Discussion Paper No Heterogeneous Conformism and Wealth Distribution in a Neoclassical Growth Model. Kazuo Mino and Yasuhiro Nakamoto

Transcription

1 Dicuion Paper No Heterogeneou Conformim and Wealth Ditribution in a Neoclaical Growth Model Kazuo Mino and Yauhiro Nakamoto

2 Heterogeneou Conformim and Wealth Ditribution in a Neoclaical Growth Model Kazuo Mino y and Yauhiro Nakamoto z October, 25 Abtract Thi paper explore the role of conumption externalitie in a neoclaical growth model in which houehold have heterogeneou preference. We nd that a higher degree of average conformim accelerate the convergence peed of the economy toward the teady tate a in the cae of homogeneou conformim. Furthermore, we reveal that the wealth inequality expand or hrink in the cae of heterogeneou conformim, while it doe not expand but hrink in the cae of homogeneou conformim. Keyword: conumption externalitie, heterogeneou agent, wealth ditribution JEL Clai cation Code: D3, E3, E2, O4 We would like to thank two referee for their penetrating comment that are extremely helpful for improving our paper. We alo thank Shinuke Ikeda and Yohiyau Ono for their ueful comment on an earlier verion of thi paper. Our reearch ha been nancially upported by JSPS Kakenhi Project (No ). Kazuo Mino reearch ha been upported by JSPS Grand-in-Aid Specially Promoted Reearch Project (No. 23), and Yauhiro Nakamoto reearch ha been upported by JSPS Grand-in-Aid Specially Promoted Reearch Project (No.23249). y Intitute of Economic Reearch, Kyoto Univerity, Yohida Honmachi, Sakyo-ku, Kyoto, Japan, mino@kier.kyoto-u.ac.jp z Faculty of Informatic, Kanai Univerity, 2-- Reizanji-cho,Takatuki-hi, Oaka, , Japan, e- mail: nakamoto@kanai-u.ac.jp

3 Introduction It ha long been recognized that ocial comparion i one of the central feature of human behavior. In recent year, a number of experimental tudie challenge to invetigate whether ocial comparion a ect individual well-being. Along with the development in the neurocience and behavioral economic, there ha been a renewed interet in the role of conumption externalitie in macroeconomic dynamic. The baic aumption of thi literature i that conumer felicity depend not only on their private conumption but alo on the average conumption in the economy at large. The preence of uch a pychological external e ect may alter aving behavior of conumer and thu dynamic property of the entire economy. Baed on thi idea, the foregoing tudie have dicued variou iue uch a aet pricing (Abel 99 and Galí 994), income taxation (Ljungqvit and Uhlig 2 and Fiher and Hof 2), equilibrium e ciency (Liu and Turnovky 25, Nakamoto 29 and Arrow and Dagupta 29), belief-driven buine cycle (Alono-Carrera, et al. 28, Chen and Hu 27, Chen et al. 23 and 24, and Weder 2) and long-term economic growth (Carroll et al. 997 and 2, and Harbaugh 996). 2 While the exiting macroeconomic tudie on conumption externalitie dicu a variety of topic, they hare a common feature: all the tudie mentioned above employ repreentativeagent model. In the repreentative-agent economy, the ocial average conumption coincide with the level of private conumption and, hence, conumption behavior of all the agent are identical. A a reult, the exiting tudie employing the repreentative-agent model fail to capture the ocial comparion behavior of houehold in a atifactory manner. In thi repect, García-Peñaloa and Turnovky (28) tudy i a notable exception. Thee author contrat a neoclaical growth model with conumption externalitie in which aet holding of houehold are heterogeneou o that conumption behavior of each agent i divergent from For example, Fliebach et al. (27) examine the impact of ocial comparion on brain activity uing functional magnetic reonance imaging (fmri), howing that not only the abolute level of payment but alo relative level of payment imilarly a ect brain activity. In their paper, it i conidered that neurophyiological evidence upport the importance of ocial comparion in the human brain. Uing urvey-experimental method, Alpizar et al (25) how that, on average, both abolute and relative conumption matter for individual well-being, and conclude that mot individual are intereted in other conumption behavior. 2 Other in uential tudie on economic analye of conumption externalitie include Carlon et al. (27), Clark et al. (28), Dupor and Liu (23), Eaterlin (2), Frank (25) and Luttmer (25). 2

4 each other. 3 Their main concern i to explore how the preence of conumption externalitie a ect the pattern of wealth ditribution and the tranition dynamic of the aggregate economy. Although García-Peñaloa and Turnovky (28) make a ubtantial extenion, they till aume that houehold have identical, homothetic preference. Due to thi aumption, the aggregate behavior of the economy i independent of wealth ditribution. The purpoe of thi paper i to extend García-Peñaloa and Turnovky (28) by introducing heterogeneou preference into their etting. 4 We are particularly concerned with the ituation where houehold have di erent degree of conformim in the ene that every houehold like being imilar to other, but the degree of uch an enthuiam i agent peci c. A hown in next ection, the degree of conformim i determined by the trength of conumption external e ect a well a by the intertemporal elaticity of private conumption. It i to be pointed out that a number of behavioral economic tudie emphaize that behavior of ocial comparion i heterogeneou among conumer depending on the agent characteritic uch a income, age, race, gender, family tatu, education, occupation and urbanity: ee, for example, Burn (26), Hewtone et al. (22), Maurer and Meier (28), Mullen et al. (992), and Rubin and Willi (22). Our tudy follow uch a reearch agenda The key feature of our generalization i that we can ditinguih the e ect of average degree of conumption externalitie from thoe of individual level of conumption conformim. We how that the average degree of houehold conformim play a relevant role in determining the behavior of the economy at large, while individual degree of conformim yield a deciive impact on wealth ditribution. More peci cally, we preent three nding. Firt, an economy with a higher degree of conformim grow fater than an economy with a lower degree of conformim. Namely, a rie in the average level of conumer conformim increae the converging peed of the aggregate economy. Second, a houehold with a high degree of individual conformim accumulate her wealth fater than a houehold whoe conformim i weak. A a reult, an initially poor houehold may catch up with an initially rich houehold, if the poor ha a trong conformim in her conumption behavior. Third, the preence of 3 It i to be noted that Koyuncu and Turnovky (2) examine the e ect of income taxation in the model of García-Peñaloa and Turnovky (28). 4 While García-Peñaloa and Turnovky (28) aume that labor upply i variable, we treat a model with xed labor upply. In thi point, their model i more general than our formulation. 3

5 heterogeneou conformim may enhance wealth inequality in the long-run equilibrium. Thi reult i in contrat to the concluion of García-Peñaloa and Turnovky (28) who how that conumption externalitie tend to reduce the inequality of wealth ditribution if houehold preference are identical and homothetic. We preent the condition under which the preence of conumption externalitie enlarge wealth inequality in the teady tate. We dicu our nding baed on the general form of utility function a well a on an peci c form of utility function that i frequently employed in the macroeconomic literature on conumption externalitie. Finally, our numerical example how that thee nding can be een under the plauible parameter et. The reminder of thi paper i organized a follow. Section 2 et up the baeline framework. Section 3 characterize the teady-tate equilibrium and the equilibrium dynamic of the aggregate economy. Section 4 dicue the e ect of conumption externalitie on the dynamic behavior relative wealth and the wealth ditribution in the teady-tate equilibrium. Section 5 conclude. 2 Baeline Setting 2. Production and Conumption We conider a imple neoclaical growth model with identical rm. The aggregate production function i aumed to atify contant return to cale with repect to capital and labor; and it i expreed a Y = F ^K; L = Lf (K) where Y i output, ^K i capital, L i labor and K ^K=L denote capital intenity. The productivity function, f (K) ; i monotonically increaing, trictly concave in K and ati e the Inada condition. In competitive factor and nal good market, the real rent and real wage rate are repectively determined by r = f (K) = r (K) ; w = f (K) Kf (K) = w (K) : () There i a continuum of houehold with a unit meaure. Houehold are aumed to be heterogeneou in the ene that each houehold ha agent-peci c preference and di erent 4

6 tock of wealth. The intantaneou utility function of type i houehold i u i = u i (c i ; C) ; i 2 [; ] : Here, c i denote private conumption of type i houehold and C i the average conumption in the economy at large: C = c i di: (2) The above formulation mean that an individual houehold felicity i a ected by the preence of conumption externalitie repreented by the average conumption of an entire economy. 5 In what follow, we aume that u i (c i ; C) i monotonically increaing and trictly concave in c i : We alo aume that u i (c i ; C) i a monotonic function of C: Further retriction on the individual utility function are dicued in Section 2.3. The i-th agent maximize a dicounted um of utilitie U i = ubject to ow budget contraint e t u i (c i ; C) dt _a i = ra i + wl i c i ; (3) where a i and l i repectively denote wealth holding and labor upply. The initial holding of wealth a i () i given and each houehold i ubject to the non-ponzi game condition uch that lim exp t! Z t r d a i : (4) When olving thi problem, the houehold take the entire equence of the reference conumption, fc (t)g t= ; a given. Denoting the (private) utility price of capital by q i ; the optimization condition give the following: u i (c i ; C) = q i ; (5) _q i = q i ( r) ; (6) 5 In a general etting, the felicity function i given by u i = u i (c i; C i) : Here, C i denote the average conumption in a i-th group of agent, that i, C i = R i2n i c idi; where N i [; ] i a ubet of agent. In thi paper we focu on the impli ed cae where external e ect prevail the entire economy. 5

7 together with the tranverality condition: lim e t q i a i = : (7) t! We aume that each houehold upplie one unit of labor in each moment of time o that l i = : Since the ma of houehold i unity, the aggregate labor i alo L = : The net wealth of thi economy i the aggregate capital tock, and thu the equilibrium condition of the aet market i given by K = Since K i only real aet, we may aume that houehold directly own real capital, o that we et a i = k i in the ubequent dicuion. a i di: Finally, the equilibrium condition of the nal good market i Y = _ K + C: (8) For notational implicity, we ignore capital depreciation. 2.2 Characterizing Competitive Equilibrium We have aumed that the houehold have heterogeneou preference, o that the dynamic behavior of the aggregate conumption, C; and capital, K; are not independent of the behavior of individual variable, c i and k i : Moreover, the houehold contitute a continuum, meaning that we hould treat a dynamic ytem that involve an in nite number of endogenou variable. Hence, it i not trivial to con rm whether we can obtain a well-de ned, tractable dynamic ytem that characterize the perfect-foreight competitive equilibrium of our economy. In thi repect, it i ueful to analyze a peudo planning economy whoe behavior mimic the decentralized economy. In what follow, we et up a peudo planning problem whoe olution exactly correpond to the competitive equilibrium of our model economy. Then we how that the olution of the planning problem provide u with a well de ned dynamic ytem. Denoting by! i a weight of individual i utility, we aume that the planner olve the following problem: max fc i g t= e t! i u i (c i ; C) di dt;! i > ; i 2 [; ] 6

8 ubject to the reource contraint _K = f (K) and a given initial level of aggregate capital, K () : In thi problem the planner maximize a weighted um of individual welfare. Here, the key aumption i that in olving thi problem, the planner take the external e ect, i.e. the equence of aggregate conumption, fc t g t= ; involved in the individual utility function a given. To olve the problem, we et up the Hamiltonian function in uch a way that H =! i u i (c i ; C) di + f (K) c i di ; where denote the hadow value of aggregate capital evaluated by the ocial welfare. The optimization condition include c i di! i u i (c i ; C) = ; (9) _ = f (K) ; () lim e t (t) K (t) = : () t! Due to the aumption of trict concavity of u i (:) with repect to c i ; condition (9) how that the optimal level of conumption of agent i i uniquely written a c i = c i ; C ; (2)! i where the conumption demand function c i (:) monotonically decreae with =! i : The reduced form of a complete dynamic ytem for thi planning problem i thu given by _K = f (K) c i ; C di;! i _ = f (K) : In addition, aggregation of individual conumption demand yield: c i ; C di = C: (3)! i A hown in Section 2.3, when houehold are conformit, their conumption demand monotonically increae with the average conumption, C: We will focu on thi cae and aume i (c i ; C) di < 7

