Growth, Income Inequality, and Fiscal Policy: What are the Relevant Tradeoffs?* Cecilia García-Peñalosa GREQAM and CNRS

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1 Groth, Income Inequalty, and Fscal Polcy: What are the Relevant Tradeoffs?* Cecla García-Peñalosa GREQAM and CNRS Stephen J. Turnovsky Unversty of Washngton Revsed verson October 2005 Abstract: We develop an endogenous groth model th elastc labor supply, n hch agents dffer n ther ntal endoments of physcal captal. In ths context, the groth rate and the dstrbuton of ncome are jontly determned. We then examne the dstrbutonal mpact of dfferent ays of fnancng an nvestment subsdy. We fnd that polces amed at ncreasng the groth rate tend to result n a more unequal dstrbuton of pre-tax ncome, consstent th the postve correlaton beteen ncome nequalty and groth observed n the recent emprcal lterature. Hoever, there seems to be no conflct beteen effcency and equty f nequalty s measured n terms of the dstrbuton of elfare. JEL Classfcaton : E62, O17, O40. Keyords: endogenous groth, ncome nequalty, fscal polcy. * García-Peñalosa ould lke to acknoledge the fnancal support receved from the Insttut d Econome Publque (IDEP), Marselle. Turnovsky s research as supported n part by the Castor endoment at the Unversty of Washngton. The paper has benefted from semnar presentatons at the Central European Unversty, Erasmus Unversty, the Unversty of Calforna, Rversde, the Unversty of Kansas, the Unversty of Munch, and the Unversty of Oregon. The constructve comments of to anonymous referees are also gratefully acknoledged.

2 1. Introducton The last decade has seen a revval of nterest n the relatonshp beteen ncome nequalty and groth. Ths research has employed a range of theoretcal frameorks and yelded conflctng emprcal results. Whle earler evdence suggested a negatve tradeoff beteen groth and nequalty, more recent studes have tended to support a postve relatonshp; see e.g. Barro (2000), Forbes (2000), and Lundberg and Squre (2003). Vrtually all of ths debate has gnored the role of dstortonary taxes. But once ther presence s acknoledged, t becomes evdent that polcymakers may face to potental tradeoffs, n that groth-enhancng polces may have conflctng effects on the pre-tax and post-tax dstrbutons of ncome. Of these to measures, t ould seem that the latter s n fact the more relevant as a gude to polcy. Indeed, one can go even further. Snce presumably, one s ultmately nterested n the effect of fscal polcy on economc elfare, a further potental tradeoff one beteen groth and elfare nequalty tself naturally arses. In ths paper e analyze the effects of fscal polcy on these varous groth-nequalty tradeoffs. To do so e develop an endogenous groth model th elastc labor supply and agents ho dffer n ther ntal endoments of physcal captal. In ths frameork the equlbrum groth rate and the dstrbuton of ncome are jontly determned, the key mechansm generatng the latter beng the postve equlbrum relatonshp e derve beteen agents relatve ealth (captal) and ther relatve lesure. Ths relatonshp has a very smple ntuton. Wealther agents have a loer margnal utlty of ealth. They therefore choose to ork less and to enjoy more lesure, and gven ther relatve captal endoments, ths generates an equlbrum ncome dstrbuton. Ths role played by endogenous labor supply s analogous to ts role n other models of captal accumulaton and groth, here t provdes the crucal mechansm by hch demand shocks, such as government consumpton expendture, ll stmulate captal accumulaton. The key factor s the ealth effect and the mpact ths has on the labor-lesure choce. Ths mechansm s also central to emprcal models of labor supply based on ntertemporal optmzaton; see e.g. MaCurdy (1981). There s substantal emprcal evdence documentng ths negatve relatonshp beteen ealth and labor supply. Holtz-Eakn, Joulfaan, and Rosen (1993) fnd evdence to support the ve that large nhertances decrease labor partcpaton. Cheng and French (2000) and Coronado 1

3 and Perozek (2003) use data from the stock market boom of the 1990s to study the effects of ealth on labor supply and retrement, fndng a substantal negatve effect on labor partcpaton. Algan, Chéron, Harault, and Langot (2003) use French data to analyze the effect of ealth on labor market transtons, and fnd a sgnfcant ealth effect on the extensve margn of labor supply. Assgnng such a central role to the adjustment n labor supply relates our analyss to another recent body of lterature. The denng gap beteen orkng hours n the Unted States and Europe has recently sparked a debate about the causes and effects of dfferences n labor supply; see Prescott (2004) and Alesna et al. (2005). Ths lterature has largely focused on hether taxes have drven these dfferences, and on the mpact of labor supply on groth. Hoever, lttle attenton has been pad to the dstrbutonal mplcatons of an endogenous labor supply. Our analyss can therefore be veed as extendng ths dscusson to focus on ths mportant, but neglected, aspect. The central role of the labor-lesure tradeoff n our model, hereby polces that ncrease labor supply also tend to ncrease nequalty s consstent th the postve correlaton beteen average hours orked n a country and the Gn coeffcent of ncome reported by Alesna et al. (2005) for OECD economes. As s ell-knon, the AK model lassez-fare mples a sub-optmally lo equlbrum groth rate so that some form of stmulus to nvestment becomes desrable. We therefore ntroduce a drect nvestment subsdy and compare the groth and dstrbutonal consequences of fnancng ths subsdy by ether a tax on captal ncome, on labor ncome, or on consumpton. Changes n tax rates are shon to have a substantal effect on the supply of labor, n lne th recent emprcal evdence; see Carda, Kozhaya, and Ruge-Murca (2003), as ell as Prescott (2004). Our results hghlght the sharply contrastng effects of these three dfferent modes of fnance. To essental results emerge. Frst, e fnd that polces that enhance the groth rate are most frequently assocated th greater pre-tax ncome nequalty. Ths s because groth s fostered by polces that ncrease the return to captal, and snce captal s more unequally dstrbuted than s labor, hgher returns to captal translate nto greater ncome nequalty. The postve correlaton beteen groth and ncome nequalty, and the fact that both varables are jontly determned, are consstent th the recent emprcal fndngs of Barro (2000), Forbes (2000), and Lundberg and Squre (2003). Hoever, our analyss also ndcates that such polces tend to reduce 2

4 elfare nequalty, thus suggestng that (gross) ncome nequalty s a poor proxy for the assessment of the effects of polcy on the dstrbuton of elfare. Second, because some polces tend to have opposte effects on the pre-tax (gross) and post-tax gross (net) dstrbutons of ncome, t s possble to nduce faster groth n conjuncton th a more equal dstrbuton of dsposable ncome. Despte the fact that ncreased groth s compatble th loer elfare nequalty, no polcy domnates n both dmensons. Tradeoffs therefore stll exst among these three modes of fnance, and the polcy maker needs to egh these carefully n evaluatng the consequences for groth and dstrbuton. Thus, hle all three are groth enhancng, consumpton tax-fnancng s superor from a groth perspectve, captal ncome tax-fnancng s superor from the standpont of reducng elfare nequalty, hle age ncome tax-fnancng may actually exacerbate elfare nequalty. Overall, the analyss provdes support for the use of ether a tax on captal ncome or a tax on consumpton to fnance a subsdy on nvestment, n that both polces ncrease the groth rate and reduce nequalty n post-tax ncome and elfare. But an even more attractve polcy conssts of adoptng a consumpton tax together th an equal-n-magntude age subsdy to fnance the nvestment subsdy, snce ths does not dstort the labor-lesure choce. The paper contrbutes to the recent lterature on the relatonshp beteen ncome dstrbuton and groth. 1 It s close to Bertola (1993), ho also examnes ho polces drected at ncreasng the groth rate affect the dstrbuton of consumpton, although hs assumpton of a constant labor supply mples that the dstrbuton of ncome s ndependent of polcy choces. It s also related to Bénabou (2002), here the tradeoffs engendered by dfferent fscal polces are examned. Bénabou consders a model th rsky human captal nvestment, and compares drect ncome redstrbuton th redstrbutve educaton fnance. He fnds that the latter s preferable n terms of groth, but nferor from an nsurance pont of ve. We propose an alternatve scenaro, but share the concluson that even f t s possble to enhance both groth and equty, dfferent polces affect these to objectves to dfferent extents, and hence a careful analyss of polcy optons s necessary. Our approach has to man lmtatons. Frst, the assumpton that agents dffer only n ther 1 See, for example, Galor and Zera (1993), Alesna and Rodrk (1994), Persson and Tabelln (1994), and Aghon and Bolton (1997), as ell as the overve n Aghon, Carol, and García-Peñalosa (1999). 3

5 ntal endoments of captal coupled th an AK technology mples that there are no ncome dynamcs. Whle ths s restrctve, t has the compensatng advantage of enablng us to examne the dstrbutonal consequences of fscal polcy analytcally. 2 The other s that e are gnorng other mportant elements central to the groth-ncome nequalty relatonshp, most notably human captal and educaton. These aspects are emphaszed by Galor and Zera (1993) and Vaene and Zlcha (2003), among others. We choose to focus on a dfferent source of ncome dfferences, the role of the return to captal, hch has been largely gnored by the recent lterature. The argument that the behavor of captal returns s essental to understandng dstrbutonal dfferences has, hoever, been emphaszed by Atknson (2003) and s supported by recent emprcal evdence for the OECD (see Checch and García-Peñalosa, 2005). The paper s organzed as follos. Sectons 2 and 3 present the structure of the model and derve the macroeconomc equlbrum. Secton 4 employs the frameork to address the mpact of taxaton on both groth and nequalty. Secton 5 supplements our theoretcal analyss th some numercal smulatons, used to llustrate some of the dstrbutonal mplcatons of the varous polces. Secton 6 concludes, hle techncal detals are provded n the appendx. 2. The Model 2.1 Descrpton of the decentralzed economy Technology and factor payments Frms shall be ndexed by j. We assume that the representatve frm produces output n accordance th the Cobb-Douglas producton functon 3 Y = A( L K) K 0< α < 1 (1a) α 1 α j j j 2 Ths assumpton s common to a large part of the lterature on dstrbuton and groth, eg. Bertola (1993), Alesna and Rodrck (1994), Persson and Tabelln (1994). Hoever, n García-Peñalosa and Turnovsky (2005) e examne the smultaneous determnaton of groth and the dstrbuton of ncome hen there are dmnshng returns to captal, n hch case both the dstrbuton of ealth and ncome exhbt transtonal dynamcs. See also Casell and Ventura (2000) for a more general study of heterogenety and the dynamcs of dstrbuton n groth models. 3 We note that the use of the Cobb-Douglas producton functon, although convenent, s actually less restrctve than may appear. All of our analytcal results contnue to hold f e generalze the producton functon (1a) to Y = F( L K, K ) here F s homogeneous of degree one n the to arguments, 4 LK j and K j. j j j

6 here K j denotes the ndvdual frm s captal stock, L j denotes the ndvdual frm s employment of labor, K s the average stock of captal n the economy, so that L j K measures the effcency unts of labor employed by the frm. The producton functon exhbts constant returns to scale n the prvate factors -- labor and the prvate captal stock. All frms face dentcal producton condtons. Hence they ll all choose the same level of employment and captal stock. That s, K j = K and L j = L for all j, here L s the average economy-de level of employment. Furthermore, e assume that the aggregate labor market clears, so that L= 1 l, here l s the average lesure tme. The economy-de captal stock yelds an externalty such that n equlbrum the aggregate (average) producton functon s lnear n the aggregate captal stock, as n Romer (1986), namely α Y = AL K Ω ( L) K (1b) here Ω( L) AL α and Ω / L > 0. We assume that the age rate, ω, and the return to captal, r, are determned by ther respectve margnal physcal products. Dfferentatng the producton functon and gven that frms are dentcal, e derve equlbrum factor prces as a functon of lesure tme, 1 α 1 ω = αωl K = αa(1 l) K ( l) K (2a) r ( A l α = 1 α ) Ω = (1 α) (1 ) (2b) These expressons mply that the equlbrum return to captal s ndependent of the stock of captal hle the age rate s proportonal to the average stock of captal, and therefore gros th the economy. 4 In addton, r / l < 0 and / l > 0, reflectng the fact that more employment (less lesure) rases the productvty of captal but loers that of labor. Consumers There s a mass 1 of nfntely-lved agents n the economy. Consumers are ndexed by and 4 Intutvely, n a grong economy, th the labor supply fxed, the hgher ncome earned by labor s reflected n hgher returns, hereas th captal grong at the same rate as output, returns to captal reman constant. 5

7 are dentcal n all respects except for ther ntal endoments of captal, K 0. Snce the economy s grong, e are nterested n the share of ndvdual n the total stock of captal, k, defned as k K K. Relatve captal has a dstrbuton functon Gk ( ), mean k = 2 1, and varance σ k. All agents are endoed th a unt of tme that can be allocated ether to lesure, l or to ork, 1 l L. A typcal consumer maxmzes expected lfetme utlty, assumed to be a functon of both consumpton and the amount of lesure tme, n accordance th the soelastc utlty functon 1 η γ βt max ( C( t) l ) e dt, th < γ < 1, η > 0,1> γ(1 + η) (3) 0 γ here ε 1/(1 γ ) equals the ntertemporal elastcty of substtuton. 5 The preponderance of emprcal evdence suggests that ths s relatvely small, certanly ell belo unty, so that e shall assume γ < 0. The parameter η represents the elastcty of lesure n utlty. Ths maxmzaton s subject to the agent s captal accumulaton constrant (1 sk ) = (1 τ ) rk+ (1 τ )(1 l) K (1 + τ ) C (4) k c here s denotes a subsdy to nvestment n physcal captal, hle τ k, τ, and τ c denote the tax rates on captal ncome, labor ncome, and consumpton, respectvely. Wth the equlbrum age rate beng ted to the aggregate captal stock, e observe from (4) that the ndvdual s rate of captal accumulaton depends on the aggregate stock of captal, hch the ndvdual takes as gven. Government polcy The objectve of ths paper s to examne the varous trade-offs faced by the polcymaker. The macroeconomc equlbrum therefore needs to take explct account of the constrants to hch the government s subject. Gven the four polcy nstruments defned above, the government balances the publc budget at each nstant t n accordance th the constrant sk = τ (1 ) cc + τkrk + τ l K (5) here C denotes aggregate consumpton, and l s the average economy-de average lesure tme, so 5 The restrctons n (3) are requred to ensure the concavty of the utlty functon n ts to arguments. 6

8 that (1 lk ) denotes the aggregate age bll. 2.2 Consumer optmzaton The consumer s formal optmzaton problem s to choose her rate of consumpton, lesure, and rate of captal accumulaton to maxmze (3) subject to the accumulaton equaton (4). The correspondng frst-order condtons are C γ ηγ = τ λ (6a) 1 s 1 1+ c l ηcl γ ηγ 1 1 τ K = λ (6b) 1 s 1 k r τ λ = β 1 s λ (6c) here λ s agent s shado value of captal, together th the transversalty condton t lm λ Ke β = 0 (6d) t These optmalty condtons are standard. Together th the ndvdual s accumulaton equaton (4), they yeld the ndvdual savng and lesure decsons. In the Appendx e sho that the economy s alays on ts balanced groth path. 6 To key relatonshps that e establsh nclude 1 τ k rl () C C K K β 1 s = = = ψ = C C K K 1 γ η l l = l k 1+ η ( 1) (7) (8) The frst equaton gves the rate of consumpton groth of the ndvdual, hle the second yelds the ndvdual s lesure choce. Equaton (7) asserts that n equlbrum the rates of groth of consumpton and captal are equal and the same for all ndvduals, beng equal to the tax-adjusted return to captal less the rate of tme preference all multpled by the ntertemporal elastcty of 6 Ths s also the case n the representatve agent model; see Turnovsky (2000). 7

9 substtuton. Observe that the return to captal depends upon aggregate lesure, hch as e sho belo, s jontly determned n conjuncton th the groth rate by the rate of return equlbrum condton (11a) [.e. (7)] and the product market equlbrum condton (11b). The macroeconomc equlbrum, ncludng both the groth rate and aggregate labor supply (lesure), s therefore ndependent of the dstrbuton of ealth. Furthermore, the captal stock of all agents gros at the same rate, mplyng that at any pont n tme, the share of agent, k, remans equal to her ntal share k,0, say. That s, the relatve ealth poston of agents, k, s unchangng over tme. Equaton (8) represents the relatve labor supply functon and s the crucal mechansm that equates groth rates across ndvduals. In (A.10b) n the Appendx e sho that the transversalty condton (6d) mples η l >, (9) 1 + η so that (8) yelds a postve equlbrum relatonshp beteen relatve ealth and lesure, such that the relatve ealth poston of agent, s unchangng over tme. Ths relatonshp provdes the lnk beteen the agent s ntal relatve endoment of captal and the equlbrum dstrbuton of ncome. Wealther agents have a loer margnal utlty of ealth. They therefore choose to supply less labor and to buy more lesure. In effect, they compensate for ther larger captal endoment, and the hgher groth rate t ould support, by provdng less labor, thereby havng an exactly offsettng effect on the groth rate. Ths role that the elastcty of labor supply s playng n the determnaton of ncome dstrbuton s analogous to the role t plays n other smlar groth models. For example, government consumpton expendture ll stmulate captal accumulaton n the Ramsey model, and groth n the Romer model, f and only f labor s suppled elastcally. In both cases the underlyng responses are drven by ealth effects. 2.3 Macroeconomc equlbrum Wth the economy alays beng on ts balanced groth path, the key aggregate equlbrum relatonshps can be summarzed by the follong equatons: 8

10 Equlbrum groth rate 1 τ k r β 1 s ψ = 1 γ (10a) Aggregate consumpton-captal rato C K 1 τ = l η 1+ τ c (10b) Goods market equlbrum C ψ =Ω() l (10c) K Government budget constrant C τ kr+ τ(1 l) + τc = sψ (10d) K Recallng the defntons of r (l), (l), and Ω(l), and gven k, these equatons jontly determne the aggregate consumpton-captal rato, C K, the average lesure tme, l, the average (common) groth rate, ψ, and one of the fscal nstruments gven the other three polcy parameters. Gven l, (8) determnes the ndvdual lesure tme, l, hle the ndvdual consumpton-captal rato can be derved by dvdng (6a) by (6b) and expressed as C K 1 τ l = η 1+ τ k c (10b ) Substtutng (10b) nto (10c), and recallng (2a) and (2b), the macroeconomc equlbrum of the economy can be summarzed by the follong par of equatons that jontly determne the equlbrum mean groth rate, ψ, and average lesure tme, l: RR PP (1 α) Ω( l)(1 τk ) /(1 s) β ψ =, 1 γ (11a) α 1 τ l ψ =Ω() l 1. η 1+ τc 1 l (11b) 9

11 The frst equaton descrbes the relatonshp beteen ψ and l that ensures the equalty beteen the rsk-adjusted rate of return to captal and return to consumpton. The second descrbes the combnatons of the mean groth and lesure that ensure product market equlbrum holds. We shall focus our attenton on solutons that are not only vable, n the sense of satsfyng the transversalty condton, but also generate postve equlbrum groth. From (11b) and (9), the equlbrum soluton for l must therefore le thn the range: η(1 + τ c ) η > l > α(1 τ ) + η(1 + τ ) 1+ η c (12) 2.4 The lassez-fare economy Settng the tax rates and the subsdy to zero, the equlbrum mean groth rate and lesure n the lassez-fare economy are determned by the follong par of equatons: RR: (1 α) Ω( l) β ψ =, (11a ) 1 γ α l PP: ψ =Ω() l 1 η 1 l, (11b ) These RR and PP locuses are depcted n Fgure 1. Frst, note that equaton PP s alays decreasng n l, reflectng the fact that more lesure tme reduces output, thus ncreasng the consumpton-output rato and havng an adverse effect on the groth rate of captal. In addton, the RR curve s also decreasng n l. Intutvely, a hgher fracton of tme devoted to lesure reduces the productvty of captal, requrng a fall n the return to consumpton. Ths s obtaned f the groth of the margnal utlty of consumpton rses, that s, f the balanced groth rate falls. Both schedules are concave, and suffcent condtons for a unque equlbrum to exst, at pont Q, say, are β α γ + > 0 ; A 1 αl 1 α 1+ > η(1 l) 1 γ (13) hch are certanly met f γ 0, and hold under much eaker condtons as ell. 10

12 3. The Dstrbuton of Income and Welfare We no consder the relatve ncome of agent, havng captal stock K. Her gross ncome s Y = rk + K(1 l ), hle average economy-de ncome s Y = rk + K(1 l). Usng equaton (8) to substtute for the ndvual s labor supply, e can rte her relatve ncome, y Y Y, as y(, l k ) = k + α (1 k) k (1 k) (1 + η) Ω = + (1 + η)(1 l) (14) hch e may express more compactly as: α y(, l k) 1 = ρ()( l k 1), here ρ() l 1, (14 ) (1 + η )(1 l ) Equaton (14 ) emphaszes that the dstrbuton of ncome depends upon to factors, the ntal (unchangng) dstrbuton of captal, and the equlbrum allocaton of tme beteen labor and lesure, nsofar as ths determnes factor reards. The net effect of an ncrease n ntal ealth on the relatve ncome of agent s gven by ρ(l). As long as the lassez-fare equlbrum s one of postve groth, t s straghtforard to sho that 7 0 < ρ( l) < 1 (15) Thus relatve ncome s strctly ncreasng n k, ndcatng that although rcher ndvduals choose a loer supply of labor, ths effect s nsuffcently strong to offset the mpact of ther hgher captal ncome. Consequently, the standard devaton of ncome across the agents, σ y, hch provdes a convenent measure of ncome nequalty, s less than ther (unchangng) varablty of captal, σ k. To see ths more ntutvely, note that the relatve labor supply functon, equaton (8), mples that the standard devaton of labor supples can be expressed as = = l ( (1 + )) the term ( l η /(1 η) ) σ L σl η η σk. From (9), + les beteen 0 and 1 (1 + η), ndcatng that labor supples are less unequally dstrbuted than are captal endoments, thus reducng the varablty of ncome across agents 7 The fact that ρ(l) < 1 s mmedate from ts defnton n (14 ). It s straghtforard to sho that f the equlbrum s one of postve groth, (11b) suffces to ensure that ρ(l) > 0. We should also note that the polcy maker could set tax rates so drve ρ () l to zero, f he shes to ensure that all agents have the same pre-tax ncome. But ths ould requre a negatve equlbrum groth rate to offset the dfferental n the ntal captal endoments. 11

13 relatve to that of ther underlyng captal endoments. The standard devaton of relatve ncome, σ y, serves as an analytcally convenent measure of (gross) ncome nequalty. The DD locus n the loer panel of Fg. 1 llustrates the relatonshp beteen σ y and the standard devaton of captal endoments, σ k, namely DD σ y = ρ(l)σ k (16) Gven σ k, σ y s a decreasng and concave functon of aggregate lesure tme. Ths s because as lesure ncreases (and labor supply declnes) the age rate rses and the return to captal falls, compressng the range of ncome flos beteen the ealthy th large endoments of captal and the less ell endoed. Thus, havng determned the equlbrum allocaton of labor, denoted by Q, from the upper panels n Fg. 1, (16) determnes the correspondng unque equlbrum varablty of ncome across agents, denoted by the pont M. Snce taxes also have drect redstrbutve effects, e must dstngush beteen the before-tax and after-tax ncome dstrbutons. We therefore defne the agent s after-tax relatve ncome as N r(1 τ k) k + (1 τ )(1 l) N y (, l k, τk, τ) = 1 ρ (, l τ, τk)(1 k) r(1 τ ) + (1 τ )(1 l) k (17a) N here ρ summarzes the dstrbuton of after-tax (net) ncome and s related to the correspondng before-tax (gross) measure, ρ(l), by N ( τ τ k) ρ ( l, τ, τk) = ρ( l) + ( 1 ρ( l) )(1 α) α (1 τ ) + (1 α )(1 τ ) k (17b) th the standard devaton of after-tax ncome gven by 8 σ = ) (17c) N N y ρ ( l, τ, τ k σ k The dsperson of pre-tax ncome across agents exceeds post-tax dsperson f and only f τ k > τ. 9 8 Agan, e rule out 0 N ρ < as a perverse case mplyng a negatve relatonshp beteen ealth and after tax ncome. It can effectvely be ruled out for any plausble tax confguraton. 9 In affluent OECD countres, such as US and Canada, pre-tax ncome nequalty typcally exceeds post-tax ncome nequalty (as measured by the Gn coeffcents) by about 2-4 percentage ponts, reflectng the progressvty of the tax structure characterstc of such economes. 12

14 From (17a) and (17b) e see that the ncome tax rates, τ k and τ, both exert to effects on the after-tax ncome dstrbuton. Frst, by nfluencng the equlbrum supply of labor, l, they nfluence gross factor returns, and therefore the before-tax dstrbuton of ncome, as summarzed by ρ () l. In addton, they have drect redstrbutve effects, hch are summarzed by the second term on the rght hand sde of (17b): a hgher tax on captal ncome reduces ths component, hle a hgher tax on labor ncome rases t. From (14) and (17) e establsh the follong partal effects: N N ρ 1 ρ ρ l (1 ρ)(1 α)(1 τ) = τ k 1 ρ l τk α(1 τ) + (1 α)(1 τk) [ ] N N ρ 1 ρ ρ l (1 ρ)(1 α)(1 τk ) = + τ 1 ρ l τ α(1 τ) + (1 α)(1 τk) [ ] (18a) (18b) here the frst term n each of these expressons captures the ndrect effect of taxaton on net ncome nequalty, and the second term the drect effect. In contrast, an nvestment subsdy, s, or a consumpton tax, τ c, affect after-tax nequalty only ndrectly, through ther mpact on labor supply. Fnally, e compute ndvdual elfare. By defnton, ths equals the value of the ntertemporal utlty functon (3) evaluated along the equlbrum groth path. Thus, the optmzed level of utlty for an agent startng from an ntal stock of captal, K,0, can be expressed as η (( C / K) l ) 1 X ( K ) = γ β γψ K γ 0,0 γ (19a) The elfare of ndvdual relatve to that of the ndvdual th average ealth s then ( ) ( CK) ( ) C K l ηγ xk k l γ = γ ηγ = l l γ (1 + η ) γ (19b) here the second term s obtaned by substtutng for (10b) and (10b ) and usng equaton (8) yelds η 1 xk ( ) = 1+ 1 ( k 1) 1+ η l γ (1 + η ). (19c) Consder no to ndvduals havng relatve endoments k 2 > k 1. Indvdual 2 ll have a 13

