HARVARD JOHN M. OLIN CENTER FOR LAW, ECONOMICS, AND BUSINESS

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1 HARVARD JOHN M. OLIN CENTER FOR LAW, ECONOMICS, AND BUSINESS ISSN (prnt) ISSN (onlne) TAXING LEISURE COMPLEMENTS Lous Kaplow Dscusson Paper No /2008 Harvard Law School Cambrdge, MA Ths paper can be downloaded wthout charge from: The Harvard John M. Oln Dscusson Paper Seres: The Socal Scence Research Network Electronc Paper Collecton:

2 Taxng Lesure Complements Lous Kaplow * Abstract Ever snce Corlett and Hague (1953), t has been understood that t tends to be optmal on second-best grounds to (relatvely) tax complements to lesure and subsdze substtutes because dong so helps to offset the dstortng effect of taxaton on labor supply. Yet n the context of smultaneous optmzaton of a nonlnear ncome tax and commodty taxes, Atknson and Stgltz (1976) clam to have demonstrated the opposte, that goods complementary wth lesure should face lower tax rates, whereas substtutes face hgher tax rates. Dervatons n leadng texts on optmal taxaton seem to yeld opposng conclusons regardng the sgn of optmal devaton of commodty taxes from unformty. It s demonstrated that the optmalty of relatvely taxng lesure complements s ndeed correct, and conflctng results are explaned. JEL Classes: H21, H24 Keywords: ncome taxaton, commodty taxaton, optmal taxaton, lesure complements Lous Kaplow. All rghts reserved. * Harvard Unversty and Natonal Bureau of Economc Research. I am grateful to Tm Brennan (the edtor), the referees, Robn Boadway, Peter Damond, James Hnes, and Dan Slverman for helpful comments and dscussons, and to the John M. Oln Center for Law, Economcs, and Busness at Harvard Unversty for fnancal support.

3 1. Introducton Income taxaton dstorts labor effort downward. Accordngly, t tends to be optmal on second-best grounds to set other polcy nstruments so as to ntroduce addtonal dstortons f (though only f) dong so helps to mtgate the excessve consumpton of lesure. Ths prncple has been consdered most extensvely wth regard to dfferental commodty taxaton. In a Ramsey settng, Corlett and Hague (1953) showed that t s optmal to tax complements to lesure and to subsdze substtutes relatvely speakng whch serves to offset the dstortng effect of taxaton on labor supply. Consder a smple example for concreteness. Begn wth a labor ncome tax, whch as s well known nduces ndvduals to reduce labor effort, ncreasng lesure. (Ths s the substtuton effect; even f the ncome effect s so large that uncompensated labor supply does not fall, t s the substtuton effect that matters for effcency.) Ideally, we would lke to tax lesure drectly or otherwse adjust the ncome tax to avod ths neffcency, but ths s assumed to be mpossble because the government can only observe ncome and not the amount of labor effort that contrbuted to producng t. Ths preexstng neffcency rases the possblty that other, more ndrect nstruments may have a role to play, and ndeed they do, specfcally wth regard to commodty taxaton. To develop ths possblty further, suppose that dsposable ncome s spent on two goods, pzza and swmsuts. Thnk of pzza as a benchmark good that produces utlty equally well n all settngs, whether at work or at play, ncludng at the beach or swmmng pool. Swmsuts are classc lesure complements: havng more swmsuts enhances utlty to a greater extent the more tme one has to swm, whereas swmsuts produce no addtonal value whle at work. In ths settng, the Corlett-Hague argument suggests the optmalty of a relatve tax on swmsuts. To be sure, ths wll mpose an effcency cost by dstortng ndvduals consumpton (too few swmsuts, too much pzza), but that cost s second-order as the tax s mposed startng at a level of zero. The other effect of the tax wll be to make lesure less attractve, now that ndvduals consume fewer swmsuts. Ths, n turn, wll rase labor effort. Snce labor s dstorted downward by the ncome tax, ths effect entals a frst-order effcency gan. Therefore, t s ndeed the case that mposng some postve relatve tax on swmsuts, the lesure complement, s optmal. Analyss of these matters became more formal wth Atknson and Stgltz s (1976) semnal examnaton of the smultaneous optmzaton of a nonlnear ncome tax and commodty taxes. The artcle s most famous for ts unformty result: f labor s weakly separable n the utlty functon, the optmum s characterzed by no dfferental commodty taxaton. 1 Although many Ramsey prncples (for example, the nverse elastcty rule) do not survve the ntroducton of an ncome tax, the Corlett-Hague result that t tends to be optmal to tax lesure complements 1 Kaplow (2006) and Laroque (2005) generalze ths result to the case n whch the ncome tax need not be optmal and n other respects, and Kaplow (2008) dscusses a varety of other qualfcatons to the unformty result and how the analyss relates to that n famlar models of Ramsey taxaton

