Chemistry Reference
In-Depth Information
and
1
a
1
j
.2/
!
:
J
.2/
D
J
.6/
D
12
01
a
2
D
a
1
.N
1/
!
:
where j
.2/
12
.N
1/x
2
2
3
3
q
.N 1/
2
x
2
C3.Ac
1
/
The other Jacobians are obtained by changing the off-diagonal elements of
J
.1/
to zero. The interior equilibrium
x D
.x
1
; x
2
/ in the semi-symmetric case is
2A
.N
C
1/c
1
C
.N
1/c
2
2
p
.N
C
2/.NA
c
1
.N
1/c
2
/
;
x
1
D
2A
3c
2
C
c
1
2
p
.N
C
2/.NA
c
1
.N
1/c
2
/
x
2
D
;
(2.37)
.1/
. Due to the algebraic complexity of the expressions
involved, the study its local stability analytically is quite difficult. Moreover, when
the equilibrium
provided it is in region
D
x
crosses the boundaries of the region
D
.1/
(or other periodic points
.i /
) border collision bifurcations may occur that
cause the creation, destruction, or modification of the qualitative properties of the
attractors. In what follows we employ a combination of analytical and numerical
methods to gain some information about the global dynamical behavior of this non-
linear piece-wise differentiable model and about the global bifurcations occurring
as some parameters are varied.
Let us assume that firm 1 has higher unit costs than the rest of the industry, so
that c
1
>c
2
, and we shall study the properties of the equilibrium as the number of
firms varies. Note that 2A
C
c
1
3c
2
>0is guaranteed by the assumptions c
1
>c
2
and A>c
k
(k
D
1;2). As long as 2A
C
.N
1/c
2
.N
C
1/c
1
>0,allfirmsare
active in the market. Profits are then given by
move across different regions
D
.2A
C
.N
1/c
2
.N
C
1/c
1
/
2
2.N
C
2/
p
.N
C
2/.NA
c
1
.N
1/c
2
/
'
1
D
;
.2A
C
c
1
3c
2
/
2
2.N
C
2/
p
.N
C
2/.NA
c
1
.N
1/c
2
/
'
2
D
:
However, if N>.2A
c
2
c
1
/=.c
1
c
2
/ then firm 1 stops producing. In this case
x
1
D
0 and the other N
1 identical firms select their symmetric equilibrium
A
c
2
p
.N
C
1/.N
1/.A
c
2
/
x
2
D
;
(2.38)
which can be obtained from relation (1.14) with N being replaced by N
1 and all
c
k
by c
2
. The profit of the active firms reads
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