Essays in Optimal Taxation. Matthew C. Weinzierl

Transcription

1 Essays n Optmal Taxaton A thess presented by Matthew C. Wenzerl to The Department of Economcs n partal fulllment of the requrements for the degree of Doctor of Phlosophy n the subject of Economcs Harvard Unversty Cambrdge, Massachusetts Aprl, 2008

2 c 2008 by Matthew C. Wenzerl All rghts reserved.

3 Abstract Matthew C. Wenzerl Thess Advsors: Professor Aleh Tsyvnsk (char) Professor N. Gregory Mankw Professor Carolne M. Hoxby Essays n Optmal Taxaton Ths thess conssts of four essays that span a wde range of topcs n, and perspectves on, the domnant modern framework for optmal taxaton research. The rst analyzes a potental tax reform based on ths framework: age-dependent taxaton. Usng modern dynamc optmal tax methods, I provde a comprehensve theoretcal and quanttatve examnaton of age dependence and compare t to two alternatve polces: an age-ndependent polcy and a dynamc optmal polcy. Despte ts smplcty, age dependence yelds a large welfare gan equal to between one and three percent of aggregate annual consumpton, and t captures a substantal porton of the gan from reform to the dynamc optmal polcy. The second ncorporates a factor nto ths framework that s generally neglected: preference heterogenety. I avod techncal dfcultes that arse wth preference heterogenety n general by focusng on a specc but mportant class of

4 Abstract v preferences: those aganst whch ndvduals do not want to be nsured. Promnent crtcs of redstrbutve taxaton have long stressed the normatve mportance of ths class, and I examne ts effects on the optmal desgn of taxaton. In addton, I present cross-country evdence that s consstent wth a key predcton of the model, suggestng preference heterogenety's mportance n a postve analyss of taxaton. The thrd apples ths framework to a new ssue: the optmal polcy response to heterogenety n parents' altrusm toward ther chldren. I derve a normatve crteron for optmalty by modfyng the standard framework to apply to an economy wth parents and chldren. I argue that the socal welfare functon n ths settng fully offsets the effects of heterogenety n parental altrusm on chldren, contrary to the conventonal economc approach to optmal ntergeneratonal polcy. The fourth challenges ths framework by pontng out that ts theory, combned wth emprcal evdence, recommends a tax credt for short taxpayers and a tax surcharge for tall ones. Wth my coauthor, N. Gregory Mankw, I show that the optmal heght tax n the Unted States s substantal. We argue that f socety rejects ths polcy, we must revst the standard framework for optmal taxaton that recommended t.

5 Acknowledgments It has been my great fortune to study wth my three advsors. Carolne Hoxby has shown me how to combne deep economc nsght wth an emphass on accuracy that has made my research substantally better and that wll always nform my work. Greg Mankw, both through hs teachng and by example, has helped me to develop my ablty to thnk as a creatve and rgorous economst, and workng wth hm has been a great prvlege. Aleh Tsyvnsk, who ntroduced me to the subject of ths thess, has trelessly pushed me to develop my deas further, and he has been a true mentor. In addton, Alberto Alesna, Robert Barro, Davd Cutler, Emmanuel Farh, Martn Feldsten, Ncola Fuchs-Schündeln, Davd Labson, Erzo F.P. Luttmer, Andrew Oswald, Amartya Sen, and Jeff Wllamson provded especally valuable gudance along the way. Frends Yael Aufgang, Jesse Barnes, Flpe Campante, Vrgna Edwards, Alex Gelber, Noam Krson, Ilyana Kuzemko, Joe Mazor, Brent Neman, Kartn Shastry, Ncole Tateosan, Dan Tortorce, and Dan Wood helped make the journey fun as well as enrchng. My parents, Sondra and Charles Wenzerl, have supported me durng these last several years, as always, wth an understandng for whch I am truly grateful. Fnally, my heartfelt thanks to my wfe, Coventry Edwards-Ptt, for both the nsghts she brngs to my work and the joy she brngs to my lfe. v

6 Contents Contents v Preface The Surprsng Power of Age-Dependent Taxes Introducton Baselne economy Setup Socal planner's problem n three polcy scenaros Analytcal results Numercal results Summary of baselne model Case 2: Model wth prvate savng and borrowng Socal planner's problem n three polcy scenaros Analytcal results Numercal results Case 3: Model wth stochastc wage paths Socal planner's problem n three polcy scenaros Analytcal results Numercal results v

7 Contents v.5 Case Numercal results Dscusson and Specal Topcs Endogenety of wage paths Elastcty of labor supply by age Extensve margn Pareto-mprovng age dependence Concluson Incorporatng Preference Heterogenety nto Optmal Tax Models Introducton Modelng preference heterogenety The problem wth heterogeneous preferences "Pure preferences" Implcatons for labor ncome taxaton The cardnalzaton of pure preferences Optmal polcy mplcatons Suggestve smulaton results Internatonal data on preferences and polcy Preferences as an alternatve to belefs Implcatons for captal taxes and socal nsurance Dsablty nsurance and means-testng wth preference heterogenety Captal taxaton wth heterogenety n dscountng Concluson

8 Contents v 3 Optmal Polcy wth Heterogeneous Parental Altrusm Introducton The Orgnal Poston The orgnal poston thought experment wth ndvduals The Orgnal Poston wth famles The economy The orgnal poston thought experment wth famles The Equal Weghts Socal Welfare Functon Alternatve approaches Combnaton of Equal and Dluted Weghts Optmal Polcy Optmal nequalty across chldren Optmal ntergeneratonal allocatons Innte horzon model wth a one-tme preference shock The Equal Weghts SWF polcy The Unequal Weghts SWF polcy The Dluted Weghts SWF polcy The Combnaton SWF polcy Numercal llustraton and dscusson Concluson The Optmal Taxaton of Heght: A Case Study of Utltaran Income Redstrbuton (Jont wth N. Gregory Mankw) Introducton

