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Optimal cordon pricing in a non-monocentric city
Transportation Research Part A 39 (2005) 723–736 www.elsevier.com/locate/tra
Optimal cordon pricing in a non-monocentric cityq Se-il Mun a
a,*
, Ko-ji Konishi b, Kazuhiro Yoshikawa
b
Graduate School of Economics, Kyoto University, Yoshida Hon-machi, Sakyo-ku, Kyoto 606-8501, Japan b Department of Civil Engineering, Kansai University, Yamate-cho, Suita, Osaka 564-8680, Japan Received 28 October 2003; received in revised form 14 December 2004; accepted 22 February 2005
Abstract This paper examines the effect of cordon pricing based on an urban spatial model of a non-monocentric city where trips may occur between any pair of locations in the city. The model describes the spatial distribution of trip demand and traffic congestion under alternative pricing schemes. We evaluate the efficiency of resource allocation by comparing three schemes: no-toll equilibrium, first-best optimum, and optimal cordon pricing. Optimal cordon pricing is defined as a combination of cordon location and toll level that maximizes the social surplus in a city. Simulations show that cordon pricing is not always effective for congestion management: cordon pricing tends to be effective as the urban structure is more monocentric. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Recent years have seen increasing interest in cordon pricing as a policy instrument to control traffic congestion in urban areas. Implementations were successful in Singapore and three Norwegian cities (Oslo, Bergen, Trondheim).1 Policy makers in cities suffering heavy traffic q
An earlier version of this paper was presented at the Urban Economics Workshop in Tokyo, Applied Micro Economics Workshop in Kyoto, and International Symposium on the Theory and Practice of Congestion Charging in London. * Corresponding author. Tel.: +81 75 753 3447; fax: +81 75 753 3492. E-mail address: [email protected] (S.-i. Mun). 1 London began collecting congestion charges for the central area in 2003. That system is ‘‘area pricing’’, which differs slightly from cordon pricing. 0965-8564/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tra.2005.02.020
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congestion now consider cordon pricing as a promising policy alternative. This situation has encouraged many researchers to evaluate the effects of cordon pricing, and to develop methods to obtain optimal designs of the pricing system (e.g., May and Milne, 2000; Santos et al., 2000; Verhoef, 2002; Zhang and Yang, 2004). These studies were based mainly on network models. Consequently, their results depend on the network structures that are specified for the simulations examined. Furthermore, they did not address the effect of land use structure of cities. Santos et al. (2000) calculated optimal cordon tolls for eight English towns. They reported that the effects of cordon pricing differed considerably among the eight towns. These differences in effectiveness could be attributed to differences in network structures and land use patterns among the cities. It is worth examining the effects of these factors on the effectiveness of pricing policies in an idealized setting, such as continuous space models in the urban economics literature (e.g., Kanemoto, 1980; Sullivan, 1983; Fujita, 1989; Kraus, 1989; Anas and Xu, 1999). Mun et al. (2003) investigated the effect of cordon pricing based on an urban spatial model of a monocentric city. That study showed that cordon pricing attains an economic welfare level very close to the first-best optimum.2 The system works as follows: under optimal cordon pricing the urban area is divided into three zones with respect to the distance from the center. Trips from locations inside the cordon are under-priced, those just outside the cordon are over-priced, and those in the urban area fringe are under-priced. Cordon location and tolls result in a fine tuning of the trip rate in each zone to minimize the deviation of traffic volume from the first-best case. This model is based on a number of assumptions to obtain an analytical solution; these include assumptions on monocentricity, linear demand, uniform density, uniform road capacity, etc. It remains unclear how relaxing these assumptions affects the results. This paper extends the previous analysis to deal with the situation in a non-monocentric city. Our definition of a non-monocentric city is one in which trips may occur between any pair of locations in a city. This situation differs qualitatively from that of a monocentric city in which all trips are destined to the CBD.3 We develop a model to describe the spatial distribution of trip demand and traffic congestion under alternative pricing schemes. The model also allows variable density of land use and road capacity across space. We evaluate economic welfare under three schemes: a notoll equilibrium, first-best optimum, and optimal cordon pricing. Optimal cordon pricing is defined as the combination of cordon location and a toll level that maximizes the social surplus in a city. We examine the effects of urban spatial structures and various parameters on traffic patterns and the effectiveness of cordon pricing. Such information may provide useful insights for policy makers.
2
Ho et al. (2003) obtained a similar result in a more general setting, i.e., a two-dimensional continuum traffic network. Our definition differs somewhat from that of geographers or urban planners. According to their definition, the city is monocentric if it has a single peak density distribution. Our definition is consistent with the urban economic literature such as that of Solow and Vickrey (1971), Beckmann (1976), and Ogawa and Fujita (1980). While the literature in urban economics are concerned primarily with land use structure, our paper focuses on the spatial variation of traffic congestion. 3
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2. The model 2.1. No-toll equilibrium We assume a city developed on a one-dimensional space such as the long-narrow city in Solow and Vickrey (1971), in which only trips in the lengthwise direction are considered.4 Each location is represented by a coordinate value in one-dimensional space, with the center specified as its origin (see Fig. 1). Individuals with homogenous characteristics are located according to a given density R B function, n(x), of which the density is highest at the center (x = 0). The total city population is nðxÞ dx, where B and B are the right and left fringes of the city measured by the distance from B the center. Each individual makes trips to various locations. In other words, the trip destinations are not confined to the CBD, but are dispersed across urban space. Trip demand is elastic and trip frequency depends on trip costs that consist of time and toll costs. Congestion levels along the course of the trip affect time costs. Let q(x, y) denote the number of trips that an individual makes from location x to location y. The no-toll equilibrium of trip distribution is characterized as the situation in which private marginal benefit equals the trip time cost for every O–D pair: P ðqðx; yÞÞ ¼ Cðx; yÞ for all x; y;
ð1Þ
where P(q) is the inverse demand function that represents the private marginal benefit of trips, and C(x, y) is the time cost for a trip from x to y. C(x, y) is formulated as Z y QðzÞ t dz; ð2Þ Cðx; yÞ ¼ LðzÞ x where t(Q(z)/L(z)) is the time required to drive a unit distance around location z, and Q(z) and L(z) are, respectively, the traffic volume and road capacity at z. The function t(Æ) is increasing with respect to the volume–capacity ratio. Traffic volume is the sum of trips passing location z along the trip route. Z BZ z Z z Z B nðxÞqðx; yÞ dy dx þ nðxÞqðx; yÞ dy dx ð3Þ QðzÞ ¼ B
z
z
B
The first term on the RHS represents the sum of trips passing z from left to right, where the second term represents the sum of those from right to left.5
4
Although this setting is adopted for analytical convenience, it is applicable to some places in the real world (see, e.g., Tong and Wong, 1997). 5 This paper addresses traffic conditions for one day. While q(x, y) represents outward trips from x to y, the same number of trips must be made from y to x within the same day. The formulation of Eq. (3) implies that Q(z) includes both outward and homeward trips for one direction. For example, on the lane for rightward traffic, the first and second terms on the RHS of (3) represent outward and homeward trips, respectively. The opposite is true for the lane for leftward traffic. This treatment implies that our model ignores interaction between two opposing movements; the congestion level is not affected by that on the opposite lane.
