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NotesECN741-page52 - μ and φ be multipliers on the IC and...

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ECN 741: Public Economics Fall 2008 Note that P m ( w ) is strictly concave, P m ( w ) P ( w ) . Also, note that only the first term depends on w . Therefore P 0 m ( w ) = 1 - ˆ λ E 1 v 0 ( v ( y ( w, θ 0 )) + w 0 - w ) and lim w →-∞ P 0 m ( w ) = 1 . Therefore, lim w →-∞ P 0 ( w ) lim w →-∞ P 0 m ( w ) = 1 . Hence, we proved that lim w →-∞ P 0 ( w ) = 1 . In the next lemma we show that 1 - P ( w 0 ( w, θ )) can be bounded above and bellow for all θ . Let’s rewrite the problem ( 51 ) again 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 )) (52) 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 (
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Unformatted text preview: μ and φ be multipliers on the IC and promise keeping (and let π ( θ H ) = π ) FOC w.r.t c ( w,θ ) π ³ 1-ˆ λ 1 u ( c ( w,θ H )) ´ + φπ + μ = 0 (1-π ) ³ 1-ˆ λ 1 u ( c ( w,θ L )) ´ + φπ-μ = 0 FOC w.r.t w ( w,θ ) π ˆ βP ( w,θ H ) + φπβ + βμ = 0 (1-π ) ˆ βP ( w,θ L ) + φ (1-π ) β-βμ = 0 52...
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