CES Production Function

Another popular neo-classical production function is the constant elasticity of substitution (CES) production function. The CES production function was developed by Arrow, Chenery, Minhas and Solow as a generalisation of the Cobb-Douglas production function that allows for non-negative and constant elasticity of substitution.

Functional Form:

The standard CES production function can be written as follows:

Y = A [δ𝐾^−𝜌 + (1−δ) 𝐿^−𝜌]^−1/𝜌

Where,

Y is the output,

K and L are capital and labour inputs, and

A, δ, ρ are the parameters.

• A, technology determines the productivity and A𝜖[0, ∞).
• δ determines the optimal distribution of inputs and δϵ[0,1].
• Ρ determines the elasticity of substitution (σ) where ρϵ[-1,0) U(0,∞) and 𝜎 = 1/1+𝜌

Special Cases under CES Production Function

1. For ρ→0, σ approaches 1» Cobb Douglas Production Function.
2. For ρ→+∞, σ approaches 0» Leontief Production Function.
3. For ρ→-1, σ approaches ∞» Perfect Substitutes.

The CES production function is linearly homogeneous and therefore exhibits constant returns to scale. It is non-linear in parameters, so it cannot be estimated using the least squares method.

Properties of CES Production Function

(i). Constant Returns to Scale:

The Cobb-Douglas production function exhibits constant returns to scale.

Y = A [δ𝐾^−𝜌 + (1−δ) 𝐿^−𝜌]^−1/𝜌

𝑓(𝜆𝐾, 𝜆𝐿, 𝐴) = A [δ (𝜆𝐾)^−𝜌 + (1−δ) (𝜆𝐿)^−𝜌]^−1/𝜌

= A λ[δ𝐾^−𝜌 + (1−δ) 𝐿^−𝜌]^−1/𝜌

=λY

(ii). Positive and Diminishing Returns to Inputs:

The marginal products of the input are:

𝑀𝑃_𝐾 = ∂Y/∂K = 𝛿/𝐴𝜌 (𝑌/𝐾)^𝜌+1

𝑀𝑃_𝐿 = ∂Y/∂L = 1−𝛿/𝐴𝜌 (𝑌/𝐿)^𝜌+1

They both are positive for K, L> 0. Any small increase in capital or labour increases the output but at a diminishing rate.

(iii) Inada Conditions

(iv). The Elasticity of Substitution is 𝝈 = 𝟏/𝟏+𝝆

The elasticity of substitution is calculated using the formula:

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