Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 169

that is,

ˇ S .2N 4 C a/<4 .N C 1/a:
Since the multiplier of ˇ S is positive for N 2 and a>0, this relation can be
rewritten as

ˇ S < 4 .N C 1/a

2N 4 C a :

The right hand side is positive if

4
N C 1 ;

a<

so both a and ˇ S have to be sufficiently small. Notice that the above bound on a
decreases in N and converges to zero as N !1 . So increasing the number of firms
reduces the stability region.
Models with isoelastic price functions can be examined similarly, the details are
not given here but are illustrated in the next subsection.
The stability of equilibria in multiproduct oligopolies with intertemporal demand
interaction was first examined in Szidarovszky (1990). These results with some
extensions are also reported in Okuguchi and Szidarovszky (1999). The model
and results presented in this section are slight generalizations of those given in
Szidarovszky and Zhao (2004).

4.3.2 Discrete Time Models and Global Stability

The global asymptotic stability of oligopolies with intertemporal demand interaction
can be discussed in a similar fashion to the case of concave Cournot models in
Chap. 2. In the following example we illustrate some global dynamic properties and
complex asymptotic behavior by using the methods applied in earlier chapters.
Example 4.11. In this example we will consider N firms, isoelastic price function,
f.Q;S/ D A=.Q C ˇ S S/, and linear cost functions, C k .x k / D d k

C c k x k for

k D 1;2;:::;N. Then the profit (4.22) of firm k becomes

Ax k
x k C Q k C ˇ S S .d k C c k x k /:

Assuming an interior optimum, the first order condition shows that at the optimum,

A.x k C Q k C ˇ S S/ Ax k
.x k C Q k C ˇ S S/ 2

c k D 0;

implying that the solution is

s A
c k .Q k C ˇ S S/ .Q k C ˇ S S/:

z k D