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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−γ MPH = MPL γ µ AH (t) AL (t) ả H (t) L (t) ả σ1 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 662