A Representative Consumer Theory of Distribution: A Simple Characterization of the Ramsey Model * Cecilia García-Peñalosa CNRS and GREQAM

Transcription

1 A Representatve Consumer Theory of Dstrbuton: A Smple Characterzaton of the Ramsey Model * Cecla García-Peñalosa CNRS and GREQAM Stephen J. Turnovsy Unversty of Washngton, Seattle June 26 Abstract: It has been shown that the Ramsey growth model wth agents that dffer n ther ntal wealth endowments has an aggregate behavor that mmcs that of the model wth a representatve consumer. The dynamcs of aggregate magntudes and ther steady state values are then ndependent of dstrbuton; however, the evoluton of the dstrbuton of wealth depends on the dynamcs of the aggregate captal stoc. In ths note we provde a smple characterzaton of the dynamcs of the dstrbuton of wealth and of ncome across agents n the Ramsey model. We show that whether wealth and ncome nequalty ncrease or decrease durng the transton to the steady state depends on smple and ntutve condtons on parameter values. Standard values for these parameters ndcate that t s more lely that wealth nequalty decreases as the economy accumulates captal. JEL Classfcaton Numbers: D31, O41 Key words: growth; wealth dstrbuton; ncome dstrbuton; transtonal dynamcs. * García-Peñalosa s research was supported n part by the Insttut d Econome Publque (IDEP) at GREQAM. Turnovsy s research was supported n part by the Castor endowment at the Unversty of Washngton.

2 1. Introducton The representatve-agent Ramsey growth model has been wdely used by macroeconomsts for almost eghty years, and n recent years, n partcular, has become the standard framewor for addressng a number of mportant ssues n contemporary macrodynamcs and growth theory. 1 In a recent paper, Casell and Ventura (2) have characterzed relatvely mld condtons under whch varous sources of heterogenety are nevertheless compatble wth vewng the aggregate (average) economy behavng as f t s populated by a sngle representatve consumer (RC). 2 Ths s an mportant contrbuton, snce t provdes the RC model wth a much frmer theoretcal underpnnng and counters some of the crtcsm to whch ths framewor has been subjected; see e.g. Krman (1992), and others. As Casell and Ventura stress, the RC assumpton does not rule out heterogenety and ndeed t has mportant dstrbutonal consequences that depend on the evoluton of aggregate magntudes. But ther man pont s that the RC model permts one to study aggregate behavor wthout needng to consder the detals of dstrbuton, thus preservng analytcal tractablty. 3 In the second part of ther paper, Casell and Ventura address some of the consequences of the RC model for the dynamcs of wealth dstrbuton. They show that durng the transton to the steady state the cross-secton of (relatve) wealth could become more or less unequal, and provde a number of numercal examples showng that both ncreasng and decreasng wealth nequalty are possble. Ther examples are based on a logarthmc utlty functon and they dentfy three cases: () wth a Cobb-Douglas producton functon, the dstrbuton of wealth becomes more compressed (dspersed) durng the transton to the steady state from below (above), () wth a Constant Elastcty of Substtuton (CES) technology havng an elastcty of substtuton less than 1, and a low 1 The model dates bac to Ramsey (1928) and s dscussed n a number of contemporary macroeconomcs textboos begnnng wth Blanchard and Fscher (1989). 2 The sources of heterogenety they consder are: () ntal endowments of captal, () tastes, and () slls. 3 One mportant assumpton of Casell and Ventura (2) s to follow the Ramsey model and assume an nelastc labor supply. Sorger (2) endogenzes labor supply and shows how ths requres the dstrbuton and aggregate behavor to be determned smultaneously, renderng the explct soluton for the dynamcs ntractable. Turnovsy and García- Peñalosa (26) show how the RC model can be restored wth endogenous labor supply provded the utlty functon s homogeneous n lesure and consumpton. 1

