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Traffic rationing and pricing in a linear monocentric city

Authors

  • Wei Liu,

    1. Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong
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  • Hai Yang,

    Corresponding author
    1. Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong
    • Correspondence to: Hai Yang, Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. E-mail: cehyang@ust.hk

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  • Yafeng Yin

    1. Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL, U.S.A
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SUMMARY

This paper presents a simple spatial equilibrium model for a linear monocentric city to investigate the effects of rationing and pricing on morning commuters' travel cost and modal choice behavior in each location. Under rationing and pricing, every day in the morning peak hour, each commuter is classified as either “free” or “rationed”. “Free” commuters are allowed to use the highway without paying the toll, whereas “rationed” commuters can avoid the toll only if they travel by transit. Each day, a fraction of commuters are rationed in their free use of the highway, and the rationing fractions are determined systematically so that everyone is equally rationed in a given period. It is found that Pareto-improving rationing and pricing scheme might be obtained as a combination of the rationing degree and the toll associated with rationing. Extension to the rationing and pricing scheme with cordon and park-and-ride service has been made. Cordon and park-and-ride might help in improving the efficiency of rationing and pricing strategy although remains its Pareto-improving property. Copyright © 2012 John Wiley & Sons, Ltd.

1 INTRODUCTION

Congestion pricing has been advocated to reduce traffic congestion since 1920 (see, [1] for a recent review). Furthermore, the advanced electronic tolling system makes congestion pricing more practical and successful implementations (e.g., Singapore and London). However, getting the public to accept congestion pricing is still a major obstacle. Among reasons of public opposition, social inequality is an important one. Many have mentioned the social inequality issue of congestion pricing [2, 3]; Levinson [4]; Ecola and Light [5].

It has been suggested [6] that the strategy proposed as “hybrid between rationing and pricing” has the potential to reduce traffic congestion without penalizing anyone. In [7], Daganzo and Garcia have explored the similar strategy under a dynamic setting with single bottleneck congestion. Guo and Yang [8] investigate congestion pricing and revenue refunding schemes that reduce total travel time and make every user better off. Nie and Liu [9] examine the impacts of traveler's value of time on the existence of a pricing-refunding scheme that is both self-financing and Pareto-improving. In [10], Lawphongpanich and Yin have investigated the Pareto-improving tolls on general transportation networks that do not require revenue distribution. In Wu et al. [11], Pareto improvement is achieved by appropriately charging tolls on highway links and adjusting the fares of transit lines.

After Daganzo proposed the rationing and pricing (RP) scheme in 1995, Nakamura and Kockelman [12] presented an empirical application of the “hybrid between rationing and pricing” strategy to San Francisco Bay Bridge corridor. Han et al. [13] has analyzed the efficiency of the plate number-based traffic rationing in general networks. Wang et al. [14] have investigated the effects of road space rationing from both short-term and long-term perspectives. (long-term equilibrium under road rationing takes into account the public's self-adjustment activities such as car consumption and car disposal) This paper follows a similar strategy with Daganzo [6] and considers the Pareto-improving RP scheme without revenue refunding. Under the RP scheme, every day, each commuter is classified as either “free” or “rationed”. “Free” commuters are allowed to use the highway without paying the toll, whereas “rationed” commuters have to pay the toll if they drive. Each day a fraction of commuters are rationed in their free use of the highway, and the rationing fractions are determined systematically so that everyone is equally rationed in a given period. Different from Daganzo's single bottleneck model, we take a continuum modeling approach and conduct a general bi-modal analysis in a linear monocentric city with a competitive railway and highway system. Continuum modeling approach has been applied to explore the general tendencies and patterns of commuters' behaviors and their responses to policy changes in the transportation system at a macroscopic level [15, 16]. Ho and Wong [17] provide a recent review of the development and applications of the two-dimensional continuum traffic equilibrium modeling approach.

Unfortunately, the complexity of the two-dimensional space often makes it difficult to obtain analytical properties. By considering a simplified one-dimensional continuum corridor, analytical solution and properties of the spatial traffic equilibrium with multimodal choices can be obtained. Haring et al. [18] are the first to examine a simple solution of the user equilibrium on a traffic corridor with several congested modes. Jehiel [19] proves that the simple solution would exist if capacities of two congested modes are constant. In recent years, many have conducted analysis of various problems [e.g., road pricing and park-and-ride (P&R) facility] with linear monocentric city model [20-28]. In [29], Arnott and de Palma have considered the no toll equilibrium on a traffic corridor that connects a continuum of residential locations to a point central business district (CBD). De Palma and Arnott [30] examined a single-lane road with Lighthill-Whitham-Richards (LWR) flow congestion and Greenshields' relation.

The remainder of this paper is organized as follows. Section 2 makes some assumptions and formulates the continuum user equilibrium model without RP in a linear city. Section 3 proposes the RP scheme and explores some properties of the user equilibrium under this scheme. Extension to RP with cordon and P&R service has been made in Section 4. Numerical results are presented in Section 5, and the concluding remarks and suggestions for further researches are provided in Section 6.

