The size-growth relationship: Exploring the rank order of city populations in mainland Britain
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.
Section snippets
City-size rules?
Restating the observed regularity as Zipf's law, this relates the log of rank-size of cities to the log of the corresponding population. The rank-size rule, a sister of Zipf's law, asserts that other city sizes are a function of the first city. Specifically, the second city is expected to be half the size of the first, and the third city, one third.
Gabaix (1999) proposes that Zipf's law can be linked to the law of Gibrat, positing that the growth of the city is independent of its size.
Path-dependency
An important dimension in this slow-dynamics environment is development is not subject to random shocks, rather it is path-dependent (Martin et al., 2014; Storper, 1995). Storper and Walker (1989) argue that long-term, unbalanced regional growth is inherent in the capitalist system. Cities and regions are based on industrial agglomerations, exploiting external economies of scale. Growth comes from the exploitation of new products or processes in the Schumpeterian sense. Urban and regional
Method
To generate the share of the population living in each city, the urban populations are divided by the UK's total population for each year. The populus could shift between cities, and between cities and third-party areas, so a path reflects a city's relative prominence or the proportion of the UK's population it serves at a given time. To standardise them further, aligned ranks are taken. This entails ranking populations for each of the T periods from the 62 cities. This is done to prevent
Data
The data is taken from the Centre for Cities' website.4 This is a repository of census-based population data from 1981 to 2018. The definition of a city as a primary urban area (PUA) is drawn from the work of CURDS.5 The method for defining PUAs was originally devised as a foundation step of the research incorporated in the State of the English
Analysis of aligned ranks
Averaging the first and last 4 years for each city to avoid over or under inflating growth rates due to random shocks, over 1981–2018, there is a rise in the UK population of 16.8%, with the 62 cities growing slightly smaller rate (15.2%). The proportion that cities account for is stable at around 53%.
Reviewing the evolution paths by eye of the 62 the two most common paths are of continuous decline (22%) and continuous growth (18%). The most populous urban centres are declining. London,
Shuffling
Fig. 4 shows the path of W2 (Levels) which suggests the agreement between the rank order of populations in 1981 and those of subsequent years confirms an almost rigid size-structure. The coefficient falls steadily to a value of 0.988, almost identical to the global value of W1 of 0.9938 [0.000]. To explore shuffling among a variety of sizes, subgroups are analysed. Cities are divided into Large (city average of around 700,000 to 2.4 million, Small (up to 300,000), Medium-sized (300,000 to
Consistent growth?
Shifting to growth, a Gibrat thesis implies no agreement in the order of city-growth over time. Boyle and McCarthy's two versions of W point to a structure in growth. W1 = 0.334 [0.000] implies persistently different growth rates over 38 years across the cities. Comparing the growth rates across 1981/82 with future years, the pairwise W2 (82) value for that with 2013/14 of 0.76 [0.000] points to a very similar growth orders for samples 32 years apart. Moreover, this latter value is above the
Page test
In keeping with the above, a Page test, when using the mean-ranks of the city-sizes as a structure to the growth-order, suggests that the null of equal growth rates across the 62 is rejected (6.47 [0.000]). The alternative indicates that growth is inversely related to size; cities at the lower end of the size hierarchy grow more rapidly. Bifurcating the city sample into the 29 larger (2.67 [0.004]) and 32 Small (3.11 [0.000]) cities, the Page test results for the cities replicate the global
The regional pattern
Fothergill and Houston (2016) conclude that UK city performance is an issue of location rather than size. Pike et al. (2016) emphasise deindustrialisation. Here, history and location combine to paint a picture of an urban crisis of a deconcentration of jobs and population. Of the 23 British cities with declining profiles with sluggish growth (CD, RR3 and LD3), 14 are located in the north and west of England, three are in Scotland and two in Wales. Note that there are only 9 other cities in the
Analysis
By comparing the mean-rank in the first and last four years, there are 36 decliners and 26 advancers in the relative size order. The average shift downwards [upwards] is 3 [3.4] places. Twenty-seven [19] shift the equivalent of one place or more downwards [upwards]. Once the changes of less than one place are removed (which could reflect non-reranking of large cities, such as London), the values become 3.81 [4.4]. In other words, ascendance is quicker and descent is slower, but there are more
Deviations at the top
Estimating the Pareto exponent q = −0.906 (R2 = 0.963) for 1981. Performing this for every year the path of values is illustrated in Fig. 4 as Pareto. Over the 38 years, the exponent drifts downwards. Using Batty's (2015) combination of the rank 1981 with the city-size for subsequent years presents an almost identical drift (not shown). A variant of Batty's approach entails using the log of the aligned mean-ranks as the explanatory variable for each year. This is shown as ParetoB. Although the
Conclusion
This paper sets out to examine size of population centres and the associated growth rates. The relative prominence of 62 centres examined over 1981–2018. It evaluates whether common growth rates can be revealed among various sizes of city, consistent with a test for Zipf's law, and whether the size-order of cities across Britain is stable.
Turok and Mykhnenko (2007) profiles of city prominence are augmented so that a more finely grained picture can emerge. Using Euclidean distances as a means of
CRediT authorship contribution statement
The work submitted and resubmitted is mine and mine alone.
Declaration of competing interest
There are no funders and no conflicts of interests.
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David Gray is a Lecturer in Economics and the University of Lincoln.
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His interests include housing and labour markets, especially measures of spillover between regional and local markets.