Pareto efficient strategies for regulating public transit operations

Public Transp (2012) 3:199–212 DOI 10.1007/s12469-011-0047-8 O R I G I N A L PA P E R Pareto efficient strategies for regulating public transit operations Qiong Tian · Hai Yang · Hai-Jun Huang Published online: 20 December 2011 © Springer-Verlag 2011 Abstract This paper investigates how the local authorities could efficiently regulate the public transit, which is operated by a private firm. Both the waiting time at stops and the in-vehicle congestion costs are taken into account to reflect the transit service quality. The Pareto-efficient frontier is derived and three types of regulation strategies, namely Price-cap, Return-on-output and Quantity control, are analyzed and compared. On one hand, although the Price-cap regulation can attract more demand effectively, the private firm will inefficiently supply a lower frequency to keep the cost down. On the other hand, both the Return-on-output (ROO) and Quantitycontrol regulations are Pareto efficient that can keep the transit system operating along the Pareto-efficient frontier. Especially, Quantity-control regulation seems to be more attractive than ROO as there is no need for the firm’s accounting information. In addition to the investigations on regulation, a new optimal demand-frequency correspondence is also derived that extends the Mohring’s “Square Root Principle” in incorporating transit in-vehicle congestion effects. Keywords Transit regulation · Pareto · efficient frontier · Quantity control · Congestion Q. Tian () · H.-J. Huang School of Economics & Management, Beijing University of Aeronautics and Astronautics, Beijing, P.R. China e-mail: tianqiong@buaa.edu.cn H.-J. Huang e-mail: haijunhuang@buaa.edu.cn H. Yang Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, P.R. China e-mail: cehyang@ust.hk 200 Q. Tian et al. 1 Introduction Urban mass transit systems play a vital role in reducing traffic congestion, offering alternative means of travel, and contributing greatly to the quality of urban life. Its operations, planning and economics have become an important issue of long standing interest to economists and transportation scientists. Because of the belief of economies of scale in transit service, governments or local authorities usually owned and operated the public transit systems by themselves. Since the end of the 1970s and the early years of the 1980s, great changes have been made by governments in the transport policy of most West European countries and the United States of America. Deregulation of transport service into a market-oriented system has been carried out in many urban regions in these countries (Andersen 1992). Notable earlier studies include Pedersen (1994, 1995) who studied the optimal regulation policies in situations where private profit-seeking firms operate the markets. His works considered asymmetric information about operating cost and demand, respectively. Recently, there has been an increasing interest in the study of Public-private Partnership (PPP) projects in transportation management. PPP is a mechanism for financing large infrastructure development such as transportation projects, hospitals, schools and public facilities (Maskin and Tirole 2008). Most notably, PPP provides the opportunity for more efficient project management, proficient risk mitigation, and enhanced technological innovation (Grasman et al. 2008). In the case of transport infrastructure, due to its public nature, projects must often comply with regulations established by public authorities in order to address environment, safety and, sometimes, social considerations. Then, the public sector must become involved because a purely privately-funded project would tend to maximize revenues to a level below the optimal dictated by the maximization of economic development. Pedersen (2003) investigated the welfare maximizing transit policy in which both the positive externalities from the route supplier to the passengers and the negative externalities within the passengers are taken into consideration. To analyze the optimal fare and optimal quality of supply from the perspective of a transit operator for profit maximization, Jorgensen and Pedersen (2004) adopted a special utility function, which is a weighted average of the profit and the passenger surplus, and investigated how the operator’s objectives influence the quality of transport and the optimal transit fare. Later, Jorgensen and Preston (2007) established the relationship between the weighting parameter and the shadow price of raising public funds under the Ramsey rule. The social optimum and monopoly solution are obviously two extreme configurations of Pareto-efficiency. These two extreme situations can be hardly