CPB Discussion Paper No 130 Wefare anaysis in transport networks Pau Besseing and Maarten van 't iet Te responsibiity for te contents of tis CPB Discussion Paper remains wit te autor(s) Association for European Transport and contributors 010 1
CPB Neterands Bureau for Economic Poicy Anaysis Van Stokweg 14 P.O. Box 80510 508 GM Te Hague, te Neterands Teepone +31 70 338 33 80 Teefax +31 70 338 33 50 Internet www.cpb.n ISBN Association for European Transport and contributors 010
Abstract in Engis Soud one cacuate user benefits from canges in door-to-door journeys or from canges in te use of separate inks of te network? Quite often te second approac is deemed wrong, as consumers are supposed to demand journeys, not parts of journeys. However, we sow tat for a quite genera economic mode and under fairy genera assumptions regarding te network bot approaces are equivaent. Te cost-benefit anaysis practitioner can expoit tis resut. Te inks approac reveas on wat part of te networks user benefits and/or osses are generated. Tis additiona piece of information migt ep optimize te project design. Keywords: Cost-benefit anaysis, transport networks JE codes: D61, H54, 4 Abstract in Dutc Moet men wevaartsbaten afmeten aan veranderingen in deur-tot-deur verpaatsingen of aan veranderingen in et gebruik van afzonderijke deen van et netwerk? De tweede benadering wordt doorgaans as fout bestemped omdat verondersted worden dat de vraag van consumenten betrekking eeft op integrae verpaatsingen, niet op afzonderijke onderdeen van een verpaatsing. We aten ecter zien dat, voor een tameijk agemeen economisc mode en onder tameijk agemene verondersteingen ten aanzien van de eigenscappen van et netwerk, beide benaderingen tot een identieke uitkomst eiden. In de KBA-praktijk kan men ier andig gebruik van maken. De tweede benadering aat immers zien op weke deen van een netwerk de winsten of veriezen voor gebruikers ontstaan. Deze informatie kan men aanwenden om et ontwerp van et project te optimaiseren. Steekwoorden: Kosten-batenanayse, transport netwerken 3 Association for European Transport and contributors 010 3
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Contents Summary 7 1 Introduction 9 Mode and network 11.1 Network 11. Consumers 13.3 Producers and congestion 14.4 Government 15.5 Equiibrium 16 3 Wefare measurement 18 3.1 Wefare anaysis in a distorted economy 18 3. Decompositions of wefare cange 0 3.3 Wefare measurement & approximations 4 Practica issues 5 4.1 Te rue-of-a-af for a new ink 5 4. Aggregating routes into OD-reations 6 5 Concuding remarks 8 eferences 30 5 Association for European Transport and contributors 010 5
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Summary Soud one cacuate user benefits on te basis of te cange in door-to-door journeys or derive tem from te cange in te use of eac and every separate ink of te networks affected? Economists mosty favour te first approac since consumers tink in terms of door-to-door journeys. However, since poicy measures typicay invove a cange in te user costs of parts of te networks, one is tempted to use te second approac. To investigate tis question we first deveop a concise genera equiibrium mode eaborating on Kidokoro (004, 006). His mode consists of consumers, producers, a government, a network and a congestion externaity. We extend is mode in two directions. First, we repace is eementary network of two routes between two nodes by a fu-fedged network consisting of an undetermined number of inks between an undetermined number of nodes for an undetermined number of modaities. Second, we repace is representative consumers iving at a particuar node by eterogeneous consumers iving somewere. In tis mode, te door-to-door journeys are caed routes, te separate parts of te networks are caed inks. Utiity is defined as a function of te consumption of routes, not inks, as it soud be. Neverteess, it turns out tat in equiibrium bot te demand for routes and te demand for inks and te costs of using te routes and te costs of using te inks a pays a roe. For measuring wefare we foow te standard procedure of first deriving te expression for indirect utiity. We sow tat indirect utiity migt be eiter cast in terms of equiibrium prices for te use of routes or cast in terms of equiibrium prices for te use of inks. Ten we proceed by deriving te rue-of-a-af (OH) as an approximation for te cange in te user benefits. We sow tat te OH can bot be evauated in terms of te use of routes and in terms of te use of inks. Te two approaces yied exacty te same outcome. Tis resut impies tat te coice for eiter approac does not depend on teory but on practica considerations. Te ink approac migt give ess precise outcomes in case of a new ink. In tat case, we propose a step-by-step approac for te new ink and te most affected competing inks. Te routes approac acks precision as we. Since te number of routes is seer endess, routes aways ave to be aggregated into origin-destination-matrices. However, te conditions tat aow for aggregation are ardy ever being met. From a rea-ife case, a cost-benefit anaysis for a ig-speed rai ink between Amsterdam and Brusses, we sowed tat aggregation of journeys wic are not perfect substitutes does indeed yied considerabe measurement errors. Hence, we recommend te inks approac as a compement, or even as a substitute, for te OD-matrix approac. 7 Association for European Transport and contributors 010 7
