Economics Dynamics Problems 243

Economics Dynamics Problems 243 - Discrete systems of...

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Discrete systems of equations 227 or 2 . 85078 v r 1 v r 2 = 0 v r 1 + 0 . 35078 v r 2 = 0 Let v r 1 = 1 then v r 2 =− 2 . 8508. The second eigenvector is found from ( A 0 . 5 I ) v s = 0 i.e. ± 0 . 35078 1 12 . 85078 ²± v s 1 v s 2 ² = ± 0 0 ² or 0 . 35078 v s 1 v s 2 = 0 v s 1 + 2 . 85078 v s 2 = 0 Let v s 2 = 1 then v s 1 2 . 8508. Hence v r = ± 1 2 . 8508 ² , v s = ± 2 . 8508 1 ² One eigenvector represents the stable arm while the other represents the unstable arm. But which, then, represents the stable arm? To establish this, convert the system to its canonical form, with z t + 1 = V 1 u t + 1 . Now take a point on the Frst eigenvector, i.e., point (1 , 2 . 8508), then z 0 = V 1 u 0 = ± 1 2 . 8508 2 . 8508 1 ² 1 ± 1 2 . 8508 ² = ± 1 0 ² Hence, z 1 t = 2 t (1) z 2 t = ( 0 . 5) t (0) = 0 Therefore z 1 t →+∞ as t →∞ . So the eigenvector v r must represent the unstable arm. Now take a point on the eigenvector v s , i.e., the point ( 2 . 8508 , 1), then z 0 = V 1 u 0 = ± 1 2 . 8508 2 . 8508 1 ² 1 ± 2 . 8508 1 ² = ± 0 1 ² Hence, z 1 t = 2 t (0) = 0 z 2 t = (
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