Imperfect Commitment, Altruism, and the Family - Andrew D. Foster PDF
Preview Imperfect Commitment, Altruism, and the Family - Andrew D. Foster
Forthcoming:ReviewofEconomicsandStatistics ImperfectCommitment,AltruismandtheFamily:EvidencefromTransfer BehaviorinLow-IncomeRuralAreas AndrewD.Foster BrownUniversity MarkR.Rosenzweig UniversityofPennsylvania July2000 TheresearchforthispapersupportedinpartbygrantsNIHHD30907,NIHHD28687,andNSFSBR93- 0405 I.Introduction Altruismormutual“caring”amongfamilymembershasbeenhypothesizedtoplayanimportant roleinfacilitatingrisksharing,particularlyinenvironmentswithlessdevelopedmarkets(Becker,1991). Indeed,recentstudieshaveprovidedevidencethatfamily-basedincometransfersdocontributeto consumptionsmoothing(Rosenzweig,1988;RosenzweigandStark,1989;RosenzweigandWolpin, 1994).However,thereisalsoevidencesuggestingthatwithinfamiliesidiosyncraticriskisnotfully insured(Altonjietal.,1992),andthatthetransferofresourcesamongfamilymembersinresponseto incomefluctuationsdoesnotconformfullytosimplemodelsofaltruism(Cox,1987;Altonjietal., 1997). Aproblemwithexistingempiricalstudiesofaltruismisthattheydonotmodelexplicitlythe well-knownconstraintsoninsurancearrangementsthataltruismmaynotfullyovercome,suchasbarriers toinformationthatgiverisetomoralhazardandthecommitmentproblemsthatarisewhenindividuals cannotenterintobindingcontracts.Theimplicationsderivedfromtheassumptionofaltruisminsuch modelsthusmaybemisleading.Moreover,if,forexample,costlystateverificationistheprincipal probleminattainingperfectinsurance,itmaybethatthefamilyisprominentinrisksharingnotbecause ofaltruismbutbecauseoffamilialinformationadvantages. Recentworkonself-enforcingcontracts(CoateandRavallion(1993),ThomasandWorrall (1995))hasprovidedinsightintothestructureofconstrained-efficientinsurancecontractsinthepresence ofimperfectcommitment.However,thisworkdoesnotexploretheinteractionsbetweencommitment problemsandaltruismindeterminingrisk-sharingarrangementsandfailstoexplainthepervasivefact thattransfers,inbothhighandlow-incomecountries,frequentlyoccurbetweenfamilymembers.Ina worldinwhichcontractsareeitherself-enforcingorenforceablebyathirdparty,therewouldseemtobe noparticularreasonforthistobethecase:itseemslikelythathouseholdsotherthanthosetowhichone isrelatedwouldingeneralbeabletoprovideonewithmorecompleterisksharingthanwouldthoseto 1 whichoneisconnectedthroughfamilytiesif,forexample,theincomesofnon-kinarelesscorrelated thanthoseofkin.Inaworldinwhichcontractsarenotenforceable,however,altruismmayplayan importantrole.Totheextentthathouseholdsenteringintoarisksharingarrangementcareabouteach others’welfarewewouldexpectthattheywillgainmorefrominsurancethantheywouldotherwiseand thusthatthescopeofrisk-sharingcontractswillbegreater. Inthispaper,weexamineempiricallywhetherriskpoolingismoreadvantageousamong altruisticcomparedtoselfishagentsinaframeworkwhereindividualscannotmakebinding commitments. Weincorporatealtruismintotherecentmodelofrisksharingunderimperfect commitmentofThomasandWorrall(1995)toestablishtestsoftherolesofbothaltruismand commitmentproblemsindeterminingtheextentofinsuranceandtheintertemporalmovementsin interhouseholdtransfers.1 Weshowthatitispossibletoimplementthetestsintermsofdirectly measurablevariables,onlysomeofwhichhavebeenusedinpriorstudiesassessingalternativeincome smoothingmechanisms.Inparticularweshowthathowmuchpasttransfersaffectcurrenttransfers-the “transferasseteffect”-andhowmuchofacontemporaneousincomeshockissharedbytransferpartners -the“incomeshockeffect”-differinspecificwaysbybothwhetherthetransferpartnersarealtruistic andbypartnerincomecovariancesinthepresenceofcommitmentproblems.Thetestsarecarriedout usingthreepaneldatasetsfromtwocountriesofruralSouthAsiathatprovidedetailedinformationon 1ThomasandWorrall(1995)useanapproachdevelopedinthecontextofimplicitlaborcontracts (ThomasandWorrall(1988))toexaminethequestionofwhetheroptimalimplementablerisksharing contractsarenon-stationary.ThisquestionwasraisedbyCoateandRavallion(1993)who,buildingon workbyFoster(1988)andKimball(1988),examinedoptimalstationaryrisk-sharingcontracts.Ligon, WorrallandThomas(forthcoming)provideanempiricaltestofthismodelusingoneofthethreedata setsemployedinthispaper.Whiletheirresultssupporttheimperfectcommitmentmodelrelativeto perfectinsuranceandautarchy,theirapproach(inparticularthefactthatexplicitinformationontransfers bypartnersarenotused)doesnotpermitanassessmentoftheroleofaltruismasacommitmentdevice.It shouldalsobenotedthattheLigon,WorrallandThomas(forthcoming)approachfocusesonintra-village risksharing,andthusdoesnotpermitanassessmentof inter-villagerisksharingarrangements,which playanimportantroleasshownbelow. 2 transfersandenablethemeasurementofincomeshocks.2 InsectionIIwecharacterizetheoptimalimplementablerisksharingcontractunderimperfect commitmentandillustrateusingmodelsimulationhowthedegreeofaltruismamongtransferpartners andthemagnitudeoftheirincomecorrelationinfluencethiscontract.Themodelsimulationsshowthat althoughsufficientlyaltruisticfamilieswithlessthanperfectlycorrelatedincomeswilloptimallyengage inriskpoolingarrangements,non-altruisticindividualswillnotengageinriskpoolingwhenincome correlationsaresufficientlypositive.Wealsoshowthatiftransfersplayaninsurancerolebutdonot achievefullinsurancethentransferswillrespondtobothcontemporaryincomeshocksandtothehistory ofprevioustransfers,withthelatterarisingasaresultoftheinabilityofhouseholdstocommittomake first-beststate-contingenttransfers.Moreover,thehistorydependenceoftransfersandtheirdegreeof responsivenesstoincomeshocksdependonthedegreeofaltruism.Inparticular,wedemonstratethati) theexistenceofbindingimperfectcommitmentconstraintsimpliesthathouseholdsthathavemadenet transfersinpreviousperiodsarelesslikelytoprovidesubsequenttransfers,giventhecurrentstateofthe world,thanarehouseholdsthathavebeennetrecipientsoftransfers,butii)contemporaneoustransfers arelessresponsivetotransfersinthepastandmoreresponsivetocurrentincomeshocks,forgiven partnerincomecorrelations,whenpartnersarealtruisticallylinkedcomparedtowhentheyarenot. InsectionIII,weshowthatestimatesoflinearapproximationstothetransferfunctionsimplied bythemodelarecapableofidentifyingtherolesofimperfectcommitment,altruismandincome covariancesindeterminingtransferbehavior.Wealsousethesimulationstoinvestigatetherobustnessof theresultstolackofsymmetryamongtransferpartnersandtoassumptionsabouttherelationships betweenkinship,incomecorrelationandinformation.Thedatafromthreeruralpanelsurveysthatare 2Wolpin(1984),Paxson(1992),Behrman,Foster,andRosenzweig,(1997),Udry(1994,1996), Rose(forthcoming)usevariousmeasuresofincomeshockstotestforconsumptionsmoothingbehavior andtoassesstheimportanceofparticularsmoothingmechanisms,includingfinancialsavings,asset divestiture,andlaborsupply. 