cons3-sl - Stochastic income-uctuations problem Focus on...

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Stochastic income- f uctuations problem Focus on the case where shocks are iid through time Look at the cases β (1 + r )=1 and β (1 + r ) > 1 Enduplook ingatthetwoshockcase e { e l ,e h } with constant relative risk aversion preferences Assume we have made su cient assumptions to ensure principle of optimality applies can apply the contraction mapping theorem value function v is strictly increasing (in all arguments), strictly concave
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The Recursive formulation of this problem is V ( a, e )=max c,a 0 u ( c )+ β P e 0 { e l ,e h } π ( e 0 ) V ( a 0 ,e 0 ) subject to c + a 0 = e +(1+ r ) a a 0 ≥− φ c 0
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If productivity shocks are i.i.d we can take a transformation of the state variables to eliminate e as a state variable. Let z = e +(1+ r ) a + φ Thus z denotes maximum disposable resources (maximum possible consump- tion given e and a, the interest rate r, and the borrowing constraint φ ). In terms of the single state variable z the household’s budget set is given by c + a 0 z φ If the probability distribution over e 0 is independent of e, then provided the household knows z he does not need to know e to solve his optimization prob- lem. In other words, the household does not care whether the resources he has
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at his disposal come from savings he made in the previous period or from his current period endowment shock. We can therefore rewrite the value function as follows. V ( z )=max c,a 0 u ( c )+ β P e 0 { e l ,e h } π ( e 0 ) V ( z 0 ) c = z φ a 0 z 0 = e 0 +(1+ r ) a 0 + φ a 0 ≥− φ Denote the decision rules that solve this problem c ( z ) and a 0 ( z ) .
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The f rst order condition is u 0 ( c ) β (1 + r ) P e 0 { e l ,e h } π ( e 0 ) V 0 ( z 0 )= if a 0 > φ (1) Note that the transition probabilities do not depend on e.
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cons3-sl - Stochastic income-uctuations problem Focus on...

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