Public education and income inequality

European Journal of Political Economy Vol. 19 (2003) 289 – 300 www.elsevier.com/locate/econbase Public education and income inequality Gerhard Glomm a,*, B. Ravikumar b a Department of Economics, Indiana University, Wylie Hall, Room 105, Bloomington, IN 47405-6620, USA b Department of Economics, University of Iowa, W 210 John Pappajohn Business Building, Iowa City, IA 52242-0000, USA Received 10 November 2000; received in revised form 12 August 2002; accepted 29 September 2002 Abstract This paper examines the evolution of inequality in an overlapping generations model where each individual’s human capital investment depends on quality of schools. We consider an education regime where the quality of schools is a publicly provided input financed by an income tax. We show that the income gap between the rich and the poor may widen even when the quality of public education is the same across all individuals. Thus, in the short run, public education may not be the great equalizer as intended by its proponents, though it is in the long run. We also show that the effect of taxes on inequality is ambiguous. D 2003 Elsevier Science B.V. All rights reserved. Keywords: Public education; Income inequality; Income tax 1. Introduction In these days, it is doubtful that any child may reasonably be expected to succeed in life if he is denied the opportunity of an education. Such an opportunity where the state has undertaken to provide it, is a right which must be made available to all on equal terms. U.S. Supreme Court, Brown vs. Board of Education, 19541. In most industrialized countries, public education has been the dominant mode of providing educational services for the last century. In the U.S., for instance, over the last 100 years, the fraction of students at the elementary and secondary level who attend public * Corresponding author. E-mail address: [email protected] (G. Glomm). 1 Reprinted in Kurland (1968). 0176-2680/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0176-2680(02)00178-7 290 G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 schools has been around 90%. For OECD countries, public school enrollment typically exceeds 70%. As early as 1848, Horace Mann argued that public provision of education is ‘‘beyond all other devices of human origin, the great equalizer of the condition of men. . .’’ (Cremin, 1957, p. 87). Coons et al. (1970) argue that the purpose of public schools is to provide equality of opportunity and, therefore, unequal access to education ought to be eliminated. The essential idea behind their argument is that, under equal access to public education, some inputs to the learning technology are constant across all children, so the common input would make incomes converge.2 The main purpose of our paper is to study the evolution of income inequality in a model with public education and to evaluate whether equal access to public education yields income convergence. A secondary goal is to examine how such an economy behaves over time under different funding levels for public education. In Section 2, we construct an overlapping generations model with heterogeneous individuals. The basic framework is similar to that in Glomm and Ravikumar (1992). All individuals live for two periods. Their preferences over leisure in youth and consumption in old age are described by a CES utility function. Individuals within a generation are differentiated by the stock of human capital of their parents. This is the only source of heterogeneity in our model. Human capital of each individual depends on time allocated to learning, quality of schools and the stock of human capital of the individual’s parents. Quality of schools is a publicly provided input financed by tax revenues from a uniform tax on income. The publicly provided input is common across all agents. Our model, by construction, has the forces suggested by proponents of public education. All agents have equal access to the public expenditures on education. All agents use the same learning technology. The quantity of the publicly provided input to the learning technology is the same for all agents. That is, we have eliminated, by assumption, the concern in Bowles (1978) or Wälde (2000). Thus, the common publicly provided input should potentially yield income convergence. Yet, we show for reasonable parameters that exactly the opposite occurs. Furthermore, we demonstrate the possibility of adverse distributional consequences without appealing to elitism. Our model is also different from Glomm and Ravikumar (1992), where we did not examine how an economy with public education behaves over time under different funding levels for public education. Rather, we compared public and private funding regimes, and endogenized public policy on education. Thus, there was exactly one resulting public policy. That model, therefore, cannot deliver comparative dynamic statements on how different public policies influence the evolution of income distribution. Since the model assumed Cobb – Douglas preferences and technology, time allocated to learning was constant over time and independent of parental human capital and the level of funding for public education. In this paper, time allocated to learning is a nontrivial 2 Bowles (1978) suggests that the structure educational policy contributes to economic inequality because the resources are allocated disproportionately to the rich. More recently, in Wälde (2000), the degree of elitism in educational policy (measured by public spending per student in tertiary relative to