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224
Economic Dynamics
considering the general solution
u
t
=
ar
t
v
r
+
bs
t
v
s
where
u
t
=
±
x
t
−
x
∗
y
t
−
y
∗
²
If

r

<
1 and

s

<
1 then
ar
t
v
r
→
0
and
bs
t
v
s
→
0
as
t
→∞
and so
u
t
→
0
and
consequently the system tends to the Fxed point, the equilibrium point.
Return to example 5.4 where
r
=
0
.
7262 and
s
=−
0
.
8262. The absolute value
of both roots is less than unity, and so the system is stable. We showed this in terms
of Fgure 5.1, where the system converges on the equilibrium, the Fxed point. We
pointed out above that the system can be represented in its canonical form, and
the same stability properties should be apparent. To show this our Frst task is to
compute the vector
z
0
. Since
z
0
=
V
−
1
u
0
, then
±
z
10
z
20
²
=
±
0
.
5793
5
.
7537
11
²
−
1
±
−
4
.
4
−
12
.
8
²
=
±
−
13
.
3827
0
.
5827
²
and
z
1
t
=
(0
.
7262)
t
(
−
13
.
3827)
z
2
t
=
(
−
0
.
8262)
t
(0
.
5827)
Setting this up on a spreadsheet, we derive Fgure 5.4. The canonical form has
transformed the system into the (
z
1
,
z
2
)plane, but once again it converges on the
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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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