Willingness to spend and road pricing rates

ARTICLE IN PRESS Transport Policy 17 (2010) 160–172

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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 in f o

a b s t r a c t

Available online 8 February 2010

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.

Keywords: Urban transport system Road pricing Willingness to spend Congestion level Environmental effects

1. Introduction Road pricing, that is, charging motorists to travel along the streets of an urban network, is widely recognized as a highly effective instrument for maintaining congestion and pollution within acceptable limits. The classical theory of road pricing (see e.g., Beckmann et al., 1956; Hau, 1992; Hearn and Yildirim, 2002; Rietveld and Verhoef, 2000; Yang and Huang, 1998) is based on the consideration that the cost perceived by a driver travelling along a road is less than the social costs generated; the difference is due, in part, to the greater cost that the presence of one more driver on the road network imposes on the other drivers and, in part, to the greater environmental costs of the pollution from the extra vehicle (Johansson-Stenmann and Sterner, 2000). Charging drivers a fee equal to the difference between the marginal social cost, which they do not perceive, and the private cost, of which they are instead aware, can bring about a condition of social optimum by removing from the road network all those for whom the utility of travelling by car is less than the consequent social costs. Such people can choose a different way to travel, or simply forego their journey. Calculating road-pricing rates by applying this theory is very difficult, if not impossible. In effect, it would be necessary to know both the demand function, that is, the number of drivers

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travelling the road network as a function of the cost they perceive, and the function of the marginal social cost due to both congestion and pollution. Moreover, both these costs must be expressed in monetary terms, so as to be able to compare them with the road pricing cost perceived by drivers. Some authors (see e.g., Bell, 1995; Ferrari, 1995, 1997; Larson and Patriksson, 1995; Yang and Bell, 1997) have therefore proposed a different approach to the theory of road pricing, the aim being to avoid the need to know the marginal social cost function. This approach is based on defining thresholds, that is, the values not to be exceeded, for both congestion and pollution for different city areas. Then, some or all links of the road network are assigned capacity constraints, which are the maximum traffic volumes so as to not exceed the above-mentioned thresholds. The road-pricing rates charged to drivers travelling along network links for which a capacity constraint has been set are then calculated as the additional costs that must be imposed on these links in order to attain a network traffic equilibrium that respects the capacity constraints. However, this second approach also requires knowing the demand function, which is quite difficult to determine. Other authors (Li, 1999, 2002; Yang et al., 2004) have therefore proposed a method for calculating road-pricing rates that does not require knowing the demand function, but only the cost function associated to each network link, that is, the relation between vehicle flow and the private cost that it produces, which is generally used in procedures for the demand assignment to road networks. Limiting themselves to considering only the

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contribution of congestion to social costs, these authors derive the marginal social cost function from the private cost function. They then propose a method based on experimental measurements of the travel speeds or times consequent to application of a given road-pricing rate, which enables determination, through a trialand-error procedure, of the rate that gives rise to a condition of social optimum. A trial-and-error procedure had been also proposed previously by Downs (1993) and Vickrey (1993), neither of whom, however, suggested any technique for implementation. All the above-mentioned methods are based on the assumption that the monetary value attributed to each component of transport cost is the same for all drivers, and that this value can be taken equal to the mean of its actual distribution in the population in question. The underlying assumption is that, as long as the mean value results to be the same, any differences in the distribution of the willingness to spend within the population would have no influence on road-pricing rates. This paper will instead prove that such differences do indeed have a considerable influence on the effects of road-pricing rates. The proof stems from the application of a model of urban transport similar to that considered in Ferrari (2005), in which a square urban area, served by a road grid travelled by private cars and city buses, experiences a fixed transport demand, uniformly distributed in time and space. Overall demand is split between the two modes of transport as a function of the respective costs perceived by users. The hypotheses that overall demand is inelastic with respect to road pricing is founded on the fact that most trips in the charging zone during the period of road pricing imposition have purposes people cannot give up; thus some people shift to other modes of transport if they cannot afford the burden of road pricing, but the overall transport demand remains substantially unchanged. This fact is confirmed by the data recorded in some European cities, where a road pricing scheme has been implemented. For instance congestion charging in Central London (Transport for London, 2004, p. 58) has produced a daily average reduction between 65,000 and 70,000 in the number of drivers entering the charging area. Less than 5000 of them have travelled to other destinations or have reduced their trip frequency, so that the overall transport demand has remained substantially unchanged. Less than 5000 have travelled outside the charging hours, while for the most part, between 40,000 and 50,000, have shifted to other modes of transport, essentially to public transport. Similar results have been obtained in Milan (Comune di Milano, 2008), where the imposition of road pricing on some particular categories of cars has caused a daily average decrease in 20,540 passenger entering by car the charging zone and an increase in 19,083 people using public transport. The model supposes that the same toll is imposed on each kilometer of trip, independently of the period of the day and the streets that are travelled. This kind of road pricing scheme is due to the fact that the purpose of road pricing in urban areas is to maintain both average congestion and average concentration of pollutants produced by road traffic within acceptable thresholds. In fact there is no punctual correspondence between vehicle emissions and concentrations of the most dangerous pollutants (see e.g., Horowitz, 1982, pp. 63–77); high pollutant concentrations can be measured in points distant from streets of high congestion some hours after congestion disappeared. The only pollutant for which such a correspondence exists is carbon monoxide, whose contribution to urban pollution is at present very scarce, because most cars are equipped with catalytic converters, which convert carbon monoxide to carbon dioxide and water. Two very dangerous pollutants, nitrogen dioxide and ozone, are secondary pollutants, i.e. they are not emitted from cars, but they are the effect of chemical reactions of some primary

