Chemistry Reference
In-Depth Information
Implicitly differentiating this equation with respect to Q
k
and S
k
, and consider-
ing
z
k
D
R
k
.Q
k
;S
k
/ we have
f
0
.R
0
kQ
C
1/
C
R
0
kQ
f
0
C
.
z
k
C
S
k
/f
00
.R
0
kQ
C
1/
C
0
k
R
0
kQ
D
0
(4.91)
and
f
0
R
0
kS
C
.R
0
kS
C
1/f
0
C
.
z
k
C
S
k
/f
00
R
0
kS
C
0
k
R
0
kS
D
0;
(4.92)
where R
0
kQ
D
@R
k
=@Q
k
and R
0
kS
D
@R
k
=@S
k
. Therefore the derivatives of the best
response function are given by
f
0
C
.
z
k
C
S
k
/f
00
2f
0
C
.
z
k
C
S
k
/f
00
C
0
k
R
0
kQ
D
(4.93)
and
f
0
2f
0
C
.
z
k
C
S
k
/f
00
C
0
k
R
0
kS
D
;
(4.94)
implying that
1<R
0
kQ
0 and R
0
kS
<0: (4.95)
If in addition, f
0
C
.
z
k
C
S
k
/f
00
C
0
k
0; then
1
R
0
kS
<0. The payoff
function ‰
k
of each firm is concave in x
k
, continuous, and if each firm has a finite
capacity limit L
k
, then the Nikaido-Isoda theorem (see for example Forgo et al.
(1999)) implies the existence of at least one Nash equilibrium.
Before examining the dynamic extensions and investigating the asymptotic
behavior of the resulting systems we will briefly discuss the effect of partial coopera-
tion on the equilibrium quantities. Given that R
k
.Q
k
;S
k
/ denotes the best response
of firm k with partial cooperation, then clearly R
k
.Q
k
;0/is its best response in the
standard oligopoly with no cooperation. One can also consider the best responses as
functions of the total production level Q, as is usual in oligopoly theory and which
was also introduced in Chap. 2. Then clearly
8
<
if f.Q/
C
S
k
f
0
.Q/
C
k
.0/
0;
0
R
k
.Q;S
k
/
D
if f.Q/
C
.L
k
C
S
k
/f
0
.Q/
C
k
.L
k
/
0;
(4.96)
L
k
:
z
k
otherwise;
where
z
k
is the unique solution of the equation
f.Q/
C
.
z
k
C
S
k
/f
0
.Q/
C
k
.
z
k
/
D
0
(4.97)
inside the interval (0;L
k
).
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