INFRAMARGINAL TRAVELERS AND TRANSPORTATION POLICY

• Published date: 01 August 2024
• Author: Jonathan D. Hall
• Date: 01 August 2024
• DOI: http://doi.org/10.1111/iere.12692

INTERNATIONALECONOMIC REVIEW

Vol. 65, No. 3, August 2024 DOI: 10.1111/iere.12692

INFRAMARGINAL TRAVELERS AND TRANSPORTATION POLICY∗

By Jonathan D. Hall

University of Alabama, USA

Structural models of traffic congestion, such as the bottleneck model, are used to answer important, policy-

relevant questions. However, existing models typically assume that no travelers are inframarginal regarding

when to travel; that is, given equilibrium travel times, no travelers strictly prefer their ex ante departure time

to all others. In this article, I address this shortcoming by incorporating inframarginal travelers into these mod-

els. This change significantly improves these models’ ability to fit the data and changes policy prescriptions. In

the case of congestion pricing, it typically changes the optimal toll by at least 25% and significantly worsens the

distributional impacts.

1. introduction

The share of the world’s population living in urban areas continues to grow dramatically

(U.N., 2019), imposing significant challenges on transportation systems worldwide. It is thus

unsurprising that nearly half of all U.S. mayors name traffic congestion as one of their top

three problems (Bloomberg Philanthropies, 2018). The impacts of congestion are enormous,

costing the average American commuter 99 hours each year (INRIX, 2020). This extra time

spent in traffic wastes fuel, creating additional pollution where people live and work, lead-

ing to serious health problems (Currie and Walker, 2011). These problems are even worse in

many developing countries (INRIX, 2020). As a result, cities worldwide are considering con-

gestion pricing and other transport policies to address congestion.1

Structural models of congestion have proven themselves a powerful tool for analyzing ur-

ban transportation policy. These models have been used extensively to provide insight on a

variety of important questions, including optimal highway capacity (Arnott et al., 1993), the

value of travel time information (Arnott et al., 1999; Khan and Amin, 2018), how tolls affect

urban spatial structure (Gubins and Verhoef, 2014; Takayama and Kuwahara, 2017), which

road segments should be tolled (de Palma et al., 2004), nonprice mechanisms for addressing

congestion (Nie, 2015), and the distributional consequences of road tolls (van den Berg and

Verhoef, 2011; Hall, 2018; Kreindler, 2018).2

∗Manuscript received December 2021; revised January 2024.

I am grateful for feedback from Richard Arnott, Karen Bernhardt-Walther, Gustavo Bobonis, Gilles Duran-

ton, Chelsea Hall, Stephan Heblich, Lewis Lehe, Robin Lindsey, Mark Phillips, Robert McMillan, Peter Morrow,

Erik Verhoef, William Strange, and audiences at the National University of Singapore, Tinbergen Institute, Uni-

versity of Alabama, University of Copenhagen, University of Toronto, International Transport Economics Associ-

ation, Urban Economics Association, and the North American Meetings of the Regional Science Association In-

ternational. This article previously circulated as “Improving Structural Models of Congestion.” Please address cor-

respondence to: Jonathan D. Hall, University of Alabama, Box 870224, Tuscaloosa, Alabama 35487, USA. E-mail:

jonathan.hall@ua.edu.

1New York City, Los Angeles, Beijing, São Paulo, Vancouver, Toronto, Portland, and Seattle are considering con-

gestion pricing. Since 2015, new toll roads have opened in Colorado, California, Florida, Georgia, North Carolina,

Texas, and Virginia. Other policies that address congestion include expanding road capacity, land-use planning, build-

ing public transit, and introducing driving restrictions.

2Other important questions addressed include the effects of second-best pricing (Lindsey et al., 2012; van den

Berg, 2014), how tolls affect unemployment and the spatial distribution of activity across a region (Vandyck

1519

© 2024 the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association.

