Chemistry Reference
In-Depth Information
8
<
:
z
1
<0;
0
if
R
1
..N
1/x
2
/
D
z
1
>L
1
;
L
1
if
z
1
otherwise;
D
A
c
1
B.N
1/x
2
=
2.B
C
e
1
/
and
with
z
1
8
<
z
2
<0;
0
if
R
2
.x
1
C
.N
2/x
2
/
D
z
2
>L
2
;
L
2
if
:
z
2
otherwise;
with
z
2
D
A
c
2
B.x
1
C
.N
2/x
2
/
=
2.B
C
e
2
/
:
The interior equilibrium is independent of a
k
, k
D
1;2, but depends on the
number of firms N.ItisgivenbyE
D
.x
1
.N/; x
2
.N// with
A.B
C
2e
2
/
2c
1
e
2
C
B.c
2
.N
1/
c
1
N/
2B.N
2/.B
C
e
1
/
C
4.B
C
e
1
/.B
C
e
2
/
B
2
.N
1/
;
x
1
.N/
D
2.B
C
e
1
/.A
c
2
/
B.A
c
1
/
2B.N
2/.B
C
e
1
/
C
4.B
C
e
1
/.B
C
e
2
/
B
2
.N
1/
:
x
2
.N/
D
The Jacobian matrix computed at the interior equilibrium is
1
a
1
!
;
B.N
1/
a
1
2.B
C
e
1
/
B.N
2/
2.B
B
2.B
a
2
1
a
2
a
2
C
e
2
/
C
e
2
/
from which the stability conditions can be obtained by applying conditions (1.40).
Interesting stability results are obtained for the boundary equilibria, in the case when
B
2
>4.B
C
e
1
/.B
C
e
2
/ (illustrated in Fig. 1.3 for one possible situation). The
Jacobian evaluated in the neighborhood of E
1
is either
1
a
1
!
1
a
1
0
01
B.N
1/
2.BCe
1
/
a
1
or
a
2
0
1
a
2
or both, if the equilibrium is on the boundary between the two regions, since R
2
0
here. The Jacobian evaluated in the neighborhood of E
2
is either
1
a
1
!
1
a
1
!
0
0
or
B
2.B
B.N
2/
B.N
1/
a
2
1
a
2
a
2
0
1
a
2
a
2
C
e
2
/
2.B
C
e
2
/
2.B
C
e
2
/
or both, because R
2
0 here.
Search WWH ::
Custom Search