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pollutants (oxides of nitrogen and hydrocarbons), directly emitted from cars, with some components of the atmosphere. These reactions produce their effects in a wide area and in a long period of time, thus also in points very distant from the point of emission of the primary pollutants and some hours after the emission. Another very dangerous pollutant is the particulate matter with size less that 10 mm (1 mm= 10  6 m), usually named PM10, which has many sources; among them there are the emissions from the diesel engines and the actions of the wheels of vehicles on the road surface. The particulate matter is dispersed by wind in a wide area, so that we can measure high concentrations of PM10 very far from the points where it was produced. The thresholds of pollutant concentrations are set by the environmental legislation. In Europe these thresholds, computed as averages of various measures recorded during a day, are 200 mg/m3 (1 mg=10  6 g) for nitrogen dioxide, 120 mg/m3 for ozone, and 50 mg/m3 for the particulate. The average pollutant concentration can be measured by using detectors distributed in the urban area. We define the level of congestion due to traffic as the ratio of the increase in unit distance travelling time under the actual conditions with respect to the free flow conditions, to the average time spent to travel a unit distance under free flow conditions. Thus we can measure the average congestion level in an urban area through the measure of average unit distance travelling time obtained using, e.g., the moving car observer method (Transport for London, 2003a, p. 49). Measuring the average congestion level in the urban area in different days, characterized by different transport demands, and the corresponding pollutant concentrations, an experimental relation can be obtained between congestion and pollutant concentration, which enables us to know the congestion level corresponding to the pollutant concentration threshold. The congestion level that must not be exceeded is the minimum between that corresponding to the threshold of pollutant concentration and that due to other causes (e.g., noise, excessive traffic density in the streets, etc.). In order to maintain average congestion below this level, we cannot consider different tolls on the various streets of a town in the various periods of the day, but we have to impose a toll independently of the streets and the time of the trip. This scheme of road imposition is similar to that implemented in London, where paying the road pricing fee enables drivers to travel in the urban area during the entire day. Since we refer to the average level of congestion, independently of the travel demand characteristics and of road network pattern, we propose a theoretical model in which the road network and the travel demand are such as to cause a constant congestion level in the urban area. We denote as travel cost in an urban area, by whatever means of transport, the disutility perceived by users due to the time and money spent travelling. We have considered that both the costs of time and of money are random variables distributed within the urban population according to certain laws of probability. The distribution of the cost of money depends on the distribution of the willingness to spend. This term is different from willingness to pay, which in economics denotes the maximum amount a person would be willing to pay for a good, and is often used in the welfare evaluation of a policy that alters the quality of a resource (see e.g., McFadden, 1995). Instead the willingness to spend has a more general meaning; it refers to the consumer behaviour in the activities of saving and spending, and thus it depends on the income and also on some psychological attitudes, e.g., on optimism on the state of the economy, and determines the cost, i.e. the disutility, which people attribute to the money spent anyway (Mackie et al., 2001). With the aim of evaluating the effects of different distributions of the willingness to spend on the road-pricing rate necessary to

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maintain congestion and pollution within acceptable limits, the model has been applied to two situations characterized by two distributions of the cost of money exhibiting the same mean, but with very different skewnesses. The results obtained reveal the heavy dependence of road-pricing rates on the distribution of the willingness to spend, and therefore the impossibility of reliably calculating such rates based on the mean cost value attributed to a unit of monetary expenditure. Moreover, the results enable proposing a computation method that avoids the need for any restrictive hypotheses. They reveal that a relation exists between the road-pricing rate and consequent congestion level; such relation depends on the congestion level in the absence of any road pricing and on the distribution of the willingness to spend. The relation can be obtained experimentally in any city area by successively imposing three different rates and measuring the corresponding congestion levels. The rate resulting in the desired level of congestion can then be determined directly from the relation, without having to resort to any trial-and-error procedure. This paper is organized as follows. Section 2 describes the model, while Section 3 is devoted to its application to two urban areas whose populations are assumed to have two very different distributions of willingness to spend, though with the same mean. This is with the aim of evaluating the influence on road-pricing rates of both the congestion level in the absence of road pricing and the distribution of willingness to spend. Section 4 presents an analysis of the results obtained, whence a method for computing road-pricing rates follows. An example of application of the model to a real world case is reported in Section 5. Lastly, in Section 6 a summary of the main points is presented and some conclusions advanced.

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. We denote by d the overall transport demand in the urban area, that is to say, the overall number of trips made in the area during a unit time using both means of transport. We assume d to be rigid, that is independent of transport costs, constant over time, and uniformly distributed over the urban area. Let g = d/b2 be the number of trips originating in any unit area during a unit time, and g/b2 the number of trips coming from, and directed to, a unit area. Since all other characteristics of the urban transport system (frequency and ticket price of buses, road-pricing rate) are also assumed to be constant over time, the transport phenomenon is stationary. Let us suppose that the urban area is served by a square grid of roads, along which run the bus lines, sharing the same roads with private cars. Let da and db, with da +db = d, be the numbers of trips in the area over unit time using, respectively, private and public transport. Since the transport phenomenon is stationary, da and db are also constant over time and we suppose that they are uniformly distributed over space. Let us now consider a rectangular strip of urban area of unit width, parallel to one of the sides of the square, for example, the horizontal one (Fig. 1). Let us suppose that all users of both means of transport travelling from origin O to reach a point A at coordinates (x1, y1), located within the strip, follow the same itinerary, for instance, by proceeding from the origin along a vertical road up to the horizontal road of ordinate y1, and then continuing along this latter to destination. In this case, the overall flow of users on both means of transport who, during a unit time, travel West to East across section Z of the rectangular strip situated at distances db and (1  d)b (0 o d o1) from the two vertical sides of the area, is given by the demand originating in the

Fig. 1. Computation of the vehicle flow across a section Z.

rectangle with sides db and b and whose destination is the portion of the rectangular strip of length (1  d)b to the right of section Z. Since the demands for private and public transport originating in the unit area are, respectively, ga =da/b2 and gb =db/b2, it follows that flows fa and fb, that is, the numbers of users travelling, respectively, on private and public transport who cross Z during a unit time in each of the two directions, are da dð1dÞ b b b g d dð1dÞ f b ¼ 2 b2 dð1dÞb ¼ b b fa ¼

ga

b2

b2 dð1dÞb ¼

ð1Þ

The same result would follow if the users moved from one point to the other in the urban area by first proceeding along a horizontal road and then continuing along a vertical one. Let Za and Zb be the mean numbers of passengers, respectively, a in a car and on a bus. The flow of cars across Z is f^ ¼ f a =Za ; if Zc is the car equivalent of a bus, the flow of buses across Z measured in b

equivalent cars is f^ ¼ f b Zc =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 b approximately Zb/Zc =cost= 1/Z. Therefore we have f^ ¼ f b =Z, for which the overall flow across Z in terms of the number of cars is a b f^ ¼ f^ þ f^ . In view of (1), and by putting db = d  da, we obtain  a  d ðdda Þ dð1dÞ f^ ¼ a þ ð2Þ Z Z b

