EVALUATING TRANSPORT USER BENEFITS: ADJUSTMENT OF THE LOGSUM DIFFERENCE FOR CONSTRAINED TRAVEL DEMAND MODELS

Transcription

1 EVALUATING TRANSPORT USER BENEFITS: ADJUSTMENT OF THE LOGSUM DIFFERENCE FOR CONSTRAINED TRAVEL DEMAND MODELS Chrstan Wnkler German Aerospace Center (DLR), Insttute of Transport Research Rutherfordstr., Berln, Germany Tel: , Fax: , Emal: Word count: word text + tables x 0 words (each) = Submsson Date //0

2 Wnkler 0 ABSTRACT Transport user benefts are of great mportance n cost beneft analyss when apprasng transport proects. Generally they have the greatest mpact on the results of cost-beneft analyses. It s common to adopt the consumer surplus for calculatng transport user benefts. The consumer surplus measure s based on the underlyng demand model and follows from the ntegraton of the demand curve. If the popular logt model s used for forecastng travel demand, consumer surplus measure takes a closed form (the logsum ) that s easy to calculate. Furthermore, n cost-beneft analyses the change n consumer surplus between an ntal and fnal state s needed, whch can be easly derved by the dfference of the logsums of the two states. Ths logsum approach s proven and correct for travel demand models based on the logt model wthout multple constrants. However, for travel demand models dealng wth two or more sets of constrants the logsum approach fals. In ths paper we descrbe a mathematcal approach for a transport user benefts measure that corresponds to the consumer surplus and s unversal for all travel demand models wth constrants. We exemplarly derve the measure for a doubly constraned trp dstrbuton. The applcablty of the derved approach s shown by a small example.

3 Wnkler INTRODUCTION Transport nvestments and polces are often assocated wth hgh nvestment costs for socety and cost changes for transport users, whch mght lead to welfare changes. It s therefore essental to antcpate welfare mpacts ex ante and gve decson makers robust decson support. There are dfferent concepts for evaluatng proects and polces; however, cost-beneft analyss (CBA) s normally the method of choce. Generally, n transport proect and polcy apprasals, transport user benefts have (n addton to nvestment costs) the most mpact on welfare changes. User benefts result from the changes of user costs such as travel tme and travel cost between the ntal state (the most lkely transport stuaton over the course of the apprasal perod f no nterventon were to occur) and the fnal state (the ntal state wth transport nterventon). Economc theory as the theoretcal bass of the CBA provdes the concept of the (change n) consumer surplus for the calculaton of the transport user benefts. Ths measure can be derved by the mathematcal ntegral of the demand functon (see, for example, Varan, ). Accordngly, there s a drect lnk between user benefts, costs, and travel demand changes. The appled method of ntegratng the travel demand functon defnes the accuracy of the resultng user benefts. The standard approach n practce s the so called Rule of a Half (RoH). It has been wdely used n proect nvestments and polcy analyss snce ts frst ntroducton n the 0s (Wllams, ). The concept assumes a lnear path between the user costs n the ntal and fnal state, whereas the real shape of the underlyng demand functon s neglected. For ths reason, the RoH only provdes a precse beneft measure for small user cost changes and only f the demand curve s nearly a straght lne, otherwse t can be only consdered as a rough approxmaton (De Jong et al. 00). Demand curves of travel demand models are generally not straght lnes, and the RoH often provdes nexact transport user benefts. However, there are other approaches to obtan more precse results. In partcular, the antdervatve of the appled travel demand functon could be used for calculatng a mathematcal exact consumer surplus. Ths approach presupposes that the antdervatve s known. If that s the case, drect ntegraton s possble and would be the best soluton. The wdely used multnomal logt model (MNL) provdes the antdervatve that s called logsum term (or short logsum ). The MNL s based on the concept of random utlty maxmzaton and was mcroeconomcally derved by McFadden (). In context of proect evaluaton, the consumer surplus of the ntal and fnal states can be provded by logsums. Hence, the change n consumer surplus s calculated as the logsum dfference. The logsum approach s a theoretcally well-founded and easly calculable method for provdng transport user benefts. It s appled by more and more researchers and transportaton planners (Zhao et al., 0). However, t strongly depends on the use of the MNL (or nested logt) as the underlyng travel demand functon. There are many travel demand models appled n research and practce that are not apparently congruent to the MNL. Ths s especally the case for travel demand models wth constrants. These models are tradtonally used for trp dstrbuton and regard orgn and destnaton constrants, whch ensure that gven total numbers of trps produced n and attracted to each traffc zone equal resultng numbers of trps. If such a model s used for calculatng trp dstrbuton, the logsum approach fals and as a rule transport user benefts are approxmated by the RoH. In ths paper we want to transfer the logsum approach for these wdely used constraned travel demand models and provde a mathematcal exact transport user beneft measure for these models. For ths purpose an adustment of the logsum dfference s necessary. We wll carry out the analyss by the EVA model (Vrtc et al., 00) as a representatve of constraned models, whch s characterzed by dfferent constrant types. For ths reason the paper frst ntroduces travel demand models wth constrants, n partcular the EVA model and ts logt verson. It then goes on wth beneft measures and problems of the logsum dfference approach wth constrant models. After that, the mathematcal adustment of the

