The optimal accumulation of human capital over the life cycle

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Faculty Working Papers

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

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

-2-

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

-3-

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:

-4-

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,

-5-

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

-6-

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>

-7-

* *

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

-8-

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.

-9-

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)

-10-

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)

-11-

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

-12-

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*„

-13-

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

-14-

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)

-15-

*

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 £- _< &„

-16-

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 £, _> £_

-17-

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.

-18-

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.

-19-

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.

-20-

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

-21-

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)

-26-

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)

-27-

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).