Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 153

R k

L

2 d k
LN

Q k

0 LN

2 d k
L

LN −

Fig. 4.6 Example 4.4; the discrete time model of an N-firm labor-managed oligopoly. Linear
price function and quadratic labor unit functions. Best response of firm k

From the first order condition we have

z k .LN Q C z k / C 2d k

D 0;

so that

z 2
k

D 2d k z k .LN Q/;

implying that

2d k z k .LN Q/
2d k

r k D <1:

Hence Case 1 (shown in Fig. 4.3) occurs with all j >0, so the equilibrium is locally
asymptotically stable if for all k,

a k .1 C r k /<2

and

X

N

r k
1 C r k

<1:

k D 1

Since

r k
1 C r k D

1
1 C

< 1

2 ;

1
r k