OLG_SL - Overlapping Generations Prof Lutz Hendricks August...

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Unformatted text preview: Overlapping Generations Prof. Lutz Hendricks August 5, 2009 L. Hendricks () Overlapping Generations August 5, 2009 1 / 45 Overlapping Generations What do we want in a macro model? Capital accumulation, growth, ... things the Solow model has But we also want microfoundations. The simplest type of model assumes that households live forever. ... but we need tools for that Now we study models in which households live for 2 periods . L. Hendricks () Overlapping Generations August 5, 2009 2 / 45 What we do in this section How to set up and solve an OLG model Show that the world is not e¢ cient : households may save too much. &Social security¡can prevent overaccumulation We can make households "in¢nitely lived" by adding altruistic bequests . L. Hendricks () Overlapping Generations August 5, 2009 3 / 45 What we don&t do in this section We sidestep some technical issues: why is there a representative household? why is there a representative ¡rm? We come back to those later. L. Hendricks () Overlapping Generations August 5, 2009 4 / 45 An OLG Model Without Firms L. Hendricks () Overlapping Generations August 5, 2009 5 / 45 An OLG Model Without Firms Demographics At each date a cohort of size N t = N ( 1 + n ) t is born. Each person lives for two periods. Therefore, at each date there are N t young and N t & 1 old households. An important missing market: there cannot be any intergenerational borrowing and lending . For example, the young at t cannot borrow from the old because the old won&t be around at t+1 to have their loans repaid. If households live for more periods, the problem becomes weaker, but does not go away. L. Hendricks () Overlapping Generations August 5, 2009 6 / 45 An OLG Model Without Firms Endowments Young households receive endowments w t . Technology: Endowments can be stored. Storing s t today yields f ( s t ) tomorrow. Markets: Goods are traded in spot markets. Households can issue one period bonds with interest rate r t + 1 . L. Hendricks () Overlapping Generations August 5, 2009 7 / 45 Household Problem The utility function is u ( c y t ) + β u ( c o t + 1 ) . The budget constraints are w t = c y t + s t + 1 + b t + 1 c o t + 1 = f ( s t + 1 ) + b t + 1 ( 1 + r t + 1 ) Lifetime budget constraint: w t & c y t & s t + 1 = [ c o t + 1 & f ( s t + 1 )] / [ 1 + r t + 1 ] L. Hendricks () Overlapping Generations August 5, 2009 8 / 45 Household Problem Lagrangian Γ = u ( c y t ) + β u ( c o t + 1 ) + λ f [ w t & c y t & s t + 1 ] & [ c o t + 1 & f ( s t + 1 )] / [ 1 + r t + 1 ] g FOCs: u ( c y t ) = λ β u ( c o t + 1 ) = λ / ( 1 + r t + 1 ) f ( s t + 1 ) = 1 + r t + 1 L. Hendricks () Overlapping Generations August 5, 2009 9 / 45 Household Problem Euler equation β f ( s t + 1 ) u ( c o t + 1 ) = u ( c y t ) Bond holdings follow residually from the period 1 budget constraint....
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OLG_SL - Overlapping Generations Prof Lutz Hendricks August...

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