Question

Consider a firm which produces according to the following production function by using labor and capital: f(l,k) = K^1/2 L^1/2

(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.

Answer 1