9 A dicued below, in thi paper we treat the cae where every houehold i conformit: the houehold increae it private conumption, c i ; a the reference level of ocial conumption, C; rie. Thu the above retriction mean that houehold average conformim i not trong enough to make the aggregate conumption reponded more than the rie in the reference level of conumption. Given thi aumption, (3) yield a monotonic, negative relation between C and under a given welfare weight pro le, f! i g i= : We expre uch a relation a C = C () ; C () < : Conequently, the aggregate behavior of the planning economy i decribed by _ = f (K) ; (4) _K = f (K) C () ; (5) together with a given K () and the tranverality condition: lim t! e t (t) K (t) = :Since the aggregate dynamic ytem derived above i eentially the ame a that of the tandard one-ector optimal growth model, the planning problem ha a unique optimal path that converge to the teady tate equilibrium. Now de ne! i q i ; i 2 [; ] ; (6) which evaluate the marginal value of capital from agent i private perpective, that i q i =q j =! j =! i o that q i =q j tay contant over time. Furthermore, ince _q i =q i = _ = = f (K) for all i 2 [; ] ; it hold that from (6)! i i a contant weight. A to the tranverality condition, (7) and q i! i = yield Aggregating both ide of the above give! i lim t! e t (t) k i (t) = : lim e t (t) K (t) = : Therefore, the tranverality condition correponding planning problem i ati ed a well. In view of the equilibrium condition of the nal good market and the determination of factor price, we ee that the aggregate behavior of the competitive economy mimic the optimal trajectory of the peudo planning economy de ned above. Therefore, the aggregate behavior 8

10 of our economy i completely characterized by a pair of di erential equation of K and given by (4) and (5) : Finally, to complete our analyi we hould determine the welfare weight,! i : Notice that the q j =q i tay contant over time for all i; j 2 [; ] : A a reult, at the outet of planning it hold that u j (c j () ; C ()) u i (c i () ; C ()) = q j () q i () =! i for all i; j 2 [; ] : (7)! j The optimal choice of the initial conumption level, c i () i determined to make the optimal trajectory tarting from the initial conumption, and ati e the intertemporal budget contraint uch that exp Z t r () d c i (t) dt = k i () + exp Z t r()d w (t) dt; i 2 [; ] ; where r (t) = f (K (t)) and w (t) = f (K t ) f (K (t)) K (t) : 6 Given the initial holding of capital, k i () ; the level of c i () i uniquely i determined, o that C () = R c i () di alo take a unique value. Hence, if! i (i 2 [; ]) i elected to atify (7) ; then the olution of the peudo-planning problem coincide with the competitive equilibrium. Note that q i i proportional to and from (2), c i depend on C and q i : In addition, the dynamic behavior of k i depend on K and c i : Therefore, once the optimal path of (K; C; ) in the planning problem are etablihed, behavior of q i ; c i and k i are determined a well. 2.3 Conformim and Conumption Behavior The condition (5) and (6) yield We expre thi equation a _c i = ui (c i; C) u i (c i; C) (r ) u i 2 (c i; C) u i (c i; C) _ C: _c i = i (c i ; C) (r ) + i (c i ; C) _ C; (8) where i (c i ; C) = ui (c i; C) u i (c i; C) > ; (9a) 6 The intertemporal budget contraint hold a an equality due to the non-ponzi-game contraint and the tranverality condition. 9

11 i (c i ; C) = ui 2 (c i; C) u i (c i; C) : (9b) In the above, = i (c i ; C) repreent the degree of abolute rik averion of type i houehold. Following Gollier (24), we call i (c i ; C) the degree of conformim of type i houehold. Thi function how how the private conumption repond to a change in the average conumption to keep the marginal utility of private conumption contant. If i > ; the houehold i i a conformit in the ene that he change her own conumption in the ame direction of the change in the average conumption. In contrat, if i (c i ; C) < ; the houehold change her conumption in the oppoite direction: the houehold i an anti-conformit. Moreover, when i (c i ; C) > ; the houehold i an over-conformit, becaue the houehold change her conumption more than a change in the average conumption to keep her marginal utility of private conumption contant. In thi paper we focu on the cae each houehold i a conformit but i not over-conformit, o that we aume < i (c i ; C) < for all i 2 [; ] : The peci cation of the utility function that ha been frequently ued in the literature i the following multiplicative form of externalitie: 7 u i (c i ; C) = i (c i ) i (C) ; i 2 [; ] ; where i (c i) > ; i (c i) < and i (C) > : Given thi functional form, we obtain i (c i ) = i (c i) i (c i) > ; i (c i ; C) = i (c i ) i (C) i (C) > : Note that if the external e ect are introduced in the multiplicative form, the abolute rik averion depend only on the private conumption, while the degree of conformim i a ected by private a well a ocial level of conumption (the cae of non-eparable confrmim). 8 A imple example i to et i (c i ) = c i = ( ) and i (C) = C i( ):, o that the intantaneou utility function i: u i (c i ; C) = c ic i ; > ; 6= ; < i < : (2) 7 For intance, thi type of utility function i given in Gali (994) and Carroll et al (997). 8 In contrat to the non-eparable conformim, we can ee the eparable conformim in the ene that the degree of conformim depend on the average conumption alone by uing the ubtractive form of conumption externalitie a follow: where i the degree of rik averion. u i (c i; C) = (ci i (C)) ; 6= ; > :

12 where how the common degree of abolute rik averion among houehold and i i the degree of external e ect of type i houehold. 9 preference only tem from the di erence in i : Here, we obtain i (c i ; C) = c i ; i (c i ; C) = In thi peci cation, the heterogeneity of c i i C ; implying that if the houehold i a conformit, we hould aume > becaue < i <. Hence, in thi well employed functional form, the degree of individual conformim depend on the intertemporal elaticity of private conumption, =, the individual degree of external e ect, i ; a well a on the private conumption relative to the ocial average, c i =C: Notice that even if the preference parameter are identical ( i = for all i) ; the degree of individual conformim may di er each other unle c i = C for all i: It i alo to be noted that in thi example the Euler equation (8) i rewritten a which lead to _c i = (r ) + c i _c i c i _c j = c j _C i C ; C ( i j ) _ C : A a reult, under the condition of > and i > ; while each houehold change her conumption in the ame direction of the change in the average conumption, the relative peed of conumption change between two individual depend on the ranking of the degree of external e ect, that i, the ign of i j : Since the peed of conumption adjutment a ect the peed of wealth accumulation, the example clearly demontrate that the heterogeneity of conumption conformim may yield a deciive e ect on the long-run wealth ditribution among houehold. 9 The introduction of heterogeneou rik averion mean that the economy ha two type of conformim: i > and ( = i ) > o that ( = i ) i > ; i < and ( = i ) < o that ( = i ) i >. A hown later, ince the key element of eeing our nding i the degree of conformim, the introduction of heterogeneou rik averion may make our paper verboe in the ene that the above two type of conformim lead to the ame nding. Therefore, we omit the heterogeneity of rik averion. If the heterogeneity of rik averion i alo preent, then the conumption growth path i more complicated in the ene that the growth rate of private conumption without the external e ect i di erent among houehold.