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S.-i. Mun et al. / Transportation Research Part A 39 (2005) 723–736 q( x, y )
q( x, y ')
-B
y’
0
x
y
B
Fig. 1. Setting of the model.
2.2. First-best optimum The first-best optimum is attained when the marginal social benefit equals the marginal social cost for each O–D pair, as follows: P ðqðx; yÞÞ ¼ Cðx; yÞ þ Eðx; yÞ for all x; y Z y 0 QðzÞ QðzÞ dz. Eðx; yÞ ¼ t LðzÞ LðzÞ x
ð4Þ ð5Þ
In these equations, E(x, y) represents the congestion externality that an additional trip from x imposes on all drivers using the road between x and y. The first-best solution is achieved by levying tolls that are equal to congestion externalities, but implementing such a tolling scheme is not feasible because it requires a prohibitive amount of information. This case does not provide an alternative policy, but serves as a reference point for evaluating the performance of cordon pricing as a second-best policy.
3. Optimal cordon pricing in a non-monocentric city Suppose that the cordon is located at xm and xm so as to surround the central area, and that a toll equal to s is levied on each vehicle passing the cordon. The equilibrium under cordon pricing is characterized by the following condition: P ðqðx; yÞÞ ¼ Cðx; yÞ þ J ðx; yÞs
for all x; y;
ð6Þ
where J(x, y) is the number of times crossing the cordons along the route of a trip from x to y, which is defined as J ðx; yÞ ¼ jhðxÞ hðyÞj;
ð7Þ
where h(x) = 1 for B 6 x < xm, h(x) = 0 for xm 6 x < xm, and h(x) = 1 for xm 6 x < B. For some O–D pairs, such as trips from the LHS of xm to the RHS of xm, drivers must pay the toll twice.6 Optimal cordon pricing is obtained by solving the problem of maximizing the total social surplus with respect to the toll level and cordon location,
6 Because we assume a one-dimensional space, it is impossible to choose routes that avoid the toll payment, such as detours around the cordon line. This fact engenders underestimation of the congestion level around the cordon line.
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Z max xm ;s
Z
B
B
Z
nðxÞ B
B
qðx;yÞ
P ðqÞ dq Cðx; yÞqðx; yÞ dy dx;
727
ð8Þ
0
subject to equilibrium conditions (6) and (7). Considering that there are only two control variables, we obtain the optimal solution by a grid search.
4. Simulations 4.1. Specifications of functional forms and parameters The trip demand function is specified as7 qðx; yÞ ¼ amðyÞ expðbP ðx; yÞÞ;
ð9Þ
where a, b are positive constants, m(y) is the intensity of economic activities (e.g., population, employment) at location y that represents the attractiveness of a destination. P(x, y) is the full price (generalized cost) of a trip from x to y: it consists of the time cost and toll. The aggregated number of trips from x to y is obtained as an(x)m(y) exp(bP(x, y)), which is equivalent to an exponential type gravity model. The above specification implies that the elasticity of substitution between trip destinations is zero. Substitutions between destinations are described implicitly by changes in shares: trips to some destinations may be increased while trips to other destinations may be decreased in response to changes in trip costs. We obtain the marginal benefit of trips by inverting the demand function (9), as follows: 1 qðx; yÞ ð10Þ P ðqðx; yÞÞ ¼ ln b amðyÞ We estimate the parameter values using trip survey data for Osaka prefecture, Japan. Thereby, we have a = 1.48 · 107, b = 0.0312. The population density function n(x) is specified as a negative exponential form, which is widely applied in the literature (e.g., McDonald, 1989). nðxÞ ¼ d expðgjxjÞ
ð11Þ
In that equation, d = 24,500, g = 0.0854. Similarly, the attractiveness of destinations, m(y), is expressed as a density function by mðyÞ ¼ d expðcjyjÞ;
ð12Þ
where parameter values d, c are assumed to be equal to d and g, respectively for the base case. Road capacity L(x) is also specified as the exponential function LðxÞ ¼ k expðvjxjÞ.
ð13Þ
We used the ratio of road area to the land area as a proxy for road capacity. Thereby, we obtained estimates of parameters as k = 0.24 and v = 0.0446. 7
The demand function (9) is derived from individuals utility maximization. See Appendix A for the details.
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Time to travel unit distance is assumed to be linear with respect to the volume–capacity ratio.8 QðxÞ QðxÞ t ¼f þc ; ð14Þ LðxÞ LðxÞ where f = 1.2, c = 0.00000552. We estimated these values based on a similar method reported in Mun et al. (2003). In that equation, f is interpreted as the time to travel 1 km under free-flow conditions; f = 1.2 implies that the free-flow speed is 50 km/h. Finally, the distance from the center to the city fringe, B, is assumed to equal 50 km. 4.2. Results for base case We conduct numerical simulations to evaluate the relative performance of the three schemes. For the base case, it is assumed that m(y) = n(y). This setting implies that trip destinations are dispersed completely across the urban space, as discussed by Beckmann (1976). The result for the base case is summarized as follows: xm ¼ 13 km
s ¼ 19 min
SSNo toll ¼ 7.731E þ 08 min
SSCordon ¼ 8.008E þ 08 min
Gain from cordon pricing ¼
SSCordon SSNo toll ¼ 0.036 SSNo toll
Maximal gain ¼ Relative gain ¼
SSFirst best ¼ 8.397E þ 08 min
SSFirst best SSNo toll ¼ 0.086 SSNo toll
SSCordon SSNo toll ¼ 0.415 SSFirst best SSNo toll
where SSNo toll, SSCordon, and SSFirst best are the time equivalent values of social surpluses for the no-toll equilibrium, optimal cordon pricing, and first-best optimum, respectively. The optimal cordon location is 13 km from the center and the toll in time equivalent units is 19 min. The toll is 633.3 yen (or $5.81), if we adopt 2000 yen/h as the value of travel time, which is the estimate from an empirical study in Japan. The effectiveness of cordon pricing is evaluated by two indexes: ‘‘Gain from cordon pricing’’ represents improvement from the no-toll equilibrium; ‘‘Relative gain’’ represents the ratio of gain from cordon pricing to the maximal gain achievable in the first-best optimum. In the base case, cordon pricing improves the social surplus by 3.6% compared with the no-toll equilibrium, which accounts for 41.5% of the maximal gain. The relative gain of cordon pricing is much lower than that reported in Mun et al. (2003),9 which assumes a monocentric city, uniform density, linear demand, etc. 8
The linear travel time function is regarded as a reduced form. Arnott et al. (1990) showed that trip cost in a dynamic equilibrium with a bottleneck is proportional to the total number of trips during rush hours divided by the road capacity. Q(z) in (14) is compatible with the definition of that number because it represents the traffic volume for one day, as discussed in footnote 5. 9 The value of relative gain in Mun et al. (2003) is equal to 94%, as calculated from Table 2 in that paper.
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729
1600000 1400000
No toll First-best Cordon pricing
Congestion level
1200000 1000000 800000 600000 400000 200000 0 -50 -40 -30 -20 -10
0
10
20
30
40
50
Location
Fig. 2. Spatial variation of congestion levels for three schemes.
