problem2sol - Problem Set 2 Econ 210 Core Macro Monika...

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Unformatted text preview: Problem Set 2 Econ 210 Core Macro Monika Piazzesi Stanford University Due Wednesday, October 6 in class 1. Math warmup (a) D = R and f ( x ) = x 2 D is not compact (closed but not bounded), f is continuous, and f does not attain a max on D (b) D = (0 ; 1) and f ( x ) = x D is not compact (bounded but not closed), f is continuous, and f does not attain a max on D (c) D = [ & 1 ; 1] and f ( x ) = & if x = & 1 or x = 1 x if & 1 < x < 1 D is compact (bounded and closed) , f is not continuous at x = & 1 and x = 1 , f ( D ) is thus the open interval ( & 1 ; 1) , and so f does not attain a max on D (d) D = R ++ and f ( x ) = & 1 if x is rational otherwise D is not compact (not bounded, not closed), f is discontinuous at every point in R , nevertheless f attains a max (at every rational number). 1 2. Guess and Verify A consumer maximizes 1 X t =0 & t c 1 & & t 1 & subject to initial wealth w 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) The state variable is initial wealth w t . The control variable is c t : (b) A variational argument leads to the following Euler equation: u ( c t ) = &u ( c t +1 ) R c & & t = &Rc & & t +1 The LHS represents the marginal utility cost of lowering consump- tion at time t by one unit. The RHS represents the discounted marginal utility gain, multiplied by the additional consumption at t + 1 achieved by saving one more unit of consumption at time t: (c) The Bellman equation is v ( w ) = max c u ( c ) + &v ( R ( w & c )) With our guess, the RHS becomes c 1 & & 1 & + &AR 1 & & ( w & c ) 1 & & 1 & The &rst order condition becomes c & & = &AR 1 & & ( w & c ) & & 2 which can be solved for the policy function c = g ( w ) = w & 1 + [ &AR 1 & & ] 1 =& (1) Substituting the policy function back to the Bellman equation gives A w 1 & & 1 & = 1 1 & w (1 + B ) 1 & & + &AR 1 & & 1 1 & w 1 & 1 (1 + B ) 1 & & where B = ( &AR 1 & & ) 1 =& : Since this has the correct functional form, the guess is veri&ed. Moreover, we can use the last equation to solve for A . A = (1 + B ) & & 1 + B & 1 & 1 1 + B 1 & & This implies A (1 + B ) 1 & & = 1 + B & B 1 & & = 1 + B A 1 =& = 1 + &AR 1 & & 1 =& A = 1 1 & ( &R 1 & & ) 1 =& ! & Remark: Instead of guessing the value function, we could have also guessed the policy function g ( w ) = Xw: The Euler equation is then Xw = ( &R ) & 1 =& XR ( w & Xw ) which can be solved for w & 1 =& R (1 & & ) =& = 1 & w Again, since the conjectured functional for is recovered, the guess is veri&ed....
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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.

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problem2sol - Problem Set 2 Econ 210 Core Macro Monika...

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