New properties of the Cournot duopoly with isoelastic demand and constant unit costs.

The object of the work is to perform the global analysis of the Cournot duopoly model with isoelastic demand function and unit costs, presented in Puu (1991). The bifurcation of the unique Cournot fixed point is established, which is a resonant case of the Neimark-Shacker bifurcation. New properties associated with the introduction of horizontal branches are evidenced. These properties di¤er significantly when the constant value is zero or positive and small. The good behavior of the case with positive constant is proved, leading always to positive trajectories. Also when the Cournot fixed point is unstable, stable cycles of any period may exist.Cournot duopoly, isoelastic demand function, multistability, border-collision bifurcations.

Mathematical Properties of a Combined Cournot-Stackelberg model.

The object of this work is to perform the global analysis of a new duopoly model which couples the two points of view of Cournot and Stackelberg. The Cournot model is assumed with isoelastic demand function and unit costs. The coupling leads to discontinuous reaction functions, whose bifurcations, mainly border collision bifurcations, are investigates as well as the global structure of the basins of attraction. In particular, new properties are shown, associated with the introduction of horizontal branches, which di¤er significantly when the constant value is zero or positive and small. The good behavior of the model with positive constant is proved, leading to stable cycles of any period.Cournot-Stackelberg duopoly, Isoelastic demand function, Discontinuous reaction functions, Multistability, Border-collision bifurcations.

Bertrand oligopoly revisited

This paper reconsiders Bertrand duopoly and oligopoly in the spatial formulation due to Hotelling, 1929. The equilibrium configurations of price and location structure are considered, given elastic demand, and a full dynamics is formulated in order to check for stability of equilibrium and the possibilities of complex dynamics, such as occurs easily with Cournot oligopoly. The main discussion concerns Hotelling's original case of two sellers on a given interval, though results for different cases, such as three firms on a circle, and lattices in 2D are indicated

The dynamics of a triopoly Cournot game when the competitors operate under capacity constraints

Oligopoly theory, i.e., the economic theory for competition among the few, goes back to 1838 and Augustin Cournot [7]. See also [11]. Quite early it was suspected to lead to complex dynamic behaviour and chaos. See Rand 1978 [13]. The probably simplest case under which this happens with reasonable economics assumptions was suggested by one of the present authors in 1991, see [9]. It assumes an isoelastic demand function, which always arises when the consumers maximize utility functions of the Cobb-Douglas type, combined with constant marginal costs. The particular layout was a duopoly, the case of only two competitors. The model was shown to produce a period doubling sequence of ip bifurcations ending in chaos for the outputs of each of the two competitors. Later the triopoly case under these assumptions was studied. See [2], [3], and [4] for examples. An interesting fact is that with three competitors the main frame becomes the Neimark-Hopf bifurcation, which provides new and di erent scenarios. The main reason for economists to study increasing numbers of competitors is to nd out whether it is the number of competitors that uniquely decides a road from monopoly over duopoly, oligopoly, and polypoly, to perfect competition, a state where each rm is so small that its actions cannot in uence the market at all. To nd out about this it is of primary interest to know whether the number of competitors stabilizes or destabilizes the equilibrium state. Some authors have questioned the assumption, to which most economists adhered, that increasing numbers of competitors bring stabilization. However, we must be clear about what we compare. If we study increasing numbers of competitors with constant unit production costs, we are not reducing the size of the rms when their number increases. Constant marginal cost means that potentially each rm has in nite capacity, and adding such rms is not what we want for comparison. It is therefore interesting to combine an increased number of rms with decreasing size of each rm, but in order to do so we have to introduce capacity limits. Already Edgeworth [8] insisted on the importance of capacity limits. It is not so easy to nd non-constant marginal cost functions which allow us to solve for the reaction functions for the rms in explicit form, but one of the present authors, see [12], found one type of function, which models the capacity limit by letting marginal cost go to in nity at a nite output. That paper discussed the competition between two duopolists. The objective of the present paper is to nd out the facts when there are three competitors, and we still keep the assumption of capacity limit

The Cournot-Theocharis Problem Reconsidered

In 1959 Theocharis [10] showed that with linear demand and constant marginal costs Cournot equilibrium is destabilized when the competitors become more than three. With three competitors the Cournot equilibrium point becomes neutrally stable, so, even then, any perturbation throws the system into an endless oscillation. Theocharis's argument was in fact proposed already in 1939 by Palander [4]. None of these authors considered the global dynamics of the system, which necessarily becomes nonlinear when consideration is taken of the facts that prices, supply quantities, and profits of active firms cannot be negative. In the present paper we address the global dynamics.The authors 1 and 3 are partially supported by the grant FS 00684/PI/04 from Fundación Séneca (Comunidad Autónoma de la Región de Murcia, Spain

A mathematics refresher for students of economics

Freely downloadable as pdf

Attractors, bifurcations, & chaos: nonlinear phenomena in economics

The present book relies on various editions of my earlier book "Nonlinear Economic Dynamics", first published in 1989 in the Springer series "Lecture Notes in Economics and Mathematical Systems", and republished in three more, successively revised and expanded editions, as a Springer monograph, in 1991, 1993, and 1997, and in a Russian translation as "Nelineynaia Economicheskaia Dinamica". The first three editions were focused on applications. The last was differ­ ent, as it also included some chapters with mathematical background mate­ rial -ordinary differential equations and iterated maps -so as to make the book self-contained and suitable as a textbook for economics students of dynamical systems. To the same pedagogical purpose, the number of illus­ trations were expanded. The book published in 2000, with the title "A ttractors, Bifurcations, and Chaos -Nonlinear Phenomena in Economics", was so much changed, that the author felt it reasonable to give it a new title. There were two new math­ ematics chapters -on partial differential equations, and on bifurcations and catastrophe theory -thus making the mathematical background material fairly complete. The author is happy that this new book did rather well, but he preferred to rewrite it, rather than having just a new print run. Material, stemming from the first versions, was more than ten years old, while nonlinear dynamics has been a fast developing field, so some analyses looked rather old-fashioned and pedestrian. The necessary revision turned out to be rather substantial