Fig. 2 shows the spatial variation of congestion levels that are represented by the volume-capacity ratio.10 That figure shows that cordon pricing lowers congestion levels around the cordon location (x = 13 and x = 13). In other locations, congestion levels under cordon pricing do not differ greatly from those under a no-toll equilibrium. This result suggests that trip-makers reduce trips that cross the cordon and increase those trips with no crossing. To examine this fact further, we take a closer look at changes in O–D trips. Table 1(a) and (b) shows differences between O–D trips under cordon pricing and those under no-toll equilibrium and the first-best optimum. In those tables, signs in each cell indicate the following. Table 1(a): sgn(q**(x, y)q*(x, y)). Table 1(b): sgn(q**(x, y)q0(x, y)). Therein, q*(x, y), q0(x, y), q**(x, y) are O–D trips from x to y under no-toll equilibrium, firstbest optimum, and optimal cordon pricing, respectively. The table shows O–D pairs between representative locations, which are selected as: 1 (the center), 6 (inner area), 12 (just inside the cordon), 14 (just outside the cordon), 25 (outer area), and 35 (far outer area).
10
Values on the vertical axis are very large because the units of traffic volume and road capacity are different. Traffic volume is measured in the number of trips per day, while the ratio of road area (road area/land area) is used as a proxy for road capacity. The ratio takes a value of 0.24 at maximum, according to (13).
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Table 1 (a) Difference in O–D trips between cordon pricing and the no-toll equilibrium and (b) differences in O–D trips between cordon pricing and the first-best optimum Panel (a) Cordoned area D O
-35
-35
Cordoned area
-25
-14
-12
-6
1
6
12
14
25
35
+
+
-
-
-
-
-
-
-
-
+
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+
+
+
+
-
-
-
+
+
+
-
-
-
+
+
-
-
-
+
-
-
-
-
-
-
+
+
-25
+
-14
+
+
-12
-
-
-
-6
-
-
-
+
1
-
-
-
+
+
6
-
-
-
+
+
+
12
-
-
-
+
+
+
+
14
-
-
-
-
-
-
-
-
25
-
-
-
-
-
-
-
-
+
35
-
-
-
-
-
-
-
-
+
+
+
Panel (b) Cordoned area
O
D -35
-35
Cordoned area
-25
-14
-12
-6
1
6
12
14
25
35
+
+
+
+
+
+
+
+
+
+
+
-
+
+
+
+
+
+
+
-
-
- +
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-
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-
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-
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-25
+
-14
+
+
-12
+
-
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-6
+
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-
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1
+
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6
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12
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35
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+ + + +
S.-i. Mun et al. / Transportation Research Part A 39 (2005) 723–736
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m(x) m(x)
n(x)
City A
n(x)
City B
Fig. 3. Difference in urban spatial structure.
Table 1(a) shows differences between cordon pricing and a no-toll equilibrium. Those differences describe the direction of changes in O–D trips when cordon pricing is introduced to an initially no-toll situation. Positive signs indicate that the number of trips under cordon pricing is greater than that under a no-toll case for corresponding O–D pairs. All O–D trips crossing the cordon are reduced by the introduction of cordon pricing, whereas those trips not crossing the cordon are increased. These changes demonstrate that trip makers switch destinations to avoid paying the toll. The sum of these individual responses reduces traffic volumes near the cordon, and they become even less than under the first-best optimum (see Fig. 2). Table 1(b) shows deviations of trips under cordon pricing from those under the first-best optimum. Positive signs show that trips are under-priced for the O–D pairs. The table shows that short trips crossing the cordon are over-priced while other trips are under-priced. This property is consistent with that obtained in the monocentric case (Mun et al., 2003). Note that those trips crossing the cordon twice are underpriced, even though they are charged twice. The cordon pricing system in this paper is unconventional in that travelers passing through the cordoned area are charged twice. Most areas with tolls (e.g., Singapore and London) charge only once. The above result suggests that charging twice is more effective because efficiency is improved by discouraging long-distance trips. Next we investigate the effects of urban structure and various parameters to determine situations in which cordon pricing performs better or worse. 4.3. Effects of urban spatial structure In our model, spatial distributions of trip generation and destination are represented by the density functions n(x) and m(x), respectively. We examine the effects of urban spatial structure via differences in the gradients of the density functions. Assuming that the gradient of m(x) becomes steeper while holding the gradient of n(x) unchanged, this change can be interpreted as indicating that the urban spatial structure becomes more monocentric (see Fig. 3). The gradients of n(x) and m(x) are represented, respectively, by parameters g and c in (11) and (12). Therefore, we consider that the city becomes more monocentric as c/g increases. At the extreme, the city is completely monocentric when all trips are destined for the CBD, as assumed by Mun et al. (2003). This is represented by the situation in which c is extremely large.
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Table 2 Effects of urban spatial structure (1): gradient of m(x) Cordon location Toll SSNo toll SSCordon SSFirst best Maximal gain Gain from cordon pricing Relative gain
Basic case
10c
50c
13 19 7.731E+08 8.008E+08 8.397E+08 0.086 0.036 0.415
3 28 3.985E+08 4.603E+08 4.864E+08 0.220 0.155 0.703
0 25 1.621E+08 1.917E+08 1.939E+08 0.196 0.182 0.931
We compute the solutions for various values of c in m(x) holding the parameters in n(x) fixed at the level in the base R B case. When c in m(x) is changed, another parameter value, d, is adjusted so that the value of B mðxÞ dx is constant. This allows a focus on the pure effects of spatial distribution while avoiding the scale effect. Table 2 and Fig. 4 show the results of simulations. Cordon pricing performs better as trip destinations become more concentrated around the center. This fact suggests that cordon pricing is effective in cities where the urban structure is close to monocentric. While people can adjust their choice of trip destinations to avoid crossing a cordon in a non-monocentric city, such responses are infeasible in a monocentric city in which all trips are destined to the center. Next we examine the overall gradient of population density. Gradients of n(x) and m(x) are changed in parallel. Table 3 shows these results. Both the gain from cordon pricing and the relative gain increase as the density gradient increases. The optimal cordon location is closer to the center as the density gradient is higher. This adjustment captures more vehicles crossing the cordon.
1 0.887
0.931
Relative gain
0.703
0.415
0
Base case
10γ
30γ
50γ
Fig. 4. Monocentricity and relative gain of cordon pricing.
