The size-growth relationship: Exploring the rank order of city populations in mainland Britain

Highlights

• A majority of cities are continuously declining or growing in prominence, implying a persistent cause.

• With the notable exception of London, large centres are declining in prominence.

• Movement down the size hierarchy more common than movement up but the latter is more rapid than the former.

• There is prominence-convergence among the smaller cities.

• There is a U-shape in the size-growth relationship, obscured by a decline in prominence among very large cities.

Abstract

This paper considers the relationship between city-size and growth in a UK context. This has a number of elements. Using population data from 62 cities with an extended Turok and Mykhnenko (2007) classification of time paths, aligned ranks of population shares reveals that persistent advantageous or disadvantageous trends are in the majority. These persistent paths are at odds with the proposition that Zipf's law for cities is perpetuated by shocks being randomly distributed across the size spectrum.

Three contributions in the field of convergence that can be applied here are: intra-distributional mobility; convergence, where there are persistent trends, implies eventual leapfrogging and divergence; which is explored by Boyle and McCarthy's (1997) concordance-based tests for beta-convergence.

As population growth is found to be inversely related to city-prominence, there is evidence for convergence, but only in part. There is a U-shape in the size-growth relationship that is obscured by a decline in prominence among the very large cities whilst the primate city, London, is pulling away from the rest of UK urban system. It is argued that ‘path-dependency’ that reflects the inheritance of deindustrialisation is more appropriate than a Gibrat-type stochastic model of growth for the north and west.

Introduction

An observed regularity in urban economics is that cities in a system are of unequal sizes, and that the number of cities inversely related to size. Mainstream urban economics appeals to scale economies to explain the inequality and to underwrite greater nominal wages and day-time and resident populations. However, the spatial general equilibrium framework (Roback, 1982) projects that higher wages and house prices reflect greater productivity in larger centres. As such, productivity inequalities exist but, through mobile agents, relocating within and between nodes to maximise their utility, there is a mechanism to achieve equilibrium across space.

Central place theory suggests a size hierarchy evolves in part because, where services are provided to the local hinterland, there are minimum efficient scales that require demand thresholds for services to be met. This restricts the number of larger nodes. Hsu (2012) argues that central place theory can be used to generate a city-size distribution that follows Zipf's law. This is based on heterogeneity in economies of scale: larger cities can support greater, as well as a wider range of industries. Interestingly, Hsu's analysis predicts fractal structures, which leads to the power law. Storper and Walker (1989) argue that cities hold their position in the size-hierarchy by capturing propulsive industries, which predicts slow, disorderly change. Failure to adapt to a declining industrial environment can place cities some on a relatively ineluctable path to decreasing prominence and shrinkage, notably with deindustrialisation (Martin et al., 2014).

Challenging theoretical positions on the rank and city-size is Gabaix (1999), who proposes that city-growth can be linked to the law of Gibrat. That is, the growth of the city is independent of its size. City-size increases (and possibly shrinks) stochastically. Zipf's power law holds true because the majority of economic shocks decline with city-size (Gabaix, 1999: 760). Batty (2015) observes that city-growth volatility is inversely related to rank-order, and that growth differences are ironed out by the logarithmic scaling.

Beta-convergence is said to exist when a region with a relatively low-income per head experiences a faster growth rate than the average of the group of regions, and the gap between it and richer regions is reduced; low-income regions have a higher growth rate than high-income ones. Among the variety of convergence concepts, Sala-i-Martin (1996) includes intra-distributional mobility. Quah (1996) proposes that a logical implication of such convergence analysis is leapfrogging; low-income regions continue on persistent growth paths so that, eventually, their position in the rank-order changes, or there is shuffling. By implication, if trends persist, there can be convergence followed by divergence, so differential growth and reshuffling may not undermine a hierarchy, just reorder the elements. In a consideration of the flattening of the city-size distribution, or size-convergence, Black and Henderson (2003) explore the rate of distributional mobility among US city populations over 1900–1990. Although they find that movement up the rankings is much faster than downwards, they report cities tend to revert to a long-run size-order.

This paper classifies population trajectories in a style pioneered by Turok and Mykhnenko (2007), contrasting the outcomes against the analysis of city-size and growth in mainland Britain. Although the rank-size rule and Zipf's law provide predictions about the distribution of city-sizes they do not say how stable this is. The notion of shuffling is found in Boyle and McCarthy's (1997) approach to beta-convergence. The paper reworks Boyle and McCarthy's use of ranks in the world of unequal levels. It evaluates, first, whether the city rank-order across Britain is stable; and second, whether growth rates and general sizes of city are associated. Both reflections relate to mainstream tests of the rank-size rule and Zipf's law, but using non-parametric methods.

The paper is constructed as follows. First, there is a review of Zipf's law and the rank-size rule. Second, Storper and Walker's (1989) view of history, structure and the notion of the path-dependency is explored. This is followed by a discussion of Turok and Mykhnenko (2007) nine paths of population evolution. Converting populations into proportions then aligned ranks, patterns are classified by eye, grouped by hierarchical cluster analysis and then by Euclidean distances. Two non-parametric techniques are applied to the population data. Kendall's concordance is used to establish whether the size-order is stable. A Page test is used to reveal any links between size-order and growth. It is concluded that history and location are key explanations of the size-growth relationships. Deindustrialisation, particularly among the largest cities, explains sluggish growth and, indirectly, why the rank-size is not undermined.