Why Is Mobile Capital Taxed? Kangoh Lee. Department of Economics San Diego State University San Diego, CA

Transcription

1 Why Is Moble Captal Taxed? by Kangoh Lee Department of Economcs San Dego State Unversty San Dego, CA emal : klee@mal.sdsu.edu December 2011 I am grateful to Sam Bucovetsky and John Wlson for ther comments on an earler verson of ths paper, and I also thank two anonymous referees for ther helpful comments that mprove the paper sgnfcantly. Ths research was n part supported by a crtcal thnkng (assgned tme) grant from the College of Arts and Letters at San Dego State Unversty.

2 Abstract The government for a jursdcton has both captal and labor taxes at ts dsposal. It taxes moble captal to fnance the publc good despte the desre to attract captal and despte the tax ommoble labor f captal ncomes are dstrbuted more unequally than labor ncomes among the resdents of the jursdcton. The result extends to progressve taxes and to pure redstrbuton. JEL classfcaton: H71, H72, H73 Key words: moble captal; captal tax; labor tax

3 Why Is Moble Captal Taxed? 1 Introducton As economes have become ncreasngly more open due to market ntegraton such as the formaton of the European Unon and ts expanson, captal moves more freely among countres and competton for moble captal has ntensfed. Accordngly, researchers and polcymakers both have pad a good deal of attenton to the effects of market ntegraton. In partcular, they have expressed a concern that attempts of jursdctons to attract captal lower the taxes on captal and lead to underprovson of the publc goods (OECD 1998, 2000; Wlson and Wldasn 2004). In addton, tax competton models suggest that the tax on moble captal s dstortonary and that moble captal should not be taxed f mmoble factors can be taxed (Zodrow and Meszkowsk 1986; Wldasn 1989). Whle the concers a logcal consequence of the standard model of tax competton, t does not seem to be consstent wth realty, as the effectve captal tax rates for European Unon countres range from 11.1% n Ireland to 53.1% n Sweden (Haufler 2001) and the average effectve captal tax rate for OECD countres s 31.5% (Chen et al. 2007). The goal of the paper s to consder a more realstc envronment that can explan such captal taxes. In partcular, the analyss demonstrates the possblty that moble captal s taxed even though the tax on moble captal s dstortonary and mmoble factors can be taxed. A jursdcton conssts of heterogeneous resdents n terms of ther factor endowments, and the objectve of the jursdctos to maxmze a Benthamte socal welfare functon. Ths gves the jursdcton the ncentve to redstrbute from rch to poor resdents. The rch own more captal and more labor than the poor. In addton, the rch own relatvely more captal than labor whle the poor own relatvely more labor than captal, or captal ncomes are dstrbuted more unequally than labor ncomes, as observed n practce (Daz-Gmenez et al. 1997; Goodman et al. 1997). The government for a jursdcton has both captal and labor taxes at ts dsposal to fnance the publc good. Captal s freely moble between jursdctons whle labor s of fxed supply and mmoble. If mmoble labor s taxed, moble captal s not taxed n the standard model wth homogeneous resdents, because the tax on moble captal dstorts the allocaton of resources whle the tax on mmoble labor amounts to lump-sum taxaton. However, wth heterogeneous resdents, captal s 1

4 taxed. The reasos that ancrease n the captal tax ncreases the tax revenues and hence the publc good but decreases the captal ncomes. The jursdctonal welfare gan from the ncreased publc good depends on the dstrbuton of labor endowments as the labor tax s chosen to satsfy a Samuelson-type condton for an optmal level of the publc good n a jursdcton. The jursdctonal welfare cost from the decreased captal ncome depends on the dstrbuton of captal endowments. If the rch own relatvely more captal whle the poor own relatvely more labor, then the ncreased publc good, resultng from the ncrease n the captal tax, benefts the poor relatvely more whle the costs of the decreased captal ncomes are borne relatvely more by the rch. The jursdcton that desres to maxmze ts jursdctonal welfare thus taxes captal despte the well-known dstortonary effect of the tax on moble captal and despte the avalable tax ommoble labor. The result that moble captal s taxed n equlbrum despte the tax ommoble labor extends to progressve labor taxes and to lump-sum redstrbuton. Wth progressve labor taxes, a jursdcton taxes 100% of the rch s labor ncomes although t desres to tax them more, because the tax rate cannot exceed 100%. The taxes and hence the publc good level are thus lower than the maxmzaton of the jursdctonal welfare dctates, ncreasng the margnal beneft from the ncreased publc good. Progressve labor taxes thus ncrease the jursdctonal welfare gan from the ncreased publc good, caused by ancrease n the captal tax, makng t more lkely that captal s taxed than wth a sngle labor tax rate. If the tax revenues are used to provde the publc good and to redstrbute ncome n the form of a lump-sum transfer for all resdents of the jursdcton, t agan makes t more lkely that captal s taxed. Ths occurs because the rch pay more captal taxes to fnance the transfers whle everyone receves the same transfer. The next secton surveys a few strands of related lterature that study why moble captal s taxed n the presence of non-dstortonary taxes. Secton 3 descrbes a smple model of captal taxaton wth heterogeneous resdents n terms of factor endowments. Secton 4 analyzes the labor and captal taxes chosen by a jursdcton, and Secton 5 dscusses neffcency of captal taxaton. Secton 6 extends the analyss to progressve labor taxes and to lump-sum redstrbuton. The last secton concludes. 2 Related Lterature 2

