___ IS (070 Pollcy Research
WORKING PAPERS
Public Disclosure Authorized Infrastructureand Urban Development Department TheWorld Bank December1992 WPS1070 EconomicFundamentals of RoadPricing Public Disclosure Authorized
A DiagrammaticAnalysis Public Disclosure Authorized TimothyD. Hau
Most economists agree that road pricing benefits society by curtailingcongestion. Efficiency analysis demonstrates why the Public Disclosure Authorized public rejects congestionpricing. A dedicatedroad or transport fund is more viable whenthe road users are chargednot only for the damagecaused by heavyvehicles but also for congestion.
PolicyRearchWoding PapsdismLataeth rmdingsofwoiprogrcsatncountgethccxchangeofideaunangB nkstaffand til othas iMacenedin de-volopnelttssues. Them papcgs cary thenanm of die autho, reflactealy theirviews, znd shel be usedand citedaccordingly. hefindn.inu cprcsttion, andconduws aretheauthoz' own Tley shouldnoto atttbutedtotheWeWd Bank, its Boardof Directos,its managment,or any of itsmember counties. PolicyResearchl
Transport|
WPS1070
This paper - a product of the TransportDivision, Infrastructure and Urban DevelopmentDepartment'- is part of a larger effort in the departmentto evaluateoptions for chargingfor roaduse. Copiesof the paper are availablefree from the WorldBank, 1818H StreetNW,Washington, DC 20433.Please contact Jennifer Francis,room S 10-063,extension 31005 or 35205 (December1992, 96 pages). Hau presentsa conceptualframework for road On urban roads with indivisibilitiesand pricing based on a rigorousdiagrammatic - but diseconomiesof scale, efficient pricing may nonmathematical- frameworkderived from curtail the extent of profitableundertakings. On first (economic)principles. His analysisof rural roads with indivisibilitiesand economiesof traditionalarguments about road pricing shows scale, marginalcost pricing can produce short- why implementingcongestion pricing as prac- run profits.Economic efficiency is enhancedby ticed in the past has encounteredobstacles. pursuingoptimal pricing in the short run and Partly, it is because both types of road users- optimal investmentin capacity in the long run. the tolled and the tolled off (thosewho avoid the The rule is to implementshort-run marginalcost road to shun a toll) - are shown to be worse off pricing while varying road capacity over the long under a constantvalue of time, exceptfor the run. government.And when differencesin time valuationare taken into account,primarily those Insightsby Newbery, Small,and Winston- with very high time values are better off. Unless about the economicimplications of the extensive congestiontoll revenues are earmarkedand damage that heavy vehiclescause to roads - travelersperceive that the money is channeled enrich the basic Mohring model. Chargingfor back in reduced taxes, lower user charges,or both the external and variablecost of road improvedtransport services, neither the tolled damage,by assigninga fee basedon vehicle nor the tolled off will support road pricing. Only weight per axle, can help cover deficits arising where thcre is hypercongestionis everyone from road congestion. better ofi with congestionpricing. Even if a road networkis broadly character- In the absenceof scale economiesor ized by increasingreturns to scale in building diseconomies,the level of economicprofits- and strengtheningroads, the deficit could be toll revenuecollections less a road's fixed and closedby diseconomiesof scope.A road net- non-use-relatedcosts - serves as a surrogate work that accommodatesboth cars and trucks market mechanismindicating that a road should costs more than the sum of an autos-onlyand a be expandedor downsized.The decision to let (smaller)trucks-only road system.So the surplus roads deteriorateover time is iiself an act of associatedwith diseconomiesof scope offsets d;sinvestment. the potential loss associatedwith scale-specific economies.A dedicated road or transportfund is Hau shows that if a road authoritylevies all the more viable because road users are economicallyefficient chargesfor congestion,it chargednot only for the damagecaused by is possibleto make money on a road. Roads can trucks and heavy vehiclesbut also for conges- be profitablein urban areas in the long run tion. becauseland rents are high; congestiontolls reflect the associatedhigh opportunitycosts.
T'he Policy Research Working PaperSeries disseminates the fidings of workbyder way in teBank. An objective of the seDies is to get these findings out quickly, even if presentations are less than ftuly polished. 'Me fndings, interpretations, and conclusionls in these papers do not necessarily represent of ficial Bank policy.
Produced by the Policy Research DissemiinationCenter Economic Fundamentals of Road Pricing: A Diagrammatic Analysis
by
TimothyD. Hau
Tansport Division Infiratructureand Urban DevelopmentDepartment The WorldBank Economic Fundamentals of Road Pricing: A Diagrammatic Analysis
by
Timothy D. Hau
CONTENTS
ACKNO.lEDGvMENn ...... iv
ABSTRACT ...... , v
I. CoNCEnUAL GiDELES ...... 3
H. CAusE OF CONGeBsTION:PuBLIC/PiVATE OWNERSP OF RoAS ...... 4
HI. FOuNDATIONSOF RoAD CONGESTION- THE CLASSICALCASE (IN TH SHORT R.UN) ...... 8
IV. DEI@DAND AM SUPPLY...... 10
V. VERYCONoEsTED RoA AMD BorLNEcKS ...... 12
VI. THL WELFAE PACTOF RoAD PRicIN ...... 13
VI. THE EFFEC ON TH TOLLD, TH ToED OFF AND THEUNTOLLD ...... 13 Policy nIplications ...... 17
V L. SHORTRUN EQULMRIUM...... 18
IX. CONVERGENCE TOWARDS LONG-RUN EQUILBRUM UNDER CONSTANT RiIU.Ns ...... 20
X. Op*M>L NvsT ...... * *.. 22
Xl. LoNo-RUN VS. SHORT-RUN MARGNAL COSTPRICING...... 24
XI. TRADE-OFFBETWEEN INDiDUAlS' Tm AND EASURY AccouNTs ...... 25 xm. FnsT-BEsT OPTMAL PRICING AND INvSTmEN RULES...... 28 Empnjri1Considerations...... 28 The First Rule -- The Optimal Pricing Rule ...... 28 The Second Rule -- The Optimal Capacity Rule ...... 29 Perspective on the Result ...... 29
XlV. RELAXATIONOFASSUMPTIONS ...... 31 1) Differences in Time Valuation .. 31 2) Demand Variability and Peak-Load Pricing ...... 32 3) Indivisibilities ...... 4...... Optimal Pricing and Investment with Indivisibilities: An Example ... . 35 4) Returns to Scale ...... 37 A. Economies of Scale and RuraRoads ...... 38 B. Diseconomies of Scale and Urban Roads ...... 39 C. Diseconomies of Scale and ndivisibilities ...... 41 D. Economies of Scale and Indivisibilities ...... 42 E. The Extent of Indivisibilities vs. Divisibility and Their Effects on Scale (Dis)economies ...... 44 F. Empirical Evidence on the Scale Economy Issue ...... 45 G. Recent Results on Cost Recovery ...... 47 5) Vanability of Road Thickness .. 51 The first rde -- the optimal pricirg rule ...... 52 The second rule -- the optimal capacity;,xle...... 52 The third rle -- the optimal durability rue ...... 52 Economies of Scope vs. Diseconomies of Scope...... 53
XVh. SUMMAY AMDCONCLUSIONS...... 56 A. The Role of a Road Fund ...... 61 B. The Role of a Transport Fund .. 62
APPElDMI: ...... 65 Measurement of the Welfare Impactof Road Pricing ...... 65
FIGuE ...... 71
REFRENCES ...... 89
LIT OF FIGURS
Figure 1 Derivation of a Travel Time-Flow Curve of an Urban Highway .73
Figure 2 Derivation of the Marginal Cost Curve and Congestion Tol .... 74
Figure 2(a) 'Dynamic' Phenomenon of Traffic Growth: The Relaxation Effect ... 75
Figure 3 Welfare Impact due to the Itroduction of Road Pricing in the Peak Period: Short-Run Marginal Cost Pricing .76 Figure 3(a) Welfare Impact due to the Introductionof Road Pricing in the Peak Period: Short-RunMarginal Cost Pricing 'Hypercongestion' Case ...... 77
Figure 4 Effect of the Introductionof Road Pricing in the Peak Period on the Off-Peak Period ...... 78
Figure 5 Introducing the (Short-RunAverage) Fixed Cost, SRAFC, of a Road, Short-RunOptimal Toll with Economic Profit ...... 79
Figure 6 Long-RunEquilibrium of an Optimally DesignedRoad with Both Optimal Pricing and Optimal Investment ...... 80
Figure 7(a) ConstantReturns to Scale with Road Divisibility: Doubling Optimal Road Capacity, K, and Traffic, Q, Result in Doubling Fixed Cost, Variable Cost and Total Cost (FC, VC, TC) and Toll Revenues (t*- Q*) ...... 81
Figure 7(b) The Relationshipbetween Short-RunAverage Total Cost and Long-Run Average Total Cost and Marginal Cost with Perfect Road Divisibility and Constant Returns to Scale ...... 81
Figure 8(a) Road lndivisibilitiesunder Constant Retums ...... 82
Figure 8(b) Optimal Pricing and Investmentwith Indivisibilities: Expansion from a 2-Lane Road to 4-Lane Road ...... 82
Figure 9 Economies and Diseconomiesof Scale in the Provision of Road Capacity with the Growth of Travel Demand ...... 83
Figure 10 Doubling the Number of Streets - Road Capacity -- Results in Quadruplingthe Number of Intersectionsand Traffic Lights and DoublingWaiting Time .84
Figure 10(a) Original Scenario-Existing Street Configuration .84
Figure 10(b) Final Scenario - Number of Streets are Doubled .84
Figure 11 Diseconomiesof Scale: Urban Road Network with Perfect Divisibility ...... 85
Figure 12 Economiesof Scale: Rural Roads with Perfect Divisibility . .86
Figure 13 Dereasing Returs to Scale and Extent of Indivisibilities . .87
Figure 14 Increasig Returns to Scale and Extent of Indivisibilities. 87 ACKNOWLEDEMENTS
A rudimentaryversion of this resech paper was pented at the 2nd InternationalConference on Privatizationand Deegulaton, Tampere,Finland, June 16-20, 1991. Part of the findingsof this researckwas presentedat the 6th WorldConference on TransportResearch, Lyon, France,June 29 July 3, 1992; Seminaiiode TarificacionVial 1992 (Road Pricing Seminar), sponsored by theMinistr* of Trnsport, Santiago,Chile, July 20-21, 1992; the Symposiumon RoadPricing, sponsoredby the SwedishNational Road Adminisation and the Departmentof RegionalPlanning, Royal Institute of Technology,Sigtuna, Sweden, November9-10, 1992; and the Southwest CongestionPricing Conference,sponsored by the Universityof Houston's Center for Publih Policy and Citzens AdvocatingReponsibl Tansportation, Houston, Te.as, January 4-5, 1993. Commentsby participantsat theseplaces are gratefullyacknowledged. I have benefittedfrom the discussionsand commentsof manypeople on the controversalsubject of road pricingand I take this opportunityto thankWilliam. Vickrey, Michael Beedey, Kiran Bhatt, Richard Bird, Andrew Evans, Stephen Glaister, PhilGoodwin, Jose G6mez-Bdflz, Chris Hendrckson,Peter J. Hills,Jake Jacoby,Jan Jansson,Peter Jones, TheodoreKeeler, DamienKulash, Odd Larsen, HenningIauridsen, DouglassB. Lee, Peter Mackie,Anthony May, RobertMcGillivray, Herbert Mohring, Richw-d Musgave, YewKwang Ng, Esko Niskanen,Tae Oum, Farideh Ramjerdi,Gabniel Roth, KennethSmall, Antti Talvitie,Tom Tietenberg,Burkhard von Rabeaauand Sir AlanWalters. I am also gratefulto World Bank staff, includingEsra Bennathan,Philip Blackshaw, Jose Carbajo, Anthony Churchill, Shanta Devarajan, Asif Faiz, JeffreyGutman, Clell Harral, Ian Heggie,Chris Hoban, Gregory Ingram, FridaJohansen, Kyu Sik Lee, WilliamMcCleary, WilliamPaterson, Richard Scurfield, Zmarak Shalii and Larry Summers.
Supportsevices from Marat Callan,Pam Cook, Barbar Gregory,Beatrice Sito and Gabriella St_imetz are gratefullyackmowledged.
Theseacknowledgements do not meanthat they sharein the viewsexpressed in this paper. I alone etainfull responsibilityfor its contents.
TimothyD. Hau v -
ABSTRACT Economic Fundamentalsof Road Pricing: A DiagrammaticAnalysis by Timothy D. Hau
This paper presents a conceptual framework for road pricing based on a rigorous diagrammatic -- but non-mathematical-- framework derived from first (economic)principles. It throws light on congestion pricing systems and issues surrounding short-run and long-nm marginal cost priing, scale economies and diseconomies,indivisibilities, road durability, the peak-loadproblem in urban transport and the financial viability of the public provision of road services. The paper integrates the ideas of Mohring, Strotz, Vickrey, Walters, Keeler, Small, Winston and Newbery into a single analytical framework. Analysisof the traditionalroad pricing argumentsdemonstrates why congestionpricing as practiced in the past has understandablyencountered obstacles to implementation. This is partly because both types of road users, the tolled and the tolled off (those who avoid a road in order to shun the toll), are shown to be worse off - with the exception of the governL -nt - under a constantvalue of time. Even if differences in time valuation are taken into account, it is still essentially the case that primarily those with very high time values are made better off. Unless congestion toll revenues are earmarked and travellers perceive that the money is channeledback in the form of reduced taxes, lower user charges or improved transport services, neither the priced nor the pnced off would support road pricing. It is only in the case of hypercongestioncan congestionpricing be shown to make everyone better off.
In the absence of scale economies or diseconomies,the level of economic profits, i.e., toll revenue collectionsless the fixed and non-use related costs of a road, serves as a surrogate market mechanismindicating that a road ought to be expanded or downsized. The decision to let roads deteriorate over time in and of itself is an act of disinvestment. Further, it is shown graphically that if a road authority were to efficiently charge for congestion, it is possible to make money on the road. Profitable roads arise in urban areas in the long run because land rents are high and congestion tolls reflect the high opportunitycosts. Yet, efficient pricing in the presence of both indivisibilitiesand diseconomiesof scale in urban roads may curtail the xent of profitable undertakings,whereas pursuing marginal cost pricing under the restrictive - vi - conditionsof bothindivisibilities and scaleeconomies in rural roadscou d resultin profitsin the short run. Economicefficiency would be enhancedif optimalpricing were pursued in the short run and optin'l investmentin capacitywere pursuedin the long run. The rule is thereforeto implementshort-run marginal cost pricingwhile varying road capacityover the long run.
Recentextensions by Newbery,Small and Wimstonhave enrichedthe basic Mohrng modelthat this paperdevelops diagrammatically by incorpomtingthe fact that heavy vehicles are responsiblefor extensiveroad damages. Chargingfor both the extemaland variablecost of road damageson a vehicleweight per axle basis can help cover the deficit that may arise from congestiontolling. Evenif a road networkis broadlycharacterized by increasingreturns to scalein road construction'suse and strengthening,the deficitcould be closedby diseconomies of scope. The existenceof scopediseconomies in highwaysmeans that a road networkthat accommodatesboth loadingand trafficvolume, as founduniversally, costs more than the sum of an autos-onlyand a (smaller)trucks-oniy road system. Hence, the surplusassociated with diseconomiesof scopeoffsets the potentialloss associatedwith scale-specificeconomies. The viabilityof a dedicatedroad or transportfund is enhancedby the fact that the roadpavement is chargedin two dimensions:once when traffic flow creates congestion and anotherwhen traffic loadingscause road damages. EconomicFundamentals of Road Pricing: A DiagrammatiAnalysis
1. is major conurbationsof both developedand developingcountries, congestionis inctssantduring the peak periodsand oftentimesthe interpeak.Y/Yet the traditionalmethods of effectivelycurtailing congestion are few, and their usefulnesslimited. On the supplyside, the expansionand improvementof roadsis restrictedby increasinglytight fiscaland physical onstraints. On the demand side, however, the problem can be addressedby pricing or regulation. This paperuses the pricing(or market-based)approach to grapplewith congesrion becauseof its inherentflexibility and power of discrimination31 The aim of road pricing is to internalizethe externalitiesgenerated by road use. My focushere is on removingthe external effects caused by motorists by charging for congestionand road damage externalities. Congestionis recognizedas an importanttype of externalityfrom vehicle usage in both developedand developingcountries in thatit representsa large shareof total esumatedroad use costs (Newbery(1988bc, 1990) and Newbery,Hughes, Paterson and Bennathan(1988).I
2. This researchbegins with a seriesof twopapers on road pricingin theoryand practice. The first paper presentsa conceptualframework for road pricing. It gives an interpretative abridgmentof the literatureon the theoryof optimalpricing, investment and durabilityof roads
By the tum of the centuy, moryo tan four-fifthsof the world's most heavily-ppulted agglomeratonsare proect to be in tho developingwodd (WorldBank (1991), p. 3 andp. 22).
ZI Thoregulaoy approachon quanity - the so-calledcommand and controlmeasures - suffersfrom its inability to provide correct market sipals to induce the most efficient trips to be undertaken. In contrast to priing incentives,it gfferates virbtallyno revenuesfor the publicsector (see Pozdena, Schmidt, and Martin (1990)).
3/ The totalcost of undetkWnga trip - both 'private'and 'social' - involves: 1) congestion(which is bomeby roadusers); 2) pavementwear (which is coveredby the roadauthority); 3) air and noisepollution; and 4) costs of accidents(both of whichare bomeby societyat large);in additionto 5) vebieleoperating costs. 'Social cost' (i.e., bothinta andextenal costs)here are distinguishedfrom 'pivate costs'(i.e., intanal costs),the ltter of whichare self-financd. Thosetaveller-borne interal costsinclude: 1) vehicleopeating costsas weol as 2) tim costs and delayin conested traffic(Newbery's definition of roaduse costsrefers only to the extel cost.) This paper dealsmainly with the marginalcost pricig of congestionand pavementwear chargesas opposedto margial socialcost picig, whichis definedto include- in additionto privatecosts - all th arnal costsof coagwestion,air polluton,noise pollution, accidents, rod damagesand externalities.Hence, stictly speing, 'congestionpricing' refers solely to the pricingof congestionextenality whereas 'road pricing' refersmore genally to the priing of al externalitiesfrom mobile sources. - 2 - based on the works of Herbert Mohring, Robert Strotz, William Vickrey, Alan Walters, Theodore Keeler, Kenneth Small, Clifford Winston and David Nowbery.4/ It aims to integratetheir ideasand principlesinto a singleanalytical framework. I am convincedthat the presentation of a rigorousand unified diagrammatic -- but non-mathematical -- framework derivedfrom first economicprinciples casts important light en congestionpricing systems and on issues surroundingshort-run and long-runmarginal cost pricing, scale economiesand diseconomies,indivisibilities, the poak-loadproblem in urbantransport, optimal road durability, financialviability and cost recoveryin the publicprovision of road services. Whilethere are severalautomatic road user chargingand electronictoll collectionsystems in use in parts of Norway,Italy, France, and the UnitedStates, as wellas billsin parliamentto implementvarious forms of road pricing in Santiagoand Stockholm,one congestionpricing systemis currently operating(Hau (1992)). That is Singapore'sArea LicensingScheme, which is now in the processof beingconverted into the ElectronicRoad Pricing System, to be operationalin 1995 (see Watsonand Holland(1976, 1978),Behbehani, Pendakur and Armstrong-Wright(1984)). Even so, the charging of vehiclesby daylight hours in Trondheim,Norway (with further differentialpricing for electronictag users)since 1991could be regardedas a rudimentaryform of congestionpricing.
3. Recent technologicalbreakthroughs in automaticroad user charging have brought electronicroad pricing nmuchcloser to reality. Thus, the secondpaper (Hau (1992))in this researchpresents a taxonomyof alternativetechnologies of congestionpricing. It drops the crucialassumption that the implementationof short-runmarginal cost pricingis costless(Hau (1990)). In particular,it comparesand contraststhe methodsof manualversus electronic chargingschemes. Electronicapproaches are analyzedprogressively by discussingfirst, the increasinglypopular electronic toll collection mechanisms which incorporate recent technological advances *n automatic vehicle identification- commonly known as AVI - as well as smart cards. The ariateness and applicabilityof electronictoll collectionmechanisms to
I owe much intellcul debt to theseauthors and I am very gratefil for the opportunityof discussingmy work with severalof them -3- electronicroad pricing are thendealt with. The companionpaper also analyze%the relative cost° effectivenessof each technologyand performsbenefit-cost analysis where data permits. The implicationsof usingeach of themetechnologies for relievingcongestion are discussedand policy recommendationsare drawn.
I. CoNcEPTuALGuIDELNS
4. Rising real incomes result in increased aspirationsfor the ownershipof private automobiles.Barring major restraint measures, an increasingnumber of motorvehicles means that travel demandswells concomitantly. Becausemunicipalities are findingit increasingly difficultto financenew road construction and improvements, the rate of growthof trvel demand outstripsthe growthof road capacity(Hau (1988)). Transportplanners, with limitedoptions available,find it very difficultto combateffectively mounting traffic problems in the face of increasingurbanization. The resultingtraffic explosion is an illustrationof Parkinson'sLaw or Downs'law of peak-hourexpressway congestion, in whichcommuter traffic ascends rapidly to the level of new capaity in urban areas (Downs(1962), Mohring (1965)). Traffic engineers have long been familiarwith this "fundamentallaw of highwaycongestion" in whichlatent demandexpands to fill the gap createdwhenever highway capacity is improved(Small, Winston and Evans (1989), p. 85). In this section,I set up the conceptualguidelines that allow authoritiesto curtailtraffic congestion in an efficaciousmanner at the same time as satisfying the WorldBank's general guidelines for publicsector projects and urban transport policy (World Bank's OperationalManual Statement No. 2.25 (1977),World Bank'sUrban TransportPolicy Paper (1986),and Julius and Alicbusan(1989)).
5. In a nutshell,the essentialprinciples include: 1) implementingshort-run marginal cost pricing (short-runefficiency) to generate maximumnet benefits for society: efficiencypricirg; 2) undeaking investmentin infrastructurewhenever the additionalbenefits exceed the costs (long-ran efficiency)to society of doing so: econonuc iabilty; - 4 - 3) investingin transport serviceswhen benefitsexceed costs; promotingpublic transportservices especially when revenues exceed costs: financialviability; 4) maintaining'fairness' amongbeneficiaries, for example,via benefittaxation - equity -- wherepossible; and 5) usingpricing and cost recovery policies to improvethe efficiencyof managingthe public sector - cost-effectveness and managerial efficiency - if possible.
H. CAuSEOF CONGESTION:PUBLIC/PRIVATE OWNERSHP OF RoADs
6. Roadswhich are infrequentlyutilized possess the characteristicof nonrivalconsumption amongusers and are traditionalexamples of publicgoods. Joint consumptionmeans that roads yield services that are simultaneouslyenjoyed by more than one user, without substantial detrimentto the satisfactionof others. If roads are totally nonrivalrous,then neoclassical economicprinciples dictate that roadsought to be providedfor by the publicsector and financed from generalrevenue taxation (and perhapsland valuetaxation), fully takinginto accountthe socia opportunitycost of publicfunds. Onthe otherhand, roads which are heavilyutilized have the nature of rival consumptionamong users and are called congestedpublic goods. These varisble-usecongested public facilities then approximateto varyingdegrees the rival natureof privategoods. Privategoods are of courseprovided contingent upon payment, excluding those whoare not willingto pay for them. Alas, withfree accessto roads,people are not banredfrom the use of scarce services,resulting in overuse. Hencemarket failure due to nonexcludabiliy cals for governmentalintervention in the form of better designedroad user charges and motivatingcharges which correct for externalities. (the failure of the voluntary pricing mechanismdue to nonexcludabilityis referredto as the 'free-riderproblem' in thepublic finance literature(Boadway and Wlldasin(1984), Chapter 3)). Note that it is the traditionalinability to exclude motorsts from the use of crowded streets that is die cause of market failure. However,when refinementsin automaticvehicle identfication technology become even more cost-effective,automatic road user chargingmeans that marketfailure is somewhatovercome. This is becausepriccs are thenable to reflectboth the intensityof demandand the true economic cost of road use. The problemsof nonexcludabilityand preferencerevelation would then be - 5 - virtualy eliminatedand the advantagesof price incentivesreaped. Thus the standardpublic financetext (see,for example,Musgrave and Musgrave(1989, Chapter 4)) arguesthat both the
nonrval consumptioncharacteristic of publicgoods and the nonexcludability- or rather, costly excludability-- of congestedpublic goodsare causesof marketfailure, calling for government intervention.Hence, roads which possess the attributeof congestedpublic goods, and thushave a partally rival consumptioncharacteristic, ought to be treatedby the relevantgovernmental authoritiesas mixedor impurepublic goods,if not privateor clubgoods (Buchanan(1965)).
7. The foregoingdiscussion leads naturally to the transport economist'sdefinition of congestion. That is, as more and more vehiclesjoin a traffic stream, the travel times of all motoristsmaking trips are raised, resulting in delay to all. In essence, the congestion phenomenonis one of excessdemand, given a fixedroad network.
8. Yet the fundamentalreason why congestionoccurs so ubiquitouslyis becauseproperty nghts are not clearlydelineated, yielding market failure. If roadsare both privatelyowned and competitionprevails in the provisionof roads, usage wouldbe (Pareto)optimal in the absence of scale economies(Knight's conjecture (1924))Y If roads are not privatelyowned, access to and usage of roads are effectivelyfree to the traveller,resulting in excessiveuse of those roadswhere taffic is heavyenough to producesignificant adverse interaction among vehicles.
5/ Weknow that comption rsuls in conomcefficiency. in the contextof roads,owners maxim the return to thir landby settingdo pic equato theshort-nm marinal cost of production,where the difference between the short-n maginadcost adthe averg vaiable cost represa the quasi-entto the fixedfactor of produon. CM notio of qi-ent is discussedin footnote29.) Thisis the analogto societymimicking the decetalized competitiveladowner' no maximizg behaviorby chari the optimal(Pigouvian) toil - the divergencebeween maI cost d averg variablecost (seedio n in SectionVm and the proofof this conecturein Vwkrey(1968)). Kniht furter arguesthat the caseof increasingreturs is dealtwith by the maket alowingany one gentto rmin boind to exhaustscale economies (see Waltors' (1954) interpretation of Knight'scojectu (1924),which does not accord with my intpretaion here). I arguethat the government is left wi the rol of being the aget tt char the congetiontoll, in order to .Jfeventmonopoly pricing. Afterall, it is uncla a pori thatthe biformationalreqirments and transactioncosts of publicownership wouldexceed the socil cost (sch as rn-sekig) of reguatig prvate ownersip. In other word, it is debatale whethr gove _mentfailure is neemily gter than marketfailure, especially in developing countios. Forinstae thogovenmet's powerof eminentdomain over public projects may reduce its land acquisitionand road cadmact cost whers the commonproperty natur of roadswould encourage rent dissiain - 6 - (rhe overcrowdingoccurs despitethe fact that motoristsalready have to pay fuel taxes and registration/licensefees and purchase taxes for owninga vehicle,which can be regardedas entry fees only.)V On the other hand, it is unlikelythat private ownershipof roads would yield a perfectly competitivemarket structure becausealternative routes are in general not perfect substitutes.Hence, those road ownersmight exploit their locational attributes and monopolistic prerogativesby raisingcharges above marginalcost. In addition,incentives would arise for collusion. Thus competitionwould at best be imperfectin the case of road provisionwhere lumpinessis found. The relevantcomparison, based on efficiencyconsideration, is the welfare cost associatedwith governmentintervention vis-a-vis the dead-weightloss associatedwith the monopolisticor oligopolisticpricing practicesof private ownership. In addition, it seems reasonable to argue that there are other grounds - such as equity - for regarding landowners' super-normalprofits, or collusivetendencies, as undesirable.
9. In practice, virtuallyall roads belongto the governmentand not to private owners. Because of free-accessroads, one should not be surprised to be confronted with the pervasivenessof congestion.This is the commonproperty resource problem, which yields an externality.2 / One might ask whether or not some congestion-proneroads ought to be privatizedto internalizethe externality.The key factoris whether(sufficient) competition could be madeto prevail to ensure,in the absenceof regulatoryrestraints, that these roads are not overpriced. A testable hypothesisin contestablemarkets theory is to check whether the
fi/ The fuel taxeswhich motorists pay are generallyused to contibute to: 1) the maitenance and operating(and in part the investment)costs of roads, 2) debt service on highwaybonds, and/or 3) generaltax revenues. (Independentof howfuel tax revenuesare disposedof, the mainproblem of the uniformfuel tax is its inability to varyby timeand placeof use (see Hau (1992)).) The combinationof a fuel tax, first registationtaxes and annuallicense fees can be regardedas, to a roughorder of approximation,a two-partor multi-parttaiff. 2/ As longas propertyrights are well-defined,exchangeable and enforceable,private bargaining would internalize all 'extenalities' and yield an efficientallocation of resourcesindependent of who is held liablefor creatn externaleffects in the first place(Cosae (1960, 1988)). SinceCoae's so-calledtheorem holds only if the costS of transactionsare nil and incomeeffects are negligible,it appliesmainly to the case of smallnumbers. With congestionand mobile-sourceexternalities, I argue that the large numbersof peopleaffected cleady result in largetrnsactions costs,and hencemarket failure, beckoning govenment itervention. Further,Cooter (1982) arguesthat Coae does not distnguishbetween zero tansactionscosts and zero bargainingcosts, and that the generait of the CoaseTheoom is oveted. - 7 -
conditionof potentialcompetition and not actualcompetition is satisfied(Baumol, Panzar and Willig (1982,Chapter 1)). One surmisesthat it may be difficultto assumethat it wouldbe effectivein practice,especially in the case of roads whereirreversibility and lumpinessexists (Baumoland Lee (1991)). Also, noncomparabilitymay occur because a competitorwould find it difficultto duplicatea road on the same alignmentand would likely settle for inferior alignment.Economic efficiency could be enhancedin somecases if therewere a mixtureof both privateand public roads, sincethe existence of publicly-providedfireewaysmight limit the degree to whichthose who use privateexpressways would be chargedmonopoly prices. Similarly, private toll roads could serve to over-zealousgovemments from overchargingon public toll roads.A/
10. In short, the congestedpublic good nature of roads suggeststhat these facilitiesshould be treatedmore like club goods or private goods, yet the status quo appearsto suggestthat publicownership of roads is insdtutionallypreferable. If so, municipalitiesought to simulate the worldngsof a competitive,private firm and industryby settingcongestion tolls on public roads to internalizethe congestionexternality. This thereby deals with the problemsof nonexcludabilityand preferencerevelation explored earlier. If a privateownership arrangement is deemedbeneficial at times, the governmentcould exercise its power of eminentdomain in reverseby alteringthe structureof propertynghts of the relevantfacilities from public to private ones. This couldbe done, for example,by auctioningoff govenmmentland to the highestbidder (and simultaneouslyreducing general taxes and/or providing more public goods), or, alternatively,by designatingthe competitiveprovision of highwayservices to corporationsor privatecontractors via 'franchisebidding" (Mills (1981)). I haveargued, on both efficiencyand equity grounds - followingour conceptualguidelines - that the facility's ownershiparrangement
8/ Note tbat a relativelyuncongeated toil road would yield inefficint tc allocatiao if its effectis to plice traffic off to alternativepublic roads whichar eady congted. SooJohan (1989)and Catlingand Roth (1987) for a discussionof tollrads and roadprivadition. -8-
ought to reside with the government,with the proviso of setting cost-effectivecongestion tolls.2/ I next showhow a congestiontoll can be derivedfrom a transportationengineering speed-flowcurve. In the proces, the equivalenceof efficientcongestion tolling and the short- run marginalcost pricingof vehicleflow is established.
IIm. FOUNDATIONSOF ROAD CONGESTION - THE CLASSICAL CASE (IN TME SHORT RUN)
11. Road congestionis well founded in the economicsand transportationengineering literature(Dupuit (1844, 1849),Pigou (1920)and Waltes (1961a, 1961b)).p It beginsby consideringa rpresentativedriver cruisingunder low traffic conditionsalong a given stretch of urbanroad with fixedbeginning and end points. The representativedriver wouldbe able to achievea meanfree speedthat balancesthe benefitsto him of a fastertrip againstthe costs to him of higherenergy requirements and a greaterrisk of an accident. CeterLsparibus, as other vehiclesenter the road thereafter,density increases, speed drops and travel time (or delay) lengthens(and accident probability rises). The causalityis as follows: trafficdensity determines speedand not vice versa. Paalleling the theoryof fluid dynamics,tffic flow is the product of density, in vehicles per kllometer,and speed, in kilometersper hour. Note that the rectangular area in Fig. l(a) - the speed-concentrationreationship - is equivalent to traffic flow,expressed in vehiclesper hour (seeGerlough and Huber(1975, Fig. 4.12) andMay (1990, Chapters7, 10), for example). Hence,taffic flow is the productof trafficdensity and speed,
with trafficflow attaininga maximumat F" with speedat S' in Fig. l(b) - the speed-flow relationship(see Haight's (1963) 'fundamental diagram of road traffic' - a flow-densitycurve -
2/ WhiloI am convncd of to advuto s of _arkt hoes, I bhwesornis reevatons abouttdir extnt. For eample, sono have arued dot may 1sowlem shul be prvided for by the mt, and aso that tansport infrastructursoud ao be pivavtd (Raoth(1987, aptr 6) nd Cating and Roth(1987)). It is difficultto sme,in to abso of raod divisibilityand competitivefoae, howmainal costpricing would be pursuedand _maaed by pnvat toad owws Frm a publicchoico perpctvo, the uno argumnt couldbe reversodand applihdwithp repc to govemmta authores. This explain why I arue hat perps a mixed systemof public/pivateo wouldbe able to nap the dteh of both.
IQ/ WhileWale (1968,pp. 31-34,aC._ 3 Annex)diaees aptma investnt, hiswork emphasiethe short run cdaacter of roadping. Mb alpproh tk intspais to lhihlihtboth the bort-and-g-ru natm of die congestio probm in ordw to explore de issue of coat rcovery. -9 - - and similarfigures in Monison(1986)). (Toutedmaximum flow figures of the 'capacity' for a typicalexpressway are about 1800-2000vehicles per lane-hourat 50-55kilometers per hour (30-35mph) (see Gerloughand Huber (1975,Chapter 4) and any HighwayCapacity Manual, for example,Transportation Research Board (1985, Chapters2-3));w
12. Givena fixeddistance of say a kilometerof road,the trafficengineer's speed-flow curve can be straightforwardlyconverted to a travel time-flowcurve as travel time is the reciprocal of speed,with vehicles-kilometerper lane-kilometer-houron the horizontalaxis (see Fig. l(c)). Usinga constantvalue of time as a shadowprice for the representativedriver, tr,--eltime is then convertedto a moneybasis whichyields time cost, calledthe averagevariable cost, AVC (see Fig. 2). Low trafficvolume corresponds with relatively high speed,so fuel cost wouldbe high. Withhigh trafficflow and low speed,however, fuel costwould also be kepthigh becauseof fuel inefficienciescaused by the alternateacceleration and decelerationassociated with densetraffic. Thesetwo factorsroughly cancel one anotherout, leadingto the plausibleassumption that the costs of opeating an automobile(which include fuel, oil, maintenanceand depreciationcosts) are approximatelyindependent of the level of taffic flow (Mohring (1976, Chapter 3), AASHO's(1960) 'Red Book'). A fixedmoney cost for the vehicleoperating cost can therefore be added to the time cost portion to form the generalizedcost - an accpted constructof transport economists (Nash (1976) and Button (1982)). Similarly, the road's variable maintenancecost, whichis assumedto be proportionalto the traffic level, followingWalters (1968,p. 24), can be added up also (see Fig. 2). 1 / So it is the time cost elementthat is mainlyresponsible for the upward-slopingportion of the AVC curve. The AVCcurve climbs upwardsbecause significant negative interactions occur before trafficreaches maximumbasic capacity,Q; it is variablein the sensethat as trffic flow, Q, is inased, congestiondelay
JJ Economists'notion of 'cpcity' is whever congeio dday bogins,which is considerablyless t traffic engineer'concepL Furte, vehiculaflow is sometimesnormaized by the capacityof a crtain roadto yield a 'volun_eocapacityqratio. 12/ Hencethe margi co curve,MC, when summed up verticalywith the vehicle openg costand vaiable road maintenancecot, yieldsa anlad mginal os curve,MC. Thesm notaions pplyto the averagevariable co curve. - 10-
actually sets in rapidly at substantiallybelow the traffic level Qr0 1 (contrary to the engineering notion of a constantaverage variable cost curve extendingup to the point Qu i FP). After the engineering or basic capacity, Q"" is reached, AVC becomes an 'inverse supply' curve. Note that the standard supply curve is nonexistentin the context of roads.ia/
IV. DEMANDAND SUPPLY
13. The 'supply' side can be made to be congruent with the demand side when a conventionaldemand curve is specifiedto depend on the travel cost (price) facing a traveller for a single trip. When an initial demand function, Q",intersects the AVC curve at point U (Fig. 2), a (stable) equilibrium is said to exist at Q°. This is an equilibrium point because travellers' willingness-to-paycurve, i.e., the inverse demand function, equals the average variable cost curve -- the function upon which travellers base their travel decisionc.. After a small excursion of the demand in the neighborhood of the equilibrium point, unfettered use results in a return to the observed traffic level, Q°, hence the equilibrium is considered stable.
14. Basic price theory says that whenever the average variable cost rises, it means the marginal cost curve lies above it.15/ The vertical difference between the two cost curves is
13/ The solid portion of the speed-flowcurve and travel time-flowcurves in Figs. 1 and 2 denote the 'normal' part of the curves. The non-soLidbackward-bending portion of the cost curve (Fig. 2) meanstat time cost increases because traffic flow is reduced after the gineering capacityis reached. The backward-bendingcurve, far from being fictitious, has been substantiatedin the litrature (Gerloughand Huber (1975, Chapter 4), Keeler, Smal and Associates(1975, Fig. 1)). Schiff (1991) explores this backward-bendingcase.
14/ The demand fimction,Qd, is an observed, constant-money-incomeMarshallian dmand curve, with the usual regularity conditions. It approximatesthe exact Hicksian demand curve, which yields the true reflection of willingness-to-payand marginalbenefit. Formally,at equilibriumGC(Q) = Q'P), and Q"(GC(QO))= Q°,where GC is the generalizedcost. Since GC is simply a tra tion of AVC, the interpretationof one is synonymous with the other.
15/ Marginal cost is obtained as fTollows:MC * AC(Q)/AQ = AVC(Q) + Q *AAVC(Q)/AQ = AVC(Q) *(1 + e) where C(Q) is the cost function, e is the elasticityof the AVC curve, i.e., the rate at which time cost rises with respectto a one percent rise in traffic flow (Walters (1961)). The first term composesof only time cost and the second term is the marginal(external) congestioncost, set equal to the congestiontoll. Marginal cost pricing of a trip, P, is achievedby setdng P = MC. This is known as the first-bestoptimal pricing rule: our first optimalityrule. (Note that AVC dependsparametrically on the capacitylevel K, and can be expressed - 11 -
the marginal (external)congestion cost - the additionaldelay that one driver imposes on the rest -- which is not taken into account by the last driver who joins the traffic stream. In fact, since each driver chooses whether or not to travel according to the AVC curve -- being the decision curve -- he or she totally ignores the resulting external congestion cost imposed on felow motorists. We thus have the optimal point Y at which the marginal cost curve intersects the (peak) demand curve in Fig. 2. In other words, Q' is the associatedoptimal output in the sense that the generalized cost which includes extemal congestioncost and other variable costs (i.e., constant (unit) operating cost of a vehicle and variable maintenancecost of a road), is equated to the price. The congestioncost is the additional time cost that a motorist imposes on others, calculatedby taking the increment in average time cost caused by the added trips and multiplied by the number of vehicles in the traffic stream. The Pigouvian tax applied to roads is that optimal toll which closes the wedge betweenthe marginalcost and average variable cost curves by emitting the correct signal wrd creating appropriate (dis)incentives.
15. This Pigouvian toll-tax is equal to the marginal external congestion cost. It is also known as the net-benefitmaximizing, economicaUy efficient, (Pareto) optimal and marginalcost toll. (If one prefers, it can be regarded in graphic terms as a sin tax, even though it has not yet included the cost of environmentaland other externalities.) Hence the marginal cost pricing of trips in the short run (given that a road is fixed) yields a first-best Pareto optimal allocationof resources. The optimal road user charge is then comprised of a congestion toll and another componentwhich covers the variable road maintenancecost (see Fig. 2's legend).
as AVC (Q, K). WithoutloSS of generality,the inclusionof the averge vehicleoperating cost and variableroad maintenanccost-both beingconstant with respect to taffic -simply alters both the left-hand side and the right-hand side by the same amount). Implementingmarginal cost pricing in this case means pricing the differencebetween marginal cost and the average variablecost of a trip, plus a componentwhich covers the cost which the motorist imposeson the community(see Fig. 2's legend). We shall return to an intuitive discussion of this subtle but important distinction. Note further that marginalcost rises asymptoticallyto the engineering capacity level of Q and is undefined for the AVC curve at points beyond V. - 12 -
V. VERYCONGESTED RoAS ANDBoTrLENEcKs
16. Observethat at the equilibriumpoint U in Fig. 2, the resultant throughputis significantly less than the road's maximal flow capacity of point V. The 'ackward-bending 'supply curve' exists because a one percent increase in density results in more than a one percent decrease in speed when very dense traffic is reached. (That point X is a stable equilibrium point can be seen intuitively by perturbing the price level while point w is unstable in that any disturbance will result in a movement to U or X.) The point here is that quite a few cities, for example, Bangkok, Budapest, Buenos Aires, Hong Kong, Jakarta, Medico City, Santiago, Sao Paulo, Seoul, Singapore and Taipei are faced with extremely congested situations such as point X, cerainly during peak of the peak.
17. Consider the dynamic phenomenonof traffic growth as shown in Fig. 2 (a). Suppose the initial demand curve intersectsthe average vanable cost curve at point 1. As traffic grows, the observed number of vehicles per lane-hour increases from point 1 to 2 to 3 (which is identical to point U of Fig. 2). A further increase in demand beyond point 4 would result in a discontinuousjump to the backward-bendingpart of the unit variable cost curve at point 5. Intuitively, traffic congestionworsens rapidly and 'jams up' all of a sudden at times. After a while at this point, travel demand would start to slack off to points 6 and 7 (which is identical to point X of Fig. 2). Note that traffic would be moving at a snail's pace as queuing develops. This corresponds diagrammaticallyto the positive sloping part - as opposed to the relatively smooth trffic of the 'normal' downward sloping portion - of the speed-flow curve, Fig. l(b). As travel demand continues to diminish towards the end of a rush hour, say, the traffic level would touch point 2 briefly but would then end up at point 8. Thereafter, further slackeningof demand results in traffic retumningto the upward sloping portion of the unit cost curve at point 1, say. This is known as a relaxation phenomenon in an engineering and physics context. Therefore, the points between 4 and 8, such as the equilibriumpoint W, are never reached in realityl 1 We conclude that it is precisely the daily recufrent peaking characteristicof travel
16/ In othercontexts such as Waltms(1987), point W is desuibedu an unstableequiIibriumL - 13 - demandthat calls for innovativesolutions. That the trffic level at those nsh hour periods seems to be near gridlock -- an illustrationof 'hypercongtion' - is too important a case to be ignored(Walters (1987)).
VI. TBEWELFARE IMPACT OF ROADPRICING
18. The purposeof this sectionis to highlightone of the mainpoints of thispaper, the proof of whichis relegatedto theappendix. Economists know that road pricingresults in improvement in welfareto society,yet politicians and the public almost unanimously regard it with skepdcism. Why? To economists,the increasein welfarecomes about becauseof the impositionof an externality-corrective(toll-)tax. Yet, for those motorists remaining on the road, the congestion toll is similar to a tax increase. Under conditionsof 'normal' traffic (or non-hypercongested situations),note that the toll-taxpaid by the motoristexceeds the valuationof time savings resultingfrom road pricingon average,so that the 'tolled' is worse off.1 Those who are pricedoff the road to an inferiormode or timeof travelin orderto avoidpaying the toll are also worseoff, whilethose who remain on other indirecdyimpacted roads are either worse off, if congestionarises there, or just as welloff, if there is no resultingcongestion. It tums out that the government,in collectingtoll revenues,becomes the main party that is better off.
