Chemistry Reference
In-Depth Information
‰
k
.x
1
;:::;x
N
/
D
.x
k
f.Q/
C
k
.x
k
//
C
X
l
kl
.x
l
f.Q/
C
l
.x
l
//; (4.88)
¤
k
in the case of the classical Cournot model. Other model variants can be examined in
a similar manner.
We can rewrite the payoff function of firm k as
‰
k
.x
1
;:::;x
N
/
D
.x
k
C
S
k
/f.x
k
C
Q
k
/
C
k
.x
k
/
X
l
kl
C
l
.x
l
/; (4.89)
¤
k
where
D
X
l
S
k
kl
x
l
:
¤
k
Notice that
@‰
k
@x
k
D
f.x
k
C
Q
k
/
C
.x
k
C
S
k
/f
0
.x
k
C
Q
k
/
C
k
.x
k
/
and
@
2
‰
k
@x
k
D
2f
0
.x
k
C
Q
k
/
C
.x
k
C
S
k
/f
00
.x
k
C
Q
k
/
C
0
k
.x
k
/:
Assume that a slightly more restrictive set of conditions than that assumed for
concave oligopolies (see Sect. 2.1) is satisfied, that is,
(A) f
0
.Q/<0,
z
f
00
.Q/
C
f
0
.Q/
0,
(B)
(C) f
0
.Q/
C
0
k
.x
k
/<0,
for all k, all feasible values of x
k
and Q,and0
z
P
lD1
L
l
.Then‰
k
is strictly
concave in x
k
with fixed values of Q
k
and S
k
,sincex
k
C
S
k
Q and so @
2
‰
k
=@x
k
is negative. As earlier, let L
k
denote the finite capacity limit of firm k,thenithasa
unique best response function, which depends on both Q
k
and S
k
and is given by
8
<
if f.Q
k
/
C
S
k
f
0
.Q
k
/
C
k
.0/
0;
0
R
k
.Q
k
;S
k
/
D
if f.L
k
C
Q
k
/
C
.L
k
C
S
k
/f
0
.L
k
C
Q
k
/
C
k
.L
k
/
0;
L
k
:
z
k
otherwise;
where
z
k
is the unique solution of the equation
f.
z
k
C
Q
k
/
C
.
z
k
C
S
k
/f
0
.
z
k
C
Q
k
/
C
k
.
z
k
/
D
0
(4.90)
inside the interval .0;L
k
/.
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