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problem2 - Problem Set 2 Econ 210 Core Macro Monika...

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Problem Set 2 Econ 210 Core Macro Monika Piazzesi Stanford University Due Wednesday, October 6 in class 1. Math warmup The Weierstrass theorem guarantees the existence of a maximum for a problem with a continuous objective function and a compact constraint set. The following are a number of cool examples that illustrate how the theorem works. For all examples, the maximization problem is max x 2 D f ( x ) For all examples, answer the following three questions: (i) is the func- tion f continuous? (ii) is the set D compact? (if not, why not) and (iii) does a maximum exist? (a) D = R and f ( x ) = x 2 (b) D = (0 ; 1) and f ( x ) = x (c) D = [ ° 1 ; 1] and f ( x ) = ° 0 if x = ° 1 or x = 1 x if ° 1 < x < 1 (d) D = R ++ and f ( x ) = ° 1 if x is rational 0 otherwise 1
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2. Guess and Verify A consumer maximizes 1 X t =0 ° t c 1 ° ° t 1 ° ± subject to initial wealth w 0 which is given, and w t +1 = R ( w t ° c t ) : The gross interest rate R = 1 + r is constant. The budget equation is written recursively: the consumer comes into the period with wealth w t and consumes c t , and thus saves w t ° c t for next period. The wealth w t +1 next period is savings times interest earned on the savings. (a) What is the state variable? What is the control variable? (b) Derive the Euler equation. (c) Guess that the value function has the same form as the utility function v ( x ) = A x 1 ° ° 1 ° ± ; where x is the state variable you de°ned in a: Verify that this value function is indeed valid, and provide an expression for A: (d) How does the policy function look like? (e) So far, we have been considering the savings problem for a rentier who does not have any labor income. Suppose the household receives income y t +1 in period t + 1 and so the wealth dynamics is w t +1 = R ( w t ° c t ) + y t +1 Is the guess for the value function in ( c ) still valid?
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