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Increasing returns, human capital, and the Kuznets curve
JOURNAL
OF
Development Journal of DevelopmentEconomics Vol. 55 (1998) 353-367
ELSEVIER
ECONOMICS
Increasing returns, human capital, and the Kuznets curve Gerhard Glomm a,*, B. Ravikumar
b
a Michigan State University, MI, USA b University of Iowa, 10, USA
Received 31 May 1996; accepted 31 December 1996
Abstract We present a simple model of human capital accumulation which generates the Kuznets curve as an equilibrium outcome. The central ingredient that helps generate the Kuznets curve in the model is what we call short-run increasing returns to scale in the learning technology. The learning technology exhibits increasing returns to scale, but only in the short run, since one of the factors of production, time, is bounded above by the endowment. We show that short-run increasing returns to scale is necessary to obtain the Kuznets curve, but not sufficient. © 1998 Elsevier Science B.V. Keywords: Human capital; Increasingreturns; Kuznets curve
1. Introduction In 1955 Kuznets published his first of a by now famous sequence of papers on the relationship between per capita income and income distribution. He suggested that income inequality exhibited a secular decline for a few North American and European countries. In a paper in 1963 Kuznets showed that income inequality might actually follow an inverted U pattern along the development path: at first income inequality rises as economies grow, but eventually income inequality declines. This hypothesis that income inequality might follow an inverted U pattern over the development process sparked a wave of investigation both in the
* Corresponding author. 0304-3878/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0304-3 87 8(98)00040-6
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Development Economics literature and in Economic History. Researchers in both fields documented with more and more, and over time with presumably better data sets, if the Kuznets hypothesis of an inverted U relationship between income inequality and economic development could be confirmed or rejected. For surveys of this literature see Adelman and Robinson (1989) and Brenner et al. (1991). In recent years, there have been a few dynamic general equilibrium models in which inequality affects economic growth and the evolution of income inequality is endogenous. Examples include Tamura (1991, 1992) and Glomm and Ravikumar (1994) where income inequality declines over time due to diminishing returns to capital investment and Alesina and Rodrik (1994) where income inequality stays constant over time. Other general equilibrium models deliver the Kuznets inverted U as an equilibrium relationship. In Greenwood and Jovanovic (1990), there is a fixed cost to joining a financial network. The financial network guarantees higher returns to investment relative to a backyard technology. Individuals with less wealth expect to join the network later than those with more wealth. Greenwood Jovanovic show that savings rate, for those outside the network, increases with the level of wealth and, hence, there is an initial increase in inequality. Eventually, everyone joins the network and the savings rate is independent of wealth, so everyone grows at the same rate. Aghion and Bolton (1993) and Khan (1993) provide alternative models of financial intermediation with credit market imperfections that generate the Kuznets curve. Galor and Tsiddon (1992), Perotti (1993), and Tamura (1996) present models of human capital accumulation which generate the inverted U relationship. In Banerjee and Newman (1993) the Kuznets curve arises in a model of occupational choice. Models of migration from agriculture to manufacturing such as Glomm (1992) and Ranch (1993) also give rise to the Kuznets curve. In this paper we present a simple model of human capital accumulation which generates the Kuznets inverted U relationship between income inequality and development. We deliberately impose simple functional forms on preferences and technology so that we can expose the fundamental economic factors at work in generating the inverted U. In the model, which is described in Section 2, human capital is accumulated by overlapping generations of individuals. All individuals value leisure when young and consumption when old; the utility function is CES. Consumption when old is a function of human capital. Human capital is accumulated using a Cobb-Douglas learning technology with time and parental human capital as inputs. Unlike many of the papers that generate the Kuznets curve, our model neither has any exogenous thresholds nor relies on income redistribution through political institutions. The central ingredient that helps generate the Kuznets curve in our model is what we call short-run increasing returns to scale in the learning technology. Each factor in the learning technology exhibits weakly diminishing returns, but since one of the factors is bounded above by the time endowment, we have increasing returns only in the short-run and not in the long-run.
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In a closely related paper Galor and Tsiddon (1992) obtain the Kuznets curve in an overlapping generations model where technical progress depends upon intragenerational as well as intergenerational externalities. At low levels of technology, their economy has multiple steady states, one unstable and two stable. Around the unstable steady state, the poor experience a decline in wealth whereas the rich experience an increase in wealth, so wealth inequality increases initially. As technology advances the unstable steady state is eliminated and the economy exhibits a decline in wealth inequality as incomes converge to the unique stable steady state. In their economy and ours, the equilibrium law of motion for human capital exhibits short run increasing returns to scale. Our model, however, does not rely on intragenerational externalities or technical progress to deliver the Kuznets curve. Short run increasing returns to scale are also present in the other models (mentioned above) generating the Kuznets curve. In Greenwood and Jovanovic (1990), Aghion and Bolton (1993), and Khan (1993), the short run increasing returns to scale arise because of fixed investment required to access better investment opportunities. In Perotti (1993), the short run increasing returns to scale enter through exogenous thresholds in the learning technology. In Tamura (1996), the rate of return to investment in skills initially rises with the level of skill, but in the long run it is constant. In the occupational choice models of Banerjee and Newman (1993) and in the migration models of Glomm (1992) and Ranch (1993), the equilibrium mapping from present wealth to future wealth displays short run increasing to scale due to the discrete nature of occupational or sectoral choice. We show in Section 3 that the short-run increasing returns to scale is necessary to obtain the Kuznets curve. Due to the short-run increasing returns, the equilibrium relationship between future and current human capital in our model is convex for low levels of current human capital and concave for high levels of current human capital, i Given this shape we show that the relationship between per capita income and income inequality will follow the Kuznets curve for some initial income distributions. In the Appendix, we extend our results to economies that exhibit sustained growth from exogenous technological progress.
2. The Model The economy is populated by overlapping generations where individuals live for two periods and where population is constant over time. Each generation consists of a large number of agents. Population size is normalized to unity. At
1 An early example of a model with a convex-concave production technology is Skiba (1978). His focus, however, is not on the evolution of income distribution but on optimal growth.
