JOURNAL OF ECONOMIC THEORY 58, 355-376 (1992) Efficient Equilibrium Convergence: Heterogeneity and Growth ROBERT TAMURA * University Received of Iowa, Iowa City, Iowa 52242 October 10, 1990; revised June 11. 1992 An endogenous growth model arising from task specialization is used to examine the effects of differing initial human capital distributions. Two macroeconomic phenomena are produced: income convergence and leapfrogging of living standards. Income convergence occurs for all countries in the same market, arising from diminishing returns to individual human capital. Leapfrogging occurs by comparing countries in two separate markets. Income leapfrogging can occur because the stationary growth rate increases with market size, and human capital heterogeneity reduces growth. The introduction of coordination costs of task assignment can produce gains in welfare and growth from human capital heterogeneity. Journal of ‘!? 1992 Academic Press, Inc. Economic Literarure Classification Numbers: 015. 041. In this paper I develop an endogenous growth model based upon specialization of task production. Task specialization is the microfoundation of the CES aggregate production function studied here. I use this model to examine the macroeconomic effects of differing initial human capital distributions.’ Two macroeconomic phenomena are identified: income convergence and leapfrogging of living standards, i.e., a change in rank order. Income convergence is predicted for all countries in the same market and is brought about by the convergence of human capital. Leapfrogging * I thank Kevin Murphy, Sherwin Rosen, Robert E. Lucas, Jr., Forrest Nelson, Paul Romer, Nancy Stokey, Steve Trejo, Kei-Mu Yi, John Kennan, Narayana Kocherlakota, Barbara McCutcheon, George Neumann, an anonymous referee, and an associate editor for helpful comments. I am particularly indebted to these individuals for making this manuscript readable! All remaining errors are mine. ’ This paper concentrates only on the macroeconomic implications of heterogeneous agents and will be referenced as heterogeneity. A market is identified by all agents specializing in the production of different tasks. In this paper a market is a collection of countries trading the output of specialized tasks to produce an aggregate consumption good. Throughout the paper individuals or agents are synonyms for countries. Although important, this paper ignores the industrial organization implications of industry market structure. 355 0022-0531/92 $5.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved. 356 ROBERT TAMURA occurs by comparing countries in two separate markets. For expository ease, I refer to effects of a heterogeneous human capital distribution as arising from heterogeneity. The aggregate production function contains all three returns to scale. The aggregate production function exhibits constant returns to scale in the human capital distribution, diminishing returns in each individual’s human capital, and it displays increasing returns in the number of agents in the market. Strict concavity of individual human capital in production and convex preferences imply that heterogeneity reduces output in any period and retards growth over time. In the dynamic model the competitive equilibrium produces efficient human capital convergence, and hence, income convergence. Income convergence. applies to all agents in the same market. Convergence is monotone; richer agents remain richer. Constant returns in production and accumulation provide for endogenous growth. Growth is increasing in market size. Income convergence is controversial. Some work provides evidence of income convergence, while other work disputes the existence of income convergence. Abramowitz [ 1) and Baumol [6, 71 utilize the Maddison data set to illustrate income convergence for a subset of the OECD countries. Using Summers and Heston [27] data, Dowrick and Nguyen [13], Mankiw et al. [lS], and Tamura [29] report income convergence for some countries. Barro and Martin [3] and Mieszkowski [ 193 identify income convergence for regions in the United States. Becker and Tomes [9] identify convergence in intergenerational data sets. O’Neill [21] finds evidence that education convergence predicts income convergence. Some of these studies are critiqued in DeLong [12] and Quah [22], who show that convergence may arise from selection bias and