In an effort to encourage human capital development and spur economic growth, many governments throughout the world heavily subsidize the education of their citizens. These subsidies—particularly those provided to university students—are often objected to on equity grounds, because their benefits are alleged to accrue predominantly to students from more privileged backgrounds. However, this paper raises a second and possibly more fundamental objection: in countries with a limited pool of resources from which to finance investments in physical and human capital, a subsidy to education financed by a general income tax can lead to a long-run reduction in the fraction of the population that chooses to become educated. Thus, the subsidies may yield a result precisely the opposite of what they are intended to achieve. In addition, in many countries heavy subsidies to education have put substantial pressure on fiscal budgets and may not be sustainable. This paper identifies conditions under which cuts in government subsidies to education undertaken for reasons of fiscal adjustment are less likely to lead to substantial declines in the long-run educational attainment of the population.
The paper examines a two-sector aggregative growth model where a spillover exists between the sector producing human capital and the sector producing a consumption good. In the model, the current output of the consumption good sector is increasing in the current output of the education sector. However, firms in the education sector are unable to capture the benefits of this spillover. As a result, the level of investment in education and the long-run fraction of the population that is educated are lower than the social optima. One way for the equilibrium to be moved toward the social optimum could be for the government to collect a lump sum tax from producers of the consumption good and use the proceeds to subsidize the production of education. If there are legal or practical prohibitions against such a course of action, however, the government may instead attempt to increase the output of the education sector by providing subsidies on the demand side: that is, by offering to pay a portion of each student’s tuition if he chooses to enroll in school. As noted above, the analysis of this model indicates that under certain circumstances a positive tuition subsidy will lead to a long-run reduction in the fraction of the population that is educated. Put another way, if the government wishes to increase the long-run fraction of the population that is educated the correct subsidy may be negative.
Although positive spillovers are a common feature in endogenous growth models, 1/ to date their use has been restricted to models with full employment of human and physical capital. Moreover, agents in these models typically determine how much human capital to acquire not by weighing the return on education against the return on holding an alternative asset but rather by equilibrating the extra earnings they will get from increasing human capital with the loss of wages they will suffer from taking time away from production of physical capital. The model in this paper therefore differs from its predecessors both in assuming that the market for educated labor does not clear and in modelling the decision to acquire education as depending specifically on the returns on other available assets. Thus, the decision whether to invest in additional human capital in this model depends explicitly on the current and future rate of interest.
The paper focuses on the effects of government tuition subsidies to education, which are substantial in many parts of the world. Table 1, below, provides information on the percentage of central government spending dedicated to education in several African countries. The data indicate that education consumes a sizable portion of government budgets throughout the continent. Moreover, as cost-recovery efforts in education are at a preliminary stage in most countries in Africa, virtually all of this spending represents a subsidy.
The model in this paper also features unemployment among the educated, which remains a serious problem in many countries. Although the phenomenon is well-documented in many developing countries—for example by Blaug et. al. (1969) in India—it occurs in developed countries, as well. For example, one study of Italian labor markets in the 1970s found that unemployment rates correlated positively with educational achievement. 1/ The presence of educated unemployment is frequently attributed to government subsidies to education that encourage excessive investment in human capital. In the model considered in this paper government subsidies to education may lead to an increase in educated unemployment, but they do so not by increasing the fraction of the population that chooses to become educated but rather by decreasing the equilibrium per capita stock of physical capital.
This paper extends a two-sector aggregative growth model developed by Findlay and Rodriguez (1981) and Chaudhuri and Khan (1984). Upadhyay (1993) modifies these models to include increasing returns to scale in the production of education and finds that a positive tuition subsidy increases the demand for education and the educated unemployment rate. By contrast, this paper demonstrates that when the education sector produces not only newly educated labor units but also new knowledge that increases productivity in the composite output good sector, tuition subsidies can reduce the demand for education and the educated unemployment rate. Moreover, this paper introduces to the model rational expectations in the form of perfect foresight, which allows for a richer analysis of the two asset problem.
The following Section describes the model, solves for the temporary and long-run equilibria and examines the impact of an educational subsidy on the long-run equilibrium. Section 3 offers some conclusions.
II. The Model
The economy consists of two sectors, one producing a consumption/capital good as r. function of educated labor, uneducated labor and capital, and the other producing newly educated labor as a function of educated labor and capital. However, there exists a spillover between the production of educated labor and the production of the consumption good so that, certeris paribus, the greater is the current output of the education sector the greater will be the current output of the consumption good sector. Specifically, assume the following production functions (with time subscripts suppressed for clarity):
where X is the output of the education sector (the instantaneous increment to the existing stock of educated labor), Q is the output of the consumption good, Ei and Ki are the stocks of educated labor and of capital, respectively, that are used to produce good i, L is the stock of uneducated labor, δ is a parameter and N is the current population. Thus, for given stocks of educated labor, uneducated labor and capital employed in producing the consumption good, the larger is the per capita output of the education sector the larger will be the output of the consumption good sector. 1/ Per capita rather than absolute output of the education sector is chosen as a normalization to indicate that the spillover depends on the relative, not absolute, size of the education sector in the economy.
This formulation attempts to capture the idea that the education sector does more than simply create newly educated labor but also in the process creates new knowledge that can be applied to the production of output (as in the endogenous technological change models of Romer (1990), Bartelsman (1990) and others). In addition, it attempts to capture the idea that there are networking effects with respect to knowledge, so that the greater the fraction of educated people in a given society the more they are able to learn from each other and the more productive they will be. 2/ At any instant t, the total stock of educated labor and of capital available to the economy, and the total population, are predetermined. Accordingly, the economy faces the constraints
Newly educated workers are added instantaneously to the existing stock of educated workers, meaning that the stock of educated labor grows according to the equation
Capital is simply stored consumption good. Hence if C(t) is consumption at t,
Consumption good that is stored in one period may be consumed in any subsequent period, and there is no depreciation of capital.
Both the education and consumption/capital goods sectors consist of competitive, profit-maximizing firms. In maximizing their profits, firms face the standard costs of rental on capital and wages for educated and uneducated labor. While the rental rate, the uneducated wage and the price of education are all determined endogenously, we assume that the wage paid to educated labor,
If we take the price of the consumption good as the numeraire and denote the price of education as P, then total profits in each sector are given as
where r is the rental rate for a unit of capital and wL is the wage paid to uneducated labor. If each firm takes the price of education, the uneducated wage and the rental rate for capital as given when making its output decision, and if the spillover from the education sector is a pure public good that the education sector cannot capture, then the first order conditions for profit maximization in the output sector are
while those in the education sector are
To close the model an investment efficiency condition is developed that along with the assumption of perfect capital markets governs the decision of risk-neutral individuals to seek education. 1/ Consider an average uneducated individual deciding whether to become educated at instant t. If he chooses to become educated he pays tuition P(t) and then earns a wage of
Before going on to solve for the temporary and long-run equilibria of this economy we demonstrate that the output of the education sector is lower in the market equilibrium than would be the case in a social planning problem (or, equivalently, if education firms could perfectly capture the spillover to the output sector). First order conditions for profit maximization when the education sector can capture the benefits of the spillover are
Compared to those for the market equilibrium these first order conditions suggest that the physical marginal products of educated labor and capital are relatively lower when education firms can capture the spillover, and therefore that the output of the education sector is relatively higher. Moreover, since the per capita fraction of the population that is educated grows according to
where e = E/N, x = X/N and n equals the rate of population growth, this suggests that if a long-run equilibrium exists then the long-run fraction of the population that is educated will be higher under the social planner’s solution than under the market equilibrium.
