willingness to spend and road pricing rates
TRANSCRIPT
ARTICLE IN PRESS
Transport Policy 17 (2010) 160–172
Contents lists available at ScienceDirect
Transport Policy
0967-07
doi:10.1
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Willingness to spend and road pricing rates
Paolo Ferrari n
University School of Engineering of Pisa, Via Diotisalvi 2, 56126 Pisa, Italy
a r t i c l e i n f o
Available online 8 February 2010
Keywords:
Urban transport system
Road pricing
Willingness to spend
Congestion level
Environmental effects
0X/$ - see front matter & 2010 Elsevier Ltd. A
016/j.tranpol.2010.01.003
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a b s t r a c t
Using a theoretical model of urban transport system the paper examines the influence of distribution of
willingness to spend within the urban population on road pricing rates. It shows that the rates that
must be imposed in an urban area in order to maintain pollutant concentration and congestion due to
traffic within acceptable levels is heavily dependent on the distribution of the urban population’s
willingness to spend. This fact severely limits the reliability of any method for calculating road pricing
rates based on theoretical analysis, so that an experimental approach seems necessary. The paper shows
that a relation exists between the toll rate per kilometer of trip and the average traffic congestion,
which is typical of each urban area and can be determined experimentally by successively imposing
three different rates and measuring the corresponding congestion levels. The relation can then be used
to determine the pricing scheme when the purpose of road pricing is to maintain both the congestion
and the environmental effects due to urban traffic below acceptable thresholds. An example shows how
the model can help policymakers in decision-making processes.
& 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Road pricing, that is, charging motorists to travel along thestreets of an urban network, is widely recognized as a highlyeffective instrument for maintaining congestion and pollutionwithin acceptable limits. The classical theory of road pricing (seee.g., Beckmann et al., 1956; Hau, 1992; Hearn and Yildirim, 2002;Rietveld and Verhoef, 2000; Yang and Huang, 1998) is based onthe consideration that the cost perceived by a driver travellingalong a road is less than the social costs generated; the differenceis due, in part, to the greater cost that the presence of one moredriver on the road network imposes on the other drivers and, inpart, to the greater environmental costs of the pollution from theextra vehicle (Johansson-Stenmann and Sterner, 2000). Chargingdrivers a fee equal to the difference between the marginal socialcost, which they do not perceive, and the private cost, of whichthey are instead aware, can bring about a condition of socialoptimum by removing from the road network all those for whomthe utility of travelling by car is less than the consequent socialcosts. Such people can choose a different way to travel, or simplyforego their journey.
Calculating road-pricing rates by applying this theory is verydifficult, if not impossible. In effect, it would be necessary to knowboth the demand function, that is, the number of drivers
ll rights reserved.
travelling the road network as a function of the cost theyperceive, and the function of the marginal social cost due to bothcongestion and pollution. Moreover, both these costs must beexpressed in monetary terms, so as to be able to compare themwith the road pricing cost perceived by drivers.
Some authors (see e.g., Bell, 1995; Ferrari, 1995, 1997; Larsonand Patriksson, 1995; Yang and Bell, 1997) have thereforeproposed a different approach to the theory of road pricing, theaim being to avoid the need to know the marginal social costfunction. This approach is based on defining thresholds, that is,the values not to be exceeded, for both congestion and pollutionfor different city areas. Then, some or all links of the road networkare assigned capacity constraints, which are the maximum trafficvolumes so as to not exceed the above-mentioned thresholds. Theroad-pricing rates charged to drivers travelling along networklinks for which a capacity constraint has been set are thencalculated as the additional costs that must be imposed on theselinks in order to attain a network traffic equilibrium that respectsthe capacity constraints.
However, this second approach also requires knowing thedemand function, which is quite difficult to determine. Otherauthors (Li, 1999, 2002; Yang et al., 2004) have thereforeproposed a method for calculating road-pricing rates that doesnot require knowing the demand function, but only the costfunction associated to each network link, that is, the relationbetween vehicle flow and the private cost that it produces, whichis generally used in procedures for the demand assignment toroad networks. Limiting themselves to considering only the
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P. Ferrari / Transport Policy 17 (2010) 160–172 161
contribution of congestion to social costs, these authors derive themarginal social cost function from the private cost function. Theythen propose a method based on experimental measurements ofthe travel speeds or times consequent to application of a givenroad-pricing rate, which enables determination, through a trial-and-error procedure, of the rate that gives rise to a condition ofsocial optimum. A trial-and-error procedure had been alsoproposed previously by Downs (1993) and Vickrey (1993), neitherof whom, however, suggested any technique for implementation.
All the above-mentioned methods are based on the assump-tion that the monetary value attributed to each component oftransport cost is the same for all drivers, and that this value can betaken equal to the mean of its actual distribution in thepopulation in question. The underlying assumption is that, aslong as the mean value results to be the same, any differences inthe distribution of the willingness to spend within the populationwould have no influence on road-pricing rates.
This paper will instead prove that such differences do indeedhave a considerable influence on the effects of road-pricing rates.The proof stems from the application of a model of urbantransport similar to that considered in Ferrari (2005), in which asquare urban area, served by a road grid travelled by private carsand city buses, experiences a fixed transport demand, uniformlydistributed in time and space. Overall demand is split between thetwo modes of transport as a function of the respective costsperceived by users.
The hypotheses that overall demand is inelastic with respect toroad pricing is founded on the fact that most trips in the chargingzone during the period of road pricing imposition have purposespeople cannot give up; thus some people shift to other modes oftransport if they cannot afford the burden of road pricing, but theoverall transport demand remains substantially unchanged. Thisfact is confirmed by the data recorded in some European cities,where a road pricing scheme has been implemented. For instancecongestion charging in Central London (Transport for London,2004, p. 58) has produced a daily average reduction between65,000 and 70,000 in the number of drivers entering the chargingarea. Less than 5000 of them have travelled to other destinationsor have reduced their trip frequency, so that the overall transportdemand has remained substantially unchanged. Less than 5000have travelled outside the charging hours, while for the most part,between 40,000 and 50,000, have shifted to other modes oftransport, essentially to public transport. Similar results havebeen obtained in Milan (Comune di Milano, 2008), where theimposition of road pricing on some particular categories of carshas caused a daily average decrease in 20,540 passenger enteringby car the charging zone and an increase in 19,083 people usingpublic transport.
The model supposes that the same toll is imposed on eachkilometer of trip, independently of the period of the day and thestreets that are travelled. This kind of road pricing scheme is dueto the fact that the purpose of road pricing in urban areas is tomaintain both average congestion and average concentration ofpollutants produced by road traffic within acceptable thresholds.In fact there is no punctual correspondence between vehicleemissions and concentrations of the most dangerous pollutants(see e.g., Horowitz, 1982, pp. 63–77); high pollutant concentra-tions can be measured in points distant from streets of highcongestion some hours after congestion disappeared. The onlypollutant for which such a correspondence exists is carbonmonoxide, whose contribution to urban pollution is at presentvery scarce, because most cars are equipped with catalyticconverters, which convert carbon monoxide to carbon dioxideand water. Two very dangerous pollutants, nitrogen dioxide andozone, are secondary pollutants, i.e. they are not emitted fromcars, but they are the effect of chemical reactions of some primary
pollutants (oxides of nitrogen and hydrocarbons), directly emittedfrom cars, with some components of the atmosphere. Thesereactions produce their effects in a wide area and in a long periodof time, thus also in points very distant from the point of emissionof the primary pollutants and some hours after the emission.Another very dangerous pollutant is the particulate matter withsize less that 10 mm (1 mm=10�6 m), usually named PM10, whichhas many sources; among them there are the emissions from thediesel engines and the actions of the wheels of vehicles on theroad surface. The particulate matter is dispersed by wind in awide area, so that we can measure high concentrations of PM10very far from the points where it was produced. The thresholds ofpollutant concentrations are set by the environmental legislation.In Europe these thresholds, computed as averages of variousmeasures recorded during a day, are 200 mg/m3 (1 mg=10�6 g) fornitrogen dioxide, 120 mg/m3 for ozone, and 50 mg/m3 for theparticulate.
The average pollutant concentration can be measured by usingdetectors distributed in the urban area. We define the level of
congestion due to traffic as the ratio of the increase in unitdistance travelling time under the actual conditions with respectto the free flow conditions, to the average time spent to travel aunit distance under free flow conditions. Thus we can measure theaverage congestion level in an urban area through the measure ofaverage unit distance travelling time obtained using, e.g., themoving car observer method (Transport for London, 2003a, p. 49).Measuring the average congestion level in the urban area indifferent days, characterized by different transport demands, andthe corresponding pollutant concentrations, an experimentalrelation can be obtained between congestion and pollutantconcentration, which enables us to know the congestion levelcorresponding to the pollutant concentration threshold. Thecongestion level that must not be exceeded is the minimumbetween that corresponding to the threshold of pollutantconcentration and that due to other causes (e.g., noise, excessivetraffic density in the streets, etc.). In order to maintain averagecongestion below this level, we cannot consider different tolls onthe various streets of a town in the various periods of the day, butwe have to impose a toll independently of the streets and the timeof the trip. This scheme of road imposition is similar to thatimplemented in London, where paying the road pricing feeenables drivers to travel in the urban area during the entire day.
Since we refer to the average level of congestion, indepen-dently of the travel demand characteristics and of road networkpattern, we propose a theoretical model in which the roadnetwork and the travel demand are such as to cause a constantcongestion level in the urban area.
We denote as travel cost in an urban area, by whatever meansof transport, the disutility perceived by users due to the time andmoney spent travelling. We have considered that both the costs oftime and of money are random variables distributed within theurban population according to certain laws of probability. Thedistribution of the cost of money depends on the distribution ofthe willingness to spend. This term is different from willingness to
pay, which in economics denotes the maximum amount a personwould be willing to pay for a good, and is often used in the welfareevaluation of a policy that alters the quality of a resource (see e.g.,McFadden, 1995). Instead the willingness to spend has a moregeneral meaning; it refers to the consumer behaviour in theactivities of saving and spending, and thus it depends on theincome and also on some psychological attitudes, e.g., onoptimism on the state of the economy, and determines the cost,i.e. the disutility, which people attribute to the money spentanyway (Mackie et al., 2001).
With the aim of evaluating the effects of different distributionsof the willingness to spend on the road-pricing rate necessary to
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Fig. 1. Computation of the vehicle flow across a section Z.
P. Ferrari / Transport Policy 17 (2010) 160–172162
maintain congestion and pollution within acceptable limits, themodel has been applied to two situations characterized by twodistributions of the cost of money exhibiting the same mean, butwith very different skewnesses. The results obtained reveal theheavy dependence of road-pricing rates on the distribution of thewillingness to spend, and therefore the impossibility of reliablycalculating such rates based on the mean cost value attributed to aunit of monetary expenditure. Moreover, the results enableproposing a computation method that avoids the need for anyrestrictive hypotheses. They reveal that a relation exists betweenthe road-pricing rate and consequent congestion level; suchrelation depends on the congestion level in the absence of anyroad pricing and on the distribution of the willingness to spend.The relation can be obtained experimentally in any city area bysuccessively imposing three different rates and measuring thecorresponding congestion levels. The rate resulting in the desiredlevel of congestion can then be determined directly from therelation, without having to resort to any trial-and-error procedure.
This paper is organized as follows. Section 2 describes themodel, while Section 3 is devoted to its application to two urbanareas whose populations are assumed to have two very differentdistributions of willingness to spend, though with the same mean.This is with the aim of evaluating the influence on road-pricingrates of both the congestion level in the absence of road pricingand the distribution of willingness to spend. Section 4 presents ananalysis of the results obtained, whence a method for computingroad-pricing rates follows. An example of application of the modelto a real world case is reported in Section 5. Lastly, in Section 6 asummary of the main points is presented and some conclusionsadvanced.
2. Urban transport system model
Let us considers a square urban area, with sides of length b,served by two means of transport: private cars and city buses. Wedenote by d the overall transport demand in the urban area, that isto say, the overall number of trips made in the area during a unittime using both means of transport. We assume d to be rigid, thatis independent of transport costs, constant over time, anduniformly distributed over the urban area. Let g=d/b2 be thenumber of trips originating in any unit area during a unit time,and g/b2 the number of trips coming from, and directed to, a unitarea. Since all other characteristics of the urban transport system(frequency and ticket price of buses, road-pricing rate) are alsoassumed to be constant over time, the transport phenomenon isstationary.
Let us suppose that the urban area is served by a square grid ofroads, along which run the bus lines, sharing the same roads withprivate cars. Let da and db, with da+db=d, be the numbers of tripsin the area over unit time using, respectively, private and publictransport. Since the transport phenomenon is stationary, da and db
are also constant over time and we suppose that they areuniformly distributed over space.
Let us now consider a rectangular strip of urban area of unitwidth, parallel to one of the sides of the square, for example, thehorizontal one (Fig. 1). Let us suppose that all users of both meansof transport travelling from origin O to reach a point A atcoordinates (x1, y1), located within the strip, follow the sameitinerary, for instance, by proceeding from the origin along avertical road up to the horizontal road of ordinate y1, and thencontinuing along this latter to destination. In this case, the overallflow of users on both means of transport who, during a unit time,travel West to East across section Z of the rectangular stripsituated at distances db and (1�d)b (0odo1) from the twovertical sides of the area, is given by the demand originating in the
rectangle with sides db and b and whose destination is the portionof the rectangular strip of length (1�d)b to the right of section Z.Since the demands for private and public transport originating inthe unit area are, respectively, ga=da/b2 and gb=db/b2, it followsthat flows fa and fb, that is, the numbers of users travelling,respectively, on private and public transport who cross Z during aunit time in each of the two directions, are
f a ¼ga
b2b2dð1�dÞb¼
da
bdð1�dÞ
f b ¼gb
b2b2dð1�dÞb¼
db
bdð1�dÞ
ð1Þ
The same result would follow if the users moved from one pointto the other in the urban area by first proceeding along ahorizontal road and then continuing along a vertical one.
