General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 104

of the rest of the industry at the equilibrium, therefore r k >0is possible for at most
one firm. Similarly to the concave case we assume that a k D ˛ 0 k .0/ > 0 for all k.
Number the firms in such a way that the different a k .1 C r k / values are

a 1 .1 C r 1 />a 2 .1 C r 2 /> >a s .1 C r s /;

and these values are repeated m 1 ;m 2 ;:::;m s times, respectively, among the N
firms. By adding the terms with identical denominators in the bracketed factor of
(2.23) we obtain (2.24), where at most one j can be positive. If all j values are
non-positive, then the problem remains the same as in the concave case with the
same stability results. Therefore assume now that there is a j 0 such that j 0 >0.
If j ¤ 0 and m j D 1,then1 a j .1 C r j / is not an eigenvalue of the Jacobian.
Otherwise it is, and the other eigenvalues are the roots of the equation

s

X

j
1 a j .1 C r j / D 0;

1 C

j D1

where we assume that all j ¤ 0.
Let g./ denote again the left hand side of the last equation. Then clearly

lim g./ D 1;

!˙1

( 1 if j D j 0 ;
˙1 if j ¤ j 0 ,
however in contrast to the concave case, g 0 ./ has no definite sign, that is, g is not
necessarily monotonic. All poles are less than unity. Depending on the value of j 0
we have the following cases:-
Case 1. j 0 D 1.
The graph of g./ for this case is shown in Fig. 3.1. There are s 2 real roots
between each pair of poles 1 a j .1 C r j / and 1 a j C1 .1 C r j C1 / for j D
2;:::;s 1. If the other two roots are real and they are between 1 a 1 .1 C r 1 /
and 1 a s .1 C r s /, then the equilibrium is locally asymptotically stable if 1
a 1 .1 C r 1 /> 1.
Case 2. j 0 D s.
The graph of g./ in this case is shown in Fig. 3.2. All roots are real, one is
before the smallest pole, one after the largest pole, and one between each pair of
poles 1 a j .1 C r j / and 1 a j C1 .1 C r j C1 / for j D 1;:::;s 2. All roots are
between -1 and 1 if 1 a 1 .1 C r 1 /> 1 and g. 1/ > 0 and g.1/ > 0.
Case 3. 1<j 0 <s.
The graph of g./ is shown in Fig. 3.3. There are s 2 real roots. If we assume
that the remaining two roots are real and are between 1 a 1 .1 C r 1 / and
1 a s .1 C r s /, then all roots are between 1 and1if1 a 1 .1 C r 1 /> 1
and g. 1/ > 0.

lim
!1a j .1Cr j /˙0

g./ D