In addition, note that by totally differentiating the production functions and adding up constraints for the economy, making use of the first order conditions, and manipulating, the transformation curve for the economy can be solved for as
where Γ = Q/PX, the ratio of the value of the output of the consumption/capital good sector to the value of the output of the education sector. Regardless of the signs of dX/dP (which will be shown below to be positive) and du/dP (which will be shown to be ambiguous), when the value of the education sector’s output is relatively small the transformation curve may have a positive slope. The intuition behind this result is that when the output of the education sector is low, shifting resources away from the production of Q, where their marginal product is low, toward the production of X, where their marginal product is high, can lead to a net increase in the output of both sectors if the spillover effect on the consumption/capital good from the production of additional education exceeds the direct effect of reduced resources employed in the consumption/capital good sector. Because of the unemployment of educated labor, production will always occur within the production possibilities frontier. The social planner will never choose an equilibrium allocation on the upward-sloping portion of the transformation curve, because for any such allocation there exists another allocation that is both feasible and Pareto improving, offering more of both Q and X. However, because the spillover cannot be captured by education firms the relative prices used in the consumption and production decisions in the decentralized equilibrium differ from those used by the social planner and an inferior equilibrium may be realized.
2.1 Temporary equilibrium
At any instant t the price level and the total stocks of educated labor, uneducated labor and capital are all predetermined. Therefore, we first consider equilibrium under the market solution for a given K(t), E(t) and P(t) before looking at the long-run behavior of the economy.
Perfect competition and constant returns to scale in the production of education gives
where aiX is the input-output ratio in the production of x (e.g. aEX = Ex/X). Total differentiation and rearrangement yields
where θKX is the share of capital in the price of education (i.e. θKX = rKX/PX and ẑ = dz/z and where we take advantage of the fact that cost minimization in the education sector implies
Constant returns to scale in F(Kq, Eq, L) yields the following equation
where aKQ = Kq/F(Kq, Eq, L) and aEQ and aLQ are similarly defined. Total differentiation and rearrangement yields
where this equation is again simplified by taking advantage of the first order conditions for cost minimization by firms.
Taking the adding-up conditions for each of capital, uneducated labor and educated labor and dividing each by N to put them in per capita terms yields the following three equations:
Total differentiation and rearrangement of the first and third of these equations yields
where λki is the fraction of K that is used to produce output in sector i (so that λKQ = Kq/K, for example) and where ρ = e/(1 − e). These two equations can then be used to solve explicitly for
Thus, with constant factor prices (and hence constant input-output ratios) an increase in the per capita stock of capital or in the fraction of the population that is educated will raise the per capita output of the education sector. An increase in the per capita stock of capital will also raise the output of the consumption good sector, while an increase in the fraction of the population that is educated may increase or decrease the output of the consumption good sector depending on whether production of the consumption good is more or less capital intensive than is production of education. The intuition behind this last result is that with the total population fixed, an increase in the fraction of the population that is educated can only occur via a decrease in the fraction of the population that is not educated. With uneducated labor fully employed producing the consumption/capital good, this will tend to decrease output of the consumption/capital good (with an elasticity of ρ). However, the increase in the fraction of the population that is educated tends to increase the output of the education sector (with an elasticity of ρλKQ/λKX) which, through the spillover, will tend to increase output of the composite good as well. Accordingly, the net effect on output of the consumption/capital good will depend on the relative sizes of the two elasticities.
Each of the âij represent changes in the techniques of production employed in each sector. Of course, these will depend on factor prices, so
Totally differentiating we get
If we assume that both types of labor are complements and that capital is a substitute for both types of labor in both production functions, and if we assume λKQ > λKX then we can determine that ηK > 0 and ηL < 0 unambiguously.
The determinant of the left-hand-side matrix is clearly positive. A sufficient condition for the coefficient on ê in the bottom row of the right hand-side column vector to be positive is λKX > λEX (which, given the earlier assumption that λKQ > λKX would imply the ordering λEQ > λKQ > λKX > λEX). Alternatively, if ρ is sufficiently small or if the share of capital going to education is not much less than the share of employed educated labor doing so, the coefficient on è will be positive. Assume that one of these conditions holds. We also assume that βK is negative. This allows us to determine the following partial derivatives:
The effect of an increase in the price of education on the educated unemployment rate is ambiguous because it increases both the reward to capital (which is a substitute for educated labor) and the reward to uneducated labor (which is a complement to educated labor). Thus, the changes in factor rewards resulting from an increase in P move the unemployment rate in opposing directions. If βKηL > βLηk then an increase in the price of education tends to reduce the unemployment rate for educated labor. Remarkably, the partial of the interest rate with respect to the per capita stock of capital is zero: the interest rate is determined solely by the price of education, which is predetermined at any t.
