Economics Dynamics Problems 118

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online