The Effect of Endogenous Human Capital Accumulation on Optimal

60 Pages·2013·0.33 MB·English

The Effect of Endogenous Human Capital Accumulation on
Optimal Taxation
WilliamBPeterman∗
May13,2013
Abstract
This paper considers the impact of endogenous human capital accumulation on optimal tax policy
in a life cycle model. Including endogenous human capital accumulation, either through learning-by-
doing or learning-or-doing, is analytically shown to create a motive for the government to use age-
dependent labor income taxes. If the government cannot condition taxes on age, then a capital tax is
used in order to mimic such taxes. Quantitatively, in my benchmark model, introducing learning-by-
doingcausestheoptimalcapitaltaxtoincreaseby7.3percentagepoints,whileintroducinglearning-or-
doingincreasestheoptimalcapitaltaxbyonly0.7percentagepoint. Overall,theoptimalcapitaltaxis
6.6percentagepointshigherinthemodelwithlearning-by-doingcomparedtothemodelwithlearning-
or-doing implying that how human capital accumulates is of significant importance when determining
theoptimaltaxpolicy.Ifindthattheeffectsoflearning-by-doingonoptimaltaxationareenhancedwhen
oneconsidersamodelwithwithincohortheterogeneity.
JEL:E24,E62,H21.
KeyWords: OptimalTaxation,CapitalTaxation,HumanCapital.
∗20thandCStreetNW,WashingtonDC20551. Tel: [phone]. E-mail: [email protected]. Viewsexpressed
onthissitearemyownanddonotreflecttheviewoftheFederalReserveSystemoritsstaff. Forextensivediscussionsandhelp-
fulcomments,Ithanktheanonymousreferee,VasiaPanousi,IrinaTelyukova,ValerieRamey,RogerGordon,andScottBorger,
aswellasseminarparticipantsatUniversityofCaliforniaatSanDiego,MadridMacroeconomicWorkshop,theFederalReserve
BoardofGovernors,theFederalReserveBankofPhiladelphia,theEasternEconomicsAssociationConference,theMissouriEco-
nomicsConference,theMidwesternMacroeconomicsConference,andtheConferenceinComputinginEconomicsandFinance.
Apreviousversionofthispaperwasdistributedunderthetitle“TheEffectofLearning-by-DoingonOptimalTaxation”
1
1 Introduction
In their seminal works, Chamley (1986) and Judd (1985) determine that it is not optimal to tax capital in
an infinitely-lived agent model. In contrast, previous studies using simplified life cycle models (Atkeson
et al. (1999), Erosa and Gervais (2002), and Garriga (2001)) demonstrate that if the government cannot
conditionlaborincometaxesonage,thenitwillgenerallybeoptimaltotaxcapitalatanon-zerorate.1 The
government ideally wants to condition tax on age since variation in age-specific human capital causes the
agenttovaryhisconsumptionandlaboroverthelifecycle.2 Whenthegovernmentcannotconditionlabor
incometaxesonage,itisgenerallyoptimaltotaxcapitalinordertomimicanage-dependenttax. Recently,
quantitativeexercisesbyConesaetal.(2009)andPeterman(2013)demonstratethatinacalibratedlifecycle
model,theinabilitytoconditiontaxesonagecanbeastrongmotiveforapositivecapitaltax.
Even though age-specific human capital is a driving mechanism for the positive optimal capital tax,
previous life cycle studies typically incorporate human capital variation in models exogenously through
age-specificproductivitylevels. Byincludinghumancapitalaccumulationexogenously, themodelsignore
any effect that endogenous accumulation may have on the optimal tax policy. This paper analytically and
quantitativelyassessestheimpactofincludingendogenousage-specifichumancapitalaccumulationonthe
optimalcapitaltaxinalifecyclemodel.
