Measuring Time Preference and Parental Altruism

32 Pages·2000·0.25 MB·English

TechnicalPaperSeries
CongressionalBudgetOf¿ce
Washington,D.C.
Measuring Time Preference and Parental Altruism
ShinichiNishiyama
CongressionalBudgetOf¿ce
October2000
2000-7
Technical papers in this series are preliminary and are circulated to stimulate discussion and
criticalcomment. ThesepapersarenotsubjecttoCBO¶sformalreviewandeditingprocesses.
Theanalysis and conclusions expressed in them are thoseof theauthor and should not be in-
terpreted as those of the Congressional Budget Of¿ce. Any reference to this paper in other
publicationsshouldbeclearedwiththeauthor. Papersinthisseriescanbeobtainedbysending
[email protected].
(cid:6)
Measuring Time Preference and Parental Altruism
Shinichi Nishiyama
Congressional BudgetOf¿ce
Washington,D.C.20515
e-mail: [email protected]
October2000
Abstract
This paper extends a heterogeneous agent overlapping generations model by im-
plementing bequests† both altruistic and accidental † and measures the degrees of
timepreferenceand parentalaltruismthrough the calibration of themodeltotheU.S.
economy. Inthismodel,aparenthouseholdanditsadultchildhouseholdsbehavecon-
sistentlyandstrategicallytodeterminetheiroptimalconsumption,workinghours,and
savings. Basedontheobtainedparameters,thepaperalsoshowstheindividualeffects
ofaltruisticandaccidentalbequestsonwealthaccumulationbyexaminingtheimpacts
of a 100 percent estate tax and a perfect annuity market in the model. To match the
economy¶scapital-outputratioandtherelativesizeofbequestsobservedintheUnited
States, the parent household would have to discount its children¶s utility by about 30
percentrelativetoitsownutility,accordingtothemodel. Themodelsuggeststhatto-
tal bequests contribute about 14 percent to wealth accumulation in a closed economy
and 21 percentin asmall open economy. Also, theeffect of a perfect annuity market
dependsonthedegreeofparentalaltruismintheeconomy.
1 Introduction
Most existing macroeconomic models assume either that people care about their descen-
dantsasmuchastheycareaboutthemselves(in¿nitehorizonmodels)orthattheydon¶tcare
abouttheirdescendantsatall(life-cyclemodels). Butwheneconomistsevaluategovernment
policiesthatinvolveincomeredistributionbetweengenerations,theeffectsofthesepolicies
critically depend on the degree of the altruism between parents and children. If people are
mostly altruistic, in¿nite horizon models are appropriate. But if they are mostly sel¿sh,
overlappinggenerationsmodelsarebetter.
Previousanalysesthatevaluatethedegreeofbequestmotivesusingpaneldata,however,
show several different conclusions, and it is still inconclusive to what extent parents are
W
IwouldliketothankAndyAbel,BobDennis,DougHamilton,RichardRogerson,andseminarparticipants
attheUniversityofPennsylvaniaandtheCongressionalBudgetOf¿ceforvaluablecommentsandsuggestions.
I am especially indebted to Víctor Ríos-Rull for his guidance from the early stage of this paper. The views
expressedhereindonotnecessarilyre(cid:192)ectthoseoftheCongressionalBudgetOf¿ce.
1
altruistic toward their children. For example, Hurd (1987) and Wilhelm (1996) show that
parental altruism is insigni¿cant and that bequests are likely to be accidental. In contrast,
Menchik and David (1983) and Bernheim (1991) show that the bequest motives of parents
arestrong,althoughthesearenotnecessarilycausedbyaltruism. Infact,HurdandBernheim
actuallyusedthesamedataset,theLongitudinalRetirementHouseholdSurvey,butreached
opposingconclusionsusingdifferentapproaches.