sustainable in practice. On one hand, the socially optimal configuration aims for maximizing total social welfare including the passengers’ travel cost and the transit operating cost. This usually needs large amount of subsidies for the operation of the transit system to break even. On the other hand, the monopoly operator generally uses its market power to seize extra profit, and the resulting service might be socially undesirable. As a result, the second-best transit regulating policies have been considered for PPP projects to compromise the public and private’s interests. In this case, we have to Pareto efficient strategies for regulating public transit operations 201 take into account multiple objectives of a PPP contract simultaneously for public transportation services. Relevant to the current study is a stream of literature on the optimization of transit service quality. Mohring (1972, 1976) developed a microeconomic foundation for public transportation services with fixed demand. Transit service quality is measured by the expected waiting time at stops, a half of the headway. He proposed the well known “Square Root Principle” for the determination of optimal bus service frequency. Namely, the optimal bus frequency is proportional to the square root of the number of commuters. This principle is in line with the work of Vickrey (1955) on the implications of marginal cost pricing for public utilities. The classical model proposed by Mohring (1972, 1976) is quite useful for transit service planning in a static environment. In subsequent works, Jansson (1980) extended the square root principle to a model in which service frequency is simultaneously optimized with bus size. Ahn (2009) extended Mohring’s work to the situation where buses share the congestion interaction road with other automobiles. Most recently, an argument was raised for discussing whether a profit-maximizing monopolist would supply higher frequencies than those of social optimum or not. Van Reeven (2008) and Karamychev and Van Reeven (2009) insist on that a monopolist in public transport may oversupply frequency relative to the social optimum. However, Basso and Jara-Díaz (2010) and Savage and Small (2010) argued that Van Reeven (2008)’s results depend on the reduction or elimination of the effect of fares on demand, which cause the optimal prices to be indeterminate within broad ranges. Unfortunately, all the above-mentioned researches did not incorporate the invehicle congestion into the passengers’ travel cost. It is recognized that the in-vehicle crowding conditions critically affect the behavior of commuters, which in turn affects the use of the line capacity (Vuchic 2005). Lam et al. (1999) used stated preference surveys to investigate the effects of crowding at the Light Rail Transit (LRT) in Hong Kong. They found that passengers are more sensitive to the crowding condition in a vehicle for a journey with longer travel time. A few recent studies have incorporated the crowding effects into the passenger’s generalized cost as an important factor that affects transit service quality (Huang et al. 2004; Hamdouch and Lawphongpanich 2008). Tian et al. (2009), based on an empirical study in Beijing, found that there is a linear relationship between the passenger number and the in-vehicle congestion cost under the linear schedule delay assumption. In this paper, we incorporate both waiting time at stops and in-vehicle congestion into transit service modeling and investigate the properties of Pareto-efficient solutions and regulations for multi-objective optimization of transit services. Our analysis is limited to a single public transit line in a medium-term period for a simple illustration purpose. Section 2 describes the notations and formulates the theoretical model. We extend the “Square Root Principle” to the case when in-vehicle congestion is considered. In Sect. 3, we derive the Pareto-efficient frontier, where no further Pareto improvement can be made at any point on both social welfare and private profit. Three traditional regulations, including Price-cap regulation, Return-on-output regulation and Quantity regulation are analyzed in Sect. 4. Section 5 concludes the study. 202 Q. Tian et al. 2 Model formulation Consider a transit line connecting a residential area with the Central Business District (CBD). For simplicity, we assume the line’s on-trip travel time is constant, denoted by T . For focusing on the regulation in a medium term, we ignore the fixed costs of purchasing such capital equipment as vehicles. Because whatever the initial construction projects are invested by the local authority or a PPP contract, these costs are sunk cost and could not be the decision variables in medium term operating. Then, there are only two variables, the frequency and the fare, for operators to set the service quality in operating a transit system. The demand is elastic and the passengers are homogeneous with the same valuation of waiting time, travel time and in-vehicle congestion. 