Volume 38, Issue 2. Optimal capital income taxation in the case of private donations to public goods

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1 Volume 38, Issue 2 Optmal captal ncome taxaton n the case of prvate donatons to publc goods Shgeo Morta Osaka Unversty Takuya Obara Htotsubash Unversty Abstract In ths study, we nvestgate optmal nonlnear labor and captal ncome taxaton and subsdes for contrbuton goods n a dynamc settng. We show that when ndvduals can contrbute to a publc good--even f addtve and separable preference between consumpton and labor supply s assumed and ndvduals dffer only n earnng ablty--margnal captal ncome tax rate for low-ncome earners s not zero, ndcatng that the Atknson-Stgltz theorem does not hold. We thank our supervsors Motohro Sato and Yukhro Nshmura for ther nvaluable comments, suggestons, and support. We also thank Yoshhro Takamatsu for agreeng to be a dscussant for the presentaton of our paper and the partcpants at the Japanese Economc Assocaton, June 19, Fnancal support from Htotsubash Unversty s gratefully acknowledged. Ctaton: Shgeo Morta and Takuya Obara, 2018) ''Optmal captal ncome taxaton n the case of prvate donatons to publc goods'', Economcs Bulletn, Volume 38, Issue 2, pages Contact: Shgeo Morta - ngp024ms@student.econ.osaka-u.ac.jp, Takuya Obara - ed142003@g.ht-u.ac.jp. Submtted: June 05, Publshed: Aprl 30, 2018.

2 1. Introducton Optmal tax theory plays an mportant role n desgnng ncome redstrbuton polces and mplementng publc projects. For several years, researchers have explored aspects of the feld: n partcular, economsts have been concerned wth the queston, Should captal ncome be taxed? Ths queston arses from the fact that the government can renforce redstrbutve polces by levyng taxes on savngs; however, ths s a form of double taxaton. Although the lterature has dscussed whether taxaton on captal ncome s justfed, t remans an ongong research ssue. Ths study ams to nvestgate the desrablty of captal ncome tax from the vewpont of economc behavors, specfcally when ndvduals can contrbute to publc goods. The motvaton stems from several emprcal studes whch fnd that a hgh tax rate on captal gans generally leads to taxpayers choosng chartable gvng as strateges to avod recognzng taxable gans. Indeed, Hood et al. 1977) fnd that Canada s 1971 Tax Reform, whch ntroduced a 50% captal gan tax, brought about a decrease n ndvdual chartable donatons. More recently, Auten et al. 2002) estmate the prce elastcty of donatons, where prce s a weghted average of the prce of gvng cash and apprecated propertes. Ther estmatons show that the value of the prce elastcty of donaton s negatve. These fndngs mply that ndvduals account for captal ncome tax when donatng toward publc goods. These suggest that ndvduals chartable gvng cannot be dscussed ndependently of captal ncome taxaton. Therefore, our study takes a step toward theoretcally clarfyng how captal ncome tax schemes should be desgned when ndvduals can contrbute to a publc good. Our analyss comprses a dynamc settng n whch ndvduals lve for two perods. We assume that n the frst perod, ndvduals can spend a part of ther savngs on donatons to a charty. Conceptually, we regard the prvate donaton to the publc good as the chartable gvng and utlze the framework of the optmal taxaton n the presence of a publc good. Ths setup s n lne wth Andreon 1988), Saez 2004), and Damond2006). For smplcty, there are two types of ndvduals: hgh- and low-sklled ndvduals. The government desgns three types of tax schedules: nonlnear taxes on labor and captal ncome and nonlnear subsdes for contrbutons to a publc good. We demonstrate that although a utlty functon s represented by the preference that prvate goods are addtvely separable from lesure, the margnal tax on captal s zero for the hgh sklled but not low sklled when prvate contrbutons are made to publc goods. The amount of donaton to a publc good dffers between hgh- and low-sklled ndvduals, whch affects the margnal rate of substtuton between consumpton n the frst and second perod. If a hgh-sklled ndvdual s valuaton of future consumpton s hgher than a low-sklled ndvdual s one, the dstorton on savngs behavor for the latter relaxes the self-selecton constrant for the former. Thus, the relatonshp between prvate consumpton and donaton to a publc good plays an mportant role n characterzng the optmal tax rates for margnal captal ncome.

