Inequality and Growth: What are the Tradeoffs?

Transcription

1 Inequalty and Growth: What are the Tradeoffs? Stephen J. Turnovsky Unversty of Washngton Address to NZAE, Palmerston North, June 2012

2 Background and Overvew Growth and nequalty relatonshp dates back to Kuznets (AER, 1955) --Inverted-U between nequalty and development --Controversal and nconclusve --Kuznets orgnally assocated t wth dual economy dynamcs --transton from agrcultural to ndustral economy Key feature of recent emprcal research has focused on the tradeoffs between ncome nequalty and growth Addressed by runnng regressons of growth rates on measures of nequalty (and other varables) 1

3 Anand and Kanbur (JDE, 1993), Alesna and Rodrk (QJE, 1994), Persson and Tabelln (AER, 1994), Perott (JEG, 1996), and others obtan negatve relatonshp. The varous explanatons for ths nclude: -- Poltcal economy consequences of nequalty -- Negatve mpact of nequalty on educaton -- Captal market mperfectons and credt constrants, etc Other studes fnd postve, or more ambguous, relatonshp; see e.g. L and Zou (RDE, 1998), Forbes (AER, 2000), and Barro (JEG, 2000). Explanatons nclude -- Relatve savngs propenstes of rch vs. poor -- Investment ndvsbltes -- Incentves. Lundberg and Squre (EJ, 2003) ntroduce a number of factors that may potentally nfluence both nequalty and growth; test for jont sgnfcance. They dentfy two common varables [openness and cvl lbertes] that mpact both varables n the same drecton, thereby mplyng a postve relatonshp between ncome nequalty and growth. 2

4 From theoretcal perspectve, emprcal controversy not surprsng. Growth rate and ncome dstrbuton are both endogenous outcomes. Income nequalty-growth relatonshp wll reflect the underlyng set of forces to whch both are smultaneously reactng. Ths can be understood only wthn the context of a consstently specfed general equlbrum growth model. Assocaton rather than causalty s more approprate characterzaton.. 3

5 Key element s heterogenety Many sources (tastes, endowments, technology.) In most general context to solve for growth and nequalty smultaneously s ntractable (see Sorger, ET, 2000) But f assume that underlyng utlty functon s homogeneous, then aggregaton s possble (Gorman, Econom.,1953) Problem becomes tractable Assumpton of homogenety s routne throughout modern growth theory Leads to representatve agent theory of dstrbuton (Casell and Ventura, AER 2000) Macroeconomc equlbrum has a smple recursve structure: --Aggregate equlbrum s ndependent of dstrbuton, dstrbuton depends upon aggregate. 4

6 My Research My work n ths area has followed ths approach As the source of heterogenety I have taken the ntal endowments of captal (assets) and n some cases ablty Qute an extensve lterature takng ths approach Alternatve approach s to assume that people start out dentcal and that heterogenety s endogenzed through dosyncratc random shocks (Krusell and Smth, JPE 1998, and others). 5

7 Buld models of ncreasng realsm (complexty) Fall nto three categores, structurally 1. Economes always on balanced growth path (One-sector endogenous growth model model) --No dynamcs ether n aggregates or dstrbutons. 2. Models wth transtonal dynamcs (e.g. neoclasscal growth models; twostate varable endogenous growth model) --Have transtonal dynamcs of aggregates and dstrbuton, but no moblty (nequalty may expand or contract but no changng n rank) --Introduces ntertemporal as well as ntratemporal tradeoffs. 3. Two sources of heterogenety n endowments (e.g. captal and sklls) --Permts both wealth and ncome nequalty dynamcs and moblty. These need not move together. 6

8 Applcatons Effects of structural changes (e.g. productvty ncreases) Tax and expendture polces (e.g. government nvestment vs. government consumpton and fnancng) Extensons to nternatonal economy (e.g. dstrbutonal consequences of nternatonal transfers) Path dependence of structural changes and ther dstrbutonal consequences (partcularly relevant for ssues relatng to foregn ad) Human captal vs. physcal captal Role of externaltes Heterogenety vs. demographc dstrbuton 7

9 Remander of Talk A smple canoncal model A dynamc model Role of path dependence and consequences for dstrbuton Sklls and captal endowments 8

10 A Canoncal Model Frms Frms face dentcal producton condtons; make dentcal decsons. Y A( L K) K 0 1 (1a) 1 j j j Economy-wde captal stock yelds externalty such that n equlbrum the aggregate (average) producton functon s lnear n aggregate captal stock Y AL K ( L) K (1b) Wage rate and the return to captal, r, are determned by margnal products F L j K KL, L j j 1 1 L K AL K wk (2a) F K j KK, LL (1 ) (1 ) AL r (2b) 9

11 Consumers Consumers are dentcal except for ntal endowments of captal, K 0. Agent s share of total captal stock s k K K. Relatve captal has ntal dstrbuton functon G ( ) 0 k, mean k 1, and coeffcent of varaton k. Agents endowed wth a unt of tme that can be allocated ether to lesure, l or to work, 1l L. The agent s decson problem s 1 t max C ( ), wth 1, 0, 1 0 t l e dt (3) subject to the captal accumulaton constrant K rk (1 l ) wk C (4) 10

12 Optmalty condtons are C l 1 Cl (5a) wk (5b) 1 r (5c) together wth the transversalty (ntertemporal solvency) condton 11