2.1 Notations Let N denote the number of passengers served, p the transit fare and f the frequency of dispatching buses on the route. The full price for a bus trip, G(N, p, f ), contains three parts as follows: the transit fare, the expected waiting time cost at stops and the in-vehicle travel cost (including the congestion effect). This follows,    N α + βT 1 + g G(N, p, f ) = p + . (1) 2f f In the above equation, the first term, p, is the transit fare. The second term is the waiting time cost which equals the half of the headway with an assumption that the arrival rate of bus passengers to the stop is distributed uniformly, and α is the monetary cost of unit waiting time. The third term is the monetary cost of journey time and the invehicle congestion, where β is the monetary cost of unit in-vehicle time, and g(N/f ) reflects the in-vehicle congestion effects. Empirical studies in North America, Europe and Asia (Mohring et al. 1987) have generally found α to be two or three times of β. g(N/f ) is assumed to be a strictly increasing function of the in-vehicle passenger density, which is represented by the ratio of total demand over transit frequency, N/f . Let D(N) be the inversed demand function with the property D ′ (N ) < 0 (D ′ (N ) denotes the derivative of D(N) with respect to N ), which represents demand decreasing with the generalized costs. Thus, at user equilibrium, we have D(N) = G(N, p, f ). (2) Substituting (2) into (1) yields p(N, f ) = D(N) −     N α . + βT 1 + g 2f f (3) Let p(N, f ) denote the transit fare which leads the total demand to be N when the transit frequency is f . Given the transit frequency, the correspondence between the demand N and the transit fare p is one to one. Originally, there are three decision variables: price, frequency and the number of passengers served. Here, we find two Pareto efficient strategies for regulating public transit operations 203 of these are independent while the third one is determined by (3). Without loss of generality, the decision variables are reduced from (N, p, f ) to (N, f ) hereafter. The total passenger surplus is π(N) =  N D(n)dn − D(N) · N. 0 (4) The transit operator’s total revenue can be written as R(N, f ) = p(N, f ) · N. (5) C(N, f ) = h1 · f + h2 · N, (6) Total transit operating cost is where h1 denotes the operating cost of each transit run and h2 is the average cost of the transit for one passenger. The operator’s profit, denoted by F (N, f ), is F (N, f ) = R(N, f ) − C(N, f ). (7) Finally, the total social welfare of the system is defined as: W (N, f ) = π(N) + F (N, f ), (8) including both passenger’s surplus and operator’s profit. 2.2 Monopoly solution Without government regulation, the transit operator will seek to maximize the profit through setting a typical decision pair (N, f ) as follows: max F (N, f ). N,f (9) The first order conditions for the above mathematical programming problem (9) are ∂F (N, f ) = 0 and ∂N From ∂F (N,f ) ∂N ∂F (N, f ) = 0. ∂f (10) = 0, we have M D(N ) + N M   M  α N · D (N ) = + βT 1 + g M 2f fM  M N · NM, + h2 + βT g ′ fM ′ M (11) where the superscript M is used to denote the solution associated with profit maximization and g ′ (x) represents the derivative of a function g(x) with respect to x. 204 Q. Tian et al. The Left Hand Side (LHS) is the marginal revenue of the transit operator, in which the second term reflects the monopolist’s market power. The Right Hand Side (RHS) is the marginal cost caused to the system for an additional passenger, including the in-vehicle congestion externality, βT g ′ (N M /f M ) · N M . ) = 0, we have From ∂F (N,f ∂f f M =    M  N NM M ′ . ·N α/2 + βT g h1 fM (12) Substituting N M and f M into (3) yields the transit fare p M = h2 + βT g ′  NM fM  · N M − N M · D ′ (N M ). (13) The above equation states that the fare for implementing profit maximization includes three terms, namely the variable cost per passenger h2 , the congestion external cost βT g ′ (N M /f M ) · N M and the consumer surplus caused by additional passenger −N M · D ′ (N M ). 