Optimal Income Taxation with Career Effects of Work Effort

Transcription

1 Optimal Incom Taxation with Carr Effcts of Work Effort Michal Carlos Bst & Hnrik Jacobsn Klvn Fbruary 203 Abstract Th litratur on optimal incom taxation assums that wag rats ar gnratd xognously by innat ability and thrfor do not rspond to bhavior and taxation. This is in stark contrast to a larg mpirical litratur documnting a strong ffct of currnt work ffort on futur wag rats. W xtnd th canonical Mirrlsian optimal tax framwork to incorporat such carr ffcts and provid analytical charactrizations that dpnd on stimabl ntitis. Bsids th standard static arnings lasticity with rspct to th marginal tax rat, th optimal tax schdul also dpnds on th lasticity of futur wags with rspct to currnt work ffort. W xplor th mpirical magnitud of this carr lasticity in a mta-analysis of th litratur on th rturns to work xprinc and tnur, concluding that a rasonabl valu for this lasticity lis btwn 0.2 and 0.4. Calibrating th modl to US micro data (undr rasonabl valus of th carr lasticity), w prsnt numrical simulations of optimal nonlinar tax schduls that dpnd on pr-priod arnings and potntially on ag. In th cas of ag-indpndnt taxation, th prsnc of carr ffcts mak th tax schdul substantially lss progrssiv than in standard modls with xognous wag rats. In th cas of ag-dpndnt taxation, carr ffcts crat a strong argumnt for lowr taxs on th old, opposit th rcommndation in th rcnt litratur on ag-dpndnt taxation. This rsult rflcts both a carr incntiv ffct and an quity ffct, whr th lattr ffct ariss bcaus incrasing arnings ovr th carr path for ach ability lvl imply that, conditional on arnings, ag and ability ar ngativly corrlatd. JEL Cods: H2, H24, J22, J24. W would lik to thank Robin Boadway, Ptr Diamond, Mik Golosov, Bas Jacobs, Viln Lipatov, Alan Manning, Emmanul Saz and Johanns Spinnwijn for discussions and commnts. All rrors rmain our own. Bst: STICERD & Dpartmnt of Economics, London School of Economics, Houghton Strt, London WC2A 2AE, UK (m.c.bst@ls.ac.uk). Klvn: STICERD & Dpartmnt of Economics, London School of Economics, Houghton Strt, London WC2A 2AE, UK (h.j.klvn@ls.ac.uk)

2 Whatvr muscls I hav ar th product of my own hard work and nothing ls -Evlyn Ashford: Olympic 00m champion Whn I was young, I obsrvd that nin out of tn things I did wr failurs. So I did tn tims mor work. -Gorg Brnard Shaw: Nobl laurat in litratur Introduction Th modrn litratur on optimal incom taxation is cast in th Mirrlsian framwork in which innat ability gnrats a wag rat that is xognous and thrfor unrlatd to individual bhavior and taxation. This holds both for static vrsions of th framwork (.g. Mirrls, 97; Diamond, 998; Saz, 200) and for rcnt dynamic vrsions (.g. Golosov t al., 2007, 20; Farhi & Wrning, 202) in which th wag rat is allowd to chang ovr tim in a potntially non-dtrministic fashion, but nvr dpnds on bhavior. In this litratur, arnings in any priod of th lif cycl rspond to taxation only through contmporanous changs in hours workd. This assumption stands in sharp contrast to a larg body of work in labor conomics studying th various ways in which currnt bhavior including work ffort affcts futur wags. Motivatd by this rsarch, w xplor th optimal tax implications of braking th simpl mapping btwn abilitis and wags by allowing currnt hours workd to affct futur productivity and wags. Th link btwn work ffort and futur wags is widly documntd in a vast litratur in labor conomics. This litratur studis th rlationship btwn th wag rat and various masurs of work xprinc, including potntial xprinc (ag minus schooling), actual xprinc, tnur in an individual s currnt job, and xprinc lost as a rsult of job losss (s Blundll & MaCurdy, 999 and Farbr, 999 for survys). Concptually, a varity of mchanisms ar likly to b in opration such as improvmnts in gnral and firmspcific human capital (Bn-Porath, 967), improvmnts in mployr-mploy matchs (.g. Manning, 2000) and ability signaling ffcts (Holmström, 999). In this papr, w captur all th channls through which currnt labor supply affcts futur wag rats in a simpl rducd-form rlationship, which kps th othrwis vry complicatd dynamic optimal taxation problm tractabl and allows us to obtain transparnt analytical rsults that dpnd on mpirical ntitis.

3 To xplor th mpirical magnitud of ths ffcts, w conduct a mta-analysis of svntn mpirical studis that prmit th drivation of an stimat of th lasticity of futur wags with rspct to currnt work ffort th paramtr that w show is crucial for optimal incom taxation. W find that 80% of th 08 stimats of this carr lasticity li btwn 0.9 and 0.38, implying that an additional 0% of work ffort whn young raiss wags whn old by btwn 2 and 4%. Ths ffcts ar strong nough to hav important qualitativ and quantitativ implications for optimal tax schduls. Our papr also contributs to th rcnt dbat about ag-dpndnt taxation, as rviwd by Banks & Diamond (20) in th rcnt Mirrls Rviw. This work argus that ag constituts a usful tagging dvic (Akrlof, 978), which can b usd to rlax th incntiv compatibility constraints of th optimal incom tax problm. For instanc, applying th static Mirrls modl sparatly to diffrnt ag groups, Krmr (200) argus that arnings distributions and labor supply lasticitis ar so diffrnt across ags that th implid pattrn of optimal tax rats would vary gratly by ag. Mor rcntly, th dynamic optimal tax litratur considrs this qustion (Winzirl, 20; Golosov t al., 20; Farhi & Wrning, 202) and finds that ag-dpndnt tax schduls with highr tax rats on oldr workrs ar wlfar-improving and abl to raliz most of th gains from a fully optimizd historydpndnt tax schdul. A ky rason for th powr of ag-dpndnc in this litratur is th fact that th obsrvd wag distribution of oldr workrs faturs both a highr man and a highr varianc than th wag distribution of youngr workrs (Winzirl, 20). Sn through th lns of th Mirrls modl, this translats to diffrncs in th man and varianc of th ability distribution that crats an quity and insuranc argumnt for highr taxs on th old. What this argumnt nglcts is that th diffrnc in th wag distributions of th young and th old rflcts, not diffrncs in xognous ability, but th fact that th young and th old ar obsrvd at diffrnt stags of thir (ndognous) carrs. This is th issu that forms th basis of our papr, and w show that it can rvrs prvious conclusions in th litratur. As our framwork of analysis, w considr a two-priod Mirrls modl in which th wag rat as young quals innat ability whil th wag rat as old is a gnral function of innat ability and hours workd as young. Th young and th old hav drawn thir abilitis from th sam undrlying ability distribution, but fac diffrnt wag rats for two rasons. On rason is that ffort as young srvs as an invstmnt in labor productivity as old (bhavioral carr ffct). Th othr rason is that, indpndntly of individual bhavior, a givn innat Winzirl (20) also discusss th importanc of modling th ndognity of wag paths in ordr to fully valuat th cas for ag-dpndnt taxation. 2

