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FACULTY WORKING PAPERS College of Commerce and Business Administration University of Illinois at Urbana-Champaign March 22, 1979
THE OPTIMAL ACCUMULATION OF HUMAN CAPITAL OVER THE LIFE CYCLE
John Graham, Assistant Professor, Department of Economics
#553
Summary;
This paper summarizes the important contributions of the new life cycle human capital literature and demonstrates that many of these results can be derived more simply than in their original presentations. Within three period discrete- time framework it is demonstrated how the optimal pattern of human capital in- vestment over the life cycle depends upon the choice of the objective function, the life cycle of leisure, and the extent of nonmarket benefits of human capital. The paper offers sufficient conditions for the optimality of a profile of monotonically declining investment activity over the life cycle.
The Optimal Accumulation of Human Capital Over the Life Cycle I. Introduction
Although formal modeling of the process of human capital accu- mulation has been going on for some 15 years now, the development of such models within an explicit life cycle context has a far shorter history. Even as late as 1976, Blaug's comprehensive survey of the field offers no discussion of human capital accumulation over the life cycle. Only since then have several articles appeared that
attempt to solve for the optimal life time pattern of human capital
2 investment. This paper summarizes the important results of this new
literature and demonstrates that most of these results can be derived
more simply than in their original presentations and without some
special assumptions previously imposed. Finally, this paper offers
sufficient conditions for the optimality of a profile of declining
investment time ever the life cycle.
Early models of human capital accumulation were either static, or
3 if dynamic, ignored the constraints imposed by the life cycle. The
development of models within a life cycle context represent a major advance for several reasons. First, the life cycle lends added realism to models of optimal accumulation. It forces the model builder to incor- porate several essential characteristics of human capital investment that set it apart from other forms of consumer saving ;as well as from other types of physical investment. Unlike many other forms of consumer saving, acquisition of human capital is irreversible. Bonds acquired today can be sold tomorrow, but human capital acquired today cannot be so easily liquidated. By incorporating the process of aging, life cycle
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models make explicity the necessarily sequential and one-directional nature of human capital accumulation. Unlike investment in most physical capital, investment in human capital is subject to constraints on the magnitude of accumulation. With perfect financal markets, firms
can acquire as much capital as they want to during any period, but
4 consumers cannot always acquire as much human capital as they desire.
The reason is that human capital is not purchased in the market at a
constant price, but "produced" by the consumer with his finite resource
of time. Since there exists no market in which more of this time input
can be acquired immediately, the only alternative is to postpone some
investment until the future.
One benefit of studying human capital within a life cycle context is that an examination of the pattern of accumulation that emerges can provide a check on the internal consistency of the model. For example, one should become wary of putting too much faith in the policy prescriptions that follow from a model in which monotonically increasing investment over the life cycle emerges as the optimal pattern of investment.
Casual observation suggests that the most common pattern of human capital accumulation is a declining profile of investment time vrtrt the life cycle. Therefore, it should be interesting to investigate the conditions under which such a pattern is indeed optimal. Additionally, one would hope that a model might provide insight into other patterns of accumulation. How can one explain the not uncommon pattern of a return to full-time schooling after a period of full-time work? Is
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such behavior simply irrational, or is it possible to provide a choice- theoretic framework in which this behavior emerges as both rational and optimal?
It is important at this stage to interject a word of caution. While it is tempting to equate human capital accumulation with school- ing, it is wrong to do so. The models developed in this paper and in the related literature take a broader view. Human capital accu- mulation is defined as all non-costless current activity that enhances the value of future time. Thus, it includes not only schooling, but also on-the-job training and investment in health or information. Viewed in this wider context, it is not even clear that the "normal" life cycle pattern has investment time declining with age.
