Larger Cities are More Congested

Yes, the headline looks like it was written by Captain Obvious. Today, I want to offer an introduction to urban scaling as it relates to traffic congestion and discuss why large cities tend to be more congested.

I have written several posts about some of the drawbacks of larger cities. Most recently, I discussed how the incidence of infectious disease scales with city size, and there I found that clear evidence of a relationship exists only for sexually transmitted diseases. Last year, I discussed how many types of crime, on a per capita basis, become more prevalent within larger cities. Today’s post on cities and traffic congestion can be treated as part of that group.

This post is done under a Living Literature Review grant from Coefficient Giving. It is part of the Scaling in Human Societies project. All conclusions are my own and do not reflect the views of the funding organization.

We will first look at two papers related to urban scaling and traffic congestion.

Let us start with Depersin and Barthelemy (2018). It is not the most advanced or logically airtight result out there, but it is a good starting place. This paper approaches the question of traffic congestion and city size by finding a scaling exponent with regression. Long-time followers of the Scaling project has now seen this technique many times, going back to the foundational Bettencourt, Lobo, and Helbing (2007). The paper does acknowledge some wrinkles, which we will get to.

Depersin and Barthelemy (2018) 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—as usual, metropolitan region rather than a city as defined 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 materially affected either way.

One of the headline numbers 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, which means 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.

By now, I hope that some readers have seen problems with the headline number of 1.36. A few months ago, I discussed the issue of conflating temporal (one city over time) data with cross-sectional (many cities at a single point in time) data. The two perspectives often give different measures of scaling exponents, and that is true with Depersin and Barthelemy (2018) as well. Yet the authors amalgamate the two perspectives into a single dataset. To make matters more complicated, individual cities are highly path dependent. A city with high traffic congestion today will most likely have high congestion next year, and so they are not independent measurements. For these reasons, I do not consider the 1.36 exponent to be reliable. In fairness, the authors do explicitly highlight these problems.

There are a few other 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. I also have some doubts about the TTI-based metric for congestion, which is total delay, and I think that delay divided by total driving distance/time would be a more relevant metric.

Our second, somewhat older paper, Louf and Barthelemy (2014), offers a somewhat 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 area 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 assumed. For now, the important feature is that it 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.

Coming back to the density issue, 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.

Empirically and for theoretical reasons discussed in Betterncourt (2013), larger cities tend to be denser and have less roadway per capita. This is an efficiency of large cities, but it comes at a cost of congestion. Depersin and Barthelemy (2018), as well as the lived experience of just about anyone who is old enough, show that traffic congestion, and the resulting waste of time and fuel, has gotten worse over time.

It can be argued1 that, for all the attention that deservedly goes to restrictive zoning, traffic congestion is also a major barrier to infill housing in large, prosperous cities and thus a contributor to high prices. Even if one would not go that far, good urbanism, which has the goal of unlocking the benefits of agglomeration that large cities offer, need to tackle the downsides of cities. Crime is one such downside that I discussed before. Congestion might be the most potent.