9 Contents x 4.2 The model A general framework Analytcal results for a smple example Calculatons based on the emprcal dstrbuton The data What explans the heght premum? Baselne results Senstvty to parameters The taxaton of heght n other approaches to optmal taxaton Perspectves from poltcal phlosophy Poltcal economy constrants Costs mssng from the conventonal model Crtques of the basc framework Concluson A Appendx to Chapter A. Implementaton of the optmal allocatons A.2 Multple generatons A.3 Hamltonan approach to baselne model A.4 Proofs of Propostons A.5 Specc expressons omtted from text A.6 Numercal llustratons omtted from the man text B Appendx to Chapter B. General utlty functon analyss

10 Contents x B.2 Intertemporal applcatons C Appendx to Chapter C. Applcaton of approach to ablty heterogenety C.2 More general utlty functons C.3 Proofs References

11 Preface Taxaton s socety's fundamental economc polcy tool, and ts analyss has always been a prmary focus of economsts, socologsts, and phlosophers. Recent economc research on taxaton centers on a model economy, ntroduced n Mrrlees (97), that can be used to derve tax polcy that s "optmal" n a normatvely appealng sense. The research ncluded below works wth ths framework to suggest polcy reforms, extends ths framework to address factors neglected n the lterature, and rases fundamental questons about ths framework's use as the bass for optmal tax analyss. Ths set of analyses may seem contradctory: how can one both use a framework to generate polcy recommendatons and challenge the use of that framework as a tool for polcy analyss? The perspectve emboded by ths dssertaton s that the Mrrleesan framework provdes an mportant tool for both normatve and postve analyss, rgorously capturng a key aspect of the tradeoff between efcency and equalty that has long played a central role n debates over taxaton. Moreover, the framework's exblty suggests that contnued research based on t wll reveal addtonal nsghts wth meanngful effects on polcy. At the same tme, the framework s subject to mportant crtques that, f left unaddressed, wll lmt ts nuence on both researchers and practtoners. To realze the potental of our current approach to optmal taxaton and to develop addtonal, complementary approaches, we must engage the full range of factors that matter for the formaton and effects of tax polcy. Ths dssertaton s a small contrbuton to that challengng but mportant effort.

12 Chapter The Surprsng Power of Age-Dependent Taxes. Introducton Indvduals' wages change over ther lfecycle. Startng wth Golosov, Kocherlakota, and Tsyvnsk (2003), ths fact has nspred a recent surge of research on optmal taxaton n dynamc economes that bulds on the classc, statc Mrrlees (97) framework. Unfyng the mportant nsghts of ths research s the prncple that optmal taxaton n dynamc economes depends, other than n specal cases, 2 on a taxpayer's hstory. Whle ths prncple ponts the way toward potentally powerful reforms, ts near-term mpact may be lmted by the complexty of the reforms to exstng polces that t mples. Ths paper studes a smple reform to tax polcy that responds to changes n wages over the lfecycle: age-dependent labor ncome taxes. Though not often recognzed as such, age dependence s a component of Mrrleesan dynamc optmal taxaton and thus can be I am grateful to Stefana Albanes, Robert Barro, Mchael Boskn, V.V. Char, Davd Cutler, Emmanuel Farh, Martn Feldsten, Carolne Hoxby, Oleg Itskhok, Larry Jones, Emr Kamenca, Larry Katz, Patrck Kehoe, Narayana Kocherlakota, Erzo F.P. Luttmer, Greg Mankw, Ellen McGrattan, Thomas Mertens, Brent Neman, Chrs Phelan, Jm Poterba, Robert Shller, Larry Summers, Aleh Tsyvnsk, and Ivan Wernng for ther comments and suggestons. 2 Saez (200) s the leadng recent study of the classc Mrrlees framework. In the dynamc Mrrlees framework, Albanes and Sleet (2006) shows that hstory dependence can be replaced wth dependence on current wealth f shocks to sklls are..d.. Golosov and Tsyvnsk (2006) shows that asset-testng can replace hstory dependence f shocks are permanent:.e., an absorbng state. For general shock processes, Kocherlakota (2005) shows that hstory dependence s requred. 2

13 . Introducton 3 analyzed as a partal reform on the path toward optmal polcy. Takng that approach, ths paper makes three contrbutons. Frst, ts status as a partal reform allows me to provde a comprehensve theoretcal characterzaton of age dependence usng the tools of modern dynamc Mrrleesan optmal tax research. I derve theoretcal results for age-dependent polcy that connect to and extend results and ntuton from two alternatve Mrrleesan polces, an age-ndependent optmal polcy and a dynamc optmal polcy. In a baselne model wth determnstc wage paths and no prvate savng or borrowng, I show that age-dependent polcy avods dscouragng the labor supply of the top earner at each age whle age-ndependent polcy cannot, extendng a classc result from statc Mrrlees research. Then, I analytcally characterze the ntertemporal consumpton margn, where age-dependent polcy satses a condton that mproves on age-ndependent polcy but falls short of the full optmum. I characterze how age dependence affects these margns n economc envronments wth stochastc wages and prvate savng and borrowng, as well. Second, I provde a detaled quanttatve study of age dependence usng ndvdual wage data from the U.S. Panel Study of Income Dynamcs to calbrate and smulate polcy. Ths s n contrast to most of the dynamc optmal taxaton lterature, n whch llustratve numercal smulatons are the rule, and t resembles the realstc calbratons by Saez (200) of the statc Mrrlees model and Golosov and Tsyvnsk (2006) of optmal dsablty nsurance. The numercal smulatons generate specc mplcatons for polcy desgn. Two results largely robust across envronments are that age dependence: () lowers margnal taxes on average and especally on hgh-ncome young workers, and (2) lowers average tax

14 . Introducton 4 rates for young workers relatve to older workers when prvate savng and borrowng are restrcted. These results capture key features of the optmal dynamc polcy. Fnally, the numercal smulatons allow me to quantfy and decompose the welfare gan from ths partal reform. Age dependence yelds a large welfare gan equvalent to between and 3 percent of aggregate annual consumpton, and ths gan represents a substantal share of the potental gan from full reform to the dynamc optmal polcy. A detaled decomposton reveals three man components of the gan. Efcency and equty gans each account for a bt less than half of the total welfare gan, whle ntertemporal consumpton smoothng accounts for approxmately ten percent n the baselne model. Age dependence allows tax polcy to be talored to changes n the dstrbuton of wages at each age and to transfer resources between age groups, avodng nefcent dstortons to labor effort and enablng more redstrbuton. 3 Age dependence s a compellng example of the potental value of partal reforms. The noton of partal reform was formalzed by Guesnere (977) n response to concerns rased by Feldsten (976) about the practcalty of statc optmal tax desgn. Partal reforms are most valuable when they yeld substantal welfare gans whle rasng fewer realworld concerns than fully optmal polcy. A partcularly strong case can be made for partal reforms that yeld Pareto mprovements, a characterstc stressed by Guesnere. Thus, to further motvate the practcalty of ths reform, I consder a model n whch I add the constrant that age dependence must be ex post Pareto mprovng. Ths constraned reform 3 A related polcy s lfetme ncome taxaton, where an ndvdual pays taxes or receves refunds n each year based on ther (currently) expected lfetme ncome. The age-dependent polcy studed here subsumes that polcy and adds the mportant ablty to talor margnal taxes by age. Moreover, lfetme ncome taxaton also uses age as an argument n the tax functon, so t provdes lttle or no advantage n smplcty relatve to fully age-dependent taxaton.

15 . Introducton 5 generates nearly as large a welfare gan as does the Utltaran-optmal polcy, underscorng the potental power of age dependence. Though ths paper s the rst to gve the partal reform of age dependence a comprehensve treatment n a modern dynamc Mrrleesan tax model, 4 the dea of havng taxes depend on age has been around for some tme and was even mentoned n passng by Mrrlees (97). Despte ths long hstory, age dependence was rst rgorously analyzed n the mportant work of Kremer (2002), who demonstrated that margnal ncome taxes not condtoned on age are unlkely to be optmal and suggested that they should be lower for young workers, consstent wth the results below. Kremer's analyss dd not provde a full characterzaton or smulaton of age dependent taxes and was lmted to a statc settng, lmtatons hs paper acknowledges. Followng on that work, Blomqust and Mcheletto (2003) and Judd and Su (2006) provde results on age-dependent taxaton n llustratve dynamc economes usng theoretcal and numercal approaches, respectvely. The need for a more comprehensve analyss s underscored by the suggeston n the upcomng Mrrlees Revew that age dependence s one of the most promsng areas for the near-term reform of developed-country tax polcy (see Banks, Damond, and Mrrlees, 2007). 5 The lterature on age dependence may be relatvely sparse because t has been assumed that age s merely another "tag", followng Akerlof (978), and that standard taggng 4 Age dependence has been studed n the Ramsey tax framework. For example, Erosa and Gervas (2002) study the effect of prvate asset accumulaton and changes n the elastcty of labor supply wth age on optmal lnear taxes. As a representatve agent framework, the conventonal Ramsey approach neglects the redstrbutonal role of taxaton. I have smulated a Ramsey verson of age-dependent taxaton (.e., age-specc lnear taxes and lump-sum taxes or grants) n the heterogeneous-agent model economy dscussed below, and the welfare gans from age-dependence are approxmately one-tenth ther sze n the Mrrlees approach. The Ramsey approach cannot talor margnal taxes to varaton n the dstrbuton of wages wth age. 5 The Mrrlees Revew s the modern counterpart to the Meade Report of 978, the nuental revew of taxaton.

16 . Introducton 6 analyss teaches us all we need to know about age dependence. That assumpton s mstaken. A standard tag, such as gender, provdes nformaton on an ndvdual's expected place n the dstrbuton of lfetme ncome, but age reveals no nformaton by tself. The dfference s that a tag dvdes the populaton nto mutually exclusve groups, whle the entre populaton moves through all age groups. In other words, each age group has the same dstrbuton of lfetme ncomes. To provde useful nformaton to the tax authorty, age must be combned wth data on an ndvdual's current ncome and how ncomes at each age relate to lfetme ncome. Therefore, age dependence cannot be fully understood wth ntuton from conventonal taggng analyss. Consstent wth the small academc lterature on age dependence, current polcy n developed countres ncludes only uncoordnated and often unntentonal age dependence. 6 In the Unted States, for example, the man dependences on age work at cross purposes. Socal Securty and Medcare payroll taxes are leved on all ndvduals but, gven the pay-as-you-go structure of these programs, these taxes place a larger effectve burden on the young than on the old (see Feldsten and Samwck, 992). On the other hand, the exstng dsablty system effectvely places a larger effectve burden on the old than on the young, snce the latter would receve a longer strng of benets f dsabled. Some transfer programs, such as the chld tax credt or Earned-Income Tax Credt, are more lkely to benet the young than the old, but they also mean hgher margnal rates on the young who earn enough to be n the "phase-out" regon of these benets. Fnally, the deductblty of mortgage nterest and chartable donatons are more lkely to benet the mddle-aged, for 6 One excepton s Sngapore's Provdent Fund, a natonal retrement savngs program that talors personal contrbuton rates to age.