13 Macroeconomc equlbrum Economy s n fact always on ts balanced growth path. All agents face same real wage and same rate of return on captal and thus choose same growth rate for consumpton and lesure (although levels dffer). So that summng over all agents we get C C l l ; l l C C l l and s constant for each, and yet to be determned. for l () t all (6) Macroeconomc equlbrum reduces to dfferental equaton n l, of form: dl Gl () dt where G() l 0 (7) Only soluton that s ntertemporally vable s that l be at ts steady-state value at all ponts of tme. 12

14 The next step: show that the change n the relatve stock of captal of agent s descrbed by the followng equaton 1 1 k () t w() l 1 l1 1l 1 k() t (8) Transversalty condton (ntertemporal solvency) mposes the restrcton r whch can be further wrtten as l. (9) 1 13

15 Transversalty condton mples only vable soluton s k 0 for all tme. Snce k reflects captal stocks that evolve gradually over tme, ths s accomplshed by agents selectng ther respectve lesure, l, n accordance wth the relatve labor supply functon l l l k 1 1 Ths condton mples a postve relatonshp between relatve wealth and lesure. Settng l n accordance wth ths equaton mples that the relatve wealth poston of agents, k, s unchangng over tme. (10) Ths s key mechansm generatng the endogenous dstrbuton of ncome. Intuton: Wealther agents have lower margnal utlty of wealth. They work less, enjoy more lesure, and gven ther relatve captal endowments, ths translates to an endogenously determned dstrbuton of ncome. Supported by emprcal evdence from a range of sources (e.g. data from stock market boom, effect of nhertance) 14

16 Macroeconomc equlbrum Economy s always on ts balanced growth path. Equlbrum as n Romer (1986) model, but have aggregated over agents havng dfferent endowments and supplyng dfferent labor and havng dfferent consumpton levels: Equlbrum growth rate: rl () 1 (11a) Aggregate consumpton-captal rato: C K w() l l (11b) Goods market equlbrum: () l C (11c) K 15

17 Summarze by (12a,b): jontly determne growth rate and lesure tme, l: RR (1 ) ( l), (12a) 1 l PP ()1 l 1 l. (12b) RR ensures equalty between return to captal and return to consumpton. PP descrbes combnatons of and l for product market equlbrum. Focus attenton on equlbra that are () vable, [.e. satsfy transversalty condton], () also generate postve growth. l 1 (13a) Both schedules are concave; suffcent condtons for a unque equlbrum: 0; A 1l 1 1 (1 l) 1 (13b) 16

18 Dstrbuton of Wealth, Income and Welfare Wealth dstrbuton remans unchanged at ts orgnal level Income of ndvdual : Y rk wk(1 l ), Average economy-wde ncome s: Y rk wk(1 l). Relatve ncome of ndvdual, y Y Y y(, l k) 1 ()( l k 1), where () l 1 (1 )(1 l, (14) ) Relatve ncome depends upon two factors, () the ntal (unchangng) relatve holdng of captal, and () the equlbrum allocaton of tme between labor and lesure 17

19 As long as the equlbrum s one of postve growth, 0 ( l) 1 (15) Relatve ncome s ncreasng n k ; rcher ndvduals supply less labor, but ths effect s nsuffcently strong to offset mpact of ther hgher captal ncome. The DD locus llustrates the relatonshp between the standard devaton of relatve ncome, y, [our measure of ncome nequalty], and the standard devaton of captal endowments, k, DD () l (12c) y k Gven k, the standard devaton of ncome s a decreasng and concave functon of aggregate lesure tme. Havng determned equlbrum allocaton of labor from the upper panel n Fg. 1, (12c) determnes correspondng varablty of ncome across agents. 18

20 Welfare Measures Indvdual welfare equals the value of the ntertemporal utlty functon (3) evaluated along the equlbrum growth path. Startng from K,0, ths s X ( K ) 1 ( C / K) l 0 K,0 (16) Welfare of ndvdual relatve to that of the ndvdual wth average wealth (1 ) C K l l 1 xk ( ) k 1 1 k 1 CK l l 1 l (1 ) The better endowed agent wll have the hgher absolute level of welfare. 19

21 We can now compute the followng natural measure of welfare nequalty. xk uk l k where 1 (1 ) ( ) ( ) 1 ( )( 1) 1 () l 1 1 l Welfare nequalty, expressed n terms of equvalent unts of captal, can be measured by the standard devaton of relatve utlty UU () l (17b) u In the absence of taxes, and assumng postve growth k k y u (17c) (17a) Fg 1 provdes smple framework for analyzng effects of structural changes n growth-nequalty relatonshp. 20

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

23 Applcaton to Taxes: (Garca-Penalosa-Turnovsky, JMCB, 2007) Need to dstngush before-tax and after-tax ncome nequalty The general effects of fscal polcy on these dstrbutonal measures can be summarzed by the followng proposton: Gven the ntal dstrbuton of captal across agents: () Fscal polcy nfluences the before-tax dstrbuton of ncome and the dstrbuton of welfare through effect on labor supply. () Any fscal polcy that ncreases (decreases) the supply of labor ncreases (decreases) before-tax ncome nequalty and decreases (ncreases) welfare nequalty. () A labor ncome tax and a captal ncome tax both nfluence after-tax dstrbuton n two ways; --Drect redstrbutve mpact, --Indrect effect through changes n the labor supply. (v) A consumpton tax or nvestment subsdy has only latter effect. 21

24 Proposton hghlghts how, when labor supply s endogenous, ncome nequalty s a poor measure of welfare nequalty. In fact, changes n pre-tax nequalty are nversely related wth those of welfare nequalty. Post-tax ncome nequalty not necessarly better measure of welfare nequalty. Fscal polcy has two effects on the dstrbuton of post-tax ncome, Net ncome nequalty need not move together wth welfare nequalty. 22