Previous studies also differ greatly with regard to estimating the importance of transfer
wealthversuslife-cyclewealth. Ontheonehand,Modigliani(1988)andothersestimatethat
theshareofbequeathedwealthamongtotalwealthisatmost20percent.1 Ontheotherhand,
Kotlikoff and Summers (1981) show that transfer wealth (which includes inter vivos trans-
fers) accounts for more than 80 percent of total wealth. The difference between those two
conclusions is signi¿cant, although it is due in large part to different de¿nitions of transfer
wealth.
Thispaperconstructsameasureoftimepreference(howmuchpeoplediscounttheirown
future utility) and a measure of parental altruism (how much they discount their children¶s
utility), and it answers the same questions using an extended dynamic general equilibrium
model.
Morespeci¿cally,thepaperusesaheterogeneousagentoverlappinggenerationsmodel,
extendedtoincludealtruisticandaccidentalbequests.2 Inthismodel,aparenthouseholdand
itsadultchildhouseholdsplayaCournot-Nashgameanddecidetheiroptimalconsumption,
workinghours,andsavingsineachperiod. ThemodeliscalibratedtotheU.S.economy. In
that calibration, the two main parameters, time preference and parental altruism, are deter-
mined simultaneously so that the steady-state equilibrium of the model replicates the U.S.
economy in termsof the following two key statistics †the capital-output ratio and the rel-
ative size of bequests. Finally, based on the parameters obtained, the paper calculates the
individualcontributionofaltruisticandaccidentalbequeststowealthaccumulation.
Forsimplicity,themodelabstractsfromintervivostransfersaswellaseducationspend-
ingbyparents. So,inthismodel,intergenerationaltransfersoccuronlyatthetimeofdeathin
theformofbequests.3 But,inthecalibration,bothbequestsandpartofintervivostransfers
inthedataareregardedasbequestsinthemodel.4 Altruisminthismodelisone-sided, i.e.,
parents care about their children but children don¶t care about their parents. In the baseline
economy,thereareassumedtobenoannuitymarketsotherthantheSocialSecuritypensions(cid:30)
later, the model is extended to include a private annuity market to analyze the contribution
ofprecautionarysavings.
1SeeModigliani(1988)forseveralestimatesofothers.
2Intentionalbequestsarenotnecessarilymotivatedbyparentalaltruismbutalsobyrisksharingandbygift
exchange. Accordingtotheempiricalstudies,however,thoselasttwomotivesarenotverylarge. Forexample,
Wilhelm(1996)showedthatparentstendtoleaveanequalamountofbequeststoeachoftheirchildreneven
if the children¶s earnings are signi¿cantly different. Also, Behrman and Rosenzweig (1998) showed that the
relationshipbetweentheamountofbequestsandthenumberofvisitsacrosschildrenisnotsigni¿cant.
3This assumption is partly justi¿ed because uncertainty about earnings and lifetime implies incentives for
altruistic parents to defer transfers to their children. But, in the presence of borrowing constraints, there are
incentivesforparentstomakeintervivostransfers.
4Inthecalibration,educationspendingpaidbyparentsandpartofintervivostransfersarenotregardedas
bequestsinthismodel. Instead,Iassumetheparent-childworkingabilitycorrelation,partlyduetoschooling,
usingaMarkovtransitionmatrix.
2
The ¿ndings in this paper are as follows. With a coef¿cient of relative risk aversion of
2.0, the annual time preference parameter turns out to be 0.934, and the degree of parental
altruism is 0.51 for each child household and 0.69 in total.5 The last-mentioned number
indicatesthataparenthouseholdcaresaboutitschildhouseholds31percentlessthanitcares
aboutitself. Ifa100percentestatetaxwasintroducedtoeliminateparentalbequestmotives,
national wealth would be reduced by 10 percent in a closed economy and by 15 percent in
a small open economy. In addition, if a perfect annuity market was introduced to eliminate
precautionary savings and accidental bequests, national wealth would be reduced, in total,
by 14.3 percent and 20.5 percent in a closed and a small open economy, respectively. The
effect of a perfect annuity market depends critically on whether parents are altruistic. With
parental altruism, the introduction of the perfect annuity market would increase national
savingsslightly.