II. Human Capital in the Life Cycle Literature
The first dynamic theories of optimal human capital accumulation due to Ben-Porath and Becker represented significant advances over the existing static models, but even they ignored any explicit life cycle consideration. In other words, neither assumed the consumer to be making choices that would maximize lifetime utility subject to the constraints of age, lifetime income and intertemporal prices. Instead, the under- lying hypothesis maintained by both Becker and Ben-Porath was that indi- viduals would select a lifetime plan for human capital investment to maximize total lifetime income, and then at some later stage would maxi- mize lifetime utility subject to this value of lifetime income. It is exactly this recursive two-step maximization that the recent life cycle human capital literature demonstrates to be invalid. According to Ryder, Stafford and Stephen:
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The criterion of maximal present value for investment decisions is usually justified by what Hirshleifer terms the Separation Theorem. That is, 'given perfect and com- plete markets, the productive decision is to be governed solely by the objective market criterion represented by attained wealth — without regard to the individual's subjec- tive preferences that enter into their consumption deci- sions.' Unlike physical capital, however, human capital is embodied in the human being and, hence, in the unit that makes decisions about the life-cycle allocation of time. One implication of this is that whenever time has an alternative use that produces utility the separation- theorem holds only if restrictive assumptions are made.
Unfortunately the gain in realism provided by utility maximization in a life cycle context has not been coscless. When consumption and investment decisions are interrelated, the consumer's maximization prob- lem is very difficult to solve in general and even particular functional forms rarely provide closed form solutions. To obtain any meaningful results, the literature has been forced to impose potentially severe simplifying assump* ■,~~s. For example, to obtain interpretable life cycle paths for leisure, invertnient and work time, Ryder, Stafford, and Stephen assume that the lifetime utility function can be written in logarithmic form. Cinder and Weiss do not choose particular functional forms, but they do assume that their utility function exhibits separ- ability of time ani roods at- erch io.stauc of time. Furthermore, their production function for human capital is assumed homogeneous of degree one in the input of time; in other words, doubling the time devoted to investment doubles the output of human capital produced. Finally, in Heckman's papers the existing stock of human capital enters both the utility and production functions in a very special way which Heckman dubs the "neutrality hypothesis." According to this specification,
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additions to the stock, of human capital equally augments the efficiency of all uses of time — work, leisure, and investment.
In this study we first extend the work of Ryder, Stafford and Stephen to examine how the optimal lire cycle profile of human capital accumulation depends upon the objective function of the consumer. In Section III we derive the optimal investment profile for a wealth maxi- mizer, and in Section IV the optimal profile for a utility maximizer. Unlike earlier work our utility function is general in form and does not assume separability of commodities within a given period. Our pro- duction function exhibits diminishing returns to the time input. In Section V we extend Heckman's results by allowing for non-neutral in- fluences of human capital on the efficiency of alternative uses of time by introducing a leisure technology parameter. We examine what influence this parameter has on the profile of Investment time. Throughout the paper the results are expressed In terms of sufficient conditions for the optimality of a "normal" declining investment time profile over the life cycle.
III. Optimal Accumulation Within a Wealth-Maximization. Model
Suppose an individual expects with certainty to live for three identical and consecutive discrete time periods. During each per- iod he must allocate his predetermined nonleisure time between two activities — work time which earns a monetary reward and human capi- tal investment time which enhances the value of future work time. Let K. denote the fraction of time during period i devoted to in- vestment. The individual Is endowed with an initial earnings potential
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Hn which can be augmented through human capital production according to the transition equation
H± - H^ - F(Kt) (1)
For simplicity, assume the production function F satisfies the Inada conditions. Assume that work time yields a return to the existing stock of human capital at the constant rental rate w. Therefore first period earnings equal wH^l-K.), that iss the return on the initial stock of human capital, wH», times the length of noninvestment time, (1-K.). Second period earnings equal wH..(l-K_), where the new stock of human capital is obtained from equation (1) as determined by the first period's investment activity. Finally, third period earnings equal wH- • K- equals zero since returns to any third period invest- ment would not accrue until after the terminal date. The choice problem for the individual is to select K. and K? to maximize wealth, or discounted lifetime income:
(l+r)2wH()(l-K1) + (l+rJwH^l-Kp + wH2
where H^ and H2 are defined in equation (1), and r is the interest rate in the perfect financial capital market. Assuming interior so- lutions, wealth is maximized when
§; - (l+r)Hl (2)
(1^)(1-V% + %=<14t>2ho ' <3>
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* *
The equations can be solved, in principle, for ]L. and K_, the optimal
fractions of nonleisure time devoted to human capital accumulation. Together these equations represent the familiar requirement that in- vestment time be chosen to equate the marginal cost of production (on the right hand side) with its discounted lifetime marginal benefits (on the left hand side) .
