A Discrete-Choice Model Approach to Optimal Congestion Charge

Transcription

1 Dscrete-Choce Model pproach to Optmal Congeston Charge Stenar Strøm Jon Vsle

2 Ths seres s publshed by the Unversty of Oslo Department of Economcs P. O.ox 1095 lndern N-0317 OSLO Norway Telephone: Fax: Internet: e-mal: econdep@econ.uo.no In co-operaton wth The Frsch Centre for Economc Research Gaustadalleén 21 N-0371 OSLO Norway Telephone: Fax: Internet: e-mal: frsch@frsch.uo.no Last 10 Memoranda No 8/08 No 7/08 No 6/08 No 5/08 No 4/08 No 3/08 No 2/08 No 1/08 No 23/07 No 22/07 M. I. D Tomasso, Stenar Strøm and Erk Magnus Sæther Nurses wanted. Is the job too harsh or s the wage too low? tle Seerstad Exstence of optmal nonantcpatng controls n pecewse determnstc control problems Halvor Mehlum and Karl Moene Kng of the Hll Postonal Dynamcs n Contests Zheng Song, Kjetl Storesletten and Fabrzo Zlbott Rotten Parents and Dscplned Chldren: Poltco-Economc Theory of Publc Expendture and Debt Mchael Hoel Envronmental taxes n an economy wth dstortng taxes and a heterogeneous populaton Fedor Iskhakov Dynamc programmng model of health and retrement tle Seerstad and Sgve D. Stabrun Dscontnuous control systems Kjell rne rekke, Rolf Golombek and Sverre. C. Kttelsen Is electrcty more mportant than natural gas? Partal lberalzaton of the Western European energy markets Ger. shem and Taoyuan We Sectoral Income Ger. shem and Seung Han Yoo Coordnatng under ncomplete nformaton complete lst of ths memo-seres s avalable n a PDF format at:

3 Dscrete-Choce Model pproach to Optmal Congeston Charge by Stenar Strøm 1 and Jon Vsle 2 prl 10, 2008 bstract: We model the choce of transportaton mode n a smplfed Hotellng-lke cty, wth a fxed number of total travellers, fxed road capacty and wth no trade-off between when to travel and the tme spent n a ueue. person that chooses to take her own car wll nflct a congeston cost on all travellers. To get the travellers to nternalse these external costs, a congeston charge has to be mposed. We derve an optmal congeston charge wthn n a dscrete-choce framework, wth a benevolent government maxmsng expected tax-adjusted socal surplus. The congeston charge to be mposed on prvate drvng, beyond the opportunty cost eual to the fare on publc transportaton s shown to be a weghted average of a Ramsey-lke term (capturng the goal to rase publc revenue) and a Pgou-term capturng the envronmental cost of a person s prvate drvng. Ths property s smlar to the optmal envronmental tax derved by Sandmo (1975). However, the behavoural assumpton underlyng the present framework s ute dfferent from the standard theory of consumer choce adopted by Sandmo. JEL classfcaton: D11, H23, L13, L91 Keywords: Dscrete choce, urban transport, congeston, congeston charges 1 Unversty of Turn, Italy 2 Unversty of Oslo, Norway

4 1 1. Introducton and man fndngs Ever snce Duput and Pgou, economsts have been nterested n prcng congested facltes, as the phenomenon s both socally mportant nvolvng socal gans f beng mplemented, whle also provdng a very nce example of a negatve consumpton externalty. Sound economc prncples are rarely mplemented by the poltcal system. However, congeston charges or road prcng, correctng for the negatve externaltes or tme delays, are from tme to tme brought onto the poltcal agenda. Durng the last few decades a number of ctes have mposed such road charges or congeston tolls n order to reduce a costly over-utlzaton of restrcted road or drvng capacty durng peak perods of the day. Despte very strong opposton ex ante, most hard-boled opponents have admtted ex post that ths measure has been a success, nvolvng rather substantal economc benefts. In ths paper we characterse a second-best congeston toll wthn the context of a Hotellng-lke lnear cty, wth a fxed number of total travellers, lvng at the one extreme of the cty whle workng at the other. Each traveller has a bnary choce and nelastc demand ether takng her own car or takng publc transportaton. There are no substtuton possbltes related to when to travel, because we do not model the off-peak perods. Choosng the prvate alternatve (car) may gve rse to congeston or ueues, nflctng a tme cost on any other traveller, ncludng those takng publc transportaton (bus) as well. We employ a structural dscrete choce model wth random utlty and extreme value dstrbuted taste shfters. The model s smple but suffcently rch so as to offer a set of nterestng conclusons and testable hypothess. n optmal congeston charge s determned by a benevolent planner maxmsng expected tax-adjusted socal surplus, wthn a partal eulbrum framework. The company offerng transportaton s publc, and the revenue from chargng drvers wll be collected by the government, as well. Our model s formulated so as to allow us only to determne explctly a congeston charge on a fxed road capacty, whereas the opportunty cost the fare on publc transportaton can be charactersed by makng use of the remanng eulbrum condtons of the model. The second-best optmal congeston charge mposed on prvate drvng, beyond the opportunty cost eual to the fare on publc transportaton, s a weghted average of a Ramsey-lke term

5 2 (capturng the goal to rase revenue due to dstortonary taxaton elsewhere) and a Pgou-term capturng the congeston cost of a person s prvate drvng. Ths property s smlar to the optmal envronmental tax derved by Sandmo (1975), but here we adopt a set-up based on dfferent behavoural assumpton rather than the one underlyng standard theory of consumer choce. The paper s organsed as follows: In Secton 2 we revew some of relevant lterature. In Secton 3 the model, n partcular the demand sde, s outlned, whereas we n Secton 4 derve the optmal congeston charge. Secton 5 concludes. 2. bref revew of related lterature The ssue of desgnng congeston charges (or more general road charges) has a long tradton n the economc lterature. Even though the ueston has always been to correct for socal costs or negatve consumpton externaltes, the modellng approach dffers substantally among the large group of contrbutors. Early contrbutors, n addton to Duput and Pgou, are Vckrey (1969) and Walters (1961), whereas Newbery (1988a, 1988b) has provded nsght nto varous cost components, ncludng the congeston cost, to be ncluded n a general road charge. Second-best optmal congeston tolls have been derved wthn a general eulbrum settng by Marchand (1968) and Sherman (1971), both nfluenced by a paper, publshed n French, by Levy-Lambert (1968), where each traveller can choose between dfferent modes of transportaton, as n the present paper, but ther approach s based on standard theory of consumer choce. rnott, de Palma and Lndsay (1993) analyse a rch structural model of peak-perod congeston, wth specal emphass on the user s behavoural decson, but wth only one mode of transportaton. Ther model s related to varous prcng regmes wth bottleneck congeston n the sprt of Vckrey (1969). Contrary to our approach, they offer a choce as to when to travel, whereas our approach consders only the congeston toll wthout ntroducng off-peak perods, and hence no trade-off between when to travel and the tme stuck n a ueue. Our approach to modellng the choce of transportaton mode s ute dfferent from those early papers. We adopt a dscrete choce model; see en-kva and Lerman (1985). Each traveller has a choce between

6 3 two modes of transportaton. One of the modes (prvate use of cars) wll nflct external congeston costs on all travellers, ncludng those choosng publc transportaton as well. The cost due to congeston wll be ncreasng n the number of cars on the road. 3. Transportaton demand For expostory reasons we assume that the agents demandng transportaton n an urban area have the same determnstc utlty. traveller can choose between drvng her own car or use publc transportaton when gong from home to work and back. Let Un be the random utlty of traveller n when choosng transportaton mode or alternatve, where, ; = automoble, = bus or publc transportaton, whereas n denotes the ndvdual n 1,2,..., N, s supposed to be: (1) Un P gt n (2) Un Q gt n Here s a mode-specfc constant, P s the cost of drvng own car, whle Q s the publc transportaton fare. 3 Travel tme for alternatve s denoted T. Unobserved taste-shfters affectng utlty are n. These taste-shfters are assumed to be..d.extreme value dstrbuted wth standard devaton, and gven by n n, wth n beng ndependently and dentcally extreme value dstrbuted wth zero expectaton and unt varance. Dvdng through wth yelds U g n (3) P T n : a P T n U g n (4) Q T n : a Q T n 3 These prces are measured per unt dstance, whch for smplcty s assumed to be the same for all travellers n our lnear cty.

