City Size Distribution Research Articles

The evolution of Zipf's Law for U.S. cities

AbstractExploiting the hierarchical structure of cities and based on a dataset for U.S. cities between 1840 and 2016, the aim of this paper is to analyze the evolution of the U.S. city size distribution. For that purpose we estimate a general three‐parameter Zipf model, which can be traced back to Mandelbrot (1982), and validate our results by means of the hierarchical scaling law. Especially in the second half of the twentieth century, we find a pronounced departure from the exact Zipf's law. The city size distribution has become more equally distributed over time. Besides, the applied estimation method reveals evidence for leading cities dominating the remaining largest cities. Thus, the growing equality of the city sizes can be explained rather by growing smaller cities than by a loss of importance of the largest ones.

A theory of heterogeneous city growth

AbstractWe consider an economy with three cities producing different outputs. Two cities produce intermediate goods, a type 1 city producing an intermediate “agricultural” good with capital and labor only, and a type 2 city producing an intermediate “industrial” good with capital, labor, and human capital. A type 3 city produces the final good which is obtained from the two intermediate goods and labor. The asymmetric introduction of human capital allows us to prove that the three cities experience, at equilibrium, heterogeneous endogenous growth rates which are proportional to the growth rate of human capital. We show that the “industrial” type 2 city is characterized by the larger growth rate while the “agricultural” type 1 city experiences the lower growth rate, and thus the type 3 city is characterized by a growth rate which is a convex combination of the two former growth rates. This implies that the relative size in terms of output of the “agricultural” city decreases over time. This property allows us to recover the empirical fact that most non‐agricultural production occurs in growing metropolitan areas. But, simultaneously, as we prove that total labor employed in each city is proportional to the total population, the relative population size distribution of cities is constant over time, as shown in empirical studies.

Development of Optimal City Size Theory: A Critical View

Nowadays more and more people reside in metropolitan areas, and the impact of this phenomenon is clearly creating challenges for the environment, the economy and the society at large. It is therefore useful and instructive to consider the issue of optimal city size and review the literature over the past decades on optimal city size theory. Based on the progress in recent years of comparative analysis and inductive research, from both theoretical and empirical perspectives, this paper outlines the original definitions proposed for optimal city size and discusses some of the advantages and disadvantages of the various theories. The two prevailing models in the classical analysis of optimal city size, the Henry George Theorem (HGT) and the neoclassical approach, are examined later. This paper then introduces the supply oriented dynamic model (SOUDY model) and offers a plausible model for optimal city size based on the general equilibrium analysis, which is related to sustainable development. The results show that: (1) Simple optimal city size theories come from the cost-benefit analysis developed by main stream economics, and therefore cannot overcome the defect that the optimal size of cities is the same since the same production function; (2)The Henry George Theorem, which is one of the classical analysis paradigms of optimal city size, has been used more frequently in recent years to research the issues of optimal tax and public goods and has gradually lost its connection to the topic of optimal city size; (3) The neoclassical approach to optimal city size consists of externalities influencing optimal city size, the equilibrium and optimal models for city size, the optimal town, product variety and city size distribution. This kind of research is still in the mainstream of the study of optimal city size. Compared to single optimal city size, more attention is paid to the optimal distribution of city size; and (4) The supply-oriented dynamic model allows for the integration of a spatial dimension into the economic research of optimal city size, and the constraint between city level and city size is relaxed and this has more power to explain real-world problems. At the same time, the general equilibrium analysis, as a significant economic research method, provides a new perspective for the study of optimal city size. The supply-oriented dynamic model and general equilibrium analysis are two new research paradigms that deserve more attention in the analysis of optimal city size.

The Impact of Migration and Innovations on the Life Cycles and Size Distribution of Cities

We present a comprehensive agent-based model of a closed system of cities. The model includes two types of agents - employees and firms. Firms compete for workers and make decisions concerning what to produce and whether to adopt innovations. Individual employees make migration decisions. Some migrants become intrapreneurs when their employers adopt production process innovations that they propose. Some migrants become entrepreneurs when the product innovations that they propose are implemented by their employers in new subsidiary firms. These firms tend to be technological leaders. The decisions of individuals and of firms generate innovation-migration dynamics that generate a variety of city sizes. A city that is home to firms that are currently relatively attractive to migrating innovators experience moderate or fast growth. Because of particular decision patterns by individuals and firms this growth may decline and stop and the city may stagnate and loose workers as its relative attractiveness decreases. Cities that remain unattractive or long periods can stop growing and shrink. We model explicitly the extent to which cities attract immigrants and innovators and demonstrate that the size distribution of cities is defined by the ability of its resident firms to adopt the innovations and to let the product innovators establish technologically advanced enterprises. These decisions result in high market value of the most productive firms, of the entire industrial system the city where the firm is located and of the entire urban system.