Let Za and Zb be the mean numbers of passengers, respectively,
in a car and on a bus. The flow of cars across Z is fa¼ f a=Za; if Zc is
the car equivalent of a bus, the flow of buses across Z measured in
equivalent cars is fb¼ f bZc=Zb. We suppose that the bus frequency
n is fixed, so that the bus capacity, and thus Zb, increases with db.But if Zb increases, Zc increases too, so that we can consider that
approximately Zb/Zc=cost=1/Z. Therefore we have fb¼ f b=Z, for
which the overall flow across Z in terms of the number of cars is
f ¼ faþ f
b. In view of (1), and by putting db=d�da, we obtain
f ¼da
Zaþðd�daÞ
Z
� �dð1�dÞ
bð2Þ
Let us denote by
d¼da
Zaþðd�daÞ
Z ð3Þ
the overall demand in the urban area, expressed as the number ofequivalent car trips. From Eq. (2), we have
f ¼d
bdð1�dÞ ð4Þ
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Let K be the overall capacity of all road lanes crossing Z in eachdirection, that is, the maximum number of cars that can traverse Z
during a unit time and give rise to conditions of circulationconventionally held to be at the limits of tolerability. Setting f ¼ K ,and accounting for Eqs. (2) and (3), we obtain the expression forthe demand d that results in a traffic flow in each direction acrossZ equal to the capacity:
d¼K
dð1�dÞb ð5Þ
Under the assumption that capacity K is a function of thelocation of Z, and therefore of d as well, K=K(d), so that
KðdÞdð1�dÞ
¼H ð6Þ
where H is a constant, we have that the demand yielded by Eq. (5)leads to capacity at the same time on all the roads in the area. Letus thus define Hb as the capacity of the urban area and call H itscapacity factor. The fact that K(d)/(d(1�d)) is constant means thatthe capacity K(d) of the cross sections of a rectangular strip growswith d(1�d), that is the road density increases on proceedingfrom the outskirts of the area toward the centre, a situationgenerally encountered in urban areas.
Dividing f by capacity K, and taking expressions (4) and (6)into account, it can be seen that the flow/capacity ratio is constantthroughout the entire area and is given by
f
KðdÞ¼
d
Hbð7Þ
Now let t0 be the mean travel time by car along a road stretchof unit length under conditions of free circulation; this is held tobe constant throughout the entire urban area. We assume that thecar travel time ta along the same road stretch when the flowacross Z is f , is given by
ta ¼ t0 1þa f
KðdÞ
!b24
35 ð8Þ
where a and b are two parameters that depend on the geometricaland operating characteristics of the road network.
Substituting f=KðdÞ in Eq. (8) with the expression given in Eq.(7), it follows that ta is constant throughout the entire area and isgiven by
ta ¼ t0 1þa d
Hb
!b24
35 ð9Þ
or, by substituting expression (3) for d:
ta ¼ t0 1þa da
Zaþðd�daÞ
Z
� �1
Hb
� �b !ð10Þ
Since t0 is constant throughout the entire area (as also ta forany given values of d and da), we take as a measure of the urbanarea congestion level CL the ratio of the increase in unit-distancetravelling time under actual conditions with respect to the freeflow conditions to the time spent to travel a unit distance underfree flow conditions. We therefore have
CL¼ a da
Zaþðd�daÞ
Z
� �1
Hb
� �bð11Þ
We have supposed that all users travelling between the twopoints, O and A, of the area follow the same path of minimumlength; they proceed along a vertical line originating from O andcontinue along the horizontal line directed to A. Since the time ta
to travel a unit length is constant in the area, as well as the othercomponents of travel cost, the cost of this used path is equal tothat of the unused paths with the same length connecting the
same pair of points, and less than the cost of the unused paths ofgreater length. Thus the flow pattern is a Wardrop equilibrium.
Now let la and lb be the lengths of the pedestrian routes takenduring a trip with private and public transport, respectively, andsp the walking speed of a pedestrian. We consider all threequantities to be the same for all users at their mean values.
Having assumed that trips from any origin are equallydistributed in all directions, only a small fraction of bus userswill have origins and destinations near the same bus line. It canthus be assumed that all bus users will need to change lines onceduring a trip (Newell, 1979), for which they must sustain twowaits of duration w each, as well as the inconvenience of havingto transfer. We reckon that such inconvenience is perceived bythe user as an increase t in travel time of between 5 and 10 min,once again following the suggestion of Newell (1979). If n is thearrival frequency of buses, which reach stops at regular intervals,while passengers arrive randomly, the mean waiting time at a busstop is w=1/(2n).
As we said before, the travel cost (it will be denoted simply by cost
in the following) is the disutility due to the expense in terms of timeand money that a person attributes to a trip of any given length witha given means of transport in the urban area. It is a dimensionlessvariable that varies randomly from one person to another, and it isgiven by the sum of two random fractions. The first represents thatpart of the cost that is a function of the measurable burdens of travel(time and money spent) and depends on each individual’s socio-economic characteristics (sex, age, job, etc.), which are assumed to berandomly distributed in the urban population. The second randomfraction has zero mean and represents the shift from the first variabledue to each individual’s own perceptions of the cost of a trip, whichdepends on personal sensibilities. It should be noted that such adivision of cost into two random fractions is similar to that used insome mixed-logit models (see e.g., Hensher and Greene, 2003). Thefirst random fraction is in its turn the sum of two random variables:the first represents the cost of time, the second the cost of monetaryexpenditures. The distribution of the cost of monetary expendituresdepends, as it was seen in the previous section, on the distribution ofthe willingness to spend. The substitution ratio between time andmoney measures the monetary value of time; it is a random variablewhose mean is the value of time used in many problems of transport.Many methods have been proposed for its estimation, which arebased on different models of discrete choice (see e.g., Gaudry et al.,1989; Hensher, 2001a, 2001b; Hess et al., 2005). However, sincedifferent values of the costs of time and of money expenditures cangive rise to the same monetary value of time, it is not possible toderive the cost distributions of either time or money expendituresfrom the distribution of the value of time.
We assume that the cost of travel time is proportional to thetime spent travelling according to a random coefficient B1, andthat all users perceive the ratios between the costs of the differentcomponents of the travel time in the same way, so that it ispossible to measure the overall cost due to travel time as if it weredue to the in-vehicle time. Following the suggestions of theEnglish Department for Transport (Ortuzar and Willumsen, 1996,pp. 414–415), it has been assumed that the cost of a unit of timespent walking or waiting for a bus is perceived as equal to twicethat actually spent on board. Thus, if l is the length of a routetravelled on board, the cost of the travel time by car is
Cat ¼ B1 talþ
la
sp
� �ð12Þ
where ta is given by Eq. (10).In comparison with a car trip of equal length, the duration of a
bus trip must be increased by multiplying by a coefficient r41, inorder to account for the idle time at bus stops. Therefore, the cost
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P. Ferrari / Transport Policy 17 (2010) 160–172164
of the time spent on a bus trip of length l on board is given by
Cbt ¼ B1 talrþtþ lb
spþ2w
� �ð13Þ
We assume that, for each unit length of route on board, eachuser of public transport pays a ticket price of X1 for bus, whileeach user of private transport pays a price p for fuel and, whenapplicable, a price X2 for road pricing. The cost of monetaryexpenditures is proportional to these prices according to arandom coefficient B2, whose probability law is independent ofB1 and is a function of the distribution of the willingness to spendin the urban area’s population.
