Collective Reputation and Market Structure

Pierre Fleckinger

Description

Niveau: Supérieur, Licence, Bac+1
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.

• no traceability


• quality


• market structure


• also differ


• certification


• when quality


• firms choose


• average quality offered


• welfare-maximizing quantities

Sujets

• Bourgeon
• Tanguy
• Lecocq
• École polytechnique
• Parker's
• Quality
• Market structure

Informations

• Publié par: profil-ontoe-2012
• Nombre de lectures: 16
• Langue: English

Extrait

CollectiveReputationandMarketStructure: RegulatingtheQualityvsQuantityTrade-o! PierreFleckinger ! Thisversion:December2007 Abstract Betweenmarketunravelingandindividualreputationbuilding,marketsforexpe- riencegoodsoftenexhibitintermediatepatterns.Thispaperexploresthesituation inwhichconsumersknowtheaveragequalityo!eredbyasetofproducers,butnot thequalityofonegivenproduct.Afirstissueishowsuchanaggregatesignalshapes competitionwhenstrategicvariablesarequantityandquality.Theequilibriumwelfare functionisconvexinthenumberofcompetingfirmsasaconsequenceofdecreasing qualityandincreasingquantity.Asecondissueisregulation.Itispossibletotrade-o! quantityagainstqualitythroughanumberofregulatorytools.Amongone-instrument policies,entryandquantityregulationperformbetterthanprice-basedregulation. Policyimplicationsforprofessionalregulationinservicemarketsandproducers’or- ganisationsinagriculturearebrieflydiscussed. JEL: D4,L1,L43,Q13 Keywords: UncertainQuality,MarketStructure,CollectiveReputation,Minimum QualityStandard. ! ColumbiaUniversity,BusinessSchool&EcolePolytechnique,Laboratoired’Econome´trie,1rue Descartes,75005Paris.email: [e-mail] . Thispaperwasmostlywrittenwhentheauthorwasa"liatedtoINRA. IamgratefultoJean-MarcBourgeon,IgorMouraviev,Se´bastienLecocq,Herve´TanguyandEricGiraud- He´raudforcommentsanddiscussions.Thispaperalsobenefitedfromcommentsbyseminarparticipantsat INRA(Paris-Ivry)andAgroParisTechaswellasbyparticipantstothe23e`mesJMA(Nantes),EARIE2006 (Amsterdam)andEEA2007(Budapest).PartialsupportfromtheEcolePolytechniqueChairinBusiness Economicsisgratefullyacknowledged. 1

1Introduction SinceAkerlof(1970),economiststendtopredictthatwhenqualityisunknown,lowquality productsdrivehighqualityonesoutofthemarket.Consideraconsumer’s(she)buying decisionwhensheisnotsureaboutthequalityofthegood:sheisnotreadytopaymore thantheexpectedquality.Butsinceshedoesnotobservethequalitychoice,theseexpecta- tionsdonotdependonwhatthebuyeractuallydoes.Thenhehasnoincentivetoproduce (costly)quality.Thereforetheonlyequilibriumexpectationstheconsumercanholdisthat theproducersellsthelowestquality:marketforhighqualitythusunravels.Inthispaper,we studyhowmarketfunctionswhenconsumersdoknowaveragequalitypresentonthemar- ket,butnotthatofonegivenproduct.Thisrepresentsanintermediatesituationbetween theperfectinformationsettingandtheasymmetricinformationsetting.Theinformation publiclyavailableconsistsofaglobalassessmentofquality.This(true)averagequalityis theresultofthestrategicchoicesbymanyproducers. Undoubtfully,unravelingmaybepreventedwhentheconsumerrepeatedlypurchasesfrom thesamesupplier:afterexperiencingabadqualityproduct-orservice,shecandecideto stopbuyingit(andpotentiallybuyanotherone),whichdisciplinesthesupplier.Thisisthe essenceofbrandbuildingbyfirms,thatcanbemodeledbyarepeatedpurchasegame(Heal, 1976)andisreferredtoas’reputation’ingametheoreticalstudies(seeforexampleShapiro, 1982,1983).However,thissolutionrequiresfirstthattherelationshipislonglasting,and secondthattheproducingfirmisidentifiedateachpurchase.Thelatterconditionfailswhen identityoftheproducerislostintheretailingchain(e.g.thereisnotraceability),when thefrequencyofpurchasedislowenough(inparticularforspecializedservices)orwhenthe consumerhasboundedmemoryandlimitedcognitivecapacities.Asaleadingexampleof productsubjecttosuchrestrictions,considerthecaseofFrenchwines.Exceptforfamous Chateaux,therelevantinformationforthestandardconsumerboilsdowntotheregionand yearofproduction 1 .Whattheconsumerreferstoinsuchacaseisratherapublicknowledge thatthekindofwinehassomeaveragequality,afactthathasreceivedstrongempiricalsup- port(seeforexampleLandonandSmith,1998).Howsuchapublicsignalisgeneratedisnot thesubjectofthepaper,anditisalreadyathoroughlyinvestigatedtopic.Thissignalcan 1 Onthis,seetheevidenceinCombrisetal.(1997).IntheUnitedStatesandemergingwinecountries, thisinformationconsistsratherofbrandandtypeofvine. 2

