Zipf’s Law and the Gibrat’s Law: What Do the Facts Have to Say about the Brazilian Cities? (original) (raw)
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A Comparative Analysis of Gibrat's and Zipf's Law on Urban Population
SSRN Electronic Journal, 2015
The regional economics and geography literature on urban population size has in recent years shown interesting conceptual and methodological contributions on the validity of Gibrat's Law and Zipf's Law. Despite distinct modeling features, they express similar fundamental characteristics in an equilibrium situation. Zipf's law is formalized in a static form, while its associated dynamic process is articulated by Gibrat's Law. Thus, it is likely that both Zipf's Law and Gibrat's Law share a common root. Unfortunately, empirical investigations on the direct relationship between Gibrat's Law and Zipf's Law are rather rare and not conclusive. The present paper aims to answer the question whether (a generalisation of) Gibrat's Law allows us to infer Zipf's Law, and vice versa? In our conceptual and applied framework, particular attention will be paid to the role of the mean and the variance of city population as key indicators for assessing the (non-) validity of the generalised Gibrat's Law. Our empirical experiments are based on a comparative analysis between the dynamics of the urban population of four countries with entirely mutually contrasting spatial-economic and geographic characteristics: Botswana, Germany, Hungary and Luxembourg. We arrive at the following results: if (i) the mean is independent of city size (first necessary condition of Gibrat's law) and (ii) the coefficient of the rank-size rule/Zipf's Law is different from one, then the variance is dependent on city size.
Physica A: Statistical Mechanics and its Applications, 2006
This work studies the Zipf Law for cities in Brazil. Data from censuses of 1970Data from censuses of , 1980Data from censuses of , 1991Data from censuses of and 2000 were used to select a sample containing only cities with 30,000 inhabitants or more. The results show that the population distribution in Brazilian cities does follow a power law similar to the ones found in other countries. Estimates of the power law exponent were found to be 2.22 ± 0.34 for the 1970 and 1980 censuses, and 2.26 ± 0.11 for censuses of 1991 and 2000. More accurate results were obtained with the maximum likelihood estimator, showing an exponent equal to 2.41 for 1970 and 2.36 for the other three years. PACS: 89.75Da; 89.65.Cd; 89.75.-k; 05.45.Df
Zipf’s law for cities: an empirical examination
2003
We use data for metro areas in the United States, from the US Census for 1900–1990, to test the validity of Zipf's Law for cities. Previous investigations are restricted to regressions of log size against log rank. In contrast, we use a nonparametric procedure to estimate Gibrat's Law for city growth processes as time-varying geometric Brownian motion and to calculate local Zipf exponents from the mean and variance of city growth rates. Despite variation in growth rates as a function of city size, Gibrat's Law does hold.
Zipf’s law and city size distribution: A survey of the literature and future research agenda
Physica A: Statistical Mechanics and its Applications, 2018
This study provides a systematic review of the existing literature on Zipf's law for city size distribution. Existing empirical evidence suggests that Zipf's law is not always observable even for the upper-tail cities of a territory. However, the controversy with empirical findings arises due to sample selection biases, methodological weaknesses and data limitations. The hypothesis of Zipf's law is more likely to be rejected for the entire city size distribution and in such case the alternative distributions have been suggested. On the contrary, the hypothesis is more likely to be accepted if better empirical methods are employed and cities are properly defined. The debate is still far from to be conclusive. In addition, we identify four emerging areas in Zipf's law and city size distribution research including the size distribution of lower-tail cities, the size distribution of cities in sub-national regions, the alternative forms of Zipf's law, and the relationship between Zipf's law and the coherence property of the urban system.
Gibrat's Law and the Growth of Cities in Brazil: A Panel Data Investigation
Urban Studies, 2004
The paper builds on the results of Clark and Stabler who associated Gibrat's law on the independence of growth rate and city size with unit root tests. The paper proposes a direct test of the unit root hypothesis for firm size based on recently developed panel data unit root tests. The results for a sample of Brazilian cities over the period 1980-2000 favour Gibrat's law. Moreover, the results are robust when one considers sub-samples defined for different population sizes and age of municipality.
