Doespropitiousselectionexplainwhyriskierpeoplebuylessinsurance?Philipp - PDF document

Doespropitiousselectionexplainwhyriskierpeoplebuylessinsurance?PhilippeDeDonder,UniversityofToulouse(GREMAQ&IDEI)JeanHindriks,DepartmentofEconomicsandCORE,UniversitéCatholiquedeLouvain.March200621alléedeBrienne,31000Toulouse,France.Email:dedonder@cict.fr.Tel.:+33.561.128.603.Fax:+33.561.128.637 1IntroductionMosteconomicpapersoninsuranceassumesomeformofasymmetricinforma-tion.Theinsuredisassumedtoeitherhaveinformationthatisrelevanttothecontractbutthatisunknowntotheinsurer(adverseselection)ortobeabletoperformsomerelevantactionthatishiddentotheinsurer(moralhazard).Ineithercase,asymmetricinformationhasbeenshowntoproduceapositivecorrelationbetweenriskandinsurancedemandinequilibrium.Inthecaseofadverseselection,highriskindividualsbuymoreinsurancewhile,inmoralhazardsettings,agentswho,forsomeunexplainedreasons,pickupinsurancewithlesscoverageareencouragedtoexertmoreeort,resultinginalower(endogenous)risk.ChiapporiandSalanié(2000)stressthatthepositivecorrelationbetweenriskandinsurancedemandisfairlyrobustintheory.Forinstance,itdoesnotdependonthemarketstructure(perfectcompetitionormonopoly),noronspecicpropertiesofpreferences(suchassinglecrossing).However,thereisscantempiricalevidenceofsuchapositivecorrelation.ChiapporiandSalanié(2000)usevariousparametricandnonparametricmethodsandndnoevidenceofapositivecorrelationinaFrenchsurveyofautomobileinsurancecontracts.Saito(2003)comestothesameconclusionwithaJapanesedatasetalsoonautomobileinsurance.Onthecontrary,variousauthorsndevidenceofanegativecorrelation:deMezaandWebb(2001)forcreditcardpurchaseinsurance(alowerproportionofcreditcardsarereportedstolenamongthosethatareinsuredthanamongtheuninsured),CawleyandPhilipson(1999)forthelifeinsurancemarket(thosewhopurchaselifeinsurancehavealowerdeathratethanthosewhodonot,evenaftercontrollingforfactorssuchassmokingstatusandincome)andFinkelsteinandMcGarry(2003)forhealthcare(peoplewhospendmoreontheirownhealthbothbuymoreinsuranceandarelesslikelytouselong-termnursinghomecare).Inaseminalpaper,Hemenway(1990)hasproposedanexplanationforthisnegativecorrelation.Histheory,whichhecalls“propitiousselection”assumesthatpotentialinsurancebuyershavedierenttastesforriskandthattheyareconsistentintheirtasteforriskacrossphysicalandnancialdimensions.Toquotehim,“Theimplicationisthatindividualswhoarehighlyriskavoidingaremorelikelybothtotrytoreducethehazardandtopurchaseinsurance”(p.1064),resultinginanegativecorrelationbetweenriskandinsurancepurchases.ChiapporiandSalanié(2000)alsosuggestthatthislineofargument,whichtheycall“cherrypicking”,mayexplainwhythey SeealsoHemenway(1992). precautioncanproduceanincreasingmarginalwillingnesstopayforinsurance.WeprovideintheAppendixanexample,basedonYaari(1987)’sdualtheoryofchoiceunderriskforwhichtheregularity,single-crossingandmonotonicitypropertiesareallsatised,butwherethemarginalwillingnesstopayforinsuranceisincreasing(sincemarginalutilityofincomeisconstant)sothatweobtainapositivecorrelationbetweenriskandinsurancepurchases.ThemainmessageofthispaperisthusthatitisverydicultforthepropitiousselectionargumentsuggestedbyHemenway(1990)toproduceanegativecorrelationbetweenriskandinsurancepurchasesatequilibrium.Finally,thepropitiousselectionargumenthasalsoveryimportantnorma-tiveandpolicyimplicationsforsocialinsurance,evenwhenitgeneratesapositivecorrelationbetweenriskandinsurancepurchases.WeillustratethisbyshowingthattheintroductionofamandatorypoolingsocialinsuranceschemethatagentscancomplementwithprivateinsurancemaybeParetodamaging.Theintuitionforthisresultissimple:lessriskaverseagentsexhibitlowerriskinequilibrium,duetothefactthattheybuylessinsurance,andthussuerfrombeingpooledwithmoreriskyindividuals.Thisinturninducestheprivateinsurancemarkettoincreaseoverprovisiontoseparatethemoreriskaverseindividuals.Ifmoreriskaverseagentsarenumerousenough,theextracostofoverinsurancedominatesthecross-subsidybenesothattheyalsolosefromtheintroductionofsocialinsurance.2PropitiousselectionandprivateinsuranceWemodelaneconomywhereallindividualsfacethesamepotentialmonetaryloss.Westudyathree-stagegamewithtwotypesofindividualswhodionlyinriskaversion,whichisnotobservablebyinsuranceproviders.Intherststage,zero-protinsurancecontractsareoeredtotheagents.Theyconsistofapremiumandofacoveragerate,i.e.thefractionofthedamagethatwillbereimbursedbytheinsurerincasethedamageoccurs.Inthesecondstage,individualschoosewhich,ifany,privatecontracttobuy.Inthelaststage,theydecidehowmuchprecautiontoexert.Asusual,wesolvethisgamebackwards,startingwiththeprecautionchoice.Wemakeexplicitsomekeypropertiesunderlyingthepropitiousselectionargument,inordertoshowthattheyarenotsucienttoguaranteeanegativecorrelationbetweenriskandinsurancepurchases.Inthatsense,weidentifya(fatal)awinthisargumentwithoutevendisputingthekeypropertieswhicharethemselvescontroversial. exertmoreprecaution.Thisrequiresinparticularpartialinsurancebecausecompleteinsuranceinducesminimumpreventionandmaximumriskforallindividualsregardlessofriskaversion.Jullienetal.(2001)arguethatthemostappropriatewaytomodelthecostofprecautionistouseamonetarycostlike,sothatthemarginalrateofsubstitutionbetweenprecautionandwealthdoesnotdependontheshapeoftheutilityfunction.Withthismonetaryformulation,theyshowthattheregularitypropertydoesnotnecessarilyhold.Thereasonisthatprecautionreduceswealthevenintheeventofaloss,sothatmoreriskaverseagentsmaychoosetosaveonthecostofprecautiontoincreasetheirwealthinthebadstate.Obviously,forthiseecttoarise,thelossprobabilityhastobesucientlyhigh.2.2ThedemandforinsuranceWenowlookatindividualpreferencesoverinsurancecontracts.Plug-gingtheoptimalprecautionchoice(2)intheutilityfunction(1),weobtaintheindirectutilityfunctionoftypeforaninsurancecontractAcrucialquestionintermsofindirectpreferencesiswhethertheysatisfythesingle-crossingproperty,(i.e.whethermarginalwillingnesstopayforinsuranceismonotoneinrisk-aversion).Therearetwooppositeeectsatplay.Ontheonehand,ahigherrisk-aversionresultsinahigherwillingness-to-payforinsurancegiventhesamerisklevel.Ontheotherhand,moreriskaverseagentsmayexertmoreprecautionandendupbeinglessrisky,whichlowerstheirwillingnesstopayforinsurance.Thepropitiousselectionargumentassumesthat:Property2(Singlecrossing):Giventheoptimalprecautionchoice,foranyinsurancecontract,themarginalwillingnesstopayforinsuranceishigherforthemoreriskaverseindividual.Inwords,thepreferenceeectdominatestheriskeectonthewillingnesstopayforinsurance,foranygiveninsurancecontract,sothatthemoreriskaverseindividualsaremorewillingtopayforinsuranceeventhoughtheyfacelowerriskbybehavingmorecautiously.Jullienetal.(2001)showthatthesingle-crossingpropertyalwaysholdswiththemonetaryformulationofthecostofprecaution.DeMezaandWebb(2001)showthatthesingle-crossingconditionbetweenrisk-neutralandriskaverseindividualsmaynotbesatisedwithnon-monetarycostsofprecaution.Thelastpropertyunderlyingthepropitiousselectionargumentrelatestotheequilibriumdemandforinsuranceforeachtypewhentheinsurance EeckhoudtandGollier(2002)focusontheimpactofprudence(i.e.,apositivethirdderivativeoftheutilityfunction)onprecaution. Likewise,Figure1alsomakesclearthatapoolingcontractwithfullcoveragecannotbeanequilibrium,sinceitwouldbepossibletooeracontractwithlesscoveragethatattractsthemoreriskaversetype(possiblytogetherwiththelessriskaverse)andmakesapositiveproFigure2illustratestheexistenceofaseparatingequilibrium.Inthisequilibrium,thelessriskaverseindividualsobtaintheirmost-preferredcontract(i.e.point).Whenevertheincentivecompatibilityconstraintoftype-isbindingatequilibrium(whichisthecaseinFigure2sinceallfeasiblecontractspreferredbytype--theshadedarea-arealsopreferredbytype-individualstocontract),type-individualsareproposedafaircontractwithmorecoveragethantheywouldwish.Theintuitionforthisequilibriumisthatitisnecessarytooverprovideinsurancetothemoreriskaverseindividualsinordertoseparatethemfromthelessriskaverse.[InsertFigure2:Separatingequilibriumwithoverinsurance]Wearenowinpositiontomakeourcentralpointthatthecorrelationbetweenriskandinsurancedemandisaprioriambiguousatequilibrium.Ononehand,moreriskaverseindividualsarelessriskythanlessriskaverseindividualsiftheyallfacethesamecoveragerate.Ontheotherhand,intheseparatingequilibriummoreriskaverseagentsgetmoreinsurance,whichreducestheirrelativelevelofprecaution.Figure3showsthatmoreriskaverseindividualsareinfactmorerisky,inanyseparatingequilibrium,iftheindierencecurveofthelessriskaverseindividualsthroughthiscontractisconvex.