7 4 where : (Moreover g shows the wllngness to pay for one mnute reduced travel tme. b a: for,.) Defne also U U :. * n The cost of drvng own car s P c, where c s the prvate cost whereas s a congeston charge or the toll. Q s the net prce (net after margnal costs) of publc transportaton. For expostory reasons we set the total number of daly travellers, N, eual to 1. We wll assume that tme spent on travellng s a gven tme t plus travel tme due to congestons, whch s beng proportonal to the expected number of travellers usng own car. ll travellers wll be adversely affected by congeston. Expected travellng tme s gven by (5) T t h (6) T t where s the probablty that car s the preferred mode of transportaton. (The observed counterpart to ths probablty s the share of the populaton usng own car when travellng.) y assumng h, we get that prvate transportaton s more affected by congeston than publc transportaton. The probablty that agent n prefers to use her own car rather than takng the bus, n, s gven by v * * n n n v v (7) Pr(U U ) e e e where (8) v a P T (9) v a Q T and v s the expected utlty from choosng alternatve. The structure of the probablty n (7) follows from the assumpton that the taste shfters are..d. extreme

8 5 value across alternatves and ndvduals. The probablty of usng publc transport s gven by (10) 1 The model mples that traveller n s probablty of travellng mode depends on the aggregate probablty of travellng. ecause we have assumed (for expostory reasons only) that agents have the same determnstc utlty functon, all probabltes wll be eual. The model suffers from the assumpton of..d., whch can be avoded, for example, by assumng that the parameter s random across ndvduals. Ths wll be mportant when takng the model to data, but here we wll stck to the smplfyng assumpton that s the same for all travellers. Defne r : h 0. Then we get the followng dervatves and prce elastctes of the demand probablty, wth respect to the prces and Q, denoted e and e Q, respectvely: (11) 0 1 r (12) 0 Q 1 r (13) e : 0 1 r Q (14) e Q : 0 Q 1 r Q Q We note from the structure of the model that eq e 0. hgher toll rate wll, as we should expect, reduce the use of cars and a hgher fare rate on publc transportaton wll ncrease the use of cars. The smplfed demand structure s only for expostory reason. gan, when the model s taken to data, one has to allow for a more flexble pattern of response to prce changes.

9 6 4. Optmal congeston charge To derve the optmal toll rate or congeston charge, we let the benevolent government maxmse expected tax-adjusted socal surplus, gven by the sum of expected consumer surplus, expressed as V(,Q), and producer surplus, (,Q), from both publc transportaton and tax revenues, wth respect to the toll rate, gven the prce of publc transport Q. The objectve functon, wth m as a postve (exogenous) margnal cost of publc funds, s then: (15) W V(, Q) (1 m) (, Q) * v v (16) V(, Q) E[max (,) U ] ln(e e ) (17) (, Q) Q F Prces ( and Q) are net of varable margnal costs, whle F s a fxed cost related to toll statons and the publc transport system. From (7) (17) we then obtan m (18) Q 1 m ( / ) 1 m or m (19) Q 1 m ( e ) 1 m or Q m 1 1 (20) 1 m ( e ) 1 m where (21) : (h ) 0 We can nterpret as expected congeston cost. The frst term on the rght hand sde of ether (18) or (19) s the opportunty cost of car, whch s here the publc transportaton fare. The second term s a Ramsey prcng term, whch s due to the second best nature of the problem (dstortonary taxes). The thrd term s one related to the margnal externalty cost due to congeston. Ths last term may be denoted a Pgovan term. 4 In (20) we have the assocated Lerner-ndex, showng how the 4 See Sandmo (op.ct.).

