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The Production of Human Capital and the Life Cycle of Earnings
Author(s): Yoram Ben-Porath
Source: The Journal of Political Economy, Vol. 75, No. 4, Part 1, (Aug., 1967), pp. 352-365
Published by: The University of Chicago Press
Stable URL: http://www.jstor.org/stable/1828596
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THE PRODUCTION OF HUMAN CAPITAL AND
THE LIFE CYCLE OF EARNINGS
YORAM BEN-PORATH*
University of Chicago
T HE applicationof capital theory to
decisions on individual improvement,
and in particular improvement
of earning capacity, has provided a
framework for the understanding of
many aspects of observed behavior regarding
education, health, occupational
choice, mobility, etc., as rational investment
of present resources for the purpose
of enjoying future returns. The formulation
by Friedman and Kuznets (1945)
and the significant development of the
theory by Becker (1962, 1964) and
Mincer (1958, 1962) provided a novel
view of the life cycle of earnings by
linking it to the time profile of investment
in human capital: People make
most of their investments in themselves
when they are young, and to a large
extent by foregoing current earnings.
Observed earnings are therefore relatively
low at early years, and they rise as
investment declines and as returns on
past investments are realized. The main
reason why investment is undertaken
mostly by the young is that they have a
longer period over which they can re*
This paper is based on part of my Ph.D. dissertation
submitted to the Department of Economics,
Harvard University, February, 1967. I
benefited from discussing earlier drafts with my
adviser, Professor Simon Kuznets, and with Professor
Rodney A. Dobell, Gur Ofer, and Christopher A.
Sims; I received many useful comments from Professors
Gary S. Becker, Zvi Griliches, Theodore W.
Schultz, Lester G. Telser, and other friends. Financial
support was provided by a Ford Dissertation
Fellowship and a grant from the Ford Foundation
to the University of Chicago for Research on Investment
in Human Capital.
ceive returns on their investment. The
purpose of this paper is to combine that
part of the argument concerning the
demand for human capital with a more
explicit treatment of the supply, or cost
conditions, facing the individual.
It is hard to think of forms of human
capital that the individual can acquire as
final goods-he has to participate in the
creation of his human capital. His own
abilities, innate or acquired, the quality
of co-operating inputs, the constraints
and opportunities offered by the institutional
setup-all determine the "technology,"
or the production function.' Together
with the relevant factor prices,
the properties of the production function
determine the optimal way in which any
quantity of human capital is to be produced
and determine the cost of production.
I shall show how the production
function (through supply or cost conditions)
enters into the determination of
the optimal path of investment, analyze
some of the implications for the individual's
allocation of time, and demonstrate
how the life cycle of earnings can be
affected by various properties of the
production function.
The basic model generates some of the
qualitative characteristics of the observed
life cycle of earnings-typically,
an initial period of no earnings followed
by a period in which earnings rise at a
declining rate and, eventually, decline.
1 Such a production has been recently introduced
also by Becker (1966).
352
HUMAN CAPITAL AND THE LIFE CYCLE OF EARNINGS 353
Actual, or observed, earnings turn out to
be always lower, to change faster, and to
peak at a later age than the attainable
maximum, the earning capacity of the
individual. The reverse is true of the
relation between observed earnings and
earnings net of all investment costs: the
former are always higher, change slower,
and peak at an earlier date than the
latter. The model thus specifies the
nature of the bias that may exist when
earnings are used, as they often have
been, to infer changes in productive
capacity with age.
I. THE MODEL
In addition to the assumptions incorporated
in the production function, to
be discussed later, I assume:
1. Individual utility is not a function
of activities involving time as an input.
2. There is a fixed amount of time to
be allocated every period to activities
that produce earnings and additions to
the stock of human capital.
3. The stock of human capital, K, of
which every individual has some initial
endowment, is homogeneous and subject
to an exogenously given rate of deterioration,
3.
4. The stock of human capital is not
an argument in the individual's utility
function.
5. Unlimited borrowing and lending
take place at a constant rate of interest,
r.
The first two assumptions express the
fact that leisure is ignored in this analysis.
In conjunction with the other assumptions
they allow the partition of
the individual's decision-making into
two stages:
a) The individual allocates the given
periods of time between earning and producing
human capital and finds the corresponding
outlays on investment that
maximize the discounted value of any
time t of disposable earnings (defined
below) from t to T, where T is assumed
with certainty to be the end of life.
b) Given the optimal time path of disposable
earnings, the individual decides
on the timing of the consumption. This
is the point of departure of the life-cycle
theories of consumption, which take the
stream of earnings as given and explain
consumption as dependent upon it.
