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Economics Dynamics Problems 244

Economics Dynamics Problems 244 - Fxed point and these are...

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228 Economic Dynamics Figure 5.6. canonical representation. Similarly, point ( 2 . 8508 , 1) on the eigenvector associ- ated with s = − 0 . 5 became point (0 , 1) in its canonical representation. In other words, the arms of the saddle point equilibrium became transformed into the two rectangular axes in z -space. This is a standard result for systems involving saddle point solutions. So long as we have distinct characteristic roots these results hold. Here, however, we shall confine ourselves to the two-variable case. To summarise, if r and s are the characteristic roots of the matrix A for the system u t = Au t 1 and derived from solving | A λ I | = 0, then (i) if | r | < 1 and | s | < 1 the system is dynamically stable (ii) if | r | > 1 and | s | > 1 the system is dynamically unstable (iii) if, say, | r | > 1 and | s | < 1 the system is dynamically unstable. In the case of (iii) the system will generally be dominated by the largest root and will tend to plus or minus infinity depending on its sign. But given the fixed point is a saddle path solution, there are some initial points that will converge on the
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Unformatted text preview: Fxed point, and these are values that lie on the stable arm of the saddle point. As we shall see in part II, such possible solution paths are important in ratio-nal expectations theory. Under such assumed expectations behaviour, the system ‘jumps’ from its initial point to the stable arm and then traverses a path down the stable arm to equilibrium. Of course, if this initial ‘jump’ did not occur, then the trajectorywouldtendtoplusorminusinFnityandbedrivenawayfromequilibrium. 5.6.2 Repeating roots When there is a repeating root, λ , the system’s dynamics is dominated by the sign/value of this root. If | λ | < 1, then the system will converge on the equilibrium value: it is asymptotically stable. If | λ | > 1 then the system is asymptotically unstable. We can verify this by considering the canonical form. We have already...
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