Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 60

and

2 a k .1 C r k / > 1 X
l¤k

r k a k r l a l

2 a l .1 C r l / :

Simple algebra shows that this last inequality holds if and only if

2.1 C S k /
1 C S k .1 C r k / ;

a k < (2.28)

where

D X
l¤k

r l a l
2 a l .1 C r l / 2 . 1;0:

S k

It is easy to see that the right hand side of relation (2.28) is always positive, since
r k 2 . 1;0.
Consider next the case of a duopoly when N D 2. In this special case conditions
(2.21) and (2.22) can be rewritten as

a 1 .1 C r 1 /<2; a 2 .1 C r 2 /<2

and

a 2 r 2
2 a 2 .1 C r 2 / > 1:
With fixed values of a 1 2 .0;2=.1 C r 1 //; firm 2 has to select a sign-preserving
adjustment function with ˛ 0 2 .0/ D a 2 satisfying condition (2.28) in order to stabilize
the system. In the case of a duopoly this condition has the special form

a 1 r 1
2 a 1 .1 C r 1 / C

2 1 C

a 1 r 1
2 a 1 .1 C r 1 /

2.2 a 1 /
2 a 1 .1 r 1 r 2 / :

a 2 < D (2.29)

a 1 r 1 .1 C r 2 /
2 a 1 .1 C r 1 /

1 C

Notice that for a 1 ;a 2 2 .0;1 this relation is always satisfied, so the equilibrium
is always locally asymptotically stable. The stability region of this condition in the
.a 1 ;a 2 / plane is illustrated in Fig. 2.2.
Several generalizations of the above analysis, including multiproduct models,
are discussed in Okuguchi and Szidarovszky (1999). In addition, the existence and
uniqueness of the equilibrium is proved without imposing the conditions of differ-
entiability of the price and cost functions, and in the linear cases several alternative
sufficient and necessary stability conditions are derived. The very first stability result
in discrete time dynamic oligopolies dates back to Theocharis (1960) and follow-
ing in his footsteps many researchers have worked intensively on this topic, a task
which continues even to the present day. For an extensive literature review, see
Kopel (2009).