The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 19

most simple expectation scheme is the one in which the firms use the latest available
information,

Q k .t C 1/ D X
l¤k

x l .t/; (1.17)

which is sometimes called the static ,or naive ,or Cournot expectation .
The firms are also able to develop certain learning procedures based on earlier
data. The most popular such learning scheme is obtained when the firms adjust their
expectations adaptively according to

0
@ X
l¤k

1

Q k .t C 1/ D Q k .t/ C a k x l .t/ Q k .t/

A ;

(1.18)

with a k being a positive constant known as the speed of adjustment of firm k.Itis
usually assumed that 0<a k 1 for all k. The interpretation of this dynamic learning
scheme is that, if firm k underestimated (overestimated) the output of the rest of the
industry in the previous time period, then in the next time period this firm wants
to increase (decrease) its estimate. This increase (decrease) is represented by the
second term, and the coefficient a k determines the speed (or rate) of adjustment. If
the expectation of a firm were correct in the previous time period, then there would
be no need to change the expectation, in this case the second term would be zero.
Notice that the special case of a k D 1 reduces to the static or Cournot expectation.
Mathematically, the dynamic process (1.16), together with naive expectations
(1.17) form the N-dimensional dynamical system

0
@ X

1

A

x k .t C 1/ D R k x l .t/ .k D 1;2;:::;N/; (1.19)

l ¤ k

to which we will refer as best response dynamics with naive expectations .
Under the adaptive expectations scheme (1.18), the dynamic process (1.16)
becomes the 2N-dimensional dynamical system

0
@ a k X
l

1

A ;

x l .t/ C .1 a k /Q k .t/ x k .t C 1/ D R k (1.20)

¤ k

Q k .t C 1/ D a k X
l

x l .t/ C .1 a k /Q k .t/; (1.21)

¤ k

for k D 1;2;:::;N. We will refer to this process as the best response dynamics
with adaptive expectations .
In the latter formulation we have formally 2N state variables, however it is easy
to show that the best response dynamics with adaptive expectations are actually
driven by the N expectation variables and the production outputs can be computed