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Economics Dynamics Problems 135

Economics Dynamics Problems 135 - ± 1 1 a ² t x − x ∗...

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Discrete dynamic systems 119 then y t + 1 = (1 + a ) y t by t y t + 1 Solving we obtain y t + 1 = (1 + a ) y t 1 + by t This can be transformed by dividing both sides by y t + 1 y t 1 y t = 1 y t + 1 1 + a 1 + by t i.e. 1 y t + 1 = 1 + by t (1 + a ) y t = 1 (1 + a ) 1 y t + b 1 + a Let x t = 1 / y t , then x t + 1 = 1 1 + a x t + b 1 + a In equilibrium x t + 1 = x t = . . . = x , hence x = 1 1 + a x + b 1 + a Solving for x we obtain the fixed point x = b a Subtracting the equilibrium equation from the recursive equation we obtain x t + 1 x = 1 1 + a ( x t x ) which has the general solution
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Unformatted text preview: ± 1 1 + a ² t ( x − x ∗ ) or x t = b a + (1 + a ) − t ± x − b a ² Substituting back x t = 1 / y t for all t 1 y t = b a + (1 + a ) − t ± 1 y − b a ² Hence, y t = 1 ± b a ² + (1 + a ) − t ± 1 y − b a ² or y t = ay by + (1 + a ) − t ( a − by ) (3.22)...
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