Chemistry Reference
In-Depth Information
and
X
N
r
k
a
k
h
k
2
a
k
.1
C
r
k
/
>
1:
(5.12)
kD1
(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
h
k
2
a
k
.1
C
r
k
/
<
1:
k
D
1
Proof.
The structure of matrix
H
is the same as (E.4) shown in Appendix E. There-
fore the eigenvalue equation of
H
can be written as (E.5), which here has the special
form
"
1
C
#
Y
N
X
N
r
k
a
k
h
k
1
a
k
.1
C
r
k
/
.1
a
k
.1
C
r
k
/
/
D
0:
(5.13)
k
D
1
k
D
1
If we assume that both f
0
and f
k
are negative, then H
k
is positive, so h
k
>0.
Conditions (B) and (C) imply that the believed price functions satisfy the con-
ditions of concave oligopolies stated at the beginning of Sect. 2.1. Under these
conditions
1<r
k
0
for all k, which can be proved similarly to (2.7) when f
0
and f
00
are replaced by
f
k
and f
0
k
. Assume that 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 they 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 corre-
sponding numerators r
k
a
k
h
k
, we can rewrite (5.13) as (2.24). Therefore the proof
of Theorem 2.1 can be applied to show the assertion.
In the full information case f
k
f for all k,soH
k
is the identity function,
and h
k
D
1 for all k. In this special case, the matrix (5.10) reduces to (2.20), and
Theorem 5.1 specializes to Theorem 2.1.
Example 5.2.
Assume as in Example 5.1 that the price function is isoelastic. Let
f.Q/
D
A=Q be the true price function and assume that firm k believes that the
price function is f
k
.Q/
D
A
k
=Q,whereA
k
>0is a constant. The believed best
response function of firm k has the same derivative as given by (3.3) with the only
difference being that A is replaced by A
k
, so it has the same properties as in the
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