Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 161

Both roots have negative real parts if all coefficients are positive, which occurs
if 1 .N 1/r >0.Thatis,ifr <1=.N 1/, then the equilibrium is locally
asymptotically stable, and if r >1=.N 1/, then it is unstable.
In the case when m D 1, the cubic equation (2.59) is obtained. All coefficients are
positive if r <1=.N 1/, and the Routh-Hurwitz criterion shows that the eigenval-
ues have negative real parts if (2.61) is satisfied. Since all coefficients on the left
hand side are positive, this inequality always holds implying the local asymptotic
stability of the equilibrium. If r >1=.N 1/, then the equilibrium is unstable.

The special case of Example 4.1 has been examined in Okuguchi and Szidarovszky
(1999) with continuous time scales. The further special case of Example 4.3 with
infinitely many equilibria was investigated by Li et al. (2003), in which the theory
of differentiable manifolds was used to prove that the equilibrium ray is a strongly
attracting set.

4.3 Oligopolies with Intertemporal Demand Interaction

In this section we consider an N firm oligopoly without externalities, but with
intertemporal demand interaction. As in the earlier chapters, let f denote the market
price function and C k the cost function of firm k.1 k N/. Intertemporal demand
interaction is often a realistic assumption, since previous consumption might satu-
rate the market, or might contribute to taste and habit formation for the consumers,
to mention only some of the most common phenomena.
Okuguchi and Szidarovszky (2003), Szidarovszky and Zhao (2006) and Chiarella
and Szidarovszky (2008b) introduced and analyzed various dynamic models that
extend the classical oligopoly models to include intertemporal demand interaction.
The special case of market saturation was examined by Szidarovszky et al. (2006).
Consider first discrete time scales, and let S.t/ represent the cumulative effect
of the earlier consumptions up to time period t. If for example, market saturation is
considered, then after each time period a certain proportion of goods already in use
by the consumers remains in usable condition, while the rest has to be replaced. It
is assumed that variable S.t/ follows the dynamic rule

X

N

S.t C 1/ D ˇ S S.t/ C x k .t C 1/; (4.21)

k D 1

where 0 ˇ S <1is a given constant. This constant represents how past experience
with the product affects current demand, and in the case of market saturation it
shows the fraction of goods remaining in usable condition after each time period.
If we assume that the price depends on the current value of the variable S,then
the profit of firm k at time period t C 1 can be written as

x k f.x k C Q k .t C 1/ C ˇ S S.t// C k .x k /

(4.22)