204A-slides04c

204A-slides04c - (4c)-P.1 Overlapping Generations &...

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(4c)-P.1 Overlapping Generations & Dynamic Inefficiency Motivating example : Why inefficiency may be empirically relevant and deserves analysis. • Assume log-utility, Cobb-Douglas production, depreciation δ >0 r t + 1 = f ' ( k t + 1 ) = α k t + 1 1 δ • Steady state: k * = [ β /(1 + ) (1 + n ) + g ) )] => r * = [ /(1 + ) + n ) + g ) )] 1 = + n ) + g ) 1 1 + • Suppose most capital depreciates ( δ≈ 1 ), time preference near zero ( β≈ 1 ): => 1 + r * = + n ) + g ) 2 1 => 1 + r * < + n ) + g ) < • Illustration in capital market diagram: Is the intersection between f 1 ( k ) and a ( w ( k ), r ) above or below + n ) + g ) ? Evidence • Long-run U.S. Data: GDP-growth ~ 3%/p.a (n~1%/p.a., g~2%/p.a.) Real return on T-bills r~1%; return on capital r~(5%-8%). Which return is relevant? • Abel-Mankiw-Summers-Zeckhauser: Dynamic efficiency applies if (Capital income) - (Gross investment) > 0 Evidence: difference is positive.
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(4c)-P.2 Why does Pareto-efficiency impose a lower bound on r? • Pareto efficiency : Maximize utility of some cohort s.t. constraint that other cohorts’ utility is not reduced => Lagrangian with cohort utility + sum of other cohort utilities * multipliers => Equivalent to maximizing a weighted average of utilities • Social planning problem : Maximize welfare function s.t. resource constraint. Suppose: - Period t=1 old are endowed with K 1 . - Period t=T young all die at the end of period T. (Finite endpoint to start.) - Planner discounts future generations’ per-capita utility with factor γ (unit weight on t=0) - Abstract from productivity growth (to avoid clutter). • Maximize welfare function: W = L 0 β u ( C 21 ) + γ t L t t = 1 T 1 [ u ( C 1 t ) + u ( C 2 t + 1 )] + T L T u ( C 1 T ) , s.t. L t C 1 t + L t 1 C 2 t + K t + 1 = F ( K t , L t ) + (1 δ ) K t , or equivalently C 1 t + 1 1 + n C 2 t + + n ) k t + 1 = f ( k t ) + k t • Consider Lagrangian with multipliers λ t and take FOC: a. consumption of workers: t L t u '( C 1 t ) L t λ t = 0 => t = t u '( C 1 t ) b. consumption of retirees: t 1 L t 1 u '( C 2 t ) L t 1 t = 0 => t = t 1 u '( C 2 t ) c. capital accumulation: t ( F K ( K t , L t ) + 1 ) t 1 = 0 , where F K ( K t , L t ) + 1 = 1 + f '( k t ) = 1 + r t
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(4c)-P.3 • Combined the optimality conditions: 1. Period t workers &retirees
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204A-slides04c - (4c)-P.1 Overlapping Generations &amp;...

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