Economics Dynamics Problems 115

Economics Dynamics Problems 115 - Discrete dynamic systems...

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Unformatted text preview: Discrete dynamic systems 99 The fixed points can be found from a = 3.2a − 0.8a2 0.8a2 − 2.2a = 0 a(0.8a − 2.2) = 0 a=0 a = 2.75 and To establish stability let y = f (x) = 3.2x − 0.8x2 then f (x) = 3.2 − 1.6x f (0) = 3.2 and f (2.75) = −1.2 Since f (0) > 1 then a = 0 is unstable Since f (2.75) > 1 then a = 2.75 is unstable. Although a = 2.75 is unstable, knowledge about f (x) does not give sufficient information to determine what is happening to the sequence {yn } around the point a = 2.75. 3.5 Solving first-order difference equations For some relatively simple difference equations it is possible to find analytical solutions. The simplest difference equation is a first-order linear homogeneous equation of the form yt+1 = ayt (3.8) If we consider the recursive nature of this system, beginning with the initial value y0 , we have y1 = ay0 y2 = ay1 = a(ay0 ) = a2 y0 y3 = ay2 = a(a2 y0 ) = a3 y0 . . . n yn = a y0 The analytical solution is, therefore, yn = an y0 satisfying the initial value y0 . The properties of this system depend only on the value and sign of the parameter a. There is only one fixed point to such a system, y∗ = 0. For positive y0 , if a exceeds unity, then the series gets larger and larger over time, tending to infinity in the limit. If 0 < a < 1, then the series gets smaller (3.9) ...
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