In our analysis of scaling effects in cities, we review the model of Bettencourt (2013), which holds that socioeconomic quantities such as per-capita gross domestic product of a city increase with the 1/6 power of the city’s population. However, most empirical results we examine in that same article find weaker scaling relationships with smaller exponents, and in some cases, there is no clear evidence that per-capita GDP grows with population at all.
One possible explanation for the discrepancy is that, in Bettencourt (2013), one of the model’s assumptions is that interaction is local. A city is implicitly treated in isolation. In reality, though, economically significant interactions can clearly be nonlocal. Especially in the era of the Internet, economically significant interactions take place through telecommunications. Furthermore, airports, highways, and intercity rail foster important interactions.
Complementary Interactions and Flattened Scaling
We may consider two separate questions.
Are local and remote interactions complements, rather than substitutes?
To say that local (within the same city) and remote (between different cities) are substitutes means that, as one increase, the other generally decreases. To say that they are complements means that, as one increases, the other generally increases as well.
The second question is,
Do remote interactions flatten urban scaling?
It is possible that both answers are ‘yes’ simultaneously. Let us consider a mathematical model for how this may be possible. Suppose that socioeconomic quantities, such as gross domestic product, are proportional to total interactions. Suppose also that that local interactions per capita scale with the 1/6 power of the city’s population, as in Bettencourt (2013). Suppose also that remote interactions can be expressed as a function r(N) of the city population N, and total interactions is the sum of local and remote interactions. Then socioeconomic quantities y(N) can be expressed as
\[y(N) \propto aN^{1/6} + r(N).\]As we will see, we do not have strong evidence to express r(N) with precision. If a city is an isolated system with no remote interaction, as implicitly modeled by Bettencourt (2013), then r(N) = 0. If local and remote interactions are substitutes, then since local interactions grow with N, it follows that remote interactions r(N) decrease with N. Now suppose, hypothetically, that local and remote interactions are complements, and r(N) is proportional to the 1/12 power of N. Then
\[y(N) \propto aN^{1/6} + bN^{1/12}.\]Since the scaling factor is determined as the slope of the natural log of the output curve relative to the natural log of population, we can calculate the observed factor as follows.
\[m = \frac{d ln(y)}{d ln(N)} = \frac{1}{6}\frac{aN^{1/6}+\frac{b}{2}N^{1/12}}{aN^{1/6}+bN^{1/12}} \lt \frac{1}{6} .\]If b is large relative to a, then the scaling exponent is close to 1/12 rather than 1/6. If b increases relative to a over time as a result of advancing telecommunications technology, but in a way that is independent from N, then we have that local and remote interactions are complements at any fixed moment in time, and also that remote interactions are increasingly suppressing the strength of urban scaling over time. However, if r(N) is proportional an exponent of N that is greater than 1/6, then the effect of remote interaction is to strengthen urban scaling, rather than suppress it. It should be emphasized, though, that we do not have strong evidence to describe r(N) other than that it increases with N.
A testable prediction of the model is that, for smaller cities, the remote interaction term dominates and causes a lower measured scaling exponent, while for larger cities, the local interaction term dominates and causes a higher measured scaling exponent.
Do Telecommunications Flatten Scaling?
In our review of the pace of life, we examine Sapienza et al. (2023) and Schläpfer et al. (2014), both of which find that cell phone call volume per capita tends to increase in larger cities.
Superlinear scaling has been found by other researchers. Bokányi, Kondor, and Vattay (2019) examine a dataset of Twitter (now X) corpora and find that the number of Tweets per user grows with the 1.02 power of city size. The number of words tweeted also tends to grow with the 1.02 power, negating the possibility that users in large cities are decreasing the size of tweets to compensate for volume. They also examine the prevalence of individual words and find that some grow superlinearly with city size and some grow sublinearly. The confidence interval of Bokányi, Kondor, and Vattay (2019) contains 1.0, and so they cannot rule out that there is a linear relationship between city size and Twitter activity. In a similar vein, Arthur and Williams (2019) examine the relationship between the number of Tweets from a region and the region’s population density. Excluding some types of Tweets, such as from bots and weather stations, they find much stronger relationship: the number of Tweets from a region grows in an exponent of the population density as high as 1.7, depending on the spatial resolution at which density is measured.
There are several methodological differences between Bokányi, Kondor, and Vattay (2019) and Arthur and Williams (2019) that makes the results not directly comparable. The geographic scopes differ; the former spans the United States, while the latter is based in the South-West region of the United Kingdom. The former study regresses against city sizes, while the latter study regresses against population density. To further illustrate the sensitivity of the results to the geographic unit of analysis, Arthur and Williams (2019) find a scaling exponent of about 1.7 for a spatial resolution of around 30×30 kilometers to 80×80 kilometers; the exponent drops to around 1.3 to 1.4 for resolutions less than 20×20 kilometers, and it also shrinks for higher resolutions. Arthur and Williams (2019) also consider geo-tagged Tweets, which remove the majority of Tweets that are generated by bots, weather stations, and third party services, while the other study does not make such a modification. A moral of the story is that some scaling results, such as those related to social media usage, are not robust to methodological choices and should not be extrapolated far beyond the context in which the result was measured.
Arvidsson, Lovsjö, and Keuschnigg (2023) examine several socioeconomic metrics–related to connectivity, economic output, and innovation–to identify scaling properties. Their key insight is that examining city averages to identify scaling offers an incomplete picture, and we need to understand the dynamics of these metrics within cities. They define the tail to be those individuals who are above the 90th percentile for a given metric within their city. They find that, depending on the metric, individuals within the tail are responsible for 36-80% of the observed scaling. Variations in the behavior of individuals at the tail explain 34% of the variation between cities. Firm revenue per employee drops the most when the tail is excluded, though all six metrics drop. This implies that, when the top 10% of individuals are excluded, cities vary less by size than scaling laws would imply. This may imply that the tails are disproportionately conducted remote interaction.
One important caveat of these studies, as well as the studies of Sapienza et al. (2023) and Schläpfer et al. (2014), is that they do not distinguish between local and remote interaction. It is thus unclear how much they contribute to the remote r(N) term in our model.
Evidence from Air Travel
Distant interactions can be mediated by telecommunications, as discussed above, or by intercity transportation such as commercial aviation. Matsumoto (2007) hypothesize that a gravity model explains the volume of air travel, both passenger and cargo, between cities. Modeled from Isaac Newton’s law of gravitation, a gravity model holds that the volume of interaction between two cities should be proportional to the product of the “masses” of the two cities (here, measured both as GDP and as population) and inversely proportional to the square of the distance.
For intercontinental flights in 2000, Matsumoto (2007) finds scaling exponents with respect to population of 0.25 and 0.33 for per capita passenger air travel and per capita cargo transport respectively. These numbers suggest that aviation increases, rather than diminishes, the importance of size in explaining a city’s prowess. Scaling exponents greater than Bettencourt’s (2013) theoretical 1/6 are found for per capita passenger and cargo air transport for flights within the Americas, within Europe, and within Asia as well. The paper also finds strong scaling with respect to GDP–exponents of 0.31 and 0.32–and similarly high values for the regional flights, with the exception of an exponent of -0.10 for per capita air cargo within the Americas.
O’Connor (2003) argues that the impact that commercial aviation has on cities depends greatly on the particulars of aircraft technology. He finds that prior to 1990, aviation generally favored GDP growth in the largest and wealthiest cities. However, from 1999 to 2000, second tier cities (defined on a four tier scale) gained the greatest number of air passengers, while first tier cities lost the greatest number. Among the top 100 airports by traffic, the top 10 declined from 36.3% to 31.0% of passengers from 1990 to 2000, and those ranked 21-50 increased from 25.2% to 29.4%. He attributes this change to the introduction of mid-sized long haul aircraft–the Boeing 777 and Airbus 340 particularly–the refinement of engine technology to enable 13-15 hour flights, and the introduction of regional jets, though the latter’s impact may be ambiguous by increasing the catchment area of top tier airports. Additional factors include deregulation of the airline industry and congestion at top tier airports. O’Connor and Fuellhart (2013) perform a similar analysis and find that from 2005 to 2010, the 41 Alpha cities saw a 13.0% increase in the number of passengers, while in 40 Beta cities, that increase was 16.4%.
Wong et al. (2019) also perform an analysis similar to that of O’Connor (2003), and they too find that second tier airports gain passengers relative to top tier airports, as well as an increasing phenomenon of hub bypassing. However, their analysis extends to the phenomenon of large cities that host multiple airports. On the airport-city level, they find a slight increase in total concentration.
Oliveira et al. (2020) find ambiguous effects on the concentration of air travel in Brazil. When examining cities by population, they find a decoupling from 1995 to 2012, driven by the technological feasibility of smaller jets and by deregulation. However, when examining cities by GDP, they find an increased concentration over that time, indicating that new routes were concentrated heavily in wealthier cities.
One important distinction between these studies is the cross-sectional analysis versus the time trends. Matsumoto (2007) finds strong cross-sectional scaling of air travel with city size, while O’Connor (2003) and the follow-up analyses find dispersal over time. Thus the effect of aviation on city concentration depends on which perspective we take.
Conclusion
It is plausible that communication and transportation technology are leading to a “death of distance” and suppressing the importance of city agglomeration as a driver of economic activity. The evidence reviewed in this article, however, paints an ambiguous picture as to whether this is the case.
References
Bettencourt, L. “The Origins of Scaling in Cities”. Science 340(6139), pp. 1438-1441. June 2013.
Sapienza, A., Lítlá, M., Lehmann, S., Alessandretti, L. “Exposure to urban and rural contexts shapes smartphone usage behavior”. PNAS Nexus 2(11): pgad357. November 2023.
Schläpfer, M., Bettencourt, L., Grauwin, S., Raschke, M., Claxton, R., Smoreda, Z., West, G.B., Ratti, C. “The scaling of human interactions with city size”. Journal of the Royal Society Interface 11(98). September 2014.
Bokányi, E., Kondor, D., Vattay, G. “Scaling in words on Twitter”. Royal Society Open Science 6(10): 190027. October 2019.
Arthur, R., Williams, H.T. “Scaling laws in geo-located Twitter data”. PloS one 14(7): e0218454. July 2019.
Arvidsson, M., Lovsjö, N., Keuschnigg, M. “Urban scaling laws arise from within-city inequalities”. Nature Human Behaviour 7(3), pp. 365-374. March 2023.
Matsumoto, H. “International air network structures and air traffic density of world cities”. Transportation Research Part E: Logistics and Transportation Review 43(3), pp. 269-282. May 2007.
O’Connor, K. “Global air travel: toward concentration or dispersal?”. Journal of Transport Geography 11(2), pp. 83-92. June 2003.
O’Connor, K., Fuellhart, K. “Change in air services at second rank cities”. Journal of Air Transport Management 28, pp. 26-30. May 2013.
Wong, W.H., Cheung, T., Zhang, A., Wang, Y. “Is spatial dispersal the dominant trend in air transport development? A global analysis for 2006–2015”. Journal of Air Transport Management 74(1-12). January 2019.
Oliveira, R.P., Oliveira, A.V., Lohmann, G., Bettini, H.F. “The geographic concentrations of air traffic and economic development: A spatiotemporal analysis of their association and decoupling in Brazil”. Journal of Transport Geography 87:102792. July 2020.