Chemistry Reference
In-Depth Information
8
<
:
z
k
0;
0
if
R
k
.Q
k
/
D
z
k
(4.107)
L
k
if
L
k
;
z
k
otherwise.
Notice that the best response functions have the same shape as those considered,
for example, in Example 1.5 and Example 3.4.
Example 4.16.
In order to investigate the effect of a change in the cooperation
levels
k
on the equilibrium values and on the global dynamics, following Exam-
ple 3.4 we consider the semi-symmetric case. That is, we consider (4.106) with
Q
1
D
.N
1/x
2
and Q
2
D
x
1
C
.N
2/x
2
, that is, the production decisions made
by firm 1 and the identical firms 2;:::;N are captured 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
/;
x
2
.t
C
1/
D
.1
a
2
/x
2
.t/
C
a
2
R
2
.x
1
C
.N
2/x
2
/:
T
W
The unique positive equilibrium of this system may be written as
.N
1/A.1
1
/.1
2
/Œ.N
1/.1
1
/c
2
.N
2/.1
2
/c
1
.c
1
.1
2
/
C
.N
1/.1
1
/c
2
/
2
x
1
D
;
Ac
1
.N
1/.1
1
/.1
2
/
2
.c
1
.1
2
/
C
.N
1/.1
1
/c
2
/
2
:
D
x
2
(4.108)
It is interesting to note that if one of the cooperation levels equals 1 (full coop-
eration), then both equilibrium quantities vanish. This is due to the fact that a fully
cooperative firm behaves like a profit maximizing monopolist (and therefore, due to
the particular form of the isoelastic demand function, it selects a quantity close to
zero, as mentioned in Example 1.5, whereas the other firm selects a small (close
to zero) quantity as best reply. Notice also that if the degrees of cooperation of all
firms are identical, so that
k
D
for k
D
1;2;:::;N, then the firms' equilibrium
quantities become
.N
1/A.1
/Œ.N
1/c
2
.N
2/c
1
.c
1
C
.N
1/c
2
/
2
x
1
D
;
(4.109)
c
1
.N
1/A.1
/
.c
1
C
.N
1/c
2
/
2
:
x
2
D
:::
D
x
N
D
These expressions coincide with the expression we derived earlier in Example 4.3
if we let
D
0 (no cooperation). Note that in the case of identical cooperation levels
of all firms, from the expression of the individual equilibrium values (4.109) we can
easily deduce that not only the total industry output decreases for increasing cooper-
ation levels, but also the
individual
equilibrium quantities. However, because of the
more complicated expressions of individual equilibrium output quantities (4.108)
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