Let us denote by da ðdda Þ d^ ¼ a þ

Z

Z

ð3Þ

the overall demand in the urban area, expressed as the number of equivalent car trips. From Eq. (2), we have d^ f^ ¼ dð1dÞ b

ð4Þ

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Let K be the overall capacity of all road lanes crossing Z in each direction, that is, the maximum number of cars that can traverse Z during a unit time and give rise to conditions of circulation conventionally held to be at the limits of tolerability. Setting f^ ¼ K, and accounting for Eqs. (2) and (3), we obtain the expression for the demand d that results in a traffic flow in each direction across Z equal to the capacity: d¼

K

dð1dÞ

b

ð5Þ

Under the assumption that capacity K is a function of the location of Z, and therefore of d as well, K = K(d), so that KðdÞ ¼H dð1dÞ

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. Let us thus define Hb as the capacity of the urban area and call H its capacity factor. The fact that K(d)/(d(1 d)) is constant means that the capacity K(d) of the cross sections of a rectangular strip grows with d(1 d), that is the road density increases on proceeding from the outskirts of the area toward the centre, a situation generally 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 constant throughout the entire area and is given by f^ d^ ¼ KðdÞ Hb

ð7Þ

Now let t0 be the mean travel time by car along a road stretch of unit length under conditions of free circulation; this is held to be constant throughout the entire urban area. We assume that the car travel time ta along the same road stretch when the flow across Z is f^ , is given by 2 !b 3 f^ a 04 5 t ¼ t 1þa ð8Þ KðdÞ where a and b are two parameters that depend on the geometrical and 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 is given by 2 !b 3 d^ a 04 5 t ¼ t 1þa ð9Þ Hb ^ or, by substituting expression (3) for d: !   a  b d ðdda Þ 1 þ t a ¼ t 0 1þ a a Z Z Hb

ð10Þ

Since t0 is constant throughout the entire area (as also ta for any given values of d and da), we take as a measure of the urban area congestion level CL the ratio of the increase in unit-distance travelling time under actual conditions with respect to the free flow conditions to the time spent to travel a unit distance under free flow conditions. We therefore have  b  a d ðdda Þ 1 þ ð11Þ CL ¼ a Za Z Hb We have supposed that all users travelling between the two points, O and A, of the area follow the same path of minimum length; they proceed along a vertical line originating from O and continue 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 other components of travel cost, the cost of this used path is equal to that of the unused paths with the same length connecting the

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same pair of points, and less than the cost of the unused paths of greater length. Thus the flow pattern is a Wardrop equilibrium. Now let la and lb be the lengths of the pedestrian routes taken during a trip with private and public transport, respectively, and sp the walking speed of a pedestrian. We consider all three quantities to be the same for all users at their mean values. Having assumed that trips from any origin are equally distributed in all directions, only a small fraction of bus users will have origins and destinations near the same bus line. It can thus be assumed that all bus users will need to change lines once during a trip (Newell, 1979), for which they must sustain two waits of duration w each, as well as the inconvenience of having to transfer. We reckon that such inconvenience is perceived by the 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 the arrival frequency of buses, which reach stops at regular intervals, while passengers arrive randomly, the mean waiting time at a bus stop 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 time and money that a person attributes to a trip of any given length with a given means of transport in the urban area. It is a dimensionless variable that varies randomly from one person to another, and it is given by the sum of two random fractions. The first represents that part of the cost that is a function of the measurable burdens of travel (time and money spent) and depends on each individual’s socioeconomic characteristics (sex, age, job, etc.), which are assumed to be randomly distributed in the urban population. The second random fraction has zero mean and represents the shift from the first variable due to each individual’s own perceptions of the cost of a trip, which depends on personal sensibilities. It should be noted that such a division of cost into two random fractions is similar to that used in some mixed-logit models (see e.g., Hensher and Greene, 2003). The first random fraction is in its turn the sum of two random variables: the first represents the cost of time, the second the cost of monetary expenditures. The distribution of the cost of monetary expenditures depends, as it was seen in the previous section, on the distribution of the willingness to spend. The substitution ratio between time and money measures the monetary value of time; it is a random variable whose mean is the value of time used in many problems of transport. Many methods have been proposed for its estimation, which are based on different models of discrete choice (see e.g., Gaudry et al., 1989; Hensher, 2001a, 2001b; Hess et al., 2005). However, since different values of the costs of time and of money expenditures can give rise to the same monetary value of time, it is not possible to derive the cost distributions of either time or money expenditures from the distribution of the value of time. We assume that the cost of travel time is proportional to the time spent travelling according to a random coefficient B1, and that all users perceive the ratios between the costs of the different components of the travel time in the same way, so that it is possible to measure the overall cost due to travel time as if it were due to the in-vehicle time. Following the suggestions of the English Department for Transport (Ortuzar and Willumsen, 1996, pp. 414–415), it has been assumed that the cost of a unit of time spent walking or waiting for a bus is perceived as equal to twice that actually spent on board. Thus, if l is the length of a route travelled on board, the cost of the travel time by car is   la Cta ¼ B1 t a l þ 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 r 41, in order to account for the idle time at bus stops. Therefore, the cost

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of the time spent on a bus trip of length l on board is given by   lb Ctb ¼ B1 t a lr þ t þ þ 2w sp

ð13Þ

We assume that, for each unit length of route on board, each user of public transport pays a ticket price of X1 for bus, while each user of private transport pays a price p for fuel and, when applicable, a price X2 for road pricing. The cost of monetary expenditures is proportional to these prices according to a random coefficient B2, whose probability law is independent of B1 and is a function of the distribution of the willingness to spend in 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   la þ B2 lðp þ X2 Þ þ ea C a ¼ B1 t a l þ sp

