y): x 0, y 0, x b( y, a), y b( x, a)}, Ba ( ) = {( x, y): x qa ( ), y qa ( )}, and Ca ( ) = {( x, y): x 0, y 0,( x, y) A( a),( x, y) B( a)}. In region A(a) the subgame equilibrium is in pure strategies, with eah firm setting a prie equal to the market learing prie. The market learing prie is determined by the intersetion of demand with a vertial supply at quantity x + y. In region B(a) the subgame equilibrium is a Bertrand equilibrium with prie equal to zero (marginal prodution ost) for both firms. If apaities are in region C(a) then there is no pure strategy subgame Nash equilibrium in pries (exept for the trivial ase in whih one firm has zero apaity); there is a unique mixed strategy equilibrium in pries ( [0, Theorem ]). Our analysis fouses on the expeted revenue in subgame equilibria rather than on the form of subgame equilibrium strategies. The following lemma summarizes results from [7, Lemma 5] and [9, Theorem, Table ]. j 8
LEMMA Expeted revenue for firm one in a subgame equilibrium is the funtion r ( x, y, a), whih satisfies the following onditions: r( x, ya, ) = Px ( + yax, ), if ( x, y) A( a) x rxya (,, ) = Rxa (, ) y, if( xy, ) Ca ( ), x yandrxa, (, ) ypya (, ) r( x, y, a) = ~ s x, if ( x, y) C( a), x y, and R( x, a) > yp( y, a), where s satisfies s D s a = R x a rxya (,, ) = Rya (, ), if( xy, ) Ca ( ), x> y rxya (,, ) = 0, if ( x, y) B( a) Expeted revenue for firm 2 is given by r ( y, x, a). The funtion r( ) is ontinuous in x and y for eah value of a. In region C(a) the expeted revenue of the large firm depends on the level of demand and the apaity of the smaller firm. The large firm s expeted revenue is independent of its own apaity. The small firm s expeted revenue in region C(a) takes one of two forms. For apaities in what we refer to as Region I, expeted revenue for a small firm is the fration y/x of the larger firm s expeted revenue (where y is the small firm s apaity). For apaities in Region II, the small firm s expeted revenue is ~ sy, where ~ s is an impliit funtion of the small firm s apaity and the level of demand. 3. Main Results The first theorem onerns existene of symmetri equilibria for apaity hoies when there is a ontinuous distribution of demand levels. A orollary provides an easily interpretable version of the result for the speial ase of additive demand shoks. The seond theorem haraterizes equilibrium apaity hoies for a linear demand funtion and a two-point distribution of demand levels. 9
THEOREM. Suppose that the probability density for the demand shift variable is positive and ontinuous on its support, [ aa, ], where0 a< a<. (i) If q ˆ( a) q~ then there is a symmetri equilibrium in apaity hoies, with apaity equal to q ~ for eah firm. Capaity is fully utilized and pries are set equal to the market learing level for every demand realization in this equilibrium. (ii) If q ˆ ( a) < q~ then a symmetri equilibrium in pure strategies for apaity hoies does not exist. Proof: See the appendix. Part (i) is similar to the well-known result of Kreps and Sheinkman [7]. They show that the equilibrium for a game with ertain demand involves pries that are set to lear the market, fully utilized apaity, and apaity levels that orrespond to Cournot output levels. Part (i) applies to situations in whih the Cournot output level for the game with no osts and the lowest possible demand level ( qa $( )) is greater than or equal to the Cournot output level for the game with marginal ost > 0 and unertain demand q ~ ). The payoffs for apaity investment are then essentially equivalent to the payoffs assoiated with the game of output hoie in whih output is hosen prior to the realization of the demand level and firms have marginal ost. The symmetri equilibrium apaities orrespond to Cournot duopoly outputs for suh a game of output hoie. Capaities are fully utilized and pries are set to lear the market for every demand realization in equilibrium. ( 0
