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The effect of uncertain returns on human capital investment patterns.

Introduction

The recent trend of the graying of the average college student has not received the attention it warrants in economics literature. Adult students, those 25 years old and over, constitute an ever increasing percentage of the college population. In 1990, it is estimated that approximately 43 percent of all students over the age of 18 were 25 years old or over. Additionally, 21 percent of all full-time students over the age of 18 were 25 years old or over. This is up from only 11 percent of all full-time students in 1970 [American Council on Education, 1989-90, Table 47]. This increase is not simply the result of more graduate students who are, by definition, older. Rather, this is due to an increase in the number of adults returning to undergraduate classes. From 1980 to 1990, the number of undergraduate students age 25 to 34 grew by 29 percent while the percentage of the population age 25 to 34 grew by approximately 5 percent [U.S. Department of Commerce, Current Population Reports, 1970-1990, Social and Economic Characteristics of Students, 1970-90]. This trend is not just the product of drawn-out college attendance but, rather, as pointed out by Light [1995], due in part to interrupted or delayed college enrollment.

The increased number of older college students is widely discussed in education circles but has received limited attention in the economics profession. The primary emphasis of the existing economics literature on college timing is on the role of differences in individual characteristics. In their seminal work, Rosen and Willis [1979] address differences in unobserved ability in the schooling decision. Spletzer [1990, 1992] addresses liquidity constraints in the enrollment process over time. Light [1995] investigates preenrollment employment differences among individuals and their impact on the decision to reenroll in college and Altonji [1993] develops a model of changing college majors in response to uncertain returns to an individual's field of study.

The current literature does not address the role of changing expectations of the returns to education and the effect it may be having on the decision for an individual to resume schooling. This paper fills that void in literature by providing a theoretically plausible hypothesis of school delay motivated by updates in the expectations of the returns to human capital. This theoretical construct is consistent with the empirically well-established, stylized fact of increasing returns to education during the 1980s. The standard model of human capital investment, formalized by Ben-Porath [1967], results in monotonically decreasing investments in human capital. The recent trend of postponing or resuming higher education after a period of declining investment in education warrants a reexamination of the standard human capital model and its results. The Ben-Porath model provides for a static rental rate of human capital and, thus, an investment pattern that is unaffected by changes in the rental rate. However, the model developed in this paper incorporates a dynamic and uncertain rental rate of human capital and results in patterns of increasing investment in human capital. If expectations of the future rental rate of human capital increase, then the marginal revenue of human capital may not monotonically decrease over time. So investment in human capital may not steadily decrease either. As a result, it may be optimal for an individual to increase investment in human capital if expectations of the rental rate of human capital increase enough. Intuitively, the shorter the time horizon remaining to reap the higher returns to human capital, the greater the increase in the expected rental rate of human capital necessary to prompt an increase in investment.

The Model

The following human capital model formalizes the hypothesis of changing expectations of the returns to education resulting in a resumption of schooling. The model is one in which an individual maximizes discounted life-cycle utility, subject to certain restrictions, and presents an equation of motion for human capital. Additionally, it is assumed that the individual is risk-neutral and free to borrow against future earnings so that the problem of maximizing the expected life-cycle utility as a function of consumption simply becomes maximizing the expected present discounted value of lifetime wealth. The standard Ben-Porath human capital model is adapted in this case to include a time-varying and uncertain rental rate of human capital, [Pi](t).

The following expressions outline the model:

[Mathematical Expression Omitted], (1)

subject to:

[Mathematical Expression Omitted], (2)

0 [less than or equal to] S(t) [less than or equal to] 1, (3)

K(0) = [K.sub.0]. (4)

Equation (1) represents the problem of maximizing expected life-cycle utility of consumption, C(t), under the assumption of perfect capital markets. In this case, maximizing expected utility is the same as maximizing expected lifetime wealth, the right-hand side of (1). The individual must now decide what percentage of his human capital to devote to investment in additional human capital, S(t). This control variable can be thought of as the percentage of time devoted to human capital accumulation. S(t) = 1 is equivalent to full-time investment in human capital. K(t) is accumulated human capital and [Pi](t) is the rental rate of human capital. Equation (2) is the equation of motion for human capital. It represents a production function for human capital accumulation. [Alpha] and b are parameters of this production function that translate investment in human capital to changes in the capital stock and [Phi] is the depreciation rate of human capital. Finally, r and 6 are the interest rate and subjective discount rate of time, respectively.

Solving for the optimal investment behavior necessary to maximize expected life-cycle wealth subject to the equation of motion for human capital requires solving the Hamiltonian equation. The Hamiltonian for this model is:

[Mathematical Expression Omitted], (5)

where [E.sub.t] indicates expectations taken at time t and [Lambda](t) is the marginal revenue of human capital investment. The necessary and sufficient canonical equation with respect to the percent of human capital devoted to investment in human capital, S(t), is:

[Mathematical Expression Omitted]. (6)

Solving (6) for the marginal revenue of human capital, [Lambda](t), and the amount of human capital investment, S(t)K(t), yields:

[Lambda](t) = [E.sub.t][Pi](t)/[Alpha]b [(S(t)K(t)).sup.1-b], (7)

[Mathematical Expression Omitted], (8)

Minus the partial derivative of the Hamiltonian with respect to the capital stock is:

[Mathematical Expression Omitted]. (9)

Substituting for [Lambda](t) from (7) above yields:

[Mathematical Expression Omitted]. (10)

Further, solving this differential equation for the marginal revenue of human capital in terms of the present discounted value of the expected future rental rate of human capital yields:

[Mathematical Expression Omitted]. (11)

