Chemistry Reference
In-Depth Information
Proof.
Now let g.
z
k
;Q;
k
/ denote the left hand side of (4.99), then for all feasible
z
k
and Q,
@g
@
z
k
D
.1
k
/f
0
.Q/
C
0
k
.
z
k
/<0;
@g
@Q
D
.1
C
k
/f
0
.Q/
C
..1
k
/
z
k
C
k
Q/f
00
.Q/
0;
and if 0
z
k
Q,then
@g
@
k
D
.Q
z
k
/f
0
.Q/
0:
We will first prove that R
k
.Q;
k
/ is decreasing in Q. Assume first that
Q
.1/
<Q
.2/
, and fix the value of
k
. If with Q
D
Q
.1/
the first case of (4.98) occurs,
then condition .B
0
/ implies that the same holds for Q
D
Q
.2/
,soR
k
remains the
same. Assume next that the second case of (4.98) occurs with Q
D
Q
.1/
,thenR
k
cannot increase further. If with Q
D
Q
.1/
, the third case of (4.98) occurs, then with
Q
D
Q
.2/
the second case of (4.98) is impossible, so either R
k
.Q
.2/
;
k
/ is zero or
is the solution of (4.99). In the first case, R
k
decreases, and it will easily be proven
that this is the case even if the third case of (4.98) occurs with Q
D
Q
.2/
. Assume
not, that is, R
k
.Q
.1/
;
k
/<R
k
.Q
.2/
;
k
/.Then
0
D
g.R
k
.Q
.1/
;
k
/;Q
.1/
;
k
/>g.R
k
.Q
.2/
;
k
/;Q
.1/
;
k
/
g.R
k
.Q
.2/
;
k
/;Q
.2/
;
k
/
D
0;
which is a contradiction. Next it will be shown that R
k
.Q;
k
/ is also decreasing in
k
in the domain
f
R
k
.Q;
k
/
Q
g
. Assume next that
.1/
k
<
.2/
k
and fix the value
of Q. If the first case of (4.98) occurs with
.1/
k
, then the negativity of f
0
implies
that the same holds for
.2/
k
, and if the second case occurs with
.1/
k
,thenR
k
cannot
D
.1/
increase further. Assume next that with
k
k
the third case occurs. Then the
second case is impossible for
k
D
.2
k
, since if firm k selects its maximal output
level L
k
,thenQ
L
k
. So either R
k
.Q;
.2/
k
/ is zero or the solution of (4.99). It
will also be proven that even in this case, R
k
.Q;
.2/
k
/
R
k
.Q;
.1/
k
/. Assume not,
then
0
D
g.R.Q;
.1/
k
/;Q;
.1/
k
/>g.R.Q;
.2/
k
/;Q;
.1/
k
/
g.R
k
.Q
k
;
.2
k
/;Q;
.2
k
/
D
0;
which is again a contradiction.
Let .x
1
;:::; x
N
/ be an equilibrium of the oligopoly with partial cooperation
levels
N
1
;:::;
N
N
. Assume that at least one player increases its cooperation level, so
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