[InsertFigure3:Relativerisksinseparatingequilibria]Thestraightlinegoingthroughpointshowsallfaircontractsattheequilibriumriskleveloftype-individuals.Theslopeofthislineislowerthantheslopeofatpoint(duetothemoralhazardinducedconvexity),whichisequaltotheslopeof’sindierencecurve(becausetype’smost-preferredfaircontract).Theconvexityoftogetherwiththefactthatliesbelowforallensurethattheindierencecurvethroughcrosses(atpoint)abovetheintersectionbetweenthelinethroughand.Itfollowsfromthefactthatanyequilibriumcontractfortypeisactuariallyfairandabove(orat)pointthattypeismoreriskythantypeatanyseparatingequilibrium,andthatrisksanddemandofinsurancearepositivelycorrelated.Wethenobtainthefollowresult. socialandcomplementaryprivateinsurance)onthehorizontalaxisandtotalpremiumontheverticalaxis.Weassumethat,withonlyprivateinsurance,type’sincentivecompatibilityconstraintisbindingatequilibrium,sothatthereisoverprovisionontheprivatemarket,withtype-’scontractatpointandtype-’scontractat[InsertFigure4:Paretoworseningpublicinsurance]Considerthatthesocialinsurancecoverageislessthanorequaltotheamountofprivateinsurancethetype-wouldbuyonthemarketinabsenceofanysocialinsurance.Sowedonotforcethelessriskaversetobuymoreinsurancethanhewouldlike(whichwouldotherwisehaveanextranegativewelfareeect).Therefore,type-individuals’totalcoverageisunchangedaftertheintroductionofsocialinsurance(perfectcrowdingout).However,socialinsuranceispushingupthepriceofinsuranceforthelessriskaverseindividualsbecauseinequilibriumthistypeislessriskyandthuscross-subsidizesthemoreriskytype(i.e.themoreriskaverseindividuals).InFigure4,type-’sequilibriumpointmovesfromwhensocialinsuranceisintroduced,sotheseindividualsaremadeworseo.Furthernotethatthisutilitylossincreaseswiththeextentofsocialinsuranceandwiththeproportionoftypeindividuals.Thisutilitylosswillincreasetheincentivefortype-toclaimthecontractintendedfortype-.Asaresult,theseparationofthetwotypesismademoredicult,whichincreasestheoverprovisiontothemoreriskaverseindividualsrequiredtoseparatethemout.Atthesametime,type-agentsbenetfromthecross-subsidizationfromtype-.Graphically,thenewequili-briumpointfortype-agents(point)islocatedtotherightofpoint(becauseofincreasedoverprovision),ontype-’sindierencecurvethrough(indicatingthattype-’sincentiveconstraintisbinding)andbelowtheactuariallyfaircontractcurvefortype-(becauseofthecross-subsidization).Notethattheverticaldistancebetweenisdecreasingwiththeproportionoftype-individuals.ItisclearfromFigure4that,iftheproportionoftype-islargeenough,thentheywillalsolosewhenintroducingsocialinsurance.Wethenhavethefollowingresult.Result2UnderProperties1-3,ifthemarginalwillingnesstopayforinsuranceofthelessriskaverseindividualsisincreasingwithcoveragerates,thentheintroductionofsocialinsuranceisParetoinferiorwhentheproportionofmoreriskaverseindividualsissucientlyhigh.Theintuitionforthisresultissimple:lessriskaverseagentshavealowerriskinequilibriumduetolowercoverageandthussuerfrombeingpooledwithmoreriskyindividuals.Thisinturninducestheprivateinsurance Appendix:Exampleofpropitiousselectionwiththedualtheoryofchoiceunderrisk.Wemodelindividuals’riskpreferencesusingYaari(1987)’stheory,whichisdualtotheexpectedutilitytheoryinthesensethatitislinearinwealthbutnonlinearinprobabilities.Thisformulationallowsustoseparateattitudetowardsriskfromattitudetowardswealth:withYaari’sapproach,riskavers-ionisentirelydrivenbyatransformationofprobabilitieswherebybadoutcomesaregivenmoreweightwhilegoodoutcomesaregivenlessweight.FollowingDeDonderandHindriks(2003),theutilityfunctionofanindividualfacedwithalossprobabilityandwhobuystheinsurancecontract)=(1++(1(1+(1+wheredenoteshisexogenousincomeandishisriskaversionparame-ter.Inwords,theindividualoverestimatesbyafractionhisexpectednancialdamage.Theutilitywithoutinsuranceis(1+andthereservationpremiumis)=(1+Thereforeriskaversioninourmodeltakestheformofarelativemarkupovertheactuariallyfairprice.rstsolvefortheoptimalprecautionchoicegiventheinsurancecontractbought:(1+Weassumethat 21 forr,1]and=(1+(with ).Theoptimalprecautionchoicefortype 21 andthecorrespondingriskfortype 2) 4[1 4,1 2]. 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Figure2:Separatingequilibriumwithoverinsurance 1•HvLv • A Figure4:Paretoworseningsocialinsurance 1•Lv•A Lv •