The stock of human capital is defined
as a concept analogous to "machines" in
the case of tangible capital. There is a
market in which the services of human
capital are traded, and a rental, ao, is
determined for the services of a unit of
human capital, K, per unit of time. The
sum of the services offered in the market
by various individuals is an input into
the production of other goods and services.
It may well have a diminishing marginal
productivity, which will cause
a downward-sloping aggregate demand
curve for the services of human capital.
Each individual is assumed, however,
to possess only a small fraction of the
total homogeneous stock of human capital
in the economy and is regarded as a
perfect competitor facing a given rental
ao, which is independent of the volume
of services that he offers in the market.
Earning capacity at time t, Yt, is therefore
the maximum services of human
capital the individual can offer in the
market valued by the rental ao.
Yt = aoKt.(1
Let Et be disposable earnings in period
it-the portion of current earnings disposable
for purposes of consumption or
the purchase of non-human assets. It
may be smaller than earning capacity if
the individual engages in production of
human capital; other uses for time are
excluded by assumption. The difference
354 YORAM BEN-PORATH
(Y, - Et) is the cost of investment It;
it depends on the production function
and on input prices.
Let (2) be the production function of
human capital:
Qt= 0( sKt)P1DO2, (2)
where Al, 12 > 0 an d 31 + /2 < 1; Qis
the flow of human capital produced. D is
the quantity of purchased inputs, the
price of which is denoted by Pd; St is the
fraction of the available stock of human
capital allocated to the production of
human capital, so that stKt is the quantity
of human capital allocated for the
purpose. If activities are not "mixed,"
that is, if there is no joint production of
earnings and of human capital, then st
is also the proportion of time devoted
to the production of human capital. The
fraction st is constrained by the condition
O< st:< 1 . (3)
(The properties of the production function
including the assumption of decreasing
returns to scale are discussed in
a subsequent section.) The rate of change
of the capital stock is given by (4):
Kt=Qt- SKt (4)
(a dot above any variable indicates a
derivative with respect to time), where a
is the rate by which the stock of human
capital deteriorates. Equations (1) and
(4) imply that a unit of capital can be
used from the moment that it is pro-
duced.
Investment costs have two compo-
nents:
It = aostKt+ PdDt, (5)
that is, (a) opportunity costs, or "foregone
earnings" (the value of the productive
services withdrawn from the market),
and (b) the direct costs of purchased
goods and services. Minimizing
It with respect to st and Dt, subject to
(2) and ignoring (3), we get (6) as a
condition for a minimum of (5):
aosKt 1 6
PdDt f2(
From (2) and (6) substituted into (5) we
get investment costs as function of out-
put:
f1 + 032 (lPd\ 2'(01+2)
it= ao )t
01
?
#2a0 (7)
The objective of the individual at any
time t is to maximize the present value
of his disposable earnings:
T
Wt =| e-rv[aoK(v)-I (v) ]dv. (8)
The objective expressed by (8) is treated
as applicable to an individual from birth;
up to a given age actual decisions are
made by his parents, and if they were to
take into account the full future life of
their child, (8) still expresses the relevant
maximand under the assumptions of the
model. If the parents do not take into
account the full economic life of their
child, or in the extreme case where only
the period of attachment to the present
household is considered, less investment
would be undertaken, and the age of
entry into the labor force would be lower.
The problem posed here is suitable
for treatment by the techniques of optimal
control. Three phases are suggested
by the constraints on s, the fraction of
human capital, or time, allocated to the
production of human capital: (i) The
available stock of human capital Kt,
even when fully allocated to produce
human capital, is not large enough to
provide the flow of services demanded
given the relevant prices. The upper
bound on s is thus an effective constraint.
(ii) The available stock is large
enough to supply the services demanded
HUMAN CAPITAL AND THE LIFE CYCLE OF EARNINGS 355
and more, so that 0 < s < 1 and the
services of human capital are truly a
variablefactor. (iii) The stock of capital
is too big so that the optimal policy
requiresmore disinvestmentthan is feasible
through deterioration, that is, to
produce negative quantities of human
capital. Here st is constrained by its
lowerbound.2
The firstphaseis by definitiona period
in whichno humancapitalis allocatedto
2 In terms of the techniques developed by Pontryagin
et al., the maximization of (8) subject to (2),
(3), and (4) involves the maximization of the Hamil-
tonian
H:e-r[(l - s)aoK - PdD]
(1')
+ q Q - K),
where qis the discounted shadow price of investment
in human capital
aH
= -e-rt(l- )ao
In terms of current prices ,6- qert
i'= -( 1-s )ao
- &[I: i
Q-(a6+r) ('
Transversality condition:
(3')
41(T)K(T) = 0.
