Congestion Scaling
It is generally understood that larger cities tend to experience greater traffic congestion than smaller cities. Several papers have, following Bettencourt et al. (2007), have attempted to derive scaling laws to describe this behavior. In Bettencourt et al. (2007), a major foundational work in urban scaling, scaling laws are derived by performing a linear regression of the quanity in question against the logarithm of a city’s population. For some quantities, such as income per capita, it is generally found to increase with the city’s population, while other quantities, such as road space per capita, generally decrease.
Depersin and Barthelemy (2018)
One of the simpler attempts to derive a scaling law for congestion is performed in Depersin and Barthelemy (2018). They analyze a traffic dataset from the Texas A&M Transportation Institute. The dataset consists of 101 cities and annual traffic data spanning 1982 to 2014, or 33 years, for a total of 3333 data points. They define congestion as the time delay due to traffic, or in other words, the difference between actual travel time and what the travel time would be under free flowing traffic. For each city, defined as a metropolitan region rather than a city as determined by political boundaries, they also have the population from the same data set. Here, population is defined as the number of drivers, rather than the total number of people, though the results would not be similar either way.
The headline number is that total traffic delay scales with an exponent of β ≈ 1.36±0.01 (per capita delay exponent of 0.36) when applied to the full set of 3333 data points.
Considering temporal trends, the authors divide the 101 cities into three categories, based on how they develop over time. Type-1 cities themselves follow a power law pattern over time reasonably well. For individual cities of this category, the scaling exponent over time is usually greater than that of the whole data set, with only two of the 31 cities having an exponent of less than 1.5 and 13 with an exponent greater than 2.5.
Type-2 cities, comprising 43 cities in the dataset, are best described by a two-piece power law function. That is, for one time period, traffic delays are described by a power law with exponent β₁, and for the other time period, traffic delays are described by a power law with a different exponent β₂. For all but two (Jackson, MS and Lancaster-Palmdale, CA) of those cities, β₁ > β₂, and in some cases, β₂ < 1, meaning that per capita traffic delays actually declines with further city growth. The authors’ interpretation of this fact is that, as a city grows earlier in the dataset, delays increase rapidly up to a saturation point, beyond which motorists respond by driving less rather than suffering more delays.
A third category, Type-3 cities, comprise the rest of the dataset and are those cities that do not fit into the other two types.
Perhaps the biggest weakness of Depersin and Barthelemy (2018) is the way that they aggregate cross-sectional (many cities at a single point in time) data with temporal (one city over a period of time) data into a single data set that is the basis for regression. Often, scaling relationships as determined from cross-sectional and temporal data are quite different. Additionally, individual cities are highly path-dependent, meaning that for a given city, congestion at two nearby points in time is highly correlated. Consequently, the 1.36 scaling exponent should not be regarded as a reliable figure.
There are some additional issues. Over the 32 year span considered, U.S. metros generally grew by no more than 60%, and often much less than that, and so the scaling exponents, especially a two-piecewise model, hold over too small a range to draw firm conclusions. The relevance of the TTI-based metric for congestion, which is total delay, can be questioned. Perhaps a more relevant metric is given by dividing delay time by total driving distance or total driving time.
Louf and Barthelemy (2014)
A somewhat older paper, Louf and Barthelemy (2014), offers a more theoretical approach to scaling and congestion. The authors start with an idealized model of a city. It is reasonable (though not quite accurate as we’ll see) to suppose that the population density of a city remains constant as it grows. In one extreme, the monocentric model applies, in which all commuters travel to the city center each day. Then, since radius grows like the square root of population, so does the per-capita driving. In the other extreme, the city remains entirely disbursed, per-capita driving remains constant, and the large city functions like many small, autonomous cities that happen to be close to each other. Where between these extremes actual cities land is determined empirically and not theoretically.
A mathematical form of congestion, which is again defined as excess commute time over free-flowing traffic, is given. The authors find that this metric of congestion grows with the amount of traffic on a given road and is superlinear in city size. Both assumptions are eminently reasonable, but again, the precise form of the equation is assumed and not shown.
