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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/22/2011 for the course EECS 501 taught by Professor Yunding during the Fall '09 term at University of Florida.

Page1 / 7

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online