The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 23

8
<
:

>0 if x>0,
D 0 if x D 0,
<0 if x<0.

˛.x/ (1.27)

Assume now that for all k, ˛ k is a sign-preserving function, then the dynam-
ical system (1.20)-(1.21) for the best response with adaptive expectations can be
extended to

0
@ Q k .t/ C ˛ k . X
l

1

A ;

x l .t/ Q k .t//

x k .t C 1/ D R k (1.28)

¤ k

0
@ X
l¤k

1

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

A :

(1.29)

Similarly the discrete time dynamical system (1.23) for the dynamics of partial
adjustment towards the best response with naive expectations becomes

0
@ R k . X
l¤k

1

A ;

x k .t C 1/ D x k .t/ C ˛ k x l .t// x k .t/ (1.30)

whilst the continuous time dynamical system (1.24) for the same process becomes

0
@ R k . X
l

1

A :

x k .t/ D ˛ k x l .t// x k .t/ (1.31)

¤ k

Another important class of adjustment processes that has been investigated in the
literature on dynamic oligopolies by many authors is that of the gradient adjustment
process. This adjustment process is based on the observation that if for firm k at a
certain time period, @' k =@x k is positive, then it is in firm k's interest to increase
the output level, if @' k =@x k is negative, then the firm wants to decrease it, and if
@' k =@x k D 0,thenfirmk believes that it is already at its maximum level, so it wants
to maintain the same output level. This idea can be mathematically realized in the
gradient adjustment processes

x k .t C 1/ D x k .t/ C ˛ k @' k .x 1 .t/;:::;x N .t//
@x k

.1 k N/; (1.32)

in discrete time and
x k .t/ D ˛ k @' k .x 1 .t/;:::;x N .t//
@x k

.1 k N/; (1.33)