Economics Dynamics Problems 123

Economics Dynamics Problems 123 - b t Y Y t = b t ² Y −...

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Discrete dynamic systems 107 and so on. The general result emerging is Y t = a t 1 + ba t 2 + b 2 a t 3 + · · · + b t 1 a 0 + b t Y 0 or Y t = t 1 k = 0 b t 1 k a k + b t Y 0 Having derived the general result two cases are of interest. The first is where a k = a for all k ; the second is where a k = a and b = 1 for all k . Case (i) a k = a for all k In this case we have Y t = a + bY t 1 with the general result Y t = a t 1 k = 0 b t 1 k + b t Y 0 or Y t = a 1 b t 1 b + b t Y 0 It is useful for the economist to see this result from a different perspective. In equilibrium Y t = Y for all t . So Y = a + b Y Y = a 1 b Re-arranging the result for case (i), we have Y t = a 1 b ab t 1
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Unformatted text preview: + b t Y Y t = b t ² Y − a 1 − b ³ + a 1 − b Itisclearfromthisresultthatif | b | < 1thentheseriesconvergesontheequilibrium. If 0 < b < 1, there is steady convergence; while if − 1 < b < 0, the convergence oscillates. If | b | > 1, the system is unstable. Case (ii) a k = a and b = 1 for all k In this case Y t = a + Y t − 1 with result Y t = a t − 1 ± k = (1) t − 1 − k + Y i.e. Y t = at + Y...
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