704. Information Economics

LectureNote4.InformationEconomicsApril21,2010NSun(SHUFE)InformationEconomicsApril21,20101/278.1AdverseSelectionInformationandtheEfficiencyofMarketOutcomesTheBasicSettingsConsideramarketforautoinsuranceinwhichmanyinsurancecompaniessellinsurancetomanyconsumers.Eachconsumeri(=1,2,···,m)hasaaccidentprobabilityπi∈[0,1]oflosingLdollars,aninitialwealthωandacontinuous,strictlyincreasing,strictlyconcavevNMutilityofwealthfunctionu(·).IdenticalInsuranceCompaniesoffersforsalefullinsuranceatthepricep.Theexpectedprofitisp−πiLNSun(SHUFE)InformationEconomicsApril21,20102/27SymmetricInformationWheninformationissymmetric,theinsurancecompanycanidentifyeachconsumer’saccidentprobability.PoliciesbenefitingdistinctconsumersaredistinctcommoditiesandcouldcommanddistinctpriceLetpidenotethepriceofpolicypayingLdollarstoconsumeriwhenhehasanaccident.RefertothisastheithpolicyWewishtodeterminethecompetitiveequilibriumpricep∗iofpolicyi.NSun(SHUFE)InformationEconomicsApril21,20103/27TheEquilibriuminthecaseofsymmetricinformationOnthesupplyside,pi=πiListhecasewheninsurancecompaniesbrokeevenoneachpolicyiandarewillingtosupplyanynumberofsuchpoliciesOnthedemandside,wheneverinsuranceisfair(i.e.pi=πiL)risk-averseconsumerswillpreferfullinsurance.Ifpi πiLconsumeriwillpurchaseatmostonepolicyi.(MWGCh.6)Puttingdemandandsupplytogether,theonlypossibilityforequilibriumiswhenpi=πiL,eachconsumerconsumeonepolicy.Consumersarefullyinsuredandinsurancecompaniesearnzeroprofit.NSun(SHUFE)InformationEconomicsApril21,20104/27TheParetoEfficiencyofEquilibriumAnallocationisanassignmentofwealthtoconsumersandinsurancecompaniesineachstate,anallocationisfeasibleifineverystate,thetotalwealthassignedisequaltothetotalconsumerwealth.NofeasibleallocationParetodominatesthecompetitiveallocation:Bywayofcontradiction,supposeWLOGthecompetitiveallocationisdominatedbyafeasibleallocationinwhicheachconsumer’swealthisthesamewhetherornothehasanaccident:i.e.thewealthofconsumeriisfixedat¯wi.Fortheallocationtodominatethecompetitiveone,itmustbethecasethat¯wi≥w−πiLforeachi.NSun(SHUFE)InformationEconomicsApril21,20105/27TheParetoEfficiencyofEquilibriumWeconsideraninsurancecompany’sexpectedprofitfromconsumeriis(1−πi)(w−¯wi)+πi(w−L−¯wi)=w−πiL−¯wi.TheRHSoftheequationisnonpositive(asthelastslidesays).LetEPjidenotecompanyj’sexpectedprofitsfromconsumeri,thenwehave:w−πiL−¯wi=PjEPji≤0Sincecompaniesmustearnnonnegativeprofits,PiEPji≥0foreverycompanyj,soPiEPji=0,whichcontradictthedefinitionofadominatingallocation.So,thecompetitiveallocationisefficient.NSun(SHUFE)InformationEconomicsApril21,20106/27AsymmetricInformationandAdverseSelectionBasicSettings:Interval[π,π]containsthesetofaccidentprobabilities,Fisadistributionfunctionon[π,π]representingtheinsurancecompany’sinformation.foreachπ∈[π,π],F(π)denotesthefractionofconsumershavingaccidentprobabilitylessorequaltoπ.Allothersettingsareidenticaltothecaseofsymmetricinformation.Insurancecompaniessellonlyfullinsurance..NSun(SHUFE)InformationEconomicsApril21,20107/27TheBasicConclusionThereisasingleequilibriumpriceofthefullinsurancepolicyforallconsumersLetpdenotethesinglepriceofthefullinsurancepolicy,wenowwishtodeterminetheequilibriumpricep∗ANaturalGuessofSolution:p∗=E(π)L,whereE(π)=RπππdF(π)istheexpectedaccidentprobabilityOnlythosewithrelativelyhighaccidentprobabilitieswillchoosetopurchaseinsurance.E(π)willunderestimatetheaccidentprobability;Theconsumerbuysapolicyforpricepiif:u(w−p)≥πu(w−L)+(1−π)u(w),orπ≥u(w)−u(w−p)u(w)−u(w−L)≡h(p)NSun(SHUFE)InformationEconomicsApril21,20108/27p∗isacompetitiveequilibriumpriceunderasymmetricinformationifp∗=E(π|π≥h(p∗))LE(π|π≥h(p∗))L=Rπh(p∗)πdF(π)istheexpectedaccidentprobabilityconditionalonπ≥h(p∗)Aconsumerwithaccidentprobabilityπwillpurchasethefullinsurancepolicyatpricepaslongasπ≥h(p).Theconditionp∗=E(π|π≥h(p∗))Lensuresfirmsearnzeroexpectedprofitconditionalontheaccidentprobabilitiesofconsumerswhoactuallypurchasethepolicy.Sotheconditionabovesetthesupplyofpoliciesequaltothedemand,i.e.,itdescribeanequilibrium.NSun(SHUFE)InformationEconomicsApril21,20109/27TheExistenceofEquilibriumLetg(p)=E(π|π≥h(p))Lforeveryp∈[0,π],whereπisthehighestaccidentprobabilityamongallconsumers.Theconditionalexpectationiswell-definedbecauseh(p)≤πforeveryp∈[0,π].Inaddition,gmapstheinterval[0,πL]intoitself.Finally,sincehisstrictlyincreasinginp.So,gisnondecreasinginp.Thus,gisanondecreasingfunctionmappingaclosedintervalintoitself.Itmusthasafixedpointp∗∈[0,π].Thefixedpointisanequilibrium.NSun(SHUFE)InformationEconomicsApril21,201010/27TheEfficiencyPerspectiveoftheEquilibrium:Theequilibriummaybenotefficient:e.g.,ifFisuniformlydistributedover[π,π]=[0,1],theng(p)=(1+h(p))L2isstrictlyincreasingandstrictlyconcave.Sothere’sauniqueequilibriumpricep∗=g(p∗).Sinceh(L)=1,p∗=L,whichcombinedwithp∗=E(π|π≥h(p∗))L,impliesE(π|π≥h(L))=1,i.e.,allconsumerswillbeuninsuredexceptthosewhoarecertaintohaveanaccident.Clearlytheoutcomeisinefficient.Theinefficiencyliesinthefactthatraisingpricewillmakethepoolofconsumerscontinuingtobuyinsuranceriskierinaverage.NSun(SHUFE)InformationEconomicsApril21,201011/27SignalingIntheabove