3 rate of tme dscount, the dstrbuton of wealth becomes more compressed (dspersed) durng the transton to the steady state from below (above), () wth a CES technology havng an elastcty of substtuton less than 1, and a hgh rate of tme dscount, the dstrbuton of wealth becomes more dspersed (compressed) durng the transton to the steady state from below (above) f the economy s not too far from the steady state. The objectve of ths note s to provde a more systematc analyss of the dstrbutonal consequences of the RC model, snce ths too s mportant. We focus on a sngle, but mportant source of heterogenety, namely, dfferences n the ntal wealth across ndvduals. 4 We derve a smple proposton summarzng the relatonshp between the taste parameters and producton parameters that determne whether nequalty ncreases or decreases durng the transton towards the steady state. Most of our attenton focuses on the dstrbuton of wealth, and ncludes the examples of Casell and Ventura. In addton, we also show how t s straghtforward to derve the consequences for ncome dstrbuton, whch may not evolve n the same way as the wealth dstrbuton when we move beyond the Cobb-Douglas producton functon. One of the objectves of ths note s pedagogc: to provde a smple ntutve dscusson of the underlyng dstrbutonal consequences of a standard worhorse macroeconomc model. The paper s organzed as follows. Secton 2 descrbes the economy and derves the macroeconomc equlbrum. Secton 3 characterzes the dstrbuton of wealth, and secton 4 derves the man results of the paper pertanng to the condtons under whch wealth nequalty ncreases or decreases. We next loo at the dstrbuton of ncome, whle secton 6 concludes. 2. The Ramsey model wth heterogeneous wealth endowments We begn by settng out the components of the model. The economy s populated by L ndvdual consumers, each of whom provdes a unt of labor nelastcally, so that L s also total labor supply. 4 Bertola, Foellm, and Zwemüller (26) survey the lterature on the Ramsey model wth heterogeneous captal endowments. 2

4 2.1 Technology and factor payments For convenence we assume that there are M frms, ndexed by j. Each representatve frm produces output usng a standard neoclasscal producton functon 5 where (, ) Y = F K L (1a) j j j K j and L j denote the ndvdual frm s captal stoc, and employment of labor, respectvely. All frms face dentcal producton condtons and hence they all choose the same level of employment and captal stoc. Summng over the M dentcal frms, the aggregate producton functon s Y (, ) = F K L (1b) where MK j = K, MLj = L and MYj = Y, for all j and L, K, and Y denote the correspondng aggregate quanttes. The wage rate, wt (), and the return to captal, rt (), are determned by ther respectve margnal physcal products, Yj Y w() t = = = FL ( K(), t L) w( K(), t L) L L j Yj Y r() t = = = FK ( K(), t L) r( K(), t L) K K j (2a) (2b) where w = F > ; w = F < ; r = F < ; r = F >. K KL L LL K KK L KL 2.2 Consumers The L ndvdual consumers are represented as a contnuum, each ndexed by and dentcal n all respects except for ther ntal endowments of captal, K,. We shall defne the share of 5 That s both factors of producton have postve, but dmnshng, margnal physcal products and the producton functon exhbts constant returns to scale (lnear homogenety). 3

5 captal owned by agent as () t K ()[ t K() t L]. Note that relatve captal has mean 1. We denote ts ntal dstrbuton functon by H ( ), the ntal densty functon by h ( ), and the ntal (gven) standard devaton of relatve captal by σ,. Agent maxmzes lfetme utlty, assumed to be a functon of consumpton, n accordance wth the soelastc utlty functon 1 γ βt max C ( ), wth 1 t e dt < γ < (3) γ where ω 1(1 γ ) equals the ntertemporal elastcty of substtuton. Ths maxmzaton s subject to the agent s captal accumulaton constrant K & ( t) = r( t) K ( t) + w( t) C ( t) (4) and yelds the famlar Euler equaton 6 C& C rkl (, ) β = 1 γ for each (5) The mportant pont about (5) s that each agent, rrespectve of captal endowment, faces the same rate of return on captal, and thus chooses the same growth rate for consumpton, mplyng C & C C& C h = for all, h. (6) h 2.3. Macroeconomc equlbrum and aggregate equlbrum dynamcs We frst derve the macroeconomc equlbrum and the dynamcs of the aggregate economy. Havng determned these, we shall then obtan the dynamcs of the dstrbuton of captal. In order to obtan the macroeconomc equlbrum we wrte the ndvdual s accumulaton equaton, (4), n the 6 Tme dependence of varables wll be omtted whenever t causes no confuson. 4

6 form K& wkl (, ) C = rkl (, ) + K K K (7) L Summng over (7) and notng that ( t) d = 1, yelds the aggregate accumulaton equaton K& wk (, LL ) C = rkl (, ) + K K K In addton, (6) mples that wth a fxed populaton, L, aggregate consumpton, (7 ) C L Cd, also grows at the common ndvdual growth rate, namely C& C C& = for all (6 ) C We may then wrte C θ ( C/ L) L =, where θ d = L, and θ s constant for each, and yet to be determned. Thus defned, θ denotes agent s consumpton relatve to the mean. The equlbrum dynamcs of aggregate consumpton and captal are then gven by C& C = FK ( K, L) β 1 γ (8a) K& = FKL (, ) C (8b) These two equatons embody the essence of the RC model. Wth all ndvduals followng the same Euler equaton, the aggregate economy evolves ndependently of dstrbutonal consderatons. Under qute general condtons, the economy proceeds as f there s a sngle representatve agent. Ths s the case as long as the producton functon has the standard neoclasscal propertes, and agents have the same tastes represented by a utlty functon homogeneous n ts sngle argument, consumpton. 7 Invong the standard transversalty condtons, these aggregate quanttes converge 7 In Turnovsy and García-Peñalosa (26) we address the ssue n the more general case where the agent also faces a labor-lesure choce. The condtons for the RC model to apply n that case become more complex. 5

7 to a steady state characterzed by a constant captal stoc and consumpton level that we denote by K % and C %, respectvely, and can be expressed as 8 F (, ) K K% L = β (9a) C % = F( K %, L) (9b) Wth no long run growth, the captal stoc s determned by equatng the margnal product of captal to the rate of tme dscount. The second smply ndcates that n steady state all output s consumed. Lnearzng equatons (8a) and (8b) around the steady states (9a) and (9b) yelds the local aggregate dynamcs FKK F C& C C (1 γ ) % = K & K K 1 F % K (1) from whch we can see that the aggregate dynamcs s characterzed by saddle-pont behavor. The stable paths for K and C can be expressed as K() t = K% + ( K K% ) e µt (11a) FKK F( K%, L) µ t µ t C() t = C% + ( K K% ) e = C% + ( β µ )( K K% ) e (11b) (1 γ ) µ where µ < s the stable egenvalue and s the soluton to: FKK F + = 1 γ 2 µ βµ The slope of stable arm s postve; accumulatng captal s therefore assocated wth ncreasng consumpton, a standard property of the Ramsey model. 8 t The transversalty condtons are lm λ K ( t) e β = for each. t 6

8 3. The dynamcs of the relatve captal stoc To derve the dynamcs of ndvdual s captal stoc, relatve to the economy-wde average, () t K () t ( K()/ t L), we combne (7 ) and (7) to obtan wkll (, ) C & () t = ( 1 () t ) ( () t ) K K θ (12) where K, C evolve n accordance wth (11a, b) and the ntal relatve captal, s gven from the ntal endowments. To solve for the tme path of the relatve captal stoc, we frst note that agent s steady-state shares of captal and consumpton satsfy ( % ) = % ( θ % ) wkll ( %, ) 1 C for each (13) To analyze the evoluton of the relatve captal stoc, we lnearze equaton (12) around the steady-state K, C, defned by (9) and (13). Ths yelds (, ) ( ) ( ) () ( () ) ( 1 ) wk % LL Ct C % t r t ( 1 ) FKL( K, L) L Kt K % & = % % % + % % K% C% K% (14) whch we can wrte n the more convenent form K K% & () t = β ( () t % ) + h( K% )1 ( % ) e K% µ t (14 ) where [see Appendx] (1 ) hk ( % β ε ) (1 s) + µ % ε (15) and ε ( FF K L) ( FKLF), s FK K F denote the elastcty of substtuton and the captal share, respectvely, and denotes steady state. 9 As we wll see, ths term plays a crucal role n 9 We wrte h = h( K% ) to reflect the fact that n general t depends upon the steady-state aggregate stoc of captal through both ε, s, and µ. 7

