ECN 741: Public Economics Fall 2008 P ( w ) = max c,y,w0 X θ π ( θ ) ± u ( c ( θ,w ))-v ( y ( θ,w )) θ-ˆ λc ( θ,w ) + ˆ λy ( θ,w ) + ˆ βP ( w0 ( θ,w )) ² (51) sub. to u ( c ( θ,w ))-v ( y ( θ,w )) θ + βw0 ( θ,w ) ≥ u ( c ( θ0 ,w ))-v ( y ( θ0 ,w )) θ + βw0 ( θ0 ,w ) ∀ θ,θ0 X θ π ( θ ) ± u ( c ( θ,w ))-v ( y ( θ,w )) θ + βw0 ( 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 P0 ( w ) =-∞ and lim v →-∞ P0 ( 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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This note was uploaded on 12/10/2011 for the course MAT 121 taught by Professor Wong during the Fall '10 term at SUNY Stony Brook.