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AN ESTIMATION OF EFFICIENT TIME-VARYING TOLLSFOR CROSS HARBOR TUNNELS IN HONG KONG

TIMOTHY D. HAU*,¶,‡‡, BECKY P. Y. LOO†,||,K. I. WONG‡,** and S. C. WONG§,††

*School of Economics and Finance, University of Hong KongPokfulam Road, Hong Kong

†Department of Geography, University of Hong KongPokfulam Road, Hong Kong

‡Department of Transportation Technology & ManagementNational Chiao Tung University 1001 Ta Hsueh Road

Hsinchu, 30010, Taiwan§Department of Civil Engineering

University of Hong Kong, Pokfulam Road, Hong Kong¶[email protected]

||[email protected]**[email protected]††[email protected]

This work estimates the distribution of a time-varying toll over a 24-hour period that minimizes thecombined queue length of the three tunnels that traverse Hong Kong’s Victoria Harbour, taking intoaccount institutional constraints. Our results reveal that switching from a flat toll to a time-varyingtoll scheme would eliminate all existing tunnel queues. We argue that optimal tunnel tolling, coupledwith the nonstop electronic toll collection mechanism already in place, could be the first step towardthe implementation of electronic road pricing in Hong Kong. Optimal tolling would obviate the needto build a fourth harbor crossing in the near future.

Keywords: Congestion; congestion pricing; time-varying toll.

JEL Classification: R41, R48

1. Introduction

We investigate the efficiency aspects of tolling for parallel tunnel traffic in Hong Kong byfollowing William Vickrey’s basic idea of equating effective demand with the space thatis available at congested toll facilities (Vickrey, 1967, 1996). Lam et al. (1996) optimizetunnel traffic across the harbor in Hong Kong by using a Land Use Transport Optimizationmodel and a Comprehensive Transport Study model to optimize the total network traveltime in a hypothetical network using two tunnel tolls. Yang and Lam (1996) use a bi-level

‡‡Corresponding author.

The Singapore Economic Review, Vol. 56, No. 4 (2011) 467–488© World Scientific Publishing CompanyDOI: 10.1142/S0217590811004432

467

http://dx.doi.org/10.1142/S0217590811004432

programming approach to determine the optimal road toll pattern under conditions ofqueuing and congestion, and illustrate their approach with some network examples.

Our research builds on a series of studies on the estimation of the queue-minimizing flattoll of the three harbor crossings in Hong Kong (Shum, 2001), with the case study locationshown in Map 1.1 Here we estimate the distribution of the time-varying toll over a 24-hourperiod that minimizes the combined queue length for the three tunnels, taking into accountinstitutional constraints. The underlying objective of the time-varying toll scheme is toinduce the efficient utilization of the total capacities of the three parallel tunnels that runnorth-south — the Cross Harbour Tunnel (CHT), the Eastern Harbour Crossing (EHC),and the Western Harbour Crossing (WHC) — for each hour, while exacting the lowest tollfee from motorists. We take into account that all three tunnels were constructed as privatelyfinanced 30-year Build-Operate-Transfer (BOT) projects. However, CHT’s 30-year leaseexpired on 31 August 1999, and full control of toll setting reverted to the Hong KongGovernment thereafter. The Eastern Harbour Crossing, constructed in 1986, is also a dualtwo-lane tunnel (with two lanes in each direction). The new Western Harbour Crossing, the

Map 1. Case Study LocationSource: Shum (2001)

1The modeling is driven principally and parametrically by the toll fee for private cars. The toll for the dual two-lane EasternHarbour Crossing is HK$15, for the dual two-lane Cross Harbour Tunnel it is HK$20, and for the dual three-lane WesternHarbour Crossing it is HK$37 (up to July 3, 2004). The rates for taxis, light goods, medium goods vehicles, and heavy goodsvehicles for the toll fee structure in 2002 are shown in Table 1. These parameters constitute the flat toll case. Thisconfiguration is assumed and scaled parametrically for the time-varying toll that is discussed later.

468 The Singapore Economic Review

construction of which began in 1993, is a dual three-lane tunnel with plenty of sparecapacity, and is almost always queue-free even at the peak of the peak periods, which is awaste of society’s resources. This phenomenon of perverse utilization due to non-optimaltolling is fairly common the world over. Table 1 gives the salient characteristics of the threeparallel tunnels that traverse the Victoria Harbour.

“In the idealized bottleneck-and-queue situation, where the capacity of thebottleneck is sharply defined, the ideal toll is clearly that which will justeliminate the queue [authors’ emphasis]. If traffic flows are stable andpredictable from day to day, there will be some pattern of tolls over timethat will accomplish this: the toll is zero whenever traffic is below thecapacity of the bottleneck, and, whenever traffic at zero toll would exceedcapacity, the toll is just sufficient to keep traffic at the capacity levelwithout the formation of a queue… To impose a toll at times when trafficis less than capacity would restrict the use of the facility to no purpose,assuming that below capacity no significant delay occurs. To allow aqueue to develop causes wasteful delay without producing any increasedflow through the bottleneck or enhancing the usefulness of the facility inany way. Thus the appropriate pattern of toll can be developed by trial anderror for any given facility; the levels of toll so arrived at will in turnprovide information essential for estimating when and to what extentexpansion of the facility is justified.”

Nobel Laureate William S. Vickrey (1967, p. 127)2

Table 1. Characteristics of the Three Harbor Tunnels

Eastern HarbourCrossing (EHC)

Cross HarbourTunnel (CHT)

Western HarbourCrossing (WHC)

Tunnel CharacteristicsStart date of Build-Operate-Transfer franchise 1986 1969 1993Opening date 1989 1972 1997Capacity Dual two-lane Dual two-lane Dual three-lane

Toll Fee Structure in 2002 (in HK dollars)Private cars 15 20 37Taxis 15 10 35Light goods vehicles 23 15 50Medium goods vehicles 30 20 70Heavy goods vehicles 45 30 100

2The Vickrey (1969) bottleneck model is one of two principal models of traffic flow in the transportation literature and wasfurther developed by Arnott et al. (1990). The other model is the standard continuous-flow model with smooth cost functionsthat derives from the fundamental diagram of road traffic (see Hau, 2005a,b; Mohring, 1999). Both models are systematicallytreated in McDonald et al. (1999).

Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 469

2. A Simple Model

The approach that is taken is to estimate by hour the toll that eliminates the ever-presentqueues at the entrances to the tunnels that cross Victoria Harbour. We aim to characterizethe traffic flow between Hong Kong Island and the Kowloon Peninsula through the threeharbor crossings using a simple model. We further assume that the exits of the three tunnelsare fully connected with sufficient capacity to handle the east–west traffic movementon either side of the harbor. This stringent assumption is plausible because two east-west strategic highway links — the Central-Wanchai Bypass and the Central-KowloonRoute — on either side of the harbor will be built and operating in the near future. Thisstudy focuses on the elimination of queues at the tunnel entrances through optimal tolling.

Hong Kong is divided into six regions (with three on Hong Kong Island and three on theKowloon Peninsula) such that there are 18 origin–destination (O-D) pairs that involvecross-harbor movements. Five vehicle types are considered in the model: private cars,occupied taxis,3 light goods vehicles, medium goods vehicles, and heavy goods vehicles.Public buses with fixed routes are treated as background traffic. The O-D matrices areobtained from the 2002 daily matrices and are disaggregated into 24 hourly matrices basedon information from the Travel Characteristics Survey (TCS) of 1992 that best matches theobserved traffic patterns (MVA Asia, 1993).

For a particular hour, each O-D pair, and each vehicle class, the proportions of the flowsthat pass through a particular harbor crossing are determined by a multinomial logit modelthat takes the standard form4:

Pi ¼expðUiÞPm2I expðUmÞ

, ð1Þ

where I is the tunnel choice set, Ui is the mean utility of tunnel i, and the mean utilityfunction of tunnel i that crosses the harbor is given by

Ui ¼ θTFðTFiÞ þ θQVi ðQViÞ þ θOD

i , ð2Þwhere TFi is the toll fee in Hong Kong dollars (1 US$¼ 7.80HK$), QVi is the number ofqueued vehicles at the entrance to the tunnel at that hour, and θTF , θQV

i , and θODi are the

utility coefficients that correspond to the toll fee, the number of queued vehicles, and therespective O-D pair dummy variable, respectively. The utility coefficients for the toll feesare tunnel independent, whereas the utility coefficients for the number of queued vehiclesare tunnel dependent, because the number of queued vehicles may induce different mag-nitudes of waiting times depending on the capacity of the tunnel and the characteristics ofthe network. Changes in vehicle mix due to toll fees and queue lengths are not consideredhere.

3Occupied taxis represent taxis that are carrying passengers. Vacant taxis are less likely to cross the harbor in Hong Kong tosearch for customers due to the toll charges.4Nobel Laureate Daniel McFadden has shown that assuming individual utility maximization and a stochastic taste term thatis Gumbel-distributed yields the logit model (see, for instance, McFadden, 1974; Domencich and McFadden, 1975; andTrain, 2003).


The utility coefficients are estimated from a stated preference survey in which 426people were interviewed with five hypothetical scenarios involving the three variablesabove.5 The maximum likelihood method is employed to estimate the utility coefficients,with an acceptable goodness of fit measure of over 66% being correctly predicted for all ofthe models that we calibrated. McFadden’s generalized coefficient of determination fordiscrete choice models, or R-squared as it is now commonly known, is 0.32300 for privatecars [see Table 2(a)], 0.54096 for occupied taxis [see Table 2(b)], 0.41692 for light goodsvehicles [see Table 2(c)], and 0.45958 for medium goods and heavy goods vehicles[see Table 2(d)]. Even though the sample size is limited for the large number of variables in

