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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 the present paper we study endogenous price leadership in the context of a homogeneous product Bertrand duopoly model in which the firms have different, strictly convex cost functions. In such a framework it is well known that a simultaneous move price choice game does not have an equilibrium in pure strategies, but it has an equilibrium in mixed strategies. In the Stackelberg games with an exogenous price leader, we show that a pure strategy subgame perfect Nash equilibrium (SPNE) always exists. Although the SPNE might not be unique, the payoffs are the same across all SPNE. Finally, we analyze the issue of endogenous price leadership using the continuous version of the Robson (1990) timing game. The result is unexpected. One would expect the more efficient firm to emerge as the endogenous price leader. This is not always true. In most cases the endogenous leader is the firm with the highest "threshold" price. However, we also provide conditions under which the more efficient firm emerges as the leader. Our paper essentially complements Yano (2001), which is based on the Hamilton and Slutsky (1990) framework.

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.

Solves von Stackelberg equilibria in a Bertrand‐Edgeworth duopoly game. Shows that, initially, the environment is characterized by efficient rationing and capacity constraints. Since interest lies in sustaining monopoly outcomes from non‐co‐operative behaviour, introduces the buyout option, where rivals can absorb one another’s output before any consumer. Reveals that the outcomes change drastically in that players together are able to reach the monopoly profits.

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.

Non-cooperative game theory has in recent years become one of the main tools in the social sciences and economics. It deals with decision problems of more than one (rational) player. The actions of those players result in outcomes that are the ‘best’ that may have happened to a player, given the actions of the other players. This is precisely the content of the 'Nash equilibrium' concept. It would be nice when each game had a unique Nash equilibrium. Unfortunately, both non-existence and multiple equilibria occur. An important role in game theory is played by the ‘information‘ available to the players. Among the economic applications are: the ‘Theory of Industrial Organisation,’ ‘Bargaining’ and the ‘Economic Theory of Information.’

This paper analyzes Stackelberg price leadership in a duopoly in which firms are capacity constrained and products are imperfect substitutes. Assuming symmetric substitutes, linear demand, and efficient rationing, we characterize the equilibria with an exogenously specified leader. Using the equilibrium profits derived from these games, we argue that over certain ranges of asymmetric capacities an endogenous price leader will emerge. When endogenous leadership does arise, it is the large capacity firm which is the leader. We thus provide a game theoretic model of dominant firm price leadership.

We show that both cooperative and non-cooperative game models can substantially increase our understanding of the functioning of actual markets. In the first part of the paper, we provide a brief historical sketch of the differences and complementary between the two types of models, by going back to the work of the founding fathers, Von Neumann, Morgenstern and Nash. In the second part, we illustrate our main point by means of examples of bargaining, oligopolistic interaction and auctions.

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 the location models studied in the literature, consumers buy one unit of two, or more, horizontally differentiated products. As such, those models are for durable goods. For ordinary consumption goods like oranges, a model is developed, in which consumers may buy more than one unit, possibly different units, of a horizontally differentiated product. Demand and different rationing rules are derived for such a model, based on reservation prices that depend on quantities bought and the quality of both goods.

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.

An (n n)-matrix T is in class Mk if and only if T can be written as the sum of a diagonal matrix and a matrix of rank k. It is shown that The paper concentrates on the class M 1 and the eigenvalues of its members. For example, it is shown that in case all the main diagonal elements have the same sign then all the eigenvalues, except one, are in the interval determined by the maximum and minimum main diagonal element of the corresponding diagonal matrix. Applications for the input-output Leontief model and oligopoly-like games are given. The paper concludes with four research problems.

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.