Taxing Capital? The Importance of How Human Capital is Accumulated

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Taxing Capital? The Importance of How Human Capital is Accumulated William B. Peterman 2015-117 Please cite this paper as: Peterman, William B. (2015). “Taxing Capital? The Importance of How Human Capital is Accumulated,” Finance and Economics Discussion Series 2015-117. Washington: Board of Governors of the Federal Reserve System, http://dx.doi.org/10.17016/FEDS.2015.117. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Taxing Capital? The Importance of How Human Capital is Accumulated WilliamBPeterman∗ December17,2015 Abstract Thispaperconsiderstheimpactofhowhumancapitalisaccumulatedonoptimalcapitaltaxpolicyin alifecyclemodel. Inparticular,itcomparestheoptimalcapitaltaxwhenhumancapitalisaccumulated exogenously, endogenously through learning-by-doing, and endogenously through learning-or-doing. Previous work demonstrates that in a simple two generation life cycle model with exogenous human capitalaccumulation,iftheutilityfunctionisseparableandhomotheticineachconsumptionandlabor, then the government has no motive to condition taxes on age 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, if the government cannotconditiontaxesonage,thenacapitaltaxcanbeoptimalinordertomimicsuchtaxes. Thispaper quantitativelyexploresthestrengthofthischannelandfindsthat,includinghumancapitalaccumulation withlearning-by-doing,asopposedtoexogenously,causestheoptimalcapitaltaxtoincreasebybetween 7.3and14.5percentagepoints.Incontrast,introducinglearning-or-doingcausesamuchsmallerincrease intheoptimalcapitaltaxofbetween0.7and3.7percentagepoints. Takenasawhole, thispaperfinds that the specific formulation by which human capital is accumulated can have notable implications on theoptimalcapitaltax. JEL:E24,E62,H21. KeyWords: OptimalTaxation,CapitalTaxation,HumanCapital. ∗20thandCStreetNW,WashingtonDC20551.Tel:202-452-3703.E-mail:william.b.peterman@frb.gov.Viewsexpressedon thissitearemyownanddonotreflecttheviewoftheFederalReserveSystemoritsstaff. Forextensivediscussionsandhelpful comments,Ithanktheanonymousreferee,VasiaPanousi,IrinaTelyukova,ValerieRamey,RogerGordon,andScottBorger,aswell asseminarparticipantsatUniversityofCaliforniaatSanDiego, MadridMacroeconomicWorkshop, theFederalReserveBoard ofGovernors,theFederalReserveBankofPhiladelphia,theEasternEconomicsAssociationConference,theMissouriEconomics Conference,theMidwesternMacroeconomicsConference,andtheConferenceinComputinginEconomicsandFinance. 1

1 Introduction Intheirseminalworks,Chamley(1986)andJudd(1985)determinethatitisnotoptimaltotaxcapitalinan infinitely-livedagentmodel. Incontrast,Peterman(2013)andConesaetal.(2009)demonstratethatinalife cyclemodeltheoptimaltaxoncapitalispositive. Theauthorsshowthat,inpart,thenon-zerooptimalcapital taxisdrivenbythegovernmentwantingtoconditiontaxesonageduetovariationinconsumptionandlabor over the life cycle.1 This variation in consumption and labor is partially due to fluctuations in an agent’s productivityoverhislifecycle,orage-specifichumancapital. Despitetheimportanceofage-specifichuman capitalforthenon-zerooptimalcapitaltaxresultinlifecyclemodels,previousresearchtendstoassumethat it is accumulated exogenously. One exception, Peterman (2015) demonstrates that incorporating a specific form of endogenous human capital accumulation can cause considerable effects on the optimal capital tax. However, less is known about how different forms of endogenous human capital accumulations affect the optimal capital tax. Thus, this paper revisits optimal capital taxation by, analytically and quantitatively, assessing the effect of various different human capital accumulation processes on the optimal capital tax. Overall, this paper finds that the way in which human capital is accumulated can have considerable effects ontheoptimalcapitaltax. 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 acquires human capital by working. Alternatively, in LOD, which is also referred to as Ben Porath type skill accumulationoron-the-jobtraining,anagentacquireshumancapitalbyspendingtimetraininginperiodsin whichheisalsoworking.2 Thus,withLBD,anagentdetermineshislevelofage-specifichumancapitalby choosingthehoursheworks,whilewithLOD,anagentdetermineshishumancapitalbychoosingthehours hetrains. Ianalyzetheeffectsofallthreeformssinceeachiscommonlyemployedinquantitativelifecycle modelssounderstandingtheeffectontheoptimalcapitaltaxofthedifferenthumancapitalassumptionsis important.3 First I analytically assess the implications of how human capital is accumulated in simple overlapping 1Atkesonetal.(1999),ErosaandGervais(2002),andGarriga(2001)demonstratethisresultanalyticallyinasimplelifecycle model. 2Thispaperdoesnotevaluatetheeffectofformaleducationonoptimaltaxpolicybutinsteadfocusesonhumancapitalacquired afteranindividualsbeginsworking.Although,thequantitativemodeliscalibratetoexcludetimespentinschool,themechanisms bywhichLODchangestheoptimaltaxpolicywouldbesimilarforformaleducation. Foradiscussionoftheeffectsofformal educationonoptimaltaxationseeJacobsandBovenberg(2009). 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). 2

generations model (OLG) model where the utility function is both separable and homothetic with respect to consumption and hours worked. Garriga (2001) finds that in this type of model with exogenous human capital accumulation the optimal tax policy does not include age-dependent taxes on labor income and the optimalcapitaltaxiszero.4 Incontrast,IfindaddingLBDorLODcausestheoptimaltaxpolicytoinclude age-dependent taxes. Moreover, if age-dependent taxes are not available then a non-zero capital tax can be used to mimic 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 relativelylarger(smaller)taxonyounglaborincome. 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. The importance of the human capital benefit, which is unique to LBD, decreases as an agent approaches retirement. Thus, adding LBD 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, since it implicitly taxes this less elastically supplied labor income from agentswhentheyareyoungeratarelativelyhigherratethanwhentheyareolder. Irefertothischannelas theelasticitychannelsinceitistransmittedthroughchangesinthelaborsupplyelasticityprofile. AddingLODtothemodelalsocauseschangestotheoptimalcapitaltax. Therearetwochannelsthrough whichLODaffectstheoptimaltaxpolicy: theelasticitychannelandthesavingschannel. First,addingLOD changesanagent’selasticityprofile. Trainingisanimperfectsubstituteforlaborasbothinvolveforfeiting 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 youngagenttosupplylaborrelativelymoreelastically. WithLOD,theelasticitychannelcausestheoptimal capitaltaxtobelowerinordertodecreasestheimplicittaxesonlaborincomewhenagentsareyounger. The secondchannel,thesavingschannel,arisesbecausetrainingisanalternativemethodofsaving,asopposed toaccumulatingphysicalcapital. Whenthegovernmenttaxeslabortheyimplicitlydecreasethedesirability of saving via training as opposed to ordinary capital. In order to mitigate this distortion, the government increasesthecapitaltaxandreducesthelabortax. Sincethesetwochannelshavecounteractingeffects,one cannotanalyticallydeterminethecumulativedirectionoftheirimpactontheoptimaltaxpolicy.5 4Ahostofworkdemonstratesasimilarsetofresultsinatwogenerationmodelwithasinglecohort. Twoexamplesofthese worksincludeAtkinsonandStiglitz(1976),andDeaton(1979). 5Itisassumedthatthegovernmentcannotdirectlytaxhumancapitalsinceitisunobservable. 3

Next, I quantitatively assess the effect of the form by which age-specific human capital is accumulated on the optimal capital tax in a calibrated life cycle model which includes exogenously determined retirement, areducedformsocialsecurityprogram, lifetimelengthuncertainty. Ifindthataddingeitherformof endogenoushumancapitalincreasestheoptimalcapitaltaxcomparedtotheoptimalcapitaltaxwithexogenous human capital accumulation.6 When LBD is included, I find that the optimal tax on capital increases between 7.3 and 14.5 percentage points.7 LBD has In contrast, when LOD is included, the optimal tax on capital increases only between 0.7 and 4.7 percentage points compared to the rates with exogenous human capitalaccumulation. Thus,theoptimaltaxoncapitalvariesbyupto14.5percentagepointsdependingon how human capital accumulation is modeled. Therefore, this modeling choice is of first order importance whendeterminingoptimalcapitaltaxpolicy. 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.8 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 (2013) 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 6Unliketheanalyticallytractablemodel,Ifindthatinthemodelwithexogenoushumancapitalaccumulationtheoptimaltax oncapitalisbetween18.2and31.8percentdependingontheformoftheutilityfunction. SeePeterman(2013)foranindepth discussionofmotivesforapositivecapitaltaxincalibratedOLGmodelwithexogenoushumancapitalaccumulation. 7TherangeoftheincreasesintheoptimalcapitaltaxisbecauseIfindthatthesizeoftheeffectisdifferentdependingonthe utilityfunction. 8ThisismodelM4inConesaetal.(2009). IrefertoCKK’smodelthatabstractsfromidiosyncraticearningsriskandwithincohortheterogeneitybecausetheyfindthatthesefeaturesdonotaffecttheleveloftheoptimalcapitaltax. Moreover,Peterman (2015)findsthattheeffectofLBDontheoptimalcapitaltaxwithidiosyncraticearningsriskissimilartotheeffectinthispaper. However, both studies find that the within cohort heterogeneity from idiosyncratic productivity shocks does affect the optimal progressivity of the labor tax. Therefore, I exclude these features in my benchmark analysis in order to focus on the effects of humancapitalaccumulationontheoptimalcapitaltaxandabstractformtheeffectofendogenoushumancapitalaccumulationon theoptimalprogressivity. 4

ratefromordinarycapitalincomeisalsoalargecontributiontothepositiveoptimalcapitaltax.9 Moreover, CespedesandKuklik(2012)findthatwhenanon-linearmappingbetweenhoursandwagesisincorporated intoamodelsimilartoCKKhoursbecomemorepersistentandtheoptimalcapitaltaxfallsignificantly. All ofthesestudiesassumehumancapitalisaccumulatedexogenously. Thus,thispaperextendstheseprevious life cycle studies of optimal tax policy by determining how all three forms of human capital accumulation affecttheoptimalcapitaltaxpolicy. Thispaperisrelatedtoasecondstrandoftheliteraturethatanalyzestheeffectofhowhumancapitalis accumulated on the tradeoff between labor and capital taxes 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 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 model with exogenous human capital accumulation, the optimal capital tax is still zero as long as the level of capital does not influence the relative productivity of human capital. Chen et al. (2010) find in an infinitely lived agent model with labor search, that including endogenous human capital accumulation causestheoptimalcapitaltaxtoincrease, relativetoamodelwithexogenoushumancapitalaccumulation, becauseahighercapitaltaxunravelsthelabormarketfrictions.11 Thissecondstrandofliteratureisunableto accountfortheeffectsofendogenoushumancapitalaccumulationthroughlifecyclechannels. SinceCKK and Peterman (2013) demonstrate that these life cycle channels are quantitatively important for motivating a positive capital tax this paper includes them. Thus, this paper combines both strands of the literature anddeterminestheeffectonoptimalcapitaltaxpolicyofhowhumancapitalisaccumulatedinalifecycle model.12 9Furtherwork,suchas,Karabarbounis(2012)andPeterman(2012),demonstratethatincorporatingendogenousfluctuationsin laborsupplyontheextensivemargincanenhancethismotiveforthegovernmenttouseacapitaltaxtomimicage-dependenttaxes onlaborincome. 10Inarelatedpaper,BestandKleven(2012)examinehowintroducingLBDchangestheoptimalgeneralincometaxinamodel withoutsavings. BestandKleven(2012)showthatintroducingLBDcausesthegovernmenttochangetheprogressivityofthetax ratessuchthattherelativetaxonyoungincomeincreases. Thisresultissimilartotheresultinthispaper. However,inthispaper when the government can use either a progressive tax on labor or a non-zero tax on capital to mimic age-dependent taxes they choosethetaxoncapital. 11ThelabormarketfrictionsinChenetal.(2010)causealowerlevelofemploymentintheireconomy. Acapitaltaxcausesthe wagediscounttoincrease,thuscausingfirmstopostmorevacancieswhichinturncausesanincreaseinworkerparticipation. 12TwoexceptionsthatexaminetheeffectsofendogenoushumancapitalaccumulationinalifecyclemodelarePeterman(2015) anddaCostaandSantos(2015)ontheoptimaltaxpolicy. However,theseothertwostudiesincludewithincohortheterogeneity andfocusontheoptimalprogressivityofthelabortaxandoptimallevelofthecapitaltaxthatbalancesequityversusefficiency.In contrast,thisstudyexcludesthewithincohortheterogeneityandfocusesonhowtheoptimalcapitaltaxthatmaximizesefficiency 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). In particular including either form of endogenous human capital accumulation in a model withautilityfunctionthatisseparableandhomotheticineachconsumptionandlaborthegovernmentcreates an incentive to condition labor income taxes on age. In contrast, Garriga (2001) finds that with exogenous human capital accumulation the government does not want to condition labor income taxes on age.13 It is useful to determine if the government wants to use age-dependent taxes because both Garriga (2001) and Erosa and Gervais (2002) show that if the government wants to condition taxes on age and cannot do so, thentheoptimalcapitaltaxwillgenerallybenon-zeroinordertomimicthisage-dependenttax. 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,inordertofocusonthelifecycleelementsofthemodelI assumethatthemarginalproductsofcapitalandlaborareconstantandthusfactorpricesareconstant. Since the factor prices do not vary, I suppress their time subscripts in this section. All of these assumptions are relaxedinthecomputationalmodel. 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 changeswithdifferentformsofhumancapitalisaccumulation. 13AsimilarsetofresultsfortheexogenousandLBDmodelareinPeterman(2015).Iincludetheminthispaperforcompleteness. 6