4 Wnkler logsum approach follows. Then, a short and very smple example wll serve as an llustraton of the approach. Fnally, a summery s provded. It has to be noted that the man am of the paper s to provde a theoretcal dervaton of an adusted logsum approach for travel demand models wth constrants rather than to show specfc applcatons and comparsons. These questons wll be dscussed n the future. TRAVEL DEMAND MODELS WITH CONSTRAINTS There are lots of dfferent concepts and approaches for modellng travel demand, and the well-known - step-algorthm s of partcular mportance. These steps can be calculated successvely or partly combned, whereby a combnaton of trp dstrbuton and mode choce s most common. Here we focus on ths approach and gve models wth constrants specal attenton. Tradtonally, constrants are used for trp dstrbuton. The best-known model s the doubly constraned gravty model (see Wlson, ), whch satsfes the gven total numbers of trps orgnatng n and attracted to a travel zone. The total numbers of trps are results of trp generaton and nput varables of trp dstrbuton. EVA Model Lohse et al. () extended the doubly constraned gravty model to mode choce and to less restrctve knds of constrants. The model s called EVA model from the German terms for producton (Erzeugung), dstrbuton (Vertelung) and mode choce (Auftelung). It s more unversal and flexble than tradtonal approaches, whch can also be formulated by the EVA approach. The algorthm contans trp generaton, ont trp dstrbuton, and mode choce. However, n the followng we do not consder trp generaton and mode choce, snce t has no nfluence of the topc dscussed n ths paper. For a more comprehensve dscusson of the EVA model see Vrtc et al. (00). The EVA approach models travel behavor based on a detaled segmentaton of behavorally homogenous groups of persons, ther actvtes and so forth. For the sake of clarty, we omt dfferentaton, but all followng analyses wll be applcable to all groups. The man focus here s on constrants, and the EVA model dstngushes nelastc (or hard) and elastc (or soft) constrants. Inelastc constrants are used f the total number of trps of an alternatve (orgn or destnaton) can be derved exclusvely from zone characterstcs,.e. transport supply and subsequent competton between zones do not have any effect on trp producton. In contrast, elastc constrants are used f transport supply and zone competton have an addtonal nfluence. Snce t s not possble to show all facets, n the followng the model specfcaton wth a set of nelastc constrants for the orgns and a set of elastc constrants for the destnatons wll be used. The doubly constraned model s then: T =f g p ( ) abd T =O T =D D max constrants () wth a, b f O D max D balancng factors to satsfy the constrants deterrence functon wth regard to the generalzed costs g k from zone to zone by mode k total number of trps orgnatng at zone total number of trps attracted to zone maxmal total number of trps attracted to zone

