An Extensive Look at Taxes: How does endogenous retirement affect optimal taxation?

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. An Extensive Look at Taxes: How does endogenous retirement affect optimal taxation? William B. Peterman 2012-28 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.

An Extensive Look at Taxes: How does endogenous retirement affect optimal taxation? WilliamBPeterman FederalReserveBoardofGovernors∗ April17,2012 Abstract Thispaperconsiderstheimpactonoptimaltaxpolicyofincludingendogenouslydeterminedretirement in a life cycle model. Allowing individuals to determine when they retire causes the optimal tax oncapitaltoincreaseby75%becauseoftwoimplicitchangesintheaggregatelaborsupplyelasticity. First,includingendogenousretirementcausesanincreaseintheoverallaggregatelaborsupplyelasticity sinceagentscanchangetheirlaborsupplyonboththeintensiveandextensivemargins. Inresponse,the governmentlimitsthedistortionsfromthetaxpolicybyloweringthetaxonlaborandincreasesthetax oncapital. Second,giventhatthechoicetoretireismorerelevantforolderindividuals,endogenousretirementdisproportionatelyincreasesolderagent’selasticitycomparedtoyoungerindividuals. Ideally, the government would decrease the relative labor income tax on individuals when they are older and supplylabormoreelastically. However,intheabsenceofage-dependenttaxes,thegovernmentmimics suchataxpolicybyfurtherincreasingthetaxoncapital. Ifindthatthewelfarelostfromnotaccounting for endogenous retirement when solving for the optimal tax policy is equivalent to approximately one percentoflifetimeconsumption. JEL:E24,E62,H21. KeyWords: OptimalTaxation,CapitalTaxation,EndogenousRetirement. ∗E-mail: william.b.peterman@frb.gov. Views expressed on this site are my own and do not reflect the view of the Federal ReserveSystemoritsstaff. 1

1 Introduction Previous work demonstrates that in a life cycle model the optimal tax on capital is directly related to the representative cohort’s Frisch labor supply elasticity (see Garriga (2001), Atkeson et al. (1999), Erosa and Gervais(2002),Conesaetal.(2009),andPeterman(2011)). Thiselasticityisgovernedbydecisionsonboth theintensiveandextensivemargins. However,previousstudiesthatdeterminetheoptimaltaxoncapitalina lifecyclemodeltypicallyonlyallowindividualstomakelaborsupplydecisionsontheintensivemarginand assume agents are completely inelastic on the extensive margin. The aim of this paper is to determine the effectonoptimaltaxpolicyofrelaxingthisassumptionbyincludingendogenouslydeterminedretirement. In order to assess the quantitative effect of this assumption I solve computationally for the optimal tax policyintwosimilarlifecyclemodelswherethegovernmentisrequiredtoraiseafixedamountofrevenue. In the first model, individuals are forced to retire at an exogenously set age. In the second model, agents endogenously determine when they retire. Both models includes a reduced form social security program that is in a similar spirit to the program in the United States. In the exogenous model agents are forced to retire at the normal retirement age and then receive a social security benefit. Similar to the U.S. social security program, the reduced form social security program is set up in the endogenous model such that if agentschoosetoretireanytimewithinanineyearwindow,centeredaroundthenormalretirementage,then theyreceiveanactuariallyequivalentamountofbenefitsfromsocialsecurity. Ifindthattheoptimaltaxon capitalisseventyfivepercentlargerinthemodelwithendogenouslydeterminedretirementcomparedtothe exogenous model. These results indicate that this modelling assumption is of first order importance when determiningtheoptimaltaxpolicy. Including retirement endogenously increases the optimal tax on capital because it alters the implied agent’s aggregate Frisch labor supply elasticity in two ways.1 First, by removing the restrictions on labor participation, the agent now has two margins (intensive and extensive) by which they can alter their labor supply. Therefore, the aggregate Frisch elasticity is larger than in the exogenous model where the agent canonlyaltertheirlaborsupplyontheintensivemargin. Accordingly,thegovernmentwouldprefertorely less on a labor tax and more heavily on a capital tax in order to minimize the distortions imposed by the tax code.2 Second, since the choice to retire is more relevant for older individuals, endogenous retirement disproportionately increases the elasticity when an agent is old compared to when they are young. The government would like to respond by decreasing the relative tax on the labor income when agents are old 1IdefinetheaggregateFrischelasticitytoincorporatechangesonboththeintensiveandextensivemargin. 2Astandardresultinpublicfinanceisthat,ifitisnecessarytousedistortionarytaxes,itisoptimaltotaxinelasticallysupplied factorsatrelativelyhigherratessincethispolicywillminimizethedistortionstotheeconomy. 2

and more responsive. When the government cannot use age-dependent taxes, it will use a larger tax on capitalinordertomimicsuchataxpolicy. Examinetheeffectofthisexogenousretirementdecisionisrelevantbecauseotherstudiesindicatethat decisionsontheextensivemarginhavealargeimpactonboththeleveloftheaggregateFrischelasticityand theslopeofthelifetimeFrischelasticityprofile. Importantly,thelifetimeFrischlaborsupplyelasticityprofilefromasimulatedmodeltendstobemoreconsistentwiththedataifendogenousretirementisincluded. Using simulated data from a simple model, Rogerson and Wallenius (2009) demonstrate that ignoring the extensive margin could significantly lower estimates of the aggregate Frisch elasticity. Erosa et al. (2011) calibratealifecyclemodeloflabormarketswhichincorporateschoicesonboththeintensiveandextensive margin. They find that the extensive margin accounts for approximately half of the aggregate labor supply response to a temporary wage change. Finally, Chetty et al. (2011), Peterman (2012), and Fiorito and Zanella (2008) find that incorporating fluctuations on the extensive margin causes estimates of the Frisch elasticity to increase substantially. These larger estimates are more consistent with the parameters used in macro models that are calibrated such that the model’s fluctuations in aggregate hours over the business cyclematchthedata. Additionally,thereissomeempiricalevidencethatolderindividualshavearelatively higherFrischelasticitycomparedtoyoungerindividuals. French(2005)estimatealifecyclemodelinwhich heallowsforabreakpointintheinFrischelasticityparameterattheageofforty. Theauthorfindsthatthe Frisch elasticity is over three times larger for the older individuals compared to younger individuals.3 A modelthatincludesendogenouslydeterminedretirementismoreconsistentwiththelargerestimatesofthe Frischlaborsupplyelasticityforolderindividuals. I choose to focus on endogenously determined retirement, as opposed to examining both the entry and exit from the labor force, since the empirical evidence is consistent with the extensive margin having a larger impact on the labor supply elasticity of older individuals.4 Moreover, Jacobs and Bovenberg (2009) demonstrate that including the decision of when to stop education and enter the workforce could further enhance the motive for a positive tax on capital. The authors analyze the trade off between a labor and capitaltaxinalifecyclemodelwithpre-workeducation. Theyfindthatinatwo-periodmodelwhereagents acquire education in the first period and work in the second period the optimal tax on capital is generally positive if educational investment is not verifiable. The tax on capital reduces the tax on labor income, 3OnecounterexampleisClarkandSummers(1981)whichfindsevidencethattheextensivemargincausesteenageindividuals to be more responsive to the business cycle. My study is primarily focused on individuals after they have finished school and enteredtheworkforcewhichexcludesmanyoftheseteenagers. 4Itispossiblethatindividualscouldmakemorethanjustoneentryandexitdecisionovertheirworkinglifetime. However, Erosaetal.(2011)documentthatfromtheagesoftwentytwothroughfiftyonlyasmallfractionofindividualsworklessthanone hundredhours.Thesmallfractionnotworkingoverthenormalworkinglifeimpliesthatgenerallyonceaheadofhouseholdstarts workinghetendstocontinueuntilheretires. 3

whichinturnreducesthedistortionsonthebenefittoeducation. Therefore,includinganendogenousentry decisionmayfurtherincreasetheoptimaltaxoncapital. This paper is organized as follows: Section 2 introduces the computational model, and presents the competitiveequilibrium. Section3describesthefunctionalformsandcalibrationparameters. Section4sets upthecomputationalexperimentandsection5reportstheresultsofthecomputationalexperiment. Section 5concludes. 2 Computational Model In this section, I describe the computational model and present the definition of a stationary competitive equilibriumwhichisusedtodeterminetheeffectofendogenousretirementonoptimaltaxpolicy. 2.1 Demographics In the model, time is assumed to be discrete and there are J overlapping generations. Ψ is the probability j of an agent living to age j+1 conditional on being alive at age j. All agents who live to an age of J die the next period. If an agent dies with assets, the assets are confiscated by the government and distributed equallytoallthelivingagentsastransfers(Tr ). t An agent retires at the age j . Once they retire they receive a social security benefit and can no longer r work. In the exogenous model, agents are forced to retire at the age j (j = j ). In the endogenous exog r exog modelagentschoose j and j isconsideredthe“normalretirementage”. Agentscanchoosetoretireup r exog tofouryearsbeforeorafterthenormalretirementage(j ∈(j −4,j +4))andreceiveanactuarially r exog exog equivalentsocialsecuritybenefit.5 Ineachperiodacontinuumofnewagentsisborn. Thepopulationofnewagentsborneachperiodgrows at rate n. Given the population growth rate and conditional survival probabilities, the time invariant cohort shares,{µ }J ,aregivenby j j=1 Ψ j−1 µ = µ ,fori=2,....,J, (1) j j−1 1+n whereµ isnormalizedsuchthat 1 J ∑µ =1 (2) j j=1 5In the U.S. economy individuals can choose to retire outside of this nine year window however, the change in their social securitybenefitwillnotbeactuariallyfaircomparedtothebenefittheywouldhavereceivediftheychosetoretireatthenormal retirementage. Inthismodel,sincetherearenoidiosyncraticshockstoearningsalargeportionofallindividual’sconsumption afterretirementisfinancedbysocialsecurity.Therefore,noindividualswouldchoosetoretireoutsideofthenineyearwindow. 4

