The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 6

Notice that with zero fixed cost the equilibrium profit of firm k is non-negative, and
if x k >0 and d k is sufficiently small, then ' k is necessarily positive. If capacity
limits are present and this “unconditional” equilibrium becomes infeasible, then
the “conditional” equilibrium can still be computed, but cannot be represented by
simple equations. Okuguchi and Szidarovszky (1999) discuss algorithms to compute
such equilibria.

If nonlinearity (which was in the form of capacity constraints in the above exam-
ple) is introduced into the models, then usually numerical methods are required to
compute the equilibrium in the general case. Analytical methods are available in
only very special cases, for example by assuming symmetric or semi-symmetric
firms. If all firms have identical capacity limits and cost functions, and their initial
outputs are also the same, then the oligopoly is called symmetric .If.N 1/ firms
are identical in this sense and one firm is different, then we have a semi-symmetric
case. We will frequently make use of such special cases in later chapters.

Example 1.2. Assume again a linear price function f.Q/ D max f 0;A BQ g but
quadratic cost functions C k .x k / D c k x k C e k x k : The profit of firm k now has the
form

( x k .A Bx k BQ k / .c k x k C e k x k / if x k C Q k

A

B ;

' k .x 1 ;:::;x N / D

.c k x k C e k x k / otherwise:

For the sake of simplicity we assume again that P kD1 L k

A=B, that is, the zero

segment of the price function cannot occur.

(i) Assume first that for all k, 0<e k . Then the cost function is convex, so that
marginal costs are increasing in x k , and the profit is concave in x k .Since

@' k
@x k D A 2Bx k BQ k c k 2e k x k ;

the best response is unique and has the form

8
<

0 if A BQ k c k 0;
L k if A 2BL k BQ k c k 2e k L k 0;
.A BQ k c k /=.2.B C e k // otherwise;

R k .Q k / D

:

which is piece-wise linear, similar to the case of the previous example where
both demand and cost were linear. Notice that if A c k ,thenR k .Q k / D 0
regardless of the value of Q k , so we assume that A>c k for all firms. In the
case of duopoly the x 1 intercept of R 1 .x 2 / is the monopoly output x 1 of firm
1, and the x 2 intercept of R 2 .x 1 / is the monopoly output x 2 of firm 2. It can
be proved (see Chap. 2) that there is always a unique Nash equilibrium in this
case.