Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 93

with

A k ./ D C a k S k
q k

C 1 .l k C1/

;

and

B k ./ D a k r k T k
p k

C 1 .m k C1/ ;

where

( 1 if l k

D 0;

q k D

l k if l k >0;

and

( 1 if m k D 0;
m k if m k >0:
The set of equations (2.53) have non-trivial solution if and only if

p k D

0 1

A 1 ./ B 1 ./ B 1 ./
B 2 ./ A 2 ./ B 2 ./
: : : : : : : : :
B N ./ B N ./ A N ./

@ A D 0:

det (2.54)

Notice that this determinant is the same as (E.2) discussed in Appendix E, where
it is shown that this equation can be rewritten as

" 1 C #

N N

Y X

B k ./
A k ./ B k ./

.A k ./ B k .// D 0: (2.55)

k D 1 k D 1

Since A k ./ and B k ./ are all rational functions, this equation is equivalent to a
polynomial equation showing that there is a finite number of eigenvalues.
Equation (2.55) generally reduces to a very complicated high order polynomial
equation, so no general analytic results can be derived. However in the special case
of symmetric firms we will be able to derive simple stability conditions and exam-
ine the complex asymptotic behavior of the system. For this purpose assume that
a k a;r k r;T k T;S k S;m k m;l k l; so q k q and p k p that
is the firms are identical with respect to speeds of reaction and slopes of their reac-
tion functions at the steady state and furthermore they use the same time weighting
schemes. Assume in addition that the initial output levels of the firms are identi-
cal. Then system (2.52) reduces to a one-dimensional integro-differential equation,
since the assumed symmetry implies that the output trajectories of the firms are also
identical. Therefore in (2.53) we set, v k

v ;A k ./ A./ and B k ./ B./,so

that it simplifies to

A./ C .N 1/B./ D 0; (2.56)