Urban scaling laws

This theme issue is dedicated to urban scaling laws which are one of the cornerstones in the search for a science of cities. Such search is an interdisciplinary quest, which can only be advanced through cumulative theories that build on ideas about how the form of cities determines their functions and vice versa as well as localised knowledge that defines the particular characteristics of individual cities and their cultures (Mumford, 1938). These come together to explain the underlying mechanisms which give rise to the observed generic patterns in urban systems, despite their distinct historical and political trajectories, socio-demographic and geographical constraints. For centuries now, different disciplines have been aware of some of the emergent characteristics that cities display; nevertheless, it was not until the middle of the last century that a bottom-up approach integrating the different fields began to shape the way urbanism might be approached. Location theory developed in the late 19th century which became crystallised in the works of Christaller (1966 [1933]) and Losch (1954 [1940]) generated theories of cities embodying spatial economics, and the evolution of urban economics tied together ideas about transport, cost, income, rent, and density. Notions about how cities grow and are planned from the bottom up build on ideas about systems, emergence, path dependence and a host of related ideas that had come to form the science of complexity by the millennium (Alexander, 1964; Jacobs, 1961; Weaver, 1948). Ideas from statistical physics also found analogies in city systems and out of this came notions about scaling (Berry, 1964; Simon, 1955; Zipf, 2012 [1949]). Complexity science has thus become the overarching umbrella under which different systems are considered as co-evolving entities whose characteristics are outcome of the interdependencies and interactions at the local level.

Among observed universal characteristics, the well-known Zipf’s law (Zipf, 2012 [1949]) defines the way the sizes of settlements in a country relate to one another and it is thus the key signature measuring the fact that a hierarchy of city sizes exists. In its pure form, the law suggests that in a coherent system of cities, there is only one big city whose size is twice the size of second biggest, three times bigger than the third city, and so forth. In mathematical terms, this forms a power law distribution of city sizes (Auerbach, 1913), usually holding only for cities above some minimum size threshold. This has generated considerable controversy, as to whether or not the full distribution of city sizes could be better modelled by a log-normal distribution rather than a power-law (Corral et al., 2019; Eeckhout, 2004). Moreover this also focuses the debate on how we define cities for in a world where ultimately we will all be living in cities of one size or another, the very definition of a city itself is brought into focus. It is important to take such limitations on size into consideration when constructing a science of cities where universal behaviour is under scrutiny: Is there a science for all cities, or can it only be devised for the largest cities? Or for just the richest cities? Or for cities in the developed world? These are key open questions and there are many.

Another universal observation pertaining to cities is the agglomeration phenomenon that encapsulates economies and diseconomies of scale. Since the 18th century, increasing returns were first conceived implicitly through the division of labour by Adam Smith (1778), and then increased productivity associated with cities came to occupy centre-stage within economics as first articulated by Marshall (1890). Such regularities entail a non-linear relationship between the size of a city and its productivity. In biology, non-linear relationships between size and form for organisms have been largely studied as part of allometry (Huxley, 1932; Snell, 1892; Thompson, 1917). In the 1960s, such concepts were extended to urban systems particularly by the Michigan School of Mathematical Geographers, and along with a concern for urban growth and population density, a whole issue of the journal Ekistics edited by Dutton and Woldenberg (1973) was devoted to these ideas but only in a casual fashion. Ideas about dimension particularly through the emergence of fractal geometry quickly emerged and to an extent were popularised by Benoit Mandelbrot (1967, 1983). Scaling was then taken to the next level being used to model urban growth in cities through self-similarity (Batty et al., 1989; Batty and Longley, 1994; Frankhauser, 1998). Form in cities could thus be seen as being reflected as self-similarity through universal scaling properties which serve to establish their fractal structure.

In addition to physiological structures (form), processes (function) measured by the metabolic rate, R, also determine non-linear relationships with respect to the size M of the animal as R ∼ M^β. Kleiber determined that the metabolic rate is efficient by empirically finding that the scaling coefficient was β = 3/4 (Kleiber, 1932, 1947) but it is worth mentioning that in relating body surface and volume, one would expect an exponent of 2/3. This phenomenon was then analysed as a branching process based on fractal geometry and in the 1990s, a mathematical derivation confirmed an exponent of 3/4 (West et al., 1997). The idea of the city as a metabolic system dates back to Wolman (1965) but the usual approach to urban metabolism either investigates energy equivalents or more broadly flows of mass, in particular water, materials and nutrients (Kennedy et al., 2011).

In the last 15 years, the broader relationships between the size of a city and its urban indicators have been reintroduced under the rubric of ‘Urban Scaling Laws’ particularly by Bettencourt et al. (2007) who were searching for a universal taxonomy of urban indicators, within the same spirit that Kleiber’s law suggested for the metabolic rate. On the other hand, an earlier working paper by Pumain (2004) made the point that size was not a sufficient quantity to establish such relationships. She pointed out that the way cities were defined together with how the speed of transportation would affect such measures. In particular, she suggested that indicators based on economic activity would depend on the level of maturity of the sector in question; hence for the same industry, different exponents would likely be obtained for developed and developing countries.