4 ability may b associatd with an ag-varying wag profil rathr than a constant wag ovr th carr (what w call a mchanical carr ffct). 2 W show that th prsnc of bhavioral carr ffcts provids a plausibl micro-foundation for th wll-documntd mpirical fact that labor supply lasticitis ar largr for oldr workrs than for youngr workrs (.g. Blundll & MaCurdy, 999). Sinc th young ar working to rais futur wags as wll as for consumption in th prsnt whil th old ar working only to financ consumption in th prsnt, th labor supply of th young is naturally lss lastic than th labor supply of th old undr th sam prfrncs. Bsids ths implications for th own-tax lasticitis of th young and th old, carr ffcts hav implications for th cross-tax lasticitis as, for xampl, lowr taxs on th old induc th young to work hardr du to th ffort invstmnt ffct, what w labl th aspiration ffct in th papr. W considr a prfrnc structur allowing us to bypass issus rlatd to savings and capital taxation, and provid analytical charactrizations of th optimal taxation of labor arnings that rlat in intuitiv and transparnt ways to xisting rsults without carr ffcts. 3 Ths charactrizations show that th optimal tax schdul can b xprssd as a function of long-run arnings lasticitis for th young and th old that incorporat th implications of ndognous carr paths. Sinc such long-run arnings rsponss ar not what is capturd by th mpirical labor supply and taxabl incom litraturs using shortrun variation in micro data (as pointd out by,.g., Piktty & Saz, 203), w show that th rlvant long-run lasticitis dpnd on two undrlying sufficint statistics: th standard static arnings lasticity with rspct to th marginal nt-of-tax rat (as stimatd in th normous taxabl incom lasticity litratur) and th lasticity of futur wags with rspct to currnt work ffort (th magnitud of which can b infrrd from th larg litratur on th rturns to work xprinc and tnur). For th cas of ag-dpndnt taxation, this framwork brings to th for two important ffcts that hav bn ignord in prvious optimal tax analyss. First, in th mpirically rlvant cas of incrasing wag profils ovr th carr, an old workr of a givn ability lvl has a highr wag rat and highr arnings than a young workr of th sam ability lvl. As a consqunc, an old workr at a givn arnings lvl must b of lowr ability than young workrs at th sam arnings lvl. Thrfor, conditional on arnings, ag is ngativly corrlatd with ability which crats a classical tagging argumnt for supplmnting 2 Hr w will assum that this wag profil is dtrministic, but thr is no rason that our framwork could not b xtndd to allow for it to b stochastic. 3 Consistnt with ral-world tax policy, w focus on annual tax schduls that involv sparat taxation of arnings in diffrnt tim priods but may b ag-dpndnt rathr than fully history-dpndnt tax schduls. 3

5 an arnings-basd incom tax with a tax brak to oldr workrs. Scond, th prsnc of bhavioral carr ffcts crat an fficincy argumnt for lowr taxs latr in th carr, an ffct that oprats through th own-tax and cross-tax lasticitis of labor supply dscribd abov. In particular, lowr taxs on th old ar dsirabl both bcaus oldr workrs ar rlativly lastic with rspct to thir contmporanous tax rat and bcaus youngr workrs ar lastic with rspct to thir futur tax rat via th aspiration ffct. In summary, both th ag-ability corrlation ffct and th lasticity ffcts call for ag-dpndnt taxation with lowr incom tax rats on oldr workrs. 4 This is dirctly opposit to th policy rcommndation in th rcnt optimal tax litratur, but is consistnt with th policy dbat outsid conomics in which ag-dpndnc is typically discussd in th contxt of tax braks for oldr workrs. 5 Whn taxs ar constraind to b ag-indpndnt, w show that th optimal schdul of marginal tax rats can b writtn as a wightd avrag of th two optimal ag-dpndnt marginal tax rat schduls. Sinc arnings incras ovr th carr path, at highr incom lvls a gratr fraction of th population is old and so th wight placd on th old rlativ to th young in th optimal marginal tax schdul is incrasing in incom. As th optimal agdpndnt marginal tax rats ar lowr on old workrs, th incrasing wight on th old maks th optimal ag-indpndnt marginal tax rat schduls flattr lss progrssiv than in th standard modl with xognous wag paths. In ordr to ascrtain th quantitativ implications of th nw ffcts w hav idntifid, w carry out numrical simulations basd on data for th Unitd Stats, xtnding th simulation mthod st out by Saz (200) to a stting with carr ffcts. Th simulations for agdpndnt tax schduls raffirm th thortical argumnts mad abov. In a stting with no bhavioral carr ffcts (but mchanical carr ffcts gnrating an incrasing wag profil ovr th lif cycl at a givn ability), th optimal tax systm faturs a wak dgr of ag dpndnc with slightly highr taxs on oldr workrs. Howvr, vn vry modst bhavioral carr ffcts ar sufficint to rvrs this rsult and gnrat lowr taxs on oldr workrs. Undr ralistic assumptions about th strngth of carr ffcts (basd on our mpirical mta-study), it is possibl to gnrat vry strong ag dpndnc with much lowr 4 Bsids ths two ffcts, a third offstting ffct is drivn by th diffrnt hazard ratios of th arnings distributions of young and old workrs. Optimal marginal tax rats on arnings dpnd positivly on such hazard ratios (s.g. Saz, 200, in th contxt of th standard Mirrls modl), and th mpirical fact that arnings distributions of old workrs fatur highr hazard ratios than arnings distributions of young workrs maks it mor fficint to tax th old than th young, othr things qual. This is prcisly th ffct that is cntral to th rsults in Krmr (200) and Winzirl (20), as discussd abov, but in our analysis it is not sufficintly strong to ovrturn th othr argumnts calling for lowr taxs on th old. 5 For xampl, th UK tax systm involvs limitd ag-dpndnc favoring old workrs, and th Mirrls Rviw proposs to go furthr in this dirction. 4

