The Dynamics of Wealth and Income Distribution in a Neoclassical Growth Model * Stephen J. Turnovsky. University of Washington, Seattle


 Lee West
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1 The Dynamcs of Weath and Income Dstrbuton n a Neocassca Growth Mode * Stephen J. Turnovsy Unversty of Washngton, Seatte Ceca GarcíaPeñaosa CNRS and GREQAM March 26 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 n consumpton and esure mpes that the macroeconomc equbrum s ndependent of the dstrbuton of weath and aows us to fuy characterze ncome and weath dynamcs. We fnd nondegenerate ongrun dstrbutons of weath and ncome. The mode shows that () the nta eve of aggregate capta s an essenta determnant of whether nequaty ncreases or decreases durng the transton to the steady state; () temporary shocs to the stoc of capta have ongrun effects on the dstrbuton of weath even f they do not affect the statonary aggregate varabes; () ncome nequaty need not move together wth weath nequaty f factor shares change durng the transton. 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. We are gratefu to Carne Nourry for her comments.
2 1. Introducton Begnnng wth Stgtz s (1969) semna contrbuton, the evouton of the dstrbuton of weath n neocassca growth modes has been extensvey studed. A centra queston to emerge from ths terature concerns the crcumstances under whch the neocassca mode wth an unequa dstrbuton of weath mmcs the representatveagent mode nsofar as the macroeconomc aggregates are concerned. Case and Ventura (2) examne the Ramsey mode wth exogenous abor suppy and heterogeneous agents that dffer n ther nta weath (capta) endowment. They show that n ths case, the macroeconomc equbrum s ndependent of the dstrbuton of weath, whe the dstrbuton of weath depends on the evouton of the aggregate economy. They term ths a representatve consumer theory of dstrbuton. In ths paper we determne the condtons under whch the Ramsey mode wth endogenous abor suppy may aso exhbts ths feature. A Ramsey mode wth endogenous abor and heterogeneous weath endowments has been studed by Sorger (2). He shows that for hs chosen specfcaton of utty over consumpton and esure, the evouton of aggregate varabes depends on the entre dstrbuton of weath at each pont n tme. A ey feature of hs specfcaton s that the utty functon s nonhomogeneous n ts two arguments. In ths paper we consder an aternatve specfcaton for the utty functon, namey that t s of the famar constant eastcty for n consumpton and esure, mang t s member of a more genera cass of homogeneous utty functons. Indeed, homogenety turns out to pay a cruca roe n agents abor suppy choces, mang the ndvdua suppy of abor not ony a functon of the agent s weath, as n Sorger, but a near functon. As a resut, aggregate varabes are ndependent of the dstrbuton of weath, and we recover the ndependence of the macroeconomc equbrum from dstrbuton of the Ramsey mode wthout endogenous abor. Ths resut 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, wth 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 1
3 dstrbuton renders the anayss of the dynamcs ntractabe. 1 Our formuaton aows us to 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 dstrbuton. Then ndvdua quanttes are obtaned as a functon of the aggregate magntudes, mpyng that we are abe to characterze the transtona dynamcs of the dstrbutons of weath, and utmatey, ncome. We fnd that weath hodngs converge to a steady state dstrbuton such that agents hod dfferent amounts of capta. Weather ndvduas reman weather, but the steady state dstrbuton depends both on parameters of the economy and on nta condtons. In fact, the dynamcs of the dstrbuton of weath depend on whether the economy converges to the steady state from beow or from above. If the nta stoc of capta s beow the steady state one, then weath nequaty w decrease durng the transton. Ths contrasts wth the resuts obtaned by Case and Ventura (2) n the Ramsey mode wth exogenous abor suppy, where weath nequaty fas durng the transton from beow for certan parameterzatons, whe t ncreases for others. We show that t s abor suppy responses that prevent weath dvergence. We aso characterze the steady state dstrbuton of ncome. We fnd that the dynamcs of the dstrbuton of ncome can be compex, as they depend both on the dynamcs of ndvdua weath and on the evouton of factor rewards. If the shares of abor and capta n aggregate output are not constant, t s possbe for the dstrbuton of ncome to exhbt epsodes of ncreasng and epsodes of decreasng ncome nequaty durng the transton to the steady state. Two ey eements drve our resuts. The frst 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 1 In fact, Sorger aso obtans a near reatonshp between ndvdua capta and abor suppy for the partcuar parameter vaue for whch hs utty functon s homogenous. However, snce he s nterested n how dstrbuton affects output eves he does not focus on ths case, and hence does not derve the dynamcs of weath. 2