beprovidedbyexperts 2 ,bycertificationbodies 3 orbydiscussionwithotherbuyersthrough variouskindsofconsumersnetworks,atopicyetunderstudiedbyeconomists 4 .Thispublic signalisdi!erentfroma”reputation”inthesensethatitisnotbasedonpastconsump- tionsofasimilarproduct,butinsteadreliesonexpertjudgment,orontheconsumption ofthesamegenerationofproduct(insteadofapastproductofthesamebrand)byother consumers.Inshort,theemergenceofsuchapublicknowledgeregardingaveragequality isnotinter-temporal,buthastodowiththeformationofqualityexpectationregardinga one-shotproduction 5 .Asconcernsexpertratings,theyevenpreemptpublicconsumption: Itisindeedtheveryroleofpremieresandjournalistsofspecializedpresstoprovidethe publicwithanevaluationofproductsbeforepurchasetakesplace.Again,thecaseofwine whereexpertsarethefirsttotasteandtogiveanoverallappreciationforagivenregionfor thecurrentyearisillustrative.Weassumeherethattheexpertsarereliable 6 ,sothatthe customerscanrelyonanobjectivereferencewhenbuyingtheexperiencegood. ThesettingunderstudyhereisaCournotoligopolywithendogenouslydi!erentiatedex- periencegoods.Producerschoosebothqualityandquantity,andconsumersknowaverage qualitydespitethefactthattheydonotidentifyproducersindividually.Animportantfocus isonthecomparativestaticswithrespectton,thenumberofcompetitors:Underrelatively mildassumptions,wedemonstrateequilibriumexistenceanduniquenessforanynumberof competitors.Whiletotalmarketedquantityisincreasinginn,qualityisdecreasing,yield- ingaU-shapedwelfareasafunctionofn.Thisimpliesthateitherperfectcompetitionor monopolyistheoptimalmarketstructure.Thisissobecauseofafree-ridinge!ectonaver- agequality:Essentially,theassumptionontheinformationstructuretransformstheadverse selectionproblemintoamoralhazardproblema`laHolmstro¨m(1982).Ontheonehand,a monopoly,orawell-functioningproducerorganisation,choosesthesociallyoptimalquality levelbecauseitisnotsubjecttothismoralhazardproblem:averagequalityrevealsdirectly theinvestmentofthemonopoly.Butontheotherhand,competitionincreasesthequantities 2 SeeinparticularAlietal.(2005)foranempiricalestimationofRobertParker’sgradesimpacton Bordeauxprices,andthereferencesthereinonthetopicofexpertratinginwine,bothatthebrandandat theregionallevel. 3 Amongothers,LernerandTirole(2006)andPeyracheandQuesada(2004)developmodelsinwhichthe certificationprocessisendogenous. 4 Forexample,Curienetal.(2004)emphasizetheimportanceofconsumers’onlinenetworks. 5 Itishoweverperfectlypossibletointerpretthesettingasacompactreputationmodel,embeddingthe dynamicse!ectsintheone-periodequilibrium. 6 Ofcourse,theseexpertsmaybesubjecttocapture(seeStrausz,2005,forarepresentativerecent contribution),butwewillabstractherefromthispossibility. 3

sold.Asecondresultisthatthequantitye!ectismorethano!setbythequalitye!ect: competitionhappenstobeharmfultotheconsumersiftheenforceablequalitystandardis relativelylow.Inthelimit,perfectcompetitioncandestroyallpotentialsurpluswhenno qualitystandardisenforced.Thisresultcanexplainthecreationofagriculturalsyndicates thatarenotfoughtbygovernments,orevenlegallyencouraged 7 .Thisresultmightalsoshed lightonprofessionalregulationssuchasinthemedicineandlawsectors. Withthetwobenchmarksoffirst-bestandcompetitivesituationsinmind,weundertake theanalysisofselectiveregulationofsuchmarkets.Thereisaroomforregulationunder anymarketstructure.Whenqualitystandardsareenforceableandstringentenough,perfect competitionisclosetotheParetooptimum-theusualconvergenceresultofCournotequi- libriumtowelfare-maximizingquantitiesholdwhenqualityis(almost)notaconcern.There ishoweveraroomforentryregulationwhenthestandardislow.Animportantmessage hereisthatqualitystandardsandcompetitionarecomplementaryundercollectivename. Finally,itisshownthatamongone-instrumentpolicies,thebestregulationtoolisquantity regulation,whichperformsbetterthanpriceregulationforanymarketstructure. 2Relationtotheliterature ThemodelisrelatedtothecollectivereputationmodelsofTirole(1996),WinfreeandMc- Cluskey(2005)andBourgeonandCoestier(2007),despitethestaticnatureofthepresent analysis.Indeed,thosepapersareconcernedwiththedynamicsoftheproblem,inabsence ofpublicsignal.Itturnsouthoweverthatsomeconclusionsareverysimilar,namelyunder- provisionofquality,althoughinadi!erentform.Whatmainlydistinguishesthecurrent analysisfromtheprecedingonesisthatwefocushereonmarketstructureandwelfareanal- ysis,thefirsttopicbeingalmostcompletelyabsentfromthementionedliterature 8 .The assumptionstherealsodi!erfromoursinmanyways.Tirole(1996)considersacasewhere agentsareofdi!erenttypes(honest,dishonest,orstrategic):asisstandardinreputation model(e.g.KrepsandWilson,1982),agentsareexogenouslydi!erent.Regardingthelit- eratureonexperiencegoods,MoavandNeeman(2005)areinterestedintheroleofthe inspectiontechnologywithtwoclassesofproducers.Finally,theinformationsettingstudied 7 SeeTitleIVoftheEuropeanCouncilRegulationNo1493/1999.Furtherdevelopmentsonthecaseof winecanalsobefoundinGiraud-He´raudetal.(2003). 8 Whiledealingwithadi!erentproblem-operatingaregulatednetwork-themodelofAuriol(1998)also featuresafree-ridinge!ectthatcanmake(regulated)duopolyworsethan(regulated)monopoly.However, theparallelendshere,becausethecoststructuresandregulationproblemsarequitedi!erent. 4

