Taxing Capital? The Importance of How Human Capital is Accumulated

54 Pages·2014·0.29 MB·English

Taxing Capital? The Importance of How Human Capital is
Accumulated
WilliamBPeterman∗
January20,2014
Abstract
Thispaperconsiderstheimpactofhowhumancapitalisaccumulatedonoptimalcapitaltaxpolicy
inalifecyclemodel. Inparticular,itcomparestheoptimalcapitaltaxwhenhumancapitalisexamined
exogenously, endogenously through learning-by-doing, and endogenously through learning-or-doing.
Previous work demonstrates that in a simple two generation life cycle model, if the utility function is
homotheticinbothconsumptionandlabor,thenthegovernmenthasnomotivetoconditiontaxesonage
or tax capital. In contrast, this paper demonstrates analytically that adding either form of endogenous
human capital accumulation creates a motive for the government to use age-dependent labor income
taxes.Moreover,ifthegovernmentcannotconditiontaxesonage,thenacapitaltaxcanbeusedtomimic
such taxes. This paper quantitatively explores the strength of this channel and finds that, compared to
thebenchmarkmodelwithexogenoushumancapital,introducinglearning-by-doingcausestheoptimal
capitaltaxtoincreaseby7.3percentagepoints.Incontrast,introducinglearning-or-doingcausesamuch
smallerincreaseintheoptimalcapitaltax,0.7percentagepoints,comparedtothebenchmarkmodelwith
exogenoushumancapitalaccumulation. Takenasawhole,thispaperfindsthatthespecificformulation
bywhichhumancapitalisaccumulatedcanhavelargeimplicationsontheoptimalcapitaltax.
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
Intheirseminalworks,Chamley(1986)andJudd(1985)determinethatitisnotoptimaltotaxcapitalinan
infinitely-lived agent model. In contrast, Peterman (2013b) and Conesa et al. (2009) demonstrate that in a
lifecyclemodeltheoptimaltaxoncapitalispositive. Theauthorsshowthat, inpart, thenon-zerooptimal
capital tax is driven by the government wanting to condition taxes on age due to variation in consumption
and labor over the life cycle.1 This variation is partially due to variation in an agent’s productivity over
hislifecycle,orage-specifichumancapital. Despiteage-specifichumancapitalbeingpartiallyresponsible
for the non-zero optimal capital tax result in life cycle models, previous research tends to assume that it
is accumulated exogenous. This paper revisits optimal capital taxation by, analytically and quantitatively,
assessingtheeffectofthehumancapitalaccumulationprocessontheoptimalcapitaltax.
Specifically,thispaperexploresthechangeintheoptimalcapitaltaxwhenhumancapitalisaccumulated
exogenously,endogenouslywithlearning-by-doing(LBD),orendogenouslywithlearning-or-doing(LOD).
As opposed to being pre-determined with exogenous human capital accumulation, with LBD an agent ac-
quires human capital by working. Moreover, in LOD, which is also referred to as Ben Porath type skill
accumulation or on-the-job training, an agent acquires human capital by spending time training in periods
inwhichheisalsoworking.2 Thus,withLBD,anagentdetermineshislevelofage-specifichumancapital
by choosing the hours he works, while with LOD, an agent determines his human capital by choosing the
hourshetrains. Ianalyzetheeffectsofallthreeformssinceeachiscommonlyemployedinquantitativelife
cyclemodels.3
I analytically assess the implications of how human capital is accumulated in simple overlapping gen-
erations model (OLG) model where the utility function is both separable and homothetic with respect to
consumption and hours worked. Similar to Garriga (2001), I find that in this model the optimal tax policy
does not include age-dependent taxes on labor income and the optimal capital tax is zero.4 In contrast, I
find adding LBD or LOD causes the optimal tax policy to include age-dependent taxes. Moreover, if age-
dependenttaxesarenotavailablethenanon-zerothenanon-zerocapitaltaxcanbeusedtomimicthewedge
1Atkesonetal.(1999),ErosaandGervais(2002),andGarriga(2001)demonstratethisresultanalyticallyinasimplelifecycle
model.
2Thispaperdoesnotevaluatetheeffectofformaleducationonoptimaltaxpolicysincethepaperfocusesontrainingoncean
individualsbeginsworking.Therefore,thequantitativemodeliscalibratetoexcludetimespentinschool.However,themechanisms
bywhichLODchangestheoptimaltaxpolicywouldbesimilarifindividualsworkwhileattendingschool.
3ExamplesoflifecyclestudiesthatincludethesethreeformsofhumancapitalaccumulationareConesaetal.(2009),Conesa
andKrueger(2006),Huggettetal.(2007)Hansenand˙Imrohorogˇlu(2009),ImaiandKeane(2004),Changetal.(2002),Jonesetal.
(1997),JonesandManuelli(1999),Guvenenetal.(2009),Kuruscu(2006),Kapicka(2006),andKapicka(2009).
4Ahostofworkdemonstratesasimilarsetofresultsinatwogenerationmodelwithasinglecohort. Twoexamplesofthese
worksincludeAtkinsonandStiglitz(1976),andDeaton(1979).
2
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) tax on
younglaborincome.
Adding LBD alters the optimal tax policy because it alters an agent’s incentives to work over his life
cycle. In a model with exogenous skill accumulation, an agent’s only incentive to work is his wage. In a
model with LBD, the benefits from working are current wages as well as an increase in future age-specific
human capital. I refer to these benefits as the “wage benefit” and the “human capital benefit,” respectively.
Theimportanceofthehumancapitalbenefitdecreasesasanagentapproachesretirement. Thus,addingLBD
causes the agent to supply labor relatively less elastically early in his life compared with later in his life.
Relying more heavily on a capital tax reduces the distortions that this tax policy imposes on the economy,
sinceitimplicitlytaxesthislesselasticallysuppliedlaborincomefromyoungeragentsatahigherratethan
olderagents. Irefertothischannelastheelasticitychannelsinceanalterationtothelaborsupplyelasticity
profileisresponsibleforthechangeintheoptimalcapitaltax.
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,
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.5
Iquantitativelyassesstheeffectoftheformbywhichage-specifichumancapitalisaccumulatedonthe
optimalcapitaltaxinacalibratedlifecyclemodelusingthespecificutilityfunctionfromGarriga(2001). In
acalibratedOLGmodelwhichincludesexogenouslydeterminedretirement,areducedformsocialsecurity
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
5Itisassumedthatthegovernmentcannotdirectlytaxhumancapitalsinceitisunobservable.
3
includeadditionalfeaturesinthismorerealisticmodel.6 Ifindthataddingeitherformofendogenoushuman
capital increases the optimal capital tax. In the model with LBD the optimal tax rates are 25.5 percent on
capital and 22.1 percent on labor. The optimal tax rates in the model with the LOD framework are 18.9
percent on capital and 23.6 percent on labor. The optimal tax on capital varies by up to 7.3 percentage
points(approximatelyfortypercent)dependingonhowhumancapitalaccumulationismodeled. Therefore,
thismodelingchoiceisoffirstorderimportancewhenexaminingoptimalcapitaltaxpolicy.
I test the sensitivity of these results with respect to the utility function. Using an alternative utility
functionthatisneitherseparablenorhomotheticwithrespecttoconsumptionandhoursworked,Ifindthat
the optimal capital tax is much larger in all three models. 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
humancapitalaccumulation. Nevertheless,Ifindthattherangeofoptimalcapitaltaxesinthethreemodels
is even larger, 14.5 percentage points (approximately 45 percent). Therefore, evenin this set up which has
a large motive for a capital tax in all three models, how human capital accumulation is included has large
implicationsfortheoptimalcapitaltax.