1520 hall

The use of structural congestion models continues to grow. The most commonly used struc-

tural model of congestion, the bottleneck model of Vickrey (1969) and Arnott et al. (1993),

was used in 149 papers between 2010 and 2019, more than triple its use in the previous decade

(Li et al., 2020).

This article’s contributions are, first, showing that standard structural models of congestion

fail to accurately predict travel times when using reasonable, commonly used values for pref-

erence parameters; second, showing how to improve the models so they fit the data, arguing

that the most important change is to allow travelers to be inframarginal regarding the choice

of when to arrive; and third, showing that making these improvements changes policy counter-

factuals quantitatively and qualitatively.

I start by comparing the predicted travel times from standard structural models of conges-

tion to data, finding that they fit observed travel times poorly when using reasonable, com-

monly used parameter values for traveler preferences. In Section 3, I use travel time data for

7.6 million origin–destination (O–D) pairs in 41 cities worldwide, and find that for essentially

all of the O–D pairs these models’ predictions for either peak travel times or length of the

peak period to be off by an order of magnitude (or some combination of the two). Further-

more, the parameter values required for the model to fit the data imply that in essentially all

of the O–D pairs, travelers with even moderately inflexible schedules do not exist. The poor

fit suggests that these models are abstracting away from features that have a first-order impact

on model predictions, limiting our ability to use them to evaluate counterfactuals.

The source of the poor fit lies almost entirely with how we model traveler preferences over

a continuum of arrival times, and has little to do with the model of congestion itself. Thus,

these problems affect all structural models of congestion, including the bottleneck model

(Vickrey, 1969; Arnott et al., 1993), the no-propagation model (Henderson, 1974; Chu, 1995),

the instantaneous propagation (also known as the bathtub or macroscopic fundamental dia-

gram) model (Arnott, 2013; Vickrey, 2019), the cell transmission model (Daganzo, 1994), and

the hydrodynamic traffic flow model (Lighthill and Whitham, 1955; Richards, 1956).3Since

this same model of preferences is used in models of transportation by train and airplane, these

issues are likewise relevant to those models.4These problems do not apply to all alterna-

tive ways of modeling traveler preferences; for example, they do not apply to discrete-choice

models.5

Sections 4 and 5 explore multiple possible solutions to the poor fit, concluding there are

two improvements to make to how we model traveler preferences. First and most importantly,

these models should allow for travelers who are inframarginal with respect to the choice of

when to arrive. Such travelers strictly prefer their chosen arrival times to all others, given

equilibrium prices, and so are not indifferent between multiple arrival times. More formally,

these are travelers for whom the marginal benefit of a small change in their arrival time is not

equal to the marginal cost of such a change, and so they are at a corner solution regarding

the choice of when to arrive. To be concise, I call these travelers inframarginal, suppressing

and Rutherford, 2018), how teleworking impacts congestion (Gubins and Verhoef, 2011), optimal parking prices

(Inci, 2015), carpooling (Yu et al., 2019), and the cost of travel time variability (Noland and Small, 1995). Further-

more, the time-of-use decision at the heart of structural models of traffic congestion is also used in models of trans-

portation by train (Kraus and Yoshida, 2002; de Palma et al., 2017) and airplane (Silva et al., 2014; Blondiau et al.,

2018; de Palma et al., 2018).

3These models differ only in how they model congestion. In the bottleneck model, travelers slow down only those

who come after them; in the no-propagation model, travelers slow down only those departing (or arriving) at the

same time as them; in the instantaneous propagation model, travelers slow down those who come before, after, and

traveling at the same time; and in the cell transmission and Lighthill and Whitham (1955)–Richards (1956) models,

travelers slow down both those who come after and those traveling at the same time.

4For example, de Palma et al. (2017) and Silva et al. (2014).

