Urban Scaling Reconsidered
First off, a good number of new readers have subscribed recently, and so welcome. This blog, once strictly for my personal use, is now a companion to my Scaling in Human Societies project. This is a venture supported by a Living Literature Review grant from Coefficient Giving, although CG exercises no editorial control over the project.
About a year and a half ago, shortly after I started this project, I wrote a post about how cities scale with size. There is strong evidence of superlinear scaling in some socioeconomic quantities, such as incomes and patents. This means that as cities grow, per capita wages tend to increase, and there are more patents per capita. Conversely, some quantities such as road area show sublinear scaling, which means that the total road area of a city tends to decrease per capita as the population increases.
This week, I am revisiting the topic with several studies that challenge the results and the methodology behind urban scaling theory. After considering the challenges, the results are weaker and less certain than I had previously thought. However, I still believe that scaling effects are real. This is an area that continues to merit attention from policymakers and advocates, but also one that should be treated cautiously.
I have already updated the post on the formal Scaling in Human Societies site, and it can be seen here.
The Theory of Scaling
I assume that most readers have not read my post from a year and a half ago, or if they did, do not remember the details. Therefore, we will briefly review the theory and some of the leading results.
Bettencourt et al. (2007) is the foundational work on the subject of urban scaling, but I recommend Bettencourt (2013) for an introduction to the theory. They derive the well-known (in the scaling literature) 7/6 rule for socioeconomic quantities. This means that if the population of a city increases, certain quantities such as total economic output and the number of patents should grow with the 7/6 power of population. In more concrete terms, for example, if the population doubles, all else being equal, we expect average wages to increase by 12%.
The aforementioned Bettencourt et al. (2007) offers some empirical results. Finding results is usually as simple as building a dataset of cities with their population and the desired socioeconomic quantity, such as GDP; taking the logarithm of both quantities; fitting a least-squares line through the data; and calling the result the scaling exponent. Except it isn’t always so simple. Bettencourt et al. (2007) present quite a few results; a sampling of them are as follows. They find that GDP tends to grow with the 1.15 (China, 2002 data), 1.26 (European Union, 1999-2003), or 1.13 (Germany, 2003) power of population. Serious crimes (U.S., 2003) grow with the 1.16 power of population.
As part of the theory, Bettencourt (2013) demonstrates how certain infrastructural quantities tend to grow with the 5/6 power of population. This is sublinear scaling, which means that as cities get larger, there tends to be less infrastructure per capita. Bettencourt et al. (2007) again present several relevant results, such as that road area grows with the 0.83 power of population (Germany, 2002).
There have been a ton of results along these lines, and it would be impossible to review them all. Instead, let us turn to some of the major critiques.
The Challenge of Defining Cities
I’ve touched on this issue so many times that it may be worth its own dedicated post someday, but the task of defining a city is far from straightforward, especially when we are attempting international comparisons. The most common working definition is that a city is a unified labor market, in which anyone can commute to and from the center on a daily basis and in a reasonable amount of time. Historically, a “reasonable amount” has been 30 minutes each way, a regularity known as Marchetti’s constant.
This definition obviously leaves much wiggle room. Marchetti’s constant is only about averages, and so it is reasonable to include outer fringes, 45 minutes or even an hour of commuting time, as part of the city. Major cities are generally polycentric, meaning that they have several separate employment centers, or they may have such job dispersal that most jobs are not in a readily identifiable center. Some continuous built-up urban areas, such as the Nile Delta, the Pearl River Delta, the Blue Banana, and the Northeast Corridor, are far too large to be considered single cities in and of themselves and too polycentric to be divided into cities in a way that would be universally agreed upon, and so there is much room for debate as to which cities constitute the units of analysis for a scaling hypothesis. And the problem is that the results are sensitive to the way that cities are defined.
Cottineau et al. (2017) illustrates the problem well. They consider about 5000 ways in which cities in France may be defined, based on combinations of variables on density, commuting patterns, and population cutoffs. As an example of how the definition matters, more restrictive definitions of cities replicate the finding that total road area grows sublinearly in city size. However, they find that with more expansive definitions of cities, which include outer fringes of metropolitan areas, road area tends to grow superlinearly in population.
