assessing transport investments – towards a multi-purpose tool

16
Assessing transport investments – Towards a multi-purpose tool André de Palma a , Stef Proost b , Saskia van der Loo b, * a Ecole Normale Supérieure de Cachan and Ecole Polytechnique, France b Center for Economic Studies, KULeuven. Naamse straat 69 3000 Leuven, Belgium article info Article history: Received 26 November 2009 Accepted 1 December 2009 Keywords: CBA Dynamic Congestion Passenger Freight transport abstract This paper presents a multi-purpose tool to assess transport investments in congestible facilities. The model can handle any combination of passenger and freight transport modes in a simplified network. Within each mode, there can be competing operators. It is cali- brated to a given traffic forecast and can be used to assess the benefits and costs of com- binations of strategic pricing behavior and investment. The use of the model is illustrated with examples. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Cost-Benefit analysis calls upon several disciplines such as transportation economics, engineering and finance, including public finance (see Layard and Glaister, 1994). Several models have been developed and are used in practice, but there have been very few attempts to offer the different dimensions needed to help the decision makers. Decision makers face the choice of which infrastructure should be developed, when it should be built, how its use should be priced and how it should be financed. This paper presents the methodology and analytical description of the MOLINO-II model. 1 It is a multi-purpose model that allows to assess investments as well as strategic pricing behavior by operators in a simplified network. The network can con- tain different types of modes and can deal with freight and passenger transport simultaneously. A beta version of the model has been used to assess investment and pricing of a bridge, and four large infrastructure projects that applied for EU funding. What is the role of a model like MOLINO-II? Take a private investor or a government agency (World Bank, European Investment Bank, European Commission, etc.) that subsidizes transport investments in different regions and is confronted with demands for project subsidies that deal with roads, rail, canals, etc. Let’s consider two cases. In the best case, the agency has a detailed cost benefit assessment for each project and a more crude assessment at a broader network level that gives the possible interaction between projects. In this best case, our model can have a double function: first it can serve as second opinion tool to verify the individual and mode specific assessments and second, it allows to analyze possible conflicts and strategic pricing behavior between operators. Unfortunately, in most cases, there are only incomplete cost benefit analyzes of one particular investment available, using a specific modelling tool for the region at hand. In this case our model can fulfill the second opinion function and serve to study strategic pricing but can also be used to analyze the possible interaction with 0191-2615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2009.12.006 * Corresponding author. E-mail addresses: [email protected] (A. de Palma), [email protected] (S. Proost), [email protected] (S. van der Loo). 1 For a less technical description, see de Palma et al. (2008). A first version (MOLINO-I) containing only two parallel links is documented in de Palma et al. (2007b). Transportation Research Part B 44 (2010) 834–849 Contents lists available at ScienceDirect Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

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Transportation Research Part B 44 (2010) 834–849

Contents lists available at ScienceDirect

Transportation Research Part B

journal homepage: www.elsevier .com/ locate / t rb

Assessing transport investments – Towards a multi-purpose tool

André de Palma a, Stef Proost b, Saskia van der Loo b,*

a Ecole Normale Supérieure de Cachan and Ecole Polytechnique, Franceb Center for Economic Studies, KULeuven. Naamse straat 69 3000 Leuven, Belgium

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 November 2009Accepted 1 December 2009

Keywords:CBADynamicCongestionPassengerFreight transport

0191-2615/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.trb.2009.12.006

* Corresponding author.E-mail addresses: andre.depalma@ens-cachan.

(S. van der Loo).1 For a less technical description, see de Palma et a

(2007b).

This paper presents a multi-purpose tool to assess transport investments in congestiblefacilities. The model can handle any combination of passenger and freight transport modesin a simplified network. Within each mode, there can be competing operators. It is cali-brated to a given traffic forecast and can be used to assess the benefits and costs of com-binations of strategic pricing behavior and investment. The use of the model is illustratedwith examples.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Cost-Benefit analysis calls upon several disciplines such as transportation economics, engineering and finance, includingpublic finance (see Layard and Glaister, 1994). Several models have been developed and are used in practice, but there havebeen very few attempts to offer the different dimensions needed to help the decision makers. Decision makers face thechoice of which infrastructure should be developed, when it should be built, how its use should be priced and how it shouldbe financed.

This paper presents the methodology and analytical description of the MOLINO-II model.1 It is a multi-purpose model thatallows to assess investments as well as strategic pricing behavior by operators in a simplified network. The network can con-tain different types of modes and can deal with freight and passenger transport simultaneously. A beta version of the modelhas been used to assess investment and pricing of a bridge, and four large infrastructure projects that applied for EU funding.

What is the role of a model like MOLINO-II? Take a private investor or a government agency (World Bank, EuropeanInvestment Bank, European Commission, etc.) that subsidizes transport investments in different regions and is confrontedwith demands for project subsidies that deal with roads, rail, canals, etc. Let’s consider two cases. In the best case, the agencyhas a detailed cost benefit assessment for each project and a more crude assessment at a broader network level that gives thepossible interaction between projects. In this best case, our model can have a double function: first it can serve as secondopinion tool to verify the individual and mode specific assessments and second, it allows to analyze possible conflicts andstrategic pricing behavior between operators. Unfortunately, in most cases, there are only incomplete cost benefit analyzesof one particular investment available, using a specific modelling tool for the region at hand. In this case our model can fulfillthe second opinion function and serve to study strategic pricing but can also be used to analyze the possible interaction with

. All rights reserved.

fr (A. de Palma), [email protected] (S. Proost), [email protected]

l. (2008). A first version (MOLINO-I) containing only two parallel links is documented in de Palma et al.

A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849 835

nearby projects. Of course the model we present cannot solve all questions. Its ambition is not to redo the analysis startingfrom scratch but rather to perform an extra check or a more strategic appraisal of a project that has been evaluated usingexisting models.

There is a huge fixed cost in constructing and maintaining state of the art models that contain an explicit description ofthe whole multi-modal transport network of an entire region or country. The network models often have a high degree ofdetail and describe user’s behavior via discrete choice techniques. Our modelling approach differs from the traditionalone in three respects. First, we focus on a simplified network that is directly relevant for the investment. Second, we useaggregate modules to represent behavior as they can be calibrated with a minimum of data. Third, we allow different typesof strategic pricing behavior by network operators and governments. The latter feature is becoming increasingly importantwhen all transport can be priced and the funding of investment relies more and more on user pricing. Most network modelsare so detailed that they focus on the correct computation of a user equilibrium but are incapable of analyzing the strategicinteractions between operators.

We start the paper with an example to show how we represent the network and the supply. We describe the modelling ofthe users, operators and infrastructure managers as well as the objective functions of local and federal governments. Section3 discusses the calibration and simulation use of the model. In Section 4 we describe the software. Section 5 uses the modelto illustrate Nash equilibria in a simple parallel network example and for a real investment project. In the conclusion webriefly discuss possible extensions of this model.

2. Model

We describe the model starting with the network representation and supply. Next we discuss the analytical representa-tion of the users of the transportation network where we distinguish passenger and freight transport behavior.

2.1. Network representation and supply

The model we present is mainly used for investment analysis and this will guide the network definition. An investmentproject aims to improve the quality or speed on a given link (or set of links) that connects some origins and destinations. Westart from the links that are improved, examine what type of trips are affected and add to the network definition the com-peting links that may be affected by the project.

Before we start with the formalization, an example can clarify the concepts. Consider the Gdansk–Vienna/Bratislava cor-ridor, which is a roughly 900 km long North–South corridor, linking the Baltic Sea port of Gdansk to the Central Europeancapital cities Vienna and Bratislava (see Fig. 1). The Northern part of the corridor, which stretches from the Baltic sea to

Fig. 1. An example of a network in MOLINO-II.

836 A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849

the heavily industrialized region of Silesia around Katowice and Krakow, comprises two alternative routes, one of which in-cludes the Polish capital Warsaw. South of Katowice, the corridor splits in two branches, one to Vienna via the Eastern CzechRepublic and the other to Bratislava via Slovakia. Besides linking major centres of the countries it crosses, the corridor alsoprovides access to the sea port of Gdansk. Within the TEN-T priority projects a new road project (denoted by NR1–NR5 inFig. 1) and a upgrade of an existing rail axis (denoted by NT1 and NT2 in Fig. 1) are planned in this corridor.

The projects will have an impact on the existing road and rail network and so these links are also included in the networkdescription. In order to model the improvements on this corridor a network with seven nodes needs to be set up. Each pair ofnodes is linked to each other by a sequence of links. Such a sequence of links will be denoted as a path, for example:the originnode Gdansk is connected to the destination node Katowice, amongst others, via the path [NR1, NR2, NR3] or [NT1,NT2]. Inthis way 18 OD pairs linked by 38 paths were constructed.

To describe the model more formally we need the following notation and concepts:Let O be the set of origins, D the set of destinations and d ¼ fO;Dg the set of OD pairs, L ¼ f1; . . . ; l; . . . ; LÞ is the set of links

and P ¼ f1; . . . ; r; . . . ;RÞ is the set of paths. Moreover, the notation fr; lgmeans: all paths r which contain link l, similarly fd; lgstands for the OD pairs where there is at least one path of which l is part of, l 2 r stands for the links l that are part of the pathr and l 2 d stands for the links l that are part of a path which connects the origin and destination node of d.

The model uses a horizon of T periods of one year t ¼ 1; . . . ; t; . . . ; T; each year is represented by a typical day and that dayhas m subperiods, to simplify one distinguishes between peak and off-peak subperiods. Between each origin and destination,different types of users can travel; we distinguish between k ¼ 1; . . . ;K classes of users. Classes of users can represent dif-ferent types of freight transport (container, bulk, etc.), different types of motives of passenger transport and/or different lev-els of income. The behavior of every class is given by a representative individual or shipper. The total transport volume of arepresentative user of type k travelling between OD pair d, on a given link l (or a path r) during the subperiod m in year t isrepresented by Xdk

mlðtÞ (or XdkmrðtÞ).

The generalized price for a trip for a user of type k using path r during period m, is the sum of a monetary term TkmlðtÞ plus

the time cost, ttkmlðtÞ:

pkmrðtÞ ¼

Xl2r

TkmlðtÞ þ ttk

mlðtÞh i

: ð1Þ

The monetary term is the sum of a resource cost, taxes and a toll skmlðtÞ

� �for link l, these can further differ among sub-

periods and user types. The time cost for link l expresses the value of the time needed for a user of type k to make a trip onthe link, given the volume of the different users and the available capacity:

ttkmlðtÞ ¼ VOTk

mlðtÞdlðtÞ

vmaxlk ðtÞ

1þ AlðtÞP

k0 ;fd;lglk0

mlXdk0

ml ðtÞslðtÞ

!BlðtÞ24 35; ð2Þ

where VOTkmlðtÞ is the value of time of a user of type k travelling during subperiod m on link l in year t. For every link l and

every year t we define a length dlðtÞ, a capacity slðtÞ and the maximum speed vmaxlk ðtÞ. The parameter lk

ml is the relative con-tribution to congestion of user class k during subperiod m on link l and AlðtÞ and BlðtÞ are parameters defining the appropriatespeed–flow relation for link l. As can be seen, the congestion on a link is the result of the use made by different types of usersof this link. This is the most elementary representation of congestion, one could expand the model to include congestion atcrossings, make the monetary cost dependent on total traffic flow, etc.

The network structure defines OD pairs and the alternative paths. For each OD pair we define the behavior of users: howmany trips each category of users make and what path they select.

2.2. User behavior

For both passenger and freight transport we opt for an aggregate representation of behavior. Disaggregate choice behavioris clearly superior if a representative sample is available for simulation. Our experience is that this sample is often not avail-able if one wants to go for a quick check of an investment project, so we rely on more aggregate data. We distinguish be-tween passenger transport and freight transport.

2.2.1. Passenger transportPassenger transport is modeled in an aggregate way by making use of a nested Constant Elasticity of Substitution (CES)

utility function. For every user class and every OD pair we calibrate a nested CES utility function with four levels (see Fig. 2):We allow consumers on each OD pair to choose the aggregate transport level, the period in which they travel and the

path. The path can contain combinations of modes. This means we rule out substitution between destinations. The elasticityof substitution chosen for every branching of the tree will determine the ease of substitution and the cross price elasticitybetween different paths and, implicitly, also modes. For perfect substitution possibilities between paths we end up in theWardrop equilibrium but substitution can also be imperfect. So we do not necessarily use the Wardrop principle for thechoice of paths. In this sense, we rely on the concept of Stochastic User Equilibrium, which is well known in transportation,when the user choice on a network is described by a discrete choice model (see Sheffi, 1985). Here we use the same concept,

Utility

Transport Other consumption

Rn…

Peak

R1 R2 R6… Rn…

Off-Peak

R1 R2 R6…

Level 3

Level 2

Level 1

Level 0

Fig. 2. Nested Utility tree for passenger transport.

A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849 837

but the discrete choice approach is replaced by a CES model. The Logit and the CES model are both demand models for dif-ferentiated goods . The Logit model is a disaggregate demand model, while the CES model is a representative consumer mod-el. The Logit model has a representative consumer formulation, where the direct utility has an entropic form. The CES modelcan be derived as a two step disaggregate model, where the first stage is which good to buy, while the second stage is con-cerned with how much to buy. In this case, the indirect utility function is logarithmic in income, i.e. there are income effects.The two models are related, and can be given a common denominator, either at the aggregate level or at the disaggregatelevel (see Anderson et al., 1992) . The main advantage of the CES is that it can be easily calibrated , while the Logit approach(the discrete choice approach), is more amenable to the individual data. As such data are harder to find for several transpor-tation studies, we have preferred to use the aggregate CES form. The calibration of a CES requires the elasticities of substi-tution at each branch plus the total quantities and prices at the lowest level of the utility tree. Finally note that the Logit hasbeen criticized in the literature, since it requires that the demands satisfy the IIA (independence of irrelevant alternative)property.

We define the nested utility function by specifying the different utility components. Following Keller (1976, Eqs. (4), (6),(8) and (10)), we assume that all the utility components are linear homogeneous CES functions of the associated componentsat the next lower level.2 Formally, at the nth level, the utility component qn;i ð8n > 1Þ is given by

2 Theutility cunique.

qn;i ¼Xj2i

ðan�1;jÞ1�qn;i ðqn�1;jÞqn;i

" # 11�qn;j

; qn;i ¼rn;i � 1

rn;i; ð3Þ

where an�1;j P 0; 0 6 rn;i 61 andP

j2ian�1;j ¼ 1. Note that we suppress here the superscripts d and k corresponding to theOD pair and user type in order to lighten the notation. The parameter rn;i in Eq. (3) is the elasticity of substitution at level n ofthe tree and the parameter an�1;i is a share parameter at the next lower level. The notation ‘‘j 2 i” indicates those j’s for whichqn�1;j 2 qn;i.

For each utility component, an aggregate quantity index and corresponding aggregate price index can be computed. It is aproperty of CES functions that the utility component, qn;i, is itself a consistent quantity index and that the correspondingprice index, pn;i, takes similar functional forms, i.e.:

pn;i ¼Xj2i

an�1;jðpn�1;jÞq0

n;i

" # 1q0

n;j

; q0n;i ¼r0n;i � 1

r0n;i; r0n;i ¼

1rn;i

; 8n; i: ð4Þ

At each level the sum of the expenditures of the lower level equals the total income computed with the price and quantityindexes at that level,

yn;i ¼Xj2i

yn�1;j ¼ pn;iqn;i; 8n; i: ð5Þ

It can be shown that the demand functions are

q0;i ¼yp3

Y3

n¼1

an�1;ipn;i

pn�1;i

!rn;i

; 8i: ð6Þ

utility components at different levels are said to be associated if the higher-level component is a function of the component at the lower level. If twoomponents qn;i and qm;j (with m < n) are associated we write qm;j 2 qn;i . Notice that for utility components at level n P 1 the notation is not necessarily

Production Cost

Transport Services Other Inputs

Rn…

Peak

R1 R2 R6… Rn…

Off-Peak

R1 R2 R6…

Level 3

Level 2

Level 1

Level 0

Fig. 3. Nested Cost tree for freight transport.

838 A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849

The demand for transport services q0;i will correspond to the aggregate number of trips on a path r during a period m for acategory of users k on OD pair d, also denoted by Xdk

mr . The lowest price index p0;i corresponds with the generalized price ofmaking a trip using path r during period m for a category of users k, previously denoted by pk

mr given in Eq. (1).The main advantage of the nested CES formulation is its ease of calibration. The drawbacks of a limited number of param-

eters are the implied restrictions. First there are unitary income elasticities for all transport goods – this can be mitigated byrecalibrating the utility function when large income variations are foreseen in future years. Second the compensated crossprice elasticities between goods of the same nest are only a function of the initial budget share and the elasticity of substi-tution for that nest.

2.2.2. Freight transportFor freight transport we use a similar approach. We assume that the production function of each firm that needs transport

services is a nested CES function of the different production inputs: labor, capital and different transport services. For eachfirm, keeping production levels constant, the minimization of the firm’s production cost, generates demand functions for in-puts including the demand for different transport services. In practice we aggregate the non-transport goods into one nestand transport into another nest. The transport nest is then disaggregated into peak/off peak and into paths. The structure isthen similar to the one used for passenger transport (see Fig. 3).

The demand functions for inputs are conditional on the production level of the firm and the prices of all the inputs, includ-ing the prices of non transport inputs. MOLINO-II is a partial equilibrium model that concentrates on the transport marketand takes the prices of all other inputs, as well as all product prices other than transport services, as given.

A major assumption is that we take production locations and production levels as given. This allows us to derive and cal-ibrate demand functions easily. If we want to integrate endogenous location we need a spatial general equilibrium model.This is beyond the model scope we have in mind: a flexible model that can be calibrated to an existing study. This leaves uswith the question: how important are the changes in production and location compared to changes in transport and distri-bution over paths and modes in response to changes in the prices generated by a particular infrastructure project? As trans-port costs are for most firms maximum 5–10% of total costs , a new project may strongly affect the modal choice but in a newequilibrium total production costs may change by 1–2% . This is in most cases a second order effect compared to the trans-port changes for given production levels.

2.3. Behavior of operators, infrastructure managers and governments

2.3.1. Behavior of operators and infrastructure managersThere are two types of agents involved for each link of the network: the manager of the infrastructure and the operator of

the transport services. The manager of the infrastructure decides upon (and pays for) the capacity maintenance and invest-ments. He receives a fee (or infrastructure-use charge) from the transport services operator. The operator sets the level oftolls, collects the toll revenues and pays for the operation cost and the infrastructure-charge. Fig. 4 presents this schemat-ically, arrows stand for payments.

The model also traces the position of the infrastructure fund that is fed by subsidies and taxes and which can be used tofinance the investments of the infrastructure manager (see Fig. 5).

The profit of the operator of link l; ðPorl Þ, can be computed. We have:

Porl ¼ Tollrev l � INFCl � hor

l � OPCl þ suborl ; ð7Þ

where Tollrev l are the tollrevenues on link l collected by the operator, INFCl denotes the infrastructure-use charge of link lwhich consists of the sum of a fixed infrastructure-use charge and a variable infrastructure-use charge per vehicle paid tothe infrastructure manager by the operator, OPCl, the operation cost on link l, which again is a sum of a fixed operation cost

Final user

CompetitiveSupplier

OperatorTransport services

Centralgovernt

Local governt

Infrastructure manager

Localtax

Centraltax

Resource costs

Tolls, charges, tickets

Infrastructure use charge

Maintenanceand

Investments

operation

profit tax

profit tax

Fig. 4. Money flows.

subsidy

taxorsubsidy

OperatorTransport services

Centralgovernt

Local governt

Infrastructure manager

Infrastructure Fund

taxorsubsidy

subsidy

Fig. 5. The Infrastructure Fund.

A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849 839

and a variable cost depending on the number of travelers and is paid by the operator. The variable suborl denotes possible

subsidies received by operator of link l (in the case that the operator has to pay taxes to provision the fund then subopl is neg-

ative). The (exogenous) parameter horl , captures the efficiency of the agent as a function of the market organization. This

parameter will depend on the type of contract between the principal (in casu the operator) and the agent (e.g. the firmresponsible for operating the link). With tendering the parameter will be close to 1 (this means at the minimum technolog-ically feasible costs) while without tendering they will be higher since efficiency will then decrease (see Engel et al., 2001). Ifthe same operator is in charge of different links, its profits will be the sum of the profits on each of its links.

In the same way, the profit of infrastructure manager of link l, denoted by Pinfl is given by:

Pinfl ¼ INFCl � hinf

l � INVCl � hinfl �MCl þ subinf

l þ Salvage value ð8Þ

where INVCl denotes the investment costs of link l, MCl denotes the maintenance costs of infrastructure on link l, which con-sists of the sum of a fixed maintenance cost per unit capacity and a variable cost per vehicle, subinf

l are the possible subsidiesreceived by infrastructure manager of link l and ‘‘Salvage value” is the salvage value of the investments made by the managerof link l. The parameter hinf

l is similar to the parameter horl in Eq. (7).

Operators pay infrastructure charges to the infrastructure managers and set prices for transport services that maximizetheir objective function. We foresee two types of behavior: either profit maximization and setting monopoly or regulatedprices or welfare maximization and setting prices equal to marginal social costs. The marginal social marginal costs is theresource cost plus the marginal external cost. We assume that the behavior is static (one maximizes the profit or welfarefor the given year) and takes all other prices as given (Nash behavior).

Similarly, infrastructure managers charge fees for the use of their infrastructure and can decide on the investments. Againwe foresee two types of behavior for user fees: profit maximization or setting prices equal to the marginal social cost.

The investment behavior is either exogenous or can be substituted by a naive rule where each year the benefit of capacityextension is compared to the annuity of the total cost of extension. As the model is solved year by year and is not forwardlooking, the benefit of capacity extension for the future years is based on an extrapolation of the current benefits. We call ittherefore a naive investment rule.

840 A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849

2.3.2. Local and federal governmentsEvery link of the network falls under the jurisdiction of a local and of a central (or federal) government. Both can charge

taxes and tolls (if the government is the operator of the link) and do so with different objectives in mind. Local governmentsmay be interested in the user benefits of the local voters and in the toll revenue raised from the transit transport users, whilea central government will also be interested in the benefits of the transit transport user. This can give rise to different pricingand investment behavior (see De Borger et al., 2007). We represent the objective function of each government by a socialwelfare function (SWF) that is a weighted sum of the welfare of its voters. If the welfare of all voters receives an equal weight,we obtain the total benefits and costs, traditionally used in cost-benefit analysis. If the weights are different, we have a polit-ical economy approach where the objective function translates the weight of different voters or lobby groups as in a commonagency model (see Dixit et al., 1997).

The welfare function SWF is a weighted sum of different terms:

SWF ¼Xk2p

wkUk �Xk2f

wkPCk þ fCðwkÞCC Taxcen þ fLðwkÞCLTaxloc

þX

l

f orl ðwkÞPor

l þX

l

f infl ðwkÞPinf

l �wextEXTC þ Fund: ð9Þ

In the two first terms, Uk is the utility of an individual of type k and PCk represents the production costs of a firm of type k,both terms are computed by the model using the nested CES utility functions and production costs. The weights wk representthe social weight the decision maker gives to the different passenger user types or different types of freight. If the differentuser types correspond to different income groups, one may want to give the lower income group a higher weight per person.In the case of freight one can, for example, make the distinction between local and non local freight. For a local firm, an in-crease in transport costs will affect the local households via a higher cost of consumption goods or via lower profits on ex-ports. In order to allocate this effect over different groups of voters one can, for instance, allocate the benefits in proportion tothe relative consumption of the different income groups of the output of that firm or via the ownership shares. Summing theweighted consumption or ownership shares over individuals gives the weight wk for that firm. Changes in the productioncosts of non local transport firms do not have a direct effect on the local population, only total tax and toll revenues matterfor them. In the case of a local government which is only interested in the welfare effects on its local population the weight ofthe production costs of the non local transport firms is set to zero. This is only one of the many possibilities, every case studywill need its own interpretation.

The 3rd and 4th terms in Eq. (9) represent the total transport tax revenues collected by central (C) and local (L) govern-ments. Each of these terms receives two weights: CC:or:L:, the marginal cost of funds (MCPF) and a distributional weight func-tion fC:or:L:ðwkÞ. The marginal cost of funds parameter stands for the marginal welfare cost of one unit of public revenue raisedby a given tax. This parameter can be larger than one when distortive labor taxes are the marginal source of tax revenue.Extra government tax revenues raised on transport then allow to decrease distortions in the rest of the economy. The dis-tributional weight function f:C:or:LðwkÞ translates tax revenues into utility changes for different income groups by using anassumption for the final incidence of the redistributed tax revenue. Both functions are of the following form:

f:C:or:LðwkÞ ¼X

k

bkC:or:L:w

k; withX

k

bkC:or:L ¼ 1:

The way the transport tax revenues are redistributed is often more important for the final incidence of a transport chargethan the relative use of the transport facility itself (see Mayeres and Proost, 2001).

The 5th and 6th term in Eq. (9) compute the ultimate incidence on voter groups of the profits of operators and infrastruc-ture managers. This again requires the specification of some distributional weight functions f or

1 ðwkÞ and f inf1 ðwkÞ. To deter-

mine these functions we need to know whether link l is privately or publicly managed (operated). If it is privatelymanaged (operated), whether profit taxes go to local or central government, or if it is publicly managed (operated), whetherit is managed (operated) by local or central government. If a link is publicly operated (managed) then the weight functionswill be equal to the MCPF times the distributional weight function f...ðwkÞ of the previous paragraph.

f or=infl ðwkÞ ¼ f...ðwkÞC...:

If a link is privately operated (managed) then the profit taxes will be collected by some government (local or central) andwill receive a weight similar to the weight of the tax revenues. The profits after tax will be allocated between different usertypes such that:

f or=infl ðwkÞ ¼ f...ðwkÞC...sprofit þ

Xk

bkor=inf ;lw

k

!ð1� sprofitÞ; with

Xk

bkor=inf;l ¼ 1:

The 7th term captures the effect of external costs EXTC other than congestion, congestion is already accounted for in thetravel time. The last term is the net account of the infrastructure fund.

Summing up, in Eq. (9), the weights ðwkÞ allow to measure the impact of a policy on political acceptability. Using uniformweights wk comes down to a standard cost-benefit analysis.

A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849 841

The model allows to compute cooperative equilibria where the two government levels agree to maximize a commonobjective and where the tolling and investment are fully controlled by the government. Alternatively, one can also computenon-cooperative equilibria where two or more private or public operators of parallel or serial links maximize profits. Oftenreality is much more complex as governments do not have full control on operators’ behavior and there are institutional con-straints in place.

3. Calibration and convergence

Once the network characteristics, the alternative paths and the type of users have been specified, one needs to calibratethe utility and cost functions for each user type to complete the demand side of the model.