10 7 optmal tax rate or the congeston charge rate should be stpulated. The optmal margnal cost of drvng one s own car, Q, s a weghted average of the Ramseyterm and the Pgovan term, wth weghts m 1 m and 1 1 m. We observe that Q 1 (22) lm m ( e ) Thus when the margnal cost of publc funds becomes extremely hgh, due to a costly tax system, the toll company should behave as a proft maxmsng monopolst, settng the toll rate so as to maxmse publc revenue from taxng prvate car use. The government wll care only about tax revenue, n addton to the tcket revenues from publc transportaton. (Note that ths rule corresponds to the nverse-elastcty-rule the less elastc s demand, the hgher can the charge be set.) On the other hand, when the margnal cost of publc funds becomes very low n the extreme case the government can use lump-sum taxes, then Q (23) lm or lmm 0 Q m 0 In ths case the toll rate should be eual to the sum of the opportunty cost, Q, and the expected margnal cost of congeston, as gven by the expected margnal congeston cost, evaluated for the socal optmum. The amount pad by taxpayers depends on expected profts, evaluated at optmal prces. Whether one should have publc transportaton and toll system depends on whether, evaluated at optmal prces, wll exceed zero. s an llustraton of what emprcs may yeld, let m 0,2, e 2 and 5 (per unt of trp per unt dstance). Then we can calculate the optmal toll n to be 1,091 Q 4,55.

11 8 Note that to fnd a closed form soluton for the optmal toll rate, say from (19), one has to take nto account that e and both depend on the congeston charge. ecause we have 1, we can also solve for the prce, Q, of publc transportaton that s consstent wth the optmal tollng. From (19) we observe that the optmal congeston charge or toll rate wll always exceed the prce of publc transportaton. To allow for the wllngness to pay for one mnute shorter travel to vary among travellers s straghtforward. Let n be the wllngness that ndvdual n s wllng to pay for one mnute shorter travel. The expected number of travellers usng own car, E[N ] euals n n E[N ] wth respect to the prce. Thus, E[N ]. Then, n. Let E denote the elastcty of n E[N ] E 0 E[N ]. Moreover the expected margnal congeston cost,, s now gven by / / n n (24) E[N ] nh E[N ] n 0 n E[N ]/ n E[N ]/ The euvalent to (19) s then m (25) Q 1 m ( E ) 1 m 5. Conclusons We have modelled the choce of transportaton mode n a very smplfed Hotellnglke cty, wth a fxed number of travellers, fxed road capacty and wth no trade-off between when to travel and the tme spent n a ueue. person that chooses to take her own car wll nflct a congeston cost on all travellers. To get the travellers to nternalse these external costs, a congeston charge has to be mposed.

12 9 We have derved an optmal congeston charge n a dscrete-choce framework, when a benevolent government maxmses expected tax-adjusted socal surplus, n an mperfect envronment wth dstortonary taxaton. The congeston charge to be mposed on prvate drvng, beyond the opportunty cost eual to the fare on publc transportaton s shown to be a weghted average of a Ramsey-lke term (capturng the goal to rase publc revenue due to dstortonary taxaton elsewhere n the economy) and a Pgou-term capturng the envronmental cost of a person s prvate drvng. Ths property s smlar to the optmal envronmental tax derved by Sandmo (1975). We get a smlar result as Sandmo dd, but our set-up s based on a dfferent behavoural assumpton than standard theory of consumer choce. References: rnott, R,. de Palma and R. Lndsay (1993), Structural Model of Peak-Perod Congeston: Traffc ottleneck wth Elastc Demand, mercan Economc Revew 83 (May), en-kva, M., and S. Lerman (1985), Dscrete Choce nalyss: Theory and pplcaton to Travel Demand, The MIT-Press, Cambrdge, Mass. Levy-Lambert, H. (1968), Tarfcaton des Servces à Qualté Varable pplcaton aux Peages de Crculaton, Econometrca 36 (July October), Marchand, M. (1968), Note on Optmal Tolls n an Imperfect Envronment, Econometrca 36, (July October), Newbery, D.M. (1988a), Road User Charge n rtan, Economc Journal 98 (2), Newbery, D.M. (1988b), Road Damage Externaltes and Road User Charge, Econometrca 56 (March), Sandmo,. (1975), Optmal taxaton n the presence of externaltes, The Swedsh Journal of Economcs 77 (1), Sherman, R. (1971), Congeston Interdependence and Urban Transt Fares, Econometrca 39 (May), Vckrey, W. (1969), Congeston Theory and Transport Investment, mercan Economc Revew 59 (May), Walters,.. (1961), The Theory and Measurement of Prvate and Socal Cost of Hghway Congeston, Econometrca 29 (October),