Chemistry Reference
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x
2
L
2
D
(3)
D
(4)
D
(5)
A
−
c
1
D
(2)
D
(1)
D
(6)
2
1
L
+
A
−
c
1
−2
L
1
D
(9)
D
(8)
D
(7)
x
1
L
1
2
L
+
A
−
c
2
−2L
2
A
−
c
2
Fig. 2.15
Example 2.5; quadratic price and linear cost functions. The regions of the piece-wise
map in the duopoly case
The strategy space, given by the trapping region
D D
Œ0;L
1
Œ0;L
2
, can be sub-
.i /
, i
D
1;:::;9, (see Fig. 2.15), similar to the
divided into nine different regions
D
previous examples.
Some of these regions may be empty when one or both the
capacit
y limits are too
small. For example
p
A
c
1
. Moreover, the
.3/
,
.4/
and
.5/
do not exi
st if L
2
D
D
D
profit of firm k is positive as long as x
1
C
x
2
<
p
A
c
k
, hence some of the regions
in Fig. 2.15 may involve negative profits. For example, if c
1
.3/
,
D
c
2
then regions
D
.7/
all involve negative profits and if an attractor is completely
included inside these regions, it should be considered as economically infeasible.
Instead, trajectories that pass through such regions and then exit it to enter other
regions characterized by positive profits can be considered as economically feasible.
The Jacobians of the regions are
.4/
;
.5/
;
.6/
and
D
D
D
D
0
!
1
x
2
3
q
x
2
C
2
3
1
a
1
a
1
c
1
/
@
A
3.A
!
J
.1/
D
;
x
1
2
3
a
2
3
q
x
1
C3.Ac
2
/
1
a
2
and matrices which can be obtained from this Jacobian by changing one or both
off-diagonal elements to zero. That is,
0
!
1
x
2
2
3
1
a
1
a
1
3
q
x
2
C3.Ac
1
/
@
A
J
.2/
D
J
.6/
D
;
0
1
a
2
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