Economics Dynamics Problems 245

# Economics Dynamics Problems 245 - Discrete systems of...

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Discrete systems of equations 229 showed that the canonical form of u t = Au t 1 is z t = J t z 0 In the case of a repeated root this is z t = λ t t λ t 1 0 λ t z 0 Hence z 1 t = λ t z 10 + t λ t 1 z 20 z 2 t = λ t z 20 Therefore if | λ | < 1, then λ t 0 as t → ∞ , consequently z 1 t 0 and z 2 t 0 as t → ∞ . The system is asymptotically stable. If, on the other hand, | λ | > 1, then λ t → ∞ as t → ∞ , and z 1 t → ±∞ and z 2 t → ±∞ as t → ∞ . The system is asymptotically unstable. We can conclude for repeated roots, therefore, that (a) if | λ | < 1 the system is asymptotically stable (b) if | λ | > 1 the system is asymptotically unstable. Example 5.11 Consider the following system x t + 1 = 4 + x t y t y t + 1 = − 20 + x t + 3 y t Then x = 12 and y = 4. Representing the system as deviations from equilibrium,
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