Theremainderofthepaperislaidoutasfollows. Theextendedoverlappinggenerations
modelisdescribedinsection2. ThemodeliscalibratedtotheU.S.economy,andtwomain
parameters are obtained in section 3. The contribution of bequests to wealth accumulation
is analyzed in section 4, and section 5 concludes the paper. The algorithm to calculate the
steady-state equilibria and an optimal annuity holding are explained in the appendix to this
paper.
2 Model
Thissectiondescribesafour-periodheterogeneousagentoverlappinggenerationsmodelwith
altruistic and accidental bequests. The model considers both parental altruism and lifetime
uncertainty,andhouseholdsinthiseconomyplayaCournot-Nashgametomakeanoptimal
decisionabouttheirconsumption,workinghours,andsavings.
2.1 Economy
Themodelisbasedonastandardgrowtheconomythatconsistsofalargenumberofhouse-
holds, a perfectly competitive ¿rm, and a government. Each household is assumed to act
as a single person. In the calibration of the model, a household is assumed to be a married
couple, but all of their decisions are made jointly. Also, there is assumed to be no strategic
interactionbetweensiblings.
The Life Cycle of Households. The life of a household is shown in Figure 1. In each
period, new households are born without any wealth. The life span of each household is
eitherthree orfour periods. Oneperiod inthis modelcorrespondsto 15 yearsstartingfrom
the actual age of 30. So, age 1 corresponds to 30-44 years old, age 2 corresponds to 45-59
yearsold,age3correspondsto60-74yearsold,andage4correspondsto75-89yearsold.6 A
householddieseitherattheendofage3orattheendofage4. Themortalityrateisknown,
5Inthecalibration,Iassumethepopulationgrowthrateas1percentperyearand35percentpergeneration.
So,eachparenthouseholdhas1.35childhouseholds.
6Becauseofthissetting,householdsinthismodelareassumedtoretireatthebeginningofage60,contrary
tothefactthatmostpeopleretireateitherage62orage65intheUnitedStates.
3
Figure1: TheLifeCycleofaHousehold
Age 1 Age 2 Age 3 Age 4
(30-44) (45-59) (60-74) (75-89)
A household It becomes It dies with It dies with
is ‘born.’ Its parent dies a ‘parent’ prob. 1-(cid:28). prob. 1.
with prob. 1- (cid:28). (its children
are ‘born’).
Its parent dies
with prob. 1.
but for each household its own life span is uncertain. When a household reaches age 3, its
childhouseholdsofage1are‡born,·andtheformerbecomesaparenthousehold.
Labor Income and Capital Income. When a household is age 1 or 2, its working abil-
ity (labor productivity) at each age is stochastically determined. It receives labor income
(earnings) according to the market wage rate, its working hours, and its working ability. A
household of age3 or4 is assumedtoberetired. Thougha household can work at home to
producealimitedamountofconsumptiongoodsandservices,itsworkingabilityisassumed
to be low and deterministic. There are only one kind of assets (which are supposed to be a
mixture of bonds, stock, and real estate) a household can hold. It receives capital income
accordingtoitswealthlevelandthemarketinterestrate. Thereisassumedtobeaborrowing
constraint,andthewealthofeachhouseholdmustbenonnegative.
Taxes and Social Security Bene¿ts. A household pays federal income tax according to
its total income (the sum of labor income and capital income). A household that inherits
any wealth from its parent also pays federal and state estate taxes. In addition, a household
of age 1 or 2 pays payroll tax for Social Security and Medicare based on its labor income.
A household of age 3 or 4 receives Social Security bene¿ts. The Social Security system is
assumedtobeoneofde¿nedbene¿ttype. Everyhouseholdofage3or4iseligibleforSocial
Securitybene¿tsand,forsimplicity,thesizeofthebene¿tisassumedtobethesameforall
households.
