Time Preference and the Distributions of Wealth and Income. Richard M. H. Suen University of Connecticut

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1 Tme Preference and the Dstrbutons of Wealth and Income Rchard M. H. Suen Unversty of Connectcut Workng Paper January 202

2 Tme Preference and the Dstrbutons of Wealth and Income Rchard M. H. Suen y Ths Verson: January 202 Abstract Ths paper analyzes the connecton between tme preference heterogenety and economc nequalty. To acheve ths, we extend the standard neoclasscal growth model by ntroducng three addtonal features, namely () heterogenety n consumers dscount rates, () drect preferences for wealth, and () human captal formaton. The second feature prevents the wealth dstrbuton from collapsng nto a degenerate dstrbuton. The thrd feature generates a strong postve correlaton between earnngs and captal ncome across consumers. A calbrated verson of the model s able to generate patterns of wealth and ncome nequalty that are very smlar to those observed n the Unted States. Keywords: Inequalty, Heterogenety, Tme Preference, Human Captal JEL class caton: D3, E2, O5. I would lke to thank James Daves, Jang-Tng Guo, Karen Kopecky, Jm MacGee, Perre-Danel Sarte, Png Wang, semnar partcpants at the Unversty of Western Ontaro, conference partcpants at the 200 Mdwest Macro Meetngs and the 200 CEA Annual Conference for helpful comments and suggestons. y Department of Economcs, 34 Mans eld Road, Unt 063, Unversty of Connectcut, Storrs CT Emal: rchard.suen@uconn.edu. Phone: (860) Fax: (860)

3 Introducton Emprcal studes show that ndvduals do not dscount future values at the same rate. Snce ndvduals asset accumulaton and schoolng choces are strongly a ected by the way they dscount the future, ths type of heterogenety would naturally lead to cross-sectonal d erences n wealth and ncome. To examne the connecton between tme preference heterogenety and economc nequalty, ths study develops a dynamc compettve equlbrum model n whch consumers only d er n terms of ther dscount rates. It s shown that a calbrated verson of the model can generate patterns of wealth and ncome nequalty that are very smlar to those observed n the Unted States. The mportance of tme preference heterogenety n explanng wealth nequalty s well recognzed n exstng studes. There s now a vast lterature n macroeconomcs that uses the ncomplete markets model of Huggett (993, 996) and Ayagar (994) to explan wealth and ncome nequalty. 2 The standard ncomplete markets model, however, has d culty n explanng certan features of the wealth dstrbuton n the Unted States. In partcular, t fals to generate a hgh concentraton of wealth at the top end of the wealth dstrbuton. 3 Krusell and Smth (998) show that ntroducng tme preference heterogenety can sgn cantly mprove the Ayagar (994) model n ths regard. Smlarly, Hendrcks (2007) shows that ntroducng ths type of heterogenety nto the lfe-cycle model of Huggett (996) can mprove the model s ablty to account for wealth nequalty. In both Krusell and Smth (998) and Hendrcks (2007), cross-sectonal varaton n ncome s manly drven by unnsurable dosyncratc earnngs rsk, whch s exogenous and ndependent of the heterogenety n dscount rates. These two sources of consumer heterogenety are then used to account for the wde dsperson n wealth. Ths approach, however, gnores the e ects of tme preferences on lfetme earnngs. Intutvely, more patent ndvduals are more wllng to nvest n nancal assets as well as human captal than less patent ones. A hgher level of human captal then leads to a hgher level of lfetme earnngs for those who are more patent. Ths ntuton s consstent wth emprcal ndngs. Lawrance (99) and Warner and Pleeter (200) nd that more-educated households and ndvduals tend to have lower dscount rates than less-educated ones. Ths connecton between patence and educatonal attanment suggests that human captal formaton may provde an addtonal channel through whch tme preference heterogenety can gve rse to wealth and ncome nequalty. A detaled revew of these studes can be found n Frederck et al. (2002) Secton 6. 2 An excellent revew of ths lterature can be found n Heathcote et al. (2009). 3 See Castañeda et al. (2003) for a detaled dscusson of ths problem. 2

4 The man objectve of ths study s to explore the quanttatve mplcatons of ths addtonal channel. To acheve ths, we generalze the standard determnstc neoclasscal growth model to allow for three mportant features, namely () heterogenety n tme preference, () human captal formaton, and () consumers drect preferences for wealth. The assumpton of drect wealth preference has long been used n economc studes. In an early paper, Kurz (968) ntroduces wealth preference nto the optmal growth model and explores the long-run propertes of the model. Zou (994) nterprets ths type of preference as re ectng the captalst sprt, or the tendency to treat wealth acquston as an end n tself rather than a means of satsfyng materal needs. Cole et al. (992) suggest that the ncluson of nancal wealth n consumers preferences can be vewed as a reduced-form spec caton to capture people s concern for ther wealth-nduced status wthn socety. Subsequent studes have followed these tradtons and nterpreted ths type of preference as ether capturng the sprt of captalsm or re ectng the demand for wealth-nduced status. In ths paper, we refer to ths feature smply as wealth preference. There s now a rapdly growng lterature that explores the mplcatons of wealth preference on a wde range of ssues, such as asset prcng, economc growth, expectatons-drven busness cycles, e ects of scal polcy and wealth nequalty. 4 The man purpose of ntroducng wealth preference n our model s as follows. It s now well known that the standard neoclasscal growth model has d culty n generatng realstc wealth dstrbuton based on d erences n dscount rates alone. Becker (980) shows that when consumers have tmeaddtve separable preferences and d erent constant dscount rates, all the wealth n the neoclasscal world wll eventually be concentrated n the hands of the most patent consumers. In other words, the wealth dstrbuton s degenerate and extremely unequal n the long run. Several exstng studes have dent ed condtons under whch the long-run wealth dstrbuton s non-degenerate. 5 In ths study, we show that a non-degenerate wealth dstrbuton can be obtaned by assumng that consumers 4 Studes that explore the mplcatons of wealth preference on asset prcng nclude Baksh and Chen (996), and Boleau and Braeu (2007) among others. Studes on economc growth nclude Zou (994) and Smth (999) among others. Karnzova (200) ntroduces ths type of preference nto a neoclasscal growth model wth captal adjustment costs and shows that the model can generate expectatons-drven busness cycles. Gong and Zou (2002) and Nakamoto (2009) examne the welfare mplcatons of scal polcy when consumers value wealth drectly. Fnally, Luo and Young (2009) explore the mplcatons of wealth preference on wealth nequalty. Ths study wll be dscussed n greater detal later on. 5 Boyd (990) shows that Becker s result s no longer vald when consumers have recursve preferences. Sarte (997) establshes the exstence of a non-degenerate wealth dstrbuton by ntroducng a progressve tax structure nto Becker s model. Sorger (2002) shows that Becker s result cannot be extended to the case where consumers are strategc players, rather than prce-takers, n the captal market. Espno (2005) establshes a non-degenerate wealth dstrbuton by assumng that consumers have prvate nformaton over an dosyncratc preference shock. Except for Sarte (997), none of these studes have explored the quanttatve mplcatons of ther model. Sarte shows that a calbrated verson of hs model can replcate the ncome dstrbuton n the Unted States. However, unlke the current study, he does not attempt to explan wealth and ncome nequalty smultaneously. 3

