Chemistry Reference
In-Depth Information
Let now .x
1
;:::;x
N
/ be an equilibrium of the oligopoly without partial cooper-
ation, and let
Q
D
P
kD1
x
k
. Assume that .x
0
1
;:::; x
0
N
/ is an equilibrium with
D
P
kD1
D
P
l¤k
kl
x
l
. Then the
Q
0
S
k
x
0
k
partial cooperation, and let
and
following theorem holds:
Q
0
Q
, that is, partial cooperation
Theorem 4.4.
Under conditions (A)-(C),
decreases the total production level of the industry.
Q
0
> Q.Then
Proof.
Assume in contrary, that
N
N
X
X
Q
0
D
R
k
. Q
0
; S
k
/
R
k
. Q
0
;0/
k
D
1
k
D
1
X
N
R
k
. Q;0/
D
Q;
kD1
which is a contradiction.
In order to obtain more interesting results assume that each firm has identical
cooperation levels towards its competitors, that is,
kl
k
for all l
¤
k. We can
then rewrite (4.96) as
4
8
<
0; if f.Q/
C
k
Qf
0
.Q/
C
k
.0/
0;
L
k
; if f.Q/
C
..1
k
/L
k
C
k
Q/f
0
.Q/
C
k
.L
k
/
0;
z
k
; otherwise,
R
k
.Q;
k
/
D
:
(4.98)
where
z
k
is the unique solution of the equation
f.Q/
C
..1
k
/
z
k
C
k
Q/f
0
.Q/
C
k
.
z
k
/
D
0:
(4.99)
For the current situation we modify conditions (B) and (C) to read
(B
0
) .1
C
k
/f
0
C
v
f
00
0,
(C
0
) .1
k
/f
0
C
0
k
<0
h
0;
P
kD1
L
k
i
and
z
k
for all Q;
v
2
2
Œ0;L
k
.
Lemma 4.2.
The reaction function
R
k
.Q;
k
/
defined above is a decreasing func-
tion of
Q
, and a decreasing function of
k
in the domain defined by
R
k
.Q;
k
/
Q:
4
Here in the notation we emphasize the dependence of R
k
on Q and
k
, since in this case
S
k
D
k
.Q
x
k
/, and use the same notation for this new form of the reaction function.
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