5 The queston of why moble captal s taxed n the presence of non-dstortonary taxes s related to a number of strands of lterature. 1 Frst, even though labor or proft taxes are avalable, moble captal may be taxed due to labor market mperfectons. Rchter and Schneder (2001) consder a monopsony model of labor market, and demonstrate that t s benefcal to tax captal f proft ncome cannot be taxed although wage ncome s taxable. In a model of dual labor markets wth compettve and unonzed sectors, Koskela and Schöb (2005) show that taxng captal s effcent despte non-dstortonary proft taxes, as t reduces the compettve wage that serves as an outsde opton for unonzed workers, ncreasng employment and output n the unonzed sector. Ogawa et al. (2006) study tax competton wth unemployment, and argue that a jursdcton taxes captal even wth a head tax, because captal taxaton decreases the amount of captal and ncreases employment n the jursdcton when captal and employment are substtutes. Second, recognzng that free moblty of captal s not optmal when foregn-source ncomes cannot be taxed, jursdctons mpose captal controls. In such models wth captal controls, t may be effcent to tax moble captal. The reasos that captal controls by a jursdctoncrease, relatve to free moblty of captal, the amount of captal and hence the captal tax base n the jursdcton, enablng the jursdcton to lower the captal tax rate to acheve the same tax revenues (Razn and Sadka 1991). Huber (1997) consders transactons costs of makng nternatonal fnancal transactons, and shows that whether captal s taxed depends on the revenue requrement of a jursdcton. In partcular, when the requrement s hgh, jursdctons both tax captal and mpose captal controls. Thrd, when frms n a jursdcton are partly owned by foregners, the jursdcton has an ncentve to tax profts earned by foregners. Snce profts depend on captal, captal taxaton serves as anstrument to tax profts. Jursdctons thus tax captal to capture part of profts earned by foregners, as n Huznga and Nelsen (1997) and Sörensen (2004). In Burbdge et al. (2006), frms move n response to the dfference n proft taxes between jursdctons, and proft taxes are dstortonary. Nevertheless, the governments of jursdctons mpose a tax on profts n order to collect part of profts that accrue to foregners evef non-dstortonary labor taxes are avalable. Fourth, Huber (1999) consders an optmum labor and captal ncome tax problem n a tax 1 Most of the papers examne the possblty of taxng or subsdzng captal, lke n ths paper, but the dscusson below focuses on taxaton to conserve space. 3

6 competton model wth unobservable wages or abltes. Ancrease n the captal tax of a jursdcton decreases captal, alterng the wages of both hgh-sklled and low-sklled ndvduals and hence affectng the self-selecton constrant. Snce the socal-welfare maxmzng government wshes to weaken the self-selecton constrant, t taxes captal f the outflow of captal due to the captal tax lowers the rato of the hgh-sklled ndvdual s wage to the low-sklled ndvdual s wage whle t subsdzes captal otherwse. Fuest and Huber (2001) analyze a smlar model wth a lump-sum tax, and fnd that captal s subsdzed n order to weaken the self-selecton constrant. Ffth, as n ths paper, heterogeneous factor ownershp may be a reason for captal taxaton. In Braulke and Corneo (2004), ancrease n the captal tax of a country () lowers the wage ncomes of the country due to the exodus of captal, () ncreases the captal tax revenues, and () decreases the captal ncomes, net of the taxes, (ncome from captal owned by the country s ctzens). Effect () outweghs effect (), because a 1 dollar ncrease n the tax ncreases the tax revenues by the same 1 dollar per unt of captal, but t decreases the wage ncomes less than 1 dollar. 2 In addton, the more captal s located n the country or equvalently the more labor the country has, the more tax revenues t collects from the ncrease n the tax. 3 At the same tme, the loss of captal ncome by the country decreases f ther ctzens own less captal. As a result, the country benefts from captal taxaton and wll tax moble captal when the country has more labor and less captal relatve to other countres. In ths paper, countres are dentcal, and there s no world-wde redstrbuton between countres. Rather, redstrbuton occurs among the ctzens of a country. Borck (2003) consders a votng equlbrum on labor and captal taxes wth tax competton. If the medan voter owns less captal and more labor than the average of a jursdcton, he prefers to tax moble captal even though mmoble labor can be taxed. The reasos that for the medan voter, the ncome loss resultng from captal taxatos less than the average whle the gan from the ncreased publc good depends on the average tax base or the average amount of captal. That s, the medan voter of a jursdcton prefers to tax captal, because captal taxaton redstrbutes to hmself from the rest of the jursdcton. Ths medan voter s desre to redstrbute to hmself s analogous to that n Meltzer 2 The return to captal equals f (k) = r (nterest) + τ (captal tax), and total dfferentaton gves f (k)( k/ τ) = 1 + r/ τ < 1. The change n the wage ncomes equals (f kf )/ τ = kf ( k/ t) > k whle the change n the tax revenue equals k. 3 Due to perfectly moble captal and dentcal producton functons, the captal-labor rato s dentcal across countres, meanng that more captal of a country means a more labor of the country. 4

7 and Rchard (1981). Ths paper takes a jursdctonal welfare approach, and whether a jursdcton prefers to tax captal depends on the dstrbuton of factor endowments among the resdents of the jursdcton. As a result, the condton for captal taxaton ths paper dffers from that n Borck (2003), as wll be dscussed n Secton4-B. 4 3 The Model The economy conssts of q jursdctons, and jursdcto has resdents, = 1, 2,..., q. The resdents of jursdcto dffer only n ther factor endowments, captal and labor, where labor s measured n effectve unts. 5 Resdent h of jursdcto owns a fracton α h [0, 1] of the jursdcton s exogenous captal stock k, and a fracton β h [0, 1] of the jursdcton s exogenous total labor supply l, wth α h = 1, and β h = 1, for = 1, 2,..., q. (1) Assume α h > α h, and β h > β h, for h > h, h = 1, 2,..., 1. (2) The nequaltes say that resdent h of jursdcto owns more captal and more labor than resdent h, and that resdent h s rcher than resdent h. Ths assumptos much stronger than needed but smplfes the presentaton, as dscussed below. Nevertheless, ths assumpton does not seem unreasonable, because there appears to be a postve correlaton betweencome from captal and ncome from labor (resdent h earns more ncome from captal and more ncome from labor than resdent h does). That s, hgh wage earners can save and nvest more, leadng to more ncome from captal. Ths explans a postve correlaton. To see the other drecton that hgher ncome from captal means hgher ncome from labor, note that those famles wth hgher captal ncome can better afford educaton for ther chldren (Loury 1981; Banerjee and Newman 1993; Galor and Zera 1993), 6 and those chldren wth more educaton can earn hgher wages (Card and Krueger 1992; 4 Lee (2004) explores another reason for captal taxaton a model wth producton uncertanty. Ancrease n the captal tax drves captal out of the jursdcton and hence moderates the fluctuaton of the uncertan wages due to complementarty between labor and captal. As a result, the dstortonary tax on moble captal acts as nsurance and captal s taxed even though a lump-sum tax s avalable. Ths result s analogous to that n Eaton and Rosen (1980a, 1980b). 5 Labor endowments thus dffer across ndvduals. In partcular, ndvduals supply the same quantty of labor nelastcally, but they dffer n ther productvty so that the number of effectve unts of labor supply dffers, as n human captal theory. 6 Lqudty constrants that low ncome famles face lmt ther ablty to provde educaton for ther chldren, and those chldren wth less educaton earn lower wages (Card and Krueger 1992; Murnane et al. 1995). 5

