cons2-sl - Consider the problem we discussed before sup...

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Consider the problem we discussed before sup { c t ,a t +1 } E 0 P t =0 β t u ( c t ) such that c t + a t +1 = y t +(1+ r ) a t a t +1 ≥− φ c t 0 a 0 given and where y t Y = { y 1 ,y 2 ,...,y N } follows a stochastic process de f ned by the transition probability matrix Π . Let us suppose that there is a unique ergodic distribution over y t associated with Π de f ned by p = Π p . Assume that p de f nes the time zero probability distribution over y 0 .
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We can write this problem in the general form outlined by Stokey and Lucas as sup { a t +1 } E 0 P t =0 β t F ( a t ,a t +1 ) such that a t +1 Γ ( a t ,y t )= { a t +1 R ; φ a t +1 y t +(1+ r ) a t } where F ( a t ,y t ,a t +1 )= u ( y t +(1+ r ) a t a t +1 ) is the one-period return function.
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The Recursive formulation of this problem is v ( a, y i )=max a 0 ( u ( c )+ β P j Π ij v ( a 0 ,y j ) ) subject to c + a 0 = y i +(1+ r ) a a 0 ≥− φ c 0 or, in general form, v ( a, y i )= max a 0 Γ ( a,y i ) ( F ( a, y i ,a 0 )+ β P j Π ij v ( a 0 ,y j ) )
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Recall that in the non-stochastic case we showed that as long as the constraint set was non-empty, and in the limit lifetime utility was well-de f ned for any feasible plan, then 1. (SL Th 4.2) The function v de f ning the supremum for lifetime utility in the sequence problem (the ‘true’ problem) for di f erent values for ini- tial wealth satis f es the corresponding functional equation (the recursive formulation of the problem) 2. (SL Th 4.3) The converse: If we have a function v that solves the functional equation and satis f es lim n →∞ β n v ( x n )=0 where x n could belong to any feasible sequence for x ,then v = v .
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There are a bunch of additional results that we didn’t yet discuss for the non-stochastic case 3. (SL Th 4.4) Plans that are optimal in the sequence problem satisfy
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This note was uploaded on 11/10/2011 for the course ECON 601 at Cornell.

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cons2-sl - Consider the problem we discussed before sup...

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