11Equation Section 1

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: v.a.c.vanden.berg@vu.nl, tel: +31 20 598 6049#: Affiliated to the Tinbergen Institute, Roetersstraat 31, 1018 WB Amsterdam.

AbstractMost 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 acceptabilityJEL 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-and-practice/

1. IntroductionTheoretical 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

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.

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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 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,

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).

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Section 7 concludes.

2. The general step-toll modelThis 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 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 symbolsSymbol Descriptiont Arrival timeθ Time-invariant part of the tollρi The ith step part of the toll (which level depends on t):[t] Total toll at tcSD Schedule delay cost cTD Travel delay costc (Generalised) cost: c≡cSD+cTD

P (Generalised price: P≡+ct* Preferred arrival time α Value of time (i.e. monetary value of an hour of travel time)β Value of schedule delay earlyγ Value of schedule delay lateδ 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*)

ts Time the peak startste Time the peak endsD Inverse demandN Number of users (which is endogenous due to the price-sensitive demand )TC Total cost: TC≡c∙NW 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):

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3

t*t1t

1t

2t

1mtmt

st et2t

1mt

mt

m

2 1 1( )

1 1( )

2

m

θ

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 and ends at ; the times for the ith late

part are, respectively, and With step tolling, the peak starts at and ends at . 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 ( ) multiplied by the number of users (N):

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To find the optimal time-invariant part (θ) for a given pattern of the step toll, the following Lagrangian is maximised:

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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

55\* MERGEFORMAT () (4a)

(4b) (4c)

These equations imply that the optimal time-invariant part of the toll equals

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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

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.

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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 model3.1. 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 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*):

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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:

88\* MERGEFORMAT ()(7a)

(7b)

In these equations, s is the capacity of the bottleneck. Equilibrium user-cost is given in 9. 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 10 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:

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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.

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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:

This FB toll halves total cost, average Marginal External Cost (E[MEC]), and average user-cost (E[c]), since all travel delays are converted into toll payments; while the price is unaffected:

E[cFB]=½ δ ∙N/s, 1111\*MERGEFORMAT ()PFB=δ ∙N/s, 1212\* MERGEFORMAT ()E[MECFB]= ½ δ ∙N/s. 1313\* MERGEFORMAT ()

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 tollThe 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.

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Toll

U n ifo rm Tol l

FB

2 .0 1 .5 1 .0 0 .5 0 .0 0 .5

2

4

6

8

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 tollingOf 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 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 , where the users who will arrive after

start waiting in front of the tolling point before . In the ADL model, there are no separate

queues. Instead a mass departure at equalises expected prices before and after . 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 tollThe 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

7

t

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

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1515\* MERGEFORMAT ()

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 (Lindsey et al., 2012). The step part of the toll follows

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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

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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

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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 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

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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.

Step To l l

U n ifo rm To ll

FB

2 .0 1 .5 1 .0 0 .5 0 .0 0 .5

2

4

6

8

Fig. 3: A single-step toll3.6. 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 there is the mass departure of the users who pay the i−1th 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 16):

9 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.

9

Toll

t

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Conversely, the late tolls are not simple fractions of the central toll, ρ1:

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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 tollThe 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 . The first

users to pay the i−1th level arrive just after The last users to pay the ith toll arrive at

For the prices at these two arrival times to be equal, the ∆ti has to equal

The total time the bottleneck is idle,10 , is the sum of all the . It depends only on the level of ρ1 and the preference parameters α and γ:

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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:

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The idle time 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 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):

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10 The bottleneck is ‘idle’ when users stand still in front of the tolling point in order to prevent paying the toll.10

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From 24 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:

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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:

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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 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 exampleThis 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

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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 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 price-sensitive 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).

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A D LBrak ing

L aih

0 2 4 6 8 10m8

9

10

11

12

P

Fig. 5: Generalised price as a function of the number of steps

Brak ing

L aihA D L

0 2 4 6 8 10m

0 .5

0 .6

0 .7

0 .8

0 .9

1 .0

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.

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

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

ω

P

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)

(a) ADL toll (b) Laih toll10 s tep s5 s teps

2 s teps

1 s tep

1.6 0 .8 0 .4 0 .2 00 .5

0 .6

0 .7

0 .8

0 .9

1

E las ticity

Relat

ivee

fficie

ncy

Rel

ativ

eef

ficie

ncy

1

0.9

0.8

0.7

0.6

0.5

Elasticity

−1.6 −0.8 −0.4 −0.2 0

1 0 s t e p s

5 s t e p s

2 s t e p s

1 s t e p

1 . 6 0 . 8 0 . 4 0 . 2 00 . 5

0 . 6

0 . 7

0 . 8

0 . 9

E l a s t i c i t y

Relat

ive

effici

ency

Rel

ativ

eef

ficie

ncy

0.9

0.8

0.7

0.6

0.5

Elasticity−1.6 −0.8 −0.4 −0.2 0

(c) Braking toll

14

1 0 s tep s

5 s teps

2 s teps

1 s tep

1 .6 0 .8 0 .4 0 .2 00 .4

0 .5

0 .6

0 .7

0 .8

Elas tic ity

Rel

ativ

eeff

icie

ncy

Rel

ativ

eef

ficie

ncy

0.8

0.7

0.6

0.5

0.4

Elasticity−1.6 −0.8 −0.4 −0.2 0

Fig. 9: Relative efficiency and price sensitivity for (a) the ADL, (b) Laih, and (c) Braking models

(a) ADL toll (b) Laih toll

10 s teps5 s teps

2 s teps

1 s tep

1.6 0.8 0.4 0.20

0.5

1

1.5

E las t icity

av.

CS∆

av.C

S0

−0.5

−1

−1.5

Elasticity

−1.6 −0.8 −0.4 −0.2

10 s teps5 s teps

2 s teps

1 s tep

1 . 6 0 . 8 0 . 4 0 . 20

0 . 5

1

1 . 5

Elas t ic ity

av

.CS

(c) Braking toll

1 0 s tep s5 s tep s2 s tep s

1 s tep

1 .6 0 .8 0 .4 0 .20

0 .5

1

1 .5

Elas tic ity

av

.CS

∆ av

.CS

0

−0.5

−1

−1.5

Elasticity

−1.6 −0.8 −0.4 −0.2

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

15

Elasticity −1.6 −0.8 −0.4 −0.2

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. DiscussionThis 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 real-world 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

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

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 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 time-invariant 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:

2828\* MERGEFORMAT ()

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

where is the slope of the inverse demand, and 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 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. ConclusionModels 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

18

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 Koster, Hugo Silva, and Mrs. Ellman for valuable suggestions. The usual disclaimer applies.

Appendix: Analytics for one- and two-step ADL tollingSingle-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, , and end

of the peak, . 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:

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

(28a)

2929\* MERGEFORMAT () (28b)

These equations only differ from the equations for the NT and Laih equilibria by the terms in

brackets. For the term is smaller than 1 for relevant parameter values (i.e. γ>α>β>0), while

the term for 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

19

3030\* MERGEFORMAT ()

3131\* MERGEFORMAT ()

Marginal External Cost is on average

3232\* MERGEFORMAT ()

This equation shows that, for a given N, the ADL externality is lower than in the Laih model

in 17, since when γ>α>β>0. Surprisingly, given the complex 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:

3333\* MERGEFORMAT ()

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 tollThe 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:

3434\* MERGEFORMAT ()

3535\* MERGEFORMAT ()

3636\* MERGEFORMAT ()

Hence, the early toll is half the central toll while the late toll is somewhat higher. Total costs follow

3737\* MERGEFORMAT ()

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Unlike with a single step, the formula for the time-invariant part of the toll now differs from the one in the Laih model:

3838\* MERGEFORMAT ()

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.

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