9 determnng the dynamcs of wealth, and consequently ncome, dstrbuton. Solvng (14) and mposng the condton that the relatve share of captal s bounded, (for each ), the stable soluton to ths equaton s () t 1 = δ ()( t % 1) (16) where δ h( K + ) K ( t ( t) 1 ) 1 β µ K, (17) Settng t = n (16) and (17), we have ( ) 1 = δ h K K ()( 1) 1 1 = + ( 1) β µ K (18), where, s gven from the ntal dstrbuton of captal endowments. The evoluton of agent s relatve captal stoc s determned as follows. Frst, gven the tme path of the aggregate economy, and the dstrbuton of ntal captal endowments, (18) determnes the steady-state dstrbuton of captal, ( % 1), whch together wth (16) then yelds the entre tme path for the dstrbuton of captal. Usng (16) (18), and equatons (11), descrbng the evoluton of the aggregate economy, we can express the tme path for () t n the form δ ( t) 1 t) = ( δ () 1 K( t) K ) = ( K K ) = e ( ) µ t (,,, from whch we see that () t also converges to ts steady state value at the rate µ. Then, as has been shown by Casell and Ventura (2), the cross-secton of wealth converges to a long-run dstrbuton n whch wealth s unequally dstrbuted, and the ranng of agents accordng to wealth s the same as n the ntal dstrbuton. We can also determne the tme path for the ndvdual s relatve ncome as follows. We defne the ncome of ndvdual at tme t as Y() t = r() t K () t + w() t, whle average economy-wde (19) 8

10 ncome s Yt ()/ L= rtkt () ()/ L+ wt (). Relatve ncome, y () t Y() t ( Y()/ t L) can be expressed as FK ( K, L) K y() t 1 = s()( t () t 1) ( () t 1) (2) FKL (, ) In the case of the Cobb-Douglas producton functon, when factor shares are constant, the dstrbuton of relatve ncome mrrors that of relatve wealth, as determned by (19). For more general producton functons t wll also depend upon the evoluton of factor shares, st (), nsofar as these respond to the evolvng captal stoc. Fnally, we can also derve the tme path for the ndvdual s consumpton. Frst, havng determned ( % 1), (13) yelds agent s (constant) relatve consumpton, θ, namely FL ( K%, L) L s% θ 1= 1 1 = 1 = 1 F( K%, L) δ () ( % ) s% ( % ) (, ) (21) Thus, nowng the tme path for aggregate consumpton, C (t), the tme path for ndvdual s consumpton, C ( t) = θ C( t), mmedately follows. 4. The long-run change n the dstrbuton of wealth Havng establshed the exstence of a long-run dstrbuton of wealth, we can compare t to the ntal dstrbuton. It s convenent to measure dstrbuton by the standard devaton of the captal stoc (wealth), although t can be show that the same analyss apples n terms of more conventonal Gn coeffcents. 1 Because of the lnearty of (16), and (18), we can mmedately transform these equatons nto correspondng expressons for the standard devaton of the dstrbuton of captal. Specfcally, σ ( t) = δ( t) % σ and σ, = δ() % σ, mplyng h K K K σ σ ( δ ) ( ) 1 () σ, σ β µ K = = (22) Consder a permanent structural change that leads to a change n the aggregate captal stoc. 1 See Turnovsy and García-Peñalosa (26). 9

11 Wealth nequalty then ncreases durng the transton to the steady state f and only f 11 ( K K ) h ( K) > (23) There are then two factors that determne whether nequalty ncreases or decreases durng the transton, the ntal condton (relatve to the long run) and the value of h (K). In order to assess the latter we recall (15), whch s certanly negatve for values of the elastcty of substtuton greater than or equal to one, but could be postve for ε less than one. In partcular, h (K) s postve f and only f ε 1 and the followng condton s satsfed, ε β > µ (24) 1 ε To proceed further, we assume that the producton functon s of the CES form: ρ ρ ( αk + (1 α) ) 1/ ρ Y = F( K, L) = L (25) As demonstrated n the Appendx, we can express (24) as β 1 ε ε εα >. (26) 1 γ (1 ε ) We can summarze these results n the followng proposton. Proposton 1: The dynamcs of the dstrbuton of wealth crucally depend on the elastcty of substtuton, and the ntal condton K. () For ε 1, an economy that starts below (above) the steady state,.e. K < K ( K > K ), wll experence a reducton (ncrease) n wealth nequalty durng 11 For example, (9a) mples that a decrease n the rate of tme preference leads to an ncrease n K % and hence, f the economy starts from an ntal steady state, mples K % K >. It s also clear that & σ () () t = & δ t % σ = ( hk ( % )( β µ ))( Kt & () Kt ()) % σ, mplyng that the dstrbuton of wealth evolves monotoncally durng the transton. Ths s, however, due to the fact that we have lnearzed the system around the steady state. When the economy s far from the steady state, t s possble for dstrbuton to evolve non-monotoncally as Casell and Ventura (2) show n one of ther examples. 1

