3456econ

5 Question 5: Caballero & Krishnamurthy (2006) - Welfare and Bubbles Consider the model in Caballero and Krischnamurthy (JME, 2006) that we studied in class. Compute equilibrium welfare in the case where there is no bubble and in the case where there is a bubble. Compare and determine whether: (a) it is always better to have a bubble; (b) it is always better to have no bubble; 16 or (c) it depends on parameters. Throughout, you must assume the parameter restrictions imposed in the paper are satisfied.Problem 1 (Eat-the-Pie Problem): Consider the following sequence problem: max >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 1 (Eat-the-Pie Problem): Consider the following sequence problem: max >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 4: Consider an optimal investment problem. Every period you draw a cost c (distributed uniformly between 0 and 1) for completing a project. If you undertake the project, you pay c, and complete the project with probability 1 - p. Each period in which the project remains uncompleted, you pay a late fee of 1. The game continues until you complete the project. a. Write down the Bellman Equation assuming no discounting. Why is it OK to assume no discounting in this problem. b. Derive the optimal threshold: c* = #2/. Explain intuitively, why this threshold does not depend on the probability of failing to complete the project, p. c. How would these results change if we added discounting to the framework? Redo steps a and b, assuming that the agent discounts the future with discount factor 0 B(1 + c). Assume too that commitment is not available (except for question 3.)