VII. THB EFF S ON THETOLLED, TH TOLLEDOFF AD TH UNTOLLED
19. In the appendix, I establish the propositionthat marginal cost pricing of trips maximizes the net benefit to society in the sense that a Pareto-efficientsituation is attained, that is, no one can be made better off without making someoneelse worse off. The quantity approach (the 'American' approach) yields the welfare gain due to road pricing of area I (shown by the trangle UYZ in Fig. 3). The equivalent area, e+f-d, is the welfare gain using the change in total benefits and costs approach (the 'Britsh' approach). The latter area is the cost saving of area iZ/ Mm 'toiled,the toled off, ad td u-toiled' an cnn coined by Zeds ad Call (1964). My approach difl_tiatee f*omboth Zettoland Carl and Woldad Hamidckacs (1984,pp. 114-116)in that I applythe standardnoesoai methodof wdfre onalym ysthe appdix to detivemy reaut and policyimplications. - 14 - e+f+g+k less the loss of the use valueof area g+k+d. We notethat there are two groupsof travellersthat are clearly adverselyaffected by road priing: the tolled and the tolled off. Briefly,the Britishapproach to the calculationof welfaregain says that those who remain on theroad after roadpricing is introducedincur a costby makinga toll paymentof the rectangular area b+c+e+f. On the other hand, the travellerbenefits in the form of reducedtravel time -- 'forcibly'induced -- and the resultanttime savingsis the smallerrectangular area e+f. Hence the consumer-travellerwould regard this exchangeas not gettinggood value for moneybecause the travelleris still facedwith a net paymentof area b+c relativeto the no toll situation: this area is his loss of consumer'ssurplus. Despitesuch a trade, the travellerwould undertake the trip becausehis willingness-to-paystill exceeds the price.
20. The other identifiablegroup includes all thosemarginal users whose willingness to pay is not high enoughand henceare tolled off the road. As a group, their loss in valuation,the vertical trapezoidalarea d+g+k, exceedstheir savingin time cost of the area g+k by the (welfare)loss in consumer'ssurplus of area d. So both groupsare necessarilyworse off vis-a- vis the originalsituation. This is shownto be true despitean argumentthat those whoremain benefitfrom reducedtime cost! They do benefitfrom reducedtravel time, but they have to exchangemoney for time. In addition,the "tolledon" is eitherjust as well off, or worse off, dependingon whetheror not congestionarises. So it appearsthat, a move from a Pareto inferiorposition to a Paretooptimal state leaveseveryone worse off! How couldthis happen? It comesabout becausewe have not yet accountedfor one agent: the government. The toll revenuecollected by the governmentis consideredan actualgain to societyin that it is counted onceonly. Indeed,it mayhave a greatervalue than the dollaramount itself if it replacesother (income)revenue sources with an excessburden.
21. Supposethe toll revenueis collectedand thenput aside. An 'efficient'amount of traffic
and congestionwould still exist on the road in question,yet society- an aggregationof both
gainersand losers- wouldbe definitelyworse off. Unlessthe publicin general,or road users
II/ It is as if a blakmie - thegovernmt - ca ou pat of t motoris'sconsumers suplus. - 15 - in particular,can partake in the tax proceedseither in the provisionof public goods and/or reductionin tax revenues,they will definitelybe worseoff from this impositionof an 'optimal' congestiontoll. If fundsare not channeledback to road users, the government,or the rest of society, gains, but only at the expenseof those faced with road pricing. However, because of the nature of the Pigouviantax, which possessesan asymmetricprice signal, generatorsof externalitiesare taxedbut thoseaffected by the externalityare not supposedto be. compensated.This is a requirementof optimalityin the case of both publicand privategood- type externality-correctivetaxes (Baumoland Oates(1988, Chapter4))?Ql
22. Twochoices to thosewho are tolledoff theroad are to use the road duringthe off-peak periodor switchover to publictransport during the peak. The analysisis similar in spirit for both classes, so I will illustrateit for the off-peakperiod, for simplicity,following the same notationused thus far. Substitutionof travel demandfrom peak to off-peakhas the effect of shiftingthe off-peakdemand curve to the right, from Q0,, to Qdqt(Fig. 4). It is easy, as is sometimesdone, to regard the area m as an additional'benefit'. This procedureis incorrect becausethe area boundedby the demandcurves in the substitutedoff-peak period is in fact a pseudo-benefitand is alreadyaccounted for by the welfaregain to road pricing in the peak period, i.e., the triangulararea 1, in Fig. 3 (Mishan(1988, Chapter 8)). Intuitively,since there are no changesin consumer'sor producer'ssurpluses, there are no net changesin benefitsor costs: the additionaltrips madeduring the off-peak period are entirelyself-financing in the short run. If congestionwere to set in during an off-peakperiod with relativelymedium levels of congestion,say, duringthe inter-peak,efficiency suggests that anothercongestion toll level be set to internalizethe congestion externality. In thisway, trafficwill settleto anotherequilibrium
1/ The Smeed Report of 1964 and others effectivelyasswne away the problem by stating tbat the congestiontoll revenues will be retuned to the popuaion in a lump sum nondistorionary manner.
2QI Note tat Baumol & Oates (1988, 2nd edition, Chapter 4) correcs the mistake made in the earlier edition(1975, 1st ed., Chapter 3), which says that only the victimsof private good-ype externalitiesought to be compensated for after the impositionof a Pigouvian tax. This result is contray to acceptednotions of justice. However, Baumol and Oates (1988, pp. 236-240)point out that if the tax revenueswere funelled bacwkeabwrcy, then ther would perhaps be only insignificantdivergences fiom Paeto optimality. The idea is that wedth effects an consmption ought to be minimized. - 16 -
level with a smallercongestion toll in the inter-peakpeiod. 'Ais is the idea behindpeak-load or differentialpdcing. A dynamicprocess takes placeamong the peak and inter-peakperiods untilan equilibriumis settledupon. Propercost-benefit analysis requires at the changesin net benefits be calculatedonly in the periods which encounterchanges in travel time due to congestionor decongestion.Wit}. a two marketmodel, the welfareeffects of road pricingin thepeak period are simplyrepeated in the inter-peakperiod whenever congestion is encountered.
23. It turns out that there is a case often overlookedin which everyone,including the government,can be shownto be betteroff. This is the caseof 'hypercongestion',where density is beyondthe point of maximumflow. Here the trafficdensity is so high thatboth trafficflow and speeddiminish, with the generalizedcost POand trafficQ 0occurringat point A (Fig. 3(a)). (Eventhough it violateseconomic rationality to end up at sucha pointA, this typeof trafficjam doesin fact transpirefairly regularly, though limited to peak periodssuch as peak-of-the-peak.) The implementatonof a marginalcost toll wouldresult in travellersreverting to thenormal non- hypercongestedportion of the speed-flowcurve (such as may be observeddownstream of a bottleneck),which corresponds to the lowerbranch of the AVC curveat point C in Fig. 3 (a). In addition, the travellershave to pay a unit toll paymentof distanceBC, resultingin a genemlizedcost to the motoristof P', whichis still lower thanP0. Becauseof the price decrease from PI to P', traffictherefore increases from Q to Q' correspondingly.In this case, literally everyoneis made better off: the tolled, what I call the "tolled on," and the government.31 If the speed-flowcurve is as depictedin Fig. 1 (b), then hypercongestionappears more often than is commonlyrealized: it occurswhenever speed drops to half (approximately60%) of the maximumspeed limit.
2/ It can alsobe shown,using te techniqueof wdlfr, anDaysiselaborated in the appendix,tbat netai fit to societyfrom such a ntional move is equal to area o+p+q+r+a+t+u+v (- ar o+p+q+r+a+t+w+x). - 17 -
Policy ImDlicatios:
24. Ever since the Frenchengineer, Jules Dupuit (1844), introduced the powerfiulconcept of consumer'ssurplus to analyzeissues of the efficientpricing of (toll)roads, economistshave embracedthat notion,extending and deepeningknowledge in that area. Efficiencyanalysis carefullyshown here indicatesthat societywould unequivocally gain from road pncing. The questionthat arises naturallyis why road pricing, with one single exceptionworldwide, has failed in the sixties,seventies and eightiesto get off the ground. I show,using the analytical frameworkdeveloped above, that road pricing as sold in the past is most likely doomedto politicalfailure. Thisis becausealmost all motorists,including both the ones whoare tolledand tolled off, find that they are invariably worse off as a result, except in the case of hypercongestion(see footnote for qualification). The sole unmistakablegainer is the government. If road users do not perceiveor are not persuadedthat they benefit from the government'snewly collectedrevenues in the form of provisionof transport servicesand worthwhilepublic expenditures or receivetransfers in the formof reducedgeneral tax payments, albeitindirectly, it is highlyunlikely that these groups would acquiesce to thepricing of existing roads. Cooperationwould be more likely if they are guaranteeda reductionin motorvehicle- relatedtaxes suchas importduties, first registrationtaxes, annual license fees and/or fuel taxes. In particular, the replacementof - rather than an addition to - existing vehicle-relatedtaxes by cost-effectivecongestion tolls would especially be welcomeby road users.
25. A naturalquestion then is as follows:is therea theoreticalargument for dedicatedfunds or earmarkingso that societyas a wholewould benefit from theimplementation of road pricing? Restatedanother way, is there a way in which road users can act as beneficiariesand be
22/ The policyimplications discud beroa basedon the assumptionof costat vduo of tme - the as_mpm used by Walteosand othr n derivingthe avegp variable coat curvo(see sectionXIV, subsaecon , below). My reuk that roadpcing makeseveayone wor off exceptthe govenmnt is meantin an averg sune. People'stim valuationsdiffer in reaity, so thatthis resultis modifiedto say that oly thosewith fairlyhg valuationof time wouldbe betteroff, and eveyone elsostill worse off. Te conditionis that the weighted valuationassciated witht timesavimgs rectagle of area e+f mustexceed their total mony 'ayment of ae b+c+e+f basedon a weightedcongestion toll. Otrwis, even thos who remainon the toiledroad wold continueto pay but would actuallybe worse off relativo to their status quo. - 18 - indirectly'compensated for' the paymentof tolls by satisfyinga commonlyaccepted notion of fairness,while carefully skirting the first-bestpricing rule? I think that the answer is 'yes', althoughnot entirelywithout qualifications.
VI. SHORTRuN EQUILIRIUM
26. The propositionthat optimalpricing of and investmentin a highwaysystem parallels the shortand longrun equilibriumconditions of a competitiveindustry of a textbookcommodity was first shownby HerbertMohring (Mohring and Harwitz(1962), (Mohring(1965, 1976)).3/ Economicanalysis of transportproblems is simplifiedconsiderably by explicitlyrecognizing the travelleras both consumerand producer,and as producer-travellerhe purchasesfactor inputs such as travel time from himself. This short-run/long-runapproach is advancedboth cogently andlucidly by Mohringand usedby establishedtransportation economists such as Keeler,Small, Kraus, Glaister,Morrison, Winston and Oum.
27. In the short run, some inputs for producinga textbookcommodity are regarded as fixed. Under competition,an economicallyefficient output level is achievedwhen the market-determinedprice equals the short-run marginal cost of producing that good. A competitiveproducer purchases variable inputs by hiringlabor and procuringraw materials,in additionto investingin a fixedinput, capital. Thereafter,the firm combinesthe inputsvia the productionprocess and creates commoditiesto be sold to a consumer. In other words, the produceruses the revenuewhich he obtainsfrom sellingthe good at the givenprice to pay for the variableinputs of labor and raw materials,plus the fixedinput in the form of quasi-renton the capitalequipment, normally regarded as the accountingprofit. The graphsfor the textbook commodityare similarto (but not exactlythe sameas) the caseof roadsconsidered in Figures
ZV See surveyby Winston(1985).
4/ Mhmstandard definition of shortrun is a situaion in which some productive inputs are regarded as fixed, and hencecetain costswould be fixed. However,the definiton of longrun refers to a sitationin whichall npUtS vary. In the roadcontext, the durationof thelong run depends on therate at which,say, the size of a roadand hence the basic or enginering capacity,can be varied. - 19 -
2 and 3.2/ In transport, the short-run marginal cost of a trip, which we derived rigorously from an engineering speed-flow curve, is to be set equal to the 'price' of a trip. Recall that transport is unusual in that the traveller is both a producer and a consumer. Analogousto the paallel case of a textbook commodity, the road user, when undertaldng a trip, supplies some of his own variable inputs, which include vehicle operating cost and time cost. We have seen that the competitivelevel of trips exceeds the efficient quantity in the presence of congestion. Hence, becausethe quasi-rentof a highway facility would be dissipateddue to free competition, an optimal toll should be imposed to capture this quasi-rent. Clearly, even though the dictum of short-run marginal cost pricing prevails in both cases, the optimal toll does not equal the short-run marginal cost of producing an output but is equivalent to the difference between marginal cost and average variable cost. This is a subtle but important distinction between transport and widgets.
28. The optimal toll is the efficient charge referred to in Mohring and Harwitz's (1962) mathematicalstatement of this problem and Newbery's (1989) Proposition 1. The optimal user charge is then the optimal toll plus another component required to cover the variable maintenancecost of a road discussedin Walters' (1968, p.24) and Newbery's (1989) Proposition 2 (see Fig. 2's legend).X/ As discussed before, to focus our efforts here, the optimal user charge shouldultimately include air, noisepollution, accidentcost and road damageexternalities.
29. Walters (1968, Chapter 2) defines the term 'user charge' as the money charge that governmentalauthorities levy on travellers for the congestioncost they impose on others and for the variable maintenancecost of the road incurred due to their use of that road. In the absence of congestion, the user charge covers the unit road maintenancecost component only and would be independentof the traffic level. Walters (1968, Chapter 4) then coins the term -economic
251 With sandard goods, both dh short run nmrginaland averagevariable cost curves canldecline and swing upwards,whers I haveshown that both the shortrun marginaland average variable cost curvesi transpt newc declinebut only rs upwards.
2I Moreprecisely, Newbery's (1989) Propositions1 and 2 include the invariatemaintenace cost due to weathr. - 20- user charge (EUC)"to be equivalentto the generalizedcost conceptor user price employed here. This latter usagemight lead to a possiblemisunderstanding about the demandand supply side or even double-counting and is the-eforeavoided here.
30. To recapitulate,short run equilibriumin transportoccurs when the government,in the form of a highwayagency, behaves in an optimizingfashion just as a privatecompetitive firm would were it possibleto organizethe industryin a competitivefashion. The optimaluser charge shouldnot be set equal to the price but wothe differencebetween the marginalcost and the averagevariable cost of a trip.
IX. CONVRGENCETOWARDS LONG-RUN EQunITMBR UNDER CONSTANT RETURS
31. So far we have confinedourselves to short-runequilibrium. The fixedcost component has been deliberatelyleft out of the analysisof the marginalcost of a trip.= The motorist is obliviousto the capitalcost of a road, and his behavioris independentof it. However,from the highwayagency's planning point of view, the capitalcost of a road is very muchtaken into account. Oncea highwayis built, however,it is regardedas sunk. The sunk cost of a road, onceincurred, is irrelevantto a planner: only currentand futurecosts, not historicalcost, serve as a correctguide to planningfuture investment. Sincethe variableroad maintenancecost is assumedto be constant,the marginalcost of a trip thus remainsthe same.
32. In the long run, however, a highwayagency can vary the fixed capital input by expresswayexpansion, if theinvestment is deemedjustifiable. On theother hand, if a rural road has been built as a result of past planningerrors, it can be allowed to deteriorateor be downgraded(or be evenauctioned oft. Expandinga road untilthe additionalbenefit equals the additionalcost of buildingit wouldyield maximalnet benefit to the community.How might this be done withoutresortg to a full-scalecost-benefit study?
Z1/ Ih fid cost of a firmis definedto be Xt minum amountof outay necessaryto str production. - 21 - 33. To see how this might be done, we introduce the fixed cost, i.e., the cost of construction,together with the 'invariate' maintenance,depreciation and operatingcosts of a road that are incunredby a governmentalauthority in Figure5.W1 We thenconvert the entire fixedcost into the cost per time period of a unit of capitalfor utilizingthe flow of highway services. This is donein order to makeit commensuratewith the averagevariable cost of a trip discussedthus far. The summationof the short-runaverage fixed cost and the averagevariable cost curve yieldsthe averagetotal cost curve. Chargingthe optimaltoll of the distancet' in Figure 5 seems to be more than sufficientto cover the short-runaverage fixed cost of the facility. In this case, the optimaltoil, t', exceedsthe short-runaverage fixed cost of the facility, SRAFC', by the unit profitdifference of u'. In general,there is no a priori reasonwhy toll revenuecollections based on short-runmargnal cost pricingcannot cover the non-use-related costs of a givenhighway facility.
34. In the caseof a textbookcommodity, whenever the quasi-rentbeing earned by a firm's existing capital equipmentexceeds its cost, there is an incentive to expand production. Ultimately,the quasi-rentearned by the existingcapital equipment would then be equal to its (fixed)cost.2 Puttingit another way, upon seeingthe existenceof economicprofits, other firms enter the industry also, shiftingthe industry supply outward, increasingoutput and loweringprice as a result. The unrestrictedmobility of resourcesand theentry and exit of firms serveas the instument by whichprofits would be competedaway in due course. Whencapital is freely varying, long-run equilibriumis reached when zero economic profit occurs. Equivalentlysated, the quasi-rentearned on the firm's capitalequipment equals its cost, i.e.,
71/ hmtwm lnvarlaw, ie., non-taffiC rlated nit c cot is found in Walftr (1968, p. 23). aI/ e quasim-reitto a firm is the acounti profitsp1w interestexpens on borrowedfiuds, if any. The acomting profit is the finr's totalrevu lesa its contwua costs, includinginterest expse on borrowed fiuds, wags to labor, cost of vw maters, and rentalcoat of leasedbuildings. Accountingprofit less the madmtrun of the owner-mupli.dma_s i t oconomicprofit Altenaively, th qua-rent on imvest capit is te total evau lm the variabb costs, La., includingwages, cost of raw mateials and rental cost, but ewludig interet expens an borrowedfunds. Totalreve lesototal vaiable and fixedcost yields economicprofits. Tbe fixedcost of investedcapital includes tih entireopportunity cost of capital,regardless of wheter funds bonowedor not. (Seete thiWfootnote of Appendix(footnote 71) andMoaring (1976, pp. 8-11)). I am gratful to HEbrt Mahringfor clarifyingthese points. - 22 - the market return of the cost of reproducingthe investedcapital. This conditionholds under constantreturns to scale, wherea proportionateincrease in all inputsis compatiblewith the same proportionateincrease in outputs. Givenfixed factor prices, total cost also doubles, so marginalcost remainsconstant in the long run. With a slightbut crucialmodification, this analysiscarries over to the case of roads. When the quasi-rentof the existingcapital stock exceedsthe normal market return on the cost of reproducingthe investedcapital plus the highwayfacility's invariate maintenance and depreciationcosts, new investmentis expectedto find its way in that road segmentof the highwayindustry if the appropriateprice signalsare given. Equivalently,in the long run, if toll revenues- whichrecover quasi-rents throughout
time periods- exceedthe entirefixed costof the existingfacility, the highwayauthority woull have the appropriateincentive to expanda stretchof that road until aU economicprofits are erodedaway. As we have seen in the caseof roads, the variablecost is composedof the user- suppliedtime and operating costs and are fuUlyself-financing. The non-userelated costs are then financedseparately by the road agencyvia toll revenuecollections. In this way, full costs are coveredand there is no need to raise chargeswhen there are constantretums to scale.
X. OPrwAL INVSTMNT
35. In the long run, toll revenues would then exactly cover the amorized cost of construction,invaniate maintenance and depreciationcosts of roads - a powerfulresult first shownby Mohringand Harwitz(1962, Chapter 2) and Mohring(1965) - under the technical conditionsof constantreturns to scalein roadconstruction, maintenance and road use. Constant returns to scale intuitivelymeans that the cost of buildingand maintainingan expresswayis proportionalto the capacity. Constantreturns to road use yields an intuitiveinterpretation: travel time dependssolely on the volume-cacity ratio. If the engineenngcapacity and the
QI The(opportunity) cost of a resourceused hereis thehighest reatun of it elsewhere,hence the cost is the maket returnon reproductioncost and not historicalcost. - 23 - traffic flow were doubled,unit travel times wouldremain the same.#'1 The final long-run equilibriumis shownin Fig. 6. By faithfullypursuing the policyof marginalcost pricingof a trip by charginga congestiontol - the differencebetween the marginalcost and the unit variable(time) cost -- and by expandingor appropriatelyreducing the capacityof the road until there is zero economicprofit, the output (of vehicle-kilometersper lane-km per hour) is consideredoptimal. At a momentin time for an existingroad, outputis optimalin the sense that, giventhe marginal-costprice, the efficientlevel of trips is achieved. Undertakingeither more or less trips wouldinvolve lowering the net benefitto the community. In the long run, outputwould be 'doubly' optimalif it is the efficientlevel of trips for that link of road which has been optimallybuilt. Diagrammatically,not only does the implementationof a congestion toll internalizethe externalcongestion cost, it can be seen that the toll covers the short-run averagefixed cost of the road in a stationarystate. Recall that for homogeneoustraffic the averagefixed cost of a roadis simplydefined to be the differencebetween the averagetotal cost and the averagevariable cost. Clearly,collecting a unit toll wouldcover the entire average fixedcost of the road andyield zero profit only because the existenceof economicprofit or loss servesas a quasi-marketmechanism in the investmentdecision of whetherto expandor contract the highwaycapacity. With zero profit, the minimumpoint of a short-runaverage total cost curveis obtained. For any givenlevel of output,the minimumtotal costof yieldingthis output wouldbe obtainedonly if the optimalinvestment level in capacityhad been chosen. Looldng at it the other way, the optimumsize of a road is obtainedby drawinga locusof all the minima of the varous short-runaverage total cost curves of different sizes (and capital costs) of
3I/ i 1) the capital and invariatemaintenance cost of highwaycapacity, KC, is directlypreportional to tbe engineeringcapacity, K, i.e., KC (K) = aK, wher a is a constant,dhen there exists constat trns to scale in highwayconstruction (and invarate road _mnce). (In mtematical jargon, KC is homogeneousof degroeone in capacity.) The engiering capacityis measiredby lane-widthand is treatedas a continuous variable.Further, tf2)(a) trafficcan be expressedin termsof a homogeneousunit, Q, in vehicle per lne-hour, and the timecost function AVC(Q,K) depends direcdy on e trafficflow but is invetselyrelated to the capacity and (b) {(doublingboth highway capacity and trafficflow resut in the traveltime of a tt4premain the same, then thereexists constant returns to roaduse. (Ma cally, the AVCfuncticn is homogeneousof degree zero in trafficvolume and capacity.) Withconstant retrns to road uw, AVC(Q,K)can be formallyrewritten as AVC(QIK),where Q/K is the volume-capacityratio. Sinceunit vehicle opeating and variable road maintenace costs are both indepdent of the level of output,and capital cost, KC, is proportionalto lane expansion, ATC(Q,K)= ATC(Q/K)holds also. Thesetwo technical conditions are crucialto Morng andHaMritz's (1962, pp. 85-90)so-called theorem and to Keele and Small's(1977) extsion of Mohring'stheorem. - 24 - highwaysand choosingthat particularsize of the road associatedwith the point where the demandcurve intersectsits marginalcost curve. We do this becausethe demand reflects motorists'maximum willingness-to-pay and hencethe incrementalbenefit of the last trip. Since the minimumpoints of the short-runaverage total costs underconstant returns are of the same height,the long-runaverage total cost is a horizontalline tangentto all the minima. With the long-runaverage total cost beingconstant, so also is the long-runmarginal cost. Hence, it is only at the minimumSRATC point that long-rn margin cost pricing holds.