Part (ii) stems from the properties of the expeted revenue funtion for apaities in region C(a). If qa $( ) is less than q ~ then any viable andidate for a symmetri equilibrium will be in a subgame in region C(a) with positive probability. In region C(a) there is an asymmetry in the effet of a hange in apaity on a firm s expeted revenue for a subgame. A hange in apaity for a large firm has no effet on its expeted revenue. The expeted revenue for a small firm delines for an inrease in its apaity, when its apaity is nearly as large as its rival s apaity (apaities are in Region I of C(a)). This asymmetry in expeted marginal revenue due to a apaity hange leads to a kink in a firm s expeted profit funtion at symmetri apaity pairs, as long as there is a positive probability that apaities are in region C(a), for some realizations of a. Part (ii) of Theorem One rules out existene of symmetri equilibria in pure strategies for apaity hoies. Lemma Seven of Dasgupta and Maskin [2] may be applied to prove existene of a symmetri mixed strategy equilibrium in apaity hoies. The Corollary shows that the hypotheses of Theorem One have a simple interpretation for the ase of additive demand shoks. The existene of a symmetri equilibrium in pure strategies for apaities turns on the extent to whih demand may deviate below its mean value. COROLLARY. Suppose that the demand shok is additive, so that inverse demand ( may be expressed as, P ( q, a) = P( q) + a, for q q( a). If a E( a) then there is a symmetri equilibrium in pure strategies for apaity hoies, in whih apaity is fully utilized and pries are set equal to the market learing level for every demand realization. If a < E( a) then a symmetri equilibrium in pure strategies for apaity hoies does not exist.
The Corollary is proved by translating the omparisons of qa $( ) and q ~ in Theorem One into omparisons of a and Ea ( ). 4 THEOREM 2. Suppose that the demand level is a > 0 with probability θ ( 0, ) and a > a with probability θ, and that inverse demand is Pqa (, ) = a q, forq a, and Pqa (, ) = 0, forq> a. (i) If a E( a) then there is a symmetri equilibrium in apaity hoies, with apaity equal to ~ = ( E( a) ) / 3 for eah firm. Capaity is fully utilized and q pries are set equal to the market learing level for every demand realization in this equilibrium. (ii) If 2 a a < E ( a ) then a symmetri equilibrium in pure strategies for apaity hoies does not exist; there are pure strategy asymmetri equilibria in apaity investments. In the low demand subgame firms play asymmetri mixed priing strategies and set pries above the market learing level. Pries are set at the market learing level and apaity is fully utilized in the high demand subgame. Proof: See the appendix. Part (i) is quite similar to part (i) of Theorem One and the Corollary. The only differene is that uniqueness of equilibrium apaities is established for the model with a two-point demand distribution. The result in part (ii) is illustrated in Figure Two, whih provides a diagram of apaity reation funtions, for x > y. The reation funtion for firm two, R ~ 2, has a downward jump aross the 45 degree line beause of the disontinuity in marginal revenue 2
of apaity (when demand is low) at symmetri apaities and another downward jump aross the boundary of Regions I and II. The ondition, a < 2 a, insures that R ~ 2 rosses the reation urve, R ~, of the large firm in Region I. What sustains the asymmetri position of firms as an equilibrium outome? Note that in the high demand state the marginal revenue for apaity for the larger firm is smaller than the marginal revenue for apaity for the smaller firm. Sine the firms have the same marginal ost of apaity investment, there must be a orresponding differene in the marginal revenue for apaity in the low demand state. Speifially, marginal revenue for apaity for the smaller firm in the low demand state must be negative, sine marginal revenue for apaity for the larger firm is zero in the low demand state. When demand is low, additional apaity for the small firm lowers the distribution of subgame equilibrium pries in a way that redues expeted revenue for the large firm and may