Asymptotic Efficiency in Stackelberg Markets with Incomplete Information. ** Wissenschaftszentrum Berlin für Sozialforschung

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1 discussio papers FS IV 99-7 Asymptotic Efficiecy i Stackelberg Markets with Icomplete Iformatio Jiabo Zhag Zhetag Zhag Uiversity of Kasas Wisseschaftszetrum Berli für Sozialforschug May 999 ISS r Forschugsschwerpukt Marktprozeß ud Uterehmesetwicklug Research Area Market Processes ad Corporate Developmet

2 Zitierweise/Citatio: Jiabo Zhag, Zhetag Zhag, Asymptotic Efficiecy i Stackelberg Markets with Icomplete Iformatio, Discussio Paper FS IV 99-7, Wisseschaftszetrum Berli, 999. Wisseschaftszetrum Berli für Sozialforschug ggmbh, Reichpietschufer 50, 0785 Berli, Tel. (030)

3 ABSTRACT Asymptotic Efficiecy i Stackelberg Markets with Icomplete Iformatio by Jiabo Zhag ad Zhetag Zhag This paper examies the asymptotic (i)efficiecy of Stackelberg markets with icomplete iformatio. Firms who are early i the queue make their quatity choices based o limited iformatio ad their output choices are likely to deviate from those optimal uder complete iformatio. Due to the presece of both payoff exterality ad iformatio exterality, the output deviatios of early firms have a lastig effect o all subsequet output decisios. Cosequetly, the total market output diverges from the competitive equilibrium output eve as the umber of firms goes to ifiity. That is, Stackelberg markets with icomplete iformatio are asymptotically iefficiet with probability oe. ZUSAMMEFASSUG Asymptotische Effiziez i Stackelberg-Märkte mit uvollstädiger Iformatio I diesem Beitrag wird die asymptotische (I-)Effiziez vo Stackelberg-Märkte mit uvollstädiger Iformatio utersucht. Uterehme, die frühzeitig auf de Markt komme, bestimme ihre Ausbrigugsmege uter uvollstädiger Iformatio. Aus diesem Grude sid ihre Megeetscheiduge im allgemeie verschiede vo de optimale Ausbrigugsmege uter vollstädiger Iformatio. Auszahlugswirksame Exteralitäte ud Iformatiosexteralitäte bewirke, daß die Megeetscheiduge der frühzeitig auf de Markt treffede Uterehme zu pfadabhägige Megeetscheiduge achfolgeder Uterehme führe. Im Ergebis ist da die gesamte Ausbrigugsmege aller Uterehme verschiede vo dem Kokurrezgleichgewicht - selbst da, we die Azahl der Uterehme gege uedlich strebt. Das heißt, Stackelberg-Märkte mit uvollstädiger Iformatio sid asymptotisch ieffiziet mit der Wahrscheilichkeit eis. We would like to thak Larry Samuelse for his isightful discussios o the topic of this paper.

4 . Itroductio It is well kow that markets may fail because of limited competitio or the existece of icomplete iformatio. The iefficiecy due to these two sources, however, ca be respectively elimiated i a large market as the umber of firms teds to ifiity. Wilso (977) demostrates that i a sealed bid teder auctio where each bidder has private iformatio, the wiig bid will coverge i probability to the true value of the object as the umber of bidders grows large. This result is exteded ad geeralized by Milgrom (979), who obtais ecessary ad sufficiet coditios for covergece. 2 Regardig the iefficiecy from market power, ovshek (980) ad Robso (990) ivestigate Courot ad Stackelberg markets respectively ad show that uder geeral demad ad U-shaped average cost curves, the iefficiecy arisig from market power will disappear asymptotically as the miimum efficiet scale teds to zero. That is, both Courot ad Stackeberg equilibria uder complete iformatio coverge to the competitive equilibrium as the umber of firms teds to ifiity. A atural questio to raise is whether iefficiecy i markets with both market power ad icomplete iformatio is also elimiated as the umber of firms teds to ifiity? The asymptotic property of the Courot model with icomplete iformatio has bee ivestigated i the literature. Palfey (985) shows that, uder certai assumptios o the iformatio structure, a Courot market with ukow demad becomes efficiet as the umber of firms grows large. Li (985) obtais the same result by edogeizig firms decisio to share iformatio. 3 Vives (988) demostrates that Palfey ad Li s result depeds o the productio techology exhibitig costat returs to scale. However, so far i the literature, o studies have looked at whether the iefficiecy i Stackelberg markets with icomplete iformatio could be elimiated as the umber of firms teds to ifiity? That is, whether large Stackelberg markets aggregate iformatio efficietly? This paper attempts to fill this gap. 2 3 See, for example, Akerlof (980) for classical discussio of the problem. Swikels (996) shows that discrimiatory private value auctios for multiple objects are asymptotically efficiet as the umber of players grows large. The icetives for Courot oligopolists to share iformatio have bee studies extesively i the literature, for example, by Li (985), Shapiro (986) ad Vives (984). They coclude that whe the ucertaity is about a firm-specific parameter, perfect revelatio is the uique equilibrium. O the other had, whe the ucertaity is about a commo parameter, o iformatio sharig is the uique equilibrium.

