Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 154

the local asymptotic stability of the equilibrium is guaranteed for N D 2 and a k 1
for k D 1;2. Since the best response functions are continuous, the N-dimensional
best response mapping

R 1 X ! ;:::;R N X !!

R .x 1 ;:::;x N / D x l x l

l ¤ 1 l ¤ N

maps the convex compact set X kD1 Œ0;L k into itself, the Brouwer fixed point the-
orem guarantees the existence of at least one equilibrium. The uniqueness of the
equilibrium however cannot be guaranteed as the following example shows.

Example 4.5. In Example 4.4 sele ct N D 3, L D 4, d k D 8 :5.k D 1;2;3/. We can

easily show that both x .1 k D 3 p 0:5 and x .2 k D 3 C p 0:5 are symmetric equilibria
by verifying that both satisfy the best response relations. Clearly 0 x .i /
k

L for

all k and i,furthermore

2d k
LN Q .i /
k

2d k
LN 2x .i /
k

17
12 2.3

R k . Q .i k / D

D D p 0:5/

p 0:5 p 0:5/

9 0:5

3 8:5.3

8:5
3 ˙

8:5
3 ˙

D p 0:5 D p 0:5 p 0:5 D

3

p 0:5 D x .i k :

D 3

Okuguchi and Szidarovszky (1999) discussed the discrete time dynamic model of
the linear case given in Example 4.1. The general nonlinear case with local stability
analysis has been examined in Li and Szidarovszky (1999 b ). The equilibrium anal-
ysis of Example 4.3 has been presented in Li et al. (2003) but no stability analysis
was given in the discrete time case.

4.2.2 Discrete Time Models and Global Dynamics

The global dynamic behavior in discrete-time labor-managed oligopolies is first
illustrated with the following example.

Example 4.6. Consider again the situation of Example 4.4. We now consider a semi-
symmetric oligopoly, that is, firms k, with k 2, have identical fixed costs, d k D
d 2 (k 2), identical constant speeds of adjustment a k D a 2 (k 2), as well as
identical initial outputs. Then their entire trajectories are identical. In this case as
before Q 1 D .N 1/x 2 and Q 2 D x 1 C .N 2/x 2 . If we assume that the capacity
limits of all firms are identical and equal to L, and the firms use partial adjustment
towards the best response to update their quantity selections, then the adjustment
process is represented by the two-dimensional dynamical system