redue expeted revenue for the small firm. Reall from the Lemma that the small firm s revenue in Region I is a fration of the large firm s revenue; firm two's expeted y revenue is R( y, a) x for y < x. Given the assumptions of Theorem Two, as the small firm expands its apaity y beyond its equilibrium apaity, y*, the revenue funtion R( y, a) dereases faster than the ratio of apaities y / x * inreases, and so the small firm s marginal revenue is negative in the low demand state. Thus, it is the possibility that demand will be low that deters the small firm from expanding (in equilibrium) beause extra apaity will atually lower its (expeted) revenue. The other fore that maintains the asymmetri equilibrium outome may be traed to the fat that the expeted marginal revenue of apaity for the large firm is zero in the 3
low demand state. If demand turns out to be low then a large firm would have gained nothing by having less apaity, provided its apaity remains above its rival s apaity. At an asymmetri equilibrium, the expeted marginal revenue of apaity due to the high demand state is balaned by the marginal ost of apaity. If the large firm redues its apaity below its equilibrium level then the expeted revenue loss will exeed the apaity ost savings. A haraterization of pure strategy equilibria for general demand funtions and demand shok distributions is a hallenging problem. As our analysis has demonstrated, payoff funtions are not differentiable everywhere and best response orrespondenes may have jumps. It is possible that no pure strategy equilibria exist for some demand and distribution onfigurations. One possible approah for a more general analysis would be to apply the theory of submodular games, as in Topkis [5]. This approah requires that expeted payoffs exhibit non-inreasing differenes in apaities 5 ; differentiability is not required. Pure strategy equilibria exist for submodular games, even though best response orrespondenes may have jumps. However, this approah annot be applied diretly to our model, sine payoffs do not satisfy non-inreasing differenes for all apaity pairs. For example, if apaities are in Region I (see Figure One) then expeted subgame revenue for firm two in the low demand subgame an have a positive ross partial derivative in apaities. This an yield a positive ross partial derivative for total expeted payoff for firm two, whih would be inonsistent with the submodular approah. There may be restritions on parameters or funtional forms that would allow one to apply the submodular games approah. 4
4. Related Literature and Disussion The model analyzed by Gabszewiz and Poddar [4] is similar to ours exept that firms hoose outputs rather than pries in the final stage subgame. They prove that if firms fully utilize apaity in all demand states in equilibrium then equilibrium apaities orrespond to ertainty equivalent Cournot apaities (in terms of our notation, eah firm hooses apaity, q ~ ). This is analogous to our result in part (i) of Theorem One. We believe that our formulation has the advantages of providing an expliit model of prie formation and of generating riher impliations for the impat of demand flutuations on priing ompared to the model of Gabszewiz and Poddar. Our analysis predits that the variane of output pries in industries with apaity-onstrained, priesetting firms is higher during low demand periods (due to mixed strategy priing) than in high demand periods. Wilson and Reynolds [6] examine this predition using observations of industrial output and pries over the business yle. They find that the estimated variane of pries is signifiantly higher when demand is low than when demand is high in almost all U.S. manufaturing industries. Several related oligopoly analyses have formulated output or apaity hoie models that yield asymmetri outomes. Saloner [2] analyzes a Cournot duopoly model in whih firms may spread prodution over two periods. The opportunity to observe and reat to a rival s initial output hoie indues multiple subgame perfet Nash equilibria (SPNE), inluding all outomes on the outer envelope of the best-response funtions between the firms Stakelberg outomes. Pal [] generalizes Saloner s model by allowing prodution osts to differ over time (say, due to time disounting). If osts fall 5