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time t --- 0, there is an initial generation of old agents in which the human capital endowment of the i th member is hi0. Individuals differ only in the stock of human capital of their parents. We assume a simple kind of heterogeneity: initial old belong to one of two types, rich or poor. The initial old of the poor type are each endowed with h, while the rich type are endowed with h. In this case inequality can be measure-d simply by the ratio of the incomes of the two types. 2 All agents born at time t = 0, 1, 2 . . . . have identical preferences over leisure when young, n t, and consumption when old, ct+ r Their utility function is given by i - o" .~_ l-o"
nt
ct+ 1
, o r > O.
1--o"
We will interpret tr = 1 as logarithmic utility; 1 / t r is the elasticity of substitution between consumption and leisure. Consumption when old is equal to the stock of human capital that the agent possesses when old. That is, there is a technology which transforms human capital one for one into output. (Hence, we use the terms 'human capital' and 'income' interchangeably throughout the paper.) Each agent consumes his entire output when old. Each individual is endowed with one unit of time while young. He spends part of the endowment toward leisure and the rest toward human capital accumulation. His human capital when old depends on time, 1 - n,, and his parents' human capital, h t, i.e., h,+, = Oh~t(l - nt)~, 0 > 0, 6 e ( 0 , 1 ) , fie(0,1].
(1)
Notice that in Eq. (1) the returns to investing in human capital depend positively on the human capital of ones own parent. There is, thus, an intergenerational externality. When fl + ~ > 1, the learning technology exhibits increasing returns. Since 1 - n t is bounded above by one we can think of the case fl + ~ > 1 as one of short-run increasing returns. Each individual at time t chooses n t and c1+ 1 to solve the problem l-o-+
max
nt
l-o"
ct+ 1
1--o"
s.t. c,+ l = 0(1 - nt)t3h~. An equilibrium for this economy is a set of sequences {nt}t~=o, {ct+ l}t~__0 , and
2 Our arguments below are valid for more general income distributions with compact support as well.
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{ht+ 1}t=0 for each dynasty, given their initial human capital, such that for t = 0, 1,2 . . . . (i) {nt, ct+ 1} solves the individual's maximization problem in period t, and (ii) hi+ 1 = 0(1 - nt)t3h~t. Two features of our model are worth noting at this stage. First, there is no trade between agents, i.e., there are no labor markets or loans markets. Second, there is no intergenerational income mobility: if a member of a particular generation is in the rich group, all of his descendants will be in the rich group. We thus have two dynasties which are distinguished by their initial human capital. We will refer to these as the rich and the poor dynasty. We comment on both these features in Section 4.
3. T h e K u z n e t s curve
Before we explore the relationship between inequality and development, a few preliminaries will be useful. Consider a young agent born in period t whose parents' human capital is h t. The unique interior solution to the agent's optimization problem is given by n~-~ = {0h~}1-'~/3(1 - nt) ~1 - ~ ) - '
(2)
Notice that the time allocated to human capital investment is a decreasing function of the parental human capital when o- > 1; it is independent of the parental human capital when o- = 1; and, when 0 < ~ < 1, it is an increasing function of the parental human capital. In our model, an increase in parental human capital raises the relative price of leisure. For 0 < o-< 1, the substitution effect dominates the income effect and hence, the higher parental human capital increases the time devoted to human capital accumulation; for ~r = 1, the two effects cancel each other and for ~r > 1, the income effect dominates. We henceforth restrict attention to the case 0 < cr < 1 since the evidence at the aggregate level as well as the micro level seem to point in that direction. At the aggregate level, average years of schooling increase with per capita income. At the micro level, using data from Heckmann and Hotz (1986) report that parental background is positively related to returns from additional years of schooling, so children born in high human capital households have more years of schooling. We first establish two useful results. In Proposition 1, we show that short-run increasing returns to scale are necessary for our economy to exhibit the Kuznets curve. In Proposition 2, we show that short-run increasing returns to scale alone is not sufficient to generate the Kuznets curve. Proposition 1, Short-run increasing returns to scale ( ~ + 6 > 1) are n e c e s s a ~ f o r our economy to generate the Kuznets inverted U.
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Proof. To establish this result, we will show that income inequality declines monotonically if/3 +/~ < 1. Let n(h t) be the leisure choice of a young agent with parental human capital h t. Then, the growth rate of human capital for the agent is -
h t- + l ht
=
0{1
-
n(ht)}~h~ -l
.
(3)
Taking logs of both sides of Eq. (3) and differentiating with respect to ln(h t) yields Oln(T,) /3
Oln( h,)
Oln(1 - n ( h t ) ) 01n(h,)
+6-1.
Now, Tt is decreasing in h t if and only if the elasticity 01n(1 - nt)/Oln(h t) < (1 - 6)~ft. Take logs of both sides of Eq. (2) and differentiate with respect to In(h): 01n(1 - n , ) 01n(h,)
6(1 - o-) =
o-(1 - n , )
(4)
+1-/3(1-o")
nt
Denote the elasticity as e(h,). Then,
e(ht)
3 <
o-(1 - n z )
(since0 < o- < 1)
+1-/3
nt
<
1
(for 6 < 1 - / 3 ) .
For /3+ 6 < 1, we have ( 1 - 6 ) / / 3 > 1 and hence, the elasticity is less than ( l - ~ ) / f l . Thus, the growth rate is negatively related to the current level of income. As a consequence, income inequality, measured by the ratio of incomes in the two dynasties, declines monotonically, and the Kuznets curve cannot occur. Short-run increasing returns, in our framework, delivers a convex-concave shape to the dynamic map of the future state as a function of the current state. When the production technology exhibits a very small elasticity of substitution between factors, Galor (1996) shows that the equilibrium dynamical system in an overlapping generations economy has a convex-concave shape; Caselli and Ventura (1996) demonstrate a similar result in a Ramsey economy. Proposition 2. Short-run increasing returns to scale (/3 + 6 > 1) are not sufficient
to generate the Kuznets inverted U.