Regarding density, Louf and Barthelemy (2014) show with the data that the population density is not in fact constant as population grows. They find that area grows with roughly the 0.85 power of population, meaning that density increases with the 0.15 power of population. On the question of polycentricity, they find that the number of activity centers in a city grows with the 0.64 power of population, and so the commuting time and distance of individuals necessarily grows with population. Note that people do not necessarily work in the nearest activity center. The paper models that workers seem to maximize their salary minus transportation costs, and so sometimes they will choose to work in a center that is farther but at which there is a higher paying job.
The paper also estimates an ideal city size, which is determined by minimizing the sum of two costs: maintenance of road infrastructure and congestion. Since larger cities are denser, as we saw above, road maintenance cost minimization would steer cities toward larger sizes. However, since larger cities have less road per capita but require the same driving per capita, they become more congested. Ideal (and presumably, actual) city size is determined by a tradeoff between maintenance and congestion costs.
Agglomeration economies, in the sense of higher wages and GDP per capita, do not significantly figure into Louf and Barthelemy (2014). The one agglomeration economy they do focus on is, as discussed above, shared infrastructure and thus less per-capita infrastructure.
With all this additional structure, Louf and Barthelemy (2014) do fit a power law estimate to congestion in terms of population in a manner similar to that of Depersin and Barthelemy (2018). Based on 97 urban ares in the United States, they find a scaling exponent of 1.270 ± 0.067, a bit less than the finding of Depersin and Barthelemy (2018) but within the same ballpark.
Road Space and City Size
Bettencourt (2013), one of the foundational works in urban scaling, presents as one of its theoretical results that the length of infrastructure grows with the 5/6 power of population. Here, “infrastructure” refers to lane-miles of roadway, as well as length of water pipes and electrical transmission lines. Bettencourt et al. (2007) find a scaling exponent of 0.83, close to the theoretical prediction, by examining 29 cities in Germany in 2002.
Louf and Barthelemy (2014) derive a scaling relationship of several transportation quantities in terms of population. They finds that area grows with the 0.85 power of population and total lane miles with the 0.86 power of population. Although it therefore appears that the portion of city land dedicated to roads is constant with size, one caution for drawing this conclusion is that the two power laws are not formed on the same data sets.
Guerra, Duranton, and Ma (2025) estimate that in the United States, 21.7% of urban land is dedicated to roadways. Their approach is first to build a dataset of roads segments from the U.S. Highway Performance Monitoring System, the Topologically Integrated Geographic Encoding and Referencing system, and several other public data sources. To convert these segments to area, it is necessary to multiply the lengths of roadways by their width. The authors have lane and shoulder width for 10% of the roads in the HPMS dataset, and they extrapolate those widths to the rest of the roads using a predictive model. They subtract the area at intersections to avoid double-counting.
In the data of Guerra, Duranton, and Ma (2025), there is no obvious relationship between city population and the share of urban area that goes to roads. As we saw above, larger cities tend to have both a higher population density and less roadway per capita, so that roadway per hectare remains relatively constant with growing population. Nevertheless, there is substantial variation in the roadway/area quotient across cities. The authors find that New York City has the smallest number of lane-miles per capita, at 0.8, while the highest value is in Dallas, Texas at 1.7 lane-miles per capita. These figures are again with the caveat that lane-miles are an imperfect proxy for roadway area.
Guerra, Duranton, and Ma (2025) do find, however, that the roadway portion varies by distance from the city center. They construct a standardized distance metric for each metropolitan area. This is defined as the difference between a census block’s distance from the city center and the average block distance, divided by the standard deviation. The authors find a U-shaped pattern for road area. Blocks with a standardized distances of -1.5 to -1.0 (i.e. very close to the city center) have road portion of around 30%, dropping to less than 20% at around -0.7, and then rising to as high as 50% at a standardized distance greater than 3.0 (i.e. very far from the city center).
To measure total built-up area, Guerra, Duranton, and Ma (2025) estimate from Landsat data. Area within the Census-defined metropolitan area can be misleading, as it may includes large parks or otherwise undeveloped land, and thus this data would providing a misleadingly low assessment of road area. The authors estimate that the road share is 16.9% on the city level, less than the 17.5% in the urban core and 24.2% in the metropolitan area. These numbers confirm the U-shaped pattern of road area.
