Economic growth and economic development 502

Introduction to Modern Economic Growth
closed economy version of the current model, capital per firm is fixed by bequest
decisions from the previous period. The main economic forces we would like to
emphasize here are seen more clearly when physical capital is not predetermined.
For this reason, let us imagine that the economy in question is small and open,
so that R (t) = R∗ is pinned down by international financial markets (the closed
economy version is further discussed in Exercise 10.18). Under this assumption, the
equilibrium level of capital per firm is determined by
³ ´´
³
Z 1 ∂F k,
ˆ h
ˆ i kˆ
di = R∗ .
(10.39)
(1 − λ)
∂k
0
Proposition 10.3. In the open economy version of the model described here,
there exists a unique positive level of capital per worker kˆ given by (10.39) such that
ˆ Given k,
ˆ the human capital
the equilibrium capital per worker is always equal to ³k.
´
ˆ i kˆ such that
investment of worker i is uniquely determined by h
(10.40)

λai

³ ´´

³
ˆ
ˆ
∂F k, hi kˆ
∂h

³ ´⎞
ˆhi kˆ
⎠.
= γ0 ⎝
ai
⎛

³ ´
ˆ and a decline in R∗ increases kˆ and h
ˆ i for
ˆ
We have that hi kˆ is increasing in k,

all i ∈ [0, 1].

In addition to this equilibrium, there also exists a no-activity equilibrium in which
ˆ i = 0 for all i ∈ [0, 1].
kˆ = 0 and h
³ ´
ˆ i kˆ is a concave
Proof. Since F (k, h) exhibits constant returns to scale and h
³ ´´
´
R1³ ³

ˆ
ˆ
ˆ
ˆ
function of k for each i, 0 ∂F k, hi k /∂k di is decreasing in kˆ for a distribu³ ´
ˆ (10.40) determines h
ˆ i kˆ
tion of [ai ]i∈[0,1] . Thus kˆ is uniquely determined. Given k,
³ ´
ˆ i kˆ is
uniquely. Applying the Implicit Function Theorem to (10.40) implies that h
ˆ Finally, (10.39) implies that a lower R∗ increases k,
ˆ and from the
increasing in k.
ˆ i for all i ∈ [0, 1] increase as well.
previous observation h

The no-activity equilibrium follows, since when all firms choose kˆ = 0, output is
ˆi = 0
ˆ i = 0, and when h
equal to zero and it is best response for workers to choose h
for all i ∈ [0, 1], kˆ = 0 is the best response for all firms.
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