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.14 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.15 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 14Timeworkingismeasuredasapercentageofendowmentandnotinhours.However,forexpositionalconvenience,Ialsorefer tohj,t ashours. 15Agentsonlylivefortwoperiodsintheanalyticallytractablemodelsotheychoosenottosavewhentheyareold.Therefore,in thismodeltherestrictiononthecapitaltaxpolicyisnotbinding. 7

2.1.2 TaxonCapitalMimicsAge-DependentTaxonLabor Whenexaminingtheoptimalcapitaltaxitusefultodetermineifitisoptimaltoconditionlaborincometaxes onage. ExaminingtheintertemporalEulerequation, U (t) 1−τ h1 h,1 ε =β(1+r(1−τ )) . (8) 2 k U (t+1) 1−τ h2 h,2 it is clear that if the government wants to create a wedge on the marginal rate of substitution by varying the labor income tax rate by age, then τ is an alternative option. A positive (negative) capital tax induces k a wedge on the marginal rate of substitution that is similar to a relatively larger tax on young (old) labor income. Thus,throughouttheanalyticalanalysis,itwillofimportancetodeterminewhetherage-dependent laborincometaxesareoptimal. 2.1.3 OptimalTaxPolicy Next, 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+σ 1 2 . I solve for the 1−σ1 1+ σ 1 2 optimaltaxpolicyusingtheprimalapproachwhichimpliesIsolvefortheoptimalallocation(seeAppendix A.1 for details of the approach). From this optimal allocation, one can determine the optimal tax policy. In particular, I find that the optimal allocation implies the following ratio for the optimal labor taxes (see AppendixA.2fortheformulationoftheproblem), 1−τ h,2 = 1+λ t (1+ σ 1 2 ) =1. (9) 1−τ h,1 1+λ t (1+ σ 1 2 ) Equation9demonstratesthatinthismodelthegovernmenthasnoincentivetoconditionlaborincometaxes onagewhenage-specifichumancapitalisexogenous.16 Moreover, using the primal approach, the optimal allocation of consumption is represented by the followingexpression, (cid:32) (cid:33)−σ1 c 1,t =β(1+r). (10) c 2,t+1 16λistheLagrangemultiplierontheimplementabilityconstraint. SeeAppendixA.1formoredetails. Thisresultisspecificto thisutilityfunction.SeeGarriga(2001)forfurtherdetails. 8

Assumingthebenchmarkutilityfunction,theoptimalallocationindicatedbytheprimalapproachis, (cid:32) (cid:33)−σ1 c 1,t =β(1+r(1−τ )). (11) k c 2,t+1 Thus,theoptimalcapitaltaxiszero. AsGarriga(2001)pointsout,sincethereisnodesiretoconditionlabor income taxes on age in this exogenous model, the optimal tax on capital is zero regardless of whether the governmentcanconditionlaborincometaxesonage.17 2.2 Learning-by-Doing 2.2.1 IncludingLBDCreatesMotiveforAge-DependentTaxesonLaborIncome Next, I examine the LBD model. In the LBD model, age-specific human capital for a young agent is normalizedtoone. Age-specifichumancapitalforanoldagentisdeterminedbythefunctions (h ). The 2 1,t function s (h ) is a positive and concave function of the hours worked when young. In this model agents 2 1,t maximizethesameutilityfunctionsubjectto, c +a =(1−τ )h w, (13) 1,t 1,t h,1 1,t and c =(1+r(1−τ ))a +(1−τ )s (h )h w. (14) 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))], (15) h,1 h,2 2,t+1 h1 U (t) U (t) c1 c1 U (t+1) h2 =−ws (h )(1−τ ), (16) 2 1,t h,2 U (t+1) c2 and U (t) c1 =β(1+r(1−τ )). (17) k U (t+1) c2 17WhenthegovernmentcannotconditionlaborincometaxesonagethentheLagrangianincludesanadditionalconstraint, U (t) U (t+1) ε2U h c1 1 (t) = U h c2 2 (t+1) . (12) However,intheanalyticallytractablemodelwithexogenoushumancapitalaccumulation,thisconstraintisnotbindingandthus theLagrangemultiplieronthisconstraintwouldbeequaltozero. 9

The first order conditions with respect to h and a are similar in the LBD (equations 16 and 17) and 2 1 exogenous models (equations 6 and 7). However, the first order condition with respect to h is different in 1 the two models (equations 15 and 5) because working has the additional human capital benefit in the LBD model. The formulation for the government’s problem and the resulting first order conditions (utilizing the benchmarkutilityfunction)areinappendixA.3. Theoptimalallocationisrepresentedby, 1−τ h,1 = 1−τ h,2 (cid:16) (cid:17)(cid:16) (cid:17) 1+λ (1+ 1 )−λ (1+ 1 ) h1,tsh1 (t+1) 1+ h2,t+1s2 (18) t σ2 t σ2 s2 1+r(1−τk) − h 2,t+1 s h2 (t+1) , 1+λ (1+ 1 )+h 1+ σ 1 2h 1+− σ2 1 λt (cid:16) sh1 (t+1) −s (t+1) (cid:17) 1+r(1−τ k ) t σ2 2,t+1 1,t s2 s2 h1,h1 where s represents the partial derivative of the skill function for an older agent with respect to hours h1 worked when young. Equation 18 demonstrates that generally in the LBD model the government has an incentive to condition labor income taxes on age. Moreover, they will generally want to tax labor income at a relatively higher rate when agents are young.18 This result contrasts with the exogenous model, in whichthegovernmenthasnoincentivetoconditionlaborincometaxesonage(seeequation9). AsGarriga (2001),ErosaandGervais(2002),andPeterman(2013)demonstrate,ifthegovernmentwantstocondition laborincometaxesonagebutage-dependenttaxesarenotallowedthenthegovernmentwilltypicallyusea non-zerocapitaltaxtomimicthistypeofage-dependenttaxpolicy. 2.2.2 LBDEnhancesMotiveforPositiveTaxonCapital In order to get a sense of why the government wants to tax labor income when an agent is young at a relatively higher rate, I examine the intertemporal Euler equation (determined by combining equations 15, 16and17), U (t) 1−τ h1 h,1 s (h ) =β(1+r(1−τ )) +βh s (t+1). (19) 2 1,t k 2,t+1 h1 U (t+1) 1−τ h2 h,2 IncludingLBDcausestheintertemporalEulerequationtohaveanextrapositivetermontherighthandside (see equation 8 and equation 19). Therefore, holding all else equal and setting ε =s , the tax on young 2 2 laborincomewouldneedtoberelativelyhigherinordertoinducethesamewedgeonthemarginalrateof substitutionintheLBDmodel. ExaminingtheFrischelasticitiesintheexogenousandLBDmodels,providestheintuitionwhyadding 18Inparticular,therelativetaxonyounglaborishigherthanthetaxonoldlaborincomeaslongasλispositive. 10

LBDincreasestheoptimalrelativetaxonyounglaborincomeortaxoncapital. Sincethefunctionalforms oftheseelasticitiesextendtoamodelwhereagentsliveformorethantwoperiods, Idenoteanagent’sage with i. In the exogenous model, the Frisch elasticity simplifies to Ξ =σ . The Frisch elasticity in the exog 2 LBDmodelis,Ξ = σ2 . 19 LBD 1− hi+1,t+1wt+1(hi,tσ2shi,hi(t+1)−shi(t+1)) si,t(1+rt(1−τk))wt TheFrischelasticityintheexogenousmodelisconstantandvaluedatσ . IntheLBDmodel,theextra 2 terms in Ξ increase the size of the denominator, thus holding hours and consumption constant between LBD the two models, Ξ >Ξ . Intuitively, the inclusion of the human capital benefit makes workers less exog LBD responsive to a one-period change in wages since the wage benefit is only part of their total compensation for working in the LBD model. Moreover, the human capital benefit does not have a constant effect on an agent’s Frisch elasticity over his lifetime. The relative importance of the human capital benefit decreases as an agent ages because he has fewer periods to use his human capital.20 Therefore, adding LBD causes an agent to supply labor relatively less elastically when they are young than when they are old. This shift in relative elasticities creates an incentive for the government to tax the labor income when agents are younger at a relatively higher rate. Thus, if the government cannot condition labor income taxes on age, thentheoptimalcapitaltaxwillbehigherintheLBDmodeltomimicthisage-dependenttax. Iusetheterm “elasticity channel” to describe the effect on optimal tax policy caused by a change in the Frisch elasticity from including endogenous human capital. The elasticity channel is responsible for the change in optimal taxpolicyfromincludingLBD.21 2.3 Learning-or-Doing 2.3.1 IncludingLODCreatesMotiveforAge-DependentTaxesonLaborIncome Next, I examine the LOD model to demonstrate that this form of endogenous age-specific human capital accumulation also creates a motive for the government to condition labor income taxes on age. Similar to the other models, age-specific human capital for a young agent is normalized to one. Age-specific human capital for an old agent is determined by the function s (n ) which is a positive and concave function 2 1,t 19SincethisistheFrischelasticitywithrespecttoatemporaryincreaseinthewage,onemustdistinguishbetweenwt andw t+1 . 20Forthehumancapitalbenefittodeclineoverthelifetime,itissufficienttoassumeagentsworkforafinitenumberofperiods. 21Alternative intuition for this result can be demonstrated in the commodity tax framework of Corlett and Hague (1953). In theirstaticframework,thegovernmentwantstotaxleisure. However,iftheycannotdirectlytaxleisure,thegovernmentwilltax commoditiesthataremorecomplementarytoleisureatahigherrate.Viewingthissimpletwogenerationmodelinthatframework, adding LBD raises the relative opportunity cost of leisure when agents are young so young labor is less of substitute (more of a complement) with leisure. This change leads the government to want to increase the tax on young labor. Moreover, if the governmentcannotuseage-dependenttaxesthentheywillincreasethetaxoncapitaltoimplicitlytaxconsumptionfromtheold atarelativelyhigherratesinceLBDmakesconsumptionandleisuremorecomplementaryfortheolderagentsthantheyounger agents. 11

of the hours spent training when an agent is young (n ). In the LOD model, I need a utility function 1,t that incorporates training. I alter the benchmark utility specification so that it consistently incorporates the disutility of non-leisure activities, c1−σ1 −χ (h+n) 1+σ 1 2 . In this model agents maximize their utility function 1−σ1 1+ σ 1 2 subjectto, c +a =(1−τ )h w, (20) 1,t 1,t h,1 1,t and c =(1+r(1−τ ))a +(1−τ )s (n )h w. (21) 2,t+1 k 1,t h,2 2 1,t 2,t+1 Theagent’sfirstorderconditionsaregivenby, U (t) h1 =−[w(1−τ )], (22) h,1 U (t) c1 U (t+1) h2 =−ws (n )(1−τ ), (23) 2 1,t h,2 U (t+1) c2 U (t) c1 =β(1+r(1−τ )), (24) k U (t+1) c2 and U (t) n1 =−βw(1−τ )s (n )h . (25) h,2 n1 1,t 2,t+1 U (t+1) c2 The first order conditions with respect to h , h , and a are similar in the LOD model (equations 22, 23, 1 2 1 and24)andtheexogenousmodel(equations5,6,and7). However,sinceagentshavetheadditionalchoice variablen intheLODmodel,thereisanadditionalfirstordercondition(equation25). 1 The formulation of the government’s problem and resulting first order conditions are provided in appendixA.4.22 Combingthefirstorderconditionsyieldsthefollowingrelationshipforoptimaltaxesonlabor income, (cid:16) (cid:17) 1+λ 1+ h1,t + ηts2 1−τ h,2 = t σ2(h1,t+n1,t) σ2(h1,t+n1,t) . (26) (cid:16) (cid:17) (cid:16) (cid:17) 1−τ h,1 1+λ 1+ 1 −η s (t+1) 1+ 1 t σ2 t n1 σ2 Equation 26 demonstrates that the government generally has an incentive to condition labor income taxes onagewhenLODisintroducedintothemodel. Although equation 26 shows that including LOD creates an incentive for the government to condition labor income taxes on age, it is unclear at which age the government wants to impose a relatively higher labor income tax. Comparing equations 9 and 26, there are two channels through which introducing LOD 22ηt istheLagrangemultiplieronanadditionalconstraintthatisincludedtoensurethatintheoptimalallocationsboth23and 25arerespected. 12

changes the optimal tax policy. The first channel results from using a utility function that is non separable intrainingandlabor. Thenonseparabilityaffectstheoptimaltaxpolicythroughtheelasticitychannelsince it causes LOD to alter the Frisch elasticity. This channel causes the numerator of the ratio to include the additionalterm h1,t . Asaresultofthisnewterm,theexpressiondecreases. h1,t+n1,t The second channel results from the intertemporal link created because agents can save not only with ordinarysavingsinthismodelbutalsocansaveviatraining. Irefertothischannelasthesavingschannel. (cid:16) (cid:17) This second channel causes the inclusion of the additional terms −η s (t+1) 1+ 1 and ηts2 in t n1 σ2 σ2(h1,t+n1,t) the denominator and numerator, respectively.23 Assuming that η is positive, these additional terms cause t the expression to increase.24 Thus, the two channels may have opposing effects on the optimal tax policy, andtheoveralleffectisunclear. Examining the Frisch labor supply elasticities provides intuition for how the first channel affects the optimal tax policy. Since the altered utility function is not additively separable in time spent working and training, the Frisch labor supply elasticity is not constant in the LOD model. The Frisch elasticity for the altered utility function is Ξ = σ2(h+n) . This functional form implies that an agent supplies labor LOD h relatively more elastically with LOD than with exogenous human capital accumulation because the agent hasasubstituteforworkingintheformoftraining. Additionally,theeffectontheFrischelasticityislarger when he spends a larger proportion of his non-leisure time training (or when training is a better substitute for generating lifetime income). Therefore, if an agent spends less time training as he ages, then he will supply labor relatively more elastically when he is young, and the government would want to tax the labor income from agents when they are young at a relatively lower rate. Decreasing the tax on capital mimics thistypeofage-dependenttax. Thus,theelasticitychannelfromLODcancausesadecreaseintheoptimal capitaltax. Examininganagent’sfirstorderconditionwithrespecttotrainingdemonstrateshowthesavingschannel affects the optimal tax policy. An agent optimizes his choices such that the marginal disutility of training when he is young equals the marginal benefit of training (U (t)= Uc1(t)w(1−τh,2 )h2,t+1sn1(t+1) ). The marginal n1 1+r(1−τk) benefitisincreasedbyraisingthetaxoncapitalorbydecreasingthetaxonolderlaborincome. Byadopting either of these changes, the government makes it relatively more beneficial for the agent to use training to saveasopposedtoordinarycapital.25 23Theterminthenumeratorcomesfromboththeintertemporallinkandthenonseparabilityoftheutilityfunction. However,I groupbothtermsinthesavingschannelbecausetheimpactontheoptimaltaxpolicywillbeinthesamedirectionastheotherterm. 24Thesignofηwilldependonwhetherthegovernmentwantstoincreasetherelativeincentivetosavewithtrainingorcapital. Ifηispositive,itimpliesthatthegovernmentwantstoincreasetherelativeincentivetosavewithtraining. Igenerallyfindinthe computationalsimulationsthatηispositiveandthereforetreatitaspositiveintheexposition. 25Anadditionalreasonthatthegovernmentwantstoincreasethecapitaltaxistounwindthedistortiontosavingsbehaviorthat areinducedbyapositivelaborincometax. 13

3 Computational Model Next,Ideterminethedirectionandmagnitudeoftheeffectofhowhumancapitalisaccumulatedonoptimal capitaltaxpolicyinalessparsimoniousmodel. Isolvefortheoptimaltaxpolicyinseparateversionsofthe model with exogenous human capital accumulation, LBD and LOD. CKK and Peterman (2015) find that idiosyncratic earnings risk and heterogenous ability types can affect the optimal progressivity of the labor taxbutdonotaffecttheoptimalcapitaltax. Thus,Iexcludethesesourcesofheterogeneityinmymodelin ordertofocusonthemechanismsthatmayaffecttheoptimalcapitaltax. 3.1 Demographics Inthecomputationalmodel,timeisassumedtobediscrete. Agentsenterthemodelwhentheystartworking at the age of 20, and can live to a maximum age of J. Thus, the model is populated with J-19 overlapping generations. Conditionalonbeingaliveatage j, Ψ istheprobabilityofanagentlivingtoage j+1. Ifan j agentdieswithassets,theassetsareconfiscatedbythegovernmentanddistributedequallytoalltheliving agentsastransfers(Tr ). Allagentsarerequiredtoretireatanexogenouslysetage j . t r In each period a cohort of new agents is born. The size of the cohort born in each period grows at rate n. Given a constant population growth rate and conditional survival probabilities, the time invariant cohort shares,{µ }J ,aregivenby, j j=1 Ψ j−1 µ = µ ,fori=2,....,J, (27) j j−1 1+n whereµ isnormalizedsuchthat 1 J ∑ µ =1. (28) j j=20 3.2 Individual An individual is endowed with one unit of productive time per period that he divides between leisure and non-leisure activities. In the exogenous and LBD models the non-leisure activity is providing labor. In the LODmodelthenon-leisureactivitiesincludetrainingandworking. Anagentchoosesconsumptionaswell ashowtospendhistimeendowmentinordertomaximizehislifetimeutility, J−j−1 s u(c ,h +n )+ ∑ βs∏(Ψ )u(c ,h +n ), (29) j j j q s+1 s+1 s+1 s=20 q=1 wherec istheconsumptionofanagentatage j,h isthehoursspentworking,n isthetimespenttraining, j j j andβisthediscountfactorconditionalonsurviving. 14

In the exogenous model, an agent’s age-specific human capital is ε . In the endogenous models, an j agent’s age-specific human capital, s , is endogenously determined. In the LBD model, s is a function of j j askillaccumulationparameter(Ω ), previousage-specifichumancapital(s ), andtimeworkedinthe j−1 j−1 previous periods(h ), denoted by s =S (Ω ,s ,h ). In the LOD model, s is a function of a j−1 j LBD j−1 j−1 j−1 j skill accumulation parameter (Ω ), previous age-specific human capital (s ), and time spent training j−1 j−1 (n ), denoted by s =S (Ω ,s ,n ). The sequence of skill accumulation parameters {Ω } jr−1 j−1 j LOD j−1 j−1 j−1 j j=20 arecalibrationparameterssetsothatintheendogenousmodel,underthebaseline-fittedU.S.taxpolicy,the agent’s choices result in an agent having the same age-specific human capital as in the exogenous model. Individualscommandalaborincomeofh ε w intheexogenousmodelandh s w intheendogenousmodel. j j t j j t Agentssplittheirincomebetweenconsumptionandsavingusingarisk-freeasset. Anagent’slevelofassets isdenoteda ,andtheassetpaysapre-taxnetreturnofr . j t 3.3 Firm Firmsareperfectlycompetitivewithconstantreturnstoscaleproductiontechnology. Aggregatetechnology isrepresentedbyaCobb-Douglasproductionfunction. Theaggregateresourceconstraintis, C +K −(1−δ)K +G ≤KαN1−α, (30) t t+1 t t t t where K, C, and N represent the aggregate capital stock, aggregate consumption, and aggregate labor t t t (measuredinefficiencyunits),respectively. Additionally,αisthecapitalshareandδisthedepreciationrate forphysicalcapital. Unliketheanalyticallytractablemodel,Idonotassumealinearproductionfunctionin thecomputationalmodel,sopricesaredeterminedendogenouslyandfluctuatewithregardtotheaggregate capitalandlabor. 3.4 GovernmentPolicy Thegovernmenthastwofiscalinstrumentstofinanceitsunproductiveconsumption,G .26 First,thegovernt menttaxescapitalincome,y ≡r (a+Tr ),accordingtoacapitalincometaxscheduleTK[y ]. Second,the k t t k governmenttaxeseachindividual’staxablelaborincome. Partofthepre-taxlaborincomeisaccountedfor bytheemployer’scontributionstosocialsecurity,whichisnottaxableundercurrentU.S.taxlaw. Therefore, thetaxablelaborincomeisy ≡w s h (1−.5τ ),whichistaxedaccordingtoalaborincometaxschedule l t j j ss 26AsopposedtoassumingGtisunproductive,includingGtsuchthatitenterstheagent’sutilityfunctioninanadditivelyseparable mannerwillresultinthesameoptimaltaxpolicies. 15

Tl[y ]. Iimposefourrestrictionsonthelaborandcapitalincometaxpolicies. First,Iassumehumancapital l is unobservable, meaning that the government cannot tax human capital accumulation. Second, I assume theratescannotbeage-dependent. Third, bothofthetaxesaresolelyfunctionsoftheindividual’srelevant taxableincomeinthecurrentperiod. Finally,Iexcludetheuseoflumpsumtaxes. Inadditiontoraisingresourcesforconsumptionintheunproductivesector,thegovernmentrunsapayas-you-go (PAYGO) social security system. I include a simplified social security program in the model because Peterman (2013) demonstrates that excluding this type of program in a model with exogenously determined retirement causes unrealistic life cycle profiles and can alter the optimal tax policy. In this reduced-form social security program, the government pays SS to all individuals that are retired. Social t security benefits are determined such that retired agents receive an exogenously set fraction, b , of the t average income of all working individuals.27 Social security is financed by taxing labor income at a flat rate,τ . Thepayrolltaxrateτ issettoassurethatthesocialsecuritysystemhasabalancedbudgeteach ss,t ss,t period. Thesocialsecuritysystemisnotconsideredpartofthetaxpolicythatthegovernmentoptimizes. 3.5 DefinitionofStationaryCompetitiveEquilibrium InthissectionIdefinethecompetitiveequilibriumfortheexogenousmodel. SeeappendixBforthedefinitionofthecompetitiveequilibriumsintheendogenousmodels. Givenasocialsecurityreplacementrateb,asequenceofexogenousage-specifichumancapital{ε } jr−1 , j j=20 governmentexpendituresG,andasequenceofpopulationshares{µ }J ,astationarycompetitiveequilibj j=20 riumintheexogenousmodelconsistsofthefollowing: asequenceofagentallocations,{c ,a ,h }J ,a j j+1 j j=20 productionplanforthefirm(N,K),agovernmentlabortaxfunctionTl:R →R ,agovernmentcapitaltax + + functionTk :R →R ,asocialsecuritytaxrateτ ,autilityfunctionU :R ×R →R ,socialsecurity + + ss + + + benefitsSS,prices(w,r),andtransfersTrsuchthat: 1. Givenprices,policies,transfers,andbenefits,theagentmaximizesequation29subjectto c +a =wε h −τ wε h ,+(1+r)(a +Tr)−Tl[wε h (1−.5τ )]−Tk[r(a +Tr)], (31) j j+1 j j ss j j j j j ss j for j< j ,and r c +a =SS+(1+r)(a +Tr)−Tk[r(a +Tr)], (32) j j+1 j j for j≥ j . r Additionally, c≥0,0≤h≤1,a ≥0,a =0. (33) j 20 27Althoughanagent’ssocialsecuritybenefitsareafunctionoftheaverageincomeofallworkers,sinceallagentsarehomogenous withinacohort,thebenefitsaredirectlyrelatedtoanindividual’spersonalearningshistory. 16

2. Priceswandrsatisfy (cid:18) N (cid:19)1−α r=α −δ (34) K and (cid:18) K (cid:19)α w=(1−α) . (35) N 3. Thesocialsecuritypoliciessatisfy wN SS=b (36) ∑ j jr = − 2 1 0 µ j and τ = ss∑ J j=jr µ j . (37) ss w∑ j j r = − 2 1 0 µ j 4. Transfersaregivenby J Tr= ∑ µ (1−Ψ )a . (38) j j j+1 j=20 5. Governmentbalancesitsbudget J jr−1 G= ∑ µ Tk[r(a +Tr)]+ ∑ µ Tl[wε h (1−.5τ )]. (39) j j j j j ss j=20 j=20 6. Themarketclears J K = ∑ µ a , (40) j j j=20 J N = ∑ µ ε h , (41) j j j j=20 and J J ∑ µ c + ∑ µ a +G=KαN1−α+(1−δ)K. (42) j j j j+1 j=20 j=20 4 Calibration and Functional Forms To determine the optimal tax policy, it is necessary to choose functional forms and calibrate the model’s parameters. Calibratingthemodelsinvolvesatwo-stepprocess. Thefirststepischoosingparametervalues forwhichtherearedirectestimatesinthedata. TheseparametervaluesareinTable1. Second,tocalibrate theremainingparameters, valuesarechosensothatunderthebaseline-fittedU.S.taxpolicycertaintargets inthemodelmatchthevaluesobservedintheU.S.economy.28 ThesevaluesareinTable2. 28Since these are general equilibrium models, changing one parameter will alter all the values in the model that are used as targets.However,Ipresenttargetswiththeparameterthattheymostdirectlycorrespondto. 17

Table1: CalibrationParameters Parameter Value Target Demographics RetireAge: j 65 ByAssumption r MaxAge: J 100 ByAssumption Surv. Prob: Ψ BellandMiller(2002) Data j Pop. Growth: n 1.1% Data FirmParameters α .36 Data δ 8.33% I =25.5% Y A 1 Normalization Adding endogenous human capital accumulation to the model fundamentally changes the model. Accordingly, if the calibration parameters are the same, then the value of the targets will be different in the endogenousandexogenousmodels. Thus,Icalibratethesetofparametersbasedontargetsseparatelyinthe threemodels. Thiscalibrationimpliesthattheseparametersaredifferentinthethreemodels. 4.1 Demographics Agentsenterthemodelatageof20whentheybegintoworkandareexogenouslyforcedtoretireatareal worldageof65. Ifanindividualsurvivesuntiltheageof100,theydiethenextperiod. Isettheconditional survival probabilities in accordance with the estimates in Bell and Miller (2002) and assume a population growthrateof1.1percent. Table2: CalibrationParameters Parameter Exog. LBD LOD Target CalibrationParameters ConditionalDiscount: β 0.995 0.993 0.997 K/Y =2.7 UnconditionalDiscount: Ψ β 0.982 0.980 0.984 K/Y =2.7 j Riskaversion: σ 2 2 2 CKK 1 FrischElasticity: σ 0.5 0.73 0.47 Frisch= 1 2 2 DisutilityofLabor: χ 61 46 80 Avg. h +n = 1 j j 3 GovernmentParameters ϒ .258 .258 .258 GouveiaandStrauss(1994) 0 ϒ .768 .768 .768 GouveiaandStrauss(1994) 1 G 0.137 0.136 0.13 17%ofY b 0.5 0.5 0.5 CKK 18

