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Subsidies to higher education and income-tax progression.

ABSTRACT

The impact of fiscal activity on human-capital formation has often been analyzed with regard to the expenditure side of the budget; i.e., the subsidization of (higher) education. However, recent contributions have increasingly focused on the effect of taxation on human-capital formation. Less attention has been given to the simultaneous effect of both subsidization and taxation. As subsidies to higher education may offset the distortionary effect of taxation, this paper aims to analyze the amount of subsidization that is necessary to counteract the discouraging effect of various kinds of income-tax progression on human-capital accumulation. An important policy advice derived from the analysis is that a country's educational policies should be evaluated, not just by the levels of educational spending, but also by the level and structure of taxation.

The framework we use to illustrate our point is a two-period cohort model with heterogeneous agents, who endogenously decide on higher education with respect to taxation and subsidization. In this framework, we also demonstrate that subsidies may fail to offset existing tax distortions, limiting the optimistic view of some previous literature. In particular, we show that a second-best subsidy rate must be lower than unity. Otherwise an optimal subsidy rate leads to the problem of overeducation.

1. INTRODUCTION

In the last decades, advocates of public activities in the education sector have particularly referred to externalities, credit constraints, and distributional issues. (1) The discussion about externalities gained more importance in the 1980s and 1990s, particularly because of the seminal paper of Haveman & Wolfe (1984) and because of new developments in growth theory, following the dismissal of earlier explanations based on neoclassical marginal productivity theory (cf. Blaug, 1970, pp. 112ff). However, the empirical evidence for positive externalities is scant at best (see Acemoglu & Angrist (2000); Bils & Klenow (2000); Krueger & Lindahl (2001) for recent contributions).

The importance of credit constraints is disputable as well. Capital-market imperfections, so the argument goes, may hinder poor agents financing the costs of obtaining higher education (see Saint-Paul & Verdier (1993); Perotti (1993); Benabou (2000); Benabou (2002)). However, there is little empirical evidence (see, e.g., Carneiro & Heckman, 2002; Cameron & Heckman, 2001; Keane & Wolpin, 2001). Friedman (1962) and others (see Epple & Romano (1998) for an overview) have persuasively argued that vouchers or student loans, for example, are a better means to compensate for unwanted effects that result from credit constraints. However, even if all classical arguments in favor of public subsidization cannot be dismissed as a whole, most economists argue that these arguments cannot justify the wide prevalence of education subsidies in many countries, in particular in Europe.

There is a large literature on the effect of taxation on human capital formation. Previous examinations are, e.g., Heckman (1976), and Eaton & Rosen (1980). In both works, labor-income taxation was found to have a neutral effect if the educational outcome is certain, but in both papers only the opportunity costs of obtaining higher education are considered. Boskin (1975) focuses on the effect of taxation on different kinds of human-capital investment. He argues that from a lack of educational expenditure depreciation the tax system alters the composition of human investment, e.g. biases investments away from higher education to on-the-job training. Beside, Boskin showed that a progressive income-tax system creates a disincentive to accumulate human capital in general. In this paper, we do not focus on the role of fiscal policy on different kinds of human-capital formation, but we consider explicitly the effect of income-tax progression by combining its effect with the effect resulting from subsidization. This way of looking at educational policy is important because, as (Steuerle, 1996, p. 335) pointed out, "when government provides educational benefits, many analysts incorrectly assess their value by looking solely at direct spending. In fact, relative value depends also upon the tax treatment."

The impact of taxes on human-capital accumulation has become the central element in the recent literature. Trostel (1993, 1996) has shown that taxation has a negative impact on human capital investments and that education subsidies should primarily be seen and justified as a compensation for this tax distortion. In making this argument, Trostel uses an econometric model with a proportional tax rate, and it is assumed that the direct costs of obtaining higher education are not tax-deductible.

Dupor et al. (1998) analyzed the distorting impact of progressive taxation based on US tax law in 1970. The findings show that progressivity led to an approximately 5-percent decline in human-capital investment in 1970. Based on data from 1990, the impact differed considerably depending on the choice of schooling, and lay between close to zero and -22%. Sturn & Wohlfahrt (2000) referred to the foregone smoothing benefit. Due to tax progression, combined with annual tax assessment, graduates pay more taxes than nongraduates with the same net lifetime earnings because graduates accumulate their income in a shorter period of time. (2)

Our paper is also related to the studies by Heckman et al. (1998); Heckman et al. (1999). They assess the effects of tax reform and tuition subsidies on human-capital formation in a dynamic overlapping-generations general equilibrium framework. Starting from a progressive tax on labor income and a flat tax on capital income they focus on the effects of two different tax reforms on human-capital formation. The first tax reform aims at flattening the labor tax schedule to a flat tax on labor income holding the initial flat tax on capital income constant. The second reform considered is a uniform flat tax on consumption. In both tax-reform scenarios, tuition fees are not treated as deductible. The first reform leads to an increase in human-capital formation whereas the second reform is less favorable to human capital but more pro-capital. Next, Heckman et al. use their model to evaluate proposals to subsidize tuition. In their general-equilibrium setting, this proposal leads to a slight decrease in university attendance. The reason is that the university-high-school wage ratio falls as more individuals enroll in universities. Individuals, who foresee this effect of subsidized tuition on the wage ratio adjust their enrollment behavior accordingly. (3)

However, the combined effect of (progressive) taxation and subsidization on human-capital formation has been neglected so far. Agents may be encouraged to invest in their human capital even though taxation discourages such an investment. This is the case when subsidization offsets the distorting effect of taxation. However, to what extent should education be subsidized, given some degree of income-tax progression? The analysis of this question is a purpose of the present paper.

In this paper, we give a rationale for this relationship between the rate of subsidization and the income-tax progressivity. Furthermore, we demonstrate that subsidies that offset existing tax distortions may be in league with the devil: by counteracting distortions, new inefficiencies may arise so that subsidies may fail to offset for tax distortions. The reason is that in the model used in this paper government does not--in contrast to the framework outlined by Steuerle (1996, p. 354f)--grant greater subsides to those students with greater abilities, but also to those with the lowest ability.

By showing this, we moderate the optimistic view of subsidies found in some of the related literature, noted above. A consequence of our analysis is that international comparisons of education policy, as carried out e.g. by the OECD (2002, Ch. B), should not focus exclusively on the expenditure volume for educational institutions. Rather, they should take into account the comprehensive effect of public policy on human-capital formation, which clearly includes the tax system (see Steuerle, 1996). Considering this, it seems that differences among OECD countries are smaller than a first glance at subsidies might suggest.

The paper is organized as follows. In Section2, we introduce our basic model structure and deal with the simplest case of a proportional income-tax system in Section 3. We extend then in Section 4 the case of proportional taxation to the case of income-tax progression and derive a necessary condition for an optimal subsidy rate. In Section 5, we emphasize some limits of distortion-offsetting subsidies. We demonstrate that under proportional taxation subsidies may lead low-ability individuals to over-invest in education. Finally, the last section concludes the paper.

2. THE MODEL

To make our point, we use an amended version of the model presented by Creedy & Francois (1990). Their model consists of a population of agents who differ with respect to their innate endowment. It is a two-period model. In the first period, all agents face the decision of whether to enroll in a degree program or not. In the second period, all agents work, either as graduates or as non-graduates. The government is assumed to raise taxes. The entire public revenue is spent financing subsidies to higher education, and for a publicly-provided good. The graduation rate depends on the tax rate, the rate of subsidization, and on an externality created by those who attend higher education.

Our framework differs from the model of Creedy & Francois (1990) in two particulars. Firstly, we neglect the existence of externalities. A justification for fiscal activities is given by a distortion created by income taxation according to the aforementioned studies. Secondly, the resulting revenue from taxation is spent on redistribution and subsidization purposes. Each agent receives an identical lump-sum transfer, denoted by N[member of][R.sub.+], whose amount depends on the tax base, the structure of taxation, and the amount devoted to financing higher-education subsidies. If no subsidization takes place, however, the entire revenue is distributed uniformly among all individuals.