2.3 Social optimum Without participation of private sectors, the local authority solely operates the transit line for social welfare maximization. The operating strategy is equivalent to solving the following optimization problem: max W (N, f ). N,f (14) The first order conditions are ∂W (N, f ) = 0 and ∂N ∂W (N, f ) = 0. ∂f ∂W (N, f ) = 0 leads to ∂N   1 α D(N W ) = W + 2aT N W + T + h2 , f 2 (15) (16) where the superscript W is used to denote the solution associated with welfare maximization. The LHS is the marginal utility of the passenger while the RHS is the marginal cost of the system. (N,f ) ) = ∂F (N,f . This leads to From (8), we have ∂W ∂f ∂f f W =     W N NW W . · N α/2 + βT g ′ h1 fW (17) Pareto efficient strategies for regulating public transit operations Substituting N W and f W into (3) yields the transit fare  W N · NW . p W = h2 + βT g ′ fW 205 (18) Thus, the fare for social welfare maximization includes two terms, i.e., the variable W ) · NW . cost per passenger h2 and the congestion external cost βT g ′ ( N fW 2.4 Demand-frequency relationship It can be found that whatever the transit operates for profit-chasing or welfaremaximizing, the relationship between the demand, N , and the frequency, f , is the same, although the realized demand may be different. If we ignore the in-vehicle  congestion cost, i.e. g(x) = 0, we then have f M = (N M α)/(2h1 ), which is exactly the Mohring’s “Square Root Principle”. Note that the expected waiting time of passengers may be constant in some situations. When the frequency is too low and all passengers have perfect knowledge about the transit schedule, the waiting time tends to be constant (Lam and Morrall 1982). On the other hand, if the frequency is so high that even the time variance of getting the transit stop may be larger than the waiting time at stops, the expected waiting time may be left aside while considering the trip cost. Since our model is for long term planning purpose and the schedule cost is not involved, we can set the constant waiting time zero.  We still take the in-vehicle congestion cost into consideration and then have f = βT g ′ (N/f )/ h1 · N . According to Tian et al. (2009)’s empirical study in Beijing, the congestion cost is linearly increasing with the in-vehicle passenger density under the linear schedule delay cost assumption. For simplicity, here we assume that the in-vehicle congestion function takes the form of g(x) = a ·√x, where a > 0 (Tian et al. 2007; Hamdouch and Lawphongpanich 2008), then f = βT a/ h1 · N is reached. This indicates a linear relationship between frequency and demand exists. According to the above discussion, we have the following proposition. Proposition 1 If the expected waiting time can be ignored during a passenger’s trip, both local government and private operator will linearly increase the service frequency with the total passenger demand to achieve their goals. 3 Properties of Pareto-efficient frontier Negotiating the PPP contract for transportation service procurement can be regarded as a non-zero sum game between the local authority and the private firm. In this case, both social welfare and private profit should be considered jointly. Therefore, we are facing a bi-objective optimization problem of the transit services. We first give a definition about the Pareto-efficient service as follows. Definition (Pareto-efficient service) A transit operating pair (N ∗ , f ∗ ) is said to be a Pareto-efficient service if there is no other transit operating pair (N, f ) such that F (N, f ) ≥ F (N ∗ , f ∗ ) and W (N, f ) ≥ W (N ∗ , f ∗ ) with at least one strict inequality. 206 Q. Tian et al. Using the Pareto efficiency to assess a multi-objective economic system is a popular method in mechanism design and attracts more and more attentions of transportation researchers (Tan et al. 2010; Guo and Yang 2010, among others). From the above definition of Pareto-efficient service, we have the following proposition, which characterizes the Pareto-efficient operating pair. Proposition 2 A Pareto-efficient transit operating pair (N ∗ , f ∗ ) has the following relations:  N∗ ∗ ∗ ∗ (α/2 + aβT N ∗ ), (19) f = f (N ) = h1     N∗ α ∗ ∗ ∗ ∗ ∗ ,(20) +T 1+a· p = p(N , f (N )) = D(N ) − 2f ∗ (N ∗ ) f ∗ (N ∗ ) where N ∗ ∈ [N M , N W ]. Proof According to Miettinen (1999), the weighted sum method can be used to find every Pareto optimum when the multi-objective problem is a convex problem. Since both W (N, f ) and F (N, f ) are convex in the region {N ≥ 0, f ≥ 0}, we can maximize the following weighted objective function to derive the Pareto optimum set. U (N, f ) = W (N, f ) + λ · F (N, f ), (21) where λ ≥ 0. The first order condition of maxN,f U (N, f ) is sufficient to guarantee the transit operating strategy to be Pareto-efficient. Substituting (8) into (21), we have U (N, f ) = π(N) + (1 + λ)F (N, f ). The first order condition of maxN,f U (N, f ) with respect to f is equivalent to ∂F (N, f )/∂f = 0. Thus, the optimal frequency has the following relation with the total demand:  N∗ (α/2 + aβT N ∗ ). (22) f ∗ (N ∗ ) = h1 Given the total demand from the interval N ∗ ∈ [N M , N W ], Equation (19) is achieved.  