A competitive system with transit and highway: Revisiting the political feasibility of road pricing

Journal Pre-proof A competitive system with transit and highway: Revisiting the political feasibility of road pricing Ren Tao, Hai-Jun Huang PII:

S0967-070X(19)30615-8

DOI:

https://doi.org/10.1016/j.tranpol.2020.01.011

Reference:

JTRP 2284

To appear in:

Transport Policy

Received Date: 18 August 2019 Revised Date:

7 December 2019

Accepted Date: 17 January 2020

Please cite this article as: Tao, R., Huang, H.-J., A competitive system with transit and highway: Revisiting the political feasibility of road pricing, Transport Policy (2020), doi: https://doi.org/10.1016/ j.tranpol.2020.01.011. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Reversed version, submitted to Transport Policy

A competitive system with transit and highway: Revisiting the political feasibility of road pricing Tao Ren, Hai-Jun Huang*1 School of Economics and Management, Beihang University, Beijing 100191, China

Abstract: This paper analyzes the political feasibility of road pricing. In a monocentric city, we assume that every morning, there are a fixed number of commuters going to work by car or by bus. A flat toll is levied on commuters who drive to work and the toll revenues are redistributed to all commuters or bus riders only, or not at all. If more than half commuters support or benefit from the pricing scheme, it is considered politically feasible, and vice versa. We show that no one supports implementing road pricing if the toll revenues are not returned, and that the two revenue redistribution measures can produce the identical modal split and departure pattern, but their politically feasible toll intervals are different. Furthermore, if commuters without cars are the majority, they will benefit from the pricing scheme while commuters with cars suffer, regardless of the redistribution measures. However, all commuters may benefit from the pricing scheme when commuters with cars account for an overwhelming majority. This research provides a method for the government to evaluate under what conditions implementing road pricing is politically feasible.

Keywords: Road pricing, Political feasibility, Bottleneck congestion, Monocentric city

* Corresponding author. Email: [email protected]

Revised version, submitted to Transport Policy

A competitive system with transit and highway: Revisiting the political feasibility of road pricing

Abstract: This paper analyzes the political feasibility of road pricing. In a monocentric city, we assume that every morning, there are a fixed number of commuters going to work by car or by bus. A flat toll is levied on commuters who drive to work and the toll revenues are redistributed to all commuters or bus riders only, or not at all. If more than half commuters support or benefit from the pricing scheme, it is considered politically feasible, and vice versa. We show that no one supports implementing road pricing if the toll revenues are not returned, and that the two revenue redistribution measures can produce the identical modal split and departure pattern, but their politically feasible toll intervals are different. Furthermore, if commuters without cars are the majority, they will benefit from the pricing scheme while commuters with cars suffer, regardless of the redistribution measures. However, all commuters may benefit from the pricing scheme when commuters with cars account for an overwhelming majority. This research provides a method for the government to evaluate under what conditions implementing road pricing is politically feasible.

Keywords: Road pricing, Political feasibility, Bottleneck congestion, Monocentric city

1

1. Introduction Economists consider road pricing as an efficient way to alleviate traffic congestion because it encourages commuters to adjust their behaviors (de Palma and Lindsey, 2011). Successful cases, London, Stockholm, etc., suggest that well-designed pricing schemes did improve the local traffic condition. However, road pricing is actually introduced only in a limited number of cities. This raises the question of why such an efficiency-enhancing policy is rarely adopted by governments. Many studies attribute this to lack of political feasibility (Gärling and Schuitema, 2007; Russo, 2013; Westin et al., 2016). This is because road pricing is not merely an economic policy; implementing such a policy is also a political problem (King et al., 2007). In spite of much importance of the political feasibility of road pricing, however, attention on it is quite scarce, markedly less than that on its economic efficiency from a normative standpoint. More broadly, research from a political economy perspective (more positive) seems to be overlooked for a long time. Fortunately, economists have arisen their interests to study urban traffic policy from the political economy angle (Borck and Wrede, 2005; Brueckner and Selod, 2006; De Borger and Proost, 2012, 2015, 2016 a,b; Russo, 2013; Westin et al., 2016; De Borger and Russo, 2017, 2018; Glaeser and Ponzetto, 2017), of which road pricing is, of course, one of the key topics. Inspired by De Borger (2018), in this paper we study road pricing from the political economy standpoint. Recently, the legislature and government of Beijing have listed road pricing as one possible policy to combat congestion. Unlike Edinburgh, Stockholm, etc., whether a Chinese city implements road pricing does not depend on the referendum, but mainly on the decision of policymakers due to different social systems. Therefore, it is important for the government to evaluate the political feasibility of road pricing before it is really implemented. Implementing a pricing scheme has different effects on people from different groups (Borck and Wrede, 2005; De Borger and Proost, 2015; De Borger and Russo, 2017, 2018). Every morning, many people who live in Xianghe, Yanjiao, etc., small towns around Beijing, go to work from their home to the workplaces. These small towns and Beijing are usually connected by expressways. Commuters who own cars travel by car or by bus. However, commuters without cars can only take buses. This phenomenon is ubiquitous. People in Beijing cannot get a car partly because the price is unaffordable for the staff in the bottom and partly because the limited ration of the license plate to limit the car ownership (license 2

plate lottery). Consequently, commuters can be divided into two different groups, i.e., commuters with and without cars, hereafter, called “car owners” and “non-car owners” respectively. Commuters with and without cars may differ in the value of time and sensitivity towards bus in-vehicle crowding. Once a pricing scheme is implemented, it may produce different effects on car owners and non-car owners. For commuters from a certain group, whether they support the pricing scheme or not is essentially determined by the potential reform outcome, or more precisely, dependent on the net benefits they perceived. Politicians focus on the potential change of welfare of commuters before and after introducing a pricing scheme. They expect that commuters benefitting from the pricing scheme are the majority because they seek for winning the next election (commuters vote in accordance with the reform outcome). Accordingly, a pricing scheme is considered politically feasible when a majority of commuters, i.e., more than a half, benefit from such reform compared to the status quo (no toll situation), where commuters’ generalized travel costs are reduced. With these observations in mind, we use a monocentric model to study the political feasibility of road pricing, where a monocentric city consists of a central business district (CBD) and a suburb. Commuters with an exogenously fixed number go to work from the suburb to the CBD either by car or by bus each morning. Commuters by car will be levied a flat toll. The toll revenues are used for lump-sum redistribution or for subsidizing to bus riders or not redistributed at all. We first demonstrate that no one supports levying a toll if the toll revenues are not returned, which is a trivial result and consistent with the literature (Marcucci et al., 2005; De Borger and Proost, 2012; De Borger and Russo, 2017). This result could be used to explain why city governments often promise earmarking the toll revenues for improving the local traffic to reduce oppositions before introducing a pricing scheme. We then show that driving takes less travel time than taking buses is a sufficient condition to ensure road pricing feasible. Because such a condition is quite mild, this result does provide support for implementing road pricing, especially in cities being confronted severe congestion. Moreover, both lump-sum redistribution and subsidizing public transport can produce the identical modal split and departure pattern, even though the politically feasible toll intervals associated with them are different. In particular, the politically feasible toll interval for the former is wider than that for the latter. This result gives a new understanding of why the toll revenues should be mainly used for subsidizing public transport, since doing so can in general lessen commuters’ out-of-pocket costs. Finally, if non-car owners are the majority, they will always benefit from the pricing scheme, while car owners lose no matter how the toll revenues are 3

used. However, all commuters may benefit from the pricing scheme when car owners account for an overwhelming majority. This is because car owners are exploited by non-car owners, and therefore, they are not willing to pay for the travel and vice versa for non-car owners. With the increase of the proportion of car owners, however, the extent of exploitation is limited for the reason that car owners share the relatively larger weight of toll revenues. Note that this result is different from Arnott et al. (1994), where users with higher value of time benefit while the others with lower value of time lose. Our paper is related to two strands of the literature. First, it relates to the small but growing literature on the political economy model of road pricing. Marcucci et al. (2005) analyzed the political acceptability of road pricing with a citizen-candidate framework. In their paper, users are divided into three types according to their income level and no one group forms the absolute majority. The political equilibrium is attained when a majority of users vote for the toll with and without redistribution. De Borger and Proost (2012) used a simple majority voting model to study how uncertainty regarding revenue use and modal substitution cost influence the benefits of travelers. We neglect such uncertainty and focus on under what conditions implementing road pricing is politically feasible, especially when there is a significant difference between users with and without cars. Classifying commuters into two different groups in accordance with whether or not they own cars was considered in De Borger and Proost (2015); however, they studied the political economy of public transport pricing and quality decisions rather than road pricing. Russo (2013) also used a monocentric model to study the political economy of a road toll and parking fees. Unlike us, his study focused on the effects of the institutional setup on the choice of traffic policy and the role of additional financial support. Using a two-region model, Borger and Proost (2016a) showed that decentralized decisions on road pricing are often better than centralized decisions in terms of expected welfare, in the absence of restrictions on federal decision making. However, after imposing some institutional constraints, e.g., uniform pricing or legislative bargaining, the efficiency of centralized decision making can be greatly improved and may outperform decentralization (Borger and Proost, 2016b). Their studies are mainly used to explain why some decisions on road pricing are determined at the central level, while others are not. Most recently, De Borger and Russo (2017, 2018) analyzed the political economy of parking fees and cordon tolls. In the two papers, the authors focus on the interest conflicts between different groups (e.g., downtown and suburb retailers, drivers and passengers, landowners and renters), and therefore compare political equilibrium with system optimum. Different from them, we only consider the interest conflicts between commuters with and without cars. 4

