Collective Reputation and Market Structure

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Collective Reputation and Market Structure: Regulating the Quality vs Quantity Trade-off Pierre Fleckinger? This version: December 2007 Abstract Between market unraveling and individual reputation building, markets for expe- rience goods often exhibit intermediate patterns. This paper explores the situation in which consumers know the average quality offered by a set of producers, but not the quality of one given product. A first issue is how such an aggregate signal shapes competition when strategic variables are quantity and quality. The equilibrium welfare function is convex in the number of competing firms as a consequence of decreasing quality and increasing quantity. A second issue is regulation. It is possible to trade-off quantity against quality through a number of regulatory tools. Among one-instrument policies, entry and quantity regulation perform better than price-based regulation. Policy implications for professional regulation in service markets and producers' or- ganisations in agriculture are briefly discussed. JEL: D4, L1, L43, Q13 Keywords: Uncertain Quality, Market Structure, Collective Reputation, Minimum Quality Standard. ?Columbia University, Business School & Ecole Polytechnique, Laboratoire d'Econometrie, 1 rue Descartes, 75005 Paris. email: . This paper was mostly written when the author was affiliated to INRA. I am grateful to Jean-Marc Bourgeon, Igor Mouraviev, Sebastien Lecocq, Herve Tanguy and Eric Giraud- Heraud for comments and discussions.

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CollectiveReputationandMarketStructure:RegulatingtheQualityvsQuantityTrade-o!PierreFleckinger!Thisversion:December2007AbstractBetweenmarketunravelingandindividualreputationbuilding,marketsforexpe-riencegoodsoftenexhibitintermediatepatterns.Thispaperexploresthesituationinwhichconsumersknowtheaveragequalityo!eredbyasetofproducers,butnotthequalityofonegivenproduct.Aﬁrstissueishowsuchanaggregatesignalshapescompetitionwhenstrategicvariablesarequantityandquality.Theequilibriumwelfarefunctionisconvexinthenumberofcompetingﬁrmsasaconsequenceofdecreasingqualityandincreasingquantity.Asecondissueisregulation.Itispossibletotrade-o!quantityagainstqualitythroughanumberofregulatorytools.Amongone-instrumentpolicies,entryandquantityregulationperformbetterthanprice-basedregulation.Policyimplicationsforprofessionalregulationinservicemarketsandproducers’or-ganisationsinagriculturearebrieﬂydiscussed.JEL:D4,L1,L43,Q13Keywords:UncertainQuality,MarketStructure,CollectiveReputation,MinimumQualityStandard.!ColumbiaUniversity,BusinessSchool&EcolePolytechnique,Laboratoired’Econome´trie,1rueDescartes,75005Paris.email:pierre.fleckinger@polytechnique.edu.Thispaperwasmostlywrittenwhentheauthorwasa"liatedtoINRA.IamgratefultoJean-MarcBourgeon,IgorMouraviev,Se´bastienLecocq,Herve´TanguyandEricGiraud-He´raudforcommentsanddiscussions.ThispaperalsobeneﬁtedfromcommentsbyseminarparticipantsatINRA(Paris-Ivry)andAgroParisTechaswellasbyparticipantstothe23e`mesJMA(Nantes),EARIE2006(Amsterdam)andEEA2007(Budapest).PartialsupportfromtheEcolePolytechniqueChairinBusinessEconomicsisgratefullyacknowledged.1