Table 2(a). Utility Coefficients of Private Cars




TF �0.040218 (�7.206)*

QV �0.000512 (�4.819)* �0.000723 (�7.750)* �0.000439 (�6.451)*

KLE!HKE OD01 2.561952 (4.825)* 0.470709 (0.734) —

KLE!HKC OD02 2.271719 (4.683)* 1.460368 (2.774)* —

KLE!HKW OD03 0.918831 (2.985)* 0.125322 (0.345) —

KLC!HKE OD04 0.845889 (2.112)* 1.774600 (4.526)* —

KLC!HKC OD05 �1.876428 (�3.963)* 0.659457 (2.198)* —

KLC!HKW OD06 �3.014829 (�5.575)* �0.383958 (�1.425) —

KLW!HKE OD07 �0.844502 (�2.687)* �0.290674 (�0.967) —

KLW!HKC OD08 �2.828347 (�6.564)* �1.163173 (�3.910)* —

KLW!HKW OD09 �4.938279 (�4.826)* �2.268323 (�5.953)* —

HKE!KLE OD10 3.586124 (5.832)* 1.330858 (1.970)* —

HKE!KLC OD11 0.868603 (2.536)* 1.304215 (4.198)* —

HKE!KLW OD12 0.179474 (0.552) 0.567792 (1.932) —

HKC!KLE OD13 3.017218 (5.495)* 2.151825 (3.930)* —

HKC!KLC OD14 �1.609727 (�3.748)* 0.437243 (1.796) —

HKC!KLW OD15 �2.884609 (�3.918)* �0.005762 (�0.022) —

HKW!KLE OD16 1.044872 (3.126)* 0.085567 (0.222) —

HKW!KLC OD17 �2.734729 (�4.482)* �0.249905 (�0.981) —

HKW!KLW OD18 �3.535209 (�4.832)* �1.951641 (�4.954)* —

No. of observations¼ 1510Log likelihood function¼ �1121:880Restricted log likelihood¼ �1657:1313

R-squared¼ 0.32300Percent correct¼ 66.6%

Notes: TF : Toll fee; QV : Queued vehicles; (.): t-statistic; *: Significant at the 0.05 level; —: Dummyvariable at reference level. The left column lists the locations of the OD pairs, for example, KLE standsfor Kowloon East and HKC stands for Hong Kong Central.

5Several pilot and main surveys were first conducted in various locations in Kowloon and Hong Kong Island, involving thedrivers of different vehicle classes in the fall of 1999 (Shum, 2001).


the estimation, the results look good. Although there are 18 pairs of zone-specific variablesin Tables 2(a)–2(d), these variables are not correlated. This is because only one pair ofzone-specific variables appears in each observation, as seen in the utility function Equa-tion (2). Therefore it allows us to use a smaller sample size for good estimation. It is alsonoted that in the survey questionnaire the number of queued vehicles at the entrance to atunnel is expressed as the location of the end of the queue (i.e., the queue length) for atunnel in each direction. This mapping takes into account the road characteristics andtopology of the network, and the selection of that location accords with the informationabout queue lengths that is provided by regular morning radio broadcasts. As thesemorning radio broadcasts regarding the exact location of the ends of the queues to cross theharbor are made frequently, it is reasonable to expect that motorists have almost perfectinformation regarding all of the alternative choice sets that were made available to them inthe stated preference survey. Furthermore, we only estimate the utility coefficients for the

Table 2(b). Utility Coefficients of Occupied Taxis




TF �0.059475 (�5.365)*

QV �0.001780 (�4.550)* �0.001600 (�4.575)* �0.000908 (�3.609)*

KLE!HKE OD01 34.047383 (0.000) 0.180995 (0.000) —

KLE!HKC OD02 2.829680 (2.004)* 1.979783 (1.349) —

KLE!HKW OD03 1.808169 (2.092)* �0.238566 (�0.187) —

KLC!HKE OD04 32.224861 (0.000) 32.507978 (0.000) —

KLC!HKC OD05 �0.463913 (�0.397) 1.817815 (1.855) —

KLC!HKW OD06 �1.757513 (�1.295) 0.548534 (0.557) —

KLW!HKE OD07 �0.573950 (�0.590) �0.114768 (�0.125) —

KLW!HKC OD08 �32.011095 (0.000) �0.409553 (�0.481) —

KLW!HKW OD09 �33.310924 (0.000) �2.076405 (�1.722) —

HKE!KLE OD10 33.533167 (0.000) 30.912759 (0.000) —

HKE!KLC OD11 33.912954 (0.000) 31.803152 (0.000) —

HKE!KLW OD12 0.868286 (1.009) �0.408513 (�0.467) —

HKC!KLE OD13 3.163182 (2.338)* 3.283148 (2.594)* —

HKC!KLC OD14 1.100473 (0.718) 3.264933 (2.752)* —

HKC!KLW OD15 �30.596190 (0.000) 0.905409 (1.045) —

HKW!KLE OD16 1.314831 (1.176) 1.605543 (1.655) —

HKW!KLC OD17 �1.287057 (�0.869) �1.481721 (�1.124) —

HKW!KLW OD18 �31.586454 (0.000) �1.366082 (�1.954) —



Notes: TF : Toll fee; QV : Queued vehicles; (.): t-statistic; *: Significant at the 0.05 level; —: Dummyvariable at reference level. The left column lists the locations of the OD pairs, for example, KLE standsfor Kowloon East and HKC stands for Hong Kong Central.


