Chemistry Reference
In-Depth Information
We also proved that the best response function of firm k is a piece-wise linear
function
8
<
if A
BQ
k
c
k
0;
0
R
k
.Q
k
/
D
L
k
if A
2BL
k
BQ
k
c
k
0;
:
1
2
Q
k
C
.A
c
k
/=.2B/ otherwise
(2.12)
by assuming that
P
kD1
L
k
A
B
.
In the first two cases R
0
k
.Q
k
/
D
0 and in the third case R
0
k
.Q
k
/
D
1
2
showing
that (2.7) is always satisfied.
In Example 1.2 we examined oligopolies with linear price and quadratic cost
functions. In case (ii) of that example we observed the possibility of multiple equi-
libria, however it is easy to see that under the stated assumptions condition (C) is
violated.
Let us turn our attention next to the general case when the cost of firm k
is C
k
.x
k
;Q
k
/, perhaps because of the presence of externalities as discussed in
Sect. 1.1. In this more general case the profit of firm k is given as
'
k
.x
1
;:::;x
N
/
D
x
k
f.x
k
C
Q
k
/
C
k
.x
k
;Q
k
/
with derivatives
@'
k
@x
k
D
f.x
k
C
Q
k
/
C
x
k
f
0
.x
k
C
Q
k
/
C
kx
.x
k
;Q
k
/
and
@
2
'
k
@x
k
D
2f
0
.x
k
C
Q
k
/
C
x
k
f
00
.x
k
C
Q
k
/
C
0
kxx
.x
k
;Q
k
/;
where C
kx
and C
0
kxx
denote the first and second order partial derivatives of C
k
with
respect to x
k
: Assume that conditions .A/;.B/ are satisfied, furthermore assume
(C') f
0
.x
k
C
Q
k
/
C
0
kxx
.x
k
;Q
k
/<0,
for all k and feasible values of x
k
and Q
k
.
Under conditions (A), (B) and (C'), the profit '
k
of firm k is strictly concave in
x
k
, therefore there is a unique best response function of firm k given by
8
<
if f.Q
k
/
C
kx
.0;Q
k
/
0;
0
R
k
.Q
k
/
D
if f.L
k
C
Q
k
/
C
L
k
f
0
.L
k
C
Q
k
/
C
kx
.L
k
;Q
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
f
0
.
z
k
C
Q
k
/
C
kx
.
z
k
;Q
k
/
D
0;
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