Taxation and Income Distribution Dynamics in a Neoclassical Growth Model * Cecilia García-Peñalosa Aix-Marseille University

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1 Taxaton and Income Dstrbuton Dynamcs n a Neoclasscal Growth Model * Cecla García-Peñalosa Ax-Marselle Unversty Stephen J. Turnovsy Unversty of Washngton, Seattle Revsed verson May 211 Abstract: We examne how changes n tax polces affect the dynamcs of the dstrbutons of wealth and ncome n a Ramsey model n whch agents dffer n ther ntal captal endowments. The endogenety of the labor supply plays a crucal role, as tax changes that affect hours of wor wll affect the dstrbuton of wealth and ncome, renforcng or offsettng the drect redstrbutve mpact of taxes. We consder dfferent ways of fnancng government expendture and fnd that polces that reduce the labor supply are assocated wth lower output but also wth a more equal dstrbuton of after-tax ncome. We llustrate these effects by examnng the mpact of recent tax changes observed n the US and n European economes. JEL Classfcaton Numbers: D31, O41 Key words: taxaton; wealth dstrbuton; ncome dstrbuton; endogenous labor supply; transtonal dynamcs. * The paper has benefted from semnar presentatons at the Free Unversty of Berln, at GREQAM, and at Washngton State Unversty, as well as presentatons at the conference Growth wth Heterogeneous Agents: Causes and Effects of Inequalty, Marselle, June 28, the 14 th Conference on Computaton n Economcs and Fnance, Pars, June 28 the 3 rd Worshop on Macroeconomc Dynamcs, held n Melbourne, July 28, the 29 PET conference n Galway. Comments receved at these varous presentatons are gratefully acnowledged. In partcular, we than Jess Benhabb, Julo Davla, and Roger Farmer for ther comments. The paper has also benefted from the constructve suggestons of two referees and the journal s edtor. García-Peñalosa s a CNRS researcher at GREQAM (Ax-Marselle Unversty), and would le to acnowledge the support receved from the French Natonal Research Agency Grant ANR-8-BLAN Turnovsy s research was supported n part by the Castor endowment at the Unversty of Washngton.

2 1. Introducton The role of taxaton n the neoclasscal growth model has been extensvely studed, and the mpact of dfferent taxes on both the long-run equlbrum and transtonal dynamcs s well documented. 1 However, the mplcatons of the tax structure for the dstrbutons of ncome and wealth have receved much less attenton. Two notable exceptons are Krusell, Quadrn, and Ros- Rull (1996) and Correa (1999). Krusell, Quadrn, and Ros-Rull examne the effcency and dstrbutonal effects of swtchng from an ncome tax to a consumpton tax. Correa proposes a methodology to ran alternatve aggregate equlbra n terms of ther dstrbutonal mplcatons by examnng the effect on factor prces. In ths paper we examne the dstrbutonal effects of taxes leved on captal ncome, labor ncome, and consumpton. We characterze both the steady-state dstrbutons of ncome and wealth, as well as ther respectve transtonal dynamcs, and dscuss how these responses depend upon the allocaton of the tax revenues rased. Ths s mportant, snce as we shall demonstrate below, fscal polcy typcally nvolves sharp tradeoffs between ts effects on the level of actvty and ts dstrbutonal consequences. In García-Peñalosa and Turnovsy (27) we dscuss the tradeoffs between growth and dstrbuton generated by tax polces usng a smple endogenous growth model. The AK-technology employed has the ey feature characterstc of that class of models: there are no transtonal dynamcs, and hence the dstrbuton of captal remans constant over tme. As a result, the full mpact of any polcy shoc on the dstrbuton of ncome occurs mmedately and only through changes n factor prces and the aggregate labor supply. Ths rases the queston of the effect of polcy on ncome dstrbuton under more general technologcal condtons, whch nevtably cause transtonal adjustments n the economy, resultng n the dstrbuton of wealth evolvng over tme. The mportance of changes n relatve wealth for ncome dynamcs was establshed n our prevous wor, Turnovsy and García-Peñalosa (28). There we show that the dstrbutonal effects of mprovements n technology were drven by a combnaton of changes n factor prces and 1 See, for example, Judd (1985), Chamley (1986), Lucas (199), Stoey and Rebelo (1991), Ladrón-de-Guevara et al. (22), and Turnovsy (24). 1

3 changes n the dstrbuton of wealth, whch tended to move n opposte drectons durng the transton, mang ther overall effect dffcult to assess. Moreover, the short-run and long-run changes n the dstrbuton of ncome were n opposte drectons, as the ntal response n nequalty to a productvty shoc was then partally or totally offset by changes n the dstrbuton of wealth durng the transton to the new steady state. Hence, our prevous wor rases two questons concernng the mpact of taxaton on ncome dstrbuton n a neoclasscal growth model: to what extent does a conflct exst between output and dstrbuton (as n the AK model we have prevously examned), and whether the trade-off between the short- and the long-run responses of nequalty caused by a productvty ncrease also occurs as a result of tax changes. In ths paper we employ the one-sector Ramsey model, developed n Turnovsy and García- Peñalosa (28), n whch agents dffer n ther ntal endowments of captal (wealth). 2 Representng preferences by a utlty functon that s homogeneous n consumpton and lesure allows aggregaton as n Gorman (1953) or Esenberg (1961), and generates a representatveconsumer characterzaton of the macroeconomc equlbrum. Ths homogenety assumpton whch n any event s the domnant one n contemporary growth theory yelds a substantal payoff n terms of analytcal tractablty, enablng us to examne dstrbutonal ssues sequentally. 3 Frst, the dynamcs of the aggregate stoc of captal and labor supply are jontly determned, ndependently of dstrbutonal consderatons. The dstrbutons of wealth and ncome and ther dynamcs are then characterzed n terms of the aggregate magntudes. In ths context, there are three mechansms that we must consder to understand the dstrbutonal mplcatons of a fscal polcy change. Frst, a polcy shoc wll n general cause a change n the steady-state captal stoc. Durng the transton to the new steady state, agents wth dfferent ntal wealth levels accumulate captal at dfferent rates so that wealth nequalty may ether ncrease or decrease over tme. In Turnovsy and García-Peñalosa (28) we showed that ths aspect was mportant when consderng the dstrbutonal effect of an ncrease n productvty. 2 There are of course many potental sources of heterogenety gvng rse to dstrbutonal ssues, but dfferental ntal wealth endowments seems to us to be arguably one of the most sgnfcant. For evdence of the mportance of nherted wealth see Petty (21). 3 Wthout the homogenety assumpton, aggregate behavor and dstrbuton become smultaneously determned and analyss of the transtonal dynamcs becomes ntractable. 2

