VES Production Function

In the last section, we read about the CES production function. Once we relax the assumption of constant elasticity of substitution, we arrive at variable elasticity of substitution (VES). The VES production function can be expressed as:

𝑌 = 𝛾𝐾^{𝛼(1−𝛿𝜌)} [𝐿 + (𝜌−1) 𝐾]^𝛼𝛿𝜌

Where,

Y: Output

K: Capital

L: Labor

α, ρ, δ and γ: Parameters

• 𝛾 > 0, α > 0
• 0 < δ <1
• 0 ≤ ρδ ≤ 1
• 𝐿/𝐾 > 1−𝜌/1−ρδ

Here, α is the parameter of returns of scale. If the value of α is 1, the production function exhibits constant returns to scale.

Elasticity of Substitution:

The elasticity of substitution σ for the VES production function is-

σ = σ (K, L) = 1 + (𝜌−1)/(1 − 𝛿𝜌) 𝐾/𝐿

• The elasticity of substitution σ varies with the capital-labour ratio around the intercept term of unity.
• The elasticity of substitution is greater than zero over the relevant range of K/L.
• For σ>0 requires that 𝐿/𝐾 > (𝜌−1)/(1 − 𝛿𝜌).

Properties of VES Production Function

(i). Positive and Diminishing Returns to Inputs:

The VES production function is increasing in labour and capital, i.e. positive marginal products.

Here any small increase in capital will lead to a decrease in the marginal product of capital. Any small increase in capital cause output to rise but at a diminishing rate. The same is true for labour.

(ii). VES Production Function reduced to Cobb Douglas and Linear Production Function

• For 𝜌 = 0 » Harrod Domar Fixed Coefficient Model
• For 𝜌 = 1 » Cobb Douglas Production Function
• For 𝜌 = 1/𝛿 (>1) » Linear Production Function

(iii). The Elasticity of Substitution can vary along an Isoquant:

The VES requires that the elasticity of substitution be the same only along a ray through the origin.

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