Galton’s fallacy. Thus convergence depends on sample selection criteria, time period, and measure of dispersion. Unlike the income convergence controversy, a change in rank order is undeniable. For example the United States and Germany leapfrogged the United Kingdom early in the 20th century. Recent work has focused on differences in technologies, preferences, and public policies to explain both leapfrogging and convergence. Public policy differences are identified by Barro [2], Glomm and Ravikumar [14], Jones and Manuelli [ 151, and Rebel0 * [23]. This paper provides an alternative explanation of convergence and leapfrogging that is based on differing initial human capital distributions. A static model of task specialization is developed in Section II. It forms the microfoundation for aggregate production. In Section III the dynamic model is studied, where investment in human capital leads to convergence in human capital and income. In contrast to Tamura [28,29-J, where an external effect of human capital in the accumulation technology produced EFFICIENT EQUILIBRIUM CONVERGENCE 357 convergence, convergence in this model is efficient. Prices are found to decentralize the efficient solution in Section IV. In Section V I analyze the balanced growth path and present simulations of the effects of heterogeneity. Heterogeneity reduces growth and welfare. In Section VI I modify the model to include a cost of coordinating task assignments. In contrast to the model without coordination costs, heterogeneity can increase net output. Furthermore, in a two-period coordination cost model heterogeneity can increase growth and welfare compared to homogeneity. II. MODEL In this section I provide a microeconomic model of task specialization, building on the work of Rosen [25], Baumgardner [4,5], and Becker [8]. Task specialization generates a reduced form aggregate production function that I use to analyze the growth process. The model exhibits decreasing returns to individual inputs of human capital, but increasing returns in the number of market participants. Diminishing returns to an individual’s human capital makes homogeneity desirable. Because of this, a social planner chooses dynamics such that the human capital distribution collapses over time. There are a continuum of “tasks” involved in the production of final output, and these tasks are indexed by a E [0, 11. There are N agents in the market. Let m,(a) denote the level of skill of worker i applied to task a, and let ti(a) be the amount of time worker i spends at task a. I assume that the amount of output of task a is z(a)=Cy=, m,(a) t,(a); alternatively, z(u) can be interpreted as an intermediate good. Let x{ a 1 ti(u) > 0} be the characteristic function that is 1 if ti(u) > 0, and 0 otherwise; and Hi is the amount of human capital owned by i. An individual allocates skill across tasks, but the geometric mean of his or her skill use cannot exceed his or her human capital. Thus the individual’s skills constraint is (J @‘(a) ~{a 1 ti(a) > 0} d~)“~ = Hi, where OE [0, 11. Likewise, an individual can only spend time on task production up to his or her time endowment: s t,(a) da = 1. Final output is produced by a Leontief production function : y = min[z(a)], a UE [O, 11. (1) I assume that labor is inelastically supplied. With these assumptions the output maximizing social planner chooses the allocation of workers to tasks and the optimum quantity of tasks produced in order to 358 ROBERT TAMURA max y m(o).1(u) s.t. J’ = min [~(a)] u z(a)= f m,(u) t,(a) ;= I I/<lJ my(u) x{u 1t,(a) >0)da 1 (2) =Hi 5t;(u) da = 1. The solution to this maximum problem involves complete specialization of tasks; thus, no two individuals work on the same task. It follows that the level of time and human capital skill devoted is the same for all tasks that an individual works on: m,(u) = ~,(a’) = m, and ti(u) = t,(u’) = ti for all a, a’ in [0, 11.’ Only the number of tasks that an agent provides remains to be solved. Let Ai be the number of tasks that i provides. The efficient number of tasks for agent i can be determined from the resource constraints. The skills constraint produces mid f’” = Hi = mj = Hi/A!‘“. From the time constraint, ti= l/Ai. Therefore, output is y = Hi/AI1 +o)‘o. Since the total task length is 1, the equilibrium task assignments are H”“‘+o) Ai= C”= ’ H+(‘+W)’ !I iE {1,2, ...) N}. (3) J Hence agents with large amounts of human capital do more tasks than agents with small amounts of human capital. From (3) output is If agents are endowed with the same amount of human capital, they do the same number of tasks, A = l/N, and output is N” +““““H. This shows the increasing returns to participation. Kydland and Prescott [ 161 and Neumann and Topel [20] identify this feature. Holding Cy’, H, constant, maximum output occurs when all agents are identical. The constant elasticity of substitution production function contains all types of returns to scale. Output y is homogeneous of degree 1 in {Hi} ,“=1. In Section V I show that perpetual growth can exist in this model. Diminishing returns in H, is the motivation for equalization of human capital across agents. ‘This is the same result as in Rosen [25] and Baumgardner [4, 51 EFFICIENT EQUILIBRIUM III. HETEROGENEITY CONVERGENCE 359 AND CONVERGENCE A dynamic version of the static model is introduced and analyzed in this Section. I characterize the solution to the associated social planner’s problem and show that there is convergence of the distribution of human capital levels in the limit. The intuition for convergence is the same as in the static model: for fixed worker size and fixed total resources, maximum output occurs when workers are identical. The efficient dynamic growth path eliminates the differences across individuals. Assume that agents have identical preferences given by ,zo P’$3 (5) where 0 < 1; preferences are logarithmic if CJ= 0. The law of motion of the stocks of human capital is Hi,+, = AX;Hf,-P, (6) where Xi, is the amount of investment in the stock of the ith individual’s human capital in period t. Assume that B[AN” + W)p’w]u < 1. Anticipating the development of the model below, this restriction imposes the transversality condition, and the finiteness of the value function. For CTQ 0, B < 1 is sufficient. If {1,}:= i 2 0 are the Pareto weights, then the social planner’s problem becomes max { i i=l & f B’ $}, 1=0 (P) Define aggregate consumption, c, - ({Cy= i H$(‘tW’}‘l+W)‘W - Cy=, Xi,). Efficient allocation of consumption has each agent consuming a fixed fraction of aggregate consumption: The last step solves for the efficient allocation of investment resources. 360 ROBERTTAMURA Let {H,) = { Hir}y=, , and [A’,), = { Xir}r=, , t 3 0. The social planner’s optimization problem is to choose investment resources X, to satisfy The following theorem shows that the social planner eliminates heterogeneity. Without loss of generality; order the N agents from lowest to highest human capital stock, thus agent N has the most human capital. THEOREM. Define h, = Hil/HN,, then ‘the ratios of human capital converge to 1, i.e., lim hi,= 1, t-m iE { 1, 2, .... N}. Proof: The proof uses three features of the problem. The value function is strictly concave and symmetric in each agent’s human capital and is homogeneous of degree 0 in human capital. Strict concavity and symmetry of the value function imply that low human capital agents grow faster than high human capital agents. Homogeneity of the value function implies that the ratio of agent i’s human capital to agent N’s human capital, hi,, converges to a stationary solution, h* = 1. Using (6) to replace (X,1, c”,/ (T is strictly concave in ((Hi,, H, + ,)} f”=I. Therefore from the theorems of Chapter 4 of Stokey and Lucas [26], the value function is strictly increasing, strictly concave., and differentiable in {H,, H, + , }. The first-order condition for agent i is (1 -PVP Hit+1 [ Hit1 +@/PpC;-- The derivative of the value function theorem. This produces (8) au:HH”‘). If+ is calculated 1 from the envelope (9) Equation (9) shows that the derivative of the value function is symetric; if H ,r+,=Hjr+~, then &,GH, + 1= &/aH,, + 1. Since the value function is strictly concave and its derivatives are symmetric, (8) shows that H, + 1 is EFFICIENT EQUILIBRIUM CONVERGENCE 361 strictly increasing in Hi,. Thus if H, < Hi,, then H, + 1 < Hi, + 1, Vt. Therefore if H, < Hi,, (8) also shows that the growth rate of human capital for agent i exceeds the growth rate of human capital of agent j. Agents with lower human capital grow faster than agents with higher human capital, but never surpass higher human capital agents. The previous arguments imply that for each agent, hi,<hi,+l < 1, v’t. Each sequence converges to a limit, say h:. By symmetry h* = hi* = h*, for