OPTIMAL TRANSIT PRICING WITH CROWDING AND TRAFFIC CONGESTION: A DYNAMIC EQUILIBRIUM ANALYSIS 1

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1 OPTIMAL TRANSIT PRICING WITH CROWDING AND TRAFFIC CONGESTION: A DYNAMIC EQUILIBRIUM ANALYSIS Introducton André de Palma, Ecole Normale Supéreure de Cachan Robn Lndsey, Unversty of Brtsh Columba Gullaume Monchambert, KU Leuven Economsts have long advocated congeston prcng as the best way to tackle traffc congeston. Yet congeston prcng s stll farly rare, and varous second-best polces for congeston relef contnue to gan attenton. A leadng canddate s to subsdze transt fares n order to attract people out of ther cars. Subsdzaton s poltcally popular but t has several lmtatons. Frst, reducng fares below margnal socal cost creates a deadweght loss from nduced trps and t contrbutes to crowdng whch s a serous problem n many ctes 2. Second, f transt s a poor substtute for drvng large fare reductons are needed to make a dent n traffc congeston. Thrd, f the own-prce elastcty of car trps s large then any potental benefts from congeston relef wll be largely offset by latent demand (Duranton and Turner, 20). Fnally, lowerng fares exacerbates transt defcts. Ctes vary wdely n ther fare polces. Many levy fares that are constant throughout the day. Others have adopted some degree of tme varaton ether as peak-perod surcharges (e.g., London and Washngton, D.C.) or off-peak dscounts (e.g., Sngapore and Melbourne). The man goal of ths paper s to analyze optmal fare polces when traffc congeston and transt crowdng are both present. We use a dynamc model that accounts for trp-tmng decsons and the evoluton of transt crowdng and traffc congeston over the course of a peak travel perod. The focus s on how transt fares should be set to smultaneously address traffc congeston and transt crowdng externaltes, and how the level and tme structure of fares affect overall effcency of the two-mode system. Lterature Revew There are many studes of second-best transt prcng n the presence of traffc congeston. 3 One of the frst s Glaster (974) who used a model featurng cars and buses, peak and off-peak tme perods, and parametrc cross-prce demand elastctes between each of the four mode-tme perod choces. Glaster showed that peak and off-peak fares should both be set below margnal socal cost. The peak fare may be below the off-peak fare, and ether fare can be zero or even negatve. Glaster and Lews (978) extended Glaster's (974) model to nclude a ral mode and congeston nteracton between cars and buses. They explored the potental benefts from second-best transt prcng n the Greater London area. Proost and Van Dender (2008) conducted a smlar analyss for London and Brussels usng a more elaborate model. These and other studes reveal the role of own-prce and cross-prce demand elastctes n governng optmal fare polcy. Nevertheless, ther approach s lmted by the use of dscrete peak and off-peak tme perods and parametrc elastctes, and neglect of transt crowdng. Tabuch (993) advanced the treatment of tme by usng the bottleneck model to descrbe travelers' trptmng decsons and the evoluton of traffc congeston on the road. However, he assumed that transt servce s provded by a ral system wth suffcent capacty to delver all passengers to the destnaton on tme and wthout crowdng. Hs model therefore features only a sngle fare, and cannot be used to study de Palma, Lndsey, Monchambert

2 tme-of-day fare varatons. Huang (2000) bult on Tabuch (993) by addng crowdng costs, but retaned the assumpton that transt delvers users on tme. Huang et al. (2007) relaxed ths assumpton by supposng that ral servce s provded on multple trans accordng to a tmetable. However, they dd not analyze optmal prcng for ether mode. Kraus (202) uses a smlar model to examne how transt usage depends on the prcng of roads. He gnores crowdng costs and assumes that tran fares are set accordng to frst-best prcng prncples. de Palma, Klan and Proost (205) and de Palma, Lndsey and Monchambert (205) do allow for crowdng, but assume that transt s the only travel mode so that frstbest transt prcng s de facto optmal. The Model The model ncorporates components of the models n Huang (2000), Huang et al. (2007) and de Palma, Lndsey and Monchambert (205). One orgn s connected to one destnaton by a road and a tran servce wth a separate rght of way. Utlty from travel s descrbed by a quas-lnear utlty functon U N, N g N s the number of car (automoble) trps, N s the number of ral trps, and g R A, where A s a composte numerare consumpton good. Functon ral trps are mperfect substtutes. R U s strctly quasconcave so that car trps and As n the Vckrey (969) model, trp-tmng preferences are descrbed by a pecewse lnear schedule delay cost functon. A traveler departng at tme t and arrvng at tme t a ncurs a combned travel tme and schedule delay cost of t a t t ta ta t, where t s desred arrval tme at the destnaton, s the unt cost of tme spent travelng, s the unt cost of arrvng early, and s the unt cost of arrvng late. Congeston on the road takes the form of queung behnd a bottleneck. The cost of a car trp departng at t and arrvng at t a s: where A0 tme t. A A0 C s the free-flow cost of a car trp, C t C q t t t q t t q t t t, qt s queung delay and To smplfy analyss, travel tme by tran s normalzed to zero so that t a be provded contnuously and at a constant capacty rate over a fxed tme nterval t t t. The cost of a tran trp at t s: 0 e C t C n t t t t t t, R R0 t s the road toll (f any) at t. Tran servce s assumed to t, 0 t e where where C R0 s the fxed cost of a tran trp (e.g., the tme cost of access and egress tme), nt s the number of users takng the tran at tme t, s a parameter measurng dsutlty from crowdng, and t s the fare at tme t. Users have heterogeneous preferences. 4 There are two user groups 5, and 2, that dffer wth respect to parameters,, and, but have the same values of / and t. Parameter values satsfy four condtons. Frst, 2 so that group 2 has stronger on-tme preferences than group. Second, 2 de Palma, Lndsey, Monchambert

