Economic growth and economic development 676

Introduction to Modern Economic Growth
CES production function takes the form
i σ
h
σ−1
σ−1 σ−1
,
Y (t) = γ (AL (t) L (t)) σ + (1 − γ) (AH (t) H (t)) σ

where AL (t) and AH (t) are two separate technology terms, γ ∈ (0, 1) is a distribu-

tion parameter which determines the importance of the two factors in the production
function, and σ ∈ (0, ∞) is the elasticity of substitution between the two factors.

When σ = ∞, the two factors are perfect substitutes, and the production function

is linear. When σ = 1, the production function is Cobb-Douglas, and when σ = 0,

there is no substitution between the two factors, and the production function is
Leontieff. When σ > 1, we refer to the factors as gross substitutes, and when σ < 1,
we refer to them as gross complements.
Clearly, by construction, AL (t) is L-augmenting, while AH (t) is H-augmenting.
We will also refer to AL as labor-complementary. Interestingly, whether technological change is L-biased or H-biased depends on the elasticity of substitution, σ.
Let us first calculate the relative marginal product of the two factors (see Exercise
15.1):
(15.1)

1
MPH
=
MPL

à

AH (t)
AL (t)

à
ả 1


H (t)
L (t)

ả 1

.

The relative marginal product of H is decreasing in its relative abundance, H (t) /L (t).
This is the usual substitution effect, leading to a negative relationship between relative supplies and relative marginal products (or prices) and thus to a downwardsloping relative demand curve (see Figure 15.3). The effect of AH (t) on this relative
marginal product depends on σ, however. If σ > 1, an increase in AH (t) (relative
to AL (t)) increases the relative marginal product of H. In contrast, when σ < 1, an
increase in AH (t) reduces the relative marginal product of H. Therefore, when the
two factors are gross substitutes, H-augmenting (H-complementary) technological
change is also H-biased. In contrast, when the two factors are gross complements,
the relationship is reversed, and H-augmenting technical change is now L-biased.
Naturally, when σ = 1, we are in the Cobb-Douglas case, and neither a change
in AH (t) nor in AL (t) is biased towards any of the factors. Note also for future
reference that by virtue of the fact that σ is the elasticity of substitution between
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