3.1. Calibration

In the utility functions (Eq. (3)), the unknown parameters are the shares ðaÞ of the components at each level of the nestedCES utility functions. We assume that the elasticities of substitution are known.3 Making use of the known market demandquantities and generalized prices for each of the transport modes and preferences (as captured in the elasticities of substi-tution or price elasticities), the shares can be computed. This means that, given observed quantities, generalized prices, thenest-structure and the elasticities of substitution, it is possible to calibrate the demand functions and the utility function.

We use following procedure in five steps:

1. Calculate the commodity level expenditures ðy0;iÞ using Eq. (5).2. For every cluster of nodes (utility elements associated with a common utility element one level higher up), calculate

the share parameters a0;i using the following expression:

3 Theliteratu

an�1;j ¼yn�1;j

yn;j

pn�1;j

pn;j

!rn;j�1

; ð10Þ

and knowing that for every cluster, the sum of the a0;i’s is 1 (see example below).

3. Calculate expenditures of the next higher level ðy1;iÞ using Eq. (5).4. Use these results to calculate price indices of the higher level ðp1;iÞ using Eq. (4).5. Repeat steps 2–4 to the highest level.

3.1.1. Example of the calibration procedureAs an example, consider following simplified decision tree: (see Fig. 6):This cluster has two lower level elements ðq0;p1 and q0;p2Þ which correspond to observed quantities (respectively during

the peak on option 1 and during the peak on option 2). Together with the observed generalized prices for these commoditiesðp0;p1 and p0;p2Þ one can easily compute the expenditures ðy0;iÞ (step 1). Using Eq. 10 we can write a0;i as:

a0;i ¼y0;i

y1;i

p0;i

p1;i

!r1;i�1

; i ¼ 1;2: ð11Þ

Divide a0;p1 by a0;p2 to obtain

a0;p1

a0;p2¼

y0;p1

y1;p1

p0;p1

p1;p1

!r1p1�1y1;p2

y0;p2

p1;p2

p0;p2

!r1;p2�1

: ð12Þ

The two elements q0;p1 and q0;p2 have a common next higher level and thus q1;p1 ¼ q1;p2. This implies thatp1;p1 ¼ p1;p2; y1;p1 ¼ y1;p2 and r1;p1 ¼ r1;p2. The equation then simplifies to:

a0;p1

a0;p2¼

y0;p1

y0;p2

p0;p1

p0;p2

!r1p1�1

: ð13Þ

Knowing thatP

i¼p1;p2a0;i ¼ 1, the parameters can be calculated knowing the expenditures and prices at this level. For theexample considered the procedure stops here. If, however, there are several higher levels, we compute the next higher levelparameters a1;i in similar way using the total expenditure of that particular level; y1;i ¼ y0;p1 þ y0;p2 (using Eq. (5)) and priceindexes p1;i which are computed using Eq. (4).

elasticities of substitution are not common information but they can be easily related to price and cross-price elasticities for which one can rely on there.

Peak

Option 2Option 1

Fig. 6. Example of a simplified decision tree.

842 A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849

The cost functions are calibrated with the same type of procedure using the shares of transport in total transport, prices oftransport and other goods as well as the total quantities of the different goods.

3.2. Simulations

Once the demand parameters are calibrated and the time cost parameters in Eq. (2) are specified for each link, we cancalculate the behavioral response in quantities Xdk

mr (or q0;i) after a change of the generalized prices pdkmr (or p0;i). A change

in the generalized price can be due to a change in the level of the toll, taxes or other components of the monetary termTk

mlðtÞ in Eq. (1) or can be due to a change of the speed–flow relation defined in Eq. (2) that specifies the time cost. A changein the generalized price on a specific link can have an effect on the demand for all links in the network. The change in volumeimplies, in turn, a change in congestion conditions on the different links which on their turn influences demands. It is there-fore clear that we need an iterative procedure to compute the new equilibrium.

3.2.1. Simulation of the effects of a change in the toll levelsWe will consider, for example, a change in the level of the toll on some link l:

skml ! sk

ml;

which implies a change in the monetary term Tkml ! bT k

ml. We use the following algorithm:

1. First compute the new generalized prices holding the quantities constant pkmrð0Þ (where the argument 0 does not

stand for the time but should be interpreted here as an iteration index):

pkmrð0Þ ¼

Xl2r

bT kml þ ttk

mlð�1Þh i

; ð14Þ

where ttkmlð�1Þ is the time cost when the quantities are the equilibrium quantities bXdk

mrð0Þ� �

for a toll equal to the ori-ginal level sk

ml.

2. Using equation Eq. (4) one computes the induced price indices pn;ið0Þ for the higher levels all the way up to p3ð0Þ.3. Starting from the top and knowing p3ð0Þ, one calculates the expenditures using the expression given in Eq. (10 ) where

now the an;i parameters are known. This requires one additional assumption: we assume that for passengers, incomeis constant , so that y3ð0Þ ¼ y3. In the case of freight, it is not the income but the total production quantity (the quan-tity index at the highest level, q3) that is constant, so that in this case y3ð0Þ ¼ q3 � p3ð0Þ. Using the new y3ð0Þ we canthen compute the lower level expenditures all the way down to y0;ið0Þ. Once these lowest level expenditures areknown we can easily calculate the new demands q0;ið0Þ (or bXdk

mrð0Þ) using the fact that y0;ið0Þ ¼ p0;ið0Þq0;ið0Þ.4. Since the time cost is a function of the quantities, these will change and we thus have to compute the time cost bttk

mlð0Þby substituting bXdk

mrð0Þ into Eq. (2)Define the new time costs as:

ttkmlð0Þ ¼ ~k0ttk

mlð�1Þ þ ð1� ~k0Þbttkmlð0Þ;

where

~kn ¼~k

nþ 1; 0 6 ~k 6 1:

5. Compute the new generalized prices:

pkmrð1Þ ¼

Xl2r

bT kml þ ttk

mlð0Þh i

:

One repeats steps 2–5 until convergence, i.e.:

ttkmlðnÞ � ttk

mlðn� 1Þttk

mlðnÞ< e:

When the last inequality holds, we have the new equilibrium quantities and generalized prices.

A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849 843

3.2.2. Improvement of an existing linkInvestments on an existing link aim to improve the quality or the speed on that given link. This means a change in the

parameter values of the speed–flow relation. Denote the new speed–flow relation by ettkml. To compute the new equilibrium

after the improvement we use very much the same procedure as described above. As before we first compute the new gen-eralized prices holding the quantities fixed but now using the new speed–flow relation ettk

ml:

pkmrð0Þ ¼

Xl2r

Tkml þ ettk

mlð�1Þh i

:

The remainder of the procedure stays the same.

3.2.3. Adding a new linkMOLINO-II is not a forecasting model, so we assume that the effects on demand flows and generalized prices once the new

link is added are known. The model is then recalibrated using the new paths and quantities. The ‘‘without new link” equi-librium is simulated by setting extremely high generalized prices on the new link.

4. Software implementation

The MOLINO-II model was reprogrammed in a user-friendly way using WinDev that contains an appropriate user inter-face. The user interface operates as follows.

It first asks to define types of agents (users, operators and governments), number of subperiods and number years in theplanning horizon. This is followed by the construction of the network. First the nodes are labeled. Next one selects the linksbetween nodes that make sense, and the user help is also required to define the paths of interest as an automatic generationof all possible paths quickly gives a very high number of paths to deal with.

For every link one needs information on length, speed–flow relation, capacity of the link, variable operating cost, numberand type of users as well as the type of operator (private/public and what type of public operator), resource costs, values oftime, local and central taxes on use.

When elasticities of substitution are provided by type of user, the model is automatically calibrated for every year bycomputing users’ costs and combining these with quantity of trips to construct utility and demand functions as explainedin Section 3.1.1. Once the model is calibrated one can specify the policy simulations of interest.

Solving a case study as described in Section 5. takes of the order of a couple of minutes on a PC.

5. Illustrations of the use of MOLINO-II

Our model can be used for cost-benefit analysis of real world investments in a transport network but it also allows for amore general analysis like the simulation of Nash equilibria between competing profit maximizing operators who each con-trol part of the network. We first illustrate this feature for a simple parallel network, then we discuss a real world case-study:the investment in the Brenner tunnel.

5.1. Nash equilibrium tolls

We illustrate how the model can be used to analyze Nash toll competition between operators on a network. Operatorstypically compete along several dimensions. Among these, the price is a key one, others can be capacity, quality or the levelof maintenance. When a new operator comes into the market, the incumbent operators react, and usually cut their price,since the market is now more competitive. The failure to take this reaction into account can be a disaster for a new invest-ment, as we have observed in the past, for instance, in the case of the Chunnel (tunnel between France and UK) that did com-pete ferry services who cut their prices in response to the arrival of the Chunnel (see Kay et al., 1989). We use the Nash non-cooperative equilibrium concept: at equilibrium, every agent (operator) takes the prices of the other players as given and noagent can increase his profit level by unilaterally changing his price. Computing Nash equilibria can, even for very simplenetworks, quickly become a difficult analytical exercise if one wishes to use general functional forms for the demand func-tions. So one needs to rely on numerical simulations for which MOLINO-II is very well suited.

Consider a very simple network consisting of two nodes, A and B connected by two links as depicted in Fig. 7.For simplicity we assume only one type of user and one subperiod. Each link is operated by a different private operator

which has following objective function Porl :

Porl ¼ ðsl � OPClÞql; l ¼ 1;2; ð15Þ

where sl is the toll on link l; cl the operating cost and ql the demand on link l. Demand is described by the nested CES func-tions as in Eq. (6). Here we write them in an abbreviated form: ultimately the demand function can be written as a functionof the generalized prices pl on both links:

q1 ¼ Fðp1;p2Þ; ð16Þq2 ¼ Gðp1;p2Þ: ð17Þ

q1

q2

q=q1+q2

A B

q=q1+q2

Fig. 7. Parallel network.

844 A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849

For the sake of clarity, we assume a simple form for the generalized prices: we use a linear time cost function so thatgeneralized prices become a linear function of the volume: pl ¼ al þ blql þ sl where al; bl > 0:

In a Nash equilibrium, operators will set their tolls such that their profits are maximized taking the toll of the competitoras given. Solving the two first order conditions simultaneously gives us the Nash toll:

Table 1Parame

veh/a (frOperElastElastAll o

s�l ¼ OPCl �qldqldsl

:

If we simplify the problem even further and restrict ourselves to the symmetric case where b1 ¼ b2 ¼ b, the derivative ofdemand with respect to the toll reduces to (see Appendix A for details):

dq1

ds1¼

F1 þ b ðF2Þ2 � ðF1Þ2h i

ð1� bF1Þ2 � ðbF2Þ2¼ dq2

ds2; ð18Þ

where Fl � @F@pl

and Gl � @G@pl

(in the symmetric case at hand the direct- (resp. cross-) price derivatives are equal: F1 ¼ G2 andF2 ¼ G1). If there is no congestion ðb ¼ 0Þ the decrease in demand due to an increase in the toll will be equal to F1, the directprice effect. For b > 0, the decrease in demand will be smaller because the volume on the own route will decrease, so thetime cost decreases and total generalized cost increase is lower. In addition, some of the traffic is deviated to the other routeso the generalized price on the other route increases and this tends to also decrease the effect. We see that, even with all thesimplifying assumptions, the expressions remain highly implicit and no general result can be derived analytically. To test thesensitivity of the Nash tolls to the congestion function parameter ðbÞ and to the substitutability between the two routes weneed to rely on our numerical model. The values used for the main parameters are reported in Table 1.

The Nash equilibrium tolls s� in function of the parameter b are depicted in Fig. 8:A higher b corresponds to more congestion. If the free-flow speed is 120 km/h, then a b of 0.25 corresponds to a peak-

period speed of 60 km/h, a b of 1.75 with a peak-period speed of 15 km/h and a b of 0.08333 corresponds with a peak-periodspeed of 90 km/h. The fact that for higher levels of congestion, the equilibrium toll increases is in line with the theory, e.g. DeBorger et al. (2005) and de Palma and Leruth (1989).

In Fig. 9 we show the sensitivity of the Nash equilibrium toll to the substitutability between the two routes. The Y-axisreports the toll charged by an operator. The X-axis reports the substitution elasticity between the two routes. We assume thecongestion parameter bl for both links to be equal to 0.25.

As can be seen, a higher substitutability between the two routes reduces the toll. If the substitution elasticity tends tozero, we approach the monopoly solution. At the other end of the spectrum, with perfect substitutability, the toll convergesto the marginal external congestion cost as this is a Bertrand equilibrium: the marginal external congestion cost equals herebq, or approximately (0.25) (2) = 0.5.

This simple numerical example has demonstrated the need and the capacity of a numerical model to compute networkequilibriums with endogenous toll setting. In the next section we show how the model can be used for a real world casestudy.

ter values for the parallel network.

h per link when s ¼ 0 2ee flow time cost) 0.5ation cost per veh (OPC) 0icity of subst between transp and other consumption 3icity of subst between links 1.1ther user costs 0

Fig. 8. Nash toll in function of the congestion parameter for a parallel network.

Fig. 9. Nash toll in function of the elasticity of substitution between the two routes.

A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849 845

5.2. Evaluation of a large-scale investment: the Gdansk–Vienna corridor

The improvement of both the road and rail network in the Gdansk–Vienna corridor is one of the TEN-T priority projectsand has been studied using our model (see Haller and Emberger, 2008 for full details). The main arguments brought forwardin favour of the projects are their relevance on a European scale due to a high share of international traffic, a desired shifttowards rail for long-distance transport and the improvement of road safety as well as an alleviation of congestion. The net-work used for this case study has been depicted in Fig. 1. The resulting network is a simplification of the real transport net-work, since it is restricted to the links on the Polish territory. Nonetheless, the network reflects the most importantcharacteristics of the transport going from Gdansk to Vienna or Bratislava. The transport model covers passenger and freighttransport and distinguishes domestic, incoming/outgoing and transit flows, resulting in a total of six user types.