5 have drect preferences for wealth. The ntuton behnd ths result can be explaned as follows. In the orgnal Becker (980) model where there s no drect wealth preference, a consumer wll choose to hold a constant postve level of nancal wealth only when the equlbrum nterest rate s dentcal to hs dscount rate. Snce there s only one nterest rate n the neoclasscal model, t s not possble for consumers wth d erent dscount rates to mantan constant postve levels of wealth smultaneously. In the long-run equlbrum, nterest rate s equated to the lowest dscount rate n the populaton. Thus, only the most patent consumers would have postve asset holdngs. All other consumers wth dscount rate greater than the equlbrum nterest rate wll deplete ther wealth untl t reaches zero. Thus, the long-run wealth dstrbuton n the Becker (980) model s extremely unequal. Introducng drect preferences for wealth changes ths result by creatng some addtonal bene ts of holdng nancal assets. These addtonal bene ts fundamentally change the consumers savng behavor. In partcular, consumers are now wllng to mantan constant postve levels of wealth even f the nterest rate s lower than ther dscount rates. These addtonal bene ts not only prevent the consumers from depletng ther wealth to zero, they also nduce d erent types of consumers to hold d erent levels of wealth. Thus, the equlbrum wealth dstrbuton s non-degenerate. In the quanttatve analyss, we nd that consumers drect preferences for wealth play a crucal role n explanng wealth nequalty. In partcular, our results show that a model wth tme preference heterogenety and wealth preference alone can replcate some key features of the wealth dstrbuton n the Unted States (see Secton 5.2). Such a model, however, cannot produce large varatons n earnngs across consumers. Ths type of varaton s mportant n explanng ncome nequalty because earnngs s the most mportant source of ncome n the model economy. Consequently, a model wth only tme preference heterogenety and wealth preference cannot explan the observed patterns of wealth and ncome nequalty smultaneously. Introducng human captal formaton helps mprove ths result n two ways. Frst, consumers earnngs are now ted to ther dscount rates through the nvestment n human captal. Ths provdes a channel va whch tme preference heterogenety can lead to sgn cant varatons n earnngs across consumers. Second, ntroducng human captal helps create a strong postve correlaton between earnngs and captal ncome. Ths happens because more patent consumers have hgher earnngs and more nancal wealth than less patent ones. Ths n turn generates a substantal degree of ncome nequalty n our model. A calbrated verson of the model wth all three features s able to replcate the observed patterns of wealth and ncome nequalty n the Unted States. 6 6 We do not clam that other factors, such as lfe-cycle factors, ncome uncertanty, precautonary savngs, redstrbutve 4

6 The current study d ers from Krusell and Smth (998) n three mportant ways: Frst, the current study ams to explan both wealth and ncome nequalty usng only one source of consumer heterogenety, namely d erences n dscount rates. Second, the current study takes nto account the endogenous components of labor ncome, namely endogenous labor hours and human captal formaton. Thrd, nstead of assumng that ndvduals dscount rates are stochastc and dosyncratc n nature, the current study focuses on xed, predetermned d erences n dscount rates across ndvduals. 7 Ths study s also close n sprt to Luo and Young (2009) n the sense that both studes analyze wealth and ncome nequalty n the presence of wealth preference. There are two major d erences between the two studes. Frst, the source of consumer heterogenety s d erent n the two models. In Luo and Young (2009), consumers share the same dscount rate but face dosyncratc uncertanty n labor productvty as n the Ayagar (994) model. Thus, ths study does not consder the e ects of tme preference heterogenety on wealth and ncome nequalty. Second, the earnngs dstrbuton n the two models are determned by d erent factors. In Luo and Young (2009), earnngs are jontly determned by labor productvty shock and consumers labor-lesure choces. In partcular, human captal formaton s not consdered n ther model. Despte these d erences n model spec caton, both studes nd that wealth preference s a force that tends to reduce wealth nequalty. In our model, ths tendency s manfested n two ways. Frst, the equlbrum wealth dstrbuton s no longer extremely unequal once we ntroduce wealth preference nto Becker s model. Second, n the quanttatve analyss, we nd that the degree of wealth nequalty decreases as we ncrease the coe cent that controls the strength of wealth preference. Smlar results are also reported n Luo and Young (2009). The rest of ths paper s organzed as follows. Secton 2 descrbes the model envronment. Secton 3 descrbes the benchmark parameter values that we use n the quanttatve exercse. Secton 4 presents the benchmark results. Secton 5 dscusses the man determnants of wealth and ncome nequalty n the benchmark economy. Ths s followed by some concludng remarks n Secton 6. taxaton and transfer programs, are not mportant n understandng economc nequalty. The man purpose of the numercal exercse s to llustrate the quanttatve relevance of the mechansm captured by ths model n explanng economc nequalty. 7 Exstng studes show that predetermned factors (or ex ante heterogenety) are at least as mportant as dosyncratc shocks (or ex post heterogenety) n explanng cross-sectonal varaton n lfetme utlty. Keane and Wolpn (997) argue that as much as 90 percent of the dsperson n lfetme utlty can be attrbuted to predetermned, xed factors. The remanng ten percent s attrbuted to exogenous dosyncratc shocks. More recently, Huggett, Ventura and Yaron (20) nd that predetermned factors are more mportant n explanng the dsperson n lfetme earnngs and lfetme wealth than dosyncratc shocks. 5