5 2 The iformatio structure arisig i Stackelberg competitio is similar to the oe i iformatio cascade literature itroduced by Baerjee(992), ad Bikchadai, Hirshleifer ad Welch (992). Uder such a iformatio structure, agets take actios sequetially after observig the actio history ad a private sigal. A iformatio exterality occurs sice each aget s private iformatio is revealed, perfectly or imperfectly, through its actio to the followig agets ad may thereby alter their believes about the uderlyig ucertaity. This iformatio exterality may give rise to iformatio cascades 4. Although the literature o iformatio cascade illumiates how iefficiecies ca be geerated i a sequetial actio model through iformatio exterality, it is ot well suited to aalyze the iformatio aggregatio problem i large Stackelberg markets. The reaso for this is that iformatio cascade models assume that there is o strategic value for a player early i the queue to maipulate its actio i order to ifluece the actios of the followig players. That is, they assume that there are o strategic iteractios betwee players ad thereby o payoff exterality. By assumig away the strategic iteractios, the iformatio cascade literature captures the iefficiecies resultig oly from iformatio exteralities but ot from payoff exteralities. 5 I Stackelberg markets with icomplete iformatio, the effects of every firm s actio o the payoffs of its successive firms are two-folded: First, the strategic iteractios betwee leaders ad followers create payoff exterality. Secod, every firm s actio coveys its private iformatio ad thereby affects its followig firms belief about the ukow state, which creates iformatio exterality. It is this payoff exterality etwied with the iformatio exterality, as show below, that drives the efficiecy loss eve as the umber of firms goes to ifiity. The mai result of the paper ca be illustrated by cosiderig the followig sceario: a umber of firms egage i Stackelberg competitio makig their productio choices sequetially. The ature of demad is ukow to the firms. I additio to receivig a private sigal, each firm observes all the actios of the precedig firms ad tries to ifer their private iformatio through these actios. Based o the private sigal ad the iferred public iformatio, every firm makes its quatity choice. I geeral, this game is a exteded sigalig game with may players where the quatity choice of each 4 5 A iformatio cascade is defied, as by Lee (993), as the covergece of the sequece of actios. A fully revealig iformatio cascade is said to occur if the limit is optimal uder the true state. Otherwise, a o-fully revealig iformatio cascade occurs. Similarly, Vives (993) studies the speed of covergece to the ratioal expectatios equilibrium i a simple dyamic model of ratioal learig betwee agets. However, he assumes that the actio of oe player does ot affect the profits of other periods. That is, he assumes that there are o strategic iteractios betwee players across periods.

6 3 leader is a sigal about its private iformatio to all its followers. 6 Recall that dyamic games with icomplete iformatio ted to have multiple equilibria ad there is o exceptio i this game. 7 The refiemet of Perfect Bayesia equilibrium adopted i this paper is called the exteded ituitive criterio, which is a exteded versio of the ituitive criterio of Cho ad Kreps (987). More precisely, the ituitive criterio is applied to every cotiuatio game of the exteded sigalig game. Sice every cotiuatio game satisfies the sigle crossig property, the exteded ituitive criterio leaves us with a uique separatig equilibrium. This implies that every firm s quatity choice fully reveals its private iformatio to all its followers. Therefore, accordig to the strog law of large umbers, the true state of demad is evetually revealed to the firms who are sufficietly late i the queue. The basic poit is as follows: i the iformatio cascade models with o payoff exterality, the revelatio of the truth ecessarily forces the later players to take actios which are optimal uder complete iformatio. This caot happe i the curret model precisely due to the payoff exterality that is preset. Ituitively, the first firm ca get either a high or a low sigal with a positive probability regardless of what the true state of demad is. As a cosequece, its quatity choice may be differet from the quatity choice uder complete iformatio. The quatity choice of the first firm will the affect the quatity choices of all the firms later i the queue, due to the payoff exterality. As a result, output deviatios of early firms have a lastig effect o all subsequet output decisios ad the total market output does ot coverges to the competitive equilibrium. This is true despite the fact that firms sufficietly back i the queue have almost complete iformatio about demad. 8 The rest of the paper is orgaized as follows. I Sectio 2 we sets up the model. I Sectio 3 we ivestigates the asymptotic (i)efficiecy of the Stackelberg market. I Sectio 4 commets ad cocludes are give I this exteded sigallig game, every player has private iformatio while i the stadard sigallig games, oly the first player has private iformatio. To obtai a uique equilibrium a refiemet is ecessary. I our set-up, we do ot assume that there is a desiged mechaism which solicit iformatio before implemetig the trasactios as i Gul ad Postlewaite (992). either do we assume there is a market auctioeer who pools iformatio ad sets market clearig price as i Rutichii, Satterthwaite ad Williams (994). Istead, the private iformatio is revealed through the quatity choices of the firms.

7 4 2. Model Cosider a Stackelberg market with firms makig productio choices sequetially. Firms are assumed to have costat margial cost; i.e., cq ( )= cq, = 2,,.... There are o fixed costs. Followig ovshek ad Soeschei (982) ad Vives (988), firms private iformatio is about the demad itercept: p = a+ S bq; where Q is total market output ad it is assumed that a > c ad b > 0. State S is a radom variable which is distributed over a fiite state space Ω R. The ukow state could be iterpreted as a parameter affectig cosumers taste, so that a higher state would result i a higher market demad ad vice versa. Firms do ot kow the realizatio s of S, but have commo iitial prior distributio µ 0 ( s) = Pr ob( S = s), which is assumed to be o-degeerate. The iformatio structure of the game is as follows: at the begiig of the game, a state s is draw radomly from the fiite state space Ω ad remais fixed throughout. Each firm makes output choice based o its private sigal ad the public iformatio i order to maximize its profits. Followig Lee (993), the private sigal of firm, x, is draw radomly from a Beroulli distributio x { 0, }. The draw of a sigal is coditioally i.i.d. give the state. 9. The fact that A is compact ad covex guaratees the existece of a equilibrium. Give the state of ature s, the iformatio set of firm is give by Ω = { h, x }, where h = ( q, q2,..., q ) (h = φ ) is the history of actios ad x is the private sigal of firm. The behavior strategy of firm, σ ( q ( h, x )) is a mappig from the iformatio set to the actio set. Each firm, before makig its output choice, has a Firm, chooses q from its actio set A, where A = [ 0, Q] idetical iitial prior belief µ 0 ( s ) over the states. After observig the output choices of its precedig firms as well as receivig a private sigal, the firm updates its prior belief accordig to Bayes rule. The output choice is optimal with respect to the posterior belief. Let µ = prob( s( h, x )) deote the posterior belief of firm. Firm s expected profit fuctio is give by Eµ π = E{(a + S c- bq)q Ω }, where Q is the total market output. We solve for Perfect Bayesia equilibria. 9 The upper boud of firm s output is give by Q = a c b + [ E( SΩ )] =. This is because ay choice of quatity greater tha Q results i a strict expected loss, regardless of other firms choices.