4 and subsdze substtutes s wdely understood to reman ntact. 2 Surprsngly, ths generally accepted understandng seems to be contradcted by Atknson and Stgltz (1976). Ther central analytcal contrbuton n the relevant part of the artcle s ther dervaton of an expresson for optmal taxaton of any gven commodty (relatve to the tax rate on commodty 1, arbtrarly set equal to zero as a benchmark). Frst, they use ther dervaton to establsh ther unformty result n the case of weak separablty. When the utlty functon has the stated feature, one of the terms n ther expresson for the optmal commodty tax rate on good equals zero (and ths s true for all ), so no commodty taxaton s optmal. However, regardng the more general case, Atknson and Stgltz (1976) state: From the results gven above, t follows that goods whch are complementary... wth lesure... wll face lower tax rates, whereas substtutes face hgher tax rates (p. 68, emphass added). 3 Myles s (1995) text presents a smlar dervaton that produces an analogous formula, after whch t states: Therefore, those goods relatvely preferred by the consumers supplyng most labour should be taxed more (p. 164, emphass added) that s, the government should tax labor complements, equvalent to lesure substtutes, rather than lesure complements. Not all statements n the lterature are n accord. For example, Salané s (2003) text begns wth essentally the same set-up as the others use, derves a formula that (when adjusted to conform wth those n other sources) has the opposte sgn, and states In more economc terms, government should tax more heavly goods that are more complementary to lesure (p. 116). Although he states that hs unformty result (for the case of weak separablty) s due to Atknson and Stgltz), he fals to menton the dfference n sgn or n concluson for the more general case. The more famlar prescrpton also appears n Myles (1995), who, after the opposng language quoted above, goes on to state: goods for whch demand ncreases f more lesure s obtaned, but wth no change n ncome, should have postve commodty taxes ntroduced (p. 165). What s the rght answer? And how can we explan the contradctory results? The Corlett-Hague suggeston that t s optmal (relatvely) to tax lesure complements and to subsdze substtutes s ndeed correct, and for just the reason presented above. The explanaton for (most of) the contradctory results s smple, yet subtle. Authors dervatons vary n how they present the Hamltonan (the analogy n optmal control theory to the Lagrangan). In a common formulaton used, for example, n Atknson and Stgltz (1976) the relevant multpler (the costate varable) turns out to be negatve. Ths multpler, moreover, appears n the varous formulas for the optmal commodty tax rate on each commodty. It seems, however, that authors (and one supposes readers) n ths lterature have nterpreted the resultng expressons on the mplct assumpton that the multpler s postve. Ths mssng mnus sgn reverses all nterpretatons (ncludng those that confrm the standard 2 Ths vew s reflected, for example, n emprcal work that consders the optmalty of taxng forms of consumpton that may be sgnfcant lesure complements. See Iorwerth and Whalley (2002), West and Wllams (2007), and Parry and West (2007). 3 Omtted by the ellpses s the pont that complementary s used n the Edgeworth, not the more usual Hcksan, sense, whch means n Atknson and Stgltz s notaton that U L < 0. The precse concept s elaborated below, n the nterpretaton of expresson (6)

5 vew), except of course when another term equals zero, whch arses n the case of weak separablty that produces the unform commodty tax result. 4 The remander of ths note brefly presents the typcal formulaton of the problem and the standard result, explans why the pertnent costate varable n that set-up s negatve, and thus reconcles exstng dervatons wth the ntuton dervng from Corlett and Hague. 2. Analyss 2.1. Model Indvduals utltes u(x 1,..., x n, l) depend on ther level of consumpton of n commodtes, x 1,..., x n, and on ther level of labor effort, l. Indvduals vary n ther wage rates, w, whch have densty functon f(w). Indvduals before-ncome-tax ncome s wl. There s a nonlnear ncome tax T(wl) whch, followng conventon, may be negatve, as tends to be optmal for low-ncome ndvduals and commodty taxes on each good x are J (whch may be subsdes, n whch case they are negatve). 5 Producer prces for each commodty are assumed to be constant and may be normalzed to 1. Accordngly, an ndvdual s budget constrant s n () 1 x ( 1+ τ ) = wl T( wl). = 1 The government s problem s to maxmze socal welfare, ( 2) Wuw ( ( )) f( wdw ), 0 where W s a concave welfare functon and u(w) denotes the level of utlty acheved by an ndvdual wth wage level (of type) w. In maxmzng expresson (2), the government faces two constrants. The resource constrant (whch sometmes s equvalently formulated as a constrant on government revenue 6 ) s n ( 3) wl( w) x ( w) f ( w) dw R, 0 = = 1 4 For statements n accord wth conventonal wsdom that are reversed once the multpler s correctly nterpreted, the explanaton s ether a dscrepancy between a verbal statement and the correspondng dervaton or, n one nstance, an apparent sgn error (see note 11). 5 Atknson and Stgltz (1976) allow commodty taxes to be nonlnear, but ths complcaton s gnored here because t s not central for present purposes, s omtted n others analyses, and s often thought to be nfeasble (due to arbtrage). 6 The term n large parentheses n expresson (3) s ndvduals producton mnus ther consumpton. Ths dfference also equals the sum of taxes (net of subsdes) that ndvduals pay, so the left sde of expresson (3) also measures total tax revenue

6 where l(w) and x (w) refer to levels of labor effort and consumpton chosen by ndvduals of type w, and R s the level of resources requred by the government. In addton, there are ncentve constrants. These arse because the government, although able to observe an ndvdual s ncome, wl, and commodty purchases, s assumed to be unable to observe ndvduals w s (or l s). Under standard assumptons, the bndng constrants of concern are that ndvduals mght wsh to mtate types just below ther own, specfcally by reducng labor effort slghtly so as to earn the same ncome as the lower type. (That s, snce the mmckers w s hgher than the type whom they are mtatng by earnng the same ncome whch s all the tax authorty can observe they are accomplshng ths by reducng ther labor effort l, whch, ceters parbus, rases ther utlty). To prevent such mmckng behavor and thus for the government s scheme to be ncentve compatble, t s necessary that utlty rse wth w at a suffcent rate, whch condton s usually expressed as ( ) ( 4) du w dw ul l = w gw ( ). See, for example, Mrrlees (1971). Ths expresson for the rate at whch u(w) must ncrease wth w (redefned as g(w) for convenence) can be obtaned by takng the dervatve of u wth respect to w, makng the assumpton that all of an ndvdual s response s channeled through a downward adjustment n l (whch yelds a vald result because, at the ndvdual s optmum, all margns of response have the same effect on utlty). Snce dsposable ncome s therefore constant, so s earned ncome, wl; hence, the change n l s gven by!l/w. To obtan the effect of ths change n l on the ndvdual s utlty, the change s weghted by u l, the dervatve of the ndvdual s utlty functon wth respect to labor effort. Thus, the government chooses T(wl) and the J to maxmze expresson (2) for socal welfare subject to the revenue constrant (3) and ncentve constrants (4). It has become standard to use optmal control theory n solvng ths problem. There are varous choces for the control varables: the formulaton here wll follow Atknson and Stgltz (1976) n havng the control varables be labor effort l and commodtes x 2,..., x n (commodty x 1 s omtted because, from the budget constrant (1), t s automatcally determned when the other varables are chosen). Indvduals utlty levels u(w) are taken as the state varable, so that expresson (4) ndcates the rate of change n the state varable (here, over type, rather than over tme, as n many famlar control problems n economcs). The Hamltonan s ordnarly expressed as follows: n ( 5) Hux (, 2,..., xn, l, λμ,, w) = Wuw ( ( )) + λ wlw ( ) x( w) R f( w) + μ( wgw ) ( ), = 1 where 8 s a conventonal Lagrange multpler (here ndcatng the shadow value of a dollar n resources or, equvalently, government revenue) and :(w) s the costate varable, a shadow prce concernng the ncentve constrant n expresson (4). (More wll be sad about the nterpretaton - 4 -