7 2 The Model Consder an economy nhabted by N > groups of n ntely-lved consumers. Each group s ndexed by a subjectve dscount factor ; for 2 f; 2; :::; Ng : The dscount factors can be ranked accordng to 0 < 2 ::: N < : Consumers wthn the same group are dentcal n all aspects. The share of type- consumers n the populaton s gven by 2 (0; ) : The sze of total populaton s constant and s normalzed to one, hence P N = = : There s a sngle commodty n ths economy whch can be used for consumpton and nvestment. The consumers preferences are represented by X t u (c ;t ; k ;t ) ; t=0 where c ;t s the consumpton of a type- consumer at tme t; and k ;t s the stock of physcal captal owned by the consumer at the begnnng of tme t: The (perod) utlty functon u (c; k) s dentcal for all types of consumers and s gven by u (c; k) = c + k ; () wth > 0 and 0: 8 The parameter captures the mportance of wealth preference n the utlty functon. In partcular, a hgher value of means that the same ncrease n wealth would generate a larger gan n utlty. If = 0; then the consumers are sad to have no drect preference for wealth. In each perod, each consumer s endowed wth one unt of tme whch can be dvded between market work and on-the-job tranng. Consder a type- consumer wth human captal h ;t at the begnnng of tme t: If he spends a fracton l ;t 2 [0; ] of tme on market work durng the perod, then hs earnngs are gven by w t l ;t h ;t : We refer to l ;t h ;t as e ectve unt of labor hours. The varable w t s the market wage rate for an e ectve unt of labor. The consumer also receves ' ( l ;t ) h & ;t unts of newly produced human captal, where ' > 0; 2 (0; ) and & 2 (0; ) : Hs human captal at tme t + s then gven by h ;t+ = ' ( l ;t ) h & ;t + ( h) h ;t ; (2) where h 2 (0; ) s the deprecaton rate of human captal. 8 Ths type of utlty functon s commonly used n the lterature of wealth preference. See, for nstance, Zou (994), Gong and Zou (2002), and Luo and Young (2009) among others. 6

8 Besdes labor ncome, consumers also receve nterest ncome from ther prevous savngs. All savngs are held n the form of physcal captal, whch s the only asset n ths economy. The e ectve rate of return from savngs s (r t k ) ; where r t s the rental rate of physcal captal at tme t and k 2 (0; ) s the deprecaton rate of physcal captal. As n Becker (980), consumers are not allowed to borrow n every perod. Gven a sequence of wage rates and rental rates, the consumers problem s to choose a sequence of consumpton, labor hours, physcal captal and human captal so as to maxmze ther lfetme utlty, subject to the sequental budget constrants, borrowng constrants and the law of moton for human captal. For each type- consumer, ths problem can be expressed as max fc ;t ;l ;t ;k ;t+ ;h ;t+ g t=0 X t u (c ;t; k ;t ) t=0 (P) subject to c ;t + k ;t+ ( k ) k ;t = w t l ;t h ;t + r t k ;t ; (3) h ;t+ = ' ( l ;t ) h & ;t + ( h) h ;t ; k ;t+ 0; l ;t 2 [0; ] ; and the ntal condtons: k ;0 > 0 and h ;0 > 0: The optmal choces are determned by (2), (3) and the followng condtons, whch can be derved from the rst-order condtons, (c ;t ) (k;t+ ) + ( + r t+ k ) (c ;t+ ) ; (4) wth equalty holds f k ;t+ > 0; w t h ;t (c ;t ) = ;t ' ( l ;t ) h & ;t ; (5) n o ;t = t+ ' ( l ;t+ ) h & ;t+ [& ( l ;t+) + l ;t+ ] + ( h ) : (6) Equaton (4) s the Euler equaton for consumpton. Introducng drect preferences for wealth creates some addtonal bene ts for holdng assets. These addtonal bene ts are captured by the expresson (k ;t+ ) > 0 n (4). If consumers have no drect preference for wealth (.e., = 0), then the 7

9 Euler equaton s dentcal to the one n Becker (980). Equaton (5) s the optmalty condton for choosng labor hours. Ths equates the margnal bene ts of workng longer hours, w t h ;t (c ;t ) ; to the margnal costs, ;t ' ( l ;t ) h & ;t : The varable ;t s the Lagrangan multpler for (2), whch s often nterpreted as the shadow value of human captal. Equaton (6) determnes the dynamcs of ;t : 2. Producton Output s produced accordng to a Cobb-Douglas producton functon: Y t = K t (X t L t ) ; wth 2 (0; ) ; where Y t denote aggregate output at tme t, K t s aggregate captal, L t s aggregate labor hours, and X t s the level of labor-augmentng technology. We wll refer to b L t X t L t as e ectve unt of aggregate labor. The technologcal factor s assumed to grow at a constant exogenous rate so that X t t for all t; where s the exogenous growth factor and X 0 s normalzed to one. 9 Snce the producton functon exhbts constant returns to scale, we can focus on a representatve rm whose problem s gven by n o max Kt (X t L t ) w t L t r t K t ; K t;l t for any t 0: The soluton of ths problem s completely characterzed by the rst-order condtons: w t = ( ) X t K t (X t L t ) ; (7) r t = K t (X t L t ) : (8) 2.2 Compettve Equlbrum Let c t = (c ;t ; c 2;t ; :::; c N;t ) be a dstrbuton of consumpton across groups at tme t: Smlarly, de ne k t ; h t and l t as the dstrbutons of physcal captal, human captal and labor hours at tme t, respectvely. Gven the ntal dstrbutons k 0 and h 0 ; a compettve equlbrum for ths economy conssts of a 9 Unlke the endogenous growth model consdered n Lucas (988), human captal accumulaton does not serve as the engne of growth n ths model. Ths s due to the condton & 2 (0; ), whch means there are dmnshng margnal returns to h ;t n the producton of human captal. The man purpose of ntroducng human captal n ths model s to ncrease the cross-sectonal varaton n earnngs. 8