4 demand shocs nfuence the rate of capta accumuaton. 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 aboresure choce, as emphaszed by both Ortguera (2) and Turnovsy (2). 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 the negatve reatonshp between weath and abor suppy. HotzEan, 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 weathesure mechansm beng emphaszed n ths paper. The other cruca eement n our anayss s the assumpton of that utty s homogeneous n consumpton and esure. 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. 2 It s therefore an mportant queston to understand whether ths type of preferences aow a representatveconsumer representaton of socetes wth heterogeneous agents. Homogenous utty functons are aso commony used by abor economsts, and are generay consstent wth emprca evdence on abor suppy responses; see Hecman (1976) and Zabaza (1983). In our anayss we focus on a constant eastcty (CobbDougas) specfcaton for utty, athough the ndependence of the aggregate equbrum from dstrbuton woud hod for any homogenous utty functon. Ths specfcaton, whch mpes that utty s nonseparabe n 2 See Kng and Rebeo (1999) for a survey on busness cyce theory, and Rebeo (1991), LadróndeGuevara et a. (1997, 1999), Turnovsy (2) and GarcíaPeñaosa and Turnovsy (26a,b) for endogenous growth modes. 3
5 consumpton, s frequenty used n pocy anayses of endogenous growth modes. 3 Athough emprca evdence on ths aspect of preferences s scarce, two recent studes ndcate that the eastcty of substtuton between esure and consumpton s dfferent from zero. Latner and Sverman (25) use changes n consumpton over the fecyce to estmate the substtutabty between esure and consumpton, and show that separabe preferences are not consstent wth the observed drop n consumpton tang pace at retrement. Jacobs (25) uses nvestment data to estmate the varous parameters n the utty functon and cannot reject the assumpton of nonseparabty. Ths evdence ndcates that our choce of preferences s a natura setup n whch to study weath and ncome dynamcs. Our paper provdes a brdge between the representatve consumer theory of dstrbuton of Case and Ventura (2) and the mode wth endogenous abor of Sorger (2). It s aso reated to Ghgno and Sorger (22), who consder a Ramsey mode wth endogenous abor, but focus on the mpact of the dstrbuton of weath on the possbty of ndetermnacy, and to Bss (24), who generazes the resut of nonconvergence of weath n the basc Ramsey mode to the case of nontme separabe preferences. Chatterjee (1994) has aso examned the dynamcs of the dstrbuton of weath n the Ramsey mode, and fnds that under reasonabe parameter specfcatons weath nequaty ncreases durng the process of deveopment. The mechansm drvng ths resut s, however, very dfferent from ours. He assumes 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. Lasty, there s an extensve terature focusng on the Ramsey mode wth agents that dffer n ther rate of tme preferences. In ths framewor, the most patent agent ends up hodng a the capta n the ongrun, 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). 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 3 See, for exampe, Rebeo (1999) and Turnovsy (2). 4
6 derves the man resuts of the paper. Comparatve statcs are estabshed n secton 4, and these are then ustrated by a number of numerca exampes. Secton 6 concudes. 2. The Anaytca framewor We begn by settng out the components of the mode. 2.1 Technoogy and factor payments We assume that there are a fxed number of frms, M say, ndexed by j. Each representatve frm produces output usng a standard neocassca producton functon 4 where (, ) Y = F K L (1a) j j j K j and L j denote the ndvdua frm s capta stoc, and empoyment of abor, respectvey. A frms face dentca producton condtons. Hence they a choose the same eve of empoyment and capta stoc. That s, K j = K, L j = L, and Y j = Y for a j, where L, K, and Y denote the correspondng economywde average quanttes, per popuaton. The economywde average producton functon s therefore Y (, ) = F K L (1b) The wage rate, w, and the return to capta, r, are determned by ther respectve margna physca products, Yj Y w= = = FL ( K, L) w( K, L) L L j Yj Y r = = = FK ( K, L) r( K, L) K K j (2a) (2b) where w = F > ; w = F < ; r = F < ; r = F >. 5 K KL L LL K KK L KL 4 That s both factors of producton have postve, but dmnshng, margna physca products and the producton functon exhbts constant returns to scae (near homogenety). 5 The sgns F <, F < are a consequence of dmnshng margna product, whe F > s a consequence of the LL KK assumpton of near homogenety, an mpcaton of whch s rk + wl = FKK + FLL = Y. 5 KL
7 2.2 Consumers At tme, the economy s popuated by N ndvduas, represented as a contnuum, each ndexed by 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 famy at tme t has grown to e nt and the tota popuaton of the economy has grown to Nt () Ne nt. 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 () Tota amount of capta per famy s T K () t 1 = N N N nt K () t e d (v) Average stoc of capta per capta s thus K () t 1 1 Kt K ted K td T N N nt () = () () nt N Ne = N whch s aso the average among the fames. Snce the economy s growng we need to be carefu n defnng the dstrbuton of the capta stoc. We sha defne the share of capta owned by famy as 6