elementary and secondary education) provides incentives to develop technologies that allow skilled labor to replace unskilled labor and, hence, generates higher income inequality. G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 291 function of parental human capital and public expenditures on education, and hence the implications for future income distributions are much more far reaching than in our 1992 paper. Moreover, the predictions concerning time allocations in the present paper are consistent with microevidence. In Section 3, we illustrate the possibility that, even under equal access to public education, the income gap between the rich and the poor may widen in the short run, even though public provision of education decreases inequality in the long run. (The short run in this framework may last for a few generations.) In our model, the evolution of inequality depends on the elasticity of substitution between consumption and leisure, and the elasticity of parental human capital in the learning technology. Under certain conditions on these two parameters, incomes below a critical level exhibit divergence, while those above the critical level exhibit convergence. The limiting distribution depends upon the evolution of critical income. In a growing economy, the critical income declines over time. Thus, income inequality increases for some periods, but eventually declines. The increase in income inequality occurs, although quality of public education is the same across all agents. We then numerically examine the effect of initial income distribution on the evolution of income inequality. We find that higher initial per capita income reduces future inequality and also reduces the number of periods over which the income gap widens. Moreover, higher initial income inequality increases future inequality and the income gap widens for more periods. The effect of taxes on the evolution of income inequality is ambiguous. For sufficiently small tax rates, an increase in the tax rate lowers the income inequality. This result is reversed for high tax rates. In our model, the tax rates affect the quality of public education directly. However, there is also an indirect effect because changes in tax rates affect the incentives to accumulate human capital. Section 4 contains concluding remarks. 2. The model We consider an overlapping generations economy with constant population where individuals live for two periods. Each generation consists of a continuum of agents. At time t = 0, there is an initial generation of old agents in which the jth member is endowed with human capital hj0. Agents in each period are differentiated by the stock of human capital of their parents. Every individual born at t = 0, 1, 2. . . has identical preferences over leisure when young and consumption when old. Formally, the preferences of an individual j born at time t are represented by: 1
r n1
r j;t þ cj;tþ1 1
r ; r > 0; ð1Þ where nj,t is leisure at time t and cj,t + 1 is consumption at time t + 1. If r = 1, then the corresponding component of the utility function is logarithmic. 292 G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 Individuals are endowed with one divisible unit of time in their youth. Young individuals in period t allocate their time between leisure and learning. Agent j born at time t accumulates human capital according to: hj;tþ1 ¼ hEtc hdj;t ð1
nj;t Þ; h > 0; c; dað0; 1Þ; c þ d < 1; ð2Þ where Et is the quality of public schools and hj,t is the stock of human capital of the parent. Note that the quality of public schools is not specific to individual j. The learning technology specified in Eq. (2) is consistent with a number of empirical studies. For instance, Heyneman (1984) provides cross-country evidence that school specific factors are positively related to various educational output measures. Coleman et al. (1966) find a positive correlation between parental education and performance on standardized tests in U.S. data. The learning technology is assumed to be linear in 1
n purely for convenience; it helps us solve for the human capital investment decisions analytically. The other two factors, quality and parental human capital, satisfy the diminishing returns assumption. The restriction that c + d < 1 guarantees that the economy has a steady state. At time t + 1, each individual’s income depends on his or her human capital: yj;tþ1 ¼ hj;tþ1 : ð3Þ The individual’s earnings at time t + 1 are taxed at a constant rate s. Tax revenues determine the quality of public schools faced by each young agent at time t + 1 according to: Etþ1 ¼ sYtþ1 ; ð4Þ where Yt + 1 is the per capita income at time t + 1. Because Et is the same for all young individuals in period t, Eqs. (2) and (3) seem to imply that, ceteris paribus, the heterogeneity vanishes in the long run. The young agent’s problem at time t is essentially one of choosing nj,t. The choice of nj,t pins down hj,t + 1, yj,t + 1 and cj,t + 1. Formally, given Et and hj,t, the young agent’s problem at time t is to choose nj,ta[0,1] to maximize Eq. (1) subject to: cj;tþ1 ¼ ð1
sÞhEtc hdj;t ð1
nj;t Þ: ð5Þ An equilibrium for this economy is a set of sequences {nj,t}tl= 0, {hj,t + 1}tl= 0, l l l {cj,t}l t = 0, { yj,t}t = 0, for ja[0,1], and { Yt}t = 0 and {Et}t = 0 such that (i) for ja[0,1], nj,t l l solves agent j’s problem at time t; (ii) the sequences {hj,t + 1}l t = 0, { yj,t}t = 0 and {cj,t}t = 0 are determined according to Eqs. (2), (3) and (5), respectively; (iii) Yt is the mean of the income distribution in period t and Et = sYt; and (iv) given the distribution of human capital at time t, each agent’s human capital at time t + 1 is determined by the transformation (2). It can be verified that there is a unique solution to agent j’s problem. Assuming an interior solution, the schooling choice is given by: 1