3 An mportant contrbuton to the lterature on captal ncome taxaton s Ordover and Phelps 1979). They examne optmal nonlnear taxaton on ncome and savngs n an overlappng generatons model n the case of unobservable earnngs ablty. Ther man concluson states that f preferences are weakly separable between prvate goods and lesure, taxes on savngs are redundant. Ths s consstent wth the Atknson and Stgltz 1976) theorem. On the other hand, Saez 2002) nvestgates condtons necessary to obtan the Atknson Stgltz theorem and shows that f ndvduals have heterogeneous taste n prvate consumpton, the results are volated, even f the utlty functon s weakly separable between prvate goods and lesure. The present study s closely related to the model explanng the desrablty of captal ncome taxes on the bass of heterogeneous tastes for goods between hgh- and low-ncome earners, whch stems from Saez 2002). The extant lterature treats dfferentaton n taste based on ntal endowments and dscount rates as an assumpton Boadway et al. 2000), Cremer et al. 2001), Damond and Spnnewjn 2011)). In contrast, we show that taste dfferentaton can also result from ndvduals behavor wthout explctly assumng addtonal characterstcs. Thus, we present the desrablty of captal ncome taxes by establshng the theoretcal foundaton that taste dfferentaton occurs. To the best of our knowledge, ths study provdes new evdence justfyng captal taxaton snce prvate donatons have not been consdered n the context of ndvdual behavors n captal ncome taxaton theory. In our model, the government offers subsdes for contrbutons to publc goods. Pror studes have nvestgated optmal tax polcy assumng the presence of chartable gvng Andreon 1988), Saez 2004)). The study most closely related n terms of tax treatment of prvate donatons s Damond 2006), who shows that the welfaremprovng effect s acheved by ntroducng a subsdy on prvate donatons toward a publc good through nonlnear ncome taxes on labor. However, Damond 2006) adopts a statc model that does not allow the government to mpose ncome taxes on captal and attempts to derve an optmal tax treatment formula. We extend the Damond model as a two-perod model to nvestgate the desrablty of captal ncome taxes. The remander of ths paper s organzed as follows. Secton 2 descrbes the framework of the basc model. Secton 3 characterzes optmal tax formulas. Secton 4 provdes addtonal analyses. Secton 5 concludes the paper. 2. Model 2.1 Envronment We consder an economy n whch ndvduals lve for two perods and work only n the frst. There are two types of ndvduals: low-sklled ndvduals whose earnng wage rate s w 1 and hgh-sklled ones wth earnng wage rate w 2, where w 2 > w 1. Ther before-tax ncome s y w l, where l denotes the labor supply of type ndvduals.

4 We consder a fnte number of ndvduals as n Damond 2006). The number of type ndvduals s defned by π, whch s a natural number greater than two and for now, s nvarant. 1 The utlty functon of type ndvduals s U c,x,g,l ) = uc,x,g) vl ) 1) where c denotes consumpton of a prvate good n the frst perod, x s consumpton n the second perod, and G s the amount of publc good. Indvduals can contrbute to a publc good; then, the aggregate amount of publc good s G = π g +g G 2) where g denotes type s prvate donatons to the publc good and g G the publc donaton to the publc good. Contrbutons by a partcular economc agent mprove the servce and then the amount of publc good s an argument n the utlty functon of ndvduals. 2 The sub-utlty functon, u ), s strctly ncreasng, concave, and twce dfferentable and satsfes the Inada condton and v ) s strctly ncreasng, convex, and twce dfferentable. Let s denote savngs of type ndvduals and r be the nterest rate. The budget constrants type ndvduals face can be wrtten as follows: c +s +g τg ) = y Ty ) 3) s ) Φs r) = x 4) where τg ) s a subsdy for prvate donatons by type ndvduals to a publc good, Ty ) s an ncome tax payment, and Φrs ) s the captal ncome tax payment, whch respectvely, are nonlnear functons of g, y, and rs. Indvduals choose c, x, s, g, andl tomaxmzetheutltyfunctonequaton1))subjecttotherbudgetconstrants equatons 3) and 4)). Combnng ths wth the frst-order condtons yelds u c c,x,g)+{) rφ rs )}u x c,x,g) = 0 5) 1 Usng a fnte number of ndvduals, Pketty 1993) and Hamlton and Slutsky 2007) show that the frst-best allocaton can be acheved f an ndvdual s tax schedule depends on the behavor of other ndvduals. Followng tradtonal optmal taxaton lterature, the present study restrcts an ndvdual s tax schedule to a functon of the value of hs/her labor ncome, captal ncome, and prvate donaton to a publc good. 2 We consder publc goods fnanced by not only ndvduals but also the government such as health, educaton, and socal servces. Accordng to Chartable Gvng Statstcs by Natonal Phlanthropc Trust, n the Unted States, ndvduals chartable gvng accounts for 71% of total gvng and majorty of donatons are made to relgous, educatonal, and healthcare organzatons. A donaton to relgous organzatons s a sutable example for the outcomes of Lemma 1 because the US government cannot contrbute to them.

