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Unformatted text preview: 210 Economic Dynamics
and we can ﬁnd a and b for t = 0 from
u0 = avr + bvs
where u0 is known. This can either be done by direct substitution, or using
the fact that
avr + bvs = vr
i.e.
or
(4) V vs a
= u0
b a
= u0
b a
= V−1 u0
b Write the solution
ut = art vr + bst vs But we can do this whole process in one step. First we note
ut = art vr + bst vs = vr
= VDt vs rt
0 0
st a
b a
b But
a
= V−1 u0
b
Hence
ut = VDt V−1 u0
which is the result we proved above. The gain, if there is one, in doing the four steps
is the need to solve for a and b. Since this can often be done by direct substitution,
then the four steps involve no inverse matrix computation.
Example 5.4
Let
xt+1 = −8 − xt + yt
yt+1 = 4 − 0.3xt + 0.9yt
setting xt+1 = xt = x∗ and yt+1 = yt = y∗ for all t, the ﬁxed point is readily shown
to be (x∗ , y∗ ) = (6.4, 20.8).
Now consider the system in terms of deviations from equilibrium, then
xt+1 − x∗ = −(xt − x∗ ) + ( yt − y∗ )
yt+1 − y∗ = −0.3(xt − x∗ ) + 0.9( yt − y∗ )
or
ut = Aut−1 ...
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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.
 Fall '11
 Dr.Gwartney
 Economics

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