8 Murnane et al. 1995). In addton to these arguments, usng a set of U. S. data from the 1992 Survey of Consumer Fnances and the and Panel Study of Income Dynamcs, Daz-Gmenez et al. (1997) fnds that there s a postve correlaton between wealth and labor earnngs n the Unted States. Employng captal k and labor l, each jursdcton produces a sngle output that serves as a numerare accordng to a constant-returns-to-scale technology, f (k, l ). The government for jursdcton mposes a source-based proportonal tax on the returns to captal (captal ncomes), t 1, and a proportonal tax on the returns to labor (labor ncomes), τ 1, employed ts jursdcton to fnance the publc good, z, and/or lump-sum transfers (redstrbuton) T. Captal s perfectly moble between jursdctons, and the employment of captal n each jursdcton, k, s endogenously determned by the tax polces of jursdctons, (t 1, t 2,..., t q ). The net returns to captal must then be equalzed, and (1 t )f (k ) = ρ, = 1, 2,..., q, (3) where ρ s the economy-wde net return to captal. The total captal supply to the economy s fxed at q =1 k, so that q q k = k. (4) =1 =1 Labor s assumed mmoble, and l = l, so l = l s omtted n the producton functon f (k ). The q equatons n (3), along wth (4), mplctly determne the q +1 endogenous varables, (k 1, k 2,..., k q, ρ), gven the tax polces of q jursdctons, (t 1, t 2,..., t q ). Total dfferentaton of the q + 1 condtons n (3) and (4) can show 7 k (.) = k (.) t j ρ = 1 f D (1 t )f f D(1 t )f j = < 0, D 1 D(1 t )f q j=1 1 (1 t j )f j f j (1 t j )f j 1 (1 t j )f j < 0, > 0, < 0, j,, j = 1, 2,..., q, (5) 7 Total dfferentaton of (1 t )f = ρ wth respect to t gves k / = (f + ρ/)/(1 t )f whle total dfferentaton of (1 t j)f j = ρ wth respect to t gves k j/ = ( ρ/)/(1 t j)f j, j. Total dfferentaton of (4) wth respect to t, along wth the expressons for k / and k j/, gves the desred results n (5). 6

9 where the arguments of the producton functons are omtted for smplcty. Intutvely, ancrease n t lowers the net return to captal employed n jursdcto, drvng captal out of jursdcton. Ancrease n t j drves captal from jursdcton j, ncreasng k. Ancrease n the tax of a jursdcton decreases the net return to captal. As ndcated n (5), the analyss assumes that a jursdcton s tax polcy affects the net return to captal, ρ/ < 0, so that the jursdcton can alter the captal ncomes of ts resdents and hence ts jursdctonal welfare. If ρ/ = 0, as s the case wth nfntely many jursdctons, then a jursdcton of course would have no ncentve to tax captal, as n Bucovetsky and Wlson (1991). Factor markets are compettve, and each factor earns ts margnal product. A unt of captal n jursdcto earns the gross return f (k ), and the net return ρ. A unt of labor earns the gross return [f (k ) k f (k )]/l, and the net return equals (1 τ ) tmes the gross return. Gven the factor endowments descrbed above, resdent h of jursdcto earns the ncome net of taxes, x h = α h k ρ + β h (1 τ )[f (k ) k f (k )], h = 1, 2,...,, and = 1, 2,..., q. (6) The frst term of the equaton shows the captal ncome net of taxes, and the remanng terms represent the net labor ncome. The tax revenues of jursdcto equal R = t k f + τ (f k f ). (7) The tax revenues are used to provde the publc good z. Assumng for smplcty that one unt of the prvate good can be transformed nto one unt of the publc good, z = R. (8) In (6) through (8), a lump-sum transfer T was not consdered, but Secton 6-B wll consder t. The resdents of jursdcto have the same preferences, represented by a concave utlty functon U (x h, z ). In studyng the characterstcs of the captal taxes that the governments for jursdctons choose, much of the tax competton lterature has focused on the case wth homogeneous resdents of a jursdcton, and the benevolent government s assumed to maxmze the utlty of a representatve resdent. In the current setup, a jursdcton conssts of heterogeneous resdents, and the government needs to aggregate the heterogeneous preferences when selectng the taxes. Ths paper assumes 7

10 that the benevolent government for a jursdcton maxmzes the sum of the utltes of resdents. In partcular, the government s assumed to maxmze a utltaran jursdctonal welfare functon, V U (x h, z ). 8 Snce k enters the utltes of the resdents of jursdcto, and snce k depends on the tax polces of all jursdctons, (t 1, t 2,..., t,..., t q ), the tax polces are jontly determned. A Nash equlbrum n the tax rates s consdered for the subsequent analyss. The maxmzaton of jursdctonal welfare s usually consdered a normatve approach. At the same tme, a normatve approach n the tax competton lterature typcally examnes the determnaton of captal taxes from the economy s (all jursdctons) perspectve, rather than from a sngle-jursdcton s perspectve. Indeed, Secton 5 consders effcency of tax competton, a normatve queston, and shows that tax competton results neffcency assocated wth the moblty of captal between jursdctons. For ths reason, the maxmzaton of jursdctonal welfare wthn a jursdctos called a postve approach that helps n aggregatng heterogeneous preferences. The jursdctonal welfare approach, taken ths paper, s consstent wth and comparable to the standard models of tax competton that focus odentcal resdents n a jursdcton, n the sense that there s no neffcency from a jursdcton s perspectve but there s from the economy s perspectve. As mentoned n the prevous secton, an alternatve postve approach s to consder majorty votng on taxes, and the dfference between two approaches wll be dscussed n Secton 4-B. Another postve approach s to assume that taxes are determned by lobbyng actvtes, and welfare maxmzaton here may be nterpreted as a closed-form soluton to a lobbyng game, such as the one n Grossman and and Helpman (1994). In partcular, Grossman and Helpman (1994) show that the equlbrum of the lobbyng game may concde wth the effcent soluton that maxmzes socal welfare. 9 For the analyss of tax competton to be meanngful, a Nash equlbrum n the tax rates must exst. However, t has been known to be notorously dffcult to establsh the exstence. The dffculty s that evef the utlty functon U (x h, z ) s concave n x h and z and the producton functon f (k ) s concave, the utlty functos not necessarly quas-concave n the tax rates due to 8 Secton 4-B consders other objectves. 9 Grossman and Helpman (1994) postulate that the government or a poltcan chooses a polcy to maxmze the sum of poltcal contrbutons and socal welfare, gven the contrbuton schedules determned by the lobbes. Snce the poltcan desres to receve poltcal contrbutons, the polcy n a poltcal equlbrum s not necessarly set to maxmze socal welfare. However, they show that the equlbrum polcy maxmzes socal welfare and hence s effcent when all voters are represented by lobbyng groups. Ths paper does not argue that the equlbrum of the lobbyng game s effcent. Rather, the effcent equlbrum makes the analyss consstent wth the tax competton lterature, as t makes the polcy effcent wthn a jursdcton. 8

