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Highway congestion and congestion tolls
JOURNAL
OF URBAN
ECONOMICS
4, 340.-359 (1977)
Highway Congestion and Congestion Tolls’ ANTHONYE. BOARDMAN School of Public and Urban Policy, University of Pennsylvania, Philadelphia, Pennsylvania I9104
LESTER B. LAVE Graduate School of Industrial Administration, Carnegie-MelIon University, Pittsburgh, PennsyIvania 15213
Received October 1, 1975 Highway congestion is ubiquitous. We model the speed-flow relationship, identifying private and social costs, and the implied congestion toll for a number of proposed formulations. Using data for a limited access highway, we estimate these speed-flow relationships and find that flow as a quadratic function of speed fits best. The unit of observation is the individual vehicle and flow is measured in terms of the vehicles which passed a point shortly before or after the reference vehicle. Maximum possible flow occurs at 30-35 mph and the congestion toll is infinite for slower speeds. Rush-hour drivers have greater speeds for a given volume of traffic and thus congestion tolls must vary by time of day.
1. INTRODUCTION Congestion may be the most common nonpecuniary externality. Congestion has been studied for inland waterways [I 11, airports [15], and highways [I, 8, 10, 16, 18-22, 24-261. In this paper we focus on highway congestion. Since the speed at which a person drives on a highway decreasesas traffic flow increases, an additional driver will lower the speeds of other users. Reduced speedsincrease total travel time. The value of this additional time reflects the congestion imposed by the additional user. Economic theory dictates that the user act as if he paid the social marginal cost, which is greater than the private cost. Many authors have suggested fascinating methods for collecting a congestion tax.2 The main problem, however, is 1We thank William M. Whitby of the Office of Research and Development, Federal Highway Administration for supplying the data for this analysis. 2 See, for example, [4, 171. 340 Copyright @ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
ISSN 0094-l 190
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to determine the correct tax. In this paper we develop a model of social costs which we combine with an estimated speed-volume relationship to determine the optimal congestion toll. 2. A MODEL OF SOCIAL COSTS Commuters call upon a transportation network for quick and cheap mobility. The output of a mode equals the number of commuters serviced per unit time. In order to dispense with all unnecessary complications, we assumeq vehicles are uniformly spaced over 1 hr, one person occupies each vehicle, and all have the same origin and the same destination. Each vehicle takes a time of 6/v (distance divided by speed) hours to reach its destination. We assumethat all costs (variable vehicle running costs plus travel time costs) can be put on an hourly basis ($c/hr). Thus, the cost to one driver of reaching his destination is $c@/v). The exposition follows more clearly if we assume distance, 6, equals unity. Thus, a traveler experiences an average cost of $c/v per mile. The total cost per mile, T, of q vehicles reaching their destination equals the average cost multiplied by the flow: T = qc,‘v.
If an additional vehicle usesthe highway, the increase in total cost equals the social marginal cost, SMC, where qc dv
SMC=dT=!---, 4 ZZ
v
v2 dq
YC- qc(dv/dq) v2
*
If an individual takes to the highway he experiences a (private) cost of $c/v per mile. Society as a whole experiencesthe social marginal cost, SMC, which exceedsthe private cost. In order to internalize this cost, the individual must behave as if he faces a cost equal to the marginal social cost, rather than the private cost. One way of making an individual behave in the desired fashion is to levy a “congestion toll,” 7, equal to the difference between the social and private costs: 7 = SMC - private costs, 7 = SMC - (c/v), (3) -qc dv 7=----20 v2 &
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TABLE
1
An Example of Congestion Tolls Flow, q (vehicles/hr)
Speed, v (mph)
0 500
104.335 100.779 97.019 93.014 88.709 84.025 78.840 72.946 65.938 56.746 50.146 35.339
1000 1500 2OcKl 2500 3cKlO 3500 4ooo 4500 4750 4979.3
Private cost, $6/v ($/vehicle-mile) 0.0575 0.0595 0.0618 0.0645 0.0676 0.0714 0.0761 0.0823 0.0910 0.1057 0.1197 0.1698
Social marginal cost, SMC ($/vehicle-mile)
Congestion toll, 7 ($/vehicle-mile)
0.0575 0.0617 0.0668 0.0731 0.0813 0.0923 0.1079 0.1324 0.1772 0.2930 0.4855 m
0.0 0.0022 0.0049 0.0086 0.0137 0.0209 0.0319 0.0502 0.0862 0.1872 0.3659 co
Assume that variable running costs and travel time costs amount to $6 per hour. Also consider the following quadratic speed-flow relationship: q=a+bv+d+, which is estimated by
(4)
q = 3673. + 73.93~ - 1.0469. The various relationships are shown in Table 1. One measure of highway output is the number of vehicles served multiplied by the time it takes each to go a given distance: qv. This output rises as flow increasesfrom zero, but reachesa maximum at about 4050 vehicles per hour, which is less than the maximum flow of 4979.3 vehicles3per hour. Another way of looking at this phenomenon is to note that when only one traveler uses the highway he pays no toll. Since nobody else uses the highway, the first user has no other user’s speed to affect. For flows of less than 2000 vehicles per hour, the toll is small. As the flow increases,the SMC rises faster than the private cost and the toll rises accordingly. In practice, the toll would not become large. Travelers faced with high tolls would look for transport alternatives, such as off-peak travel, car pools, or public transport. The total revenue from the toll amounts to $rq per mile, for a constant flow at q vehicles per hour. This revenue may exceed, equal, or fall short of road-building and maintenance costs 1221. Widening the highway will lead to less congestion, higher speeds, and lower tolls. A highway should be 3This figure is higher than the generally accepted capacity flow of between 3800 and 4400 vehicles per hour (see [3]).
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widened if the value of the savedtime equals or exceedsthe total costs (private plus social) of expansion and maintenance. A congestion toll is unconventional in that it is highest before a construction program and is likely to fall drastically after completion of the construction. 3. SPEED-FLOW FORMULATIONS The model of social costsrequires knowledge of (a) variable vehicle running costs plus travel time costs and (b) the relationship between speed and flow. Readers may recompute easily the estimated, optimal congestion tolls (Table 1) for different cost figures. Knowledge of the speed-flow relationship is much more fundamental and controversial. The remainder of this paper concerns primarily the determination of the best relationship among speed, flow, and density. This section considers the relationship between speed and flow. Later sections consider the relationships between speedand density, and between flow and density. Some formulations include road width. Our work applies only to highways with two lanes in each direction. The most common, simple formulation used by Kraft and Wohl [9] and many others, assumes a linear relationship between speed and flow, thus: v -t aq = b.
(5)
Smeed [20] gives the two following formulations and claims that both equations fit his data equally well except for speedsunder 10 mph: q + bv2 = a,
(6)
q + bv3 = a.
(7)
Vickrey [24] provides an alternative formulation : a + bq” = l/v.
(8)
Vickrey estimates that k equals 4.5 for the Holland Tunnel. Figure 1 shows the graphs of these speed-flow relationships. Each of the speed-flow relationship has an associated toll that depends only on the flow. Assuming unit distance and unit cost per hour, the tolls follow for Eqs. (4)-(g), respectively:4 7 = -(a+bv+W (b + 2dv)v2
’
b-v 7-=-y V2
4 Usually, the formulas include q as the argument, not v. We use v because it facilitates the exposition and there is a 1: 1 transformation between q and v (in the relevant range).
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L
1-1 Boardman
and Lave
l-3
AND LAVE
\\\\
q
Smeed
“ix
#l
q
“L 1-4
Smeed
X2
q
1-5
Vickrey
4
FIG. 1. Graphical representations of five speed-flow formulations, Eqs. (4)-(8), respectively.
a - bv2 -Z-=-j 2bv3
(11)
a - bv3 7=-J
3bv4
T = (k/v) - ak.
(12) (13)
4. LIMITATIONS OF PREVIOUS EMPIRICAL STUDIES Previous studies observe a highway during some period and then divide the data into time intervals of about 10 min. Flow equals the number of vehicles that pass the observation point during the time interval. Instead of using individual vehicle speed, these studies use an average speed, which may be an arithmetic mean or harmonic mean of all the vehicles that passed during the interval. This procedure results in high R2, about 0.9, almost by definition. However, aggregation is likely to obscure the underlying relationship. To estimate the speed-flow relationship in greater detail, particularly to discriminate among various proposals, we treat the individual vehicle as the unit of analysis.
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When the focus is on the individual vehicle, individual vehicle speedis the relevant datum. Not so with flow; several alternative measuresseemrelevant, depending on how a driver perceives and reacts to flow. The vehicles in front of and behind a vehicle influence its speed (those in front may slow it directly, while those behind may increase congestion and slow it indirectly). But, how far in front and in back do vehicles still exert an influence ? Does a vehicle which passed the same point half an hour before influence a current driver? Having chosen the individual vehicle as the unit of observation, we must consider these questions explicitly. Previous research assumedimplicitly that drivers have identical preferences and habits. However, casual observation suggests that Sunday afternoon drivers travel slowly, independent of flow. Commuters are more experienced at driving under conditions of high flow than people who miss the rush-hour traffic. This experience probably translates into higher speed, for the same number of vehicles per lane mile. Thus, the speed-flow curve may shift depending on the time of day. We will test the stability of speed-flow relationships as a function of time to seewhat has been ignored as a consequence of assuming that drivers are homogeneous. In addition to differences in experience with driving under conditions of high flow, other differences characterize drivers. Independent of experience, some drivers desire and do drive faster than others under identical conditions. Although important, such preferences are difficult to measure. We can, at least, distinguish between drivers in the fast lane and drivers in the slow lane. Finally, one must remember speed, flow, and density are related by the expression D = q/v. Perhaps speed depends more meaningfully on density rather than flow. Density should be calculated independently of speed, for example, by an aerial photograph.5 By necessity, we estimate density by dividing volume by speed. For such a ratio, an estimated speed-density relationship has speed as the dependent variable and as the denominator of an explanatory variable; thus, the estimated R2 is inflated. The surrogate measure of density and inflated R2 should be kept in mind in interpreting our estimated speed-density relationships. 5. THE DATA Lerch [12] observed speed for over 10,000 southbound vehicles on I 495 near 170s on August 20, 1968, from 3:48 PM until 6:30 PM. The road was dry and visibility was good; there were no entrances or exits near the observation point to affect speed or flow. For each vehicle, Lerch recorded the time a vehicle passedthe observation point, in thousandths of an hour, individual vehicle speed,and whether the vehicle traveled in the slow or fast lane. Speed 5See [14].
BOARDMAN
346 245
-
275
-
305
-
AND LAVE
0 &O 00 +z 0
;395 I3 z ‘425
iy sfp
0
455
8
8 8,
485
8
515
545
L
0
I
I
I
f0
20
30
I
I
I
I
I
40 50 Speed
60
70
80
FIG. 2. Speed-flow observations.
was calculated from the time taken to travel across bands placed 30 feet apart (15 for heavy traffic). Speed limits were 70 mph for cars and 65 mph for trucks. Figure 2 shows the relationship between speed and flow. The horizontal axis measuresspeed, in miles per hour, of vehicles in the fast lane only. The vertical axis measuresflow, in vehicles per 6 min, based on vehicles in both lanes. The reason for this speed-flow combination becomes apparent later. Because of the quantity of data, we grouped observations to make the graph more easily readable. This procedure leaves the graph qualitatively similar to a more densegraph that contains all observations. Table 2 explains our notation. The graph demonstrates that breakdown occurred during the observation period; for speeds below approximately 30 mph, speed and volume decreasesimultaneously.
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TABLE 2 Key to Fig. 2 Symbol
Description
+ *
Three or fewer vehicles Four, five, or six vehicles Seven, eight, or nine vehicles Ten or more vehicles
6. CALCULATION
OF THE BEST MEASURE OF FLOW
The data include speed, lane number, and time of observation, We must calculate our own flow figures. Two considerations are: (i) the time interval for flow, and (ii) the speed lane-flow lane combination. We discuss these issuesin turn. We estimate flow by taking a time interval, t, around each vehicle, and counting the number of vehicles in that time interval. This procedure gives the number of vehicles per unit time. Vehicles in back of and in front of a driver constitute perceived flow. We assigned two-thirds of the interval to flow in front and one-third of the interval to flow behind the driver. We considered time intervals of t = 0.1,0.5, 1, 2, 3, 5, 6, 10, 14, 20, and 60 min. To exemplify these points, consider the chosen time interval of 6 min. For each observation, we count the number of cars which passedthe checkpoint not more than 4 min before the observation or not more than 2 min after, the observation, The number of vehicles in this interval equals the flo~.~ TABLE 3 The Goodness of Fit for Different Time Intervals
t (min)
0.5 6.0 20.0 60.0
ii+ for v=a+bq specification 0.1269 0.2408 0.2741 0.1316
Z? for
q=a+bv+& specification 0.3494 0.4178 0.4323 0.2844
6This measure of flow is appropriate when one considers each vehicle as the unit of analysis. Of course, the average of these flow figures overestimates the actual number of vehicles that passed during a longer time interval of, for example, and hour. Notice that the mean of our flow measure equals 455.9 vehicles per 6 min (4560 vehicles per hour), while, in fact, approximately 10,ooO vehicles were observed in approximately 2.60 hr (3846 vehicles per hour). The fact that we compute flow only when a vehicle passes that travels in the fast lane makes no difference to the argument. The difference stems from frequently recomputing flow when there are many observations, i.e., when flow is high.
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TABLE 4 Possible Speed Lane and Flow Lane Combinations Combination
Speed
Flow
SSFS SSFB SFFF SFFB SBFB
Slow lane only Slow lane only Fast lane only Fast lane only Both lanes
Slow lane only Both lanes Fast lane only Both lanes Both lanes
Table 3 shows how the goodness of fit varies as the time interval varies. Instead of presenting all time possibilities that we tried, we present only four, for t = 0.5, 6.0, 20.0, and 60.0 min. We present the E2 for two model specifications. Observations on speed come from the fast lane only, observations on flow come from both lanes. Table 3 shows that as the time interval increases,the explanatory power of the equations increases, reaches a maximum, and eventually declines. The general flow of traffic near a vehicle servesas a better predictor of speedthan does flow in the immediate vicinity of a vehicle or flow far from a vehicle. The issue remains to choose an interval between 6 and 20 min. In 6 min an average driver will travel almost 4.5 miles; two-thirds of this is 3 miles; this distance is so great that we choose the 6-min time interval. Most analyses make no distinction between vehicles that travel in the fast or slow lanes. Certainly, one possibility is to combine all the data together. However, a driver in the slow lane may vary his speed more in accordance with the flow in the slow lane than the flow in both lanes. Also, a driver in the fast lane may vary his speedmore in accordance with the flow in the fast lane than the flow in both lanes. These considerations suggest five possible combinations (seeTable 4). TABLE 5 Goodness of Fit for Different Speed Lane-Flow Lane Combinations Combination
SSFS SSFB SFFF SFFB SBFB
v = a + bq specification
q = a + bv + dv2 specification
a
b
23
n
b
d
Rz
92.83 82.82 54.73 101.6 91.00
-0.1926 -0.09683 -0.05939 -0.1260 -0.1098
0.4098 0.2331 0.0096 0.2400 0.2286
280.2 415.9 110.4 367.3 420.2
2.725 5.398 3.856 7.393 4.607
-0.06156 -0.09902 -0.04523 -0.1046 -0.07870
0.4889 0.3240 0.1905 0.4178 0.3170
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TABLE 6 EstimatedSpeed-Flow Equations with Speedas the DependentVariable0 l/v = 0.000925+ o.OOoO5941q (18.88) (0.W v = 101.6 - a.1269 (62.55) (35.77) v2= 8156.0 - 12.94 (58.32) (42.56) v3= 604500.0 - 1043.Oq (58.77) (46.75)
I? = 0.0807
(14)b
B= = 0.2400
(15)
@ = 0.3089
(16)
R2 = 0.3504
(17)
d In all of thesetables, speed,v, is measuredin miles per hour, and flow, q, is measured in vehiclesper 6 min. A detailed description of the variable namesappearsin Appendix A. For each coefficient, the absolute value of the r-statistic appearsbelow it. There are 4051 observationsfor all equations. bActually, we tried different valuesof k, including k = 4.5, and found that k = 1 fit best.
In order to determine the best speed lane-flow lane combination, we obtained the goodness of fit of the different combinations for two model specifications. In accordance with the above conclusion, the time interval equals 6 min. The results appear in Table 5. Consider first vehicles in the slow lane. Table 5 shows that these drivers consider the flow in the slow lane only, rather than the flow in both lanes. Once a driver has chosen the slow lane, he may feel inclined to stay in that lane, and he adjusts his speed to the flow in that lane only. The opposite result obtains for drivers in the fast lane. They adjust more to the flow in both lanes than to the flow in the fast lane only. Both the SSFS and the SFFF combinations fit better than SBFB, where all drivers are considered together. An F test rejects the null hypothesis that the same parameters hold for both slow and fast lanes. The best fits obtain for the speed in the slow lane-flow in the slow lane combination. Initially, this combination appears the best one to use, Notice, however, that the increase in the R2 from the linear specification to the quadratic specification is least for the SSFS combination, while the increase is high for the SFFB combination, and is highest for the SFFF combination. The reason is that breakdown is much more pronounced for vehicles in the fast lane than for vehicles in the slow lane. This point favors the use of the SFFB combination over the SSFS combination for further analysis. Finally, one must remember that SSFS considers vehicles in only one lane, while SFFB considers vehicles in both lanes. Since statements about capacity flow on a two-lane highway require consideration of the flow in both lanes, we decided on the SFFB (speed fast lane only, Aow both lanes) combination for our empirical analyses of different speedflow models, and for the estimation of alternative model specifications.
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TABLE 7 Estimated Speed-Flow Equations with Flow as the Regressand q = 510.4 0.02396~~ (324.59) (42.56) q = 499.4 0.0003362~~ (389.12) (46.75) q = 367.3 + 7.393~ - 0.1046~~ (68.11) (27.55) (35.18)
&? = 0.3089
(18)
P = 0.3504
(19)
82 = 0.4178
(20)
7. RESULTS Speedand flow are modeled as causally related, but the dependent variable is not evident. To determine whether speedis to be a function of flow or vice versa one might look to estimation difficulties. Most formulations are estimated straightforwardly because speed or a transform of speed is a polynomial function of flow. Equation (4) is an exception, because flow is a quadratic function of speed. Also, Vickrey’s formulation does not express flow as a polynomial function of speed. Thus, we cannot estimate all formulations with the same dependent variable without going to nonlinear estimation procedures-an added difficulty with equivocal benefits. However, note that the calculated measureof speedcontains few errors of measurement. In contrast, we must searchfor someapproximation for flow. This observation suggeststhat flow ought to be the dependent variable with the error term reflecting some of the errors of observation. We begin by presenting those formulations which can be estimated by ordinary least squares with transformations of speed or flow as dependent variables. Table 6 has speed as the dependent variable, while Table 7 has flow as the dependent variable. The Vickrey formulation fits rather poorly; speed does not appear to be a decreasing convex function of flow. As we have seen before, the simple linear formulation explains almost a quarter of the variance in speed, but other formulations explain much more. Smeed’s two formulations fit best. Smeed seems correct in asserting that his two formulations fit almost equally well. The principal result of Table 7, where flow is the dependent variable, is that the quadratic formulation fits best. Table 6 servesto rule out the Vickrey formulation as uninteresting; Table 7 indicates the superiority of the quadratic form.’ 7A numberof other simple functional forms were tried, including piecewise linear and circular forms; none fit better and all tended to confirm the appropriateness of the quadratic formulation.
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Estimated Speed-Density Equations with Speed as the Regressand v= In v = v= In v = In v =
74.38 - 174.844DDM (348.16) (161.67) 4.059 9.457DDM2 (956.26) (131.37) -15.078 31.055 In DDM (25.04) (245.18) 4.510 4.742DDM (1402.97) (291.45) 4.695 6.924DDM + 4.924DDMa (1052.29) (153.02) (50.27)
R= = 0.8658
(21)
@ = 0.8099
(22)
@ = 0.9369
(23)
I3 = 0.9545
(24)
R= = 0.9720
(25)
An alternative set of formulations involves using density as an independent variable in explaining either speed or flow. Here, density (DDM) equals flow divided by speed, times a constant. Equation (21), which we attribute to Greenshields [6], shows that density explains much more of the variance in speedthan does flow. Of course, the method of calculating density accounts for some of the high fit. Speed or a function of speed appears on both sides of the equation. A consideration of the relationship between flow and density provides an estimate of the effect of this phenomenon. We shall consider this relationship soon. Meanwhile, consider the other speed-density equations in Table 8. Equation (23), which Greenberg [SJ derived from an analogy with fluid dynamics, fits well but not best; we have estimated a constant term, even though Greenberg does not postulate one. A convenient feature is the ease of reading the speed at which flow is maximized: It is the regression coefficient, 3 1.055 mph. Equation (24), which we attribute to Underwood [23], demonstrates that taking the natural logarithm of speed may improve the fit (we have not adapted the B2 for the log transformation). May’s [13] idea of squaring density (Eq. (22)) actually reduces the goodness of fit. Equation (25) has the highest explanatory power. This formulation appears to have more
Density FIG. 3. The fundamental diagram of road traffic.
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BOARDMAN AND LAVE TABLE 9 EstimatedFlow-Density Equations with Flow as the Regressand;Estimated Equations of the Fundamental Diagram of Road Traffice Flow q = 288.2
(99.40)
+
2.1730 -
0.005207D~
I8 = 0.4778
(26)
I?? = 0.7243
(27)
(55.83)
(46.70) q = -730.5 I.7950 + 301.0In D (56.64) (67.74) (88.02)
curvature than Underwood’s formulation. However, Eq. (25) may have a
positive slope at the highest densities,s but not for densities within the observed range. We turn now to flow-density relationships. The approximate functional relationship between flow and density is well known; it is called the Fundamental Diagram of Road Traffic. Usually, the figure has a positive skew (see Fig. 3). We estimated the two flow-density relationships given in Table 9.g A comparison of Eqs. (20) and (26) demonstrates that replacing speed by density increasesthe explanatory power slightly. This result implies that little of the increase in the explanatory power of the speed-density relationships over the speed-flow relationships can be ascribed to allowing functions of speed on both sides of the equation; this result increases interest in speeddensity relationships. The positively skewed function, Eq. (27) has much more explanatory power than Eq. (26). If we differentiate Eq. (27), set the derivative equal to zero, and solve for density, we determine that a density of 167.69 vehicles per mile corresponds to capacity flow. At this density, flow equals 5104 vehicles per hour. The maximum flow according to this formulation is higher than the capacity flow derived from the formulation where flow is a quadratic function of speed (4980 vehicles per hour at 35 mph; seeEq. (20) and Table 1). The differences, however, are not great. Furthermore, consideration of the effect of time of day reduces the differences. We turn now to these considerations. 8. THE RESULTS FOR EQUATIONS INCLUDING
TIME
(a) The Relationship betweenSpeedand Flow From 3~30 PM to 4~30 PM traffic flowed lightly. The flow increased and reached a maximum between 4:30 PM and 5:00 PM and then fell off. Thus, 8For a more thorough statistical comparisonof alternative speed-densityformulations, see[Z]. 9For other theoretically derived flow-density relationships, see[7, pp. 67-851.
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a quadratic function of time may approximate flow: q = 139.0 + 0.04517T 0.000001342T2 (38.95) (96.84) (98.43)
i?2 = 0.7052. (28)
Indeed, Eq. (28) fits extremely well. The following equations demonstrate that time also explains much of the variance in speed, where T is a linear measure of time and TI, Tz, T3, T4, Ts, and TSare dummy variables for 30-min periods between 3:30 and 6:30 (T3 is excluded from the regression). Y = 119.1 0.00902T + 0.000000231ST2 (103.69) (60.08) (52.81) v=
82 = 0.5378
(29)
49.25 + 19.531; + 12.95T2- 16.34T4- 20.571; - 1.256To (I 14.28) (11.68) (21.01) (26.90) (33.54) (1.82) B2 = 0.4995. (30)
Now consider introducing time into Eq. (20), the best speed-flow equation; dummy time variables are introduced in Eq. (31), a linear time variable is added in Eq. (32), and a third-order polynomial of time is used in Eq. (33). q = 445.7 + 3.792~ 0.05752~~- 144.21; - 34.30T2 (87.86) (16.42) (21.84) (24.58) (15.35) - 6.663Td- 8.934Ts - 101.5Ts E2 = 0.6232 (31) (2.98) (3.82) (41.35) q = 408.9 + 3.539~ 0.05465~~+ 0.003365T - 119.0T1- 18.95T2 (55.64) (15.22) (6.88) (20.61) (17.30) (6.02) - 24.OOT4- 43.12Ts - 152.1TG @ = 0.6275 (32) (7.14) (7.86) (19.65) q = 174.5 + 3.302~ 0.03829~~+ 0.02586T - 0.00000021T2 (21.34) (17.09) (17.04) (15.69) (1.91) - 0.000000000020U5T3 R2 = 0.7313. (33) (8.94) Even with the inclusion of time-related variables, the regression coefficients for speedand speedsquared remain highly significant, although both decrease slightly. Drivers who regularly travel during the 4:30 to 5:00 PM maximum flow appear to be able to handle heavy traffic flows better than can drivers who travel usuahy under light flow conditions. According to Eq. (31) during the time period with highest flow, a speed of 32.96 mph maximizes flow. At this speed, flow equals 508.1968 vehicles per 6 min (5082 vehicles per hour). These figures are close to the maximum capacity flow of 5100 vehicles per hour and an associated speed of 30 mph,
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calculated from Eq. (27). Thus, we may consider these figures as reasonably reliable estimates for the capacity (and its associated speed) for a two-lane limited accesshighway. The flow cannot exceed 5100 vehicles per hour and vehicle speeds below 30 mph are in the range where both speed and flow would increase if cars left the highway (see Fig. 4). (b) The Relationship between Speed and Density, and Flow and Density Adding time-related variables to the speed-density relationships only slightly improves the explanatory power of these equations. Consider, for example, the increases in the explanatory powers of Eq. (25), when time variables are included. In v =
7.434DDM + 5.999DDM2 + 0.OOOOO3797T 4.729 (59.63) (5.02) (447.79) (145.34) 0.124Tl - 0.006099T~ - 0.04362Td - 0.06195Tb (1.24) (8.42) (7.34) (11.54) 0.18747, ii!2 = 0.9796 (34) (15.79)
In v =
8.072DDM + 6.991DDM2 + 0.00004047T 4.546 (32.57) (699.5) (149.49) (67.75) 0.000000001257T2 R2 = 0.9813. (35) (36.3)
A comparison of the @ for these equations and with Eq. (25) shows that the addition of time variables increases the R2 by less than 1%. Another interesting feature of these equations is that the coefficients for time have reversed signs between Eq. (29) and Eq. (35). The explanation comes from differences in ability to cope with high flow conditions. Consider tkst Eqs. (28) and (29). Equation (28) reflects the fact that during the observation period, flow is initially small, it increases,and then declines. Since, in general, speed decreasesas flow increases, Eq. (29) reflects this same phenomenon: Speedsare high early and late in the period when flow is light, and speeds are lowest in the middle of the period, when flow is heavy. Equation (35) shows that when we control for density, speed initially increases, reaches a maximum, and then decreases.Theseresults are explained by the experienceddriver hypothesis: Independent of density or flow, those people who are experienced at driving under congestedconditions will drive faster at a given density or flow than people who are inexperienced at driving under congested conditions. For completeness,let us consider two flow-density equations that include time variables. The following results obtain from adding time variables to a
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365
425
485
cn
v = 445.7 + 3.792q
o”o
- o.0575q2 (at time 3) I 0
FIG.