This paper is generally related to a class of research which demonstrates that in a model where the
governmenthasanincompletesetoftaxinstrumentsanon-zerocapitaltaxmaybeoptimalinordertomimic
the missing taxes (see Correia (1996), Armenter and Albanesi (2009), and Jones et al. (1997)). This paper
combines two related strands of the literature within this class of research. The first strand examines the
optimalcapitaltaxinacalibratedlifecyclemodelbutdoesnotassesstheimportanceofhowhumancapital
is accumulated. Conesa et al. (2009), henceforth CKK, solve a calibrated life cycle model to determine
the optimal capital tax in a model with exogenous human capital accumulation. They determine that the
optimal tax policy is a flat 34 percent capital tax and a flat 14 percent labor income tax.7 They state that a
primary motive for imposing a 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, and the authors use a utility specification in which the agent’s
Frisch labor supply elasticity is a negative function of hours worked. Peterman (2013b) confirms that this
is an economically significant motive 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
ratefromordinarycapitalincomeisalsoalargecontributiontothepositiveoptimalcapitaltax.8 However,
6SeePeterman(2013b)foranindepthdiscussionofmotivesforapositivecapitaltaxincalibratedOLGmodel.
7ThisismodelM4inConesaetal.(2009). IrefertoCKK’smodelthatabstractsfromidiosyncraticearningsriskandwithin-
cohortheterogeneitybecausetheyfindthatthesefeaturesdonotaffecttheleveloftheoptimalcapitaltax.Therefore,Ialsoabstract
fromthesefeaturesinmybenchmarkanalysis.
8Furtherwork,suchas,Karabarbounis(2012)andPeterman(2012),demonstratethatincorporatingendogenousfluctuationsin
4
CespedesandKuklik(2012)findthatwhenanon-linearmappingbetweenhoursandwagesisincorporated
intoamodelsimilartoCKKhoursbecomemorepersistentandtheoptimalcapitaltaxfallsignificantly. One
exceptionisPeterman(2013a)whichcomparesoptimaltaxpolicyinamodelwithexogenoushumancapital
accumulationandLBD,butdoesnotassesstheeffectofaddingLOD.Thispaperextendsthesepreviouslife
cyclestudiesofoptimaltaxpolicybydetermininghowallthreeformsofhumancapitalaccumulationaffect
theoptimalcapitaltaxpolicy.
This paper is related to a second strand of the literature that analyzes the effect of how human capital
isaccumulatedonthetradeoffbetweenlaborandcapitaltaxesbutnotinalifecyclemodel.9 Forexample,
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 similar to ordinary capital. They
findthatifthegovernmentcandistinguishbetweenpureconsumptionandhumancapitalinvestment, then,
similar to a model with exogenous human capital accumulation, it is not optimal to distort either human
or physical capital accumulation in the long run. Moreover, Reis (2007) shows in a similar model that if
the government cannot distinguish between consumption and human capital investment, then similar to a
modelwithexogenoushumancapitalaccumulation,theoptimalcapitaltaxisstillzeroaslongasthelevelof
capitaldoesnotinfluencetherelativeproductivityofhumancapital. Chenetal.(2010)findinaninfinitely
lived agent model with labor search, that including endogenous human capital accumulation through both
LBDandLODcausestheoptimalcapitaltaxtoincrease,relativetoamodelwithexogenoushumancapital
accumulation, because a higher capital tax unravels the labor market frictions.10 This second strand of
literature does not account for the effects of endogenous human capital accumulation through life cycle
channels. Since CKK and Peterman (2013b) demonstrate that these life cycle channels are quantitatively
important for motivating a positive capital tax this paper includes them. Overall, this paper combines both
strands of the literature and determines the effect on optimal capital tax policy of how human capital is
accumulatedinalifecyclemodel.
laborsupplyontheextensivemargincanenhancethismotiveforthegovernmenttouseacapitaltaxtomimicage-dependenttaxes
onlaborincome.
9OnepaperthatcombinesbothstrandsisJacobsandBovenberg(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.