15 hgher mean ncome. The transversalty condton (9) mples that f γ > 0, then ther relatve elfares satsfy x(k 2 ) > x(k 1 ) > 0, hle f γ < 0, x(k 1 ) > x(k 2 ) > 0. Hoever, n the latter case absolute elfare, as expressed by (19a) s negatve. Thus n ether case, the better endoed agent ll have the hgher absolute level of elfare. We can no compute a measure of elfare nequalty. A natural metrc for ths s obtaned by applyng the follong monotonc transformaton of relatve lfetme utlty, enablng us to express the relatve utlty of ndvdual as xk = uk = + l k here 1 γ(1 + η) ( ) ( ) 1 ϕ( )( 1) η 1 ϕ() l 1 1+ η l (20a) From (9), 0 < ϕ( l) < 1, and s an ncreasng, concave functon n l. Welfare nequalty, expressed n terms of equvalent unts of captal, can then be measured by the standard devaton of relatve utlty σ = ϕ( l) σ (20b) u k It s straghtforard to sho that n the absence of taxes, and assumng postve groth σ y > σu > σk (20c) so that elfare nequalty exceeds the underlyng ealth nequalty, but s less than (gross) ncome nequalty. Welfare nequalty s plotted as UU n the loer part of Fg Havng determned the equlbrum lesure at Q, (20b) yelds the correspondng degree of elfare nequalty, denoted by N, hch by vrtue of (20c) les above M. From ths fgure, t s evdent that any structural shft or polcy change that causes a change n the equlbrum lesure affects equlbrum gross ncome nequalty, M, and elfare nequalty, N, n conflctng ays. We are nterested n the mpact of taxes and subsdes on the follong three measures of nequalty: pre-tax ncome nequalty, post-tax ncome nequalty, and elfare nequalty. 11 Before examnng the mpact of specfc polces, e summarze ho fscal polcy ll affect these dstrbutonal measures th the follong proposton: 10 The transversalty condton (9) mples that the pont U lyng on the l axs les to the left of the equlbrum pont Q. 11 Note that consumpton nequalty, C / C 1 = kc / c 1 = (1 η /(1 + η )(1/ l ))( k 1) and s dentcal to that of elfare. 14

16 Proposton 1: Gven the ntal dstrbuton of captal across agents: () Fscal polcy nfluences the before-tax (gross) dstrbuton of ncome and the dstrbuton of elfare through ts effect on the labor supply. () Any fscal polcy that ncreases (decreases) the supply of labor ncreases (decreases) before-tax ncome nequalty and decreases (ncreases) elfare nequalty. () A labor ncome tax and a captal ncome tax both nfluence the after-tax (net) dstrbuton of captal n to ays; each has a drect redstrbutve mpact, n addton to an ndrect one through changes n the labor supply. A consumpton tax or nvestment subsdy has only the latter effects. Proposton 1 hghlghts ho, hen labor supply s endogenous, ncome nequalty s a poor measure of elfare nequalty. In fact, changes n pre-tax nequalty are nversely related th those of elfare nequalty. Is post-tax ncome nequalty therefore a better measure of elfare nequalty? The anser s not necessarly. As e have seen, fscal polcy has to effects on the dstrbuton of post-tax ncome, and net ncome nequalty need not move together th elfare nequalty. To see ths, note that elfare nequalty s a eghted average of consumpton nequalty and lesure nequalty, both of hch ll fall as the labor supply ncreases. A hgher labor supply (loer l) rases the return to captal, ncreasng the ncentves to save, and reducng the consumpton/captal rato. Ths effect s stronger for those th greater ealth holdngs, so that the dstrbuton of consumpton becomes less unequal. The dstrbuton of lesure also becomes less unequal, [as can be observed from σ l ], and thus the overall effect s to render the dstrbuton of elfare more equal. 4. Fscal Polcy and the Relatonshp beteen Inequalty and Groth A famlar feature of the Romer (1986) model s that by gnorng the externalty assocated th the aggregate captal stock, the decentralzed economy generates a sub-optmally lo groth rate. Ths suggests that by ncreasng the groth rate, an nvestment subsdy ll move the equlbrum closer to the socal optmum. Wth heterogeneous agents, to questons arse. Frst, ho to fnance ths subsdy f the government s concerned th both average elfare and elfare 15

17 nequalty. An nvestment subsdy rases the return to captal and ll thus favor those th large captal holdngs. Are there ays n hch ths reverse redstrbuton can be avoded? Second, e ant to kno the consequences of dfferent polces for the groth-nequalty relatonshp. 4.1 Fnancng an nvestment subsdy In ths secton e nvestgate these questons n some detal, by consderng the effect of fnancng an nvestment subsdy, s, usng one of the three dstortonary taxes. Subsdy to nvestment fnanced by a tax on captal ncome Suppose that the fscal authorty decdes to fnance the subsdy to nvestment by mposng a tax on captal ncome, alone. The mpact of fscal changes on the equlbrum groth rate and labor supply (lesure) can be llustrated graphcally n Fg. 1. Holdng all other taxes constant, an ncrease n the subsdy rate, s, shfts the RR schedule upards, movng the equlbrum Q to the left along the PP curve, and ncreasng the groth rate and reducng lesure. An ncrease n the tax on captal, τ k, has the opposte effect, shftng the RR schedule donards, so that the overall effect of ths mode of fnancng depends upon the sze of the captal ncome tax needed to fnance the subsdy. tax s: Settng τ = τ c = 0 n the government budget constrant, (10d), the necessary captal ncome s τ k = (1 α) 1 α l η 1 l (21a) From equaton (11a) e fnd that the drect effect of the subsdy domnates, so that the RR schedule shfts upards, movng the equlbrum Q to the left, ncreasng the groth rate and reducng lesure. As a result M and N move to the left along the DD and UU curves, respectvely, so that before-tax ncome nequalty, σ () l, ncreases, hle elfare nequalty, σ ( l ), declnes. Recallng the net y u dstrbuton of ncome as characterzed by (17b), e see that taxng captal ncome ensures that N ρ (, l ττ) < ρ() l. If the redstrbutve effect domnates, as our smulatons belo suggest may k plausbly occur, the after-tax nequalty actually declnes, relatve to the lassez-fare dstrbuton. 16