I 10
I 20
I 30
I 40 Speed
I 50
I 60
I 70
I 80
4. Some estimated quadratic speed-flow relationships.
variant of Eq. (26) (DDM
= O/755.59):
cj = 313.3 + 1169.ODDM - 2014.0DDM2 + 0.003867T - 102.3T1 (49.32) (38.0) (33.27) (15.83) (8.50) - 10.53T, - 32.88Tb- 60.35T, - 157.1Ts @ = 0.6750 (36) (10.56) (11.88) (3.57) (22.01) q = 156.8 + 636.9DDM (42.44) (20.75)
- 1191.0DDM2 + 0.03472T (20.31) (49.16) 0.000001057T2 82 = 0.7334. (53.69)
(37)
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While time explains more of the variance in flow than it does of the variance in speed, the inclusion of time variables does not contribute nearly so much to the flow-density relationships as they contribute to the speed--flow relationships. 9. IMPLICATIONS
FOR CONGESTION TOLLS
Regressions (28) through (37) make it clear that the speed-flow relationship changesduring the day. Indeed, the increased skill of commuters is only one of several factors that would tend to shift the speed-flow relationship; driver personality, the weather, amount of moisture on the road surface, the amount of light, and the state of repair of the highway are factors that can be expected to shift the equilibrium speedfor any given flow. All factors would affect the congestion toll. Thus, the speed-flow relationship must be modeled as being dynamic, rather than static and the calculation of the congestion toll is more difficult than is shown in Table 1. Equations (28) to (37) assumea particularly simple form of the dynamic speed flow relationship: Only the intercept is assumed to change. Comparing Eqs. (31) and (20), the addition of the time dummy variables has resulted in lowering the coefficients of v and v2 by half; thus, the congestion tolls in Table 1 would have to be recalculated. However, if we accept Eq. (31) as a good characterization of the conditions then prevailing, including the skill of the drivers at each time, it is an easy matter to take account of the time shifts, since only a constant need be added to the recalculated congestion toll. This speed-flow relationship would have to be estimated for different conditions, and reestimated after the tolls had been put into effect. However, Eqs. (20) and (31) give what we regard as reasonable estimates of the speedflow relationship and show the feasibility of estimating them. 10. CONCLUSION In this paper we have developed a model for determining the optimal congestion tax on a limited access two-lane highway. This tax depends critically on the value of time and the relationship between speed and flow. Readers may recompute easily the optimal tolls for their value of time. The more controversial issue concerns the speed-flow formulation. Unlike other empirical studies, we have used each vehicle as the unit of analysis. Aggregation tends to hide some important characteristics of the true underlying relationship between individual vehicle speed and flow. We have shown that one can obtain different fits depending on the time interval one usesto calculate flow and the chosen speed lane-flow lane combination. A time interval around the driver of between 6 and 20 min fits best. Drivers
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in the slow lane adjust their speedsaccording to the flow in the slow lane, while drivers in the fast lane adjust their speedsaccording to the flow in both lanes. From among all simple, speed-flow relationships, we found that an equation in which flow is a quadratic function of speedworks best. However, much better fits are obtained for the speed-density relationships. We find that the maximum capacity for a two-lane highway is about 5000 vehicles per hour. Speed should never fall below 30 mph. Additionally, we find that experienced drivers can travel faster under high-flow conditions that can inexperienced drivers. The speed-flow relation must be modeled as dynamic, since it depends on driver skill, light conditions, weather conditions, etc. These more general speed-flow relationships require recalculation of the congestion toll, but add no conceptual problems. APPENDIX A: DESCRIPTION OF THE VARIABLES Abbreviation
V
V2
b-7
In v l/v 4 D 02
In D DDM In DDM DDM2 T
Tz
Mean (standard deviation) 44.16 (17.98) 2274. (1623.) 129100. (123100.) 3.690 (0.4645) 0.02801 (0.01461) 455.9 (69.93) 130.6 (72.31) 22290. (25240.) 4.720 (0.5604) 0.1729 (0.0957) - 1.907 (0.560) 0.03903 (0.0442) 16560. (6848.) 321200000. (2342OOOOO.)
Description
Code (units)
Velocity = speed
mph
Velocity**2
(mph)**2
Velocity**3
(mph)**3
In (velocity)
mph
(Velocity)-l
hr/mile
Flow = volume
vehicles/O. 1 hr
Density = flow*lO/velocity
vehicles per mile
Density**2
(vehicles per mile)**2
In (density)
vehicles per mile
Densitya/755.59
(1/755.59)*vehicles per mile
In (density/755.59) DDM**2 Timeb = time of day (in tenthousandths of an hour) after 3:30 PM Time**2
hours/lO,OOO
358
BOARDMAN
Abbreviation
T3 Tl
0.1228 0.2056 (0.4042) 0.2150 (0.4109) 0.2180 (0.4129) 0.2098 (0.4072) 0.1363 (0.3431)
TZ 7.3
T4 TS T6
LAVE
Description
Mean (standard deviation) 6.894E 12 (6.782E 12) 0.0153
AND
Code (units)
Time**3 Tl = = Tz = = T3 = = Ta = = Tj = = T6 = =
1 if time 5 5000 0 otherwise 1 if < 5000 time < 10,000 0 otherwise 1 if 1000 < time 5 15,000 0 otherwise 1 if 15,000 < time 5 20,000 0 otherwise 1 if 2000 < time _< 25,000 0 otherwise 1 if 25,000 < time 0 otherwise
T = 0 = 3:30PM T= 5000=4:00PM T = 10,000 = 4:30 PM T = 15,000 3 5:00 PM T = 20,000 3 5:30 PM T = 25,000 3 6:00 PM
(1We divide by 755.59 because this number is our initial best estimate of the maximum density. bThus, for example, T = 16560 means 16,560 ten-thousandths of an hour after 3:30 PM or 9.36 min after 5:00 PM.
REFERENCES 1. M. E. Beesley and G. J. Roth, Restraint of traffic in congested areas, The Town Pluming Rev. 33 (1962). 2. J. S. Drake, J. L. Schafer, and A. D. May, Jr., A statistical analysis of speed density hypotheses, Highway Res. Record 154, 53-87 (1967). 3. D. R. Drew and C. J. Keese, Freeway level of service as influenced by volume and capacity characteristics, Highway Res. Record 99, l-39 (1965). 4. D. Friedman, “The Machinery of Freedom,” Harper & Row, New York (1973). 5. H. Greenberg, An analysis of traffic flow, Operations Rex 7, 79-85 (1959). 6. B. D. Greenshields, A study of highway capacity, Highway Res. Board Proc. 14, 46 (1935). 7. F. A. Haight, “Mathematical
8. 9. 10.
11. 12. 13.
Theories of Traffic Flow,” Academic Press, New York (1963). M. Johnson, On the economics of road congestion, Econometrica 32, 137-150 (1964). G. Kraft and M. Wohl, New directions for passenger demand analysis and forecasting, Transportation Res. 1, 205-230 (1968). L. B. Lave, “Transportation, City Size and Congestion Tolls,” Rand Corp. (1969). L. B. Lave and J. S. DeSalvo, Congestion, tolls, and the economic capacity of a waterway, J. Polit. Econ. 76, 375-391 (1968). G. W. Lerch, A study of the speed-volume relationship on high speed highways, unpublished Master’s dissertation, The Catholic University of America, Washington, D. C. (1970). A. D. May, Jr., Discussion of “freeway level of service as influenced by volume and capacity characteristics” by D. R. Drew and C. J. Keese, Highway Res. Record 99, 3943
(1965).
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14. W. R. McCasland, Comparison of two techniques of aerial photographs for application in freeway traffic operations studies, Highway Res. Record 65, 95-115 (1965). 15. R. E. Park, Congestion tolls for commercial airports, Econometrica 39, No. 5, 683 (1971). 16. G. J. Roth, An economic approach to traffic congestion, Town Planning Rev. 36, 49-61(1965). 17. G. J. Roth, “Paying for Roads: The Economics of Traffic Congestion,” Penguin, Baltimore (1967). 18. C. Sharp, Congestion and welfare-an examination of the case for a congestion tax, Econ. J. 76, 816-817 (1966). 19. C. Sharp, Congestion and welfare reconsidered, J. Transport Econ. Policy 2, 33-70 (1968). 20. R. J. Smeed, Traffic studies and urban congestion, J. Transportation Econ. Policy 2, 33-70 (1968). 21. R. J. Smeed et al., “Road Pricing: The Economic and Technical Possibilities,” H. M. Stationary Office No. 55-411, London (1964). 22. R. Strotz, Urban transportation parables, in “The Public Economy of Urban Communities” (Margolis, Ed.), Resources for the Future, New York (1965). 23. R. T. Underwood, Speed, volume and density relationships: Quality and theory of traffic flow, Yale Bureau of Highway Traffic, 141-188 (1961). 24. W. Vickrey, Optimization of traffic and facilities, J. Transport Econ. Policy 1, 123-135 (1967). 25. W. Vickrey, Congestion charges and welfare, J. Transport Econ. Policy 2, 107-118 (1968). 26. A. A. Walters, The theory and measurement of private and social cost of highway congestion, Econometrica 29 (1961).