25 Specfc ssue they address relates to fnancng an nvestment subsdy Compare fnancng usng () captal ncome tax, () labor ncome tax, () consumpton tax () Investment subsdy fnanced by a captal ncome tax: ncreases the growth rate, labor supply, and before-tax ncome nequalty. It reduces welfare nequalty and has an ambguous effect on after-tax ncome nequalty. () An nvestment subsdy fnanced by a labor ncome tax: ncreases the growth rate. Although t has an ambguous effect on labor supply, and therefore on before-tax ncome nequalty and welfare nequalty, t ncreases after-tax ncome nequalty. () An nvestment subsdy fnanced by a consumpton tax: ncreases the growth rate, labor supply, and both before-tax and aftertax ncome nequalty, whle t reduces welfare nequalty. 23

26 Hghlghts the dfferent growth-nequalty tradeoffs generated by ths polcy. 1. Changes n fscal polcy that ncrease the growth rate wll also ncrease pretax ncome nequalty, n lne wth some recent emprcal evdence. Reason: Faster growth greater supply of labor, reducng the wage rate, rasng the return to captal, the dstrbuton of ncome more unequal. Ths apparent conflct between effcency and equty dsappears when nequalty s measured n terms of utlty. 2. Pre-tax and post-tax nequalty need not move together. Indrect effect of subsdy s always the same (ncreasng the growth rate, labor supply, and hence pre-tax nequalty), Drect effect depends upon the tax used to fnance t. Consumpton tax has no drect redstrbutve mpact, labor ncome tax redstrbutes toward those wth hgher ncomes, captal ncome tax redstrbutes n the reverse drecton. These drect effects can be large enough to domnate the ndrect effect. 24

27 Can carry out smlar analyss for model of human captal If human captal s only form of captal then economy s always on balanced growth path (Turnovsky, JHC 2011) 25

28 A Dynamc Model Two dfferences: () () Populaton s growng at constant rate n Neoclasscal producton functon Y F K, L (18) Yj Y w FL ( K, L) w( K, L) L L j (19a) Yj Y r FK ( K, L) r( K, L) K K j (19b) where w F 0; w F 0; r F 0; r F 0. K KL L LL K KK L KL Otherwse, framework remans as before 26

29 Aggregate Equlbrum Dynamcs Summng over ndvduals, aggregate macroeconomc equlbrum s descrbed by the followng par of equatons n Kt ( ) and lt: ( ) FL (, ) ( KLl K F K L, ) nk (20a) FKL ( K, L) FL ( K, L) l FK ( KL, ) n(1 ) FKL (, ) nk FL ( K, L) l (20b) Gl () where Gl 1 (1 ) (1 ) F (20c) l F () LL L Ths s conventonal Ramsey model wth endogenous labor supply Involves dynamc adjustment Aggregate behavor s ndependent of dstrbuton 27

30 Steady State F (, ) K KL n (21a) FL (, ) ( KLl F K L nk, ) (21b) L l 1 (21c) These equatons mply the key relatonshp: l 1 (22) 28

31 Local Aggregate Dynamcs K a a K K l a21 a 22 l l where a11a22 a12a21 0. (23) The stable path for K and l s K() t K ( K K ) e t (24a) 0 a a lt l Kt K Kt K (24b) () () () a22 a12 In general lt ( ) vares nversely wth Kt ( ) The evoluton of aggregate lesure over tme s an essental determnant of the tme path of wealth nequalty, and thus of ncome nequalty. Suppose economy s subject to an expansonary structural shock ( K0 wll lead to a jump n average lesure, such that l(0) ncrease monotoncally durng the transton. K ). Ths l ; thereafter, lesure wll 29

32 The dstrbuton of ncome and wealth The dynamcs of relatve captal stock wkl (, ) 1 1 k () t 1 l 1 1l 1 k K (25) where K, l evolve n accordance wth (24a, b). To solve for tme path of k () t, note that agent s steady-state share of captal s l 1 1l 1 k 0 for each (26) or, equvalently [cf equaton (10)] l l l ( k 1) 1 for each (27) 30

33 Next, lnearze (25) around the steady-state K, l, k, wkl (, ) 1 1 k () t 1 ( k )(() l t l) l 1 1 k () t k K The bounded soluton to ths equaton s (28) where k () t 1 ()( t k 1) (19) 1 FL ( K, L) lt ( ) () t 1 1, (30) K l 1 FL ( K, L) l(0) k,0 1 (0)( k 1) 1 1 ( k 1) K l and k,0 s gven from the ntal dstrbuton of captal endowments. (31) 31

34 The evoluton of agent s relatve captal stock s determned as follows. () Gven the tme path of aggregate economy, and dstrbuton of ntal captal endowments, (31) determnes the steady-state relatve captal, ( k 1), () Together wth (29) ths yelds entre tme path for the dstrbuton of captal. () We can express the tme path for k () t n the form () t 1 l() t 1 t k( t) k ( k,0 k ) ( k,0 k ) e ( k,0 k ) (0) 1 l(0) 1 (32) 32

35 Because of lnearty, we can transform these equatons nto correspondng results for the coeffcent of varaton of the dstrbuton of captal: () t () t (29 ) k k k,0 (0) k (31 ) () t 1 l() t 1 t k( t) k ( k,0 k) ( k,0 k) e ( k,0 k) (0) 1 l(0) 1 (32 ) Two crtcal factors: () how agent s relatve wealth evolves over tme and () the consequences of ths for the dstrbuton of ncome. A crtcal determnant of ths s the magntude of ( t). From (30) ths s seen to depend upon lt () l, whch n turn depends upon how the underlyng shock affects the evoluton of the aggregate captal stock, ( Kt ( ) K ). 33