X. LoNG-RuN VS. SHORT-RUNMARGINAL COST PICINGRI
36. Intuitively,the long-runmarginal cost of producinga trip yields the total cost of undertakinga trp to the communitywhen all fixed and variable inputs can be varied continuouslyin thelong run. Proponentsof long-mnmarginal cost pricing argue that the market return to capitalinvestment would presumably be fullycovered. Yet the equivalenceof short- un and long-runmarginal cost pricingholds only in certaincases, including the static demand and singleperiod case consideredhere. As shownin Fig. 6, long-runmarginal cost pricing wouldcover aUthe variablecosts, including time cost, vehicleoperatig cost and variableroad maintenancecost, phw the fixedconstruction, invariate maintenance, depreciation and opeating costs of the road. In fact, short-runmarinal cost pricingcovers the entire capitalcost of the facilityjust as muchas long-runmarginal cost pricingdoes, as can be seen diagrammaticalyin Fig. 6. After all, both the short-runand long-runmarginal and averagecosts are equalin the long mn, with both sets of cost curves intersectingthe demandcurve at the same point. However,if a road is not optimallyconstucted but underbuilt,then long-runmarginal cost pricing would bendout too low a price signal, thereby exacerbatdngcongestion. Short-run marginalcost pricing,on the otherhand, wouldgive the corect signalof higherwllingness-to-
321 Pro (1969,p.8), Water (1968,p.33), BDunahn sad Waltn (1979,p.33) and Bird(1976, pp. 33 -3 9) argue for ihort-runrqmaurl coatpricng whbwruothes argue f1r loun murgnaslcomt pncins. Sincethe msueof long-rmvs. shost-mu maqprnl cot pncnighas beanwith u for SmO tinum,a claification in order (swetfe ntw debate beiwehmJordan (1983a, 1983b, 1985) ad Vwikoy (198S)). Vikry poinls out tht th concept of log-nm maurginlcoat beoome. obfuscated wbh swvealdemand priods, e.g., peak, mitepeakand off-peak, occurdiunmaUy, givan dlo came of a an sportaionbifrstrucur dot hs aleudy beanconstructed - 25 - pay and also yield positivetoll revenuesand economicprofits as a by-product. Short-run marginalcost pricingis the rule to use wheneverlong-run equilibrium is not reached(and of course when it is). Lookingat it another way, if short-runmarginal cost is below long-run marginalcost at the currentoutput, it meansthat the road has been overbuilt. But, of course, this does not mean that the size of the expresswayshould be or indeed can be varied instantaneouslywhenever demand fluctuates daily. Rather, it meansthat the price ought to be variedaccording to demandpatterns using short-run marginal cost pricing.
37. Indeed,Professor WfIliam Vickrey has emphaticallyargued that therecan be no solution to the urban transportationproblem without peak-load pricing. Proper time-of-daypricing can be implementedonly usingshort-run marginal cost. (Weshall explore this point furtherin the sectionon demandvariability.) Pursuingshort-run marginal cost pricingperiod by period by varyingroad capacityincrementally over time wouldnot onlyguarantee the best use of society's
resourcesbut wouldalso enableroad agenciesto recoverall costs - as an incidentalby-product
- in the long run. It is thereforerecommended that short-runmarginal cost pricingbe used sincethe conceptof long-runmargnal cost cannotbe unambiguouslydefined whenever cyclical vaiatons in demandare involved.
XII. TRADE-OFFBETWEEN INDBVDUALS' TIM ANDTREASURY AccouNTs
38. Another way of obtainingthe optimal investmentlevel for roads is to answer the followingquestion: what is the minimumcost to the communityof road building,taking into accountboth the highwayagency's desire to minimizethe fixedcost of capitalfacilities and the
travellingpublic's desire to save time' By minimizingthe total cost - the sum of these two costs: the variable (time) cost of trip make and the fixed cost of the governmental
authority- a twade-offis foundbetween individuals' time and the teasury's accounts. Given a non-optimalcapital stock (K') assodatedwith a particularhighway, as in the previousgraph,
3/ Solvingtho problem of codtmininiutios quivalet to solviugtbe problemof maximiton of net benefitto the community. - 26 -
Fig. 5, it can be seen that the least cost for the communityinvolves having a road that is too small, for the level of demanddepicted. Expandingthe capacityof the road may reduce the user's trip cost evaluatedat a giventraffic level. Long-runequilibrium is reachedwhen the minimumpoint of the short-runaverage total cost curve (equalsthe short-runmarginal cost curve)intersects the demandcurve. For the governmentalauthority, road capacityis a choice variable. By increasingits size, the volume-capacityratio dropsin the short run, and so does time cost. However,the cost of road capacityincreases. Intuitively,the highwayagency continuesto expandthe road until the marginalbenefit from savingusers' time costs is just offset by the marginalcost of one unit of capacity. It is at the output, Q* with an optimallybuilt road K*, in Fig. 6 that the valuationof the last trip takenjust equalsits marginal cost, that is, the incrementalcost of the trip to others,the motorist'sown time cost in congested traffic, plus the vehicle operatingcost and the road maintenancecost.3' The highway agency,by settingan optimalroad user charge whichis equal to the congestiontoll and the variableroad maintenancecost component,would be able to inducethe motoristto travel up to the point wherethe price of a trip equalsits short-runmarginal cost. By pursuingthis pricing policyfor each stretchof road, the use of a non-optimal,existing highway network would be optimized.Further, by expandinghighway capacity up to the pointwhere the quasi-rentof each capitalfacility just coversthe costof reproducingit, with zero (economic)profit remaining, the net benefitto the communitywould be maximized.By symmety, abandoningor downgrading roads is necessarywhen economiclosses occur. The decisionof not maintainingroads is tantamountto the act of disinvestingroads.
AII Formally,given a particularlevel of output,the cost-iniming autority wouldexpand the roadup to the poiat wherethe marginalvaluation in timesavings due to a unit increasein capacity, - Q *AAVC(Q,K)/IK, equals to the marginalcost of a unit of capacity,R. R is the rentalcost per timeperiod of capacity,which includes the invariatemaintenance and otheropeting costsof a road, depreciationand imputed interest on investedcapital. The negativesign would offset the inverserelationahip of AVCand K, yieldinga positivemagnitude for the entireterm. Alternatively,the road is to be expandedup to the pointwhere the marginalextenal congestioncost just offsetsthe marginal cost of investmentin capacity.This is the secondoptimality le: the optimalinvestmt in capacitynle.
D1/ The superscript* symbolindicates that that variableis optimized. - 27 -
39. Henceforth, to simplifyboth our discussionand the diagrams, we ignore the individual's vehicle operating cost and the variable road maintenancecost since they are self-financing.2/ Note that our conclusions thus far hold under the assumption of constant returns to scale and perfect divisibilityof roads. Consider a three-lane road with capacity K3 in Fig. 7(a), where output is now measured in vehicles per hour.=/ Since the highway authority has efficiently
priced the road by setting the congestiontoll t3* and optimaly built the road by investing at K3*, it can be seen that the toll revenue covers the fixed cost of the road. So far we have simply translated Fig. 6 into Fig. 7(a) with the costs borne by motorists convenientlyleft out but not forgotten. Assume that both traffic volume and road width, i.e., the number of lanes, are doubled and that the inputs to each of the componentcosts under constant returns are doubled, then i) the fixed costs of construction, maintenanceand depreciation,ii) the variable time costs, and fii) the total costs are all doubled.38/ Intuitively, the geometric doubling of the rectangularareas of road constructionand maintenancecosts is synonymouswith the condition of zero economies of scale (given fixed input prices) in road construction and invariate maintenance. Similarly, the horizontal doubling of the rectangular areas of the time costs suppliedby individualusers is akin to the technicalcondition of zero economies of scale in road use.
36/ I have tbereforegrouped the non-trafficrelated ma_itc and operting costs as part of dte short-runaverage fixed coat curve. Bear il mind that the variableroad maitenance cost (not drawn in Fig. 7(a)) is being recoveredby a separateroad user charge componentas shown in Fig. 2. With the condition of zero economies of scalein totalroad mainltnce, the total mintenancecost is thusexactly doubled. Also, economicprofits fromnow on willbe refenredto as 'profits'. n/ Three lane roads are foundin Australiawhere they are referredto as, 'two-and-a-halflane roads' by Hoban (1987).
Lg/ The stringentassumption of a roadbeing finely divisible will be relaxedand indivisibilitiesintroduced later on. - 28 -
XH. FIRST-BBSTOPTIMAL PRICING AM INVESIMNTRuLE
Empda Considerations:
40. Byestimatng behavioral travel demand functions using state-of-the-art logit mode choice models, one couldobtain empirical esdmates of marginalvaluation of time (as a functionof incomelevels) (Hau (1986)). When combinedwith a fine-grained,parametric transportation corridorsupply model of the San FranciscoBay Area (Talvitieand Associates(1978)), multi- marketdemand and supplycould be equilibratedand cost-benefitanalysis of alternativepolicies performed(Hau (1987)). Heuristically,the adjustmentprocess towardsthe final equilibium parallelsthat of the cobwebequilibrium model of adjustment. Optimaltolls and welfaregains and losses could thus be simulatedwith apprpriat specificationof the marginaltravel time function. The resultsobtained are for a short-runequilibrum model of demandand supply.
41. Even with poor data, one could make some progress in empiricalwork. Given an estimateof a speed-flowcurve and the correspondingtravel-time flow curve, we know how these engineeringcurves can be convertedto a short-runaverage variable cost curveof a trip, usingan estimateof the valueof time. A 'supply' elasticityestimate together with a value for unit variable cost would yield a one-to-onecoaepondence betweenthe short-runaverage variablecost and the marginalcost (see footnote15 for formula). A rmughestimate of the demand elasticity and the traffic level of a particular road would yield a first order approximationof the propercongestion toll. Now, in order to maximizeaggregate net benefit, two operatingrules shouldbe followedby the road authority.
The First Rule - The OptimalPridng Rule: For each stretchof road, short-runmarginal cost pricing is fulfilledby settinga toll at the excess of short-runmarginal cost over short-nm averagevariable cost. The intuitionis that this congestiontoll wouldserve to internalizethe congestioncost that a driver imposeson others. In addition,the motoristis chargedanother componentwhich covers the variablemaintenance and opeating costs of a road which he imposeson the road authority. Thus the publicauthority's imposition of an optimalroad user -29 -
chargewould cover both the extrnal costof congestionas wellas thevariable road maintenance and operatingcosts.
The SecondRule - The OptimalCapacit Rule: Underconstant returns to scale and optimal pricing, whenevereconomic profit is foundin the operton of one road link, the procedure wouldbe to expandthe capacityof that stretchof road. The existenceof a loss undershort-run marginalcost pricing suggests that the road has beenoverbuilt. By alteringthe capacityof each road in the long run accordingto the quasi-marketsignal of profits and losses, the entire highwaynetwork's investment level in capacitywould be aptimized,with the fixedcost of each road covered. Alternatively,the road authority,by trading its direct resourcecosts against individuals'travel time, followsthe rule of settingthe marginaltravel time savingsequal to the matginal cost of investment for an additional unit of capacity. The capacity of a road is expandeduntil the marginalcapital cost equalsthe marginal(external) congesdon cost.
erWpectiveon the Result:
42. What we havedescibed is the long-runequilibrium of an optimallydesigned capacity of a road network under constantren. If the road authoritywere: 1) to pursue the efficiency-enhancingpolicy of pricing accordingto the margnal cost of a trip, and 2) to minimizethe sum of the directresource costs of providinga road and thevalue of user-supplied travel time inputs, then the road wouldbe both efficientlyutlized and optimallyexpanded. Noticethat the optimallydesigned road has a positiveamount of externalcongestion cost. This results from the road agency's desire to minimizeboth the sum of the direct cost of the investmentin capacityand individualdrivers' travel time cost. In our simple framework, congestiondelay wouldnever be entirelyabsent, contrary to what environmentalistsand road users wouldprefer, becauseachieving zero congeson is very costy to the community.in other words,an optimalamount of congestionextenality is a validconcept, just as an optimalamount of pollutionhas long beenrecognized in the envinmental economicsliterature. Whatif there is no congestionon a particularroad? Zero congestionmeans that that stretchof road has been overbuilt(or priced non-optimally)and shouldperhaps be downgradedor even abandoned. If - 30 -
excesscapacity occurs all the time, the road possessesthe non-rivalconsumption characteristic of a pure publicgood. Thenwe are facedsquarely with the standardtask of provisionof public goods. If resourcesare plentiful,financing the shortfallvia generalrevenue taxation has been theconventional dictum. Similarly,if publicresources are scarce,the opportunity cost of public funds must be accountedfor.& If lump sum taxationis infeasibleand resources are severelyinadequate because of politicalconstraints, then it is possiblefor one to considerthe feasibilityof financingvia Ramseypricing.0 If it is uncertainwhether the (marginal) deadweightlosses from general revenue financing exceed those obtained from Ramsey taxation, financialand other considerationssuch as equitymay then have to be appealedto in order to justify the potentialuse of Ramseypricing in the road sector.
43. Bycontrast, a road wouldpossess the .ival consumptioncharacteristic of a privategood whenexcess demand occurs. Hencea congestedroad is also regardedas a congestedvariable- use publicfacility. Becauseof this mixedgood nature,and based on the theory derivedhere from first principles,the provisionof road servicesought to reside with the public sector. Under tL ;onditionof constantreturns, the optimaltoll revenue,which captures the quasi-rent earnedfrom the investedcapital, would cover the entire fixedcost of the road in the long run. No residualor overheadcost need be allocated. If profit exists, then it is becausethere is insufficientroad capacity(or pricingat a levelabove marginal cost). The road is thereforenot
9/ In the comion paper (Hau (1991)), I present calclatios of the opporunity cost of finacing alternative chang mechianms.
40I Frank Ramsey's(1927) inverse elasticity formula under the case of independentdemands was discoveredin rens to Pigou's question of how to set tax rates in order to minimize the welfare losses associatedwith meeftinga tax revenuerequirem The problemof the choiceof optimaltax rates whichare subjectto a revenueconshaint is formallyequivalent to thatof the settingof optimalprices which are budget-constrained. Ramseyprcing in the presenceof axtalities reqires thatit be computedon the basisof maginal costand a facdonof the marginalextemd cost (Oumand Trethaway (1988)). Clearly,in the presenceof leakages,non- excludabilityand partid rivalryin consumption,the uirementsfor the implemtaton of Ramseypricing are considerablymmre stingent than those of pursuingmarginal cost pricing. On conceptual,empirical and implementationgrounds, marginal cost pricingis superiorto even the simplestform of Ramseypricing: the invene elasticitynrle. The computatonof Ramseyprices, for instance,requires the estimationof margial cost as a prerequisiteand the determinationof revenuetargets. Despiteall the problemswith which Ramsey pricing is fraught,research into this itring issuois potentiallyuseful. Afterall, increasinglytight fiscal constains, countrieswill demandalteative fundingmachaims. - 31 -
in long-runequilibrium. The existenceof profitserves as a surrogatemarket signal to expand capacity. The mottois as follows: "If a road makesmoney (i.e., economicprofit), expandit, else not." Similarly,if a road loses money,it suggeststhat plannersmade the wrongdecision or were given over-opdimisticforecasts of travel demand. In that case, marginalcost pricing is still to be adheredto, with the congestiontoll set close to nil. A user chargecomponent is still neededto coverthe variableroad maintenancecost. Thusit may even be worthwhileto abandona money-losingroad and save on any annualinvariate maintenance costs that might arise. Efficientpricing, financialviability and cost recoveryare thereforeentirely consistent with one anotherunder constant returns to scale in long-runequilibrium.
XNV. RELAXAnONOF AssuMPTIONs
44. The above discussionassumes that the governmentaims to maximizewelfare of the communityby simulatingthe workingsof a competitiveindustry and pricinghighway services at marginalcost. Thereare a few majorassumptions that needto be relaxed: 1) constantvalue of time, 2) staticdemand, 3) perfextdivisibility, 4) constantreturns to scale and 5) variability of road thickness. We considerthe relaxationof each assumptionin turn.
1) Differencesin Time Valtion
45. The traditionalpresentation of road pricingand my ensuingcritique assume a constant value of time (Walters(1961a))A' 1 The diagrammaticanalysis in Figs. 2 and 3 implicitly assumesthat every driver is identicaland maintainsthe same time valuation. What happens when there are heterogeneousmotorists, with different time valuation and tastes? A mathematicalproof that generalizesthe aboveresult for homogeneousdrivers to heterogeneous ones with differentvalues of time is shownby Mohring((1975), (1976, Chapter 4 Appendix) and Strotz(1964a, 1964b), but theintuition behind it is not difficult. Insteadof the optimaltoll
41/ In his pionering work on road pricing, Walters (1961) (and authors thereafter) assues that traffic is homogeneous,with all vehiclesand drivers being the same, with the resultant identicalvaluation of time for all. - 32 -
beingbased on a representativedriver's valueof ime, the timevalue is nowa weightedaverage of the differentmotorists' valuation of time, weightedby the numberof trips taken by those motoristswho actuallyuse the facility. If a traveller'stime valueand the numberof trips are close to the average,he will pay the averagetoll payment. If anothermotorist's time valueis higher [lower]than average, he would be willing to pay more (lessi than the average toll payment for taking a trip. He thus would be willing to, though begrudgingly,pay the difference.The congestiontoll, P' - P", in Fig. 3 thencan be labeled the wighted congestion toll, and the constantvalue of time is re-interpretedas the weightedaverage valuation of time. For a trip witha sufficientlyhigher than average time value,the time savingof a trip, (P - PI% can be even higherthan the weightedcongestion toll, P' - P", thus makingthe motoristbetter off. On the other hand, for a (shopping)trip havinga lower time valuation,the user still has to makethe averagepayment and thereforewould be madeworse off. Nevertheless,they both remainon the tolledroad, as opposedto beingtolled off, becausetheir individual trips' marginal valuationor maximumwillingness-to-pay still exceedsthe generalizedcost of their respective journeys. The use of alternativevalues of time wouldrelax the pointI madeearlier that road pricing wouldmake aUl groups except the governmentworse eff. By relaxingthe assumption of a constantvalue of time for everyone,those people with high valuesof time wouldbe made better off at the expenseof thosewith low valuesof time. This intuitiveanalysis assumes that everyoneis facedwith the sametoll, as in the workingsof a competitiveeconomy, and that a perfectly discriminatingmonopolistic authority is non-existent.
46. We concludethat a singletransportation facility with differencesin valuesof time would not alter fundamentallyour derivedresult, uing the standardassumpton of constantreturns. Again,with efficientpnrcng, financialviability and full cost recoveryare achievable.
2) DemandVariabilit and Peak-LoadPricing
47. We havein fact consideredthe caseof variabledemands, Iwr ala, whenwe discussed the welfareimpact of road pricingon the peak and the off-peakperiods. There it was shown that a congestiontoll is neededduring the peak whenthere is excessdemand but not duringthe - 33 -
off-peakwhen there is excesscapacity. Noticethat becausethere is free-flowingtraffic in the off-peak,no tollingis requiredbecause no externalcost of congestionis generated. Therefore no quasi-rentis beingearned on the investedcapital and only the variablecosts are paid for by the traveller. On the other hand, during the peak period, a positive quasi-rentis earned. (Supposefurther that there is an inter-peakperiod, somequasi-rents are also generated.) With highwaycapital stock remaining unchanged, the systematic,diurnal nature of travel demand(as opposedto the static,invariant demand case) meansthat the sumof quasi-rents(rather than just the quasi-rentfrom the singularpeak period itself)of the investedcapital should be compared with the cost of the highwayfacility. In other words, whenall the quasi-rentsover the entire demandcycle are summedup and comparedwith the capitalcost, expansionof the highwayis either warrantedor not, under constantreturns. The same conclusionsobtained thus far again hold.F
48. P intereting implicationis that the entire capitalcost of the highwayis 'allocatedto' and bome by peak travelers, mainlynsh-hour commuters. This surprisingresult may seem 'inequitable',yet it is perfectlyconsistent with efficiencyanalysis. After all, it is peak users themselvesthat createcongestion and theythat demandthe useof heavilycongested expressways whichrequire massive infrastructure developments. Without them, the optimalsize of the road wouldbe considerablysmaller. The result of allocatingall capitalcosts to users of the peak period has long been recognizedin the literatureon the pricingof publicutilities a la Boiteux (1960). Followingour earlier example, the extent to which there is another period - the interpeakor shoulderperiod - with even a modestamount of congestion,would allow for differentialpricing and thus the allocationof somecapital costs to these interpeaktravellers. The optimalinvestment rule is then to expanda road until the sum of the quasi-rentsover the demand cycle equals the entire capital cost of the facility under constant returns. By inplementingpeak-load pricing and altering the investmentlevel of the highwayfacility, dependingon whetherprofits are positiveor negative,the highwaynetwork is againoptimized.
