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

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

AN ESTIMATION OF EFFICIENT TIME-VARYING TOLLS FOR 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 Kong Pokfulam Road, Hong Kong †Department of Geography, University of Hong Kong Pokfulam Road, Hong Kong ‡Department of Transportation Technology & Management National 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 the combined queue length of the three tunnels that traverse Hong Kong’s Victoria Harbour, taking into account institutional constraints. Our results reveal that switching from a flat toll to a time-varying toll scheme would eliminate all existing tunnel queues. We argue that optimal tunnel tolling, coupled with the nonstop electronic toll collection mechanism already in place, could be the first step toward the implementation of electronic road pricing in Hong Kong. Optimal tolling would obviate the need to 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 by following William Vickrey’s basic idea of equating effective demand with the space that is available at congested toll facilities (Vickrey, 1967, 1996). Lam et al. (1996) optimize tunnel traffic across the harbor in Hong Kong by using a Land Use Transport Optimization model and a Comprehensive Transport Study model to optimize the total network travel time in a hypothetical network using two tunnel tolls. Yang and Lam (1996) use a bi-level

‡‡ Corresponding author.

467 468 The Singapore Economic Review programming approach to determine the optimal road toll pattern under conditions of queuing 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 flat toll of the three harbor crossings in Hong Kong (Shum, 2001), with the case study location shown in Map 1.1 Here we estimate the distribution of the time-varying toll over a 24-hour period that minimizes the combined queue length for the three tunnels, taking into account institutional constraints. The underlying objective of the time-varying toll scheme is to induce the efficient utilization of the total capacities of the three parallel tunnels that run north-south — the Cross Harbour Tunnel (CHT), the Eastern Harbour Crossing (EHC), and the Western Harbour Crossing (WHC) — for each hour, while exacting the lowest toll fee from motorists. We take into account that all three tunnels were constructed as privately financed 30-year Build-Operate-Transfer (BOT) projects. However, CHT’s 30-year lease expired on 31 August 1999, and full control of toll setting reverted to the Hong Kong Government thereafter. The Eastern Harbour Crossing, constructed in 1986, is also a dual two-lane tunnel (with two lanes in each direction). The new Western Harbour Crossing, the

Map 1. Case Study Location Source: Shum (2001)

1The modeling is driven principally and parametrically by the toll fee for private cars. The toll for the dual two-lane Eastern Harbour Crossing is HK$15, for the dual two-lane Cross Harbour Tunnel it is HK$20, and for the dual three-lane Western Harbour Crossing it is HK$37 (up to July 3, 2004). The rates for taxis, light goods, medium goods vehicles, and heavy goods vehicles for the toll fee structure in 2002 are shown in Table 1. These parameters constitute the flat toll case. This configuration is assumed and scaled parametrically for the time-varying toll that is discussed later. Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 469

Table 1. Characteristics of the Three Harbor Tunnels

Eastern Harbour Cross Harbour Western Harbour Crossing (EHC) Tunnel (CHT) Crossing (WHC)

Tunnel Characteristics Start date of Build-Operate-Transfer franchise 1986 1969 1993 Opening date 1989 1972 1997 Capacity Dual two-lane Dual two-lane Dual three-lane Toll Fee Structure in 2002 (in HK dollars) Private cars 15 20 37 Taxis 15 10 35 Light goods vehicles 23 15 50 Medium goods vehicles 30 20 70 Heavy goods vehicles 45 30 100

construction of which began in 1993, is a dual three-lane tunnel with plenty of spare capacity, and is almost always queue-free even at the peak of the peak periods, which is a waste of society’s resources. This phenomenon of perverse utilization due to non-optimal tolling is fairly common the world over. Table 1 gives the salient characteristics of the three parallel tunnels that traverse the Victoria Harbour. “In the idealized bottleneck-and-queue situation, where the capacity of the bottleneck is sharply defined, the ideal toll is clearly that which will just eliminate the queue [authors’ emphasis]. If traffic flows are stable and predictable from day to day, there will be some pattern of tolls over time that will accomplish this: the toll is zero whenever traffic is below the capacity of the bottleneck, and, whenever traffic at zero toll would exceed capacity, the toll is just sufficient to keep traffic at the capacity level without the formation of a queue… To impose a toll at times when traffic is less than capacity would restrict the use of the facility to no purpose, assuming that below capacity no significant delay occurs. To allow a queue to develop causes wasteful delay without producing any increased flow through the bottleneck or enhancing the usefulness of the facility in any way. Thus the appropriate pattern of toll can be developed by trial and error for any given facility; the levels of toll so arrived at will in turn provide information essential for estimating when and to what extent expansion of the facility is justified.” Nobel Laureate William S. Vickrey (1967, p. 127)2