herealsodi!ersfromthatconsideredbyWolinsky(1983),whereconsumersgetindependent privatesignalonthequalityofeachdi!erentproduct. Intheliteratureconcerningagriculturalproducersorganisation,manypapersdealswith closeissues.Maretteetal.(1999)andMaretteandCrespi(2003)studycertificationandthe roleoftheseorganisations,modeledascartels 9 .Undertheassumptionofexogenousdiscrete quality,withconsumersformingexpectationsaboutthequalityofuncertifiedproducts,they showthatcartelssharingthecertificationcostandcolludingonquantitiescandobetter thatcompetitionfromasocialpointofview(intuitively,ithappenswhencertificationis individuallytoocostly).Also,AuriolandSchilizzi(2003)studiestheroleofthe(fixed)cer- tificationcostonmarketstructure.Firmschoosethesociallyoptimumqualitylevelassoon astheyseekcertification(beitpublicorprivate),andthusqualitydistortionscomesonly fromnon-certifiedfirm,thatproducethelowestquality.Intheirmodel,fixedcertification costhasthefollowingimplications:whenfirmsself-certifiedthemarketisoligopolistic,as expectedwithdecliningaveragecosts.Next,theyassumethatitispossiblenotoduplicate thesecertificationcosts,andcomparetopossiblecertificationarrangements:sharingcosts proportionatelytothequantitysoldforeachfirm,orpubliclyfunded.Concerningproduc- ersorganisations,WinfreeandMcCluskey(2005)dealwithcollectivereputationanduse Shapiro(1983)setting,sotheydonotmodelendogenouslearningofqualitybythecon- sumers:consumersdonotformBayesianexpectations.Insteadtheirbeliefsevolvethrough apre-specifiedMarkovianprocessratherthanthroughBayesianrevision.Inaddition,their paperonlydealswithproducerssurplus,withoutreferencetothewelfareimpact. Theassumptionofapublicsignalregardingaveragequalityallowsbothtoavoidadhoc learningbyconsumersandtotacklethestudyofoligopoly,that,asMilgromandRoberts mention,”involvesignificantadditionalproblems”(1986,footnote9)withrespecttothe monopolycasewhenqualityisendogenous.Thus,beyondtherealismembeddedinit,this assumptionalsoallowstogoonestepfurther,whilekeepinginareducedformtheproblem ofcollectivereputation.Italsoallowstotreatinaunifiedframeworkalltoolsaregulator maywanttouseandcomparetheire"ciency. Finally,onthetechnicalside,theresultonequilibriumuniquenessobtainedinthefirst 9 Zago(1999)developsamechanismdesignmodeltostudycollectivedecisionwithinproducersorganisa- tionsinarelatedcontext. 5

partofthispaperinvolvessomedi"cultiesthatexistingapproachescannotovercome.The rootofthisdi"cultiesliesontheonehandinthetwo-dimensionalstrategiesoffirmsandon theotherhandinthequalityexternality,afeaturewhichisknowntomakequasi-concavity breakdown.Infact,thestandardexistenceresultssimilartothatofRosen(1965),rely- ingondi!erentialcalculus 10 donotapplyduemainlytothelackquasi-concavityofthe profitfunctions.Thecontractionmappingapproachisnotappropriateherebecauseoftwo- dimensionalstrategyspace(seeLongandSoubeyran,2000,foranelegantresultregarding purequantitycompetition),neitherarethetechniquesofsupermodulargamesandrelated tools(seeVives,1999).Therefore,theproofsrequirespecifictreatmentsofthecaseunder study,butbythesametokenprovidespecificinsightsonthecollectivereputationmechanics inoligopoly. Inthenextsection,themodelisstatedanditsessentialfeaturesarepresented.Thefourth sectioncharacterizesthecompetitiveequilibrium.Thefifthsectionisinterestedinthewelfare propertiesofdi!erentmarketstructures.Finally,thesixthsectiondiscussregulationand policyimplicationswhilethelastoneconcludes.Allproofsarerelegatedtotheappendix. 3Model 3.1Theconsumers Weconsiderapopulationofconsumersthatdi!erthroughtheirtastetforquality!,where ! " [!,!] # (0,+ $ ).Consumers’tastes 11 aredistributedover[0,t]accordingtothecumu- lativepopulationweightF(t).FollowingMussaandRosen(1978),aconsumerwithtastet facingapricepderivesthefollowingutilityfrombuyingoneunitofagoodwithquality!: u(!,p;t)=!t%p andshewillbuy(exactlyoneunitof)thegoodifu(!,p;t)&0.Thequantitysoldistherefore p Q=F(t)%F( ! ),correspondingtothefollowinginversedemand: p=!F !1 F(t)%Q "! 10 SeealsoKolstadandMathiesen(1987);Novshek(1985)andmorerecentlyGaudetandSalant(1991). 11 Settingthelowesttasteat0yieldsaninversedemandfunctionthatdoesnotdiscontinuouslyvanishes atsomestrictlypositivequantity.MilgromandRoberts(1986),forexample,useasimplifiedversionofthis settingintheirsectionII. 6

