Chemistry Reference
In-Depth Information
For the sake of simplifying the notation let
;
L.N
C
1/
2T
C
LT
2
1
C
2LT
T
2
˛
2
D
D
then
1;2
D˙
i˛ and at these values
˙
2T˛
3
i
C
˛
2
.2
C
2LT/
2˛Li
3˛
2
T
2
P
D
˙
2˛i.2T
C
LT
2
/
C
.1
C
2LT/
˛
2
.2
C
2LT/
˙
2i.T˛
3
˛L/
2.1
C
2TL/
˙
2˛i.2T
C
LT
2
/
D
with real part
4˛
2
.1
.LT/
2
/
4.1
C
2TL/
2
Re
P
D
¤
0;
C
4˛
2
.2T
C
LT
2
/
2
since it is easy to show that
.LT/
1
<1<.LT/
2
based on the fact that .LT/
1
; and .LT/
2
are reciprocals of each other. Hence all
conditions of the Hopf bifurcation theorem are satisfied.
Larger values of m lead to higher order polynomial equations and therefore the
stability analysis becomes more complicated and requires the use of computational
methods.
5.3.3
Unknown Slope with Known Reservation Price
In this section we assume that the firms know the value of the reservation price,
f.0/
D
A, but they are uncertain about the slope B of the price function. In this
case firm k believes that the price function is f
k
.Q/
D
A
"
k
Q wherethevalueof
"
k
is estimated and updated at each time period.
As in the previously discussed cases let us examine the way firm k reasons in
this situation. It believes that the profit of each firm (including its own) is
x
l
.A
"
k
Q
l
"
k
x
l
/
.c
l
e
e
x
l
C
d
l
/;
(5.137)
so the believed best response of firm l is
Q
l
A
c
l
2"
k
e
x
l
D
2
;
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