8 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 devaton of reatve capta by σ,. We now focus on a partcuar agent. Each such agent 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. 6 The preponderance of emprca evdence suggests that ths s reatvey sma, certany we beow unty, so that we sha restrct γ <. 7 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) and yeds the correspondng frstorder condtons 8 C γ 1 ηγ = λ (5a) ηc r = wλ (5b) γ ηγ 1 n λ λ = β (5c) 6 Ths utty functon s homogeneous of degree γ (1 + η) n C and. The utty functon n genera empoyed by 11/ θ Sorger (2) s of the form ( c 1) (1 1/ θ) + β n, whch s nonhomogeneous. 7 See e.g. the dscusson of the emprca evdence summarzed and reconced by Guvenen (forthcomng). 8 Tme dependence of varabes w be omtted whenever t causes no confuson. 7
9 where λ s agent s shadow vaue of capta, together wth the transversaty condton t m λ Ke β = (5d) t These optmaty condtons are standard and requre no further comment. Taen together wth the ndvdua s accumuaton equaton and 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 economywde aggregates (averages) as 1 N () = () N Z t Z t d Summng over frms and househods, equbrum n the capta and abor marets s descrbed by K K K () t e d K () t d j M N e N (6a) N N nt j = = = nt L L e d d j M N e N (6b) N N nt j = = 1 = (1 ) = (1 ) nt Equaton (6b) gves the reatonshp between aggregate esure and the aggregate 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 (6b), we can equay we wrte them as functons of aggregate esure tme, (1 ), namey, w= w( K, ) and r = r( K, ). In order to obtan the macroeconomc equbrum we begn by dvdng (5b) by (5a) to yed C η = wk (, ) (7) and usng ths expresson we may wrte the ndvdua s accumuaton equaton, (4), n the form 9 As a sma technca pont we assume that the number of frms s represented by dscrete unts, whe househods are represented by a contnuum. 8
10 K K w( K, ) 1+ η = r( K, ) n + 1 K η (8) Tang the tme dervatve of (5a) and combnng wth (5c) mpes C λ ( γ 1) + ηγ = = β + n r( K, ) C λ for each (9) Ths s the Euer equaton modfed to tae nto account the fact that esure changes over tme. The mportant pont about (9) s that each agent, rrespectve of capta endowment, chooses the same growth rate for the shadow vaue of capta. We can then show (see Appendx) that C C = ; = C C for a, (1) That s, a agents w choose the same growth rate for consumpton and esure. Now turn to the aggregates. Summng (7) over a agents and notng that 1 N 1 N K () t () t d = d = 1 N N K() t, 1 N d N = (11) the aggregate economywde consumptoncapta rato s C η = w( K, ) (7 ) whe summng over (8) yeds the aggregate accumuaton equaton K K w( K, ) 1+ η = r( K, ) n + 1 K η (8 ) In addton, (1) mpes that average consumpton, C, and esure,, aso grow at ther respectve common ndvdua growth rates, namey C C = ; = C C for a (1 ) 2.4. Aggregate equbrum dynamcs We can now derve the dynamcs of aggregate capta and esure, Kt () and t () as foows. 9
11 Frst, substtutng for the equbrum rate of return on capta, rk, (, ) and the wage rate, wk (, ), nto (8 ) and recang the near homogenety of the producton functon, yeds the aggregate goods maret cearng condton FL (, ) ( K, L ) K = F K L nk (12a) η Second, substtutng (1 ) nto (9) and tang the tme dervatve of (7 ) to substtute for C / C, yeds the aggregate modfed Euer equaton w ( K, ) wk ( K, ) K K ( γ 1) ηγ + ( γ 1) = β + n r( K, ) wk (, ) wk (, ) K Fnay, substtutng (12a) nto ths equaton, and recang the expressons w L etc. mpes FKL( K, L) FL ( K, L) FK ( K, L) β n (1 γ) F( K, L) nk FL ( K, L) η = (12b) G () where G( ) 1 γ (1 + η) (1 γ ) F F LL (12c) L Equatons (12a) and (12b) are autonomous equbrum dynamc equatons n the economywde average quanttes of capta and esure (or empoyment). Assumng that the economy s stabe, these aggregate quanttes converge to a steady state characterzed by a constant average per capta capta stoc, abor suppy, and esure tme, whch we denote by K, L and, respectvey. Settng K = = n (12a) and (12b), we can then express the steady state as F K ( K, L) = β + n (13a) FL (, ) ( K, L ) F K L nk = (13b) η L + = 1 (13c) 1
12 where the frst two equatons come from (12a) and (12b), and the thrd s smpy the abor maret cearng condton. These equbrum reatonshps are standard. Equaton (13a) 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 frstorder condton (7 ) equatng the margna rate of substtuton of consumpton and esure to the prce of esure (the rea wage), where the efthand sde captures the fact that n steady state consumpton s equa to output mnus the amount needed to eep per capta capta constant wth a growng popuaton. Usng (13a) and (13b), whe recang the homogenety of the producton functon, we mmedatey nfer that: 1 η > (14) 1 +η Ths nequaty yeds a ower bound on the steadystate 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. Lnearzng equatons (12a)  ( 12b) around steady state yeds the oca dynamcs K a a K K = a21 a 22 (15) 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 (16a) a ( t) = + µ a 22 µ a11 ( K( t) K ) = + ( K( t) K ) 21 a 12 (16b) where µ < s the stabe egenvaue. From the sgn pattern estabshed n the Appendx, ( a 22 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 the capta stoc on the evouton of esure. 1 We can combne (13a) and (13b) to yed: ( FL K )( (1 ) ) η + η η = β 11