6 taxs on oldr workrs. This rsult is drivn by th ag-ability corrlation and lasticity ffcts discussd abov. 6 Th simulation rsults for ag-indpndnt tax schduls show that vn modst carr ffcts can hav substantial impacts on optimal marginal tax rats, which ar lowr and flattr than in th absnc of carr ffcts. Th ida that work ffort rprsnts an invstmnt in highr futur wags (for xampl via larning by doing) is rlatd to th larg litratur on human capital invstmnts. Sinc th implications of standard human capital invstmnts (formal ducation) for optimal taxation hav bn xplord in arlir work (.g. Eaton & Rosn, 980; Bovnbrg & Jacobs, 2005), it is important to not that th tax implications of larning by doing ar fundamntally diffrnt from th implications of ducation. First, ducation and work rprsnt two substitutabl uss of tim, and th ky cost of ducation is thrfor th opportunity cost of forgon nt-of-tax arnings during ducation. This implis that ducation costs ar ffctivly tax dductibl in which cas incom taxation nd not distort human capital invstmnts at all (Eaton & Rosn, 980). By contrast, sinc larning by doing is a byproduct of work ffort, incom taxation will always distort this form of human capital invstmnt. Scond, formal ducation is an activity that can b obsrvd and thrfor dirctly subsidizd or taxd by th govrnmnt, whras larning by doing cannot b sparatd from labor supply and so cannot rciv a sparat tax tratmnt. For both of ths rasons, modls of optimal taxation with ndognous ducation ar concptually vry diffrnt from our framwork and do not shd light on th issus that w highlight in this papr. As far as w ar awar, th only prvious papr that allows for larning-by-doing ffcts in th contxt of optimal incom taxation is Kraus (2009), who focuss on th implications of such ffcts for th no-distortion-at-th-top rsult in th contxt of a two-typ Stiglitz (982) modl. W will procd as follows. Sction 2 prsnts th stting and shows th implications of carr ffcts for arnings lasticitis. Sction 3 charactrizs optimal incom tax schduls and discusss th implications of carr ffcts for both ag-dpndnt and ag-indpndnt taxation. Sction 4 invstigats mpirically th carr ffct of work ffort basd on a mtaanalysis of th litratur on xprinc and tnur ffcts. Sction 5 prsnts numrical simulations that dmonstrat th quantitativ importanc of carr ffcts for optimal tax policy, and finally sction 6 concluds. 6 In particular, ths two ffcts dominat th ffct coming from th diffrnc in th arnings distributions of th young and th old (what w will call th hazard ratio ffct), which is what drov th prvious findings that ag dpndnc should fatur highr taxs on oldr workrs. 5

7 2 Th Stting 2. Individuals W analyz th simplst possibl stting that allows us to xplor th implications of carr ffcts for optimal tax schduls. Individuals liv for 2 priods, i {y, o}, work in both of thm and at any point in tim thr is a continuum of mass of individuals of ach ag aliv. Thy hav tim sparabl prfrncs with no discounting and thir pr-priod utility is quasi-linar and givn by u (c i, l i ) = c i +/ l+/ i. This formulation has th virtu that individuals will not sav and so w can focus on th analysis of wag ffcts without th additional complication of saving ffcts. In th first priod of lif, individuals ar paid according to thir innat ability n, th distribution of which is givn by th cdf F (n). Thrfor, arnings whn young ar z y = nl y (n). Our ky innovation is to allow th scond-priod wag to dpnd both on innat ability and on th first priod s ffort choic. W allow this ffct to manifst itslf in a vry gnral way, mrly positing that th wag rat whn old ω is a gnral function of innat ability and first-priod ffort, i.. ω = ω (n, l y ). Earnings whn old ar thn givn by z o = ω (n, l y ) l o. Th rsponsivnss of th wag rat whn old to innat ability may b capturd by th lasticity η = ω n and rflcts th mchanical carr ffct of highr ability n ω on th lif-cycl profil of wags. Th rsponsivnss of wags whn old to ffort whn young is capturd by th lasticity δ = ω l y = ω z y l y ω z y and rflcts th bhavioral carr ffct du ω to th invstmnt componnt of work ffort as young. Whn w turn to simulations of th optimal tax schduls in sction 5, w will assum that w ar in th mpirically plausibl cas whr η, δ 0, howvr not that this rstriction is not ncssary for our drivations of th optimal tax schduls. Sinc thr ar no savings, consumption at ag i is simply qual to arnings nt of incom taxs at that ag, i.. c i = z i T i (z i ). Th incom tax liability at ag i, T i (z i ), dpnds on arnings at that ag (but not on arnings at othr ags) and possibly on ag itslf (as th T i (.) function is allowd to vary with i). This is consistnt with ral-world tax schduls, which ar always basd on annual incom and somtims fatur aspcts of ag-dpndnc (s, for xampl, th Mirrls Rviw for a dscription of ag-dpndnc in th UK tax systm). Liftim utility is givn by U (z y, z o ) = z y T y (z y ) + / ( ) zy +/ + zo T o (z o ) n ( + / z o ω (n, z y /n) ) +/ () 6

8 which has first-ordr conditions for arnings chosn whn young and whn old givn by and τ y (z y ) ( ) ( ) zy n n + zo + ω τ o (z o ) δ z y = 0 (2) ( ) zo ω ω = 0 (3) whr τ i (z) T i (z) is th marginal tax rat on arnings in priod i. 2.2 Earnings Elasticitis To facilitat intrprtation of our main rsults, this sction starts by charactrizing th rlationship btwn th strngth of carr ffcts and arnings lasticitis for th young and th old. 7 At th xtrm, whn thr ar no bhavioral carr ffcts (δ = 0), this modl rducs to a simpl two-priod vrsion of a standard optimal incom tax modl lik that studid in Diamond (998). In particular, th young ar rsponsiv only to th tax schdul thy fac whn young vn though thy know th tax schdul thy will fac whn old, and similarly for th old. This is bcaus thir bhavior whn young dos not affct th dcision-making problm as old, and vic vrsa. Morovr, it is asy to s from th first-ordr conditions (2) and (3) that th lasticity of arnings at ag i with rspct to th marginal nt-of-tax rat at that ag, τ i, is givn by th utility paramtr for both ag groups. Howvr, whn w introduc carr ffcts through δ > 0, this changs. W dfin th lasticity of arnings at ag i with rspct to th marginal nt-of-tax rat at ag j as E ij dz i τ j d( τ j ) z i. Applying th implicit function thorm to th pair of first-ordr conditions (2) and (3), Appndix A shows that th arnings lasticitis can b xprssd as E yy E oy E yo E oo = κ ( + ) δ ( + ) δ zo( τo) z y( τ y) [ ] + δ ( + δ) ( + ) zo( τo) z y( τ y) (4) whr κ + δ ( δ) ( + ) zo( τo) z y( τ y). Th lasticitis E yy and E oo ar contmporanous arnings lasticitis of th young and th old with rspct to th marginal nt-of-tax rats facd at thos rspctiv ags, whil E yo and E oy ar intrtmporal arnings lasticitis of th young and th old that rflct th prsnc of carr ffcts. Th lasticity E yo rflcts what w rfr to as th aspiration ffct: sinc part of th rturn to currnt work ffort is 7 Throughout th papr, w focus on arnings lasticitis (including hours-workd and wag-rat ffcts) rathr than hours-workd lasticitis, bcaus it is th formr lasticity concpt that mattrs dirctly for optimal tax schduls. Howvr, th main qualitativ proprtis of arnings lasticitis that w charactriz in this sction also applis to hours-workd lasticitis. 7

