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NotesECN741-page36 - y θ H> y θ L therefore u c θ L-v...

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ECN 741: Public Economics Fall 2008 Example : one period problem Suppose T = 1 and there are only two types, θ H and θ L with θ H > θ L . Consider the utilitarian planer’s problem max π ( θ H ) u ( c ( θ H )) - v y ( θ H ) θ H + π ( θ L ) u ( c ( θ L )) - v y ( θ L ) θ L sub. to. u ( c ( θ H )) - v y ( θ H ) θ H u ( c ( θ L )) - v y ( θ L ) θ H u ( c ( θ L )) - v y ( θ L ) θ L u ( c ( θ H )) - v y ( θ H ) θ L π ( θ H ) [ c ( θ H ) - y ( θ H )] + π ( θ H ) [ c ( θ H ) - y ( θ H )] = 0 For a moment suppose there is no private information. Then the optimal allocation must satisfy (note that v ( · ) is convex): u 0 ( c ( θ H )) = u 0 ( c ( θ L )) = λ c ( θ H ) = c ( θ L ) 1 θ H v y ( θ H ) θ H = 1 θ L v 0 y ( θ L ) θ L = λ y ( θ H ) > y ( θ L ) Note that there is no distortion u 0 ( c ( θ )) = 1 θ v 0 y ( θ ) θ We will show that this allocation does not satisfy I.C. constraints. Note that
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Unformatted text preview: y ( θ H ) > y ( θ L ) , therefore u ( c ( θ L ))-v ² y ( θ L ) θ H ³ = u ( c ( θ H ))-v ² y ( θ L ) θ H ³ > u ( c ( θ H ))-v ² y ( θ H ) θ H ³ the I.C. for type H is violated. So we know that when individuals have private information about their type, at least of the the I.C. constraints is binding at the optimal solution. 36...
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