Distributional Dynamics in a Neoclassical Growth Model: The Role of Elastic Labor Supply * Stephen J. Turnovsky University of Washington, Seattle

Transcription

1 Dstrbutona Dynamcs n a Neocassca Growth Mode: The Roe of Eastc Labor Suppy * Stephen J. Turnovsy Unversty of Washngton, Seatte Ceca García-Peñaosa CNRS and GREQAM Revsed verson June 27 Abstract: We examne the evouton of the dstrbutons of weath and ncome n a Ramsey mode n whch agents dffer n ther nta capta endowment and where the abor suppy s endogenous. The assumpton that the utty functon s homogeneous mpes that the macroeconomc equbrum s ndependent of the dstrbuton of weath and aows us to characterze fuy ncome and weath dynamcs. We fnd that athough the dynamcs of the dstrbuton of weath are smar under fxed and fexbe abor, those of the ncome dstrbuton are not. In response to a structura change, ncome nequaty may move n opposte ways dependng on whether or not the abor suppy s fxed. JEL Cassfcaton Numbers: D31, O41 Key words: weath dstrbuton; ncome dstrbuton; endogenous abor suppy; transtona dynamcs. * Turnovsy s research was supported n part by the Castor endowment at the Unversty of Washngton. The paper has benefted from presentatons at GREQAM, McG Unversty, the Unversty of Yor, the 26 Conference of the Socety for Computatona Economcs, hed n Lmasso, Cyprus, as we as from the comments of two referees. We are aso gratefu to Carne Nourry for her comments.

2 1. Introducton The Ramsey (1928) mode has been wdey empoyed by macroeconomsts for amost eghty years, and over tme has become the standard framewor for addressng a number of mportant ssues n macrodynamcs and growth theory. Whe t typcay treats a agents as dentca (the representatve agent paradgm), f preferences are homothetc the framewor can ready ncorporate certan forms of agent heterogenety; see Chatterjee (1994), Case and Ventura (2), and Maar and Maar (21). An mportant queston whch then naturay arses concerns the consequences of ths heterogenety for the dynamcs of weath and ncome dstrbuton. Ths s partcuary pertnent n ght of observed ong-run changes n weath and ncome nequaty. 1 Whe an extensve terature, datng bac to Stgtz s (1969) semna contrbuton, exsts on the dynamcs of weath dstrbuton, the evouton of ncome nequaty has receved much ess attenton. In ths paper we anayze the dynamcs of weath and ncome dstrbuton n a Ramsey mode wth eastc abor suppy, examnng n partcuar the sgnfcance of the endogenety of abor for ncome dstrbuton. The source of the heterogenety we ntroduce s the agent s nta endowment of capta (weath). Assumng that preferences are represented by a utty functon that s homogeneous n consumpton and esure, aows aggregaton as n Gorman (1953) or Esenberg (1961), and generates the representatve-consumer characterzaton of the macroeconomc equbrum. 2 We can then represent the macroeconomc equbrum n terms of a smpe recursve structure. Frst, the dynamcs of the aggregate stoc of capta and abor suppy are jonty determned, ndependenty of dstrbutona concerns. Then, the cross-sectons of ndvdua ncome and weath and ther dynamcs are characterzed n terms of the aggregate magntudes, thereby enabng us to address dstrbutona ssues n a tractabe way. We characterze both the tme paths of the dstrbutons of weath and ncome, as we as ther steady-state dstrbutons, and compare ther responses to a number of structura changes under 1 For exampe, Petty and Saez (23) document dramatc ong-run changes n ncome dstrbuton for the Unted States over the perod Ths type of utty functon s common n many areas of macroeconomcs, most notaby n busness cyce theory but aso n the endogenous growth terature. See Kng and Rebeo (1999) for a survey on busness cyce theory, and Ladrónde-Guevara et a. (1997, 1999, 22), Turnovsy (2), and García-Peñaosa and Turnovsy (26a, 27) for endogenous growth modes wth esure. 1

3 both fxed and fexbe abor suppy. We fnd that athough for pausbe parameter vaues the dynamcs of weath nequaty are reatvey nsenstve to the nature of abor suppy, whether abor suppy s fxed or fexbe can have a dramatc mpact on the dynamcs of ncome dstrbuton. A cruca mechansm determnng the evouton of ncome nequaty s the reatonshp we derve between agents reatve weath (capta) and ther reatve aocaton of tme between wor and esure. Ths reatonshp s very basc and has a smpe ntuton. Weather agents have a ower margna utty of weath. They therefore choose to ncrease consumpton of a goods ncudng esure, and reduce ther abor suppy. Indeed, the roe payed by abor suppy n ths mode s anaogous to ts roe n other modes of capta accumuaton and growth, where t provdes the cruca mechansm by whch demand shocs nfuence the rate of capta accumuaton. 3 Ths mechansm s aso centra to emprca modes of abor suppy based on ntertempora optmzaton; see e.g. MaCurdy (1981). There s substanta emprca evdence documentng ths negatve reatonshp between weath and abor suppy. Hotz-Ean, Joufaan, and Rosen (1993) fnd evdence to support the vew that arge nhertances decrease abor 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 weath on abor suppy and retrement, fndng a substanta negatve effect on abor partcpaton. Agan, Chéron, Haraut, and Langot (23) use French data to anayze the effect of weath on abor maret transtons, and fnd a sgnfcant weath effect on the extensve margn of abor suppy. Overa, these studes and others provde compeng evdence n support of the weath-esure mechansm beng emphaszed n ths paper. Our anayss shows that endogenzng the abor suppy has three substanta consequences for the dynamcs of ncome dstrbuton. Frst, whereas wth fxed abor suppy the dstrbuton of ncome evoves contnuousy, when abor s supped eastcay, the nta jump n abor suppy foowng a structura change mpes that ncome dstrbuton w aso jump, eadng to much arger movements n the dstrbuton. As a resut, ncome nequaty may adjust non-monotoncay, 3 For exampe, n the standard Ramsey mode, government consumpton expendture w generate capta accumuaton f and ony f abor s supped eastcay. Wth neastc abor suppy t w smpy crowd out an equvaent amount of prvate consumpton. The ey factor s the weath effect and the mpact ths has on the abor-esure choce, as emphaszed by both Ortguera (2) and Turnovsy (2). 2

