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ECON 401 Hartman Autumn 2010 ANSWERS FOR PROBLEM SET VII 1. Substitute for t I to see that we want to maximize 1 0 () [ ( 1)] (1 ) ttt t t t t pfK q K K Z r  222 3 2 111 2 1 000 1 0 2 () [ ( ( ( ) [ ) ] ... 1( 1 ) K K K rr  1 1 1 [ ( 1 )] ( )[ ( 1 ) ] ... ) ) t t t t t t t tt K p fK qK K    . Now, 0 K was determined in the past, but we are free to choose all future K s starting with 1 K . For any of those future K s, say t K , the first order condition is 1 1 () ( 1 ) 0 ) ) t t t qp f K q Z Kr r    . This condition can be rewritten as 1 ( 1 ) ( t pf K q r q   . The user cost of capital for period t is 11 1 ) ) ( ) t t t t qr q q r q q q    ; the first term after the equality is foregone interest, the second term is the cost of depreciation over the period, and the last term is the net capital gain which is subtracted. 2.a. Substitute for I to see that we want to maximize 1/3 0 [3 ( ) ] rt ep K q KK d t . If we let 1/3 (, , ) [ 3 ( rt GtKK e p K qK K   , then 2/3 /( ) rt GKe p K q   , / rt GK eq  , and rt rt dG re q e q dt K  

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