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1 Introduction 1 Network projects raise many questions for te cost-benefit anaysis practitioner. One of te ardest nuts to crack regards te unit of anaysis: soud one cacuate user benefits on te basis of te cange in door-to-door journeys or derive tem from te cange in te use of eac and every separate ink of te networks affected? Economists mosty favor te first approac since consumers tink in terms of door-to-door journeys. However, since poicy measures typicay invove a cange in te user costs of parts of te networks, one is tempted to use te second approac. A number of questions are utimatey reated to tis same issue. Soud a cange in te out-of-pocket costs of car trave be treated as a wefare benefit? And wat if te cange in out-of-pocket costs is due to moda sift? And in case of moda sift, soud one use te vaue of trave time (VOT) of te od or te new transport mode? We wi sow tat te journey, or route approac and te ink approac are teoreticay equivaent: weter one derives te cange in wefare from a cange in te demand for routes or from a cange in te demand for inks, te outcome wi be te same. To proof tis we deveop a concise genera equiibrium mode eaborating on Kidokoro (004, 006). His mode consists of consumers, producers, a government, a network and a congestion externaity. We extend is mode in two directions. First, we repace is eementary network of two routes between two nodes by a fu-fedged network consisting of an undetermined number of inks between an undetermined number of nodes for an undetermined number of modaities. Second, we repace is representative consumers iving at a particuar node by eterogeneous consumers iving somewere. Kidokoro sows tat one ougt to measure user benefits from canges in te demand for eac of te two routes separatey, rater tan from te cange in te average cost of trave between te node of origin and te node of destination. If bot routes are perfect substitutes one migt sette for te origin-destination approac, but Kidokoro sees no surpus vaue in doing tat. We carry te anaysis one step furter by considering an undetermined number of inks on eac route between a node of origin and a node of destination. We argue tat one migt cacuate user benefits bot on te basis of te use of te routes and on te basis of te use of te separate inks. Sugden (1979) pointed aready to tis way of making wefare assessments of transport networks more down to eart, but in day-to-day practice is suggestion was not foowed. Textbook modes invariaby focus on door-to-door journeys between origins and destinations, te OD-matrix approac. Tis appies to bot te oder iterature (e.g. Jones, 1977) and te more recent (e.g. Jara-Díaz, 007). Te same is true for CBA guideines. Mackie et a (003, Note 6, p. 7) even write expicity: User benefits soud be cacuated on a matrix basis and not a ink basis. Unfortunatey, tey do not expain wy. Teir recommendation migt stem from 1 We benefited from numerous discussions over te ast few years wit quite some CBA practitioners. We tank Care Eijgenraam, Toon van der Hoorn, Gerard de Jong, Sytze ienstra, Bas Turpijn, Erik Veroef, No Verster and Peter Zwaneved for teir encouragements and comments on earier versions. 9 Association for European Transport and contributors 010 9
practica considerations. Possiby tey favor te OD-matrix approac as a way to dea wit probems tat arise if te project at and introduces a new ink in te network. Terefore, after aving estabised te teoretica equivaence of te two main approaces, we wi refect on a number of practica issues one encounters in appying te teory. In section we wi first mode te network. We define routes as a combination of adjacent inks. Tis is a wider concept of routes tan usuay empoyed in te sense tat one particuar route migt comprise inks tat beong to different transport modes. We wi estabis te precise reation between te use of routes and te use of inks and we wi make fairy genera assumptions about te reation between te costs of using te routes and te costs of using te inks. Ten we incude te network in a genera equiibrium mode. Utiity is a function of te consumption of routes, not inks. Neverteess, in equiibrium bot te demand for routes and te demand for inks and te costs of using te routes and te costs of using te inks a pay a roe. Section 3 deas wit wefare measurement. We foow te standard procedure of first deriving te expression for indirect utiity, wic migt be eiter in terms of equiibrium prices for te use of routes or in terms of equiibrium prices for te use of inks. By empoying oy s identity, tis yieds a measure for te cange in wefare as te sum of te cange in income pus te canges in demand induced by canges in prices. Again, te canges in demand coud be eiter te demand for routes induced by te canges in te user cost of routes or te demand for inks induced by te canges in te user cost of inks. Ten we proceed by deriving te rue-of-a-af (OH) as an approximation for te cange in te user benefits. We sow tat te OH can bot be evauated in terms of te use of routes and in terms of te use of inks. Te two approaces yied exacty te same outcome. Tis impies tat te coice for eiter approac does not depend on teory but on practica considerations. In section 4 we expore some practica issues. egarding te ink approac, we note tat te approximation by te OH migt be particuary imprecise in case of a new ink. In te guise of Netorp and Hyman (001) and Kato et a (003) we advocate a step-by-step metod for cacuating user benefits on suc a ink. Tis requires some extra work. egarding te route approac, we observe tat no practitioner can ever cacuate wefare canges on eac and every route, because of te seer endess number of routes. Terefore, one as to aggregate routes into a manageabe number, e.g. by reducing te network to a syntetic OD-matrix, before starting te wefare measurement. We discuss te imitations of tis approac sigty more extensivey tan didkidokoro (004). We iustrate it wit a rea-ife case of a CBA for a new ig speed rai track, were overooking te imitations of te OD-matrix approac ed to serious errors. Te concusions are summarized in section 5. 10 Association for European Transport and contributors 010 10
Mode and network We present a mode buiding on Kidokoro (004). Trips are te economic goods, consumed and produced, of wic equiibrium prices come about on markets. Congestion makes tat a gap arises between te private and socia costs of a trip. Tis externaity is modeed on te production side of te economy. Government intervenes in te markets for trips wit indirect taxes, suc as tos, and wit investment in te capacity of te transportation network. Toug wefare anaysis of tese investments can be done in te context of partia equiibrium te fu genera equiibrium mode deveoped in tis section provides a compete reference. It wi serve to demonstrate te equivaence between genera equiibrium anaysis and cost benefit anaysis. As te goods invoved are trips over routes and inks of a transportation network we eaborate on tis first..1 Network We consider a transportation network comprised of te actuay avaiabe infrastructure for different modes of transportation. Tis abstract network is