To evaluate the life cycle pattern of investment time in this model it is sufficient to compare K. to L. At this point it should be clear why we have selected a three period life cycle. With K =0, three periods represent the minimum life cycle needed to compare two unconstrained periods. In principle the analysis could be extended to any number of discrete periods.
We shall prove and then discuss the following theorem.
Theorem One; YL > K~ is the only feasible investment path that satisfies (2) and (3).
We proceed with a proof by contradiction. Suppose K=K =K. Then
dF dF dF
~- - •—- = —;. Combine (2) and (3) to obtain:
7 (1-H:)2(1-K)K1 + (l-fr)^ = (l+r)2H0 (4)
It is straightforward to show that (4) cannot hold with equality. To see this, differentiate the left hand side of (4) with respect to K obtaining
(l+r)2(l-K) j[|
which is nonnegative for 0 <_ K <_ 1. Therefore, the left hand side of (4) is an increasing function of K. Evaluating this expresion at
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K-0, its smallest value, yields (l+r)Tl0 + (l+r)HQ. Since this al- ready exceeds the value of the right hand side, (4) cannot hold with equality for any 0 <_ K <_ 1. Therefore, the assumption K^*^ ^as *e(* t0
a contradiction.
dF dF Now suppose K? > 1L. . This implies that -tjt- < ^7~. Now when (2)
dF is substituted into (3), -rrr- must be replaced by even more than (14r)H.. .
This only makes the left hand side of (4) larger, so the contradition
stands.
dF dF Finally, suppose K- > K„. This means -tz- < -rr-, so when substituting
(2) into (3), -Tjr- will be replaced by less than (l+r)H.. This is the
only case for which equality in (4) can possibly hold. For some K,
K. > K_ does not imply a contradiction. Q.E.D.
The importance of Theorem One is that it demonstrates that aside
from an ad hoc change in the form of production function for human
capital over the life cycle, there is no way to explain a nonde-
creasing investment profile when individuals select investment time
to maximize lifetime wealth. Intuitively, investments are made
early rather than late in life for two reasons. First, the earlier
the investment is undertaken, the greater are the remaining number of
periods for the benefits to accrue. Investment late in the life
cycle leaves insufficient time to reap the benefits. The other reason
why earlier investment i& more profitable is that is when the costs of
production are the lowest. Investment not only increases the value of
future work time, but it also increases the cost of future investment.
These rising costs and declining benefits reinforce each other to make
a dedllning life cycle of investment activity the only feasible profile.
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IV. Optimal Accumulation Within a Utility-Maximization Kodel
Assume the identical set-up to the previous problem, except that now the individual selects K to maximize lifetime utility. This means he has several additional choices to make. Let I be the fraction of total time devoted to leisure in period i, where leisure is defined as all nonwork, noninvestment time, and the only use of time that yields utility directly. In addition to his allocation of time, the individual must decide upon his pattern of purchases of goods over the life cycle. Let X. be a composite nondurable commodity purchased in period i at a price of one. The individual's choice problem can be summarized as selecting X.. , X_, X„, £. , l~, £», IC. , and K„ to maximize lifetime utility
u(X1,l1) +v(X2,£2) + z(X3,£3),
subject to: (l+r^U-JLj-K^wHg + (l-+r) (l-J^-K^wl^
+ wH2(l-£3) - (l+r)^ - (l+r)X2 - ^ - 0
The only assumption imposed on preferences is that the lifetime utility function exhibits intertemporal separability. The complete set of first-order conditions appears in Appendix A.
Reproduced below are the two first order conditions from Appendix A for choosing K. and K_:
it jv (l-*3) ||-= (1+r)^ (5)
1K2 (l-hrMl-**-^) 3|-+ <l-*3) f^-« (l+r)<-H0
dF • - -** dF - (l+r)2Hn (6)
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Like (2) and (3), these equations say that investment time should be chosen to equate foregone marginal costs with marginal benefits. But unlike in the earlier pair of equations, benefits now depend upon the planned life cycle consumption of leisure time. This means that the optimal profile of human capital investment cannot be derived solely from the pair of equations above, but requires the two additional first order conditions for leisure in the second and third periods. However, we shall see that equations (5) and (6) by themselves are sufficient to obtain a sense of the lifetime profile of investment as long as we assume that the future time path of leisure is always being chosen optimally (and denoted Jt„ and £„).