ð14Þ

  lb C b ¼ B1 t a lr þ t þ þ 2w þ B2 lX1 þ eb sp

ð15Þ

the traveller will choose the car if Ca oCb, that is, if   2la B1 t a l þ þ B2 lðp þX2 Þ þ ea sp   2lb o B1 t a lr þ t þ þ 2w þ B2 lX1 þ eb sp

Thus, the probability pa that a person chooses to travel by car is given by the probability of event (16): 2     2la 2lb 6 þ B2 lðp þ X2 Þ þ ea o B1 t a lr þ t þ þ 2w pa ¼ P4B1 t a l þ sp sp þB2 lX1 þ eb

 ð17Þ

With given values of the area size b, capacity factor H, demand d, prices p, X1, and X2, pedestrian route lengths la and lb, walking speed sp, and coefficients Za, Z, and r, and knowing the probability laws for B1, B2, ea, eb, and l, the right side of Eq. (17) is a function of da. Thus, Eq. (17) can be rewritten as pa ¼ Cðda Þ

where ea and eb are two random variables with zero mean, which measure, as mentioned before, the difference between the trip cost perceived by each person and that be ascribed to the measurable burdens of travel; ea and eb depend on each traveller’s particular sensibilities and are independent of each other and of B1 and B2. The distributions of B1 and B2 have been derived from these considerations. Generally speaking, there is a proportion of a city’s population that leads a rather inactive life style, and therefore has ample time to devote to moving about. For such people, the cost due to a unit of time spent travelling is modest, varies little from one person to the next, and its value is therefore contained within a rather restricted range. Other people, instead, lead very active lives, must often go through trip chains, and have a rather limited time budget to do so (see for example Kitamura et al., 1990; McGuckin and Murakami, 1999; Nishii et al., 1990). These people therefore attribute high cost to a unit of time spent travelling, and because such perceived cost also varies widely from person to person, it is therefore distributed over a broad range of values. Bearing such considerations in mind, it can be assumed that the distribution of B1, which is defined only for positive values, is characterized by positive skewness. The distribution of coefficient B2 depends on the distribution of the willingness to spend, and hence also that of income, within the urban population. If the willingness to spend does not differ greatly among the various segments of the population, coefficient B2, which can only take on positive values, would be distributed in quite a symmetrical way around the mean, and its values would in any event be contained within a rather limited range. If instead the willingness to spend varies widely within the population, a certain portion of the population would attribute rather low cost to the money spent for travelling, and its values would be contained within a small range, while the remaining part would perceive the cost as high, and its values for 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 from such considerations that very different distributions of B2 may exist in the real life, despite their giving rise to the same mean value. Any person intending to take an urban-area trip, whose invehicle length l varies randomly from one person to the next, chooses the means of transport perceived as less expensive. Thus,

ð16Þ

ð18Þ

Considering that at equilibrium pa is constant over time and that, for the strong law of large numbers (Feller, 1968, pp. 202– 204), da = pad, we have from Eq. (18) the following equilibrium equation: pa ¼ Cðpa dÞ ¼ Fðpa Þ

ð19Þ

From Eq. (19), it is impossible to derive a closed-form expression for paas a function of the variables on which it depends. Expression (19) can nevertheless be solved via a a simulation procedure, as follows. Given a value p^ of pa in the a a a range (0, 1), and the corresponding d^ ¼ p^ d, the value Fðp^ Þ can be calculated by the Montecarlo method. A set of five values is extracted, each independently, from the distributions of B1, B2, ea, eb, and l, and the corresponding values of the two members of expression (16) are calculated. The operation is repeated n times; if expression (16) turns out to be satisfied n1 number of times, we a ^ ðpa Þ ¼ pa Fðpa Þ is strictly convex set Fðp^ Þ ¼ n1 =n. Since function F (as can be easily verified empirically), the solution to problem (19) can be obtained by using the bisection method (see for ^ ðpa Þ example, Zangwill, 1969, p. 122) to calculate the nil point of F a ^ in the range (0, 1), by repeating the calculation of Fðp Þ for a ^ ðpa Þ deviates from zero by a quantity different values of p^ until F deemed negligible.

3. Application of the model The model has been applied to a square urban area of side b=5 km. Private transport is represented by cars, while public transport is provided by a square grid of bus lines running at a distance of 0.5 km from one another. The mean length of the pedestrian route for public transport riders, given by the sum of the mean distance from the point of departure to the starting bus stop 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 of private transport, given by the sum of the mean distance from the point of departure to the starting car park and the mean distance from the arrival park to destination, is la = 0.1 km. The mean walking speed of pedestrians is assumed to be sp = 4 km/h. The travel time under conditions of free circulation has been attributed a value of t0 = 1.9 min/km, which has been drawn from recent measurements on traffic in Central London (Transport for London, 2003a, p. 52). By convention, the road capacity has been

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assumed to be that value of car flow that yields a travel time per unit length of road twice of what would occur in free flow conditions, and therefore a congestion level of CL =1. We moreover assume that users of public transport perceive the cost of transferring between two bus lines as equivalent to a t = 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 the value a =1 is consistent with the given definition of road capacity; a in effect, when f^ ¼ KðdÞ, we have ta = 2t0. The road network in the city centre is assumed to be made up of grid of two-lane, one-way roads arranged alternately at a distance of 250 m from one another (Fig. 2). Taking into account the presence of intersections, a capacity of 1150 car/h has been attributed to each road, whence the capacity in each direction of a 1 km-wide strip turns out to be 1150/0.5=2300 car/h. Since d = 0.5 in the central urban area, Eq. (6) yields a capacity factor of H= 2300/0.25= 9200 car/(h km), which in turn gives the area capacity Hb= 46,000 car/h. We assume the expenditure for fuel for private vehicles to be p = 0.1 h/km, and assign a frequency of n = 12 bus/h for each public transport line and a bus ticket price of X1 = 0.1 h/km. With a given trip length, the in-vehicle travel time is greater in buses than in cars, due to the stops. In order to account for this, in computing the costs we consider a virtual trip length for buses that is 30% greater than the length of a car journey, thereby attributing a value of 1.3 to coefficient r in Eq. (13). We set Za = 1.4 as the mean number of travellers in a car, and we consider that, when the mean number of travellers in a bus is Zb = 50, we have Zc = 3, so that Z = Zb/Zc = 16.67. We have considered trip cost as a dimensionless quantity; thus, by measuring time in minutes and price in Euros, the dimensions of B1 and B2 are, respectively, [min  1] and [h  1]. We have measured trip costs on a scale by which the mean cost of a 1 h outlay is equal to 1; we therefore have E(B2)=1 h  1. We assume that B1 and B2 are both lognormal random variables, while both ea and eb are normal random variables with zero mean and variance Var(e)= 0.25. The probability density function of a lognormal random variable x is (see e.g., Larson, 1974, p. 386) "  2 # 1 1 lnxm pffiffiffiffiffiffi exp  fx ðxÞ ¼ ð20Þ s 2 xs 2p 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 in mind the scale assumed for the costs measurement, the value of

Fig. 2. Grid of urban streets in the central part of the area.

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E(B1) indicates that the mean cost of 1 min spent travelling is equal to the mean cost of a monetary outlay of 0.15 h. The probability density function of B1 is represented in Fig. 3, where it can be seen that B1 has a positive skewness, equal to 1.014, with a median of 0.135, quite below the mean. This means that about half the population that we are dealing with attributes a rather low value to travel time and that such value is contained within a rather narrow range, while the value ascribed to time by the other half varies across a much broader range and also reaches very high values. Two different distributions have been considered for variable B2; their probability density functions are shown in Fig. 4. The parameters 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 skewness Sk(B2)=0.778 and a median of 1, the same as the mean, while distribution 2 presents skewness Sk(B2)=6.18 and a median of 0.607, quite below the mean. Such distributions are representative of urban populations with very different socio-economic characteristics. B2 distribution 1 is representative of a population in which the perceived cost of monetary expenditures is distributed rather symmetrically around the mean and is contained within a rather narrow range of values. B2 distribution 2 is instead characteristic of a population in which the cost of monetary expenditures is distributed quite asymmetrically; half the population attributes a low cost to monetary expenditures, and such cost is contained within a very narrow range, while for the other half, such costs vary widely and also attain very high values. The maximum length of a trip in the urban area in question is 2b, while the mean length, given the grid arrangement of the road network, is 2b/3 (Newell 1979). It therefore seems reasonable that the random variable l (trip length) can, with sufficiently close approximation, be assigned a triangular distribution defined over the range (0, 2b), with a maximum value of 1/b for l =0, and a minimum of zero for l= 2b. Different values of demand d have been considered, so as to cover a wide spectrum of congestion levels in the absence of any road pricing. The model has been applied, with the given distributions of B1 and l, for each of the two B2 distributions, using the simulation technique illustrated in Section 2 above. For each assigned value a p^ of probability pa, the calculation of each of the two members of expression (16) has been repeated n= 100,000 times in order to determine n1, the number of times that inequality (16) is satisfied, a and thereby arrive at a good approximation for the value of Fðp^ Þ, given by ratio n1/n. In a first stage the model has been applied for each value of demand d and for each distribution of B2, considering different values of the road-pricing rate X2, thereby computing the congestion level CL at equilibrium corresponding to each case. It has 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 the value of rate X2 that results in a congestion level, CL= 0.9, deemed acceptable for an urban area (under the assumption that the corresponding pollution level is tolerable as well). The model has been then applied again separately for each of the two B2 distributions, by setting the pricing rate X2 that yields CL= 0.9 for each value of the demand d, with the aim of calculating the corresponding means Ba1 and Bb1 of the B1 values associated to users of private and public transport, respectively. In addition, we have calculated the means, Ca and Cb, of the trip costs incurred by car and bus users, respectively, the mean cost C of a journey for ^ C^ of the increase in the the entire population, and the ratio ðCCÞ= mean cost for the entire population due to the imposition of road pricing to the mean cost C^ in the absence of road pricing. The results of the calculations are summarized in Tables 1 and 2, which, respectively, refer to each of the two B2

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8 7

Probability density

6 5 4 3 2 1 0 0.1

0

0.2

0.3 Coefficient B1

0.4

0.5

0.6

Fig. 3. Distribution of coefficient B1.

1.8 1.6

Probability density

1.4 1

2

1.2 1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5 Coefficient B2

2

2.5

3

Fig. 4. Two distributions of coefficient B2.

Table 1 Effects of road pricing when coefficient B2 has distribution 1. d

CL

X2 (h/km)

Ba1 ðmin

64,000 66,800 69,600 72,400 75,200 78,000

0.974 1.205 1.478 1.798 2.172 2.605

0.263 0.425 0.522 0.602 0.675 0.745

0.152 0.155 0.158 0.160 0.162 0.164

1

Þ

Ca

Cb

C

C^

CC^ C^

0.066 0.080 0.087 0.093 0.097 0.101

3.484 3.912 4.110 4.237 4.324 4.373

4.114 5.188 5.622 5.930 6.152 6.322

3.496 3.998 4.272 4.482 4.654 4.792

2.678 2.898 3.158 3.463 3.819 4.232

0.305 0.380 0.353 0.294 0.219 0.132

1

Bb1 ðmin

Þ

Table 2 Effects of road pricing when coefficient B2 has distribution 2. d

CL

X2 (h/km)

Ba1 ðmin1 Þ

Bb1 ðmin1 Þ

Ca

Cb

C

C^

CC^ C^

64,000 66,800 69,600 72,400 75,200 78,000

0.974 1.205 1.478 1.798 2.172 2.605

0.160 0.346 0.484 0.616 0.765 0.894

0.151 0.153 0.155 0.156 0.158 0.159

0.090 0.105 0.111 0.114 0.117 0.119

3.024 3.300 3.455 3.571 3.637 3.692

7.513 7.423 7.368 7.378 7.328 7.314

3.102 3.565 3.857 4.101 4.293 4.460

2.667 2.887 3.147 3.453 3.809 4.222

0.163 0.235 0.226 0.188 0.127 0.056

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167

1 1

2

Road pricing (euro/km)

0.8

0.6

0.4

0.2

0 1

1.25

1.5 1.75 2 Congestion level in the absence of road pricing

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.

