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Unformatted text preview: Econ 100A December 7, 2008 Problem Set 6 Solutions Michael Schihl 1 Problem 8.4 Suppose you are the manager of a watchmaking firm operating in a cometitive market. Your cost of production is C ( q ) = 200 + 2 q 2 , where q is the level of output and C is your total cost. (The marginal cost of production is d dq T ( q ) = 4 q ; the fixed cost of production is $200, the portion that does not vary with q ). (a) If the price of watches is $100, how many watches should you produce to maximize profit? Answer: 1.1 Method 1—Maximize Profits Directly Write the profit function as π ( q ) = pq- C ( q ) = 100 q- (200 + 2 q 2 ). As usual, take the first derivative and set it equal to zero. d dq π ( q ) = 0 (1) d dq 100 q- (200 + 2 q 2 ) = 0 (2) 100- 4 q = 0 (3) 4 q = 100 (4) q = 25 (5) Thus, the profit maximizing level of output is 25. We should, technically, verify that the second derivative is negative, to ensure that we are finding maximum, rather than a minimum. d 2 dq 2 π ( q ) =- 4, so since the second-order condition is satisfied, we can be confident that our solution using the first-order condition is a maximum. 1.2 Method 2—Set MR Equal to MC Before doing this, note that π- TR- TC , so when you set the first derivative of profit equal to zero, you get MR- MC = 0, which is the same as MR = MC . Thus, this method is starting from Equation (3), noting that, for a firm in a perfectly competitive market, marginal revenue is equal to price ( MR ( q ) = d dq TR ( q ) = d dq pq = p ), since the firm is a price taker. Write the profit function as π ( q ) = pq- C ( q ) = 100 q- (200 + 2 q 2 ). As usual, take the first derivative and set it equal to zero. MR = MC (6) 100 = 4 q (7) q = 25 (8) (b) What will the profit level be? Answer: See Figure 2. Simply plug in 25 into the profit function. π = pq- TC ( q ) = 100 × 25- ( 200 + 2 × 25 2 ) = $1050. (c) At what minimum price will the firm produce a positive output? Answer: The firm’s short-run supply curve is its marginal cost curve above average variable cost. We know that the marginal cost curve intersects average variable cost at average variable cost’s minimum, so this question is essentially asking for the level of average variable cost at average variable cost’s minimum. We can find this minimum in at least two ways. 1 Figure 1: Problem 8.4a (a) TR ( q ) ,TC ( q ) , and π ( q ) (b) AV C ( q ) ,ATC ( q ) ,MC ( q ) and π ( q ) Figure 2: Problem 8.4(b) and 8.4(c) 1.3 Method 1: Minimum of AV C —Directly The direct method is to set the first derivative of AV C to zero. AV C ( q ) = V C ( q ) q = 2 q 2 q = 2 q . d dq AV C ( q ) = 0 (9) d dq 2 q = 0 (10) q = 0 (11) Now plug this q into AV C ( q ) to find the minimum of AV C ; AV C (0) = 2 × 0 = 0. Thus, the firm will produce positive output as long as the price is positive. We typically assume that the firm can produce fractional units....
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- Spring '07