First-order conditions for the maximization of (1'):
aH
=-aoK e-rt + V/e-r
3Q >0. ( 4')
C)S ~~~~~S
-H -rt D 2
Pd ere+ lIertI2 O. (5')
O9D D
A systematic analysis of a variant of this problem,
using the techniques of optimal control, is pursued
in an unpublished note by Eytan Sheshinski.
(t =00
00 )(01+i2)/(l-i,-iO) 0j2ao 02/ 19112
[ I1-e -(r+b)( T- t) ] (102/ -1112
Q-j -lPde 1 1 )
= NE1 - e (r+S)( T-0)(8i+ft2)/(1-flif2) ? 0
the market so that no earnings are realized.
Using conventional tools, we shall
first analyze the later phases in which
s < 1 and then shall return to discuss
the first.
Differentiating (7) with respect to Q
we get (9), the marginal cost of producing
human capital; it is a rising function
of the quantity produced starting from
the origin and is independent of the size
of the existing stock of capital of the in-
dividual.
MCt = 9
_
(I3Pd2(_1?#2)
00i3 0y32ao/ (9)
(Qt It1/(t6+02) The
value at time t of acquiring an additional
unit of human capital is the discounted
value to that time of the additions
to earnings that the undepreciated
part of this unit will bring about. This is
the "demand price" of human capital,
given by (10). This price is
T
Pt= aof e-(r+5)^dv
(10)
ao [1-(+)(T-t)
r +
independent of the number of units added
or the existing stock. It is a declining
function of time because of the presence
of the finite horizon.3 The optimal production
of human capital is determined
by equating the marginal cost to the
price. Equating (9) and (10) we can solve
for Q as shown in equation (11).
3 In terms of the framework of n. 2 we are operating
in the phase where (4') holds exactly as equality.
Substituting (4'), (5'), and (2) into (2'), and given
(3'), we get a differential equation to which (10) is a
solution.
356 YORAM BEN-PORATTI
The production of human capital, Q,
the gross additions to the stock, are always
positive except when t = T. At
the date of compulsory retirement, T,
the human capital loses its value, P(T)
= 0 (see eq. [10]). No production is undertaken
at period T, and the stock of
human capital is reduced by sK (see eq.
[3]). The point I = T lies in the third
phase. The reason why only this point
lies in this phase is that for any earlier
age, t < T, the demand price of human
capital is positive, and the fact that the
marginal cost curve starts from the origin
insures that some positive quantity will
always be produced. This would be true
of any production function homogeneous
of a degree less than 1 in these inputs.
MC
i-P2
vP3
.: "P4
Q4 Q3"Q1
Q2
FIG. 1
The market prices that enter into the
determination of the optimal production
of human capital Q are: r, the rate of
interest; ao, the rental on human capital;
and Pd, the price of purchased inputs.
Differentiating (11) with respect to these
shows that Q varies inversely with the
rate of interest and with the price of
purchase inputs and directly with the
rental on human capital. It is, however,
only the ratio between the last two that
is important. The elasticity of Q with
respect to the relative price ao/Pa is 32/
(1 - /1 - 12)- When 12 = 0, that is,
when purchased inputs do not enter into
the production of human capital, changes
in ao, the price of the services of human
capital, do not affect the quantity produced.
The obvious reason is that in this
case, when the services of human capital
are the only input, a rise in aoraises marginal
cost by exactly the same amount
that it raises the value of a unit of human
capital. Needless to say, all this discussion
refers to alternative stationary price
levels when the current prices are expected
to prevail throughout the individual's
lifetime. Qt is larger the larger
the parameters of the production function
Oo,A1, 02; the greater the length of
economic life T; and the lower the rate
of deterioration, which has exactly the
same effect as the rate of interest.
As t rises, the flow of Q produced declines,
as indicated by (12), the derivative
of Q with respect to t:
Q= 1-A1-
e-(r+5)( T-t) ] [(IIl-)/(1O2)I1 ( 1 2)
X e(r+5)(T[-(r-+t ) 1? 0
What we have here is a marginal cost
curve as a function of Q starting from
the origin and remaining stationary
through time, and a perfectly elastic
demand curve which slides down with
time, thus creating a pattern of positive
but declining optimal quantities of Q to
be produced (Fig. 1). If the horizon were
infinite, the demand curve would not
slide down, and there would be at this
phase one stationary rate of production
of human capital (given by N in eq. [11}).
Note that we are saying something about
the gross additions to stock of human
capital but not about the net additions,
which depend also on the rate of depreciation
and the size of the existing stock.
If the initial stock is very high, net additions
may be negative from an early age.
If we think of the normal case as one
11UMAN CAPITAL AND THE LIFE CYCLE OF EARNINGS 357
where the capital stock does rise over a
period, eventually as gross additions become
very small and the stock becomes
large this must be reversed, and toward
the end of life, T, the stock will decline,
if there is any deterioration.
