7.29. Suppose that the firm uses three inputs to produce its output: capital K, labor L, and materials M. The firm's production function is given by Q=K³L³M³Q=K²L³M³. For this production function, the marginal products of capital, labor, and materials are MPK = K-L³M³MPK = }K¹L³M³, MPL = K³L³M³MP₂-K³L-M³, and MPM = K L M-MPM-KLM-I. The prices of capital, labor, and materials are r=1, w=1, and m = 1, respectively. a) What is the solution to the firm's long-run cost-minimization problem given that the firm wants to produce Q units of output? b) What is the solution to the firm's short-run cost-minimization problem when the firm wants to produce Q units of output and capital is fixed at KK? c) When Q=4, the long-run cost-minimizing quantity of capital is 4. If capital i fixed at K = 4R-4 in the short run, show that the short-run and long-run cost-minimizing quantities of labor and materials are the same.

7.29. Suppose that the firm uses three inputs to produce its output: capital K, labor L, and materials M. The firm's production function is given by Q=K³L³M³Q=K²L³M³. For this production function, the marginal products of capital, labor, and materials are MPK = K-L³M³MPK = }K¹L³M³, MPL = K³L³M³MP₂-K³L-M³, and MPM = K L M-MPM-KLM-I. The prices of capital, labor, and materials are r=1, w=1, and m = 1, respectively. a) What is the solution to the firm's long-run cost-minimization problem given that the firm wants to produce Q units of output? b) What is the solution to the firm's short-run cost-minimization problem when the firm wants to produce Q units of output and capital is fixed at KK? c) When Q=4, the long-run cost-minimizing quantity of capital is 4. If capital i fixed at K = 4R-4 in the short run, show that the short-run and long-run cost-minimizing quantities of labor and materials are the same.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Concept explainers

Isocost and Isoprofit Curve Analysis

An isocost curve in the production theory that is a crucial part of economics depicts different combinations of factor inputs at a given market cost. In other words, this shows all the combinations of production factors that cost the same for the firm to employ. Usually, this can be depicted as a downward sloping straight line when the firms act as the price-takers in the factor markets. The slope of this mainly depicts the relative prices of different production factors.

Question

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7.29. Suppose that the firm uses three inputs to produce its output: capital K, labor L, and materials M. The firm's production function is given by Q = KL3 M³Q = K*L*M*. For this production function, the marginal products of capital, labor,
and materials are MPK = KL$M$ MPg = }K¯³L>M*, MPL = KLM3 MP, = }K}L¬}M³, and MPM = KL$M¯MPM = }K*L}M¬}. The prices of capital, labor, and materials are r = 1, w = 1, and m = 1, respectively.
a) What is the solution to the firm's long-run cost-minimization problem given that the firm wants to produce Q units of output?
b) What is the solution to the firm's short-run cost-minimization problem when the firm wants to produce Q units of output and capital is fixed at KR?
c) When Q = 4, the long-run cost-minimizing quantity of capital is 4. If capital is fixed at K = 4K = 4 in the short run,
* MPx = }K÷L}M³, MPL = KLM MP, = †K}L=M$, and MPM = KLM MPM = }K}L}M¯i. The prices of capital, labor, and materials are r=
= 1, w= 1, and m =
%3D
show that the short-run and long-run cost-minimizing quantities of labor and materials are the same.

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