17 . Introducton 7 whom rentng s less common and ncomes are hgher, than the young or old. Together, these largely unntentonal age dependences are unlkely to mmc, and may often work n the oppostve drecton of, the optmal age-dependent polcy as studed n ths paper. Therefore, the analyss of ths paper contrbutes to several lteratures. Most drectly, t provdes a new, emprcally-drven applcaton of the tools of the dynamc Mrrleesan optmal tax lterature surveyed n Golosov, Tsyvnsk, and Wernng (2006). It also extends the lterature on partal reform, concdng wth recent work of Farh and Wernng (2007) on a dfferent partal reform based on dynamc optmal polcy. The paper's consderaton of both determnstc and stochastc wages relates to the mportant ongong debate over the determnants of lfecycle wage nequalty and the approprate polcy responses to t: e.g., Storesletten, Telmer, and Yaron (200) and Keane and Wolpn (997). Fnally, t adds to a recent lterature that emprcally estmates the value of condtonng taxes on personal characterstcs, such as Alesna and Ichno (2007). The paper proceeds as follows. Secton.2 descrbes the socal planner's problem n three polcy scenaros for a baselne economy,.e., wth determnstc wage paths and no prvate savng or borrowng. For each polcy, I analytcally characterze the ntratemporal and ntertemporal dstortons on prvate behavor and numercally smulate the structure of taxes. I also quantfy the welfare mplcatons of reform from the statc optmal polcy to age-dependent polcy and the dynamc optmal polcy. Secton.2 serves as the reference pont for the followng three sectons, n whch I vary the assumptons about the economy to test the robustness of the baselne results. Secton.3 allows ndvduals to save and borrow, Secton.4 ncorporates stochastc wage paths, and Secton.5 combnes these

18 .2 Baselne economy 8 two varatons. largely robust. Throughout these extensons, the results from the baselne economy are Secton.6 dscusses a range of specc topcs that fall outsde the man analyss of the paper, and Secton.7 concludes. Appendx A, whch gves further detals related to ths paper, can be found at the end of ths dssertaton..2 Baselne economy In ths secton I analyze age-dependent labor ncome taxes for a baselne economy characterzed by two smplfyng assumptons. Frst, ndvduals cannot transfer resources across perods: that s, they can nether save nor borrow. Second, each ndvdual's lfetme wage path s determnstc, so that each ndvdual knows n advance the exact path of wages t wll have over ts lfetme (.e., there are no stochastc shocks to wages). These smplfyng assumptons allow for cleaner analytcal results, but I generalze the model to relax them n later sectons of the paper. One assumpton made throughout ths paper, and throughout the dynamc optmal tax lterature n general, s that wage paths are exogenous to ndvduals; I dscuss the potental mplcatons of ths assumpton n Secton.6. I start by settng up the economy and then specfyng the socal planner's problem n three polcy scenaros.

19 .2 Baselne economy 9.2. Setup All ndvduals lve and work for T perods, 7 ndexed by t = f; 2; :::; T g, and are members of the same generaton. 8 Indvduals are heterogeneous n ther ablty to earn ncome over ther lfetmes. Ths ablty comes n I types, ndexed by = f; 2; :::; Ig ; wth probabltes so that P I = =. At each age t, an ndvdual of type can earn a wage w t for each unt of ts labor effort, and each ndvdual knows ts full lfetme path of wages fwtg T t= at tme t =. Wages are not publcly observable. I refer to the present value of wages for type as type 's lfetme ncome-earnng ablty, dened as P T wt t= where R R t s the exogenous gross rate of return. Types are sorted so that = I refers to the type wth the hghest lfetme ncome-earnng ablty. Producton s structured as follows. Labor ncome y s the product of the wage and labor effort l, so y = wl. There s no captal n the economy. All ndvduals have the same separable preferences over consumpton c and labor effort l, where l = y w. The utlty U t for ndvdual of age t s U t (c; y) = u (c) y v ; (.) wt where I assume u 0 () > 0; u 00 () < 0, v 0 () > 0; v 00 () > 0. For an ndvdual ; lfetme utlty V s the dscounted sum of ts utlty at each age: V = TX t Ut (c; y) ; (.2) t= 7 I assume an exogenous date of entry nto the labor market. Ths matches the quanttatve analyses below, whch consder age-dependent taxes desgned to have mnmal effects on ncentves to obtan hgher educaton or tranng. Age dependence could generate even larger welfare gans f t were desgned to operate effectvely on these ncentves. 8 In the Appendx, I dscuss how the analyss generalzes largely unchanged to a settng wth overlappng generatons.

20 .2 Baselne economy 0 where ndvduals dscount utlty ows wth the factor. Socal welfare W s a weghted sum of ndvdual lfetme utltes: IX W = V ; (.3) = where ndcates a scalar Pareto weght on ndvdual. In general, the form of (.3) allows us to consder any pont along the Pareto fronter. I assume a form for Pareto weghts (speced n the numercal smulatons) n whch an ndvdual's weght s postve, bounded above by one, and not ncreasng n ts lfetme ncome-earnng ablty..2.2 Socal planner's problem n three polcy scenaros Now I derve optmal taxes n three polcy scenaros usng the technques of modern dynamc optmal tax analyss. In ths approach, the tax problem s recast as a problem for a cttous socal planner that uses a drect mechansm to allocate resources (see Golosov, Tsyvnsk, and Wernng, 2006 for a revew). 9 The socal planner maxmzes socal welfare (.3) by offerng a menu of ncome and consumpton pars to ndvduals. Indvduals choose optmally from the menu, earn the assgned ncome, and receve the assgned consumpton. Knowng ths, the planner desgns ts menu of fc; yg pars ntendng each par to be chosen by a specc ndvdual. Because ndvduals dffer n ther lfetme ncome-earnng ablty and age, I wrte c t and y t for the par ntended for the ndvdual of type and age t. 0 9 I show the connecton between these allocatons and taxes n the Appendx: an approprate nonlnear labor ncome tax system mplements each polcy scenaro's allocatons. In the Statc Mrrlees polcy, the labor ncome tax s a drect functon of ncome only; n the Partal Reform t s a drect functon of ncome and the taxpayer's current age; whle n the Full Optmum t s a drect functon of the lfetme path of ncomes and the taxpayer's current age. 0 In the language of the drect mechansm, each ndvdual makes a report to the planner about ts personal

