Chemistry Reference
In-Depth Information
1
a
1
0
01
;
J
.3/
D
J
.5/
D
J
.7/
D
J
.9/
D
a
2
0
1
1
a
1
0
!
@
A
J
.4/
D
J
.8/
D
:
x
1
2
3
a
2
3
q
x
1
C3.Ac
2
/
1
a
2
The interior equilibrium is
2A
3c
1
C
c
2
4
p
2A
c
1
c
2
;
2A
3c
2
C
c
1
x D
. x
1
; x
2
/
D
4
p
2A
c
1
c
2
and it is an equilibrium of the dynamical system provided it belongs to the region
D
.1/
. Unfortunately, even in the duopoly case, explicit conditions for the local stabil-
ity of the interior equilibrium are not easy to obtain. However, numerical simulations
indicate that whenever
x
exists then it appears to be globally asymptotically stable.
Now the questions arises under which conditions is stability lost in the oligopoly
case, that is for N>2. Clearly, the general case is hard to analyze. However, to
get some insight into the effect of an increasing number of firms or an increase in
the speeds of adjustment on the stability of the equilibrium we can consider the
semi-symmetric
case. Hence, we assume c
2
D
:::
D
c
N
, a
2
D
:::
D
a
N
,and
L
2
D
:::
D
L
N
: Under the further assumption of identical initial conditions for
firms 2;:::;N,thatisx
2
.0/
D
:::
D
x
N
.0/, the production decisions of firm 1 and
the identical firms 2;:::;N are governed by the two-dimensional dynamical system
x
1
.t
C
1/
D
.1
a
1
/x
1
.t/
C
a
1
R
1
..N
1/x
2
.t//;
x
2
.t
C
1/
D
.1
a
2
/x
2
.t/
C
a
2
R
2
.x
1
.t/
C
.N
2/x
2
.t//;
T
W
(2.36)
where
8
<
p
A
c
1
N 1
;
0 if x
2
q
L
1
C
A
c
1
2L
1
L
1
if x
2
;
R
1
..N
1/x
2
/
D
q
.N
1/
2
x
2
C
3.A
c
1
/
2.N
1/x
2
N
1
:
1
3
otherwise;
and
R
2
.x
1
C
.N
2/x
2
/
8
<
p
A
c
2
;
0 if x
1
C
.N
2/x
2
q
L
2
C
A
c
2
2L
2
;
L
2
if x
1
C
.N
2/x
2
D
q
.x
1
C
.N
2/x
2
/
2
C
3.A
c
2
/
2.x
1
C
.N
2/x
2
/
otherwise:
:
1
3
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