slightly over time then asymmetri equilibria remain. Otherwise, the model generates a unique, symmetri SPNE. More reently, Maggi [8] utilizes the sequential deision struture of [] and [2] to analyze the role of sequential apaity investment in generating asymmetri duopoly outomes. Maggi introdues demand unertainty into the model, with the realization of demand revealed between the first and seond investment stages. The nature of outomes depends on the demand level. If demand is low then asymmetri outomes emerge, but if demand is high then firms invest up to a symmetri outome. Our analysis is similar to Maggi [8] in that we analyze the interation of apaity investment and demand unertainty in generating asymmetri outomes. We also use a sequential deision struture. However, we assume that a single stage of apaity investment is followed by a single stage of priing, as in Kreps and Sheinkman [7], rather than utilizing a sequential apaity or output hoie formulation. Asymmetri outomes are generated in our model by the possibility of low demand realizations and the orresponding impliations for the marginal revenue of apaity. A related result appears in Davidson and Denekere [3]. They modify the Kreps and Sheinkman analysis to allow random demand rationing, rather than effiient rationing. This hange in the rationing rule an reate inentives for apaity investment suh that exess apaity emerges in equilibrium. For suffiiently small osts of apaity, Davidson and Denekere find that reation urves (in apaity spae) are disontinuous and that only asymmetri equilibria exist. We view our model as a omplement to the sequential output/apaity approah in [8], [] and [2] and to the random rationing approah in [3], rather than as a substitute 6
for these formulations. We onjeture that the inlusion of sequential apaity hoie and random demand rationing in our formulation would expand the range of parameters over whih asymmetri outomes emerge and perhaps inrease the magnitude of asymmetries. Staiger and Wolak [4] formulate and analyze a model that is a speial ase of the Corollary (with, Pqa (, ) = a q and a = 0 < E( a) ). Their analysis of the two-stage game is then used to haraterize ollusion when the game is infinitely repeated. Staiger and Wolak identify a symmetri apaity outome for our two-stage model as the equilibrium to whih the firms revert following a deviation from the ollusive path in the repeated game. Their proof, however, fails to reognize the lak of differentiability of the revenue funtion at symmetri apaity pairs. In terms of our notation, they speify the derivative of r( x, ya, ) with respet to x at a symmetri apaity pair to be zero in region C(a). This is orret for the right hand derivative, but not for the left-hand derivative. As a onsequene, Staiger and Wolak inorretly identify a symmetri outome in apaities as an equilibrium of the two-stage game. 6 Why does this diffiulty matter for the analysis of ollusion? Staiger and Wolak find symmetri ollusive apaities and equal ollusive profits for the two firms based on the symmetri reversionary outome that they identify. These ollusive apaities are not supported as part of a nonooperative equilibrium if the reversionary outome is not itself an equilibrium for the stage game. There would appear to be several ways to remedy this diffiulty. One approah, whih is in fat suggested in Staiger and Wolak [4], would be to use symmetri optimal punishment strategies of the type onsidered by Abreu []. This approah would yield results on optimal ollusion that are qualitatively similar to the results reported in [4]. 7 Another approah would be to utilize a symmetri mixed strategy 7
equilibrium in apaity hoies, whih does exist, as noted earlier. This would yield equal expeted profits for the two firms in the reversionary equilibrium and ould support symmetri ollusive apaities (though not the ollusive apaities identified by Staiger and Wolak). 8 A third approah would be to haraterize an asymmetri equilibrium for the two-stage game, as we do in Part (ii) of Theorem Two, and use this as the basis for an asymmetri ollusive mehanism. Asymmetri ollusion of this type may be an interesting topi for further researh. 8
Notes. We show that marginal revenue is negative for a small firm if its apaity is suffiiently lose to the larger firm s apaity. 2. See Lemma Six in KS for the speifi form of a firm s demand as a funtion of firms pries. This assumption is not innouous. The rationing rule is related to the issue of whether symmetri equilibria exist in the model without demand unertainty, as pointed out in Davidson and Denekere [3]. 3. This is more restritive than KS, who allow apaity ost to be an inreasing, onvex funtion of apaity. 4. A related result is derived by Hviid [6] in an analysis of our two stage model with linear demand and zero apaity osts. He shows that the equilibrium involves pure strategies only if there is no demand unertainty. In this ase equilibrium apaities equal the Cournot output, q ~. This result is a speial ase of our Corollary. If = 0 then the ondition a E( a) is satisfied only if the support of the demand shok distribution is a single point. 5. If the payoff funtion is twie differentiable then non-inreasing differenes orresponds to a non-positive ross partial derivative with respet to the two apaities. 6. A symmetri equilibrium in apaity hoies an exist if the lower bound for demand levels is positive and the variane of the distribution of demand levels is small. This ase is desribed in part (i) of Theorem One and its Corollary. However, in suh an equilibrium, exess apaity never arises; firms set market learing pries in eah possible subgame. 9
7. In their working paper [3], Staiger and Wolak derive results based on symmetri optimal punishment strategies. 8. Some would argue that reliane on mixed strategies, whether for apaities or pries, is a symptom of model mis-speifiation. We argue that mixed strategies are easier to justify for pries than for apaities. For example, if there is ex ante unertainty about short run marginal prodution ost and firms have private information about their own osts, then an equilibrium may exist in whih firms utilize pure strategies for pries, but for whih pries exhibit substantial variability. Holt [5] shows that equilibrium pure strategies for pries for a game with ost unertainty onverge to equilibrium mixed strategies as the amount of ost unertainty vanishes. It is diffiult to think of a orresponding interpretation for mixed strategies for apaity hoies. 20
Referenes. D. Abreu, Extremal equilibria of oligopolisti supergames, J. Eon. Theory 39 (986), 9-225. 2. P. Dasgupta and E. Maskin, The existene of equilibrium in disontinuous eonomi games, I: Theory, Rev. Eon. Studies 53 (986), -26. 3. C. Davidson and R. Denekere, Long-run ompetition in apaity, short-run ompetition in prie, and the Cournot model, RAND J. Eon. 7 (986), 404-45. 4. J. Gabszewiz and S. Poddar, Demand flutuations and apaity utilization under duopoly, Eonomi Theory 0 (997), 3-46. 5. C. Holt, Oligopoly prie ompetition with inomplete information: onvergene to mixed-strategy equilibrium, University of Virginia working paper, 994. 6. M. Hviid, Capaity onstrained duopolies, unertain demand and non-existene of pure strategy equilibria, European J. of Politial Eonomy 7 (99), 83-90. 7. D. Kreps and J. Sheinkman, Quantity preommitment and Bertrand ompetition yield Cournot outomes, Bell J. Eon. 4 (983), 326-337. 8. G. Maggi, Endogenous leadership in a new market, RAND J. Eon. 27 (996), 64-659. 9. M. Osborne and C. Pithik, Prie ompetition in a apaity onstrained duopoly, Working Paper, Marh 983. 0. M. Osborne and C. Pithik, Prie ompetition in a apaity onstrained duopoly, J. Eon. Theory 38 (986), 238-260.. D. Pal, Cournot duopoly with two prodution periods and ost differentials, J. Eon. Theory 55 (99), 44-448. 2