5 Wnkler T p d trps from zone to zone destnaton specfc attractor/potental Thereby the trps T are calculated takng nto account generalzed costs, destnaton specfc attractors and the two sets of constrants. The constrants are satsfed by the balancng factors, whch are solved teratvely. For solvng ths optmzaton problem dfferent methods can be used, for example, the Furness algorthm (Furness, ). The gven O s are an nelastc set of constrants and are calculated n the trp generaton a pror. The same apples to D max, whch, however, defne only upper lmts of what zones can accommodate. The resultng D s are calculated wthn the trp dstrbuton under consderaton of D max and the spatal competton of destnaton zones. Ths competton s reflected by dfferent generalzed costs and destnaton specfc attractors, whch could be, for nstance, populaton n zones or D max. There are dfferent combnatons of constrants that are possble and reasonable. For example, mposng nelastc constrants for work or school trps are necessary, because we expect to fnd workers or students at ther places of work and educaton respectvely. Elastc constrants are used for substtutable trps such as for lesure or shoppng. In addton to constrants, the above-mentoned deterrence functon has a hgh mportance for modellng results. The EVA model s of hgh flexblty and nearly every functon type can be used. Probably the exponental functon s the most mportant functon and wll be appled below. In addton to generalzed costs, orgn and destnaton specfc characterstcs can be consdered. In equaton (), d P represents such a varable. Consderng zone specfc values would only have an mpact on modellng results n the case of elastc, but not nelastc constrants. The reason s that nelastc constrants are not nfluenced by zone competton and the resultng total numbers of trps per zone are guaranteed by modfyng the balancng factors. Travel demand models are used for estmatng demand changes due to transport nterventons. For ths reason, forecastng capablty s of great relevance. An mportant queston s how constrants should be handled. There are dfferent approaches to answerng ths queston. If nformaton about the (maxmal) number of trps n each zone s known for the fnal state, the same procedure as n the ntal state should be executed. If nformaton s not avalable or vald, balancng factors defned n the ntal state could be used. From ths t follows that the total number of trps per zone are not restrcted by constrants. However, the most common approach s that orgn and destnaton constrants are gven and new balancng factors are calculated. EVA Logt Model For detaled economc analyses and welfare evaluaton, t s reasonable and possble to formulate the EVA model n terms of an MNL. Other authors, such as Anas () and Erlander and Stewart (0), have already shown how a constraned travel demand model may be defned as an MNL. However, they demonstrated t for doubly constraned gravty models whch exclusvely consder nelastc constrants. Here we apply the approach for the EVA model wth dfferent sets of constrants. Startng pont s the well-known MNL. Ths model s grounded on the economc concept of random utlty maxmzaton (RUM) and the ndvduals are assumed to choose the most benefcal (optmal) alternatve (McFadden, ). Unlke the constraned models, the MNL respects per se a maxmum of one set of constrants (sngly constraned model) and calculates probabltes for the dscrete alternatves and not the trps drectly. These are defnable by multplyng the probabltes wth the total number of trps (total demand). The comparson of alternatves s on the bass of the alternatve specfc utltes. Therefore utlty functons ncludng decson-relevant varables have to be defned. In addton, the utlty s decomposed nto an observed and an unobserved (random) component. The reason s that not all decson-relevant

6 Wnkler factors are observable for researchers. The utlty that a sngle decson maker or representatve of a homogenous group obtans from alternatve can be descrbed by (Ben-Akva and Lerman, ): U =V +ε wth U V ε, () utlty of an alternatve observable part of utlty (ndrect utlty) unobservable part of utlty In a standard MNL the unobserved utlty s assumed to have..d. extreme value dstrbuton wth standard varance (Π²/). Then the choce probabltes P are gven by: 0 P = ' e V e ' V'' () 0 0 It represents the demand for a representatve person. Trps T result from a globally fxed MNL: T =PT () wth T total number of trps over all zones (total demand) For defnng the EVA MNL, the observable part of utlty s the central element. Agan, we show the case wth nelastc orgn and elastc destnaton constrants, but other cases can be easly defned as well. Frstly, the utlty functon s defned by the observable alternatve specfc attrbutes (negatve) generalzed costs and destnaton utltes. In addton we nclude orgn and destnaton shadow prces (Neuburger and Wlcox, ). The term shadow s used, snce these prces are not defnable a pror but dervable from the balancng factors of the EVA model a posteror (after the teraton process). The gven numbers of orgnatng and attractng trps contan addtonal nformaton about the underlyng decson process of transport users. Ths nformaton s mplemented n the utlty functon by the shadow prces, whch serve the emendaton of the a pror utlty. Then, by means of the shadow prces, the utlty maxmzaton process provdes the probabltes of alternatves producng a result wth satsfed constrants. The utlty functon can then be wrtten: V = -g + z + + τ wth, τ z, () shadow prces of orgn and destnaton observable destnaton utlty By mplementng ths utlty functon n the MNL we get the EVA MNL for estmatng trps T : T =PT= (-g +z + +τ) e T ( '' ' ' ' ) e -g +z + +τ T =O T =D max ' D ' ()

7 Wnkler The EVA MNL s n fact a doubly constraned MNL and defned by ths equaton, although shadow prces are unknown. However, t can be shown that shadow prces are easly dervable from the EVA model by mathematcal redrafts as follows (Wnkler, 0): ( ) a = e ln a = ( ) τ b = e ln b = τ Hence, the dervaton of the shadow prces s straghtforward by logarthmc transformatons of the balancng factors. The same apples to destnaton utltes that are defned by logarthmc transformaton of the destnaton attractors. LOGSUM DIFFERENCE AND ITS FAILURE IN CASE OF CONSTRAINED MODELS Measurng transport user benefts s often necessary for evaluaton transport nvestments and polces by CBA. For ths purpose the concept of change n consumer surplus s often used. When applyng an MNL for estmatng change n travel demand, change n consumer surplus can be easly derved by the logsum dfference. It s a smple mathematcal formula and consstent wth mcroeconomcs. The logsum dfference as a measure for change n consumer surplus has been nvestgated and used n transport for some tme (see, for a comprehensve lterature synthess, Ma et al., 0). For dervng the change n consumer surplus on the base of MNL, expected maxmum utlty of a decson maker or representatve has to be defned and t s: ( ) ( ) = ( ) E U E Max U,, V () = ln e + C where C s an unknown constant that represents that the absolute level of utlty s not measurable. Ths mathematcal expresson corresponds wth the antdervatve of the MNL (see, for example, Small and Rosen, ). The argument n parentheses s the denomnator of the MNL and the term ln exp( V ) s called logsum, whch s smply the log of the denomnator. However, for evaluatng transport nvestments and polces we are more nterested n change n utlty that s defned by the dfference between ntal (0) and fnal () state condtons. For ths purpose a dfference n logsum terms s appled: E ( U ) = ln e - ln e 0 V V () The unknown constant C enters expected maxmum utlty both before and after change and drops out. On the bass of ths change n expected maxmum utlty, the requred change n consumer surplus can be derved, whch s the monetzed equvalent. Hence we get the change n consumer surplus by normalzng the logsum dfference by the margnal utlty of ncome λ (of a representatve traveler of a homogenous group): E ( CS ) = ln e - ln e λ 0 V V (0) Ths normalzaton presumes that λ s constant and ncome effect s not accounted for (see, for example, Wllams, and Small and Rosen, ). Ths assumpton s reasonable for many problems. However, ()

8 Wnkler f ncome effects occur, dfferent and more complex approaches for dervng monetary user benefts have to be used. Mcroeconomcs apples the compensatng varaton (CV) as the Hcksan (or compensated) verson of consumer surplus (see, for example, Varan, ). There has been extensve research for dervng CV for dscrete choce models, see, for example, McFadden,, Karlström and Morey, 00 and Dagsvk and Karlström, 00, but ths s not the topc here. In the present analyss we suppose a constant margnal utlty of ncome, whch, however, has to be defned. Usually, a prce or cost varable c enters the ndrect utlty V and the negatve of ts coeffcent β c s λ per defnton (Tran, 00). The derved change n consumer surplus apples for a representatve traveler. It can be nterpreted as the average consumer surplus n the group under consderaton. The change n total consumer surplus (sum over all travelers of the group) can then be calculated by multplyng the monetzed logsum dfference by total demand and t s: ΔE ( TCS ) = - ln e - ln e T βc 0 V V () After ths bref ntroducton of the logsum dfference, we shall dscuss the problem wth applyng t n the EVA MNL. The bases for changes n consumer surplus are proect-nduced changes of varables n the ndrect utlty functon. If generalzed costs decrease after nterventon, ndrect utlty and consumer surplus ncrease. However, the EVA MNL comprses shadow prces as addtonal bonus components and shadow prces change as a consequence of changng costs. Ths means that f costs decrease, (postve) shadow prces also decrease. The queston arses whch consequence does t have? It can be shown (see Wnkler, 0) that t s: ( -g +z + +τ ) T= e () In the standard case T 0 = T t follows: (-g +z + +τ) (-g +z + +τ ) ΔE ( U ) = ln e - ln e 0 ( ) ( ) = ln T - ln T =0 From ths t follows that, regardless of the changes of generalzed costs, the logsum dfference provdes no (or ncorrect) benefts. However, due to cost decreases, transport users n fact receve real beneft. Therefore, the logsum dfference obvously fals for constraned models. There s only a lttle lterature concernng ths queston. Wllams () and later Martnez and Araya (000) showed approaches for dervng transport user beneft measure on the bass of doubly constraned gravty models. These dervatons are based on the maxmum entropy model and t s not easly transferable to other models. Furthermore, a rgorous relatonshp between the transport user beneft measure wthn the entropy framework and the logsum beneft measure wthn random utlty theory has not been establshed so far (Geurs et al., 00). Here we wll close the gap and show a unversal approach, whch adusts the logsum dfference to multple-constraned models. ADJUSTMENT OF THE LOGSUM DIFFERENCE We have shown that the logsum dfference fals for the EVA MNL. For dstngushng the real change n ΔE U and the ncorrect measure, we defne the logsum dfference appled to expected maxmum utlty ( ) ()