4/ Theouput vuiable needs to be rodefinedas vehiclespe laneper cycle, with the cycle being the durton of a particularchugin period. - 34 - Hence,the consideration of demandvariability and peak-load pricing would not changethe status of our conclusions,in the presenceof differencesin valuationof time. The fact that the fluctuatingdemands over the variouspeak, off-peakand inter-peakperiods of a demandcycle are linkedby a fixed capitalfacility and the observationthat the consumptionof trips must be satisfiedby the productionof trips duringthat particulartime periodcombine to yielda simple modificationof our result. Pricing, financialviability and cost recoveryare again consistent with one another.
49. Keeler and Small (1977) show rigorouslyhow the Mohring-Harwitzframework developedhere is extendedto the case of variabledemands using peak-loadpricing in the presenceof independentdemands and no indivisibilities.43By assumingthe demandin each periodin fact dependson other periods,i.e., the caseof dependentdemands, the derivedresults still go through(Mohring (1970))AY'
3)ndisiaik
50. Whilestill retaining the assumptionof constantreturns, but accountingfor differences in valuesof time and demandvariability, we proceedto drop the assumptionof a road being finely divisible. Road construction,in fact, involvessignificant indivisibilities that cannotbe ignored. For example,a road mustpossess the minimumwidth for accommodatinga standard- sizedautomobile and shouldalso, ideally,be bi-directional.In the perfectlydivisible case, the long-runaverage total cost carve which envelopesa continuwmof closely-packedshort-run averagetotal cos; cur. es at their minimumpoints is madehorizontal. A flat LRMCcurve also coincideswith the correspondingLRATC curve (see Fig. 7(b)). Due to the presence of
431 It is due to the asumptionof independet dems Ihat long-rn marginalcost pricing (equalsto short-nm margnalcost pricing) stil holdsat each timeperiod. The conceptof long-nmmarginal cost pricingis blurrd in the case of jointnessof demand
44/ Usingthe distibutionof currentdemand distribution as given,which is synonymouswith asumig independent demands,would result in upwardbias in peakperiods and downwardbias in off-peakpenods becauseof the possibilityof substitution(Keeler and Small(1977)). - 35 - indivisibilities,however, the formerlyneat and continuouspattern of theLRMC curve is broken (Neutze(1966), Kraus (1981b)). The new long-runaverage total cost curve is now composed
of a seriesof short-runaverage total cost curves,where SRATC 2, SRATC4 and SRATC6 denote a two-lane,four-lane and six-laneroad respectively.The long-nmaverage total cost curve is a seriesof short-runaverage total cost curvesconnected together in a scalloped-likepattem (see the solid LRATCcurve labelled ABCDEFG in Fig. 8).AV The long-runmarginal cost curve takes on the various short-runmarginal cost curves in the respectiveregions, resultingin a discontinuousshark's tooth-shapedLRMC curve (see the thick LRMC curve in Fig. 8(a)). Henceone is alwaysworking with the short-runcurves themselves since one couldonly operate with a capitalfacility which is given. By now, it shouldbe clear that whenevera short-run marginalcost curve rises abovea short-runaverage total cost curve, profits can be obtained under short-runmarginal cost pricing. Thus, if demandhappens to intersectthe short-run
marginalcost curvesin the regionof Q 2Q24, Q 4 Q4 and Q6Q&, and multiplesthereof, then the road makesmoney in the long run underconstant returns. Per contra,to the left of the outputs,
Q 2 , Q%and Q6, namely in the region of °Q2, Q2 4Q4 and Q44Q6, the road loses money. With a 2-laneroad, as trafficincreases, the road's large fixedcost is spreadover the additionaltraffic, and as congestionsets in, the road beginsto make money. In other words, as travel demand continuesto grow alongthe trend,adherence to short-runmarginal cost pricing suggests that the road wouldgo throughan unavoidablecyclical pattern of deficit,surplus, deficit, surplus, etc. Whetheror not one undertaks a road expansionproject fromtwo to four lanesdepends on the magnitudeof the net benefitpie, takinginto accountthe welfaregain and loss triangles.
OptimalPricing and Investmentwith Indivisibilities:An Example
51. Consider the demand as depicted in Fig. 8(b), with the actual considerationof indivisibilities.First, we followthe first-bestpricing rule of tolling the differencebetween
4S/ A similr set of discontinuouscurves is foundin, for example,Bennathan and Walters(1979, Fig. 2.2). Note thatthe scalloped-like pattern is asymmetricbecause the unit cost curves for four andsix lane roads are horizontal multiplesof thoseof the two-laneroad. In otherwords, under constant retns to scale, the AVCand AFC curves,and henceATC curves fan outhorizontally - a mistakethat is quitefrequently made in the literature (see, for example,Hayutin (1984, Fig. 2.8 and2.16)). - 36 -
short-runmarginal cost and short-runaverage variable cost, t*. This yieldsthe optimaltraffic levelof Q* and a positiveprofit. Withoutthe indivisibilityconstraint, the existenceof profit would indicate that the road is underbuilt. With indivisibilities,the direct one-to-one correspondencebetween economic profit and road expansionis lost. One is thereforeleft with the binary choiceof, say, expandingthe road from a 2-laneroad to a 4-laneroad. Usingthe welfareapparatus that we have developedin the Appendixand Figs. 3 and 4, however,the net benefitof such a moveis shownto be the sumof the welfaregain of going from Qz to Q**, as indicatedby the triangularareas b+c, and the welfareloss of movingfrom Q* to Qz4,as shownby the triangulararea a. Sucha movewould clearly be desirable. Witha 4-lane road, however,the optimaltoll of t** wouldbe insufficientto cover the fixedcost of this new road, resultingin a shortfall. Thus, even thoughthe movefrom Q* to Q** is a beneficialone, it wouldmean that the road authontywould switch from a profitableregime to a loss regimeafter incurringthe investmentcost.
52. Thus the optimal sequenceof decision-makingis first to establishthe policy of implementingmarginal cost pricing and then to plan future adjustmentsof the road network accordingto exeed futuredemand and establishedpricing policies. Whendemand fluctuates, pursuingshort-run marginal cost pricingat presentwould mean setting different prices, or tolls, in responseto expectedcurrent conditions.
53. Suppose the govermmentwere faced with a tight budget constraint. It would understandablythen be unableto undetak all publicprojects with positivenet benefits. If this road authoritywere mainlyconcened aboutcash flow and financialviability, it couldopt not to expanda road, i.e., under-invest,but stll chargea congestiontoll on the built-uptraffic and satisfy economicefficiency in the short run. This option however would not lead to the maximizationof society'swelfare in the long run. - 37 - * Returmsto Scale
54. The issue of whetherconstant returns to scale exists or not in road transportis a controversialand imporant one. Uldmately,it can be answeredonly via careful statistical analysis. The availableevidence in road transportationindicates that aU three cases exist: decreasing,constant and increasingreturns to scale (see Fig. 9) - paraleling the case of a
competitiveprivate firm and industry- withprofit, zero profitand loss, respectively.(This is but a well-knownresult of economictheory applied with slightmodification to the highway.) It is importantto realizeat the outsetthat the case of scale economies,or increasingreturns to scale with fixed factor pnces under least cost combinations,is merely a case of insufficient
demandwith respectto the marketsize in the long run - a point that is sometimesneglected. The implicationis that if trafficwere to growto a point wherethe capacityof a road is reached, congestiondelay would set in, and congestiontoll revenuescould be collected. After all, the short-runmarginal cost curveis alwaysnon-decreasing, as shownin Fig. 8. Profitsmay occur despiuethe fact that the long-runaverage and costcurve may be decliningand theconesponding long-mnmarginal cost curve lies below the averagecost curve.46 Then if traffic were to continueto grow as real incomesand auto ownershiprise, concomitantwith expressway expansion,the decreasingretun region wouldthen be encountered(see Fig. 9). In the case of incresn returnswith perfectdivisibility (where natural monopoly arguments lie), the long- run marginalcost curve below the long-runaverage total cost curve pulls it downwards, resultingin losses, beckoninggovernment subsidization. On the other hand, if travel demand is sufficientlyhigh relative to the engineeringcapacities of roads, the money-maldngroad enteprises wouldprovide muchsought-after funds which could be used to financeefficiently priced but money-losingroads - only if theseroads yield positivenet benefitsto society. We turn nextto a disussion of thetheory underlying the economiesvs. diseconomiesissue, together with perfect divisibilityvs. indivisibilities,and end with a review of the empirical evidenceand recentwork. di/ Thispoiut wM be e_bid laer in Fig. 12 and 13(a). Basically,with paefct or lmostperfect divisibilityand scaleoe_ondes, only losskswill occur, wh with indiviibilies, eithe profitsor losses will mlt depadu athM level of travel demand - 38 -
A. Economiesof Scale and Rural Roads
55. There is a preponderance of evidence -- geometric, engineering and otherwise - supporting the case that there are significant economies of scale in the construction of rural 47' roads (Walters (1968, pp. 180-82),Mohring (1976, pp. 140-42)).- The illustrations below follow Mohring's scale economy analysis using the geometry of transport right-of-way. In particular, a two-lane road requires a minimumof a twelve-feetwidth for each lane and a few feet for shoulders and drainage ditches. What this means is that a non-trivialproportion of the provision of a road's right-of-way involves dead space. These indivisibilities- required to contribute to the buildingof a minimumacceptable standard such as a given pavementthickness
and road size - help contribute to economiesof scale as the large fixed cost of constructionand invariate maintenancecosts are shared over greater amountsof traffic. Thus doublingthe width of a two-lane road more than doubles its traffic capacity, the so-called 'shoulder-effect' (Hayutin (1984), pp. 106 and 154). Further, we know that the engineeringor basic capacity of a two-lane road is about 2000 vehicles per hour. Since the standard four-lane road has an average engineeringcapacity of 1800 - 2000 vehiclesper lane-hour, doublingthe width of a two- lane road almost quadruples its capacity (see any Highway Capacity Manual, for example, Transportation Research Board (1985, Tables 2-1 and 2-2), yielding economies of scale associatedwith road use. However, capacity per lane remainsconstant beyond four-lane roads, resulting in zero economiesof road use thereafter. Further, in order to level hilly terrain and/or fill valley for transportationpurposes, the earth moving costs rise less than proportionately. In fairly flat or rolling country selected as sites for road building, doubling the width of a two-lane road generallyinvolves less than doublingthe earth movingcosts. (On steep hillsides, however, the reverse may be true.) Hence, for these three reasons of: 1) the existenceof large fixed costs due to indivisibilities,2) the technologyof road capacity, and 3) the possible reduction in earth moving costs, we can claim that there are economiesof scale associated with the construction of a two-lane to a four-lane road. Nevertheless,despite the fact that typical four-lane highways
471 Meyer,lain andWohl (1965, pp. 200-204)also found some evidence of scaleeconomies of ur*a roads. However,Keeler and Smal (1977, p. 5) queryMeyer, Kain and Wohl's findings, stadng that theirresults arein factbased on their inital assumptions. - 39 - possesstwo-thirds dead spaceand eight-lanehighways have only half the dead space, it is not clear from the geometryof highwayrights-of-way that economiesof scale in urban highway constructionexist. This is becauseit is ratherdifficult to controleconometrically for the effects of urbanizationand separateit from the effectsof size. For example,four-lane roads tend to be builtin ruralareas, where interchanges and overpasses are widelydispersed, and right-of-way costs are low. On the otherhand, six-laneor eight-laneroads are built mainlynear metropolitan conurbations,where expressway interchanges and overpassesare closelyspaced together, and landacquisition costs are high. In practice,the road authoritytends to tradeoff (andavoid) high right-of-waycosts with increasedtunnelling and flyoverconstruction costs. Lane expansion from a six-laneto an eight-laneexpressway at the margin, for example, would increasingly encounteralignment constraints associated with the terrain. (rhese constraintsmight explain whycapacity per lane is reducedwith six- to eight-laneexpansion (Mohring and Harwitz(1962, p. 97)).) This argumentis quite independentof whetherthe expresswayis locatednear urban areas. Henceall three casesof returnsto scaleoccur, resulting in theclassic, U-shaped long-run averagecost curve, parallelingthat of a firm withina competitiveindustry as we have seen in Fig. 9.
B. Diseconomiesof Scaleand Urban Roads
56. The discussionthus far has centeredon economiesof scale to road width for single roads, as opposed to a system of roads. Strotz (1964a) conjectures, and Vickrey argues convincingly,that there are considerabledisecononr -- of scale associatedwith an urban road networks.48/ The reasoningis based on the geometryof rc I network. Givena rectangular grid for an urban road networkspaced 2 kilometersapart, as in Fig. 10(a),there are 9 sets of (space-intensive)intersections and trafficlights in a 6 kilometer-widearea. Supposethe number of streetsis doubledin order to doubleroad capacity,yielding a grid of a one kilometer-wide spacing(Fig. 10(b)). Quiteapart fromthe possibleincrease of constructionor landacquisitions
Al/ I am extremelygratefil to WilliamVickrey for pointingout the subtletiesof theseargumets. See Vickrey(1965, pp. 287-288)and Mohring (1976, pp. 144-145)also for thebasic line of reasoning. - 40 -
costs, the numberof intersections- and the requiredland area and tffic light installations- is quadrupledto 36. (If no taffic lightsare installed,each intersection requires an evencostlier overpassor perhapseven a fullinterchange, a caseconsidered by Mohring(1976, pp. 144-145).) Moreover,despite the fact that trip lengthremains unchanged between the origin, 0, and the destination,D, the numberof trafficlights encountered, and hence waitingcosts, wouldthen doublefrom 4 to 8 (unlessoverpasses or interchangesare constructed).Either the installation of traffic lights or the buildingof overpassesand interchangeswould serve to bid up the opportunitycost of land (becauseless non-highwayspace would be left for businessor other activity)as wellas to increasesubstantially the sumtotal of the costsof undertakinga trip to the community,resulting in a risinglong-run average cost and also a long-runmarginal cost curve. The resultantlong-run equilibrium for an urban road networkin the presenceof diseconomies of scaleis depictedin Fig. 11. The analysisof the rlationship betweenlong- and short-runcost curvesis similarto the case of constantreurns to scale and is not repeatedhere. However, becausethe long-runaverage cost curve is rising, the short-runaverage total cost curve for a two-laneroad, SRATC2, is tangentto the long-runaverage cost curve to the right of the
minimumSRATC 2 point. Becausethe SRMC2 lies above the SRATC2, SRMC2 intersectsthe LRMCfrom below. Short-runmarginal cost pricing is equivalentto long-runmarginal cost pricingat the efficientoutput level Q2*. Noticethat the optimaltoll, tV*,follows from pricing at short-runmarginal cost and tollingthe differencebetween short-run marginal cost and short- run averagevariable cost. The differencebetween SRATC 2 and SRAVC2 is, by definition, SRAFC2. The optimaltoll, t2*, cLearlyexceeds the SRAF4 by the unit profit of 12*, with the correspondingrectangular area of profit wa*Q2*. As the urbanroad networkexpands from an existingsingle two-lane road to a doubletwo-lane road, say, substantiallycostlier construction, tunnellingand land acquisitioncosts are encountered.10 We have also establishedthat time costsrise becauseof additionalinterons and wait time required. Eitherof the abovetwo factorswould serve to pushthe SRAFCand SRAVCcurves up, togetherwith theSRATC curve. Just as thecongestion toll filledin the wedgecaused by the divergenceof short-runmarginal and
49/ raing finai cost of otuctin via tu ig aNd/orflyover., together with high land r_smptioncod, ae comdmt withthe findiypof Hau(1989) for Hong Kog. - 41 - averagevariable cost curves,the divergencebetween long-run marginal and averagetotal cost curves serves as an indicatorof the unit profit (or loss). In competitiveequilibrium, all economicprofits are competedaway in the long run, so the questionsthat followare: in what senseis the case of diseconomiesof scale a 'long-run'concept, and whatis the interpretation 4 of theprofit areas of r2* Q2* and v4* Q 4 * (for demandcurves Q 2 and Q, respectively)?The existenceof economicprofits in the long run is attributableto the rents earnedby an invaluable
fixed factor of production- land.5Q/ Even thoughboth the SRAFCand the SRAVCcurves reflectthe increasein costs mentionedabove, the existenceof the risingopportuniy cost of the fixedfactor of the remainingparcels of landis stillleft unaccountedfor. Intuitively,just as the driver, in the short run, by imposingexternal congestion cost on othersdue to his presenceon the road, is chargedfor it, so also shouldthe urbancommunity, in the long run, chargefor the use of scarce urban land in a market economy. Putting it another way, if all factors of
production -- includingland - were doubled, so that a scarcity value could be imputed to land, all economicprofits would be competedaway and vanishin the long run. Clearly,the supply of land cannot be doubled, so it is the existenceof land rents which yields (the areas of) economicprofits in Fig. 11. Noticethat we couldno longeruse of the existenceof profits as a surrogatemarket signal because of decreasingreturns to scale. Sincethe urban road network is supposedto generatesubstantial sums of moneybecause of high land values, relyingsolely on the profitmechanism and incautiouslyinvesting on urbanroads until aUeconomic profits are competedaway would resultin over-investmentin road capacity. With diseconomiesof scale and divisibility,roads geneate positiveprofits. Performingproper projectappraisal of roads cannottherefore be circumvented.
C. Diseconomiesof Scale and Indivisibilities
57. The case of disoconomiesof scale combinedwith indivisibilitiesis similarin spirit to the analysisof the constantreturns case with indivisibilitiesin Fig. 8. For instance,at any
IQ/ A puristMh agu ttthes rs actullyacorns to lanonesmd thusualudoly of longnmsuper- profitsbeg zeo sotillholds i tb pnpo of risnglog-nm aveag costs. - 42 -
momentin time, the regionsto the right of Qz,, and Qg in Fig. 11 would yield profits. In the rise of decreasingreturns to scale, with or withoutindivisibilities, the correct recourseis to investonly if the road projectin questionpasses a stringentcost-benefit test and only if rising costs of roads are comparedto the quasi-rentsgenerated L the variable demand periods diurnally. In other wordsshort-run marginal cost pricing,or congestiontolling, as opposedto long-runmarginal cost pricing, shouldbe implementedfor each period over the demandcycle and the quasi-rentssummed up, regardlessof whetherthe facilityis optimallybuilt. When demandis not knownwith certainty, probabilistic or expecteddemand could be usedinstead and all real benefitsand costs over the economiclife and time periods of a project should be properlymeasured, discounted and comparedwith the capitalcost of implementingthe change. If finelydivisible projects are available,we have seen that the procedureis to invest until the marginal benefit of a project -- in the form of time savings -- equals the marginal cost of constructingthe capital facility. If indivisibilitiesabound, then traditionalpublic investment criteria using the net presentvalue criterionor Mishan's (1988, Chapters35-38) normalized intemal rate of return procedurecould then be complementedwith the diagrammaticwelfare analysispresented in this paper. In sucha case, unfortunately,the neatlinkage between profits and lossesas a 'market' mechanismand guideto investmentis severed. Thus it is possiblefor a project with high net benefit to yiew a financialdeficit. As mentionedbefore, if a road authoritywere faced with a severefiscal constraint, then the road agencycould choose to under- invest, rather than over-invest,when faced with an all-or-nothingsituation of indivisibilites. Still, it shouldpursue marginal cost pricing in the short run, whilefully takdnginto accountthe opportunitycost of congestiontoll financing(see Hau (1992)). In this way, short-runmarginal costpricing would yield botheconomic efficiency and profit, eventhough an optimalinvestment strategywould generate an even higherlevel of net benefitfor the communityin the long run.
D. Economiesof Scaleand Indivisibilites
58. Per contra,the aboveanalysis for the case of decreasingretums to scale of an urban road networkcarries over in reverseto the caseof increasingreturns to scale for rural roads. There the sumtotal of thequasi-rents captured via short-runmarginal cost pricingis insufficient - 43 -
to cover the entire fixed cost of a particularrurl road. Scale economiesabound in the
constructionof rural roads, so that unit losses,12* and L4*, resultin the case of an optimized two-laneand four-lanerural road, respectively(see Fig. 12). Standardneoclassical arguments then call for subsidizationout of the public treasuryfor the case of scale economies. The presenceof both indivisibilitiesand scale economies:iuld alter the calculationof optimaltolls and subsidiessubstantially (Kraus (1981b)). It turnsout, perhapssurprisingly, that theexistence of indivisibilitiesserves to improvethe stateof affairsvis-a-vis the governmentbecause, as was shownin the constantand decreasingreturns to scale cases with indivisibilities,both surpluses and deficitswould occur alternately, depending on the levelof traveldemand. Similarly,in the case of rural roads with both scale economiesand indivisibilities,there are regions(such as those to the right of QO,, and Q4.1.in Fig. 12) where short-runmarginal cost pricing yields profitsrather thanlosses. This is because,with indivisibilities, the long-runmarginal cost curve - composed of joined segmentsof the short-run marginal cost curves -- is no longer declining all the waybut possessesa sawtoothedpattern, alternately exceeding its correspondinglong-run (and short-run)average total cost curvesand risingat an even fasterrate. Thus,just as in the caseof constantreturns withindivisibilities, whenever a SRMCcurve exceeds a SRATCcurve, profitexists and vice versa. It is thereforequite conceivable to havea congestedtwo-lane road whichgenerates profits even when subjectto increasingreturns to scale for sufficientlylarge changesin capacity.The existenceof lossesdoes not meanthat the road agencyshould cut back on the provisionof highwayservices. On the contrary,it merely meansthat other sourcesof funds ought to be soughtin order to financea worthyproject. The governmentauthority's decisionto cut back servicesjust becauseof losses in such a situationwould yield under- investmentand possiblystifle economic development and growthin the longerrun. The neat linkagebetween road expendituresand toll revenueshas disappeared.Again, the passageof a tough cost-benefitcriterion is thena prerequisitefor a projectto result in maximizingsociety's welfare. To repeat,only (fcongestiontoll financingis all thatis soughtby theroad agency,and not optimnlinvestment, is it possble to let a two-laneroad becomecongested in the face of rising urbanizationand motorization,while differentially pricing it via congestiontolls. In this way, efficientuse of a given road is enhancedvia short-runmarginal cost pricing, but the efficientlevel of road capacityis not beingachieved in the long run. - 44 - E. The Extentof Indivisibilitiesvs. Divsibilty and Their Effectson Scak Qs)=coomies
59. How often do we encountersurpluses in the presenceof scale economiesand deficits in the presenceof diseconomies?The answerdepends on the extentof thepresence or absence of indivisibilities.There are two viewson this issue. The fist perspectivea la Keeler, SmaU and Starkieargues that the aggregateroad networkcould be regardedas divisible. The other view, presentedby Walters (1968, Chapter 3) and Kraus (1981b), contendsthat roads are indivisiblebecause the main measureof highwaycapacity involves the discreetnessof the numberof lanes.