Thus, the overall costs perceived by users for a trip of length l
in a car and on a bus is, respectively, given by
Ca ¼ B1 talþla
sp
� �þB2lðpþX2Þþea ð14Þ
Cb ¼ B1 talrþtþ lb
spþ2w
� �þB2lX1þeb ð15Þ
where ea and eb are two random variables with zero mean, whichmeasure, as mentioned before, the difference between the tripcost perceived by each person and that be ascribed to themeasurable burdens of travel; ea and eb depend on each traveller’sparticular sensibilities and are independent of each other and ofB1 and B2.
The distributions of B1 and B2 have been derived from theseconsiderations. Generally speaking, there is a proportion of a city’spopulation that leads a rather inactive life style, and therefore hasample time to devote to moving about. For such people, the costdue to a unit of time spent travelling is modest, varies little fromone person to the next, and its value is therefore contained withina rather restricted range. Other people, instead, lead very activelives, must often go through trip chains, and have a rather limitedtime budget to do so (see for example Kitamura et al., 1990;McGuckin and Murakami, 1999; Nishii et al., 1990). These peopletherefore attribute high cost to a unit of time spent travelling, andbecause such perceived cost also varies widely from person toperson, it is therefore distributed over a broad range of values.Bearing such considerations in mind, it can be assumed that thedistribution of B1, which is defined only for positive values, ischaracterized by positive skewness.
The distribution of coefficient B2 depends on the distribution ofthe willingness to spend, and hence also that of income, withinthe urban population. If the willingness to spend does not differgreatly among the various segments of the population, coefficientB2, which can only take on positive values, would be distributedin quite a symmetrical way around the mean, and its valueswould in any event be contained within a rather limited range.If instead the willingness to spend varies widely withinthe population, a certain portion of the population wouldattribute rather low cost to the money spent for travelling,and its values would be contained within a small range, whilethe remaining part would perceive the cost as high, and its valuesfor this segment would be spread over a broad range. In this case,coefficient B2 would certainly not be distributed symmetrically,but would present marked positive skewness. It is clear fromsuch considerations that very different distributions of B2 mayexist in the real life, despite their giving rise to the same meanvalue.
Any person intending to take an urban-area trip, whose in-vehicle length l varies randomly from one person to the next,chooses the means of transport perceived as less expensive. Thus,
the traveller will choose the car if CaoCb, that is, if
B1 talþ2la
sp
� �þB2lðpþX2Þþea
oB1 talrþtþ 2lb
spþ2w
� �þB2lX1þeb ð16Þ
Thus, the probability pa that a person chooses to travel by car isgiven by the probability of event (16):
pa ¼ P B1 talþ2la
sp
� �þB2lðpþX2ÞþeaoB1 talrþtþ 2lb
spþ2w
� �264þB2lX1þeb
�ð17Þ
With given values of the area size b, capacity factor H, demandd, prices p, X1, and X2, pedestrian route lengths la and lb, walkingspeed sp, and coefficients Za, Z, and r, and knowing the probabilitylaws for B1, B2, ea, eb, and l, the right side of Eq. (17) is a function ofda. Thus, Eq. (17) can be rewritten as
pa ¼CðdaÞ ð18Þ
Considering that at equilibrium pa is constant over time andthat, for the strong law of large numbers (Feller, 1968, pp. 202–204), da=pad, we have from Eq. (18) the following equilibriumequation:
pa ¼CðpadÞ ¼FðpaÞ ð19Þ
From Eq. (19), it is impossible to derive a closed-formexpression for paas a function of the variables on which itdepends. Expression (19) can nevertheless be solved via asimulation procedure, as follows. Given a value p
aof pa in the
range (0, 1), and the corresponding da¼ p
ad, the value Fðpa
Þ canbe calculated by the Montecarlo method. A set of five values isextracted, each independently, from the distributions of B1, B2, ea,eb, and l, and the corresponding values of the two members ofexpression (16) are calculated. The operation is repeated n times;if expression (16) turns out to be satisfied n1 number of times, weset Fðpa
Þ ¼ n1=n. Since function FðpaÞ ¼ pa�FðpaÞ is strictly convex(as can be easily verified empirically), the solution to problem(19) can be obtained by using the bisection method (see forexample, Zangwill, 1969, p. 122) to calculate the nil point of FðpaÞ
in the range (0, 1), by repeating the calculation of FðpaÞ fordifferent values of p
auntil FðpaÞ deviates from zero by a quantity
deemed negligible.
3. Application of the model
The model has been applied to a square urban area of sideb=5 km. Private transport is represented by cars, while publictransport is provided by a square grid of bus lines running at adistance of 0.5 km from one another. The mean length of thepedestrian route for public transport riders, given by the sum ofthe mean distance from the point of departure to the starting busstop and the mean distance from the end bus stop to destination,is lb=0.5 km. The mean length of the pedestrian route for a user ofprivate transport, given by the sum of the mean distance from thepoint of departure to the starting car park and the mean distancefrom the arrival park to destination, is la=0.1 km. The meanwalking speed of pedestrians is assumed to be sp=4 km/h. Thetravel time under conditions of free circulation has beenattributed a value of t0=1.9 min/km, which has been drawn fromrecent measurements on traffic in Central London (Transport forLondon, 2003a, p. 52). By convention, the road capacity has been
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P. Ferrari / Transport Policy 17 (2010) 160–172 165
assumed to be that value of car flow that yields a travel time perunit length of road twice of what would occur in free flowconditions, and therefore a congestion level of CL=1.
We moreover assume that users of public transport perceivethe cost of transferring between two bus lines as equivalent to at=5-min increase in the in-vehicle travel time. In Eqs. (8)–(11),we have set a=1 and b=5. From Eq. (8) it can be seen that thevalue a=1 is consistent with the given definition of road capacity;in effect, when f
a¼ KðdÞ, we have ta=2t0.
The road network in the city centre is assumed to be made upof grid of two-lane, one-way roads arranged alternately at adistance of 250 m from one another (Fig. 2). Taking into accountthe presence of intersections, a capacity of 1150 car/h has beenattributed to each road, whence the capacity in each direction of a1 km-wide strip turns out to be 1150/0.5=2300 car/h. Since d=0.5in the central urban area, Eq. (6) yields a capacity factor ofH=2300/0.25=9200 car/(h km), which in turn gives the areacapacity Hb=46,000 car/h.
We assume the expenditure for fuel for private vehicles to bep=0.1 h/km, and assign a frequency of n=12 bus/h for each publictransport line and a bus ticket price of X1=0.1 h/km. With a giventrip length, the in-vehicle travel time is greater in buses than incars, due to the stops. In order to account for this, in computingthe costs we consider a virtual trip length for buses that is 30%greater than the length of a car journey, thereby attributing avalue of 1.3 to coefficient r in Eq. (13). We set Za=1.4 as the meannumber of travellers in a car, and we consider that, when themean number of travellers in a bus is Zb=50, we have Zc=3, sothat Z=Zb/Zc=16.67.
We have considered trip cost as a dimensionless quantity;thus, by measuring time in minutes and price in Euros, thedimensions of B1 and B2 are, respectively, [min�1] and [h�1]. Wehave measured trip costs on a scale by which the mean cost of a1 h outlay is equal to 1; we therefore have E(B2)=1 h�1. Weassume that B1 and B2 are both lognormal random variables, whileboth ea and eb are normal random variables with zero mean andvariance Var(e)=0.25. The probability density function of alognormal random variable x is (see e.g., Larson, 1974, p. 386)
fxðxÞ ¼1
xsffiffiffiffiffiffi2pp exp �
1
2
lnx�ms
� �2" #
ð20Þ
where m and s are two parameters.Values of �2.0 and 0.45 have been, respectively, assigned to
parameters m and s of B1, so that we have E(B1)=0.15; bearing inmind the scale assumed for the costs measurement, the value of
Fig. 2. Grid of urban streets in the central part of the area.
E(B1) indicates that the mean cost of 1 min spent travelling is equalto the mean cost of a monetary outlay of 0.15 h. The probabilitydensity function of B1 is represented in Fig. 3, where it can be seenthat B1 has a positive skewness, equal to 1.014, with a median of0.135, quite below the mean. This means that about half thepopulation that we are dealing with attributes a rather low value totravel time and that such value is contained within a rather narrowrange, while the value ascribed to time by the other half variesacross a much broader range and also reaches very high values.
Two different distributions have been considered for variableB2; their probability density functions are shown in Fig. 4. Theparameters values assigned for distribution 1 are m=0 and s=0.25,while the values for distribution 2 are m=�0.5 and s=1.0. Both B2
distributions have mean E(B2)=1. Distribution 1 exhibits skewnessSk(B2)=0.778 and a median of 1, the same as the mean, whiledistribution 2 presents skewness Sk(B2)=6.18 and a median of0.607, quite below the mean. Such distributions are representativeof urban populations with very different socio-economiccharacteristics. B2 distribution 1 is representative of a populationin which the perceived cost of monetary expenditures isdistributed rather symmetrically around the mean and iscontained within a rather narrow range of values. B2 distribution2 is instead characteristic of a population in which the cost ofmonetary expenditures is distributed quite asymmetrically; halfthe population attributes a low cost to monetary expenditures, andsuch cost is contained within a very narrow range, while for theother half, such costs vary widely and also attain very high values.
The maximum length of a trip in the urban area in question is2b, while the mean length, given the grid arrangement of the roadnetwork, is 2b/3 (Newell 1979). It therefore seems reasonable thatthe random variable l (trip length) can, with sufficiently closeapproximation, be assigned a triangular distribution defined overthe range (0, 2b), with a maximum value of 1/b for l=0, and aminimum of zero for l=2b.
Different values of demand d have been considered, so as tocover a wide spectrum of congestion levels in the absence of anyroad pricing.
The model has been applied, with the given distributions of B1
and l, for each of the two B2 distributions, using the simulationtechnique illustrated in Section 2 above. For each assigned valuep
aof probability pa, the calculation of each of the two members of
expression (16) has been repeated n=100,000 times in order todetermine n1, the number of times that inequality (16) is satisfied,and thereby arrive at a good approximation for the value of Fðpa
Þ,given by ratio n1/n.
In a first stage the model has been applied for each value ofdemand d and for each distribution of B2, considering differentvalues of the road-pricing rate X2, thereby computing thecongestion level CL at equilibrium corresponding to each case. Ithas thus been possible to obtain the relation between X2 and CL
for each value of d and for each distribution of B2, and arrive at thevalue of rate X2 that results in a congestion level, CL=0.9, deemedacceptable for an urban area (under the assumption that thecorresponding pollution level is tolerable as well).
The model has been then applied again separately for each ofthe two B2 distributions, by setting the pricing rate X2 that yieldsCL=0.9 for each value of the demand d, with the aim of calculatingthe corresponding means B1
a and B1b of the B1 values associated to
users of private and public transport, respectively. In addition, wehave calculated the means, Ca and Cb, of the trip costs incurred bycar and bus users, respectively, the mean cost C of a journey forthe entire population, and the ratio ðC�CÞ=C of the increase in themean cost for the entire population due to the imposition of roadpricing to the mean cost C in the absence of road pricing.
The results of the calculations are summarized in Tables 1and 2, which, respectively, refer to each of the two B2
ARTICLE IN PRESS
0
1
2
3
4
5
6
7
8
0Coefficient B1
Pro
babi
lity
dens
ity
0.1 0.2 0.3 0.4 0.5 0.6
Fig. 3. Distribution of coefficient B1.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0Coefficient B2
Pro
babi
lity
dens
ity
1 2
0.5 1 1.5 2 2.5 3
Fig. 4. Two distributions of coefficient B2.
Table 1Effects of road pricing when coefficient B2 has distribution 1.
d CL X2 (h/km) Ba1 ðmin�1
Þ Bb1 ðmin�1
Þ Ca Cb C C C�CC
64,000 0.974 0.263 0.152 0.066 3.484 4.114 3.496 2.678 0.305
66,800 1.205 0.425 0.155 0.080 3.912 5.188 3.998 2.898 0.380
69,600 1.478 0.522 0.158 0.087 4.110 5.622 4.272 3.158 0.353
72,400 1.798 0.602 0.160 0.093 4.237 5.930 4.482 3.463 0.294
75,200 2.172 0.675 0.162 0.097 4.324 6.152 4.654 3.819 0.219
78,000 2.605 0.745 0.164 0.101 4.373 6.322 4.792 4.232 0.132
Table 2Effects of road pricing when coefficient B2 has distribution 2.
d CL X2 (h/km) Ba1 ðmin�1
Þ Bb1 ðmin�1
Þ Ca Cb C C C�CC
64,000 0.974 0.160 0.151 0.090 3.024 7.513 3.102 2.667 0.163
66,800 1.205 0.346 0.153 0.105 3.300 7.423 3.565 2.887 0.235
69,600 1.478 0.484 0.155 0.111 3.455 7.368 3.857 3.147 0.226
72,400 1.798 0.616 0.156 0.114 3.571 7.378 4.101 3.453 0.188
75,200 2.172 0.765 0.158 0.117 3.637 7.328 4.293 3.809 0.127
78,000 2.605 0.894 0.159 0.119 3.692 7.314 4.460 4.222 0.056
P. Ferrari / Transport Policy 17 (2010) 160–172166
ARTICLE IN PRESS
0
0.2
0.4
0.6
0.8
1
1Congestion level in the absence of road pricing
Roa
d pr
icin
g (e
uro/
km)
1 2
1.25 1.5 1.75 2 2.25 2.5
Fig. 5. Relationships between the congestion level in the absence of road pricing and the road pricing rate that gives rise to a congestion level equal to 0.9, for distributions
1 and 2 of coefficient B2.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1Congestion level in the absence of road pricing
Incr
ease
in c
ost d
ue to
road
pric
ing
1 2
1.2 1.4 1.6 1.8 2 2.2 2.4
Fig. 6. Relationships between the congestion level in the absence of road pricing and the increase in trip cost due to the imposition of a road pricing rate that gives rise to a
congestion level equal to 0.9, for distributions 1 and 2 of coefficient B2.