8 5 Give the umber of firms ad the realized sigal vector x r = ( x, x2,..., x ), a perfect Bayesia equilibrium (hereafter, PBE) is defied as follows: Defiitio: A PBE is a set of strategies (σ,..., σ ) ad posterior beliefs ( µ,..., µ ) such that for ay µ 0, h ad =,2,,, ( P ) σ ( q ( h, x )) argmax E π ( h, q, q + ( q ),..., q ( q )); q µ ( P 2 ) σ+ i( q) arg max Eµ π+ i( h+ i, q+ i, q+ i+ ( q+ i),..., q( q+ i)), for ay i=, 2,,-; q+ i + i ( B ) µ ( s( h, x )) is derived from the prior µ 0 ( s ), σ ( q ( h, x )) ad ( q ) accordig to Bayes rule, whe applicable. 0 σ + i That is, a PBE of a exteded sigalig game with players requires that the strategies yield a PBE for every cotiuatio game. To simplify the aalysis, we make the followig assumptios. Assumptio : The sigals are ubiased estimators of the state. That is, E( x s)= s. From Assumptio, we have that s = prob( x = s) ad s= prob( x = 0s). That is, it is more likely to get a good sigal ( x = ) whe the state of ature is good. Assumptio 2: The coditioal probability of sigal, x =, is strictly betwee 0 ad. That is, 0 < prob( x = s ) <. Assumptio 2 implies that for ay give state, there is a positive probability that each firm gets either sigal x = or sigal x = 0. Assumptio 2 rules out degeerate cases. To exted the spirit of subgame perfectio to this game, we would like to require that the strategies yield PBE for every cotiuatio game startig from every possible 0 ote that if q is ot part of firm (-) s optimal strategy for some private sigal, observig q is a probability 0 evet, ad Bayes rule does ot pi dow posterior beliefs. Ay posterior beliefs are the admissible.

9 6 history h. We make the followig assumptios o players beliefs at the start of each cotiuatio game. Assumptio 3: For ay history h, player + to player have the same beliefs about the state of ature give h. Recall that dyamic games with icomplete iformatio ted to have multiple equilibria because Bayes rule has o bite i out-of-equilibrium evets ad ay posterior beliefs are admissible at out-of-equilibrium iformatio sets. Cosequetly, the game above will potetially have multiple equilibria, uless we make use of a refiemet. Usig the ituitive criterio itroduced by Cho ad Kreps (987), we propose a exteded ituitive criterio which meas that we apply the ituitive criterio of Cho ad Kreps (987) to every cotiuatio game. I other word, we exted the ituitive criterio to our exteded sigalig game with may players. 3. Stackelberg Competitio with icomplete Iformatio We ca ow ivestigate whether Stackelberg markets with icomplete iformatio are asymptotically efficiet i the presece of both iformatio ad payoff exteralities. There are firms makig their output choice sequetially i a hierarchical Stackelberg framework. Without loss of geerality, the order of firms actios is assumed to be exogeously give by,2,...,. Let us examie poolig equilibria first. At a poolig equilibrium, firm produces the same output o matter what sigal he gets (0 or ). The followers of firm, firm + to firm, therefore update their posterior believes oly based o their private sigals. This implies that for ay, the iformatio set of firm is simply Ω = { x }. Hece, firm s expected payoff fuctio ca be simplified to µ π = E{(a + S c- bq)q x }. E It is a iterestig problem to edogeize the order of firms actios. Mailath (993) studies edogeous sequecig of firm decisios i a duopoly setup with asymmetric iformatio betwee firms. Chamley ad Gale (994) ad Zhag (997) edogeize the sequetial choice of agets i a iformatio cascade cotext.

10 7 The existece ad refiemet of the poolig equilibria are give i the followig propositio: Propositio : Give the umber of firms ad the realized sigal vector r x = ( x, x2,..., x ), there exists a cotiuum of poolig PBE. Furthermore, the exteded ituitive criterio elimiates all the poolig equilibria. Proof: See Appedix. ext, we study separatig equilibria. For a separatig equilibrium, every firm s quatity choice perfectly reveals its private iformatio. Therefore, firm s iformatio set Ω ca be reduced to Ω = { x, x2,..., x, x }. Cosequetly, the expected payoff fuctio of firm is give by E µ π = E{(a + S c- bq)q ( x,..., x )}. The existece ad the refiemet of the separatig equilibria are give by the followig propositio: Propositio 2: Give the umber of firms ad the realized sigal vector r x = ( x, x2,..., x ), there exists a cotiuum of separatig PBE. Furthermore, the extesive ituitive criterio elimiates all but oe separatig equilibrium which is give as follows: r a c E( SΩ ) Q(, x) = ( ) +, 2 b b = 2 () r r px (, ) = a bqx (, ) + S. (2) Proof: See Appedix. Propositio ad 2 imply that the exteded ituitive criterio equilibrium refiemet leaves us with a uique PBE. Before proceedig to ivestigate the asymptotic (i)efficiecy of Stackelberg markets i the presece of both iformatio ad payoff exteralities, we preset the followig useful lemmas.

11 8 Lemma : Alog the uique PBE path, the best resposes of firm s followers, firm + to firm, satisfy respectively { + = 2, Ω = x x2 x x },,...,,. + 2 =,..., 4 = 2 ; where = 2,,..., ad Proof: See Appedix. From Lemma, it is trivial to show that dq = ( ) 2. This is a importat result as it implies that the impact of a firm s output choice o total output is smaller the later back i the game the firm is. Coversely, the output choices of early firms have a lastig effect o all subsequet output decisios. I order to have a base for compariso, we use the perfect competitive equilibrium outcome with complete iformatio as a bechmark. 2 The followig lemma is trivial to obtai. Lemma 2: Let ( Q 0 s p 0 s ) ( ), ( ) be the competitive equilibrium outcome with complete a c+ s iformatio. The Q 0 ( s)= ad p 0 ( s )= c. b We are ow ready to state our mai result regardig the asymptotic (i)efficiecy of Stackelberg markets with icomplete iformatio. Theorem: For ay realizatio s of S, Let ( Q s p s) ( ), ( ) be the vector of radom variables which represets the uique (stochastic) PBE give by () ad (2). The Q s p s Q ( s), p ( s) as goes to ifiity. For almost ( ( ), ( )) coverges to some ( ) all realizatios s, ( Q ( s), p ( s) ) ( Q 0 s p 0 s ) ( ), ( ). 3 2 Our mai results would ot chage if we use the competitive equilibrium outcome uder icomplete iformatio as the bechmark sice the stochastic competitive equilibrium outcome coverges i probability to a degeerate distributio, as the umber of firms goes to ifiity (see Vives (988)).