2The Vickrey (1969) bottleneck model is one of two principal models of traffic flow in the transportation literature and was further developed by Arnott et al. (1990). The other model is the standard continuous-flow model with smooth cost functions that derives from the fundamental diagram of road traffic (see Hau, 2005a,b; Mohring, 1999). Both models are systematically treated in McDonald et al. (1999). 470 The Singapore Economic Review

2. A Simple Model The approach that is taken is to estimate by hour the toll that eliminates the ever-present queues at the entrances to the tunnels that cross Victoria Harbour. We aim to characterize the traffic flow between Hong Kong Island and the Kowloon Peninsula through the three harbor crossings using a simple model. We further assume that the exits of the three tunnels are fully connected with sufficient capacity to handle the east–west traffic movement on either side of the harbor. This stringent assumption is plausible because two east- west strategic highway links — the Central-Wanchai Bypass and the Central-Kowloon Route — on either side of the harbor will be built and operating in the near future. This study 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 the Kowloon Peninsula) such that there are 18 origin–destination (O-D) pairs that involve cross-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 are obtained from the 2002 daily matrices and are disaggregated into 24 hourly matrices based on information from the Travel Characteristics Survey (TCS) of 1992 that best matches the observed traffic patterns (MVA Asia, 1993). For a particular hour, each O-D pair, and each vehicle class, the proportions of the flows that pass through a particular harbor crossing are determined by a multinomial logit model that takes the standard form4:

P expðUiÞ Pi ¼ , ð1Þ m2I expðUmÞ where I is the tunnel choice set, Ui is the mean utility of tunnel i, and the mean utility function of tunnel i that crosses the harbor is given by θ TF θ QV θ OD, Ui ¼ ðTFiÞþ i ðQViÞþ i ð2Þ where TFi is the toll fee in Hong Kong dollars (1 US$ ¼ 7.80 HK$), QVi is the number of θ TF θ QV θ OD queued vehicles at the entrance to the tunnel at that hour, and , i , and i are the utility coefficients that correspond to the toll fee, the number of queued vehicles, and the respective O-D pair dummy variable, respectively. The utility coefficients for the toll fees are tunnel independent, whereas the utility coefficients for the number of queued vehicles are 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 of the network. Changes in vehicle mix due to toll fees and queue lengths are not considered here.

3Occupied taxis represent taxis that are carrying passengers. Vacant taxis are less likely to cross the harbor in Hong Kong to search for customers due to the toll charges. 4Nobel Laureate Daniel McFadden has shown that assuming individual utility maximization and a stochastic taste term that is Gumbel-distributed yields the logit model (see, for instance, McFadden, 1974; Domencich and McFadden, 1975; and Train, 2003). Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 471

The utility coefficients are estimated from a stated preference survey in which 426 people were interviewed with five hypothetical scenarios involving the three variables above.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 of the models that we calibrated. McFadden’s generalized coefficient of determination for discrete choice models, or R-squared as it is now commonly known, is 0.32300 for private cars [see Table 2(a)], 0.54096 for occupied taxis [see Table 2(b)], 0.41692 for light goods vehicles [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

Eastern Harbour Cross Harbour Western Harbour Crossing (EHC) Tunnel (CHT) Crossing (WHC)

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 observations ¼ 1510 Log likelihood function ¼1121:880 Restricted log likelihood ¼1657:1313 R-squared ¼ 0.32300 Percent correct ¼ 66.6%

5Several pilot and main surveys were first conducted in various locations in Kowloon and Hong Kong Island, involving the drivers of different vehicle classes in the fall of 1999 (Shum, 2001). 472 The Singapore Economic Review

Table 2(b). Utility Coefficients of Occupied Taxis

Eastern Harbour Cross Harbour Western Harbour Crossing (EHC) Tunnel (CHT) Crossing (WHC)

