The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 14

Obviously, if firm k's unit costs c k are too high (for a given number of firms N),
production might not be feasible (so that firm k offers x k D 0). Furthermore, for
increasing N (and given unit costs) some (high-cost) firms might drop out of the
market. The equilibrium price is given by

2A C P l D 1 c l
N C 2

p D >0

and the equilibrium profit of firm k is

.2A C P lD1 c l .N C 2/c k / 2

' k D

2.N C 2/ q .N C 2/.NA P lD1 c l /

:

Example 1.5. Assume again linear cost functions, C k .x k / D d k C c k x k , but isoe-
lastic (hyperbolic) price function, f.Q/ D A=Q. The form of the profit of firm k
depends on whether Q k is positive or zero. If Q k >0;then

Ax k
x k C Q k .d k C c k x k /;

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

and if Q k D 0; then

( A .d k C c k x k / if x k >0;
d k

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

D 0;

if x k

where we assume that firm k cannot exit the market, so with zero production level
it must face fixed costs. Notice that if Q k D 0, then with any x k >0, the revenue
of firm k is always A. In this case firm k has no best response and its interest is to
select a very small output level, since the supremum of its profit occurs at x k D 0.
Assume next that Q k >0:In maximizing ' k , the first order condition is

AQ k
.x k C Q k / 2

c k D 0:

Since ' k is strictly concave in Q k , the best response of firm k is

if q AQ k
c k

8
<

0 Q k 0;

q AQ k
c k

R k .Q k / D

L k if Q k L k ;

:

p AQ k =c k Q k

otherwise:

This function is illustrated in Fig. 1.9. We note that the best response is first increas-
ing and then decreasing. This is in contrast to the examples considered previously,
where the best responses were decreasing everywhere. Some authors consider