12 9 Proof: From Propositio 2, for ay ad r x, there exists a uique PBE give by equatio () ad (2). For ay realizatio s of S, it is trivial that a c Q( s) = ( ) + 2 b b = [ E( SΩ ) s] 2, ad p( s ) = a bq( s ) + s ; where x, x,..., x, x }. { Ω = 2 As, accordig to the strog law of large umbers, [ ( Ω ) s] a c EES Q( s) + Q b b 2 = [ ( Ω ) s] EES ps ( ) c+ s p 2 = ( s). ( s), ad Let s be the uique solutio to the followig equatio: [ Ω s] EES s = ( ) 2 =. It is the trivial that ( Q s p 0 0 ( ), ( s) ) ( Q ( s), p ( s) ) uless s = s. Q.E.D. The Theorem shows that the Stackelberg output is isufficietly low ( Q ( s) < Q 0 ( s) ) whe the true state of demad is good ( s> s ), while it is isufficietly high ( Q ( s) > Q 0 ( s) ) whe the true state of demad is bad ( s< s ). I sum, the Stackelberg output is asymptotically iefficiet with probability oe. 4 The ituitio behid the above result is as follows. Firms make their productio decisios sequetially based o their private iformatio as well as the iferred public iformatio of the precedig firms. Sice every cotiuatio game of the Stackelberg 3 4 We are grateful to a aoymous referee for the way this theorem is ow stated. The fact that the stochastic Stackelberg equilibrium outcome does ot coverge to the competitive equilibrium outcome is equivalet to statig that there is o covergece to a degeerate distributio sice the degeerate Stackelberg equilibrium outcome coverges to the competitive equilibrium outcome (See Robso (990)).

13 0 game satisfies the sigle crossig property, the exteded ituitive criterio selects a separatig equilibrium, which implies that the leaders quatity choices fully reveal their private iformatio. Cosequetly, the firms who are sufficietly back i the queue have almost complete iformatio, accordig to the strog law of large umbers. However, the firms who are early i the queue have very limited iformatio about the ukow demad ad their quatity choices ted to be differet from the choices uder complete iformatio. I additio, these early firms productio choices affect the output choices of the later firms due to the payoff exterality existig i the game. Therefore, the deviatios of the early firms output choice have a lastig effect o all subsequet output decisios ad causes the total market output to be divergig from the competitive equilibrium output eve as the umber of firms goes to ifiity. The above theorem ca be further illustrated i the followig example. Example: Suppose the iitial prior distributio µ o ( s ) is the uiform distributio over (0,). The expected value of posterior distributio i the uique separatig PBE ca be simplified as: E( SΩ ) = + x + 2, where Ω = x, x2,..., x, x }. 5 = Therefore, [ ( Ω ) ] EES s s + = by Assumptio. 2 2 ( + 2) = = a c s Hece, Q( s) + + ; ps ( ) c+ s b b 2 ( + 2) = { = s +. 2 ( + 2) 2 r r From Taylor expasio l( r) = ( r ) 2 for < r <, we have that = 2 s + = ( 6 8 l 2 ) s+ ( 4 l ) 0. 46s a c Therefore, Q( s) + ( 046. s ) Q ( s), ad b b ps ( ) c+ ( 054. s 027. ) p( s). 5 See Welch (992).

14 It is trivial that Q ( s) < Q 0 ( s) if s > 2, Q ( s ) > Q 0 ( s ) if s < 2 Q ( s) = Q 0 ( s)., ad otherwise 4. Cocludig Remarks I this paper, we have demostrated that large Stackelberg markets do ot aggregate iformatio efficietly, eve if the techology exhibits costat retur to scale. That is, i the presece of icomplete iformatio, Stackelberg markets are asymptotically iefficiet with probability oe. This is because the early firms make their productio choices based o the very limited iformatio ad cosequetly ted to over- or uderproduce. I additio, the payoff exterality esures that the quatity choices of the early firms have a lastig effect o the output decisios of all subsequet firms. As a result, the over- or uder-productio of the early firms gets carried over ad drives the efficiecy loss. The exteded ituitive criterio selects a uique separatig equilibrium, which esures that each firm s private iformatio is fully revealed to the successive firms ad accordigly the uderlig ucertaity is gradually resolved alog the queue as the umber of firms becomes large. Therefore, firms who are sufficietly far back i the queue have almost complete iformatio about the demad. I this sese, there is o efficiecy loss from the iformatio exterality per se, ad there is o possibility for a o-fully revealig iformatio cascade to occur. It would be iterestig to ivestigate a class of games where agets actios do ot fully reveal their private iformatio. 6 I this case, a o-fully revealig iformatio cascade may arise as discussed i Baerjee(992), Bikchadai, Hirshleifer ad Welch (992), ad Zhag ad Zhag (995). The efficiecy loss i these games is therefore expected to be larger due to the additioal iefficiecy from iformatio exteralities. 6 Oe example of this class of games is a Stackelberg game where each firm ca oly observe some but ot all its precedig firms actios.

