Problem Set 5

Problem Set 5
Deadline: January 7, 2015 in class
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1. A firm’s production function is F (L, K) = L 2 K 2 . The wage rate is w = 8 and the rent is r = 4 per
unit of capital. The firm is a price-taker.
(a) Suppose the price is p = 2, how much output should the firm produce? How many units of labor
and capital should it use?
(b) Derive the long-run supply function of the firm.
2. A firm’s cost function is T C(q) = 400 + 5q + q 2 . The firm is a price-taker.
(a) Derive the firm’s short run supply function (assuming fixed costs are sunk).
(b) Suppose that in the short run the firm has no sunk costs, the cost function is T C(q) = 400 + 5q + q 2
if q > 0 and T C(q) = 0 if q = 0. Derive the firm’s supply function.
3. A firm’s cost function is T C(q) = 200 + 50q − 10q 2 + q 3 . The firm is a price-taker.
(a) Derive the firm’s short run supply function.
(b) Derive the firm’s short run supply function assuming the fixed cost of 200 is not sunk.
4. A firm can use two factories to produce its output. Factory A operates with a cost function T Ca (qa ) =
200 + 8qa2 and factory B operates with a cost function T Cb (qb ) = 200 + 2qb2 . Fixed costs are non-sunk.
(a) Derive each factory’s AC(q), AV C(q) and M C(q) functions.
(b) Is one of the factories more eﬃcient than the other? If yes, which one and why?
(c) If the firm wants to produce q = 15, what is the best way of allocating production between factories?
(d) If the firm wants to produce q = 40, what is the best way of allocating production between factories?
5. Two firms constitute the entire industry. One has a long-run cost function T C1 (q1 ) = 625 + 0.25q12
and the other has a long-run cost function T C2 (q2 ) = 5000 + 0.5q22 . Fixed costs are non-sunk.
(a) Derive each firm’s AC(q), AV C(q) and M C(q) functions and plot all the curves in one diagram.
(b) Derive each firm’s supply function qi (p) and show the corresponding curves in the diagram in (a).
(c) Suppose no new firms can enter the industry. What is the industry’s supply function? Plot the
industry’s supply curve in the diagram in (a). Indicate price ranges at which no firms will operate,
one firm will operate, both firms will operate.
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