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# lecture10 - Lecture 10 Dynamic Inefficiency in the...

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Lecture 10 Dynamic Inefficiency in the Overlapping Generations Model

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ECON 7240 X - 2 Capital Dynamics The fraction of wage income that is saved is Capital difference equation: ) 1 )( 1 ( ) ( ) ( 1 1 g n k w k s k t t t + + = + + θ θ ρ δ / 1 1 1 1 1 ) ) ( ' 1 ( 1 1 ) ( - - + + + - + + = t t k f k s
ECON 7240 X - 3 Discrete Solow Model The analogous capital difference equation for the Solow model is Given our assumptions about f , there is a unique steady state with positive capital per effective labor k *. ) 1 )( 1 ( ) 1 ( ) ( 1 g n k k sf k t t t + + - + = + δ *) ( * ) ( k sf k ng g n = + + + δ

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ECON 7240 X - 4 Steady State in the OLG Model In the OLG model, k t +1 is a function of wages at t rather than output at t . Given the assumptions we have made about f , w ( k ) need not be a concave function that satisfies Inada conditions. Weird things can happen: Multiple steady states A positive steady state may not exist? k t +1 may be indeterminate?
ECON 7240 X - 5 Slope of Capital Map The derivative of the capital map is where η s ( k ) is the elasticity of s ( k ) at k . The sign of η s ( k ) is ambiguous. The capital map could decrease. With CRRA utility, η s ( k ) > 0 if θ 1. With Cobb-Douglas production the capital map is always strictly increasing since η s ( k ) < 1.

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lecture10 - Lecture 10 Dynamic Inefficiency in the...

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