The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 15

R k .Q k /

L k

Q k

P i ¤k L i

A
4c k

A
c k

Fig. 1.9 Example 1.5; the best response function ( thick line )ofatypicalfirmk with a linear cost
function and hyperbolic price function

x 1 D D x N D 0 as a trivial equilibrium in a limiting sense. 3 In a non-trivial
equilibrium, when N Q>0;still some equilibrium outputs might be zero, when the
marginal costs, c k , for some firms are very large. By assuming that the value of
L k is sufficiently large for all firms, the positive equilibrium can be computed as
follows. Since for all k,

s A.Q x k /
c k

x k D .Q x k /;

we have

c k Q 2

D A.Q x k /;

implying that

AQ c k Q 2
A

x k D :

Summing this equation over all N firms, we obtain

NAQ Q 2 P k D 1 c k
A

Q D :

So the total output of all firms is

.N 1/A
P kD1 c k

Q D

;

3 See Agliari et al. (2005, 2006), Agliari (2006) and Matsumoto and Serizawa (2007).