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

Updated On: Feb 10, 2025
  • 2
  • 1
  • 4
  • 0.5
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is B

Solution and Explanation

Calculating Elasticities from the Production Function 

Step 1: Given Production Function

The production function is:

Q(K, L) = 2K1/2 + 3L1/2

Step 2: Elasticity Formula

The elasticity of output with respect to capital (Ξ·K) and labor (Ξ·L) is given by:

Ξ·K = (βˆ‚Q / βˆ‚K) Γ— (K / Q)

Ξ·L = (βˆ‚Q / βˆ‚L) Γ— (L / Q)

Step 3: Calculating Partial Derivatives

  • βˆ‚Q / βˆ‚K = 2 Γ— (1/2) K-1/2 = 2K-1/2
  • βˆ‚Q / βˆ‚L = 3 Γ— (1/2) L-1/2 = 3L-1/2

Step 4: Computing Elasticities

Substituting the derivatives into the elasticity formulas:

Ξ·K = (2K-1/2 / (2K1/2 + 3L1/2)) Γ— K

Ξ·L = (3L-1/2 / (2K1/2 + 3L1/2)) Γ— L

Step 5: Summing the Elasticities

The total elasticity is:

Ξ·K + Ξ·L = 1

Conclusion

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

Was this answer helpful?
0
1

Top Questions on Production Function

View More Questions

Questions Asked in IIT JAM EN exam

View More Questions