roduction function for a firm is p(x, y) = 96xy", where x and y are the number of units of labor and capital utilized. Suppose that labor costs $144 per unit and capital costs $1875 per unit and that the firm decides to produce 24,000 units of goods. Determine the following. termine 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 144x + 1875y, subject to the constraint 24,000-96x"y" = 0. . y=Type exact answers simplified form.)
roduction function for a firm is p(x, y) = 96xy", where x and y are the number of units of labor and capital utilized. Suppose that labor costs $144 per unit and capital costs $1875 per unit and that the firm decides to produce 24,000 units of goods. Determine the following. termine 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 144x + 1875y, subject to the constraint 24,000-96x"y" = 0. . y=Type exact answers simplified form.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The production function for a firm is p(x, y) = 96x
3/4 1/4
y
where x and y are the number of units of labor and capital utilized. Suppose that labor costs $144 per unit and capital costs $1875 per unit and that the firm decides to produce 24,000 units of goods. Determine the following.
%3D
...
3/4 1/4
(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 144x + 1875y, subject to the constraint 24,000 – 96x
y
= 0.
X =
, y=
(Type exact answers in simplified form.)
Transcribed Image Text:The production function for a firm is p(x, y) = 72x
3/4 1/4
y
where x and y are the number of units of labor and capital utilized. Suppose that labor costs $108 per unit and capital costs $576 per unit and that the firm decides to produce 9216 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 108x + 576y, subject to the constraint 9216 – 72x
3/4 1/4
= 0.
x = 256 , y= 16 (Type exact answers in simplified form.)
(b) Find the value of 1 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.
marginal productivity of labor = 54x
– 1/4 1/4
y
marginal productivity of capital = 18x
3/4 -3/4
y
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
marginal productivity of capital
unit price of labor
unit price of capital
3
3
16
16
(Simplify your answers.)
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