In contrast to the lump-sum transfer, the effect of income taxation is twofold. It allows the described redistribution policy, but it distorts the choice between education and work in the first period. This distortion calls for efficiency-enhancing subsidies. The efficiency gains created by a (partial) subsidization are potentially Pareto-superior. We do not ask why a distortionary taxation exists. We instead assume that a non-distortionary tax system is politically not feasible, so that policy aim is to implement a second-best means to offset for distortion.

Assume that a population is heterogeneous with respect to the innate endowment [y.sub.i], with 0 < [y.sub.i] < [??]. Population size is normalized to unity. As in Creedy & Francois (1990), we consider that the cohort lives for two periods. In the first period, each agent can choose between higher education and work. In the second period, the entire population works. An individual's gross income is determined by his or her individual innate endowment and his or her return from higher education (if obtained). The distribution of the initial endowments is represented by the twice differentiable density function, f(y), and its corresponding distribution function, F(y). A constant and exogenously given tax rate, t [member of] [0,1) [subset] R, is levied on all income.

An individual chooses higher education if his or her net lifetime earnings with an university degree would exceed the lifetime earnings if he or she did not invest in higher education. The degree causes direct (and non tax-deductible) costs, c[member of][R.sub.++], for each individual, where a proportion [rho][member of][0,1][subset]R is borne by the taxpayers. In this model, the subsidy is directed to the students in form of reductions in tuition, not directed to institutions of higher education. The government knows only the distribution of the innate abilities, but cannot observe the endowment of each agent. Accordingly, the government can not establish individual-specific subsidies.

It is important to note that the costs of higher education, c, are not tax-deductible. (4) The total costs, therefore, consist of the direct costs, such as teaching aids and tuition fees, and earnings foregone. Basic incomes equal the innate endowment, [y.sub.i]. Students have the opportunity to work even in the first period and, thus, earn the portion h [member of] [0,1] [subset] R of the income earned without higher education. Therefore, the total cost of obtaining higher education amounts to

(1) (1--h)[y.sub.i](l--t) + c(l--r).

Individuals who have completed a degree in the first period will raise their income in the second period because of the rate of return to education. To simplify matters, we adopt the assumption made in Creedy & Francois (1990) that the individual rate of return to education, [s.sub.i], is proportional to the individual endowment:

(2) [s.sub.i] [equivalent to] u x [y.sub.i]

As noted above, in the first period each individual faces the decision of whether to enroll in a degree program or, alternatively, to start working without a university degree. The share of those choosing higher education depends on the exogenously given distribution of y.

The present values of the net lifetime income of educated agents, [V.sup.E], and of non-educated ones, [V.sup.N], are given by

(3) [V.sup.E.sub.i] = (l--t)h [y.sub.i]--c(1--[rho]) + (1--t)[y.sub.i](1 + u x [y.sub.i])/(1 + r) + N

and by

(4) [V.sup.N.sub.i] = (l--t)[y.sub.i] + (l + t)[y.sub.i]/(1 + r) + N.

It is straightforward to find an ability level corresponding to that of an agent who is indifferent to investing in his or her higher education by setting (3) = (4). The agent's endowment is denoted by [[??].sup.[p]] and is henceforth referred to as the educational-choice margin (ECM). It is

(5) [[??].sup.[p]] = [PSI] 9 [square root of ([[PSI].sup.2] + [omega] x (1--[rho])/(1-t))],

where [PSI] = (1- h)(1--r)/2 x u and [omega] = c/u (1 + r). In other words, the educational-choice margin is the ability level at which an agent is indifferent between attending higher education and not doing so. It splits the population by a self-selection mechanism in (at least) two groups. We assume that agents behave atomistically, neglecting the impact of their investment on aggregate income and total tax revenue.

As can be seen, the lump-sum transfer has no impact on the educational-choice margin. This is because the lump-sum transfer is granted to both types of agents uniformly and, therefore, does not distort the choice of educational investment. The only reason why we introduce it is to close up the model.