The major result of this section is that Theorem One which guaran- teed a declining life cycle of investment time will no longer hold uncon- ditionally. When individuals maximize lifetime utility and not wealth, for certain parameter values any investment profile is feasible. At best we can offer the following sufficient condition for the optimality of the "normal" decreasing pattern of investment time over the life cycle .
Theorem Two: K- > K_ is the only feasible investment path that
* *
satisfies (5) and (6) if «, <_ I .
We again offer a proof by contradiction. Suppose K.=!K_=K?. Then
dF dF dF dF
-Ty~ = -T7T- ■ -j-jr • Substituting from (5) into (6) to eliminate — , we
obtain:
*
(1— & -K) ?
(1+r)2 2_ ^ + (i+r)^ = (l+r)2!^ (7)
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Dlfferentiation of the left hand side of (7) shows that the expression
Is a nondecreasing function of K:
*
„ ,_x2 ^-y*0 dF „ n
(1+r) ; — Hkl°-
Now evaluate (7) at K=0, where the left hand side will be the smal- lest:
, (l-O ?
(l+rr =~ H_ + (14r)Hn =* (l+r)ZHn (8)
(1-**) ° ° °
JL J,
Equality in (8) cannot hold if £„ _< £„, but it can possibly hold for some £_ < £„. Therefore, as long as £„ <_ £„, the assumption that Ki"Ko leads to a contradiction. Just as in the proof to theorem one,
another contradiction is reached (given £. <_ £ ) if it is assumed that
dF K. < K_ since -r—- in equation (6) will be replaced by even more than
jj — from equation (5), making the left hand side of (7) even
u-V
larger. Only IL < K2 produces no contradiction. Q.E.D.
Theorem Two can be explained intuitively. The first order condi- tions (5) and (6) for selecting the investment profile say that an important determinant of human capital investment is planned consumption of leisure. Since returns to human capital investment accrue only when the individual is working, the individual's preferences over leisure will influence the extent of future benefits from current investment. It is easy to see how an "abnormal" investment profile might occur. If consumption of leisure is falling and work time is increasing over the life cycle, the discounted benefits from investing during period two might be greater than the discounted benefits from investing during
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period one if the returns in period three (large relative to those in period two because of the declining life cycle of leisure time) are sufficiently heavily discounted.
But, of course, the life cycle of leisure itself is not indepen- dent of the individual's production of human capital. Therefore, it is necessary to re-examine the first-order conditions to see if a declining life cycle of leisure (which could produce an increasing investment profile) is likely or even feasible. From Appendix A, the first order conditions for £_ and £„ are:
8vU9,X9)
^ - (Ht)wH1A = 0 (9)
3z(£ ,X )
^ - wfi2X = 0 (10)
where A is the lagrangian multiplier associated with the budget con- straint. Without imposing further assumptions on the form of the utility function, very little can be said about the life cycle of leisure. However, if we are willing to assume separability of leisure and goods within any period, and further to assume that the utility function for leisure differs across periods only by the individual's subjective rate of discount which also equals the market interest rate (i.e.,
rr-3 (1+r) -rr— ) , then we can divide (9) by (10), obtaining the 3£2 3*2
condition:
3z
at, a nr = ST (11)
3*„
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Now as long as the second order conditions for utility maximization
32z hold ( — j < 0), then according to equation (11), JU is less than £,
whenever TJ_ exceeds H. . Whenever K„ is nonzero, £L will exceed H. ,
so leisure consumption will fall from period two to period three. This
likely occurance is exactly the circumstances under which the investment
profile can be increasing over some portion of the life cycle.
V. Optimal Accumulation Within A Utility-Maximization Model With Consumption Benefits From Human Capital
Robert Michael formalized the notion that investment in human capital might not only enhance future market earnings, but also provide direct consumption benefits. We shall investigate how the intensity of such consumption benefits might influence the optimal profile of lifetime human capital investment. Our particular speci- fication of consumption benefits of human capital is due to James Heckman.
