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

This article is part of the ongoing Special Collection 8 "Formal Relationships"

Abstract

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

Higher incomes are increasingly associated with higher fertility: Evidence from the Netherlands, 2008–2022
Volume 51 - Article 26    | Keywords: fertility, income, inequalities, Netherlands, parenthood

Differences in mortality before retirement: The role of living arrangements and marital status in Denmark
Volume 50 - Article 20    | Keywords: inequalities, living arrangements, marital status, mortality, retirement

Gender and educational inequalities in disability-free life expectancy among older adults living in Italian regions
Volume 47 - Article 29    | Keywords: disability, gender, health, inequalities, Italy, mortality, older adults, regions

Do tenants suffer from status syndrome? Homeownership, norms, and suicide in Belgium
Volume 46 - Article 16    | Keywords: housing tenure, inequalities, life course, mental health, mortality, social norms, socioeconomic disparity, suicide

Parental education, divorce, and children’s educational attainment: Evidence from a comparative analysis
Volume 46 - Article 3    | Keywords: divorce, education, inequalities, parental divorce