NotesECN741-page49

NotesECN741-page49 - ECN 741: Public Economics () u(c(, w)...

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ECN 741: Public Economics Fall 2008 P ( w ) = max c,y,w 0 X θ π ( θ ) ± u ( c ( θ,w )) - v ( y ( θ,w )) θ - ˆ λc ( θ,w ) + ˆ λy ( θ,w ) + ˆ βP ( w 0 ( θ,w )) ² (51) sub. to u ( c ( θ,w )) - v ( y ( θ,w )) θ + βw 0 ( θ,w ) u ( c ( θ 0 ,w )) - v ( y ( θ 0 ,w )) θ + βw 0 ( θ 0 ,w ) θ,θ 0 X θ π ( θ ) ± u ( c ( θ,w )) - v ( y ( θ,w )) θ + βw 0 ( θ,w ) ² w We want to show that in this problem in the long-run the promised utility cannot be at misery ( w > -∞ ). We use two lemmas to show this Lemma 2 The value function P ( w ) is strictly concave and continuously differentiable on ( -∞ , ¯ w ) . Moreover, lim v →-∞ P ( w ) = lim v ¯ w P ( w ) = lim v ¯ w P 0 ( w ) = -∞ and lim v →-∞ P 0 ( w ) = 1 Proof. We are not going to prove concavity and differentiability. We take them as given. Warning: This proof is not complete! There are some steps that needs to be filled in or reformulated. I present it to provide the core idea of the proof as I understand it. Warning: The proof that I presented in class is wrong! The mistake is in constructing the upper and lower bound value function (
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