1. A perfectly competitive firm produces output q with capital K and labor L according to the production function:

q = f(K,L) = 4K^1/4 L^ 1/4

The price of labor is w = 4, and the price of capital is r = 4. The firm has fixed costs equal to 8. The current market price is 8.

(a) Prove that this production function has decreasing returns to scale.

(b) Find the optimal (cost-minimizing) ratio of capital to labor inputs K/L for any level of output (use scale expansion path).

(c) For any q, and the cost-minimizing inputs as functions of output: i.e., derive functions L(q) and K(q).

(d) Find cost functions TC(q), ATC(q), and MC(q), i.e. as functions of output. At what value of q does the minimum of ATC occur?

(e) Find total and marginal revenue as functions of output, i.e. TR(q) and MR(q). (f) Find the pro t maximizing output q∗ and use that too and TR(q*), TC(q*), π(q*), L(q*), and K(q*).

(g) Will other rms have an incentive to enter or exit this market? What will be the new market price that results from the entry/exit of firms? At the new price, what is the rm's new output? What is the rm's profit?

2. Consider a competitive rm with total costs given by TC(q) = 100 + 10q + q^2 The firm faces a market price p = 50.

(a) Write expressions for total revenue TR and marginal revenue MR as functions of output q.

(b) Write expressions for average total cost ATC, average variable cost AVC, and marginal cost MC as functions of output q.

(c) For what value of output is AT C minimized?

(d) Find the profit-maximizing level of output q∗. At this level of output, what are TR, TC, ATC, and π?

(e) Graph the ATC, AV C, MC, and MR curves in a single graph, and indicate the profit-maximizing level of output. If there are profits, shade the region corresponding to profit and label it. (f) If fixed costs increase from 100 to 500, what happens to the pro t maximizing level of output, TR, TC, and π?

(g) If fixed costs increase from 100 to 500, should the rm continue to operate in the short run? What about the long run?