18 Subsdy to nvestment fnanced by a tax on age ncome Alternatvely, the subsdy may be fully fnanced by a age tax s τ = α ( 1 ( αη)( l (1 l) )) 1 ( s η )( l (1 l) ) ( ) (21b) In ths case, both the RR and PP schedules shft up, resultng n a hgher groth rate and greater or loer lesure, dependng on the relatve shfts. The ambguous mpact on lesure arses because the age tax tends to reduce the supply of labor, hle the hgher groth rate nduced by the subsdy tends to ncrease t. The ambguous response of labor complcates the mpact on the nequalty of ncome. Frst, the ncrease (decrease) n lesure tme ll reduce (ncrease) the dsperson of gross ncomes, as seen from (14 ). Hoever, the requred (postve) age tax mples taxng the factor that s more equally dstrbuted, and for any gven dstrbuton of gross ncomes ths rases the varablty of net ncomes (see (17b) above). If the polcy reduces lesure t ould then unambguously ncrease pre-tax and post-tax ncome nequalty. But hen lesure ncreases the to effects ork n opposte drectons: there ll be a reducton n the varablty of gross ncome, hle net ncome nequalty may ncrease or decrease as compared to the equlbrum thout taxes. 12 Subsdy to nvestment fnanced by a tax on consumpton As a thrd example, the subsdy may be fnanced by settng the consumpton tax equal to τ = s c ( 1 ( αη)( l (1 l) )) ( αη)( l (1 l) ) s ( ) (21c) N n hch case ρ (, l τ, τ ) = ρ() l. Agan both schedules shft upards, ncreasng the groth rate. k In ths case t can be shon that lesure declnes, so that gross ncome nequalty ncreases. Snce the consumpton tax has no drect redstrbutve effect, the gross and the net dstrbutons of ncome are dentcal and hence net ncome nequalty ncreases as ell. 12 N We can, hoever, see that hen the subsdy rate matches the externalty, s = α, τ = 1 and ρ = 1 mplyng that the net ncome nequalty s ncreased to that of the ntal endoment of captal. 17

19 We summarze our results th Proposton 2: Consder the effects of an nvestment subsdy. () An nvestment subsdy fnanced by a captal ncome tax ncreases the groth rate, labor supply, and before-tax ncome nequalty. It reduces elfare nequalty and has an ambguous effect on after-tax ncome nequalty. () An nvestment subsdy fnanced by a labor ncome tax ncreases the groth rate. Although t has an ambguous effect on labor supply, and therefore on before-tax ncome nequalty and elfare nequalty, t ncreases after-tax ncome nequalty. () An nvestment subsdy fnanced by a consumpton tax ncreases the groth rate, labor supply, and both before-tax and after-tax ncome nequalty, hle t reduces elfare nequalty. Table 1 provdes formal expressons for the effects of the nvestment subsdy on groth and dstrbuton, under the dfferent fnancng modes. We see that for each of the three methods of fnance there are to effects that may ether renforce or offset each other. All three taxes tend to reduce the groth rate. Hoever, f used to fnance an nvestment subsdy, the postve effect of the latter ll alays domnate and the groth rate ll rse. Proposton 2 hghlghts the dfferent tradeoffs beteen nequalty and groth generated by ths polcy. Frst, changes n fscal polcy that ncrease the groth rate ll also ncrease pre-tax ncome nequalty, n lne th the emprcal evdence n Forbes (2000) and others. The reason for ths s smply that faster groth nduces a greater supply of labor, reducng the age rate, and rasng the return to captal, thus makng the dstrbuton of ncome more unequal. 13 Hoever, ths apparent conflct beteen effcency and equty dsappears hen nequalty s measured n terms of utlty. Second, pre-tax and post-tax nequalty need not move together. Whle the ndrect effect of the subsdy s alays the same (ncreasng the groth rate, labor supply, and hence pre-tax 13 One can also sho that structural changes such as an ncrease n productvty or a change n the rate of tme preference ll generate a postve correlaton beteen nequalty and groth; see García-Peñalosa and Turnovsky (2006). 18

20 nequalty), the drect effect vares, dependng upon the tax used to fnance t. The consumpton tax has no drect redstrbutve mpact, the labor ncome tax redstrbutes toard those th hgher ncomes, hle the captal ncome tax redstrbutes n the reverse drecton. These drect effects can be large enough to domnate the ndrect effect and n Secton 5, belo, e present numercal examples here pre-tax and post-tax ncome respond n opposte ays to changes n τ and τ. k 4.2 Polcy rankngs The three modes of fnancng the subsdy can be ranked n terms of ther relatve mpacts on the groth rate and varous nequalty measures. These rankngs are reported n Table 2, here e see ho they can be sharpened by mldly strengthenng the restrctons, as ndcated, but consstent th our smulaton results. Ths table offers an alternatve perspectve on the tradeoffs nvolved for the dfferent modes of fnancng the nvestment subsdy. Whle, as noted, all methods of fnancng the subsdy rase the groth rate, consumpton tax fnancng s the most effectve, and captal ncome tax fnancng the least so. At the same tme, captal ncome tax fnancng has the most adverse effect on gross ncome nequalty, folloed by the consumpton tax. Both these cases yeld a postve relatonshp beteen groth and gross ncome nequalty, consstent th the recent emprcal evdence. By contrast, the labor ncome tax may qute plausbly reduce gross ncome nequalty, leadng to a negatve groth-gross ncome nequalty relatonshp n that case. Second, the relatve rankngs of the three modes of fnance are precsely reversed nsofar as net ncome nequalty s concerned, beng essentally the same as those of elfare nequalty. Ths suggests that hle gross ncome nequalty s a poor ndcator, net ncome nequalty s a good ndcator of the relatve rankngs of elfare nequalty. 14 Thrd, hle each mode of fnancng may be superor n terms of some crteron, t s domnated n others. The table thus hghlghts the need for a polcy maker shng to stmulate nvestment n ths ay to take careful account of the tradeoffs along these varous dmensons. In partcular, captal ncome tax fnancng, although nferor from the standpont of stmulatng groth, may n fact be desrable f the 14 Hoever, the effect on net ncome nequalty s not necessarly a good ndcator of the effect on elfare nequalty n all cases, and example of here these to measures move n opposte ays s gven n Table 3 belo. 19