nj;t ¼ fð1
sÞhEtc hdj;t g 1
r r 1 þ fð1
sÞhEtc hdj;t g 1
r r : ð6Þ G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 293 Note that the schooling choices decrease with s for r < 1, holding E and h constant. An increase in the tax rate not only reduces the individual’s net income, but also lowers the relative price of leisure. For r < 1, the substitution effect dominates the income effect and, hence, the higher tax rate reduces the time devoted to human capital accumulation. Similarly, 1
nj,t increases with hj,t when r < 1, i.e. young agents born in a high human capital family spend more time learning than those born in a low human capital family. Finally, better quality of public schools implies higher time input to learning when r < 1. Income/human capital of the agent when old is given by: 1 yj;tþ1 ¼ fð1
sÞ1
r hEtc ydj;t g r 1 þ fð1
sÞhEtc ydj;t g 1
r r : ð7Þ Because yj,t + 1 is an increasing function of yj,t, there is no intergenerational income mobility.3 We trace the evolution of the entire income distribution as follows. At t = 0, the initial income distribution is exogenous. We use Eq. (4) to determine E0 and Eq. (7) to determine yj,1 for each j. This pins down the equilibrium income distribution for t = 1. We can repeat this procedure to determine the income distribution at any time t. 3. Divergence of income The preference parameter r and the coefficient on parental human capital d are the keys to analyzing the evolution of income inequality. We shall concentrate on the case r < 1 to make our model consistent with the microevidence on human capital accumulation. For instance, Card and Krueger (1992) find that, for the U.S., men educated in states with high average school quality have a higher return to additional years of schooling. Using data from Panama, Heckman and Hotz (1986) report that parental background is positively related to schooling returns. Suppose that 0 < d < r < 1. A simple way to determine whether income inequality declines over time is to compare the income growth rate for a poor family with the income growth rate for a rich family. Using Eq. (7), the gross growth rate of a family with income yt is given by: 1 ytþ1 fð1
sÞ1
r hEtc g r : ¼ 1
r 1
d yt yt r þ fð1
sÞhEtc g r y1
r t ð8Þ We have suppressed the index j for convenience. It is evident from Eq. (8) that yt + 1/yt is decreasing in yt. That is, the growth rate of income is higher for families with low incomes 3 If agents born in each period have different innate abilities, as in Owen and Weil (1998), then there will be some intergenerational mobility. In this case, the limiting distribution of income is not degenerate. Eckstein and Zilcha (1994) avoid a degenerate limiting distribution by assuming that there are random differences in bequest motive across households. Allowing for non-degenerate limiting distributions does not change the qualitative nature of our results. 294 G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 Fig. 1. Growth rate of household income. than for those with high incomes. Thus, incomes converge over time and income inequality declines. For 0 < r < d < 1, we can show, using Eq. (8), that the gross growth rate, yt + 1/yt, is not monotonic in yt. It is a strictly decreasing function of yt only if yt>yct where: yct ¼
d
r rð1
dÞ  r dð1
rÞ
1 ð1
sÞhEtc  1d ð9Þ is the critical income level. To illustrate the nonmonotonicity, we graph yt + 1/yt against yt in Fig. 1. Define yt to be the lowest income such that yt + 1/yt = 1 whenever yt = yt. At this stage, suppose that the poorest household’s income in period t exceeds yt so that no household’s income is declining in absolute terms.4 Consider two families, j and k, whose period t incomes, yj,t and yk,t, respectively, are below yct. If family j is poorer than family k in period t, then the growth rate of family j’s income is lower than that of family k. Hence, in this region, income inequality increases. If both incomes are above yct, then income inequality declines. Thus, family incomes converge if they are above the critical level, but they diverge below the critical level. The limiting distribution of income depends on what happens to the critical income over time. In a growing economy, under our assumption of constant tax rates, the quality of 4 Formally, we cannot make this assumption in period t because the income distribution in period t is endogenous. However, we can make this assumption in period 0 and, in equilibrium, it will indeed be the case that the income distribution in period t is bounded away from yt. G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 295 schools increases over time. From Eq. (9), it is easy to see that the critical income must decline over time. Eventually, both family incomes exceed the critical income. Thus, the income gap between families j and k widens for some periods, but, in the long run, the gap narrows down to zero. The intuition behind the nonmonotonic relationship in Fig. 1 is as follows. In a given period, the differences in growth rates across families stem from two sources: (i) the time input to learning, 1
nt, and (ii) the level of income, yt. It is easy to see from Eqs. (2) and (3) that the growth rate is decreasing in the level of income, holding 1