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Table 3 Effects of urban spatial structure (2): overall density gradient Cordon location Toll SSNo toll SSCordon SSFirst best Maximal gain Gain from cordon pricing Relative gain
0.5g
Basic case
1.5g
23 18 7.505E+08 7.743E+08 8.093E+08 0.078 0.032 0.405
13 19 7.731E+08 8.008E+08 8.397E+08 0.086 0.036 0.415
8 19 8.119E+08 8.452E+08 8.878E+08 0.093 0.041 0.438
4.4. Effects of parameters We examine the effect of demand elasticity. For the functional form specified as (9), demand elasticity e is defined as follows: e¼
dq P ¼ bP dP q
The demand is more elastic as parameter b takes a larger value. Table 4 shows results for different values of b. The table shows that both maximal gain and gain from cordon pricing decrease with b. In other words, pricing policies are more effective as demand becomes less elastic. This result is contrary to that obtained by Santos et al. (2000). In fact, the effects of b on values of welfare gain is ambiguous. The values of welfare gain may be either increasing or decreasing with b. Moreover, those values are very small when b is extremely small or large. Therefore, it is no wonder that opposite results are obtained. In this case, the values of gain from cordon pricing change more than maximal gain. Consequently, the index of relative gain decreases with b. Table 5 shows the effects of road capacity. We examine the uniform expansion of road capacity across the city area, which is represented by increasing the value of parameter k in Eq. (12).11 Gains from pricing policies, represented by maximal gain and gain from cordon pricing, are smaller as road capacity becomes larger. This result is consistent with intuition: the welfare loss of congestion is larger as road capacity is small because the congestion level in the no-toll case is greater. The difference in values of social surplus between different road capacities represent the benefits of road expansion. We observe that the benefit of road expansion under cordon pricing is larger than that in either the no-toll equilibrium or first-best situation. Table 5 shows that benefits of road expansion from the base case to 1.5k, (50% increase) are obtained as the following: No-toll equilibrium: 8.627E+8 7.731E+8 = 0.896E+8. Cordon pricing: 8.923E+8 8.008E+8=0.916E+8. First-best optimum: 9.284E+8 8.397E+8 = 0.887E+8.
11
Note that an increase in the value of k has an identical effect to that of a decrease in c in Eq. (14).
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Table 4 Effects of demand elasticity Cordon location Toll SSNo toll SSCordon SSFirst best Maximal gain Gain from cordon pricing Relative gain
0.5b
Basic case
1.5b
10 39 2.055E+09 2.148E+09 2.247E+09 0.093 0.045 0.485
13 19 7.731E+08 8.008E+08 8.397E+08 0.086 0.036 0.415
13 12 4.270E+08 4.395E+08 4.598E+08 0.077 0.029 0.380
0.5k
Basic case
1.5k
14 21 6.313E+08 6.547E+08 6.964E+08 0.103 0.037 0.360
13 19 7.731E+08 8.008E+08 8.397E+08 0.086 0.036 0.415
11 17 8.627E+08 8.923E+08 9.284E+08 0.076 0.034 0.451
Table 5 Effects of road capacity Cordon location Toll SSNo toll SSCordon SSFirst best Maximal gain Gain from cordon pricing Relative gain
The above implies that the benefit of road expansion is enhanced by the introduction of cordon pricing. As a consequence, as shown in Table 5, relative gains from cordon pricing are larger as road capacity is larger.
5. Conclusion This paper presents a spatial model of traffic congestion in a non-monocentric city, where trips occur between any pair of locations in a city. We evaluate the economic welfare for three schemes: no-toll equilibrium, first-best optimum, and optimal cordon pricing. Numerical simulations have demonstrated that the effectiveness of cordon pricing depends on various factors, including a citys spatial structure and other parameters. The simulation results are summarized as follows. Welfare improvement from introducing cordon pricing is relatively larger when: • • • •
the urban spatial structure is close to monocentric; the density gradient is steeper; trip demand is less elastic; and road capacity is larger.
Cordon pricing is not always effective. The results obtained above suggest that cordon pricing is likely to be effective in small cities. According to the urban economics literature, smaller city size
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engenders a tendency toward a monocentric spatial structure and a steeper density gradient. Furthermore, trip elasticity tends to be higher because of the greater availability of alternative travel modes that are typically found in larger cities. We do not have sufficient empirical evidence regarding the relation between city size and road capacity. Policy makers should consider the suitability of environments in choosing whether or not to adopt cordon pricing for congestion management.
Acknowledgments We thank Takanori Ida, Tatsuhiko Nariu, Agachai Sumalee, Takatoshi Tabuchi, Hai Yang, and the participants at those conferences for valuable comments. We are also grateful to three anonymous referees for a number of detailed comments and suggestions to improve the presentation. Trip survey data were provided by the Keihanshin Transport Planning Council; Kiyoshi Kobayashi assisted in data use; Gen-en Nishimura helped with computer work. Those assistances are gratefully acknowledged. This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid for Scientific Research (No. 13630008, 13851002) and for 21st Century COE Program ‘‘Interfaces for Advanced Economic Analysis’’.
Appendix A. Demand function and utility maximization The demand function (9) is derived from the following utility function: Z Z a B mðyÞ sðx; y; jÞ sðx; y; jÞ sðx; y; jÞ ln dj dy; U ¼Xþ b B 0 a a where X represents the consumption of a composite good that does not involve trips; s(x, y, j) is the number of trips that an individual located in x makes to the jth destination located in y. The above utility function implies that individuals enjoy higher utility as they visitR various B R mðyÞ destinations. Utility maximization subject to the budget condition, I ¼ X þ B 0 P ðx; yÞsðx; y; jÞ dj dy, yields the following relation: sðx; y; jÞ ¼ a expðbP ðx; yÞÞ. The number of trips that an individual in x makes to y, q(x, y) is obtained as Z mðyÞ sðx; y; jÞ dj ¼ amðyÞ expðbP ðx; yÞÞ. qðx; yÞ ¼ 0
The above is equivalent to (9).
References Anas, A., Xu, R., 1999. Congestion, land use, and job dispersion: a general equilibrium model. Journal of Urban Economics 45, 451–473.
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Arnott, R., de Palma, A., Lindsey, R., 1990. Economics of a bottleneck. Journal of Urban Economics 27, 111–130. Beckmann, M.J., 1976. Spatial equilibrium in the dispersed city. In: Papageorgiou, G.J. (Ed.), Mathematical Land Use Theory. Lexington Books, Lexington, Mass, pp. 117–125. Fujita, M., 1989. Urban Economic Theory: Land Use and City Size. Cambridge University Press, Cambridge, New York and Melbourne. Ho, H.W., Wong, S.C., Yang, H., Loo, B.P.Y., 2003. Cordon-based congestion pricing in a continuum traffic equilibrium system. Presentation at International Conference on the Theory and Practice of Congestion Charging, London. Kanemoto, Y., 1980. Theories of Urban Externalities. North Holland. Kraus, M., 1989. The welfare gains from pricing road congestion using automatic vehicle identification and on-vehicle meters. Journal of Urban Economics 25, 261–281. May, A.D., Milne, D.S., 2000. Effects of alternative road pricing systems on network performance. Transportation Research, Part A 34, 407–436. McDonald, J.F., 1989. Econometric studies of urban population density: a survey. Journal of Urban Economics 26, 361–385. Mun, S., Konishi, K., Yoshikawa, K., 2003. Optimal cordon pricing. Journal of Urban Economics 54, 21–38. Ogawa, H., Fujita, M., 1980. Equilibrium land use patterns in a non-monocentric city. Journal of Regional Science 20, 455–475. Santos, G., Newbery, D., Rojey, L., 2000. Static vs. Demand Sensitive Models and the Estimation of Efficient Cordon Tolls: An Exercise for Eight English Towns, Paper No. 01-0246, Department of Applied Economics, University of Cambridge. Solow, R., Vickrey, W., 1971. Land use in a long narrow city. Journal of Economic Theory 3, 430–447. Sullivan, A.M., 1983. Second-best policies for congestion externalities. Journal of Urban Economics 14, 105–123. Tong, C.O., Wong, S.C., 1997. The advantages of a high density, mixed land use, linear urban development. Transportation 24, 295–307. Verhoef, E.T., 2002. Second-best congestion pricing in general networks: heuristic algorithms for finding second-best optimal toll levels and toll points. Transportation Research, Part B 36, 707–729. Zhang, X., Yang, H., 2004. The optimal cordon-based network congestion pricing problem. Transportation Research Part B 38, 517–537.