2.2 Individual An individual is endowed with one unit of productive time per period which he splits between providing laborservicesandleisurepriortohisretirement. Afterretirementheusesallhistimeforleisure. Anagent maximizehislifetimeutility J j−1 {∑βj∏Ψ u(c ,h )}, (3) q−1 j j j=1 q=1 wherec istheconsumptionofanagentatage jandh isthehoursspentprovidinglaborservices. Theagent j j faces a fixed utility cost to working which implies the disutility from working discontinuously increases whenanagentgoesfromzerotopositivehoursworked.6 Thediscountfactorconditionalonsurvivingisβ. Anagentreceiveslaborincomeofh ε w whereε istheagent’sage-specifichumancapital. Thislabor j j t j income is split between consumption and saving using a risk free asset. An agent’s stock of assets are denotedbya andhereceivesapre-taxnetreturnofr ontheassetsperperiod. j t 2.3 Firm Firmsareperfectlycompetitivewithconstantreturnstoscaleproductiontechnology. Aggregatetechnology isrepresentedbyaCobb-Douglasproductionfunction. Theaggregateresourceconstraintis, C +K −(1−δ)K +G ≤KαN1−α, (4) t t+1 t t t t whereK,C,N,G ,α,andδrepresenttheaggregatecapitalstock,aggregateconsumption,aggregatelabor t t t t (measured in efficiency units), government consumption, the capital share, and the depreciation rate for physicalcapital,respectively. 2.4 GovernmentPolicy The government is exogenously forced to consume in an unproductive sector. The government uses two fiscalinstrumentstofinancetheirconsumption, G , inanunproductivesector.7 First, thegovernmenttaxes t capitalincome,y ≡r (a+Tr ),accordingtoacapitalincometaxscheduleTK[y ]. Second,thegovernment k t t k taxes part of each individual’s 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. Therefore, the 6Thefixedcostisnecessaryinordertoprovideenoughdegreesoffreedomtocalibratethemodeltomatchboththeretirement ageandhoursworked. 7A formulation that induces the same optimal tax policy is if Gt enters the agents utility function in an additively separable manner. 5

taxable labor income is y ≡ w ε h (1−.5τ ), which is taxed according to a labor income tax schedule l t j j ss Tl[y ]. Iimposethreerestrictionsonthelaborandcapitalincometaxpolicies. First,Iassumeanonymityof l the tax code so the rates cannot be personalized, nor can they be age-dependent. Second, both of the taxes arefunctionsonlyoftheindividual’srelevanttaxableincomeinthecurrentperiod. Finally, asistypicalin aRamseyproblem,Iassumethegovernmentcannotuselumpsumtaxation. The government also runs a pay-as-you-go social security system. I include a reduced form social security program because Peterman (2011) demonstrates that in a model with retirement, a social security program is necessary to produce realistic life cycle profiles. In the reduced form program included in the model,thegovernmentpaysanannualbenefit,SS ,toallindividualsthatareretired. Thesebenefitsareset t suchthatretiredagentsreceiveanexogenouslydeterminedfraction,b ,oftheaverageincomeofallworking t individuals. Anagent’ssocialsecuritybenefitsareproportionaltotheirlifetimeearnings.8 However,inthe endogenousmodel,ifanagentchoosestoretireearlierthan j thentheyarechargedalumpsumpenalty exog at the time of retirement which is the equivalent of the net present value of the actuarially fair amount of extra benefit he will receive in the years prior to j . Alternatively, if he chooses to retire later than j exog exog then he receives an actuarially fair rebate. This formulation mimics the spirit of the US social security program.9 Social security is financed by taxing labor income at a flat rate, τ . The payroll tax rate τ is set ss,t ss,t to ensure the social security system has a balanced budget each period. The social security system is not consideredpartofthetaxpolicythatthegovernmentoptimizes. 2.5 DefinitionofStationaryCompetitiveEquilibrium In this section I define the competitive equilibria for the computational model. Given a social security replacement rate b, government expenditures G, and a sequence of population shares {µ }J , a stationary j j=1 competitive equilibrium is a sequence of agent allocations, {c ,a ,h ,j }, a production plan for the firm j j+1 j r (N,K), a government labor tax function Tl :R →R , a government capital tax function Tk :R →R , + + + + asocialsecuritytaxrateτ ,autilityfunctionU :R ×R →R ,socialsecuritybenefitsSS,prices(w,r), ss + + + andtransfersTrsuchthat: 8Althoughbenefitsarenotdirectlylinkedtoanindividual’searningshistory,sinceallagentsareidenticalwithinacohort,the socialsecuritypayoutwillbedirectlyproportionaltotherepresentativeagent’slifetimelaborearnings. 9TheUSsocialsecurityprogramchangestheannualbenefitasopposedtoprovidingalumpsumtransferatthetimeofretirement toaccountforearlyorlateretirement. Providingaonetimelumpsumtransferorchangingtheannualbenefitareequivalentifan individual is not liquidity constrained at the time of retirement. However, using a lump sum transfer to account for early or lateretirementiscomputationallylessburdensomebecauseitdoesnotrequireonetotracktheyearofretirementthroughoutan individualslifeasanadditionalstatevariable. 6

