Economics Dynamics Problems 186

Economics Dynamics Problems 186 - 170 Economic Dynamics the...

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170 Economic Dynamics the saddle point, which are derived from the eigenvectors associated with the characteristic roots. Because of the importance of saddle points in economics, we shall consider two examples here. Example 4.13 Let ˙ x = x + y ˙ y = 4 x + y then ± ˙ x ˙ y ² = ± 11 41 ²± x y ² with A = ± ² , det( A ) =− 3 , A λ I = ± 1 λ 1 λ ² giving det( A λ I ) = λ 2 2 λ 3 = ( λ 3)( λ + 1) = 0. Hence, λ = r = 3 and λ = s 1. For λ = r = 3 then ( A λ I ) v r = ± 21 4 2 ² v r = 0 i.e. 2 v r 1 + v r 2 = 0 4 v r 1 2 v r 2 = 0 Let v r 1 = 1, then v r 2 = 2. Hence, one solution is u 1 = e rt ± 1 2 ² and v r = ± 1 2 ² For λ = s 1, then ( A λ I ) v s = ± 42 ² v s = 0 i.e. 2 v s 1 + v s 2 = 0 4 v s 1 + 2 v s 2 = 0 Let v s 1 = 1, then v s 2 2. Hence, a second solution is
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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