Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 190

This value is larger than x k .t/ if and only if

LN Q k c k >2x k .t/:

Notice also that the profit function of firm k has two parabolic segments, both are
concave, and they have identical derivatives at x k .t/. Consequently, define

8
<

LN Q k c k
2

if LN Q k c k 2x k .t/;

z k D

:

LN Q k c k C 2 k x k .t/
2 C 2 k

otherwise;

then the best response of firm k is given by

8
<

z k

0 if 0;

R k .Q k ;x k .t// D

z k L if L;

:

z k otherwise.

It is easy to see that in both cases the derivative of the best response with respect
to Q k is between 0 and

1

2 so the local stability properties of the corresponding
dynamical system are similar to the concave oligopoly case. Note that the best
responses in this model are piece-wise linear. Therefore, the dynamical system
based on partial adjustment towards the best response belongs to the same class
as the models with linear and quadratic cost functions. Since we have analyzed the
latter models in detail in Chap. 1, we abstain from presenting the details of a global
analysis of the present model. Instead we leave such an analysis to the reader. Con-
sider finally the symmetric case, when a k

a, c k c, d k d, L k D L, k D

and the initial outputs are identical. Then Q k D .N 1/x,and

8
<

LN .N 1/x c
2

if LN .N 1/x c 2x;

z

D

: LN .N 1/x c C 2x

2 C 2

otherwise:

Then the common best response of the firms is

8
<

z 0;

0 if

R.x/ D

z L; L if

:

z otherwise: