Suppose a firm has the following production function: Y = A[ln K + ln L]. A is a variable representing the overall productivity of capital and labor. A profit maximizing firm will choose labor and capital so that their marginal products are equal to their marginal costs. Assume that the marginal cost of labor is the wage rate, w, and the marginal cost of capital is the real interest rate, r. a. Find the profit maximizing level of K and L for the firm. b. Does the firm use more or less capital and labor when productivity rises? (Show this mathematically and explain). c. Compute the second derivative with respect to K d. Compute the second derivative with respect to L e. If you were to graph the production function with K on the horizontal axis (and holding L fixed), would the graph be concave or convex? f. What is the partial cross derivative of this production function. Interpret the economic meaning of the result.
Suppose a firm has the following production function: Y = A[ln K + ln L]. A is a variable representing the overall productivity of capital and labor. A profit maximizing firm will choose labor and capital so that their marginal products are equal to their marginal costs. Assume that the marginal cost of labor is the wage rate, w, and the marginal cost of capital is the real interest rate, r. a. Find the profit maximizing level of K and L for the firm. b. Does the firm use more or less capital and labor when productivity rises? (Show this mathematically and explain). c. Compute the second derivative with respect to K d. Compute the second derivative with respect to L e. If you were to graph the production function with K on the horizontal axis (and holding L fixed), would the graph be concave or convex? f. What is the partial cross derivative of this production function. Interpret the economic meaning of the result.
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 7E
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Suppose a firm has the following production function: Y = A[ln K + ln L]. A is a variable representing the overall productivity of capital and labor. A profit maximizing firm will choose labor and capital so that their marginal products are equal to their marginal costs. Assume that the marginal cost of labor is the wage rate, w, and the marginal cost of capital is the real interest rate, r.
a. Find the profit maximizing level of K and L for the firm.
b. Does the firm use more or less capital and labor when productivity rises? (Show this mathematically and explain).
c. Compute the second derivative with respect to K
d. Compute the second derivative with respect to L
e. If you were to graph the production function with K on the horizontal axis (and holding L fixed), would the graph be concave or convex?
f. What is the partial cross derivative of this production function. Interpret the economic meaning of the result.
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