Cross-sectional evolution of the U.S. city size distribution

We report nonparametrically estimated nonlinear stochastic transition kernels for the evolution of the distribution of populations of metropolitan areas, for the period 1900 to 1990, based on US Census data. Comparison of kernels across successive time periods with the kernel for a pooled sample suggests a fair amount of uniformity in the patterns of mobility during the study period. The distribution of city sizes is predominantely characterised by persistence. Comparison of the kernel for the pooled sample with the kernel for city sizes relative to their own regional average does not reveal any stark differences in intra-region mobility patterns. We then develop measures that allow us to characterise the nature of intra-distribution dynamics for the city size distribution: one is the first-order "serial" (across the ranking) correlation coefficient of the differences in relative sizes of cities with successive rankings; the second is the mean squared variation of the differences in relative sizes of cities with successive rankings. These measures have the major advantages that they do not require discretization of the city size distribution, nor do they obscure subtle changes within the distribution. We employ these measures to study the degree of mobility within the US city size distribution and, separately, within regional and urban subsystems. We find that different regions show different degrees of intra-distribution mobility. In addition, in contrast to received wisdom, second-tier cities show more mobility than top-tier cities.