1. Givenprices,policies,transfers,andbenefitstheagentmaximizesthefollowing J j−1 ∑Max βj−1[∏Ψ ]u(c ,h ) cj,hj,aj+1,jr q j j j=1 q=0 subjectto c +a =wε h −τ ws h ,+(1+r)(a +Tr)−Tl[wε h (1−.5τ )]−Tk[r(a +Tr)]for j< j , j j+1 j j ss j j j j j ss j r c +a =SS+(1+r)(a +Tr)−Tk[r(a +Tr)], for j> j j j+1 j j t r jexog s c +a =SS+(1+r)(a +Tr)−Tk[r(a +Tr)]−SS ∑[ ∏ Ψ (1+r)j−s]for j= j j j+1 j j t q r s=j q=j+1 h =0, for j≥ j j r c≥0,0≤h≤1,a ≥0, anda =0. j 1 2. Priceswandrsatisfy: (cid:18) N (cid:19)1−α (cid:18) K (cid:19)α r=α −δandw=(1−α) K N 3. Thesocialsecuritypoliciessatisfy: SS=b wN andτ = ss∑J j=jr µ j . ∑ j j r = − 1 1µ j ss w∑ j j r = − 1 1 ε j µ j 4. Transfersaregivenby: J Tr= ∑µ (1−Ψ )a . j j j+1 j=1 5. Governmentbudgetbalance: J jr−1 G= ∑µ Tk[r(a +Tr)]+ ∑ µ Tl[wε h (1−.5τ )]. j j j j j ss j=1 j=1 6. Marketclearing: J J K= ∑µ a , N= ∑µ ε h and j j j j j j=1 j=1 J J ∑µ c +∑µ a +G=KαN1−α+(1−δ)K. j j j j+1 j=1 j=1 7

3 Calibration and Functional Forms To determine the optimal tax policy it is necessary to choose functional forms and calibrate the model’s parameters. IbasemylifecyclemodelonConesaetal.(2009)andPeterman(2011). Calibratingthemodel involvesatwo-stepprocess. Thefirststepischoosingparametervaluesforwhichtherearedirectestimates inthedata. Second,theremainingparametersaredeterminedsothatunderthebaseline-fittedU.S.taxpolicy certain targets in the model match the values observed in the U.S. economy.10 I calibrate all of the second groupofparametersinamodelwithendogenouslydeterminedretirementwhichimpliesthattheparameter valuesarethesameinbothmodels. Theparametervaluesarelistedintable1. Table1: CalibrationParameters Parameter Value Source Demographics: NormalRetirementAge: j 65 ByAssumption exog MaxAge: J 100 ByAssumption Surv. Prob: Ψ BellandMiller(2002) j Pop. Growth: n 1.1% Data FirmParameters: α .36 Data δ 8.33% I =25.5% Y A 1 Normalization CalibrationParameters: ConditionalDiscount: β 0.995 K =2.7 Y Riskaversion: σ 2 Conesaetal.(2009) 1 FrischElasticity: σ 0.5 IntensiveFrisch= 1 2 2 DisutilitytoLabor: χ 60.9 Avg. h = 1 1 j 3 FixedCosttoWorking: χ 0.5 NormalRetirement=65 2 GovernmentParameters: ϒ .258 GouveiaandStrauss(1994) 0 ϒ .768 GouveiaandStrauss(1994) 1 G 0.137 17%ofY b 0.5 Conesaetal.(2009) 10Since these are general equilibrium models, changing one parameter will alter all the values in the model that are used as targets.However,Ipresenttheparameterswiththetargetsthathasthemostdirectcorrespondence. 8