13 3 The Aggregate Economy 3. Aggregate Dynamic and the Steady-State Equilibrium A hown in the previou ection, the aggregate dynamic of our economy are decribed by the total capital, K; and it utility price, : Since the aggregate conumption, C; i a function of ; the aggregate dynamic can be conidered in term of K and C a well. To ee thi, it i to be noted that from (8) ; the average (aggregate) conumption follow: which lead to _C = (r ) i (c i ; C) di + i (c i ; C) di _C = (r ) ; R _C; i (c i ; C) di R i (c i ; C) di ; (2) where =C repreent the elaticity of intertemporal ubtitution in ocial conumption. In the following, we aume that the average degree of conformim of the ociety doe not exhibit over-conformim o that i (c i ; C) di < ; (22) implying that ha a poitive value. Equation (2) how that, other thing being equal, a higher degree of average level of conformim make the average conumption more enitive to a change in the real interet rate, r: Subtituting (2) into (8) ; we nd that the conumption of individual houehold follow _c i = i (r ); i 2 [; ]; (23) where i = i (c i ; C) + i (c i ; C). Thi expreion mean that when the houehold of type i ha a higher degree of conformim, i (c i ; C i ) ; her private conumption i more enitive to a change in the real interet rate. The di erence of i among houehold are derived by the heterogeneou conformim and In the cae of (2), (22) i given by R icidi=c < =( ): 2

14 the di erence of initial level of capital holding. Uing (2), the i can be given by: 2 where = R ic i di C i = the average degree of conformim. c i 8 >< >: + 9 >= ( i ) {z } >; ; (24) (#) repreent the average degree of external e ect, while repreent Even if the preference are homogeneou, (24) can be reduced to the elaticity of intertemporal ubtitution in the repreentative-agent model; alternatively, the heterogeneity of conumption externalitie yield the di erence of ( i ) in the term (#). Noting that >, other thing being equal, an increae in the degree of conformim of type i houehold ( =) i and the average degree of conformim ( =) lead to the increae in the value of i. From (8) the dynamic behavior of the aggregate (average) capital follow _K = f (K) C: (25a) A a conequence, a complete dynamic ytem of the entire economy conit of (23) ; (25a) ; _k i = rk i + w c i ; i 2 [; ]; (25b) together with (), (2) and a given initial level of capital ditribution among the houehold. The teady-tate level of average conumption and capital tock, C and K ; are uniquely determined by f (K ) = C ; f (K ) = : (26a) (26b) The above teady-tate condition demontrate that ditribution of wealth and the preence of conumption externalitie fail to a ect the teady-tate level of average variable a in the repreentative-agent model with the xed labor upply. In addition, when all the houehold are conformit o that i (c i ; C) > for all i; and if the the average degree of conformim 2 The long-run level of private conumption i given by: Z t c i(t) = c i() exp i()(r() )d ; and furthermore, the initial jump of private conumption i characterized by the degree of conformim and the initial holding of capital of type i houehold. 3

15 ati e (22) ; then the average Euler equation exhibit the familiar pattern of dynamic: the average conumption increae (decreae) when the average capital i lower (higher) than it teady tate level, K : Therefore, K and C follow a table addle path converging to the teady tate given above. 3.2 Convergence Speed We rt examine local dynamic of the aggregate ytem around the teady tate equilibrium. The linearly approximated ytem of (2) and (25a) at the teady tate conit of the following dynamic equation _K = (K K ) (C C ) ; _C = f (K ) (K K ) ; where denote the teady tate level of given by the following: and c i = R i (c i ; C ) di R i (c i ; C ) di denote the teady-tate level of the individual conumption. Noting that K = f () and C = f(f ()), we can rewrite K = K (), C = C () and w = w (). Therefore, the teady-tate value of c i ati e (27) c i = k i + w () : (28) Furthermore, we nd that the table root of the above ytem i = 2 h 2 4 f (K ()) =2 i < ; (29) where i given by (27) : Since the abolute value of the table root repreent the peed of convergence on the aggregate economy on the table addle path, we immediately ee that the economy with a higher degree of average conformim, and hence a higher value of exhibit a higher peed of convergence toward the teady tate. More peci cally, the peed of convergence i fater a the poitive value of R i di approache the unity and the poitive value of R i di i larger. Furthermore, and more importantly, becaue of the heterogeneity of conformim, the 4

16 teady-tate ditribution of capital itelf a ect the value of. Uing (2), the following hold. 3 i (c i ; C )di = C ; i (c i ; C )di = R ic i di C : (3) A een in (3), the value of R i(c i ; C )di i uniquely given, but the value of R i(c i ; C )di i a ected by the wealth ditribution. In detail, looking at R ic i di, we can argue that when the relatively wealthier houehold hold tronger degree of conformim, the value of R ic i di and hence R i(c i ; C )di are greater o that the peed of convergence in the unequal economy become fater. Dynamic behavior of individual conumption and wealth are decribed by _k i = rk i + w c i = f (K) (k i K) + f (K) c i ; _c i = [ i (c i ; C) + i (c i ; C) ] f (K) : On the table addle path of the aggregate ytem, it hold that C C = ( ) (K K ). Hence, the approximated behavior of individual capital, individual conumption and the aggregate capital repectively follow _k i = (k i k i ) (c i c i ) + f (K ) (k i K ) (K K ) ; _c i = i f (K ) (K K ) ; _K = (K K ) ; where i = i + i > : (3) Note that the table root of thi ytem i till ; which mean that on the table addle path of the entire economy each relation between individual capital (or individual conumption) and the aggregate capital ati e k i k i = i f (K ) f (K )(k i K ) (K K ); (32) c i c i = i f (K ) (K K ) : (33) 3 Alternatively, uing the ubtract form of conformim, ince the conformim i eparable from the individual capital, the wealth ditribution doe not have any impact on the peed of convergence if the degree of conformim are heterogeneou alone. 5

