The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 4

equilibria may be finite or infinite. In the case of multiple equilibria, the problem
of equilibrium selection arises. In such situations, the non-negativity of the profits
and the dynamic evolution of the oligopoly game, determined by the adjustment
processes and the degree of bounded rationality of the players, can be used to deter-
mine which equilibria are realistic and which are not. We will return to this problem
in later chapters.

Example 1.1. Consider the case of a linear oligopoly where the price function has
the form f.Q/ D max f 0;A BQ g with Q D P kD1 x k and C k .x k / D d k C
c k x k (1 k N) with A, B, c k , d k being all positive. Note that the max operation
ensures that the price is zero for total output above the market saturation point A=B.
In this case ' k is strictly concave in x k with derivative

( A BQ k 2Bx k c k

if Q k C x k < B ;

@' k
@x k D

if Q k C x k > B ;

c k

and this derivative does not exist if Q k C x k D A=B.
If for any firm k it is the case that A c k 0,then@' k =@x k is always negative, so
the best response of this firm is always zero, and hence entry for this firm is blocked.
Hence such firms do not participate in production, and therefore we can ignore them
in all further discussions. If for firm k, the capacity limit L k is sufficiently large, then
with A>c k , its monopoly quantity is x k D .A c k /=.2B/, which can be obtained
from the first order condition with Q k D 0.
In order to determine the best response of the firms, consider firm k and assume
that the total production level Q k of the rest of the industry is fixed. Notice first that
the best response of this firm cannot exceed A=B Q k , that is, the total industry
output cannot be larger than the market saturation point. In contrast, assume that
x k >A=B Q k , then the price is zero, and by decreasing the value of x k by a small
amount, the price will be still zero and the cost decreases. So the payoff of this
firm would increase contradicting the assumption that x k is the firm's best response.
Therefore with fixed values of Q k the best response of firm k is selected in the
interval Œ0; L k with

L k

D min f L k ;A=B Q k g . If the capacity limits of the firms

are sufficiently small, that is, when P kD1 L k A=B, then the zero segment of the
price function cannot occur, so L k D L k for all k and Q k . For the sake of simplicity
in the following discussion we will assume that this is the case. Since ' k is strictly
concave in x k , the best response of firm k is unique and is given as

8
<

@' k

0 if

@x k j x k D0 0;

@' k

R k .Q k / D

L k if

@x k j x k DL k 0;

:

z k otherwise;

where z k is the solution of

@' k
@x k D 0;