General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 108

have stability if the product j r 1 r 2 j is sufficiently small, which holds if c 1 and c 2 are
sufficiently close to each other.

Example 3.3. Next we examine an N-firm semi-symmetric oligopoly with linear
cost functions, so we assume that firms 2;3;:::;N have identical marginal costs,
c k D c 2 .k D 2;3;:::;N/, identical capacity limits and common linear adjustment
functions, and their initial outputs are also the same, so that x 2 .0/ D ::: D x N .0/.
Given these assumptions the entire output trajectories of these firms are the same.
Therefore we get a two-dimensional system with state variables x 1 and x 2 where
x k D x 2 for k 2. In this case Q 1 D .N 1/x 2 and Q 2 D x 1 C .N 2/x 2 .
Assuming that the capacity limits L k are sufficiently large the general expressions
for the equilibrium quantities given in Example 3.1 imply for the semi-symmetric
case that

1

.N 1/A
c 1 C .N 1/c 2

.N 1/c 1
c 1 C .N 1/c 2

x 1 D

.N 1/c 2 .N 2/c 1
c 1 C .N 1/c 2

.N 1/A
c 1 C .N 1/c 2

D

1

.N 1/A
c 1 C .N 1/c 2

.N 1/c 2
c 1 C .N 1/c 2

x 2 D ::: D x N D

:

.N 1/A
c 1 C .N 1/c 2

c 1
c 1 C .N 1/c 2

D

For the total industry output in equilibrium we obtain

.N 1/A
c 1 C .N 1/c 2 :

Q D x 1 C .N 1/x 2 D

The derivatives of the best replies are obtained from (3.3) as

Q

A 2c 1 .N 1/c 2 C .3 2N/c 1

2.N 1/c 1

r 1 D R 0 1 . Q 1 / D

Q D

;

2c 1

and

Q

A 2c 2 c 1 .N 1/c 2

2.N 1/c 2

r 2 D R 0 2 . Q 2 / D

Q D

:

2c 2

Conditions (3.5) for k D 1 and k D 2 are of the form

c 1 C .N 1/c 2
N 1

c 1 C .N 1/c 2
N 1

c 1 ; 2 ;

where the second inequality always holds and the first one can be written as

c 2
c 1

N 2
N 1 ;

WD