Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 140

In the existing literature, market share attraction models are predominantly used
in a static framework. In this literature, similarly to that on oligopolistic competition,
the emphasis of the investigation lies on demonstrating the existence and uniqueness
of Nash equilibria (see Friedman (1958), Mills (1961) and Schmalensee (1976))
and on studying the properties of these Nash equilibria (see Monahan (1987) and
Karnani (1985)). Only a few papers have addressed the problem of local stability
of these equilibria (Schmalensee (1976) and Balch (1971)), but issues concerning
the global dynamics of these types of models have been completely disregarded.
This is quite surprising, since Schmalensee (1976) remarked: “Ideally, analysis of
the dynamic behavior of a model of this sort away from equilibrium can perform
two services. First, if it turns out that additional parameter restrictions are needed
to ensure global stability, more comparative static information may be obtained.
Second, such analysis can provide a further test of the model's plausibility, since
systems that go to equilibrium only if they begin life in a neighborhood thereof are
unattractive.” (p. 502). One reason for a lack of understanding of the global proper-
ties of dynamic models is apparently that there has been a lack of appropriate meth-
ods to carry out such an analysis. In this subsection we will introduce a dynamic
version of a market share attraction model. We will assume that competitive effort
allocations for the two brands are adaptively adjusted over time, and characterize
the global properties of this model. Our main concern here is to provide a rigorous
description of the set of initial effort allocations which leads to convergence to the
Nash equilibrium, and the changes of this set if parameters of the model are varied.
A dynamic version of a market share attraction model can, for example, be
obtained on the basis of marginal profits. We assume that at time t the marketing
efforts of the next period, x 1 .t C 1/ and x 2 .t C 1/, are determined according to the
adjustment process
x 1 .t C 1/ D x 1 .t/ C 1 .x 1 .t// @' 1 .x 1 .t/;x 2 .t//
@x 1

;

x 2 .t C 1/ D x 2 .t/ C 2 .x 2 .t// @' 2 .x 1 .t/;x 2 .t//
@x 2

: (4.4)

Notice that this dynamic process is a generalization of the gradient adjustment pro-
cess, since in this case the constant speeds of adjustment of each firm are replaced
by speeds of adjustment dependent on the marketing effort of the particular firm. In
Sect. 1.3.3 a similar model was examined.
The expressions i . / determine by how much efforts can vary from period to
period and they can be interpreted as the “speeds of reaction.” Obviously, the steady
states of the dynamical system (4.4) are given as solutions of the equations

1 .x 1 / @' 1 2 .x 2 / @' 2

@x 1 D 0; @x 2 D 0:

Any interior Nash equilibrium of the underlying market share attraction game
is obtained as the positive solution of the first order conditions @' 1 =@x 1 D 0;