4.2 Preferences Agents have time-separable preferences over consumption and labor services. I use the benchmark utility function for the exogenous and LBD models, c1−σ1 −χ (h) 1+σ 1 2 , and an altered form of this utility function 1−σ1 1+ σ 1 2 fortheLODmodel, c1−σ1 −χ (h+n) 1+σ 1 2 .29 1−σ1 1+ σ 1 2 I determine β such that the capital-to-output ratio matches U.S. data of 2.7.30 One reason that the reduced form social security program is included is to capture the relevant savings motives that affect the capital to output ratio. I determine χ such that under the baseline-fitted U.S. tax policy, agents spend on average one third of their time endowment in non-leisure activities. Following CKK, I set σ =2, which 1 controls the relative risk aversion.31 Past micro-econometric studies (such as Altonji (1986), MaCurdy (1981), and Domeij and Flode´n (2006)) estimate the Frisch elasticity to be between 0 and 0.5. However, morerecentresearchhasshownthattheseestimatesmaybebiaseddownward. Reasonsforthisbiasinclude: utilizing weak instruments; not accounting for borrowing constraints; disregarding the life cycle effect of endogenous-age specific human capital; omitting correlated variables such as wage uncertainty; ignoring secondaryearners; and notaccounting forlabor marketfrictions.32 Therefore, Isetσ suchthat theFrisch 2 elasticityisattheupperboundoftherange(0.5). Thepreferenceparametersaresummarizedintable2. 4.3 Age-SpecificHumanCapital Theage-specific humancapital parametersare differentinthe threemodels. Inthe exogenousmodel, Iset {ε } jr−1 sothatthesequencematchesasmoothedversionoftherelativehourlyearningsestimatedbyagein j j=20 Hansen(1993). Intheendogenousmodels,IusethefunctionalformsfromHansenand˙Imrohorogˇlu(2009). Specifically, in the LBD model, agents accumulate age-specific human capital according to the following process, s =Ω sΦ1hΦ2, (43) j+1 j j j wheres istheage-specifichumancapitalforanagentatage j,Ω isanage-specificcalibrationparameter, j j Φ controls the importance of an agent’s current human capital on LBD, and Φ controls the importance 1 2 of time worked on LBD. In the LOD model, agents accumulate human capital according to the following 29Using this benchmark utility function for the exogenous, which is homothetic and separable, implies that the Frisch labor supplyelasticityisconstantasopposedtobeingafunctionoftheleveloflaborsupply. Thisflexibilityallowsmetoisolatethe effectsofeachofthechannelsontheoptimaltaxpolicy. 30Thisistheratiooffixedassetsandconsumerdurablegoods,lessgovernmentfixedassetstoGDP(CKK). 31EventhoughCKKuseadifferentutilityspecification,theirspecificationhasaparameterthatcorrespondstoσ1 . 32SomeofthesestudiesincludeImaiandKeane(2004),DomeijandFlode´n(2006),Pistaferri(2003),Chetty(2009),Peterman (2016),andContrerasandSinclair(2008). 19

process, s =Ω sκ1nκ2, (44) j+1 j j,t j wheren isthepercentofanagent’stimeendowmenthespendstraining. Inthisformulation,κ controlsthe j 1 importanceofanagent’scurrenthumancapitalonLODandκ controlstheimportanceoftimetrainingon 2 LOD.Intheendogenousmodels,Icalibratethesequence{Ω } jr−1 suchthattheagent’sequilibriumlabor j j=20 or training choices cause {s } jr−1 under the baseline-fitted U.S. tax code to match the age-specific human j j=20 capitalcalibratedintheexogenousmodel({ε } jr−1 ).33 j j=20 To calibrate the rest of the LBD parameters, I rely on the estimates in Chang et al. (2002), setting Φ =0.407 and Φ =0.326. Following Hansen and ˙Imrohorogˇlu (2009), I set κ =1 and κ =0.004 in 1 2 1 2 the LOD model. Both functional forms imply full depreciation of skills if individuals choose not to work or train at all in the LBD and LOD models, respectively. In the case of the LBD model, full depreciation willneverbebindingbecauseagentschoosetoworklargequantitiesinallperiodsintheexogenousmodel which does not include the additional human capital incentive for working . In the LOD model, I find that if I include skill accumulation with a function form that is separable in past skills and training time, so as tonotimplyfulldepreciationwhenagentsdonottrain,thenthelife-cycleprofilesaremoreconsistentwith formaleducationasopposedtotraining.34 Therefore,Iusethisnonseparablefunctionalformwiththevalue of κ =1 which implies that there is little depreciation of human capital as long as agents use just a small 1 amountoftheirtimeendowmentfortraining.35 Thevaluesofκ and{Ω } jr−1 implythatatthestartofan 2 j j=20 agent’s career the ratio of time spent training to working is approximately 10 percent and declines steadily until retirement. Through the agent’s entire working life, the ratio of the average time spent training to market hours is about 6.25 percent. This average value is in line with the calibration target in Hansen and ˙Imrohorogˇlu(2009).36 4.4 Firm IassumetheaggregateproductionfunctionisCobb–Douglas. Thecapitalshareparameter, α, issetat.36. Thedepreciationrateissettotargetaninvestmentoutputratioof25.5percent. 33Icalibratethesesetsofparameterssuchthattheyaresmoothoverthelifecycle. 34Guvenenetal.(2009)useanalternativeLODaccumulationspecificationthatisadditivelyseparableinpastskillsandtraining. I find that when I use this specification an agent does not accumulate any assets for the first 10-15 years of their working life, andinsteadtendstosaveusingskillaccumulation. Inaddition,duringthistimeagentsworkonlythenecessaryhourstofinance consumptioncausingtheirlaborsupplyprofiletobelowandflat(seeFigure5inGuvenenetal.(2009)). Sincetheshapeofthese lifecycleprofilesdoesnotmatchthedata,Ichoosenottousethisfunctionalform. 35SeeKuruscu(2006)andHeckmanetal.(1998)forotherexamplesofquantitativestudiesthatassumelittledepreciation. 36Mulligan(1995)providesempiricalestimatesofhoursspentonemployerfinancedtrainingthataresimilartothecalibration target. 20

4.5 GovernmentPoliciesandTaxFunctions Before calibrating the parameters so that the model matches targets in the data, I need to set a baseline tax function that mimics the U.S. tax code. I use the estimates of the U.S. tax code in Gouveia and Strauss (1994)forthistaxpolicy,whichIrefertoasthebaseline-fittedU.S.taxpolicy. TheauthorsmatchtheU.S. taxcodetothedatausingathreeparameterfunctionalform, T(y;ϒ 0 ,ϒ 1 ,ϒ 2 )=ϒ 0 (y−(y−ϒ1+ϒ 2 ) − ϒ 1 1), (45) where y represents the sum of labor and capital income. The average tax rate is principally controlled by ϒ , and ϒ governs the progressivity of the tax policy. To ensure that taxes satisfy the budget constraint, 0 1 ϒ is left free. Gouveia and Strauss (1994) estimate that ϒ =.258 and ϒ =.768 when fitting the data. 2 0 1 Theauthorsdonotfitseparatetaxfunctionsforlaborandcapitalincome. Accordingly,Iuseauniformtax systemonthesumofbothsourcesofincomeforthebaseline-fittedU.S.taxpolicy. Icalibrategovernment consumption, G, so that it equals 17 percent of output under the baseline-fitted U.S. tax policy, consistent with CKK. In particular, ϒ is determined as the value that equates government spending to 17 percent of 2 GDP. Whensearchingfortheoptimaltaxpolicy,Irestrictmyattentiontorevenueneutralchangesthatimply that government consumption is equal under the baseline-fitted U.S. tax policy and the optimal tax policy. However,whensearchingfortheoptimaltaxpolicy,Iallowthetaxratesoncapitalandlabortodiffer. Thegovernmentalsorunsabalanced-budgetsocialsecurityprogram. Socialsecuritybenefitsaresetso that the replacement rate, b, is 50 percent.37 The payroll tax, τ , is determined so that the social security ss systemisbalancedeachperiod. 5 Computational Experiment Thecomputationalexperimentisdesignedtodeterminethetaxpolicythatmaximizesagivensocialwelfare functionholdinggovernmentrevenueconstant. Ichooseasocialwelfarefunction(SWF)thatcorrespondsto aRawlsianveilofignorance(Rawls(1971)). Whensearchingfortheoptimaltaxpolicy,Isearchoverboth flat and progressive tax policies. However, I determine that the optimal tax policy consists of separate flat taxes on capital and labor income. For notational convenience, I present the computational experiment as 37ThereplacementratematchestherateinCKKandConesaandKrueger(2006).TheSocialSecurityAdministrationestimates thatthereplacementratioforthemedianindividualis40percent(seetableVI.F10inthe2006SocialSecurityTrusteesReport; availableatwww.ssa.gov/OACT/TR/TR06/). ThisestimateislowerthanthereplacementrateIuse;however,ifonealsoincludes thebenefitspaidbyMedicare,thentheobservedreplacementratiowouldbehigher. 21

choosingtheoptimalflattaxratesoncapitalandonlabor.38 Sincelivingagentsfacenoearningsuncertainty, thesocialwelfareisequivalenttomaximizingtheexpectedlifetimeutilityofanewborn, J−j−1 s SWF(τ ,τ )=u(c ,h )+ ∑ βs ∏(Ψ )u(c ,h ), (46) h k j j q s+1 s+1 s=20 q=20 whereτ istheflattaxrateonlaborincomeandτ istheflattaxrateoncapitalincome. h k 6 Results Inthissection,Iquantitativelyassesstheeffectsontheoptimalcapitaltaxpolicyofhowage-specifichuman capitalisaccumulatedinmybenchmarklifecyclemodel. Ideterminetheoptimaltaxpoliciesintheexogenous,LBD,andLODmodelsandthenhighlightthechannelsthatcausethedifferences. Tofullyunderstand theeffectsofhumancapitalaccumulation,Ianalyzetheaggregateeconomicvariablesandlifecycleprofiles inallthreemodelsunderthebaseline-fittedU.S.taxpolicy,aswellasthechangesinducedbyimplementing theoptimaltaxpolicies. 6.1 OptimalTaxPoliciesinExogenous,LBD,andLODModels Table3describestheoptimaltaxpoliciesinthethreemodels. Theoptimaltaxpolicyintheexogenousmodel isan18.2percentflatcapitalincometax(τ =18.2%)anda23.7percentflatlaborincometax(τ =23.7%). k h Unliketheanalyticallytractablemodel,theoptimalcapitaltaxinthecomputationalexogenousmodelisnot zero due to: the inability of the government to borrow; agents being liquidity constrained, the government not being able to tax transfers at a separate rate from ordinary capital income, and exogenous retirement coupledwithsocialsecurity.39 Including LBD, I find that the optimal capital tax increases 7.3 percentage points (forty percent) to 25.5% and the optimal labor tax decreases to 22.1%.40 The alteration in the Frisch labor supply elasticity 38ThisfindingissimilartoCKKwhofindthattheoptimaltaxpoliciesareflatintheirmodelwithoutwithincohortheterogeneity. However,incontrast,Gervais(2010)findsthatthegovernmentpreferstousebothataxoncapitalandaprogressivetaxonlabor incometomimicanage-dependenttax. 39Iincludesomeofthesefeaturesthatmotivateapositivecapitaltaxsothatincentivesinthemodelcorrespondtotheincentives intheU.S.economy. Forexample,thereducedformsocialsecurityprogramisnecessarysothatthelevelofindividualsavings arerealistic. Otherofthesefeaturesareincludedtoclosethemodelinatractablemanner. SeePeterman(2013)forathorough discussionoftherelativestrengthsofeachofthesemotivesinamodelsimilartotheexogenousmodel. 40AlthoughtheirarelargedifferencesbetweentheoptimaltaxpolicyintheLBDandexogenousmodelsthewelfarelossesinthe LBDfromadoptingtheoptimaltaxpolicysolvedforintheexogenousmodelcausesawelfarelossthatisequivalenttoonly0.1% oflifetimeconsumption. ThesmallwelfareeffectsfromadoptingthewrongoptimaltaxpolicyarenotsurprisingsincePeterman (2015)showsthatthewelfarelossesfromadoptingsub-optimallevelsofthecapitalandlabortaxaremuchsmallerthanadopting asub-optimallevelofprogressivity. 22

Table3: OptimalTaxPoliciesinBenchmarkModels TaxRate Exog LBD LOD τ 18.2% 25.5% 18.9% k τ 23.7% 22.1% 23.6% h τk 0.77 1.16 0.8 τh profile is the principal reason that the optimal capital tax increases in the LBD model. The left panel of Figure 1 plots the lifetime Frisch labor supply elasticities in the LBD model and the exogenous model.41 The lifetime labor supply elasticity is flat in the exogenous model and upward sloping in the LBD model. AddingLBDcausesagentstosupplylaborrelativelymoreelasticallyastheyagebecausethehumancapital benefitdecreases. TheoptimalcapitaltaxishigherintheLBDmodelinordertoimplicitlytaxagentswhen theyareyounger,andsupplylaborlesselastically,atahigherrate. ToconfirmthattheelasticitychannelisresponsibleforthechangeintheoptimaltaxpolicyintheLBD model, I vary σ by age in a counterfactual exogenous model (LBD elasticity) so that the shape of the 2 lifetime Frisch labor supply elasticity profile is the same as it is in the LBD model under the optimal tax policy. Ifindthattheoptimaltaxpolicyinthiscounterfactualexogenousmodel(LBDelasticity),τ =25.6% k andτ =22.1%,isalmostidenticaltotheoptimaltaxpolicyintheLBDmodel. Thus,theelasticitychannel h isprimarilyresponsibleforthechangeintheoptimalcapitaltaxintheLBDmodel. The optimal tax policy in the LOD model is τ =18.9% and τ =23.6%. The optimal capital tax in k h theLODmodelisonly0.7percentagepointlarger(approximatelyfivepercent)comparedtotheexogenous modeland6.6percentagepointssmaller(approximatelytwentyfivepercent)comparedtotheLBDmodel. Insection2.3.1,Ishowanalyticallythatboththeelasticitychannelandthesavingschannelaffecttheoptimal capitaltaxintheLODmodel. TherightpanelofFigure1plotstheFrischelasticityprofileintheexogenous and LOD models. Compared to the exogenous model, adding LOD to the model causes agents to supply laborrelativelymoreelasticallywhentheyareyoungversuswhentheyareold. Theelasticitychannelcauses adecreaseintheoptimalcapitaltaxsothatagentsareimplicitlytaxedatalowerratewhentheyareyoung. Additionally, the inclusion of LOD allows individuals to use training to save, which activates the savings channel. To quantify the effect of the channels, I solve for the optimal tax policy in an alternative version of the 41Theprofilesaredeterminedundertheoptimaltaxpolicies. 23