3.1 Demographics In the model, agents are born at a real world age of 20 that corresponds to a model age of 1. The current population in the U.S. faced a normal retirement age of between 65 and 66.11 Since I am calibrating the model to the current U.S. economy, I choose the normal retirement age to be 65 which implies that in the exogenous model agents are forced to retire at a real world age of 65. In the endogenous model, there is a fixed cost to working (χ ), which is calibrated such that individuals choose to retire at age 65.12 If an 2 individualsurvivesuntiltheageof100,hediesthenextperiod. Isettheconditionalsurvivalprobabilitiesin accordancewiththeestimatesinBellandMiller(2002). Iassumeapopulationgrowthrateof1.1percent. 3.2 Preferences Agents have time-separable preferences over consumption and labor services, and conditional on survival, they discount their future utility by β. I use a utility function that is separable and homothetic in both consumptionandlabor. IchoosetousethistypeofutilityfunctionsincebothGarriga(2001)andPeterman (2011)demonstratethatviolatingtheseassumptionswillleadtoalargeoptimaltaxoncapitalandIdonot want this motive to be confounded with a motive from endogenous retirement. The utility function I use is c1−σ1 −χ (h) 1+σ 1 2 −χ . I determine β such that the capital-to-output ratio matches U.S. data of 2.7.13 I 1−σ1 1+ σ 1 2 2 determine χ such that under the baseline-fitted U.S. tax policy, agents work on average one third of their 1 time endowment prior to retiring.14 As mentioned, I determine χ such that under the baseline-fitted U.S. 2 taxpolicyindividualschoosetoretireatthenormalretirementage(65)intheendogenousmodel. Following Conesaetal.(2009),Isetσ =2,whichcontrolstherelativeriskaversion.15 1 Theparameterσ controlstheFrischelasticityontheintensivemargin(intensiveFrischelasticity). The 2 intensiveFrischelasticityisdifferentfromtheaggregateFrischelasticityinthatitonlyincorporateschanges inhoursontheintensivemarginwhiletheaggregateFrischelasticityincorporateschangesinhoursonboth the intensive and extensive margins. Unlike the intensive Frisch elasticity, which is determined by σ , the 2 aggregateFrischelasticitydoesnotequalacalibrationparameterbutinsteadisimplicitlydeterminedwithin the model. Past micro-econometric studies estimate that the intensive Frisch elasticity is between 0 and 11Thispopulationincludesindividualswhoarecurrentlyretired. 12An alternative formulation that would induce agents to make decisions on the extensive margin is to include a non-linear mappingbetweenhoursandproductivity. Althoughbothmodellingoptionscreateanactiveextensivemargin,Ifoundthatafixed costwasmorestablewhencalibrating. 13Thisistheratiooffixedassetsandconsumerdurablegoods,lessgovernmentfixedassetstoGDP(Conesaetal.(2009)). 14Usingatargetofone-thirdisstandardinquantitativeexercises.Forexamples,seeConesaetal.(2009),Nakajima(2010),and Garriga(2001). 15EventhoughConesaetal.(2009)useadifferentutilityspecification,theirspecificationhasaparameterthatcorrespondstoσ1 . 9

0.5.16 However, more recent research has shown that these estimates may be biased downward. Reasons for this bias include: utilizing weak instruments; not accounting for borrowing constraints; disregarding the life cycle effect of endogenous-age specific human capital; omitting correlated variables such as wage uncertainty;andnotaccountingforlabormarketfrictions.17 Therefore,Isetσ suchthattheintensiveFrisch 2 elasticityisattheupperboundoftherange(0.5).18 Iusethevaluesfor{ε } jr−1 fromConesaetal.(2009) j j=0 whichareasmoothedversionoftherelativehourlyearningsestimatedbyageinHansen(1993).19 3.3 Firm IassumetheaggregateproductionfunctionisCobb–Douglas. Thecapitalshareparameter, α, issetat.36. Thedepreciationrateissettotargettheobservedinvestmentoutputratioof25.5percent. 3.4 GovernmentPoliciesandTaxFunctions In order to calibrate the second set of parameters that imply the targets in the models match the values in thedata,itisnecessarytoincludeabaselinetaxfunctioninthemodelthatmimicstheU.S.taxcode. Iuse Gouveia and Strauss (1994) estimates of the U.S. tax code, which I refer to as the baseline-fitted U.S. tax policy. TheauthorsmatchtheU.S.taxcodetothedatausingathreeparameterfunctionalform, T(y;ϒ 0 ,ϒ 1 ,ϒ 2 )=ϒ 0 (y−(y−ϒ1+ϒ 2 ) − ϒ 1 1), (5) where y represents labor or capital income. The average tax rate is principally controlled by ϒ , and ϒ 0 1 governs the progressivity of the tax policy. The third parameter, ϒ , is set so that taxes satisfy the budget 2 constraint. Gouveia and Strauss (1994) estimate that ϒ = .258 and ϒ = .768. The authors do not fit 0 1 separate tax functions for labor and capital income. Accordingly, I use the same parameter values on both sources of income for the baseline-fitted U.S. tax policy. I calibrate government consumption, G, so that it equals 17 percent of output under the baseline-fitted U.S. tax policy, as observed in the U.S. data.20 More specifically,ϒ isdeterminedasthevaluethatcausestaxestobeequalto17percentofGDP.Whensearching 2 fortheoptimaltaxpolicy,Irestrictmyattentiontoflattaxesoncapitalandlaborthatinducerevenueneutral 16ForexamplesseeAltonji(1986),MaCurdy(1981),andDomeijandFlode´n(2006). 17SomeofthesestudiesincludeImaiandKeane(2004),DomeijandFlode´n(2006),Pistaferri(2003),Chetty(2009),andContrerasandSinclair(2008). 18Variousstudieshavedemonstratedthataddinganendogenousdecisionabouttheextensivemarginmeansthattherepresentative agent’sFrischelasticitywillbelarger(forexamplesseeChetty(2009),Chettyetal.(2011),RogersonandWallenius(2009),and Erosaetal.(2011)). 19TheseriesinConesaetal.(2009)endsatthenormalretirementage. Iextendproductivityafterthenormalretirementage assumingasteadyrateofdeclineinhumancapitaloverthelastdecadeofworking. 20Todeterminetheappropriatevalueforcalibration,Ifocusongovernmentexpenditureslessdefenseconsumption. 10

