Economic efficiency of second-best congestion pricing schemes in urban highway systems

TRANSPORTATION RESEARCH PART B

Transportation Research Part B 33 (1999) 157±188

Economic eciency of second-best congestion pricing schemes in urban highway systems Louie Nan Liu a, John F. McDonald b,* b

a Louis Berger and Associates, 100 Halsted Street, East Orange, NJ 07019, USA Department of Economics, University of Illinois at Chicago, 601 S. Morgan St., Chicago, IL 60607, USA

Received 15 February 1996; received in revised form 13 May 1998

Abstract This paper examines urban highway congestion pricing in the instance in which it is not possible to levy a congestion toll on a major portion of the urban road system. This case is pertinent because of technical and/or political constraints. The paper uses economic theory of the second best and a simulation model to compare ®rst-best, second-best and no-toll solutions for a model with two routes and two time periods (peak and pre-peak). The main ®ndings from the simulation results are: (1) the second-best scheme is e€ective but less ecient than the ®rst-best scheme in reallocating trac volumes; (2) the second-best tolls are appreciably smaller than the ®rst-best tolls; (3) the welfare gains from the second-best policy are much smaller than the welfare gains that are possible with a complete set of ®rst-best tolls. It is also shown that the nature of the simulation results is not sensitive to reasonable cost and demand parameters. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Trac congestion; Congestion pricing

1. Introduction Trac congestion in many urban areas is one of the most serious problems challenging transportation policy makers. Conventional strategies such as road capacity expansion have been less e€ective in reducing trac congestion. Recently, the idea of using congestion pricing to improve the eciency of existing urban highway systems is attracting increasing interest. Downs (1992) provides an overview of congestion pricing. Gomez-Ibanez and Small (1994) focus on foreign experience with the use of congestion pricing for congestion management. The economic principle * Corresponding author. Tel.: +1-312-996-2980; fax: +1-312-413-7948. 0191-2615/99/$Ðsee front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0191-2615(9 8 ) 0 0 0 2 5 - 3

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of congestion pricing, known as marginal cost pricing, is to impose higher charges on travelers who travel at times and places where a road system is congested because travel in congested periods or places imposes high costs on other travelers. A congestion fee can thus be viewed as a user charge based on the di€erence between the marginal or social cost and the average cost perceived by the traveler. Congestion pricing aims at alleviating congestion by altering travel behaviorÐ by encouraging people to travel during less congested time periods, by less congested routes, by alternative modes, or not at all. This paper focuses on eciency issue associated with congestion pricing, i.e. the e€ectiveness of congestion pricing in reallocating trac volumes to maximize social welfare. The problems of technical complexity, trac spillover, and the tradition of free access to local streets make the implementation of congestion pricing inherently dicult. Under these technical and/or political constraints, it may not be possible to have an electronic toll-charging system to cover all the relevant roads and streets in a metropolitan area; hence, there will be portions of the urban road network that cannot not be subjected to ecient tolls. In this paper we assume that tolls will be collected using the best electronic system available. Does a congestion pricing scheme under these constraints (a so-called second-best scheme) still yield welfare gains from societal point of view? What would be the di€erences between the second-best toll scheme and two other regimes, the case in which no toll is imposed and the case in which ecient tolls are imposed on all roads and streets (i.e. a ®rst-best scheme)? These issues are critical in designing a toll system. This paper considers issues related to the second-best toll for an urban highway system. First, the paper will study the impacts of the congestion tolls on the trac volume allocations. Optimal trac volumes under di€erent toll regimes (no-toll, second-best, and ®rst-best) will be computed for each route and each time period. In particular, the diversion of peak period trac to untolled routes and the shift of peak period trac to o€-peak periods will be studied. Second, the paper will analyze the di€erences between the second-best and the ®rst-best tolls. Optimal tolls will be computed by route and time period (peak vs o€-peak period). The di€erence between the secondbest tolls and the ®rst-best tolls will be examined. Third, the paper will examine the welfare implications of the second-best congestion pricing scheme. Social welfares (social bene®ts minus social costs) yielded from the second-best toll regime will be compared to the two other regimes. The comparison will give the change in the social welfare and will demonstrate whether there is a gain from the imposition of the second-best congestion tolls. In addition, the ratio of the gains in the second-best to the gains in the ®rst-best will be computed to show the relative welfare improvement of the second-best toll regime. The pricing of roads has been studied by many researchers; see Small (1992) for a survey of this work. Most studies assume that tolls can be levied on every road of a network, which leads to marginal-cost pricing or the ®rst-best solution. This assumption, however, may not be held in real world when part of the network are not subject to tolls due to technical and/or political constraints. Marchand (1968) is an exception. He studied the theory of second-best tolls using a general equilibrium model. Marchand considered a network with two routes in parallel (a toll route and a free route) for a ®xed period. However, because of the static nature, his model is not able to study temporal aspects of congestion tolls such as peak shifting phenomenon. Arnott et al. (1990b) analyze user equilibrium, system optimum, and various pricing regimes for a network of two routes in parallel. The main idea in the paper is based on Vickrey (1969),

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159

which models roads as bottlenecks through which trac ¯ow is either uncongested or ®xed at a capacity independent of trac density. The paper considers both travelers' departure time and route choices which are assumed to be governed by the tradeo€s between travel time and schedule delay (the di€erence between actual and desired arrival time). The main weakness of the paper is that the demand is inelastic; hence, gross bene®ts cannot be evaluated using demand functions. In addition, the analysis belongs to ®rst-best analysis and does not consider the case in which ecient tolls cannot be imposed on both routes. McDonald (1995) uses economic theory and numerical examples to investigate the question of optimal congestion pricing in the short run for the case in which it is not possible to impose a congestion toll on a major portion of the urban road system due to technical and/or political reasons. The paper studies the second-best congestion pricing problem for a two-route network and a single peak period. The main result of the paper is that the second-best tolls can vary appreciably from the optimal tolls in the ®rst-best regime in which ecient tolls can be imposed on all routes. The paper also indicates that the failure to levy a congestion toll on a major portion of the system is costly in terms of potential welfare gains of congestion pricing that are lost. Liu and McDonald (1995) develop a theoretical model of second-best congestion pricing by combining the theory of peak-load pricing with the pricing of a transportation system. The paper examines the eciency of various congestion toll regimes on urban highways as a problem in the theory of the second-best. The main contribution of the paper is to introduce traveler's departure time choice into the route choice models for a single peak period. This paper presents an empirical extension and a thorough simulation study of the above model. The paper explores the ®rst-best and the second-best congestion tolls in an urban highway system under a variety of cases for cost and demand functions. The simulation study provides some policy implications on the implementation of congestion pricing schemes. The following section of the paper describes the model formulation of congestion pricing problems. Section 3 speci®es the cost and demand functions and describes the solution methods for the simulation study. Section 4 presents the simulation results with the base parameters and the results of sensitivity analyses of some key cost and demand parameters. The ®nal section summarizes the major conclusions and suggests further research topics. 2. Model formulation of congestion pricing problems 2.1. Description of the transportation system 2.1.1. The setting We consider an urban highway network consisting of two routes connecting an origin (e.g. home) and a destination (e.g. workplace). This network has been used by other researchers, e.g. Arnott et al. (1990b), Ben-Akiva et al. (1986), and Marchand (1968). The main reason for choosing such a system is that the network is simple enough to conduct an economic analysis of traveler's route choice behavior. The routes considered here have no speci®c engineering con®guration and are assumed to be perfect substitutes. It is assumed that all trips in the system are made by automobile only; hence, there is no mode choice. In the system, one route is a toll route and, due to technical and/or political constraints, the other route must remain untolled.

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In addition to the spatial framework, the study considers two time periods, peak and o€-peak periods, to describe traveler's departure time choice. To be more speci®c, the study considers the problem of the morning commute. The capacity for each route is assumed to be ®xed and to be the same for two periods. Each traveler can travel either early during the pre-peak period, or during the peak period. During each period, trip from the origin to the destination can be made on either route. 2.1.2. Travel costs In this paper, only short run costs relating to congestion will be considered since the focus of the study is on congestion. The cost borne by an individual traveler who travels on a route during a time period, consists of travel time cost and schedule-related cost. 1. Travel time costs are caused by congestion in either peak or pre-peak period. The travel time cost on a route in a period is assumed to be a function of the trac volume on the route in the period. 2. Schedule-related cost is a penalty for a traveler who makes a trip in the pre-peak period. This concept is a special case of schedule delay costs used in the previous studies. The schedule delay (the di€erence between actual and desired arrival time) was initially introduced by Vickrey (1969) and is normally related to endogenous scheduling. Small (1982) further develops this concept and estimates the value of schedule delay. Arnott et al. (1990a ,b) use the concept in models of trac bottlenecks and endogenous scheduling. Unlike the endogenous formulation in the previous studies, the schedule-related cost considered in this study is assumed to be an exogenous value, which represents an average monetary value of schedule-related times borne by every commuter traveling in the pre-peak period. Hence, an individual traveler can travel either in the peak period to incur potentially higher congestion costs and no schedule-related costs, or in the pre-peak period to avoid higher congestion costs but to bear schedule-related costs. These two costs together are called the average cost, which is denoted by cir …vir †; i ˆ 1; 2; r ˆ t; f

…1†

where index i represents time period, i=1 for the peak, i=2 for the pre-peak; index r represents route, r=t for the toll route, r=f for the free route; vir is trac volume on route r in period i. The average cost is assumed to be a monotonically increasing function of the trac volume: c0ir …vir † > 0; i ˆ 1; 2; r ˆ t; f:

…2†

In addition to the average cost, two other concepts are important for the study of congestion pricing problems: marginal cost and total cost. From Eq. (1), the marginal cost is de®ned as:

MCir ˆ

d‰vi r^ci r^…vi r^†Š ˆ cir …vir † ‡ vir c0ir …vi r^†; i ˆ 1; 2; r ˆ t; f: dvir

…3†

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The total cost consists of the costs borne by all travelers (for each route and for each time period) in the system and is de®ned as: C ˆ ‰v1t c1t …v1t † ‡ v1f c1f …v1f †Š ‡ ‰v2t c2t …v2t † ‡ v2f c2f …v2f †Š:

…4†

Under the assumption that the two periods are equal in duration, the duration factors in the total cost can be ignored. 2.1.3. Demand characteristics The aggregate demand for one period, called period demand, is the total trac volume from the origin to the destination in that period. The demand in one period is a function of trip prices in both peak and pre-peak periods. The income e€ect is assumed to be negligible. The demand functions for the peak (i=1) and pre-peak (i=2) periods are given by: v1 ˆ f1 …P1 ; P2 † v2 ˆ f2 …P1 ; P2 †

…5†

where vi is aggregated trac volume and Pi is trip price for period i. For demand functions Eq. (5), the following assumptions are made regarding dependency: 1. Negative own-price e€ect: @v1 @v2 < 0; and < 0: @P1 @P2

…6†

2. Positive cross-price e€ect: @v1 @v2 > 0; and > 0: @P2 @P1

…7†

The purpose of introducing the dependency of period demands is to study the peak shifting problem, i.e. the diversion of the peak period trips to the pre-peak period, by considering the response of the peak period demand to the pre-peak trip price, and vice versa. From Eq. (5), the inverse period demand function, i.e. the trip price for one period, can be derived as a function of the trac volume in both the peak and pre-peak periods, or P1 ˆ P1 …v1 ; v2 †; andP2 ˆ P2 …v1 ; v2 †:

…8†

Given the inverse demand functions Eq. (8), the gross bene®t for the system, denoted by B, can be expressed as a line integral

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L.N. Liu, J.F. McDonald/Transportation Research Part B 33 (1999) 157±188 …v1…;v2 †

Bˆ

P1 …v1 ; v2 †dv1 ‡ P2 …v1 ; v2 †dv2 :

…9†

…0;0†

Under the assumption that the two periods are equal in duration, the duration factors in the gross bene®t can be ignored. However, there are two major problems associated with the expression Eq. (9). The ®rst problem relates to the de®nition of the line integral because it depends on the particular path on which the integral is calculated and is thus not unique. The second one relates to the di€erentiability of the line integral. As stated in Pressman (1970), both problems are solved by assuming the following integrability condition: @P1 @P2 ˆ @v2 @v1

…10†

This condition says that the e€ect on the peak period trip price resulting from a change in the prepeak trac volume is the same as the e€ect on the pre-peak trip price resulting from a change in the peak trac volume. Condition Eq. (10) is one of the fundamental properties associated with the compensated (or Hicksian, i.e. cost-minimizing demand function) demand function. Since this study assumes the demand function has an income e€ect of zero, the demand function can be regarded as the Hicksian demand function which satis®es integrability condition Eq. (10). Under condition Eq. (10), the line integral is uniquely de®ned and is independent of the path chosen; and the ®rst derivative of the line integral with respect to the trac volumes for one period is equal to the inverse period demand function, i.e. the trip price for that period. Pressman (1970) provides a complete discussion of this integrability condition; the results are given by the following theorem. Theorem 1. Suppose Pi(v1,v2) (i=1,2), and @Pi/@vj (i,j=1,2) are continuous and single valued at every point of a simply connected region. Then, if and only if @P1 @P2 ˆ @v2 @v1 will the line integral Eq. (9) a. be independent of the path from (0,0) to (v1,v2); b. be zero around every closed curve in the region; c. there exists a function B(v1,v2) such that: dB ˆ P1 dv1 ‡ P2 dv2 ; and

…11†

d. the following equations are satis®ed: @B…v1 ; v2 † @B…v1 ; v2 † ˆ P1 …v1 ; v2 †; ˆ P2 …v1 ; v2 † @v1 @v2

…12†

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This theorem implies that if integrability Eq. (10) is satis®ed, then line integral Eq. (9) is uniquely de®ned and thus can be calculated along any path from (0,0) to (v1,v2). For example, two special paths can be selected to calculate the line integral: v…1

…v1…;v2 †

P1 …w1 ; w2 †dw1 ‡ P2 …w1 ; w2 †dw2 ˆ

Bˆ

v…2

P1 …w1 ; 0†dw1 ‡ 0

…0;0†

P2 …v1 ; w2 †dw2

…13†

0

which is calculated along the path I: (0,0)!(v1,0) and (v1,0)!(v1,v2); or v1… ;v2 †

Bˆ

v…2

P1 …w1 ; w2 †dw1 ‡ P2 …w1 ; w2 †dw2 ˆ

v…1

P2 …0; w2 †dw2 ‡ 0

…0;0†

P1 …w1 ; v2 †dw1

…14†

0

which is calculated along the path II: (0,0)!(0,v2) and (0,v2)!(v1,v2). 2.2. A second-best congestion pricing model 2.2.1. Formulation Given the theoretical framework in the previous sections, a second-best congestion pricing problem (called Model SB) can thus be formulated as a constrained optimization program. The problem is to maximize net bene®ts, or maxW ˆ B ÿ C …v1…;v2 †

P1 …v1 ; v2 †dv1 ‡ P2 …v1 ; v2 †dv2

ˆ …0;0†

…15†

ÿ ‰v1t c1t …v1t † ‡ v1f c1 f…v1f †Š ÿ ‰v2t c2t …v2t † ‡ v2f c2 f…v2f †Š; subject to P1 …v1 ; v2 † ˆ c1f …v1f † ˆ c1t …v1t † ‡ 1t ;

…16†

P2 …v1 ; v2 † ˆ c2f …v2f † ˆ c2t …v2t † ‡ 2t ;

…17†

v1 ˆ v1t ‡ v1f ; v2 ˆ v2t ‡ v2f ;

…18†

v1t  0; v1f  0; v21  0; v2f  0; v1  0; v2  0

…19†

where congestion tolls on the toll route are denoted by  1t and  2t. Eq. (16) is the constraint on the pricing of the free route in the peak period. In the peak period the equilibrium price of a trip on either route is equal to the average cost on the free route. The equilibrium price of the trip on the

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toll route is the average cost plus the congestion toll in the peak period. Similar to the condition for the peak period, Eq. (17) is the constraint on the pricing of the free route in the pre-peak period. Eq. (18) states that the total trac volume in the peak (pre-peak) period is equal to the sum of the trac on each route in the peak (pre-peak) period. Eq. (19) is the nonnegativity condition for trac volumes. By solving the model for optimal trac volume allocation (v1t,v1f,v2t,v2f,v1,v2), the second-best congestion tolls ( 1t, 2t) on the toll route for the peak and pre-peak periods are determined by: 1t ˆ P1 …v1 ; v2 † ÿ c1t …v1t †; and2t ˆ P2 …v1 ; v2 † ÿ c2t …v2t †:

…20†

2.2.2. First-order conditions The Lagrangian of the second-best problem is written as: L…v1t ; v1f ; v2t ; v2f ; v1 ; v2 † ˆ B…v1 ; v2 † ÿ C…v1t ; v1f ; v2t ; v2f † ÿ l1 ‰P1 …v1 ; v2 † ÿ c1t …v1f †Š ÿ l2 ‰P2 …v1 ; v2 † ÿ c2f …v2f †Š

…21†

ÿ 1 ‰v1 ÿ v1t ÿ v1f Š ÿ 2 ‰v2 ÿ v2t ÿ v2f Š By Theorem 1, the ®rst-order conditions for Eq. (21) are derived as follows: @L ˆ ÿMC1t ‡ 1 ˆ 0 @v1t

…22†

@L ˆ ÿMC1f ‡ l1 c01f …v1f † ‡ 1 ˆ 0 @v1f

…23†

@L ˆ ÿMC2t ‡ 2 ˆ 0 @v2t

…24†

@L ˆ ÿMC2f ‡ l2 c02f …v2f † ‡ 2 ˆ 0 @v2f

…25†

@L ˆ P1 ÿ l1 P11 ÿ l2 P21 ÿ 1 ˆ 0 @v1

…26†

@L ˆ P2 ÿ l1 P12 ÿ l2 P21 ÿ 2 ˆ 0 @v2

…27†

@L ˆ ÿP1 …v1 ; v2 † ‡ c1f …v1f † ˆ 0 @l1

…28†

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165

@L ˆ ÿP2 …v1 ; v2 † ‡ c2f …v2f † ˆ 0 @l2

…29†

@L ˆ ÿv1 ‡ v1t ‡ v1f ˆ 0 @1

…30†

@L ˆ ÿv2 ‡ v2t ‡ v2f ˆ 0 @2

…31†

v1t  0; v1f  0; v2t  0; v2f  0; v1  0; v2  0

…32†

where MCir is the marginal cost on route r in period i [see Eq. (3)], and Pij represents the partial derivative of Pi(v1,v2) with respect to vj: Pij ˆ

@Pi ; i; j ˆ 1; 2: @vj

…33†

By eliminating v1, v2, l1, l2, 1, 2, conditions Eqs. (22)±(32) can be simpli®ed to the following system of equations for (v1t, v1f, v2t,v2f): P1 …v1 ; v2 † ˆ MC1t ‡ l1 P11 ‡ l2 P21

…34†

P1 …v1 ; v2 † ˆ c1f …v1f †

…35†

P2 …v1 ; v2 † ˆ MC2t ‡ l1 P12 ‡ l2 P22

…36†

P2 …v1 ; v2 † ˆ c2f …v2f †

…37†

v1t  0; v1f  0; v2t  0; v2f  0

…38†

where v1, v2, l1, l2 are substituted respectively by: v1 ˆ v1t ‡ v1f ; v2 ˆ v2t ‡ v2f ; and l1 ˆ

MC1f ÿ MC1t t MC2f ÿ MC2t ; l2 ˆ : 0 c1f …v1f † c02f …v2f †

…39† …40†

Eqs. (34)±(38) are the optimality conditions of Model SB for each route and for each period. Among them, Eqs. (34) and (36) are the optimality conditions of the trip prices for the toll route in the peak and pre-peak periods, respectively. Eq. (34) [(36)] says that the trip price on the toll route for the peak (pre-peak) period is equal to the marginal cost plus two adjustment terms for the peak (pre-peak) period.

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2.3. Two alternative congestion pricing models In order to evaluate the second-best congestion pricing scheme, it is necessary to study two other regimes for the system: the ®rst-best problem in which congestion tolls can be imposed on both routes; the no-toll problem in which congestion tolls cannot be imposed on any route. The two problems can be formulated as follows. For the ®rst-best problem (called Model FB), the pricing constraints Eq. (16) and Eq. (17) do not exist since congestion tolls can be imposed on both routes: …v1…;v2 †

P1 …v1 ; v2 †dv1 ‡ P2 …v1 ; v2 †dv2

maxW ˆ …0;0†

ÿ ‰v1t c1t …v1t † ‡ v1f c1f …v1f †Š

…41†

ÿ ‰v2t c2t …v2t † ‡ v2f c2f …v2f †Š ˆ B…v1 ; v2 † ÿ C…v1t ; v1f ; v2t ; v2f † subject to trac volume constraints Eq. (18) and nonnegativity condition Eq. (19). The Lagrangian of the ®rst-best problem is written as: L1 …v1t ; v1f ; v2t ; v2f ; v1 ; v2 † ˆ B…v1 ; v2 † ÿ C…v1t ; v1f ; v2t ; v2f † ÿ 1 ‰v1 ÿ v1t ÿ v1f Š ÿ 2 ‰v2 ÿ v2t ÿ v2f Š

…42†

By Theorem 1, the ®rst order conditions for Eq. (42) are derived as follows: @L1 ˆ ÿMC1t ‡ 1 ˆ 0 @v1t

…43†

@L1 ˆ ÿMC1f ‡ 1 ˆ 0 @v1f

…44†

@L1 ˆ ÿMC2t ‡ 2 ˆ 0 @v2t

…45†

@L1 ˆ ÿMC2f ‡ 2 ˆ 0 @v2f

…46†

@L1 ˆ P 1 ÿ 1 ˆ 0 @v1

…47†

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167

@L1 ˆ P 2 ÿ 2 ˆ 0 @v2

…48†

@L1 ˆ ÿv1 ‡ v1t ‡ v1f ˆ 0 @1

…49†

@L1 ˆ ÿv2 ‡ v2t ‡ v2f ˆ 0 @2

…50†

v1t  0; v1f  0; v2t  0; v2f  0; v1  0; v2  0

…51†

By simpli®cation, conditions Eqs. (43)±(51) are reduced to the following system of equations for (v1t, v1f,v2t, v2f): P1 …v1 ; v2 † ˆ MC1t

…52†

P1 …v1 ; v2 † ˆ MC1f

…53†

P2 …v1 ; v2 † ˆ MC2t

…54†

P2 …v1 ; v2 † ˆ MC2f

…55†

v1t  0; v1f  0; v2t  0; v2f  0

…56†

where v1, v2 are equal to: v1 ˆ v1t ‡ v1f ; v2 ˆ v2t ‡ v2f :

…57†

Eqs. (52)±(56) are the optimality conditions of Model FB for each route and for each period. These conditions says that the trip price for each route and for each period is given by its marginal cost. From Eqs. (52) to (55), and the notation in Eq. (3), the ®rst-best congestion tolls can be derived as follows: 1t ˆ P1 …v1 ; v2 † ÿ c1t …v1t † ˆ v1t c01t …v1t † 1f ˆ P1 …v1 ; v2 † ÿ c1f …v1f † ˆ v1f c01f …v1f †

…58†

2t ˆ P2 …v1 ; v2 † ÿ c2t …v2t † ˆ v2t c02t …v2t † 2f ˆ P2 …v1 ; v2 † ÿ c2f …v2f † ˆ v2f c02f …v2f † where cir(vir) (i=1,2; r=t,f) is the average cost for a traveler; vir c0ir …vir † is called the marginal congestion cost, which represents the cost that the traveler imposes on all other travelers by adding to the level of congestion.