7 of the costate varable n a moment.) Before proceedng, t wll be helpful for readers who may wsh to compare expresson (5) wth correspondng expressons n other work to note some ponts that relate to the sgn on the costate varable. Frst, some authors use a mnus sgn rather than a plus sgn for the fnal term. As s famlar from Lagrange multplers, there s nothng wrong wth ths, but the sgn of the resultng costate varable wll be reversed. Below, when I refer to the costate varable beng negatve, I have the presentaton n expresson (5) specfcally n mnd. Second, some authors use expresson (4) to substtute for g(w) when formulatng the Hamltonan. Ths too s equvalent, but notce that the substtuted value from expresson (4),!u l l/w, has a leadng mnus sgn; hence, when a Hamltonan of ths alternatve form has a mnus sgn before the last term, t s equvalent to expresson (5) (and, when a plus sgn s used n the alternatve case, t s equvalent to usng a mnus sgn here). 7 Ths Hamltonan (5) s maxmzed wth respect to all of the x (exceptng x 1, as noted) and substtutons are made usng ndvduals frst-order condtons for ther choce of commodtes. Furthermore, t s common to set J 1 equal to zero as a normalzaton. Ths s mportant n understandng the result because all assessments of complementarty are relatve to good x 1, and accordngly a postve (negatve) value for any partcular J ndcates that the good x should be taxed (subsdzed) relatve to good x 1. Wth ths normalzaton n mnd, the maxmzaton yelds the followng condton, whch holds for = 2,... n (and also, trvally, for = 1): 8 7 There s yet another complcaton: some analysts state utlty to be a functon of lesure rather than of labor, whch s often the source of an addtonal negatve sgn (or one less): n the analogue to expresson (4), there s no mnus sgn, but u l s replaced by the dervatve of utlty wth respect to lesure, whch has the opposte sgn. 8 Expressed n the present notaton, the dervatve s H x x = λ x 1 μ wl x f w u u + ( ) 1 1 ( ) l + = 0. l w 1 x The partal dervatve that appears twce n parentheses can be understood as the mechancal change n x 1 as x changes, as dctated by the budget constrant (1) (recallng that x 1 s not a control varable, but nstead s thus determned as the control varables l and the other x are adjusted). Usng ths fact to substtute for the partal dervatve where t frst occurs (the dervatve equals!(1+j )/(1+J 1 )); further takng advantage of ndvduals frst-order condtons to note that ths rato n turn equals!u /u 1, whch may be used n the second nstance; usng the fact that J 1 s normalzed to zero; and movng the second term to the rght sde yelds λτ f μl uu l1 = + ul. w u1 Agan, usng ndvduals frst-order condton and the normalzaton, we can multply the left sde by 1/(1+J ) and (equvalently) multply the rght sde by u 1 /u ; then, dvdng both sdes by 8f and makng use of the defnton of log, we can produce expresson (6)

8 u d log τ μlu u 1 1 () 6 = 1 + τ λwf dl. Wth weak separablty between labor and all commodtes (as a group) that s, when the utlty functon can be stated as u(v(x 1,..., x n ), l), where v s a subutlty functon the dervatve on the rght sze of expresson (6) s zero for all. Hence, n ths case, optmal commodty taxes are all equal to zero (whch was the benchmark rate stpulated for x 1 ). Suppose now that some good (relatve to good 1) s complementary to lesure. Ths means that, as l ncreases and accordngly the ndvdual enjoys less lesure, good becomes relatvely less valuable. In ths case, the dervatve on the rght sde of expresson (6) s negatve. If all the other terms were postve, ths mples that the optmal J s negatve, correspondng to a subsdy on the lesure complement. Before returnng to the other terms, let us brefly revst the example from the ntroducton wth two goods, pzza (x p ) and swmsuts (x s ). To capture the dea that pzza s valuable wthout regard to labor effort whereas swmsuts value depends on the extent of lesure tme avalable, suppose that utlty s gven by u(x p,x s,l) = v p (x p ) + (1!l)v s (x s )! z(l), where each subutlty functon has the usual propertes. It s straghtforward to show that, n ths case, the dervatve on the rght sde of expresson (6) equals!1/(1!l), whch s negatve. Now consder elements of the frst term on the rght sde of (6), skppng for the moment the costate varable :: l s postve (we are consderng only nteror solutons, as s standard n the lterature); u 1 s the margnal utlty of the frst commodty, whch s postve; 8 s the multpler on resources (or, equvalently, on the government s budget), whch s postve; w s the ndvdual s wage rate, taken to be postve; and f s the densty of the dstrbuton (evaluated at w), also taken to be postve. Hence, the sgn of the dervatve ndeed ndcates the sgn of the optmal commodty tax, suggestng that we should subsdze lesure complements (swmsuts n the example) f and only f the sgn of the costate varable, :, s postve. If, nstead, the sgn of : s negatve, the Corlett-Hague ntuton s correct. Accordngly, the next step s to explan why the sgn of : s ndeed negatve (contrary to the apparently standard mplct assumpton when nterpretng condtons lke expresson (6), as llustrated by the quotatons n the ntroducton). 9 9 There s a further, generally overlooked complcaton n nterpretng expresson (6). Most terms on the rght sde n general, all of them except the multpler 8 are themselves functons of w. (Ths ncludes the costate varable, :(w), as wll be apparent n secton 2.2.) Accordngly, the optmal level of each commodty tax s determned, n prncple, for each type w. However, t s ordnarly supposed that commodty taxes are anonymous (and must be so, for the most part, due to the possblty of arbtrage). Ths problem s nconsequental for Atknson and Stgltz s (1976) result concernng unformty n the case of weak separablty. Because each type s optmal commodty tax vector s zero, the nablty to dfferentate by type does not dsrupt the posted optmum. For the general case, however, whch s pertnent to analyses mplyng the optmalty of taxng lesure complements and subsdzng substtutes, ths problem s materal. If some good (say, swmsuts, n the above example) s a lesure complement for everyone, then t s optmal for everyone to face a postve relatve tax on that good. Even so, the optmal level of that tax would n general depend on an ndvdual s type w. Accordngly, the optmal anonymous commodty tax on that good wll reflect some sort of weghted - 6 -

