homework3 - where d (0 , 1). Using your Euler equation...

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ECO 475 Homework 3 Jay H. Hong Due: Sep 27, 2010 1 Properties of Value Function Let’s define the following operator T : C → C ( Tv )( x ) = max a Γ( x ) { u ( x,a ) + βv [ f ( x,a )] } , where C is the set of continuous and bounded real-valued functions on X . Show that Under certain assumptions (specify those), T maps increasing function into increasing function. Under certain assumptions (specify those), T maps strictly increasing function into strictly increasing function. Under certain assumptions (specify those), T maps concave function into concave function. Under certain assumptions (specify those), T maps strictly concave function into itself. (optional) Under certain assumptions (specify those), T maps differentiable function into itself. 2 Policy Function Iteration Consider the same environment as in Q1 from Homework 2. (1) Derive the Euler Equation. Be precise when you get V 0 ( k ). (2) Let your η 0 ( k ) = dAk α
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Unformatted text preview: where d (0 , 1). Using your Euler equation derive 1 ( k ) and show that 1 ( k ) = 1 + -d Ak . (3) Derive 2 ( k ) , n ( k ) and ( k ) (by n ). (4) Discuss whether we need d (0 , 1) or not. Why? (5) Discuss whether n ( k ) ( 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 } t =0 X t =0 t [log( c t ) + B log( c t-1 )] subject to an initial k and c-1 and a law of motion k t +1 = Ak t-c t , where A > , (0 , 1) , (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 . 1...
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