Economics Dynamics Problems 143

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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 ε such that when | x − x ∗ | < ε 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....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online