9 2.2. The Sgn of the Costate Varable Understandng the ntuton behnd the costate varable can be a bt trcky, a fact confrmed by many dscussons I have had n preparng ths artcle (n whch most quered were unsure, many stated t was ndeed trcky, and some offered conjectures that were ncorrect). For smpler optmal control problems, lke choosng a path of consumpton over tme gven an expresson for how the captal stock evolves (and a startng and endng value for the captal stock), the costate varable at tme t ndcates how much the objectve functon s ncreased by a unt ncrement to the state varable (the captal stock) at tme t. (The economc nterpretaton s presented n Dorfman (1969); a helpful exposton appears n the text by Léonard and Long (1992, 4.5), whch s n turn based on the formal dervaton n Léonard (1987); see also Kamen and Schwartz (1981, II.4).) In these famlar problems, the costate varable s postve: addtonal captal at tme t s valuable because one can consume more at some pont n the future whle stll satsfyng the pertnent constrants. The challenge s n carryng ths ntuton over to the optmal ncome tax problem. Here, the state varable s utlty (u); nstead of tracng an evoluton over tme, we have an evoluton across types (w). Moreover, the relatonshp governng that evoluton derves not from technology (for example, how captal s converted nto output or how captal deprecates) but nstead from ncentve constrants. Specfcally, the equaton of moton, speakng metaphorcally, s expresson (4). To understand what s nvolved n the present settng, consder what happens f, at some wage level w, g(w) s ncreased by a small ncrement, t beng as f a small unt of utlty s deposted lke manna from heaven. At frst, ths would seem to be a good thng. But t sn t. The drect effect s second order, for the benefted group ncludes solely those who are precsely of type w, whch s a group of measure zero (ths problem has a contnuum of types, whch s typcal n optmal control problems of ths form). The ndrect effect relates to the equaton of moton, expresson (4). Ths equaton presents the ncentve constrants, whch are tghtened for hgher types by the hypotheszed ncrement to g(w) at the chosen w, whch results n a loss n achevable socal welfare. The heurstc argument s as follows. When utlty at the precse pont w s rased slghtly n the manner ndcated, ndvduals of slghtly hgher types wll now want to mmc those of type w. After all, at the postulated optmum, where the constrants emboded n expresson (4) were bndng, such ndvduals were ndfferent. But now, by stpulaton, mmckng yelds slghtly hgher utlty, so ndfference s broken. Accordngly, to restore the constrant, the controller s requred to rase utlty of ths potental mmcker. To do ths, the type s taxes must be reduced. But now, what about the next hgher type? That ndvdual, who was prevously ndfferent, now faces a better outcome from mmckng the type whose taxes were just reduced and thus must also have taxes reduced to average of these optmal type-specfc rates. (The same reasonng apples f some good s a lesure complement for some types of ndvduals but a substtute for others, whch n general s possble.) More precsely, the proper formulaton of the optmal control problem needs to reflect that the commodty tax rates are constraned to be unform across types: they may vary across goods, but that exhausts the avalable flexblty

