f981a-key - 1 Selected answers to practice final 1 Spring...

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1 Selected answers to practice f nal 1 Spring 1998 (A) Question 1 . The pro f t function: π = pq wL rK = p (60 K +34 L 4 KL 6 K 2 3 L 2 ) wL rK The FOC’s give: 34 4 K 6 L w p =0 60 4 L 12 K r p which can be solved for K and L. i.e. by Cramer’s rule · 64 41 2 ¸· L K ¸ = · 34 w/p 60 r/p ¸ where | A | =60 L = ¯ ¯ ¯ ¯ 34 w/p 4 60 r/p 12 ¯ ¯ ¯ ¯ 60 K = ¯ ¯ ¯ ¯ 63 4 w/p 46 0 r/p ¯ ¯ ¯ ¯ 60 The Hessian is | H | = ¯ ¯ ¯ ¯ 6 4 4 12 ¯ ¯ ¯ ¯ | H 1 | = 6 < 0 | H | =6 0 > 0 Question Two: The Lagrangian is Z = xy 2 + λ 1 (100 x y )+ λ 2 (120 2 x y ) and the KT conditions: Z x = y 2 λ 1 2 λ 2 0 Z y =2 xy λ 1 λ 2 0 Z 1 = 100 x y 0 Z 2 = 120 2 x y 0 Solution: x =20 and y =80 . The second constraint is Binding. 1
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Question Three The lagrange equation is Z = 200 Q 1 0 . 5 Q 2 1 +190 Q 2 0 . 5 Q 2 2 10( Q 1 + Q 2 ) 20 K + λ 1 ( K Q 1 )+ λ 2 ( K Q 2 ) The Kuhn-Tucker conditions are Z 1 = 200 Q 1 10 λ 1 0 Q 1
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f981a-key - 1 Selected answers to practice final 1 Spring...

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