12 the transton. () For ε < 1, an economy that starts below (above) the steady state,.e. K < K ( K > K ), wll experence a reducton (ncrease) n wealth nequalty durng the transton f and only f β 1 ε ε εα < 1 γ (1 ε ) holds. If ths condton does not hold, then an economy that starts below (above) the steady state,.e. K < K ( K > K ), wll experence an ncrease (reducton) n wealth nequalty durng the transton. To understand why the evoluton of nequalty depends on the ntal condton consder two ndvduals havng dfferent captal endowments. Homothetc preferences mply that they both spend the same share of total wealth at each pont n tme and have the same rate of growth of total wealth. Total wealth has two components, physcal captal and the present value of all future labor ncome. Snce wages are growng at the same rate for both agents but represent a hgher share of total wealth for the poorer ndvdual, then hs captal must be changng more rapdly than that of the wealther agent. When the economy s accumulatng captal, ths means that hs captal stoc s growng faster and nequalty s dmnshng. When the economy s convergng from above,.e. when the stoc of captal s fallng, he wll dsave faster and nequalty wll ncrease. Ths result may however be reversed f the elastcty of substtuton s less than one, and the dstrbuton of wealth could wden as the economy converges from below. The reason for ths s that a low elastcty of substtuton mples fast wage growth as the economy accumulates captal. Consumers calculate ther total wealth and choose a constant consumpton-to-total-wealth rato. If wages are growng slowly, poor consumers wll need a hgh rate of captal accumulaton to sustan ther consumpton path. However, f wages are growng fast, a lower rate of captal accumulaton s optmal. Wth suffcently hgh wage growth, poor consumers may choose to dsave early n ther 11

13 lfe-tmes and fnance current consumpton wth ther (hgh) future wages. As a result, the dstrbuton of captal becomes more unequal. The strng feature of (26) s that t provdes a crteron for whether wealth nequalty ncreases or decreases wth an accumulaton of the aggregate captal stoc, expressed n terms of four exogenous parameters, α, βεγ.,, Emprcal evdence on the elastcty of captal n producton and the rate of tme preference are consstent at around α =.4, β =.4, respectvely. In contrast, emprcal estmates of the elastcty of substtuton n producton, ε, and the ntertemporal elastcty ω 11 γ, are much more far-rangng. 12 Fgure 1 plots the of substtuton n consumpton, ( ) tradeoff between ε and ω, for set values of α =.4, β =.4, that would mae (26) hold wth equalty. For ponts lyng below the XY locus, an ncrease n the aggregate captal stoc s assocated wth an ncrease n the nequalty of wealth, whle for ponts lyng above ths locus wealth nequalty decreases. For values of the elastcty of substtuton just below unty (say around.8) the ntertemporal elastcty of substtuton must be very small (less than.2) for wealth nequalty to ncrease, whle for hgher values of the ntertemporal elastcty of substtuton (say around.75), the elastcty of substtuton n producton would have to be extremely low (less than.1). Overall, ths fgure suggests that decreasng wealth nequalty assocated wth ncreasng captal stoc s the more plausble outcome, although ncreasng wealth nequalty cannot be ruled out. 13 Casell and Ventura (2) emphasze the mportance of wage growth, n determnng the effect of an ncrease n the aggregate captal stoc on the dstrbuton of wealth. To consder ths aspect, we can rewrte hk ( % ) as a functon of the evoluton of wages (see Appendx), 12 The preponderance of emprcal evdence suggests that ω s relatvely small, certanly below unty. Guvenen (25) reconcles the estmates derved from consumpton data, whch are typcally smaller (often around.2) wth larger estmates based on fnancal data (often around.75). Estmates of ω > 1 also exst; see e.g. Vssng-Jørgensen and Attanaso (23). An early nfluental study by Berndt (1976) reconcles alternatve estmates of ε obtaned usng dfferent functonal forms and data sets. He fnds that for hs preferred data sets and estmaton methods the estmates are tghtened and generally le n the range.7 to 1.3. More recently, Duffy and Papageorgou (2) suggest that ε exceeds 1 for rch countres, but s less than 1 for poor countres. 13 The examples of Casell and Ventura (2) can be easly dentfed n Fg 1 as follows. Frst, the Cobb-Douglas producton functon, ε = 1, les above the XY curve for any ω > and s therefore assocated wth less wealth nequalty as wealth ncreases. Second, snce an ncrease n the rate of tme preference, β, shfts the XY curve up, t s clear that a hgher rate of tme preference wll rase the lelhood of an ncrease n wealth beng assocated wth more nequalty and vce versa. 12