5 where u c c,x,g) u denotes the margnal utlty of consumpton n the frst perod, c u x c,x,g) u s the margnal utlty of consumpton n the second perod, and x Φ rs ) dφ s the margnal captal ncome tax rate functon correspondng to returns drs of savngs rs. The producton sector utlzes labor and captal. Producton technology exhbts constant return to scale. Ths means that each unt of effectve labor w l s requred to produce one unt of prvate good and each unt of prvate good saved n the frst perod produces ) unts of a prvate good n the second perod Plannng problem The objectve of the government s represented by the followng utltaran socal welfare functon: W = π U c,x,g,l ) 6) The budget constrant for the government s π Ty )+) 1 π Φrs ) = g G + π τg ) 7) Followng Damond 2006), we assume that there s no response of government budget constrants to a devaton from ndvduals antcpated revealng strateges. Usng the budget constrants that ndvduals face, equaton 7) can be equvalently wrtten as y π c + x +g )π g G 0 8) The nformatonal assumptons are conventonal: the government can observe ndvduals donaton, labor ncome, and captal ncome, whle ther ablty s never observable. We focus on the case n whch the government attempts to redstrbute from type-2 to type-1 ndvduals. Ths means the followng ncentve compatblty constrant s bndng at the socal optmum: U 2 c 2,x 2,G, y2 w 2) U2 c 1 1 y1,x,ĝ, w2) 9) 3 Prttlä and Tuomala 2001) show that captal ncome taxaton s justfed when wages are endogenously determned and the relatve wage rate s affected by the amount of savngs. By contrast, we assume no general-equlbrum effects of nput prces. Ths s because we never obtan the novel effect even f we endogenze nput prces, that s, the optmal tax formula for captal ncome just nvolves the endogenous wage term proposed by Prttlä and Tuomala 2001). Thus, for smplcty, our model can be seen as the two-perod, partal equlbrum verson of Prttlä and Tuomala 2001) model. At the optmum, where only the government contrbutes to a publc good, our model s outcome s consstent wth that of ther model.

6 where Ĝ G g2 + g 1 denotes the aggregate level of a publc good acheved when type-2 ndvduals mmc. The socal plannng problem s to maxmze the socal welfare functon equaton 6)), subject to the equatons for a publc good equaton 2)), resource constrants equatons8)), and ncentve compatblty constrantsequaton9)). The Lagrangean correspondng to ths plannng problem can be formulated as follows: [ [ L = W +µ g π +g G G +γ y π c + x +g )π g G 10) +λ [U 2 c 2,x 2,G, y2 c 1 1 y1,x,ĝ, w 2) U2 w 2) where µ, γ, and λ are the Lagrange multplers. 3. Characterzng optmal captal taxaton Here, we present the key features of our model s outcomes. The results mply that the government should desgn taxes on captal ncome such that t supplements the tax treatment of prvate donatons to a publc good. 4 Let MRS cx u cc,x,g) u x c,x,g) and MRS ˆ cx u cc 1,x 1,Ĝ) u x c 1,x 1,Ĝ) denote the margnal rate of substtuton between prvate consumpton n the frst and second perod faced by type ndvduals and the correspondng margnal rate of substtuton that the mmcker faces. Combnng the optmalty condton regardng c and x yelds the optmal captal ncome tax rate for type ndvduals: Φ rs 1 ) = λu xc 1,x 1,Ĝ) ) MRS 1 rπ 1 cx MRS γ ˆ cx 11) Φ rs 2 ) = 0 12) The dervaton s presented n Appendx A. Equaton 11) mples that the devaton of the optmal tax rate on captal ncome from the Atknson Stgltz theorem depends on the term n the brackets on the rght-hand sde. These equatons gve the followng proposton: Proposton 1. 4 In the present paper, we omt to characterze the optmal tax treatment of prvate donatons for a publc good. Morta and Obara 2016), whch s the workng paper verson, derves the optmal tax treatment formula.

7 When a publc good has a more complementary substtutonary) relatonshp wth the consumpton good n the frst perod than n the second perod, even f ndvdual preferences can be separated between labor and consumpton, the margnal captal ncome tax rate s postve negatve) for type-1 ndvduals and zero for the type-2 ndvduals. When a publc good has no relatonshp wth both the consumpton good n the frst perod and n the second perod, the margnal captal ncome tax rate s zero for both types of ndvduals. The result of Proposton 1 s crucally related to the dfference between G and Ĝ. At the optmum, the level of a publc good s hgher when a type-2 ndvdual chooses a truth-tellng strategy than a mmckng-one, that s, G > Ĝ. As shown n the Appendx B, t s optmal that only type-2 ndvduals contrbute to the publc good, g 1 = 0, g 2 > 0, and g G = 0. 5 Ths suggests that nducng type-2 ndvduals to contrbute mproves ther level of socal welfare from the allocaton, where no one makes a prvate donaton to publc goods. Ths s consstent wth Damond 2006). At the optmum, the level of publc good s hgher when a type-2 ndvdual chooses a truth-tellng strategy than a mmckng one, that s, G > Ĝ. Assumng that the publc good has a stronger complementary relatonshp wth the prvate good n the frst perod than n the second, the ntertemporal margnal rate of substtuton for the mmcker s lower than the correspondng margnal rate of substtuton for the mmcked, that s, MRScx 1 > MRS ˆ cx. In other words, the mmcker values the consumpton n the second perod more than the mmcked type-1 ndvduals). Ths mples that dstortng the captal ncome of type-1 ndvduals downward hurts the mmcker more than the mmcked and thus relaxes the ncentve compatblty constrant. Therefore, the margnal captal ncome tax rate should be postve. Consequently, ndvduals behavor n terms of prvate donatons to a publc good creates an nformatonal advantage for the government. Note that ths outcome depends on the assumpton of fnte number of ndvduals. Ths s because when there s an nfnte populaton, the level of publc good does not change even f a type-2 ndvdual acts as a mmcker, that s, G = Ĝ. Ths means that captal ncome taxaton s redundant. On the other hand, equaton 12) shows that the government should not dstort type-2 ndvduals savng behavor, makng zero margnal captal ncome tax rate desrable. 4. Robustness So far, we have assumed that the amount of savngs are observable. In ths secton, 5 The ntuton s as follows: type-1 ndvduals do not have ther ncentve compatblty constrant tghtened by prvate donaton to a publc good by type-2 ndvduals and thus nducng type-2 ndvduals to donate to a publc good allows the government to reduce mmcker s utlty due to G > Ĝ, that s, t relaxes the bndng ncentve constrant.

8 we explore the robustness of our results by assumng that the amount of savngs s unobservable, that s, the government s not allowed to employ nonlnear captal ncome taxes, and show that our man concluson s robust as long as prvate donatons to a publc good are observable, otherwse t s ambguous. We also analyze the model wth the endogenous nput prces. 4.1 Lnear captal ncome taxaton and nonlnear subsdes for prvate donatons Frst, we examne a case n whch the government observes both prvate donatons to a publc good and labor ncome for each type, but s unable to observe captal ncome for each type. Therefore, the government can only levy lnear tax on savngs at rate t s. Followng the tradtonal lteratures n optmal ncome taxaton, we decompose the ndvdual s problem nto two stages. Frst, each ndvdual chooses the amount of labor supply and prvate donatons to a publc good, whch determnes dsposable ncome R y Ty ) g + τg ), gven nonlnear labor ncome taxes and subsdes for prvate donatons. Second, dsposable ncome s allocated nto prvate consumpton n the frst and the second perod. We suppose that ndvduals antcpate the outcome of the second stage at the frst stage. In the second stage, type ndvduals choose c and x to maxmze the sub-utlty functon uc,x,g) subject to ndvdual s budget constrant whch s gven by c +s = R 13) s = q s x 14) where q s 1/1 t s )). The frst-order condton s gven by u c c,x,g) u x c,x,g) = 1 q s 15) On ths occason, we defne the sub-ndrect utlty functon for type ndvduals as V Vq s,r,g) uc,x,g), where c cq s,r,g) denotes the optmal soluton wth respect to the consumpton n the frst perod and x xq s,r,g) the optmal soluton wth respect to the consumpton n the second perod. The government chooses q s, G, g 1, g 2, g G, R 1, R 2, y 1, and y 2 to maxmze the socal welfare subject to the government s budget constrant, and the ncentve compatblty constrant, whch s expressed by Vq s,r 2,G) v y2 w 2) Vq s,r 1,Ĝ) vy1 w2) 16)

9 where ˆV 1 Vq s,r,ĝ) ndcates the mmcker s sub-ndrect utlty. Here, we defne the optmal soluton for the mmcker wth respect to the consumpton n the frst and the second perod as ĉ cq s,r 1,Ĝ) and ˆx 1 xq s,r,ĝ), respectvely.6 The frst-order condtons are shown n Appendx C. By rearrangng these results, we can derve the optmal lnear captal ncome tax rate as follows: rt s q s = λ ˆV R 1 ˆx x 1) γ π x 17) Equaton 17) s consstent wth the formula for optmal lnear tax rate proposed by Edwards et al. 1994) and Nava et al. 1996). The devaton form the Atknson-Stgltz theorem reles on the numerator, that s, the dfference of the demand for the consumpton n the second perod between the mmcker and the person beng mmcked. Because only type-2 ndvduals contrbute to a publc good as wth nonlnear captal taxnstrumentssee, AppendxD),thats, wehaveg > Ĝ. Thus, theatknson-stgltz theorem s not vald. Therefore, even though the government cannot observe savngs for each type, weak-separablty of preferences between consumpton and labor supply s not suffcent condton to make captal ncome taxaton redundant. Note that when there s an nfnte populaton, we obtan G = Ĝ, and then t s = Lnear captal ncome taxaton and lnear subsdes for prvate donatons Next, ths sub-secton extends the model n whch the government cannot observe boththeamountofprvatedonatonstoapublcgoodandsavngsforeachtype. Therefore, the only nonlnear tax nstrument s labor ncome taxaton, and the government can only levy lnear subsdes on prvate donatons at rate t g and lnear tax on savngs at rate t s. We now turn to the analyss of ndvdual s behavor. In the second stage, type ndvduals choose c, x, and g to maxmze the sub-utlty functon uc,x,g) subject to ndvdual s budget constrant whch s gven by c +s +q g g = y Ty ) R 18) s = q s x 19) where q g 1 t g and q s 1/1 t s )). The frst-order condtons are gven by u c c,x,g +G ) u x c,x,g +G ) = 1 q s 20) 6 We omt the frst-order condtons wth respect to G, y 1, and y 2 snce we focus on the characterzaton of optmal lnear captal ncome tax rates.

