The market for sweet potatoes consists of 1,000 identical firms. Each firm has a short-run total cost curve of STC = 100 + 100 q + 100q2, and a short-run marginal cost curve of SMC=100+200q where q is output. Suppose that sunk costs are 75 and nonsunk costs are 25. What is the equation of an individual firm's short-run supply curve?

Question

Quizplus · Accepted Answer

To find the firm's short-run supply curve, we need to compare the firm's short-run marginal cost (SMC) with the market price (P). If P is greater than or equal to the lowest point on the firm's average variable cost (AVC) curve, then the firm will produce where SMC = P. Otherwise, the firm will shut down and produce nothing.

First, let's find the lowest point on the firm's AVC curve:
AVC = STC / q = (100+100q+100q^2) / q = 100/q + 100 + 100q  
AVC reaches its minimum value at q = 1, which is equal to 300.

Therefore, the firm will produce where SMC = P as long as P is greater than or equal to 200.

When P is greater than or equal to 200:
SMC = 100 + 200q = P
q = (P-100)/200

Thus, the firm's short-run supply curve is:
q = (P-100)/200 for P >= 200, and q = 0 otherwise.

Therefore, the correct answer is D. To derive an individual firm's short-run supply curve, we need to compare its marginal cost (MC) curve with its average variable cost (AVC) curve. In this case, the AVC is given by AVC = STC / q = 100/q + 100 + 100q. To minimize AVC, we take its derivative with respect to q and set it equal to zero: dAVC/dq = -100/q2 + 100 = 0, which implies q = 1. We plug q = 1 back into AVC to get a minimum average variable cost of 300. Since AVC is increasing as q increases beyond 1, we know that the shutdown point is at a price lower than 300, specifically at P = 75, where AVC = SMC + sunk cost = 75 + 100 + 75 = 250. Therefore, the firm's short-run supply curve consists of three segments: (1) zero output as long as P < 75, (2) output q = (P - 250) / 200 as long as 75 ≤ P < 450, and (3) maximum output q = 1 as long as P ≥ 450. Solving for q in the second segment yields q = P / 200 - 1.25, which simplifies to q = P / 200 - .5 because the nonsunk costs of 25 are irrelevant for determining the firm's supply decisions. Therefore, the correct answer is D.

The Market for Sweet Potatoes Consists of 1,000 Identical Firms q=P/200−.5q = P / 200 - .5q=P/200−.5

The Market for Sweet Potatoes Consists of 1,000 Identical Firms $q = P / 200 - .5$