36 Proposton (Wealth dynamcs): () () The long-run dstrbuton of wealth converges to a non-degenerate steady-state dstrbuton, whch s proportonal to ntal dstrbuton. If economy experences an expanson (contracton) n ts aggregate captal stock,.e. K 0 < K ( K 0 > K ), then wealth nequalty wll decrease (ncrease) durng transton, and the long-run dstrbuton of wealth wll be less (more) unequal than s the ntal dstrbuton. 34

37 Income Dstrbuton Income of ndvdual and average economy-wde ncome at tme t are Y() t r() t K() t w()(1 t l()) t, Yt () rtkt () () wt ()(1 lt ()), Relatve ncome y () t Y()/ t Y() t. s() t F K/ Y denotes the share of output gong to captal, K lt () y() t 1 st ()( k() t 1) (1 st ()) (1 ) 1 lt ( ) (33) The relatve ncome of agent has two components, () relatve captal ncome, () relatve labor ncome. 35

38 We can express (33) as y () t 1 ()( t k () t 1), (34) lt () 1 1 where () t 1 (1 s())1 t 1 1 lt ( ) l 1. (35) ( t) Because of lnearty of (28 ) n ( k () t 1) we can express ths relatonshp between relatve ncome and relatve captal n terms of standard devatons () t () t () t () t 1 (28 ) y k mplyng that ncome s more equally dstrbuted than s captal. 36

39 In steady state: (28 ) y k 1 1 s 1 FL ( KL, ) lm ( t) 1 1 t 11 l 1 F( KL, ) FL K L F K L FL K 0 L 0 F K 0 L (1 ( ) ( ) y k k 1 1 (1 ( ) ( ) y,0 0 k,0 k,0 Whether long-run dstrbuton, followng a structural change, s more or less unequal than ntal dstrbuton depends on long-run change n the dstrbuton of captal, as reflected n k k,0, and factor returns, as reflected n 0. (36) 37

40 Dynamcs of relatve ncome dy() t dk() t 1 dl() t l() t ds() t st () (1 st ()) k() 1 2 t 1 dt dt 1 lt ( ) dt 1 l ( t ) dt The evoluton of the relatve ncome of agent depends upon two factors, () Evoluton of relatve captal ncome (frst term), () Evoluton of relatve labor ncome. [Ths can be expressed as a functon of the evoluton of aggregate lesure, and of the relatve rewards to captal and labor, as reflected by the captal share, s(t)]. 38

41 For Cobb-Douglas producton functon, s( t ) remans constant. Whether ncome nequalty ncreases or decreases depends on whether the economy converges to the steady state from below or from above. If economy that starts below the steady state, so thatk0 lesure s rsng, whle wealth nequalty s decreasng. K. Then l(0) l and Consder an agent wth above average wealth, ( k 1) 0, then dk / dt 0 and ( 1) 0, mplyng that the frst two terms are negatve and that the relatve ncome of the agent s decreasng durng the transton. The evoluton of factor shares may renforce or offset these effects. 39

42 Proposton (Income dynamcs): The evoluton of ncome nequalty for an economy that converges to ts steady state from below,.e. K0 K, (respectvely, from above,.e. K0 K ) s drven by three factors: () Decreasng (ncreasng) wealth nequalty, whch tends to reduce (rase) ncome nequalty; () Increasng (decreasng) lesure, whch tends to ncrease (decrease) the relatve labor ncome of the captal-poor and hence reduce(rase) ncome nequalty; () Change n the share of captal n ncome, whch depends both on whether the economy s convergng from below or above, and on the elastcty of substtuton n producton. If the share of captal s constant, ncome nequalty wll decrease (ncrease) durng the transton to the steady state from below (above). 40

43 Elastc labor supply s mportant Wth nelastc labor supply, there s no change n lesure tme. Hence, the evoluton of the ncome dstrbuton s drven by two forces, the change n wealth nequalty and that of labor share. Wth a hgh elastcty of substtuton n producton, these two forces have opposte sgns, and t s possble that the second domnates, makng the dstrbuton of ncome more dspersed. When the labor supply can respond, the changes n average lesure wll change the dstrbuton of work-tme and tend to renforce the wealthdstrbuton effect. The presence of ths effect mples that ncome nequalty may move n opposte drectons dependng on the elastcty of lesure. 41

44 Numercal Smulatons Producton functon: 1 Y A( K (1 ) L ) Utlty functon: U 1 Cl Basc parameters: A 1, 0.4 1/ 3, 0, 0.2 (elast of sub 0.75, 1, 1.25) 0.04, 1.5, n Endogenous labor 1.75 Exogenous labor L 0.316, 0.278, correspondng to 0.75, 1, 1.25 Economy s ntally n a steady state n whch aggregate fracton of tme devoted to lesure s l 0 and the average stock of captal s K 0. For Cobb-Douglas economy wth endogenous labor supply, l , L To preserve comparablty, n the case of nelastc labor supply we normalze the fxed labor supply to the same level; mplyng that all aggregate magntudes wll be the same n the two cases. 42