10 u G c,x,g +G ) u c c,x,g +G ) = q g 21) where G π 1)g + π j g j + g G, j = 1,2. Here, let c = cq s,q g,r,g ), = 1,2, be the best response functon of the consumpton n the frst perod, x = xq s,q g,r,g ), = 1,2, the best response functon of the consumpton n the second perod, and g = gq s,q g,r,g ), = 1,2, the best response functon of prvate donatons to a publc good. We can defne the sub-ndrect utlty functon for type as V V q g,q s,r 1,R 2,g G ) uc,x,g +G ) where the superscrpt *) refers to the Nash equlbrum outcome. Under lnear tax polcy, the ncentve constrant preventng hgh-type ndvduals from mmckng low-type ones s expressed by: V 2 q g,q s,r 1,R 2,g G ) v y2 w 2) ˆVq g,q s,r 1,R 2,g G ) v y1 w2) 22) where ˆVq g,q s,r 1,R 2,g G ) ndcates the mmcker s sub-ndrect utlty. Here, we defne the best response functon for the mmcker wth respect to the consumpton n the frst perod as ĉ cq g,q s,r 1, G 2 ), the best response functon for the mmcker wth respect to the consumpton n the second perod as ˆx xq g,q s,r 1, G 2 ), and the best response functon for mmckers wth respect to the prvate donaton to a publc good as ĝ gq g,q s,r 1, G 2 ), where G 2 = π 1 g 1 +π 2 1) g 2 +g G and g s type s prvate donatons n the presence of the mmcker. Here, we defne Ĝ ĝ+ G 2 as the aggregate amount of publc good faced by the mmcker. Note that t s not necessarly that g1 = g 1 or g2 = g 2 holds snce these realze as a Nash equlbrum n contrast wth the case n whch the government can observe prvate donatons to a publc good and thus desgn the allocaton for each type. To sum up, the government chooses q s, q g, R 1, R 2, g G, y 1, and y 2 to maxmze the socal welfare subject to the government s budget constrant, and the ncentve compatblty constrant. 7 Solvng the plannng problem yelds: ) tg rt sq s = λ 1 γ 1 γ 1 j=1,2 π u G j=1,2 π u G u 2 G u 2 G G 2 + G 2 G + G R g + G R j g j G + G R g + G R j g j g R G 2g R 2 2 G 2 + G 2x R G 2x R 2 2 ) ) û G G 2 ) û G ) λû c + G 2 γ 1 ĝ ) g 1 ˆx x 1 R 1 g 1 + G 2 R 2 g 2 G 2 + G 2 R 1 x 1 + G 2 R 2 x 2 ) ) 23) 7 Followng the argument n footnote 6, we omt the frst-order condtons wth respect to g G, y 1, and y 2.

11 where 1 s the nverse matrx of whch denotes the 2 2 matrx as follows: π g + π g g R + j=1,2 π g g R j j π x + π x g R + j=1,2 π x g R j j π g + π g R x + j=1,2 π g R j x j π x + π x R x + j=1,2 π x R j x j When the government cannot observe prvate donatons to a publc good for each types, the optmal tax formula conssts of three terms. The frst and second terms reflect well known effects respectvely: the Pgouvan and non-pgouvan elements dscussed by Cremer et al. 1998). 8 The thrd term s the novel term, whch comes from the dfferent mpact of the response of the other ndvduals to the perturbaton n parameters to the mmcker and the hgh-sklled agent who does not behave as a mmcker. Ths s an addtonal nformaton for the government to relax the bndng ncentve constrant. The thrd term stems from the assumpton of a fnte populaton snce ndvduals prvate donatons can affect the total amount of a publc good. If there s an nfnte populaton, the term vanshes, whch s consstent wth Cremer et al. 1998). The condton to restore Atknson Stgltz theorem crucally depends on prvate donatons to a publc good among ndvduals. If g1 = g 1 = g2 = g 2, we have G 1 = G 2, and then g1 = ĝ and x 1 = ˆx. In addton, snce t causes G = Ĝ, the thrd term n the rght hand sde vanshes. Therefore, Atknson Stgltz theorem remans. However, n contrast wth the observablty of prvate donatons to a publc good, we cannot analytcally compare the amount of prvate donatons among ndvduals snce the government cannot drectly control ther prvate donatons. As the same wth the prevous sub-secton, f there s an nfnte populaton, no person donates to a publc good snce the prvate donaton does not affect the aggregate amount of publc good. Thus, Atknson Stgltz theorem s vald. 5. Concludng Remarks Ths study s largely relevant to debates on the desrablty of captal ncome taxes. Snce Ordover and Phelps 1979) semnal work, a large body of lterature has accumulated on whether captal ncome taxes are requred from the vewpont of heterogeneous ) 8 If the publc good s addtvely separable n the utlty functon, the margnal utlty of the publc good of a low-sklled ndvdual concdes wth that of a hgh-sklled ndvdual, that s, u G u 1 G = u2 G. In ths case, the Pgouvan term s smplfed as follows. ) tg rt sq s = u G π 1 +π 2 1 γ 0 λ γ u G G 2 u G G 2 ) λû c γ ĝ ) g 1 ˆx x 1 + G 2 R 1 g 1 + G 2 R 2 g 2 + G 2 R 1 x 1 + G 2 R 2 x 2 ) û G G 2 + G 2 R 1 g 1 + G 2 R 2 g 2 ) û G 2 G + G 2 R x G 2 R x 2 2 ) ) 24)