Different combinations of infrastructure and pricing scenarios are analyzed using the model. As to infrastructure the fol-lowing infrastructure policies are distinguished: (i) building nothing, (ii) building the road projects only, (iii) building the railprojects only and (iv) building both the road and rail project. For pricing policy, the options are: (a) the pricing regime cur-rently in force in Poland , (b) implementing a marginal social cost pricing scheme or (c) to toll the new or upgraded links on acost recovery basis.4 Marginal cost pricing is applied to the whole network of the case study, whereas the cost recovery tollsin scenario (c) are limited to new or upgraded links. Table 2 reports the main results of the different scenarios. The first col-umn gives the values for the baseline. All other columns give changes compared to the baseline.

All scenario’s lead to an increase of welfare but the welfare gain will be much higher when marginal cost pricing is imple-mented (scenario’s B1, C1 and D1 perform far better than their counterparts with existing pricing, namely B0, C0 and D0). Itshould be noted, however, that with MSC pricing, foreign users will be worse of since they do not benefit from toll revenuesredistribution.

4 In this scenario it is assumed that the link manager/operator completely skims the generalized cost savings from transport users. This leaves demandunchanged compared to the reference scenario without project implementation.

Table 2Aggregate cost benefit results for scenarios.

Scenario Infrastructure No TEN-T project Road project only Rail project only Road and rail project

Pricing Existing (baseline) MSC toll Existing MSC toll Max. cost recovery Existing MSC toll Max. cost recovery Existing MSC toll Max. cost recoveryCode A0 A1 B0 B1 B2 C0 C1 C2 D0 D1 D2

Passenger transport users’ surplusDomestic 1,507,500 �10,300 1200 �9200 0 700 �9600 0 1900 �8500 0Inbound/outgoing 157,000 �1000 100 �900 0 0 �1000 0 100 �900 0Transit 37,300 �200 0 �200 0 0 �200 0 0 �200 0

Freight transport users’ costsDomestic 557,500 �8500 300 �8300 0 300 �8200 0 600 �7900 0Import/export 259,300 �2700 100 �2700 0 0 �2700 0 100 �2600 0Transit 24,100 �300 0 �300 0 0 �300 0 0 �300 0

Passenger transport users’ surplus minus freight transport users’ costsDomestic 898,800 �20,600 1600 �19,300 0 1000 �19,600 0 2600 �18,200 0Foreign �37,900 �2400 100 �2300 0 100 �2300 0 200 �2200 0Domestic + foreign 860,900 �23,000 1700 �21,600 0 1100 �22,000 0 2800 �20,400 0

Toll revenuesRoad (PP25) 0 19,000 0 19,100 1700 0 19,000 0 0 19,100 1700Rail(PP24) 2900 2200 0 2200 0 100 2300 1000 100 2300 1000

Infrastructure costsInvestment – road (PP25) 0 0 �2200 �2200 �2200 0 0 0 �2200 �2200 �2200Investment – rail (PP23) 0 0 0 0 0 �1400 �1400 �1400 �1400 �1400 �1400Salvage value – road (PP25) 0 0 1200 1200 1200 0 0 0 1200 1200 1200Salvage value – rail (PP23) 0 0 0 0 0 800 800 800 800 800 800Operation costs – road (PP25) 1200 300 �100 200 �100 0 300 0 �100 200 �100Operation costs – rail (PP23) 500 0 0 0 0 0 0 0 0 0 0

Tax revenuesPoland (national government) 3800 �600 0 �600 0 0 �600 0 0 �600 0

Users’ surplus after allocation of tax revenues and profitsDomestic 903,800 200 500 700 600 400 600 400 900 1000 900Foreign �37,900 �2400 100 �2300 0 100 �2300 0 200 �2200 0Total 866,000 �2200 600 �1700 600 500 �1700 400 1100 �1200 900

External costs (excl. congestion)External cost 27,000 4200 �300 4000 0 0 4200 0 �300 4000 0

Total welfareDomestic 876,900 4400 200 4700 600 400 4800 400 600 5100 900Foreign �37,900 �2400 100 �2300 0 100 �2300 0 200 �2200 0Total 839,000 2000 300 2400 600 500 2500 400 800 2800 900

846A

.dePalm

aet

al./TransportationR

esearchPart

B44

(2010)834–

849

A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849 847

6. Extensions and conclusions

The MOLINO-II model presented in this paper is a multi-purpose model to assess transport investments using a minimumof data. It can be used for two purposes. First, as second opinion tool to verify the modelling results obtained using differentmode specific model assessments. Second, because it has a lighter network structure, it allows to analyze possible conflictsand strategic pricing behavior between operators. The model has been tested successfully on a number of investment pro-jects. Applications included investment projects like the Brenner-tunnel, the Betuwe rail line, the canal Seine–Escaut (seeProost et al., 2007) as well as applications to pricing issues (see de Palma et al., 2007a). There are different avenues forextension.

First, the model does not allow for substitution between destinations: for a given OD pair, consumers and firms decideonly on number of trips, paths and modes. One can redefine destinations in a broader sense and define them as goods.The physical destination then becomes part of the path. Take as example a Belgian exporter who wants to export to China,and has a choice of using the train or the truck to bring his goods to the harbour, say the port of Antwerp. One could redefinethe destination as China and include alternative ports (Antwerp, Rotterdam) in the definition of the path. For consumers onecould change the destination ‘‘London” for a shopping trip by ‘‘shopping” where the paths contain London, Paris, etc. Thisbroader definition of transport goods allows to study the interaction between imperfect competition and transport accessas in Dunkerley et al. (2009).

Second there is a need to develop a consistent approach to deal with the huge uncertainty in long term developmentsin demand and costs. One can of course use a Monte Carlo approach using a probability distribution for each of theuncertain parameters. This generates a distribution of the welfare and financial indicators. Project mistakes are oftendue to missing joint distributions of uncertain parameters: growth in demand for a particular link may be correlatedto exogenous changes in cost parameters (e.g. oil prices). The challenge is to recognize these correlations and to includethem.

Third, when checking existing project assessments, these are often incomplete. They are only made for a representativeyear, some categories of traffic or some of the competing modes are not included, etc. For models like MOLINO-II who needexisting data and forecasts for the initial calibration, this is a problem. There is no easy remedy, on the basis of experiencewith previous studies one could build a range of default parameters for the missing modes and traffic so that one can at leasttest the sensitivity of the omissions.

Acknowledgement

We thank N. Adler, E. Calthrop, B. De Borger and three referees for helpful comments. We would like to acknowledge theFUNDING consortium (EU – VIth Framework Programme) and Policy research Center Budgetary and Tax Policy, Governmentof Flanders (B) for financial support.

