The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 20

directly from them. In fact, for all k, (1.18) can be written as
Q k .t C 1/ D a k X
l

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

l

C .1 a k /Q k .t/:

¤ k ¤ k

(1.22)
The dynamic process now reduces to an N-dimensional dynamical system in the
expected variables Q 1 .t/;:::;Q N .t/, and at each time period t the output of firm
k is given as

x k .t/ D R k Q k .t/ ;

which is a static mapping from beliefs to realizations in the sense that both sides of
the mapping are computed at the same time t.
In most industries any increase of the output level of any firm requires time, new
hirings, purchase of new machinery, or sometimes even the opening up of a new
plant. Therefore output changes are made gradually. For example, in the case of
the dynamic process (1.19) instead of selecting the best response directly, the new
output level of firm k is selected somewhere in between the current level and the best
response to ensure that the output level change occurs in the right direction. This
concept of partial adjustment towards the best response with naive expectations can
be described by the modified N-dimensional dynamical system

0
@ X
l

1

A C .1 a k /x k .t/;

x k .t C 1/ D a k R k x l .t/ (1.23)

¤ k

for some a k 2 .0;1. In the case of a k D 0 the output level would never change,
therefore this value is excluded. Notice that in the case of a k D 1, the partial adjust-
ment towards the best response with naive expectations (1.23) reduces to best
response dynamics with naive expectations (1.19).
In the special case of two firms .N D 2/ both dynamical systems (1.22) and
(1.23) have the common form

y 1 .t C 1/ D a 1 R 1 .y 2 .t// C .1 a 1 /y 1 .t/;
y 2 .t C 1/ D a 2 R 2 .y 1 .t// C .1 a 2 /y 2 .t/

with y 1 D x 1 and y 2 D x 2 in (1.23), and y 1 D Q 2 , y 2 D Q 1 and a 1 and a 2 being
interchanged in (1.22). If N>2, then systems (1.22) and (1.23) are equivalent if

R k P l¤k y l .t/ D P l¤k R l .y l .t// holds for all k. In the symmetric case (when

R k R), this condition holds if R.Q k / D rQ k with some constant r.
It is important to realize that dynamic adjustment processes of the kind con-
sidered above are defined on the action space … kD1 Œ0;L k and incorporate only
the firms' quantity decision. In order to obtain economically feasible trajectories,
we need to keep in mind the fact that prices (and profits) have to be non-negative
in the long run, though it is possible (as we shall indeed find) that over some