Macroeconomics Exam Review 89

Macroeconomics Exam Review 89 - c 2001 Michael Carter All...

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4. Since ? ? +1 ± ( ? ? ) for every ² , x satisFes Bellman’s equation, that is ³ ( ? ? )= ´ ( ? ? ,? ? +1 )+ µ³ ( ? ? +1 ) =0 , 1 , 2 ,... Therefore x is optimal (Exercise 2.17). 2.126 1. In the previous exercise (Exercise 2.125) we showed that the set ± of solu- tions to Bellman’s equation (Exercise 2.17) is the solution correspondence of the constrained maximization problem ± ( ? ) = arg max ± ² ( ³ ) { ´ ( ?,¶ ( ) } This problem satisFes the requirements of the monotone maximum theorem (The- orem 2.1), since the objective function ´ ( ( ) supermodular in displays strictly increasing di±erences in ( )s inceforevery ? 2 ? 1 ´ ( ? 2 ( ) ´ ( ? 1 ( ´ ( ? 2 ) ´ ( ? 1 ) · ( ? ) is increasing. By Corollary 2.1.2, ± ( ? ) is always increasing. 2. Let x =( ? 0 1 2 ) be an optimal plan. Then (Exercise 2.17) ? ? +1 ± ( ? ? ) , 1 , 2 Since ± is always increasing ? ? ? ? 1 = ? ? +1 ? ? for every ² =1 , 2 . Similarly ? ? ? ? 1 = ? ? +1
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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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