14 1 wt ( ) w hk ( % ε & % ε ) = (1 s% ) β + ε ( K( t) K% ) K% s% (15 ) Ths expresson wll be postve only f wages are growng suffcently fast, for a gven accumulaton of captal. Note also from (26) that nequalty wll ncrease durng the transton from below f, gven ε < 1 the dscount rate β s hgh enough. That s, wages must be growng fast and consumers must care suffcently lttle about future consumpton, so that those wth low ntal wealth fnance current consumpton through borrowng aganst future wages. 5. The long-run change n the dstrbuton of ncome Analogous to (22) we may express the change n the dstrbuton of ncome across steady states as [(1 ()) s ( s s )] ( s s ()) % σ σ = δ + % % σ = % δ % σ (27) y y, Ths comprses two components, the frst due to the change n wealth dscussed n Secton 4, and the second due to the change n factor returns, reflected n the factor share. In the case of the Cobb- Douglas producton functon, s% = s, and the behavor of the long-run ncome dstrbuton wll smply mrror that of wealth. Otherwse, t s straghtforward to establsh: K% K hk ( %) ε 1 hk ( %) K% K % σ y σ y, = s% % σ + (1 s% ) 1 K% β µ ε β µ K% (28) where hk ( % ) s gven n (15). The frst term s the wealth effect (22), whle the second s the effect due to the change n factor returns. Regroupng terms we obtan Notng from (2) that ncome nequalty can be wrtten as σ () t = s() t δ() t % σ, we can show: ( ) ( ) ( ) & σ y () t = s() t (1 s()) t ( ε 1) ε 1 h( K)( β µ ) K() t K 1 ( h( K)( β µ )) K() t K + + ( K() t K()) t σ % % % % & % In the case of the Cobb-Douglas producton functon, ncome nequalty mrrors the monotonc adjustment of wealth. In other cases the adjustment may be non-monotonc and the model can generate a Kuznets curve. y 13

15 s% (1 s% ) % σ K% K K% K % σ ( ) y σ y, = µ ε 1 hk ( % ) ε( β µ ) K% K% (29) From (28) and (29) we may deduce the followng proposton: Proposton 2: The dynamcs of the dstrbuton of ncome depends crucally on the sze of the long-run change n the aggregate captal stoc. () If the economy starts suffcently close to ts steady state,.e. K % K s suffcently small, then an economy that starts below (above) the steady state,.e. K < K ( K > K ), wll experence a reducton (ncrease) n ncome nequalty durng the transton, rrespectve of ε. () For larger changes K % K, the sze of ε becomes mportant. If 1 ε <, and hk ( % ) <, then an economy that starts below ts steady state,.e. K < K, wll experence a reducton n ncome nequalty durng the transton that exceeds the reducton n wealth nequalty. If K > K, t s possble for ncome nequalty to declne, n contrast to wealth nequalty. () If ε > 1, and K > K the economy wll experence an ncrease n ncome nequalty, whle f t starts from below equlbrum ncome nequalty wll declne less than wealth nequalty and may even ncrease. 6. Concludng comments The objectve of ths note has been to provde a systematc analyss of the dstrbutonal consequences of the representatve consumer optmal growth model. It has been prevously shown that the Ramsey model s compatble wth a wde range of dstrbutonal outcomes, yet t s dffcult to characterze under whch crcumstances the dstrbuton of wealth becomes more or less unequal. 14

16 Our analyss of the transtonal dynamcs of ndvdual ncome and wealth allows us to derve a smple set of condtons under whch wealth and ncome nequalty ncrease or decrease n a growng economy. We show how both producton and preference parameters affect the dstrbutonal outcome. In partcular, we fnd that ether a low elastcty of substtuton n producton or a small ntertemporal elastcty of substtuton n consumpton are requred for wealth nequalty to ncrease durng the growth process. Exstng estmates of these parameters ndcate that the most lely scenaro s one n whch the dstrbuton of wealth becomes more compressed as captal accumulates. The exstence of substantal changes n the degree of wealth nequalty has been recently documented by Petty, Postel-Vnay, and Rosenthal (26), who show that there was a substantal reducton of wealth nequalty durng the 2 th century. Part of t was due to accdental causes such as war and taxaton- but n the post war perod, the reducton n captal ncomes was a major equalzng force. Our analyss ndcates that ths s a lely outcome once an economy s captal marets are wellfunctonng and agents are not constraned n ther savng and nvestment decsons. 15

