Question

Consider the production function:
$ Q(K, L) = (2 \sqrt {K} + 3 \sqrt{L})^2$ where Q is the output, K is capital, and L is labour. If $\eta_K$ and $\eta_L$ denote the output elasticities with respect to capital and labour, respectively, then the value of ($\eta_K$ + $\eta_L$) is

• A. 2
• B. 1
• C. 4
• D. 0.5

Answer 1

The Correct Option is B

Calculating Elasticities from the Production Function 

Step 1: Given Production Function

The production function is:

Q(K, L) = 2K^1/2 + 3L^1/2

Step 2: Elasticity Formula

The elasticity of output with respect to capital (η_K) and labor (η_L) is given by:

η_K = (∂Q / ∂K) × (K / Q)

η_L = (∂Q / ∂L) × (L / Q)

Step 3: Calculating Partial Derivatives

• ∂Q / ∂K = 2 × (1/2) K^-1/2 = 2K^-1/2
• ∂Q / ∂L = 3 × (1/2) L^-1/2 = 3L^-1/2

Step 4: Computing Elasticities

Substituting the derivatives into the elasticity formulas:

η_K = (2K^-1/2 / (2K^1/2 + 3L^1/2)) × K

η_L = (3L^-1/2 / (2K^1/2 + 3L^1/2)) × L

Step 5: Summing the Elasticities

The total elasticity is:

η_K + η_L = 1

Conclusion

Since the sum of the elasticities equals 1, this production function exhibits constant returns to scale.