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Proof. To see this, assume /3 = 1.3 The time allocated to human capital investment is 1
1 -n,=
~_~
{ohm)
(5)
+ 1
The equilibrium law of motion for human capital is I
(Ohm} ht+l
=
l-,r
l+{0h
}
so that the growth rate of human capital is I ht+ i
ht
0~ l-,~
+ 0
(6)
h',
Notice that when 6 < ~ , the growth rate is decreasing in the level of human capital, i.e., growth rate for the rich dynasty is smaller than that for the poor dynasty. Hence, inequality declines over time and the Kuznets curve cannot occur. The natural question to ask at this stage is, is it possible for our economy to exhibit the Kuznets curve? The answer is yes. Before we illustrate the possibility, the following intuition is helpful to understand the mechanism that generates the Kuznets curve. In a given period, the differences between the rich and the poor stem from two sources: (i) the investment in human capital, 1 - n , , and (ii) the level of income, h t. It is easy to see from Eq. (1) that the growth rate of income is decreasing in the level of income, holding 1 - n constant. However, for ~r < 1, the investment in human capital is increasing in the level of income. Thus, the net effect on growth rate depends on which one of these two effects is stronger. It is easy to see from Eq. (4) that the elasticity, e(h), is decreasing in h. That is, for low values of h, the substitution effect is strong so that 1 - n is sensitive to changes in h. The magnitude of this substitution effect depends on o- : low values of o- imply very strong substitution effects. Hence, the growth rate is increasing in the level of income when current income is low and ~ is low. For larger levels of income, the growth rate is decreasing since 1 - n is not very sensitive to changes in h and since 1 - n is bounded above. From propositions 1 and 2 we know that the region in the parameter space to look for the Kuznets curve is /3 + 6 > 1 and 6 > ~r. Again, for simplicity, assume that /3 = 1. When 6 > o-, Eq. (6) implies that the growth rate is non-monotonic in
3 The /3 = 1 case helps us solve for the h u m a n capital i n v e s t m e n t analytically.
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ht+l
-~
I hu
~
hs
ht
b. Growth rate of income
ht+l ht
r
hu
~
hs
ht
Fig. 1. Evolutionof income. the level of income. In Fig. 1, panel a, we illustrate the future income as a function of current income. For low values of current income this function is convex; for large values of current income this function is concave. In panel b, we illustrate the gross growth rate of income as a function of the current income. The growth rate is the highest at h where 1
t3r
=
For current incomes below h u the gross growth rate is less than one so that incomes shrink over time to zero. The same is true for current incomes above h s. In our economy, h u is the unstable steady state and h s is the stable steady state.
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We are now in a position to show that this model is capable of generating the Kuznets inverted U.
Proposition 3. Assume /3+ 6 > 1 and 6 > o'. Let the initial distribution of income satisfy h, < h < h < h. Then this economy exhibits the Kuznets inverted U.
Proof. Since initial incomes lie above h u, incomes in both dynasties grow (see Fig. 1). But since the initial incomes are in the region of increasing returns, income in the rich dynasty grows faster than that in the poor dynasty. Hence, income inequality rises. At some finite time /~, income in the rich dynasty surpasses the level ,~. From that point on, the gross growth rate within the rich dynasty starts to decline to unity and, asymptotically, income within the rich dynasty converges to the steady state level h s. At some time T > 7~. the poor dynasty surpasses the income level h and thereafter, income in the poor dynasty grows faster than in the rich dynasty. Hence, income inequality declines over time. Eventually the income distribution becomes degenerate, as incomes in both dynasties converge to the same steady state level h s. Thus, the economy follows the Kuznets inverted U over time: inequality is low initially, then rises, and finally falls. It is certainly not the case that even when /3 + 6 > 1 and 6 > tr our economy exhibits the Kuznets inverted U for all initial distributions. For instance, suppose that the initial distribution is such that h < h < h. Then incomes in both dynasties grow, but income in the rich dynasty grows more slowly than in the poor dynasty (see Fig. 1). Hence, income inequality declines monotonically. 4 We have so far established that, depending on the initial income distribution, the inequality-development relationship over time may either look like the Kuznets inverted U or look monotonically declining. The time series evidence also seems to suggest both possibilities. The Kuznets curve is consistent with the data for Prussia, the U.K. and the U.S. Dumke (1991), using Prussian wealth data from tax returns, shows that the inequality rises until about 1870 and then falls. Lindert (1986) and Williamson and Lindert (1980) document the inverted U in the U.K. and the U.S. respectively. 5 The declining income inequality result is consistent with the experience of Norway: Soltow (1965) provides evidence that income inequality in Norway declined monotonically over the 19th and 20th century.
4 We have concentrated on the 'interesting' initial distributions where the entire support lies above h u. It is easy to see the evolution of income for other configurations. Suppose h < h u < h. Then income in the rich dynasty increases, and converges to h~; income in the poor dynasty declines to zero. Thus, inequality increases monotonically. For h < h < h u, incomes in both dynasties decline to zero, so the limiting distribution is degenerate. Notice, however, that inequality first rises and then declines. 5 The Kuznets curve also shows up in cross section regressions for the less developed countries after World War II even though not all of the countries in the sample exhibit the Kuznets curve in the time series data (see Fields and Jakubson (1993)).
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3.1. Kuznets curve in years of schooling Our model is also capable of generating the findings of Ram (1990) on schooling for some initial human capital distributions. In a cross-sectional sample o f 100 countries, he finds that there is an inverted U relationship between inequality in years of schooling and average years of schooling. 6 Since the time input to learning in our model is 1 - n , we need to show that our model can generate an inverted U relationship between inequality in 1 - n and average 1 - n. As an example, consider the human capital investment decision Eq. (5). W e can show that 1 - n is convex in h for sufficiently small h and that 1 - n is concave in h for sufficiently large h, provided 6 > o-/(1 - o-). To see this, rewrite Eq. (5) as l-~r
~(1 - ~r)
0~h 1 --n
=
1-o"
1+0
~ h
~5(I - o ' )
"
'~
Define 0' ~ 0 (1-~')/'~ and o~ - 6(1 - o - ) / o - , so 0,h,~ 1 --n
=
-
-
1 + O'h ~"
Thus, 02(1 - n ) 0h2
oth~-2(ol-l-O'hC'(l+ot)} {1 + O'ha} 3
Since cr < 1, we know c~ > 0. A necessary condition for the above derivative to be positive is a > 1, i.e., 6 > ~ r / ( 1 - o'). It is easy to see that the derivative is positive for small h and negative for large h. To obtain the Kuznets curve in 1 - n, we have to show that 1 - n takes on a c o n v e x - c o n c a v e shape in the region where h is growing; only then will we know that average 1 - n is increasing. W e already know that 1 - n is increasing in h, i.e., the rich dynasty chooses a higher learning time than the poor dynasty. In the convex region, learning time grows at a higher rate for the rich dynasty than for the poor dynasty. Thus, both dispersion and mean learning time grows in the convex region. In the region where 1 - n is concave in h, dispersion in learning time falls. This generates the Kuznets curve. What remains to be verified is whether 1 - n is convex in h in part of the region where h is growing. W e verify this numerically. W e set the parameters for our economy as follows: 0 = 9, o- = 0.45, and 6 = 0.6. The initial distribution o f
6 Using time series data on education of the labor force in the U.S., he also finds an inverted U relationship in schooling between 1952 and 1981.
