NotesECN741-page46 - ECN 741 Public Economics Fall 2008 at...

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ECN 741: Public Economics Fall 2008 at w = w * t ( ¯ θ t ) (note: I imposed the βR = 1 assumption here) K ( w ) = min c,y,W X θ π ( θ ) [ c ( θ, w ) - y ( θ, w ) + βK ( w 0 ( θ, w ))] 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 ) w The second constraint is called “promise keeping” constraint. Proposition 10 K ( w ) is strictly increasing and strictly convex (assumptions on v ( · ) is needed). Also. Let w and ¯ w be the lowest and highest possible values for promised utility. Then, lim w w K 0 ( w ) = 0 and lim w ¯ w K 0 ( w ) = lim w ¯ w K ( w ) = 0 . Let μ ( θ, θ 0 ) be multiplier on incentive constraint and φ the multiplier on promise keeping. First order condition with respect to c ( θ, U ) is u 0 ( c ( θ, w )) X θ 0 μ ( θ, θ 0 ) - X θ 0 μ ( θ 0 , θ ) + π ( θ
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