RUHRECONOMIC PAPERS - PDF document

RUHRECONOMIC PAPERS Like Father, Like Son:Inheriting and Bequeathing Lars Kunze Imprint Ruhr Economic Papers Published byRuhr-Universität Bochum (RUB), Department of EconomicsUniversitätsstr. 150, 44801 Bochum, GermanyTechnische Universität Dortmund, Department of Economic and Social SciencesVogelpothsweg 87, 44227 Dortmund, GermanyUniversität Duisburg-Essen, Department of EconomicsUniversitätsstr. 12, 45117 Essen, GermanyRheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI)Hohenzollernstr. 1-3, 45128 Essen, GermanyEditors Prof. Dr. Thomas K. BauerRUB, Department of Economics, Empirical EconomicsPhone: +49 (0) 234/3228341, e-mail: thomas.bauer@rub.deProf. Dr. Wolfgang LeiningerTechnische Universität Dortmund, Department of Economic and Social SciencesEconomics – MicroeconomicsPhone: +49 (0) 231/755-3297, email: W.Leininger@wiso.uni-dortmund.deProf. Dr. Volker ClausenUniversity of Duisburg-Essen, Department of EconomicsInternational EconomicsPhone: +49 (0) 201/183-3655, e-mail: vclausen@vwl.uni-due.deProf. Dr. Christoph M. SchmidtRWI, Phone: +49 (0) 201/81 49-227, e-mail: christoph.schmidt@rwi-essen.deEditorial O ce Joachim SchmidtRWI, Phone: +49 (0) 201/81 49-292, e-mail: joachim.schmidt@rwi-essen.deRuhr Economic Papers # Responsible Editor: Wolfgang LeiningerAll rights reserved. Bochum, Dortmund, Duisburg, Essen, Germany, 2012ISSN 1864-4872 (online) – ISBN 978-3-86788-367-2The working papers published in the Series constitute work in progress circulated to stimulate discussion and critical comments. Views expressed represent exclusively the authors’ own opinions and do not necessarily re ect those of the editors. Ruhr Economic Papers #318Lars KunzeLike Father, Like Son:Inheriting and Bequeathing sche Informationen der Deutschen Nationalbibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der deutschen National-bibliogra e; detaillierte bibliogra sche Daten sind im Internet über: http://dnb.d-nb.de abru ar.http://dx.doi.org/10.4419/86788367ISSN 1864-4872 (online)ISBN 978-3-86788-367-2 Lars KunzeLike Father, Like Son:Inheriting and BequeathingAbstractEmpirical evidence suggests that parents who have themselves inherited from their own parents are more likely to leave an estate to their children even after controlling for income, wealth and education. This implies an indirect reciprocal behavior between three generations by transmitting the attitude towards bequeathing from one generation to the next. We incorporate such an intergenerational chain into an overlapping generations model and show that the economy might be characterized by multiple steady states involving poverty traps. Individuals will not leave bequests unless per capita income levels exceed a threshold level. In such a situation, an unfunded social security security programme may help to overcome poverty by providing additional old age income out of which to bequeath.JEL ClassiÞ cation: D64, D91, H55Keywords: Capital accumulation; indirect reciprocity; overlapping generations; unfunded social securityFebruary 2012 1 TU Dortmund. – I am grateful to Sebastian Garmann, Falko Jüssen, Wolfram F. Richter and to Nicolai Suppa for helpful comments and suggestions. Any remaining errors are mine. – All correspondence to: Lars Kunze, Economics Department (Public Economics), Technische Universität Dortmund, Vogelpothsweg 78, 44227 Dortmund, Germany, E-Mail: lars.kunze@tu-dortmund.de. 1IntroductionThereisalongstandingdebateintheeconomicliteratureabouttherelativeim-portanceofbequestsincapitalaccumulation.Empiricalevidencesuggeststhatatleast80%ofUScapitalstockisaccumulatedthroughintergenerationaltrans-fers(KotlikoandSummers,1981),andthat50%ofindividualssaveinordertoleaveanestate(LaitnerandJustner,1996).Fromatheoreticalpointofview,severalbequestmotiveshavebeenproposedtoexplainsuchaltruisticallymoti-vatedtransfers,e.g.parentsmaycareabouttheirdirectdescendantsutility(Barro,1974),theirdescendantsincomelevel(Lambrechtetal.,2006)oraboutthebequest”owitself(Andreoni,1989).Inthesemodels,altruismistypicallyexogenouslygiven,thusneglectingitssourceandevolutionarydevelopmentHowever,itisbynowwellacceptedthatpreferences,normsandculturalattitudesarepartlyformedastheresultofheritablegenetictraits(seee.g.Heckman(2006)),andpartlytransmittedthroughgenerationsbyalearningandsocializationprocessortheimitationofrolemodels(seee.g.BisinandVerdier(2006,2011)