Answered: shown to be a special case of a larger…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Question

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.

QQ	= =	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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