The Limit-Price Dynamics — Uniqueness, Computability and Comparative Dynamics in Competitiive Markets

In this paper, a continuous-time price-quantity trading process is defined for exchange economies with differentiable characteristics. The dynamics is based on boundedly rational agents exchanging limit-price orders to a central clearing house, which rations infinitesimal trades according to Mertens (2003) double auction. Existence of continuous trade and price curves holds under weak conditions and in particular even if there is no long-run competitive equilibrium. Every such curve converges towards a Pareto point, and every Paretian allocation is a locally stable rest-point. Generically, given initial conditions, the trade and price curve is piecewise unique, smooth and computable, hence enables to effectively perform comparative dynamics. Finally, in the 2 x 2 case, the vector field induced by the limit-price dynamics is real-analytic.Non-tatonnement, price-quantity dynamics, limit-price mechanism, myopia, computable general equilibrium.

Two Generalizations of Homogeneity in Groups with Applications to Regular Semigroups

Let $X$ be a finite set such that $|X|=n$ and let $i\leq j \leq n$. A group G\leq \sym is said to be $(i,j)$-homogeneous if for every $I,J\subseteq X$, such that $|I|=i$ and $|J|=j$, there exists $g\in G$ such that $Ig\subseteq J$. (Clearly $(i,i)$-homogeneity is $i$-homogeneity in the usual sense.) A group G\leq \sym is said to have the $k$-universal transversal property if given any set $I\subseteq X$ (with $|I|=k$) and any partition $P$ of $X$ into $k$ blocks, there exists $g\in G$ such that $Ig$ is a section for $P$. (That is, the orbit of each $k$-subset of $X$ contains a section for each $k$-partition of $X$.) In this paper we classify the groups with the $k$-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the $(k-1,k)$-homogeneous groups (for $2<k\leq \lfloor \frac{n+1}{2}\rfloor$). As a corollary of the classification we prove that a $(k-1,k)$-homogeneous group is also $(k-2,k-1)$-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the $k$-universal transversal property have the $(k-1)$-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank $k$ transformation on $X$ generate a regular semigroup (for $1\leq k\leq \lfloor \frac{n+1}{2}\rfloor$). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.Comment: Includes changes suggested by the referee of the Transactions of the AMS. We gratefully thank the referee for an outstanding report that was very helpful. We also thank Peter M. Neumann for the enlightening conversations at the early stages of this investigatio

Continuity argument revisited: geometry of root clustering via symmetric products

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $\Omega$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in control theory: root clustering ($D$-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., $D$-decomposition(Yu.I. Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A. Fam, J. Meditch, J.Ackermann). Our approach is based on the interpretation of correspondence between roots and coefficients of a polynomial as a symmetric product morphism. We describe the topology of strata up to homotopy equivalence and, for many important cases, up to homeomorphism. Adjacencies between strata are also described. Moreover, we provide an explanation for the special position of classical stability problems: Hurwitz stability, Schur stability, hyperbolicity.Comment: 45 pages, 4 figure

The Limit-Price Dynamics — Uniqueness, Computability and Comparative Dynamics in Competitiive Markets

URL des Documents de travail :http://ces.univ-paris1.fr/cesdp/CESFramDP2007.htmDocuments de travail du Centre d'Economie de la Sorbonne 2007.20 - ISSN : 1955-611XIn this paper, a continuous-time price-quantity trading process is defined for exchange economies with differentiable characteristics. The dynamics is based on boundedly rational agents exchanging limit-price orders to a central clearing house, which rations infinitesimal trades according to Mertens (2003) double auction. Existence of continuous trade and price curves holds under weak conditions and in particular even if there is no long-run competitive equilibrium. Every such curve converges towards a Pareto point, and every Paretian allocation is a locally stable rest-point. Generically, given initial conditions, the trade and price curve is piecewise unique, smooth and computable, hence enables to effectively perform comparative dynamics. Finally, in the 2 x 2 case, the vector field induced by the limit-price dynamics is real-analytic.On définit un processus d'échanges en prix et en quantités et en temps continue pour des économies différentiables. La dynamique est fondée sur la rationalité limitée d'agents myopes qui adressent des ordres de prix-limites qu'ils adressent à une agence de clearing centrale, laquelle rationne les échanges infinitésimaux en fonction de l'enchère double de Mertens (2003). L'existence de courbes de prix et d'échanges est vérifiée sous de faibles conditions, en particulier en l'absence d'équilibre concurrentiel de long terme. Toute courbe d'échange converge vers un optimum de Pareto et inversement tout optimum est un point stationnaire localement stable de la dynamique. Génériquement, à conditions initiales données, la courbe d'échange et de prix est unique par morceaux, lisse et calculable, ouvrant la possibilité d'une dynamique comparative effective. Enfin, dans le cas 2 x 2, le champ de vecteurs associé à la dynamique est réel-analytique