11 the presence of k / n (5). A few attempts have been made to establsh the exstence of a Nash equlbrum (Bucovetsky 1991; Laussel and Le Breton 1998; Bayndr-Upmann and Zad 2005), but the condtons for the exstence n these studes appear to be demandng, ndcatng the dffculty of provng the exstence more generally. For these reasons, t has been standard to not dscuss the exstence ssue n the tax competton lterature, and ths paper assumes the exstence of a Nash equlbrum Basc Analyss 4-A Captal Taxes The government for jursdcto chooses τ and t, takng τ j and t j as gven, n order to maxmze V U (x h, z ). The frst-order condtons for anteror maxmum of V read as dv dτ = (f k f ) [Uh z Uhβ x h ] = 0 = Uh z = Uhβ x h, dv dt = [U x ρ h(α h k (1 τ )β h k f t k ) + U z h(k f + (t f + t k f τ k f ) k )] = 0, (9) where superscrpts x and z n the utlty functons denote partal dervatves such as U x h U (x h, z )/ x h, and the arguments of the utlty functons are omtted for smplcty. The frst condton of (9), n U h z = U h x β h, s a Samuelson-type condton for an optmal level of the publc good z, assocated wth the use of a labor tax and wth the Benthamte welfare objectve. As the queston concerns the possblty of taxng captal despte the taxes on the mmoble factor, t suffces to evaluate the sgn of dv /dt at t = 0, whch equals dv dt (t =0)= [U x ρ h(α h k (1 τ )β h k f k ) + U t t h(k z f τ k f = [Uh(α x ρ h k (1 τ )β h k f k ) + U t hβ x h (k f τ k f k )] = Uh[α x ρ h k + β h (k f k f k )] = Uh[ α x h k f 1 Df + β h k f 1 Df ] k )] 10 It s possble to establsh the exstence of a Nash equlbrum wthout relyng on the quas-concavty of the payoff functon (Vves 2001, Ch. 2). However, ths ssue s beyond the scope of ths paper. 9

12 = f Df Uh[β x h k α h k ]. (10) The second equalty uses the Samuelson-type condton, U z h = U x h β h, n (9). 11 The fourth equalty comes from (5). The sgn of (10) depends on the relatonshp between k and k, and on the relatonshp between α h and β h. To see the frst relatonshp, consder the case of dentcal resdents n the standard models wth α h = β h = 1/ and x h = x for all h. Eq. (10) then reduces to (f /Df )U x(k k ), whose sgs the same as the sgn of k k. The last expresson can be consdered a terms-of-trade effect. If k > k, (10) becomes postve and t > 0 wth denotng an equlbrum. Intutvely, n ths case, jursdcto employs captal more thats endowments by mportng captal. As a consequence, ancrease n t benefts jursdcto, as the ncrease n t lowers the net return to captal that jursdcto pays whemportng captal. Jursdcto thus taxes captal. If k < k, the opposte holds and jursdcto subsdzes captal. As the paper focuses on the effect of heterogenety n factor endowments of resdents on captal taxes, ths terms-of-trade effect wll be gnored to sharpen the result. To that end, jursdctons are assumed dentcal for the subsequent analyss. The analyss of dentcal jursdctons has been much of the focus of the lterature. 12 More mportantly, ths paper concerns the effects of the heterogenety of resdents n terms of factor endowments wthn a jursdcton, and the assumpton of dentcal jursdctons helps solatng the effects of the heterogenety of resdents n a clear manner. Wth the dentcal-jursdcton assumpton, (10) becomes dv dt (t =0)= f Df k Uh(β x h α h ), (11) where k k = k. Assume that ncome from captal s more unequally dstrbuted thancome from labor, so that there s a crtcal resdent, called resdent ĥ (1, ), and δ h α h β h < (=, >) 0 for h < (=, >) ĥ, h = 1, 2,...,. (12) The assumpton states that the poor wth h < ĥ own more labor than captal whle the rch wth h > ĥ own more captal than labor.13 Observe that assumpton (12) does not requre that δ h > δ h 11 Due to the Samuelson-type condton, the terms nvolvng the labor tax rate, τ β h k f ( k /), vansh, as n the thrd equalty. 12 For the analyss of non-dentcal jursdctons, see for example Wlson (1991) and Brueckner (2000). 13 Recallng that α and β are fractons, the poor own more labor than captal n terms of ther fractons of the total labor ncome and the total captal ncome of the jursdcton, but not necessarly n terms of dollar amounts of ther labor ncome and ther captal ncome. An analogous comment apples to the rch. 10