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) —

No. of observations ¼ 200 Log likelihood function ¼100:6924 Restricted log likelihood ¼219:3522 R-squared ¼ 0.54096 Percent correct ¼ 74.0%

Notes: TF : Toll fee; QV : Queued vehicles; (.): t-statistic; *: Significant at the 0.05 level; —: Dummy variable at reference level. The left column lists the locations of the OD pairs, for example, KLE stands for Kowloon East and HKC stands for Hong Kong Central. the estimation, the results look good. Although there are 18 pairs of zone-specific variables in Tables 2(a)–2(d), these variables are not correlated. This is because only one pair of zone-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 also noted that in the survey questionnaire the number of queued vehicles at the entrance to a tunnel is expressed as the location of the end of the queue (i.e., the queue length) for a tunnel in each direction. This mapping takes into account the road characteristics and topology of the network, and the selection of that location accords with the information about queue lengths that is provided by regular morning radio broadcasts. As these morning radio broadcasts regarding the exact location of the ends of the queues to cross the harbor are made frequently, it is reasonable to expect that motorists have almost perfect information regarding all of the alternative choice sets that were made available to them in the stated preference survey. Furthermore, we only estimate the utility coefficients for the Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 473

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

Eastern Harbour Cross Harbour Western Harbour Crossing (EHC) Tunnel (CHT) Crossing (WHC)

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)* —

No. of observations ¼ 205 Log likelihood function ¼130:1694 Restricted log likelihood ¼223:2438 R-squared ¼ 0.41692 Percent correct ¼ 74.6%

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 of demand with respect to each of the tunnel tolls for private cars, occupied taxis, light goods vehicles, medium goods vehicles, and heavy goods vehicles are shown in Table 3, and the formulae are reported in the Appendix on Elasticities. The empirical estimates all possess the expected signs, with all of the own-price elasticities of demand being negative and almost all being inelastic. The cross-price elasticities of demand for each of the tunnel facilities are all positive, which means that all of the tunnel facilities serve as substitutes for one 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, the own-price elasticities are consistent with the empirical estimates obtained for mostly 474 The Singapore Economic Review

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

Eastern Harbour Cross Harbour Western Harbour Crossing (EHC) Tunnel (CHT) Crossing (WHC)

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) —

No. of observations ¼ 215 Log likelihood function ¼127:6147 Restricted log likelihood ¼236:1414 R-squared ¼ 0.45958 Percent correct ¼ 74.4%

[see TFi in Tables 2(a)–2(d)], and are therefore more sensitive to the toll prices of the tunnels. 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 goods and heavy goods vehicles. The asymmetry of the cross-price elasticities of demand as reflected by the off-diagonal elements in Table 3 is consistent with the fact that the commonly observed Marshallian consumer’s surplus measures are dependent on the path of integration (Hau, 1985). The travel demand of each vehicle class for the 18 O-D pairs is multiplied with the corresponding proportion using each tunnel in Equation (1) to obtain the corresponding tunnel incoming flow. The total incoming traffic heading to a tunnel during a particular Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 475

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 0.112 CHT 0.184 0.430 0.250 WHC 0.101 0.220 0.328

Note that the diagonal elements refer to the estimates of the aggregate own-price elasticities of demand and the off-diagonal elements refer to the estimates of the cross- price elasticities of demand. The positive off-diagonal terms indicate that all of the tunnels are substitutes for one another (and not complements). For example, the own-price elasticity of demand for private cars of 0.297 means that the number of vehicles using the Eastern Harbour Crossing decreases by 0.297% when the toll for the tunnel is increased by 1%. (Hence the price elasticity of demand is more precisely termed the toll elasticity of demand.) The cross-price elasticity of demand of +0.250 means that the number of vehicles using the Cross Harbour Tunnel increases by 0.250% when the toll for the Western Harbour Crossing is increased by 1%, whereas the number of vehicles using the Western Harbour Crossing increases by 0.220% when the toll for the Cross Harbour Tunnel is increased by 1%.

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

Δ Demand (%) Δ Toll Fee (%) EHC CHT WHC

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

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

Δ Demand (%) Δ Toll Fee (%) EHC CHT WHC

EHC 0.853 0.512 0.390 CHT 0.401 0.827 0.414 WHC 0.411 0.573 1.023 476 The Singapore Economic Review

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

Δ Demand (%) Δ Toll Fee (%)

EHC CHT WHC

EHC 0.976 0.501 0.382 CHT 0.548 1.061 0.562 WHC 0.419 0.523 0.897 hour can then be obtained by summing the flows to the tunnel for all O-D pairs and vehicle classes as well as the background traffic of public buses. A queue builds up when the total inflow (the incoming flow in that hour plus the residual queue from the previous hour) is greater than the capacity of the tunnel. The queue that accumulates in an hour will remain as part of the total inflow for the next hour. Thus, the queue length for each hour can be represented by the following equation (May and Keller, 1967). 0 QVi ¼ max {0, ðQV i þ ITiÞCTig, ð3Þ 0 where QVi and QV i 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 the capacity of the tunnel. The period between 4 and 5 a.m. is chosen as the starting time of the 24-hour queuing analysis, as the traffic flow at this time is minimal and the queue length is zero. We can then simulate the hourly exit flow by relating the flows in an hour to the flows in the previous hour by using the accumulated queues (if any). This process is repeated until the estimated exit 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 the Eastern Harbour Crossing, the Cross Harbour Tunnel, and the Western Harbour Crossing in 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 Harbour Crossing, the Cross Harbour Tunnel, and the Western Harbour Crossing in the southbound and northbound directions over the course of the day is shown in Figure 2. Note that Figure 1 indicates that more traffic heads southward to Hong Kong Island during the morning peak than those heading northward to the Kowloon Peninsula, and the reverse pattern is observed during the afternoon peak. Vehicular traffic in both peak periods is truncated, however, by the respective binding capacities of the two dual two-lane Eastern Harbour Crossing and Cross Harbour Tunnel, as can be seen in Figures 1 and 2. The truncation at 3200 vehicles accords with a standard highway capacity manual result, which states that a dual two-lane carriageway with limited access has a capacity of 1600 vehicles per lane per hour. The number of queued vehicles expressed as the queue length in distance from the entrances of the three tunnels in the southbound and northbound directions based on the queue transformation of the two queue representations are shown in Figures 3 and 4, respectively, for the flat toll case. Note that, consistent with the incoming traffic flow Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 477

Flat toll case (Base case, 2002) 5000

4000

3000 EHC-SB EHC-NB CHT-SB 2000 CHT-NB WHC-SB WHC-NB Incoming flow (veh) 1000

0 00:30 01:30 02:30 03:30 04:30 05:30 06:30 07:30 08:30 09:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 18:30 19:30 20:30 21:30 22:30 23:30 Hour

Figure 1. Incoming Flow for the Flat Toll Case

Flat toll case (Base case, 2002) 5000

4000

3000 EHC-SB EHC-NB CHT-SB 2000 CHT-NB

Exit flow (veh) WHC-SB WHC-NB 1000

0 00:30 01:30 02:30 03:30 04:30 05:30 06:30 07:30 08:30 09:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 18:30 19:30 20:30 21:30 22:30 23:30 Hour

Figure 2. Exit Flow for the Flat Toll Case and exit flow figures in Figures 1 and 2, the queues at the entrances to the Cross Harbour Tunnel depict a similar pattern during the morning and afternoon peak periods. There is some 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, the Western Harbour Crossing is always queue free. 478 The Singapore Economic Review

Flat toll case- Southbound (Base case, 2002) 2500

2000

1500 EHC

CHT

1000 WHC Queued vehicles (veh)

500

0 00:30 01:30 02:30 03:30 04:30 05:30 06:30 07:30 08:30 09:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 18:30 19:30 20:30 21:30 22:30 23:30 Hour

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

Flat toll case - Northbound (Base case, 2002) 2500

2000

1500 EHC

CHT

1000 WHC Queued vehicles (veh)

500

0 00:30 01:30 02:30 03:30 04:30 05:30 06:30 07:30 08:30 09:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 18:30 19:30 20:30 21:30 22:30 23:30 Hour

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

We now compare the combined exit flows (both southbound and northbound) of each of the 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 exit flow is underestimated during the morning peak and interpeak periods (see Figure 5). The Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 479

Flat toll case (Base case, 2002) Exit flow pattern of the Eastern Harbour Crossing (Southbound + Northbound) 8000

7000

6000

5000 Modeled Observed 4000

Exit flow (veh) 3000

2000

1000

0 0123456789101112131415161718192021222324 Hour

Figure 5. Exit Flow Pattern for the Eastern Harbour Crossing, Observed Versus Modeled 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 modeled exit flow for the Western Harbour Crossing is slightly overestimated during the morning and 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 the predictive test. We observe that our modeled exit flows for the combined directional flows track 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 the total capacities of the three parallel tunnels (the Eastern Harbour Crossing, Cross Harbour Tunnel, and Western Harbour Crossing) within each hour, while collecting the lowest toll fee from tunnel users. As mentioned, all three tunnels that traverse the harbor are privately financed 30-year BOT projects. However, since the Hong Kong Government resumed control of the Cross Harbour Tunnel on 1 September 1999 upon the expiry of the BOT project, flexibility in toll setting comes into play. In the time-varying toll scheme, we assume the following:

(1) The (high) toll level for the newer Western Harbour Crossing is fixed, because if the younger toll operator is to recoup its investment, it will be unable to lower its toll. 480 The Singapore Economic Review

(It also turns out that the Western Harbour Crossing’s dual three-lane tunnel has spare capacity at all hours of the day, which suggests that the toll should not rise above its present toll rate). (2) The toll level for the Eastern Harbour Crossing is allowed to rise, that is, the tunnel is allowed to recoup its investment and excess traffic at this crossing indeed pushes up the 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 of the day because the Hong Kong Government regained its right to set tolls for all traffic upon the expiry of the BOT project.

Generally there are three cases in the scheme as follows. Case 1 During 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, if there is excess capacity, the toll is set at a nominal amount, to cover tunnel maintenance for instance (but not zero). Case 2 When 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 particular direction at a certain hour is less than the total capacities of the three tunnels, there are two possible outcomes.

(i) If queues build up at the Eastern Harbour Crossing, we increase the toll level for that tunnel to ward off excess traffic. (ii) If queues are built up at the Cross Harbour Tunnel, we increase the toll level for that tunnel to ward off excess traffic.

In this situation (Case 2), no queue will be carried over to the next hour. Case 3 In principle, it is possible that the total traffic in a particular direction during a certain hour will be greater than the total combined capacities of the three tunnels put together due to growth in traffic. However, our data indicate that the total traffic in one direction is always within the total capacities of the three tunnels, especially given that the newest tunnel, the Western 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 for tunnel maintenance to avoid a toll-free charging period. Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 481

Flat toll case (Base case, 2002) Exit flow pattern of the Cross Harbour Tunnel (Southbound + Northbound) 8000

7000

6000

5000 Modeled 4000 Observed

Exit flow (veh) 3000

2000

1000

0 0123456789101112131415161718192021222324 Hour

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

Flat toll case (Base case, 2002) Exit flow pattern of the Western Harbour Crossing (Southbound + Northbound) 8000

7000

6000

5000 Modeled Observed 4000

Exit flow (veh) 3000

2000

1000

0 0123456789101112131415161718192021222324 Hour

Figure 7. Exit Flow Pattern for the Western Harbour Crossing, Observed Versus Modeled 482 The Singapore Economic Review

Time-varying toll case - Southbound 70

40 EHC CHT 30

Toll fee ($) WHC

0 00:30 01:30 02:30 03:30 04:30 05:30 06:30 07:30 08:30 09:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 18:30 19:30 20:30 21:30 22:30 23:30 Hour

Figure 8. Time-Varying Toll (Southbound)

Time-varying toll case - Northbound 70

40 EHC CHT 30 Toll fee ($) WHC

0 00:30 01:30 02:30 03:30 04:30 05:30 06:30 07:30 08:30 09:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 18:30 19:30 20:30 21:30 22:30 23:30 Hour

Figure 9. Time-Varying Toll (Northbound) difference in this case (Figures 8 and 9) is that we follow the initial institutional set-up of charging a high toll of HK$37 for private cars that use the Western Harbour Crossing 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 Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 483

Time-varying toll case 5000

4000

3000 EHC-SB EHC-NB CHT-SB 2000 CHT-NB WHC-SB WHC-NB

1000 Incoming flow = Exit (veh)

0 00:30 01:30 02:30 03:30 04:30 05:30 06:30 07:30 08:30 09:30 10:30 11:30 12:30 13:30 14:30 15:30 16:30 17:30 18:30 19:30 20:30 21:30 22:30 23:30 Hour