LODmodelthatusesanalternativeutilityfunction, c1−σ1 (h) 1+ σ 1 2 (n) 1+ σ 1 2 −χ −χ , (47) 1−σ 1 1 1+ 1 2 1+ 1 σ2 σ2 which is separable in training and hours worked. Using this alternative utility function eliminates the elasticity channel since the Frisch labor supply elasticity in this LOD is no longer a function of time spent training.42 Eliminatingtheelasticitychannelmeansthatonlythesavingschannelremainsinthisalternative LODmodel(separableutility)withthisalternativeutilityfunction.43 Theoptimaltaxpolicyinthisalternative LOD model (separable utility) is τ =19.9% and τ =23.3%. These results indicate that the savings k h channel causes a 1.7 percentage point increase in the optimal capital tax in order to encourage agents to saveviahumancapitalasopposedtophysicalcapital. Thetotalincreaseintheoptimalcapitaltaxisjust0.7 percentagepointswhenbothchannelsareincludedinthebenchmarkLODmodelandtheoptimalcapitaltax increases 1.7 percentage points in the alternative LOD model (separable utility) with just the saving channel. Theseincreasesimplythattheelasticitychannelcausestheoptimalcapitaltaxtodecrease1percentage point,cancelingjustoverhalfofthesavingschannel’seffect. Figure1: LifeCycleFrischLaborSupplyElasticityinEndogenousModel LBDModel Frisch Elasticity 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 20 40 60 80 100 yticitsalE hcsirF LODModel Frisch Elasticity 0.53 0.52 Exogenous LBD 0.51 0.5 0.49 0.48 0.47 0.46 20 40 60 80 100 Age yticitsalE hcsirF Exogenous LOD Age Fromtheagesof20to63,addingLODhasanopposite,butsimilarlysized,effectontheFrischelasticity profileasaddingLBD.However,theelasticitychannelhasamuchsmallereffectontheoptimaltaxpolicy intheLODmodel. Therearetwopotentialreasonsthattheelasticitychannelmayhavealargereffectinthe LBDmodel. First,theFrischelasticityincreasesrapidlyoverthelasttwoworkingyearsintheLBDmodel making the overall magnitude of the slope much steeper in the LBD model compared to the LOD model. 42InparticulartheFrischelasticitysiσ2 sincetheutilityfunctionisseparableinallthreearguments. 43Thisalternativeutilityfunctionalsoeliminatespartoftheimpactofthesavingschannelsotheseresultsarealowerboundon theimpactofboththesavingsandelasticitychannel.Seethesection2.3.1formoredetails. 24

Toconfirmtheslopeoverthelastfewyearsisnotthereasonfortheelasticitychannelhavingalargereffect ontheoptimalcapitaltaxintheLBDmodelversustheLODmodel,Ideterminetheoptimaltaxinanother counterfactualexogenousmodel(smoothedLBDelasticity)whichmatchestheslopeoftheFrischelasticity in the LBD model only from ages 20 - 63.44 Again I find that the optimal tax policy in this counterfactual exogenousmodel(smoothedLBDelasticity)isalmostidenticaltotheoptimaltaxintheLBDmodel. Thus themagnitudeoftheeffectoftheelasticitychannelisnotstrongerintheLBDmodelversustheLODmodel because the Frisch elasticity in the LBD model increases rapidly over the last few years of the working lifetime. Instead, the reason that the elasticity channel in the LOD model has a smaller effect on the optimal tax policy is that it causes young agents to be more liquidity constrained. Figure 2 plots the labor supply profile for a young agent. The solid line represents the labor supply profile in the exogenous model under the optimal tax policy. The dashed line represents the labor supply profile, under the same tax policy, but inacounterfactualexogenousmodel(LODelasticity)calibratedsuchthatthelaborsupplyelasticityprofile matches the LOD model.45 In all the models, young agents tend to work less compared to middle aged agents since their lower human capital implies their total wage is lower. In this counterfactual exogenous model(LODelasticity), thelaborsupplyelasticityishigherforyoungeragentscomparedtothelaborsupply elasticity in the unaltered exogenous model. Therefore, in this counterfactual exogenous model (LOD elasticity)agentstendtoworkrelativelylesshourswhentheyareyoungcomparedtothebenchmarkmodel. Becausetheysupplylesslabor,theseagentsaremoreliquidityconstrainedwhentheyareyounginthecounterfactual exogenous model (LOD elasticity). The optimal tax policy includes a larger capital tax in order to alleviate these binding liquidity constraints by shifting some of the tax burden to later in an agent’s life, whenheisnolongerliquidityconstrained.46 Thus,thedecreaseintheoptimalcapitaltaxintheLODmodel due to the downward sloping elasticity profile is somewhat offset by the desire to increase the capital tax duetoliquidityconstraintsbeingexacerbatedintheLODmodel.47 44Inthiscounterfactualexogenousmodel(smoothedLBDelasticity),IsettheFrischelasticityatages64and65equaltoage63 intheLBDmodel. 45Forthemostrelevantcomparison,IchoosetomatchthelaborsupplyelasticityprofileintheLODmodelundertheoptimaltax policy. 46Furthermore,themotivetoshiftthetaxburdenawayfromtheseearlieryearswhenagentsareliquidityconstrainedisenhanced because,intheLODmodel,theseyoungerliquidityconstrainedagentsprovidelabormoreelasticallywhichenhancesthedistortions frombindingliquidityconstraints. 47Foradetaileddiscussionofmagnitudeoftherelationshipbetweenliquidityconstraintsandtheoptimalcapitaltax,seePeterman (2013)andCKK. 25

Figure2: AffectofLODElasticityonYoungLaborSupply 0.385 0.38 0.375 0.37 0.365 0.36 0.355 0.35 0.345 20 25 30 35 tnemwodnE emiT fo % Exog. Alt. Exog. (LOD Elast.) Age 6.2 ComparisonofModeltoData In this section, I examine how well the exogenous model fits compared to the observed data.48 Figure 3 plots the life cycle profiles from the exogenous model under the baseline-fitted U.S. tax policy and in the actual data.49 Overall, the model does a decent job matching the data; the general shapes of the profiles aresimilar. However,therearesomediscrepanciesbetweentheprofilespredictedbythemodelandthelife cycleprofilesfromthedata. Theupperleftpanelcomparestheaveragepercentofthetimeendowmentthatisspentworkingoverthe lifetimeandtheupperrightcomparesthelaborincome. Theactualprofilesareconstructedfromthe1967- 1999 waves of the Panel Survey of Income Dynamics (PSID). I focus my attention on the earnings for the headofthehouseholdbetweenages20and80.50 Starting by focusing on the labor supply profiles, the model generated profiles have a similar hump shapeastheprofilesfromthedata. Additionally,bothprofilesdeclinerapidlyaftertheageofsixty. Despite the general shapes being similar, there are two main differences. First, in the data, households work 30 percentoftheirtotallaborendowmentatage20,whichgrowsrapidlyoverthenextfewyearsuntilitpeaks at around 35 percent of the time endowment. In contrast, in the model, agents work 35 percent of the total 48Thedifferencesinthelifecycleprofilesbetweenthethreemodelsarenotlarge,thusIchoosetoonlycomparethedatatothe exogenousmodel(seesection6.3) 49Earnings,consumption,andsavingsfromthemodelareconvertedtoreal2012dollarsbyequatingtheaverageearningsinthe modelandthedata. 50Ifindthatthedataforindividualsolderthan80wereextremelyvolatile. 26

laborendowmentatage20. Althoughthemodelcontinuestooverpredictlaborsupply,theincreaseinlabor supply over the next few subsequent years is more gradual than in the data. This difference in the labor supply of young households is primarily due to liquidity constraints. In the model agents cannot borrow againstfutureearnings. Thus, agentstendtoworkmoreearlyintheirlifetimeinthemodelbecausewages riserapidlyinthebeginningofthelifecycle. Incontrast,inthedata,younghouseholdsmayhaveameans to borrow, decreasing the relative wealth effect for young households from working. The second major differencebetweentheprofilesisthatthemodelgeneratedprofilestartstoslopedownwardsjustbeforethe age of forty while the profile from the data does not start to slope downwards until households are in their fifties. However, Ifindinsection7.1thatamorerapidlydeclininglaborsupplyprofiledoesnotmaterially affecttheoptimaltaxpolicy.51 Focusingontheupperrightpanel,theearningsprofileinthedataissimilartotheonegeneratedbythe model. Both profiles are humped shaped with a peak around forty years old. However, since in the model agents are forced to retire at 65, but in the U.S. economy some households retire after the age of 65, the earningsprofilefortheseolderhouseholdsishigherinthedata. The lower left panel compares the consumption profile in the model to the per-capita expenditures on food in the PSID. I find that both profiles are hump-shaped; however, consumption on food tends to peak earlierinthedatathantotalconsumptioninthemodel. Additionally,comparingthegrowthinconsumption fromtheage20tothepeak,themodelexhibitsmoregrowthinconsumptionoverthelifetime. Onepossible reason for these differences is that the PSID data are limited to just expenditures on food while the model generatedconsumptionrepresentsallconsumption. Finally,thelowerrightpanelexaminessavingsinthemodelandmediantotalwealthinthe2007Survey ofConsumerFinances(SCF)forindividualsbetweentheagesof20and80.52 Ismooththroughsomeofthe volatilityinthewealthdatabyusingfiveyearagebins. Evenaftersmoothing,thedataforindividualsafter age80isnotincludedbecausetherearefewobservationsinthesampleleadingtheestimatestobeextremely volatile. Ifindthattheprofilesaresimilarinthemodelandthedata. Botharehump-shaped,peakingaround $300,000 at the age of 60. One discrepancy between the two profiles is that the model predicts that agents will deplete their savings more quickly than they do in the data. This discrepancy could arise because the 51Thelackofrelationshipbetweenthelaborsupplyprofileandoptimaltaxpolicyisnotsurprising. Peterman(2013),Garriga (2001), Erosa and Gervais (2002) all demonstrate, when using a utility function that is homothetic and separable in labor and consumption,suchastheoneinthebenchmarkmodel,thatregardlessofthelaborsupplyprofilethegovernmentdoesnotwantto conditiontaxesonage. However,iftheutilityfunctionisnothomotheticandseparablethenthegovernmentwantstocondition laborincometaxesonageandintheabsenceoftheabilitytouseage-dependenttaxesadownwardslopinglaborsupplywilllead toapositiveoptimaltaxoncapital. 52Inordertopreventtheuppertailofthewealthdistributionfromskewingthestatisticforcomparison,Ichoosetofocusonthe medianlevelofwealthasopposedtotheaverage. 27

modeldoesnotincludeanymotiveforindividualsleavingabequestforyoungergenerations. Overall,Ifind thatthemodeldoesfairjobmatchingthedata. Figure3: ActualandExogenousLifeCycleProfiles Labor 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 20 40 60 80 100 tnemwodnE emiT fo % Earnings 60 Model 50 Actual 40 30 20 10 0 20 40 60 80 100 Age 000,1$ Model Actual Age Consumption 45 40 35 30 25 20 20 40 60 80 100 )ledoM( 000,1$ 20 18 16 14 12 10 8 6 4 Age )dooF lautcA( 000,1$ Savings 3.5 Model Actual 3 2.5 2 1.5 1 0.5 0 20 40 60 80 100 000,001$ Model Actual Age Note: Theseplotsarelifecycleprofilesoftheexogenousmodelmodelsunderthebaseline-fittedU.S.taxpolicyandtheactual profiles in the data. The units of the consumption, earnings, and capital profiles are converted to real dollars by matching the averageearningsinthemodelandinthedata. 6.3 TheEffectsofAddingEndogenousAge-SpecificHumanCapital This section analyzes the effect on the aggregate economic variables and life cycle profiles of changing fromexogenoushumancapitalaccumulationtoeitherLBDorLODunderthebaseline-fittedU.S.taxpolicy. Figure4plotsthelifecycleprofilesofhours,consumption,assets,andage-specifichumancapitalinallthree models. Table4describestheoptimaltaxpoliciesandsummarizestheaggregateeconomicvariablesunder both the baseline-fitted U.S. tax policy and optimal tax policies. The first, fourth, and seventh columns are theaggregateeconomicvariablesunderthebaseline-fittedU.S.taxpolicyintheexogenous,LBD,andLOD models, respectively. The second, fifth, and eighth columns are the aggregate economic variables under the optimal tax policies. The third, sixth, and ninth columns are the percentage changes in the aggregate economicvariablesinducedfromadoptingtheoptimaltaxpolicies. 28