changes.21 These policies imply that government consumption is equal under the baseline-fitted U.S. tax policyandtheoptimaltaxpolicy. Inadditiontoconsuming,thegovernmentrunsabalanced-budgetsocialsecurityprogram. Socialsecurity benefits are set so that the replacement rate, b, is 50 percent.22 In the endogenous model, an agent can choosetoretireatanyagewithinfouryearsofthenormalretirementage. Onceanagentretirestheyreceive theannualsocialsecuritybenefit. Ifanagentchoosestoretirepriortothenormalretirementagethenthey must pay a fixed amount when they retire which is actuarially equivalent to the expected benefits they will receive before the normal retirement age. If they choose to retire after the normal age of retirement then theyreceiveanactuariallyequivalenttransfer. Thepayrolltax,τ ,isdeterminedsothatthesocialsecurity ss systemisbalancedeachperiod. 4 Computational Experiment Thecomputationalexperimentisdesignedtodeterminethetaxpolicythatmaximizesagivensocialwelfare function. Ichooseasocialwelfarefunction(SWF)thatcorrespondstoaRawlsianveilofignorance(Rawls (1971)). Sincelivingagentsfacenoearningsuncertainty,thesocialwelfareisequivalenttomaximizingthe expectedlifetimeutilityofanewborn, J j−1 SWF(τ ,τ )= ∑βj−1[∏Ψ ]u(c ,h ), (6) h k q j j j=1 q=0 whereτ istheflattaxrateonlaborincomeandτ istheflattaxrateoncapitalincome.23 h k When I determine the optimal tax policy, I search over a grid of values of the tax on labor income (τ ) h and determine the corresponding values for the tax on capital (τ ) which implies a revenue neutral change. k Therefore,theexperimentistofindτ thatsatisfies h maxSWF(τ ,τ ) (7) h k τh 21IrestrictmyattentiontoflattaxesbecauseConesaetal.(2009)andPeterman(2011)showthatinmodelswhereagentsare homogenouswithinthecohort,flattaxesaretypicallyoptimal. 22ThereplacementratematchestherateinConesaetal.(2009)andConesaandKrueger(2006). TheSocialSecurityAdministrationestimatesthatthereplacementratioforthemedianindividualis40percent(seetableVI.F10inthe2006SocialSecurity TrusteesReport;availableatwww.ssa.gov/OACT/TR/TR06/). ThisestimateislowerthanthereplacementrateIuse;however,if onealsoincludesthebenefitspaidbyMedicare,thentheobservedreplacementratiowouldbehigher. 23FormoredetailsonwhyIrestrictmyattentiontoflattaxes,seeConesaetal.(2009)andPeterman(2011). 11

subjectto, J jr−1 G= ∑µ τ r(a +Tr)]+ ∑ µ τ [ws h (1−.5τ )]. (8) j k j j h j j ss j=1 j=1 5 Results Table2describestheoptimaltaxpoliciesinbothmodels. Theoptimaltaxpolicyintheexogenousmodelis an 18.2 percent tax on capital income (τ =18.2%) and a 23.7 percent tax on labor income (τ =23.7%). k h Including endogenously determined retirement has a large impact on the optimal tax policy. The optimal taxoncapitalincreasedbyalmostseventyfivepercentto31.8percent(τ =31.8%andτ =20.6%)when k h endogenous retirement is included. Using the endogenous model, I find that the welfare is reduced by an amount equivalent to one percent of total lifetime consumption if the planner adopts the tax system that is optimalfortheexogenousmodel. Table2: OptimalTaxPolicies TaxRate Exogenous Endogenous τ 18.2% 31.8% k τ 23.7% 20.6% h τk 0.77 1.54 τh The differences in the optimal tax policy are primarily driven by two key changes in the aggregate Frisch labor supply elasticity. Note that the aggregate Frisch elasticity is determined within the model and incorporates labor changes on both the intensive and extensive margin. In contrast, the intensive Frisch elasticity only incorporates changes in hours on the intensive margin and is equal to σ . First, introducing 2 a retirement decision in the model provides the agents with an additional margin on which to change their labor supply. The result is an overall increase in the level of the aggregate Frisch labor supply elasticity. In order to minimize the distortion that the tax policy induces, the government relies more heavily on a capital income tax as opposed to a labor income tax. Second, since the choice to retire is more relevant forindividualswhentheyareolder,endogenousretirementdisproportionatelyincreasestheelasticitywhen agents are older compared to when they are younger. This change in the slope of the aggregate Frisch elasticity profile causes the government to want to reduce the relative tax rate on the labor income from whenagentsareolder. Sincethegovernmentcannotconditionlaborincometaxesonage,theyincreasethe taxoncapitalinordertomimicthistypeofage-dependenttaxpolicy. 12