What are the implications of this pattern
of investment for the life cycle of
earnings? Disposable earnings were defined
as the difference between earning
capacity and investment outlays; their
rate of change over time is given by (13).
E1t-aoKt-It . (13)
Let us define "observed earnings" E
to be actual earnings realized in the labor
market (14).
flt::-:aoKt- AIt (14)
/31?102
They are larger than disposable earnings
by the direct costs PdD and smaller than
earning capacity by foregone earnings.
Their rate of change over time is given
by (15).
?aokt - - It = aoQt
(15)
- +:-(AICt)Qt-aobKt
I is always negative (except when t =
T), so that the change in observed earnings
is always algebraically larger than
the change in earning capacity and
smaller than the change in disposable
earnings
(:1 + 02)
Thus, the curve of observed earnings exaggerates
the rate of increase of earning
capacity when the latter increases and
understates its decline when it declines.
In particular, when we observe an individual
at the peak of his earnings he is
already past the peak of his productive
capacity. The existence of a downturn in
observed earnings is here only a consequence
of depreciation (recall that interactions
with "leisure" are excluded by
assumption). If depreciation is zero,
there is always, except at T, an increase
in the three types of earnings, and at
each point in time their rank by rate of
change will be the reverse of their rank
by level.
From the second derivative of observed
earnings with respect to time (16),
we learn more about the shape of the
life cycle of earnings:
C)
i (+ 3)AICt 01
=- e-(r+)( T-t)
031+132
| 1 - 1 e- (r+b) (T+t)0
16)
- ao6Qt + aoa2Kt
If there is no deterioration, that is, a = 0,
the ever rising curve of observed earnings
is always concave from below. This is also
true of the rising portion of the curve
when a > 0. If /1 + 02 > 2, there will be
a certain range in the vicinity of t = T
where the curve is convex from below.4
All these qualitative results hold for any
production function homogeneous of a
degree smaller than 1 in sK and in D.
'Q is negative, and the term in braces is positive:
from the equality between (11) and (12)
ao= Arc.
I1- e-(r+b)( T-0)Al
The right-hand side of this equality is here multiplied
by a term always smaller than + 1 and deducted
from the left side so that the difference is
positive and the first term is negative. If 6 = 0 and
there is no deterioration, the other two terms vanish,
and the curve of observed earnings is always concave
from below. This is also true of the curve of disposable
earnings with 1 substituted for 01/(O3 + 32).
When a > 0, concavity is assured for the range
where K = Q - 6K > 0 If Al + 02 > a, the first
term tends to zero as t -* T; in the vicinity of t =T.
K < 0, so in this case the earnings curve will have
there a shape that is convex from below.
358 YORAM BEN-PORATH
The preceding discussion relates to the
phase where the optimal s, the fraction
of human capital or time allocated to the
production of human capital, is smaller
than 1. As indicated before, the maximization
of (8) may, however, require
that there will be a phase of complete
specialization in the production of human
capital in which s = 1 and the size of
the stock is an effective constraint.
Movement along the marginal cost
curve (7) by increasing Q implies an increase
in the required sK. Once s = 1 is
reached, larger rates of production can be
achieved only by combining more purchased
inputs with a fixed flow of services
of human capital, thereby increasing
costs at a higher rate than is implied by
(7). The marginal cost curve described
by (7) is a long-run envelope from which
steeper marginal cost curves rise up,
corresponding to alternative levels of the
available stock of human capital. When
people are young the value of a unit of
human capital is only negligibly affected
by the finite horizon T, so demand for
human capital is relatively high. On the
other hand, the available stock is still
small; therefore the marginal cost curve
rises up from the long-run curve, (7), at
relatively small output, so that at a young
age production is likely to occur at outputs
where s = 1.
Note that the demand price given by
(10) is now only the lower limit of the
true shadow price of human capital. An
increase in the stock of human capital
implies a reduction in future costs of production
of human capital. The precise
value of this depends on the future
quantities of Q that will be produced
between t and t*, the date of transfer
from the first to the second phase. While
in phase (ii), where 0 < s < 1, we were
able to determine the optimal Qt by the
intersection of demand and supply schedules
independent of the future levels of
Q, here only a complete, dynamic program
in which the effects of present actions
on future condition are explicitly
taken into account will generate an optimal
path of investment.