4 ncreasng on mpact and decnng thereafter, somethng that s not possbe wth fxed abor. Second, t s possbe for a structura change that eads to a decne n ncome nequaty wth fxed abor suppy to ead to greater ncome nequaty when abor s supped eastcay (and vce versa). The reason for ths s the change n the dstrbuton of esure tme, and hence of wor hours. Snce the reducton n the wage due to a hgher (average) abor suppy nduces poorer agents to wor (reatvey) harder, an aggregate abor suppy change w resut n a change n the dstrbuton of abor ncomes. Ths effect can be strong enough to offset the mpact of decnng weath nequaty, whch s the ony effect to operate when abor suppy s fxed. The extent to whch ths may occur depends upon the form of the producton functon, as we as the nature of the shoc. For exampe, for a Cobb-Dougas producton functon, a reducton n the rate of tme preference from.4 to.2 w ead to a ong-run decne n ncome nequaty of around 7% wth fxed abor suppy, but t w yed a ong-run ncrease n ncome nequaty of 1% f abor suppy s fexbe. Thrd, f abor suppy s fxed, a three structura changes we consder a 5% ncrease n productvty, a 1.5 percentage pont decne n popuaton growth, and a 2 percentage pont decne n the rate of tme preference have amost dentca mpacts on ncome nequaty. In contrast, the responses of ncome nequaty wth eastc abor suppy can vary sharpy, dependng on the source of the shoc. For exampe, wth a hgh eastcty of substtuton n producton, a reducton n the rate of popuaton growth or n the rate of tme preference woud resut n a sma ncrease n ong-run ncome nequaty wth fxed abor; wth fexbe abor the decrease n the popuaton growth rate woud reduce nequaty by over 3%, whe the preference shoc woud ncrease nequaty by amost 24%. The concuson to emerge from these exampes s that to treat abor suppy as fxed, when n fact t s supped eastcay, may yed serousy mseadng concusons wth respect to the dynamc evouton of ncome nequaty. Our paper contrbutes to the recent terature characterzng dstrbutona dynamcs n growth modes. 4 Ths queston was frst examned n Stgtz (1969) usng a form of the Soow mode. Eary wor examnng the evouton of the dstrbuton of weath n the Ramsey mode assumed agents that dffer n ther rate of tme preferences. In ths framewor, the most patent agent ends up 4 See Bertoa, Foem, and Zwemüer (26) for a survey. 3

5 hodng a the capta n the ong-run, athough the presence of progressve taxaton or capta maret mperfectons can prevent a degenerate dstrbuton of weath; see Becer (198), Becer and Foas (1987), Sorger (22). An extensve terature has examned weath dynamcs when agents have the same rate of tme preference but dffer n ther nta capta endowments. Case and Ventura (2) consder the dynamcs of weath and ncome dstrbuton n the Ramsey mode, though they restrct ther anayss to exogenous abor suppy. Startng wth Chatterjee (1994), severa papers have examned the mpcaton of assumng a mnmum consumpton requrement. 5 Chatterjee fnds that under reasonabe parameter specfcatons weath nequaty ncreases durng the process of deveopment. The mechansm drvng ths resut s the exstence of a mnmum consumpton requrement whch tends to mae the propensty to save ncreasng n ndvdua weath, and hence exacerbates nequates as the economy accumuates capta. Ths terature has, however, consdered ony modes wth fxed abor, and has examned the evouton of weath but not ncome nequaty. Cosest to our anayss are Sorger (2) and Maar and Maar (21). Sorger studes the Ramsey mode wth endogenous abor and heterogeneous weath endowments. Hs specfcaton assumes that the utty functon s non-homogeneous n ts two arguments, and as a resut the evouton of aggregate varabes depends on the entre dstrbuton of weath at each pont n tme. Ths dfference has two mportant mpcatons. Frst, Sorger fnds a correaton between per capta ncome eves and the dstrbuton of weath, whch depends on the eastcty of ntertempora substtuton. In contrast, for our chosen utty specfcaton, a partcuar eve of per capta output s compatbe wth any dstrbuton of weath, dependng on the nta dstrbuton. Second, Sorger focuses on the statonary state, as the nterdependence of the macroeconomc equbrum and dstrbuton renders the anayss of the dynamcs ntractabe. Maar and Maar (21) examne a neocassca mode wth endogenous abor and weath heterogenety. As n our anayss, homothetc preferences permt a representatve-consumer characterzaton of the macroeconomc equbrum. However, they as a dfferent queston snce they focus on the abor maret mpcatons of busness cyces when agents are heterogeneous, rather 5 See aso Chatterjee and Ravumar (1999), Avarez-Peáez and Díaz (25), and Obos-Homs and Urruta (25). 4

6 than on the dynamcs of the weath dstrbuton. Maar, Maar, and Mora (25) examne the busness-cyce behavor of the dstrbutons of ncome and weath, and n an earer verson of ther paper consdered the case of endogenous abor. Ther concuson that endogenous and exogenous abor have the same mpcatons for dstrbutona dynamcs s, however, due to the fact that they consder a Cobb-Dougas producton functon and a technoogy shoc. We aso fnd that under these assumptons both modes ndeed yed smar dstrbutona dynamcs. However, by generazng the producton functon and extendng the range of shocs we fnd that the endogenety of abor has mportant consequences for the dynamcs of ncome dstrbuton. The paper s organzed as foows. Secton 2 descrbes the economy and derves the macroeconomc equbrum. Secton 3 characterzes the dstrbutons of weath and ncome and derves the man resuts of the paper. Secton 4 derves the effects of aternatve structura changes on the ong-run dstrbutons of weath and ncome, whch are then ustrated n Secton 5 wth a number of numerca exampes. Secton 6 concudes, whe nsofar as possbe the technca detas are reegated to an Appendx. 2. The Anaytca Framewor We begn by settng out the components of the mode. 2.1 Technoogy and factor payments Aggregate output s produced by a snge representatve frm accordng to a standard neocassca producton functon, so that 6 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 capta, abor suppy and output. The wage rate, w, and the return to capta, r, are determned by the margna physca products of abor and capta, w( K, L) = F ( K, L) w = F, w = F (2a) L K LK L LL 6 That s, both factors of producton have postve, but dmnshng, margna physca products and the producton functon exhbts constant returns to scae, mpyng F F 2 LL KK LK F =, from whch F > mmedatey foows. 5 LK

