lecture8 - Lecture 8 Transition Dynamics and Public Policy...

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Unformatted text preview: Lecture 8 Transition Dynamics and Public Policy in the Cass-Koopmans- Ramsey Model ECON 7240 VIII - 2 Intensive Equations of Motion Instantaneous budget constraint: Euler equation: ) ( ) ( ) ( )) ( ( ) ( t k g n t c t k f dt t dk + +-- = g t k f dt t c d--- = )) ( ( ' ) ( ln ECON 7240 VIII - 3 Balanced Growth Path For any k (0), there is a unique c ( k (0)) such that the solution to the equations of motion satisfies Starting the system at k (0) and c (0) = c ( k (0)) , k ( t ) will converge to k * and c ( t ) to c * where g k f + + = *) ( ' * ) ( *) ( * k g n k f c + +- = ( 29 . *) ( exp ) ( lim = + +- t r g n t k t ECON 7240 VIII - 4 Difference Equations There is no analytic solution to this system of differential equations. We can solve them numerically by approximating them by difference equations. For small t , define k i = k ( i t ) and c i = c ( i t ). t k g n c k f k k i i i i i + +-- + = + ) ) ( ) ( ( 1 --- + = + t g k f c c i i i ) ( ' 1 1 ECON 7240 VIII - 5 Shooting Method Given k , choose a large T at which we believe k ( T ) k *. We have to find c = c ( k ). If c > c ( k ), household will not save enough and k T < k *. If c is too much larger than c ( k ), k i will be negative for some i and k T is undefined. If c < c ( k ), household will save too much and k T > k *. ECON 7240 VIII - 6 Dynamics of k ( = 1) 4 4.2 4.4 4.6 4.8 5 5.2 5.4 10 20 30 40 50 60 70 80 90 100 CKR Solow t k ECON 7240 VIII - 7 Dynamics of c ( = 1) 1.1 1.15 1.2 1.25 1.3 1.35 10 20 30 40 50 60 70 80 90 100 CKR Solow t c ECON 7240 VIII - 8 Balanced Growth Welfare Suppose the economy is at the steady state at T ....
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lecture8 - Lecture 8 Transition Dynamics and Public Policy...

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