45 Startng from ntal steady state, we nvestgate tme paths n response to: () An ncrease n the level of technology A from 1 to 1.5 (Fg. 1); () A decrease n the rate of populaton growth rate, n, from 1.5% to 0 (Fg. 2). () A decrease n the rate of tme preference from 0.04 to 0.02 (Fg 3). Plot the tme paths for the dstrbuton of wealth and ncome, relatve to ther respectve ntal values, [ ()/ t and () t ], where normalze 1. k k,0 y y,0 k,0 Left-hand panels plot the tme paths of the wealth and ncome nequalty when the labor supply s exogenous, Rght-hand panels present the case wth endogenous labor, for case =

46 Inelastc labor supply Fg 1: Increase n A from 1 to 1.5 Elast of Subs =0.75 Flexble labor supply dst. dst t t Elast of Subs =1 dst. dst t t Elast of Subs =1.25 dst. dst t t captal ncome

47 Inelastc labor supply Fg 2: Decrease n n from to 0 Elast of Subs =0.75 Flexble labor supply dst t dst t Elast of Subs =1 dst t dst t Elast of Subs =1.25 dst. dst t t captal ncome

48 Fg 3: Decrease n β from 0.04 to 0.02 Inelastc labor supply Flexble labor supply Elast of Subs =0.75 dst. dst t t Elast of Subs =1 dst. 1.3 dst t t Elast of Subs =1.25 dst dst t t captal ncome

49 Path Dependence and Inequalty Effect of structural change on dynamcs of wealth and ncome nequalty, depends crtcally upon ntal response of lesure (labor supply) to shock. Ths ntal response, n turn, depends upon tme path that structural change s expected (known) to follow. Impacts of a productvty ncrease of a gven magntude on the long-run dstrbutons of both wealth and ncome are crucally dependent upon the tme path that the productvty ncrease s assumed to follow. Sharp contrast to the aggregate economy. In that case, the path followed by the productvty ncrease affects only the transtonal path of the aggregate economy; t has no mpact on steady-state aggregates. 44

50 Model thus far adopts standard procedure of specfyng structural changes as an unantcpated dscrete permanent ncrease. For example a 50% ncrease n the level of productvty, whch nduces an nstantaneous ncrease n labor supply and decrease n lesure. But f same (50%) productvty level ncrease s acqured gradually over a known tme path, and s therefore antcpated after the frst nstant, the ntal response of lesure s exactly the opposte. It ncreases rather than declnes, leadng to profoundly dfferent dstrbutonal consequences. Important aspect --Model can admt a dverse set of dstrbutonal equlbra for countres havng smlar levels of aggregate development (per capta ncome), n accordance wth emprcal evdence. --Important n foregn ad whch typcally s granted over tme 45

51 Gradual mplementaton of productvty ncrease leads to dramatc dfferences along the transtonal path, from case where ts full mpact s mmedate. 1. Dscrete productvty ncrease leads to steady declne n wealth nequalty. 2. Gradual ncrease leads to ntal ncrease n wealth nequalty, followed by eventual declne. 3. In long run, lkely to lead to more rather than less wealth nequalty, unless flexblty of producton s very hgh. Even f long-run wealth nequalty declnes, wll stll be greater than f productvty ncrease s dscrete. 4. Dscrete productvty ncrease leads to ntal ncrease n ncome nequalty, followed by a steady declne to below ts ntal level. 5. Gradual ntroducton generates essentally the opposte tme profle. 6. But the most strkng aspect of the transtonal behavor s that a gradual productvty ncrease can generate a form of the Kuznets-type nverted-u relatonshp between nequalty and per-capta ncome. 46

52 Analytcal framework Level of productvty, A( t ) s assumed to ncrease gradually from ts ntal level, A 0, to ts enhanced long-run level, A, (both known to the frm). 0 A() t A A A e t (37) defnes the tme path followed by the ncrease n productvty. The conventonal approach s approxmated by lettng. Contrast between how affects the aggregate behavor of the economy and ts consequences for dstrbuton. Whle t affects the transtonal path of the aggregate economy, t has no effect on the aggregate steady state. But the choce of has profound mpacts on both the tme paths and the steadystate equlbra of wealth and ncome dstrbutons. 47

53 Macrodynamc equlbrum s descrbed by the followng equatons: A() tfl( KLl, ) K A() t F( K, L) nk (38a) 1 FKLK, L) A( t) FL ( K, L) l l AtF () K ( KL, ) n(1 ) AtFKL () (, ) nk GKl (, ) F L (38b) A () t A A () t (38c) where 1 (1 ) FLL GKl (, ) (1 ) 0, ll 1 l F L 48

54 The steady-state equlbrum s FL (, ) ( KLl AF K L nk, ) (39a) AF (, ) K K L n (39b) L l 1 (39c) Snce the steady state s ndependent of, the long-run effects of an ncrease n productvty are ndependent of the tme path by whch t s acheved. However, the transtonal responses may be sgnfcantly dfferent (as llustrated later n the numercal experments). More mportantly, ths has fundamental consequences for wealth and ncome nequalty, both n transton and across steady states. 49

55 3. Dstrbutonal dynamcs Dynamc equaton for the relatve captal stock s now: At () FL ( Kl,) 1 1 k () t 1 l 1 1l 1 k K (40) where K,, la evolve n accordance wth (38a)-(38c). The bounded soluton s k () t 1 ()( t k 1) (41a) where AF () 1 1 K L ( t ) t e d t l( ) l 50 AF L l( ) k,0 1 (0)( k 1) 1 1 e d ( k 1) 0 K l (41b) (41a ) The crucal dfference between ths analyss from prevous les n the form of the productvty shock A(t), whch s reflected n the tme path of ( t).