Increase in cost due to road pricing

0.4 0.35 1

2

0.3 0.25 0.2 0.15 0.1 0.05 0 1

1.2

1.4

1.6

1.8

2

2.2

2.4

Congestion level in the absence of road pricing 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.

distributions. Each table shows the values of the overall demand d in the urban area measured in trips per hour, the corresponding congestion level CL in the absence of road pricing, the road-pricing rate X2 necessary to obtain a congestion level of CL= 0.9, the means Ba1 and Bb1 of coefficient B1 for users of private and public transport, the mean costs Ca and Cb perceived by users of the two means of transport, the mean costs C and C^ of a journey for the entire population in the presence and absence of road pricing, ^ C. ^ respectively, and the ratioðCCÞ= To better illustrate the results obtained, Figs. 5 and 6 show the road-pricing rate X2 necessary to achieve a congestion level of ^ C^ in the mean trip cost due to the CL =0.9 and the increase ðCCÞ= imposition of this road pricing rate, as a function of the congestion level in the absence of road pricing, for each of the two distributions of coefficient B2.

4. Analysis of the results Examining the results presented in Tables 1 and 2 it can be seen that road pricing produces substantially different effects for

the two distributions of coefficient B2, that is, for different distributions of the cost that people attribute to monetary expenditures, despite their having the same mean value. In the case of B2 distribution 2, the mean value Bb1 of coefficient B1 for public transport users is always significantly higher than in the case of B2 distribution 1, particularly for the lowest values of CL. The opposite occurs for mean Ba1 for private transport users. Bearing in mind that B1 measures the cost that people attribute to 1 min spent travelling, such a result underscores the fact that when a certain number of people in the population attribute a high cost to monetary expenditures, imposing road pricing drives them to switch to public transport, despite the high perceived cost of travel time. On the contrary, people for whom travel time has a low cost continue using their cars, despite the imposition of road pricing, if they attribute scant value to the monetary expenditures. This fact can be confirmed by examining the mean trip costs, Ca and Cb, perceived by the users of the two means of transport. In the case of B2 distribution 2, Cb is considerably higher than for distribution 1, while the contrary occurs for the cost Ca for users of private transport.

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Such results have important consequences on the road-pricing rate X2 that must be imposed to achieve the desired level of congestion. As can be seen in Fig. 5, there is a critical congestion level in the absence of road pricing, at about CL=1.70, below which X2 for B2 distribution 2 turns out to be lower than for distribution 1, 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 when congestion levels in the absence of road pricing are not very high, it is sufficient to transfer a small proportion of the demand da onto public transport to reduce congestion to the desired level. In the event of B2 distribution 2, this proportion of demand is made up of those people who attribute a very high cost to monetary expenditures, and thus switch to public transport on the imposition of an even modest rate X2, despite the high cost that some of them attribute to travel time. If B2 instead exhibits distribution 1, rate X2 must be higher, because no part of the demand 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 substantial fraction of demand da must be made to switch to public transport in order to reduce congestion to the desired level. In the case of B2 distribution 2, a significant portion of the population attributes a very low cost to monetary expenditures. Since a proportion of these people must be transferred to public transport, a high X2 rate will be needed, greater than in the case of distribution 1, in which the population fraction attributing very low value to monetary expenditures is smaller. Nevertheless, although a higher X2 must be imposed in the case of distribution 2 when CL overcomes the critical value, the mean value of the trip cost for the entire population remains lower than in the case of distribution 1, as can be inferred by comparing the data in Tables 1 and 2. In effect, even if X2 is high, in the case of distribution 2 those people who continue to travel by car attribute a very low value to monetary expenditures. It follows, as can be inferred from Fig. 6, that the percentage increase in mean trip cost for the entire population due to the imposition of road pricing is considerably lower in the case of distribution 2 than in distribution 1, regardless of the congestion level in the absence of road pricing. Fig. 6 reveals that for both B2 distributions, the increase in the mean trip cost due to road pricing as a function of the congestion level in the absence of road pricing at first rises, then falls. This

can be explained by considering that with increasing CL, ever greater proportions of the demand da must be transferred to public transport by increasing the road-pricing rate X2. When the congestion level is rather low, rate X2 is also low, because all those people who value monetary expenditures highly while attributing low worth to travel time readily switch to public transport. With increasing CL it becomes necessary to increase X2, which drives those people who attribute high cost to both travel time and monetary expenditures to switch to public transport; this behaviour determines the rising mean trip cost. As CL increases further, the mean trip cost begins to decline, even if X2 increases; this is due partly to the fact that high road pricing leads to a reduction in congestion, and partly to the fact that the increase in X2 causes people with a low estimation of both the cost of monetary expenditure and travel time to switch to public transport. Fig. 7 shows the variations in the congestion level as a function of rate X2 for two different congestion levels in the absence of road pricing and for the two distributions of B2. These two congestion levels are 1.0 and 1.5, which define a range including most of the congestion levels deemed intolerable occurring in urban areas and for which the imposition of road pricing is therefore called for. By way of example, in Central London in 2002, before the application of road pricing, the mean daytime travel time was 4.2 min/km, as opposed to a time of 1.9 min/km under 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 was believed that the transport system had reached equilibrium, the mean 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 acceptable threshold for an urban area. The curves shown in Fig. 7 are third-order polynomials interpolating the calculation results with an R2 statistic of nearly 1. They are all very regular, approximately linear curves throughout a wide range of values of X2, especially for distribution 2. Analysis of the plots reveals that the relation between the roadpricing rate X2 and the associated level congestion CL for a given urban area and a given period of the day, characterized by a certain 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

1.6 1.4 1

2

Congestion level

1.2 1 0.8 0.6 0.4 0.2 0 0

0.1

0.2

0.3 0.4 Tariff of road pricing (euro/km)

0.5

0.6

0.7

Fig. 7. Relationships between the road pricing rate and the corresponding congestion level for distributions 1 and 2 of coefficient B2, and for two different situations in which the congestion level in the absence of road pricing is 1 and 1.5.

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X2 = 0. The other three can be determined by progressively increasing pricing rates, waiting a certain time period after fixing each rate in order to allow the transport system to reach equilibrium, and then measuring the resulting congestion level at the end of the period. Interpolating the points thus determined with a third-order polynomial yields the relation between X2 and CL, which enables direct computation of the rate resulting in the congestion level deemed tolerable, without having to resort to any trial-and-error procedure.