13 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 reducng the desre to ncrease esure. At the same tme, a hgher capta stoc reduces ts growth rate, and the growth rate of the wage rate, thus reducng the growth of abor and ncreasng that of esure. As we show n the Appendx, whch effect domnates 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 CobbDougas 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. However, f ε s suffcenty sma, a 21 > and the reatonshp between capta and esure mped by (16b) becomes negatve 11 As we w see beow, the evouton of aggregate 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, (16b). Snce ths reatonshp hods at a tmes, we have a µ a () = = (16b ) ( ) ( ) K K K K µ a22 a12 Thus consder a stuaton n whch the economy s subject to a structura shoc that resuts n an ncrease n the steadystate 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 (8 ) and (8) to obtan 11 See (A.7) of the Appendx. 12
14 wk (, ) () t = 1 1 (17) K η η where K, evove n accordance wth (16a, b) and the nta reatve capta, s gven from the nta endowment. Snce = we may wrte = θ where 1 N N θ d = 1 and θ s constant for each, and yet to be determned. Thus, we may wrte (17) as wk (, ) 1 1 () t = 1 θ (18) K η η To sove for the tme path of the reatve capta stoc, we frst note that agent s steadystate share of capta satsfes θ = for each (19) η η or, equvaenty η = ( 1) 1+ η for each (19 ) Recang (14), ths equaton mpes that the hgher an agent s steadystate reatve capta stoc (weath) ncreases, 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 economywde esure. As a consequence, aggregate abor suppy s ndependent of the dstrbuton of capta. It s mportant to note here that ths resut hods for any utty functon that s homogenous of degree b n consumpton and esure, and n the Appendx we show that ths s ndeed the case. 13
15 To anayze the evouton of the reatve capta stoc, we nearze equaton (18) around the steadystate K,,, n (19). Ths yeds wk (, ) 1 1 () t = 1 + ( θ)(() t ) () t K η η ( ) (2) In the Appendx we show that the stabe souton to ths equaton s () t 1 = δ ()( t 1) (21) where 1 FL ( K, L) t ( ) δ () t 1 + 1, (22) β µ K Settng t = n (21) and (22), we have 1 FL ( K, L ) (), 1 = δ ()( 1) = 1+ 1 ( 1) β µ K (23) 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, (23) determnes the steadystate dstrbuton of capta, ( 1), whch together wth (21) then yeds the entre tme path for the dstrbuton of capta. Usng (21) (23), and equatons (16), descrbng the evouton of the aggregate economy, we can express the tme path for ( t ) n the form δ ( t) 1 ( t) µ t ( t) = ( ) = (, ) = e (, () 1 () δ ), from whch we see that ( t ) aso converges to ts steady state vaue at the rate µ. (24) We can aso determne the tme path for the ndvdua s esure (abor suppy). Frst, havng determned ( 1), (19 ) yeds agent s steadystate esure aocaton,, whch, nowng the economywde average,, determnes hs constant reatve esure tme θ, namey 14
16 1 η 1 1 η θ 1= 1 ( 1) = 1 (, 1) 1 + η δ() 1+ η (25) Thus, nowng the tme path for the aggregate esure aocaton, t ( ), the tme path for ( t) θ ( t) mmedatey foows. We see from (25), n conjuncton wth (14), that any agent whose steadystate capta stoc exceeds the economywde average w enjoy above average esure tme throughout the transton. Because of the nearty of (21), (23), and (24), 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. Specfcay, correspondng to these three equatons we obtan σ () t = δ() t σ (21 ) σ, = δ() σ (23 ) δ () t 1 () t 1 µ t σ ( t) σ = ( σ, σ) = ( σ, σ) = e ( σ, σ) (24 ) δ () 1 () 1 The crtca factors upon whch we wsh to focus are: () how agent s reatve weath evoves over tme and () the consequences of ths for the dstrbuton of weath. A crtca determnant of ths s the magntude of δ () t. From (22) ths s seen to depend upon t (), whch n turn depends upon how the underyng shoc affects the evouton of the aggregate capta stoc, ( Kt ( ) K ). From (14), (16), and (22) we can deduce the foowng, where we restrct our focus to what we have dentfed as the norma case of a postve adjustment between aggregate esure and capta; see (16b). () If K > K δ t >, for a t () If K < K δ t <. As ong > K 2, so that the decne n the capta stoc s ess than 5%, we can show that δ () t >, mpyng that < δ ( t) < 1, for a t Ths s estabshed n the Appendx. 15
17 () sgn δ ( t) = sgn ( K K ) We can then state: Proposton 1 (The ongrun dstrbuton of weath): The aocaton of weath converges to a ongrun dstrbuton n whch () a agents hod postve amounts of weath, () weath s unequay dstrbuted, () the ranng of agents accordng to weath s the same as n the nta dstrbuton. Havng estabshed the exstence of a ongrun dstrbuton of weath, we can compare t to the nta dstrbuton. For the moment t s convenent to measure dstrbuton by the standard devaton of the capta stoc (weath), athough n Secton 3.4 beow we express ths n terms of more conventona Gn coeffcents. From equatons (21 ) and (23 ) 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 (16b ) and snce esure s monotoncay ncreasng or decreasng aong the transton path, ths mpes the foowng Proposton 2 (Weath dynamcs): The nta condton K affects the ongrun dstrbuton of weath. 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 ongrun dstrbuton of weath w be ess (more) unequa than s the nta dstrbuton. The ntuton for ths resut can be easy obtaned by notng, from equaton (A.1) n the Appendx, that, ( )( ) sgn( ) = sgn 1 () Reca that f the economy converges to the steady state from beow, then () 16 <. 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.
