UnravelingversusUnraveling:AMemoonCompetitiveEquilibriumsandTradeinIns - PDF document

UnravelingversusUnraveling:AMemoonCompetitiveEquilibriumsandTradeinInsuranceMarketsNathanielHendrenJanuary,2014AbstractBothAkerlof(1970)andRothschildandStiglitz(1976)showthatinsurancemarketsmayun-ravel.Thismemoclariesthedistinctionbetweenthesetwonotionsofunravelinginthecontextofabinarylossmodelofinsurance.Ishowthatthetwoconceptsaremutuallyexclusiveoccurrences.Moreover,Iprovidearegularityconditionunderwhichthetwoconceptsareexhaustiveofthesetofpossibleoccurrencesinthemodel.Akerlofunravelingcharacterizeswhentherearenogainstotrade;RothschildandStiglitzunravelingshowsthatthestandardnotionofcompetition(purestrategyNashequilibrium)isinadequatetodescribetheworkingsofinsurancemarketswhentherearegainstotrade.1IntroductionAkerlof(1970)andRothschildandStiglitz(1976)havecontributedgreatlytotheunderstandingofthepotentialproblemsposedbyprivateinformationontheworkingsofinsurancemarkets.Akerlof(1970)showshowprivateinformationcanleadtoanequilibriumofmarketunraveling,sothattheonlyuniqueequilibriumisoneinwhichonlytheworstqualitygood(i.e.thelemons)aretraded.RothschildandStiglitz(1976)showthatprivateinformationcanleadtoanunravelingofmarketequilibrium,inwhichno(purestrategy)competitiveequilibriumexistsbecauseinsurancecompanieshavetheincentivetomodifytheircontractstocreamskimthelower-riskagentsfromotherrms.Whilethetermunravelinghasbeenusedtodescribebothofthesephenomena,thedistinctionbetweenthesetwoconceptsisoftenunclear,arguablyaresultofeachpaper'sdierentapproachtomodelingtheenvironment.Akerlof(1970)worksinthecontextofasupplyanddemandenvironmentwithaxedcontractorasset(e.g.ausedcar),whereasRothschildandStiglitz(1976)workinthecontextofendogenouscontractsinastylizedenvironmentwithonlytwotypes(e.g.highandlowtypes).Thismemodevelopsageneralizedbinarylossinsurancemodelthatincorporatestheforceshigh-lightedinbothAkerlof(1970)andRothschildandStiglitz(1976).Usingthisuniedmodel,Ishowthattheequilibriumofmarketunraveling(inAkerlof)isamutuallyexclusiveoccurrencefromtheunravelingofmarketequilibrium(inRothschildandStiglitz).Moreover,undertheregularitycondi-tionthatthetypedistributioneither(a)containsacontinuousintervalor(b)includesp=1,oneof1 thesetwoeventsmustoccur:eitherthereisaCompetitive(Nash)Equilibriumofnotrade(Akerlofunraveling)oraCompetitive(Nash)Equilibriumdoesnotexist(RothschildandStiglitzunraveling).Thus,notonlyarethesetwoconceptsofunravelingdierent,buttheyaremutuallyexclusiveandgenericallyexhaustiveofthepotentialoccurrencesinaninsurancemarketwithprivateinformation.Themutualexclusivityresultismoreorlessobviousinthecanonicaltwo-typebinarylossmodel.ThemarketunravelsalaRothschildandStiglitzwhenthelowtypehasanincentivetocross-subsidizethehightypeinordertoobtainamorepreferredallocation.ThiswillingnessofthegoodrisktosubsidizethebadriskispreciselywhatensuresthemarketwillnotunravelalaAkerlof.Conversely,ifthemarketunravelsalaAkerlof,thenthegoodriskisnotwillingtosubsidizethebadrisk,whichimpliesanabsenceoftheforcesthatdrivenon-existenceinRothschildandStiglitz.Theintuitionfortheexhaustiveresultisalsostraightforward,butperhapsmorediculttoseeinthecontextofthestylizedtwo-typemodel.Whenthesupportofthetypedistributioneither(a)containsanintervalor(b)containsthepointp=1,thentradenecessarilyinvolvescross-subsidizationoftypes.1But,RothschildandStiglitz(1976)showthatacompetitive(Nash)equilibriumcannotsustainsuchcross-subsidization.Hence,ifagentsarewillingtoprovidetradethenthemarketunravelsalaRothschildandStiglitz.Incontrast,ifnoagentsarewillingtocross-subsidizetheworserisksinthepopulation,thenthereexistsauniqueNashequilibriumattheendowment:nooneonthemarginiswillingtopaytheaveragecostofworserisks,andanypotentialcontract(ormenuofcontracts)unravelsalaAkerlof(1970).Thelogiccanbeseeninthecanonicaltwo-typescase.Here,theregularityconditionrequiresonetoassumethatthebadriskwillexperiencethelosswithcertainty.Theonlywayforthelowtype(goodrisk)toobtainanallocationotherthanherendowmentistosubsidizethehightype(badrisk)awayfromherendowment.Ifthelowtypeiswillingtodoso,theequilibriumunravelsalaRothschildandStiglitz.Ifthelowtypeisunwillingtodoso,theequilibriumunravelsalaAkerlof.Inthetwotypemodel,theassumptionthatthebadriskexperiencesthelosswithcertaintyisclearlyrestrictive.However,formoregeneraltypedistributionsbeyondthetwo-typecasetheregularityconditionisquiteweak.Anydistributioncanbeapproximatedquitewellbydistributionsthathavecontinuouslydistributedregionsorbydistributionswithanarbitrarilysmallamountofmassatp=1.Inthissense,theexistenceofpurestrategycompetitiveequilibriaofthetypefoundbyRothschildandStiglitz(1976)thatyieldoutcomesotherthantheendowmentisaknife-edgeresult.Thishighlightstheimportanceofrecentandfutureworktoaidinourunderstandingofhowbesttomodelcompetitionininsurancemarkets. 1Asdiscussedbelow,Riley(1979)showsthisistrueinthecasewhenthesupportcontainsaninterval;Ishowbelowthisisalsothecasewhenthesupportisdiscretebutincludesthepointp=1.2 2ModelAgentshavewealthwandfaceapotentiallossofsizelwhichoccurswithprobabilityp,whichisdistributedinthepopulationaccordingtothec.d.f.F(p)withsupport .2IncontrasttoRothschildandStiglitz(1976),IdonotimposeanyrestrictionsonF(p).3Itmaybecontinuous,discrete,ormixed.IletPdenotetherandomvariablewithc.d.f.F(p),sothatrealizationsofParedenotedwithlower-casep.AgentsoftypephavevNMpreferencesgivenbypu(cL)+(1�p)u(cNL)whereuisincreasingandstrictlyconcave,cL(cNL)isconsumptionintheeventof(no)loss.Ideneanallocationtobeasetofconsumptionbundles,cLandcNL,foreachtypep2 ,A=fcL(p);cNL(p)gp2 .Iassumethereexistsalargesetofrisk-neutralinsurancecompanies,J,whicheachcanoermenusofcontractsAj=ncjL(p);cjNL(p)op2 tomaximizeexpectedprots4.FollowingRothschildandStiglitz(1976),IdeneaCompetitiveNashEquilibriumasanequilibriumofatwostagegame.Intherststage,insurancecompaniesoercontractmenus,Aj.Inthesecondstage,agentsobservethetotalsetofconsumptionbundlesoeredinthemarket,AU=[j2JAj,andchoosethebundlewhichmaximizestheirutility.Theoutcomeofthisgamecanbedescribedasanallocationwhichsatisesthefollowingconstraints.Denition1.AnallocationA=fcL(p);cNL(p)gp2 isaCompetitiveNashEquilibriumif1.Amakesnon-negativeprotsp2 [p(w�l�cL(p))+(1�p)(w�cNL(p))]dF(p)02.Aisincentivecompatiblepu(cL(p))+(1�p)u(cNL(p))pu(cL(~p))+(1�p)u(cNL(~p))8p;~p2 3.Aisindividuallyrationalpu(cL(p))+(1�p)u(cNL(p))pu(w�l)+(1�p)u(w)8p2 2ThemodelisadaptedfromHendren(ming),whichderivestheno-tradeconditionanalogueofAkerlofinthebinarylossenvironmentbutdoesnotprovideanydiscussionofcompetitiveequilibriums.3Tomyknowledge,Riley(1979)wastherstpapertodiscussthisenvironmentwithacontinuumoftypes.4IncontrasttoRothschildandStiglitz(1976),Iallowtheinsurancecompaniestooermenusofconsumptionbundles,consistentwiththereal-worldobservationthatinsurancecompaniesoerapplicantsmenusofpremiumsanddeductibles.3 4.Ahasnoprotabledeviations:Forany^A=f^cL(p);^cNL(p)gp2 ,itmustbethatp2D(^A)[p(w�l�cL(p))+(1�p)(w�cNL(p))]dF(p)0whereD^A=p2 jmax^pfpu(^cL(^p))+(1�p)u(^cNL(^p))g�pu(cL(p))+(1�p)u(cNL(p))TherstthreeconstraintsrequirethataCompetitiveNashEquilibriummustyieldnon-negativeprots,mustbeincentivecompatible,andmustbeindividuallyrational.Thelastconstraintrulesouttheexistenceofprotabledeviationsbyinsurancecompanies.ForAtobeacompetitiveequilibrium,therecannotexistanotherallocationthataninsurancecompanycouldoerandmakepositiveprotsonthe(sub)setofpeoplewhowouldselectthenewallocation(givenbyD^A).2.1MutuallyExclusiveOccurrencesIrstshowthat,inthismodel,theinsurancemarkethasthepotentialtounravelinthesenseofAkerlof(1970).Theorem1.Theendowment,f(w;w�l)gp2 ,istheuniqueCompetitiveNashEquilibriumifandonlyifp 1�pu0(w�l) u0(w)E[PjPp] 1�E[PjPp]8p2 nf1g(1)Proof.Theno-tradetheoremofHendren(ming)showsthatCondition(1)characterizeswhentheendowmentistheonlyallocationsatisfyingincentivecompatibility,individualrationality,andnon-negativeprots.Now,supposeA=f(w�l;l)gandconsideranyallocation,^A6=f(w�l;l)gp2 .Suppose^Adeliverspositiveprots.BecauseAistheendowment,IcanWLOGassumeallagentschoose^A(since^Acanprovidetheendowmenttotypespatnocost).Butthen^AwouldbeanallocationotherthanAsatisfyingincentivecompatibility,individualrationality,andnon-negativeprots,contradictingtheno-tradetheoremofHendren(ming). ThemarketunravelsalaAkerlof(1970)ifandonlyifnooneiswillingtopaythepooledcostofworserisksinordertoobtainsomeinsurance.ThisispreciselythelogicofAkerlof(1970)butprovidedinanenvironmentwithanendogenouscontractspace.WhenCondition(1)holds,nocontractormenuofcontractscanbetradedbecausetheywouldnotdeliverpositiveprotsgiventhesetofrisksthatwouldbeattractedtothecontract.ThisispreciselytheunravelingintuitionprovidedinAkerlof(1970)inwhichthedemandcurvelieseverywherebelowtheaveragecostcurve.Noticethatwhenthisno-tradeconditionholds,theendowmentisindeedaNashequilibrium.Sincenooneiswillingtopaythepooledcostofworseriskstoobtaininsurance,thereexistnoprotabledeviationsforinsurancecompaniestobreaktheendowmentasanequilibrium.4 Theorem1alsoshowsthatwhenevertheno-tradeconditionholds,theremustexistaCompetitiveNashEquilibrium.Thus,wheneverthemarketunravelsalaAkerlof(1970),thecompetitiveequilib-riumcannotunravelalaRothschildandStiglitz(1976).UnravelinginthesenseofAkerlof(1970)isamutuallyexclusiveoccurrencefromunravelinginthesenseofRothschildandStiglitz(1976).Two-typecaseTorelatetopreviousliterature,itishelpfultoillustratehowTheorem1worksinthecanonicaltwo-typemodelofRothschildandStiglitz(1976).So,let =pL;pH withpH�pLdenotethetypespaceandletdenotethefractionoftypespH.WhenpH1,Corollary1ofHendren(ming)showsthatthemarketcannotunravelalaAkerlof.5Hence,themutualexclusivityofAkerlofandRothschildandStiglitzholdstrivially.But,whenpH=1,thesituationisperhapsmoreinteresting.Toseethis,Figure1replicatesthecanonicalRothschildandStiglitz(1976)graphsinthecasewhenpH=1.Theverticalaxisisconsumptionintheeventofaloss,cL;thehorizontalaxisisconsumptionintheeventofnoloss,cNL.Point1istheendowmentfw�l;wg.BecausepH=1,thehorizontallinerunningthroughtheendowmentrepresentsboththeindierencecurveoftypepHandtheactuariallyfairlinefortypepH.NoticethattypepHprefersanyallocationbundlethatliesabovethisline(intuitively,shecaresonlyaboutconsumptionintheeventofaloss).Thelowtypeindierencecurverunsthroughtheendowment(point1)andintersectsthe45-degreelineparalleltoheractuariallyfairline.AsnotedbyRothschildandStiglitz(1976),theoutcomesinthisenvironmentdependcruciallyonthefractionoflowversushightypes.Figure1illustratesthetwocases.IftherearefewpHtypes(issmall),thenpoint2isafeasiblepoolingdeviationfromtheendowment.Whensuchadeviationisfeasible,unravelingalaAkerlofdoesnotoccur:thelowtypeiswillingtopaythepooledcostoftheworserisks.But,theexistenceofsuchadeviationispreciselywhatbreakstheexistenceofacompetitiveequilibriuminRothschildandStiglitz(1976).Point2involvespoolingacrosstypesandcannotbeacompetitiveequilibrium.Hence,ifthemarketunravelsalaAkerlof,theendowmentistheuniquecompetitiveequilibrium.IfthemarketunravelsalaRothschildandStiglitz,thereexistsimplementableallocationsotherthantheendowmentandAkerlof'snotionofunravelingdoesnotoccur.AsonemightgatherfromFigure1,whentypesarearbitrarilycloseto1,theonlyfeasiblecom-petitiveequilibriumistheendowmenti.e.thereisnopossibilityofapairofseparatingcontractswiththepL-typereceivingpartialcoverageinequilibrium.InowmakethispointinthemoregeneralsettingthatdoesnotrequireanymassoftypesatpH=1. 5IfpH1,thenequation(1)wouldbeviolatedatpH=1bytheassumptionofstrictconcavityofu.5 2.2ExhaustiveOccurrencesInowshowthatnotonlyarethesetwonotionsofunravelingmutuallyexclusive,buttheyarealsoexhaustiveofthepossibilitiesthatcanoccurinmodelenvironmentswhenthetypedistributionsatisesthefollowingregularitycondition.Assumption1.Either(a)thereexistsabsuchthat[a;b] or(b)12 (i.e.F(p)1forallp1).Assumption1assumesthatthesupportofthetypedistributionincludeseither(a)acontinuousintervalor(b)thepointp=1.Noteanydistributioncanbeapproximatedarbitrarilycloselybydistributionssatisfyingthisregularitycondition.Inowshowthatcompetitiveequilibriumscannotsustaincross-subsidization,aninsightinitiallyprovidedinRothschildandStiglitz(1976).Lemma1.