IZA DP No. 3928 Zipf’s Law for Cities in the Regions and the Country Kristian Giesen Jens Suedekum DISCUSSION PAPER SERIES DISCUSSION PAPER January 2009 Forschungsinstitut zur Zukunft der Arbeit Institute for the Study of Labor Zipf’s Law for Cities in the Regions and the Country Kristian Giesen University of Duisburg-Essen Jens Suedekum University of Duisburg-Essen and IZA Discussion Paper No. 3928 January 2009 IZA P.O. Box 7240 53072 Bonn Germany Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: [email protected] Any opinions expressed here are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but the institute itself takes no institutional policy positions. The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of communication between science, politics and business. IZA is an independent nonprofit organization supported by Deutsche Post World Net. The center is associated with the University of Bonn and offers a stimulating research environment through its international network, workshops and conferences, data service, project support, research visits and doctoral program. IZA engages in (i) original and internationally competitive research in all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research results and concepts to the interested public. IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. A revised version may be available directly from the author. IZA Discussion Paper No. 3928 January 2009 ABSTRACT Zipf’s Law for Cities in the Regions and the Country* The salient rank-size rule known as Zipf’s law is not only satisfied for Germany’s national urban hierarchy, but also for the city size distributions in single German regions. To analyze this phenomenon, we build on the insights by Gabaix (1999) that Zipf’s law follows from a stochastic growth process. In particular, Gabaix shows that if the regions follow Gibrat’s law, we should observe Zipf at both the regional and the national level. This theory has never been addressed empirically. Using non-parametric techniques we find that Gibrat’s law holds in each German region, irrespective of how “regions” are defined. In other words, Gibrat’s law and therefore Zipf’s law tend to hold everywhere in space. JEL Classification: R11, O4 Keywords: city size distributions, city growth, Zipf’s law, Gibrat’s law, rank-size rule Corresponding author: Jens Suedekum Mercator School of Management University of Duisburg-Essen Lotharstrasse 65 47057 Duisburg Germany E-mail: [email protected] * We thank Marcus Berliant, Maarten Bosker, Steven Brakman, Xavier Gabaix, Harry Garretsen, Wen- Tai Hsu, Stefan Hupfeld, Volker Nitsch, Henry Overman, Kurt Schmidheiny, Hiroki Watanabe, and seminar participants at the 2008 Annual Meeting of the North American Regional Science Council (NARSC) in New York City for helpful comments and suggestions. All errors and shortcomings are solely our responsibility. 1 Introduction Zipf’s law is one of the most striking empirical laws in economics, describing a remarkably stable regularity in the spatial structure of market economies (Krugman, 1996). As is well known, it states that the upper tail of the city size distribution within an area (say, the US) and at any point in time can be approximated by a Pareto distribution with shape parameter equal to -1. This implies a unique rank-size relationship according to which the area’s largest city (New York) is twice as large as the second-largest city (Los Angeles), three times as large as the third-largest city (Chicago), and so on. Following the seminal contributions by Auerbach (1913) and Zipf (1949) this alleged empirical rule has been addressed dozens of times.1 Virtually all of these studies are concerned with the city size distribution of entire countries, however. The starting point of our paper is the observation that Zipf’s law for city sizes also tends to hold in single regions of a country. FIGURE 1 HERE Figure 1 illustrates this observation with an example from Germany. We provide a standard Zipf plot for the 20 largest cities from one German Federal State only, the State of Hessen. More precisely, we plot the log of the city’s rank in Hessen’s urban hierarchy (#1 for Frankfurt, #2 for Wiesbaden, #3 for Kassel, and so on) against log city size measured by the number of inhabitants. When running a standard Zipf regression of the type log(Rank) = log(a) − ζ log(Size) we estimate ζ = 1.027 with std.error 0.325 and R2 = 0.99. As suggested by Zipf’s law, the rank-size relationship in log scales can be approximated very accurately by a linear curve. Secondly, the slope of this linear curve is very close to −1, which is also what Zipf’s law suggests. In other German regions (including Germany as a whole) this picture looks similar, even though the slope coefficients are not as close to ζ = 1 as in Hessen (see below). The main aim of this paper is to shed light on why Zipf’s law tends to hold on a regional level.2 The contribution by Gabaix (1999) is central to us in this respect. In proposition 1 of that paper Gabaix proves, that if cities grow stochastically with the same expected growth rate and same variance (a property that is known as Gibrat’s law), then subject to frictions a steady state city size distribution is implied that obeys Zipf’s law.3 In other words, Gabaix establishes Gibrat’s law as a statistical explanation for Zipf’s 1Comprehensive studies on Zipf’s law are provided by Rosen and Resnick (1980) and Soo (2005). Nitsch (2005) conducts a meta-study, and Gabaix and Ioannides (2004) present a survey of the literature. 2We do not claim to be the first to ever estimate a regional Zipf coefficient, but the more recent literature has completely neglected this dimension. Neither Gabaix and Ioannides (2004) nor Nitsch (2005) mention any study that systematically addresses intra-national city size distributions. Furthermore, we focus on the relationship between Gibrat’s law and Zipf’s law from a regional perspective. This aspect has not been studied empirically so far. 3Eeckhout (2004) shows that the pure form of Gibrat’s law will generate a log-normal city size distribu- tion, not the Pareto. Eeckhout (2004) furthermore shows that a log-normal distribution fits the overall size distribution of all places (cities and small towns) better than the Pareto. Zipf’s law is a feature pertaining only to the upper tail of the size distribution. In this paper we concentrate on this upper tail in order to address the salient Zipf’s law from a regional perspective. 1 law, thereby opening new avenues for theoretical and empirical research on the rank-size rule. Theoretical papers on the economic microfoundations of Zipf’s law typically aim at theories about Gibrat’s law or generalized versions thereof (see e.g. Rossi-Hansberg and Wright 2007; Eaton and Eckstein 1997). On the empirical side, Gabaix’s proposition is the basis for studies such as Ioannides and Overman (2003), who test Zipf’s law indirectly by investigating if Gibrat’s law holds at the national level in the US. Yet an even more important insight for our purpose is proposition 2 of Gabaix (1999). There he shows that if a country is composed of several regions, and if Gibrat’s law holds in each region, then Zipf’s law will be satisfied for all regional and also for the national city size distribution.4 To the best of our knowledge this theoretical insight has never been addressed empirically. In this paper we provide a first test of Gibrat’s law on a regional level, contemplating different concepts of a "region". Most naturally we analyze the German Länder, which can be thought of as politically well-defined clubs of spatially adjacent cities. Moreover, since Gabaix’s theory is not confined to regions with historical or administrative content, we also consider two other concepts. Firstly, we construct random regions, i.e., random draws from the population of large German cities that need not be adjacent to one another. Secondly, we build spatial clubs of large cities that are geographically adjacent, but that need not belong to the same Federal State. It turns out that Gibrat’s law holds not only at the national level in Germany, but in each region regardless of which type. According to Gabaix (1999) Zipf’s law should then also be valid in each region and in the country - and in fact, this appears to be true. These findings empirically confirm the theory on the close correspondence of random city growth at the regional and the Zipfian rank-size rule at the regional and national level. Moreover, our findings suggest that the regular pattern for urban growth that arises at the national level holds more generally. Urban growth is scale independent not only in the aggregate across all cities (or in random draws from this aggregate), but also in non-random samples. Gibrat’s law actually seems to hold everywhere in space. 2 Gibrat’s law and Zipf’s law: A reminder A casual illustration of Gibrat’s rule of proportionate growth is to say that the size of a city contains no information about its future growth rate. More precisely, cities grow randomly with the same expected growth rate and same variance. Gabaix (1999) shows that under certain conditions this growth process will converge to a Pareto distribution with exponent equal to -1: Zipf’s law.