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NotesECN741-page51 - ECN 741 Public Economics sub to Fall...

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ECN 741: Public Economics Fall 2008 sub. to. T X t =1 β t - 1 X θ T D π ( θ t ) u ( c t ( θ t )) - v ( y t ( θ t )) θ t w Note that ˜ K ( w ) is strictly convex and differentiable and lim w →-∞ ˜ K 0 ( w ) = 0 . Let P max ( w ) = w - ˜ K ( w ) + m . Then P max ( w ) P FI ( w ) and both are strictly con- cave. Also, lim w →-∞ P FI ( w ) lim w →-∞ P max ( w ) = -∞ . Therefore, lim w →-∞ P 0 FI ( w ) lim w →-∞ P 0 max ( w ) = 1 . Also, Note that lim w →-∞ P ( w ) lim w →-∞ P FI ( w ) = -∞ and therefore (since both are strictly concave) lim w →-∞ P 0 ( w ) lim w →-∞ P 0 FI ( w ) = 1 . lim w →-∞ P 0 ( w ) 1 Next, consider allocations ( c ( w 0 , θ t ) , y ( w 0 , θ t )) that solve the original problem. Suppose they attain the value P ( w 0 ) . Define new allocations c ( w, θ t ) , ˜ y ( w, θ t )) for w w 0 as ˜ c ( w, θ t ) = c ( w 0 , θ t ) θ t t ˜ y ( w, θ t ) = y ( w 0 , θ t ) θ t t > 1 ˜ y ( w, θ 1 ) = v - 1 ( v ( y ( w 0 , θ 1 )) + w 0 - w ) Now define P m ( w ) for w w 0 as P m ( w ) = T X t =1 ˆ β t - 1 X θ T D π ( θ t ) u c ( w, θ t )) - v y ( w, θ t )) θ t - ˆ λ ˜ c ( w, θ t ) + ˆ λ ˜ y ( w, θ t ) = X θ
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