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handoutramseysp08 - The Optimal Mix of Distortionary...

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The Optimal Mix of Distortionary Taxation with Commitment Draws on Atkeson, Chari, and Kehoe (1999, FRBMinn QR). See also p.478— 88 of L-S. Question: Should the government tax capital? 1 First Best Planner solves max { c t ,n t ,k t +1 } t =0 X t =0 β t U ( c t , n t ) subject to c t + g + k t +1 = F ( k t , n t ) + (1 δ ) k t (1) where U c > 0 , U n < 0 , and F is CRS. First order conditions n t : U n,t = U c,t F n,t (2) k t +1 : U c,t = βU c,t +1 [ F k,t +1 + 1 δ ] (3) First best has no distortion to the MB of working F n,t or saving F k,t +1 + 1 δ . 2 Statement of the Competitive Equilibrium Prob- lem Govt expenditure g is fi nanced through proportional taxes on income from capital (rate θ t ) and labor (rate τ t ). The government’s intertemporal budget constraint is X t =0 p t g = X t =0 p t ( τ t w t n t + θ t ( r t δ ) k t ) (4) 1
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HH problem max { c t ,n t ,k t +1 } t =0 X t =0 β t U ( c t , n t ) subject to X t =0 p t ( c t + k t +1 ) = X t =0 p t ((1 τ t ) w t n t + R kt k t ) (5) where R kt = 1 + (1 θ t )( r t δ ) and p 0 = 1 . The fi rst order conditions for a HH are c t : β t U c,t = λp t (6) n t : β t U n,t = λp t (1 τ t ) w t (7) k t +1 : p t = R k,t +1 p t +1 (8) where λ is the multiplier on (5). The fi rm’s problem is max k t ,n t F ( k t , n t ) r t k t w t n t The fi rst order conditions for the fi rm are: r t = F k,t (9) w t = F n,t (10) From these sets of conditions it is possible to see “tax wedges”. In partic- ular, (6), (7), (8), (9), (10) imply that for the houshold: U n,t U c,t = (1 τ t ) F n,t and U c,t = βU c,t +1 [1 + (1 θ t +1 )( F k,t +1 δ )] These two conditions di
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