OF URBAN
ECONOMICS
4, 340.-359 (1977)
Highway Congestion and Congestion Tolls’ ANTHONYE. BOARDMAN School of Public and Urban Policy, University of Pennsylvania, Philadelphia, Pennsylvania I9104
LESTER B. LAVE Graduate School of Industrial Administration, Carnegie-MelIon University, Pittsburgh, PennsyIvania 15213
Received October 1, 1975 Highway congestion is ubiquitous. We model the speed-flow relationship, identifying private and social costs, and the implied congestion toll for a number of proposed formulations. Using data for a limited access highway, we estimate these speed-flow relationships and find that flow as a quadratic function of speed fits best. The unit of observation is the individual vehicle and flow is measured in terms of the vehicles which passed a point shortly before or after the reference vehicle. Maximum possible flow occurs at 30-35 mph and the congestion toll is infinite for slower speeds. Rush-hour drivers have greater speeds for a given volume of traffic and thus congestion tolls must vary by time of day.
1. INTRODUCTION Congestion may be the most common nonpecuniary externality. Congestion has been studied for inland waterways [I 11, airports [15], and highways [I, 8, 10, 16, 18-22, 24-261. In this paper we focus on highway congestion. Since the speed at which a person drives on a highway decreasesas traffic flow increases, an additional driver will lower the speeds of other users. Reduced speedsincrease total travel time. The value of this additional time reflects the congestion imposed by the additional user. Economic theory dictates that the user act as if he paid the social marginal cost, which is greater than the private cost. Many authors have suggested fascinating methods for collecting a congestion tax.2 The main problem, however, is 1We thank William M. Whitby of the Office of Research and Development, Federal Highway Administration for supplying the data for this analysis. 2 See, for example, [4, 171. 340 Copyright @ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
ISSN 0094-l 190
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to determine the correct tax. In this paper we develop a model of social costs which we combine with an estimated speed-volume relationship to determine the optimal congestion toll. 2. A MODEL OF SOCIAL COSTS Commuters call upon a transportation network for quick and cheap mobility. The output of a mode equals the number of commuters serviced per unit time. In order to dispense with all unnecessary complications, we assumeq vehicles are uniformly spaced over 1 hr, one person occupies each vehicle, and all have the same origin and the same destination. Each vehicle takes a time of 6/v (distance divided by speed) hours to reach its destination. We assumethat all costs (variable vehicle running costs plus travel time costs) can be put on an hourly basis ($c/hr). Thus, the cost to one driver of reaching his destination is $c@/v). The exposition follows more clearly if we assume distance, 6, equals unity. Thus, a traveler experiences an average cost of $c/v per mile. The total cost per mile, T, of q vehicles reaching their destination equals the average cost multiplied by the flow: T = qc,‘v.
If an additional vehicle usesthe highway, the increase in total cost equals the social marginal cost, SMC, where qc dv
SMC=dT=!---, 4 ZZ
v
v2 dq
YC- qc(dv/dq) v2
*
If an individual takes to the highway he experiences a (private) cost of $c/v per mile. Society as a whole experiencesthe social marginal cost, SMC, which exceedsthe private cost. In order to internalize this cost, the individual must behave as if he faces a cost equal to the marginal social cost, rather than the private cost. One way of making an individual behave in the desired fashion is to levy a “congestion toll,” 7, equal to the difference between the social and private costs: 7 = SMC - private costs, 7 = SMC - (c/v), (3) -qc dv 7=----20 v2 &
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TABLE
1
An Example of Congestion Tolls Flow, q (vehicles/hr)
Speed, v (mph)
0 500
104.335 100.779 97.019 93.014 88.709 84.025 78.840 72.946 65.938 56.746 50.146 35.339
1000 1500 2OcKl 2500 3cKlO 3500 4ooo 4500 4750 4979.3
Private cost, $6/v ($/vehicle-mile) 0.0575 0.0595 0.0618 0.0645 0.0676 0.0714 0.0761 0.0823 0.0910 0.1057 0.1197 0.1698
Social marginal cost, SMC ($/vehicle-mile)
Congestion toll, 7 ($/vehicle-mile)
0.0575 0.0617 0.0668 0.0731 0.0813 0.0923 0.1079 0.1324 0.1772 0.2930 0.4855 m
0.0 0.0022 0.0049 0.0086 0.0137 0.0209 0.0319 0.0502 0.0862 0.1872 0.3659 co
Assume that variable running costs and travel time costs amount to $6 per hour. Also consider the following quadratic speed-flow relationship: q=a+bv+d+, which is estimated by
(4)
q = 3673. + 73.93~ - 1.0469. The various relationships are shown in Table 1. One measure of highway output is the number of vehicles served multiplied by the time it takes each to go a given distance: qv. This output rises as flow increasesfrom zero, but reachesa maximum at about 4050 vehicles per hour, which is less than the maximum flow of 4979.3 vehicles3per hour. Another way of looking at this phenomenon is to note that when only one traveler uses the highway he pays no toll. Since nobody else uses the highway, the first user has no other user’s speed to affect. For flows of less than 2000 vehicles per hour, the toll is small. As the flow increases,the SMC rises faster than the private cost and the toll rises accordingly. In practice, the toll would not become large. Travelers faced with high tolls would look for transport alternatives, such as off-peak travel, car pools, or public transport. The total revenue from the toll amounts to $rq per mile, for a constant flow at q vehicles per hour. This revenue may exceed, equal, or fall short of road-building and maintenance costs 1221. Widening the highway will lead to less congestion, higher speeds, and lower tolls. A highway should be 3This figure is higher than the generally accepted capacity flow of between 3800 and 4400 vehicles per hour (see [3]).
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widened if the value of the savedtime equals or exceedsthe total costs (private plus social) of expansion and maintenance. A congestion toll is unconventional in that it is highest before a construction program and is likely to fall drastically after completion of the construction. 3. SPEED-FLOW FORMULATIONS The model of social costsrequires knowledge of (a) variable vehicle running costs plus travel time costs and (b) the relationship between speed and flow. Readers may recompute easily the estimated, optimal congestion tolls (Table 1) for different cost figures. Knowledge of the speed-flow relationship is much more fundamental and controversial. The remainder of this paper concerns primarily the determination of the best relationship among speed, flow, and density. This section considers the relationship between speed and flow. Later sections consider the relationships between speedand density, and between flow and density. Some formulations include road width. Our work applies only to highways with two lanes in each direction. The most common, simple formulation used by Kraft and Wohl [9] and many others, assumes a linear relationship between speed and flow, thus: v -t aq = b.
(5)
Smeed [20] gives the two following formulations and claims that both equations fit his data equally well except for speedsunder 10 mph: q + bv2 = a,
(6)
q + bv3 = a.
(7)
Vickrey [24] provides an alternative formulation : a + bq” = l/v.
(8)
Vickrey estimates that k equals 4.5 for the Holland Tunnel. Figure 1 shows the graphs of these speed-flow relationships. Each of the speed-flow relationship has an associated toll that depends only on the flow. Assuming unit distance and unit cost per hour, the tolls follow for Eqs. (4)-(g), respectively:4 7 = -(a+bv+W (b + 2dv)v2
’
b-v 7-=-y V2
4 Usually, the formulas include q as the argument, not v. We use v because it facilitates the exposition and there is a 1: 1 transformation between q and v (in the relevant range).
BOARDMAN
344
L
1-1 Boardman
and Lave
l-3
AND LAVE
\\\\
q
Smeed
“ix
#l
q
“L 1-4
Smeed
X2
q
1-5
Vickrey
4
FIG. 1. Graphical representations of five speed-flow formulations, Eqs. (4)-(8), respectively.
a - bv2 -Z-=-j 2bv3
(11)
a - bv3 7=-J
3bv4
T = (k/v) - ak.
(12) (13)
4. LIMITATIONS OF PREVIOUS EMPIRICAL STUDIES Previous studies observe a highway during some period and then divide the data into time intervals of about 10 min. Flow equals the number of vehicles that pass the observation point during the time interval. Instead of using individual vehicle speed, these studies use an average speed, which may be an arithmetic mean or harmonic mean of all the vehicles that passed during the interval. This procedure results in high R2, about 0.9, almost by definition. However, aggregation is likely to obscure the underlying relationship. To estimate the speed-flow relationship in greater detail, particularly to discriminate among various proposals, we treat the individual vehicle as the unit of analysis.
