NotesECN741-page28

NotesECN741-page28 - ) U c t, + λ t ( U c t, + U cc t, c...

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ECN 741: Public Economics Fall 2008 Exercise: Show that a feasible allocation is implementable if and only if it satisfy ( 38 ). Ramsey problem Ramsey problem is the following max X t =0 γ t [ U ( c t, 0 ,l t, 0 ) + βU ( c t, 1 ,l t, 1 )] subject to U c t, 0 c t, 0 + U l t, 0 l t, 0 + β ( U c t, 1 c t, 1 + U l t, 1 l t, 1 ) = 0 ; γ t λ t c t + k t +1 = f ( k t ,l t ) + (1 - δ ) k t ; γ t φ t c t = c t, 0 + c t - 1 , 1 l t = l t, 0 + l t - 1 , 1 k t = a t - 1 , 1 First order conditions are γ t U c t, 0 + γ t λ t ( U c t, 0 + U cc t, 0 c t, 0 + U lc t, 0 l t, 0 ) = γ t φ t (39) γ t βU c t, 1 + γ t βλ t ( U c t, 1 + U cc t, 1 c t, 1 + U lc t, 1 l t, 1 ) = γ t +1 φ t +1 γ t U l t, 0 + γ t λ t ( U l t, 0 + U ll t, 0 l t, 0 + U lc t, 0 c t, 0 ) = γ t φ t f lt (40) γ t βU l t, 0 + γ t βλ t ( U l t, 1 + U ll t, 1 l t, 1 + U cl t, 1 c t, 1 ) = γ t +1 φ t +1 f lt +1 γ t φ t = γ t +1 φ t +1 (1 - δ + f kt +1 ) (41) Combine ( 39 ) and ( 41
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Unformatted text preview: ) U c t, + λ t ( U c t, + U cc t, c t, + U lc t, l t, ) U c t, 1 + λ t ( U c t, 1 + U cc t, 1 c t, 1 + U lc t, 1 l t, 1 ) = β (1-δ + f kt +1 ) (42) Steady State: In the steady state ( c t, ,c t, 1 ,l t, ,l t, 1 ,a t, 1 ) = ( c ,c 1 ,l ,l 1 ,a 1 ) and λ t = λ . Therefore, 28...
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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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