Suppose in the long-rung the production function of a competitive firm is Q=f(L,K)= L2/³K¹/4, where L is the amount of labor and K is the amount of capital. The cost per unit of labor is w and the cost of capital is r, which is the interest rate. The price per unit of output is p.
Suppose in the long-rung the production function of a competitive firm is Q=f(L,K)= L2/³K¹/4, where L is the amount of labor and K is the amount of capital. The cost per unit of labor is w and the cost of capital is r, which is the interest rate. The price per unit of output is p.
Chapter10: Cost Functions
Section: Chapter Questions
Problem 10.4P
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Please see the attached photo.
Only solve part d, e, f and g
Transcribed Image Text:Suppose in the long-rung the production function of a competitive firm is Q=f(L,K)= L2/³K¹/4, where L
is the amount of labor and K is the amount of capital. The cost per unit of labor is w and the cost of
capital is r, which is the interest rate. The price per unit of output is p.
a. Write down the profit as a function of K and L.
b. Find the marginal products of K and L: MPK and MPL
C. Now multiply MPK and MPL by the output price, p, to get the marginal revenues of
capital and labor, respectively.
d.
e.
The optimal levels of inputs satisfy the condition: marginal revenue of each input equals
its marginal cost. The marginal cost of capital is r, and the marginal cost of labor is w.
Solve the resulting system of two equations for the profit-maximizing levels of inputs.
What is the profit maximizing level of output?
Does this production function exhibit increasing, decreasing or constant returns to
scale?
f. If instead of above production function, we have f(L,K)= L2/³K²/3 for a competitive firm,
what will be the profit-maximizing level of the output? What are the returns to scale of
this new production function?
g. Finally, solve the profit-maximization problem for a competitive firm if the production
function is exhibiting constant returns to scale, and has the form f(L,K)= L2/3K¹/3. How
many solutions does this problem have? What is the maximum profit that the
competitive firm can earn?
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