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400ps3-11-soln

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

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ECON 400 Hartman Winter 2011 Solutions for Problem Set III 1. Since ( ) ( ) a b a b a b tK tL t K L , the production function is homogeneous of degree a b . 2. 2 2 2 / ( 1) 0 a b Q K a a K L and 2 2 2 / ( 1) 0 a b Q L b b K L 3. The first order necessary conditions are / 0 p Q K r and / 0 p Q L w . The second order necessary conditions are 2 2 0 Q p K , 2 2 0 Q p L , and 2 2 2 2 2 2 0 Q Q Q p p p K L L K   . Since, 2 1 1 / a b Q K L abK L , the last condition is 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 [ ( 1) ][ ( 1) ] [ ] [ ( 1) ( 1) ] [ (1 )] a b a b a b a b a b Q Q Q p p p pa a K L pb b K L pabK L K L L K a a b b a b p K L ab a b p K L   and therefore 2 2 2 2 2 2 0 Q Q Q p p p K L L K   if 1 a b . This second order necessary condition is NOT satisfied if 1 a b . 4. The second order sufficient conditions are 2 2 0 Q p K , 2 2 0 Q p L , and 2 2 2 2 2 2 0 Q Q Q p p p K L L K   . The results from 3 and 4 above imply that these conditions are satisfied if 1 a b . 5. The first order conditions imply that 1 a b paK L r and 1 a b 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.

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

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