Specifically, this paper explores the effect on optimal tax policy of including either of two different
forms of endogenous age-specific human capital accumulation: learning-by-doing (LBD) and learning-or-
doing (LOD). In LBD, an agent acquires human capital by working. In LOD, which is also referred to as
BenPorathtypeskillaccumulationoron-the-jobtraining,anagentacquireshumancapitalbyspendingtime
training in periods in which he is also working.3 With LBD, an agent determines his level of age-specific
human capital by choosing the hours he works, while with LOD, an agent determines his human capital
by choosing the hours he trains. I analyze the effects of both forms, since there is empirical evidence that
each form is responsible for age-specific human capital accumulation, and each is commonly employed in
quantitativelifecyclemodels.4
1Atkesonetal.(1999)demonstratearelatedresult. Theyshowconditionsunderwhichtheoptimalcapitaltaxiszeroifage-
dependenttaxesonlaborincomeareallowed.Gervais(2010)demonstratesthataprogressivelaborincometaxcanalsomimicking
age-dependenttaxesonlaborincome.
2Idefineage-specifichumancapitalaschangesinanindividual’slaborproductivitythroughouthisworklife.Someothermodel
features, includingliquidityconstraints, retirement, andstochasticearningsmayalsocausevariationsinconsumptionandlabor
overthelifecycleandgeneratearationalefortaxesoncapitalincome.
3Thispaperdoesnotevaluatetheeffectofformaleducationonoptimaltaxpolicysincethepaperfocusesontrainingoncean
individualsbeginsworking.Therefore,thequantitativemodeliscalibratetoexcludetimespentinschool.However,themechanisms
bywhichLODchangestheoptimaltaxpolicywouldbesimilarifindividualsworkwhileattendingschool.
4ExamplesoflifecyclestudiesthatincludevariationsofLBDorLODincludeHansenand˙Imrohorogˇlu(2009),ImaiandKeane
(2004),Changetal.(2002),Jonesetal.(1997),JonesandManuelli(1999),Guvenenetal.(2009),Kuruscu(2006),Kapicka(2006),
2
I analytically demonstrate that including either LBD or LOD causes the optimal tax policy to include
age-dependent taxes. I demonstrate this result in a model with a utility function that is both separable and
homothetic with respect to consumption and hours worked. In contrast, Garriga (2001) demonstrates in
a similar model with exogenous age-specific human capital accumulation the optimal tax policy does not
include age-dependent taxes on labor income.5 Determining whether age-dependent taxes are optimal is
importantbecauseiftheyarenotinthefeasiblepolicyset,thenanon-zerocapitaltaxcanbeusedtomimic
the wedge created by conditioning labor income taxes on age. Specifically, a positive (negative) tax on
capital can be used to impose the same wedge on the marginal rate of substitution as a relatively larger
(smaller)taxonyounglaborincome.
Furthermore, the form by which one includes endogenous human capital accumulation qualitatively
changes the properties of the optimal capital income tax. In a model with exogenous skill accumulation,
anagent’sonlyincentivetoworkishiswage. InamodelwithLBD,thebenefitsfromworkingarecurrent
wages as well as an increase in future age-specific human capital. I refer to these benefits as the “wage
benefit”andthe“humancapitalbenefit,”respectively. Theimportanceofthehumancapitalbenefitdecreases
as an agent approaches retirement. Thus, adding LBD causes the agent to supply labor relatively less
elasticallyearlyinhislifecomparedwithlaterinhislife. Relyingmoreheavilyonacapitaltaxreducesthe
distortionsthatthistaxpolicyimposesontheeconomy,sinceitimplicitlytaxesthislesselasticallysupplied
laborincomefromyoungeragentsatahigherratethanolderagents. Irefertothischannelastheelasticity
channelsinceanalterationtothelaborsupplyelasticityprofileisresponsibleforthechangeintheoptimal
capitaltax.
Adding LOD to the model also causes the government to use a non-zero capital tax. There are two
channelsthroughwhichLODaffectstheoptimaltaxpolicy: theelasticitychannelandthesavingschannel.
First,addingLODchangesanagent’selasticityprofile. Trainingisanimperfectsubstituteforlaborasboth
involve forfeiting leisure in exchange for higher lifetime income. The substitutability of training decreases
as an agent ages since he has less time to take advantage of the accumulated skills. Therefore, introducing
LOD causes a young agent to supply labor relatively more elastically. Because of this elasticity channel,
theoptimalcapitaltaxislowerinordertodecreasestheimplicitrelativetaxesonlaborincomeofyounger
agents. Thesecondchannel,thesavingschannel,arisesbecausetrainingisanalternativemethodofsaving,
andKapicka(2009). Topel(1990),Cossaetal.(1999),Altug˘ andMiller(1998)provideempiricalevidenceofLBD.Theauthors
showthatpasthoursworkedandlengthofcurrentjobtenureimpactcurrentwages.WithregardstoLOD,numerousstudiesprovide
evidencethatindividualspartakeintrainingandthatthetrainingisresponsibleforwagegrowth.ForexamplesseeMulligan(1995),
FrazisandLoewenstein(2006),Kuruscu(2006),andMincer(1989).
5Ahostofworkdemonstratesasimilarsetofresultsinatwogenerationmodelwithasinglecohort. Twoexamplesofthese
worksincludeAtkinsonandStiglitz(1976),andDeaton(1979).
3
as opposed to accumulating physical capital. When the government taxes labor they implicitly decrease
the desirability to save via training as opposed to ordinary capital. In order to mitigate this distortion,
the government increases the capital tax. Since these two channels have counteracting effects, one cannot
analyticallydeterminethecumulativedirectionoftheirimpactontheoptimaltaxpolicy.6
I quantitatively assess the effect of adding each form of endogenous age-specific human capital ac-
cumulation on optimal tax policy in a calibrated life cycle model using the specific utility function from
Garriga (2001). In a calibrated overlapping generations model (OLG) model which includes exogenously
determined retirement, a reduced form social security program, lifetime length uncertainty, and exogenous
age-specific human capital (exogenous model), I find that the optimal tax rates are 18.2 percent on capital
and 23.7 percent on labor. Unlike the analytically tractable model, the optimal capital tax is non-zero with
exogenous human capital accumulation because I include additional features in this more realistic model.7
I find that adding either form of endogenous human capital increases the optimal capital tax. In the model
withLBDtheoptimaltaxratesare25.5percentoncapitaland22.1percentonlabor. Theoptimaltaxrates
in the model with the LOD framework are 18.9 percent on capital and 23.6 percent on labor. Adding en-
dogenous age-specific human capital accumulation raises the optimal capital tax by 7.3 percentage points
(approximatelyfortypercent)intheLBDframeworkand0.7percentagepoint(approximatelyfourpercent)
in the LOD framework. Overall, the optimal capital tax is 6.6 percentage points (35 percent) higher in
the model with LBD compared to the model with LOD indicating that the form by which human capital
accumulatesisquantitativelyimportantfordeterminingtheoptimaltaxpolicy.
Itestthesensitivityoftheseresultswithrespecttotheutilityfunction. Usinganalternativeutilityfunc-
tion that is neither separable nor homothetic with respect to consumption and hours worked, I find that the
optimal capital tax is much larger in the exogenous model. The optimal capital tax is larger because, with
this utility function, the Frisch labor supply elasticity profile is upward sloping regardless of the form of
human capital accumulation. Nevertheless, I find that including either form of endogenous human capital
accumulationwiththisutilityfunctioncausesanevenlargerincreaseintheoptimalcapitaltax. Theoptimal
capitaltaxincreasesbyapproximately14.5percentagepoints(fortysixpercent)and4.7percentagepoints
(fifteenpercent)whenLBDandLODareincluded,respectively. Theoptimalcapitaltaxisstill9.8percent-
agepoints(twentysevenpercent)higherintheLBDmodelcomparedtotheLODmodel. Therefore,evenin
thissetupwhichhasalargemotiveforacapitaltaxintheexogenousmodel,includingendogenoushuman
capitalaccumulationscausestheoptimalcapitaltaxtoincrease.
6Itisassumedthatthegovernmentcannotdirectlytaxhumancapitalsinceitisunobservable.
7SeePeterman(2013)foranindepthdiscussionofmotivesforapositivecapitaltaxincalibratedOLGmodel.
4
InthebenchmarkmodelincludingLBDhasalargeeffectontheoptimaltaxpolicy. However,thisspec-
ification may not capture all of the effects of LBD on optimal tax policy because it abstracts from within
cohort heterogeneity. Therefore, in this framework, the optimal tax policy only considers efficiency as op-
posed to determining the optimal tradeoff between efficiency and equity. To assess whether LBD has an
effect on this tradeoff, I examine a model that includes inequality through idiosyncratic labor productivity
shocks. IfindthatinthismodelincludingLBDcauseslargerdifferencesintheoptimaltaxpolicies. Specif-
ically, I find that the efficiency costs of a redistributive tax policy are larger in the LBD model leading to a
lessprogressiveoptimallabortax. Furthermore,IfindthattheincreaseintheoptimalcapitaltaxwhenLBD
isincludedislargerinthisspecification;theoptimalcapitaltaxincreasesfrom12.5intheexogenousmodel
to36percentintheLBDmodel.
This paper is generally related to a class of research which demonstrates that in a model where the
government has an incomplete set of tax instruments a non-zero capital tax may be optimal in order to
mimicthemissingtaxes(seeCorreia(1996),ArmenterandAlbanesi(2009),andJonesetal.(1997)). This
papercombinestworelatedstrandsoftheliteraturewithinthisclassofresearch. Thefirststrandexamines
theoptimalcapitaltaxinacalibratedlifecyclemodelwithexogenoushumancapitalwherethegovernment
cannot condition labor income taxes on age. Conesa et al. (2009), henceforth CKK, solve a calibrated life
cycle model to determine the optimal capital tax. They determine that the optimal tax policy is a flat 34
percentcapitaltaxandaflat14percentlaborincometax.8 Theystatethataprimarymotiveforimposinga
high capital tax is to mimic a relatively larger labor income tax on younger agents when they supply labor
relatively less elastically. An agent supplies labor more elastically as he ages because his labor supply is
decreasing,andtheauthorsuseautilityspecificationinwhichtheagent’sFrischlaborsupplyelasticityisa
negativefunctionofhoursworked. Peterman(2013)confirmsthatthisisaneconomicallysignificantmotive
for the positive capital tax in a model similar to CKK’s model, but concludes that the restriction on the
government from being able to tax accidental bequests at a different rate from ordinary capital income is
alsoalargecontributiontothepositiveoptimalcapitaltax.9 However,CespedesandKuklik(2012)findthat
when a non-linear mapping between hours and wages is incorporated into a model similar to CKK hours
becomemorepersistentandtheoptimalcapitaltaxfallsignificantly. Thispaperextendsthesepreviouslife
cycle studies of optimal tax policy by determining how the optimal tax policy changes when endogenous
8ThisismodelM4inConesaetal.(2009). IrefertoCKK’smodelthatabstractsfromidiosyncraticearningsriskandwithin-
cohortheterogeneitybecausetheyfindthatthesefeaturesdonotaffecttheleveloftheoptimalcapitaltax.Therefore,Ialsoabstract
fromthesefeaturesinmybenchmarkanalysis.
9Furtherwork,suchas,Karabarbounis(2012)andPeterman(2012),demonstratethatincorporatingendogenousfluctuationsin
laborsupplyontheextensivemargincanenhancethismotiveforthegovernmenttouseacapitaltaxtomimicage-dependenttaxes
onlaborincome.
5
age-specifichumancapitalaccumulationisincludedinthisstandardlifecyclemodel.
This paper is related to a second strand of the literature that analyzes the tradeoff between labor and
capital taxes in a model that includes endogenous age-specific human capital accumulation but not in a
life cycle model.10 For example, both Jones et al. (1997) and Judd (1999) examine optimal capital tax in
an infinitely lived agent model in which agents are required to use market goods to acquire human capital
similartoordinarycapital. Theyfindthatifthegovernmentcandistinguishbetweenpureconsumptionand
human capital investment, then it is not optimal to distort either human or physical capital accumulation
in the long run. Reis (2007) shows in a similar model that if the government cannot distinguish between
consumption and human capital investment, then the optimal capital tax is still zero as long as the level of
capitaldoesnotinfluencetherelativeproductivityofhumancapital. Chenetal.(2010)findinaninfinitely
lived agent model with labor search, that including endogenous human capital accumulation through both
LBD and LOD causes the optimal capital tax to increase because a higher capital tax unravels the labor
market frictions.11 This second strand of literature does not account for the effects of endogenous human
capital accumulation through life cycle channels. Since CKK and Peterman (2013) demonstrate that these
lifecyclechannelsarequantitativelyimportantformotivatingapositivecapitaltaxthispaperincludesthem.
