Imperfect Commitment, Altruism, and the Family - Andrew D. Foster

57 Pages·2000·0.22 MB·English

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