Total city area is a metric for which Cottineau et al. (2017) find extreme variations in scaling behavior, depending on how the city is defined. City area was found to vary with population with an exponent as low as 0.334 and as high as 1.291. The high exponent implies that area grows faster than population, or that larger cities also provide more space per capita. The low exponent implies that density rises sharply with population.
Arcaute et al. (2015) examine several socioeconomic characteristics against population size for cities in England and Wales. They find that most per-capita characteristics, including several measures of incomes, do not show a relationship with size. Of those that do, including patents per capita, the relationship is sensitive to the precise way in which cities are delineated. The authors determine city boundaries through aggregation of statistical areas with different density cut-offs. Unlike most other results, this procedure does not make explicit reference to commuting patterns.
Questioning the Statistical Methods
I said above that the regression method frequently used to derive scaling laws isn’t as simple as the procedure sounds. Leitão et al. (2016) in particular challenge that method.
Most models apply a least squares error estimate to a plot of the logarithm of the city’s population and the socioeconomic quantity under consideration. These models then find a scaling exponent β for which the 95% confidence interval does not contain 1, providing strong evidence of superlinear (β>1) or sublinear (β<1) scaling. However, this approach tacitly assumes that the deviations from the best fit are independent and normally distributed. If this assumption is false, it may lead us to overestimate the model’s goodness of fit and our level of confidence in the estimate of β.
Leitão et al. (2016) apply several statistical models to 15 datasets spanning five countries around the world. They find that the data is rarely compatible with the proposed models, and the Gaussian model, a term for the assumption that errors are independent and normally distributed and which is commonly used in scaling research, is not the best fit for any of the data sets. Depending on specification, models are usually dominated by small cities, which comprise the majority of cities, or by large cities, which are home to the majority of the population.
Nevertheless, Leitão et al. (2016) find that, in some cases, the findings of superlinear or sublinear scaling are robust to the model specification. This is true specifically for GDP in the United States and in Brazil, as well as museum usage in the European Union. Road area in the U.S., library usage in the EU, and AIDS cases in Brazil show robust evidence of sublinear scaling. In contrast with previous results, patents in the United Kingdom and in the OECD appear to scale linearly with city size. In other cases, the evidence was inconclusive.
When Does Scaling Break Down?
Where scaling laws can be established, it doesn’t seem natural to expect that they will hold for arbitrarily small or arbitrarily large cities. Furthermore, it is not clear that we should expect a single exponent to govern the growth of cities for all size ranges.
Sutton et al. (2020) examine 67 indicators related to crime, age, property, and mortality in cities in England and Wales. For most indicators, the authors find a median critical density of 27 people per hectare—typical of a low-density single-family suburb—below which the indicators do not show signs of scaling. Furthermore, they show that the apparent scaling for denser cities is the result of denser cities attracting more young adults, rather than a genuine scaling effect. The thresholds for the indicators vary and are generally between 10 and 70 people per hectare. The areas analyzed consist of 348 regions in England and Wales and are based on density rather than overall size.
By contrast, Gomez-Lievano, Youn, and Bettencourt (2012) examine crime rates in cities in Brazil, Colombia, and Mexico. They argue that since some events such as homicides are rare, the apparent failing of scaling laws for very small cities is a result of statistical noise rather than an actual failing.
For large cities, Crosato, Nigmatullin, and Prokopenko (2018) find that the urban form may undergo a phase transition, rather than showing continuity. They build a model, exemplified by Greater Sydney, that shows a phase transition as a city transforms from being monocentric–with a single major center of economic activity–to polycentric, or with multiple centers of activity. This has unclear implications for how scaling develops.