5Using a discrete-choice model comes with its own set of trade-offs, as Arnott et al. (1993) argue that any model of

traffic congestion that divides time into intervals is poorly specified.

inframarginal travelers and transportation policy 1521

the specification of which dimension of choice travelers are inframarginal on.6Travelers with

inflexible schedules—including workers with fixed start times, such as factory, retail, and con-

struction workers, and those who need to pick up or drop off their children at specific times—

are likely to be inframarginal. However, whether a traveler is inframarginal depends on both

their preferences and equilibrium prices, so even travelers with somewhat flexible schedules

can be inframarginal.

Empirical evidence suggests that approximately half of all travelers are inframarginal. The

best evidence comes from Kreindler (2018), who uses a smartphone application to implement

tolling for 497 commuters in Bangalore, India. As this group is too small to affect equilib-

rium travel times, his tolls are an exogenous increase in the cost of travelers’ chosen arrival

time. He finds that 46% of drivers respond to time-varying tolls, consistent with the other

54% being inframarginal. The 2017 U.S. National Household Travel Survey reports that 57%

of workers cannot choose their start times, and Hall (2021) reports that 57% of travelers on

California State Route 91 leave early or late to avoid traffic, consistent with the other 43% be-

ing inframarginal.

The intuition for why allowing travelers to be inframarginal has such large effects on model

fit starts with recognizing that these models have at their core travelers’ decisions of when to

travel. Travelers choose between arriving on-time and facing, in equilibrium, long travel times,

or arriving early or late in exchange for shorter travel times.7The rates at which travel times

rise and fall are thus determined by travelers’ willingness to pay in travel time to reduce how

early or late they arrive. As Menger (1871) and others explained, prices (or, in this case, travel

times) depend on the marginal, instead of average, consumers’ willingness to pay. Since ex-

isting models do not allow travelers to be inframarginal, their predictions of the rates travel

times rise and fall are based on the average, instead of marginal, willingness to pay. As a re-

sult, they predict travel times that are too high.

The second way structural congestion models should be improved is by recognizing that

many travelers prefer to be late instead of early, where late and early are defined relative to

the traveler’s desired arrival time in the absence of traffic congestion. Although this clashes

with our intuition that arriving late is much worse than arriving early, arriving later than de-

sired need not imply arriving literally late. Ideally, a traveler might wish to arrive at a doctor’s

appointment at 8 am. However, given traffic congestion, he schedules the appointment for 10

am and arrives later than desired, but not late. This helps fit the data because, empirically,

the reduction in travel time from arriving slightly later (conditional on arriving after the peak

of rush hour) is typically less than the reduction in travel time from arriving slightly earlier

(conditional on arriving before the peak of rush hour).

Finally, Section 6 shows that making these improvements has important consequences when

designing policy. This occurs because errors in predicting travel times directly affect policy.

For example, if predicted travel times are too high, recommended tolls are also too high. Al-

lowing travelers to be inframarginal affects policy design in an additional way: since infra-

marginal travelers strictly prefer their chosen arrival times, any policy that changes these trav-

elers’ arrival times likely causes them significant harm. This changes both the aggregate and

distributional effects of proposed policies. I use congestion pricing as an example, finding that

allowing travelers to be inframarginal changes social welfare gains, the socially optimal toll,

and the distributional impacts. I quantify these effects using three numerical examples, finding

that it reduces social welfare gains by over 15%, changes the maximum toll by over 25%, and

6Travelers can also be inframarginal regarding other choices, such as the choice of whether to travel, where to

travel, or which route to take.

7This trade-off is similar to that between the price of housing per square foot and commuting costs in models of

urban structure, such as the monocentric city model of Alonso (1964), Mills (1967), and Muth (1969). Just as mod-

els of urban structure explain the decline in the price of housing (per square foot) with distance to the city center as

a compensating differential for commuting costs, structural models of congestion explain the decline in travel times

with distance to the peak of rush hour as a compensating differential for the cost of arriving early or late.