9 highr futur wags, and individuals anticipat th rat at which thos futur wags will b taxd, a highr tax rat latr in lif rducs th carr invstmnts mad through work ffort arlir in lif. Th lasticity E oy rflcts what w rfr to as th accumulation ffct: a highr tax rat on th young rducs work ffort and thrfor arnings by th young, which has a ngativ knock-on ffct on th wag rat and labor supply of thos individuals whn thy bcom old. In th following, w prsnt thr lmmas that clarify th prcis link btwn th carr ffct δ and th siz of arnings lasticitis. Th proofs of ths lmmas ar providd in appndix A. Th first lmma shows how th contmporanous rsponsivnss of th two ag groups is affctd by th prsnc of carr ffcts: Lmma. In th absnc of bhavioral carr ffcts, δ = 0, th contmporanous arnings lasticitis of th young and th old ar givn by E yy = E oo =. In th prsnc of bhavioral carr ffcts, δ > 0, th contmporanous arnings lasticity of th young is lowr whil that of th old is largr than in th absnc of such ffcts, i.. E yy < and E oo >. Intuitivly, th young ar working both for currnt wags (taxd at rat τ y ) and to rais thir wags whn old (taxd at rat τ o ), and so thir arnings ar naturally lss lastic to thir tax rat as young than is implid by th standard static lasticity. Manwhil, th arnings of th old rspond to th tax rat whn old both through a standard static hours-of-work rspons govrnd by th -paramtr and through a dynamic wag-rat rspons coming from th ffct of th tax rat whn old on th incntiv whil young to invst in highr wags as old. Notic that ths arnings lasticitis (and thos discussd blow) rflct full dynamic ffcts on arnings at diffrnt ags by taxpayrs who plan thir ntir lif cycl profil of arnings, prfctly anticipating th tax schdul facd in ach priod. Ths ar, of cours, th rlvant lasticitis to considr for th optimal tax analysis that follow, which focuss on th optimal tax policy by a govrnmnt that can fully commit to futur tax rats. Nxt, w turn to th implications of carr ffcts for th aspiration and accumulation lasticitis: Lmma 2. In th absnc of bhavioral carr ffcts, δ = 0, th aspiration and accumulation lasticitis ar zro, i.. E yo = E oy = 0. In th prsnc of bhavioral carr ffcts, δ > 0, th aspiration and accumulation lasticitis ar positiv and always incrasing in th strngth of th carr ffct, i.. Eyo δ > 0 and Eoy δ > 0. Th intuition bhind ths rsults follows naturally from th fact that, in this modl, it is prcisly th ffct of currnt work ffort on futur wag rats that crats an intrtmporal link btwn taxation and arnings across diffrnt priods. With positiv carr ffcts of 8

10 work ffort, arnings in on priod rspond positivly to th nt-of-tax rat in anothr priod, and th siz of this rspons is incrasing in th siz of th carr ffct δ. Th lasticitis considrd so far masur arnings rsponss as young or old to th tax rat in on priod of lif taking as givn th tax rat in th othr priod. It is usful to also considr total arnings rsponss by th young and th old to a chang in th tax rat in both priods of lif. Dfining th total lasticity of arnings at ag i as E i E iy + E io, w can stat th following: Lmma 3. In th prsnc of bhavioral carr ffcts, δ > 0, th total lasticity of arnings in ach priod is largr than th standard static lasticity, i.. E y E yy + E yo > E o E oy + E oo > Morovr, with δ > 0, th total lasticity of arnings as old is largr than th total lasticity of arnings as young, i.. E o > E y Ths rsults dmonstrat that carr ffcts incras th ovrall rsponsivnss of arnings to incom taxation and thrfor xacrbat th fficincy costs of taxation. Morovr, th dgr to which carr ffcts incras th rsponsivnss of arnings is strongr for th old than for th young. This last rsult not only provids an intrsting and plausibl microfoundation for th oftn rportd finding that labor supply and arnings lasticitis ar largr for th old than for th young (s, for xampl Blundll & MaCurdy, 999), 8 it also has potntially important implications for optimal tax structur and in particular th dsirability and dsign of ag-dpndnt taxs. 2.3 Th Govrnmnt W considr a govrnmnt imposing an annual incom tax that may or may not dpnd on th ag of th taxpayr. That is, an individual s tax liability in a givn priod dpnds xclusivly on within-priod incom and possibly on ag. This is analytically and concptually diffrnt from considring a govrnmnt choosing fully history-dpndnt tax schduls in which an individual s tax liability whn old may dpnd dirctly on incom arnd whn young. W focus on annual ag-dpndnt tax schduls rathr than fully history-dpndnt 8 Th rlationships in Lmma 3, which ar statd in trms of arnings lasticitis, also apply to hours-ofwork lasticitis. 9

11 schduls, bcaus th formr is mpirically mor rlvant: ral-world incom tax systms ar basd on annual tim-sparabl tax liability and occasionally involvs som ag-dpndnc, but ar in gnral not history dpndnt. 9 Whil at prsnt, ag-dpndnc in th incom tax systm is usd ithr in a vry limitd fashion in som countris (.g. Unitd Kingdom) or not at all in othr countris (.g. Unitd Stats), it is intrsting to analyz bcaus of svral rcnt proposals to introduc ag as a tagging dvic in tax systms. W charactriz optimal tax policy both whn full ag dpndnc is allowd (gnral schduls T y (z), T o (z)) and whn no ag dpndnc is allowd (schduls T y (z) = T o (z) = T (z) z). W assum throughout that th govrnmnt can fully commit to futur tax rats. In th cas of ag-dpndnc, th govrnmnt chooss tax schduls for th young and th old T y (z), T o (z) to maximiz social wlfar subjct to incntiv compatibility constraints and a rvnu-raising constraint, i.. max T y(z),t o(z) ˆ 0 Ψ [U (z y (n), z o (n))] df (n) ˆ 0 T y (z y (n)) df (n) + s.t. {z y (n), z o (n)} arg max U (z y, z o ) n ˆ 0 T o (z o (n)) df (n) R whr Ψ [ ] is an additivly sparabl social wlfar function dfind ovr th liftim utility of individuals, R is an xognous rvnu rquirmnt, and th siz of ach gnration has bn normalizd to. Th govrnmnt s rdistributiv tasts may b capturd by social wlfar wights qual to th social marginal utility of incom for diffrnt individuals xprssd in trms of th marginal valu of public funds. For an individual of ability n, th social wlfar wight is dfind as g (n) Ψ [U (z y (n), z o (n))] /λ whr λ is th Lagrang multiplir on th govrnmnt budgt constraint, th marginal valu of public funds. It will b usful to translat this wlfar wight from bing a function of ability to bing a function of incom, so w also dfin g y (z) Ψ [U (z, z o (z))] /λ and g o (z) Ψ [U (z y (z), z)] /λ, whr z o (z) ar th quilibrium arnings whn old of an individual who arns z whn young and z y (z) ar th quilibrium arnings whn young of an individual who arns z whn old. z y (z), z o (z) ar incrasing functions of z as long as z y (n), z o (n) ar incrasing functions of n. As in th standard Mirrls modl, th condition that z y (n), z o (n) ar incrasing in n is ncssary and sufficint to nsur that a givn path for z y (n), z o (n) can b implmntd by a truthful 9 Thr is som history dpndnc in social scurity systms, which mattrs for rtirmnt dcisions. But hr w focus on incom taxation and do not modl rtirmnt. 0

