A model of public education and income inequality with a subsistence constraint.

• Author: Sylwester, Kevin

1. Introduction

   "Deep in a citrus grove, law officers crept up on Dionatan Rocha and caught him redhanded. Dionatan, a 12-year-old wearing a baseball cap and T-shirt, was picking oranges, in plain violation of Brazil's child-labor code.... Says Mr. Grajew: 'Brazil's chronically unequal wealth distribution has one root cause: Millions of children are working instead of studying.'" (Moffett 1998).

   To reduce income inequality and spur economic development, many have advocated increased support for public education. However, public education may not be a silver bullet to eliminate poverty or lower income inequality. Fields (1980) reports that income inequality in several developing countries did not lessen even after an increased allocation of resources to public education. Sylwester (2002) considers how education expenditures are associated with subsequent changes in income inequality within a cross-section of countries. After dividing the sample into OECD and less-developed-country subsamples, he finds that education expenditures are more strongly associated with falling income inequality in the former group. One possible reason for these differing outcomes is that even when public education is freely provided, attendance is not guaranteed, especially in poorer nations. As in the above account, families may place their children in the labor force even if this is illegal, since the income that these chi ldren generate is nontrivial to the welfare of the family. Consequently, a subset of the population may exist that has little if any formal schooling, thereby limiting opportunities for these individuals later in life and possibly creating a permanent underclass. Poor children work rather than attend school and thus remain poor as adults. These adults then need their children to work to help support the family, and this poverty persists through another generation. Therefore, a better understanding of this cycle and of the policies that can break this cycle is needed to help alleviate poverty in these nations.

   This is not the first paper to examine how public education affects income inequality or poverty. Glomm and Ravikumar (1992) develop a model in which agents vote for a public or private education system and in which a public education system benefits low-income agents relative to high-income agents. Consequently, a public education system unambiguously decreases the level of income inequality. Saint-Paul and Verdier (1992) and Zhang (1996) also conclude that public education helps to lower income inequality over time. A critical assumption of these models is that attending public school is costless and so all agents attend. However, Jimenez (1986) finds that this assumption is questionable in many developing countries. Although public education is often provided freely, it is not necessarily costless. Those attending school forgo income that could have been used to help support a family, and in some cases this income is necessary for the family's survival. Jimenez (1986) also reports that taxation in many dev eloping countries is regressive. Taken together, these assertions imply that in some countries poor families are hurt by the taxation used to support public education but do not receive many of the benefits. In fact, these public education systems might even be contributing to an increase in the level of income inequality.

   This paper creates a model that explores some of these issues. Like those of Perotti (1993) and Galor and Zeira (1993), this model does not assume that all agents partake in public education. Instead, agents face an opportunity cost associated with going to school and thus choose how much schooling to acquire. It is possible that the level of income inequality does not decline even in the presence of a public education system. Fernandez and Rogerson (1995) also capture some of these concerns in their model of public education with endogenous subsidies. In their model, only agents attending school receive education subsidies, and so high-income agents (who have the means to attend school) then have an incentive to vote for high subsidies. High subsidies require high taxes, which might prevent lower-income agents from attending school, thereby leaving larger subsidies for those agents who do undertake schooling. Over time, a stratified society can develop as some agents partake in public education while others do not.

   The model presented in this paper also has the potential to lead to a stratified income structure. One difference, however, is that income is not taken to be exogenous here but is endogenously determined by an agent's behavior. Therefore, this model shows not only how the various determinants of income can affect long-run income levels but also why agents with the same income at a point in time might have very different long-run outcomes. This result can have important implications, especially in designing policies to alleviate long-run poverty. First, it might be more difficult to predict which agents are in danger of falling into some sort of poverty trap if only income levels, and not the determinants of income, are examined. Second, if aid is limited, then it becomes more important to identify those for whom aid is necessary for an escape from current poverty in order to maximize the long-run benefits of such aid.