a grap G caracterised by a set V of vertices, or nodes, and a set A of arcs, or te direct connections between te nodes, caed inks. A ink is typicay a road segment but it is aso te service of a bus company between two successive stops of a given ine. A ink may even be a pat for pedestrians between a parking ot and a raiway station. A common representation of grap G ( V, A) = is by means of its adjacency matrix wic indicates for eac pair of nodes weter tey are adjacent or not, and if so, by ow many distinct inks. Toug tere wi not be a ink between eac arbitrary pair of nodes we do assume tat a nodes can be reaced from a oter nodes: te grap is connected. Even an isand tat cannot be reaced by car wi be serviced by boat or pane. A route is a set of connected inks. A route is aso a set of adjacent nodes. Te inks of a network temseves are routes too. A given network as an infinite number of routes. Simpe routes are tose witout cyces: a ink is at most once in a route. A simpe grap is defined by te absence of mutipe inks and te absence of sef-oops. A compete grap as a ink between eac pair of nodes. et N be te number of nodes i of a given network. et te network ave a finite set of inks. If te network were simpe and compete it woud ave exacty N( N 1) inks. Te number of simpe routes r over te network is aways finite too. It must be reaized tat tis number soon becomes enormous. A simpe and connected network wit N = 10 as some 10 miion simpe routes. However, in an economic context wit cost minimisation, most of tese routes wi never be eigibe for use. Te economic goods in reation to te network are trips, or journeys, over tis network: movements from one node to anoter via a route r. et x r be te number of trips over suc a 11 Association for European Transport and contributors 010 11
specific route r of te network in a predetermined period. Use of a route impies te use of one or more inks in te same period. Te next fow variabe is movements over a specific ink. x, representing te number of et M be te matrix indicating from wic inks a route is composed. Eement M r is 1 wen ink is part of route r and is 0 oterwise. Te reation between ink and route usage is as foows: tota ink use is equa to te sum of te use of tose routes it is part of. x = M x (.1) r r r Next consider te cost of trips over inks, c, and over routes, c r. For some components of tese costs addition over inks may be straigtforward. Tink of distance, or time. However tis additivity is ess obvious for te vauation of time traveed. Neverteess suc a inear reation is often assumed (see for instance Wo en Hendrickson, 1985, p.49): te cost of a route is te sum of te cost of te composing inks. c = c M (.) r r Wit te above definitions and reations te economic mode can be deveoped, as we wi do beow. Te point of departure is a genera and muti-moda network. Equations (.1) and (.) wi be te recurring reations between te routes and te inks of te given network. And a information required of te structure of te network is contained in te route-composition matrix M. Observe tat in vector notation equation (.) impies a matrix pre-mutipication, making use of te convention among economists tat prices and costs are represented by rowvectors. x Mx = en c = c M Origin and destination pairs (OD) As an extension of te notation consider x ijk, te number of trips from node i, an origin, to node j, a destination, over a certain route k. OD-pairs pay an important roe in traffic modes and ence we sometimes empoy te ijk notation in stead of te more genera index r for routes. Exampe Finay consider as an exampe te foowing minima network of 3 nodes, 3 inks and 4 routes. Tis deviates from te cassica exampe of two nodes and mutipe inks. For simiar modeing see for instance Van Dender (004). 1 Association for European Transport and contributors 010 1
Figure.1 Network wit 3 inks and 4 routes B C A Te nodes are A, B and C. Te inks are te one-way connections AB, AC and CB. Te four routes are AB, AC, AC + CB and CB. Te tird route is composed of two connecting inks. Tis can aso be seen in te tird coumn of te route-composition matrix. M 1 0 0 0 = 0 1 1 0 0 0 1 1 (.3) Te one-way-traffic makes tat tere are just tree OD-pairs: (A,B), (A,C) and (C,B). Te OD-pair (A,B) knows two aternative routes, wic are AB and AC + CB.. Consumers At te eart of te deiberations of te consumers are routes: wi I make a nice tour on my motorcyce today or visit my aunt, ong overdue, by car? outes, or rater trips over routes, are te arguments in te utiity functions and are wat is demanded by te consumers. inks are mere intermediate goods, necessary for te production of routes. inks are fina consumption wen considered as a simpe route. Kidokoro (004) introduces N different ocations i as te nodes of a network. He associates wit eac ocation i a coection of consumers, modeed as a singe, representative, consumer i. Tis agrees wit te practice of transportation modeing. We generaize by considering a set of eterogeneous ouseods, not necessariy associated wit a given ocation. Tese different categories of ouseods can make use of eac part of te network. Utiity of te -t consumer, consumption of a vector U, is determined by consumption of a composite good x of trips z and x r. 3 Tese trips are caracterised by nodes i and j, and a 3 Superscripts are mainy used to distinguis between outes and inks. Tey are avoided for partia derivatives. 13 Association for European Transport and contributors 010 13
route k tat connects tem. Most of te time we denote tis by a route r. Substitutabiity and compementarity of trips is contained in te utiity function wic is assumed to be concave. U = U ( z, x ) wit x (..., x,... ) = (.4) r Income y consists of te vaue of te factors of production in te ands of te consumer (endowment of resources) y. In addition tere may be a transfer T. For ease of exposition we assume te existence of ony a singe productive factor tat corresponds one-on-one wit te composite good. et p be te vector of te individua route prices p r pertaining to te consumers. Tese prices are normaised wit te price for te singe factor. Tis gives te foowing budget constraint. z + p x = y = y + T were p x = p x (.5) r r r For an interior soution, 0 x >, we wi ave tat reative trip prices equa te margina r utiities. Tis amounts to standard micro-economics appied to trips as commodities. U x r = λ p r and U z = λ were λ is te margina utiity of income (.6) Wardrop principe as a specia case A specia case arises wit te Wardrop-principe (Wardrop, 195), were te functiona form of te utiity function is suc tat aternative routes k between given ocations i and j are taken to be perfect substitutes. Consumers, given tis specification, ony care about reacing a given destination from a given point of departure. Tis can be accompised by aving consumption of trips over te different routes between i and j additivey in te utiity function. Tis impies tat te margina utiities of a routes 1 and from i to j are aways equa. Te important consequences of tis specia case for transportation modeing are discussed beow. U = U ( z,..., x + x +...,...) impying ij1 ij U x U = ij1 xij (.7).3 Producers and congestion Te producer mode first of a concerns inks as tey are te appropriate eve of suppy of transportation services and tereby to identify te costs. Consider terefore production of trips over a singe ink. et te tota resource cost be a function of tis fow and of a capacity parameter C of producing te tota fow I. As tese tota costs contain X of ink use 14 Association for European Transport and contributors 010 14
monetized time cost, wic may differ per individua consumer, individua components wi be incuded. 4 Producers simpy carge te consumers teir individua average ink use costs c = c ( X, I ). Tey make no profits, nor osses. Furter, observe te simpification tat ink costs are independent of te fows and capacities of oter inks. = wit C ( X, I ) c ( X, I ) x X = x (.8) Average ink costs increase in bot tota and individua use. Costs decrease wit more capacity. c x c = > 0 X c I, < 0 (.9) Tere wi be a gap between te margina socia cost mc of producing trips x and te average cost per unit carged to te private consumer. et te congestion externaity per unit ext use t be defined as tat difference. And since te source of an additiona car on a road segment wi not matter for te degree of congestion it causes on tat ink, te externaity wi be equa for a consumers: ext t. Tis resut was aready contained in te assumption of aving te tota fow of trips as argument in te cost function. Beow subscript is a ep index for summation. C c mc c X I x ' = = (, ) + ' x ' x c c t = mc c = x = x = t > 0 ext ' ' ext ' ' ' x ' X (.10) (.11) For onger routes, covering severa inks, te assumption is tat ink costs just can be added. c = c M ( in vector notation: r r c = c M ) (.1).4 Government Government intervenes in te transportation markets wit indirect taxes t ind per unit ink use, possiby to counter congestion. Consumer prices ten consist of te individua average ink cost pus tis tax, taken equa for a. 4 Individua time preferences are modeed as costs. Tus it can be avoided to introduce time as a separate commodity. 15 Association for European Transport and contributors 010 15
ind ext ind p = c ( X, I ) + t ( p = mc t + t ) (.13) oute prices ten contain a te tos on te composing inks. ( ind p = p M = ( c + t ) M r r r p = p M ) (.14) Indirect tax revenue is ump sum redistributed. Finay, te tota outays on pubic investment I are added to te government account. Tus te description of te accounting of te mode is competed. t x = T + I (.15) ind.5 Equiibrium Consistent accounting is a prerequisite for a genera equiibrium mode. Next is te optimaity of te decisions of te agents given prices. For te consumers tis as been described above. For production te mode is just too simpe to consider optimaity. Government is exogenous. Finay markets must cear and on tese market equiibrium prices, endogenous variabes of te mode, must come about. We ave modeed markets, and ence prices, for ink and route use. Bot pay a roe. Toug consumer preferences are formuated in terms of routes, inks are te intermediate goods needed to produce tem. Te mode is based on fairy genera assumptions: no restrictions on te network ave been imposed, nor as a specific functiona form of utiity been cosen. Wit tis genera framework we wi examine te practica impications for wefare measurement. Wardrop In recognition of its practica importance, we return briefy to te specific mode of perfect substitutabiity of te aternative routes of a given origin-destination pair (i,j). To satisfy optimaity condition (.5) te endogenous route prices must be equa for te different routes between i and j. Tis is te Wardrop-principe. U x U = ij1 xij pij1 pij = (.16) Te number of trips over te aternative routes wi adjust suc tat its costs wi be equa. Tis ods for tose routes actuay used. Aternative routes wit iger costs wi not be used. 16 Association for European Transport and contributors 010 16
Congestion makes tat tere is indeed scope for adjustment. Wen costs no onger can adjust corner soution wi arise. Corner soutions wi be seen to ave practica consequences. 17 Association for European Transport and contributors 010 17
3 Wefare measurement Based on te genera equiibrium framework described above we wi derive expressions for wefare cange. We wi do so in terms of routes as we as inks. Next we wi sow tat Kidokoro s 004 decompositions of wefare cange sti od. Tese decompositions ep in estabising exacty wat te wefare effects are, and wat are not, foowing investment in a transportation network. Te discussion of wefare measurement, in particuar of te rue of af, prepares for te part on practica issues. 3.1 Wefare anaysis in a distorted economy Te economy of our mode as two distortions, 5 congestion and indirect taxes. Wen te atter canceed te former tey are corrective, Pigouvian, taxes. ind ext ind ext p mc = c + t ( c + t ) = t t (3.1) Consider a inear socia wefare function W wit wefare weigts α. Wit te weigts set to te inverse of te margina utiities of income, e.g. α = 1/ λ, te wefare function represents te principe of a doar is a doar. Moreover, wit tese weigts te socia optimum can be decentraized as te equiibrium outcome of te economy, aso in te case of our distorted economy. 6 W = α U ( z, x ) wit α 1/ λ = (3.) Using budget equations (.4) and (.14) and te fundamenta reation between te costs of routes and ink use, equation (3.) is rewritten. Socia wefare is broken down in overa consumer surpus (CS), producer surpus (PS) and net government income (BS). 7 8 (3.3) W = { ( α U ( z, x ) y ) } + { y } + { t x I } Producer surpus (PS) consists of te vaue of endowments and profit. Te atter is zero. However, wen canges are invoved, margina cost, and ence te congestion externaity, wi be visibe again. Net government income (BS) equas tota transfers. Te above breakdown 5 A distortion is an inequaity of te margina rate of substitution and te margina rate of transformation between a pair of commodities, Bagwati (1971). 6 Wit quasi-inear utiity te wefare weigts are one and inconsequentia. 7 See, for instance, Ginsburg & Keyzer, 1997, paragrap.4.. 8 Te breakdown gives overa consumer surpus. Consumer surpus at market eve, possiby better recognized, wi appear sorty. 18 Association for European Transport and contributors 010 18
sows tat working wit consumer and producer surpus impies tat eiter indirect taxes or transfers need to be incuded. Pubic investment projects are modeed as a vector of canges of te capacity parameters: Tey are scaed suc tat tey coincide wit investment costs. Te projects are financed wit ump sum transfers from te consumers so tat te government budget remains baanced. Since exogenous di are te triggers of cange it figures to first consider price canges tat cange demand tat subsequenty cange te eve of wefare. Tis cain of causaity requires a reformuation of socia wefare in terms of indirect utiity. 9 di. First take indirect utiity as a function of income and route prices as an aternative for eq. (3.). W = α V ( p, y ) wit α 1/ λ = (3.4) We now derive an expression for wefare cange by taking te tota differentia of W and appying oy s identity. Te investment is assumed to ony invove a singe ink. dw = α V dp V dy + r i di di r pr di y di (3.5) ( x (, ) r p y dpr dy ) = + r Next indirect utiity can be taken as a function of income and ink prices to reconsider (3.). W = α V ( p, y ) wit α 1/ λ = (3.6) Again appying oy s identity eads to an identica expression in te demand of ink use. (3.7) dw = ( x (, ) p y dp + dy ) Direct substitution of te reations between routes and inks, (.1) and (.13), in eq. (3.5) eads to te same resut. Te important practica significance of tis resut wi be discussed beow. Expressions (3.5) and (3.7) deserve some specia attention as wat foows is based upon tem. 9 One can aso work wit te expenditure function and define te wefare measures of compensating and equivaent variation. 19 Association for European Transport and contributors 010 19
First of a, we are sti in te ream of genera equiibrium. Tis can be seen from te use of te Marsaian demand functions x ( p, y ), or x ( p, y ), wic as as arguments income r and te prices on a markets. Secondy, we see tat te wefare weigts ave disappeared in a natura way. Terefore it is unnecessary restrictive to avoid tem beforeand by coosing a quasi-inear utiity function, as Kidokoro does. Next, it is reevant to point out tat wefare cange, by definition, is a net concept. In te word of CBA of infrastructure projects owever, tere is an empasis on benefits, wit costs taken as given. Here we consider canges. Te cange in CS, PS and government income we denote as te cange of tota benefits dtb. Te cange in tota costs, dtc, are te investment expenditures in te capacities of a inks, di. Beow a distinction wi be made between a ink wit investment and a oter inks. dw = dtb dtc = dtb di wit di = di (3.8) A ast preparatory comment concerns te consumer surpus visibe in equations (3.5) and (3.7) wic is now at market eve. ower consumer prices wi ead to a iger surpus. Te equations can be represented as beow. Te distortions wi reappear in te decompositions tat foow. dw dcsr dy = dcs + dy (3.9) r = + 3. Decompositions of wefare cange Investment in te capacity of part of a network wi cange te entire pattern of movements over tat network. Moreover, trave cost wi not ony cange for tose inks and routes wit investment but aso for a oter inks and routes. Wit a tese effects it matters cruciay to determine wic of tese, or not, contribute to wefare cange. Kidokoro (004) presents to tis purpose tree decompositions of wefare cange. Tey invove te oss of wefare as a consequence of te distortions, te deadweigt oss. We wi estabis tat te decompositions sti od in te context of a genera network, wit te assumption tat ink costs are ony a function of own capacity and its use. Tota benefits consist of, te canges in: i) consumer surpus of a inks pus indirect tax revenue, or ii) consumer surpus of te ink wit investment pus indirect taxes on tat ink pus te net deadweigt oss of a oter inks, or iii) tota cost saving of te ink wit investment pus te net deadweigt oss of a inks. We wi now derive tese decompositions. 0 Association for European Transport and contributors 010 0
Canges in income depend on te size of te investment and te effect on indirect tax revenue. Te atter wi cange because te pattern of trips canges. Summation over consumers makes te ump sum redistribution disappear. dy = t dx di (3.10) ind Substitution in (3.7) and (3.8) gives Kidokoro s first resut: tota benefits, dw + di canges in te consumer surpus of a inks pus te cange in indirect tax revenue., are te (3.11) dtb = dcs + t dx ind For te second resut canges in quantities and prices are written as functions of te investments at te root of te canges. Here a distinction is made between te ink wit investment in its capacity: 0 and a oter inks. Aso, for summation over a oter inks we use. Finay, te set of a inks, tat is incuding te one wit investment, is sti denoted by te simpe subscript. An investment on a singe ink can affect te capacity of a ost of routes. We aim to derive an expression for te cange in trave cost of a given ink, given te investment in anoter ink 0. Take te differentia of price equation (.1). 10 We assume tat te tariffs of te indirect taxes do not cange. c ( X, I ) dx c ( X, I ) di dp di di = 0 + 0 X di 0 I 0 di 0 (3.1) Te second term on te rigt-and side of te equation is zero for a inks but tose wit investment. Tis is because costs of inks ave been assumed to be independent. dp c ( X, I ) dx " " " " " = di 0 X '' di 0 (3.13) Te price cange mutipied wit te number of trips and subsequenty summed over a consumers makes te externaity on te oter ink appear. See definition (.10). c ( X, I ) dx dx " " " " ext " x " dp" = x" di 0 = t " di 0 X '' di 0 di 0 (3.14) 10 Tis is te crucia step. See pag. 303 of Kidokoro (004). 1 Association for European Transport and contributors 010 1