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For the no-toll problem (called Model NT), no maximization is involved; therefore, the optimal trac volume allocation (v1t, v1f, v2t, v2f) is determined by the following equilibrium conditions: P1 …v1 ; v2 † ˆ c1f …v1f † ˆ c1f …v1t †; and

…59†

P2 …v1 ; v2 † ˆ c2f …v2f † ˆ c2t …v2t †:

…60†

with trac volume constraints Eq. (18) and nonnegativity condition Eq. (19). 3. Design of the simulation study In general, it is dicult to solve the three congestion pricing models analytically. Therefore a simulation study will be conducted and the models will be solved numerically. This section speci®es the cost and demand functions and describes the solution methods for the simulation study. 3.1. Speci®cation of the cost functions The average cost cir(vir) includes travel time costs and schedule-related costs: cir …vir † ˆ Tir ‡ Si ; i ˆ 1; 2; r ˆ t; f:

…61†

where  and are values of travel time and schedule-related time, respectively. The values of  and are obtained fromSmall (1982):  ˆ 11…cents= min†; ˆ 6:5…cents= min†:

…62†

The travel time Tir is obtained from the BPR (Bureau of Public Roads) function [seeBranston (1976) for an overview]: "  4 # vir ; i ˆ 1; 2; r ˆ t; f: …63† Tir ˆ T0r 1 ‡ 0:15 Kr where Tr0 is the uncongested (or free ¯ow) travel time (minutes) on route r, and Kr is a certain level of capacity (vehicles per h) of route r. Since Kr is less than the maximum ¯ow of route r, trac volume vir (vehicles per h) may exceed Kr. For the schedule-related time, S1=0 and S2 is assumed to be a constant S: S ˆ 5…min†:

…64†

By substituting Eq. (63) and Si into Eq. (61), the average cost function cir(vir) may be written as: 

vir cir …vir † ˆ ir ‡ r Kr

4

; i ˆ 1; 2; r ˆ t; f:

…65†

L.N. Liu, J.F. McDonald/Transportation Research Part B 33 (1999) 157±188

where

ir ˆ T0r ; if i ˆ 1  if i ˆ 2 ˆ T0 ‡ S;

169

…66†

r

and r ˆ 0:15T0r ; r ˆ t; f:

…67†

From Eq. (65) to Eq. (67), the derivative of the average cost function c0 ir(vir) and the marginal cost function MCir, can be calculated by Eq. (3) as follows:   3 r vir 0 cir …vir † ˆ 4 ; i ˆ 1; 2; r ˆ t; f: …68† Kr Kr  MCir ˆ ir ‡ 5r

vir Kr

4

; i ˆ 1; 2; r ˆ t; f:

…69†

In order to conduct the simulation study, the parameters (Tt0, Tf0, Kt, Kf) need to be assigned values. The study is designed to solve the models for di€erent cases, which represent di€erent scenarios in the study network (i.e. the supply side). The cases are characterized by the travel time functions for the toll route and the free route: 1. The toll route "  4 # vit 0 ; i ˆ 1; 2: …70† Tit ˆ Tt 1 ‡ 0:15 Kt 2. The free route " Tif ˆ T0f

 4 # vif 1 ‡ 0:15 ; i ˆ 1; 2: Kf

…71†

where each function applies to both the peak (i=1) and the pre-peak (i=2) periods. A basic criterion is that the toll route has a free ¯ow travel time lower than or equal to the free route, i.e., Tt0Tf0. Given this criterion, four cases are considered: 1. Case 1: The toll route has a lower free ¯ow travel time than the free route; and two routes have identical capacities: T0t ˆ 1…min†; T0f ˆ 2…min†; Kt ˆ Kf ˆ 2000…veh=h†: 2. Case 2: The toll route has a lower free ¯ow travel time and a larger capacity than the free route:

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T0t ˆ 1…min†; T0f ˆ 2…min†; Kt ˆ 2000…veh=h†; Kf ˆ 1000…veh=h†: This case represents the scenario in which congestion tolls can be imposed in a major portion of the network. 3. Case 3: The toll route has a lower free ¯ow travel time and a smaller capacity than the free route: T0t ˆ 1…min†; T0f ˆ 2…min†; Kt ˆ 1000…veh=h†; Kf ˆ 2000…veh=h†: This case represents the scenario in which congestion tolls can be imposed only in a small portion of the network. 4. Case 4: This is a special case in which the two routes have identical free ¯ow travel times and capacities: T0t ˆ T0f ˆ 2…min†; Kt ˆ Kf ˆ 2000…veh=h†: For Cases 1±4, the two routes are assumed to have the same lengths which are equal to one mile. 3.2. Speci®cation of the demand functions The demand (vehicles per hour) functions for the peak and pre-peak periods are given by the linear forms: v1 ˆ Q1 ÿ 11 P1 ‡ 12 P2 ; and v2 ˆ Q2 ‡ 21 P1 ÿ 22 P2 :

…72†

where the parameters satisfy the assumptions: 1. 2. 3. 4.

Q1>Q2>0. The potential demand in the peak period is higher than the pre-peak. bij>0, i,j=1,2. The negative own-price e€ect and positive cross-price e€ect hold. b11b22ÿb12b21>0. The own-price e€ects outweigh the cross-price e€ects. b12=b21. The integrability condition Eq. (10) for the inverse demand functions holds (see the derivation below).

From Eq. (72), the inverse demand functions can be derived as: P1 …v1 ; v2 † ˆ A1 ÿ b11 v1 ÿ b12 v2 ; and P2 …v1 ; v2 † ˆ A2 ÿ b21 v1 ÿ b22 v2 : where Ai, i=1,2 is given by

…73†

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A1 ˆ

22 Q1 ‡ 12 Q2 21 Q1 ‡ 11 Q2 ; A2 ˆ d d

171

…74†

and bij, i,j=1,2 is given by: b11 ˆ

22 12 21 11 ; b12 ˆ ; b21 ˆ ; b22 ˆ : d d d d

…75†

where d ˆ 11 22 ÿ 12 21 > 0

…76†

by assumption 3. Based on the above assumptions 1±4, it is easy to see that bij, i,j=1,2 in Eq. (75) satis®es: bij > 0; i; j ˆ 1; 2

…77†

b12 ˆ b21

where b12=b21 implies that the inverse demand functions in Eq. (73) satisfy the integrability condition @P1 @P2 ˆ @v2 @v1

…78†

The base demand parameters are based onWohl and Hendrickson (1984) and are given by: Q1 ˆ 7500…veh=h†; Q2 ˆ 4000…veh=h†; 11 ˆ 21…veh2 =…cent  h††;

…79†

2

12 ˆ 21 ˆ 15…veh =…cent  h††; 22 ˆ 25…veh2 =…cent  h††:

3.3. Simulation results with the base parameters Given the speci®ed cost functions Eqs. (65)±(67) and the demand functions Eq. (73), the optimal trac volume allocations are obtained by solving the nonlinear systems of Eqs. (34)±(38) for Model SB, Eqs. (52)±(56) for Model FB, and Eq. (59) and (60) for Model NT, respectively. The solutions to the three models: second-best (SB), ®rst-best (FB), and no-toll (NT), can be computed numerically by applying Newton's method, for each of the four cases described in Section 3.1. For each case, the output includes:

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1. optimal trac volume allocations; 2. congestion tolls, equilibrium average cost, and trip prices; 3. social welfares, welfare gains, and relative welfare improvement. for Model SB, FB and NT, respectively. The results of Cases 1±4 with the base parameters Eqs. (62), (64), and (79) are presented in Tables 1 and 2. 4. Analysis of the simulation results In this section we ®rst describe the simulation results with the base parameters and then presents the results of sensitivity analyses of some key cost and demand parameters.

Table 1 Results of Cases 1 and 2 with the base parameters Case 1: Tt0=1, Tf0=2; Kt=Kf=2000 Model NT FB SB

v1t 4212 3555 3809

v1f 3193 2884 3443

v2t 3214 2379 2273

v2f 76 1521 1121

v1 7405 6439 7252

v2 3289 3900 3394

V 10 694 10 340 10 645

Model NT FB SB

(t1t,t1f) (0, 0) (65.9, 57.1) (18.3, 0)

(t2t,t2f) (0, 0) (13.2, 4.4) (8.6, 0)

(c1t,c1f) (43.4, 43.4) (27.5, 36.3) (32.7, 51)

(c2t,c2f) (54.5, 54.5) (46.8, 55.6) (46.3, 54.8)

P1 43.4 93.4 51

P2 54.5 60 54.8

± ± ± ±

Model NT FB SB

W ($) 38813.17 39530.14 39141.28


W($) 0 716.97 328.11

RW(%) ± 100 45.8

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

Case 2: Tt0=1, Tf0=2; Kt=2000, Kf=1000 Model NT FB SB

v1t 4893 4024 4261

v1f 1954 1656 2154

v2t 3225 2963 2935

v2f 461 1149 1012

v1 6847 5680 6415

v2 3686 4111 3947

V 10532 9791 10361

Model NT FB SB

(t1t,t1f) (0, 0) (108.1, 99.3) (48.1, 0)

(t2t,t2f) (0, 0) (31.8, 23) (6.8, 0)

(c1t,c1f) (70.1, 70.1) (38, 46.8) (45, 93.1)

(c2t,c2f) (54.6, 54.6) (51.4, 60.2) (51.1, 58)

P1 70.1 146.1 93.1

P2 54.6 83.2 58

± ± ± ±

Model NT FB SB

W($) 36907.46 38232.15 37507.66


W($) 0 1324.69 600.2

RW(%) ± 100 45.3

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

Units: Trac volumes=veh/h. Tolls, costs and trip prices=cents/mile/veh.

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173

4.1. Analysis of the trac volume allocations Tables 1 and 2 show that the introduction of the congestion tolls in the FB and SB models has three major impacts on the trac volume allocations: (1) diversion of the peak period trac to the free route in Model SB; (2) shift of the peak period trac to the pre-peak period; and (3) reduction in total trac volumes. These impacts are further illustrated in Figs. 1±4 for Cases 1±4, respectively. 4.1.1. Allocation of trac volumes on each route Under the SB regime, the peak period trac is diverted to the free route because there is a toll on the toll route and the free route is untolled. In Tables 1 and 2, by comparing the (v1t,v1f) for the SB with that for the NT, it is shown that the peak trac goes down on the toll route and goes up on the free route for all four cases. For instance, Fig. 2 demonstrates the diversion for Case 2. The horizontal movements from point B0 to B2, and from point A0 to A2, show the diversion of Table 2 Results of Cases 3 and 4 with the base parameters Case 3: Tt0=1, Tf0=2; Kt=1000, Kf=2000 Model NT FB SB

v1t 2553 2073 2244

v1f 4115 3421 4264

v2t 1724 1579 1568

v2f 2041 2500 2282

v1 6668 5494 6508

v2 3765 4079 3850

V 10433 9573 10358

Model NT FB SB

(t1t,t1f) (0, 0) (121.8, 112) (37.3, 0)

(t2t,t2f) (0, 0) (41, 32.2) (6.6, 0)

(c1t,c1f) (81.1, 81.1) (41.4, 50.2) (52.8, 90.2)

(c2t,c2f) (58.1, 58.1) (53.8, 62.6) (53.5, 60.1)

P1 81.1 163.2 90.2

P2 58.1 94.8 60.1

± ± ± ±

Model NT FB SB

W($) 36036.66 37446.42 36304.79


W($) 0 1409.76 268.13

RW(%) ± 100 19

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

Case 4: Tt0=Tf0=2; Kt=Kf=2000 Model NT FB SB

v1t 3584 3076 3329

v1f 3584 3076 3721

v2t 1717 1979 1849

v2f 1717 1979 1671

v1 7168 6153 7049

v2 3433 3958 3520

V 10601 10110 10570

Model NT FB SB

(t1t,t1f) (0, 0) (73.9, 73.9) (14.2, 0)

(t2t,t2f) (0, 0) (12.7, 12.7) (-0.8, 0)

(c1t,c1f) (56, 56) (40.5, 40.5) (47.3, 61.5)

(c2t,c2f) (56.3, 56.3) (57.7, 57.7) (56.9, 56.1)

P1 56 114.4 61.5

P2 56.3 70.3 56.1

± ± ± ±

Model NT FB SB

W($) 37836.48 38479.74 37909.69


W($) 0 643.26 73.2

RW(%) ± 100 11.4

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

Units: Trac volumes=veh/h. Tolls, costs and trip prices=cents/mile/veh.

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the trac on the toll route to the free route under the SB regime. Under the FB regime, however, the peak trac goes down on both routes compared to the NT regime because of the imposition of the tolls on the two routes. In Fig. 2, the horizontal movements from point B0 to B1, and from point A0 to A1, show the changes in the trac on the two routes under the FB regime for Case 2. The similar diversion also occurs in the pre-peak period for Cases 1±3. For instance, Fig. 2 describes the diversion for Case 2 under the SB (FB) regime: the vertical movement from point B0 to B2 (B1), and from point A0 to A2 (A1). However, the diversion does not exist for Case 4. It is worth noting that in Case 4, trac volume is split evenly between the toll route and the free route in both the peak and the pre-peak periods for the NT and FB regimes. This property can be derived analytically and illustrated by Fig. 4 as follows. From the equivalent system of Eqs. (52)±(55), and (59), (60), the following equations hold for the FB and the NT, respectively.

Fig. 1. Trac volume allocations for Case 1.

L.N. Liu, J.F. McDonald/Transportation Research Part B 33 (1999) 157±188

175

i. For the FB MC1t ˆ MC1f ; MC2t ˆ MC2f

…80†

ii. For the NT c1t …v1t † ˆ c1f …v1f †; c2t …v2t † ˆ c2f …v2f †

…81†

Substituting the marginal cost function Eq. (69) into (80), and the average cost function Eq. (65) with Eq. (66) and (67) into (81), the above equations become:

Fig. 2. Trac volume allocations for Case 2.

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i. For the FB 

4

 4 v1f ˆ 1f ‡ 5f Kf



4

 4 v2f ˆ 2f ‡ 5f Kf

v1t

1t ‡ 5t Kt

2t ‡ 5t

v2t Kt

Fig. 3. Trac volume allocations for Case 3.

…82†

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177

ii. For the NT 

v1t

1t ‡ t Kt 

v2t

2t ‡ t Kt

4

4



v1f ˆ 1f ‡ f Kf 

v2f ˆ 2f ‡ f Kf

4

…83†

4

From Eqs. (66) and (67), Tt0=Tf0 leads to:

1t ˆ 1f ; 2t ˆ 2f ; t ˆ f

Fig. 4. Trac volume allocations for Case 4.

…84†

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From Eq. (84), Eqs. (82) and (83) become: v1t v2t Kt ˆ ˆ ; for the FB v1f v2f Kf

…85†

v1t v2t Kt ˆ ˆ ; for the NT v1f v2f Kf which means that the ratio of the trac between the two routes in each period is proportional to the ratio of the capacities in Cases 1 and 2 where the two routes have equal free ¯ow travel times. Finally, by Kt=Kf, Eq. (85) gives: v1t ˆ v1f ; v2t ˆ v2f ; for the FB

…86†

v1t ˆ v1f ; v2t ˆ v2f ; for the NT which indicates that the two routes attract the same trac in each period if the two routes have identical free ¯ow travel times and capacities. Graphically, Fig. 4 shows that points A0 (A1), B0 (B1), and C0 (C1) all lie on the same straight line for the NT (FB) regime; in particular, point A0 (A1) coincides with point B0 (B1). 4.1.2. Allocation of total volumes in each period In the presence of the tolls (SB and FB), there is a shift of trac from the peak to the pre-peak period. Tables 1 and 2 show that for each of the SB and FB, the peak trac v1 (=v1t+v1f) falls and the pre-peak trac v2 (=v2t+v2f) rises in comparison with the NT. In each case, the FB induces a larger shift of trac from the peak to the pre-peak than the corresponding SB. The reason is that since the FB charges the tolls on both routes, the peak trac which would be switched to the free route in the SB regime is now forced to the pre-peak period. The shifting peak phenomenon are depicted in Figs. 1±4 for Cases 1±4, by the movement from point C0 to C2 and C1, under the SB and FB scheme respectively. Note that each decrease in the peak trac may not be equal to the corresponding increase in the pre-peak because the total demand V is not ®xed. 4.1.3. Change in total trac volumes over the two periods The imposition of the congestion tolls also reduces the total trac volume. Tables 1 and 2 demonstrate the reduction in the total trac V (=v1+v2) for the SB and the FB compared to the NT regime. In each case, the FB generates a larger reduction than the SB because the FB allows the tolls on both routes. This indicates that the FB is more e€ective than the SB in alleviating congestion. The fall in total trac volume, under the SB and FB scheme, are also shown in Figs. 1±4 for Cases 1±4, by the inward shift of the line TV0 to TV2 and TV1 respectively. Note in the ®gures, the total trac volume V is represented by the intercept of the line TV: v1+v2=V at v1-axis.

