400ps3-11-soln - ECON 400 Winter 2011 Solutions for Problem...

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E C O N 4 0 0 H a r t m a n Winter 2011 Solutions for Problem Set III 1. Since ( ) ( ) ab a b a b tK tL t K L , the production function is homogeneous of degree . 2. 22 2 /( 1 ) 0 QK a a KL   and 2 1 ) 0 QL b b KL   3. The first order necessary conditions are /0 pQ K r  and / 0 pQ L w   . The second order necessary conditions are 2 2 0 Q p K , 2 2 0 Q p L , and 2 2 0 QQ Q pp p KL L K     . Since, 21 1 / Q K L abK L  , the last condition is 2 2 1 1 2 2 2 222 [( 1 ) ] [(1 ) ] [ ] )( 1 ) ] [ ( 1 ) ] a b Q p p p pa a K L pb b K L pabK L L K aa bb ab pK L a b pK L  and therefore 2 2 0 Q p L K  if 1 . This second order necessary condition is NOT satisfied if 1  . 4. The second order sufficient conditions are 2 2 0 Q p K , 2 2 0 Q p L , and 2 2 0 Q p L K  . The results from 3 and 4 above imply that these conditions are satisfied if 1  . 5. The first order conditions imply that 1 paK L r and 1 pbK L w . There are several ways to solve the first order conditions to get the optimal K and L in terms of the output and input prices. One possibility is to divide the first of the first order conditions by the second to get 1 1 paK L r pbK L w or aL r bK w after simplifying. This implies that brK L aw .
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This note was uploaded on 04/04/2011 for the course ECON 400 taught by Professor Ellis,g during the Spring '08 term at University of Washington.

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400ps3-11-soln - ECON 400 Winter 2011 Solutions for Problem...

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