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University of Southern California Dissertations and Theses
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Three essays on strategic commuters with late arrival penalties and toll lanes
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Three essays on strategic commuters with late arrival penalties and toll lanes
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Content
THREE ESSAYS ON STRATEGIC COMMUTERS WITH LATE ARRIV AL
PENALTIES AND TOLL LANES
by
Tal Roitberg
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ECONOMICS)
August 2022
Copyright 2022 Tal Roitberg
Acknowledgements
Thanks to Professor Antonio Bento for his mentorship and supervision throughout this dissertation.
Thanks to Professor Paulina Oliva for many helpful comments and suggestions.
Thanks to Professors Antonio Bento, Paulina Oliva, and Matt Kahn for their support on the job
market.
Thanks to Professors Arie Kapteyn, James Moore and Giorio Coricelli for helpful feedback on
my research.
ii
Table of Contents
Acknowledgements ii
List of Tables vii
List of Figures viii
Abstract xiii
Chapter 1: Can’t Wait? Urgency with Strategic Commuters and Tolled Express Lanes 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Preview of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Main Model Setup and Some Theoretical Results . . . . . . . . . . . . . . . . . . 5
1.2.1 Road Structure and Timing . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Information Known to Commuters . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Commuter Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 Exogenous Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.5 Congestion: The Endogenous Delays . . . . . . . . . . . . . . . . . . . . 9
1.2.5.1 Defining Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.6 Optimal Choice of Lane and Time: Theoretical Results . . . . . . . . . . . 10
1.2.7 Accounting for Random Variation in Travel Time between Home and High-
way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.8 Spreading Express Lane Commuters Over Time . . . . . . . . . . . . . . . 12
1.2.9 Spreading Commuters Between Lanes . . . . . . . . . . . . . . . . . . . . 13
1.2.10 Significance of Theoretical Results . . . . . . . . . . . . . . . . . . . . . 13
1.3 Empirical Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Calibrating Parameters of Simulation: An Initial Approach . . . . . . . . . 14
1.3.2 Empirical Toll Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Obtaining Original Parameter Estimates . . . . . . . . . . . . . . . . . . . 15
1.3.4 Simulation Model Can Be Used to Estimate Likelihood of Using Express
Lane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.5 Threats to Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Agent-Based Simulation of Endogenous Congestion . . . . . . . . . . . . . . . . . 17
1.4.1 Technical Contributions of the Current Work . . . . . . . . . . . . . . . . 19
1.5 Parameters of Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 Baseline Results: Simulation of Road with Four Toll-Free and Two Tolled Lanes . 21
iii
1.6.1 Costs of Commuting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6.2 Who Uses the Express Lanes? . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Comparison to Policy Counterfactals using Calibrated Parameter Values . . . . . . 22
1.7.1 Primary Result: Estimated Benefits of Tolled Express Lanes . . . . . . . . 23
1.7.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7.3 Effects of Urgency Show Up in Arrival Time . . . . . . . . . . . . . . . . 26
1.8 Welfare Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8.1 Simulated Benefit of Express Lane with Observed Tolls . . . . . . . . . . 27
1.8.2 Commuters Who Experience Urgency Benefit More . . . . . . . . . . . . 28
1.9 Is Urgency More Determinative of Express Lane Use than Value of Time? . . . . . 28
1.10 Analysis with Original Parameter Estimates . . . . . . . . . . . . . . . . . . . . . 31
1.10.1 Estimation of Road-Specific Parameter Values . . . . . . . . . . . . . . . 31
1.11 Priorities for Related Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.11.1 More complete theory of a simpler model . . . . . . . . . . . . . . . . . . 33
1.11.2 Endogenous demand for trips . . . . . . . . . . . . . . . . . . . . . . . . 34
1.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.12.1 Relation of This Paper to the Literature . . . . . . . . . . . . . . . . . . . 35
1.13 Links to Online Sources Referenced In Text . . . . . . . . . . . . . . . . . . . . . 38
1.14 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Chapter 2: Distribution of Urgency: Recovering Parameters from Simulated Data 51
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.1 Situating this paper in the literature . . . . . . . . . . . . . . . . . . . . . 52
2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.2.2 Theoretical Model of Commuter Behavior . . . . . . . . . . . . . . . . . . 56
2.2.3 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2.3.2 Details of Simulated Commuters . . . . . . . . . . . . . . . . . 59
2.2.3.3 Details of Simulated Choice . . . . . . . . . . . . . . . . . . . . 61
2.2.3.4 Variations on the Commuter Choice Function . . . . . . . . . . 63
2.2.3.5 Challenges encountered . . . . . . . . . . . . . . . . . . . . . . 63
2.2.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.2.4.1 Estimation by Choice of Route and Time From Home . . . . . . 65
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3.1 Preferred Specification: Mixed Logit . . . . . . . . . . . . . . . . . . . . 66
2.4 Other Regression results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.4.1 A model of commuter utility . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.4.2 Conditional logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.4.3 Linear mixed effects model . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4.4 Nested logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.4.5 Additional estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5.1 Negative Results: Significance of Failure of Conventional Regressions to
Recover Choice Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 76
iv
Chapter 3: Welfare Effects of Several Toll Profiles: A Preliminary Investigation into Optimal
Toll Policy 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.1.1 Some relevant literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2 Review of the Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 Methods in the Current Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4.1 Welfare results, all commuters . . . . . . . . . . . . . . . . . . . . . . . . 83
3.4.2 Welfare results, toll lane users only . . . . . . . . . . . . . . . . . . . . . 88
References 92
Appendices 92
A Exploratory Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.1 An Elementary Model with Discrete Penalties for the Event of Late Arrival 94
A.2 Optimal Toll in Exploratory Model . . . . . . . . . . . . . . . . . . . . . 95
A.3 Models in which Urgency Can Be Relevant . . . . . . . . . . . . . . . . . 95
B Theory of Optimal Choice of Lane and Time . . . . . . . . . . . . . . . . . . . . . 96
B.1 Choice of lanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
B.2 Choice of Departure Time . . . . . . . . . . . . . . . . . . . . . . . . . . 98
B.3 Accounting for Random Variation in Travel Time between Home and High-
way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
B.4 Spreading Express Lane Commuters Over Time . . . . . . . . . . . . . . . 101
B.5 Spreading Commuters Between Lanes . . . . . . . . . . . . . . . . . . . . 102
B.6 Significance of Theoretical Results . . . . . . . . . . . . . . . . . . . . . 102
C Details of Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.1 Building the Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.2 Simulating Commuters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.3 Commuter Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.4 Commuter Choice Function . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.5 Determining the Toll Regime for the Express Lane . . . . . . . . . . . . . 105
C.6 Constructing Congestion . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C.7 Iteration and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.8 Computing Welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.9 Comparison of Welfare With and Without a Toll Lane . . . . . . . . . . . 108
D Sensitivity of Welfare Results to Distribution of Parameters . . . . . . . . . . . . . 109
E Extension: Theory of Urgency Lanes versus High-Value of Time Lanes . . . . . . 109
E.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
E.2 General Requirements for Incentive Compatibility . . . . . . . . . . . . . 116
E.3 Incentive Compatibility Requirements to Risk Late Arrival . . . . . . . . . 117
E.4 Incentive Compatibility Requirements to Take Free Lane Ex Ante, Ur-
gency Lane if Urgency is Realized . . . . . . . . . . . . . . . . . . . . . . 119
E.5 Making the Fast Lane Less Attractive than the Urgency Lane to Com-
muters Experiencing Urgency for the First Time . . . . . . . . . . . . . . 120
E.6 Intended Behavior of Rich Commuters . . . . . . . . . . . . . . . . . . . 121
v
E.7 Social Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
F Exploration of Alternative Toll Profiles . . . . . . . . . . . . . . . . . . . . . . . . 123
vi
List of Tables
1.1 Parameters of main model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2 Welfare results by commuter’s urgency type . . . . . . . . . . . . . . . . . . . . . 25
1.3 Revised parameters with heterogeneous value of time . . . . . . . . . . . . . . . . 29
1.4 Express lane use predicted by value of time and urgency . . . . . . . . . . . . . . 30
1.5 Head start predicted by VOT and urgency . . . . . . . . . . . . . . . . . . . . . . 31
2.1 Parameters, Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.2 Mixed logit results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.3 Conditional logit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4 Ratio of coefficients to toll (Conditional logit) . . . . . . . . . . . . . . . . . . . . 71
2.5 Linear mixed effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.6 Ratio of coefficients to toll (Mixed effects) . . . . . . . . . . . . . . . . . . . . . . 72
2.7 Estimated variance of random coefficients . . . . . . . . . . . . . . . . . . . . . . 73
2.8 raw output from nlogit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.9 Coefficient estimates from nlogit . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.1 Parameters, Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 High-urgency commuter welfare results . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Medium-urgency commuter welfare results . . . . . . . . . . . . . . . . . . . . . 84
3.4 Low-urgency commuter welfare results . . . . . . . . . . . . . . . . . . . . . . . 84
vii
List of Figures
1.1 Main lane congestion with the initial conditions studied here falls into a cycle
of period 3. 50 repetitions are more than adequate to characterize the long-term
behavior of this congestion profile. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.2 Express lane delays show the same periodicity as main lane delays. . . . . . . . . . 40
1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.4 Distribution of arrival times. The red vertical lines denote 7:00 AM, 8:00 AM, and
9:00 AM. Commuters fall into three arrival time types, each of which prefers to
arrive at one of these three times. . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.5 Mean cost by type of cost. ”Classical” arrival time disutility is given by b(time
early) for commuters who arrive early, andg(time late) for commuters who arrive
late. Travel time cost is the product of an individual’s value of time and the total
time spent traveling. Urgency disutility is equal to a commuter’s penalty for late
arrival if they arrive late, and zero otherwise. . . . . . . . . . . . . . . . . . . . . . 42
1.6 Breakdown of costs by type, and by commuter’s ideal arrival time. . . . . . . . . . 43
1.7 The probability of late arrival in (a) the main road (4 toll-free lanes, 2 express
lanes), (b) the high-capacity alternative road (6 toll-free lanes), and (c) the low-
capacity alternative road (4 toll-free lanes). . . . . . . . . . . . . . . . . . . . . . 44
1.8 Express lane use is plotted against q, the cost of slightly-late arrival and thus the
sensitivity to urgency. Vertical axis values are 1 for commuters that used the ex-
press lane, 0 otherwise. The blue trend line shows a linear model of the probability
that a commuter with any givenq will use the express lane. We observe that com-
muters more sensitive to urgency are somewhat less likely to use the express lane.
As Figure 3 shows, this is because they give themselves more lead time. . . . . . . 45
1.9 Panel (a) shows that commuters more sensitive to urgency leave themselves more
lead time to get to work. Panel (b) shows that they are less likely to arrive late. . . . 46
viii
1.10 The horizontal axis represents time remaining once the commuter has entered the
highway, i.e. after experiencing their realized pre-highway delay. Commuters
here face a choice between a more reliable tolled express lane and a less reliable
untolled main lane. We observe that commuters who have less time remaining
at this stage (left side) are more likely to use the express lane. This is consistent
with the express lane being a means for commuters to escape urgency. Attention
is restricted to commuters who are not already late when they enter the highway.
Panel (a) fits a linear model, and panel (b) a loess fit on a subsample. The loess fit
shows that commuters are particularly sensitive to a reduction in time remaining
when they have very little, i.e. when they experience urgency. . . . . . . . . . . . 47
1.11 On the horizontal axis, the left column shows commuters who use the toll-free
lanes, and the right column shows commuters who use the tolled express lanes.
The height of the bars indicates their probability of arriving late. We see that
express lane users are somewhat less likely to arrive late than main lane users.
Note that individuals with less time remaining until they are late are more likely to
use the express lane. The travel time advantage of the express lane is thus sufficient
to overcome the initial disadvantage that leads commuters to select into it. . . . . . 48
1.12 Benefits from the express lanes turn out not to depend on an individual’s cost of
urgency. In all panels, commuters are sorted in order of increasingq, the discrete
penalty for slightly late arrival, and thus the cost they face when experiencing
urgency. Vertical axes represent welfare improvements when a road with four toll-
free lanes and two tolled express lanes is compared to one with six toll-free lanes
(i.e. the gains from converting two lanes to express, a pure effect of tolls), or
four toll-free lanes (i.e. the gains of adding two toll lanes, combining the effect
of tolls with that of expanding capacity.). In panel (a), we see the improvement
in their travel time disutility, proportional to the reduction of time in traffic, when
two express lanes are added to four toll-free lanes. The trend is flat, showing
that the travel time saved by adding an express lane is not correlated with the
cost an individual faces from urgency. In panel (b), the vertical axis shows the
improvement in arrival time cost, which includes penalties for late arrival, from
adding the express lane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
ix
1.13 Benefits from the express lanes turn out not to depend on an individual’s cost of
urgency. In all panels, commuters are sorted in order of increasingq, the discrete
penalty for slightly late arrival, and thus the cost they face when experiencing
urgency. Vertical axes represent welfare improvements when a road with four toll-
free lanes and two tolled express lanes is compared to one with six toll-free lanes
(i.e. the gains from converting two lanes to express, a pure effect of tolls), or four
toll-free lanes (i.e. the gains of adding two toll lanes, combining the effect of tolls
with that of expanding capacity.). In panel (a), we see the improvement in their
travel time disutility, proportional to the reduction of time in traffic, when two
lanes out of a six-lane road are converted to tolled express lanes. In panel (b), the
vertical axis shows the improvement in arrival time cost, which includes penalties
for late arrival, from adding the express lane. Panels (c) and (d) show the analysis
of (a) and (b), respectively, but for converting two toll-free lanes to express. The
tolls produce little improvement in travel time, but they do improve arrival time,
with greater benefits for commuters more sensitive to urgency: as shown in panel
(d), benefits to arrival time cost are positive, and trend upward withq. . . . . . . . 50
3.1 Comparison of toll profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Empirical tolls vs highway entry time . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3 Uniform tolls vs highway entry time . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.4 PeakOnly tolls vs highway entry time . . . . . . . . . . . . . . . . . . . . . . . . 86
3.5 Triangular tolls vs highway entry time . . . . . . . . . . . . . . . . . . . . . . . . 86
3.6 Quadratic tolls vs highway entry time . . . . . . . . . . . . . . . . . . . . . . . . 87
3.7 Gaussian tolls vs highway entry time . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.8 Empirical tolls vs highway entry time (toll lane only) . . . . . . . . . . . . . . . . 88
3.9 Uniform tolls vs highway entry time (toll lane only) . . . . . . . . . . . . . . . . . 89
3.10 PeakOnly tolls vs highway entry time (toll lane only) . . . . . . . . . . . . . . . . 89
3.11 Triangular tolls vs highway entry time (toll lane only) . . . . . . . . . . . . . . . . 90
3.12 Quadratic tolls vs highway entry time (toll lane only) . . . . . . . . . . . . . . . . 90
3.13 Gaussian tolls vs highway entry time (toll lane only) . . . . . . . . . . . . . . . . 91
x
14 This figure shows that the welfare benefits of tolled express lanes are increasing
in the average commuter’s susceptibility to urgency, which seems to be driven
largely by what happens at lower values. q
i
, commuter i’s penalty for late arrival (in
dollars), is normally distributed with a standard deviation of 1, with negative values
replaced with zero. In this figure, its mean is varied in increments of 0.6, from
0.6 to 9.6. 16 increments are used, with the fifth corresponding to the literature-
derived value employed in the main results. In each case, traffic is computed for
50 iterations on the road of main interest, which has four free lanes and two toll
lanes, as well as a ”converting” comparison, where the two toll lanes are obtained
by converting two out of six free lanes, and an ”adding” comparison, where the
lanes are obtained by adding. That is, the ”converting” and ”adding” scenarios
contain simulated roads with six and four toll-free lanes, respectively. Results
shown here are the welfare gained by the mean commuter in the road with toll
lanes, relative to their welfare in the comparison road. In all cases, tolls at each
time of day are taken from I10-W in Los Angeles. The upper panel depicts the
gains to commuters versus six toll-free lanes, the lower panel versus four. Welfare
results have a similar shape in both cases: the express lanes perform far worse when
commuters’ average susceptibility to urgency is low. Surprisingly, converting free
lanes to express appears to be bad for commuters. It’s possible there is an error in
the techniques I employed to speed up the simulation for this sensitivity analysis,
and this will be avoided in future revisions. . . . . . . . . . . . . . . . . . . . . . 110
15 q
i
, commuter i’s penalty for late arrival (in dollars), is normally distributed with
a mean of 3. The standard deviation is varied in 16 increments of 0.2, from 0.2
to 3.2, with the fifth value corresponding to the literature-derived value employed
in the main results. In both panels, the vertical axis depicts the welfare gained by
the average commuter in the main road (four free lanes, two toll lanes) versus an
alternative road. The upper panel . . . . . . . . . . . . . . . . . . . . . . . . . . 111
16 a, the value of time, is here assumed to be the same for all commuters. The main
analysis uses a literature-derived value of 60 cents per minute. a is varied in 16
increments of 0.2 times the literature-derived value, with the fifth corresponding to
that value. The upper panel compares the welfare of the average commuter on a
road with four toll-free and two toll lanes to one with six toll-free lanes. The lower
panel compares to a road with four toll-free lanes. . . . . . . . . . . . . . . . . . . 112
17 The mean value of time is 60 cents per minute, corresponding to the literature-
derived value used in the paper’s main analysis. The standard deviation is increased
in 12 increments, with the largest corresponding to the interval from 0 to twice the
mean value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
18 The mean value of time is 60 cents per minute, corresponding to the literature-
derived value used in the paper’s main analysis. The standard deviation is increased
in 12 increments, each matching the standard deviation of the corresponding incre-
ment in the uniform distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
xi
19 When both the penalty for late arrival and the value of time are heterogeneous, the
effect of the lateness penalty on express lane use is small, but the effect of the value
of time is even smaller. We see that a high value of time is not the principal driver
of which commuters use the express lane. . . . . . . . . . . . . . . . . . . . . . . 115
xii
Abstract
The first essay, which I used as my job market paper, develops an agent-based model of commuters’
choice of routes and departure times on a road with a toll lane, when commuters are susceptible to
”urgency,” i.e. experience discrete penalties when arriving late, on trips from home to work. I find
that the commuters most susceptible to urgency leave home earliest and are therefore less likely to
use the express lane, but they nonetheless benefit most from the option to pay to avoid late arrival.
The second essay studies methods of empirical work to recover commuter preferences, includ-
ing urgency, from choice data. I obtain empirical tolls and travel times on two highways with
tolled express lanes in Los Angeles County. In lieu of real observations on individual commuters,
I simulate their responses using a simulation model adapted from Essay 1. On this data, I esti-
mate several discrete choice models, hoping in particular to recover the distribution of the value of
urgency. I wish to express it in willingness-to-pay terms, by dividing estimates by the coefficient
on toll paid. My preferred specification is mixed logit, which allows for agents’ choice to depend
on continuously-distributed parameters. Since this estimator is very computationally intense, I
provide results for some faster estimators as well.
My third essay explores the question of the optimal toll profile when commuters experience
urgency. I simulate several populations of commuters with heterogeneous late arrival penalties, the
populations differing in the mean of these penalties. Commuters respond to several toll profiles. I
record and visualize commuters’ highway entry times, arrival times and welfare, and rank the toll
profiles by welfare for each group of commuters.
xiii
Chapter 1
Can’t Wait? Urgency with Strategic Commuters and Tolled
Express Lanes
1.1 Introduction
Traffic congestion is familiar to commuters the world over, and its costs are large. For instance,
a report by the Texas A&M Transportation Institute estimates that in 2017, the average commuter
in the Los Angeles-Long Beach-Anaheim commuting region lost 119 hours per year to traffic
delay, and losses per commuter increased by several hours each year. This time is neither labor
nor leisure, but rather pure loss, and the effects are substantial: valued at the metro area’s mean
hourly wage of about $30 an hour, the mean commuter loses about $3570 a year to commute
time alone, far more than the mean monthly rent downtown, and over twice the monthly rent in
the less expensive neighborhoods of this notoriously expensive city. This estimate of the cost of
congestion is conservative, as this cost does not include the costs of suboptimal arrival time, either
chosen intentionally in exchange for less congestion or unintentionally due to the unreliability of
travel time. A system of traffic control that reduces congestion can mitigate the substantial costs
of travel and arrival time, freeing up time for more enjoyable or productive activities. In addition,
congestion is a disamenity of agglomeration, one of the factors that limit the growth of large and
highly productive cities. A reduction in congestion can therefore add to social welfare, well in
excess of the already substantial raw value of time saved in traffic.
1
In recent years, many world cities have implemented various forms of congestion pricing to al-
leviate this problem. In addition to predictable delays, unreliability in travel time forces commuters
to begin their trips earlier to compensate. Tolling to improve reliability may have large benefits, as
studied by, for instance, Hall and Savage (2019) Jonathan D. Hall and Savage 2019. Since 2013,
Los Angeles has high-occupancy toll lanes (HOT) on I-10, in which tolls are adjusted at intervals
of several minutes in order to maintain a reliable speed of traffic in the toll lanes. Existing esti-
mates for the value of reliability tolling assume that while commuters face linear disutility in time
late, there is no additional discrete penalty for the event of late arrival. Being ten minutes late once
is equivalent to being one minute late ten times. However, this is unrealistic when slightly-late ar-
rival incurs a large penalty, such as arriving at an airport gate shortly after the plane departs. Such
a situation differs from merely having a very high value of time: the final minutes that allow you
to catch your flight are exceptionally valuable, but once you have missed your flight, small savings
of time are worth little. Even for ordinary commutes, there is evidence that urgency, the steep cost
of slightly late arrival after an unanticipated delay, accounts for a large part of commuters’ willing-
ness to pay for express lanes with reliability tolling (Bento et al 2020) Bento, Roth, and Waxman
2020.
How does urgency matter to transportation policy? The current paper investigates by means of
a rational agent-based simulation. Commuters choose between a less reliable toll-free route and a
more reliable tolled express lane. Along each route, they face bottleneck congestion, which they
anticipate by observing past traffic. Commuters care about their travel time and their arrival time;
in contrast to most literature on endogenous congestion, this includes a discrete cost for all late
arrivals, allowing commuters to experience urgency.
Penalties for late arrival are not absent from the literature, but to my knowledge, no one has yet
modeled endogenous traffic with these penalties. Papers as early as Small (1982) have treated the
lateness penalty as a parameter to estimate. Noland and Small (1995) consider the optimal head
start of a commuter with a lateness penalty, given two distributions of travel time; and Bento et al
(2020) attempt to estimate commuter-specific lateness penalties from panel data on the repeated
2
use of a toll lane by a sample of Southern California commuters. One feature shared by these
papers is that traffic is taken as given: they do not model the production of traffic from the choices
of individual commuters who face lateness penalties, an analysis which would be necessary if we
want to evaluate the effects of a transportation policy in light of the policy’s ramifications on traffic
patterns.
