A firm faces the following production function, Y = AKa L¹-a (1) Here Y is output, K is capital, L is fixed labour, and A is a measure of technology. The firm uses an optimal amount of capital determined by the condition,
A firm faces the following production function, Y = AKa L¹-a (1) Here Y is output, K is capital, L is fixed labour, and A is a measure of technology. The firm uses an optimal amount of capital determined by the condition,
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 8E
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Question
![A firm faces the following production function,
Y = AKL¹-a
(1)
Here Y is output, K is capital, L is fixed labour, and A is a measure of technology. The firm
uses an optimal amount of capital determined by the condition,
MPK = r + 8
(2)
Where MPK is the marginal productivity of capital, r is the real interest rate, and 8 is the
depreciation rate.
(a) Using equations (1) and (2) find an expression for K*, the optimal amount of capital the
firm should use.
(b) Referring to you result from part (a), comment on what happens to K* when each of the
following variables change (holding other variables constant),
The measure of technology (A) falls
The depreciation rate (8) increases
The real interest rate (r) increases
(i)
(ii)
(iii)
[Hint: For each of parts (i)-(iii) you are being asked to comment on what happen to K* if just
the variable mentioned in the question changes. Your answers should state whether K*
increases, decreases, or stays the same. This should be accompanied for each part by a
sentence or two of explanation.]
(c) Suppose that a = 0.5, L = 25, and A = 20. Further, the real interest rate, r, is 2% and
capital depreciates at a rate (8) of 18% per year. What is the optimal amount of capital
this firm should install?
(d) Now suppose that A increases to 25, while a, Land & remain unchanged. Further, the
optimal capital stock remains at the amount you found in part (c). What does this mean
must have happened to the real interest rate? Find the new real interest rate and draw a
clear diagram showing MPK and user cost of capital lines before and after the changes
to A and r.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e210e3b-7b89-45ac-81ce-3e543ee87969%2F9bb75a81-56ed-4144-aa70-19aa329da882%2F7deeaws_processed.png&w=3840&q=75)
Transcribed Image Text:A firm faces the following production function,
Y = AKL¹-a
(1)
Here Y is output, K is capital, L is fixed labour, and A is a measure of technology. The firm
uses an optimal amount of capital determined by the condition,
MPK = r + 8
(2)
Where MPK is the marginal productivity of capital, r is the real interest rate, and 8 is the
depreciation rate.
(a) Using equations (1) and (2) find an expression for K*, the optimal amount of capital the
firm should use.
(b) Referring to you result from part (a), comment on what happens to K* when each of the
following variables change (holding other variables constant),
The measure of technology (A) falls
The depreciation rate (8) increases
The real interest rate (r) increases
(i)
(ii)
(iii)
[Hint: For each of parts (i)-(iii) you are being asked to comment on what happen to K* if just
the variable mentioned in the question changes. Your answers should state whether K*
increases, decreases, or stays the same. This should be accompanied for each part by a
sentence or two of explanation.]
(c) Suppose that a = 0.5, L = 25, and A = 20. Further, the real interest rate, r, is 2% and
capital depreciates at a rate (8) of 18% per year. What is the optimal amount of capital
this firm should install?
(d) Now suppose that A increases to 25, while a, Land & remain unchanged. Further, the
optimal capital stock remains at the amount you found in part (c). What does this mean
must have happened to the real interest rate? Find the new real interest rate and draw a
clear diagram showing MPK and user cost of capital lines before and after the changes
to A and r.
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