An Equilibrium Model in Urban Transit Riding and Fare Polices

An Equilibrium Model in Urban Transit Ridingand Fare Polices

Hai-Jun Huang1, Qiong Tian1, and Zi-You Gao2

1 School of Economics and Management,Beijing University of Aeronautics and Astronautics, Beijing 100083, China

[email protected] School of Traffic and Transportation, Beijing Jiaotong University,

Beijing 100044, China

Abstract. This paper deals with the riding behavior of commuters whotake trains from a living place to a work place during morning rushhours. The total travel cost include early and late arrival penalties aswell as carriage body capacity. An equivalent mathematical program-ming model which generates equilibrium riding behavior is presented.The number of actually chosen transit runs, passenger flow distribu-tions, and fares resulted from three system configurations namely socialoptimum, monopoly by one company and duopoly competition, are in-vestigated by numerical examples.

1 Introduction

One of the most insightful and tractable approaches used to study the trip-timingbehavior is the bottleneck model proposed by Vickrey [14]. This model considersthe commuting congestion on a highway with a single bottleneck between aresidential area and a workplace. Each commuter is confronted with a trade-offbetween travel time, cost of queuing and schedule delay cost of early or latearrival at work. The travel cost experienced by a commuter is then determinedby his or her departure time from home.

Using the deterministic queuing theory, Vickrey [14] first developed a de-parture time choice model which leads to a cost-equilibrium on all commuters.Later, this model has been extended by many others (see a reviewing paper,Arnott et al. [4]). Particularly, some researchers have successfully applied thebottleneck model to investigate the changes of individual’s commuting behaviorand the system’s performances under various road-use pricing policies.

Glazer [7] and Cohen [6] investigated the welfare effects of road pricing ondifferent commuting population. Braid [5] conducted the equilibrium analysisin the case of elastic demand. Arnott et al. [1, 2, 3, 4] made contributions inmany aspects associated with bottleneck modeling, including the first-best andsecond-best tolling in networks with parallel routes. Verhoef [13] explored theeconomics of various road transportation regulations like road pricing. However,these studies are restricted to those transportation systems involving privatecars (auto mode) only.

N. Megiddo, Y. Xu, and B. Zhu (Eds.): AAIM 2005, LNCS 3521, pp. 112–121, 2005.c© Springer-Verlag Berlin Heidelberg 2005

An Equilibrium Model in Urban Transit Riding and Fare Polices 113

Tabuchi [12] studied a competitive transportation system which containstransit and highway modes. He concluded that road pricing can be regardedas a measure for restraining auto use and providing revenue for mass transportimprovement. Unlike the auto mode, the transit mode mainly depends on itsfare level and service quality for attracting commuters. But Tabuchi [12] didnot consider the body congestion in carriages and the time headways betweentrains or buses. It is obvious that these two factors affect people’s travel be-havior greatly. The body congestion also leads to the loss of independence andprivacy (Horowitz and Sheth [8]). Recently, Huang [9, 10] extended Tabuchi’swork through introducing the concept of body congestion and stressing the het-erogeneity of commuters and the overall demand elasticity.

Progress has been made along the above studies in transit system modeling;nevertheless, a flaw still exists, i.e., the transit system’s capacity is not consid-ered, this means all transit commuters can arrive on time no matter how crowdedthe carriages may be and how long the time headway is. In a real transit system,the capacity of transporting passengers is reflected by the discrete time head-ways. It may be difficult for explicitly formulating a bottleneck model for transitsystems, but there obviously exist a trade-off between the in-carriage congestionand the schedule delay caused by the time headways of dispatching transit runs.The purpose of this paper is to model this trade-off and investigate the systemand user performances caused by different system configurations.

This paper is organized as follows. Section 2 formulates the equilibrium ridingmodel through developing an equivalent mathematical program. The solutionmethod of the problem is given in Section 3. Section 4 investigates three systemoperation configurations by comparing the system and user performances fromnumerical calculations. Section 5 concludes the paper.