60. The constructionof a road or an additionallane may not be finelydivisible in and of itself, but taking the road networkas a whok, a single newly constructedfacility can be regardedas an icremental additionto the network,resulting in theapplicability of the foregoig marginalanalysis (Keeler, SmaU and Associates (1975, Chapter 2)). Also, oftentimes, varying somedimensions of road features,other than the numberof lanes, increasesthe capacityof the road network. For example,the lane width, the provisionof auxiliarylanes, horizontaland verticalalignments, and the surfacingof road shoulderscan all be variedincrementally (Starkie (1982)). Onecould characterize this viewby treatingthe lane capacityas a continuousvariable rther than a discreteone (Small,Winston and Evans (1989, p. 103)). If the road authority pursuesthe twin optimizingrules of pricingand investmentin capacity,then the road network wouldbe in long-runequilibrium. So with constantreturns and a divisibleroad network,roads wouldbreak even. However,some individual roads wouldmake moneyand somewould lose money. On the whole,if the economiesand diseconomiasof scale are "probablyroughly offsetting"as Meyerand G6mez-Ibgflez (1981, pp. 191-192)concluded, then the highway budget wouldbe balanced. Whetheror not scale economiesand diseconomiesare counter-balancing woulddepend on the degreeof urbaniation. Withincreasng urbanizaton,profits would tend to predominateeven aftercharging land rents as a cost. Withindivisibilities, the profit (or loss)
A/ Consnt or even decreang raturns wouldbe satified if two bi-directionalroadwys ae built in proximityto on another. -45 - regimeoccurs about half the time but it is unclearwhat the relative weightswould be when travel demandis reasonablyassumed to grow over time.
61. Underdecreasing returns and perfect divisibility, we haveshown in Fig. 11 thatprofits alwaysoccur. The essenceof that figureis combinedwith Figs. 13(a)through 13(c). Perfect (Fig. 11)and almostperfect divisibility (Fig. 13(a))and an urbanroad networkwould mean that the marginalcost priing of trips wouldalways be profitable. Note that the continuousregime of profit will not be uniformonce a thresholdlevel of indivisibilitiesis reached, resultingin regionswhere losses also occur. If the extent of indivisibilitiesprogresses from small but significant(Fig. 13(b))to severe(Fig. 13(c)),the regionswhich yield potentiallosses become even larger. The symmetrycaries over to the increasingreturns to scale case. Again, the substanceof Fig. 12 is culledand combined with Figs. 14(a)through 14(c). Withperfect (Fig. 12)and almostperfect divisibility (Fig. 14(a))in thepresence of scaleeconomies, losses would always occur. With scale economiesand a 'significant'level of indivisibilities(Fig. 14(b)), smallerregions of profit wouldbecome available but would disappearwhen approachingthe neighborhoodof thelimit (Fig. 14(a)). Nevertheless,even if one wereto acceptWalters' (1968, Chapter6) argumentthat there are sgnificantindivisibilities and scaleeconomies in ruralroads, we have demonstratedthat profits (and losses, of course) would neverthelessarse under congestiontolling. Scaleeconomies in thepresence of indivisibilitiesand financialviability are notnecessarily incompatible.
F. Empirica Evidenceon the ScaleEconomy Issue
62. Fitch and Associates(1964, p. 131)give some numericalsupport for the case of scale diseconomiesin the UnitedStates. Walters(1968, pp. 184-185),using Meyer, Kain and Wohl's (1965,p. 205)data, showsthat there are diseconomiesof scalein the constructionof four-lane, six-laneand eight-lane urba road segments.(By employing Walters' straightforward approach, Hau (1989)demonstrates that there are increasingcosts assocated with a sampleof four-lane, five-laneand six-laneroads in Hong Kong.) Withoutimposing any prior specificationsabout the extentof return to scale - Keelerand Small (1977)find evidenceof constantreturns to - 46 -
scale for a sampleof San FranciscoBay Area roads. Their often cited econometricstudy is importantbecause of the balancebudget implication for congestionpricing, a result whichwas also quotedby Newbery(1990).F' By contrast,using engineering specifications to estimate each of thecost componentsof an urbanhighway network model, Kraus (1981a) finds that there are increasingreturns to scale in road constructionin terms of length of freeway and interchangesbut not for overpassesand lengthof arterials. He makesthe crucialobservation that factor prices (such as right-of-wayprices) are to be held constantfor makingrelevant comparisonsof scale - specific(dis)economies. The reciprocalof his best "pseudo-empirical" estimateof returnsto scalein urbanhighway network capital costs is 0.84, whichtranslates to the economiesof scale degree of 1.19. Meyer and Gomez-lbgilez(1981, pp. 191-192),in assessingthe availableestimates in the conflictingliterature, conclude that economiesand diseconomiesof scale are "probablyroughly offsetting." Hayutin(1984), in an unpublished dissertation,refined the Keeler-Smallmodel and applied it to a sample of U.S. Interstate Highways.By includingboth intercity(or rural)and urbanroutes in her data set, she estimates that there are clearlyincreasing returs with respectto the numberof lanes. Her x,aultsbear out Mohring'sscale economy implications of highwaygeometry with respectto the "shoulder
5I Keelerand Small's(1977) result is one of the morerigorous econometric analyses bearing on the scale economyissue, iDthat they are able to separatethe confoundingeffects of size and urbanization. In particular,they regressedconstuction cost per lane-mileon the numberof lanes and variousdiscrete variableswhich capture the dfes betweenurban, suburbanand rural-suburbanareas. It is the inclusionof the lattervariables which enable them to ecomometricallycontrol for the effectsof expressway capacityon constructioncost. The selectedsample of 57 roadswas basedon all state-miainnedroads for nine San FranciscoBay Area counbes,including arterials, expressways and rural roads. They estimatedboth a non-linearand log-linearspecification, with both yieldingstatistically insigificant stastics for the esfimaeddegree of homogeneity.Based on the slightlybetter fit for the log-linear specification,the rturms to scale parmeter of (-)0.0305translates itself into the cae of ^mildly- increasingrtuns to scaleof 1 - 0.0305= 0.9695. By taing the reciprocalof the estimaedretwns to scale puamete, the economiesof scde degree of 1.0315is obtained. Since 1.03 is staiscally indiinpuisable^ fiom 1, they concludethat there is no firm evidencefor scale economiesin road conruction masured in termsof lanes. Thefr-reaching policy ramifications of thisresult could perbaps be buttesd if there were calculationsor discussionof the power of their hypothesistests. This is becausenot beingable to trect a null hypothesisis not necessarilyequivalent to acceptingthe null hypothesis.One hopesthat the typoII error is relativelysmall. However,to be fair to their important contribution,the powerfunction for publishd papersare seldomto be foundin the literaue. -47 -
effect" More recently, in reassessing the earlier work of Keeler and Small (1977), Small, Wimstonand Evans (1989, Chapter 6) use the result of zero economies (corresponding to the scale economy parameter of 1.00) as well as the mildly increasing returns case of 0.97 (corresponding to the scale economy parameter of 1.03) (see Small (1992, Chapter 3) for a review also). Fully cognizant of the possibilityof the existence of increasing returns to scale in urban highway travel a la Kraus, they presented their simulation results on highway finance based on three encompassingparameters for the degree of scale economy: namely 1.00, 1.03 and 1.19.
G. Recent Results on Cost Recovery
63. As part of a major World Bank research project, Newbery (1988abc, 1989, 1990) and Newbery, Hughes, Paterson and Bennathan (1988) extend Mohring's now classic result of highwayfinance by relaxing the assumptionof an infinitely durable highway whosecapacity can be continuouslyadjusted. It is common knowledgethat pavementswear out after extended use, but it is less well known that the damage of a vehicle on the road pavement is related to the weight per axle, as opposed to the gross vehicle weight per se. Common Practice (Highway Research Board (1962))is to measurethis damagingpower by the number of equivalentstandard axle loads (ESAL), where one ESAL is equivalentto the load of 18,300 lbs. (8.2 tons or 80 kilo newtons) singleaxle. The AmericanAssociation of State %ighwayand TransportationOfficials' milestoneroad test indicates that the damagingpower ot a vehicle on pavement is proportional
53/ Hayutin(1984, Chapter 4) regressedconstuction investment per mileol the numberof lanesand other variables simila to thoseused by Keer and Small (1977) (see previous footnote). Further, by including the paved width per lane variable aeoconomiesto width measuredin feet. rather than in lanes, result. Hayutin(1984, ChapterS) conludes that the strongereffect of scaPleconomies with respectto the number of lanes dominates the effect of dmiecomies associatedwith building wider footage. In an excellent review of the litentue surroundingthe scale economyiasue, Hayutin(1984, pp. 158-159)observes that Meyer, Kain and Wohl's (1965, pp. 200-214)conclusion that there are increasing returns to scale in feeway constuction stems fom thdir engeing assumptions about costs, as opposed to actaul estimation. Her stistical results, however, ae surive of their conclusions. My comment regarding the power of the hypotbesistests applies to her study also. - 48 -
to the fourth power of individual axle loads, acquiring the title of the "fourth-power law. For instance, the rear axle of a 13-ton light truck with two axles can result in over 300 to 2400 times more pavement damage than that of a large car weighing 2 tons.0 One inescapable conclusion is that almost aU damage is caused by heavy vehicles such as trucks and buses: relatively little is due to automobiles. Recently, Newbery (1988a) characterized another type of externality caled a road damage externality: the emenoal damages imposed upon the pavement by heavy vehicles result in increased vehicle operatingcosts to subsequentmotorsts for the rest of the periodic life cycle of a road. (Thus, the various costs of undertakinga trip mentionedin the introduction of this paper (footnote 3) should now be more narrowly defined as eermal costs, namely: 1) marginal exteral congestioncost; 2) road wear cost and damage exteralities (grouped under the rubric of road damage cost); 3) environmentalcosts; and 4) costs due to increased risk of accidents not borne by private parties.),W
/AI Formally,the damagig power(in EASLe)of an axleload, 1,in tons,is approximatelyequal to: (U8.2y), hece we say tht the damagingpower is proportionalto the fourth power of the axle load. As a hypotheicaleample, a 24 tontruck whose weight is distributedevenly among 2 axleswould cause more than 3 timesas muchdamage as the s truckwhose weight is distributedevenly among 3 axles:(2/3) -(12V8)'= 3.37S.
S/ Mheexample assumes that more of thoweight -8 tns - is distributedon the rear axle of the lighttbuck whers a fily loaded automobilehas its weightuniformly distrbuted among the two axles. Smalland yag (1988)and Small,Winston and Evans (1989, Chapter2 Appendix)have recentlydsput the fourth-pwe law and perfimd a statisticaanalysis of AASHO'sroad test data to show that the equivalencefactor for anaxle loadrises steeply to dte thirdpower, thereby proposiAg a Zthird-powerlaw instead. Whetherthe powerterm is actually3, 3.5 or 4 (as respectivelyclaimed by Small,Pateron, or engineersconventionally) is not as crucialas the significantmantenanc cost savingsto be made if vehicle damageis efficientlycharged for. Conceivably,an increasein the power term wouldenlarge the small deficittht Small,Winston and Evans obtained by charging for congesionand road damage costs. I argue tht the chargi of aU extraities includingenvirmentl and acciet costswould very likelyclose any rmining ap.
6/ Basedaon empirical ovidenc for the whole oad networkof Tuna, Nwboery,Hughes, Paterson and Beanathan(1988, Table 7, 23 andpp. 58-59) and Newbery (198&, Table 1) deamnstrutthat the extenal urbancogesion costsalon accountfor tho 'ovewhelmng fiaciaon(about nin4enths or more)of total extenal coss of roaduse (excludig environmea and accdet cost) for autmobiles and ulities. For heavyvehicles, the reverseappear to applyin both Tuniia and Ghana,with road damagecosts doin_ating instead (8e Gronau (1991) also). (Heggie and Fan (1991, Annex 1) argue that some of Newbory'scalcuaos of congestioncosts for TunisiaarM overtdmates.) Cog restioncosts constitut a argo shae of totalroad costs for boththe UnitedKingdom (nine-tens) andthe UnitedStates (one-filfh) in the klon run, whlera road damagecosts a about 3 to 3.5% in the United Kingdomand only 2% in the UnitedStates (Neowbory (1988a), Small, Winston and Evans(1989), Chapter 6). Pollutioncosts - 49 - 64. Newbery's(1988a) "Fundamental Theorem of RoadUser Charges' saysthat if. 1) the maintenancepolicy pursued by a highway authority is condition-responsive,i.e., road maintenanceis carriedout whenevera road's pre-determinedroughness level is reached,without being optimaUyset (Paterson(1987)); 2) the age distributionof roads over the life cycle is uniformlydistributed; 3) there is no trafficgrowth, i.e., trafficflow is constantover time; and 4) all road damageis causedsolely by vehicles,dten the externalroad damagecost is nil on average (Newbery (1988a), Proposition1). The variable road maintenanceuser charge component,applied on a per ESALbasis, wiUfully cover the averagecost of repair (Newbery (1988a),Proposition 2). Whenviewed diagrammatically, this resultis analogousto charging for maintenancecost in anotherdimension, as opposedto chargingsolely for the congestive effect of automobileson a per passengercar equivalent(PCE) basis in Fig. 3. Intuitively,the damagingforce of an additionalESAL has the externaleffect of raisingthe vehicleoperatng costsof subsequentvehicles over time,just as the externalcongestive effect of an additionalcar has on other vehiclesbehind it within a traffic stream. in addition,the indirect impactof accumulatingmore ESALsis to reach the preset, maximumallowable roughness level of the road earlier than expected,thereby precipitatng an overlayand a correspondinglowering of vehicleoperating costs. It is uncleara priorl whichof these two magnitudesdominate. Hence Newbery's(1988a, 1989) breakthrough was to prove, in a varietyof ways, that the additional vehicleoeaing costsattibuted to road damageexnalities just cancel out with the vehicle operatingcost savingswhen averaging over roads of differentvintages.
65. In the generalcase, the fundamentaltheorem relaxes Assumption 4) aboveby allowing for road damagesthat are independentof use, suchas weather-relateddeterioration. In thiscase of a condition-responsivemaintenance policy, the road damageexternality is no longerzero but is quantitativelynegligible. While the use-relatedpart of maintenancecosts is chargeable becauseof the road damagecost, the fracdon due to weatheris not allocable. Hence the presenceof weather-relatedroad damageresults in onlypartal recoveryof all maintenancecosts
appear to account for less tdan o _enhof total road cost in the United State, wheres accident costs ae an te sam= order as extera conWestioncoss for the United Kindom. (Newbery (1988b, 1990)). - so -
when only charging by road damage costs. Therefore, in order to close the highway budget deficit, Newbery (1989) needs to price for the marginal external congestion cost within a standard long run framework (see Newbery (1988b) also). That is, under the conditions of constantreturns to scale and road use, the efficientcongestion toll will yield revenues that cover the capital cost of the highway but only the invariant (or non-allocable) portion of road maintenance cost attributable to weather, see Fig.5 for definition of short-run fixed cost (Newbery (1989), Proposition 1). (FollowingMohring (1965), we have seen how this is done in our simple investment rule of comparing optimal toll revenues with fixed costs, be they constructionor maintenancecosts, see Fig. 5 and the surroundingtext.) It then naturally follows that the optimal road user charge, i.e., the optimalcongestion toll-cum-road damage charge, will then yield revenues that cover the capital cost of the road and the total road maintenancecost in a constant returns world (Newbery (1989), Proposition 2). In other words, the optimal congestion toll covers both the capital cost of the road and the non-allocable fraction of road maintenance, whereas the allocable fraction of road maintenance is still chargeable to traffic loadings via the average variable road maintenancecost component per ESAL. This explains why the total expenditureon road maintenanceis fully recoverable. Ifthe same constantreturns conditions and assumptions 1) through 4) again apply, and if we firther accept that 5) heavy vehicles predominantly use the slow lanes and are confined there, then the optimal road user charge will recover the capital cost of the highway and twice the total maintenancecosts of the road (Newbery (1989), Proposition 3). If all heavy vehicles are confinedto the slow lane, then the damage costs are accumulated in a shorter time span than if they were to be spread out evenly over all lanes. This has the effect of raising the maintenancecost of the slow lane and the effective cost of widening the whole road, since all lanes are typically resurfaced together once the performance service index of the road dips below the trigger point. Cost-benefit analysis of lane expansion to reap time savings suggests that the marginal cost of investmentin capacity would now have to account for both the increased road maintenance cost for road strengthening and the annuitized capital cost on a PCE basis. Tis is a slightly modified version of our earlier optimal investmentrule.) Moreover, heavy vehicles are charged for the traffic loadinigsthey create. While stronger roads are cheaper to maintain, investmentin road strengtheningis costly, resulting in costlier upkeep and higher capital cost. The larger fixed - 51 - costs translatethemselves, in the long run, into a requirementfor higher congestioncharges, which heal contributeto an overall budgetsurplus for the road authority,despite "massive" increasingreturns to road strengthening(Newbery (1989)). Another way of lookingat thisresult is that the congestiontoll effectivelycovers the road's entirecapital cost as before, whereasthe maintenancecost is recoveredtwice -- once via the variableroad maintenanceuser charge componentapplied on a PCE basis and the secondtime around via heavyvehicles per ESAL.
66. The mostserious drawback in the propositionsabove is the conditionof constantreturns to scale.S7/ We have reviewedthe empiricalevidence and find that there are increasingas wellas decreasingreturns to scaleoperating on differentparts of theroad network,meaning that deficitsas well as surpluseswould mostlikely co-exist.
5O Mariabili of RoadThickness
67. We are now in a positionto formallyrelax the implicitlyused assumptionof a given road thicknessand to incorporatethe latest work into our model. In Road Work: A New HighwayPricing and InvestmentPolicy, Small, Winston and Evans recently provided a comprehensiveexposition of their technicalextension of Mohringand Harwitz'slong-run result on highway economics with optimal durability, within a standard neoclassicalwelfare maximizationframework.a/ Insteadof taking current highwaydesign standards as given,
57/ Newbey (1988a)regards the lack of the pursuitof a condition-responsivemainnteace strategyas the most serious limitadon. Further,Neowbey (1989) cites Keelerand Small's (1977)result of constnt reuns to scale,which is basedon a sampleof SanFrancisco Bay Area roads. Newbery(1990) also cites Kraus(1981a) as evidencesuwporting constant reatuns when Kraus demonstates lightly cicsing retuss (seeprevious section). Based on an earlierversion of Newbery(1989), Heggie and Fon(1991) take issue with manyof Newbery'sassumptions. 58 Newbery(1989), for example,derives opimal durability(or strenglh,rather) and confinesthe extentof his analysispricipally to the contant reus to scaleworld. By coDtra, Small,Winston and Evans (1989, Chapter6) explicitlyexplore the caseof incrasing retuns to scale of road consruction,as well as durability,and preet simulationreslts of anurban expressway and arteial - withsensitivity analysis - witdina long-rn equilibum famewor Ihe firstorder conditions of theiroptimization problem yield the optimalpricing, investment and durability rles. -52 -
another characteristic of a road -- its thickness - can be varied. A thicker pavement would serve to withstandthe damagingpower of trucksmore and therebyprolong the life of a road. Despite the fact that there are tremendouseconomies assocated with road durability, the additionalstrengthening of the road wouldstill substantiallyincrease the total investmentcost of an overlay. Usingthe standardoptimization technique of consumer'ssurplus and producer's surplusmaximization, the intuitionof the three allocationrules derivedare againbased on the simplenotion of settingmarginal benefit equals to marginalcost. That is:
1) Thefirst rde - the optimalpricing rie - says that the traveller should undertake a trip only up to thepoint wherethe incrementalbenefit just offsetsthe incremental cost to the community. A vehicle'sentry into a transportcorridor results in two effects: the congestiveeffect which dependson the numberof PCEs, and thedamaging effect based on the numberof ESALs. Thus our optimalpricing rule derived earlier is adjustedto includean extra component- a road damagecharge - whichbalances out any short-fall. 2) Thesecond rude- the optimal capacity rde - says that, with a condition-responsive maintenancestrategy built in, optimalinvestment for highwaycapacity is then to expandthe widthof the road up to the point wherethe marginalcost of capacityplus maintenancecost is equivalentto the resultanttime savings.
3) Thethird rile - die optmal durability rie - says that the road is optimallystrengthened by investingup to the pointwhere the marginalcost of durabilityjust equalsthe savingsin vehicle operatingcosts to other motoristsplw the associatedsavings in maintenancecosts due to a thickerpavement. The two latter rules could be regarded as an investmentrule extendedinto two different dimensions. Thus the three rules of optimalpricing, opimal capacityand optimaldurability yield efficientprices and trips, as well as the optimalnumber of lanes and inchesof pavement thickness.
68. Thereis evidencethat there existeconomies assodated with road strengthening(Small, Winstonand Evans (1989, Chapters 2-3), Winston(1991)). By reanalyzingAASHO's road test data within an economicoptimization frmwork, Smalland Winstonreport that the opimal thicknessfor a rigid pavementis an inch and a half higher than the currentten inch standard, -53 - whichfollows AASHO's guidelines. The remarkablefinding is that a mere increasein thickness of 15% would lead to a doublingof pavementlife from 13 to 26 years (Smalland Winston (1986, 1988), SmaUand Zhang (1988). The logical implicationof massiveeconomies to
increasingroad thickness - a fact long known to highway engineers - is losses for the road authority. We thereforeask: is there a theoreticalreason that wouldallow us to still adhereto the impeccablemarginal cost pricingprinciple and yet achievethe goal of cost recovery? To answerthis questionwe turn to the notionof diseconomiesof scope.
Economiesof Scopevs. Diseconomiesof SMcpe
69. The conditionof constant,increasing or decreasingreturns to scaleyields the respective resultof breakeven, loss or profit, respectively(see sectionXIV - 4 and Fig. 9). Tbe empirical findingsof returns to scale evaluatedearlier suggestthat decreasingreturns to scale in road constructionin urbanareas mayperhaps offset the increasingreturns to scale in rual areas, but that the highwaybudget may stiUbe in deficitdue to economiesof road strengthening.The answercan only be establishedusing econometric analysis. Even if we were to acceptKeeler and Small's (1977)carel findingof constantreturns, wouldit not seemplausible to arguethat the pavementdeficit due to significanteconomies to road durbility necessarilyyield an overall deficitfor the road authority? The answeris no.