).Withregardtoaltruismandbequeathingbehavior,em-piricalevidencesuggeststhatinheritancesandintendedbequestsareindeedpositivelyandsigni“cantlyrelatedevenaftercontrollingforanumberofhouse-holdcharacteristics,mostimportantlyhouseholdnetworth(CoxandStark,2005;ArrondelandGrange,2007).ThispointisillustratedinFigure1,whichpresentsnonparametricregressionresultsofthebinaryvariableforbequestin-tentiononhouseholdnetworth.Clearly,thereisastrongerintenttobequeathamonginheritorsthanamongnon-inheritors[InsertFigure1here]Havingreceivedaninheritancefromonesparentsmaythereforenotonlyincreasetheprobabilityofleavingabequestbyenhancingthedisposablenetwealthbutalsobyshapingtheattitudetowardsthepleasureofgivingandtowardsleavingabequesttoonesownchildren.Therelationshipbetweenre-ceivedinheritanceandonesownbequesthaslikewisebeenstudiedbyArrondeletal.(1997)andArrondelandGrange(2006)who“ndthattheexistenceoftransfersreceivedfromonesparentsincreasestheprobabilitytomakethesamekindoftransfertooneschildren.Consequently,transmissionpatternsandthe Note,however,thatrecentestimatesaresomewhatmoremoderate,seee.g.DeLong(2005,Fig.2-1)whoestimatestheshareofbequestintotalwealthtobe43%.ForacomprehensivesurveyofdierentaltruisticbequestmotivesseeMicheletal.(2006).AnotableexceptionisthestudybyRapoportandVidal(2007)whichexaminestherelationshipbetweenendogenousintergenerationalaltruismandlongrungrowth.Obviously,thisconclusiondoesnotholdforindividualsinthelowestquartile.However,forthissubgroup,thereareonlyfewobservationsofindividualshavingactuallyinheritedanddierencesinthebequeathing-intentionarenotstatisticallysigni“cant.Furthermore,ArrondelandMasson(2006)andWilhelmetal.(2008)showthatgenerosity,whichiscloselylinkedtoanyformofaltruisticallymotivatedbehavior,ispositivelycorrelatedwithinfamiliesacrossgenerations.Forexample,ArrondelandMasson(2006)demonstratethatindividualswhohavereceivedadonationaretwiceaslikelytogiveonethemselves,andthattheprobabilityofhelpingoneschildren“nanciallyis50%higherforindividualswho attitudestowardsthemseemtobestronglycorrelatedacrossgenerationsandtendtobereproducedfromonegenerationtothenext.Suchabehaviorisdiculttoexplainwithstandardmodelsoffamilytrans-ferswhichtypicallyconsideronlytwogenerations(parentsandchildren).Rather,theintergenerationaltransmissionofattitudestowardsleavingbequestrepre-sentsanexampleofindirectreciprocity,whereanindividualdoesnotdirectlyreciprocatetothebehaviorofanotherperson,butinsteadreciprocatesindirectlytoathirdparty.Thiskindofinteractionhasbeenfoundtobeparticularlyimportantwithinfamilyrelations(KohliandKünemund,2003).Inthecontextofbequeathingbehaviorthismeansthattheculturaltransmissionofattitudes,valuesandnormsacrossgenerationscreatesapatternofbehaviorinwhichpar-entssupporttheirownchildreninawaysimilartothewaytheirparentstreatedthem.Peopledonotdirectlyreciprocatefortheinheritancetheyhavereceivedfromtheirownparents,butratherrepayitbyleavinganestatetotheirownchildren.Hence,themoreinheritanceparentshavereceivedthemselves,themoretheyarewillingtobequeathtotheirchildren.Inthiswayleavingabe-questdoesaecttheimmediaterecipient,i.e.thenextgeneration,aswellasfuturegenerations.Thepreviousargumentisfurthersupportedbyempirical“ndingsrelatedtowealthinequalityanditspersistence,seee.g.BowlesandGintis(2002).Eco-nomicoutcomesintermsofpersonalwealthaccumulationarenotonlyverysimilaracrossgenerationsbutalsotendtobehighlypersistentovertime.Re-centestimatesofintergenerationalelasticitiesofwealthareintheorderof0.22,0.37or0.32to0.43forco-existinggenerations,i.e.beforeanytransmissionofwealthhastakenplace,seeArrondel(2011),CharlesandHurst(2003)andMulligan(1997),respectively.However,whilestandardexplanatoryfactorssuchaseducationalattainment,incomeorpersonalwealthmayaccountforalargefractionofvariation,still,almost35percentoftheintergenerationalwealthelasticityremainsunexplainedafterincome,propensitytoownassets,education,gifts,andexpectedbequestsarecontrolledfor(CharlesandHurst,2003,p.1157).Similarly,Arrondel(2011)“ndsthatthosestandardexplana-toryfactorsaccountforonly73%oftotalvariationinintergenerationalwealthelasticities.Therefore,takingintoaccountthetransmissionofattitudesandtraitstowardsbequeathingbehaviormayhelptograspabetterunderstandingofindividualbehaviorandalsocontributestoexplainingpersistenceofwealthinequalityacrossgenerations.Despitethisoverallpersuasiveempiricalevidence,therearealmostnotheo-reticalmodelscapturingtheideaofpreferenceandattitudetransmissionwithinfamilies.AnotableexceptionistherecentstudybyKirchsteigerandSebald