Theory of Zipf's Law and Beyond
Zipf's law is one of the few quantitative reproducible regularities found in economics. It states that, for most countries, the size distributions of city sizes and of firms are power laws with a specific exponent: the number of cities and of firms with s
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Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, 33615 Bielefeld, Germany Editorial Board: H. Dawid, D. Dimitrov, A. Gerber, C.-J. Haake, C. Hofmann, T. Pfeiffer, R. Slowinksi, H. Zijm _
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Alexander Saichev Yannick Malevergne Didier Sornette
Theory of Zipf’s Law and Beyond
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Professor Alexander Saichev Mathematical Department Nizhni Novgorod State University Gagarin Prospekt 23 603950 Nizhni Novgorod Russia [email protected]
Professor Yannick Malevergne ISEAG University of Saint-Etienne 2 rue Tréfilerie 42023 Saint-Etienne cedex 2 France and
Professor Didier Sornette Department of Management Technology and Economics ETH Zürich Kreuzplatz 5 8032 Zurich Switzerland [email protected]
EMLYON Business School – Cefra 23 avenue Guy de Collongue 69134 Ecully Cedex France [email protected]
ISSN 0075-8442 e-ISBN 978-3-642-02946-2 ISBN 978-3-642-02945-5 DOI 10.1007/978-3-642-02946-2 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009932422 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Zipf’s law is one of the few quantitative reproducible regularities found in economics. It states that, for most countries, the size distributions of cities and of firms (with additional examples found in many other scientific fields) are power laws with a specific exponent: the number of cities and firms with a size greater than S is inversely proportional to S. Most explanations start with Gibrat’s law of proportional growth but need to incorporate additional constraints and ingredients introducing deviations from it. Here, we present a general theoretical derivation of Zipf’s law, providing a synthesis and extension of previous approaches. First, we show that combining Gibrat’s law at all firm levels with ran
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