NotesECN741-page44 - ECN 741 Public Economics If V ar 1...

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Unformatted text preview: ECN 741: Public Economics If V ar 1 |θt u (ct+1 (θT )) Fall 2008 > 0 for some θt , then V ar 1 u (ct (θT )) < V ar 1 u (ct+1 (θT )) and therefore V ar ct (θT ) < V ar ct+1 (θT ) So inequality grows. And it is efficient. How about mobility? In short-run there is mobility. What about long-run? Note that 1 u (ct ) is a martingale (βR = 1). We also know that (by feasibility) E 1 u (ct+1 ) <∞ Martingale Convergence Theorem: If {xt }∞ is stochastic process adapted to filteration t=1 ∞ {Ft }t=1 such that xt = E [xt+1 |Ft ] and E [xt ] < ∞ for all t, then a.s. lim xt = x∞ < ∞ t→∞ where x∞ is a random variable with E[x∞ ] < ∞. Therefore, 3.1.2 1 u (ct ) converges to a finite number and hence there is no mobility in the long-run. Long-run properties of efficient allocations This part is mostly based on Farhi and Werning (2006, 2007, 2005), PHELAN (2006) and Atkeson and Lucas (1992). In what follows we maintain the following assumptions: • T =∞ • Θ = {θH , θL } • θt i.i.d over time Very important note: In what follows I skip some detailed steps. Most of the arguments are heuristic and have loose ends. For more rigorous proofs please look at the references above. 44 ...
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