Reading History Sideways
In Reading History Sideways, Arland Thornton (2005), working specifically on family structure, highlights the discredited notion in development economics that societies all pass through the same phases, and modern Western societies are merely farther ahead on the common development pathway. I use this as a general term for a manner of thinking that uses cross-sectional data (many data points at a single moment of time) as a stand-in for longitudinal (a single subject over a period of time) trends. Sometimes we have to understand reality in this way, such as with stellar evolution. But very often it causes misleading results.
With scaling laws, we can readily see the reading history sideways phenomenon at work. We want to understand what will happen to City X if its population doubles, and since this cannot be done in a rigorous experiment, we can instead compare City X to City Y, which has twice the population of City X.
To illustrate the weaknesses of this approach, Keuschnigg (2019) examines wages in 73 labor market areas in Sweden from 1990 to 2012. He finds, as scaling theory predicts, that at a given point in time, average wages grow by 9.4% as population doubles. However, with the exception of the largest markets, he does not find that population growth predicts wage growth in a given market over time. Cross-sectional analysis finds the scaling law, and longitudinal analysis, which is probably of greater interest to policymakers, mostly does not.
Unreviewed as of this writing, Marquis and Barthelemy (2026) also argue that policymakers derived flawed conclusions about the evolution of individual cities over time by examining cross-sectional data. They observe, based on the world dataset on city area and population from Angel et al. (2012), that in recent years, cross-sectional analysis suggests that cities decrease in density with higher population, but longitudinal analysis suggests that they increase in density with population.
Marquis and Barthelemy (2026) also find, considering data on metropolitan areas in the United States from Bettencourt (2013), that longitudinal analysis finds much stronger wage growth with rising population than cross-sectional analysis.
As highlighted by Bettencourt et al. (2010), a city’s residual tends to remain fairly constant over time, even when population changes drastically. Here, the residual is the amount by which the socioeconomic quantity in question is above or below the value predicted by the scaling law. In other words, a city that is particularly wealthy for its size will tend to remain wealthy for its size even when it grows several times over.
The Problem of Causation
I briefly touched on this issue in the last post, but for the sake of making policy, it is very important to understand that, just because large cities tend to be wealthier per capita than smaller cities, it does not follow that population causes higher wealth. A plausible story can be told the other way. Some cities are wealthier for unrelated reasons, such as geographic advantages and good governance, and the more prosperous attract migrants. In other words, wealth causes population rather than population causing wealth.
Melo et al. (2017) note that the most common solution to this problem is an instrumental variable analysis between long-lagged measures of urban agglomeration and geographic or geological variables. Combes et al. (2010) apply this approach in a measurement of the elasticity of wages against density in France. Without attempting to correct for unobserved variables or reverse causality, Combes et al. find that the elasticity is about 0.05. They consider possible endogenous sources of a relationship between density and wages: more productive workers may prefer to live in denser areas, and more productive cities may attract more workers. Controlling for all these sources of bias reduces the observed elasticity to 0.027. The authors find a further reduced elasticity of 0.02 when they account for the spillover of agglomeration effects across spatial boundaries.
After all the corrections, Combes et al. (2010) do still find an agglomeration effect, albeit one much more modest than that of the theoretical model of Bettencourt (2013). However, it is not clear that Combes et al. account for every conceivable endogenous variable.
Conclusions
This post has considered several methodological challenges toward urban scaling theory. Those challenges include the ambiguous definition of a “city” for analysis, the appropriateness of various statistical models, whether scaling properties change or break down at very small or very large sizes, conflation of cross-sectional and longitudinal results, and how to discern causation.
This post has been fairly negative about the research related to urban scaling, partly to serve as a complement to the previous post, which was much more positive. Despite these challenges, I remain of the belief that urban scaling is real, albeit with effects that are more modest than those found in early research.
For policymakers and for advocates, the difficulty in quantifying scaling effects is a great frustration. It seems very reasonable to me to believe that, for instance, a national measure to deregulate zoning in highly productive cities will cause population growth in those cities and thus economic growth. But how much economic growth? Based on my survey of the literature, I don’t think it is possible to answer this question with any confidence even to within an order of magnitude, and this makes it very difficult to craft wise policy.