Price Competition with Population Uncertainty

Transcription

1 Price Compeiio wih Populaio Uceraiy Klaus Rizberger firs versio: July 2008, his versio: Jauary 2009 Absrac The Berrad paradox holds ha price compeiio amog a leas wo firms elimiaes all profis i equilibrium, whe firms have ideical cosa margial coss. This assumes ha he umber of compeiors is commo kowledge amog firms. If firms are ucerai abou he umber of heir compeiors, here is o pure sraegy equilibrium. Bu i mixed sraegies a equilibrium exiss. I his equilibrium i akes a large marke o wipe ou profis. Thus, wih populaio uceraiy, wo are o eough for compeiio. Keywords: Berrad paradox, populaio uceraiy, price compeiio. JEL classificaio: C62, C72, D2, D43. Iroducio The Berrad 883 model of price compeiio o a marke for a homogeeous produc is oe of he basic models of imperfec compeiio. Is predicio appears somewha couer-iuiive, hough: Whe firms have ideical cosa margial coss, price equals margial cos ad equilibrium Helpful commes by a referee ad a associae edior are graefully ackowledged. Remaiig errors ad shorcomigs are mie. Viea Graduae School of Fiace ad Isiue for Advaced Sudies, Deparme of Ecoomics ad Fiace, Sumpergasse 56, A-060 Viea, Ausria. Tel , Fax , . rizbe@ihs.ac.a

2 profis are zero. Sice his is idepede of he give umber of firms, wo are eough for compeiio. More geerally, his also holds, for all bu he mos efficie firm, if firms have differe, bu cosa margial coss. For, ay price above he secodsmalles margial cos ca be udercu by he mos efficie firm. A ay price below he secod-smalles margial cos he mos efficie firm ca slighly raise he price wihou riskig a loss of cusomers. Therefore, i equilibrium he mos ad he secod-mos efficie firm will boh ask he secod-smalles margial cos ad cosumers will edogeously decide o buy a he mos efficie firm, as oherwise he laer would udercu. Hece, he equilibrium markup is limied by he cos advaage of he mos efficie firm. I paricular, if a leas wo firms have he smalles margial cos, equilibrium profis are agai zero. This is i sark coras o he oher popular model of oligopolisic compeiio, he Couro model, where equilibrium profis are always posiive. The Berrad paradox is derived uder he assumpio ha all releva parameers, like cos ad demad fucios, are commoly kow amog he compeiors. I paricular, i assumes ha he umber of compeiors i he releva marke is commo kowledge amog firms. Wih free ery his correspods o a siuaio, where cos fucio for all poeial eras are commoly kow, so ha i is perfecly predicable who will eer. I his sese, wo are eough for compeiio, provided he umber of acive compeiors is commoly kow. The assumpio of a give ad commoly kow umber of firms o he marke does o always seem plausible, hough. I may be appropriae i a maure marke wih high barriers o ery, where all firms o he marke have deailed iformaio abou heir rivals. I emergig idusries wih high urover raes ad low ery ad exi coss, o he oher had, here may be cosiderable uceraiy abou compeiors. If cos fucios of poeial eras are ucerai or ery ad exi coss are privae iformaio, for isace, he umber of compeiors is also ucerai. If firms cao predic wih ceraiy whe heir rivals will defaul, he perceived umber of acive compeiors is ucerai agai. So, here is reaso o cosider a oligopolisic markes, where he umber of acive firms is ucerai. This argume reas cusomers as players, whose behavior a idifferece adjuss so as o suppor equilibrium; see Simo ad Zame 990. If he marke spli a ies is exogeous, mixed equilibria obai; see Blume O he case of sricly covex cos fucios see Thepo 995 or Hoerig

