Homework Help is Here – Start Your Trial Now!
Business
Economics
Show that if the production function is concave, the cost function is convex (in y).

Show that if the production function is concave, the cost function is convex (in y).

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
See similar textbooks
icon
Related questions
Question

i need Q solution. other pic is just for help

Show that if the production function is concave, the cost function is convex (in y).
Transcribed Image Text:Show that if the production function is concave, the cost function is convex (in y).
1. The Cobb-Douglas production function is f(x1,x2) = Axx. To get the Lagrange multiplier of the cost minimization problem, you will need to solve the following cost minimization problem: min w,x1 + W2*2 X1,X220 subject to f(x1, x2) = Ax*x 2y You will set up the Lagrangian, take first-order conditions, and use them to solve for x, (W, W2, y) and x2 (W1, W2, y). Then you can substitute these values back into one of your first-order conditions to solve for your Lagrange multiplier. To show that this is equal to the derivative of the cost function, you will need to get the cost function by substituting your calculations for x, (w, W2, y) and x2 (W1, Wz2, y) into w1x1 + W1x2. This will give you the cost function, c(w1, W2, y). You can then take its derivative with respect to output, y, and note that it is equal to your Lagrange multiplier. 2. Focus on just getting the cost function and the conditional demands for inputs. You do not need to solve an additional cost minimization problem if you notice that this is similar to questions 1, but with additional fixed costs. That is, your cost functions will just be the cost function from #1 plus w,ł, + wzł2, and your conditional demands for inputs will be the same as in #1 but with +x, for x1 (W1, W2. y) and +žz for x2(W1,W2, y). Helping material 3. I did not write the problem sets (or quizzes, tests, lecture notes, etc.), so I don't have the ability/authority to change this question, but I find it very confusing and unclear as to what it is asking for. Feel free to skip this problem. 4. You can get the conditional demands for inputs and the cost function in the same manner as in problem 1, except now with the CES production function. 5. It may be helpful to use the equation s'(wy) _ dinc(wy) and the fact that xg = c'(w,y). Even though c(w.y) this one says "lots of algebra", it really shouldn't be nearly as much as some of the previous problems. 6. same process as #5
Transcribed Image Text:1. The Cobb-Douglas production function is f(x1,x2) = Axx. To get the Lagrange multiplier of the cost minimization problem, you will need to solve the following cost minimization problem: min w,x1 + W2*2 X1,X220 subject to f(x1, x2) = Ax*x 2y You will set up the Lagrangian, take first-order conditions, and use them to solve for x, (W, W2, y) and x2 (W1, W2, y). Then you can substitute these values back into one of your first-order conditions to solve for your Lagrange multiplier. To show that this is equal to the derivative of the cost function, you will need to get the cost function by substituting your calculations for x, (w, W2, y) and x2 (W1, Wz2, y) into w1x1 + W1x2. This will give you the cost function, c(w1, W2, y). You can then take its derivative with respect to output, y, and note that it is equal to your Lagrange multiplier. 2. Focus on just getting the cost function and the conditional demands for inputs. You do not need to solve an additional cost minimization problem if you notice that this is similar to questions 1, but with additional fixed costs. That is, your cost functions will just be the cost function from #1 plus w,ł, + wzł2, and your conditional demands for inputs will be the same as in #1 but with +x, for x1 (W1, W2. y) and +žz for x2(W1,W2, y). Helping material 3. I did not write the problem sets (or quizzes, tests, lecture notes, etc.), so I don't have the ability/authority to change this question, but I find it very confusing and unclear as to what it is asking for. Feel free to skip this problem. 4. You can get the conditional demands for inputs and the cost function in the same manner as in problem 1, except now with the CES production function. 5. It may be helpful to use the equation s'(wy) _ dinc(wy) and the fact that xg = c'(w,y). Even though c(w.y) this one says "lots of algebra", it really shouldn't be nearly as much as some of the previous problems. 6. same process as #5
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Risk Aversion
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.
Recommended textbooks for you
ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
Economics
ISBN:
9780190931919
Author:
NEWNAN
Publisher:
Oxford University Press
Principles of Economics (12th Edition)
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (MindTap Course List)
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-…
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS