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NotesECN741-page22

# NotesECN741-page22 - i U i y c i,y i and therefore in any...

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ECN 741: Public Economics Fall 2008 First observe that in any equilibrium U i y ( s t ) U i c ( s t ) = U j y ( s t ) U j c ( s t ) = - w ( s t )(1 - τ ( s t )) (32) U i c ( s t ) U i c ( s 0 ) = U j c ( s t ) U j c ( s 0 ) = p ( s t ) β t Pr ( s t ) p ( s 0 ) i, j Therefore, given the aggregate consumption and labor output ( C ( s t ) , L ( s t )) , the assignment of allocation of consumption and labor output { c i ( s t ) , y i ( s t ) } are efficient. In other words, given any sequence of aggregate output ( C ( s t ) , L ( s t )) , there are weights ϕ = ϕ 1 , . . . , ϕ N such that i π i ϕ i = 1 and { c i ( s t ) , y i ( s t ) } is the solution to U m ( C ( s t ) , L ( s t ); ϕ ) max { c i ,y i } X π i ϕ i U i ( c i , y i ) sub. to X i π i c i = C ( s t ) , X i π i y i = L ( s t ) Denote the solution by c i = h i c ( C, L ; ϕ ) , y i = h i y ( C, L ; ϕ ) (33) therefore ( c i ( s t ) , y i ( s t )) = h i ( C, L ; ϕ ) in which h i = ( h i c , h i y ) . Note also that U m C ( C ( s t ) , L ( s t ); ϕ ) = ϕ i U i c ( c i , y i ) (34) U m L ( C ( s t )
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Unformatted text preview: i U i y ( c i ,y i ) and therefore, in any equilibrium U m L ( s t ) U m C ( s t ) =-w ( s t )(1-τ ( s t )) (35) U m c ( s t ) U m c ( s ) = p ( s t ) β t Pr ( s t ) p ( s ) ∀ i,j Now let’s look at individual i ’s implementability constraint X t,s t β t ³ U i c ( c i ( s t ) ,y i ( s t )) c i ( s t ) + U i y ( c i ( s t ) ,y i ( s t )) y i ( s t ) ´ = U i c ( c i ( s ) ,y i ( s )) ³ R k i-T ´ 22...
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