Phase (i) is identified by the absence
of earnings. Another distinguishing characteristic
of this phase is that the relaxation
of the constraint Kt which lowers
the marginal cost curve may induce a
pattern of increasing investment with
age, both through the allocation of the
services of a growing stock of human
capital for the purpose and through
higher direct expenditure on D, justified
by the higher marginal productivity of D
that the increased K brings about. A
phase of increasing direct costs in the
early period when no earnings are realized
is certainly not inconsistent with
the real world.5
The age t* when the individual enters
the second phase and leaves the first is
of great importance, because this is the
date at which positive earnings begin to
be realized. From the preceding discussion
it should be clear that K(t*) is a declining
function of t*.6 Ko is the initial
endowment of human capital, which,
following the definition of K, is proportional
to the maximum earnings the individual
can realize when he "starts."
All other parameters given, the larger
this initial endowment, the earlier will
be the date t* at which specialization
stops, noting that Ko is an initial endowment
only in terms of the capacity to
earn, while we hold constant the capacity
5An explicit solution of our problem for the case
in which 02 = 0 is simple but not interesting. The
interesting aspects of the problem derive from the
possibility of using purchased inputs.
6 This can be verified by substituting s = 1 into
(4'), letting it hold as an equality, and substituting
into it also the value of Q and Qkusing (2), (5'), and
the solution of (2') referredto in n. 3.
HUMAN CAPITAL AND THE LIFE CYCLE OF EARNINGS 359
to increase human capital thus defined,
which is reflected in the O's of the production
function.
II. THE PRODUCTION FUNCTION
ROLE AND IMPLICATIONS
The technology which the individual
faces when he makes decisions about investing
in himself is a complicated system
of technical and institutional relationships
covering a wide spectrum of
activities including formal education, acquisition
of skills on the job, child care,
nutrition, health, etc. By writing down a
simple production function of the sort
used here we are attempting, not to reproduce
this system, but only to provide
a framework within which some of the
possible characteristics of the technology
can be considered and their implications
studied.
A. PURCHASED INPUTS, TIME, AND THE
SERVICES OF HUMAN CAPITAL
Both the theory and the measurement
of investment in human capital (see
Schultz, 1963) emphasize the importance
of foregone earnings alongside direct
cost. A composite surrogate for direct
cost here is D, the index of purchased
inputs which stands for anything from
tuition to vitamin pills. The presence of
opportunities for some substitution between
purchased and own inputs has a
"smoothing" effect on behavior and
helps the individual overcome the constraints
of his limited time and sometimes
modify the effects of its increasing
cost.
The nature of and the role played by
own inputs require some further discussion.
In the production function so far
considered, own inputs are represented
by the product sK. K is the total stock
of human capital; s is described as a
fraction that can take any value between
o and 1. We shall first regard s as a
measure of the allocation of time to the
production of human capital, although
this interpretation is not necessary.7
At any time t the stock of capital Kt
is given. If we were to assume that any
activity the individual engages in produces
either human capital or earnings
but not both, then the allocation of time
between these two types of activities is
also the allocation of the services of the
existing stock of human capital. The
larger the stock of human capital, the
larger the earnings per unit of time that
the individual could get in the market
and therefore the higher the foregone
earnings from diverting a unit of time
away from the market (see Becker, 1965).
Whether this should or should not affect
the relative attractiveness of non-market
activities hinges on whether the change
that made an hour in the market more
rewarding also made more productive an
hour outside the market, in our case an
hour of producing human capital.
The question is whether the real production
relation involved is stable in
terms of time or in terms of some other
variable. The way we defined human
capital was to make the production of
earnings stable in terms of its services,
rather than in terms of time as such. By
making (sK) the relevant input in the
production function of human capital
we are also asserting that this other
process is stable in terms of the services
of human capital rather than in terms of
time as such. The main implication of
this formulation in terms of the alloca-
7 Note that time enters in two completely different
ways-I, which moves the individual along his
life cycle and which is being treated as continuous,
and se, which is a measure of the allocation of time
at any point in t. If we were to treat t as an index of
discrete periods, and if s were understood as referring
to the allocation of each such period, this dual meaning
probably would have posed no problem.
360 YORAM BEN-PORATH
tion of time is that, when K is higher, the
higher costs of an hour diverted away
from the market (because of the higher
foregone earnings) are exactly matched
by the greater productivity of time in
the production of human capital, so that
the marginal cost curve of producing the
latter is independent of the stock of
human capital in the range where the
constraint on s is not effective.
By arguing that in the production of
human capital the services of human
capital rather than time as such are
relevant we gain some analytical simplicity,
because beyond the period of
complete specialization in the production
of human capital the benefits associated
with acquiring a unit of human capital,
the addition to future disposable earnings,
do not depend on the future allocation
between the market and the production
of human capital. This is the reason
why the dynamic programing problem
that was referred to in relation to phase
(i) degenerated into a much simpler decision
problem in phase (ii). Analytical
convenience is, however, no substitute
for relevance. In a general production
function both time as such and human
capital would appear as inputs, and if
we consider again the Cobb-Douglas
case, s and K may appear with different
coefficients. Thus, consider (17):
Q= o'i' D
= /3oSY172 ( sK)72DO2 ; (17)
0 < Yl Y2,2, < 1 .
