Chemistry Reference
In-Depth Information
We can now calculate the best response of firm k. Assume first that Q
k
D
0,sothat
the other firms do not produce. Then
(
C
k
.0/; if x
k
D
0;
A
C
k
.x
k
/; if x
k
>0:
'
k
.x
1
;:::;x
N
/
D
In this case firm k has no best choice, however it is in its interest to select a
positive value of x
k
that is as small as possible. In other words, firm k does not have
a maximum profit for Q
k
D
0, its profit has only a supremum at x
k
D
0.IfQ
k
>0,
so that the other firms produce, then
@
@x
k
'
k
.x
1
;:::;x
N
/
D
AQ
k
.Q
k
C
x
k
/
2
C
k
.x
k
/;
(3.1)
and
@
2
@x
k
2AQ
k
.Q
k
C
x
k
/
3
C
0
k
.x
k
/<0;
'
k
.x
1
;:::;x
N
/
D
showing that '
k
is strictly concave in x
k
with fixed positive values of Q
k
.Ifwe
assume again that each firm has a finite capacity limit, L
k
, then the best response
exists and is unique for each firm and is given by
8
<
A
Q
k
C
k
.0/
0;
0;
if
AQ
k
.Q
k
C
L
k
/
2
R
k
.Q
k
/
D
C
k
.L
k
/
0;
L
k
; if
:
z
k
; otherwise;
where
z
k
is the unique solution of the strictly monotonic equation
AQ
k
.Q
k
C
z
k
/
2
C
k
.
z
k
/
D
0
(3.2)
in the interval .0;L
k
/. The derivative of the best response function is obtained by
implicit differentiation of the equivalent equation
AQ
k
C
k
.
z
k
/.Q
k
C
z
k
/
2
D
0;
from which we have
A
C
0
k
R
0
k
.Q
k
C
z
k
/
2
2C
k
.Q
k
C
z
k
/.1
C
R
0
k
/
D
0
implying that
A
2C
k
Q
C
0
k
Q
2
R
0
k
.Q
k
/
D
C
2C
k
Q
:
(3.3)
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