3 Ieresigly, resuls chage whe here is uceraiy abou he umber of rivals. This paper embeds a geeralized Berrad model io a game wih populaio uceraiy à la Myerso 998, Tha is, a radom umber of firms is draw o compee i prices o a marke for a homogeeous commodiy. Each firm lears wheher or o i was draw ad wha is cos fucio is. Bu firms do o observe he umber of compeiors, or he coss of compeiors. Wih his se-up here is o equilibrium i pure sraegies. A mixed equilibrium sill exiss, bu i eails posiive expeced profis ad price dispersio. I lie wih ecoomic iuiio, profis declie as he expeced umber of compeiors grows. If he expeced umber of compeiors diverges fas eough, as i is he case i some ieresig special cases, profis are squeezed o zero ad approximaely compeiive oucomes obai i he limi. The framework is geeral eough so as o ecompass a umber of ieresig models as special cases. Firs, he disribuio of he umber of compeiors is arbirary. A deermiisic umber of rivals is, herefore, a special case, as is a fiie maximum umber of rivals. Secod, firms may have differe ad radom cos fucios ypes ha are weakly covex. Puig a deermiisic umber of firms ogeher wih radomly assiged cos fucios yields a model of price compeiio uder uceraiy abou rivals coss, where coss may be sricly covex. This special case correspods o he model by Spulber 995, who already observed ha uceraiy abou cos fucios of rival firms miigaes he Berrad paradox. 3 Wih wo firms, wo ypes, ad liear coss his gives a sochasic varia of he model by Blume Third, as he disribuio of ypes is arbirary, i may be coceraed o a sigle cos fucio. Puig ogeher a fiie maximum umber of rivals wih a sigle ype ad cosa margial coss yields he model by Jasse ad Rasmuse The combiaio of a deermiisic umber of rivals wih a sigle ype ad cosa margial coss, of course, yields he classical Berrad 883 model. Fially, whe he disribuio of he umber of compeiors is Poisso as suggesed by Myerso, 998, 2000, a model is obaied, where i ca be show ha he disribuio of prices approaches a aom a margial cos pricig as he mea populaio size diverges. 2 For a more geeral model of games wih populaio uceraiy see Milchaich Spulber 995 allows also for fixed coss ad a coiuum of ypes, boh of which are o icluded i he prese model. 3

4 The res of he paper is orgaized as follows. Secio 2 iroduces he model. Secio 3 saes he resuls, ad Secio 4 discusses special cases. Secio 5 summarizes. 2 The Model There is a couably ifiie pool of poeially acive firms from which a radom draw selecs some o compee o he marke. Each firm lears wheher or o i is draw, bu o which ad how may oher firms are draw. Moreover, each firm ha has bee draw ges radomly assiged a ype ha deermies is cos fucio. The cos fucio ype is privae iformaio o acive players. Oce o he marke, firms compee for cusomers by simulaeously seig prices. Cosumers observe all prices se by acive firms ad he decide o buy from he firms ha offer he bes price. As cosumers care oly abou he price, bu o abou he ideiy of he firm from which hey buy, ad firms produce whaever quaiy is ordered wihou capaciy cosrais, he demad side of he marke ca be modelled by a demad fucio: a wice coiuously differeiable fucio Q : R + R + ha is sricly decreasig whe i is posiive, weakly log-cocave, ad saisfies Q p =0for some fiie p R ++. A firm s cos depeds o is oupu ad is ype. Aype =,..., T ges radomly assiged o each firm ha has bee draw accordig o a probabiliy disribuio ρ =ρ =,...,T 0 idepedely across acive firms, bu possibly depedig o he umber of firms draw o play. Hece, ρ =Pr is he codiioal probabiliy ha a firm ges assiged ype give ha firms are draw. The depedece of coss o oupu ad he ype is described by a family of wice coiuously differeiable fucios C : R + R + ha are sricly icreasig ad weakly covex, saisfy C 0 = 0 ad C 0 0, ad ha boh oal coss C q ad margial coss C q sricly decrease as he ype icreases, for all q R +.Hece,higher uambiguously correspods o more efficiecy i producio. Accordigly, he moopoly profi fucio of a acive firm of ype =,..., T is give by V p =pq p C Q p for all prices p R +. The assumpios o Q ad C imply ha V is a 4

5 quasi-cocave fucio of p. For, because Q < 0, C i V p = Q + pq C Q =0 2 V p V = 2Q + pq C Q 2 C =0 Q Q C Q 2 < 0 0, ad log-cocaviy of Q implies QQ Q 2. Ay exremum of V is, herefore, a maximum. For, V is coiuous, V 0 > 0, ad V p < 0, so ha 2 has a soluio by he iermediae value heorem ad he desired coclusio follows. Moreover, ha he secod-order codiio holds wih sric iequaliy, implies ha 2 has a uique soluio. For each ype =,..., T deoe he soluio o 2 by p, viz. he moopoly price where margial reveue equals margial cos. 4 Because V is dowward slopig a p by he secod-order codiio ad C decreases wih, he moopoly price p is a sricly decreasig fucio of he ype =,..., T,sop T < p T <... < p. Furhermore, a ay fixed price p R + he profi V p sricly icreases wih, ad he margial profi V p sricly decreases wih. To model he firms uceraiy abou he umber of rivals, le π = π,,... deoe he codiioal probabiliy disribuio over he umber of remaiig acive firms give ha a firm of ype has bee draw, i.e., π 0 for all =0,,... ad π =.Thais,π capures he beliefs of a firm of ype o he umber of is compeiors. Sice he acual umbers of players of each ype are geerally ucerai, here is ohig beyod he ype ha a firm ca base is beliefs o. Bu, because he ype assigme is here assumed idepede of he umber of compeiors, codiioig o he ype does o chage a firm s forecas abou he umber of oher firms i he marke. Therefore, he subscrip o beliefs ca be dropped, i.e., π = π =π,,... for all ypes =,..., T. 2. A Example To fix ideas, cosider a simple example wih T =ad π = α, α, 0,... for some α 0,, i.e., cos fucios are commo kowledge ad here ca be a mos wo firms o he marke. Also assume ha Q p = max{0, p} 4 If fixed coss are sricly smaller ha he mooply profi V p, hey could be easily icorporaed i he prese model. 5