O-D pair dummy variable for the Eastern Harbour Crossing and the Cross Harbour Tunnel,as there is one redundant set of coefficients of tunnel alternatives for each O-D pair.

The estimates of the aggregate own-price elasticities and cross-price elasticities ofdemand with respect to each of the tunnel tolls for private cars, occupied taxis, light goodsvehicles, medium goods vehicles, and heavy goods vehicles are shown in Table 3, and theformulae are reported in the Appendix on Elasticities. The empirical estimates all possessthe expected signs, with all of the own-price elasticities of demand being negative andalmost all being inelastic. The cross-price elasticities of demand for each of the tunnelfacilities are all positive, which means that all of the tunnel facilities serve as substitutes forone another. For private cars, the own-price elasticities of the toll range from �0:297 to�0:430 for the three tunnels, which are comparable to the previous finding of Loo (2003)for Hong Kong, and the cross-price elasticities are between 0.101 and 0.250. Further, theown-price elasticities are consistent with the empirical estimates obtained for mostly

Table 2(c). Utility Coefficients of Light Goods Vehicles




TF �0.071400 (�7.269)*

QV �0.000948 (�3.139)* �0.001152 (�3.687)* �0.000771 (�3.541)*

KLE!HKE OD01 0.834410 (0.698) �1.501994 (�0.981) —

KLE!HKC OD02 1.652304 (1.676) 1.751794 (1.725) —

KLE!HKW OD03 �0.395707 (�0.475) �1.144881 (�1.152) —

KLC!HKE OD04 �0.342439 (�0.242) 1.596113 (1.121) —

KLC!HKC OD05 �30.692579 (0.000) 1.579983 (1.496) —

KLC!HKW OD06 �2.949677 (�2.363)* �0.350112 (�0.392) —

KLW!HKE OD07 0.982259 (1.051) 0.655017 (0.583) —

KLW!HKC OD08 �2.230406 (�2.078)* 0.080632 (0.084) —

KLW!HKW OD09 �2.978562 (�3.199)* �1.074105 (�1.303) —

HKE!KLE OD10 3.126244 (3.217)* 1.525478 (1.692) —

HKE!KLC OD11 1.146956 (0.875) 1.478750 (1.246) —

HKE!KLW OD12 �0.046404 (�0.039) 0.126993 (0.103) —

HKC!KLE OD13 31.419953 (0.000) 30.484203 (0.000) —

HKC!KLC OD14 0.522907 (0.461) 2.873250 (2.922)* —

HKC!KLW OD15 0.363779 (0.319) 0.374207 (0.409) —

HKW!KLE OD16 �0.933517 (�1.065) �0.251340 (�0.312) —

HKW!KLC OD17 �30.581410 (0.000) �0.223026 (�0.180) —

HKW!KLW OD18 �3.109439 (�2.579)* �2.864565 (�2.378)* —



Notes: TF: Toll fee; QV : Queued vehicles; (.): t-statistic; *: Significant at the 0.05 level; —: Dummyvariable at reference level. The left column lists the locations of the OD pairs, for example, KLE standsfor Kowloon East and HKC stands for Hong Kong Central.


developed North American cities in the literature (Goodwin, 1992; Oum et al., 1992;Hirschman et al., 1995; Burris, 2003). The elasticities for the commercial vehicle types aresomewhat higher due to their higher dollar magnitude compared to that of private cars[see TFi in Tables 2(a)–2(d)], and are therefore more sensitive to the toll prices of thetunnels. The own-price elasticities range from �0:550 to �0:820 for occupied taxis, from�0:827 to �1:023 for light goods vehicles, and from �0:897 to �1:061 for medium goodsand heavy goods vehicles. The asymmetry of the cross-price elasticities of demand asreflected by the off-diagonal elements in Table 3 is consistent with the fact that thecommonly observed Marshallian consumer’s surplus measures are dependent on the pathof integration (Hau, 1985).

The travel demand of each vehicle class for the 18 O-D pairs is multiplied with thecorresponding proportion using each tunnel in Equation (1) to obtain the correspondingtunnel incoming flow. The total incoming traffic heading to a tunnel during a particular

Table 2(d). Utility Coefficients of Medium Goods Vehicles and Heavy Goods Vehicles




