The Effect of Endogenous Human Capital Accumulation on Optimal Taxation

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. The Effect of Endogenous Human Capital Accumulation on Optimal Taxation William B. Peterman 2012-003 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.

The Effect of Endogenous Human Capital Accumulation on Optimal Taxation WilliamBPeterman∗ October13,2015 Abstract Thispaperconsiderstheimpactoflearning-by-doingonoptimaltaxpolicyinageneralequilibrium heterogenousagentlife-cyclemodel.Analytically,itidentifiestwomainchannelsbywhichlearning-bydoingalterstheoptimaltaxpolicy. First,learning-by-doingcreatesamotiveforthegovernmenttouse age-dependentlaborincometaxes. Ifthegovernmentcannotconditiontaxesonage, thenacapitaltax orprogressive/regressivelaborincometaxcanbeusedinordertomimicage-dependenttaxes. Second, aprogressive/regressivelaborincometaxispotentiallymoredistortionaryinamodelwithlearning-bydoingsincethedistortionispropagatedthroughtheadditionalintertemporallinkbetweencurrentlabor andfuturehumancapital. Quantitatively,Ifindthatbothofthesechannelsareimportantfortheoptimal tax policy. Adding learning-by-doing leads to a notably flatter optimal labor income tax due to the secondchannel. Moreover,includinglearning-by-doingcausesanincreaseintheoptimalcapitaltaxdue tothefirstchannel. Ifindthatwhensolvingfortheoptimaltaxpolicyinthelearning-by-doingmodel, thewelfareconsequencesofnotaccountingforendogenoushumancapitalaccumulationareequivalent to around one percent of expected lifetime consumption, a majority of which are due to adopting too progressiveofataxpolicy. JEL:E24,E62,H21. KeyWords: OptimalTaxation,CapitalTaxation,ProgressiveTaxation,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,theSocietyforEconomicsandDynamics,theConferenceinComputing inEconomicsandFinance,andtheBarcelonaGSESummerForum. 1

1 Introduction Previous research documents that variation in consumption and labor, in part due to fluctuations in agespecific human capital, can have important implications for optimal taxation.1 In particular, variation in consumption and labor may cause the optimal labor income tax to be age-dependent. Moreover, if agedependent taxes are disallowed then it tends to be optimal to use either a non-flat labor income tax (i.e. a progressiveorregressivetax)oranon-zerotaxoncapitaltomimictheseage-dependenttaxes.2 Therefore, variation in human capital over the lifetime can have implications for two fundamental questions in the optimal taxation literature.3 First, should the income tax be progressive? Second, should capital be taxed? However,previousresearchthatexaminesbothofthesequestionsinalifecyclemodeltendstoassumethat humancapitalisaccumulatedexogenously. Inthispaper,Iexaminetheeffectofendogenoushumancapital accumulationonboththeoptimalshapeofthelaborincometaxpolicyandtheoptimaltaxationofcapital. Inparticular,Ideterminetheeffectbothanalyticallyandquantitativelyonoptimaltaxpolicyofincluding endogenousage-specifichumancapitalaccumulationthroughlearning-by-doing(LBD)inwhichanagent’s future human capital is affected by the hours worked today.4 Despite being commonly used in life-cycle models, and empirical evidence supporting the relationship between current work and future productivity, previous research has not simultaneously examined the effect of LBD on both tax questions.5 Therefore, thispaperdeterminestheeffectofLBDonbothpartsoftheoptimaltaxpolicy. Overall,thispaperfindsthat endogenizing human capital accumulation has significant qualitative and quantitative implications for both partsoftheoptimaltaxpolicy,operatingthroughtwoimportantchannels. I begin by analytically demonstrating these two channels in a simple model. First, I demonstrate that adding LBD changes the relative incentives to work over an agent’s lifetime. In the LBD setting there are twobenefitstoworking. Workingprovidesanagentwithawage(“wagebenefit”)andanincreaseinfuture human capital (“human capital benefit”). The human capital benefit only exists in the LBD model and not 1Idefineage-specifichumancapitalashumancapitalthatisaccumulatedafteranagentbeginsworking. 2ExamplesofthisresearchincludesAtkesonetal.(1999),ErosaandGervais(2002),Garriga(2001),andGervais(2012). 3SeeDiamondandSaez(2011)andMirrleesetal.(2010)foradiscussionoftheimportanceofthesequestionsandageneral summaryofpreviousfindings. 4Analternativeformofendogenoushumancapitalaccumulationthatissometimesusedislearning-or-doing(LOD).InLOD, whichisalsoreferredtoasBenPorathtypeskillaccumulationoron-the-jobtraining,anagentacquireshumancapitalbyspending timetraininginperiodsinwhichheisalsoworking.ThispaperignoresthisformofhumancapitalaccumulationbecauseMulligan (1995)findsthatonceindividualsstartworkingtheyspendlessthan7percentoftheirtimeendowmentinformaltraining(Peterman (2014)findsthattherearemuchsmallereffectsontheoptimalcapitaltaxwhenendogenoushumancapitalisaddedwithLOD). Therefore,thisformofhumancapitalaccumulationismorerelevanttopre-workskillformulationthanage-specifichumancapital accumulation.Moreover,JacobsandBovenberg(2010)findsthatincorporatingendogenouspre-workskillaccumulationhassimilar effectsontheoptimaltaxpolicyasLBD. 5ExamplesofstudieswhichshowthatpasthoursworkedandlengthofcurrentjobtenureimpactcurrentwagesincludeTopel (1990),Cossaetal.(1999),andAltug˘ andMiller(1998). Moreover,examplesoflifecyclestudiesthatincludeLBDareHansen and˙Imrohorogˇlu(2009),ImaiandKeane(2004),andChangetal.(2002). 2

in the exogenous model leading agents to be less responsive to temporary changes in wages in the LBD model. Moreover,theimportanceofthehumancapitalbenefitdecreasesasanagentapproachesretirement. Thus,addingLBDcausestheagenttosupplylaborrelativelylesselasticallyearlyinhislifecomparedwith laterinhislife. Optimally,thesocialplannerwouldtaxlaborincomefromagentswhentheyareyoungand supplylaborlesselasticallyatarelativelyhigherratethanwhentheyareolder. Ifthesocialplannercannot use age-dependent taxes, then a tax on capital or a progressive/regressive labor income tax can be optimal inordertomimictheage-dependenttaxes.6 Irefertothisfirstchannelastheelasticitychannel. Second, I demonstrate that including LBD enhances the distortions from a non-flat labor income tax. In both the exogenous and LBD model a progressive tax distorts an agent’s labor decisions because a it causes the marginal after-tax wage benefit to decline as labor income increases. However, the distortion is magnified in the LBD model because it is propagated through the additional intertemporal link between currentlaborandfuturehumancapital. Inparticular,intheLBDmodelaprogressivelaborincometaxalso leadsthemarginalafter-taxhumancapitalbenefittodeclineasfuturelaborincomeincreases. Thusaflatter labor income tax policy is optimal with LBD. I refer to this second channel as the intertemporal distortion channel. Next,IquantitativelyassesstheimpactofLBDonoptimaltaxpolicyinarigorousgeneralequilibrium overlappinggenerationsmodel(OLG)thatincludesheterogeneityduetoidiosyncraticshockstolaborproductivity. To explore the effect of LBD, I solve for the optimal tax policies in two different cases – first in a model with no LBD (the exogenous model) and then again in a model with LBD (the LBD model). In the LBD model I find that the optimal tax policy is a 36 percent flat tax on capital income, a 22.3 percent tax on labor income with a fixed deduction of $10,901, and a lump-sum transfer of $365. In contrast in the exogenous model I find that the optimal tax policy is a 30 percent tax on capital, a 32.5 percent tax on laborincomewithafixeddeductionof$6,218,andalump-sumtransferof$3,683. Thus,addingLBDhas considerablequantitativeimplicationsonbothtaxquestions. Inparticular,addingLBDreducestheoptimal progressivity of the labor income tax policy and raises the optimal capital tax by 6.0 percentage points. Throughaseriesofcounterfactualexperiments,Iconfirmthattheintertemporaldistortionisresponsiblefor the flatter optimal tax policy and that the elasticity channel is responsible for the increase in the optimal capitaltax. Overall,IfindthatthewelfareconsequencesofnotaccountingfortheeffectsofLBDwhendetermining theoptimaltaxpolicyarenotable. Inparticular,IfindthatintheLBDmodelimplementingtheoptimaltax 6Incontrast, Garriga(2001)demonstratesthat, inaspecificsetofmodelswithexogenoushumancapitalaccumulation, itis notoptimaltoconditionlabortaxesonagenortotaxcapital. Moreover,ahostofworkdemonstratesasimilarsetofresultsina two-generationmodelwithasinglecohort.TwoexamplesoftheseworksincludeAtkinsonandStiglitz(1976),andDeaton(1979). 3

policy from the exogenousmodel – which includes a moreprogressive labor tax and lower capitaltax – as opposed to the actual optimal tax policy – which includes a flatter tax on labor and larger tax on capital – results in a welfare reduction equivalent to between 0.7 and 1.2 percent of expected lifetime consumption depending on the utility function. I find that a majority of these welfare consequences are due to the suboptimal level of progressivity as opposed to the sub-optimal capital tax. Thus for welfare purposes, the change in the optimal progressivity from adding LBD is more significant than the change in the optimal capital tax. Furthermore, I find that the change in the optimal tax policy from adding LBD is not sensitive toeithertheutilityfunctionortheparametervaluesusedtocalibratetheLBDskillaccumulationfunction. Takenasawhole,theseresultsdemonstratethatincludingLBDhasquantitativelyimportanteffectsonboth themagnitudeandtheshapesoftheoptimaltaxesoncapitalandlabor. Thispapercontributestothegeneralclassofliteraturethatexplorestheoptimaltaxpolicywhentheset of available tax instruments are restricted. Correia (1996), Armenter and Albanesi (2009), and Jones et al. (1997),demonstratethatcertaintaxinstruments,thatotherwisewouldnotbeoptimal,maybecomeoptimal when the government’s set of tax instruments are restricted. This paper combines two related strands of the literature within this class of research that quantitatively determine the optimal capital tax and optimal progressivityoftheincometaxwhenthegovernmentisrestrictedfromusingage-dependenttaxes.7 Thefirststrandsimultaneouslyexaminesbothtaxquestionsinacalibratedlife-cyclemodelbutincludes humancapitalaccumulationexogenously.8 Atkesonetal.(1999),ErosaandGervais(2002),Garriga(2001), and Gervais (2012) determine that generally it is optimal to condition labor income taxes on age in a lifecycle model.9 Moreover, they demonstrate that if age-dependent taxes are not allowed, then it is possible to mimic the optimal tax policy with either a non-flat labor income tax (i.e. a progressive or regressive tax) or a non-zero tax on capital. Conesa et al. (2009), Peterman (2013), and Gervais (2012) demonstrate quantitatively in a life-cycle model that the inability to condition taxes on age can be a strong motive for a positive capital tax and a progressive/regressive tax on labor income. In particular, Conesa et al. (2009), henceforth CKK, find in a life-cycle model that is similar to my exogenous model, the optimal tax policy includes both a progressive labor income tax and a sizeable tax on capital. Although the authors find that a primary reason for the large optimal tax on capital is to mimic an age-dependent tax, they find that the primaryreasonfortheoptimalprogressivelaborincometaxistoprovideex-anteinsuranceforidiosyncratic 7Foradiscussionoftheoptimalage-dependenttaxpolicyinthenewdynamicpublicfinanceframeworkseeFarhiandWerning (2013),Kremer(2002),andWeinzierl(2011). 8Thereisastrandofliteraturethatexaminesthesequestionsinaninfinitelylivedagentmodelasopposedtoalife-cyclemodel. SeeDiamondandSaez(2011)forareviewofthisliterature. 9Atkesonetal.(1999)demonstratearelatedresult. Theyshowconditionsunderwhichtheoptimalcapitaltaxiszeroifagedependenttaxesonlaborincomeareallowed. 4