all i, j # A? Homogeneity of the value function allows it to be expressed in terms of the ratios of human capital, hit, and the growth rate of human capital of agent N, cN. Define h, = {h,};N_ i. Ignoring H&, the social planner’s problem is to choose next periods relative human capital distribution and the growth rate of the largest human capital agent: ([C;=“=, o(h,)= max ,+(I+m) II 1(I+W)/OJ -Cj”‘l {(hi,+,I,,)IAh~,-P}“P)” The first-order conditions from this problem produce a system of equations for h,, 1 which only depend on h,. Thus h,, , = 4(h,), where 4 is continuous. Hence the limit of the sequence is lim h (+ 1 = h* = ,‘i\ t-m d(h,) = $li; h,) =&h*). Stationarity of h, implies a common growth rate of human capital. From (8) this only occurs for h* = 1. Therefore the limit of human capital ratios isl. 1 The theorem shows that human capital converges. Over time, agents become alike and the number of distinct tasks each agent performs becomes equal. Rich agents remain richer than their poorer cousins, although the gap shrinks. No leapfrogging is possible within the market. Equation (8) shows why individuals care whom their colleagues are. Having higher human capital colleagues raises welfare, i.e., 2Jv(H,)/iTH, > 0 VH,. Since output is strictly increasing in N, welfare is strictly increasing in the number of market participants. ROBERTTAMURA 362 IV. DECENTRALIZATION In this section I find prices to decentralize the efficient solution to the planner’s problem in Section III. Assume that complete markets in consumption loans exist. Assume that agents are hired in a competitive labor market. Human capital is accumulated using one’s human capital and purchased investment goods, X,,. Since X, can be consumed, the relative price of Xi, and cir is 1. The individual’s problem, (P,), is to purchase investment goods and consumption goods, given competitive equilibrium prices (p,, w,) to s.t. Hi, +,=AX”H!-p (Pi) II) ,go PIICit +xit) Gf ‘i*Wit Hiz. 1=0 Let pi be the Lagrangian multiplier individual’s Euler equations are Brczp ’ = xif In a competitive i’s wealth constraint. The PiPr7 (10) 1 (11) workers are paid their marginal product; thus PI+1 __ - Hit+1 on individual p equilibrium l-P Wir+l Xir+l +-- P Pt Hit+, ’ Equation (12) shows that wages per unit of human capital differ across agents. High human capital agents are paid less per unit of human capital than low human capital agents. However, since each agent supplies one unit of time, earnings are wiHi. These earnings are rising in Hi. This agrees with what economists observe. Substituting the equilibrium wage into (11) yields xi* Hit+1 Comparing -p P~+I 1-P J’ir+l P, ( 13) and (9) requires PlCl -= Pr 4 - Cl c,+ 1> I-o ’ t 2 0, (13) EFFICIENT EQUILIBRIUM where p,, = 1. Agent i’s initial Recall that the consumption CONVERGENCE consumption 363 is of agent i in the efficient solution is Therefore, in order to support the efficient solution, wealth transfers may be necessary. Alternatively, the original Pareto weights, li, can be those arising from the initial distribution of wealth in the market. Prices exist to decentralize the efficient allocation. V. DYNAMICS: GROWTH AND LEAPFROGGING In this section I analyze the stationary aggregate growth rate. The most interesting finding is that the aggregate growth rate is increasing in N: larger markets grow faster. I provide a finite time approximation to the infinite time model. The finite time model is solved numerically, and the simulations show that heterogeneity lowers growth during the transition from heterogeneity to homogeneity. However, despite slower growth in the transitional phase, an initially heterogeneous economy has the same asymptotic growth rate as an otherwise identical homogeneous economy. Solving the Euler equation in the homogeneous agent market implicitly defines the stationary investment rate, s: s = [pp + /3(1 - p)s] A”(sN l’w)Op. Comparative statics reveal that the aggregate growth rate is increasing in 0, A, /I, and N. It is decreasing in o. For log preferences, s = BP/( 1 - /I( 1 - p)); thus the aggregate growth rate is ,4[sN(1+“)‘“]P. This is a tractable model for analyzing the effect of heterogeneity on the aggregate growth rate. Consider two markets, identical except for the distribution of the fixed total human capital. Solving each dynamic problem through numerical simulations allows for calculation of the