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68
Economic Dynamics
Figure 2.18.
Hence
f
(
k
)
=−
(
n
+
δ
)(1
−
α
)(
k
−
k
∗
)
Since 0
<α<
1 and
n
and
δ
are both positive, then this has a negative slope
about
k
∗
2
and hence
k
∗
2
is a locally stable equilibrium. The situation is shown in
fgure 2.18.
The frstorder linear approximation about the nonzero equilibrium is then
˙
k
=
f
(
k
)
=−
(
n
+
δ
)(1
−
α
)(
k
−
k
∗
)
with the linear approximate solution
k
(
t
)
=
k
∗
2
+
(
k
(0)
−
k
∗
2
)
e
−
(
n
+
δ
)(1
−
α
)
t
As
t
→∞
then
k
(
t
)
→
k
∗
2
.
What we are invoking here is the Following theorem attributed to Liapunov
THEOREM 2.1
If
˙
x
=
f
(
x
)
is a nonlinear equation with a linear approximation
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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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