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Unformatted text preview: 92 Economic Dynamics
Given this system, the ﬁrst few terms in the sequence are readily found to be:
yt+1 = −yt + k
yt+2 = −yt+1 + k = −(−yt + k) + k = yt
yt+3 = −yt+2 + k = −yt + k
yt+4 = −yt+3 + k = −(−yt + k) + k = yt
It is apparent that this is a repeating pattern. If y0 denotes the initial value, then we
y0 = y2 = y4 = . . . and y1 = y3 = y5 = . . . We have here an example of a two-cycle system that oscillates between −y0 + k
and y0 . There is still a ﬁxed point to the system, namely
y∗ = −y∗ + k
but it is neither an attractor nor a repellor. The situation is illustrated in ﬁgure 3.4,
where again the line L denotes the difference equation and the line E gives the
equilibrium condition. The two-cycle situation is readily revealed by the fact that
the system cycles around a rectangle.
Return to the linear cobweb model given above, equation (3.5). Suppose the
slope of the (linear) demand curve is the same as the slope of the (linear) supply Figure 3.4. ...
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- Fall '11