Substitution of tis equation in (3.7), using (3.10) and rearranging gives: (3.15) " dtb = dcs 0 + t ind 0 dx 0 + ( t ind ext " t" ) dx " Te cange in tota benefits foowing an investment in a singe ink 0 consists of te cange in te consumer surpus of tat ink, pus te cange of indirect tax revenue of tat same ink, pus te cange in te net deadweigt osses of a oter inks. Te deadweigt oss is caused by bot te externaity and te indirect tax. Tis is Kidokoro s second wefare decomposition. Te tird resut comes about by a comparabe derivation wic does use te second term on te rigt and of equation (3.1). Tis decomposition empasizes te distortions in a inks. It sows te distinction between first best and second best positions. C 0 ind ext dtb = X 0 0 ( ) di + t t dx (3.16) I 0 Te first term on te rigt-and side denotes te cost saving for a traffic, i.e. not just increase, over te ink wit investment. In te absence of net distortions tis woud be te ony wefare effect. 3.3 Wefare measurement & approximations Aas, utiity and wefare cannot be measured directy. Te expressions above, owever, do contain, not quite accidentay, variabes tat can in principe be measured. Toug measuring congestion externaities may be difficut, it is not impossibe. Anoter issue remains probematic: te expressions above are based on sma (infinitesima) canges. For arger and discrete canges te differentia canges of above soud be integrated over an initia and a new situation. 11 Departing from equation (3.7) we ave te foowing. 1 (3.17) W = dw = x ( p, y ) dp + y Tis measure is not unique since it depends on te order of integration. Even if it is a first-order approximation, demand functions do depend on a prices. Tis is te probem of pat dependency. On top of tat te demand function wi not competey be known. Wit a few observations tey may be approximated. An observation of te new situation, i.e. after 11 See f.i. Boadway and Bruce (1984), par. 7.. 1 Consumer surpus now appears as integras over Marsaian demand functions. Association for European Transport and contributors 010
investment, may be suppied by a traffic mode. Suc a mode coud be a partia equiibrium mode. Te vaues of te variabes in te initia situation soud be avaiabe. We proceed wit a second-order approximation of socia wefare, using a inear approximation of te demand functions. Tus te so-caed rue of af wi be derived. Te wefare function in (3.6) is expanded around te point of te initia situation. Tis impies tat aso te margina utiities of income of te od situation are used to convert utiity to a money measure. V dp V dy dw = α di + di + p di y di 1 V dp dp + di di + ' α ' ' p p ' di di ' (3.18) 1 V dp 1 V dy + α di dy + α di p y di y di oy s identity is appied making demand functions appear and te wefare weigts disappear. (3.19) dw = x ( p, y ) dp + dy + 1 x ( p, y ) + 1 1 + dpdp ' ' p ' x ( p, y ) λ dpdy ( dy ) y y egrouping of terms gives te foowing. dw = x ( p, y ) dp + dy + 1 x ( p, y ) x ( p, y ) dp ' + dy dp + ' p ' p ' (3.0) 1 + λ ( dy ) y Te expression between accoades is exacty a inear approximation, i.e. first order, of demand wit a cross price effects and te income effect. 3 Association for European Transport and contributors 010 3
x ( p, y ) x ( p, y ) dx dp dy = ' + ' p ' p ' (3.1) Tus te expression for wefare cange wi ugey be simpified. In addition, it is common practice to assume te margina utiity of income to remain constant over te woe trajectory of utiity cange. 13 Tis makes te ast term of (3.0) disappear. Te fina measure reads: 1 dw = x dp dx dp + dy (3.) In practice one works wit observations: 1 1 1 x p, ( x, p, ), were superscript 0 0 0 (,, ) 0 denotes te initia situation and superscript 1 te new one. ewrite equation (3.). 1 dw = x ( p p ) ( x x )( p p ) + ( y y ) 0 c 1 0 1 0 1 0 1 0 i 1 dw = x + x p p + y y (3.3) 1 0 1 0 1 0 ( )( ) ( ) Tis ten is te rue of af contained in a measure for tota wefare cange. It is important to reaize tat te same expression coud ave been derived in terms of routes. Te rue of af for routes obviousy as te same form as te one expressed in inks. Under te assumption of inear additive ink costs te approximation wi ave exacty te same vaue. We demonstrate tis wit observations in routes: rewriting (3.3) using te route-ink reations. x p, 0 0 0 ( r, r, ) x p and 1 1 1 ( r, r, ) dw 1 = + + 1 0 1 0 1 0 ( x x )( p p ) ( y y ) 1 = + + (3.4) 1 0 1 0 1 0 ( p p ) M r ( xr xr ) ( y y ) r 1 = + + 1 1 0 1 0 1 0 = ( pr pr )( xr + xr ) + ( y y ) 1 0 1 0 1 0 ( p p ) M r ( xr xr ) ( y y ) r r 13 Wit tis assumption our fina measure and compensation and equivaent variation wi be te same. 4 Association for European Transport and contributors 010 4
4 Practica issues 4.1 Te rue-of-a-af for a new ink In principe one migt appy te OH in case of a new ink by simpy assuming tat te ink aready did exist, but tat te costs of using te ink were proibitivey arge. E.g. if te new ink is a road crossing fieds, te costs of crossing tese fieds by car in te absence of te new ink were incrediby ig and te number of cars actuay crossing te fieds was zero. In fact it suffices to assume tat te cost of crossing te fieds was just a itte bit iger tan te cost of using te cosest substitute. Tere can sti be doubts about te appropriateness of appying te OH in tis case. Te OH is a good approximation of user benefits ony in case of sma curvature of demand and itte variation of perceived costs (Jara-Díaz, 007, p. 88). ooking at figure 4.1a, te inearization by OH woud do we for price drops from p 0 to p 1, p 1 to p or p to p 3. But in case of a price drop from p 0 to p OH woud overestimate, and in case of a price drop from p 1 to p 3 underestimate te user benefits. Figure 4.1a A demand curve wit strong curvature Figure 4.1b: A step-by-step approac for te OH Te Marsaian demand curve depicted in figure 4.1a migt appy to a new ink being a cose substitute for an existing ink (or combinations of inks), te cost of using tat existing ink aying somewere between p 1 and p. If te cost of using te new ink remains iger tan p 1 ardy any users wi be attracted. If te costs are reduced from p 1 to p many users substitute te existing ink for te new one. Furter cost reductions beyond p wi not attract existing traffic anymore, but ony generate some new traffic. 5 Association for European Transport and contributors 010 5