2 Model Formulation

Consider a simplified corridor network in which a mass transit (e.g., subway)provides transportation service between H (a residential area) and W (a work-place). There are N identical commuters who must travel by transit mode fromH to W, on every morning. To facilitate the presentation of the essential ideasof this paper, all commuters are assumed to be identical in perceiving the timevalue and schedule delay penalty. Let the moving time from H to W be a con-stant τ which covers the in-carriage time and access (egress) time from H tosubway station (from subway station to W).

The total cost experienced by a commuter who travels from H to W can beformulated as C = p + ατ + cb + δ, where p is the transit fare, α is the unitcost of travel time, cb is the body congestion cost occurring in carriages, andδ is the cost associated with schedule delay of early or late arrival at W. Thebody congestion cost is formulated by cb = τg(n), where g(n) is a monotonicallyincreasing function of the passenger flow n in the train unit, and g(0) = 0. Whena commuter arrives at W early, let δ = βT1, where β is the unit cost of early

114 H.-J. Huang, Q. Tian, and Z.-Y. Gao

arrival, T1 equals the official work start time minus the arrival time; when he/shearrives late, let δ = γ T2, where γ is the unit cost of arrival late, T2 equals thearrival time minus the official work start time.

Kraus and Yoshida [11] recently presented a railway commuting model inwhich the waiting time cost spent at stations is covered. Considering such afact that in Beijing’s subway system, commuters at each station always packthemselves into one coming train rather than wait for the next run, we henceset τ a constant.

Suppose that in the morning rush hours, all transit runs are equally dis-patched from origin H to destination W with the headway t and move alongthe line by a constant speed. This means these trains can arrive at W by thefixed interval of t. We assume that every commuter always minimizes his/herown total cost when selecting a transit run. Because of the body congestioncost, some people prefer to take runs which can arrive at W earlier or later forpreventing from high body congestion. In the equilibrium state, the N identicalcommuters are divided into l batches, each corresponding one transit run. LetZ = {a, · · · , 2, 1, 0,−1,−2, · · · ,−b} denote the set of dispatched runs, i(∈ Z)represent a specific run. That i > 0 means this train arrives at W early, i < 0 fora train arrival late, and i = 0 for a train arrival on time. In this study, overtakingbetween runs is not allowed, so there is only one train arrival on time. Let ni

be the number of passengers who ride on the ith run (note that the integer ni isregarded as the traffic volume on train i and is hence treated as a real numberin the following analyses). Their schedule delay cost is

δ(i) =

⎧⎨

⎩

i β t i > 00 i = 0−i γ t i < 0

(1)

The total travel cost paid by one commuter who rides on the ith run is

δ(i) =

⎧⎨

⎩

p + α τ + τ g(ni) + i β t i > 0p + α τ + τ g(ni) i = 0p + α τ + τ g(ni) − i γ t i < 0

(2)

An equilibrium state is reached when no commuter can reduce his/her totaltravel cost by unilaterally changing his/her transit run choice. This is calledthe user-equilibrium (UE) condition of transit riding behavior, mathematicallystated below {

C(i) = C0 if ni > 0C(i) ≥ C0 if ni = 0 i ∈ Z (3)

where C0 is a constant representing the identical equilibrium cost.We here show that with a given p and fixed τ , finding a passenger flow

distribution {ni | i ∈ Z} satisfying the UE condition (24) is equivalent to solvethe following mathematical program problem

min L(n) =∑

i∈Z

(

τ

∫ ni

0

g(ω)dω + niδ(i))

(4)


subject to the conservation condition of passenger flows and non-negative re-quirement

∑

i∈Z

ni = N (5)

ni ≥ 0 i ∈ Z (6)

As in the classical user-equilibrium traffic assignment, the objective function (4)doesn’t have any intuitive economic or behavioral interpretation. It is easy toshow that the first-order conditions of the minimization problem (4)-(6) are

ni (τg(ni) + δ(i) − λ) = 0 i ∈ Z (7)

