714s10_hw3_ans - ECON 714 MACROECONOMIC THEORY II TA TIM...

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Unformatted text preview: ECON 714: MACROECONOMIC THEORY II TA: TIM LEE MARCH 4, 2010 Outline to HW3 Solutions Mind you, I might be wrong...open to any disputes! 1. The invariant distribution is ( π 1 π 2 π 3 ) = psq psq + rq + ( 1- p ) rs rq psq + rq + ( 1- p ) rs ( 1- p ) rs psq + rq + ( 1- p ) rs . Condition (ii) should say π 1 < 1/2 < π 1 + π 2 . (a) We want 1- r > ps 1- s ( 1- p ) > q psq < psq + rq + ( 1- p ) rs 2 psq + rq > psq + rq + ( 1- p ) rs 2 a 2 ( psq + rq + ( 1- p ) rs ) < psqa 1 + rqa 2 + ( 1- p ) rsa 3 psa 1 + ( 1- s ) a 2 + ( 1- p ) sa 3 > psqa 1 + rqa 2 + ( 1- p ) rsa 3 psq + rq + ( 1- p ) rs which can be rewritten as 1- r > ps 1- s ( 1- p ) > q s max ([ pq- ( 1- p ) r ] , [( 1- p ) r- pq ]) < rq a 2 ( psq + ( 1- p ) rs ) < psqa 1 + ( 1- p ) rsa 3 psa 1 + ( 1- s ) a 2 + ( 1- p ) sa 3 > psqa 1 + rqa 2 + ( 1- p ) rsa 3 psq + rq + ( 1- p ) rs . Depending on which is the min and which is the max, we could come up with various simplified conditions. (b) I didn’t do the calculations, but it seems like it will hold... 2. (a) V ( h ) = max u u ( α hu ) + β V ( δ h ( 1- u )) where u ( c ) = c γ / γ . 1 (b) It’s bad notation but I’ll just use u ( · ) for the utility function and u ∈ [ 0, 1 ] for the choice variable. F.o.c. and envelope condition is α u ( c P ) = βδ V ( h P ) V ( h ) = α u ( c P ) ⇒ u ( c P ) = βδ u ( c P ) .............(Euler Equation)....
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714s10_hw3_ans - ECON 714 MACROECONOMIC THEORY II TA TIM...

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