Next, I quantify the relative impact of the change in the level and the slope of the aggregate Frisch elasticity profile. I determine the effect of the increase in the aggregate Frisch elasticity on the optimal tax policy by solving for the optimal tax policy in the exogenous model using an alternative value for the intensiveFrischelasticitywhichequalstheaggregateFrischelasticityintheendogenousmodel. Iestimate thattheaggregateFrischelasticityintheendogenousmodelis2.18.24 ThislargeraggregateFrischelasticity in the endogenous model is in line with estimates of the aggregate Frisch elasticity in Peterman (2012). In the altered exogenous model, where the Frisch elasticity parameter matches the aggregate Frisch elasticity in the endogenous model, I find that the optimal tax on capital is 25.8%.25 This result indicates that, by itself,thelargeraggregateFrischelasticityisresponsibleforroughlyhalfoftheincreaseintheoptimaltax oncapitalincomeintheendogenousmodel. Table3: AggregateEconomicVariables Exogenous Endogenous %Change %Change fromBaseline fromBaseline Aggregate Baseline Optimal toOptimal Baseline Optimal toOptimal Y 0.81 0.82 1.2% 0.81 0.81 0.1% K 2.17 2.25 3.5% 2.17 2.12 -2.5% N 0.46 0.46 0% 0.46 0.47 1.5% AvgHours 0.33 0.34 0.9% 0.33 0.33 -0.1% w 1.12 1.13 1.3% 1.12 1.1 -1.5% r 0.05 0.047 -5.9% 0.05 0.054 7% tr 0.026 0.027 3.7% 0.026 0.026 -0.1% Value -160.2 -159.9 0.2% -160.2 -158.3 1.2% CEV 0.3% 1.8% AverageTaxRate Baseline Optimal Baseline Optimal Capital 15.6% 17.6% 15.6% 31.8% Labor 23.7% 24.1% 23.7% 20.6% Ratio 0.66 0.73 0.66 1.54 MarginalTaxRate Baseline Optimal Baseline Optimal Capital 19.4% 17.6% 19.4% 31.8% Labor 25.5% 24.1% 25.5% 20.6% Ratio 0.76 0.73 0.76 1.54 24IcalculatetheaggregateFrischelasticityintheendogenousmodelbyregressingthepercentchangeinhoursonwagesand marginalutility.Additionally,Irestricttheestimationsuchthattheconstantequalszero.Inordertoconfirmthatthisisanaccurate measurement,Iestimatetheregressiononagesinwhichtheagentdoesnotconsiderretirement.IfindthattheestimateoftheFrisch forthissmalleragerangeisveryclosetotheparametervaluewhichdeterminetheintensiveFrischelasticity(0.48versus0.5). 25Note,Irecalibratethemodelsuchthatitmatchestheothertargets. 13

Finally,Ifocusonthedifferencesintheunderlyingeconomieswithandwithoutendogenousretirement. Table3detailstheaggregateeconomicvariables;thefirst,secondandthirdcolumnsdescribethemodelwith exogenousretirementandthefourth,fifth,andsixthcolumnsdetailthemodelwithendogenousretirement. The first and fourth columns describe the aggregates under the baseline-fitted US tax policy. Since the twomodelshavethesamecalibrationparametersandtargetstheyareidenticalunderthebaseline-fittedUS tax policy. The second and fifth columns describe the exogenous and endogenous models when I impose each models optimal tax policies, respectively. Finally, the third and sixth columns describe the percent changeintheaggregatesthatoccurswhenIswitchfromthebaseline-fittedUStaxpolicytotheoptimaltax policies. Figure 1 plots the life cycle profiles for labor, consumption, and savings. The black lines are the profiles under the baseline-fitted US tax policy. As previously mentioned, by construction the endogenous andexogenousmodelsareidenticalunderthebaseline-fittedUStaxpolicy. Theredlinesaretheprofilesin theexogenousmodelundertheoptimaltaxpolicy. Thebluelinesaretheprofilesintheendogenousmodel undertheoptimaltaxpolicy. In the exogenous model adopting the optimal tax policy causes the average marginal tax rate on both capital and labor fall.26 In response to changing from the baseline to optimal tax policy, capital increases and labor stays relatively constant. The changes in aggregate labor and capital cause output to increase, the pre-tax wage rate to increase, and the pre-tax return to capital to fall. In order to assess the impact on welfareIcalculatetheconsumptionequivalentvariation(CEV)whichistheuniformpercentageincreasein consumption required to make an individual indifferent between the baseline and optimal two tax policies intheexogenousmodel. IfindtheCEVissmallintheexogenousmodel(0.3%). In the endogenous model adopting the optimal tax policy causes a large increase in the tax on capital income and a decrease in the tax on labor income. In response to changing from the baseline-fitted US tax policy to the optimal tax policy, aggregate capital falls two and a half percent and aggregate labor increasesoneandahalfpercent. Thechangesincapitalandlaboroffseteachotherandoveralloutputstays approximatelyconstant. Thedropinaggregatecapitalandriseinaggregatelaborleadstoanincreaseinthe pre-taxreturntocapitalanddecreaseinthepre-taxwage. Overall,Ifindthatthewelfaregainsfromadopting the optimal tax policy is more than six times larger in the endogenous model compared to the exogenous model(1.8%). Figure 1 plots the life cycle profiles for the exogenous model under the baseline-fitted US tax policy and the optimal tax policy. Adopting the optimal tax policy in the exogenous model causes a decrease in 26Notethatthebaseline-fittedUStaxpolicyisprogressivesoindividualsmakingdifferentlevelsofincomewillfacedifferent marginaltaxrates. 14