Optimal cordon pricing in a non-monocentric cityq Se-il Mun a
a,*
, Ko-ji Konishi b, Kazuhiro Yoshikawa
b
Graduate School of Economics, Kyoto University, Yoshida Hon-machi, Sakyo-ku, Kyoto 606-8501, Japan b Department of Civil Engineering, Kansai University, Yamate-cho, Suita, Osaka 564-8680, Japan Received 28 October 2003; received in revised form 14 December 2004; accepted 22 February 2005
Abstract This paper examines the effect of cordon pricing based on an urban spatial model of a non-monocentric city where trips may occur between any pair of locations in the city. The model describes the spatial distribution of trip demand and traffic congestion under alternative pricing schemes. We evaluate the efficiency of resource allocation by comparing three schemes: no-toll equilibrium, first-best optimum, and optimal cordon pricing. Optimal cordon pricing is defined as a combination of cordon location and toll level that maximizes the social surplus in a city. Simulations show that cordon pricing is not always effective for congestion management: cordon pricing tends to be effective as the urban structure is more monocentric. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Recent years have seen increasing interest in cordon pricing as a policy instrument to control traffic congestion in urban areas. Implementations were successful in Singapore and three Norwegian cities (Oslo, Bergen, Trondheim).1 Policy makers in cities suffering heavy traffic q
An earlier version of this paper was presented at the Urban Economics Workshop in Tokyo, Applied Micro Economics Workshop in Kyoto, and International Symposium on the Theory and Practice of Congestion Charging in London. * Corresponding author. Tel.: +81 75 753 3447; fax: +81 75 753 3492. E-mail address: [email protected] (S.-i. Mun). 1 London began collecting congestion charges for the central area in 2003. That system is ‘‘area pricing’’, which differs slightly from cordon pricing. 0965-8564/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tra.2005.02.020
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congestion now consider cordon pricing as a promising policy alternative. This situation has encouraged many researchers to evaluate the effects of cordon pricing, and to develop methods to obtain optimal designs of the pricing system (e.g., May and Milne, 2000; Santos et al., 2000; Verhoef, 2002; Zhang and Yang, 2004). These studies were based mainly on network models. Consequently, their results depend on the network structures that are specified for the simulations examined. Furthermore, they did not address the effect of land use structure of cities. Santos et al. (2000) calculated optimal cordon tolls for eight English towns. They reported that the effects of cordon pricing differed considerably among the eight towns. These differences in effectiveness could be attributed to differences in network structures and land use patterns among the cities. It is worth examining the effects of these factors on the effectiveness of pricing policies in an idealized setting, such as continuous space models in the urban economics literature (e.g., Kanemoto, 1980; Sullivan, 1983; Fujita, 1989; Kraus, 1989; Anas and Xu, 1999). Mun et al. (2003) investigated the effect of cordon pricing based on an urban spatial model of a monocentric city. That study showed that cordon pricing attains an economic welfare level very close to the first-best optimum.2 The system works as follows: under optimal cordon pricing the urban area is divided into three zones with respect to the distance from the center. Trips from locations inside the cordon are under-priced, those just outside the cordon are over-priced, and those in the urban area fringe are under-priced. Cordon location and tolls result in a fine tuning of the trip rate in each zone to minimize the deviation of traffic volume from the first-best case. This model is based on a number of assumptions to obtain an analytical solution; these include assumptions on monocentricity, linear demand, uniform density, uniform road capacity, etc. It remains unclear how relaxing these assumptions affects the results. This paper extends the previous analysis to deal with the situation in a non-monocentric city. Our definition of a non-monocentric city is one in which trips may occur between any pair of locations in a city. This situation differs qualitatively from that of a monocentric city in which all trips are destined to the CBD.3 We develop a model to describe the spatial distribution of trip demand and traffic congestion under alternative pricing schemes. The model also allows variable density of land use and road capacity across space. We evaluate economic welfare under three schemes: a notoll equilibrium, first-best optimum, and optimal cordon pricing. Optimal cordon pricing is defined as the combination of cordon location and a toll level that maximizes the social surplus in a city. We examine the effects of urban spatial structures and various parameters on traffic patterns and the effectiveness of cordon pricing. Such information may provide useful insights for policy makers.
2
Ho et al. (2003) obtained a similar result in a more general setting, i.e., a two-dimensional continuum traffic network. Our definition differs somewhat from that of geographers or urban planners. According to their definition, the city is monocentric if it has a single peak density distribution. Our definition is consistent with the urban economic literature such as that of Solow and Vickrey (1971), Beckmann (1976), and Ogawa and Fujita (1980). While the literature in urban economics are concerned primarily with land use structure, our paper focuses on the spatial variation of traffic congestion. 3
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2. The model 2.1. No-toll equilibrium We assume a city developed on a one-dimensional space such as the long-narrow city in Solow and Vickrey (1971), in which only trips in the lengthwise direction are considered.4 Each location is represented by a coordinate value in one-dimensional space, with the center specified as its origin (see Fig. 1). Individuals with homogenous characteristics are located according to a given density R B function, n(x), of which the density is highest at the center (x = 0). The total city population is nðxÞ dx, where B and B are the right and left fringes of the city measured by the distance from B the center. Each individual makes trips to various locations. In other words, the trip destinations are not confined to the CBD, but are dispersed across urban space. Trip demand is elastic and trip frequency depends on trip costs that consist of time and toll costs. Congestion levels along the course of the trip affect time costs. Let q(x, y) denote the number of trips that an individual makes from location x to location y. The no-toll equilibrium of trip distribution is characterized as the situation in which private marginal benefit equals the trip time cost for every O–D pair: P ðqðx; yÞÞ ¼ Cðx; yÞ for all x; y;
ð1Þ
where P(q) is the inverse demand function that represents the private marginal benefit of trips, and C(x, y) is the time cost for a trip from x to y. C(x, y) is formulated as Z y QðzÞ t dz; ð2Þ Cðx; yÞ ¼ LðzÞ x where t(Q(z)/L(z)) is the time required to drive a unit distance around location z, and Q(z) and L(z) are, respectively, the traffic volume and road capacity at z. The function t(Æ) is increasing with respect to the volume–capacity ratio. Traffic volume is the sum of trips passing location z along the trip route. Z BZ z Z z Z B nðxÞqðx; yÞ dy dx þ nðxÞqðx; yÞ dy dx ð3Þ QðzÞ ¼ B
z
z
B
The first term on the RHS represents the sum of trips passing z from left to right, where the second term represents the sum of those from right to left.5
4
Although this setting is adopted for analytical convenience, it is applicable to some places in the real world (see, e.g., Tong and Wong, 1997). 5 This paper addresses traffic conditions for one day. While q(x, y) represents outward trips from x to y, the same number of trips must be made from y to x within the same day. The formulation of Eq. (3) implies that Q(z) includes both outward and homeward trips for one direction. For example, on the lane for rightward traffic, the first and second terms on the RHS of (3) represent outward and homeward trips, respectively. The opposite is true for the lane for leftward traffic. This treatment implies that our model ignores interaction between two opposing movements; the congestion level is not affected by that on the opposite lane.