7 r( K, L) = F ( K, L) r = F <, r = F > (2b) K K KK L KL 2.2 Consumers At tme, the economy s popuated by N ndvduas, represented as a contnuum, each ndexed by I, and dentca n a respects except for ther nta endowments of capta, K,. Each ndvdua defnes a famy. Popuaton grows unformy across a fames at the exponenta rate, n, so that at tme t famy has grown to nt nt e and the tota popuaton of the economy s Nt () Ne. Each member of a gven famy has the same capta stoc, athough the dstrbuton of capta dffers across fames. From a dstrbutona perspectve we are nterested n the share of famy s capta stoc of the tota capta stoc n the economy. To ths end we dentfy the foowng quanttes: () Indvdua hods K ( t ) unts of capta at tme t, so that the amount hed by famy s K ( ) nt t e. Ths depends upon the capta of each representatve member of the famy pus the fact that the sze of the famy s growng exponentay over tme. () The tota amount of capta n the economy at tme t s the tota capta stoc owned by the N fames and can be expressed as N T nt () = () K t K t e d () The average per capta amount of capta s 1 1 Kt K ted K td N N nt () = () = () nt Ne N. Snce the economy s growng we must be carefu n defnng the dstrbuton of the capta stoc. We sha defne the share of capta owned by famy as K () t e K () t e K () t K () t nt nt () = = = T K () t N 1 N 1 N nt K () () K() t t e d K t d N N t Wth a agents n the dfferent fames growng at the same rate, we can express the dstrbuton n terms of reatve famy shares, (t). Note that reatve capta has mean 1. We denote ts nta dstrbuton functon by H ( ), the nta densty functon by h ( ), and the nta (gven) standard 6

8 devaton of reatve capta by σ,. We now focus on a partcuar agent. Each ndvdua s endowed wth a unt of tme that can be aocated ether to esure,, or to wor, 1 L. The agent maxmzes fetme utty, assumed to be a functon of both consumpton and the amount of esure tme, n accordance wth the soeastc utty functon 1 η γ βt max ( C( t) ) e dt, wth < γ < 1, η >, γη < 1 (3) γ where 1 (1 γ ) equas the ntertempora eastcty of substtuton. 7 The preponderance of emprca evdence suggests that ths s reatvey sma, certany we beow unty, so that we sha restrct γ <. 8 The parameter η represents the eastcty of esure n utty. Ths maxmzaton s subject to the agent s capta accumuaton constrant K ( t) = ( r( t) n) K ( t) + w( t)(1 ( t)) C ( t) (4) The frst-order condtons from the consumer probem are standard and are reported n the Appendx; see (A.1) - (A.4). 9 From these equatons we obtan C η = wk (, ) (5) C λ ( γ 1) + ηγ = = β + n r( K, ) C λ (6) where λ s agent s shadow vaue of capta. 1 Equaton (5) equates the margna rate of substtuton between consumpton and esure to the prce of esure, whe (6) s the Euer equaton modfed to tae nto account the fact that esure changes over tme. The mportant pont about (6) s that each agent, rrespectve of capta endowment, chooses the same growth rate for the shadow vaue of capta. Usng (5) we may wrte the ndvdua s accumuaton equaton, (4), n the form 7 For convenence we focus on a constant eastcty utty functon whch s homogeneous of degree γ (1 + η) n C and. In the Appendx, we ndcate how the anayss s easy adapted to any arbtrary homogeneous utty functon. 8 See e.g. the dscusson of the emprca evdence summarzed and reconced by Guvenen (26). 9 Tme dependence of varabes w be omtted whenever t causes no confuson. 1 We are assumng an nterna souton exsts, and hence wrte the Euer equaton wth equaty. On the roe of borrowng constrants and corner soutons for the dynamcs of weath see Hernández (1991). 7

9 K K w( K, ) 1+ η = r( K, ) n + 1 K η (7) Taen together wth the correspondng condtons for the aggregate economy we can derve the macroeconomc equbrum and the dynamcs of the aggregate economy. Havng determned these, we sha then obtan the dynamcs of the dstrbuton of capta and ncome Dervaton of the macroeconomc equbrum In genera, we sha defne economy-wde averages as 1 1 Z t Z t e d Z t d N N nt () = () () nt Ne = N Summng over househods, equbrum n the capta and abor marets s descrbed by 1 N K = K ( t) d (8a) N 1 L= = t d (8b) N 1 (1 ( )) N Equaton (8b) gves the reatonshp between average esure and the average abor suppy. Note that n equatons (2) we have defned the wage and the nterest rate, w, r, and expressed them as functons of average empoyment, L. From (8b), we can equay we wrte them as functons of aggregate esure tme, (1 ), namey, w= w( K, ) and r = r( K, ). The ey eement aowng aggregaton s that a agents choose the same growth rate for the shadow vaue of capta, as seen n (6). As a resut of ths, we can then show (see Appendx) that C C = ; = C C for a (9) That s, a agents w choose the same growth rate for consumpton and esure, mpyng further that average consumpton, C, and esure,, w aso grow at the same common growth rates. Now turn to the aggregates. Summng (5) over a agents, the aggregate economy-wde consumpton-capta rato s 8

10 C η = w( K, ) (5 ) whe summng over (6) and (7) yeds the aggregate Euer and capta accumuaton equatons C ( γ 1) + ηγ = β + n r( K, ) C (6 ) K K w( K, ) 1+ η = r( K, ) n + 1 K η (7 ) In the Appendx we show how ths can be reduced to a par of dynamc equatons n K and, [(A.8) and (A.9)], whch are ndependent of the dstrbutona aspects and dentca to those n the representatve agent economy Aggregate equbrum dynamcs The dynamc behavor of the aggregate economy as represented by (A.8) and (A.9) has been studed esewhere; see, for exampe Turnovsy (22). Assumng that the economy s stabe, the aggregate quanttes converge to a steady state characterzed by a constant average per capta capta stoc, abor suppy, and esure tme, denoted by K, L and, respectvey. Settng K = =, the steady state s summarzed by F K ( K, L) = β + n (1a) FL (, ) ( K, L ) FKL nk = (1b) η L + = 1 (1c) These equbrum reatons are standard. Equaton (1a) s the modfed goden rue, equatng the margna product of capta to the dscount rate, adjusted for popuaton growth. The second s smpy a reformuaton of the frst-order condton (5 ) equatng the margna rate of substtuton of consumpton and esure to the prce of esure (the rea wage), where the eft-hand sde captures the fact that n steady state consumpton s equa to output mnus the amount needed to eep per capta 9

