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AbstractWe provide a comprehensive account of fundamental properties of a truncated discrete Zipf distribution, complementing the results available in the literature. In particular, we obtain results on existence and uniqueness of maximum likelihood parameter estimators and propose new testing methodology for the shape parameter. We also include data examples illustrating applicability of this stochastic model.

Les constantes de la distribution canonique de Mandelbrot et le nombre moyen de mots différents T (types) d'un texte de N mots (signes) sont calculés explicitement. On en déduit que la température informationnelle θ d'un langage à lexique infini est égale à l'ordre de croissance du nombre de types en fonction du nombre de signes. Une nouvelle méthode de calcul de θ est proposée. L'étude du comportement asymptotique des « échelons » de la courbe de Zipf conduit à une dérivation plus satisfaisante de la forme approchée de celle-ci.

An avalanche of empirical studies has addressed the validity of the rank-size rule (or Zipf's law) in a multi-city context in many countries. City size in most countries seems to obey Zipf's law, but the question under which conditions (e.g. sample size, spatial scale) this 'law' holds remained largely underinvestigated. Another complementary question is whether socio-economic networks in space also show a similar hierarchical pattern. Against this background, the present paper investigates – from a methodological viewpoint – the relationship between network connectivity and the rank-size rule (or Zipf's law) in an urban-economic network constellation. After a review of the literature, we address in particular the following methodological issues: (i) the (aggregate) behavioural foundation underlying the rank-size rule/Zipf's law in the light of spatial-economic network theories (e.g. entropy maximization, spatial interaction theory, etc.); (ii) the nature of the analytical relationship between social-spatial network analysis and the rank-size rule/Zipf's law. We argue that the rank size rule is compatible with conventional economic foundations of spatial network models. Consequently, a spatial-economic interpretation – as well as a network connectivity interpretation – of the rank-size rule coefficient is provided. Our methodological contribution forms the foundation for the subsequent empirical analysis applied to spatial networks in a socio-economic context. The aim here is to test the sensitivity of empirical findings for changes in scale, functional forms, time periods, and network structures. Our application is concerned with an extensive spatio-temporal panel database related to the evolution of urban population in Germany. We test the relevance of the rank-size rule/Zipf's law, and its evolution over the years, and – in parallel – the related 'socio-economic' connectivity in these urban networks. In particular, we will show that Zipf's law (i.e., with the rank-size coefficient equal to 1) is only valid under particular conditions of the sample size. The paper concludes with some retrospective and prospective remarks.

En esta nota se examina la hipótesis de que las ciudades mexicanas siguen la ley de Zipf, la cual establece que si se ordenan las ciudades de un país (o región) de acuerdo con su tamaño poblacional, entonces el rango de cada ciudad multiplicado por su tamaño produce siempre la misma constante. En este trabajo se afirma que tal ley no es aplicable al caso mexicano, y también se arguye que muchos estudios que pretenden mostrar que tal ley se aplica para otros países carecen de fundamentos estadísticos sólidos.

"Öffentliche Bildungsfinanzierung ist hinsichtlich ihrer Verteilungswirkung nicht unsozial, wie von ihren wettbewerbsorientierten Kritikern immer wieder behauptet wird. Die These von der regressiven, also die oberen Einkommensschichten begünstigenden Verteilungswirkung der Bildungspolitik ist so nicht zu belegen. Für die Forderung nach einem Rückzug des Staates aus der Bildungsfinanzierung gibt es damit aus volkswirtschaftlicher Sicht keine empirische Grundlage." (Autorenreferat)

This article analyzes the legacy of one of the 'classics' of 20th centurymigration studies— George Zipf. Particular attention is paid to his work "Human behavior and the principle of last effort. An introduction to human ecology". The authors focus not so much on the results of Zipf's research, but on the questions he posed, and make the claim that these questions constitute the true value of Zipf's scientific heritage. His inquiry on the criteria for describing migration process efficiency has become one of the key questions when it comes to theories of migration behavior, social adaptation and integration of migrants. The question about the influence of collective and group subjectivity on migration decision-making had a significant impact on the push and pull theory as formulated by E. Lee, while Zipf's reflections on how migration behavior is influenced by the division of labor and the diversity of professional skills became the foundation for D.Massey's synthetic theory of migration. Step by step the article's authors analyze Zipf's reflections in relation to such questions as: are migrants a social group? How can migration efficiency be measured? Is there a relationship between the movement of people and the movement of capital? How do the division of labor and the law of supply and demand affect migration? In conclusion, the authors formulate those theses brought forth by G.Zipf that are still relevant to this day.