4 Second, changes n taxes nduce labor supply responses that wll have dstrbutonal mplcatons. As n our prevous wor, we derve a negatve relatonshp between agents relatve wealth (captal) and ther relatve allocaton of tme between wor and lesure. 4 Wealther agents have a lower margnal utlty of wealth, and hence choose to consume more of all goods, ncludng lesure, thus reducng ther labor supply. 5 Thrd, the dynamcs of the response of ncome nequalty (both before-tax and after-tax) are drven by two effects: the ntal jump n lesure - whch wll affect relatve labor ncomes- and the contnuous changes n the dstrbutons of captal and factor prces that occur durng the transton to the new steady state. These two responses may move n opposte drectons, mplyng non-monotonc responses to polcy changes. We characterze the tme paths of the dstrbutons of wealth and ncome, as well as ther steady-state dstrbutons, and examne ther responses to changes n tax rates. In general we show how the evoluton of wealth nequalty s drven by the dynamcs of aggregate labor supply (lesure) through ts response to the accumulaton of aggregate captal stoc. In partcular, economy-wde accumulaton of captal s assocated wth a reducton n wealth nequalty, as wealther agents, choosng to enjoy more lesure, accumulate captal at slower rates. Ths n turn drves the evoluton of pre-tax ncome dstrbuton through ts mpacts on the relatve captal ncome and relatve labor ncome. In addton to beng drven by these same determnants, post-tax ncome nequalty s also dependent on the drect redstrbutve effects assocated wth taxes on labor and captal ncome. To llustrate these channels we analyze the comparatve effects of rasng alternatve tax rates to fnance a gven ncrease n government spendng on goods, on the one hand, versus fnancng transfers, on the other. In all cases we fnd that the ncome dstrbuton effects domnate the wealth accumulaton effects, whch are n general very wea and ental only mnor changes n the dstrbuton of wealth. One consequence of ths s that the effects of alternatve ncome taxes on both pre-tax and post-tax ncome dstrbuton depend crtcally upon how the resultng revenue s spent. 4 See García-Peñalosa and Turnovsy (26, 27) and Turnovsy and García-Peñalosa (28). 5 There s substantal emprcal evdence n support of ths negatve relatonshp between wealth and labor supply. Holtz- Ean, Joulfaan, and Rosen (1993) provded evdence to suggest that large nhertances decrease labor partcpaton. Cheng and French (2) and Coronado and Peroze (23) use data from the stoc maret boom of the 199s to study the effects of wealth on labor supply and retrement, fndng a substantal negatve effect on labor partcpaton. Algan, Chéron, Harault, and Langot (23) employ French data to analyze the effect of wealth on labor maret transtons, and fnd a sgnfcant wealth effect on the extensve margn of labor supply. 3

5 As a specfc example, consder the mpact of a hgher tax on labor ncome. The aggregate effect s to reduce labor supply, the captal stoc, and total output and therefore to ncrease wealth nequalty. Whle t reduces the aggregate labor supply, the tax on labor ncome reduces that of the captal-rch by more than that of the captal-poor. Ths results n a more dspersed dstrbuton of labor ncomes, and snce these are negatvely correlated wth captal endowments, t tends to decrease pre-tax ncome nequalty. But n addton, the drect redstrbutve effect from a hgher tax on labor ncome tends to ncrease post-tax ncome nequalty for a gven degree of pre-tax ncome nequalty. If the revenues are rebated neutrally, post-tax ncome nequalty stll declnes, but f they are spent on a utlty-enhancng publc good, ths effect may domnate, and post-tax ncome nequalty may actually rse. The responses of pre-tax and post-tax ncome nequalty are reversed n the case of a tax on captal ncome. Our paper contrbutes to the recent lterature characterzng dstrbutonal dynamcs n growth models, an ssue frst examned by Stgltz (1969) usng a form of the Solow model 6 One approach has consdered economes wth ex-ante dentcal agents and unnsurable, dosyncratc shocs. 7 An alternatve approach has been to assume that agents dffer n ther ntal captal endowments, as we do n our model. 8 We follow Casell and Ventura (2), who study the dynamcs of wealth and ncome dstrbuton n the Ramsey model, though they restrct ther analyss to exogenous labor supply, and do not consder the mpact of tax polces. Closely related s Benhabb and Bsn (28), who also examne the dstrbutve mpact of taxes and derve explct expressons for the dstrbuton of wealth. The analytcal framewors are, however, rather dfferent. They consder an overlappng generatons setup n whch agents dffer n ther degree of altrusm, and focus on the role of the ntergeneratonal transmssons of wealth and state taxaton n generatng an emprcally plausble dstrbuton of wealth. In contrast, we abstract from these aspects by consderng nfntely-lved agents, and center our analyss around the endogenety of the labor supply, the way n whch t s affected by taxaton, and ts mpact on both wealth and ncome dstrbutonal dynamcs. 6 See Bertola, Foellm, and Zwemüller (26) for a survey. 7 See, for example, Krusell and Smth (1998), Castañeda, Díaz-Gménez, and Ros-Rull (1998), Díaz, Pjoan-Mas, and Ros-Rull (23), and Wang (27). 8 See Chatterjee (1994), Chatterjee and Ravumar (1999), Sorger (2), Malar and Malar (21), Alvarez-Peláez and Díaz (25), and Obols-Homs and Urruta (25) 4

6 Followng ths ntroducton, Secton 2 descrbes the economy and derves the macroeconomc equlbrum. Secton 3 characterzes the dstrbutons of wealth and ncome and derves the man analytcal results. Secton 4 derves the effects of changes n tax rates on the long-run dstrbutons of wealth and ncome, whch are then llustrated n Secton 5 wth a number of numercal examples. Secton 6 concludes, whle nsofar as possble techncal detals are relegated to an Appendx. 2. The Analytcal Framewor The analytcal framewor we employ s developed by Turnovsy and García-Peñalosa (28) where t s dscussed n greater detal. 9 Aggregate output s produced by a sngle representatve frm n accordance wth the neoclasscal producton functon (, ) Y = F K L F >, F >, F <, F <, F > (1) L K LL KK LK where, K, L and Y denote the per capta stoc of captal, labor supply and output. The wage rate, w, and the return to captal, r, are determned by the margnal physcal products of labor and captal, w( K, L) = F ( K, L) w = F >, w = F < (2a) L K LK L LL r( K, L) = F ( K, L) r = F <, r = F > (2b) K K KK L KL 2.1 Heterogeneous consumers We assume a constant populaton, N. Indvdual owns K () t unts of captal at tme t, so that the total amount of captal n the economy at tme t s N T K () t = K () t d and the average per capta amount of captal s 1 N K() t = K () t d N. We defne the relatve share of captal owned by agent to be () t K () t K() t the mean of whch 9 We should emphasze that Turnovsy and García-Peñalosa (28) focuses on very dfferent ssues, beng concerned wth structural aspects such as the role of flexble labor supply and ndeed abstracts from tax ssues beng addressed here. 5

