Chemistry Reference
In-Depth Information
equation of the Jacobian has the form
r
kx
u
k
C
r
kQ
a
k
X
l
u
l
C
r
kQ
.1
a
k
/
v
k
D
u
k
;
.k
D
1;2;:::;N/; (4.73)
¤
k
a
k
X
l¤k
u
l
C
.1
a
k
/
v
k
D
v
k
:
(4.74)
Subtract the r
kQ
-multiple of the second equation from the first one to obtain
r
kx
u
k
D
.
u
k
r
kQ
v
k
/;
implying that
r
kx
D
r
kQ
v
k
u
k
;
where it is assumed that
¤
0. Note that a zero eigenvalue cannot destroy asymp-
totic stability. Substituting this relation into (4.73) it is found that
r
kx
C
u
k
r
kQ
a
k
X
l
.1
a
k
/.
r
kx
/
u
l
C
D
0;
(4.75)
¤
k
for all k. A non-trivial solution exists if and only if the determinant of this system is
zero. Notice that by introducing the notation
.1
a
k
/.
r
kx
/
A
k
./
D
r
kx
C
; B
k
./
D
a
k
r
kQ
;
this determinant has the same structure as the one given by (E.2) in Appendix E.
Therefore by using (E.3), the resulting determinantal equation becomes
r
kx
C
a
k
r
kQ
N
Y
.1
a
k
/.
r
kx
/
k
D
1
2
3
N
X
a
k
r
kQ
4
5
D
0:
1
C
(4.76)
.1
a
k
/.
r
kx
/
r
kx
C
a
k
r
kQ
k
D
1
It is very complicated in general to find conditions that guarantee that the roots of
(4.76) lie inside the unit circle, so instead of a general analysis we will here consider
a particular example.
Example 4.12.
Consider the case of symmetric firms, when a
1
D
:::
D
a
N
D
a;r
1Q
D
:::
D
r
NQ
D
r
Q
;r
1x
D
:::
D
r
Nx
D
r
x
. The eigenvalues in this case are
the roots of the equations
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