Question

Consider a production function of the form:
$ Y = 𝑎 log L + (1 − 𝑎) log K , 𝑎 ∈ (0, 1), 𝑎 ≠ 0.5 $
where, Y is output, L is labour, and K is capital. Then, the absolute value of elasticity of substitution is

• A. 1
• B. a
• C. (1 − a)
• D. $ \infty $

Answer 1

The Correct Option is A

Elasticity of Substitution in Logarithmic Cobb-Douglas Production Function

The given production function is:

Y = a log L + (1 − a) log K, where a ∈ (0,1) and a ≠ 0.5 

Where:

• Y is the output,
• L is the input of labor,
• K is the input of capital,
• a is a parameter indicating the weight of labor in the production process.

This production function is a logarithmic form of a Cobb-Douglas production function, which is known to have a constant elasticity of substitution (CES).

Step 1: Understanding Elasticity of Substitution

The elasticity of substitution (σ) measures how easily one input (e.g., labor) can be substituted for another input (e.g., capital) while keeping the level of output constant. Mathematically, it is given by:

σ = d ln(K/L) / d ln(MRTS)

Where:

• K/L is the ratio of capital to labor,
• MRTS (Marginal Rate of Technical Substitution) measures the rate at which one input can be substituted for another while maintaining the same level of output.

Step 2: Marginal Rate of Technical Substitution (MRTS)

For the given production function:

Y = a log L + (1 − a) log K

The marginal products of labor and capital are:

• MPL = ∂Y/∂L = a / L
• MPK = ∂Y/∂K = (1 − a) / K

The MRTS is the ratio of the marginal products:

MRTS = MPL / MPK = (a / L) / ((1 − a) / K) = aK / (1 − a)L

Step 3: Substitution Elasticity in Logarithmic Cobb-Douglas

The elasticity of substitution (σ) for a Cobb-Douglas production function is always equal to 1, regardless of the values of the parameters a and (1 − a). This is because the Cobb-Douglas function inherently exhibits a constant elasticity of substitution.

Step 4: Verifying the Options

• Option (A): 1 — Correct. The Cobb-Douglas production function has a constant elasticity of substitution equal to 1.
• Option (B): a — Incorrect. The parameter a determines the share of labor in the production process but does not influence the elasticity of substitution.
• Option (C): 1 − a — Incorrect. Similar to a, this parameter determines the share of capital in the production process but does not influence the elasticity of substitution.
• Option (D): ∞ — Incorrect. An infinite elasticity of substitution is a property of linear production functions, not Cobb-Douglas functions.

Final Answer:

The absolute value of elasticity of substitution is 1.