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Unformatted text preview: Ecn 100 - Intermediate Microeconomic Theory University of California - Davis November 16, 2010 John Parman Problem Set 5 - Solutions This problem set will be graded and is due by 5pm on Wednesday, December 1st in my mailbox in the economics department. You may turn in problem sets early by putting them in my mailbox in the economics department or by dropping them off in lecture. No late problem sets will be accepted. You are welcome to work in groups. If working in a group, everyone in the group must still submit an individual problem set. 1. Short Run vs. Long Run Costs A firm uses labor ( L ) and machines ( M ) to produce output and has the following production function: f ( L,M ) = 10 L 1 2 M 1 2 (1) The wage for a worker is $25 and the price of a machine is $100. In the short run, the firms workers have unbreakable contracts so the firm cannot hire or fire workers but it can change the number of machines it uses. In the long run, both the number of workers and the number of machines can be adjusted. Currently the firm is employing 100 workers. (a) Suppose that the firm wants to produce y units of output in the short run. Derive an expression for the number of machines the firm will need to use as a function of the desired level of output ( M ( y )). In the short run, L is fixed at 100. Plugging this into the production function gives us: f (100 ,M ) = 10 100 1 2 M 1 2 f (100 ,M ) = 100 M 1 2 For any particular level of output y , there will only be one level of M that, when plugged in to the above equation, produce that amount y : y = 100 M 1 2 Rearranging this equation will give us M as a function of y : y 2 = 100 2 M M ( y ) = y 2 10000 (b) Based on your answer to part (a), derive the costs for the firm as a function of output in the short run ( C SR ( y )). Graph this cost function on a graph with output on the horizontal axis and costs on the vertical axis. 2 Problem Set 5 - Solutions Costs in the short run are just the amount of money the firm has to spend on its fixed number of workers and the amount of money it has to spend on machines: C SR ( y ) = w L + p M M ( y ) C SR ( y ) = 25 100 + 100 y 2 10000 C SR ( y ) = 2500 + y 2 100 The graph is given show at the end of the problem. (c) Derive expressions for the number of workers hired by the firm ( L ( y )) and the number of machines used by the firm ( M ( y )) in the long run if the firm wants to produce y units of output and minimize costs. In the long run, the firm can vary both the number of workers and the number of machines. To find the cost-minimizing combination of workers and machines, the firm will set the ratio of the prices of the inputs equal to the technical rate of substitution (this is the same and finding the point where the isocost line is tangent to the isoquant):- p M w = TRS- p M w =- MP M MP L p M w = 5 L 1 2 M- 1 2 5 L- 1 2 M 1 2 p M w = L M L = p M w M This tells us the number of workers relative to the number of machines but doesnt yet give us workers as a function of output and machines as...
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