11 capta constant wth a growng popuaton. The thrd s just the abor maret cearng condton. Usng (1a) and (1b), whe recang the homogenety of the producton functon, we mmedatey nfer that: 11 > η (11) 1 +η Ths nequaty yeds a ower bound on the steady-state tme aocaton to esure that s consstent wth a feasbe equbrum. As we w see beow, ths condton pays a crtca roe n characterzng the dynamcs of dstrbuton. In order to descrbe the dynamcs of the dstrbuton of capta and ncome, we frst need to obtan the dynamcs of the aggregate magntudes. Lnearzng equatons (A.8) and (A.9) around steady state yeds the oca dynamcs for K and, K a a K K = a21 a 22 (12) where a 11, a22, a12, a21 are defned n the Appendx. There we show that a 11 a22 a12a21 <, mpyng that the steady state s a sadde pont. The stabe path for K and can be expressed as Kt () = K + ( K Ke ) μt (13a) a ( t) = + μ a 22 μ a11 ( K( t) K ) = + ( K( t) K ) 21 a 12 (13b) where μ < s the stabe egenvaue. From the sgn pattern estabshed n the Appendx, ( a22 μ) >, mpyng that the sope of stabe arm depends nversey upon the sgn of a 21. The sgn of ths expresson refects two offsettng nfuences of capta on the evouton of esure. On the one hand, an ncrease n capta owers the return to capta and hence the return to consumpton, thereby reducng the growth rate of consumpton and rasng the desre to ncrease esure. At the same tme, the hgher capta stoc, by reducng the productvty of abor, rases the benefts from ncreasng abor, thus reducng the growth of esure. As we show n the Appendx, whch effect domnates 11 We can combne (1a) and (1b) to yed: ( F ηk )( (1 + η) η) = β from whch (11) foows. L 1

12 depends upon the underyng parameters and n partcuar upon the eastcty of substtuton n producton. There we demonstrate that for pausbe cases [ncudng the conventona case of Cobb- Dougas producton and ogarthmc utty ( ε = 1, γ = )] a 21 <, n whch case the stabe ocus s postvey soped; accumuatng capta s therefore assocated wth ncreasng esure. 12 As we w see beow, the evouton of average esure over tme s an essenta determnant of the tme path of weath and ncome nequaty. For expostona convenence we sha restrct ourseves to what we vew as the more pausbe case of a postve soped stabe ocus, (13b). Snce ths reatonshp hods at a tmes, we have ) = a μ a ( K K ) 21 ( 22 (13b ) Thus consder a stuaton n whch the economy s subject to a structura shoc that resuts n an ncrease n the steady-state average per capta capta stoc reatve to ts nta eve ( K < K ). The shoc w ead to an nta jump n average esure, such that () <, so that, thereafter, esure w ncrease monotoncay durng the transton; an anaogous reatonshp appes f K > K. 3. The dstrbuton of ncome and weath 3.1. The dynamcs of the reatve capta stoc To derve the dynamcs of ndvdua s reatve capta stoc, () t K () t K() t, we combne (7) and (7 ) to obtan wk (, ) 1 1 () t = 1 θ (14) K η η where K, evove n accordance wth (13a, 13b) and the nta reatve capta, s gven from the nta endowment. Snce = we may wrte = θ where 1 N N θ d = 1 12 See Appendx for further dscusson. 11

13 and θ s constant for each, and yet to be determned. To sove for the tme path of the reatve capta stoc, we frst note that agent s steady-state share of capta satsfes θ = for each η η or, equvaenty η = ( 1) 1+ η for each (15) Recang (11), ths equaton mpes that the hgher an agent s steady-state reatve capta stoc (weath), the more esure he chooses and the ess abor he suppes. Ths reatonshp s a crtca determnant of the dstrbutons of weath and ncome and expans why the evouton of the aggregate quanttes such as K and are unaffected by dstrbutona aspects. There are two ey factors contrbutng to ths: () the nearty of the agent s abor suppy as a functon of hs reatve capta, and () the fact that the senstvty of abor suppy to reatve capta s common to a agents, and depends upon the aggregate economy-wde esure. As a consequence, aggregate abor suppy depends ony on the aggregate amount of capta but not on ts dstrbuton amongst agents. It s mportant to note here that ths resut hods for any utty functon that s homogenous of degree, b say, n consumpton and esure, and n the Appendx we show that ths s ndeed the case. To anayze the evouton of the reatve capta stoc, we nearze equaton (14) around the steady-state K,,,, n (15). In the Appendx we show that the stabe souton to the resutng equaton s () t 1 = δ ()( t 1) (16) where 1 FL ( K, L) t ( ) δ () t 1 + 1, (17) β μ K 12

14 Settng t = n (16) and (17), we have 1 FL ( K, L ) (), 1 = δ ()( 1) = 1+ 1 ( 1) β μ K (18) where, s gven from the nta dstrbuton of capta endowments. The evouton of agent s reatve capta stoc s determned as foows. Frst, gven the tme path of the aggregate economy, and the dstrbuton of nta capta endowments, (18) determnes the steady-state dstrbuton of capta, ( 1), whch together wth (16) then yeds the entre tme path for the dstrbuton of capta. Usng (16) (18), and equatons (13), descrbng the evouton of the aggregate economy, we can express the tme path for ( t ) n the form δ ( t) 1 ( t) μt ( t) = ( ) = (, ) = e (, () 1 () δ ), from whch we see that ( t ) aso converges to ts steady state vaue at the rate μ. (19) We can aso determne the tme path for the ndvdua s esure (abor suppy). Frst, havng determned ( 1), (15) yeds agent s steady-state esure aocaton,, whch, nowng the economy-wde average,, determnes hs constant reatve esure tme θ, namey 1 η 1 1 η θ 1= 1 ( 1) = 1 (, 1) 1 + η δ() 1+ η (2) Thus, nowng the tme path for the aggregate esure aocaton, t ( ), the tme path for ( t) θ ( t) mmedatey foows. We see from (2), n conjuncton wth (11), that any agent whose steady-state capta stoc exceeds the economy-wde average w enjoy above average esure tme throughout the transton. Because of the nearty of (16), (18), and (19), we can mmedatey transform these equatons nto correspondng resuts for the standard devaton of the dstrbuton of capta, whch serves as a convenent measure of weath nequaty. 13 Specfcay, correspondng to these three equatons we obtan 13 Because of the nearty of (t) n, the same anayss appes n terms of more conventona Gn coeffcents., 13

15 σ () t = δ() t σ (16 ) σ, = δ() σ (17 ) μt σ t) σ = e ( σ σ ) (19 ) (, The aocaton of weath then converges to a ong-run dstrbuton. Moreover, from (19) t foows that the ranng of agents accordng to weath s the same as n the nta dstrbuton. Havng estabshed the exstence of a ong-run dstrbuton of weath, we can compare t to the nta dstrbuton. From equatons (16 ) and (19 ) we see that σ () t > σ, f and ony f δ () < δ ( t),.e. f and ony f ( t) < (), and that σ > σ, f and ony f δ () < 1. Together wth (13b ), and snce esure s monotoncay ncreasng or decreasng aong the transton path, ths mpes the foowng 14 Proposton 1 (Weath dynamcs): The ong-run dstrbuton of weath converges to a steady state dstrbuton. If the economy starts beow (above) the steady state,.e. K < K ( K > K ), then weath nequaty w decrease (ncrease) durng the transton, and the ong-run dstrbuton of weath w be ess (more) unequa than s the nta dstrbuton. The ntuton for ths resut can be easy seen by notng, from equaton (A.16) n the Appendx, that, ( )( ) sgn( ) = sgn 1 () Reca that f the economy converges to the steady state from beow, then () <. Then for peope who end up above the mean eve of weath, ther weath w have decreased durng the transton, >, whe for peope who end up beow the mean eve of weath, ther weath w have decreased,, >, mpyng a narrowng of the weath dstrbuton. Ths resut contrasts wth the evouton of the dstrbuton of weath n the Ramsey mode 14 We restrct our focus to what we have dentfed as the norma case of a postve adjustment between aggregate esure and capta 14