70. Up til now, we have confinedourselves to the singleproduct world of trafficvolume. But road transportinvolves two products: namelytraffic volume and loadings,which carries us into the literatureof multiproductindustries and returns (Baileyand Friedlaender(1982)). The issue is whethera multproductfirm canjoindy manufacturethe variousproducts cheaper thanif each firm were to producean outputseparately. If it is cheaperto combineoperaions and share in thejoint costs, thereare economiesof scope. The notionof diseconomiesof scope can best be grsped by ilusating the designof a railwaytrack (Kim(1987)). A railroadtrack that is built to withstandthe axle loadingsof freightwould need greater strength and thickness, whichconflict with the requirementof securinga smoothride for train passengers. Ensuring both thicknessand smoothnessof tracksis morecostly, resultingin diseconomiesof scope. The -54 - analogycanries over to roads, whereit wouldcost more to producea highwaythat is thick enoughto handleheavy vehicles and yet wideenough to accommodatethe considerablylarger numberof automobiles.Thus there are no economiesof scopeeven thoughone product -- the numberof loadings-- is clearlysubject to increasingreturns to durability,whereas the other product - the traffic volume -- is potentially subject to increasing returns to scale due to road construction. In other words, it is the other characteristic of production -- the scope -- that wouldtip the budgetbalance away from a deficit(Small, Winston and Evans (1989),Chapter 6).
71. Automobilescause virtually no road damagescompared with heavyvehicles but trucks are fewer in number and therefore cause lesser amount of congestion. Hence one could separate, roughly speaking,traffic flow from loadingsby identifyingautomobiles with the formerand truckswith the latter. Trafficflow, in tum, requiresroad capacitywhereas loadings requireroad strengthening.The empiricalfinding of "modestbut significant"diseconomies of scope by Small, Winstonand Evans (1989, Chapter6) of about 6 to 10% tips the product- specificeconomies of scale closer to one - the constantreturns case. (The multi-product economiesof scale rangefrom 1.00 to 1.06.) Their bottomline simulationresults of an urban expresswayand arterial and sensitivityanalysis demonstrates that a budgetbalance and hence cost recoveryare achievable. The shortfallof a few percentof total road costs can still be recoveredby maintainingsome level of first registrationfees, annual licenses and/or fuel taxes. Finally, since all externalitiesought to be internalizedin principle,air, noise pollutionand accidentcosts shouldalso be appropriatelycharged for (Carbajo(1990) and Cameron(1991)). In this way, the highwaybudget would mostlikely makea profit (see footnote56).
72. The basic intuitionbehind this remarkableresult is as follows: becausemost roads are currentlybuilt to accommodateboth automobilesand heavy vehicles, a neat dichotomyof allocatingpavement wear costs betweenautomobiles and truckscannot be achieved. Thus 1) the marginalcost pricing of trafficflow requires a congestioncharge - whicheffectively covers the capitalcost of investmentin the long run - and 2) the marginalcost pricing of pavement wear - associated with the investmentcost of strengtheningthe road -- results in a relatively -S5 - steep road damagecharge. The marginalcost-based road user charges combineto yield the double-chargingof roads, depictingdiseconomies of scope.nj One logical implicationof the diseconomiesof scope argumentis that savingscould be reapedfrom buildinga thinner autos-onlyroad system(the savingsof whichare estimatedto be on the order of 23% by Keeler and Small (1977)). With a universalroad system,roads shouldthen be built or resurfaced durably on the slow lane only and heavy vehiclesshould be confinedto that lane. (This experimentis beingcarried out in Californiaand Florida(Small, Winston and Evans(1989, p. 15)).
S9/ Ibe theorecal findingof tho double- ng of roads is dso deived by Nevwery(1989) for the cae of consantreturns to scaleand road use. However,he neithercites nor employsspexficaily the concept of multiproductrtums in his wor - 56 -
XV. SUMRY AND CONCLUSIONS
73. One of the earliestcontributions to the economicanalysis of road pricing was from a French engineer, Jules Dupuit (1844). He was the one (and not Alfred Marshall)who introducedthe conceptof consumer'ssurplus -- the comerstoneof the welfareanalysis of any
publicproject - and broughtit to bear on the subjectof toll roads. It is a similartack that this paper has taken, in the belief that a picture wouldperhaps speak a thousandwords. I have synthesizedthe dominantthinking to date on the topicof road pricingby the mainprotagonists and integratedthem into a consolidated,analytical framework.
74. Accordingto my analysis of the traditionalroad pricing arguments,it is hardly surprisingthat road pricingas advancedin the past encounteredits share of difficulties. This is becausethe congestiontoll has the effectof a tax (increa) on trip-makers,despite the fact that it is an externality-correctivetax. The gist of the argumentis as follows: people are againstcongestion pricing because: 1) those whoare tolled wouldface a higherprice relative to a no tax situationon average;§/ 2) those who are priced off the road in order to circumventpaying the toll are clearlyworse off as a resultof the 'forced' switchonto a different modeor time of day; and 3) the other road users who are not tolled-the tolled on - are no betteroff and, indeed,may even be worseoff if congestionis encountered.My appliedwelfre analysisactually takes into accountthe increasein revenueto the governmentand the issue of transferpayment. If each of the partes is separatedout as: 1) the tolled, 2) the tolled off, 3) the tolled on, and 4) the government,the groupthat standsto gain the mostis the govemment
(and the untolled- the rest of society),unless the toll revenuesare earmarked. The otherparty that is primarilybetter off are thosewith very highvalues of time. Only in the hypercongestion case couldaU groupsbe madebetter off on average.
6Q0 Considora mro alisic siuatiaonwhr the exitence of a uifom fiel tax offtively tdsates ibslf ino a tip pricetat is higher(buotno a high a th peak-perodpnice), and exactlytho sam analysis follows. -57 - 75. It has been argued that the dispositionof the revenuesof externalitycorrective (toll-) taxes should accrue to the public treasury (Baumol and Oates (1988), p. 29)AY' Conventionalcost-benefit analysis treats a dollaras a dollar to whomsoeverit accrues,and also implicitlyassumes that only consumersderive satisfaction from revenues. Hence, unless toll re;'enuesare channeledback throughreduced transportationrelated taxes, user charges or improvedpublic services,neither the tollednor the tolled off wouldendorse road pricing.
76. In this paper, I havealso shownstep-by-step how to implementshort-run marginal cost pricing in transport followingWalters and others. In particular, I have establishedthat implementingthe optJmalpricing rule -- the first rule - is equivalentto settingan optimalroad user charge, where: 1) a congestiontoll on the differencebetween the marginalcost and the averagevariable cost of a trip is imposed,and 2) the maintenancecost of road use is also charged. Further,I haveshown that the processof determiningan optimallypriced and invested road systemis similar,albeit with a coupleof importantdifferences, to theprocess of achieving long-runequilibrium of a typicalproduct within a competitiveenvironment, inspired by the basic Mohring-Harwitzmodel. The issue of short-runvis-a-vis long-run marginal cost pricing is clarifiedinter alia. For instance,implementing short-run marginal cost pricingis equivalentto pursuinglong-nm marginal cost pricingin a steady-stateworld in the long run. However,only short-runmarginal cost pricing would be able to capture the peak/off-peaknature of travel
demand. In the absenceof scaleeconomies, the optimal capacityrude - the secondrule - says that the existenceof economicprofit, i.e., toll revenuecollection less the fixed and non-use- relatedcosts of a road, wouldserve as a surrogatemarket mechanism indicating that the road ought to be expanded. Puttingit anotherway, the investmentrule says that a road oughtto be expandedto the point wherethe additionalcost of investmentin capacityequals the additional savingsin travel time. In the long run, toil revenueswould cover the intereston the capital investment,invariate maintenance, depeciation and operatingcosts of the road. Maximizing society's welfaredictates that one shouldimplement short-run margnal cost pricing over the
.J/ Thisis beas a Pigouvin tax is onewhich impose a positiveprice on a producerof an externalityand a zro pice on a consum_rof X exterality. - 58 -
long run by varyingthe size of the highwaycapital stock. In this way, the pursuitof efficient pricingand a self-financingroad systemwould be compatiblewith one anotherand no residual costneed be covered. In one stroke,the samecongestion tolling mechanism solves the pricing- cum-investmentproblem, satisfying the conceptualguidelines of efficiencypricing, economic and financialviability as set out in the introductorysection of this paper.
77. Relaxingand flushingout major assumptionsindicate that the result derived here is robust and applicableto: 1) a multiplicityof roads, 2) a road which is subject to diurnal variationof demandand the peak-loadproblem, and 3) differencesin valuesof time. If constant returns to scale can be shownto hold on averagefor a particularcity with severecongestion, it couldpotentially aid in greatlysimplifying the planningof highwayinvestment. It couldalso be used as a yardstick,against which scale economiesor diseconomiescould be measured.
78. Economicefficiency would be enhancedif marginalcost pricing of a trip were donein the short run and optimalinvestment in capacitywere pursuedover the long run. I have establishedthat, if governmentalauthorities were to charge correctly for congestion,it is possiblefor themto make moneyon a road whilesatisfying economic efficiency. Profitable roads arise in heavilyutilized or urban areas becauseland rents of real estate are high and congestiontolls reflect the rising opportunitycosts. Yet, it is possiblethat congestionpricing in the presenceof both indivisibilitiesand diseconomiesof scale in urban roadsmay curtailthe extentof profitableundertakings. Similarly, pursuing marginal cost pricing under the restrictive conditionsof both indivisibilitiesand scaleeconomies of rural roadscould also resultin profits in the short run. Thus far the pointsI have madeare based on first principles.
79. In the long run, the simplepricing and investmentmodel implies that the marginalcost pricingof trips coversall the fixedcosts of the road and the congestiontoll (whichcaptures the quasi-rent)behaves as if it is a capitalcharge. This analysisalso mea that an optimally investedroad systemis in fact one wherea road shouldnot alwaysbe uncongestedduring the peak period. An optimallycongested road is akin to the commonlyaccepted notion of an optimalpollution level in the field of environmentalexternalities. An uncongestedroad for - 59 -
every time period of the day would suggest that that road is over-invested,either because of indivisibilities or nonmarginal cost pricing. If a road is indeed overbuilt, abandoning or downsizing it in the long run may be unavoidableon narrow cost-benefit criteria. The act of downgradinglightly used roads in order to save on the costs of maintaininghigher standardsof road pavement is a form of disinvested.u/ Alas, given that almost all existing road systems are non-optimallydesigned and that costs are considered sunk in the short run, the efficient usage of such a network would still call for marginal cost tolls. Any increase above the road user charge should then be regarded as a 'pure tax element' or surcharge, whose contribution to general revenues should perhaps be made on either fiscal or non-economicgrounds.
80. One may ask: startingoff with an overbuilt road system, say, is there a way in which road pricing based on the marginal cost concept can be implemented within an institutional context where severe fiscal constraints on public expendituresprevail? To answer this question requires that we go beyond first principles.
81. Recentextensions a la Newbery-Small-Winstonhave enriched the basic modeldeveloped diagrammatically here by incorporating the fact that heavy vehicles are the cause of road damage. Charging for both the external and variable cost of road damage on a vehicle weight per axle basis would help close the deficit that may arise from congestion tolling. Further, a road needs to be strengthenedto the point where the additionalcost of investing in durabilityjust balances out the incrementalsavings from maintenanceand vehicle operating costs. Thus the third rule - the optimal durability rule - is born. As a natural extension of pricing for externalities, air, noise pollution and accidentcosts ought to be charged for. Surely in this way the highway budget would more likely involve profits than losses if the issue of cost recovery of the sector cannot be ignored.
621 The commonpractce of downgradngroads is constent with the findngs of road detenorationin developingcountries (Hal andFaiz (1988,p.32)). -60- 82. Givenpoint estimates (or preferably,functional specifications) of speed-flows,demands and the value of time, one can estimateand simulatesome of the analyticalresults developed here. Whencombined with the associatedoptimal pricing and investmentrules, the efficient l-vel of prices, user charges, speed,volume-pacity ratios, and trips, as well as the optimal numberof lanes and inchesof pavementthicknesses can be obtained.
83. The fact that in the transportcontext, the consumer-produceris both a willing'victim' as well as a 'beneficiary'has policy implications. As 'victims' of congestionexternalities, perhapstravellers ought to be compensated.Note, however,that Pigouviantoll-tax revenues are not supposedto be usedto compensate'victims' of extmalities (Baumoland Oates (1988), p. 23). Also, intuitively,motorists would be inducedto drive more becausethe level of compensatorypayments would dependon their car usage, so economicefficiency would be violated.In thiscontext, a road fundwould be consistentwith first-bestpricing only if the funds were usedin an indirectmanner. Travellersare also 'beneficiaries'of road transportby virtue of theirbeing present on congestedroads, and their contributionsto the toll revenuecomponent of 'user charges' reveal their willingnessto pay. In the absenceof lump sum transfers, earmarkingof toll revenuescould serve as a usefuldevice in principleto approximatingbenefit axation as a way of satisfyinga commonlyaccepted notion of 'fairness.' Similarly,heavy vehiclesought to incur their 'fair' shareof hety pavementwear fees. Combiningthese plausible argumentsand our earlier results of optimalpricing and investmentprinciples, suggests that some form of dedicatedfunds is perhapsnecessary - either in the form of a road fund or a transportfund - if road pricingis to gain politicalacceptance.w/
63/ Recet developmentsin electonic toil collectionand electrnic road picing in Norway,Sweden and Cabridge, (England)point to thefct thattraveller do not objectto roadpricing when the toilrevemu ar earmakedtor bothroad construction and i1provement andor the provisionof betterpublic tnort (Withoptimal tolling, however, high purchase taxe and rgsation/licens feesof vehiclesought to be reducedto a levelsufficient to coverthe administraveand enforcmet coatsof colection. If the road maintenancct is constmt with reapec to the trffic level,a sppopriat fue tx could perhapsbe usd to aoximate uwe.) Inded, a recent natonal m8uveyconducted in Englandidicates thit when peoplewere asked whetherthey an for or againstroad pricing, about 57% sr aainst iL However,when the questionwas posed in a differentway: wouldthey be supportiveof apac*ageapproach to roadpricing, withthe revenuesfrom roadpricing used only to fimmcepublic tranpoit, 57%of the same surveyedpopulation were in favor - 61 - A. The Role of a RoadFund
84. If a road fund were to be set up,compensaion wouldneed to be made sufficiently indirectto satisfyPareto efficiency. Thus the fundsfor road constructionshould be used both to maintainexisting roads and to financenew roads, and theprofits generated from urbanroads couldbe used to financethe fixed capitalcost of worthwhilerural roads in a nondistortionary manner.o To what extentcan the profits collectedfrom heavily used roads offset the losses arisingfrom the constructionof lightlyused roads? The answerdepends on the extent of the interacdonof both scale economiesand indivisibilitiesand shouldbe examinedon a countryby countryand case by case basisusing econometric analysis.
85. A road fund is attive becauseof the high marginalcost of raising a tax dollar. Moreover,a road fund run by an autonomousauthonty would increase the linkagebetween revenuesand expenditures,currently lacldng in a politically-basedbudgeting process, thereby improvingmanagerial efificiency. Without the settingup of sucha fund,however, deficits from lightlyused rural roads (withincreasing reuns to scale)would demand subsidization from the
of od pricing rdtr twn against it (Jon (1991), Goodwin (1989)). (May and Gardner's (1990) imulation results also confinm th cas for an integted approach to road pricing, and buttre the alyticalresut preseted he.) Using an eqiirum trvel demandmodel of modal choice applied to tie San Fancisco Bay Aea, Smal (1983) concludestt congestionpricing combinedwith the redistnrbutioof toll revenueswould rudt in benefitsto aUincom groups. As an ilustrion, the OsloToil Ring, cumently i operatio, chargesprivat cars (lightvehicles) going into the citycenter a tollof 10 kronos(approximately US$1.50). Roughly 60% of motoristsapt for the subscriberlanes in 1991,which are opeated via electric tollcollection, rather thanmanually-opeated toil blaes. About80% of the eamarke fiunds r used to fimmceroad consuction and 20% for busways,buses imd tms. The optio natureof thosucesfl OsloToll Ring raises the inguing idea that congestiontolling dsould pedrps be implemeatedan a voluntarybasis: with the choiceof a combinationregistiton fee/fueltax basedan 'averao' usge, or a subscriptionto the use of electronic devices j4/ The commonsense notion of uing the profitsfrom eavilyued roadsto fiance lighly used roadshas its intelleal rots in AlfredMasal's 'tax andbounty' syaem of pticing: 0no simplepln wouldbe th levyingof a ta by the communityon their own incomes,or on thoproduction of Soodswhich obey the lawof diminsing retun, and devotingtho tax to a bountyon the productionof thosegoods with road to whichthe law of incrai retum acS sharply. AlfredMasrlA, c o 1920,p. 392 Mua alsoobsvwes dat it is noeoo_y toconider to caotof adminiseringsuh a tax-subsidysysteLm - 62 - publictreasury, and would thus competefor tax moneyvalued at a high opportunitycost. By symmetry,surpluses that acue in heavilyutiliwd urban areas (with decreasing returns to scale) should then be priced at a pre,mwn. These welfare losses and premiums would presumablyoffset one anotherif viewedwithin the same (transport)sector - with a nominal value of a dollar being treatedat its face value - so that we are back to the case of pure efficiencyconcerns.
86. Even if a certaincity in a developingcountry, say, is found to be faced with mainly increasingreturns to scale, the deficitcould be closed,in principle,by appealingto the notion of diseconomiesof scope. Meetingthe requirementthat a road networkbe both large enough, in terms of capacity,and strongenough in termsof pavementthickness, can be quite costly. Scopediseconomies in highwaysmean that a road networkthat accommodatesboth loadingand trafficvolume found universallyis more costly than the sum of an autos-onlyand a tailored trucks-onlyroad system. Hence,the surplusassociated with diseconomiesof scope offsetsthe potentialdeficits associated with scle-specific economiesof road constructionor use. The viabilityof the fund is enhancedby the fact that the maintenancecost of the road pavementis recoveredtwice: once whentraffic flow createscongestion, and the second time whentraffic loadingscause road damage. Thus, the idea of a trust fund administeredby an independent agencyaccording to strict cost-benefitprnciples is likey to be feasible.
B. The Role of a Transort Fund
87. Alternatively,taking the transportsector as a whole,a transportationfind oughtto be set up.i/ If dedicatedfunds are set up in this way, indirect 'compensatory'payments can be achievedand would not depart far from optimalty. I recommendthis both becausethe
65/ I am indebtedto Sir AlanWalus for this inight. f&I Usig an entirelydiffrent o th the on usedhere, Vickrey(1977) establises thoresult that cities shoulduse landrent taxreveues aiing fwomagglomeration economies to financemass transit and public trnportaion vwhichae subjectto increasingremus. Indeed,he arguesforcefilly and provesthe case thatno subsidizingtes fixedcoats woud be inefficient - 63 - problemof highwaycongestion is tied intrinsicallyto the provisionof poor transit alternatives and becausepublic transport encompasses a substantialif not the lion's shareof trips undertaken in both newly industrializingand developingcountries (Deaton (1987)).R Typically,the productionof bus servicesis subjectto consumer-sidebus route and frequencyeconomies of scale. Hence additionalfunds in the form of user-sidesubsidies are required to meet the financialshortfall arising from the capitalequipment, if bus usage is priced at marginalcost. Road pricing would result in more crowdedand inferiorpublic transportservices unless bus companieswere to offer more bus services(and hence lower generalizedprices) as a supply response. Then 'untolled' public transport users or captive riders would be made better offWM Here the doublecharging of automobilesvia traffic volumeand heavy vehiclesvia loadingswould help to closethe deficitgap. Increasingby popularrapid masstransit and light rail systems - both of whichare subjectto significantscale economies- also require capital funds,the constructionof whichshould be basedon economicviability. Unlessa globalview is takenof the congestionproblem and morerational time-of-day pricing practiced in all modes (in contrastto tacing individual,non-optimally priced modes), the urban tWansportation problem will continueto be pervasive.
88. Even withoutdedicated funds, it is essentialto pursue efficientpricing and stringent benefit-costanalysis link by link and modeby modeon both a volumeand loadingdimension. Thereafter, the results can be presentedfor public scrutiny, thereby improvingmanageral efficiencyand publicaccountability. The competitivetendering and privateprovision of certain transportservices could also serveto enhancemanageral efficiency in the publicsector. Issues warrantingfurther investigation include the corporatizationof certaintransport agencies.
89. Subjectto furtherresearch, the idea of settingup a transportationor road fundand the pursuitof marginalcost pricingin aU its dimensionswould enable us to satisfythe quinpartite
Z/ Theprovision of publictraort mentionedin SectionViI, umed to operateunder constant retms was ued merelya an iliustrativeconvenience but thisassumption does not resultin loss of generality. 68/ Notably,captive bw pasage wouldbenefit ftom roadpncing if equilibrumtramnsit travel times m lowerd. - 64 - principlesof the World Bank'sgeneral guideines, as statedat the outsetof this paper, namely to: 1) implementefficiency pricing, 2) meet economicviability, 3) meet (to a considerable extent)financial viability, 4) achieve(some degree of) 'fairness' amongbeneficiaries, and 5) attain(somewhat) managerial efficiency of thepublic authority. The conceptionof a fundpasses manyof the criteriafor a 'good' earmarldngarrangement as presentedin McCleary(1991). The implementationof marginalcost pricingin boththe trafficand loadingdimension could be done with the adventof recenttechnological breakthroughs in automaticroad user chargingulizing automaticvehicle identification and classification,all of whichare subjectto remarkablescale economies(Elau (1992)). Alternatively,less powerfil road pricing instuments such as area licensing,simple cordon pricing schemesand the monitoringof vehicleand axle loadingvia weigh-inmotion scales can be used.
TimothyD. Hau -65- Appendix Page 1 of 6
APPENDix:
Measurementof the WelfareImyact of Road Pricing
1. A brief analysisof the measurementof the welfareimpact of road pricing would help explainwhy road pricing is unpopular. We will considerseveral acceptableapproaches to measuringthe net benefitsoffered by the introductionof road pricing on a non-perturbed equilibrium.Each casts different light and insightson thecontroversy surrounding road pricing.