17 Therefore, on the approximated addle path both individual conumption and capital move into the ame direction a the aggregate capital change. In addition, it i een that, other thing being equal, a higher level of individual conformim (a higher value of i (c i ; C)) raie the repone of c i and k i to a change in the aggregate capital. To um up, we have een the following reult a to the local dynamic of the economy: Propoition (i) The peed of convergence of the aggregate economy increae with the degree of average conformim in the economy at large; (ii) the peed of convergence of capital and conumption of each conumer increae with her own degree of conformim; and (iii) uing (2) with the heterogeneou conformim, the wealth ditribution a ect the peed of convergence a een in (3). Reult (i) in the above propoition mean that even though the degree of conformim di er each other, we till have the ame outcome etablihed in the model with homogeneou preference: a higher degree of conumption conformim raie the convergence peed of an entire economy. Reult (ii) tate that the degree of conformim of each houehold i one of the relevant determinant of long-run wealth ditribution among houehold. Reult (iii) i a natural conequence of our etting in which the behavior of aggregate variable are not independent of wealth ditribution. In um, we have con rmed that the individual degree of conformim a ect wealth ditribution, which in turn yield impact on the aggregate behavior of the aggregate economy. In the next ection, we examine the relation between individual conformim and long-run wealth ditribution in detail. 4 Wealth Ditribution 4. Behavior of Relative wealth We now examine the role of heterogeneou conformim in determining wealth ditribution at the teady tate. Let u denote the relative capital holding of agent i by k ~ i = k i =K: Uing the capital accumulation equation (25a) and (25b), we derive the dynamic of relative wealth a follow: _~k t = K f (K)K f(k) ( ~ k i ) + C K ~ki c i : (34) C 6

18 García-Peñeloa and Turnovky (28) aume that the utility function of each agent i not only identical but alo homothetic both from private and ocial perpective. Given thoe aumption, conumption of each houehold change at the ame rate o that the relative conumption of each agent, c i =C; tay contant over time and the level of c i =C i determined by the initial ditribution of wealth among the houehold. In contrat, the relative conumption in our model change during the tranition proce, which may yield ubtantial e ect on wealth ditribution. For the purpoe of comparion, let u rt conider the cae of identical and homothetic preference where c i =C doe not change over time. Oberve that the relative wealth along the table addle path ati e the following: 4 where ~k i (t) = ~ k i + ( ~ k i )Z K K() e t ; (35) A = + f (K )K f (K ) Z = (K K())A ( )K ; K f (K ) f(k ) K f (K ) f(k : ) Notice that Z and A depend only on the teady-tate level of aggregate variable except for the table root. In thi etting, García-Peñeloa and Turnovky (28) conclude that the elaticity of ubtitution between labor and capital in the production function, which a ect the ign of A ; i a key element when determining the wealth ditribution in the teady tate. To implify our dicuion, in what follow we aume that (36) f (K )K f(k ) f (K )K f (K ; (37) ) and, hence, A in (36) ha a poitive value. For example, if f (K) i a Cobb-Dougla production function f (K) = K ( < ) ; then condition (37) i ati ed. Uing (35) ; we ee that the di erence of capital tock between the houehold i and j i: ~k i (t) ~ kj (t) = ( ~ k i ~k j ) + Z K K() e t : (38) The above expreion demontrate that a long a K () < K ; if ~ k i > (<)~ k j, then ~ k i (t) > (<) ~ k j (t) for all t : That i, the catching-up doe not arie. The intuitive explanation 4 See Appendix A with repect to the derivation. 7

19 i a follow. From (5) and (6) u (c i ; C)=u (c j ; C) i contant over time. Since u (c i ; C) i monotonically decreaing in c i for all C (> ) ; if c i () > c j () ; then c i (t) > c j (t) for all t > : In view of the intertemporal budget contraint for individual houehold, the identical preference mean that if k ~ i () > k ~ j () ; then c i () > c j () : A a conequence, if the initial capital ditribution ati e that k ~ i () > k ~ j () ; then it hold that k ~ i > k ~ j and c i > c j : Namely, regardle of the preence of conumption externalitie, the initial pattern of wealth ditribution i kept in the long run equilibrium. Now conider the cae of heterogeneou preference. In our general etting, while the relative marginal utility of private conumption, u i (c i; C)=u j (c j; C); tay contant over time, c i =c j generally change during the tranition. The relative wealth in our etting i given by ~k i (t) = ~ k i + Z i K K() e t ; (39) where Zi = B ( k ~ i ) K + K B = f (K )K + (> ): i ; (4) In the above, i i given by (3) : It i to be noted that the ign of B i poitive under (37). In view of (39), we nd that the di erence in capital tock between the houehold i and j under the heterogeneou preference i: ~k i (t) ~ kj (t) = ~ k i () ~ kj () + (Z i Z j )(e t )(K K()) : (4) Here, the term (Zi Zj ) tem from the preence of heterogeneou conformim. If thi expreion how that Z i < Z j ; then ~ k i () > ~ k j () doe not necearily etablih ~ k i > ~ k j : If the initially le wealthy houehold j catche up with the initially richer houehold i at time ^t, then we can how that ^t = k ~ j () ki ~ () + (K K())(Z i Zj ) (K K())(Z i Z j ) A : (42) The catching-up arie if and only if ^t ha a poitive value. More peci cally, we conclude: Propoition 2 Suppoe that K > K() and k i () > k j (): Then, (i) the initially poorer houehold j will never catch up with the other under the identical and homothetic preference; 8

20 and (ii) the initially poorer houehold j will (will not) catch up in wealth if the following inequality i ati ed: j i > (<) k i() k j () K()=K + A ( k ~ j ~k i ) : (43) ( ) Proof. Since the catching-up arie if and only if ^t >, from (42) we can derive the following: < ~ k j () ~ ki () (K K())(Z i Z j ) + < ; which lead to the condition (43) with repect to the catching-up. The condition (43) how that the catching-up would occur if and only if an initially poorer agent j ha a u ciently large value of j. Thi i plauible becaue the greater elaticity of intertemporal ubtitution mean that the houehold plan to increae own aving, which lead to a higher level of wealth in the future. To combine thi nding with the heterogeneou conformim, let u conider (2). For expoitional implicity, we aume that the initial holding of capital atify k i () > k j () and it hold that ki = k j o that c i = c j : Given thee condition, we nd: j (c j ; C ) i (c j ; C ) = C ( j i ) ; Thu if houehold i and j have the ame magnitude of conformim uch that i = j, it hold that j(c j ;C ) i (c j ;C ) =, thereby being unable to ee the catching-up (i.e., Propoition 2(i)). Next, if the degree of conformim between houehold i and j di er each other, then j i ha a poitive value if and only if j > i ; which mean that the catching-up may arie. The above condition mean that if houehold j ha a tronger conformim toward the ocial average conumption than houehold i, the initial dicrepancy of conumption and wealth may be eliminated in the long run. In word, thee reult ugget that if an initially poor houehold who ha maller wealth than the ocial average level ha a trong apiration of catching up the ocial average, he may overtake the wealth held by an initially rich houehold with a weak level of conumption conformim. 9