weregiftbene“ciariesthemselves.SeeArrondelandMasson(2006)forasurveyoftheliterature.Theynotethatindirectreciprocitiesprovideadynamicsynthesisofaltruismandexchangeallowing[...]tointroduceintermediatemotivationsfortransferswhichbetter“tthedata.Earlystudieshoweverprovideevenlargerintergenerationalelasticitiesofwealth,seee.g.,HarburyandHitchins(1979)andMenchik(1979)who“ndestimatesof0.50fortheUKand0.70fortheUS,respectively. (2010)whoincorporateindirectreciprocitieswithregardtoeducationalinvest-mentsintoanoverlappinggenerationsmodelandexaminetheirimpactonhu-mancapitalformation,well-beingandeducationpolicies.Bycontrast,theaimofthispaperistoexploretheoreticallytheroleofintergenerationaltransmissionofattitudestowardsleavingbequest.Todoso,weincorporatetheideathathavingreceivedaninheritancefromonesparentspositivelyshapestheattitudetowardsleavingabequesttoonesownchildrenintoanotherwisestandardoverlappinggenerationsmodelwhereparentsareconcernedaboutthe”owofbequestitself,asinAndreoni(1989).Usingthisframework,weshowthatthetransmissionofattitudesmaygiverisetotheexistenceofmultiplesteadystatesinvolvingpovertytraps.Bequeath-ingbehaviordependsbothonincomeandontheexperienceofinheritingwealth:Ifpercapitaincomelevelsarelow,individualsareprimarilyfocussedonsatisfy-ingtheirownconsumptionneedsandbequestsareabsent.Thisinturnshapestheattitudeofthedescendantsinasimilarway,therebyreproducingthelowpercapitaincomelevelinthenextgenerationassavingsand,inparticular,wealthaccumulationarecompletelyneglected.Bycontrast,ifpercapitaincomelevelsaresucientlyhigh,individualssavetoleaveabequestinordertorepaytheirchildrenforhavinginheritedfromtheirownparents.Hence,theeconomyischaracterizedbyhighlevelsofcapitalaccumulationandpercapitaincome.Insuchasituation,theprovisionofpublicpensionbene“tsmayhelptoovercomeapovertytrapasitprovidesadditionaloldageincomeoutofwhichtobequeath.However,itisalsoshownthatapay-as-you-gopensionprogrammeisatbestneutraltocapitalaccumulationandgrowthinthelongrun.Speci“cally,pub-licpensionsarediametraltocapitalaccumulationifbequestsareinoperativewhereasthereisanadditionalpositiveeectstemmingfromthetransmissionofattitudestowardsbequeathingbehaviorifbequestsareoperative:Ahigherpensionlevelincreasestheamountofbequestindividualsreceivefromtheirpar-ents,whichinturnpositivelyaectsthechildrensattitudetowardsleavingabequesttotheirownchildrentherebyincreasingindividualssavingsandcapi-talaccumulation.Asaresult,theRicardianequivalenceholdsinthepresentmodel,despitethefactthatthebequestmotiveitselfdoesnottakeintoaccountthedescendantseconomicsituation.Finally,weconsideraheterogenouspopulationandshowbymeansofsimu-lationthatanunequaldistributionofinitialinheritanceendowmentsmayleadtoapermanentsplitofsocietyforotherwiseidenticalindividuals.Ourmodelmaythereforeendogenouslyexplainwhysomehouseholdssaveinordertoleaveabequestswhileotherdonot(LaitnerandJustner,1996)andalsowhyincomeinequalityishighlypersistentacrossgenerations(BowlesandGintis,2002).Inabroadersense,ourresearchisrelatedtoanemerging“eldofeconomicsthatseekstounderstandwherepreferencescomefrom.Infact,theissueof Inthiscasetheprovisionofpublicpensionsnegativelyaectstheyoungsmotivationtosave,sincepartoftheiroldageconsumptionis“nancedbynextperiodspensions.Conse-quently,capitalaccumulationandeconomicgrowthdeclines,ashas,forexample,beenarguedbyFeldstein(1974).SeeforexampletheseminalpaperbyBarro(1974)forafurtherdiscussionofthisissue. preferenceformationhasrecentlyreceivedmuchattentioninthecontextoftime-preference,consumptionexpenditure,laborhoursorriskandtrustat-titudesBeckerandMulligan(1997);Waldkirchetal.(2004);Toledo(2006);Dohmenetal.(2008).Forexample,Dohmenetal.(2008)provideevidencesuggestingthatparentswhoaremorewillingtotakerisks,ormorewillingtotrustothers,havechildrenwhoaresimilarlyrisktolerantandtrusting.Conse-quently,attitudesandtraitswithregardtodierenteconomickeyfactorsaredeterminedtoasubstantialdegreebyanindividualsparents.Thecurrentpa-percontributestothisstrandofliteraturebyexploringtheoreticallytheroleofindirectreciprocitieswithregardtobequeathingbehavior.Theremainderofthispaperisorganizedasfollows.Section2presentsthebasicmodel.Section3characterizesthedynamicsoftheeconomy,studiestheexistenceofsteadystatesandexaminestheimpactofanunfundedsocialsecurityprogrammeonbequeathingbehaviorandcapitalaccumulation.Section4introducesheterogeneityintothemodelandshowsthatinitialendowmentsofbequestsmayleadtoapermanentsplitofsocietyinthelongrun.Section52ThemodelThebasicframeworkisanoverlapping-generationmodelinthetraditionofDi-amond(1965),inwhichparentshaveanaltruisticconcernfortheirchildren.Speci“cally,weassumethatindividualsarejoy-of-givingaltruistsandderiveutilityfromtheamountofbequestitself(Andreoni,1989).However,incon-trasttomostoftheexistingliterature,wefurtherassumethatthealtruisticconcernispositivelyrelatedtotheamountofinheritanceparentshavereceivedfromtheirownparents.Suchaspeci“cationallowsonetostudytheroleofintergenerationaltransmissionofattitudestowardsleavingbequests.Moreover,marketsarecompetitive,andthesizeofpopulationisassumedtobeconstant.2.1FirmsOntheproductionsideofthemodel,perfectcompetitionbetweenalargenum-berofidentical“rmsisassumed.Arepresentative“rminperiodproducesahomogenousoutputgoodaccordingtoa(normalized)Cobb…Douglasproductionfunctionwithcapitalandhomogeneouslabourasinputs:istheshareparameterofcapital.Each“rmmaximizespro“tsunderperfectcompetition,implyingthat,inequilibrium,productionfactorsarepaidtheirmarginalproducts:=(1=(1 Itshouldbeclear,however,thatourresultsarepositive,notnormative,sinceweonlyexaminethelongruneectsofattitudeformationwithrespecttobequeathingbehaviorandtheimpactofpublicpensionsanddonotassesstheirwelfareeects. isthecapitalintensity.2.2ConsumersAteachperiodintime,thereexistanumberofyoungandanumberofold.Whenyoung,eachindividualinelasticallysuppliesoneunitoflabourandreceivesthewage.Shealsoreceivesanonnegativebequest,Incomeisspentonconsumptionandsavingsisthecontributionratetothepay-as-you-gopensionprogramme.Whenold,eachindividualallocatesthereturntosavings(plusthepensionbene“t()tosecond-periodconsumption()andtoanonnegativebequesttotheospring().ThesecondperiodsbudgetconstraintisthusIndividualpreferencesareoftheCobb-Douglastypeanddependon“rst-andsecond-periodconsumptionandontheamountofbequestdevotedtothechildren.Consequently,thelife-cycleutilityfunctionofanindividualbornin)=ln)lnisadiscountfactor,andthedegreeofaltruismwhichmeasuresindividualsattitudetowardsleavingbequest.Inordertoformalizethetrans-missionofattitudesandtraitswithregardtoleavingbequests,webuildontheempirical“ndingsdescribedintheintroductionthatparentswillingnesstoleavebequesttotheirchildrenstronglydependsontheirownexperiencewithregardtohavingreceivedinheritance.Ifparentshavereceivedbequestfromtheirpar-ents,theyattachmoreimportancetoandfeelmoreresponsibleforpassingonbequesttotheirownchildren.Wecapturethisrelationbyanattitudefunctionwhichdescribeshowtheinheritanceaparenthasreceivedfromhisownparentsrelatestotheimportanceheattachestohisownbequeathingbehavior::[0(7)denotingtheattitudeofaparenthavingreceivedabequesthisownparent.Weassumethatiscontinuousanddierentiablewith(0)=0 �0. Forreasonsofsimplicity,weassumethatcapitaldepreciatescompletelyinoneperiod. Ifaparenthasnotreceivedanyinheritancehimself,heisnotwillingtoleaveabequesttohischild.Furthermore,theattitudefunctionismonotonicallyincreasingintheamountofinheritanceaparenthasreceivedhimself.Eachindividualmaximizestheutility(6),subjecttotheconstraints(4),(5)andtothenonnegativityofbequests(),bychoosing,and.The“rst-orderconditionsofthismaximizationproblemare =Š1 ct+ t Š dt(bt) The“rstequationisthestandardconditionoverthelifecycle,determiningoptimalsavings.Thesecondonegivestheoptimalamountofbequests.3DynamicsandsteadystatesWearenowabletode“netheintertemporalequilibriumoftheeconomy.Givena“scalpolicy(thecontributionrate)andinitialvaluesofthecapitalstockandtheamountofbequest,aperfect-foresightintertemporalequilibriumischaracterizedbyasequenceofquantitiesandprices:Individualsmaximizeutilityandthegovernmentsbudgetisbalancedineachperiod,i.e.Factormarketsarecompetitive,andallmarketsclear.Moreover,populationsizewillbenormalizedtoonethroughoutthissubsection.Thedynamicsofthemodelarefullycharacterizedbythe“rstordercon-ditions(10)and(11).Insertthehouseholdsbudgetconstraints((4)and(5)),factorprices((2)and(3))andthecapitalmarketclearingcondition(intothe“rstorderconditionsandrearrangeterms,toobtain (1+ k +(1Š)]kt+1(14)Itiswellknownthatfora“xedandstrictlypositivealtruisticdegree,i.e.foranexogenouslygiven,thereexistsauniqueinteriorsteadystatewhichisgloballystable: (1+ 1Š