TF �0.063792 (�7.480)*

QV �0.001160 (�3.445)* �0.000759 (�2.423)* �0.000807 (�3.870)*

KLE!HKE OD01 2.125896 (1.836) 0.356171 (0.262) —

KLE!HKC OD02 2.287605 (1.596) 0.893112 (0.597) —

KLE!HKW OD03 �0.918039 (�0.995) �2.127007 (�1.679) —

KLC!HKE OD04 1.994946 (2.082)* 1.332053 (1.249) —

KLC!HKC OD05 �2.055301 (�1.377) �0.468262 (�0.388) —

KLC!HKW OD06 �32.004876 (0.000) �0.136352 (�0.133) —

KLW!HKE OD07 �0.099289 (�0.118) �0.726945 (�0.861) —

KLW!HKC OD08 �1.825145 (�1.479) �1.210428 (�1.059) —

KLW!HKW OD09 �4.438962 (�3.227)* �2.344564 (�2.566)* —

HKE!KLE OD10 33.163745 (0.000) 30.398395 (0.000) —

HKE!KLC OD11 2.792597 (2.394)* 2.261628 (2.187)* —

HKE!KLW OD12 1.080599 (0.897) �0.618987 (�0.604) —

HKC!KLE OD13 1.964106 (2.092)* 1.460668 (1.533) —

HKC!KLC OD14 0.163650 (0.103) 2.724750 (2.268)* —

HKC!KLW OD15 �30.127813 (0.000) �0.473866 (�0.655) —

HKW!KLE OD16 0.856129 (1.055) �1.681044 (�1.202) —

HKW!KLC OD17 �0.965818 (�0.941) �0.589994 (�0.690) —

HKW!KLW OD18 �1.332022 (�1.111) �1.457245 (�1.223) —



Notes: TF: Toll fee; QV : Queued vehicles; (.): t-statistic; *: Significant at the 0.05 level; —: Dummyvariable at reference level. The left column lists the locations of the OD pairs, for example, KLE standsfor Kowloon East and HKC stands for Hong Kong Central.


Table 3(a). Aggregate Own-Price and Cross-Price Elasticities of Demand for Private Cars

Δ Demand (%) Δ Toll Fee (%)

EHC CHT WHC

EHC �0.297 0.187 0.112CHT 0.184 �0.430 0.250WHC 0.101 0.220 �0.328

Note that the diagonal elements refer to the estimates ofthe aggregate own-price elasticities of demand and theoff-diagonal elements refer to the estimates of the cross-price elasticities of demand. The positive off-diagonalterms indicate that all of the tunnels are substitutes forone another (and not complements). For example, theown-price elasticity of demand for private cars of�0.297 means that the number of vehicles using theEastern Harbour Crossing decreases by 0.297% whenthe toll for the tunnel is increased by 1%. (Hence theprice elasticity of demand is more precisely termed thetoll elasticity of demand.) The cross-price elasticity ofdemand of +0.250 means that the number of vehiclesusing the Cross Harbour Tunnel increases by 0.250%when the toll for the Western Harbour Crossing isincreased by 1%, whereas the number of vehicles usingthe Western Harbour Crossing increases by 0.220%when the toll for the Cross Harbour Tunnel is increasedby 1%.

Table 3(b). Aggregate Own-Price and Cross-Price Elasticities of Demand for Occupied Taxis


EHC CHT WHC

EHC �0.550 0.340 0.184CHT 0.351 �0.820 0.466WHC 0.235 0.538 �0.739

Table 3(c). Aggregate Own-Price and Cross-PriceElasticities of Demand for Light Goods Vehicles


EHC CHT WHC

EHC �0.853 0.512 0.390CHT 0.401 �0.827 0.414WHC 0.411 0.573 �1.023


hour can then be obtained by summing the flows to the tunnel for all O-D pairs and vehicleclasses as well as the background traffic of public buses. A queue builds up when the totalinflow (the incoming flow in that hour plus the residual queue from the previous hour) isgreater than the capacity of the tunnel. The queue that accumulates in an hour will remainas part of the total inflow for the next hour. Thus, the queue length for each hour can berepresented by the following equation (May and Keller, 1967).

QVi ¼ max {0, ðQV 0i þ ITiÞ � CTig, ð3Þ

where QVi and QV 0i are the number of queued vehicles that accumulate in that hour and

the previous hour, respectively; ITi is the incoming flow in that hour; and CTi is thecapacity of the tunnel.

The period between 4 and 5 a.m. is chosen as the starting time of the 24-hour queuinganalysis, as the traffic flow at this time is minimal and the queue length is zero. We can thensimulate the hourly exit flow by relating the flows in an hour to the flows in the previoushour by using the accumulated queues (if any). This process is repeated until the estimatedexit flow variations by hour throughout the day in the tunnels can be obtained.

The vehicular incoming traffic flow for the flat toll case — the base case — for theEastern Harbour Crossing, the Cross Harbour Tunnel, and the Western Harbour Crossingin the southbound and northbound directions over a 24-hour period is shown in Figure 1.The corresponding vehicular exit traffic flow for the flat toll case for the Eastern HarbourCrossing, the Cross Harbour Tunnel, and the Western Harbour Crossing in the southboundand northbound directions over the course of the day is shown in Figure 2. Note thatFigure 1 indicates that more traffic heads southward to Hong Kong Island during themorning peak than those heading northward to the Kowloon Peninsula, and the reversepattern is observed during the afternoon peak. Vehicular traffic in both peak periods istruncated, however, by the respective binding capacities of the two dual two-lane EasternHarbour Crossing and Cross Harbour Tunnel, as can be seen in Figures 1 and 2. Thetruncation at 3200 vehicles accords with a standard highway capacity manual result, whichstates that a dual two-lane carriageway with limited access has a capacity of 1600 vehiclesper lane per hour. The number of queued vehicles expressed as the queue length in distancefrom the entrances of the three tunnels in the southbound and northbound directions basedon the queue transformation of the two queue representations are shown in Figures 3and 4, respectively, for the flat toll case. Note that, consistent with the incoming traffic flow

Table 3(d). Aggregate Own-Price and Cross-Price Elasticities of Demand for Medium GoodsVehicles and Heavy Goods Vehicles


EHC CHT WHC

EHC �0.976 0.501 0.382CHT 0.548 �1.061 0.562WHC 0.419 0.523 �0.897


and exit flow figures in Figures 1 and 2, the queues at the entrances to the Cross HarbourTunnel depict a similar pattern during the morning and afternoon peak periods. There issome excess traffic that results in queues in the northbound direction between 6 and 7 p.m.and between 7 and 8 p.m. for the Eastern Harbour Crossing. Due to its excess capacity, theWestern Harbour Crossing is always queue free.

Flat toll case(Base case, 2002)

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Figure 1. Incoming Flow for the Flat Toll Case


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Figure 2. Exit Flow for the Flat Toll Case


We now compare the combined exit flows (both southbound and northbound) of each ofthe three tunnels as reported in the Annual Traffic Census 2002 (Transport Department,2002) with our modeled exit flows. For the Eastern Harbour Crossing, the modeled exitflow is underestimated during the morning peak and interpeak periods (see Figure 5). The

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Figure 3. Queued Vehicles for the Flat Toll Case (Southbound)

Flat toll case - Northbound(Base case, 2002)

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Figure 4. Queued Vehicles for the Flat Toll Case (Northbound)


modeled exit flow for the Cross Harbour Tunnel tracks the observed figures quite well,except for the unimportant midnight to early morning hours (see Figure 6). The modeledexit flow for the Western Harbour Crossing is slightly overestimated during the morningand afternoon peak periods and underestimated during the interpeak period (see Figure 7).

According to Nobel Laureate Milton Friedman, the ultimate test of a model is thepredictive test. We observe that our modeled exit flows for the combined directional flowstrack the observed counterpart figures in the Annual Traffic Census 2002 quite well.

3. Case Study: Setting the Optimal Time-Varying Toll

3.1. Time-varying toll scheme

The objective of the time-varying toll scheme is to induce the efficient utilization of thetotal capacities of the three parallel tunnels (the Eastern Harbour Crossing, Cross HarbourTunnel, and Western Harbour Crossing) within each hour, while collecting the lowest tollfee from tunnel users. As mentioned, all three tunnels that traverse the harbor are privatelyfinanced 30-year BOT projects. However, since the Hong Kong Government resumedcontrol of the Cross Harbour Tunnel on 1 September 1999 upon the expiry of the BOTproject, flexibility in toll setting comes into play. In the time-varying toll scheme, weassume the following:

(1) The (high) toll level for the newer Western Harbour Crossing is fixed, because if theyounger toll operator is to recoup its investment, it will be unable to lower its toll.


Exit flow pattern of the Eastern Harbour Crossing (Southbound + Northbound)

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7000

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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ow (

veh)

ModeledObserved

Figure 5. Exit Flow Pattern for the Eastern Harbour Crossing, Observed Versus Modeled


(It also turns out that the Western Harbour Crossing’s dual three-lane tunnel has sparecapacity at all hours of the day, which suggests that the toll should not rise above itspresent toll rate).

(2) The toll level for the Eastern Harbour Crossing is allowed to rise, that is, the tunnel isallowed to recoup its investment and excess traffic at this crossing indeed pushes upthe time-varying toll for the periods in which it occurs.

(3) The toll level for the Cross Harbour Tunnel is allowed to rise or fall over the course ofthe day because the Hong Kong Government regained its right to set tolls for all trafficupon the expiry of the BOT project.

Generally there are three cases in the scheme as follows.

Case 1During off-peak hours, when the utilization of the Cross Harbour Tunnel is below capacity,we decrease the toll level to attract traffic until the tunnel’s capacity is reached. However, ifthere is excess capacity, the toll is set at a nominal amount, to cover tunnel maintenance forinstance (but not zero).

Case 2When the utilization of either the Eastern Harbour Crossing or the Cross Harbour Tunnel(or both) just exceeds their respective capacities but the total traffic in one particulardirection at a certain hour is less than the total capacities of the three tunnels, there are twopossible outcomes.

(i) If queues build up at the Eastern Harbour Crossing, we increase the toll level for thattunnel to ward off excess traffic.

(ii) If queues are built up at the Cross Harbour Tunnel, we increase the toll level for thattunnel to ward off excess traffic.

In this situation (Case 2), no queue will be carried over to the next hour.

Case 3In principle, it is possible that the total traffic in a particular direction during a certain hourwill be greater than the total combined capacities of the three tunnels put together due togrowth in traffic. However, our data indicate that the total traffic in one direction is alwayswithin the total capacities of the three tunnels, especially given that the newest tunnel, theWestern Harbour Crossing, has a 50% greater capacity than either of the other two tunnels.

4. Main Results and Conclusion

The resulting economically efficient toll, or the time-varying toll distribution over a 24-hour period, for the Cross Harbour Tunnel for southbound and northbound traffic,respectively, approximates the graphs for the queued vehicles (Figures 3 and 4).6 The

6Note that in the time-varying toll case we nominally set a minimum toll fee of HK$1 for the Cross Harbour Tunnel fortunnel maintenance to avoid a toll-free charging period.



Exit flow pattern of the Cross Harbour Tunnel (Southbound + Northbound)

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Figure 6. Exit Flow Pattern for the Cross Harbor Tunnel, Observed Versus Modeled


Exit flow pattern of the Western Harbour Crossing (Southbound + Northbound)

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Figure 7. Exit Flow Pattern for the Western Harbour Crossing, Observed Versus Modeled


difference in this case (Figures 8 and 9) is that we follow the initial institutional set-upof charging a high toll of HK$37 for private cars that use the Western HarbourCrossing and a low toll of HK$15 for private cars that use the Eastern Harbour Crossing,with the provison that these tolls can be freely adjusted upward as queues begin to

Time-varying toll case - Northbound

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Figure 9. Time-Varying Toll (Northbound)

Time-varying toll case - Southbound

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Figure 8. Time-Varying Toll (Southbound)


form.7 All queuing is eliminated with this toll set-up, and the incoming flow correspondsto the exit flow for all three tunnels regardless of whether or not the respective capacityis binding (see Figure 10).

In short, the impact of switching from a flat toll to a time-varying toll results in theelimination of all of the existing tunnel queues. Society is made unequivocally better off astime that would otherwise be lost is saved by queue reduction as expressed in vehicle-hours(see Table 4). The value for travel time is obtained from the 2001 behavioral value of timefrom Table 4.3 of the Third Comprehensive Transport Study — Technical Report(Transport Department, 1999a,b,c) and extended to 2002 using the Composite ConsumerPrice Index. This means that the 2001 behavioral value of time of HK$0.71 per minute(in 1992 prices), becomes HK$56.09 per hour when factored up to 2002 prices. When

Table 4. Changes in Queued Vehicles Per Day (From the Perspective of the Tunnel User)

Queued Vehicles Time Savings

Flat Toll (veh-hr) Time-Varying Toll (veh-hr) Time Unit (veh-hr) Monetary Unit (HK$)

EHC 996 0 996 55,866CHT 19329 0 19329 1,084,162WHC 0 0 0 0Total 20325 0 20325 1,140,029

Time-varying toll case

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Figure 10. Incoming Flow and Exit Flow for the Time-Varying Toll Case

7For the southbound direction, the time-varying toll would climb quickly from a $1 toll at 6:30 a.m. at day break to a $44 tollat 7:30 a.m. to a $57 toll at 8:30 a.m. — being the peak of the morning peak — and declining to a $15 toll during thenoontime hours before rising again to a $46 toll at 6:30 p.m. — being the peak of the afternoon peak — and falling again atnightfall.


multiplied by the 2002 value of time of HK$56.09 (US$7.19) per hour, the aggregate valueof time that is saved is HK$1.14 million a day. Note that the total travel time savingspresented in Table 4 exclude the travel time changes due to the changes in routes. Thetravel time change is small. With the flat toll, some vehicles actually detour eastward to theCross Harbour Tunnel for its lower toll and the Eastern Harbour Crossing for its shorterqueue. Under the time-varying toll, these vehicles will choose the tunnel that suits thembest. Overall, this will balance out the extra travel time of the other vehicles detouringunder the time-varying toll scheme.

Motorists as a group are roughly just as well off in the time-varying toll case as with theflat toll scheme (see Table 5). Note that motorists as a group pay slightly more in terms ofthe toll payments; putting it another way, the road authority collects a little more in tollrevenues: HK$0.05 million a day (1%) more than the total toll revenues of HK$5.70million a day for the base case. Interestingly, the privately operated toll operator of theEastern Harbour Crossing collects slightly more revenue, because the principal impact ofimplementing the time-varying toll is to divert the (long) queue from the centrally locatedCross Harbour Tunnel to the two adjacent tunnels. Further, the fact that the EasternHarbour Crossing’s time-varying toll for private cars increases from the flat toll of HK$15 toHK$20 between 5 and 6 p.m. and to HK$25 between 6 and 7 p.m. for northbound trafficmeans that there is excess demand for the Eastern Harbour Crossing, which further leads toincreased revenues. Because the toll for the Western Harbour Crossing is fixed uniformlyover time by assumption, the lower toll revenue collections reflect the fact that some of thetraffic that previously used the Western Harbour Crossing is diverted to the Eastern Har-bour Crossing and a small amount of traffic may even be diverted back to the CrossHarbour Tunnel. This suggests that the private operators of the Eastern Harbour Crossingand the Cross Harbour Tunnel should object less to the policy proposal of switching to atime-varying toll scheme.

For all practical purposes, as there is no significant increase in toll revenues for theHong Kong Government — the owner of the Cross Harbour Tunnel — the time-varyingtoll scheme should in principle encounter less political resistance from the public. Havingsaid that, the successful adoption of the time-varying scheme would hinge on the distri-bution of gains and losses.