Appendix A. A.1. Nash equilibrium tolls in a parallel network

The Nash toll is

s�l ¼ OPCl �qldqldsl

:

To derive dqldsl

, we define Fl � @F@pl

and Gl � @G@pl

, and derive the equations q1 � F ¼ 0 and q2 � G ¼ 0 w.r.t. s1. This gives us:

dq1

ds1� F1 � b1

dq1

ds1þ 1

� �� F2 � b2

dq2

ds1

� �¼ 0 ð19Þ

dq2

ds1� G1 � b1

dq1

ds1þ 1

� �� G2 � b2

dq2

ds1

� �¼ 0 ð20Þ

From the last equation we get:

dq2

ds1½1� G2 � b2� ¼ G1 � ½b1

dq1

ds1þ 1� ð21Þ

Substituting this into Eq. (19) yields:

dq1

ds11� b1F1 �

b1b2F2G1

1� b2G2

� �¼ F1 þ

b2F2G1

1� b2G2ð22Þ

In the symmetric case we have b1 ¼ b2 ¼ b, and the direct- (resp. cross-) price derivatives are equal: F1 ¼ G2 and F2 ¼ G1.The equations then simplify to:

Table 3Sub- an

dlrkmt

Table 4Supply

Xdkmlð

Tkmlðt

ttkmlð

pkmrð

VOTkm

skmlðt

dlðtÞslðtÞvmax

lk

lkml

AlðtÞ

Table 5Variabl

qn;i

an;j

rn;i

pn;i

yn;i

Table 6Variabl

Porl

Pinfl

TollrINFCOPCl

INVCMCl

subol

horl ;

848 A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849

dq1

ds1¼

F1 þ b ðF2Þ2 � ðF1Þ2h i

ð1� bF1Þ2 � ðbF2Þ2ð23Þ

the expression for dq2ds2

is derived in exactly the same way.

Appendix B. B.1. Variables and parameters tables

In the Table 3 we present all sub- and superscripts used in the paper. Next we define all variables and parameters used todescribe supply (Table 4), demand (Table 5), operators and infrastructure managers (Table 6) as well as the social welfarefunction (Table 7).

d superscripts.

OD pairLinkPathType of userSubperiodYear

variables.

tÞ Total transport volume of a representative user of type k

Travelling between OD pair d on link l during the subperiod m in year t

Þ Monetary cost for a user of type k during subperiod m on link l at year t

tÞ Time cost for a user of type k during subperiod m on link l at year t

tÞ Generalized price for a trip for a user of type k

Using path r during period m at year t

lðtÞ Value of time of a user of type k during subperiod m on link l at year t

Þ Toll for a user of type k during subperiod m on link l at year t

Length of link l at year tCapacity of link l at year t

ðtÞ Maximum speed for a user of type k on link l at year tPassenger car equivalent for a user of type k

During subperiod m on link l at year t;BlðtÞ Speed–flow parameters of link l at year t

es of the nested-CES utility functions.

Utility component at level nShare parameters at level nElasticity of substitution at level nPrice index at level nExpenditure at level n

es and parameters related to operators and infrastructure managers.

Profit of the operator of link lProfit of the infrastructure manager of link l

ev l Toll revenues on link l

l Total Infrastructure use charge on link lTotal operation costs on link l

l Total investments costs for link lTotal maintenance costs on link l

r; subinf

lSubsidies to or from the operator or infrastructure manager of link l

hinfl

Tendering parameter associated to the operator/infrastructure manager of link l

Table 7Variables and parameters related to the social welfare function.

Uk Utility of a user of type k

PCk Production costs of a firm of type k

Taxcen; Taxloc Total transport tax revenues collected by central/local governmentEXTC External costs other than congestionwk Social weight given to users of type k

wext Social weight given to external costsCC ; CL Marginal cost of public funds of the central/local government

bkC: ; bk

L: ; bkor;l; bk

inf ;lDistribution parameters

A. de Palma et al. / Transportation Research Part B 44 (2010) 834–849 849

References

Anderson, S., de Palma, A., Thisse, J.F., 1992. Discrete Choice Theory of Product Differentiation. The MIT Press, Cambridge, MA.De Borger, B., Proost, S., Van Dender, K., 2005. Congestion and tax competition in a parallel network. European Economic Review 49 (8), 2013–2040.De Borger, B., Dunkerley, F., Proost, S., 2007. Strategic investment and pricing decisions in a congested transport corridor. Journal of Urban Economics 62 (2),

294–316.de Palma, A., Leruth, L., 1989. Congestion and game in capacity: a duopoly analysis in the presence of network externalities. Annales d’Economie et de

Statistique (15/16), 389–407.de Palma, A., Proost, S., Lindsey, R. (Eds.), 2007. Investment and the Use of Tax and Toll Revenues in the Transport Sector. Research in Transportation

Economics, vol. 19. Elsevier Science 5.de Palma, A., Lindsey, R., Proost, S., Van der Loo, S., 2007. Comparing alternative pricing and revenue use strategies with the MOLINO model (Chapter 5 in de

Palma). In: Proost, Lindsey (Eds), Investment and the use of tax and toll revenues in the transport sector, Elsevier Science 5.de Palma, A., Proost, S., Van der Loo, S., 2008. MOLINO-II: model for assessing pricing and investment strategies for transport infrastructure. Transportation

Research Record: Journal of the Transportation Research Board 2079.Dixit, A., Grossman, G.M., Helpman, E., 1997. Common agency and coordination: general theory and application to government policy making. Journal of

Political Economy 104 (4), 752–769.Dunkerley, F., de Palma, A., Proost, S., 2009. Spatial asymmetric duopoly with an application to Brussels’ airports. Journal of Regional Science 49 (3), 529–

554.Engel, E.M., Fisher, R., Galetovic, A., 2001. Least-present-value-of-revenue auctions and higway franchising. Journal of Political Economy 109 (5), 993–1020.Haller, R., Emberger, G., 2008. Task Report – Molino-II Case Study: Gdansk–Vienna Case Study.Kay, J., Manning, A., Szymanski, S., 1989. The economics benefits of the channel tunnel. Economic Policy 4 (8), 212–234.Keller, W.J., 1976. A nested CES-type utility function and its demand and price-index functions. European Economic Review 7 (2), 175–186.Layard, R., Glaister, S., 1994. Cost-Benefit Analysis, second ed. Cambridge University Press.Mayeres, I., Proost, S., 2001. Marginal tax reform, externalities and income distribution. Journal of Public Economics 79 (2), 343–363.Proost, S., Dunkerley, F., Van der Loo, S., Adler, N., Nash, C., Pels, E., Johnson, D., Whiteing, T., Wright, C., Koetse, M., Rouwendal, J., Brenck, A., Jäkel, K., Winter,

M., Emberger, G., Haller, R., 2007. Deliverable 5: Case Studies: How Do the Infrastructure Fund Scenarios Affect Existing TEN-T Projects? EuropeanCommission Project Funding.

Sheffi, Y., 1985. Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods. Prentice-Hall, Englewood Cliffs, NJ, USA.