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Unformatted text preview: Ecn 100 - Intermediate Microeconomic Theory University of California - Davis November 27, 2010 John Parman Problem Set 6 - Solutions This problem set will not be graded and does not need to be turned in. It is intended to help you review the material from the last two weeks of lectures. Solutions to the problem set are available on Smartsite. 1. Industry Supply Suppose that there are two types of firms in a perfectly competitive market for widgets ( w ). Firms of type A have costs given by C A ( w ) = 5 w 2 + 2 w + 10. Firms of type B have costs given by C B ( w ) = 3 w 2 + 5. There are 100 firms of type A and 180 firms of type B . (a) What are the individual firm supply functions for each type type of firm ( S A ( p ) and S B ( p )? Are there any prices at which no firms produce? Are there any prices at which some firms produce but others do not? We know that a firms supply curve will be the same as the portion of the marginal cost curve that lies above the average variable cost curve. So to get the supply function we need to find the marginal cost curve and also the price at which the marginal cost curve crosses the average variable cost curve. For firm type A : MC A ( w ) = 10 w + 2 AV C A ( w ) = 5 w + 2 Now we set MC A equal to AV C A to find the shutdown point: MC A ( w ) = AV C A ( w ) 10 w + 2 = 5 w + 2 10 w = 5 w The only value of w that will solve this equation is zero. So the shutdown quantity is zero. To get the shutdown price, we just plug this quantity back into the marginal cost function: MC A (0) = 10 · 0 + 2 = 2. So the shutdown price for firm type A is $2. Now for firm type B : MC B ( w ) = AV C B ( w ) 6 w = 3 w w = 0 2 Problem Set 6 - Solutions MC B (0) = 6 · 0 = 0 So firm type B will shut down only when price hits zero. At prices in between $0 and $2, firms of type B produce but firms of type A do not. Given everything we’ve calculated we can now write out the supply functions: p = 10 S A ( p ) + 2 if p ≥ 2, S A ( p ) = 0 otherwise S A ( p ) = 1 10 p- 1 5 if p ≥ 2, S A ( p ) = 0 otherwise p = 6 S B ( p ) for all p > S B ( p ) = 1 6 p for all p > (b) What is the industry supply function? Graph the industry supply function and be certain to label any kinks and all relevant slopes. For prices less than $2, industry supply comes entirely from the 180 firms of type B . For prices greater than $2, both firm types supply widgets. So our supply function has two parts: S ( p ) = 180 S B ( p ) = 30 p if p < 2 S ( p ) = 100 S A ( p ) + 180 S B ( p ) = 40 p- 20 if p ≥ 2 (c) Suppose that in the long run, when firms can adjust all inputs, all firms have the following cost function: C ( w ) = w 3- 20 w 2 + 110 w . The market demand for widgets is still given by D ( p ) = 1000- p . What must the price of widgets be in the long run equilibrium? How many firms will there be producing widgets?...
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This note was uploaded on 09/11/2011 for the course ECON 100 taught by Professor Parman during the Winter '08 term at UC Davis.
- Winter '08