2 THE MODEL

2.1 Linear monocentric city

We consider a linear monocentric city with a railway and highway serving for two travel modes, transit and auto, as shown in Figure 1. Suppose commuters are continuously distributed along the city corridor, and they can choose either railway or highway to reach the CBD every morning. The characteristic of each residential location is represented by distance from the CBD. It is assumed that all commuters in the city are homogeneous, and the demand density is given and fixed. Commuters choose their preferred travel mode solely based on the generalized travel cost. Note that commuters' choice of departure times is not considered here. Let x be the distance from the CBD, d the city boundary, and q0(x) the demand density function at the morning peak hour.

Figure 1.

A linear monocentric city with a highway and a railway.

2.2 Generalized travel cost function

Travel cost is a combination of monetary cost and travel time cost. The generalized travel cost by transit mode from location x to the CBD is

display math(1)

where math formula is the fixed cost that includes access time, fixed component of transit fare, and cT is the variable cost consisting of train travel time per unit distance and variable component of transit fare. Similar with that considered in Mun et al. [20], Wang et al. [22], Chu and Tsai [24], and Liu et al. [25], the generalized travel cost by auto mode from location x to the CBD is

display math(2)

math formula is the traffic volume at x, that is, the cumulative incoming demand from all locations further away from the city center. qA(x) denotes the travel demand density of automobile at location x. math formula is the fixed cost that includes access time, parking fee, and fixed component of vehicle operating cost. cA(QA(x)) is the variable cost consisting of travel time per unit distance at location x and variable component of vehicle operating cost fare, which is a function of traffic volume at location x. We assume that cA(Q) is twice differentiable, non-decreasing, and convex function with respect to Q.

As shown in Wang et al. [22], there are several possible equilibrium mode choice patterns. The cases with one travel mode dominating the transportation market (zero transit or auto commuters) are relatively uninteresting. The analysis by Wang et al. is thus focused on the situation where (i) the railway without congestion has the lower fixed cost than a congested highway, and (ii) traveling on highway from the city boundary is cheaper than on railway if the highway is empty. Similarly, in this study, we have two assumptions about the generalized travel cost functions. First, we assume that the fixed part of travel cost for auto mode is larger than that of transit. It follows

display math(3)

This is reasonable because for auto commuters, the money spent on car maintenance and parking in CBD is considered to be included in the fixed part of daily travel cost. The second assumption is that for commuters at the city boundary, travel to the CBD by car with free flow speed is cheaper than by railway. Therefore,

display math(4)

This assumption implies if there is no congestion on the highway, it provides very good service (e.g., short travel time and low gasoline consumption) for auto commuters. These two assumptions guarantee an interior equilibrium (under no policy) in which both the number of auto and transit commuters is strictly greater than zero.

2.3 Original user equilibrium

Let qA(x) and qT(x) be the demand densities of commuters who choose automobile and transit at location x, respectively. Then, we have

display math(5)

The traffic volume on the highway and on the railway at location x are math formula and math formula, respectively. Deterministic user equilibrium is achieved when no commuter can reduce his or her travel cost by changing to an alternative travel mode at all locations. Mathematically, the user equilibrium conditions are

display math(6)

where x ∈ [0,d]. The aforementioned costs for transit mode and auto mode are given by Equations (1) and (2), respectively.

When RP is not considered, our model reduces to the one by Wang et al. [22] in the absence of P&R facilities. Under the previous two assumptions, there exists a watershed phenomenon of the equilibrium modal split along the corridor, which can be shown by Figure 2. All commuters living inside location x0 will travel by transit to avoid high fixed cost of auto mode (e.g., high parking fee), whereas other commuters travel by private car because the highway outside of location x0 is less uncongested and commuters can enjoy faster speed and short travel time there. The solution x0 can be determined by solving CA(x0) = CT(x0), that is,

display math(7)
Figure 2.

Travel cost under original user equilibrium.

Note that the previous is due to our deterministic consideration of choice behaviors of commuters. If stochastic choice behaviors are considered, commuters at the same location may choose different travel modes.

2.4 First best pricing (System Optimum: SO)

For comparison purpose, we provide the first best pricing scheme here. Under the first best pricing scheme, travel from location x to the CBD, the toll τso(x) can be given by

display math(8)

or equivalently can be given by

display math(9)

where x ∈ [0,d], and math formula is the first order derivative of cA(Q) with respect to Q. Equation (8)represents the congestion externality that an additional trip originating from x impose on all travelers between location x and the CBD (these travelers include individuals live both inside and outside location x). Equation (9) is indeed equivalent to Equation (8), but it splits the congestion externality of an additional trip originating from location x into two parts. The first and second terms in Equation (9) are the externality imposed on travelers live outside and inside location x, respectively. For commuters who live outside x, this additional trip at x has impact on them all the way from x to CBD; for commuters who live inside x, it only has impact on them from the location they live to CBD.

Lemma 1. Given cA(Q) is twice differentiable, convex and non-decreasing in Q, the first best location based toll, τso(x), is non-decreasing and concave in x.

math formula and math formula are the first and second order derivatives of cA(Q) with respect to Q, respectively. Given cA(Q) is convex and non-decreasing in Q, we have math formula and math formula. It is easy to find out that τso(x) is non-decreasing and concave because the first order derivative, that is, math formula, and the second order derivative, that is, math formula

As shown in Figure 3, under the first best pricing scheme, the equilibrium modal split along the corridor is quite similar to that of original user equilibrium (OUE), the only difference is that the watershed point goes outward, that is, xso > x0. And xso can be determined by solving the equation CA(xso) + τso(xso) = CT(xso).

Figure 3.

Travel cost under first best pricing scheme.

3 RATIONING AND PRICING WITHOUT CORDON

3.1 Rationing and pricing scheme

Under the RP scheme, every day each commuter is classified as either “free” or “rationed”. The classification method is such that: (i) in a long period the fraction of days, ϵ, that a commuter is rationed is the same for all commuters, and (ii) the fraction of “rationed” commuters is ϵ every day. “Free” commuters are allowed to use the highway without paying the toll, whereas “rationed” commuters can avoid the toll only if they travel by transit. ϵ is the rationing degree, and the toll associated with rationing is denoted by τ. For convenience, every pair of rationing degree and the associated toll, that is, (ϵ,τ), is an RP scheme.

For commuters at location x classified as free, travel cost by transit is math formula, and travel cost by automobile is math formula. For commuters at location x classified as rationed, travel cost by transit is math formula, and travel cost by automobile is math formula. For convenience, we denote travel cost for free commuters and rationed commuters by Cf(x) and Cr(x), respectively.

3.2 User equilibrium under rationing and pricing

Let math formula and math formula be the demand densities of free commuters who choose automobile mode and transit at location x, respectively. And let math formula and math formula be the demand densities of rationed commuters who choose automobile mode and transit at location x, respectively. It follows

display math(10)

where x ∈ [0,d]. The traffic volume on the highway and on the railway at location x are math formula and math formula, respectively. Similar to the OUE, the user equilibrium conditions are

display math(11)

where i = f, r and x ∈ [0,d].

In Figure 4, the travel cost curves along the traffic corridor are presented which show the travel cost of each travel mode at each location for both rationed and free commuters in the presence of RP, and the corresponding horizontal bars show the modal choices of all commuters. From Figure 4, we can see that when the toll is not too high in the city boundary, there always exist some rationed commuters who choose automobile even they have to pay the toll. When the toll is higher than a certain value, no rationed commuter in the city will choose auto mode.

Figure 4.

Travel cost under different RP schemes.

There exists an upper bound of the toll associated with rationing, math formula. When the toll is smaller than the upper bound of the toll, that is, math formula, under the previous two assumptions, there exists a watershed phenomenon of the equilibrium modal split along the corridor for free and rationed commuters, respectively in Figure 4(a). All free commuters living inside location xf, that is, math formula, travel by transit, whereas other free commuters, that is, math formula, travel by private car. All rationed commuters living inside location xr, that is, math formula, travel by transit, whereas other rationed commuters, that is, math formula travel by private car. The solution (xf,xr) can be determined by solving

display math(12)

where math formula and math formula are defined in Section 3.1.

When math formula, the toll is too high and no rationed commuter will choose auto mode. In other words, all the rationed commuters, that is, math formula, travel to the CBD by transit. However, there still exists a watershed phenomenon of the equilibrium modal split along the corridor for free commuters in Figure 4(b). All free commuters living inside location xϵ, that is, math formula, travel by transit, whereas other free commuters, that is, math formula travel by private car. The solution xϵ can be determined by solving CA(xϵ) = CT(xϵ), that is,

display math(13)

For a given rationing degree, ϵ, math formula. Further, the upper bound of the associated toll can be determined by math formula. The toll bound, math formula, increases with the rationing degree and math formula.

Proposition 1. The modal split pattern along the corridor under the first best pricing scheme is identical to that under the RP scheme (ϵ,τ), where ϵ = 1 and τ = τso(xso).

This result can be interpreted as follows: suppose the first best pricing scheme is implemented, thus commuters living inside location xso will all choose transit mode, whereas commuters living outside location xso will choose auto mode, and commuter travel from x to CBD will experience a toll equal to τso(x). Now, we implements the RP scheme (ϵ,τ), where ϵ = 1 and τ = τso(xso). As mentioned in Lemma 1, τso(x) is non-decreasing, thus for x < xso, we have τso(x) ≤ τso(xso), a higher toll would prohibit commuters living inside xso to change from transit to auto; and for x > xso, we have τso(x) ≥ τso(xso), a lower toll would not attract commuters living outside xso to change from auto to transit. Therefore, the modal split pattern under this RP scheme is identical to that under the first best pricing scheme. Indeed, under RP scheme (ϵ,τ) where ϵ = 1 and τ = τso(xso), we have xf < xr = xso. Because the rationing degree is unity, all commuters living inside xr = xso will choose transit, whereas all commuters living outside xr = xso will choose auto mode. This implies that in a linear monocentric city, it is possible to approach system optimum by a coarse toll.

3.3 Cost comparison

In the OUE case, the travel cost of commuters is

display math(14)

where math formula and math formula are calculated under original user equilibrium traffic flow pattern. After introducing the RP scheme, from a long-run perspective, travel cost of commuters is the average of cost in rationed days and free days, E(Crp(x)) = ϵCr(x) + (1 − ϵ)Cf(x), that is,

display math(15)

where math formula, math formula and math formula are calculated under the traffic flow pattern after implementing the RP scheme.

Definition 1. A Pareto-improving rationing and pricing scheme is a pair (ϵ,τ) that satisfies the inequalities: E(Crp(x)) ≤ Ce(x), x ∈ [0,d].