6 cdf p Figure : The fucio σ from he example α =0.8 ad C q =0, i.e., liear demad ad cosa ad ideical margial coss ormalized o zero. The moopoly price is p = /2. Clearly, i equilibrium o firm will se a price above he moopoly price. Each of he wo firms believes ha by seig a price p 0, /2] wih probabiliy α she will obai he moopoly profi V p =p p ad ha wih probabiliy α she will face a ideical compeior. Therefore, i ay equilibrium each firm mus expec o ear a leas α V p = α /4 > 0, because his is wha she ca make by beig o he eve ha here will be o compeiio. Therefore, firms will se prices sricly above margial coss i equilibrium, leavig room for udercuig. This implies ha here ca be o equilibrium i pure sraegies. For, if here were a pure sraegy equilibrium, where oe firm ses a sricly smaller price ha he oher, he former could gai by slighly icreasig is price, because his would raise is profi i boh eves, wihou riskig a loss of marke share whe boh are acive. If here were a pure sraegy equilibrium, where boh firms se he same price, he firm ha obais a marke share of less ha i he eve ha boh are acive could gai by slighly udercuig, sice is marke share whe boh are acive would jump o. Bu here is a mixed sraegy equilibrium. Defie he fucio σ :0, ] 0, ] by { σ p =max 0, α α } 4αp p 6

7 for all p 0, /2] ad by σ p =for p>/2 see Figure. We claim ha boh firms mixig accordig o he cumulaive disribuio fucio σ cosiues a equilibrium. For, if he oppoe mixes accordig o σ a firm expecs o ear p p α + α σ p] = α 4 > 0 by seig ay price p /2 α/2, /2], less ha α /4 by seig ay price above /2, adp p < α /4 by seig ay price below /2 α/2. Therefore, he firm is idiffere amog all profi maximizig prices p /2 α/2, /2] adbohfirmsusigσ cosiues ideed a equilibrium i which boh firms expec o ear posiive profis. 2.2 Sraegies Turig o firms sraegies, le deoe he se of all probabiliy disribuios o he ierval P 0,p ], i.e., he se of all righ-coiuous icreasig fucios from P o he ui ierval ha ake he value a p. I a game wih populaio uceraiy a sraegy fucio σ : T subsiues for wha i a sadard game is a sraegy combiaio see Myerso, 998, p For such a sraegy fucio σ, deoebyσ p 0 he codiioal probabiliy ha a firm will ask p P orlessgivehaiisofype. To abbreviae oaio, deoe by σ he evaluaio map defied by σ p =σ p for all p P ad all =,..., T. Codiioal o facig compeiors a acive firm will expec o see oher firms of ype =, 2 compeiors of ype =2,..., ad T oher firms of ype = T, wih probabiliy! T ρ T / = =!, because of he idepedece assumpio o he ype assigme where = T =. The probabiliy ha a paricular price p P is smaller ha he price se by afirmofype accordig o a sraegy fucio σ T is σ p. For oaioal breviy defie, for each vecor x 0, ] T ad each =, 2,..., he vecor f x =f x,..., f x T T 0, ] T by f x =ρ x for all =,..., T. The, give a sraegy fucio σ, he codiioal probabiliy ha a paricular price p P does o exceed he prices asked by oher acive firms, give ha oher firms have bee draw, is give by he muliomial expressio 7

8 F T σ p = T T =0 T =0... T =3 2=0 =2!f σ p T T =2! T =2! T =2 f σ p where σ p =σ p,..., σ T p 0, ] T. By he biomial law, F x = f x, ad iducio yields, for all =0,,... ad each x 0, ] T, F x = k=0 k f x k F k x = f x +F x] = f x τ = τ τ ] 3 for all =2,..., T.Hece,F x = T. x] f T = The family of fucios F =,...,T has he useful propery ha F τ x =F 0 for all τ =,..., T if ad oly if x τ =0for all τ =,..., ad x τ =for all τ =,..., T 4 which follows direcly from 3. Sice F σ p is he probabiliy ha p is he bes price amog all compeiors, codiioal o acive firms o op of he oe uder scruiy, he T expeced payoff of a acive firm of ype ca ow be defied: A acive firm of ype obais V p wheever i maages o ask he lowes price amog all compeiors or whe i is he oly firm o he marke. If a leas oe acive compeior asks a lower price, he firm obais zero. If i ies wih a acive compeior a he lowes price i he marke, i obais some fracio of V p. Assumig for he mome he absece of such ies, he expeced profiofafirmofype =,..., T is he give by ] U p, σ =V p π F σ p 5 T I follows ha i equilibrium o acive firm ca be forced below π 0 V p 0, where i bes o beig he oly supplier ad asks he moopoly price. 3 Resuls The laer observaio, ha i ay equilibrium a firm of ype mus a leas obai π 0 V p, is he key o he followig resuls. I paricular, i implies 8

