Consider a production function of three inputs, labor, capital, and materials, given by Q = LKM. The marginal products associated with this production function are as follows: MPL = KM, MPK = LM, and MPM = LK. Let w = 5, r = 1, and m = 2, where m is the price per unit of materials.a) Suppose that the firm is required to produce Q units of output. Show how the cost - minimizing quantity of labor depends on the quantity Q. Show how the cost- minimizing quantity of capital depends on the quantity Q. Show how the cost - minimizing quantity of materials depends on the quantity Q. b) Find the equation of the firms long-run total cost curve.c) Find the equation of the firms long-run average cost curve.d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a quantity of 50 units (ie., K 50). Show how the cost- minimizing quantity of labor depends on the quantity Q. Show how the cost- minimizing quantity of materials depends on the quantity Q. e) Find the equation of the short-run total cost curve when capital is fixed at a quantity of 50 units (i.e., K 50) and graph it along with the long-run total cost curve.f) Findtheequationoftheassociatedshort-runaverage cost curve.
Consider a production function of three inputs, labor, capital, and materials, given by Q = LKM. The marginal products associated with this production function are as follows: MPL = KM, MPK = LM, and MPM = LK. Let w = 5, r = 1, and m = 2, where m is the price per unit of materials.a) Suppose that the firm is required to produce Q units of output. Show how the cost - minimizing quantity of labor depends on the quantity Q. Show how the cost- minimizing quantity of capital depends on the quantity Q. Show how the cost - minimizing quantity of materials depends on the quantity Q. b) Find the equation of the firms long-run total cost curve.c) Find the equation of the firms long-run average cost curve.d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a quantity of 50 units (ie., K 50). Show how the cost- minimizing quantity of labor depends on the quantity Q. Show how the cost- minimizing quantity of materials depends on the quantity Q. e) Find the equation of the short-run total cost curve when capital is fixed at a quantity of 50 units (i.e., K 50) and graph it along with the long-run total cost curve.f) Findtheequationoftheassociatedshort-runaverage cost curve.
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.1P
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Transcribed Image Text:Consider a production function of three inputs, labor, capital, and materials, given by Q = LKM. The marginal products
associated with this production function are as follows: MPL = KM, MPK = LM, and MPM = LK. Let w = 5, r = 1,
and m = 2, where m is the price per unit of materials.a) Suppose that the firm is required to produce Q units of output.
Show how the cost - minimizing quantity of labor depends on the quantity Q. Show how the cost- minimizing quantity
of capital depends on the quantity Q. Show how the cost - minimizing quantity of materials depends on the quantity Q.
b) Find the equation of the firms long-run total cost curve.c) Find the equation of the firms long-run average cost
curve.d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a quantity of 50
units (ie., K 50). Show how the cost- minimizing quantity of labor depends on the quantity Q. Show how the cost-
minimizing quantity of materials depends on the quantity Q. e) Find the equation of the short-run total cost curve when
capital is fixed at a quantity of 50 units (i.e., K 50) and graph it along with the long-run total cost curve.f)
Findtheequationoftheassociatedshort-runaverage cost curve.
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