Now, we focus on the case that math formula. math formula and math formula are the traffic volume at location x under the original user equilibrium and an RP scheme, respectively. x0, xf and xr are determined by Equations (7) and (10).

Lemma 2. At equilibrium, whether under the RP scheme, (ϵ,τ), or not, it follows that the variable cost on the highway is less than that on the railway for all locations, that is, cA(QA(y)) < cT, for all x ∈ [0,d].

Lemma 3. Under RP scheme, (ϵ,τ), with ϵ > 0 and τ > 0, (i) the watershed points for free and rationed commuters satisfy the condition: xf < x0 < xr, and (ii) and the traffic volume is always less than that in OUE, that is, math formula, for all x ∈ [0,d].

The proofs of Lemma 2 and Lemma 3 are provided in APPENDIX PROOF OF LEMMA 2 and AppendixPROOF OF LEMMA 3, respectively. With Lemmas 2 and 3, and under the assumption that cA(Q) is non-decreasing and convex with respect to Q, we have the following Proposition 2.

Proposition 2. The sufficient and necessary condition for the RP scheme (ϵ,τ) to be Pareto-improving is

display math(16)

where 0 < ϵ ≤ 1 and math formula.

We only consider 0 < ϵ ≤ 1 and math formula, otherwise, it is not an RP scheme.

Proof. For the commuters inside location xf, because they always choose transit, their travel does not change after introducing the RP scheme, thus E(Crp(x)) = Ce(x) for x ∈ [0,xf]. For commuters living between xf and x0, because CA(x) < CT(x) (this is easy to verify with Lemmas 1 and 2), their average travel cost after introducing the RP scheme will always be less than that in OUE.

Now, we turn to the commuters living between x0 and xr. For convenience, denote E(Crp(x)) − Ce(x) by Δ(x), that is,

display math(17)

For x0 ≤ x < xr, we have math formula, thus the first and second order derivatives of Δ(x) are given by

display math(18)

where math formula is the derivative with respect to Q. Under our assumption that cA(Q) is non-decreasing and convex in Q, and when x < xr we have Qe(x) > Qrp(x), we always have d2Δ(x)/dx2 > 0. This implies that Δ(x) is strictly convex, that is, Δ(αx0 + (1 − α)xr) < αΔ(x0) + (1 − α)Δ(xr), 0 ≤ α ≤ 1. Because we have already show that Δ(x0) ≤ 0, if Δ(xr) ≤ 0, we have Δ(x) ≤ 0.

For x ≥ xr, because Qe(x) = Qrp(x), we have cA(Qrp(x)) = cA(Qe(x)), then we have the equation: math formula. This implies that E(Crp(x)) − Ce(x) is identical for all x ∈ [xr,d], which is equal to Δ(xr).

We now conclude that if and only if travel cost of commuters at location xr is less than that in OUE, the RP scheme is Pareto-improving. More specifically, we can easily derive the condition Equation (16).

For commuter at location xr, math formula is indeed the travel cost increase in the rationed days compared with the OUE, whereas τ − δr is the travel cost saving in the free days. For the RP scheme to be Pareto-improving, we need (1 − ϵ)(τ − δr) ≥ ϵδr, which is actually identical to Equation (16), that is, (1 − ϵ)τ ≥ δr.

4 RATIONING AND PRICING WITH CORDON

4.1 RP scheme with cordon and park-and-ride

The RP scheme with cordon is quite similar to the scheme without cordon. Every day each commuter is still classified as either “free” or “rationed”. But rationed commuters need to pay the toll only if they drive inside the cordon location, xc. Besides, we introduce one more travel mode, that is, P&R mode. With this new travel mode, commuters outside the cordon can drive to the cordon and park their cars, then take the transit. The triple of the rationing degree, associated toll and the cordon location, that is, (ϵ,τ,xc), constitute an RP scheme with cordon (RPC scheme).

The generalized travel cost by P&R mode from location x to the CBD is

display math(19)

where x ≥ xc, and math formula is the fixed cost that includes access time, parking fare, fixed component of transit fare and vehicle operating cost, transfer cost, cT, and cA(QA(x)) is the variable cost associated with transit and auto, respectively. It is assumed that math formula, which implies parking fare at the cordon is relatively low compared with the CBD, and the transfer from auto to transit is quite convenient. When math formula, for commuters inside cordon location, xc, travel by transit is always cheaper than P&R; therefore, no users inside cordon will choose P&R mode and there is no outflow.

4.2 User equilibrium with P&R facility

Let math formula, math formula, and math formula be the demand densities of free commuters who choose automobile, transit and P&R at location x, respectively. Let math formula, math formula, and math formula be the demand densities of rationed commuters who choose automobile, transit, and P&R at location x, respectively. It follows

display math(20)

where x ∈ [0,d]. The traffic volume on the highway at location x (inside cordon) is math formula, whereas traffic volume on the highway at location x (outside cordon) is math formula. Similar to the OUE, the user equilibrium conditions are

display math(21)

where i = f, r, x ∈ [0,d].