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Table 1 Learning time Mean × 1O- 3
Standard deviation × 1O- 5
1.06 1.58 2.70 2.70 5.51 47.98 208.43 675.88 944.75 987.22 987.22
0.02 0.05 0.10 0.10 0.28 4.16 19.48 30.54 6.10 0.94 0.94
income is assumed to be lognormal; log of income is normal with mean - 13 and standard deviation 0.0003. We draw a random sample of 500 households from this distribution and generate the time series implied by our model. The results are reported in Table 1 below. It is clear that as the mean learning time increases the standard deviation of the learning time increases initially but declines later, i.e., the learning time displays an inverted U. Note that while the above initial distribution of human capital generates the Kuznets curve in learning time, other initial distributions can generate monotonic convergence.
4. Concluding remarks We have presented a simple model of human capital accumulation which can deliver the Kuznets curve as an equilibrium outcome. One crucial element for obtaining the Kuznets curve in our model is that the learning technology exhibits short run increasing returns to scale. We show that the short run increasing returns to scale is necessary but not sufficient to obtain the Kuznets inverted U. Whether we do indeed obtain the Kuznets curve in the presence of short run increasing returns to scale depends upon the initial distribution of human capital. We have kept the model here very simple to highlight the role of short run increasing returns to scale in generating the Kuznets curve. In our model, there is only one avenue for savings and that is through human capital investment. It remains an open question, if the Kuznets curve can be generated in a version of our model which permits accumulation of both physical and human capital. While Galor and Tsiddon (1992) have both physical and human capital in their model, the physical capital stock is determined by the world interest rate which is exogenous and assumed constant over time.
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Another simple feature of our model is that average income converges to a steady state level. This convergence is inconsistent with one of the development facts, namely that average income is growing persistently. One may wonder whether the Kuznets curve can arise when the economy exhibits persistent growth. 7 In the Appendix A, we extend our model to include exogenous technological progress and show that our results on the Kuznets curve are robust to this extension. Our model also has the feature that there is no mobility of income across generations, i.e., the relative position of a dynasty does not change over time. One way to introduce income mobility is to let 0 be different across agents born in each period. Suppose we interpret 0 as innate ability and let each agent get a random draw from a given ability distribution. Agents with high ability will allocate more time to learning than those with low ability. This has a tendency to widen the income inequality initially. 8 In the long run, however, diminishing returns to human capital accumulation will induce convergence to a time invariant distribution. There are no markets in our economy. We could relax this assumption and allow the old in each period to sell their labor to a firm in a competitive labor market. If all types of labor are perfect substitutes, then the wage income when old is a linear function of the human capital. In this case all of our results survive. The model, in its current form, has no role for loans markets since the key determinant of wealth here is time input to learning. If the agents cared about consumption while young instead of leisure, some of the young may borrow for current consumption. Since the agents live only for two periods, the mutually beneficial trades in the loans market must be between members of the young generation. To this end, it may be interesting to modify our production technology to one where both skilled and unskilled labor are factors of production, so the young allocate their time between earning (unskilled) wages and accumulating human capital. We leave this for future research.
Acknowledgements We would like to thank two anonymous referees for their valuable suggestions.
7 Tamura (1996) obtains the Kuznets curve in a model where both growth and fertility are endogenous. 8 The degree of intergenerational income mobility has important ramifications for the interpretation of income inequality. As noted by Kuznets (1966), " I f the groups of family clusters originally in the upper brackets have moved, with the passage of a generation, to the bottom of the array and have been replaced by groups of clusters whose immediate forebears began at the bottom, and if the identity of the groups in the size distribution of income has changed markedly, the differences revealed by the latter have no cumulative impact; there is no persistent economic class consciousness; and there is little meaning to the question whether the poor are getting poorer and the rich richer."
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Appendix A. Exogenous technological progress Let the learning technology be h~+ i --- 0t(1 - nt)h~t, where Or+ l = (1 + / ~ ) 0 t, with 00 > 0 and /z > 0. Similar to Section 3, we can derive the e q u i l i b r i u m law of m o t i o n for h u m a n capital as 1
h,+l =
1-c~
(7)
and the growth rate as 1
ht+ I
Ot'~ 1
ht h)--q-Or
o"
hit -~
Again, note that the growth rate is n o n - m o n o t o n i c in current i n c o m e if ~ > o- and m o n o t o n i c a l l y declining in current i n c o m e if ~ > o-.
hl
huO
ho
Fig. 2. Evolution of income under exogenous technological progress.
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Proposition. Assume 6 > or. Let huo be the unstable fixed point o f Eq. (7) assuming 0_, = 0 o. Let the initial distribution satisfy either (i) h < huo < h or (ii) huo < h < h. Then this economy exhibits the Kuznets inverted U. P r o o f . T o see this, c o n s i d e r Fig, 2. U n d e r condition (i), i n c o m e in the rich dynasty g r o w s and i n c o m e in the p o o r dynasty shrinks. Thus, initially, i n c o m e inequality widens. E x o g e n o u s t e c h n o l o g i c a l progress implies the c u r v e g o v e r n e d by Eq. (7) rotates upward. In finite time, the slope o f the c u r v e at zero e x c e e d s unity. T h e n i n c o m e in the p o o r dynasty starts to g r o w and, eventually, this g r o w t h rate exceeds the one in the rich dynasty, so the i n c o m e inequality declines. Thus, the e c o n o m y exhibits the Kuznets inverted U. The p r o o f under c o n d i t i o n (ii) is essentially the s a m e as in Proposition 3.
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Ram, R., 1990. Educational expansion and schooling inequality: international evidence and some implications. Rev. Econ. Stat. 72, 266-274. Rauch, J.E., 1993. Economic development, urban underemployment, and income inequality. Can. J. Econ. 26, 901-918. Skiba, A.K., 1978. Optimal growth with a convex-concave production fnnction. Econometrica 46, 527-539. Soltow, L., 1965. Toward Income Inequality in Norway, Madison, University of Wisconsin Press. Tamura, R., 1991. Income convergence in an endogenous growth model. J. Polit. Econ. 99, 522-540. Tamura, R., 1992. Efficient equilibrium convergence: heterogeneity and growth. J. Econ. Theory' 58, 355-376. Tamura, R., 1996. From decay to growth: a demographic transition to economic growth. J. Econ. Dynamics Control 20, 1237-1261. Williamson, J.G., Lindert, P.H., 1980. American Inequality: A Macroeconomic History, Academic Press, New York.
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Development Journal of DevelopmentEconomics Vol. 55 (1998) 353-367
ELSEVIER
ECONOMICS
Increasing returns, human capital, and the Kuznets curve Gerhard Glomm a,*, B. Ravikumar
b
a Michigan State University, MI, USA b University of Iowa, 10, USA
Received 31 May 1996; accepted 31 December 1996
Abstract We present a simple model of human capital accumulation which generates the Kuznets curve as an equilibrium outcome. The central ingredient that helps generate the Kuznets curve in the model is what we call short-run increasing returns to scale in the learning technology. The learning technology exhibits increasing returns to scale, but only in the short run, since one of the factors of production, time, is bounded above by the endowment. We show that short-run increasing returns to scale is necessary to obtain the Kuznets curve, but not sufficient. © 1998 Elsevier Science B.V. Keywords: Human capital; Increasingreturns; Kuznets curve
1. Introduction In 1955 Kuznets published his first of a by now famous sequence of papers on the relationship between per capita income and income distribution. He suggested that income inequality exhibited a secular decline for a few North American and European countries. In a paper in 1963 Kuznets showed that income inequality might actually follow an inverted U pattern along the development path: at first income inequality rises as economies grow, but eventually income inequality declines. This hypothesis that income inequality might follow an inverted U pattern over the development process sparked a wave of investigation both in the
* Corresponding author. 0304-3878/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0304-3 87 8(98)00040-6
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Development Economics literature and in Economic History. Researchers in both fields documented with more and more, and over time with presumably better data sets, if the Kuznets hypothesis of an inverted U relationship between income inequality and economic development could be confirmed or rejected. For surveys of this literature see Adelman and Robinson (1989) and Brenner et al. (1991). In recent years, there have been a few dynamic general equilibrium models in which inequality affects economic growth and the evolution of income inequality is endogenous. Examples include Tamura (1991, 1992) and Glomm and Ravikumar (1994) where income inequality declines over time due to diminishing returns to capital investment and Alesina and Rodrik (1994) where income inequality stays constant over time. Other general equilibrium models deliver the Kuznets inverted U as an equilibrium relationship. In Greenwood and Jovanovic (1990), there is a fixed cost to joining a financial network. The financial network guarantees higher returns to investment relative to a backyard technology. Individuals with less wealth expect to join the network later than those with more wealth. Greenwood Jovanovic show that savings rate, for those outside the network, increases with the level of wealth and, hence, there is an initial increase in inequality. Eventually, everyone joins the network and the savings rate is independent of wealth, so everyone grows at the same rate. Aghion and Bolton (1993) and Khan (1993) provide alternative models of financial intermediation with credit market imperfections that generate the Kuznets curve. Galor and Tsiddon (1992), Perotti (1993), and Tamura (1996) present models of human capital accumulation which generate the inverted U relationship. In Banerjee and Newman (1993) the Kuznets curve arises in a model of occupational choice. Models of migration from agriculture to manufacturing such as Glomm (1992) and Ranch (1993) also give rise to the Kuznets curve. In this paper we present a simple model of human capital accumulation which generates the Kuznets inverted U relationship between income inequality and development. We deliberately impose simple functional forms on preferences and technology so that we can expose the fundamental economic factors at work in generating the inverted U. In the model, which is described in Section 2, human capital is accumulated by overlapping generations of individuals. All individuals value leisure when young and consumption when old; the utility function is CES. Consumption when old is a function of human capital. Human capital is accumulated using a Cobb-Douglas learning technology with time and parental human capital as inputs. Unlike many of the papers that generate the Kuznets curve, our model neither has any exogenous thresholds nor relies on income redistribution through political institutions. The central ingredient that helps generate the Kuznets curve in our model is what we call short-run increasing returns to scale in the learning technology. Each factor in the learning technology exhibits weakly diminishing returns, but since one of the factors is bounded above by the time endowment, we have increasing returns only in the short-run and not in the long-run.
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In a closely related paper Galor and Tsiddon (1992) obtain the Kuznets curve in an overlapping generations model where technical progress depends upon intragenerational as well as intergenerational externalities. At low levels of technology, their economy has multiple steady states, one unstable and two stable. Around the unstable steady state, the poor experience a decline in wealth whereas the rich experience an increase in wealth, so wealth inequality increases initially. As technology advances the unstable steady state is eliminated and the economy exhibits a decline in wealth inequality as incomes converge to the unique stable steady state. In their economy and ours, the equilibrium law of motion for human capital exhibits short run increasing returns to scale. Our model, however, does not rely on intragenerational externalities or technical progress to deliver the Kuznets curve. Short run increasing returns to scale are also present in the other models (mentioned above) generating the Kuznets curve. In Greenwood and Jovanovic (1990), Aghion and Bolton (1993), and Khan (1993), the short run increasing returns to scale arise because of fixed investment required to access better investment opportunities. In Perotti (1993), the short run increasing returns to scale enter through exogenous thresholds in the learning technology. In Tamura (1996), the rate of return to investment in skills initially rises with the level of skill, but in the long run it is constant. In the occupational choice models of Banerjee and Newman (1993) and in the migration models of Glomm (1992) and Ranch (1993), the equilibrium mapping from present wealth to future wealth displays short run increasing to scale due to the discrete nature of occupational or sectoral choice. We show in Section 3 that the short-run increasing returns to scale is necessary to obtain the Kuznets curve. Due to the short-run increasing returns, the equilibrium relationship between future and current human capital in our model is convex for low levels of current human capital and concave for high levels of current human capital, i Given this shape we show that the relationship between per capita income and income inequality will follow the Kuznets curve for some initial income distributions. In the Appendix, we extend our results to economies that exhibit sustained growth from exogenous technological progress.