Optimal tolling — despite the institutional constraints that tolls for tunnels which areprivately run are sticky downwards — would obviate the need to build a fourth harbor

Table 5. Changes in Toll Collection Per Day (From the Perspective of the TunnelAuthority)

Toll Revenues fromFlat Toll

Toll Revenues fromTime-Varying Toll

Changes inToll Revenues

% Difference

EHC $1,331,048 $1,337,335 $6,287 0.47%CHT $2,419,457 $2,508,603 $89,146 3.68%WHC $1,946,687 $1,899,391 �$47,296 �2.43%Total $5,697,192 $5,745,329 $48,137 0.84%


crossing in the near future at an estimated capital cost of HK$6.6 billion (in 1998 prices)(Transport Department, 1999b). Further, the elimination of queues at the entrances to thetunnels would have a substantial positive external effect on the east-west traffic on bothsides of the harbor. We argue that optimal tunnel tolling, coupled with the non-stopelectronic toll collection that is already in place, could be the first step toward theimplementation of electronic road pricing in Hong Kong (Hau, 2001, 2006a,b; TransportDepartment, 2001; Ho et al., 2005, 2007; Wong et al., 2005; Council for SustainableDevelopment, 2008).

5. Caveats and Extensions

“One of the most dramatic market failures in the public sector has beenthe failure to use market principles to reduce the waste involved in thequeuing at congested toll facilities, such as the San Francisco BayBridge and the Lincoln Tunnel. Applying market principles here involvescollecting rents for the use of the facility at various times. Such rentsshould equate effective demand with the space available [authors’emphasis].

In the case of the Lincoln Tunnel, for example, delays ranging up to ahalf hour and more are common. Persistent queuing can prevail overnearly every peak period for three to five hours. Most of the queuing couldbe eliminated by a surcharge (on top of the existing toll) which risesgradually to a brief peak and then declines gradually. By trial and error thesurcharge would be increased at times when there has been regular sig-nificant queuing [authors’ emphasis] (and few or no significant gaps in theflow at that time of day over the previous period) and lowered when therehave been significantly frequent gaps in the flow at that time. The onlyremaining queuing would be that which results from unpredictable fluc-tuation in traffic demand from one day to another.”

Nobel Laureate William Vickrey (1996, p. 225)

Ideally, with data expressed in finer time intervals, such as 10- or 15-minute periods, anda time-varying tunnel toll that “rises gradually to a brief peak and then declines gradually,”the cost of rescheduling a trip to an earlier or later time would be much less onerousand the corresponding benefits much greater. It would be considerably easier if allmotorists shifted their journeys by a few minutes, rather than a few motorists shifting theirtrips by an hour at a time. Our model essentially takes trip schedules as given, and thusefficient tunnel tolling effectively forces motorists to take an alternative route. It is likelythat commuters, for instance, would react to a high toll by rescheduling, either by risingearlier or staying later in the office. Thus, our model does not take the morning departuretime into account as in the model of Arnott et al. (1990). Extensions in this naturaldirection would involve the design and implementation of a new stated preference survey(Loo et al., 2008).


Further, both household and work locations are assumed to be fixed in the short run,with the O-D trip matrices taken as given. Over time, relocation in response to higher tollswould free up resources and effectively increase the estimated time saving. The distancetraveled and its corresponding costs (such as fuel costs) could be further modeled andcalibrated by using the location of the zone centroids and route lengths. Finally, the valueof time could be differentiated by user class and trip purpose.

Acknowledgments

The work that is described in this paper was partially supported by a grant from theResearch Grants Council of the Hong Kong Special Administrative Region (Project No:HKU7126/04E). The authors would like to gratefully acknowledge the contributions ofYin Yu Choi, Wai Hung Chung, and Wai Yip Cheng with the data collection in the statedpreference survey. Comments from Herbert Mohring on an earlier version of this paper aregratefully acknowledged.

Appendix A. Appendix on Elasticities

The responsiveness of demand with respect to an attribute is defined as the percentagechange in the quantity demanded (or in this case the choice probability) that results from a1% change in an attribute (see McFadden, 1974; Domencich and McFadden, 1975). Thedisaggregate own-price elasticity of demand can be shown to be

EPnðiÞxink ¼ ð1� PnðiÞÞxinkβk, ðA:1Þ

and the disaggregate cross-price elasticity of demand can be shown to be

EPnðiÞxjnk ¼ �Pnð jÞxjnkβk, j 6¼ i, ðA:2Þ

where PnðiÞ is the probability of respondent n choosing alternative i, xink is the attribute (inthis case the toll fee) that is faced by respondent n for alternative i, and βk is the coefficientthat corresponds to the independent variable (in this case the toll). The expected share ofthe group choosing alternative i for a group of N respondents is simply

P ðiÞ ¼PN

n¼1 PnðiÞN

: ðA:3Þ

Aggregate elasticities summarize the responsiveness of all of the respondents, rather thanthat of an individual respondent. Using the derived values, the aggregate elasticities canthen be calculated by

E P ðiÞxjk ¼

PNn¼1 PnðiÞEPnðiÞ

xjnkPN

n¼1 PnðiÞ, ðA:4Þ

which is a weighted average of the individual level elasticities using the choice prob-abilities as weights, where EPnðiÞ

xjnk is obtained from Equations (A.1) and (A.2). This rep-resents the direct elasticities for i ¼ j and the cross elasticities for the others.


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