shockstolaborproductivityandnottomimicage-dependenttaxes.10 Inaddition,Gervais(2012)findsthat in some cases, even with a large tax on capital, a mild amount of progressivity in the labor income tax is optimalinordertomimicanage-dependenttaxpolicy. Althoughthesestudiesexaminingboththeoptimal tax on capital and the optimal shape of the labor income in a life-cycle model, they include human capital accumulation exogenously ignoring any affects of endogenous human capital accumulation on the optimal tax policy. In contrast, this paper both analytically and quantitatively assesses the effects of including endogenoushumancapitalaccumulationontheoptimaltaxpolicy. This paper is related to a second strand of the literature that includes LBD but only focus on its effect on one of the tax questions.11 For example, focusing on optimal capital taxation, Chen et al. (2011) finds that,inaninfinitelylivedagentmodelwithlaborsearch,includingendogenoushumancapitalaccumulation causes the optimal capital tax to increase because a higher capital tax unravels the labor market frictions in their model.12 Since Chen et al. (2011) only examine the effect of endogenous human capital on the optimal capital tax in an infinitely lived agent model, they are unable to assess whether LBD affects the motivetouseage-dependenttaxes,orwhetherLBDaffectstheefficiencyofaprogressivelaborincometax versusataxoncapitaltomimicage-dependenttaxes. FocusingontheeffectofLBDontheoptimalamount of progressivity, both Best and Kleven (2012) and Krause (2009) demonstrate that a flatter income tax, as opposedtoaprogressivetax,isoptimalinatwo-generationmodelwithLBDsoastonotdiscouragehuman capital accumulation. However, Best and Kleven (2012) and Krause (2009) do not incorporate savings so they do not determine the effect of LBD on the optimal capital tax. Since both a tax on capital and a progressive/regressive labor income tax can be used to mimic an age-dependent tax policy, it is important toexaminetheeffectofLBDonbothquestionssimultaneouslyinalife-cyclemodel. Thispapercombines bothstrandsoftheliteratureanddeterminesthatalthoughincludingLBDinalifecyclemodelchangesthe answertobothquestions,thechangeintheoptimalprogressivityisthedominateeffectforwelfarepurposes. 10ForadiscussionofthechannelleadingtotheprogressivetaxpolicyseeMirrlees(1971),Stiglitz(1982),Mirrlees(1974),and Varian(1980).Peterman(2013)demonstratesthatanadditionalmotiveforapositivetaxoncapitalisthatthegovernmentisunable todistinguishbetweenaccidentalbequestsandordinarycapitalincome.Furtherwork,suchasKarabarbounis(2012)andPeterman (2012),demonstratethatincorporatingendogenousfluctuationsinlaborsupplyontheextensivemargincanenhancethismotivefor thegovernmenttouseacapitaltaxtomimicage-dependenttaxesonlaborincome. Incontrast,CespedesandKuklik(2013)find thatwhenanon-linearmappingbetweenhoursandwagesexiststhenhourstendtobecomemorepersistentandtheoptimalcapital taxfallssignificantly,howeverisstillpositive. 11This paper primarily focuses on the tax studies that use a Ramsey approach. There is a parallel strand of the literature in dynamic public finance that also that examines optimal taxation with endogenous human capital. Examples of this strand that examines a model with endogenous human capital accumulation is Golosov et al. (2003), Stantcheva (2015a), and Stantcheva (2015b). 12TheauthorsincludeendogenouslyhumancapitalaccumulationthroughbothLBDandalsotraining.Thelabormarketfrictions in Chen et al. (2011) cause a lower level of employment in their economy. A capital tax causes the wage discount to increase, thuscausingfirmstopostmorevacancieswhichinturncausesanincreaseinworkerparticipation. Anumberofstudiesexamine theoptimaltaxpolicyinaninfinitelylivedagentmodelwithotherformsofendogenoushumancapitalaccumulation. Examples includeJonesetal.(1997),Judd(1999),andReis(2007). 5

This paper is organized as follows: Section 2 examines an analytically tractable version of the model to demonstrate the two channels by which LBD alters the optimal tax policy. Section 3 describes the full model used in the quantitative exercise (see Appendix B for the competitive equilibrium). The calibration and functional forms are discussed in section 4. Section 5 describes the computational experiment, and section6presentstheresults. Section7examinesthesensitivityoftheresults,whilesection8concludes. 2 Analytical Model In this section, I demonstrate the two channels by which adding LBD alters the optimal tax policy. First, I show that adding LBD introduces new channels that cause the government to want to condition labor incometaxesonage. Ifthegovernmentcannotuseage-dependenttaxes,thenataxoncapitaloraprogressive/regressivelaborincometaxcanbeoptimalinordertomimictheseage-dependenttaxes.13 However,I showthatintroducingLBDenhancesthedistortionsassociatedwiththeprogressivetaxandthereforemakes it less likely that a progressive/regressive labor income tax would be optimal to mimic the age-dependent taxes. I derive these analytical results in a tractable two-period version of the computational model that nests both cases when human capital is accumulated exogenously or through LBD. For tractability purposes, the features I abstract from include retirement, population growth, idiosyncratic labor productivity shocks, and conditional survivability. Additionally, I assume that the marginal products of capital and labor are constant so factor prices do not vary.14 Since changes to the tax system do not affect the pre-tax wage or rate of return, I am able to focus on the life-cycle elements of the model. Also, because I exclude idiosyncraticlaborproductivityshocks,thereisnowithin-cohortheterogeneityintheanalyticallytractable model. Therefore,withoutthiswithin-cohortheterogeneitythesocialplannerfocusesonlyonefficiencyand ignoresthetradeoffbetweenequityandefficiency. Alloftheseassumptionsarerelaxedinthecomputational model. 13Thelinkbetweenage-dependenttaxesandthesealternativetaxinstrumentsareexploredinamodelwithexogenoushuman capitalinGarriga(2001),ErosaandGervais(2002),andGervais(2012). 14Sincethefactorpricesdonotvary,Isuppresstheirtimesubscriptsinthissection. 6

2.1 ElasticityChannel In the analytically tractable model agents live with certainty for two periods, and their preferences over consumptionandleisurearerepresentedby U(c ,1−h )+βU(c ,1−h ) (1) 1,t 1,t 2,t+1 2,t+1 where U() is a utility function that is increasing with respect to both arguments, β is the discount factor, c is the consumption of an age j agent at time t, and h is the percent of the time endowment the j,t j,t agentworks(implyingthat1-h isthepercentofthetimeendowmentconsumedasleisure). Age-specific j,t human capital is normalized to unity when the agent is young. At age two, age-specific human capital is s (h ). InthecaseofLBD,s (h )isafunctionofhoursworkedinthepreviousperiodandIassumethat 2 1,t 2 1,t ∂s2(h1,t) >0, ∂s2 2 (h1,t) <0. This first assumption, ∂s2(h1,t) >0, implies that an agent working more when they ∂h1,t ∂h2 1,t ∂h1,t areyoungwillincreasetheirskillswhentheyareold(thehumancapitalbenefit). Inthecaseofexogenous human capital accumulation, s is exogenously predetermined and thus is no longer dependent on hours 2 worked ( ∂s2(h1,t) =0, ∂s2 2 (h1,t) =0).15 Thus, the human capital benefit only exists with LBD and not in the ∂h1,t ∂h2 1,t exogenous model. The agent chooses consumption and hours worked in order to maximize equation 1 subjecttothefollowingstandardbudgetconstraints c +a =(1−τ )h w (2) 1,t 1,t h,1 1,t and c =(1+r(1−τ ))a +(1−τ )s (h )h w, (3) 2,t+1 k 1,t h,2 2 1,t 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,andr istherentalrateoncapital. I begin by assuming that the tax rate on labor income is flat but can be conditioned on age. Moreover, I assumethatthetaxrateoncapitalincomeisflatandcannotbeconditionedonage.16 Theagent’sfirst-order conditionsare U (t) w(1−τ )h s (t+1) h1 h,2 2,t+1 h1 =w(1−τ )+ , (4) h,1 U (t) (1+r(1−τ )) c1 k 15Generally, I solve for results with both forms of human capital accumulation nested. Thus, I continue to represent s as a 2 functionofhoursworkedunlessspecificallydescribingtheexogenouscase. 16Agentsonlylivefortwoperiodsintheanalyticallytractablemodelsotheychoosenottosavewhentheyareold.Therefore,in thismodel,restrictingcapitaltaxpolicytonotbeage-dependentisnotbinding. 7

U (t+1) h2 =ws (h )(1−τ ), (5) 2 1,t h,2 U (t+1) c2 and U (t) c1 =β(1+r(1−τ )), (6) k U (t+1) c2 where U (t) ≡ ∂U(c1,t,1−h1,t) . Throughout the analytical section I highlight in red the portions of the exc1 ∂c1,t pressions that are specific to LBD and do not exist when human capital is accumulated exogenously. In particular, equation 4 has an additional term in the case of LBD since the agent receives the human capital benefitatage2fromworkingatage1. Using the primal approach to solve for the optimal tax policy in this model and assuming that the utilityfunctionisseparableinconsumptionandlabor(U =0),equation(seeappendixAfortheproblem’s ch formulationandfurtherdetails)representstherelationshipbetweentheoptimalage-dependenttaxes, (cid:16) (cid:17)(cid:16) (cid:17) (1−λ)+λh2Uh2h2− λsh2 h h U −h U 1+ h2sh2 (1−τ h1 ) = Uh2 s2Uh2 2 1 h2h2 1 h2 1+r(1−τk) − h 2 s h2 . (7) (cid:16) (cid:17) (1−τ h2 ) (1−λ)+λh1Uh1h1+h2βλUh2 s (s +h s )−h s2 1+r(1−τ k ) Uh1 s2 2 Uh1 2 h2 1 h2h2 1 h2 Equation 7 demonstrates that the optimal age-dependent taxes will tend to be different with LBD and exogenoushumancapitalaccumulation. ThusincorporatinghumancapitalaccumulationwithLBDcreatesan additionalmotiveforage-dependenttaxes. Previous work (see Atkeson et al. (1999), Erosa and Gervais (2002), Garriga (2001), and Peterman (2014))demonstratesthatifthesocialplannerwantstoconditionlaborincometaxesonagebutisdisallowed fromusingtheseage-dependenttaxesthenanon-zerocapitaltaxwillbeoptimalinthistypeofmodel. For example,Garriga(2001)demonstratesthatiftheutilityfunctionisseparableinconsumptionandlaborand also homothetic in each individual argument then, in this type of simple model with exogenous human capital accumulation, the social planner does not want to condition labor income taxes on age and as such does not want to tax capital regardless of whether they can use age-dependent taxes.17 However, even if theutilityfunctionmeetstheseconditionsincorporatingLBDcreatesanewmotiveforthesocialplannerto conditionallaborincometaxesonageandthusalterstheGarriga(2001)result.18 In order to determine the direction of the effect on optimal taxes from adding LBD, I make three sufficientassumptions: (i)h U <U ,(ii)h s =κs ,and(iii)h s =(κ−1)s ,whereκissome 2 h2h2 h2 2 h2 2,t+1 2 h2h2 2,t+1 17Inparticular,iftheutilityfunctionissuchthat hUhh isequaltoaconstant. Autilityfunctionthatishomotheticinlaborwould Uh implythatthisratioisconstant. 18UndertheGarriga(2001)utilityfunctiontheblacktermsontherighthandsideofboththedenominatorandnumeratorwill simplifytothesameconstant.ThuswithoutLBDtheratioequalsone.However,onceLBDisincludedtheredtermsareintroduced totheexpressionfortheoptimaltaxpolicyandwiththeseadditionaltermstheratiodoesnotsimplifytoone. 8

arbitrary constant.19 Under these assumptions, Equation 7 implies that the optimal tax policy includes a higherlabortaxonagentswhentheyareyoung. TheintuitionforwhyaddingLBDwilltendtoincreasetheoptimalrelativetaxonyounglaborincome comesfromexaminingtheFrischelasticity.20 InparticulartheFrischelasticityis, U U h cc Frisch= (8) (cid:34) (cid:35) h U2 −U U +h(cid:48)w(cid:48)(s(cid:48)(U2 −U U )−U U s(cid:48) ) ch cc hh rws h ch cc hh cc h hh wherethenextperiodisdenotedwithaprimefornotationalconvenience(s(cid:48)=s ).2122 Theadditional j+1,t+1 expressionsfromLBDcausethedenominatortoincrease,thusholdinghoursandconsumptionconstantthe FrischelasticityislowerwhenLBDisincluded. Intuitively,theinclusionofthehumancapitalbenefitmakes workerslessresponsivetoaone-periodchangeinwagessincethewagebenefitisnowonlypartoftheirtotal compensation for working when LBD is included. Moreover, the relative importance of the human capital benefit decreases over an agent’s lifetime because he has fewer periods to use his higher human capital as heages. Inthestylizedcasewhereagentsonlylivefortwoperiods, theeffectofthehumancapitalbenefit wouldonlyexistforanagentwhentheyareyoung. Therefore,addingLBDcausesanagenttosupplylabor relativelylesselasticallywhentheyareoldcomparedtowhentheyareold. Thisshiftinrelativeelasticities creates an incentive for the social planner to tax the labor income of younger agents at a relatively higher rate. I use the term “elasticity channel” to describe the effect on optimal tax policy caused by a change in theFrischelasticityfromincludingLBD.23 In such a case when the social planner would like to use age-dependent taxes but is disallowed, then a taxoncapitalcanmimicsuchatax. TheintertemporalEuluerequationdemonstrateswhythetaxoncapital 19ThesearenotstrongassumptionsincethestandardfunctionalformsandcalibrationparametersIchoosefortheutilityfunction andtheLBDprocessinthecomputationalmodelalladheretotheseassumptions. 20TheseresultscanbederivedundermoregeneralconditionsthanthoseneededtosigntheeffectofLBDontheoptimaltax policy.Inparticular,theonlyassumptionnecessaryisthattheutilityfunctionisseparableinconsumptionandlabor. 21ThisistheFrischelasticitywithrespecttoatemporaryincreaseinthewage. Therefore,onemustdistinguishbetweenwt and w t+1 . 22IprovidetheexpressionfortheFrischelasticityinamoregeneralmodelwhereagentsliveformorethantwoperiodsinorder toseehowtheeffectofLBDvariesoverthelifecycle.However,thisexpressionalsomapsintothistwoperiodmodel.Inparticular, inatwoperiodmodeltheadditionalexpressionfromLBDonlyexistsforyoungagentsandnotoldagents. 23AlternativeintuitionforthisresultcanbedemonstratedinthecommoditytaxframeworkofCorlettandHague(1953).Intheir staticframework,thesocialplannerwantstotaxleisure. However,iftheycannotdirectlytaxleisure,thesocialplannerwilltax commoditiesthataremorecomplementarytoleisureatahigherrate.Viewingthissimpletwogenerationmodelinthatframework, addingLBDraisestherelativeopportunitycostofleisurewhenagentsareyoungsoyounglaborislessofsubstitute(moreofa complement)withleisure. Thischangeleadsthesocialplannertowanttoincreasethetaxonyounglabor. Moreover,ifthesocial plannercannotuseage-dependenttaxesthenincreasingthetaxoncapitalimplicitlytaxesconsumptionfromtheoldatarelatively higherratesinceLBDmakesconsumptionandleisuremorecomplementaryfortheolderagentsthantheyoungeragents. 9

