Chemistry Reference
In-Depth Information
The main result of this section is the following, in which we use the notation of
Sect. 2.2.
Theorem 2.2.
Assume that
a
k
>0
for all
k
D
1;2;:::;N
. Then the equilib-
rium with respect to the continuous adjustment process (1.31) is always locally
asymptotically stable.
Proof.
Using linearization, the eigenvalues of the Jacobian have to be examined.
Similarly to the discrete case, the Jacobian of system (1.31) has the special structure
0
@
1
A
a
1
a
1
r
1
a
1
r
1
a
2
r
2
a
2
a
2
r
2
;
(2.46)
:
:
:
:
:
:
:
:
:
a
N
r
N
a
N
r
N
a
N
which is a special case of the form (E.4) (studied in Appendix E). The characteristic
equation of this matrix can also be given as a special case of equation (E.5), namely
"
1
C
#
Y
N
X
N
a
k
r
k
a
k
.1
C
r
k
/
.
a
k
.1
C
r
k
/
/
D
0:
(2.47)
k
D
1
k
D
1
We will now proceed similarly to the discrete case examined earlier. Assume again
that a
k
>0 for all k and the firms are numbered in such a way that the different
a
k
.1
C
r
k
/ values are
a
1
.1
C
r
1
/>a
2
.1
C
r
2
/>
>a
s
.1
C
r
s
/
and these values are repeated m
1
;m
2
;:::;m
s
times, respectively, among the N
firms. By adding the terms with identical denominators in the bracketed expression
and denoting by
j
the sum of the corresponding numerators a
k
r
k
, we can rewrite
(2.47) as
2
3
Y
s
X
s
a
j
.1
C
r
j
/
m
j
j
a
j
.1
C
r
j
/
C
4
1
5
D
0;
(2.48)
j
D
1
j
D
1
with
j
0.1
j
s/. So we can reach the following conclusion. If
j
D
0 or
m
j
2,then
a
j
.1
C
r
j
/ is an eigenvalue, and this value is always negative. All
other eigenvalues are the roots of the equation
s
X
j
a
j
.1
C
r
j
/
C
D
0;
1
(2.49)
j D1
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