The Pivotal Role of Causality in Local Quantum Physics

In this article an attempt is made to present very recent conceptual and computational developments in QFT as new manifestations of old and well establihed physical principles. The vehicle for converting the quantum-algebraic aspects of local quantum physics into more classical geometric structures is the modular theory of Tomita. As the above named laureate to whom I have dedicated has shown together with his collaborator for the first time in sufficient generality, its use in physics goes through Einstein causality. This line of research recently gained momentum when it was realized that it is not only of structural and conceptual innovative power (see section 4), but also promises to be a new computational road into nonperturbative QFT (section 5) which, picturesquely speaking, enters the subject on the extreme opposite (noncommutative) side.Comment: This is a updated version which has been submitted to Journal of Physics A, tcilatex 62 pages. Adress: Institut fuer Theoretische Physik FU-Berlin, Arnimallee 14, 14195 Berlin presently CBPF, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazi

Relational Structure Theory: A Localisation Theory for Algebraic Structures

This thesis extends a localisation theory for finite algebras to certain classes of infinite structures. Based on ideas and constructions originally stemming from Tame Congruence Theory, algebras are studied via local restrictions of their relational counterpart (Relational Structure Theory). In this respect, first those subsets are identified that are suitable for such a localisation process, i. e. that are compatible with the relational clone structure of the counterpart of an algebra. It is then studied which properties of the global algebra can be transferred to its localisations, called neighbourhoods. Thereafter, it is discussed how this process can be reversed, leading to the concept of covers. These are collections of neighbourhoods that allow information retrieval about the global structure from knowledge about the local restrictions. Subsequently, covers are characterised in terms of a decomposition equation, and connections to categorical equivalences of algebras are explored. In the second half of the thesis, a refinement concept for covers is introduced in order to find optimal, non-refinable covers, eventually leading to practical algorithms for their determination. Finally, the text establishes further theoretical foundations, e. g. several irreducibility notions, in order to ensure existence of non-refinable covers via an intrinsic characterisation, and to prove under some conditions that they are uniquely determined in a canonical sense. At last, the applicability of the developed techniques is demonstrated using two clear expository examples.:1 Introduction 2 Preliminaries and Notation 2.1 Functions, operations and relations 2.2 Algebras and relational structures 2.3 Clones 3 Relational Structure Theory 3.1 Finding suitable subsets for localisation 3.2 Neighbourhoods 3.3 The restricted algebra A|U 3.4 Covers 3.5 Refinement 3.6 Irreducibility notions 3.7 Intrinsic description of non-refinable covers 3.8 Elaborated example 4 Problems and Prospects for Future Research Acknowledgements Index of Notation Index of Terms BibliographyDiese Dissertation erweitert eine Lokalisierungstheorie für endliche Algebren auf gewisse Klassen unendlicher Strukturen. Basierend auf Ideen und Konstruktionen, die ursprünglich der Tame Congruence Theory entstammen, werden Algebren über lokale Einschränkungen ihres relationalen Gegenstücks untersucht (Relationale Strukturtheorie). In diesem Zusammenhang werden zunächst diejenigen Teilmengen identifiziert, welche für einen solchen Lokalisierungsprozeß geeignet sind, d. h., die mit der Relationenklonstruktur auf dem Gegenstück einer Algebra kompatibel sind. Es wird dann untersucht, welche Eigenschaften der globalen Algebra auf ihre Lokalisierungen, genannt Umgebungen, übertragen werden können. Nachfolgend wird diskutiert, wie dieser Vorgang umgekehrt werden kann, was zum Begriff der Überdeckungen führt. Dies sind Systeme von Umgebungen, welche die Rückgewinnung von Informationen über die globale Struktur aus Kenntnis ihrer lokalen Einschränkungen erlauben. Sodann werden Überdeckungen durch eine Zerlegungsgleichung charakterisiert und Bezüge zu kategoriellen Äquivalenzen von Algebren hergestellt. In der zweiten Hälfte der Arbeit wird ein Verfeinerungsbegriff für Überdeckungen eingeführt, um optimale, nichtverfeinerbare Überdeckungen zu finden, was letztlich zu praktischen Algorithmen zu ihrer Bestimmung führt. Schließlich erarbeitet der Text weitere theoretische Grundlagen, beispielsweise mehrere Irreduzibilitätsbegriffe, um die Existenz nichtverfeinerbarer Überdeckungen vermöge einer intrinsischen Charakterisierung sicherzustellen und, unter gewissen Bedingungen, zu beweisen, daß sie in kanonischer Weise eindeutig bestimmt sind. Schlußendlich wird die Anwendbarkeit der entwickelten Methoden an zwei übersichtlichen Beispielen demonstriert.:1 Introduction 2 Preliminaries and Notation 2.1 Functions, operations and relations 2.2 Algebras and relational structures 2.3 Clones 3 Relational Structure Theory 3.1 Finding suitable subsets for localisation 3.2 Neighbourhoods 3.3 The restricted algebra A|U 3.4 Covers 3.5 Refinement 3.6 Irreducibility notions 3.7 Intrinsic description of non-refinable covers 3.8 Elaborated example 4 Problems and Prospects for Future Research Acknowledgements Index of Notation Index of Terms Bibliograph