Economic growth and economic development 501

Introduction to Modern Economic Growth
is the cost of making irreversible capital investment at the market price R (t). This
investment is made before the firm knows which worker it will be matched with.
The important observation is that the objective function in (10.38) is strict concave
in kj (t) given the strict concavity of F (·, ·) from Assumption 1. Therefore, each
firm will choose the same level of physical capital, kˆ (t), such that
³
´
Z 1 ∂F kˆ (t) , hi (t)
di = R (t) .
(1 − λ)
∂k (t)
0
Now given this (expected) capital investment by firms, and following (10.33) from
the previous section, each worker’s objective function can be written as:
ả
à


hi (t)

,
F k (t) , hi (t) + R (t) bi (t − 1) − γ
ai

where we have substituted for the income mi (t) of the worker in terms of his wage
earnings and capital income, and introduced the heterogeneity in human capital
decisions. This implies the following choice of human capital investment by a worker
i:
λai

³
´
∂F kˆ (t) , hi (t)
hi (t)

=

0

à

ả
hi (t)
.
ai





This equation yields a unique equilibrium human capital investment hi k (t) for

each i. This human capital investment directly depends on the capital choices of all
the firms, kˆ (t) (since this affects the marginal product
human capital) and also
³ of ´
ˆ i kˆ (t) is strictly increasing in
depends implicitly on ai . Moreover, given (10.29), h
³
´

ˆ i kˆ (t) is a strictly concave function of
kˆ (t). Also, since γ (·) is strictly convex, h
kˆ (t). Substituting this into the first-order condition of firms, we obtain
³
³
´´
Z 1 ∂F kˆ (t) , h
ˆ i kˆ (t)
(1 − λ)
di = R (t) .
∂k (t)
0
Finally, to satisfy market clearing in the capital market, the rate of return to capital,
R (t), has to adjust, such that
kˆ (t) =

Z

0

1

bi (t − 1) di,

which follows from the facts that all firms choose the same level of capital investment
and that the measure of firms is normalized to 1. This equation implies that in the
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