Conversely, there is an extensive literature on endogenous traffic in the absence of late arrival
penalties, such as several papers by Arnott et al (1990, 1992, 1994, 1994, 1999) Arnott, Palma,
and Lindsey 1994, as well as more recent work such as Hall, Liu, and Nie (2015), and even
field experiments such as Kreindler (2020). To my knowledge, however, all these papers, while
modeling endogenous congestion, omit discrete late-arrival penalties from the commuter’s utility
function. My paper is therefore the first to integrate late-arrival penalties into the framework of
endogenous congestion, with strategic commuters and congestion due to a bottleneck.
The current work contributes to the theory of endogenous traffic under urgency; provides a
simulation to complete and illustrate the theory; and applies the theory to examine the implications
of real-world policy options such as obtaining toll lanes through expansion of road capacity or
through conversion of existing capacity.
The theory and simulation developed in this paper can further be applied to estimate com-
muters’ value of urgency by maximum likelihood from administrative billing data on tolled express
lanes, provided individual commuters are observed using the express lane repeatedly. A strategy
for such analysis is described below.
1.1.1 Preview of Results
The main result of this paper is the welfare effect of the tolled express lane. I study a road like
I10-W, with four toll-free lanes and two express lanes with more reliable travel times. This road
gains a simulated revenue of $1.18 per commuter per trip. I compare the simulated traffic, arrival
times, and commuter welfare on this road to corresponding values on two comparison roads: one
contains four toll-free lanes and allows me to investigate the effect of obtaining the two toll lanes by
3
adding capacity, while the other contains six toll-free lanes, enabling me to investigate the effect of
obtaining the two toll lanes by converting existing capacity. If the partly-tolled road was obtained
by adding two toll lanes to a road with only four toll-free lanes, I find that commuter welfare is
increased by $69.21 per commuter per trip. Even if the partly-tolled road is obtained by converting
two lanes to toll on a six-lane toll-free road, commuters gain $10.13 per commuter per trip, showing
that reliability and reduced congestion contribute significantly to the value of the express lanes.
These estimated benefits are surprisingly large. Notably, even though conversion reduces the toll-
free capacity of the road, I find that the gains to aggregate welfare from the opportunity to escape
urgency outweigh these losses. Furthermore, when I focus analysis on commuters who experience
urgency, the benefit of tolls versus a six-lane untolled road rises to $13.71 per commuter.
I also simulate the same scenario when commuters have no sensitivity to urgency, i.e. under
the standard model of their preferences. Here, surprisingly, the benefits of the express lane are
larger: $86.69 when the two toll lanes are additional to a four-lane road, and $11.18 when they are
obtained by converting two lanes of a six-lane road. It is not clear to me why the simulated benefits
are larger in the absence of urgency, but I suspect this speaks to the complex nonlinear behavior of
endogenous traffic with respect to commuter preferences.
The economics profession has ignored urgency. Without it, welfare analyses of transportation
are misspecified, as is the social planner’s problem. In particular, if urgency is ignored, we under-
estimate the benefits of policies that improve travel time reliability, as well as policies that allow
those commuters currently experiencing urgency to pay to have their urgency relieved. A social
planner or policy maker unaware of urgency will thus underinvest in reliability of arrival time and
in means to escape urgency, compared to the social optimum. In addition, the dynamics of traffic
will be different, with theoretically ambiguous effects on the value of policy interventions. A richer
model of commuter preferences can allow for more accurate modeling of the response of traffic to
interventions that affect commuters’ incentives.
For decades, economists have relied on Gary Becker’s notion of a circumstance-independent
value of time. The current paper, by contrast, explores a value of time which may briefly become
4
quite high when a person experiences urgency. We see that commuters who experience urgency
gain additional benefits from the option to pay for more reliable arrival time.
The current paper studies agents that differ in their individual sensitivity to urgency (the pa-
rameter q) and in the realization of which trips present them with urgency. This allows for an
exploration of the role of urgency without requiring unmanageable complexity or raising much
worry of overfitting.
In this paper, a novel rational agent-based simulation of traffic is presented. The simulation
allows welfare analysis of the effects of policies such as the introduction of toll lanes and changes
to the toll regime, where the analysis accounts for endogenous changes in traffic. To my knowl-
edge, this is the first traffic simulation in which agents are susceptible to urgency. In addition,
the simulation provides the tools required to perform maximum-likelihood estimation of the value
of urgency using administrative billing data from tolled express lanes. A similar analysis can be
extended to toll roads with a toll-free alternative route.
I compare a simulated road similar to I10-W, with four toll-free lanes and two toll lanes, to two
counterfactual roads: one with only four toll-free lanes, to study the effects of pure addition of toll
lanes, and another with six toll-free lanes, to study the effects of converting the lanes from free
to tolled. I find that the road of interest gains commuters about $69 of welfare apiece versus the
lower-capacity road, and about $10 apiece relative to the equal-capacity road. This estimate does
not account for toll rebates, nor does it account for the cost of construction.
1.2 Main Model Setup and Some Theoretical Results
Commuters travel from A (home) to B (work) on a single road with two lane types. The road
consists partly of toll-free ”main lanes,” and partly of ”express lanes” with finely time-varying
tolls that prevent congestion. Commuters care about their travel time and their arrival time.
5
1.2.1 Road Structure and Timing
Commuters travel from home to a highway by a pre-highway road. Once they enter the highway,
they are faced with a choice of two routes: a toll-free main lane and a tolled express lane, both
of which lead the commuter to their workplace. The express lane offers more reliable travel time;
this is modeled by adding a random delay to the main lane. In addition, all commuters receive
idiosyncratic delays on the pre-highway road, corresponding to any unexpected event that might
delay a person’s entry to the highway, or even their departure from home, beyond the iplanned
moment.
The model is iterative. On each day, commuters know the traffic conditions that were realized
on the previous day, and respond to them. Each day’s timing is as follows.
(1) Commuters observe yesterday’s congestion and the distribution of random delays.
(2) Commuters commit to a departure time, anticipating their lane choice policy and responding
to the distribution of random delays and the observation of yesterday’s endogenous congestion.
(3) After leaving home, commuters experience idiosyncratic pre-highway delays.
(4) Commuters enter the highway and choose the tolled or toll-free route.
(5) Commuters’ entry into the routes creates bottleneck congestion, delaying subsequent com-
muters in the route. These congestion delays will be known to commuters tomorrow. The toll-free
lane suffers random additional delays.
(6) Commuters arrive at work.
I assume that commuters commit to a departure time based on the previous day’s realization
of endogenous congestion, on each lane at each time of day as well as the long-run distribution
of exogenous delays. This is easier to incorporate into an iterative model than Bayesian learning
based on an ever-growing series of vectors of past congestion, and it may not be as unrealistic
as it first appears. Empirical results such as Gallagher’s (2014) finding of recency bias in flood
insurance, or Busse et al’s (2015) finding that atypical weather events affect consumers’ choice
of which car to purchase in ways inconsistent with rational explanations, show that people may
respond more to recent events, and learn less from the long-term record, than a fully rational agent.
6
In light of such findings, the assumption that recent experience would have a disproportionate
effect is not unreasonable.
1.2.2 Information Known to Commuters
All agents know their own type. They know the distribution from which delays on the pre-highway
road and toll-free lane are drawn, but not the realization they will receive today. They know the
bottleneck delays experienced yesterday in each lane, at each point in time.
1.2.3 Commuter Preferences
Commuters face a discrete choice of two lanes, and a continuous choice of departure times. Com-
muter preferences resemble the exploratory model, but with heterogeneity in parameters and id-
iosyncratic preferences over lanes. The commuter’s cost function takes the form,
C
i
(t,r)=a
i
(travel timejr)+b
i
(time earlyjr)+g
i
(time latejr)+q
i
1(latejr)+z
i
X
i
+T(t
0
). (1.1)
Here idiosyncratic lane preference z
i
is applied to X
i
, an indicator for commuter i using the
express lane. TollT (t
0
) depends on the time t
0
at which the commuter enters the toll lane, if they
do. Travel time, and thus arrival time, depends on the route taken, r, as well as the time t at which
the trip begins.
Each commuter has unit demand for trips. Commuter i gets a utility
¯
U
i
from making the trip
to work. In this paper, I assume that
¯
U
i
is high enough to justify any trip, and thus that demand is
inelastic and that the commuter’s utility maximization problem Commuters choose departure time
t and route r to maximize their utility,
U
i
(t,r)= max
t,r
(
¯
U
i
C
i
(t,r)).
7
The choice of time is continuous, and the choice of route is discrete.
In principle, all cost parameters can be heterogeneous and follow an arbitrary joint distribution.
In this paper, however, I primarily analyze a model in which a, b, and g are homogeneous and
drawn from the literature, and q
i
is heterogeneous. Each commuter has a fixed value of urgency,
assumed to be the same in all of this commuter’s trips. However, the impact of the value of urgency
on the commuter’s cost is variable, depending on their current risk of arriving late, which in turn
depends on the interaction of current time, current congestion, and the distribution of random
delays. The anticipated evolution of this risk over time, in turn, drives commuter behavior.
1.2.4 Exogenous Delays
Each commuter experiences idiosyncratic random delays after leaving home, but before entering
the highway (pre-highway delays). Commuters know the distribution of these delays when they
choose their departure time from home, but they do not know the realized value of the delay they
receive today until they enter the highway. Once they have entered the highway, knowing their
pre-highway delay, they can anticipate their expected arrival time and choose between the tolled
express lanes and the toll-free main lanes. In the simulation, pre-highway delays are calibrated to
be lognormally distributed, with a log-mean of 0.1 and a log standard deviation of 1.
Travel time in the express lane is more reliable than in the main lane. I assume that commuters
know their travel time deterministically in the tolled express lane, but face uncertainty, correspond-
ing to a random delay with mean zero, in the main lane. When choosing lanes, commuters know
the distribution of this delay, but they do not know today’s realization until they arrive at work.
Ideally, the difference in reliability between lanes is due to the toll policy: the road as a whole
faces exogenously variable demand, but the toll policy responds to offset perfectly in the express
lane. For feasibility, I found it more practical for the commuter to simply take the lesser reliability
of travel time in the toll-free lane as given.
8
1.2.5 Congestion: The Endogenous Delays
Congestion is endogenous and generated by a bottleneck, as in Vickrey (1969) Vickrey 1969 or
Arnott, de Palma and Lindsey (1987, 1990, 1993, 1994).
Throughput of lane l2fm,eg is limited by a bottleneck with capacity s
l
, the maximum number
of vehicles that can cross the bottleneck of lane in a minute. Commuters enter the highway at a
rate r
l
(t). If r
l
(t) exceeds s
l
, a queue forms before the bottleneck, and its length Q(t) is given by
Q
l
t)=
Z
t
t
o
r
l
(u)du s
l
(tt
0
), (1.2)
where t
0
is the most recent time with no queue. The time spent in the queue is
T
v
l
(t)=
Q
l
(t)
s
=
Z
t
t
0
r
l
(u)
s
dut+ t
0
, (1.3)
with time derivative
dT
v
(t)
dt
=
r(t)
s
1.
Both prior to highway entry and after crossing the bottleneck, commuters face idiosyncratic
random delays, R
b
i
before the bottleneck and R
a
i
after.
Total travel time is therefore
T(t)= R
b
i
+ T
v
l
(t+ R
b
i
)+ R
a
i
. (1.4)
1.2.5.1 Defining Equilibrium
Each commuter chooses their own departure time. We say that a departure time profile r(t) is an
equilibrium if it satisfies the following:
(1) The number of commuters departing at each time t is equal to r(t).
(2) Congestion is generated from r(t) by the specified congestion technology.
(3) Given equilibrium congestion, no commuter could gain by unilaterally altering their depar-
ture time.
9
An equilibrium departure time profile describes a Nash equilbrium of the game specified above.
Note that if equilibrium is reached, commuters’ expectation that today’s traffic will be like yester-
day is correct. Other learning processes will also tend to converge on correct beliefs in the long
run, since once an equilibrium congestion profile is reached, commuters will not choose to depart
from it.
It was my initial hope that a sequence of commuter responses would converge to equilibrium;
however, this turns out not to be the case.
1.2.6 Optimal Choice of Lane and Time: Theoretical Results
A commuter’s optimal departure time depends on the lane they will take. Commuters must there-
fore begin their decision problem with a lane policy, which tells a commuter, for any time t
0
in
which they enter the highway, whether to take the tolled or toll-free route.
Commuters will choose the express lane if its utility advantage over the main lane is positive.
This advantage is given by
z+a(ET
m
(t
0
) T
e
(t
0
))+bT
e
(t
0
)bET
E
m
(t
0
)
+ p
L
(t
0
)
bET
E
m
(t
0
)+gET
L
m
(t
0
)(b+g)(t
t
0
)+q
T (t
0
).
Here ET
m
(t) js the expected travel time of a commuter who enters the toll-free main lanes at
time t, and T
e
(t
0
) is the known travel time of a commuter who enters the tolled express lane at t
0
.
Superscripts E and L denote expectations conditioned on early and late arrival, respectively, which
truncate the distribution of main lane delays in a manner that depends on t
0
.
In the absence of pre-highway delays, the commuter would identify their optimal departure
time for each lane, then choose the lane-time pair that offers the greatest expected utility. In the
presence of pre-highway delays, however, the commuter must anticipate their choice of lane and
the corresponding expected utility for each possible highway entry time. If the commuter leaves
10
home at time t, they experience a pre-highway delay d which follows a known distribution F
d
, and
enter the highway at time t+ d. It is therefore possible to compute a commuter’s expected utility
for each home departure time t.
Since the express lane advantage depends on an endogenous distribution of travel times, com-
puting it ex ante is infeasible. Subsequent analysis will use the expedient of basing tolls on the
real prices used on I10-W in Los Angeles County, California, a road chosen because its toll policy
is adjusted in real time, at intervals of several minutes, with the intent to produce reliable travel
times, much like the toll policy modeled here.
Interested readers can find details in the appendix.
1.2.7 Accounting for Random Variation in Travel Time between Home and
Highway
The discussion of optimal lane and time above describes the strategy of a commuter who can
control their exact time of entry to the highway. Importantly, lane and time are chosen together,
with the commuter using the tolled express lane if they enter the highway after a critical time, and
the toll-free lane otherwise. If, however, commuters must commit to a home departure time before
learning the realization of random delays between home and the entrance to the highway, then
their choice of home departure time merely determines the probability of entering the highway
early enough that it is unnecessary to use the toll lanes.
The commuter’s expected utility is given by U(t) = P
x
(t)U
x
(t)+(1 P
x
(t))U
m
(t), where
P
x
(t) is the probability of having to use the express lane, U
x
(t) is the expected utility of using the
express lane conditional on leaving home at time t and receiving a realized pre-highway delay long
enough to induce the commuter to use the express lane, and U
m
(t) is the expected utility of using
the main lane conditional on leaving home at time t and receiving a short enough pre-highway
delay to use the main lane. These expected utilities depend on the interaction of the departure
time, the distribution of pre-highway delays, the distribution of exogenous delays on the highway,
the time profile of endogenous congestion, and the commuter’s ideal arrival time.
11
In particular, P
x
(t) is the probability of receiving a pre-highway delay greater than˜ tt. U
x
(t) is
the expected utility of using the express lane conditional on receiving a pre-highway delay greater
than ˜ tt, a function of t which also depends on the joint distribution of delays before the highway
and in the express lane.
1.2.8 Spreading Express Lane Commuters Over Time
An ideal toll policy under bottleneck congestion is one in which commuters are willing to enter
the tolled lane at a uniform rate of s
e
, the bottleneck capacity of the lane, for the duration of the
rush hour. Whereas commuters in a toll-free lane trade arrival time against travel time, those in an
ideally-tolled lane trade arrival time against tolls.
For simplicity, we assume that commuters share a commonb andg.
Since the express lane has a perfectly preditcable travel time, commuters face no risk of unex-
pected late arrival. A commuter departing at time t on the express lane has a utility of
U
o
(t)=f
aT
o
(t)+b(t
t T
o
(t))+T (t)z , if t+ T
o
(t) t
aT
o
(t)+g(t+ T
o
(t)t
)+q+T (t)z , if t+ T
o
(t) t
The marginal cost of an additional minute of delay is
dU
o
(t)
dt
=f
aT
0
o
(t)+b(1 T
0
o
(t))+T
0
(t), if t+ T
o
(t) t
aT
0
o
(t)+g(1+ T
0
o
(t))+q+T
0
(t), if t+ T
o
(t) t
By construction, a queue is prevented in this lane, and T
o
(t) is a constant function; thus T
0
o
(t)=
0, and we have
T
0
(t)=f
b, if t+ T
o
(t) t
g, if t+ T
o
(t) t
,
12
with a discontinuous increase of q at t
. If toll prices are observed, b and g can easily be
inferred.
Ifq
i
is heterogeneous, our choice ofq
T
allows us to choose how many express lane commuters
arrive early or late. Any commuter withq
i
q
T
will choose to arrive early in the express lane. Thus,
by makingq
T
small, we can make all express lane commuters arrive early.
More generally, F
q
(q
T
), the fraction of commuters withq
i
q
T
, is the fraction of express lane
commuters who arrive late, while 1 F
q
(q
T
) is the fraction of commuters who arrive early.
1.2.9 Spreading Commuters Between Lanes
The expected utility advantage of the express lane is
z
i
+a
i
(ET
m
(t) T
e
(t))+bT
e
(t)bET
E
m
(t)
+ p
L
(t)
bET
E
m
(t)+gET
L
m
(t)(b+g)(t
t)+q
i
T (t).
We wantT (t) to have a time-invariant component that will discourage excessive commuters
using the express lane. This must adapt to the probability of arriving late in the main lane, as this
is an important component of the express lane’s advantage.
1.2.10 Significance of Theoretical Results
The theoretical results for the commuter’s choice of lane and departure time illustrate the com-
muter’s choice problem, and show that the optimal departure time depends on how this time affects
the distribution of arrival times in each lane relative to the commuter’s ideal.
The theoretical results for tolls that spread commuters over time and between lanes are infor-
mative about ideal tolling. We find that the observed tolls in the I-10 express lanes differ from
these results.
13
1.3 Empirical Strategy
1.3.1 Calibrating Parameters of Simulation: An Initial Approach
To test the simulation, parameter values are drawn from the literature.
The value of travel time is estimated at 60 cents per minute, adjusting the estimates of Lam and
Small (2001) for inflation. The per-minute cost of early and late arrival are set at 2/3 and twice this
level, respectively (i.e. 40 cents fora, 120 cents forg), in accordance with Small (1982).
California’s Department of Transportation estimates that in 2018, the average lane-mile of road
in the Los Angeles - Long Beach - Anaheim MSA saw approximately 4,250 vehicle-miles traveled
per day. The road on which the simulation is modeled, Interstate 10 from the San Gabriel Valley
to downtown Los Angeles, has four toll-free and two tolled lanes in the indicated direction. At
the MSA’s average traffic density, we would expect 22,500 vehicles to use a road of this width in
a typical day. Engineering estimates give a throughput capacity of approximately 30 vehicles per
minute per lane.
Results shown in Section 2.4 are given for the parameter values characterized here.
1.3.2 Empirical Toll Policy
Toll data comes from the Los Angeles Metropolitan Transportation Authority. I use pricing data
on all segments of the tolled express lanes along I10-W from El Monte to downtown Los Angeles,
for the time period from January 1, 2019 to March 31, 2020 and restrict analysis to weekdays. The
toll at each time of day is obtained by simply summing the tolls along all segments of the road at
that time, not accounting for commuters’ travel time.
14
1.3.3 Obtaining Original Parameter Estimates
Some parameters can be identified from publicly available data, such as PeMS traffic detector data
and the billing schedule of the tolled express lanes from the Los Angeles Metropolitan Transporta-
tion Authority (Metro). However, others require proprietary data, such as the administrative data
used by Bento et al (2020), in which repeated trips by individual commuters can be observed.
Individual commuters are often observed repeatedly using the express lanes at a particular time
of day, such as 7:00 AM. From this, we can infer the commuter’s ideal arrival time t
i
.
For reasons discussed in sections A.1 and B.4, if commuters are safely early or late, a congestion-
avoiding toll is expected to compensate them for the value of the additional time early or late in-
duced by the toll. Thus, the rate at which the toll increases for safely-early commuters can be
used to estimate b, the cost per minute of early arrival; and the rate at which it decreases for
predictably-late commuters can be used to estimateg, the cost per minute of late arrival. Knowing
g and road capacity s, traffic volume in the main lane is predicted to decline at a rategs/(a+g),
which identifiesa, the per-minute cost of travel time.
Independently of their arrival time, some commuters may be willing to pay to use the toll lane,
perceiving it to be superior or gaining psychological utility from it. By observing tolls at times
when the toll-free lanes are uncongested, we can infer that commuters observed to use the toll
lanes at such times have a lane preference at least equal to the toll at the time.
The final parameter to identify is q
i
, an individual commuter’s penalty for the event of late
arrival. The advantage of the express lane is equal to
z
i
+a
i
(ET
m
(t) T
e
(t))+bT
e
(t)bET
E
m
(t)
+ p
L
(t)
bET
E
m
(t)+gET
L
m
(t)(b+g)(t
t)+q
i
T (t),
whereq
i
is the only parameter still unknown. This can be fitted to data using simulated method
of moments. However, identifying q in this manner is not feasible until I obtain data in which I
15
can observe repeated trips for the same commuter, in which I know their time of arrival and the toll
they pay for each trip.
1.3.4 Simulation Model Can Be Used to Estimate Likelihood of Using Express
Lane
In the process of choosing their lane and departure time, commuters formulate a policy function,
L(t
h
), which specifies the lane L that a commuter would use, as a function of th time t
h
at which
this commuter enters the highway. This function depends on the commuter’s preference parameters
and the distribution of travel times in each lane, at each point in time.
Armed with the ability to generate this policy function, we can produce a likelihood function
for the parameters based on the commuter’s choice of lane. Given the place l and time t
i
at which
the commuter intends to leave home, their distribution of highway entry times is determined by the
distribution of pre-highway delays from their home.
t
hi
= t
i
+ D
pre,l
(t
i
)
The likelihood of using the express lane corresponds to the likelihood that the commuter’s entry
time is one in which their policy function prescribes the use of the express lane.
Pr(L= 1)= Pr(L(t
hi
)= 1).
1.3.5 Threats to Identification
One important difficulty we may encounter is measurement error in the time saved by using the
express lane. We can deal with this in a similar manner to Bento et al (2020), using lead travel
times one hour, one week, and two weeks after the event as instruments for variation in travel time.
