Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 187

.1 a/. r x /

r x C ar Q D 0; (4.77)

and

.1 a/. r x /

r x C ar Q C Nar Q D 0: (4.78)

Both of these equations can be written as the quadratic equations

2

C . 1 C a.1 C r Q / r x / C r x .1 a/ D 0;

and

2

C . 1 C a.1 C .1 N/r Q / r x / C r x .1 a/ D 0:
By using the result of Lemma F.1 (from Appendix F) and some simple algebra it
can be seen that all roots are inside the unit circle if

r x .1 a/<1; (4.79)

a.1 C r Q r x />0; (4.80)

2.1 C r x / a.1 C r x C r Q .1 N//>0; (4.81)

where the first two inequalities always hold for a>0, and the third is satisfied if
a is a sufficiently small positive value, that is, if the firms select a small common
constant speed of adjustment, or have a small common derivative value ˛ 0 .0/.

To compare the results just obtained for the best reply dynamics with adaptive
expectations, we now turn to partial adjustment towards the best response with naive
expectations, namely

0 0

@ X

1 1

@ R k A x k .t/ A ; 1

x k .t C 1/ D x k .t/ C ˛ k x l .t/;x k .t/ k N/;

l ¤ k

(4.82)
which is a straightforward extension of the system (1.30) to take into account
production adjustment costs. Conditions for the local asymptotic stability of the
equilibrium are given in the following theorem.

Theorem 4.3. Assume that a k >0 for all k , C 0 k 0 for all k and x k , assumptions
(A)-(C) of Sect. 2.1 of concave oligopolies hold, furthermore the conditions

2
1 r jx C r jQ

0<a j < .j D 1;2;:::;N/

are satisfied. Then the equilibrium of the system (4.82) is locally asymptotically
stable, if

N

X

a k r kQ
2 a k .1 r kx C r kQ / >0:

1 C

k D 1