1IntroductionSinceAkerlof(1970),economiststendtopredictthatwhenqualityisunknown,lowqualityproductsdrivehighqualityonesoutofthemarket.Consideraconsumer’s(she)buyingdecisionwhensheisnotsureaboutthequalityofthegood:sheisnotreadytopaymorethantheexpectedquality.Butsinceshedoesnotobservethequalitychoice,theseexpecta-tionsdonotdependonwhatthebuyeractuallydoes.Thenhehasnoincentivetoproduce(costly)quality.Thereforetheonlyequilibriumexpectationstheconsumercanholdisthattheproducersellsthelowestquality:marketforhighqualitythusunravels.Inthispaper,westudyhowmarketfunctionswhenconsumersdoknowaveragequalitypresentonthemar-ket,butnotthatofonegivenproduct.Thisrepresentsanintermediatesituationbetweentheperfectinformationsettingandtheasymmetricinformationsetting.Theinformationpubliclyavailableconsistsofaglobalassessmentofquality.This(true)averagequalityistheresultofthestrategicchoicesbymanyproducers.Undoubtfully,unravelingmaybepreventedwhentheconsumerrepeatedlypurchasesfromthesamesupplier:afterexperiencingabadqualityproduct-orservice,shecandecidetostopbuyingit(andpotentiallybuyanotherone),whichdisciplinesthesupplier.Thisistheessenceofbrandbuildingbyﬁrms,thatcanbemodeledbyarepeatedpurchasegame(Heal,1976)andisreferredtoas’reputation’ingametheoreticalstudies(seeforexampleShapiro,1982,1983).However,thissolutionrequiresﬁrstthattherelationshipislonglasting,andsecondthattheproducingﬁrmisidentiﬁedateachpurchase.Thelatterconditionfailswhenidentityoftheproducerislostintheretailingchain(e.g.thereisnotraceability),whenthefrequencyofpurchasedislowenough(inparticularforspecializedservices)orwhentheconsumerhasboundedmemoryandlimitedcognitivecapacities.Asaleadingexampleofproductsubjecttosuchrestrictions,considerthecaseofFrenchwines.ExceptforfamousChateaux,therelevantinformationforthestandardconsumerboilsdowntotheregionandyearofproduction1.Whattheconsumerreferstoinsuchacaseisratherapublicknowledgethatthekindofwinehassomeaveragequality,afactthathasreceivedstrongempiricalsup-port(seeforexampleLandonandSmith,1998).Howsuchapublicsignalisgeneratedisnotthesubjectofthepaper,anditisalreadyathoroughlyinvestigatedtopic.Thissignalcan1Onthis,seetheevidenceinCombrisetal.(1997).IntheUnitedStatesandemergingwinecountries,thisinformationconsistsratherofbrandandtypeofvine.2

beprovidedbyexperts2,bycertiﬁcationbodies3orbydiscussionwithotherbuyersthroughvariouskindsofconsumersnetworks,atopicyetunderstudiedbyeconomists4.Thispublicsignalisdi!erentfroma”reputation”inthesensethatitisnotbasedonpastconsump-tionsofasimilarproduct,butinsteadreliesonexpertjudgment,orontheconsumptionofthesamegenerationofproduct(insteadofapastproductofthesamebrand)byotherconsumers.Inshort,theemergenceofsuchapublicknowledgeregardingaveragequalityisnotinter-temporal,buthastodowiththeformationofqualityexpectationregardingaone-shotproduction5.Asconcernsexpertratings,theyevenpreemptpublicconsumption:Itisindeedtheveryroleofpremieresandjournalistsofspecializedpresstoprovidethepublicwithanevaluationofproductsbeforepurchasetakesplace.Again,thecaseofwinewhereexpertsaretheﬁrsttotasteandtogiveanoverallappreciationforagivenregionforthecurrentyearisillustrative.Weassumeherethattheexpertsarereliable6,sothatthecustomerscanrelyonanobjectivereferencewhenbuyingtheexperiencegood.ThesettingunderstudyhereisaCournotoligopolywithendogenouslydi!erentiatedex-periencegoods.Producerschoosebothqualityandquantity,andconsumersknowaveragequalitydespitethefactthattheydonotidentifyproducersindividually.Animportantfocusisonthecomparativestaticswithrespectton,thenumberofcompetitors:Underrelativelymildassumptions,wedemonstrateequilibriumexistenceanduniquenessforanynumberofcompetitors.Whiletotalmarketedquantityisincreasinginn,qualityisdecreasing,yield-ingaU-shapedwelfareasafunctionofn.Thisimpliesthateitherperfectcompetitionormonopolyistheoptimalmarketstructure.Thisissobecauseofafree-ridinge!ectonaver-agequality:Essentially,theassumptionontheinformationstructuretransformstheadverseselectionproblemintoamoralhazardproblema`laHolmstro¨m(1982).Ontheonehand,amonopoly,orawell-functioningproducerorganisation,choosesthesociallyoptimalqualitylevelbecauseitisnotsubjecttothismoralhazardproblem:averagequalityrevealsdirectlytheinvestmentofthemonopoly.Butontheotherhand,competitionincreasesthequantities2SeeinparticularAlietal.(2005)foranempiricalestimationofRobertParker’sgradesimpactonBordeauxprices,andthereferencesthereinonthetopicofexpertratinginwine,bothatthebrandandattheregionallevel.3Amongothers,LernerandTirole(2006)andPeyracheandQuesada(2004)developmodelsinwhichthecertiﬁcationprocessisendogenous.4Forexample,Curienetal.(2004)emphasizetheimportanceofconsumers’onlinenetworks.5Itishoweverperfectlypossibletointerpretthesettingasacompactreputationmodel,embeddingthedynamicse!ectsintheone-periodequilibrium.6Ofcourse,theseexpertsmaybesubjecttocapture(seeStrausz,2005,forarepresentativerecentcontribution),butwewillabstractherefromthispossibility.3

sold.Asecondresultisthatthequantitye!ectismorethano!setbythequalitye!ect:competitionhappenstobeharmfultotheconsumersiftheenforceablequalitystandardisrelativelylow.Inthelimit,perfectcompetitioncandestroyallpotentialsurpluswhennoqualitystandardisenforced.Thisresultcanexplainthecreationofagriculturalsyndicatesthatarenotfoughtbygovernments,orevenlegallyencouraged7.Thisresultmightalsoshedlightonprofessionalregulationssuchasinthemedicineandlawsectors.Withthetwobenchmarksofﬁrst-bestandcompetitivesituationsinmind,weundertaketheanalysisofselectiveregulationofsuchmarkets.Thereisaroomforregulationunderanymarketstructure.Whenqualitystandardsareenforceableandstringentenough,perfectcompetitionisclosetotheParetooptimum-theusualconvergenceresultofCournotequi-libriumtowelfare-maximizingquantitiesholdwhenqualityis(almost)notaconcern.Thereishoweveraroomforentryregulationwhenthestandardislow.Animportantmessagehereisthatqualitystandardsandcompetitionarecomplementaryundercollectivename.Finally,itisshownthatamongone-instrumentpolicies,thebestregulationtoolisquantityregulation,whichperformsbetterthanpriceregulationforanymarketstructure.2RelationtotheliteratureThemodelisrelatedtothecollectivereputationmodelsofTirole(1996),WinfreeandMc-Cluskey(2005)andBourgeonandCoestier(2007),despitethestaticnatureofthepresentanalysis.Indeed,thosepapersareconcernedwiththedynamicsoftheproblem,inabsenceofpublicsignal.Itturnsouthoweverthatsomeconclusionsareverysimilar,namelyunder-provisionofquality,althoughinadi!erentform.Whatmainlydistinguishesthecurrentanalysisfromtheprecedingonesisthatwefocushereonmarketstructureandwelfareanal-ysis,theﬁrsttopicbeingalmostcompletelyabsentfromthementionedliterature8.Theassumptionstherealsodi!erfromoursinmanyways.Tirole(1996)considersacasewhereagentsareofdi!erenttypes(honest,dishonest,orstrategic):asisstandardinreputationmodel(e.g.KrepsandWilson,1982),agentsareexogenouslydi!erent.Regardingthelit-eratureonexperiencegoods,MoavandNeeman(2005)areinterestedintheroleoftheinspectiontechnologywithtwoclassesofproducers.Finally,theinformationsettingstudied7SeeTitleIVoftheEuropeanCouncilRegulationNo1493/1999.FurtherdevelopmentsonthecaseofwinecanalsobefoundinGiraud-He´raudetal.(2003).8Whiledealingwithadi!erentproblem-operatingaregulatednetwork-themodelofAuriol(1998)alsofeaturesafree-ridinge!ectthatcanmake(regulated)duopolyworsethan(regulated)monopoly.However,theparallelendshere,becausethecoststructuresandregulationproblemsarequitedi!erent.4

herealsodi!ersfromthatconsideredbyWolinsky(1983),whereconsumersgetindependentprivatesignalonthequalityofeachdi!erentproduct.Intheliteratureconcerningagriculturalproducersorganisation,manypapersdealswithcloseissues.Maretteetal.(1999)andMaretteandCrespi(2003)studycertiﬁcationandtheroleoftheseorganisations,modeledascartels9.Undertheassumptionofexogenousdiscretequality,withconsumersformingexpectationsaboutthequalityofuncertiﬁedproducts,theyshowthatcartelssharingthecertiﬁcationcostandcolludingonquantitiescandobetterthatcompetitionfromasocialpointofview(intuitively,ithappenswhencertiﬁcationisindividuallytoocostly).Also,AuriolandSchilizzi(2003)studiestheroleofthe(ﬁxed)cer-tiﬁcationcostonmarketstructure.Firmschoosethesociallyoptimumqualitylevelassoonastheyseekcertiﬁcation(beitpublicorprivate),andthusqualitydistortionscomesonlyfromnon-certiﬁedﬁrm,thatproducethelowestquality.Intheirmodel,ﬁxedcertiﬁcationcosthasthefollowingimplications:whenﬁrmsself-certiﬁedthemarketisoligopolistic,asexpectedwithdecliningaveragecosts.Next,theyassumethatitispossiblenotoduplicatethesecertiﬁcationcosts,andcomparetopossiblecertiﬁcationarrangements:sharingcostsproportionatelytothequantitysoldforeachﬁrm,orpubliclyfunded.Concerningproduc-ersorganisations,WinfreeandMcCluskey(2005)dealwithcollectivereputationanduseShapiro(1983)setting,sotheydonotmodelendogenouslearningofqualitybythecon-sumers:consumersdonotformBayesianexpectations.Insteadtheirbeliefsevolvethroughapre-speciﬁedMarkovianprocessratherthanthroughBayesianrevision.Inaddition,theirpaperonlydealswithproducerssurplus,withoutreferencetothewelfareimpact.Theassumptionofapublicsignalregardingaveragequalityallowsbothtoavoidadhoclearningbyconsumersandtotacklethestudyofoligopoly,that,asMilgromandRobertsmention,”involvesigniﬁcantadditionalproblems”(1986,footnote9)withrespecttothemonopolycasewhenqualityisendogenous.Thus,beyondtherealismembeddedinit,thisassumptionalsoallowstogoonestepfurther,whilekeepinginareducedformtheproblemofcollectivereputation.Italsoallowstotreatinauniﬁedframeworkalltoolsaregulatormaywanttouseandcomparetheire"ciency.Finally,onthetechnicalside,theresultonequilibriumuniquenessobtainedintheﬁrst9Zago(1999)developsamechanismdesignmodeltostudycollectivedecisionwithinproducersorganisa-tionsinarelatedcontext.5

partofthispaperinvolvessomedi"cultiesthatexistingapproachescannotovercome.Therootofthisdi"cultiesliesontheonehandinthetwo-dimensionalstrategiesofﬁrmsandontheotherhandinthequalityexternality,afeaturewhichisknowntomakequasi-concavitybreakdown.Infact,thestandardexistenceresultssimilartothatofRosen(1965),rely-ingondi!erentialcalculus10donotapplyduemainlytothelackquasi-concavityoftheproﬁtfunctions.Thecontractionmappingapproachisnotappropriateherebecauseoftwo-dimensionalstrategyspace(seeLongandSoubeyran,2000,foranelegantresultregardingpurequantitycompetition),neitherarethetechniquesofsupermodulargamesandrelatedtools(seeVives,1999).Therefore,theproofsrequirespeciﬁctreatmentsofthecaseunderstudy,butbythesametokenprovidespeciﬁcinsightsonthecollectivereputationmechanicsinoligopoly.Inthenextsection,themodelisstatedanditsessentialfeaturesarepresented.Thefourthsectioncharacterizesthecompetitiveequilibrium.Theﬁfthsectionisinterestedinthewelfarepropertiesofdi!erentmarketstructures.Finally,thesixthsectiondiscussregulationandpolicyimplicationswhilethelastoneconcludes.Allproofsarerelegatedtotheappendix.3Model3.1TheconsumersWeconsiderapopulationofconsumersthatdi!erthroughtheirtastetforquality!,where!"[!,!]#(0,+$).Consumers’tastes11aredistributedover[0,t]accordingtothecumu-lativepopulationweightF(t).FollowingMussaandRosen(1978),aconsumerwithtastetfacingapricepderivesthefollowingutilityfrombuyingoneunitofagoodwithquality!:u(!,p;t)=!t%pandshewillbuy(exactlyoneunitof)thegoodifu(!,p;t)&0.ThequantitysoldisthereforepQ=F(t)%F(!),correspondingtothefollowinginversedemand:p=!F!1F(t)%Q"!10SeealsoKolstadandMathiesen(1987);Novshek(1985)andmorerecentlyGaudetandSalant(1991).11Settingthelowesttasteat0yieldsaninversedemandfunctionthatdoesnotdiscontinuouslyvanishesatsomestrictlypositivequantity.MilgromandRoberts(1986),forexample,useasimpliﬁedversionofthissettingintheirsectionII.6