Uncertainty in arrival time must also be accounted for when estimating b. We must exclude
commuters at risk of arriving late, accounting for likely measurement error in travel time.
16
1.4 Agent-Based Simulation of Endogenous Congestion
To study congestion, arrival time and welfare, I construct an agent-based simulation. A population
of commuters is modeled. A choice function maps each commuter’s preference parameters to their
optimal choice of lane and departure time, given the prior state of traffic and the distribution of
exogenous stochastic delays. Congestion is computed from commuters’ choices. The simulation
is the primary contribution of this paper.
An overview of the simulation is as follows: There is a road from point A (home) to point
B (work), with a toll-free lane and a tolled express lane. There are exogenous random delays in
each lane, as well as before entering the road; commuters know the distribution of these delays. In
addition, there is endogenous congestion in each lane due to a bottleneck. Time is discrete, with
a total duration of D minutes, each divided into M steps. The road may be initialized with some
initial profile of congestion, i.e. deterministic delays incurred by commuters leaving at each time
period, in addition to the random delays.
A population of N commuters observe the deterministic congestion of the previous day and the
long-run distribution of stochastic delays. Each commuter is characterized by an ideal arrival time
T
i
, a disutility of time in traffica
i
, a per-minute cost of early arrivalb
i
corresponding, for instance,
to the foregone value of time at home; a per-minute cost of late arrivalg
i
, and a discrete penaltyq
i
incurred any time the commuter arrives after T
i
. In addition, commuters may have a preference
for one lane over another, quantified by z
i
. Each commuter travels in their own vehicle; thus,
their departure times may vary independently, and traffic depends on the number of commuters
currently on the road.
Given a commuter’s parameters, their expected utility can be computed for each lane at each
possible departure time, making it possible to identify their optimal choice, and from this, the
queue in each lane at each point in time.
Given each commuter’s departure time, the queue at that time, and a realization of the stochastic
delay, we can find their arrival time and corresponding utility.
17
The process can be iterated, with the congestion output by one iteration being used as the ”ini-
tial” congestion input into the next iteration. If a congestion profile is a Nash equilibrium, this
iteration process will recreate the same congestion profile, however many times it is repeated. If
the equilibrium is stable, then iteration from slight perturbations of this equilibrium traffic profile
will converge back to the equilibrium. Initial hopes that the best-response process would be a
contraction mapping, and that any initial conditions would eventually converge to a single unique
equilibrium traffic profile, have come up negative: most initial conditions do not converge to equi-
librium. An equilibrium is identified, but I have yet to characterize the set of equilibria more
generally.
In this simulation, commuters respond only to the previous day’s endogenous congestion, and
to the known long-term distribution of random delays. They do not perform Bayesian updates on
the endogenous portion of congestion or the process from which it is generated. Such Bayesian
updates would be tremendously complicated, as the profile of congestion is a function of time
in two lanes, and its correspondence to commuters’ structural parameters is not straightforward.
In addition, people have been behaviorally observed to exhibit recency bias, as for instance in
Gallagher’s (2014) finding that purchases of flood insurance are driven far more by homeowners’
recent experience than the statistical risks indicated by the historical record.
The simulation allows various configurations of the road’s parameters. I investigate the benefits
to commuters of the tolled express lanes by simulating a road of main interest, and comparing it
to counterfactuals where the express lanes represent added or converted capacity. In particular, the
road of main interest, modeled after Interstate 10 West towards downtown Los Angeles, contains
four toll-free lanes and two toll lanes, each capable of a throughput of 30 vehicles per minute. To
study the effects of conversion, the counterfactual is a road of the same total capacity, but with
all lanes toll-free, i.e. six free lanes. To study the effects of addition, the counterfactual is a road
containing only the capacity of the main road’s toll free lanes, i.e. four free lanes. In all cases,
traffic is initialized in accordance with the theoretical equilibrium of a related model in Arnott
et al (1994); fifty iterations of commuter response are simulated, with commuters’ departure and
18
arrival times computed, and welfare results for each iteration are calculated in accordance with
commuters’ preferences over tolls, travel time and arrival time. Similar analysis is later done for
subsamples of the population, such as those whose realized delays made them experience urgency;
and for counterfactual parameter values, to study the sensitivity of the results.
1.4.1 Technical Contributions of the Current Work
To my knowledge, the current work is the first model, simulated or otherwise, which is able to study
the effects of urgency on traffic and welfare. Urgency, or the risk of steep costs from brief delays at
the wrong moment, alters the dynamics of traffic drastically enough to require a simulation which
explicitly models commuters’ utility optimization problem, their resulting choice of departure time
and route, and the traffic produced thereby. Since traffic is endogenous, and since commuters must
consider traffic in their own choice, the simulation is iterative, with commuter choices generating
the traffic to which commuters respond in the next iteration.
The simulation presented here follows in the tradition of several other traffic simulations, but
the necessity of a novel model presented opportunities to reach levels of modeling flexibility not
found in the literature. For instance, Hall and Savage (2019) use simulated traffic with calibrated
parameters to illustrate the results of a toll to improve traffic reliability. For tractability, Hall and
Savage work with the commonplace assumption that all commuters have identical preferences,
including an identical ideal arrival time, and exclude any possibility of urgency. The impressive
work of Kreindler (2020), in which a field experiment is used to elicit the preferences of hundreds
of drivers in Bangalore, uses a simulation of the behavior of these particular commuters as part
of a strategy of structural estimation. By contrast, the simulation presented in the current paper
allows these assumptions to be relaxed: it is possible to simulate an arbitrary number of commuters
capable of experiencing urgency, who are characterized by an arbitrary joint distribution of cost
of minutes in traffic, early, or late, as well as heterogeneous ideal arrival times. This flexibility
exceeds the needs of the current application, but allows my simulation to be applied with modest
changes to a wide variety of real and hypothetical scenarios.
19
Parameter Interpretation Mean St Dev Shape Source
a Value of Time / min $0.60 0
Lam & Small (2001)
adjusted for inflation
b Cost per Min Early $ 0.40 0
Ratio witha
based on Small (1982)
g Cost per Min Late $1.20 0
Ratio witha
based on Small (1982)
q Cost of Late Arrival $3 $1 Normal Bento et al (2020)
z Express Lane Preference $ 1.684 $2.275 Lognormal
S
M
Main Lane Capacity
(Veh/Min)
120
30 veh/min/lane
(engineering estimate)
S
E
Express Lane Capacity
(Veh/Min)
60
30 veh/min/lane
(engineering estimate)
Table 1.1: Parameters of main model
In addition, the commuter choice function in my simulation includes the necessary machinery
to estimate the likelihood of using a tolled express lane as a function of a commuter’s value of
urgency, given administrative billing data from such toll lanes. With the appropriate proprietary
data, this method can be used to derive novel structural estimates of the value of urgency. The
method is described below.
1.5 Parameters of Simulation
This table characterizes the parameters of the main model.
20
1.6 Baseline Results: Simulation of Road with Four Toll-Free
and Two Tolled Lanes
Figures 1 and 2 illustrate congestion in the main and express lanes, respectively. The initial con-
ditions I use do not result in an equilibrium pattern of congestion that does not vary over time, but
they suffice to produce a brief cycle: the pattern of congestion repeats every three iterations.
Figures 3 and 4 illustrate the distribution of travel times and arrival times, respectively. The
vertical red lines represent 7:00, 8:00, and 9:00 AM: commuters are divided into three arrival time
types, with each of these types preferring one of these three arrival times. There are spikes in
density of commuters who arrive right before 8:00 and 9:00 AM.
1.6.1 Costs of Commuting
Figure 5 shows the average cost incurred by a commuter, subdivided into travel time disutil-
ity, ”classical” arrival time disutility, and urgency disutility. Travel time disutility is given by
a
i
(travel time), where a
i
is commuter i’s value of time. Classical arrival time disutility is the
arrival time disutility commonplace in the literature, given byb (time early) for commuters who
arrive early, and g (time late) for commuters who arrive late. Urgency disutility is given by the
penalty for late arrival if the commuter arrives late, and zero otherwise. This is the smallest com-
ponent, since most commuters do not arrive late, but nonetheless has an average value of $1.34 per
commuter. Restricting analysis to commuters who arrive late, the average late commuter suffers a
penalty of $2.95. This is very slightly less than the mean lateness penalty of $3.00, consistent with
the most urgency-sensitive commuters being less likely to arrive late.
Figure 6 decomposes the mean value of each type of cost by commuters’ ideal arrival time.
Figure 7 shows that in both the main road and the alternative roads, commuters commonly arrive
late.
21
1.6.2 Who Uses the Express Lanes?
Figures 8, 9, and 10 investigate which commuters use the express lane. Figure 8 shows a result
that, on its own, is rather puzzling: there is a negative correlation between a commuter’s cost of
urgency and their likelihood of using the express lane. This result becomes less puzzling when one
considers a mediating factor: commuters more sensitive to late arrival will leave home earlier, as
shown in Figure 9, and are thus less likely to find themselves in a situation where the express lane
is the only way to avoid late arrival. Since one’s late arrival penalty is irrelevant to one’s choice of
lane if one will arrive early in any event, the effects of leaving home earlier dominate.
Figure 10 shows that commuters in an ”urgent” situation, where they have little time left to
reach work on time, are more likely to use the express lane. Commuters on the left end of the
figure enter the highway with almost no time left to avoid late arrival, whereas those on the right
end have much time remaining. The trend line slopes downward: commuters with much time left
are likely to use the toll-free lanes, while those with little time left are likely to use the express lane.
A loess fit (on a subsample of commuters, for reasons of computation time) shows that the effects
are strongest when commuters have very little time remaining, i.e. when they are experiencing
urgency.
We saw in Figure 10 that commuters experiencing urgency are particularly likely to use the
express lane. Nonetheless, as Figure 11 shows, express lane commuters are less likely to arrive
late than main lane commuters, despite their initial disadvantage.
1.7 Comparison to Policy Counterfactals using Calibrated Parameter
Values
The main simulation is compared to two policy-relevant counterfactuals. One counterfactual stud-
ies the effects of obtaining the two express lanes of the main road by adding capacity, that is, it
compares the four toll-free lanes and two express lanes of the main road to a hypothetical road
with only four toll-free lanes. The second studies the effect of obtaining the main road’s express
22
lanes by converting capacity, that is, converting a road with six toll-free lanes to one with four free
and two express lanes. It is not a priori obvious that this conversion will improve net commuter
welfare, since it comes at the expense of toll-free capacity. Nonetheless, I find that the net welfare
effect of conversion is positive.
The results in this section use the calibrated parameter values discussed above. The results
shown are for initial conditions described by Arnott et al (1994), with welfare averaged over 50
iterations, and with a ”speedup factor” of 1, i.e. with 22,500 commuters’ choices simulated in-
dividually. However, results change little when a speedup factor of 10 is used (corresponding to
a tenth of the number of commuters, on roads a tenth of the capacity, with ten times the dollar
value for a minute in traffic, a minute early or late, the event of late arrival, and tolls paid); nor do
results change much when the initial conditions are the complete absence of congestion, or when
100 iterations are used. For simplicity, 50 iterations and a speedup of 1, with initial conditions
based on the equilibrium predictions of Arnott et al (1994), is considered the canonical version of
the model, and other results are omitted. Other results are available upon request.
1.7.1 Primary Result: Estimated Benefits of Tolled Express Lanes
Commuters’ arrival time depends on their departure time and the queue at that time. This arrival
time is used to calculate the utility for each commuter. Results are compared to a similar model
where the use of the tolled express lane is impossible.
Using parameter values drawn from the literature, I calibrate the model and determine the
welfare added, in aggregate and per commuter, by converting an existing toll-free lane into a tolled
express lane or by adding a separate tolled express lane.
I find large benefits per commuter. The simulated road with tolled express lanes consists of
four toll-free lanes and two tolled lanes, each of which can allow 30 vehicles through in a minute.
This is compared with two counterfactual settings: first, a road with four toll-free lanes and no toll
lanes; and second, a road with six toll-free lanes and no toll lanes. Compared to the four-lane road,
the road with express lanes produces a simulated benefit of $69.21 per commuter, due in part to
23
added capacity. When compared to the six-lane road, the road with express lanes yields a benefit
of $10.13 per commuter. This ten-dollar benefit is due entirely to the tolls, since the latter two
roads have identical capacity.
Benefits per commuter increase when we focus attention on those who experience urgency. In
this case, commuters are said to experience urgency if their pre-highway delay is large enough to
prompt them to switch lanes: they left home intending to use the toll-free lane, but were delayed
enough to switch to the express lane. These commuters’ outcome is compared to the counterfactual
outcome they would have had on the six-lane toll-free road if they experienced the same pre-
highway delays. These commuters’ benefit rises to $13.71, an increase of $3.58 over the general
population.
All benefits discussed here are without reimbursement. If toll revenue is used to reimburse
commuters or provide a public service they value, benefits from the toll lanes would be higher. I
find that the toll lanes produce a revenue of about $1.18 per commuter; benefits of reimbursement
can be approximated by adding this figure to the benefits indicated above. However, benefits dis-
cussed in this section also omit the cost of constructing the roads, and thus overstate the advantage,
if any, of expanding capacity versus converting existing lanes.
In the primary simulation, commuters have a late arrival penaltyq which is nonzero, and based
on the estimates of Bento et al (2020). However, I also simulated the three roads described above,
with the one change that all commuters have zero discrete penalty for late arrival, i.e. that com-
muters have the preferences which are standard in the literature. Intuitively, this should decrease
the benefits of the express lane, as avoiding late arrival is no longer quite so valuable. Surpris-
ingly, my simulation results were the opposite: versus the four-lane road, where the toll lanes are
additional, their benefit was $86.69 per commuter. Versus the six-lane road, where toll lanes are
obtained by conversion without increasing capacity, their benefit was $11.18 per commuter.
Returning to commuters sensitive to urgency, i.e. those with q 0, I focus attention on those
who actually experienced urgency: the commuters who initially intended to use the toll-free main
lane, but whose pre-highway delay was large enough to induce them to switch to the express lane.
24
Scenario Adding Lanes Converting Lanes
Sensitive to Urgency $69.21 $10.13
Insensitive to Urgency $86.69 $11.18
Experienced Urgency $13.72
Table 1.2: Welfare results by commuter’s urgency type
I compare the welfare they receive on the main road to the counterfactual in which they face the
endogenous delays of the six-lane toll-free road, but with the same urgency-inducing pre-highway
delays they received on the road with express lanes. The benefit per commuter is now $13.72 for
the partly-tolled road over the equal-capacity toll-free road.
These results are summarized in the table below:
1.7.2 Initial conditions
It is possible to induce a stable congestion profile that resembles equilibrium under certain initial
conditions, as illustrated below.
25
I obtained these figures using initial conditions based on the equilibrium results of Arnott et
al (1994) with tolls that deter the use of the express lane, but was unable to reproduce the results
in later attempts. However, such initial delay profiles do produce congestion that, while not truly
an equilibrium, changes relatively little from one iteration to the next. This initial delay profile,
illustrated below, is used as my initial condition in all simulations.
1.7.3 Effects of Urgency Show Up in Arrival Time
It might be expected that commuters with a large penalty q for late arrival would benefit more
from the express lane. In particular, these benefits show up in their arrival time utility: as shown in
Figures 12 and 13, commuters with highq gain more from the option value of an express lane that
allows them to compensate for unexpected delays and avoid late arrival.
’
26
1.8 Welfare Implications
Each congestion profile has social losses due to the time spent in traffic by all commuters; the
cost of early arrival aggregated over all commuters who arrive early; and the cost of late arrival,
including discrete penalties, aggregated over all commuterw sho arrive late. This can be expressed
as
Social loss=a
(time in traffic)+b
å
early
(time early)+
å
late
(g
(time late)+q).
A toll regime is defined as optimal if it minimizes social loss among all possible toll regimes.
Even if the optimal regime is unknown, we can compare social losses between alternative toll
regimes. The toll regime studied above is one of quality-of-service pricing. It is known that in a
simpler setting, such as Arnott et al (1993), quality-of-service pricing that avoids all queueing is
socially optimal. This contrasts with the frequent result that Pigouvian pricing of goods with ex-
ternalities is optimal, though use of a road susceptible to congestion is such a good. By identifying
the marginal social cost imposed by a commuter, we can construct a Pigouvian toll regime and
compare its welfare results to those of quality-of-service tolling.
1.8.1 Simulated Benefit of Express Lane with Observed Tolls
I study the welfare effects of a road with 4 toll-free lanes and 2 express lanes, like I10-W. Using
empirically observed tolls, I find that adding two express lanes to a simulated road with four toll-
free lanes yields a benefit of $69.21 per commuter per trip. Converting a simulated road from 6
toll-free lanes to 4 toll-free and 2 express yields a benefit of $10.13 per commuter per trip. The
simulated road gains a revenue of $1.18 per commuter per trip, averaging over both commuters
who use the toll road and those who do not.
Oddly, if commuters are simulated without the discrete late arrival penaltyq, the welfare ben-
efits of the express lane increase, to $11.18 per commuter from converting lanes, and $86.69 per
commuter for adding lanes.
27
1.8.2 Commuters Who Experience Urgency Benefit More
Figure 13 shows the results of restricting analysis only to commuters who experienced urgency.
These commuters are defined as those who left home intending to use the toll-free main lane, but
experienced a large enough pre-highway delay to induce them to use the express lane instead. Their
arrival time and welfare on the road with express lanes, are compared to those they would have
incurred on the six-lane road with no express lanes, if they had experienced the same pre-highway
delays. These commuters gain about $13.71 from the conversion of some of the highway’s capacity
to tolled express lanes. Benefits are greater for commuters more sensitive to urgency, i.e. those
with greaterq.
1.9 Is Urgency More Determinative of Express Lane Use than
Value of Time?
This section returns to the baseline specification of the road, with one change: the value of time
is now heterogeneous, calibrated based on the results of Goldszmidt et al (2020) Goldszmidt, List,
Metcalfe, Muir, Smith, and Wang 2020. The table of parameters is updated to illustrate this.
Changes from the baseline simulation are shown in bold.
By making the value of time heterogeneous, we can compare the effect of variation in VOT to
the effect of variation in the penalty for late arrival.
Table 1.4 shows the results of estimating by OLS the regression model
Express
it
= K+a
i
+q
i
+ T
R
pre,it
+e
it
, (1.5)
where Express
it
is an indicator for commuter i using the express lane in simulation iteration t,
and T
R
pre,it
is the random delay that commuter i received in iteration t between departing from home
and entering the highway, and K is the intercept term. We see that commuters with larger penalties
28
Parameter Interpretation Mean St Dev Shape Source
a Value of Time / min $0.37 $0.17 Lognormal Goldszmidt et al (2020)
b Cost per Min Early $ 0.40 0 Same value as main model
g Cost per Min Late $1.20 0 Same value as main model
q Cost of Late Arrival $3 $1 Normal Bento et al (2020)
z Express Lane Preference $ 1.684 $2.275 Lognormal
S
M
Main Lane Capacity
(Veh/Min)
120
S
E
Express Lane Capacity
(Veh/Min)
60
Table 1.3: Revised parameters with heterogeneous value of time
are significantly less likely to use the express lane. This is consistent with the results of the main
model, where highly urgency-sensitive commuters respond by leaving home significantly earlier,
making them less likely overall to encounter urgency and use the express lane.
Table 1.5 estimates, by OLS, the regression model
T
H
it
= K+a
i
+q
t
+e
it
, (1.6)
where T
H
it
is the commuter’s ex ante head start – how much time they left themselves to get to
work. We see that the effect of a dollar of penalty for late arrival exceeds the effect of a dollar per
minute of the value of time spent traveling.
Figure 14 illustrates the relationship of express lane use to the value of time and the penalty for
late arrival, showing that use of the express lane is not driven mainly by express-using commuters
having a higher value of time than other commuters.
29
Table 1.4: Express lane use predicted by value of time and urgency
Dependent variable:
1(Use Express Lane Lane)
(1) (2)
a (VOT) 0.0001
0.0001
(0.00003) (0.00003)
q (urgency) 0.003
0.003
(0.0005) (0.0005)
Pre Hwy Delay 0.0004
(0.0002)
Constant 0.462
0.462
(0.002) (0.002)
Observations 1,125,000 1,125,000
R
2
0.00004 0.00005
Adjusted R
2
0.00004 0.00004
Residual Std. Error 0.498 (df = 1124997) 0.498 (df = 1124996)
F Statistic 24.760
(df = 2; 1124997) 17.810
(df = 3; 1124996)
Note:
p0.1;
p0.05;
p0.01
30
Table 1.5: Head start predicted by VOT and urgency
Dependent variable:
Head Start
a 0.064
(0.002)
q 0.185
(0.035)
Constant 24.319
(0.135)
Observations 1,125,000
R
2
0.001
Adjusted R
2
0.001
Residual Std. Error 37.398 (df = 1124997)
F Statistic 484.496
(df = 2; 1124997)
Note:
p0.1;
p0.05;
p0.01
1.10 Analysis with Original Parameter Estimates
1.10.1 Estimation of Road-Specific Parameter Values
I attempt to estimate commuter parameters for the road studied in this paper using the strategies
described above. The toll profile on the I10-W express anes is designed to prevent congestion in
those lanes. If it succeeds in doing so, then by the reasoning in section B.4, the period where
all commuters will arrive early should be characterized by tolls that increase by b per minute,
offsetting the tradeoff in arrival time; and for similar reasons, the period where all commuters
arrive late should be characterized by tolls that decrease byg per minute.
Figure 15 shows the observed toll profile, averaged over weekdays in 2019. Vertical lines
indicate 7:00, 8:00, and 9:00 AM, assumed to be the ideal arrival times of the three classes of
commuters. Toll price data was requested from metro.net, the website of Los Angeles County’s
transit authority.
31
The steep rising portion corresponds to about 5:30 to 6:30 AM and has a slope of about 0.02908
(in dollars per 5-minute interval, the resolution of the data), while the steeply falling portion cor-
responds to 9:00 to 9:30 AM and has a slope of -0.08777. If the tolls shown are in dollars, we
estimate b to be about 0.6 cents per minute and g to be about 1.8 cents per minute. This is con-
siderably less than the literature-derived values of 40 cents and $1.20s, respectively, that we used
above, but they are in the same ratio.
The identifying assumption in these estimates is that the toll successfully prevents congestion
in the express lane, and therefore that tolls are sufficient to do so. However, if tolls do not suffice
to prevent congestion in the express lane, estimates of the cost per minute of early and late arrival
will be underestimates.
The value of time,a, can be estimated by studying commuters who trade travel time for arrival
time. If late arrival is assured and tolls are absent, these are the only considerations, and equilibrium
congestion decreases at a rate ofgs/(a+g), where s is the capacity of the toll-free lanes.