4.3 Cordon location (P&R facility location)

With given ϵ and τ, the equilibrium modal choice pattern under an RPC scheme depends on the cordon location. Table 1 identifies the various possibilities by ranking the cordon location in the column and the regions of the city in the rows. Taking the case, 0 < xc ≤ x1, as an example, inside location xmr, both free and rationed commuters choose railway, whereas outside the location, both free and rationed commuters choose P&R mode. For cases I and II, xmt are determined by CM(xmt) = CT(xmt), for cases III and IV, xat and xmt are determined by CA(xat) = CT(xat) and CM(xmt) = CT(xmt) together, and for case V, xat and math formula are determined by CA(xat) = CT(xat) and math formula together. All of the generalized cost functions are calculated under corresponding traffic pattern. x1, x2, x3, and x4 are the critical location when modal choice pattern changes qualitatively; and with given ϵ and τ, they can all be determined by solving groups of equations, respectively.

Table 1. Classification of equilibrium cases with different cordon location.
RegionCase ICase IIRegionCase IIICase IVRegionCase V
0 < xc ≤ x1x1 < xc < x2x2 ≤ xc < x3x3 ≤ xc < x4x4 ≤ xc ≤ d
  1. f, free commuters; r, rationed commuters; T, transit; A, automobile; M, P&R.

  2. For case II: xat = xmt; for case IV: math formula; for case V: xat = xf and math formula

0 − xmtf and r: Tf and r: T0 − xatf and r: Tf and r: T0 − xatf and r: T
xat − xmtf: Af: Amath formulaf: A
r: Tr: Tr: T
xmt − df: Mf: M and Axmt − df: Af: Amath formulaf: A
r: Mr: Mr: Mr: M and Ar: A

Case I (0 < xc ≤ x1) and Case II (x1 < xc < x2): For both free and rationed commuters inside xmt, transit is always the cheapest travel mode. In Case I, for both free and rationed commuters outside xmt, P&R is the cheapest travel mode even with free flow speed inside cordon location. In Case II, for free commuters outside xmt, both P&R and auto mode are used, whereas for rationed commuters outside xmt, only P&R is used. With the outward of cordon location (also the P&R facility location), the P&R mode becomes less attractive. (Figure 5)

Figure 5.

Travel cost under RPC schemes: Case I and II.

Case III (x2 ≤ xc < x3) and Case IV (x3 ≤ xc < x4): For commuters inside xat, transit is always the cheapest travel mode. For free commuters outside xat, because the cordon location is far away from the city center, P&R is no longer attractive, only auto mode is used. For rationed commuters living in the middle of the city (xat ≤ x < xmt), only transit mode is used, which is also because of the cordon location is far away from their destination. For rationed commuters living near the city boundary (xmt < x ≤ d), if cordon location is still not too far away (Case III), P&R mode will be chosen to avoid the toll. In Case IV, some of the rationed commuters living near the city boundary will give up P&R mode and choose auto mode even if they have to pay the toll. (Figure 6)

Figure 6.

Travel cost under RPC schemes: Case III and IV (xc ≤ xat and xat < xc)

Case V (x4 ≤ xc < d): The equilibrium flow pattern of this case is identical to the equilibrium under the RP scheme without cordon. That is because parking far away from destination is inconvenient, and the P&R facility is extremely unattractive, which will be given up by all commuters.

4.4 Cost comparison

Similar to the RP scheme without cordon, the travel cost of commuters under the RPC scheme is also the average of cost in rationed days and free days, E(Crpc(x)) = ϵCr(x) + (1 − ϵ)Cf(x).

Definition 2. A Pareto-improving RPC scheme is a combination of rationing degree, the associated toll and cordon location, (ϵ,τ,xc), that satisfies the inequalities: E(Crpc(x)) ≤ Ce(x), x ∈ [0,d].

Because Case V (P&R facility is not used) is identical to the equilibrium under the RP scheme, now we focus on Case I, II, III, and IV in which the P&R facility is used by some commuters.

Lemma 4. At equilibrium, under the RPC scheme, (ϵ,τ,xc), it follows that the variable cost on the highway is less than that on the railway for all locations, that is, math formula, for all x ∈ [0,d].

Lemma 4 is an extension of Lemma 2, the situation here is much more complicated than that mentioned in Lemma 2 because the cordon location varies. The proof is provided in APPENDIX C.

Lemma 5. Under the RPC scheme, (ϵ,τ,xc), for case I and case II, the inequality xmt ≤ x0 always holds.

Lemma 5 can be easily proved by contradiction. The inequality xmr ≤ x0 implies that compared with the OUE, the RPC scheme with attractive P&R facility (case I and case II: when the P&R facility is near to the CBD) will lead more commuters to drive (some of them choose P&R service).

Proposition 3. In Case I and Case II, no commuters will pay the toll, and the PRC scheme, (ϵ,τ,xc), is always Pareto-improving.

With Lemmas 4 and 5, it is easy to find that E(Crpc(x)) ≤ Ce(x) always holds, thus the RPC scheme is Pareto-improving. The providence of P&R facility helps to enhance system performance in two aspects: reduced parking fee and congestion near the CBD. In these two cases, the toll has no impact on commuters because the P&R facility is too attractive and individual commuter can enjoy auto mode outside the cordon location and avoid congestion and higher parking fee in the CBD.

Lemma 6. In case III and IV, when xmt > x0, we must have xat < x0.

This result is similar to that mentioned in Lemma 3 that xf < x0 < xr.

Proposition 4. In Case III, the RPC scheme, (ϵ,τ,xc), is Pareto-improving if and only if

display math(22)

For commuter at location xmt, math formula is indeed the travel cost increase in the rationed days compared with the OUE, whereas math formula is the travel cost difference between free and rationed days.

In case III and IV, when xmt ≤ x0, the RPC scheme is always Pareto-improving. This result is similar to that in Proposition 3. We now show that Equation (22) must hold when xmt ≤ x0. It is easy to find that the left-hand side of Equation (22) is greater than zero, and the right-hand side is less than or equal to zero when xmt ≤ x0, thus Equation (22) holds.

Now, we turn to the situation when xmt > x0. As Lemma 6 mentioned, we must have xat < x0 < xmt. The argument here is similar to the proof of Proposition 2. For convenience, again we let Δ(x) = E(Crpc(x)) − Ce(x). For the commuters living inside location xat, because they always choose transit, thus Δ(x) = 0. For commuters living between xat and x0, because CA(x) < CR(x) (this is easy to verify with Lemma 4), their average travel cost after introducing the RPC scheme will always be less than that in OUE, thus Δ(x) < 0. For commuters living outside xmt, because Qe(x) = Qrpc(x), Δ(x) is identical for all x ∈ [xmt,d], which is equal to Δ(xmt).

For commuters living between x0 and xmt, in the situation with xc ≤ x0, then the proof is identical to that in Proposition 2, and we have Δ(x) that is strictly convex, Equation (22) is the sufficient and necessary condition for the RPC scheme to be Pareto-improving. In the situation with xc > x0, the proof is slightly different because the second order derivative of Δ(x) at xc may not exist (traffic volume Qrpc(x) is not continuous at xc). However, it can be shown that Δ(x) is strictly convex for x0 ≤ x < xc and xc ≤ x < xmt, respectively, and Δ(xmt) ≤ 0 ⇒ Δ(xc) ≤ 0. Then we also can conclude that the RPC scheme is Pareto-improving if and only if Δ(xmt) ≤ 0. More specifically, we can derive Equation (22). Similarly, we also have the following Proposition 5 for case IV.

Proposition 5. In case IV, the RPC scheme, (ϵ,τ,xc), is Pareto-improving if and only if

display math(23)

Actually, in case IV, math formula, thus Equation (23) is identical to the condition mentioned in Proposition 4.

5 NUMERICAL EXAMPLE

In this section, we employ an example to facilitate the presentation of essential ideas. For RP schemes without cordon, we examine its Pareto-improving property and system performance, and how they vary with the rationing degree or toll level. For RP schemes with cordon, we also explore its Pareto-improving property and system performance, and especially how they vary with cordon location.

5.1 Data input

In order to utilize the conventional discrete modeling solution techniques in this continuum model, we discretize the city corridor into a finite number of nodes as shown in Figure 7, which is similar to the discrete approximation proposed by Liu et al. [25]. For a commuter at node i, he or she can travel to the highway or railway first and then go to the city center. After introducing the RP scheme with cordon, for a commuter on the highway, he or she can get off the highway at the cordon location and travel to the parking spot, and then take transit to travel to the city center. It is straightforward to observe that as the number of nodes increases, the discretized network will approach the continuum traffic corridor.

Figure 7.

The discretized city with a highway and a railway.

The length of the city is assumed to be 100 km, that is, d = 100(km), and the corridor is discretized into 100 nodes, thus n = 100, and the distance between the nearest two nodes is b = 1(km). The demand density is q0(x) = 1500(commuters/hour/km), and the demand for every node is q0 = q0(x) ⋅ b = 1500(commuters/hour). The unit cost per distance of auto mode is given by cA(Q) = 1 + 0.3(Q/30000)5(HK$/km). The inputs of other parameters are: math formula, math formula, math formula, cT = 2.2(HK$/km).

5.2 Comparison of RP schemes without cordon

First, we look into the situation in which cordon and P&R service is not considered. For an RP scheme with given rationing degree, the total travel cost and total toll revenue both vary with the toll level. Figure 8 shows how total cost and toll revenue vary under given rationing degree equal to 0.25, 0.50, 0.75, and 1.0, respectively. When the rationing degree is relatively small (0.25 and 0.5), the optimal toll level (in terms of minimizing total social cost) occurs at the toll upper bound, which implies the optimal situation requires all the rationed commuters give up the highway and take transit. When the rationing degree is relatively large (0.75 and 1.0), the optimal toll level occurs between zero and the upper bound, thus there should be some rationed commuters using highway and paying the toll. This implies the rationing degree is too high, and we should let some rationed commuters to use auto mode. These results are not surprising because too many or too few commuters using highway might all lead to system inefficiency. Also, it is evident in Figure 8 that the total toll revenue starts at zero when toll level equal to zero and ends at zero when toll level is too high and no rationed commuter drive and pay the toll. In addition, we can see that with the increase of rationing degree (0.25 → 0.5 → 0.75 → 1), the corresponding minimum total social cost decreases (192. 27 → 189. 55 → 183. 67 → 178. 48) and approaches the system optimal value. When rationing degree equal to one, the optimal toll level is around 65, and as mentioned in Proposition 1, the equilibrium under this RP scheme without cordon is identical to the first best pricing situation.

Figure 8.

Total cost and toll revenue vary with toll level.

Similarly, for the RP scheme with given toll level, the total travel cost and total toll revenue both vary with rationing degree. Note that in Figures 9 and 10, PI means Pareto-improving and non-PI means not Pareto-improving. Figure 9 shows how total cost and toll revenue vary under given toll level equal to 25 and 35, respectively. When the toll equals 25, total social cost decreases with the rationing degree, thus the optimal rationing degree is one. However, if we want the RP scheme to be Pareto-improving, the total social cost reaches its minimum when rationing degree equals 0.6 (minimum under Pareto-improving constraint). The results are quite similar for the case with toll equal to 35. The main difference is that when rationing degree is less than about 0.3, 35 is greater than the upper bound of the associated toll, thus all these scenarios are purely rationing, and all rationed commuters will choose transit and would not pay the toll.

Figure 9.

Total cost and toll revenue vary with rationing degree.

Figure 10.

Total cost vary with rationing degree (purely rationing).

Now, we consider the situation when the toll is too large (larger than all the upper bound of toll), thus the RP scheme becomes purely rationing. As mentioned before, in this case, no rationed commuters will pay the toll because it is too costly. As shown in Figure 10, there exists an optimal rationing degree equal to about 0.75 in terms of minimizing total social cost. If we want the purely rationing scheme to be Pareto-improving, the rationing degree can be no greater than 0.55, thus the optimal purely Pareto-improving scheme occurs when rationing degree equals 0.55.

5.3 Comparison of RPC schemes

If cordon and P&R service is considered, with given rationing degree and toll level, the total travel cost varies with cordon location. Figure 11 shows how the total travel cost varies with cordon locations for four combinations of rationing degree and toll level. In Figure 11, the two real lines stand for total travel cost includes and excludes toll revenue, respectively; and the two dotted lines stand for average individual travel cost of commuters living at the city boundary under the RPC scheme and in the OUE, respectively. It can be seen that P&R facility location (cordon location) near to the CBD is of high probability lead to inefficiency. For the case when cordon location is extremely near to the CBD, one can imagine that government provides more parking spots in the CBD and lows down the parking fee, then there will be more highway users because they can enjoy the parking fee reduction provided by government, and this will lead to more congested traffic on some highway sections, which in return partly offset the savings from parking fee reduction. If emission is considered, the situation is even worse.

Figure 11.

Total cost vary with cordon location.

As shown in Figure 11, for the first two cases, no matter where one puts the cordon location, the RPC scheme is always Pareto-improving because the average travel cost of commuters living in the city boundary is less than that in OUE. For the last two cases, the cordon location cannot be too far away from the CBD if we want the RPC scheme to be Pareto-improving. As we know that when cordon location equals to 100 (km), the RPC scheme simplifies to an RP scheme without cordon. Thus, we can find in Figure 11 that introducing cordon and P&R facility not only can help to reduce total social cost but also help the non-Pareto-improving RP scheme to be Pareto-improving.

5.4 Performance comparison of various schemes

Table 2 shows the solutions under various schemes. OUE leads to the highest total social cost, 193.81(105 HK$/h), whereas the first best pricing scheme leads to the lowest social cost, 178.48(105 HK$/h). When the toll is too high (higher than the toll upper bound), the RP scheme becomes a purely rationing scheme. The purely rationing scheme, R(2), with rationing degree equal to 0.75 is the optimal purely rationing scheme, whereas the purely rationing scheme, R(1), with rationing degree equal to 0.55 is the optimal solution under Pareto-improving constraint. Given the toll level equal to 25 (HK$), the optimal rationing degree is equal to 0.6, which is the first RP scheme. Under the same rationing degree and toll level as RP(1), by minimizing total social cost with respect to cordon location, we have the scheme RPC(1). Given the rationing degree equal to 0.6, the optimal toll level is equal to 58 (HK$), which is the second RP scheme. Under the same rationing degree and toll level as RP(2), by minimizing total social cost with respect to cordon location, we have the scheme RPC(2). The third RP scheme is the one under which the modal split along the corridor is identical to that under the first best pricing scheme.

Table 2. Comparison of solutions under various schemes.
SolutionSchemes
OUER(1)R(2)RP(1)RPC(1)RP(2)RPC(2)RP(3)FB
Rationing degree 0.550.750.60.60.60.61.0 
Toll level (HK$) InfinityInfinity2525585865 
Cordon location (km)    70 76  
Social cost (105 HK$/h)193.81188.65185.49189.67188.03187.63186.40178.48178.48
Cost reduction (%)(0.00)(2.66)(4.29)(2.14)(2.98)(3.19)(3.82)(7.91)(7.91)
Revenue (103 HK$/h)   303.1118.6104.401701.71751.7
Pareto-improvingNoYesNoYesYesNoYesNoNo

It is obvious that the first best pricing scheme is the most efficient in minimizing total social cost, although it is not Pareto-improving. However, as expected, the third RP scheme (ϵ = 1.0, τ = 65) gives the same minimum total social cost as the first best pricing scheme. Further, this RP scheme, RP(3), leads to less total toll revenue. From Table 2, it can be seen that RP scheme may or may not be Pareto-improving and generally less efficient than first best pricing. Nonetheless, the introduction of cordon location and P&R facility helps to regain Pareto-improving property (RP(2) → RPC(2)) and also helps to improve efficiency (RP(1) : 189.67 → RPC(1) : 188.03 and RP(2) : 187.63 → RPC(2) : 186.40).

