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on the initial beliefs. These observations are interesting, since they once again stress
the fact that we need to study the global dynamics of the market game, in particular
the characteristics of its possible long run outcomes and their respective basins of
attraction.
For further results on continuous time models we refer the reader to Chiarella
and Szidarovszky (2004), where firms may also misspecify the shape of the demand
function and not only its scale.
5.1.1
Discrete Time Models and Local Stability
We consider first the model (5.4)-(5.5), and note that it is the mathematically equiv-
alent to the dynamical system (1.28)-(1.29) introduced in Chap. 1. In Chap. 2 we
have determined the Jacobian of this system as the matrix given in (2.15), which in
the current situation assumes the particular form
J
11
;
J
12
J
21
J
22
where
0
@
1
A
r
1
a
1
.h
1
1/ r
1
a
1
h
1
:::
r
1
a
1
h
1
r
2
a
2
h
2
r
2
a
2
.h
2
1/:::
r
2
a
2
h
2
J
11
D
;
:
:
:
:
:
:
:
:
:
r
N
a
N
h
N
r
N
a
N
h
N
:::r
N
a
N
.h
N
1/
J
12
D
diag.r
1
.1
a
1
/;r
2
.1
a
2
/;:::;r
N
.1
a
N
//;
0
@
1
A
a
1
.h
1
1/ a
1
h
1
::: a
1
h
1
a
2
h
2
a
2
.h
2
1/::: a
2
h
2
:
:
:
J
21
D
;
:
:
:
:
:
:
:
:
:
a
N
h
N
a
N
h
N
:::a
N
.h
N
1/
J
22
D
diag.1
a
1
;1
a
2
;:::;1
a
N
/;
with r
k
D
R
0
k
, h
k
D
H
k
D
. f
k
of /
0
at the steady state of the system and a
k
D
˛
0
k
.0/. Since both f and f
k
are strictly decreasing, H
k
is strictly increasing, so it
is reasonable to assume that h
k
>0. The eigenvalue equation of the Jacobian can be
written similarly to (2.17)-(2.18) as
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