21 .2 Baselne economy The planner maxmzes socal welfare subject to two types of constrants: a feasblty constrant and ncentve constrants. The feasblty constrant s: IX = TX R T t yt ct = 0: (.4) t= whch says that the lfetme paths of ncome must fund the lfetme paths of consumpton across all types. The planner can transfer resources across tme and earn or pay the gross rate R. I assume R = for smplcty. Incentve constrants reect that ndvduals choose from the planner's menu of fc; yg pars to maxmze ther utlty. In ths approach, these constrants take the form of nequaltes ensurng that each ndvdual chooses the allocaton of c t and yt that the planner ntended. 2 Importantly, varatons n the set of ncentve constrants allow us to succnctly dstngush the planner's problem n three polcy scenaros: Statc Mrrlees, Partal Reform, and Full Optmum. I now state these planner's problems formally and dscuss the dfferences between them. The Statc Mrrlees planner n the baselne model solves the followng problem: Problem : (Statc Mrrlees: Age-Independent) max fc;yg IX V ; = characterstcs. The planner then assgns to each report a consumpton and ncome par, fc; yg. Note that taxaton s purely redstrbutve. A postve net revenue requrement would mply larger tax dstortons on average, lkely ncreasng the welfare gan from age dependence calculated below. 2 These ncentve constrants reect ths approach's applcaton of the Revelaton Prncple, by whch we can restrct attenton to ncentve-compatble drect mechansms,.e., where ndvduals reveal ther true types to the planner.

22 .2 Baselne economy 2 subject to the feasblty constrant (.4) and the ncentve constrants u t c t y v t t u c j wt s y j v s wt (.5) for all ; j 2 f; 2; :::; Ig and t; s 2 f; 2; :::; T g. These ncentve constrants mean that the Statc Mrrlees planner must guarantee that each ndvdual of type and age t chooses the allocaton ntended for t over that ntended for any other ndvdual of type j and any age s. To see ths, note that each sde of the nequalty (.5) equals perod utlty for an ndvdual of type and age t. The left-hand sde s the utlty ths ndvdual obtans by earnng y t and consumng c t, whle the rghthand sde s the utlty t obtans by earnng y j s and consumng c j s. The nequalty guarantees that ths ndvdual weakly prefers ts fc; yg allocaton. I denote the multplers on these n o ;j constrants wth, where jj corresponds to the constrant preventng ndvdual jj sjt s;t sjt of age t from preferrng the allocaton ntended for ndvdual j of age s. The ncentve constrants n (.5) capture the restrcton on the Statc Mrrlees polcy that each ndvdual can choose among the same menu of ncome and consumpton pars, regardless of age. Ths s dfferent from the requrement that two ndvduals wth the same wage but dfferent ages receve the same allocatons of c and y. The latter s a stronger condton and restrcts the Statc Mrrlees planner more than s justed. Age may affect an ndvdual's optmal choce of ncome and consumpton, even f ts wage s unchanged, and the Statc Mrrlees planner can take advantage of ths wthout makng taxes age-dependent. 3 3 In the language of the drect mechansm, ndvduals n the Statc Mrrlees scenaro report only ther current wage, not ther age. But, the planner can assgn up to T dfferent fc; yg pars to each reported wage, knowng that ndvduals of dfferent ages may choose dfferent allocatons. The stronger alternatve would

23 .2 Baselne economy 3 The Partal Reform planner n the baselne model solves the followng problem: Problem 2: (Partal Reform: Age-Dependent) max fc;yg IX V = subject to the feasblty constrant (.4) and the ncentve constrants!! y u t c t v t t u c j y j t wt t v wt (.6) for all ; j 2 f; 2; :::; Ig and t 2 f; 2; :::; T g Because of the Partal Reform planner's ablty to restrct ndvduals to age-specc allocatons, these ncentve constrants say that the planner must guarantee only that each ndvdual of age t chooses the allocaton ntended for t over that ntended for any other ndvdual j of the same age t. To see ths, notce that the rght-hand sde of (.6) depends on c j t and y j t, so that both sdes of the nequalty are specc to age t (compare ths to the Statc Mrrlees planner, where the rght-hand sde depended on c j s and y j s). Formally, the set of constrants (.6) s a subset of (.5). Ths makes the set of ncentve constrants n the Partal Reform planner's problem weakly easer to satsfy than the set n the Statc Mrrlees n o ;j planner's problem. I denote the multplers on these constrants wth, where jj t corresponds to the constrant preventng ndvdual of age t from preferrng the allocaton ntended for ndvdual j of age t. In practcal terms, age dependence means that taxes can be talored to the wage dstrbuton at each age and that transfers can be made between age groups. As we wll see n jj t t further constran the Statc Mrrlees problem, magnfyng the results of the paper.

24 .2 Baselne economy 4 the numercal results below, these turn out to be valuable tools because the dstrbuton of wages vares wth age. Fnally, the Full Optmum planner n the baselne model solves the followng problem: Problem 3: (Full Optmum: Age-Dependent and Hstory-Dependent) max fc;yg IX V () subject to the feasblty constrant (.4) and the ncentve constrants TX t= u t c t for all ; j 2 f; 2; :::; Ig. = y v t wt TX t= t u c j t v y j t wt!! : (.7) These ncentve constrants reect the Full Optmum planner's ablty to make, and commt to, hstory-dependent allocatons. 4 Hstory dependence allows the planner to hold an ndvdual to the lfetme path of allocatons ntended for a sngle type at all ages. Thus, the Full Optmum planner must guarantee only that each ndvdual chooses the lfetme path of allocatons ntended for t over that ntended for any other type j. Ths s apparent from (.7) n that each sde of the nequalty s a dscounted sum of perod utltes over ndvdual 's lfetme. The left-hand sde s 's lfetme utlty f t chooses ts ntended allocatons (c t; y t) at each age t, whle the rght-hand sde s 's lfetme utlty from clamng the allocatons ntended for type j at each age. I denote the multplers on these 4 The assumpton that the planner can commt to a path of allocatons s standard n the dynamc optmal tax lterature. Bsn and Rampn (2005) study the mpacts of relaxng that assumpton.

25 .2 Baselne economy 5 constrants wth jj ;j, where jj corresponds to the constrant preventng ndvdual from preferrng the lfetme allocaton ntended for ndvdual j. Usng hstory dependence to satsfy ncentves on a lfetme bass can be a powerful tool for the planner. For example, suppose the planner wants to gve ndvdual a generous allocaton later n lfe n exchange for a "bad" allocaton early n lfe. In the Full Optmum, the planner can offer that path of allocatons to the ndvdual because t can make later allocatons dependent on earler ones. In the Statc Mrrlees or Partal Reform scenaros, such a path s not sustanable. In those scenaros, ndvduals know that the planner cannot reward early sacrce because t cannot use hstory dependence, so they wll not accept the bad allocaton early n lfe. These three sets of ncentve constrants, and ther correspondng polcy scenaros, le along a contnuum of sophstcaton, Least sophstcated Most sophstcated Statc Mrrlees Partal Reform Full Optmum where the precse meanng of "sophstcaton" depends on the characterstcs of the economy beng consdered. In ths baselne model, where there s no prvate savng or borrowng, a polcy's sophstcaton depends on two features: age dependence and hstory dependence. 5 The Statc Mrrlees polcy s nether age-dependent nor hstory-dependent. At the other extreme, n the tradton of recent work on optmal dynamc taxaton, the Full Optmum polcy s both age-dependent and hstory-dependent. 6 Partal Reform polcy strkes 5 Ths nformal dscusson of sophstcaton does not apply to the models of the economy (dscussed later) that nclude prvate savng and borrowng. Rather than specfyng a varant on sophstcaton for each model economy, we rely on the formal ncentve constrants to clarfy the dstnctons. 6 Hstory dependence cannot be avoded n ths baselne economy by usng wealth to encode past wages,

26 .2 Baselne economy 6 a mddle ground between the Statc Mrrlees and Full Optmum, beng age-dependent but not hstory-dependent. We can see ths relatonshp between the three polces formally. The transton between the Statc Mrrlees ncentve constrants (.5) and the Full Optmum ncentve constrants (.7) can be decomposed nto two steps. Frst, the planner replaces all the s subscrpts n (.5) wth t subscrpts, lmtng ndvduals to clamng only those allocatons ntended for ndvduals of the same age. Ths adds age dependence. Second, the planner adds the dscountng and summatons over t that appear n (.7), usng ts ablty to commt to lfetme allocatons. Ths adds hstory dependence. The Partal Reform polcy takes only the rst of these two steps. Before analyzng these models n detal, I note that the Statc Mrrlees polcy s not desgned to match the detaled structure of exstng tax polcy on labor ncome. Rather, t s the optmal polcy constraned by two characterstcs of exstng tax polcy: age ndependence and hstory ndependence. Our comparson of the Partal Reform to ths Statc Mrrlees polcy rather than to exstng tax polces allows us to solate the potental for age dependence to generate welfare gans by tself, relatve to an age-neutral benchmark Analytcal results Now, I compare the allocatons chosen by the socal planner n the Statc Mrrlees, Partal Reform, and Full Optmum polces along two margns: the ntratemporal margn between such as n Albanes and Sleet (2006), because ndvduals cannot accumulate wealth. 7 In addton to the smulatons of optmal polcy presented below, I have smulated a polcy approxmatng the current U.S. tax system. Usng the data and parameterzaton descrbed n the paper, I apply the 999 U.S. ncome tax schedule to labor earnngs and smulate ndvduals' behavor. The welfare gan from that polcy to the Statc Mrrlees polcy s equvalent to a 6.5 percent ncrease n aggregate annual consumpton.