16 wth neastc abor suppy. In ths case, f the eastcty of substtuton s greater than or equa to one,.e. ε 1, the dstrbuton of weath w become more equa durng the transton from beow, but for ε < 1, the dstrbuton coud wden. 15 The reason for ths s that a ow eastcty of substtuton mpes fast wage growth as the economy accumuates capta. Wth suffcenty hgh wage growth, poor consumers may choose to ds-save eary n ther fe-tmes and fnance current consumpton wth ther (hgh) future wages. As a resut, the dstrbuton of capta becomes more unequa. Wth endogenous esure, ths effect s offset by abor suppy responses, as hgher future wages tend to ncrease both current consumpton and future esure. The desre to ncrease esure n the future prevents the reducton n the rate of capta accumuaton of capta-poor agents, and hence the weath dvergence, that occurs when ndvduas cannot change wor-hours. Nevertheess, capta accumuaton w be assocated wth ncreasng weath nequaty ony for unreastc vaues of the eastcty of substtuton n producton and the ntertempora eastcty of substtuton n consumpton, 16 and as we w see beow, for pausbe parameter vaues, weath dstrbuton behaves n a smar way wth fxed and wth fexbe abor. The standard devaton of the dstrbuton of esure tme can be obtaned from (2), and usng the fact that σ, = δ() σ, can be expressed as η σ θ 1 = 1 σ (21) 1+ η whch n conjuncton wth (11) mpes that there s a postve correaton between the steady state dstrbutons of weath and esure tme. That s, there s a negatve correaton between the dstrbuton of abor suppes and that of weath, as rcher ndvduas chose to consume more esure than poorer ones. Note aso that the term n bracets n (21) s smaer the ower the steady state eve of average esure, mpyng that a hgher average abor suppy w be assocated (for gven σ ) wth a more unequa dstrbuton of wor-tme. 15 See Case and Ventura (2). 16 See García-Peñaosa and Turnovsy (26b). 15

17 3.2. Income Dstrbuton We defne the ncome of ndvdua at tme t as Y ( t) = r( t) K ( t) + w( t)(1 ( t)), average economy-wde ncome as Y ( t) = r( t) K( t) + w( t)(1 ( t)), and we are nterested n the evouton of reatve ncome, defned as y ( t) Y ( t) / Y ( t). Lettng s( t) FK K / Y denote the share of output gong to capta, and recang that = θ, reatve ncome may be expressed as t () y() t 1 = st ()( () t 1) + (1 st ()) (1 θ) 1 t ( ) (22) The reatve ncome of agent has two components, reatve capta ncome, captured by the frst term n (22), and reatve abor ncome, refected n the second term. The capta share determnes the reatve contrbuton of capta and abor to overa ncome, for gven ndvdua endowments. The endogenety of abor mpes that for a gven capta share and dstrbuton of capta, there w be ess nequaty than under exogenous abor. To see ths note that wth fxed abor, reatve ncome s gven by y ( t) 1 = s( t)( ( t) 1) + (1 s( t)) as a agents suppy the same amount of abor. When abor s fexbe, poor agents suppy more abor than do the rch (see equaton (2)), whch partay offsets the effect of the unequa dstrbuton of capta, as we can see f we rewrte (22) as ( t) 1 1 η y ( t) 1 = s( t)( ( t) 1) (1 s( t)) ( 1). (22 ) 1 ( t) 1+ η Usng equaton (15) to substtute for θ and (16), we may wrte (22) n the form y ( t) 1 = ϕ ( t)( ( t) 1), (23) where t () 1 η 1 ϕ() t 1 (1 s()) t (24) 1 t ( ) 1 + η δ( t) Agan, because of the nearty of (23) n ( ( t) 1) we can express the reatonshp between reatve ncome and reatve capta n terms of correspondng standard devatons of ther respectve dstrbutons, namey 16

18 σ () t = ϕ() t σ () t (23 ) y From nequaty (11) the term n square bracets n equaton (24) s postve and hence ϕ ( t) < 1, mpyng that ncome s more equay dstrbuted than s capta. Lettng t, we can express the steady-state dstrbuton of ncome as σ = ϕσ (23 ) y where 1 1 s 1 FL ( K, L ) ϕ = m ϕ( t) = 1 = 1 t 1+ η η F( K, L ). From (23) we can compare the ong-run dstrbuton of ncome to the nta one, namey σ σ y y, ϕ σ ) 1+ η FL ( K, L) / F( K, L σ = = (25) ϕ σ 1+ η F ( K, L ) / F( K, L ) σ, L, where the subscrpt dentfes the nta dstrbuton, from whch we nfer that n genera ( y ) = ( y, ) sgn 1 sgn 1. The dstrbuton of ncome hence converges to a ong-run dstrbuton such that the reatve ranng of agents accordng to ncome s the same as that of capta, as we as that of the nta ncome dstrbuton. Whether the ong-run dstrbuton, foowng a structura change, s more or ess unequa than the nta dstrbuton depends on the ong-run change n the dstrbuton of capta, as refected n σ,, and factor returns, as refected n ϕ ϕ σ. As we w ustrate n Secton 5 beow, any shoc eads to an nta jump n the dstrbuton of ncome, after whch t evoves contnuousy, n response to the evouton of the dstrbuton of capta and factor returns. These dynamcs can be seen most convenenty by consderng the tme dervatve of equaton (22), namey dy() t d() t 1 θ d() t () t ds() t = st () + (1 st ()) + () 1 2 t + ( θ 1) dt dt dt 1 ( t ) dt ( 1 t ( )) (26) The equaton ndcates how the evouton of the reatve ncome of agent depends upon two factors, the evouton of reatve capta ncome, refected n the frst term n (26), and that of reatve abor 17

