Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 49

Chapter 2
Concave Oligopolies

In the previous chapter we have seen that except in very special cases oligopoly
models have nonlinear features and therefore can generally exhibit a vast array of
dynamical behavior ranging from simple to complicated. Under special conditions
however the uniqueness of the equilibrium can be guaranteed, simple conditions can
be derived for the local asymptotic stability of the equilibrium with both discrete
and continuous time scales, and the global dynamics are less complicated and can
be handled with some of the standard tools of nonlinear dynamical systems. In this
chapter we will consider concave oligopolies, which are the straightforward gener-
alizations of linear oligopolies and are the most frequently discussed cases in the
literature (see for example, Okuguchi and Szidarovszky (1999) and the references
therein).
In the first section we consider oligopolies both without and with cost external-
ities, derive the best response functions in both cases and their various properties
that will be invoked in the ensuing analysis of the dynamics. In Sect. 2.2 we exam-
ine the local stability of discrete time oligopolies using the adjustment processes
introduced in Sect. 1.2. In Sect. 2.3 we then consider the global stability of the dis-
crete time oligopoly, bringing to bear the tools developed in Sect. 1.3. Section 2.4
gives a brief description of the local stability of the dynamics in both discrete and
continuous time where firms use gradient adjustment processes. The local stabil-
ity of continuous time oligopolies using certain types of best response dynamics
is studied in Sect. 2.5. Finally in Sect. 2.6 we study the impact of various kinds of
information delays on the local stability of continuous time oligopolies using best
response dynamics.

2.1 Introduction

We will first consider oligopolies without cost externalities. As in the previous chap-
ter, let N be the number of firms, x k the output of firm k.k D 1;2;:::;N/,and

Q D P kD1 x k the total output of the industry. If p D f.Q/denotes the inverse
demand function and C k .x k / is the cost of firm k, then the profit of this firm can be