Economics Dynamics Problems 210

Economics Dynamics Problems 210 - 194 Economic Dynamics...

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Unformatted text preview: 194 Economic Dynamics Figure 4.38. The resulting phase line is shown in figure 4.38. This goes beyond the possibilities of a spreadsheet, and figure 4.38 should be compared with the three twodimensional plots given in figure 4.35. It is worth noting that figure 4.38 is the default plot and the orientation can readily be changed by clicking on the figure and revolving. Appendix 4.1 Parametric plots in the phase plane: continuous variables A trajectory or orbit is the path of points {x(t), y(t)} in 2-dimensional space and {x(t), y(t), z(t)} in 3-dimensional space as t varies. Such plots are simply parametric plots as far as computer programmes are concerned. There are two methods for deriving the points (x(t), y(t)) or (x(t), y(t), z(t)): (1) (2) Solve for these values Derive numerical values by numerical means: (a) by solving numerically, or (b) deriving by recursion. Method 2(a) is used particularly in the case of differential equations, while method 2(b) is used for difference (or recursive) equations. In each of these cases initial conditions must be supplied. 4A.1 Two-variable case Consider the solution values for x and y in example 4.1, which are x(t) = 2e2t and y(t) = 3et Both x and y are expressed in terms of a common parameter, t, so that when t varies we can establish how x and y vary. More specifically, if t denotes time, then (x(t), y(t)) denotes a point at time t in the (x,y)-plane, i.e., a Cartesian representation of the parametric point at time t. If the differential equation system which generated x(t) and y(t) is autonomous, then there is only one solution curve, and we can express this in the form y = φ (x), where y0 = φ (x0 ) and (x0 , y0 ) is some initial point, i.e., x(0) = x0 and y(0) = y0 at t = 0. In the present example this is readily ...
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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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