Table4: AggregateEconomicVariables Exogenous LBD LOD % Change from % Change from % Change from Baseline to Baseline to Baseline to Aggregate Baseline Optimal Optimal Baseline Optimal Optimal Baseline Optimal Optimal Y 0.81 0.82 1.8% 0.80 0.81 1.0% 0.76 0.77 1.6% K 2.17 2.25 3.6% 2.17 2.17 0.2% 2.06 2.12 3.0% N 0.46 0.46 0.8% 0.46 0.46 1.5% 0.44 0.44 0.8% Avg Hours 0.33 0.34 0.7% 0.33 0.34 0.8% 0.33 0.34 0.7% w 1.12 1.13 1.0% 1.12 1.12 -0.5% 1.12 1.13 0.8% r 0.05 0.05 -4.6% 0.05 0.05 2.2% 0.05 0.05 -3.6% tr 0.03 0.03 4.2% 0.02 0.02 2.8% 0.03 0.03 3.8% Value -139.26 -138.46 0.6% -159.01 -158.10 0.6% -155.14 -154.36 0.5% CEV 0.7% 0.9% 0.6% Average Tax Rate Baseline Optimal Baseline Optimal Baseline Optimal Capital 15.5% 18.2% 15.6% 25.5% 15.3% 18.9% Labor 23.7% 23.7% 23.7% 22.1% 23.7% 23.6% Ratio 0.65 0.77 0.66 1.16 0.65 0.80 Marginal Tax Rate Baseline Optimal Baseline Optimal Baseline Optimal Capital 19.4% 18.2% 19.6% 25.5% 19.1% 18.9% Labor 25.5% 23.7% 25.5% 22.1% 25.5% 23.6% Ratio 0.76 0.77 0.77 1.16 0.75 0.80 Note:Theaveragehoursreferstotheaveragepercentoftimeendowmentworkedintheproductivelaborsector.Boththemarginal andaveragetaxratesvarywithincomeunderthebaseline-fittedU.S.taxpolicy.Themarginaltaxratesarethepopulationweighted averagemarginaltaxratesforeachagent. Starting by comparing the exogenous and LBD models, the first and fourth columns of table 4 demonstrate that the aggregate level of hours, labor, and capital are similar in the two models. The calibrated parametersaredeterminedsothatunderthebaseline-fittedU.S.taxpolicythemodelsmatchcertaintargets fromthedata. Sincemanyoftheaggregateeconomicvariablesaretargetsandthesecalibrationparameters aredeterminedseparatelyineachmodel,theaggregatesaresimilarintheexogenousandLBDmodels. Although adding LBD does not have a large effect on the aggregate economic variables, it does cause changesinthelifecycleprofiles. AddingLBDcausesagentstoworkrelativelymoreatthebeginningoftheir workinglifewhenthehumancapitalbenefitislarger,andlesslaterwhenthebenefitissmaller(seethesolid blackanddashedredlinesintheupper-leftpanelofFigure4). Theupper-rightpanelshowsthatthelifetime consumption profile is steeper in the exogenous model compared to the LBD model. The intertemporal Eulerequationcontrolstheslopeofconsumptionprofileoveranagent’slifetime. Therelationshipis (cid:32) (cid:33)σ1 c j+1 =Ψ βr , (48) j (cid:101)t c j wherer isthemarginalafter-taxreturnoncapital. Inordertoinducethesamecapitaltooutputratiointhe (cid:101)t LBD model, β is lower which is the primary cause of the flatter consumption profile. The lower value of 29

Figure4: LifeCycleProfilesunderBaseline-FittedU.S.TaxPolicy Hours Under Baseline 0.4 0.35 0.3 0.25 0.2 20 40 60 80 100 tnemwodnE fo tnecreP Consumption Under Baseline 0.65 Exogenous LBD 0.6 LOD (working+training) LOD (working) 0.55 0.5 0.45 0.4 0.35 0.3 0.25 20 40 60 80 100 Age noitpmusnoC Exogenous LBD LOD Age Asset Holdings Under Baseline 5 4 3 2 1 0 20 40 60 80 100 stessA Age−specific Human Capital Under Baseline 2 1.8 1.6 1.4 Exogenous 1.2 LBD LOD 1 20 40 60 80 100 Age ytivitcudorP Exogenous LBD LOD Age Note:Theseplotsarelifecycleprofilesofthethreecalibratedmodelsunderthebaseline-fittedU.S.taxpolicy.Therearetwolabor linesfortheLODmodel,onesolelyforhoursworkedandtheotherforhoursworkedplushoursspenttraining. 30

β in the LBD model decreases the value an agent places on their consumption in future periods so agents’ savingsarealsorelativelysmallerforthesecondhalfoftheirlifetime(seethelower-leftpanel). Thelifetime age-specifichumancapitalprofilesaresimilarinthetwomodelssincethesequenceofparameters{Ωj} jr−1 j=20 iscalibratedsothatage-specifichumancapitalmatches(seethelower-rightpanelofFigure4). Next, comparing the exogenous and LOD models, although the parameters values are calibrated such that the targets match, the size of the economy is smaller in the LOD model because agents must spend partoftheirtimeendowmenttraining. Comparingthefirstandseventhcolumnsoftable4,aggregatelabor supply, and physical capital are smaller in the LOD model compared with the exogenous model, however, therelativeratiosoftheaggregatesaresimilar. Adding LOD also affects the life cycle profiles. Figure 4 plots two labor supply profiles for the LOD model — the first is solely hours spent working, and the second is the sum of hours spent working and training (see the blue lines in the upper-left panel). The LOD labor supply profile, including training, is similar to the labor supply profile in the exogenous model; however the profile that excludes training is smaller. ThedifferencebetweenthetwoLODprofilesistheamountoftimespenttraining. Thisgapshrinks as an agent ages, representing a decrease in the amount of time spent training. Agents spend less time training as they age because the benefits decrease since they have fewer periods to take advantage of their human capital. Adding LOD causes the size of the economy to decrease, causing a shift down in the life cycle profile for consumption. In the LOD model, agents can use their time endowment to accumulate human capital, which acts as an alternative form of savings from assets. Therefore, during their working lives,agentsholdlessordinarycapitalandopttousehumancapitaltosupplementtheirsavings. Asanagent approachesretirementthevalueofthehumancapitaldecreasesandtheordinarysavingsprofileintheLOD model converges to the profile in the exogenous model. Finally, similar to LBD, the lifetime age-specific human capital profiles are similar in the exogenous and LOD models since the profiles are a calibration target. 6.4 TheEffectsoftheOptimalTaxPolicyintheExogenousModel Thissectionexaminestheeffectsontheeconomyofadoptingtheoptimaltaxpolicyintheexogenousmodel. Intheexogenousmodel,theoptimalcapitaltaxissmallerthantheaveragemarginaltaxunderthebaselinefittedU.S.taxpolicy,soadoptingtheoptimaltaxpolicycausesanincreaseinaggregatecapital(seecolumns one and two of table 4). The average marginal labor tax is also less under the optimal tax policy than the 31

baselinesothelaborsupplyincreases.53 Theincreaseinlaborsupplyisrelativelysmallerthantheincrease incapitalsotherentalrateoncapitaldecreasesandthewagerateincreases. Figure5: LifeCycleProfilesintheExogenousModel Hours Exogenous Model 0.4 0.35 0.3 0.25 0.2 20 40 60 80 100 tnemwodnE fo tnecreP Consumption Exogenous Model 0.65 Baseline 0.6 Optimal 0.55 0.5 0.45 0.4 0.35 0.3 0.25 20 40 60 80 100 Age noitpmusnoC Baseline Optimal Age Asset Holdings Exogenous Model 5 4 3 2 1 0 20 40 60 80 100 stessA Baseline Optimal Age Note: Sincetheskillsarethesameintheexogenousmodelsunderthebaseline-fittedU.S.taxpolicyandoptimaltaxpolicy,they arenotplotted. Figure 5 plots the life cycle profiles for time worked, consumption, and assets in the exogenous model underthebaseline-fittedU.S.taxpoliciesandtheoptimaltaxpolicies. Thesolidlinesaretheprofilesunder the baseline-fitted U.S. tax policies, and the dashed lines are the profiles under the optimal tax policies. Althoughthechangesintheprofilesfromadoptingtheoptimaltaxpoliciesaresmall,Istillinterpretthemin ordertoprovidethereaderwithabetterunderstandingofthedynamicsinthemodel. Adoptingtheoptimal taxpolicyintheexogenousmodelcauseschangesinallthreelifecycleprofiles: (i)earlyintheirlife,agents workrelativelymore;(ii)agentssavemore,especiallyduringperiodswhentheyarewealthier;and(iii)the lifetime consumption profile steepens. The first change, agents working more early in their life, is a result ofthelowerimplicittaxonyounglaborincomeduetoadecreaseinthetaxrateoncapitalincome. Implementing the optimal tax policy causes a decrease in both the capital tax and the rental rate on 53Arevenueneutraltaxchangecanincludeadecreaseinboththeaveragemarginaltaxrateonlaborandcapitalsincethebaseline isprogressiveandtheoptimalisflat. Additionally,agentsgenerallyworklongerundertheoptimaltaxpolicysothetaxbaseis larger. 32

capital leading to shifts in both the capital and savings profiles. These declines have competing effects on the marginal after-tax return on capital. Furthermore, the change in the tax rate has an uneven effect on agent’snetreturnoverhislifetimesincethebaseline-fittedUStaxoncapitalisprogressiveandtheoptimal tax is flat. The decrease in the tax rates is larger for agents who hold more savings since their marginal taxratewasrelativelyhigherundertheprogressivebaseline-fittedUStaxpolicy. Overall,thechangeinthe tax rate dominates for these middle-aged agents and the after tax return increases. The converse is true for younger agents who experience a decrease in the after tax return when the optimal tax policy is adopted. In response to these changes, middle-aged individuals increase their savings under the optimal tax policy, while younger and older agents decrease their savings (see the lower left panel of Figure 5). In addition, since the after tax return to capital controls the slope of the consumption profile, adopting the optimal tax policycausesasteeperconsumptionprofileformiddle-agedagents(Figure5,upper-rightpanel). 6.5 TheEffectsofOptimalTaxPolicyintheLBDModel Figure6: LifeCycleProfilesintheLBDModel Hours LBD Model 0.4 0.35 0.3 0.25 0.2 20 40 60 80 100 tnemwodnE fo tnecreP Consumption LBD Model 0.7 Baseline 0.65 Optimal 0.6 0.55 0.5 0.45 0.4 0.35 20 40 60 80 100 Age noitpmusnoC Baseline Optimal Age Asset Holdings LBD Model 5 4 3 2 1 0 20 40 60 80 100 stessA Age−specific Human Capital LBD Model 2 1.8 1.6 1.4 Baseline 1.2 Optimal 1 20 40 60 80 100 Age ytivitcudorP Baseline Optimal Age AdoptingtheoptimaltaxpolicyintheLBDmodelcausesanincreaseinthecapitaltaxandadecreasein thelabortax(seecolumnfour,five,andsixoftable4). Thechangesinthetaxpolicycauseasmallincrease in the capital stock and a large increase in aggregate labor supply in the LBD model. The relatively larger 33

riseinlabortranslatesintoadecreaseinthewagerateandanincreaseintherentalrateoncapital. ImplementingtheoptimaltaxpoliciesintheLBDmodelcausesthelifecycleprofilestochangesomewhatdifferentlythanintheexogenousmodel(seeFigure6). Agentsshifttimeworkedfromearliertolater yearsinresponsetothelargercapitaltax, whichimplicitlytaxeslaborincomefromearlyyearsatahigher rate (upper-right panel of Figure 6). Because agents work more in their middle years, age-specific human capital is also higher for middle aged agents (see the lower-right panel). Applying the optimal tax policy introduces two opposing effects on the agent’s lifetime asset profile. First, agents increase their savings undertheoptimaltaxpolicybecausetheeconomyislarger. Second,thelargercapitaltaxundertheoptimal tax policy decreases the average marginal after-tax return on capital, causing agents to hold fewer assets. Thefirsteffectisconstantforallagents. Thesecondeffectisnotconstantforallagents,butitisnegatively proportional to an agent’s capital income because the baseline-fitted U.S. tax policy is progressive and the optimal tax policy is flat. This second effect dominates for younger and older agents and they tend to save lessundertheoptimaltaxpolicy. Conversely,thefirsteffectdominatesformiddle-agedagentsandtheytend to save more. I find that the first effect has the dominate impact on consumption leading the consumption profiletouniformlyshiftupwards(seetheupper-rightpanel).54 6.6 TheEffectsofOptimalTaxPolicyintheLODModel Although the optimal capital tax is larger in the LOD model than in the exogenous model, the direction of the changes in the tax rates from adopting the optimal tax policy are similar in the two models: a decrease in the average marginal tax on capital and labor. Therefore, the aggregate economic variables respond to adoptingtheoptimaltaxpolicyinasimilarfashioninbothmodels: capitalincreases,laborincreases,wages increase, and the rental rate decreases. Adopting the optimal tax policy in the LOD also induces changes in the life cycle profiles much like those in the exogenous model (see Figures 5 and 7): (i) agents work moreearlierintheirlife,(ii)agentsincreasetheirsavingsduringthemiddleoftheirlifetime,and(iii)agents increasetheirconsumptionatafasterratethroughouttheirlife. OneadditionalfeatureoftheLODmodelis thatagentschoosehowmuchtotrain. Ifindthatadoptingtheoptimaltaxpolicyhasminimaleffectsonthe trainingprofile(seethelower-leftpanelofFigure7). 54Althoughadoptingtheoptimaltaxpolicydoesnotcauseauniformchangeintheafter-taxreturntocapitalintheLBDmodel, liquidityconstraintscancelouttheireffectontheslopeoftheconsumptionprofile. 34

Figure7: LifeCycleProfilesintheLODModel Hours Working + Training LOD Model 0.4 0.35 0.3 0.25 20 40 60 80 100 tnemwodnE fo tnecreP Consumption LOD Model 0.65 Baseline 0.6 Optimal 0.55 0.5 0.45 0.4 0.35 0.3 0.25 20 40 60 80 100 Age noitpmusnoC Baseline Optimal Age Asset Holdings LOD Model 5 4 3 2 1 0 20 40 60 80 100 stessA Age−specific Human Capital LOD Model 2 1.8 1.6 1.4 Baseline 1.2 Optimal 1 20 40 60 80 100 Age ytivitcudorP Baseline Optimal Age Hours Training LOD Model 0.05 0.04 0.03 0.02 0.01 0 20 40 60 80 100 tnemwodnE fo tnecreP Hours Working LOD Model 0.36 Baseline 0.34 Optimal 0.32 0.3 0.28 0.26 0.24 0.22 20 40 60 80 100 Age tnemwodnE fo tnecreP Baseline Optimal Age Note:Theupper-leftpanelisaplotoflaborandthesumoflaborandtraining. 35

