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Chapter4_SocialPlanner

Chapter4_SocialPlanner - Social Planner Juan Rubio-Ramrez...

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Social Planner Juan Rubio-Ram°rez Duke University and Federal Reserve Bank of Atlanta January 25, 2009 Juan Rubio-Ram°rez (DUKE) Social Planner January 25, 2009 1 / 21

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Motivation How do we show that an equilibrium exist? How do we characterize an equilibrium? How do we evaluate its e¢ ciency? An allocation is Pareto Optimal if there is no way to rearrange production or reallocate goods so that someone is made better o/ without making someone else worse o/. Pareto Optimality 6 = perfect state of the world or any concept like that. Juan Rubio-Ram°rez (DUKE) Social Planner January 25, 2009 2 / 21
E¢ ciency and the Social Planner Let us imagine we have a powerful dictator, the Social Planner, that can decide how much the households consume and work and how much the ±rms produce. The Social Planner is benevolent. It searches for the best possible allocation. The Social Planner does not follow prices. But it understands opportunity cost. The only constraints the social planner faces are the physical resource constraints of the economy. By construction, the solution of the Social Planner²s problem will be Pareto optimal. Juan Rubio-Ram°rez (DUKE) Social Planner January 25, 2009 3 / 21

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Social Planner Problem Problem: max f c t , k t + 1 , n t g T t = 0 T t = 0 β t u ( c t ) subject to c t + k t + 1 = k α t ( A t l t ) 1 ° α + ( 1 ° δ ) k t c t ± 0 , 0 ² l t ² 1 , k 0 > 0 given Using the fact that l t = 1 and, to save on notation assume that A t = 1 : max f c t , k t + 1 g T t = 0 T t = 0 β t u ( c t ) subject to c t + k t + 1 ° ( 1 ° δ ) k t = k α t + ( 1 ° δ ) k t c t ± 0 and k 0 > 0 given Juan Rubio-Ram°rez (DUKE) Social Planner January 25, 2009 4 / 21
Lagrangian and FOCs Lagrangian L = T t = 0 β t u ( c t ) + T t = 0 λ t [ k α t ° c t ° k t + 1 + ( 1 ° δ ) k t ] FOCs: L c t = β t u 0 ( c t ) ° λ t = 0 L c t + 1 = β t + 1 u 0 ( c t + 1 ) ° λ t + 1 = 0 L k t + 1 = ° λ t + λ t + 1 ° α k α ° 1 t + 1 + ( 1 ° δ ) ± = 0 Juan Rubio-Ram°rez (DUKE) Social Planner January 25, 2009 5 / 21

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Characterization of Solution I Rewriting these conditions yields β t u 0 ( c t ) = λ t β t + 1 u 0 ( c t + 1 ) = λ t + 1 λ t + 1 ° α k α ° 1 t + 1 + ( 1 ° δ ) ± = λ t and thus u 0 ( c t ) =
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Chapter4_SocialPlanner - Social Planner Juan Rubio-Ramrez...

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