Economics Dynamics Problems 218

Economics Dynamics Problems 218 - 202 Economic Dynamics...

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202 Economic Dynamics (ii) ± x t y t ² = ± 40 3 2 ²± x t 1 y t 1 ² + ± 0 3 ² (or u t = Au t 1 + b ) (iii) x t y t z t = 23 4 0 . 500 00 . 70 x t 1 y t 1 z t 1 (or u t = t 1 ) In general, therefore, we can write Frst-order linear homogeneous systems as u t = t 1 (5.1) and a Frst-order linear nonhomogeneous system as u t = t 1 + b (5.2) where u is a n × 1 vector, A a n × n square matrix and b a n × 1 vector. Consider the system ± x t y t ² = ± ab cd x t 1 y t 1 ² Then in equilibrium x t = x t 1 = x for all t and y t = y t 1 = y for all t . Hence ± x y ² = ± x y ² or u = An equilibrium solution exists, therefore, if u = 0 i.e. ( I A ) u = 0 or u = ( I A ) 1 0 = 0 An equilibrium for a Frst-order linear homogeneous system is, therefore,
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