(RothschildandStiglitz(1976))SupposeAisaCompetitiveNashEquilibrium.ThenpcL(p)+(1�p)cNL(p)=w�pl8p2 Proof.SeeRothschildandStiglitz(1976)forafulldiscussion.Clearly,competitionrequireszeroprotsonanyconsumptionbundle.Hence,itsucestoshowthatnoallocationcanpooltypesintothesameconsumptionbundleotherthantheendowment.Supposemultipletypesareallocatedtothesameconsumptionbundle(distinctfromtheendowment).Then,aninsurancecompanycouldoeranewallocation,^A,arbitrarilyclosetothecurrentallocationbutthatisonlypreferredbythelowestpinthepool.Hence,thisallocationwillprovidestrictlypositiveprotsandwillrendertheoriginalconsumptionbundleunprotable,therebybreakingtheNashequilibriumwithpooling.Therefore,pcL(p)+(1�p)cNL(p)=w�plforallp. 6 Now,considerthetwocasesinAssumption1.Ifp=1isinthesupportofthetypedistribution,itisstraightforwardtoseethattherecannotexistanyCompetitiveNashEquilibriumotherthantheendowment,sincetraderequirescross-subsidizationtowardtypesnearp=1.Now,supposethatp=1isnotinthesupportofthetypedistributionbutthatthetypedistributioncontainsaninterval.Here,thenon-existenceofaNashequilibriumisperhapslessstraightforward,butaproofisactuallycontainedinRiley(1979)6.Giventheinterval[a;b] ,Riley'sderivationsshowthatthereexistsaprotabledeviationwhichpoolstypesnearp=b;hencetherecanbenoNashequilibriumotherthantheendowment.Theorem2follows.Theorem2.SupposeAssumption1holds.Then,thereexistsaCompetitiveNashEquilibriumifandonlyifCondition(1)holds.WhenAssumption1holds,traderequiresrisktypestobewillingtoenterriskpoolswhichpoolex-anteheterogeneoustypes.Suchex-antepoolingisnotpossibleinaCompetitiveNashEquilibrium.So,whentheno-tradecondition(1)doesnothold,theredoesnotexistanyCompetitiveNashEquilibrium:theequilibriumunravelsalaRothschildandStiglitz(1976).3ConclusionThismemousesageneralizedbinarymodelofinsurancetohighlightthedistinctionbetweenAkerlof'snotionofunraveling,inwhichanequilibriumexistsinwhichnotradecanoccur,andRothschildandStiglitz'notionofunraveling,inwhichastandardnotionofcompetitiveequilibrium(purestrategyNash)cannotexist.Inthelattercase,thereare(Pareto)gainstotrade;butinagenericsensedescribedinAssumption1,therealizationofthesegainstotraderequirecross-subsidizationoftypes.Suchcross-subsidizationcannotbesustainedunderthecanonicalnotionofcompetition.7Hence,Akerlofunravelingshowswhenprivateinformationcanleadtotheabsenceoftradeininsurancemarkets.RothschildandStiglitzunravelingshowsthatthecanonicalmodelofcompetition(Nashequilibrium)isinadequatetodescribethebehaviorofinsurancecompaniesinsettingswheretherearepotentialgainstotrade.ReferencesAkerlof,G.(1970).Themarketforlemons:Qualitativeuncertaintyandthemarketmechanism.Quarterlyjournalofeconomics84(3),488500.Hendren,N.(Forthcoming).Privateinformationandinsurancerejections.Econometrica.Miyazaki,H.(1977).Theratraceandinternallabormarkets.TheBellJournalofEconomics8(2),394418. 6SeeTheorem3inSection4,pages341-37Inthemodiedmodelsofcompetition,proposedbyMiyazaki(1977),Wilson(1977),orSpence(1978),suchgainstotradewillberealizedinequilibrium.7 Riley,J.(1979).Informationalequilibrium.Econometrica47(2),331359.Rothschild,M.andJ.Stiglitz(1976).Equilibriumincompetitiveinsurancemarkets:Anessayontheeconomicsofimperfectinformation.TheQuarterlyJournalofEconomics,629649.Spence,A.(1978).Productdierentiationandperformanceininsurancemarkets.JournalofPublicEconomics10(3),427447.Wilson,C.(1977).Amodelofinsurancemarketswithincompleteinformation.JournalofEconomicTheory16(2),167207.8