Figure 10. Incoming Flow and Exit Flow for the Time-Varying Toll Case form.7 All queuing is eliminated with this toll set-up, and the incoming flow corresponds to the exit flow for all three tunnels regardless of whether or not the respective capacity is binding (see Figure 10). In short, the impact of switching from a flat toll to a time-varying toll results in the elimination of all of the existing tunnel queues. Society is made unequivocally better off as time 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 time from Table 4.3 of the Third Comprehensive Transport Study — Technical Report (Transport Department, 1999a,b,c) and extended to 2002 using the Composite Consumer Price 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 55,866 CHT 19329 0 19329 1,084,162 WHC 0 0 00 Total 20325 0 20325 1,140,029

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 toll at 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 the noontime hours before rising again to a $46 toll at 6:30 p.m. — being the peak of the afternoon peak — and falling again at nightfall. 484 The Singapore Economic Review multiplied by the 2002 value of time of HK$56.09 (US$7.19) per hour, the aggregate value of time that is saved is HK$1.14 million a day. Note that the total travel time savings presented in Table 4 exclude the travel time changes due to the changes in routes. The travel time change is small. With the flat toll, some vehicles actually detour eastward to the Cross Harbour Tunnel for its lower toll and the Eastern Harbour Crossing for its shorter queue. Under the time-varying toll, these vehicles will choose the tunnel that suits them best. Overall, this will balance out the extra travel time of the other vehicles detouring under the time-varying toll scheme. Motorists as a group are roughly just as well off in the time-varying toll case as with the flat toll scheme (see Table 5). Note that motorists as a group pay slightly more in terms of the toll payments; putting it another way, the road authority collects a little more in toll revenues: HK$0.05 million a day (1%) more than the total toll revenues of HK$5.70 million a day for the base case. Interestingly, the privately operated toll operator of the Eastern Harbour Crossing collects slightly more revenue, because the principal impact of implementing the time-varying toll is to divert the (long) queue from the centrally located Cross Harbour Tunnel to the two adjacent tunnels. Further, the fact that the Eastern Harbour Crossing’s time-varying toll for private cars increases from the flat toll of HK$15 to HK$20 between 5 and 6 p.m. and to HK$25 between 6 and 7 p.m. for northbound traffic means that there is excess demand for the Eastern Harbour Crossing, which further leads to increased revenues. Because the toll for the Western Harbour Crossing is fixed uniformly over time by assumption, the lower toll revenue collections reflect the fact that some of the traffic 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 Cross Harbour Tunnel. This suggests that the private operators of the Eastern Harbour Crossing and the Cross Harbour Tunnel should object less to the policy proposal of switching to a time-varying toll scheme. For all practical purposes, as there is no significant increase in toll revenues for the Hong Kong Government — the owner of the Cross Harbour Tunnel — the time-varying toll scheme should in principle encounter less political resistance from the public. Having said that, the successful adoption of the time-varying scheme would hinge on the distribution of gains and losses. Optimal tolling — despite the institutional constraints that tolls for tunnels which are privately 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 Tunnel Authority)

Toll Revenues from Toll Revenues from Changes in % Difference Flat Toll Time-Varying Toll Toll Revenues

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% Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 485 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 the tunnels would have a substantial positive external effect on the east-west traffic on both sides of the harbor. We argue that optimal tunnel tolling, coupled with the non-stop electronic toll collection that is already in place, could be the first step toward the implementation of electronic road pricing in Hong Kong (Hau, 2001, 2006a,b; Transport Department, 2001; Ho et al., 2005, 2007; Wong et al., 2005; Council for Sustainable Development, 2008).

5. Caveats and Extensions “One of the most dramatic market failures in the public sector has been the failure to use market principles to reduce the waste involved in the queuing at congested toll facilities, such as the San Francisco Bay Bridge and the Lincoln Tunnel. Applying market principles here involves collecting rents for the use of the facility at various times. Such rents should equate effective demand with the space available [authors’ emphasis]. In the case of the Lincoln Tunnel, for example, delays ranging up to a half hour and more are common. Persistent queuing can prevail over nearly every peak period for three to five hours. Most of the queuing could be eliminated by a surcharge (on top of the existing toll) which rises gradually to a brief peak and then declines gradually. By trial and error the surcharge would be increased at times when there has been regular significant queuing [authors’ emphasis] (and few or no significant gaps in the flow at that time of day over the previous period) and lowered when there have been significantly frequent gaps in the flow at that time. The only remaining 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, and a 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 onerous and the corresponding benefits much greater. It would be considerably easier if all motorists shifted their journeys by a few minutes, rather than a few motorists shifting their trips by an hour at a time. Our model essentially takes trip schedules as given, and thus efficient tunnel tolling effectively forces motorists to take an alternative route. It is likely that commuters, for instance, would react to a high toll by rescheduling, either by rising earlier or staying later in the office. Thus, our model does not take the morning departure time into account as in the model of Arnott et al. (1990). Extensions in this natural direction would involve the design and implementation of a new stated preference survey (Loo et al., 2008). 486 The Singapore Economic Review