In case of doubt, Netorp and Hyman (001) propose a procedure wic migt be caed numerica integration. Suppose (p 3, x 3 ) is te point tat indicates te projected costs and projected use of te new ink, wie p 0 is a cost eve tat is definitey iger tan te cost of using te cosest substitute. In order to correcty cacuate te user benefits one as to find te form of te demand curve between te points (p 0, x 0 = 0) and (p 3, x 3 ). One migt ask te traffic anaysts to provide some extra observations, i.e. projections for severa more modest dimensions of te new ink. Point (p, x ) coud represent te projection for a two ane road, rater tan te proposed four ane road, and (p 1, x 1 ) te projection if additionay a speed imit woud be imposed on te two ane road. Having tese extra observations, one can cacuate te tota user benefits for tis ink as te sum of te resuts of te OH cacuations for, consecutivey, te price drop from p 0 to p 1, from p 1 to p and from p to p 3. Kato et a (003) did appy tis procedure. Notice, first, tat tis procedure of numerica integration is needed ony for tose inks for wic tere are suspicions tat te conditions for te OH are not being met. For a oter inks, one step wi suffice. Notice furter tat even in te case of a new ink, te Marsaian demand curve coud be muc smooter tan te one depicted in figure a and te one step cacuation woud suffice as we. E.g. in case tere are a number of existing inks tat are pysicay substitutes in varying degrees for te new ink, or in case consumers ave eterogeneous preferences (incuding ove of variety ). 4. Aggregating routes into OD-reations In practice it is impossibe to cacuate user benefits for routes, simpy because of te seer endess number of routes. A simpe and compete network of ony 10 nodes as 90 inks but no ess tan 10 miion routes, even excuding tose wit cyces. Usuay te number of routes is sized down to manageabe numbers by aggregating tem into origin-destination reations. First, a routes between a node of origin and a node of destination are aggregated by summing a te traffic fows over te routes concerned and by cacuating te weigted average of te user costs. Ten, a nodes in a zone are aggregated ikewise in order to arrive at a manageabe number of OD-pairs. As pointed out by Kidokoro (004) in an appendix to is paper, te user benefits cacuated on te basis of te OD-matrix data are usuay not equa to te true user benefits cacuated on te basis of te use of te separate routes. However, if te Wardrop principe appies, te cacuated benefits on te basis of aggregated routes do correspond to te true benefits. Te Wardrop principe assumes tat a used routes concerned are perfect substitutes and tus ave te same user costs, wie a not-used routes are more costy. Goods tat are perfect substitutes can, from an economic point of view, aways be aggregated. So, if te traffic forecasts are produced by a mode empoying te Wardrop 6 Association for European Transport and contributors 010 6
principe, and if te preferences of te peope making trips on eac OD-reation can be assumed to be omogeneous, no arm is done by cacuating benefits on te basis of te OD-matrix data. Notice, owever, tat some modes tat do empoy te Wardrop principe in te route coice modue, revert tereafter to a stocastic tecnique for assigning te traffic fows to te various parts of te pysica network. Stocastic assignment impicity assumes ess tan perfect substitutabiity and/or some degree of eterogeneity of preferences. Ten again, te resuting OD-matrix data do not produce a correct figure of te true user benefits. If ess tan perfect substitutabiity of routes and/or some degree of eterogeneity in preferences are considered to be predominant, one wi from te outset opt for empoying a ogit mode. In fact tere are many reasons wy routes are imperfect substitutes, suc as differences in reiabiity, comfort, safety, scenery etc. Even in te case of freigt te degree of eterogeneity is considerabe (De Jong, 000). Te ogit mode is a very ric and powerfu too for deaing wit tese differences. And besides cacuating user benefits from te canges in te use of te separate routes and/or inks, one migt cacuate tem for te so-caed ogsums (De Jong et a, 006). But for wefare assessments one soud not use te OD-matrix data resuting from te ogit mode, since a te reevant information about te differences in prices and preferences is being ost in te process of aggregation. We can iustrate te possibe errors by ooking at te CBA of te ig speed rai track Amsterdam-Brusses (Van Hasseen & Van Scijnde-Pronk, 1994). First, te cange to be expected in traffic fows (by road, air, conventiona train and ig speed rai) was anayzed by a dedicated ogit mode. Ten, te user benefits were cacuated from te resuting OD-matrix data for 5 zones. Separate OD-matrices for business and non-business, domestic and internationa trave were cacuated, but a modes of transport (a routes ) were aggregated. Tis ceary resuted in errors. According to te traffic anaysis af a miion business peope woud move on a yeary basis from air transport to train, for teir trips between Amsterdam on te one and and ondon or Paris on te oter. Tey woud trade te reative advantages of air transport for te reativey ceaper ticket price of te train. Since te CBA was based on ODmatrix data, te savings on out-of-pocket cost was among te user benefits, te oss of comfort was negected. Consequenty, te tota benefits for tis group of traveers were cacuated as te equivaent of 117 euro per trip, wie appying te OH to eac and every route apart woud ave resuted in a benefit of ony 0 euro. Tis figure is muc ower since, impicity, te OH takes a reasons wy peope migt ave preferred air trave over rai into account. And te fact tat a arge number of business traveers were expected to stick to air trave, notwitstanding te iger out-of-pocket costs, does point ceary to unobserved advantages of air transport. 7 Association for European Transport and contributors 010 7
5 Concuding remarks We sowed tat for a quite genera economic mode and under fairy genera assumptions regarding te properties of te network, user benefits coud be derived in two equivaent ways. Wefare canges coud be derived eiter from canges in door-to-door journeys, wic we caed te route approac, or from canges in te use of eac and every separate part of te pysica networks, wic we caed te ink approac. Te route approac, campioned in textbooks and CBA guideines, fits in nicey wit te teory of consumer demand. However, te inks approac maintains a tigt reation to te actua poicy measures wic usuay entai a cange in te capacity or te user costs of some specific inks of a network. Bot approaces confront te CBA practitioner wit a measurement probem. Te ink approac migt give ess precise outcomes in case of a new ink. In tat case, we propose a step-by-step approac for te new ink and te most affected competing inks. Te routes approac acks precision since routes aways ave to be aggregated into an OD-matrix, wie te conditions tat aow for aggregation are ardy ever being met. It is recommended to do te wefare cacuations at te owest possibe eve of aggregation, i.e. on te basis of te most detaied OD-matrix. From a rea-ife case we sowed tat aggregation of journeys wic are ceary not perfect substitutes yieds considerabe measurement errors. From a practica point of view tere are some additiona considerations tat pead for te ink approac as a suppement to, or in some cases even instead of, te route approac. Te most important advantage of te ink approac is tat it sows, on a map, were benefits of te cange in costs or capacity occur. Even more important, it sows were new bottenecks emerge. Tis is usefu information for te design of te project. It wi suggest ow one migt optimize te design by adding some extra capacity at specific points esewere in te network. Te ink approac is neaty connected wit te actua traffic predictions, since it uses tese canges in traffic fows as inputs. Tis down-to-eart approac makes it easier to unrave seemingy impausibe CBA outcomes if tey occur. Te number of cacuations for te CBA can be kept to a minimum. Wie for te OD-matrix approac in principe a traffic fows soud be considered, since a part of eac OD-fow migt use inks in te project area, te ink approac can be imited to canges in te project area and its direct surroundings. Particuary for a sma oca project, one migt safey assume tat te user costs on inks farter away wi not cange. 8 Association for European Transport and contributors 010 8
As ong as te engt of inks is not atered, one migt eiter use te number of trips or te number of kiometers traveed over te ink as te unit of measurement. Doing a te cacuations in terms of te number of kiometers traveed is convenient, since oter items in te CBA suc as te cange in poution, use tese data as we. Te ink approac is particuary suitabe for ex post evauations. Te evauation of te Stockom congestion carge by Eiasson (009) iustrates tis. Using te ink approac, Eiasson ony ad to coect data on actua canges in traffic fows and speeds on te inks in and around te project area, witout aving to ask te motorists for te origin and destination of teir journey. By deriving te appropriateness of te ink approac from a quite genera economic mode and under fairy genera assumptions regarding te network, we sowed in fact tat te CBA practitioner does not ave to know te mode tat generates te canges in te traffic fows. Tis point was made aready by Kidokoro (004) and stressed more forcefuy in Kidokoro (006). Not ony for ex post evauations, aso for ex ante assessments it suffices tat te CBA practitioner is tod wat canges actuay wi take pace on te network, not wat drives tem. Tus, te division of abor one encounters in practice, between te traffic engineer and te CBA economist, is justified. 9 Association for European Transport and contributors 010 9
eferences Bagwati, J.N., 1971, Te generaized teory of distortions and wefare. In: Bagwati, J.N. et a, eds., 1971. Trade, Baance of Payments and Growt. Amsterdam: Nort-Hoand. Boadway,.W. and B. Nei, 1984, Wefare Economics. Oxford: Basi Backwe. De Jong, G., 000, Vaue of freigt trave-time savings. In: Henser, D.A. & Button, K.J. eds., Handbook of Transport Modeing. Amsterdam: Esevier. C. 34. De Jong, G., A. Day, E. Kroes and T. van der Hoorn, 006, Using te ogsum in project appraisa. In: 11t Internationa Conference on Trave Beaviour esearc. Kyoto, Japan 16-0 August 006. Eiasson, J., 009, A cost-benefit anaysis of te Stockom congestion carging system. Transportation esearc Part A, 43(4), pp.468-480. Ginsburg, V. and M. Keyzer, 1997, Te Structure of Appied Genera Equiibrium Modes. Cambridge: Te MIT Press. Jara-Díaz, S., 007, Transport Economic Teory. Amsterdam: Esevier. Jones, I.S., 1977, Urban Transport Appraisa. ondon: MacMian Press. Kato, Hironori, Kaneko,Yuiciro & Inoue, Masasi, 003, Measurement of transport investment benefit: empirica comparisons between OD-based approac and route-based approac. Journa of te Eastern Asian Society of Transportation Studies, 5, pp.96-971. Kidokoro, Yukiiro, 004, Cost-Benefit Anaysis for Transport Networks: Teory and Appication. Journa of Transport Economics and Poicy, 38(), pp.75-307. Kidokoro, Yukiiro, 006, Benefit estimation of transport projects - a representative consumer approac. Transportation esearc Part B, 40, pp.51-54. Mackie, P., J. Netorp, J. aird and A. Farad, 003, Tookit for te economic evauation of Word Bank transport projects. Avaiabe at: ttp://www.its.eeds.ac.uk/projects/wbtookit/index.tm [Accessed 9 June 009]. Netorp, J. and G. Hyman, 001, Aternatives to te rue of a af in matrix-based appraisa. In: European Transport Conference. Cambridge, UK 10-1 September 001. 30 Association for European Transport and contributors 010 30
Sugden,., 1979, Te measurement of consumers surpus in practica cost-benefit anaysis. Appied Economics, 11(), pp.139-146. Van Dender, K., 004, Pricing transport networks wit fixed residentia ocation. egiona Science and Urban Economics, 34, pp.89-307. Van Hasseen, H.W.J. en M.Y. van Scijnde-Pronk, 1994, Kosten-Baten-Anayse Hoge Sneeidsijn. otterdam: NEI. Varian, H.., 1984, Microeconomic anaysis, nd ed. New York: W.W. Norton & Company. Wardrop, J.G., 195, Some teoretica aspects of road traffic researc. Proceedings of te Institute of Civi Engineers, 1, pp.35-78. Wo, M. and C. Hendrickson, 1985, Transportation Investment and Pricing Principes: Introduction for Engineers, Panners and Economists. Hoboken: Jon Wiey & Sons Inc. 31 Association for European Transport and contributors 010 31