10 sequence of dstrbutons, fc t; k t ; l t ; h t g t=0 ; a sequence of aggregate nputs, fk t; L t g t=0 ; and a sequence of prces, fw t ; r t g t=0 ; so that () Gven the prces, the sequence fc ;t ; k ;t ; l ;t ; h ;t g t=0 solves each type- consumer s problem. () Gven the prces, the sequence fk t ; L t g t=0 solves the representatve rm s problem n each perod,.e., (7) and (8) are sats ed for all t 0: () All markets clear n every perod,.e., K t = NX k ;t and L t = = NX l ;t h ;t ; for all t 0: = In the followng analyss, we con ne our attenton to balanced-growth equlbra whch are ndependent of the ntal condtons. Thus, the ntal dstrbutons k 0 and h 0 are rrelevant to our analyses. A balanced-growth equlbrum s a sequence S = fc t; k t ; l t ; h t ; K t ; L t ; w t ; r t g t=0 such that () S s a compettve equlbrum as de ned above. () The rental rate of physcal captal s statonary over tme,.e., r t = r for all t: () The dstrbutons of labor hours and human captal are statonary over tme. (v) Indvdual consumpton and asset holdngs, aggregate output, aggregate captal and wage rate are all growng at the same constant rate. In partcular, the common growth factor s : De ne b k d t K t = (X t L t ) as aggregate captal per e ectve unt of aggregate labor at tme t: Usng (8) we can express ths as a functon of r; b k d (r) r : (9) Substtutng (9) nto (7) gves! t (r) ( ) t : (0) r Intutvely, f r s the equlbrum rental rate, then! t (r) s the correspondng equlbrum wage rate at tme t. De ne the transformed varables: bc c ;t = t ; b k k ;t = t and bw (r)! t (r) = t : Along any 9

11 n balanced growth path, the values of bc ; b o N k ; l ; h and the rental rate = r are determned by bc = bw (r ) l h + r b k bk ; () whch holds wth equalty f b k > 0; ( k ) r bc ; (2) b k l = ( h ) l h & ; (3) h = ' h ( l ) & ; (4) NX b k = =! NX l h b k d (r ) ; (5) = where b k + k : Equatons () and (2) can be obtaned from (3) and (4), respectvely, after mposng the balanced-growth condtons. Smlarly, (4) can be obtaned from (2) after mposng these condtons. Equaton (3) then follows from (6) and (4). The detals of ths can be found n the appendx. Fnally, (5) s the captal market clearng condton. Under some mld regularty condtons, there exsts a unque balanced-growth equlbrum for ths economy. Ths unque equlbrum has two mportant propertes. Frst, the borrowng constrant s not bndng for all types of consumers. Thus, the Euler equaton n (2) would hold wth equalty for all. Second, unlke the Becker (980) model, the equlbrum wealth dstrbuton n ths economy s non-degenerate. The formal proof of these results are shown n the appendx. Here we focus on the ntutons behnd the non-degenerate wealth dstrbuton. To explan the d erences between our result and Becker s result, t su ce to contrast the Euler equatons n the two models. Set = for the moment. In the orgnal Becker (980) model, where the consumers have no drect preference for wealth, the Euler equaton s gven by r k ; (6) wth equalty holds f b k > 0: The parameter s the dscount rate or rate of tme preference for a type- consumer. Ths equaton suggests that a consumer wth no drect preference for wealth wll 0

12 nvest accordng to the followng rules: () accumulate assets nde ntely f the e ectve rate of return (r k ) exceeds hs rate of tme preference, () deplete the stock of assets untl t reaches zero (the lower bound) f the e ectve rate of return s lower than hs rate of tme preference, and () mantan a constant postve amount of assets f the two are equal. Snce there s only one e ectve rate of return from savngs, t s not possble for d erent types of consumers to mantan a constant amount of assets smultaneously. In addton, no one can accumulate assets nde ntely n a statonary equlbrum. Thus, the e ectve rate of return must be equated to the lowest rate of tme preference n the populaton. In other words, only the most patent group of consumers wll have postve asset holdngs n any statonary equlbrum. All other groups of consumers wll deplete ther wealth untl t reaches zero. Introducng drect preferences for wealth breaks ths spell by creatng some addtonal bene ts of holdng wealth. These addtonal bene ts fundamentally change the consumers savng behavor. In partcular, a consumer s now wllng to mantan a constant postve level of assets even f the e ectve rate of return s lower than hs rate of tme preference. Ths s agan evdent from the Euler equaton for consumpton. Substtutng () nto (2) gves bw (r (r ) l h k ) = + r b k > 0: (7) b k It s now possble to obtan a non-degenerate wealth dstrbuton because d erent types of consumers wll choose d erent values of b k based on (7). By comparng (6) and (7), t s obvous that drect preferences for wealth (.e., > 0) s the man factor that prevents the wealth dstrbuton from collapsng nto a degenerate dstrbuton. The accumulaton of human captal s not essental for ths result. In partcular, a non-degenerate wealth dstrbuton can be obtaned n a model wth > 0 but wthout human captal formaton. 0 Before concludng ths secton, we want to hghlght several mportant features of the dstrbutons of labor hours and human captal. In the unque balanced-growth equlbrum, the values of fl ; h g N = can be obtaned by solvng (3) and (4). These equatons show that the dstrbutons of labor hours and human captal are non-degenerate, and are completely determned by two factors: () the dstrbuton of subjectve dscount factors and () the parameters n the human captal accumulaton process. Ths has two mportant mplcatons. Frst, the values of fl ; h g N = are ndependent of the preference parameters 0 In a companon paper, Suen (202), we begn by ntroducng wealth preference nto the orgnal Becker (980) model, where there s no human captal formaton and labor supply s exogenous. It s shown that a unque balanced-growth equlbrum exsts, and the equlbrum wealth dstrbuton s non-degenerate.

13 and : Thus, changng these parameter values would have no mpact on the dstrbutons of labor hours, human captal and earnngs. Second, the values of fl ; h g N = are ndependent of the equlbrum n o N rental rate r and the consumers asset holdngs bk : Thus, n the statonary equlbrum, the dstrbuton of earnngs s not a ected by the consumers savngs decsons. = 3 Parameter Values To examne the extent of ncome and wealth nequalty that can be generated by the model, we solve t numercally usng the parameter values ndcated n Table. Most of these values are chosen based on emprcal ndngs. Others are chosen to match certan real-world statstcs. The detals of ths procedure are explaned below. One perod n the model s a year. The share of captal ncome n total output () s The growth rate of per-capta varables ( ) s 2.2 percent, whch s the average annual growth rate of real per-capta GDP n the Unted States over the perod In the benchmark scenaro, the parameter n the utlty functon s set to one. Results obtaned under d erent values of are reported n Secton 5.5. The range of subjectve dscount factors s chosen based on the estmates n Lawrance (99). Usng data from the Panel Study of Income Dynamcs over the perod , Lawrance (99) estmates that the average rate of tme preference for households n the bottom ve percent of the ncome dstrbuton s 3.5 percent, after controllng for d erences n age, educatonal level and race. Ths mples an average dscount factor of /(+0.035)= The estmated rate of tme preference for the rchest ve percent s 0.8 percent, whch gves a dscount factor of In the benchmark scenaro, we consder a hypothetcal populaton of one thousand groups of consumers and assume that the subjectve dscount factors are unformly dstrbuted between mn = 0:966 and max = 0:992: In other words, we set N = ; 000 and = =N for all : 2 The mean dscount factor s The man deas of usng a unform dstrbuton are as follows. Take wealth nequalty To obtan these results, Lawrance (99) estmate the Euler equaton for a model wthout drect preference for wealth. Ths range of values, however, encompasses the values of dscount factors that are typcally used n quanttatve studes (wth or wthout wealth preference). In Secton 5.4, we wll examne the e ects of changng these endpont values on the benchmark results. 2 The choce of N s mmateral for our benchmark results. In partcular, changng the number of groups to ether 500 or 5,000 has vrtually no mpact on the benchmark results. 2

14 Table Benchmark Parameters Values Share of captal ncome n total output 0.33 Common growth factor.022 k Deprecaton rate of physcal captal Inverse of ntertemporal elastcty of substtuton.0 Strength of wealth preference mn Mnmum value of subjectve dscount factor max Maxmum value of subjectve dscount factor N Number of groups of consumers,000 ' Parameter n human captal producton.0 Parameter n human captal producton & Parameter n human captal producton 0.87 h Deprecaton rate of human captal * Ths gure has been rounded o to the fourth sgn cant gure. as an example. In the statonary equlbrum, wealth nequalty s drven by two types of varatons: () varatons n populaton share across groups, captured by f g N = ; and () varatons n the equlbrum n o N level of asset holdngs across groups, captured by bk : By adoptng a unform dstrbuton, we can rule out the rst type of varatons. Thus, wealth nequalty n the benchmark results s entrely drven by the cross-sectonal varatons n asset holdngs. The same argument apples to nequalty n earnngs and ncome. Our benchmark results then provde a clear llustraton of how much nequalty can be generated by the key features of the model, namely wealth preference, human captal accumulaton and heterogenety n dscount factors. In Secton 5.4, we wll examne the e ects of relaxng the unform dstrbuton assumpton and changng the values of mn and max : As for the parameter values n the human captal producton functon, we normalze ' to unty and set the values of and & accordng to the estmates reported n Heckman et al. (998). Usng data from the Natonal Longtudnal Survey of Youth for the perod , these authors nd that the values of and & for people who have completed at least one year of college educaton are and 0.87, respectvely. For those who do not have any college educaton, the correspondng values are and We use the rst set of parameter values n the numercal analyss because workers wth college = 3

15 educaton account for a larger share of U.S. labor force than those wthout college educaton. 3 As for the deprecaton rate of human captal, Heckman et al. (998) assume that t s zero. Other studes n the exstng lterature typcally nd that ths rate s greater than zero. 4 In the benchmark scenaro, we set h = 0:037 whch s consstent wth the estmate reported n Heckman (976). The two remanng parameters, and k ; are calbrated so that the model can match two realworld statstcs. The rst parameter plays a crucal role n explanng wealth nequalty. Ths pont s dscussed fully n Secton 5.. In the benchmark scenaro, we choose the value of so that the Gn coe cent of wealth predcted by the model s 0.86, whch matches the value reported n Díaz-Gménez et al. (20). 5 The requred value of s As explaned earler, the dstrbuton of earnngs s not a ected by the preference parameters and : Thus, the calbraton of has no mpact on the extent of earnngs nequalty generated by the model. The second parameter k s calbrated so that the captal-output rato generated by the model s Benchmark Results Table 2 summarzes the characterstcs of the earnngs, ncome and wealth dstrbutons obtaned under the benchmark parameter values. The rst three columns show the Gn coe cents, the coe cents of varaton and the mean-to-medan ratos for the three varables. The mean-to-medan rato s ntended to measure the degree of skewness n these dstrbutons. The rest of Table 2 shows the share of earnngs, ncome and wealth held by consumers n d erent percentles of the correspondng dstrbuton. Under the benchmark parameter values, the wealth dstrbuton n the model economy s hghly concentrated wth a large group of wealth-poor consumers and a small group of extremely wealthy ones. For nstance, the share of total wealth owned by consumers n the second quntle of the wealth dstrbuton s merely.3 percent, whereas the share owned by the wealthest ve percent s 58.5 percent. These gures are very close to the actual values observed n the Unted States. As for the ncome dstrbuton, the model s able to generate a Gn coe cent and a mean-to-medan rato that 3 Over the past twenty years, workers wth at least some college educaton have accounted for an ncreasngly larger share of U.S. labor force. In 992, ths type of worker represented 5.8 percent of cvlan labor force (over 25 years old). Ths ncreased to 62. percent by the year 200. These gures are based on the data reported n the U.S. Statstcal Abstract. 4 See Brownng et al. (999) Table 2.3 for a summary of these studes. 5 Smlar calbraton strategy s also used n Krusell and Smth (998), Erosa and Koreshkova (2007), and Hendrcks (2007) to determne the parameter values n the Markov process of the random dscount factor. In both cases, a set of unobserved, undetermned parameters s chosen so that the model can match certan key features of the U.S. wealth dstrbuton. 4

