Chemistry Reference
In-Depth Information
x
2
f
00
Q
2
f
0
. Q/
e
2
D
3e
2
p
e
1
3e
2
p
e
1
C
e
2
p
e
2
C
2e
1
p
e
2
;
J
22
D
and the off-diagonal entries are
J
12
D
x
1
f
00
Q
C
f
0
. Q/
e
1
p
e
2
e
2
p
e
1
3e
1
p
e
2
C
e
1
p
e
1
C
2e
2
p
e
1
;
2
f
0
. Q/
e
1
D
J
21
D
x
2
f
00
Q
C
f
0
. Q/
e
2
p
e
1
e
1
p
e
2
3e
2
p
e
1
C
e
2
p
e
2
C
2e
1
p
e
2
:
2
f
0
. Q/
e
2
D
Hence, the trace of the Jacobian matrix at the equilibrium is
3e
1
p
e
2
3e
1
p
e
2
C
e
1
p
e
1
C
2e
2
p
e
1
C
3e
2
p
e
1
3e
2
p
e
1
C
e
2
p
e
2
C
2e
1
p
e
2
Tr
D
and the determinant is
e
1
e
2
.7
p
e
1
e
2
C
e
1
C
e
2
/
3e
1
p
e
2
C
e
1
p
e
1
C
2e
2
p
e
1
3e
2
p
e
1
C
e
2
p
e
2
C
2e
1
p
e
2
Det
D
:
A set of sufficient conditions for the stability of
x
(that is for the eigenvalues to be
located inside the unit circle of the complex plane) is given by
1
C
Tr
C
Det > 0; 1
Tr
C
Det > 0; Det < 1
(5.63)
(see Appendix F). These conditions become trivial in our case. In fact, given that
Trand Det are both positive, the first condition is always satisfied. Moreover
2
p
e
1
e
2
e
1
C
e
2
C
6e
1
e
2
.e
1
C
e
2
/
3e
1
p
e
2
C
e
1
p
e
1
C
2e
2
p
e
1
3e
2
p
e
1
C
e
2
p
e
2
C
2e
1
p
e
2
>0
1
Tr
C
Det
D
and Det < 1 since
e
1
e
2
.7
p
e
1
e
2
C
c
1
C
c
2
/<
3e
1
p
e
2
C
e
1
p
e
1
C
2e
2
p
e
1
3e
2
p
e
1
C
e
2
p
e
2
C
2e
1
p
e
2
:
Example 5.7.
We now turn back to the case of isoelastic price and linear cost func-
tions and consider the case of N firms. As mentioned in previous chapters, one
question that is often discussed in the literature on oligopoly games deals with the
effect of the number of players on the stability properties of the equilibrium. In
general it is not a straightforward matter to find an answer to this question, since
increasing the number of players implies increasing the dimension of the dynamical
system. To obtain some insight into this problem, let us consider the model (5.53)
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