2. The Model The economy is populated by overlapping generations where individuals live for two periods and where population is constant over time. Each generation consists of a large number of agents. Population size is normalized to unity. At
1 An early example of a model with a convex-concave production technology is Skiba (1978). His focus, however, is not on the evolution of income distribution but on optimal growth.
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time t --- 0, there is an initial generation of old agents in which the human capital endowment of the i th member is hi0. Individuals differ only in the stock of human capital of their parents. We assume a simple kind of heterogeneity: initial old belong to one of two types, rich or poor. The initial old of the poor type are each endowed with h, while the rich type are endowed with h. In this case inequality can be measure-d simply by the ratio of the incomes of the two types. 2 All agents born at time t = 0, 1, 2 . . . . have identical preferences over leisure when young, n t, and consumption when old, ct+ r Their utility function is given by i - o" .~_ l-o"
nt
ct+ 1
, o r > O.
1--o"
We will interpret tr = 1 as logarithmic utility; 1 / t r is the elasticity of substitution between consumption and leisure. Consumption when old is equal to the stock of human capital that the agent possesses when old. That is, there is a technology which transforms human capital one for one into output. (Hence, we use the terms 'human capital' and 'income' interchangeably throughout the paper.) Each agent consumes his entire output when old. Each individual is endowed with one unit of time while young. He spends part of the endowment toward leisure and the rest toward human capital accumulation. His human capital when old depends on time, 1 - n,, and his parents' human capital, h t, i.e., h,+, = Oh~t(l - nt)~, 0 > 0, 6 e ( 0 , 1 ) , fie(0,1].
(1)
Notice that in Eq. (1) the returns to investing in human capital depend positively on the human capital of ones own parent. There is, thus, an intergenerational externality. When fl + ~ > 1, the learning technology exhibits increasing returns. Since 1 - n t is bounded above by one we can think of the case fl + ~ > 1 as one of short-run increasing returns. Each individual at time t chooses n t and c1+ 1 to solve the problem l-o-+
max
nt
l-o"
ct+ 1
1--o"
s.t. c,+ l = 0(1 - nt)t3h~. An equilibrium for this economy is a set of sequences {nt}t~=o, {ct+ l}t~__0 , and
2 Our arguments below are valid for more general income distributions with compact support as well.
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{ht+ 1}t=0 for each dynasty, given their initial human capital, such that for t = 0, 1,2 . . . . (i) {nt, ct+ 1} solves the individual's maximization problem in period t, and (ii) hi+ 1 = 0(1 - nt)t3h~t. Two features of our model are worth noting at this stage. First, there is no trade between agents, i.e., there are no labor markets or loans markets. Second, there is no intergenerational income mobility: if a member of a particular generation is in the rich group, all of his descendants will be in the rich group. We thus have two dynasties which are distinguished by their initial human capital. We will refer to these as the rich and the poor dynasty. We comment on both these features in Section 4.
3. T h e K u z n e t s curve
Before we explore the relationship between inequality and development, a few preliminaries will be useful. Consider a young agent born in period t whose parents' human capital is h t. The unique interior solution to the agent's optimization problem is given by n~-~ = {0h~}1-'~/3(1 - nt) ~1 - ~ ) - '
(2)
Notice that the time allocated to human capital investment is a decreasing function of the parental human capital when o- > 1; it is independent of the parental human capital when o- = 1; and, when 0 < ~ < 1, it is an increasing function of the parental human capital. In our model, an increase in parental human capital raises the relative price of leisure. For 0 < o-< 1, the substitution effect dominates the income effect and hence, the higher parental human capital increases the time devoted to human capital accumulation; for ~r = 1, the two effects cancel each other and for ~r > 1, the income effect dominates. We henceforth restrict attention to the case 0 < cr < 1 since the evidence at the aggregate level as well as the micro level seem to point in that direction. At the aggregate level, average years of schooling increase with per capita income. At the micro level, using data from Heckmann and Hotz (1986) report that parental background is positively related to returns from additional years of schooling, so children born in high human capital households have more years of schooling. We first establish two useful results. In Proposition 1, we show that short-run increasing returns to scale are necessary for our economy to exhibit the Kuznets curve. In Proposition 2, we show that short-run increasing returns to scale alone is not sufficient to generate the Kuznets curve. Proposition 1, Short-run increasing returns to scale ( ~ + 6 > 1) are n e c e s s a ~ f o r our economy to generate the Kuznets inverted U.
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Proof. To establish this result, we will show that income inequality declines monotonically if/3 +/~ < 1. Let n(h t) be the leisure choice of a young agent with parental human capital h t. Then, the growth rate of human capital for the agent is -
h t- + l ht
=
0{1
-
n(ht)}~h~ -l
.
(3)
Taking logs of both sides of Eq. (3) and differentiating with respect to ln(h t) yields Oln(T,) /3
Oln( h,)
Oln(1 - n ( h t ) ) 01n(h,)
+6-1.
Now, Tt is decreasing in h t if and only if the elasticity 01n(1 - nt)/Oln(h t) < (1 - 6)~ft. Take logs of both sides of Eq. (2) and differentiate with respect to In(h): 01n(1 - n , ) 01n(h,)
6(1 - o-) =
o-(1 - n , )
(4)
+1-/3(1-o")
nt
Denote the elasticity as e(h,). Then,
e(ht)
3 <
o-(1 - n z )
(since0 < o- < 1)
+1-/3
nt
<
1
(for 6 < 1 - / 3 ) .
For /3+ 6 < 1, we have ( 1 - 6 ) / / 3 > 1 and hence, the elasticity is less than ( l - ~ ) / f l . Thus, the growth rate is negatively related to the current level of income. As a consequence, income inequality, measured by the ratio of incomes in the two dynasties, declines monotonically, and the Kuznets curve cannot occur. Short-run increasing returns, in our framework, delivers a convex-concave shape to the dynamic map of the future state as a function of the current state. When the production technology exhibits a very small elasticity of substitution between factors, Galor (1996) shows that the equilibrium dynamical system in an overlapping generations economy has a convex-concave shape; Caselli and Ventura (1996) demonstrate a similar result in a Ramsey economy. Proposition 2. Short-run increasing returns to scale (/3 + 6 > 1) are not sufficient
to generate the Kuznets inverted U.