16 are smlar to the observed values. It s also able to replcate reasonably well the share of aggregate ncome owned by d erent quntles of the ncome dstrbuton. As for earnngs, the model predcts a more equal dstrbuton than that observed n the data. In the model economy, earnngs-poor consumers own a larger share of total earnngs than ther real-world counterparts. Consequently, the Gn coe cent predcted by the model s much lower than the actual value. 6 The bg d erence between the model s predcton and the actual value can be explaned by two factors. Frst, n the actual data, a large number of households have reported negatve earnngs. Accordng to Díaz-Gménez et al. (20), the average earnngs of households n the bottom quntle of the U.S. earnngs dstrbuton are negatve due to szable busness losses. In the model economy, earnngs must be strctly postve. Ths restrcton reduces the range and dsperson of the earnngs dstrbuton, whch n turn lowers earnngs nequalty n the model. Second, and more mportantly, almost all the households n the bottom quntle of the U.S. earnngs dstrbuton are not workers. As shown n Díaz-Gménez et al. (20) Table 4, retrees and nonworkers represent 96.9 percent of these households, and labor ncome only account for 0.2 percent of ther total ncome. If we consder only households headed by employed worker, then the Gn coe cent for earnngs n the Unted States s Ths value s much closer to the one predcted by the model whch assumes that all consumers are employed. 7 5 Dscusson The benchmark results n Table 2 show that our model s able to generate realstc patterns of wealth and ncome nequalty. To acheve ths, we have extended the standard neoclasscal growth model to allow for () drect preferences for wealth, () human captal formaton, and () heterogenety n subjectve dscount factors. In the above analyss, we assume that the utlty functon s logarthmc and addtvely separable, and the dstrbuton of dscount factors s unform. In ths secton, we examne the sgn cance of each of these features n explanng wealth and ncome nequalty. The man objectve of ths exercse s to better understand the determnants of wealth and ncome nequalty n our model. 6 Our results on earnngs nequalty, however, are comparable to those obtaned by Pjoan-Mas (2006) and Erosa and Koreshkova (2007). In the benchmark model of Pjoan-Mas (2006), the Gn coe cent and the coe cent of varaton for the earnngs dstrbuton are 0.33 and 0.65, respectvely. In the benchmark model of Erosa and Koreshkova (2007), the Gn coe cent of earnngs s As shown n Díaz-Gménez et al. (20), the Gn coe cents of ncome and wealth for employed workers n the Unted States are 0.48 and 0.78, respectvely. These values are stll qute close to the ones generated by the model. 5

17 5. Strength of Wealth Preference The purpose of ths subsecton s to llustrate the e ects of wealth preference on wealth and ncome nequalty. To acheve ths, we compute a seres of balanced-growth equlbra usng d erent values of rangng from to 0.5. For each value of ; the deprecaton rate k s recalbrated so that the captal-output rato s mantaned at 3.0. All other parameters are xed at ther benchmark values. The results of ths exercse are shown n Panel (A) of Table 3. 8 These results show a strong negatve relatonshp between wealth nequalty and the value of : As approaches zero, the Gn coe cent of wealth ncreases toward unty. Ths result s consstent wth theoretcal predcton: when = 0; the wealth dstrbuton s degenerate as n the Becker (980) model. As the value of ncreases, the wealth dstrbuton becomes more and more equal. The ntuton behnd ths result s as follows. An ncrease n means that the same ncrease n asset holdngs would now generate a larger gan n utlty. Ths has two opposng e ects on wealth nequalty. Frst, a stronger preference for wealth encourages all types of consumers to accumulate more assets. Ths e ect tends to be larger for the wealth-rch than for the wealth-poor. Thus, holdng other thngs constant, an ncrease n would make the wealth dstrbuton more unequal. Second, snce aggregate savngs ncrease as ncreases, the e ectve rate of return from savngs (r k ) needs to be adjusted downward n order to mantan the same captaloutput rato. Snce more patent consumers are more responsve to nterest rate changes than less patent ones, ths would lower the share of total wealth owned by the wealthest consumers and make the wealth dstrbuton more equal. The overall e ect of on wealth nequalty then depends on the relatve magntude between these two forces. Our results show that the second e ect domnates under the benchmark parameter values. An ncrease n also makes the ncome dstrbuton more equal. But the declne n ncome nequalty s much smaller than the declne n wealth nequalty. Ths happens because () consumers earnngs are not a ected by the parameter ; and () for most of the consumers n ths economy, earnngs account for a large fracton of ther ncome. 9 Thus, changng has only a mld mpact on the ncome dstrbuton. 8 As explaned earler, the earnngs dstrbuton s ndependent of : Thus, for all the cases consdered n ths panel, the earnngs dstrbuton s the same as n the benchmark scenaro. 9 When s 0.05 or less, earnngs represent more than 70 percent of ncome for those n the bottom four quntles (.e., the bottom 80 percent) of the wealth dstrbuton. 6