7 Sensitivity Analysis This section examines the sensitivity of two different aspects of the model. First, I demonstrate that the general shape of the labor supply profile does not affect the optimal tax policy in the exogenous model. Second, I determine that using a different utility function does not weaken the relationship between how humancapitalisaccumulatedandtheoptimalcapitaltax. 7.1 TheEffectofShapeofLaborSupplyProfileonOptimalTaxPolicy In this section, I test the relationship between the shape of the labor supply profile and the optimal tax. I examinethisrelationshipbecausetherearedifferencesbetweentheprofileinthedataandthemodels. Moreover, there are differences in the labor supply profiles between the three models. For example, comparing the labor supply profile in the actual data and the exogenous model (Figure 3), the exogenous model predictsthatthelaborsupplyprofilewillbedownwardslopingoveramajorityofthelifetimewhiletheactual profile from the data tends to be much flatter. Moreover, comparing the labor supply profile predicted by theexogenousandLBDmodels(Figure4),thelaborsupplyprofileintheLBDmodeldeclinesmuchmore rapidly over the second half of the working lifetime than it does in the exogenous model. In order to test whethertheshapeofthelaborsupplyprofileaffectstheoptimaltaxpolicy,Ifindtheoptimaltaxpoliciesin twoalternativeexogenousmodelsinwhichIvarythevaluesofχoverthelifetimesuchthatthelaborsupply profilematcheseitherthedataortheprofilepredictedbytheLBDmodel. First, I determine whether the flatter labor supply profile in the actual data has an affect on the optimal taxpolicy. Figure8plotsthelaborsupplyprofilegeneratedintheexogenousbenchmarkmodel(solidblack line)andtheaveragehoursworkedinthedata(dashedblackline). Additionally,thesolidredlineplotsthe labor supply generated in an alternative exogenous model (labor supply match data) which is calibrated to morecloselymatchthedata.55 Ifindthattheoptimaltaxpolicyinthisalternativeexogenousmodel(labor supplymatchdata),τ =23.8%τ =17.9%,isalmostidenticaltotheoptimaltaxpolicyinthebenchmark h k exogenous model (τ =23.7% and τ =18.2%). This result indicates that the steeper labor supply profile h k predictedbythemodelhasonlyanegligibleaffectontheoptimaltaxpolicy. Next, Iexaminewhetherthemorerapiddeclineinthelaborsupplyprofileovertheendoftheworking lifetime in the LBD model affects the optimal tax policy. In this experiment, I calibrate an alternative exogenousmodel(matchLBDlabor)suchthatthelaborsupplyprofilemorecloselymatchestheprofilein the LBD model. Figure 9 plots the labor supply profiles in the benchmark exogenous model (solid black), 55Thelaborsupplyprofilesareallunderthebaseline-fittedU.S.taxpolicy. 36

Figure8: FlatterLaborSupplyProfile 0.4 0.35 0.3 0.25 0.2 20 40 60 80 100 tnemwodnE fo tnecreP Exogenous Actual Alt. Exogenous I Age the LBD model (dashed black), and the new alternative exogenous model (match LBD labor) (solid red). Comparingthedashedblacklineandtheredline,thelaborsupplyprofileoverthesecondhalfoftheworking lifetime in the alternative exogenous model (match LBD labor) matches the rapid decline predicted in the LBD model. I find that the optimal tax policy in this alternative exogenous model (match LBD labor) is τ =23.6% and τ =18.9%. Again, the optimal tax policy in this altered exogenous model (match LBD h k labor)isalmostidenticaltotheoptimaltaxpolicyinthebenchmarkexogenousmodel. These results indicate that the optimal tax policy in my benchmark exogenous model is not related to thegeneralshapeofthelaborsupplyprofile.56 Theseresultsarenotsurprisingsincethebenchmarkutility functionishomotheticandseparableinlaborandconsumption. Therefore,laborsupplyisnotrelatedtothe Frisch labor supply elasticity. This utility function eliminates the most active channel by which the labor supplyprofileaffectstheoptimaltaxpolicy. Somepreviousworks,suchasPeterman(2013)andCKK,use autilityfunctioninwhichlaborsupplyaffectstheFrischlaborsupplyelasticity. Thenextsectionexamines whether the relationship between endogenous human capital accumulation and optimal taxation changes whenthistypeofutilityfunctionisused. 56Oneexceptiontothisresultisdescribedinsection7.1,whereIdemonstratethatalowerlaborsupplyinthefirstfewyearsof workingleadsagentstohavebindingliquidityconstraintsformoreyearsandcanaltertheoptimaltaxpolicy. 37

Figure9: LaborSupplyinAlternativeExogenous 0.4 0.35 0.3 0.25 0.2 20 40 60 80 100 tnemwodnE fo tnecreP Exogenous LBD Alt. Exogenous II Age 7.2 UtilityFunction In this section, I determine the effect of human capital accumulation on the optimal capital tax in a model withanalternativeutilityfunction,U(c ,1−h )= (cγ 1,t (1−h1,t)1−γ)1−ς . Thisutilityfunctionisthebenchmark 1,t 1,t 1−ς specification in CKK. I refer to this utility function as the nonseparable utility function. This function includes additional motives for a positive capital tax since it is no longer both separable and homothetic in eachconsumptionandlabor. 7.2.1 Calibration Thenonseparableutilityfunctionrequirescalibratingtwonewparameters. Thenewparametersareγ,which determinesthecomparativeimportanceofconsumptionandleisure,andς,whichcontrolsriskaversion. Itis nolongerpossibletoseparatelytargettheFrischelasticityandaveragetimeworkedsinceγcontrolsbothof thesevalues. Therefore,Icalibrateγtomatchthepercentageofthetimeendowmentworkedandnolonger usetheFrischelasticityasatarget. Table 5 lists the calibration parameters for the nonseparable utility parameters. The Frisch elasticity in the exogenous model for this utility function is (1−h)1−γ(ς−1) . The average Frisch elasticity implied by the h ς calibrationintheexogenousmodelis1.13,whichismorethantwiceaslargeaswiththebenchmarkutility specificationintheexogenousmodel. 38

Table5: PreferenceParameters Parameter Exog LBD LOD Target β 1.012 1.009 1.013 K/Y =2.7 Ψ β 0.998 0.996 1.000 K/Y =2.7 j γ 0.35 0.27 0.34 Avg. h +n = 1 j j 3 ς 4 4 4 CKK 7.2.2 OptimalTaxPoliciesinNonseparableModels There is a larger motive for a positive capital tax in all the models with nonseparable preferences for two reasons. First, the nonseparable utility implies that the Frisch elasticity profile is negatively related to the laborsupplyprofile. Sincethelaborsupplyprofileisdownwardslopingoveramajorityofthelife,theFrisch elasticityprofileisupwardslopinginallthemodels. TheupwardslopingFrischelasticityprofilemotivates a large positive capital tax.57 Second, there are less degrees of freedom when calibrating the nonseparable modelsotheFrischelasticityislargerinthenonseparablemodel. Therefore,thegovernmentwouldprefer torelyonacapitaltax,asopposedtoalaborincometax. Table 6 lists the optimal tax policies for the nonseparable models. Even with the nonseparable utility function — which motivates a large capital tax on its own — there is still a large range of optimal capital tax rates depending on how human capital is accumulated. Compared to the exogenous model, adding LBDcausestheoptimalcapitaltaxtoincreaseby14.5percentagepoints(approximatelyfortyfivepercent). Moreover,addingLODcausesa4.7percentagepoint(approximatelyafifteenpercent)increaseintheoptimalcapitaltaxcomparedtotheexogenousmodelanda9.8percentagepointdecreasecomparedtotheLBD model (approximately thirty percent). The range of the optimal capital taxes is even larger in this model indicatingthattheimportanceofhowhumancapitalisaccumulatedonoptimalcapitaltaxationisrobustto thischangeintheutilityspecification. Table6: OptimalTaxPoliciesinNonseparableModels TaxRate Exog LBD LOD τ 31.8% 46.3% 36.5% k τ 20.2% 15.0% 18.7% h τk 1.57 3.09 1.95 τh 57IfindthataddingLBDcausestheFrischelasticitytobeevensteeperandfurtherenhancesthismotiveforapositivetaxon capital.Incontrast,IfindthattheFrischelasticityisstillupwardslopingwhenIaddLOD,howeveritislesssteep. 39

8 Conclusion In this paper, I characterize the optimal capital and labor tax rates in three separate life cycle models in which age-specific human capital is accumulated exogenously, endogenously through LBD, and endogenously through LOD. Analytically, I demonstrate that compared to the exogenous model, including either form of endogenous human capital accumulation creates a motive for the government to condition labor income taxes on age and in their absence, it will use a non-zero capital tax to mimic these age-dependent taxes. Quantitatively, I find large variation in the optimal capital tax depending on whether human capital isaccumulatedendogenouslyorexogenously. Moreover, Ifindthattheformofendogenoushumancapital accumulation matters, the optimal tax rate is between 6.6 and 9.8 percentage points larger with LBD comparedtoLODdependingontheutilityfunction. Thesefindingsdemonstratethattheformbywhichhuman capitalisassumedtoaccumulatehaslargeimpactsontheoptimalcapitaltax. LBD increases the motive for a capital tax since it alters the lifetime labor supply elasticity profile. AddingLBDtothemodelcausesyoungeragentstosupplylaborrelativelylesselasticallysincethehuman capitalbenefit decreasesoveran agent’slifetime. Alarger capitaltaxis optimalbecauseit implicitlytaxes youngerlaborsupplyincome,whichissuppliedlesselastically,atahigherrate. AddingLODtothemodel has two counteracting affects on the optimal tax policy. Including LOD causes younger agents to supply laborrelativelymoreelasticallybecausetrainingisanimperfectsubstituteforworking. Thischangeinthe elasticity motivates the government to decrease the capital tax and raise the labor tax. However, a tax on laborintheLODmodeldecreasestheincentiveforagentstosavewithhumancapitalasopposedtophysical capital. Therefore, thegovernmenthasanincentivetoincreasethetaxoncapitalinordertopromotemore training. Overall, I find that this second effect dominates and adding LOD also causes the optimal capital taxtoincreaseinnumericalsimulations. Inastandardlifecyclemodel,Ifindalargeboundontheestimatesoftheoptimalcapitaltaxdepending onthemodel’sassumptionswithregardtohowhumancapitalisaccumulated. Moreover,thewayinwhich humancapitalisaccumulatedeffectstheshapeofthelifetimeFrischlaborsupplyelasticity. Foreconomists toreachmorepreciseconclusionsfromlifecyclemodels,theymustdeterminetheprocessbywhichagents acquireage-specifichumancapitaloncetheystartworking. Determiningtheshapeofthelaborsupplyelasticity profile could provide helpful guidance as to which form of human capital accumulation is consistent withthedata. 40

A Analytical Derivations ForOnlinePublication A.1 PrimalApproach I use the primal approach to determine the optimal tax policy.58 I use a social welfare function that maximizes 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 )]. (49) 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.59 Theimplementabilityconstraintistheagent’sintertemporalbudget constraint,withpricesandtaxesreplacedbyhisfirstorderconditions(equations5,6,and7) c U (t)+βc U (t+1)+h U (t)+βh U (t+1)=0. (50) 1,t c1 2,t+1 c2 1,t h1 2,t+1 h2 Includingthisconstraintensuresthatanyallocationthegovernmentchoosescanbesupportedbyacompetitiveequilibrium. Theresourceconstraintis c +c +K −K +G =rK +w(h +h ε ). (51) 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 (52) 1−σ 1 1+ 1 1−σ 1 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 whereρistheLagrangemultiplierontheresourceconstraintandλistheLagrangemultiplierontheimplementabilityconstraint. 58SeeLucasandStokey(1983)orErosaandGervais(2002)forafulldescriptionoftheprimalapproach. 59Thegovernmentbudgetconstraintisathirdconstraint. DuetoWalras’Law,Ionlyneedtoincludetwoofthreeconstraintsin theLagrangianandleaveoutthegovernmentbudgetconstraint. 41

A.2 Exogenous TheLagrangianforthisspecificationis 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 (53) 1−σ 1 1+ 1 1−σ 1 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 whereρistheLagrangemultiplierontheresourceconstraintandλistheLagrangemultiplierontheimplementabilityconstraint. Thefirstorderconditionswithrespecttolabor,capitalandconsumptionare 1 1 wρ =χhσ2(1+λ (1+ )) (54) t 1,t t σ 2 1 1 wρ θε =βχhσ2 (1+λ (1+ )) (55) t+1 2 2,t+1 t σ 2 ρ =θ(1+r)ρ (56) t t+1 ρ =c−σ1+λ (1−σ )c−σ1 (57) t 1,t t 1 1,t and θρ =βc−σ1 +βλ (1−σ )c−σ1 . (58) t+1 2,t+1 t 1 2,t+1 Combiningthefirstorderequationsforthegovernmentsproblemwithrespecttocapitalandconsumptionyields (cid:16)c 2,t+1 (cid:17)σ1 βρ t = (59) c ρ θ 1,t t+1 where ρ is the Lagrange multiplier on the resource constraint and λ is the Lagrange multiplier on the implementabilityconstraint. Takingtheratiooftheagent’sfirstorderconditions,equations5and6underthe benchmarkutilityspecificationgives 1−τ h,2 = 1 (cid:16) c 1,t (cid:17)−σ1 (cid:16)h 2,t+1 (cid:17) σ 1 2. (60) 1−τ ε c h h,1 2 2,t+1 1,t Combiningequation59and60yields 1−τ 1 (cid:16) βρ (cid:17)(cid:16)h (cid:17) 1 h,2 = t 2,t+1 σ2. (61) 1−τ ε ρ θ h h,1 2 t+1 1,t Theratiooffirstorderequationsforthegovernmentwithrespecttoyoungandoldhoursis ρ t β (cid:16)h 2,t+1 (cid:17) σ 1 2 = 1+λ t (1+ σ 1 2 ) . (62) ε 2 ρ t+1 θ h 1,t 1+λ t (1+ σ 1 2 ) 42