τg(ni) + δ(i) − λ ≥ 0 i ∈ Z (8)∑

i∈Z

ni = N (9)

ni ≥ 0 i ∈ Z (10)

where λ is the Lagrange multiplier associated with (5). Equation (7) shows thatthe travel cost of any run (excluding the pre-determined constant p + ατ whichdoesn’t affect the equilibrium riding behavior) with positive passenger flow, i.e.,ni > 0, must equal a constant, λ, the Lagrange multiplier. This constant can beregarded as the minimum travel cost among all possible transit runs, as statedby (8) where the constant p + ατ is not included. With this interpretation, it isclear that the first-order conditions are equivalent to the UE condition (24): anyactually chosen transit run has the same and minimum travel cost.

The region of feasible solutions to mathematical program (4)-(6) is made bya linear equality and inequalities. This region is then a convex set. The objectivefunction is strictly convex since all body congestion functions are monotonicallyincreasing and the terms associated with schedule delay are linear, continuous.Hence, the problem (4)-(6) is a strictly convex mathematical program which hasonly one solution satisfying the UE condition.

3 Solution Method

We now further analyze the properties of the problem (4)-(6) and find a methodto determine the number of transit runs that are actually chosen by passengersin equilibrium riding state. The number of actually chosen runs is the sum ofall runs having positive passenger flows. This number can be obtained throughsolving the mathematical program (4)-(6) with a pre-determined, large enoughset Z. Let (ni, i ∈ Z) be the solution and define θi = 1 if ni > 0 and θi = 0otherwise. Then, the number of actually chosen transit runs, denoted by l, is

l =∑

i∈Z

θi (11)


In addition, that l can be derived from the minimum travel cost λ defined in(7)-(8). It is known that in equilibrium, τg(ni) + δ(i) − λ = 0 for all ni > 0. Asg(ni) > 0 for ni > 0, then δ(i) < λ. This states that the ith transit run doesn’tneed to be put into operation when δ(i) ≥ λ holds. Referring to Equation (1),we first consider the case of arrival at W early. Let k and k +1 be two successivepositive integers such that the inequality k < λ/βt ≤ k + 1 holds. We thenhave i β t ≥ λ for i ≥ k + 1 and i β t < λ for i ≤ k, which implies that theruns numbered by (k, k − 1, · · · , 1, 0, · · ·) are chosen by passengers and the runsnumbered by (· · · , k + 1) are unnecessary. Hence, the number of actually chosenruns which arrive at W early is [λ/βt] − 1 = k, where [x] is the smallest integernot less than x. Denote the numbers of transit runs which are actually chosen bypassengers, and arrive at W early and late, by a(≤ a) and b(≤ b), respectively.We then have

a(≤ a) = [λ/β t] − 1 (12)

b(≤ b) = [λ/γ t] − 1 (13)

The total number of actually chosen transit runs in equilibrium is

l = a + b + 1 (14)

From the condition, τg(ni) + δ(i) − λ = 0 for all ni > 0, the passenger flowdistribution is given by

ni =

⎧⎨

⎩

g−1 ((λ − δ(i))/τ −b ≤ i ≤ a0 −b ≤ i < −b0 a < i ≤ a

(15)

Substituting (15) into (9), we have

∑

−b≤i≤a

g−1

(λ − δ(i)

τ

)

= N (16)

Solving the above equation, we get the value of the Lagrange multiplier λ. Inequilibrium, therefore, λ and l can be written as the functions of N and t, i.e.,

λ = U(N, t) (17)

l = V (N, t) (18)

These two functions cannot be expressed explicitly in most cases, except boththe body congestion functions and the demand functions are linear. The finalminimum travel cost for each commuter, including the constant p + α τ , is

C0 = p + α τ + λ (19)


4 Three System Configurations and Pricing Polices

Up to now, we have known that given a fare p and the time headway t, thenumber of actually used transit runs and passenger flow distribution can beendogenously determined by the equilibrium riding principle. Under differentsystem configurations, the transit manager will set different service parameters(such as the fare and the headway) in order to achieve some targets. In thissection, three types of system configurations, namely social optimum, monopolyand duopoly competition are investigated.

Let D−1(N) denote the marginal benefit from trip making or the inverseof demand function, dD−1(N/dN < 0). Let G be the fixed cost of initiatingthe transit system, F be the train-associated variable cost (e.g., the expensesfor acquiring trains/carriages and employing drivers) and f be the passenger-associated variable cost (e.g., the electric power assumption of a transit run isdependent upon the loading passengers).

The system optimum configuration designed by government is in general tomaximize the social welfare, i.e.,

maxN,t

SW =∫ N

0

D−1(ω)dω − [(α τ + λ + f)N + F l + G] (20)

where λ and l are determined by (16) and (17), respectively, representing theoutputs of equilibrium riding behavior with given N and t. When N and t aresolved from (20), one obtain the transit fare

p = D−1(N) − (α τ + λ) (21)

The monopoly configuration assumes that the transit system is operated byone company which aims at maximizing the company’s net profit, i.e.

maxN,p,t

NF = pN − (f N + F l + G) (22)

subject toD−1(N) = α τ + λ + p (23)

where λ and l are given by (16) and (17), respectively. Equation (26) states theequilibrium relationship between marginal trip benefit and individual’s travelcost.

In a two-duopoly market, the transit line is operated by two independencecompanies which maximize their net profit simultaneously. The problem becomes

{maxN1,p1,t1 NF1 = p1N1 − (f N1 + F l1 + G)maxN2,p2,t2 NF2 = p2N2 − (f N2 + F l2 + G) (24)

subject toD−1(N) = α τ + λi + pi i = 1, 2 (25)


where λi and li, i = 1, 2, are given by (16) and (17), respectively.For nonlinear body congestion function and demand function, it is hard to

analytically investigate the solutions of the above three system configurations.In this paper, we conduct numerical experiments with following input data:(β,γ)=(10,50) (HK$/hr), τ=1 hr, α=20 (HK$/hr), F=1000 (HK$/train), f=1(HK$/person), G=0 (HK$). Clearly, all data are projected into one morningrush hour. The inverse of a linear demand function is D−1(N) = − ln(N/N0)/K,where the potential demand is N0=20,000 passengers. This function implies thatfor a larger K-value the demand is more sensitive to the marginal trip benefitand the final realized demand will be lower. The body congestion function isg(n) = −1.0 × ln(1 − n/nc)), where the capacity of a transit run is nc = 500passengers per run.

Fig. 1. Transit fares in three configs Fig. 2. Individual costs in three configs

Fig. 3. Required transit runs Fig. 4. Demands in three configs

Figs. 1-4 shows the fares, the individual travel costs, the number of actuallyused runs, and the realized travel demands by the three system configurations,

N = N1 + N2 (26)


respectively. It can be seen that the fare in system optimum is the lowest, thenthat in duopoly competition and that in monopoly is the highest; the same orderis found for the individual travel costs; the number of actually chosen transit runsin system optimum is the most, then that in duopoly competition and that inmonopoly is the fewest; the demand realized by system optimum is the mostand that by monopoly is the fewest. When the value of K becomes larger, eachof these four indexes comes down.

Fig. 5. Net profit in three configs Fig. 6. Social welfare in three configs

Fig. 5 shows the net profit gained in the three system configurations. Thesystem optimum configuration gives the lowest net profit, nearly K-value inde-pendent. The monopoly configuration results in more net profit for the companythan the duopoly competition configuration though the differences are not signif-icantly large. Fig. 6 shows the social welfare generated in the three configurations.

Fig. 7. Time headways in three configurations


It can be seen that, as expected, the system optimum configuration contributesthe highest social welfare, then the duopoly competition, and the monopoly thelowest. The social welfare by the three configurations converge to the same valuewhen the K-value approaches infinite.

The above numerical results show that the duopoly competition configurationcan lead to lower bus fare, higher realized demand, less individual travel cost andmore used runs than the monopoly configuration, although the two companieshave the same variable costs. This should be contributed to the competition asstated in Microeconomics.

Finally, we check the time headways generated by the three system configu-rations, see Fig. 7. It can be seen that the system optimum dispatch the transitruns most frequently, then the duopoly competition and the monopoly in order.Oscillations can be observed in Fig. 7. This is because the number of transit runsis required to be integer and the headway is continuously adjusted to optimizethe objective function designed for some system configuration. Note that someheadways shown in Fig. 7 are less than 0.6 minutes, which may not match withreality since the input parameters of the example are selected without necessarilyrepresenting reasonable values.

5 Conclusions

In this paper we studied the equilibrium riding behavior for commuters whotake trains from a living place to a work place during morning rush hours.With the aid of equilibrium riding model, in the elastic demand case we firstformulated three system configurations and then compared their transit fares,numbers of actually used runs, demands, net profit and social welfare, by nu-merical experiments. We found that the monopoly configuration generates thehighest transit fare and individual travel cost, then the duopoly competition andthe system optimum configurations in order; the system optimum configurationneeds the most transit runs and realizes the highest demands, then the duopolycompetition and the monopoly configurations in order; the monopoly configu-ration produces more net profit for company but less social welfare than theduopoly competition configuration. The value of our study lies in the method-ology adopted in this paper for analyzing different pricing mechanisms on thebasis of equilibrium riding behavior. The approach presented in this paper canbe extended to consider the continuously distributed value of time for dealingwith riding behavior with heterogeneous commuters (Yang et al. [15]).

Acknowledgements

This research is supported by the National Natural Science Foundation of China.


References

1. Arnott, R., De Palma, A., Lindsey, R.: Schedule delay and departure time decisionswith heterogeneous commuters. Transportation Research Record. 1197 (1988) 56-67

2. Arnott, R., De Palma, A., Lindsey, R.: Economics of a bottleneck. Journal of UrbanEconomics. 27 (1990) 111-130

3. Arnott, R., De Palma, A., Lindsey, R.: The welfare effects of congestion tolls withheterogeneous commuters. Journal of Transport Economics and Policy. 28 (1994)139-161

4. Arnott, R., De Palma, A., Lindsey, R.: Recent developments in the bottleneckmodel. In: Button, K. J., Verhoef, E. (Ed.), Road Pricing, Traffic Congestion andthe Environment: Issues of Efficiency and Social Feasibility. Aldershot, EdwardElgar. (1998) 79-110

5. Braid, R. M.: Uniform versus peak-load pricing of a bottleneck with elastic demand.Journal Urban Economics. 26 (1989) 320-327

6. Cohen, Y.: Commuter welfare under peak-period congestion tolls: Who gains andwho loses? International Journal of Transport Economics. 14 (1987) 239-266

7. Glazer, A.: Congestion tolls and consumer welfare. Public Finance. 36 (1981) 77-838. Horowitz, A., Sheth, J.: Ride sharing to work: An attitudinal analysis. Transporta-

tion Research Record. 637 (1977) 1-89. Huang, H. J.: Fares and tolls in a competitive system with transit and highway:

The case with two groups of commuters. Transportation Research E. 36 (2000)267-284

10. Huang, H. J.: Pricing and logit-based mode choice models of a transit and high-way system with elastic demand. European Journal of Operational Research. 140(2002) 562-570

11. Kraus, M., Yoshida, Y.: The commuter’s time-of use decision and optimal pricingand service in urban mass transit. Journal of Urban Economics. 51 (2002) 170-195

12. Tabuchi, T.: Bottleneck congestion and modal split. Journal of Urban Economics.34 (1993) 414-431

13. Verhoef, E.: The Economics of Regulating Road Transportation. Edward Elgar,Brookfield. (1996)

14. Vickrey, W. S.: Congestion theory and transport investment. American EconomicReview. 34 (1969) 414-431

15. Yang, H., Kong, H. Y., Meng, Q.: Value-of-time distributions and competitive busservices. Transportation Research E. 37 (2001) 411-424

An Equilibrium Model in Urban Transit Riding and Fare Polices

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