Determining the Toll and Capacity of a Highway to Be Constructed in Parallel with Subway

Abstract

This paper considers the problem of toll and capacity choice of a new highway with a bottleneck added onto an existing transit network under four ownership/tolling regimes: public fine toll, public flat toll, private fine toll, and private flat toll. Whenever fine toll and flat toll are imposed, in a competitive highway/transit network with constant returns to scale in road construction, an optimally designed and priced privately owned highway would produce positive net benefit justly equal to the total markup with respect to all autocommuters, whereas an optimally designed and priced publicly owned highway would lead to a deficit; that is, the toll revenues are insufficient to cover its all costs. The economic conditions to invest a new road are investigated under different ownership/tolling regimes.

1. Introduction

As a means to relieve congestion and improve transportation system efficiency, congestion pricing lends itself to two main purposes. First, congestion pricing seeks to remedy transport externalities by imposing a charge that reflects the monetized value of the externalities associated with driving. This leads to better utilization of transportation capacity as well as the need to build new roadway capacity. Second, congestion pricing has the potential to generate significant revenues; if widely adopted, it can become an important source of funding for transportation infrastructure. In developing or developed countries, government may suffer shortage of funds for public roads in the face of ever-growing congestion on urban and interurban traffic infrastructure. The toll revenues can be used to pay the maintenance, to pay existing debts, and to pay the investments in the extension of the network [1, 2].

In recent years, the joint optimization of toll and capacity, as decision variables in the problem, has been studied in order to produce significant benefits to the design and operation of both congestible urban and expandable road networks. One of the most famous findings (usually called self-financing theorem or Mohring and Harwitz rule) in transport economics proposed by Mohring and Harwitz is that under certain technical conditions the revenues from congestion pricing will be just sufficient for financing the costs associated with the optimal capacity supply [3]. The certain conditions state that (1) capacity can be adjustable in continuous increments, (2) capacity can be expandable at constant marginal cost, and (3) trip costs are homogeneous of degree zero in usage and capacity [4]. The Mohring and Harwitz rule was regarded as one of foundations of transportation economics [5]. In the context of Vickrey’s bottleneck model of morning rush-hour autocongestion, a few researchers investigated the joint optimization of optimal toll and capacity under the circumstances where the first-best or second-best pricing applied and then examined whether it could be self-financing [2, 6]. Arnott et al. stated that, whatever the tolling regime, the revenue collected from optimal toll on an isolated road is exactly sufficient to recover the costs of constructing optimal capacity if there are constant returns to scale in capacity provision [6]. More generally, the recovery ratio of the revenue to the costs with respect to optimal capacity equals the elasticity of capacity costs. This is because the generalized travel costs in the bottleneck model are homogeneous of degree zero in usage and capacity.

Nevertheless, there are still some aspects needed to launch an in-depth investigation. First, government may suffer shortage of funds for building more new roads to meet growing demand for travel. This has stimulated a rising interest in private supply of road capacity. Many private roads are in business around the world [7]. Recent research in transportation economics has focused on the road pricing and road network ownership that defines pricing and investment objectives (see [8–11]). A recent and probably the most complete analysis of private tolling on a parallel route network is given by [12], in which a variety of ownership regimes such as private and mixed duopolies are investigated. So, compared to a public traffic bottleneck, what if the traffic bottleneck is constructed and run by a private firm? Second, it is an important socioeconomic issue under what condition introducing a road into an existing transportation network is valuable in the sense of the total social welfare or net benefit [13]. Specifically, the efficiency of the whole network must be improved when introducing a new road by the government; namely, the total social welfare (social benefit minus total cost) is enhanced for the case with elastic demand. The net benefit (total revenue minus total cost) must be positive when introducing a new road by a private agency. It is worth investigating the relations between the entry conditions under various regimes and the parameters of the system. Third, the optimal road capacity provision has been mostly viewed as a standalone road network design problem conducted separately from congesting pricing (e.g. [14, 15]). But for the strategic consideration of long-term optimal design of the capacity, the road designer should take into account both the level of capacity and prices in order to predict the volume of road users and toll revenues. If the bottleneck capacity is optimized separately from congestion pricing, how will it influence the investment choice and usage that determines the efficiency of the system? Finally, it seems desirable to highlight the impact of cost structure with respect to capacity on the Mohring and Harwitz result when the optimal tolling scheme is implemented and the capacity level is optimized. As a matter of fact, a few studies have paid attention to the impact of the cost structure on the Mohring and Harwitz result (e.g., [16–18]). If there are constant returns to scale in roads construction and strictly constant returns to road use, Newbery concluded that the optimal road user charge will recover all road costs [19]. Even if there are substantial economies of scale in providing pavement, Small et al. found that the road system as a whole comes close to self-financing because of the diseconomies of scope involved in the simultaneous production of road capacity for passenger cars and trucks [20]. More recently, assuming that transportation systems have, for a given capacity, the same total infrastructure cost but vary in the proportion of fixed costs and variable capacity cost, De Borger et al. found out that, compared to a benchmark infrastructure which has zero fixed costs, an infrastructure which has (ceteris paribus) a higher share of fixed costs leads to higher welfare for the regional government building it [21]. Moreover, even for capacity characterized by very high shares of fixed costs, financing of infrastructure is generally not an important issue as long as regions are allowed to toll through traffic.