Dynasty and Altruism. Since each household lives either three or four periods, at any
period of time there are two types of dynasties † the dynasties with both a parent house-
hold and its child households (Type I), and the dynasties with age 2 households only (Type
II). Figure 2 shows the two types of dynasties in this economy. Every parent household
caresaboutitschildhouseholdsandisassumedtobeequallyaltruistic. (Sinceaparentalso
4
Figure2: TwoTypesofDynasties
Type I (Age 4, 2)
Type II (Age 2 Only)
Generation 0:
Type I (Age 3, 1)
Generation 1:
Type I (Age 4, 2)
Generation 2:
Type I (Age 3, 1)
Generation 3:
Time
knows its children are altruistic toward its grandchildren, it actually cares about all its de-
scendantsindirectly.) Thus,aparenthouseholdchoosesend-of-periodwealth,whichwillbe
bequeathed to its child households if it dies. The wealth choice is made so as to maximize
theweightedsumofitsownutilityanditschildren¶sutility.
StrategyofaParentandaChild. Beginning-of-periodwealthofaparenthouseholdand
its child households, the working ability of the child households, and the mortality of the
parent household (at the end of age 3) are known to each other. A parent and its children
choose, simultaneously, their own optimal consumption, working hours, and end-of-period
wealth. Since a parent household is altruistic toward its child households, and the children
knowtheir parents arealtruistic, the decisions of a parent and its childrenare dependent on
eachother. Forexample,ifaparentknewitschildren¶swealthwasgoingtobehigherinthe
next period, it would reduce the amount of bequest since the marginal value of the bequest
would be smaller. Also, if a child knew the bequest of its parent was going to be higher, it
wouldreduceitsownsavingsandconsumemore.7
2.2 TheHouseholds¶Problem
2.2.1 ThePreferenceofaHousehold
ConsiderahouseholdthatliveseitherthreeorfourperiodsandcallitaGeneration0house-
hold. Ifthishouseholdissel¿sh,thehousehold¶sproblemisshownas
(cid:12)
pd{ xf @ H (cid:15)(cid:16)3(cid:129)x ff>kf .(cid:24)(cid:15)(cid:12)x ff>kf
iSf(cid:16)c(cid:5)f(cid:16)je(cid:16)’(cid:129) %[(cid:16)’(cid:129) (cid:19) (cid:16) (cid:16)(cid:20) (cid:19) e e(cid:20)&
7In this model, a parent household and its child households behave strategically. But, this model doesn¶t
considerso-calledstrategicbequests(orgiftexchanges)inwhichthechildoffersservicetotheparentinexchange
forfuturebequests.
5
subject to some budget constraint, where xf is the lifetime utility of Generation 0(cid:30) x+=>=,
is an instantaneous utility function(cid:30) ff and kf are consumption and working hours, respec-
(cid:16) (cid:16)
tively, at age l(cid:30) (cid:15) is a time preference factor(cid:30) and (cid:24) is the survival rate at the end of age 3.
Thishouseholdchoosesitsoptimalconsumptionandworkinghourstomaximizeitslifetime
utilityxf.
Suppose that Generation 0 considers not only the utility from its own consumption and
leisure(say,k (cid:3)kf)butalsotheutilityofGeneration1,andthatGeneration1considers
4@ (cid:16)
theutilityofGeneration2aswell,andsoon. ThenthetotalutilityofGeneration0(including
theutilityfromitsdescendants)Xf isshownas
Xf @ pd{ H xf.*X(cid:129)
iSfc(cid:5)fje
(cid:16) (cid:16) (cid:16)’(cid:129) k l
@ pd{ H xf.* pd{ x(cid:129).* pd{ x2. === >
iSfc(cid:5)fje 5 ;iS(cid:129)c(cid:5)(cid:129)je ;iS2c(cid:5)2je <<6
(cid:16) (cid:16) (cid:16)’(cid:129) ? (cid:16) (cid:16) (cid:16)’(cid:129) ? (cid:16) (cid:16) (cid:16)’(cid:129) @@
7 8
where*denotesthediscountfactorb=ytheparenthouseho=ldontheutilityofit>s>childhouse-
holds. Thispaperassumesthat* @ (cid:15)2(cid:20)q ? 4, whereqisthenumberofchild households
per parent household and (cid:20) is the degree of parental altruism. Since the consumption and
leisure of the child household occur two periods later, the utility is discounted by (cid:15)2. The
parent cares about its children proportionally to the number q of its child households. The
degreeofparentalaltruism(cid:20) shows howmuch the parent household cares about each of its
adultchildhouseholdsrelativetohowmuchitcaresaboutitselfinthesameperiod.