Substituting (19) into (3), we get (20). Figure 1 depicts Proposition 2 intuitively. With the assumption D ′ (N ) < 0, the marginal revenue, D(N) + N · D ′ (N ), is always below the passenger’s marginal utility, D(N). The marginal social cost caused by an additional passenger is f ∗1(N ) (α/2+ 2aT N) + T + h2 , where f ∗ (N ) is given by (9). As explained in the poof of Propo1 α sition 2, the Pareto-efficient frontier should be on the line, F (N ∗ ) = f ∗ (N ∗) ( 2 + 2aT N ∗ ) + T + h2 . The intersection between F (N) and D(N), point B, corresponds to the welfare maximization strategy, while the intersection between F (N) and D(N) + N · D ′ (N ), point A, is the profit maximization strategy. Although each pair of curves cross at two points, it is easy to verify that only one point is stable. Pareto efficient strategies for regulating public transit operations 207 Fig. 1 The Pareto-efficient frontier In comparison with the social optimum, the monopoly operator will set higher fare and lower frequency for profit maximization. Higher fare will lead to decrease in total demand. Lower frequency will increase the passenger waiting time. This result consists with the traditional Mohring effect (Mohring 1972). 4 Regulation strategies In this section, three types of regulations are compared to find which one is the best in helping the local authority keep the transit system working well during the PPP contract period. These regulations are Price-cap, Return-on-output and Quantity-control. 4.1 Price-cap Under the Price-cap regulation, the regulator sets a price, called the price cap. The regulated agent can choose a price below or equal to this cap. This mechanism is very easy to implement and has been widely used in various PPP service contracts of telecommunications, natural gas and electricity areas. If the local authority adopts a Price-cap at P̃ , the transit operator’s decision should be maxN,f F (N, f ) s.t. p(N, f ) ≤ P̃ (23) The first order conditions of the above optimization problem (23) are:    (Ñ − κ) α ˜ + aβT · Ñ , (24) f (Ñ) = h1 2   α 1 ′ + aβT (2Ñ − κ) + βT + h2 , (25) D(Ñ) + (Ñ − κ) · D (Ñ ) = f˜(Ñ) 2 where κ ≥ 0 is the shadow demand of raising transit fare. 208 Q. Tian et al. Fig. 2 The optimal decision couple under Price-cap regulation There are two possibilities: the price cap is either below or above the price that the transit operator would charge if it were not regulated. If the price cap is high enough such that p ∗ (Ñ , f˜) ≤ P̃ , the operator will charge the passengers at the price p ∗ (Ñ , f˜) and exploit the monopoly profit, meanwhile κ is zero. If the price cap acts effectively, i.e., p(Ñ , f˜) = P̃ , κ will be larger than zero. Figure 2 depicts the optimal decision under Price-cap regulation. The output under the Price-cap is the intersection between the LHS and the RHS of (25). Combining with the (24), we can intuitively observe that the optimal output under the effective price cap is point C, which induces higher demand than the monopoly solution for the lower transit fare. From (24), we find that the frequency chosen by the operator is smaller than the Pareto-efficient frequency at the output of Ñ . Obviously, the Price-cap regulation can prevent the private operator from seeking the profit maximization, but it can not induce the operator to operate along the Pareto-efficient frontier. As the restricted low price cap attracts more passengers into transit mode, the firm supplies less transit runs to keep the cost down. Thus, the Price-cap regulation is not Pareto-efficient. Proposition 4 Under the Price-cap regulation, the profit-maximizing operator will provide a service frequency lower than the Pareto-efficient level. Thus Price-cap regulation cannot achieve Pareto-efficiency. Proof For κ > 0, the operator’s decision set is on the line, P (Ñ ) = 1 (α/2 f˜(Ñ ) + aβT (2Ñ + κ)) + βT + h2 , which is above the Pareto efficient frontier P (N) = 1 f (N ) (α/2 + 2aβT · N ) + βT + h2 as shown in Fig. 2. According to (24), the optimal transit frequency under Price-cap regulation is lower than the Pareto efficient frequency with the same total demand Ñ .  