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

A non-parametric independence test using permutation entropy

In the present paper we construct a new, simple and powerful test for independence by using symbolic dynamics and permutation entropy as a measure of serial dependence. We also give the asymptotic distribution of an affine transformation of the permutation entropy under the null hypothesis of independence. An application to several daily financial time series illustrates our approach

A Non-Parametric Approach to Spatial Causality

The purpose of this paper is to show the capacity of a new non-parametric test based on symbolic entropy and symbolic dynamics to deal with the detection of linear and non-linear spatial causality. The good performance of the new test in detecting spatial causality and causal weighting matrix is notable and gives rise to an expectation that it may form a adequate tool for constructive specification searches.Causality; Spatial Dependence; Spatial Weight Matrices

Computing large direct products of free groups in integral group rings

We construct explicitly a subgroup of finite minimal index and minimal rank in which is a direct products of free groups for each finite group G for which this is possibleThe first author has been partially supported by the DGI of Spain and Fundación Seneca of Murci

Free groups and subgroups of finite index in the unit group of an integral group ring

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-abelian group G for which every non-linear irreducible complex representation is of degree 2 and with commutator subgroup G0 a central elementary abelian 2-group.Research partially supported by the Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium) and Bilateral Scientific and Technological Cooperation BWS 05/07 (Flanders-POland). Postdoctoraal Onderzoeker van het Fonds voor Wetenschappelijk Onderzoek- Vlaanderen. Research partially supported by the Fundación Séneca of Murcia and D.G.I. of Spain

A non-parametric spatial independence test using symbolic entropy

In the present paper, we construct a new, simple, consistent and powerful test for spatial independence, called the SG test, by using symbolic dynamics and symbolic entropy as a measure of spatial dependence. We also give a standard asymptotic distribution of an affine transformation of the symbolic entropy under the null hypothesis of independence in the spatial process. The test statistic and its standard limit distribution, with the proposed symbolization, are invariant to any monotonuous transformation of the data. The test applies to discrete or continuous distributions. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. We include a Monte Carlo study of our test, together with the well-known Moran’s I, the SBDS (de Graaff et al, 2001) and (Brett and Pinkse, 1997) non parametric test, in order to illustrate our approach

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

Groups generated by two bicyclic units in integral group rings

In [5] Ritter and Sehgal introduced the following units, called the bicylic units, in the unit group U(ZG) of the integral group ring ZG of a finite group G: ¯a;g = 1 + (1 ¡ g)abg; °a;g = 1 + bga(1 ¡ g); where a; g 2 G and bg is the sum of all the elements in the cyclic group hgi. It has been shown that these units generate a large part of the unit group of ZG. Indeed, for most finite groups G, the bicyclic units together with the Bass cyclic units generate a subgroup of finite index in U(ZG) [3, 6]. The Bass cyclic units are only needed to cover a subgroup of finite index in the centre and the group B generated by the bicyclic units contains a subgroup of finite index in a maximal Z-order of each non-commutative simple image Mn(D) of the rational group algebra QG. In particular, if n > 1, then B contains a subgroup of finite index in SLn(O), where O is a maximal order in D; and hence B contains free subgroups of rank two. A next step in determining the structure of U(ZG) is to investigate relations among the discovered generators. Presently this is beyond reach. Hence a more realistic goal is to study the structure of the group generated by two bicyclic units. In [4] Marciniak and Sehgal proved that if ¯a;g is a non trivial unit in ZG (here G is not necessarily finite) then the group h¯a;g; °a¡1;g¡1i is free of rank 2. Clearly, bicyclic units are of the form 1 + a with a2 = 0. Salwa, in [7], used the ideas of Marciniak and Sehgal to prove that if x and y are two elements of an additively torsion-free ring such that x2 = y2 = 0 and xy is not nilpotent then h(1 + x)m; (1 + y)mi is free of rank 2 for some positive integer m. In particular, if b1 and b2 are two bicyclic units and (b1 ¡ 1)(b2 ¡ 1) is not nilpotent, then hbm 1 ; bm 2 i is free of rank 2 for some positive integer m. In this paper we investigate the minimum positive integer m so that hbm 1 ; bm 2 i is free provided that b1 and b2 are two bicyclic units so that (b1 ¡1)(b2 ¡1) is not nilpotent. We prove the following theorem which indicates that if b1 and b2 are of the same type then frequently m = 1.The first author has been partially supported by the Onderzoeksraad of Vrije Universiteit Brussel and the Fonds voor Wetenschappelijk Onderzoek (Vlaanderen) and the second by the D.G.I. of Spain and Fundación Séneca of Murcia.We would like to express our gratitude to Victor Jiménez for some helpful conversation on inequality

Comparison of thematic maps using symbolic entropy

Comparison of thematic maps is an important task in a number of disciplines. Map comparison has traditionally been conducted using cell-by-cell agreement indicators, such as the Kappa measure. More recently, other methods have been proposed that take into account not only spatially coincident cells in two maps, but also their surroundings or the spatial structure of their differences. The objective of this paper is to propose a framework for map comparison that considers 1) the patterns of spatial association in two maps, in other words, the map elements in their surroundings; 2) the equivalence of those patterns; and 3) the independence of patterns between maps. Two new statistics for the spatial analysis of qualitative data are introduced. These statistics are based on the symbolic entropy of the maps, and function as measures of map compositional equivalence and independence. As well, all inferential elements to conduct hypothesis testing are developed. The framework is illustrated using real and synthetic maps

Groups of units of integral group rings of Kleinian type

We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16.D.G.I. of Spain and Fundación Séneca of Murcia. AMS classification index: Primary 16U60, Secondary 11R27, 16A26