General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 102

Here the denominator is always positive but the sign of the numerator is indeter-
minate. Hence, R k .Q k / is not necessarily monotonic, which stands in contrast to
the concave case discussed in the previous chapter. If we express the best response
functions in terms of the total output of the industry, then the resulting modified
best response function R k .Q/ will not be monotonic either. Therefore the existence
and uniqueness of the equilibrium cannot be examined in the same way as was done
for concave oligopolies. However, by using a different approach, the existence of a
unique equilibrium is proved in Szidarovszky and Okuguchi (1997), and this result
is also presented with further details in Okuguchi and Szidarovszky (1999).
Consider now an interior equilibrium, then from (3.2),

A Q k C k .x k / Q 2

D 0

for all k. The numerator of (3.3) at the equilibrium becomes

2A Q k
Q D

A
Q . Q 2 Q k /;

A

so R 0 k . Q k / 0 if and only if

Q 2 Q k .

Notice in addition that

R 0 k .Q k /> C 0 k Q 2

2C k Q

C 2C k Q D 1:

(3.4)

C 0 k Q 2

It is interesting to note that this is exactly the same lower bound as in the concave
case. If N D 2, then at a symmetric equilibrium R 0 k D 0 for k D 1;2: If the equilib-
rium is asymmetric, then R 0 k is positive for one firm and is negative for the other, so
R 0 1 R 0 2 <0. Assume next that N 3; and for all firms, x k Q k . This condition
means that there is no large firm dominating the rest of the industry. In this case
Q 2Q k for all k,so 1<R 0 k 0 which is similar to the concave case. Notice
that in the general case the condition Q 2Q k at the equilibrium can be violated
by at most one firm, so there is at most one firm with positive derivative R 0 k

at the

equilibrium.

Example 3.1. In Example 1.5 we have already considered the isoelastic case with
p D f.Q/ D A=Q and linear cost functions C k .x k / D d k C c k x k . There we derived
the equilibrium quantities of the firms which are given by

.N 1/ 2 Ac k
. P l c l / 2

.N 1/A
P l c l

x k D ;

for k D 1;2;:::;N, and the total industry output

.N 1/A
P l c l

Q D

: