1. Introduction

Public basic education is mandatory, non-religious and free in Mexico by law. Article 3 in the Political Constitution establishes that all individuals in the country have the right to receive an education. In addition, it says that kindergarten, elementary and middle school education will be provided by the State and that these levels of education are mandatory. In order to fulfill this obligation, the government spent about $385 000 million pesos during 2014. Given that there are about 23.4 million students in the public system at these education levels, the average direct cost of providing basic public education is about $16 450 pesos per student per year.

Although every child in the country is supposed to have access to free public education, the demand for private education has been growing in the last years. According to data reported by the Ministry of Education (^{SEP, 2014}), student enrollment in private schools grew by about 98% at the kindergarten level, by 12% for elementary schools, and by 37% for middle schools, respectively, during the last 15 years. Furthermore, private school enrollment has been growing faster than public school enrollment. Hence, the share of students in private schools increased from 8.1% to almost 9.8% during this period.

Several reasons explain why parents demand private education despite a free public alternative. One of the arguments is that private education is better than the available public education. For instance, private education usually offers longer school days, a lower student/teacher ratio or instruction in a second language, among other things. Another argument is that private education can offer certain characteristics that may be appreciated by some parents but not necessarily by others. For example, some private schools offer religious instruction or single-sex education. However, even if private and public education were identical, the differences in locations of schools and transportation costs may be sufficient to explain why some individuals are willing to pay for private education instead of using an alternative that is “free”.

The convenience of taxing or subsidizing private education has been debated in Mexico a couple of times during the last years. At the end of his term, President Calderon issued a decree that allowed individual tax deductions for tuition expenses from kindergarten through high school. This policy received mixed reviews in the media. A couple of years later, President Peña proposed a tax reform that considered eliminating the education value-added tax (VAT) exemption. In other words, the reform intended to tax private education. This was one of the most controversial points in the proposal and was heavily criticized in the media. Indeed, the education tax was dropped in Congress almost immediately after the tax proposal became public.

There are several arguments in favor of taxing private education or equivalently against subsidizing it. For instance, some analysts in the media claim that the tax deduction (that is, a subsidy on private education): (a) worsens government finances, (b) increases the quality gap between private and public education and (c) worsens income distribution because it helps the rich.¹ Of course, other analysts in the media say in favor of the deduction that a subsidy is fair because parents who send their children to private schools are paying for education twice:² First they pay taxes to finance public education and then they pay to obtain education services in the private sector.

In this article, I study the effects of an education tax (or subsidy) on enrollment at private and public schools, on the quality gap between public and private schools, and on government finances. In order to achieve these goals, I model the behavior of a private school that competes for students with a public school. The private school has to choose the quality and price of its service in order to attract students who can choose to attend a public school of given quality at price zero. It turns out that a per-unit tax (or subsidy, respectively) on private education reduces (increases) enrollment in the private school, increases (reduces) the quality gap between private and public education and has an ambiguous effect on government finances. Therefore, contrary to the common belief, a tax (or subsidy) on private education can increase (reduce) the quality gap between private and public education and worsen (improve) government finances.

There are several articles in the economic literature that study the choice between public and private education from a theoretical point of view. ^{Stiglitz (1974)}, ^{Sonstelie (1979)}, ^{Hoyt and Lee (1998)}, and ^{Epple and Romano (1998)} develop models where quality is a key variable in the education choice problem. Each of these articles focuses on particular aspects of education. The most recent articles concentrate on the effects of voucher programs in the US. For example, Hoyt and Lee (1998) study whether a voucher program has political support considering its potential effect on taxes and the quality of public education. On the other hand, Epple and Romano (1998) study the effect of vouchers on the quality of public and private education.