Underthestatedpreferences,thedemandfunctionisthusmultiplicativelyseparablebetween thequalityandquantitye!ects.Wemakethefurtherassumptionthatthedistributionof tastesisuniform,withweightKateachpoint,sothat: p(!,Q)=!(a%bQ))1( witha'tandb' K 1 . AnattractivepropertyoftheMussaandRosenspecificationofutilityisthattheinverse demandfunction(1)remainsalsovalidwhenqualityisimperfectlyknown,inwhichcase !representsthentheexpectedquality 12 .Thisistruebecauseconsumers’utilityislinear in!.Inaddition,theassumptionofuniformdistributionalsohelpssimplifyingtheinverse demand-althoughthelinear-in-quantityformulationisillusorywhenitcomesthecaseof expectedquality,aswillbecomeclear.Muchoftheresultscouldbederivedwithoutthis linearfunctionalform,althoughatthecostsofmuchheaviertechnicalities. Weassumeasthoroughlyexplainedintheintroductionthattheconsumershavesomehard informationabouttheactualqualitysoldinthemarket,moreprecisely: Assumption1Averagequalityandtotalquantitypresentonthemarketarepublicsignals. Thisimpliesthatconsumersknowforsuretheaveragequalityonthemarket(”strawberries aregoodthisyear”,”mostlawyersinthiscityareo!erlowqualityservice”,andsoon), buttheyarestillunabletodistinguishthequalityofonepreciseproduct.Notethatthis assumptionliessomewherebetweenthecaseofperfectinformationoneachproductsoldand thepureBayesiancase,inwhichconsumersonlyformexpectations,basedonlyonstrategic considerationsandnotonpublicsignal. Thefactthatconsumersknowthetotalquantity(ratherthan,morerealistically,theprice), canbeseenasashortcutaccountingforaretailingstagethattransformsthequantity informationintoapriceinformation.Infact,onecoulddispensewiththeassumptionof publiclyknownquantity,atthecostofmoresophisticationinthepriceformationmechanism. Onecannoticethatisalsoequivalenttoassumethattheconsumersknowexactlywhateach producerdid,butcannotidentifyafterwardswhereagivenproductcomesfrom.Wenow turntotheproductionside,wherequalityandquantityarechosen. 12 Leland(1979)studiesarelatedmodel,butthemicro-foundationsarenotexplicit:heratherdirectly postulatesaprice-dependenceonexpectedquality,andinadditionhedoesnotuseimperfectcompetition asequilibriumconcept. 7

3.2Theproducers Therearenidenticalproducers,indexedbyi=1..n,thatchoosetheirquantityq i andquality ! i ,ataunitcostc(! i )q i ,wherecisstrictlyincreasingandstrictlyconvex,andsatisfiesthe followingconditions:c(0)=0,c " (0)=0andc " (!)=+ $ ,whichwillensurethatsome interiorqualitylevelisoptimal.Weimposealsothetechnicalassumptionc """ & 0,whose roleisexplainedwhenrelevant.Notethatwedefinethecostfunctionindependentlyofthe minimumqualitylevel,!,sothatwecanstudythee!ectofchangingtheminimumquality standardwhileholdingtheproductiontechnologyfixed.Thisdoesnotmeanthatitislegally feasibletoselltheuselessproductwithquality0;infactthelowestpossiblequality,!,which hastobeviewedasaminimumqualitystandard,isafundamentalpolicytoolwhoseimpact isstudiedbelow. Wechoosedeliberatelytoconsidercostsfunctionsthatarelinearinquantity.Indeed, tostudytheinterplayofqualityandquantityonthemarketstructure,anyothershapeof costs 13 withregardtoquantitieswouldbiastheoptimalmarketstructuretowardsoneor theotherdirection,i.e.monopolyorperfectcompetition.Withconcavequantitycosts, forexampleinthepresenceofafixedcost,amoreconcentratedmarketwouldbesocially preferred,whilewithconvexquantitycosts,spreadingthemamongaverylargenumberof producerswouldbedesirable.Asthefocusisontheinteractionbetweenquality,quantity andmarketstructure,wemustgetridofsuchbiasesbyneutralizingthosetechnological e!ects,andtheonlyfunctionalformallowingthatisthechosenmultiplicativeone. ThetotalquantityisdenotedQ=q i ,andtheaveragequalityonthemarketis: # != ! i q i (2) # Q Givenassumption1,thereisonesinglemarketprice.Becausetheconsumersareinformed ofthetrueaveragequality,thisequilibriumpricewillbegivenbyequation(1).Producer i’sprofitistherefore: " i ( { ! j ,q j } )=p(!,Q)q i % c(! i )q i (3) Inthefollowingwedenoteby#theproducersprofit,with#=" i . # 13 Forexample,KleinandLe#er(1981)havefixedcostsdependingonqualityandAllen(1984)hasboth convexmarginalcostsandfixedcosts,whichmakesthecostfunctionneitherconcavenorconvex.The presentcostfunctionisfoundforexampleinBesankoetal.(1987)andisratherstandardindi!erentiated oligopoliesmodels. 8