Aswillbeshownbelow,however,thisresultstandsinsharpcontrastwiththoseofaneconomywithanendogenousbequeathingattitude,i.e.ofaneconomywhereparentsdegreeofaltruismdependsontheinheritancetheyhavereceivedfromtheirownparents.Inordertoillustratethedierentpossibleoutcomes,weassumeasimplelinearattitudefunctioninthefollowing,namely measuresthestrengthofthetransmissionofattitudesfromparentstochildrenwithrespecttotheirbequeathingbehavior.Giventhissimpleattitudefunction,wederivethefollowingconditionsforpossibleinteriorsteadystates(ignoretheperiodindicesinequations(13)and(14)andrearrangeterms): (1+ (17))+(1Š)]kŠ(18)Furtheranalysisofequation(18)reveals:Proposition1Bequestswillbeoperativeinthelongrunifandonlyifthecapital-labourratioissucientlylarge,i.e. +)1 Theoccurrenceofaltruisticallymotivatedbequeststurnsouttobesensitivetoconditionsrelatedtothestateofeconomicdevelopment:Bequestswillbepositiveinthelongrunonlyifpercapitaincomeexceedsacertainthresholdlevel.Thisthresholdcanbeinterpretedassomeminimallevelofincomebeyondwhichindividualshavesatis“edtheirownconsumptionneedsandstartdevotingresourcestotheirdescendants.ThenatureofthisresultissimilartotheonefoundbyRapoportandVidal(2007)wheretheaccumulationofaltruismoccursonlyifparentsdisposableincomeissucientlylarge.However,intheirmodelindividualsmaychoosetoacquireandaccumulatealtruismwhereasourmodelconcernsthetransmissionofattitudestowardsbequeathing.Furthermore,theaboveresultcontrastssharplywiththatofaneconomywithanexogenousaltruisticdegree.Insuchaframework,bequestswillalwaysbeoperativeinthelongrunaslongasparentsdegreeofaltruismispositiveImportantly,thethresholdlevelisendogenouslydeterminedandcanbetracedbacktoeconomicfundamentals:Itispositivelyrelatedtoindividuals Notethatpopulationsizeisnormalizedtoone.Therefore,percapitaincomeequalsthecapitallaborratiointhepresentmodel. patience(k/�)andnegativelytothestrengthofthetransmissionpro-cess(k/�).Moreover,ahigherpensionlevelreducesthethresholdlevelk/)asitprovidesindividualswithadditionalretirementincomeoutofwhichtobequeath.Inanextstep,westudytheexistenceofsteadystatesandtheirstabilityproperties.Ifpercapitaincomelevelsaresucientlylow,i.e.,bequestsareinoperativeinthelongrunanditiseasytoseefromequation(17)thatthereexistsatleastonesteadystate: (1+ Suchasteadystateischaracterizedbyasituationwherebequestsareabsentsothatthetransmissionofwealthacrossgenerationsiscompletelyneglected:Eco-nomicconditionspreventgrandparentsfromleavingabequesttotheirchildren,whichinturnshapestheattitudeofthegrandchildreninasimilarway.Asare-sult,thereisastickinessofpovertyacrossgenerationswhichpartlyresultsfromtheabsolutepovertyoftheparentsintermsoflowpercapitaincomelevels,andpartlyfromtheparentsowninheritanceexperience.However,whethertheeconomyinfactconvergestothissteadystatecruciallydependsontheexistenceofothersteadystatesanditsstabilitypropertiesaswellastheeconomysinitialendowmentsofcapitalandbequest.Thefollowingpropositionestablishesthattheeconomymayindeedbechar-acterizedbymultiplesteadystates:Proposition2Thereexistsacriticalstrengthofthetransmissionprocess,i.e. ) (1+ suchthatthefollowingholds.,thenthereexistsauniqueinteriorsteadystatewithoperativebequests.Bycontrast,if,thenthereeitherexistsauniquesteadystatewithinop-erativebequests(apovertytrap)ortherearethreesteadystates,onefeaturinginoperativebequestswhereasintheremainingtwobequestsarepositive.Proof:SeetheAppendix.Simulationresultssuggestthatifthereexistsauniquesteadystate(witheitheroperativeorinoperativebequests),thissteadystateisgloballystable.Bycontrast,iftherearetwointeriorsteadystates,oneofthemandthesteadystatewithinoperativebequestsarelocallystable.Forexample,assumethat.Figure2showssimulationresultsforvaryinginitiallevelsofpercapitaincomeandbequestwith.Fortheseparam-eterchoicesthecriticalvalueofthestrengthofthetransmissionprocessequals,sothatandthesystemconvergesgloballytowardsthesteadywithinoperativebequests, [InsertFigure2here]Bycontrast,assuming,themodelexhibitstwointeriorsteadystates.InFigure3wereportsimulationresultsfordierentinitialconditions.Clearly,dependingontheinitiallevelofpercapitaincomeandtheendowmentofbequest,thesystemeitherconvergestowardsthesteadystatewithoperativebequests,,ortothesteadystateinoperativebequests,[InsertFigure3here]Finally,forasucientlystrongtransmissionprocess,i.e.,thesystemconvergestothegloballystableinteriorsteadystateascanbeinferredfromFigure4.[InsertFigure4here]Asaresult,themultiplicityofsteadystatesmaybeusedtoexplainwhysocietiesthatdierwithregardtotheirinitiallevelofpercapitaincomeand/ortheirinitialendowmentofbequestsendupatdierentsteadystateswherebe-questsareeitheroperativeandthelevelofcapitalaccumulationisthereforerelativelylargeorbequestsareinoperativeandindividualsareprimarilyfo-cussedonsatisfyingtheirownconsumptionneedswhichinturnimplieslowlevelsofsavingsandcapitalaccumulation.Inthelattersituation,successivegenerationsaretrappedinpoverty.Accordingtoproposition1,apay-as-you-gopensionprogrammemayhelptoovercomepovertytrapsasitreducesthethresholdlevelofpercapitaincomeabovewhichbequestswillbeoperativeinthelongrunbyprovidingindividualswithadditionaloldageincomeoutofwhichtobequeath.However,apublicpensionprogrammealsotendstoreducecapitalaccumulation.Infact,therearetwoopposingeects:Ontheonehand,apublicpensionprogrammereducesprivatesavingsandcapitalaccumulationasitnegativelyaectstheyoungsmo-tivationtosave.Ontheotherhand,ahigherpensionlevelincreasestheamountofbequestindividualsreceivefromtheirparents,providedpercapitaincomelevelsaresucientlylarge.Thisinturnpositivelyaectsthechildrensatti-tudetowardsleavingabequesttotheirownchildrenwhichincreasesindividualsavingsandwhichisthereforebene“cialforcapitalaccumulationThefollowingpropositionestablishesthatapublicpensionprogrammeiseitherneutralwithrespecttocapitalaccumulationorreducestheaccumulationdependingonwhetherbequestsareoperativeorinoperativeinthelongrun:Proposition3Supposethatbequestsareoperativeinthelongrunbeforeandafterthepolicychange.Then,anincreaseinthepublicpensionprogrammehasnoimpactoncapitalaccumulation.Bycontrast,ifbequestsareinoperativebeforeandafterthepolicychange,an increaseinthepublicpensionprogrammeunambiguouslyreducescapitalaccu-Proof:Ifbequestsareinoperative,thesteadystatecapitalstockisgivenbyequation(20).Straightforwardcalculationsshowthat/.Tostudythecaseofoperativebequests,insert(18)into(17)andrearrangetermstoobtain (1+ (1+k (1+whichisequivalentto.Consequently,thesteadystatecapitalstockwithoperativebequestsisindependentofthesocialsecuritytaxrate.Proposition3demonstratesthat,inthepresentmodel,Ricardianequiva-lencemayholdevenifparentsareconcernedaboutthe”owofbequestitselfandnotaboutthewell-beingoftheirdescendants.These“ndingscontrastsharplywiththosederivedfromthestandardjoy-of-givingmodelbyAndreoni(1989)suggestingthatpublictransfershaverealeects.Hence,takingintoac-counttheconnectionbetweenreceivedinheritanceandownbequestmayleadtointerestingnewpolicyconclusionswithregardtotheimpactof“scalpolicyoncapitalaccumulationandlongrungrowth.Sofar,however,ourresultsrefertoaneconomywithahomogenouspop-ulationwhereeachmemberofthe“rstgenerationisendowedwiththesameamountofphysicalcapitalandbequest.Consequently,descendantsindividualchoicesareidenticalandthepossiblemultiplicityofstablesteadystatesreferstotheeconomyasawhole.Yet,onemightwonder,whetherdierencesininitialendowmentsforotherwiseidenticalmembersofthe“rstgenerationleadtodif-ferenteconomicoutcomesforthedescendantsinthelongrunorwhethersuchdierenceswillbewashedout.Toanswerthisquestion,thefollowingsectionintroducesheterogeneityintothemodel.4HeterogenousIndividualsInthissectionweassumethatindividualsdierwithregardtotheirinitialendowmentofinheritancetheyhavereceivedfromtheirparents.Morespecif-ically,weconsidertwotypesofindividuals,,withinitialendowmentofbequest.Exceptforthesedierentendowments,individualsareassumedtobeidentical.Letdenotetheshareoftypeinthepopulation.Then,theproductionfunctionisgivenby+(1 Notethatonemayeasilyintroduceadditionalsourcesofheterogeneityasforexampledierentstrengthsofthetransmissionprocess,i.e.,orwagedierentialsduetoskilldierentials.Weconjecturehoweverthatthiswouldonlystrengthenourresults. andperfectcompetitionimpliesthateachindividualreceivesthesamewageandinterestrate:=(1+(1+(1Thedynamicsofthemodelwithaheterogenouspopulationarefullychar-acterizedbythehouseholds“rstorderconditionsforeachofthetwotypes.Speci“cally,inserttheattitudefunction(16),thehouseholdsbudgetconstraints((4)and(5))andfactorprices((24)and(25))into(10)and(11),makeuseofA,B,andrearrangeterms,toobtain +(1 +(1 (1++(1 +(1+(1(27) +(1 +(1 (1++(1 +(1+(1Foridenticalinitialendowmentsofbequest,i.e.,itiseasytoseethatthedynamicsareequivalenttothoseofahomogenouspopulationmodel.Therefore,anysteadystateinthelattermodelalsoconstitutesasteadystateoftheheterogenouspopulationmodel.However,thereareadditionalsteadystateswherethepopulationispermanentlysplitintotwosubgroups:Onegroupinwhichindividualsleavebequestsandexperiencehighlevelsofpercapitaincomeandasecondgroupinwhichindividualsaretrappedinpovertyandwherepercapitaincomelevelsarelow.Assumeforexamplethefollowingparametrizationofthemodel:.Fortheseparameters,themodelwithahomogenouspopulationexhibitsthesteadystates,ashasbeenshownabove.Figure5presentssimulationresultsforvaryinginitialendowmentsofbequestsforbothtypesandaninitialcapitalstockofInsertFigure5here.Clearly,whenevertheinitialamountofbequestsoftypeAissucientlylargeandexceedsthecorrespondingamountoftypeB,theeconomyconverges