Traffic data was obtained from PeMS, the Performance Measurement System of the California
Departument of Transportation. Figure 16 shows the profile of weekday delays over time at a
traffic detector near California State University, Los Angeles, which is located near the end of the
toll road. Commuters reaching this detector between 8:55 and 9:30 are presumed to be late for a
9:00 AM arrival, but the queue is observed to increase in this interval, meaning its behavior is not
dominated by commuters attempting to arrive at 9:00 AM. In a simple model such as Arnott et al
(1993), the queue would peak at exactly the time necessary for commuters to reach the bottleneck
in order to arrive on time, then decline from there. In this case, the queue is observed to peak at
8:15 and decline steeply until 8:50, at a rate of 0.2154 cars per minute. Using s = 112 vehicles
per minute from PEMS and g = 9 cents per minute estimated above, and denoting the rate of
decline from the pre-9AM peak by ROD9, we would estimatea =g(s ROD9)/ROD9, or about
$9.11 per minute. If we believed this estimate, commuters would be very sensitive to travel time,
but very insensitive to time early – they would sooner arrive 25 hours early than spend a minute
in traffic! However, we should not believe this estimate. Given uncertainty, let alone urgency,
32
commuters plan to arrive, in expectation, at least a few minutes early. Thus, the peak congestion
is not the time marking the transition from early arrival to late arrival, but sometime earlier. The
commuters we see in the congestion spike at 8:15 AM expect to arrive early. However, their risk
of late arrival increases steeply with every minute they delay, and this accounts for much of their
very high reluctance to enter the queue a bit later, despite the savings in travel time. This estimate
ofa is an overestimate, since it was based on the steepest observed decline in congestion.
If the simulation is calibrated to the estimates of this subsection, we find that relative to a four-
lane road with no express lane, the road that adds two tolled express lanes benefits commuters by
$276.39 per commuter. Relative to a six-lane road with no express lanes, our road, which converts
two of those toll-free lanes to tolled express lanes, benefits commuters by $111.26. However, the
estimates in the literature should be preferred.
1.11 Priorities for Related Future Work
To expand upon the work described here, I intend to investigate the following extensions, in the
order presented below.
1.11.1 More complete theory of a simpler model
Future versions of this work will include a complete theoretical characterization of a model fea-
turing urgency, but without the choice of routes. I will find analytic expressions and diagrams to
represent the profile of congestion over time; the costs experienced by commuters due to travel
time, conventional schedule delay, and urgency; and the ideal toll profile. The resulting paper
will be similar to Arnott et al (1990, 1994) and focus on the minimum necessary changes to add
urgency.
33
1.11.2 Endogenous demand for trips
To endogenize demand for trips, some commuters’ benefit from the trip will be modest enough that
a high cost will deter them. Models that remain tractable will be solved analytically, while those
that require it will use numerical methods.
1.12 Conclusion
This paper studies the effects of urgency on traffic congestion and policies to mitigate it, expanding
the scope of commuter preferences that can be studied. I solve the problem of a commuter with
urgency who has the option of a reliability-tolled lane. I complete the model with numerical
simulations and study the benefits to commuters of a policy intervention, namely obtaining toll
lanes by expanding or converting road capacity.
The toll lanes provide commuters with a way to insure against negative shocks in their travel
time, a form of insurance that does not exist without such lanes. Commuters cannot self-insure in a
decentralized manner, as the difficulty of coordination prevents Coasian bargaining solutions such
as paying other commuters to clear a path when one experiences urgency.
Future technology can both facilitate optimal tolling and reduce its benefits. GPS-linked traffic
apps such as Waze, which allow commuters to learn about current traffic conditions, reduce the
uncertainty about travel time and the risk of arrriving late, thereby partially replacing the benefits
of reliability tolling schemes like the express lanes considered here. On the other hand, such real-
time traffic data will make it easier for policymakers to anticipate upcoming congestion and adapt
their tolls accordingly. Further ahead, self-driving vehicles and carpooling may make it possible
to avoid congestion, with road prices instituted to make it individually rational for commuters to
take their assigned routes at the assigned time (Ostrovsky & Schwarz 2018). Even with such futur-
istic technology, an analysis of optimal tolling will remain valuable, as the incentive compatibility
conditions depend on commuters’ value of time and the travel times on each route, rather than the
technology used to make the trip.
34
However, these upcoming diminutions to urgency apply primarily to trips that can be scheduled
in advance, such as work trips. Ride share services and self-driving cars cannot prevent medical
emergencies, and urgency will thus remain relevant, if perhaps for a smaller fraction of trips.
1.12.1 Relation of This Paper to the Literature
The contribution of the current paper is, first, producing a computer simulation of traffic produced
by the utility-maximizing behavior of commuters who may experience urgency; and second, ap-
plying this simulation to the policy-relevant analysis of tolled express lanes. To my knowledge, my
simulation is the first to allow for urgency; in addition, its modular nature allows for arbitrary joint
distributions of commuter parameters, and its iterative nature more realistically captures commuter
behavior than a model which starts in equilibrium.
Endogenous traffic is studied in classic papers such as Arnott, de Palma and Lindsey (1993,
1994), as well as more recent work such as Hall, Liu, and Nie (2015) Liu, Nie, and J. Hall 2015.
However, all of these papers rely on a particular functional form for the commuter’s objective
function. Throughout the traffic economics literature, it is assumed that commuters’ disutility is
linear in travel time and in the difference between actual and ideal arrival times. While notation for
time itself may differ between Arnott et al (1993), Hall (2020) Jonathan D Hall 2021, and Kreindler
(2020) Kreindler 2020, these papers concur in having a value of travel time, denoteda, values of
time early and time late denoted b and g, but no separate penalty for the event of arriving late
itself. I call this the standard objective function. It implies trip costs which are continuous in the
commuter’s arrival time, unlike models that add a separate penalty for all late arrivals, which would
constitute a jump discontinuity in the cost of the trip, jumping upward at the moment in time when
the traveler becomes late. Discrete penalties for late arrival are not absent from the literature, but I
am aware of no paper prior to the current one that incorporates these discrete penalties into a model
of endogenous traffic. While Small (1982) Small 1982 includes these penalties as a parameter to
be estimated, Noland & Small (1995) includes them in calculations of a commuter’s optimal head
start under certain traffic distributions, and Bento et al (2020) regards them as the main parameter
35
of interest to the exclusion of b and g, these papers all resemble each other in taking traffic as
given.
It is precisely the steep penalty for slightly-late arrival that constitutes what this paper calls
urgency. The peculiarly high value of time that a person briefly experiences in urgent moments is
driven precisely by the risk that a small delay will make the person late, and that this will be very
costly. Therefore, it is necessary to incorporate these late-arrival penalties if we want to study the
effects of urgency on traffic, or the welfare effects of policies that aim to mitigate this urgency.
The omission of discrete lateness penalties from prior endogenous traffic literature is under-
standable, as their addition complicates the model significantly. Notably, the standard objective
function is continuous in arrival time, and allows congestion profiles in which a commuter has an
interval of indifference in departure time. To my knowledge, no published paper yet solves the
less tractable problem of discontinuous objective functions, including the particular discontinuity
of utility relative to arrival time induced by the risk of literally or metaphorically ”missing one’s
flight.” Such discontinuities may be important, however: intuitively, missing a flight by half an
hour is hardly worse than missing it by a minute, but both are far worse than just making it. More
prosaically, papers from Small (1982) to Bento, Roth and Waxman (2020) find evidence that com-
muters on ordinary trips have a significant willingness-to-pay to avoid the event of slightly late
arrival. Even without such ”urgency,” papers like Hall (2018) and Hall and Savage (2019) find that
tolls to improve reliability of arrival time on part or all of the road may have large benefits. Ur-
gency would be expected to make such benefits even larger, as one of the costs of unreliable travel
time in the face of urgency is the risk of ”missing one’s flight” and incurring a discrete penalty,
which is purely additional to per-minute penalties for time late. With the continuous cost functions
in the literature, a risk of arriving slightly late has only a modest effect on a commuter’s expected
utility; but effects are much larger when the penalty depends on the event of being late, rather than
just the number of minutes one arrives late. The novel simulation presented in the current paper
is, to my knowledge, the first study of how urgency affects traffic, and of the resulting effects on
commuter welfare.
36
Urgency can resolve some apparent paradoxes in the literature. For instance, Goldszmidt et al
(2021), using data from a field experiment involving millions of ride-share trips, find that users’
value of time increases with their wait time. By contrast, Bento et al (2020), using commuters’
revealed willingness-to-pay for a tolled express lane, find that the value of time is greatest when
commuters save only a small amount of time. The apparent contradiction is resolved when we
realize that the value of time is neither inherently convex nor concave, but depends on the risk
of late arrival. When commuters wait for a ride-share vehicle, they have chosen their intended
departure time, and are subject to random delays. As their wait time increases, they are in a situ-
ation analogous to commuters in my simulation who experience ever longer pre-highway delays:
as their risk of late arrival increases, so does their willingness to pay to avoid it. Commuters who
know how much time they save on the express lane, however, are in the opposite situation. They
enter the express lane when they expect it to help them avoid lateness, and thus a large portion of
their value of urgency is included in their willingness-to-pay, even if the amount of time saved is
small: by self-selection, the time saved in the express lane is likely to make the difference between
a commuter arriving on time versus late.
Urgency in commuter trips is also relevant to the value of time (VOT): in particular, it provides
circumstances in which an individual’s VOT may change in time, becoming briefly much higher
when urgency is experienced. While historical economic work on the VOT, such as Becker’s
(1965) seminal paper Becker 1965, have treated the VOT as time invariant, more recent work such
as Goldszmidt et al (2021) uses the setting of transportation (in their case, willingness-to-pay for
ride share services) to investigate the change in an individual’s VOT with changing circumstances.
The currrent work aims to contribute to this latter thread in the literature. In particular, pure
”urgency,” with a discontinuous penalty for late arrival, is a value of time that behaves like a point
mass rather than a ”density” of dollars per hour, since the value is realized entirely in the moment
that distinguishes early from late arrival. Even if the true functional form of the VOT is not a point
mass, but a very steep section, where, for instance, disutility increases rapidly in the final minute
37
before one is late, it is still an example of a time-varying value of time: a commuter’s willingness-
to-pay for those final moments that distinguish on-time from late arrival is far higher than their
value for an equal amount of time under more ordinary circumstances.
This paper also fits into the transportation economics literature on the effects of infrastructure,
along the lines of ongoing work such as Allen and Arkolakis (2020) Allen and Arkolakis 2019.
The current paper investigates the effects of a particular infrastructure improvement, namely the
installation of tolled express lanes, and distinguishes the effects of capacity expansion from the
effects of the tolls.
The current paper uses an agent-based simulation, but one in which agents solve a complicated
utility-maximization problem. This complexity is necessary to address the research question: the
effects of urgency on traffic, and on the benefits of mitigation strategies, depend on the effects of ur-
gency on commuters’ decisions and on the resulting traffic. No model without rational commuters
and endogenous congestion would suffice.
1.13 Links to Online Sources Referenced In Text
The congestion report from Texas A&M can be found at https://mobility.tamu.edu/umr/
congestion-data/. The Bureau of Labor Statistics wage report can be found at https://www.
bls.gov/regions/west/news-release/occupationalemploymentandwages_losangeles.
htm.
38
Figure 1
Figure 1.1: Main lane congestion with the initial conditions studied here falls into a cycle of period
3. 50 repetitions are more than adequate to characterize the long-term behavior of this congestion
profile.
1.14 Figures
39
Figure 2
Figure 1.2: Express lane delays show the same periodicity as main lane delays.
40
Figure 3
Figure 1.3
Figure 4
Figure 1.4: Distribution of arrival times. The red vertical lines denote 7:00 AM, 8:00 AM, and
9:00 AM. Commuters fall into three arrival time types, each of which prefers to arrive at one of
these three times.
41
Figure 5
Figure 1.5: Mean cost by type of cost. ”Classical” arrival time disutility is given byb(time early)
for commuters who arrive early, and g(time late) for commuters who arrive late. Travel time cost
is the product of an individual’s value of time and the total time spent traveling. Urgency disutility
is equal to a commuter’s penalty for late arrival if they arrive late, and zero otherwise.
42
Figure 6
Figure 1.6: Breakdown of costs by type, and by commuter’s ideal arrival time.
43
Figure 7: Probability of Late Arrival on Each Road
Figure 1.7: The probability of late arrival in (a) the main road (4 toll-free lanes, 2 express lanes),
(b) the high-capacity alternative road (6 toll-free lanes), and (c) the low-capacity alternative road
(4 toll-free lanes).
44
Figure 8
Figure 1.8: Express lane use is plotted against q, the cost of slightly-late arrival and thus the
sensitivity to urgency. Vertical axis values are 1 for commuters that used the express lane, 0
otherwise. The blue trend line shows a linear model of the probability that a commuter with
any given q will use the express lane. We observe that commuters more sensitive to urgency
are somewhat less likely to use the express lane. As Figure 3 shows, this is because they give
themselves more lead time.
45
Figure 9
Figure 1.9: Panel (a) shows that commuters more sensitive to urgency leave themselves more lead
time to get to work. Panel (b) shows that they are less likely to arrive late.
46
Figure 10
Figure 1.10: The horizontal axis represents time remaining once the commuter has entered the
highway, i.e. after experiencing their realized pre-highway delay. Commuters here face a choice
between a more reliable tolled express lane and a less reliable untolled main lane. We observe that
commuters who have less time remaining at this stage (left side) are more likely to use the express
lane. This is consistent with the express lane being a means for commuters to escape urgency.
Attention is restricted to commuters who are not already late when they enter the highway. Panel
(a) fits a linear model, and panel (b) a loess fit on a subsample. The loess fit shows that commuters
are particularly sensitive to a reduction in time remaining when they have very little, i.e. when they
experience urgency.
47
Figure 11
Figure 1.11: On the horizontal axis, the left column shows commuters who use the toll-free lanes,
and the right column shows commuters who use the tolled express lanes. The height of the bars
indicates their probability of arriving late. We see that express lane users are somewhat less likely
to arrive late than main lane users. Note that individuals with less time remaining until they are
late are more likely to use the express lane. The travel time advantage of the express lane is thus
sufficient to overcome the initial disadvantage that leads commuters to select into it.
48
Figure 12
Figure 1.12: Benefits from the express lanes turn out not to depend on an individual’s cost of ur-
gency. In all panels, commuters are sorted in order of increasingq, the discrete penalty for slightly
late arrival, and thus the cost they face when experiencing urgency. Vertical axes represent wel-
fare improvements when a road with four toll-free lanes and two tolled express lanes is compared
to one with six toll-free lanes (i.e. the gains from converting two lanes to express, a pure effect
of tolls), or four toll-free lanes (i.e. the gains of adding two toll lanes, combining the effect of
tolls with that of expanding capacity.). In panel (a), we see the improvement in their travel time
disutility, proportional to the reduction of time in traffic, when two express lanes are added to four
toll-free lanes. The trend is flat, showing that the travel time saved by adding an express lane is not
correlated with the cost an individual faces from urgency. In panel (b), the vertical axis shows the
improvement in arrival time cost, which includes penalties for late arrival, from adding the express
lane.
49
Figure 13
Figure 1.13: Benefits from the express lanes turn out not to depend on an individual’s cost of ur-
gency. In all panels, commuters are sorted in order of increasingq, the discrete penalty for slightly
late arrival, and thus the cost they face when experiencing urgency. Vertical axes represent welfare
improvements when a road with four toll-free lanes and two tolled express lanes is compared to one
with six toll-free lanes (i.e. the gains from converting two lanes to express, a pure effect of tolls),
or four toll-free lanes (i.e. the gains of adding two toll lanes, combining the effect of tolls with
that of expanding capacity.). In panel (a), we see the improvement in their travel time disutility,
proportional to the reduction of time in traffic, when two lanes out of a six-lane road are converted
to tolled express lanes. In panel (b), the vertical axis shows the improvement in arrival time cost,
which includes penalties for late arrival, from adding the express lane. Panels (c) and (d) show
the analysis of (a) and (b), respectively, but for converting two toll-free lanes to express. The tolls
produce little improvement in travel time, but they do improve arrival time, with greater benefits
for commuters more sensitive to urgency: as shown in panel (d), benefits to arrival time cost are
positive, and trend upward withq.
50
Chapter 2
Distribution of Urgency: Recovering Parameters from
Simulated Data
2.1 Introduction
Traffic congestion is both an important disamenity of large urban areas, and a laboratory for study-
ing commuters’ preferences over time. Papers such as Goldszmidt et al (2021) and Kreindler
(2022) have studied commuters’ willingness to pay for time savings in the United States and India,
respectively. A finding of Goldszmidt et al is that the value of time is context-dependent.
An important contextual element that affects the value of time is its relation to the time of a
sensitive event. Most dramatically, a few minutes that make the difference between catching and
missing a flight are more valuable than an equal amount of time on another occasion. There is
evidence that avoidance of late arrival constitutes a significant fraction of commuters’ willingness
to pay for toll lanes, as in Bento et al (2020). Separate from the costs of late arrival, papers such
as Kreindler (2022), Hall and Savage (2019), and Hall (2021) find benefits to tolls that spread out
peak-period congestion and reduce the variability in travel time. The welfare effects of tolls are
complicated, since they affect the timing of trips, the distribution of travel times (and consequently
the risk of late arrival), and transfer money from travelers to the agency collecting the tolls. Trip
51
timing, congestion, and toll revenue depend on all parameters of commuters’ preferences, includ-
ing their penalty for the event of arriving late. To account for all these effects, we require an agent-
based structural model of endogenous congestion. Such a model is explored in Roitberg (2021).
The model should be calibrated according to observations of a sample of commuters representing
a population of interest.
The current paper uses an agent-based simulation adapted from Roitberg (2021) to simulate
commuter behavior from data on travel times and tolls on two highways with toll lanes in Los An-
geles County, California. Discrete-choice models are fitted to the resulting profile of highway entry
times, arrival times, and tolls paid, to estimate the preference parameters of these commuters. A
model that correctly identifies the parameters of the simulation can then be applied to observations
of real commuters.
2.1.1 Situating this paper in the literature
This paper builds on the work of Bento et al (2020), Roitberg (2021) and Tarduno (2021).
Using billing data from toll lanes in Southern California, Bento et al study the relationship
between commuters willingness to pay for the toll lanes and the time saved. These particular toll
lanes are unusual in having dynamic tolls, adjusted almost in real time to maintain a consistent
level of service in the toll lane; in addition, commuters can see the current toll prices and the
estimated travel time to downtown in the toll and toll-free lanes at the time they enter the toll lane.
Commuters reveal a willingness to pay for small time savings which is surprisingly high. When
regressing a hedonic price function on time saved, the slope can be interpreted as the value of time,
whereas the intercept represents the limit of willingness-to-pay as the time saved approaches zero,
for instance, if the time saving makes the difference between arriving just in time and being very
slightly late. Bento et al find that this intercept is significantly greater than zero, even controlling
for the value of travel-time reliability, and that it accounts for the majority of observed willingness-
to-pay for the toll lane. These results show the value of toll lanes that allow commuters to escape
slightly late arrival.
52
Roitberg (2021) studies congestion and welfare consequences of tolled express lanes when
commuters have a penalty for late arrival. That paper finds the theory of the commuter’s strategy,
endogenizes traffic via simulation, and characterizes the arrival time and welfare of the resulting
simulated commuters. They find that commuters with a large penalty for late arrival benefit most
from the presence of a tolled express lane, even though these commuters, who leave home early,
are not the most likely to use the express lane: their benefit to escaping urgency is large enough to
outweigh its infrequency.
Tarduno (2021) studies road pricing in the presence of both externalities from driving, and
leakage of drivers from tolled roads to alternate roads. He finds theoretically optimal second-
best tolls for many heterogeneous externality-generating goods with leakage; estimates drivers’
preference parameters and substitution patterns from billing data on Bay Area toll bridges; and
uses these results to recommend a toll policy. Tarduno’s use of a mixed logit model influenced my
choice of estimation model in this paper. An important difference of this paper from Tarduno’s
is that I do not have billing data from toll roads, but I do have the ability to simulate commuters’
choice of route and time in response to road conditions.
Like much transportation economics literature, this paper draws on classic work in the field. I
use the bottleneck model of congestion, as proposed in Vickrey (1969). I study commuters who
get disutility from travel time, time early, and time late, a model characterized in a series of papers
by Arnott, de Palma and Lindsey; in particular, congestion, welfare and tolls are characterized
in Arnott et al (1990) and (1993), while Arnott et al (Transp Res B, 1990) study the choice of
departure time and route, and Arnott et al (1994) study congestion with heterogeneous commuters.
My model differs from that of Arnott et al in that commuters also incur a discrete penalty
for the event arriving late, in addition to a cost per minute late. I am hardly the first to do so –
this specification appears as early as Small (1982); and Noland and Small (1995) characterize the
optimal head start for two distributions of travel times, as a function of this late arrival penalty.
However, late arrival penalties are rarely combined with endogenous traffic. Evidence from the
nonzero coefficient on the late penalty in Small (1982) to the results of Bento et al (2020) indicate
53
that commuters have a significant willingness to pay to avoid slightly late arrival; this creates a
specification error in models that omit this willingness to pay, resulting in an opening to improve
the accuracy of traffic models and welfare calculations by incorporating late arrival penalties.
The value of time has been of interest to economists at least since Becker (1965). Transporta-
tion is an excellent setting in which to study it, as commuters make tradeoffs between travel time,
arrival time, and monetary price paid. In recent years, economists have studied the value of time
through field experiments such as Kreindler (2022) or Goldsmidt et al (2020).
However, it is likely the value of time is situational. For instance, a commuter’s willingness
to pay for time savings depends not only on the general value they put on a minute of their time,
but also on how it affects their risk of being late to their destination, and how costly it would be to
arrive late.
2.2 Methods
I estimate discrete choice models on simulated commuter choices and the welfare-relevant coun-
terfactuals in their choice set. Commuters care about travel time, whether they arrive late, and the
difference between their ideal and actual arrival time, with different per-minute costs for early ver-
sus late arrival. A commuter’s choice set is the Cartesian product of points in time, consisting of
5-minute intervals (the temporal resolution of the data) in the AM hours, with a choice set of four
routes, corresponding to the main and express lanes of the two highways in Los Angeles County
that had toll lanes as of 2019. I obtain toll prices and travel times for each route and time step
corresponding to an arbitrary week before the coronavirus pandemic.
Simulated commuters begin at a randomly chosen distance from each of the two highways, and
choose a route and departure time for their trip to work. Each commuter’s choice minimizes their
expected disutility from among the available options. This disutility depends on the commuter’s
cost of travel time, time early, time late, and the event of being late – preference parameters which
are to be estimated – as well as the necessary statistics of the distributions of travel time, risk of
54
late arrival, and time early or late conditional on arriving early or late. For each commuter, at each
starting location, it is necessary to compute these statistics for each element of their choice set, and
to use the resulting information as data on which to fit my discrete choice model.