mimicstheage-dependentlaborincometax, U (t) 1−τ h1 h,1 s (h ) =β(1+r(1−τ )) +βh s (t+1). (9) 2 1,t k 2,t+1 h1 U (t+1) 1−τ h2 h,2 In particular, a positive (negative) capital tax induces a wedge on the marginal rate of substitution that is similartoarelativelylargertaxonyoung(old)laborincome.24 Thusifage-dependenttaxesaredisallowed thenaddingLBDwillcausealargertaxoncapitaltobeoptimalinordertomimicarelativelylargertaxon laborincomewhenagentsareyoung.2526 2.2 DistortionsfromProgressive/RegressiveTax Next,Iexaminewhyaprogressive/regressivelaborincometaxcanmimicanage-dependenttaxandhowthe relativeefficiencyofaprogressive/regressivelaborincometaxversusataxoncapitalchangeswhenLBDis introduced.27 With a progressive/regressive tax on labor income the average tax rate is no longer a function of age; instead it is a function of labor income T(h ws). This change in the tax function leads to a change in the i,t i agent’sconstraints(equations2and3) c +a =(1−T(h w))h w (10) 1,t 1,t 1,t 1,t and c =(1+r(1−τ ))a +(1−T(s (h )h w))s (h )h w, (11) 2,t+1 k 1,t 2 1,t 2,t+1 2 1,t 2,t+1 where T is the average tax rate on labor income which is assumed to be increasing and concave in labor income. Theagent’sfirst-orderconditionswiththisnewtaxfunctionare (cid:32) (cid:33) U U h1 ( ( t t ) ) =w 1−T(h 1,t w)−h 1,t T h1 (t)w+β U c U 2 (t ( + t) 1) s h1 (t+1)h 2,t+1 (cid:16) 1−T(h 2,t+1 ws 2 (h 1,t ))−s 2 (h 1,t )T s2 (t+1)wh 2,t+1 (cid:17) , c1 c1 (12) 24ExaminingtheimplicationsofLBDonthisrelationship,theadditionaltermispositive. Therefore,holdingallelseequal,the taxonyounglaborincomewouldneedtoberelativelyhigherinordertoinducethesamewedgeonthemarginalrateofsubstitution intheLBDmodel. 25Although τk may act as a substitute for age-dependent taxees, τk is not a redundant instrument because it also distorts the intertemporalmarginalrateofsubstitutionbetweenconsumption. 26Inaddition,aprogressive/regressivelaborincometaxcanalsobeusedtomimicage-dependentlaborincometaxeswhichis discussedinthenextsection. 27Whenallowingthegovernmenttouseaprogressive/regressivelaborincometaxthesameprimalapproachdoesnotyieldan analyticalsolutionfortheoptimaltaxpolicybecauselaborchoicesaffecttheaveragelabortaxrate. Therefore,pricescannotbe removedfromtheintertemporalbudgetconstraintusingthefirst-orderconditionsinordertocreatetheimplementabilityconstraint. Thus,Iamunabletosolvefortheoptimalpolicy. 10

U (t+1) (cid:16) (cid:17) h2 =ws (h ) 1−T(h ws (h ))−h T (t+1) , (13) 2 1,t 2,t+1 2 1,t 2,t+1 h2 U (t+1) c2 and U (t) c1 =β(1+r(1−τ )), (14) k U (t+1) c2 whereT (t)≡ ∂T(h1,tw) ,orthemarginaltaxrate. Thefirst-orderconditionswithrespecttolabor(equations h1 ∂h1,tw 12and13)changecomparedtothecaseofflatlaborincometaxes(equations4and5). Includingaprogressive/regressivetaxchangesthefirstorderconditionbecauseifanagentchangesthehoursheworksthenhis marginallabortaxratealsochanges. Similar to a capital tax, a progressive/regressive tax can mimic age-dependent labor taxes. Examining theEulerequationwithaprogressive/regressivetax(forexpositionalconvenienceIusethesimplercaseof exogenoushumancapitalaccumulation), (cid:32) (cid:33) U (t) 1−T(wh )−h wT (t) h1 1,t 1,t h1 s =β(1+r(1−τ )) . (15) 2 k U (t+1) 1−T(wh s )−h wT (t+1) h2 2,t 2 2,t h2 if the social planner wants to condition taxes on age but is disallowed then a progressive/regressive labor income tax can create a similar wedge in the marginal rate of substitution. In particular, if labor income increases over an agent’s lifetime then a regressive labor income tax would create a similar wedge as an age-dependentflattaxwitharelativelyhigherrateonincomeearnedatyoungages. Although both a progressive/regressive tax on labor income or a non-zero tax on capital can create a wedge on the marginal rate of substitution, there are several reasons why a tax on capital may be more desirable, especially in a less parsimonious model. First, if the social planner wants the implicit labor income tax to monotonically decrease with age, a positive tax on capital may be ideal since it mimics a monotonicallydecreasinglabortaxbyage. Incontrast,aprogressivetaximplicitlytaxeslaborincomeata higherrateatageswhenanagentearnsmore. Therefore,iflaborincomeisnotmonotonicallyincreasingor decreasingoveraworkingagent’slifethenthereisnowayforaprogressive/regressivetaxpolicytomimic amonotonicallydecreasingage-dependenttaxpolicy. The second reason a tax on capital may be preferable to a progressive/regressive labor income tax is a capital tax imposes a wedge on the marginal rate of substitution that is independent of the agent’s labor choice. Incontrast,thesizeofthewedgefromaprogressive/regressivelaborincometaxwilldependonthe amountoflaborincome. Inalessparsimoniousmodelthatincludeswithin-cohortheterogeneity,agentsof the same age may have different labor income, making it even more difficult for the social planner to use 11

a progressive/regressive labor income tax to mimic an age-dependent tax. In contrast, the wedge from a capitalincome tax willbe afunction ofage butnot laborsupply soit willbe thesame for all agents ofthe sameageregardlessifthereiswithin-cohortheterogeneityinlaborincomes. The general desirability of a progressive/regressive labor income tax may be weakened with LBD. AddingLBDaltersanagent’stradeoffsbecauseitintroducesanadditionalintertemporallinkbetweencurrent labor and future productivity (see Equation 12). Since productivity is linked to the level of income, the distortions from the progressive tax are magnified through this channel. In the LBD model, a progressive tax policy reduces an agent’s incentives to work since the progressive tax implies that the marginal human capital benefit declines as future labor income increases. Since the additional intertemporal link in theLBDmodelmagnifiesthedistortionsfromaprogressive/regressivetax,Irefertothissecondchannelas theintertemporaldistortionchannel. 3 Computational Model Next, I determine the quantitative effect of adding LBD on optimal tax policy in a rigorous version of the model that includes other channels that affect the optimal capital tax and progressivity of the labor income tax. One notable channel arises from the inclusion of within-cohort heterogeneity which causes the social plannertoconsidernotonlyefficiencybuttoweighthetradeoffbetweenefficiencyandequity. Inparticular, the social planner may use a progressive labor income tax to redistribute and provide insurance against labor income risk. I solve for the optimal tax policy in separate versions of the computational model with exogenoushumancapitalaccumulationandLBD.TheexogenousmodelisadaptedfromCKK.28 3.1 Demographics Timeisassumedtobediscrete,andthemodelperiodisequaltooneyear. Agentsentertheeconomywhen they start working, at age 20, and live for up to J years. The economy is populated with J overlapping generationsofages20,21,...,J+20. Thesizeofeachnewcohortenteringtheeconomygrowsataconstant raten. Lifetimelengthisuncertainwithmortalityriskvaryingoverthelifetime. Conditionalonbeingalive atage j,Ψ istheprobabilityofanagentlivingtoage j+1. Sinceagentsarenotcertainhowlongtheywill j live, they may die while still holding assets. If an agent dies with assets, the assets are confiscated by the government and distributed equally to all the living agents as accidental bequests (beq ).29 All agents are t 28Although I use their alternative utility function which is separable, I find qualitatively similar results with their benchmark utilityfunctionwhichisnon-separable.Ichoosetousetheseparableutilityfunctiontofollowtheanalyticallytractablemodel. 29Includingthisformulationincreasestheoptimaltaxoncapitalfortworeasons. First,Peterman(2013)demonstratesthatwith accidentalbequeststheoptimalcapitaltaxincreaseswhenthegovernmentcannotdistinguishbetweengainsonaccidentalbequests 12

requiredtoretireatanexogenouslysetage j . r 3.2 Individual An individual is endowed with one unit of productive time per period that he divides between labor (h ) i,j and leisure (1−−h ). An agent earns wω h for their labor where ω is the idiosyncratic productivity i,j i,j i,j i,j of agent i at age j. Agents split their income between saving with a one-period risk-free asset (a ) and i,j consumption (c ). Agents choose labor, savings, and consumption in order to maximize their lifetime i,j utility J−j−20 s u(c ,h )+ ∑ βs∏(Ψ )u(c ,h ). (16) i,j i,j q i,s+1 i,s+1 s=20 q=1 Agentsdiscountthenextperiod’sutilitybytheproductofΨ andβ. βisthediscountfactorconditionalon j surviving,andtheunconditionaldiscountfactorisβΨ . j The log of an agent’s idiosyncratic productivity ω in the exogenous model can be split into four i,j additivelyseparablecomponents, logω =ε +α +ν +θ . (17) i,j j i t t andintheLBDmodel, logω =s +α +ν +θ . (18) i,j i,j i t t In this specification, based on the estimates in Kaplan (2012) from the Panel Study of Income Dynamics (PSID), ε or s governs age-specific human capital. Moreover, α∼NID(0,σ2) is an individual-specific j i,j α fixedeffect(orability)thatisrealizedwhenanagententerstheeconomyandstaysfixedoverthelife-cycle, θ ∼ NID(0,σ2) is a transitory shock to productivity received every period, and ν is a persistent shock, t ε t whichfollowsafirst-orderautoregressiveprocess: ν =ρν +ψ withψ ∼NID(0,σ2)andν 0=0.30 (19) t t−1 t t ν 2 In the exogenous model an agent’s age-specific human capital ε is exogenously determined. In the LBD j model,anagent’sage-specifichumancapitalisdeterminedbys =S (Ω ,s ,h )where{Ω } jr−1 i,j LBD j−1 i,j−1 i,j−1 j j=20 isasequenceofcalibrationparametersthataresetsothatintheLBDmodel,underthebaseline-fittedU.S. andordinarycapitalincome.Second,ifnon-accidentalbequestsareincludedinsteadofaccidentalbequeststhenFusteretal.(2007) demonstratesthatthemodelismorelikeaninfinitelylivedagentmodelwheretheoptimalcapitaltaxtendstobesmaller. 30Setting ν1 =0 implies thatthis shock equals 1 whenagents enter the model since expν is how theshock enters the agents idiosyncraticproductivity.Moreoverthissettingimpliesthat,consistentwiththedata,thevarianceintheidiosyncraticshocksgrow withage. 13

taxpolicy,theagents’choicesresultinthesameaverageage-specifichumancapitalageprofileintheLBD andexogenousmodels. 3.3 Marketstructure The markets are incomplete and agents cannot fully insure against the idiosyncratic labor productivity and mortality risks by trading state-contingent assets. They can, however, partially self-insure against these risks by accumulating precautionary asset holdings, a. These savings are also used to fund consumption after retirement. The stock of assets earns a market return r . I assume that households enter the economy t withnoassetsandarenotallowedtoborrowagainstfutureincome,sothata =0anda ≥0foralliand i,20 i,j j. 3.4 Firm Firmsareperfectlycompetitivewithconstantreturnstoscaleproductiontechnology. Aggregatetechnology is represented by a Cobb-Douglas production function. Unlike the analytically tractable model, I do not assume a linear production function in the computational model, so prices are determined endogenously. Theaggregateresourceconstraintis, C +K −(1−δ)K +G ≤KζN 1−ζ, (20) 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,ζisthecapitalshareandδisthedepreciationrate forphysicalcapital. 3.5 GovernmentPolicy Thegovernmentrunsapay-as-you-go(PAYGO)socialsecuritysystem. Inthisreduced-formsocialsecurity program, the government pays SS to all individuals that are retired, independent of the individual agent’s t earnings history. To finance the system, labor income is taxed at the flat rate τ up to a maximum labor ss incomelevely, asintheactualsystem. Thepayrolltaxrate, τ , fundsthebalancedbudgetprogram. The ss,t socialsecuritysystemisnotconsideredpartofthetaxpolicythatthegovernmentoptimizes. Iincludethis simplifiedsocialsecurityprogrambecauseexcludingthisprogramwouldcauseanagenttooveremphasize savingssincealloftheirpost-retirementconsumptionwouldneedtobefinancedwithprivatesavings.31 31Peterman(2013)demonstratesthatexcludingsocialsecuritycanhavenotableeffectsontheoptimaltaxpolicy. 14