15 2 Appedix Proof of Propositio : At a poolig equilibrium, each firm updates its posterior belief oly based o its ow private sigal, i.e. { x } Ω = for =,..,. For ay give ad history h, a poolig equilibrium is said to survive the exteded ituitive criterio if it survives ituitive criterio of Cho ad Kreps (987) i every cotiuatio game. We solve this game backward.. Cotiuatio game For ay give h, this cotiuatio game cosists of oly the th firm whose equilibrium output is give by, [ ] q arg max E q ( a c + S bq ) x (A) q 2. Cotiuatio game - For ay give h 2, this cotiuatio game cosists of firm (-) ad firm. Let q deote a poolig equilibrium for firm -. Firm (-) s expected payoff is give by (A). [ ( + ) ] Eq a c S bq bq x, where Q = q i= i ad q ( q ) is Thus the best way to sustai q as a poolig equilibrium is to assume that firm believes that firm - gets sigal x = whe it observes q q. So q will ideed be a poolig equilibrium if ad oly if the followig coditios are satisfied. (M): E[ π( q, q ) x = ] max E[ ( q, q ) x = ] q π, (M2): E[ π( q, q ) x = 0 ] max E[ ( q, q ) x = 0] q π, where q ( q ) is give by (A) ad q ( q ) is give as follows: [ ] q arg max E q ( a c+ S bq)( x =, x ). q,, where q ad q are the lower ad upper boud of q which satisfies (M) ad (M2). Therefore, there exists a cotiuum poolig equilibria q [ q q ]

16 3 I order to elimiate this cotiuum poolig equilibria, we use ituitive criterio of Cho ad Kreps (987) to this cotiuatio game. Defie q < q by the smallest root of [ π(, ) = ] = E[ π( q, q ) x =, ] where arg max [ ( + )( =, )] E q r x r E r a c S bq 2 bq br x 0 x. r ow, playig q ε (for ε > 0 ) is equilibrium domiated for firm - with sigal x = but ot for firm - with sigal x = 0 sice [ π(, ) = 0 ] E[ π( q, q ) x = 0] E q q x = E( S( x =, x )) E( S( x = 0, x )) > 0. 2b Therefore, firm s posterior belief should put all the weight o x = 0 followig output q ε. However, firm - who gets x = 0 prefers to playig q ε to q. Thus, q is ot poolig output aymore. 3. Cotiuatio game -2 For ay give h 3, this game cosists of firm -2, firm - ad firm. From assumptio 3, firm - ad firm have the same believes after observig q 2. Applyig the similar argumet ad techique used i last cotiuatio game to this cotiuatio game, we ca elimiate poolig equilibria i this cotiuatio game. Cotiuig this process for every cotiuatio game, we will the elimiate all the poolig equilibria. Q.E.D. Proof of Propositio 2: For a separatig equilibrium, every firm s quatity choice perfectly reveals its private iformatio. Hece, the iformatio set of firm ca be simplified to Ω = { x, x2,..., x, x }. I additio, a separatig equilibrium is said to survive the exteded ituitive criterio if it survives ituitive criterio of Cho ad Kreps (987) i every cotiuatio game. We solve this game backward.. Cotiuatio game For ay give h, this game cosists of oly firm whose equilibrium output is give by

17 4 [ ] q arg max E q ( a c+ S bq) Ω, where Ω = x, x2,.., x }. (A2) q { 2. Cotiuatio game - For ay give h 2, this game cosists of firm - ad firm. At a separatig equilibrium, the private iformatio of firm - is fully revealed to firm through its quatity choice. Therefore, for firm - with sigal x =, it chooses the followig optimal quatity: H [ Ω ] H q arg max E q ( a c + S bq bq ( q ))(, x = ), (A3) 2 q where Q = q i= i ad q H ( q ) is give by (A2) with x =. That is, [ ] H q ( q ) arg max E q ( a c+ S bq)( Ω, x =, x ). (A4) 2 q O the other had, for firm - with sigal x = 0, a separatig equilibrium q is such that the followig coditios are satisfied joitly: H (S): max E[ ( q, q )(, x = π Ω 2 ) ] E[ π( q, r )( 2, x = ) ] q Ω, H (S2): E[ π( q, r )( Ω 2, x = 0 )] max E[ ( q, q )(, x = 2 0 )] q π Ω, H where q is give by (A4) & r arg max E[ q( a c+ S bq)(, x =, x )] q Ω 2 0. (S) says that whe firm - gets sigal x =, it does ot wat to produce the output which correspods to sigal x = 0. (S2) says that whe firm - gets sigal x = 0, it does ot wat to produce output which coveys sigal x =. L L L Therefore, there exists a cotiuum separatig equilibria q [ q q ] q L L ad q,, where are the lower ad upper boud of q L which satisfies (S) ad (S2). Therefore, there exists a cotiuum separatig equilibria. The firm with sigal H x = prefers playig q while the firm with sigal x = 0 prefers playig L L [ ] L L q q, q. From (S), it is clear that playig q is equilibrium domiated for the firm with sigal x =, but ot for the firm with sigal x = 0. So firm s L L posterior belief should put all the weight o sigal x = 0 followig q. Let q deote

18 5 [ π Ω 0 ] L q arg max E ( q, r ) (, x = ) 2 q L The q is the uique separatig equilibrium survivig the elimiatio of weakly domiated strategies for firm with sigal x = 0. Hece, for ay h 2, the equilibrium refiemet of this cotiuatio game leaves us with a set of uique PBE strategy (, q q ), where q is give by (A2) ad q is give by the followig (A5): [ ] q arg max E q ( a c + S bq bq )Ω. (A5) q 3. Cotiuatio game -2 For ay give h 3, this game cosists of firm -2, firm - ad firm. From assumptio 3, firm - ad firm have the same beliefs after observig q 2. By the similar reasoig as i last cotiuatio game, the elimiatio of weakly domiated strategies of this cotiuatio game leave us with a set of uique separatig PBE profile ( q 2, q, q ), where q is give by (A2), q is give by (A5) ad q 2 is give by the followig (A6) [ ] q arg max E q ( a c + S bq bq bq )Ω. (A6) q 2 Cotiuig this process for every cotiuatio game, the exteded ituitive criterio leaves us with a uique separatig PBE satisfyig [ ] q arg max E q ( a c+ S bq) Ω, for =, 2,,. q Therefore, alog the uique separatig PBE path, we have a - c - bq - bq - bq i=+ i + E( S Ω ) =0; where Q = j= q j ad =, 2,,. 7 Applyig Lemma (which is proved below) ad rearragig the above equatio, we have 7 We have implicitly assumed that firms have ratioal expectatios, i.e., E( q ( Ω ) Ω ) = q ; i>. i i i

