NotesECN741-page50

NotesECN741-page50 - ECN 741: Public Economics sub. to....

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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 ² w P FI ( w ) is the value to the planner from delivering utility w to individual, if we ignore incentive constraint. Note that P FI ( w ) > P ( w ) . Also, P FI ( w ) is strictly concave and differentiable. Next consider the following maximization problem m = max c,y,θ u ( c ) - v ( y ) θ - ˆ λc + ˆ λy The above problem has a solution ( u 0 ( c ) = ˆ λ , v 0 ( y ) = ˆ λθ , and θ belongs to a compact set). Next, note that P FI ( w ) = T X t =1 ˆ β t - 1 X θ T D π ( θ t ) ± u ( c t ( θ t )) - v ( y t ( θ t )) θ t - ˆ λc t ( θ t ) + ˆ λy t ( θ t ) ² = T X t =1 ˆ β t - 1 X θ T D π ( θ t ) ± u ( c t ( θ t )) - v ( y t ( θ t )) θ t - ˆ λc t ( θ t ) + ˆ λy t ( θ t ) ² + 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 - ˆ λc t ( θ t ) + ˆ λy t ( θ t ) ² - 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 - ˆ λc t ( θ t ) + ˆ λy t ( θ t ) ² = w + T X t =1 β t - 1 X θ T D π
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