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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 = 1 1 4 1 x y with A = 1 1 4 1 , det( A ) = − 3 , A λ I = 1 λ 1 4 1 λ 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 = 2 1 4 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 and v r = 1 2 For λ = s = − 1, then ( A λ I ) v s = 2 1 4 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
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