Macroeconomics Exam Review 235

Macroeconomics Exam Review 235 - c 2001 Michael Carter All...

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5.24 The Lagrangean for this problem is ? ( x )= ² 2 1 + ² 2 2 + ² 2 3 ± (2 ² 1 3 ² 2 +5 ² 3 19) A necessary condition for x to solve the problem is that the Lagrangean be stationary at x ,thatis ³ ? 1 ? =2 ² 1 2 ± =0 ³ ? 2 ? ² 2 +3 ± ³ ? 3 ? ² 3 5 ± which implies ² 1 = ±² 2 = 3 2 2 = 5 2 ± (5.89) It is also necessary that the solution satisfy the constraint, that is 2 ² 1 3 ² 2 ² 3 =19 Substituting (5.89) into the constraint we get 2 ± + 9 2 ± + 25 2 ± ± which implies ± = 1. Substituting in (5.89), the solution is x =(1 , 3 2 , 5 2 ). Since the constraint is affine and the objective ( ´ ) is concave, stationarity of the Lagrangean is also sufficient for global optimum (Corollary 5.2.4). 5.25 The Lagrangean is ? ( ² 1 2 ² ± 1 ² 1 ± 2 ± ( µ 1 ² 1 + µ 2 ² 2 ) The Lagrangean is stationary where ³ ? 1 ? = ·² ± 1 1 ² 1 ± 2
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