4.2 Return-on-output regulation Under Return-on-output (ROO) regulation, the operator is free to choose its input (frequency) and output (trip demand) levels, but is not allowed to earn economic profit in excess of a “fair” return per unit of output. The fair return is set by the regulator and Pareto efficient strategies for regulating public transit operations 209 stated in terms of dollars of profit per unit of output. The profit constraint is expressed as F (N, f ) ≤ k · N , where k is the allowed profit per unit of output. The first order conditions for the operator’s optimization problem with respect to frequency and demand are: (1 + ρ) ∂F (N, f ) = 0, ∂f (26) (1 + ρ) ∂F (N, f ) = −k · ρ, ∂N (27) where ρ ≥ 0 is the Lagrange multiplier associated with the profit constraint. The proof of Proposition 2 has shown that the Pareto-efficient conditions (19) and (20) are eventually derived from ∂F (N, f )/∂f = 0. As 1 + ρ > 0, we confirm that (26) is equivalent to the Pareto-optimal condition. Thus, we have the following proposition. Proposition 5 The ROO regulation by setting an upper bound on the profit earned from each served passenger is Pareto-efficient. According to (27), if the profit constraint is not effective, i.e., F (N, f ) < k · N, ρ (N,f ) equals zero, which is just the monopoly optimality should be zero and then ∂F ∂N condition. Otherwise, if k is not so large and the profit constraint is tight, ρ is be larger (N,f ) than zero and ∂F ∂N , being upper convex there, must be less than zero, which means that the operator is forced to increase the demand. 4.3 Quantity regulation In a variety of situations, the Quantity regulation is used to help government regulate the undesirable activities by controlling output rather than charging taxes. Such as anti-smoking and anti-trust laws restrict rather than tax particular conduct and pollution controls impose limits rather than tax emissions (Glaeser and Shleifer 2001). The Quantity regulation usually means setting an upper bound for the output of a typical good. However, in this subsection we investigate the welfare effect of introducing a lower bound on the quantity. The quantity constraint is expressed as N ≥ N , where N is the allowed smallest output. Under the Quantity regulation, the first order conditions for the operator’s optimization problem with respect to frequency and demand are: ∂F (N, f ) = 0, ∂f (28) ∂F (N, f ) = −δ, ∂N (29) where δ ≥ 0 is the Lagrange multiplier associated with the quantity constraint. Equation (28) is just the Pareto-efficient condition (19). Thus, the decision pair (N, f ) is on the Pareto efficient frontier. We thus have the following proposition. 210 Q. Tian et al. Proposition 6 The Quantity regulation by setting a lower bound on the number of served passengers is Pareto-efficient. If N is too small for the quantity constraint to be active, the Lagrange multi(N,f ) plier δ = 0. From (29), we have ∂F ∂N = 0, which represents that the operator can achieve monopoly solution. Otherwise, if the quantity constraint is tight, the La(N,f ) < 0 means that the private grange multiplier δ will be larger than zero, then ∂F ∂N firm has to serve N passengers, although they prefer to serve less total demand for more profit. Actually, combining (27) and (29), we can derive the following relationship between the lower bound of quantity and the ROO ratio k:  D(N ) − 2 h1 N(α/2 + aβT N ) − βT − h2 = k. (30) As the LHS is a monotonically decreasing function of N , there is a one-to-one correspondence between the ROO rate k and the quantity control variable N . In other words, given a ROO ratio k, there would be a quantity lower bound N , which can achieve the same effect through quantity regulation. Since there is no need for other parameters than the quantity bound N , Quantity regulation seems more attractive than ROO, which needs the firm’s accounting information. It is worthwhile noticing that a similar demand regulation, formulated by Tan et al. (2010), has also been proved to be Pareto-efficient in a privately provided BOT project. 5 Conclusions This paper investigated how the local authority could efficiently regulate the public transit operation by a private firm for achieving a Pareto-efficient service. Three types of regulation strategies, namely Price-cap, Return-on-output and Quantity control, are analyzed and compared. Although the Price-cap regulation can attract more demand effectively, the private firm tends to set the frequency at an inefficient low level to keep the cost down. In contrast, both the Return-on-output and Quantity regulation are the Pareto-efficient strategies, which can induce the private firm to provide Paretoefficient service. Especially, Quantity regulation seems more attractive than ROO for there is no need for the firm’s accounting information. In our analysis, both the waiting time at stops and the in-vehicle congestion costs are taken into account to reflect the transit service quality. The derived new demandfrequency correspondence extends the Mohring’s “Square Root Principle”. If we ignore the in-vehicle congestion cost, the Mohring’s “Square-root Principle” obtains. If the waiting time is assumed to be zero under some special conditions, both the local government and the private operator will linearly increase the frequency with the total demand to achieve their goals. Finally, one limitation associated with our work is that the demand formulation used is identical