A. OuantityApproach
2. The first approach,which is more popularin the U.S. literature,is to measure the so-calledwelfare gain or loss areas (seeKraus, Mohringand Pinfold (1976), for example).This standardmethod is labelledthe quantityapproach or the 'Amencan'approach (see Fig. 3). The loss in valuationto the consumer-travellerfrom a reductionin trips from Ql to Q' as a result of increasingthe generalizedtravel cost to him from P to P' is the verdcal, tWapezoidalarea d+g+k. The savingin resourcecost to travellersfrom the reductionin traffic, togetherwith the savingof congestionin the formof externalityreduced, is the verticalarea l+d+g+k. The net benefitto societyof the introductionof road pricingis givenby the trangular area 1.
Net Benefit Aroach
3. A variantof this approachis the net benefitapproach (see Fig. 3). The net benefitin the case of the optimaltraffic level of Q' is typicallya large trangle (the pie area a+b+e+h betweenthe demandfunction and the marginalcost curve),with thepie tiangle emanatingfrom the point of optimum. Similarly,the net benefitin the case of the non-optimallevel of Q°is givenby the differenceof the pie area a+b+e+h and the small triangulararea 1. The latter area is of coursethe welare cost savedwhen the trafficlevel is inducedto be loweredfrom Q' to Q'. This variant is intuitivelyappealing as it graphicallyillustrates that net benefit is Appendix -66 - Page 2 of 6
maximized with marginalcost pricing. Any departure from the point Q', either in a positive or negative direction, would slice into this maximal net benefit pie. To the left [or right] of Q', travellers' marginal valuation would exceed [or be less than] the marginal cost.
B. Change in Total Benefits and Total Costs Approach
4. The above procedure, and its variant, is an impeccable one. However, there is an alternative intuitive method to calculating the net benefit of introducing road pricing. This approach is widely used in the British literature (Ministryof Transport (1964), Tanner (1963, p. 318); Gwilliamand Mackie (1975, pp. 105-106),Thomson (1970) and Thomson (1974, pp. 142-145)). The findings of the Ministry of Transport, known as the Smeed Report, present a different calculationof the areas of gains and losses indicatedabove, yielding different insights into the problem (see their Appendix3). The 'British' approach uses the change in total benefits and change in total costs. The change in total benefits accruing to those who are tolled off the road are negativebecause they suffer a loss in valuationequivalent to the vertical area d +g+k. The change in total costs - expressed as the reduction in the total expenditureon travel in the form of savings in time cost - accrues to aU motorists and is given by AVCOQ- AVC"Q' (or PV - P"Q'), that is, the area e+f+g+k. The net gain to society is the area e+f-d. HeuristicaUy,the remainingusers find that they derive satisfactionfrom the savings in time cost of the area e+f. The losers - those tolled off the road - would clearly experience a welfare loss of the area d.
5. The discussion thus far gives the conclusionand mistaken impression that those who remain behind are in fact better off by the entire savings in time cost of area e+f In fact, drivers who remain on the road have to make toi paymentsof the area b+c+e+f, which in turn become a gain to the governmentin the form of toUrevenues. (This is the notion of a transfer payment excluded in cost-benefit calculations using the British approach, see Gwilliam and Mackde(1975, pp. 105-106).) Yet, paradoxically,it is precisely the imposition of this tax - resulting in a transfer payment - which enables those who remain on the road to benefit the time - 67 - Appendix Page 3 of 6
savings of an additional area e+f.01 Without the tax, motorists are not properly induced to save valuabletime resources: the time is completelylost. The ones who remain on the road, however, actually suffer a loss of consumer's surplus of the rectangular area b+c. It is as if a discriminatingmonopolist -- in the guise of the government's tax department -- carves away oart of the users' consumer surplus. Also, to 'benefit' from time savings of the area e+f, drivers are in fact trading a dollar of money for a dollar's worth of time, implying that both a standard and constant value of time and efficiency analysis are assumed. (The rest of the transfer payment of area e+f also accrues to the governmentin the form of tax revenues.)
6. Primafacie, whether or not the net benefitsof introducing road pricing using this latter approach (i.e., area e+f- and the former approach (i.e., area i) are equal is not at all obvious. The latter procedure gives less indicationof the notion of optimalitywhen compared to the first approach, especially with regard to its variant. In the quantityapproach, one could move to the left or right of Q' and observe that the net benefit pie to society would clearly be eroded, suggesting that Q' yields maximal net benefit. Using the latter approach, however, as Q increases past Q', a welfare loss area would increase. This would have to be offset with a new rectangular area of saving in resource cost. The point is that it is unclear whether Q' can be shown to be optimal, at least diagrammically, becausethe new rectangular area may not offset the new (trapezoidal)welfare loss area. Formally, the proof is as follows: the move from Q' to Q0 yields a change in cost to society of area l+d+g+k because the vertical area below the marginal cost curve is a proper measure of cost. Equivalently, the change in variable cost of
9/ The rectagla area b+c+e+f ahouldbe countedas accruingeither to the govenmentin termsof toil revenuesor returnedto consumers(via a hypotheticallump sum transfermechanism). A lessonto be leamedregarding the isue of transferpayment is to avoiddouble-counting. if a doilaris treatd as a dollarto whomsoeverit accruesand the tansfer mnchanismis implem , thenthe moveto roadpricing resultsin positivenet benefitto societyof area e+f-d Puttng it anotherway, the standardnotion of a transferpayment of the area b+c+e+f says that moneygoes from the consumer'spocket into the govenmeot's.It is importantto view the timesavings of area e+f as an additionallayer on top of the transfr paymentitself. The bottomlayer goes fromthe motorist'spocket to the roadagency's; tho top layeris obtainedbocause the motoristswho are toiledae forcedto trademoney with time. Remarkably, it is thiscoerced payment of the tax revenueaea e+fwhichbrings about a real savingof timeof an area of equalsize. Appendix - 68 - Page 4 of 6 going from Q' to Q° is the invertedL-shaped area e+f+g+k. By definition,these two areas mustbe equal, implyingthat e+f=l+d or l=e+f-d. Diagrammatically,it mayappear as if the changein total benefitsand total costs approachyields larger net benefitarea-use. However, it needs to be clearlyshown here.2'
C. Consumer'sSurplus and Producer'sSurplus Aproach
7. The third approach using the summationof changes in consumer's surplus and producer's surplusinvolves the term quasi-rent.2/ The travelleris both a consumer(in the sensethat he derivesbenefits from purchasing a transportservice) and a producer(in the sense that he himselfhas to purchasethe inputswith both his own time and operatingcosts). In the absenceof road pricing, becausedrivers travc.lup to the point where averagevariable cost intersectsthe demand(at outputlevel Q°), the entire receipt (from the consumertraveller's expenditure)goes to cover the 'payment'of user-suppliedfactor inputs, so zero quasi-rentis therebygenerated. However,it couldbe equivalentlystated that the nil area can be expressed as thedifference of two triangles,i.e., area (e+h) - (c+d+l), by simplyexploiting the meaning and geometricrelationship of averageand marginalcost curves. In the adventof road prcing,
the quasi-rent - the return to a fixed factor of production - is essentally the amount which the traveller-as-producer'receives' over and abovehis totalvariable costs. This quasi-rent,instead of accruingto the driversas such,is capturedby the governmentin the form of toll revenueor a user charge, and henceshould be accountedfor properlyin benefit-costcalculus. Note that the quasi-rentof area b+c+e+f can be re-expressedas the area b+e+h. Clearly,the change in the quasi-rentwould be equal to the area b+c+d+l. Coupledwith the loss in consumer's
2QI Lee (1982)claims that the two ares basedon the diffart mdhodologiesare the samebut doesnot prove it. Ihe spiritof the analysisI howhare underlies vAwilliamandNash's (1972) commet on Beesleyand Walte' (1970)evaluation of uba road investmnts.
1/ Mhenotion of rent is a slpperyone and warrantsclarification REmtis the analogof producer'ssurplus in the iput marlt. Reantis a p*mawetpayment to a factorover and abovethat whichis requiredto draw forthits resourc. Quasi-rentis a tpnporarypaymurt and would contnue only untilthe capital asset is depreciaedor possiblytansferred to anotheruse (see footnote29 and Mohring (1976, Chapter 2) also). Notethat a high priceand willingnesspy yied high quasi-rent,and niot the reverse. -69- Appendix Page 5 of 6 surplusof area b+c+d, the net benefitarea I emerges. Hence, we have shownin different ways that the three approachesare identical.3 / Note, also, however,the secondapproach is used and extendedbecause it graphicallyillustrates the broad distributionalimplications of road pricing.
8. Perhapsone reason why there is confusionregarding the two approachesabove is becauseof Walters'(1961a, 1961b)treatment of the MC and AVC curves as marginalsocial cost and margtnalprivate cost curvesrespectively. (My interpretion here is at variancewith the commonuse of the latter term sinceI regardit as somewhatof a misnomer.)Walters' use of the AVC as MC curveimmediately brings to mind standarddiagrams of an externalitysuch as the classiceconomics text exampleof a pollutingfactory, with the consequentchanges in consumer'ssurplus, producer's surplus and extrnality valuation. As carefullyshown above, thisexample is not validin our analysisbecause the marginalprivate cost curveis only marginal with respectto the driverhimself. The individualperceives and bears the averagevariable cost only: it is merelya decisioncurve and no more. Sincethe area belowthe AVC curve is not the totalprivate cost, only an incorrectintepretation can be drawnby mathematicallyintegrating the area under the marginalprivate cost curve, whichturs out to be an averagevariable cost curve. (Further,producer's surplus should really be interpretedas quasi-rentto avoidpossible confusion,especially in undstanding the rlationship betweenpricing and investment.)In the absence of he opdmal pnrcingof trips, average (variable)cost pricing prevails with the associated . Tis illusates strongly the need to reserve the term 'short-run marginalcost pricing' to be consstent with the World Bank's policy guideline(World Bank OperationalManual Statement (1977)). I employthe term marginalsocial cost pricingwhen
a In fact,if Q'is verycls to Q', do anal os is equivalntto dhohng in tota varae coaL By evaluangth diffeieo of dl clug in vaiableco withthe mail valuaon, to twomet&ods discuad aov (t qu approachnd the dcge in totalbenefits and tol coot poch) ar seam to be equaL Appendix - 70 - Page 6 of 6 accountingfor aU the other externalitiessuch as environmentalpollution and accident costs.L/
9. A technicalpaper by consultantshired by the HongKong Government indicates that the net benefitdue to introducingroad pricingcorresponds to the area e+f+d+g+k (Transpotech, (1983,Fig. 4)). The consultants'explanation of the extra area g+k is eitherbased on a possible misunderstandingof the first two approaches,or simplya matterof double-ountingareas g +k. The authorsstate that "this money[referring to the areas g+k] is availableto be spentin other ways,perhaps on other modesof travelling". Havingalready included the resourcesaving as a reductionin the expenditureon travel, PQ - pftQP,the resourcesaving from the tolled off driversof the verticalarea g+k shouldnot be countedtwice. The pointhere is that unlesscare is takento ensurerigorous cost-benefit analysis, the benefit (or cost) figureswould be biased, as has been the case with the evaluationof the electronicroad pricing experimentin Hong Kong.7/
73/ For examle, Glaister's(1981, Chapter 5) use of the term 'marginalsocial cost picing' is synomymous withthe margnal cost prcng conceptemployed here.
7A1 Basedon the mnuber presentedfor an illusative cas, the bias is 40% upwards. It shouldbe stressed, however,that it is unclearfrom a readig of th HongKong Goverment's MainReport ote Elecunic RoadingPricing Pilot Schemewhher the final reportfollowed the methodologyoutined in Technical Paper 1 (Traspotech(1983, 1985)). FIGURES
EconomicFundamentals of RoadPricing: A Diagammatc Analysis
by
TimothyD. Hau
TransportDivision Infastructureand Urban DevelopmentDepartment The WorldBank -73 -
Figure1 Derivatlonof a WavelTime-Flow Curve of an UrbanHighway
Figure1 (a) Figure1 (b)
Speed, Speed, In km in km per hour per hour
Sm ------1 Sm ----- s~~ \ aa------
I\/
0m Density, F IrLax Flow, Invehicles EngineeringCapacity in vehicles per krn per hour . . ~~~perlane
Figure 1 (c)
Traveli Time Dip
or Delay§e\99egate \Ime
t_rr_n L__--.I
Fmax FRow, In vehicles-km per hourper larne-km - 74 -
Figure2 Derivationof the MarginalCost Curveand CongestionToll
MC AVC MarginalCost Curve (MC) AverageVariable Cost Curve(AVC) I plusvehicle operating cost and plus vehicleoperating costand variableroad maintenance cost (Full) variableroad maintenance cost Priceotf a trip, P, I Generalized \ Cost,GC, I of atrip X I
<~~~~~~ /
\ ~~~~/I w \ ~/ II
II / < o
/1\Optimal V
AVC E
o _ . Timel. DemandCurve nme cos t Costsborne 1 Qd …~~~~~~~bymotorist1--i-4- privately l Vehicle I lI Copertin I _ _ -.. _ _…__ _ __-_L_-- 4.-- VariablARoad I Maintu.nanceCost
Q' QCf Flow,In o max ~~vehicles perlane-hour
Optimal User Charge = Costs Imposed on Other Motorists + Road Author ty = External Congestion Cost + Variable Road Maintenance Cost | ~~~~=Optimal Toll + Variable Road Maintenance Cost User Charge Component - 75 -
Figure 2(a) 'Dynamic' Phenomenonof Traffic Growth: The RelaxationEffect
(Full) Price or a trip, P.
Generalized Cost. GC, of a trip \AVC Average VariableCost Curve (AVC) pius vehidceoperating cost and vadiablerc ,d maintenancecost
\7Ex
AVC ~ ~ Initial Demand Curve Od Time Cs
} VehicleOperaUng Cost
| Varlableroad maintenancecost
Q Flow,In vehicles per lane4iour - 76 -
Figure 3 Welfare Impact due to the Introduction of Road Pricing In the Peak Period: Short-RunMarginal Cost Pricing
\ AVC \ AverageVariable Cost Curve (AVC) plusvehicle onerating cost and X \ vadableroad maintenance cost
\\~~~~~~A MarginalCost Curve hMC) plusvehide operating cost and variableroad maintenance cost (Full) PriceOf a trip, P Generalized Cost,GC, Z of a trip
a *A
Efficientp, Price =AVC"+ t'
Price 0 =AVC J Pu AVC h VCi Td DemandCurve
I VehideOperating Cost
_ VariabeRoad Maintenance Cost
0 Q0-' C( dQ Effient Observed Vehicles Output Output perlane-hour -77 -
Figure3(a) WelfareImpact due to the introductionof RoadPricing In the PeakPeriod: Short-RunMarginal Cost Pricing 'Hypercongestlon'Case
(FUN) a*tap, P MarginalCost Curve (MC) AVC pkusvhide operatincost and cookCC,o AverageVauible Cost Curve (AVC) vatl roadmaintenance coat d b*iP t pkJ vehicleoperaUng co and vartablroad mnteancecoast
* AC -. < Primce Efficentp'\> ! *AVC"i.t' \
0d
VeryHigh' I nandCurve
nAVC" z AVCE TimeCost Timecost Costsborne …___O___……__ _-* by motorist __ _ _ _ privately Vohicleoperaing cost
Varlable IRoad Mainterance Cost Q Vehicles perlane-hour - 78 -
Figure 4 Effect of the Introduction of Road Pricing In the Peak Period on the Off-Peak Period
p(Full Original Price0 Demand a trip,P Curve Inthe Cost,rizC OPeaik MC MarginalCost Curve(MC) of a trip \ plusvehicle operating cost and 0110/,d' / valiable roadmaintenance cost ad>\ Q0 DemandCurve In the Off-PeakPeriod °P OPfollowing Introduction ot CongestionToll in the PeakPerlod AVC AverageVariable Cost Curve(AVC) plus vehicleoperating cost and variableroad maintenancecost
I % Time cost
Vehideoperating cost
J Variableroad maintenance cost
0 Vehiclesper lane-hour - 79 -
Figure 5 Introducing the (Short-Run Average) Fixed Cost, SRAFC, of a Road, Short-Run Optimal Toll with Economic Profit
SRMC(K') Short-RunMarginal Cost \ Qd Z plus vehicleoperating cost and variableroad maintenance cost (Full) 1 Price of S (K a trip, P. SRAVC(K ) $ Short-RunAverage Variable Cost Unit Cost,u. plus vehicleoperating cost and variableroad maintenance cost
SRAFC (K1 \
fr SRATC(K SRT (K/ Efficent P Short-RunAverage Total Cost P:= OptimalToll t1 / \ / / plus vehicleoperating cost and OptimalToll t varable road maintenancecost
|: ~~~~~~~~~~Time Cost\
-v - __ __- - - Od DemandCurve Vehicleoperating cost
__------_ Variable Road MaintenanceCost SRAFC(K 1) Shdrt-RunAverage Fixed Cosi
Q'. Vehicles EfficientOutput per lane-hour
K1 = (Non-optimal)Road capacity t 1= Optimaltoll for a non-optimallybuilt road n'1=Economic profit for a non-optimallybullt road - 80 -
Figure 6 Long-Run Equilibrlum of an Optimally Designed Road With Both Optimal Pricing and Optimal Investment
SRMC*(K) Short-RunMarginal Cost P. \ | plus vehicleoperating cost and variableroad maintenance cost
$ SRAVC*(K) a, \ / \ Short-RunAverage Variable Cost plus vehicleoperating cost and variableroad maintenance cost
SRAFC (K) SRATC (K*) Short-RunAverage \ / Short-RunAverage Total Cost OlUmal \ FixedCost \ / / plus vehide operatlngcost and Price variableroad maintenanoe cost P > LRATC_4 J = LRMC Optmal Tollt' Long-RunAverage Total Cost = Long-RunMarginal Cost SRAFC (K)
SRAVCE Time Cost d DemandCurve
Vehide operatingcost
SRAFC (K) Variableroad maintenance cost
a* Vehicles OptimalOutput per lane-hour
t =OpUmal toll K* Optimalroad capacity - 81 -
Figure7(a) Constant Returns to Scale with Road Divisibility: Doubling Optimal Road Capacity (K ) and Traffic (0 ) Result in Doubling Fixed Cost, Variable Cost and Totai Cost (FC, VC, TC) and Toll Revenues(t. ° L)
$ SRMr3(K3*) SRMC6(K 6 ) SRATC3(K 3 *) SRATC6(Ko*) SRAVCO(K6*)
LRATC- LRRTC *LRMC
SRAFC3* /SRAFCO*
s RAFC(K3*) SRF() SRAFC3* SRAFCS*
03* * 06* 0 Vehiclesper lane-hour
Figure 7(b) The Relationship between Short-Run Average TotalCost and Long-Run Average Total Cost and Marginal Cost with Perfect Road Divisibility and Constant Returns to Scale
SRATCm(Km*)
L.ATC* LRMC LRATC* LRMC
0 Vehilesper lane-hour
() Denotesthat that vaiiable Ls optinized (L) Denotesan L-lane road Forexample, K3*denotes the optimal capdity of an optimally-built3-lane road 13*denotesthe optimal toN associated wlh an optimally-built3-lane road - 82 -
Figure 8(a) Road Indivisibllties under Constant Returns
LRATCo ABCDEFGLine LRMC- HIJKLMLine tS. A SRMC2/
- - - -~~~~~~~~~~~ ------
0 0LossRegbnRegon024*-Region-4 2 Region 446 R°eSgion 06 RteOgion* 68
Figure8(b) Optimal Pricing and Investment with Indivisibilltles: Expansion trom a 2-Lane Road to 4-Lane Road
a,~~~~a
5 s Un-
lIl Demand Curve I
Q Q24 Q * Q04.6 0 - 83 -
Figure 9 Economies and Diseconomies of Scale In the Provislon of Road Capacity with the Growth of Travel Demand
od 3
od~~ 0
Long-Run Marginal Cost
LR LRM\ALAC
Ca; \ Lon~~~~~~~~~~~~~~~~~~~~g-Run Unit Loss _ Average Total Cost I \ ~~~~~IUnit Profit
Inital Demand Curve Q1 3 I t
l l I e l I
j I Vehicles per lane-hour
Economies of Scale Constant Diseconomies of RegloRegion ------~~~~toScaleSlRetuns Region Scale Region - 84 - Figure 10 Doublingthe Numberof Streets-RoadCapacity-Results In Quadruplingthe Numberof Intersectionsand Traffic Lightsand DoublingWaiting Time
Figure 10(a) OriginalScenario - Existing Street Configuration
______D
0 = Odgin D = Destination o A B
Figure10(b) Final Scenario- Numberof Streets are Doubled
X I ~~~~~~~~IFD
6
5
0 1 2 3 4 - 85 -
Figure 11 Diseconomles of Scale: Urban Roads Network with Perfect Divisibility 7r= EconomicProfit
P, d S Q4 LRMC 0, ~~~~~~~~~~~~~~SRMC4
S4 / SRMC2 LRATC
SRAFC2SRACSRATO
SR 2 IXSRAFC~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Tr4 4
l r : RAFC2 ! | > 4S2RARAFC
Q2nn 02 024 0 4mIn 04 Vehicles per lane-hour - 86 -
Figure 12 Economies of Scale: Rural Roads with Perfect Divisibility L Loss
LRATC SRMC2
SRATC2
s VC 2 /SRMCSRATC ^~~~~~~/RA4C
SRAFC2 | I \l P4 -4.~~~~~~Q2i 0 nVeiLespRlAne-ou SAFC I ) | RFcSr...... 41
Q*2 Q2mIna .4 Q;1 4 aSmR Vehiciesper lan huRA - 87 -
Figure 13(a) Figure 14(a) Decreasing Returns to Scale and Extent of Increasing Returns to Scale and Extent of Indivisibilitles Indivisibilitles
$S Q LRMC ,
LATC
LRATC LRMC
ProfitRegion -0 0 Vehicles - LossRegion - 0 Vehicles per lane-hour per lane-hour
Figure 13(b) Figure 14(b) $ LRATC $ 0,0
LRATC
*4w-.- /p. 0\..-..~.. Q.
Profit Profit Profit Profit Profft Profit Region Region Region Region Region Region
; Figure 13(c) Figure 14(c)
LRATO
LRATC
Profit...- .. Profit Profi Region Region Region Region - 89-
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