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S.-i. Mun et al. / Transportation Research Part A 39 (2005) 723–736 q( x, y )
q( x, y ')
-B
y’
0
x
y
B
Fig. 1. Setting of the model.
2.2. First-best optimum The first-best optimum is attained when the marginal social benefit equals the marginal social cost for each O–D pair, as follows: P ðqðx; yÞÞ ¼ Cðx; yÞ þ Eðx; yÞ for all x; y Z y 0 QðzÞ QðzÞ dz. Eðx; yÞ ¼ t LðzÞ LðzÞ x
ð4Þ ð5Þ
In these equations, E(x, y) represents the congestion externality that an additional trip from x imposes on all drivers using the road between x and y. The first-best solution is achieved by levying tolls that are equal to congestion externalities, but implementing such a tolling scheme is not feasible because it requires a prohibitive amount of information. This case does not provide an alternative policy, but serves as a reference point for evaluating the performance of cordon pricing as a second-best policy.
3. Optimal cordon pricing in a non-monocentric city Suppose that the cordon is located at xm and xm so as to surround the central area, and that a toll equal to s is levied on each vehicle passing the cordon. The equilibrium under cordon pricing is characterized by the following condition: P ðqðx; yÞÞ ¼ Cðx; yÞ þ J ðx; yÞs
for all x; y;
ð6Þ
where J(x, y) is the number of times crossing the cordons along the route of a trip from x to y, which is defined as J ðx; yÞ ¼ jhðxÞ hðyÞj;
ð7Þ
where h(x) = 1 for B 6 x < xm, h(x) = 0 for xm 6 x < xm, and h(x) = 1 for xm 6 x < B. For some O–D pairs, such as trips from the LHS of xm to the RHS of xm, drivers must pay the toll twice.6 Optimal cordon pricing is obtained by solving the problem of maximizing the total social surplus with respect to the toll level and cordon location,
6 Because we assume a one-dimensional space, it is impossible to choose routes that avoid the toll payment, such as detours around the cordon line. This fact engenders underestimation of the congestion level around the cordon line.
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Z max xm ;s
Z
B
B
Z
nðxÞ B
B
qðx;yÞ
P ðqÞ dq Cðx; yÞqðx; yÞ dy dx;
727
ð8Þ
0
subject to equilibrium conditions (6) and (7). Considering that there are only two control variables, we obtain the optimal solution by a grid search.
4. Simulations 4.1. Specifications of functional forms and parameters The trip demand function is specified as7 qðx; yÞ ¼ amðyÞ expðbP ðx; yÞÞ;
ð9Þ
where a, b are positive constants, m(y) is the intensity of economic activities (e.g., population, employment) at location y that represents the attractiveness of a destination. P(x, y) is the full price (generalized cost) of a trip from x to y: it consists of the time cost and toll. The aggregated number of trips from x to y is obtained as an(x)m(y) exp(bP(x, y)), which is equivalent to an exponential type gravity model. The above specification implies that the elasticity of substitution between trip destinations is zero. Substitutions between destinations are described implicitly by changes in shares: trips to some destinations may be increased while trips to other destinations may be decreased in response to changes in trip costs. We obtain the marginal benefit of trips by inverting the demand function (9), as follows: 1 qðx; yÞ ð10Þ P ðqðx; yÞÞ ¼ ln b amðyÞ We estimate the parameter values using trip survey data for Osaka prefecture, Japan. Thereby, we have a = 1.48 · 107, b = 0.0312. The population density function n(x) is specified as a negative exponential form, which is widely applied in the literature (e.g., McDonald, 1989). nðxÞ ¼ d expðgjxjÞ
ð11Þ
In that equation, d = 24,500, g = 0.0854. Similarly, the attractiveness of destinations, m(y), is expressed as a density function by mðyÞ ¼ d expðcjyjÞ;
ð12Þ
where parameter values d, c are assumed to be equal to d and g, respectively for the base case. Road capacity L(x) is also specified as the exponential function LðxÞ ¼ k expðvjxjÞ.
ð13Þ
We used the ratio of road area to the land area as a proxy for road capacity. Thereby, we obtained estimates of parameters as k = 0.24 and v = 0.0446. 7
The demand function (9) is derived from individuals utility maximization. See Appendix A for the details.
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Time to travel unit distance is assumed to be linear with respect to the volume–capacity ratio.8 QðxÞ QðxÞ t ¼f þc ; ð14Þ LðxÞ LðxÞ where f = 1.2, c = 0.00000552. We estimated these values based on a similar method reported in Mun et al. (2003). In that equation, f is interpreted as the time to travel 1 km under free-flow conditions; f = 1.2 implies that the free-flow speed is 50 km/h. Finally, the distance from the center to the city fringe, B, is assumed to equal 50 km. 4.2. Results for base case We conduct numerical simulations to evaluate the relative performance of the three schemes. For the base case, it is assumed that m(y) = n(y). This setting implies that trip destinations are dispersed completely across the urban space, as discussed by Beckmann (1976). The result for the base case is summarized as follows: xm ¼ 13 km
s ¼ 19 min
SSNo toll ¼ 7.731E þ 08 min
SSCordon ¼ 8.008E þ 08 min
Gain from cordon pricing ¼
SSCordon SSNo toll ¼ 0.036 SSNo toll
Maximal gain ¼ Relative gain ¼
SSFirst best ¼ 8.397E þ 08 min
SSFirst best SSNo toll ¼ 0.086 SSNo toll
SSCordon SSNo toll ¼ 0.415 SSFirst best SSNo toll
where SSNo toll, SSCordon, and SSFirst best are the time equivalent values of social surpluses for the no-toll equilibrium, optimal cordon pricing, and first-best optimum, respectively. The optimal cordon location is 13 km from the center and the toll in time equivalent units is 19 min. The toll is 633.3 yen (or $5.81), if we adopt 2000 yen/h as the value of travel time, which is the estimate from an empirical study in Japan. The effectiveness of cordon pricing is evaluated by two indexes: ‘‘Gain from cordon pricing’’ represents improvement from the no-toll equilibrium; ‘‘Relative gain’’ represents the ratio of gain from cordon pricing to the maximal gain achievable in the first-best optimum. In the base case, cordon pricing improves the social surplus by 3.6% compared with the no-toll equilibrium, which accounts for 41.5% of the maximal gain. The relative gain of cordon pricing is much lower than that reported in Mun et al. (2003),9 which assumes a monocentric city, uniform density, linear demand, etc. 8
The linear travel time function is regarded as a reduced form. Arnott et al. (1990) showed that trip cost in a dynamic equilibrium with a bottleneck is proportional to the total number of trips during rush hours divided by the road capacity. Q(z) in (14) is compatible with the definition of that number because it represents the traffic volume for one day, as discussed in footnote 5. 9 The value of relative gain in Mun et al. (2003) is equal to 94%, as calculated from Table 2 in that paper.
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729
1600000 1400000
No toll First-best Cordon pricing
Congestion level
1200000 1000000 800000 600000 400000 200000 0 -50 -40 -30 -20 -10
0
10
20
30
40
50
Location
Fig. 2. Spatial variation of congestion levels for three schemes.
