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 η 1 ( k ) and show that η 1 ( k ) = αβ 1 + αβ-d Ak α . (3) Derive η 2 ( k ) ,η n ( k ) and η ( k ) (by n → ∞ ). (4) Discuss whether we need d ∈ (0 , 1) or not. Why? (5) Discuss whether η n ( k ) → η ( 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 β 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 α t-c t , where A > ,α ∈ (0 , 1) ,β ∈ (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 α . 1...
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