NotesECN741-page23

NotesECN741-page23 - 33 and 35 Then it is immediate that...

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ECN 741: Public Economics Fall 2008 Now we can replace individual i ’s allocations in terms of aggregate allocations using ( 33 ) and ( 34 ) X t,s t β t ± U m C ( C ( s t ) ,L ( s t ); ϕ ) h i c ( C ( s t ) ,L ( s t ); ϕ ) = (36) + U m L ( C ( s t ) ,L ( s t ); ϕ ) h i y ( C ( s t ) ,L ( s t ); ϕ ) ² = U i c ( C ( s 0 ) ,L ( s 0 ); ϕ ) ± R 0 k i 0 - T ² i (37) Note that ( 36 ) is expressed entirely in terms of aggregate allocations, weights ϕ and initial endowments. Proposition 7 Given initial wealth R 0 k i 0 , an aggregate allocation { C ( s t ) ,L ( s t ) ,K ( s t ) } can be implemented in a competitive equilibrium if and only if 1. It is feasible 2. There exists weights ϕ and lump-sum T such that implementability constraint ( 36 ) holds for all i = 1 ,...,N Proof. Any equilibrium allocation is feasible and we just showed that it satisfy ( 36 ) . Suppose there is a feasible aggregate allocation that satisfies ( 36 ) for sum weights and lump-sum taxes. Then individual allocations and prices can be constructed using (
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Unformatted text preview: 33 ) and ( 35 ). Then it is immediate that ( 32 ) (consumer optimality) holds. The individual allocations constructed as such are also feasible since they satisfy ( 36 ) . A Panning Problem Suppose λ i is planer’s weight on type i . ∑ i π i λ i = 1 . Consider the following planning problem max X t,s t ,i λ i π i β t Pr ( s t ) U i ( h i ( C ( s t ) ,L ( s t ); ϕ )) sub. to X t,s t β t ± U m C ( C ( s t ) ,L ( s t ); ϕ ) h i c ( C ( s t ) ,L ( s t ); ϕ ) + U m L ( C ( s t ) ,L ( s t ); ϕ ) h i y ( C ( s t ) ,L ( s t ); ϕ ) ² = U i c ( C ( s ) ,L ( s ); ϕ ) ± R k i-T ² ∀ i ; μ i π i 23...
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This note was uploaded on 12/10/2011 for the course MAT 121 taught by Professor Wong during the Fall '10 term at SUNY Stony Brook.

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