Overall, a surge of publications in the field arose providing renewed interest, and momentum was further generated through its potential applications to policy. Given ongoing urbanisation, the claim that city size could determine whether cities are more productive, more innovative, more creative – that is, whether or not the total wages per capita in large cities are higher than in small ones, can have profound consequences. On the other hand, findings such as crime scaling superlinearly in contrast to urban GDP support the common perception that larger cities are both undesirable and desirable, that large cities are part of the crucible for greater wealth but also the crucible for greater poverty (Bettencourt and West, 2010). From the scientific perspective, a set of follow-up questions emerge. In particular, urban scaling is an empirical finding, and a systematic understanding which would explain the processes leading to non-linear scaling is lacking. Certainly, agglomeration economies have been studied earlier (e.g. Sveikauskas, 1975), but the challenge is to unify scaling of the many diverse features of cities and their generalisation across countries.

As mentioned above, there are some open issues deserving attention and we will list some of these:

While in the case of metabolic systems, the organism (e.g. an animal) is a well-defined entity, this is no longer the case for cities, whose boundaries might be defined in terms of administrative delimitations, or the extent of their urban form, or in terms of their functional economic areas if commuters are considered, as is the case of metropolitan areas (Arcaute et al., 2015; Batty and Ferguson, 2011).

In 2016, we organised a symposium Cities as Complex Systems: Structure, Scaling, and Economics (CTCS2016) (Note: The workshop took place in Herrenhausen Palace, Hanover, 13–15 July 2016, and was funded by Volkswagen Foundation. http://www.pik-potsdam.de/ctcs2016/) aiming at facilitating critical and constructive discussions around some of the above-mentioned problems within a multidisciplinary perspective. Forty-five researchers from different disciplines as diverse as physics, geography, architecture and economics were brought together. The discussions were framed within three overarching topics: urban allometry, urban economics and urban morphology, all examined within the framework of complexity science. This theme issue is a sample of some of the work on urban scaling that emerged from the symposium.

The problem of the city definition for measuring the scaling exponent was considered in the first paper that follows this editorial: in Clementine Cottineau, Olivier Finance, Erez Hatna, Elsa Arcaute and Michael Batty’s paper (2019) ‘Defining urban clusters to detect agglomeration economies’ the authors used a method similar to that in Arcaute et al. (2015). Different definitions of cities in France were then used to explore agglomeration economies. The authors find that ‘larger cities tend to concentrate the jobs rather than paying higher wages for the same job’. Moreover, they report evidence for agglomeration economies at a regional scale, a productive advantage of local concentrations in terms of works, and conclude more generally that agglomeration economies depend on the scale of observation. The paper also looked at inequality, and found that size was not a sufficient quantity to predict inequality, and that spatial structure needs to be much more completely integrated into the picture.

The latter problem was further analysed by Somwrita Sarkar (2019) in a second paper ‘Urban scaling and the geographic concentration of inequalities by city size’. Instead of using aggregated quantities, she examines their distributions. Through her analysis she identifies factors for the geographic distributions and spatial inequalities of income and housing costs. In detail, she finds that income distributions and housing costs in the largest cities scale superlinearly so that the highest income earners as well as the highest housing costs agglomerate in these cities, while low-middle income earners and low-medium housing costs are underrepresented. She further argues that the concentration of wealth may push out lower and medium income earners and calls for urban policies addressing affordability, diversity and socio-spatial justice.

In the third paper ‘Two metropolisation gradients in the European system of cities revealed by scaling laws’, Denise Pumain and Celine Rozenblat (2019), instead of using urban scaling laws as a means to uncover agglomeration economies, posit that if globalisation can be considered as the diffusion of an innovation, the exponent would reveal the level of metropolisation. This refers to the level of globalisation attained given by the position of the country in question in the hierarchical structure of the diffusion process. The authors consider functional areas and show that Western and Eastern countries can both be divided in two subsystems characterised by different metropolisation gradients. They attribute such a differentiation to the way cities in Eastern countries are organised, where diffusion takes place in a stronger hierarchical way, hampering adaptation.

The final article by Olivier Finance and Clementine Cottineau (2019) looks at the statistical deficiencies of the methods used to explore urban scaling laws. In particular, they investigate the problem of zero-values for urban indicators, which are mostly present in small cities. As urban scaling is usually analysed by linear regressions to the logarithmic values, a divergence is obtained if the logarithm is applied to zero. However, those values cannot simply be omitted as they still carry information. The authors show how erroneous conclusions can be drawn from ordinary least squares and propose an approach to avoid misinterpretations.

Last but not least we introduce a short paper, a commentary by Ramana Gudipudi, Diego Rybski, Matthias Lüdeke, and Jürgen Kropp (2019), who discuss the limits and perspectives of urban scaling in general as well as of carbon emissions. Some of the issues mentioned in this editorial are considered in more detail there. In particular, the authors argue that depending on the regression method used, a range of scaling exponents – crossing super- and sub-linear regimes – can be obtained, in particular if the correlations are low to modest but still significant.

The work compiled in this special theme demonstrates that the problems identified within urban scaling laws are still in need of further investigation, and that the basic underlying mechanisms giving rise to the observed phenomenology are still poorly understood. These ideas suggest that we must be careful in drawing generalisations from these aggregate characteristics of cities such as income, wealth, crime, GDP and a host of other indicators vary with respect to their size, and we must be particularly astute about suggesting that bigger cities are richer, more creative, and more innovative than smaller ones.