effects of heterogeneity on the aggregate growth rate of output. The solutions show that heterogeneity can dramatically affect the aggregate growth rate. Each simulation solves a finite time social planning problem. Each has 48 time periods and 35 agents. The simulations are a finite time approximation to the infinite time horizon model, or an exact solution to a finite time 364 ROBERT TAMURA world. The simulations come from solving the Euler equations. These equations form a second-order difference equation system; therefore, two initial conditions must be specified. These are my specifications of the final period human capital stocks for all agents, and the zero amount of investment in the final period. Identical homothetic preferences allows for aggregation, thus simplifying the problem. Knowing {H,, X,} allows me to solve for \(X, _ 1} and hence (H,_ , }, for t 6 48. In the simulations aggregate human capital in the heterogeneous market is equal to aggregate human capital in the homogeneous market at t = 48. The top curve in Fig. 1 is the time path of the coefficient of variation of human capital. The lower curve is the time series of the ratio of the growth rate with heterogeneity to the growth rate with homogeneity. The heterogeneous market growth rate approaches the homogeneous market growth rate after 24 periods. However, in the first 24 periods, the heterogeneous market growth rate is small compared to the homogeneous market growth rate. At t =0 the growth rate in the heterogeneous agent market is 4% of the growth rate in the homogeneous agent market. Figure 2 presents the human capital growth rates of some of the agents in the market. The lowest human capital agent has a growth rate about live times that of the highest human capital agent in period 24. Figure 3 illustrates the convergence in human capital by each agent to agent 35. Until the human capital of agent 30 is more than 25 % of the human capital of agent 35, t = 24, the individual growth rates differ from each other dramatically. I 16 FIG. u(c) = In(c); p = 0.175; I 24 Time w = 0.175; I 32 A = 1.05; r I I 41B 40 fi = 0.75; N= 35. EFFICIENT EQUILIBRIUM I J 365 CONVERGENCE I I I I 75 45 15 1 I 24 FIG. I 32 28 2. Human capital Tim? growth I I I 40 44 rates of agents. 1 1 .75 75 .5 .5 .25 .25 0 24I 28I 321 36I 40I Time FIG. 3. Human capital ratios. 44I I 48 I 0 366 ROBERT TAMURA Figures 4 and 5 contain the individual investment rates of some agents and the aggregate investment rate in both markets. With log preferences, the fraction saved is independent of the distribution of human capital in the market and only depends on the number of periods remaining. Savings and growth rates, individual and aggregate, converge to zero by period 48, since this is the final period. I simulated the model for different values of C. The qualitative results are similar to the results for log preferences. However, one difference is the effect of heterogeneity on the aggregate savings rate. For (T= - 1.5 (Fig. 7) the aggregate savings rate is increasing in the degree of heterogeneity, while the opposite holds for c = 0.05 (Fig. 6). In all solutions, as the degree of heterogeneity is reduced, growth accelerates.’ The human capital convergence theorem of Section III applies to all countries in the same market. No leapfrogging exists within a market. Leapfrogging requires comparing countries in two separate markets. Assume that there are two separate markets, one large heterogeneous market and a small homogeneous market. Further assume that initially the .03 .0225 ,015 .OO?!i 0 -0 I 16 1 20 FIG. I 24 4. I 28 I 32 Time , 36 5 40 I 44 Investment rates of agents. 3 Romer [24] presents evidenceof acceleratinggrowth. Using five countries reject a nonpositive trend in growth rates at the other countries reject a nonpositive trend at the 10% level reject a nonpositive trend at the 15% level of significance. nonpositive trend. Maddison’s 1 l-country sample, 5 % level of significances. Three of significance. Two countries Only Sweden fails to reject a EFFICIENT EQUILIBRIUM 367 CONVERGENCE .32 .32 .24 .24 .I6 .16 .oa .08 0 0 FIG. 5. Aggregate investment rate, log utility. ,625 .625 5 .5 .375 ,375 .25 .25 ,125 .125 I 0 I 6 1 16 I 24 lime I 32 I 40 FIG. 6. Aggregate investment rate, u = 0.05. I 48 368 ROBERTTAMURA .12 .I2 .09 .06 .03 0 0 1 d i FIG. 7. I Ii5 Aggregate I 2’4 Time investment I I 32 rate, (r= 