7 s1. The ntal (gven) relatve endowment of agent s the ntal (gven) standard devaton of whch s σ,. To ensure that all ndvduals supply some labor we further suppose that ntal relatve captal s bounded above by. 1 Each ndvdual s endowed wth a unt of tme that can be allocated ether to lesure, l, or to wor, 1 l L, so that average (aggregate) labor and lesure can be expressed as L l = 1 N (1 l ( t)) d. The agent maxmzes lfetme utlty, assumed to be a functon of = 1 ( ) N consumpton, the amount of lesure tme, and government expendture, G (taen as gven), n accordance wth the soelastc utlty functon 1 η θ γ βt max ( C ( ) ), wth 1,,1> (1+ ) t l G e dt < γ < η > γ η (3) γ where 1 (1 γ ) equals the ntertemporal elastcty of substtuton. The preponderance of emprcal evdence suggests that ths s relatvely small, certanly well below unty, so that we shall restrct γ <. 11 The parameter η represents the elastcty of lesure n utlty, whle θ measures the relatve mportance of publc consumpton n prvate utlty. Ths maxmzaton s subject to the agent s captal accumulaton constrant K & () t = [(1 τ ) r() t δ] K () t + (1 τ ) w()(1 t l ()) t (1 + τ ) C () t + T (4) w c where τ τ, τ, w c and T are, respectvely, the tax rates on captal ncome, labor ncome, consumpton, and lump-sum transfers that the agent taes as exogenously gven, and to the nonnegatvty constrants on tme use, l < 1 and L < Our assumpton of perfect captal < marets mples that poorer ndvduals may potentally hold negatve captal stocs, and ths could be the case f such agents antcpate a hgh future wage. 13 The agent s optmalty condtons are < 1 As we wll see below, rcher ndvduals enjoy more lesure and ths could mply that wth a suffcently hgh wealth endowment an ndvdual chooses not to wor. To ensure nteror solutons to the allocaton of tme, we suppose that the ntal dstrbuton of wealth s such that all ndvduals supply a strctly postve amount of labor. Usng the ndvdual budget constrant and our results on the steady state stoc of captal derved below, t s possble to show that around the steady state L > f 1 ()( a (, ) a (, )) /( a < + δ sl K % L % ηls % K K % L % ηls % K( K %, L % )), where, a, a sk s L denote the equlbrum after-tax shares of captal and labor, respectvely, tlde denotes steady-state values, and δ () s defned n (18) below. 11 See Guvenen (26) for a summary and dscusson of the exstng emprcal evdence on the elastcty of substtuton. 12 Snce we are focusng on nteror solutons, there s no need to append the nequalty constrants on labor/lesure. 13 See the dscusson n Turnovsy and García-Peñalosa (28, p. 1411). Whether ths occurs depends also upon the nature of the structural change generatng the change n relatve wealth. 6

8 C η l 1 τ w = w( K, l) (5) 1+ τ c C& l& G& & λ ( γ 1) + ηγ + γθ β = = δ r( K, l)(1 τ ) C l G λ (6) where λ s agent s shadow value of captal, and we have used the fact that L =1 l to express the wage and nterest rate, w, r, as functons of average lesure, l, rather than of average employment, L. Equaton (5) equates the margnal rate of substtuton between consumpton and lesure to the [after-tax] prce of lesure, whle (6) s the Euler equaton modfed to nclude the fact that lesure changes over tme. A crucal aspect of the latter equaton s that t mples that each agent chooses the same growth rate for the shadow value of captal, rrespectve of captal and labor endowment. As dscussed ntally by Casell and Ventura (2), ths specfcaton of preferences wll result n the ndependence of the aggregate equlbrum from dstrbuton. Usng (5) we may wrte the ndvdual s accumulaton equaton, (4), n the form K& wkl (, )(1 τ w) 1+ η T = rkl (, )(1 τ) δ + 1 l K K η K (7) Equatons (5) and (7) ncorporate well-nown effects of taxaton. From (5) we see that the effectve tax rate on labor, ( τ τ ) 1+ τ ) w + c /( c, s a crucal determnant of the allocaton of tme, as n Prescott (24), whle equaton (7), n turn, establshes that, for a gven labor supply, hgher taxes on captal ncome reduce the rate of accumulaton of captal. 2.2 Government We assume that the government sets ts expendture and transfers as fractons of aggregate output, n accordance wth G = gy(), t T = τy() t, so that g and τ become the polcy varables along wth the tax rates. We also assume that t mantans a balanced budget expressed as τ rklk (, ) + τ wkl (, )(1 l) + τ C= G+ T= ( g + τ) FKL (, ) (8) w c Ths means that, fτ w, τ, τ c, and g are fxed, as we shall assume, then along the transtonal path, 7

9 as economc actvty and the tax/expendture base s changng, the rate of lump-sum transfers must be contnuously adjusted to mantan budget balance. 14 In order to abstract from any drect dstrbuton effects arsng from lump-sum transfers (whch are arbtrary), we shall assume T() t T() t K () t K() t N = whch ensures that ( ) N Td = T K Kd T =, consstent wth the government budget constrant. Whle ths assumpton s restrctve n that t does not capture the major redstrbutve mpact that drect transfers have, t has the analytcal advantage of allowng us to focus on the dstrbutve mpact of dstortonary taxes The macroeconomc equlbrum The ey relatonshp n dervng the aggregate dynamcs s equaton (6) whch mples that each agent chooses the same growth rate for the shadow value of captal, rrespectve of captal endowment. Usng ths equaton t can be shown that all agents wll also choose the same rate of growth of lesure, mplyng that average consumpton, C, and lesure, l, grow at the same common rates. 16 The macroeconomc equlbrum can then be descrbed by the par of dynamc equatons: FL ( K, L) l 1 w K& τ = (1 gfkl ) (, ) δ K (9a) η 1+ τc FKL F K FL ( K, L) l 1 τ w FK(1 τ) ( β + δ) (1 γ) θγ (1 g) F( K, L) K FL F δ η 1 τc l& + = (9b) 1 γ(1 + η) FLL F L (1 γ) θγ l FL F whch determne the jont dynamcs of average (aggregate) captal and lesure. The mportant 14 In our smulatons we tae τ > ; τ < therefore corresponds to a rate of lump-sum taxaton. It s mportant to emphasze that our results on nequalty do not depend on the presence of these transfers nor on the form they tae. In fact, the same analytcal results would be obtaned wth no transfers. We ntroduce them n order to focus on the mpact of dscretonary taxes whch we can set and eep fxed durng the transton to a steady state, whle the transfer rate vares endogenously so as to balance the government s budget. 15 Le all lump-sum transfer schemes, we assume that ndvdual agents are unaware of the rule adopted by the government. An alternatve procedure would be to ntroduce debt fnancng along the transtonal path, but gven perfect marets, Rcardan Equvalence mples that ths s essentally equvalent to what we are dong. The one dfference s that the relatve captal stoc dscussed n Secton 3.1 s replaced by relatve wealth (captal plus government bonds). But results are essentally dentcal to those we obtan. 16 Tang the tme dervatve of (5) and combnng t wth (6) we readly see that l & l = l & l, C & C = C & C for each ; see Turnovsy and García-Peñalosa (28) for more detals. In partcular, l = ρl where ρ s constant over tme. 8

