Modified and Extended Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 191

The dynamical system is therefore

x.t C 1/ D aR.x.t// C .1 a/x.t/:

This is again a piecewise linear model and can be analyzed in a similar way to the
one-dimensional examples studied before. We leave the analysis as an exercise for
the reader.

We mention here that Szidarovszky and Yen (1995) have introduced stability
conditions in the linear case. These results and some extensions are presented in
Okuguchi and Szidarovszky (1999). This latter topic also discusses the continuous
case where it is shown that the dynamical system is equivalent to a classical Cournot
dynamics with modified speeds of adjustment.

4.5 Oligopolies with Partial Cooperation

It is well known that by selecting the Nash equilibrium quantities, firms in an
oligopoly are trapped in a prisoner's dilemma situation, and a common way to
increase their payoffs is for the firms to reach some sort of cooperation or collu-
sion amongst each other. In this section we introduce partial cooperation into the
framework of the oligopoly models studied earlier in the topic.
The idea of partial cooperation was introduced and first explored by Cyert and
DeGroot (1973). The survey paper of Szidarovszky et al. (2008) contains some
special results.
As before, let P D f.Q/denote the price of the common product that is pro-
duced by N firms, and let C k denote the cost function of firm k. Then the profit of
firm k can be obtained as

' k .x 1 ;:::;x N / D x k f.Q/ C k .x k /; (4.87)

where x k is the output of firm k, Q D P lD1 x l is the total production of the industry
and no cost externalities are considered. Cooperation among firms can be achieved
if each firm takes the profits of its competitors into account. If, for example, a firm
has equity positions in the other firms, it would certainly also care about the other
firms' profits. In this case, cooperation among firms can be achieved, since each
firm's objective function includes the profits of its competitors (see for example
Clayton and Jorgensen (2005)). A similar effect occurs if firms are linked by partial
equity interests and joint ventures (see for example Reynolds and Snapp (1986) and
Bresnahan (1986)).
Let the parameters kl 2 Œ0;1 denote the degree of cooperation of firm k toward
firm l (k;l 2f 1;2;:::;N g ;k ¤ l), then we assume that firm k maximizes ' k C

P l¤k kl ' l instead of its own profit (see also Kopel and Szidarovszky (2006)).
Thus the payoff function of firm k becomes