Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 59

Notice that conditions (A)-(C) (or (A),(B),(C') and (D)) were assumed for all
feasible values of x k and Q k , and they imply the existence of a Nash equilibrium.
However they need to be satisfied only in a neighborhood of an interior equilibrium
in order to guarantee the local asymptotic stability of that equilibrium. Observe first
that with given price and cost functions the best responses are fixed, so the r k values
are uniquely determined. The firms' choices are only the adjustment mechanisms,
which are characterized by the functions ˛ k . Clearly, conditions (2.21) and (2.22)
are satisfied, if all a k D ˛ 0 k .0/ values are sufficiently small.
Example 2.2. Consider again the case of a linear inverse demand and linear cost
functions as in Example 2.1, where r k

1
2

D for all k. Then (2.21) holds if a k <4

for all k, and (2.22) holds if

X

N

a k
4 a k <1:

(2.25)

kD1

In the further special case when the firms select identical adjustment schemes (that
is, when ˛ 0 k .0/ D a k a), then (2.25) can be rewritten in the form

4
N C 1 :

a< (2.26)

If a 2 .0;1, then this condition always holds for duopolies (N D 2). If N 3,then
this condition is violated with naive expectations .a D 1/. This is the result derived
by Theocharis (1960). The equilibrium can still be stabilized however by selecting
sufficiently small values of a.

a and

Consider next the nonlinear case with identical firms. In this case a k

r. Condition (2.22) can now be rewritten as

r k

2
1 r.N 1/ :

a< (2.27)

In this case we do not assume that the initial outputs of the firms are the same, so
the system cannot be reduced to a one-dimensional one. Notice that in the special
linear case with r D

1

2 , (2.27) reduces to (2.26).
It is also interesting to analyze condition (2.22) from the point of view of a single
firm k. If for any other firm l, a l .1 C r l / 2,or
X

r l a l
2 a l .1 C r l / 1;

l¤k

then the equilibrium becomes unstable regardless of the adjustment scheme of firm
k.Firmk is able to stabilize the equilibrium alone merely by selecting an adjustment
function ˛ k such that its derivative at zero is sufficiently small. That is, the equilib-
rium becomes locally asymptotically stable when a k satisfies the two relations

2
1 C r k

a k <