Step-tolling with price-sensitive demand:
Why more steps in the toll make the consumer better off
Version of 12 July 2012
Vincent A.C. van den Berga,*,#
a: Department of Spatial Economics, VU University, De Boelelaan 1105, 1081HV Amsterdam, The Netherlands
*: Corresponding author: email: [email protected], tel: +31 20 598 6049
#
: Affiliated to the Tinbergen Institute, Roetersstraat 31, 1018 WB Amsterdam.
Abstract
Most dynamic models of congestion pricing use fully time-variant tolls. However, in
practice, tolls are uniform over the day, or at most have just a few steps. Such uniform and
step tolls have received surprisingly little attention from the literature. Moreover, most models
that do study them assume that demand is insensitive to the price. This seems an empirically
questionable assumption that, as this paper finds, strongly affects the implications of step
tolling for the consumer. In the bottleneck model, first-best tolling has no effect on the
generalised price, and thus consumer surplus remains the same as without tolling. Conversely,
under price-sensitive demand, step tolling increases the price, making the consumer worse off.
The more steps the toll has, the closer it approximates the first-best toll, thereby increasing the
welfare gain and making consumers better off. This indicates the importance for real-world
tolls to have as many steps as possible: this not only raises welfare, but may also increase the
political acceptability of the scheme by making consumers better off.
Key words: Congestion pricing, step tolls, bottleneck model, price-sensitive demand, consumer surplus, political acceptability
JEL codes: D62, R41, R48
†This is a post-print version of the paper published in Transport Research part A, in press,
see
http://dx.doi.org/10.1016/j.tra.2012.07.007 and http://www.journals.elsevier.com/transportation-research-part-a-policy-andpractice/
1. Introduction
Theoretical models of dynamic congestion pricing generally use a fully time-variant toll.
However, in practice, there are no such tolls. In practice, tolls are constant over the day, or at
most have just a few steps in them. For example, the Oslo toll ring has a uniform toll that is
constant over the day (Odeck and Bråthen, 1997), and the London scheme has a uniform toll
that is constant between 7:00 and 18:00.1 In contrast, Singapore uses step tolls: the toll is at its
lowest level in the early morning, and increases in steps up to its highest level in the middle of
the morning peak; thereafter, it decreases again in steps (see Fig. 1 in Section 2 for an
example of a step toll). For the evening peak a similar pattern holds, but this paper will ignore
the evening peak. At the “Bugis-Marina Centre (Nicoll Highway)” in Singapore there are
seven steps in the toll during the weekday morning peak.2 The Stockholm pricing scheme has
five steps in the morning.3 But step tolls are also used in the USA: for example, on the SR91
and San Francisco-Oakland Bay Bridge in California, and the SR520 and SR16 Tacoma
Narrows bridges in Washington State.4 Such uniform and step tolls have received surprisingly
little attention in the literature. Moreover, models of step tolls generally assume that demand
is fixed and thus insensitive to price.5,6 This seems an implausible assumption, as empirical
research shows that transport demand varies with the generalised price (or price for brevity).
For a review of price elasticities, see, for example, Brons et al. (2002) and Graham and
Glaiser (2004).
In the bottleneck model, first-best pricing changes the departure rate of drivers (i.e. it
changes behaviour), thereby halving marginal social cost and generalised user cost (hereafter
referred to as user-cost) for a given number of users. For the social optimum, marginal social
cost should equal demand. Due to the halving of marginal social cost, this occurs when the
number of users in the first-best equilibrium is the same as in the no-toll equilibrium.
Consequently, the price and consumer surplus are unchanged by the tolling. A uniform toll is
constant throughout the peak, and causes no change in the departure rate in the bottleneck
model. It can only limit congestion cost by reducing demand. The optimal uniform toll equals
the marginal external cost (i.e. marginal social cost minus user-cost) when the queue is not
eliminated (Arnott et al., 1993). Uniform tolling hence raises the price, and lowers the number
of users and consumer surplus. Accordingly, this scheme is comparable to tolling in the
textbook static-congestion model, where tolling lowers consumer surplus, and has a
substantially lower gain than the first-best bottleneck toll.
Step tolling is in between uniform and first-best tolling: it somewhat changes the departure
pattern, but also raises the price. This makes it important to control for price sensitivity of
1 This follows www.tfl.gov.uk/tfl/roadusers/congestioncharge/whereandwhen/ as retrieved on 13 June 2012.
2 Rates for 2 June to 2012 to 5 August 2012, as retrieved on 13 June 2012 from
www.onemotoring.com.sg/publish/onemotoring/en/on_the_roads/ERP_Rates.html
3 http://en.wikipedia.org/wiki/Stockholm_congestion_tax as retrieved on 13 June 2012.
4 Respectively www.octa.net/91_schedules.aspx, bata.mtc.ca.gov/tolls/schedule.htm, and www.wsdot.wa.gov/Tolling/TollRates.htm as
retrieved on 13 June 2012.
5 Interesting exceptions are Arnott et al. (1993), Chu (1999), de Palma et al. (2004, 2005), Lindsey (2004) and Stewart and Ge (2010a). Here,
the latter five have fixed overall demand. Nevertheless, Chu (1999) has a logit distribution of users over driving alone, carpool and bus; de
Palma et al. 2004, 2005) and Lindsey (2004) have an untolled and uncongestible alternative; and in Ge and Stewart (2010a,b) only two of
the three routes are tolled.This makes the number of tolled users dependent on the tolls in all five of these studies.
6 Still, although demand is not fixed, it is often rather inelastic. This is especially true when only a single link is priced and there is an
unpriced alternative (as is the case with the HOT express-lanes in the USA) or when a single road is widened, since then it also attracts
users from alternative routes and modes.
1
demand when considering step tolling. As this paper finds, in the bottleneck model, the more
steps there are, the lower the marginal social cost and price, and the higher consumer surplus.
As the number of steps goes to infinity, the step toll generally approaches the first-best toll,
and consumer surplus approaches that without tolling.
It may not optimal, or even possible, to use an infinite number of, or very many, steps: in
practice, this might be too costly to implement or too difficult for the users to understand.
Moreover, users might be insensitive to the very small changes in toll that occur with so many
steps. However, experience of, for example, the schemes on the SR91 in California and in
Singapore show that users can handle a large number of steps, and in the bottleneck model
such schemes already approach the first-best toll very closely. Hence, the practical policy
advice from this paper is that it seems better to give a system a substantial number of steps (as
is the case in Singapore or on the SR91) than to have a uniform toll or only a few steps (e.g.
London and at the Bay Bridge): more steps not only typically raises welfare, but may also
increase the acceptability of congestion pricing by making it less harmful for the consumer.
This paper investigates step tolling in three different models that use bottleneck
congestion: first, the ADL model following Arnott et al. (1990, 1993); second, the Laih model
of Laih (1994, 2004); and, third, the Braking model of Lindsey et al. (2012). In the Laih
model, an m-step toll lowers total cost by a fraction ½∙m/(1+m). Consequently, with a single
step, total cost is reduced by a quarter (or half the gain of the first-best toll); with two steps,
the reduction is a third; and, as the number of step goes to infinity, the toll approaches the
first-best toll (Laih, 2004). See also Fosgerau (2011) on the Laih toll under general scheduling
preferences. In the ADL model, the gain is larger for a finite number of steps, while the toll
also approaches the first-best toll as m goes to infinity. The Braking model takes into account
that drivers have an incentive to wait to pass the tolling point until just after the toll is
lowered: this lowers the toll they pay while only marginally increasing travel time and
schedule delay. A consequence of this is that the bottleneck capacity will go unused for some
time during the peak, and this inefficiency raises total cost. The inefficiency only increases
with the number of steps, and thus the Braking toll never approaches the first-best toll, and
always has a lower gain. Furthermore, the other two models are only stable if the government
can prevent the braking (Lindsey et al., 2012).7
The paper is structured as follows. Next, Section 2 presents a general model of step tolling
for any model of dynamic congestion. Then, Section 3 turns to the bottleneck model, and
discusses the equilibria without tolling, with first-best tolling, and with step tolling. Section 4
provides a numerical example, and Section 5 carries out sensitivity analyses. Section 6
discusses some limitations of the research and makes suggestions for future research. Finally,
Section 7 concludes.
2. The general step-toll model
This section derives the optimal level of the time-invariant part of the step toll for a
congestion model and a certain form of the step part. The solution assumes that there is a
7
This braking behaviour has been observed in Singapore (Png et al., 1994; Chew, 2008), at the San Francisco-Oakland Bay Bridge (Lee and
Frick, 2011), and in Stockholm (Fosgerau, 2011).
2
formula for the time-variant part, and is based on the results of Arnott et al. (1993) on a
uniform toll and a single-step toll. Table 1 explain the symbols used in this paper.
Table 1: Variables used and their symbols
Symbol
t
θ
ρi
[t]
cSD
cTD
c
P
t*
α
β
γ
δ
ti
ti
ts
te
D
N
TC
W
Description
Arrival time
Time-invariant part of the toll
The ith step part of the toll (which level depends on t):
Total toll at t
Schedule delay cost
Travel delay cost
(Generalised) cost: c≡cSD+cTD
(Generalised price: P≡+c
Preferred arrival time
Value of time (i.e. monetary value of an hour of travel time)
Value of schedule delay early
Value of schedule delay late
Preference variable used for short hand δ≡(β∙γ)/(β+γ).
Start of the ith early toll (before t*)
End of the ith late toll (after t*)
Time the peak starts
Time the peak ends
Inverse demand
Number of users (which is endogenous due to the price-sensitive demand )
Total cost: TC≡c∙N
Welfare (which equals consumer surplus minus total cost)
Fig. 1: General set-up for multi-step tolls
The toll τ for an arrival time t consists of the time-invariant part θ and a time-variant step
part ρi (where the level of the step part depends on t and i indicates the ith toll level):
 [t ]  i   .
(1)
As Fig. 1 shows, the central toll (ρ1) is centred on the preferred arrival time (t*) and is
indicated by 1. The further out a step part is, the higher its indicator, and the lower its level.
The levels of the ith early (before t*) and the ith late toll (after t*) are allowed to differ, but the
number of early and late tolls are the same. An early step part is indicated by the superscript +,
and a late step part by −. The ith early part starts at ti and ends at ti1 ; the times for the ith late
3
part are, respectively, ti1 and ti . With step tolling, the peak starts at ts' and ends at te' . These
times generally differ from those without tolling (ts and te), since the number of users differs.
Welfare equals the integral of (inverse) demand D minus total cost (TC), which equals
average user-cost ( E[c] ) multiplied by the number of users (N):
N
N
W   D[n]dn  TC   D[n]dn  E[c]  N .
0
(2)
0
To find the optimal time-invariant part (θ) for a given pattern of the step toll, the following
Lagrangian is maximised:
L  W    D  E[c]  E[  ]      D[n]dn  E[c]  N    D  E[c]  E[  ]    ,
N
(3)
0
where E[∙] indicates an average.8 The expectance operator is used because the user-cost and
the step toll vary over time. The first-order conditions are
L / N  0  D  E[c]  N E[c]/ N    D / N  E[c]/ N  E[  ]/ N  ,
(4) (4a)
L /   0   ,
(4b)
L /   0  D  E[c]  E[  ]   .
(4c)
These equations imply that the optimal time-invariant part of the toll equals
  MSC  E[c]  E[  ]  E[ MEC ]  E[  ]  N  E[c] / N  E[  ].
(5)
Here, the MSC is marginal social cost, which is the derivative of total cost w.r.t. the number
of users. The MEC is marginal external cost, which at t equals the difference between MSC[t]
and user-cost c[t]. Just as Arnott et al. (1993) showed for a single-step toll, in general the θ is
set such that the price equals the average MSC (or alternatively the average toll equals the
average marginal externality), and, accordingly, the average user internalise the external cost.
Since a uniform toll is a zero-step toll, the above discussion implies that this toll should
equal the average externality. Step tolling changes the equilibrium departure pattern and
lowers the externality and price for a given number of users. Nevertheless, the average step
toll, E[], again equals the average marginal externality.
3. The bottleneck model
3.1. No-Toll (NT) equilibrium
The discussion of the no-toll (NT) and first-best (FB) equilibria is kept brief since these are
extensively discussed in, for example, Arnott et al. (1990, 1993), as well as in textbooks such
8
This solution assumes that the dynamic congestion model has a reduced form that only depends on the total number of users: hence, just as
in the bottleneck model, the costs and step part of the toll at time t can be expressed as a function of the total number of users. Furthermore,
the system is in user equilibrium, and hence the price is constant throughout the peak. This allows the use of Lagrangian optimisation
instead of optimal control theory, which is difficult to use for the bottleneck model (see Yang and Huang, 1997). Because of the price is
constant over time, only a single constraint is needed that states that the average price (E[c]+E[ρ]+θ) equals inverse demand.
4
as Small and Verhoef (2007). The N identical users travel alone by car from the origin to the
destination, which are connected by a road that is subject to bottleneck queuing congestion.
Free-flow travel time is normalised to zero. Without a queue, a user thus departs from the
origin, passes the bottleneck, and arrives at the destination simultaneously. User-cost for an
arrival at t is the sum of travel-delay (cTD) and schedule-delay costs (cSD) incurred from
arriving at a different time than the common preferred arrival time (t*):
c[t ]  cTD  c SD    T [t ]    Max 0, t *  t     Max 0, t  t *  .
(6)
The α is the value of travel time or the cost of an hour of travel delay, β is the value of
schedule delay early (i.e. the cost of arriving an hour before t*), and γ is the value of schedule
delay late.
The peak starts at ts and ends at te; at these moments travel delay is zero while schedule
delay cost is at its highest:

N
,
  s

N
te  t * 
 .
  s
ts  t * 

(7)(7a)
(7b)
In these equations, s is the capacity of the bottleneck. Equilibrium user-cost is given in (8).
Since the toll is zero, the generalised price PNT equals cNT. Here, superscript NT indicates the
No-toll equilibrium. Total cost equals c NT∙N; half of this total is travel delay cost, while the
other half is schedule delay cost. Marginal social cost is twice the user-cost, which makes
marginal external cost (MEC) in (9) equal the user-cost. The preference parameter δ is used to
shorten the algebra and equals (β∙γ)/(β+γ). Hence, we have the following two equations:
c NT  p NT    N / s;
MEC NT    N / s.
(8)
(9)
In user-equilibrium, demand equals user-cost. But at this point the MSC is above the
equilibrium price, because part of the marginal social cost is external to the user.
3.2. First-Best (FB) equilibrium
Travel delays are a pure deadweight loss: drivers could arrive at the same arrival times
(hence having the same schedule delays), but with zero travel delays, if their rate of arrival at
the bottleneck would equal capacity. Moreover, as ts and te are unchanged, the equilibrium
price remains equal to −β∙ts=δ∙N/s. The first-best toll, τFB[t], that achieves this equals, at each
arrival time (t), the travel-delay cost at t in the NT case:
 FB [t ]    N / s    Max 0, t *  t     Max 0, t  t *  .
This FB toll halves total cost, average Marginal External Cost (E[MEC]), and average user5
cost (E[c]), since all travel delays are converted into toll payments; while the price is
unaffected:
E[cFB]=½ δ ∙N/s,
PFB=δ ∙N/s,
E[MECFB]= ½ δ ∙N/s.
(10)
(11)
(12)
Since the price is unaffected, the total number of users and consumer surplus remain the same.
Welfare increases, since half of the total cost (of E[cFB]∙N) is converted into toll revenue. At
each point in time, the toll equals the MEC, and thus the externality is fully internalised.
3.3 Uniform toll
The uniform toll does not affect the departure rate. It can only limit the congestion cost by
reducing demand. Hence, the formulas for marginal social cost and user-cost remain the same
as in the NT case. To ensure that marginal social cost equals demand, the number of users has
to be reduced from NNT to NU (superscript U indicates the Uniform toll). This is done by
setting a time-invariant toll equal to MEC=MSC−c, as is done in the textbook static model.
Moreover, also in accordance with the textbook model, uniform tolling lowers the number of
users and consumer surplus.
Toll
8
Uniform Toll
6
4
2
FB
t
2.0
1.5
1.0
0.5
0.0
0.5
Fig. 2: Level of the uniform toll
As Fig. 2 shows, the uniform toll is generally higher than the average first-best toll because
the marginal external cost is higher for a given number of users (this figure is based on the
calibration of the numerical example). The MEC in the uniform case is δ∙NU/s. Hence, the
MEC is higher with a uniform toll if the NFB is not more than twice the NU. Uniform tolling
raises the price if demand is not perfectly elastic (i.e. a flat curve D); and, hence, uniform
tolling lowers consumer surplus, even though it does raise welfare.
3.4. General aspects of step tolling
Of the three step-toll models for the bottleneck model, the Laih (1994; 2004) model is the
simplest to solve; the Braking and ADL models are more tedious. The three models differ in
6
how they achieve user-equilibrium. Before t*, when the toll only increases, the three models
have the same set-up. However, after t* they differ in how they equalise prices before and
after a toll decrease. In the Laih model, there are separate queues for drivers who pass the
tolling point before and after a toll decrease at ti , where the users who will arrive after ti
start waiting in front of the tolling point before ti . In the ADL model, there are no separate
queues. Instead a mass departure at ti equalises expected prices before and after ti . In the
Braking model, there is a single queue and no mass departure. Instead, users start waiting to
pass the tolling point well before the toll is lowered, since this reduces the toll they pay. It is
this waiting that means that the bottleneck capacity goes unused for some time during the
peak, and this inefficiency raises costs.
3.5. The Laih step toll
The Laih model is easiest to solve because the step part of the toll does not alter the arrival
window and price. If the number of users were fixed, at each t, the sum of the user-cost and
the step part of the toll would equal the price without tolling. However, because the optimal
time-invariant part is positive (as we will see below), Laih tolling increases the price and
lowers the number of users.
Following Laih (1994, 2004), the start and end time of the peak follow the same formulas
as for the NT and FB equilibria in (7a,b). Nevertheless, total cost is lower for a given N,
because part of the travel-time costs are converted into toll payments. Total cost and total toll
revenue can be shown to equal
N2  1 m 
1 
,
s  2 1 m 
1 m
N2
TR  TR step  TR fixed 

   N.
2 1 m s
TC  
(13)
(14)
In the optimum, the step parts of the toll are symmetric in the Laih model: i.e. the ith early
and late toll are equal, and i  i  i (Lindsey et al., 2012). The step part of the toll follows
m
N
  ,
m 1
s
m 1 i
i  i  i 
 1 , i  2,..., m.
m
1 
(15)
The step part has a simple pattern: the second toll ρ2 is a fraction (m−1)/m of the central ρ1,
and the third toll a fraction (m−2)/m. This pattern is this same as with a fixed number of users,
since this minimises total cost for a given N. Conversely, the θ is set to optimise N, such that
demand equals the MSC for a given cost structure.
The average step part of the toll, E[ρ], equals TRstep/N. Average marginal external cost is
E[ MEC ]  
N
s
m 

1 
.
2  2m 

(16)
7
This externality decreases with the number of steps. For a given N, a single step toll reduces
the average MEC by a quarter; with 2 steps, this is a third; and as m goes to infinity, the MEC
approaches the first-best MEC. Using the conclusion from Section 2 that the average step toll
should equal the average MEC, the optimal time-invariant part of the toll has to equal
  E[ MEC ]  E[  ]  
N m
.
s 1 m
(17)
This θ approaches zero as the number of steps m approaches infinity, because then the step
part of the toll approaches the first-best toll that at each t equals the MEC.
The price at the start of the peak at ts' is the sum of scheduling cost and the time-invariant
part of the toll; the time-variant toll and travel time are zero. In user-equilibrium, the price at
other used arrival times has to be the same, but travel time and toll are generally non-zero.
Accordingly, the price at all used arrival times is
P  MSC  E[c]  E[  ]      N   .
s
(18)
Fig. 3 compares the single-step Laih toll with the FB and the uniform toll and is based on
the numerical example of Section 4. Step tolling tilts down the cost curves, which means that
the price increases less, and the average toll is lower than with uniform tolling. For a finite
number of steps, the price with Laih tolling will be higher than with FB tolling. Nevertheless,
the average step-toll is lower than the uniform toll, and the price is also lower.
Toll
8
Uniform Toll
6
4
Step Toll
2
FB
t
2.0
1.5
1.0
0.5
0.0
0.5
Fig. 3: A single-step toll
3.6. The ADL step toll
With the ADL toll of Arnott et al. (1990, 1993) and fixed demand, step tolling lowers the
price, and shifts the peak to later (i.e. the start and the end times of the peak are later). Each
time the toll drops a level there is a mass departure of users. If α<γ, then, just after the last
user of the ith mass arrives at ti , there is the mass departure of the users who pay the i−1th
8
toll.9 Without the shift of the peak, the price for a user in a mass departure would be lower
than for a user who travels during the rest of the peak. By having more drivers in the masses
and fewer drivers outside, expected prices are made constant over time; and it is this that
shifts the peak to later, and lowers the equilibrium price.
Lindsey et al. (2012) find that generalising the ADL model to m steps is harder than it is
for the other models. Already for two steps, the formulas are very complex. Therefore, the
analytical discussion in the Appendix will focus on single- and two-step tolls; the numerical
example goes up to ten steps. The ADL toll has, for a finite m, a larger welfare gain than the
Laih toll, but also approaches the FB toll as m goes to infinity. Due to the shift of the peak and
mass departures, the ADL toll is asymmetric, with the ith late toll being higher than the ith
early toll.
Following Lindsey et al. (2012), the early step parts of the toll follow the same formula as
in the Laih model (see (15)):
 i 
m 1 i
 1 ,
m
i  1,..., m.
(19)
Conversely, the late tolls are not simple fractions of the central toll, ρ1:
i   i  2 3  i  3 2 ,
i  2,..., m.
(20)
It is due to this more complex formula that there are no simple solutions for the ADL toll. To
give at least some insight, the Appendix provides the analytics for one and two steps.
3.7. The Braking step toll
The ADL and Laih models overlook that drivers have an incentive to delay reaching the
tolling point when the toll is about to drop if the waiting cost they incur is less than by the
money they save. The Braking model of Lindsey et al. (2012) takes this incentive into account
(see their paper for a detailed discussion of the model with fixed demand). Users stop passing
the tolling point a time ∆ti before the ith level decreases to the i−1th level at ti . The first
users to pay the i−1th level arrive just after ti . The last users to pay the ith toll arrive at
tib  ti  ti . For the prices at these two arrival times to be equal, the ∆ti has to equal
 i  i1  /     .
The total time the bottleneck is idle,10 t , is the sum of all the ti . It
depends only on the level of ρ1 and the preference parameters α and γ:
t  1 /(   ) .
(21)
The step part of the toll follows the same formula as in the Laih model, but the levels are
generally different, as the number of users differs:
Lindsey et al. (2010) and Daniel (2009) show that, with α>γ, there are normal departures after the ith mass that still pay the ith toll. This
then ensures that there is no shift in the peak; and, therefore, the price and toll formulas are the same as in the Laih model, and hence the
ADL model simplifies to the Laih model. The focus of this paper is on α<γ, as this seems more likely for car travel.
10
The bottleneck is ‘idle’ when users stand still in front of the tolling point in order to prevent paying the toll.
9
9
1 
m
N
  ,
m 1
s
i  i  i 
m 1 i
 1 , i  2,..., m.
m
(22)
The idle time t is an inefficiency and pure deadweight loss that raises costs and makes
step tolling more harmful for the user. The idle time does not disappear as m becomes larger.
Actually, it only increases with m, since ρ1 increases with m, and t is an increasing function
of ρ1. This implies that the Braking toll does not approach the first-best: even for an infinite m,
its gain will be lower. The formulas for total cost and toll revenue are more complex in the
Braking model than in the Laih model, since they contain the fraction δ/(α+γ):
N2 
m 1
 
1 
1 
,
s  1 m 2      
N2 m 1 

step
fixed
TR  TR  TR

1 
s 1 m 2    
TC  
(23)

    N.

(24)
From (23) the average MEC can be derived, and it turns out to be similar to that in the Laih
model except for the addition of the δ/(α+γ) term:
E[ MEC ]  
N
m 1

1 
1 
s  1 m 2    

 .

(25)
For a given N, the average MEC in the Braking model is higher than in the Laih model due
to the extra costs caused by the time that the bottleneck is idle. Interestingly, since the δ/(α+γ)
term is in the user-cost and in the average step toll (E[ρ]=TRstep/N), the fixed part of the toll, θ,
simplifies to the same formula as in the Laih model:
  E[ MEC ]  E[  ]  
N 1
s 1 m
(26)
For a given N, the time-invariant toll is thus the same in the Laih and the Braking model,
while the step-part is higher with the Braking model since it ensures that the average user
internalises the extra marginal external cost due to the braking. Braking tolling is more
harmful for the consumer than Laih tolling for two reasons: 1) the time the bottleneck is idle
raises costs, and 2) the average toll is higher. This makes preventing braking even more
important with price-sensitive demand than with fixed demand, where only the first effect
occurs.
The model assumes that there is no direct cost to the user to braking. This seems
unrealistic: standing still on a road can be very dangerous, which means that there are costs
from the increased risk of an accident. Furthermore, this standing still is likely to be a traffic
violation, meaning that there is also the risk of a fine. With such costs of braking, introducing
more steps in the toll might solve the braking problem: the toll saving becomes ever smaller,
while the extra cost from the risk of an accident and fine remain. Limiting braking was one of
10
the reasons why Singapore introduced extra steps in 2003 (Chew, 2008). It also seems
important for the government to actively control for cars standing (needlessly) still, and fine
those that do, since this behaviour is not only dangerous for both the driver and other drivers
(i.e. it imposes an accident externality), but also reduces the gain from step tolling.
4. Numerical example
This section illustrates the effects of step tolling with price-sensitive demand using a
numerical example. The section looks at tolls with one to ten steps. The following preference
parameters are used: the unit cost of an hour of travel delay (i.e. the value of time) is α=8, the
value of schedule delay early is β=4, and the value of schedule delay late is γ=15.6. The
bottleneck capacity is s=3600 cars an hour. The no-toll equilibrium has 9000 drivers, and,
accordingly, the peak lasts 2.5 hours. The inverse demand follows a linear function, with the
elasticity with respect to generalised price being equal to −0.4 in the NT equilibrium.
Fig. 4 shows the tolls with five steps. The ADL toll is the solid (blue) line, the (red) striped
curve is the Braking toll, the (green) dot-dashed curve the Laih toll, and the (black) dotted
curve is the FB toll. The five-step toll is on average lower than a single-step toll: in Fig. 4, the
five-step tolls hug the first-best toll; in Fig. 3, the single-step toll is substantially higher than
the FB toll. With price-sensitive demand, the toll at the start and end of the peak equals the
time-invariant term, and is well above the FB toll. The peak lasts longer with the Braking toll
than with the other step tolls due to the time the bottleneck is idle, even though it has the
lowest number of users. The ADL and Laih peak both have a shorter duration than the FB
peak, since these tolls lower the number of users. Before t* the ADL and the Laih toll are very
similar in their levels, after t* the ADL toll tends to be higher than the Laih toll.
Fig. 4: Five-step toll for the ADL, Laih and Braking models
Fig. 5 compares the prices in the three regimes for different number of steps. Fig. 6 looks
at the relative efficiencies (i.e. welfare gain from the NT case relative to the FB gain). The
11
more steps there are, the better off the consumer: the price is lower, while consumer surplus
and the number of users are higher. A uniform toll or a step toll with few steps is a crude
instrument to reduce the congestion problem; such tolls (primarily) equate the private price
with marginal social cost by lowering the number of users. The fully-time-variant FB toll
equalises MSC and the private price by halving the MSC, while keeping the number of users
the same. The more steps a step toll has, the closer it approximates the FB toll, and the more it
alters the departure pattern, shortens total travel delay, and lowers the MSC.
The price development over the number of steps differs strongly between fixed and pricesensitive demand. With price sensitivity, the price decreases with m, since the time-invariant
toll becomes lower. With fixed demand, there is no time-invariant toll (or, more precisely, it is
undefined, and therefore arbitrarily set to zero), and the price is independent of m in the Laih
model and increases with m in the ADL and Braking models (Lindsey et al., 2012).
P P
12
11
10
Braking
ADL
Laih
9
m
8
0
2
4
6
8
10
Fig. 5: Generalised price as a function of the number of steps
1.0
ω
0.9
ADL
Laih
0.8
Braking
0.7
0.6
0.5
m
0
2
4
6
8
10
Fig. 6: Relative efficiency as a function of the number of steps
5. Sensitivity analysis
When doing (numerical) research it is important to test how sensitive the results are to the
parameter values. This section first looks at the effects of the preference parameters (α, β and
γ), and then turns to the effects of the price sensitivity which are the focus of the paper.
When changing the preference parameters, the example is recalibrated such that the
number of users and price elasticity in the NT equilibrium remain the same. Just as with fixed
demand in Lindsey et al. (2012), changing all three parameters in fixed proportions has no
effect on the relative efficiency, since the NT, FB and step-toll equilibria are proportionally
affected. Moreover, this type of change also has no effect on the percentage changes in
consumer surplus.
12
Therefore, here the focus is on the effects of the relative sizes of the parameters, and in
particular on the relative sizes of the scheduling parameters with respect to the value of time.
These relative changes have no effect on the relative efficiency (Ω) and percentage change in
consumer surplus (%∆CS) of the Laih toll, which is in accordance with Lindsey et al. (2012).
But they do affect the Braking and ADL tolls; see, respectively, Figs. 7 and 8 for the effects
on a five-step toll.
The Braking toll has a lower relative efficiency and larger decrease in consumer surplus
when β/α is higher, since this makes the extra schedule delays due to the braking more costly.
For γ>α, a higher γ/α leads to a lower relative efficiency and higher consumer surplus loss,
because this again makes the braking more costly. When α<γ, the effects are non-monotonic.
This is again similar to Lindsey et al. (2012).11
The gain from ADL tolling is always above that of the Laih toll, and consumer surplus is
also higher. With a higher β/α, the relative efficiency and percentage consumer surplus gain
are higher, because the shift of the travel window to a later time becomes more valuable. The
effect of γ/α is again non-monotonic, but now a maximum is attained at an intermediate γ/α.
Fig. 7: Effect of the relative sizes of the preference parameters on the five-step Braking toll’s relative
efficiency (left) and change in consumer surplus from the NT case (right)
Fig. 8: Effect of the relative sizes of the preference parameters on the five-step ADL toll’s relative
efficiency (left) and change in consumer surplus from the NT case (right)
11
Although different from their case with fixed demand, the minimum relative efficiency is, now, not at γ/α=0.75 for all β/α, but the
minimising γ/a increases with β/α.
13
(a) ADL toll
(b) Laih toll
11
10 steps
0.9
0.9
10 steps
5 steps
5 steps
0.9
0.9
0.8
0.8
0.7
0.7
0.8
0.8
2 steps
0.7
0.7
1 step
0.6
0.6
0.6
0.6
−1.6
1.6
−0.8
0.8
−0.4
0.4
−0.2
0.2
00
0.5
0.5
−0.8
0.8
−1.6
1.6
Elasticity
Elasticity
−0.4
0.4
Relativeefficiency
efficiency
Relative
1 step
Relativeefficiency
efficiency
Relative
2 steps
0.5
0.5
−0.2
0.2
00
Elasticity
Elasticity
(c) Braking toll
10 steps
0.8
0.8
0.7
0.7
2 steps
0.6
0.6
1 step
0.5
0.5
−1.6
1.6
−0.8
0.8
−0.4
0.4
−0.2
0.2
Relativeefficiency
Relative
efficiency
5 steps
0.4
0.4
00
Elasticity
Elasticity
Fig. 9: Relative efficiency and price sensitivity for (a) the ADL, (b) Laih, and (c) Braking models
(a) ADL toll
(b) Laih toll
Elasticity
1.6
−1.6
Elasticity
0.8
−0.8
0.4
−0.4
0.2
−0.2
Elasticity
Elasticity
−1.6 1.6
−0.8 0.8
00
10 steps
5 steps
−0.4 0.4 −0.2 0.2
0
10 steps
5 steps
2 steps
2 steps
0.5
−0.5
0.5
av.CS
1 step
∆ av.CS
av.CS
1 step
1
−1
1
1.5
1.5
−1.5
(c)
Braking toll
Elasticity
Elasticity
1.6
−1.6
0.8
−0.8
0.4
−0.4
0.2
−0.2
0
10 steps
5 steps
−0.5
0.5
−11
∆ av.CS
av.CS
2 steps
1 step
1.5
−1.5
Fig. 10: Change in average consumer surplus due to step tolling and price sensitivity for (a) the ADL,
(b) the Laih, and (c) the Braking models
14
15
Fig. 9 compares the relative efficiencies of the three step tolls over different price
elasticities: panel (a) does this for the ADL toll, (b) for the Laih toll, and (c) for the Braking
toll. Fig. 10 compares the change in average consumer surplus. The figures show that not only
is welfare higher with more steps, but the consumer is also better off: with a price elasticity of
−0.4, a single-step toll decreases average consumer surplus by 1.17 (or a 22% lower total
consumer surplus), while a ten-step toll by only 0.25 (or a 5% lower total surplus).
The effects of step tolling depend strongly on the price sensitivity. The gain of step tolling
is higher, and the absolute consumer surplus loss is lower with more price-sensitive demand:
this is because it becomes easier for users to adapt their demand, and toll revenue becomes
larger relative to the consumer surplus loss. Note that this effect of price sensitivity on
consumer surplus also occurs in a static model of congestion.12
6. Discussion
This paper makes some assumptions that may seem restrictive. This section discusses some
of these assumptions and explores what effects relaxing these could have. These relaxations
also seem interesting avenues for future research.
It is assumed that tolling is costless, but, in reality, tolling schemes typically have
substantial operating and set-up costs (see, e.g., Santos, 2005). There may be costs for the
operator in implementing more steps. There may also be costs for the users: many steps may
be confusing or hard to understand. As long as the toll schedule follows a regular pattern,13
these costs might be limited, because the users only have to learn the schedule once. Indeed,
the experience from Singapore and the SR91 suggests that users can handle complex
schedules. Users might also be insensitive to small steps in the toll (e.g. from S1.00 to $1.10).
Such insensitivity and costs to steps would imply that there is an optimal number of steps, and
that increasing the number of steps further would have no effect or even harm welfare.14
The bottleneck congestion used in this paper might not be an accurate description of realworld congestion. More realistic might be the dynamic “Bureau of public roads” congestion
of Chu (1999) or the kinetic wave model of Ge and Stewart (2010a,b) that combines flow and
queuing congestion. The downside of such more complex models is that they are typically
less tractable, and have to be solved numerically, which makes their results harder to
interpret.15 With flow congestion, it is generally found that tolling increases the price, which
is not the case with bottleneck congestion. Still, even with flow congestion, it can be expected
that there is some gain for the consumer from time-variant (step) tolling, since also then a
dynamic toll increases efficiency. In line with this view is that in Chu (1999) the FB toll raises
the price but far less than the uniform toll. Moreover, there may also be hypercongestion: i.e.
the speed-flow relation also has a backward-bending part where the flow decreases with extra
12
Still, with more sensitive demand, step tolling lowers the number of users more, and the percentage loss in consumer surplus is larger
(because the NT surplus (which is in the denominator) decreases; in the limit, as the demand becomes perfectly elastic, consumer surplus
becomes zero in all regimes).
13
For example, that it has the same pattern each week, where the schedule can differ between days: e.g. between Monday and Friday.
14
Such insensitivity to small price changes could actually be beneficial if braking would occur, since then, if there are sufficiently small
steps, users would no longer see any point in braking, and thus this welfare harming behaviour would stop.
15
Furthermore, in Stewart and Ge (2010a,b), the step toll has to be approximated at points where the level discretely changes by a rapid, but
finite, change, and they study a pre-specified toll schedule instead of one that is optimised.
16
traffic.16 Then, optimal and step tolling may actually decrease the price for the users by
eliminating the pure waste that is hypercongestion (see Fosgerau and Small (2012), who
model hypercongestion with a bottleneck model where capacity depends on the queue length).
This paper studies a simple setting with just one link; whereas, in reality, there are complex
networks with many links and many origins and destinations. If all links have bottleneck
congestion and are subject to first-best pricing, the results of this paper should hold. De Palma
et al. (2005) study a network of concentric ring roads with 33 nodes and 128 links, using the
METROPOLIS simulation model which has bottleneck congestion. Their optimal dynamic
toll goes in steps of five minutes:17 i.e. it is a step toll with very many steps. Consistent with
the results in this paper for a single link, there remains some queuing and the users are on
average slightly worse off. It is also consistent that a uniform toll leaves users substantially
worse off than the optimal toll, and that a step-toll is better for the users than a uniform toll.
However, if part of the network remains untolled things might change. Suppose that there
are two parallel links with bottleneck congestion, but only one is tolled.18 As Braid (1996) and
de Palma and Lindsey (2000) show, with a fully time-variant toll, the time-variant part
eliminates the queuing on the tolled link for a given number of users. The time-invariant part
is negative and balances two effects: 1) a lower time-invariant part attracts users away from
the untolled link—which is beneficial, as the MSC on the untolled link is above the MSC on
the tolled link—but 2) it also increases total demand—which harms welfare because the
average MSC is above the inverse demand. Note that this second-best toll follows the same
format as in the static model of Verhoef et al. (1996).
With a step-toll, the step part would remain the same, as this minimises social cost for
given number of users, while the time-invariant addition has an extra term to correct for the
untolled link. Following de Palma and Lindsey (2000) and Verhoef et al. (1996), the timeinvariant part would equal the average marginal eternal cost on the tolled road (E[MECT])
minus the mean step toll (E[ρ]) minus the MECU of the untolled road multiplied by a term
(which is between 0 and 1) that depends on the price sensitivity:
T  E[ MECT ]  E[  ]  MECU
D / N
;
CU / NU  D / N
(27)
where D / N is the slope of the inverse demand, and CU / NU   / s is the derivative of the
untolled road’s usage cost w.r.t. its number of users. See Fosgerau (2011) on an express-lane
where part of the capacity is reserved for certain users, while the other users can use the
reserved capacity when it would otherwise be idle; and de Palma et al. (2004) on pricing of a
subset of the auto-roads with all roads also having uncongestible alternatives.
For future research it is interesting to include heterogeneous preferences. This would allow
16
That is, there is a critical density of cars, and if more cars enter the system the flow will decrease. The part where travel time and flow
increase with the density is referred to as ‘congested’, and the backward-bending part as ‘hypercongested’. In the engineering literature,
these two states tend to be referred to as ‘uncongested’ and ‘congested’.
17
A finer toll was not possible since 5 minutes is “the smallest time interval at which information on occupancies can be extracted from
METROPOLIS after a simulation” (de Palma et al., 2005, p. 597).
18
This set-up can be interpreted as an express-lane next to an untolled free-way (as is common in the US), or as a tolled expressway with an
untolled parallel secondary road (which is, for example, common in France (Gómez‐Ibáñez and Meyer, 1993)).
17
studying of aggregate welfare effects as well as distributional effects. Xiao et al. (2011) study
the ADL single-step toll under fixed demand and the heterogeneity from Vickrey (1973) that
varies the values of time and schedule delay in fixed proportions. They find that step tolling
decreases user-cost more with this heterogeneity than with homogeneity, since users with high
values self-select to the tolled period, while those with low values to the untolled period.
It would also be interesting to add heterogeneity in the ratio of value of time and schedule
delay early or in the ratio of values of schedule delay. Van den Berg and Verhoef (2011)
study fully time-variant tolling under two dimensions of heterogeneity: (1) the heterogeneity
from Vickrey (1973), and (2) in the ratio of value of time to value of schedule delay. They
find that whether a certain user wins or loses depends on her values of time and schedule
delay, the extent of both types of heterogeneity, and all the price sensitivities. The
heterogeneity in the ratio of values of schedule delay should also have important effects, as
this ratio affects when a user arrives with and without step tolling (see Section 4 of Arnott et
al. (1988, 1994) on the no-toll and first-best equilibria with this heterogeneity).
Lindsey (2004) adds preference heterogeneity to the large network set-up of de Palma et al.
(2005). There are four types of drivers that differ in their values of time and schedule delay.
Aggregate consumer surplus increases due to optimal tolling. However, the two types with a
low ratio of value of time to value of schedule delay lose on average. There is also substantial
spatial heterogeneity in the distributional impacts, where drivers who drive from the outer
rings to the centre are most affected by the toll.
Börjesson and Kristoffersson (2012) use a mesoscopic simulation model to evaluate the
Stockholm pricing scheme. In their setting, the average user actually gains from the scheme
for two reasons: 1) less blockage of upstream traffic links, which also benefits users who stay
outside the charging zone; and 2) the self-selection of users by value of schedule delay, as is
also found for a single link in Vickrey (1973), Van den Berg and Verhoef (2011) and Xiao et
al. (2011).
7. Conclusion
Models of (step) tolling usually assume that demand is price insensitive. This assumption
seems empirically questionable, and has, as this paper has found, important implications for
the effects of step tolling. In the bottleneck model, a first-best toll that is fully time-variant
leaves the price unchanged, and thus price sensitivity has no effect. Conversely, step-tolling
raises the price, and thus reduces consumer surplus and the number of users, and these
reductions depend strongly on the price sensitivity. The more steps there are in the toll, the
closer it approximates the first-best toll, and the better off the consumer. This makes it
particularly important for real-world tolling systems to have as many steps in the toll as
possible: this not only raises the welfare gain of tolling, but may also raise the political
acceptability of tolling.
Acknowledgements
Financial support from the ERC (AdG Grant #246969 OPTION) is gratefully acknowledged. I thank
the reviewer for the very helpful comments. I thank Erik Verhoef, Alexandros Dimitropoulos, Paul
18
Koster, Hugo Silva, and Mrs. Ellman for valuable suggestions. The usual disclaimer applies.
Appendix: Analytics for one- and two-step ADL tolling
Single-step ADL toll
To solve the model, I start by finding the timing of the peak and step toll for given number
of users, N, and step part of the toll, ρ. It is optimal for the queue length to be zero at the start,
t+, and end, t−, of the tolling period; and the queue length is also zero at the start, ts' , and end
of the peak, te' . For the prices to be equal at t+ and t−, −β∙t+=γ∙t− has to hold. Moreover, in
optimum, the N users have to just be able to pass the bottleneck during the peak, and thus N/s
equals ts−te. At ts and t+ the travel time is zero. For the price to be equal, the difference in
schedule delay costs, β(t+−ts), should equal the step part of the toll, ρ. Accordingly, t+ equals
ts+ρ/β. Similarly for the price of the last toll payer to arrive at t- to equal the expected price of
travel in the mass, the ρ should equal the expected extra travel time and schedule delay cost in
the mass of (α+γ)(te−t−). This implies that t− equals te−ρ/(α+γ).
Using these timings, it is possible to write total cost as a function of ρ by subtracting total
revenue for the step part of the toll, ρ∙s∙(t−−t+), from the total price (i.e. the price at, for
instance, ts multiplied by N). Interestingly, minimising the total costs gives the same step part
of the toll as in the single-step Laih case:
 
N
;
2s
although, as we will see, this similarity will break down with more steps.
Using these timings and the ρ, it start and end of the peak can be simplified to
N   (   2 )   (3  2 ) 
,
   s 
2(   )(    )




N

(


2

)


(3


2

)
(28)
te'  t * 
.
   s 
2(   )(    )

ts'  t * 

(28a)
(28b)
These equations only differ from the equations for the NT and Laih equilibria by the terms in
brackets. For ts' the term is smaller than 1 for relevant parameter values (i.e. γ>α>β>0), while
the term for te' is above 1. Hence, the peak starts and ends later for a given N than with Laih
tolling. Total costs and toll revenue in optimum can be written as
N 2  1  (  2   ) 
1 
,
s  4 (   )(    ) 
N2 


step
fixed
TR  TR  TR


1 
s  2   
TC  
(29)

    N.

(30)
Marginal External Cost is on average
E[ MEC ]  
N  1  (  2   ) 
1 
.
s  4 (   )(    ) 
(31)
19
This equation shows that, for a given N, the ADL externality is lower than in the Laih model
1  (  2    )
m
1
in (16), since

 when γ>α>β>0. Surprisingly, given the complex
4 (   )(    ) 2  2m 4
formulas for the revenue of the step part of the toll and average MEC, the time-invariant part,
θ, again follows a simple formula which is the same as in the Laih case:
1
2
  E[ MEC ]  E[  ]  
N
.
s
(32)
Since the MEC and total cost for a given N are lower than in the Laih case (which due to the
larger downward tilt of the cost curves), the consumer is better off with a single-step ADL toll
than with a single-step Laih toll.
Two-steps ADL toll
The formulas with two steps are more difficult than with a single step, and the more steps
there are, the more complex the formulas become. Still, the solution procedure is the same as
before. Different from with Laih tolling, the early and late tolls are now asymmetric:

2 N
2 (   )
1  3
,
3
2
3 s  3  3    8  9    16  9  

 2  1 ,
2


 3  5 2  3 3   16  7 
.
2  1 
 4  3   3  3  2      2  2  3  


1  

(33)
(34)
(35)

Hence, the early toll  2 is half the central toll 1 , while the late toll  2 is somewhat higher.
Total costs follow

  2 3   2  3  2    2 11  6   2 8 2  9   3 2 
TC   1 

2       3 3  3 3   2 8  9    16  9  

   N .


(36)
s
Unlike with a single step, the formula for the time-invariant part of the toll now differs
from the one in the Laih model:
  E[ MEC ]  E[  ] 

4    
1
N
 1  3
.
1  2 s  3  3 3   2  8  9    16  9  
(37)
In the Laih model, the term between brackets equals 1. In this ADL model, the term is below
1 when γ>α>β>0. Accordingly, the θ and average toll are lower for a given N.
References
Arnott, R., de Palma, A., Lindsey, R., 1988. Schedule delay and departure time decisions with heterogeneous
20
commuters. Transportation Research Record 1197, 56–67.
Arnott, R., de Palma, A., Lindsey, R., 1990. Economics of a bottleneck. Journal of Urban Economics 27(1), 111–
130.
Arnott, R., de Palma, A., Lindsey, R., 1993. A structural model of peak-period congestion: a traffic bottleneck
with elastic demand. American Economic Review 83(1), 161–179.
Arnott, R., de Palma, A., Lindsey, R., 1994. The welfare effects of congestion tolls with heterogeneous
commuters. Journal of Transport Economics and Policy 28(2), 139–161.
Börjesson, M., Kristoffersson, I., 2012. Estimating welfare effects of congestion charges in real world settings.
CTS Working Paper, nr. 2012:13.
Braid, R.M., 1996. Peak-load pricing of a transportation route with an unpriced substitute. Journal of Urban
Economics 40(2), 179–197.
Brons, M, Nijkamp, P., Pels, E., Rietveld, P., 2002. A meta-analysis of the price elasticity of gasoline demand: a
SUR approach. Energy Economics 30(5), 2105–2122.
Chu, X., 1999. Alternative congestion technologies. Regional Science and Urban Economics 29(6), 697–722.
Chew, V., 2008. Electronic road pricing: developments after phase I. National Library Singapore, Singapore
Infopedia. Retrieved on 17 August 2010, from http://infopedia.nl.sg/articles/SIP_1386_2009-01-05.html
Daniel, J.I., 2009. The deterministic bottleneck model with non-atomistic traffic. Working paper of the
Department of Economics, Alfred Lerner College of Business & Economics, University of Delaware, no.
2009-08. Retrieved on 25 July 2010, from
www.lerner.udel.edu/sites/default/files/imce/economics/WorkingPapers/2009/UDWP2009-08.pdf
de Palma, A., Kilani, M., Lindsey, R., 2005. Congestion pricing on a road network: a study using the dynamic
equilibrium simulator METROPOLIS. Transportation Research Part A 39(7–9), 588–611.
de Palma, A., Lindsey, R., 2000. Private toll roads: competition under various ownership regimes. The Annals of
Regional Science 34(1), 13–35.
de Palma, A., Lindsey, R., Quinet, E., 2004. Time-varying road pricing and choice of toll locations. Research in
Transport economics 9, 107–131.
Fosgerau, M., 2011. How a fast lane may replace a congestion toll. Transportation Research Part B 45(6), 845–
851.
Fosgerau, M., Small, K.A., 2012. Hypercongestion in downtown metropolis. In: Kuhmo Nectar Conference and
Summer School on Transportation Economics 2012 – Annual conference of the ITEA, Berlin, Germany, 18–
23 June 2012.
Ge, Y.E., Stewart, K., 2010a. Investigating boundary issues arising from congestion charging in a bottleneck
scenario. In: C. Tampre, F. Viti and L.H. Immers (Eds.), New Developments in Transport Planning:
Advances in Dynamic Traffic Assignment. Aldershot: Edward Elgar, pp. 303–326.
Ge, Y.E., Stewart, K., 2010b. Investigating boundary issues arising from congestion charging. In: The Fifth
Travel Demand Management Symposium, Aberdeen, Scotland, U.K., 26–28 October 2010.
Graham, D.J., Glaiser, S., 2004. Road traffic demand elasticity estimates: a review. Transport Reviews, 24(3),
261–274.
Gómez‐Ibáñez, J.A., Meyer, J.R., 1993. Going private: the International experience with transport privatization.
The Brookings Institution, Washington, D.C., U.S.A.
Laih, C.H., 1994. Queuing at a bottleneck with single and multi-step tolls. Transportation Research Part A,
28(3), 197–208.
Laih, C.H., 2004. Effects of the optimal step toll scheme on equilibrium commuter behavior. Applied
Economics, 36(1), 59–81.
Lee, L., Frick, K., 2011. San Francisco-Oakland Bay Bridge Congestion Pricing. In Congestion pricing projects
in the San Francisco Bay Area: I-680 express lanes and Bay Bridge time-of-day pricing. Retrieved on 15
July 2011, from
www.fhwa.dot.gov/ipd/pdfs/revenue/resources/webinars/webinar_congestion_pricing_033011.pdf
Lindsey, R., 2004. The welfare-distributional impacts of congestion pricing on a road network. In: Proceedings
of the 39th Annual Conference of the Canadian Transportation Research Forum: 2004 Transportation
Revolutions, Calgary, Canada, 9–12 May 2004, pp. 149–163.
Lindsey, R., van den Berg, V.A.C., Verhoef, E.T., 2012. Step tolling with bottleneck queuing congestion.
Journal of Urban Economics, 72(1), 46–59.
Odeck, J., Bråthen, S., 1997. On public attitudes toward implementation of toll roads: the case of Oslo toll ring.
Transport Policy, 4(2), 73–83.
Png, G.H., Olszewski, P., Menon, A.P.G., 1994. Estimation of the value of travel time of motorists. Journal of
the Institution of Engineers, Singapore, 34(2), 9–13.
Santos, G., 2005. Urban congestion charging: a comparison between London and Singapore. Transport Reviews,
25(5), 511-534.
Small, K.A., Verhoef, E.T., 2007. The Economics of Urban Transportation. Routledge, Abingdon, England.
21
van den Berg, V.A.C., Verhoef, E.T., 2011. Winning or losing from dynamic bottleneck congestion pricing? The
distributional effects of road pricing with heterogeneity in values of time and schedule delay. Journal of
Public Economics, 95(7–8), 983–992.
Verhoef, E.T., Nijkamp, P., Rietveld, P., 1996. Second-best congestion pricing: The case of an untolled
alternative. Journal of Urban Economics, 40(3), 279–302.
Vickrey, W.S., 1973. Pricing, metering, and efficiently using urban transportation facilities. Highway Research
Record, 476, 36–48.
Xiao, F., Qian, Z., Zhang, H.M., 2011. The morning commute problem with coarse toll and nonidentical
commuters. Networks and Spatial Economics, 11(2), 343–369.
Yang, H., Huang, H.J., 1997. Analysis of the time-varying pricing of a bottleneck with elastic demand using
optimal control theory. Transportation Research Part B, 31(6), 425–440.
22