10 observaton about ths par of equatons s ts mplcaton that aggregate behavor s ndependent of wealth and/or ncome dstrbutons. Assumng that the economy s stable, the aggregate dynamc system, (9), converges to a steady state characterzed by a constant average captal stoc, labor supply, and lesure tme, denoted by K ~, L ~ and ~ l, respectvely. Settng K & = l & =, the steady state s summarzed by (1 τ ) F ( K %, L % ) = β + δ (1a) K FL ( K%, L% ) l% 1 τ w (1 gfkl ) ( %, %) δ K% = η 1+ τc (1b) L% + l% = 1 (1c) Note that (1b) mples ~ ~ ~ ~ ~ ~ 1+ η ( 1 g) F δk FL L(1 τ l ) + FL (1 τ l ) 1 l =. (11) η If we assume that the share of prvate consumpton expendture, [ (1 g) δ ( K F) after-tax share of labor ncome, ( FL % % F % )(1 τ ), then (11) mposes the restrcton 17 L l % % ], exceeds the l% > η (12) 1 +η Ths nequalty yelds a lower bound on the steady-state tme allocaton to lesure that s consstent wth a vable equlbrum. As we wll see below, ths condton plays a crtcal role n characterzng the dynamcs of the wealth dstrbuton. In order to descrbe the dynamcs of the dstrbuton of captal and ncome, we frst need to obtan the dynamcs of the aggregate magntudes. Lnearzng equatons (9a) and (9b) around steady state yelds the local dynamcs for K and l, 17 Ths condton s lely to be met. For example n our smulatons (1 g) δ ( K% / F% ) =.69 and clearly exceeds (1 τ )( FL % %/ F% ) =.5. w L 9

11 K& a a K K% = l& a21 a22 l l % (13) where a 11, a12, a21 and a 22 are defned n the Appendx. 18 There we show that, under the assumpton that θ s not too large, a a a a, mplyng that the steady state s a saddle pont. The stable < paths for K and l can be expressed as K() t = K% + ( K K% ) e μt (14a) a μ a l t = l% + K t K% = l% + K t K% (14b) ( ) ( ) () () () μ a22 a12 where μ < s the stable egenvalue. In the Appendx we establsh that ( a 22 μ) >, mplyng that the slope of the stable arm has the opposte sgn to that of a 21. The sgn of ths expresson reflects two offsettng nfluences of captal on the evoluton of lesure. On the one hand, a greater captal stoc reduces the return to captal and hence to future consumpton, thus ncreasng desred lesure. On the other, greater K ncreases wages and thus reduces the growth rate of lesure. Whch effect domnates depends upon the underlyng parameters, and n partcular upon the elastcty of substtuton n producton, ε. In the Appendx we demonstrate that for plausble cases [ncludng the conventonal case of Cobb-Douglas producton and logarthmc utlty ( ε = 1, γ = )] a, n whch case the stable locus s postvely sloped; n equlbrum accumulatng captal s therefore assocated wth ncreasng lesure and a21 ( μ a22) reflects the strength of ths postve assocaton. For expostonal convenence we shall restrct ourselves to what we vew as the more plausble case of a postvely sloped stable locus, (14b). The evoluton of average lesure wll be shown below to be an essental determnant of the dynamcs of wealth and ncome nequalty, hence t s mportant to consder n detal how lesure 21 < 18 The relablty of employng lnearzaton to evaluate the dynamc adjustments s examned by Atola, Chatterjee and Turnovsy (21), who consder responses to changes n government expendture n an aggregate one-sector growth model. Ther results ndcate that lnearzaton yelds a satsfactory approxmaton n a context le ths, where the stable manfold s frst order, and n addton the transtonal dynamcs reman close to steady state. As s seen n Secton 3.1 below, relatve wealth dynamcs s then studed by substtutng the soluton from (13) nto (15). However, snce the resultng equaton s lnear n the relatve captal stoc,, the qualty of the approxmaton obtaned at the aggregate level s mantaned n characterzng the dstrbutonal dynamcs. 1

12 reacts to a polcy shoc. Snce (14b) holds at all tmes, we have % a ( % ). (14b ) μ a 21 l() l = K K 22 Whenever the economy s subject to a polcy shoc that results n an ncrease n the steady-state ~ average per capta captal stoc relatve to ts ntal level ( K < K ), there wll be an ntal jump n average lesure, such that l() transton; an analogous relatonshp apples f K < l %, so that, thereafter, lesure wll ncrease monotoncally durng the > K %. We should pont out, however, that n our smulatons the net mpact of the two offsettng nfluences of the accumulatng captal stoc on the transton of lesure [as measured by the slope of (14b)] s small. Consequently the response of lesure followng the fscal shoc s completed almost mmedately, on mpact. 3. The dstrbuton of ncome and wealth 3.1 The dynamcs of the relatve captal stoc To derve the dynamcs of ndvdual s relatve captal stoc, () t K () t K() t, we sum (7) over the agents to obtan an expresson for the evoluton of K () t and combne t wth the ndvdual captal dynamcs n (7). Wth transfers set such thatt / K = T K ths leads to 19 / w( K, l)(1 ) 1 1 & τ w ( t) = 1 l l 1 + ( t) (15) K η η where K, l evolve n accordance wth (14a, 14b) and the ntal relatve captal, s gven from the ntal endowment. Frst, we note from (15) that agent s steady-state lesure, l %, and share of captal stoc, %, are related by ~ ~ ~ η ~ l l = l ( 1) 1+ η for each (16) Ths equaton captures one of the crtcal elements determnng the evoluton of the dstrbutons of 19 For more of the detals see Turnovsy and García-Peñalosa (28). We have also consdered an alternatve lump-sum transfer rule T = T, wth very small dfferences n results from those we are reportng here. 11