P. Ferrari / Transport Policy 17 (2010) 160–172 167
distributions. Each table shows the values of the overall demand d
in the urban area measured in trips per hour, the correspondingcongestion level CL in the absence of road pricing, the road-pricingrate X2 necessary to obtain a congestion level of CL=0.9, themeans B1
a and B1b of coefficient B1 for users of private and public
transport, the mean costs Ca and Cb perceived by users of the twomeans of transport, the mean costs C and C of a journey for theentire population in the presence and absence of road pricing,respectively, and the ratioðC�CÞ=C.
To better illustrate the results obtained, Figs. 5 and 6 show theroad-pricing rate X2 necessary to achieve a congestion level ofCL=0.9 and the increase ðC�CÞ=C in the mean trip cost due to theimposition of this road pricing rate, as a function of the congestionlevel in the absence of road pricing, for each of the twodistributions of coefficient B2.
4. Analysis of the results
Examining the results presented in Tables 1 and 2 it can beseen that road pricing produces substantially different effects for
the two distributions of coefficient B2, that is, for differentdistributions of the cost that people attribute to monetaryexpenditures, despite their having the same mean value. In thecase of B2 distribution 2, the mean value B1
b of coefficient B1 forpublic transport users is always significantly higher than in thecase of B2 distribution 1, particularly for the lowest values of CL.The opposite occurs for mean B1
a for private transport users.Bearing in mind that B1 measures the cost that people attribute to1 min spent travelling, such a result underscores the fact thatwhen a certain number of people in the population attribute ahigh cost to monetary expenditures, imposing road pricing drivesthem to switch to public transport, despite the high perceivedcost of travel time. On the contrary, people for whom travel timehas a low cost continue using their cars, despite the imposition ofroad pricing, if they attribute scant value to the monetaryexpenditures.
This fact can be confirmed by examining the mean trip costs,Ca and Cb, perceived by the users of the two means of transport. Inthe case of B2 distribution 2, Cb is considerably higher than fordistribution 1, while the contrary occurs for the cost Ca for users ofprivate transport.
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P. Ferrari / Transport Policy 17 (2010) 160–172168
Such results have important consequences on the road-pricingrate X2 that must be imposed to achieve the desired level ofcongestion. As can be seen in Fig. 5, there is a critical congestionlevel in the absence of road pricing, at about CL=1.70, below whichX2 for B2 distribution 2 turns out to be lower than for distribution1, a difference that is quite marked especially for low values of CL.The opposite occurs when CL exceeds this critical value.
Such a pattern can be explained by considering that whencongestion levels in the absence of road pricing are not very high,it is sufficient to transfer a small proportion of the demand da ontopublic transport to reduce congestion to the desired level. In theevent of B2 distribution 2, this proportion of demand is made up ofthose people who attribute a very high cost to monetaryexpenditures, and thus switch to public transport on theimposition of an even modest rate X2, despite the high cost thatsome of them attribute to travel time. If B2 instead exhibitsdistribution 1, rate X2 must be higher, because no part of thedemand attributes a very high cost to monetary expenditures.When instead the congestion level without road pricing is great,that is, above the aforementioned critical value, a substantialfraction of demand da must be made to switch to public transportin order to reduce congestion to the desired level. In the case of B2
distribution 2, a significant portion of the population attributes avery low cost to monetary expenditures. Since a proportion ofthese people must be transferred to public transport, a high X2
rate will be needed, greater than in the case of distribution 1, inwhich the population fraction attributing very low value tomonetary expenditures is smaller. Nevertheless, although ahigher X2 must be imposed in the case of distribution 2 whenCL overcomes the critical value, the mean value of the trip cost forthe entire population remains lower than in the case ofdistribution 1, as can be inferred by comparing the data inTables 1 and 2. In effect, even if X2 is high, in the case ofdistribution 2 those people who continue to travel by car attributea very low value to monetary expenditures. It follows, as can beinferred from Fig. 6, that the percentage increase in mean trip costfor the entire population due to the imposition of road pricing isconsiderably lower in the case of distribution 2 than indistribution 1, regardless of the congestion level in the absenceof road pricing.
Fig. 6 reveals that for both B2 distributions, the increase in themean trip cost due to road pricing as a function of the congestionlevel in the absence of road pricing at first rises, then falls. This
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0Tariff of road
Con
gest
ion
leve
l
0.1 0.2 0.3
Fig. 7. Relationships between the road pricing rate and the corresponding congestion
which the congestion level in the absence of road pricing is 1 and 1.5.
can be explained by considering that with increasing CL, evergreater proportions of the demand da must be transferred topublic transport by increasing the road-pricing rate X2. When thecongestion level is rather low, rate X2 is also low, because all thosepeople who value monetary expenditures highly while attributinglow worth to travel time readily switch to public transport. Withincreasing CL it becomes necessary to increase X2, which drivesthose people who attribute high cost to both travel time andmonetary expenditures to switch to public transport; thisbehaviour determines the rising mean trip cost. As CL increasesfurther, the mean trip cost begins to decline, even if X2 increases;this is due partly to the fact that high road pricing leads to areduction in congestion, and partly to the fact that the increase inX2 causes people with a low estimation of both the cost ofmonetary expenditure and travel time to switch to publictransport.
Fig. 7 shows the variations in the congestion level as a functionof rate X2 for two different congestion levels in the absence ofroad pricing and for the two distributions of B2. These twocongestion levels are 1.0 and 1.5, which define a range includingmost of the congestion levels deemed intolerable occurring inurban areas and for which the imposition of road pricing istherefore called for. By way of example, in Central London in2002, before the application of road pricing, the mean daytimetravel time was 4.2 min/km, as opposed to a time of 1.9 min/kmunder conditions of free circulation (Transport for London, 2003a,p. 52), with a corresponding congestion level of CL=1.20. By 2004,about 2 years after institution of road pricing and when it wasbelieved that the transport system had reached equilibrium, themean travel time had fallen to 3.6 min/km (Transport for London,2004, p. 13), with a corresponding congestion level of CL=0.9,which, as previously mentioned, represents an acceptablethreshold for an urban area.
The curves shown in Fig. 7 are third-order polynomialsinterpolating the calculation results with an R2 statistic ofnearly 1. They are all very regular, approximately linear curvesthroughout a wide range of values of X2, especially for distribution2. Analysis of the plots reveals that the relation between the road-pricing rate X2 and the associated level congestion CL for a givenurban area and a given period of the day, characterized by acertain average congestion level in the absence of road pricing,can be approximated closely by determining four paired values(X2, CL). One of these is the congestion level corresponding to
pricing (euro/km)
1 2
0.4 0.5 0.6 0.7
level for distributions 1 and 2 of coefficient B2, and for two different situations in
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P. Ferrari / Transport Policy 17 (2010) 160–172 169
X2=0. The other three can be determined by progressivelyincreasing pricing rates, waiting a certain time period after fixingeach rate in order to allow the transport system to reachequilibrium, and then measuring the resulting congestion levelat the end of the period. Interpolating the points thus determinedwith a third-order polynomial yields the relation between X2 andCL, which enables direct computation of the rate resulting in thecongestion level deemed tolerable, without having to resort toany trial-and-error procedure.