13 for h > h. That s, t s possble, for example, that δ 1 < δ 3 < δ 2 <... < δ ĥ = 0 <..., so that resdent 2, poorer than resdent 3, can own relatvely more captal than resdent 3 although both resdents are poor (ther δ s are negatve) and hence own more labor than captal. That s, all t says s that the poor wth h < h have more labor than captal and ther δ s are negatve, but ther rankngs n terms of δ do not matter among the poor. Lkewse, the rch wth h > h have more captal than labor and ther δ s are postve, but ther rankngs do not matter among the rch. 14 By contrast, f ncome from labor s more unequally dstrbuted thancome from captal, the nequaltes n (12) are reversed, so that δ h α h β h > (=, <) 0 for h < (=, >) ĥ, h = 1, 2,...,. (13) If (12) holds, the sgn of (11) s postve, because 15 Uh(β x h α h ) = Uh[ δ x h ] = ĥ Uh[ δ x h ] + Uh[ δ x h ] ĥ > U x [ δ ĥ h ] + U x [ δ ĥ h ] ĥ ĥ = U x [ δ ĥ h ] = 0. (14) The nequalty follows because [ δ h ] > 0 and U x h > U x ĥ for all h < ĥ, and because [ δ h] < 0 and U x h < U x ĥ for all h > ĥ. The last equalty uses (1), [ δ h] = (β h α h ) = 0. Intutvely, snce [ δ h ] > 0 for the poor whle [ δ h ] < 0 for the rch, and snce the poor s margnal utlty of prvate good consumpton, U x h, carres more weght than the rch s due to the concavty of U (.), the poor s postve values of U x h [ δ h] outwegh the rch s negatve values of U x h [ δ h]. 16 As a result, n U x h [ δ h] > 0, and t > 0. If (13) holds, the nequalty n (14) s reversed, and t < 0 and 14 The poor (the rch) are used as a relatve concept and not specfed throughout whenever ther dentty s clear. 15 ĥ s assumed anteger. If not, take the largest nteger smaller than ĥ (denoted h ) and the smallest nteger greater than ĥ (denoted h+), and wrte the summaton (14) as = h +. The same result as (14) h+ then holds. Thus, for expostonal convenence, the nteger assumptos mantaned throughout. 16 For (14) to hold, the assumpton (2) s not necessary. All that s needed s that the maxmum of x h for h [1, ĥ] s smaller than or equal to the mnmum of x h for h [ĥ, n], so that U h x for all h [1, ĥ] exceeds U h x for all h [ĥ, n]. That s, unlke n (2), resdent h does not have to be rcher than resdent h for all h > h, or x h does not have to ncrease n h. 11

14 captal s subsdzed. Ths result can be stated as: Proposton 1 Assume dentcal jursdctons. () If δ h < (=, >) 0 for h < (=, >) ĥ, t > 0 (the equlbrum captal tax rate s postve). () If δ h > (=, <) 0 for h < (=, >) ĥ, t < 0 (the equlbrum captal tax rate s negatve). The proposton stands n contrast wth the standard result n the lterature (Zodrow and Meszkowsk 1986; Bucovetsky and Wlson 1991) that f mmoble factors are taxed to fnance the publc goods, moble factors are not taxed. The reason for the standard result s that the taxaton of moble factors dstorts the allocaton of resources whle the taxaton of mmoble factors amounts to lump-sum taxaton. However, n ths paper, evef mmoble labor s taxed, moble captal may be taxed or subsdzed f the nequaltes n (12) or (13) hold. Ths dfference from the lterature arses from the dfferent factor-endowment dstrbuton. The lterature typcally consders homogeneous resdents n a jursdcton wth α h = β h = 1/ and δ h = 0 for all h, so that each resdent owns the same amount of captal and labor. Wth homogeneous resdents, (11) becomes zero and the equlbrum captal tax s zero. The ntuton can be ganed as follows. Ancrease n the captal tax drectly ncreases the captal tax revenues and hence the publc good by k f.17 The ncrease n t also drves out captal, decreasng labor tax revenues by τ k f ( k / ). Snce k f > τ k f ( k / ), 18 ancrease n t ncreases the tax revenues and hence the publc good by k f τ k f ( k / ). The margnal welfare gan to the jursdcton from ths ncreased publc good s U z h (k f τ k f ( k / )) = U x h β h (k f τ k f ( k / )) once the labor tax s chosen to satsfy the Samuelson-type condton. At the same tme, ancrease n the captal tax decreases prvate good consumpton x h n two ways. It decreases the net return to captal and hence captal ncome of resdent h by α h k ( ρ/ ). It also drves out captal and decreases the net labor ncomes of resdent h by β h (1 τ )k f ( k / ). The margnal welfare loss to the jursdcton from ths decreased ncomes s then U x h [α hk ( ρ/ ) β h (1 τ )k f ( k / )]. The change n jursdctonal welfare from the ncrease n the captal tax s thus U x h [α hk ( ρ/ ) + β h (k f k f ( k / ))], whch reduces to (10) and to (11) wth dentcal jursdctons. The change s thus proportonal to U x h (β h α h ). Ths change s negatve 17 Although the ncrease n the captal tax drves out captal and decreases the captal tax revenues by t (f + k f )( k /), ths dstortonary effect of the captal tax on the publc good vanshes at t = Wth dentcal jursdctons, τ k f ( k /) = τ k f (q 1)/Df = τ k f (q 1)/q < k f at t = 0 from (5). 12

15 for the rch whle t s postve for the poor f the rch own relatvely more captal whle the poor own relatvely more labor, as n (12). As the rch s margnal utlty counts less, the welfare loss for the rch s outweghed by the welfare gan for the poor and the net welfare gan from the ncrease n the captal tax s postve. As a consequence, ancrease n t, startng from t = 0, mproves the jursdctonal welfare, mplyng that t > 0 and captal s taxed n equlbrum. By contrast, f the rch own relatvely more labor whle the poor own relatvely more captal, as n (13), the opposte holds and t < 0. 4-B Dscusson As majorty votng s an alternatve way to determne the tax rates of a jursdcton, ths secton dscusses the dfference from the votng outcome. A decson-makng unt s andvdual, and voter or resdent h of jursdcto prefers to choose (τ, t ) to maxmze hs utlty U (x h, z ). The frstorder condtons would be the same as those n (9), except that U x h and U z h replace U x h and n U z h, respectvely. As n (9), ancrease n t startng from t = 0 decreases hs prvate ncome by α h k ( ρ/ ) (1 τ )β h k f ( k / ), but ncreases the publc good by k f τ k f ( k / ). Once the labor tax rate τ s chosen to maxmze hs utlty, the value of the publc good equals β h tmes that of the prvate good. The net gan from ancrease n t to resdent h becomes then (f k/df )U h x (β h α h ), equaton (11) wth U h x replaced by U h x. Ths equatos postve and resdent h prefers to tax captal f β h > α h. Interpretng the ncrease n the publc good by an ncrease n t as a redstrbuton to resdent h from the rest (the average resdent) of jursdcto, the condton means that the redstrbuton exceeds resdent h s lost ncome due to the captal tax. Intutvely, when resdent h owns less captal relatve to the jursdcton (the average resdent), the lost ncome s smaller whle the redstrbutos ndependent of hs captal ownershp, and resdent h prefers to tax captal (Borck 2003). 19 Ths desre of resdent h to redstrbute to hmself from the rest of the jursdctos common votng models wth publc goods, as n Meltzer and Rchard (1981). If a majorty of the resdents of jursdcto prefer to tax captal, then captal s taxed under majorty votng. 19 There s a large lterature on votng on captal taxaton wth heterogeneous captal ownershp (for example, Persson and Tabelln 1992; Kessler et al. 2002). However, ther focus dffers and do not consder why moble captal s taxed evef non-dstortonary taxes are avalable. In addton, evef the utlty functos quas-concave n the tax rates, votng equlbrum n general does not exst when there are multple tax nstruments, as n ths paper. Ths non-exstence may provde a justfcaton for the jursdctonal-welfare approach adopted n ths paper. 13