The assumptions already made about absolute factor intensities are sufficient to ensure that increases in the fraction of the population that is educated or in the per capita stock of capital will—with constant factor rewards—increase the output of both the education and the consumption good sectors. In addition, it can be shown that an increase in the price of education will lead to an increase in the output of the education sector. To see this, solve Equation 1 for
The effect of an increase in the price of education on the output of the consumption good sector is more difficult to determine, however. As noted above, the increase in P tends to raise the wage paid to uneducated labor, which is the extreme factor in the production of the consumption good. This tends to reduce the output of the consumption good sector. However, the increase in the output of the education sector spills over into the consumption good sector, which tends to increase Q. The following section concentrates on the case where the effect of the spillover is sufficiently small that
2.2 Capital accumulation
The equations of motion for the three predetermined variables in the model are
If we assume that investment in physical and human capital is financed by saved revenues, and that all individuals save a constant fraction s of their incomes, then
This and the accumulation equation for educated labor can be rewritten in per capita terms as
To evaluate the stability of the model we need to take the derivatives of each of the equations of motion with respect to k, e and P:
These equations can be arranged in matrix form as follows:
Among the partials, ∂k/∂P is unambiguously negative and ∂ê/∂k, ∂ê/∂P, and
then the determinant of the left hand-side matrix is unambiguously positive. If the trace is negative, which is more likely the larger is the rate of population growth, then the model will feature a saddle-plane. 1/
As noted earlier, the inability of firms in the education sector to capture the benefits of the spillover from their sector into the consumption good sector makes the output of the education sector—and the long run fraction of the population that chooses to acquire education—lower than that which would be chosen by a social planner. The optimal solution to this problem may be for the government to apply a lump-sum tax to the consumption good sector and to distribute the benefits to the education sector so that the first order conditions of the market solution match those of the social planning problem (that is, offer to pay each firm in the education sector δF(Kq, Eq, L)XE/N for each educated worker that it hires and δF(Kq, Eq, L)XE/N for each unit of capital that it employs). 2/ Suppose, nowever, that for administrative, political or legal reasons the government is unable to adopt this course. In that case, governments may instead seek to increase the output of the education sector by working through the demand side. For example, the government may offer to pay a fraction of the tuition of each student who chooses to enroll in school, hoping that this will raise demand for education and thus the long-run fraction of the population that is educated.
To analyze the impact of educational subsidies on the long-run fraction the population that is educated, the long-run per capita stock of capital and the long-run price of education, assume that the government will pay a fraction 0 < r < 1 of each student’s tuition costs and that the subsidy is financed by a general income tax. 3/ In that case, the per capita stock of capital grows according to
while the equilibrium condition for the change in the price of education can be rewritten as
since the cost of education for the student is lowered to P(1 − τ). At any instant t the price of education, the per capita stock of capital and the fraction of the population that has chosen to become educated are predetermined. Accordingly, an increase in the subsidy to education can affect the equilibrium only through its impact on the time path of these three variables and on their long-run equilibrium values. Totally differentiating Equations 11, 12, and 13 at the equilibrium given by
where the asterisks on each of k, e, and P indicate that they are the long-run equilibrium values of those variables.
In order to draw any conclusions about the comparative static properties of this model we need to determine the signs of ∂k/∂τ,∂ê/∂τ and
A rise in the subsidy to education reduces the per capita volume of private sector savings used to finance education by Pxdτ but reduces per capita private sector savings by only sPxdτ. Thus, the volume of savings available to finance physical capital tends to increase when the subsidy increases. A rise in the subsidy to education also tends to reduce the benefit of deferring education, which makes the expected benefits of acquiring an education exceed those of waiting to do so. In order to return the economy to an equilibrium where individuals are indifferent between acquiring an education in the current period or waiting until a subsequent period to do so, the price of education must be expected to fall (or to grow less quickly) in the future. Typically,
It is easy to demonstrate the following propositions:
Proposition 1 An increase in the subsidy to education will have an ambiguous effect on the long-run per capita stock of capital. Other things being equal, the larger the rate of subsidy to education and the rate of savings, the more likely is the stock of capital to fall.