T1he corresponding marginal cost curve
is (18):
MCt = ao (I/ Pd )2/(7+1?2)
do71 02 ao
(18)
x Qt +
If y1 = 7Y2, we get an expression identical
to (2), with yYreplacing 01. If -Yj>
72, the marginal cost curve is shifting upward
as K increases. This can be viewed
as a case where time plays an independent
part in the production of human
capital or, more directly, as a case where
the larger productivity of time in the
market indicated by an increase in K is
not completely matched by a larger productivity
in the production of human
capital, so that the cost of the latter in
terms of the former rises. The case where
71 = 72 implies, for example, that the
more highly educated person is also better
equipped for learning, so that his higher
opportunity cost is matched by the
greater amount of skills that he can acquire
per hour. If this is not so, then the
situation that we are describing now is
relevant. A limiting case would be one
in which human capital does not at all
affect the ability to produce more human
capital, 72 = 0.
In the basic model considered before,
the decline over time in investment
beyond the period of complete specialization
is brought about by the downward
drift, due to the approaching horizon 7'
of the demand function, along a stationary,
upward-sloping marginal cost curve.
In the situation now described, as K increases
the cost curve shifts upward. In
young age, when T - t is large, the
quantitative effect of changes in I on the
demand price is small, and the shifts in
the cost function may be more important
for changes in investment than in the
downward drift of the demand curve.
The role of purchased inputs here is also
clear-the larger the 02, the smaller the
effect of K on the marginal cost.
The other case, in which Y2> -y, also
cannot be ruled out. Here capital accumulation
reduces the cost of producing
human capital, and it is possible even in
HUMAN CAPITAL AND THE LIFE CYCLE OF EARNINGS 361
phase (ii) to have a stretch of time over
which investment rises rather than declines
(the downward shift in the costs
being more important than the declines
in demand). The corresponding life cycle
may then have an early convex portion.
Eventually the declining marginal productivity
of K (due to 72 < 1) and the
growing effect of the approaching horizon
T will turn the rise in investment into
a decline, and the life cycle will have the
familiar S-shape.
B. THE ROLE OF RISING COSTS
With a homogeneous stock of human
capital, the services of which are sold at
a fixed price in a competitive market,
both the determination of a finite desired
stock and the speed in which the
available stock is adjusted to the desired
level depend on the cost of acquiring
human capital. If human capital could
be acquired at a fixed (or declining) cost
without limitation, that is, if the supply
schedule were perfectly elastic or declining,
the desired stock would be either
zero or infinite, and the optimal adjustment
would be instantaneous. In the case
of the demand for tangible capital, costs
of adjustment are sometimes introduced
to explain investment as a function of
the interest rate. In the case of the aggregate
economy, it is the rising cost of investment
goods in terms of consumption
goods that provides the negative slope
of the aggregate demand function of investment
(the marginal efficiency of investment
curve) and a finite rate of in-
vestment.
The nature of human capital, the fact
that it has to be produced by the individual,
makes for some similarity in the
considerations involved and provides a
natural basis for dealing with the question
of what makes the rate of investment
and the attained stock of human
capital finite. One form in which constraint
on the rate of investment could
come about is from individual capacity
limitations solely. Thus, consider the
case in which (2) 01 + 12 = 1, j 1 > 0,
and 12 > 0. In every period the available
stock of human capital Kt is given. Costs
would be constant up to that level of
output which implies s - 1, that is,
Q (= Pd 01
For larger outputs marginal costs would
rise (if 12 > 0) or become infinite (if
12 = 0). In either case, given the nature
of the demand for human capital, the
optimum rate of investment would always
imply a value of 1 or 0 for s > n,
the allocation of all the services of the
available stock of human capital either
to the production of human capital or to
the labor market, but not to both. Because
of the declining demand over time
for human capital, a period of complete
specialization in the production of human
capital (phase [i]), if it comes, must precede
the period of complete specialization
in market work (phase [iii]). The
resulting life cycle of (observed) earnings
would therefore have a portion of zero
earnings and then a jump to some positive
level, which would be stationary in
the absence of depreciation and declining
in its presence.
Neither this implied life cycle of earnings
nor the behavior to which it is related
is supported by what we know of
the real world. The basic model allows
for an initial period of complete specialization
in the production of human capital
and, correspondingly, a period of
zero observed earnings. But if we assume
that the sum of the production elasticities
of the variable inputs is smaller than
1 (01 + 12 in eq. [2] and yl + 12 in
eq. [17], the more general formulation),
362 YORAM BEN-PORATH
marginal cost rises continuously even
before the capacity constraint is reached.