19 6 q = 2 a-c+e( S Ω ) Q b ; =,2,,.8 - Summig over =,2,...,, we have that for every ad the realized sigal vector r x = ( x,..., x ), the uique stochastic PBE which survives the exteded ituitive criterio is give as follows: r a c Q(, x) = ( ) + 2 b b = E( SΩ ). 2 Thus, p(, X ) = a bq(, X ) + S. Q.E.D. Proof of Lemma : We prove this lemma by mathematical iductio. Alog the uique separatig PBE path, we have that a-c-2bq -bq - + E( sω ) =0; where Q = q Thus, the best respose of firm to firm (-) s output is satisfies that j= j. = 2. ow suppose that the lemma holds for firm +. That is, alog the uique separatig PBE path, the best resposes of firm (+) s followers, firm +2 to firm, satisfy that = 2, = 4, + = + (L) 2 ( ) From the above, it is trivial that q q j + = 2 for ay j +2. (L2) ow we wat to show the lemma also holds true for firm. 8 We therefore assume that a c> 2 mi( E( SΩ ),..., E( SΩ )) that q > 0. = E( SΩ ) 2 i order to guaratee

20 7 From the proof of Propositio 2, firm (+) s best respose alog the uique separatig PBE path ca be derived as follows:, j a c 2 bq+ bq b q j bq+ + ES = 0 ( Ω +) ; where Q = q j= + 2 j= j= j. Rearrage it ad applyig (L), we have + ( ) 2 bq+ + b q j = a c bq + E( S Ω+ ) j= + 2 Takig derivative with respect to q, we have + + ( ) 2 j b + b = b ; j= + 2 (L3) where +2 + = 2 2 from (L2). Similarly, + 3 = = 4 4,, ad + = 2 2. Therefore, j + = ( ) ( + ). 2 j= + 2 Substitutig the above back ito (L3) ad rearrage it, we have that + = 2.

21 8 Hece, = = , = = ,, ad 8 + = = Q.E.D.

22 9 Refereces Akerlof, G.A. (970) "A Market for Lemos: Quality Ucertaity ad the Market Mechaism. " Q.J.E., 84, Baerjee, A. (992) "A Simple Model of Herd Behavior." Q.J.E., 07, Bikchadai, S. Hirshleifer, D. ad Welch, I. (992), "A theory of Fads, Fashio, Custom, ad Cultural Chages as Iformatio Cascades." J.P.E. 00, Chamley, C. ad D. Gale, (994), "Iformatio Revelatio ad Strategic Delay i a Model of Ivestmet. " Ecoometrica, Vol. 62, Cho, I. K. ad D. Kreps (987), "Sigalig Games ad Stable Equilibria." Q.J.E. 502, Gul, F. ad A. Postlewaite (992), "Asymptotic Efficiecy i Large Exchage Ecoomies with Asymmetric Iformatio." Ecoometrica, 60, Lee, I Ho (993) "O the covergece of Iformatioal Cascades." Joural of Ecoomic Theory, 6, Li, L (985), "Courot Oligopoly with Iformatio Sharig." Rad Joural of Ecoomics, 6, Mailath, G. (993), "Edogeous Sequecig of Firm Decisios." Joural of Ecoomic Theory, 59, Milgrom, P. R. (979), "A Covergece Theorem for Competitive Biddig with Differetial Iformatio." Ecoometrica 47, ovshek, W. (980), "Courot Equilibrium with Free Etry." Review of Ecoomic Studies, 47, ovshek, W. ad H. Soeschei (982), "Fulfilled Expectatios Courot Duopoly with Iformatio Acquisitio ad Release." Bell Joural of Ecoomics, 3, Palfrey, T. R. (985), "Ucertaity Resolutio, Private Iformatio Aggregatio ad the Courot Compeittive Limit." Review of Ecoomic Studies, 52, Robso, A. J. (990), "Stackelberg ad Marshall," The America Ecoomic Review 80, Shapiro, C. (986), "Exchage of Cost Iformatio i Oligopoly." Review of Ecoomic Studies, 53,

23 20 Swikels, J. (996), "Asymptotic Efficiecy for Discrimiatory private value Auctios," orthwester Uiversity discussio paper o. 73. Vives, X. (988), "Aggregatio of Iformatio i Large Courot Markets." Ecoometrica, 56, Vives, X. (993), "How Fast do Ratioal Agets Lear?" Review of Ecoomic Studies, 60, Welch, I. (992) "Sequetial Sales, Learig ad Cascades." Joural of Fiace 47, Wilso, R. (977), "A Biddig Model of Perfect Competitio." Ecoometrica 44, Zhag, J. (997) "Strategic Delay ad the Oset of Ivestmet Cascades." The Rad Joural of Ecoomics, Vol 28, o., Zhag, J. ad Zhag, Z. (995), "Iformatio Exterality, Limited Observatios ad the Emergece of Truth i Sequetial Decisios." WZB Workig Papers, FS IV