12 tastes n prvate consumptons, even though the utlty functon s weakly separable between prvate goods and lesure. For nstance, Boadway et al. 2000), Cremer et al. 2001), and Damond and Spnnewjn 2011) consder a multdmensonal heterogenety settng n whch ndvduals dffer n not only earnng abltes but also other characterstcs such as ntal endowments bequest or nhertance) and dscount rates, whch are assumptons. By contrast, ths study provdes addtonal economc ratonale for captal ncome taxes from the vewpont of economc behavor that s, n realty, ndvduals deduct chartable contrbutons. Under the standard optmal tax approach, we show that the government should desgn taxes on captal ncome to supplement ts redstrbuton polcy when ndvduals can contrbute to a publc good. Ths perssts even f the addtve and separable preference between consumpton and labor supply s satsfed and ndvduals dffer n only earnng abltes. The theoretcal contrbuton of ths paper s as follows. Although we show that Atknson Stgltz theorem breaks down as a result of heterogeneous preferences, as n the case of Saez 2002), we justfy captal ncome taxes by clarfyng the source of heterogenety on the bass of ndvdual behavor and not assumptons. It s worth notng that our justfcaton s based on the assumpton there s a fnte populaton. Prttlä and Tuomala 1997) examne the commodty taxaton on an externalty-generatng good and nonlnear taxaton on labor ncome under the condton of an nfnte populaton. They show that the optmal tax formula reflects the two types of terms, that s, the externalty nternalzng effect and the nfluence through the ncentve compatblty constrant. When the preference s assumed to be addtve and separable between consumpton and lesure, polcy outcomes n ther paper are consstent wth standard Pgouvan taxes. However, under the settng of a fnte populaton as our study, the correspondng tax formula ncludes the novel term, the nteracton between the Pgouvan term and the self-selecton term, whch s the rght hand sde of equaton 11). Ths s true even f the addtve and separable preference s assumed. Our paper derves a condton accordng to whch captal ncome of low-ncome earners should be taxed or not. Ths polcy mplcaton depends on the shape of utlty functon, n partcular, the sgns of 2 u and 2 u. Ths s an mportant ssue of c G x G emprcal study. References Andreon, J., Prvately provded publc goods n a large economy: The lmts of altrusm. Journal of Publc Economcs 35, Atknson, A. B., Stgltz, J. E., The desgn of tax structure: Drect versus ndrect taxaton. Journal of Publc Economcs 6, Auten, G. E., Seg, H., Clotfelter, C. T., Chartable gvng, ncome, and taxes: An analyss of panel data. Amercan Economc Revew 92,

13 Boadway, R., Marchand, M., Pesteau, P., Redstrbuton wth unobservable bequests: A case for taxng captal ncome. Scandnavan Journal of Economcs 102, Caputo, M. R., The envelope theorem and comparatve statcs of Nash equlbra. Games and Economc Behavor 13, Cremer, H., Gahvar, F., Ladoux, N., Externaltes and optmal taxaton. Journal of Publc Economcs 70, Cremer, H., Pesteau, P., Rochet, J.-C., Drect versus ndrect taxaton: The desgn of the tax structure revsted. Internatonal Economc Revew 42, Damond, P. A., Optmal tax treatment of prvate contrbutons for publc goods wth and wthout warm glow preferences. Journal of Publc Economcs 90, Damond, P. A., Spnnewjn, J., Captal ncome taxes wth heterogeneous dscount rates. Amercan Economc Journal: Economc Polcy 3, Edwards, J., Keen, M., Tuomala, M., Income tax, commodty taxes and publc good provson: A bref gude. FnanzArchv 51, Hamlton, J., Slutsky, S., Optmal nonlnear ncome taxaton wth a fnte populaton. Journal of Economc Theory 132, Hood, R. D., Martn, S. A., Osberg, L., Economc determnants of ndvduals chartable donatons n Canada. The Canadan Journal of Economcs 10, Morta, S., Obara, T., Optmal captal ncome taxaton n the case of prvate donatons to publc goods, Dscusson papers n Economcs and Busness, Graduate School of economcs and Osaka School of Internatonal Publc Polcy OSSIP), Osaka Unversty, No Nava, M., Schroyen, F., Marchand, M., Optmal fscal and publc expendture polcy n a two-class economy. Journal of Publc Economcs 61, Ordover, J. A., Phelps, E. S., The concept of optmal taxaton n the overlappnggeneratons model of captal and wealth. Journal of Publc Economcs 12, Pketty, T., Implementaton of frst-best allocatons va generalzed tax schedules. Journal of Economc Theory 61, Prttlä, J., Tuomala, M., Income tax, commodty tax and envronmental polcy. Internatonal Tax and Publc Fnance 4,