56 If the complete productvty ncrease occurs nstantaneously, l( ) l l(0) l e and ( t), (0) smplfy to 1 AF L ( K, L) lt ( ) () t 1 1 ; K l 1 AF L ( K, L) l(0) (0) 1 1 K l Only current allocaton of tme to lesure relatve to steady-state allocaton s relevant n determnng current wealth nequalty relatve to ts long run. When the productvty ncrease occurs gradually over tme, the entre tme profle of A( t ), reflected n lt, ( ) needs to be taken nto account. If durng transton l( ) l, so that lesure approaches ts long-run steady state from below, then ( t) 1 and wealth nequalty wll declne over tme. As smulatons show, ths s the case for the dscrete productvty ncrease, when followng an ntal drop, lesure ncreases monotoncally over tme. 51

57 Smulatons also show that a gradual productvty ncrease leads to an ntal ncrease n lesure, takng t ntally above ts new steady-state level. Snce transtonal path s of nverted U-shaped form eventually approachng l from below, whether nequalty rses or falls over tme depends upon the extent to whch l( ) l durng the early phases of the adjustment. Closer l( ) s to ts steady state, l, the smaller the subsequent adjustment n lt, () and hence smaller s the overall change n the dstrbuton of wealth. Ths s because f economy and therefore all ndvduals fully adjust ther respectve lesure tmes nstantaneously, they wll all accumulate wealth at the same rate, causng the wealth dstrbuton to reman unchanged. 52

58 Numercal analyss Compare aggregate and dstrbutonal consequences of a 50% ncrease n productvty, whch we allow to take effect n two alternate ways: () An mmedate unantcpated jump n productvty from 1 to 1.5. () A 50% ncrease n A that occurs gradually over tme, [ A( t ) adjusts at the (known) rate 10 % per perod (year)]. In the latter case, the enhanced productvty level s acheved asymptotcally. As a result, the nstant t starts to ncrease, the subsequent growng levels of productvty are fully antcpated along the transton path. 53

59 Fgure 1 : Aggregate Dynamcs.75 K l Years Years 1.00 K l Years Years 1.25 K l Years Years Dscrete Contnuous

60 Wealth Fgure 2 : Dstrbutonal Dynamcs.75 Income Years Years Years Years Years Dscrete Years Contnuous

61 Fgure 3 : Robustness of Dstrbutonal Dynamcs Wealth Income Years Years Fgure 4 : Kuznets' Curve Wealth Income Normalzed Percapta Income Normalzed Percapta Income

62 The growth-nequalty relatonshp Consequences of the nature of productvty change for the Kuznets curve? Such a relatonshp cannot be generated by a dscrete ncrease n productvty. Ths s because after ntal jumps n output and ncome nequalty, subsequent ncreases n ncome are assocated wth a unform declne n nequalty. The more flexble specfcaton where the level of productvty ncreases gradually s capable of generatng such a pattern for both wealth and ncome nequalty along the transton path, after ntal adjustments. The nverted U-shaped relatonshp between nequalty and per-capta ncome generated by ths model reles on a mechansm that s very dfferent from the tradtonal explanatons of the Kuznets curve. Whle the sectoral composton of captal, labor, and knowledge s the tradtonal mechansm behnd the Kuznets curve, we focus on the role of consumptonsmoothng n response to a gradual, but antcpated, change n productvty, and ts effects on nvestment and the choce of labor and lesure. 54

63 Heterogeneous Sklls Consumers Two sources of heterogenety: agents ntal endowments of captal, K,0, and fxed relatve skll levels, a. Relatve captal (wealth) s k() t K()/ t K() t, has mean 1, and dsperson across agents, k () t, wth ntal (gven) dsperson beng k,0. Relatve sklls across agents has mean zero and constant dsperson a. --prevous model corresponds to a 1 k,,0 a possbly correlated across agents. 55

64 Technology and factor payments Effectve labor employed by the frm: L 1 al N Frms pay captal and labor accordng to ther margnal physcal products, Average wage rate rt () rkl (, ) F( KL, ) (22a) K w() t w( K, L) F ( K, L) a F ( K, L) (22b) L L wt () wk (, L) F( K, L) (22b ) L Dstrbuton of relatve wage rates, w () t w() t, s gven, and reflects the gven dstrbuton of skll levels across agents. 56

65 Macroeconomc equlbrum (1 L) K F( K, L) FL ( K, L) K 1 FKL (1 L) L (1 ) F FL K ( ) F ZL ( ) F L 1 (1 ) (1 ) FLL Z( L) 0 1 L F Two dfferences from prevous model --no populaton growth, but have deprecaton of captal -- more convenent to work n terms of L rather than l L K 57

66 Assumng stablty, aggregate quanttes converge to steady state F (, ) K K L (43a) (1 L) FKL (, ) K FL ( KL, ) (43b) L l 1 (43c) Agan get constrant 1 L 0 1 (44) (Locally) stable paths for K () t and Lt () as before. 58

67 The dynamcs of relatve wealth and ncome Relatve captal stock (wealth) Dynamcs of ndvdual s relatve captal stock, k () t K () t K() t : wkl (, ) 1 1 k () t a a l 1 l k() t K (45) (45) mples steady-state relatonshp: 1 k L L L 1 1 a for each (46) An analogous relatonshp holds at all ponts of tme lt () 1 k L () t L() t L 1 l 1 a (46 ) 59