19 ncome. The atter can be expressed as a functon of the evouton of aggregate esure (.e. abor suppy), and of the reatve rewards to capta and abor, as refected by the capta share, s(t). Ceary, the term d ( t) / dt s not present when the abor suppy s exogenous. It s usefu to start by examnng what happens for a Cobb-Dougas producton functon. In ths case the capta share remans constant, and whether ncome nequaty ncreases or decreases depends on whether the economy converges to the steady state from beow or from above. Consder an economy that starts beow the steady state, so that K < K. Then () < and esure s rsng, d / dt >, whe weath nequaty s decreasng. Consder an agent wth above average weath, ( 1) >, then d / dt < and ( θ 1) >, mpyng that the frst two terms n (26) are negatve and that the reatve ncome of the agent s decreasng durng the transton. The opposte woud be true for an agent wth weath beow average, ( 1) <, and hence ncome nequaty w decne durng the transton to the steady state from beow. For an economy that starts above the steady state,.e. for K > K, then () >, and together wth the fact that weath nequaty s ncreasng (see Proposton 1) ncome nequaty w be rsng durng the transton. The evouton of factor shares may renforce or offset these effects. For an economy that converges from beow, a fang capta share, ds / dt <, woud renforce the mpact of the dstrbuton of weath and ncome nequaty w decne over tme. If the capta share rses over tme, ds / dt >, and ths effect w be offsettng. If ths atter effect domnates, the dstrbuton of ncome woud become ess equa over tme. Proposton 2 (Income dynamcs): The evouton of ncome nequaty for an economy that converges to ts steady state from beow,.e. K < K, (respectvey, from above,.e. K > K ) s drven by three factors: () decreasng (ncreasng) weath nequaty, whch tends to reduce (rase) ncome nequaty; () ncreasng (decreasng) esure, whch tends to ncrease (decrease) the reatve abor ncome of the capta-poor and hence reduce(rase) ncome nequaty; () the change n the share of capta n ncome, whch depends both on whether the economy s convergng from beow or above, and on the eastcty of substtuton 18

20 n producton. If the share of capta s constant, ncome nequaty w decrease (ncrease) durng the transton to the steady state from beow (above). Proposton 2 has two mpcatons. Frst, an economy may experence epsodes of ncreasng and epsodes of decreasng ncome nequaty. To see ths, consder an economy whch s beow ts steady state and has an eastcty of substtuton greater than one. As we saw above, the frst two terms n (26), mpy that nequaty tends to fa. However, the hgh eastcty of substtuton mpes that as capta accumuates the share of capta ncreases, tendng to mae the dstrbuton of ncome more dspersed. At dfferent stages, one or the other effect may domnate, mpyng epsodes of rsng or fang nequaty. The second mpcaton concerns the dfferences between the cases of neastc and eastc abor suppy. Wth exogenous abor, there s no change n esure tme. Hence, the evouton of the ncome dstrbuton s drven by two forces, the change n weath nequaty and that of abor share. Wth a hgh eastcty of substtuton n producton, these two forces have opposte sgns, and t s possbe that the second domnates, mang the dstrbuton of ncome more dspersed. When the abor suppy can respond, the changes n average esure w change the dstrbuton of wor-tme and tend to renforce the weath-dstrbuton effect. The presence of ths effect mpes that ncome nequaty may move n opposte drectons dependng on the eastcty of esure, as we w see n the numerca exampes n Secton Long-run adjustments of weath and ncome nequaty To ustrate the dynamc adjustments of weath and ncome dstrbuton we anayze three shocs that are of nterest: () an ncrease n productvty, () a decrease n the popuaton growth rate, () a decrease n the dscount rate. In ths secton we report the forma expressons for the steady-state responses, and w smuate the dynamc adjustments n the Secton Aggregate and dstrbutona effects of an ncrease n productvty We begn by recang the steady state condtons for the aggregate economy, (1a) (1c), 19

21 where we modfy the producton functon Y = AF( K, L), and parameterze the productvty ncrease by an ncrease n A. Snce the dstrbutons of weath and ncome depend upon the aggregate economy, we frst derve the steady-state responses as foows: d K K 1 L (1 + η) L(1 + η) ε = s 1 L(1 + η) dl 1 L (1 + η) da = (1 ε ) L 1 s A da A d Y 1 L (1 + η) εk L β (1 s) n da Y 1 s (1 s )( AF nk ) A (27a) (27b) (27c) where ε denotes the eastcty of substtuton between capta and abor n producton. 17 An ncrease n productvty w rase both the steady-state average per capta capta stoc, K, as we as output, Y. Its effect on steady-state abor suppy depends upon the eastcty of substtuton,ε, rasng abor suppy f ε < 1 and reducng t f ε > 1. To consder the consequences of ths for the ong-run weath dstrbuton we reca Proposton 1. Snce an ncrease n productvty rases the ong-run capta stoc (weath) t eads to a decrease n the ong-run nequaty of weath. To see what ths mpes for ong-run ncome nequaty reca (23), namey σ σ y y, ϕ σ = ϕ σ, 1+ η F, ) L ( K, L) / F( K L σ = 1+ η F ( K, L ) / F( K, L ) σ L, Ths breas down the steady-state change n ncome nequaty nto: () effect due to the change n weath nequaty (weath effect), whch we have just shown to be negatve, and () effect due to the change n abor suppy (abor suppy effect). The atter depends upon what happens to: FL ( K, L ) 1 ϕ 1 F( K, L ) 1+ η 17 Note that 1 L(1 + η) = (1 + η) η > by the transversaty condton; s denotes steady-state share of capta. 2

22 Dfferentatng ths wth respect to A we obtan 18 d ϕ K β L (1 s) n ( ε 1) + = da AL (1 + η) AF nk (27d) so that d ϕ sgn = sgn( ε 1) da An ncrease n productvty eads to a reducton n ong-run weath nequaty. Ths w ead to a arger, equa, or smaer decne n ong-run ncome nequaty accordng to whether the eastcty of substtuton s smaer than, equa to, or arger than, unty. For a suffcenty arge eastcty of substtuton ong-run ncome nequaty may actuay ncrease. 4.2 Aggregate and dstrbutona effects of a decrease n the popuaton growth rate Second, we consder a decrease n the growth rate of popuaton, yedng the effects d K K β L (1 + η) ε = + s dn< Dε 1 L (1 + η) dl β = ( ε s ) dn L Dε d Y Y β s ε = ( ε s) dn Dε 1 L(1 + η) (28a) (28b) (28c) where Dε ( β + n) AF nk (1 s ) K (1 L ) >. The effect of a reducton n the growth rate of popuaton on abor suppy depends crtcay on the reatve szes of the eastcty of substtuton and the share of ncome gong to capta. Despte the ambguty of ths response, the reducton n the popuaton growth rate ncreases the steady state stoc of capta, K, and hence reduces the nequaty of weath. The net effect on per capta ncome depends upon the reatve szes of ε and s. 18 The dervatons of a number of these expressons, such as d ϕ da, d ϕ dn nvoves a ot of deta, mang extensve use of equbrum condtons. They can be expressed n a number of equvaent ways, and we have chosen what we vew as the most convenent form. Snce these cacuatons do not have any ntrnsc nterest, we do not report them, but they are avaabe from the authors on request. 21

