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for k
D
1;2;:::;N, where the firm's adaptive expectations are based on its belief
Q
k
on the output of the rest of the industry. Here ˛
k
is a sign-preserving function
for all k.
The corresponding discrete time model with partial adjustment towards the best
response has the form
. f
k
of /.
x
l
.t//
x
k
.t/
!
x
k
.t/
!
N
X
R
k
x
k
.t
C
1/
D
x
k
.t/
C
˛
k
(5.6)
l
D
1
for k
D
1;2;:::;N.
Under the assumption of continuous time scales and that each firm adjusts its
output in the direction toward its believed best response we have the model
. f
k
of /.
x
l
.t//
x
k
.t/
!
x
k
.t/
!
:
X
N
R
k
x
k
.t/
D
˛
k
(5.7)
lD1
Notice that in the case of full knowledge of the price function we have R
k
D
R
k
and f
k
D
f for all k,sothat(f
k
of ) is the identity function and the models (5.4)-
(5.7) formally reduce to the models (1.28)-(1.31) introduced earlier in Chap. 1. For
the sake of simplicity we introduce the notation H
k
D
. f
k
of ). If f
k
is a good
approximation of f ,thenH
k
is a good approximation of the identity function. In
the subsequent parts of this section, the asymptotic behavior of systems (5.4)-(5.7)
will be examined. Since usually f
k
of differs from the identity function, and the
best response functions R
k
are different from the full information best responses
R
k
, the steady states of these systems are usually different from the Nash equilibria
of the full information case. Even if any of these systems is asymptotically stable,
the outputs of the firms will not converge to the Nash equilibria. The trajectories will
instead converge to the steady state of the system, which can be called a
believed
or
subjective equilibrium
.
Example 5.1.
Assume that the true price function is isoelastic, f.Q/
D
A=Q,but
firm k believes that it is f
k
.Q/
D
A
k
=Q,whereA
k
¤
A.Sincefirmk does not
know the true price function f , it is not able to derive the true value of Q
k
.After
having observed a particular price p, it is only able to estimate this quantity by using
the relationship given in (5.2), which in the present case becomes
A
k
Q
k
C
x
k
D
p
D
A
Q
k
C
x
k
:
So firm k believes that the output of the rest of the industry is
A
k
px
k
p
Q
k
D
:
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