3 / /. Ths mples that group 2 tolerates queung more than group. Group 2 arrves by car 2 2 closer to t than group, and creates a hgher margnal external congeston cost when queung occurs. Thrd, / 2 / 2. Ths mples that group 2 tolerates crowdng more than group, and arrves by tran closer to t than group. Fnally, / 2 / 2 so that the two groups value late arrval relatve to early arrval by the same rato. Tran loads are determned by the numbers of users n each group. Results Prelmnary results have been derved for the regmes shown n Table : Table : Admnstratve and prcng regmes Fare polcy Road toll Transt fare Frst-best optmum Frst-best Optmal tme-varyng Tme-varyng Second-best None Tme-varyng Thrd-best None Tme-varyng Flat fare None Flat Free transt None None The frst-best optmum and free transt regmes serve as benchmarks aganst whch the effcency of the other regmes s measured. In the frst-best, second-best and thrd-best prcng regmes, the fare can be vared freely over tme but t s anonymous n the sense that t cannot depend on whether a user belongs to group or group 2. In the frst-best prcng regme the road s prced optmally to elmnate queung at the bottleneck and the fare schedule s chosen to optmze welfare. In the second-best prcng regme the road s not tolled and the transt operator takes traffc congeston nto account when settng the fare schedule. By contrast, n the thrd-best prcng regme the operator behaves myopcally and neglects traffc congeston. Fnally, n the flat-are regme the fare s restrcted to be the same for all trans but the level of the common fare can be optmzed. Several general propertes of the regmes have been establshed. Frst-best optmum: In the frst-best optmum, for each group the numbers of trps taken by each mode are chosen to equalze ther margnal socal costs. Passenger loads are also dstrbuted across trans to equalze the margnal socal costs of each trp by members of the same group. Trans arrvng closer to t carry hgher loads. Frst-best prcng: In ths regme the road s optmally tolled and the fare can be vared freely over tme. Nevertheless, the frst-best optmum stll cannot be acheved unless 2. To see why, consder an early arrval perod and suppose group travels durng the nterval t ˆ 0, t and group 2 durng the nterval tt ˆ,. If 2, tran loads must decrease at ˆt n order to provde less crowded condtons for group 2, but users n group can then reduce ther trp costs by devatng from the optmum and takng a tran just 3 de Palma, Lndsey, Monchambert

4 after ˆt. Conversely, f 2 tran loads must ncrease at ˆt but group 2 users can then reduce ther costs by takng a tran just before ˆt. Reschedulng trps ths way can be deterred by ntroducng a sutable upward or downward jump n the fare at ˆt, but dong so upsets optmalty condtons for numbers of trps and modal splts. 6 Second-best prcng: When the road s not tolled, second-best prcng calls for a transt subsdy. The sze of the subsdy for each group depends on the traffc congeston externalty t creates, ts own-prce demand elastcty, and the cross-prce elastcty between modes. Because group 2 creates a larger traffc congeston externalty than group, the subsdy s hgher ceters parbus for group 2. Snce group 2 travels on peak-perod trans ths requres lowerng peak-perod fares more than off-peak fares. However, ths polcy s constraned by trp reschedulng ncentves as wth the frst-best prcng regme. Thrd-best prcng: Thrd-best prcng entals settng fares as f frst-best condtons apply. Fares are thus set as n the frst-best prcng regme, and too many car trps are made. Flat fare: In the flat-fare regme t s mpossble to prce dscrmnate ether between trans or between groups. The level of the fare s chosen to balance the costs of traffc congeston (whch calls for a low fare) and the costs of excessve transt trps (whch calls for a hgh fare). More precsely, the fare level s set to balance overprcng off-peak trps and trps by group, and underprcng peak-perod trps and trps by group 2. Although the model s farly smple, the presence of user heterogenety complcates the analytcs and precludes analytcal solutons. Numercal analyss reveals that, when car trps and transt trps are good substtutes, thrd-best prcng can be much less effcent than second-best prcng and t can also perform less well than an optmal flat fare. Dfferences between the regmes narrow usng more realstc assumptons about the degree of substtutablty between modes. A Numercal Example In ths secton we present a numercal example that s calbrated to yeld results broadly consstent wth emprcal evdence. Aggregate travel demand by each group s descrbed by a representatve ndvdual wth a lnear-quadratc utlty functon:, 2 a, b and d are postve parameters, and d U N N a N N b N N d N N, where R A R A R A R A b,,2. The and parameters affect the equlbrum only through the composte parameter and numercal values are assgned drectly to ths composte. Parameter values / for the varous components of the model are gven n Table 2. Group 2 dffers from group n havng a hgher choke prce on trps (.e., a2 a), and stronger on-tme preferences (.e., 2 ). Other preference parameters are assumed to be the same for the two groups. 4 de Palma, Lndsey, Monchambert

5 Table 2: Base-case parameter values a $60/trp cr cr2 $8/trp a 2 $75/trp ca ca2 $5/trp b b2 $0.02/trp 2 2 $0.0005/passenger d d2 $0.0667/trp 2 t mmateral $20/h g mmateral 2 $8/h s 5000 vehcles/h 2 $6/h Servce nterval, 0 e t t h Results are shown n Table 3. Table 3: Equlbra Regmes n order of ncreasng effcency Free transt (n) Flat fare (f) Thrd-best (3) Second-best (2) Frst-best (o) N R,690,69,603,693,603 N A,858,858,882,857,882 N R2 2,233 2,234 2,06 2,95 2,06 N A2 2,247 2,247 2,282 2,259 2,282 NR NR2 3,923 3,925 3,709 3,887 3,709 NA NA2 4,06 4,05 4,65 4,5 4,65 Fxed toll for group $0 -$0.04 $0 -$.82 $0 Fxed toll for group 2 $0 -$0.04 $0 -$.73 $0 ˆt Full prce elastctes (group, group 2) Auto, Auto , , , , , Ral, Ral , , , , , Auto, Ral 0.39, , , , , 0.45 Ral, Auto 0.28, , , , , 0.36 Welfare components Total costs $3,205 $3,22 $07,982 $09,865 $89,940 CS $84,023 $84,047 $8,26 $84,07 $8,26 CS 2 $33,844 $33,875 $28,480 $32,26 $28,480 Fare revenue $0 -$55 $0,795 $4,502 $2,408 Toll revenue $0 $0 $0 $0 $8,042 Surplus $27,867 $27,867 $220,537 $220,84 $238,579 Rel. effcency r de Palma, Lndsey, Monchambert

6 CS denotes the aggregate consumers' surplus of group. The relatve effcences of the regmes are compared usng the ndex r W W n / W O W n where W s socal surplus or welfare, ndexes the regme, n denotes the free-fare regme and o denotes the frst-best optmum. Free transt: In the free-transt regme there s no fare and no toll so that transt crowdng and queung congeston both mpose external costs. About 4,000 trps are made by each mode. Group 2 takes more ral trps than group, but group 2 restrcts ts trps to the 20 percent of trans that arrve closest to on-tme. The own-prce elastcty of demand for automoble trps s a lttle over one thrd: n lne wth estmates for short-run elastctes. The own-prce elastcty of demand for transt trps s a lttle hgher. A rule of thumb s that the elastcty s about one thrd. However, long-run elastctes can be consderably larger (Schmek, 206). Smlar elastctes obtan n the other four regmes. Thus, the elastctes can be nterpreted as applcable over an ntermedate tme nterval of perhaps -2 years. Cross-prce elastctes are about one thrd the magntude of the own-prce elastctes reflectng the fact that automoble and ral trps are rather mperfect substtutes. Frst-best optmum: At the opposte extreme to free transt s the frst-best optmum n whch optmal numbers of trps are chosen or each group by each mode. Because 2, the frst-best optmum can be supported by frst-best anonymous (.e., no dscrmnatory) prcng. Snce queung congeston s more severe than transt crowdng n the free-transt regme, the frst-best optmum entals somewhat fewer ral trps by each group and more automoble trps. Both groups end up worse off as apparent from the declne n ther consumers' surplus, but the losses are outweghed by transt revenue and substantal toll revenue. The overall gan amounts to $2,325 or about $3 per trp. Flat fare: The optmal flat fare turns out to almost zero ($0.04) and yelds no perceptble welfare gan. The reason for ths s that the benefts from settng a negatve fare to allevate queung congeston almost exactly balance the benefts from settng a postve fare to reduce excessve ral trps. Both externaltes are hgher or group 2, but wth a flat fare t s not possble to dscrmnate between the two groups usng the prcng mechansm. Thrd-best fare: In the thrd-best fare regme the fare s vared over tme to fully nternalze ral crowdng costs wthout consderng congeston on the road. The fare structure s therefore the same as for the frstbest optmum and no flat-fare component s added to or subtracted from the schedule. Consequently, the numbers of trps by each group usng each mode, prce elastctes of demand and consumers' surplus are dentcal to those n the frst-best optmum. Second-best fare: In the second-best fare regme the fare s vared over tme as n the thrd-best regme but the fare level s decreased to reduce automoble travel. Because group 2 creates a larger negatve traffc congeston externalty than group, the optmal downward shft s larger for group 2 than group. However, trp reschedulng ncentves prevent the unrestrcted second-best fare from beng mplemented and the overall welfare gan compared to the thrd-best prcng s very lmted. moreover, the second-best fare only yelds about one-seventh of the welfare gan acheved from frst-best prcng. The reason for ths s that (gven the parameter values chosen for the example) traffc congeston s more costly than transt crowdng and can only be allevated drectly by levyng a tme-varyng toll. 6 de Palma, Lndsey, Monchambert