68 Bounded soluton for k () t can be expressed as 1 ( t) (0) ( t) k () t k a 1 (0) 1 (0),0 FL ( K, L)/ K L(0) L (0) 1L (47) (48) () t (0) e t (48 ) At any pont of tme, agent s relatve captal s weghted average of k,0 and a. Weghts of the two endowments change over tme. As economy converges to new steady state, factor prces change, alterng relatve contrbutons of wealth and skll endowments to ndvdual s ncome, and hence to savngs. In expandng economy, L(0) L, and (0) ( t) 0, () t () t 0 ;.e. relatve weght shfts from the endowment of captal toward sklls. 60

69 Dfference between an agent s relatve captal and the mean s 1 ( t) (0) ( t) k() t 1 ( k,0 1) ( a 1) 1 (0) 1 (0) (49) 1 (0) 1 k 1 ( k,0 1) ( a 1) ( k ( t) 1) ( t)( a 1) 1 (0) 1 (0) 1 ( t) Expressons llustrate potental for agents to change relatve wealth postons. If agent begns wth above-average captal ( k,0 1), but s endowed wth belowaverage sklls ( a 1), he may end up wth below-average captal. Two offsettng forces drvng the accumulaton of captal. 1. Those wth large ntal wealth accumulate captal more slowly (durng an expanson), whch tends to deterorate ther relatve poston. 2. Those wth more ablty have hgher ncomes, and hence accumulate more captal, whch tends to mprove ther relatve poston. Ths contrasts wth the prevous model where ( a 1). 61

70 The dynamcs of relatve ncome Agent s relatve ncome can be expressed as y() t () t k() t (1 ()) t a (50) where lt () 1 1 () t sk() t sl() t L ll () t 1 (51) 1 ( t) represents weght n current relatve ncome due to current relatve captal. Dynamcs of y () t are drven by those of the aggregate varables, K (), t L() t, (both drectly and through effect on factor shares), as well as by agent s relatve rate of captal accumulaton, k () t. We can express current relatve ncome as a weghted average of k,0 and a 1 ( ) 1 ( ) () () t t y t t k 1 () t a 1 (0) 1 (0),0 (52) 62

71 4.3 Wealth and ncome moblty Compare two ndvduals, j, and express ther wealth gap at tme t as 1 ( t) (0) ( t) k() t kj() t k a 1 (0) 1 (0) where a a aj and k k,0 kj,0. Two offsettng forces nfluencng ths gap, the dfferences n ntal captal and the dfferences n ablty. In a growng economy, (0) ( t), () t 0, mplyng that the term multplyng the captal gap s less than one and declnng over tme. (53) 63

72 Wealth moblty: Possblty that ndvdual havng an ntal small wealth endowment overtakes some other ntally rcher agent. Proposton: In an economy that s accumulatng captal [ (0) 0], () f ndvdual j s ntally endowed wth less wealth than s ndvdual, the poorer agent wll catch up n wealth f and only f a (0) k; () f ndvdual j s ntally endowed wth both less wealth and less ablty than ndvdual, the poorer agent wll never catch up. Poorer agent wll catch up n wealth f and only f he has suffcently superor ablty. 64

73 Income moblty: May occur on mpact, or along subsequent transtonal path. Proposton: Indvdual may ntally be rcher than j because of hgher ntal wealth, hgher ablty, or both. If that s the case, then () f has a larger endowment of both ablty and wealth, j cannot catch up to s ncome level; () f j s ntally endowed wth less wealth than s, the poorer agent wll catch up n ncome along the transtonal path f and only f a (0) 1 (0) k 1 (0) and the economy satsfes (0) / (1 (0)) ; (54a) () f j s ntally endowed wth less skll than s, the poorer agent wll catch up n ncome f and only a (0) 1 (0) k 1 (0) and the economy satsfes (0) / (1 (0)). (54b) 65

74 As the economy converges to new steady state, ncome moblty s possble only for one type of agent, ether skll-rch or the captal-rch, but not both. If wages are growng fast, then skll-rch agents wll be able to catch-up but captal-rch ndvduals wll not, and vce versa. The behavor of factor prces depend on both the structure of aggregate economy and the nature of the shock, whch s captured by sgn (0) / (1 (0)). For Cobb-Douglas producton functon, n growng economy (0) / (1 (0)) always holds. Then skll-rch that may catch up n ncome; n contrast to contractng economy t s captal-rch for whom ths s possble. 66

75 Measure of wealth moblty Defnton: Letâ be the mnmum ablty gap requred for j to catch up to s wealth, gven ther ntal wealth gap, k. We then defne the extent of wealth moblty, denoted k, as ˆ / 1 k a k. Measure of wealth moblty s nverse of the mnmum ablty gap for catch-up. k FL( K)(1 w)/ K L(0) L (0) FL( K )(1 w) L L K 1 (55) The degree of wealth moblty depends on both structural characterstcs of the aggregate economy and specfc change generatng the ntal jump n aggregate labor supply. 67

76 Measure of ncome moblty Defnton: Let a ( k ) be the mnmum ablty (wealth) gap requred for j to catch-up n ncome when t s the ablty-rch (captal-rch) that may experence ncome moblty. Whenever the skll-rch can catch-up wth the captal-rch, we a defne measure of ncome moblty to be / 1 y a k ; whenever t s the k captal-rch that are catchng up, we defne t to be / 1 y k a. We measure degree of ncome moblty by endowment gap requred for poorer agent to be able to catch up to the rcher one durng the transton, where ncome moblty depends on whch agent s dong the catchng up. From the defntons a y k y 1 (0) 1 (0) (56a) (56b) Hgher value of a k y or y mples that, for gven dstrbutons of ntal wealth and sklls, a greater fracton of the populaton wll change ther relatve poston along the dstrbuton of ncome. 68