Underthestatedpreferences,thedemandfunctionisthusmultiplicativelyseparablebetweenthequalityandquantitye!ects.Wemakethefurtherassumptionthatthedistributionoftastesisuniform,withweightKateachpoint,sothat:p(!,Q)=!(a%bQ))1(witha'tandb'K1.AnattractivepropertyoftheMussaandRosenspeciﬁcationofutilityisthattheinversedemandfunction(1)remainsalsovalidwhenqualityisimperfectlyknown,inwhichcase!representsthentheexpectedquality12.Thisistruebecauseconsumers’utilityislinearin!.Inaddition,theassumptionofuniformdistributionalsohelpssimplifyingtheinversedemand-althoughthelinear-in-quantityformulationisillusorywhenitcomesthecaseofexpectedquality,aswillbecomeclear.Muchoftheresultscouldbederivedwithoutthislinearfunctionalform,althoughatthecostsofmuchheaviertechnicalities.Weassumeasthoroughlyexplainedintheintroductionthattheconsumershavesomehardinformationabouttheactualqualitysoldinthemarket,moreprecisely:Assumption1Averagequalityandtotalquantitypresentonthemarketarepublicsignals.Thisimpliesthatconsumersknowforsuretheaveragequalityonthemarket(”strawberriesaregoodthisyear”,”mostlawyersinthiscityareo!erlowqualityservice”,andsoon),buttheyarestillunabletodistinguishthequalityofonepreciseproduct.NotethatthisassumptionliessomewherebetweenthecaseofperfectinformationoneachproductsoldandthepureBayesiancase,inwhichconsumersonlyformexpectations,basedonlyonstrategicconsiderationsandnotonpublicsignal.Thefactthatconsumersknowthetotalquantity(ratherthan,morerealistically,theprice),canbeseenasashortcutaccountingforaretailingstagethattransformsthequantityinformationintoapriceinformation.Infact,onecoulddispensewiththeassumptionofpubliclyknownquantity,atthecostofmoresophisticationinthepriceformationmechanism.Onecannoticethatisalsoequivalenttoassumethattheconsumersknowexactlywhateachproducerdid,butcannotidentifyafterwardswhereagivenproductcomesfrom.Wenowturntotheproductionside,wherequalityandquantityarechosen.12Leland(1979)studiesarelatedmodel,butthemicro-foundationsarenotexplicit:heratherdirectlypostulatesaprice-dependenceonexpectedquality,andinadditionhedoesnotuseimperfectcompetitionasequilibriumconcept.7

