The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 12

different ways. At any equilibrium, Q D 1:5; so the equilibrium price is f. Q/ D 1;
and therefore the profit of firm k is always positive, being given by

' k .x 1 ; x 2 / D x k 1 0:5x k

D 0:5x k :

Example 1.4. In this example we assume linear cost functions, C k .x k / D c k x k
with some positive constant c k , and a quadratic price function where

( A Q 2

p A;

if 0 Q

f.Q/ D

if Q> p A:

0

It is also assumed that A>c k f o r all k. Notice that at the best response of firm k
it is the case that Q k C x k

p A, otherwise the value of x k can be decreased by a
small amount, when the price is still zero and the cost would decrease. There fo re at

the best response of all firms the total output has t o b e less than or equal to p A.For
the sake of simplicity assume that P kD1 L k

p A, the other case can be discussed
in a similar way. By assuming an interior optimum, the first order condition implies
that

@
@x k Œx k .A .x k C Q k / 2 / c k x k D A 3x k 4x k Q k Q k c k

D 0:

If c k A,then' k is strictly decreasing in Q k , so the best response of firm k is
always zero. Therefore we may assume that c k <Afor all k. The solution of the
above quadratic equation is

q Q k C 3.A c k / 2Q k :

1
3

z k

D

Since the payoff function of firm k is strictly concave in x k , the best response
assumes the form

8
<

z k <0; 0 if

R k .Q k / D

z k >L k ;

L k if

:

z k

otherwise:

This function is illustrated in Fig. 1.8. Simple differentiation shows that z k is strictly
decreasing and convex in Q k . It can be proved that there is always a unique equi-
librium. Since at x k D 0 the profit of firm k is zero, the profits at the best responses
and therefore the equilibrium profits must be non-negative for all firms. In the case
of an interior equilibrium the equilibrium quantities can be derived in closed-form.
The first order condition may be rewritten as

A Q 2

C x k . 2Q/ c k D 0;