9 ha all equilibria mus be mixed. For, if he equilibrium sraegy fucio assigs pure choices o all ypes, he a acive firm ca discoiuously improve is payoff by slighly udercuig rivals of is ow ype, provided he sraegy fucio yields posiive payoff for is ow ype. Ad he laer follows from π 0 V p > 0 wheever π 0 > 0. Of course, if π 0 =, he firm kows ha i is a moopolis. Proposiio If 0 <π0 <, here exiss o equilibrium i pure sraegies. Proof. Suppose o he corary ha each firm of ype =,..., T asks p P wih ceraiy i equilibrium. The π 0 > 0 implies ha equilibrium payoffsforallypescaofallbelowπ 0 V p > 0, sohav p π 0 V p > 0 for all =,..., T. Wih σ T deoig he sraegy fucio ha saisfies σ p =0for all p<p ad σ p =for all p p a aom a p forall =,..., T i follows ha αv p π F σ p T V p π F σ p T for some α 0, ] for raioig a ies, all p P,adall =,..., T. Sice by slighly udercuig p T afirmofype = T caobaiapproximaely he payoff lim ε 0 U T p T ε, σ avoidig ies wih oher ypes, his implies from he equilibrium codiio above, 3, f T σ p T T = 0, ad f T σ p T T ε = ρ T for ε>0 ad all =0,..., ha T lim U T p T ε, σ =V T p T π ρ T + f τ σ p τ T ] ε 0 τ = αv T p T π F σ p T T = αv T p T π F T σ p T This implies from V p > 0 ha βπ 0 + π βf T σ p T + = k= ] ρ k T k F k T σ p T 0 where β = α, icoradiciooα 0, ], π 0 for all =0,,..., F T σ p T 0, ad π ρ k T k F k T σ p T π ρ T > 0 = k= 9 =

10 because F 0 σ p = for all p, σ P T ad all =,..., T,ad π 0 <. As a corollary, eve i he symmeric case wih T =, here exiss o symmeric pure sraegy equilibrium as log as 0 <π0 <, eveiffirms have ideical cosa margial coss. For he case wihou populaio uceraiy, where firms have cosa, bu differe margial coss, i is kow ha mixed equilibria may obai Blume, 2003, as wih sricly covex cos fucios Hoerig, Wih edogeous ie-breakig ad differe cosa margial coss, pure equilibria exis wihou populaio uceraiy Simo ad Zame, 990, bu o wih. I geeral, a pure Berrad equilibrium does o exiss uder populaio uceraiy, eve if i exiss wihou. This raises a exisece issue for Nash equilibria i discoiuous games wih populaio uceraiy. Sadard exisece proofs for discoiuous games e.g. Dasgupa ad Maski, 986a,b do o apply whe he payoff fucio, as i 5, is o liear i mixed sraegies. No aemp is made here o supply a geeral exisece proof for equilibria i discoiuous games wih populaio uceraiy. Isead, a mixed equilibrium is cosruced for he game a had. Theorem There exiss a Nash equilibrium i mixed sraegies. Proof. The proof employs a recursive cosrucio, where firms of ype mix over compac iervals p 0,p]. Se p = p ad p =mi{ p,p } 0 for all =2,..., T,where { } p 0 =mi p P V p π F 0 V p π F 0 for all =,..., T oe ha F = ad F 0 0 = 0 for all =, 2,..., ad deermie σ p =x as he soluio o he equaios ] V p π x+ ρ τ = V p π ρ τ] 6 f τ = for each p p 0,p]. For all p<p0 se σ p =0ad for all p>p se σ p =.Thesum 0 ρ τ is zero by coveio, so for =he righ τ = had side of 6 becomes V p π 0. τ = 0