16 The desre of resdent h to redstrbute to hmself from the rest of the jursdcton does not apply to the jursdctonal welfare approach, taken ths paper. To see ths, note that the decsonmakng unt s the government of jursdcto and the approach takes nto account the utltes of all resdents of jursdcto. Snce the analyss rules out a terms-of-trade effect that occurs between jursdctons, captal taxaton of a jursdcton has no effect on the aggregate ncome of the jursdcton. Ths can be seen by settng Uh x = 1 for all h n (11), so that the aggregate ncome gan by ancrease n t, (f k/df ) β h = f k/df, equals the loss, (f k/df ) α h = f k/df at t = 0. That s, the redstrbuton beneft from ancrease n the publc good by ancrease n t just equals the jursdcton s lost ncomes, or a jursdcton cannot redstrbute to tself from tself. A jursdcton that attempts to maxmze ts aggregate ncome would then set t = 0 and would not tax or subsdze captal. For taxaton or subsdzaton of captal to occur, the value or utlty of ncome should dffer across ndvduals of the jursdcton. Wth dmnshng margnal utlty of ncome, the poor s ncome counts more than the rch s, and the jursdcton that wshes to maxmze the jursdctonal welfare desres to redstrbute from the rch to the poor. Thus, redstrbuton plays amportant role n the determnaton of captal taxes both n the votng approach and n the jursdctonal welfare approach, but the scope of redstrbuton dffers. Redstrbuton occurs between a resdent and the rest of the jursdcton the former approach, but between the rch and the poor of the jursdcton the latter approach. Analytcally, resdent h prefers to tax captal f β h > α h, but the jursdcton does f U x h β h > U x h α h. The above dscussomples that captal n jursdcto s taxed at a votng equlbrum f a majorty of resdents of the jursdcton own more labor than captal, regardless of who, the rch or poor, own more labor than captal. That s, all that matters s that the number of resdents wth β h > α h exceeds /2. By contrast, the government of jursdcto that ams to maxmze ts jursdctonal welfare taxes captal f condton (12) holds, so that the poor of the jursdcton own more labor than captal (or the rch own more captal than labor), regardless of how many resdents own more labor than captal. As a consequence of ths dfference, captal may be taxed at a votng equlbrum when the maxmzaton of jursdctonal welfare dctates that captal should be subsdzed, and vce-versa. Thus, n prncple, the condton for the taxaton of captal under majorty votng can be more general or restrctve than that wth maxmzaton of jursdctonal 14

17 welfare. However, under a plausble assumpton that the poor own more labor than captal, (12) s satsfed and captal s taxed wth maxmzaton of jursdctonal welfare, but captal s not necessarly taxed under majorty votng unless the poor wth β h > α h are a majorty. In ths regard, captal s more lkely to be taxed wth maxmzaton of jursdctonal welfare than under votng equlbrum. To see ths dfference through an example, suppose that β h = 1/ for all h and α h > α h, h > h. In ths case, δ h = α h (1/ ) < (>) 0 for the poor (rch), and (12) s satsfed. Captal s thus taxed wth maxmzaton of jursdctonal welfare. However, at a votng equlbrum, captal may not be taxed unless δ h < 0 for a majorty of resdents. An analogous dscusson apples to the dfference between the two approaches n the condtons under whch captal s subsdzed. The desre of jursdcto to redstrbute from the rch to the poor, arsng from the dmnshng margnal utlty of ncome, plays amportant role n the taxaton of captal. However, such desre does not necessarly lead to captal taxaton. That s, jursdctons tax captal, not smply because they want to redstrbute from the rch to the poor. To see ths pont, consder two specal cases. Frst, suppose that the frst nequalty of (2) becomes a weak nequalty, so that α h = 1/ for all h and β h > β h. Resdent h s thus stll rcher than resdent h, but captal s not taxed but subsdzed, because δ h = (1/ ) β h > (<) 0 for the poor (rch) and hence part () of the proposton apples to ths case. Intutvely, wth α h = 1/ for all h, everyone loses the same ncome from captal taxaton, but the poor lose more n utlty. In addton, snce the poor own less labor, and snce the beneft from captal taxaton (the ncreased publc good) s proportonal to labor ownershp, β h, the beneft s lower for the poor. As the poor s utltes count more, the jursdcton subsdzes captal. Second, suppose that (2) holds, but α h = β h or δ h = 0 for all h. A rcher resdent then owns more captal and more labor than a poorer resdent, but the government for jursdcto does not tax captal, because the captal tax has no advantage or dsadvantage over the labor tax. In fact, (11) becomes zero and t = 0 wth δ h = α h β h = 0 for all h, and nether part () nor part () of the proposton apples to ths case. Ths specal case shows that evef the rch own more captal than the poor, captal s not necessarly taxed. To sharpen the argument that redstrbutos nether necessary nor suffcent for the taxaton of moble captal, consder a dfferent objectve of the government for jursdcto, other than the utltaran socal-welfare functon wth a dmnshng margnal utlty of ncome. Suppose for 15

