ECN 741: Public Economics Fall 2008 also from envelope condition we have K0 ( w ) = φ therefore K0 ( w ) = X θ π ( θ ) K0 ( w0 ( θ,w )) Start from a given w0 , construct a stochastic process w t as w t +1 = w0 ( θ t ,w t ) then K0 ( w t ) = E t [ K0 ( 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 K0 ( w ∞ ) >0 . Note that convergence implies that w0 ( θ,w ∞ ) = w0 ( θ0 ,w ∞ ) ∀ θ,θ0 and therefore c ( θ,w ∞ ) = c ( θ0 ,w ∞ ) ∀ θ,θ0 and then incentive compatibility implies y ( θ,w ∞ ) = y ( θ0 ,w ∞ ) ∀ θ,θ0 but we know from our two type static example that the planer can do better by diﬀerentiating
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