OLG_SS_SL - OLG E ciency and Social Security Prof Lutz...

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OLG: E¢ ciency and Social Security Prof. Lutz Hendricks August 5, 2009 L. Hendricks () E¢ ciency and Social Security August 5, 2009 1 / 26
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Issues The OLG model can have ine¢ cient equilibria . We solve the problem of a °cticious social planner This yields a Pareto optimal allocation by construction. We learn from this: 1 Solving the planning problem may be an easy way of characterizing CE (if it is optimal). 2 Comparing it with the CE points to sources of ine¢ ciency. L. Hendricks () E¢ ciency and Social Security August 5, 2009 2 / 26
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The Social Planner±s Problem L. Hendricks () E¢ ciency and Social Security August 5, 2009 3 / 26
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Planner±s problem Imagine an omnipotent social planner. She can assign actions to all agents (consumption, hours worked, ...). She maximizes some average of individual utilities. She only faces resource constraints . L. Hendricks () E¢ ciency and Social Security August 5, 2009 4 / 26
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Welfare function The planner±s objective function is assumed to be a weighted average of individual utilities: ω 0 β u ( c o 1 ) + t = 1 ω t [ u ( c y t ) + β u ( c o t + 1 )] Utility comparisons across persons don±t make sense, but it is often used. By varying the weights ( ω t ) we can obtain all Pareto optimal allocations. To ensure that the objective function is °nite, suitable conditions need to be imposed on the weights. L. Hendricks () E¢ ciency and Social Security August 5, 2009 5 / 26
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Planner±s problem The planner only faces feasibility constraints: ( 1 ° δ ) k t + f ( k t ) = c y t + c o t / ( 1 + n ) + ( 1 + n ) k t + 1 Lagrangian: Γ = ω 0 β u ( c o 1 ) + t = 1 ω t [ u ( c y t ) + β u ( c o t + 1 )] + t = 1 λ t ° ( 1 ° δ ) k t + f ( k t ) ° c y t ° c o t / ( 1 + n ) ° ( 1 + n ) k t + 1 ± L. Hendricks () E¢ ciency and Social Security August 5, 2009 6 / 26
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Planner±s problem FOCs: ω t u 0 ( c y t ) = λ t ω t ° 1 β u 0 ( c o t ) = λ t / ( 1 + n ) λ t [ 1 ° δ + f 0 ( k t )] = λ t ° 1 ( 1 + n ) L. Hendricks () E¢ ciency and Social Security August 5, 2009 7 / 26
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Planner±s problem Static optimality: ω t u 0 ( c y t ) = ω t ° 1 ( 1 + n ) β u 0 ( c o t ) Euler equation: ω t u 0 ( c y t )[ 1 ° δ + f 0 ( k t )] = ω t ° 1 u 0 ( c y t ° 1 )( 1 + n ) L. Hendricks () E¢ ciency and Social Security August 5, 2009 8 / 26
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