Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 150

1− a 1 (1+ r 1 ) 1− a 2 (1+ r 2 ) 1− a S -1 (1+ r S -1 ) 1− a S (1+ r S )

λ

Fig. 4.5 The discrete time model of labor-managed oligopolies. The determination of the
eigenvalues that are roots of the graph of g./, plotted here with a sign change in the j values

which is clearly strictly concave. The first order condition

d k
p k x k

B
p k D 0

implies that

r d k
B ;
and this is independent of the outp ut sele ctions of the competitors. Hence there is

x k D

a unique equilibrium with x k D p d k =B for all k, and since R 0 k 0, the matrix
(2.20) becomes diagonal with diagonal elements 1 a 1 ;:::;1 a N , which are the
nonzero eigenvalues of the Jacobian. So the equilibrium is locally asymptotically
stable if a j <2for all j and is unstable if for at least one j, a j >2.

Example 4.3. Let us modify the payoff function (4.13) of Example 4.2 by assuming
an isoelastic price function and that the labor-independent cost is a linear function
of the output x k . In this case we have

A
Q ; k .x k / D p k x k ; C k .x k / D d k C c k x k ;

f.Q/ D

so that

A
x k C Q k .d k C c k x k /
p k x k

x k

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

A
p k .x k C Q k / W

c k
p k

d k
p k x k :

D (4.15)