12 mchanism or, quivalntly, by a nonlinar tax systm. Th analytical charactrization in sction 3 assums that this condition is satisfid whil sction 5 vrifis this numrically. 3 Optimal Tax Schduls This sction charactrizs analytically th implications of carr ffcts for th optimal nonlinar tax schdul in th ag-dpndnt and ag-indpndnt cass. W driv optimal tax formulas using both Hamiltonian and tax prturbation approachs, whr th lattr is particularly usful for facilitating conomic intuition about th rol of diffrnt ffcts. Th optimal marginal tax rats ar xprssd in trms of ntitis that ar obsrvabl or stimabl in th mannr of Diamond (998) and Saz (200), which lnds itslf naturally to a calibration xrcis as considrd in sction 5. As w dscrib in dtail blow, th implications of bhavioral carr ffcts for optimal incom taxation can b split into lasticity ffcts coming from how carrs affct th rsponsivnss of arnings by th young and th old to taxs, a wlfar wight ffct coming from how carrs affct th social marginal utilitis of incom of th young and th old, and a hazard ratio ffct coming from how carrs gnrat diffrnt arnings distributions for th young and th old. 3. Optimal Ag-Dpndnt Taxs In this sction w charactriz th optimal ag-dpndnt, nonlinar incom tax schdul {T y (z), T o (z)} with corrsponding marginal tax rat schduls {τ y (z), τ o (z)}. W can show: Proposition. Th optimal ag-dpndnt tax schdul, T i (z) at ag i {y, o}, is associatd with marginal tax rats whr (for i j) w hav τ i (z) τ i (z) = A i (z) B i (z) C i (z) (5) { τ j (z j (z)) z j (z) A i (z) = E ii + E ji τ i (z) z [ g z i (z )] dh i (z ) B i (z) = H i (z) C i (z) = H i (z) zh i (z) }

13 Figur : Non Linar Tax Rat Prturbation Post-Tax Earnings zi Ti(zi) Pr-rform Post-rform Slop = τ i dzdτ i Slop = τ i dτ i z z + dz Earnings at ag i, z i at any arnings lvl z. In ths xprssions, H i (.) and h i (.) dnot th quilibrium cdf and pdf, rspctivly, of arnings at ag i. Proof. Hr w prov th rsult dirctly using a tax prturbation mthod (as first dvlopd by Piktty, 997; Saz, 200), first for th young and thn for th old as this illustrats th intuition for th rsults bttr. A tchnically mor rigorous proof basd on th Hamiltonian approach is found in appndix B alongsid a proof that th two mthods produc quivalnt rsults in th contxt of our modl. For th young and th old sparatly, considr a small prturbation around th optimal tax schdul as dpictd in Figur. Th prturbation incrass th marginal tax rat by a small amount dτ i at ag i on incoms falling in a small band (z, z + dz) but is othrwis lft unchangd. Th tax schdul of th young W first considr th prturbation in th tax schdul of th young. Th marginal tax rat incras dτ y in th small band (z, z + dz) has a mchanical ffct on tax rvnu and wlfar for all young individuals abov z as wll as two bhavioral ffcts on thos with arnings btwn z and z + dz as young. W procd to analyz th thr ffcts sparatly: 2

14 Mchanical Wlfar Effct All young taxpayrs with arnings abov z pay dτ y dz mor in taxs (holding bhavior constant), which crats a mchanical rvnu gain for th govrnmnt but rducs th utility of thos individuals. Th nt social wlfar ffct of th mchanical tax incras of a young individual with incom z is givn by dτ y dz [ g y (z )]. Hnc, th total mchanical ffct on social wlfar is givn by M y = dτ y dz Contmporanous Earnings Effct ˆ z [ g y (z )] dh y (z ) Using th dfinition of th contmporanous arnings lasticity of th young E yy in sction 2.2, ach young prson in th band (z, z + dz) rducs arnings by E yy dτ y τ y(z) z. Multiplying th arnings rspons by th marginal tax rat τ y (z), w gt th chang in tax liability by ach individual in this band. As thr ar h y (z) dz young individuals in th band, th total ffct of contmporanous arnings rsponss on tax rvnu is givn by Accumulation Effct E y = dτ y dz zh y (z) E yy τ y (z) τ y (z) Th labor supply rspons of young workrs locatd in th band (z, z + dz) affcts human capital accumulation and thrfor th wag rat and arnings of thos young workrs whn thy bcom old. As stablishd arlir, a givn tax systm is associatd with a mapping btwn arnings as young and arnings as old, so that a prson with arnings z as young has arnings z o (z) as old. This implis that changing th tax rat on young workrs at incom lvl z has an accumulation ffct on old workrs at incom lvl z o (z). Using th dfinition of th accumulation lasticity E oy, an old prson at z o (z) rducs dτ arnings by E oy y z τ y(z) o (z). Th numbr of old workrs whos arnings chang as a rsult of this accumulation ffct (thos in th band (z o (z), z o (z + dz)) of th distribution h o (z o )) is qual to th numbr of young workrs who changd thir labor supply in rspons to th highr tax rat on th young (thos in th band (z, z + dz) of th distribution h y (z y )), i.. w hav h o (z o (z)) dzo dz dz = h y (z) dz, and thrfor th total ffct on tax rvnu du to th accumulation ffct on all old workrs affctd is givn by Optimality AC = dτ y dz z o (z) h y (z) E oy τo (z o (z)) τ y (z) At th optimal tax schdul, thr should b no first-ordr wlfar ffct of this prturbation, and so w hav 3

15 M y + E y + AC = 0 Insrting th abov xprssions and rwriting givs th following optimality condition on th tax schdul for th young { } τ y (z) τ y (z) = E yy + E oy τo (z o (z)) z o (z) [ g z y (z )] dh y (z ) τ y (z) z zh y (z) which, aftr multiplying and dividing by H y (z), is quivalnt to th xprssion in Proposition for i = y. Th tax schdul of th old As in th tax prturbation for th young, th marginal tax rat incras on th old dτ o in th band (z, z + dz) givs ris to a mchanical wlfar ffct abov z along with two bhavioral ffcts on thos btwn z and z + dz as old. Th mchanical wlfar ffct on th old is analogous to th xprssion for th young: M o = dτ o dz ˆ z [ g o (z )] dh o (z ) Thr is also a contmporanous arnings ffct on th old taking th sam form as for th young: E o = dτ o dz zh o (z) E oo τ o (z) τ o (z) Finally, instad of th accumulation ffct of th tax prturbation for th young, w hav an aspiration ffct of th tax prturbation for th old. Aspiration Effct Th highr tax rat on old workrs in th arnings band (z, z + dz) discourags young workrs who anticipat bing in this band whn old from invsting in futur productivity and arnings. Using th mapping btwn arnings as young and arnings as old, this bhavioral ffct on th young occurs in th arnings band (z y (z), z y (z + dz)). dτ Th chang in arnings by ach young workr who is affctd quals E yo o z τ o(z) y (z). Th numbr of young workrs affctd (thos in th band (z y (z), z y (z + dz)) of th distribution h y (z y )) is qual to th numbr of old workrs facing a highr marginal tax rat (thos in th band (z, z + dz) of th distribution h o (z o )), so that h y (z y (z)) dzy dz dz = h o (z) dz. This implis 4

16 that th total ffct on tax rvnu du to th aspiration ffct can b writtn as AS = dτ o dz z y (z) h o (z) E yo τ y (z y (z)) τ o (z) Optimality At th social optimal, w hav M o + E o + AS = 0 which givs th xprssion in Proposition for i = o. W hav thus charactrizd th optimal tax schdul in trms of two xprssions that shar svral qualitativ faturs with th standard formulas in Diamond (998) and Saz (200), but with som important diffrncs that bar flshing out. W will discuss ths in th contxt of thir implications for th optimal form and dgr of ag dpndnc in th tax systm. 3.2 Ag Dpndnc in th Optimal Tax Systm Th xistnc of carr ffcts of work ffort has implications for all thr trms in th optimal incom tax formula (5): th invrs lasticity trm A i, th wlfar wight trm B i and th hazard ratio trm C i. Considring ach of ths trms sparatly, w now discuss th implications of carr ffcts for optimal incom tax structur. W mphasiz how carr ffcts chang th thr ky trms in diffrnt ways for th young and th old, and thrfor hav important ffcts on th optimal form and dgr of ag dpndnc in th tax systm. Th lasticity ffct of carrs oprats through A y (z) and A o (z). For th taxation of old workrs (A o (z) trm), Lmmas & 2 show that carr ffcts δ > 0 giv ris to a contmporanous arnings lasticity for th old that is largr than th standard static lasticity, E oo >, as wll as a positiv aspiration lasticity for th young, E yo > 0. Th combination of ths ffcts imply A o (z) < /, so that th invrs lasticity trm for old is always smallr than in standard modls without carr ffcts. This calls for lowr taxs on th old, othr things qual. For th taxation of young workrs (A y (z) trm), Lmmas & 2 show that δ > 0 implis a contmporanous arnings lasticity for th young that is smallr than th standard lasticity, E yy <, along with a positiv accumulation lasticity on th old, E oy > 0. Hnc, dpnding on th magnituds of ths lasticitis, A y (z) may b ithr blow or abov /. Du to fact that th lasticitis E yy, E oy (s quation 4) and th wighting trm on E oy in th optimal tax formula ar ndognous to th tax systm itslf, it is not possibl to analytically dtrmin if A y (z) is smallr or gratr than /. Nvrthlss, 5

17 our numrical simulations (discussd in sction 5) show that A y (z) / undr a wid rang of rasonabl paramtr assumptions, so that th lasticity ffct of carrs calls for ithr unchangd or highr taxs on th young, othr things qual. Th combination of ths insights imply that th lasticity ffct on its own calls for ag-dpndnt taxs with lowr taxs on th old than on th young, conditional on arnings. Th wlfar wight ffct of carrs oprats through th trms B y (z) and B o (z). In th discussion, it is usful to dnot by G i (z) th avrag social wlfar wight on individuals of ag i with arnings abov z, so that w may writ B i (z) = G i (z). Whn considring th ffct of ag on th avrag social wlfar wight G i (z), notic first that th social wlfar wight on any givn individual is a function of hr liftim utility which dpnds on hr innat ability, but not on hr ag. Howvr, th avrag social wlfar wight ovr th arnings sgmnt (z, ) is not indpndnt of ag, bcaus this arnings sgmnt is associatd with diffrnt ability sgmnts for th young and th old du to carr ffcts. Sinc arnings profils ar incrasing ovr th lif cycl, th pool of old workrs with arnings abov z consists of all thos whos arnings wr abov z as young and also som individuals whos arnings wr blow z as young. Givn that arnings ar incrasing in ability n conditional on ag (th condition for implmntability of th dirct mchanism), thos blow z as young must b of lowr ability than thos abov z as young. Dnoting th avrag wlfar wight among workrs who ar blow z as young but abov z as old by G y (z ), it follows that G y (z ) > G y (z) undr concav social prfrncs. Th avrag social wlfar wight on old workrs abov z can thn b writtn as G o (z) = s G y (z) + ( s) G y (z ) > G y (z) for s (0, ). Intuitivly, with incrasing arnings profils ovr th carr path at ach ability n, oldr workrs in a givn arnings rang ar, on avrag, of lowr ability than young workrs in th sam arnings rang (ag and ability ar ngativly corrlatd, conditional upon arnings), and thrfor th avrag social wlfar wight on th old is largr than on th young. This ffct implis B o (z) < B y (z), and so th wlfar wight ffct, lik th lasticity ffct discussd abov, calls for ag-dpndnt taxs with lowr taxs on th old than on th young, conditional on arnings. Finally, th hazard ratio ffct of carrs oprats through th trms C y (z) and C o (z). Ths hazard ratios can b sn as masurs of th thicknss of th arnings distribution abov a cutoff z for th young and th old, rspctivly. As an xampl, if arnings ar distributd according to th Parto distribution, ths ratios ar qual to th invrs of th Parto paramtr and masur th thicknss of th uppr tail. In our modl, th prsnc of incrasing arnings profils ovr th carr crat an arnings distribution for oldr workrs with a thickr uppr tail than for youngr workrs, which implis C o (z) > C y (z) at last 6