The only modification from the previous problem is in the form of the utility function. Let lifetime utility be written as
uCXj^Hj) + v(X2,A2H£) + z(X3,£3H^) (12)
where intertemporal separability is again assumed to prevail, but not separability of time and goods within any period. Michael suggests that if human capital enhances efficiency in market production, it may also increase the efficiency of nonmarket production. In (12), the parameter a_ captures the relative change in efficiency of work time compared to leisure time. For a»=0, (12) reduces to the utility function of the previous section in which human capital provided no
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consumption benefits. For a=l, human capital accumulation increases the efficiency of both work and leisure equally. To see this, notice that the marginal cost of leisure is constant in all three periods when a=l. In other words, even though the market wage increases as a result of investment, the opportunity cost of leisure time remains unchanged because leisure increases in efficiency at exactly the same rate as the market wage. Heckman has dubbed this case "Michael- neutrality". For 0<a<l, efficiency still increases with human capital accumulation, but the increase is not as fast as the increase in market efficiency, so the opportunity cost of leisure increases with investment. For a>l, the increase in the efficiency of leisure time is greater than that in work time, so the opportunity cost of leisure actually falls with investment in human capital. We want to investigate how this leisure-technology parameter influences the optimal profile of human capital accumulation.
The formal choice-problem for the individual is to select a life cycle profile for consumption of goods and leisure and invest- ment in human capital to maximize the lifetime utility function in (12) subject to the lifetime budget constraint and production func- tion for human capital. The formal solution to this problem appears in Appendix B. As before, our interest focuses upon the two first- order conditions for investment time:
(1 - £^ + a£*) ^|- = (l+OR, (13)
. , * * dF * * dF 2_
(1-Hr)(l - l2 - K2 + a£2) ||- + (1 - ^ + a^) ^- = (1+r)^ (14)
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*
where £. again means that leisure is being chosen optimally.
These equations differ from the marginal conditions In the pre- vious section by the addition of one (or more) extra term of benefits
* dF on the left hand side. For example, a£_ ~r—~ in (13) represents the con-
.5 dK„
sumption benefits received in period three (in terms of increased efficiency of leisure time) from investing in human capital during period two. Ceteris paribus, the presence of nonmarket benefits in- creases the lifetime production of human capital.
Two interesting cases should be noted. When a=0, (13) and (14) reduce to the marginal conditions of the previous section, as should be expected. Perhaps less expected, when a=l, (13) and (14) reduce to the marginal conditions of the wealth-maximization model. In this case planned lifetime consumption of leisure disappears from the calculation of marginal benefits of Investment, because even in leisure the individual captures the returns from human capital investment.
Again we want to examine the life cycle profile of investment time that emerges from this model. The following theorem summarizes the influence of the leisure technology parameter on the investment profile:
Theorem Three: K. > K„ is the only feasible investment profile that satisfied (13) and (14) when (a) a=l or (b) 0 < a < 1 and I* >_ SL*
* *
or (c) a>l and £- _< &„
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Proof : We proceed with a proof by contradiction. Suppose K=K =K,
AV AV ATI AT?
so — — - ■ t— - = — . Combining (13) and (14) to eliminate -r=i
diC. or.,, aK QK.
'2
^(1-£2-K2 + aV _ 1. ^
<i+r> (i - % ;,i3) hi + (i+r)Hi = (i+r)^0 <i5>
Differentiation of the left hand side of (15) with respect
to K shows it is a nondecreasing function of K since
(1+r)2 (1 - £„ - K + a£ ) 3H-
■ > 0 .
(1 - £3 + a£3) 9K
Now evaluate the left hand side of (15) at K=0, where the expression
attains its smallest value:
(1 + (a-l)£ ) (1+r) H0 (1 -f (a-l)£3) + ^"o (16)
It is evident that the value of (16) varies both with the value of a^ and relation between £_ and £..
Suppose a=l, as in part (a) of the theorem. (15) reduces to
1
(l+r)2H0 + (l+r)H0 = (l+r)2HQ
which cannot hold with equality. And since the left hand side of (15) only increases as K increases, the assumption K=K„ always produces a contradiction. Following the previous two proofs, the assumption 1C > K2 only serves to make matters worse, so KL. > K» is the only possible assumption that does not produce a contradiction.
Now as in part (b) of the theorem, suppose 0<a<l. For K=0 and £_ sufficiently less than £„, the. first term of (16) can be made arbitrarily small, so (15) can possibly hold with equality. However, for £, _> £_
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equality would be again impossible, and only worse for K > 0 or 1L > L. Thus l~ > L is a sufficient condition for K_ > K„ given 0 <_ a < 1.
