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problem2sol

# problem2sol - 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 (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 ) = ° 0 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 0 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

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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) 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 0 ( c t ) = °u 0 ( 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 ² (1) Substituting the policy function back to the Bellman equation gives A w 1 ° ° 1 ° ± = 1 1 ° ± ³ w (1 + B ) ´ 1 ° ° + °AR 1 ° ° 1 1 ° ± µ w µ 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 ° µ 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. (d) The policy function is given by equation 1. 3

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(e) With time varying income, there would be two state variables, w t and y t . So the guess in ( c ) is de°nitely not valid because it only has one state variable.
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