Economics Dynamics Problems 248

Economics Dynamics - 232 Economic Dynamics Considering the canonical form zt = Vzt1 then zt = Jt z0 = i)t 0 0 z i t 0 To investigate the stability

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232 Economic Dynamics Considering the canonical form z t = Vz t 1 , then z t = J t z 0 = ± ( α + β i ) t 0 0( α β i ) t ² z 0 To investigate the stability properties of systems with complex conjugate roots, we employ two results (see Simon and Blume 1994, appendix A3): (i) α ± i β = R (cos θ ± i sin θ ) (ii) ( α ± i β ) n = R n [cos( n θ ) ± i sin( n θ )] (De Moivre’s formula) where R = ³ α 2 + β 2 and tan θ = β α From the canonical form, and using these two results, we have z 1 t = ( α + β i ) t z 10 = R t [cos( t θ ) + i sin( t θ )] z 2 t = ( α β i ) t z 20 = R t [cos( t θ ) i sin( t θ )] It follows that such a system must oscillate because as t increases, sin( t θ ) and cos( t θ ) range between + 1 and 1. Furthermore, the limit of z 1 t and z 2 t as t →∞ is governed by the term ´ ´ R t ´ ´ = | R | t .If | R | < 1, then the system is an asymptotically stable focus ;if | R | > 1, then the system is an unstable focus ; while if | R | = 1, then we have a centre . 6 We now illustrate each of these cases. Example 5.13 | R |
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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