The size distribution of cities in China: Evolution of urban system and deviations from Zipf’s law

Abstract How are urban residents distributed and agglomerated across different-sized cities? This question can be addressed by studying urban system or Zipf’s law. Using Chinese data for the period of 1990–2017, this paper contributes to the literature by exploring the evolution of urban system, and more importantly deviations from Zipf's law. It is found that (1) The size distribution of Chinese cities, as expected, did not follow Zipf’s law until very recently. Both large cities and small towns were under-sized relative to the Zipf’s law, implying possible loss of agglomeration economies; (2) When the top 10% large cities are excluded, the Chinese data fit the Zipf's law better, demonstrating the adverse role of government interference in containing the growth of mega-cities; (3) By and large, the distribution has been moving closer or further away from Zipf’s law, corroborating with policy directions in the relevant five-year plans of the central government; and (4) Government interferences helped enlarge while market forces helped reduce the numerical deviations of individual cities from Zipf’s law. The last three analytical findings are the first in the literature since no previous studies have focused on the issue of why urban system in China does not follow Zipf’s law. In particular, no earlier efforts have been made to model numerical deviations from Zipf’s law, as attempted in this paper. The results appeal for removal of government interferences and furthering market-based reforms in order to reap economies of agglomeration.

Importancia económica de las ciudades medias en México

El crecimiento urbano en México ha conducido a una participación económica más equitativa de las ciudades de acuerdo con el tamaño de su población. En particular, las ciudades de tamaño medio de población han ganado peso económico. El objetivo del presente trabajo es analizar la importancia de este tipo de aglomeraciones urbanas en la economía del país durante 2000 y 2010. Esto se prueba a través de la metodología planteada por la ley de Zipf, que mide la distribución del tamaño ―económico― de las ciudades por el coeficiente de Pareto. En términos econométricos, se emplean mínimos cuadrados ordinarios (MCO) con errores estándar robustos y se incluye toda la gama de ciudades considerando regresiones de rangos móviles. Los resultados permiten aceptar la hipótesis de un coeficiente de Pareto menor a la unidad en ambos años para submuestras de mayor tamaño, rechazando, por tanto, la ley de Zipf. En este sentido, se argumenta que las ciudades de tamaño medio de población tienen mayor peso económico relativo que el de cualquier otro tipo de aglomeración urbana.

On the city size distribution: A finite mixture interpretation

Previous studies have shown that city size is lognormally distributed but have not reached a consensus on the shape of the upper tail. Using three datasets of U.S. cities and empirical distribution function statistics, I show that (1) the entire city size distribution is not lognormally distributed; (2) Zipf’s law does not hold generally; and (3) the power-law tail is robust. I then provide an alternative explanation for the observed fat tail: A mixture of lognormal distributions can generate a power law tail. In fact, this fat tail has its statistical origin in Shaked’s theorem. Finally, I provide an urban growth theory to explain how heterogeneous growth factors form a mixture that shapes the aggregate city size distribution.

City Size Distribution, Export-Oriented Economies, and Regional Technical Efficiency: The Case of China

ABSTRACTThis study examines two empirical considerations related to the distribution of cities of different sizes in China. First, the evolution of the distribution of Chinese cities of different sizes is examined across 22 provinces from 1995 to 2015. Second, the determinants of technical efficiency of individual regions within the size distribution are analyzed, as well as how regional growth is influenced by changes in human capital, regional city size distribution, foreign direct investment (FDI), and other factors. A panel data analysis is conducted to examine the potential determinants of the technical efficiency level in Chinese provinces for the period 1995 to 2015 and of regional technical efficiency. It is shown that a region’s urban concentration affects its effective economic growth; however, FDI should not be the engine of future economic growth.

Demography, Symmetry and the Emergence of Universal Patterns in Urban Systems

Urban areas exist in a wide variety of population sizes, from small towns to huge megaci- ties. The statistics of city size distributions have been under scrutiny for over a century. No proposed form has received more attention than Zipf’s law, a Pareto distribution with power value one. Data from many urban systems approximately support Zipf’s distribution while typically also violating its expectations for small and large cities and showing variations in ex- ponents. No theory presently exists to derive city size distributions from demographic choices while also explaining consistent variations. Here we show how the structure of migration flows between cities together with the differential magnitude of their vital rates determine a variety of size distributions. These results provide a powerful framework for deriving Zipf’s law and other size distributions under specific additional conditions, and to resolve puzzles associated with their deviations in terms of concepts of symmetry, information, and selection.