27 .2 Baselne economy 7 consumpton and lesure and the ntertemporal margn between consumpton n one perod and the next. In partcular, I derve theoretcal results on these margns that connect to and extend classc results and ntuton from statc and dynamc Mrrleesan tax analyss. The mportance assgned to these margns s due to ther relaton, conceptually and formally, to margnal taxes on labor ncome and taxes on captal ncome. I evaluate average taxes, a key dmenson along whch the three polcy scenaros dffer, n the secton usng numercal smulatons. The Mrrlees model, and ths extenson of t, are not well-suted to dscussng average taxes analytcally. Intratemporal dstortons Frst, I compare the dstortons to ndvduals' choces of how much ncome to earn:.e., dstortons on ther margnal choces between consumpton and lesure. I begn wth a denton: Denton The ntratemporal dstorton for an ndvdual of type and age t s denoted (; t) and equals (; t) = v 0 y t wt wtu 0 (c t) : (.8) Denote SM (; t), P R (; t), and F O (; t) as the ntratemporal dstortons for an ndvdual of type and age t n the solutons to the Statc Mrrlees (SM), Partal Reform (PR), and Full Optmum (FO) planner's problems. In expresson (.8), postve (; t) dstorts the ndvdual's choce away from work (and consumpton) and toward lesure. If (; t) = 0, the ndvdual sets the margnal

28 .2 Baselne economy 8 utlty from an extra unt of consumpton equal to the margnal dsutlty of earnng t, so there s no dstorton on ths margn. Wth these prelmnares, we can now turn to the results. The followng proposton serves as a benchmark: Proposton (Intratemporal Benchmark) Let the utlty functon for all 2 f; 2; :::; Ig and t 2 f; 2; :::; T g be dened by (.), and let R =. If w s = w t for all 2 f; 2; :::; Ig and s; t 2 f; 2; :::; T g, then. SM (; s) = SM (; t), P R (; s) = P R (; t), and F O (; s) = F O (; t) for all 2 f; 2; :::; Ig and s; t 2 f; 2; :::; T g. 2. SM (; t) = P R (; t) = F O (; t) for each 2 f; 2; :::; Ig and t 2 f; 2; :::; T g. Proof. In Appendx. Ths proposton s about an economy n whch each ndvdual has a constant wage throughout ts lfetme: w s = w t for all 2 f; 2; :::; Ig and s; t 2 f; 2; :::; T g. Ths means that the dstrbuton of wages n the populaton s the same at every age. In such an economy, each ndvdual faces the same dstorton at all ages n each polcy. 8 For example, n the Statc Mrrlees polcy, SM (; s) = SM (; t) for all 2 f; 2; :::; Ig and s; t 2 f; 2; :::; T g. In addton, each ndvdual faces the same dstorton at a gven age n all three polcy scenaros, so SM (; t) = P R (; t) = F O (; t) for each 2 f; 2; :::; Ig and t 2 f; 2; :::; T g. 8 Wernng (2007a) proves a smlar result for the optmal dynamc polcy.

29 .2 Baselne economy 9 Intutvely, an unchangng wage dstrbuton means that each age s a replca of the others, so that the planner's best soluton for one age wll be the best soluton for all ages. Ths means constant ntratemporal dstortons for each ndvdual over tme. It also means that the three planners are solvng equvalent problems, so they choose the same pattern of ntratemporal dstortons. As ths proposton mples, any dfferences between the three polcy scenaros' ntratemporal dstortons rely on ndvduals' lfecycle wage paths not beng constant. Before characterzng these dstortons n general, I focus on a specc dstorton: that at the top of the ncome dstrbuton. The top margnal dstorton In ths subsecton, I focus on a classc result from statc optmal tax analyss: the top earner n the economy should face no ntratemporal dstorton. 9 The ntuton for the classc result s that an ntratemporal dstorton has both a cost and a benet. The cost s that t causes ndvduals to work and consume dfferently than they would wthout taxes, leadng to ether lower utlty or less efcently-provded utlty for these ndvduals. The benet s that t enables the planner to collect more tax revenue from hgher earners, ncreasng the extent of redstrbuton. 20 At the top of ncome dstrbuton, ths benet 9 Damond (998) and Saez (2002) show that ths result depends on the shape of the wage dstrbuton. Wth a bounded wage dstrbuton such as that used throughout the paper, the zero top rate result always holds. In the Appendx, I show that the results presented here have analogues n a model wth a wage dstrbuton more smlar to that whch they use. 20 It does ths by takng advantage of dfferences n the margnal dsutlty of ncome across ndvduals wth dfferent wages. A dstorton on a gven ndvdual harms less than t harms a hgher-sklled ndvdual j clamng 's allocatons. Whle suffers lower consumpton due to the dstorton, t also enjoys workng less. For j, the loss n consumpton exacerbates the man dsadvantage to clamng 's allocaton, whle the decrease n requred ncome s of less value because j s already workng much less than t would to earn ts own allocaton. Thus, the planner can assgn a larger lumpsum transfer to whle dscouragng j from

30 .2 Baselne economy 20 s zero (there are no hgher earners from whom to collect more tax revenue). Thus, a dstorton on the top earner solely dscourages effort, and t s avoded n the optmal polcy. The followng proposton descrbes how ths classc result apples to a dynamc economy under ths paper's three polcy scenaros: Proposton 2 (Top Margnal Dstorton) In the baselne economy,. f w t w j t for all j 2 f; 2; :::; Ig ; then P R (; t) 0 and F O (; t) 0, 2. f w t w j t for all j 2 f; 2; :::; Ig and for all t 2 f; 2; :::; T g, and f a a j for all j 2 f; 2; :::; Ig, then P R (; t) = 0 and F O (; t) = 0 for all t 2 f; 2; :::; T g, and SM (; t) = 0 for t such that w t w s for all s 2 f; 2; :::; T g. Proof. In Appendx. The rst part of ths proposton states that the hghest wage earner at each age faces a nonpostve ntratemporal dstorton whenever the socal planner can condton taxes on age: that s, n the Partal Reform and Full Optmum polcy scenaros. In partcular, the dstorton on the top earner at each age could be negatve n these scenaros. The second part of ths proposton states that an ndvdual who s the hghest wage earner at all ages faces no ntratemporal dstorton at any age n the Partal Reform and Full Optmum scenaros, but only at ts peak-earnngs age n the Statc Mrrlees scenaro. The rst part of ths proposton s a surprsng theoretcal devaton from the classc statc optmal tax result because t allows for the possblty that, wth age dependence, clamng 's allocaton.