18 To understand why the evouton of nequaty depends on the nta condton consder two ndvduas havng dfferent capta endowments. Homothetc preferences mpy that they both spend the same share of tota weath at each pont n tme and have the same rate of growth of tota weath. Tota weath has two components, physca capta and the present vaue of a future abor ncome. Snce wages are growng at the same rate for both agents but represent a hgher share of tota weath for the poorer ndvdua, then hs capta must be changng more rapdy than that of the weather agent. When the economy s accumuatng capta, ths means that hs capta stoc s growng faster and nequaty s dmnshng. When the economy s convergng from above,.e. when the stoc of capta s fang, he w dsave faster and nequaty w ncrease. Ths resut contrasts wth the evouton of the dstrbuton of weath n the Ramsey mode wth neastc abor suppy. In ths case, f the eastcty of substtuton s greater or equa to one,.e. ε 1, the dstrbuton of weath w become more equa durng the transton from beow. However, for ε < 1, the dstrbuton coud wden. 13 The reason for ths s that a ow eastcty of substtuton mpes fast wage growth as the economy accumuates capta. Consumers cacuate ther tota weath and choose a constant consumptontototaweath rato. If wages are growng sowy, poor consumers w need a hgh rate of capta accumuaton to sustan ther consumpton path. However, f wages are growng fast, a ower rate of capta accumuaton s optma. Wth suffcenty hgh wage growth, poor consumers may choose to dsave eary n ther fetmes 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. Hgher future wages have two effects, as they 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 captapoor agents, and hence the weath dverge, that occurs when ndvduas cannot change worhours. As a resut, there s an unambguous narrowng of the weath dstrbuton durng the transton from beow. Proposton 2 has a number of mpcatons for the dstrbuton of weath that can be summarzed as foows: 13 See Case and Ventura (2). 17
19 Coroary 2: () Consder two economes dentca n a respects except for ther nta capta stoc. They w have the same steady state macroeconomc equbrum, but the poorer country (the one wth the ower nta K ) w have a more equa ongrun weath dstrbuton. () Temporary shocs to the stoc of capta have ongrun effects on the dstrbuton of weath, wth any temporary reducton n the stoc of capta resutng n a permanent reducton n weath nequaty. () Suppose an economy s subject to a pocy shoc or structura change that generates a ongrun ncrease n the capta stoc, so that () run, the weath dstrbuton of that economy w be narrower. <. Then n the ong 3.2. Income Dstrbuton We turn now to the dstrbuton of ncome. We defne the ncome of ndvdua at tme t as Y ( t) = r( t) K ( t) + w( t)(1 ( t)), average economywde 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 ( ) (26) The reatve ncome of agent has two components, reatve capta ncome, captured by the frst term n (26) 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. Usng equaton (19) to substtute for θ and (21), we may wrte (26) n the form t () 1 η 1 y () t 1 = st () (1 st ()) 1 ( () t 1), (27) 1 t ( ) 1 + η δ( t) whch we can express more compacty as 18
20 y ( t) 1 = ϕ ( t)( ( t) 1), (28) where t () 1 η 1 ϕ() t 1 (1 s()) t (29) 1 t ( ) 1 + η δ( t) Agan, because of the nearty of (28) 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 σ () t = ϕ() t σ () t (28 ) y From nequaty (14) the term n square bracets n equaton (29) s postve and hence ϕ ( t) < 1, mpyng that ncome s more equay dstrbuted than s capta. Lettng t, we can express the steadystate dstrbuton of ncome as σ = ϕσ (28 ) y where 1 1 s 1 FL ( K, L ) ϕ = m ϕ( t) = 1 = 1 t 1+ η η F( K, L ) In the Appendx we use the steadystate equbrum condtons (13) n conjuncton wth (14) to show that ϕ es n the bounds n s < ϕ < s (3) β + n mpyng n partcuar that the steadystate dstrbuton of ncome s ess unequa than that of capta. From (28) we can compare the ongrun dstrbuton of ncome to the nta one, namey ( + η )( FL K L F K L ) ( η )( FL K L F K L ) σ 1 1 (1 ( ) ( ) y ϕ σ σ = = σ ϕ σ 1 1 (1 + ( ) ( ) σ y,,, (31) 19
21 where the subscrpt dentfes the nta dstrbuton, from whch we nfer that n genera ( y ) = ( y, ) sgn 1 sgn 1. We may summarze these resuts n: Proposton 3 (The ongrun dstrbuton of ncome): The dstrbuton of ncome converges to a ongrun dstrbuton such that: () ncome s unequay dstrbuted, and the reatve ranng of agents accordng to ncome s the same as that of capta, as we as that of the nta ncome dstrbuton. () The rato of the standard devaton of the dstrbuton of ncome to that of capta s ess than the share of capta n output. Whether the ongrun dstrbuton, foowng a structura chang, s more or ess unequa than the nta dstrbuton depends on the ongrun 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 (26), 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 ( )) (32) Ths 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 (32), and that of reatve abor ncome. The atter can be expressed as a functon of the evouton of aggregate esure, and of the reatve rewards to capta and abor, as refected by the capta share, s(t). It s usefu to start by examnng what happens for a CobbDougas 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, 2