Empirical Investigation of City Size Distribution in Iran

This study investigates the evidence on the city-size distribution in Iranian urban system. The aim of this paper is to undertake the empirical testing of Zipf and Gibrat’s law based on the national population in 2006, 2011 and 2016, considering 337 cities, which range from 20,010 to 8,737,510 inhabitants. The findings show that the process of city growth in Iran is not follows by Zipf’s law and confirm the uneven city growth dynamism and heterogeneous distribution of population in urban structure of Iran. The main reason behind that is the fact that the Capital of Iran, Tehran, exercise a primate city with extreme population concentration in comparison to other urban structure elements due to gigantic economic activity compared to other areas and centralized administrations that serves beyond their national boundaries.

Statistical analysis of urbanization and an application towards a Pareto model

Background: India is only 23.7 percent urban at the present time and this cannot be considered a high level of urbanization. However, when we look at the number of people living in the cities (around 156 million out of the total population of 684 million in 1981) the significance of urban population assumes importance. Aims and objective: To fit Pareto model to the urban population data of Tamil Nadu state and to compare the urbanization in Tamil Nadu over the 3 decades (1981, 1991, 2001). Materials and method: we were collected three years Indian census population data of Tamil Nadu state (1981, 1991, and 2001) As city size models are in skew nature the Pareto model has been fitted to the Tamil Nadu population data for the above said three years and the fitness of the model were tested using chi-square statistics. Result: The estimates of the parameters a, k for the years 1981, 1991 and 2001 are 1.784, 6039.4224, 1.3901, 5807.60 and 1.79572, 6688.082, the χ2 –statistics for the years 1981, 1991 and 2001 are 171.64, 104.737 and 304.819, i.e., for all the three years the χ2 0.05 < χ2 calculated value hence the Pareto model is not good fit for the all the population data. Conclusion: The city size distribution is skewed distributions. The less variation between the observed and expected number of cities in all the 3 years. Thus the Pareto model may be proposed and we found that the Pareto model is not a good fit for all the population data.

Testing Heaps\u2019 law for cities using administrative and gridded population data sets

Since 2008 the number of individuals living in urban areas has surpassed that of rural areas and in the next decades urbanisation is expected to further increase, especially in developing countries. A country’s urbanisation depends both on the distribution of city sizes, describing the fraction of cities with a given population (or area), and the overall number of cities in the country. Here we present empirical evidence suggesting the validity of Heaps’ law for cities: the expected number of cities in a country is only a function of the country’s total population (or built-up area) and the distribution of city sizes. This implies the absence of correlations in the spatial distribution of cities. We show that this result holds at the country scale using the official administrative definition of cities provided by the Geonames dataset, as well as at the local scale, for areas of 128 times 128 km2 in the United States, using a morphological definition of urban clusters obtained from the Global Rural-Urban Mapping Project (GRUMP) dataset. We also derive a general theoretical result applicable to all systems characterised by a Zipf distribution of group sizes, which describes the relationship between the expected number of groups (cities) and the total number of elements in all groups (population), providing further insights on the relationship between Zipf’s law and Heaps’ law for finite-size systems.

Spatiotemporal Pattern Analysis of China’s Cities Based on High-Resolution Imagery from 2000 to 2015

The urbanization level in China has increased rapidly since beginning of the 21st century, and the monitoring and analysis of urban expansion has become a popular topic in geoscience applications. However, problems, such as inconsistent concepts and extraction standards, low precision, and poor comparability, existing in urban monitoring may lead to wrong conclusions. This study selects 337 cities at the prefecture level and above in China as research subjects and uses high-resolution images and geographic information data in a semi-automatic extraction method to identify urban areas in 2000, 2005, 2010, and 2015. City size distribution patterns, urban expansion regional characteristics, and expansion types are analyzed. Results show that Chinese cities maintained a high-speed growth trend from 2000 to 2015, with the total area increasing by 115.79%. The overall scale of a city continues to expand, and the system becomes increasingly complex. The urban system is more balanced than the ideal Zipf distribution, but it also exhibited different characteristics in 2005. Urban areas are mostly concentrated in the eastern and central regions, and the difference between the east and the west is considerable. However, cities in the western region continuously expand. Beijing, Shanghai, Tianjin, and Guangzhou are the four largest cities in China. Approximately 73.30% of the cities are expanding in an extended manner; the urban form tends to be scattered, and land use efficiency is low. The new urban areas mainly come from cultivated land and ecological land.

