The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 9

R k .Q k /

L k

Q k

P i ¤k L i

A c k

B

Fig. 1.4 Example 1.2; the Cournot model with linear price function and quadratic cost function
in the case of duopoly .N D 2/. The figure shows the reaction function of a typical firm in case

(iii) when e k D B. The number of equilibria may be 1;3 or infinite

the two straight lines either coincide or are parallel. Therefore there are either
infinitely many equilibria, or a unique boundary equilibrium.
(iii) In the case where e k D B for all k, the profit function assumes the linear
form

' k D x k .A BQ k c k /;

therefore

8
<

0 if A BQ k c k <0;

R k .Q k / D

L k if A BQ k c k >0;

:

if A BQ k c k

D 0:

arbitrary x k

We can assume again that c k <A,otherwiseR k .Q k / D 0 for all Q k .Thisbest
response function is illustrated in Fig. 1.4 in the case when

< X
i

A c k
B

L i :

¤ k

In the case when the last inequality becomes an equality, the vertical seg-

ment moves to Q k D P i ¤k L i . If however the above relation is violated with
strict inequality, then R k .Q k / D L k for all Q k . Depending on the values of
.A c k /=B and L k , in the duopoly case the number of equilibria can be 1, 3
or infinite; Fig. 1.5 shows a case where three equilibria exist.
(iv) Assume finally that for all k, e k < B. In this case ' k is convex in x k ,sothe
best response is located at an endpoint of the feasible interval [0;L k ] and is of
the form