13 wealth and ncome, and explans why the evoluton of the aggregate quanttes are unaffected by dstrbutonal aspects. On the one hand, the agent s labor supply s a lnear functon of hs relatve captal. On the other, the senstvty of labor supply to relatve captal s common to all agents, and depends upon the aggregate economy-wde lesure. As a result, aggregate labor supply depends only on the aggregate amount of captal but not on ts dstrbuton across agents. Moreover, recallng (12), equaton (16) mples that the greater an agent s steady-state relatve wealth, the more lesure he consumes and the less labor he supples. Ths generates an equalzng effect that partly offsets the mpact of wealth nequalty on the dstrbuton of ncome. In order to analyze the evoluton of the relatve captal stoc, we lnearze equaton (15) ~ around the steady-states K L ~ ~ ~,,, l. 2 From (12), the coeffcent n (15) on s postve, mplyng that n order for agent s relatve stoc of captal to reman bounded, and therefore yeld a fnte steady-state wealth dstrbuton, () t must follow the stable path: () t 1 = δ ()( t % 1) (17) where FL( K%, l% )(1 τw) lt () FL( K%, l% )(1 τw) 1+ η δ () t 1+ 1 l% 1 μ K% l% K% η 1, (18) Settng t = n (17) and (18) yelds ~ 1 1 = ( δ (), 1). (19) Ths s, the steady state relatve captal of agent, whch depends both on,.e. on the ntal (gven) dstrbuton of captal endowments and on the value at t = of the coeffcent drvng the dynamcs of wealth dstrbuton, δ (t) Note that equatons (1) determne K %, l %. Gven these aggregate quanttes, the remanng two varables, %, l% are determned by (16) together wth (19) below and depend upon the agent s ntal relatve stoc of captal. 21 Equaton (19) enables us to address whether or not ndvduals always hold postve captal. From (19) we see that % f and only f, ( 1 δ() ) δ(). If all agents begn wth non-negatve holdngs of captal (, ) and the structural change generates an expanson n the aggregate long-run captal stoc, (14b) and (18) mply δ () > 1, so that ths condton s always met. Moreover, snce the transton n () t s monotonc, () t > along the entre transtonal 12

14 Usng (17) (19), and equatons (14), descrbng the evoluton of the aggregate economy, we can express the tme path for () t n the form ~ μt ( t) = e ( ~ ), (2), from whch we see that ( t ) also converges to ts steady state value at the rate μ. Note that equatons (15) to (18) mply that of all the polcy nstruments only the tax on labor ncome has a drect effect on relatve captal accumulaton. Other nstruments wll mpact ( t ) only through the changes they nduce n aggregate magntudes, ncludng the rate of adjustment, μ. In partcular, government expendture has no drect effect, despte the fact that t affects the margnal utlty of consumpton (see equaton (5)) and hence affects both savngs and hours of wor. The reason for ths s that, wth homothetc preferences, the relatonshp between the growth rate for the shadow value of captal and for government expendture s the same for all agents, rrespectve of ther endowments [see (6)]. As a result t has no drect mpact on the relatve rate of accumulaton and hence on the dstrbuton of wealth. 3.2 The dynamcs of relatve ncome An mportant aspect of ncome taxes s ther drect redstrbutve effect, causng us to dstngush between before-tax and after-tax ncome dstrbutons, wth the latter measure beng arguably of greater sgnfcance. We defne the relatve before-tax and after-tax ncome of ndvdual at tme t as y rk + w(1 l) =, (21) rk + w(1 l) y rk (1 τ ) + w(1 l )(1 τ ) =, (22) rk(1 τ ) + w(1 l)(1 τ ) a w w respectvely. Note that the after-tax ncome measure gnores the drect dstrbutonal mpacts of lump-sum transfers, whch are arbtrary. path. For () on each agent s ntal holdng of captal. t > to hold followng a contractonary shoc, ( δ ) 13, 1 () δ() mposes a postve lower bound,

15 The relatve before- and after-tax ncome of agent may then be expressed as y ( t) 1 = ϕ ( t)( ( t) 1), (23) ( ( t) 1) a y ( t) 1 = ψ ( t) (24) where lt () 1 η 1 ϕ() t s() t (1 s()) t 1. (25) 1 lt ( ) l% 1 + η δ ( t) st ()( τw τ) ( ) ( ) ( ) τ + st τ ψ () t ϕ() t + 1 ϕ() t st ()1 (1 ())1 w (26) Consder, frst the expresson for before-tax ncome nequalty, ϕ () t. Ths measure has two components, the share of captal ncome, s() t, and relatve labor ncome, reflected n the second term of (25) and whch captures the fact that less wealthy agents supply more labor. In the case of after-tax relatve ncome, we see that both ncome tax rates, τ and τ w, exert two effects on the after-tax ncome dstrbuton. Frst, by nfluencng ϕ () t, they nfluence gross factor returns, and therefore the before-tax dstrbuton of ncome. But, n addton, they have drect redstrbutve effects that are captured by the second term on the rght hand sde of (26). Post-tax ncome nequalty wll be less than pre-tax ncome nequalty f and only f τ w < τ. In contrast, the redstrbutve effects of a consumpton tax on post-tax nequalty are only ndrect and operate to the extent that τ τ w. 3.3 The dstrbutons of wealth and ncome We can now compute ndexes of nequalty. Because of the lnearty of (17), (23) and (24) we can mmedately transform these equatons nto correspondng relatonshps for the standard devatons of the dstrbutons of captal and ncome across agents, whch therefore serve as convenent (and tractable) measures of wealth and ncome nequalty. From (17) we can express the standard devaton of the dstrbuton of relatve captal at tme t as 14

16 ( ) ) ~ δ t σ ( t) = δ ( t σ = σ,. (17 ) δ () Equaton (2) allows us to wrte ~ μt σ t) σ = e ( σ ~ σ ), from whch we see that the degree of (, wealth nequalty does ndeed vary durng the transton, convergng to the long-run dstrbuton ~ σ. 22 In addton, (17 ) together wth (18)and (14b ) mply that σ > σ > % σ f and only f K < Kt () < K %. (27), () t Thus we conclude that f the economy undergoes an expanson (contracton) n ts captal stoc then wealth nequalty wll decrease (ncrease) durng the transton, and the long-run dstrbuton of wealth wll be less (more) unequal than s the ntal dstrbuton. Further ntuton for ths result can be easly seen by notng from (17) - (19), that, ( )( ) sgn( % ) = sgn % 1 l() l% Recall that f the economy converges to the steady state from below, then l() 15 < l %. For people who end up above the mean level of wealth, ther relatve wealth wll have decreased durng the transton % <, whle for people who end up below the mean level of wealth, ther relatve wealth wll have, ncreased, % >, mplyng a narrowng of the wealth dstrbuton. Equatons (18) and (19) further, mply that the closer l () jumps to ts steady state, l %, the smaller the subsequent adjustment n lt (), and hence the smaller s the overall change n the wealth dstrbuton. Ths s because f the economy and therefore all ndvduals fully adjust ther respectve lesure tmes nstantaneously, they wll all accumulate wealth at the same rate, causng the wealth dstrbuton to reman unchanged. Smlarly, because of the lnearty of (23) and (24) n (t), we can express the relatonshp between before-tax and after-tax relatve ncome and relatve captal (wealth) n terms of correspondng standard devatons of ther respectve dstrbutons, namely σ () t = ϕ() t σ () t (23 ) y a σ () t = ψ() t σ () t (24 ) y 22 However (2) mples that the ranng across agents accordng to wealth remans unchanged throughout the transton. To obtan changes n these ranngs requres an addtonal source of heterogenety, such as dfferental abltes.

