A firm faces the following production function, Y = AKα 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, MPK = r +8 (2) Where MPK is the marginal productivity of capital, r is the real interest rate, and ♪ is the depreciation rate. (a) Using equations (1) and (2) find an expression for K*, the optimal amount of capital the firm should use. [3 marks] (b) Referring to your result from part (a), comment on what happens to K* when each of the following variables change (holding other variables constant), (i) The measure of technology (A) falls (ii) The depreciation rate (8) increases (iii) The real interest rate (r) increases [2 marks] [2 marks] [2 marks] [Hint: For each of parts (i)-(iii) you are being asked to comment on what happens to K* if just the variable mentioned in the question part changes. Your answer 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, & A = 20. Further, the real interest rate, r, is 2% and capital depreciates at a rate (8) of 18%. What is the optimal amount of capital this firm should install? [2 marks] (d) Now suppose that A increases to 25, while a, L and 8 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. [4 marks]
A firm faces the following production function, Y = AKα 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, MPK = r +8 (2) Where MPK is the marginal productivity of capital, r is the real interest rate, and ♪ is the depreciation rate. (a) Using equations (1) and (2) find an expression for K*, the optimal amount of capital the firm should use. [3 marks] (b) Referring to your result from part (a), comment on what happens to K* when each of the following variables change (holding other variables constant), (i) The measure of technology (A) falls (ii) The depreciation rate (8) increases (iii) The real interest rate (r) increases [2 marks] [2 marks] [2 marks] [Hint: For each of parts (i)-(iii) you are being asked to comment on what happens to K* if just the variable mentioned in the question part changes. Your answer 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, & A = 20. Further, the real interest rate, r, is 2% and capital depreciates at a rate (8) of 18%. What is the optimal amount of capital this firm should install? [2 marks] (d) Now suppose that A increases to 25, while a, L and 8 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. [4 marks]
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 1.5CE
Related questions
Question
![A firm faces the following production function,
Y = AKα 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,
MPK = r +8
(2)
Where MPK is the marginal productivity of capital, r is the real interest rate, and ♪ is
the depreciation rate.
(a) Using equations (1) and (2) find an expression for K*, the optimal amount of
capital the firm should use.
[3 marks]
(b) Referring to your result from part (a), comment on what happens to K* when
each of the following variables change (holding other variables constant),
(i)
The measure of technology (A) falls
(ii)
The depreciation rate (8) increases
(iii)
The real interest rate (r) increases
[2 marks]
[2 marks]
[2 marks]
[Hint: For each of parts (i)-(iii) you are being asked to comment on what happens to
K* if just the variable mentioned in the question part changes. Your answer 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, & A = 20. Further, the real interest rate, r, is 2%
and capital depreciates at a rate (8) of 18%. What is the optimal amount of capital
this firm should install?
[2 marks]
(d) Now suppose that A increases to 25, while a, L and 8 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.
[4 marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe67b9373-07fc-4c99-ae45-ace7cb60cf24%2F6da75f62-bbbb-47de-a986-47b55d5de9cb%2Feuy1ch7_processed.png&w=3840&q=75)
Transcribed Image Text:A firm faces the following production function,
Y = AKα 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,
MPK = r +8
(2)
Where MPK is the marginal productivity of capital, r is the real interest rate, and ♪ is
the depreciation rate.
(a) Using equations (1) and (2) find an expression for K*, the optimal amount of
capital the firm should use.
[3 marks]
(b) Referring to your result from part (a), comment on what happens to K* when
each of the following variables change (holding other variables constant),
(i)
The measure of technology (A) falls
(ii)
The depreciation rate (8) increases
(iii)
The real interest rate (r) increases
[2 marks]
[2 marks]
[2 marks]
[Hint: For each of parts (i)-(iii) you are being asked to comment on what happens to
K* if just the variable mentioned in the question part changes. Your answer 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, & A = 20. Further, the real interest rate, r, is 2%
and capital depreciates at a rate (8) of 18%. What is the optimal amount of capital
this firm should install?
[2 marks]
(d) Now suppose that A increases to 25, while a, L and 8 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.
[4 marks]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Managerial Economics: Applications, Strategies an…
Economics
ISBN:
9781305506381
Author:
James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris
Publisher:
Cengage Learning
Microeconomics: Principles & Policy
Economics
ISBN:
9781337794992
Author:
William J. Baumol, Alan S. Blinder, John L. Solow
Publisher:
Cengage Learning
Managerial Economics: Applications, Strategies an…
Economics
ISBN:
9781305506381
Author:
James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris
Publisher:
Cengage Learning
Microeconomics: Principles & Policy
Economics
ISBN:
9781337794992
Author:
William J. Baumol, Alan S. Blinder, John L. Solow
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Economics (MindTap Course List)
Economics
ISBN:
9781337617383
Author:
Roger A. Arnold
Publisher:
Cengage Learning