This paperdecomposes *and measuresthe degree oftime preference (cid:15) andthe degree
ofparentalaltruism(cid:20)(perchild)or(cid:20)q(intotal)throughthecalibrationusingtheaggregate
statisticsofnationalwealthandintergenerationaltransfers.
2.2.2 TheStateofaDynasty
Since the utility maximization problem of a household is nested as shown above, in the
followingsectionsthepreferenceofhouseholdsisdescribedbyvaluefunctions.
ForTypeI dynasties, the state ofeach dynasty is shownby the ages of parentand child
householdsi+6>4,>+7>5,j,thebeginning-of-periodwealthoftheparentd 5D@^3>d ‘
R 4@
andthatofitschildrend 5D>andthelaborproductivity(whichdetermineshourlywage)of
&
thechildrenh 5 H @ ^H >H ‘=Inthecalibration,h isamemberofih(cid:129) >h2 >h(cid:12) j
& 4(cid:16)? 4@ &c(cid:16) &c(cid:16) &c(cid:16) &c(cid:16)
for age l @1 or 2, and it follows a Markov process. For Type II dynasties, the state of each
dynasty is simply shown by the age i5j, the beginning-of-period wealth, and the working
abilityoftheage2households+d>h,.
Fornotationalsimplicity,letv andv denotethestatesofaTypeIdynastyandaType
U UU
IIdynasty,respectively,where
v @+d >d >h , and v @+d>h,=
U R & & UU
ThenthevaluefunctionofaTypeIhouseholdofagelisdenotedasy +v ,,andthevalue
Uc(cid:16) U
functionofaTypeIIhouseholdofage2isdenotedasy +v ,.
UUc2 UU
6
2.2.3 TypeIHouseholds
AnAge3ParentandItsAge1Children. Thevaluefunctionofanage3parenthousehold
isshownas
y +v >(cid:11) , @ pd{ ix+f >k ,.(cid:20)qx+f >k ,
Uc(cid:12) U (cid:129) SRc(cid:5)Rc@(cid:30)R R R & &
.(cid:15)H (cid:24)y +v(cid:30),.+4(cid:3)(cid:24),(cid:20)qy +v(cid:30) ,mh (1)
Uce U UUc2 UU &
subjectto (cid:5) (cid:6)(cid:12)
4
(cid:30)
d @ izh k .+4.u,d .wu (cid:3)(cid:31) +ud ,(cid:3)f j> (2)
R 4.(cid:25) R R R 77 8 R R
wherev isthestateofthisdynasty,
U
v @+d >3>h ,>
U R &
(cid:11) istheparent¶sconjectureofitsage1childhouseholds¶decision,
(cid:129)
(cid:11) +v ,@+f >k >d(cid:30),>
(cid:129) U & & &
andthelawofmotionofthestateofthisdynastyis
v(cid:30) @ d(cid:30)>d(cid:30)>h(cid:30) >
U R & &
(cid:19) (cid:20)
v(cid:30) @ d(cid:30) .d(cid:30)@q(cid:3)(cid:31) d(cid:30)@q >h(cid:30) = (3)
UU & R . R
(cid:19) (cid:19) (cid:20) (cid:20)
The parent household chooses its optimal consumption f , working hours (housework
R
(cid:30)
only) k , and end-of-period wealth level (normalized by the economic growth) d , taking
R R
the decision of its child households (cid:11) as given. It discounts the utility of each of q child
(cid:129)
households by (cid:20). At the end of age 3, the parent household dies with probability +4(cid:3)(cid:24),.