while maximizing the social welfare. This may lead to underestimation of the total welfare for ignoring different demand characteristics (Glaister 1979). Incorporating traveler heterogeneity is one of our on-going works. Pareto efficient strategies for regulating public transit operations 211 Acknowledgements The authors would like to thank two anonymous reviewers and the conference participants at CASPT 2009 (Hong Kong) for their helpful comments. The work described in this paper was supported by the joint research scheme between the National Natural Science Foundation of China (70801002, 70931160447, 71071011) and the Research Grant Council of the Hong Kong Special Administrative Region (N_HKUST607/09) and a project from the Fundamental Research Funds for the Central Universities (YWF-10-01-A26). References Ahn K (2009) Road pricing and bus service policies. J Transp Econ Policy 43:25–53 Andersen B (1992) Factors affecting European privatisation and deregulation policies in local public transport: the evidence from Scandinavia. Transp Res, Part A, Policy Pract 26:179–191 Basso L, Jara-Díaz S (2010) The case for subsidization of urban public transport and the Mohring effect. J Transp Econ Policy 44:365–372 Glaeser EL, Shleifer A (2001) A reason for quantity regulation. Am Econ Rev 91:431–435 Glaister S (1979) On the estimation of disaggregate welfare losses with an application to price distortions in urban transport. Am Econ Rev 69:739–746 Grasman SE, Faulin J, Lera-Lopez F (2008) Public-private partnerships for technology growth in the public sector. In: IEMC—Europe 2008: international engineering management conference proceedings, pp 241–244 Guo X, Yang H (2010) Pareto-improving congestion pricing and revenue refunding with fixed demand. Transp Res, Part B, Methodol 44(8–9):972–982 Hamdouch Y, Lawphongpanich S (2008) Schedule-based transit assignment model with travel strategies and capacity constraints. Transp Res, Part B, Methodol 42:663–684 Huang HJ, Tian Q, Yang H, Gao ZY (2004) Modeling the equilibrium bus riding behavior in morning rush hour. In: Proceedings of the 9th annual conference of the Hong Kong society of transportation studies, Hong Kong, pp 434–442 Jansson JO (1980) A simple bus line model for optimization of service frequency and bus size. J Transp Econ Policy 14:53–80 Jorgensen F, Pedersen PA (2004) Travel distance and optimal transport policy. Transp Res, Part B, Methodol 38:415–430 Jorgensen F, Preston J (2007) The relationship between fare and travel distance. J Transp Econ Policy 41:451–468 Karamychev VA, Van Reeven P (2009) A monopolist in public transport. Tinbergen Institute, Discussion Paper TI 2009-77/1 Lam WHK, Morrall J (1982) Bus passenger walking distances and waiting times: a summer-winter comparison. Transp Q 36:407–421 Lam WHK, Cheung CY, Lam CF (1999) A study of crowding effects at the Hong Kong light rail transit stations. Transp Res, Part A, Policy Pract 33:401–415 Maskin E, Tirole J (2008) Public–private partnerships and government spending limits. Int J Ind Organ 26:412–420 Miettinen KM (1999) Nonlinear multiobjective optimization. Kluwer Academic, Boston Mohring H (1972) Optimization and scale economies in urban bus transportation. Am Econ Rev 62:591– 604 Mohring H (1976) Transportation economics. Ballinger, Cambridge Mohring H, Schroeter J, Wiboonchutikula P (1987) The values of waiting time, travel time, and a seat on a bus. Rand J Econ 18:40–56 Pedersen PA (1994) Regulating a transport company with private information about costs. J Transp Econ Policy 28:307–318 Pedersen PA (1995) Public regulation of a transport company with private information about demand. J Transp Econ Policy 29:247–251 Pedersen PA (2003) On the optimal fare policies in urban transportation. Transp Res, Part B, Methodol 37:423–435 Savage I, Small K (2010) A comment on “Subsidization of urban public transport and the Mohring effect”. J Transp Econ Policy 44:373–380 Tan ZJ, Yang H, Guo XL (2010) Properties of Pareto efficient contracts and regulations for road franchising. Transp Res, Part B, Methodol 44:415–433 212 Q. Tian et al. Tian Q, Huang HJ, Yang H (2007) Equilibrium properties of the morning peak-period commuting in a many-to-one mass transit system. Transp Res, Part B, Methodol 41:616–631 Tian Q, Huang HJ, Lam WHK (2009) How do transit commuters make trade-off between schedule delay penalty and congestion cost: an empirical study in Beijing. Transp Res Rec 2134:164–170 Van Reeven P (2008) Subsidization of urban public transport and the Mohring effect. J Transp Econ Policy 42:349–359 Vickrey WS (1955) Some implications of marginal cost pricing for public utilities. Am Econ Rev 45:605– 620 Vuchic VR (2005) Urban transit: operations, planning and economics. Wiley, New York