2. G. Saloner, Cournot duopoly with two prodution periods, J. Eon. Theory 42 (987), 83-87. 3. R. Staiger and F. Wolak, Collusive priing with apaity onstraints in the presene of demand unertainty, Working Paper in Eonomis E-90-5, Hoover Institution, 990. 4. R. Staiger and F. Wolak, Collusive priing with apaity onstraints in the presene of demand unertainty, RAND J. Eon. 23 (992), 203-220. 5. D. Topkis, Equilibrium points in nonzero-sum n-person submodular games, SIAM J. Control and Optimization 7 (979), 773-787. 6. B. Wilson and S. Reynolds, Prie movements over the business yle in U.S. manufaturing industries, Federal Trade Commission Working Paper #29, June 998. 22
APPENDIX A: PROOF OF THEOREM Part (i). Consider x, y ) = ( q ~, ~ q ) as a andidate for equilibrium apaities. Note that ( ( x, y ) A( a ) for all a [ a, a] sine qa $( ) > qa $( ) x = y for a > a. Is x a best response for firm one to y? If x < x then ( xy, ) Aa ( ) for all a [ a, a]. By the Lemma firm one s expeted profit is, π( xy, ) = EPx [ ( + y, ax ) x]. This funtion is stritly inreasing in x for x ~ < x = q sine the RHS is stritly onave in x, and q ~ is the optimal output response to y = q~. If x > x then there are three ases to onsider. Case. ( x < b( y, a)): In this ase ( xy, ) Aa ( ) for all a [ a, a]. The marginal payoff for apaity for firm one is, EPx [ ( + y, a) + xp ( x+ y, a)]. This marginal payoff is negative sine it is the derivative of, EPx [ ( + y, ax ) x], whih is maximized at x = q~ < x. Case 2. (b( y, a) x< by (, a) ): In this ase ( xy, ) is in C(a) for some a-values and in A(a) for other a-values. Let a ( x, y ) be the value of a that satisfies x = b( y, a ( x, y )). Then ( xy, ) Ca ( ) for a < a ( x, y ) and ( xy, ) Aa ( ) for a a ( x, y ). By the Lemma the marginal payoff for apaity for firm one is, a π a ( x, y ) ( xy, ) = [ Px ( + y, a ) + xp ( x + y, a )] dfa ( ). () If x = b( y, a) then this marginal payoff is, a [ Px ( + y, a) + xp( x+ y, a)] dfa ( ) a, whih is negative sine it orresponds to the marginal payoff of output in a Cournot game with output in exess of the best response. The derivative of the marginal payoff in () with respet to x is, 23
π a a ( x, y ) 2 ( xy, ) = [ P ( x + y, a ) + xp ( x + y, a )] dfa ( ) a ( x, y ) + { Px ( + y, a ( xy, )) + xp ( x+ y, a ( xy, ))} x The integrand is negative sine P is dereasing and onave in output and the term in urly brakets is zero. So, the marginal payoff for apaity for firm one is negative in ase two, sine π ( xy, ) is negative at x = b( y, a) and π ( xy, ) is dereasing in x for by (, a) x< by (, a). Case 3. ( x π ( xy, ) = < 0. b( y, a)): In this ase ( xy, ) Ca ( ) for all a [ a, a] and This demonstrates that firm one s expeted profit is inreasing in x for x dereasing in x for x best response to x. < x and > x, so x is a best response to y. By idential arguments, y is a Part (ii). Let ( x, y ) be any pair of apaities suh that x = y. Consider ( x, y ) as a andidate for equilibrium apaities. If x = y q$( a) then ( x, y ) is in A(a) for all a and, by the Lemma, the expeted payoff for firm one is, EPx [ ( + y, ax ) x ]. The hypothesis of part (ii), q ˆ ( a) < q~, implies that x = y < q~, so that the marginal payoff for apaity for firm one is stritly positive; x annot be a best response to y. If x = y q(a) then ( x, y ) is in B(a) for all a and, by the Lemma, the expeted revenue is zero for every possible demand realization, and firm one earns payoff, x. Any apaity x < x would yield higher expeted payoff for firm one. 24
