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Unformatted text preview: Noah Williams Economics 714 Department of Economics Macroeconomic Theory University of Wisconsin Spring 2010 P ROBLEM S ET 2 Solutions by Tim Lee 1. (a) Refer to code on TA website. Let us normalize N = 1. The problem we want to solve is V ( k ) = max k c 1 1 + V ( k ) s.t. c + k = k + ( 1 ) k with parameter values as given in the homework. The policy function crosses the 45 line at the steady state. 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 k g(k) Deterministic Optimal Policy Function Optimal Policy 45 deg line Deterministic Policy Function 1 (b) The = 1.01 path is below the the = 3 path. Intertemporal elasticity is higher (or less risk averse), so Im more willing to save today for more consumption tomorrow. Convergence is also faster, as can be seen by the distance between the dots. 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1 1.2 1.4 k c Saddlepaths delc=0 delk=0 gamma=3 saddle gamma=1.01 saddle Saddle Paths 2 (c) Now the problem is V ( k , z ) = max k c 1 1 + z { z H , z L } P ( z , z ) V ( k , z ) s.t. c + k = zk + ( 1 ) k , with the values of ( z H , z L ) and P ( ) as given in the homework. Obviously, the z H ( z L ) policy is higher (lower) than the deterministic policy. 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 k g(k,z) Stochastic Optimal Policy Function z H z L 45 deg Stochastic Policy Function 3 (d) As above, the z H ( z L ) consumption function is higher (lower) than the deter ministic consumption function. Furthermore, expected consumption is lower with uncertainty there is a precautionary savings motive that decreases ex pected consumption for fixed levels of the state. 0.5 1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1 1.2 1.4 k c(k,z) Consumption Functions Deterministic z H z L zmean Consumption Functions 4 (e) Equilibrium bond prices are P ( k , z ) = E [ u ( c ( k , z ))  k , z ] u ( c ( k , z )) = E " c ( k , z ) c ( k , z ) k , z # ....
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This note was uploaded on 06/19/2011 for the course ECON 714 taught by Professor Staff during the Spring '08 term at Wisconsin.
 Spring '08
 Staff
 Economics

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