5. An example The model proposed in this paper can be used to determine a reference value for the road pricing rate that should be imposed in an urban area, e.g., one among the three values needed to the experimental computation of road pricing illustrated in the previous section. The data needed to the application of the model are the dimension of the zone to which road pricing should be imposed, the total number of cars entering and exiting from this zone during the daily charging period, the characteristics of the public transport system, the monetary cost per kilometre borne by each car driver, and the socio-economic characteristics of the populations on which the distributions of B1 and B2 coefficients depend. The actual problem is transformed into an equivalent one that satisfies the hypotheses of the model; it is supposed that the charging zone has square shape with the same area as the real one, and that the transport demand uniformly distributed in space and time during the charging period is that entering and exiting from the zone. By using the aforementioned data and the proposed model, the value of toll per kilometre is computed, and then converted into that imposed on drivers depending on the charging technique used. If tolls are collected at a cordon around the charging zone each time a car crosses the cordon, the toll is obtained by multiplying the value per kilometer by the average length of a trip inside the zone, approximately two third of the side of the square. Instead, if a driver is charged only once a day, the toll is obtained by multiplying the value per kilometer by the average length travelled by a driver during a day inside the charging zone. The tolls so obtained are only reference values.

169

The actual value to impose should be determined by an experimental method, as was said above. This method has been applied to the central area of London, for which we have a lot of data recorded before and after the charging imposition (Transport for London, 2003a, 2003b, 2004), which enable us to test the validity of the proposed method. We refer to the zone of about 22 km2 on which congestion charging was introduced in 2003; it is equivalent to a square of side 4.7 km. Cars are charged on the weekdays, from 7h30 to 18h30, when they cross the first time during a day the cordon around the charging zone. Our computations refer to an epoch in which the toll was 5 £, equivalent to 7.2 h, but the average charge paid was actually lower, because of exemptions and reduced charges for some people. The total traffic crossing the cordon during the charging period before the charging imposition was 752,000 vehicles. Considering that there are on average 1.34 persons per vehicle, the equivalent hourly passenger demand, uniformly distributed in the square area, is 91,607 passenger/h; when road pricing was imposed, a portion of this demand shifted to public transport. The value of 0.1 h/km has been attributed to the average operating monetary cost borne by car drivers. The value of time has been supposed to be 0.20 h/min, equal to that computed by Mackie (2005), considering the proportion of business travels in Central London. As in Section 3 we measure the trip cost in a scale by which the mean cost of a 1 h outlay is equal to 1. Thus a lognormal distribution has been attributed to coefficient B1, whose parameters are m = 1.70 and s = 0.45, with a mean E(B1) =0.20 and a skewness Sk(B1)= 1.014, the latter being the same as that of the distribution represented in Fig. 3. The equivalent hourly vehicle demand, supposed uniformly distributed in the square area, before the toll imposition is 68.364 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 as those in Section 3. Using the model presented in this paper, and considering the two distributions of coefficient B2 represented in Fig. 4, the congestion level in the square area equivalent to the charging zone in London has been computed for various values of the road pricing rates, ranging from 0 to 1 h/km. The relation between

1.3 1.2 1.1 1

1

2

Congestion level

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4 0.6 Road pricing (euro/km)

0.8

1

Fig. 8. Relationship between road pricing and congestion level in Central London for distributions 1 and 2 of coefficient B2.

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Table 3 Values 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)

Cam (h)

Cbm (h)

Cm (h)

Ta (min)

Tb (min)

T (min)

Ca

Cb

C

1

0 0.43 0 0.35

0.312 1.570 0.312 1.345

– 0.535 – 0.497

0.312 1.503 0.312 1.291

14.564 12.172 14.564 12.269

– 44.576 – 42.775

14.564 14.256 14.564 14.224

3.317 4.213 3.317 3.612

– 5.280 – 7.779

3.317 4.282 3.317 3.879

2

congestion level and road pricing is reported in Fig. 8 for each of the two distributions of B2. Congestion level in London after the imposition of road pricing decreased, reaching at equilibrium the value CL= 0.90. Fig. 8 shows that the tolls computed according to the model that would give rise to this congestion level are 0.43 h/km and 0.35 h/km for distributions 1 and 2 of B2, respectively. The corresponding hourly values of demand by car and by bus are, respectively, da = 85,790 passenger/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 for London, 2003b, Section 4.9). Considering that weekends and other holidays are free of charge, road pricing is imposed on 250 days per year, so that the daily charge payment is 165,000,000/ 250= 660,000 £. Since the number of car trips after the road pricing imposition in the charging zone is 630,000, the average toll paid for a trip is 660,000/630,000 =1.05 £. Considering that the average length of a trip is 2  4.70/3= 3.13 km, and the average number of passengers per car is 1.34, we have that the toll paid by a 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 for distribution 2 of coefficient B2. This result shows that the cost of monetary expenses is rather dispersed in the population travelling in Central London, with a percentage of it that assigns a rather high value to the monetary expenses. Table 3 reports some characteristics of the transport system computed 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. The characteristics that have been computed are the average monetary expenses Cam and Cbm borne by the car and the bus users, respectively, for a trip in the charging zone, along with the average monetary expense Cm borne by the overall user population, the corresponding average times Ta, Tb, and T spent for a trip by each category of users, and the overall costs Ca, Cb, and C. These numbers show that, for both B2 distributions, as a consequence of the road pricing imposition the monetary expense is greater than in the absence of road pricing for both car users and for people who moved to public transport; the trip time is less than in the absence of road pricing for car users, but it is substantially greater for people who decided to shift to public transport; the overall cost is greater than in the absence of road pricing for both user categories. From the latter result we argue that road pricing is useful from the social point of view, because it reduces the pollution due to traffic by reducing the congestion, along with the transfer of money from car drivers to society as a whole, but it increases the disutility of all people who used the car before the road pricing imposition. It is interesting to note in Table 3 that the generalized cost of users who moved to public transport is greater for distribution 2 of B2 than for distribution 1, even if both monetary expenses and the trip times are less. This result depends on the fact that in the case 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 of distribution 1.