There is another area of literature not directly related to education that studies the impact of taxes on quality. Some of the most notable theoretical articles in this literature include ^{Barzel (1976)}, ^{Leffler (1982)} and ^{Krishna (1990)}. While ^{Barzel (1976)} theorizes that a per-unit tax increases quality, ^{Leffler (1982)} builds a model that shows that this type of tax has an ambiguous effect on quality in the context of monopoly. On the other hand, ^{Krishna (1990)} models the behavior of a monopolist in the context of international trade and finds that specific taxes increase average quality. In a sense, the model that I build in this article links the theoretical literature on the choice of public and private schools with the literature that studies the impact of taxes on quality.

The rest of the article is organized as follows. In section 2, I develop the model where a private school chooses price and quality to compete with a “free” public school. In section 3, I study the effects of a per-unit tax or subsidy on enrollment at public and private schools, on the quality gap between them and on government finances. I present conclusions in the last section.

2. Model

There is a group of children that must receive basic education. I normalize the size of this group to be one. Their parents choose between private and public schools. In the absence of transportation costs, peer effects or religious considerations, quality is the key to determine parents’ choice.³ Assume that the public school provides education of quality s ^o normalized to be zero, while the private school provides education of quality s > s ^o ≥ 0. Therefore, variable s represents the quality gap between private and public schools. Assume that the inverse of the demand function for private education is p(q, s). The variable q ≤ 1 represents students who attend the private school. Willingness to pay for private education depends crucially on the private school providing higher quality education than that of the available public school (i.e., p (q, s ^o ) = 0). This assumption is not really necessary to obtain the main results of this article but is relatively standard in the literature. For instance, similar considerations are made by ^{Stiglitz (1974)}, ^{Sonstelie (1979)}, ^{Hoyt and Lee (1998)}, and ^{Epple and Romano (1998)}.⁴

Next, I will formalize some notations and assumptions about the form of the demand for private education. Hereafter, I will use subscripts to denote partial derivates. The basic assumptions with regard to the demand function are the following. First, demand has negative slope (A1: p _q ≡ ∂p∂q < 0). Second, willingness to pay for private education increases with the quality gap between the private and the public school (A2: p_s ≡∂p∂s > 0). Third, parents who are willing to pay more for private education are also willing to pay more for a marginal increase in quality (A3: p _qs = p _sq ≡ ∂2p∂q∂s =∂2p∂s∂q <0). It is important to note that some of these assumptions are fairly standard. Furthermore, these assumptions allow for a wide variety of demand functions.

Figure 1 depicts demand for private education. The solid line is the demand for private education of quality s ^’ , while the dotted line is the demand for private education of quality s ^’’ > s ^’ . Note that an increase in the quality of private education tilts demand to the right. However, enrollment in private and public education is bounded by the size of the group of students that attend school. The demand for public education is a residual demand of private education and vice versa. Given that the size of the group of students that attend school is normalized to be one, the demand for public education is 1 −q ≥ 0.

Figure 1 Demands for private and public education 

Providing either public or private education is costly. The cost of this service depends on the technical efficiency of schools as well as on the quality of education they provide. Assume that the public school provides education at constant marginal cost c ^o > 0, while the private school provides it at constant marginal cost c(s). I will assume that the marginal cost function of the private school is increasing (A4: c _s > 0) and convex (A5: c _ss ≥ 0) in quality.

I should make a few additional comments about the costs of providing education. First, several articles in the literature, such as ^{Stiglitz (1974)}, ^{West (1985)}, and ^{Hoyt and Lee (1998)} explain that the marginal cost of education is approximately equal to the average cost from a policy perspective. The argument that supports this idea is the following. Public policy generates relatively large movements of students between private and public schools. Therefore, schools have to open new classrooms and hire teachers to serve students moving from one system to the other. Second, some articles, such as ^{Stiglitz (1974)} and ^{Hoyt and Lee (1998)}, consider that the quality of education depends on expenditures. These comments are consistent with the assumption that the marginal cost of education is constant for a given quality but increases if the school decides to provide higher quality.