14 Prttlä, J., Tuomala, M., On optmal non-lnear taxaton and publc good provson n an overlappng generatons economy. Journal of Publc Economcs 79, Saez, E., The desrablty of commodty taxaton under non-lnear ncome taxaton and heterogeneous tastes. Journal of Publc Economcs 83, Saez, E., The optmal treatment of tax expendtures. Journal of Publc Economcs 88, Appendx A The frst order condtons assocated wth c 1, x 1, c 2, and x 2 are L c 1 = π1 u c c 1,x 1,G) γπ 1 λu c c 1,x 1,Ĝ) = 0 L x 1 = π1 u x c 1,x 1,G) γπ1 ) λu xc 1,x 1,Ĝ) = 0 A.1) A.2) L c 2 = π2 u c c 2,x 2,G) γπ 2 +λu c c 2,x 2,G) = 0 L x 2 = π2 u x c 2,x 2,G) γπ2 ) +λu xc 2,x 2,G) = 0 A.3) A.4) Combnng equaton A.1) wth equaton A.2) yelds: π 1 {u c c 1,x 1,G) )u x c 1,x 1,G)} = λ{u c c 1,x 1,Ĝ) )u xc 1,x 1,Ĝ)}A.5) Combnng equaton 5) wth equaton A.5) yelds: π 1 u x c 1,x 1,G) λu x c 1,x 1,Ĝ))rΦ rs 1 ) = λu x c 1,x 1,Ĝ)u cc 1,x 1,G) u x c 1,x 1,G) u cc 1,x 1,Ĝ) u x c 1,x 1,Ĝ)) A.6) SubsttutngequatonA.2)ntothetermnthebracketsofthelefthandsde, weobtan equaton 11). Smlarly, combnng equaton A.3) wth equaton A.4) yelds: π 2 {u c c 2,x 2,G) )u x c 2,x 2,G)} = λ{u c c 2,x 2,G) )u x c 2,x 2,G)}A.7) Ths can be rewrtten as follows: π 2 +λ)u x c 2,x 2,G)rΦ rs 2 ) = 0 A.8)

15 EquatonA.3)mplesthatπ 2 +λspostve. Then, equatona.8)mplesthatφ rs 2 ) s zero. Appendx B Dfferentatng L wth respect to g G, g 1, and g 2 mples L = γ +µ g B.1) G L g 1 = γπ1 λu G c 1,x 1,Ĝ)+µπ1 B.2) L g 2 = γπ2 +λu G c 1,x 1,Ĝ)+µπ2 B.3) If equaton B.1) s equal to zero, equaton B.3) s as follows: L g 2 = λu Gc 1,x 1,Ĝ) > 0 B.4) In ths case, the optmal soluton does not exst because of dvergng. Therefore, at the optmum, we must have L < 0 and g G = 0 to satsfy Kuhn-Tucker condtons. g G Gven ths condton, from equaton B.2), no contrbuton to a publc good of type 1 ndvduals s optmal, that s, g 1 = 0. On the other hand, the prvate donaton to a publc good of type 2 ndvduals s not zero because the second term n equaton B.3) s suffcently larger than the sum of the frst and thrd term by the Inada condton when g 2 s close to zero gven g 1 = g G = 0. Therefore, g 2 s postve. In addton, g 2 s an nteror soluton. As g 2 L s close to nfnty, converges to γπ 2 +µπ 2 whch s g 2 negatve. Ths mples that g 2 must not be corner soluton at the optmum. Appendx C The correspondng Lagrangan s formulated as follows: [ L = W + µ g π +g G G [ + γ π y g R )+ 1 π q s ) 1)xq s,r,g) g G + [Vq λ s,r 2,G) v y2 w 2) Vq s,r 1,Ĝ)+vy1 w 2) C.1)

16 where γ, λ, and µ are the Lagrange multplers. The frst order condtons assocated wth q s, R 1, and R 2 are L = π V + γ π x + q ) s) 1 x + q λ V 2 ˆV ) 2 = 0 C.2) s L 1 R = π1 V 1 R 1 γπ1 + γπ 1q s) 1 L 2 R = π2 V 2 R 2 γπ2 + γπ 2q s) 1 We now combne these constrants by takng x 1 2 ˆV λ R1 x 2 R 1 = 0 2 V + λ R2 R 2 = 0 C.3) C.4) L + L R x C.5) From the Roy s dentty and the Slutsky decomposton, we can get the followng relatonshps: V = V R x C.6) x = x x R x C.7) where x ndcates the compensated demand functon of type ndvduals for the consumpton n the second perod. Usng Roy s dentty and Slutsky decomposton, ths gves γ q s) 1 Thus, we can obtan equaton 17). Appendx D π x ˆV + λ R 1ˆx x 1) = 0 Dfferentatng L wth respect to g G, g 1, and g 2 mples C.8) L = γ + µ g D.1) G L g = 1 γπ1 λu 1 G q s,r,ĝ)+ µπ1 D.2)