InadditiontorunningtheSocialSecuritysystem,thegovernmenthastwofiscalinstrumentstofinance itsconsumption,G ,whichisdoneinanunproductivesector.32 First,thegovernmenttaxescapitalincome, t y ≡r (a+beq ), according to a capital income tax schedule TK[y ]. Second, the government taxes each k t t k individual’s taxable labor income. Part of the pre-tax labor income is accounted for by the employer’s contributions to social security, which is not taxable under current U.S. tax law. Let pyl be the pre-tax i,j laborincomewhichisequalto pyl ≡w s h . Sincepartofthesecontributionsarenottaxable,theagent’s i,j t i,j i,j taxable labor income is y ≡ pyl −.5τ min{pyl ,y}, which is taxed according to a labor income tax i,j i,j ss i,j schedule Tl[y ]. I impose three restrictions on the labor and capital income tax policies. First, I assume l human capital is unobservable and cannot be taxed directly. Second, I assume the tax rates cannot be agedependent. Third, both of the taxes are solely functions of the individual’s relevant taxable income in the currentperiod. 4 Calibration and Functional Forms Priortosolvingthemodels,itisnecessarytochoosefunctionalformsandcalibratethemodels’parameters. Calibrating the models involves a two-step process. The first step is choosing parameter values for which there are direct estimates in the data. These parameter values are in Table 1. Second, to calibrate the remaining parameters, values are chosen so that under the baseline-fitted U.S. tax policy certain targets in themodelmatchthevaluesobservedintheU.S.economy(SimulatedMethodofMoments).33 Thesevalues areinTable2. 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 LBDandexogenousmodels. Toassurethatboththemodelsmatchthetargetsunderthebaseline-fittedU.S. tax policy, I calibrate the set of parameters based on targets separately in the two models implying that the valuesoftheseparametersvarybetweenthetwomodels. 4.1 Demographics Agents enter the model at age 20 and are exogenously forced to retire at 66. If an individual survives until the age of 100, he dies the next period. I set the conditional survival probabilities in accordance with the estimatesinBellandMiller(2002). Iadjustthesizeofeachcohort’sshareofthepopulationtoaccountfor 32IncludingGt suchthatitenterstheagent’sutilityfunctioninanadditivelyseparablemannerisanequivalentformulation. 33Since these are general equilibrium models, changing one parameter will alter all the values in the model that are used as targets.However,Ipresenttargetswiththeparameterthattheymostdirectlycorrespondto. 15

Table1: CalibrationParameters Parameter Value Target Demographics RetireAge: j 66 ByAssumption r MaxAge: J 100 ByAssumption Surv. Prob: Ψ BellandMiller(2002) Data j Pop. Growth: n 1.1% Data FirmParameters ζ .36 Data δ 8.33% I =25.5% Y A 1 Normalization ProductivityParameters: PersistenceShock: σ2 0.017 Kaplan(2012) ν Persistence: ρ 0.958 Kaplan(2012) PermanentShock: σ2 0.065 Kaplan(2012) α TransitoryShock: σ2 0.081 Kaplan(2012) ε GovernmentParameters: PayrollTax: τ 0.124 CKK ss ϒ .258 GouveiaandStrauss(1994) 0 ϒ .768 GouveiaandStrauss(1994) 1 LBDParameters: Φ .407 Changetal.(2002) 1 Φ .326 Changetal.(2002) 2 16

apopulationgrowthrateof1.1percent. Table2: CalibrationParameters Parameter Exog. LBD Target CalibrationParameters DiscountFactor: β 0.994 0.992 K/Y =2.7 Riskaversion: σ 2 2 CKK 1 FrischElasticity: σ 4 3.3 Frisch= 1 2 2 ValueofLeisure: χ 1.08 2.2 Avg. h +n = 1 j j 3 GovernmentParameters G 15.6 15.3 17%ofY 4.2 Preferences Agentshavetime-separablepreferencesoverconsumptionandleisure. Iuse c1−σ1 +χ (1−h)1−σ2 asthebench- 1−σ1 1−σ2 mark utility function. In Section 7.1 I check the sensitivity of the results with regards to a different utility functioninwhichlaborinsteadofleisuredirectlyenterstheutilityfunction. I determine β such that the capital-to-output ratio is 2.7, in accordance with U.S. data.34 I determine χ such that under the baseline-fitted U.S. tax policy, agents spend on average one-third of their time endowment working.35 Following CKK, I set σ =2, which controls the relative risk aversion. Past micro- 1 econometricstudies(suchasAltonji(1986),MaCurdy(1981),andDomeijandFlode´n(2006))estimatethe Frischelasticitytobebetween0and0.5. However,morerecentresearchhasshownthattheseestimatesmay bebiaseddownward. Reasonsforthisbiasincludeutilizingweakinstruments,notaccountingforborrowing constraints,disregardingthelife-cycleeffectofendogenous-agespecifichumancapital,omittingcorrelated variablessuchaswageuncertainty,andnotaccountingforlabormarketfrictions.36 Therefore,Isetσ such 2 thatifagentsworkonethirdoftheirtimeendowmentthentheFrischelasticityisattheupperboundofthe range(0.5).37 34Thisistheratiooffixedassetsandconsumerdurablegoods,lessgovernmentfixedassetstoGDP(CKK). 35Usingatargetofone-thirdisstandardinquantitativeexercises.Forexamples,seeCKK,Nakajima(2010),andGarriga(2001). 36SomeofthesestudiesincludeImaiandKeane(2004),DomeijandFlode´n(2006),Pistaferri(2003),Chetty(2012),andContrerasandSinclair(2008). 37ThisvalueisconsistenttheestimatesinKaplan(2012). 17

4.3 IdiosyncraticProductivity TheidiosyncraticlaborproductivityshocksarecalibratedbasedontheestimatesfromthePSIDdatainKaplan(2012).38 Thesepermanent,persistent,andtransitoryidiosyncraticshockstoindividuals’productivity aredistributednormalwithameanofzeroandtheshockparametersaresetinaccordancewiththeestimates inKaplan(2012): ρ=0.958,σ2 =0.065,σ2=0.017andσ2=0.081. Idiscretizeallthreeoftheshocksin α ν ε ordertosolvethemodel,usingtwostatestorepresentthetransitoryandpermanentshocksandsevenstates forthepersistentshock.39 Forexpositionalconvenience,Irefertothetwodifferentstatesofthepermanent shockashighandlowability. 4.4 Age-SpecificHumanCapital The age-specific human capital calibration parameters are different in the exogenous and LBD models. In theexogenousmodel, Iset{ε } jr−1 tobeconsistentwiththevaluesestimatedinKaplan(2012)whichare j j=20 basedoffoftheaveragehourlyearningsbyageinthePanelSurveyofIncomeDynamics.40 In the LBD model I use the same functional form for human capital accumulation as in Hansen and ˙Imrohorogˇlu(2009), s =Ω sΦ1hΦ2, (21) j+1 j i,j i,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. I do not set {s } jr−1 directly, rather I calibrate the sequence {Ω } jr−1 such that i,j j=0 j j=20 the agents’ equilibrium labor choices lead the average {s } jr−1 under the baseline-fitted U.S. tax code to i,j j=20 match the age-specific human capital calibrated in the exogenous model ({ε } jr−1 ).41 Similar to Hansen j j=20 and ˙Imrohorogˇlu (2009), I calibrate the rest of the LBD parameters based on the estimates in Chang et al. (2002),settingΦ =0.407andΦ =0.326. 1 2 38Fordetailsonestimationofthisprocess,seeAppendixEinKaplan(2012). 39I use the Rouwenhorst method to discretize the persistent shock since Kopecky and Suen (2010) demonstrate with highly persistentprocessesthismethodispreferred. 40Imakethreeadjustmentstotheprocess. Theprofileissmoothedbyfittingaquadraticfunctioninage,normalizedsuchthat thevalueequalsonewhenanagententerstheeconomy,andisextendedtocoverages20through66. 41Icalibrate{Ωj} j jr = − 2 1 0 suchthatthesequenceissmoothoverthelife-cycle. Although{Ωj}j j r = − 2 1 0 hasanaffectonsomeofthe otherlabortargetsliketheFrischelasticityandaveragehoursworked,Ifindthattheeffectisfairlyminimalanditmostlyaffects theskillsprofile. 18

4.5 Firm I assume the aggregate production function is Cobb–Douglas. The capital share parameter, ζ, is set at .36. Thedepreciationrateissettotargettheobservedinvestmentoutputratioof25.5percent. Theseparameters aresummarizedinTable1. 4.6 GovernmentPoliciesandTaxFunctions WhilecalibratingparametersbymatchingcertaintargetsinthemodelsandthedataitisimportanttoincorporateataxfunctionthatissimilartotheU.S.taxcode. IusetheestimatesoftheU.S.taxcodeinGouveia and Strauss (1994) for this tax policy, which I refer to as the baseline-fitted U.S. tax policy. The authors matchtheU.S.taxcodetothedatausingathreeparameterfunctionalform, T(Ty;ϒ 0 ,ϒ 1 ,ϒ 2 )=ϒ 0 (Ty−(Ty−ϒ1+ϒ 2 ) − ϒ 1 1), (22) where Ty represents the sum of the taxable labor and capital income. The average tax rate is principally controlledbyϒ ,andϒ governstheprogressivityofthetaxpolicy. Toensurethattaxessatisfythebudget 0 1 constraint, ϒ is left free. Gouveia and Strauss (1994) estimate that ϒ =.258 and ϒ =.768 when fitting 2 0 1 thedata. Theauthorsdonotfitseparatetaxfunctionsforlaborandcapitalincome. Accordingly, Iuseone tax system on aggregate income for the baseline-fitted U.S. tax policy. However, when searching for the optimaltaxpolicy,Iallowfordifferenttaxratesoncapitalandlaborincome. I calibrate government consumption, G, so that it equals 17 percent of output under the baseline-fitted U.S.taxpolicytobeconsistentwithCKK.Whensearchingfortheoptimaltaxpolicy,Irestrictattentionto revenue-neutralchangesthatimplythatgovernmentconsumptionisequalunderthebaseline-fittedU.S.tax policyandtheoptimaltaxpolicy. In addition to government consumption, the government also runs a balanced-budget social security program. To be consistent with CKK, Social security tax rates are set at 12.4% and the maximum taxable incomeforthesocialsecurityprogramissetat2.5timestheaverageearningsintheeconomy. 5 Computational Experiment Thecomputationalexperimentisdesignedtodeterminethetaxpolicythatmaximizesagivensocialwelfare function. Ichooseasocialwelfarefunction(SWF)thatcorrespondstoaRawlsianveilofignorance(Rawls (1971)). The social welfare is equivalent to maximizing the ex-ante expected lifetime utility of agents 19

before entering the model. When searching for the optimal tax policy I determine an optimal tax policy where capital and labor income are taxed at separate rates. I determine the optimal flat capital income (τ ) k taxbutallowforaprogressivelaborincometaxpolicy. Isearchoverthreedifferentfunctionalformsforthe progressive labor income tax policy.42 First, I use the three parameter functional form from Gouveia and Strauss(1994)suchthatlabortaxesequal, τ g h0 s (yl−((yl)−τg h s 1+τ g h2 s ) − τ g h 1 s 1, (23) whereyl isthetaxablelaborincome. Second,IusethetwoparameterfunctionalformfromBe´nabou(2002) wherelabortaxesequal, yl−τb (yl)1−τh1b , (24) h0 wheregenerallyτb governstheleveloftaxationandτb governstheprogressivity. Finally,Iexamineathree 0 1 parameterpiecemealtaxfunctionwherelabortaxesequal, τ p min{0,yl−τ p }−τ p (25) h0 h1 h2 p p p whereτ isthetaxrate,τ isafixeddeduction,andτ isalump-sumtransfer. Thereforethecomputational h0 h1 h2 experimentismaximizingtheexpectedutilityforanewborn, (cid:34) (cid:35) J−j−1 s SWF(τ ,τ ,τ ,τ )=E u(c ,h )+ ∑ βs∏(Ψ )u(c ,h ) , (26) h0 h1 h2 k 1 1 q s+1 s+1 s=1 q=1 whereτ ,τ ,andτ aretherelevantlaborincometaxparametersforthefunctionalformofinterest,andτ h0 h1 h2 k istheflattaxrateoncapitalincome.43 Todeterminetheeffectsofendogenoushumancapitalaccumulation, IcomparethetaxpoliciesthatmaximizetheSWFinthetwomodels. 6 Results Inthissection,IquantitativelyassesstheeffectsontheoptimaltaxpolicyofincludingLBDinacalibrated life-cyclemodel. InordertoassessLBD’seffect,Ideterminetheoptimaltaxpoliciesintheexogenousand LBDmodels,highlightthechannelsthatcausethedifferences,anddeterminetheeffectsontheeconomyof theseoptimalpolicies. 42Allofthesefunctionalformsnestaflatlaborincometaxrate. 43Thesearchisdoneseparatelyforeachtaxfunction.Inthecaseofthetwoparameterfunctionalformτh2 canbeignored. 20