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When the focus is on the individual vehicle, individual vehicle speedis the relevant datum. Not so with flow; several alternative measuresseemrelevant, depending on how a driver perceives and reacts to flow. The vehicles in front of and behind a vehicle influence its speed (those in front may slow it directly, while those behind may increase congestion and slow it indirectly). But, how far in front and in back do vehicles still exert an influence ? Does a vehicle which passed the same point half an hour before influence a current driver? Having chosen the individual vehicle as the unit of observation, we must consider these questions explicitly. Previous research assumedimplicitly that drivers have identical preferences and habits. However, casual observation suggests that Sunday afternoon drivers travel slowly, independent of flow. Commuters are more experienced at driving under conditions of high flow than people who miss the rush-hour traffic. This experience probably translates into higher speed, for the same number of vehicles per lane mile. Thus, the speed-flow curve may shift depending on the time of day. We will test the stability of speed-flow relationships as a function of time to seewhat has been ignored as a consequence of assuming that drivers are homogeneous. In addition to differences in experience with driving under conditions of high flow, other differences characterize drivers. Independent of experience, some drivers desire and do drive faster than others under identical conditions. Although important, such preferences are difficult to measure. We can, at least, distinguish between drivers in the fast lane and drivers in the slow lane. Finally, one must remember speed, flow, and density are related by the expression D = q/v. Perhaps speed depends more meaningfully on density rather than flow. Density should be calculated independently of speed, for example, by an aerial photograph.5 By necessity, we estimate density by dividing volume by speed. For such a ratio, an estimated speed-density relationship has speed as the dependent variable and as the denominator of an explanatory variable; thus, the estimated R2 is inflated. The surrogate measure of density and inflated R2 should be kept in mind in interpreting our estimated speed-density relationships. 5. THE DATA Lerch [12] observed speed for over 10,000 southbound vehicles on I 495 near 170s on August 20, 1968, from 3:48 PM until 6:30 PM. The road was dry and visibility was good; there were no entrances or exits near the observation point to affect speed or flow. For each vehicle, Lerch recorded the time a vehicle passedthe observation point, in thousandths of an hour, individual vehicle speed,and whether the vehicle traveled in the slow or fast lane. Speed 5See [14].
BOARDMAN
346 245
-
275
-
305
-
AND LAVE
0 &O 00 +z 0
;395 I3 z ‘425
iy sfp
0
455
8
8 8,
485
8
515
545
L
0
I
I
I
f0
20
30
I
I
I
I
I
40 50 Speed
60
70
80
FIG. 2. Speed-flow observations.
was calculated from the time taken to travel across bands placed 30 feet apart (15 for heavy traffic). Speed limits were 70 mph for cars and 65 mph for trucks. Figure 2 shows the relationship between speed and flow. The horizontal axis measuresspeed, in miles per hour, of vehicles in the fast lane only. The vertical axis measuresflow, in vehicles per 6 min, based on vehicles in both lanes. The reason for this speed-flow combination becomes apparent later. Because of the quantity of data, we grouped observations to make the graph more easily readable. This procedure leaves the graph qualitatively similar to a more densegraph that contains all observations. Table 2 explains our notation. The graph demonstrates that breakdown occurred during the observation period; for speeds below approximately 30 mph, speed and volume decreasesimultaneously.
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TABLE 2 Key to Fig. 2 Symbol
Description
+ *
Three or fewer vehicles Four, five, or six vehicles Seven, eight, or nine vehicles Ten or more vehicles
6. CALCULATION
OF THE BEST MEASURE OF FLOW
The data include speed, lane number, and time of observation, We must calculate our own flow figures. Two considerations are: (i) the time interval for flow, and (ii) the speed lane-flow lane combination. We discuss these issuesin turn. We estimate flow by taking a time interval, t, around each vehicle, and counting the number of vehicles in that time interval. This procedure gives the number of vehicles per unit time. Vehicles in back of and in front of a driver constitute perceived flow. We assigned two-thirds of the interval to flow in front and one-third of the interval to flow behind the driver. We considered time intervals of t = 0.1,0.5, 1, 2, 3, 5, 6, 10, 14, 20, and 60 min. To exemplify these points, consider the chosen time interval of 6 min. For each observation, we count the number of cars which passedthe checkpoint not more than 4 min before the observation or not more than 2 min after, the observation, The number of vehicles in this interval equals the flo~.~ TABLE 3 The Goodness of Fit for Different Time Intervals
t (min)
0.5 6.0 20.0 60.0
ii+ for v=a+bq specification 0.1269 0.2408 0.2741 0.1316
Z? for
q=a+bv+& specification 0.3494 0.4178 0.4323 0.2844
6This measure of flow is appropriate when one considers each vehicle as the unit of analysis. Of course, the average of these flow figures overestimates the actual number of vehicles that passed during a longer time interval of, for example, and hour. Notice that the mean of our flow measure equals 455.9 vehicles per 6 min (4560 vehicles per hour), while, in fact, approximately 10,ooO vehicles were observed in approximately 2.60 hr (3846 vehicles per hour). The fact that we compute flow only when a vehicle passes that travels in the fast lane makes no difference to the argument. The difference stems from frequently recomputing flow when there are many observations, i.e., when flow is high.
348
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TABLE 4 Possible Speed Lane and Flow Lane Combinations Combination
Speed
Flow
SSFS SSFB SFFF SFFB SBFB
Slow lane only Slow lane only Fast lane only Fast lane only Both lanes
Slow lane only Both lanes Fast lane only Both lanes Both lanes
Table 3 shows how the goodness of fit varies as the time interval varies. Instead of presenting all time possibilities that we tried, we present only four, for t = 0.5, 6.0, 20.0, and 60.0 min. We present the E2 for two model specifications. Observations on speed come from the fast lane only, observations on flow come from both lanes. Table 3 shows that as the time interval increases,the explanatory power of the equations increases, reaches a maximum, and eventually declines. The general flow of traffic near a vehicle servesas a better predictor of speedthan does flow in the immediate vicinity of a vehicle or flow far from a vehicle. The issue remains to choose an interval between 6 and 20 min. In 6 min an average driver will travel almost 4.5 miles; two-thirds of this is 3 miles; this distance is so great that we choose the 6-min time interval. Most analyses make no distinction between vehicles that travel in the fast or slow lanes. Certainly, one possibility is to combine all the data together. However, a driver in the slow lane may vary his speed more in accordance with the flow in the slow lane than the flow in both lanes. Also, a driver in the fast lane may vary his speedmore in accordance with the flow in the fast lane than the flow in both lanes. These considerations suggest five possible combinations (seeTable 4). TABLE 5 Goodness of Fit for Different Speed Lane-Flow Lane Combinations Combination
SSFS SSFB SFFF SFFB SBFB
v = a + bq specification
q = a + bv + dv2 specification
a
b
23
n
b
d
Rz
92.83 82.82 54.73 101.6 91.00
-0.1926 -0.09683 -0.05939 -0.1260 -0.1098
0.4098 0.2331 0.0096 0.2400 0.2286
280.2 415.9 110.4 367.3 420.2
2.725 5.398 3.856 7.393 4.607
-0.06156 -0.09902 -0.04523 -0.1046 -0.07870
0.4889 0.3240 0.1905 0.4178 0.3170
HIGHWAY CONGESTION AND CONGESTION TOLLS
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TABLE 6 EstimatedSpeed-Flow Equations with Speedas the DependentVariable0 l/v = 0.000925+ o.OOoO5941q (18.88) (0.W v = 101.6 - a.1269 (62.55) (35.77) v2= 8156.0 - 12.94 (58.32) (42.56) v3= 604500.0 - 1043.Oq (58.77) (46.75)
I? = 0.0807
(14)b
B= = 0.2400
(15)
@ = 0.3089
(16)
R2 = 0.3504
(17)
d In all of thesetables, speed,v, is measuredin miles per hour, and flow, q, is measured in vehiclesper 6 min. A detailed description of the variable namesappearsin Appendix A. For each coefficient, the absolute value of the r-statistic appearsbelow it. There are 4051 observationsfor all equations. bActually, we tried different valuesof k, including k = 4.5, and found that k = 1 fit best.
In order to determine the best speed lane-flow lane combination, we obtained the goodness of fit of the different combinations for two model specifications. In accordance with the above conclusion, the time interval equals 6 min. The results appear in Table 5. Consider first vehicles in the slow lane. Table 5 shows that these drivers consider the flow in the slow lane only, rather than the flow in both lanes. Once a driver has chosen the slow lane, he may feel inclined to stay in that lane, and he adjusts his speed to the flow in that lane only. The opposite result obtains for drivers in the fast lane. They adjust more to the flow in both lanes than to the flow in the fast lane only. Both the SSFS and the SFFF combinations fit better than SBFB, where all drivers are considered together. An F test rejects the null hypothesis that the same parameters hold for both slow and fast lanes. The best fits obtain for the speed in the slow lane-flow in the slow lane combination. Initially, this combination appears the best one to use, Notice, however, that the increase in the R2 from the linear specification to the quadratic specification is least for the SSFS combination, while the increase is high for the SFFB combination, and is highest for the SFFF combination. The reason is that breakdown is much more pronounced for vehicles in the fast lane than for vehicles in the slow lane. This point favors the use of the SFFB combination over the SSFS combination for further analysis. Finally, one must remember that SSFS considers vehicles in only one lane, while SFFB considers vehicles in both lanes. Since statements about capacity flow on a two-lane highway require consideration of the flow in both lanes, we decided on the SFFB (speed fast lane only, Aow both lanes) combination for our empirical analyses of different speedflow models, and for the estimation of alternative model specifications.
