Economic growth and economic development 704

Introduction to Modern Economic Growth
Then the next proposition shows a negative result on the possibility of purely laboraugmenting technological change.
Proposition 15.13. Consider the baseline model of directed technological change
with the knowledge spillovers specification and state dependence. Suppose that δ < 1
and capital accumulates according to (15.46). Then there exists no BGP.
Ô

Proof. See Exercise 15.22.

Intuitively, even though technological change is more labor augmenting than
capital augmenting, there is still capital-augmenting technological change in equilibrium, and this, combined with capital accumulation, is inconsistent with balanced
growth. In fact, even a more negative result can be proved (see again Exercise
15.22): in any asymptotic equilibrium, the interest rate cannot be constant, thus
consumption and output growth cannot be constant.
In contrast to these negative results, there is a special case that justifies the basic
structure of the neoclassical growth model. This takes place when we have extreme
state dependence, so that δ = 1. In this case, it can be verified that technology
market equilibrium implies the following relationship in BGP (see Exercise 15.23):
r (t) K (t)
= η −1 .
wL (t) L

(15.47)

Thus, directed technological change ensures that the share of capital is constant in
national income. This already gives the intuition for why steady capital accumulation should lead to purely-labor augmenting technological change (from our analysis
in Chapter 2). This is indeed the case. More specifically, recall from (15.19) that
à
ả à
ả 1 à
ả 1

K (t) σ
1 − γ σ NK (t) σ
r (t)
=
,
wL (t)

NL (t)
L
therefore,
r (t) K (t)
=
wL (t) L (t)

à

1


ả à

NK (t)
NL (t)

à
ả 1


In this case, (15.47) combined with (15.46) implies that
(15.48)

N˙ L (t) N˙ K (t)
−
= sK .
NL (t) NK (t)
690

K (t)
L

¶ σ−1
σ

.