Give or take? Rewards versus charges for a congested bottleneck

Regional Science and Urban Economics 42 (2012) 166–176

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Give or take? Rewards versus charges for a congested bottleneck☆ Jan Rouwendal ⁎, 1, Erik T. Verhoef 1, Jasper Knockaert Department of Spatial Economics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands

a r t i c l e

i n f o

Article history: Received 2 September 2010 Received in revised form 25 August 2011 Accepted 26 August 2011 Available online 3 September 2011 JEL classification: R41 R48 D62 Keywords: Traffic congestion Road pricing Subsidies Rewards Bottleneck model

a b s t r a c t This paper analyzes the possibilities to relieve traffic congestion using subsidies instead of Pigouvian taxes, as well as revenue-neutral combinations of rewards and taxes (‘feebates’). The model considers a Vickrey–ADL model of bottleneck congestion with endogenous scheduling. With inelastic demand, a fine (time-varying) reward is found to be equivalent to a fine toll, and to a continuum of combinations of time-varying tolls and rewards, including fine feebates. When demand is price-sensitive, a reward becomes less attractive from the efficiency viewpoint, because it attracts additional users to the congested bottleneck. As a result, both the second-best optimal fraction of rewarded travelers in the scheme, and the relative efficiency that can be achieved with it, decrease when demand becomes more elastic. Our analytical and simulation results for coarse schemes reveal that a coarse reward is less effective than a coarse feebate, which is itself less effective than a coarse toll. The most efficient coarse system is the step toll, which is also allowed to be positive in the fringes of the peak. Despite the smaller efficiency gains, rewards and feebates may be attractive to use in circumstances where public and political acceptability of tolling is especially low, so that its implementation is unlikely, including the temporary use of price incentives in case of road works and large-scale events. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Although the idea of road pricing has been around for a long time, and its popularity seems to be on the rise with successful introductions in cities like London, there often still exists fierce resistance against introducing a price for a commodity – access to public roads – that has been freely available for such a long time. Consequently, there are many examples of road pricing proposals that do not survive the planning stage, and that are dropped for political reasons before implementation commences. In cases where temporary changes in supply or demand conditions (e.g., road works, large-scale events) cause severe congestion on a more short-lived basis, the public and hence political acceptability of road pricing may even be smaller than otherwise, notwithstanding the greater potential gains, as it may be perceived to be especially unfair to regular users who are already confronted with longer travel times. Despite the clear economic case for marginal cost pricing, it may therefore be helpful to explore alternative, more acceptable possibilities to alleviate congestion, other than archetypical pricing solutions. One possibility would be the use of subsidies or rewards, instead of

☆ We thank two anonymous reviewers for helpful comments. Financial support from TRANSUMO is gratefully acknowledged; Verhoef furthermore acknowledges support from the ERC (AdG Grant # 246969 OPTION). ⁎ Corresponding author. Tel.: + 31 20 5986093; fax + 31 20 5986093. E-mail address: [email protected] (J. Rouwendal). 1 Affiliated to the Tinbergen Institute, Gustav Mahlerplein 117, 1082 MS Amsterdam, The Netherlands. 0166-0462/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.regsciurbeco.2011.08.011

the penalty of tolling. Even though most economists would hasten to point out that positive marginal external costs would call for a positive Pigouvian tax, that subsidies may induce all sorts of perverse incentives, that taxes needed to raise revenues to finance rewards are likely to cause or aggravate distortions elsewhere in the economy, and that a net reward is likely to attract additional users whereas a net reduction is desired, there may be reasons to consider rewards nevertheless. Ultimately, a reward system may be less effective and efficient in combating congestion than a tolling system, but if the latter is unfeasible for political reasons, a more relevant comparison is between a reward system and the absence of any financial incentives. Such considerations have motivated proposals for, for example, combinations of pricing and rationing (Daganzo, 1995), revenue neutral ‘credit-based’ congestion pricing (Kalmanje and Kockelman, 2005), tradable driving permits (Verhoef et al., 1997), exemptions from paying tolls (Daganzo and Garcia, 2000) and so-called Fast and Intertwined Regular (FAIR) lanes (De Corla-Souza, 2000). It is the purpose of the present paper to explore the possibilities of rewards in traffic congestion management. We do so by studying various types of rewards in the context of the well-known bottleneck model (Vickrey, 1969; Arnott et al., 1990, 1993). The idea for this paper stems from a Dutch policy experiment called ‘Spitsmijden’ (Avoiding the Peak), documented in Knockaert et al. (2009) and Bliemer and Van Amelsfort (2010). The purpose of this experiment was to gain insight into the potential of positive financial incentives in the management of peak traffic congestion. To that end, regular users of a given highway (the A12 between Zoetermeer and The Hague) could earn rewards

J. Rouwendal et al. / Regional Science and Urban Economics 42 (2012) 166–176

ranging from 3 to 7 Euros for avoiding the A12 during the morning peak (7:30–9:30 am). Given that participation was voluntary, it is no surprise that relatively strong behavioral impacts from rewarding were found among participants: one might expect a greater probability of participation, the more easily an individual can change behavior and thus earn rewards. According to the automated vehicle observations, the participants reduced their aggregate number of peak hour trips by some 50% when doing so was rewarded. Given that departure time adjustments were, by far, the most popular behavioral response to the incentive (roughly accounting for four-fifths of all adjusted trips), it seems appropriate to use a dynamic model, with endogenous departure time choices, for the present paper. Having rewards in place when taxes would be first-best optimal makes the policy second-best in nature. In addition, a second and quite different type of second-best considerations will arise in our analyses because we will not only consider smoothly time-varying ‘fine’ tolls and rewards, but also more realistic ‘coarse’ systems.2 These involve a limited number of discrete toll or reward levels during the peak. Our analysis thus fits in a much wider literature on second-best road congestion pricing reviewed in, among many others, Small and Verhoef (2007). Probably closest to our paper are earlier studies that looked into the efficiency of so-called ‘coarse tolls’ and ‘multi-step tolls’ in the bottleneck model. Arnott et al. (1990, 1993) were the first to consider this type of problem, and found that the relative efficiency of such measures is above that of ‘uniform tolling’, which entails a constant toll level throughout the peak, but considerably lower than that of fine tolling. The welfare gains are around 50% for the coarse (single-step) toll, a result that was later confirmed by Chu (1999) in a more elaborate model. Quite intuitively, Laih (1994) showed that the efficiency of step-tolls increases with the number of steps. Xiao et al. (2009) explore coarse tolling with heterogeneous users, and find that this improves the relative efficiency of coarse tolling. An important aspect of dynamic equilibria with step tolls in the bottleneck model, already identified by Arnott et al. (1990, 1993), is that it entails mass departures in the second part of the peak, since the equilibrium condition of constancy of generalized prices over times requires that a discrete drop in the toll level is matched by a discrete increase in (expected) travel time. The implied mass departures complicate the analytical treatment of step tolls. Lindsey et al. (2010) show that when drivers can halt before passing a toll gantry, in anticipation of a drop in the toll level, there will be no such mass departures in equilibrium. In the present paper, we will ignore this ‘braking’ behavior, and stick to the original model of coarse tolling in the bottleneck model, as developed by Arnott et al. (1990). The main difference with earlier studies of step tolling is our focus on rewards. More specifically, we will be looking at rewards, involving non-negative subsidies for all users; ‘feebates’, involving budgetneutral combinations of taxes and subsidies producing a zero net revenue for the regulator; and, as an important benchmark, tolls, involving non-negative taxes for all users. For all cases to be considered, we assume that tolls and rewards will be faced only by users of the bottleneck, typically in function of the moment of passing it. This means that no rewards will be assumed to be given to individuals who refrain from using the bottleneck altogether, e.g. by staying at home or using a public transport option. Although such subsidies would be possible in reality, and would have an impact if demand for the use of the bottleneck is pricesensitive, we dismiss the possibility for two reasons. The practical reason is that such policies would usually be considered to be intolerably expensive in reality, at least if all non-users would qualify for a reward. The

