Chemistry Reference
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and linear costs C
k
.x
k
/
D
c
k
x
k
, which was introduced in Example 1.4. We assume
that A>c
k
for all k
D
1;:::;N. As shown in Example 1.4, the best response of
firm k is given by the continuous and piecewise differentiable function
8
<
:
p
A
c
k
;
0 if x
k
0 i.e.,
Q
k
q
L
k
C
A
c
k
2L
k
;
L
k
if x
k
L
k
i.e.,
Q
k
R
k
.Q
k
/
D
otherwise i.e.,
q
L
k
C
A
c
k
2L
k
<Q
k
<
p
A
c
k
;
z
k
where
q
Q
k
C
3.A
c
k
/
2Q
k
:
It is easy to see that
q
L
k
C
A
c
k
2L
k
<
p
A
c
k
holds given our assumption
that A>c
k
and L
k
>0. In the case of N firms, the unique equilibrium, see (1.14),
is
1
3
z
k
D
2A
C
P
lD1
c
l
.N
C
2/c
k
x
k
D
2
q
.N
C
2/.NA
P
lD1
c
l
/
;
under the assumption that it is interior.
Let us first consider the case of duopoly, that is N
D
2, with partial adjustment
towards the best response. The sequence of production quantities in this case is
obtained by the repeated application of the piecewise differentiable map
x
1
.t
C
1/
D
.1
a
1
/x
1
.t/
C
a
1
R
1
.x
2
.t//;
x
2
.t
C
1/
D
.1
a
2
/x
2
.t/
C
a
2
R
2
.x
1
.t//;
T
W
where the reaction functions are given by
8
<
p
A
c
1
;
0
if x
2
q
L
1
C
A
c
1
2L
1
;
L
1
if x
2
R
1
.x
2
/
D
q
x
2
C
3.A
c
1
/
2x
2
:
1
3
otherwise;
and
8
<
p
A
c
2
;
0
if x
1
q
L
2
C
A
c
2
2L
2
;
L
2
if x
1
R
2
.x
1
/
D
q
x
1
C
3.A
c
2
/
2x
1
:
1
3
otherwise:
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