17 If labor s suppled nelastcally, ϕ () t = s() t so that σ () t σ () t = s() t. However, when labor s flexble, poor agents supply more labor than do the wealthy, whch partally offsets the effect of the unequal dstrbuton of captal. From nequalty (12) the term n square bracets n equaton (25) s postve and hence ϕ () t < s() t, mplyng that the rato of wealth nequalty to ncome nequalty s less than the share of captal. Fnally, lettng t, we can express the steady-state dstrbutons of ncome as y % σ = % ϕ % σ (23 ) y % σ = ψσ % %, (24 ) a y where ~ ~ ~ ~ 1 1 s 1 FL ( K, L) ϕ lmϕ( t) = 1 ~ = 1 ~ ~, (25 ) t 1+ η 1 l 1+ η F( K, L) ~ ~ ~ s ( τ w τ ) ψ lmψ ( t ) = ϕ + (1 ~ ~ ~ ) (1 τ ) + (1 )(1 τ ) ϕ. (26 ) t s s w From (12) and (25 ) we see that f η + s% η > l% > (12 ) 1+ η 1+ η these expressons mply that sgn ( y 1) = sgn ( y, 1) %. In that case, the dstrbuton of ncome converges to a long-run dstrbuton such that the relatve ranng of agents accordng to ncome s the same as that of captal, as well as that of the ntal ncome dstrbuton. However, f s l% η + % η > >, (12 ) 1+ η 1+ η n the long run, agents havng above average wealth wll have below-average ncome. Ths wll occur f the tax on labor ncome s suffcently hgh so that ndvduals are nduced to devote a very small fracton of ther tme to labor. A plausble example of ths s gven n Secton 5.4 below. Note that the dynamcs of the dstrbuton of ncome dffer from those of the dstrbuton of captal n that ncome nequalty (both before-tax and after-tax) jumps n response to a shoc. To see 16

18 that we rewrte (23 ) as () 1 () () () [ 1 ()] lt η σ y t = s t σ t s t 1 σ 1 lt ( ) l 1+ η % % (28) whch ndcates that although the dstrbuton of wealth, σ ( t ) evolves gradually, the ntal jump n lesure, l (), means that ncome dstrbuton s subject to an ntal jump followng a structural or polcy shoc. Thereafter, t evolves contnuously n response to the evoluton of the dstrbuton of captal and factor returns. To assess the long-run effect of polcy on ncome dstrbuton, note that ~ σ ~ ~ ~ a y ϕ σ σ ~ ~ = y ψ σ and a = σ ϕ σ σ y, ψ σ, y,, where σ y, and σ denote the ntal dstrbutons of before and after-tax ncome. Whether the a y, long-run dstrbuton s more or less unequal than the ntal dstrbuton depends on the long-run change n the dstrbuton of captal, as reflected n % σ σ,, factor returns, as reflected n % ϕ % ϕ, and n the case of the after-tax dstrbuton the sze of the wage tax relatve to the captal tax rate. 3.4 Dstrbuton of welfare Recallng (3), agent s welfare at tme t s ( γ ) and the fact that l ( t) = ρ l( t) [see footnote 16], mples: Ω () t 1 C () t γ l () t γη G θγ. Usng equaton (5) Ω t Ω( t) ( ) γ (1+ η ) = ( ρ ) (29) where Ω () t s the average welfare level at tme t, and from (16) ρ η ~ 1+ 1 ~ ( 1) (1 ) + η l = Substtutng (29) nto (3), yelds an analogous relatonshp for the relatve ntertemporal welfare, evaluated along the equlbrum growth path: W, 17

19 W W t Ω( t) Ω ( ) γ (1+ η ) = = ( ρ ) (3) At each nstant of tme, agent s relatve welfare remans constant, so that hs ntertemporal relatve welfare, W W, remans constant as well. We can now compute a measure of welfare nequalty. A natural metrc for ths s obtaned by applyng the followng monotonc transformaton of relatve utlty, enablng us to express the relatve utlty of ndvdual as 1/(1 + η ) 1/(1 + η ) W Ω() t η = = 1+ 1 ( % 1) W Ω () t (1 + η) l% Both nstantaneous and ntertemporal welfare nequalty, expressed n terms of equvalent unts of wealth, can then be measured by the standard devaton of relatve utlty, η % σ u = 1 % σ % χσ% (1 + η ) l% (31) Ths expresson s dentcal to that obtaned n our earler wor, García-Peñalosa and Turnovsy (27). Note that, n contrast to the dynamcs assocated wth ncome nequalty, welfare nequalty remans constant durng the transton, reflectng the fact that relatve lesure s constant. 4. Steady state effects of fscal polcy Our objectve s to examne the effect of dfferent fscal polces on the tme paths of the dstrbutons of wealth and ncome. We begn by consderng the steady-state effects, snce wth forward-loong agents, these long-run responses are crtcal for determnng the transtonal dynamcs. In all cases we assume that the economy starts from an ntal equlbrum n whch government expendture s fully fnanced by lump-sum transfers, so that g + τ = n equaton (8), and ntal dstortonary tax rates are all zero,.e. τ = τ = τ =. c w In Table 1 we summarze and compare alternatve modes of fnancng a specfed ncrease n government spendng. The change n tax rates requred by the ncrease n government expendture s gven by the government budget constrant (8), whch we can wrte n the form 18