4b I 4i3 -1.5. large heterogeneous market is richer than the small homogeneous market. As shown in the simulations, the poor, small market can have a faster rate of growth initially. Therefore the small market might leapfrog the richer market because of the rich, large market’s heterogeneity. Over time the large market becomes homogeneous, and its growth rate increases. Since the large market has a faster stationary growth rate, eventually the income of the large market will leapfrog back over the income of the small market. VI. COORDINATION COSTS This section modifies the model by introducing coordination costs of task specialization. Coordination costs can prevent all agents from working in the same market. Holding C;“= 1 Hi constant, if more than one market exists, then heterogeneity may raise welfare relative to homogeneity. In the static model this occurs because heterogeneity may support larger markets, increasing output net of coordination costs. In a two-period model, market heterogeneity may raise growth and welfare by creating larger markets in the second period, compared to market homogeneity. In the short run, therefore, coordination costs can reverse the desirability of homogeneity. 369 EFFICIENT EQUILIBRIUM CONVERGENCE Assume the costs are incurred to coordinate task specialization. Specialized inputs must be organized to produce the right compatible parts and avoid task duplication. These costs are modelled in reduced form by assuming that costs increase with the number of market participants: costs = c(N), c’(N) > 0, c”(N) > 0, c(l)=O. (14) These coordination costs are similar to the coordination costs in Becker and Murphy [lo].” Assume that N is the population of the market. In a static homogeneous agent model the solution to the social planner’s problem, (PC), is to maximize net output: i=l The optimal market size, ni, chosen by a social planner satisfies: n!‘+“)iwH-c(n,)>(ni+ l)(‘+O)‘V-c(ni+ nl’+O’/~~-c(ni)~(ni-l)(‘+m)‘“H-c(ni- l), 1). (15) When (15) holds for ni = N, the equilibrium is identical to the static problem without coordination costs, except that net output replaces gross output. Under this specification of coordination costs, income growth in a dynamic model implies that eventually (15) holds as a strict inequality for ni = N. However, if (15) holds as a strict inequality for n, < N, then there will be more that one market. Suppose that there exist m markets. Markets can be ordered in the following manner: Net output in the homogeneous agent case is it1 w+a)‘wH- c(q)}. (16) 4 Coordination costs are assumed to only depend on the number of market participants. If coordination costs also depend on the scale of output, {H,}, then the implication of full market integration in the long run can be overturned. I thank the associate editor for bringing this to my attention. 642/58/2-17 370 ROBERT TAMURA Observe that a redistribution of human capital from the small markets to the large markets increases output. If human capital were mobile across agents, then the maximum net output is at least n1 (1+0)/w N (17) -~(i~~)=n:/wlvH-~(t~,). (7 n1 This is a lower bound on the gains from heterogeneity, because the size of the largest market may increase. Equation (17) is the maximum net output possible by only reallocating human capital, i.e., working on the intensive margin. Net output might rise by increasing market size, the extensive margin. Therefore introducing coordination costs in the static model can make heterogeneity more desirable than homogeneity. This gain from heterogeneity in the static market provides the intuition into possible benefits of heterogeneity in the dynamic context. I now extend the static coordination cost model into a dynamic model by introducing an additional period. Intuitively, a gain from heterogeneity can arise because there are two margins to use. In a dynamic coordination cost model, heterogeneity may create larger markets earlier compared to homogeneity. By increasing market size, it is possible that in the short run heterogeneity is beneficial to growth and welfare. In both periods the social planner maximizes net output. Define maximum net output as (18) m s.t. c ni= N. i= 1 Welfare in the second period is u2((H2))= y2(‘H2’)u, 0 (19) The social planner chooses investments given the first period distribution human capital in order to (~l({Hl))-CiN,l ~d{Hd)= max IXL, St. . ff. r2 =AXPH!