Combiningequation62and61generatesthefollowingexpressionforlabortaxes 1−τ h,2 = 1+λ t (1+ σ 1 2 ) =1. (63) 1−τ h,1 1+λ t (1+ σ 1 2 ) A.3 LBD The Lagrangian for this LBD specification is modified from the exogenous model. In particular,human capital benefit alters the implementability constraint. Suppressing the arguments of the skills function, the implementabilityconstraintintheLBDmodelis βh U (t+1)h s (t+1) 1,t h2 2 h1 c U (t)+βc U (t+1)+h U (t)− +βh U (t+1)=0, (64) 1,t c1 2,t+1 c2 1,t h1 2,t+1 h2 s 2 wheres (t+1)representsthepartialderivativeoftheskillfunctionforanolderagentwithrespecttohours h1 workedwhenyoung. ThustheLagrangianfortheLBDmodelis, 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 (65) 1−σ 1 1+ 1 1−σ 1 1+ 1 σ2 σ2 −ρ (c +c +K −K +G −rK −w(h +h s )) t 1,t 2,t t+1 t t t 1,t 2,t 2,t −ρ θ(c +c +K −K +G −rK −w(h +h s )) t+1 1,t+1 2,t+1 t+2 t+1 t+1 t+1 1,t+1 2,t+1 2,t+1 1+ 1 +λ (c1−σ1+βc1−σ1−χh 1+ σ 1 2 + χβh 2,t+ σ 1 2h 1,t s h1 (t+1) −βχh 1+ σ 1 2) t 1,t 2,t+1 1,t s 2,t+1 2 whereρistheLagrangemultiplierontheresourceconstraintandλistheLagrangemultiplierontheimplementabilityconstraint. Thefirstorderconditionswithrespecttolabor,capitalandconsumptionare 1 1 wρ =χhσ2(1+λ (1+ ))−θρ h s (t+1) t 1,t t σ t+1 2,t+1 h1 2 (cid:34) (cid:35) (66) +λ χh 1+ σ 1 2βh s h1 (t+1)2 − s h1,h1 (t+1) t 2,t+1 1,t s2 s 2 2 (cid:34) (cid:35) 1 1 1 h s (t+1)λ wρ θs =βχhσ2 1+λ (1+ )+(1+ ) 1,t h1 t (67) t+1 2 2,t+1 t σ σ s 2 2 2 ρ =θ(1+r)ρ (68) t t+1 ρ =c−σ1+λ (1−σ )c−σ1 (69) t 1,t t 1 1,t and θρ =βc−σ1 +βλ (1−σ )c−σ1 . (70) t+1 2,t+1 t 1 2,t+1 The first order conditions with respect to capital and consumption are the same in the exogenous (56, 57, and58)andLBDmodels(68,69,and70). Thereforeequation10stillholdsforthismodelandthereforethe optimaltaxoncapitalisstillzerowhenthegovernmentcanconditionlaborincometaxesonage. Combiningthefirstorderequationsforthegovernmentsproblemwithrespecttocapitalandconsump- 43

tionyields (cid:16)c 2,t+1 (cid:17)σ1 βρ t = (71) c ρ θ 1,t t+1 Taking the ratio of the agent’s first order conditions, equations 15 and 16 and combining with equation 71 yields 1−τ (cid:16) h (cid:17) 1 (cid:16)ρ θs (cid:17) h s (t+1) h,1 = 1,t σ2 t+1 2 − 2,t+1 h1 . (72) 1−τ h βρ 1+r(1−τ ) h,2 2,t+1 t k Combiningequations72,66and67theratiooftheoptimaltaxesonlaboris, 1−τh,1 = 1−τh,2 (cid:16) (cid:17)(cid:16) (cid:17) 1+λt(1+σ 1 2 )−λt(1+σ 1 2 ) h1,tsh s 1 2 (cid:16) (t+1) 1+ 1 h + 2 r ,t ( + 1− 1 τ s2 k) (cid:17)− h2,t+1sh2(t+1) . (73) 1+λt(1+σ 1 2 )+h 1 2 + ,t+ σ 1 1 2h 1 1 + ,t − σ2 1 s λ 2 t sh1 s ( 2 t+1) −sh1,h1(t+1) 1+r(1−τk) A.4 LOD Since agents have the additional choice variable n in the LOD model, the have an additional first order 1 condition with respect to this variable (equation 25). This new first order condition requires an additional constraint in the government’s Lagrange that ensures that the allocation the government chooses properly equates an individual’s disutility of training when young and working when old (see equations 23 and 25). This constraint simplifies toU (t)s =βU (t+1)h s (t+1). I use η as the Lagrange multiplier on n1 2 h2 2,t+1 n t thisnewconstraint. TheLagrangianfortheLODmodelis L = c1 1 − ,t σ1 −χ (h 1,t +n 1,t ) 1+ σ 1 2 +β c1 2 − ,t+ σ 1 1 −χ h 1 2 + ,t+ σ 1 1 2 (74) 1−σ 1 1+ 1 1−σ 1 1+ 1 σ2 σ2 −ρ (c +c +K −K +G −rK −w(h +h s )) 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 s )) 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 1+ 1 1 +η t (χh 2,t+ σ 1 2s n1 (t+1)−χ(h 1,t +n 1,t )σ2s 2 ) where ρ is the Lagrange multiplier on the resource constraint, λ is the Lagrange multiplier on the implementabilityconstraintandηistheLagrangemultiplierontheconstraintequatingthefirstorderconditions withrespecttotrainingandwork. Thefirstorderconditionswithrespecttolabor,capital,consumptionand trainingare, (cid:34) (cid:35) 1 (cid:16) h 1,t (cid:17) η t s 2 wρ t =χ(h 1,t +n 1,t )σ2 1+λ t 1+ + (75) σ (h +n ) σ (h +n ) 2 1,t 1,t 2 1,t 1,t (cid:34) (cid:35) 1 (cid:16) 1 (cid:17) (cid:16) 1 (cid:17) wρ θs =βχhσ2 1+λ 1+ −η 1+ s (t+1) (76) t+1 2 2,t+1 2 σ t σ n1 2 2 ρ =θ(1+r)ρ (77) t t+1 ρ =c−σ1+λ (1−σ )c−σ1 (78) t 1,t t 1 1,t θρ =βc−σ1 +βλ (1−σ )c−σ1 (79) t+1 2,t+1 t 1 2,t+1 44

and θρ h s (t+1)= (80) t+1 2,t+1 n2 1 (cid:16) (cid:17) 1+ 1 χ(h 1,t +n 1,t )σ2 λ t h 1,t +η t s 2 +σ 2 (h 1,t +n 1,t )(1+η t s n2 (t+1)) −βχη t σ 2 h 2,t+ σ 1 2(h 1,t +n 1,t )s n2,n2 (t+1) σ (h +n ) 2 1,t 1,t (81) The first order conditions with respect to capital and consumption are the same in the exogenous (56, 57, and58)andLODmodels(77,78,and79). Thereforeequation10stillholdsforthismodelandthereforethe optimaltaxoncapitalisstillzerowhenthegovernmentcanconditionlaborincometaxesonage. Combiningthefirstorderequationsforthegovernmentsproblemwithrespecttocapitalandconsumptionyields (cid:16)c 2,t+1 (cid:17)σ1 βρ t = (82) c ρ θ 1,t t+1 Taking the ratio of the agent’s first order conditions, equations 22 and 23 and combining with equation 82 yields 1−τ (cid:16) h (cid:17) 1 (cid:16) βρ (cid:17) h,2 = 2,t+1 σ2 t . (83) 1−τ h +n ρ θs h,1 1,t 1,t t+1 2 Takingtheratioofequations75and76yields, (cid:16) (cid:17) (cid:16) h 2,t+1 (cid:17) σ 1 2 (cid:16) βρ t (cid:17) = 1+λ t 1+ σ2(h1 h , 1 t+ ,t n1,t) + σ2(h η 1, t t s + 2 n1,t) . (84) (cid:16) (cid:17) (cid:16) (cid:17) h 1,t +n 1,t ρ t+1 θs 2 1+λ 1+ 1 −η s (t+1) 1+ 1 t σ2 t n1 σ2 Combiningequations83and84generatesthefollowingexpressionfortheratiooftheoptimallabortaxes, (cid:16) (cid:17) 1+λ 1+ h1,t + ηts2 1−τ h,2 = t σ2(h1,t+n1,t) σ2(h1,t+n1,t) . (85) (cid:16) (cid:17) (cid:16) (cid:17) 1−τ h,1 1+λ 1+ 1 −η s (t+1) 1+ 1 t σ2 t n1 σ2 45

B Competitive Equilibrium ForOnlinePublication B.1 LBDModel Givenasocialsecurityreplacementrateb,asequenceofskillaccumulationsparameters{Ω } jr−1 ,governj j=20 mentexpendituresG,andasequenceofpopulationshares{µ }J ,astationarycompetitiveequilibriumin j j=20 theLBDmodelisasequenceofagentallocations,{c ,a ,h }J ,aproductionplanforthefirm(N,K), j j+1 j j=20 a government labor tax function Tl :R →R , a government capital tax function Tk :R →R , a social + + + + securitytaxrateτ ,aage-specifichumancapitalaccumulationfunctionS:R ×R ×R →R ,autility ss + + + + functionU :R ×R →R ,socialsecuritybenefitsSS,prices(w,r),andtransfersTrsuchthat: + + + 1. Givenprices,policies,transfers,andbenefits,theagentmaximizesequation29subjectto c +a =ws h −τ ws h ,+(1+r)(a +Tr)−Tl[ws h (1−.5τ )]−Tk[r(a +Tr)], (86) j j+1 j j ss j j j j j ss j s =S (Ω ,s ,h ), (87) j+1 LBD j j j for j< j ,and r c +a =SS+(1+r)(a +Tr)−Tk[r(a +Tr)], (88) j j+1 j j for j≥ j . r Additionally, c≥0,0≤h≤1,a ≥0,a =0. (89) j 20 2. Priceswandrsatisfy (cid:18) N (cid:19)1−α r=α −δ (90) K (cid:18) K (cid:19)α w=(1−α) (91) N 3. Thesocialsecuritypoliciessatisfy wN SS=b (92) ∑ j jr = − 2 1 0 µ j τ = ss∑ J j=jr µ j (93) ss w∑ j jr = − 2 1 0 µ j 4. Transfersaregivenby J Tr= ∑ µ (1−Ψ )a (94) j j j+1 j=20 5. Governmentbudgetbalance: J jr−1 G= ∑ µ Tk[r(a +Tr)]+ ∑ µ Tl[ws h (1−.5τ )] (95) j j j j j ss j=20 j=20 46

6. Marketclearing: J K = ∑ µ a (96) j j j=20 J N = ∑ µ s h (97) j j j j=20 J J ∑ µ c + ∑ µ a +G=KαN1−α+(1−δ)K (98) j j j j+1 j=20 j=20 B.1.1 LODModel Givenasocialsecurityreplacementrateb,asequenceofskillaccumulationsparameters{Ω } jr−1 ,governj j=20 mentexpendituresG,andasequenceofpopulationshares{µ }J ,astationarycompetitiveequilibriumin j j=20 theLBDmodelisasequenceofagentallocations,{c ,a ,h }J ,aproductionplanforthefirm(N,K), j j+1 j j=20 a government labor tax function Tl :R →R , a government capital tax functionTk :R →R , a social + + + + securitytaxrateτ ,aage-specifichumancapitalaccumulationfunctionS:R ×R ×R →R ,autility ss + + + + functionU :R ×R →R ,socialsecuritybenefitsSS,prices(w,r),andtransfersTrsuchthat: + + + 1. Givenprices,policies,transfers,andbenefits,theagentmaximizesequation29subjectto c +a =ws h −τ ws h ,+(1+r)(a +Tr)−Tl[ws h (1−.5τ )]−Tk[r(a +Tr)], (99) j j+1 j j ss j j j j j ss j s =S (Ω ,n ,h ), (100) j+1 LOD j j j for j< j ,and r c +a =SS+(1+r)(a +Tr)−Tk[r(a +Tr)], (101) j j+1 j j for j≥ j . r Additionally, c≥0,0≤h≤1,a ≥0,a =0. (102) j 20 2. Priceswandrsatisfy (cid:18) N (cid:19)1−α r=α −δ (103) K (cid:18) K (cid:19)α w=(1−α) (104) N 3. Thesocialsecuritypoliciessatisfy wN SS=b (105) ∑ j jr = − 2 1 0 µ j τ = ss∑ J j=jr µ j (106) ss w∑ j jr = − 2 1 0 µ j 4. Transfersaregivenby J Tr= ∑ µ (1−Ψ )a (107) j j j+1 j=20 47

5. Governmentbudgetbalance: J jr−1 G= ∑ µ Tk[r(a +Tr)]+ ∑ µ Tl[ws h (1−.5τ )] (108) j j j j j ss j=20 j=20 6. Marketclearing: J K = ∑ µ a (109) j j j=20 J N = ∑ µ s h (110) j j j j=20 J J ∑ µ c + ∑ µ a +G=KαN1−α+(1−δ)K (111) j j j j+1 j=20 j=20 48

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