18 5.2 Importance of Human Captal Formaton In ths subsecton, we present a smpl ed verson of our model n whch there s no human captal formaton. The man objectve of ths exercse s to llustrate the mportance of ths feature n generatng wealth and ncome nequalty. Suppose now each consumer n the model economy s endowed wth one unt of tme whch s suppled nelastcally to the market. Labor ncome at tme t s dentcal for all types of consumers and s gven by w t : The consumer s problem s now gven by max fc ;t ;k ;t+ g t=0 X t u (c ;t; k ;t ) t=0 (P2) subject to the sequental budget constrant, c ;t + k ;t+ ( k ) k ;t = w t + r t k ;t ; the borrowng constrant, k ;t+ 0; and the ntal condton, k ;0 > 0: The utlty functon s addtvely separable as n (). The rest of the model s the same as n Secton 2. A balanced-growth equlbrum for ths economy can be de ned smlarly as n Secton 2.2. The detals of ths can be found n Suen (202). Here we focus on the quanttatve mplcatons of ths smpl ed model. We use the same values of f; ; ; N; mn ; max g as shown n Table. The dstrbuton of dscount factors s agan assumed to be unform. Smlar to Secton 5., we solve the model under d erent values of rangng from to 0.5. For each value of ; the deprecaton rate of physcal captal s recalbrated so that the captal-output rato s mantaned at 3.0. The man ndngs of ths exercse are summarzed n Panel (B) of Table 3. In the absence of human captal, our model can stll generate a hghly concentrated wealth dstrbuton wth a large group of wealth-poor consumers and a small group of extremely wealthy ones. When comparng the results n Panel (A) and Panel (B), we can see that removng human captal formaton from the model only lowers the Gn coe cent of wealth by.5 percent when = 0:0202: In other words, wealth nequalty n the benchmark results s manly drven by wealth preference and the heterogenety n dscount factors. However, these two features alone cannot generate a substantal degree of ncome nequalty. Ths happens because () earnngs are dentcal for all consumers n the smpl ed model, and () earnngs represent a szable porton of ncome for most of the consumers n ths economy. Introducng human captal formaton helps mprove ths result n two ways. Frst, the 7

19 dstrbuton of earnngs s now endogenously determned and s closely ted to the dstrbuton of dscount factors. Ths ntroduces sgn cant varatons n earnngs across consumers. Second, snce more patent consumers tend to nvest more n both physcal and human captal than less patent ones, there exsts a strong postve correlaton between earnngs and captal ncome n the benchmark economy. Ths n turn generates a substantal degree of ncome nequalty n our model Non-Separable Utlty Functon In the exstng lterature, t s also common to use the followng non-separable utlty functon, 2 8 >< u (c; k) = >: e c + ( ) k e for < and 6= 0; e c k e ; for = 0; wth e > 0 and 2 (0; ) : Under ths spec caton, the Euler equaton n the balanced-growth equlbrum s gven by e ( k ) r = bw (r) l h + r b k : (8) b k A drect comparson between (7) and (8) suggests that they can be made dentcal by a sutable choce of parameter values. When ths s mposed, the equlbrum wealth dstrbuton and the equlbrum rental rate wll be dentcal under these two spec catons of utlty functon. 22 Formally, let b k = bk ; :::; b k N be the dstrbuton of physcal captal obtaned under the addtvely separable spec caton and a common growth factor : Then the same dstrbuton can be obtaned under a non-separable utlty functon wth = ; = =(+); and a common growth factor e = e : 23 Ths observaton suggests that these two forms of utlty functon are lkely to yeld quanttatvely smlar results n the balanced-growth equlbrum. 24 The addtvely separable form s preferred because t nvolves fewer parameters. 20 Another way to endogenze the dstrbuton of earnngs s to allow for endogenous labor supply decsons n (P2). However, due to the wealth e ect on labor supply, wealth-rch consumers would tend to work less than wealth-poor ones. Consequently, earnngs and captal ncome are negatvely correlated n a model wth endogenous labor supply but wthout human captal. Because of ths negatve correlaton, ntroducng endogenous labor n (P2) would actually lower the Gn coe cents of ncome n Panel (B) of Table 3. The detals of ths can be found n the workng paper verson. 2 See, for nstance, Boleau and Braeu (2007) and Karnzova (200). The second study assumes that wealth e ect s derved from the stock of physcal captal owned by the consumer at the end of the current perod,.e., k ;t+: 22 Snce the values of fl ; h g N = are ndependent of the utlty functon, the dstrbutons of labor hours, human captal and earnngs are also dentcal under these two spec catons of utlty functon. 23 In partcular, our benchmark results can be obtaned from a non-separable utlty functon wth = 0; e = and = =( + 0:0202): 24 We stress that the above argument s vald only n the balanced-growth equlbrum. The two spec catons are lkely to yeld very d erent results along any transton path. 8