Second, our work is related to the literature on Pareto-improving pricing scheme. Road pricing is considered to be Pareto-improving if it reduces travel cost or improves social welfare while no one loses compared to no toll situation (Arnott et al., 1994; Liu et al., 2009; Guo and Yang, 2010; Lawphongpanich and Yin, 2010; Nie and Liu, 2010). Admittedly, Pareto-improving pricing schemes are really politically feasible in accordance with our definition of political feasibility. Different from us, Lawphongpanich and Yin (2010) emphasized how to solve a mathematical program with complementarity constraints to get a Pareto-improving toll without pricing intervention. Others considered revenue redistribution. Although many (see, for example, Arnott et al., 1994, Liu et al., 2009; Guo and Yang, 2010; Nie and Liu, 2010) consider user heterogeneity where users’ value of time either supports a continuous distribution function or differs in different classes, the heterogeneity that users have different sensitivity towards in-vehicle crowding is usually neglected. In addition, we assume commuters within the same group are identical, which provides us the opportunity to focus on the conflicts between two groups. The rest of this paper is organized as follows. Section 2 introduces a basic structure of a monocentric model with bottleneck congestion and presents two kinds of equilibria. Section 3 derives the politically feasible conditions of a flat toll in different situations. Numerical analyses are conducted in Section 4. Section 5 concludes the paper.

2. The model A monocentric city consists of a CBD and a suburb. A job center is located at the CBD while the suburb has not workplaces. Between the two areas is an expressway, consisting of a bus exclusive lane and some normal lanes, as shown in Fig. 1. The normal lanes have a bottleneck with a fixed capacity, measured in vehicles per hour. Every morning, a fixed number of commuters go to work from the suburb to the CBD. Commuters are divided into two different groups in accordance with whether or not they own cars, namely “car owners” and “non-car owners”, respectively. Treating car and public transport as perfect substitutes, and accordingly, car owners can choose a trip mode from the two alternatives, driving or taking buses. However, non-car owners have to choose public transit. Hereafter, for simplicity, we call commuters who possess cars and drive to work “drivers” and those who choose to take buses “passengers”. Carpooling is ruled out. Each car owner is assumed to minimize her generalized travel cost by choosing a specific departure time and a travel mode, while non-car owners can only decide which bus they will take in order to minimize their 5

generalized travel costs. The main notations used throughout this paper are summarized in Table 1. There are five groups of notations in Table 1, i.e., parameter, variable, no toll situation, pricing with ϕ =0 , and pricing with ϕ >0 . For the last three groups, other corresponding notations are introduced, e.g., N d ,0 is introduced when there is no toll; N d ,1 is introduced when levying a toll for a special case of ϕ =0 ; and N d ,4 is introduced when levying a toll for a general case of ϕ >0 . An equilibrium is attained if no car owner could reduce her generalized travel cost by unilaterally changing either the departure time or the travel mode, and meanwhile no non-car owner could reduce her generalized travel cost by unilaterally changing her departure time. Let us first assume N d ,0 < (1 − ϕ ) N , i.e., not every car owner chooses to drive to work (we will discuss the case of N d ,0 = (1 − ϕ ) N in Subsection 2.3). Hence, the equilibrium number of bus riders is equal to the number of non-car owners plus the number of passengers, i.e.,

ϕ N + ( (1 − ϕ ) N − N d ,0 ) . The equilibrium costs of drivers, passengers and non-car owners will be discussed in Subsections 2.1 ~ 2.2, respectively. Subsection 2.3 will investigate two kinds of equilibria.

Bottleneck

Highway

CBD

Suburb

Transit exclusive lane

Fig. 1. A simple monocentric model.

6

Table 1 Notations.

s1

Capacity of bottleneck, measured in vehicles per hour

s2

Maximum carrying capacity of a bus

N

Total number of commuters

ϕ

Proportion of non-car owners

Tb ( Tc )

Bus (Car) travel time

α1 ( β1 , γ 1 )

Value of unit travel time (unit schedule delay early, unit schedule delay late) for car owners Value of unit travel time (unit schedule delay early, unit schedule delay late) for non-car owners

λ ( λ1 )

Value of unit in-vehicle crowding for car (non-car) owners

τb

Bus fare

α ( β ,γ

)

Parameter

h m

t

Variable No toll situation

Pricing with

ϕ =0

*

ϕ >0

Total bus number Work start time or desired arrival time for commuters

τ

Toll level levying on drivers

N d ,0 ( N br ,0 )

Equilibrium number of drivers (bus riders) with no toll

cd ,0 ( c p ,0 , cnc ,0 )

Equilibrium cost of drivers (passengers, non-car owners) with no toll

N d ,1 ( N p ,1 )

Equilibrium number redistribution

cd ,1 ( c p ,1 )

Equilibrium cost of drivers (passengers) without revenue redistribution

N d ,2 ( N p ,2 )

Equilibrium number redistribution

cd ,2 ( c p ,2 )

Equilibrium cost of drivers (passengers) with lump sum redistribution

N d ,3 ( N p ,3 )

Equilibrium number of drivers (passengers) with subsidizing public transport Equilibrium cost of drivers (passengers) with subsidizing public transport

cd ,3 ( c p ,3 )

Pricing with

Bus time headway

of

of

drivers

drivers

(passengers)

(passengers)

without

with

revenue

lump

sum

N d , so ( N p , so )

Number of drivers (passengers) under the socially optimal state

cd , so ( c p , so )

Cost of drivers (passengers) under the socially optimal state

N d ,4 ( N br ,4 )

Equilibrium number of drivers (bus riders) with lump sum redistribution

cd ,4 ( c p ,4 , cnc ,1 )

Equilibrium cost of drivers (passengers, non-car owners) with lump sum redistribution Equilibrium number of drivers (bus riders) with subsidizing public transport Equilibrium cost of drivers (passengers, non-car owners) subsidizing public transport

N d ,5 ( N br ,5 ) cd ,5 ( c p ,5 , cnc ,2 )

7

2.1. Equilibrium cost of drivers without a toll All commuters (including car owners and non-car owners) have the same desired arrival time at the workplace. However, due to the limited road supply, some must leave home early and others late for avoiding excessive rush hour congestion in front of the bottleneck. Commuters who arrive at the CBD early or late incur a schedule delay cost. According to the standard bottleneck model (Vickrey, 1969), if the incoming flow rate at the bottleneck exceeds its capacity, a queue forms. The generalized travel cost for a driver includes the travel time cost, the waiting time cost and the schedule delay cost. Mathematically, the generalized travel cost for a driver who departs at t , denoted by c(t ) , is formulated as

{

}

c(t ) = α ( w(t ) + Tc ) + max β ( t ∗ − t − w(t ) − Tc ) , γ ( t + w(t ) + Tc − t ∗ ) ,

(1)

where w(t ) denotes the waiting time in the queue.1 At equilibrium, all drivers experience the same and minimal travel cost regardless of their departure times, i.e.,

cd ,0 = α Tc + δ where δ =

N d ,0 s1

,

(2)

βγ 2 . β +γ

2.2. Equilibrium cost of bus riders without a toll We apply the model proposed by de Palma et al. (2017) with some differences to describe the equilibrium for bus riders. A bus line with a dedicated right-of-way connects two stations without intermediate stops. There are m buses, indexed in order of departure, which run on a timetable. All buses have the same capacity and are occupied during morning peak hour.3 Service is perfectly reliable. Bus i leaves the origin station (in the suburb) on schedule at time ti , i = 1,L , k * ,L , m , where k * is the bus that can arrive at the CBD punctually. Bus travel time is independent of both departure time and bus occupancy. It is assumed that bus riders know the timetable and the crowding degree on each bus and 1

In this equation, Tc is assumed to be dependent on the traffic state (i.e., a free flow state or a congested flow state). The value of Tc keeps constant under a certain traffic state but can vary as the state changes. Obviously, the value of Tc under a congested state is higher than that under the uncongested state. In some literature (see, Newell, 1988; Li and Huang, 2017), the travel time is related to the traffic flow in the corridor. 2 See Arnott et al. (1990) for the rigorous derivations. 3 If some buses are running without passengers, these buses can always be out-of-service to decrease the operating cost. 8

therefore get on buses on time. Thus, at equilibrium, bus riders never have to wait for buses (Huang et al., 2005). Notice that bus riders are inclusive of passengers and non-car owners, where passengers have different value of time and sensitivity towards in-vehicle crowding from non-car owners; therefore, the equilibrium costs of passengers and non-car owners are different. We will first derive the equilibrium cost of passengers. Under our setting, the schedule delay cost for passengers on bus i is formulated as

ηi = max {β h(k * − i ), γ h(i − k * )} , i = 1,L , k * ,L , m .

(3)

Following de Palma et al. (2017), we assume that the in-vehicle crowding cost for passengers on bus i is linearly increasing with the number of bus riders actually carried by this bus, ni . It follows4

q(ni ) = λ

ni , i = 1,L , k * ,L , m . s2

(4)

Notice that λ in (4) denotes the value of unit in-vehicle crowding for car owners. If λ is large, car owners are more sensitive towards in-vehicle crowding, and vice versa. Accordingly, we define the generalized travel cost of a passenger on bus i as the sum of travel time cost, bus fare, schedule delay cost, and in-vehicle crowding cost, i.e.,

cip ,0 = α Tb + τ b + max {β h(k * − i ), γ h(i − k * )} + λ Let η =

ni , i = 1,L , k * ,L , m . s2

(5)

1 m ∑ηi be the average schedule delay cost of all buses for passengers. It follows m i =1

η=

1 m k * ( k * − 1) β + ( m − k * )( m − k * + 1)γ η = h. ∑i m i =1 2m

(6)

From (5), we have

ni =

(c λ

s2

i p ,0

− α Tb − τ b − ηi ) , i = 1,L , k * ,L , m .

(7)

Taking the sum between both sides of (7) from i to m leads to

N br ,0 =

s2

m

∑c λ i =1

i p ,0

−

s2

m

∑η λ i =1

i

−

ms2

λ

(α Tb + τ b ) .

(8)

At equilibrium, all passengers should experience identical generalized travel cost no matter which bus they take, i.e., cip ,0 = c pj ,0 = c p ,0 , ∀i ∈ [1, m], j ∈ [1, m] . Hence, we have

4

See, Tirachini et al. (2016), a recent empirical study about transit crowding costs. 9

Nbr ,0 =

s2

λ

mc p ,0 −

s2

m

∑η λ i =1

i

−

ms2

λ

(αTb + τ b ) .

(9)

Dividing both sides of (9) by m and rearranging lead to

λ Nbr ,0 + η + α Tb + τ b . ms2

c p ,0 =

(10)

Substituting (10) into (7) leads to ni =

s2

λ

(η − ηi ) +

N br ,0 m

, i = 1,L , k * ,L , m .

(11)

One can get the minimum equilibrium cost of passengers if and only if η in (10) is at its minimum. Taking h → 0 so that the departure of bus is continuous, one can get

k * = Int ( 0.5 + mγ ( β + γ ) ) by solving dc p ,0 dk * = 0 , where “ Int ( x )” is the minimal integer large than x for guaranteeing the punctual bus k * to be an integer. Considering that commuters with and without cars differ in the value of time and sensitivity towards in-vehicle crowding, we distinguish notations by using α1 , β1 , γ 1 and

λ1 to represent the values of unit travel time, unit early delay, unit late delay and unit bus in-vehicle crowding for non-car owners. It is further assumed that β β1 = γ γ 1 = λ λ1 > 1 .5 With this assumption, the equilibrium always exists and the punctual bus number does not change (note that 0.5 + mγ ( β + γ ) = 0.5 + mγ 1 ( β1 + γ 1 ) ). Accordingly, from (6), one can immediately give the average schedule delay cost for non-car owners, i.e.,

5

We first assume

β γ equals β γ following Arnott (1994) and Huang (2000). Now, the number of 1

1

bus riders on bus i is ni = s2 (η − ηi )

λ + N br ,0 m = s2 (η − η1,i ) λ1 + N br ,0 m , 1

i = 1, L , k , L , m , *

where η1,i denotes the schedule delay penalty of non-car owners on bus i . Thus, for any i ∈ [1, m ] , we have λ λ1 = (η − ηi ) (η1 − η1,i ) . Without loss of generality, we have

λ λ1 = η η1 = β β1 = γ γ 1

∗

by taking i = k . This assumption means that car owners incur a higher schedule delay penalty and are more sensitive towards in-vehicle crowding than non-car owners. It is reasonable for two reasons: (1) Car owners have more flexible choices in terms of travel mode and departure time, compared to non-car owners. Therefore, car owners can more easily adjust their travel mode or departure time based on their preferences or traffic state, while non-car owners have to take the bus (note that the bus schedule is fixed); (2) In most of developing counties such as China, car owners usually earn more than non-car owners, which means they may have a higher value of time. 10

η1 =

k ∗ ( k ∗ − 1) β1 + (m − k ∗ )(m − k ∗ + 1)γ 1 h. 2m

(12)

Similarly, it is easy to know the equilibrium cost of non-car owners from (10), i.e.,

cnc ,0 =

λ1 N br ,0 ms2

+ η1 + α1Tb + τ b .

(13)

Note that the total bus carrying capacity and the total number of bus riders actually carried are ms2 and N br ,0 , respectively. Hence, (10) simply shows that the equilibrium cost for passengers is the sum of the average in-vehicle crowding cost, the average schedule delay cost, the travel time cost and the bus fare (the same for (13)). In addition, (11) shows that the number of bus riders on a certain bus is determined by the schedule delay cost and the total number of bus riders. Graphically, Fig. 2 depicts the equilibrium cost of passengers for seven buses ( m = 7 ).6 In this figure, the travel time cost and bus fare are denoted with blue and black lines, respectively. Notice that passengers incur the identical travel time and pay the same amount of fare regardless of their departure times; accordingly, the seven blue or black lines are of the same height. However, the in-vehicle crowding cost and schedule delay cost are different from a bus to another, which are represented by green and red lines respectively. Note that bus i = 4 arrives at the CBD on time and carries the most commuters, and accordingly, travelers on this bus incur a zero schedule delay cost but suffer the most in-vehicle crowding. Meanwhile, the schedule delay cost linearly increases when the departure time is deviating from the punctual one, whereas the in-vehicle crowding cost decreases. The dash lines in this figure are used to separate different costs. This figure also shows that passengers on different buses incur the identical travel cost at equilibrium.

6

The equilibrium cost components of non-car owners is similar to that of passengers. However, non-car

owners incur a less equilibrium cost than passengers do, since non-car owners are assumed to be less sensitive towards in-vehicle crowding and have lower value of time. 11

q (ni )

ηi

α Tb τb

t1

t2

t3

t4 t5 Departure time

t6

t7

Fig. 2. Equilibrium cost components for passengers.

2.3. Separating equilibrium and pooling equilibrium There are two possible equilibria in our competitive system, namely a separating equilibrium and a pooling equilibrium. We differentiate the two equilibria in accordance with

whether or not the car owners choose to take buses. In particular, a separating equilibrium occurs if no car owner has the incentive to take buses so that N d ,0 = (1 − ϕ ) N , while a pooling equilibrium occurs if some car owners choose to drive and the remaining choose to take buses, i.e., N d ,0 < (1 − ϕ ) N . We have investigated a pooling equilibrium in Subsections 2.1 and 2.2. Here, we give a graphical illustration of this equilibrium, shown in Fig. 3.

12

cp

cd

cnc

(1 − ϕ ) N

cd ,0

ϕN

c p ,0

cnc ,0 A

C

λϕ N ms2

N d ,0

η1 + α1Tb + τ b

Nbr ,0

α Tc

O1

O

+ η + α Tb + τ b

O2

Fig. 3. A pooling equilibrium of the competitive system.

In Fig. 3, the travel costs of drivers and passengers are denoted by a blue solid line and a red solid line, respectively, while the travel cost of non-car owners is represented by a red dash line. Note that the three equilibrium costs are computed with (2), (10) and (13), respectively. The number of drivers (passengers) is equal to zero ( (1 − ϕ ) N ) at point O ( O1 ), and increases from point O ( O1 ) to point O1 ( O ), and is equal to (1 − ϕ ) N (0) at point O1 ( O ). Similarly, the number of bus riders is equal to zero at point O2 , and increases from

point O2 to point O , and is equal to N at point O . It should be noted that the travel cost of passengers starts from

λϕ N ms2

+ η + α Tb + τ b since there are ϕ N non-car owners who

have to take buses. Moreover, the red solid line is steeper than the red dash one since λ > λ1 . The equilibrium occurs at the intersection between the blue solid line and the red solid line, i.e., point A . Solving cd ,0 = c p ,0 gives the expression of N br ,0 , N br ,0 + N d ,0 = N , i.e., N br ,0 =

s2δ mN + ms1s2 (α (Tc − Tb ) − τ b − η ) s1λ + s2δ m

.

(14)

From (14), we know that the total number of bus riders increases when the bus service is improved, e.g., decreasing the bus headway or increasing the travel speed. The equilibrium costs for car owners and non-car owners can be determined by substituting (14) into (10) and (13), respectively. 13

In contrast, a separating equilibrium occurs when N d ,0 = (1 − ϕ ) N . In this case, the equilibrium costs for car owners and non-car owners are simply computed with (2) and (13) respectively. Similar to the pooling equilibrium, we give a graphical illustration of a separating equilibrium, shown in Fig. 4.

cp

cd

cnc

cd ,0 c p ,0

N br ,0 = ϕ N

N d ,0 = (1 − ϕ ) N

cnc ,0 λϕ N ms2

+ η + α Tb + τ b

C

A η1 + α1Tb + τ b

α Tc O

O1

O2

Fig. 4. A separating equilibrium of the competitive system.

Compared Fig. 4 to Fig. 3, it is not hard to see that the proportion of non-car owners with a separating equilibrium is greater than that with a pooling equilibrium. This is because a separating equilibrium is more likely to occur when non-car owners account for a large majority. In addition, there is no intersection between the red solid line and the blue solid line in Fig. 4, suggesting that no car owner is willing to be a passenger. This is the reason why a separating equilibrium occurs.