2.2.1 Data
Data on travel times comes from the California Department of Transportation’s (Caltrans) Per-
formance and Measurement System (PeMS). Expressways in California contain traffic monitors
consisting of inductive-loop traffic detectors in each lane, allowing the authority to measure traffic
speed, traffic volume, and the composition of vehicles. I collect observations of speed at five-
minute intervals from all monitors on Los Angeles County expressways that have a toll lane,
namely, Interstate 10-W from West Covina to downtown Los Angeles, and Interstate 110-N from
the harbor to downtown LA. I require a sample of typical travel times on weekdays and week-
ends – as a preliminary exercise, I use the first week of June, 2019, an arbitrary week before the
coronavirus pandemic, but I can easily retrieve similar data for the entire year of 2019. To convert
from speed to travel time, I associate each monitor with a segment of the road, divide the length of
that segment by the speed observed to obtain a travel time for that segment associated with each
five-minute interval, and add together all travel times for segments along a route.
The raw data from CalTrans includes, for each segment, the speed along that segment (on each
lane and in aggregate) at each day and time of day. A separate file gives the length of all segments.
To obtain travel time, I match each segment to its length, then divide that length by speed for each
observed point in time to obtain a travel time associated with that point in time. To obtain the travel
time along a route at a single point in time, such as 8:00 AM on June 1, 2019, I add together the
calculated travel times of all segments on that route at 8:00 AM. This is not an accurate reflection
of real travel times, as a commuter who enters the highway at West Covina at 8:00 will reach
segments near downtown considerably later, but it is far easier to compute simultaneous travel
times than to account for the variable change in time of day during a commuter’s traversal of a
route.
55
In Los Angeles, the LA County Metropolitan Transit Authority (LACMTA) operates tolled
express lanes on several highways. Unusually, the tolls on these lanes vary in real time, with the
objective of maintaining a consistent quality of service in the toll lanes, an objective which is
complicated in practice by the ability of high-occupancy vehicles to use the lanes free of charge.
To use the lanes, a driver must have a transponder installed in their vehicle, which communicates
with toll stations to indicate the precise date and time in which the vehicle passed this station, from
which their toll can be computed. Individual records of travel are linked to the vehicle owner’s
identity and billing address, and the make and model of their vehicle, data that would make it
possible to identify repeated trips by an individual, attached to the tolls paid for each trip; however,
this billing data is confidential and not easily obtained. Simple data on the price assessed at each
toll station over time, however, is available upon request. I obtain toll prices for all toll stations in
Los Angeles County in 2019, and extract tolls corresponding to the days for which I have travel
times. Tolls along a route at any given time are obtained by simply adding the tolls, at that time,
for all segments along the given route, the same method used for travel times along a route.
2.2.2 Theoretical Model of Commuter Behavior
Commuters have a trip cost of the form,
C=a(travel time)+b(minutes early)+g(minutes late)+q(if late)z(use express)+toll.
The timing of the model is as follows:
(1) At home, anticipate travel time and tolls on each route.
(2) Stage 1 Choice: Commit to home departure time and highway, anticipating intended choice
of lane.
(3) Travel from home to highway, experience pre-highway delay.
(4) Reach highway, resolve uncertainty
56
(5) Stage 2 Choice: Opportunity to switch lanes.
(6) Experience delay on highway (and toll, depending on lane) drawn from empirical data.
(7) Arrive at work.
The expected utility of leaving home at any time, and thus the optimal departure time, depends
on the commuter’s policy function for subsequent decisions. Commuters must anticipate, at each
highway entry time, their likelihood of using the express lane conditional on entering the highway
at that time, and the resulting distribution of arrival times, tolls, and risk of late arrival. Armed with
this policy function, the expected utility of departing from home at a given time t can be obtained by
integrating the expected utility of highway entry time over the distribution of pre-highway delays
conditional on leaving home at time t.
A commuter’s choice set consists of combinations of home departure time, intended route, and
(at a later point in time, when more information is available), the option to switch lanes along that
route. The choice is best thought of as occurring in two stages. In the first stage, made at home,
commuters can choose a highway, an intended lane along that highway, and a home departure time.
The choice of highway and departure time are irrevocable for this trip, but the choice of lane may
be altered in the second stage; however, it must be considered in the first stage, as the optimal
departure time depends on one’s intended lane. An example element of the first-stage choice set is
”leave home at 8:05 AM, intending to use the main lane of interstate 110N.” The second stage of
choice occurs once commuters enter the highway. The choice set for the same commuter consists
of ”remain on the main lane of I-110N, entering it at current time ˜ t” and ”switch to the express
lane of I-110N, entering it at current time ˜ t.” Note, however, that the highway entry time ˜ t depends
on the time chosen at home and the realization of today’s pre-highway delay, and that the lanes
available depend on the route chosen from home; thus, the choice set in Stage 2 depends both on
the choice made in Stage 1 and the realization of delays between the stages.
Commuters’ expectations depend on their beliefs over the distribution of travel times and tolls
at each time of day. Commuters live at a location with a fixed distance from each highway. They
know this distance and the resulting distribution of travel times to each highway, but each day’s
57
realization is an independent draw from this distribution. On the highway itself, the travel time and
toll on each lane vary by time of day. At home, commuters experience considerable uncertainty,
knowing the distribution of travel times to and along each highway route at each time of day,
but not the realization. Commuters learn the realization of their travel time to the highway when
they experience it, i.e. when they reach the highway. In addition, entering the highway reveals
today’s realization of highway travel times and tolls – we can think of commuters seeing signage
announcing this information.
Upon entering the highway and choosing a lane, commuters’ arrival time at work is determined
by the travel time in that lane at the moment they enter it. Their welfare, in turn, depends on this
arrival time relative to the time they left home (determining travel time), relative to their ideal
arrival time (determining whether they arrive late, and how early or late), and the toll they pay
along the way.
In this paper, I take congestion as exogenous, with travel times drawn from the empirical dis-
tribution of travel times by time of day on certain Southern California roads. In subsequent work,
congestion can be endogenized using the bottleneck model, set forth in Vickrey (1969) and charac-
terized in Arnott et al (1993, 1994). Since endogenous congestion depends on commuter behavior,
which depends in turn on congestion, endogenous travel times must be determined iteratively. This
paper’s exogenous travel times, however, do not.
2.2.3 Simulation Methods
2.2.3.1 Overview
Simulated commuters choose from a menu of routes and home departure times on their trip to
work. Their choice set of departure times consists of 5-minute intervals between midnight and
noon (the AM hours). Their choice set of routes consists of two expressways, Interstate I-10 West
(from the beginning of the express lane at Francisquito near West Covina to the downtown terminus
at Central), and Interstate I-110 North (from the beginning of the express lane at Redondo Beach to
58
the downtown terminus at 3rd Street). Along each expressway, commuters face a choice between
a tolled express lane and a toll-free main lane.
Travel times and tolls along each route vary by day and time of day. At home, commuters
know, for each time of day, the multi-day average of the tolls and travel times along each route at
this time of day, as well as the standard deviation of travel times during rush hour (defined as 7:00
AM to 9:00 AM). They commit to a departure time and an expressway (I-10W or I-110N) using
only this information. After leaving home and entering the highway, commuters learn today’s
realization of the travel time and toll on the highway they have selected; at this point, they are free
to choose between the express lane and the main lane on their selected highway.
The choice of routes is made in two stages. In the first stage, commuters are at home. They
know the average travel time from home to work along each route at each time of day, as well as
the average toll along each route at each time of day, but not today’s realization. In the second
stage, made once commuters enter the highway, they are locked into a highway and time, but have
the opportunity to enter or exit that highway’s express lane.
Commuter welfare is assumed to depend on their travel time from home, their departure time
at work in relation to their ideal departure time, and the toll paid. Any choice function must, at a
minimum, output these values for the alternative selected and some suitable choice set.
2.2.3.2 Details of Simulated Commuters
Commuters travel from home to work, choosing from a menu of routes and departure times which
vary in their travel time to work, the uncertainty of this time, and the toll paid. Commuter i suffers
a disutility a
i
for each minute they spend traveling. If they arrive at work at their ideal time t
i
,
there is no arrival time cost; if they arrive earlier, they have a disutilityb
i
for each minute early. If
they arrive after t
i
, they suffer a penaltyq
i
for the event of being late, plusg
i
for each minute late.
This specification is standard in the literature, though papers that endogenize traffic typically force
q
i
to be homogeneously zero for reasons of tractability.
59
Parameter Interpretation Mean St Dev Shape Source
a Value of Time / min $0.37 $0.17 Lognormal
Goldszmidt et al
(2020)
b Cost per Min Early $ 0.40 0
Same value
as main model
g Cost per Min Late $1.20 0
Same value
as main model
q Cost of Late Arrival $3 $1 Normal Bento et al (2020)
z Express Lane Preference $ 1.684 $2.275 Lognormal
D
10
Commuter’s initial distance
to interstate 10W
6 classes spaced
5 miles apart
D
110
Commuter’s initial distance
to interstate 110N
6 classes spaced
5 miles apart
Table 2.1: Parameters, Chapter 2
As is typical in the transportation economics literature, commuters have a cost per minute for
travel time and difference (early or late) between their ideal and actual arrival time. A feature not
commonly combined with endogenous traffic is the discrete penalty, q, incurred every time the
commuter arrives later than their ideal arrival time t
. In addition, commuters may gain utility z
from using the express lane, independent of their arrival time – this may correspond to subjective
preference for the lane, or to unobserved differences in road quality. Since toll enters linearly, this
cost function is specified as quasilinear in money, and coefficients (divided by the coefficient on
toll, if applicable) can be interpreted as willingness-to-pay per unit.
Commuters are characterized by the preference parameters in the equation above, as well as
their ideal arrival time and the distance from their home to each highway.
My preferred specification for commuter parameters is given in the table below:
Commuters are as in Roitberg (2021), section 9, with one new feature. As in that paper, com-
muters have heterogeneous values of time, values of urgency, and preference for the express lane;
and homogeneous costs per minute early and per minute late. Changed from that paper, commuters
60
now have two routes to choose from, and are assigned an initial location consisting of one of six
bins of distance from one expressway, and independently, one of six bins of distance from the other
expressway.
2.2.3.3 Details of Simulated Choice
The ideal choice of route and departure time depends on the distribution of anticipated arrival times,
tolls paid, and likelihood of arriving late, as functions of both choice variables. To anticipate these,
in turn, requires both knowledge of the distribution of travel times before and after entering the
highway, as well as anticipation of the commuter’s likelihood of choosing the express lane once
they face that choice.
In anticipation of their choice of lanes, commuters determine expected utility of the main and
express lanes on both highways as a function of the time the commuter enters the highway and
faces the choice, in light of the information available to commuters at home. For each lane and
each highway entry time, commuters must anticipate the probability of late arrival, the distribution
of arrival times conditional on early or late arrival given this highway entry time, and the toll paid.
With this information, commuters can anticipate which lane they are likely to choose conditional
on entering the highway at any given time, given the information they have at home. Using the
distribution of departure times from home to the highway entry point, commuters can anticipate
the likelihood of using the express lane on each highway as a function of the time they leave home.
This allows the commuter to anticipate their expected utility (ex ante) for each home departure
time, for both I-10W and I-110N. Given this, commuters depart from home at their selected time,
and embark toward their selected expressway.
Once commuters arrive at their expressway, they gain additional information. In particular,
rather than anticipating their choice of lane based on average travel times and tolls, they learn the
actual travel time and toll for each of the two lanes on their chosen expressway. Thus, for instance,
a commuter who initially chose I-10W and anticipated a 60% probability of using the main lane,
but who, upon entering the highway, learns that the travel time on the main lane of I-10W is longer
61
than usual today, but the travel time on the express lane, is free to switch to the express lane.
Alternatively, they may switch lanes if the travel time from home to the highway entry differs from
their anticipation.
A commuter’s welfare depends (if arrival time is known for sure) on their arrival time relative
to their departure time from home (giving travel time) and relative to their ideal arrival time (giving
time early or late, and whether or not the commuter is late), as well as the toll they pay. When
commuters choose between the express and main lanes after entering the highway, they know the
time at which they will arrive if they make either choice. Therefore, the choice function outputs
not only the arrival time and toll for the route taken, but also the one for the alternative lane on the
same expressway, given the same highway entry time.
However, the choice of departure times from home is done without knowing the exact arrival
time. Given a commuter’s preference parameters, computing their expected utility requires us to
know, for each route and departure time in their choice set, the probability of late arrival, several
statistics of the distribution of arrival time conditional on departing at this time and being early or
late, and the probability that they use the express lane. To estimate commuters’ preferences over
departure times, we must compute the above statistics for each route and home departure time in
the commuter’s choice set.
At home, commuters face a menu of four routes (the main and express lanes of interstates I-
10W and I-110N); for each route, they have 144 possible departure times, consisting of 5-minute
intervals throughout the AM hours. To complicate matters, when commuters leave home, they
face stochastic delays before entering each highway, the distribution of which depends on the com-
muter’s distance from this highway. At home, commuters know the distribution of pre-highway
delays to each highway; however, they learn today’s realization only upon reaching the highway.
At home, commuters commit to a home departure time and a highway; they choose with the inten-
tion of taking either that highway’s main lane or its express lane, since this intention affects their
optimal choice of time, but they will have the opportunity to change lanes later. The commuter’s
information set at home consists of the mean and (discretized) distribution of pre-highway travel
62
times to each highway; the mean travel time along each highway and lane at each time of day; the
mean toll at each time of day; and the empirical CDF of weekday travel times between 7:00 and
9:00 AM. With this information, they can compute, for each option in their choice set: the prob-
ability of arriving late; the conditional distribution of arrival time conditional on arriving early or
on arriving late; the enjoyment they will get (if any) from using the express lane; and the expected
toll they will pay.
The commuters’ choice does not end when they leave home: en route to the highway, com-
muters learn today’s realization of their travel time to the highway, their travel time downtown
from the highway entrance, and the toll, resolving their uncertainty. Knowing their arrival time
and the toll they will pay to change it, commuters are free to switch between the main and express
lanes of their chosen highway.
2.2.3.4 Variations on the Commuter Choice Function
Depending on the analysis we wish to conduct, we may require different sets of counterfactual
outcomes, requiring modification to the commuter choice function. Analysis of welfare requires
only the knowledge of each commuter’s home departure time, ideal and realized arrival time, and
toll paid (if any), along with this commuter’s preference parameters. However, to estimate a dis-
crete choice model, we must know the utility-relevant attributes of each element of the commuter’s
choice set.
I obtain these counterfactuals by slightly modifying my simulation: commuter behavior is un-
changed, but for each commuter, I must also record their information set from home, for each
option in their choice set. This provides sufficient information to estimate the commuters’ prefer-
ences based on their choice of route and time from home.
2.2.3.5 Challenges encountered
The foremost challenge of my analysis is the complexity of my model, in which commuters first
choose an ex-ante route and departure time from a menu, then gain additional information, and
63
finally face a second choice of route (main and express lane on their chosen highway) in light of
their new information set. The difficulty is that the attributes of each option in the second stage
depend on the choice made in the first stage, as well as the information learned after the first stage.
It is difficult to construct a dataset with all the relevant information to estimate the model in its
entirety, without simplification. As of this writing, I am attempting to do so. Until I succeed in
this, I have analyzed simpler models, focusing on the first-stage choice made at home.
Prior to the analysis, I require a simulation of commuters making the choice specified in my
model. The greatest difficulty has been in correctly accounting for time: commuters value their
time in minutes, and travel time data is likewise in minutes, but the temporal resolution of the data
is in 5-minute steps. At all stages, I must be attentive to the distinction between minutes and time
steps. This delayed my progress, but I believe I have dealt with it successfully. In the process, I
allow my code to include quadratic costs for early arrival, as well as additional costs for leaving
home too early.
2.2.4 Analysis
The full model involves commuters who anticipate their expected utility from each route and de-
parture time; upon leaving home, they commit to a departure time and one of two expressways,
but not to the choice of tolled or toll-free lane; and once entering the highway, they gain addi-
tional information about realized travel times and choose between the tolled and toll-free lane. For
feasibility, I estimate related but simpler models.
In my simulation model, commuters face a choice in two stages. The first stage has them choose
from a menu of routes and home departure times, given information about the expected travel time,
likelihood of late arrival, and conditional distribution of arrival times if early and if late, but not
knowing today’s realization of travel time to the highway or travel time on the highway in their
chosen route. In the second stage, commuters learn the realization, and are free to switch between
the main and express lanes of their chosen highway.
64
Time constraints restrict me to use estimators already available in software. Since I am not
aware of any that allow me to incorporate the second-stage information, with a different choice
set and information set, albeit one which depends on the choice made in the first stage, I restrict
attention to the commuter’s choice in the first stage, where they commit to a home departure time
and a highway, and choose their initial intention to use the main or express lane. This choice of
lane can later be reversed, but it affects their choice of time and highway.
2.2.4.1 Estimation by Choice of Route and Time From Home
This version of the model considers the alternatives a commuter faces from home. I estimate a
model of the form,
C
irt
a
i
E(T
irt
)+q
i
P(late
irt
)+bE(T
e
irt
)+gE(T
l
irt
+z
i
Express
irt
+T
irt
, (2.1)
where C
it
denotes the commuter’s choice, E(T) denotes the mean travel time, T
e
and T
l
denote
the expected schedule delay early and late, respectively, andT is the toll paid. Subscript i indexes
commuters, r routes, and t time steps.
We can consider the commuter’s utility as a function of the form U
i j
= f(b
i
,X
j
), whereb
i
are
the preference parameters of commuter i, and X
j
are the attributes of alternative j. The commuter
parameters are the parameters of Equation 2.2.4.1. The alternative attributes are the expected
travel time, the probability of late arrival, the expected number of minutes early conditional on
early arrival, the expected number of minutes late conditional on late arrival, the likelihood of
using the express lane, and the expected toll paid, for each alternative in the choice set.
A model with multi-stage choice, during which the agent’s choice set and information set
change between stages, is quite complicated. I am not aware of an existing estimation package
for such a model, and coding one myself, which explicitly computes the likelihood function, gra-
dient, and convergence criteria, would be difficult. While such an approach is possible for future
65
revisions, I currently limit myself to existing estimation techniques, namely a mixed-effects logit
model describing the choice from home.
Commuters choose route and time simultaneously, as the optimal choice of departure time
depends on one’s anticipated choice of route, including choice between main and express lanes.
The choice set thus contains two dimensions, a spatial dimension (in which commuters choose
between four routes) and a temporal dimension (in which commuters choose between 144 possible
departure times, spaced 5 minutes apart); any combination of the two dimensions is a possible
choice.
2.3 Results
2.3.1 Preferred Specification: Mixed Logit
Mixed logistic regression, or mixed logit, is a highly general form of logistic regression. The theory
of this estimator can be found in Kenneth Train, ”Discrete Choice Methods with SImulation.” The
estimator can be derived from a utility-maximizing model with random coefficients: the utility of
agent n from alternative j is given by U
n j
=b
0
n
x
n j
+e
n j
, where x
n j
are attributes that can vary by
agent, by alternative, or both, and e
n j
are i.i.d. extreme value. Coefficients b
n
, representing the
importance per unit of each attribute in agent n’s utility, vary by agent. Each agent’s coefficients
b
n
are drawn from a distribution f(b). The researcher specifies a shape for this distribution, and
estimates its parameters by maximum likelihood.
In principle, this model is a good fit for my empirical analysis. The first stage of my model
involves a discrete choice between several routes and departure times, a setting for which logis-
tic regressions are far better suited than linear. I simulate commuters whose utility, as assumed
in mixed logistic regression, depend linearly on several attributes. Also corresponding to the as-
sumptions, commuters vary in some of these parameters: in particular, a
i
, the value of time, is
heterogeneous and lognormally distributed, andq
i
, the cost of the event of late arrival, is normally
66
distributed. A heterogeneous-parameter model such as mixed logit makes it possible to estimate
the distribution of a parameter of interest.
Mixed logit regression is available in Stata, but it is quite computationally intensive. A numer-
ical hill-climbing algorithm first estimates the best fit for a fixed parameter model, then adjusts
the scale parameter to improve fit. Since the likelihood function of such an estimator is not al-
ways concave, and since the best-fit fixed parameters do not necessarily correspond to the location
parameters of the best-fit mixed-effects model, the mixed logit estimator may fail to converge.
Owing to the computational difficulty of this model, I estimate it on a small subset of the
data, using fewer iterations and looser tolerances than Stata’s defaults. The table below has results
obtained in Stata using choice data on 10 simulated commuters, each with a choice of 4 routes and
144 time steps. I use the data from their choice from home to estimate their preference parameters.
Their express lane preference has been scaled up by a factor of 4 from the table in section 2.2.3.2,
but cannot be included in the estimation due to collinearity. On this data, I specify this regression
with the estimating equation,
cmmixlogit Chosen ExpectedTravelTime HowEarly HowLate Toll, casevars(ID) random(Prob Late)
vce(cluster ID) iterate(10) favor(space) difficult tolerance(1e-3) ltolerance(1e-3) nrtolerance(1e-
3)
The table is given here:
y1
o.ExpectedTravelTime 0
HowEarly -19.728092
HowLate -39.764781
o.Toll 0
Prob Late 218.41265
sd(Prob Late) .31981488
Table 2.2: Mixed logit results
Intercepts for each alternative are estimated, but not displayed.
67
This table gives me, for the first time, an estimate of both the mean and the variance of the cost
of late arrival, recovered from data, for a logistic discrete choice model. This specification is far
preferable to the linear mixed effects model, since the choices I consider are in fact discrete.
An important problem, however, is that the estimate for the toll paid is omitted due to collinear-
ity, even when I increase commuters’ preference for the express lane. Without this, I am unable to
convert my estimates into willingness-to-pay, which hinders comparison to the setup of my model.
I can note, however, that the ratio of mean to standard deviation is very high. This does not
match the setup of the model, and indicates the continuing inadequacy of the estimator. I have the
luxury of making this determination precisely because, in the current exercise, I recover parameters
from data I generated myself. Without such privileged access to the true data generating process, I
would not know that its parameters had not been successfully recovered.
2.4 Other Regression results
I estimate several less computationally demanding regressions to investigate the robustness of the
estimates.
As above, I restrict analysis to the choice made in the first stage. Since there are many alterna-
tives, I omit estimates from binomial choice models.
