Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 246

Theorem 5.4. Under assumption (5.97) the steady state of system (5.91) is locally
asymptotically stable if for all k

a k A
.N C 1/B <2

(5.98)

and

N

X

a k k .N C 1/
2.N C 1/B a k A > 1:

(5.99)

k D 1

If for at least one k ,

a k A
.N C 1/B 2

or

X

N

a k k .N C 1/
2.N C 1/B a k A < 1;

k D 1

then the steady state is unstable.

Notice that both stability conditions (5.98) and (5.99) are satisfied if the speeds
of adjustment a k are sufficiently small for all firms.
Consider now the special case of symmetric firms, when a k

a and c k c.

Notice that in this case

Nc
N C 1 c D

c
N C 1 <0;

k D

so condition (5.97) is satisfied. Relation (5.98) can be rewritten as

a< 2.N C 1/B
A

(5.100)

and (5.99) has the form

Na.N C 1/
2.N C 1/B aA > 1;

which is equivalent to

2.N C 1/B
A N.N C 1/ D

2.N C 1/B
A C cN

a< : (5.101)

This inequality is stronger than (5.100), so if (5.101) holds, then the steady state is
locally asymptotically stable, and if (5.101) is violated with strict inequality, then
the steady state is unstable.
The stability results given above express sufficient conditions for the local
asymptotic stability of the equilibrium, so they ensure the convergence of the
adjustment process provided that the initial factors selected by the firms, " k .0/,