17 Appendx Ths Appendx s devoted to the dervaton of several techncal detals A.1 Dervaton of (14 ), (15) Usng (9b), we can wrte (11b) as Ct () C% β µ Kt () K K % = % C % FKL ( %, ) K % (A.1) enablng us to rewrte (14) n the form K K% & () t = β ( () t % ) + h( K% )1 ( % ) e K% µ t where LFL ( K%, L) hk ( %) LFKL ( KL %, ) ( β µ ) FKL ( %, ) (A.2) Usng the defntons of ε and s, we may rewrte (A.2) as (15) of the text (1 ) hk ( % β ε ) (1 s) + µ ε (A.3) A.2 Evaluaton of (26) To evaluate the condton (26), note that (1) mples that the egenvalue s gven by F K 4F( K) FKK µ = 1 1 (A.4) 2 2 (1 γ ) F K Now, the CES producton functon mples 16

18 F F KK K 1 s = (A.5) ε K and 1 s 1 α = s α L K ρ (A.6) Also, the steady state condton (9a) mples that the steady state captal-labor rato s defned by ρ (1 ε) L β (1 α) = α K% α (A.7) These expressons together wth (A.4) allow us to wrte (24) as (26). A.3 Dervaton of (15) Now, recall that the wage s defned as w= F ( K, L). Lnearzng ths relatonshp around the steady state, w% F ( K%, L), yelds L ( ) L wt () = w% + F Kt () K % (A.8) LK Tang the tme dervatve, usng (11a), and notng the defnton of ε, we may rewrte (A.8) as sw wt () = % % & Kt () K ε K % µ ( % ) (A.9) Usng ths expresson to substtute for µ nto equaton (15) we have 1 wt ( ) w hk ( % ε & % ε ) = (1 s% ) β +. (A.1) ε ( K( t) K% ) K% s% 17

19 References Berndt, E.R., 1976, Reconclng Alternatve estmates of the Elastcty of Substtuton, Revew of Economcs and Statstcs, 58, Bertola, G., R. Foellm, and J. Zwemüller, 26, Income Dstrbuton n Macroeconomc Models, Prnceton Unversty Press, Prnceton and Oxford. Blanchard, O.J. and S. Fsher, 1989, Lectures on Macroeconomcs, MIT Press, Cambrdge, Massachusetts. Casell, F., and J. Ventura, 2, A Representatve Consumer Theory of Dstrbuton, Amercan Economc Revew, 9, Duffy, J. and C. Papageorgou, 2, A Cross-Country emprcal Investgaton of the Aggregate Producton Functon Specfcaton, Journal of Economc Growth, 5, Guvenen, F., 25, Reconclng Conflctng Evdence on The Elastcty of Intertemporal Substtuton: A Macroeconomc Perspectve, Journal of Monetary Economcs, forthcomng. Krman, A.P., 1992, Whom or what does the Representatve Indvdual Represent? Journal of Economc Perspectves, 6, Petty, T., G. Postel-Vnay, and J.L. Rosenthal, 26, Wealth concentraton n a developng economy : Pars and France, , Amercan Economc Revew, vol. 96,6, p Sorger, G. 2, Income and Wealth Dstrbuton n a Smple Growth Model, Economc Theory, 16, Turnovsy, S.J. and C. García-Peñalosa, 26, The Dynamcs of Wealth and Income Dstrbuton n a Neoclasscal Growth Model, mmeo. Vssng-Jørgensen, A. and O.P Attanaso, 23, Stoc-maret Partcpaton and the elastcty of Intertemporal Substtuton, Amercan Economc Revew, Papers and Proceedngs 93,

20 Fg. 1 ntertemp. elast. sub 2 X Increase n captal assocated wth decreasng wealth nequalty.5.25 Increase n captal assocated wth ncreasng wealth nequalty Y elast. sub. prod