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Corchón and Mas-Colell [1996. On the stability of best reply and gradient systems with applications to imperfectly competitive models. Economics Letters 51, 59-65] showed that in heterogeneous oligopoly (almost) everything is possible. In order to obtain a similar result for homogeneous oligopoly, either one needs an externality in the cost function, or the reaction correspondences should fulfill a special condition.Keywords: Cournot oligopoly; (In)Stability; (Limit)Cycles; Morse theoryJEL classification codes: D43; C62

In this paper, a solution concept from cooperative game theory is applied to a special kind of oligopolistic markets. Traditionally, oligopoly theory uses non-cooperative game theory. The cooperative solution concept, the Price Core, has been specially developed for this paper. In contrast to the Core, in the Price Core, different consumers may pay different prices for the same commodity. An example shows that the set of competitive allocations may be a proper subset of the Core and that the Core may be a proper subset of the Price Core.

Shortly after World War II, many articles on matrices were published in economic journals. This chapter is an incomplete review of some of the applications of the theory of matrices to the stability of the equilibria of both: exchange economies and Cournot oligopolies. When in the seventies an `(almost) everything is possible’ result was derived, there was a shift in interest from linear(ized) models towards non–linear ones and economists were not longer interested in stability properties. When economists are not longer interested in stability, they do not write articles on matrices anymore. That is why nowadays there are only a few articles on matrices in economic journals.

Many research contributions have investigated the identity of the leader in oligopolistic markets. As a general rule, this literature points to the leader being either the most efficient, the largest or the best informed player. However, there tend to be exceptions to this rule and it is unclear who will become the leader when firms at the same time are different in size, efficiency of production and/or the quality of the information they have. The present paper reveals insights regarding this last question. As such, it points to the elements “that really matter” to explain for leadership when firms are different from each other in more than one respect. It turns out that the most efficient firm becomes the leader, regardless whether or not its rival has invested in more capacity. While this result tends to be in line with the previous findings regarding the identity of the leader being the relatively more efficient firm, it contradicts the “dominant firm” scenario regarding price leadership. This is the result of enlarging the strategy space of the players, by including a buyout option. As such, the paper provides the foundations and hence a theoretical justification for price leadership out of a collusive motive. Since a priori the model also does not exclude a barometric price leadership outcome, it is appropriate to conclude that the buyout option truly is a collusive device strong enough to suppress other reasons for price leadership. Therefore, the arrangements observed in reality aiming at the implementation of buyout possibilities have to be watched very closely from an antitrust perspective.

In this paper we have derived sufficient conditions for the stability of the G. S. E. These conditions follow from the condition on the stability of the G. C. E., given by Okuguchi (1978). When the conditions for asymptotic stability are fulfilled, the existence of the G. S. E. may be demonstrated in the same way as the existence of the G. C. E. has been demonstrated by Okuguchi (1978), that is: the G. S. E. is the fixpoint of a contracting mapping. In the case of linear demand functions and quadratic cost functions we have found the rather strong result that the sufficient conditions for (asymptotic) stability of the G. C. E. imply the (asymptotic) stability of the G. S. E. as well. This conclusion may be surprising to followers of Stackelberg. Mostly the outcome of a Stackelberg oligopoly is considered as adisequilibrium. In this paper we have made clear that a firm maximizing expected profit and using a generalised Stackelberg strategy, like a firm using a generalised Cournot strategy, only correctly predicts inequilibrium the price of the other firms. (See also on this point Heertje and Furth (1979)). When the outcome of a G. S. E. gives one or more firms less profit than in a G. C. E. then they expect, any price change, under-taken by them, will lower their expected profit even more.

For regular oligopolies (both homogeneous and heterogeneous) the (local) Cournot-Nash equilibria are the (non-degenerate) critical points of a Morse-Smale vector field, defined on the feasible region of non-negative prices and outputs. When at the boundary of this feasible region this vector field points inwards, it follows from the Morse inequalities that there is at least one stable equilibrium. When there is a unique, non-stable, interior equilibrium, necessarily the vector field points outwards somewhere along the boundary of the feasible region. This raises to a stable boundary equilibrium.