6 CONCLUSION

This paper presents a deterministic continuum equilibrium model to characterize commuters' modal choice behavior with and without RP policies in a linear monocentric city. There exists an upper bound of the toll associated with rationing. When the toll is higher than or equal to the upper bound, no rationed commuters choose auto mode. Pareto-improving RP schemes without cordon might be found if congestion is significant. For the RP scheme with cordon, P&R mode might be used, which depends on the attractiveness of the P&R facility determined by its location. RP scheme is less efficient than first best pricing scheme although might be Pareto-improving. Cordon and P&R might help to improve efficiency of RP strategy while remain its Pareto-improving property.

The analysis is this paper can be further extended along several lines. First, the uniform toll for all the rationed commuters along corridor might be replaced by a fine toll associated with the location. Second, it is meaningful and challenging to consider a model that can consider the dynamic traffic patterns in a linear monocentric city.

ACKNOWLEDGEMENTS

The work described in this paper was supported by a grant from Hong Kong's Research Grants Council under project HKUST HKUST620910E. The third author would like to acknowledge the support from National Natural Science Foundation of China (71228101).

APPENDIX A: PROOF OF LEMMA 2

Under no scheme (OUE), in equilibrium we have CA(x0) = CT(x0), that is, math formula. Under assumption that math formula, it is obvious that cA(Q(x0)) < cT. In equilibrium, when x ≤ x0, Q(x) = Q(x0), thus cA(Q(x)) = cA(Q(x0)); when x > x0, Q(x) < Q(x0), thus cA(Q(x)) < cA(Q(x0)). Therefore, cA(Q(x)) < cT.

The proof for the case under RP scheme is similar. In equilibrium, we have CA(xf) = CT(xf), that is, math formula. Under assumption that math formula, it is obvious that cA(Q(xf)) < cT. In equilibrium, when x ≤ xf, Q(x) = Q(xf), thus cA(Q(x)) = cA(Q(xf)); when x > xf, Q(x) < Q(xf), thus cA(Q(x)) < cA(Q(xf)). Therefore, cA(Q(x)) < cT.

APPENDIX B: PROOF OF LEMMA 3

For the OUE and equilibrium under RP scheme, we have the following three equations:

display math(B.1)

For ϵ > 0, τ > 0, from (B.1) it is obvious that xf < xr. Now, we prove xf < x0 < xr by contradiction.

Assume that x0 ≤ xf, then we have Qe(x0) > Qrp(xf) (given that ϵ > 0, τ > 0), thus cA(Qe(x0)) > cA(Qrp(xf)). By Lemma 1, we have 0 < cT − cA(Qe(x0)) < cT − cA(Qrp(xf)). Also, we have [cT − cA(Qe(x0))]x0 = [cT − cA(Qrp(xf))]xf from (B.1), therefore, x0 > xf which contradicts the assumption that x0 ≤ xf.

Now, assume that x0 ≥ xr, then we have Qe(x0) < Qrp(xf) (given that ϵ > 0, τ > 0), thus cA(Qe(x0)) < cA(Qrp(xf)). By Lemma 1, we have cT − cA(Qe(x0)) > cT − cA(Qrp(xf)) > 0. Also, we have [cT − cA(Qe(x0))]x0 = [cT − cA(Qrp(xf))]xf from (B.1); therefore, x0 < xf which contradicts the previous result that x0 > xf. Therefore, we have xf < x0 < xr.

From the results earlier, it is easy to verify that, when x ≥ xr, we have Qe(x) = Qrp(x); when x < xr, we have Qe(x) > Qrp(x). Therefore, for x ∈ [0,d], we always have Qe(x) ≥ Qrp(x).

APPENDIX C: PROOF OF LEMMA 4

For the OUE and equilibrium under RPC scheme, we have the following three equations:

display math(C.1)

For case I and case II, Qrpc(x) ≤ Qrpc(xat), because we have the assumption that math formula, according to Equation (C.1), cA(Qrpc(x)) ≤ cA(Qrpc(xat)) < cT, for x ∈ [0,d]. For cases III and IV when xc ≤ xat, the inequality Qrpc(x) ≤ Qrpc(xat) also holds, thus by similar argument we again always have cA(Qrpc(x)) < cT. For cases III and IV when xc > xat, we always have Qrpc(x) ≤ max{Qrpc(xat), Qrpc(xc)}, again from Equation (C.1) we have the inequality cA(Qrpc(xat)) < cT holds, thus we only need to prove that cA(Qrpc(xc)) < cT. Now consider the critical scenario when xc = xat, denote this specific point by x* and the specific xmt by math formula, and we have cA(Qrpc(x*)) < cT. For cases III and IV in which xc > xat, indeed we have xc > x*, and under this condition, it is easy to check that math formula should hold, and we have the following result:

display math(C.2)

and finally we have cA(Qrpc(xc)) ≤ cA(Qrpc(x*)) < cT. For case V, the situation is identical to that under the RP scheme without cordon of which the result has been already proved in Lemma 2.

Ancillary