3 usedtoestimatethesefunctions,specificaspectsoftheempiricalimplementationandtheempirical resultsarepresentedinSectionIV. Theestimatesprovidestrongsupportforthenotionthatimperfect commitmentsubstantiallyconstrainsinformaltransferarrangements,whetherkin-basedornot,butalso provideevidencethataltruismplaysanimportantroleinamelioratingcommitmentconstraintsandthus inincreasingthegainsfromincomepooling. II.TheoryandSimulations Tocapturethebasicelementsofinterhouseholdinteractionsandyetkeepthemodelstructureas simpleaspossibleweassumethattherearetwohouseholdsi=1,2.Ineveryperiodteachhouseholdi receivesanincomey(s)wheresisthestateofnatureinperiodt.3Incomecannotbestoredorsaved i t acrossperiods.Weallowforaltruismbetweenthehouseholdsbyassumingthathouseholdsingle-period utilityisseparableinownandotherhouseholdconsumption.Inparticular,theutilityfunctionsfor households1and2areassumedtobe: γ u(c1)+ v(c2) (1) γ v(c2)+ u(c1), (2) γ whereu()andv()areincreasingandconcaveand <1reflectsthedegreeofaltruism.Householdsare δ infinitelylived,discountthefuturewithdiscountfactor ,andareexpectedutilitymaximizers.4 3Becauseincomeisexogenousinthemodel,weareassumingawayproblemsofmoralhazard. Hoff(1997)formulatesamodelwhichshowshowmoralhazardlimitstheextentofinsuranceandmakes insurancearrangementsvulnerabletowealthshocks.Inthatmodel,however,itisassumedthatcontracts areenforceableandtheshareofincomethatispooledisendogenouslychosenbythemedianvoter,but respondstoshockswithalag. Hoffpresentsevidencethatthereisconcernaboutmoralhazardinsome low-incomecountrycontexts.However,incorporatingmoralhazardconsiderationsintoamodelinwhich contractsmustbeself-enforcingandthereisnonstationarity,ashere,wouldaddconsiderablecomplexity. Moreover,theempiricalassessmentoftheadditionalroleofmoralhazardwouldrequiredataonthe behaviorofalltransferpartnersthatarenotavailable. 4AsnotedbyCoateandRavallion(1993)andThomasandWorrall(1995)theassumptionthat householdsareinfinitelylivedisnotasstringentasitwouldseem:themodelincorporatesthepossibility thattheplayersbelievethatanyinsurancearrangementwill,forreasonsexogenoustothehistoryof statesandtransfers,failwithsomepositiveprobability(e.g.,asaresultofthedeathofoneparticipant), ineachperiod.Inthiscase,thefailureprobabilityisabsorbedintothediscountfactor. 4 Becauseofriskaversionwhentheincomesofthetwohouseholdsarenotperfectlycorrelatedthe householdscan,inprinciple,benefitthroughsharingofresources.Itisassumed,however,thatcontracts arenotlegallyenforceable.Asaconsequence,enforcementofanycontractmustrelysolelyonthe potentialconsequencesforthetwopartiesofviolatingthecontract.Inparticular,itisassumedthatif eitherofthehouseholdsdoesnotmeetthetermsofthecontractthenthehouseholdsreverttoasequence ofstaticNashequilibria(SSNE),whichcanbeoftwotypes:forsufficientlylowlevelsofaltruismthe householdsoperateinautarchy,asinstandardmodelsofimperfectcommitment,butathigherlevels somelimitedrisk-sharingstillobtains.Thislatterpossibilitymeansthatwhilealtruismmayincreasethe willingnessofanindividualtomakeatransfertoapartnerwhenhisincomeisrelativelyhigh,itmayalso