23 Dfferentatng ϕ wth respect to n yeds {(1 ε) β ( β + ) + ( ) (1 )( + β ) + (1 )(1 )( )} K K n AF L s n L d s L AF L ϕ = dn AF (1 + η)( β + n) AF nk ( ) (28d) mpyng that f ε 1, then d ϕ dn>. A decrease n the popuaton growth rate eads to a reducton n ong-run weath nequaty. Ths w ead to a arger, equa, or smaer decne n ongrun ncome nequaty dependng on the eastcty of substtuton. If ε 1, the shoc resuts n a arger than proportonate decne n ncome nequaty. For a suffcenty arge eastcty of substtuton ong-run ncome nequaty may actuay ncrease. 4.3 Aggregate and dstrbutona effects of a decrease n the dscount rate Consder now the effect of a reducton n the dscount rate. In ths case, d K K β n L(1 ) (1 s) 1 ε + η = + + dβ < Dε β 1 L (1 + η) dl β n = s ε dβ L Dε (1 ) + (1 ) β d Y Y (29a) (29b) β n s ε = (1 s) 1 ε dβ Dε + + < β 1 L(1 + η). (29c) The mpact of a ower β s smar to that of a reducton n n. The reducton n the dscount rate ncreases the steady state stoc of capta, K, and hence reduces the nequaty of weath. The effect on abor suppy depends on the eastcty of substtuton, whe per capta ncome unambguousy ncreases. Dfferentatng ϕ wth respect to β yeds {( ε 1) β ( β + ) + ( )(1 )( + β ) + (1 )(1 )( ) } K K n AF L s n L s L d AF L ϕ n = dβ AF (1 + η)( β + n) AF nk ( ) (29d) mpyng that f ε 1, then d ϕ / dβ <. As before the decrease n the dscount rate eads to a reducton n ong-run weath nequaty, and ths w ead to a arger, equa, or smaer decne n ong-run ncome nequaty dependng on the eastcty of substtuton. In contrast to the effect of a 22

24 reducton n the popuaton growth rate, for an eastcty of substtuton ε 1 ong-run ncome nequaty w actuay ncrease. 5. Numerca Smuatons To obtan further nsghts nto the dynamcs of weath and ncome dstrbuton we smuate the economy n response to these shocs. The smuatons are based on the foowng functona forms and standard parameter vaues, characterzng the benchmar economy. Producton functon: Y = A( αk + (1 α) L ) η Utty functon: U = ( 1 γ )( C ) ρ ρ 1 ρ Basc parameters: A = 1, α =.4 ρ = 1/ 3,,.2 (east of sub ε =.75, 1, 1.25) β =.4, γ = 1.5, n =. 15 Endogenous abor η = Exogenous abor L =.316,.278,.193 correspondng to ε =.75, 1, 1.25 γ Preferences are specfed by a constant eastcty utty functon, wth ntertempora eastcty of substtuton1 (1 γ ) =.4, whe the eastcty of esure n utty s The producton functon s of the CES form, where we aow the eastcty of substtuton to assume the vaues.75, 1, and 1.25, whe the dstrbutona parameter s α =.4. Popuaton grows at the rate of 1.5% per annum, whe A = 1 scaes the nta eve of productvty. We assume that the economy s ntay n a steady state n whch aggregate fracton of tme devoted to esure s and the average stoc of capta s K. For the benchmar Cobb-Dougas economy wth endogenous abor suppy, =. 722, whch s pausbe and mpes a abor suppy of L =.278. To preserve comparabty, n the case of neastc abor suppy we normaze the fxed abor suppy to the same eve,.e. L =. 278 ; mpyng that a aggregate magntudes w be the same n the two cases. Smar adjustments are made for other vaues of ε, as ndcated. We defne the nta steady state dstrbutons of weath (capta) and ncome (pror to any shoc) by the quanttes FL ( K, L ) 1 σ, and σ y, = 1 σ,, respectvey. Before examnng F( K, L ) 1+ η the effects of the shocs, consder the steady-state reatonshp between ncome and weath, 23

25 σ / y σ,. Tabe 1 reports ths rato for varous vaues of the eastctes of abor suppy and of substtuton n producton and reveas that t s hghy senstve to both. The frst coumn reports the case of an neastc abor suppy,.e. η =. The tabe ndcates that as η ncreases, ncome nequaty decnes reatve to the fxed abor suppy economy, and ths s the case for a vaues of ε. For the benchmar vaue η = 1. 75, the rato of ncome to weath nequaty s about 5% of what t woud be f abor suppy s fxed. Ths s because of the negatve reaton between ndvdua weath and ndvdua abor suppy as captured by equaton (15). Snce weather ndvduas suppy ess abor than do the poorer ones, ther reatve abor ncome s ower and hence partay offsets the dstrbutona mpact of weath nequaty. The hgher η, the stronger s the (negatve) correaton between weath and abor suppy, and the more equa s the ncome dstrbuton for any gven eve of asset nequaty. Ths suggests that treatng abor suppy as fxed s ey to serousy overstate the amount of ncome nequaty. Consder now the effects of the three shocs dscussed n secton 4. Startng from the nta steady state, σ,, σ y,, we sha nvestgate the tme paths of the economy n response to three structura changes: () An ncrease n the eve of technoogy A from 1 to 1.5 (Fg. 1); () A decrease n the rate of popuaton growth rate, n, from 1.5% to (Fg. 2) () A decrease n the dscount rate, β, from 4% to 2% (Fg 3). We pot the tme paths for the dstrbuton of weath and ncome, reatve to ther respectve nta vaues, namey, σ ( t) / σ, and σ y() t σ y,, where we further normaze σ, = In the efthand panes of these fgures we pot the tme paths of the weath and ncome nequaty when the abor suppy s exogenous,.e. for η, whe the rght-hand panes present the case wth endogenous abor, under the assumpton that η = Increase n A from 1 to 1.5 From Fg. 1 we see that a productvty ncrease aways reduces ong-run weath nequaty. 19 In effect we are graphng δ () t δ () n the case of weath and ϕ() t δ() t δ () n the case of the dstrbuton of ncome. 24