3.3WelfareOptimumandbenchmarkcase Inafirst-bestworld,abenevolentsocialplannercouldassigntoeachproduceraqual- ity/quantityplan,toserveapredeterminedsetofconsumers.Thatis,itwouldputinplace adouble-sideddiscrimination,withfirmsproducingdi!erentlevelsofquality,corresponding todi!erentmarketsegments.Inthelimit,withacontinuumoffirmsor,say,withfree-entry, thisamountstoapoint-wisematchingbetweenfirmsandconsumers,whoseoptimumis derivedfromapoint-wisemaximizationforeachconsumer’staste.Thequalitymenu!(t) wouldverify: c " (!(t))=t Forafinitenumberoffirms,thefirst-bestoptimumwouldconsistofapartitionofthecon- sumersintoqualitygroups,adiscreteapproximationoftheabovemenu. Butthatisnotfeasiblewhenconsumerscannotdiscriminatebetweenproducers,asstated byassumption1.Giventhatobservabilityassumption,theconsumersdonotidentifythe di!erentqualitylevels,whichamountstoconsideringasinglequalitycategory,averagedover thedi!erentproducers,andconsequentlythereisasingleassociatedprice.Thissituation willconstituteourbenchmarksecond-bestoptimum.HencewenowconsidertheMarshallian consumersurplusdefinedby 14 : Q $ U=p(!,Q)dQ % p(!,Q)Q(4) 0 Forthedemandfunctionspecified,thisexpressionremainsvalidwhenqualityisheteroge- nous,inwhichcase!istheaveragequality.Giventheconvexityofc,thecostforagiven levelofaveragequalityisminimizedwhenallproducerschooseexactlythatqualitylevel.In turn,sinceproducersareidenticalandcostsarelinearwithrespecttoquantity,allocationof productionbetweenthedi!erentproducersisunimportant.Thesocialoptimumistherefore givenbymaximizingover!andQthefollowingexpression: Q $ W=U+#=p(!,Q)dQ%c(!)Q 0 Thecorrespondingfirstorderconditionsare: ! # (a % bQ # )=c(! # ) % ! Q 0 (a%bQ)dQ=c " (! # )Q # & t 14 Itisofcourseequivalenttodefineitas t 0 u ( t,!,p ) dt ,where p denotestheequilibriumpriceand t 0 the & consumerindi!erentbetweenbuyingornot. 9