As in part (c) of the theorem, suppose a>l. For Kc0 and £.„ suffi- ciently less than £_, the first term of (16) can be made arbitrarily small, so (15) can hold with equality. However, for Z„ <^ £_, equality is again impossible, and only worse for K > 0 or K2 > K.. Thus l~ £ £- is a sufficient condition for K. > K2 given a>l. Q.E.D.
The importance of Theorem Three is that it demonstrates that the "normal" declining investment time profile is not the only possible life cycle pattern to emerge when individuals behave as utility maximizers and human capital provides consumption benefits. When these consumption benefits are small (0<a<l) , so that human capital augments the efficiency of work more than leisure time, then a sufficient condition for the "normal" declining investment time profile to be optimal is that leisure time is nondecreasing. The intuition for the possibility of an "abnormal" increasing profile of investment time is similar to that for Theorem Two. Since most returns to investment in human capital (but not all returns, if a>0) are enjoyed only when the individual is working, if work time is most intensive late in life and if the future is heavily discounted, then later rather than earlier investment may be more profitable. A similar (but reversed) argument follows when a>l. Now, human capital augments the efficiency of leisure time more than work time. Since most returns to human capital now accrue during leisure time, if leisure time is increasing over the life cycle, the incentives to invest in human capital might increase with age if the future is sufficiently discounted.
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The most interesting result is that if human capital is "neutral," so that investment in human capital equally augments the efficiency of work and leisure, then the "normal" declining investment time profile is the only feasible pattern. And since (13), (14) and (15) are all monotonic in ji, we can say that the closer human capital is to being neutral, the more likely a normal investment profile emerges regardless of the life cycle of leisure. Again, the intuition should be clear. For a=l, the first-order conditions are formally the same as in the wealth maximization model where the "normal" investment profile was the only feasible pattern. Since human capital is neutral, the alloca- tion of time between work and leisure activities over the life cycle does not affect the returns from investing in human capital so it cannot affect the timing of such investment over the life cycle.
Theorem Three assumes that leisure time is always being chosen optimally to satisfy the system of first order conditions in Appendix B. What does this life cycle of leisure look like? Just as in Section IV, very little can be said without imposing additional structure to the utility function. If we once again assume separability of time and goods within each period and if we assume that the utility function differs across time only by the individual's subjective rate of discount, then
we can write
* dz
d£ ILa H dz a2
d£3H2a
* where z is the utility operator on leisure time.
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Notice that this condition is not quite identical to equation (11)
in Section IV because of the presence of efficiency effects on leisure.
* a
If we assume that z is homogeneous of degree e in H, , then
|
* dz |
2 |
«T |
* dz |
|
|
d£2Hl |
d*2 |
|||
|
(17) |
can be : |
rewritten as |
||
|
* dz d£2 * dz |
H^a£ |
|||
|
dJS |
t. |
(18)
Equation (18) suggests that the life cycle of leisure depends in part upon the leisure technology parameter a_. Suppose a^ is sufficiently small so that 1 - ae > 0. Since EL >_ EL, then l~ <_ £„ or in other words, leisure time declines over the life cycle. But according to part (b) of Theorem Three, these are the exact conditions under which the optimality of the "abnormal" increasing profile of investment time can not be ruled out. Now suppose a_ exceeds one by a sufficient amount so that 1 - ae < 0. For EL, >L, H*""** £ H^"ae and *3 >_ *2; or in other words, leisure time increases over the life cycle. According to part (c) of Theorem Three, these conditions also are consistent with the possible optimality of the "abnormal" increasing investment profile.
To summarize, our investigation of the life cycle of leisure permits many feasible paths for leisure time over the life cycle. It is not possible to show that the only optimal paths for leisure coincide with the sufficient conditions stated in Theorem Three for the optimality of the "normal" declining investment profile. The leisure profile can reasonably follow paths that might be consistent with nondecreasing in- vestment time.