6.1 ComparisonofModeltotheData Prior to examining the effects of LBD on the optimal tax policy, I compare both models’ predictions for the life-cycle profiles to the data. Figure 1 plots the average life-cycle profiles from the models under the baseline-fitted U.S. tax policy and in the data.44 The upper-left panel compares the average percent of the timeendowmentthatisspentworkingoverthelifetimeandtheupper-rightcomparesthelaborincome. The actual labor supply and labor earnings profiles are constructed from the 1967 - 1999 waves of the Panel SurveyofIncomeDynamics(PSID).InthedataIfocusmyattentiononthelaborsupplyandlaborearnings for the head of the household between ages 20 and 80. The lower-left panel compares the consumption profile in the model to the per-capita expenditures on food in the PSID.45 The lower-right panel examines themediansavingsinthemodelsandmediantotalwealthinthe2007SurveyofConsumerFinances(SCF) forindividualsbetweentheagesof20and80.46 Ismooththroughsomeofthevolatilityinthewealthdata byusingfiveyearagebins.47 Focusingonthelaborsupplyprofiles,themodels’predictedprofileshaveadifferentgeneralpatternthan the data. In the data, the labor supply profile is more hump shaped. In contrast, the models tend to predict the labor supply will decrease throughout the working lifetime. However, in the last few working years this decrease in labor supply is larger in the LBD model, more closely matching the data. Moreover, the modelsseverelyoverpredictstheamountoftimeyoungagentsspendworkingbecauseinthemodelsagents cannotborrowagainstfutureearnings. Therefore,inordertoaccumulateprecautionarysavingsagentswork moreearlyintheirlifetime.48 Incontrast,inthedata,someyounghouseholdsmayhaveameanstoborrow, minimizingtheseverityofthiseffect. Despitethedifferencesinlaborsupply,theprofileoflaborearningsinthedataismoresimilartotheones generatedbythemodels(upper-rightpanel). Allthreelaborearningsprofilesarehumpshapedwithapeak just after forty years old. The LBD model does a bit better job matching the earnings profile at the end of theworkinglifetimebecauseofthelargerdropinlaborsupplyforthelastfewyearsoftheworkinglifetime. Onemaindifferencebetweenthedataandbothofthemodelgeneratedprofilesishowlaborearningsevolve after the age of 66. Agents are forced to retire at 66 in the models, but in the U.S. economy some head of 44Earnings,consumption,andsavingsfromthemodelsareconvertedtoreal2012dollarsbyequatingtheaverageearningsin eachofthemodelsandthedata. 45Per-capitaexpendituresforeachhouseholdarecalculatedasthetotalfamily’sexpendituresdividiedbythetotalnumberof individualsinthehousehold(includingchildren). 46WhencomparingthesavingsdatatothemodelIchoosetousethemedianasopposedtothemeansothattheupper-tailofthe distributiondoesnotskewthecomparisonstatistic. 47Thedataforindividualsafterage80arenotincludedbecausetherewerefewobservationsinthesampleleadingthesmoothed estimatestobeextremelyvolatile. 48ForfurtherdiscussionseeHeathcoteetal.(2010) 21

householdsretireaftertheageof66. Thus,theearningsprofilefortheseolderhouseholdsarehigherinthe datathanineitherofthemodels. Figure1: ActualandExogenouslife-cycleProfiles Labor 0.5 0.4 0.3 0.2 0.1 0 20 40 60 80 100 tnemwodnE emiT fo % Earnings 60 Exog. 50 LBD Actual 40 30 20 10 0 20 40 60 80 100 Age 000,1$ Exog. LBD Actual Age Consumption 45 40 35 30 25 20 15 20 40 60 80 100 )sledoM( 000,1$ 18 16 14 12 10 8 6 4 Age )dooF lautcA( 000,1$ Savings 3.5 3 2.5 2 1.5 Exog 1 LBD Actual 0.5 0 20 40 60 80 100 000,001$ Exog. LBD Actual Age Note:Theseplotsarelife-cycleprofilesintheexogenousmodelunderthebaseline-fittedU.S.taxpolicyandtheactualprofilesin thedata. Theunitsoftheconsumption,earnings,andcapitalprofilesareconvertedtorealdollarsbymatchingtheaveragelabor earningsinthemodelandinthedata.Thelabor,earnings,andconsumptionprofilesaretheaverageacrossthecohort.Thesavings profilesarethemedianvalueswithineachcohort. Whencomparingtheconsumptionprofiles,Ifindthatalltheprofilesarehump-shaped. However,Ifind that consumption on food tends to peak earlier in the data than total consumption in either of the models. Additionally, comparing the growth in consumption from the age 20 to the peak, the model exhibits more growthinconsumptionoverthelifetime. OnepossiblereasonforthesedifferencesisthatthePSIDdataare limitedtojustexpendituresonfoodwhilethemodelgeneratedconsumptionrepresentsallconsumption. In part,thismaybeduetoavaryingpercentageoftotalexpendituresdevotedtofoodoverthelifetime.49 Finally,Ifindthatthesavingsprofilesaresimilarinthemodelsandthedata. Allofthemediansavings profiles are hump-shaped, peaking between $250,000 and $300,000 at the age of 60. One smaller discrepancy is that the models predict that agents will deplete their savings more quickly than savings declines in thedata. Thisdifferencecouldarisebecausethemodeldoesnotincludeanymotiveforindividualsleaving 49Forexample,iffoodislessofaluxurygoodthanotherexpenditurecategoriesthenexpendituresonfoodmaypeakbeforetotal expenditures. 22

a bequest for younger generations or holding savings in case of some unexpected end of life expenditures suchasmedicalexpenses. Overall,bothmodelsdoafairjobmatchingthedatawithrespecttoearningsandsavings. Moreover,if anythingtheLBDmodeldoesabitbetterjobmatchingtheearningsprofileattheendoftheworkinglifetime. However, one concern is that neither model produces a labor supply profile that matches the shape of the data. Thus, Section 7.1 examines whether the effect of LBD is consistent even when I use an alternative utilityfunctionthatimplieslessrelationshipbetweenthelaborsupplyprofileandoptimaltaxpolicy.50 6.2 OptimalTaxPoliciesinExogenousandLBDModels Whendeterminingtheoptimaltaxpolicy,IsearchoverallthreetaxfunctionsdescribedinSection5. Ifind that in both models using the piecemeal tax function of the form of a flat tax on labor income with a fixed deductioncoupledwithalump-sumtransferisoptimal.51 Thistaxfunctionimpliesthatforagentswithlow laborincomethemarginallaborincometaxrateiszeroandtheaveragerateisnegative.52 Table3describes the optimal tax policies in the two models and Figure 2 plots the average and marginal labor tax rates by income in both models. Starting with the exogenous model, the optimal tax policy is a 30 percent tax on capital, a 32.5 percent tax on labor income with a fixed deduction of $6,218, and a lump-sum transfer of $3,683. Incontrast,inthemodelwithLBD,Ifindthattheoptimaltaxpolicyisa36percenttaxoncapital,a 22.3percenttaxonlaborincomewithafixeddeductionof$10,901,andalump-sumtransferof$365. There aretwomaindifferencesbetweentheoptimaltaxpoliciesinthemodels. First,theoptimallabortaxpolicy ismuchflatterintheLBDmodelprimarilybecauseofthemuchsmallerlump-sumtransfer. Second,thetax oncapitalisnotablylargerintheLBDmodel. These changes in the optimal tax policy are due to the intertemporal distortion channel and elasticity channel. First, asSection2.2demonstrates, thedistortionsfromaprogressivelabortaxaremagnifiedwith LBD since the current labor choice affects the level of future human capital. For example, in the LBD modelworkingmoretodaywillincreasehumancapital, whichwouldimplyahigherfuturemarginallabor income tax rate with a progressive tax policy. A progressive tax provides the benefit of insurance against 50Although the labor supply profile can affect the optimal tax policy through numerous channels, the general shape does not seemrelatedtotheoptimaltaxpolicywhentheutilityfunctionisbothseparableandhomotheticineachconsumptionandlabor (seePeterman(2013)formoredetails). 51Itisnotsurprisingthattheoptimaltaxpolicyisoftheformofaflattaxsinceotherstudies(seeCKK)tendtofindthatthe optimalpoliciescanbecloselyapproximatedbyaflattaxwithafixeddeduction.Whensolvingfortheoptimalwiththisfunctional formIfoundthatitwasimportanttouseagridsearchinordertoensureIfoundaglobaloptimum. 52Allworkingagentsreceivethelump-sumportionofthelabortax. Inaddition,anyincomeunderthefixeddeductionisnot subjecttothemarginaltaxrate. Thus,ifanagent’staxablelaborincomeislessthanthefixeddeductionthentheirtotallabortax billisthenegativevalueofthelump-sum. 23

Table3: OptimalTaxPolicies Figure2: OptimalLaborIncomeTaxRates 40 TaxParameters Exog LBD 30 20 LaborTaxRate 32.5% 22.3% 10 Fixeddeduction $6,218 $10,901 0 Lump-sum $3,683 $365 CapitalTaxRate 30% 36% −10 −20 −30 −40 0 50 100 150 200 tnecreP LBD Mrg LBD Avg Exog Mrg Exog Avg Income ($1,000) Figure3: FrischElasticity 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 20 30 40 50 60 70 yticitsalE Exog. LBD Age Note: Theupper-leftpanelaretheoptimaltaxpolicies. Theupper-rightpanelplotstheaverageandmarginaltaxratesunderthe optimaltaxpoliciesineachmodel.ThelowerpanelplotstheFrischlaborsupplyelasticityprofilesineachmodel. the idiosyncratic labor productivity but distorts agents’ decisions. Thus, adding LBD changes this tradeoff betweenefficiencyandequtyleadingaflatterlaborincometaxpolicytobeoptimal. Second, as Section 2.1 demonstrates, adding LBD causes agents to supply labor even more elastically astheyagebecausethehumancapitalbenefitdecreases(elasticitychannel). ThischangeintheFrischlabor supply elasticity is apparent in the Frisch labor supply elasticity profile from the models (see Figure 3). Because the Frisch elasticity tends to increase more as an agent ages in the LBD model, the social planner wouldliketotaxlaborincomeearnedwhenanagentisyoungatarelativelyhigherratethanincomeearned when an agent is old. The social planner can mimic this type of age-dependent tax with either a tax on capital or a progressive/regressive tax on labor income. Specifically, a positive tax on capital implies that the tax on labor income is monotonically decreasing as an agent ages.53 In contrast, the effectiveness of 53Inparticular, sincethereturnsfromsavingcompound, theimplicittaxonlaborincomefromapositivetaxdecreasesatan exponentialrateasanagentages. 24

mimickingthistypeofage-dependenttaxpolicywithaprogressive/regressivelaborincometaxdependson theshapeoftheaveragelaborearningsprofile. TheupperrightpanelofFigure1demonstratesthatalthough average labor income is increasing over a slight majority of the working lifetime in the LBD model, the income profile is hump shaped. The non-monotonicity of the labor income profile implies that although a moreregressivelaborincometaxwilltendtotaxlaborincomeearnedwhenanagentisyoungatarelatively higherrate,itwillalsotaxlaborincomeearnedattheendoftheworkinglifetimeatarelativelyhigherrate. Therefore,itislikelythattheelasticitychannelisresponsibleforthechangeintheoptimalcapitaltaxandis lessresponsibleforthechangeintheoptimalprogressivity. InSection7.1, Iconfirminaslightlydifferent modelthattheelasticitychannelisprimarilyresponsibleforthechangeintheoptimalcapitaltaxleavingthe intertemporal distortion channel primarily responsible for the reduction in the progressivity of the optimal taxpolicy. 6.3 WelfareEffects In order to determine the economic significance of endogenous human capital on optimal taxation, I turn tothewelfareeffectsofnotaccountingforLBDwhensolvingforvariouspiecesoftheoptimaltaxpolicy. I measure welfare in consumption equivalent variation (CEV) which is defined as the uniform increase in consumptionanagentwouldneedateachageinordertobeindifferenttobeingbornintoaneconomywith alessoptimaltaxpolicycomparedtolivinginaneconomywiththeoptimaltaxpolicy.54 First,Idetermine thewelfarelossintheLBDmodelfromadoptingtheoptimaltaxpolicysolvedforintheexogenousmodel (whichincludesaprogressivelabortaxandalowertaxoncapital)asopposedtothetrueoptimaltaxpolicy (which includes a flat tax on labor income and a larger tax on capital). I find ignoring LBD when solving fortheoptimaltaxpolicycausesanotablewelfarelossthatisequivalentto.73percentofexpectedlifetime consumption(seeColumnIIofTable4).55 Adoptingthissub-optimaltaxpolicyhastwoeffectsontaxes. First,itentailsadoptingalabortaxpolicy that is more progressive than optimal. Second, it entails adopting a tax policy in which the ratio of the capital-to-labor tax rates is relatively lower than optimal. In order to determine the implications of each of these effects on welfare, I determine the welfare loss when the tax policy includes the second but not the first effect. In particular, I include the progressivity parameters (lump-sum and fixed deduction) from the 54Theincreaseiscalculatedastheex-anteincreasepriortoanagentrealizingtheiridiosyncraticwageshocksorageofdeath. I calculatethewelfarelossasthisincreaseinconsumptionforanagentlivingunderasub-optimaltaxpolicynecessarytomakethem indifferenttolivingundertheoptimaltaxpolicy. 55Thelabortaxrateunderthetaxpolicysolvedforintheexogenousmodel(seethefirstcolumnofTable3)isslightlydifferent fromthelabortaxratewhenapplyingthispolicytotheLBDmodel(seecolumnIIofTable4)inordertoensurethegovernment budgetconstraintissatisfied. 25

