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PS3 - iid income shocks y Utility is exponential u c...

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Economics 475 Part II Problem Set 3 1. Consider a 2 period economy. Income in period 1 is y 1 and income in period 2 is a random variable ˜ y 2 . Utility is u ( c 1 ) + β E u ( c 2 ). Assume u 0 > 0, u 00 < 0, and u 000 > 0. Agents can save at an interest rate r . (a) Derive the first order condition for consumption. (b) Describe what happens to savings in the first period if we make agents more patient ( β ). (c) Describe what happens to savings in the first period if we increase the interest rate ( r ). (d) Describe what happens to savings if we alter ˜ y 2 by a mean preserving spread (i.e., let ˜ y 0 2 = ˜ y 2 + ε , with E ( ε ) = 0). A mean preserving spread is another way of saying 2nd order stochastic dominance. Hint: Use the fact that if the distribution characterized by the cdf F ( x ) is a mean preserving spread of G ( x ) and h ( x ) is concave, then R h ( x ) dF ( x ) < R h ( x ) dG ( x ). 2. Consider an infinite horizon model in which agents face iid
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Unformatted text preview: iid income shocks y . Utility is exponential: u ( c ) =-e-γc . Note that c can be negative and the only constraint on assets is the No Ponzi condition. Agents face a constant interest rate r . (a) Show that if x t = y t + a t , then the optimal savings decision follows a t +1 = δx t + b , for some constants δ and b . (b) Use your expression for the policy function for assets to show that cash on hand is a random walk with drift. What does this imply about the boundedness of assets and consumption as t → ∞ when β (1 + r ) < 1? Give some intuition relative to the CRRA case discussed in class. 3. Prove that if lim c →∞ u 00 ( c ) u ( c ) = 0 and β (1 + r ) < 1, then assets are bounded. How does this apply to the case of CRRA utility versus exponential utility? 1...
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