31 .2 Baselne economy 2 the top earner at a gven age may face a negatve dstorton. 2 Intutvely, suppose an ndvdual has the hghest wage wthn ts current age but also has a low lfetme ncomeearnng potental. An example mght be an entertaner or athlete whose earnngs power s temporarly hgh at a young age. The planner wants to assgn ths ndvdual both hgh ncome and hgh consumpton: the former because t s a productve worker, and the latter because ts welfare weght s relatvely large. A negatve margnal dstorton makes ths possble. Wth t, the planner can levy hgh average taxes on ndvduals who earn less at the current age but more over ther lfetmes, for example consultants or lawyers, whle usng the negatve dstorton to reduce the tax burden on the top current earner. The lower earners wll not be wllng to earn enough ncome to qualfy for the negatve dstorton, so the planner s able to target ts resources. 22 The second part of the proposton hghlghts a key dfference n ntratemporal dstortons between the Statc Mrrlees and the two more sophstcated scenaros. To see why t holds, consder an ndvdual wth the hghest wage at all ages. As mentoned above, the potental benet of a postve margnal dstorton on any ndvdual s that t makes possble the collecton of more tax revenue from hgher earners. If there are no hgher earners than at each age, postve dstortons on have costs but no benets, so they are avoded 2 The optmalty of a negatve top dstorton has also been suggested by Judd and Su (2006), but for a dfferent reason. In ther model wth multple dmensons of heterogenety, the nteracton of wage and labor supply elastcty dfferences can justfy negatve top dstortons. Judd and Su also perform an llustratve smulaton of age-dependent taxaton. 22 In the language of the mechansm, a worker wth a lower current wage but a hgher lfetme ncomeearnng potental may be tempted to clam the allocaton assgned to ths top current earner. To prevent ths, the planner makes such a clam more costly for the lower-sklled worker by negatvely dstortng the top earner, ncreasng ts pre-tax ncome. Followng a smlar logc, a negatve dstorton s, n prncple, possble n the Statc Mrrlees for the top earner n the entre economy across ages f that top earner s, at other ages, a low earner.

32 .2 Baselne economy 22 by the planners wth access to age-dependent taxes. The Statc Mrrlees planner, on the other hand, faces a more dfcult problem. Whle no other ndvduals of the same age earn more than, some ndvdual of a dfferent age (perhaps tself) earns more than for each age except the age at whch 's earnngs peak. Thus, n order to collect revenue from the hghest earners across all age groups, the planner wll use dstortons on at all but ts peak-earnngs age. 23 A smple numercal llustraton of ths s provded n the Appendx for the nterested reader. Though the second part of the proposton s about a specal topc, the top margnal dstorton, t hghlghts the fundamental lmtaton of the Statc Mrrlees polcy relatve to the Partal Reform and Full Optmum: the former's nablty to hold ndvduals to agespecc tax schedules. The effects of ths lmtaton rpple throughout the wage dstrbuton, allowng the more sophstcated planners to dstort ndvduals' labor efforts less than the Statc Mrrlees planner. Later, we wll quantfy these effects and ther mpact on socal welfare n the numercal smulatons below. Next, however, we turn to characterzng ntratemporal dstortons other than at the top of the ncome dstrbuton. General characterzaton In ths subsecton, I begn by gvng formal expressons for the ntratemporal dstortons n each scenaro. As n classcal Mrrleesan analyss, characterzng ntertemporal dstortons n general s dfcult due to ther dependence on the detals of the wage dstrbu- 23 In the language of the drect mechansm, no other ndvdual of the same age wants to mmc. The only reason to dstort an ndvdual s to dscourage mmckng by others, so the age-dependent polces leave undstorted. In the Statc Mrrlees, some ndvdual of a dfferent age (perhaps tself) earns more than for each age except the one at whch 's earnngs peak. Thus, wll be a temptng target for mmckng, and the Statc Mrrlees planner must dstort t at all but ts peak-earnngs age.

33 .2 Baselne economy 23 ton. Thus, the formal expressons below are not n closed form. Nevertheless, we can use them to determne the key forces drvng dstortons and to address some natural questons about the pattern of dstortons n each polcy. 24 To smplfy the results, I assume dsutlty takes the soelastc form y v t = y t : (.9) wt wt where the parameter gves the constant-consumpton elastcty of labor supply. age t s: In the Statc Mrrlees scenaro, the ntratemporal dstorton on worker of type and SM (; t) = P T P I s= j= + P T P I s= j= jj sjt where, as stated after (.5), jj tjs w t s t jj ws j tjs P T P I w t s= j= ws j s : (.0) t jj tjs s the multpler on the ncentve constrant preventng ndvdual of age t from clamng the allocaton of any other ndvdual j of age s. where jj t In the Partal Reform scenaro, t s: P I P R j= (; t) = + P I j= jj t w t jj w j t t P I w t j= w j t : (.) jj t s the multpler on the ncentve constrant preventng ndvdual of age t from clamng the allocaton of any other ndvdual j of the same age t: In the Full Optmum scenaro, t s: P I w F O j= t w (; t) = j t + P I P I j= jj j= jj w t w j t : jj 24 Smlar conclusons to those below, as well as closed form results for ntratemporal dstortons, are obtanable for a contnuous wage dstrbuton from an analyss usng the Hamltonan methods famlar from statc Mrrlees analyss such as Saez (2002) or Salane (2003). Such an analyss for the baselne model can be found n the Appendx. The method does not extend to the dynamc settngs wth prvate savng consdered later n ths paper (Case 2 and Case 4), so I use the alternatve method, conventonal n the dynamc optmal tax lterature, throughout the paper.