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: Cengage Learning
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Chapter 7, Problem 10E
To determine
Whether the given function shows constant returns to scale or not at v=1
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Use second image for reference,
for part b here is referene;
The maximum profit is found at the tangency between the production function and the
isoprofit line.
In other words, the slope of the production function and the slope of the
isoprofit line must be the same. This is written as
MPL = w
where w is the slope of the isoprofit line. Then we get
sqrt1 / 2L = w
=>
1/2w = sqrtL
=>
L*D = 1/4w^2
For each of the following production functions Q = f(K,L),
find the marginal productivity with respect to K and L
a) Q = 5KL – 2K² – 2L²
at K = 1 and L = 1
%3D
1
b) Q = 0.03K³ – 0.4KL + 0.5L7 at K = 8 and L = 4
-
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…
Chapter 7 Solutions
Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)
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