Thevalueofthishouseholdatthebeginningofthenextperiodistheweightedaverageofits
ownfuturevaluey (whenthishouseholdisalive)anditsqchildren¶sfuturevalueqy
Uce UUc2
discountedby(cid:20)(whenthishouseholdisdeceased). ThetermH^. mh ‘denotesaconditional
&
expectationgiventhatthecurrentworkingabilityofanage1childhouseholdish ,i.e.,
&
H y +v(cid:30),mh @ y +v(cid:30),(cid:28) +h(cid:30) mh ,gh(cid:30)>
Uce U & Uce U (cid:129)c2 & & &
].
(cid:5) (cid:6)
(cid:30) (cid:30)
where (cid:28) +h mh , is a conditional probability of the working ability being h in the next
(cid:129)c2 & & &
period. The equation (2) is a budget constraint of this parent household, where (cid:25) is the
growthrateoftheeconomy,zisthewageperef¿ciencyunitoflabor,u istherateofreturn
oncapital,wu denotesSocialSecuritybene¿ts,and(cid:31) +=>=,isafederalincometaxfunction.
77 8
(cid:30)
Whentheparenthouseholddies,itsend-of-periodwealthd issplitequallyandbequeathed
R
to each of q child households. In the law of motion (3), (cid:31) +=, denotes an estate tax (both
.
federalandlocal)function.
Similarly,thevaluefunctionofanage1childhouseholdisshownas
y +v >(cid:11) , @ pd{ x+f >k ,.(cid:15)H (cid:24)y +v(cid:30),.+4(cid:3)(cid:24),y +v(cid:30) ,mh (4)
Uc(cid:129) U (cid:12) S&c(cid:5)&c@(cid:30)& & & Uc2 U UUc2 UU &
(cid:11) (cid:5) (cid:6)(cid:12)
7
subjectto
4
(cid:30)
d @ izh k .+4.u,d (cid:3)(cid:31) +zh k >ud ,(cid:3)(cid:31) +zh k ,(cid:3)f j> (5)
& 4.(cid:25) & & & 8 & & & 7 & & &
where(cid:11) isthechild¶sconjectureofitsage3parenthousehold¶sdecision,
(cid:12)
(cid:11) +v ,@+f >k >d(cid:30),>
(cid:12) U R R R
andthelawofmotionofthestateis(3).
Thechildhouseholdchoosesitsoptimalconsumptionf ,workinghoursk ,andend-of-
& &
period wealth level d(cid:30)> taking the decision of its parent household (cid:11) as given. The value
& (cid:12)
ofthechildhouseholdatthebeginningofthenextperiodistheweightedaverageofitsown
future value when its parent is alive, y , and its future value when its parent is deceased,
Uc2
y . The equation (5) is a budget constraint of this child household, and (cid:31) +=, is a Social
UUc2 7
SecurityandMedicaretax(payrolltax)function.
AnAge4ParentandItsAge2Children. Anage4parenthouseholdisassumedtodieat
theendofthisperiod,andeachofitschildhouseholdsbecomesaparentatthebeginningof
thenextperiod=So,thevaluefunctionofanage4parenthouseholdisshownas
y +v >(cid:11) , @ pd{ x+f >k ,.(cid:20)qx+f >k ,.(cid:15)(cid:20)qH y +v(cid:30),mh (6)
Uce U 2 SRc(cid:5)Rc@(cid:30)R R R & & Uc(cid:12) U &
(cid:11) (cid:5) (cid:6)(cid:12)
subjectto(2),wherethelawofmotionofthestateis
v(cid:30) @+d(cid:30) .d(cid:30)@q(cid:3)(cid:31) +d(cid:30)@q,>3>h(cid:30),= (7)
U & R . R &
Theparenthouseholdconsidersitschildren¶svalueatthebeginningofthenextperiodqy
Uc(cid:12)
discountedby(cid:20)=Similarly,thevaluefunctionofanage2childhouseholdisshownas
y +v >(cid:11) , @ pd{ x+f >k ,.(cid:15)H y +v(cid:30),mh (8)
Uc2 U e S&c(cid:5)&c@(cid:30)& & & Uc(cid:12) U &
(cid:11) (cid:5) (cid:6)(cid:12)
subjectto(5),wherethelawofmotionofthestateis(7).