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 tolls would free up resources and effectively increase the estimated time saving. The distance traveled and its corresponding costs (such as fuel costs) could be further modeled and calibrated by using the location of the zone centroids and route lengths. Finally, the value of 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 the Research Grants Council of the Hong Kong Special Administrative Region (Project No: HKU7126/04E). The authors would like to gratefully acknowledge the contributions of Yin Yu Choi, Wai Hung Chung, and Wai Yip Cheng with the data collection in the stated preference survey. Comments from Herbert Mohring on an earlier version of this paper are gratefully acknowledged.

Appendix A. Appendix on Elasticities The responsiveness of demand with respect to an attribute is defined as the percentage change in the quantity demanded (or in this case the choice probability) that results from a 1% change in an attribute (see McFadden, 1974; Domencich and McFadden, 1975). The disaggregate own-price elasticity of demand can be shown to be

PnðiÞ β , : E xink ¼ð1 P nðiÞÞxink k ðA 1Þ and the disaggregate cross-price elasticity of demand can be shown to be

PnðiÞ β , , : E xjnk ¼P nð jÞxjnk k j 6¼ i ðA 2Þ where P nðiÞ is the probability of respondent n choosing alternative i, xink is the attribute (in this case the toll fee) that is faced by respondent n for alternative i, and βk is the coefficient that corresponds to the independent variable (in this case the toll). The expected share of the group choosing alternative i for a group of N respondents is simply P N P ðiÞ P ðiÞ¼ n¼1 n : ðA:3Þ N Aggregate elasticities summarize the responsiveness of all of the respondents, rather than that of an individual respondent. Using the derived values, the aggregate elasticities can then be calculated by P N PnðiÞ P ðiÞE x E P ðiÞ Pn¼1 n jnk , : xjk ¼ N ðA 4Þ n¼1 PnðiÞ which is a weighted average of the individual level elasticities using the choice prob- PnðiÞ abilities as weights, where E 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. Estimation of Efficient Time-Varying Tolls for Cross Harbor Tunnels in Hong Kong 487

