Oligopolies with Misspecified and Uncertain Price Functions, and Learning - Nonlinear Oligopolies: Stability and Bifurcations - page 227

The numerator can be simplified to

4˛ 2 TŒT.˛ 2 a 2 .r C 1/ 2 / 2a.r C 1/:

Here the first factor is positive, and the second factor can be rewritten as

1 .aT/ 2 .r C 1/ 2
T

¤ 0;

since it is easy to see that in Case 3

1
r C 1 <.aT/ 2 :

.aT/ 1 < (5.46)

Hence all conditions of the Hopf bifurcation theorem are satisfied, and therefore
there is the possibility of the birth of limit cycles around the equilibrium.

Consider again the general equation (5.39) and assume very shallow best response
functions so that

r> 1

1 C Nh :
We can easily show that in this case all roots of this equation have negative real parts,
so the equilibrium is asymptotically stable. On the contrary assume that Re 0,
then

j C a.r C 1/ j a.r C 1/

and

T
p j 1;

j 1 C

so

ˇ ˇ ˇ ˇ ˇ

mC1 ˇ ˇ ˇ ˇ ˇ a.r C 1/ > Narh Dj Narh j ;

. C a.r C 1// 1 C

T
p

and hence cannot be a solution of (5.39).
In the general case, higher values of m in (5.39) require the use of computa-
tional methods to locate the eigenvalues. We note again that the case when for all
k, f k f (that is, the full information case) is the special case of model (5.36)
by selecting H k as the identity mapping with h k D 1 for all k. However there are
slight differences between the full information model presented in Sect. 2.6 and the
model shown in this section. In the full information case for all firms we assumed
time delays in the information on the output of the rest of the industry, and also
in the firms' own output levels. In this section we assumed that the firms receive
information only on the price, and they compute the output of the rest of the indus-
try by using the observed market price. Here they use delayed information on the
function values f.Q k C x k / (which is the actual market price), but they use their
most current output levels for x k .