Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 151

Assuming an interior optimum in x k , the first order condition is

A
p k .x k C Q k / 2

d k
p k x k

C D 0;

implying that

p d k
p A p d k

R k .Q k / D Q k : (4.16)

In order to ensure that x k >0, we have to assume that d k <Afor all k. Simple dif-
ferentiation shows that the second order conditions are satisfied at the best response.
Next we will show the existence of infinitely many equilibria under realistic condi-
tions. From (4.16) and noting that x k D R k .Q k / we find that

1 C ! x k

p A p d k
p d k

p A
p d k Q D Q k C x k D D x k ;

implying that

p d k
p A

kD1

N

kD1

N

x k
Q D

1 D :

The payoff of firm k at any equilibrium is

1 p d k

p A

A
p k

c k
p k

d k
p k x k D

A
p k

c k
p k

Q W

W ;

Q

Q is sufficiently small, in particular if Q satisfies

which is positive for all k if

( p A. p A p d k /
p k W C c k

) :

Q<min
k

Hencewehaveshownthatifd k <Afor all k, then positive equilibria exist if and
only if

X

N

p d k p A:

D (4.17)

k D 1

In this case there are infinitely many equilibria, and the set of equilibria are all points
on the ray

p d k

p A Q x k D (4.18)

Q>0. In addition, if Q is sufficiently small, then the profits of all firms are

for any

positive at the equilibrium.