Overall, this paper combines both strands of the literature and determines the optimal tax policy in a life
cyclemodelwithendogenouslydeterminedhumancapital.
This paper is organized as follows: Section 2 examines an analytically tractable version of the model
todemonstratethatincludingendogenoushumancapitalaccumulationcreatesamotiveforthegovernment
toconditionlaborincometaxesonage. Section3describesthefullmodelandthecompetitiveequilibrium
usedinthequantitativeexercises. Thecalibrationandfunctionalformsarediscussedinsection4. Section5
describesthecomputationalexperiment,andsection6presentstheresults. Section7teststhesensitivityof
theresultswithrespecttocalibrationparametersandutilityspecifications,whilesection8concludes.
10OnepaperthatcombinesbothstrandsisJacobsandBovenberg(2009),whichanalyzesthetradeoffbetweenalaborandcapital
taxinalifecyclemodelwithpre-workeducation. Theauthorsfindthatinatwo-periodmodelwhereagentsacquireeducationin
thefirstperiodandworkinthesecondperiodtheoptimalcapitaltaxisgenerallypositiveifeducationalinvestmentisnotverifiable.
Thetaxoncapitalreducesthetaxonlaborincome,whichinturnreducesthedistortionsonthebenefittoeducation. Jacobsand
Bovenberg(2009)isrelatedtothecurrentwork; howevertheyfocusonhumancapitalaccumulationpriortoworkingwhilethis
studyexaminestheresultsofendogenoushumancapitalaccumulationonceanagentbeginstowork.Inanotherrelatedpaper,Best
andKleven(2012)examinehowintroducingLBDchangestheoptimalgeneralincometaxinamodelwithoutsavings. Bestand
Kleven(2012)showsthatintroducingLBDcausesthegovernmenttochangetheprogressivityofthetaxratessuchthattherelative
taxonyoungincomeincreases. Thisresultissimilartotheresultinthispaper. However,inthispaperwhenthegovernmentcan
useeitheraprogressivetaxonlabororanon-zerotaxoncapitaltomimicage-dependenttaxestheychoosethetaxoncapital.
11ThelabormarketfrictionsinChenetal.(2010)causealowerlevelofemploymentintheireconomy. Acapitaltaxcausesthe
wagediscounttoincrease,thuscausingfirmstopostmorevacancieswhichinturncausesanincreaseinworkerparticipation.
6
2 Analytical Model
Inthissection,Idemonstratethataddingendogenoushumancapitalaccumulationoverturnstheresultfrom
Garriga(2001)thatforaspecificutilityfunctionthatisseparableandhomotheticinbothconsumptionand
laborthegovernmenthasnoincentivetoconditionlaborincometaxesonage. Itisusefultodetermineifthe
government wants to use age-dependent taxes because both Garriga (2001) and Erosa and Gervais (2002)
showthatifthegovernmentwantstoconditiontaxesonageandcannotdosothentheoptimalcapitaltaxwill
generallybenon-zeroinordertomimicthisage-dependenttax. Ibeginthissectionbysettinguptheagent’s
problem and demonstrating why a capital tax is an imperfect substitute for age-dependent taxes on labor
income. Next, using the primal approach, I solve for the optimal tax policy in the exogenous model, with
a benchmark utility function that is homothetic with respect to consumption and hours worked,U(c,h)=
c1−σ1 −χ(h)1+σ12 . I confirm the Garriga (2001) result that the government has no incentive to condition
1−σ1 1+σ12
labor income taxes on age and therefore the optimal capital tax is zero. I show that adding endogenous
human capital accumulation to this model causes the optimal tax policy to include age-dependent taxes.
Thereforetheseresults,coupledwiththepreviousfindingsinGarriga(2001)andErosaandGervais(2002),
demonstratethatifage-dependenttaxesareunavailableintheendogenousmodel,thenanon-zerocapitaltax
isoptimal. Ialsodemonstratethechannelsbywhichtheformsofendogenoushumancapitalaccumulation
affecttheoptimaltaxpolicy.
I derive these analytical results in a tractable two-period version of the computational model. For
tractability purposes, the features I abstract from include: retirement, population growth, progressive tax
policy,andconditionalsurvivability. Additionally,Iassumethatthemarginalproductsofcapitalandlabor
areconstant. Thisassumptionpermitsmetofocusonthelifecycleelementsofthemodel,inthatchangesto
thetaxsystemdonotaffectthepre-taxwageorrateofreturn. Sincethefactorpricesdonotvary,Isuppress
theirtimesubscriptsinthissection. Alloftheseassumptionsarerelaxedinthecomputationalmodel.
2.1 ExogenousAge-SpecificHumanCapital
2.1.1 GeneralSet-up
In the analytically tractable model, agents live with certainty for two periods, and their preferences over
consumptionandlaborarerepresentedby
U(c ,h )+βU(c ,h ) (1)
1,t 1,t 2,t+1 2,t+1
7
where β is the discount rate, c is the consumption of an age j agent at time t, and h is the percent of
j,t j,t
the time endowment the agent works.12 Age-specific human capital is normalized to unity when the agent
is young. At age two, age-specific human capital is ε . The agent maximizes equation 1 with respect to
2
consumptionandhourssubjecttothefollowingconstraints
c +a =(1−τ )h w (2)
1,t 1,t h,1 1,t
and
c =(1+r(1−τ ))a +(1−τ )ε h w, (3)
2,t+1 k 1,t h,2 2 2,t+1
where a is the amount young agents save, τ is the tax rate on labor income for an agent of age j, τ is
1,t h,j k
thetaxrateoncapitalincome,wistheefficiencywageforlaborservices,andristherentalrateoncapital. I
assumethatthetaxrateonlaborincomecanbeconditionedonage;however,thetaxrateoncapitalincome
cannot.13 Icombineequations2and3toformajointintertemporalbudgetconstraint:
c w(1−τ )ε h
2,t+1 h,2 2 2,t+1
c + =w(1−τ )h + . (4)
1,t h,1 1,t
1+r(1−τ ) 1+r(1−τ )
k k
Theagent’sproblemistomaximizeequation1subjectto4. Theagent’sfirstorderconditionsare
U (t)
h1
=−w(1−τ ), (5)
h,1
U (t)
c1
U (t+1)
h2
=−wε (1−τ ), (6)
2 h,2
U (t+1)
c2
and
U (t)
c1
=β(1+r(1−τ )), (7)
k
U (t+1)
c2
whereU (t)≡ ∂U(c1,t,h1,t). Given a social welfare function, prices, and taxes, these first order conditions,
c1 ∂c1,t
combinedwiththeintertemporalbudgetconstraint,determinetheoptimalallocationof(c ,h ,c ,h ).
1,t 1,t 2,t+1 2,t+1
12Timeworkingismeasuredasapercentageofendowmentandnotinhours.However,forexpositionalconvenience,Ialsorefer
tohj,t ashours.
13Agentsonlylivefortwoperiodsintheanalyticallytractablemodelsotheychoosenottosavewhentheyareold.Therefore,in
thismodeltherestrictiononthecapitaltaxpolicyisnotbinding.
8
2.1.2 TaxonCapitalMimicsAge-DependentTaxonLabor
To demonstrate why a capital tax has a similar effect to an age-dependent labor income tax, I derive the
intertemporalEulerequationbycombiningequations5,6,and7:
U (t) 1−τ
h1 h,1
ε =β(1+r(1−τ )) . (8)
2 k
U (t+1) 1−τ
h2 h,2
Equation8demonstratesthatifthegovernmentwantstocreateawedgeonthemarginalrateofsubstitution
byvaryingthelaborincometaxratebyage,thenτ isanalternativeoption. Apositive(negative)capitaltax
k
inducesawedgeonthemarginalrateofsubstitutionthatissimilartoarelativelylargertaxonyoung(old)
laborincome.
2.1.3 PrimalApproach
I use the primal approach to determine the optimal tax policy.14 I use a social welfare function that maxi-
mizes the expected utility of a newborn and discounts future generations with social discount factor θ (see
section5formoredetails),
∞
[U(c ,h )/θ]+∑θt[U(c ,h )+βU(c ,h )]. (9)
2,0 2,0 1,t 1,t 2,t+1 2,t+1
t=0
The government maximizes this objective function with respect to two constraints: the implementability
constraintandtheresourceconstraint.15 Theimplementabilityconstraintistheagent’sintertemporalbudget
constraint,withpricesandtaxesreplacedbyhisfirstorderconditions(equations5,6,and7)
c U (t)+βc U (t+1)+h U (t)+βh U (t+1)=0. (10)
1,t c1 2,t+1 c2 1,t h1 2,t+1 h2
Includingthisconstraintensuresthatanyallocationthegovernmentchoosescanbesupportedbyacompet-
itiveequilibrium. Theresourceconstraintis
c +c +K −K +G =rK +w(h +h ε ). (11)
1,t 2,t t+1 t t t 1,t 2,t 2
14SeeLucasandStokey(1983)orErosaandGervais(2002)forafulldescriptionoftheprimalapproach.
15Thegovernmentbudgetconstraintisathirdconstraint. DuetoWalras’Law,Ionlyneedtoincludetwoofthreeconstraintsin
theLagrangianandleaveoutthegovernmentbudgetconstraint.
9
Includingthebenchmarkutilityspecification,theLagrangianthegovernmentmaximizesis
1+ 1 1+ 1
c1−σ1 h σ2 c1−σ1 h σ2
L = 1,t −χ 1,t +β 2,t+1 −χ 2,t+1 (12)
1−σ1 1+ 1 1−σ1 1+ 1
σ2 σ2
−ρ (c +c +K −K +G −rK −w(h +h ε ))
t 1,t 2,t t+1 t t t 1,t 2,t 2
−ρ θ(c +c +K −K +G −rK −w(h +h ε ))
t+1 1,t+1 2,t+1 t+2 t+1 t+1 t+1 1,t+1 2,t+1 2
1+ 1 1+ 1
+λ (c1−σ1+βc1−σ1−χh σ2 −βχh σ2)
t 1,t 2,t+1 1,t 2,t+1
whereρistheLagrangemultiplierontheresourceconstraintandλistheLagrangemultiplierontheimple-
mentabilityconstraint.
2.1.4 OptimalTaxPolicy
Isolvefortheoptimaltaxpolicyintheanalyticallytractableexogenousmodel. Theformulationofthegov-
ernment’sproblemandtheirfirstorderconditionsforthismodelcanbefoundinappendixA.1. Combining
thegovernment’sfirstorderconditionsgeneratesthefollowingexpressionforoptimallaborincometaxes:
1−τh,2 = 1+λt(1+σ12) =1. (13)
1−τh,1 1+λt(1+σ12)
Equation13demonstratesthatthegovernmenthasnoincentivetoconditionlaborincometaxesonagewhen
age-specifichumancapitalisexogenous.16
UtilizingthefirstorderconditionfromtheLagrangianwithrespecttocapitalandconsumptionleadsto
thefollowingequation:
(cid:32) (cid:33)−σ1
c
1,t
=β(1+r). (14)
c
2,t+1
Applyingthebenchmarkutilityfunctiontoequation7providesthefollowingrelationship:
(cid:32) (cid:33)−σ1
c
1,t
=β(1+r(1−τ )). (15)
k
c
2,t+1
Equations14and15demonstratethatinorderforthehouseholdtochoosetheoptimalallocationindicated
bytheprimalapproach,thecapitaltaxmustequalzero.17
16Thisresultisspecifictothisutilityfunction.SeeGarriga(2001)forfurtherdetails.
17Regardlessofwhetherthegovernmentcanconditionlaborincometaxesonage,inthismodeltheydonotwanttotaxcapital
becausethereisnodesiretomimicanage-dependenttaxonlaborincome. Whenthegovernmentcannotconditionlaborincome
10