Congestion Tolling for Mixed Urban Freight and Passenger Traffic

This paper investigates the welfare effects of optimal tolling on urban traffic congestion, in a bottleneck model, with mixed freight and passenger users. The users' marginal utility of time is considered to be varying with time. Under both no-toll equilibrium and socially optimal tolling, the users are found to sort their arrival t ime according to the increasing rates of marginal utility at the destination. The optimal toll that maximizes social welfare does not change each user's indirect utilit y relative to the no-toll equilibrium, but completely removes the queue, which also removes the barrier of freight carriers to accept congestion pricing by relating their marginal utilit ies directly to the toll. When the toll is equally rebated, the proposed social optimal tolling is a Pareto improvement relative to the no-toll equilibrium. Those more productive users also suffer more in both no-toll equilibrium and optimal tolling, which indicates that a differentiated redistribution of toll revenues could be an incentive to improve productivity.


Introduction
This paper focuses on the dynamics of traffic congestion induced by two distinct transportation demands, i.e. passengers and freights.Freight vehicles are evidently found to be competing with passenger cars for the capacity of urban roads.Empirical ev idence [1] shows that the that freight traffic is a significant proportion of the total traffic volu me at the major interstate crossings of the Port Authority of New York and New Jersey (PANYNJ), and the flow of freight and passenger traffic are overlapped.GPS data of Tokyo Metropolis Area [2] indicates that trucks are concentrating in the city center during rush hours.In a survey of Italian cities [3], the peak of freight traffic often overlaps with the peak of passenger traffic, during 8AM-11AM and 3PM-5PM, due to regulatory concerns.A report of Beijing, China, indicates that over 80% of freight traffic volu me in terms of the number of trips occurs during daytime in the central area of the city, including 11.7% occurs during peak hours of passenger traffic [4].
Despite the existence of mixed freight and passenger traffic, the study of the dynamics of urban traffic congestion begins by treating the users as commuters only.As the pioneer work, [5] first depicts the scheduling preference of co mmuters as their travel cost when passing through a single bottleneck in the city center.The mod el assumes that commuters valued the travel time per se at a rate of .They are supposed to arrive at an exogenous preferred arrival time (PAT), and are subjected to a scheduling cost at a rate of per time unit fo r early arrival or per time unit for late arrival, which is wellknows as the specification.In light of this simp le and tractable scheduling preference specification, the congestion dynamics with respect to the heterogeneity of passenger traffic has been extensively discussed.
Limited nu mber of user classes and continuously distributed user classes are considered by [6] and [7] respectively, but the ratio of is fixed for the purpose of analytic convenience.[8] d iscusses equilib riu m properties by assuming a more general scheduling delay cost function, but the PAT remains exogenous.The specification of PAT with an indifference band has also been considered in a nu mber of studies to allow for flexibility of arrival time (e.g.[9], [10]).
However, it is not generally true that the preferred arrival t imes can be exogenously treated and are unrelated with scheduling cost, since the PAT itself is ambiguous without the mechanism underlying the preference decision.The seminal work of [11] first connects the scheduling preference with the utilities generated by the activities conducted before and after the travel.This specification relates the PAT with the marginal utility of being at the base and the destination, which facilitates the analyze of PAT decision behavior as well as more general time-vary ing marg inal utilities.The t ime -varying marginal utility has been empirically identified among commuters, as evidenced by [12][13][14].By elaborating the model of [11], [15] and [16] notes that if the travel duration is fixed and is irrelevant to time, then the traveler will optimally choose to travel between the times when his marginal utility of being at the base equals his marginal utility of being at the destination.[17] p rovides a useful review on the connection between the timevarying marginal utility specificat ion and the common αβ-γ preference.[18] further associates the agglomeration DOI: 10.1051/matecconf/201712402003 ICTTE 2017 of co-workers with their marginal utilit ies, as well as commuters' scheduling preferences.However, none of these literatures deal with the heterogeneity of marginal utilities, and the type of user are also limited to co mmuter.
Freight carriers differs fro m the private commuters in at least two ways.First, they tend to use vehicles with lager capacity, collecting several goods for mu ltiple receivers each trip to lower the marginal cost of delivery.Second, the fact that commuters travel time is usually not paid by their employers.Therefore, h igher value of travel time per vehicle of freight users than that of private commuters has been empirically found [19].On the other hand, a freight delivery trip to the city center often ends at an area where a number of receivers spatially close to each other.Due to the limited capacity of urban facilities and the heterogeneity of receivers' preferences, it is usually not possible for the receivers who locate at the city center arrive at their workplace simu ltaneously.As a result, the numbers of available receivers distribute unevenly over a specific period of time, which will cause the carrier's productivity to vary over t ime at the destination.This is exact ly the same mechanism as the time-vary ing productivity of commuters with the number of working dependents [20].It is therefore possible to treat the commuting and freight users by their heterogeneous marginal utilities.
Given these findings, it is now clear that the dynamics of urban road congestion involves both freight and commuting user who are heterogeneous users in terms of their marginal utilit ies, and that their marginal utilit ies is likely to vary over time.Th is naturally raise two questions: First, how the users with heterogeneous timevarying marginal utilit ies behave when queuing at a bottleneck, compared with ho mogeneous users?Second, if their behavior affects welfare distribution of the firstbest pricing strategy?
To deal with these two questions, this paper introduces the user heterogeneity to the time -varying marginal utility model [11].The scheduling preference of freight and commuting users is defined by a marginal utility function with a contentiously distributed increasing rate.This methodology is built in light of [16] that shows a way of analyzing the properties of congestion when the users' travel distances to the urban bottleneck are continuously distributed.By applying the model framework of [11], they find that users sort their arrival time at the bottleneck according to the t ravel distance to the bottleneck, and that the distant users tend to gain more in optimal tolling scheme.Our paper d iffers fro m [16] in two major respects.First, it concerns user heterogeneity in marg inal utility of time, instead of travel distance.Second, the achieved utility function of user is parametrized, which allows for a continuously distributed rate of marginal utility.We thereby find users sort their arrival time at the bottleneck by the increasing rate of marginal utilities at the destination under both no -toll equilibriu m and socially optimal tolling.The social optimal toll removes the queue completely and does not change social welfare when no toll revenue is rebated.The finding imp lies that equal redistribution to all users can be a Pareto improvement.
In the context of mixed freight and passenger traffic, extending the model of [16] by introducing the heterogeneity of marg inal ut ility also relates with the attitude of freight carrier towards congestion charging.Since the carrier calculates the cost of delivery per trip instead of the number of receivers they serve, the congestion pricing policy with a time-dependent cordon toll, wh ich is the most conventional way in practice, cannot enter the marg inal utility of freight carrier.The toll, therefore, becomes a fixed cost and does not vary with the number of receivers per trip.When the charge rate of delivery has to be set to the marginal cost under a complete co mpetit ive market, the receivers cannot get any price signal fro m the cordon toll to choose less congested delivery times [21].To cope with this fact, [22] proposes a time-distance pricing scheme to show an incentive of off-peak delivery to the carriers, wh ich, however, not applicable when receivers are densely concentrated in the city center.In the setting of the present paper, the above problems are solved by setting the optimal toll to be varying with the amount of available receivers.Indeed, this paper does not intent to discuss the market equilibriu m between the freight carriers and the receivers.Throughout the discussion, the charge rate to the receivers is assumed to be fixed.We also limit the freight traffic to light freight vehicles physical externalities (e.g.size and acceleration performance) are similar to private passenger vehicles.
In Section 2, the scheduling model of freight and passenger users is described in a unified fo rmulat ion.Section 3 shows the conditions when the no-toll equilibriu m uniquely exist.Then the socially optimal tolling that exactly removes the equilibriu m queue and the corresponding welfare effects are d iscussed in Section 4. A numerical examp le is simulated in Section 5, where the distribution of productivity increasing rate is bimodal.The setting of distribution is to reflect the largely differentiated emp irical value of travel t ime and schedule delay time estimated in freight and passenger users.Section 6 concludes the key findings and identifies the possible extension to be introduced.The proofs of propositions are all deferred to the Appendix B.

A unified model
Consider that road users consisting of freight carriers and private co mmuters travel along a single urban corridor fro m the suburb to the city center.The freight carrier loads a batch of packages to be delivered to mu ltip le receivers.Co mmuter each drives a car and travels alone.Before arriving at the destination, they travel without congestion first and then passing through a bottleneck with a constant capacity of vehicles per t ime unit.All users follow the FIFO(first-in-first-out) discipline.The freight carriers and co mmuters achieve their utility by spending their time with relating parties at their respective bases and the destination.Both freight carriers and commuters incur no further penalties other than the loss utility of time, when their arrivals at the ICTTE 2017 destination delays.The agglo meration o f their respective relating part ies varies with time of day, therefore, at either ends of the travel, users' marginal utilit ies of t ime (M UT) are time-vary ing.The exogenous agglomeration magnitudes are assumed to be decreasing at their respective base and increasing at the destination.The receivers locate very closely in the city center, then the travel time between different receivers is ignored.The commuters also do not need additional travel time when interacting with their co-workers.
Both the carriers and co mmuters desire to make maximu m use of their t ime at both the base and destination within an exogenously given time window, denoted by , which can be interpreted as the traveling time and the two periods of time adjacent to it.The achieved utility within the time window is denoted by , where and are respectively the time when the travel starts and ends.Suppose that the users have completely no gain during the t ime spending on the move and could only achieve utility gain before and after .We consider that the road is allowed to be tolled by measured in utility unit, and that the toll is not returned to the user.Then an individual user can attain utility of during .Since the travel demand is assumed to be fixed, we then define a social welfare function as the total utilities of all users.All other utility losses are assumed to be constant and thus able to be ignored without affecting the result qualitatively.
The marginal utility of t ime at the base is assumed to be and that at the destination to be Then can be described as where are constants, and are the cumulat ive utility at the base and the destination respectively.We assume that , and that .The achieved utility of a single user is therefore strictly concave.
The users have scheduling preferences that are only heterogeneous in terms o f marginal utility of t ime at the destination.The increasing rate of marginal utility at the destination follows a d istribution with density and cumulat ive distribution , both of which have support on .With the assumptions on and , there must be an intersection of the two marginal utility functions.We define the time when and intersect as the ideal arrival t ime(IAT), which is the optimal arrival t ime when the travel t ime being reduced to zero.Given we on ly consider varies among users, their IATs are identical.
In order to examine the impacts of tolling on users with heterogeneous MUT, we assume that all the users have to travel the same T time units before arriv ing at the bottleneck.The arrival time at the bottleneck is denoted by , and the arrival rate at the bottleneck by .When is larger than , a vertical queue is generated at the entrance of bottleneck.Let and be the time when the first and the last user jo in the queue respectively, then the cumulat ive arrival , and the total travel t ime of the user who arrives at the bottleneck at time will be .
A single user with fixed travels through the corridor and the bottleneck without congestion can achieve utility , where is the time he arrives at the bottleneck as well as the destination.To maximize the utility, he will choose the optimal arrival time which solves the utility maximization problem .A graphical solution to the maximization problem is illustrated in Figure 1.The shaded area gives the min imu m utility loss when the user schedule his arrival t ime at the optimal arrival t ime .The optimal arrival time is found to have the fo llo wing property.
Proposition 1 If and hold for all , and if, but not necessarily if, holds, then the optimal arrival time uniquely exists and .This proposition identifies the feature that travelers sort their optimal arrival t imes according to the increasing rates of their marginal utilities.The negative indicates that the traveler whose MUT at the destination increases faster will prefer to arrive earlier when traveling without congestion.
In particular, a freight carrier is more likely to gain a higher increasing rate of marginal utility at the destination than a single co mmuter, owing to the dense agglomerat ion of payloads in freight vehicles.In the mean while, the receivers of the payload are not likely to arrive at the city center simu ltaneously.Therefore, the agglomerat ion process of receivers leads to an increase to the marginal utility of shippers.The proposition shows us that the higher increasing rate of marg inal ut ility at the destination is one of the reason that drives carrier to arrive earlier at the bottleneck than commuters.[1] evidences that the peak hour for freight vehicles is typically one hour earlier than for a ll traffic in the mo rning with similar travel distance from suburbs to the urban area.

No-toll equilibrium
Consider each user schedules his travel by taking the decisions of all other users as given.The conditions when Nash equilibriu m exists and the basic properties can be described by the following propositions.

Proposition 2 Assu me that holds for all
. When users taking Nash equilibrium strategy, the queue begins and ends during an interval , where , such that and .The condition ensures that if the user with the highest increasing rate of marginal utilities arrive the bottleneck at his optimal arrival time, all the other users with lower increasing rate of marginal utilities arrive later than their optimal arrival t ime.The queue which starts at and ends at always exists during where the first and last user experience exactly no queue.The first user arrives at the bottleneck earlier and the last arrives later than his optimal arrival t ime.Th is phenomenon can be visualized in Figure 1, when travel time increased fro m to .We have so far shown the basic queueing property when the heterogeneity of marg inal utility exists.The following proposition shows the properties of arrival time, and the analytical solution of scheduling utility.

Proposition 3 Assu me that holds for all . The no-toll equilibrium exists and is unique. Users who derive increasing marginal utility at rate arrive at the bottleneck at time and its derivative satisfies
The scheduling utility under equilibrium is The proposition shows that the arrival time at the destination is a function of the increasing rate of marginal utility at the destination under no-toll equilibriu m.The derivative of is strict ly negative which indicates that those who gain higher increasing rate of marginal utility at the destination arrive earlier.Given the assumption of FIFO(first-in-first-out) discipline, the bottleneck preserves the arriving order at the entrance.The time of their arrival at the destination is also in the order of their at the destination.
According to the arrival schedule, the first user who arrives at bottleneck as well as the destination at time has the highest increasing rate .The user derives an increasing rate of his marginal utility at the destination arrives at the destination after users at the bottleneck at time , while those who gain lower increasing rate of the marginal utility arrive later.Lastly, the user with the lowest increasing rate of marg inal utility joins the queue, and arrives at the destination at time .

Optimal tolling
In this section, we first find the socially optimal toll , then discuss the welfare effects by the optimal toll on users with different increasing rates of MUT.
Proposition 4 A socially optimal toll exists if, but not necessarily if, .The arrival at the bottleneck begins at which solves The optimal toll completely removes the queue, which satisfies and gives one of the optimal tolls.The users arrive the bottleneck and the destination simultaneously at The order of the arrival at the destination remain unchanged relative to no-toll equilibrium.
Infin itely nu mber of optimal tolls can exist when the demand elasticity is ignored, where is one of them.The proof shows that the optimal toll keeps the sorting property of arrival under no-toll equilib riu m unchanged.The formulat ion of arrival t ime at the destination is also identical to the one under no -toll equilibrium.
Along with the co mmuters, the carriers are also facing the optimal toll that varies with the increasing rate of MUT at the destination .The toll now enters the marginal cost when delivering an additional order to a receiver.It is then of great interest to discuss if anyone is getting better or worse off by tolling when this important barrier for freight users to accept the toll is removed.The next proposition also answers the question if the arrival interval is able to be shifted by the optimal toll.
Proposition 5 With the conclusion of , consider that an socially optimal toll can be achieved by .The arrivals in social optimum are the same as that under no-toll equilibrium: . Relative to the no-toll equilibrium, the socially optimal toll does not change each user's indirect utility, when the toll revenues are not redistributed to the users.DOI: 10.1051/matecconf/201712402003

ICTTE 2017
Under the no-toll equilibriu m, the user with highest MUT increasing rate at the destination departs and arrives first to avoid the queue, and the one with lowest MUT increasing rate arrives last.If the queue were removed, the first user could arrive later, and the last one earlier.However, a social optimal toll that removes the queue requires the travelers pay exactly the same monetary cost as the scheduling utility gain.This exp lains the reason why the arrival interval under socially optimal tolling remains unchanged.This observation is very similar to the conventional bottleneck model with homogeneous user [23] and the one with heterogeneous users whose cost of arriving late differs [6].
It is also worthy to relate the current finding with that of [16].The heterogeneity of distance to the bottleneck in the model of [16] causes unequal gain and loss under the first-best tolling scheme.The first-best pricing scheme under their setting does not equal the time saving benefit.User who travels longer distance to the bottleneck whose MUT at the destination is higher than thos e short distance traveler, therefore they benefit more fro m the elimination of the queue.

Numerical example
A numerical simulat ion is presented here to visualize the theoretical model by freight and passenger users with continuously distributed increasing rates of MUT at the destination.The users are assumed to be a continuum with mass 1.The schedule preference is specified by the following achieved utility: where is assumed to be equal to without losing generality.and are arbitrarily set to 0 without affecting the result in qualitative.The capacity of bottleneck is 0.02 users per minute, wh ich means all users can pass the bottleneck in 50 minutes.Free flow travel time T before arriving at the bottleneck is arbitrarily set to 1 minute.The increasing rates of MUT at the destination follow a bimodal distribution, composed of two beta distributions, each with mass 0.5.One of the two has support on [1,2] and the other on [2,3].The density of is shown by Figure 2.
The simulat ion calcu lates for each , then performs a search of by solving .Under social optimu m, is found numerically to maximize mean scheduling utility as well as social welfare.Simu lation result of arrival times are illustrated in Figure 3.
The dot-dash line shows the situation when all users travel without congestion where only free flo w travel time is needed.As states, the optimal arrival t imes are sorted by the descending order .Since is assumed to be equal to , then .The first user with the highest schedules his individual optimal arrival time at .Under no-toll equilibriu m, as shown by the upper thick line, the user with the highest increasing rate of MUT at the destination shifts his arrival t ime to to avoid the queue, which is earlier than his optimal arrival time .He arrives at the destination as soon as he gets to the bottleneck, as there is no queue for h im.At the other end of the queue, the user gains the lowest increasing rate of M UT arrives last at time .He also experiences exactly no queue either.The thin line is the arrival time at the bottleneck, wh ile the lower thick line gives departure time fro m their bases.Since all user are supposed to travel the same time before arriving at the bottleneck, the departure time function has the same derivative as the arrival time at the bottleneck, and the traveler with lower always departs later.Next consider the situation under social optimu m.Since the functional form remain unchanged, the order of arrival at the destination is identical to that in no-toll equilibriu m.Moreover, the constant does not shift the arrival interval either earlier or later, co mpared with that of no-toll equilibriu m.Therefore, the arrival time under social optimu m shown by the upper dash line in Figure 3 coincides with the arrival time under no-toll equilibriu m.Consequently, the departure time in social optimum does not move.
Figure 5 presents the achieved utilit ies of users with different increasing rates of M UT at the destination.The indirect utility under no-toll equilibriu m shown by the lower thick curve consists only scheduling utility.The scheduling utility in social optimu m is shown by the upper thin line.The lower dash line gives the indirect utility in social optimu m, which exact ly coincides with the indirect ut ility under no-toll equilibriu m.The difference between them is thus the socially optimal toll.As stated in Proposition 5, the welfare gain in terms of scheduling utility is co mpletely offset, and no user can benefit fro m the tolling, when the toll revenues are not returned to the user.
It is noteworthy that the indirect utilities under both no-toll equilib riu m and social optimu m are a monotonically decreasing function of .The users with higher productivity increasing rate achieve less utility instead of more, as they incur higher scheduling cost.It implies that a higher rebate to those more productive users can motivate them to improve marginal utility.

Conclusion
This paper introduces the heterogeneity of preference to the time-varying marg inal ut ility scheduling model.The model is then applied to treat the dynamics of urban traffic congestion incurred by both freight and passenger traffic.By directly relating the productivity of freight carriers to the amount of toll, the proposed optimal tolling removes the fundamental barrier for freight carriers to accept congestion pricing.The first-best pricing strategy without rebate is found inducing no welfare gain or loss to any of the users, in spite of the heterogeneity of marginal utilities.Th is finding gives an important policy implication that an equal revenue rebate of a social optimal toll can generate a Pareto improvement, relat ive to the no-toll equilibrium.
Our interesting result is that the user who gains higher productivity increasing rate failed to achieve higher ut ility, which indicates that a differentiated redistribution of toll revenues can be an incentive to improve productivity.This shows the possibility of two interesting extensions to our current model.The first would be to examine the second-best pricing policy to freight and passenger traffic, for example in the form of freight dedicated lane [24] or pricing a portion of lanes [25].The second is to make the productivity increasing rate in our model endogenous.Co mpared with commuters, it is more possible for freight carriers to decide their marg inal utilit ies at the destination, as they could choose an appropriate load per trip.By capturing this responsive behavior, the dynamics of congestion and the effect of tolling policy can be more comprehensively examined.Proof of Proposition 2 Let be the minimu m arrival interval required by all N users in no-toll equilibriu m.By Proposition 1 , for all , then for all , otherwise there could be some users who are supposed to arrive within interval can shift their arrive to an earlier t ime at their optimal arrival t ime without queueing and then increase utility.At the end of the queue, one could similarly have for all .Therefore, and .Next we show that the length of the interval when notoll equilibriu m must be .If , there must be a period of time when the bottleneck is not fully utilized.Given , someone who is supposed to arrive later in the period can shift to an earlier arrival time when the bottleneck was not fully utilized, and this will lead to strict increase of utility and contradict the equilibriu m.If , there will be a residual queue at (see [26] for further discussion on the condition when the property of no residual queue holds).The user in the residual queue can arrive later at the bottleneck to reduce queueing time and arrive the destination at the same time as they wait in the residual queue.

Appendix A. Main notations
Proof of Proposition 3 To prove has a unique solution, we first assume the equilibriu m exists for the

Figure 1 .
Figure 1.Graphical solution to the utility maximization problem of users with different marginal utility increasing rates

Figure 2 .
Figure 2. The density of increasing rates of marginal utility at the destination.

Figure 3 .
Figure 3. Departure and arrival times under no-toll equilibrium and social optimum

Figure 4 .Figure 4 g
Figure 4. Queue length road users Capacity of the bottleneck in the city center M arginal utility of time A specific period of time within a day T Travel time from the base to the bottleneck Achieved utility Time-varying toll at the bottleneck M arginal utility of time at the base M arginal utility of time at the destination Constant of the marginal utility of time at the base Increasing rate(slope) of M UT at the base Constant of the marginal utility of time at the destination Increasing rate(slope) of M UT at the destination Cumulative utility at the base Cumulative utility at the destination

Figure 5 .
Figure 5. Utilities of users with different increasing rate of marginal utilities at the destination