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VI Summary and Extensions
This paper offers sufficient conditions (in terms of three theorems) under which the optimal amount of time devoted to human capital produc- tion declines over the life cycle. The results have been stated in this form since there is some casual empirical evidence that a declining profile is the "normal" life cycle — particularly when human capital production take the form of schooling. To summarize these conditions, a declining investment profile is the only feasible path if
1. Individuals maximize lifetime income, or
2. Individuals maximize lifetime utility and human capital is "neutral" in its effects on the efficiency of work and leisure time (a=l), or
3. Individuals maximize lifetime utility, human capital is "biased" towards market efficiency (0 <^ a < 1), and leisure time increases over the life cycle, or
4. Individuals maximize lifetime utility, human capital is "biased" towards leisure efficiency (a > 1), and leisure time decreases over the life cycle.
If none of these conditions is satisfied, then other investment profiles may be possible, including increasing investment time over the life cycle. How likely is it that one of these four conditions will be satisfied? The recent human capital literature dismisses the first possibility of wealth maximization as an unrealistic paradigm. If individuals behave as utility maximizers, then the question reduces to an examination of the neutrality or nonneutrality of human capital (the value of a). Unfortunately there have been very few attempts at estimating the value
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9 of the leisure technology parameter a_. If It could be demonstrated,
for example, that human capital was approximately neutral, then there would be evidence that the second condition is satisfied. This is obviously a question that cannot be resolved without a good deal more evidence.
There is another way in which the current work is unsatisfactory. Our treatment of nonmarket benefits of human capital in Section V is really assymmetric. If an increase in the stock of human capital makes both work and leisure time more efficient, then why doesn't it also raise the efficiency of the other use of time — investment time? The production function could and should be amended to include the possi- bility of neutral or nonneutral increases in the efficiency of invest- ment time. One could rewrite equation (1) as
Hi ~ Hi_i = F(KiHi_i> <19>
where b is a positive learning technology parameter. If b=l, equation (19) suggests that part of the current stock of human capital Is itself an input in the production of additional human capital. This neutral specification was introduced by Ben-Porath. If 0 < b < 1, acquisition of human capital still augments the production of future human capital but not as much as it increases market earnings — human capital is biased toward work. For b > 1, human capital is biased toward invest- ment since the efficiency of investment time rises faster than market wages .
Just as we investigated the effect of the leisure technology para- meter a_ on the life cycle profile of investment, so too could we, in
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principle, investigate the effect of the learning technology parameter b_. Unfortunately, this does not turn out to be a very easy task. It is impossible to offer any closed form conditions that summarize the influence of b_ on the investment profile. It is easy to see why this is so: raising b_ has two opposing effects. On the one hand, raising b_ Increases the incentives to invest now because it raises the marginal benefit of current human capital production since returns will accrue not only as higher future earnings, but also as lower costs of future production. On the other hand, raising b_ increases the incentives to postpone some investment since future marginal costs will be lower. While it may not be possible to explicitly derive any theorems on the effect of b_ on the investment profile, Paula Stephan has in- vestigated the influence of _b through computer simulation analysis. She shows how life cycle investment and earnings profiles will be affected when human capital has a market bias (0 <_ b < 1) and when it has a production bias (b > 1) . Unfortunately, however, she assumes that the individual selects his investment plan to maximize lifetime income, not lifetime utility. It would be interesting to investigate how Stephen's conclusions are affected by a change in the objective function. This paper suggests her results may well be very sensitive to the wealth maximization criterion. As yet, the question remains unresolved.
M/D/190
1See Blaug [3].
-23- Footnotes
2 Five articles are noteworthy. They are Blinder and Weiss [4], Ryder,
Stafford and Stephen [10], Ghez and Becker [5], and two by Heckman [7,8].
3
For example, Becker [1] and Ben-Porath [2].
4 Of course, even firms can be constrained in the presence of internal or
external adjustment costs. Blinder and Weiss point out the formal simi- larity between a firm's investment behavior faced with adjustment costs and human capital accumulation.
Ryder, Stafford, and Stephen [10], page.
2 6 dF d F
Specifically, it is assumed that — > 0, — =■ < 0, F(0) = 0,
AK Um dF ,n. . Aim dF ... n
*K> d£ (0) ~ "• and E4 dK (1) = °'
7 See Michael [9].
g See Heckman [7].
9 Graham [6] cannot reject the hypothesis that a=l based upon a panel
study of the human capital investment behavior of some 900 families
surveyed by the Michigan Survey Research Center.
