Chemistry Reference
In-Depth Information
In general, this dynamical system involves the best replies which are based on
misspecified beliefs. However, in the special case where firm k mistakenly over-
or underestimates the actual demand by a factor of "
k
,sothat
f
k
.p/
D
"
k
f
1
.p/;
(5.28)
the dynamical system can be expressed in terms of the true reaction functions
R
1
and R
2
(see again Leonard and Nishimura (1999), Bischi et al. (2004b)).
This can be seen as follows. First note that with this misspecification we obtain
f
k
.Q/
D
f."
k
Q/, and hence f
k
.Q/
D
"
k
f
0
."
k
Q/. Therefore, assuming an
interior solution, the first order condition for firm k can be written as
f."
k
x
k
C
"
k
Q
E;prior
/
C
"
k
x
k
f
0
."
k
x
k
C
"
k
Q
E;prior
/
C
k
D
0:
k
k
If we contrast this equation with the first order condition in the full information
case ("
k
D
1 ) which implicitly defines the relation x
k
D
R
k
.Q
E;prior
k
/, we can con-
clude that the first order condition together with assumption (5.28) implicitly defines
the relation "
k
x
k
D
R
k
."
k
Q
E;prior
k
/,whereR
k
denotes the reaction function in
the full information case. Obviously, it follows further that the relations between the
reaction functions R
k
and the full information reaction functions are given by
R
k
.Q
E;prior
/
D
"
k
R
k
."
k
Q
E;prior
/:
(5.29)
k
k
Consequently, if we take (5.28) and (5.29) into account, the dynamical system based
on partial adjustment towards the best response can be rewritten as
x
1
.t
C
1/
D
x
1
.t/
C
a
1
"
1
R
1
Œ
"
1
1
"
1
x
1
.t/
C
x
2
.t/
x
1
.t/
;
(5.30)
x
2
.t
C
1/
D
x
2
.t/
C
a
2
"
2
R
2
Œx
1
.t/
C
x
2
.t/
x
2
.t/
:
"
2
1
"
2
Notice that if both players know the true demand ("
1
D
"
2
D
1), then (5.30) reduces
to the partial adjustment process introduced in Chap. 1. On the other hand, our
derivations show that even if players over- or underestimate the demand by a cer-
tain factor "
k
¤
1, the dynamics of the repeated duopoly game with misspecified
demand is still governed by equations only involving the reaction functions of the
full information case. It is clear that this property makes this particular type of
misspecification quite appealing for further analysis. We will now briefly describe
the effects of mistaken beliefs on the long-run properties of the dynamical system.
More precisely, we will focus on the existence and stability of steady states if firms
misspecify the demand relationship and we will study the extent and topological
structure of the basins of attractions of these steady states. To consider a particular
example, we follow the model of Sect. 3.2 and specify the full information reaction
functions as
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