3.2TheproducersTherearenidenticalproducers,indexedbyi=1..n,thatchoosetheirquantityqiandquality!i,ataunitcostc(!i)qi,wherecisstrictlyincreasingandstrictlyconvex,andsatisﬁesthefollowingconditions:c(0)=0,c"(0)=0andc"(!)=+$,whichwillensurethatsomeinteriorqualitylevelisoptimal.Weimposealsothetechnicalassumptionc"""&0,whoseroleisexplainedwhenrelevant.Notethatwedeﬁnethecostfunctionindependentlyoftheminimumqualitylevel,!,sothatwecanstudythee!ectofchangingtheminimumqualitystandardwhileholdingtheproductiontechnologyﬁxed.Thisdoesnotmeanthatitislegallyfeasibletoselltheuselessproductwithquality0;infactthelowestpossiblequality,!,whichhastobeviewedasaminimumqualitystandard,isafundamentalpolicytoolwhoseimpactisstudiedbelow.Wechoosedeliberatelytoconsidercostsfunctionsthatarelinearinquantity.Indeed,tostudytheinterplayofqualityandquantityonthemarketstructure,anyothershapeofcosts13withregardtoquantitieswouldbiastheoptimalmarketstructuretowardsoneortheotherdirection,i.e.monopolyorperfectcompetition.Withconcavequantitycosts,forexampleinthepresenceofaﬁxedcost,amoreconcentratedmarketwouldbesociallypreferred,whilewithconvexquantitycosts,spreadingthemamongaverylargenumberofproducerswouldbedesirable.Asthefocusisontheinteractionbetweenquality,quantityandmarketstructure,wemustgetridofsuchbiasesbyneutralizingthosetechnologicale!ects,andtheonlyfunctionalformallowingthatisthechosenmultiplicativeone.ThetotalquantityisdenotedQ=qi,andtheaveragequalityonthemarketis:#!=!iqi(2)#QGivenassumption1,thereisonesinglemarketprice.Becausetheconsumersareinformedofthetrueaveragequality,thisequilibriumpricewillbegivenbyequation(1).Produceri’sproﬁtistherefore:"i({!j,qj})=p(!,Q)qi%c(!i)qi(3)Inthefollowingwedenoteby#theproducersproﬁt,with#="i.#13Forexample,KleinandLe#er(1981)haveﬁxedcostsdependingonqualityandAllen(1984)hasbothconvexmarginalcostsandﬁxedcosts,whichmakesthecostfunctionneitherconcavenorconvex.ThepresentcostfunctionisfoundforexampleinBesankoetal.(1987)andisratherstandardindi!erentiatedoligopoliesmodels.8

3.3WelfareOptimumandbenchmarkcaseInaﬁrst-bestworld,abenevolentsocialplannercouldassigntoeachproduceraqual-ity/quantityplan,toserveapredeterminedsetofconsumers.Thatis,itwouldputinplaceadouble-sideddiscrimination,withﬁrmsproducingdi!erentlevelsofquality,correspondingtodi!erentmarketsegments.Inthelimit,withacontinuumofﬁrmsor,say,withfree-entry,thisamountstoapoint-wisematchingbetweenﬁrmsandconsumers,whoseoptimumisderivedfromapoint-wisemaximizationforeachconsumer’staste.Thequalitymenu!(t)wouldverify:c"(!(t))=tForaﬁnitenumberofﬁrms,theﬁrst-bestoptimumwouldconsistofapartitionofthecon-sumersintoqualitygroups,adiscreteapproximationoftheabovemenu.Butthatisnotfeasiblewhenconsumerscannotdiscriminatebetweenproducers,asstatedbyassumption1.Giventhatobservabilityassumption,theconsumersdonotidentifythedi!erentqualitylevels,whichamountstoconsideringasinglequalitycategory,averagedoverthedi!erentproducers,andconsequentlythereisasingleassociatedprice.Thissituationwillconstituteourbenchmarksecond-bestoptimum.HencewenowconsidertheMarshallianconsumersurplusdeﬁnedby14:Q$U=p(!,Q)dQ%p(!,Q)Q(4)0Forthedemandfunctionspeciﬁed,thisexpressionremainsvalidwhenqualityisheteroge-nous,inwhichcase!istheaveragequality.Giventheconvexityofc,thecostforagivenlevelofaveragequalityisminimizedwhenallproducerschooseexactlythatqualitylevel.Inturn,sinceproducersareidenticalandcostsarelinearwithrespecttoquantity,allocationofproductionbetweenthedi!erentproducersisunimportant.Thesocialoptimumisthereforegivenbymaximizingover!andQthefollowingexpression:Q$W=U+#=p(!,Q)dQ%c(!)Q0Thecorrespondingﬁrstorderconditionsare:!#(a%bQ#)=c(!#)%!Q0(a%bQ)dQ=c"(!#)Q#&t14Itisofcourseequivalenttodeﬁneitast0u(t,!,p)dt,wherepdenotestheequilibriumpriceandt0the&consumerindi!erentbetweenbuyingornot.9