20 τ s+ τ (1 s) + τ C F = g+ τ (8 ) w c We consder frst the effects of an ncrease n government expendture, whch are obtaned when the ncrease n g s fnanced by a reducton n transfers. Startng from τ = τ = τ = the c w requred reducton n transfers s dτ = dg and the long-run responses are reported n the frst column. Snce dstrbuton s determned by aggregate behavor, we begn wth the latter, whch are summarzed n the frst four rows. These results are standard and requre lttle dscusson. The ncrease n government spendng rases the long-run captal stoc, labor supply, and therefore output, proportonately, leavng factor shares unaffected. The dstrbutonal mpacts are reported n Rows 5-8. Notng the response of captal (Row 1), n conjuncton wth (27), we mmedately conclude that an ncrease n government expendture reduces long-run wealth nequalty, σ%. As dscussed n Secton 3, pre-tax ncome nequalty s gven by % σ y = % ϕσ%, the response of whch depends upon that of both σ% and ϕ%, reported n Row 6. As already noted, durng the transton to the new steady state the dstrbuton of wealth wll become less dspersed, tendng to reduce ncome nequalty. At the same tme, recallng that 1 1 s% % ϕ = 1 1+ η 1 l% we see that wth factor shares, s%, remanng unchanged (Row 4) and wth the expanson n government spendng ncreasng labor supply (reducng l ~ ), ths causes an offsettng ncrease n ϕ%. To understand the role of the labor supply n ths adjustment, we rewrte (16 ) n the form ( 1) L% η = L% l% % (16 ) 1+ η whch mples that ndvdual labor supples are negatvely correlated wth captal endowments and have a standard devaton of % σ = ( l% η/(1 + η)) % σ, whch s also the dsperson of relatve labor L ncomes. Thus an ncrease n labor supply, that s, a declne n l ~, mples less dsperson of labor ncomes whch, snce they are negatvely correlated wth captal endowments, tends to ncrease ncome nequalty. Concernng after-tax ncome nequalty, a σ% y, we see from (26) that g does not have drect 19

21 redstrbutve effects, mplyng that ts mpact on ψ% operates entrely through ϕ ~. That s, the effect on pre-tax and post-tax ncome dstrbutons s the same. The fnal row reports the response of welfare nequalty, whch apples unformly throughout the transton and not just n steady state. The ey pont s that the adjustment n labor supply now renforces the effect due to wealth nequalty. For example, the fact that an ncrease n government expendture reduces aggregate lesure, thereby reducng the dspersons of both labor supply and lesure, mples an overall unambguous declne n welfare nequalty. In the case of dstortonary tax fnancng, we assume that the correspondng tax rate s set such that t fully fnances the long-run change n government expendture. Thus, agan startng from τ c = τw = τ =, the correspondng requred changes n the three tax rates are respectvely c ( ) %, dτ = ( 1(1 % s) ) dg, d ( 1 ) dτ = F C% dg w τ = s% dg (32) Ths means that durng the transton resdual fnancng usng lump-sum tax fnancng must also be employed to ensure that the budget remans balanced at all tmes. Thus n general, the response of varable X, say, to a dstortonary tax-fnanced ncrease n government expendture s gven by dx X X τ = + dg g τ g τ τ = τ, τ, τ c w In the case where the ncrease n government expendture s fnanced usng a consumpton tax, we see from column 2 n Table 1 that the net effects are zero. It s straghtforward to show that the mpact of an ncrease n the consumpton tax on all aggregate magntudes are opposte to those of government expendture, and can be explaned by the same ntuton. On balance ths mode of fnancng has no mpact on ether aggregate quanttes or on dstrbuton. Ths s unsurprsng. Frst, the reducton n consumpton due to the hgher tax s exactly offset by the hgher margnal utlty of consumpton nduced by the ncrease n g, thus leavng captal and labor unchanged. Second, the consumpton tax has no drect redstrbutve mplcatons. As a result nether aggregate magntudes nor dstrbutonal varables change. Aggregate effects are more complex n the case of taxes on captal and labor snce they combne the expansonary effect of ncreased expendture wth the contractonary effect of hgher 2

22 dstortonary taxaton. When the expanson s fnanced by a tax on wage ncome the dstortonary tax effect domnates the drect expendture effect. As a result aggregate captal, labor, and output all fall proportonately. An ncrease n g fnanced by the tax on captal ncome rases the long-run margnal physcal product of captal, mplyng that t reduces the captal-labor rato. Although captal declnes unambguously, labor wll do so only f the elastcty of substtuton n producton, ε, s suffcently large. 23 ( ε 1). 24 Whle output also declnes unambguously, the net effect on the factor shares depends upon In contrast to fnancng by reducng transfers, an ncrease n ether τ w or τ, by decreasng the long-run captal stoc, results n an ncrease n wealth nequalty. Wth a wage ncome tax, ϕ% also declnes thus offsettng the wealth effect. Recall that a hgher tax on labor maes the fscal system less progressve, the reason beng that labor ncome s less unequally dstrbuted than s captal ncome. As a result the mpact of the polcy on ψ% combnes the drect effect that tends to reduce post-tax nequalty (captured by the term ~ s (1 ~ ϕ ) /(1 ~ s ) ) and the ndrect on stemmng from the change n ϕ ~, so that post-tax nequalty may ncrease or decrease. In contrast, welfare nequalty s unambguously lower as the effects on ncome dstrbutons combne wth a more dspersed labor supply. The mpact of τ on ~ ϕ s more complex snce t nvolves three effects. Frst, durng the transton to the new steady state wth a lower stoc of captal the dstrbuton of wealth wll become more dspersed. Second, ϕ ~ falls or ncreases dependng on whether as the new steady state generates a lower ~ l. Thrd, the reducton n the captal-labor rato mples a change n the captal share, ~ s, the mpact of whch depends on whether ε > < 1. In the case of the Cobb-Douglas producton functon ths effect s zero and snce n ths case L% τ <, then ϕ% and also falls. For the Cobb-Douglas, the drect redstrbutve effect of the captal ncome tax renforces the pre-tax ncome nequalty effect and post-tax ncome nequalty falls relatve to wealth nequalty. 23 For example, ε > s% s a wea condton to ensure that employment declnes. 24 Ths s not surprsng snce τ s equvalent to a declne n productvty, the effect of whch on factor shares s well nown to depend upon ( ε 1). 21