This is the source of the existence of a
phase (ii), a period in which 0 < s < 1,
when the individual engages simultaneously,
or alternately, in work in the market
and in the production of his human
capital. (A more complicated functional
form could have allowed increasing returns
to scale at small outputs and eventually
decreasing returns. This may be
more plausible and still would provide
the eventual check on the rate of invest-
ment.)
In this framework, where there is not
"automatic" growth in earnings, increase
in earning can come only from the
allocation of more services of human
capital to the market. This can be either
because of a reduction in what is allocated
to investment (st) or because of an
increase in the total stock available (K).
In order for some positive earnings to be
observed there must be some work in the
market; in order for earnings to decline
there must be some investment that
either increases from period to period the
total available stock or, by itself declining,
raises the fraction allocated to the
market. The closer the sum of the production
elasticities of the variable factors
to 1, that is, the closer we get to constant
returns to scale in terms of the variable
factors, the more concave is the life cycle
of earnings; the case of constant marginal
cost in which the life cycle of earnings is
a one-step function is the limiting case.
Throughout the preceding discussions
st has been interpreted as a parameter of
the allocation of time. If there are, however,
activities in which earnings and
human capital can be jointly produced,
then Stloses this meaning. We have many
examples of learning on the job in which
the time spent on the job cannot be allocated
in any meaningful sense between
pure work and pure learning; jointness is
too prevalent to be excluded. We can
still think of the individual as being faced
with a production frontier indicating the
possible combination of flows of earnings
and of human capital that he can produce.
Movements along this frontier,
however, would not necessarily represent
shifts between pure activities but, rather,
represent movement between activities,
each offering a different mix of earnings
and additions to productive capacity.
Shifts along this frontier are represented
by the control st, and they may involve
a change in occupations, of jobs within
the occupation, or of function with the
job. The less numerous and close together
are these alternative combinations, the
smaller is the justification for the continuous
differentiable frontier implied by
our formulation. Discontinuities and
kinks in this production frontier are
translated into kinks and jumps in the
life cycle of earnings. In either case, it
is clear that st becomes an empty concept
once it has lost its link with the observable
phenomenon of the allocation
of time.
III. THE MODEL AND AGGREGATE
EARNING PROFILES
The preceding section clarifies the role
played in this framework by increasing
costs in the explanation of a gradually
rising portion in the individual life cycle
of earnings. The purpose of the present
section is to show that even within this
framework an explanation of cohort life
cycles of earnings, or of the cross-section
profile of earnings, can be provided without
the assumption of rising costs. Thus,
assume for simplicity that 02 = 8 = 0
and /3 = 1. By the argument presented
in the preceding section, the individual
life cycle will be a stepwise function.
The marginal (and average) cost of
HUMAN CAPITAL AND THE LIFE CYCLE OF EARNINGS 363
producing human capital is (ao/to), fo
differing among people. The higher is Oo,
the later comes the date t* when the
downward-drifting demand price crosses
it from above-which is the date when
people jump from observed earnings of
zero to their earning capacity. By equating
acoo with P (10) we get (19):
=T+- ln (1--J . (19)
r \13o
The time path of the stock of human
capital is given by
Kt= Koefot for 0< 1< 1*
(20)
Kt = Koefoe*for t* < t < T.
Observed earnings are zero for 0 < t < t*
and are equal to earning capacity after-
ward.
Et= aoKoedot*
- aoKoefot T+(1/r) ln[1-(r/fo)]J(21
Using (21) we can see that, given
aoK0, earnings on the time of "emergence,"
1*, and beyond it, are an increasing
convex function of 1*. If in a
given cohort Ko and i3oare not negatively
correlated, we can expect to observe a
rising curve as we follow the earnings
history of the cohort through time (or
as we examine the cross-section profile of
earnings of a completely static population,
with stationary distributions by F0
and Ko, in a static economy). The exact
form of the curve depends on the distribution
of the cohort by 1o; reasonable
distributions by jOocan generate a curve
with the familiar S-shape.
SUMMARY AND SOME OPEN QUESTIONS
To the theory of investment in man
we have added here a production function
of human capital and explored some
of the implications of its properties for
the optimal path of accumulation of
human capital and the life cycle of earnings.
The concepts and parameters introduced
raise again some familiar problems
and are used to focus attention on
some others. In lieu of a summary we
shall now refer to a few of these.
