Unformatted text preview: where d âˆˆ (0 , 1). Using your Euler equation derive Î· 1 ( k ) and show that Î· 1 ( k ) = Î±Î² 1 + Î±Î²d Ak Î± . (3) Derive Î· 2 ( k ) ,Î· n ( k ) and Î· ( k ) (by n â†’ âˆž ). (4) Discuss whether we need d âˆˆ (0 , 1) or not. Why? (5) Discuss whether Î· n ( k ) â†’ Î· ( k ) is pointwise convergence or uniform convergence. Why? (Hint: Theorem 3.8 SLP) 3 Habit Persistance Utility Consider the following problem. max { c t ,k t +1 } âˆž t =0 âˆž X t =0 Î² t [log( c t ) + B log( c t1 )] subject to an initial k and c1 and a law of motion k t +1 = Ak Î± tc t , where A > ,Î± âˆˆ (0 , 1) ,Î² âˆˆ (0 , 1) and B > 0. Formulate and solve the recursive problem. What are your control variable and state variables? Derive the closed form solutions for the value function V and the policy function Î± . 1...
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 Fall '07
 Hong
 Convex function, Concave function, certain assumptions, Jay H. Hong, max âˆž

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