23 5. Numercal Smulatons To obtan further nsghts nto the dynamcs of wealth and ncome dstrbuton we employ numercal smulatons. These are based on the followng functonal form and parameter values, characterzng the benchmar economy: α 1 α Producton functon: Y = AK L Basc parameters: A = 1.5, α =.36 β =.4, γ = 1. 5, η = 1. 5, θ =. 3, δ =.6 Fscal parameters: τ =.276, τ =.224, τ =.8, g =.15 w c Preferences reman specfed by the constant elastcty utlty functon, (3), wth the ntertemporal elastcty of substtuton1 (1 γ ) =.4, rate of tme preference of 4%, whle the elastcty of lesure n utlty s 1.5, and that wth respect to government expendture.3. The producton functon s of the Cobb-Douglas form, wth dstrbutonal parameter α =.64, and A = 1.5 scales the level of productvty. The deprecaton rate s 6% per annum. 25 These parameters are all very standard and typcal of those found n the lterature. 26 The choce of tax rates s less straghtforward and has generated debate, due to the dffculty of mappng the complextes of the real world tax structure nto a smple one-sector growth model. We use the effectve tax rates on consumpton, labor and captal constructed by McDanel (27), followng the methodology proposed by Mendoza et al. (1994). The rates lsted above are the US averages for the decade A ey feature s that for the US economy τ > τ w, a characterstc that holds unformly snce Fnally, settng government consumpton expendture rate at g =.15 approxmates the US experence n the 199 s. 27 Our measures of nequalty are reported n terms of the standard devaton measures employed n our theoretcal dscusson, wth the ntal standard devaton of relatve captal (wealth) 25 For smplcty we assume that deprecaton costs are not tax deductble. In the event that they are δ =.6 s equvalent to a pre-tax deprecaton rate of around 8.3%. 26 For example, the ntertemporal substtuton of substtuton of.4 s well wthn the range summarzed by Guvenen (26), whle the relatve weght on lesure n utlty s close to the conventonal value of the real busness cycle lterature; see Cooley (1995). The producton elastcty α =. 36 dates bac to at least Kydland and Prescott (1982) and has been adopted consstently snce; see e.g. Papageorgou and Pérez-Sebastan (27) for a recent example. 27 Ths s documented by Turnovsy (24). 22

24 beng normalzed to 1. We suppose that the economy s ntally n steady state, and consder the mpact of polcy changes on both the new steady-state dstrbutons and ther transtonal dynamcs. Lne 1 of Table 3 reports the benchmar equlbrum for the chosen benchmar parameters. There we see that γ = 1.5 together wth η = 1.5 yelds an equlbrum allocaton of labor of 3.7% and an mpled Frsch elastcty of labor supply of around 1.2, both of whch are consstent wth exstng emprcal evdence. 28 The elastcty θ =.3 mples that the optmal rato of publc to prvate consumpton s.3, whch s reasonably close to the observed rato of.28 for the US durng the 199 s. The deprecaton rate n conjuncton wth α s the crtcal determnant of the rate of convergence. Settng δ = 6% yelds a convergence rate of around 5.6%, very close to the value of 5.8% suggested by Evans (1997) as beng a more accurate estmate of the true value. 29 The mpled share of output devoted to transfers s 14.8%, whch mples that total tax revenue s 29.8%, very close to that observed n the US durng the perod One of the ssues confrontng polcymaers n both the Unted States and Europe s the problem of mantanng government spendng, gven the current level of debt. Wthout ntroducng government debt explctly, whch a detaled analyss of ths ssue would requre, our framewor can nevertheless provde some prelmnary nsght nto ths queston and the constrants that current debt levels mpose. More specfcally, the exercses that we have been descrbng formally, and that we analyze numercally, can be vewed as studyng how to fnance government spendng and transfers, whle holdng the long-run government debt to GDP rato constant The Frsch elastcty of labor supply descrbes the elastcty of hours wored wth respect to the wage rate, gven a constant margnal utlty of wealth. For the utlty functon (3) t s ( dl L) ( dw w ) = ( ll)( (1 γ) (1 γ[1 + η]) ). Emprcal estmates vary extensvely across gender and race; see Brownng, Hansen, and Hecman (1999). Our mpled value 1.2 s close to that (1.4) obtaned n the early study of Lucas and Rappng (197). It s also consstent wth more recent estmates (around 1) obtaned by Chang and Km (26) and Kmball and Shapro (28). 29 Orgnal benchmar estmates of the convergence rate of around 2% were establshed by Barro (1991), and Manw, Romer and Wel (1992), among others. Subsequent studes have suggested that these estmates may be substantally downwardly based; see e.g. Islam (1995), Casell, Esquvel, and Lefort (1996), and Evans (1997). 3 The mpled transfer share s hgher than what we tend to observe n the data. The reason for ths s that we are abstractng from publc nvestment. The dstrbutonal mpact of government nvestment expendtures s a complex queston whch s beyond the scope of ths paper; see for example Calderon and Servén (24). 31 To see ths, assume that government bonds, B say, are perfect substtutes for captal and are taxed at the same rate. Recallng (1a), the steady-state government budget constrant (8 ) becomes τ s + τ (1 s) + τ C F = g+ τ + β B F. 23 w c

25 5.1. Fnancng an ncrease n government expendture Table 2 reports the effect of an ncrease n government expendture from ts base rate of 15% of output by 5 percentage ponts to 2% of output. To consder the dstrbutonal mplcatons of the expendture ncrease, we defne the ntal steady-state dstrbutons of pre-tax ncome and post-tax ncome (pror to any shoc) by the respectve quanttes: % σ ϕσ 1 sk ( %, L% ) = 1 1 σ % σ y,,, 1 l% 1+ η % σ y, 1 τ w ψ σ = 1 1 σ s % τ + % τ a y,, ( 1 ) (1 s) ( 1 ), w, where we normalze the ntal gven dstrbuton of captal by σ, = 1.Snce τ > τ w n the benchmar economy, the fscal system entals drect redstrbuton and the ntal equlbrum mples a standard devatons of pre-tax and post-tax ncome dstrbutons of % σ y, =.165 > % σ y, =.144, both of whch overstate the degree of welfare nequalty, σ u = The next three rows report the numercal effects of fnancng an (arbtrary) ncrease n government expendture of 5 percentage ponts by a reducton n transfers (.e. a lump-sum tax), an ncrease n the wage ncome tax, and an ncrease n the captal ncome tax, respectvely. Lump-sum tax fnancng entals an ncrease n output, as government expendture encourages prvate consumpton, leadng to an ncrease n labor supply and captal stoc, whle the dstortons created by captal or wage ncome taxes result n reduced output and less captal. These results are qualtatvely dentcal to the theoretcal results reported n Table 2. The effects on dstrbuton of the three modes of fnancng contrast sharply, n lne wth Table 2. In all cases, the long-run changes n wealth nequalty are small, reflectng the fact that most of the adjustment n L occurs nstantaneously. As a result, the changes n ncome dstrbuton are domnated by the relatve ncome effects, ϕ% and ψ%. Thus, whle lump-sum taxaton results n a 32 The fnal column reports the standard devaton of labor supply. Informaton on ths s sparse. The only estmate we have been able to obtan s one of around.361 usng German data for obtaned by Heer and Maussner (29, p.381), to whch our soluton s qute close. But gven the role of labor supply t would seem that the mplcatons for the dsperson of labor supply across agents merts more detaled emprcal nvestgaton. 24