Population size, the Pareto distribution and steady growth


  • Author: John Fry
  • Date: 04 Dec 2014
  • Copyright: Image appears courtesy of iStock Photo

Chancellor George Osborne has recently announced moves to make Greater Manchester a “northern powerhouse” [1]. Does the available data suggest that Greater Manchester can be used as a model for the economic regeneration of other Northern cities? In the aftermath of the Scottish referendum it is envisaged that one of the ways that future growth may be secured is via increased devolution [2]. However, some political and economic concerns at this move have been raised [1].

thumbnail image: Population size, the Pareto distribution and steady growth

Named after the American linguist George Zipf (1902-1950), Zipf’s law describes the distribution of population sizes for large cities. Zipf’s law predicts a Pareto distribution with exponent 1. For relevant background and additional theoretical modelling see e.g. [3]. Zipf’s law dates at least as far back as the late 19th century and the work of the French stenographer Jean-Baptiste Estoup (1868-1950). Zipf’s law is one of a number of empirical power laws which hint at close links between economics and physics. Other key examples include the study of wealth distributions by the Italian economist Vilfredo Pareto at the turn of the 20th century and a plethora of more recent work (see e.g. [4] for an overview).

In a series of recent papers [5,6] large outliers contravening a Pareto distribution have been interpreted as being generated by a different underlying mechanism to the rest of the data. Thus, if say London or Greater Manchester do not obey Zipf’s law the implication would be that there may be no lessons for economic development that can usefully be applied to other areas. London and Greater Manchester may simply represent special cases.

We begin by looking at metropolitan areas rather than cities. The area of Greater Manchester with an estimated population of 2.7m from the time of the 2011 census dwarfs the City of Manchester with an estimated population of 511,000. In addition to their sheer size metropolitan areas outside of London clearly pack a hefty economic punch. Office for National Statistics (ONS) estimates are that Greater Manchester, the West Midlands and Merseyside contributed £51bn, £49bn and £25bn Gross Value Added (GVA) to the economy in 2012, respectively [1] compared with a total of £309bn for London.

Population data is available for 46 metropolitan areas across the UK from the time of the 2001 census. Although it is originally outlined as a model for city growth rather than for the development of wider metropolitan areas Zipf’s law still seems to offer a reasonable explanation of the data. A log-log plot of the data is shown below in Figure 1 and seems to be roughly linear in nature. Results from a hypothesis test (not reported) retain the null hypothesis that the exponent of the Pareto distribution is 1 both for metropolitan areas in the UK as a whole and for the 19 metropolitan areas in the North of the United Kingdom.

Figure 1: Log-log plot of population size for Metropolitan areas in the United Kingdom

Greater London is by far the largest metropolitan area in the UK as a whole. The population of Greater London is around 3.7 times larger than the second largest metropolitan area of Birmingham. In contrast, as far as the North of the United Kingdom is concerned Greater Manchester is only 10% larger than the next largest metropolitan area of Leeds-Bradford. So it appears that Greater Manchester is unlikely to play the dominant role for the North of the United Kingdom that London has played for the country as a whole. However, because the differences between Manchester and the rest of the North of England are relatively small perhaps economic lessons from Manchester can be more readily applied.

A remarkable consequence of the so-called Renyi’s Representation Theorem [7] is that if population sizes follow Zipf’s law then the ratio of successive order statistics can be shown to have a beta distribution (see e.g. [3]). The result in [3] can also be extended to a Pareto distribution with any general exponent and reflects, at least in part, the richness of the area of Extreme Value Theory. This allows us to construct relevant hypothesis tests and confidence intervals. Here, these suggest that Greater London does not seem unduly large with respect to the rest of the United Kingdom. Further, results also suggest that neither does Greater Manchester seem unduly large with respect to the North of the United Kingdom.

However, there is some suggestion that looking at city sizes may give a better view of the economic geography of inequality [8]. Zipf’s law suggests that the second largest city should be roughly half the size of the second, the third largest should be a third the size of the largest etc. According to [8] this law holds for the relative sizes of cities in most countries but it does not appear to hold in Britain where our second-tier cities appear to be too small relative to London. Following a similar approach in [6] we look at the largest 200 towns and cities in the United Kingdom based on figures available from the 2011 census. Does population data on UK cities, rather than metropolitan areas, lead to formal statistical evidence that London is unduly large?

A log-log plot of population sizes across the 200 largest UK towns and cities is shown in Figure 2. This again suggests that the Pareto distribution appears a reasonable candidate model. However, this time we have some evidence against Zipf’s law. A 95% confidence interval for the exponent of the Pareto distribution is (1.173, 1.548). In [3] Zipf’s law is interpreted as satisfying a steady-growth condition. Finally, confidence intervals constructed using methods in [3] again suggest that the City of London does not appear unduly large with respect to other cities in the United Kingdom.

Figure 2: Log-log plot for the 200 largest towns and cities in the United Kingdom

In conclusion, neither London, Greater London nor Greater Manchester seem to be markedly different from the rest of the United Kingdom in terms of population size. If these areas are seen as generally large and prosperous perhaps they do contain economic lessons that the rest of the United Kingdom can learn from. Metropolitan areas appear to be in a state of steady growth whilst cities do not. Our findings are slightly different to similar studies [6,7], reinforcing the fact that Zipf’s distributions, extreme-value statistics and even the simplest economic data sets all constitute remarkably complex systems.

1. Webber, E. (2014) Should all English cities be more like Manchester? BBC News Website. www.bbc.co.uk/news/uk-29459243
2. Blond, P. and Morrin, M. (2014) Devo Max – Devo Manc: Place-based public services. Published Report, ResPublica Trust Think Tank. www.respublica.org.uk/documents/csv_Devo%20Max%20Report.pdf
3. Gabaix, X. (1999) Zipf’s law for cities: an explanation. The Quarterly Journal of Economics 114, 739-767.
4. Voit, J. (2005) The Statistical Mechanics of Financial Markets, third edition. Springer, Berlin.
5. Malevergne, Y., Pisarenko, V. F., Sornette, D. (2011) Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities. Physical Review E 83, 036111.
6. Pisarenko, V, F. and Sornette, D. (2012) Robust statistical tests of Dragon-Kings beyond power law distributions. European Physical Journal Special Topics 205, 95-115.
7. Reiss, R. (1989) Approximate Distributions of Order Statistics. Springer, London.
8. Overman, H. (2013) The economic future of British cities. CentrePiece Summer, 2-5.

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