9 Wnkler 0 0 the EVA MNL provdes E( U Δ ) and t s ΔE( U ) ΔE( U). In the followng we want to clarfy how the logsum dfference should be adusted to the EVA MNL. Frst of all, we derve the approach for the representatve traveler. We then move over to the change n (total) consumer surplus. The startng pont of the analyss s the dfferental: 0 ( ) ( ) ( ) ( ) ΔE U =E U -E U = de U () C where de ( U ) represents the demand functon (Varan, ). Therefore t s: ( ) ( V ) () de U = P dv Here V s a vector of all alternatve-specfc determnstc utltes, snce the probabltes depend on all alternatves. Hence t can be wrtten: ( ) ( ) V () C ΔE U = P dv Ths ntegral s a lne ntegral and ts soluton depends on the path C between the fnal and ntal state. Due to the fact that the antdervatve of the MNL, and hence of the EVA MNL, s known, a path ndependent soluton of equaton () s possble. However, as shown above, n the case of constrants the logsum dfference does not provde the wanted change n expected maxmum utlty. For ths reason a more detaled analyss ncludng a segmentaton of the EVA MNL s necessary. Frst of all the total dfferental dv k can be expressed by partal dfferentals (Chang and Wanwrght, 00): V V V V dv = - dg + dz + d + dτ g z τ All components are composed addtve and wthout addtonal weghtngs. Hence, the fgures of the partal dervatves equal one and t follows: dv = -dg + dz + d + dτ () For the sake of smplcty we defne ( ) ( ) ( ) P V =P and t s: de U = P -dg + dz + d + dτ () Next we can nsert t n equaton () and obtan the ntegral wth respect to the bounds of ntegraton: () V ( ) ( ) ΔE U = P -dg + dz + d + dτ 0 V Furthermore, a separaton of the ntegral s necessary and possble (Merzger et al., ): (0) ( ) g z τ ΔE U = - P dg + P dz + P d + P dτ g z τ ΔE ( U ) results from the sum of all ntegrals. For each ntegral the antdervatve s gven by the logsum formulae. However, all ntegrals can only be solved one by one wth respect to the consdered changng ()

10 Wnkler 0 varable (g, z etc.) whle all other components have to be constant. It has to be taken nto account that beneft contrbutons of dfferent varables depend on the sequence of ntegratons (path dependency). Hence, sngly beneft contrbutons cannot be defned unambguously, but the sum over all s ndependent. The requested change n expected maxmum utlty s only a functon of real and observable changng attrbutes between the ntal and fnal state (Wllams, ). In the present analyss these are generalzed costs g and destnaton utltes z and t s: ( ) g z ΔE U = - P dg + P dz 0 0 g z Now we can reformulate equaton () and express ΔE ( U ) ndrectly: ( ) ( ) τ ΔE U = ΔE U - P d - P dτ 0 0 τ () () 0 In ths equaton the logsum dfference of the EVA MNL s adusted by the shadow prce ntegrals. However, agan, ther fgures depend on sequence of calculaton. Consequently the unambguousness of ΔE U depends on the soluton of the ntegrals of shadow prces. These ntegrals are only the requested ( ) one-dmensonal because ther dfferentals only have one dmenson (I or ). Snce a change n has no nfluence on probablty of destnaton (same apples recprocally for τ ), t can be wrtten: ( ) ( ) τ ΔE U = ΔE U - P d - P dτ 0 0 τ () 0 0 where P, and P are the probabltes of choosng orgn I and destnaton. Hence, these probabltes represent representatve traveler s demand for orgns and destnatons. However, the demand functons for these probabltes are not explctly defned, but result mplctly from the summaton of alternatvespecfc probabltes P. Consequently, antdervatves do not exst and ntegraton cannot readly be ΔE U. acheved, but t s necessary for dervng ( ) For the soluton of the ntegrals, t s of great mportance whether the ntal and fnal probabltes P and P are equal or not. Generally, they are equal n the case of nelastc and unequal n the case of elastc 0 constrants. Next we shall derve the soluton for the constant case through the example of P =P. 0 Afterwards we turn to the elastc constrants ( P P ). The followng dervatons are transferable one to one to other dmensons. Case P =P 0 The probabltes are obvously constant and accordngly ndependent from shadow prces. Hence there are no formal nterdependences of probabltes of the orgns and the shadow prces. As a consequence, the probabltes can be taken outsde the ntegrals. Thus, the ntegraton s path-ndependent because of the ndependent ntegrablty of all ntegraton varables. So we obtan: Pd = P d 0 0 ()

11 Wnkler Then the ntegraton of the dfferentals d provdes the ntegraton varables. Next we nsert the bounds of ntegraton and the soluton for the nelastc constrant case s: 0 0 ( ) Pd = P - () The probablty for choosng orgn s defned by the rato of total number of trps orgnatng at zone and total demand. These numbers are known from the trp generaton and t can be wrtten: 0 Pd = ( - T ) 0 O Ths s an unambguous soluton and all relevant varables are gven by the solved EVA MNL. () Case P P 0 0 If destnaton probabltes change between ntal and fnal state, we need another approach, snce we cannot take the probabltes outsde the ntegral readly. So, n prncpal, t s necessary to ntegrate the (unknown) functons of the resultng probabltes P s. But we only have nformaton about the P s for the two equlbrum (ntal and fnal) states. Hence, we have to make an assumpton about the shape of the functon n between. Accordng to the RoH we assume a lnear form. Wth the assumpton of a lnear path of ntegraton t can be wrtten: τ 0 τ P +P Pdτ = dτ 0 0 τ τ Now the ntegral can be solved analogcally to the nelastc constrant case and we obtan: τ 0 τ Pdτ = P +P τ -τ 0 0 ( ) Replacng probabltes by total numbers of trps gves: () () 0 τ 0 τ Pdτ = D +D τ -τ 0 0 T ( ) Agan, all necessary varables are gven by the EVA MNL. Now we are able to formulate the change n expected maxmum utlty of a representatve traveler. For the example under consderaton wth one set of nelastc constrants for orgns and one set of elastc constrants for destnatons, we obtan: (-g +z + +τ) (-g +z + +τ ) ΔE ( U ) = ln e - ln e 0 O τ -τ D +D T T 0 0 ( ) ( ) (0) ()

12 Wnkler Obvously the logsum dfference s corrected by the nfluence of the shadow prces. Furthermore, t s crucal to note that f shadow prces are unchanged, no correcton s necessary. The last two terms cancel out and the logsum dfference provdes the change n consumer surplus. That s the case when generally no constrants occur or when constrants are only compled n the ntal case. Then, n the fnal state ntal shadow prces are consdered as addtonal orgn, destnaton or mode utltes. ln exp -g + z + + τ - ln exp -g + z + + τ = logdff Fnally, wth ( ) ( ) equaton () can be wrtten as: 0 O D +D ΔE U = logdff τ - τ T T 0 0 ( ) ( ) ( ) () Total change n consumer surplus So far, we have derved the change n expected maxmum utlty of a representatve traveler. Now we ΔE U has to move on to the soluton for the needed (total) change n consumer surplus. For ths purpose ( ) be monetzed by -β c (negatve margnal utlty of the cost varable) and multpled by the total number of trps T. In case of T 0 = T,.e. total demand s fxed and no proect-nduced new generated trps occur, we have: ΔE TCS =- O - + D +D τ -τ βc ( ) ( ) ( ) ( ) () The logsum dfference cancels out, snce, as already partly shown n equaton (), t s: (-g +z + +τ) (-g +z + +τ ) 0 T ln e - ln e = ln ( T ) - ln ( T ) = ln 0 T () and log() = 0. Hence, the change n total consumer surplus results from the change n shadow prces. However, these changes are caused by changes of observable costs and (destnaton) utltes, therefore the proect-nduced transport user benefts are ndrectly measured by ths approach. If, for example, a cost reducton for trps orgnatng n zone occurs, ths zone attans better accessblty. As a result of ths, and under the assumpton of respectng constrants, the shadow prce of ths zone decreases. Conversely, f shadow prces ncrease, negatve benefts result. Hence, benefts changes are based on changes n accessblty of zones. It mght be possble that a transport nvestment or polcy leads to a change n the total number of trps and t s T 0 T. In ths case an ssue arses comparable to the shown changng probabltes of destnatons. For calculatng transport user benefts, we have to take nto account the shape of the overall demand curve between T 0 and T, but we do not have nformaton about that shape and only know total demand for the two equlbrum (ntal and fnal) states. Hence, agan, we have to make an assumpton about t and use a ΔE U (equaton ()) has to be multpled by (T 0 + T ) / and t s: lnear form. Then, ( ) T +T O +O D +D ΔE TCS = - logdff τ - τ βc 0 0 ( ) ( ) ( ) () Ths equaton provdes the unversal soluton for calculatng transport user benefts based on the MNL wth or wthout constrants. If no constrants and shadow prces occur, the logsum dfference multpled by the (mean) total demand provdes the change n consumer surplus. The same apples to the constraned

13 Wnkler case, f shadow prces are the same n the ntal and fnal state. In these cases the last two terms on the rght sde cancel out. In contrast, f constrants are respected n both calculatons, the derved adustment of the logsum dfference has to be taken nto account. Then, under consderaton of the mathematcal smplfcaton n equaton (), t can be wrtten T T +T O +O D +D ΔE TCS =- ln τ -τ βc T 0 0 ( ) 0 ( ) ( ) () If shadow prces of sngle dmensons (e.g., destnatons) are equal n the ntal and fnal state, the correspondng correcton term dsappears, whle the others stll apply. The meanngs of the beneft terms n equaton () are the same as explaned for the case wth fxed total demand. However, the frst term enters as an addtonal beneft term that represents the postve benefts of new generated trps. Negatve mpacts of these addtonal trps (lke competton for destnatons) are taken nto account by orgn and destnaton specfc shadow prces. That s, f new trps occur due to transport nvestment, shadow prces decrease less than wthout these addtonal trps. EXAMPLE The basc prncple of the developed approach can be llustrated by a smple synthetc example, whch s defned by fve travel zones one homogenous group of persons nelastc orgn and destnaton constrants More realstc examples and case studes wll be dscussed n the future. However, a smple example such as ths mght be helpful for ntal understandng. Frst of all we want to show the concept of calculaton of travel demand by the EVA MNL. For ths the costs of the ntal and fnal state are summarzed n TABLE. For the fnal state we suppose a reducton of costs for the relatons from zone to and vce versa (marked n yellow). TABLE : Costs generalzed costs [$] - ntal state generalzed costs [$] - fnal state g g,0,,,0,,0,,,,,0,0,,,00,0,0,,,00,,,0,00,00,,,0,00,00,0,,,0,0,,,,0,0,0,,00,,,0,,00,, The result of the trp calculaton (by equaton ()) for the ntal state s shown n TABLE. In ths example we consder nelastc constrants for orgns and destnatons and the resultng O s and D s equal the gven O s and D s (constrants are satsfed). The total number of trps T (00 trps) s defned by the sum of the gven O s and Z s, respectvely. The shadow prces are derved from the teratvely calculated factors a and b (see, equaton ()). Trp dstrbuton s analogously calculated for the fnal state and the results are also shown n TABLE. It s obvous that trps T and T have ncreased snce the correspondng costs have decreased. The orgn and destnaton constrants are agan satsfed, but by new shadow prces.

14 Wnkler TABLE : Resultng trps ntal state T O-resultng O-gven D-resultng D-gven τ fnal state T O-resultng O-gven D-resultng D-gven τ The results of the travel demand for the ntal and fnal states are the base of the calculaton of the change n consumer surplus. For applyng the adusted logsum dfference (see, equaton ()), all O s and D s as well as all shadow prces are necessary. All other components n equaton () cancel out. The result of the calculated change n consumer surplus s summarzed n TABLE. It should be noted that decreasng shadow prces and therefore a postve dfference nduce postve benefts. A reducton of shadow prces s prmary acheved for traffc zones that are drectly affected by the cost changes (here: zones and ). The reason s that the accessblty of these zones has been mproved and transport users need not be so forced to choose these zones (for satsfyng constrants) by shadow prces. However, all constrants have to be satsfed n the fnal state, too. Therefore, all other zones are affected ndrectly by the cost changes. Although ther accessbltes have not changed, these zones loose attractveness compared to zones and and shadow prces ncrease for satsfyng constrants. The change n consumer surplus results from shadow prce dfferences and the total numbers of trps, and t s calculated for each zone. However, t s mportant to note that only the sum over all zones s nterpretable as the transport users beneft measure needed for evaluatng transport nvestments by CBA, because the sngle values are only proxes for defnng benefts of real affected transport users. The resultng change of consumer surplus for the example s $..

15 Wnkler TABLE : Change n consumer surplus total number of trps orgnatng at zone dfference of orgn shadow prces CS [$] O 0 O - M 0,, O 00 O - M -0,00-0, O 0 O - M 0,0 0, O 00 O - M 0,0, O 00 O - M -0,0 -, Σ(), total number of trps attracted to zone dfference of destnaton shadow prces CS [$] D τ O -τ M 0,, D τ O -τ M -0,0 -, D τ O -τ M -0,0 -, D 00 τ O -τ M 0,0, D τ O -τ M -0,0 -, Σ(), 0 Now the result of the adusted logsum dfference approach should be compared wth the wdely used tradtonal RoH, whose applcablty for travel demand models wth constrants has recently been proved (Wnkler, 0). The RoH s an approxmaton of the user benefts and t depends on the change n costs and trps for each orgn-destnaton-relaton: 0 0 ΔE ( TCS ) = ( g - g )( T + T ) () Σ(,), As we can see n equaton () and TABLE, transport user benefts only occur for relatons wth changng costs. The comparson of the results shows that the approxmate soluton (RoH) overestmates the exact measure (adusted logsum dfference) wth %. However, the dfference between the RoH and an exact measure (logsum or adusted logsum dfferences) can be postve or negatve and t strongly depends on the amount of the cost changes and the shape of the underlyng demand functon (Ma et al., 0). Therefore, t s recommended to use the exact measure for dervng transport user benefts. TABLE : Change n consumer surplus calculated by the RoH g ΣT/ 0,00 0,00 0,00, 0,00, 0,,, 0, 0,00 0,00 0,00 0,00 0,00,,, 0,, 0,00 0,00 0,00 0,00 0,00 0,,,,,, 0,00 0,00 0,00 0,00,,,0,, 0,00 0,00 0,00 0,00 0,00,,, 0,, CS 0,00 0,00 0,00, 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00, 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 $,00

16 Wnkler 0 0 CONCLUSIONS In ths paper we have seen by the example of the EVA model that multple-constraned travel demand models are expressble n terms of a MNL. Ths s the bass for an economc analyss of these models and the startng pont for the dervaton of a transport user beneft measure that s usually needed for evaluatng transport nvestments by CBA. For ths purpose the change n consumer surplus s often used. Ths measure s provded by the MNL and called logsum dfference, whch s a very practcal and mathematcally exact approach. However, the logsum dfference fals for travel demand models wth constrants and only approxmatons can be used. For ths reason we have derved an approach that adusts the logsum dfference for travel demand models wth two sets of constrants. Ths approach provdes the change n consumer surplus for models wth dfferent constrants (elastc and/or nelastc). Furthermore, t contans the logsum dfference for the MNL wthout constrants as an excepton. For the sake of clarty, the analyses presented have been focused on the case wth nelastc constrants for the orgn and elastc constrants for destnaton choce. Other cases can be analogcally derved, but equaton () provdes the general soluton for the change n consumer surplus that covers all possble combnatons of constrants. The approach derved s sutable for measurng benefts caused by changng costs, but also for changng attractons. Attracton changes can be expressed by dfferent orgn or destnaton utltes or by a changed gven total number of trps orgnatng at or attracted to zones. Hence, t s possble to evaluate short-run and long-run mpacts of transport nvestments and the derved approach can be used for a wde range of evaluatons.

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