Table4: WelfareLossinLBDModelWhenUsingOptimalTaxPoliciesfromDifferentModels Opt. Exog. Opt. Exog. Level Exog. Capital I II III IV LaborTaxRate 22.3% 33.5% 26.4% 23.3% Fixeddeduction $10,901 $6,272 $10,901 $11,327 Lump-sum $365 $3,715 $365 0 CapitalTaxRate 36% 30% 24.3% 30% CEV 0.73% 0.2% 0.05% Note: WelfarelossesaresolvedforintheLBDmodelandarerelativetotheoptimaltaxpolicyinColumnI(seesecondcolumn ofTable3).ColumnIIcalculatesthewelfarelossfromlivinginaneconomywiththesub-optimaltaxpolicyinwhichtheratesare consistentwiththeoptimaltaxpolicysolvedforintheexogenousmodel(leavingthelabortaxratefreetoclearthegovernment budgetconstraint).ColumnIIIdeterminesthewelfareeffectsofthesub-optimaltaxthatincludestheoptimalratioofthelaborand capitaltaxratesolvedforintheexogenousmodelbutincludestheoptimalprogressivityparametersfromtheLBDmodel.Column IVsolvesfortheoptimaltaxpolicyintheLBDmodelrestrictingthecapitaltaxratetobetheoptimalrateintheexogenousmodel anddeterminesthewelfareloss. true optimal LBD tax policy but a ratio of the capital-to-labor tax rates that is consistent with the optimal ratio in the exogenous model. The welfare loss from including just the sub-optimal capital-to-labor tax ratioisequivalentto.20%ofexpectedlifetimeconsumption(seecolumnIII),aboutonequarterofthetotal welfare loss when both effects are included. Column IV determines how much of the welfare loss from adoptingasub-optimalcapitaltaxratecanbemitigatedwhentheprogressivityparametersarere-optimized. In particular, column IV adopts the sub-optimal lower capital tax rate but allows the social planner to reoptimizethelaborincometaxparametersconditionalonthislowercapitaltaxrate. Ifindthatcoupledwith thislowcapitaltaxratethesocialplannerpreferstoincludealargerfixeddeductionandeliminatethelumpsumtransferleadingtoonlyaminimalwelfareloss(only.05%ofexpectedlifetimeconsumption)compared to the unrestricted optimal tax policy. Taken as a whole, these results demonstrates that a majority of the welfare lost in the LBD model from not accounting for LBD is due to the inclusion of a more progressive taxpolicy(thefirsteffect)andalmostalloftheremainingwelfarelossfromthelowerthanoptimalcapital tax(thesecondeffect)canbereducedbyadjustingtheprogressivityofthelaborincometaxpolicy. I further investigate the welfare consequences of adopting sub-optimal tax parameters in both models in Table 5 in order to determine the relative sensitivity of welfare to the different parameters. Columns I-IIIexaminetheeffectsintheexogenousmodel,andcolumnsIV-VIexaminetheeffectsintheLBDmodel. ColumnIandIVdeterminetheeffectswhenthecapitaltaxrateisincreasedbytheequivalentoffiftypercent oftheoptimalrateintheexogenousmodel. Likewise,ColumnsIIandVexaminetheimplicationswhenthe fixeddeductionisincreasedbyfiftypercentandcolumnsIIIandVIdeterminethewelfareeffectswhenthe 26

lump-sumtransferisincreasedbyfiftypercent. Thegeneralsizesofthewelfarelossesfromchangingeach of the tax parameters are fairly similar in both models with a change in the lump-sum transfer causing the largest welfare losses and a change in the fixed deduction causing the smallest welfare losses. Moreover, the welfare from raising the capital tax 15 percentage points in the LBD model is equivalent to 0.35% of expected lifetime consumption. The size of this welfare loss demonstrates some of the reason for the relativelysmallereffectfromthecapitaltaxinTable4columnIIisbecauseofthesmallerdifferenceinthe optimalcapitaltaxratesinthetwomodelsthanthedifferencesintheprogressivityparameters. Table5: WelfareEffectsofMisspecifiedOptimalTaxPolicy Exogenous LBD I II III IV V VI LaborTaxRate 28.1% 37% 40% 16.3% 25.6% 31.1% Fixeddeduction $6,218 $9,327 $6,218 $10,901 $14,10 $10,901 Lump-sum $3,683 $3,683 $5,524 $365 $365 $2,207 CapitalTaxRate 45% 30% 30% 51% 36% 36% CEV 0.26% 0.12% 0.33% 0.35% 0.09% 0.44% Note: Thewelfareeffectsarethewelfarelossesfromswitchingfromtheoptimaltaxpolicyineachspecificmodeltothesenonoptimaltaxpolicies. 6.4 EffectsonMacroeconomy Tofullyunderstandtheeffectsofendogenoushumancapitalaccumulation,Ianalyzetheaggregateeconomic variables and life-cycle profiles in both models under the baseline-fitted U.S. tax policy as well as the changesinducedbyimplementingtheoptimaltaxpolicies. 6.4.1 TheEffectsofAddingEndogenousAge-SpecificHumanCapital First, Icomparetheeffectontheaggregateeconomicvariablesandlife-cycleprofilesfromaddingLBDto theexogenousmodelunderthebaseline-fittedU.S.taxpolicy. Table6summarizestheaggregateeconomic variablesunderboththebaseline-fittedU.S.taxpolicyandoptimaltaxpolicies. Figure4plotsthelife-cycle profilesofhours,consumption,assets,andage-specifichumancapitalinbothmodels. ComparingtheexogenousandLBDmodelsunderthebaseline-fittedU.S.taxpolicy,thefirstandfourth columns of Table 6 demonstrate that the levels of aggregate hours, labor supply, and aggregate capital are similar in the two models. The calibrated parameters are determined so that under the baseline-fitted U.S. 27

Table6: AggregateEconomicVariables Exogenous LBD %∆from %∆from Baseline Baseline Aggregate Baseline Optimal toOptimal Baseline Optimal toOptimal Y 0.91 0.87 -4.83% 0.90 0.88 -2.56% K 2.48 2.28 -7.93% 2.43 2.25 -7.38% N 0.52 0.51 -3.04% 0.52 0.52 0.23% AvgHours 0.33 0.32 -5.49% 0.33 0.33 -0.6% w 1.12 1.10 -1.85% 1.12 1.09 -2.79% r 0.05 0.05 9.02% 0.05 0.06 13.84% beq 0.03 0.03 -9.35% 0.03 0.03 -6.25% CEV 1.12% 0.69% AverageTaxRate Baseline Optimal Baseline Optimal Labor 0.19 0.18 0.19 0.15 Capital 0.18 0.30 0.18 0.36 Ratio 1.07 0.59 1.07 0.42 MarginalTaxRate Baseline Optimal Baseline Optimal Labor 0.2 0.32 0.2 0.22 Capital 0.19 0.30 0.19 0.36 Ratio 1.06 1.08 1.07 0.62 Note:Theaveragehoursreferstotheaveragepercentoftimeendowmentworkedintheproductivelaborsector.Sincethemarginal taxratesvarywithincome,thereportedmarginaltaxratesarethepopulationandincomeweightedaveragemarginaltaxratesfor allagents. 28

taxpolicythemodelsmatchcertaintargetsfromthedata. Sincemanyoftheseaggregateeconomicvariables aretargetsusedtocalibrateeachofthemodels,theaggregatesaresimilarinthetwomodels. Figure4: Life-CycleProfilesunderBaseline-FittedU.S.TaxPolicy Labor Supply 0.5 0.4 0.3 0.2 0.1 0 20 40 60 80 100 tnemwodnE fo % Consumption 1 0.9 0.8 0.7 0.6 0.5 Exog. 0.4 LBD 0.3 0.2 20 40 60 80 100 Age noitpmusnoC Exog. LBD Age Savings 6 5 4 3 2 1 0 20 40 60 80 100 stessA Age−specific Human Capital 2.5 2 1.5 Exog. LBD 1 20 40 60 80 100 Age ytivitcudorP Exog. LBD Age Note:Theseplotsarelife-cycleprofilesofthethreecalibratedmodelsunderthebaseline-fittedU.S.taxpolicy. Although adding LBD does not have a large effect on the aggregate economic variables, it does cause somechangesinthelife-cycleprofiles. AddingLBDcausesagentstoworkrelativelymoreatthebeginning oftheirworkinglifewhenthehumancapitalbenefitislarger,andlesslaterwhenthebenefitissmaller(see the solid black and solid red lines in the upper-left panel of Figure 4).56 The upper-right panel shows that thelifetimeconsumptionprofileisatouchflatterintheLBDmodelcomparedtotheexogenousmodel. The intertemporalEulerequationgenerallycontrolstheslopeoftheconsumptionprofileoveranagent’slifetime. Therelationshipis (cid:32) (cid:33)σ1 c j+1 =Ψ βr , (27) j (cid:101)t c j where r is the marginal after-tax return on capital. The right-hand-side of Equation 27 is lower in the (cid:101)t 56InboththeLBDandexogenousmodelstheintratemporalmarginalrateofsubstitutionbetweenconsumptionandleisureis affected by scale in such a way that high-type agents (with higher consumption) tend to work less than low-type agents. The differencebetweenhoursworkedbyhigh-typeandlow-typeagentsisofasimilarmagnitudeinbothmodelssoitisnotresponsible forthedifferencesinthetwomodels’laborsupplyprofiles. 29

LBDmodel,primarilybecauseβislower,whichleadstotheflatterconsumptionprofile. Generally,agents have a similar level of savings during their working years in both models. Thus, the flatter consumption profile in the LBD model translates into less savings for the second half of the agents life because with less consumption they do not need as high of level of savings (see the lower-left panel). The lifetime agespecific human capital profiles are similar in the two models since the sequence of parameters {Ωj} jr−1 j=20 is calibrated so that age-specific human capital in the LBD model matches the exogenous model (see the lower-rightpanelofFigure4). 6.4.2 TheEffectsoftheOptimalTaxPolicyintheExogenousModel This subsection examines the effects on the economy of adopting the optimal tax policy in the exogenous model. In the exogenous model, the optimal capital tax is larger than the tax under the baseline-fitted U.S. tax policy so adopting the optimal tax policy causes a decrease in aggregate capital (see columns one and twoofTable6). Theaveragemarginallabortaxishigherundertheoptimaltaxpolicybuttheaveragelabor taxissmaller. Thus,adoptingtheoptimaltaxpolicycausesabitlargerdecreaseinaggregatecapitalthanin aggregate labor leading the rental rate to increase and the wage rate to decrease. Adopting the optimal tax policy in the exogenous model leads to a welfare increase that is equivalent to 1.12% of expected lifetime consumption.57 Figure 5 plots the life-cycle profiles for time worked, consumption, and assets in the exogenous model under the baseline-fitted U.S. tax policies and the optimal tax policies. Adopting the optimal tax policy in theexogenousmodelcauseschangesinallthreelife-cycleprofiles: (i)agentsworkless,especiallyearlyin theirlife,(ii)agentssaveless,and(iii)thelifetimeconsumptionprofileisflatter. Overall,agentsworkless because of the higher marginal labor tax and the lower wage. Moreover, the higher implicit tax on young labor income due to the increase in the tax rate on capital is responsible for agents working considerably lessearlyintheirlife.58 Implementing the optimal tax policy causes an increase in both the capital tax and the rental rate on 57Allowingforage-dependenttaxationcouldpotentiallyleadtoabitlargerwelfaregains. Forexample, Gervais(2012)finds inasimilarmodelwithexogenoushumancapitalaccumulationthatthewelfarelossfromdisallowingage-dependenttaxationand restrictingthesocialplannertoaparticularclassoftaxfunctionsisapproximately0.4percentoflifetimeconsumption.daCostaand Santos(2015)findsinalife-cyclemodelthatthewelfaregainsfromallowingthetaxpolicytobeconditionedonageisbetween2 and4percentoflifetimeconsumptiondependingonwhetherhumancapitalisaccumulatedexogenouslyorendogenously.However, thesewelfareestimatesmaybeanupperbound.Inparticular,theyrestricttheclassofbothage-independentandage-dependenttax functionstothetwoparameterfunctionalformfromBe´nabou(2002). InmymodelIfoundthattheoptimalpoliciesfromthistax functionwerealwaysinferiortotheothertwofunctionalforms. Thus,incorporatingamoreflexiblefunctionalformthatallows foralump-sumwouldlikelyleadtowelfaregainsbothwithage-independentandage-dependenttaxpolicies.However,onewould suspectthatthesegainswouldbeevengreaterfortheage-independenttaxpolicysincetheoptimalprogressivitywouldbemore important. 58Inadditionagentsworkatleastalittlelessacrossallagesoftheirlifetimebecauseofthehighermarginaltaxonlabor. 30