Universal City-size distributions through rank ordering

We consider a two-parameter discrete generalized beta (DGB) distribution and propose its universal applications to study the size-distribution of the urban agglomerations across various countries in the world, where the urban agglomerations include the small and mid-sized cities along with the heavily populated cities. Our proposition is validated by an exhaustive study with the 3 decades’ census data for India and China and census data of USA for a time window of 8 years. Moreover, we have studied the city size distributions for many different countries, like Brazil, Italy, Sweden, Australia, Uganda etc., from all the continents around the world according to the availability of the data. The detailed analyses exhibit a unique global pattern for the city size distributions, from low to high size, across the world with various geographic and economic conditions. Further analyses based on the entropy of the distribution provide insights on the underlying randomness and spreads of the city sizes within a country. The DGB distribution, a typical rank order (RO) distribution, through its two parameters, not only fits the data on wider range of city sizes better than the well-known power law for all the countries considered, it also helps us to characterize, discriminate and study their evolution over time.

Size distribution of cities: A kinetic explanation

We present a kinetic approach to the formation of urban agglomerations which is based on simple rules of immigration and emigration. In most cases, the Boltzmann-type kinetic description allows to obtain, within an asymptotic procedure, a Fokker–Planck equation with variable coefficients of diffusion and drift, which describes the evolution in time of some probability density of the city size. It is shown that, in dependence of the microscopic rules of migration, the equilibrium density can follow both a power law for large values of the size variable, which contains as particular case a Zipf’s law behavior, and a lognormal law for middle and low values of the size variable. In particular, connections between the value of Pareto index of the power law at equilibrium and the disposal of the population to emigration are outlined. The theoretical findings are tested with recent data of the populations of Italy and Switzerland.

Lognormal city size distribution and distance

This paper analyses whether the size distribution of nearby cities is lognormally distributed by using a distance-based approach. We use data from three different definitions of US cities in 2010, considering all possible combinations of cities within a 300-mile radius. The results indicate that support for the lognormal distribution decreases with distance, although the lognormal distribution cannot be rejected in most of the cases for distances below 100 miles.

Trade, Merchants, and the Lost Cities of the Bronze Age*

AbstractWe analyze a large data set of commercial records produced by Assyrian merchants in the nineteenth century BCE. Using the information from these records, we estimate a structural gravity model of long-distance trade in the Bronze Age. We use our structural gravity model to locate lost ancient cities. In many cases, our estimates confirm the conjectures of historians who follow different methodologies. In some instances, our estimates confirm one conjecture against others. We also structurally estimate ancient city sizes and offer evidence in support of the hypothesis that large cities tend to emerge at the intersections of natural transport routes, as dictated by topography. Finally, we document persistent patterns in the distribution of city sizes across four millennia, find a distance elasticity of trade in the Bronze Age close to modern estimates, and show suggestive evidence that the distribution of ancient city sizes, inferred from trade data, is well approximated by Zipf’s law.

US city-size distribution and space

ABSTRACTThis paper focuses on spatial city-size distribution in the United States. It proposes a new distance-based approach to analyze the influence of distance on the city-size distribution parameter by considering the Pareto distribution and using data from different definitions of US cities in 2010. Considering all possible combinations of cities within a 300-mile radius, the results indicate that the Pareto distribution cannot be rejected in most cases regardless of city size. Placebo regressions validate the results, thereby confirming the significant effect of geography on the Pareto exponent.

Analysis of city size distribution in Ethiopia: Empirical evidence from 1984 to 2012

Urbanisation in Ethiopia is increasing rapidly. This paper uses city-level data to investigate the evolution of Ethiopia’s city size distribution from 1984 to 2012, using descriptive statistics and time-series Gini coefficient calculations. Over that period, the study found that overall city size distribution followed an apparent parallel growth pattern, with disparities among the country’s regional states. This demonstrates that the parallel growth rule of city size distribution found in developed countries and most emerging economies applies in Ethiopia. Findings suggest that changes in the country’s city size distribution from 1984 to 2007 resulted from changes in the political system and policy issues. The results enhance understanding of the evolution of Ethiopia’s city size distribution and have important implications for policy debates.

The principle of least effort and Zipf distribution

”Each individual will adopt a course of action that will involve the expenditure of the probably least average of his work.” This statement was named ”the principle of least effort”. The principle of least effort is often known as a ”deterministic description of human behavior”. In this paper, we present a brief introduction of this principle. Applications of the principle in different fields are also summarized. As the principle of least effort is proposed by Zipf, it is also called Zipf’s law. We then discuss the correlation between three widely considered distributions: Zipf distribution, Pareto distribution and probability distribution. With empirical investigations, it is often stated that, most social behaviors are controlled by the pure Zipf’s law that corresponds to the Zipf distribution of exponent -1. We summarily present the discovery of Zipf’s law in different social behaviors. Some empirical studies are also given as examples, verifying that, in most countries, the distribution of city size by population follows Zipf’s law, and the exponent of Zipf distribution of individual income is about -0.5, the same as Zipf predicted in theory.