If q ˆ( a) < x = y < q( a) then there is a set of a-values suh that ( x, y ) is in C(a). This set is defined as, Γ { a [ a, a]: a a min{ a ( x ), a }}, where a ( x ) satisfies, x = b( x, a ( x )). Consider the marginal revenue for apaity for firm one for any a Γ. By the Lemma the right hand partial derivative of the revenue funtion is zero sine rxy (,, a) is independent of x for x > y. That is, rx (, y, a) x + = 0. Reall that Rxa (, ) bxapbxa (, ) ( (, ) + xa, ). R is ontinuous in x given a, sine the inverse demand funtion and the best response funtion, b( xa, ), are ontinuous in x given a. The aim of this setion is to show that Rxa (, ) < ypy (, a) for x smaller than, but suffiiently lose to, x = y. Note that if a Γ then x > q$( a). This inequality implies that b( x, a) < qa $( ) < x = y, sine under Assumption One, the best response to an output x is smaller than the Cournot output ( x is greater than the Cournot output). So, given x and a, there exists ~ ε ( 0, x q$( a )) suh that b( x, a) + ~ ε < x = y. Also, Pbx ( (, a) + x, a) Px (, a) sine inverse demand is non-inreasing in output. Combining the two previous inequalities yields, P( b( x, a) + x, a)[ b( x, a) + ~ ε ] = R ( x, a ) + P ( b ( x, a ) + x, a ) ~ ε < x P( x, a) = y P( y, a). Given the ontinuity of R in apaity, for ε ~ ε Pbx ( (, a ) + x, a ) > 0 there exists δ > 0 suh that Rxa (, ) < Rx (, a) +ε, for x ( x δ, x ). Thus, for x suffiiently lose to x = y, we have Rxa (, ) bxapbxa (, ) ( (, ) + xa, ) < ypy (, a). By the Lemma, for x smaller than and suffiiently lose to x = y the expeted revenue funtion for firm one is, 25
rxy (,, x a ) = y bxapbxa (, ) ( (, ) + xa, ). Firm one s marginal revenue for apaity is, r( x, y, a) = [ bp + xbp + xbp( b + )], y where the arguments in funtions on the right-hand-side have been dropped in order to express the result more ompatly. Marginal revenue simplifies to, r( x, y, a) = [ bp + xbp + xb( P + bp)]. y The expression, P+ bp = Pbxa ( (, ) + xa, ) + bxap (, ) ( bxa (, ) + xa, ), is zero sine b( xa, ) is a revenue maximizing output response to x, given a. The left-hand derivative is, rx (, y, a) lim x x x r x y a bx (, a) = (,, ) [ Pbx ( (, a ) x, a ) xp ( bx (, a ) x, a )] = + + + y bx (, a) < [ P ( b ( x, a ) + x, a ) + b ( x, a ) P ( b ( x, a ) + x, a )] = y The final term in brakets is zero sine b( x, a) is the revenue maximizing output response to x given a. The inequality follows beause P is negative and b( x, a) < x. 0 The set Γ has positive measure and the density funtion for demand shoks is positive and ontinuous. Therefore, the non-differentiability property of the revenue funtion arries over to the expeted profit funtion. π ( x, y ) (,, ) (,, ) π = ( <, E rx y a E rx y a x y = ) x x x x + + Capaity x annot be a best response for firm one to y when this ondition holds. These arguments show that no symmetri pair of apaities satisfies the mutual best response property required for equilibrium, when q ˆ ( a) < q~. n 26
APPENDIX B: PROOF OF THEOREM 2 Part (i). Consider y [ 0, a]. If x [ 0, b( y, a)] then ( xy, ) Aa ( ), ( xy, ) Aa ( ), and 3 π( xy, ) = ( θ) xa ( x y ) + θxa ( x y ) x. π ( xy, ) is stritly onave for x 2 [ 0, b( y, a)] and reahes a loal maximum at, b ( y ) = ( θa+ ( θ ) a) y ) = ( E( a) y ). 2 Note that ( b ( y ), y ) A( a) sine b ( y ) = ( E( a) y ) ( a y ) = b( y, a) by the 2 2 hypothesis of the Theorem. Output b ( y ) is also a unique global maximum of π ( xy, ) sine π is ontinuous in x and, π( xy, ) x = ( θ)( a 2x y ) < 0, for b( y, a) < x< by (, a) and π( xy, ) x = <0, for x > b( y, a). Let ~ = ( E( a) ) / 3 be the proposed equilibrium apaity per firm. Note that q a E( a) implies that q~ 3 and that b ( q~ ) = q~. So, ( q~, ~ q ) is an equilibrium and is the only equilibrium in whih both outputs are less than or equal to be established by ruling out equilibria in whih one or both outputs exeed a. Uniqueness may 3 a. 3 Part (ii). If 2 a a < E ( a ) then there is no equilibrium involving symmetri apaities. The proof of this result is analogous to the proof of part (ii) of Theorem One. The restrition that 2 a a is suffiient to rule out symmetri equilibria in whih apaities are in the set Ba ( ) (where both firms earn zero revenue when demand is low). 27