Figure5: Life-CycleProfilesintheExogenousModel Labor Supply 0.5 0.4 0.3 0.2 0.1 0 20 40 60 80 100 tnemwodnE fo % Consumption 1 0.9 0.8 0.7 0.6 0.5 Baseline 0.4 Optimal 0.3 0.2 20 40 60 80 100 Age noitpmusnoC Baseline Optimal Age Savings 6 5 4 3 2 1 0 20 40 60 80 100 stessA Baseline Optimal Age Note: Sincetheskillsarethesameintheexogenousmodelsunderthebaseline-fittedU.S.taxpolicyandoptimaltaxpolicy,they arenotplotted. 31

capital,leadingtoshiftsinboththeconsumptionandsavingsprofiles. Overall,theincreaseinthecapitaltax dominates, so the marginal after-tax return on capital falls under the optimal tax policy, leading to a flatter consumption profile (Figure 5, upper-right panel). Moreover, because of the lower returns, agents tend to holdlesssavingsthroughouttheirlifetime(seethelowerleftpanelofFigure5). 6.4.3 TheEffectsofOptimalTaxPolicyintheLBDModel Figure6: Life-CycleProfilesintheLBDModel Labor Supply 0.5 0.4 0.3 0.2 0.1 0 20 40 60 80 100 tnemwodnE fo % Consumption 1 0.9 0.8 0.7 0.6 0.5 Baseline 0.4 Optimal 0.3 0.2 20 40 60 80 100 Age noitpmusnoC Baseline Optimal Age Savings 6 5 4 3 2 1 0 20 40 60 80 100 stessA Age−specific Human Capital 2.5 2 1.5 Baseline Optimal 1 20 40 60 80 100 Age ytivitcudorP Baseline Optimal Age Adopting the optimal tax policy in the LBD model causes a large increase in the capital tax and a flattening of the labor tax policy. Thus, adopting the optimal tax policy in the LBD causes an even larger decreaseintheaveragelabortaxrateandasmallerincreaseintheaveragemarginallabortaxratethaninthe exogenous model. These tax changes lead to a similar fall in aggregate capital as in the exogenous model. However,becausetheoptimallabortaxrateissmallerintheLBDmodelcomparedtotheexogenousmodel, adoptingtheoptimaltaxpolicycausesasmallincreaseinaggregatelabor(asopposedtothesmalldecrease intheexogenousmodel). Adoptingtheoptimaltaxpolicyleadstoawelfareincreaseacrossallagentsthat is equivalent to 0.69% of lifetime consumption, smaller than in the exogenous model. One reason for the smaller welfare gains from adopting the optimal tax policy in the LBD model is because the optimal tax 32

policyintheLBDmodelisclosertothebaseline-fittedUStaxpolicythanintheexogenousmodel.59 ImplementingtheoptimaltaxpoliciesintheLBDmodelcausesagentstoshifttimeworkedfromearlier tolateryearsinresponsetothelargercapitaltax,whichimplicitlytaxeslaborincomefromearlyyearsata higher rate. Unlike the exogenous model where there is an overall decline in labor, in the LBD model the change is more of a reshuffling of hours from earlier in their lifetime to later in their lifetime (upper-left panel of Figure 6).60 This shift in hours leads age-specific human capital to also be lower in the first half of agents’ lifetimes and higher in the second half under the optimal tax policy. Moreover, because agents worklessinthefirsthalfoftheirlifetheyalsotendtosavelessduringtheseages(lower-leftpanelofFigure 6). ThereisasimilarflatteningintheconsumptionprofilefromadoptingtheoptimaltaxpolicyintheLBD model. 7 Sensitivity of Results Inthissection,Iexaminethesensitivityoftheresultswithrespecttoanumberofdimensions. 7.1 AlternativeUtilityFunction I test the sensitivity of the effect of LBD on the optimal tax policy using an alternative utility function in which the consumption and labor components are separable and each homothetic. In particular, I use the utility function c1−σ1 −χ (h) 1+σ 1 2 . I refer to this utility function as “labor utility” since labor, as opposed to 1−σ1 1+ σ 1 2 leisure,enterstheutilityfunction.61 Inthelaborutilityfunction,σ istheFrischelasticityintheexogenous 2 model. Therearethreeadvantagestousingthisutilityfunction. First,usingthelaborutilityfunctionimpliesthat intheexogenousmodeltheFrischelasticityisnotafunctionoftheagent’slabordecision. Thisflexibilityin theFrischelasticityallowsonetodecomposetherolethateachchannelplaysinthechangetotheoptimaltax policywhenincludingLBD.62Second,usingthisfunctionallowsmetotestwhethertheresultsaresensitive totheinclusionofleisureasopposedtolaborasthedirectinputintotheutilityfunction. Third,asPeterman 59Inparticular,inbothmodelsbecauseoftheinclusionofthelump-sumtransfersintheoptimaltaxpolicies,theoptimalaverage taxratesaremoreprogressivethaninthebaseline-fittedUStaxpolicy. However,inthecaseoftheexogenousmodeltheoptimal averagelabortaxratesareevenmoreprogressivethanthebaseline-fittedUStaxpolicyortheoptimalpolicyintheLBDmodel. 60The smaller overall changes to the labor supply profile from adopting the optimal tax in the LBD model compared to the exogenous model are not due to offsetting changes across different types of agents. Instead the smaller changes in the profiles reflectthatingeneralagentsrespondlesstoadoptingtheoptimaltaxpolicyintheLBDmodelthanintheexogenousmodelsince theincreaseinthemarginaltaxrateissmallenoughthatitdoesnotcauseanoverallreductioninthelaborsupply. 61Alloftheparametersarerecalibratedinthismodelsuchthatthemodelstillmatchestherelevanttargets. 62Inthebenchmarkutilityfunctionleisureasopposedtolaborenterstheutilityfunction.Thusinthebenchmarkutilityfunction theagentslaborsupplydecisionaffectstheFrischlaborsupplyelasticity.Thusdoingasimilardecompositionwiththebenchmark utilityfunctionisnotfeasible. 33

(2013) demonstrates, the optimal tax policy tends to be less sensitive to the labor supply profile with this typeofutilityfunction. Sincetherearesomedifferencesbetweenthemodelgeneratedlaborsupplyandthe data, it is useful to confirm that the effects of adding LBD on the optimal tax policy are similar with the laborutilityinordertoconfirmthattheeffectsarenotabyproductofchangesinthelaborsupplyprofiles.63 The first two columns of Table 7 present the optimal tax policies in the exogenous and LBD models using the labor utility function, respectively. Comparing these two columns, adding LBD causes a similar changeintheoptimaltaxpolicywiththelaborutilityfunctionasitdoeswiththebenchmarkutilityfunction. Specifically,addingLBDcausestheoptimaltaxonlaborincometobelessprogressiveandtheoptimaltax oncapitaltobelarger. AlthoughthechangesfromaddingLBDaresimilar,thedecreaseintheprogressivity isabitsmallerandtheincreaseinthecapitaltaxisabitlargerwiththelaborutilityfunctionthanwiththe benchmarkutilityfunction. IcalculatethewelfarelossforanagentintheLBDmodelwhentheoptimaltax policysolvedforintheexogenousmodel(whichincludesmoreprogressivityandalowertaxoncapital)is included,ascomparedtothewelfarefortheagentifthetrueoptimaltaxpolicy(whichincludesaflatterlabor taxandahighertaxoncapital)isincluded. IfindthatthewelfarelossfromnotaccountingforLBDwiththe labor utility function is equivalent to 1.19% of lifetime consumption, a bit larger than the same calculation under the benchmark utility (0.72%-see column I of Table 4). Thus, the overall result that incorporating LBDcausesanotablechangeintheoptimaltaxpolicyisrobusttothischangeintheutilityfunction. Table7: OptimalTaxPoliciesWithLaborUtilityFunction TaxParameters Exog LBD Alt. Exog I II III LaborTaxRate 35.2% 20.1% 35.4% Fixeddeduction $1,030 $4,241 $1,272 Lump-sum $5,816 $909 $6,664 CapitalTaxRate 24.1% 33.5% 39.2% Note:Allmodelsincludethelaborutilityfunction.Theexogenousmodelincludeshumancapitalaccumulationexogenously.The LBDmodelincludeshumancapitalaccumulationwithLBD.Thealternativeexogenousmodelincludeshumancapitalaccumulation exogenouslybutisalteredsuchthattheFrischlaborsupplyelasticitymatchestheLBDmodel. Next,Idecomposetheeffectofeachofthechannelsontheoptimaltaxpolicy. Thethirdcolumndetails the optimal tax policy in an alternative exogenous model in which human capital is accumulated exogenously, but σ is set to vary by age such that the Frisch elasticity profile in the alternate exogenous model 2 63Furtherconfirmingthisresult,Ifoundthatwhenχiscalibratedsuchthatthelaborsupplyprofilesintheexogenousmodels matchthedata(witheitherthebenchmarkorlaborutilityfunction)thechangesintheoptimaltaxpoliciesarequiteminimal. 34

matches the Frisch elasticity profile in the LBD model.64 Solving the optimal tax policy in this alternative exogenous model isolates the effect of the elasticity channel since it incorporates this channel in the exogenousmodelbyincludinganage-dependentFrischelasticitybutexcludestheadditionalintertemporal link. The optimal tax policy in this alternative exogenous model (column III) includes a similar amount of progressivity as the optimal tax policy in the exogenous model with the constant Frisch elasticity (column I).However,theoptimaltaxoncapitalisevenlargerincolumnIIIthanwhenLBDisincluded(columnII). Thus,theelasticitychannelcanmorethanaccountfortheincreaseintheoptimaltaxoncapitalwhenLBD is included. Given that the average labor income profile is hump shaped, it is not surprising that the social planner finds a tax on capital, as opposed to a regressive labor income tax, to be the more effective way to mimic age-dependent taxes that monotonically decrease with age. Moreover, the similar progressivity of the labor income tax in the exogenous model and the alternative exogenous model demonstrates that the elasticitychannelisnotresponsibleforthechangeintheoptimalprogressivityofthelaborincometaxwhen LBDisincluded. Instead,theintertemporaldistortiondecreasesthedesirabilityofusingaprogressivelabor tax to redistribute in the LBD model, since the additional intertemporal link in the LBD model magnifies thedistortionfromaprogressivelabortax. 7.2 ParameterizationofLBDFunction In order to test how sensitive the results are to the LBD accumulation function, I solve for the optimal tax policy with two different variants of the parameters in the LBD model. In particular, I examine how the optimal tax policy changes if I independently increase φ or φ . I examine the effects of a one standard 1 2 deviation increase in these parameter values as measured in Chang et al. (2002).65 Table 8 describes the optimaltaxpolicyinthebenchmarkLBDmodelandthesetwovariants. Comparingtheoptimaltaxpolicyin columnItotheoptimalpoliciesincolumnsIIandIII,changingeitheroftheseparametershasonlyminimal implications on the optimal tax policy. Thus, the changes in the optimal tax policy due to incorporating LBDseemtobefairlyrobusttothechoiceofparametersfortheLBDaccumulationfunction. 64InordertokeepthealternativeexogenousmodelconsistentIusethesameothercalibrationparametersasinthebenchmark exogenous model and also hold transfers and social security benefits constant between the two models. I choose to keep these constantbecausePeterman(2013)demonstratesthatchangesinbothcanhavelargeimpactsontheoptimaltaxpolicy. 65IneachofthesevariantsoftheLBDmodelIrecalibratealloftheparameterssuchthatthemodelmatchesthespecifiedtargets inTable2. 35