Fig. 2 shows the spatial variation of congestion levels that are represented by the volume-capacity ratio.10 That figure shows that cordon pricing lowers congestion levels around the cordon location (x = 13 and x = 13). In other locations, congestion levels under cordon pricing do not differ greatly from those under a no-toll equilibrium. This result suggests that trip-makers reduce trips that cross the cordon and increase those trips with no crossing. To examine this fact further, we take a closer look at changes in O–D trips. Table 1(a) and (b) shows differences between O–D trips under cordon pricing and those under no-toll equilibrium and the first-best optimum. In those tables, signs in each cell indicate the following. Table 1(a): sgn(q**(x, y)q*(x, y)). Table 1(b): sgn(q**(x, y)q0(x, y)). Therein, q*(x, y), q0(x, y), q**(x, y) are O–D trips from x to y under no-toll equilibrium, firstbest optimum, and optimal cordon pricing, respectively. The table shows O–D pairs between representative locations, which are selected as: 1 (the center), 6 (inner area), 12 (just inside the cordon), 14 (just outside the cordon), 25 (outer area), and 35 (far outer area).
10
Values on the vertical axis are very large because the units of traffic volume and road capacity are different. Traffic volume is measured in the number of trips per day, while the ratio of road area (road area/land area) is used as a proxy for road capacity. The ratio takes a value of 0.24 at maximum, according to (13).
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Table 1 (a) Difference in O–D trips between cordon pricing and the no-toll equilibrium and (b) differences in O–D trips between cordon pricing and the first-best optimum Panel (a) Cordoned area D O
-35
-35
Cordoned area
-25
-14
-12
-6
1
6
12
14
25
35
+
+
-
-
-
-
-
-
-
-
+
-
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-
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+
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+
+
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-
+
-
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-
-
-
-
+
+
-25
+
-14
+
+
-12
-
-
-
-6
-
-
-
+
1
-
-
-
+
+
6
-
-
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+
+
+
12
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-
-
+
+
+
+
14
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25
-
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-
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-
-
-
+
35
-
-
-
-
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-
-
-
+
+
+
Panel (b) Cordoned area
O
D -35
-35
Cordoned area
-25
-14
-12
-6
1
6
12
14
25
35
+
+
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S.-i. Mun et al. / Transportation Research Part A 39 (2005) 723–736
731
m(x) m(x)
n(x)
City A
n(x)
City B
Fig. 3. Difference in urban spatial structure.
Table 1(a) shows differences between cordon pricing and a no-toll equilibrium. Those differences describe the direction of changes in O–D trips when cordon pricing is introduced to an initially no-toll situation. Positive signs indicate that the number of trips under cordon pricing is greater than that under a no-toll case for corresponding O–D pairs. All O–D trips crossing the cordon are reduced by the introduction of cordon pricing, whereas those trips not crossing the cordon are increased. These changes demonstrate that trip makers switch destinations to avoid paying the toll. The sum of these individual responses reduces traffic volumes near the cordon, and they become even less than under the first-best optimum (see Fig. 2). Table 1(b) shows deviations of trips under cordon pricing from those under the first-best optimum. Positive signs show that trips are under-priced for the O–D pairs. The table shows that short trips crossing the cordon are over-priced while other trips are under-priced. This property is consistent with that obtained in the monocentric case (Mun et al., 2003). Note that those trips crossing the cordon twice are underpriced, even though they are charged twice. The cordon pricing system in this paper is unconventional in that travelers passing through the cordoned area are charged twice. Most areas with tolls (e.g., Singapore and London) charge only once. The above result suggests that charging twice is more effective because efficiency is improved by discouraging long-distance trips. Next we investigate the effects of urban structure and various parameters to determine situations in which cordon pricing performs better or worse. 4.3. Effects of urban spatial structure In our model, spatial distributions of trip generation and destination are represented by the density functions n(x) and m(x), respectively. We examine the effects of urban spatial structure via differences in the gradients of the density functions. Assuming that the gradient of m(x) becomes steeper while holding the gradient of n(x) unchanged, this change can be interpreted as indicating that the urban spatial structure becomes more monocentric (see Fig. 3). The gradients of n(x) and m(x) are represented, respectively, by parameters g and c in (11) and (12). Therefore, we consider that the city becomes more monocentric as c/g increases. At the extreme, the city is completely monocentric when all trips are destined for the CBD, as assumed by Mun et al. (2003). This is represented by the situation in which c is extremely large.
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Table 2 Effects of urban spatial structure (1): gradient of m(x) Cordon location Toll SSNo toll SSCordon SSFirst best Maximal gain Gain from cordon pricing Relative gain
Basic case
10c
50c
13 19 7.731E+08 8.008E+08 8.397E+08 0.086 0.036 0.415
3 28 3.985E+08 4.603E+08 4.864E+08 0.220 0.155 0.703
0 25 1.621E+08 1.917E+08 1.939E+08 0.196 0.182 0.931
We compute the solutions for various values of c in m(x) holding the parameters in n(x) fixed at the level in the base R B case. When c in m(x) is changed, another parameter value, d, is adjusted so that the value of B mðxÞ dx is constant. This allows a focus on the pure effects of spatial distribution while avoiding the scale effect. Table 2 and Fig. 4 show the results of simulations. Cordon pricing performs better as trip destinations become more concentrated around the center. This fact suggests that cordon pricing is effective in cities where the urban structure is close to monocentric. While people can adjust their choice of trip destinations to avoid crossing a cordon in a non-monocentric city, such responses are infeasible in a monocentric city in which all trips are destined to the center. Next we examine the overall gradient of population density. Gradients of n(x) and m(x) are changed in parallel. Table 3 shows these results. Both the gain from cordon pricing and the relative gain increase as the density gradient increases. The optimal cordon location is closer to the center as the density gradient is higher. This adjustment captures more vehicles crossing the cordon.
1 0.887
0.931
Relative gain
0.703
0.415
0
Base case
10γ
30γ
50γ
Fig. 4. Monocentricity and relative gain of cordon pricing.
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733
Table 3 Effects of urban spatial structure (2): overall density gradient Cordon location Toll SSNo toll SSCordon SSFirst best Maximal gain Gain from cordon pricing Relative gain
0.5g
Basic case
1.5g
23 18 7.505E+08 7.743E+08 8.093E+08 0.078 0.032 0.405
13 19 7.731E+08 8.008E+08 8.397E+08 0.086 0.036 0.415
8 19 8.119E+08 8.452E+08 8.878E+08 0.093 0.041 0.438
4.4. Effects of parameters We examine the effect of demand elasticity. For the functional form specified as (9), demand elasticity e is defined as follows: e¼
dq P ¼ bP dP q
The demand is more elastic as parameter b takes a larger value. Table 4 shows results for different values of b. The table shows that both maximal gain and gain from cordon pricing decrease with b. In other words, pricing policies are more effective as demand becomes less elastic. This result is contrary to that obtained by Santos et al. (2000). In fact, the effects of b on values of welfare gain is ambiguous. The values of welfare gain may be either increasing or decreasing with b. Moreover, those values are very small when b is extremely small or large. Therefore, it is no wonder that opposite results are obtained. In this case, the values of gain from cordon pricing change more than maximal gain. Consequently, the index of relative gain decreases with b. Table 5 shows the effects of road capacity. We examine the uniform expansion of road capacity across the city area, which is represented by increasing the value of parameter k in Eq. (12).11 Gains from pricing policies, represented by maximal gain and gain from cordon pricing, are smaller as road capacity becomes larger. This result is consistent with intuition: the welfare loss of congestion is larger as road capacity is small because the congestion level in the no-toll case is greater. The difference in values of social surplus between different road capacities represent the benefits of road expansion. We observe that the benefit of road expansion under cordon pricing is larger than that in either the no-toll equilibrium or first-best situation. Table 5 shows that benefits of road expansion from the base case to 1.5k, (50% increase) are obtained as the following: No-toll equilibrium: 8.627E+8 7.731E+8 = 0.896E+8. Cordon pricing: 8.923E+8 8.008E+8=0.916E+8. First-best optimum: 9.284E+8 8.397E+8 = 0.887E+8.