In many large cities there exists a mass commuting mode and a highway with a bottleneck between home and workplace/central business district (CBD). This paper considers the toll and capacity choice for a new highway with a bottleneck added as alternative to the existing transit network. Similar to [13], the public authority (government) aims to seek welfare-maximizing, while the private firm aims to seek profit-maximizing. The time-varying toll is a completely flexible time-dependent toll for eliminating queue, and the flat toll is constant over rush hours. However, we here consider an existing transit network. Unlike the highway, the trains of a railway normally arrive on time no matter how crowded their carriages may be. Secondly, in reality, the rush-hour transit crowding is considerably serious. To a certain extent, crowding congestion will result in transit commuters’ switch from departing on time to departing early or late for avoiding congestion. Accordingly, the travel costs of commuters on trains are not homogeneous degree of zero in link flow. In addition, in previous studies, the operating and maintenance costs (OMC) of a highway varying with the level of capacity and its usage are generally omitted or assumed constant and without loss of generality set to zero (see [6, 12]). This leads to a substantial deviation from optimal solutions. In the paper, like [2, 19], taking into account the OMC, the joint optimization of toll and capacity, as decision variables concerning a highway with a bottleneck, is considered under various ownership/tolling regimes: public fine toll, public flat toll, private fine toll, and private flat toll.

The paper is organized as follows. Section 2 introduces the two basic models, respectively. The joint optimization of capacity and toll levels is studied under different ownership/tolling regimes. Section 4 investigates the conditions to invest the new highway project being investigated under various ownership regimes. And finally, Section 5 concludes the paper.

2. The Basic Models

Consider a simplified corridor network which contains two modes (railway and added-on highway) to provide transportation service between home or a resident area and a workplace or central business district. Mode “𝑎” represents an added-on highway with a single bottleneck which is located at the entering point of the highway and has a deterministic capacity of 𝑠 commuters per unit time. Mode “𝑏” represents an existing railway.

2.1. Train Mode

We first investigate the existing railway mode. Let 𝑇_𝑏 be the on-train moving time. In general, all transit costs can be classified into two categories, that is, fixed costs and variable costs. Set the fixed costs to be zero for simplicity, since they are independent of the transit usage and do not affect the road toll decision. But, the variable costs generally increase with the number of passengers. Let 𝑘_𝑏 denote the transit fare, which is mainly determined by the fixed costs and the operator’s daily expenses on labor, fuel, electricity, and routine materials as well as the expected profit rate. It is assumed in this study that the transit fare should remain unchanged during a period of considerable length; that is, 𝑘_𝑏 does not vary with the total demand. Let 𝑟_𝑏(𝑡) be the departure rate of commuters on railway, [𝑡_𝑠,𝑡_𝑒] the time window of train commuters’ departure, and 𝑟_𝑏(𝑡)=0 when 𝑡∉[𝑡_𝑠,𝑡_𝑒]. Let 𝑚 and 𝜉 be the number of trains dispatched and the length of time interval of dispatching trains during [𝑡_𝑠,𝑡_𝑒], respectively, and 𝑚𝜉=𝑡_𝑒−𝑡_𝑠. It is true that mass transit may bring people discomfort generated by body congestion at railway stations and in carriages if they are crowded. This body congestion also leads to the loss of independence and privacy by transit commuters [22]. In reality, rush-hour transit crowding is very serious. To a certain extent, crowding congestion will result in transit commuters’ switch from departing on time to departing early or late to avoid congestion. Some studies introduced body congestion for modeling urban mass transit services [23, 24]. Therefore, in this paper, we explicitly consider body congestion for modeling transit commuters’ departure time choice. Like [25], for simplicity, we assume that the body congestion cost is proportional to the number of passengers in carriage and the transit time. Thus, the generalized cost per train commuter is given by 𝑐_𝑏(𝑡)=𝛼𝑇_𝑏+𝑘_𝑏+𝛽𝑡̃−𝑡+𝜓𝜉𝑇_𝑏𝑟_𝑏(𝑡),𝑡∈𝑡_𝑠,𝑡̃,𝛼𝑇_𝑏+𝑘_𝑏+𝛾𝑡−𝑡̃+𝜓𝜉𝑇_𝑏𝑟_𝑏(𝑡),𝑡∈𝑡,̃𝑡_𝑒, (1) where 𝜓 is the body congestion parameter, 𝜓𝜉𝑇_𝑏𝑟_𝑏(𝑡) is the body congestion cost, and 𝑡̃ is the departure time at which train commuters can arrive on time 𝑡^∗; that is, 𝑡̃+𝑇_𝑏=𝑡^∗.