towardstheunegalitariansteadystate.ThissteadystateissymmetricwithrespecttotheindividualstypeHence,dependingontheinitialconditionsandontheparametervaluesofthemodel,asocietymaybepermanentlysplitintotwosubgroups,despitethefactthatallagentsfacethesameinterestandwagerate.Povertygetsinheritedfromgenerationtogeneration,preventingconvergenceofthetwopopulation5ConclusionsStandardmodelsofbequestsandfamilytransferstypicallyconsideronlytwogenerations(parentsandchildren).Theyarethusnotabletotakeintoaccountthespeci“cin”uenceofpastinheritancereceivedfromparentsonbequesttochildrenthathasbeenfoundinthedata(CoxandStark,2005;ArrondelandGrange,2007).Therefore,theaimofthepresentpaperwastosetupathe-oreticalmodelofindirectreciprocitiesbetweenthreegenerations,whichmayreplicatethesametypeoftransferfromonegenerationtothenext.Speci“cally,wehaveincorporatedtheideathathavinginheritedfromonesownparentspositivelyaectstheattitudetowardsleavingabequesttotheownchildrenintoanoverlappinggenerationsmodel.Itisthenshownthatthismechanismofindirectreciprocitygivesrisetotheexistenceofpovertytrapsasbequeathingbehaviordependsbothonincomeandontheexperienceofinher-itingwealth.Insuchasituation,povertymaybeovercomewiththehelpofpubliclyprovidedpensionbene“tswhichincreaseoldageincomeandthereforetheamountofbequestoldindividualsdevotetotheirchildren.However,anunfundedpensionprogrammeturnsouttobeatbestneutraltocapitalaccu-mulationifbequestsareoperativewhereasitunambiguouslyreducescapitalaccumulationwhenbequestsareinoperative.Inthe“rstcase,Ricardianequiv-alenceholdsinthepresentmodeldespitethefactthatthebequestmotiveisindependentofthedescendantswell-being.Allowingforheterogeneity,sim-ulationssuggestthatanunequaldistributionofinitialinheritancesmayleadtoapermanentsplitofsocietyforotherwiseidenticalindividuals.Ourmodelthereforecontributestoexplainingthepersistenceofincomeinequalityacrossgenerationsanddierencesinindividualsavingsbehaviorinordertoleaveabequest,seeBowlesandGintis(2002)andLaitnerandJustner(1996),respec-tively.Finally,thepresentanalysiscouldbeextendedbyintroducinghumancapitalaccumulationwhichwouldallowonetostudytheeectsofendogenousattitudeformationtowardsbequeathingeducatingchildren(seeKirchsteigerandSebald(2010)foraseparateanalysisofthelatterissue).Thishoweverwouldfurtherincreaseanalyticaltractability. Notealsothatthese“ndingscanbeshowntoberobustagainstsmallchangesintheparametervalues. Andreoni,J.,1989.Givingwithimpurealtruism:Applicationstocharityandricardianequivalence.JournalofPoliticalEconomy97,1447…1458.Arrondel,L.,2011.myfatherwasright:Thetransmissionofvaluesbetweengenerations.workingpaper.Arrondel,L.,Grange,C.,2006.Transmissionandinequalityofwealth:Anempiricalstudyofwealthmobilityfrom1800to1938infrance.JournalofEconomicInequality4,209…232.Arrondel,L.,Grange,C.,2007.Bequestsandfamilytraditions:Thecaseof19thcenturyfrance.mimeo.Arrondel,L.,Masson,A.,2006.Altruism,exchangeorindirectreciprocity:Whatdothedataonfamilytransfersshow?HandbookontheEconomicsofGiving,ReciprocityandAltruism2,ch.14.Arrondel,L.,Masson,A.,Pestieau,P.,1997.Bequestandinheritance:Empiricalissuesandfrenchuscomparison.InG.ErreygersandT.Vandevelde(eds.),IsInheritanceLegitimate?,Springer-Verlag,Heidelberg.Barro,R.J.,1974.Aregovemmentbondsnetwealth?JournalofPoliticalEconomy82,1095…1117.Becker,G.S.,Mulligan,C.B.,1997.Theendogenousdeterminationoftimepreference.QuarterlyJournalofEconomics112,725…758.Bisin,A.,Verdier,T.,2006.Culturaltransmission.inNewPalgrave.Bisin,A.,Verdier,T.,2011.Theeconomicsofculturaltransmissionandsocial-ization.HandbookofSocialEconomics1A,ch.9.Bowles,S.,Gintis,H.,2002.Theinheritanceofinequality.JournalofEconomicPerspectives16(3),3…30.Charles,K.K.,Hurst,E.,2003.Thecorrelationofwealthacrossgenerations.JournalofPoliticalEconomy111,1155…1182.Cox,D.,Stark,O.,2005.Bequests,inheritancesandfamilytraditions.CRRWP2005-09.DeLong,B.J.,2005.Bequests:Ahistoricalperspective.WorkingPaper.Uni-versityofCalifornia,Berkeley.Diamond,P.A.,1965.Nationaldebtinaneoclassicalgrowthmodel.AmericanEconomicReview55,1126…1150.Dohmen,T.J.,Falk,A.,Human,D.,2008.Theintergenerationaltransmissionofriskandtrustattitudes.IZADiscussionPaperNo.2380. Feldstein,M.,1974.Socialsecurity,inducedretirement,andaggregatecapitalformation.JournalofPoliticalEconomy23,905…926.Harbury,C.,Hitchins,D.,1979.Inheritanceandwealthinequalityinbritain.London,AllenandUnwin.Heckman,J.J.,2006.Skillformationandtheeconomicsofinvestingindisad-vantagedchildren.Science312(5782),1900…1902.Kirchsteiger,G.,Sebald,A.,2010.Investmentsintoeducation-doingastheparentsdid.EuropeanEconomicReview54,501…516.Kohli,M.,Künemund,H.,2003.Intergenerationaltransfersinthefamily:Whatmotivatesgiving?InBengtsonV.andLowensteinA.(eds.)(2003),GlobalAgingandChallengestoFamilies,NewYork:AldinedeGruyter,123…142.Kotliko,L.J.,Summers,L.H.,1981.Thecontributionofintergenerationaltransfersinaggregatecapitalaccumulation.JournalofPoliticalEconomy89,Laitner,J.,Justner,T.F.,1996.Newevidenceonaltruism:Astudyoftiaa-crefretirees.AmericanEconomicReview86(4),893…908.Lambrecht,S.,Michel,P.,Thibault,E.,2006.Capitalaccumulationand“scalpolicyinanolgmodelwithfamilyaltruism.JournalofPublicEconomicTheory8(3),465…486.Menchik,P.L.,1979.Intergenerationaltransmissionofinequality:Anempiricalstudyofwealthmobility.Economica46,749…762.Michel,P.,Thibault,E.,Vidal,J.-P.,2006.Intergenerationalaltruismandneo-classicalgrowthmodels.HandbookoftheEconomicsofGiving,AltruismandReciprocity2,partII,ch.15.Mulligan,C.B.,1997.Parentalprioritiesandeconomicinequality.UniversityofChicagoPress,Chicago.Rapoport,H.,Vidal,J.-P.,2007.Economicgrowthandendogenousintergener-ationalaltruism.JournalofPublicEconomics91,1231…1246.Toledo,M.,2006.Ontheintergenerationalpersistenceofworkhours.Mimeo,UniversityofRochester.Waldkirch,A.,Ng,S.,Cox,D.,2004.Intergenerationallinkagesinconsumptionbehavior.JournalofHumanResources39,355…381.Wilhelm,M.O.,Brown,E.,Rooney,P.M.,Steinberg,R.,2008.Theintergenera-tionaltransmissionofgenerosity.JournalofPublicEconomics92,2146…2156. AppendixIna“rststep,westudythepropertiesofthefunctions (1+ (1+ whichresultfromsettinginequation(17)andfrominsertingequation(18)into(17),respectively.Thedomainofbothfunctionsisrestrictedtotheintervalal,(.Itisthenstraightforwardtoshowthathasalocalminimum )+)Ž1 .Moreover,wehave(0)=0 )+)Ž1 Nowconsiderthefunction.PossiblerootsofareimplicitlygivenbythesolutionsofthefollowingequationLHSRHSLHS(0)=0LHSRHSRHS.Furthermore,LHShasalocalmaximumat RHSstrictlyconcave,i.e.RHS.Consequently,depend-ingontheparametersofthemodel,haseithernorootsortwo.Moreover,itisstraightforwardtoshowthatInasecondstep,weconsiderthepiecewisefunction,k,k�onthedomainomain,(withkbeingde“nedinproposition1.accountsforthefactthatbequestscannotbenegativeanditsrootscorrespondtothesteadystatesofourmodel.StraightforwardcalculationsshowthatGiventhepropertiesofthefunctions,itfollowsthatthereexistsauniquesteadystatewithoperativebequestswhenever.Bycontrast,theremaybeeitheronesteadystatewithinoperativebequestsorthreesteadystatesif.Thelatterinequalitycanberewrittenasfollows: +)1 Š) (1+ +)0+) ) (1+ Byexampleweshowthatbothcases,onesteadystateorthreesteadystates,areindeedfeasible.Considerthefollowingparameterizations: ,respectively.Inthe“rstcasehasauniquerootatwhereasforexhibitsthreerootsat.Finally,forthereisexactlyonerootat.TheseresultsareillustratedinFigure4.[InsertFigure6here] Figures                  Figure1:Intentionstobequeathandnetworth:inheritorsversusnon-inheritorsSource:CoxandStark(2005) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09bequestphysical capital Legend:x  starting value*  steady statesteady state:b*=0 Figure2:Endogenousbequeathingattitudewithandnointeriorsteady 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09bequestphysical capital Legend:x  starting valuesteady state: steady state: Figure3:Endogenousbequeathingattitudewithandtwointeriorsteady 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.05 0.1 0.15 0.2 0.25bequestphysical capital Legend:x  starting value*  steady statesteady state: Figure4:Endogenousbequeathingattitudewith 0 0,005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 Initial amount ofbequest: Type AInitial amount of bequest: Type BLegend:x Š egalitarian steady state (no bequests)+ Š egalitarian steady state (positive bequests)o Š unegalitarian steady state (A leaves bequest B does not) Figure5:Heterogeneousinitiallevelsofbequests: Figure6:Endogenousbequeathingattitudewith