21 polcymaker attrbutes suffcent eght to reducng post-tax or elfare nequalty. We summarze these results th Proposton 3: Suppose a polcy maker shes to stmulate nvestment through a subsdy. () From a groth perspectve, fnancng the subsdy usng a consumpton tax s superor to a age tax, and n turn to a captal ncome tax, although all are groth-enhancng. () In terms of elfare nequalty, the captal ncome tax s superor to the consumpton tax, hch n turn domnates the age ncome tax. These rankngs also apply to ther mpact on net ncome nequalty, but are reversed n terms of ther mpact on gross ncome nequalty. 5. Numercal Examples To obtan further nsghts nto the groth-ncome nequalty relatonshp e provde some numercal examples. To do so e use the follong, mostly conventonal, parameter values: Parameter Values Producton A = 0.75, α = 0.60 Preferences β = 0.04, γ = 2, η = 1.75 The choce of producton elastcty of labor measured n effcency unts mples that 60% of output accrues to labor. One consequence of the Romer technology, s that hereas ths value s realstc n terms of the labor share of output, t mples an mplausbly large externalty from aggregate captal hch mples extreme solutons for the frst-best fscal polcy, dscussed belo. The choce of the scale parameter A = 0.75, s set to yeld a plausble value for the equlbrum captal-output rato. Turnng to the preference parameters, the rate of tme preference of 4% s standard, hle the choce of the elastcty on lesure, η =1.75, s standard n the real busness cycle lterature, mplyng that about 72% of tme s devoted to lesure, consstent th emprcal evdence. Estmates of the 20

22 ntertemporal elastcty of substtuton are more varable throughout the lterature. Our choce 0.33 s ell n the range of the emprcal evdence, hch th fe exceptons les n the range (0,1). Our measures of ncome dstrbutons are reported n terms of standard Gn coeffcents, as ell as the standard devaton measures employed n our theoretcal dscusson. Ths nvolves choosng an ntal dstrbuton of ealth, hch s less straghtforard, as data on the dstrbuton of ealth are dffcult to obtan. 15 The choce e have made yelds a Gn coeffcent of ncome around 33.3%. Ths s the value observed for net ncome n Italy n 2000, and compares th 36.8% and 25.2% for the US and Seden, respectvely, n the same year (Luxembourg Income Study, 2005). The frst lne of Table 2 reports the benchmark equlbrum n the lassez-fare economy for our base parameters. We see that 72.5% of tme s allocated to lesure, yeldng a groth rate of 3.3%, a standard devaton of ncome nequalty of 0.206, and a Gn coeffcent of ncome of 33.3%. Ths s a plausble benchmark and l = les n the range [0.636, 0.745], consstent th (12). The next three ros report the numercal effects of fnancng a fxed (arbtrary) nvestment subsdy of 30% through a captal ncome, age tax, or a consumpton tax, respectvely. The results llustrate the analytcal results obtaned above. Table 3 also confrms some of the more ambguous effects dscussed. For example, t confrms the theoretcal possblty that the redstrbutve effects n both captal ncome tax and age ncome tax fnancng domnate, so that post-tax nequalty n the former declnes, hle n the latter t rses. In addton, the table reports the elfare gans for the average ndvdual, expressed as equvalent varatons n the ntal captal stock. 16 One nterestng result of these smulatons s that fnancng the subsdy by a tax on captal ncome s attractve n that t s the only polcy that ncreases groth, hle sgnfcantly reducng both elfare nequalty and net-ncome nequalty. Although usng a consumpton tax generates more groth and a hgher elfare gan for the average agent, the dstrbutons of post-tax ncome and elfare need not move 15 We have assumed that dstrbuton of ealth among the 5 quntles s 0, 0, 1.2%, 12%, 86.8%, hch are consstent th the data. For example, n the US n 1992 the bottom 40% of the populaton held 0.4% of total ealth, hle the top 20% oed 83.8% of the total; see Wolff (1998). Also, e have normalzed σ k = We have also evaluated the socal benefts of polcy n terms of a utltaran elfare functon, defned as the sum of the elfares over the heterogeneous agents n the economy. In the case that the dstrbuton of endoments s unform across agents e fnd that utltaran elfare comprses to components: the utlty of the medan ndvdual, adjusted by a term that takes account of the dsperson across agents. Snce the elfare comparsons for ths functon are qualtatvely smlar to those obtaned by consderng just the average ndvdual, e do not report the numercal results n ths case. In general, the elfare level of the medan agent overstates the utltaran level of utlty. 21

23 together n ths case. The last ro of the table consders fnancng the subsdy usng a combnaton of age and consumpton taxes. In partcular, e set τ = τ c ; that s, these to taxes are optmally set, although the subsdy s belo the frst-best level. 17 The effect of ths polcy on the groth rate s stronger than n the prevous three cases, the reason beng that ths polcy does not dstort the allocaton of tme beteen labor and lesure. Employng only a age or a consumpton tax tends to reduce the supply of labor, partally offsettng the effect of the subsdy. When both are used, ths effect s absent. Snce settng τ = τ c results n faster groth than usng only one tax, ths polcy generates larger elfare gans than any of the pure polces. The effect on dstrbuton s qute sgnfcant, as the polcy mples taxng consumpton and subsdzng and age ncomes, hch reduces substantally post-tax ncome nequalty and elfare nequalty. 6. Concludng Comments The lterature on the relatonshp beteen groth and nequalty has rased ne questons about the mpact of fscal polcy, and n partcular about hether or not there s a tradeoff beteen redstrbuton and groth. The exstng lterature has focused on the mpact of fscal polcy hen captal markets are mperfect. In ths paper e have argued that f the labor supply s endogenous and agents dffer n ther ntal captal endoments, groth and the dstrbuton of ncome are smultaneously determned. As a result, macroeconomc polces amed at ncreasng the groth rate ll have dstrbutonal mplcatons even n the absence of captal market mperfectons. The key mechansm hereby the ntal dstrbuton of captal endoments nfluences the dstrbuton of ncome s through the ealth effect, hch mples that ealther agents supply less labor, although the resultng dstrbuton of labor supples s less unequal than that of the captal endoments. As a result any polcy that tends to ncrease the supply of labor and rase the relatve return to captal rases the return to the factor that s the source of the nequalty, causng the 17 It s straghtforard to demonstrate that n the absence of externaltes n consumpton or n the labor market, the frstbest combnaton of the taxes on labor ncome and consumpton must satsfy (1 τ ) (1 + τ c) = 1 so as not to dstort the correspondng optmalty condtons (6a) and (6b). Ths relatonshp, famlar from the representatve agent model [see Turnovsky, 2000] extends to the form of heterogenety ntroduced here. 22

24 dstrbuton of ncome to become more unequal. We have llustrated the effect of polcy by examnng the dstrbutonal consequences of fnancng an nvestment subsdy through a varety of taxes. We consder the mpacts of the varous polces on four key varables: the rate of groth, the pre-tax dstrbuton or ncome, the post-tax dstrbuton of ncome, and the dstrbuton of elfare. A number of results emerge. Frst, e fnd that polces that ncrease the groth rate tend to make the dstrbuton of gross ncome more unequal, but that of elfare more equal. Second, t s often the case that fscal polcy has opposte effects on the dstrbuton of gross and net ncome. As a result, t s possble to ncrease the groth rate and to reduce net ncome nequalty, although gross ncome nequalty ould be exacerbated. Moreover, net ncome nequalty and elfare nequalty need not move together n response to a polcy change. These results hghlght the fact that usng gross ncome nequalty to assess the dstrbutonal mplcatons of fscal polcy may be msleadng hen the labor supply s endogenous. Fnally, e conclude th a caveat. Whle the smple AK model has the advantage of provdng a tractable frameork for nvestgatng the groth-nequalty relatonshp and ts polcy mplcatons, t also has the lmtaton that the economy s alays on ts balanced groth path. It therefore cannot address ssues pertanng to the dynamcs of ealth and ncome dstrbuton. In current ork e fnd that the essental structure of our analyss carres over to a neoclasscal technology havng a dmnshng margnal product of captal; see García-Peñalosa and Turnovsky, (2005). Specfcally, the key relatonshp lnkng relatve lesure to relatve captal contnues to hold and plays the same crucal role. In ths case, hoever, the relatonshp evolves over tme, thereby generatng dynamc tme paths for the dstrbutons of ealth and ncome. Hoever, further ork s needed to determne the effect of fscal polcy on these transtonal dynamc tme paths. Moreover, extendng the present model to nclude human captal, thereby generatng transtonal dynamcs as n Bond, Wang, and Yp (1996),ould not only be mportant n ts on rght, but ould also be a sgnfcant step n determnng the robustness of the results th respect to alternatve frameorks. 23

25 P R Q U D 1 l β 1 γ N M R U D P σ y, σ u Fg 1: Equlbrum Groth, Employment, and Income Dstrbuton

26 Table 1 Effects of ntroducng nvestment subsdy under alternatve modes of fnance ψ τ k 2 2 α Ω 1 l 1 αl 1 1+ > 0 D(1 γ)(1 l) η 1 l 1 l τ 2 αω 1 α 1 αl l 1+ 1 > 0 D(1 l) 1 γ η η(1 l) η(1 l) τ c 2 αω 1 α 1 > 0 D(1 l) 1 γ η l ρ N ρ ϕ αω 1 l 1 < 0 D(1 γ) η 1 l αω(1 ρ) 1 l 1 > 0 D(1 γ)(1 l) η 1 l αω(1 ρ) 1 l 1 D(1 γ)(1 l) η 1 l α l (1 ρ) 1 η 1 l αω(1 ϕ) 1 l 1 < 0 D(1 γ) l η 1 l Ω αl l 1 α Ω αl 1 α β 1 1 = < 0 D η(1 l) η(1 l) 1 γ D η(1 l) 1 γ D Ω(1 ρ) αl l 1 α 1 D(1 l) η(1 l) η(1 l) 1 γ αω(1 ρ) αl 1 α 1 > 0 D(1 l) η(1 l) 1 γ Ω(1 ρ) αl l 1 α αω(1 ρ) αl 1 α 1 1 > 0 D(1 l) η(1 l) η(1 l) 1 γ D(1 l) η(1 l) 1 γ (1 α)(1 ρ) α l + 1 > 0 α η 1 l Ω(1 ϕ) αl l 1 α 1 Dl η(1 l) η(1 l) 1 γ αω(1 ϕ) αl 1 α 1 < 0 Dl η(1 l) 1 γ αω 1 αl 1 α D 1+ (1 l) η(1 l) 1 γ > 0 from (13)

27 Table 2 Relatve Rankngs of Alternatve Modes of Fnancng Investment Subsdy ψ ψ ψ Groth rate: > > > 0 s s s τ τ τ c k ρ ρ ρ Gross ncome nequalty > > max,0 s τ s s k τc τ ρ suffcent condton for < 0 s τ η s l α γ < here η η α + η 1 γ N N N ρ ρ ρ Net ncome nequalty > > max,0 s s s τ τ c τ k N ρ suffcent condton for s τ c N ρ > s τ k α s η < 2 (1 α) ϕ ϕ ϕ Welfare Inequalty < < mn,0 s τ s s k τc τ ϕ suffcent condton for > 0 s τ η s l α γ < here η η α + η 1 γ

28 Table 3 Alternatve Fnancng of Fxed Investment Subsdy s τ k τ τ c l ψ ρ N ρ ϕ Gn Gn Pre-tax Post-tax (X)

29 Appendx: Dervaton of macroeconomc equlbrum In ths appendx e derve the macroeconomc equlbrum, shong that the economy s alays on ts balanced groth path. We begn by dvdng (6b) by (6a) to obtan C η K 1 τ K = ll () 1+ τ K c (A.1) hle e may rte the ndvdual s accumulaton equaton (4) n the form K 1 τk 1 τ K l ψ = rl () l () (1 l) K 1 s + 1 s K η (A.2) Takng the tme dervatve of (6a) and combnng th (6c) mples C l λ 1 τ k ( γ 1) + ηγ = = β rl ( ) C l λ 1 s for each (A.3) ndcatng that each agent, rrespectve of her captal endoment, chooses the same groth rate for her shado value of captal. Takng the tme dervatve of (A.1) mples C l () l l l K = + C l () l l K (A.4) Consderng equatons (A.3) and (A.4) for ndvduals and k, mmedately mples that all agents ll choose the same groth rate for consumpton and lesure. Ths n turn mples that average consumpton, C, and lesure, l, also gro at ther respectve common groth rates, namely C C k C l lk l = = ; = = C C C l l l k k for all, k (A.5) No turn to the aggregates. Summng (A.1) over all agents and notng that l = l, the aggregate economy-de consumpton-captal rato s k =1, C η K 1 τ = ll () 1+ τ c (A.1 ) A1