2

We will not be using the terminology of “third-best” or “fourth-best” problems to distinguish schemes with increasing numbers of constraints. It turns out that doing so might easily become confusing rather than helpful. The term “second-best” will therefore be used to denote any policy that seeks to optimize social surplus under at least one binding constraints on the pricing instrument.

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analytical reason is that a solid treatment would require the inclusion of a second transport mode in the model. This would harm the transparency of our results when introducing a third type of second-best considerations, in particular if this mode were assumed to be inefficiently priced as is usually the case in reality. Given that congestion entails an external cost, one would expect that tolls outperform rewards and feebates in terms of efficiency. It is therefore not primarily the ranking of the different policies that we are most interested in, but rather assessing their relative efficiency. Our analysis should thus give insight into the circumstances under which feebates or rewards may offer a worthwhile alternative to tolling in the management of congestion; and if so, how the policy should be designed to maximize its efficiency. The paper is organized as follows. In the next section we introduce the basic bottleneck model, and discuss the equivalence of fine tolling, fine feebates and fine rewards when demand is completely inelastic. In Section 3 we study the properties of fine schemes in the context of price-sensitive demand. Section 3 thus focuses on the first second-best aspect: the use of rewards where tolls would be optimal. We will find that with rewards in place instead of conventional tolls, it is not optimal to apply the financial incentive throughout the entire peak, but instead it is preferable to reward only a certain a fraction of drivers, so that rewards can be earned only during certain parts of the entire peak period. We thus derive the optimal fractions of rewarded commuters. 3 We will impose that there be a dynamic equilibrium in the sense that all rewarded and non-rewarded drivers should face the same generalized equilibrium price, so that the optimal fraction is realized without making an explicit ex ante subdivision between drivers who do or do not potentially qualify for a reward. Next, in Section 4, we move a step closer to practical policy experiments and introduce the other second-best aspect that we are interested in, by analyzing coarse systems where tolls or rewards only change in a few discrete steps during the peak. Finally, Section 5 concludes. 2. The basic bottleneck model and some variants on fine tolling The analysis in this section uses the basic bottleneck model, in which during the peak a homogeneous group of users of a given size N has to pass a bottleneck with given capacity s. Given that we consider only one single route, the free-flow travel time can be and will be set equal to zero without loss of generality, so that in absence of a queue, a traveler departs from home, passes the bottleneck, and arrives at work all at the same instant. However, a travel delay of Q(td) / s is incurred if at the moment td of departing from home and joining the queue, its length is Q(td). The rate of change of Q when positive is equal to the difference between the departure rate from home, and the capacity of the bottleneck. We consider the following, conventional, cost function; usually associated with Vickrey (1969) and Small (1982):
 
 cðt Þ ¼ α·T ðt Þ þ β· max 0; t −t þ γ· max 0; t−t :

ð1Þ

The Greek characters represent positive parameters, where α denotes the ‘value of time’; β indicates the unit shadow cost of schedule delay early, and γ that of schedule delay late. T is the travel delay incurred in the queue before passing the bottleneck, t the arrival time at the bottleneck's exit, and t⁎ the preferred arrival time, at which schedule delay costs are naturally zero.

3 This type of optimization was not present in the Spitsmijden experiment mentioned above, where rewards were given only to a limited number of participants. Another difference is that in the experiment, rewards were also given to participants who switched to alternatives such a public transport, whereas in this paper rewards will be assumed to be given only to drivers who pass the bottleneck during the time intervals when a reward applies.

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The equilibrium for this model is discussed in detail in, for example, Arnott, de Palma and Lindsey (ADL henceforth) (1990). They show that in the no-toll equilibrium, denoted with superscript 0, the equilibrium cost of the commuting trip is for all drivers equal to: 0

c ¼ δ·

N β·γ with δ≡ s βþγ

ð2Þ

where the composite schedule delay cost parameter δ is introduced for notational convenience. Total social cost N· c 0 is therefore equal to C 0 = δ · N 2/s. The peak hour starts at a moment that we will refer to as B = t* − (δ/β) · (N/s), and ends at E = t* + (δ/γ) · (N/s). At these moments, the queue has a zero length, and it is easily verified that the schedule delay costs at these two moments are equal, as required in equilibrium, both amounting to δ ⋅ (N/s). 4 The driver who passes the bottleneck at the desired arrival time t⁎, in contrast, incurs no schedule delay costs, but only travel delay costs because of the time spent in the queue. For all other drivers, the cost is a combination of positive schedule delay costs and positive travel time costs. In the no-toll equilibrium, the sum of these two must be constant over time whenever there is traveling. This requires a queue that first grows and then shrinks linearly, such that travel delays as a function of arrival time grow and fall at rates β / α and −γ / α for early and late arrivals, respectively. This makes the generalized cost in Eq. (1) constant over time. The assumed linearity of the schedule delay cost function implies that, in equilibrium, aggregate schedule delay cost and travel delay cost will be equal, both making up 50% of the aggregate social cost (recall that the exit rate of the bottleneck is constant and equal to s throughout the peak). ADL (1990) also show that the total cost can be reduced by 50% by introducing a ‘fine’ toll, meaning that the toll varies continuously over time. The optimal fine toll exactly replicates the no-toll equilibrium pattern of travel delay costs, and thereby completely eliminates the queue and therefore all travel delay costs: the sum of toll and scheduling delay is then constant over time, so that in equilibrium also travel time must be constant. Because the queue has a zero length initially, its length will remain zero throughout the peak. The optimal fine toll thus substitutes a monetary cost for a time cost. The crucial difference from the viewpoint of efficiency is that the time spent in the queue is a social loss, whereas the monetary toll is merely a transfer. This creates the welfare gain from optimal pricing; it amounts to 50% of the initial cost, exactly because the travel delay cost made up 50% in the no-toll equilibrium. The schedule delay costs – i.e., the other 50% – remain unaffected by imposition of the toll: the bottleneck operates at full capacity both with and without tolling. The arrival rate at the destination therefore does not change in the optimum, even though the queue has disappeared. Following ADL (1990), we can now write the optimal fine toll, τ f(t) with its time derivatives of β for early arrivals and −γ for late arrivals, as follows: f

τ ðt Þ ¼



  a−ðt −t Þ·β if B≤t≤t a−ðt−t  Þ·γ if t bt≤E:

ð3Þ

Here, a is a constant that defines whether we are considering a toll system, a reward system, or a combination of the two. We will restrict attention to a ≤ δ · (N/s), so that the toll will not exceed zero outside the interval [B,E]; i.e., when no one travels. 5 For a = δ · (N/s), the toll will be zero at B and E, and positive in between. However, with a strict inequality a b δ · (N/s), Eq. (3) implies that the toll will be negative,

This can be checked by multiplying t⁎ − B as implied above by β, and E − t⁎ by γ. Note that the regulator has quite some flexibility in defining the toll schedule for the periods before B and after E, when no one should travel. Any schedule that has, for these periods, tolls above the levels implied by extrapolation of Eq. (3), will in fact do. 4 5

and hence a subsidy, for at least some t during the peak period. Indeed, with a = 0, the toll will not be positive for any driver. In other words: depending on the choice of the constant a, the fine toll may in fact be a reward for some drivers, or even for all. 6 In the basic bottleneck model, a can be freely chosen, without affecting social surplus. The conventional fine toll with a = δ · (N/s) is thus equivalent in terms of effects to a fine reward with a = 0. Another potentially interesting possibility is a fine ‘feebate’, which we define as a budget-neutral fine toll that takes on both positive and negative values and generates no net revenues. This requires a ¼ 12 ·δ·ðN=sÞ. The insensitivity of welfare with respect to a in the basic bottleneck model stems from the assumed perfect inelasticity of demand. It can be shown that the price paid by each driver is equal to the marginal social cost under fine tolling if a = δ ⋅ (N/s). 7 As we will derive formally below, this is the unique optimal choice if demand is pricesensitive. Efficiency will then generally be lower under second-best schemes that put a constraint on a, as is true for rewards and feebates, rather than simply optimizing it. This motivates our choice for considering price-sensitive demand in the remainder of this paper. We will sometimes call this ‘elastic’ demand. Note that in other contexts this term is also used to denote an absolute demand elasticity above unity; we instead use it to indicate any price-sensitive demand that is not perfectly inelastic. 3. Elastic demand, fine tolling and rewarding, and endogenous participation To analyze elastic demand, we postulate a downward sloping inverse demand function D(N), and define social surplus S as the difference between aggregate (Marshallian) consumer benefit and the cost C(N) associated with using the bottleneck: N

S ¼ ∫ DðxÞdx−C ðNÞ

ð4Þ

0

The equilibrium number of users N is now endogenously determined by the condition: DðNÞ ¼ p:

ð5Þ

Here, p denotes the generalized price, which is at any arrival time t defined as the sum of the generalized cost and the toll (which may be negative): p(t) = c(t) + τ(t). Because in a dynamic equilibrium, this price will be equalized for all moments at which drivers arrive, a time-independent value p can be used in Eq. (5). In this section we consider the class of fine incentives. We restrict attention to those toll and reward schedules that, consistent with Eq. (3), have slopes β for early arrivals and − γ for late ones. These are the values that will keep the queue length constant, and the flow of vehicles at capacity s. In particular the queue length will be kept at zero provided the incentive period (the period during which a reward is given or a toll is levied) starts with a zero

6 Braid (1996) observes that it is in fact second-best optimal for a time-varying toll to start and end as a net subsidy at a bottleneck with an unpriced parallel bottleneck. This is needed to achieve equalization of marginal costs at the two bottlenecks in that second-best optimum. 7 Because optimal tolling eliminates queuing, the social cost becomes half the level in the no-toll equilibrium, and thus equals 12 ·δ·N 2 =s. The marginal cost is then δ · N/s, which is equal to the toll at t⁎ and hence the generalized price for the person arriving at t⁎ when a = δ · N/s. Because the generalized price is equalized in the decentralized optimum, this optimal price level applies for all drivers.

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queue. 8 A steeper incentive schedule would discourage traveling closer to t⁎ in the incentive periods in absence of queuing, disrupting the required constant departure rate equal to capacity s during that period. A flatter schedule would induce equilibrium queuing during the incentive period without changing the arrival rate. Neither is desirable from the perspective of social surplus, irrespective of whether tolls are used or rewards; hence our focus on incentive schedules is consistent with Eq. (3). Furthermore, for reasons of space we will restrict attention to the two extremes of fine tolls and fine rewards, postponing consideration of the intermediate case of feebates to the next section.9 We do, however, add one more dimension to the regulator's choice set, and that is to optimize the period over which the incentive applies. In particular, for rewards we allow the reward to start when the first traveler departs but to stop before t⁎; and again to start after t⁎ and then to last until the final traveler as arrived. Limiting the rewarding period in this way is relevant because with elastic demand there will be a trade-off between, on the one hand, extending the rewarding period so as to increase the period during which no queue will exist; and, on the other hand, limiting it so as to depress p by a smaller amount so that fewer additional travelers are attracted to the underpriced bottleneck. That the bottleneck will indeed be underpriced with rewards can be shown formally by determining the first-best optimal value of the coefficient a in the toll schedule in Eq. (3) when demand is elastic. To do so, first observe that in the first-best optimum, no queues should exist: eliminating queuing would reduce travel delay cost without raising schedule delay cost. Therefore, the toll period should capture the entire peak. Next, as discussed in Section 2, because a fine toll eliminates all queuing, it implies that the average generalized cost (that is, net of tolls) equals 12 ⋅δ⋅ðN=sÞ. Still, the policy maker is free to set the generalized price by choosing the parameter a. In fact, the generalized price will be exactly equal to a, because the person arriving at t⁎ will have neither travel delay nor schedule delay cost, and only faces a toll a. Because the generalized price is equal for all users in equilibrium, it equals a for everyone. We can thus write the Lagrangian: N

1 N2 Λ ¼ ∫ DðxÞdx− ·δ· þ λ·ða−DðN ÞÞ 2 s

ð6Þ

0

and derive the first-order conditions: ∂Λ N ′ ¼ DðNÞ−δ· −λ·D ðNÞ ¼ 0 s ∂N

ð7aÞ

∂Λ ¼λ¼0 ∂a

ð7bÞ

∂Λ ¼ a−DðNÞ ¼ p−DðNÞ ¼ 0 ∂λ

ð7cÞ

where D′ ≡ dD/dN. It follows from these conditions that a and hence p have a unique optimal value of δ ⋅ (N/s), so that a zero toll applies for the very first and last driver, while all other drivers face positive tolls. Other values of p and a, including all values consistent with reward 8 Because it would only raise aggregate travel delay cost to start the incentive period with a queue of positive length, without bringing any gain in schedule delay cost, we can safely restrict attention to equilibria with initial queues of a zero length in the incentive periods. Note that for an incentive period at the start of the overall peak, an initial queue length of zero is especially natural to assume: no one has departed before the incentive starts. For an incentive period that starts after a period with queuing, the regulator will time the incentive such that the first driver to face it will not depart before any earlier queue will have fully dissipated. We come back to this issue later in this section. 9 Note that a fine feebate is not at all straightforward to model as, when it involves an incentive-free central period of the peak like the fine reward does (see below), it induces a mass departure right after the early incentive period when it ends with a net toll. Given that a feebate, when implemented, would most likely take on a coarse form anyway, we consider it only for that situation, in Section 4 below.

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systems, are incompatible with the first-best outcome when demand is price-sensitive.10 This does, of course, not imply that a fine reward system can only play a useful role when demand is completely inelastic. Even though it is a second-best instrument, it may still achieve substantial welfare gains. To maximize these, we need to find the optimal time windows during which the rewards should apply — with, as argued above, slopes β for early arrivals and −γ for late arrivals. This task can be simplified by first noting that if the reward does not apply throughout the entire peak, it will apply during its fringes. That is, the reward will apply in an early time interval [B″,B′] before t⁎, and a late interval [E′,E″] after t⁎, where B″ and E″ denote the beginning and ending of the peak. This can be verified using Fig. 1. The solid lines indicate travel delays in the original untolled equilibrium, with B and E denoting the beginning and ending of the peak, while the dashed lines show those for the equilibrium with time-varying rewards, applying during the fringe periods. The longer duration of the peak with rewarding reflects the induced use when demand is price-sensitive. Note that the generalized prices for both equilibria must be equal to α multiplied by the travel delay for the person arriving at t⁎. The larger demand and longer peak duration with rewarding is therefore consistent with the lower travel delay at t⁎. The timing of the two reward periods, at the peak's fringes, can now be verified as follows. Imagine keeping the reward periods of fixed lengths, but shifting them inwards compared to the sketched situation. The absence of a time-varying reward for the very first and last drivers would lead to positive travel delays immediately after B″, and immediately before E″. The travel delay function has slope β/α between B″ and the start of the shifted rewarding period. Travel delays remain constant during the rewarding period, and then start again increasing with slope β/α. Since the duration of the rewarding period does not change, the travel delay at t′ remains the same. After t′ travel delay decreases with slope −γ/α, until the second rewarding period begins, stays constant until its end and then decreases further with the same slope until E″. Since the top of the dashed triangle would not change due to such a shift of the rewarding periods the generalized price and equilibrium demand would also not change. But the different timing of the rewarding periods would lead to higher travel delays for all travelers arriving before the end of the first rewarding period, and after the beginning of the second one. Because aggregate schedule delay cost does not change, this lowers social surplus compared to the timing of the rewarding periods at the peak's fringes. In the same way, it can be verified that it is welfare reducing to cut up the early and late reward periods in multiple sub-intervals. The timing of the rewards at the peak's fringes will thus indeed be the second-best optimal choice. This helps enormously in specifying the analytical optimization problem below. The equilibrium can therefore be described as a sequence of three phases. In the interval [B″,B′], early travelers depart at a rate equal to the bottleneck capacity s, which is supported by a reward that falls over time at a rate −β. Because there should be an equilibrium between rewarded and unrewarded drivers, and the drivers just before and after B′ have nearly equal schedule delay cost and travel delay cost, the reward should have fallen to zero at B′. Next comes the unrewarded interval bB′,E′N, during which a queue grows and shrinks as in the standard equilibrium; at rates β / α and −γ / α, respectively. And finally, the reward starts again at zero at E′, and grows at a rate γ in the interval [E′,E″], which again induces travelers to depart at a constant rate s from home. We denote the number of users in the

10 Note that this is true given our earlier assumption that only users of the bottleneck can be rewarded: if also a fixed subsidy equal to δ · (N/s) could be given to all individuals who do not use the bottleneck, and if there is no cost associated to the raising of the required public funds, the fine subsidy of (3) with a = 0 could again achieve the optimum. But since all potential travelers would need to receive the subsidy, it would be a very costly scheme.

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Fig. 1. Equilibrium with time-varying rewards for passing the bottleneck early or late.

central period as NC, and in the two fringes jointly as NF. With N still denoting the total use, we naturally have NC + NF ≡ N. The induced demand can be expected to make the reward system less efficient, and this would be the reason for having an unrewarded interval bB′,E′N of a positive duration. To optimize this duration, the relevant cost functions need to be identified. First observe that the average cost for the unrewarded central drivers is identical to what it would be in a conventional bottleneck equilibrium with NC ≡ N − NF users: cC = δ · (NC/s). With fine rewarding, the fringe drivers only experience schedule delay costs, which are on average equal to cF ¼ 12 ·δ·ðNC þ NÞ=s. 11 Because the reward itself is a transfer and we ignore the cost of public funds, 12 it is not part of the social cost C(NC, NF), which can therefore be written as: 1 2 ðN−NC Þ· ·ðN þ NC Þ NC 2 þ δ· C ðNC ; NF Þ¼ NC ·cC þ NF ·cF ¼ δ· s s 1 N 2 þ NC 2 : ¼ ·δ· 2 s

ð8Þ

If demand is completely inelastic, maximization of social surplus is equivalent to minimizing social costs, and it is easily verified in Eq. (8) that in this case it is optimal to have all drivers rewarded so that NC = 0. The time-varying reward then reproduces the first-best outcome, as argued in the previous section. When demand is pricesensitive, in contrast, there is the induced-demand effect of rewarding to take into account, because the first-best optimality conditions in Eqs. (7a) and (7b) can no longer be fulfilled. To see how this affects the second-best reward, we maximize social surplus by solving the Lagrangian:   N 2 2 1 N þ NC N Λ ¼ ∫ DðxÞdx− ·δ· þ λ· δ· C −DðNÞ : 2 s s 0

∂Λ N ¼ δ· C −DðNÞ ¼ 0 s ∂λ

ð10cÞ

These first-order conditions imply that the relative numbers of non-rewarded and rewarded drivers should be set such that: NC ¼

δ=s −D′ ðN Þ ·N⇔NF ¼ ·N: ′ δ=s−D ðNÞ δ=s−D′ ðNÞ

ð10dÞ

This reconfirms that with perfectly inelastic demand (D′(N) → − ∞) all drivers should be rewarded to maximize social surplus: it implies NC = 0 ; NF = N. In contrast, with perfectly elastic demand (D′(N) = 0) no one should be rewarded: NC = N ; NF = 0. The reason is that latent demand is then so strong that any attempt to decrease the number of queuing drivers is completely ineffective, as drivers will keep joining the unrewarded drivers up to the level where the initial equilibrium price is restored (note that this equilibrium price is determined by the constant marginal willingness to pay that applies with a perfectly elastic demand). The fine reward is then a completely ineffective instrument, and cannot produce any welfare gain. For intermediate elasticities, we expect the relative efficiency of fine rewards to decrease with demand sensitivity, along with the second-best optimal share of rewarded drivers as implied in Eq. (10d). Fig. 2 confirms this for a numerical example. The figure shows the relative efficiency of fine rewarding for various levels of demand elasticity (in the no-toll equilibrium), and for the full range of possible fractions of rewarded travelers. The latter is indicated as π = NF/N along the horizontal axis. On the vertical axis, ω denotes the relative efficiency: the gain in social surplus with the reward system, compared to the no-toll equilibrium, as a fraction of the gain in

ð9Þ

The constraint reflects that the generalized price will be equal for all drivers, and will amount to cN as derived above. The first-order conditions are as follows: ∂Λ N ′ ¼ DðNÞ−δ· −λ·D ðN Þ ¼ 0 s ∂N

ð10aÞ

∂Λ N δ ¼ −δ· C þ λ· ¼ 0 s s ∂NC

ð10bÞ

11 Because there is a linear relationship between schedule delay cost and t, and arrivals are constant over time between B″ and E″, the average schedule delay cost between E′ and E″ is the average of that at E′ and E″; and similarly for B″ and B′. 12 This means that we ignore the excess burden from other taxes that will have to be increased in order to pay for the reward, or that could be reduced when a road tax were used.

Fig. 2. Relative efficiency (ω) of fine rewarding by the fraction of rewarded travelers (π) for various elasticities of demand in the no-toll equilibrium: − 0.1 (red, solid), − 0.2 (purple, large dashing), − 0.4 (blue, medium dashing), and − 0.8 (green, small dashing).

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surplus with first-best tolling. Hence, ω = 0 corresponds to social surplus in the no-toll equilibrium, and ω = 1 to that in the first-best optimum. The results are obtained for a numerical specification with the following parameters: α = 7.5; β = 3.75; γ = 15; N = 9000; and s = 3600. The implied ratios of utility parameters reflect the usual values (e.g., Small, 1982; Arnott et al., 1993); the value for α (when expressed in Euros) is close to the current “official” value in The Netherlands; and the implied peak duration of 2.5 (hours) seems reasonable for a morning peak bottleneck. The two parameters in the assumed linear demand function are chosen such that both the target level of N and the target demand elasticity are achieved in the base (unpriced) equilibrium. The upper solid line in Fig. 2 shows that with a small absolute value of the elasticity, the efficiency of the reward system approaches that of the first-best situation: ω approaches 1. It is also optimal to have almost all drivers rewarded: the top of the curve is not far from π = 1. When demand is more elastic, represented by the lower lines, both the relative welfare gains and the optimal fraction of rewarded drivers – the π for which a maximum ω is reached – decrease. This confirms the conjecture that strong latent demand effects make a reward system less effective. Note that for the highest (absolute value of the) demand elasticity, a scheme in which more than 90% of the drivers are rewarded in fact lowers welfare compared to the no-toll equilibrium. The adverse induced demand effect then outweighs the benefit of eliminating queuing for the rewarded drivers. 4. Coarse rewards, feebates and tolls We now turn to consider simpler but probably more realistic incentive schemes. These concern ‘coarse’ schemes, in which tolls or rewards are constant over time within the periods during which they apply. One example is the coarse toll considered by Arnott et al. (1990, 1993), which is a uniform toll that is levied during a central part of the peak period only, for instance between B′ and E′ in Fig. 1, while the toll is zero outside that period. The coarse reward, in contrast, could be zero in the central part of the peak and positive in the fringes. We will discuss the similarities and differences between these two systems below. In addition, we will consider what we will call the coarse ‘feebate’, defined as a budget neutral combination of a coarse toll (in the central part of the peak) and reward (in the fringes). For completeness, we will also consider the benchmark of an unrestricted ‘step toll’, for which the peak is again divided in three periods during which differentiated fringe and central toll or reward levels can apply, but now no constraints on the signs or values of these tolls, or their net revenues, have to be met. Because tolls are piecewise constant in the regimes considered in this section, queuing will remain existent in equilibrium both during the fringes and in the central period. As before, in all equilibria that we will consider, all drivers experience the same generalized prices. The average costs for the two groups differ, and the difference in toll or reward levels compensates for this difference to establish a market equilibrium. The only difference between the four regimes will be the way in which the cost difference is compensated for: through a reward given to the high-cost fringe users, a coarse toll levied on the low-cost central drivers, a toll difference between both groups (for the step toll), or a combination of a reward and a toll (for the feebate). We start the discussion by emphasizing that, as explained by Arnott et al. (1990), an equilibrium with a coarse incentive requires a mass departure at E′, i.e. at that moment after t⁎ where the incentive level changes. The mass departure leads to the sudden emergence of a queue at that moment. Only then can the generalized price right before E′, when a higher toll or lower reward applies, be equal to the (expected) generalized price right after E′. All travelers in the mass departure have the same probability of obtaining a particular position in the queue, so that all users who pass the bottleneck between E′ and

171

E″ experience the same expected cost. The actual costs will differ. The first driver passes the bottleneck right after E′, and experiences only a schedule delay cost equal to γ · (E′ − t*). The last one passes at E″, and therefore spends time E″ − E′ in the queue, implying a generalized cost α · (E″ − E′) + γ · (E″ − t*). Because the realization of generalized cost rises linearly between these two instants, the expected travel cost of the users in the mass departure is the average of these ex     

tremes, and equals 12 ·γ· E′ −t  þ 12 · α· E″ −E′ þ γ· E″ −t  . Throughout our analysis of coarse instruments we will assume γ N α. This is generally believed to be the more relevant case empirically for car traffic in the morning peak (e.g. Small, 1982; Arnott et al., 1990, 1993). As pointed out by Arnott et al. (1990), the relative size of α and γ is important in the analysis of coarse pricing because it affects the qualitative properties of the equilibrium. In particular, in the empirically less relevant case where γ b α, there will be users departing after the mass departure, joining the queue before it has fully dissipated, and the departure time interval will not shift as it does with γ N α. Fig. 3 illustrates the workings of the coarse incentive schemes, taking the coarse reward as an example. The solid line gives travel delay as a function of arrival time in the original equilibrium, and the dashed line represents the equilibrium with a coarse reward. Since demand is price-sensitive and the generalized price decreases with a reward, the total duration of the peak increases. During the early fringe, between B″ and B′, a constant reward applies so that the dynamic user equilibrium condition implies that a queue will be growing at the familiar rate β / α. The queue and travel delay will have been longest for the last rewarded person to pass the bottleneck during the early fringe, who therefore has departed well before B′. The early-fringe queue will be completely resolved at B′, but a next queue for central users starts growing at B′. That queue starts shrinking at t*, and will have resolved at E′. At E′, there is the mass departure of rewarded drivers, causing the instantaneous emergence of the third, late-fringe queue. Depending on an individual's place in that queue, there is again a positive waiting time. The lucky user at E′ will face no delay, but the user who passes at E″ will have had both the longest travel time and the largest schedule delay late. The qualitative pattern of travel delays, as just described, will be the same for each of the four coarse schemes. The temporal separation of fringe and central travelers that is achieved with the coarse incentive reduces average queuing times, as the queue will have disappeared both at B′ and right before E′. We can now determine the generalized costs for the central drivers, cN, and fringe drivers, cF, as functions of the numbers of drivers: cC ¼ δ·

NC β·γ with δ≡ βþγ s



cF ¼ cC þ δ ·

β· 12 ·ðα þ γÞ NF  with δ ≡ s β þ 12 ·ðα þ γÞ

ð11aÞ

ð11bÞ

These functions hold true for each of the four coarse schemes. Eq. (11a) follows immediately from noting that the cost for the central drivers will be equal to the level that would apply in a conventional notoll equilibrium with NC drivers; compare Eq. (2). Eq. (11b) follows from solving the set of two equations that, first, defines the two periods of fringe arrivals to be together sufficiently long to host these drivers, and secondly, that equates the (expected) generalized cost to be equal for early and late fringe drivers:  NF  ′ ″ ″ ′ ¼ B −B þ E −E s

ð12aÞ

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Fig. 3. Travel delays with a coarse incentive scheme: rewarding.

  1 ′ ″ ″ ′ β· B −B þ cC ¼ ·ðα þ γÞ· E −E þ cC : 2

ð12bÞ

Eqs. (12a) and (12b) can be solved to obtain:  ′ ″ B −B ¼

β

1 N 2 ·ðα þ γÞ · F þ 12 ·ðα þ γÞ s

:

ð12cÞ

Next, we use that cF is equal to the sum of cC and the additional schedule delay cost at B″ compared to B′:  ′ ″ cF ¼ cC þ β· B −B :

ð12dÞ

Substitution of Eq. (12c) into Eq. (12d) then produces Eq. (11b). Note that Eq. (11b) applies whenever the toll or reward levels are equalized for the early and late fringes, as we assume. It is therefore not a condition that would only apply for optimized tolls or rewards. It is clear from Eq. (11b) that the fringe users have a higher generalized cost than the central users. User equilibrium requires that the generalized prices of both groups are equal; the difference in toll or reward levels should bridge the gap. That is, the net difference Δ between toll and/or reward levels in the central versus the fringe periods should for all schemes be equal to: 

Δ¼δ ·

NF : s

ð13Þ

In what follows, we apply these results to find the optimal numbers of users in the various coarse schemes. We do so by maximizing social surplus under the side-condition that the generalized price faced by all users will be the same. The exact formulation of this side-condition differs over the systems, as will become clear below. 4.1. Coarse reward A coarse reward is zero in the central period and positive in the fringes, and thus attracts some users to these fringe periods. In equilibrium, the generalized price will for all drivers, rewarded or not, be equal to the generalized cost for central drivers. The Lagrangian is 13: N

Λ ¼ ∫ DðxÞdx−δ· 0

  2 N·NC N  ðN−NC Þ −δ · þ λ· δ· C −DðNÞ s s s

ð14Þ

13 The formulation for total cost in Eq. (14) follows after reorganization of NC·cC + (N − NC)·cF; note that we have substituted out NF = N − NC.

Note that the constraint is the same as in problem in Eq. (9) with fine rewards. The first-order conditions are as follows: ∂Λ N  N−NC ′ −λ·D ðNÞ ¼ 0 ¼ DðNÞ−δ· C −2·δ · s s ∂N

ð15aÞ

∂Λ N δ  N−NC þ λ· ¼ 0 ¼ −δ· þ 2·δ · s s s ∂NC

ð15bÞ

∂Λ N ¼ δ· C −DðNÞ ¼ 0: s ∂λ

ð15cÞ

Eq. (15c) can be substituted in Eq. (15a) to derive the second-best equilibrium expression for the Lagrange multiplier λ: λ¼

C 2·δ · N−N s ′ −D ðNÞ:

ð15dÞ

Recall that this shadow price should give the marginal impact on second-best optimized social surplus from a relaxation of the constraint; i.e., from an increase in the generalized price. With rewarding, this generalized price is below marginal social cost, so the shadow price is positive. The denominator shows that the shadow price decreases and approaches zero as demand becomes less elastic, and induced demand is consequently less distortive. The numerator shows that the shadow price decreases as fewer users are rewarded, capacity is higher, and the composite cost coefficient δ⁎ is smaller. Substitution of Eq. (15d) in Eq. (15b) reveals, after some manipulations, that the ratio of rewarded fringe travelers to total use should amount to: NF 1 δ −D′ ¼ · · : 2 δ δ=s−D′ N

ð15eÞ

The third term gives the ratio that was found for a fine reward, in Eq. (10d), and the interpretation is similar. In particular, the ratio approaches zero as demand approaches perfect elasticity, for the same reason as with a fine reward. The upper limit of Eq. (15e), which applies with perfectly inelastic demand, however, is now not unity as it was for the fine reward, but 12 ·δ=δ . This is the same share of fringe drivers that Arnott et al. (1990) find for a coarse toll.14 This reflects that for perfectly inelastic demand, the coarse reward is as efficient as the coarse toll. The two instruments are then equally effective in affecting departure time choice, the only relevant margin of behavior if demand is perfectly inelastic. The instruments should therefore be set in a similar way, the

14 Arnott, De Palma and Lindsey do not mention this share explicitly. But using their expressions for the coarse toll (their Eq. (14a)), and the moments at which it is switched on (their 14c) and off (their 14d), this share can be computed, and it indeed turns out to be equal to the one mentioned in the main text.

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only difference being that the fringe users now receive a subsidy that is equal to the tax that central users would pay in the coarse toll schedule. But as soon as demand becomes price-sensitive, it is preferable to reduce the fraction of rewarded drivers. Finally, the equilibrium value for the coarse reward, which we denote as σ, can be found by substitution of Eq. (15e) into Eq. (13). This gives: 1 δ −D′ σ¼ · · ·N: 2 s δ=s−D′

ð15fÞ

For perfectly inelastic demand, this again simplifies to an expression that applies for the coarse toll of Arnott et al. (1990). 15 For perfectly elastic demand, the second-best optimal reward becomes zero. 4.2. Coarse toll While the coarse reward is zero in the center of the peak and positive in the fringes, the coarse toll is positive in the center and zero in the fringes. Arnott et al. (1990) derive the second-best coarse toll for perfectly inelastic demand. In a later paper, Arnott et al. (1993) consider coarse tolling with price-sensitive demand, but they do not derive the corresponding second-best toll level analytically. To do so, we solve a Lagrangian that is rather similar to the one in Eq. (14), the difference being that the generalized price will now be the generalized cost level incurred by the fringe drivers, given in Eq. (11b), rather than the central drivers, given in Eq. (11a): N

2 N·NC  ðN−NC Þ Λ ¼ ∫ DðxÞdx−δ· −δ · s s 0   N  N−NC −DðNÞ : þ λ· δ· C þ δ · s s

ð16Þ

The first-order conditions are now:    ∂Λ N δ  N−NC ′ þ λ· −D ðNÞ ¼ 0 ¼ DðNÞ−δ· C −2·δ · s s s ∂N

ð17aÞ

∂Λ N δ−δ  N−NC þ λ· ¼0 ¼ −δ· þ 2·δ · s s s ∂NC

ð17bÞ

∂Λ N  N−NC −DðN Þ ¼ 0: ¼ δ· C þ δ · s s ∂λ

ð17cÞ

Eqs. (17c) and (17a) imply the following second-best equilibrium expression for λ:

Substitution of Eq. (17d) in Eq. (17b) shows that the ratio of fringe travelers to total use should now amount to: 

ð17dÞ

For given levels of the right-hand side variables, this implies a lower shadow price than for the coarse reward in Eq. (15d): the numerator is now twice as small, whereas the denominator is larger. This lower shadow price reflects that the problem of induced demand is, for the coarse toll, smaller than for the coarse reward. But the shadow price is still positive, reflecting that an increase in the generalized price by raising the fringe toll level (now zero) would raise welfare — see also the step toll below. 15 See their Eq. (14a). Note that, given N, this coarse toll is equal to the average level of the fine toll.

′

NF 1 δ δ =s−D ¼ · · : 2 δ ðδ þ δ Þ=2s−D′ N

ð17eÞ

For perfectly inelastic demand, we again find a fraction of 12 ·δ=δ , which is again equal to the ratio implied by the results of Arnott et al. (1990) for a coarse toll and perfectly inelastic demand. This ratio is therefore the same as for the coarse reward with perfectly inelastic demand. However, when approaching perfect elasticity of demand, Eq. (17e) implies a positive ratio of δ/(δ + δ*), which for example equals ½ if α = γ. This gives an important difference with the results for the coarse reward, where this share would approach zero. It reflects that the toll has a positive effect of discouraging travelers to use the congested bottleneck, whereas the reward tends to attract users. Note that the ratio of δ/(δ + δ*) for perfectly elastic demand is equal to that of 12 ·δ=δ for perfectly inelastic demand when α = γ (so that δ = δ*). 16 Finally, the equilibrium value for the coarse toll τ implied by Eq. (13) and Eq. (17e) is:

τ¼

1 δ δ =s−D′ · · ⋅N: 2 s ðδ þ δ Þ=2s−D′

ð17fÞ

Consistent with what was just said, this toll remains positive for a perfectly elastic demand, whereas the reward in Eq. (15f) then becomes zero. Also note that for given right-hand side variables, the toll will certainly be larger than the subsidy in Eq. (15f) as long as γ N α ⇒ δ N δ*, a condition usually found and assumed to apply in reality (e.g., Small, 1982). Under those conditions, the fraction in Eq. (17e) will also certainly exceed that in Eq. (15e). Note that the condition γ N α is more than sufficient in this context. This confirms the intuitive expectation that the coarse toll is, in absolute terms, bigger than the coarse reward, because it does not attract additional users to an already congested facility. As anticipated, with perfectly inelastic demand, the toll is again equal to the level found by Arnott et al. (1990). 4.3. Step toll The step toll is quite similar to the coarse toll just discussed, but allows a positive toll to be charged also in the fringes. 17 This fringe toll may – and will – be different from the central toll. One may expect the step toll to be at least as efficient as the coarse toll, because the fringe toll may be set at zero, which would make the two policies identical. The Lagrangian now becomes: N

C δ · N−N s λ ¼ δ : ′ s −D ðN Þ

173

2

N⋅NC  ðN−NC Þ Λ ¼ ∫ DðxÞdx−δ· −δ · s s 0   NC N−N  C þ λ· δ· þδ · þ τF −DðN Þ s s

ð18Þ

16 When α b γ, δ N δ*, and the ratio of fringe travelers with perfectly elastic demand is smaller than that with perfectly inelastic demand. There does not seem to be a simple intuition behind these results. 17 Lindsey et al. (2010) show that the early fringe toll should ideally have another value than the late fringe toll. Here, we will assume that the two fringe tolls should be equal, providing a closer match with the equality of incentive levels in the early and late fringe that apply in the other coarse schemes we consider. The differences between second-best optimal early and late fringe tolls found by Lindsey, Van den Berg and Verhoef (2010) are relatively small, anyway, so the numerical importance in the example below is likely to be limited.

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where τF denotes the fringe toll. The first-order conditions are now:    ∂Λ N δ  N−NC ′ þ λ· −D ðNÞ ¼ 0 ¼ DðNÞ−δ· C −2·δ · s s s ∂N

ð19aÞ

This means that the Lagrangian becomes: N

Λ ¼ ∫DðxÞdx−δ· 0

∂Λ N δ−δ  N−NC þ λ· ¼0 ¼ −δ· þ 2·δ · s s s ∂NC

ð19bÞ

∂Λ N  N−NC þ τF −DðNÞ ¼ 0 ¼ δ· C þ δ · s s ∂λ

ð19cÞ

∂Λ ¼ λ ¼ 0: ∂τF

ð19dÞ

þλ· δ·

2 N·NC  ðN−NC Þ −δ · s s

! ! 2 NC N N N  −2· C þ C −DðNÞ : þδ · s s s s·N

ð21Þ

The first-order conditions are now: ! ! ∂Λ N δ N 2  N−NC ′ þ λ· · 1− C2 −D ðNÞ ¼ 0 ð22aÞ ¼ DðNÞ−δ· C −2·δ · s s s ∂N N

  ∂Λ N δ δ δ ·NC  N−NC þ λ· −2· þ 2· ¼0 ¼ −δ· þ 2·δ · s s s s s·N ∂NC

ð22bÞ

! 2 ∂Λ NC N NC NC  −2· þδ · þ −DðNÞ ¼ 0: ¼ δ· s s s s·N ∂λ

ð22cÞ

The shadow price λ is now zero, reflecting that the generalized price and hence overall demand can be controlled perfectly using the fringe toll level τF. The zero shadow price reflects that, among the coarse schedules considered, this step toll is the most efficient: there are no binding constraints limiting the choice of the two toll levels. The ratio of fringe users can be found after substituting Eq. (19d) into Eq. (19b):

Somewhat surprisingly, because the term between large brackets in Eq. (22b) is equal to −1 / N times the term in front of the λ-term, the fraction of fringe users can be found directly from Eq. (22b), independent of λ:

NF 1 δ ¼ · : 2 δ N

NF 1 δ ¼ · : 2 δ N

ð19eÞ

ð22dÞ

Given that overall demand can now be controlled perfectly using τF, it is no surprise that this ratio is the same as what was found above for the coarse toll and the coarse reward for perfectly inelastic demand. There is now no reason to deviate from this ratio in an attempt to restrict demand. Substitution of Eq. (19d) and Eq. (19c) into Eq. (19a) gives the level of the fringe toll:

It is the familiar expression that we already found for perfectly inelastic demand, and for the step toll. This reflects that the feebate is ineffective at addressing the problem of induced demand. That gives a somewhat different reason than for the step toll to aim for the cost-minimizing ratio of fringe users without considering induced demand effects. The equilibrium expression for λ is tedious for this regime, but again reflects that it will approach zero as demand approaches perfect inelasticity:

N 1 N τF ¼ δ · F ¼ ·δ· : 2 s s

 δ ·N· N2 −NC 2  λ¼   2 : δ · N −NC 2 −D′ ·s·N2



ð19fÞ

Adding Δ then gives the central toll τC: τC ¼ δ·

N s

Using Eq. (22d) and Eq. (20a) we can compute the following levels for τ and σ: ð19gÞ

The fringe toll is now equal to the average level of what the fine toll would be with N users, while the central toll is twice as high and therefore equal to its maximum. The total revenues from step tolling are therefore, for a given N, higher than those from fine tolling. 4.4. Feebate Finally, we consider the coarse feebate: a budget neutral combination of a toll τ that has to be paid by all central drivers, and a reward σ that is received by all fringe drivers. The revenues from the toll have to be equal to the expenditures on the reward, and Eq. (13) still applies, so we can find an expression for τ as a function of N and NF: 8 N > 2 < τ þ σ ¼ δ · F s ⇒τ ¼ δ · NF : N > s·N : σ ¼ τ· C NF

ð20aÞ

Adding cC gives the generalized price in equilibrium: p ¼ δ·

! 2 NC N N N N 2  ðN−NC Þ  −2· C þ C : þδ · ¼ δ· C þ δ · s s s·N s s s·N

ð22eÞ

ð20bÞ

τ¼

1 N 1 δ δ ·δ· F ¼ · ·  ·N 2 4 s δ s

ð22fÞ

σ¼

  1 N 1 δ 2·δ −δ ·δ· C ¼ · · ·N  2 4 s δ s

ð22gÞ

Note that Eq. (22f) and Eq. (22g) imply that the difference in monetary prices between the fringe and central periods is consistent with Eq. (22d) in the sense that it is again equal to the coarse toll level for inelastic demand: τþσ ¼

1 δ · ·N: 2 s

ð22fÞ

The reader may also verify that substitution of Eq. (22d) into Eq. (22f) confirms that Eq. (13) is satisfied. 4.5. Numerical results It is clear from the above derivations that the main factor determining the relative performance of the four coarse schedules will be the elasticity of demand. We illustrate this in the context of the same numerical model as we used in the previous section. The results are summarized in Figs. 4, 5 and 6. The horizontal axes of these figures give the absolute value of the elasticity of demand in the no-toll equilibrium.

J. Rouwendal et al. / Regional Science and Urban Economics 42 (2012) 166–176

175

Fig. 4. Relative efficiency (ω) of step tolls (red, solid), coarse tolls (purple, large dashing), coarse feebates (blue, medium dashing), and coarse rewards (green, small dashing) by elasticity of demand (ε) in the no-toll equilibrium.

Fig. 6. Second-best optimal tolls (τ) and subsidies (σ, in negative quadrant) with step tolls (red, solid; upper curve is central peak and lower curve is fringe), coarse tolls (purple, large dashing), coarse feebates (blue, medium dashing; upper curve is central peak and lower curve is fringe), and coarse rewards (green, small dashing) by elasticity of demand (ε) in the no-toll equilibrium.

Fig. 4 shows the relative efficiency, ω. Consistent with our analytical findings, the four coarse regimes are equally efficient when the price elasticity of demand in the original equilibrium is equal to 0. When demand becomes more elastic, the step toll becomes relatively more efficient, the relative efficiency of the coarse toll remains more or less unchanged, but the other two systems – which rely partly or completely on rewarding and therefore suffer from induced demand – become gradually less efficient, although the gains remain appreciable. Fig. 5 shows the optimal fraction of fringe drivers in the four systems, NF / N. There is a large discrepancy between the systems that rely partly or completely on tolling, and the ‘purest’ reward system. The optimal fraction is clearly decreasing in the elasticity of demand for the latter system, whereas it remains rather constant in the other regimes. Finally, Fig. 6 shows the optimal values of the tolls and rewards in the four regimes. The figure is drawn such that negative values in the diagram denote positive rewards. In the step toll, the optimal value of both tolls is slightly decreasing in the elasticity of demand. This is consistent with total use declining more strongly due to step tolling as demand becomes more elastic. The optimal reward decreases in magnitude when demand becomes more elastic, confirming our analytical results. The coarse toll, and the toll and reward in the feebate regime, turn out to be rather insensitive to the elasticity of demand.

shown that with inelastic demand a fine (time-varying) reward is equivalent to a fine toll, and to a continuum of combinations of time-varying tolls and rewards (including fine ‘feebates’; i.e., combinations of tolls and subsidies with a zero net revenue for the regulator). When demand is price-sensitive, a reward becomes less attractive from the efficiency viewpoint, because it attracts additional users to the congested bottleneck. As a result, both the second-best optimal fraction of rewarded travelers in the scheme, and the relative efficiency that can be achieved with it, was found to decrease when demand becomes more elastic. We also studied the properties of coarse schemes. Both our analytical and simulation results suggest that a coarse reward is less effective than a coarse feebate, which is itself less effective than a coarse toll. Moreover, the relative efficiency of the coarse reward is more sensitive to the price elasticity of demand that of the other systems. The most efficient coarse system is the step toll, which is also allowed to be positive in the fringe periods of the peak. These conclusions reflect that congestion entails an external cost, which should ideally be internalized. Our analysis nevertheless suggests that a reward system can relieve congestion problems to an appreciable extent. Especially in situations where tolls are unusually unacceptable – one may think of road works, bad weather forecasts or large scale (sport) events – a reward to stimulate people to avoid the peak may be more attractive than a toll. When demand is expected to be rather elastic, it may be preferable to look for possibilities to impose a feebate system, rather than a reward system. Many potentially important aspects of a reward system have been left undiscussed in this paper. Heterogeneity among users – possibly in their preferred arrival time, but also in their shadow costs of travel delays and schedule delays – may be particularly important, as may be uncertainty about demand and the capacity of the bottleneck. These issues are left for future work.

5. Conclusion This paper analyzed the possibilities to relieve bottleneck congestion by using rewards instead of – or in combination with – taxes. We have

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