21 4.2 Pattern of Wealth Ditribution In thi ubection we conider the dynamic of wealth ditribution. De ning the di erence between the aggregate and the individual capital tock a i (t) ~ k i, we rewrite (39) a i (t) = i + Z i K K() e t : (44a) Again, we aume that the initial level of aggregate capital ati e K () < K : We rt examine the cae of identical and homothetic preference. In thi cae, from (35) ; equation (44a) i rewritten a i (t) = i + Z (K K()) e t : (44b) Equation (44b) how the characteritic of dynamic of relative wealth under the identical wealth. Di erentiating (44b) with repect to time yield: Then, we can ee that _ i (t) > (<) if i _ i (t) = i Z (K K()) e t : < (> ) for all houehold, o that the diperion in wealth holding hrink over time under our aumption (37). For example, there i the relative-wealth rich uch a i > in the long run. Since _ i(t) <, the divergence between the level of individual wealth and the average wealth decreae over time, and i (t) converge i (> ), implying that i() > i >. Similarly, if i < in the long run, the revere can be applied o that the relative wealth become mall along time, > i > i(): Thee reult mean that if we de ne the index of wealth inequality in time t by S = 2 i di; then in the cae of identical and homothetic preference, the teady tate level of S = R ( i )2 di i le than it initial level, S () = R i () 2 di a long a (37) hold. 5 A hown in Propoition, the preence of conumption conformim raie the peed of convergence of the aggregate economy. Hence, when the aggregate capital increae toward 5 A can be eaily predicted, it hold that S < S() in the identical preference: A S (K K()) (2( S() = S() ) + A (K K())) (< ); ( + A (K K())) 2 where we raie both ide of (44b) to the double power and take account of t =. 2

22 it teady tate level, the rate of return to capital decline fater in the economy with conumption externalitie than in the economy without them. Note that in our model with xed labor upply, the income di erence among houehold only come from their capital income, r (K) k i : Thi mean that the negative impact generated by the decreae in the rate of return to capital i higher for the houehold whoe capital holding are larger. A a conequence, in the preence of conumption conformim, the dicrepancy in capital holding hrink fater, o that the teady-tate ditribution of capital among houehold i more equal than in the economy without conformim. Thi reult i a key nding of García-Peñeloa and Turnovky (28). In the cae of heterogeneou conformim, i (t) change according to _ i (t) = (K K())Zi e t : (45) In thi cae, the ign of i fail to pecify the ign of i (t) during the tranition. Equation (45) tate that if we focu on the cae K () < K ; then ign _ i (t) = ign Z i : Turning back to the de nition of Z i Z i = B i K given in (4) ; we can ee that {z } (#2) + K i : (46) Then, irrepective of the long-run poition of individual capital, the e ect in (#2) i negative (poitive) if i < (>), which implie that thi e ect make the diperion of relative wealth decreaed. Baed on the ign of (#2) in (46), the following reult immediately follow: Propoition 3 Under the aumption of (37) and K () < K, it hold that (i) if i < and i = >, then _ i (t) > for all t ; and (ii) if i _ i (t) < for all t : > and i = <, then Proof. From (46)if i < and i = > ; it hold that Zi >, implying that _ i(t) < in (39). If i > and i = <, the oppoite outcome hold. Thi propoition preent a et of condition under which the divergence between individual capital holding and the average tock of capital monotonically decreae during the 2

23 tranition a in the cae of homogeneou conformim. The key element i the ign of i =. In particular, we notice that the degree of i i larger a the degree of own conformim i increae. Furthermore, noting that R i di =, condition i = > (<) mean that the elaticity of intertemporal ubtitution in conumption from the private perpective i higher (lower) than that from ocial perpective. Auming that a houehold hold relatively le wealth in the long run, reult (i) how that the houehold whoe private intertemporal elaticity of ubtitution i relatively high accumulate her capital fater than the ocial average during the tranition toward the teady tate. A a reult, the di erence between her capital and the average one hrink during the tranition. Turning to reult (ii), uppoe that a houehold hold relatively more wealth in the long run, and that the houehold whoe private intertemporal elaticity of ubtitution i relatively low accumulate her capital lower than the average. Then, we are able to ee that the relative wealth hrink. Therefore, a rough implication of thi propoition i that if the long-run rich houehold have relatively low degree of private elaticity of intertemporal ubtitution and if the long-run poor houehold have relatively high value of intertemporal ubtitution, then the wealth ditribution in the teady tate i more equal than the initial ditribution. that To obtain a more precie implication, it i ueful to examine the de nition of i = uch i = i di i R i di + R i i di Thi expreion reveal that the relative elaticity of intertemporal ubtitution in conumption between the private and the ocial perpective i high, if at leat one of the following condition hold: (i) the degree of average abolute rik averion R i di and average conformim R i di are mall; and (ii) the degree of private abolute rik averion i and conformim i, are large. When i = take a relatively high value, the houehold i attain fater accumulation of her capital than the ocial average. Notice that the condition given in Propoition 3 are not neceary but u cient. Therefore, we may conider a more complex ituation. In particular, conidering that the e ect (#2) i negative for expanding the wealth inequality, if the e ect of key element i = on the wealth ditribution i the oppoite with the e ect (#2), and furthermore, thi e ect dominate the e ect (#2), then the diperion of relative wealth increae during the tran-! : 22

24 ition. For example, uppoe that when a houehold i relative-wealth rich in the long run (i.e., i > ) where we aume i() >. If < i = o that Z i <, then we can ee the enlargement of relative wealth _ i (t) > and < i () < i. Importantly, uch an expanion of relative wealth cannot be een in the cae of homogeneou conformim. Conidering thoe alternative poibilitie, we may preent a general condition a to the wealth ditribution in the teady tate: Propoition 4 The long-run wealth inequality i larger (lower) than the initial level of inequality if the following condition i ati ed: K() 2 R! ( i )2 di K ( R +2 i {z di)2 } (#3) where where M = K() K Proof. Following Appendix B, we derive the following: X B R ( i )2 di S S() = R i di 2 C A R i() i R di i {z di > (<)M (2 + M )S(): } (#4) (47) K() B K : (48) M(2 + M)S() + X ( + M) 2 ; (49) K() 2 K + 2 R K() i() i di K R i di : (5) If X > (<)M(2 + M)S(), which correpond to (47), then it hold that S > (<)S(). The condition in Propoition 4 are rather complex, but we may obtain an intuitive implication. Firt, (#3) how the diperion of degree of conformim in the entire economy relative to the average degree of conformim. We mut notice that (#3) alway ha a non-negative ign by the Cauchy-Schwarz inequality, which mean that the heterogeneity of conformim directly expand the wealth inequality regardle of any patial arrangement of dipered heterogeneou conformim. Moreover, when the degree of conformim are largely dipered, which correpond to a larger value of R ( i )2 di, the wealth inequality further expand. Alternatively, taking account of the homogeneity of conformim a an extreme cae, we can eaily ee that R ( i )2 di = ( R i di)2 o that (#3) =. 23

25 Next, conider the term (#4), which indicate to a correlation between the initial holding of capital tock and the heterogeneity of conformim. Aume a growing economy in the ene that K > K(). The e ect (#4) how that when the initial riche have the greater degree of conformim, it hold that R i() i di ha a poitive ign, which mean that the wealth inequality tend to expand. Intuitively, when the initial riche have the greater degree of conformim, they like to ave but the initial wealth-poor people dilike the aving, which implie that the initial riche hold more wealth over time; and hence, the wealth inequality expand. On the other ide, when the initially-wealth poor people have the greater degree of conformim, they want to ave over time, thereby eeing that the wealth inequality doe not expand but hrink. To examine R i() i di further, we make ue of the peci ed utility function (2):6 i i ()di = i () i di + ( )= C R ic i di i () i i di + i () i di! : In that cae, it generally hold that R i() di >, becaue the initially relative-wealth riche eem to keep the relative-wealth one in the long run. (5) Furthermore, if the rich people at the initial period have the greater degree of conformim, we can con rm that ( ) R i() i idi + R i() i di > : Hence, ince R i i()di > ; then wealth ditribution become more unequal in the long run. Although our main propoition ay the poibility of increaing the wealth inequality due to the heterogeneitie of conformim in an analytical way, it would be di cult to ee if the wealth inequality actually increae becaue our model ha two heterogeneitie: the initial holding of capital tock and the heterogeneou conformim. Therefore, uppoing that the di erence of initial holding of capital tock among agent do not exit o that S() =, we pay attention to the role of heterogeneou conformim for the wealth inequality. If the utility function i identical and homothetic, the identical level of initial capital tock among the agent yield the identical jump of private conumption, and furthermore the relative conumption between agent i contant, thereby concluding that S = S() =. 7 6 To derive (5), we make ue of R i()c i di = R i() i di and R i()c i idi = R i() i idi + R i()idi. 7 When the utility function i identical and homothetic, the value of X i zero in (49), which can be een that S = if S() =. 24

26 Alternatively, when the degree of conformim are not identical among agent, from (49), we can derive R ( i ) 2 di 2 S ( R K() = i di)2 K ( + M) 2 (> ); (52) which i analytically evident to ee that the long-run level of wealth inequality i greater than the initial level, S > (= S()). Finally, let u conider the relationhip between the wealth inequality and the peed of convergence from the viewpoint of heterogeneou conformim. A een in (27) and (29), the peed of convergence become fater if R i(c i ; C )di approache to the unity. When we make ue of (3), uch an economy mean that the long-run relative-wealth rich ha a greater degree of conformim. Turning our interet into Propoition 4, from (24) we can ee that the diperion of the long-run elaticity of intertemporal ubtitution in (#3) become larger, which mean that the wealth inequality tend to expand. Furthermore, from (#4) if the long-run relative-wealth rich i rich in the initial period a well, the diperion of wealth become larger. 4.3 Numerical Analyi Finally, we examine numerical example that may capture the central meage of our tudy in the implet manner. To do o, we aume that there are only two type of houehold. The intantaneou utility of each type of houehold i (2) : u (c i ; C) = c ic i ; > ; 6= ; < i < ; i = ; 2: It i aumed that type houehold contitute a continuum with a ma of 2 (; ) : They have identical initial capital, k () ; The econd group of houehold i alo a continuum with a ma of whoe initial capital i k 2 () : By de nition, the aggregate (average) conumption i C = c + ( ) c 2 : In what follow, we aume that type houehold are initially richer than type 2 houehold, and thu k () > k 2 () : The production function i given by Cobb-Dougla: Y = AK where < A and < <. Therefore, irrepective of the value of production parameter A and, the condition (37) i alway ati ed, which mean that if = 2 o that both type of houehold have an identical degree of conformim, then the level of wealth inequality in the long run i le than 25

27 the initial level, that i, S < S(): In the following, we et: A = ; = :35; = 2:5; = :4: From (26b), the interet rate i.4 and the long-run level of aggregate capital i given by K = 9:96. Furthermore, we aume the initial level of aggregate capital equal 8 % level of it teady-tate one, that i, K() = :8K, which implie that our economy i growing over time. A to the population ize and the initial ditribution of capital, we conider the following four cae: (i) = :5; () = :; (ii) = :5; () = :; (iii) = :2; () = : (iv) = :2; () = :: Cae (i) and (ii) aume that the population ize of each group i the ame. Cae (iii) and (iv) conider a more realitic ituation where the rich group (type ) ha a maller ize of population than the poor one (type 2). A for the initial ditribution of capital, Cae (i) and (iii) aume that the initial capital tock held by type houehold i % higher than the average. Since it hold that () + ( ) 2 () =, the initial capital holding of type 2 houehold i 2 () = : in Cae (i) and 2 () = :25 in Cae (iii). Similarly, the level of relative wealth in Cae (ii) and (iv), () = : are maller than in Cae (i) and (iii) where 2 () = : in Cae (iii) and 2 () = :25 in Cae (iv), meaning that the di erence of wealth between houehold in rich and poor group are maller in Cae (ii) and (iv). Given thoe parameter magnitude, we change and 2 in the range that i. All gure divide ( ; 2 ) pace according to whether the level of wealth inequality in the teady tate i larger than it initial one. 8 The area with a red triangle how the combination of 8 We derive the teady-tate level of individual capital a follow. Firt, uing (2) and C = c +( )c 2, the ratio of marginal utility u (c ; C )=u (c 2; C ) = lead to c = c () where (> ) i an unknown contant parameter, leading to k = k() under (28), and hence, = (). Subtituting k = k() and = () into (44a) at t =, we can obtain the unique relationhip between and k () where we make ue of () + ( ) 2() =. A a reult, we can obtain k = k((k ())): 26

