General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 125

(2002) and Furth (1986, 2009)). We will consider a duopoly market and we will
show that in a simple model with cost externalities we obtain several coexisting
equilibria. Since an equilibrium point can be considered as a convention that arises
among firms interacting repeatedly, stability arguments are often used to solve
this coordination problem. See for example, Van Huyck and Battalio (1998) and
Van Huyck et al. (1984, 1997). If a stability argument selects a single equilibrium,
this point can be considered as the solution of the oligopoly game. However, as we
will see, in the model with cost externalities multiple equilibria survive this type
of refinement and several (locally) stable equilibria coexist. Each of these equilibria
has its own basin of attraction and, consequently, the dynamic process becomes path
dependent. The long run outcome of the players' myopic output decisions crucially
depends on the initial production quantity. Hence, in such a situation it is not suffi-
cient to analyze the local stability properties. In order to be able to give some insight
into the long run market outcome, it is important to gain some knowledge about the
boundaries that separate the basins of attraction of the various coexisting equilibria,
and to study the role of these boundaries in the occurrence of global bifurcations
that drastically change the topological structure of the basins.
Recall from Example 1.6 that if the inverse demand function is linear, p D
f.Q/ D A BQ, and the cost functions of the oligopolists are characterized by
interfirm externalities, that is C k .x k ;Q k / D x k M k .Q k / with M k .Q k / D A B.1 C
2 k /Q k 2B k Q k , then the best response of firm k is given by

8
<

0 if k Q k .1 Q k / 0;

R k .Q k / D

k Q k .1 Q k / L k ;

L k if

:

z k

otherwise;

where z k D k Q k .1 Q k / and L k denotes the capacity of firm k. The parameters
k measure the intensity of the interfirm cost externality (see Kopel (1996)). In what
follows we consider a duopoly market (N D 2), so that Q 1 D x 2 and Q 2 D x 1 .We
let k 2 .1;4 and for simplicity we assume that L k D 1. Under these assumptions
the reaction functions reduce to

R 1 .x 2 / D 1 x 2 .1 x 2 /; R 2 .x 1 / D 2 x 1 .1 x 1 /: (3.19)

The Nash equilibria of this duopoly are located at the intersections of the two
reaction curves x 1 D R 1 .x 2 / and x 2 D R 2 .x 1 /. The reaction functions are shown
in Fig. 3.10, where the two panels illustrate that beside the trivial Nash equilibrium
O D .0;0/, multiple interior Nash equilibria can exist depending on the level of the
cost externalities. For example, for 1 D 3; 2 D 3:5 there is just one interior Nash
equilibrium E S (part (a)), whereas for 1 D 3:7; 2 D 3:5 there are two additional
interior Nash equilibria E 1 and E 2 (part (b)). Analytically, the interior equilibria are
obtained as the real solutions of the fourth degree algebraic system

D 1 x 2 .1 x 2 /; x 2 D 2 x 1 .1 x 1 /;

x 1

and this system can have up to four solutions.