decreasetheabilitytopenalizedeviantbehaviorbyreducingthecredibilityofthethreatofautarchy.5 Thetaskistospecifythesetofcontractsthatcanbesupported.Ingeneral,arisk-sharingcontract τ inthecontextofthismodelcanbecharacterizedasatransferfunction (h)thatdictatesthe(possibly t negative)transferfromhousehold1tohousehold2thatshouldoccurifthecontractisinforceinperiodt afterhistoryh ,wherethelatterisdefinedasthesequenceofstatesuptoandincludingthestaterealized t inperiodt:h ={s ,s ,...,s}.ThetransferfunctionforthestaticNashequilibriumgivenstatesisdenoted6 t 1 2 t ) & ) % 'γ ) ) γ t s.t. u (y (s) t)/v (y (s) t) if u (y (s))/v (y (s))< τ ' ) 1 & ) 2 % ' γ ) 1 ) 2 γ N(s) t s.t. u (y (s) t)/v (y (s) t) 1/ if u (y (s))/v (y (s))>1/ (3) 1 2 1 2 0 otherwise 5Ontheotherhand,ifinformationamongfamilymembersisgenerallysuperiortothatamong non-kin,thethreatofrelegationtothelimitedrisk-sharingstatemaybemorecredible,asthefamily memberwhosetransferarrangementisdiscontinuedwillfinditmoredifficulttofindanotheraltruistic partner.WeconsiderinsectionIIItheempiricalimplicationsofthispotentialinformationaladvantage forfamiliesinenforcingpunishments. 6Notethatinline1ofequation1thetransferispositive,andinline2itisnegative.Intuitively, thebetter-offhouseholdwillmakeatransfertotheworse-offhouseholdinthestaticNashequilibrium whenhecaressufficientlyaboutthathouseholdthatheisbetteroffdoingsothanhewouldbeby acceptingtheautarchicallocation. 5 DefineU(h)astheexpecteddiscountedutilitygainfromtherisk-sharingcontractrelativetotheSSNE t afterhistoryh sothat: t ' &τ & &τ %γ %τ &γ %τ U(h) u(y (s) (h)) u(y (s) N(s)) v(y (s) (h)) v(y (s) N(s)) t 1 t t t 1 t t 2 t t t 2 t t % j4 δ& &τ & &τ %γ %τ &γ %τ (4) E j t(u(y (s) (h)) u(y (s) N(s)) v(y (s) (h)) v(y (s) N(s))) '% 1 j t j 1 j t 2 j t j 2 j t j t 1 andletV(h)denotetheanalogousexpressionforhousehold2.Theassumptionthatthecontractis t enforcedthroughreversiontotheSSNEimpliesthatagivencontractcanbeimplementedonlyifthe $ expecteddiscountedutilitygainfromthecontractrelativetotheSSNEisnon-negative,U(h) 0and t $ V(h) 0, foreachhistoryh.Werefertotheseconstraintssubsequentlyastheimplementability t t constraints.7 Thesetofimplementablecontractscanbecharacterizedbyahistory-dependentconvexfunction W(U)thatdescribesthemaximal(overallimplementablecontracts)expecteddiscountedutilitygainthat s canaccruetohousehold2instatesifhousehold1receivesautilityofU.8 Further,theoptimalcontract canbecharacterizedintermsoftheevolutionoftheratioof marginalutility ) %τ %γ ) &τ λ 'v (y (s) (h)) u (y (s) (h))'& ) (h) ) 2 t &τt t %γ ) 1 t %τt t W (U(h)) (5) t s t t u (y (s) (h)) v (y (s) (h)) t 1 t t t 2 t t t λ overtime,where- istheslopeoftheParetofrontier.Inparticularforeachstatesthereisahistory λ λ independentinterval[ L, U]correspondingtothesetofimplementablepointsontheParetoefficient s s λ frontierinstatessuchthatthe (h)evolveaccordingto: t γ 7Whenever >0,eventheone-shotgame(definedin(3))cansupportsometransfers.The questionsishowmuchbettercantheplayersdounderatime-linkedcontract.Theimplementability constraintissimplythestatementthatthelattercontractmustprovideutilitynolessthanundertheone- shotgame. 8ThomasanγdWorrall(1995).AlthoughtheproofbyThomasandWorrall(1995)referstothe non-altruisticcase( =0)theextensiontothealtruisticcaseisstraightforward. 