7 Conclusons and Drectons for Future Research In ths paper we have taken a smple, frst-cut analyss at studyng the optmal level and tme structure of transt fares when transt crowdng and traffc congeston are both sgnfcant externaltes. Extensve senstvty analyss wll be requred to determne the degree to whch transt fare dscounts can be used to reduce peak-perod automoble trps that create the most congeston wthout overloadng the transt system and exacerbatng crowdng. REFERENCES de Palma, A., Klan, M., and Proost, S. (205), Dscomfort n mass transt and ts mplcaton for schedulng and prcng, Transportaton Research Part B, 7(), -8. de Palma, A., Lndsey, R., and Monchambert, G. (205), The economcs of crowdng n publc transport. Workng paper, November 7. Duranton, G., and Turner, M. A. (20), The fundamental law of road congeston: Evdence from US ctes, The Amercan Economc Revew, 0, Glaster, S. (974), Generalsed consumer surplus and publc transport prcng, Economc Journal, 84(336), Glaster, S., and Lews, D. (978), An ntegrated fares polcy for transport n London, Journal of Publc Economcs, 9(3), Huang, H.J. (2000), Fares and tolls n a compettve system of transt and hghway: The case wth two groups of commuters, Transportaton Research Part E, 36(4), Huang, H.J., Tan, Q., Yang, H., and Gao, Z.Y. (2007), Modal splt and commutng pattern on a bottleneck-constraned hghway, Transportaton Research Part E, 43(5), Kraus, M. (202), Road prcng wth optmal mass transt, Journal of Urban Economcs, 72(2-3), OECD (204), Valung convenence n publc transport. Techncal report, ITF Round Tables. Parry, I. W.H., and Small, K.A. (2009), Should urban transt subsdes be reduced?, Amercan Economc Revew, 99(3), Proost, S., and Van Dender, K. (2008), Optmal urban transport prcng n the presence of congeston, economes of densty and costly publc funds, Transportaton Research Part A, 42(9), Prud'homme, R., Konng, M., Lenormand, L., and Fehr, A. (202), Publc transport congeston costs: the case of the Pars subway, Transport Polcy, 2, Schmek, P. (206), Dynamc estmates of fare elastcty for U.S. publc transt, Transportaton Research Record, 2538, Small, K.A., and Verhoef, E.T. (2007), The Economcs of Urban Transportaton, Routledge. Tabuch, T. (993), Bottleneck congeston and modal splt, Journal of Urban Economcs, 34, de Palma, Lndsey, Monchambert

8 Vetch, T., Partrdge, J., and Walker, J. (203), Estmatng the costs of over-crowdng on Melbourne's ral system. 36th Australasan Transport Research Forum. Brsbane, Queensland, Australa. Vckrey, W. S. (969), Congeston theory and transport nvestment, Amercan Economc Revew, 59(2), Regular paper. 2 See OECD (204), Prud'homme et al. (202) and Vetch et al. (203). 3 For revews see Small and Verhoef (2007, Secton 4.5) and Parry and Small (2009). 4 Heterogeneous preferences are a crucal element of the model. Wthout heterogenety t s optmal, as n Kraus (202), to nternalze transt crowdng costs by varyng fares over tme. Devaton from frst-best prcng s lmted to applyng a unform subsdy for all trans so that the tme profle of the fare s the same as when car travel s effcently prced (or not an opton). 5 Lmtng heterogenety to two types not only smplfes the analyss but also facltates understandng the mplcatons of heterogenety n the varous preference parameters. 6 Arbtrage-lke behavour of ths sort would also occur n a model wth dscrete tran servce f the headway between trans s suffcently short. 8 de Palma, Lndsey, Monchambert