This implies that the marginal revenue of human capital is clearly an increasing function of the expected future rental rate of human capital. Taking the time derivative of this optimal shadow price of human capital results in:

[Mathematical Expression Omitted]. (12)

Equation (12) indicates that the change in the marginal revenue of human capital over time is equal to the change in marginal revenue over time, holding expected future rental rates, [E.sub.t][Pi]([Tau]), constant. It also indicates the change in the marginal revenue of human capital due to changes in the expected future rental rate of human capital.(1) If expectations of future returns increase, then the second term on the right-hand side is positive. If this term is large enough, it may imply an increase in the marginal revenue of human capital over time. This result is significant since the standard Ben-Porath human capital model implies monotonically decreasing investments in human capital over time, even if the rental rate of human capital increases. In this model, however, investments in human capital may increase over certain intervals because of changes in expected future returns to human capital. The rate of change in the marginal revenue of human capital increases with changes in the expected rental rate:

[Mathematical Expression Omitted]. (13)

Consequently, investment in human capital, S(t)K(t), will decrease over time if expectations of the returns to education are static, as in (14), but will increase with increases in the expected rental rate of human capital, as in (15):

[Mathematical Expression Omitted], (14)

[Mathematical Expression Omitted]. (15)

This result implies that increases in the expected future rental rate of human capital will increase current investment in human capital and will accelerate the investment process as well. Equation (15) indicates that rising expectations of the future returns to human capital will prompt an increase in the marginal revenue of human capital, [Lambda](t), and, thus, an increase in human capital investment, S(t)K(t), ceteris paribus.(2) Of course, expectations do not adjust instantaneously as suggested by (15) but, rather, adapt over time. If the increase in expected future returns to education are large enough to offset the negative effect of time on marginal revenue of human capital investment, then investment in human capital may increase. This is the intuition behind the time derivative of the marginal revenue of human capital in (12).

Conclusion

The results of this model imply that investment in human capital is dependent upon expectations of future rental rates of human capital. An increase in expected returns to human capital may result in an increase in investment in human capital. Clearly, a greater increase in the anticipated rental rate is necessary to increase the marginal revenue of human capital when the shorter the remaining time horizon. Similarly, otherwise identical individuals may have different investment patterns due to varying expectations of future returns to human capital investment. By introducing changes in expected rental rates of human capital, optimal full-time investment is no longer limited to early in the life cycle. An increase in expected wages paid to human capital may lead to increasing marginal revenue of human capital and, thus, an increase in investment in human capital. The decision to invest in human capital at time t is dependent on the expectations of the future benefits of that investment. The greater the increase in expected returns to human capital, the greater the investment in human capital.

This theoretical construct is consistent with the stylized fact of an increasing education premium through the 1980s. The human capital model outlined here reveals that changing expectations of the returns to human capital may result in increases in human capital investment. The human capital model formalized in this paper is consistent with recent changes in the returns to education and also with recent trends in increased enrollment among older students.

Changing expectations of the returns to human capital is not the only possible explanation for an increase in human capital investment. Using a similar approach, Polachek [1975] outlines a model of differences in labor force participation which results in non-monotonically decreasing human capital investment. Additionally, fluctuations in the depreciation rate of human capital may also lead to increases in human capital investment. Clearly, there are a number of alternative explanations for the trend of later schooling. The model here merely attempts to outline a plausible hypothesis to explain the recent patterns of increasing investment in human capital later in the life cycle which is consistent with the increasing education wage premium.

Footnotes

1. t indicates that expectations are taken in the current period regarding the rental rate of human capital in future periods, [Tau].

2. Changes in the current rental rate of human capital that are matched by exact increases in the expected future rental rate of human capital, naive expectations, do not affect investment in human capital. In this case, the model simply reverts to the standard Ben-Porath version where the future returns to human capital are exactly offset by the increased opportunity costs of using stock of capital for further investment. In this case, the second term on the right-hand side of (12) becomes zero and investment declines monotonically over time.

References

Altonji, Joseph. "The Demand for and Return to Education When Education Outcomes Are Uncertain," Journal of Labor Economics, 11, 1, 1993, pp. 48-83.

American Council on Education. Fact Book on Higher Education, New York, NY: Macmillian, 1989-90.

Ben-Porath, Yoram. "The Production of Human Capital and the Life-Cycle of Earnings," Journal of Political Economy, 75, 1967, pp. 352-65.

Light, Audrey. "Hazard Model Estimates of the Decision to Re-Enroll in School," Labour Economics, 2, 4, 1995, pp. 381-406.

Polachek, Solomon William. "Differences in Expected Post-School Investment as a Determinant of Market Wage Differentials," International Economic Review, 16, 2, 1975, pp. 451-70.

Rosen, Sherwin; Willis, Robert J. "Education and Self-Selection,"Journal of Political Economy, 87, 5, 1979, pp. S7-S36.

Spletzer, James. "Post-Secondary Education and Labor Supply in the Early Stages of the Life Cycle," unpublished mimeo, Office of Research and Evaluation, Bureau of Labor Statistics, 1990.

-----. "Testing for the Presence of Liquidity Constraints in Post-Secondary Educational Attainment," unpublished mimeo, Office of Research and Evaluation, Bureau of Labor Statistics, 1992.

U.S. Department of Commerce, Bureau of the Census. Current Population Reports, P-20 Series, Washington, DC: U.S. Government Printing Office, 1970-90.

U.S. Department of Commerce, Bureau of the Census. Social and Economic Characteristics of Students, Washington, DC: U.S. Government Printing Office, 1970-90.
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Author:Monks, James
Publication:Atlantic Economic Journal
Date:Dec 1, 1998
Words:2183
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