Unformatted text preview: 102 Economic Dynamics
yt = (3.11) c
+ at y0 −
1−a which clearly satisﬁes the initial condition.
Consider, for example, the cobweb model we developed earlier in the chapter,
equation (3.5), with the resulting recursive equation
pt = a−c
b and with equilibrium
p∗ = a−c
b+d Taking deviations from the equilibrium, we have
pt − p∗ = − ( pt−1 − p∗ )
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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