Education and Human Capital
optimal investment in training, whether general or specific. Since y(0) $ z, the planner
decides to assign all of them to the technology y(·) in use in the market rather than let
them produce z domestically. If the planner dedicates an amount i of resources to the
training of an individual, his problem is written as follows:
max y(i) 2 i
i
The solution of this problem is again given by the equality (4.1). Thus, in a perfectly competitive economy, individual choices regarding training are socially efficient.
The theory of human capital suggests that the mechanisms of competition give
individuals an incentive to become educated for the purpose of acquiring knowledge or
skills on which the market sets a premium. Moreover, it shows that individual educational choices are socially efficient if the labor market is perfectly competitive.
Evidently in reality, markets are not perfectly competitive, as we will see more
precisely in the second part of the present book. That being the case, wages and productivity differ, and educational choices are no longer efficient. For example, if wages are
lower than productivity because firms dispose of monopsony power, the investment in
human capital is less than the social optimum. We will also see at the end of this chapter that there are externalities, for the most part positive, associated with education.
In particular, better-educated persons transmit part of their knowledge, which boosts
the productivity of those around them. Education also reduces criminality for that matter. The collective return to education is thus greater than the individual return. Now,
individuals do not take the positive externalities of education into account when they
choose how much effort to put into their education. That implies that educational effort
is generally insufficient in the absence of any intervention by the public authorities.
In sum, the results obtained in this section signify, more generally, that competition on the labor market allows workers to derive value, in the form of earnings, from
knowledge that improves their productivity. Conversely, in the absence of competition,
the incentives to invest so as to improve productivity disappear. For this reason it is generally firms that invest in specific training, which wage earners cannot exploit to increase
their market value. We now examine in greater detail the determinants of educational
investment in a context in which improved productivity leads to a higher wage.
2.2
Schooling and Wage Earnings
The theory of human capital throws light on the choice of the duration of studies. It
shows that the length of time spent in school is influenced by individual characteristics
such as aptitude and inherited human capital, by the discount rate, and by the productivity achieved thanks to the accumulation of human capital.
2.2.1
The Choice Between Getting Educated and Getting Paid
To illustrate these propositions, we examine the choices of an individual who can
acquire education starting at date t 5 0 and whose life in the labor force ends at date
T . 0. The retirement period is set aside for the sake of simplification. We work with
a continuous time model in which the preferences of an agent are represented by an
instantaneous utility function equal to his current earnings and by a discount factor r . 0. At every moment it is possible to study or work but not to do both at the
201
202
P a r t One
Chapter 4
same time. Education allows the accumulation of “human capital,” that is, it allows the
agent to increase his stock of knowledge. We assume from this point forward that over
every interval of time [t, t1dt], it is possible for an individual to dedicate a fraction
s(t) ∈ [0, 1] of this interval to training. The law of motion of human capital, denoted
h(t), is defined by the differential equation:3
ḣ(t) 5 us(t)h(t)
(4.2)
The parameter u represents the efficiency of the effort made by the agent to become
educated, so it reflects his aptitude. Relation (4.2) simply means that if an individual
decides to become educated, the relative increase ḣ/h in his human capital is proportional to his individual efficiency u and to his effort in education s(t). Let us assume
that an individual endowed with a stock of human capital h(t) at date t produces a
quantity of goods Ah(t), A . 0, at this date, and that there is free entry into any type
of job. Then profits are zero and the wage received at date t by this person will simply
equal Ah(t) when that person works. It follows that if an individual dedicates a fraction
s(t) of period [t, t1dt] to education, he works during a fraction 1 2 s(t) of this period,
and so receives earnings A[1 2 s(t)]h(t)dt. Thus his gain discounted over the whole of
his life cycle is defined by:
T
V5
A[1 2 s(t)]h(t)e2rt dt
(4.3)
0
To define the optimal choice of schooling, it is useful to compute the marginal
returns to education effort at time t. We get:
äV
5 2Ah(t)e2rt 1
äs(t)
T
A[1 2 s(z)]
0
äh(z) 2rz
e dz
äs(t)
Since the integration of the differential equation (4.2) yields:
t
h(t) 5 h0 exp u s(z)dz
0
where h0 denotes the stock of human capital for zero year of schooling, we have:
äh(z)
5 0 if z , t
äs(t)
äh(z)
5 uh(z)
äs(t)
and
if z $ t
Therefore,
äV
5 2Ah(t)e2rt 1
äs(t)
3 The time derivative of h(t) is denoted ḣ(t).
t
T
uA[1 2 s(z)]h(z)e2rz dz
(4.4)
Education and Human Capital
Then we can compute the derivative of the marginal returns to education effort with
respect to t, or, formally:
d
äV
5 2Aḣ(t)e2rt 1 rAh(t)e2rt 2 uA[1 2 s(t)]h(t)e2rt
dt äs(t)
This last equation implies, together with (4.2):
äV
d
5 Ah(t)e2rt (r 2 u)
dt äs(t)
(4.5)
This equation shows that the marginal return to educational effort increases over time
if r . u. Since equation (4.4) entails that the marginal return to education is negative
at date T, the marginal return to education is necessarily negative in the interval [0, T]
if r . u. In consequence, s(t) 5 0 for all t # T if r . u. In other words, there is never
an interest in educating oneself if the discount rate r is greater than the efficiency u of
educational effort. So if one is to acquire education, one must be sufficiently patient and
the returns to education must be sufficiently high.
2.2.2
The Optimal Duration of Schooling
If r , u, relation (4.5) tells us that the marginal return to educational effort decreases over
äV
time. Since at date T, we have, according to equation (4.4), äs(T)
5 2Ah(T)e2rT , 0,
äV
d
äV
5 0. As dt
, 0, the marginal return to
there may exist a date s such that äs(s)
äs(t)
education is positive for t , s and negative for t . s. This means that educational effort
is necessarily null after date s. Before date s, the fact that äV/äs(t) . 0 entails that it
is optimal to furnish maximum effort, or s(t) 5 1. There is thus an interest in devoting
all one’s time to becoming educated, s(t) 5 1, before date s and to acquire no further
education, s(t) 5 0, after date s. In this case, we have h(s) 5 Ah0 eus and h(t) 5 h(s) for
äV
T$ t $ s. Since date s is defined by äs(s)
5 0, relation (4.4) allows us to obtain an explicit
expression of s.
We have:
⎧
⎨ T 1 1 ln
r
s5
⎩
0
u2r
u
if u $
r
12e2rT
otherwise
This equation shows that the duration of schooling increases with the duration of life
T and with the efficiency parameter u. Hence the most efficient individuals spend the
longest amount of time on education.
We can also see that the duration of schooling decreases with the discount rate
r. This means that more impatient individuals, or ones facing higher financial hurdles
that drive up the cost of borrowing, must study for shorter periods. We also note that s is
positive only if r , u(1 2 e2rT ), in other words, if the efficiency of education and the age
of retirement are sufficiently large with respect to the discount rate. Hence it might be
optimal not to get any training or education when the efficiency parameter is too small,
203