3. Political feasibility of road pricing Politicians are rational and seek to be reelected or reappointed (Biglaiser and Mezzetti, 1997; Russo, 2013). Image a two-period model for elections: in the first period, a politician decides whether or not to implement a pricing scheme; between periods, there is an election in which voters choose between the incumbent and a challenger.7 Once a pricing scheme is 7

The election might never occur. Here, we only use it to illustrate the rationality of the definition of

political feasibility. 14

implemented, commuters observe its effects on themselves. For commuters who benefit from the pricing scheme, they will vote for the incumbent in the election. For those who suffer, however, they will vote for the challenger. The candidate who gets more than a majority of support will win the election. Accordingly, politicians are devoted themselves to improving the welfare of the majority of commuters when they consider such a policy.8 We define a pricing scheme to be politically feasible if a majority of commuters (more than a half) benefit from the pricing scheme, where their generalized travel costs are reduced compared with the status-quo. Two different scenarios are considered in this study. The first is a special case that all commuters own cars so that they can go to work by car or by bus, at which commuters have the same value of time and sensitivity towards in-vehicle crowding, hereafter referring to as the homogeneous case. The second is a general case that some commuters have no cars so that they have to take buses, where commuters with and without cars differ in the value of time and sensitivity towards in-vehicle crowding, hereafter referring to as the heterogeneous case. For each scenario, drivers will be levied a flat toll.9 The toll revenues are either not redistributed or evenly redistributed to all commuters (lump-sum redistribution) or evenly redistributed to bus riders only (subsidizing public transport). Before discussing the political feasibility of levying a flat toll, two more assumptions are given as follows: (1) It is faster to drive than to take buses, i.e., Tc < Tb ; (2) At a given equilibrium state, an additional commuter is more willing to drive to work, i.e., ms2δ < s1λ . With the two assumptions, from (14) it immediately leads to N br ,0 <

N , showing that drivers are the majority, when the user 2

equilibrium is achieved with no toll. 8

Seemingly, the best response for politicians is to maintain the status-quo rather to make a reform.

However, there are at least two possible reasons which stimulate politicians to implement a new policy. Firstly, politicians are not exclusively concentrated on the re-election. They also focus on improving the social welfare (De Borger and Russo, 2017). Doing so may help to improve their “ego-rents” when traffic congestion is being alleviated after implementing a pricing scheme. On the other hand, if politicians do not take any measures at all, they might be regarded as incompetent due to the deteriorated traffic congestion. The reputational concerns will also influence their decisions (Suurmond et al., 2004). 9

We only consider a flat toll on drivers in this paper, mainly because it is quite easy to implement. Since

the step tolls (Lindsey et al., 2010) do not affect the modal split, they can also be levied simultaneously so as to reduce the queuing cost. Doing so, however, has no benefits in terms of extra insight but complicates the derivation significantly. 15

3.1. All commuters own cars (a special case with ϕ = 0 ) We first derive the condition which makes all buses being occupied. In reality, the operator can adjust the total number of buses and the departure time headway to meet the travel demand. Since we focus on the short-run effects of levying a toll, we assume the operator can only adjust the departure time headway, while the total number of buses keeps constant. That all buses are occupied requires that the first bus and the last bus have passengers, i.e.,

N N s2 s (η − η1 ) + br ,0 > 0 and 2 (η − η m ) + br ,0 > 0 (see (11)). Then we λ λ m m

have

h<

2λ (δ N + s1 g ) , 2 β y (k * − 1) − f δ s2

(15)

where g = α (Tc − Tb ) − τ b , f = k * (k * − 1) β + (m − k * )(m − k * + 1)γ and y = s1λ + s2δ m . Note that (15) implicitly requires h > 0 . We omit the intuitive requirement to be concise. If drivers are levied a toll and the toll revenues are not reimbursed, the generalized travel costs for drivers and passengers become (see, (2) and (10))

cd ,1 = α Tc + δ

N d ,1 s1

+τ ,

(16)

and

c p ,1 =

λ N p ,1 + η + α Tb + τ b , ms2

(17)

respectively. At equilibrium, all car owners should incur the identical travel cost, independent of their departure and mode choices. Hence, one can derive the expression of the number of passengers by solving cd ,1 = c p ,1 , i.e.,

N p ,1 =

s2δ mN + ms1s2 (α (Tc − Tb ) − τ b + τ − η ) s1λ + s2δ m

.

(18)

Note that the number of passengers is also computed with (14) for the homogeneous case. Obviously, N p ,1 is greater than N br ,0 , by comparing (18) with (14). Taking the derivative of c p ,1 with respect to τ , we have

10

dc p ,1 dτ

=

s1λ > 0 ,10 showing that the s1λ + s2δ m

Here, we can also have this result by taking the derivative of cd ,1 with respect to

at equilibrium. 16

τ , since cd ,1 = c p ,1

equilibrium cost increases with the toll, and all commuters suffer from the pricing scheme compared with no toll situation.11 Therefore, it is politically infeasible to levy a toll without toll revenue redistribution. Next, we will derive the politically feasible toll intervals for lump-sum redistribution and subsidizing public transport in Subsections 3.1.1 and 3.1.2, respectively.

3.1.1 Lump-sum redistribution For this moment, the total toll revenue is equal to τ N d ,2 , i.e., the product of the number of drivers and the toll. Seeing as the toll revenues are evenly redistributed to all commuters, the allowance that each commuter receives is

τ N d ,2 . According to (2) and (10), the N

generalized travel cost for passengers and drivers are

c p ,2 =

λ N p ,2 τN + η + α Tb + τ b − d ,2 , ms2 N

(19)

and

cd ,2 = α Tc + δ

N d ,2 s1

+τ −

τ N d ,2 N

,

(20)

respectively. Solving cd ,2 = c p ,2 for equilibrium yields N p ,2 = N p ,1 , showing that both N p ,2 and N p ,1 share the same expression, see (18). This is because the lump-sum redistribution does not change the modal split. Compared N p ,2 with N br ,0 , it shows that some initial drivers will switch to take buses, leading to an increase in the number of passengers. Let ∆N be the increased number of passengers compared to no toll situation, i.e., ∆N = N p ,2 − N br ,0 . It follows

∆N =

ms1s2τ . s1λ + s2δ m

For lump-sum redistribution, passengers support levying a toll if and only if

11

(21) 12

Note that the benefit from less emission caused by road pricing has no effect on the individual cost due

to the cost structure defined in this paper. This result is likely to be reversed if commuters prefer clear air very much and internalize the benefit into their costs. 12

For the homogeneous case, commuters either support or oppose implementing a pricing scheme. Hence,

if passengers benefit from the pricing scheme, so do drivers. 17

λ∆N τ ( N − N p ,2 ) < . ms2 N

(22)

The left hand side of (22) denotes the increase in the bus in-vehicle crowding cost compared with the status-quo, while the right hand side is the reimbursed subsidy. Therefore, (22) simply requires the total benefit overweight the total lose.13 Substituting (18) and (21) into (22), with some algebra, we have

τ <η − g .

(23)

Note that η is greater than g because of Tc < Tb . Therefore, (23) simply states that a politically feasible toll interval exists for lump-sum redistribution.

3.1.2 Subsidizing public transport If the total toll revenue is evenly redistributed to bus riders (here also referring to passengers) only, then the total toll revenue and the reimbursed subsidy are equal to τ N d ,3 and r =

τ N d ,3 , respectively. Again, one can give the generalized travel costs of passengers N p ,3

and drivers in accordance with (2) and (10), i.e.,

c p ,3 =

λ N p ,3 + η + α Tb + τ b − r , ms2

(24)

and

cd ,3 = α Tc + δ

N d ,3 s1

+τc ,

(25)

respectively. Solving c p ,3 = cd ,3 leads to

N p ,3 =

s2δ mN + ms1s2 (α (Tc − Tb ) − τ b − η + τ + r ) s1λ + s2δ m

,

(26)

and

r=

13

b 2 − 4ac − b , 2a

(27)

The condition is so particularly intuitive that we will use it to derive the feasible toll intervals repeatedly.

In the following analogous formulae, we will no longer explain the relation unless necessary (see, (29), (46), (49), (59), (62), (72) and (74)). 18

where a = ms1s2 > 0 , and

b = s2δ mN + a( g − η ) + 2aτ = yN br ,0 + 2aτ > 0 , c = ( a( g − η + τ ) − s1λ N )τ = y ( N br ,0 − N )τ < 0 . In this case, the increased number of passengers compared to no toll situation becomes

∆N = N p ,3 − N br ,0 , i.e., ∆N =

ms1s2 (τ + r ) . s1λ + s2δ m

(28)

Similarly, passengers support implementing the pricing scheme only when

λ∆N ms2


(29)

The left hand side of (29) denotes the increase in the bus in-vehicle crowding cost compared with the status-quo, while the right hand side refers to the subsidy for each passenger. Similar to (22), (29) states that the benefits are greater than the losses. Substituting (26) and (28) into (29), with some algebra, we have

τ<

s2δ m(η − g ) . s1λ + s2δ m

(30)

Equation (30) shows that the politically feasible toll interval does exist when the toll revenues are used to subsidize the public transport. Compared to (23), we know that the feasible toll interval under subsidizing public transport is less than that under lump-sum redistribution.

3.1.3 Optimal tolls and social optimum We refer the optimal tolls to that leading to the minimum equilibrium cost. By this definition, commuters benefit the most when levying the optimal tolls. Taking the derivatives of c p ,2 and c p ,3 with respect to τ yields

dc p ,2 dτ

=

s2δ mN + ms1s2 ( g − η ) + 2ms1s2τ ms2δ − , s1λ + s2δ m ( s1λ + s2δ m ) N

(31)

and dc p ,3 dτ

= 1−

s2δ mN b 2 − 4ac

Solving dc p ,2 dτ = 0 leads to

19

.

(32)

τ = (η − g ) / 2 .

(33)

Solving dc p ,3 dτ = 0 leads to

N  1  s2δ m + br ,0  (η − g ) . 4  s1λ + s2δ m N 

τ= 

(34)

Obviously, the two optimal tolls belong to the corresponding feasible toll intervals. Interestingly, the minimum equilibrium costs under the above two different redistribution measures are equal. To see this, let us now solve the social optimum problem. At social optimum, the total travel cost of the system is minimized, i.e, solving the following problem

min N d , so cd , so + N p , so c p , so , subject to N d , so + N p , so = N , where, c p , so =

(35)

λ N p , so N + η + α Tb + τ b and cd ,so = α Tc + δ d , so . Solving (35) leads to ms2 s1 N p , so =

2s2δ mN + ms1s2 ( g − η ) . 2 ( s1λ + s2δ m )

(36)

It is obvious that N p , so > N br ,0 , and therefore c p , so > cd , so (note that η > g ). This implies that, compared with user equilibrium, more commuters are required to take buses to minimize the total travel cost. Therefore, for lump-sum redistribution, a toll, which is equal to

(c

p , so

− cd , so ) , is required to levy to decentralize the social optimum. By substituting N p , so

into c p , so and cd , so , and rearranging

(c

p , so

− cd , so ) , we get the socially optimal toll

τ so = (η − g ) / 2 .

(37)

For subsidizing public transport, the socially optimal toll satisfies that τ so N c , so = rN p , so and τ so + r = c p , so − cd , so . Combining and solving the two equations leads to

N  1  s2δ m + br ,0  (η − g ) . 4  s1λ + s2δ m N 

τ so = 

(38)

We can see that the two tolls which can decentralize the social optimum are exactly the same as the optimal tolls given with (33) and (34). This implies that the optimal tolls can decentralize the competitive system to social optimum. Note that all toll revenues are redistributed; therefore, the equilibrium individual cost is minimized only when the social optimum is obtained, and is equal to

N p , so c p , so + N d , so cd , so N

, regardless of the toll

redistribution measures. This further suggests that any possible equilibrium pattern (different 20

modal splits and departure patterns) can be supported by two different redistribution measures (note that different equilibrium patterns correspond to different equilibrium costs). We now summarize the results obtained in the case of ϕ = 0 in Proposition 1.

Proposition 1. (i) No one supports levying a flat toll if the toll revenues are not reimbursed; (ii) If driving is faster than taking buses, the politically feasible toll always exists, regardless of the revenue redistribution measures used; (iii) Any equilibrium pattern (modal split and departure pattern) can be supported by lump-sum redistribution or by subsidizing public transport, while the politically feasible toll interval under lump-sum redistribution is wider than that under subsidizing public transport. This proposition illustrates the importance of how the toll revenues are used. It is consistent with the observation that city governments often promise earmarking the toll revenues for improving the local traffic to reduce oppositions before introducing a pricing scheme (Leape, 2006). A sufficient condition to make levying a flat toll politically feasible is that the travel time of driving is less than that of taking buses. This is a quite mild condition. Accordingly, the proposition provides support for implementing road pricing. While the two different revenue redistribution measures can produce the identical equilibrium pattern, subsidizing public transport can in general lessen commuters’ out-of-pocket costs compared to lump-sum redistribution. This can be used to explain why the toll revenues are mainly used to subsidize public transport in many cities which have successfully implemented road pricing, for example, Landon, Stockholm, and Milan.

3.2. Some commuters own cars (a general case with ϕ > 0 ) For this moment, there are two possible equilibria without a toll, namely, a separating equilibrium and a pooling equilibrium (see, Section 2.3). Particularly, a separating equilibrium occurs if and only if no car owner has the incentive to choose to take buses, which requires cd ,0 ≤ c p ,0 (see, Fig. 4). This leads to ϕ ≥ N br ,0 / N . Conversely, there is a pooling equilibrium if ϕ < N br ,0 / N . Suppose that there is a separating equilibrium without a toll. For this moment, one can give a sufficient condition under which all buses will be occupied. Notice that there are now

ϕ N non-car owners who have to choose public transit. That all buses are occupied requires s2 ϕN s ϕN (η1 − η1,1 ) + > 0 and 2 (η1 − η1,m ) + > 0 (see (11)), where η1,1 (η1,m ) represents λ1 λ1 m m 21

the schedule delay cost for non-car owners on the first (last) bus. Hence, we have

h<

2λ1ϕ N , s2 ( 2mβ1 (k * − 1) − f1 )

(39)

where f1 = k1* (k1* − 1) β1 + (m − k1* )(m − k1* + 1)γ 1 . Obviously, if there is a pooling equilibrium before levying a toll, all buses will be occupied by giving any h-value satisfying (39), since the number of bus riders increase compared to the separating equilibrium. Now let us assume (39) is satisfied. With this assumption, we will derive the politically feasible toll intervals for an initially separating equilibrium in Subsections 3.2.1 and 3.2.2, respectively. Subsection 3.2.3 will investigate the case of an initially pooling equilibrium.

3.2.1 Separating equilibrium with lump sum redistribution If drivers are levied a flat toll with lump sum redistribution, the generalized travel costs of drivers and passengers can be denoted as

cd ,4 = α Tc + δ

N d ,4 s1

+τ −

τ N d ,4 N

,

(40)

and

c p ,4 =

λ N br ,4 ms2

+ η + α Tb + τ b −

τ N d ,4 N

,

(41)

respectively. Obviously, there exists such a toll, denoted by τ ′ , that makes cd ,4 equal to c p ,4 , and meanwhile the separating equilibrium still exists. In order to find such a toll, it only requires substituting N d ,4 = (1 − ϕ ) N and N br ,4 = ϕ N into (40) and (41).

Then,

solving

cd ,4 = c p ,4 leads to

τ′ =

ϕ Ny − Nms2δ ms1s2

+η − g .

(42)

From (42), we know that a separating equilibrium still exists as long as τ ≤ τ ′ . Otherwise, i.e., τ > τ ′ , the initial separating equilibrium will turn to be a pooling equilibrium. Without loss of generality, let us now assume that τ in (40) and (41) is not less than τ ′ . Further, one can derive the equilibrium number of bus riders by solving cd ,4 = c p ,4 again, i.e, N br ,4 =

s2δ mN + ms1s2 (α (Tc − Tb ) − τ b + τ − η ) s1λ + s2δ m

.

(43)

We see that the expression of N br ,4 is the same as the expression of N p ,1 . However, the 22

meanings of the two expressions are totally different. Notice that N p ,1 is derived under the assumption that all commuters own cars; hence, N p ,1 is the equilibrium number of passengers. Differently, N br ,4 denotes the equilibrium number of bus riders, which is comprised of passengers and non-car owners. With the previous analyses, we know that the number of non-car owners can be equivalently denoted as

ϕN =

s2δ mN + ms1s2 (α (Tc − Tb ) − τ b + τ ′ − η ) s1λ + s2δ m

.

Now, the increased number of bus riders, ∆N , is equal to

∆N =

(44) (43) minus (44), i.e.,

ms1s2 (τ − τ ′) . s1λ + s2δ m

(45)

Analogous to (22), non-car owners support levying a flat toll if and only if

λ1∆N ms2 Substituting

<

τ ( N − N br ,4 ) N

.

(46)

(43) and (45) into (46) and rearranging, we have

a1τ 2 + b1τ + c1 < 0 ,

(47)

where a1 = a > 0 ,

b1 = s1λ1 N − yN d ,0 < s1λ N − yN d ,0 = a ( g − η ) < 0 , c1 = − s1λ1 Nτ ′ < 0 . Hence, the toll interval that benefits non-car owners is

b12 − 4a1c1 − b1 . 2a1

τ<

(48)

Similarly, car owners support implementing the pricing scheme only when

τ ( N − Nbr ,4 ) N Substituting

+δ

∆N >τ . s1

(49)

(43) and (45) into (49) and rearranging, we have

a2τ 2 + b2τ + c2 < 0 ,

(50)

where a2 = a > 0 , b2 = a ( g − η ) < 0 , and c2 = ms2δ Nτ ′ > 0 . Therefore, it requires b2 2 − 4a2 c2 > 0 to ensure that (50) has positive solutions. This leads to

23

ms ( 2δ N + s1 ( g − η ) ) ϕ< 2 . 4δ yN 2 2

(51)

If ϕ satisfies (51), the toll interval that benefits car owners is

 −b − b 2 − 4a c −b + b 2 − 4a c 2 2 2 2 2 2 τ ∈ 2 , 2  2a2 2a2 

 .  

(52)

There are two positive roots when taking (50) equal to zero. It is not surprising because there is a separating equilibrium before levying a toll. With lump-sum redistribution, the equilibrium cost will first increase, then decrease, and increase again with the toll.

3.2.2 Separating equilibrium with subsidizing public transport If the toll revenues are evenly redistributed to bus riders only, the generalized travel costs of drivers and passengers can be denoted as

cd ,5 = α Tc + δ

N d ,5 s1

+τ ,

(53)

and c p ,5 =

λ N br ,5 ms2

+ η + α Tb + τ b −

τ N d ,5 N br ,5

,

(54)

respectively. Similar to lump sum redistribution, there also exists such a toll, denoted by τ * , that makes cd ,5 equal to c p ,5 , and meanwhile the separating equilibrium still exists. Substituting N d ,5 = (1 − ϕ ) N

and N br ,5 = ϕ N

into (53) and (54), and then solving

cd ,5 = c p ,5 , we have

τ * = ϕτ ′ .

(55)

Equation (55) simply states that a separating equilibrium still exists if τ ≤ τ * , and the initially separating equilibrium will become a pooling equilibrium otherwise. Again, the number of non-car owners can be also denoted as

ϕN = where r * =

s2δ mN + ms1s2 (α (Tc − Tb ) − τ b − η + τ * + r * )

(1 − ϕ )τ *

ϕ

s1λ + s2δ m

,

(56)

denotes the subsidy each bus rider receives. If τ > τ * , then solving

cd ,5 = c p ,5 will give the explicit expression of the equilibrium number of bus riders

24

N br ,5 =

s2δ mN + ms1s2 (α (Tc − Tb ) − τ b − η + τ + r ) s1λ + s2δ m

.

(57)

It shows that both N br ,5 and N p ,3 share the same expression. However, N br ,5 denotes the number of bus riders, while N p ,3 denotes the number of passengers. For this moment, the increased number of bus riders (the number of passengers) becomes ms1s2 (τ + r − τ * − r * ) ∆N = , s1λ + s2δ m

(58)

which equals the number of bus riders less the number of non-car owners. Accordingly, non-car owners support levying a toll if and only if

λ1∆N ms2

<

τ ( N − N br ,5 ) N br ,5

(57) and

Substituting

.

(59) (58) into (59) and rearranging, we have

a3τ 2 + b3τ c + c3 < 0 ,

(60)

2

 y  where a3 = a   > 0,  y − s1λ1  b3 =

y  s1λ1 ′  τ  − yN < 0 ,14  yN br ,0 − 2a y − s1λ1  y − s1λ1  2

 sλ  sλ c3 = a  1 1 τ ′  − 1 1 τ ′ yN br ,0 .  y − s1λ1  y − s1λ1 Because whether c3 < 0 is uncertain, we cannot immediately obtain the politically feasible toll interval. Fortunately, the toll interval that benefits non-car owners does exist with some intuitive analyses. Reconsidering the prerequisite that a separating equilibrium exists before levying a toll, then for any toll belonging to 0,τ *  , non-car owners benefit from the pricing scheme because they receive subsides but do not suffer loss. Once the toll exceeds τ * , some car owners will switch to take buses, leading to an increase in the in-vehicle crowding cost.

14

We have b3 =

y y − s1λ1

s2δ mN +

sλ   a ( g − η ) − 2a 1 1 τ ′  − yN by substituting N br ,0 into the  y − s1λ1  y − s1λ1  y

original expression of b3 . Since λ > λ1 , we know η > g , we then have b3 <

y y − s1λ1

s2δ mN <

sλ   a ( g − η ) − 2a 1 1 τ ′  < 0 .  y − s1λ1  y − s1λ1  y

25

y y − s1λ

s2δ mN = yN . Note that

Undoubtedly, there is a toll level such that the equilibrium costs are the same as before and after levying a toll. Therefore, the toll interval that benefits non-car owners is

−b3 + b32 − 4a3c3 τ< . 2a3

(61)

Similarly, car owners support implementing the pricing scheme if and only if

δ

∆N >τ . s1

(62)

(58) and (27) into (62) and rearranging, we have

Substituting

a4τ 2 + b4τ + c4 < 0 ,

(63)

where,

a4 = a +

s14 λ 2 2s12 λ + >0, aδ 2 δ

 s 2λ  b4 =  a + 1  ( g − η + 2τ ′ ) , δ   c4 = aτ 2 + yN br ,0τ ′ > 0 . In order to guarantee that there exists politically feasible toll interval, it requires b4 < 0 and

b4 2 − 4a4 c4 > 0 .

( g − η ) 2 − 4τ ′

δN s1

The

two

conditions

are

equivalent

to

2τ ′ < η − g

and

> 0 , respectively. From 2τ ′ < η − g , we have

ϕ< From ( g − η ) 2 − 4τ ′

δN s1

N br ,0 N

+

a (η − g ) . 2 yN

(64)

> 0 , we have

N ms ( s ( g − η ) + 2δ N ) as1 = br ,0 + ϕ< 2 1 ( g − η )2 . 2 2 4δ N y 4δ yN N 2

Notice that

(65)

a (η − g ) a (η − g )( 2δ N − s1 ( g − η ) ) as1 − ( g −η )2 = > 0 . Hence, it only 2 2 yN 4δ yN 4δ yN 2

requires (65) is satisfied to ensure there exists a politically feasible toll interval. Interestingly, the condition is exactly the same as that derived under lump sum redistribution (see, (51)). When (65) holds, the toll interval that benefits car owners is

 −b − b 2 − 4 a c −b + b 2 − 4 a c 4 4 4 4 4 4 τc ∈ 4 , 4  2a4 2a4  26

 .  

(66)

From the above analyses, we have the following lemmas.

Lemma 1. If ϕ > 0.5 or

N d ,0 N

<ϕ <

N d ,0 N

+

as1 ( g − η ) 2 , the politically feasible toll 2 4δ yN

always exists.

Proof. Let us first prove this lemma for ϕ > 0.5 . When the toll revenues are evenly redistributed to all commuters (bus riders only), the separating equilibrium still exists if

τ < τ ′ ( τ < τ * ). For this moment, it is obvious that car owners suffer from the pricing scheme, while non-car owners benefit. Since non-car owners account for a majority, any toll level, which is less than τ ′ ( τ * ), is politically feasible for lump sum redistribution (subsidizing public transport). Next, let us consider the case of τ ≥ τ ′ ( τ ≥ τ * ) for lump sum redistribution (subsidizing public transport). For lump sum redistribution, differentiating cd ,4 with respect to τ leads to

dcd ,4 dτ

=

N s1λ τ ms1s2 + br ,4 + −1 . s1λ + s2δ m N N s1λ + s2δ m

(67)

Further, the second order derivative of cd ,4 with respect to τ is

d 2 cd ,4 dτ

2

=

2ms1s2 > 0. ( s1λ + s2δ m ) N

(68)

Equation (68) implies that the minimum value of (67) is obtained when τ = τ ′ . Substituting

τ = τ ′ into (67), we have dcd ,4 dτ

τ =τ ′

=

N br ,4 − N p ,0 s1λ + + ϕ −1. s1λ + s2δ m N

(69)

Note that s1λ > s2δ m and N br ,4 > N p ,0 . Therefore, the value of (67) is always greater than zero if ϕ > 0.5 . It shows that for lump-sum redistribution, the generalized travel cost for car owners will always increase with the toll, and accordingly car owners never benefit from the pricing scheme. However, non-car owners benefit from the pricing scheme as long as the toll belongs to the feasible toll interval given by (48). Similarly, if the toll revenues are used to subsidize public transport, we have

dcc ,5 dτ

= 1−

s2δ mN b 2 − 4ac

27

.

(70)

The second order derivative of cd ,5 with respect to τ is denoted as

d 2cc ,5 dτ

2

= 2as2δ myN 2 ( b 2 − 4ac )

−3/ 2

> 0.

(71)

Equation (71) shows that the minimum value of (70) occurs when τ = τ * . In order to demonstrate car owners will suffer from the pricing scheme, it only requires the minimum value of (70) is great than zero. This necessarily requires

( s2δ mN )

2

< b 2 − 4ac . Note that

b 2 − 4ac = ( s2δ mN ) + ( a ( g − η ) ) + 2a ( g − η ) s2δ mN + 4ayNτ * , 424 3 42444 3 1 1444424444 3 144 2

2

C

B

A

where, B = 2a ( g − η ) yN − 2a ( g − η ) s1λ N , and C = 4 (ϕ yN ) + 4ϕ a (η − g ) yN − 4ϕ ys2δ mN 2 . 3 14243 14 4244 3 1442443 1424 3 144244 2

B1

B2

C2

C1

C3

Hence, we have b 2 − 4ac = A + B1 + C 2 − B 2 + C1 − C 3 . If ϕ > 0.5 , we have B1 + C 2 > 0 and C1 − C 3 > 0 . Moreover, it is obvious that − B 2 > 0 since g < η . Therefore,

b 2 − 4ac > A > ( s2δ mN ) , showing that the value of (70) is always greater than zero. With 2

these analyses, one can conclude that car owners always suffer from the pricing scheme. However, non-car owners benefit from the pricing scheme as long as the toll belongs to the feasible toll interval given by (61). According to (52) and (66), we know that car owners will benefit from the pricing scheme

ϕ<

N br ,0 N

regardless

+

of

how

the

toll

revenues

are

redistributed,

as

long

as

as1 ( g − η ) 2 . In this case, all commuters (including car owners and non-car 4δ yN 2

owners) benefit from the pricing scheme. This is not surprising. Note that car owners are more sensitive towards in-vehicle crowding than non-car owners. If the subsidies are high enough to compensate the increase of in-vehicle crowding cost for car owners, the subsidies can definitely cover the increase of in-vehicle crowding cost for non-car owners. This completes the proof.

Lemma 2.

N br ,0 N

+

■

as1 ( g − η ) 2 ≤ 0.5 . 2 4δ yN

Proof. According to Lemma 1, we know: (a) if ϕ > 0.5 , non-car owners will benefit from the pricing scheme, while car owners suffer; (b) if ϕ < 28

N br ,0 N

+

as1 ( g − η ) 2 , all 4δ yN 2

commuters (including car and non-car owners) will benefit from the pricing scheme. There is a

fact

N br ,0 N

+

that

the

two

situations

never

occur

simultaneously.

Suppose

that

as1 ( g − η ) 2 > 0.5 , then, one can always find such a value of ϕ satisfying 4δ yN 2

0.5 < ϕ <

N br ,0 N

+

as1 ( g − η )2 . This implies that both (a) and (b) could be valid 4δ yN 2 N br ,0

simultaneously, which is contradictory with the fact. Hence, This completes the proof.

N

+

as1 ( g − η ) 2 ≤ 0.5 . 2 4δ yN

■

According to Lemma 1, we know that the conditions that lead to a politically feasible toll interval are the same no matter which redistribution measure is adopted. According to Lemma 2, we know that if non-car owners are the majority, car owners will always suffer from the pricing scheme no matter how the toll revenues are redistributed, and that if car owners account for a majority, all users may benefit from the pricing scheme.

3.2.3 Pooling equilibrium with two redistribution measures A pooling equilibrium exists before levying a toll if ϕ <

N br ,0 N

. For this moment, using

λ1 to replace λ in (22), we can immediately give the condition that non-car owners support implementing the pricing scheme under lump sum redistribution

λ1 ms2

∆N <

N − N br ,1

τ.

(72)

+η − g .

(73)

N

Substituting (18) and (21) into (72), we have

τ<

N ( λ − λ1 ) ms2

Taking λ1 = λ in (73) leads to τ < η − g . This is the toll interval that benefits car owners, the same as (23). For subsidizing public transport, replacing λ in (29) by λ1 , non-car owners support levying a toll if and only if

λ1 ms2

∆N <

N − N br ,3 N br ,3

Substituting (26) and (28) into (74), we have

29

τ.

(74)

N ( y − s1λ1 ) − ( y − s1λ1 ) yN br ,0 2

τ<

a ( s1λ + s2δ m )

Replacing λ1 in (75) by λ , we have τ <

.

s2δ m (η − g ) s1λ + s2δ m

(75)

. This is the toll interval that

benefits car owners when the toll revenues are used to subsidize public transport, the same as (30). Note that

N br ,0 N

< 0.5 . This implies that the politically feasible toll exists only when car

owners benefit from the pricing scheme. We now summarize the results obtained for the case of ϕ > 0 in Proposition 2.

Proposition 2. (i) If driving is faster than taking buses, the toll interval that benefits non-car owners always exists regardless of the revenue redistribution measures used, whilst it is not necessarily true for car owners; (ii) If non-car owners are the majority, car owners will always suffer from the pricing scheme no matter how the toll revenues are redistributed; (iii) If car owners account for a majority, all users may benefit from the pricing scheme. Similar to the homogeneous case, that driving is faster than taking buses is also a sufficient condition to make levying a flat toll politically feasible. To see this, we can refer to formulae (23), (30), (48), (52), (61), (66), (73) and (75). If the travel time for a car user is less than that for a bus rider, it leads to η > g , then the toll levels calculated with these formulae are positive. Otherwise, buses may be crowed even there is no toll. This is because taking buses becomes relatively attractive compared to driving so that more commuters may choose to go to work by bus. In this case, it is no wonder that implementing road pricing is not the best choice. It seems intuitive that non-car owners support levying a toll no matter how the revenues are redistributed, as they are not required to pay any toll at all but share the toll revenues. Although some car owners will switch to take buses as the toll increases, the reimbursed subsidies are high enough to compensate the increase in the in-vehicle crowding cost, provided that the toll is not too high. For car owners, however, they probably benefit from the pricing scheme when there is a significant reduction in travel cost after levying a toll. One possible situation, for instance, is that car owners are the overwhelming majority. In this case, roads might be congested while buses are not quite crowded. The travel time may be significantly reduced as some initial drivers switch to take buses after levying a toll on drivers. There is another explanation for the last two points of Proposition 2, i.e., exploitation (see, De Borger and Proost, 2016a, b). Because car owners are exploited by non-car owners, 30

they are not willing to pay for the travel and vice versa for non-car owners. With the increase of the proportion of car owners, however, the extent of exploitation is limited for the reason that car owners share the relatively larger weight of toll revenues. Coupled with the reduction in travel time, car owners have the incentive to support implementing the pricing scheme. The following demonstrates whether Proposition 1 holds or not under the case of ϕ > 0 . If there is a pooling equilibrium before levying a toll, the feasible toll interval under lump-sum redistribution is still wider than that under subsidizing public transport. For the former, it is obvious that η − g >

s2δ m (η − g ) s1λ + s2δ m

. For subsidizing public transport, we have

N ( y − s1λ1 ) − ( y − s1λ1 ) yN br ,0 2

τ<

a ( s1λ + s2δ m )

=

( y − s1λ1 ) ( yN d ,0 − s1λ1 N ) s2δ m ( yN d ,0 − s1λ1 N ) . < a ( s1λ + s2δ m ) a ( s1λ + s2δ m )

Substituting N d ,0 = N − N br ,0 into the above expression and rearranging, we have

s2δ m ( yN d ,0 − s1λ1 N ) a ( s1λ + s2δ m )

=

N ( λ − λ1 ) + s2 m (η − g )

( s1λ / δ + s2 m )

<

N ( λ − λ1 ) ms2

+η − g .

But unfortunately, if there is a separating equilibrium before levying a toll, it is very hard to prove such relation with analytical analysis, because these expressions are not tractable, see, (48), (52), (61) and (66). Still, numerical results in Section 4 show that Proposition 1 is still valid for the heterogeneous case. Finally, we summarize the politically feasible conditions and toll intervals for various situations in Table 2 to end this section.

31

1

Table 2

2

The politically feasible conditions and toll intervals for various situations. Lump-sum redistribution

τ <η − g

Homogeneous case

Heterogeneous case with an initially separating equilibrium

Heterogeneous case with an initially pooling equilibrium

Subsidizing public transport

2  b1 − 4 a1c1 − b1 , if ϕ > 0.5 τ < 2 a1   2 2 −b2 + b2 − 4a2 c2  − b2 − b2 − 4a2 c2 < < , if τ  a a 2 2 2 2  2  N br ,0 ms2 ( 2δ N + s1 ( g − η ) )  <ϕ < 2 N 4δ yN  

ϕ<

N br ,0 N

32

, τ <η − g

τ<

s2δ m (η − g ) s1λ + s2δ m

 −b3 + b3 2 − 4a3c3 , if ϕ > 0.5 τ < 2 a3   2 2 −b4 + b4 − 4a4 c4  − b4 − b4 − 4a4 c4 < < , if τ  a a 2 2 4 4  2  N br ,0 ms2 ( 2δ N + s1 ( g − η ) )  <ϕ < 2 N 4δ yN  

ϕ<

N br ,0 N

,τ<

s2δ m (η − g ) s1λ + s2δ m

4. Numerical experiments In this section, we verify our theoretical results with numerical experiments. The basic parameters required for computing are given in Table 3.15 Other parameters may vary in different situations and will be given in the corresponding places.

Table 3 Parameters setting. Parameter

α

β

γ

λ

N

s1

s2

Tc

Tb

τb

h

Value

6

4

12

10

7500

3000

150

1

1.1

1

0.083

4.1. All commuters own cars (a special case with ϕ = 0 ) We set the total number of buses as m = 20 for the homogeneous case. For this moment, the bus departure headway which makes all buses being occupied belongs to (0, 0.086], including our setting value 0.083, i.e., 5 minutes. When the toll revenues are redistributed, how the equilibrium costs vary with the toll is shown in Fig. 5. In this figure, ‘0’ refers to no toll situation; ‘1’ refers to levying a toll with lump-sum redistribution, and ‘2’

Equilibrium cost

to levying a toll and the toll revenues are returned to bus riders (the same in Fig.7~Fig. 9).

Fig. 5. The politically feasible toll intervals for the homogeneous case.

15

The units of these parameters are: $ for τ b , $/h for α , β and γ ; $ per unit bus in-vehicle crowding

for λ ; hour for Tc , Tb and h ; veh/h for s1 , passenger per bus for s2 . The work start time is 9 am. 33

In Fig. 5, the equilibrium cost with no toll is 12.715, i.e., the straight line with legend ‘0’. It can be observed that the politically feasible toll intervals are [0, 4.10] for lump-sum redistribution and [0, 0.95] for subsidizing public transport. It is also shown that the equilibrium costs first decrease and then increase with the toll, regardless of the redistribution measures adopted. In particular, when the toll is set to 2.05 (0.34) for lump-sum redistribution (for subsidizing public transport), the equilibrium cost is minimal, at which the system optimum is attained. As expected, the system optimum modal split can be decentralized by levying different tolls. Policymakers can always choose a relatively lower toll to lessen commuters’ out-of-pocket costs. When the toll revenues are evenly returned to all commuters, drivers have the gains inclusive of the subsidies and the saving time cost because the total number of drivers decreases compared to no toll situation. When the toll revenues are exclusively used to subsidize bus riders, the reductions in the travel time become the only source of gains for drivers. Passengers will receive subsidies and experience a deteriorated in-vehicle crowding, as some initial drivers switch to take buses after levying a toll. Fig. (6) shows that the gains always outweigh the losses if the toll is within its feasible interval, where Fig. 6(a) depicts the gains and losses for commuters under lump-sum redistribution, while Fig. (6)b under subsidizing public transport. However, the difference between the gains and losses first increase and then decrease with the toll. That is why the equilibrium costs will first increase

Cost

Cost

and then decrease (see Fig. 5).

