Chemistry Reference
In-Depth Information
Example 5.9.
Assume again linear cost functions and (subjective) best response
dynamics. Then a
k
D
1, r
kx
1
2
and C
0
k
r
kQ
D
D
0 for all k. So (5.81) can be
simplified to
X
Nf
0
f
00
Q
2f
0
3
2
;
D
>
r
kQ
k
or
.N
3/f
0
C
f
00
N
Q>0:
(5.82)
Notice that under conditions (A) and (B) the norm-based sufficient condition (5.76)
assumes the form
.N
3/f
0
C
.N
1/x
k
f
00
x
k
j
f
00
j
>0
(5.83)
for all k.Iff
00
. Q/ < 0, then these conditions may hold only for N
D
2, and have
the following forms
f
0
C
f
00
N
Q>0
and
f
0
C
2x
k
f
00
>0 .k
D
1;2/:
The second inequality is slightly stronger than the first one unless the x
k
values are
identical. Assume next that f
00
.Q/
0. In the case of N
D
2 both conditions (5.82)
and (5.83) are satisfied. If N
D
3, then the two conditions have the special forms
f
00
N
Q>0 and f
00
x
k
>0;
where the second inequality is stronger again. If N>3, then we have the two
conditions
.N
3/f
0
C
f
00
N
Q>0
and
.N
3/f
0
C
.N
2/x
k
f
00
>0;
where the second condition is again stronger then the first one (by taking x
k
D
mi
l
f
x
l
g
, the left hand side of the second inequality is smaller than that of the first
one). Very similar conditions can be obtained by assuming quadratic cost functions
and in comparing conditions (5.77) and (5.81).
Example 5.10.
Consider next the case of an isoelastic price function. Then f.Q/
D
A=Q and therefore
A.2x
k
Q/
Q.2A
C
Q
2
C
0
k
/
and r
kx
D
A
Q
2
;f
00
2A
Q
3
;r
kQ
2Ax
k
Q.2A
C
Q
2
C
0
k
/
:
f
0
D
D
D
If there is no dominant firm which produces more than the rest of the industry, then
2x
k
Q
0 for all k, so at the equilibrium
Search WWH ::
Custom Search