18 for a high nough z. This prdiction is born out by th data (hazard ratios undr actual tax systms) and by our numrical simulations blow (hazard ratios undr th optimal tax systm), in which th hazard ratio is largr for oldr than for youngr workrs, xcpt at vry low lvls of arnings. On its own, this ffct calls for highr taxs on th old than on th young, conditional on arnings, and thrfor works to offst th lasticity and wlfar wight ffcts dscribd abov. This hazard ratio ffct is what drivs th strong ag-dpndnc rsults in Winzirl (20). In our framwork, it is not possibl to stablish analytically whthr th hazard ratio ffct (calling for highr taxs on th old) is abl to dominat th lasticity and wlfar wight ffcts (calling for lowr taxs on th old), and so w turn to numrical simulations basd on U.S. micro data to xplor this in sction Optimal Ag-Dpndnt Top Tax Rats Assuming that th uppr tails of th arnings distributions for th young and th old ar both Parto distributd (with potntially diffrnt Parto paramtrs), th optimal top marginal tax rats dpnd on carr ffcts in a particularly simpl way. proposition W stat th following Proposition 2. Suppos that for vry high incoms, th arnings of th young and th old ar distributd according to Parto distributions with Parto paramtrs a y and a o rspctivly. Suppos furthr that th wlfar wights on th young and th old convrg to ḡ y and ḡ o and that th lasticitis E ij, i, j {y, o} convrg to constant valus dnotd by Ēij. Thn th optimal top marginal tax rats τ i on th young (i = y) and th old (i = o) ar givn by whr i, j {y, o}, i j. τ i τ i = ḡ i a i [Ēii + Ēji a ] j/(a j ) τ j (6) a i /(a i ) τ i Proof. To prov th proposition, w show that ach of th componnts of quation (5) convrgs to a constant. B i (z) and C i (z) ar straightforward. Clarly, if th wlfar wights convrg to ḡ i thn B i (z) ḡ i. It is a proprty of th Parto distribution that [ H i (z)] / [zh i (z)] = /a i so C i (z) /a i. To stablish th limiting valu of A i (z) w us th proprty of th Parto distribution with Parto paramtr a i that E [z z > x] = a i a i x. For individuals in ag group i th limiting valu of th ratio of thir arnings whn in th othr ag group j to thir currnt arnings is lim z z j (z) z = lim x E [z j z j > x] /E [z i z i > x] = a j /(a j ) a i /(a i ). Combind with th assumption that th lasticitis E ij convrg to constant valus 7

19 this implis that A i (z) Āi [ Ē ii + Ēji a ] j/(a j ) τ j. a i /(a i ) τ i Combining ths pics stablishs th rsult in quation (6). Equation (6) highlights th thr concptual ffcts discussd in sction 3.2 in a vry simpl way. Th wlfar wight ffct is capturd by th trm ḡ i (whr w hav ḡ o > ḡ y sinc incrasing carr-arnings profils imply that, conditional on arnings, th old hav lowr abilitis than th young), th hazard ratio ffct is capturd by th invrs of th Parto paramtr /a i (whr w hav a y > a o sinc incrasing carr-arnings profils crat a thickr uppr tail in th arnings distribution of th old than in th arnings distribution of th young), and finally th lasticity ffct is capturd by th bracktd trm in th dnominator (whr carr ffcts imply Ēoo > Ēyy and Ēyo > Ēoy, favoring lowr taxs on th old). Not also that, in th limit whr z, th wlfar wights on both ag groups will asymptot to zro undr standard concav social wlfar functions, and so th wlfar wight ffct would not support any ag-dpndnc at th limit. Thrfor, at vry high lvls of arnings, optimal ag dpndnc rflcts a simpl trad-off btwn th rlativ Parto paramtrs th ky mchanism in prvious work arguing for highr marginal rats on th old (Krmr, 200; Winzirl, 20) and carr incntiv ffcts which tnd to call for lowr marginal tax rats on th old as discussd abov. 3.4 Ag-Indpndnt Taxs As currnt tax systms in th world tnd to mak limitd or no us of xplicit ag-dpndnc, it is of obvious intrst to considr whthr th carr ffcts w introduc hav any bit in influncing optimal ag-indpndnt tax schduls. This sction thrfor charactrizs th optimal ag-indpndnt, nonlinar incom tax schdul T (z) with corrsponding marginal tax rat schdul τ (z). W will s that it is still possibl to xprss th optimal tax formula in trms of obsrvabl quantitis and lasticitis, and that th ky ffcts discussd abov ar still prsnt and affct th lvl and profil of marginal tax rats. In this stting, w hav Proposition 3. Th optimal ag-indpndnt tax schdul T (z) is associatd with marginal tax rats τ (z) τ (z) = α (z) B y (z) C y (z) + [ α (z)] B o (z) C o (z) α (z) A y (z) + [ α (z)] A o (z) (7) whr α (z) h y (z) / [h y (z) + h o (z)] is th proportion of individuals with incom z who ar young and A y (z), A o (z), B y (z), B o (z), C y (z) and C o (z) ar as dfind in Proposition. 8

20 Proof: Again, w prov th rsult dirctly using th prturbation mthod, laving th Hamiltonian mthod and th dmonstration of thir quivalnc for appndix B. Th prturbation that w considr is similar to th on dpictd in Figur, xcpt that it prtains to th uniqu tax schdul facd by both th young and th old. Hnc, th marginal tax rat on both th young and th old is incrasd by a small amount dτ in a small arnings band (z, z + dz). W now charactriz th social wlfar ffcts of this tax rform. Mchanical Wlfar Effct All taxpayrs with arnings abov z fac a mchanical incras in tax liability of dτdz. For a young individual with arnings z > z th social valu of this is givn by dτdz [ g y (z )], whil for an old individual at z > z th social valu of this quals dτdz [ g o (z )]. Th total mchanical wlfar ffct is thrfor givn by {ˆ M = dτdz [ g y (z )] dh y (z ) + Contmporanous Earnings Effcts z ˆ z [ g o (z )] dh o (z ) In th band (z, z + dz), ach young prson rducs dτ arnings by E yy z whil ach old prson rducs arnings by E dτ τ(z) oo z. Th τ(z) total tax rvnu implications of ths arnings rspons qual Aspiration Effct E = dτdz z {h y (z) E yy + h o (z) E oo } τ (z) τ (z) Th highr tax rat in th arnings band (z, z + dz) inducs young workrs who anticipat bing in this band whn old to invst lss in futur wag incrass. In particular, ach young prson in th band (z y (z), z y (z + dz)) rducs arnings by E yo dτ z τ(z) y (z). Sinc th total numbr of young workrs rsponding through this channl is givn by h y (z y (z)) dzy dz = h dz o (z) dz, th total tax rvnu implications of th aspiration ffct can b writtn as Accumulation Effct AS = dτdz z y (z) h o (z) E yo τ (z y (z)) τ (z) Th labor supply rspons of young workrs in th band (z, z + dz) affcts th wag rat and arnings of thos workrs whn thy bcom old. This ffct implis dτ that ach old prson in th band (z o (z), z o (z + dz)) rducs arnings by E oy z τ(z) o (z). Th numbr of old workrs affctd h o (z o (z)) dzo dz = h dz y (z) dz, and so th total accumulation } 9