28 and 2 that yield the expanion of wealth inequality in the long run. Alternatively, the area with a black cro mean that the wealth inequality hrink in the long run. Figure (a) depict Cae (i). Here, we ee that the lower-right area i occupied by the ign of red triangle. In other word, when the value of i omewhat greater than that of 2, the long-run level of wealth inequality i greater than it initial one, S > S(), meaning that when the houehold in rich group have the tronger degree of conformim than thoe in poor group, the wealth inequality tend to expand. However, if the di erence between and 2 i mall enough, the wealth inequality between both type of houehold will reduce in the long run. Thi i becaue the right-hand ide of (47) ha the larger value than the left-hand ide. In other word, the initial-rich houehold achieve the larger jump of private conumption at the initial period, and therefore, the di erence in wealth between two type of houehold will be lowered. When the di erence in the degree of conformim between the two group i u ciently mall, the e ect of initial jump of private conumption i kept over time. A a reult, the long-run level of wealth inequality decreae. In contrat, when the houehold in poor group have the greater degree of conformim than the other, the wealth inequality i reduced. Furthermore, it i to be noted that when the degree of conformim in both group are the ame ( = 2 ), the wealth inequality hrink a een in the lat ubection. Thi reult recon rm García-Peñaloa and Turnovky (28) main nding in our context. Figure (b) correpond to Cae (ii). There are two di erence from Figure (a). Firt, if the degree of conformim of initial rich i lightly larger than that of the initial poor, the wealth inequality may expand in the long run. The reaon i that in the Cae (ii) the initial jump of private conumption of the rich group i lightly larger than that in poor group becaue the initial level of capital in both group trivially di er. Conequently, the heterogeneity of conformim ha a larger impact on the wealth inequality o that the diperion of wealth become larger in the long run. Second, and more interetingly, the upper-left area i occupied by the ign of red triangle, which implie that when the degree of conformim in poor group are larger than thoe in rich group to ome extent, the level of wealth inequality in the teady tate i greater than that at the initial period. In other word, the initially poorer houehold catch up with the houehold who are initially rich a een in Propoition 2, and after the reveral of wealth arie, the di erence of wealth expand over time. A a reult, the long-run level of wealth inequality i larger than it initial level. 27

29 Figure 2(a) and 2(b) repectively depict Cae (iii) and (iv) where the population ize of the initial rich (type ) i.2. The pattern of long-run wealth ditribution in the cae of = :2 are imilar to thoe oberved in Figure (a) and (b) in which = :5: However, there are quantitative di erence between the two cae: the area which how the expanion of wealth inequality are larger in Figure 2(a) and 2(b) than thoe in Figure (a) and (b). Thi implie that even if the di erence of degree of conformim i mall, the wealth inequality may expand if there i a ubtantial gap in the population ize of two group. It i intereting to note that a the population ize of houehold who initially rich become maller, the longrun wealth inequality will expand. 9 5 Concluion Thi paper introduce preference heterogeneity into García-Peñaloa and Turnovky (28). We have tudied how the preence of heterogeneou degree of conformim among houehold a ect the aggregate dynamic a well a wealth ditribution in the long run. We have preented three main reult. Firt, Propoition of the paper how that an economy where houehold have tronger conformim on average grow fater. Thi reult con rm that the nding in the repreentative-agent model with conumption externalitie till hold even if houehold have heterogeneou preference. Second, it i hown that if houehold preference are heterogeneou, an initially poor houehold may catch up with initially rich houehold, a long a her conumption conformim i trong enough. The precie condition for uch a catch up are given in Propoition 2. Thi propoition alo demontrate that the catch up will not arie if houehold preference are identical and homothetic with repect to private and the average conumption. Third, it i revealed that becaue of the preence of heterogeneou conformim, the wealth inequality may be enhanced in the long run. Thi i in contrat to the cae of homogeneou and homothetic preference under which wealth inequality tend to be reduced during the tranition. Propoition 3 and 4 preent a et of explicit condition that determine the long- 9 We have een that the peed of convergence become fater relative to the economy in which houehold have no conformim. In other word, even if a portion of houehold ha degree of conformim but the ret houehold have no conformim, the economy which ha a degree of conformim on average ha the fater peed of convergence. Thi nding i likely to upport Propoition (i). 28

30 run pattern of wealth inequality. Finally, our numerical example how the poibilitie of catch-up by the initially le wealth peron, and the expanion of wealth inequality under the plauible parameter et. Uing our model, we can re-examine the exiting tudie on conumption externalitie with heterogeneou agent. For example, Koyuncu and Turnovky (2) analyze the role of tax policy in the context of García-Peñaloa and Turnovky (28) model. Our model with heterogenou preference may extend their policy analyi. In addition, Gollier (24) tudie a tatic, general equilibrium model of aet market with conumption externalitie and heterogenou preference. Baed on our framework, we may preent a dynamic verion of Gollier (24) dicuion. Thoe topic would deerve further tudy. 29

31 Appendice Appendix A We derive the equation (35) and (39) where the derivation i fundamentally the ame a García-Peñaloa and Turnovky (26, 28). Firt, ubtituting the individual a well a the aggregate capital accumulation equation into _ ~ ki = _ k i =K ~ k i _K=K and arranging for it, we can how _~k i = K n (f (K)K f(k))( ~ k i ) + C( ~ k i c i C ) o : (A.) Identical preference: Note that the relative conumption c i C i contant over time under the identical preference. Approximating (A:) around the teady tate, we can obtain _~k i = ( ~ k i ~ k i ) + f (K )( ~ k i )(K K ) + ~ k i c i =C K (C C ); (A.2) and nally arranging for it, we can derive (35). Heterogeneou preference: Since the relative conumption c i C approximation (A:) around the teady tate i i not contant, the linear _~k i = ( k ~ i k ~ i ) + f (K )( k ~ i )(K K ) + n o ~k K i (C C ) (c i c i ) : (A.3) Therefore, uing C C = ( )(K K ) and (33), we can how (39) where we ue = f (K ) ( ) derived by umming (32) over all houehold. Appendix B Raiing both ide of (44a) to the double power at t = and umming up for all houehold yield: where S() = S + 2(K K()) Z i i di = B S (Zi ) 2 di = (B ) 2 S 2B ( ) (K ) 2 (K ) 2 K i Zi di + K K i i di + 3 K() 2 (Zi ) 2 di; (B.) i i di; K 2 R + ( i )2 dic R 2 A : i

32 A a conequence, we can how S() = ( + M ) 2 S D ; (B.2a) where M i de ned by (48) and D = K() 2 K R ( i )2 di ( R i di)2! + 2( + M ) R K() K i i R di i di : (B.2b) Notice that = R i di. To derive (5), we can rewrite (B:2b). We make ue of (44a) at the initial time and rewrite the equation a follow: i = + M i () + and furthermore we can obtain the following: i i di = + M i () i di + K() K i R i di!! ; (B.3a) R K() (! Z! i) 2 di K ( R i di)2 i di : (B.3b) Finally, ubtituting (B:3b) into (B:2b), D correpond to X in (5). 3

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