2.4.1 A model of commuter utility
U
irt
= k+at
v
rt
+b[t
(t+ t
v
)](1 D
L
)+g[(t+ t
v
)t
]D
L
+qD
L
+z D
E
+T +e
l
irt
(Agent i faces a choice of route r and departure time t. An agent who leaves home at time t
faces a realized pre-highway delay t
p
, and a highway entry time˜ t= t+t
p
. Travel time on highway
is t
h
(˜ t). Total travel time t
v
= t
p
+ t
h
. Ideal arrival time t
. D
L
is 1 if late, 0 otherwise. D
E
is 1 if
express lane is used, 0 otherwise.)
Choice is 1 if the route and time for which values are specified is chosen, 0 if another route and
time are chosen.
68
2.4.2 Conditional logit
McFadden’s conditional logit model can be used to estimate discrete choices between several al-
ternatives. An agent’s utility from each alternative depends additively on its attributes (eg travel
time) and an extreme-value error term. The result is that log-odds ratios of each choice are linear
in attributes. We must observe variation in travel time (or other attributes), and can estimate a
homogeneous constant parameter that best fits the value of travel time in the commuter’s choice
function. This model is among the easier discrete choice models to estimate and to derive marginal
effects, due to the precise specification of the error term; however, this specification imposes a spe-
cific pattern of substitution, which features independence of irrelevant alternatives (IIA) – a pattern
that is a poor fit for many applications. More sophisticated models that allow agent-specific param-
eters and non-independent error terms can allow the econometrician to specify other substitution
patterns, but they do not match conditional logit’s simpicity.
I use data on first-stage choices and counterfactuals. In the first stage, commuters determine
the expected travel time, expected time early conditional on early arrival, expected time late con-
ditional on late arrival, probability of late arrival, and expected toll for each option in their choice
set. I therefore collect the same information and use it to predict the commuter’s choice, which
takes a value of 1 for the route-departure time combination that is chosen, and 0 otherwise.
In Stata: clogit Choice ExpectedTravelTime HowEarly HowLate Prob Late IsExpress Toll,
group(ID)
Regression results can be found in table 2.3
U
irt
= k+at
v
rt
+b[t
(t+ t
v
)](1 D
L
)+g[(t+ t
v
)t
]D
L
+qD
L
+z D
E
+T +e
cl
irt
69
Table 2.3: Conditional logit
(1)
V ARIABLES Choice
ExpectedTravelTime -0.109*
(0.0570)
HowEarly -0.299***
(0.0132)
HowLate -0.228***
(0.0195)
Prob Late -6.897***
(0.957)
IsExpress -18.97
(1,401)
Toll 0.0109
(147.0)
Observations 576,000
Standard errors in parentheses
*** p0.01, ** p0.05, * p0.1
2.4.3 Linear mixed effects model
In Stata: mixed Choice ExpectedTravelTime HowEarly HowLate Prob Late UseExpress Toll ID:
ExpectedTravelTime Prob Late UseExpress
Y
irt
= k+a
j
E[t
v
]
rt
+b
j
E[t
(t+t
v
)jL](1P
L
)+g
j
[(t+t
v
)t
jL]P
L
+q
j
P
L
+z
j
D
E
+T +e
mnl
irt
A linear mixed effects model regresses a numerical outcome on a linear combination of pre-
dictor variables. It differs from a simple linear model in allowing random effects: instead of being
a constant, a parameter such as a in the equation above can be a random variable. For a given
commuter j, a
j
is drawn from the distribution of a. Depending on specification, mixed effects
models can allow for heterogeneous and non-independent subsamples. It may be, for instance,
that observations across commuters are independent, but observations within a commuter are not,
70
h
Table 2.4: Ratio of coefficients to toll (Conditional logit)
Regressor Ratio
Time -10.00
Event late -632.7522936
how early -27.43119266
how late -20.91743119
use express -1740.366972
toll 1
due to sharing an unobserved, but estimable, commuter-specific random effect. A linear model
that fails to account for this non-independence is misspecified, and can produce entirely incorrect
estimates.
Since I am studying simulated commuters, the data-generating process is known. As described
in section 2.2.3.2, some parameters are fixed across the whole population of commuters, while
others, such as the value of time and the value of urgency, vary across commuters according to a
known distribution; in other words, the DGP is known to contain mixed effects. I wish to estimate
a discrete choice model such as mixed logistic regression, but owing to its computational intensity,
I also wish to use mixed-effects models that can be computed more quickly.
I use data on first-stage choices and counterfactuals. I specify the coefficients on expected
time early and time late as fixed, i.e. constant across the population, whereas the coefficients on
expected travel time, probability of late arrival, and using the express lane are specified as random,
i.e. able to vary by commuter. As specified here, the linear mixed effects model simply allows
the coefficients on expected travel time, probability of late arrival, and using the express lane to
vary by commuter. I estimate this model simply as a way of specifying heterogeneity that is less
computationally difficult than mixed-effects logistic regression.
71
Table 2.5: Linear mixed effects
(1) (2) (3) (4) (5) (6)
V ARIABLES Choice lns1 1 1 lns1 1 2 lns1 1 3 lns1 1 4 lnsig e
ExpectedTravelTime 0.00147***
(7.46e-05)
HowEarly -0.000157***
(4.31e-06)
HowLate -8.76e-05***
(6.78e-06)
Prob Late -0.00925***
(0.000281)
UseExpress -0.00238***
(0.000151)
Toll -0.000196***
(1.74e-05)
Constant 0.00406*** -19.43 -19.16 -19.27 -18.14 -3.183***
(0.000326) (251,636) (22.36) (22.34) (192,668) (0.000932)
Observations 576,000 576,000 576,000 576,000 576,000 576,000
Number of groups 1,000 1,000 1,000 1,000 1,000 1,000
Standard errors in parentheses
*** p0.01, ** p0.05, * p0.1
h
Table 2.6: Ratio of coefficients to toll (Mixed effects)
Regressor Ratio
Time -7.50
Event late 47.19387755
how early 8.01E-01
how late 4.47E-01
use express 12.14285714
toll 1
72
h
Table 2.7: Estimated variance of random coefficients
Regressor Coefficient Ratio
var(Time) 1.33e-17 6.68e-12
var(event late) 2.26e-17 1.01e-15
var(use express) 1.84e-17 8.22e-16
2.4.4 Nested logit
I’ve had some difficulty specifying nested logit, and making sure my data is organized appropri-
ately.
The structure of my nested logit model is as follows: On each highway, the two lanes (main
and express) share a nest.
Table 2.8: raw output from nlogit
(1)
V ARIABLES Mean
c.ExpectedTravelTime@1bn.Alternative 4.282***
(0.00172)
c.ExpectedTravelTime@2.Alternative 4.079***
(0.00254)
c.ExpectedTravelTime@3.Alternative 3.976***
(0.00256)
c.ExpectedTravelTime@4.Alternative 3.483***
(0.00311)
Observations 576,000
Standard errors in parentheses
*** p0.01, ** p0.05, * p0.1
73
Table 2.9: Coefficient estimates from nlogit
(1)
V ARIABLES Choice
ExpectedTravelTime 0.4568***
(.0661)
HowEarly -.4237***
(.0234)
HowLate -0.7918***
(0.0896)
Prob Late 0.5535
(1.030)
Toll -0.2777 **
(0.110)
Observations 39,000
Standard errors in parentheses
*** p0.01, ** p0.05, * p0.1
2.4.5 Additional estimators
There are many discrete choice estimators. Famously, Berry, Levinsohn and Pakes (1995) (BLP)
propose a model of market equilibrium in which the distributions of taste parameters can be esti-
mated from product-level and population-level data, including appropriate instruments to account
for endogeneity in supply and demand. The BLP method enables estimation when the econometri-
cian cannot match the characteristics of a consumer to the attributes of that consumer’s individual
purchase, while providing for patterns of substitution more plausible than conditional logit. How-
ever, this method is not particularly germane to the exercise conducted in the current essay, as I
have individual-level data on the choice made by each commuter.
Heterogeneity of commuter parameters can be represented, as above, with a mixed logit model:
some parameters relevant to the agent’s choice are random variables with a continuous distribu-
tion. Conditional on any given realization of the parameter, choice probabilities follow a logit
model; at the population level, however, these probabilities must be weighted by the density of the
parameter. More concretely, for instance, commuters’ penalty for being late may follow a normal
74
distribution. Each commuter’s penalty is drawn from this distribution, and each commuter, know-
ing their penalty, chooses according to a conditional logit model; but in aggregate, the choices of
a commuter of any type must be weighted by the likelihood that an arbitrary commuter is of this
type. By having commuters vary, mixed logit allows for different aggregate substitution patterns
than conditional logit.
A similar model, called latent-class conditional logit, gives the unknown parameters a discrete
distribution. For instance, the late arrival penalty may take on values of $5 or $10, and the econo-
metrician wants to estimate the frequency of each. This model can be estimated in stata using the
lclogit command (Pacifico and Yoo, 2013). I choose not to perform this estimation, because my
simulated commuters’ distribution for the value of urgency is continuous.
2.5 Discussion
To the author, the main benefit of this exercise is to further advance my skill in agent-based simu-
lation, as well as my skill in data analysis (obtaining observed tolls and travel times, determining
counterfactual outcomes for choices not taken, running regressions on realized and counterfactual
outcomes, specifying and estimating discrete choice models). In addition to the direct value of
results obtained here, the skills developed in the process are applicable to research in a wide range
of specialties.
I have yet to find a regression that successfully recovers the simulation parameters. It will
probably be necessary to specify an estimation strategy that embeds more of the structure of my
model, such as the second-stage choice set, the dependence of this set on the first-stage choice
and the information acquired in the interim, and the heterogeneity of the value of time and value
of urgency in my simulation. I have yet to determine if this can be done with existing regression
methods, or if it will require a customized estimator, perhaps developed in collaboration with an
econometrician or industrial organization economist.
75
With some additional improvement in the estimation methods and with data on commuters’
realized highway entry and choice of lanes, I can adapt these methods to estimate parameters
such as the mean and variance of the value of time and the value of urgency in real populations of
commuters. This information will be of value to engineers and policy makers looking to understand
the optimal investment in road infrastructure and toll policies.
The study of real commuters will greatly increase this paper’s value to readers.
2.5.1 Negative Results: Significance of Failure of Conventional Regressions
to Recover Choice Parameters
The regressions in section 2.4 fail to recover the parameters described in section 2.2.3.2. I attribute
this to differences between the complicated multi-stage choice process of the simulation and the
assumptions of conventional discrete choice models. It may be necessary to use customized meth-
ods to fit the simulation model to data. Since even conventional models such as mixed-effects
logistic regression are quite computationally intensive, the fully customized model might require
the use of high-performance computing.
If it is indeed the difference between the true data-generating process and the regression model
that accounts for the failure to recover the expected results, this poses significant challenges for
empirical work more generally.
However, an obvious risk of complicated simulation is that it may contain errors unnoticed by
the author. I have needed to amend my simulation on many occasions during the writing of this
paper, and can expect to do so again.
76
Chapter 3
Welfare Effects of Several Toll Profiles: A Preliminary
Investigation into Optimal Toll Policy
3.1 Introduction
Traffic congestion is a common problem in cities, and policy makers often turn to tolls on part of
the road network as a means of mitigation. The optimal toll policy depends on the structure of the
road and the behavior of the drivers using it. There is evidence that drivers care significantly about
the event of arriving late to their destination, separately from the cost per minute late; but this is
often omitted in the literature on endogenous traffic because it complicates analysis. However, as
a factor in driver behavior, these late arrival penalties are relevant to an analysis of the optimal toll
policy.
This paper conducts a preliminary investigation into the effects of late arrival penalties on the
optimal toll policy. I simulate commuters on a road network based on the highways with toll lanes
in Los Angeles County, California. Commuters travel to work under several potential toll policies,
including the ones empirically observed on these roads and several counterfactual toll policies. I
illustrate the consequences of these toll policies on commuter welfare and the profile of highway
entry times and delays.
77
3.1.1 Some relevant literature
There is an extensive literature on the theory of optimal tolls. Among economists, a certain set of
assumptions are common enough that I will refer to them as the standard setting: in this setting,
commuters experience a subjective cost for each trip which is linear in travel time, as well as in
the difference between their ideal and actual arrival time. The cost of subideal arrival time may be
asymmetric, with late arrival more costly than early arrival. Demand for trips may be specified as
perfectly inelastic, or as sloping downward in trip cost. For a given trip, if its benefits can justify
the costs, commuters choose the departure time from home that minimizes this cost. Congestion,
where present, is due to a bottleneck, as in Vickrey (1969): traffic flow is limited by a short road
segment that permits no more than v vehicles per minute. If commuters enter faster than this, a
queue forms, where the delay to a new commuter joining the queue is equal to the length of the
queue at the time they join, divided by the capacity of the bottleneck. If demand for the road is
known and fixed, the optimal toll is a smoothly time-varying one that just eliminates the queue,
making commuters willing to enter at a uniform rate that does not exceed road capacity by just
offsetting, for commuters entering at time t, the disutility they would receive by arriving at the
corresponding arrival time. The peak level of the toll would be such that it does not change the
number of trips: commuters arriving exactly on time would be charged the value of the time they
would have lost in peak-period queues, time they save since the tolls prevent queues. Toll revenue
is a pure social gain: the toll prevents wasteful queues, while making each commuter pay for the
bnefit they thereby receive.
Arnott et al (1993) explicate the theory of the bottleneck and extend it to elastic demand. They
define equilibrium congestion as a congestion profile induced by drivers’ choice of departure time
and the bottleneck, and which gives no driver an incentive to unilaterally change their departure
time. With a continuum of identical drivers, they characterize the equilibrium profile of conges-
tion, departure times, and welfare with no tolls, optimal tolls, and several alternative toll policies
(uniform and peak-period), quantifying the welfare benefits of optimal tolling. When demand is
elastic, however, the less-sensitive tolling regimes, which impose greater disutility on commuters,
78
also induce fewer trips for a road of given capacity; the optimal capacity, therefore, is larger if tolls
are coarse than if they are fine. If tolls pay for capacity expansion, the optimal toll is self-financing
if costs of capacity are constant; it maintains a surplus with increasing costs, and a deficit with
decreasing costs.
Arnott et al (1994) consider a similar analysis for heterogeneous commuters: in particular,
commuters may come in several types with regard to their value of time; their aversion to late
arrival; and their ideal arrival time. Commuters with a high value of time relative to their cost of
early or late arrival (less inflexible commuters) will use the road near the beginning and end of the
congested period, when travel times are shorter but arrival times are less desirable. (Commuters
may be inflexible, or have a highb/a, either because their travel time is cheap or because they can
afford little deviation in arrival time.) Only when the least-inflexible have paused to wait for the
late end of the congested period, do the second-least inflexible commuters enter the road. While
any given group is entering, the queue increases at a slope proportional to their cost of time early,
divided by the difference between their VOT and cost of time early. In the no-toll equilibrium,
departure order follows relative schedule delay costs (relative to VOT); however, the optimal toll
rearranges departure times if needed so that the departure order follows absolute schedule delay
costs. Thus, the optimal tolls will shift high-VOT commuters toward the peak time. This is welfare-
enhancing even without rebates, if the highest-VOT commuters travel at peak time, and is often a
Pareto improvement when toll revenue is rebated.
The papers by Arnott et al study a single road, all of it subject to the same toll policy. Real
commuters, however, are typically faced with a menu of alternative routes, necessitating a study
that accounts for substitution patterns between these routes. In the simplest case, commuters may
be faced with two routes, of which one is tolled (or equivalently, one road containing tolled and toll-
free lanes). It is often feared that such toll lanes will be harmful for many commuters. However,
Hall (2018) shows that if overuse of a road can reduce throughput (hypercongestion), toll lanes
can be Pareto improving even without rebates. In Hall’s model, congestion is due to a bottleneck
susceptible to hypercongestion: capacity is large as long as the road is open, but drops abruptly, and
79
reversibly, while a queue exists. Hall allows agents’ ideal arrival times to vary along a continuum;
this turns out to affect equilibrium outcomes. Hall finds that if there is congestion without tolls, and
if some rich (high-VOT) agents travel at the peak of rush hour without tolls, then tolling part of the
road can generate a Pareto improvement. Furthermore, if x percent of commuters using the road
at peak times are rich, then pricing x percent of the road generates a Pareto improvement. In later
work, Hall (2020) applies a similar model to traffic and survey data from an isolated bottleneck
road in California to estimate the welfare that could be gained by optimal tolling of part of this
road.
Roads may sometimes be unreliable: unfortunate chance events such as traffic accidents may
dramatically reduce thir capacity, and the risk of such an event increases the more heavily a road is
used. Hall and Savage (2019) study optimal tolling in such a setting. There exist non-negative tolls
which make all drivers better off, despite reducing the peak departure rate, if the risk of breakdown
is low enough.
While traffic is an important externality produced by driving and mitigated by tolls, it is not the
only one. Tarduno (2021) considers toll policies when trips exert externalities due to congestion
and pollution, when externalities vary by time of day and speed of travel, and where tolls on one
route cause some commuters to ”leak” into other routes. In this setting, he empirically estimates
leakage in the toll bridges of the San Francisco Bay Area, and proposes an optimal policy for
cordon tolling around San Francisco.
However, to my knowledge, the effects of late arrival penalties on optimal tolls are under-
studied. Discrete penalties for late arrival affect the optimal toll policy by altering the profile of
congestion (which is no longer single-peaked, as commuters who will arrive late choose to wait for
the queue to decline enough to offset late arrival), the number of commuters who should optimally
arrive early versus late, and the optimal start and end of the congested period. As such, it is not
obvious that many existing results will generalize to the new setting. The current work acts as an
early exploration of this topic.
80
3.2 Review of the Simulation Model
The simulation used here is nearly identical to the one used in Roitberg (2022), and similar to that
used in Roitberg (2021). Commuters have a choice of two highways, each containing a toll and
toll-free lane. Each commuter lives at some distance to each highway. Commuters choose their
departure time from home and their preferred route based on the distribution of travel times to the
highway, travel time on each lane, and expected tolls. Once they arrive at the highway, they learn
today’s realization of travel times and tolls, and have the opportunity to switch between the toll
and express lane.
3.3 Methods in the Current Paper
Simulated commuters are characterized by the table below. Their preference for the express lane
has been increased compared to Roitberg (2021) in order to ensure that commuters use the toll
lane.
81
Parameter Interpretation Mean St Dev Shape Source
a Value of Time / min $0.37 $0.17 Lognormal Goldszmidt et al (2020)
b Cost per Min Early $ 0.40 0 Same value as main model
g Cost per Min Late $1.20 0 Same value as main model
q Cost of Late Arrival $3 $1 Normal Bento et al (2020)
z Express Lane Preference $ 12 $ 10 Lognormal
D
10
Commuter’s initial distance to
interstate 10W
6 classes spaced 5 miles apart
D
110
Commuter’s initial distance to
interstate 110N
6 classes spaced 5 miles apart
Table 3.1: Parameters, Chapter 3
82
Figure 3.1: Comparison of toll profiles
I generate three variations: ”high-urgency” commuters, where mean(theta) = 6, ”medium-
urgency” commuters, where mean(theta) = 3, and ”low-urgency” commuters, where mean(theta)
= 0.
I generate several toll profiles, constructed to have the same mean and different shapes.
3.4 Results
3.4.1 Welfare results, all commuters
The following tables show the welfare results for all commuters. For reference, 5000 commuters
are simulated, each of which makes 7 trips, with realized tolls and delays corresponding to the
first week of June, 2019. Out of the 35,000 total trips, I report the number in which the commuter
arrives late and the number in which they use the express lane.
High-urgency commuters (tolls in order from best toll to worst)
83
name MeanDisutility NumLate NumExpress
Quadratic High 71.2837381872706 707 17413
Triangular High 71.4389765469836 684 14980
Uniform High 72.3546545026146 909 15144
Empirical High 72.5127982491911 1067 10662
Gaussian High 73.1911172317282 897 11327
PeakOnly High 75.6564291418708 844 10826
Table 3.2: High-urgency commuter welfare results
name MeanDisutility NumLate NumExpress
Quadratic Medium 71.1145719397393 707 17385
Triangular Medium 71.2647860661957 684 14935
Uniform Medium 72.1558265481087 909 15139
Empirical Medium 72.2661529899846 1067 10651
Gaussian Medium 72.9755893765222 897 11326
PeakOnly Medium 75.4709747112568 844 10826
Table 3.3: Medium-urgency commuter welfare results
Medium-urgency commuters
Low-urgency commuters
Figures 2-7 illustrate the timing of each toll profile, relative to the timing of highway entry time
for commuters exposed to this toll profile. These figures show the profile of highway entry times
for all commuters, including those who choose not to use the toll lane. Commuter numbers and
tolls are normalized for easier viewing.
name MeanDisutility NumLate NumExpress
Quadratic Low 70.9628951328235 708 17363
Triangular Low 71.1023869291729 686 14899
Uniform Low 71.9430978194827 909 15136
Empirical Low 71.9717373988581 1068 10651
Gaussian Low 72.7272936062641 899 11326
PeakOnly Low 75.2565374642807 845 10828
Table 3.4: Low-urgency commuter welfare results
84
Figure 3.2: Empirical tolls vs highway entry time
Figure 3.3: Uniform tolls vs highway entry time
85
Figure 3.4: PeakOnly tolls vs highway entry time
Figure 3.5: Triangular tolls vs highway entry time
86
Figure 3.6: Quadratic tolls vs highway entry time
Figure 3.7: Gaussian tolls vs highway entry time
87
Figure 3.8: Empirical tolls vs highway entry time (toll lane only)
3.4.2 Welfare results, toll lane users only
Figures 8-14 illustrate the timing of each toll profile, relative to the timing of highway entry time
for commuters exposed to this toll profile. These figures show the profile of highway entry times
only for commuters who use the toll lane. Commuter numbers and tolls are normalized for easier
viewing.
88
Figure 3.9: Uniform tolls vs highway entry time (toll lane only)
Figure 3.10: PeakOnly tolls vs highway entry time (toll lane only)
89
Figure 3.11: Triangular tolls vs highway entry time (toll lane only)
Figure 3.12: Quadratic tolls vs highway entry time (toll lane only)
90
Figure 3.13: Gaussian tolls vs highway entry time (toll lane only)
91
References
Allen, Treb and Costas Arkolakis (2019). The Welfare Effects of Transportation Infrastructure
Improvements. Working Paper 25487. National Bureau of Economic Research. DOI:10.3386/
w25487.
Arnott, Richard, Andr´ e de Palma, and Robin Lindsey (1994). “The Welfare Effects of Congestion
Tolls with Heterogeneous Commuters”. In: Journal of Transport Economics and Policy 28.2,
pp. 139–161.
Becker, Gary S. (1965). “A Theory of the Allocation of Time”. In: The Economic Journal 75.299,
pp. 493–517.
Bento, Antonio, Kevin Roth, and Andrew R Waxman (2020). Avoiding traffic congestion external-
ities? the value of urgency. Tech. rep. National Bureau of Economic Research.
Goldszmidt, Ariel, John A List, Robert D Metcalfe, Ian Muir, V Kerry Smith, and Jenny Wang
(2020). The Value of Time in the United States: Estimates from Nationwide Natural Field Ex-
periments. Tech. rep. National Bureau of Economic Research.
Hall, Jonathan D (2021). “Can tolling help everyone? estimating the aggregate and distributional
consequences of congestion pricing”. In: Journal of the European Economic Association 19.1,
pp. 441–474.
Hall, Jonathan D. and Ian Savage (2019). “Tolling roads to improve reliability”. In: Journal of
Urban Economics 113, p. 103187. DOI:https://doi.org/10.1016/j.jue.2019.103187.
Kreindler, Gabriel (2020). “Peak-hour road congestion pricing: Experimental evidence and equi-
librium implications”. In: Unpublished paper.
Liu, Yang, Yu (Marco) Nie, and Jonathan Hall (2015). “A semi-analytical approach for solving
the bottleneck model with general user heterogeneity”. In: Transportation Research Part B:
Methodological 71, pp. 56–70. DOI:https://doi.org/10.1016/j.trb.2014.09.016.
Small, Kenneth A (1982). “The scheduling of consumer activities: work trips”. In: The American
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92
Appendices
A Exploratory Work
In this section, I outline some theoretical work exploring the model where commuters have discrete
penalties for late arrival. The size of these penalties is a measure of the commuter’s value of
urgency: commuters with a lot to lose if they arrive slightly late will be very sensitive to slight
delays at critical moments, and will pay a high price to avoid them.
This sensitivity to urgency makes commuters’ disutility discontinuous in arrival time – there is
an abrupt increase in arrival time disutility at the transition from slightly early or perfectly on-time
arrival to slightly late arrival, which cannot be offset by an equally abrupt decrease in travel time
disutility. The first subsection illustrates one important consequence of this discontinuity: unlike
in the continuous models discussed in Arnott et al (1993, 1994), once commuters have a penalty
q 0 for the event of late arrival, there is no longer an equilibrium with an uninterrupted rush hour,
in which some commuters knowingly arrive slightly late.
I wish to investigate how commuters’ sensitivity to urgency affects traffic, toll policies, and the
value of toll regimes that improve travel time reliability. I therefore require rational commuters
and endogenous congestion; as this section shows, the simplest model with these features does not
suffice.
93
A.1 An Elementary Model with Discrete Penalties for the Event of Late
Arrival
A commuter i leaving home at time t incurs a subjective cost of
C
i
(t)=a
i
(travel time)+b
i
(time early)+g
i
(time late)+q
i
1(late)+T (t) (A.1)
whereT (t) is the toll at the time t at which commuter i enters the highway.
Consider a single road with no delays except the bottleneck, as in Arnott et al (1993). The
road consists of a single lane with a capacity of s vehicles per minute, and congestion follows the
bottleneck model described below.
If all commuters are homogeneous, the equilibrium profile of congestion is such that a com-
muter is indifferent between any departure time during the congested interval. As in Arnott et al
(1993), in the absence of a toll, there exists an equilibrium in which the travel time disutility in-
duced by the queue offsets the schedule delay cost of the corresponding arrival time. Their analysis
of the equilibrium departure rates of commuters who arrive early and those who arrive late remain
valid. However, one feature of this model is importantly different: the commuter’s schedule delay
cost is discontinuous in their arrival time. As arrival time approaches the ideal time t
, schedule
delay cost approaches zero, but abruptly increases toq the moment after. This cannot be offset by
an instantaneous decrease in the queue, and as a result, in equilibrium, there must be an intermedi-
ate period during which no commuters depart, until the queue clears enough to bring total disutility
in line with that of the early commuters.
This model, the simplest with q 0, suffices to illustrate that results change when commuters
are sensitive to urgency, even though this simple model lacks urgency itself, the risk of slightly
late arrival at great cost. The analysis of tolls proves informative for models that add urgency: the
objective of the ideal time-varying toll is to ensure that the express lane is efficiently utilized at its
capacity or as nearly as can be achieved without exceeding it and causing queues, for the duration
of the period where this lane is in use. The optimal start and end times of this period depend on the
94
optimal number of commuters that arrive late. An explicit expression for the optimal time-varying
toll of the exploratory model can be found in the appendix. Considerations of the optimal toll
are described in section B.4, but since the toll and the endogenous pattern of congestion mutually
depend on each other, identifying the optimal toll profile proved infeasible.
A.2 Optimal Toll in Exploratory Model
As in Arnott et al, the optimal time-varying toll at any time t is just enough to offset the schedule
delay cost of arrival for a commuter departing at t who encounters no congestion. In particular, the
optimal fine toll has the form
t
o
(t)=
f
0, t t
q
a(t
t)b,t2[t
q
,t
]
a(tt
)gq,t2(t
,t
q
0]
0, t t
q
0
Here a is some number less than
bg
b+g
N
s
, with N the number of commuters; t
q
is the beginning
of the rush hour, t
q
0 its end, and t
the ideal arrival time.
This model, the simplest with q 0, suffices to illustrate that results change when commuters
are sensitive to urgency, even though this simple model lacks urgency itself, the risk of slightly late
arrival at great cost. The analysis of tolls proves informative for models that add urgency.
A.3 Models in which Urgency Can Be Relevant
Commuters can experience urgency as soon as they face a distribution of travel times. The choice
problem of a commuter with the preferences given above, who takes as given a distribution of travel
95
times and has only one route, is solved for particular distributions in Noland and Small (1995). Of
note, each commuter has a unique optimal departure time, which depends on all the commuter’s
preference parameters and the rate at which congestion changes over time. To obtain a continuum
of departure times therefore requires heterogeneous commuters, who vary continuously in at least
one parameter. However, commuters need not have a continuum of ideal arrival times: as Noland
and Small find, a commuter’s optimal departure time depends on all their preference parameters,
including their penalty for the event of late arrival.
The current paper considers commuters who vary in their penalty for late arrival (aka their
sensitivity to urgency), which drives variation in their departure time.
B Theory of Optimal Choice of Lane and Time
Commuters of type i have a cost function given by
C
i
(t)=a
i
(travel time)+b
i
(time early)+g
i
(time late)+q
i
D
L
+T (t). (B.1)
Whereq = 0, results can be found in Arnott et al (1994). Forq 0, we examine the commuter’s
choice of departure time.
There are a continuum of commuters, each taking congestion as given. Commuters mini-
mize cost functions of the form cost=a(travel time)+b(time early)+g(time late)+qD
L
+toll,
where D
L
is an indicator for late arrival.
For simplicity, all commuters go from home to work on a single road. The commuter’s choice
variable is departure time, and conditional on departure time, whether to use a congested lane with
no toll (the main lane) or a tolled lane with no congestion (the express lane).
Traffic is determined in equilibrium – in particular, all commuters take traffic as given and best-
respond, and traffic is determined by commuters’ choice of departure time and lane, as well as the
congestion technology. An equilibrium is a profile of commuter strategies such that no commuter
can gain from unilateral deviation – in essence, a Nash equilibrium.
96
Commuters take as given the traffic in each lane. In particular, they know the distribution of
travel times in the main and express lanes, but not today’s realization. Travel time in the express
lane has a lower mean and less dispersion than in the main lane, but commuters using this lane
must pay a toll. If it proves helpful, I may assume that travel time in the express lane is known
with certainty. A commuter who arrives at the entrance to the express lane knows their time of
arrival and the toll they will pay if they use the express lane, and the distribution of arrival times if
they use the main lane.
B.1 Choice of lanes
A commuter receives linear disutility from time in traffic and number of minutes early or late, as
well as a discrete penalty q if they arrive late. In addition, commuters may have heterogeneous
preferences over lanes. In addition, it will be assumed throughout that travel time in the tolled
express lanes is more reliable than travel time in the toll-free main lanes.
We will want convenient notation here. A commuter at time t faces travel time T
o
(t) in an
optimally tolled express lane, and T
m
(t), a random variable, in the main lane. Their desired time
of arrival is t
. The express lane has a toll oft. A commuter on the main lane has a probability p
L
of arriving late, i.e. after t
. On the express lane, we assume the commuter cannot arrive late, i.e.
t
t T
o
(t).
Commuters will choose the express lane if their utility from this lane is greater than their
expected utility in the main lane. In the express lane, we assume that the comsumer has no risk
of arriving late, and therefore their utility is given by a(T
o
(t))+b(t
t+ T
o
(t)). In the main
lane, their expected utility is given by aE[T
m
(t)]+(1 p
L
)bE[time early if early]+ p
L
(q +
g(E[time late if late]))
Commuters have an idiosyncratic preference z
i
for the express lane, plus preferences of the
form discussed above. The utility advantage of the express lane is equal to
U
it
=z
i
+a
i
(DT
it
)+(change in schedule delay
it
)t
it
+e
it
, (B.2)
97
whereDT is the advantage in travel time of the express lane over the main lane and t
it
is the
difference in toll.
B.2 Choice of Departure Time
Commuters travel from home to work on a single road with two lanes indexed by l=f0,1g – a toll-
free main lane, in which there are deterministic delays caused by congestion as well as stochastic
delays, and a tolled express lane, in which the toll adjusts to prevent congestion. Commuters have
a shared ideal arrival time t
and a disutility of the form
C
i
(t,l)=a
i
travel time(t,l)+b(time early(t,l))+g(time late(t,l))+q
i
(if late). (B.3)
If we add lane preference, we get cost = a(travel time)+b(time early)+g(time late)+
q
i
D
L
+z
i
D
E
+ toll, where D
L
is an indicator for late arrival, D
E
for using the express lane (en-
dogenous), andz
i
is a commuter’s idiosyncratic lane preference.
A commuter departing at time t on the express lane has a utility of
U
o
(t)=f
aT
o
(t)+b(t
t T
o
(t))+T (t)z , if t+ T
o
(t) t
aT
o
(t)+g(t+ T
o
(t)t
)+q+T (t)z , if t+ T
o
(t) t
(B.4)
In the main lane, arrival time is uncertain, and the commuter’s expected utility depends on the
probability of arriving late.
EU
m
(t)=aET
m
(t)+(1 p
L
(t))b(t
t E[T
m
(t)jT
m
(t) t
t])
+p
L
(t)g(t+ E[T
m
(t)jT
m
(t) t
t])+ p
L
(t)q.
(B.5)
where p
L
(t) is the probability of arriving late if you depart at time t and use the main lane.
The advantage of the express lane is therefore
98
z+a(ET
m
(t) T
e
(t))+bT
e
(t)bET
E
m
(t)
+ p
L
(t)
bET
E
m
(t)+gET
L
m
(t)(b+g)(t
t)+q
T (t).
Commuters use the express lane if its advantage, accounting for tolls, is positive. Simulations
support the conclusion that the express lane advantage is monotonically increasing in time, and
that there is therefore a critical time ˜ t, before which commuters use the main lane, and after which
they use the express lane. This time may differ from one commuter to the next.
Given the choice of lane, the commuter chooses their optimal time to maximize
U(t)=f
za(T
e
(t))b(t
t T
e
(t))t(t), if t ˜ t
a(ET
B
m
(t))(1 p
L
(t))b(t
t ET
EB
m
(t)) p
L
(t)g(ET
LB
m
(t)+ tt
) p
L
(t)q, if t˜ t.
where the B superscript denotes conditioning the expectation on departure before ˜ t, the E su-
perscript on early arrival, and the L superscript on late arrival.
This can be solved by taking first-order conditions with respect to t. Before ˜ t, the commuter
uses the main lane and faces the FOC
0=a
dET
m
dt
(t)
dP
L
dt
b(t
t ET
B
m
(t))
(1 p
L
(t))b(1+
dET
B
m
dt
(t)) (B.6)
+
p
L
dt
(t)g(ET
LB
m
(t)+ tt
)+ p
L
(t)g[1+
dET
LB
m
dt
(t)]+q
d p
L
dt
(t). (B.7)
After ˜ t, the commuter uses the express lane and faces the FOC
0=a
dT
e
dt
(t)bb
dT
e
dt
(t)+
dt
dt
(t). (B.8)
99
A commuter can identify the optimal time to use each lane with the above FOCs, compute their
corresponding utility, and choose lane and departure time simultaneously to maximize expected
utility.
B.3 Accounting for Random Variation in Travel Time between Home and
Highway
The discussion of optimal lane and time above describes the strategy of a commuter who can
control their exact time of entry to the highway. Importantly, lane and time are chosen together,
with the commuter using the tolled express lane if they enter the highway after a critical time, and
the toll-free lane otherwise. If, however, commuters must commit to a home departure time before
learning the realization of random delays between home and the entrance to the highway, then
their choice of home departure time merely determines the probability of entering the highway
early enough that it is unnecessary to use the toll lanes.
In the absence of pre-highway delays, the commuter would identify their optimal departure
time for each lane, then choose the lane-time pair that offers the greatest expected utility. In the
presence of pre-highway delays, however, the commuter must anticipate their choice of lane and
the corresponding expected utility for each possible highway entry time. If the commuter leaves
home at time t, they experience a pre-highway delay d which follows a known distribution F
d
, and
enter the highway at time t+ d. It is therefore possible to compute a commuter’s expected utility
for each home departure time t.
The problem becomes more tractable in cases when the express lane advantage above is nega-
tive before some critical time ˜ t, and positive afterward. The commuter’s expected utility is given
by U(t)= P
x
(t)U
x
(t)+(1 P
x
(t))U
m
(t), where P
x
(t) is the probability of having to use the ex-
press lane, U
x
(t) is the expected utility of using the express lane conditional on leaving home at
time t and receiving a realized pre-highway delay long enough to induce the commuter to use the
express lane, and U
m
(t) is the expected utility of using the main lane conditional on leaving home
at time t and receiving a short enough pre-highway delay to use the main lane. These expected
100
utilities depend on the interaction of the departure time, the distribution of pre-highway delays, the
distribution of exogenous delays on the highway, the time profile of endogenous congestion, and
the commuter’s ideal arrival time.
In particular, P
x
(t) is the probability of receiving a pre-highway delay greater than˜ tt. U
x
(t) is
the expected utility of using the express lane conditional on receiving a pre-highway delay greater
than ˜ tt, a function of t which also depends on the joint distribution of delays before the highway
and in the express lane.
B.4 Spreading Express Lane Commuters Over Time
For simplicity, we assume that commuters share a commonb andg.
A commuter departing at time t on the express lane has a utility of
U
o
(t)=f
aT
o
(t)+b(t
t T
o
(t))+T (t)z , if t+ T
o
(t) t
aT
o
(t)+g(t+ T
o
(t)t
)+q+T (t)z , if t+ T
o
(t) t
The marginal cost of an additional minute of delay is
dU
o
(t)
dt
=f
aT
0
o
(t)+b(1 T
0
o
(t))+T
0
(t), if t+ T
o
(t) t
aT
0
o
(t)+g(1+ T
0
o
(t))+q+T
0
(t), if t+ T
o
(t) t
By construction, a queue is prevented in this lane, and T
o
(t) is a constant function; thus T
0
o
(t)=
0, and we have
T
0
(t)=f
b, if t+ T
o
(t) t
g, if t+ T
o
(t) t
,
with a discontinuous increase ofq at t
.
101
Ifq
i
is heterogeneous, our choice ofq
T
allows us to choose how many express lane commuters
arrive early or late. Any commuter withq
i
q
T
will choose to arrive early in the express lane. Thus,
by makingq
T
small, we can make all express lane commuters arrive early.
More generally, F
q
(q
T
), the fraction of commuters withq
i
q
T
, is the fraction of express lane
commuters who arrive late, while 1 F
q
(q
T
) is the fraction of commuters who arrive early.
B.5 Spreading Commuters Between Lanes
The expected utility advantage of the express lane is
z
i
+a
i
(ET
m
(t) T
e
(t))+bT
e
(t)bET
E
m
(t)
+ p
L
(t)
bET
E
m
(t)+gET
L
m
(t)(b+g)(t
t)+q
i
T (t).
We wantT (t) to have a time-invariant component that will discourage excessive commuters
using the express lane. This must adapt to the probability of arriving late in the main lane, as this
is an important component of the express lane’s advantage.
B.6 Significance of Theoretical Results
The theoretical results for the commuter’s choice of lane and departure time illustrate the com-
muter’s choice problem, and show that the optimal departure time depends on how this time affects
the distribution of arrival times in each lane relative to the commuter’s ideal.
The theoretical results for tolls that spread commuters over time and between lanes are infor-
mative about ideal tolling. We find that the observed tolls in the I-10 express lanes differ from
these results.
102
C Details of Simulation Model
This section describes the simulation I use to give the main results of the current paper.
C.1 Building the Setting
Time is discrete, with MT time periods, representing T minutes and M steps per minute. The
timing of the model is as follows: first, commuters anticipate congestion and the distribution of
exogenous delays, and commit to a time to depart from home. Second, they travel from home to
highway, receiving a realization of a lognormally distributed delay in the process. Third, they enter
the highway and choose to use a tolled express lane or a toll-free main lane, knowing the current
tolls and the profile of queues over lane and time from the previous iteration. Fourth, depending
on the lane they choose, commuters receive an additional lognormally distributed delay. Fifth and
finally, commuters arrive at work, receiving utility that depends on the time they arrive relative to
the commuter’s ideal arrival time, the travel time taken, the lane used, and the toll paid.
C.2 Simulating Commuters
There are N commuters, calibrated to 22,500 (see below). Value-of-time parametersa
i
, b
i
, andg
i
are made homogeneous (a,b,g) for simplicity, but can easily be made heterogeneous.
Lateness penaltyq
i
is i.i.d. normally distributed, while lane preference parameterz
i
is lognor-
mal and independent ofq.
Commuters fall into a finite number of arrival time classes, depending on their ideal arrival
time. These correspond, in the calibrated model, to ideal arrival at 7 AM, 8 AM, and 9 AM. Since
most commuters begin work at one of a small set of start times, I regard this as an appropriate sim-
plification; compared to allowing a unique ideal arrival time for each commuter, the simplification
substantially reduces run time.
The cost experienced by a commuter arriving early is equal toa times their travel time plusb
times the time early, plus any toll they pay; for a commuter arrriving late, the corresponding cost is
103
a times travel time plusg times the time late, plus a penaltyq for the event of late arrival, plus any
toll paid. Execution of the code can therefore be accelerated by dividing the number of commuters
by a constant (the ”speedup factor”) and multiplying their utility parameters and the tolls they pay
by the same constant. If computation time is abundant, the user can specify a very fine temporal
resolution, a speedup of 1, and arbitrarily many ideal arrival times; if faster computation is desired,
the user can specify a very coarse temporal resolution and a substantial speedup and expect similar
results.
C.3 Commuter Heterogeneity
Results shown here are for commuters who are homogeneous in a, b and g, varying in their dis-
crete penalty for late arrival q, their lane preference z , and their ideal arrival time t
. q varies
continuously and follows a normal distribution with mean 3 and standard deviation 1, approximat-
ing the findings of Bento et al (2020).
While results for other joint distributions of commuuter parameters are not shown here, the
simulation can be fed with an arbitrary number of commuters, whose parameters follow an arbi-
trary joint distribution.
C.4 Commuter Choice Function
Commuters choose a departure time and a lane (tolled express or toll-free main lane) to minimize
their cost, C
i
(t)=a
i
(travel time)+b
i
(time early)+g
i
(time late)+q
i
D
L
+T (t).
Calculations of expected utility require us to find the distribution of arrival times conditional
on early or late arrival, since the value of time changes at the late-arrival threshold. For each
available value of T
i
and each possible departure time t, we find the probability of late arrival,
the probability distribution of travel times conditional on departing at time t and arriving early
(i.e. at T
i
or earlier), and the corresponding distribution conditional on arriving late. Since this is
computationally intensive, it is done once for each arrival time class.
104
Commuters face the choice problem described in the theory section. They must identify their
optimal choice of lane as a function of the time they enter the highway, then choose the optimal
time to enter the highway.
An important complication is added by the exogenous random delays between leaving home
and entering the highway. The timing of the model is as follows: commuters commit to a departure
time from home, knowing the distribution of all delays; they then realize a delay before entering the
highway, after which they enter the highway and choose a lane. In the absence of these exogenous
delays, a choice of home departure time commits the commuter to a highway entry time and a
lane. However, once pre-highway delays are added, the commuter’s choice of home departure
time merely determines the probability that they will enter the highway late enough to induce them
to use the express lane.
C.5 Determining the Toll Regime for the Express Lane
Results shown in this paper are based on the observed toll profile for Interstate I-10 Westbound in
Los Angeles County, as described above in section 1.3.1. In addition, I partially characterize the
ideal toll regime, as described in the current section.
The model has enough flexibility to study a variety of toll regimes, but the one of primary
interest is one that aims to lead to efficient utilization of the toll lane without queueing. Such a toll
regime must spread out express lane users over time, and must induce just enough commuters to
use the toll lane (in contrast to the toll-free lane) to utilize the toll lane’s capacity during the rush
hour. Theoretical considerations can be found above. The details of the optimal toll regime must
be adapted to the population of commuters under consideration, so as to induce them to enter the
toll lane at a uniform rate equal to the lane’s capacity.
Commuter i wishes to arrive at time t
i
, for which they must enter the road at time ˆ t
i
, which
depends on congestion. In the simple case where there are no delays before entering the road,
where all commuters share a common ideal arrival time t
, a common value of time early b,
and a common value of time late g, tolls can simply increase by b dollars per minute before ˆ t
105
and decrease by g dollars per minute after ˆ t. If commuters are homogeneous in the value q of
urgency, the toll can discontinuously jump downward byq at ˆ t to render homogeneous commuters
indifferent to their arrival time, and thus willing to depart at a uniform rate, determined by road
capacity, during the congested period. Without pre-highway delays, such indifference suffices
to create an equilibrium in which there is no congestion in the toll lane, and in which ˆ t = t
.
These results are as in Arnott et al (1990), but with a discontinuity in tolls to compensate for the
discontinuity in commuter preferences over arrival time; some details appear above in section A.1.
If commuters share t
but have idiosyncraticq
i
, the toll will control which commuters arrive late:
by choosing a toll profile of the form described above, with a discontinuity of
¯
q, the policymaker
induces all commuters with aq
i
smaller than
¯
q to arrive late, while the rest arrive early.
If commuters vary in t
i
as well asq
i
, the optimal toll regime is more difficult, as commuters can
no longer be made indifferent over the whole congested interval, particularly if t
i
follows a discrete
distribution rather than a continuous one. A starting point for such a toll regime can be found in the
results of Arnott et al (1987) for equilibrium congestion with several classes of ideal arrival times.
It is possible to prevent congestion by imposing on commuters a continuous dispersion of t
i
.
Notably, the observed toll regime differs from any of these theoretical results, suggesting that
some gains remain to be made from adjusting tolls.
C.6 Constructing Congestion
Each commuter, upon entering a lane, joins the queue to get through its bottleneck. This queue
clears at a rate of s
l
vehicles per minute in lane l and has nonnegative length; thus, the length of
the queue is a function of cumulative departures and time since the last time there was no queue. If
commuter i enters lane l and encounters a queue of length q
i
, the delay they face due to this queue
is equal to q
i
/s
l
minutes.
106
C.7 Iteration and Equilibrium
A congestion profile is a correspondence from points in time t to the delay due to queueing incurred
in each lane by a commuter entering the queue at time t. A Nash equilibrium of the traffic model
is a profile of departure times for each commuter such that, given the congestion profile generated
by this profile of departure times, no commuter has an incentive to unilaterally alter their own
departure time.
Since commuters’ utility depends only on travel and arrival times, a commuter’s best response
is determined by the congestion profile to which they respond. The full information about each
commuter’s departure time is not required: since any commuter, once they join the queue, imposes
an identical delay on the commuters behind them, a commuter has no need to know the identities
of the commuters departing at a particular time, only their number and the resulting delay.
In the simulation, in iteration 1, commuters choose a departure time in response to an initial
profile of congestion. In iteration n (n 1), the initial profile of congestion is replaced by the
profile of congestion generated in iteration n 1. The iterative process in the simulation provides
commuters with a congestion profile P and has them respond optimally with a choice of lane
and home departure time. Commuters’ departure time profile thus represents a best response to
congestion profile P, and if P is a congestion profile corresponding to a Nash equilibrium, then
commuters’ best response to P will recreate it.
Results shown are for 50 iterations, but the number of iterations can be varied without dra-
matically changing the results. When the initial conditions from Arnott et al (1994) are used, and
when the number of commuters in the main lane is correctly anticipated, equilibrium congestion is
reached in fewer than 50 iterations.
C.8 Computing Welfare
Each commuter’s utility depends on their travel time and their arrival time relative to their own ideal
arrival time T
i
. Each commuter’s home departure time, chosen above, is recorded. They are given
a realization of the pre-highway delay, assigned to a lane, wait in the endogenously determined
107
queue, and are given a realization of their random delay on the highway. With these delays, each
commuter’s arrival time can be known, from which is calculated the disutility incurred by each
commuter due to late arrivals, time late or early, travel time, and use of the non-preferred lane.
Since each type of utility is computed for each commuter and available alongside the commuter’s
individual preference parameters, a wide variety of analysis is possible.
For traffic congestion profiles that are in equilibrium, it suffices to compute the utility of each
commuter in the equilibrium profile. If iteration produces cyclic behavior in the congestion profile,
which has been observed, each commuter’s utility can be averaged over the cycle.
C.9 Comparison of Welfare With and Without a Toll Lane
To study the welfare effects of the tolled express lane, a similar simulation is created in which all
lanes are toll-free. By varying road capacity, it is possible to study the effects of adding toll lanes,
either as an addition or a replacement for some of the toll-free lanes. In particular, congestion and
arrival times are computed for the primary scenario of interest, a road with two toll lanes and four
toll-free lanes, analogous to I10-W near Los Angeles, and for two comparison scenarios, both with
no express lanes: a road with four toll-free lanes, showing the benefit of tolled express lanes as an
addition, and one with six toll-free lanes, showing their benefit as a replacement.
In addition, I perform some analysis on subsamples of the population, such as those whose
realized pre-highway delays are large enough to make them experience urgency. I consider the
welfare effects on a commuter of being switched from the choice set of one scenario to another.
Commuters in a subsample take traffic as given: they respond to a counterfactual scenario such
as the six-lane toll-free road by choosing a departure time in response to period t 1 congestion,
whereas their arrival time and welfare are determined by the period t congestion already computed
for that road.
108
D Sensitivity of Welfare Results to Distribution of Parameters
Figures D1 to D6 show the sensitivity of welfare results to several commuter parameters. In all
figures, welfare gains for the average commuter on the main road (four free lanes, two toll lanes)
are shown versus a toll-free road of capacity equal to all lanes of the main road (upper panel) and
a toll-free road of capacity equal to only the toll-free roads (lower panel). Thus, the upper panel of
each figure shows the gains to commuters if the two toll lanes are obtained by conversion, and the
lower panel shows the gains if the two toll lanes are additional to a four-lane road.
The mean value of urgency, the standard deviation of the value of urgency, and the mean value
of time are all varied in 16 increments, with the fifth corresponding to the literature-derived value
used in the main results. For each counterfactual parameter value, commuters’ arrival time and
welfare was simulated for 50 iterations on the main and counterfactual roads. Owing to the com-
putationally intense nature of this analysis and the time constraints faced in obtaining it, the results
shown are for a version of the simulation that employs a technique described above to shorten its
runtime: the number of simulated commuters is divided by 10, with all road capacities divided by
the same amount to produce similar congestion profiles. While this should not significantly alter
results, imperfections in its execution may have unintended effects, which account for differences
between the results shown in the sensitivity analysis and those shown for corresponding parameter
values in the main results.
Figures D1 to D6 belong to this section.
E Extension: Theory of Urgency Lanes versus High-Value of
Time Lanes
In this section, I consider the problem of a policy maker who offers three types of lanes: free
lanes (subscript f ), which have no tolls; urgency lanes (subscript u), which should be used only
by commuters experiencing urgency, and high-value of time lanes (subscript h), to be used by a
109
Figure D1: Sensitivity to Mean Theta
Figure 14: This figure shows that the welfare benefits of tolled express lanes are increasing in the
average commuter’s susceptibility to urgency, which seems to be driven largely by what happens
at lower values. q
i
, commuter i’s penalty for late arrival (in dollars), is normally distributed with
a standard deviation of 1, with negative values replaced with zero. In this figure, its mean is
varied in increments of 0.6, from 0.6 to 9.6. 16 increments are used, with the fifth corresponding
to the literature-derived value employed in the main results. In each case, traffic is computed
for 50 iterations on the road of main interest, which has four free lanes and two toll lanes, as
well as a ”converting” comparison, where the two toll lanes are obtained by converting two out
of six free lanes, and an ”adding” comparison, where the lanes are obtained by adding. That is,
the ”converting” and ”adding” scenarios contain simulated roads with six and four toll-free lanes,
respectively. Results shown here are the welfare gained by the mean commuter in the road with
toll lanes, relative to their welfare in the comparison road. In all cases, tolls at each time of day
are taken from I10-W in Los Angeles. The upper panel depicts the gains to commuters versus
six toll-free lanes, the lower panel versus four. Welfare results have a similar shape in both cases:
the express lanes perform far worse when commuters’ average susceptibility to urgency is low.
Surprisingly, converting free lanes to express appears to be bad for commuters. It’s possible there
is an error in the techniques I employed to speed up the simulation for this sensitivity analysis, and
this will be avoided in future revisions.
110
Figure D2: Sensitivity to Variance Theta
Figure 15: q
i
, commuter i’s penalty for late arrival (in dollars), is normally distributed with a mean
of 3. The standard deviation is varied in 16 increments of 0.2, from 0.2 to 3.2, with the fifth value
corresponding to the literature-derived value employed in the main results. In both panels, the
vertical axis depicts the welfare gained by the average commuter in the main road (four free lanes,
two toll lanes) versus an alternative road. The upper panel
111
Figure D3: Sensitivity to Mean Alpha
Figure 16: a, the value of time, is here assumed to be the same for all commuters. The main
analysis uses a literature-derived value of 60 cents per minute. a is varied in 16 increments of
0.2 times the literature-derived value, with the fifth corresponding to that value. The upper panel
compares the welfare of the average commuter on a road with four toll-free and two toll lanes to
one with six toll-free lanes. The lower panel compares to a road with four toll-free lanes.
112
Figure D4: Sensitivity to Variance Alpha (Uniform distribution)
Figure 17: The mean value of time is 60 cents per minute, corresponding to the literature-derived
value used in the paper’s main analysis. The standard deviation is increased in 12 increments, with
the largest corresponding to the interval from 0 to twice the mean value.
113
Figure D5: Sensitivity to Variance Alpha (Normal distribution)
Figure 18: The mean value of time is 60 cents per minute, corresponding to the literature-derived
value used in the paper’s main analysis. The standard deviation is increased in 12 increments, each
matching the standard deviation of the corresponding increment in the uniform distribution.
114
Figure D6
Figure 19: When both the penalty for late arrival and the value of time are heterogeneous, the
effect of the lateness penalty on express lane use is small, but the effect of the value of time is even
smaller. We see that a high value of time is not the principal driver of which commuters use the
express lane.
115
small number of rich commuters with a high value of time. The policy maker must impose a toll
regime that maintains incentive compatibility, such that all lane types are used, and such that rich
commuters do not use the urgency lane for every trip.
E.1 Model
There are N drivers. For simplicity, all drivers have the same value of urgency q, and experience
random delays of size T
D
with probability p, realized after leaving home but before choosing lanes,
which are the source of urgency. There are two types of drivers: N
p
”poor” drivers, with a value of
timea
p
, and N
r
”rich” drivers, with a value of timea
r
a
p
. Independently of their type, commuters
also have idiosyncratic ideal arrival times t
i
.
There are three lane types: toll-free lanes, with capacity s
f
, high-value of time lanes, with
capacity s
h
, and urgency lanes, with a capacity of s
u
. Tolls are 0 in the toll-free lane;t
u
for the first
use of the urgency lane, and ¯ t for subsequent uses; andt
h
for the high-VOT lane.
Commuter i anticipates their ex-ante best lane and commits to a home departure time t
i
. After
departing home, they experience a delay of size T
D
with probability p. Once they experience the
delay, they have the option of remaining in their intended lane or switching to another lane. As in
this paper’s main model, commuters face a cost function of the form,
C(t)=a
i
T(t)+b(time early)+g(time late)+q(if late)+ toll.
Unlike the main model,a
i
consists of two types, and commuters face two types of toll lanes.
E.2 General Requirements for Incentive Compatibility
Commuters take traffic and tolls as given. Incentive compatibility requires tolls and travel times
that achieve the following:
1. When poor commuters leave home, they will anticipate using the free lane.
116
2. Commuters with a delay will use the urgency lane. This requires them not to have so long a
head start that late arrival is impossible.
3. Tolls on the urgency lane will deter poor commuters who do not experience a delay, but will
not deter poor commuters who do experience a delay.
4. Tolls on the high-VOT lane will deter poor commuters but not rich commuters.
5. Even if rich commuters sometimes use the urgency lane, the toll for repeated use will be
high enough that they do not use it for every trip.
For simplicity, in the following subsections, the travel time in each lane is assumed to be
constant over time: T
f
for all trips in the free lane, T
u
for all trips in the urgency lane, and T
h
for
all trips in the high-VOT lane. This can be achieved by dispersing ideal arrival times enough to
prevent the formation of a queue at peak hours: for instance, ideal arrival times may be uniformly
distributed over the interval[T
f
,T
f
+ N
p
/s
f
], such that the lanes are used exactly to capacity, and
each commuter arrives exactly at their ideal arrival time. This simplification allows the probability
of late arrival to be exogenous. A commuter with ideal arrival time t
i
will enter the highway at
time t
i
= t
i
T
f
and arrive at t
i
, incurring a cost ofa
p
T
f
.
E.3 Incentive Compatibility Requirements to Risk Late Arrival
If there is a risk of delay, commuters have the choice of leaving home before t
i
T
f
T
D
, with no
risk of arriving late, and leaving home after that time, with a probability p of arriving late. If they
choose to risk late arrival, the optimal time to leave home is t
i
T
f
, as leaving any earlier will add
to the predictable early-arrival penalty without reducing the risk of arriving late.
Under what circumstances is the second time preferable?
Expected cost of leaving at the second time is
C
2
=(1 p)a
p
T
f
+ p[a
p
(T
f
+ T
D
)+gT
D
+q],
117
which can be rewritten
C
2
=a
p
T
f
+ p[(a
p
+g)T
D
+q].
By contrast, the cost of leaving at the first time is
C
1
=a
p
T
f
+(1 p)bT
D
We assume that p is small.
Incentive compatibility means the second departure time is, ex ante, less costly, i.e. C
2
C
1
.
Therefore,
p[(a
p
+g)T
D
+q] (1 p)bT
D
.
Takingq as a fixed characteristic of commuters, this imposes constraints on T
D
.
T
D
p
(1 p)b p(a+g)
q.
Alternatively, taking T
D
as a fixed feature of the congestion environmment, we get constraints
onq such that commuters will, ex ante, risk late arrival:
q
1 p
p
b+a
p
+g
T
d
My preferred approach is to take q as given, in which case our results are a constraint on T
D
:
delays must be long enough or infrequent enough that avoiding the risk of exposure is not sufficient
reason to incur the extra time early.
118
E.4 Incentive Compatibility Requirements to Take Free Lane Ex Ante, Urgency
Lane if Urgency is Realized
The random delay T
D
must be at least a certain length, determined by commuter valuations on
travel and arrival time, to make it unprofitable for commuters to leave home so early that they have
no risk of arriving late.
Assume T
D
is long enough that the optimal time to enter the toll-free lane is t
i
T
f
.
I want conditions on the tolls in the urgency lane such that when urgency is incurred, commuters
will use the urgency lane.
In the urgency lane, travel time is T
u
. For this lane to be useful, we require T
u
T
f
T
D
, so
that if a delay is experienced, it is possible for commuters to avoid being late.
Let’s suppose that indeed T
u
T
f
T
D
, and that a commuter leaves home at time t
i
T
f
. In
the case where they are not delayed, they will use the toll-free lane and arrive exactly in time.
When they are delayed, their choice is to remain in the toll-free lane, arrive T
D
minutes late,
and incur a disutility of
C
D f
=a
p
(T
f
+ T
D
)+gT
D
+q
or to use the urgency lane and incur a disutility of
C
Du
=a
p
T
u
+b(T
f
T
u
T
D
)+t
u
,
provided this is their first use of the urgency lane.
From this we derive a constraint ont
u
such that the urgency lane will ever be used:
t
u
a(T
f
+ T
D
T
u
)+b(T
u
+ T
D
T
f
)+gT
D
+q.
119
This gives us an upper bound on toll relative to T
D
, the duration of the random delay. Previously,
we obtained a lower bound on T
D
consistent with commuters sometimes arriving late. Note that
we have so far not ruled out arbitrarily long delays, and that longer delays allow for higher tolls.
E.5 Making the Fast Lane Less Attractive than the Urgency Lane to Commuters
Experiencing Urgency for the First Time
We want poor commuters experiencing urgency not to use the fast lane, but instead the urgency
lane. We will consider two possibilities: first, that the fast lane is faster than the urgency lane, and
second, that it is slower. Later consideration may rule out one of these possibilities.
Case 1: T
h
T
u
.
In this case, poor commuters who experience delays and switch to the fast lane will arrive early.
Their cost will be
C
Dh
=a
p
T
h
+b(T
f
T
h
T
D
)+t
h
.
To make this less attractive to poor commuters than the urgency lane, we require
t
h
t
u
+a
p
(T
u
T
h
)b(T
u
T
h
),
where by assumption, T
u
T
h
0. The difference in tolls must offset both the benefit of spending
less time in traffic and the detriment of arriving longer in advance of the commuter’s ideal time.
Case 2a: T
u
T
h
T
f
T
D
In this case, commuters who use the high-VOT lane when delayed still arrive early. The in-
equality above still holds, though now T
u
T
h
0.
Case 2b: T
u
T
f
T
D
T
h
In this case, commuters who use the high-VOT lane when delayed will still arrive late. This
lane now competes with the free lane. Commuters who use the fast lane will arrive less late (saving
120
g(T
f
T
h
)), and spend less time on the road (saving a
p
(T
f
T
h
)). To make the high-VOT lane
less attractive to poor commuters, we require
t
h
(a
g
)(T
f
T
h
).
E.6 Intended Behavior of Rich Commuters
We want the rich commuters to use the high-VOT lane.
I can imagine two cases: in one case, the high-VOT lane is faster than the urgent lane, and rich
commuters will always use it. In the other case, the urgent lane is faster, and rich commuters will
use it for their first trip, but the high toll on repeated use will deter them.
Consider the first case first: T
h
T
u
.
In this case, rich commuters who are not delayed can take the urgency lane at time t
i
T
u
,
arriving exactly on time with a disutility ofa
r
T
u
+t
u
; or they can take the high-VOT lane at time
t
i
T
h
, arriving exactly on time with a disutility ofa
r
T
h
t
h
.
To make the high-VOT lane attractive to rich commuters, we must impose an upper bound on
t
h
,
t
h
t
u
+a
r
(T
u
T
h
),
which resembles the lower bound needed to deter poor commuters, but differs in the value of
time, and in the fact that rich commuters at this juncture are free to choose their home departure
time.
Rich commuters must also prefer the high-VOT lane over the free lane, which gives us another
upper bound,
t
h
a
r
(T
f
T
h
).
121
If the high-VOT lane is the fastest option, and if the time saved justifies the tolls for a rich
commuter, then rich commuters will use the fast lane exclusively.
Case 2: T
h
T
u
In this case, high-VOT commuters will use the urgency lane for their first trip. They will not
be deterred by any tolls that will not deter poor commuters.
To deter subsequent use, the toll ¯ t for reusing the urgency lane must be high enough to deter
its use ex ante by rich commuters:
¯ tt
u
+a
r
(T
h
T
u
).
Since the travel time of rich commuters is particularly valuable, a social planner who can
choose travel times would prefer the first case over the second.
E.7 Social Optimality
Here I consider the circumstances under which it is socially optimal to offer all three lane types
described above. A social planner aims to minimize aggregate social cost. Utility is quasilinear
in money, and all toll revenue is rebated to commuters as a lump-sum, neutralizing its effect on
social welfare. Thus, the social welfare effects of toll policies, including the effects of offering
different lanes with different toll policies, depend only on the resulting travel time and arrival time
of commuters. Intuitively, it is optimal to offer a product for each type of commuter, and to allocate
to each product sufficient lane capacity to meet the incentive compatibility constraints.
In contrast to my main model, which uses bottleneck congestion, in this case, I find a different
congestion technology more practical. It simplifies analysis to consider a steady-state relationship
between traffic volume and travel time; however, this does not exist in the bottleneck model, in
which travel time depends on the cumulative number of departures since the road was last free
of congestion. Henderson (1974) provides a framework for considering other congestion tech-
nologies, and Kreindler (2020) finds that a linear model provides a good fit to the congestion he
122
observes in Bangalore. Following Kreindler, I assume travel time in lane L is given by T
L
= Q
L
/s
L
,
where Q
L
is the number of commuters using lane L, and s
L
is the lane’s capacity.
Lane demand Q
h
, Q
u
, and Q
f
are determined by the number of rich commuters, poor com-
muters experiencing urgency, and poor commuters not experiencing urgency, respectively. The
planner can choose s
h
, s
u
, and s
f
such that T
h
T
u
T
f
, s
h
+ s
u
+ s
f
= 1, and the incentive compati-
bility constraints are satisfied.
F Exploration of Alternative Toll Profiles
I have not yet identified a toll profile which is provably optimal for the model discussed here.
However, the simulation discussed here can simulate the effects of any hypothetical toll regime,
optimal or otherwise, helping policymakers to determine whether a proposed alteration to the toll
structure would improve commuter welfare or toll revenue.
I suspect the limited efficacy of the toll lanes in preventing late arrival is because the toll policy
simulated does not respond to the simulated traffic, and because the toll policy fails to induce
an equilibrium pattern of congestion. In the results shown here, I use a fixed toll profile, based
on that observed on I10-W. The fixed profile is dictated by data availability: historical prices
on toll roads are publicly available, but the algorithm by which tolls are updated is confidential.
Commuters’ departure time profile, however, varies across iterations. This also complicates the
commuters’ choice problem, as their belief that today’s traffic will follow the same pattern as
yesterday’s is often incorrect. Future work will continue to explore means by which the toll profile
can be optimized in response to observed congestion, to guide it towards an equilibrium profile in
which the express lane is used, but suffers no endogenous delays.
For purposes of exploration, I simulated the model described above, with the change that the
toll profile just offsets the incentive to queue that commuters would experience in the no-urgency
model (see Arnott et al 1994 for a description of the relevant queue). By the theoretical results
of Arnott et al, this toll should be ideal for the no-urgency model. To my surprise, I found my
123
simulated commuters crowding into the express lane despite not benefiting from it. In the case of
commuters sensitive to urgency, this results in a welfare loss of $146 per commuter when lanes
were added, and $202 when converted. For commuters insensitive to urgency, the numbers were
$126 and $201, respectively. It may seem impossible for rational commuters to be harmed by
the addition of an option, especially since their welfare does not include the cost of constructing
additional lanes. However, these commuters look backward, not forward, when anticipating traffic.
Commuters choose their lane and departure time each day in response to yesterday’s traffic. This
process, as we have seen, does not usually converge to equilibrium, and thus its results may be
very different from equilibrium results. And since commuters are mistaken about the choice set
they face, they can fail to maximize their own utility.
An ideal toll regime would be such that the express lane is used to its capacity but no more,
with no congestion, for the entire period during which it is in use. This period would be chosen
to balance the disutility of commuters induced to arrive late with those induced to arrive early.
Lengthy exploration of this problem has shown that it is very difficult to achieve: offsetting the
theoretical incentive to queue in the equilibrium described by Arnott et al (1994) does not achieve
this, nor do other intuitively appealing ideas. For instance, ex ante, the number of commuters who
can use the express lane is equal to its capacity s times the number of minutes for which it is in use,
T . It seems intuitively appealing to identify each commuter’s express lane advantage at the time
they use the lane, identify the sT
th
highest, and impose a toll to offset that. When this is attempted,
though, congestion remains in the simulated express lane.
In future work, I intend to explore optimal time-varying tolls in the presence of urgency. The
problem has been solved in the absence of urgency, but my work has shown that urgency makes
the problem of determining the ideal toll regime more difficult.
124
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Asset Metadata
Creator
Roitberg, Tal
(author)
Core Title
Three essays on strategic commuters with late arrival penalties and toll lanes
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Economics
Degree Conferral Date
2022-08
Publication Date
07/20/2022
Defense Date
06/16/2022
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
bottleneck,OAI-PMH Harvest,transportation,urban economics,urgency
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application/pdf
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Language
English
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Electronically uploaded by the author
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Advisor
Bento, Antonio (
committee chair
), Coricelli, Giorgio (
committee member
), Kapteyn, Arie (
committee member
), Moore, James (
committee member
), Oliva, Paulina (
committee member
)
Creator Email
roitberg@usc.edu,troitberg@gmail.com
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Tags
bottleneck
transportation
urban economics
urgency