The production function for a firm is p(x, y) = 68x"y"", where x and y are the number of units of labor and capital utilized. Suppose that labor costs $102 per unit and capital costs $544 per unit and that the firm decides to produce 8704 units of goods. Determine the following. (a) Determine the amounts of labor and capital that should be utilized in order to minimize the cost. That is, find the values of x and y that minimize 102x + 544y, subject to the constraint 8704 - 68x*y" = 0. x= 256, y= 16 (Type exact answers in simplified form.) (b) Find the value of at the optimal level of production. A= 4 (Type an exact answer in simplified form.) (c) Find the expressions for marginal productivity of labor and marginal productivity of capital. Then find the ratio of marginal productivity of labor to marginal productivity of capital at the optimum level of production and the ratio of unit price of labor to unit price of capital and determine the relationship between the ratios. 11 marginal productivity of labor = 51x y* 3 3 marginal productivity of capital = 17x*y Determine the marginal productivity labor and marginal productivity capital at the optimum level of production and then compare the ratios. marginal productivity of labor marginal productivity of capital unit price of labor unit price of capital 16 (Simplify your answers.)
The production function for a firm is p(x, y) = 68x"y"", where x and y are the number of units of labor and capital utilized. Suppose that labor costs $102 per unit and capital costs $544 per unit and that the firm decides to produce 8704 units of goods. Determine the following. (a) Determine the amounts of labor and capital that should be utilized in order to minimize the cost. That is, find the values of x and y that minimize 102x + 544y, subject to the constraint 8704 - 68x*y" = 0. x= 256, y= 16 (Type exact answers in simplified form.) (b) Find the value of at the optimal level of production. A= 4 (Type an exact answer in simplified form.) (c) Find the expressions for marginal productivity of labor and marginal productivity of capital. Then find the ratio of marginal productivity of labor to marginal productivity of capital at the optimum level of production and the ratio of unit price of labor to unit price of capital and determine the relationship between the ratios. 11 marginal productivity of labor = 51x y* 3 3 marginal productivity of capital = 17x*y Determine the marginal productivity labor and marginal productivity capital at the optimum level of production and then compare the ratios. marginal productivity of labor marginal productivity of capital unit price of labor unit price of capital 16 (Simplify your answers.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:The production function for a firm is p(x, y) = 68x
3/4 1/4
y
where x and y are the number of units of labor and capital utilized. Suppose that labor costs $102 per unit and capital costs $544 per unit and that the firm decides to produce 8704 units of goods. Determine the following.
(a) Determine the amounts of labor and capital that should be utilized in order to minimize the cost. That is, find the values of x and y that minimize 102x + 544y, subject to the constraint 8704
68x°7y'
3/4 1/4
= 0.
x= 256 , y= 16 (Type exact answers in simplified form.)
(b) Find the value of A at the optimal level of production.
1 = 4 (Type an exact answer in simplified form.)
(c) Find the expressions for marginal productivity of labor and marginal productivity of capital. Then find the ratio of marginal productivity of labor to marginal productivity of capital at the optimum level of production and the ratio of unit price of labor to unit price of capital and determine the
relationship between the ratios.
1
1
marginal productivity of labor = 51x
4.4
y
4
marginal productivity of capital = 17x* y
4
Determine the marginal productivity of labor and marginal productivity of capital at the optimum level of production and then compare the ratios.
marginal productivity of labor
unit price of labor
marginal productivity of capital
unit price of capital
16
(Simplify your answers.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