22 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 (32) 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 2) 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. Moreover, at dfferent stages one or the other effect may domnate, mpyng epsodes of rsng or fang nequaty. Proposton 4 (Income dynamcs): Consder an economy that converges to ts steady state from beow,.e. K < K. Then: () f the share of capta n ncome s constant or decreases durng the transton, ncome nequaty w decrease durng the transton to the steady state, () f the share of capta n ncome ncreases durng the transton, then the ongrun dstrbuton of ncome may be more or ess unequa than the nta one, and the economy may experence epsodes of ncreasng and epsodes of decreasng ncome nequaty. Consder an economy that converges to ts steady state from above,.e. K > K. Then: () f the share of capta n ncome s constant or ncreases durng the transton, ncome nequaty w ncrease durng the transton to the steady state, (v) f the share of capta n ncome decreases durng the transton, then the ongrun dstrbuton of ncome may be more or ess unequa than the nta one, and the economy may experence epsodes of ncreasng and epsodes of decreasng ncome nequaty. 21
23 3.3 Reatve consumpton Fnay, we examne the dstrbuton of consumpton. From the ndvdua s frst order condtons we have C = w η, whch mpes that reatve consumpton s gven by c () t C () t C() t = () t () t = θ, that s 1 η 1 η 1 σ c() t = 1 σ = 1 σ 1+ η 1 + η δ() (33) The dstrbuton of consumpton s then constant over tme, and depends on the nta dstrbuton of capta, together wth the parameters that mpact on the esure decson. It s straghtforward to estabsh that σ c < σ y, so that consumpton s more equay dstrbuted than both steadystate ncome and capta. The reason for ths s that weather ndvduas choose to have both a hgher eve of consumpton and a hgher eve of esure, so that ony part of the dfference n capta transates nto dfferences n consumpton. 3.4 Gn Coeffcents The conventona measures of nequaty are gven by Gn coeffcents and n ths secton we brefy consder the Gn coeffcents for weath and ncome. Reca that H ( ) denotes the nta dstrbuton functon of reatve capta. Then, the Gn coeffcent of the nta dstrbuton of weath s (see Cowe, 2): 1 G, = j dh( ) dh( j ) 2 Then at tme t we have ( t) ( t) 1 G ( t) = ( t) j ( t) dh t ( ) dh t ( j ) (34) 2 ( t) ( t) where H ) s the dstrbuton functon of reatve capta at t, and (t) and (t) ts upper and t ( ower bounds. Wrtng the frst equaton n (24) n the form 22
24 () t () t () t = δ δ 1 δ() + δ(), we can transform (34) to get 1 δ( t) δ( t) G () t = dh ( ) dh ( ) = G (34 ) j j, 2 δ() δ() Smary, usng the reatonshp (28), the Gn coeffcent form ncome can be expressed as δ () t Gy() t = ϕ() t G() t = ϕ() t G, (35) δ () The man pont to observe s that both Gn coffcents evove exacty as does the reatve capta stoc, so that comparng the standard devatons of the reatve capta stoc and ncome, as we have been dong, carres over to the Gn coeffcents 4. Longrun adjustments of weath and ncome nequaty To ustrate the dynamc adjustments of weath and ncome dstrbuton we anayze two shocs that are of nterest: () an ncrease n productvty, () a decrease n the popuaton growth rate. In ths secton we derve the forma expressons for the steadystate responses, and w smuate the dynamc adjustments n the next secton. We begn by recang the steady state condtons for the aggregate economy, (13a) (13c). Because of the homogenety of the producton functon t s convenent to wor n terms of ntensve quanttes,.e. Y = AF( K, L) = ALf( z) where z K L s the average stoc of capta per manhour and A denotes the eve of productvty. 14 Usng ths notaton, we can rewrte steady state condtons n ntensve form Af ( z ) = β + n (13a ) 14 That s, T z K L = K NL and sha be referred to as the average captaempoyment rato. 23
25 [ ] [ ] (1 L L Af( z ) nz = A f( z ) zf ( z ) ) η (13b ) L + = 1 (13c) Snce the dstrbutons of weath and ncome depend upon the aggregate economy, we frst derve the steadystate responses as foows: 4.1 Aggregate and dstrbutona effects of ncrease n productvty, A From (13a ), (13b ) and (13c) we can derve the foowng expressons for the aggregate steadystate effects of an ncrease n productvty dz f da () = > z zf A (36a) () dl L 1 L(1 + η) da = (1 ε ) (36b) 1 s A and hence () dk K dl dz 1 L(1 + η) L(1 + η) ε = + = + L 1 z 1 s 1 L(1 + η) dy da dl dk 1 L(1 + ) z [ L + (1 s ) n] (v) η ε β Y = + (1 s ) + s = A L K 1 s da A da ( 1 s )( Af nz) A (36c) (36d) where ε denotes the eastcty of substtuton between capta and abor n producton, s. An ncrease n productvty rases the steadystate captaempoyment rato, whe ts effect on steadystate abor suppy depends upon the eastcty of substtuton, ε, rasng abor suppy f ε < 1 and reducng t f ε > 1. Irrespectve of the response of abor suppy, an ncrease n productvty w rase both the average per capta capta stoc, K, as we as output, Y. To consder the consequences of ths for the ongrun weath dstrbuton we reca Proposton 2. Snce an ncrease n productvty rases the ongrun capta stoc (weath) t eads to a decrease n the ongrun nequaty of weath. To see what ths mpes for ongrun ncome nequaty reca (31), namey 24
26 ( + η )( FL K L F K L ) ( η )( FL K L F K L ) σ 1 1 (1 ( ) ( ) y ϕ σ σ = = σ ϕ σ 1 1 (1 + ( ) ( ) σ y,,, Ths breas down the steadystate 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 Dfferentatng ths wth respect to A we obtan 15 d ϕ z = ( ε 1) da AL(1 + η) [ βl + (1 s ) n] [ Af nz ] (37) so that d ϕ sgn = sgn( ε 1) da Thus we can derve the foowng proposton: Proposton 5: () An ncrease n productvty eads to a reducton n ongrun weath nequaty. () Ths w ead to a arger, equa, or smaer decne n ongrun 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 ongrun ncome nequaty may actuay ncrease. () If the eastcty of substtuton s ess than one the decne n ncome nequaty w be assocated wth a hgher per capta eve of ncome. If ε s suffcenty arge, a hgher per capta eve of ncome may be assocated wth greater ncome nequaty. 15 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. 25
27 4.2 Aggregate and dstrbutona effects of decrease n popuaton growth rate effects () As a second exampe we consder a decrease n the growth rate of popuaton, yedng the dz 1 = dn < z Af z d L β () ( = ε s )dn L Dε () (v) where dk dl dz = + K L z d Y Y Dε β L(1 + η) ε = + s dn < Dε (1 + η) η (38a) (38b) (38c) β s ε = ( ε s ) dn (38d) Dε (1 + η) η Af ' [ Af nz] (1 s ) z(1 L) Whe a reducton n the growth rate of popuaton rases the captaempoyment rato, ts effect 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 capta empoyment effect domnates and thus the reducton n 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. Dfferentatng ϕ wth respect to n yeds [ βkf + f (1 s n + L )( )] d ϕ z (1 ε ) = β dn ALf (1 + η) f mpyng that f ε 1, then d ϕ dn> (1 s L )(1 ) f + ( Af nz) f ( Af nz). Thus we can derve the foowng proposton: (39) Proposton 6: () A decrease n the popuaton growth rate eads to a reducton n ongrun 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 ths eads to a arger than 26
28 proportonate decne n ncome nequaty. For a suffcenty arge eastcty of substtuton ongrun ncome nequaty may actuay ncrease. () The decne n weath nequaty may be assocated wth ether a rse or fa n average per capta ncome. 5. Numerca Smuatons To obtan further nsghts nto the dynamcs of weath and ncome dstrbuton we smuate the economy n response to these two shocs. The smuatons are based on the foowng functona forms and parameters, characterzng the benchmar economy. Producton functon: Utty functon: ( ) Y = A( αk + (1 α) L ) 1 η γ U = C γ Basc parameters: A = 1, α =.4 ρ ρ 1 ρ ρ = 1/ 3,,.2 (east of sub =.75, 1, 1.25) γ = 1.5, n =.15, η = 1.75 Preferences are specfed by a constant eastcty utty functon, wth ntertempora eastcty of substtuton 1 (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. These parameter vaues are standard and noncontroversa. 16 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. We defne the nta steady state 16 For exampe, the ntertempora eastcty of substtuton.4 s we wthn the range of emprca estmates summarzed by Guvenen (forthcomng), whe the choce of η = 1.75 s standard wthn the rea busness cyce terature; see Cooey (1995). Aowng the eastcty of substtuton n producton to vary between.75 and 1.25 covers the range of most of the emprca estmates, whe α =.4 mpes that 4% of output goes to capta n a CobbDougas word, aso broady consstent wth emprca evdence. 27
29 dstrbutons of weath (capta) and ncome (pror to any shoc) by the quanttes, σ,, and FL ( K, L ) 1 σ y, = 1 σ,, respectvey. Startng from ths nta steady state, we sha be FK (, L ) 1+ η concerned wth nvestgatng the tme paths of the economy n response to two shocs: 17 () 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). In the eft hand panes of these fgures we pot the tme paths of the aggregate quanttes, capta, Kt ()/ K, output, Yt () Y, and abor suppy, Lt () L, reatve to ther respectve orgna benchmar eves. Capta aways evoves contnuousy n response to a gven shoc, whe abor suppy and output w undergo endogenous jumps at the nta pont. In the rght hand panes, 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, = 1. The dstrbuton of capta evoves contnuousy, whe the dstrbuton of ncome jumps at tme zero, due to a jump n A and n L. 18 Loong at Fgures 1 and 2 two genera features stand out. Frst, whe the two shocs affect the dstrbuton of weath n quatatvey comparabe ways, there s a sharp contrast n ther mpacts on the dstrbuton of ncome. In partcuar, the tme path for the dstrbuton of ncome n response to an ncrease n productvty s hghy senstve to reatvey md changes n the eastcty of substtuton n producton. Second, there s a sharp contrast between the tme path of weath dstrbuton and that of ncome dstrbuton, partcuary n response to an ncrease n productvty; sometmes they move together and sometmes n opposte drectons, agan dependng upon the fexbty of producton. 5.1 Increase n A from 1 to 1.5 It s convenent to focus frst on the case of the CobbDougas producton functon, ustrated by the mdde par of fgures n Fg 1. As we showed n Secton 5, for ε = 1, an ncrease n 17 For the Benchmar CobbDougas economy =.722, whch s pausbe and satsfes (14). 18 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. 28
30 productvty A w eave the ongrun aggregate (average) empoyment unchanged, but w rase aggregate capta. Steadystate aggregate output w therefore rse, n fact dong so n the same proporton as the capta stoc. 19 In the short run, the capta stoc remans unchanged. However, the hgher productvty of abor, and therefore the hgher wage rate, w ncrease abor suppy n the short run, resutng n a correspondng decne n esure. Output therefore rses by around 55%, consstng of the 5% drect ncrease n productvty, pus the contrbuton of the addtona abor suppy. Thereafter, as the aggregate stoc of capta contnues to be accumuated, the average fracton of tme devoted to esure ncreases (sghty), n accordance wth (16b), mpyng that the average fracton of tme devoted to abor decnes correspondngy, convergng bac to ts nta vaue. Whe these two responses have opposng effects on the tme path of output, the capta accumuaton effect ceary domnates and output contnues to rse, athough at a sower rate than does capta. The effect of the decne n the ongrun capta stoc s to cause the dstrbuton of capta to graduay become more equa over tme. For ths parameter set, weath nequaty as measured by the standard devaton, decreases unformy, decnng by around 7%, asymptotcay. Athough the ongrun dstrbuton of ncome decnes by the same proporton [see Proposton 5], ts transton s very dfferent. The short run decne n average esure tme [ncrease n abor suppy] eads to an ncrease n shortrun ncome nequaty. Ths can be seen most drecty from equaton (25). The correcty antcpated decne n ongrun weath nequaty, reduces (ncreases) the amount of esure tme, θ, 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 aboveaverage 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 behavor of the economywde aggregate varabes are quatatvey generay smar. The ess fexbe producton functon means that that there s ess ongrun accumuaton n the aggregate capta stoc, whe the hgher productvty 19 For the CobbDougas case we can show 1 = = (1 α), whch n ths case equas 83.3%. dy Y dk K da A 29
31 ncreases abor suppy permanenty, though st ess so than on mpact. The tme path for the dstrbuton of weath s reatvey unaffected, but as shown n Proposton 5, for ε =.75 n the ongrun, ncome nequaty decnes more than does weath nequaty. However, the decne n ongrun esure,, rases the senstvty of θ to the antcpated ongrun decne n weath nequaty, so that n the shortrun weather peope ncrease further the amount of ther abor suppy, whe poorer peope decrease ther abor suppy correspondngy. As a resut, the shortrun ncome nequaty ncreases by 2%, rather than by 12% n the CobbDougas case. Over tme, ncome nequaty decreases more rapdy than does weath nequaty, wth ts decne overtang that of the atter after about 2 years. The man dfference n the case of the hgher eastcty of substtuton (ε =1.25) s that there s now a ongrun decne n abor suppy. At the aggregate eve ths has the effect of reducng the ongrun proportonate ncrease n output to beow that of capta, athough the enhanced fexbty eads to arge proportonate ncreases n both. Agan, the reducton n the ongrun capta stoc mpes a gradua decne n weath nequaty. In contrast, the fact that average abor suppy decreases rather than ncreases has profound effects on ncome dstrbuton. In the short run, the ncrease n can offset the effect of the antcpated reducton n weath nequaty on the reatve abor suppy, θ, so that ncome nequaty actuay decnes on mpact. Over tme, as average esure ncreases, ths effect s offset and ncome nequaty ncreases over tme, to above ts orgna eve, consstent wth Proposton 5. Thus over tme, ncome and weath nequates move n opposte drectons. One further pont we see that ncome dstrbuton exhbts some md nonmontoncty durng ts transton. Ths can be understood by recang (28 ) and the fact that the ncome dstrbuton s respondng to two offsettng factors; frst the decnng weath nequaty, and second the reatve empoyment effect, θ, whch n ths case s movng n the opposte way. Fnay, Fg. 1 ustrates an nterestng dynamc reatonshp between per capta ncome and ncome nequaty. For the two cases ε =.75, ε =1, we see that on mpact the ncrease n productvty causes both per capta ncome and ncome nequaty to ncrease together. However, as the former contnues to ncrease, whe the atter decnes, over the ong run they move n opposte drectons. For a hgh eastcty of substtuton the pattern s reversed. Per capta ncome and 3
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