Chemistry Reference
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and so
s
AQ
k
c
k
z
k
D
Q
k
;
which is the best response of the firm with full information. This observation means
that even if firms misspecify the price function in this way, they still take the right
decision with their best responses.
The global asymptotic stability of the models (5.4)-(5.5) for best reply dynamics
with adaptive expectations and (5.6) for the partial adjustment dynamics can be
discussed similarly to the other cases. The Jacobian of system (5.6) has the special
form (5.10). By using Lemma B.2 of Appendix B, we see that the equilibrium is
globally asymptotically stable if for all k and all feasible values of x
1
;:::;x
N
,
j
r
k
a
k
.h
k
1/
C
1
a
k
j C
.N
1/
j
r
k
a
k
h
k
j
<1;
(5.15)
the feasible output sets are compact and all functions R
k
, ˛
k
and H
k
are continu-
ously differentiable on these sets. Under conditions (A)-(C) and by assuming that
h
k
>0, this inequality can be written as the pair of inequalities
r
k
a
k
.h
k
1/
C
1
a
k
.N
1/r
k
a
k
h
k
<1
and
r
k
a
k
.h
k
1/
1
C
a
k
.N
1/r
k
a
k
h
k
<1:
These relations can be rewritten as
a
k
r
k
h
k
.2
N/
r
k
1
<0
(5.16)
and
a
k
1
C
r
k
.1
Nh
k
/
<2:
(5.17)
Consider the first relation (5.16). Since
1<r
k
0,forN
D
2 it always holds,
and for N
D
3 it holds if
r
k
.h
k
C
1/
1<0
that is, when
r
k
>
1
h
k
C
1
:
If N becomes larger, then r
k
h
k
.2
N/becomes a large positive number, so (5.16)
no longer holds, and stability is lost. Consider next relation (5.17). Since
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