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TABLE 7 Estimated Speed-Flow Equations with Flow as the Regressand q = 510.4 0.02396~~ (324.59) (42.56) q = 499.4 0.0003362~~ (389.12) (46.75) q = 367.3 + 7.393~ - 0.1046~~ (68.11) (27.55) (35.18)
&? = 0.3089
(18)
P = 0.3504
(19)
82 = 0.4178
(20)
7. RESULTS Speedand flow are modeled as causally related, but the dependent variable is not evident. To determine whether speedis to be a function of flow or vice versa one might look to estimation difficulties. Most formulations are estimated straightforwardly because speed or a transform of speed is a polynomial function of flow. Equation (4) is an exception, because flow is a quadratic function of speed. Also, Vickrey’s formulation does not express flow as a polynomial function of speed. Thus, we cannot estimate all formulations with the same dependent variable without going to nonlinear estimation procedures-an added difficulty with equivocal benefits. However, note that the calculated measureof speedcontains few errors of measurement. In contrast, we must searchfor someapproximation for flow. This observation suggeststhat flow ought to be the dependent variable with the error term reflecting some of the errors of observation. We begin by presenting those formulations which can be estimated by ordinary least squares with transformations of speed or flow as dependent variables. Table 6 has speed as the dependent variable, while Table 7 has flow as the dependent variable. The Vickrey formulation fits rather poorly; speed does not appear to be a decreasing convex function of flow. As we have seen before, the simple linear formulation explains almost a quarter of the variance in speed, but other formulations explain much more. Smeed’s two formulations fit best. Smeed seems correct in asserting that his two formulations fit almost equally well. The principal result of Table 7, where flow is the dependent variable, is that the quadratic formulation fits best. Table 6 servesto rule out the Vickrey formulation as uninteresting; Table 7 indicates the superiority of the quadratic form.’ 7A numberof other simple functional forms were tried, including piecewise linear and circular forms; none fit better and all tended to confirm the appropriateness of the quadratic formulation.
HIGHWAY
CONGESTION
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TABLE
CONGESTION
351
TOLLS
8
Estimated Speed-Density Equations with Speed as the Regressand v= In v = v= In v = In v =
74.38 - 174.844DDM (348.16) (161.67) 4.059 9.457DDM2 (956.26) (131.37) -15.078 31.055 In DDM (25.04) (245.18) 4.510 4.742DDM (1402.97) (291.45) 4.695 6.924DDM + 4.924DDMa (1052.29) (153.02) (50.27)
R= = 0.8658
(21)
@ = 0.8099
(22)
@ = 0.9369
(23)
I3 = 0.9545
(24)
R= = 0.9720
(25)
An alternative set of formulations involves using density as an independent variable in explaining either speed or flow. Here, density (DDM) equals flow divided by speed, times a constant. Equation (21), which we attribute to Greenshields [6], shows that density explains much more of the variance in speedthan does flow. Of course, the method of calculating density accounts for some of the high fit. Speed or a function of speed appears on both sides of the equation. A consideration of the relationship between flow and density provides an estimate of the effect of this phenomenon. We shall consider this relationship soon. Meanwhile, consider the other speed-density equations in Table 8. Equation (23), which Greenberg [SJ derived from an analogy with fluid dynamics, fits well but not best; we have estimated a constant term, even though Greenberg does not postulate one. A convenient feature is the ease of reading the speed at which flow is maximized: It is the regression coefficient, 3 1.055 mph. Equation (24), which we attribute to Underwood [23], demonstrates that taking the natural logarithm of speed may improve the fit (we have not adapted the B2 for the log transformation). May’s [13] idea of squaring density (Eq. (22)) actually reduces the goodness of fit. Equation (25) has the highest explanatory power. This formulation appears to have more
Density FIG. 3. The fundamental diagram of road traffic.
352
BOARDMAN AND LAVE TABLE 9 EstimatedFlow-Density Equations with Flow as the Regressand;Estimated Equations of the Fundamental Diagram of Road Traffice Flow q = 288.2
(99.40)
+
2.1730 -
0.005207D~
I8 = 0.4778
(26)
I?? = 0.7243
(27)
(55.83)
(46.70) q = -730.5 I.7950 + 301.0In D (56.64) (67.74) (88.02)
curvature than Underwood’s formulation. However, Eq. (25) may have a
positive slope at the highest densities,s but not for densities within the observed range. We turn now to flow-density relationships. The approximate functional relationship between flow and density is well known; it is called the Fundamental Diagram of Road Traffic. Usually, the figure has a positive skew (see Fig. 3). We estimated the two flow-density relationships given in Table 9.g A comparison of Eqs. (20) and (26) demonstrates that replacing speed by density increasesthe explanatory power slightly. This result implies that little of the increase in the explanatory power of the speed-density relationships over the speed-flow relationships can be ascribed to allowing functions of speed on both sides of the equation; this result increases interest in speeddensity relationships. The positively skewed function, Eq. (27) has much more explanatory power than Eq. (26). If we differentiate Eq. (27), set the derivative equal to zero, and solve for density, we determine that a density of 167.69 vehicles per mile corresponds to capacity flow. At this density, flow equals 5104 vehicles per hour. The maximum flow according to this formulation is higher than the capacity flow derived from the formulation where flow is a quadratic function of speed (4980 vehicles per hour at 35 mph; seeEq. (20) and Table 1). The differences, however, are not great. Furthermore, consideration of the effect of time of day reduces the differences. We turn now to these considerations. 8. THE RESULTS FOR EQUATIONS INCLUDING
TIME
(a) The Relationship betweenSpeedand Flow From 3~30 PM to 4~30 PM traffic flowed lightly. The flow increased and reached a maximum between 4:30 PM and 5:00 PM and then fell off. Thus, 8For a more thorough statistical comparisonof alternative speed-densityformulations, see[Z]. 9For other theoretically derived flow-density relationships, see[7, pp. 67-851.
HIGHWAY
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353
a quadratic function of time may approximate flow: q = 139.0 + 0.04517T 0.000001342T2 (38.95) (96.84) (98.43)
i?2 = 0.7052. (28)
Indeed, Eq. (28) fits extremely well. The following equations demonstrate that time also explains much of the variance in speed, where T is a linear measure of time and TI, Tz, T3, T4, Ts, and TSare dummy variables for 30-min periods between 3:30 and 6:30 (T3 is excluded from the regression). Y = 119.1 0.00902T + 0.000000231ST2 (103.69) (60.08) (52.81) v=
82 = 0.5378
(29)
49.25 + 19.531; + 12.95T2- 16.34T4- 20.571; - 1.256To (I 14.28) (11.68) (21.01) (26.90) (33.54) (1.82) B2 = 0.4995. (30)
Now consider introducing time into Eq. (20), the best speed-flow equation; dummy time variables are introduced in Eq. (31), a linear time variable is added in Eq. (32), and a third-order polynomial of time is used in Eq. (33). q = 445.7 + 3.792~ 0.05752~~- 144.21; - 34.30T2 (87.86) (16.42) (21.84) (24.58) (15.35) - 6.663Td- 8.934Ts - 101.5Ts E2 = 0.6232 (31) (2.98) (3.82) (41.35) q = 408.9 + 3.539~ 0.05465~~+ 0.003365T - 119.0T1- 18.95T2 (55.64) (15.22) (6.88) (20.61) (17.30) (6.02) - 24.OOT4- 43.12Ts - 152.1TG @ = 0.6275 (32) (7.14) (7.86) (19.65) q = 174.5 + 3.302~ 0.03829~~+ 0.02586T - 0.00000021T2 (21.34) (17.09) (17.04) (15.69) (1.91) - 0.000000000020U5T3 R2 = 0.7313. (33) (8.94) Even with the inclusion of time-related variables, the regression coefficients for speedand speedsquared remain highly significant, although both decrease slightly. Drivers who regularly travel during the 4:30 to 5:00 PM maximum flow appear to be able to handle heavy traffic flows better than can drivers who travel usuahy under light flow conditions. According to Eq. (31) during the time period with highest flow, a speed of 32.96 mph maximizes flow. At this speed, flow equals 508.1968 vehicles per 6 min (5082 vehicles per hour). These figures are close to the maximum capacity flow of 5100 vehicles per hour and an associated speed of 30 mph,
354
BOARDMAN
AND LAVE
calculated from Eq. (27). Thus, we may consider these figures as reasonably reliable estimates for the capacity (and its associated speed) for a two-lane limited accesshighway. The flow cannot exceed 5100 vehicles per hour and vehicle speeds below 30 mph are in the range where both speed and flow would increase if cars left the highway (see Fig. 4). (b) The Relationship between Speed and Density, and Flow and Density Adding time-related variables to the speed-density relationships only slightly improves the explanatory power of these equations. Consider, for example, the increases in the explanatory powers of Eq. (25), when time variables are included. In v =
7.434DDM + 5.999DDM2 + 0.OOOOO3797T 4.729 (59.63) (5.02) (447.79) (145.34) 0.124Tl - 0.006099T~ - 0.04362Td - 0.06195Tb (1.24) (8.42) (7.34) (11.54) 0.18747, ii!2 = 0.9796 (34) (15.79)
In v =
8.072DDM + 6.991DDM2 + 0.00004047T 4.546 (32.57) (699.5) (149.49) (67.75) 0.000000001257T2 R2 = 0.9813. (35) (36.3)
A comparison of the @ for these equations and with Eq. (25) shows that the addition of time variables increases the R2 by less than 1%. Another interesting feature of these equations is that the coefficients for time have reversed signs between Eq. (29) and Eq. (35). The explanation comes from differences in ability to cope with high flow conditions. Consider tkst Eqs. (28) and (29). Equation (28) reflects the fact that during the observation period, flow is initially small, it increases,and then declines. Since, in general, speed decreasesas flow increases, Eq. (29) reflects this same phenomenon: Speedsare high early and late in the period when flow is light, and speeds are lowest in the middle of the period, when flow is heavy. Equation (35) shows that when we control for density, speed initially increases, reaches a maximum, and then decreases.Theseresults are explained by the experienceddriver hypothesis: Independent of density or flow, those people who are experienced at driving under congestedconditions will drive faster at a given density or flow than people who are inexperienced at driving under congested conditions. For completeness,let us consider two flow-density equations that include time variables. The following results obtain from adding time variables to a
HIGHWAY
CONGESTION
AND CONGESTION
TOLLS
355
365
425
485
cn
v = 445.7 + 3.792q
o”o
- o.0575q2 (at time 3) I 0
FIG.