Our analysis takes place in a self-selection model, similar to the model put forth by Borjas (1987). High-ability agents have an incentive to attend college and to increase their lifetime income. While low-ability agents have no incentive (at least under proportional taxation) to do so, the investment in education skews the income distribution to the right. Thus, the opportunity to attend college increases lifetime income inequality. Note that this result is not crucial to the assumption expressed in equation (2). Consider, for example, the special case [s.sub.i] = u, so that the return is similar for all agents independently of their initial endowment. The educational-choice margin becomes [[??].sup.0] = c(1 + r)([rho]--1)/(t--1)[(h--1)(1 + r) + u]. Similar to the case with [s.sub.i] = u x y, low-ability agents do not alter their gross lifetime income, but high-ability agents will increase their income. Consequently, the lifetime income distribution becomes more uneven compared to a situation without the opportunity to invest in higher education. However, the special case where return does not depend on ability makes the analysis more complicated, because it does not ensure the existence of an educational choice margin in the relevant range. A solution only exists if u is sufficiently high, otherwise [[??].sup.0], y is negative.

For the ongoing discussion, it is useful to define a benchmark equilibrium. For this, we take the noninterventionist, redistribution-free equilibrium, where the government does not implement any income policy, so that the educational-choice margin is fully determined by market forces. This benchmark case is determined by [rho] = t = 0. The educational-choice margin is then given by

(6) [[??].sup.[bm]] = [PSI] + [square root of ([[PSI].sup.2] + [omega]]

We compare the benchmark case with the case that considers a (flat) tax on income (0 < t < 1) and no education subsidies ([rho] = 0). As noted above, we assume that the direct cost of obtaining higher education is not effectively tax-deductible. The educational-choice margin in this case is given by

(7) [[??].sup.[bm]] = [PSI] + [square root of ([[PSI].sup.2] + [omega]/(1 + t)].

As can be seen, the higher t, the higher the educational-choice margin and, consequently, the lower the graduation rate. On the other hand, the educational-choice margin is lowered if part of the cost of obtaining higher education is borne by the state. This can be seen by comparing (5) and (7). The word graduation rate might be misleading. Basically, the graduation rate here is really an enrollment rate, measured by the population share with an ability level greater than the educational-choice margin. We do not allow for the possibility to fail examination. All enrolled students will therefore become graduates after the first period. As a consequence, enrollment rate (or matriculation rate) and graduation rate coincide in this model.

3. THE CASE OF PROPORTIONAL TAXATION

Starting from the benchmark case ([rho] = t = 0), there would be no potential for Pareto improvement through the establishment of subsides to higher education. As there are no tax distortions or other market failures, the outcome is Pareto optimal. Subsidization financed by a nondistorting tax (5) would always lead to redistribution.

However, a more realistic case is one where a distorting income tax is imposed. Hence, starting from [[??].sup.[p].sub.0] we are interested in the effect of various [rho]-values on the educational-choice margin. In particular, we wish to find the optimal rate of subsidization if [[??].sup.[p].sub.0] equals the educational-choice margin in the benchmark case. The subsidy to higher education is said to be efficient if it leads to increased aggregate income.

Proposition 1. Under proportional taxation, a fiscal activity, which consists of the combination of revenue and spending policy, is optimal if the rate of subsidization equals the tax rate. If the rate of subsidization exceeds the tax rate, the educational-choice margin falls and the graduation rate (p) rises. In the opposite case, p falls if [rho]/t < 1.

Proof. If [rho]/t <1 it follows that the term (1--[rho])/(1--t) = 1 and, hence, [[??].sup.[p]] = [[??].sup.[bm]].

[FIGURE 1 OMITTED]

Figure 1 shows the ECMs that result from various [rho]- and t-values. As can be seen along the [rho]-axis, the higher the rate of subsidization, the lower is the educational-choice margin. The opposite holds for the tax rate, except for one special case. This special case arises if the costs of obtaining higher education are totally borne by the government.

Proposition 2. If the direct costs of obtaining higher education are completely borne by the state ([rho] = 1), t has no effect on p.

Proof. If [rho] = 1, it follows that [[??].sup.[p]] = 2[PSI] = (1--h)(1 + r)/u and, thus, is independent of t.

The intuition is as follows. The only distortion under this simple case of a proportional tax system arises from the nondeductibility of the direct cost of obtaining higher education. However, if the direct costs of higher education are completely borne by the state, the distortionary effect of nondeductibility does not play any role, because the agents have nothing to deduct.

Optimality implies that aggregate net lifetime earnings--the sum of the net lifetime earnings of those who do and those who do not invest in higher education--are maximized when subsidization completely countervails the tax distortion. For the case of proportional taxation, we derive optimality when the rate of subsidization coincides with the tax rate. A proof is given in Barbaro (2005, p. 61).

The next subsections shed light on the distortive effect of various kinds of progressive taxation and determine the subsidy rate required to offset such distortions. Under proportional taxation, it would suffice to allow deductibility in order to ensure optimality. However, the analysis of progressive taxation indicates that allowing for tax deduction is not sufficient.

4. PROGRESSIVE TAXATION

We expand the tax schedule from the previous section by a tax-free threshold, [kappa], to model a progressive taxation. A lot of different groups of individuals would have to be considered under this kind of income-tax progression: two groups of individuals who invest in higher education, and two groups that are below the ECM. One subgroup of those investing in human capital pays no taxes in the first period because the [y.sub.i] of its members is below the threshold. Members of the second subgroup pay taxes in the first period as their basic income exceeds the threshold. Of those not investing in higher education, the first subgroup receives a basic income that is below the threshold. Hence, these individuals pay no taxes. The second group of individuals not investing in higher education pay taxes in both periods as their endowment exceeds the threshold ([y.sub.i] > [kappa]). Considering all these cases would certainly complicate the analysis. Therefore, it shall be assumed that the income of students during their qualification period does not exceed the threshold ([kappa] > h x [??])and that all non-graduates pay taxes in both periods.

The net lifetime earnings of non-graduates differ from those of graduates with earnings below the threshold only with regard to the double relief of the threshold (which, of course, has to be discounted in the second period). Introducing an indirect income-tax progression changes the net lifetime earnings of graduates in two ways. First, no income taxes are paid in the first period. Second, the threshold increases income in the second period by [tau] x [kappa]/1 + r.

In the second period, the relief due to the basic allowance is the same for both graduates and non-graduates. Therefore, the effect in the first period is crucial. In this case, the easing of t[kappa] for non-graduates is opposed by an easing of for graduates. As we have assumed above that the income of students during their qualification period does not exceed the threshold, h[y.sub.i] < [kappa] applies. By comparing the relief accruing to graduate and n on-graduates in the first period , (ht[y.sub.i] versus y t x [kappa]) it becomes clear that it is larger for the latter group. Therefore, it is expected that introducing a tax-free threshold will lead to a higher educational-choice margin than will proportional taxation.

With the restricting assumption made above, the net lifetime earnings of those investing in higher education are

(8) [V.sup.E[ip].sub.i] = h [y.sub.i]--c(1--[rho]) + (1--t)[y.sub.i](1 + u x [y.sub.i])+ t[kappa]/(1 + r)+ N

and that of non-educated are given by

(9) [V.sup.N.sub.i] = (1--t)[y.sub.i] + t[kappa] + (1--t)[y.sub.i] + t[kappa]/(1 + r)+ N.

The educational choice margin becomes more complicated, it is given by

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equating (10) and [[??].sup.[bm]] yields the optimal rate of subsidization:

(11) [rho] = t[1 + ([kappa]--[[??].sup.[bm]])/c].

Proposition 3. In the case of indirect income-tax progression, the distortion-correcting rate of subsidization has to be higher than the tax rate.

Proof. First, the assumption that [kappa] > h x [??] implies [kappa] > h x [[??].sup.[bm]]. Second, c is strictly positive.

By rearranging equation (11), we can state that the optimal subsidy rate exceeds unity if

(12) [kappa] > c 1--t/t + h x [[??].sup.[bm]]

In order to show that [rho] > 1 would be a relevant case, we use some proxies for t, c, h, and the educational choice margin to receive a number for the right-hand side. The left-hand side could by quantified simply by looking at the tax law. We use data from the German taxpayer survey (a description of the survey can be found at Barbaro & Sudekum (2006)).

First, we set t = .36, which is a good approximation of the average tax rate (including social security contributions) paid by German taxpayers. Furthermore, we set c = 7,000. According to the OECD's Education at a Glance (2002), the cost of expenditure on educational institutions per student amounted to USD 10,393 (~ 7,000 Euros). Obtaining a proxy for the educational choice margin is more complicated. The enrollment rate in Germany according to the OECD is roughly 40% (see OECD (2002, p. 222)). As an approximation, we use the 60th percentile income from the taxpayer survey (31,300 Euros) and set h = .2, according to the German survey on student life (see Barbaro (2003) for a more detailed description of the dataset). It turns out that the tax-free threshold must exceed 18,704 Euros. The German tax law (of 1998) assigns a threshold of 15,328 Euros (for married taxpayers) and an additional threshold of 2,400 for employees. Summing both values, the total threshold amounts to 17,728 Euros. Using this (rough) approximation, it can be shown that it is not unlikely that the distortion-correcting subsidy exceeds unity.

It is important to note that the subsidy rate expressed in equation (11) depicts only a necessary condition for optimality. As we will see in the Section, optimality can be derived only if the rate of subsidization does not exceed unity. In contrast to the case of proportional taxation, this condition is not always satisfied.

5. OPTIMAL SUBSIDIES AND OVERINVESTMENT

As the preceding chapters have shown, the optimal rate of subsidization may be very high. If taxation is heavily distorting, it may be necessary for the subsidy rate to exceed unity in order to ensure an optimal educational choice margin (in the second-best sense). However, if a subsidy rate greater than unity is required to reconstitute an educational choice margin equal to [[??].sup.[bm]], then subsidization fails to ensure optimality. The reason is that a subsidy rate greater than one means that the direct costs of obtaining higher education are totally borne by the state and agents investing in higher education receive a grant. This grant may exceed the lifetime income of less talented agents, thus encouraging them to invest in higher education, even though they are not suited to pursuing a degree. As a consequence, a fourth group of agents arises, starting from the left-hand side of the density function of y. Thus, the higher the subsidy rate, the greater the number of low-ability agents investing in higher education.

Analytically speaking, two solutions arise from equating (3) and (4): the first is given by equation (5), i.e.,

[[??].sup.[p]] = [PSI] + [square root of ([[PSI].sup.2] + [omega] x (1-[rho])/(1-t))] and a second solution is given

by

(13) [[??].sub.2] = [PSI] + [square root of ([[PSI].sup.2] + [omega] x (1-[rho])/(1-t))].

As long as [omega], [rho] and t are all nonnegative, and 0 [less than equal to] [rho] [less than or equal to] 1, 0 [less than or equal to] t < 1, this second solution is negative, because the square root exceeds [PSI]. Hence, for all [rho] [member of] [0,1], the first solution is unique. On the other hand, if [[??].sub.2] exceeds unity, the second solution is also relevant, because [[??].sun.2] can become positive.

At the beginning of the paper, we stated the individual's decision problem. An individual can choose not to invest in higher education and thus faces the lifetime income expressed in equation (4). Otherwise, lifetime income would amount to the right-hand side of equation (4). By stating the decision problem differently, we can say that an individual trades off the additional income from being educated; that is, (1--t) u x [[??].sup.2.sub.i]/1 + r versus the foregone earning in the first period, (1--t) (1--h)[y.sub.i]. Let us define

(14) Z([y.sub.i]) [equivalent to] (1--t) [u x [y.sup.2.sub.i]/1-r(1-h)[y.sub.i]]

as the difference in earned income an agent may receive if they decide to invest in higher education. It is worth investing if Z([y.sub.i]) exceeds the net costs of obtaining higher education, denoted by (1-[rho]) c. In Figure 2, Z is represented by the solid convex line. To provide an intuition for the shape of the Z(y)-curve, consider first the role of subsidies for low-ability students. Basically, they are paid to attend college. It is worth drawing on the subsidy because the opportunity cost (foregone earnings in the first period) is relatively low. For agents with intermediate endowments, these foregone earnings are too high, and therefore the benefit from subsidization does not exceed the foregone earnings. In addition, the return from college in the second period is only moderate. High-ability agents, on the other hand, are those with the highest foregone earnings but the return from investment is disproportionately high, due to the assumption made in equation (2).

[FIGURE 2 OMITTED]

The horizontal lines represent the costs for different rates of subsidization. The uppermost dashed line, for instance, represents the case without subsidization ([RHO] = 0.0). The direct costs are equal to c, whereas we set c = 2. Without subsidization and taxation, the educational choice margin is given by the intersection of Z and c. All agents with a Z value above c will invest in higher education. For all Z(y)>0 and c(1-[rho])>0, the educational choice margin is unique, because the Z line increases monotonically after having intersected the axis of the abscissa.

Next, consider the case of partial subsidization. The middle dotted line denotes the case where half of the costs c are borne by the taxpayer, ([rho] = 0.5). As in the previous case, the net costs of obtaining higher education are positive, so that the point of intersection between the Z line and either of both the horizontal lines are unique.

Thus, in both cases the following result holds: if individual 1 with an endowment does not invest in higher education, then individual 2 with an endowment [y.sub.2] < [y.sub/1] also abstains from investing.

Next, consider the conceivable case that the optimal rate of subsidization exceeds unity, as shown in the previous section. In that case, the net costs of obtaining higher education are represented by the bottom long-dashed line. Here, the rate of subsidization exceeds unity so that net costs of obtaining education are negative. Still, an individual i with endowment [y.sub.i] will invest if her Z line lies above the costs line. Figure 2, however, shows that in contrast to the previous cases, two points of intersection exist. Low-ability agents also find it worth investing, simply because the grant received from subsidization is substantial relative to lifetime income. For instance, let [rho] = 1.2 such that a student receives a grant amounting to 0.2 x c simply because she decides to attend higher education.

We can state formally two conditions under which the overinvestment result can occur. First, for some [y.sub.i], must hold. This first condition is always satisfied as long as there are some . In the second Section, we made the assumption that the initial endowment is represented by the twice-differentiable density function, f(y), and we stated that all [y.sub.i]'s are positive. With these assumptions, we can now state that a region where the derivative is negative always exists. The second condition is that the rate of subsidization exceeds unity, so that students are paid to go to college. We have already shown this by discussing the existence of a second solution for the educational choice margin (see equation (13)). In Figure 2, the axis of the abscissa coincides with the net cost line if [rho] = 1. Because the Z line starts right from the origin (because y > 0), it follows immediately from the two conditions that the Z line cannot intersect the net cost line twice if [rho] = 1.

5.1 Variation in the rate of return

How does the rate of return, u, affect the number of low-ability students? Consider the educational choice margins. Both [PSI] and [omega] decrease in increasing u values, because u is only in the denominator in both cases. Thus, the expression under the square root decreases in increasing u values, too. As a consequence, both solutions are closer to each other the higher the rate of return. The reason is that a higher u value encourages middle-ability agents to attend college and makes the investment decision of low-ability students more lucrative. A second effect is that the higher u, the smaller [[??].sub.2]. This is a second consequence of the fact that both [PSI] and [omega] decrease in increasing u values. Thus, the size of the region below the axis of abscissa depends crucially on the value of u. In Figure 2, the effect of different u values on the two educational choice margins are plotted. The convex dashed line denotes the Z curve if the return to human capital investment is lower (in this case, u = 0.1) instead of u = 0.2, which has been used to construct the solid convex line. Using such small u values is quite reasonable, as small u values lead to relatively high returns on investment. For example, u = 0.1 and y = 20 implies a threefold return. Nevertheless, the conditions for the existence of low-ability students stated above are both independent of the size of u. The return affects only the size of low-ability students given that the two conditions hold. Stated differently, the return itself is not the reason for the overinvestment result. However, if the overinvestment result holds, then the return heavily affects its magnitude.

6. CONCLUDING REMARKS

In this paper, we analyzed the combined effect of taxation and subsidization on human capital formation. In this, we have supported an example of recent literature that suggests subsidies to higher education can enhance efficiency by offsetting existing tax distortions. However, we highlighted some limits to the offsetting role of education subsidies. This insight may provide a rationale for introducing ability tests when providing grants in order to correct distortions.

