027277WY.t
$3 00  O.IWl
8 1%lPergamonPressLtd
Imperfect Capital Markets and Investment in Education EDWARD Department of Economics,
Cleveland
State
B. BELL
University.
Cleveland.
OH
ItIlS.
U.S.A.
Abstract This paper examines a theoretical issue concerning the relationship between private pecuniary returns to a college education and the fraction of highschool graduates who choose to enroll in college. If capital markets are imperfect in the sense that the rate of interest at which individuals can borrow exceeds the rate at which they can lend, then we cannot rule out the possibili!y of a paradox; namely. enrollment rates may not always be directly related to the net present value of an education. 0 model of an individual’s investment The result is developed from a twoperiod utilitymaximizin, decision.
INTRODUCTION
recognized and have served as a focus for debate. For example, researchers have discussed and debated alternative methods and data sources which have been used to estimate anticipated ageearnings profiles for highschool graduates, anticipated ageearnings profiles for college graduates and the direct costs of acquiring a college education. These issues are important because differences in methods and data sources have generated conflicting results about whether or not the financial returns to a college education have increased over certain periods of time (see, for example, Freeman. 1975, 1980; Rumberger. 1980; Witmer 1980). The discount rate is also an important element in estimating the net present value of an education. Some economists have suggested that a relatively high discount rate. the interest rate on personal loans, may be appropriate in some cases (Wilkenson, 1966). while other studies offer little or no discussion of the basis for the discount rate which is used. There have been instances in which the results of studies on the returns to education have been challenged on the basis of the discount rate used (Schwartz and Thornton. 1980). In this paper I argue that discount rates are
THERE are a number of studies on the relationship between education investment choices and the financial returns to education. A common hypothesis is that the fraction of highschool graduates who choose to attend college is directly related to the net present value of anticipated lifetime earnings (less direct costs of college) of a college graduate and the present value of anticipated lifetime earnings of a highschool graduate. When empirical evidence does not clearly support the hypothesis, there is still a general reluctance to doubt the validity of the hypothesis. Instead, researchers turn their attention to the empirical problems associated with testing the hypothesis. The purpose of this paper is to point out a theoretical issue that has been ignored by researchers. If capital markets are imperfect in the sense that the borrowing rate of interest exceeds the lending rate, then a paradox is possible; namely, we should not necessarily expect college enrollments to be directly related to the net present value of an education. Many of the important and difficult empirical problems in testing the hypothesis are widely [Manuscript
received
5 November
1960; revision
accepted
for publication
105
16 April
1982.1
106
Economics of Education Review
important in examining the relationship between enrollments and returns. The analysis which I present differs from existing studies in that the basic model of an individual’s investment decision incorporates two interest rates. The basic model, which follows Hirshleifer’s (1958, 1970, pp. 195200) earlier work, is used to show that an observed inverse relationship between the enrollment rate and the net present value of an education does not necessarily imply that one of the standard empirical problems is present. Such an observed relationship is not inconsistent with economic theory. The next section of this paper outlines the basic model of an individual’s investment choice. The last section applies the model to the question of the relationship between changes in enrollment rates and changes in financial returns.
interest at which the individual can borrow or lend, then the present value of lifetime income is represented by PV,, in Figure 1.
A MODEL OF AN INDIVIDUAL’S INVESTMENT DECISION This section outlines a model of individual decision making with respect to the choice of one of two alternative income streams. One income stream is realized if the individual attends college and the other if he foregoes college.’ The following assumptions hold: (I) Utility is a function of Ci and Cl, where C, and CZ are present and future consumption, respectively. (2) The income stream associated with an investment in a college education, YF, Y$ (where the subscripts refer to time periods and the superscript K indicates that the income stream is associated with a college education), is known with certainty. (3) If the individual does not acquire education, his future income stream, Yiv, Y;“. is also known with certainty. (3) There are only two alternative income streams available. (5) There are no nonpecuniary returns associated with either alternative. If an individual can borrow or lend at the same rate of interest, an analysis of the decision to invest is relatively simple. In Figure 1 point N represents the individual’s income stream over periods 1 and 2 if he does not invest in education. To maximize utility he lends in period 1 and his twoperiod consumption pattern is given at point M. If period 2 income is discounted by 1 + i, where i is the rate of
PVN
py(
y,c,
Figure 1 The point Kin Figure 1 represents the twoperiod income stream available with an investment in education. With K. the individual maximizes utility by borrowing in the current period and consuming at the rates in periods 1 and 2 shown by point J. Discounting stream K, we calculate the present value of K to be PV,. The usual method used to calculate the nef present value of an education is to discount each income stream and then take the difference between the two present values. In Figure 1 the net present value of the education is PVK  PVN. If the borrowing rate of interest equals the lending rate and earnings are discounted at that rate of interest, then an individual maximizes utility by investing in education whenever the net present value is positive. The Decision when the Borrowing Rate Exceeds the Lending Rate We now turn to cases where the borrowing rate of interest differs from the lending rate. In this situation the ordering of the present value of alternatives discounted by the borrowing rate of interest can differ from the ordering of present values discounted by the lending rate, as shown in the geometry by the crossover of the consumption opportunity lines in Figures 2 and 3. When the
Imperfect Capital Markets and Investment in Education lending rate is used. accumulated values [AC’ = Yr + Y? (1 + rL)] are shown in Figures 2 and 3 rather than present values. The rank of a project when projects are ordered by accumulated value is the same as its rank when projects are ordered by present value. Likewise, the rank of a project when projects are ordered by change in accumulated value is the same as its rank when projects are ordered by change in present value. Accumulated values are used in Figures 2 and 3 for the sole purpose of making the graphs easier to read. They hold no special significance and, in fact, the text of this paper refers only to present values. Thus when the lending rate is used as the discount rate. the text will refer to the present values of the projects although accumulated values are actually shown in Figures 2 and 3. y*’ c2 AV K AV N
PVK
PVN
Y,?C,
Figure 2 In Figure 2 points K and N again represent the income streams available to the individual with and without the education respectively. Income streams K and N are such that PVK > PV, if the lending rate is used as the discount rate and PVN > PVK if the borrowing rate is used as the discount rate. Generally an educational investment is such that Y: < Yf and YF > Yy. Thus it is possible that at some discount rates (where both income streams are discounted at the same rate) PV, < PVN, while for other, higher rates PV,v < PV,.’ In studies on investment in education, it is sometimes asserted that earnings should be discounted at the ‘appropriate’ discount rate. What the appropriate discount
107
rate is intended to reflect varies from one study to another and sometimes the problem is ignored. It has been suggested that the appropriate discount rate, at least for some purposes. is the rate of interest on personal loans. With preferences depicted by Ui in Figure 2, an individual maximizes utility by acquiring additional education and borrowing in the current period. Yet if the net present value of the education is measured by using the borrowing rate as the discount rate, PV,  PV, is negative. For another individual with preferences given by Z/z in Figure 2. foregoing education and lending in the current period maximizes utility. However, if earnings are discounted by the lending rate, then PVK  WV is positive. Other suggestions for an appropriate discount rate include using ‘the market rate of interest’ along with an assumption that capital markets are perfect. At the same time, explanations for some observed behavior with respect to investing in education rely on the existence of borrowing difficulties. All efforts to find the single appropriate discount rate for solving the optimal investment problem are doomed to failure if the decision context is one of imperfect markets. In particular, if the environment is such that (to a good approximation, at least) borrowing can be engaged in at the constant rate rn, and lending at the constant rate rt. (with rn > rL), then both of these rates must be incorporated into the analysis. For some choices between alternatives the ~a rate will be relevant (if both alternatives are such that the investor will in either case be borrowing to finance the investment). For some choices the rL rate will be relevant (if both alternatives are such that the investor will in either case be left with additional funds to lend even after making the investment). And finally, for some choices (where one alternative is associated with borrowing and the other with lending) it will be necessary to refer to the utility function to find the optimum. CHANGES CHANGES
