Chemistry Reference
In-Depth Information
Theorem 2.1.
Assume that
a
k
D
˛
0
k
.0/ > 0
for all
k
D
1;2;:::;N:
(i) The equilibrium is locally asymptotically stable if for all
k
,
a
k
.1
C
r
k
/<2
(2.21)
and
N
X
r
k
a
k
2
a
k
.1
C
r
k
/
>
1:
(2.22)
k
D
1
(ii) The equilibrium is unstable if for at least one
k
,
a
k
.1
C
r
k
/
2
or
X
N
r
k
a
k
2
a
k
.1
C
r
k
/
<
1:
kD1
Proof.
Notice that the structure of matrix
H
is the same as matrix
A
given in equa-
tion (E.4) of Appendix E. Therefore we can use relation (E.5) to determine that its
characteristic equation has the form
"
1
C
#
N
N
Y
X
r
k
a
k
1
a
k
.1
C
r
k
/
.1
a
k
.1
C
r
k
/
/
D
0:
(2.23)
k
D
1
k
D
1
In order to make the mathematical analysis easier assume that a
k
>0for all k
and the firms are numbered in such a way that the different a
k
.1
C
r
k
/ values sat-
isfy a
1
.1
C
r
1
/>a
2
.1
C
r
2
/>
>a
s
.1
C
r
s
/ and their values are repeated
m
1
;m
2
;:::;m
s
times. By adding the terms with identical denominators in the
bracketed expression and denoting by
j
the sum of the corresponding numerators
r
k
a
k
, we can rewrite (2.23) as
2
3
Y
s
X
s
j
1
a
j
.1
C
r
j
/
.1
a
j
.1
C
r
j
/
/
m
j
4
1
C
5
D
0;
(2.24)
j
D
1
j
D
1
where
j
0. So we conclude that if
j
D
0 or m
j
2,then1
a
j
.1
C
r
j
/ is an
eigenvalue of
H
. This eigenvalue is always less than 1, so it is inside the unit circle
if and only if a
j
.1
C
r
j
/<2. All other eigenvalues are the roots of the equation
s
X
j
1
a
j
.1
C
r
j
/
D
0;
g./
1
C
j D1
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