shown to be a special case of a larger class of linear homogeneous production functions having the following mathematical form: Q=γ[δK−ρ+(1 - δ)L−ρ]−ν/ρ�=�[δK−ρ+(1 - δ)�−ρ]−ν/ρ where γ is an efficiency parameter that shows the output resulting from given quantities of inputs; δ is a distribution parameter (0 ≤ δ ≤ 1) that indicates the division of factor income between capital and labor; ρ is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the production process; and ν is a scale parameter (ν > 0) that indicates the type of returns to scale (increasing,
shown to be a special case of a larger class of linear homogeneous production functions having the following mathematical form: Q=γ[δK−ρ+(1 - δ)L−ρ]−ν/ρ�=�[δK−ρ+(1 - δ)�−ρ]−ν/ρ where γ is an efficiency parameter that shows the output resulting from given quantities of inputs; δ is a distribution parameter (0 ≤ δ ≤ 1) that indicates the division of factor income between capital and labor; ρ is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the production process; and ν is a scale parameter (ν > 0) that indicates the type of returns to scale (increasing,
Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)
14th Edition
ISBN:9781305506381
Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris
Publisher:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris
Chapter7: Production Economics
Section: Chapter Questions
Problem 10E
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The Cobb-Douglas production function can be shown to be a special case of a larger class of linear homogeneous production functions having the following mathematical form:
Q=γ[δK−ρ+(1 - δ)L−ρ]−ν/ρ�=�[δK−ρ+(1 - δ)�−ρ]−ν/ρ
where γ is an efficiency parameter that shows the output resulting from given quantities of inputs; δ is a distribution parameter (0 ≤ δ ≤ 1) that indicates the division of factor income between capital and labor; ρ is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the production process; and ν is a scale parameter (ν > 0) that indicates the type of returns to scale (increasing, constant, or decreasing).
Complete the following derivation to show that when ν = 1, this function exhibits constant returns to scale.
First of all, if ν = 1:
| Q� | = = | γ[δK−ρ+(1 - δ)L−ρ]−1/ρ�[δK−ρ+(1 - δ)�−ρ]−1/ρ |
| = = | γ[δK−ρ(−1/ρ)+(1 - δ)L−ρ(−1/ρ)]�[δK−ρ(−1/ρ)+(1 - δ)�−ρ(−1/ρ)] | |
| = = |
Then, increase the capital K and labor L each by a factor of λ, or K* = (λ)K and L* = (λ)L. If the function exhibits constant returns to scale, then Q* = (λ)Q.
| Q*Q* | = = | γ[δ(λ)K−ρ+(1 - δ)(λ)L−ρ]−1/ρ�[δ(λ)K−ρ+(1 - δ)(λ)L−ρ]−1/ρ |
| = = | γ[δλK−ρ(−1/ρ)+(1 - δ)λL−ρ(−1/ρ)]�[δλK−ρ(−1/ρ)+(1 - δ)λL−ρ(−1/ρ)] | |
| = = | ||
| = = | ||
| = = | λQλQ |
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