Proposition 2 The effect of an increase in the subsidy to education on the long-run fraction of the population that chooses to become educated is ambiguous. Other things being equal, the relatively smaller the education sector in the economy, and the relatively larger the rate of subsidy to education, the rate of savings and the rate of population growth, the more likely is an increase in the subsidy to lead to a decline in the fraction of the population that chooses to become educated.
Proposition 3 An increase in the subsidy to education will lead to an unambiguous increase in the long-run equilibrium price of education.
The effects of an increase in the subsidy to education on the long-run equilibrium per capita stock of physical capital, fraction of the population that chooses to become educated and price of education are illustrated graphically in Figures 1, 2, and 3. In Figure 1, which is drawn for a given P, the rise in τ leads the k = 0 locus to shift outward but has no effect on the ê = 0 locus, leading to an increase in both k and θ. Starting from the long-run equilibrium per capita stock of physical capital, an increase in the subsidy to education financed by an income tax reduces the volume of private sector savings but leads (for a given P) to an even bigger decline in the volume of investment that must be financed by the private sector. Thus, the per capita stock of physical capita tends to increase. With a higher level of physical capital the output of the education sector also increases, which implies a higher equilibrium fraction of the population that chooses to become educated. Because investments in education deplete the pool of savings from which investments in physical capital can be financed, the increase in the long-run fraction of the population that chooses to become educated helps to moderate, in part, the increase in the per capita stock of physical capital. The greater the rate of savings in the economy and the faster the rate of population growth, the smaller will be the increases in k and e.
Figure 1Effects of Increase in τ P = P*
In Figure 2, which is drawn for a given k, the increase in the subsidy to education leads the
Figure 2Effects of Increase in τ k = k*
In Figure 3, which is drawn for a given e, the increase in the subsidy to education leads to a backward shift of the
Figure 3Effects of Increase in τ c = c*
Thus, the net effect of the increase in the subsidy to education on the long-run fraction of the population that chooses to become educated is ambiguous: although the subsidy reduces the fraction of a student’s tuition that he must pay himself, which encourages uneducated workers to enroll in school, the subsidy also leads to an increase in the price of education, which can result in a decline in the long-run per capita stock of physical capital. This lower stock of physical capital will tend to reduce the output of the education sector. Moreover, the lower capital stock and higher price of education will tend to raise the rate of educated unemployment, the rate of interest and the wage paid to uneducated labor, reducing the incentives to enroll in school. In the context of Figure 1, the general equilibrium effect of an increase in the subsidy to education involves three shifts of curves. First, as illustrated in the Figure the k = 0 locus shifts outward, tending to increase k and e. However, the increase in P that accompanies the increase in the subsidy leads to a backward shift of the k = 0 locus and an outward shift of the ê = 0 locus. Whether the net effect of these three shifts will be an increase or decrease in the long-run fraction of the population that chooses to become educated will depend on the relative sizes of the three shifts. Other things being equal, the larger are the subsidy to education, the rate of savings and the rate of population growth, and the smaller is the share of the stock of physical capital used in the education sector, the more likely is the long-run fraction of the population that chooses to become educated to decline as the result of an increase in the subsidy to education.
This paper has examined a model where a spillover exists between the production of human capital in the education sector and the production of output in the consumption good sector. In the model, higher levels of output in the education sector lead—for given amounts of educated labor, uneducated labor and capital devoted to the production of the consumption good—to greater levels of output in the consumption good sector. Because educational firms are not compensated for this spillover they do not take it into account when they decide how much educated labor and capital to hire. As a result, the level of educational output in the economy is lower than that which would be chosen by a social planner. If the government seeks to increase the level of educational output by offering to subsidize the tuition of any individual who chooses to enter education, we found that in the long-run this could actually lead to a decrease in the fraction of the population that chooses to be educated. Put another way, if the government seeks to increase the fraction of the population that is educated the optimal subsidy could be negative. A positive subsidy to education will lead to an increase in the price of education and possibly to a long-run decline in the per capita stock of physical capital, reducing both the output of the education sector and the incentive to enroll in school. By contrast, a negative subsidy toward the purchase of education would reduce the price of education and could encourage investment in physical capital. The negative subsidy could thereby reduce the educated unemployment rate sufficiently to increase the expected return on an investment in education, leading to a long-run increase in the fraction of the population that chooses to enroll in school.