References Arnott, R, A de Palma and R Lindsey (1990). Economics of a bottleneck. Journal of Urban Economics, 27, 111–130. Burris, MW (2003). Application of variable tolls on congested toll road. Journal of Transportation Engineering, 129, 354–361. Council for Sustainable Development (2008). Report on the better air quality engagement process. February 2008, Hong Kong, pp. 1–25. Available at http://www.susdev.org.hk/susdevorg/ archive2007/en/councilreport/councilreport.htm, Accessed 13 June 2011. Domencich, TA and D McFadden (1975). Urban Travel Demand: A Behavioral Analysis. Amsterdam: North-Holland. Goodwin, PB (1992). A review of new demand elasticities with special reference to short and long run effects of price changes. Journal of Transport Economics and Policy, 26, 155–169. Hau, TD (1985). A Hicksian approach to cost-benefit analysis with discrete choice models. Economica, 52, 479–490. Hau, TD (2001). Demand-side measures and road pricing. In Modern Transport in Hong Kong for the 21st Century, AGO Yeh, PR Hills and SKW Ng (eds), pp. 127–162. Hong Kong: Centre of Urban Planning and Environmental Management, The University of Hong Kong. Hau, TD (2005a). Economic fundamentals of road pricing: A diagrammatic analysis, part I — Fundamentals. Transportmetrica,1,81–117. Hau, TD (2005b). Economic fundamentals of road pricing: A diagrammatic analysis, part II — Relaxation of assumptions. Transportmetrica, 1, 119–149. Hau, TD (2006a). Congestion charging mechanisms for roads, part I — Conceptual framework. Transportmetrica,2,87–116. Hau, TD (2006b). Congestion charging mechanisms for roads, part II — Case studies. Trans- portmetrica,2,117–152. Ho, HW, SC Wong, H Yang and BPY Loo (2005). Cordon-based congestion pricing in a continuum traffic equilibrium system. Transportation Research Part A, 39, 813–834. Ho, HW, SC Wong and TD Hau (2007). Existence and uniqueness of a solution for the multi- class user equilibrium problem in a continuum transportation system. Transportmetrica,3, 107–117. Hirschman, I, C McKnight, J Pucher, RE Paaswell and J Berechman (1995). Bridge and tunnel toll elasticities in New York: Some recent evidence. Transportation, 22, 97–113. Lam, WHK, ACK Poon and R-J Ye (1996). Optimization of tunnel tolls in land use and transport planning. Journal of Advanced Transportation, 30, 45–56. Loo, BPY (2003). Tunnel traffic and toll elasticities in Hong Kong: Some recent evidence for international comparisons. Environment and Planning A, 35, 249–276. Loo, BPY, SC Wong and TD Hau (2008). Choice or rank data in stated preference surveys? The Open Transportation Journal,2,74–79. May, AD and HEM Keller (1967). A deterministic queuing model. Transportation Research,1, 117–128. McDonald, JF, EL d’Ouville and LN Liu (1999). Economics of Urban Highway Congestion and Pricing. Boston: Kluwer Academic Publishers. McFadden, D (1974). The measurement of urban travel demand. Journal of Public Economics,3, 303–328. Mohring, H (1999). Congestion. In Essays in Transportation Economics and Policy: A Handbook in Honor of John R. Meyer, J Gómez-Ibáñez, WB Tye and C Winston (eds.), pp. 181–221. Washington D.C.: Brookings Institution. MVA Asia (1993). Travel Characteristics Survey — Final Report, Territory Transport Planning Division, Transport Department, Hong Kong Government, Hong Kong. 488 The Singapore Economic Review

Oum, TH, WG Waters II and JS Yong (1992). Concepts of price elasticities of transport demand and recent empirical estimates: An interpretative survey. Journal of Transport Economics and Policy 26, 139–154. Shum, J (2001). The optimal toll for cross harbour traffic, M.Sc. Dissertation (unpublished), Department of Civil Engineering, The University of Hong Kong, Hong Kong. Train, KE (2003). Discrete Choice Methods with Simulation. Cambridge, England: Cambridge University Press. Transport Department (1999a). Hong Kong Third Comprehensive Transport Study — Final Report, Transport Department, Hong Kong Government, Hong Kong. Available at http://www.td.gov.hk/ en/publications_and_press_releases/publications/free_publications/index.html. Accessed 13, June 2011. Transport Department (1999b). Hong Kong Third Comprehensive Transport Study — Technical Report, Transport Department, Hong Kong Government, Hong Kong. Transport Department (1999c). Hong Kong Third Comprehensive Transport Study — Technical Report Appendices Volume 1, Transport Department, Hong Kong Government, Hong Kong. Transport Department (2001). Feasibility Study of Electronic Road Pricing: Final Report, April. Available at http://www.td.gov.hk/en/publications_and_press_releases/publications/free_publications/ index.html. Accessed 13, June 2011. Transport Department (2002). The Annual Traffic Census, TTSD Publication No. 03CAB1, Hong Kong Government, Hong Kong. Available at http://www.td.gov.hk/en/publications_and_press_releases/ publications/free_publications/index.html. Accessed 13, June 2010. Vickrey, WS (1967). Optimization of traffic and facilities. Journal of Transport Economics and Policy, 1, 123–136. Vickrey, WS (1969). Congestion theory and transport investment. American Economic Review, 59, 251–260. Vickrey, WS (1996). Privatization and marketization of transportation. In Privatizing Transpor- tation Systems, S Hakim, P Seidenstat and GW Bowman (eds), pp. 221–248. London: Praeger. Wong, WKI, RB Noland and MGH Bell (2005). Editorial, special issue: The theory and practice of congestion charging. Transportation Research Part A, 39, 567–570. Yang, H and WHK Lam (1996). Optimal road tolls under conditions of queuing and congestion. Transportation Research Part A, 30, 319–332.