Economics Dynamics Problems 118

# Economics Dynamics Problems 118 - 102 Economic Dynamics or...

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Unformatted text preview: 102 Economic Dynamics or yt = (3.11) c c + at y0 − 1−a 1−a which clearly satisﬁes the initial condition. Example 3.5 Consider, for example, the cobweb model we developed earlier in the chapter, equation (3.5), with the resulting recursive equation pt = a−c − b d pt−1 b and with equilibrium p∗ = a−c b+d Taking deviations from the equilibrium, we have d pt − p∗ = − ( pt−1 − p∗ ) b which is a ﬁrst-order linear homogeneous difference equation, with solution pt − p∗ = − d b t ( p0 − p∗ ) or (3.12) pt = a−c b+d +− d b t p0 − a−c b+d With the usual shaped demand and supply curves, i.e., b > 0 and d > 0, then d/b > 0, hence (−d/b)t will alternate in sign, being positive for even numbers of t and negative for odd numbers of t. Furthermore, if 0 < |−d/b| < 1 then the series will become damped, and in the limit tend towards the equilibrium price. On the other hand, if |−d/b| > 1 then the system will diverge from the equilibrium price. These results are veriﬁed by means of a simple numerical example and solved by means of a spreadsheet, as shown in ﬁgure 3.11. The examples we have just discussed can be considered as special cases of the following recursive equation: (3.13) yn+1 = an yn y0 at n = 0 The solution to this more general case can be derived as follows: y1 = a0 y0 y2 = a1 y1 = a1 a0 y0 y3 = a2 y2 = a2 a1 a0 y0 . . . yn = an−1 an−2 . . . a1 a0 y0 ...
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