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Proof. To see this, assume /3 = 1.3 The time allocated to human capital investment is 1
1 -n,=
~_~
{ohm)
(5)
+ 1
The equilibrium law of motion for human capital is I
(Ohm} ht+l
=
l-,r
l+{0h
}
so that the growth rate of human capital is I ht+ i
ht
0~ l-,~
+ 0
(6)
h',
Notice that when 6 < ~ , the growth rate is decreasing in the level of human capital, i.e., growth rate for the rich dynasty is smaller than that for the poor dynasty. Hence, inequality declines over time and the Kuznets curve cannot occur. The natural question to ask at this stage is, is it possible for our economy to exhibit the Kuznets curve? The answer is yes. Before we illustrate the possibility, the following intuition is helpful to understand the mechanism that generates the Kuznets curve. In a given period, the differences between the rich and the poor stem from two sources: (i) the investment in human capital, 1 - n , , and (ii) the level of income, h t. It is easy to see from Eq. (1) that the growth rate of income is decreasing in the level of income, holding 1 - n constant. However, for ~r < 1, the investment in human capital is increasing in the level of income. Thus, the net effect on growth rate depends on which one of these two effects is stronger. It is easy to see from Eq. (4) that the elasticity, e(h), is decreasing in h. That is, for low values of h, the substitution effect is strong so that 1 - n is sensitive to changes in h. The magnitude of this substitution effect depends on o- : low values of o- imply very strong substitution effects. Hence, the growth rate is increasing in the level of income when current income is low and ~ is low. For larger levels of income, the growth rate is decreasing since 1 - n is not very sensitive to changes in h and since 1 - n is bounded above. From propositions 1 and 2 we know that the region in the parameter space to look for the Kuznets curve is /3 + 6 > 1 and 6 > ~r. Again, for simplicity, assume that /3 = 1. When 6 > o-, Eq. (6) implies that the growth rate is non-monotonic in
3 The /3 = 1 case helps us solve for the h u m a n capital i n v e s t m e n t analytically.
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ht+l
-~
I hu
~
hs
ht
b. Growth rate of income
ht+l ht
r
hu
~
hs
ht
Fig. 1. Evolutionof income. the level of income. In Fig. 1, panel a, we illustrate the future income as a function of current income. For low values of current income this function is convex; for large values of current income this function is concave. In panel b, we illustrate the gross growth rate of income as a function of the current income. The growth rate is the highest at h where 1
t3r
=
For current incomes below h u the gross growth rate is less than one so that incomes shrink over time to zero. The same is true for current incomes above h s. In our economy, h u is the unstable steady state and h s is the stable steady state.
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We are now in a position to show that this model is capable of generating the Kuznets inverted U.
Proposition 3. Assume /3+ 6 > 1 and 6 > o'. Let the initial distribution of income satisfy h, < h < h < h. Then this economy exhibits the Kuznets inverted U.
Proof. Since initial incomes lie above h u, incomes in both dynasties grow (see Fig. 1). But since the initial incomes are in the region of increasing returns, income in the rich dynasty grows faster than that in the poor dynasty. Hence, income inequality rises. At some finite time /~, income in the rich dynasty surpasses the level ,~. From that point on, the gross growth rate within the rich dynasty starts to decline to unity and, asymptotically, income within the rich dynasty converges to the steady state level h s. At some time T > 7~. the poor dynasty surpasses the income level h and thereafter, income in the poor dynasty grows faster than in the rich dynasty. Hence, income inequality declines over time. Eventually the income distribution becomes degenerate, as incomes in both dynasties converge to the same steady state level h s. Thus, the economy follows the Kuznets inverted U over time: inequality is low initially, then rises, and finally falls. It is certainly not the case that even when /3 + 6 > 1 and 6 > tr our economy exhibits the Kuznets inverted U for all initial distributions. For instance, suppose that the initial distribution is such that h < h < h. Then incomes in both dynasties grow, but income in the rich dynasty grows more slowly than in the poor dynasty (see Fig. 1). Hence, income inequality declines monotonically. 4 We have so far established that, depending on the initial income distribution, the inequality-development relationship over time may either look like the Kuznets inverted U or look monotonically declining. The time series evidence also seems to suggest both possibilities. The Kuznets curve is consistent with the data for Prussia, the U.K. and the U.S. Dumke (1991), using Prussian wealth data from tax returns, shows that the inequality rises until about 1870 and then falls. Lindert (1986) and Williamson and Lindert (1980) document the inverted U in the U.K. and the U.S. respectively. 5 The declining income inequality result is consistent with the experience of Norway: Soltow (1965) provides evidence that income inequality in Norway declined monotonically over the 19th and 20th century.
4 We have concentrated on the 'interesting' initial distributions where the entire support lies above h u. It is easy to see the evolution of income for other configurations. Suppose h < h u < h. Then income in the rich dynasty increases, and converges to h~; income in the poor dynasty declines to zero. Thus, inequality increases monotonically. For h < h < h u, incomes in both dynasties decline to zero, so the limiting distribution is degenerate. Notice, however, that inequality first rises and then declines. 5 The Kuznets curve also shows up in cross section regressions for the less developed countries after World War II even though not all of the countries in the sample exhibit the Kuznets curve in the time series data (see Fields and Jakubson (1993)).
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3.1. Kuznets curve in years of schooling Our model is also capable of generating the findings of Ram (1990) on schooling for some initial human capital distributions. In a cross-sectional sample o f 100 countries, he finds that there is an inverted U relationship between inequality in years of schooling and average years of schooling. 6 Since the time input to learning in our model is 1 - n , we need to show that our model can generate an inverted U relationship between inequality in 1 - n and average 1 - n. As an example, consider the human capital investment decision Eq. (5). W e can show that 1 - n is convex in h for sufficiently small h and that 1 - n is concave in h for sufficiently large h, provided 6 > o-/(1 - o-). To see this, rewrite Eq. (5) as l-~r
~(1 - ~r)
0~h 1 --n
=
1-o"
1+0
~ h
~5(I - o ' )
"
'~
Define 0' ~ 0 (1-~')/'~ and o~ - 6(1 - o - ) / o - , so 0,h,~ 1 --n
=
-
-
1 + O'h ~"
Thus, 02(1 - n ) 0h2
oth~-2(ol-l-O'hC'(l+ot)} {1 + O'ha} 3
Since cr < 1, we know c~ > 0. A necessary condition for the above derivative to be positive is a > 1, i.e., 6 > ~ r / ( 1 - o'). It is easy to see that the derivative is positive for small h and negative for large h. To obtain the Kuznets curve in 1 - n, we have to show that 1 - n takes on a c o n v e x - c o n c a v e shape in the region where h is growing; only then will we know that average 1 - n is increasing. W e already know that 1 - n is increasing in h, i.e., the rich dynasty chooses a higher learning time than the poor dynasty. In the convex region, learning time grows at a higher rate for the rich dynasty than for the poor dynasty. Thus, both dispersion and mean learning time grows in the convex region. In the region where 1 - n is concave in h, dispersion in learning time falls. This generates the Kuznets curve. What remains to be verified is whether 1 - n is convex in h in part of the region where h is growing. W e verify this numerically. W e set the parameters for our economy as follows: 0 = 9, o- = 0.45, and 6 = 0.6. The initial distribution o f
6 Using time series data on education of the labor force in the U.S., he also finds an inverted U relationship in schooling between 1952 and 1981.