21 ffct on tax rvnu is givn by Optimality drivd abov must b zro: AC = dτdz z o (z) h y (z) E oy τ (z o (z)) τ (z) At th optimal tax schdul, th sum of th diffrnt social wlfar ffcts M + E + AS + AC = 0 By insrting th abov ffcts in this optimality and noting that by th dfinition of α (z), α(z) h y (z) = h α(z) o (z), w obtain th rsult in Proposition 3. Th optimal ag-indpndnt tax schdul in Proposition 3 dpnds on wightd avrags of th trms that wr also prsnt in th ag-dpndnt tax schduls for th young and th old. Both th numrator and th dnominator of quation (7) ar avrags of thir countrparts for th ag-dpndnt cas in Proposition, whr th wight on th young is givn by th proportion of individuals at that arnings lvl who ar young, α (z). Hnc, th sam basic ffcts that w discussd arlir in sction 3.2 ar still at play in th dtrmination of ag-indpndnt taxs. Bcaus individuals hav incrasing arnings profils ovr th lif cycl, highr incom lvls will b populatd to a largr dgr by oldr workrs than by youngr workrs, and vic vrsa at lowr incom lvls, implying that α (z) is dcrasing in z. This implis that at th bottom th optimal ag-indpndnt tax rat τ (z) puts a rlativly high wight on th young and is thrfor closr to th ag-dpndnt tax rat on th young τ y (z), whras at th top th optimal ag-indpndnt tax rat τ (z) puts a rlativly high wight on th old and is thus closr to th ag-dpndnt tax rat on th old τ o (z). This in turn implis that th arlir conclusions rgarding optimal ag-dpndnc (i.., th diffrnc btwn τ y (z) and τ o (z) at ach arnings lvl) in th ag-indpndnt cas manifst thmslvs as an ffct on th profil of th marginal tax rat with rspct to arnings (progrssivity of τ (z) with rspct to arnings). If thr is a wlfar argumnt for ag-dpndnc favoring th old (τ o (z) < τ y (z)), this would in itslf lowr marginal tax rat progrssivity in th ag-indpndnt schdul as highr arnings lvls put mor wight on τ o (z). 4 How Big Ar Carr Effcts? Having stablishd how th optimal way for govrnmnts to tax incom dpnds on th carr ffcts of work ffort, th natural nxt qustion is how larg ths carr ffcts actually ar 20

22 in practic. In sctions 2 and 3 abov w hav shown that th ky sufficint statistics for optimal incom taxation ar th long-run arnings lasticitis (including carr ffcts) of th two ag groups to th tax rat at ach ag. As argud arlir, this is not what is idntifid by th micro litratur on labor supply and taxabl incom rsponss, which mostly studis short-run arnings rsponss to contmporanous tax rats. In practic, this litratur coms closr to stimating th static lasticity in our framwork than th dynamic carr-inclusiv lasticitis E ij lasticitis (s Piktty & Saz, 203 for a similar argumnt). Nvrthlss, as quation (4) shows, th E ij lasticitis ar functions of th undrlying static lasticity and th lasticity of futur wag rats with rspct to currnt arnings δ. Whil th voluminous litratur on labor supply and taxabl incom rsponss can srv as a guid to what a rasonabl valu for th static lasticity is, thr is no such rady guidanc whn it coms to a rasonabl valu of th carr lasticity δ. A carful stimation of this paramtr is byond th scop of this papr, but thr is a vry larg litratur on xprincarnings profils in labor conomics from which w can larn somthing about th likly siz of δ. W thrfor conduct a mta-analysis of this litratur, focusing on 7 mpirical paprs studying th ffcts of xprinc, tnur and sniority on wags whos stimats prmit th drivation of an stimat of δ. In ordr to driv this stimat, w must prform a simpl transformation of th rportd stimats as most of ths paprs modl log wags as polynomials in xprinc along th lins of ln (w) = α + β EXP + β 2 EXP 2 + ε whras w want to stimat an lasticity δ = ln (w) / ln (EXP ). To driv an stimat of δ w not that by th chain rul ln(w) = ln(w) x and by th invrs rul of calculus ln(x) x ln(x) x = [ ] ln(x) ln(x) x = x and so w can driv an stimat of δ as whr ˆδ = [ ˆβ + 2 ˆβ 2 EXP ] EXP (8) EXP is th sampl man of EXP, and w can obtain standard rrors by th dlta mthod whrvr th paprs provid th ncssary variancs. W can also xtnd this to highr-ordr polynomials whr th appropriat sampl mans ar availabl. Many paprs us multipl masurs of xprinc, for xampl total labor markt xprinc EXP and tnur in th individual s currnt job T EN as in quation (9). ln (w) = α + β EXP + β 2 EXP 2 + γ T EN + γ 2 T EN 2 + ε (9) In this cas, a similar drivation to that abov suggsts that w should us variants of 2

23 Figur 2: Drivd stimats of δ Frquncy d Prcntils man s.d Nots: Th 08 stimats of δ whos distribution is shown ar drivd using variants of quation (8) whr appropriat (th vast majority of cass) and th rgrssion of prdictd wag lvls on log xprinc lvls as outlind in th txt in th rmaining cass. Th black lin is a krnl dnsity stimat, and summary statistics of th distribution ar displayd in th tabl blow th figur. ˆδ = [ ˆβ + 2 ˆβ 2 EXP ] EXP + [ˆγ + 2ˆγ 2 T EN ] T EN as our stimat. Finally, som paprs, particularly thos using mor structural mthods, prsnt tabls of prdictd wags at various lvls of xprinc rathr than polynomials in xprinc. For ths, w combin th stimatd wag lvls by simply rgrssing th prdictd log wag on th log of xprinc, and again obtaining standard rrors by th dlta mthod whr possibl. Applying ths mthods w ar abl to driv 08 stimats of δ. A full tabl of th stimats along with rfrncs to th xact locations in th paprs and th mthods usd by th authors is in th onlin appndix, but Tabl summarizs our findings. For ach of th 7 paprs, Tabl prsnts th datast(s) usd, th population(s) studid, and th mthod(s) mployd, as wll as th avrag drivd δ and its standard rror, whr th avrag δ is wightd by th numbr of obsrvations usd to stimat ach δ in th papr. Tabl shows that whil th stimats vary slightly from papr to papr, thy mostly agr that δ lis roughly btwn 0.5 and 0.4 implying that a 0% incras in xprinc is associatd with an incras in wags of btwn.5% and 4%. To rinforc this point, Figur 2 shows th distribution of all 08 stimats of δ with an ovrlaid krnl dnsity alongsid som summary statistics of th distribution which again show that 80% of th stimats li btwn 0.9 and 0.38 with a man of