24 Bücher des Forschugsschwerpukts Marktprozeß ud Uterehmesetwicklug Books of the Research Area Market Processes ad Corporate Developmet (ur im Buchhadel erhältlich/available through bookstores) Lars Bergma, Chris Doyle, Jordi Gual, Lars Hultkratz, Damie eve, Lars-Hedrik Röller, Leoard Waverma Europe s etwork Idustries: Coflictig Priorities - Telecommuicatios Moitorig Europea Deregulatio 998, Cetre for Ecoomic Policy Research Mafred Fleischer The Iefficiecy Trap Strategy Failure i the Germa Machie Tool Idustry 997, editio sigma Christia Göseke Iformatio Gatherig ad Dissemiatio The Cotributio of JETRO to Japaese Competitiveess 997, Deutscher Uiversitäts-Verlag Adreas Schmidt Flugzeughersteller zwische globalem Wettbewerb ud iteratioaler Kooperatio Der Eifluß vo Orgaisatiosstrukture auf die Wettbewerbsfähigkeit vo Hochtechologie-Uterehme 997, editio sigma Horst Albach, Jim Y. Ji, Christoph Schek (eds.) Collusio through Iformatio Sharig? ew Treds i Competitio Policy 996, editio sigma Stefa O. Georg Die Leistugsfähigkeit japaischer Bake Eie Strukturaalyse des Bakesystems i Japa 996, editio sigma Stephaie Rosekraz Cooperatio for Product Iovatio 996, editio sigma Horst Albach, Stephaie Rosekraz (eds.) Itellectual Property Rights ad Global Competitio - Towards a ew Sythesis 995, editio sigma. David B. Audretsch Iovatio ad Idustry Evolutio 995, The MIT Press. Julie A Elsto US Tax Reform ad Ivestmet: Reality ad Rhetoric i the 980s 995, Avebury Horst Albach The Trasformatio of Firms ad Markets: A etwork Approach to Ecoomic Trasformatio Processes i East Germay Acta Uiversitatis Upsaliesis, Studia Oecoomiae egotiorum, Vol , Almqvist & Wiksell Iteratioal (Stockholm). Horst Albach "Culture ad Techical Iovatio: A Cross- Cultural Aalysis ad Policy Recommedatios" Akademie der Wisseschafte zu Berli (Hg.) Forschugsbericht 9, S , Walter de Gruyter. Horst Albach Zerissee etze. Eie etzwerkaalyse des ostdeutsche Trasformatiosprozesses 993, editio sigma. Zolta J. Acs/David B. Audretsch (eds) Small Firms ad Etrepreeurship: A East- West Perspective 993, Cambridge Uiversity Press. Aette Boom atioale Regulieruge bei iteratioale Pharma-Uterehme: Eie theoretische Aalyse der Marktwirkuge 993, omos Verlagsgesellschaft. David B. Audretsch/Joh J. Siegfried (eds), Empirical Studies i Idustrial Orgaizatio 992, Kluwer Academic Publishers. Zolta J. Acs/David B. Audretsch Iovatio durch kleie Uterehme 992, editio sigma. Hafried H. Aderse, Klaus-Dirk Heke, J.-Matthias Graf v. d. Schuleburg (Hrsg.) Basiswisse Gesudheitsökoomie, Bad : Eiführede Texte 992, editio sigma. Hafried H. Aderse, Klaus-Dirk Heke, J.-Matthias Graf v. d. Schuleburg uter Mitarbeit vo Georg B. Kaiser Basiswisse Gesudheitsökoomie, Bad 2: Kommetierte Bibliographie 992, editio sigma.

25 DISCUSSIO PAPERS 997 Rabah Amir R&D Rivalry ad Cooperatio uder FS IV 97 - Joh Wooders Oe-Way Spillovers Frak Verbove Testig for Moopoly Power whe Products FS IV 97-2 are Differetiated i Quality Frak Verbove Localized Competitio, Multimarket Operatio FS IV 97-3 ad Collusive Behavior Jim Y. Ji Comparig Courot ad Bertrad Equilibria Revisited FS IV 97-4 Reihard Koma Huma Capital ad Macroecoomic Growth: FS IV 97-5 Dalia Mari Austria ad Germay Lars-Hedrik Röller Why Firms Form Research Joit Vetures: FS IV 97-6 Mihkel M. Tombak Theory ad Evidece Ralph Siebert Rabah Amir Cooperatio vs. Competitio i R&D: FS IV 97-7 Joh Wooders The Role of Stability of Equilibrium Horst Albach Learig by Doig, Spillover ad Shakeout i FS IV 97-8 Jim Ji Moopolistic Competitio Dietmar Harhoff Iovatiosareize i eiem strukturelle FS IV 97-9 Oligopolmodell Catherie Matraves Die deutsche Idustriestruktur im iteratioale FS IV 97-0 Vergleich Yair Tauma A Model of Multiproduct Price Competitio FS IV 97 - Amparo Urbao Juichi Wataabe Dalia Mari The Ecoomic Istitutio of Iteratioal FS IV 97-2 Moika Schitzer Barter William ovshek Capacity Choice ad Duopoly Icetives FS IV 97-3 Lyda Thoma for Iformatio Sharig Horst Albach Wirtschaftspolitische ud techologie- FS IV 97-4 politische Folge der Globalisierug Horst Albach Humakapitaltheorie der Trasformatio FS IV 97-5 Horst Albach Guteberg ud die Zukuft der Betriebs- FS IV 97-6 wirtschaftslehre Horst Albach Risikokapital i Deutschlad FS IV 97-7 Dieter Köster Hiroyuki Okamuro Risk Sharig i the Supplier Relatioship: ew FS IV 97-8 Evidece from the Japaese Automotive Idustry Berard Siclair-Desgagé Career Cocers ad the Acquisitio FS IV 97-9 Olivier Cadot of Firm-Specific Skills

26 Steve Casper Corporate Goverace ad Firm Strategy FS IV Catherie Matraves i the Pharmaceutical Idustry Bruce R. Lyos Idustrial Cocetratio ad Market Itegratio FS IV 97-2 Catherie Matraves i the Europea Uio Peter Moffatt Petri Lehto Cosolidatios ad the Sequece FS IV Mihkel M. Tombak of Acquisitios to Moopoly Vesa Kaiaie Project Moitorig ad Bakig Competitio FS IV Rue Stebacka uder Adverse Selectio Dalia Mari Ecoomic Icetives ad Iteratioal Trade FS IV Moika Schitzer Ila M. Semeick Alam Log Ru Properties of Techical Efficiecy FS IV Robi C. Sickles i the U.S. Airlie Idustry Dietmar Harhoff Citatio Frequecy ad the Value FS IV Fracis ari of Pateted Iovatio Frederic M. Scherer Katri Vopel Dietmar Harhoff Explorig the Tail of Pateted FS IV Frederic M. Scherer Ivetio Value Distributios Katri Vopel Jim Ji The Effect of Public Iformatio o FS IV Michael Tröge Competitio ad R&D Ivestmet Daiel A. Traca Import-Competitio, Market Power FS IV ad Productivity Chage Michael Tröge Bakig Competitio as Mixed Commo ad FS IV Private Value Auctio Lars-Hedrik Röller Capacity ad Product Market Competitio: FS IV 97-3 Robi C.Sickles Measurig Market Power i a "Puppy-Dog" Idustry Talat Mahmood Survival of ewly Fouded Busiesses: FS IV A Log-Logistic Model Approach Silke eubauer Iterdivisioal Iformatio Sharig - The Strategic FS IV Advatage of Kowig othig Silke eubauer The Cosequeces of Edogeous Timig for FS IV Diversificatio Strategies of Multimarket Firms Christoph Schek Capacity Decisios ad Subcotractig FS IV Michael Tröge Idustry Owership of Baks ad Credit FS IV Market Competitio

