Chemistry Reference
In-Depth Information
We can also rewrite the best responses of the firms in terms of the total output of
the industry. This idea will be very helpful in proving the existence and uniqueness
of the equilibrium and it can be also used to derive a simple computational method
to find the equilibrium. From (2.3) we have
8
<
:
if f.Q/
C
k
.0/
0;
0
R
k
.Q/
D
if f.Q/
C
L
k
f
0
.Q/
C
k
.L
k
/
0;
(2.8)
L
k
z
k
otherwise;
where
z
k
is the unique solution of the equation
f.Q/
C
z
k
f
0
.Q/
C
k
.
z
k
/
D
0
(2.9)
inside the interval .0;L
k
/. We point out that in the case when we consider the best
response as a function of Q (rather than Q
k
) we denote it by R
k
. Notice that in the
third case of (2.8), the left hand side is positive at
z
k
D
0,negativeat
z
k
D
L
k
,and
strictly decreasing, since it has a negative derivative given by
@
@
z
k
f
f.Q/
C
z
k
f
0
.Q/
C
k
.
z
k
/
gD
f
0
.Q/
C
0
k
.
z
k
/<0:
The derivative of R
k
.Q/ can be obtained by implicit differentiation, so that
f
0
C
R
0
k
f
0
C
R
k
f
00
C
0
k
R
0
k
D
0;
from which
f
0
C
R
k
f
00
f
0
C
0
k
R
0
k
D
0:
Since R
k
.Q/ is continuous in the interval Œ0;
P
lD1
L
l
, it is non-increasing in
Q for all Q
2
Œ0;
P
lD1
L
l
: Finally, consider the single-variable equation
X
N
1
R
k
.Q/
Q
D
0;
(2.10)
k
D
which must hold at the equilibrium. The left hand side of (2.10) is strictly decreasing
in Q, it is non-negative at Q
D
0 and non-positive at Q
D
P
kD1
L
k
.There-
fore there is a unique solution
Q, and the corresponding equilibrium outputs are
D
R
k
. Q/:
x
k
Example 2.1.
In our earlier Example 1.1 we introduced oligopolies with linear price
and cost functions,
f.Q/
D
A
BQ and C
k
.x
k
/
D
d
k
C
c
k
x
k
; 1
k
N/:
(2.11)
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