Economic growth and economic development 682

Introduction to Modern Economic Growth
Finally, the prices of the two intermediate goods are derived from the marginal
product conditions of the final good technology, (15.3), which imply
à
ả 1
1 YH (t)
pH (t)
=
p (t)
pL (t)

YL (t)
à
ả 1
1 N
1
(t)
H
H
p (t)
=

NL (t) L
à
ả à
ả
NH (t) H
1
(15.18)
=
,

NL (t) L
the first line simply defines p (t) as the relative price between the two intermediate
goods and uses the fact that the marginal productivity of the two intermediate goods
must be equal to this relative price. The second line substitutes from (15.16) and
(15.17) above. Using the latter equation, we can also calculate the relative factor
prices in this economy as:
ω (t) ≡

wH (t)
wL (t)

NH (t)
NL (t)
à
ả à
ả 1 à ả 1
1
NH (t)
H
=
.

NL (t)
L
= p (t)1/β

(15.19)
where

σ ≡ ε − (ε − 1) (1 − β) .
is the (derived) elasticity of substitution between the two factors. The first line of
(15.19) defines ω (t) as the relative wage of factor H compared to factor L. The second line uses the definition of marginal product combined with (15.16) and (15.17),
and the third line uses (15.18). We refer to σ as the (derived) elasticity of substitution between the two factors, since it is exactly equal to
ả1
à
d log ω (t)
.
σ=−
d log (H/L)
To complete the description of equilibrium in the technology side, we need to
impose the following free entry conditions:
(15.20)

η L VL (t) ≤ 1 and η L VL (t) = 1 if ZL (t) > 0.
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