18 example that the government of jursdcto selects the tax rates (t, τ ) to maxmze the sum of the utltes of resdents n group G, for example, because polcymakers of jursdcto vew the group as representng the jursdcton. In addton, assume that the utlty functos wrtten as U (x h, z) = θx h + ω(z) wth θ denotng a postve constant, so that the margnal utlty of prvate good, θ, does not depend oncomes. Applyng the steps, analogous to those leadng to (11), t s straghtforward to verfy that (11) becomes d U h (x h, z ) dt (t =0)= f k Df Uh(β x h α h ) = f k h G Df θ h α h ). h G h G(β Thus, t > (=, <) 0 f h G β h > (=, <) h G α h, and jursdcto taxes (subsdzes) captal f the resdents n G own more (less) labor than captal. The decson of the jursdcton to tax or subsdze captal has no relaton to the desre to redstrbute, because the utlty functos lnear ncomes. Ths dscussollustrates that the desre for redstrbutos not a necessary condton for a jursdcton to tax or subsdze moble captal. 4-C Factor Ownershp Ths secton dscusses f condton (12) s satsfed n realty. Table 1 borrows from Green et al. (1990, Table 2) and Daves et al. (2007, Table 1), and shows the dstrbuton of earnngs and wealth across quntles for fve countres that have comparable data. 20 Earnngs n Table 1 represent wages and salares, and can be consdered labor ncomes. Wealth represents fnancal and real assets. Snce the returns to wealth can be consdered captal ncomes, the dstrbuton of wealth would approxmate that of captal ncomes. Translatng the data n the table to factor ownershp parameters n the model, the data for Australa n the frst panel of Table 1 mples β 1 =.082, β 2 =.154, β 3 =.19, β 4 =.233, β 5 =.341, α 1 = 0, α 2 =.04, α 3 =.12, α 4 =.22 α 5 =.62, 20 A large number of data sets for ncome dstrbutons are avalable. However, such data sets are not sutable for the purpose of ths paper, as most of the data do not nclude detaled dstrbutons of ncomes such as dstrbuton across decles or quntles, as n Table 1. Rather, they have nformaton about top 5% or top 10% or medancomes. An analgous comment apples to wealth data sets. Even Table 1, earnngs data and wealth data may not be drectly comparable, because most of the earnngs data are for the md 1980 s whle most of the wealth data are for the late 1990 s. However, ths dfference does not appear to be sgnfcant, because the dstrbutons of earnngs usually change lttle over 5 or 10 years. For example, the dstrbutons of earnngs changed less than 1% pont for each quntle between the early 1980 s and the md 1980 s or between the late 1970 s and the late 1980 s (Green et al. 1990, Table 6). 16

19 where subscrpts h denote each quntle ncome group, h = 1, 2, 3, 4, 5. The dfference between α and β s then δ 1 =.082 < 0, δ 2 =.114 < 0, δ 3 =.07 < 0, δ 4 =.013 < 0, δ 5 =.279 > 0. The nequaltes above satsfy the condton (12), as the rchest group, group 5, own more captal than labor whle the remanng groups own more labor than captal. It s easy to verfy that the same pattern of δs holds for other countres n the table. 21 In addton to the data n Table 1, a dstrbuton of ncome from captal s known to be n general more unequal than a labor ncome dstrbuton. For example, Goodman et al. (1997) and Daz-Gmenez et al. (1997) document ths known fact for the Unted Kngdom and the Unted States, respectvely, although they do not document detaled dstrbutons of ncomes. These examples llustrate another possblty of the nequaltes n (12). Taken together, part of () Proposton 1 s more lkely to apply than part (), and captal s more lkely to be taxed than subsdzed n equlbrum. Ths concluson accords well wth the observaton that moble captal s taxed n realty. For nstance, the effectve captal tax rates for European Unon countres and for OECD countres are typcally about 30% (Haufler 2001; Chen et al. 2007). 5 Effcency of Tax Competton The paper manly concerns a postve queston of whether jursdctons tax moble captal n the presence of non-dstortonary taxes. However, effcency of tax competton has been extensvely dscussed n the lterature, and ths secton consders the effcency ssue. As the analyss focuses on symmetrc jursdctons, the number of jursdctons has no qualtatve effect on the analyss and two symmetrc jursdctons, and j, are assumed. Gven that neffcency of tax competton stems from the moblty of captal, effcent tax rates for jursdcto are the rates that maxmze ts welfare V wth k / = 0. Gven symmetrc jursdctons, the effcent tax rates are the same for both jursdctons. The frst-order condtons for anteror maxmum of V wth respect to the effcent 21 For Italy, no wealth data s avalable for the thrd and fourth quntles, but δ 1 < 0, δ 2 < 0 and δ 5 > 0. Thus, the same pattern holds among those three groups for whch the data are avalable. 17

20 tax rates read as dv dτ = (f k f ) [Uh z Uhβ x h ] = 0 = Uh z Uhβ x h = 0, dv dt = [Uhα x h k ( f ) + Uhk z f ] = 0 = Uh z Uhα x h = 0. (15) The condton for effcent τ s the same as that n (9), as the labor tax s non-dstortonary. The condton for effcent t assumes k / = 0 and hence ρ/ = ((1 t )f )/ = f, along wth k = k j = k = k. If α h = β h = 1/ for all h, as n the standard model wth dentcal resdents, the two condtons n (15) concde, and t does not matter whch factor s taxed. Intutvely, snce both captal and labor are fxed factors wthout moblty of captal, the tax on ether factor s essentally a lump-sum tax. However, wth heterogeneous factor ownershp, the welfare loss from taxaton dffers between the two taxes, and the two taxes are not equvalent. Assume that (12) holds, because an analogous analyss apples to the case where (13) holds. The nequaltes n (12) mply U x h β h > U x h α h, and the two condtons n (15) cannot hold smultaneously, and there are two possbltes. Frst, dv /dτ = 0, and dv /dt > 0. The effcent captal tax rate s then the maxmum rate (equal to one n the absence of any nsttutonal constrants), and t exceeds the equlbrum tax rate that satsfes (9). Second, dv /dt = 0, and dv /dτ < 0. The labor tax rate s set at a mnmum possble level however t s determned. The man concers the relatonshp between the effcent captal tax rate and the equlbrum captal tax rate. Evaluatng the second condton of (9) at the effcent captal tax rate, ˆt, that satsfes (15), n dv dt (t =ˆt ) = [U x ρ h(α h k (1 τ )β h k f k ) + U t hα x h (k f + (t f + t k f τ k f ) k )] = [Uh(α x ρ h k (1 τ )α h k f k ) + U t hα x h (k f + (t f + t k f τ k f ) k )] + Uh[(1 x τ )α h k f k (1 τ )β h k f k ] = Uhα x ρ h [k k f k + k f + (t f + t k f ) k ] + Uh(1 x τ )(α h β h )k f k 18

