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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 game the players are firms involved in a Bertrand–Edgeworth duopoly market. Payoffs to the low priced firm depend only on the own price, whereas the payoff to the high priced firm depends on both its own price and the price of the opponent. The price of the opponent enters the payoff function of the high priced firm through buyout or a first refusal contract. Only when the total capacity in the market is less than the output in a monopoly situation, there is an equilibrium in pure strategies.Journal of Economic LiteratureClassification Numbers: C72, D43, L12.

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.