   Other considerations of this model include potential effects of the size or efficiency of the public education system on the distribution of income. The model shows how it is possible that better public education systems lead to more income inequality and why a gradual allocation of resources to public education may prove more beneficial than a sudden, large shift of resources. This model also considers how various factors such as productivity parameters and the growth rate of world knowledge can affect the long-run distribution of income within a country.

   This paper is organized as follows. Section 2 describes the model and section 3 presents the solution. Section 4 considers an extension in which agents fund public education through tax payments, where taxes are endogenously determined. A conclusion follows.

2. The Model

   An agent in this model represents a single family (although I continue to use the term agent). Agents differ as to their initial levels of human capital, but I omit notation regarding agents unless it becomes necessary. This heterogeneity is assumed to arise from historical factors or because some agents have greater access to private education than do others. There is no population growth. At each point in time, an agent decides how much time to allocate to public schooling and how much time to allocate to working, from whence that agent can acquire income and consume output. (1) Relevant to this allocation is that agents must obtain at least a subsistence level of income at each point in time, and this might limit how much schooling is undertaken.

   At time 0, each agent wants to maximize lifetime utility discounted to the present. Utility at each moment in time, t, is given by the felicity function [c.sup.[sigma].sub.t]/[sigma], where [c.sub.t] denotes consumption at time t. Consumption equals income, as agents do not save. Instead, they augment future income by investing in education today. The maximization problem is more formally given as

   max [[integral].sup.[infinity].sub.0] 1/[sigma][c.sup.[sigma].sub.t][e.sup.-pt] dt with p > 0 and 0

   subject to the following constraints:

   (1) [c.sub.t] = [y.sub.t] [greater than or equal to] c for all t [greater than or equal to] 0.

   (2) [y.sub.t] = A[(1 - [u.sub.t])[h.sub.t]].sup.[alpha], with 0 [less than or equal to] [u.sub.t] [less than or equal to] 1, 0 0.

   (3) [h.sub.t] = [Bu.sub.t] - d[h.sub.t] if [h.sub.t] > h or [[u.sub.t] > dh/B; [h.sub.t] = if [h.sub.t] = h and [u.sub.t] 0, d > 0, and h > 0.

   (4) [h.sub.0] [greater than or equal to] h given.

   (5) [lim.sub.t[right arrow][infinity]] [e.sup.-pt][w.sub.t][h.sub.t] = 0.

   The parameter [rho] denotes the discount rate. Constraint 1 states that at each point in time, consumption must be greater than some subsistence level of income (c). In many maximization problems with a subsistence constraint, the felicity function is given by [([c.sub.t] - c).sup.[sigma]]/[sigma]. I do not employ such a specification here because it would add greater complexity to the dynamics of the model without changing long-run human capital levels. The steady states presented below remain the same whether [([c.sub.t] - c).sup.[sigma]]/[sigma.] or [c.sup.[sigma].sub.t]/[sigma] is used. Nor do the qualitative conclusions of the model depend on which of the two felicity functions is used. Since long-run outcomes remain the same and the dynamics are easier to analyze, I choose to proceed using the simpler functional form.

   Constraint 2 shows income to be a function of the level of human capital ([h.sub.t]), the fraction of time devoted to working (1 - [u.sub.t]), and a productivity parameter (A). Income can thus be viewed as an increasing function of effective labor input. The total time endowment for the agent is normalized to 1.

   Any time not spent working is devoted to increasing human capital, as shown in constraint 3. The variable [h.sub.t] denotes the derivative of [h.sub.t] with respect to time. It increases with the amount of time devoted to education. The parameter h denotes a minimum level of human capital. h can be thought of as the level of human capital that agents would have in the absence of any schooling. Although it is still possible for agents to augment their human capital at h by allocating sufficient time to schooling [u.sub.t] > dh/B), human capital will not decrease once it reaches h.

   Constraint 4 simply states that agents do not begin with a human capital level below this minimum. The parameter B controls for the productivity of the public education system. In section 4, I take B to be an increasing function of tax revenue going to public...