We seek a haraterization of an equilibrium with x > y. We will suppose that equilibrium apaities are in Region I of Ca ( ) and then verify that this is the ase given the hypothesized parameter restritions. If demand is low then expeted revenue in the priing subgame for firm two in Region I is, r ( y, y x, a ) = x Rya (, ) = y 4x ( a y) If demand is low then expeted revenue in the priing subgame for firm one is, 4 rxya (,, ) = Rya (, ) = ( a y) Expeted profit for firm one in stage one is, π( xy, ) = θrya (, ) + ( θ) xa ( x y) x. Optimal apaity hoie for firm one satisfies, or, 2. 2. π( xy, ) = ( θ)( a 2x y) = 0 x = ( a y). 2 θ Expeted profit for firm two in stage one is, π( yx, ) = y θ Rya (, ) ( θ) ya ( x y) y x +. Optimal apaity hoie for firm two satisfies, π θ ( yx, ) = ( )( 3 ) ( θ)( 2 ) 0 4x a y a y + a x y =. 28
The following notation will prove to be onvenient. Let ψ ondition a < E( a) is equivalent to ψ >. Also, let X apaity hoie ondition for firm one may now be expressed as, a a x = and Y a. The ( θ)a y =. The optimal a X = Y 2 ( ψ ), (2) and the optimal apaity hoie for firm two may be expressed as, θ 4 X ( Y)( 3Y) + ( θ)( ψ X 2Y) = 0. (3) Substituting for X from (2) into (3) yields, θ( Y)( 3Y) + ( θ)( ψ Y)( ψ 3Y ) = 0. (4) The LHS of (4) is a weighted sum of two quadratis in Y. One solution lies between /3 and ψ /3, the smaller roots of the two quadratis; this solution is the eonomially relevant one. Let Y* be the solution in the interval, (/3, ψ /3). Then, 3 2 X* = ( ψ Y*) > ψ / > Y*, as required for the payoff funtions used in these derivations. Let x* = ax * and y* = ay*. The next step is to verify that (x*, y*) is in Region I for firm two when demand is low. The boundary between Regions I and II in Ca ( ) satisfies, Rya (, ) = xpxa (, ) or, y = β( x) a 2 x( a x). A suffiient ondition for (x*, y*) to be in Region I is that the best response for firm one to y = a/3 be smaller than the value of x satisfying, β( x) = a / 3. In terms of Figure Two, this ondition insures that firm one s reation funtion, R ~ a a, is to the left of the boundary between Regions I and II, for y ψ,. 3 3 29
Firm one s best response to y = a /3 is x = ( aψ a/ 3) = ( 3ψ ) a/ 6. The ondition 2 a < 2 a implies that ψ < 2, whih implies that β ( a/ 3) = ( 3+ 5) a/ 6> ( 3ψ ) a / 6. So, the hypothesis of part (ii) implies that (x*, y*) is in Region I. The preeding arguments show that y* is a best response to x* when y is restrited to values in Region I. This leaves open the possibility that the best response to x* is in Region II. In terms of Figure Two, ~ R 2 may have a downward jump disontinuity before it rosses ~ R in Region I. In Region II, expeted revenue for firm two in the low demand subgame is independent of x; r y, x*, a) = y[ a y(2a y ) ]. ( 2 This funtion is maximized at, y r( yii, x*, a) = a II = 2 2 3. 8 2 3 a, and 2 The apaity pair ( x*, a / 3 ) is in Region I if demand is low. In this ase, firm two s expeted revenue in the subgame is, 3 3 a 2 a r ( a / 3, x*, a) = ( a 3 a) =, 4x * 27x * whih exeeds r( yii, x*, a), sine x* < a (the result that (x*, y*) is in Region I implies that x* < a). If demand is high then firm two has positive marginal revenue for apaity y ( 0, y*), given x* (sine ( x*, y) A( a) ). So, π( a/ 3, x*) = θr( a/ 3, x*, a) + ( θ)( a x* a/ 3) a/ 3 a/ 3 max > θr( yii, x*, a) + ( θ)( a x* β( x*)) β( x*) β( x*) π( yx, *) y [ 0, β( x*)] 30
where the strit inequality follows beause β( x*) < a / 3 and the small firm s marginal revenue for apaity is positive if demand is high. The apaity pair ( x*, a / 3 ) is in Region I for firm two and y = y* is the best response to x* for firm two within Region I. max Therefore, π( y*, x*) π( a/ 3, x*) > π( yx, *). y [ 0, β( x*)] Given that firm one hooses apaity x*, all apaity hoies for firm two in Region II of Ca ( ) yield lower expeted profit for firm two than y*. n 3
Figure One Capaity Regions for the Priing Subgame y B (a ) q (a ) y =b (0,a ) A (a ) C (a ) Region I (x > y ) Region II (x > y ) x =b (0,a ) q (a ) x A (a )= { (x, y ): x 0, y 0, x b (y, a ), y b (x, a )} B (a ) = { (x, y ): x q (a ), y q (a )} C (a )= { (x, y ): x 0, y 0, (x, y ) A (a ), (x, y ) B (a )} 32
Figure Two Capaity Reation Curves (x > y ) 45º y B (a ) 3 a ψ ~ 2 R Region I β (x ) y * ~ 2 R Region II 3 a ~ R y II A (a ) ~ 2 R 3 a 2 a x * 2 a ψ a x 33