20 5.4 Changng the Dstrbuton of Dscount Factors We now examne the e ects of changng the dstrbuton of dscount factors. In partcular, we consder two types of changes: relaxng the unform dstrbuton assumpton and changng the values of mn and max : Changng the Shape of the Dstrbuton Suppose now the share of each group n the populaton s determned by = N N ; wth > 0; for 2 f; 2; :::; Ng : The endponts of the dstrbuton are xed at ther benchmark values,.e., mn = 0:966 and max = 0:992: The benchmark scenaro then corresponds to the case when = : When > ; the sze of the most patent group s less than =N and the dstrbuton of dscount factors s more concentrated on low values of : The opposte s true when 2 (0; ) : Intutvely, a hgh value of represents an economy n whch most of the consumers share smlar values of dscount factor (close to mn ), whle a small group of consumers are relatvely more patent. To better understand the e ects of changng ; we consder two calbraton exercses. In the rst exercse, we examne the extent of economc nequalty under d erent values of : The results are shown n Panel (A) of Table 4. For each value of ; the deprecaton rate of physcal captal s adjusted so as to mantan the captal-output rato at 3.0. All other parameters (ncludng ) are xed at ther benchmark values. In the second exercse, both and k are recalbrated n each case so that the two calbraton targets (Gn coe cent of wealth and the captal-output rato) are the same as n the benchmark scenaro. The results of the second experment are summarzed n Panel (B) of Table 4. We begn by summarzng the e ects of changng on the dstrbuton of dscount factors. Increasng from.0 to 2.0 rases the sze of the least patent group ( ) from to 0.036, and reduces the sze of the most patent group ( N ) by half. Because of the skewness of the dstrbuton, the mean value of s greater than the medan value when >. These results are dentcal for the two panels. The results n Panel (A) show that the Gn coe cents produced by the model are rather robust to changes n the sze of the most patent group ( N ) : More spec cally, reducng N by half rases the Gn coe cents of earnngs, ncome and wealth by 3.6 percent, 0.8 percent and 8.5 percent, respectvely. 9

21 The share of total wealth and total ncome owned by the rchest consumers are more senstve to ths change. The ntutons behnd these results are as follows. Frst, consder the ncrease n earnngs nequalty. In the statonary equlbrum, ths type of nequalty s drven by () cross-sectonal varatons n the populaton share, f g N = ; and () cross-sectonal varatons n human captal and labor hours, fh ; l g N = : As shown n (3) and (4), the values of fh ; l g N = are ndependent of the e ectve rate of return (r k ) and the populaton shares. Ths means changng has no mpact on the values of fh ; l g N = : Thus, the ncrease n earnngs nequalty that we observed n Panel (A) s completely drven by the changes n f g N = : In partcular, an ncrease n lowers the share of very patent consumers n the populaton. Snce these consumers tend to have more human captal and hgher earnngs than the less patent ones, a large porton of total earnngs s now concentrated n the hands of fewer consumers. Thus, the earnngs dstrbuton becomes more unequal as ncreases. An ncrease n has a smlar e ect on wealth nequalty. Spec cally, such an ncrease means that a large porton of total wealth s now concentrated n the hands of fewer consumers. Ths makes the wealth dstrbuton more unequal. However, an ncrease n would also nduce changes n the e ectve rate of return from savngs. Ths creates a second e ect on wealth nequalty. More spec cally, an ncrease n the share of less patent consumers leads to a declne n aggregate savngs. In order to mantan the same captal-output rato, we need to adjust the e ectve rate of return upward as ncreases. Snce more patent consumers are more responsve to nterest rate changes than less patent ones, ths wdens the d erences n asset holdngs across groups and further ncreases wealth nequalty. As for ncome, snce t s just the sum of earnngs and captal ncome, ncome nequalty ncreases as earnngs and wealth nequalty ncrease. Next, we turn to the results n Panel (B). Snce adjustng has no e ect on the earnngs dstrbuton, the Gn coe cents of earnngs are the same as n Panel (A). When the Gn coe cent of wealth s held constant, ncreasng from.0 to 2.0 rases the Gn coe cent of ncome by 6.5 percent, whch s smaller than the ncrease n Panel (A). The most sgn cant d erence between the two panels s that, when the Gn coe cent of wealth s held constant, an ncrease n would lower the share of total wealth and total ncome held by the rchest consumers. Ths happens because we need to adjust upward as ncreases so as to mantan the Gn coe cent of wealth at the same level. As shown n Table 3, ths tends to lower the share of total wealth and total ncome held by the rchest consumers. The results n Panel (B) thus show that the qualtatve e ects of on wealth and ncome nequalty 20

22 are robust to changes n : Changng the Range of Dscount Factors We now examne the e ects of changng the range of dscount factors. We mantan the unform dstrbuton assumpton as n the benchmark scenaro, but consder ve d erent combnatons of endpont values. In the rst varaton, the benchmark values are reduced by 0.0 so that mn = 0:956 and max = 0:982: In the second varaton, the benchmark values are reduced by In these two experments, the range 4 j max mn j s the same as n the benchmark scenaro. In the thrd and fourth experments, ths range s reduced by half. We consder the upper half n the thrd experment,.e., mn = 0:979 and max = 0:992; and the lower half n the fourth one. In the nal experment, we extend the benchmark nterval to the left by 50 percent, so that mn = 0:953 and max = 0:992: Smlar to the prevous subsecton, we report two sets of results for each experment. Panel (A) of Table 5 reports the results obtaned when the captal-output rato s kept at 3.0 and s xed at Panel (B) reports the results obtaned when the two calbraton targets are kept constant. Two man observatons can be made from the results n Panel (A). Frst, shftng the dstrbuton of dscount factors whle leavng the range 4 unchanged only has a mld mpact on the Gn coe cents. For nstance, reducng the benchmark endpont values by 0.0 would rase the Gn coe cents of earnngs and ncome by 4.5 percent and 2.2 percent, respectvely. The e ect of ths on the Gn coe cent of wealth s neglgble. The share of total wealth and total ncome owned by the rchest consumers s also qute robust to ths change. The second observaton s that nequalty n all three varables are postvely related to the sze of 4: Ths s evdent from the results of the last three experments. These results suggest that our model does not rely on large values of dscount factor (.e., very patent consumers) to generate a hgh concentraton of wealth and ncome. Instead, economc nequalty n our model s largely determned by the relatve magntude between mn and max : Next, consder the results n Panel (B). When the Gn coe cent of wealth s held constant, ncome nequalty s less senstve to changes n 4: The man d erences between the two panels are the e ects of changng 4 on the share of total wealth and total ncome held by the rchest consumers. When the parameter s kept constant, these shares are postvely related to the sze of 4: When the Gn coe cent of wealth s kept constant, these shares become negatvely related to the sze of 4: Ths happens because, n order to mantan the same Gn coe cent of wealth, we need to adjust upward 2