Table8: OptimalTaxPoliciesWithAlternativeUtility TaxParameters Benchmark HigherΦ HigherΦ 1 2 I II III LaborTaxRate 22.3% 23.5% 22.8% Fixeddeduction $10,0901 $10,872 $10,806 lump-sum $365 $661 $546 CapitalTaxRate 36% 35.9% 36.3% Note:Allmodelsincludethebenchmarkutilityfunction.ColumnIarethebenchmarkresultsfromTable3.ColumnIIandIIIare resultswiththeoptimaltaxpolicieswhenφ1 andφ2 areincreasedbyonestandarddeviation,respectively. 7.3 Transition This paper focuses on computing the optimal steady state tax policy from the perspective of an agent who hasyettoentertheeconomy. Thiswelfarecriterianeglectsanyoftheextracostsorbenefitsthatmayexist during the transition to the steady state with the new tax policy. There are three reasons why one might thinkthatagentswhoarealiveduringthistransition(transitionalagents)mayexperienceadifferentwelfare effect compared to the steady state welfare effects. First, these living agents are already part way through theirlifetimewhichchangestherelativecompositionofthesourcesoftheirincome. Inparticular,thefarther alonganagentisintheirlifethelargerpercentageoftheirfuturelifetimeincomewillcomefromcapitalas opposedtolabor. Sinceadoptingtheoptimalsteadystatetaxpoliciesinbothmodelsincludesadecreasein thelabortaxandincreaseinthecapitaltax,adoptingthesepolicieswilltendtofavortransitionalagentswho areyoungeratthetimeoftheadoption. Althoughthiseffectmaycausedifferentialwelfareeffectsbetween cohorts,itwillnotcauseauniformincreaseordecreaseinthewelfareeffectsacrossallthecohorts. The second reason for a discrepancy between the transitional and steady state welfare effects is that agents may directly benefit or suffer due to a different level of aggregate capital in the initial steady state (underthebaseline-fittedUStaxpolicy)andthefinalsteadystate(undertheoptimaltaxpolicy). Aggregate capital tends to decrease over the transition in both models. In order to bring about this reduction, agents willneedtoincreasetheirconsumptionoverthetransition. Thus,thehigherlevelofconsumptionwillcause thissecondeffecttobenefittransitionalagentsandleadtoevenlargerwelfaregainsthaninthesteadystate. Finally, since it is likely that it will take many periods for capital to transition to the new steady state level under the optimal tax policies, both the rental rate and wage rate will not immediately jump to their newlevels. Unlikethefirsttwoeffects,itissomewhathardertodeterminewhetherthiswillleadtomoreor less of a welfare benefit for transitional agents. In order to get a sense of the impact of the slow transition 36

of capital, I calculate the welfare effects in both models in a counterfactual partial equilibrium steady state wheretheoptimaltaxpoliciesareadopted,butaggregatecapitalisfixedatthelevelunderthebaseline-fitted UStaxpolicy.66 Asopposedtothesteadystatewelfaregainof1.12%and.69%,Ifindthatthewelfaregains inthesecounterfactualsteadystatesare.89%and.98%intheexogenousandLBDmodels,respectively. The steady state welfare gain is a bit smaller than the counterfactual steady state in the case of the exogenous model, and a bit larger in the case of the LBD model. Although there are some differences, agents still experienceawelfaregaininthiscounterfactualsteadystatesinboththeLBDandexogenousmodels. Taken as a whole, it seems unlikely that the steady state welfare gains would be completely reversed and that on average transitional agents would not still experience a welfare gain from adoption of these optimal tax policiesinboththeexogenousandLBDmodels. 8 Conclusion Two important questions for optimal taxation are should the income tax policy be progressive and at what rate should capital be taxed? In this paper, I examine the effect of LBD on optimal taxation and find that it affects the answers to both questions. Analytically, I demonstrate that including endogenous human capital accumulation changes the Frisch elasticity over an agent’s lifetime. Thus, the elasticity channel createsamotiveforthegovernmenttoconditionlaborincometaxesonage,andifdisallowed,eitheranonzerocapitaltaxorprogressive/regressivelaborincometaxcanbeusedtomimictheseage-dependenttaxes. Althoughaprogressive/regressivetaxcanbeusedtomimicanage-dependenttax,Ishowthatthedistortions from this type of tax are magnified in the LBD model due to the additional intertemporal link. Thus, the intertemporaldistortionchannelmakesitlessappealingtouseaprogressivetax. Quantitatively, I find that these two channels cause the inclusion of LBD to substantially change the optimal tax policy. Specifically, including LBD causes the optimal tax policy to include a flatter labor income tax and higher capital tax rate. Moreover, not accounting for LBD causes a loss in welfare of between.7and1.2percentofexpectedlifetimeconsumptiondependingontheutilityfunction. Amajority ofthiswelfarelosscomesfromincludingasub-optimalamountofprogressivity. Ifindthattheseresultsare robusttotheutilityfunction,andtheparametersusedintheLBDskillaccumulationprocess. Giventhesize oftheeffectsonwelfareandtherobustnessoftheeffect,theseresultsindicatethatwhenexaminingoptimal taxation,accuratelyincorporatingtheskillsaccumulationprocesscanbeoffirst-orderimportance. 66Iallowlabortorespondimmediatelytothechangeinthetaxpolicy. Thissteadystateisinpartialequilibriumbecausethe equilibriumconditionthataggregatecapitalequalsthesummationofallagents’levelofsavingsisviolated.However,alltheother aggregatevariablesareconsistentwiththeequilibriumconditions. 37

A Analytical Derivations Whensolvingfortheoptimaltaxpolicyintheanalyticalmodel,Iusethestandardsocialwelfarefunction, ∞ [U(c ,1−h )/θ]+∑θt[U(c ,1−h )+βU(c ,1−h )]. (28) 2,0 2,0 1,t 1,t 2,t+1 2,t+1 t=0 whichmaximizestheexpectedutilitydiscountingfuturegenerationswithsocialdiscountfactorθ. Inorder todeterminetheoptimaltaxpolicyIusetheprimalapproach. Inthisapproachthesocialplannermaximizes directly over allocations. In order to ensure that the chosen allocation can be supported in a competitive equilibrium the implementability constraint is included. In particular, the implementability constraint is formedfirstbycombiningequations2and3toformajointintertemporalbudgetconstraint, c w(1−τ )s (h )h 2,t+1 h,2 2 1,t 2,t+1 c + =w(1−τ )h + . (29) 1,t h,1 1,t 1+r(1−τ ) 1+r(1−τ ) k k The implementability constraint is then determined by taking the agent’s intertemporal budget constraint andreplacingpricesandtaxeswiththeagent’sfirst-orderconditions.67 (cid:16) −βh s (t+1)U (t+1)(cid:17) 2,t+1 h1 h2 c U (t)+βc U (t+1)−h U (t)+ −βh U (t+1)=0. (30) 1,t c1 2,t+1 c2 1,t h1 2,t+1 h2 s (h ) 2 1,t Suppressing the arguments of the skill accumulation formula, the Lagrangian that the Social Planner maximizesis L =U(c ,1−h )+βU(c ,1−h ) (31) 1,t 1,t 2,t+1 2,t+1 −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 (cid:32) (cid:33) (cid:16) −βh s (t+1)U (t+1)(cid:17) 2,t+1 h1 h2 +λ c U (t)+βc U (t+1)−h U (t)+ −βh U (t+1) t 1,t c1 2,t+1 c2 1,t h1 2,t+1 h2 s 2,t+1 whereρistheLagrangemultiplierontheresourceconstraintandλistheLagrangemultiplierontheimplementabilityconstraint.68 Assumingthattheutilityfunctionisseparableinconsumptionandlabor(U =0), ch thefirst-orderconditionswithrespecttocapital,laborandconsumptionare ρ =θ(1+r)ρ (32) t t+1 wρ =(1−λ )U (t)+λ h U (t) t t h1 t 1,t h1h1 h (wθρ s2 s (t+1)−βλU (t+1)) − 2,t+1 t+1 2,t+1 h2 t h2 [s (s (t+1)+h s (t+1))−h s2 (t+1)] s2 2,t+1 h2 1,t h2h2 1,t h2 2,t+1 (33) (cid:32) (cid:33) λ t s h2 (t+1)(cid:0) (cid:1) wρ θs (t+1)=β (1−λ )U (t+1)+λ h U (t+1)− h h U (t+1)−h U (t+1) t+1 2 t h2 t 2,t+1 h2h2 2,t+1 1,t h2h2 1,t h2 s 2,t+1 (34) 67SeeLucasandStokey(1983)orErosaandGervais(2002)forafulldescriptionoftheprimalapproach. 68Inadditiontotheimplementabilityconstraint,theresourceconstraintandgovernmentbudgetconstraintareincluded.However, duetoWalras’Law,thegovernmentbudgetconstraintcanbeexcludedfromtheLagrangian. 38

ρ =(1+λ )U (t)+λU (t)c (35) t t c1 t c1c1 1,t θρ =β[(1+λ )U (t+1)+λU (t+1)c ]. (36) t+1 t c2 t c2c2 2,t+1 Combiningequations35and36yieldstheexpression βρ Uc1(t)+λ (U (t)+U (t)c ) t t c1 c1c1 1,t = (37) θρ U (t+1)+λ (U (t+1)+U (t+1)c ) t+1 c2 t c2 c2c2 2,t+1 Assuming thatthe utility functiondemonstrates constant relativerisk aversion, then thisexpression further simplifiesto, βρ Uc1(t) t = . (38) θρ U (t+1) t+1 c2 Combiningequations4,5,and6yields, (1−τ ) U (t+1)U (t)s h s (t+1) h1 c2 h1 2,t+1 2,t+1 h2 = − . (39) (1−τ ) U (t)U (t+1) 1+r(1−τ ) h2 c1 h2 k Combiningequations38and39andsimplifyingyields, (1−τ ) θρ U (t)s h s (t+1) h1 t+1 h1 2,t+1 2,t+1 h2 = − . (40) (1−τ ) βρU (t+1) 1+r(1−τ ) h2 t h2 k Inordertodeterminetheoptimallaborincometaxes, Icombineequations40, 32, 33, and34andsuppress thetimesubscripts, (cid:16) (cid:17)(cid:16) (cid:17) (1−λ)+λh2Uh2h2− λsh2 h h U −h U 1+ h2sh2 (1−τ h1 ) = Uh2 s2Uh2 2 1 h2h2 1 h2 1+r(1−τk) − h 2 s h2 (41) (cid:16) (cid:17) (1−τ h2 ) (1−λ)+λh1Uh1h1+h2βλUh2 s (s +h s )−h s2 1+r(1−τ k ) Uh1 s2 2 Uh1 2 h2 1 h2h2 1 h2 B Definition of Stationary Competitive Equilibrium In this section I define the stationary competitive equilibrium for the model. An agent’s state variables are assets a, previous periods human capital s, age j, ability α, persistent shock ν, and idiosyncratic shock θ. Foragivensetofexogenousdemographicparameters{n,Ψ },asequenceofskillaccumulationsparameters j {Ω } jr−1 ,agovernmentlabortaxfunctionTl :R →R ,agovernmentcapitaltaxfunctionTk:R →R , j j=20 + + + + a social security tax rate τ , a maximum amount of taxable income for social security y, social security ss benefits SS, a production plan for the firm (N,K), an age-specific human capital accumulation function S:R ×R ×R →R , and a utility functionU :R ×R →R , a stationary competitive equilibrium + + + + + + + consists of agents’ decision rules {c,h} for each state x, factor prices {w,r}, accidental bequests beq, and thedistributionofindividuals{µ(x)}suchthatthefollowingholds: 1. Givenprices,policies,accidentalbequests,benefits,andthatωfollowsequation17or18,theagent 39

maximizesequation16subjectto c+a(cid:48)=wωh−τ wωh,+(1+r)(a+beq )−Tl[wωh−.5τ min{wωh,y})]−Tk[r(a+beq)], (42) ss t ss for j< j ,and r c+a(cid:48)=SS+(1+r)(a+beq)−Tk[r(a+beq)], (43) for j≥ j . r Additionally, c≥0,0≤h≤1,a≥0,a =0. (44) 1 2. Priceswandrsatisfy (cid:18) N (cid:19)1−ζ r=ζ −δ (45) K and (cid:18) K (cid:19)ζ w=(1−ζ) . (46) N 3. Thesocialsecuritypoliciessatisfy ∑ ssµ(x) τ = j>=jr . (47) ss ∑ min{hwω,y}µ(x) j<jr 4. Accidentalbequestsaregivenby beq=∑(1−Ψ)a(cid:48) µ(x). (48) 5. Governmentbalancesitsbudget G=∑Tk[r(a+beq)]µ(x)+ ∑ Tl[wωh−.5τ min{wωh,y})]µ(x). (49) ss j<jr 6. Themarketclears K =∑aµ(x), (50) N =∑hωµ(x), (51) and ∑cµ(x)+∑a(cid:48) µ(x)+G=KζN1−ζ+(1−ζ)K. (52) 7. Thedistributionofµ(x)isstationary,thatis,thelawofmotionforthedistributionofindividualsover thestatespacesatisfiesµ(x)=Q µ(x),whereQ isaone-periodrecursiveoperatoronthe µ µ distribution. 40

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