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Problem 1 (Growth Model): Recall the growth model that we discussed in class. We expressed the sequence problem as v(ko) = sup Bt In(k; - ki+1) {kiti himo f=0 subject to the constraint ki+1 E [0, ki ] = [(ki). Consider the associated Bellman equation v(k) = sup In(k" - y) + Bu(y). VEr() Finally, note that 0 S a 0, there exists a value of T, such that ky T. Hint: this is true. e. In the growth problem (problem 1), lim,-x B"v(r.) S'u(c,) subject to the constraints: WHI = R(W, - CF) OSCEW, Wo > 0 given. I'm now going to ask you to analyze this problem. This analysis provides a quick review of concepts that should by now be familiar. I will put problems like this on the final exam. a. Motivate the economic problem above. Evaluate the implicit assumptions. What is economically sensible and what is not sensible about this modeling set-up? b. Explain why the Bellman equation for this problem is given by: MA {((2 -M)y)ag + (o)n} dns = (1)4 DE[0, W] Why doesn't an expectation operator need to appear in this Bellman equation? c. Using Blackwell's sufficiency conditions, prove that the Bellman operator, B, MA {((3-M))fg + (3)n} dns = (M)(fa) CE[0, W] is a contraction mapping. You should assume that u is a bounded function. (Why is this boundedness assumption necessary for the application of Blackwell's Theorem?) Explain what the contraction mapping property implies about iterative solution methods. d. Now assume that, ify e (0,co) and y # 1 Inc ify = 1 (So u is no longer bounded.) Use the guess method to solve the Bellman equation. Specifically, guess the form of the solution. v (W) = if y e (0,0) and y * 1 $ + wlnW if y = 1 Confirm that this solution works. e. Derive the optimal policy rule: c =WFW W # = 1 - (8R1- )+ Note that this rule applies for all values of y. f. When y = 1 the consumption rule collapses to c = (1 -6) W. Why does consumption no longer depend on the value of the interest rate? (Hint: think about income effects and substitution effects.)Problem 2 (Growth Model): Recall the growth model that we discussed in class. We expressed the sequence problem as v(ko) = sup B' In (K; - Kil) subject to the constraint KHI E [0,K; ] = [(K.). Consider the associated Bellman equation v(k) = sup In(k" - y) + Bv(v). Finally, note that 0 x* where, x* = (expp) (1 - [1 - exp(-2p)]1/2). Hint: use a similar conceptual approach to the one that we used in class when w = 0