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179

4.2. Analysis of congestion tolls After analyzing the impacts of the tolls on the trac volume allocations, it is necessary to examine the quantitative aspects of the tolls; the signs and the sizes of the tolls in relation to the resulting trac volume allocations. The congestion tolls, the equilibrium average costs and trip prices for Cases 1±4 are presented in Tables 1 and 2. As discussed in Section 2, the congestion tolls are endogenously determined by the equilibrium trac volume allocations. Recall that the congestion tolls are de®ned as the di€erence between the trip price and the average cost: ir ˆ Pi …v1 ; v2 † ÿ cir …vir †; i ˆ 1; 2; r ˆ t; f

…87†

where v1, v2 are give by v1 ˆ v1t ‡ v1f ; v2 ˆ v2t ‡ v2f :

…88†

Given Eq. (87), the tolls in Model NT are equal to zero due to Eqs. (59) and (60); the second-best tolls on the free route are equal to zero due to Eqs. (35) and (37). From Eqs. (58) and Eq. (65)±(67), the ®rst-best toll can be derived analytically: irFB ˆ vir c0ir …vir †; i ˆ 1; 2; r ˆ t; f  4 0 vir ˆ 0:6Tr Kr

…89†

which indicates that the ®rst-best toll is positively related to the free ¯ow travel time and the ratio of trac-volume to capacity. The ®rst property from Eq. (89) is that the FB peak toll dominates the pre-peak toll on each of the two routes. As Tr0 and Kr are the same between the peak and pre-peak period, and the trac in the peak is higher than the pre-peak, the FB toll on each route is larger in the peak than the pre-peak period: FB FB FB FB > 2t ; 1f > 2f 1t

…90†

Another property is the free ¯ow travel time is an important factor that in¯uences the ®rst-best tolls. In Case 4, the ®rst-best tolls are equal between the two routes in each period. This fact can be derived analytically. Since Tt0=Tf0 in Case 4, so Eq. (85) holds and is equivalent to: v1t v1f v2t v2f ˆ ; ˆ …91† Kt Kf Kt Kf Substituting Eq. (91) into (89), and by Tt0=Tf0, the ®rst-best tolls have the following property: FB FB FB FB 1t ˆ 1f ; 2t ˆ 2f

…92†

This property indicates that the two routes have the same ®rst-best tolls in each period as long as they have identical free ¯ow travel times.

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Unlike Case 4, for Cases 1±3, the FB tolls on the toll route are larger than those on the free route in both the peak and the pre-peak periods. Note that the FB toll in Case 2 is lower than the corresponding FB toll in Case 3 because of the lower ratio of the trac volume to the capacity. For instance, in Case 2, the ratio for the toll route in the peak period is 2.01, compared to 2.07 in Case 3. A common property shared by both Models FB and SB is that the peak toll on each route dominates the pre-peak toll on that route because each route has higher trac-volume/capacity ratio in the peak than the pre-peak period. However, the second-best tolls di€er sizably from the ®rst-best tolls. In Cases 1±3, the SB tolls on the toll route are much smaller than the FB tolls in both the peak and the pre-peak periods. For example, the peak period toll on the toll route in Case 2 is $1.08 per mile in FB compared to $0.48 per mile in SB. An interesting result in Table 2 is that for Case 4 the SB toll in the pre-peak period is negative. For example, t2t (cents per mile per vehicle) is equal to ÿ0.8. The negativity implies that under some circumstance (e.g. the two routes have equal free ¯ow travel times, etc.), people are encouraged to travel in the pre-peak period by being subsidized instead of being tolled in order to achieve the second-best goal. 4.3. Analysis of social welfare properties In addition to the e€ectiveness in reallocating the trac volumes, another important measure for assessing the second-best congestion pricing scheme is the social welfare values (social bene®ts minus social costs) generated from such a scheme. Similarly to the previous analyses, the no-toll solution is taken as a benchmark, and the welfare properties of the second-best and the ®rst-best solutions are studied in comparison with the NT solution. The basic welfare properties can be derived analytically. Denote WNT, WFB and WSB as the social welfares under the NT, FB and SB policies, respectively. The social welfare function W, given the speci®ed cost and demand functions Eqs. (65) and (73), can be calculated as: …v1…;v2 †

P1 …v1 ; v2 †dv1 ‡ P2 …v1 ; v2 †dv2

Wˆ …0;0†

ÿ ‰v1t c1t …v1t † ‡ v1f c1f …v1f †Š ÿ ‰v2t c2t …v2t † ‡ v2f c2f …v2f †Š b11 2 b22 2 ˆ A1 v1 ‡ A2 v2 ÿ b12 v1 v2 ÿ v ÿ v ÿ ‰v1t c1t …v1t † ‡ v1f c1f …v1f †Š 2 1 2 2 ÿ ‰v2t c2t …v2t † ‡ v2f c2f …v2f †Š

…93†

Thus, WNT, WFB and WSB, can be written as follows: WNT ˆ W…v 2 S0 † ˆ W…v0 †; WFB ˆ maxv2S1 W…v† ˆ W…v †; 

WSB ˆ maxv2S2 W…v† ˆ W…v †:

…94†

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181

where v=(v1t,v1f,v2t,v2f,v1,v2), the superscript 0, *, ** stand for the optimal values for the NT, FB and SB, respectively; S0, S1 and S2 represent the sets of v in which v satis®es the constraints Eqs. (59), (60), (18) and (19) for the NT; Eq. (18) and (19) for the FB; and Eq. (16±19) for the SB, respectively. It is easy to see that: S0  S2  S1 :

…95†

Thus, WNT  WSB  WFB

…96†

which means that the social welfare under the second-best scheme is at least as high as under the no-toll scheme; the social welfare under the ®rst-best scheme is at least as high as under the second-best scheme. The property Eq. (96) can be rewritten in terms of the welfare gains of the SB and FB policies from the NT policy, denoted by
Wp (p=SB,FB): 04
WSB 4
WFB ;

…97†

and the relative welfare improvement of the SB to the FB, denoted by RWSB: 04RWSB 41; if
WFB > 0:

…98†

where
Wp ˆ Wp ÿ WNT ; p ˆ SB; FB RWSB ˆ


WSB
WFB

…99† …100†

Inequalities Eqs. (97) and (98) mean that both the SB and FB policies have welfare gains against the NT, but the gains under the FB policy are at least as large as under the SB policy. The above welfare properties are tested using the simulation results. Tables 1 and 2 indicate both the second-best and the ®rst-best policies outweigh the no-toll policy; in other words, both the second-best and the ®rst-best have welfare gains (
W) against the no-toll policy. As expected, the ®rst-best generates larger welfare gains than the second-best scheme; and the relative welfare improvement (RW) of the SB to the FB varies between Cases 1±4. Tables 1 and 2 show when the two routes are similar in size, e.g. have equal capacities as in Cases 1 and 4, the social welfares (Ws) under the three policies are higher than those in Cases 2 and 3. The reason is that the two routes in Cases 1 and 4 have the highest total capacities (4000 veh/h) among the four cases and therefore can supply the most total trac volumes. However, in Cases 1 and 4, the welfare gains (
Ws) generated by the SB and the FB policies are smaller than those in Case 2. In Case 1, the welfare gain under the FB policy is $716.97 and the SB pricing policy yields a welfare gain of $328.11, or 45.8% of the possible gain, which is the highest relative welfare improvement among all cases. The situation in Case 4 is quite di€erent: the use of the FB scheme results in a welfare

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gain of $643.26, but the SB scheme only yields a welfare gain of $73.2. In this case nearly 90% of the welfare gain is lost because of the constraint that the free route cannot have a congestion toll. Unlike Cases 1 and 4, the two routes are di€erent in size in Cases 2 and 3. In Case 2, where two-thirds of the highway system is subject to toll, the welfare gain under the FB policy is $1324.69 and the SB policy yields a welfare gain of $600.2, or 45.3% of the possible gain. In Case 3, where only one-third of the highway system is subject to toll, the use of the FB scheme results in a welfare gain of $1409.76, but the SB scheme only yields a welfare gain of $268.13. In this case 81% of the welfare gain is lost because of the constraint that the free route cannot have a congestion toll. Therefore, the failure to impose a congestion toll on a major portion of the network results in a major loss of the potential welfare gains. Furthermore, in the results, the relationship between the welfare gain and the proportion of the highway system covered by a toll is nonlinear. Cases 2 and 3 show that imposing a toll on 1/3 (2/3) of the system yields 19% (45%) of potential welfare gains. These net bene®ts of a toll system must be compared to the cost of the electronic toll collection system to determine overall net bene®ts. 4.4. Sensitivity analysis of the cost and demand parameters The purpose of this section is to test whether the conclusions from the previous sections are still valid after altering the values of some cost and demand parameters in the base case. Case 2 is used to conduct the sensitivity analysis. 4.4.1. Sensitivity analysis of the cost parameter S As discussed in Sections 1 and 2, this study models travel in the pre-peak period in a di€erent way from the previous studies. The study by Arnott et al. (1990b) uses endogenous scheduling to model traveler's departure time choice by assuming the total demand is constant. While the conventional peak-load pricing model such as Pressman (1970) considers di€erent demand functions for di€erent periods and an identical cost function in each period. This study combines the above two methods by introducing an exogenous S in the average cost function to distinguish the prepeak period travel from the peak period travel. In addition, the study also considers two distinct demand functions for the peak and pre-peak periods. This section tests whether the conclusions  from the previous sections are still valid by assigning S=0, i.e. the case in which the average cost function in each period is identical. Table 3 summarizes the results of Case 2 for the sensitivity analysis of change in the cost  the penalty of traveling in the pre-peak parameter S in the base case. Note that the parameter S, period, goes down to 0 min from 5 min in the base case. The similarities and di€erences brought about by the change in S can be found by comparing Table 3 with the base results of Case 2 in Table 1. First, the three impacts of the congestion tolls on the trac volume allocations, which are  discussed in Section 4.1, still hold with S=0. Compared with Table 1, the decrease in S causes the following changes for each of the three models: The trac volumes on both routes, vit and vif, go down in the peak period (i=1) and go up in the pre-peak period (i=2). This change makes the total peak trac v1 fall and the total pre-peak trac v2 rise, but there is an increase in total trac

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183

Table 3  Results of Case 2 with S=0 (min) Case 2: Tt0=1, Tf0=2; Kt=2000, Kf=1000 Model NT FB SB Model NT FB SB Model NT FB SB

v1t 4709 3956 4188 (t1t,t1f)

v1f 1862 1626 2055 (t2t,t2f)

v2t 3382 3188 3248 (c1t,c1f)

v2f 932 1265 1228 (c2t,c2f)

v1 6571 5581 6244 P1

v2 4313 4453 4476 P2

V 10885 10034 10719 ±

(0, 0) (101, 92.2) (38.2, 0)

(0, 0) (42.6, 33.8) (7, 0)

(61.7, 61.7) (36.2, 45) (42.7, 80.9)

(24.5, 24.5) (21.6, 30.4) (22.5, 29.5)

61.7 137.3 80.9

24.5 64.2 29.5

± ± ±

W($) 38678.09 39624.7 39054.69


W($) 0 946.61 376.6

RW(%) ± 100 39.8

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

volume V because the cost of pre-peak travel has decreased. For S =0, the increase in V is: 3.35% for Model NT, 2.48% for Model FB, and 3.46% for Model SB. Second, the change in S does not alter the qualitative properties of the congestion tolls: e.g., for Model FB, the tolls on the toll route are larger than those on the free route in both the peak and the pre-peak periods; for both Model FB and Model SB, the peak toll on each route outweighs the pre-peak toll on that route; the FB tolls are always larger than the SB tolls in both the peak and the pre-peak periods.   By comparing the corresponding results for each of the three models between S=0 and S=5,  however, some variations can also be found. As a result of the reduction in S, the congestion tolls for Models FB and SB decrease in the peak period and increase in the pre-peak period. Moreover, the average cost and trip price drop for each of the three models and for each period. The changes in the peak period are due to the decrease in the trac volumes; while the reason for the changes in the pre-peak period is that the reduction in S outweighs the increase in the travel time cost induced by the increased trac volumes.   Finally, the three Models with S=0 show the same welfare properties as the base case S=5. Both the FB and SB policies generate welfare gains against the NT policy; and the welfare gains generated by the SB policy are less than half of those by the FB policy. Compared with the base  case, for S=0 the social welfare levels yielded by each of the three models increase due to the reduction in the social costs. However, the welfare gains of the SB and FB policies from the NT shrink; and the relative welfare improvement of the SB to the FB drops from 45 to 40%. The above analysis indicates that the change in the value of S does not a€ect the main conclusions obtained in Sections 4.1±4.3. 4.4.2. Sensitivity analysis of the demand parameters bijs The sensitivity analysis of the demand parameters is conducted by altering the coecients DP=(b11,b12,b21,b22) in the base case where DP=(21,15,15,25). Two sets of new coecients arechosen: DPI=(21,10,10,25) and DPII=(14,10,10,18). In DPI, b12 (=b21) is reduced and b11

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Table 4 Results of Case 2 with new demand parameters bijs Case 2: Tt0=1, Tf0=2; Kt=2000, Kf=1000 ±DPI: b11=21, b12=b21=10, b22=25 Model NT FB SB

v1t 4782 3886 4141

v1f 1899 1594 2103

v2t 3214 2629 2606

v2f 73 953 854

v1 6681 5481 6244

v2 3287 3583 3460

V 9968 9063 9704

Model NT FB SB

(t1t,t1f) (0, 0) (94.1, 85.3) (45.3, 0)

(t2t,t2f) (0, 0) (19.7, 10.9) (8.0, 0)

(c1t,c1f) (64.9, 64.9) (34.5, 43.2) (41.3, 86.6)

(c2t,c2f) (54.5, 54.5) (48.4, 57.2) (48.3, 56.3)

P1 64.9 128.6 86.6

P2 54.5 68.1 56.3

± ± ± ±

Model NT FB SB

W($) 20965.93 22263.88 21590.29


W($) 0 1297.96 624.36

RW(%) ± 100 48.1

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

DPII: b11=14, b12=b21=10, b22=18 Model NT FB SB

v1t 4993 4206 4469

v1f 2003 1738 2207

v2t 3233 3008 2943

v2f 532 1173 1016

v1 6996 5944 6676

v2 3765 4181 3959

V 10761 10125 10635

Model NT FB SB

(t1t,t1f) (0, 0) (129.1, 120.3) (48.2, 0)

(t2t,t2f) (0, 0) (33.8, 25.0) (6.8, 0)

(c1t,c1f) (75.1, 75.1) (43.3, 52.1) (52.1, 100.3)

(c2t,c2f) (54.8, 54.8) (51.9, 60.7) (51.2, 58.0)

P1 75.1 172.4 100.3

P2 54.8 85.7 58.0

± ± ± ±

Model NT FB SB

W($) 52841.07 54147.07 53345.50


W($) 0 1306.00 504.42

RW(%) ± 100 38.6

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

and b22 remain unchanged. In DPII, each coecient is reduced. Table 4 presents the results of Case 2 with DPI and DPII. The results of the trac volume allocations in Table 4 show that like the base case in Table 1, the imposition of the congestion tolls in Model FB and SB still has the three impacts on the trac volume allocations in Model NT: (1) the diversion of the peak period trac to the free route in Model SB; (2) the shift of the peak period trac to the pre-peak period; and (3) the reduction in total trac volumes. The ordering relationship in the congestion tolls remains unchanged from Table 1. For the FB tolls, for each period, the toll on the toll route dominates the toll on the free route. For both FB and SB tolls, the peak toll on each route outweighs the pre-peak toll on that route. Comparing the SB toll with the FB toll, the FB toll is larger than the SB toll for each period:

L.N. Liu, J.F. McDonald/Transportation Research Part B 33 (1999) 157±188

FB :Tit > Tif ; i ˆ 1; 2; 1r > 2r ; r ˆ t; f:

185

…101†

SB : 1t > 2t : FB and SB : itFB > itSB ; i ˆ 1; 2: The welfare results of Case 2 with the new demand parameters demonstrate similar properties to the base case in Table 1. Model FB still yields the highest social welfare among the three models. Model SB also generates welfare gains against Model NT. But the welfare gains produced by Model SB still are less than half of the gains produced by Model FB. The above analysis shows that moderate adjustment in the value of bijs does not a€ect the main conclusions obtained in Sections 4.1±4.3. More empirical research is needed to obtain estimates of those own and crossprice e€ects. 4.4.3. Sensitivity analysis of the demand parameters Qis The sensitivity analysis is also conducted for the demand parameters Q1 and Q2, the potential peak and pre-peak demands. For Case 2, parameter Q2 is ®xed at 4000 (veh/h), and Q1 is reduced from the base value Q1=7500 (veh/h) by 10% to Q1=6750 and by 20% to Q1=6000, respectively. Table 5 presents the results of Case 2 with the reduced Q1. The results of the trac volume allocations in Table 5 show that like the base case in Table 1, the imposition of the congestion tolls in Model FB and SB still has the three impacts on the trac volume allocations in Model NT: (1) the diversion of the peak period trac to the free route in Model SB; (2) the shift of the peak period trac to the pre-peak period; and (3) the reduction in total trac volumes. The ordering relationship Eq. (101) for the congestion tolls still holds. For model FB, for each period, the toll on the toll route dominates the toll on the free route. For both FB and SB models, the peak toll on each route outweighs the pre-peak toll on that route. Compared with the FB toll, the SB toll is smaller than the FB toll for each period. It is worth noting that as a result of the reduction in Q1, the FB toll falls in both the peak and pre-peak periods, while the SB toll goes down in the peak but goes up in the pre-peak period. The welfare results of Case 2 with the reduced Q1 demonstrate similar properties to the base case in Table 1. Model FB still yields the highest social welfares among the three models. Model SB also generates welfare gains against Model NT. Note the relative welfare improvement of the SB to the FB has increased from 45.3 to 54.2%. The above analysis implies that the decrease in the value of Q1 does not a€ect the main conclusions obtained in Sections 4.1±4.3. The sensitivity analysis of the demand parameter Q2 is performed as follows. As discussed in Section 4.2, the second-best tolls in the pre-peak period can negative for Case 4. By looking at the de®nition Eq. (87), the pre-peak toll t2t depends on the inverse demand function in the pre-peak period P2(v1,v2) which is given by Eq. (73): P2 …v1 ; v2 † ˆ A2 ÿ b21 v1 ÿ b22 v2 where A2 is: A2 ˆ

B21 Q1 ‡ B11 Q2 d

…102† …103†

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By substituting Eqs. (102) and (103) into (87), it is shown that t2t depends on the parameters Q1 and Q2, the potential demand in the peak and pre-peak periods. The sensitivity analysis is conducted by ®xing Q1=7500 and increasing Q2: Q2 ˆ 4400; 4800; 6000; 7000 …veh=h†:

…104†

The resulting FB and SB tolls for Case 4 are presented in Table 6. Table 6 shows that Q2 plays a role in a€ecting the SB tolls. As Q2, the potential demand in the pre-peak period goes up, other things being equal, the SB tolls in the peak and the pre-peak periods change in both the signs and the sizes, while the FB tolls rise steadily. The SB toll is positive when Q2 is extremely high compared to Q1, e.g. 7000/7500=93%. Otherwise, the SB tolls are negative, i.e. people need to be subsidized to travel on the toll route in the pre-peak period since people have little tendency to travel on the toll route. Table 5 Results of Case 2 with new demand parameter Q1 Case 2: Tt0=1, Tf0=2; Kt=2000, Kf=1000 Q1=6750 Model NT FB SB

v1t 4583 3813 4040

v1f 1798 1561 1998

v2t 3215 2823 2769

v2f 270 1071 926

v1 6381 5374 6038

v2 3485 3894 3695

V 9866 9268 9733

± ± ± ±

Model NT FB SB

(t1t,t1f) (0, 0) (87.2, 78.4) (36.1, 0)

(t2t,t2f) (0, 0) (26.2, 17.4) (7.4, 0)

(c1t,c1f) (56.5, 56.5) (32.8, 41.6) (38.5, 74.6)

(c2t,c2f) (54.5, 54.5) (50, 58.8) (49.6, 56.9)

P1 56.5 120 74.6

P2 54.5 76.2 56.9

± ± ± ±

± ± ± ±

Model NT FB SB

W($) 32334.62 33275.62 32788.16


W($) 0 941 453.55

RW(%) ± 100 48.2

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

Q1=6000 Model NT FB SB

v1t 4255 3584 3795

v1f 1621 1456 1822

v2t 3214 2689 2622

v2f 96 991 848

v1 5877 5039 5617

v2 3310 3680 3470

V 9186 8719 9088

± ± ± ±

Model NT FB SB

(t1t,t1f) (0, 0) (68, 59.2) (26, 0)

(t2t,t2f) (0, 0) (21.6, 12.8) (7.8, 0)

(c1t,c1f) (44.8, 44.8) (28, 36.8) (32.4, 58.4)

(c2t,c2f) (54.5, 54.5) (48.9, 57.7) (48.4, 56.2)

P1 44.8 96.1 58.4

P2 54.5 70.4 56.2

± ± ± ±

± ± ± ±

Model NT FB SB

W ($) 27947.31 28599.91 28301.15


W ($) 0 652.59 353.84

RW (%) ± 100 54.2

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

± ± ± ±

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187

Table 6 Congestion tolls (cent/mile/veh): Case 4 with di€erent Q2 Case 4: Tt0=Tf0=2; Kt=Kf=2000 Model FB tolls SB tolls

Q2=4400± t1t t2t 75.6 16.9 13.9 ÿ1.0

Q2=4800± t1t t2t 77.6 21.9 13.6 ÿ1.2

Q2=6000± t1t t2t 85.3 40.6 12.3 ÿ1.1

Q2=7000 t1t t2t 93.6 60.1 11.2 0.6

5. Conclusions This paper examines urban highway congestion pricing in the instance in which it is not possible to levy a congestion toll on a major portion of the urban road system. This case is pertinent because of technical and/or political constraints. The paper uses economic theory and a simulation model to show that the optimum ``second-best'' tolls can vary appreciably from the optimal tolls in a ®rst-best regime in which ecient tolls can be imposed on all routes. Second-best congestion pricing policy is found to have three major impacts on the allocation of trac volume: (1) diversion of peak period trac to the free route; (2) shift of peak period trac to the o€-peak period; and (3) reduction in total trac volume. However, the second-best tolls are less ecient than the ®rst-best tolls in reallocating trac volume. Furthermore, the optimal second-best tolls are appreciably smaller than the ®rst-best tolls. Lastly, the paper shows that the welfare gains from the second-best tolls are much smaller than the welfare gains that are possible with a complete set of ®rst-best tolls. The simulations reported in the paper lead us to the following conclusions: 1. Within the range of values tried out, overall results for various cost and demand parameters are not sensitive to these changes, but there are some di€erences worth noting. 2. Cases 2 and 3 show that coverage of system with toll from 0, 1/3, 2/3, 1 generates a nonlinear percentage of potential welfare gains: 0, 19, 45, 100. 3. Cases 1 and 4 show that the system with faster toll route yields higher percentage of welfare gains: 46 vs 11%. 4. Demand parameters consist of slopes and intercepts. For the assumed slopes, the basic conclusions hold but more empirical studies are needed to obtain estimates of the slopes. For the intercepts it is shown that with the potential peak demand reduced, percentage of possible welfare gains rises; second-best tolls fall in the peak and rise in the o€-peak period. 5. There is a possibility of negative second-best toll in the o€-peak period with low potential o€-peak demand. This paper provides a starting point for evaluating second-best congestion pricing schemes. The current research may be extended by including vehicle operating costs, income distribution e€ects, more time periods, and modal choice. References Arnott, R., de Palma, A., Lindsey, R., 1990. Economics of a bottleneck. Journal of Urban Economics 27, 111±130.

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