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Unformatted text preview: Problem Set 3 Econ 210 Core Macro Monika Piazzesi Stanford University Due Wednesday, October 13 in class Before you start this problem, read Chapter 5 of Krueger and Chapter 4 of SLP in its entirety. There is no need to go through proofs of the theorems. Here, we are interested in applying the theorems. 1. Famous example (continue what we did not &nish in class) In&nitely lived household has initial wealth x 2 X = R , borrows and lends at the gross interest rate 1 + r = 1 =& > 1 , so that the price of a one-period bond is q = &: There is a sequential budget constraint c t + &x t +1 & x t The sequential problem is w ( x ) = sup 1 X t =0 & t c t subject to & c t & x t &x t +1 x given (a) Argue that running a Ponzi scheme is optimal and &nd w ( ) as- sociated with the Ponzi scheme. 1 (b) In the recursive formulation, de&ne the state variable, control vari- able, and the functional equation. (c) Show that w ( & ) satis&es this functional equation. What is the value function v ( & ) associated with the Ponzi scheme solution? (d) Show that e v ( x ) = x is an alternative solution to the functional equation. Consider the feasible plan x n = x =& n . Show that for this feasible plan, the alternative solution e v does not satisfy the limit-condition for the Principle of Optimality (Theorem 42 in Krueger, and so we cannot conclude that e v = w ). 2. Famous example (variation, with a borrowing constraint of zero) w ( x ) = max f x t +1 g 1 t =0 1 X t =0 & t ( x t &x t +1 ) subject to x t +1 x t & x given...
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- Fall '09