Chemistry Reference
In-Depth Information
If g./ denotes again the left hand side of the above equation, then
lim
g./
D
1;
!˙1
(
1
if j
D
j
0
;
˙1
if j
¤
j
0
.
However similarly to the discrete time case, g
0
./ has no definite sign. The graph
of g./ is the same as shown earlier in Figs. 3.1-3.3 with the only difference being
that the poles are all negative and given by
a
1
.1
C
r
1
/;:::;
a
s
.1
C
r
s
/. Therefore
we have again three cases.
lim
0
g./
D
!
a
j
.1
C
r
j
/
˙
Case 1. If j
0
D
1, then there are s
2 real roots between each pair of poles
a
j
.1
C
r
j
/ and
a
j C1
.1
C
r
j C1
/ for j
D
2;:::;s
1. If the other two roots are real
and are between
a
1
.1
C
r
1
/ and
a
s
.1
C
r
s
/, then the equilibrium is locally
asymptotically stable.
Case 2. If j
0
D
s, then all roots are real and are negative if g.0/ > 0. This condition
can be rewritten as
X
N
r
k
1
C
r
k
<1:
kD1
Case 3. If 1<j
0
<s, then there are s
2 real roots, one before
a
1
.1
C
r
1
/,
and one in between each pair of poles
a
j
.1
C
r
j
/ and
a
j C1
.1
C
r
j C1
/ for
j
D
1;:::;j
0
2, j
0
C
1;:::;s
1. If we assume that the remaining two roots
are real and between
a
1
.1
C
r
1
/ and
a
s
.1
C
r
s
/, then all roots are negative.
The possibility of complex roots will be shown later in Example 3.6. If there are
complex roots, then no simple stability conditions can be given. We will next return
to the case of Example 3.3, but under the assumption of continuous time dynamics.
Example 3.5.
Consider again the N-person semi-symmetric oligopoly of Exam-
ple 3.3, now under the assumption of continuous time adjustment of the outputs of
the firms of the oligopoly. Assume again that c
2
D
:::
D
c
N
.ThenQ
1
D
.N
1/x
2
and Q
2
D
x
1
C
.N
2/x
2
by assuming that firms 2;:::;N select identical lin-
ear adjustment function and initial outputs. From Example 3.3 we know that at the
interior equilibrium
.N
1/A
c
1
C
.N
1/c
2
;
r
1
D
R
0
1
. Q
1
/
D
Q
D
.N
1/c
2
C
.3
2N/c
1
2.N
1/c
1
;
c
1
.N
1/c
2
2.N
1/c
2
r
2
D
R
0
2
. Q
2
/
D
:
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