The policy implications of these results are significant, perhaps most so for many developing countries. In many of these countries some of the assumptions of the model, for example, a relatively limited pool of resources from which to finance investment in physical and human capital, are most likely to hold. In addition, in many developing countries the education sector is likely to be both relatively small and relatively non-capital intensive, 1/ and population growth is likely to be relatively rapid, increasing the likelihood that subsidies to education will lead to decreases in the fraction of the population that chooses to become educated. Accordingly, in these countries reductions in subsidies to education may not only relieve strains on the fiscal budget and help restore equity, but may even lead to increases in the fraction of the population that chooses to become educated. The conditions necessary for a cut in the subsidy to result in an increase in the fraction of the population that chooses to become educated will of course not be present in every country at every moment. Even when these conditions do not hold, however, the results of the paper illustrate that the larger the rate of subsidy to education, of population growth and of savings, and the smaller the relative size of the education sector in a given country, the less likely are cuts in that country’s educational subsidies to yield substantial declines in the long-run educational attainment of its population.
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This paper is a revised version of Chapter 3 of my PhD dissertation [Gerson (1993)]. I thank my advisors, M. Ali Khan and Louis J. Maccini, for their assistance. I also thank Stephen Blough, William Carrington, and Se-Jik Kim for their comments on earlier drafts.
Upadhyay (1993) employs a related functional form for the production of education in order to generate increasing returns to scale in that sector.
Implicitly, this functional form assumes an instantaneous depreciation rate of 1 for new knowledge or for networking effects from the newly educated. It may be worth emphasizing that new knowledge in this model effects the physical production process for consumption goods itself, rather than, for example, encouraging the production of consumption goods through a shift in demand.
The assumption of a fixed educated wage is made for analytical convenience. Chapter 2 of Gerson (1993) examines a similar model where the educated wage is endogenous (but long-run educated unemployment persists because firms face a turnover cost when educated workers quit) and obtains results similar to those of this paper. An alternative explanation for the persistence of educated unemployment could, for example, be that firms and workers lack information about supply and demand conditions in specialized markets for educated labor. It may also be worth noting that although the educated wage is fixed, individuals’ decisions in the model depend on the expected lifetime incomes of the educated, which are endogenous because of educated unemployment.
Although human capital-based models of growth are a relatively recent phenomenon, the implications of education and training for economic growth were clear to the classical economists. In fact, the spirit of the equilibrium condition used in this paper can be attributed to Adam Smith, who noted
The wages of labour vary with the easiness and cheapness, or the difficulty and expence of learning the business. When any expensive machine is erected, the extraordinary work to be performed by it before it is worn out, it must be expected, will replace the capital laid out on it, with at least the ordinary profits. A man educated at the expence of much labour and time to any of those employments which require extraordinary dexterity and skill, may be compared to one of those expensive machines. The work which he learns to perform, it must be expected, over and above the usual wages of common labour, will replace to him the whole expence of his education, with at least the ordinary profits of an equally valuable capital.
(Smith (1976), pp.113-114.)
With a positive determinant, the product of the three roots must be positive. With a negative trace, the sum of the roots must be negative. These two conditions imply that model has two negative roots and one positive root, or that the convergent subspace is two-dimensional. If the trace of the matrix is positive, it is not possible to determine whether the model is globally unstable or features a saddle plane.
In fact, it may be the case that since universities usually deduct a fraction of faculty research grants for overhead, government-sponsored faculty research serves as one form of such a transfer.
The effects of alternative tax regimes on the long-run equilibrium of the economy could be an interesting area for additional analysis.
For example, in many African countries the higher education sector is relatively small, with many students studying at government expense in foreign universities. In addition, education in many developing countries may be less capital intensive than in more developed countries because schools are less likely to have large investments in libraries, laboratories, audio-visual equipment, etc.