5. An example
The model proposed in this paper can be used to determine areference value for the road pricing rate that should be imposed inan urban area, e.g., one among the three values needed to theexperimental computation of road pricing illustrated in theprevious section.
The data needed to the application of the model are thedimension of the zone to which road pricing should be imposed,the total number of cars entering and exiting from this zoneduring the daily charging period, the characteristics of the publictransport system, the monetary cost per kilometre borne by eachcar driver, and the socio-economic characteristics of the popula-tions on which the distributions of B1 and B2 coefficients depend.The actual problem is transformed into an equivalent one thatsatisfies the hypotheses of the model; it is supposed that thecharging zone has square shape with the same area as the realone, and that the transport demand uniformly distributed inspace and time during the charging period is that entering andexiting from the zone. By using the aforementioned data and theproposed model, the value of toll per kilometre is computed, andthen converted into that imposed on drivers depending on thecharging technique used. If tolls are collected at a cordon aroundthe charging zone each time a car crosses the cordon, the toll isobtained by multiplying the value per kilometer by the averagelength of a trip inside the zone, approximately two third of theside of the square. Instead, if a driver is charged only once a day,the toll is obtained by multiplying the value per kilometer by theaverage length travelled by a driver during a day insidethe charging zone. The tolls so obtained are only reference values.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
0Road pric
Con
gest
ion
leve
l
0.2 0.4
Fig. 8. Relationship between road pricing and congestion level i
The actual value to impose should be determined by anexperimental method, as was said above.
This method has been applied to the central area of London, forwhich we have a lot of data recorded before and after the chargingimposition (Transport for London, 2003a, 2003b, 2004), whichenable us to test the validity of the proposed method. We refer tothe zone of about 22 km2 on which congestion charging wasintroduced in 2003; it is equivalent to a square of side 4.7 km.Cars are charged on the weekdays, from 7h30 to 18h30, whenthey cross the first time during a day the cordon around thecharging zone. Our computations refer to an epoch in which thetoll was 5 £, equivalent to 7.2 h, but the average charge paid wasactually lower, because of exemptions and reduced charges forsome people. The total traffic crossing the cordon during thecharging period before the charging imposition was 752,000vehicles. Considering that there are on average 1.34 persons pervehicle, the equivalent hourly passenger demand, uniformlydistributed in the square area, is 91,607 passenger/h; when roadpricing was imposed, a portion of this demand shifted to publictransport. The value of 0.1 h/km has been attributed to theaverage operating monetary cost borne by car drivers. The valueof time has been supposed to be 0.20 h/min, equal to thatcomputed by Mackie (2005), considering the proportion ofbusiness travels in Central London. As in Section 3 we measurethe trip cost in a scale by which the mean cost of a 1 h outlay isequal to 1. Thus a lognormal distribution has been attributed tocoefficient B1, whose parameters are m=�1.70 and s=0.45, with amean E(B1)=0.20 and a skewness Sk(B1)=1.014, the latter beingthe same as that of the distribution represented in Fig. 3.
The equivalent hourly vehicle demand, supposed uniformlydistributed in the square area, before the toll imposition is68.364 vehicle/h, which gives rise to a congestion level CL=1.20(Transport for London, 2003a, p. 54). Taking into account Eq. (11),it results that the capacity factor of the charging area is H=14,000.The other characteristics of the transport system are the same asthose in Section 3.
Using the model presented in this paper, and considering thetwo distributions of coefficient B2 represented in Fig. 4, thecongestion level in the square area equivalent to the chargingzone in London has been computed for various values of the roadpricing rates, ranging from 0 to 1 h/km. The relation between
ing (euro/km)
1 2
0.6 0.8 1
n Central London for distributions 1 and 2 of coefficient B2.
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Table 3Values of road pricing X2 in Central London that give rise to a congestion level CL=0.90 for the two distribution of B2, and the corresponding transport characteristics
compared with those in the absence of road pricing.
Distr.B2 X2 (h/km) Cma (h) Cm
b (h) Cm (h) Ta (min) Tb (min) T (min) Ca Cb C
1 0 0.312 – 0.312 14.564 – 14.564 3.317 – 3.317
0.43 1.570 0.535 1.503 12.172 44.576 14.256 4.213 5.280 4.282
2 0 0.312 – 0.312 14.564 – 14.564 3.317 – 3.317
0.35 1.345 0.497 1.291 12.269 42.775 14.224 3.612 7.779 3.879
P. Ferrari / Transport Policy 17 (2010) 160–172170
congestion level and road pricing is reported in Fig. 8 for each ofthe two distributions of B2.
Congestion level in London after the imposition of road pricingdecreased, reaching at equilibrium the value CL=0.90. Fig. 8shows that the tolls computed according to the model that wouldgive rise to this congestion level are 0.43 h/km and 0.35 h/km fordistributions 1 and 2 of B2, respectively. The corresponding hourlyvalues of demand by car and by bus are, respectively, da=85,790passenger/h and db=5910 passenger/h for both distributions of B2.
The total annual charge payment in London after the 5 £charge imposition was 165,000 millions of pounds (Transport forLondon, 2003b, Section 4.9). Considering that weekends and otherholidays are free of charge, road pricing is imposed on 250 daysper year, so that the daily charge payment is 165,000,000/250=660,000 £. Since the number of car trips after the roadpricing imposition in the charging zone is 630,000, the averagetoll paid for a trip is 660,000/630,000=1.05 £. Considering that theaverage length of a trip is 2�4.70/3=3.13 km, and the averagenumber of passengers per car is 1.34, we have that the toll paid bya passenger per kilometer is 1.05/(3.13�1.34)=0.249 £=0.359 h.This value is rather close to that computed by the model fordistribution 2 of coefficient B2. This result shows that the cost ofmonetary expenses is rather dispersed in the population travel-ling in Central London, with a percentage of it that assigns a ratherhigh value to the monetary expenses.
Table 3 reports some characteristics of the transport systemcomputed by the model, for the two distributions of coefficient B2,when the road pricing gives rise to a congestion level CL=0.90,compared with those in the absence of road pricing. Thecharacteristics that have been computed are the averagemonetary expenses Cm
a and Cmb borne by the car and the bus
users, respectively, for a trip in the charging zone, along with theaverage monetary expense Cm borne by the overall userpopulation, the corresponding average times Ta, Tb, and T spentfor a trip by each category of users, and the overall costs Ca, Cb,and C. These numbers show that, for both B2 distributions, as aconsequence of the road pricing imposition the monetary expenseis greater than in the absence of road pricing for both car usersand for people who moved to public transport; the trip time is lessthan in the absence of road pricing for car users, but it issubstantially greater for people who decided to shift to publictransport; the overall cost is greater than in the absence of roadpricing for both user categories. From the latter result we arguethat road pricing is useful from the social point of view, because itreduces the pollution due to traffic by reducing the congestion,along with the transfer of money from car drivers to society as awhole, but it increases the disutility of all people who used the carbefore the road pricing imposition.