77 Proposton: In a growng economy f agent catches up to agent j s level of wealth he wll do so only after he has caught up to agent j s level of ncome. It s also possble that he wll catch up to hs level of ncome, but not to hs level of wealth. Snce agents save a fracton of ther ncome strctly less than one and gven that had a hgher ntal stock of captal, j wll manage to accumulate as much wealth as only f he has a hgher level of ncome. Hence, he must catch up s ncome level before he can catch-up to hs wealth. 69

78 5. Wealth and ncome nequalty Lnearty of expressons permts us to transform them nto measures of aggregate wealth nequalty, expressed ether as CV, [ () t ], or SCV, [ 2 () t ]. Both have qualtatvely smlar mplcatons, they have dfferent advantages; () t s dmensonally equvalent to Gn coeffcent, 2 () t s decomposable. k k() t 1 () t 2 k,0 (0) () t a 2(1 ())[ t (0) ()] t k,0a [1 (0)] where s the correlaton coeffcent between ntal captal endowments and sklls ,0 (0) 2 (0),0 k [1 (0)] k a k a k k k 70

79 Consder an economy that s accumulatng captal as a result of an expansonary external shock. In general, ths can be assocated wth an ncrease or decrease n wealth nequalty, dependng upon the relatve dspersons of the ntal endowments of captal and sklls and ther correlaton. Wealth nequalty can emerge from dfferences n skll endowments alone 71

80 Analogously, we can express ncome nequalty n terms of ts SCV () t () t (1 ()) t 2 ()(1 t ()) t y k,0 a k,0 a Consder an economy wth a Cobb-Douglas producton technology. If the economy experences an expansonary external shock that leads to an accumulaton of captal and does not cause a long-run declne n employment, we obtan the followng: () If a 0, ncome nequalty, as measured by ts SCV, ntally ncreases and then declnes unambguously durng the transtonal phase. () For,0 0 k, ncome nequalty, as measured by ts SCV, ntally declnes and then ncreases unambguously durng the transtonal phase. 72

81 Frst term captures equalzng effect of transton: captal-rch accumulate more slowly than captal-poor, then nequalty n wealth s reduced. The second result captures unequalzng effect of the transton, and reflects two forces. 1. Accumulatng captal mples that wage ncreases, magnfyng effect of unequal abltes. 2. Because those wth hgh ablty have hgher ncomes they wll also save more, addng to the nequalty n ablty an nequalty n captal. As a result nequalty ncreases durng the transton. 73

82 Relatonshp between ncome nequalty and moblty. We can express the change n ncome nequalty followng a shock as y y y y,0 a 0 a 0 k,0 a k,0 a a k,0 a Inequalty and moblty need not move together. Possble for a shock to generate substantal ncome moblty (.e. large value of a ) and yet engender small changes n nequalty, 1/(1 a ) y y 0 close to zero. 74

83 Garca-Penalosa and Turnovsky perform numercal smulatons on moblty n response to two structural changes () () Increase n productvty Changes n tax structure Crtcal element s elastcty of labor supply 75

84 Table 1: Increase n productvty Baselne: Cobb-Douglas ( 0, 1 ) and labor supply elastcty of Labor K Y Base: A L (0) A 2 L (0%) (+53.6%) (+53.6%) 2 k (-3.03%) 2 e (-2.57%) 2 y (-1.10%) k a y Low elastcty of the labor supply: 1. 0 (and Cobb-Douglas producton) Labor K Y Base: A L (0) A 2 L (0%) (+53.6%) (+53.6%) 2 k (-2.93%) 2 e (-2.05%) 2 y (-1.44%) k a y Hgh elastcty of substtuton n producton: 0.13, (and labor supply elastcty of ) Labor K Y Base: A L (0) A 2 L (-2.18%) (+70.8%) (+63.6%) 2 k (-1.28%) 2 e (+12.6%) 2 y (+3.11%) k a y

85 Table 2: Fscal changes Base: g w k Expendture/tax reducton g 0.17 w k Shft n the tax burden: Reducton n captal ncome tax 0.17, 0.245, 0.22 k w g Shft n the tax burden: Reducton n labor ncome tax 0.322, 0.17, 0.22 k w g Baselne: Cobb-Douglas ( 0, 1 ) and labor supply elastcty of Labor K Y 2 k L (0) L L (0) L L (0) L e 2 y k a y Base: g w k Expendture/tax reducton g w k Shft n the tax burden: Reducton n captal ncome tax 0.17, 0.245, g 0.22 k w Shft n the tax burden: Reducton n labor ncome tax 0.321, 0.17, g 0.22 k w Low elastcty of the labor supply: 1. 0 (and Cobb-Douglas producton) Labor K Y 2 k L (0) L L (0) L L (0) L e 2 y k a y

86 Table 2 (contnued): Fscal changes Base: g w k Expendture/tax reducton g 0.17 w k Shft n the tax burden: Reducton n captal ncome tax k 0.17, w 0.254, g 0.22 Shft n the tax burden: Reducton n labor ncome tax 0.298, 0.17, 0.22 k w g Hgh elastcty of substtuton n producton: 0.13, (and labor supply elastcty of ) Labor K Y 2 k L (0) L L (0) L L (0) L e 2 y k a y

87 Conclusons Relatonshp between growth and nequalty/dstrbuton s complcated! Multdmensonal Ths dscusson has dealt wth just one aspect and under specal condtons Hopefully has presented how one mght study the ssue n a tractable way that provdes some nsghts 76