-PI Xi)“+pV2({H CT 11 2 1, of 3 I (PC) . This two-period, three-agent model is solved numerically. The homogeneous market has period-one aggregate human capital equal to the period- EFFICIENT EQUILIBRIUM 371 CONVERGENCE one aggregate human capital of the heterogeneous market. Figures 8-12 present simulation results of the effect of heterogeneity on welfare, individual growth rates, the aggregate growth rates of output, consumption, and mean human capital. They show that binding coordination costs can reverse the results obtained in Section V. In Fig. 8 there are ranges of mean human capital, where heterogeneity increases welfare relative to homogeneity. I interpret these differences as capturing the distinction between the short run and the long run. As wealth rises over time full integration of market participants is possible. In the short run, however, coordination costs can inhibit the formation of markets. As the size of the market rises, so does the growth rate of output. Therefore changes in the extensive margin can explain increasing growth rates. This captures the dynamic gains from trade: the larger the market, the larger the gains from specialization, and the greater the specialization, the larger the growth rate. Introducing coordination costs to the model provides an alternative explanation of leapfrogging. Heterogeneous countries grow at unequal rates when the market structure moves from three small markets to two markets. In this case the two countries integrating into the intermediate market grow much faster than the left out country. Since the higher human capital countries expand the market, they diverge from the smaller country. This is shown in the growth rates of the individual countries in Fig. 9. I J I 2.25 Mean FIG. 8. Value function, I 4.5 Human Capital u(c) = In(c); 6.75 o = 0.5; p = 0.5. 1 9 ROBERT TAMURA 312 5 4 3 2 1 I 2.25 I 4.5 Mean Human Capital I 6.75 I 9 FIG. 9. Human capital growth rates of agents. 17.5 15 mogeneou 12.5 10 7.5 5 fi 2.5 -r 0 I 2.25 I 4.5 Mean Human I Caoital 6.75 FIG. 10. Growth rate of aggregate output. I 9 EFFICIENT EQUILIBRIUM 373 CONVERGENCE - 32 28 z 2 -24 - 16 -8 4 FIG. 11. I 5.5 Growth rate of aggregate I I 2.25I 4.5I consumption I I / 6.75 I 9 - 2.5 0I Mean FIG. 642/58/2-18 12. Growth Human rate of mean Caoital human capital 374 ROBERTTAMURA From Fig. 9, in the intermediate range, the smallest human capital country loses not only relative to the other two players, but absolutely as well! Thus growth need not make all countries richer, although with complete markets in consumption loans, all countries are better off, since consumption rises for all. In the fully integrated outcome, the two smaller countries gain relative to the biggest country, i.e., gl > g2 > 83. The interior range characterizes leapfrogging of countries 2 and 3 relative to country 1, and convergence between 2 and 3. In Figs. 11 and 12 there exist ranges where consumption growth and mean human capital growth is higher under heterogeneity than homogeneity. CONCLUSIONS This paper provides an endogenous growth model with efficient equilibrium convergence in human capital and income. This arises from the decreasing returns to individual human capital. The CES production function implies that agents have an incentive to equalize the human capital in society. Since there are increasing returns to participation, all agents desire to participate in the same market. In Tamura [28,29] an external effect of human capital in the accumulation technology produces convergence; therefore high human capital agents have an incentive to segregate themselves from low human capital agents. In the model of this paper, absent coordination costs, high human capital agents prefer high human capital coworkers, but are happy to have anybody participate in the market. In this model I compute the effect of heterogeneity on growth. Without coordination costs, the simulations indicate that heterogeneity reduces growth. Introducing coordination costs can reverse this negative relationship. If heterogeneity creates larger markets faster than homogeneity, then growth and welfate increase. Finally, the model provides an explanation of reversal of income rankings, or leapfrogging. Suppose there are two markets, a large rich, heterogeneous market, and a small, poor, homogeneous market. 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