You manage a firm. The output is produced according to the Cobb-Douglas production function q = f(K, L) = K¹/4 L¹/3 (1) a) At the use of inputs K=10,000 and L=8,000 calculate the rate at which output changes as each is adjusted, i.e. the marginal products of the two inputs MPK (K, L) and MPL(K, L). b) At the same use of inputs K=10,000 and L=8,000 calculate the rate at which the firm can trade between these inputs. This is the technical rate of substitution, TRSL,k(K, L). c) The firm can sell its output at the price p = $3,840. The firm pays w = $10 for each unit of labor and r = $60 for each unit of capital. Calculate the profit maximizing use of inputs and the profit maximizing output level. d) In class, we learned that a firm is producing a given level of output at the minimum cost if the rate at which the firm trades inputs, TRSL,K (K, L), is equal to the rate at which the market trades inputs, -. Show this holds at the profit maximizing output calculated in part (1c). e) A manager should be aware of all the different recipes that can produce a given level of output. For the production function given in equation (1), calculate and plot the isoquants for the output levels à = 200, ā= 400, and 600. For your graphs, please put labor on the horizontal axis and capital on the vertical axis. =
You manage a firm. The output is produced according to the Cobb-Douglas production function q = f(K, L) = K¹/4 L¹/3 (1) a) At the use of inputs K=10,000 and L=8,000 calculate the rate at which output changes as each is adjusted, i.e. the marginal products of the two inputs MPK (K, L) and MPL(K, L). b) At the same use of inputs K=10,000 and L=8,000 calculate the rate at which the firm can trade between these inputs. This is the technical rate of substitution, TRSL,k(K, L). c) The firm can sell its output at the price p = $3,840. The firm pays w = $10 for each unit of labor and r = $60 for each unit of capital. Calculate the profit maximizing use of inputs and the profit maximizing output level. d) In class, we learned that a firm is producing a given level of output at the minimum cost if the rate at which the firm trades inputs, TRSL,K (K, L), is equal to the rate at which the market trades inputs, -. Show this holds at the profit maximizing output calculated in part (1c). e) A manager should be aware of all the different recipes that can produce a given level of output. For the production function given in equation (1), calculate and plot the isoquants for the output levels à = 200, ā= 400, and 600. For your graphs, please put labor on the horizontal axis and capital on the vertical axis. =
Chapter10: Cost Functions
Section: Chapter Questions
Problem 10.4P
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Transcribed Image Text:Problem 1:
You manage a firm. The output is produced according to the Cobb-Douglas production
function
q = f(K, L) = K¹/4L¹/3
(1)
a) At the use of inputs K=10,000 and L=8,000 calculate the rate at which output changes
as each is adjusted, i.e. the marginal products of the two inputs MPK (K, L) and
MPL(K, L).
b) At the same use of inputs K=10,000 and L=8,000 calculate the rate at which the firm
can trade between these inputs. This is the technical rate of substitution, TRSL,K(K, L).
c) The firm can sell its output at the price p = $3,840. The firm pays w = = $10 for each
unit of labor and r = = $60 for each unit of capital. Calculate the profit maximizing use
of inputs and the profit maximizing output level.
d) In class, we learned that a firm is producing a given level of output at the minimum
cost if the rate at which the firm trades inputs, TRSL,K(K, L), is equal to the rate at
which the market trades inputs, Show this holds at the profit maximizing output
calculated in part (1c).
W
e) A manager should be aware of all the different recipes that can produce a given level
of output. For the production function given in equation (1), calculate and plot the
isoquants for the output levels ā = 200, ā = 400, and ā 600. For your graphs, please
put labor on the horizontal axis and capital on the vertical axis.
=
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