NotesECN741-page47

NotesECN741-page47 - ECN 741: Public Economics Fall 2008...

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ECN 741: Public Economics Fall 2008 also from envelope condition we have K 0 ( w ) = φ therefore K 0 ( w ) = X θ π ( θ ) K 0 ( w 0 ( θ,w )) Start from a given w 0 , construct a stochastic process w t as w t +1 = w 0 ( θ t ,w t ) then K 0 ( w t ) = E t [ K 0 ( w t +1 )] hence w t is a martingale. By martingale convergence theorem there must exist a w such that w t a.s. -→ w . Suppose K 0 ( w ) > 0 . Note that convergence implies that w 0 ( θ,w ) = w 0 ( θ 0 ,w ) θ,θ 0 and therefore c ( θ,w ) = c ( θ 0 ,w ) θ,θ 0 and then incentive compatibility implies y ( θ,w ) = y ( θ 0 ,w ) θ,θ 0 but we know from our two type static example that the planer can do better by differentiating
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