26 Long-run ncome nequaty aso decnes f ε < 1, though t ncreases f ε = These ong-run responses are vrtuay dentca for both eastc and neastc abor suppy, as s the entre dynamc tme path of the dstrbuton of weath. In contrast, whe η has tte effect on the change n the ong-run dstrbuton of ncome, t does have a sgnfcant mpact on ts tme path. To see ths, t s convenent to focus frst on the Cobb-Dougas producton functon, ustrated by the mdde par of fgures n Fg 1. As we showed n Secton 4, for ε = 1, an ncrease n productvty A w eave the ong-run aggregate (average) empoyment unchanged, but w rase aggregate capta. The effect of the ncrease n the ong-run capta stoc s to cause the dstrbuton of capta to become graduay more equa over tme. For ths parameter set, weath nequaty as measured by the standard devaton, decreases unformy, decnng by around 8%, asymptotcay for both eastc and neastc abor suppy. Athough ong-run ncome nequaty aso decnes by the same proporton n the two cases, wth eastc abor suppy ts transton s very dfferent. The short run decne n average esure tme (ncrease n abor suppy) eads to an ncrease n short-run ncome nequaty. Ths can be seen most drecty from equaton (2). The correcty antcpated decne n ong-run weath nequaty reduces (ncreases) the amount of esure tme, chosen by peope wth above (beow) average weath. That s, weather peope ntay ncrease ther wor tme, whe poorer peope wor ess and ncome nequaty ncreases. Over tme, as average esure ncreases the reatve ncome of agents havng above-average weath decnes, for reasons noted n Secton 3.2, and ncome nequaty decnes over tme, eventuay catchng up to the decne n weath nequaty. For a ower eastcty of substtuton (ε =.75), the ess fexbe producton functon mpes that the ong-run accumuaton n the aggregate capta stoc resuts n a arger reducton n the return to capta and ncrease n the wage rate. The tme path for the dstrbuton of weath s reatvey unaffected, but over tme ncome nequaty decnes more than does weath nequaty. In the case of the hgher eastcty of substtuton (ε =1.25) the reducton n the ong-run capta stoc agan mpes a gradua decne n weath nequaty; n contrast, the ong-run dstrbuton of ncome becomes more unequa, movng n the opposte drecton to the weath dstrbuton. The reason for ths s the decrease n the abor share nduced by a hgher capta-abor rato when ε >1, whch aways domnates the effect of the more equa dstrbuton of capta. 25

27 As n the Cobb-Dougas case, there s a sharp dfference n the dynamcs of ncome nequaty between fxed and fexbe abor suppy, due to the nta jump n the abor suppy whch generates a short-run response of ncome nequaty that s of opposte sgn to ts ong-run response. In the case where ε =.75, the short-run ncrease n ncome nequaty wth η = 1.75 means that over tme t decnes more rapdy than does weath nequaty, overtang the decne n the atter after around 2 years. In contrast, wth ε =1.25, ncome nequaty under fxed abor ncreases monotoncay to a eve 3.8% hgher that ts nta vaue; wth endogenous abor, t frst fas by 8% and then rses sharpy, ncreasng by 3.2% asymptotcay. In both cases, the possbty of a abor suppy response mpes much arger movements n ncome nequaty than wth fxed abor, as we as non-monotonc behavor. One further pont we see s that f ε =1.25, ncome dstrbuton exhbts some md nonmonotoncty durng ts transton. Ths can be understood by recang (26) and the fact that ncome dstrbuton s respondng to three factors: decnng weath nequaty and reatve empoyment effect, θ, both of whch tend to reduce nequaty, and ncreasng capta share, whch tends to rase t. Intay, the ncreasng capta share domnates and nequaty rses, but as the capta-abor rato approaches steady state vaue the effect of factor shares becomes weaer and ncome nequaty s drven by the evouton of the dstrbuton of capta, eadng to a perod of decnng nequaty. 5.2 Decrease n the rate of popuaton growth from 1.5% to When abor suppy s fxed, the effects of reducng the popuaton growth rate from 1.5% to are quatatvey smar to those of an ncrease n productvty, wth weath nequaty decreasng for a vaues of the eastcty of substtuton and ncome nequaty fang n a cases but that of a hgh eastcty n producton (ε =1.25). The response wth fexbe abor shows two sharp dfferences. Frst, as ong as ε > s, a restrcton that hods n a three cases, the abor suppy decnes both n the short-run and ong-run. As a resut, ncome nequaty exhbts an nta short-run drop due to the decne n abor suppy, and eeps fang as the weath dstrbuton becomes more equa. The consequence of the abor suppy response s that athough the reducton n weath nequaty s vrtuay dentca for fxed and for fexbe abor, ncome nequaty decnes much more n the atter 26

28 case. For exampe, for the Cobb-Dougas producton, ong-run ncome nequaty fas by 7% wth η = and by about 37% for η =1.75. The second mportant dfference s that wth a hgh eastcty of substtuton (ε =1.25) ongrun ncome nequaty may ncrease or decrease dependng on the vaue of η. To understand ths, agan reca equaton (26). There are three ways n whch a shoc affects reatve ncome: the change n the dstrbuton of weath, the change n esure, and the change n the abor share. Wth exogenous abor, there s no change n esure tme. Hence, when ε >1, there are two opposte forces drvng the evouton of the ncome dstrbuton: the reducton n the abor share tends to ncrease ncome nequaty whe the more equa dstrbuton of capta tends to reduce t. For our parameter set, the frst effect domnates, resutng n a more unequa dstrbuton of ncome. When the abor suppy can respond, the ower rate of popuaton growth w ead to more esure tme (as ong as ε > s ). Ths effect s stronger for rch ndvduas and hence they w suppy (reatvey) ess abor, whe poor ndvduas w suppy reatvey more abor, tendng to reduce ncome nequaty. For the chosen parameter set, the effects through the dstrbutons of capta and abor tme are stronger than that operatng through the reducton n the abor share, eadng to esser ncome nequaty. That s, the decne n popuaton growth w ncrease ncome nequaty f abor suppy s fxed but ncrease t when abor s supped eastcay. 5.3 Decrease n the rate of dscount from 4% to 2% Wth fxed abor suppy, the effects of reducng the rate of tme preference from 4% to 2% are agan quatatvey smar to those n the prevous cases. The dfferences n the response of ncome nequaty wth and wthout fexbe abor are hghy senstve to the eastcty of substtuton n producton. For the Cobb-Dougas producton (ε =1) ong-run ncome s reduced f abor s fxed but ncreases f abor s fexbe. Ths s because wth endogenous abor, the reducton n β eads to a reducton n average esure tme, and hence to a more equa dstrbuton of wor tme. Ths effect s suffcenty strong to offset the decne n weath nequaty, eadng to a more dspersed dstrbuton of ncome. For a ow eastcty (ε =.75) ong-run ncome nequaty decnes n both cases; however, 27