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Table 1 Learning time Mean × 1O- 3
Standard deviation × 1O- 5
1.06 1.58 2.70 2.70 5.51 47.98 208.43 675.88 944.75 987.22 987.22
0.02 0.05 0.10 0.10 0.28 4.16 19.48 30.54 6.10 0.94 0.94
income is assumed to be lognormal; log of income is normal with mean - 13 and standard deviation 0.0003. We draw a random sample of 500 households from this distribution and generate the time series implied by our model. The results are reported in Table 1 below. It is clear that as the mean learning time increases the standard deviation of the learning time increases initially but declines later, i.e., the learning time displays an inverted U. Note that while the above initial distribution of human capital generates the Kuznets curve in learning time, other initial distributions can generate monotonic convergence.
4. Concluding remarks We have presented a simple model of human capital accumulation which can deliver the Kuznets curve as an equilibrium outcome. One crucial element for obtaining the Kuznets curve in our model is that the learning technology exhibits short run increasing returns to scale. We show that the short run increasing returns to scale is necessary but not sufficient to obtain the Kuznets inverted U. Whether we do indeed obtain the Kuznets curve in the presence of short run increasing returns to scale depends upon the initial distribution of human capital. We have kept the model here very simple to highlight the role of short run increasing returns to scale in generating the Kuznets curve. In our model, there is only one avenue for savings and that is through human capital investment. It remains an open question, if the Kuznets curve can be generated in a version of our model which permits accumulation of both physical and human capital. While Galor and Tsiddon (1992) have both physical and human capital in their model, the physical capital stock is determined by the world interest rate which is exogenous and assumed constant over time.
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Another simple feature of our model is that average income converges to a steady state level. This convergence is inconsistent with one of the development facts, namely that average income is growing persistently. One may wonder whether the Kuznets curve can arise when the economy exhibits persistent growth. 7 In the Appendix A, we extend our model to include exogenous technological progress and show that our results on the Kuznets curve are robust to this extension. Our model also has the feature that there is no mobility of income across generations, i.e., the relative position of a dynasty does not change over time. One way to introduce income mobility is to let 0 be different across agents born in each period. Suppose we interpret 0 as innate ability and let each agent get a random draw from a given ability distribution. Agents with high ability will allocate more time to learning than those with low ability. This has a tendency to widen the income inequality initially. 8 In the long run, however, diminishing returns to human capital accumulation will induce convergence to a time invariant distribution. There are no markets in our economy. We could relax this assumption and allow the old in each period to sell their labor to a firm in a competitive labor market. If all types of labor are perfect substitutes, then the wage income when old is a linear function of the human capital. In this case all of our results survive. The model, in its current form, has no role for loans markets since the key determinant of wealth here is time input to learning. If the agents cared about consumption while young instead of leisure, some of the young may borrow for current consumption. Since the agents live only for two periods, the mutually beneficial trades in the loans market must be between members of the young generation. To this end, it may be interesting to modify our production technology to one where both skilled and unskilled labor are factors of production, so the young allocate their time between earning (unskilled) wages and accumulating human capital. We leave this for future research.
Acknowledgements We would like to thank two anonymous referees for their valuable suggestions.
7 Tamura (1996) obtains the Kuznets curve in a model where both growth and fertility are endogenous. 8 The degree of intergenerational income mobility has important ramifications for the interpretation of income inequality. As noted by Kuznets (1966), " I f the groups of family clusters originally in the upper brackets have moved, with the passage of a generation, to the bottom of the array and have been replaced by groups of clusters whose immediate forebears began at the bottom, and if the identity of the groups in the size distribution of income has changed markedly, the differences revealed by the latter have no cumulative impact; there is no persistent economic class consciousness; and there is little meaning to the question whether the poor are getting poorer and the rich richer."
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Appendix A. Exogenous technological progress Let the learning technology be h~+ i --- 0t(1 - nt)h~t, where Or+ l = (1 + / ~ ) 0 t, with 00 > 0 and /z > 0. Similar to Section 3, we can derive the e q u i l i b r i u m law of m o t i o n for h u m a n capital as 1
h,+l =
1-c~
(7)
and the growth rate as 1
ht+ I
Ot'~ 1
ht h)--q-Or
o"
hit -~
Again, note that the growth rate is n o n - m o n o t o n i c in current i n c o m e if ~ > o- and m o n o t o n i c a l l y declining in current i n c o m e if ~ > o-.
hl
huO
ho
Fig. 2. Evolution of income under exogenous technological progress.
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Proposition. Assume 6 > or. Let huo be the unstable fixed point o f Eq. (7) assuming 0_, = 0 o. Let the initial distribution satisfy either (i) h < huo < h or (ii) huo < h < h. Then this economy exhibits the Kuznets inverted U. P r o o f . T o see this, c o n s i d e r Fig, 2. U n d e r condition (i), i n c o m e in the rich dynasty g r o w s and i n c o m e in the p o o r dynasty shrinks. Thus, initially, i n c o m e inequality widens. E x o g e n o u s t e c h n o l o g i c a l progress implies the c u r v e g o v e r n e d by Eq. (7) rotates upward. In finite time, the slope o f the c u r v e at zero e x c e e d s unity. T h e n i n c o m e in the p o o r dynasty starts to g r o w and, eventually, this g r o w t h rate exceeds the one in the rich dynasty, so the i n c o m e inequality declines. Thus, the e c o n o m y exhibits the Kuznets inverted U. The p r o o f under c o n d i t i o n (ii) is essentially the s a m e as in Proposition 3.
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