Consider a firm that produces according to the production function:
Q = √KL
where Q is the firm's output and K and L are the quantities of capital and labor that it employs. With this production function, the slope of an isoquant at the point (L, K) is given by −K/L.
a. Suppose that the going wage rate of labor is W and the going rental rate on capital is R. What is the slope of an isocost? If the firm uses K units of capital and L of labor in long-run equilibrium, derive a formula for K in terms of L, W, and R. Derive a formula for L in terms of K, W, and R.
b. Using the production function and the result of part (a), write a formula for L in terms of Q, W, and R, and a formula for K in terms of Q, W, and R.
c. Write a formula for the total cost of producing Q units of output.
d. Describe the firm's long-run marginal cost curve.
e. In long-run equilibrium, what must the price of output be? Would you have had enough information to answer this question if your answer to part (d) had been different than it was?
f. In terms of Q, how much does the firm pay out to labor and to capital? What is its total revenue? What is its profit?