whichcanberestatedas: # Q # = 1 (a % c(!) )(5a) # !b # c " (! # )= 1 (a+ c(!) )(5b) # !2 Ofcourse,theproblemisinterestingif,first,anoptimallevelexists,thatis,equation(5b) hasasolution,and,second,thissolutionishigherthantheminimumqualitystandard!. Thenextlemmaclarifiesthis. Lemma1Assumec """ & 0andc " (!)< a .Then! # isuniqueand! # >!. 2 Onceagain,recallthatthisreferencecaseisnotafirst-bestsituation,butrathercon- stitutesthesociallyoptimalproductionplanundertheimperfectobservability.Inwhat follows,wewillkeepthisassumptions,alsotheyarenotneededforallresults. 4UnconstrainedCompetition Sincethereisacontinuumofconsumersandafinitenumberoffirms(howeverlargeitcan be),theconsumptionsideofthemarketisassumedperfectlycompetitive.Thereforethe firmsfacethedemandschedulein(1).Inturn,ontheproductionside,thereisimperfect competition:TheproducersplaytheNashequilibriumofthegamedefinedbystrategiesq i ,! i andpayo!sin(3).Anticipatingabitontheresults,weconsiderthefirst-orderconditions oftheprofit,forsomefirmi(thereare2nfirst-orderconditionsoverall). ! i Q !! (a % bQ)q i % b!q i +!(a % bQ) % c(! i )=0 % q Q i (a % bQ)q i % c " (! i )q i =0 Tograspsomeintuitiononwhatisgoingoninequilibrium,assumethattheseconditions areindeedsatisfied.Inspectionofthefirstequationisespeciallyinstructive,sowerewrite itasfollows: p(!,Q) % c(! i )=b!q i + ! % ! i q i p(!,Q)(6) Q! Theleft-handsideissimplypriceminusmarginalcost,andrepresentsthustheunitmargin. Theright-handsidepertainstomarketpower,anditdecomposesintotwoe!ects.The firsttermisclassicallyrelatedtotheelasticityofpricewithrespecttoquantity,asinany Cournotmodel.Thesecondtermisthekeystoneofthe”collectivequality”environment.It illustratesthequalitydilutione!ect,whichispositivewhenproducerichoosesalowerthan 01

averagequality,andthereforecorrespondstoafree-ridinge!ectonquality.Themagnitude ofthise!ectalsodependsonherelativesize( q Q i )oftheconsideredproducer,andonthe absolutevalueoffree-riding 15 ,namelytheprice. Asmentionedintheintroduction,thegameunderstudydoesnothaveasmoothstruc- ture.Clearly,themaindi"cultycomesfromthethetwo-dimensionalstrategies,oralter- natively,fromthepotentialdi!erentiationoftheproductsinequilibrium,whichmakesthe usualanalysiswithonesingledimensionfail.Indeed,fromonefirmpointofview,itisnot possibletodealwithonlyoneaggregatevariablesummarizingalltheotherfirms’behavior. Thisimpliesthatnogeneralresultsapplytoshowequilibriumexistenceanduniqueness.To beprecise,standardresultsforhomogeneousCournotcompetition(e.g.KolstadandMath- iesen,1987)havenobiteinthepresentcontext,norarethetechniquesinVives(1999,p. 47)applicable. Thedi"cultiesincharacterizingtheequilibriumarenumerous,andweonlymentionhere themostimportantones.First,asisoftenthecaseinCournot-likemodels,theremightexist degenerateequilibria.Thefirstlemmaintheproofisdedicatedtoshowingthatitisnot thecase.Foranynumberoffirms,theyareallactiveinequilibrium.Then,wedemonstrate thatthereareonlytwokindofcandidateequilibria(dependingonwhethertheconstraints associatedwiththeminimumqualitystandardarebinding),thathappentobesymmetric. Theycannotcoexist.Overall,weobtainexistenceanduniquenessoftheequilibrium,that, wheninterior,ischaracterizedbythetwoequations: nc(! n ) Q n = (n+1)b (a % ! )(7a) n 1ac(!) c " (! n )=(+ n )(7b) n+1n! n Withaslightabuseofnotation,wedenoteinthefollowingbyQ n and! n theequilibrium valuesforthecompetitiveequilibriumwithnfirms. Proposition1Thegamehasauniquesymmetricequilibrium.Ithasthefollowingproper- :seit (i)Qualityisdecreasinginthenumberofcompetitors,andthereexistsN(!)suchthat: 15 Notethatifqualityisexogenouslysetatsomeuniformlevel,themodelcollapsestoastandardCournot oligopolywithhomogenousgoods,andthefree-ridinge!ectdisappearsinthatcase. 11