The Strategy of a Parent and Its Children. Let g and g be the set of decisions of a
R &
parenthouseholdandeachofitschildhouseholds,respectively,i.e.,
g @+f >k >d(cid:30),> g @+f >k >d(cid:30),=
R R R R & & & &
The best response functions of an age 3 parent and an age 1 child are derived from the
valuefunctions(1)and(4)asfollows:
U +g >v , @ duj pd{ ix+f >k ,.(cid:20)qx+f >k ,
(cid:12) & U R R & &
SRc(cid:5)Rc@(cid:30)R
.(cid:15)H (cid:24)y +v(cid:30),.+4(cid:3)(cid:24),(cid:20)qy +v(cid:30) ,mh
Uce U UUc2 UU &
subjectto(2),and (cid:5) (cid:6)(cid:12)
U +g >v ,@duj pd{ x+f >k ,.(cid:15)H (cid:24)y +v(cid:30),.+4(cid:3)(cid:24),y +v(cid:30) ,mh
(cid:129) R U S&c(cid:5)&c@(cid:30)& & & Uc2 U UUc2 UU &
(cid:11) (cid:5) (cid:6)(cid:12)
8
subjectto(5),wherethelawofmotionofthestateis(3).
Similarly, the best response functions of an age 4 parent and an age 2 child are derived
fromthevaluefunctions(6)and(8)asfollows:
U +g >v ,@duj pd{ x+f >k ,.(cid:20)qx+f >k ,.(cid:15)(cid:20)qH y +v(cid:30),mh
e & U SRc(cid:5)Rc@(cid:30)R R R & & Uc(cid:12) U &
(cid:11) (cid:5) (cid:6)(cid:12)
subjectto(2),and
U +g >v ,@duj pd{ x+f >k ,.(cid:15)H y +v(cid:30),mh
2 R U S&c(cid:5)&c@(cid:30)& & & Uc(cid:12) U &
(cid:11) (cid:5) (cid:6)(cid:12)
subjectto(5),wherethelawofmotionofthestateis(7).
Solvingthesebestresponsefunctions,Nashequilibriumdecisionrulesareobtainedas
g +v , @ f +v ,>k +v ,>d(cid:30) +v ,
Uc(cid:16) U Uc(cid:16) U Uc(cid:16) U Uc(cid:16) U
(cid:19) (cid:20)
forl5 i4>5>6>7j. Theconsistencyconditionis(cid:11) +v ,@g +v ,forallv 5D2(cid:5)H and
(cid:16) U Uc(cid:16) U U
l 5i4>5>6>7j=
2.2.4 TypeIIHouseholds
Thevaluefunctionofanage2householdissimply
y +v , @pd{ x+f>k,.(cid:15)H y +v(cid:30),mh (9)
UUc2 UU Sc(cid:5)c@(cid:30) Uc(cid:12) U
(cid:11) (cid:5) (cid:6)(cid:12)
subjectto
4
(cid:30)
d @ izhk.+4.u,d(cid:3)(cid:31) +zhk>ud,(cid:3)(cid:31) +zhk,(cid:3)fj> (10)
8 7
4.(cid:25)
wherethelawofmotionofthestateis
v(cid:30)@+d(cid:30)>3>h(cid:30),=
U &
The household chooses its optimal consumption f, working hours k, and end-of-period
wealthleveld(cid:30). Thehousehold¶sdecisionrulesareobtainedas
g +v , @ f +v ,>k +v ,>d(cid:30) +v , =
UUc2 UU UUc2 UU UUc2 UU UUc2 UU
(cid:19) (cid:20)
2.3 TheMeasureofHouseholds
Let{ +v ,denotethemeasureofTypeIhouseholdsofagel 5 i4>5>6>7j,andlet{ +v ,
Uc(cid:16) U UUc2 UU
denote that of Type II households of age 2= Also, let [ +v , and [ +v , be the corre-
Uc(cid:16) U UUc2 UU
spondingcumulativemeasures.
9