Wu and Huang stated that at equilibrium state 𝑟_𝑏(𝑡_𝑠)=0 and 𝑟_𝑏(𝑡_𝑒)=0. Consequently, in equilibrium, any train commuter is unable to find a departure time which can reduce his/her travel cost [25]. Letting 𝑑𝑐_𝑏(𝑡)/𝑑𝑡=0, we get the departure rate from home by train: 𝑟_𝑏(𝑡)=𝛽𝑡−𝑡_𝑠𝜓𝜉𝑇_𝑏,𝑡∈𝑡_𝑠,𝑡̃𝛾𝑡_𝑒−𝑡𝜓𝜉𝑇_𝑏,𝑡∈𝑡,̃𝑡_𝑒. (2) Thus, for any given 𝑁_𝑏, the solutions to 𝑡_𝑠 and 𝑡_𝑒 are unique and given as 𝑡_𝑠=𝑡̃−2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏𝛽,𝑡_𝑒=𝑡̃+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏𝛾. (3) From (1)–(3), we easily get the generalized cost per train commuter 𝑐_𝑏=𝛼𝑇_𝑏+𝑘_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏. (4)

2.2. Automode

We now consider that there is an added-on highway between an origin, such as a residential area, and a destination, such as a work place. When the highway is free from congestion (i.e., the arrival rate at bottleneck is less than the capacity), the free-flow commuting time is a constant 𝑇_𝑎. If the arrival rate exceeds the capacity, a queue develops. Thus, the commuting time is given by 𝑇(𝑡)=𝑇_𝑎+𝑄(𝑡)𝑠, (5) where 𝑡 is the departure time from home, 𝑄(𝑡) is the queue length (number of automobiles in front of the bottleneck), and 𝑠 is the bottleneck capacity.

Let 𝑟_𝑎(𝑡) be the departure rate of autocommuters from home at time 𝑡; then the queue length is 𝑄(𝑡)=^𝑡_𝑡0𝑟_𝑎(𝑡)𝑑𝑡−𝑠𝑡−𝑡₀, (6) where 𝑡₀ is the most recent time at which there is no queue.

Due to the finite bottleneck’s capacity, some automobile commuters have to arrive earlier or later than the work starting time 𝑡^∗, for preventing from having very high travel cost caused by queue. With such an assumption that the travel cost is proportional to the travel time and the schedule delay, we can formulate the generalized cost of an autocommuter as follows: 𝑐_𝑎(𝑡)=𝛼𝑇(𝑡)+𝛽𝑡^∗−𝑡−𝑇(𝑡)+𝑘_𝑎,𝑡∈𝑡₀,𝑡𝛼𝑇(𝑡)+𝛾𝑡+𝑇(𝑡)−𝑡^∗+𝑘_𝑎,𝑡∈𝑡,𝑡₁, (7) where 𝑘_𝑎 is the out-of-pocket cost (including parking fee, oil consumption, and highway fare), 𝛼 is the unit cost of travel time, 𝛽 is the schedule penalty for a unit time of early arrival, 𝛾 is that for a unit time of late arrival, 𝑡₁ is the time at which the queue ends, and 𝑡 is the departure time at which an individual arrives at work at time 𝑡^∗; that is, 𝑡+𝑇(𝑡)=𝑡^∗. For rationality, the relation 𝛾>𝛼>𝛽 holds according to the estimates by [26].

Suppose that each autocommuter minimizes his/her generalized commuting cost through selecting the departure time. At a nontoll equilibrium state, any autocommuter is unable to find a departure time to reduce his/her own travel cost by unilaterally changing the departure time. In other words, 𝑐_𝑎(𝑡) in (7) should be constant for all time 𝑡∈[𝑡₀,𝑡₁]. Letting 𝑑𝑐_𝑎(𝑡)/𝑑𝑡=0, we have 𝑟_𝑎(𝑡)=𝛼𝑠𝑡𝛼−𝛽,𝑡∈₀,𝑡𝛼𝑠𝛼+𝛾,𝑡∈𝑡,𝑡₁. (8) Substituting (8) into (6), we get the queue length 𝑄(𝑡)=𝛽𝑠𝑡−𝑡₀𝑡𝛼−𝛽,𝑡∈₀,𝑡,𝛽𝑠𝑡−𝑡₀−𝛼−𝛽𝛾𝑠𝑡−𝑡𝛼+𝛾,𝑡∈𝑡,𝑡₁. (9)

Since 𝑡+𝑇(𝑡)=𝑡^∗, 𝑄(𝑡₀)=𝑄(𝑡₁)=0, and ∫^𝑡1_𝑡0𝑟_𝑎(𝑡)𝑑𝑡=𝑁_𝑎, we can solve for the three time unknowns: 𝑡₀=𝑡^∗−𝛿𝑁_𝑎𝛽𝑠−𝑇_𝑎,𝑡₁=𝑡^∗+𝛿𝑁_𝑎𝛾𝑠−𝑇_𝑎,𝑡=𝑡^∗−𝛿𝑁_𝑎𝛼𝑠−𝑇_𝑎, (10) where 𝛿≡𝛽𝛾/(𝛽+𝛾). As a result, the generalized commuting cost per autocommuter is 𝑐_𝑎=𝛼𝑇_𝑎+𝑘_𝑎+𝛿𝑁_𝑎𝑠. (11) The constancy of 𝑐_𝑎 implies that, in equilibrium, no autocommuter can decrease his/her generalized commuting cost by changing his/her departure time. In other words, all individuals must experience identical travel cost, no matter when she or he leaves home. According to [6], the total travel cost of the autosystem, 𝑐_𝑎𝑁_𝑎, is the sum of the total travel time cost 𝛼𝑇_𝑎𝑁_𝑎+0.5𝛿(𝑁_𝑎)²/𝑠, the total out-of-pocket cost 𝑘_𝑎𝑁_𝑎, and the total schedule delay cost 0.5𝛿(𝑁_𝑎)²/𝑠.

3. Toll and Capacity Optimization

Assume that, besides generalized costs, there is only one other price component of trip making for an autocommuter, namely, a toll 𝜏 (if levied). So, the perceived generalized price of the trip, 𝑝, can be written as the sum of the generalized cost 𝐶 and the toll 𝜏: 𝑝=𝐶_𝑎+𝜏. Assume that the cost for the road construction will depend on its capacity 𝑠 only and that will be written as 𝐺(𝑠). It is to be interpreted as a per-unit-of-time cost, including capital and depreciation. Note that it is not the investment cost. For simplicity, we denote it by capital cost of capacity. According to [2, 18], we assume that OMC is the sum of a term which depends on usage (road damage due to usage), a term which is dependent on capacity, and a constant which is independent of both usage and capacity. The second term depends on natural degradation, due, for example, to bad weather conditions (induced potholes, cracks, and the like), and the third term represents other operating costs such as electricity expenditure. So, the OMC can be written as 𝑀(𝑁_𝑎,𝑠)=𝑤(𝑄)𝑁_𝑎+𝜈(𝑄)𝑠+𝑓, where 𝑄 is the quality of the road, 𝑤(𝑄)>0 the maintenance cost per usage, and 𝜈(𝑄)>0 the maintenance cost per capacity, the constant 𝑓>0 is independent of both usage and capacity. Here, we omit the quality dependency, and without loss of generality, the constant 𝑓 is set equal to zero since it has no effect on our decision making. Thus, 𝑀(𝐹,𝐾)=𝑤𝑁_𝑎+𝜈𝑠, where the parameter 𝑤 can be interpreted as marginal demand (or usage) OMC, and the parameter 𝜈 is marginal capacity OMC. We indicate a downward convex inverse demand function by 𝐷(𝑛), 𝑑𝐷(𝑛)/𝑑𝑛<0, and 𝑑²𝐷(𝑛)/𝑑𝑛²≥0, which also represents marginal benefits of commuters. So, its integral between 0 and 𝑁 gives aggregate benefits of trip making (per unit of time).

Due to the fact that a growing number of toll roads have been constructed and operated privately [27], we here concern two types of ownership: public ownership and private ownership. Principally, fine (time-varying) toll reflects real-time variation in congestion levels and achieves more efficiency. As a practical matter, however, deviation between ideal and actual implementation conditions occurs because of technological and public acceptance issues that make it difficult or costly to vary tolls over time at prescribed levels. Therefore, the flat toll is still widely adopted despite its deficiency. In this paper, we consider two types of polar tolls—fine toll and flat toll. The combination of ownership and tolling produces four ownership/tolling regimes: public fine toll, public flat toll, private fine toll, and private flat toll.

3.1. Public Ownership

We first consider a publicly owned highway. A public authority would be to maximize social welfare surplus (social surplus): the difference between aggregate benefits of trip making and the aggregate social costs of making these trips possible. Thus, the social welfare maximization (SWM) problem is formulated as follows: max_{𝑁𝑎,𝑁𝑏,𝑁,𝑠,𝜏}SW𝑁_𝑎,𝑁_𝑏=,𝑁,𝑠,𝜏_𝑁0𝐷(𝑛)𝑑𝑛−𝑁_𝑎⋅𝛼𝑇_𝑎+𝐾_𝑎+𝜇𝛿𝑁_𝑎𝑠−𝐺(𝑠)−𝑤𝑁_𝑎−𝜈𝑠−𝑁_𝑏⋅𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏,s.t.𝐷(𝑁)=𝛼𝑇_𝑎+𝐾_𝑎+𝛿𝑁_𝑎𝑠+𝜏=𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏,𝑁_𝑎+𝑁_𝑏=𝑁, (12)(13)(14) where 𝜇=1 for a flat toll and 𝜇=0.5 for a fine toll which can completely eliminate queuing (please see [6]).

From (13), we have 𝜕𝑁_𝑏=𝜕𝑁2𝑁_𝑏𝛿𝜓𝜉𝑇_𝑏⋅𝑑𝐷(𝑁).𝑑𝑁 (15)

From (14), we get 𝜕𝑁_𝑎+𝜕𝑁𝜕𝑁_𝑏𝜕𝑁=1. (16) Together with (13)-(14), we obtain 𝜕𝑁_𝑎𝜕𝑁=1−2𝑁_𝑏𝛿𝜓𝜉𝑇_𝑏⋅𝑑𝐷(𝑁).𝑑𝑁 (17) With (13), the objective function becomes max_𝑁,𝑠SW(𝑁,𝑠)=^𝑁₀𝐷(𝑛)(𝑁)+⋅𝛿𝑁𝑑𝑛−𝑁𝐷1−𝜇^𝑎2𝑠+𝑁_𝑎𝐷(𝑁)−𝛼𝑇_𝑎−𝐾_𝑎−𝛿𝑁_𝑎𝑠−𝐺(𝑠)−𝑤𝑁_𝑎−𝜈𝑠. (18)

The first-order condition of (18) with respect to flow 𝑁 leads to 𝜕SW(𝑁,𝑠)𝜕𝑁=−𝑁_𝑏⋅𝑑𝐷(𝑁)𝑑𝑁−2𝜇𝛿𝑁_𝑎𝑠⋅𝜕𝑁_𝑎+𝜕𝑁𝜕𝑁_𝑎(𝑁)−𝜕𝑁⋅𝐷𝜕𝑁_𝑎⋅𝜕𝑁𝛼𝑇_𝑎+𝐾_𝑎+𝑤=0. (19) With (13), (19) can be reduced to (𝑁)𝜏=𝑑𝐷𝑁𝑑𝑁^𝑏𝜕𝑁𝜕𝑁_𝑎+𝜇𝛿𝑁_𝑎𝑠−1−𝜇𝛿𝑁_𝑎𝑠+𝑤. (20)

Equation (20) shows that optimal road use requires imposition of the so-called Pigouvian toll, which is equal to the sum of marginal total travel cost with respect to all transit commuters (𝑁_𝑏(𝑑𝐷(𝑁)/𝑑𝑁)⋅𝜕𝑁/𝜕𝑁_𝑎), marginal travel cost with respect to autocommuters (𝜇𝛿𝑁_𝑎/𝑠), and marginal usage OMC (𝑤) minus the marginal fine toll ((1−𝜇)𝛿𝑁_𝑎/𝑠 if existing).

The first-order condition of (18) with respect to 𝑠 yields 𝜕SW(𝑁,𝑠)𝛿𝑁𝜕𝑠=𝜇^𝑎2𝑠²−𝑑𝐺(𝑠)𝑑𝑠−𝜈=0. (21) Similarly, (21) can be reduced to 𝜇𝛿𝑁_𝑎²𝑠²=𝑑𝐺(𝑠)𝑑𝑠+𝜈. (22)

Equation (22) indicates that the capacity should be constructed to the point where the marginal benefits of capacity expansion (i.e., the value of aggregate travel time saving) is equal to the sum of the marginal capital cost of capacity and the marginal capacity OMC.

With (20) and (22), the total toll revenue 𝑅 is then 𝑅=𝜏𝑁_𝑎+𝛿𝑁1−𝜇^𝑎2𝑠=𝑁_𝑎𝑁_𝑏⋅𝑑𝐷(𝑁)⋅𝑑𝑁𝜕𝑁𝜕𝑁_𝑎+𝑠𝑑𝐺(𝑠)𝑑𝑠+𝜈𝑠+𝑤𝑁_𝑎. (23) Thus, the net toll revenue 𝑃𝑅 is 𝑃𝑅=𝑁_𝑎𝑁_𝑏⋅𝑑𝐷(𝑁)⋅𝑑𝑁𝜕𝑁𝜕𝑁_𝑎+𝑠𝑑𝐺(𝑠)𝑑𝑠+𝜈𝑠+𝑤𝑁_𝑎−𝐺(𝑠)−𝑤𝑁_𝑎−𝜈𝑠=𝑁_𝑎𝑁_𝑏⋅𝑑𝐷(𝑁)⋅𝑑𝑁𝜕𝑁𝜕𝑁_𝑎+𝑠𝑑𝐺(𝑠)(𝑠).𝑑𝑠−𝐺 (24)

With constant return to scale in road construction, the elasticity of capital cost with respect to capacity 𝜌 is defined by (𝑠)𝜌=𝑑𝐺⋅𝑠𝑑𝑠𝐺(𝑠)≡1. (25) Thus, we can rewrite (24) as follows: 𝑃𝑅=𝑁_𝑎𝑁_𝑏𝑑𝐷(𝑁)⋅𝑑𝑁𝜕𝑁𝜕𝑁_𝑎+𝐺(𝑠)=𝑁𝜌−1_𝑎𝑁_𝑏(𝑑𝐷(𝑁)/𝑑𝑁)1−2𝑁_𝑏/𝛿𝜓𝜉𝑇_𝑏⋅((𝑁)).𝑑𝐷/𝑑𝑁 (26) From (26), we have the following proposition.

Proposition 1. Whenever flat toll and fine toll are imposed, in a competitive highway/transit network with constant return to scale in road construction, an optimally designed and priced added-on highway for maximizing social welfare would lead to a deficit; that is, the toll revenues are insufficient to cover its all costs.

Proof. An optimally designed and priced added-on highway implies that both modes are all used; that is, 𝑁_𝑎>0 and 𝑁_𝑏>0. Since 𝑑𝐷(𝑁)/𝑑𝑁<0, 𝑃𝑅<0.
This is different from the Mohring and Harwitz rule. Mohring and Harwitz showed that, under certain conditions, an optimally designed and priced road would generate user toll revenues just sufficient to cover its all costs [3].

3.2. Private Ownership

When a private firm is in charge of the operating, maintenance, and construction of the road, the firm will seek the toll and capacity level to maximize its own net benefits (MNB). So, the MNB problem can be formulated as follows: max_{𝑁𝑎,𝑁𝑏,𝑁,𝑠,𝜏}𝑃𝑅𝑁_𝑎,𝑁_𝑏,𝑁,𝑠,𝜏=𝑁_𝑎𝛿𝑁𝜏+1−𝜇^𝑎2𝑠−𝐺(𝑠)−𝑤𝑁_𝑎()-(),−𝜈𝑠,s.t.1314 (27) where 𝜇=1 for a flat toll and 𝜇=0.5 for a fine toll which can completely eliminate queuing.

Similarly, with (13), we have max_𝑁,𝑠𝑃𝑅(𝑁,𝑠)=𝑁_𝑎𝐷(𝑁)−𝛼𝑇_𝑎−𝐾_𝑎−𝛿𝑁_𝑎𝑠+𝛿𝑁1−𝜇^𝑎2𝑠−𝐺(𝑠)−𝑤𝑁_𝑎−𝜈𝑠. (28)

The first-order condition of (18) with respect to flow 𝑁 is given by 𝜕𝑃𝑅(𝑁,𝑠)=𝜕𝑁𝜕𝑁_𝑎𝐷(𝑁)𝜕𝑁−𝛼𝑇_𝑎−𝐾_𝑎−𝛿𝑁_𝑎𝑠+𝑁_𝑎𝜕𝐷(𝑁)−𝛿𝜕𝑁𝑠𝜕𝑁_𝑎𝜕𝑁+21−𝜇𝛿𝑁_𝑎𝑠𝜕𝑁_𝑎𝜕𝑁−𝑤𝜕𝑁_𝑎𝜕𝑁=0. (29) With (13), (29) can be reduced to 𝜏=−𝑁_𝑎𝑑𝐷(𝑁)𝑑𝑁𝜕𝑁𝜕𝑁_𝑎+𝜇𝛿𝑁_𝑎𝑠−1−𝜇𝛿𝑁_𝑎𝑠+𝑤. (30)

Compared to the above welfare-maximizing toll in (20), the profit-maximizing toll excludes a marginal total travel cost with respect to all transit commuters but adds the markup with respect to each autocommuter.

The first-order condition of (18) with respect to flow 𝑠 is given by 𝜇𝛿𝑁_𝑎²𝑠^𝑤=𝑑𝐺(𝑠)𝑑𝑠+𝜈. (31)

Interestingly, while the toll rule is adapted to reflect the profit-maximizing objective, the capacity rule remains the same as the welfare-maximizing objective: the marginal benefits of capacity expansion equals the sum of the marginal capital cost of capacity and the marginal capacity OMC.

Thus, the net toll revenue 𝑃𝑅 is 𝑃𝑅(𝑁,𝑠)=−𝑁_𝑎²⋅𝜕𝐷(𝑁)⋅𝜕𝑁𝜕𝑁𝜕𝑁_𝑎+𝑠𝑑𝐺(𝑠)(𝑠)𝑁𝑑𝑠−𝐺=−_𝑎²⋅𝑑𝐷(𝑁)⋅𝑑𝑁𝜕𝑁𝜕𝑁_𝑎𝑁=−^𝑎2(𝑑𝐷(𝑁)/𝑑𝑁)1−2𝑁_𝑏/𝛿𝜓𝜉𝑇_𝑏⋅((𝑁)).𝑑𝐷/𝑑𝑁 (32)

From (26), we have the following proposition.

Proposition 2. Whenever flat toll and fine toll are imposed, in a competitive highway/transit network with constant return to scale in road construction, an optimally designed and priced added-on highway for maximizing net benefit would lead to a surplus, and the surplus is exactly equal to the total markup with respect to all autocommuters.

Proof. Similarly, as Proposition 1, 𝑁_𝑎>0 and 𝑁_𝑏>0. With 𝑑𝐷(𝑁)/𝑑𝑁<0 and (32), 𝑃𝑅(𝑁,𝑠)>0.
Of course, if a deficit result occurs, a private firm has no investment incentives when it is not subsidized by a government.

4. Conditions to Invest the New Toll Road

It is an important socioeconomic issue under what condition introducing the competitive project as an “add-on” road into an existing transportation network is valuable in the sense of the total social welfare [13]. Intuitively, the efficiency of the whole network (total social welfare) must be improved when introducing a new road by the government. Of course, the net benefit must be positive when introducing a new road by a private agency. In this section, we derive the condition to invest the new toll road under public ownership regime and private ownership regime, respectively. Without introducing a new road, in equilibrium, we have 𝐷(𝑁)=𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁. (33)

We denote 𝑁^∗_𝑜 as the solution to (32), and the social welfare maximization problem is formulated as SW𝑁^∗_𝑜=^𝑁∗𝑜₀𝐷(𝑛)𝑑𝑛−𝑁^∗_𝑜𝐷𝑁^∗_𝑜. (34)

After introducing a new road, we denote 𝑁^∗_𝐺 as the solution to the SWM problem. Clearly, 𝑁_𝑎+𝑁_𝑏=𝑁^∗. Thus, we have the following lemma.

Lemma 3. In equilibrium, under public ownership, 𝑁^∗_𝐺>𝑁^∗_𝑜>𝑁_𝑏 if 𝛼𝑇_𝑎+𝐾_𝑎+2()+𝑘+𝜈𝛿𝜇𝑁𝑑𝐷(𝑁)/𝑑𝑁1−2𝑁/𝛿𝜓𝜉𝑇_𝑏⋅(𝑑𝐷(𝑁)/𝑑𝑁)_{𝑁=𝑁∗𝑜}+𝑤<𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁^∗_𝑜. (35)

Proof. Putting (17) and (22) into (13), we have 𝛼𝑇_𝑎+𝐾_𝑎+2()+𝑁𝑘+𝜈𝛿𝜇_𝑏𝑑𝐷(𝑁)/𝑑𝑁1−2𝑁/𝛿𝜓𝜉𝑇_𝑏⋅(𝑑𝐷(𝑁)/𝑑𝑁)+𝑤=𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏. (36)
In equilibrium, if 𝑁_𝑎=0, that is, 𝑁_𝑏=𝑁^∗_𝑜, Wardrop’s first principle leads to 𝛼𝑇_𝑎+𝐾_𝑎+2()+𝑘+𝜈𝛿𝜇𝑁𝑑𝐷(𝑁)/𝑑𝑁1−2𝑁/𝛿𝜓𝜉𝑇_𝑏⋅(𝑑𝐷(𝑁)/𝑑𝑁)_{𝑁=𝑁∗𝑜}+𝑤≥𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁^∗_𝑜. (37)
Clearly (37) contradicts (35). Therefore, 𝑁_𝑎>0.
We now prove 𝑁^∗_𝐺>𝑁^∗_𝑜. If 𝑁^∗_𝐺≤𝑁^∗_𝑜, then 𝐷𝑁^∗_𝐺=𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏≥𝐷𝑁^∗_𝑜. (38)
Together with (33), 𝐷(𝑁^∗_𝐺)=𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏≥𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁^∗_𝑜. Thus, 𝑁_𝑏≥𝑁^∗_𝑜. Since 𝑁_𝑎>0, 𝑁_𝑎+𝑁_𝑏=𝑁^∗_𝐺>𝑁^∗_𝑜. This contradicts 𝑁^∗_𝐺≤𝑁^∗_𝑜. Therefore, 𝑁^∗_𝐺>𝑁^∗_𝑜.
With 𝑁^∗_𝐺>𝑁^∗_𝑜, 𝐷(𝑁^∗_𝐺)=𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏<𝐷(𝑁^∗_𝑜). Clearly, 𝑁^∗_𝑜>𝑁_𝑏.

Lemma 3 leads to the following proposition.

Proposition 4. Under public ownership, the government will invest the new highway if condition (37) holds.

Proof. With 𝑑²𝐷(𝑛)/𝑑𝑛²≥0 and 𝑁^∗_𝐺>𝑁^∗_𝑜, we have (𝑑𝐷(𝑁)/𝑑𝑁)|_{∗𝐺𝑁=𝑁}≥(𝐷(𝑁^∗_𝐺)−𝐷(𝑁^∗_𝑜))/(𝑁^∗_𝐺−𝑁^∗_𝑜).
Thus, together with (20) and (22), we have SW𝑁_𝑎,𝑁_𝑏,𝑁^∗_𝐺,𝑠^∗_𝐺,𝜏^∗_𝐺=^𝑁∗𝐺₀𝐷(𝑛)𝑑𝑛−𝑁^∗_𝐺𝐷𝑁^∗_𝐺+𝑁_𝑎𝑁_𝑏𝑑𝐷(𝑁)𝑑𝑁𝜕𝑁𝜕𝑁_{𝑎𝑁=𝑁∗𝐺}>^𝑁∗𝐺₀𝐷(𝑛)𝑑𝑛−𝑁^∗_𝐺𝐷𝑁^∗_𝐺+𝑁_𝑏𝐷𝑁^∗_𝐺−𝐷𝑁^∗_𝑜𝑁^∗_𝐺−𝑁^∗_𝑜𝑁_𝑎𝜕𝑁𝜕𝑁_{𝑎𝑁=𝑁∗𝐺}=^𝑁∗𝐺₀𝐷(𝑛)𝑑𝑛−𝑁^∗_𝐺𝐷𝑁^∗_𝐺+𝑁_𝑏𝑁_𝑎𝐷𝑁^∗_𝐺−𝐷𝑁^∗_𝑜𝑁^∗_𝐺−𝑁^∗_𝑜𝑁^∗_𝐺−𝑁^∗_𝑜𝑁_𝑎=−0_𝑁∗𝐺0𝐷(𝑛)𝑑𝑛−𝑁^∗_𝐺𝐷𝑁^∗_𝐺+𝑁_𝑏𝐷𝑁^∗_𝐺−𝐷𝑁^∗_𝑜. (39) With (34) and (39), we have SW𝑁_𝑎,𝑁_𝑏,𝑁^∗_𝐺,𝑠^∗_𝐺,𝜏^∗_𝐺−SW𝑁^∗_𝑜>^𝑁∗𝐺₀𝐷(𝑛)𝑑𝑛−^𝑁∗𝑜₀𝐷(𝑛)𝑑𝑛+𝑁^∗_𝑜𝐷𝑁^∗_𝑜−𝑁^∗_𝐺⋅𝐷𝑁^∗_𝐺+𝑁_𝑏𝐷𝑁^∗_𝐺−𝐷𝑁^∗_𝑜=^𝑁∗𝐺_𝑁∗𝑜𝐷(𝑛)𝑑𝑛+𝑁^∗_𝑜𝐷𝑁^∗_𝑜−𝑁^∗_𝐺⋅𝐷𝑁^∗_𝐺+𝑁_𝑏𝐷𝑁^∗_𝐺−𝐷𝑁^∗_𝑜. (40) With 𝑑𝐷(𝑛)/𝑑𝑛<0, mean value theorem leads to ^𝑁∗𝐺_𝑁∗𝑜𝐷(𝑛)𝑑𝑛=𝐷(𝜁)𝑁^∗_𝐺−𝑁^∗_𝑜>𝐷𝑁^∗_𝐺𝑁^∗_𝐺−𝑁^∗_𝑜,𝑁^∗_𝑜<𝜁<𝑁^∗_𝐺. (41) Together with (41) and 𝑁^∗_𝑜>𝑁_𝑏, (40) yields SW𝑁_𝑎,𝑁_𝑏,𝑁^∗_𝐺,𝑠,𝜏−SW𝑁^∗_𝑜=𝐷(𝜁)𝑁^∗_𝐺−𝑁^∗_𝑜+𝑁^∗_𝑜𝐷𝑁^∗_𝑜−𝑁^∗_𝐺⋅𝐷𝑁^∗_𝐺+𝑁_𝑏𝐷𝑁^∗_𝐺−𝐷𝑁^∗_𝑜>𝐷(𝜁)−𝐷𝑁^∗_𝐺𝑁^∗_𝐺−𝑁^∗_𝑜>0. (42) This result states that, if condition (37) holds, since introducing a new road improves the total social welfare, the government will invest the new toll road.

Under private ownership, we denote 𝑁^∗_𝑃 as the solution to the MNB problem. Similarly, we have Lemma 5.

Lemma 5. In equilibrium, under private ownership, 𝑁^∗_𝑃>𝑁^∗_𝑜>𝑁_𝑏 if 𝛼𝑇_𝑎+𝐾_𝑎+2(𝑘+𝜈)𝛿𝜇+𝑤<𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁^∗_𝑜. (43)

Proof. Putting (30) and (31) into (13), we have 𝛼𝑇_𝑎+𝐾_𝑎+2()−𝑁𝑘+𝜈𝛿𝜇_𝑎𝑑𝐷(𝑁)/𝑑𝑁1−2𝑁/𝛿𝜓𝜉𝑇_𝑏⋅(𝑑𝐷(𝑁)/𝑑𝑁)+𝑤=𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁_𝑏. (44)
In equilibrium, if 𝑁_𝑎=0, that is, 𝑁_𝑏=𝑁^∗_𝑜, Wardrop’s first principle leads to 𝛼𝑇_𝑎+𝐾_𝑎+2(𝑘+𝜈)𝛿𝜇+𝑤≥𝛼𝑇_𝑏+𝐾_𝑏+2𝛿𝜓𝜉𝑇_𝑏𝑁^∗_𝑜. (45)
Clearly (45) contradicts (43). Therefore, 𝑁_𝑎>0.
Similarly, as Lemma 3, we have 𝑁>𝑁^∗_𝑃>𝑁_𝑏.

With Lemma 5, we have the following proposition.

Proposition 6. Under private ownership, a private agency will invest the new toll road if condition (43) holds.

Proof. With Lemma 5, we have 𝑁_𝑎=𝑁^∗_𝑃−𝑁_𝑏>𝑁^∗_𝑃−𝑁^∗_𝑜>0. Equation (32) directly leads to 𝑃𝑅(𝑁^∗_𝑃,𝑠^∗_𝑃)>0.
Proposition 6 states that the demand for trips without introducing a new highway and the system parameters satisfy (43), a private firm will certainly invest in the new highway since it can earn a positive profit.

5. Conclusions

This paper investigated how and why to introduce a new “add-on” road into an existing transit network. Commuters either use their private cars on a bottleneck-constrained highway or choose regularly dispatched trains for travelling from a residential area to a workplace. Taking into account the OMC, the joint optimization of the toll and capacity level for a new toll road added to an existing transit network under four ownership/tolling regimes was investigated. We found that, whenever flat toll and fine toll are imposed, in a competitive highway/transit network with constant returns to scale in road construction, an optimally designed and priced added-on publicly owned highway would lead to a deficit; that is, the toll revenues are insufficient to cover its all costs, whereas an optimally designed and priced added-on privately owned highway would lead to a positive net benefit, and the net benefit is exactly equal to the total markup with respect to all autocommuters. The conditions to introduce a new “add-on” highway to the existing transit network were derived under different ownership regimes.

Although the transportation system studied in this paper is simple and the body congestion in train carriage is linear, the insights obtained are useful for enhancing our understanding of the dynamic and combinatorial commuting behavior and joint optimization of road toll and capacity level. The research framework adopted in this paper can be extended to consider joint optimization of road toll and transit fare, parking pricing [28], heterogeneous commuters [29], stochastic travel time [30], and even more complex transportation systems. These are our ongoing works.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by Grants from the National Natural Science Foundation of China (71071004, 71101098, 71271004, and 71371128).