Macroeconomics Exam Review 247

Macroeconomics Exam Review 247 - c 2001 Michael Carter All...

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which implies that ? ? 0 ? ( ± +1) ? ? 0 or ? 0. Similarly, let { e 1 , e 2 ,..., e ± } denote the standard basis for ± (Example 1.79). For any ² =1 , 2 ,...,³ ,thepo int( 0 e ² ) (which corresponds to decreasing resource ² by one unit) belongs to ´ and therefore (from (5.75)) ± ? ? ( 0 e ² )= ± + µ ² ± ? ? 0 = ± which implies that µ ² 0. 5.51 Let ˆ c = x ) < 0 and ˆ ± = · x ) Suppose ? = 0. Then, since ¸ is nonzero, at least one component of ? must be nonzero. That is, ? 0 and therefore ? ? ˆ ¹< 0 (5.107) But (ˆ c , ˆ ± ) º and (5.74) implies ? ˆ ± ? ? ˆ c ? ? 0 and therefore ? = 0 implies ? ? ˆ ¹ 0 contradicting (5.107). Therefore, we conclude that ?> 0. 5.52 The utility’s optimization problem is max ³,´ 0 » ( ¼,½ µ =1 ³ ? 0 ( ¾ ( ¿ ) ¹ ) À¿ ¹ 0 ½ subject to ( y ¼ ½ 0 Á , 2 ,..., The demand independence assumption ensures that the objective function » is concave, since its Hessian à · = ľ
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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