Although some articles claim that private schools are more efficient than public schools, ^{Ladd (2002)} argues that this is not necessarily true. Among other things, ^{Ladd (2002)} explains that tuition does not necessarily reflect the cost of private education in the US because many private schools receive funding from their church. Therefore, I will not assume that either the public or the private school is necessarily more efficient than its counterpart. The model allows for both possibilities. On the one hand, if c ^o < c (s ^o ) then the public school is more efficient than the private one. That is, the marginal cost of the public school is lower than the marginal cost of the private school when both provide equal quality (that is, at s = s ^o = 0). On the other hand, if c ^o > c (s ^o ) then the private school is more efficient than the public one. That is, the marginal cost of the private school is lower than the marginal cost of the public school for the same quality. Alternatively, the private school can provide higher quality than the public school at the same marginal cost (that is, c ^o = c (s) at s > s ^o ).

The public school is passive in a sense. That is, the quality of the public school is given by the budget and the price is set to zero by law. In contrast, the private school chooses price and quality to maximize profits given a per-unit tax t ≥ 0 imposed by the government. Alternatively, the private school chooses q and s in order to maximize

The necessary conditions for profit maximization are

and

Let q ^∗ (t) > 0 and s ^∗ (t) > 0 be the pair that solves equations (2) and (3) as a function of the tax. Now suppose that the profit function is strictly concave. Hence, the following conditions are satisfied:

and

Conditions (4), (5) and (6) ensure that the solution pair (q ^∗(t), s ^∗(t)) is unique. Note that condition (4) is met as long as the demand function is not very convex (that is, the condition is met even if p _qq is positive but not very large). Similarly, condition (5) is met as long as marginal costs increase with quality faster than willingness to pay (that is, as long as p _ss < c _ss ). Note also that (2), (3) and A3 imply that

Finally, I will follow the approach of ^{West (1985)} and ^{Hoyt and Lee (1998)} to evaluate the effects of a tax (or subsidy) from the perspective of government finances. Therefore, let EB denote the education budget of the government. This budget has three components: (a) resources initially assigned to public education (RIAE), (b) revenues from the tax on private education (RTPE) and (c) the cost of providing public education (CPE).⁵ Formally, for a given tax we can write the following:

3. Analysis

In this section, I analyze the impact of a per-unit tax (or subsidy) on the choice between private and public schools, on the quality gap between them and on government finances.

4. Conclusions

This article presents a relatively simple model of choice between private and public schools. This model brings together the theory of education choice with the theory that studies the effects of taxes on quality. Among other things, the results of the model indicate that a per-unit tax (or subsidy) on education can increase (reduce) the quality gap between private and public education and worsen (improve) government finances. It is important to mention that the main results of the model hold for fairly general demand and cost functions.

Rather than claiming that a particular education tax policy -like the VAT proposal or the tax deduction discussed in Mexico- affects the quality gap or government finances in a given direction, the main message of this article is that the theoretical effects of education tax policies are not straightforward. An appropriate analysis of the VAT proposal or the tax deduction would have to take into account the different theoretical effects of the tax policy explained in this article as well as the practical complexities of the Mexican tax system.

A couple of aspects that are usually neglected in the analysis of education tax policy are the elasticity of demand for private education and the cost of public education. As I show in this article, the impact of a per-unit education tax on government finances depends crucially on these two variables. A tax on education tends to improve government finances if demand for private education is relatively inelastic, or if the cost of public education is low. Alternatively, a subsidy tends to improve government finances if demand for private education is relatively elastic, or if the cost of public education is high.

Finally, it is important to note that this article does not address the effect of education taxes or subsidies on the distribution of income. It is reasonable to assume, as do ^{Epple and Romano (1998)}, that willingness to pay for private education increases with income. This would lead me to conclude that an education tax improves the distribution of income while a subsidy worsens it. However, this approach would neglect the fact that only a relatively small fraction of low-, middle- and high-income households spend on private education.⁸ On the other hand, a well-known result in the public finance literature is that education subsidies help to reduce the distortions in human capital accumulation caused by progressive income taxation. In particular, ^{Bovenberg and Jacobs (2005)} explain that the optimal policy in terms of efficiency is to make all education expenditures completely tax deductible. Therefore, a tax on private education might be a costly way to redistribute income.