Be sure to show all of your work and to identify your final answers clearly. There are 5 questions.
1. Suppose a firm has the following short run cost function:
????(????) = 0.5????2 + 5???? + 100
a. Assume the firm is a price taker in the output market, and that the price is $25. Derive the firm’s profit maximizing level of output. How much profit does the firm make at this price and quantity? (5 points)
b. Suppose the firm has to pay a lump sum tax of $50, regardless of how many units it sells or how much profit it makes. Rewrite the cost function to include this lump sum tax. What is the firm’s profit-maximizing level of q under this new cost structure (for p=$25)? (5 points)
c. Suppose that instead of a lump sum tax the firm must pay a tax on each unit of output sold, equal to $5. Rewrite the cost function to include this tax. What the profit-maximizing level of q now (for p=25)? (5 points)
d. Now suppose that, instead of either a lump sum tax or a per unit tax, the firm pays a proportional tax of 10% on all profits. Does this tax affect the profit maximizing level of q (for p=25)? (5 points)
2. Consider this production function:
q=5L + ln(K)
a) Derive the rate of technical substitution. (5 points)
b) Assume some w and v and assume the firm minimizes the cost of producing any level of output. Derive expressions for L and K in terms of w, v and q. (10 points)
c) Based on your answers to (a) and (b), describe the expansion path for this production function: Holding w and v constant, how does the ratio of K to L change as we increase quantity produced? (5 points)
3. Consider the following short-run profit function for a firm with a fixed amount of capital K1.
???? = ????????1.5????.5 − ????????1 − ????????
a) Derive the profit-maximizing value of L as a function of P, K1, v, and w. (5 points)
b) Use you answer to (a) to re-write the profit function π as a function of P, K1, v, and w. (10 points)
c) Use your answer to part (b) to derive a supply function for q as a function of P, w, v,
and K1 by noting that
= ????. (5 points)
4. Consider the short-run production function q= ????1.5????.5, with capital fixed at K1. This is the supply function that is embedded in the profit function above.
a) Use this production function to derive the amount of labor needed to produce a given amount of q. (5 points)
b) Use your answer to (a), and assume some wage w and some cost of capital v, to derive a short run cost function C(q), where cost is a function of q (as well as w, γ, and the fixed amount of capital K1). (5 points)
c) Use this cost function to derive a short run marginal cost curve. Assuming that this firm is a price taker, use this result to derive a supply curve for q as a function of P (and w, v and K1). (10 points)
5. Consider this long-run cost curve C(q) = q2-10q +100. Assume this industry is perfectly competitive and that firms operate at their minimum average cost in the long run.
a) Use the cost curve to derive the firm level of q and the price P in the long-run equilibrium. (5 points)
b) Suppose industry demand is D(P)=2000-50P. Derive the industry level of output Q and the number of firms n. (5 points)
c) Suppose the cost of renting production space, essentially a fixed cost of production, rises, so that the new long-run cost function is C(q) = q2-10q +144. Derive the new long- run equilibrium values of q, P, Q, and n resulting from this increase in fixed costs. (5 points)
d) Suppose instead that wages rise, so that variable costs of production go up. The new long-run cost function is C(q) = 4q2-10q +100. Derive the new long-run equilibrium values of q, P, Q, and n resulting from this increase in variable costs. (5 points)