IN RETURNS AND IN ENROLLMENTS
The hypothesis that changes in college enrollment rates (the fraction of individuals who choose to attend college) and the financial returns to a college education move in the same direction over time is common in the literature. The analysis in the previous section is now extended to show that we should not always expect to observe that relation
108
Economics of Education Review
ship if capital markets are imperfect. The model allows for the possibility that the fraction of individuals who choose to attend college will decrease over some period of time when the net present value of college unambiguously increases. Although no assertions are made about the frequency with which the paradox can be expected, some necessary conditions for its existence are discussed. The anticipated lifetime earnings stream with a college education and the anticipated stream without the education can change so that the net present value of college increases (i.e. dPVK  df V,v > 0) if the borrowing rate is used as the discount rate but decreases if the lending rate is used. These cases will be ignored. Instead, the analysis is limited to cases where the net present value unambiguously increases. An unambiguous increase occurs when the net present value of college increases using both the borrowing and lending rates.’ In order to examine the relationship between enrollment rates and financial returns, we shall consider the various combinations of choices which could be observed with respect to an individual’s initial education decision and what his decision would be if the net present value unambiguously increases. The three commonly expected results can be obtained. The individual may choose college under both sets of circumstances; he may forego college under both sets of circumstances; or he may choose to forego education in the initial circumstance but choose to attend at an unambiguously higher net present value. The paradox is also possible: an individual who chooses to attend college might have foregone college if the net present value had been unambiguously larger.
The Paradox Given K. N. rn and rL, assume that an individual maximizes utility if he chooses stream K (invests in education). One such case is depicted in Figure 3. Suppose earnings streams K and N shift to K* and IV’. Then the Ned present value of an education rises, i.e. PVk  PV& is greater than PI/,  PVN. regardless of whether the borrowing or lending rate is used to discount the earnings streams. But faced with income streams K” and N”. the individual maximizes utility by choosing income stream N* (foregoes education). This case is important because it shows that the paradox can exist. An individual uho originally would have chosen education chooses
Figure 3
to forego the education when its net present value is higher. There are at least two necessary conditions for the paradox. First, the increase in the net present value of education must result from a change in both income streams.’ If the net present value of education were to increase as a result of a change in the anticipated ageearnings profile of college graduates with no change in the anticipated earnings profile of highschool graduates, then a person who initially selects stream K would also select K’. The condition that both streams must change would normally hold if one is looking at the response of college enrollment rates to changes in the net present value of a college education over time. Another necessary condition for the paradox is that neither consumption opportunity line (the one associated with acquiring education and the one associated with foregoing education) can dominate the other either before or after the rise in the net present value of education. Whether or not this condition generally holds is an empirical question, but existing estimates of the private internal rate of return to a college education suggest that the condition does hold in many instances.s
109
Imperfect Capital ,Warkets and Investment in Edrccation The Standard
It
Results
clear that if the net present value rises. an individual who originally invests does not necessarily forego education at the higher net present value. With an alternative set of indifference curves (not shown), the original equilibrium position in Figure 3 could still be at point A (invest and borrow) and the second equilibrium position could. for example, be at point B with K” and borrolving. There is no paradox since the individual would choose to invest originally and continue with that choice if the net present value increased. It is also easy to generate cases where an individual originally foregoes education and then chooses education at the higher net present value. Start with K and N and assume that the individual foregoes education. If the income streams change so that the twoperiod consumptionpossibility locus associated with K” dominates the consumptionpossibility locus associated with N”, then the individual necessarily chooses education after the increase in its net present value. Finally, the fourth possible combination of choices can also occur. An individual may choose to is
forego education both before and after an unambiguous increase in the net present value of the education investment project. For example. indifference curves could be drawn so that C (in Figure 3) represents the original equilibrium point (forego education and borrow) and D the equilibrium position (forego education and borro\v) after the increase in the net present value. In summary, we can generate all possible combinations of behavior by an individual with a given set of preferences in response to a change in the net present value of an education. The main point of this discussion is that with an imperfect capital market, where the borrowing rate exceeds the lending rate, a rise in measured net returns to more education will not necessarily be positively correlated with the fraction of the population that chooses to acquire additional education.