Fig. 6. The gains and losses for drivers and passengers.

34

4.2. Some commuters own cars (a general case with ϕ > 0 ) After introducing user heterogeneity, commuters with and without cars differ in the value of time and sensitivity towards bus in-vehicle crowding. Numerically, we take α1 = 6 ,

β1 = 2 , γ 1 = 6 and λ1 = 5 for non-car owners (note that β β1 = 2 ). Two different cases are considered, i.e., case A and case B. Case A (B) refers to that non-car owners are the majority (minority). The total number of buses is set to 40 (20) for case A (B). The proportion of non-car owners is initially set to 0.60 (0.12) for case A (B), and then is set to 0.60 (0.10) for case A (B).16 Other required parameters are the same as the homogeneous case shown in Table 3. For this moment, the bus departure headway which makes all buses being occupied is h ∈ (0, 0.1) , including the setting value, 0.083. This requires ϕ > 0.5 or ϕ < 0.122 to ensure there is a politically feasible toll interval (Lemma 1).

Fig. 7. The politically feasible toll intervals for case A with an initially separating equilibrium.

Fig. 7 shows the change of the equilibrium costs with the toll. Without a toll, the equilibrium costs for non-car owners and car owners are 13.85 and 9, respectively. The toll intervals that benefit non-car owners are [0, 16.94] for lump-sum redistribution and [0, 15.12] for subsidizing public transport, see Fig. 7 (b). There is a separating equilibrium if the toll is not more than 11.10 (6.66) for lump-sum redistribution (subsidizing public transport). The equilibrium costs for non-car owners first gradually decrease with the toll because they 16

Note that N b ,0 N = 0.105 . Therefore, a pooling equilibrium occurs if the proportion of non-car owners

is less than 0.105, and a separating equilibrium occurs otherwise. For the two different sets of values, both separating equilibrium and pooling equilibrium are included. 35

receive subsidies but have not any losses. When the toll equals 11.10 (6.66) for lump-sum redistribution (subsidizing public transport), the equilibrium cost for non-car owners is minimal. After that, some car owners switch to take buses to avoid paying a high toll, leading to a pooling equilibrium. More and more car owners choose to take buses as the toll further increases so as to raise the bus in-vehicle crowding cost. Accordingly, the equilibrium costs turn to increase with the toll. If the toll exceeds 16.94 (15.12) for lump-sum redistribution (subsidizing public transport), non-car owners will suffer from the pricing scheme. As far as car owners are concerned, they always suffer from the pricing scheme no matter how the revenues are used, see Fig. 7(a). If non-car owners account for a majority, car owners will be exploited by non-car owners. From this figure, it can also be observed that the feasible toll interval under lump-sum redistribution is wider than that under subsidizing public transport. Accordingly, policymakers can still choose a relatively lower toll to decentralize a desired modal split.

Fig. 8. The politically feasible toll intervals for case B with an initially separating equilibrium.

The politically feasible toll intervals for case B (the proportion of non-car owners is 0.12) are shown in Fig. 8. With no toll, the equilibrium costs for car owners and non-car owners are 12.60 and 10.35, respectively. The politically feasible toll intervals are [1.38, 2.72] for lump-sum redistribution and [0.20, 0.51] for subsidizing public transport, see Fig. 8(a). The equilibrium cost for car owners increases with the toll if it is less than 0.50 (0.06) for lump-sum redistribution (subsidizing public transport). Once the toll exceeds 0.50 (0.06) for lump-sum redistribution (subsidizing public transport), the initial separating equilibrium becomes a pooling equilibrium. The equilibrium cost then decreases with the toll until reaching its minimum, because there are significant reductions in the travel time cost after 36

some car owners switch to take buses. However, the equilibrium cost for non-car owners is always decreasing with the toll, see Fig. 8(b), and therefore they always benefit from the pricing scheme. Again, both lump-sum redistribution and subsidizing public transport can produce the same modal split and departure pattern.

Fig. 9. The politically feasible toll intervals with an initially pooling equilibrium.

Let us now set the proportion of non-car owners to 0.10 for case B. In this time, car owners account for an overwhelming majority, and there is a pooling equilibrium before levying a toll. The politically feasible toll intervals under two different redistribution measures are totally as the same as the homogeneous case (comparing Fig. 8 with Fig. 5). The difference is that the equilibrium costs for non-car owners are always less than that for car owners in the heterogeneous case.

5. Concluding remarks With a very stylized monocentric model, this paper addressed under what conditions implementing road pricing is politically feasible. A pricing scheme is considered to be politically feasible if a majority of commuters (more than a half) benefit from it, where their generalized travel costs are reduced in comparison with no toll situation. We found no one supports levying a toll if the toll revenues are not returned. In addition, the relative relationship between the bus travel time and car travel time is closely related to the political feasibility of road pricing. In particular, as long as driving requires a less travel time cost, the politically feasible toll always exists for the homogeneous case, while exists conditionally for the heterogeneous case. The latter requires that non-car owners account for 37

a majority or that car owners are the overwhelming majority. It is perfectly normal that driving costs a less travel time than taking buses in many cities. Therefore, existing of a politically feasible toll does provide support for the implementation of road pricing in these cities. In case of an opposite case, i.e., it’s faster to take buses, however, implementing road pricing might be politically infeasible. In this case, implementing a pricing scheme has very marginal or even no effects on alleviating congestion. This gives the understanding of why some cities such as Beijing do not implement congestion pricing even it has been proposed to the legislature for many years. Even though both lump-sum redistribution and subsidizing public transport can achieve the totally same results, the toll level is rather lower under the latter than the former. That may be why the revenues are mostly used to improve bus service in some cities (see, London). Finally, car owners are probably exploited by non-car owners. If non-car owners are the majority, car owners will always suffer from pricing no matter how the toll revenues are redistributed. However, all users may benefit from implementing the pricing scheme when car owners account for an overwhelming majority. In this study, it is assumed that there are bus exclusive lanes so that cars and buses are spatially separated. Without this assumption, commuters taking buses also incur a queuing time cost since buses share the same road with cars, which will complex our analyses significantly. Actually, if no transit exclusive lane is deployed, our setting context can be analogous to a competitive system, where a road with a bottleneck is parallel to a subway system (Tabuchi, 1993). We expect that our main conclusions are still valid for that case. Moreover, we do not consider the bus capacity constraint so that the actual number of commuters carried in a bus is likely to be greater than its capacity. One more realistic situation is to add the bus capacity constraint. In that case, commuters will incur an extra penalty once the number of bus riders actually carried by a bus is equal to its capacity (note that it will never exceed the bus capacity under such circumstance). Such a penalty can be internalized by introducing a time-varying bus fare or fare-reward scheme (see, for example, Yang & Tang, 2018). Accordingly, adding bus capacity constraint will not violate our main conclusions. Several aspects are for future study. In practice, it is necessary to calibrate the parameters in the model (e.g., α , β , γ , λ , etc.) with real data. In our future work, we plan to collect data from Beijing to evaluate the political feasibility and to propose the possible toll level. The total number of commuters is assumed to be fixed in this paper. This assumption might be realistic in the short run, but one might question to what extent it is valid in the long 38

run as commuters are able to relocate or change their jobs. Moreover, from the prospective of economy-wide equilibrium, it is still unclear whether congestion pricing can produce net welfare. Vandyck and Rutherford (2018) studied the economy-wide and distributional implications of congestion pricing in the presence of agglomeration externalities and unemployment, and shown that indirect effects of time-invariant tolls can lead to welfare losses for low-skilled urban residents. When both negative traffic congestion externalities and positive agglomeration externalities are considered, Arnott (2007) shown that the optimal toll might not only be substantially lower than the Pigouvian toll, but might even be negative, entailing a subsidy to urban travel. Therefore, it is essential to study the effects of road pricing from more broad perspectives. Of course, it is hard to capture all factors that determine the political feasibility of road pricing in a single model. De Borger and Proost (2012) introduced uncertainty to explain the same problem on road pricing. However, policymakers also have the inherent uncertainty towards the reform outcome due to inaccurate parameters estimation or other white noise. Under such circumstance, the politician might hesitate or reject to implement a pricing scheme, even though others like economists recommend it. Introducing politician’s uncertainty is also our future work. Finally, we believe that the political feasibility of parking pricing is also well worth studying (Tian, et al, 2018).

Acknowledgments This research was supported by a grant from the National Natural Science Foundation of China (71890971/71890970) and a program jointly awarded by NSFC and JPI Urban Europe (71961137001).

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41

Highlights Investigate the political feasibility of road pricing Toll revenues redistributed to all commuters or bus riders only or not at all Find the politically feasible toll intervals

Author Statement The revised version of our paper “A competitive system with transit and highway: Revisiting the political feasibility of road pricing” is now submitted to Transport Policy for possible publication. We hereby declare: the work contained in this paper has not been published previously and is not under consideration for publication elsewhere. And, the paper contains/publishes all data that the study requires.

Hai-Jun Huang Corresponding author School of Economics and Management, Beihang University, Beijing 100191, China E-mail: [email protected]