In: Economics
Consider a firm which produces according to the following production function by using labor and capital: f(l,k) = K1/2 L1/2
(a) Solve the cost minimization problem of this firm for the given wage rate, w and the rental rate of capital,v. Derive the long-run total cost function of the firm.
(b) Derive the short-run total cost, the short-run average cost, the short- run marginal cost function of the firm under the assumption that capital is the fixed input.
(c) What amount of capital minimizes the short-run total cost?
(d) Is there any relation between the short-run total cost function and the long-run total cost function at the capital level that you find in part (c).
(e) Suppose the wage rate of labor is 2 $, the rental rate of capital is 2 $ and fixed capital input, k ̄, is 2 units. What amount of output minimizes short-run average cost? What is the minimum possible short-run average cost?
(f) What is the short-run marginal cost at the quantity level that you find in part (e)? How is it related to minimum possible short-run average cost?
(g) Find short-run firm supply as a function of input prices, w and v, and output price, p.
(h) Solve the profit maximization of the firm for a given price p, and derive the supply function. (i) Derive the profit function of the firm. (j) Decide whether the production function exhibits constant, increasing or decreasing returns to scale.