The implication of our analysis is that international comparisons of education policy, as carried out, for example, by the OECD (2002, Ch. B), should not focus exclusively on the volume of expenditure by educational institutions. Rather, they should take into account the comprehensive effect of public policy on human capital formation, which clearly includes the tax system. Considering this, it seems that differences among OECD countries are smaller than a first glance at subsidies may suggest.

In our model, distortion-correcting subsidies that exceed unity may lead to an overinvestment of low-ability students. This result has not yet been examined in the literature. Nevertheless, this is a result from a specific model. On the other hand, entry to higher education in many countries is limited by admission requirements. These are likely to prevent overinvestment by low-ability agents. However, in some other countries, particularly in Europe, admission requirements are rarer.

A caveat of the paper is that we only derive partial equilibrium results. In particular, the return is assumed to be independent of investment behavior as a whole, and, therefore, independent of the graduation rate (see equation (2)). In reality, educational systems as well as labor markets are more complex. A political implication of the results derived in the last section could be that low-ability students would not find an appropriate job after having completed their degree. To the extent that educational attainment is assumed to be an indicator of skill, it can signal to employers the potential knowledge and capacities of employment candidates. However, in the longer run, one can suppose that employers find out that the graduate does not have the same capacity expected from a degree holder. The low-ability graduate will, therefore, be forced to accept a job that does not require higher education. Using German data from the Socio-Economic Panel, a dataset containing 31,131 individuals in 2004, we found out that only 50% (exactly: 49.53%) of those with a university degree are currently working in a job that requires a tertiary education diploma. The remainders are employed in jobs where vocational training would suffice or are unemployed. It is, of course, often argued that higher education lowers the risk of unemployment. In virtually all OECD countries, the labor force participation rate of graduates is higher than the participation rate among those without a tertiary qualification. The country mean considering all levels of education is 60% for males and 37% for females. Considering those with a tertiary education alone, the rates are 72% and 53%, respectively (data from OECD, 2002). On the other hand, the effect that graduates may crowd out job applicants with a secondary qualification from appropriate jobs is rarely discussed. This is presumably an important general equilibrium result, the derivation of which is beyond the scope of the paper.

ACKNOWLEDGEMENTS

I am grateful to session participants in both the 18th Annual Congress of the European Economic Association, Stockholm and the 59th Annual Meeting of the International Institute of Public Finance, Prague. I am also indebted to Magnus Hoffmann, Alvaro M. Pina, Helga Pollak, Dorothee Schmidt, Robert Schwager, and Jens Sudekum for very helpful remarks on an earlier version of this paper. I also thank three anonymous referees of PFM for their helpful comments and Carolin Dobler and Janina Pahmeier for their excellent research assistance. The usual disclaimer applies.

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(1) See Barbaro (2003) for a survey of empirical works on the issue.

(2) In addition, Wigger (2004) supported the implications of the above research in the case where subsidies to higher education are combined with optimal linear income taxes a' la Sheshinski (1972), but social welfare cannot be increased by supplementing a nonlinear income tax (in the tradition of Mirrless (1971)) with a subsidization of direct costs.

(3) Building on Heckman et al. (1998); Heckman et al. (1999), Taber (2002) extends the analysis by e.g. performing welfare analysis and by adding a number of robustness checks.

(4) This assumption, which holds for a wide range of countries (see Trostel, 1993), is the driving force in Trostel (1993, 1996).

(5) Optimal-tax theory states that the optimal tax is a lump-sum tax. We can prove that a lump-sum tax, denoted by [tau], does not influence the educational-choice margin. The present value of a graduate's lifetime income is give by h [y.sub.i]--c + [Y.sub.i](1 + [s.sub.i])/1 + r -[tau] and that of a nongradute by [y.sub.i](1 + 1/(1 + r))-[tau]. By equating both, the resulting educational-choice margin is independent of [tau].

(6) [partial derivative]Z/[partial derivative][y.sub.i] < 0 [??] 2u x [y.sub.i] (1-t)/(1 + r)-(1-h)(1-t) < 0. Isolating [y.sub.i] yields [y.sub.i] < (1--h)(1-r)/2u, which is positive.

Salvatore Barbaro

Johannes-Gutenberg University Mainz
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