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Unformatted text preview: Problem Set 2 Solutions: The Partial Equilibrium Competitive Model in the Long Run Econ 100B James Rauch Problem 1 Suppose there are 1,000 identical firms producing diamonds and the total cost func- tion for each firm is given by C ( q ) = q 2 + wq , where q is the firms output level and w is the wage rate of diamond cutters. (a) If w = 10, what will be the firms short-run supply curve? What is the indus- trys short-run supply curve? How many diamonds will be produced at a price of 20 each? How many more diamonds would be produced at a price of 21? (b) Suppose the wages of diamond cutters depend on the total quantity of dia- monds produced and the form of this relationship is given by w = 0 . 002 Q , where Q represents total industry output. In this situation, show that the firms short-run supply curve depends on Q . What is the industry short-run supply curve? How much will be produced at a price of 20? How much more will be produced at a price of 21? How did the shape of the short-run supply curve change compared to part a? (c) Does a long-run competitive equilibrium exist for this industry? Why or why not? Solution (a) Differentiate the cost function with respect to q to find that find that MC ( q ) = 2 q + w . When the wage is $10 this becomes MC ( q ) = 2 q + 10. To maximize profits, set price equal to marginal cost, yielding: p = 2 q + 10. Hence the individual firms short run supply curve is q = 1 2 p- 5. Since there are 1000 identical firms, market supply is Q = 1000( 1 2 p- 5) = 500 p- 5000. At a price of $20, Q = 500(20)- 5000 = 5000. At a price of $21, Q = 500(21)- 500 = 5500. (b) Now we have w = 0 . 002 Q so that C = q 2 + (0 . 002 Q ) q . Under perfect compe- tition, each firm makes up a negligible portion of market supply, so it treats Q as a constant. Thus, MC = 2 q +0 . 002 Q . Setting marginal cost equal to price, 1 we have q = 0 . 5 p- . 001 Q as the firms short run supply curve. Aggregating over the 1000 identical producers, market supply is...
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- Fall '07