11 For fixed p p 0,p ] he lef had side of 6 is coiuous ad sricly decreasig i x. The moopoly profi V p is sricly icreasig o he ierval p 0,p ], becausep p. Sice ρ τ = τ ] = F 0 >F 0 for all =0,,... by 3, he lef had side of 6 is a leas as large as he righ had side a x =0,ad,sice ρ τ ] = F τ = 0 for all =, 2,..., he lef had side of 6 cao exceed he righ had side a x =, for all p p 0,p ]. Therefore, by coiuiy a uique soluio o 6 i he ui ierval exiss for all p p 0,p ] by he iermediae value heorem. Implicily differeiaig 6 yields dx dp = V p π f x+ ρ τ ] τ = V p +π +ρ f x+ ρ τ ] > 0 τ = Moreover, if x =0he, by k=0 k f F 0k k 0 = F 0 for all = 0,,..., 3 implies p = p 0,adifx =he, by k=0 k f F k k 0 = F 0 for all =0,,..., 3 implies p = p. Hece, σ p as cosruced from 6 is a valid coiuous disribuio fucio wihou aoms o he ierval p 0,p]. By cosrucio he iervals p 0,p ] = supp σ are ordered by p 0 T p T p0 T p... T p0 p = p. Therefore, uder he sraegy fucio σ = σ =,...,T T hus cosruced, σ τ p = for all τ = +,..., T ad σ τ p =0for all τ =,...,, for ay p p 0,p ]. I follows from 4 ha F σ p = T k=0 k f σ p k F k 0 for all p p 0,p]. Therefore, for all p p 0,p ], he equilibrium payoff U σ V p π F σ p = V T p π ρ τ] 7 τ = is cosa. For ay τ he payoff o he ierval p 0,p ] saisfies from 5 U τ p, σ = V τ p V p V ] p π ρ τ Because V τ p /V p / p =V V τ V V τ ] /V 2, V τ p sricly icreases wih τ, adv p sricly decreases wih τ, he payoff U τ τ p, σ is icreasig i p p 0,p ] wheever τ =,..., adiisdecreasigip p 0,p ] wheever τ = +,..., T. Fially, for ay p p,p 0 he erm π F σ p is cosa, implyig ha he payoff U T τ p, σ decreases τ =

12 i p for all τ =,..., T as p τ p for τ, ad i icreases i p for all τ =,..., as p 0 p for τ<, for all =,..., T. These properies τ verify ha he pure bes replies of ype are precisely he prices p p 0,p ]. Thus, a equilibrium has bee cosruced. Ulike Proposiio he heorem does o ake he hypohesis ha 0 < π 0 <. This is because mixed sraegy equilibria may occasioally be pure. For isace, if π 0 =, a simple compuaio shows ha he probabiliy disribuios σ cosruced i he proof of Theorem degeerae o aoms wih ui mass a he moopoly prices p for all ypes =,..., T.Thais, implicily he heorem covers cases, where pure sraegy equilibria exis, like whe he firm kows ha i is a moopolis. Two more pois are worh oig. Firs, as 6 ad 7 reveal, wha cous for he equilibrium payoff of a firm of ype ishemassoflessefficie ypes codiioal upo he umber of compeiors, ρ τ = τ. If here were a coiuum of ypes, his sum would be replaced by a iegral over less efficie ypes, bu oherwise lile would chage i he proof. I his sese he assumpio of a fiie umber of ypes cosiues o serious loss of geeraliy. Secod, Theorem makes o claim abou uiqueess. I fac, here are special cases for which oher equilibria are kow o exis. For isace, if here is o populaio uceraiy, i.e. π m =for some m>ad π =0for all m, he prese model is covered by Spulber s 995 model where he ype disribuio i he laer is a sep fucio. Ye, Spulber ideifies a pure sraegy equilibrium. 5 Sice he proof of Theorem cosrucs a equilibrium, is properies ca be sudied. A firs comparaive saic exercise is obvious from he proof of Theorem : Sice p + p 0 for =,..., T, he equilibrium price disribuio of more efficie firm ypes is suppored a lower prices ha he oe of less efficie ypes. Therefore, i his equilibrium more efficie ypes will ed o ask lower prices ad sell higher quaiies sice Q is decreasig i he price i he eve ha hey wi he marke. Furhermore, Proposiio already shows ha a ouside observer will see price dispersio o a marke wih populaio uceraiy, because equilibria are i mixed sraegies. The proof of Theorem qualifies he sochasic aure of prices. Firms radomize over bouded suppors compac ier- 5 I fac, Theorem qualifies Spulber s uiqueess claim 995, Proposiio 2, p. 5 as oly applyig o pure sraegy equilibria. 2

13 vals, ad differe firm ypes employ differe suppors. Therefore, observig a price offer does coai iformaio abou he firm s efficiecy ype. Tha is, eve hough prices are oisy, hey do coai iformaio. The ex resul adds furher comparaive saics. Is firs par says ha, ulike i he classical Berrad case, equilibrium profis are posiive, wheever here is a chace o be a moopolis. Is secod par says, as expeced, ha more efficie firms do beer ha less efficie oes. Is hird par is a comparaive saic exercise ha cocers he expeced umber of compeiors. To make precise wha i meas o expec more rivals, say ha a probabiliy disribuio ˆπ o he oegaive iegers firs-order sochasically domiaes aoher, π, if m m ˆπ π for all m =0,,... 8 If his is he case, he mea ˆπ wih respec o ˆπ is a leas as large as he mea π wih respec o π. Sochasic domiace is hus he appropriae oio of a larger expeced umber of compeiors. Proposiio 2 I he equilibrium cosruced i Theorem profis a are posiive for all acive firms, wheever π 0 > 0; b icrease wih he ypes of acive firms; c declie whe he expeced umber of compeiors grows. Proof. a Firs, wheever π 0 > 0, equilibrium profis for all firms are posiive, i.e. U σ π 0 V p > 0 for all =,..., T.Toseehis,observe ha if p p 0 he p = p implies U σ =V p π F 0 V p π F 0 for all p P, implyig ha U σ V p π F τ 0 π 0 V p for all p P ad all τ =0,...,, i paricular, U σ π 0 V p > 0. If p = p0 < p, he o he ierval p 0,p ] he expeced profi U ca be wrie as 2 U p, σ =V p π ρ τ+ρ σ p] = τ= V p V p V p π F 2 0 = 3 V p V p U σ

14 which is a decreasig fucio of p because V p /V p / p < 0 ad, herefore, aais is maximum a p 0. Hece, U σ = U p 0,σ U p, σ V p π F 2 0, sicev p V p for all p p 0,p ]. Sice, by he same logic, for ay p 0 τ >p 0 he expeced profi saisfies U p 0 τ,σ = V p 0 τ π F τ 0 V p V τ p τ π Fτ 0 /V τ p V p π Fτ 0 for all τ =,..., ad all p p 0 τ,p τ] agai because V p /V τ p / p < 0 for all τ<, he iequaliy U σ V p π F 2 0 holds for all p p 0, i paricular, U σ V p π F 2 0 π 0 V p > 0. b Secod, equilibrium profis rise wih he ype. To see his, observe ha by 7, for all =2,..., T, U σ = V p V p 0 V p 2 ] π ρ τ τ= V p 0 V p 0 V p 2 ] π ρ τ > τ= V p 2 ] π ρ τ = U σ τ= because eiher p = p 0 or p p 0 implies p = p,sohav p = V p V p 0, verifyig he firs iequaliy, ad V p >V p for all p p 0 T,p ], verifyig he secod iequaliy. c Third, equilibrium profis fall as he expeced umber of compeiors rises. To see his, firs oe ha, wheever ˆπ sochasically domiaes π, he g ˆπ g π holds, for ay real-valued decreasig fucio g o he oegaive iegers. As a cosequece, if ˆπ sochasically 4

15 domiaes π, he, for =, U σ =V p 0 π ρ = π 0 V p ˆπ 0 V p =V ˆp 0 ˆπ ρ = Û ˆσ deoig variables uder ˆπ by a ha. Applyig iducio, assume ow ha Û ˆσ U σ. For ypes =2,..., T he followig cases have o be disiguished. If p mi {ˆp 0,p0 },heˆp = p = p ad Û ˆσ =V p ˆπ F 0 V p π F 0 = U σ by 8 yields he desired coclusio. If ˆp 0 < p p 0,heˆp =ˆp0 <p = p implies Û ˆσ =V ˆp 0 ˆπ F 0 V p ˆπ F 0 V p by 8, as desired. If ˆp 0 <p0 p, he Û ˆσ =V ˆp 0 ˆπ F 0 < V p 0 ˆπ F 0 V p 0 π F 0 = U σ π F 0 = U σ follows from V ˆp 0 <V p 0 ad 8. The remaiig wo possibiliies are more ivolved. If p 0 ˆp0 p, he for all p ˆp 0, ˆp ] 2 ] Û p, ˆσ =V p ˆπ ρ τ+ρ ˆσ p = τ= V p Û ˆσ V p V p V p U σ =U p, σ usig Û ˆσ U σ. I paricular, Û ˆp 0, ˆσ = Û ˆσ U ˆp 0.,σ, ad he laer saisfies U ˆp 0,σ U p 0,σ = U σ, because U p, σ 5

