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
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
$ 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
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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 = (βQ / βK) Γ (K / Q)
Ξ·L = (βQ / βL) Γ (L / Q)
Step 3: Calculating Partial Derivatives
- βQ / βK = 2 Γ (1/2) K-1/2 = 2K-1/2
- βQ / βL = 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.
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