Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 152

From (4.16) we know that the derivatives of the best response functions are

p d k
p A p d k

D R 0 k .Q k / D

r k ; (4.19)

and simple substitution of D 1 into the eigenvalue equation (4.12) of the Jacobian
shows that it is always an eigenvalue. 1 Therefore we cannot establish local asymp-
totic stability in the usual sense. This result clearly should be the case, since a small
move away from any given equilibrium along the ray (4.18) would result in another
equilibrium and since the state remains there for all future time, the trajectory of the
state variable does not converge back to the original equilibrium.
Example 4.4. We will consider now a special N-firm labor-managed oligopoly.
Assume that the firms have identical capacity limits, L, and the price function is
f.Q/ D LN Q. Notice that the price is always non-negative. We also assume
that the number of labor units is a quadratic function for each firm, h k .x k / D p k x k .
Then the profit (4.9) per labor unit of firm k is given by

x k .LN x k Q k / Wp k x k d k
p k x k

LN Q k
p k x k

d k
p k x k

. 1 D

p k C W/:

Notice that the value of p k has no effect on the best response of firm k, it only
affects the optimal profit. The derivative of this profit function can be written as

LN Q k
p k x k

2d k
p k x k

1
p k x k

C D .2d k x k .LN Q k //;

implying that the profit function is increasing for x k <2d k =.LN Q k / and
decreasing if x k >2d k =.LN Q k /. So the stationary point is

2d k
LN Q k :

z k

D

Since z k

is necessarily positive, the best response of firm k is

( z k ; if z k L,
L; if z k >L,

R k .Q k / D

which is illustrated in Fig. 4.6. If LN 2d k =L 0,thenR k .Q k / D L for all Q k .
We can also show that 0<r k <1if the best response is interior. In this case

z k 2
2d k >0:

2d k
.LN Q k / 2

D R 0 k .Q k / D r k D

1 Recall the definition of j above (2.24), and make use of (4.17) and (4.19).