6 λ λ λ L if (h)< L λ ' λ s λ t0 λs λ (h% ) (h) if (h) [ L, U] (6) t 1 λt λ t λs s U if (h)> U s t s Theoptimalcontractmaythusbecharacterizedasfollows.Uponenteringthecontract,thetwo householdschooseadesireddistributionofexpectedutilitygainsgiventhecurrentstate(i.e.,apointon λ W(U)).ThenegativeoftheslopeontheParetofrontieratthispoint, (h )=-W ’(U))correspondstoa s 0 s0 ratioofmarginalutilitiesthatthehouseholdswillattempttoimplementsubsequently.Insubsequent periods,theleveloftransferisspecifiedsothat,giventheincomesofthetwohouseholds,theratioof λ single-periodmarginalutilitiesequalsthisprespecified aslongasthedistributionthatresultssatisfies theimplementabilityconstraintsgiventhecurrentstateandassumingthatthecontinuationpayoffs(i.e., whattheplayersexpecttoreceiveifthecontractismaintained)aredeterminedbythepointonthe λ frontierW(U)thatcorrespondstothe fromtheimmediatelyprecedingperiod.Ifoneofthe s implementabilityconstraintsisbinding(i.e,theoneforthehouseholdrequiredtomakeapositive transfer),however,boththecurrentallocationofresourcesandthecontinuationpayoffsareadjusted together,withtheadjustmentbeingassmallaspossibleinordertojustrelaxtheimplementability λ constraint.Theresulting andthusthenewratioofsingle-periodmarginalutilitiesisthenmaintained untilasubsequentimplementabilityconstraintbinds. Therelationbetweenthiscontractandthefirst-bestcontractisstraightforward.Asiswellknown (see,e.g.,Townsend1994),thefirstbestrisk-sharingcontractinvolvesequatingthemarginalutilitiesat allpointsintimeandinallstatesoftheworld.Thus,forexample,inthecaseofiso-elasticutility functions,thefirst-bestcontractwillinvolvesimplypoolingincomeineveryperiodandthendividingit accordingtosomefixeddistributionrule.Givenimperfectcommitmentandsufficientimpatience, however,inastateoftheworldinwhichonehouseholddoesparticularlywellthebetter-offhousehold 7 wouldhaveanincentivetorenegeonthisfirst-bestcontractbywithholdinghistransferandacceptingthe SSNEtransfersinthefuture.Theoptimalimplementablecontractspecifiedhereremovesthisincentive byallowingthedistributionruletobeshiftedinfavorofthebetter-offhousehold.Changesinthe distributionofresources,andthusthelossrelativetothefirst-bestarrangement,arekepttoaminimum byallowingbothcurrentandfuturedistributiontobeaffectedbytheimplementabilityconstraintswhen theybind.9 Itisnotpossibleexceptincaseswithverylimitedstatespacestoderiveanalyticalsolutionsfor decisionrulesfromthemodel.Wethusexplorethequalitativeimplicationsofcombinationsofdifferent degreesofaltruism,imperfectcommitmentandincomecovariancesusingsimulations.Weshowthat becauseofthenonstationaritycharacteroftheoptimalimplementableinsurancecontractforgiven currentincomes,ahouseholdthathasrecentlyreceivedtransfersislesslikelytoreceivesubsequent transfersthanahouseholdthathasrecentlyprovidedtransfers.Thus,exceptinthecasethatfullrisk- sharingisimplementable(inwhichcasetheimplementabilityconstraintsareneverbinding),transfers shouldbehavelikecreditinthesensethattheresultsofpastbehaviors(i.e.,outstandingdebt) importantlyaffectcurrentbehaviors(currentborrowing). Wealsoshowthataltruisminfluencesthe extentof commitmentproblemsasmanifestedinoptimalimplementableinsurancecontracts.However, whileweshowthataltruismmayfacilitaterisksharing,itisstillnecessarytoexplainwhytherearerisk- sharingarrangementsamongnon-kin,eveninlow-incomeenvironmentsthatlackformalrisk-sharing institutions.Animportantreasonisincomecovariance.Wethusalsoconsidertheimplicationsof 9Thefactthatboththecurrentandsubsequentallocationsoftransfersareaffectedwhenan implementabilityconstraintbindsimpliesthattheoptimalimplementablecontractdevisedbyThomas andWorrall(1995)isanimprovementupontheoptimalstationarycontractconsideredbyCoateand Ravallion(1993).Inthelattercaseonlycurrentallocationsofresourcesarealteredtomeetthe implementabilityconstraintresultinginlessperfectrisksharing.Thedifferencecanbesubstantial:ina numericalsimulationThomasandWorrall(1995)foundthatforsufficientlylowdiscountfactorsthatthe optimalstationarycontractresultedinnorisksharing,whereastheoptimalnon-stationarycontractcould achievemostofthegainfromfullinsurance. 8 covarianceinincomesforthenatureofrisk-sharingarrangements.Anadditionalimplicationisthat incomecovariancesamongnon-kincontractinghouseholdsareendogenouslydeterminedandpossibly relatedtowhetherfamily-basedrisk-sharingarrangementsarefeasible. Tocarryoutthesimulationsweassumethatu()andv()arelogrithmicandthusthattheperiod γ γ specific-utilityforhouseholds1and2areln(c )+ ln(c )andln(c )+ ln(c ),respectively.Incomesfor 1 2 2 1 eachhouseholdtakeononeoftwovalues,2or4,eachwithprobability½,withvaryingdegreesof correlation.Tocapturethesedifferencesincorrelationweletthestatestakeon4values,1,2,3,and4 withy (s)equalto2ifs=1ors=3and4otherwise.Similarly,welety (s)equal2ifs=1ors=2and4 1 2 otherwise.Then,thecaseofindependencecorrespondstoeachstateoccurringwithprobability1/4,the caseofapositivecorrelationof0.5canberepresentedbyassumingstates1and4occurwithprobability 3/8andstates2and3withprobability1/8,andthecaseofanegativecorrelationof-0.5canbe representedbyassumingstates1and4occurwithprobability1/8andstates2and3withprobability3/8. γ Inthesimulationsthatarepresentedbelowthediscountfactorissetto0.7510andthedegreeofaltruism isallowedtovaryfrom0to0.6.Wefocusonsymmetricoptimalimplementablecontracts,i.e.,thosefor whichthehouseholdsagreetohaveequalexpectedutilitygainsfromthecontractatthestart.We examinebelowhowtheresultsarealteredwhenthepartnersareengagedinanon-symmetric relationship. Thesolutionstothisproblemmaybecharacterizedasfollows.11Startingfromperiod0, 10Thediscountfactorischδosensothatthekeyfeaturesofthecontractmaybeillustrated.For moreusualdiscountfactors(e.g., =0.9)theoptimalimplementablecontractachievesclosetothefirst bestcontractevenintheabsenceofaltruism.Ourresults,presentedbelow,provideevidenceconsistent withthepropertiesofoursimulatedmodelforthelowdiscountfactorusedhere.Whilethisisconsistent withthenotionthatruralhouseholdsareimpatientor,asnoted,thattheyfaceasignificantexogenous riskofhavingtheagreementterminated,itmayalsobeattributedtoothersimplificationsofthemodel. Forexample,wemaketheextremeassumptionthatadevianthouseholdmustsubsequentlyconsumeat SSNElevels.Werelaxthisassumptionbyallowingforapositiveprobabilityofnonenforcementbelow. 11Solutionsforthevalueandtransferfunctionsforeachsetofparameterswerefoundbyreducing theproblem,throughappropriatesubstitution,tooneofsolvingtwonon-linearequationsintwo 9
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