Economics Dynamics Problems 143

Economics Dynamics Problems 143 - x t on the vertical axis....

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Discrete dynamic systems 127 3.11 Linear approximation to discrete nonlinear difference equations In chapter 2, section 2.7, we considered linear approximations to nonlinear differ- ential equations. In this section we do the same for nonlinear difference equations. A typical nonlinear difference equation for a one-period lag is x t x t 1 = g ( x t 1 ) ± x t = g ( x t 1 ) However, it is useful to consider the problem in the recursive form x t = g ( x t 1 ) + x t 1 i.e. x t = f ( x t 1 ) because this allows a graphical representation. In this section we shall consider only autonomous nonlinear difference equations and so f ( x t 1 ) does not depend explicitly on time. We have already established that a Fxed point, an equilibrium point, exists if x = f ( x ) for all t and that we can represent this on a diagram with x t 1 on the horizontal axis and
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Unformatted text preview: x t on the vertical axis. A Fxed point occurs where f ( x t 1 ) cuts the 45 -line, as shown in Fgure 3.18, where we have three such Fxed points. Since f ( x ) = x 3 then y = f ( y ) and satisFes y = y 3 or y ( y 2 1) = 0. This results in three values for y , y = , 1 and 1. It is to be noted that we have drawn x t = f ( x t 1 ) as a continuous function, which we also assume to be differentiable. We have also established that x is an attractor, a stable point, if there exists a number such that when | x x | < then x t approaches x in the limit, otherwise it is unstable. In the present illustration we can consider only local stability or Figure 3.18....
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