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the quadratic production cost functions as C
k
.x
k
/
D
c
k
x
k
C
e
k
x
k
.Inordertoavoid
trivial best responses we assume again that A>c
k
for k
D
1;2.
1.3.1
A Cournot Duopoly Game
We first consider a duopoly game (N
D
2), where the firms use partial adjustment
towards the best response. The reaction functions in this case become
8
<
z
1
<0;
0
if
R
1
.x
2
/
D
z
1
>L
1
;
(1.34)
L
1
if
:
z
1
otherwise;
and
8
<
z
2
<0;
0
if
R
2
.x
1
/
D
z
2
>L
2
;
(1.35)
L
2
if
:
z
2
otherwise;
BQ
k
2.BCe
k
/
.k
D
1;2/ with Q
1
D
x
2
and Q
2
D
x
1
. If the duopolists
partially adjust their quantities towards the best replies (based on naive expectations)
and if the speeds of adjustment are constant, the dynamical system is generated by
the iteration of the map T
a
W
Œ0;L
1
Œ0;L
2
!
Œ0;L
1
Œ0;L
2
,where
A
c
k
where
z
k
D
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
a
W
(1.36)
with 0<a
k
1. Recall from Sect. 1.2 that the best reply dynamics with naive
expectations is obtained as a special case with a
k
D
1 for k
D
1;2.Wehavealso
shown in Sect. 1.2 that in a duopoly partial adjustment towards the best response
and the best reply dynamics with adaptive expectations are equivalent. Hence, the
results obtained in this section also describe what happens if best reply dynamics
with adaptive expectations are considered. Using (1.36) together with the steady
state conditions x
k
.t
C
1/
D
x
k
.t/, k
D
1;2, leads to the equations x
1
D
R
1
.x
2
/,
x
2
D
R
2
.x
1
/, which shows that the steady states of this dynamical system coincide
with the Cournot-Nash equilibria of the underlying game and that they are located
at the intersections of the reaction curves. Clearly, the steady states do not depend
on the adjustment speeds a
k
. As demonstrated in Sect. 1.1, the number of equilibria
depends on the marginal costs. If marginal costs are increasing or even decreasing
but not too strongly such that B
C
e
k
>0and
B
2
<4.B
C
e
1
/.B
C
e
2
/;
(1.37)
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