Figure1: LifeCycleProfiles Labor 0.4 0.35 0.3 0.25 0.2 20 40 60 80 100 sruoH Consumption 0.65 Baseline 0.6 Optimal Exogenous 0.55 Optimal Endogenous 0.5 0.45 0.4 0.35 0.3 0.25 20 40 60 80 100 Age noitpmusnoC Baseline Optimal Exogenous Optimal Endogenous Age Asset Holdings 5 4 3 2 1 0 20 40 60 80 100 stessA Baseline Optimal Exogenous Optimal Endogenous Age Note:Theseplotsarelifecycleprofilesinbothmodelsunderthebaseline-fittedU.S.taxpolicyandtheoptimaltaxpolicies.Since the calibration parameters and targets are the same the economies in both models are identical under the baseline-fitted US tax policy. 15

the tax on capital which decreases the implicit tax on younger labor income. The decrease translates into younger agents working more under the optimal tax policy. The changes in the consumption and savings profilesaregovernedbythechangeintheaftertaxreturntocapital. Adoptingtheoptimaltaxpolicyhastwo counteractingeffectsontheaftertaxreturntocapital. First,thepre-taxreturntocapitaldecreases. Second, thetaxoncapitalreturnsdecreaseswhichincreasestheafter-taxreturn. Thefirsteffectisconsistentforall individuals. Sincethebaseline-fittedUStaxpolicyisprogressiveandtheoptimaltaxpolicyisflatthesecond effectislargerforindividualswithhigherincome. Overall,thesecondeffectdominatesformostindividuals and their after-tax return to capital increases causing the consumption profile to be steeper. The change in the consumption profile is more pronounced for middle-aged individuals with higher incomes since they experiencethelargestdecreaseinthetaxoncapital. Additionally,thesehigherincomeindividualsrespond tothelargerdropinthetaxoncapitalbyholdingmoreassets. Sinceadoptingtheoptimaltaxpolicyintheendogenousmodelincreasesthetaxoncapitalthechanges in the life cycle profiles are different compared to the changes in the exogenous model. The larger tax on capital implicitly taxes young labor income at a higher rate. Therefore, in response to adopting the optimal tax policy in the endogenous model individuals shifts hours worked from earlier to later years. In the endogenous model adopting the optimal tax policy causes agents to change their labor supply not only on the intensive margin but also to choose to retire one year later. The higher tax on capital translates into a lower after-tax return tocapital which causes a flattening of the consumption profile. Prior to retirement, agents choose to hold less savings under the optimal tax policy since the tax on capital is larger. However, since they choose to postpone retirement until after the normal retirement age they receive a lump sum benefit equal to the missed social security payment. This lump sum transfer results in a higher level of post-retirementsavingsundertheoptimaltaxpolicy. 6 Conclusion InthispaperIcomputationallysolvefortheoptimalcapitalandlabortaxratesinseparatelifecyclemodels with exogenously and endogenously determined retirement. I find that including endogenous retirement causes a large increase in the optimal tax on capital. In the model with exogenous retirement the optimal taxpolicyincludesa24.1%taxonlaborincomeanda17.6%taxoncapital. Inthemodelwithendogenous retirementtheoptimaltaxpolicyincludesa20.6%taxonlaborincomeanda31.8%taxoncapitalincome. Relaxingthesimplifyingassumptionthatretirementisexogenouslydeterminedcausesaseventyfivepercent increaseintheoptimaltaxoncapital. Furthermore,Ifindthatthewelfarecostofadoptingtheloweroptimal 16

taxoncapitalfromtheexogenousmodelinthemodelwithendogenousretirement,whichcallsforahigher tax on capital, is equivalent to one percent of total lifetime consumption. These result indicates that this simplifying assumption of exogenously determined retirement has large consequences when solving for optimaltaxpolicy. Including endogenously determined retirement causes the optimal tax on capital to increase because it affectstheaggregateFrischelasticityintwoways. First,itincreasestheaggregateresponsivenessofhoursto changesinthereturntolabor. Therefore,thegovernmentwouldprefertorelymoreheavilyonacapitaltax as opposed to a labor tax in order to minimize the distortions induced by the tax policy. Second, including a retirement decision causes individuals to be relatively more responsive to changes in the after-tax wage when they are old. The government would like to condition labor income taxes on age to tax agents when they are older and more responsive at a relatively lower rate. Since age-dependent labor income taxes are unavailable,thegovernmentusesahighertaxoncapitaltomimicsuchataxpolicy. Economists are constantly trying to balance realism and tractability when they model the economy. In thispaperIdemonstratethatthesimplifyingassumptionofexogenouslydeterminedretirementhasasizable impact on optimal tax policy. Therefore, future work that examines optimal taxation in a life cycle model shouldincorporateendogenousretirement. Additionally,mostofthepreviousworkanalyzingoptimaltaxation assumes that the social security program is outside of the control of the government. Given the large impactsofendogenousretirementonoptimaltaxpolicyitseemsthattheremaybewelfaregainsfromoptimizingboththetaxpolicyandsocialsecurityprogramtogetherasopposedtooptimizingtheminisolation. Ileavethisexerciseforfutureresearch. 17

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