6. Conclusions This paper has presented a theoretical model apt to evaluate the influence that a distribution of the willingness to spend in an urban population has on the values of road pricing rates that have to be imposed in order to reduce the congestion level to an acceptable threshold. The willingness to spend of persons determines the disutility, that is the cost, they attribute to the money spent anyway, and depends on their income and also on some psychological attitudes, for instance on optimism on the state of the economy. We have considered two distributions of the cost of the monetary expenditures, representative of two urban populations with very different socio-economic characteristics. Distribution 1 is representative of a population of rather homogeneous socioeconomic characteristics, in which the perceived cost of monetary expenditures is distributed rather symmetrically around the mean and is contained within a rather narrow range of values. Distribution 2 is instead representative of a population with rather dispersed socio-economic characteristics, in which the perceived costs of monetary expenditures are distributed quite asymmetrically: half the population attributes low costs to monetary expenditures, and such costs are contained within a very narrow range, while for the other half such costs vary widely and also attain very high values. We have obtained from the application of the model to a theoretical case that, even if the two distributions have the same mean, the road pricing rates X2 needed to maintain the congestion level at the desired threshold are different, and this difference depends on the level of congestion CL in the absence of road pricing. When CL is not very high, below a critical value that in the cases studied in the paper is about 1.70, X2 for distribution 2 turns out to be lower than for distribution 1, while the opposite occurs when X2 exceeds the critical value. Such a pattern can be explained by considering that, when the congestion level in the absence of road pricing is not very high, it is sufficient to transfer a small portion of car drivers onto public transport to reduce congestion to the desired level. In the event of distribution 2, this proportion of demand is made up of those people who attribute a very high cost to monetary expenditures, and thus switch to public 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 to monetary expenditures. When instead the congestion level without road pricing is large, that is above the aforementioned critical value, a substantial fraction of car drivers must be made to switch to public transport in order to reduce congestion to the desired level. In the case of distribution 2, for which the median is less than the mean, a significant portion of the population attributes a

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very low cost to monetary expenditures. Since a fraction of these people must be transferred to public transport, a high X2 rate will be needed, greater than in the case of distribution 1, in which the population fraction attributing very low values to monetary expenditures is smaller. Moreover we have seen that the X2 value needed to maintain the congestion level at a desired level gives rise to an average overall cost borne by the users of public transport that in the case of distribution 2 is higher than for distribution 1, while the contrary occurs for the cost borne by people who continue to travel by car. This is due to the fact that when a certain number of people in the population attribute a high cost to monetary expenditures (as in distribution 2), imposing road pricing drives them to switch to public transport, despite the high perceived cost of travel time. On the contrary, people for whom travel time has a low cost continue using their cars, despite the imposition of road pricing, if they attribute scant value to monetary expenditures. The consequence is that in the case of distribution 2, unlike distribution 1, there is a considerable fraction of users of public transport for whom both costs of time and money expenditures are high, so that the overall cost is high, and a considerable fraction of car users for whom both costs are low, so that the overall cost is low. The fact that the road-pricing rate that must be imposed in an urban area in order to maintain congestion within acceptable levels is heavily dependent not only on the level of previous congestion, but also on the distribution of the urban population’s willingness to spend, severely limits the reliability of any method for calculating road pricing rates based on estimates of the mean cost attributed to monetary expenditures. On the other hand, any method based on theoretical analyses must also be considered equally unreliable, given that such methods involve a large number of variables for which it is very difficult to assign realistic values. It therefore seems much more appropriate to adopt an experimental approach. This paper has shown that, when the purpose of road pricing is to maintain the pollutant concentration and the average congestion in an urban area below acceptable thresholds, we cannot consider different tolls for various streets of the town in the various periods of the day, but we have to impose a toll independently of the streets and of the time of the trip. A relation exists between the toll rate per kilometer of trip and the average congestion in the urban area, which can be obtained experimentally by successively imposing three different rates and measuring the corresponding congestion levels. The relation can then be used to directly determine the rate that results in the congestion level deemed tolerable. This paper also shows how the proposed model can be used to compute a reference value for the road pricing rate that should be imposed in an urban area, i.e. one among the three aforementioned rates. The data needed to the application of the model are the dimension of the zone to which road pricing should be imposed, the total number of cars entering and exiting from this zone during the daily charging period, the characteristics of the public transport system, the monetary cost per kilometre borne by each car driver, and the socio-economic characteristics of the populations on which the distributions of costs of time and of monetary expenses depend. The real problem is transformed into an equivalent one that satisfies the hypotheses of the model; it is supposed that the charging zone has square shape with the same area as the real one, and that transport demand uniformly distributed in space and time during the charging period is that existing and entering the zone. Using the aforementioned data and the proposed model, the value of the toll rate per kilometer is computed, and then converted to that imposed on drivers taking account of the charging technique used.

171

The model has been applied to a real world example, the central area of London, for which we have a lot of data recorded before and after the congestion charging imposition. These data enabled us to test the validity of the model. Using the two B2 distributions considered in the paper we have obtained that the rates X2 needed to maintain the congestion level at the value CL= 0.90 reached in London after the charging implementation in 2003 are X2 =0.43 h/km in the case of distribution 1 and X2 = 0.35 h/km in the case of distribution 2. The charging rate per kilometre imposed in London in 2003, computed as ratio of the average daily charge payment to the average number of kilometers travelled in the charging zone during the daily charging period, is X2 = 0.359 h/km. This value is very close to that computed by the model in the case of distribution 2; thus the model indicates that the cost of monetary expenditures is rather dispersed in the population travelling in Central London, with a portion of it that assigns a rather high value to the cost of monetary expenses. From the application of the model to the London charging area we have found that for both B2 distributions the generalized cost borne by users after the imposition of road pricing is greater than that in the absence of road pricing, for those who shift to public transport and for those who continue to travel by car as well. From this result we argue that road pricing is useful from the social point of view, because it reduces the pollution due to traffic by reducing congestion, but it increase the disutility of all people who used the car before the road pricing imposition.

Acknowledgments The author is indebted to two anonymous referees for their useful comments and suggestions on an earlier edition of this paper.

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