Economics Dynamics Problems 141

Economics Dynamics Problems 141 - A similar result can be...

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Discrete dynamic systems 125 With only one real root, r , the same conditions hold. Hence, in the case of real roots with 0 < b < 1, the path of income is damped for 0 < v < 1 and explosive for v > 1. If the solution is complex conjugate then r = α + β i and s = α β i and the general solution Y t = c 1 R t cos( t θ ) + c 2 R t sin( θ t ) + Y exhibits oscillations, whose damped or explosive nature depends on the ampli- tude, R . From our earlier analysis we know R = ± α 2 + β 2 . But α = b + v 2 and β = + ± 4 v ( b + v ) 2 2 Hence R = ² ³ b + v 2 ´ 2 + 4 v ( b + v ) 2 4 = v For damped oscillations, R < 1, i.e., v < 1; while for explosive oscillations, R > 1, i.e., v > 1. All cases are drawn in ±gure 3.17. The dividing line between real and complex roots is the curve ( b + v ) 2 = 4 v , which was drawn using Mathematica ’s Implicit- Plot command and annotated in CorelDraw
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Unformatted text preview: . A similar result can be derived using Maple . The instructions for each software are: Mathematica <<Graphics`ImplicitPlot` ImplicitPlot [(b+v)^2==4v, {v,0,5}, {b,0,1}] Maple with(plots): implicitplot ((b+v)^2=4v, v=0. .5, b=0. .1); On the other hand, the division between damped and explosive paths (given 0 < b < 1) is determined by v < 1 and v > 1, respectively. The accelerator model just outlined was utilised by Hicks (1950) in his dis-cussion of the trade cycle. The major change was introducing an autonomous component to investment, I , which grows exogenously at a rate g . So at time t , Figure 3.17....
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