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Mathematics for Economists

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ANSWERS PAMPHLET 101 and ] ] a j f ( x p ( b , a )) 5 ] L ] a j ( x p , b p , a p , l p , m p ) . 19.10 Form L ( x , a, m ) 5 f ( x ) 2 m j h j ( x , a ). Let ( x p ( a p ) , m p ( a p ) ) denote the maximizer and its multiplier when the parameter a 5 a p . a 7 → h j ( x p ( a ) , a ) is the zero function. f ( x p ( a )) 5 f ( x p ( a )) 2 X m p j ( a ) h j ( x p ( a ); a ) 5 L ( x p ( a ) , m p ( a ); a ) for all a. d da f ( x p ( a )) 5 d da L ( x p , m p ( a ); a ) 5 X i ] L ] x i ( x p ( a ) , m p ( a ) , a ) dx p i da ( a ) 1 X j ] L ]m j ( x p ( a ) , m p ( a ) , a ) d m p j da ( a ) 1 ] L ] a ( x p ( a ) , m p ( a ) , a ) 5 0 1 X j ( 2 h j ( x p ( a ) , a )) d m p j da ( a ) 1 ] L ] a ( x p ( a ) , m p ( a ) , a ) 5 ] L ] a ( x p , m p ( a ) , a ) . 19.11 max f ( x ) subject to h 1 ( x ) 5 a 1 , . . . , h k ( x ) 5 a k . Suppose the constraint qualification holds and that the solution x p depends on a 5 ( a 1 , . . . , a k ). The Lagrangian is L ( x , l ; a ) 5 f ( x ) 2 l i ( h i ( x ) 2 a i ). By Theorem 19 . 5, ] ] a j f ( x p ( a )) 5 ] L ] a j ( x p ( a ) , l p ( a ); a ) . It follows from the above formula for L that ] L ] a j 5 l j . So, ] ] a j f ( x p ( a )) 5 l j ( a ). But this is the conclusion of Theorem 19 . 1 . 19.12 max x 2 1 y 2 subject to x 2 1 xy 1 . 9 y 2 5 3 .

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102 MATHEMATICS FOR ECONOMISTS Write the constraint as x 2 1 xy 1 by 2 5 3. L 5 x 2 1 y 2 2 l ( x 2 1 xy 1 by 2 2 3) . ] L ] b 5 2 l y 2 . For b 5 1, the max is at ( x, y, l ) 5 ( 6 p 3 , 7 p 3 , 2) with f p 5 6; and the min is at ( x, y, l ) 5 ( 6 1 , 6 1 , 2 6 3) with f p 5 2. For b 5 0 . 9, f p < 6 1 ( 2 2 ? 3) ? ( 2 . 1) 5 6 . 6 at the max and f p < 2 1 ( 2 (2 6 3) ? 1) ? ( 2 . 1) 5 2 1 (2 6 30) 5 2 1 15 at the min. If we want actual distance and not distance squared, we would take the square roots of these numbers and obtain 2.569 and 1.438, respectively. 19.13 max x 2 1 x 1 ay 2 subject to 2 x 1 2 y # 1 , x \$ 0 , y \$ 0 . The Lagrangian is L 5 x 2 1 x 1 ay 2 2 l 1 (2 x 1 2 y 2 1) 1 l 2 x 1 l 3 y. For a 5 4, x p 5 0, y p 5 0 . 5, l p 1 5 2, l p 2 5 3, l p 3 5 0, and f p 5 1. At these values, ] L ] a 5 y 2 5 0 . 5 2 5 0 . 25 so f p (4 . 1) < f p (4) 1 ] L ] a ? D a 5 1 1 0 . 25(0 . 1) 5 1 . 025 . 19.14 18.2): H 5 0 2 x 1 y x 1 2 y 2 x 1 y 2 2 2 l 2 l x 1 2 y 2 l 2 2 2 l 5 0 3 3 3 2 6 3 2 2 6 3 3 2 2 6 3 2 6 3 at the minimizer (1 , 1 , 2 6 3). det H 5 2 24 , 0, the SOC for a constrained min. At the max, ( p 3 , 2 p 3 , 2), det H 5 det 0 p 3 2 p 3 p 3 2 2 2 2 2 p 3 2 2 2 2 5 1 24 . 0 , the SOC for a constrained max.
ANSWERS PAMPHLET 103 18 . 3): det H 5 det 0 2 x 2 1 2 x 2 1 2 l 0 2 1 0 2 5 2 (2 1 2 l 1 8 x 2 ) , 0 , the SOC for a constrained min. 18 . 5): 0 0 3 1 1 0 0 1 1 1 3 1 2 0 0 1 1 0 2 0 1 1 0 0 2 has positive determinant, the SOC for a constrained min when there are 3 variables and 2 constraints. 19.15 If ] h ] x ( x p , y p ) 0, then C h can be written as x 5 c ( y ) around ( x p , y p ); i.e., h ( c ( y ) , y ) ; 0 for all y near y p and c 0 ( y ) 5 2 ] h ] y ( c ( y ) , y ) ` ] h ] x ( c ( y ) , y ) . Let F ( y ) 5 f ( c ( y ) , y ) . So, F 0 ( y p ) 5 0 and F 00 ( y p ) , 0 implies that y p is a strict local max of F and that ( x p , y p ) is a strict local constrained max of f . F 0 ( y p ) 5 ] f ] x ( c ( y p ) , y p ) c 0 ( y p ) 1 ] f ] y ( c ( y p ) , y p ) 5 ] f ] x ( c ( y p ) , y p ) c 0 ( y p ) 1 ] f ] y ( c ( y p ) , y p ) 2 m p ] h ] x ( c ( y p ) , y p ) c 0 ( y p ) 1 ] h ] y ( c ( y p ) , y p ) 5 ] L ] x ( x p , y p ) c 0 ( y p ) 1 ] L ] y ( x p , y p ) 5 0 , since ] L ] x 5 ] L ] y 5 0 by FOCs .

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