1. The particular definition of human
capital K used is a measure of a quantity
of a source of productive services. It is
the stock that produces labor services in
"standard units" and is thus the analogue
of "machines" in the case of tangible
capital. This should be sharply distinguished
from Wt (defined in [8]),
which is the value of the individual as a
productive factor. It is affected by a
broader view of the individual's productivity,
by his durability, and by the rate
of interest, and it can be misleading as a
quantity measure of an input in any
given point in time. Being a part of the
individual's net worth (and the price that
he could get for himself in a competitive
slave market), it is probably better described
as "human wealth."
2. The speed of adjustment, the rate
at which the individual increases the
stock of human capital, determines the
ultimate size of this stock. The considerations
affecting the speed of adjustment
and the path of investment with age
merit an explicit analysis. The mechanism
provided here relies on the introduction
of dependence between the marginal
cost of producing human capital
and the rate of production (a justification
for this can come from the "learning
curve" considerations). Beyond a certain
point the saving in costs from postponing
investment to the next period compensates
for the loss of the returns that
would have come from getting the unit of
capital a period earlier.
Rising costs result here from a certain
specification of the technology. The
361 YORAM BEN-PORATII
properties of the production function and
the parameters introduced draw attention
to certain distinctions that have to
be made when the conglomerate of abilities
and external conditions is considered
and the implications of its properties are
explored.
3. In discussing the "initial endowment"
of individuals a different role is
played by the initial endowment in terms
of the ability to earn in the market (K9)
and the abilities to produce additions to
earning capacity. These consist of the
ability to make efficient use of inputs
purchasable in the market, D (expressed
by Oo and 32), and of one's own time
(/0, /1, or -y, in [17]). A related issue is
to what extent the individual, as he increases
his human capital, K, also raises
his efficiency in the production of K (in
eq. [17] this is expressed by the distinction
between 7y, and 72). This question
is related to the homogeneity of human
capital.
4. The possibility of producing human
capital in addition to earnings means
that the "real output" of the individual
consists of earnings plus the value of human
capital produced, when the latter is
evaluated by its shadow price. This will
be a measure of by how much total inactivity
at a given period would affect W.
5. Not much has been said of 8,
the rate of deterioration. Consideration
ought to be given to cases where a is
negative rather than positive as assumed
here and also to the dependence of deterioration
on the allocation of human
capital to different activities.
6. The horizon T is treated here as
exogenous. Opening the analysis to include
leisure will, of course, make retirement
endogenous, but even if T is the
date of death there can be types of investments
that will affect that date.
7. The three market prices-the rate
of interest r, the rental on human capital
a0, and the price of purchased inputs Pd
-affect behavior in the expected way.
These are the parameters of the models
that public policy can most directly attack.
The framework presented here may
prove convenient for mapping possible
effects of public policy and their interaction
with ability factors.
8. Various problems of measurement
are implied but not directly discussed.
One of these is the relation between the
integral of investment costs, (7), over
stretches of time and the integral of K,
or of Q. The problem of measurement in
this model hinges on the measurability
of st, which is a question about the prevalence
of activities in which human capital
and earnings are jointly produced. When
se is an observable phenomenon of the
allocation of time, the model becomes
operational, but, without it, much is lost.
9. We have abstracted here from uncertainty
and from capital rationing two
important considerations in investment
in human capital which should be incorporated
in a complete analysis.
10. A natural extension of the two-way
choice analyzed here between activities
producing current earnings and investment
in human capital would be to deal
with the three-way choice involving also
the allocation of time for activities of a
consumptive nature, following the approach
of Becker (1965).
The reader probably does not have to
be told how simplifying many of the
assumptions are and how many additional
aspects of investment in human
capital and of the determination of the
life cycle of earnings are relevant. The
main purpose of this paper was to raise
some more questions rather than provide
definite answers.
HUMAN CAPITAL AND THE LIFE CYCLE OF EARNINGS 365
REFERENCES
Becker, G. S. "Investment in Human Capital:
A Theoretic Analysis," J.P.E., LXX, No. 5,
Part 2 (October, 1962), 9-49.
. Human Capital. New York: Columbia
Univ. Press (for the National Bureau of
Economic Research), 1964.
. "A Theory of the Allocation of Time,"
Econ. J., Vol. LXXV (September, 1965).
. "Human Capital and the Personal Distribution
of Income: An Analytical Approach."
New York: National Bureau of
Economic Research, October, 1966 (mimeo-
graphed).
Friedman, M., and Kuznets, S. Income from
Independent Professions. New York: National
Bureau of Economic Research, 1945.
Mincer, J. "Investment in Human Capital
and Personal Income Distribution," J.P.E.,
LXVI, No. 4 (August, 1958), 281-302.
. "On-the-Job Training: Costs, Returns,
and Some Implications," ibid., LXX, No. 5,
Part 2 (October, 1962), 50-79.
Schultz, T. W. The Economic Value of Education.
New York: Columbia Univ. Press, 1963.