10ThelabormarketfrictionsinChenetal.(2010)causealowerlevelofemploymentintheireconomy. Acapitaltaxcausesthe
wagediscounttoincrease,thuscausingfirmstopostmorevacancieswhichinturncausesanincreaseinworkerparticipation.
5
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.
2 Analytical Model
Inthissection,Idemonstratethataddingendogenoushumancapitalaccumulationoverturnstheresultfrom
Garriga(2001)thatforaspecificutilityfunctionthatisseparableandhomotheticinbothconsumptionand
laborthegovernmenthasnoincentivetoconditionlaborincometaxesonage.11 Itisusefultodetermineif
thegovernmentwantstouseage-dependenttaxesbecausebothGarriga(2001)andErosaandGervais(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.
11AsimilarsetofresultsfortheexogenousandLBDmodelareinPeterman(2013a).Iincludetheminthispaperforcomplete-
ness.
6
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
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
12Timeworkingismeasuredasapercentageofendowmentandnotinhours.However,forexpositionalconvenience,Ialsorefer
tohj,t ashours.
13Agentsonlylivefortwoperiodsintheanalyticallytractablemodelsotheychoosenottosavewhentheyareold.Therefore,in
thismodeltherestrictiononthecapitaltaxpolicyisnotbinding.
7
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
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
14SeeLucasandStokey(1983)orErosaandGervais(2002)forafulldescriptionoftheprimalapproach.
15Thegovernmentbudgetconstraintisathirdconstraint. DuetoWalras’Law,Ionlyneedtoincludetwoofthreeconstraintsin
theLagrangianandleaveoutthegovernmentbudgetconstraint.
8
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
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
16Thisresultisspecifictothisutilityfunction.SeeGarriga(2001)forfurtherdetails.
9
Applyingthebenchmarkutilityfunctiontoequation7providesthefollowingrelationship:
(cid:32) (cid:33)−σ1
c
1,t
=β(1+r(1−τ )). (15)
k
c
2,t+1
Equations14and15demonstratethatinorderforthehouseholdtochoosetheoptimalallocationindicated
bytheprimalapproach,thecapitaltaxmustequalzero.17
2.2 Learning-by-Doing
2.2.1 IncludingLBDCreatesMotiveforAge-DependentTaxesonLaborIncome
Next,IintroduceLBDintotheexogenousmodel. IntheLBDmodel,age-specifichumancapitalforayoung
agent is normalized to unity. Age-specific human capital for an old agent is determined by the function
s (h ). The function s (h ) is a positive and concave function of the hours worked when young. In this
2 1,t 2 1,t
modelagentsmaximizethesameutilityfunctionsubjectto
c +a =(1−τ )h w (17)
1,t 1,t h,1 1,t
and
c =(1+r(1−τ ))a +(1−τ )s (h )h w. (18)
2,t+1 k 1,t h,2 2 1,t 2,t+1
Theagent’sfirstorderconditionsaregivenby
U (t) U (t+1)
h1 c2
=−[w(1−τ )+β w(1−τ )h s (t+1))], (19)
h,1 h,2 2,t+1 h1
U (t) U (t)
c1 c1
U (t+1)
h2
=−ws (h )(1−τ ), (20)
2 1,t h,2
U (t+1)
c2
and
U (t)
c1
=β(1+r(1−τ )). (21)
k
U (t+1)
c2
17Regardlessofwhetherthegovernmentcanconditionlaborincometaxesonage,inthismodeltheydonotwanttotaxcapital
becausethereisnodesiretomimicanage-dependenttaxonlaborincome. Whenthegovernmentcannotconditionlaborincome
taxesonagethentheLagrangianincludesanadditionalconstraint:
U (t) U (t+1)
ε2Uhc11(t) =Uhc22(t+1). (16)
However,intheanalyticallytractablemodelwithexogenoushumancapitalaccumulation,thisconstraintisnotbindingandthus
theLagrangemultiplierequalszero.
10