10 restore ncentve compatblty. And so on up the lne. When the process s completed, all ndvduals whose type s above w wll have hgher utlty, a good thng. However, they all have hgher utlty because they pay lower taxes than before, a bad thng, the weght of whch s gven by 8, the shadow prce on the resource (revenue) constrant. Whch effect on socal welfare s greater? It s apparent that the latter, negatve effect domnates. After all, what has just been wrought s a redstrbuton from the treasury to the upper porton of the populaton. One way to make the revenue cost concrete s to suppose that t s pad for by cuttng the unform grant to the populaton that s, the ntercept of the nonlnear ncome tax schedule, T(0), whch s typcally negatve. (Snce 8 s the shadow prce of resources or revenue, at the optmum t represents the value of any margnal use of the revenue through adjustments to the avalable nstruments, ncludng the one just descrbed.) In addton to ths heurstc explanaton, the result can be derved formally. Indeed, ths has been done (mplctly or explctly) n dervatons of the optmal nonlnear ncome tax n the case n whch there s only one type of consumpton and thus no place for dfferental commodty taxaton. The maxmum prncple used n optmal control theory ncludes the condton that the rate of change of the costate varable equals the negatve of the partal dervatve of the Hamltonan wth respect to the state varable. In the present formulaton, ths means that d:(w)/dw =!MH/Mu(w). Analysts then ntegrate ths functon to nfnty, makng use of the Fundamental Theorem of Calculus and the transversalty condton (ndcatng that :(w) = 0 at the upper lmt of the dstrbuton of w, or as w approaches nfnty, as the case may be), to yeld an expresson for :(w), whch s typcally derved for the specal case of quas-lnear preferences. 10 See, for example, Salané (2003, pp ). Those expressons ndeed match the above ntutve explanaton. Accordngly, the sgn of the costate varable :(w) s negatve (except at the endponts of the dstrbuton, as just suggested, a detal that wll be gnored here). It follows, therefore, that we must nclude n our mnds a mnus sgn precedng the dervatve term on the rght sde of expresson (6) for the optmal commodty tax. Furthermore, as noted above, f the Hamltonan (5) were nstead formed wth a mnus sgn precedng the costate varable, the costate varable would then be postve (as usually magned), but n that case the addtonal mnus sgn would carry through and appear explctly n an analogue to expresson (6) More precsely, the ntegral s from the upper lmt to the stated w, so a mnus sgn s ntroduced. Quas-lnear preferences refer (n ths one-good case) to utlty equal to consumpton mnus a functon ndcatng the dsutlty of labor. 11 The result n Salané (2003) that was mentoned n the ntroducton stll has not been fully explaned. It s stated there that he starts wth the same formulaton as the others: one can verfy ths by comparng hs Hamltonan (p. 115) wth those n Atknson and Stgltz (1976, p. 67) and Myles (1995, p. 164) note that they all follow the present approach of usng a plus sgn before the costate varable although ths s less apparent snce they use the aforementoned substtuton usng expresson (4), so a mnus sgn appears (recall that ths mnus sgn s from the rght sde of the equals sgn n expresson (4) and not part of the formulaton of the Hamltonan). Furthermore, Salané s fnal result (p. 116) seems to be of dfferent sgn from the results n the other two wrtngs (ths can only be determned after some manpulaton, snce the left sde of Salané s analogue to expresson (6) s presented dfferently). It appears that the explanaton s a sgn error n Salané s dervaton on p. 115 n movng from hs Hamltonan to hs expresson (1). Accordngly, although the ntroducton dentfed Salané (2003) as the one text that s formally n support of the famlar ntuton, once the apparent sgn error s corrected hs text seems to become yet another voce aganst the ntuton based - 8 -

11 3. Concluson Ths artcle demonstrates that the common ntuton on the second-best optmalty of taxng lesure complements s correct despte contrary results n the formal lterature on the optmal taxaton of commodtes n the presence of an ncome tax. The formal dervaton and accompanyng dscusson of ntuton help to resolve the conflct n the lterature. Ths result s not only mportant conceptually but has growng polcy sgnfcance as researchers ncreasngly dentfy mportant categores of consumpton (such as food, gasolne, and alcohol; see the lterature cted n note 1) that may be sgnfcantly complementary wth lesure. on Corlett and Hague

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