whichcanberestatedas:#Q#=1(a%c(!))(5a)#!b#c"(!#)=1(a+c(!))(5b)#!2Ofcourse,theproblemisinterestingif,ﬁrst,anoptimallevelexists,thatis,equation(5b)hasasolution,and,second,thissolutionishigherthantheminimumqualitystandard!.Thenextlemmaclariﬁesthis.Lemma1Assumec"""&0andc"(!)<a.Then!#isuniqueand!#>!.2Onceagain,recallthatthisreferencecaseisnotaﬁrst-bestsituation,butrathercon-stitutesthesociallyoptimalproductionplanundertheimperfectobservability.Inwhatfollows,wewillkeepthisassumptions,alsotheyarenotneededforallresults.4UnconstrainedCompetitionSincethereisacontinuumofconsumersandaﬁnitenumberofﬁrms(howeverlargeitcanbe),theconsumptionsideofthemarketisassumedperfectlycompetitive.Thereforetheﬁrmsfacethedemandschedulein(1).Inturn,ontheproductionside,thereisimperfectcompetition:TheproducersplaytheNashequilibriumofthegamedeﬁnedbystrategiesqi,!iandpayo!sin(3).Anticipatingabitontheresults,weconsidertheﬁrst-orderconditionsoftheproﬁt,forsomeﬁrmi(thereare2nﬁrst-orderconditionsoverall).!iQ!!(a%bQ)qi%b!qi+!(a%bQ)%c(!i)=0%qQi(a%bQ)qi%c"(!i)qi=0Tograspsomeintuitiononwhatisgoingoninequilibrium,assumethattheseconditionsareindeedsatisﬁed.Inspectionoftheﬁrstequationisespeciallyinstructive,sowerewriteitasfollows:p(!,Q)%c(!i)=b!qi+!%!iqip(!,Q)(6)Q!Theleft-handsideissimplypriceminusmarginalcost,andrepresentsthustheunitmargin.Theright-handsidepertainstomarketpower,anditdecomposesintotwoe!ects.Theﬁrsttermisclassicallyrelatedtotheelasticityofpricewithrespecttoquantity,asinanyCournotmodel.Thesecondtermisthekeystoneofthe”collectivequality”environment.Itillustratesthequalitydilutione!ect,whichispositivewhenproducerichoosesalowerthan01

averagequality,andthereforecorrespondstoafree-ridinge!ectonquality.Themagnitudeofthise!ectalsodependsonherelativesize(qQi)oftheconsideredproducer,andontheabsolutevalueoffree-riding15,namelytheprice.Asmentionedintheintroduction,thegameunderstudydoesnothaveasmoothstruc-ture.Clearly,themaindi"cultycomesfromthethetwo-dimensionalstrategies,oralter-natively,fromthepotentialdi!erentiationoftheproductsinequilibrium,whichmakestheusualanalysiswithonesingledimensionfail.Indeed,fromoneﬁrmpointofview,itisnotpossibletodealwithonlyoneaggregatevariablesummarizingalltheotherﬁrms’behavior.Thisimpliesthatnogeneralresultsapplytoshowequilibriumexistenceanduniqueness.Tobeprecise,standardresultsforhomogeneousCournotcompetition(e.g.KolstadandMath-iesen,1987)havenobiteinthepresentcontext,norarethetechniquesinVives(1999,p.47)applicable.Thedi"cultiesincharacterizingtheequilibriumarenumerous,andweonlymentionherethemostimportantones.First,asisoftenthecaseinCournot-likemodels,theremightexistdegenerateequilibria.Theﬁrstlemmaintheproofisdedicatedtoshowingthatitisnotthecase.Foranynumberofﬁrms,theyareallactiveinequilibrium.Then,wedemonstratethatthereareonlytwokindofcandidateequilibria(dependingonwhethertheconstraintsassociatedwiththeminimumqualitystandardarebinding),thathappentobesymmetric.Theycannotcoexist.Overall,weobtainexistenceanduniquenessoftheequilibrium,that,wheninterior,ischaracterizedbythetwoequations:nc(!n)Qn=(n+1)b(a%!)(7a)n1ac(!)c"(!n)=(+n)(7b)n+1n!nWithaslightabuseofnotation,wedenoteinthefollowingbyQnand!ntheequilibriumvaluesforthecompetitiveequilibriumwithnﬁrms.Proposition1Thegamehasauniquesymmetricequilibrium.Ithasthefollowingproper-:seit(i)Qualityisdecreasinginthenumberofcompetitors,andthereexistsN(!)suchthat:15Notethatifqualityisexogenouslysetatsomeuniformlevel,themodelcollapsestoastandardCournotoligopolywithhomogenousgoods,andthefree-ridinge!ectdisappearsinthatcase.11

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