16 is decreasig o he ierval p 0,p ] ad p 0 ˆp 0 by hypohesis. Fially, if p 0 p < ˆp 0,heˆp = p ad ˆπ F 0 V p 0 V ˆp 0 π ρ τ] < π F 0 τ= from p 0 < ˆp0 implies Û ˆσ =V p ˆπ F 0 V p V p 0 V ˆp 0 π F 0 < V p V p U σ V p 0 V p 0 U σ =U σ because V p /V p is decreasig o he ierval p 0,p ]. Thus, if ˆπ sochasically domiaes π, heû ˆσ U σ for all =,..., T, i.e., a higher umber of expeced rivals drives equilibrium profis dow. Eve hough equilibrium profis declie as he firm expecs more compeiors, accordig o Proposiio 2c, here is i geeral o guaraee ha hey are drive o zero as he expeced umber of rivals diverges. Ideed, hey will o, as log as π 0 > 0, accordig o Proposiio 2a. Tha growig compeiio wipes ou profis, herefore, ecessarily requires ha he chaces of beig a moopolis, π 0, goozero. Siceπ is a primiive of he model, lile ca be said abou his wihou addiioal srucure. Ye, as will be show below, i some promie isaces his is ideed he case. 4 Special Cases I his secio a few ieresig special cases of he equilibrium cosruced i Theorem are preseed. Those are cases ha have bee sudied i he lieraure o a cerai exe. The mai advaage of hese cases is ha he equilibrium sraegy ca be compued i closed form, hus allowig for more comparaive saic isighs. 4. No Populaio Uceraiy Firs, cosider he case of o populaio uceraiy, where π m =for some m>0 ad π =0for all m. The he payoff from 5 becomes 6

17 T m U p, σ =V p τ= f τ m σ τ p] ad 6 yields σ p = τ= ρ m τ ρ m V p /m ] V p wih p 0 {p =mi P V p τ= ρ m τ ] m V p τ= ρ m τ ] } m ad p =mi { p,p 0 } for all =2,..., T, p 0 =mi{p P V p 0}, ad p = p. Hece, he leas efficie firms of ype =makeoprofis ad play a pure sraegy wih σ p =0for all p<p 0 ad σ p =for all p p 0 a aom a p0. Firms wih a more efficie ype >make posiive profis, bu less so he higher m is. As m goes o ifiiy, profis are wiped ou for all ypes. Thus, wih uceraiy abou he compeiors cos fucios, bu o populaio uceraiy, equilibrium profis are posiive for fiiely may firms, bu go o zero as he umber of firms becomes arbirarily large. I fac, more ca be said abou he case of large m. Sice he erm V p /V p /m coverges o as m goes o ifiiy, he equilibrium price disribuios for all ypes coverge o a aom of ui mass a p 0. By he previous observaio, his implies ha equilibrium prices approach average coss as he umber m of firms becomes arbirarily large, for each ype =,..., T. Hece, i a large marke price dispersio will become small ad expeced prices will approximae average coss. This coclusio is i lie wih he pure sraegy equilibrium ideified by Spulber 995, Proposiio 4. Wih T = 2 = m+ ad cosa, bu differe margial coss his versio of he model yields a sochasic varia of he mixed Berrad equilibrium described i Blume Because of uceraiy abou cos fucios of rivals, however, i is here he more efficie firms ha mix, while iefficie firms play a pure sraegy. I Blume, 2003, he siuaio is he oher way aroud. Ye, he equilibrium cosruced i he proof of Proposiio remais valid eve whe cos fucios are sricly covex. This gives a mixed Berrad equilibrium for he case of sricly decreasig reurs o scale ad uceraiy abou he compeiors cos fucios. 7

18 4.2 No Cos Uceraiy Secod, cosider he symmeric case of a sigle ype, T =. The, accordig o 6, all firm mix accordig o V p π σ p] = π 0 V p o he ierval p 0, p], wherep 0 =mi{p P V p π 0 V p} suppressig he subscrip for he sigle ype. Sice he righ had side of 6 equals he righ had side of 7 ad V p > 0, i follows ha equilibrium profis will go o zero if ad oly if he probabiliy of beig a moopolis, π 0, does. This is a codiio ha i a more ecompassig model would be deermied by he decisio of firms wheher or o o eer he marke. Here, he disribuio π is a primiive, so ha lile ca be said abou i i geeral. Sill, for some promie cases from he lieraure he codiio holds rue. Wih π = m α m α for all =0,..., m ad π =0 for all = m, m +,... for some α 0, ] ad wih C q =0for all q 0 he model wih a sigle ype is exacly he model of Jasse ad Rasmuse I heir case α is he probabiliy uiform ad idepede across firms ha a firm is hi by a shock ha preves i from paricipaig i he marke. Accordigly, 6 reads V p ασ p] m = α m V p σ p = α V p m α α V p Agai, equilibrium profis are posiive, bu fall as m rises, ad approach zero as m goes o ifiiy. Moreover, lim m V p /V p /m =implies ha he equilibrium price disribuio coverges o a aom of ui mass, i.e., agai price dispersio becomes small as he umber of compeiors grows. Sice also equilibrium profis approach zero as m, expeced equilibrium prices approximae margial equal o average coss i a large marke. 6 Fially, wih m =, i.e. o populaio uceraiy, his versio yields he classical Berrad case, where all firms se a price equal o margial cos ad profis are zero. 6 I is o difficul o obai a similar coclusio if, for isace, he disribuio of he umber of compeiors is geomeric, i.e. π = δ δ for some δ 0,, raher ha biomial as i he model by Jasse ad Rasmuse