Is the Zipf law spurious in explaining city-size distributions
The Zipf law, which states that that the rank associated with some size S is proportional to S to some negative power, is a regularity observed in natural and social sciences. One popular application of the Zipf law is the relationship between city sizes and their ranks. This paper examines the rank-size relationship through Monte Carlo simulations and two examples. We show that a good fit (indicated by a high R 2 value) can be found for many statistical distributions. The Zipf law's good fit is a statistical phenomenon, and therefore, it does not require an economic theory that determines city-size distributions. D
A dynamic model for city size distribution beyond Zipf 's law
Physica A-statistical Mechanics and Its Applications, 2007
We present a growth model for a system of cities. This model recovers not only Zipf's law but also other kinds of city size distributions (CSDs). A new positive exponent a, which yields Zipf's law only when equal to 1, was introduced. We define three classes of CSD depending on the value of a: larger than, smaller than, or equal to 1. The model is based on a random growth of the city population together with the variation of the number of cities in the system. The striking result is the peculiar behavior of the model: it is only statistical deterministic. Moreover, we found that the exponent a may be larger, smaller or equal to 1, just like in real systems of cities, depending on the rate of creation of new cities and the time elapsed during the growth. It is to our knowledge the first time that the influence of the time on the type of the distribution is investigated. The results of the model are in very good agreement with real CSD. The classification and model can be also applied to other entities like countries, incomes, firms, etc
New Evidence on Gibrat’s Law for Cities
2010
The aim of this work is to test empirically the validity of Gibrat’s Law in the growth of cities, using data on the complete distribution of cities (without size restrictions or a truncation point) in three countries (the US, Spain and Italy) for the entire 20th century. For this we use different techniques. First, panel data unit root tests tend
Zipf's Law for cities: a cross-country investigation
Regional Science and Urban Economics, 2005
Several recent papers have sought to provide theoretical explanations for Zipf's Law, which states that the size distribution of cities in an urban system can be approximated by a Pareto distribution with shape parameter (Pareto exponent) equal to 1. This paper assesses the empirical validity of Zipf's Law, using new data on 73 countries and two different estimation methods -standard OLS and the Hill estimator. Using OLS, we find that, for the majority of countries (53 out of 73), Zipf's Law is rejected. Using the Hill estimator, Zipf's Law is rejected for the minority of countries (29 out of 73). Non-parametric analysis shows that the Pareto exponent is roughly normally distributed for the OLS estimator, but bimodal for the Hill estimator. Variations in the value of the Pareto exponent are better explained by political economy variables than by economic geography variables.
The Statistics of Urban Scaling and Their Connection to Zipf’s Law
PLoS ONE, 2012
Urban scaling relations characterizing how diverse properties of cities vary on average with their population size have recently been shown to be a general quantitative property of many urban systems around the world. However, in previous studies the statistics of urban indicators were not analyzed in detail, raising important questions about the full characterization of urban properties and how scaling relations may emerge in these larger contexts. Here, we build a selfconsistent statistical framework that characterizes the joint probability distributions of urban indicators and city population sizes across an urban system. To develop this framework empirically we use one of the most granular and stochastic urban indicators available, specifically measuring homicides in cities of Brazil, Colombia and Mexico, three nations with high and fast changing rates of violent crime. We use these data to derive the conditional probability of the number of homicides per year given the population size of a city. To do this we use Bayes' rule together with the estimated conditional probability of city size given their number of homicides and the distribution of total homicides. We then show that scaling laws emerge as expectation values of these conditional statistics. Knowledge of these distributions implies, in turn, a relationship between scaling and population size distribution exponents that can be used to predict Zipf's exponent from urban indicator statistics. Our results also suggest how a general statistical theory of urban indicators may be constructed from the stochastic dynamics of social interaction processes in cities.