27 Petra Kordörfer The Lik Betwee Iterest Rates o Iterbak Moey FS IV ad for Credit Lies: Are Asymmetric Iterest Rate Adjustmets Empirically Evidet? Damie J. eve Uio Power ad Product Market Competitio: FS IV Lars-Hedrik Röller Evidece from the Airlie Idustry Zhetag Zhag Horst Albach Dokumetatio der Kaisha-Datebak - Zur FS IV Ulrike Görtze Datebak der Jahresabschlüsse japaischer Tobias Miarka Idustrieaktiegesellschafte Adreas Moerke Thomas Westphal Rita Zobel Tobias Miarka ew Directios i Japaese Bak-Firm-Relatioships: FS IV Jiapig Yag Does a Relatioship Matter for Corporate Performace? Ulrike Görtze R&D Activities ad Techical Iformatio Flow FS IV 97-4 i Japaese Electroic Corporatios Adreas Moerke Japaische Uterehmesgruppe - eie FS IV empirische Aalyse Adreas Moerke Does Goverace Matter? Performace ad FS IV Corporate Goverace Structures of Japaese keiretsu Groups Rita Zobel Employee-Trasfer as a Istrumet of Iformatio- FS IV Trasfer through Vertical Relatios? Dietmar Harhoff Are there Fiacig Costraits for R&D ad FS IV Ivestmet i Germa Maufacturig Firms? Lutz Bellma Zur Aalyse vo Grüduge ud Schließuge FS IV Dietmar Harhoff auf Grudlage der Beschäftigtestatistik orbert Schulz Adreas Stepha The Impact of Road Ifrastructure o Productivity FS IV ad Growth: Some Prelimiary Results for the Germa Maufacturig Sector Jim Y. Ji Icetives ad Welfare Effect of Sharig FS IV Firm-Specific Iformatio Jim Y. Ji Iformatio Sharig about a Demad Shock FS IV 97-49

28 DISCUSSIO PAPERS 998 Horst Albach Uterehmesgrüduge i Deutschlad FS IV 98 - Potetiale ud Lücke Dietmar Harhoff Vertical Orgaizatio, Techology Flows ad R&D FS IV 98-2 Icetives - A Exploratory Aalysis Karel Cool Der Eifluß des tatsächliche ud des potetielle FS IV 98-3 Lars-Hedrik Röller Wettbewerbs auf die Retabilität vo Uterehme Beoit Leleux der pharmazeutische Idustrie Horst Albach Blühede Ladschafte? FS IV 98-4 Ei Beitrag zur Trasformatiosforschug Shiho Futagami Shukko i Japaese Compaies ad its Ecoomic FS IV 98-5 Tomoki Waragai ad Maagerial Effects Thomas Westphal Dietmar Harhoff Ledig Relatioships i Germay: Empricial FS IV 98-6 Timm Körtig Results from Survey Data Joha Lagerlöf Are We Better Off if Our Politicias Kow FS IV 98-7 How the Ecoomy Works? Justus Haucap Locatio Costs, Product Quality, ad Implicit FS IV 98-8 Christia Wey Frachise Cotracts Jes Barmbold Mafred Fleischer Patetig ad Idustrial Performace: The Case FS IV 98-9 of the Machie Tool Idustry Dieter Köster Was sid etzprodukte? - Eigeschafte, FS IV 98-0 Defiitio ud Systematisierug vo etzprodukte Adreas Blume Coordiatio ad Learig with a Partial Laguage FS IV 98 - Adreas Blume A Experimetal Ivestigatio of Optimal Learig FS IV 98-2 Uri Geezy i Coordiatio Games Adreas Blume Learig i Seder-Receiver Games FS IV 98-3 Douglas V. DeJog George R. euma atha E. Savi Has Mewis The Stability of Iformatio Cascades: How Herd FS IV 98-4 Behavior Breaks Dow Lars-Hedrik Röller The Icetives to Form Research Joit Vetures: FS IV 98-5 Mihkel M. Tombak Theory ad Evidece Ralph Siebert Christie Zuleher Ecoometric Aalysis of Cattle Auctios FS IV 98-6

29 DISCUSSIO PAPERS 999 Sucha Chae Bargaiig Power of a Coalitio i Parallel Bargaiig: FS IV 99 - Paul Heidhues Advatage of Multiple Cable System Operators Christia Wey Compatibility Ivestmets i Duopoly with Demad FS IV 99-2 Side Spillovers uder Differet Degrees of Cooperatio Horst Albach Des paysages florissats? Ue cotributio FS IV 99-3 à la recherche sur la trasformatio Jeremy Lever The Developmet of British Competitio Law: FS IV 99 4 A Complete Overhaul ad Harmoizatio

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31 Abseder/From: Versadstelle - WZB Reichpietschufer 50 D-0785 Berli BESTELLSCHEI / ORDERFORM Bitte schicke Sie mir aus der Liste der Istitutsveröffetlichuge folgede Papiere zu: Bitte schicke Sie bei Ihre Bestelluge vo WZB-Papers ubedigt eie -DM-Briefmarke pro paper ud eie a Sie adressierte Aufkleber mit. Dake. For each paper you order please sed a "Coupo- Répose Iteratioal" (iteratioal moey order) plus a self-addressed adhesive label. Thak You. Please sed me the followig papers from your Publicatio List: Paper r./o. Autor/Author + Kurztitel/Short Title

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