17 L g = 2 γπ2 + λu 1 G q s,r,ĝ)+ µπ2 D.3) Followng the same proof as n Appendx B, we conclude that g G = g 1 = 0 and g 2 > 0. Appendx E Usng Theorem 1 of Caputo 1996), we can get the followng relatonshps. V = φg +u G G E.1) V = φx +u G G V R = G φ+u G R E.2) E.3) V R = G j u G R, j j E.4) where let φ be the Lagrange multpler wth respect to ndvdual s budget constrant. The correspondng Lagrangan s formulated as follows: L = W + [V λ 2 q g,q s,r 1,R 2,g G ) v y2 w 2) ˆVq g,q s,r 1,R 2,g G )+v y1 w [ 2) + γ π y R )+ π q g 1)g + 1 E.5) π q s ) 1)x g G where γ and λ are the Lagrange multplers. The frst order condtons assocated wth q s, q g, R 1, and R 2 are as follows: L = π V [ + q λ V 2 ˆV s [ 1 ) + γ π )x +q s ) 1) x + E.6) π q g 1) g = 0 q s L = π V [ + γ [ + q λ V 2 g ˆV ) π g +q g 1) g + 1 π q s ) 1) x = 0 E.7)

18 L 1 2 R = π1 V 1 R +π2 V 1 + γ [ R + λ V 2 1 R 1 ˆV R 1 [ π 1 +π 1 q g 1) g 1 R 1 +π2 q g 1) g 2 R π1 q s ) 1) x 1 R π2 q s ) 1) x 2 R 1 [ V 2 L 1 2 R = π1 V 2 R +π2 V 2 R + λ 2 + γ R 2 ˆV R 2 [ π 2 +π 2 q g 1) g 2 R 2 +π1 q g 1) g 1 R π2 q s ) 1) x 2 R π1 q s ) 1) x 1 R 2 We now combne these constrants by takng L + L R g = 0 = 0 E.8) E.9) E.10) and These gve j=1,2 π [ V + V R g + V R jg j L + L R x [ + γq g 1) π g + q g + γ q [ s) 1 π x + π x q g R g + j=1,2 [ + λ V 2 + V 2 R 1g 1 + V 2 R 2g 2 ˆV ˆV R 1g 1 ˆV R 2g 2 and j=1,2 π [ V + V R g + V R jg j π x R jg j = 0 [ + γq g 1) π g + q s + γ q [ s) 1 π x + π x q s R x + j=1,2 [ + λ V 2 + V 2 R 1x 1 + V 2 R 2x 2 ˆV ˆV R 1x 1 ˆV R 2x 2 π x R jx j = 0 π g R g + j=1,2 π g R x + j=1,2 E.11) π g R jg j E.12) π g R jx j E.13)

19 Usng equaton from E.1) to E.4), E.12) and E.13) are transformed as [ G π u G + G R g + G R j g j j=1,2 [ + γq g 1) π g + π g q g R g + π g R jg j j=1,2 + γ q [ s) 1 π x + π x q g R g + π x R jg j j=1,2 [ ) + λ G u 2 2 G + G 2 R 1 g 1 + G 2 G 2 R 2 g 2 û G + G 2 R 1 g 1 + G ) 2 R 2 g 2 = 0 and [ G π u G + G R g + G R j g j j=1,2 [ + γq g 1) π g + π g q s R x + j=1,2 + γ q [ s) 1 π x + π x q s R x + [ ) + λ G u 2 2 G + G 2 R 1 x 1 + G 2 R 2 x 2 = 0 π g R jx j j=1,2 π x R jx j û G G 2 + G 2 R 1 x 1 + G 2 R 2 x 2 Usng matrx notaton, E.14) and E.15) can be rewrtten as ) ) tg rt sq s = 1 γ j=1,2 π u G G + G g R + G g R j j ) j=1,2 π u G λû ĝ ) c g 1 G + G g R + G g γ ˆx x 1 R j j λ γ u 2 G u 2 G G 2 + G 2g R G 2g R 2 2 G 2 + G 2x R G 2x R 2 2 ) û G G 2 ) + G 2 g R G 2 g R 2 2 ) û G 2 G + G 2 x R G 2 x R 2 2 +û c ĝ g1) E.14) +û c ˆx x 1) E.15) ) ) E.16)

20 Multplyng equaton E.16) by 1 yelds 23). If the publc good s addtvely separable n the utlty functon, the margnal utlty of the publc good concdes between a low-sklled and a hgh-sklled ndvdual. In ths case, equaton E.16) reduces to ) tg rt sq s = u G γ π1 +π 2 1)δ 1 λû ĝ ) c g 1 γ ˆx x 1 λ γ u G G 2 + G 2 R 1 g 1 + G 2 R 2 g 2 u G G 2 + G 2 R 1 x 1 + G 2 R 2 x 2 ) û G G 2 g R G 2 g R 2 2 ) û G 2 G + G 2 x R G 2 x R G 2 ) ) E.17) where δ 1 ndcates the frst column vector of. Multplyng equaton E.17) by 1 yelds 24).