11
Note that an increase in the value of k has an identical effect to that of a decrease in c in Eq. (14).
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Table 4 Effects of demand elasticity Cordon location Toll SSNo toll SSCordon SSFirst best Maximal gain Gain from cordon pricing Relative gain
0.5b
Basic case
1.5b
10 39 2.055E+09 2.148E+09 2.247E+09 0.093 0.045 0.485
13 19 7.731E+08 8.008E+08 8.397E+08 0.086 0.036 0.415
13 12 4.270E+08 4.395E+08 4.598E+08 0.077 0.029 0.380
0.5k
Basic case
1.5k
14 21 6.313E+08 6.547E+08 6.964E+08 0.103 0.037 0.360
13 19 7.731E+08 8.008E+08 8.397E+08 0.086 0.036 0.415
11 17 8.627E+08 8.923E+08 9.284E+08 0.076 0.034 0.451
Table 5 Effects of road capacity Cordon location Toll SSNo toll SSCordon SSFirst best Maximal gain Gain from cordon pricing Relative gain
The above implies that the benefit of road expansion is enhanced by the introduction of cordon pricing. As a consequence, as shown in Table 5, relative gains from cordon pricing are larger as road capacity is larger.
5. Conclusion This paper presents a spatial model of traffic congestion in a non-monocentric city, where trips occur between any pair of locations in a city. We evaluate the economic welfare for three schemes: no-toll equilibrium, first-best optimum, and optimal cordon pricing. Numerical simulations have demonstrated that the effectiveness of cordon pricing depends on various factors, including a citys spatial structure and other parameters. The simulation results are summarized as follows. Welfare improvement from introducing cordon pricing is relatively larger when: • • • •
the urban spatial structure is close to monocentric; the density gradient is steeper; trip demand is less elastic; and road capacity is larger.
Cordon pricing is not always effective. The results obtained above suggest that cordon pricing is likely to be effective in small cities. According to the urban economics literature, smaller city size
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engenders a tendency toward a monocentric spatial structure and a steeper density gradient. Furthermore, trip elasticity tends to be higher because of the greater availability of alternative travel modes that are typically found in larger cities. We do not have sufficient empirical evidence regarding the relation between city size and road capacity. Policy makers should consider the suitability of environments in choosing whether or not to adopt cordon pricing for congestion management.
Acknowledgments We thank Takanori Ida, Tatsuhiko Nariu, Agachai Sumalee, Takatoshi Tabuchi, Hai Yang, and the participants at those conferences for valuable comments. We are also grateful to three anonymous referees for a number of detailed comments and suggestions to improve the presentation. Trip survey data were provided by the Keihanshin Transport Planning Council; Kiyoshi Kobayashi assisted in data use; Gen-en Nishimura helped with computer work. Those assistances are gratefully acknowledged. This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Grant-in-Aid for Scientific Research (No. 13630008, 13851002) and for 21st Century COE Program ‘‘Interfaces for Advanced Economic Analysis’’.
Appendix A. Demand function and utility maximization The demand function (9) is derived from the following utility function: Z Z a B mðyÞ sðx; y; jÞ sðx; y; jÞ sðx; y; jÞ ln dj dy; U ¼Xþ b B 0 a a where X represents the consumption of a composite good that does not involve trips; s(x, y, j) is the number of trips that an individual located in x makes to the jth destination located in y. The above utility function implies that individuals enjoy higher utility as they visitR various B R mðyÞ destinations. Utility maximization subject to the budget condition, I ¼ X þ B 0 P ðx; yÞsðx; y; jÞ dj dy, yields the following relation: sðx; y; jÞ ¼ a expðbP ðx; yÞÞ. The number of trips that an individual in x makes to y, q(x, y) is obtained as Z mðyÞ sðx; y; jÞ dj ¼ amðyÞ expðbP ðx; yÞÞ. qðx; yÞ ¼ 0
The above is equivalent to (9).
References Anas, A., Xu, R., 1999. Congestion, land use, and job dispersion: a general equilibrium model. Journal of Urban Economics 45, 451–473.
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Arnott, R., de Palma, A., Lindsey, R., 1990. Economics of a bottleneck. Journal of Urban Economics 27, 111–130. Beckmann, M.J., 1976. Spatial equilibrium in the dispersed city. In: Papageorgiou, G.J. (Ed.), Mathematical Land Use Theory. Lexington Books, Lexington, Mass, pp. 117–125. Fujita, M., 1989. Urban Economic Theory: Land Use and City Size. Cambridge University Press, Cambridge, New York and Melbourne. Ho, H.W., Wong, S.C., Yang, H., Loo, B.P.Y., 2003. Cordon-based congestion pricing in a continuum traffic equilibrium system. Presentation at International Conference on the Theory and Practice of Congestion Charging, London. Kanemoto, Y., 1980. Theories of Urban Externalities. North Holland. Kraus, M., 1989. The welfare gains from pricing road congestion using automatic vehicle identification and on-vehicle meters. Journal of Urban Economics 25, 261–281. May, A.D., Milne, D.S., 2000. Effects of alternative road pricing systems on network performance. Transportation Research, Part A 34, 407–436. McDonald, J.F., 1989. Econometric studies of urban population density: a survey. Journal of Urban Economics 26, 361–385. Mun, S., Konishi, K., Yoshikawa, K., 2003. Optimal cordon pricing. Journal of Urban Economics 54, 21–38. Ogawa, H., Fujita, M., 1980. Equilibrium land use patterns in a non-monocentric city. Journal of Regional Science 20, 455–475. Santos, G., Newbery, D., Rojey, L., 2000. Static vs. Demand Sensitive Models and the Estimation of Efficient Cordon Tolls: An Exercise for Eight English Towns, Paper No. 01-0246, Department of Applied Economics, University of Cambridge. Solow, R., Vickrey, W., 1971. Land use in a long narrow city. Journal of Economic Theory 3, 430–447. Sullivan, A.M., 1983. Second-best policies for congestion externalities. Journal of Urban Economics 14, 105–123. Tong, C.O., Wong, S.C., 1997. The advantages of a high density, mixed land use, linear urban development. Transportation 24, 295–307. Verhoef, E.T., 2002. Second-best congestion pricing in general networks: heuristic algorithms for finding second-best optimal toll levels and toll points. Transportation Research, Part B 36, 707–729. Zhang, X., Yang, H., 2004. The optimal cordon-based network congestion pricing problem. Transportation Research Part B 38, 517–537.