Volume 44 - Article 49 | Pages 1165–1184

Measuring the concentration of urban population in the negative exponential model using the Lorenz curve, Gini coefficient, Hoover dissimilarity index, and relative entropy

By Joel E. Cohen

Print this page  Facebook  Twitter


Date received:12 Oct 2020
Date published:11 Jun 2021
Word count:2713
Keywords:Clark's exponential model, density, Gini coefficient, Hoover index, inequalities, Lorenz curve, population concentration, spatial distribution, urban population
Weblink:All publications in the ongoing Special Collection 8 "Formal Relationships" can be found at http://www.demographic-research.org/special/8/


Background: Stewart (1947) and Clark (1951) proposed that urban population density is a negative exponential function of the distance from a city’s center. This model of the spatial distribution of urban population density has been influential in urban economics, transportation planning, and urban demography. Duncan (1957) suggested characterizing the inequality in the distribution of urban population density in this model by using standard economic measures of concentration or unevenness: the Lorenz curve, the Gini coefficient, and the Hoover dissimilarity index. Batty (1974) advocated measuring concentration using relative entropy.

Objective: We execute Duncan’s (1957) and Batty’s (1974) suggestions using mathematical analysis, not simulations.

Methods: We modify the negative exponential model to recognize that any city has a finite radius.

Results: Mathematical analysis reveals that all four measures of concentration depend sensitively on the finite radius of the city in the negative exponential model. We give a numerical example of the sensitivity of the concentration measures to the boundary radius.

Contribution: In empirical applications of the negative exponential model of urban population density, it is important to have clear, consistent standards for defining urban boundaries. Otherwise, differences between cities or over time within the same city in these four and perhaps other measures of concentration could be due at least in part to differences in defining the radius or other boundaries of the city.

Author's Affiliation

Joel E. Cohen - Rockefeller University, United States of America [Email]

Other articles by the same author/authors in Demographic Research

» Gompertz, Makeham, and Siler models explain Taylor's law in human mortality data
Volume 38 - Article 29

» Taylor's power law in human mortality
Volume 33 - Article 21

» Is the fraction of people ever born who are currently alive rising or falling?
Volume 30 - Article 56

» Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy
Volume 24 - Article 11

» Life expectancy is the death-weighted average of the reciprocal of the survival-specific force of mortality
Volume 22 - Article 5

» Constant global population with demographic heterogeneity
Volume 18 - Article 14

Most recent similar articles in Demographic Research

» Introduction to the special collection on family changes and inequality in East Asia
Volume 44 - Article 40    | Keywords: inequalities

» Outsurvival as a measure of the inequality of lifespans between two populations
Volume 44 - Article 35    | Keywords: inequalities

» Trajectory of inequality of opportunity in child height growth: Evidence from the Young Lives study
Volume 42 - Article 7    | Keywords: inequalities

» Race/ethnic inequalities in early adolescent development in the United Kingdom and United States
Volume 40 - Article 6    | Keywords: inequalities

» Cause-specific mortality as a sentinel indicator of current socioeconomic conditions in Italy
Volume 39 - Article 21    | Keywords: inequalities






Similar Articles



Jump to Article

Volume Page
Volume Article ID