10See Stephen [11],
-24-
Appendix A
The individual selects X , X., X , £ , £_, £ , IL, and K~ to maximize
uCXj^,^) + v(X2,£2) + z(X3,£3)
subject to the budget constraint
(l+r)2(l-£1-K1)wH0 + (l-H:)(l-£2-K2)wH1
+ (l-£3)wH2 - (1+r)^ - (l+r)X2 - X3 =* 0
and the human capital production functions
H2 = Hx + FCi^)
The first-order necessary conditions (where X is the multiplier associated with the budget constraint) are:
Su/SXj^ - X(l-Hr)2 = 0 (A.l)
3v/9X2 - X(l+r) = 0 (A. 2)
3z/8X3 - A = 0 (A. 3)
3u/3£1 - X(l+r)2wHQ = 0 (A.4)
3v/3£2 - X(l+r)wH1 = 0 (A.5)
3u/3£3 - XwH2 = 0 (A. 6)
d+r) a-vVoTr + (1-V^k~ = (1+r>2H0 (A-7)
-25-
(i-v§
U+OHj,
(A.8)
(14r)Z(l-£1-K1)wH0 + (1-hr) (l-£2-K2)wH1 + (l-£3)wH2 - (1+r)^ - (l+r)X2 - X3 = 0
(A.9)
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Appendix B
The individual selects X , X., X , £., I , 9. ' , iL, and K to maximize
u(X1,£1H0a) + v(X2,Jl2H1a) + z(X3,£3H2a), a>0
subject to the budget constraint and production functions in Appendix A. The first order conditions are:
Su/SXj^ - A(l-fx)2 - 0 (B.l)
3v/3X2 - A(l+r) - 0 (B.2)
3z/3X3 - A = 0 (B.3)
H* - A(l+r)2wHn = 0 (B.4)
— — — H a - A(l+r)wH1 = 0 (B.5)
— — — H a - AwH„ = 0 (B.6)
3SH2a
3v . ua-l dF , 3z . ..a-1 dF a£„Hn -rr— + a£_H0 -y=-
3*2^ 2 1 dh 3£3H2a 3 2 dKl
+ K-Cl+r)2^ + (1-h:) (l-Aj-K^wg-
+ d-^3)v^|- ) = 0 (B.7)
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a^X"1 ~- + XC-U+OwH-
a*3H2a 3 2 dK2 1
+ d-A3)«^) " ° (B.8)
(l+r)2(l-£1-KL)wH0 + (l+r)(l-il2-K2)wH1
+ (l-Z3)wH2 - (1+r)^ - (l+r)X2 - X3 - 0 (B.9)
To obtain equation (13) in the text, substitute (B.5) and (B.6) into (B.7). To obtain equation (1A) in the text, substitute (B.6) intc (B.8).
-28-
References Cited
[1] Becker, Gary, Human Capital, New York: NBER, 1975.
[2] Ben-Porath, Yoram, "The Production of Human Capital and the Life Cycle of Earnings," Journal of Political Economy (August 1967).
[3] Blaug, Mark, "Human Capital Theory: A Slightly Jaundiced Survey," Journal of Economic Literature (September 1976).
[4] Blinder, Alan and Ycram Weiss, "Human Capital and Labor Supply: A Synthesis," Journal of Political Economy (June 1976).
[5] Ghez, Gilbert and Gary Becker, The Allocation of Time and Goods Over the Life Cycle, New York: NBER, 1976.
[6] Graham, John, "The Influence of Nonhuman Wealth on the Accumula- tion of Human Capital," manuscript.
[7] Heckman, James, "A Life Cycle Model of Earnings, Learning, and
Consumption," Journal of Political Economy (August 1976, part 2).
[8] Heckman, James, "Estimates of a Human Capital Production Function Embedded in a Life Cycle Model of Labor Supply," in Nestor Terleckyj, editor, Household Production and Consumption, New York: NBER, 1976.
[9] Michael, Robert, The Effect of Education and Efficiency in Con- sumption, New York: NBER, 1972.
10] Ryder, Harl, Frank Stafford and Paula Stephen, "Training and Leisure over the Life Cycle," International Economic Review (October 1976).
11] Stephen, Paula, "Human Capital Production: Life-Cycle Production with Different Learning Technologies," Economic Inquiry (December 1976).