Jack Hirshleifer, Doug Stewart. Allan Taub, Don Parsons and Vijay Mathur read earlier drafts of this paper and made several useful suggestions. The final version of this paper benefited from the comments of anonymous referees.
Acknowledgements 
NOTES 1. The analysis also can be used to explain why some individuals choose a field of study (at a given level of education) with low measured money returns over a field with higher measured returns. YF < Y,v and Yf > YF or Y:’ > Yy and Yt < YT are necessary conditions for a change in the sign of PV,  fVN when the discount rate is changed, but they are not sufficient. Thcrc are cases. for example, where (PV,  PV,v) > 0 at both discount rates. In these cases the consumption opportunity locus associated with K dominates the locus associated with N and utility is maximized by acquiring an education. If YF > Y’y and Yf > YT, then (PV,  PV,v) > 0 for all discount rates and utility is maximized by acquiring an education. 3. If r,_ is used as the discount rate, then dPV,  dPVN > 0 when
2.
dY$
If ra is used as the discount

dY”
$ 0 and (I + rl,)’
rate. then dPV,

dY$‘dY;$Oand(l
Thus an unambiguous dYf  dYT < 0 and
The and
dY” 2 IL dY$
_ dY”’ 
dY;
dPV,v > 0 when
+rrc)
,,
dYy

dYf
<
(jyc
_
dyv'
1
increase in the net present value of the education
dY”
_
d)‘K
dY:

dv.;
occurs. for example.
when
’ > (1 + rl.)‘.
sign of the change in the net present value would be ambiguous
if, for example,
dYF  dY> < 0
110
Economics of Education Review 4. I owe this point to two anonymous referees. 5. If the private internal rate of return to college (i.e. the discount rate which squares PV, and PV,) exceeds the lending rate of interest and is less than the borrowing rate. then neither consumption opportunity line will dominate the other. The value of the internal rate of return, which has been estimated for several different years (see, for example. Hansen. 1963; Welch, 1970; Freeman, 1975; Carney and Marenbach, 1975). often falls in the range of S15% per year. Because of the difficulties young people encounter when they try to borrow. it does not seem unreasonable to speculate that the borrowing rate has at times exceeded the internal rate. It also does not seem unreasonable to believe that the lending rate was less than some of the estimated values of the internal rate of return. REFERENCES CARNEY,
M.
and
MARENBACH.
Resources X, 312331.
D. (1975) The return
to schooling
in the United
States.
193969.
J. hum.
FREEMAS, R.B. (1975) Overinvestment in college training? J. hnm. Resources X. 287311. FREEMAN. R.B. (1980) The facts about the declining economic value of college. J. hm~. Resources XV, 124142. HANSEN, W.L. (1963) Total and private rates of return to investment in schooling. J. pobr. Econ. 71. 128110. HIRSHLEIFER, J. (1958) On the theory of optimal investment decision. J. polir. Econ. 66, 329352. HIRSHLEIFER, J. (1970) lnvestmenr, Interest, and Cupiful. Englewood Cliffs. NJ: PrenticeHall. RUMBERGER.R.W. (1980) The economic decline of college graduates: fact or fallacy? J. hum. Resources xv, 99112. SCHWARTZ, E. and THORNTON, R. (1980) Overinvestment in college training. J. hum. Resources XV, 121123. WELCH, F. (1970) Education in production. J. polif. Econ. 78, 3559. WILKENSON, B.W. (1966) Present values of lifetime earnings for different occupations. J. polit. Econ. 74, 556572. WITMER, D. R. (1980) Has the golden age of American higher education come to an abrupt end’? J. hum. Resources XV. 113120.