19 4.3 Poisso Game Fially, a ieresig special case emerges if π is assumed o be Poisso, π = e µ μ /! for all = 0,,... a Poisso game i he sese of Myerso, 998, 2000 ad he ype disribuio does o deped o, i.e. ρ = ρ for all =, 2,... The, π ] { ρ τ = e µ μ ρ τ] =exp μ! τ= τ= } T ρ τ ad 6 yields σ p = l V p l V p] / μρ for all p p 0,p ], ad 7 reads { } U σ =V p T exp μ ρ τ for all =,..., T. Hece, equilibrium profis are clearly decreasig i he mea μ ad approach zero as μ goes o ifiiy. Moreover, as μ becomes large, he disribuios σ become more ad more cocave puig more mass a lowerprices adapproachaaomofmass a p = p 0 as μ +. Thus, agai i he limi each firm ype plays a pure sraegy, pricig a average cos. 5 Coclusios The Berrad model holds ha wo firms are eough o drive prices o margial coss o a marke for a homogeeous commodiy provided i is commo kowledge how may firms compee ad ha all have ideical cosa margial coss. This is a paradoxical coclusio, because equilibrium profis are ideically zero ad, herefore, do o provide ay ery iceives. Droppig he commo kowledge assumpio abou eiher he umber of compeiors or abou ideical cosa margial cos or boh resolves his paradox: Expeced equilibrium profis are posiive wih more efficie firms makig more. Sill, ad i lie wih ecoomic iuiio, more firms yield more compeiio ad lower equilibrium profis. This holds boh wih populaio uceraiy ad wih uceraiy abou cos fucios. The prese paper combies hese wo sources of uceraiy i a sigle model ha also allows for a poeially ifiie umber of firms. τ= τ= 9

20 A similar model could also be applied o procureme aucios wih a variable umber of bidders. The differece is ha i reverse aucios he quaiy sold is usually o price depede, i.e., here is o coiuous demad fucio aymore. Occasioally, whe such aucios are used o source buyer-desiged goods or services, he ulimae quaiy is decided expos i he ligh of he wiig bid. I ha case he prese model could be applied direcly. Bu mos of he ime he desired quaiy is fixed ex-ae, implyig ha Q is a cosa. I ha case a moopoly price does o exis. Wih a upper boud o feasible prices he argumes i he prese paper could poeially sill be adaped o cover ha case. This is lef for fuure research, hough. Refereces ] Berrad, J. 883, Review of Theorie mahemaique de la richesse sociale & Recherche sur les pricipes mahemaique de la heorie des richesse, Joural des Savas, ] Blume, A. 2003, Berrad Wihou Fudge, Ecoomics Leers, 78, ] Dasgupa, P., ad E. Maski 986a, The Exisece of Equilibrium i Discoiuous Games, I: Theory, Review of Ecoomic Sudies, 53, ] Dasgupa, P., ad E. Maski 986b, The Exisece of Equilibrium i Discoiuous Games, II: Applicaios, Review of Ecoomic Sudies, 53, ] Hoerig, S. H. 2002, Mixed Berrad Equilibria uder Decreasig Reurs o Scale: A Embarrassme of Riches, Ecoomics Leers, 74, ] Jasse, M., ad E. Rasmuse 2002, Berrad Compeiio uder Uceraiy, Joural of Idusrial Ecoomics, 50, -2. 7] Milchaich, I. 2004, Radom-Player Games, Games ad Ecoomic Behavior, 47, ] Myerso, R. 998, Populaio Uceraiy ad Poisso Games, Ieraioal Joural of Game Theory, 27,

21 9] Myerso, R. 2000, Large Poisso Games, Joural of Ecoomic Theory, 94, ] Simo, L. K., ad W. R. Zame 990, Discoiuous Games ad Edogeous Sharig Rules, Ecoomerica, 58, ] Spulber, D. F. 995, Berrad Compeiio whe Rivals Coss are Ukow, Joural of Idusrial Ecoomics 43, -. 2] Thepo, J. 995, Berrad Oligopoly wih Decreasig Reurs o Scale, Joural of Mahemaical Ecoomics, 24,