It is interesting to note in Table 3 that the generalized cost ofusers who moved to public transport is greater for distribution 2of B2 than for distribution 1, even if both monetary expenses andthe trip times are less. This result depends on the fact that in thecase of distribution 2, as a consequence of its high skewness,people who move to public transport assign to the cost of
monetary expenses a value much greater than in the case ofdistribution 1.
6. Conclusions
This paper has presented a theoretical model apt to evaluatethe influence that a distribution of the willingness to spend in anurban population has on the values of road pricing rates that haveto be imposed in order to reduce the congestion level to anacceptable threshold. The willingness to spend of personsdetermines the disutility, that is the cost, they attribute to themoney spent anyway, and depends on their income and also onsome psychological attitudes, for instance on optimism on thestate of the economy.
We have considered two distributions of the cost of themonetary expenditures, representative of two urban populationswith very different socio-economic characteristics. Distribution 1is representative of a population of rather homogeneous socio-economic characteristics, in which the perceived cost of monetaryexpenditures is distributed rather symmetrically around themean and is contained within a rather narrow range of values.Distribution 2 is instead representative of a population withrather dispersed socio-economic characteristics, in which theperceived costs of monetary expenditures are distributed quiteasymmetrically: half the population attributes low costs tomonetary expenditures, and such costs are contained within avery narrow range, while for the other half such costs vary widelyand also attain very high values.
We have obtained from the application of the model to atheoretical case that, even if the two distributions have the samemean, the road pricing rates X2 needed to maintain the congestionlevel at the desired threshold are different, and this differencedepends on the level of congestion CL in the absence of roadpricing. When CL is not very high, below a critical value that in thecases studied in the paper is about 1.70, X2 for distribution 2 turnsout to be lower than for distribution 1, while the opposite occurswhen X2 exceeds the critical value. Such a pattern can beexplained by considering that, when the congestion level in theabsence of road pricing is not very high, it is sufficient to transfer asmall portion of car drivers onto public transport to reducecongestion to the desired level. In the event of distribution 2, thisproportion of demand is made up of those people who attribute avery high cost to monetary expenditures, and thus switch topublic transport on the imposition of an even modest rate X2,despite the high cost that some of them attribute to travel time.Instead in the case of distribution 1, rate X2 must be higher,because no part of car drivers attributes a very high cost tomonetary expenditures. When instead the congestion level with-out road pricing is large, that is above the aforementioned criticalvalue, a substantial fraction of car drivers must be made to switchto public transport in order to reduce congestion to the desiredlevel. In the case of distribution 2, for which the median is lessthan the mean, a significant portion of the population attributes a
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P. Ferrari / Transport Policy 17 (2010) 160–172 171
very low cost to monetary expenditures. Since a fraction of thesepeople must be transferred to public transport, a high X2 rate willbe needed, greater than in the case of distribution 1, in which thepopulation fraction attributing very low values to monetaryexpenditures is smaller.
Moreover we have seen that the X2 value needed to maintainthe congestion level at a desired level gives rise to an averageoverall cost borne by the users of public transport that in the caseof distribution 2 is higher than for distribution 1, while thecontrary occurs for the cost borne by people who continue totravel by car. This is due to the fact that when a certain number ofpeople in the population attribute a high cost to monetaryexpenditures (as in distribution 2), imposing road pricing drivesthem to switch to public transport, despite the high perceivedcost of travel time. On the contrary, people for whom travel timehas a low cost continue using their cars, despite the imposition ofroad pricing, if they attribute scant value to monetary expendi-tures. The consequence is that in the case of distribution 2, unlikedistribution 1, there is a considerable fraction of users of publictransport for whom both costs of time and money expendituresare high, so that the overall cost is high, and a considerablefraction of car users for whom both costs are low, so that theoverall cost is low.
The fact that the road-pricing rate that must be imposed in anurban area in order to maintain congestion within acceptablelevels is heavily dependent not only on the level of previouscongestion, but also on the distribution of the urban population’swillingness to spend, severely limits the reliability of any methodfor calculating road pricing rates based on estimates of the meancost attributed to monetary expenditures. On the other hand, anymethod based on theoretical analyses must also be consideredequally unreliable, given that such methods involve a largenumber of variables for which it is very difficult to assign realisticvalues.
It therefore seems much more appropriate to adopt anexperimental approach. This paper has shown that, when thepurpose of road pricing is to maintain the pollutant concentrationand the average congestion in an urban area below acceptablethresholds, we cannot consider different tolls for various streets ofthe town in the various periods of the day, but we have to imposea toll independently of the streets and of the time of the trip. Arelation exists between the toll rate per kilometer of trip and theaverage congestion in the urban area, which can be obtainedexperimentally by successively imposing three different rates andmeasuring the corresponding congestion levels. The relation canthen be used to directly determine the rate that results in thecongestion level deemed tolerable.
This paper also shows how the proposed model can be used tocompute a reference value for the road pricing rate that should beimposed in an urban area, i.e. one among the three aforemen-tioned rates. The data needed to the application of the model arethe dimension of the zone to which road pricing should beimposed, the total number of cars entering and exiting from thiszone during the daily charging period, the characteristics of thepublic transport system, the monetary cost per kilometre borneby each car driver, and the socio-economic characteristics of thepopulations on which the distributions of costs of time and ofmonetary expenses depend. The real problem is transformed intoan equivalent one that satisfies the hypotheses of the model; it issupposed that the charging zone has square shape with the samearea as the real one, and that transport demand uniformlydistributed in space and time during the charging period is thatexisting and entering the zone. Using the aforementioned dataand the proposed model, the value of the toll rate per kilometer iscomputed, and then converted to that imposed on drivers takingaccount of the charging technique used.
The model has been applied to a real world example, thecentral area of London, for which we have a lot of data recordedbefore and after the congestion charging imposition. These dataenabled us to test the validity of the model. Using the two B2
distributions considered in the paper we have obtained that therates X2 needed to maintain the congestion level at the valueCL=0.90 reached in London after the charging implementation in2003 are X2=0.43 h/km in the case of distribution 1 andX2=0.35 h/km in the case of distribution 2. The charging rate perkilometre imposed in London in 2003, computed as ratio of theaverage daily charge payment to the average number of kilo-meters travelled in the charging zone during the daily chargingperiod, is X2=0.359 h/km. This value is very close to thatcomputed by the model in the case of distribution 2; thus themodel indicates that the cost of monetary expenditures is ratherdispersed in the population travelling in Central London, with aportion of it that assigns a rather high value to the cost ofmonetary expenses.
From the application of the model to the London charging areawe have found that for both B2 distributions the generalized costborne by users after the imposition of road pricing is greater thanthat in the absence of road pricing, for those who shift to publictransport and for those who continue to travel by car as well.From this result we argue that road pricing is useful from thesocial point of view, because it reduces the pollution due to trafficby reducing congestion, but it increase the disutility of all peoplewho used the car before the road pricing imposition.
Acknowledgments
The author is indebted to two anonymous referees for theiruseful comments and suggestions on an earlier edition of thispaper.
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