21 = Uhα x h t f k + Uh(1 x τ )(α h β h )k f k < 0. (16) The frst equalty uses U z h = U x h α h n (15). The next one s obtaned by subtractng and addng the same term, U h x (1 τ )α h k f k, to the frst lne. The thrd equalty comes from smplfcaton of the terms n the second lne. The last one uses ρ/ = f + (1 t )f ( k / ). The nequalty comes from the assumpton that (12) holds and U x h β h > U x h α h. The nequalty n (16) mples that the equlbrum captal tax rate s lower than the effcent level. Thus, regardless of whch of (15) holds as equalty, the equlbrum captal tax rate s lower than the effcent level. Ths neffcency of tax compettomples that tax coordnaton among jursdctons would mprove effcency evef each jursdcton chooses ts tax rates to maxmze ts welfare. To understand (16), consder frst the standard homogeneous factor ownershp case wth α h = β h = 1/. The term nvolvng the second summaton vanshes, and (16) reduces to the standard term, U x h α ht f k < 0. Ths standard negatve term reflects a decrease n the tax revenue of jursdcto, caused by ancrease n t and the resultng outflow of captal from jursdcto to j. The outflow of captal also ncreases the tax revenue of jursdcton j, cancelng out the decrease n the revenue of jursdcto wth symmetrc jursdctons. The socal planner that cares about the welfare of both jursdctons and j thus does not consder ths standard negatve term when choosng the effcent tax rate. As a result, the effcent tax rate exceeds the equlbrum tax rate. Alternatvely speakng, jursdcto does not consder the external beneft that jursdcton j enjoys due to the ncreased tax revenue, and sets ts tax rate at a level lower than the effcent level. 22 It s worth relatng ths standard case to Bucovetsky and Wlson (1991). In ther model, evef jursdctons tax only labor, tax compettos stll neffcent, because labor supply s endogenous. In partcular, ancrease n the labor tax rate of a jursdcton reduces the labor supply of the jursdcton. Ths decrease n the labor supply reduces the demand for captal by a jursdcton, ncreasng the supply of captal avalable to other jursdctons and beneftng other jursdctons. The jursdcton, however, does not consder ths external beneft on other jursdctons, resultng n too low a tax rate on labor and underprovson of publc goods. In ths paper, labor taxatos non-dstortonary, but captal taxaton creates aneffcency for the reason mentoned above. 22 A smple calculaton shows that dv j/dt, the external effect of ancrease n t on jursdcton j, concdes wth (16), except that subscrpts and j are swtched. Snce k j/ > 0, ths external effect s postve and tax competton creates a postve externalty. 19

22 Wth heterogeneous factor ownershp, the term nvolvng the second summaton (16) does not vansh. By the assumpton that (12) holds, U x h (α h β h ) < 0 and hence U x h (1 τ )(α h β h )k f k < 0. To understand ths negatve term, note frst that all terms n the frst lne of (16) nvolve α h, except (1 τ )β h k f k, the loss of after-tax labor ncomes due to the outflow of captal from jursdcto to j. Snce the poor own relatvely more labor and ther utltes count more, the outflow of captal addtonally decreases the welfare of jursdcto, relatve to the case where β h = α h for all h, because n that case there would be no addtonal decrease n the welfare. The socal planner agan does not consder ths negatve term, because the outflow of captal to jursdcton j ncreases labor ncomes of the resdents of jursdcton j and hence ncreases the welfare of jursdcton j, cancelng out the decrease n the welfare of jursdcto. Alternatvely speakng, jursdcto gnores ths external beneft that jursdcton j enjoys due to the ncreased welfare from heterogeneous factor ownershp, and sets ts tax rate at too low a level. As a result, heterogeneous factor ownershp exacerbates the undertaxaton of captal by jursdcto. Intutvely, jursdcto gnores not only the standard external beneft that the captal outflow to jursdcton j ncreases ts tax base, but also the addtonal external beneft that the welfare of jursdcton j ncreases due to the poor ownng relatvely more labor. 6 Extensons 6-A Progressve Labor Taxes In the prevous sectons, all resdents of jursdcto pad the same labor tax rate, τ. It mght be argued that f the rch can be taxed more heavly, then the publc good can be provded wthout taxng moble captal, as the government for a jursdcton smply reles more on the non-dstortonary labor tax. However, as the analyss below shows, captal s stll taxed. Suppose that two labor tax rates, τ L and τ H wth τ H > τ L, exst. 23 Labor ncomes up to M are taxed at the rate of τ L, and at the rate of τ H thereafter. Wthout loss of generalty, assume that there s a resdent h (1, ) such that β h (f k f ) < (=, >) M for h < (=, >) h. (17) 23 The tax rates should be wrtten as τ L and τ H, but jursdctonal subscrpts are omtted for smplcty. In addton, the analyss and results contnue to hold wth more than two labor tax rates, as shown an earler verson of ths paper. 20

23 For later use, let L {h h h } denote the set of the poor wth labor ncomes less than M, and smlarly for H {h h h }. Then, x h becomes { x h = α h k ρ + β h (1 τ L )(f k f ) f h L α h k ρ + β h (1 τ H )(f k f ) + (τ H τ L )M f h H, (18) and z = t k f +(τ L b L +τ H b H )(f k f ) (τ H τ L )M( h ) wth b L h L β h and b H h H β h, (19) because there are ( h ) rch resdents n H. Jursdcto selects (M, τ L, τ H, t ) to maxmze ts welfare V. Usng the frst-order condtons, t can be shown the Appendx that dv dt (t =0) > dv dt (t =0,τ L =τ H =τ)= kf Df Uh[ δ x h ]. (20) Notng that the last expresson of (20) s dentcal to the one n (11) wth the sngle labor tax, the followng result can be stated: Proposton 2 Assume dentcal jursdctons. Wth progressve labor taxes, t > 0 (the equlbrum captal tax rate s postve) f δ h < (=, >) 0 for h < (=, >) ĥ. The ntuton of the result s the same as that of Proposton 1, and the result that f (12) holds, captal s taxed n equlbrum extends to the progressve labor tax case. The only dfference s that f (13) holds, captal may be stll taxed here due to the nequalty n (20) whle captal was subsdzed n Proposton 1. The progressve taxes thus make t more lkely that captal s taxed n equlbrum. The key to ths result s that wth the progressve labor tax, U z h > U x h β h, and the Samuelson-type condton (9), U z h = U x h β h, does not hold. Intutvely, jursdcton desres to tax the labor ncomes of hgher ncome resdents, h H, to the maxmum extent possble. As t cannot tax more than 100% of the labor ncomes, the labor tax for those hgh ncome resdents s set at τ H = 1, and s lower than the maxmzaton of the jursdctonal welfare dctates (that s, the constrant on the tax rate, τ H 1, s bndng). The publc good s thus lower than the maxmzaton of the jursdctonal welfare dctates, and U z h > U x h β h. To relate ths to the captal tax, recall from the dscusson of Proposton 1 that the jursdctonal net welfare gan from captal taxatos proportonal to U x h β h U x h α h once the labor tax s chosen to satsfy 21