n constant. However, for r < 1, the time input to learning is increasing in the level of income. Thus, the net effect on the growth rate depends on which one of these two effects is stronger. As noted earlier, changes in yt (or ht) change the trade-off between consumption and leisure. For low values of r, the substitution effect is strong so that 1
n is very sensitive to changes in y. Hence, the growth rate is increasing in the level of income. For larger levels of income, the growth rate is decreasing since 1
n is bounded above.5 Is our result just a theoretical possibility? One can assess the empirical plausibility of our results in two ways: (i) evaluate whether available estimates of c, d and r satisfy the parameter restriction r < d < 1 and c + d < 1, and (ii) evaluate whether the implications of the model are consistent with observations. Direct estimates of d are rare, but Wachtel (1976) estimates d = 0.8 using data on children’s wage income and father’s years of schooling. Solon (1992) and Zimmerman (1992), using children’s income and father’s income, estimate d to be close to 0.4, while earlier estimates of Becker and Tomes (1986) are in the neighborhood of 0.2. As noted earlier, microevidence from Heckman and Hotz (1986) and Card and Krueger (1992) suggests r < 1. While we cannot pin down r any further, values of r close to 0 will deliver our results. Regarding c, Harris (2000) finds that the ‘‘best’’ estimate from the empirical education production function literature is close to 0.07. Thus, both restrictions on the parameters are empirically plausible. To evaluate whether the model’s implications are consistent with the data, consider the evidence on income inequality and public expenditures on education. Rising wage inequality over the last few decades is well documented (see the symposium on Wage Inequality in the Journal of Economic Perspectives, Spring 1997). Since the 1970s, both the judiciary and the legislature in the U.S. have actively equalized the public educational expenditures across school districts. For instance, Fernandez and Rogerson (1999) document the narrowing of the expenditures in California between 1972 and 1987. This decline in variance of public expenditures has not been accompanied by a decline in variance of income; in fact, California is 1 of the 10 states where income inequality grew the most between the late 1970s and 1990s. Other states in the U.S. have carried out similar equalization policies, yet the earnings inequality has risen since.6 Thus, despite (more) ‘equal’ access to education, incomes do not exhibit a tendency to converge. 5 The divergence of income result holds even if the young agents choose between consumption/production and learning instead of choosing between leisure and learning. In a variant of our model where young agents receive utility from consumption and where there are no imperfections in the credit markets, the young agents will maximize the present value of their incomes. In such a model, it is easy to show that incomes will diverge whenever there are increasing returns to augmentable factors, i.e. whenever c + d>1. 6 For other explanations of rising wage inequality, see Johnson (1997), Topel (1997) and Fortin and Lemieux (1997). 296 G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 3.1. Comparative dynamics We conduct two comparative dynamics exercises below for the case 0 < r < d < 1 by asking: (i) What is the effect of the initial income distribution on the evolution of income inequality? and (ii) What is the effect of tax rates on the evolution of income inequality? Both exercises are numerical. We consider two individuals, rich and poor, and measure income inequality in each period by the ratio of rich income to poor income. At t = 0, suppose that the income of the rich equals 1 and the income of the poor equals 0.05. Unless otherwise stated, the baseline parameters are r = 0.5, d = 0.8, c = 0.1, h = 5 and s = 0.05. For these parameters, the initial income of the rich is close to but greater than the critical income in period 0, while the initial income of the poor is much less than the critical income. In terms of Fig. 1, the rich are close to the peak, while the poor are to the left and far below the peak. 3.2. Initial distribution To understand the effect of initial income distribution, consider two economies identical in all respects except that they have different mean incomes in period 0. The rich and the poor have incomes 1 and 0.05, respectively, in the high mean income economy, and 0.90 and 0.045 in the low mean income economy. The evolution of income inequality for the two economies is illustrated in Fig. 2. In the low mean income economy, the inequality is higher and the income gap widens for three periods, whereas, in the high mean income economy, the income gap widens only for two periods. There are two aspects of the exercise that favor convergence in the high mean income economy: (i) in period 0, the Fig. 2. Mean income and the evolution of income inequality. G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 297 quality of public education is higher and (ii) the initial distribution is such that the poor are closer to the critical income level (see Fig. 1). Now consider two economies with the same mean income in period 0 but different spreads. One economy has low initial income inequality; the rich and the poor begin with incomes 1 and 0.05, respectively. The other has high initial income inequality; the rich and the poor begin with incomes 1.02 and 0.03, respectively. As illustrated in Fig. 3, the second economy experiences higher inequality in all periods and the income gap widens for more periods. Because the mean income is the same in period 0, the quality of public education and, hence, the critical income level is the same in the two economies. The income gap widens for more periods in the second economy because the poor are farther away and the rich are closer to the critical income level. 3.3. Public policy Because quality of public education is the main force in our model that helps the poor catch up with the rich, it is natural to ask if the widening income gap may be eliminated through a higher quality of public education. One way to increase the quality of public education in our model is increase the tax rate, s. To examine the effect of an increase in s on the evolution of income inequality, we consider three economies identical in all respects except for tax rates. Fig. 4 reveals that the relationship between tax rates and the evolution of income inequality is not monotonic. The low tax rate in Fig. 4 is 0.01, the medium tax rate is 0.05 and the high tax rate is 0.6. An increase in tax rate increases the quality of public education, ceteris paribus. An increase in the quality of public education increases the time input to learning Fig. 3. Initial income inequality and the evolution of income inequality. 298 G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 Fig. 4. Tax rates and the evolution of income inequality. because r < 1. However, changes in the tax rate change the incentives for learning, which affect income. From Eq. (9), it is easy to see that the relationship between critical income and tax rate is U-shaped with the minimum occurring at a tax rate of c/ (1 + c). When the tax rate is 0.01, the critical income is very high so that income inequality increases. When the tax rate is 0.05, the critical income is lower so that both families exceed the critical income sooner and income inequality declines. When the tax rate is 0.6, the critical income is again very high and the income gap widens for more periods before it shrinks. Alternatively, consider the evolution of income from periods 0 to 1 in economies with different tax rates. Using Eq. (7), we can write the ratio of rich to poor income in period 1 as follows: 0 d 10 1 1
r r yrich;0 1 þ fð1
sÞhsc Y0c ydpoor;0 g r yrich;1 A@ A: ¼@ d 1
r ypoor;1 1 þ fð1
sÞhsc Y c yd g r yr poor;0 0 rich;0 The left-hand side reaches its peak when s = c/(1 + c). 4. Concluding remarks We have used a simple overlapping generations model to show that, in the short run, public education may not be the ‘‘great equalizer’’ as intended by its proponents, even though in the long run it is equalizing. We showed that income inequality may increase for a few generations even when the quality of schools is the same across all individuals 299 G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 Table 1 Mandatory schooling and enrollment rates Country Required years of schooling Enrollment ratios at age 14 15 16 Austria Belgium Bulgaria Czech Republic France Germany Greece Hungary Ireland Malta Netherlands Romania Spain Switzerland UK Canada USA 9 12 8 9 10 12 9 10 9 11 13 8 10 9 11 10 10 74.2 84.1 86.6 59.0 100.0 100.0 86.0 79.8 100.0 99.8 100.0 92.3 92.0 99.5 100.0 99.7 100.0 36.0 95.7 81.2 19.0 100.0 97.2 96.1 89.7 100.0 95.5 100.0 80.5 77.8 95.1 100.0 98.0 98.7 20.1 99.3 78.0 15.0 97.4 68.1 80.5 82.2 91.7 60.0 100.0 70.9 61.8 54.9 84.9 92.8 89.1 Source: UNESCO Statistical Yearbook, 1999. in an economy. Initial conditions as well as public policy affect the evolution of income inequality. Here, we have abstracted completely from self-selection by households into districts of differing public school quality. Such self-selection increases the tendency toward inequality, because the rich typically select better school districts. In our model, income inequality is transmitted from one generation to the next through human capital investment decisions of individuals. Individuals whose parents have low human capital allocate less time to learning than those whose parents have high human capital. One may conjecture that a policy of mandatory schooling would induce income convergence. See Eckstein and Zilcha (1994) for a model with compulsory schooling. It is true that many countries have mandatory schooling requirements. In European and North American countries, where mandatory schooling typically covers children from ages 6 to 15 or 16, enrollment ratios fall short of 100% (see Table 1). It seems that simply imposing mandatory schooling requirements does not in and of itself guarantee that the time input to learning is constant across individuals. We conclude from the data in Table 1 that divergence of income in a regime of equal public provision of educational expenditures is a possibility that should be taken seriously. The impact of public provision of education combined with imperfectly enforced mandatory schooling is an interesting extension of our paper. Acknowledgements We thank Alex Wolman for excellent research assistance. Financial support from the Bankard Fund for Political Economy is gratefully acknowledged. 300 G. Glomm, B. Ravikumar / European Journal of Political Economy 19 (2003) 289–300 References Becker, G., Tomes, N., 1986. Human capital and the rise and fall of families. Journal of Labor Economics 4, S1 – S39. Bowles, S., 1978. Capitalist development and educational structure. World Development 6, 783 – 796. Card, D., Krueger, A., 1992. Does school quality matter? Returns to education and the characteristics of public schools in the United States. Journal of Political Economy 100, 1 – 40. 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