I 10
I 20
I 30
I 40 Speed
I 50
I 60
I 70
I 80
4. Some estimated quadratic speed-flow relationships.
variant of Eq. (26) (DDM
= O/755.59):
cj = 313.3 + 1169.ODDM - 2014.0DDM2 + 0.003867T - 102.3T1 (49.32) (38.0) (33.27) (15.83) (8.50) - 10.53T, - 32.88Tb- 60.35T, - 157.1Ts @ = 0.6750 (36) (10.56) (11.88) (3.57) (22.01) q = 156.8 + 636.9DDM (42.44) (20.75)
- 1191.0DDM2 + 0.03472T (20.31) (49.16) 0.000001057T2 82 = 0.7334. (53.69)
(37)
356
BOARDMAN
AND LAVE
While time explains more of the variance in flow than it does of the variance in speed, the inclusion of time variables does not contribute nearly so much to the flow-density relationships as they contribute to the speed--flow relationships. 9. IMPLICATIONS
FOR CONGESTION TOLLS
Regressions (28) through (37) make it clear that the speed-flow relationship changesduring the day. Indeed, the increased skill of commuters is only one of several factors that would tend to shift the speed-flow relationship; driver personality, the weather, amount of moisture on the road surface, the amount of light, and the state of repair of the highway are factors that can be expected to shift the equilibrium speedfor any given flow. All factors would affect the congestion toll. Thus, the speed-flow relationship must be modeled as being dynamic, rather than static and the calculation of the congestion toll is more difficult than is shown in Table 1. Equations (28) to (37) assumea particularly simple form of the dynamic speed flow relationship: Only the intercept is assumed to change. Comparing Eqs. (31) and (20), the addition of the time dummy variables has resulted in lowering the coefficients of v and v2 by half; thus, the congestion tolls in Table 1 would have to be recalculated. However, if we accept Eq. (31) as a good characterization of the conditions then prevailing, including the skill of the drivers at each time, it is an easy matter to take account of the time shifts, since only a constant need be added to the recalculated congestion toll. This speed-flow relationship would have to be estimated for different conditions, and reestimated after the tolls had been put into effect. However, Eqs. (20) and (31) give what we regard as reasonable estimates of the speedflow relationship and show the feasibility of estimating them. 10. CONCLUSION In this paper we have developed a model for determining the optimal congestion tax on a limited access two-lane highway. This tax depends critically on the value of time and the relationship between speed and flow. Readers may recompute easily the optimal tolls for their value of time. The more controversial issue concerns the speed-flow formulation. Unlike other empirical studies, we have used each vehicle as the unit of analysis. Aggregation tends to hide some important characteristics of the true underlying relationship between individual vehicle speed and flow. We have shown that one can obtain different fits depending on the time interval one usesto calculate flow and the chosen speed lane-flow lane combination. A time interval around the driver of between 6 and 20 min fits best. Drivers
HIGHWAY
CONGESTION
AND CONGESTION
TOLLS
357
in the slow lane adjust their speedsaccording to the flow in the slow lane, while drivers in the fast lane adjust their speedsaccording to the flow in both lanes. From among all simple, speed-flow relationships, we found that an equation in which flow is a quadratic function of speedworks best. However, much better fits are obtained for the speed-density relationships. We find that the maximum capacity for a two-lane highway is about 5000 vehicles per hour. Speed should never fall below 30 mph. Additionally, we find that experienced drivers can travel faster under high-flow conditions that can inexperienced drivers. The speed-flow relation must be modeled as dynamic, since it depends on driver skill, light conditions, weather conditions, etc. These more general speed-flow relationships require recalculation of the congestion toll, but add no conceptual problems. APPENDIX A: DESCRIPTION OF THE VARIABLES Abbreviation
V
V2
b-7
In v l/v 4 D 02
In D DDM In DDM DDM2 T
Tz
Mean (standard deviation) 44.16 (17.98) 2274. (1623.) 129100. (123100.) 3.690 (0.4645) 0.02801 (0.01461) 455.9 (69.93) 130.6 (72.31) 22290. (25240.) 4.720 (0.5604) 0.1729 (0.0957) - 1.907 (0.560) 0.03903 (0.0442) 16560. (6848.) 321200000. (2342OOOOO.)
Description
Code (units)
Velocity = speed
mph
Velocity**2
(mph)**2
Velocity**3
(mph)**3
In (velocity)
mph
(Velocity)-l
hr/mile
Flow = volume
vehicles/O. 1 hr
Density = flow*lO/velocity
vehicles per mile
Density**2
(vehicles per mile)**2
In (density)
vehicles per mile
Densitya/755.59
(1/755.59)*vehicles per mile
In (density/755.59) DDM**2 Timeb = time of day (in tenthousandths of an hour) after 3:30 PM Time**2
hours/lO,OOO
358
BOARDMAN
Abbreviation
T3 Tl
0.1228 0.2056 (0.4042) 0.2150 (0.4109) 0.2180 (0.4129) 0.2098 (0.4072) 0.1363 (0.3431)
TZ 7.3
T4 TS T6
LAVE
Description
Mean (standard deviation) 6.894E 12 (6.782E 12) 0.0153
AND
Code (units)
Time**3 Tl = = Tz = = T3 = = Ta = = Tj = = T6 = =
1 if time 5 5000 0 otherwise 1 if < 5000 time < 10,000 0 otherwise 1 if 1000 < time 5 15,000 0 otherwise 1 if 15,000 < time 5 20,000 0 otherwise 1 if 2000 < time _< 25,000 0 otherwise 1 if 25,000 < time 0 otherwise
T = 0 = 3:30PM T= 5000=4:00PM T = 10,000 = 4:30 PM T = 15,000 3 5:00 PM T = 20,000 3 5:30 PM T = 25,000 3 6:00 PM
(1We divide by 755.59 because this number is our initial best estimate of the maximum density. bThus, for example, T = 16560 means 16,560 ten-thousandths of an hour after 3:30 PM or 9.36 min after 5:00 PM.
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8. 9. 10.
11. 12. 13.
Theories of Traffic Flow,” Academic Press, New York (1963). M. Johnson, On the economics of road congestion, Econometrica 32, 137-150 (1964). G. Kraft and M. Wohl, New directions for passenger demand analysis and forecasting, Transportation Res. 1, 205-230 (1968). L. B. Lave, “Transportation, City Size and Congestion Tolls,” Rand Corp. (1969). L. B. Lave and J. S. DeSalvo, Congestion, tolls, and the economic capacity of a waterway, J. Polit. Econ. 76, 375-391 (1968). G. W. Lerch, A study of the speed-volume relationship on high speed highways, unpublished Master’s dissertation, The Catholic University of America, Washington, D. C. (1970). A. D. May, Jr., Discussion of “freeway level of service as influenced by volume and capacity characteristics” by D. R. Drew and C. J. Keese, Highway Res. Record 99, 3943
(1965).
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14. W. R. McCasland, Comparison of two techniques of aerial photographs for application in freeway traffic operations studies, Highway Res. Record 65, 95-115 (1965). 15. R. E. Park, Congestion tolls for commercial airports, Econometrica 39, No. 5, 683 (1971). 16. G. J. Roth, An economic approach to traffic congestion, Town Planning Rev. 36, 49-61(1965). 17. G. J. Roth, “Paying for Roads: The Economics of Traffic Congestion,” Penguin, Baltimore (1967). 18. C. Sharp, Congestion and welfare-an examination of the case for a congestion tax, Econ. J. 76, 816-817 (1966). 19. C. Sharp, Congestion and welfare reconsidered, J. Transport Econ. Policy 2, 33-70 (1968). 20. R. J. Smeed, Traffic studies and urban congestion, J. Transportation Econ. Policy 2, 33-70 (1968). 21. R. J. Smeed et al., “Road Pricing: The Economic and Technical Possibilities,” H. M. Stationary Office No. 55-411, London (1964). 22. R. Strotz, Urban transportation parables, in “The Public Economy of Urban Communities” (Margolis, Ed.), Resources for the Future, New York (1965). 23. R. T. Underwood, Speed, volume and density relationships: Quality and theory of traffic flow, Yale Bureau of Highway Traffic, 141-188 (1961). 24. W. Vickrey, Optimization of traffic and facilities, J. Transport Econ. Policy 1, 123-135 (1967). 25. W. Vickrey, Congestion charges and welfare, J. Transport Econ. Policy 2, 107-118 (1968). 26. A. A. Walters, The theory and measurement of private and social cost of highway congestion, Econometrica 29 (1961).