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Unformatted text preview: Econ 300  Solutions to Problem Set 1  Julien Bengui 10.1.8 First we write the firms profit function using the factor and output price information as ( K,L ) = 4 * 9 L 1 / 3 K 1 / 3 12 L 6 K = 36 L 1 / 3 K 1 / 3 12 L 6 K. To find the optimal levels for each input, we write the first order conditions: K = 12 L 1 / 3 K 2 / 3 6 = 0 , L = 12 L 2 / 3 K 1 / 3 12 = 0 . Solving the second equation for K yields K = L 2 . This can be substituted into the first equation: 2 L 1 / 3 ( L 2 ) 2 / 3 = 1 , which can be simplified to 2 L 1 = 1, or L = 2. Plugging this back into K = L 2 yields K = 4. Thus, the stationary point of the profit function is ( K,L ) = (4 , 2). To check that this stationary point is a maximum, we compute the second order derivatives of the profit function and evaluate them at the stationary point: KK = 8 L 1 / 3 K 5 / 3 = 8 * 2 1 / 3 4 5 / 3 = 8 * 2 1 / 3 2 10 / 3 = 8 * 2 9 / 3 = 8 * 2 3 = 1 < , LL = 8 L 5 / 3 K 1 / 3 = 8 * 2 5 / 3 4 1 / 3 = 8 * 2 5 / 3 2 2 / 3 = 8 * 2 3 / 3 = 8 * 2 1 = 4 < . We also calculate the cross partial derivative KL = 4 L 2 / 3 K 2 / 3 = 4 * 2 2 / 3 4 2 / 3 = 4 * 2 2 / 3 2 4 / 3 = 4 * 2 6 / 3 = 4 * 2 2 = 4 / 4 = 1 , from which we see that KK LL = 4 > 1 = ( KL ) 2 . We can thus conclude that the point ( K...
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This note was uploaded on 05/18/2008 for the course ECON 300 taught by Professor Cramton during the Spring '08 term at Maryland.
 Spring '08
 cramton

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