UN-Habitat (2013) finds an even larger variation when taking a global perspective. The report estimates the street area of 60 cities worldwide through satellite data, a tool that allows relatively consistent measurements across countries. They find that roadway/area varies from as low as 6% in Bangui, Central African Republic to as high as 36% in Manhattan. The report distinguishes between cities and suburbs and find that the former tend to devote a larger share of area to roadways. Generally, cities in lower-income countries have less roadway, as expansion area is often taken by informal housing. UN-Habitat recommends that 30-35% of area goes to roadways, a target that is met by few cities.
Do Cities Need More Roads?
UN-Habitat (2013) argues that most cities in the world should have more road area, but Guerra, Duranton, and Ma (2025) argue that that the costs of building more roads in the United States exceed the benefits by more than a factor of four. Their costs include the opportunity cost of the land, since if it is not used for roadways, it can be used for something else, and they amount to $216 billion for a hypothetical 10% in urban road capacity. Costs also include $211 billion in externalities, including greenhouse gas emissions, local pollution, oil dependency, congestion, and traffic collisions. They also include $159 billion of government spending. These costs are compared against $110 billion of benefits in the form of time savings—taking into account the rebound effect of more driving associated with more road capacity, as well as $25 billion in benefit to the freight trucking industry.
There are some weaknesses with the cost-benefit calculations of Guerra, Duranton, and Ma (2025), though. It is unclear whether the additional trips generated through induced/latent demand are a net positive, given that there are unpriced externalities in the form of congestion and pollution, but the paper counts only costs and not the benefits of extra trips. The paper assess the opportunity cost of land as being the average land value of existing roads, but road expansion is more likely to occur in the outer regions of a city, where land values are lower. Perhaps most significantly, the paper does not consider the possibility that road expansion brings new land into the urban system, and so additional roadway does not necessary come at the one-to-one expense of urban land for other uses.
It may still be the case that road expansion in the United States fails to pass cost-benefit analysis, though some projects may be worthwhile even if road expansion is not sensible in the general case.
As a counterpoint, Allen and Arkolakis (2022) ask the same question—what are the costs and benefits of road expansion?—using a general equilibrium spatial model in which the new or expanded road segment affects transportation costs and traffic congestion. They apply the model to the U.S. highway network and to the Seattle road network. In both cases, they find that the average return for road expansion is positive, with an average rate of return on investment of 108% for the U.S. highway system and 16% for the Seattle road network. Some high-value segments have much higher returns, with the most valuable highway networks being connectors just outside of major metropolitan areas, and in Seattle, the most valuable roads being links surrounding downtown. Some highways and Seattle roads show negative returns on investment, underscoring the need to be judicious even in the face of high average returns overall.
Unlike Guerra, Duranton, and Ma (2025), Allen and Arkolakis (2022) considers spatial reallocation and agglomeration economies. However, also unlike Guerra, Duranton, and Ma (2025), Allen and Arkolakis (2022) does not consider the opportunity cost of road space, possibly leading to a major underestimate of costs. To my knowledge, no cost-benefit analysis does both.
References
Bettencourt, L.M., Lobo, J., Helbing, D., Kühnert, C., West, G.B. “Growth, innovation, scaling, and the pace of life in cities”. Proceedings of the National Academy of Sciences 104(17), pp. 7301-7306. April 2007.
Depersin, J., Barthelemy, M. “From global scaling to the dynamics of individual cities”. Proceedings of the National Academy of Sciences 115(10), pp. 2317-2322. March 2018.
Louf, R., Barthelemy, M. “How congestion shapes cities: from mobility patterns to scaling”. Scientific Reports 4(1): 5561. July 2014.
Bettencourt, L. “The Origin of Scaling in Cities”. Science 340(6139), pp. 1438-1441. June 2013.
Guerra, E., Duranton, G., Ma, X. “Urban Roadway in America: The Amount, Extent, and Value”. Journal of the American Planning Association 91(1), pp. 102-116. January 2025.
UN-Habitat. “Streets as Public Spaces and Drivers of Urban Prosperity”. November 2013.
Allen, T., Arkolakis, C. “The Welfare Effects of Transportation Infrastructure Improvements”. The Review of Economic Studies 89(6), pp. 2911-2957. November 2022.