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The demand curve under road pricing and the problem of political feasibility: A comment
Transpn. Res.-A. Vol. 29A, No. 6, pp. 459-465, 1995 Coovriaht 0 1995 Elsevier Science Ltd Printed’& &ea~Britain. All rights reserved 0965~8564/95 .$9.50+0.00
Pergemon
THE DEMAND CURVE UNDER ROAD PRICING AND THE PROBLEM OF POLITICAL FEASIBILITY: A COMMENT ERIK VERHOEF* Department of Spatial Economics, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands (Received I December 1994)
Abstract-This paper contains a comment on an article by Professor Lave, recently published in this journal (Trunspn Res. UIA, 83-91, 1994). Lave’s approach towards analyzing the political feasibility of road pricing is challenged on several grounds. In a simple setting, where individual road users are identical in terms of private cost of driving and valuation of time, Lave’s approach is seen to be, although less clear, in essence equivalent to the traditional textbook analysis. In a more complex setting, where differences across individuals are allowed for, his approach is seen to suffer from lacking recognition of differences in individual marginal utilities of income.
1. INTRODUCTION
In a recent article in this journal, Lave presents what he claims to be “a new way to depict the demand for travel on a priced road” (Lave, 1994, p. 83). Using this approach, an analysis is made of the political feasibility of road pricing schemes, essentially focusing on the number of ‘winners’ and ‘losers’ of such a scheme, and the intensity of welfare change for individual members in both groups. Although Lave’s article is intriguing and points out some very important topics, it may be challenged on several grounds. In Section 2, I will demonstrate that Lave’s approach is not at all that much different from traditional textbook analyses of road pricing. Then, in Section 3, I will argue that the potentially strong side of Lave’s way of analysis, being the explicit recognition of differences in individual preferences of time and money, is unfortunately not dealt with in an entirely satisfactory way in his paper. Section 4 concludes. 2. LAVE’S VERSUS THE TEXTBOOK
ANALYSIS OF ROAD PRICING
Whereas traditional diagrammatic representations of the problem of road pricing make use of one single demand curve, a rising marginal private cost curve and a rising marginal social cost curve for road use, Lave (1994) presents an analysis in which the latter two are left out. Instead, a family of demand curves is used, each of which represents what may be called the ‘net’ willingness to pay for using the road considered at a particular speed. The prefix ‘net’ indicates that the horizontal axis of Lave’s diagram actually corresponds to the private cost of road use at a certain speed, so that the different demand curves give the ‘willingness to pay a user fee’ for different speeds. Apparently, Lave interprets the congestion externalityt as a negative external benefit rather than as a positive external cost. At first glance, there is of course not much that can be said against such an approach; however, the question is whether it really yields important new insights. In this section, I will argue that it does not, at least not in the simplest case where all car drivers are iden*The author is affiliated to the Tinbergen Institute in Amsterdam and participates in the VSB Foundation sponsored research project “Transport and Environment”. tcongestion costs may include time costs as well as any other cost component negatively related to speed. Like Lave (1994), I will merely consider the time component in this comment. 459
Erik Verhoef
460
$
I
I
Q, Q,
MPB
Q
Fig. 1. Equivalence between the traditional diagrammatic representation of congestion (left) and Lave’s approach N&t).
tical as far as their private cost and value of time is concerned, and where they only differ with respect to the willingness to pay to make a trip. A discussion of the case where individual valuations of time differ among road users is postponed to Section 3. Figure 1 compares the traditional approach to Lave’s representation. The left panel shows the traditional diagrammatic representation of congestion, where the MPB (marginal private benefit) curve gives the demand, and the MPC gives the marginal private cost, which are rising because of congestion. The market outcome will be at Qt. As the identical individual road users perceive the average social cost (ASC) as their marginal private cost, the marginal social cost (MSC) will be higher than MPC; the vertical distance between these two represents the external congestion cost. The socially optimal level of road use is at Q2, where marginal benefits are equal to marginal social cost and the welfare loss represented by triangle bzc is avoided. The optimal road price therefore amounts to r*. Clearly, with road users identical in terms of costs and values of time, everyone is worse off if the tax revenues are not somehow redistributed: drivers between Q2 and Q, are priced off the road and have to choose an option that is inferior to them (otherwise they would not have been on the road before road pricing was implemented), and the QZ drivers that remain on the road see their individual surplus decline by f - a, which can be broken down into a cost advantage of a - g due to decreased congestion on the one hand, and the road price f - g they have to pay in exchange for this on the other. The right panel of Fig. 1. shows Lave’s representation of the same problem (see Lave, 1994, for more details). Whereas in the left panel the MPB was solely related to benefits associated with making the trip, the marginal net private benefit (MNPB) curves depicted here already contain a discount depending on the private cost of making the trip at the speed given in the subscripts. Therefore, for a certain speed X, MNPB, in the right panel is equal to MPB-MPC(Q,) in the left panel, where QX denotes the usage Q consistent with speed X, and MPC(Q,) the marginal private cost at this speed. Both represent the individual surplus of making a trip under given conditions (that is: at a certain speed), and are therefore equal to the marginal ‘willingness to pay a user fee’. As said before, time costs are one of the elements in MPC, and therefore different MNPB curves are drawn for different speeds (we assume the market outcome Qt to correspond to a speed of 45 mph, and the optimal outcome Qz to correspond to a speed of 50 mph).* *By connecting points such as b and c in the right panel, the ‘general’ MNPB curve can be found, which is equivalent to the vertical distance between MPB.and MPC in the left panel. This curve [which Lave (1994) calls the ‘composite demand curve] gives the equilibrium levels of road usage for each possible road price. In the sketched case, this curve (not included in the diagram) would be concavely shaped.
The demand curve under road pricing
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At a higher speed, one is prepared to pay a higher fee, hence the positions of the curves. However, the (relative) positions of the MNPB curves are of course by no means coincidental. The vertical distance represents the private valuation of speeding up from 45 mph to 50 mph, and it should make no difference whether one interprets this as a decline in private costs (as in the left panel), or as an increase in private benefits (as in the right panel). In other words, given the assumption of road users being identical in terms of costs and values of time, the vertical distance d - e in the right panel should be equal to a - g in the left panel.* Given the equivalence of the two approaches sketched above, it should be no surprise that welfare measures as discussed in Lave (1994) can easily be found in the left panel as well. That is, the private users’ surplus associated with unpriced road use at 45 mph (abe), or at 50 mph with the associated necessary road price r* (fed) appear in both diagrams. However, finding the optimal road price is a bit harder in the right panel. As a matter of fact, it is remarkable that Lave (1994) does not mention optimality at all when discussing his approach, but rather sort of postulates the regulator’s wish to decrease congestion and increase speed. This is illustrated by phrases like: ‘Suppose we want to decrease congestion enough to increase the speed to 40 mph” (Lave, 1994, p. 84). Still, the optimal level of road use can be found in the right panel by searching that particular speed for which the sum of the users’ surplus and the tax revenues (areas such as ghcd) is maximized. Likewise, the welfare loss bzc cannot easily be found in the right-hand panel, since in that approach it is actually given by ghcd - abe. Finally, Lave suggests that one might construct a “composite demand curve that incorporated the three-way relationship between price, usage and speed” (Lave, 1994, p. 85) by connecting points such as b and c in the right panel (see also footnote * on p. 2). It may now be clear that the need for this ‘composite’ demand curve follows directly from the fact that, for each different speed level, the horizontal axis in the right panel corresponds to a different level of private costs. Consequently, whereas in the left panel it is obvious that a restriction of road use from Qi to Q2 requires a road price of f-g rather than f-a, Lave needs two paragraphs to convince us on that (Lave, 1994, pp. 84-85). Although the two diagrammatic representations may be formally equivalent, in my opinion the traditional one deserves preference. A main reason is that the interdependence between the private cost and the congestion externality is completely clouded in Lave’s diagram, as information on the former is hidden along the horizontal axis, whereas the latter is modelled as a negative external benefit rather than as a positive external cost. Apart from that, in the traditional approach it is much easier to show the optimal road price and the welfare gain resulting from it. On the contrary, in Lave (1994) the question of optimality tends to be neglected. Therefore, a diagram as Fig. 2 in Lave (1994) does not really say much, as the question of political feasibility of road pricing is likely to become more pregnant the closer one gets to its optimal level. It may well be that Qi and Q2 mentioned there are both way above the optimal level of road use. In that case, the conclusion would be that for very modest, far below optimal levels of road pricing, it may be easy to find social support. However, in this section I had to make a very restrictive assumption, namely that the private cost of driving and the value of time are equal for all road users. As Lave rightly points out, this need certainly not be the case. It is indeed difficult to see how this phenomenon could be captured in the traditional approach, as in that case the average social cost and marginal private cost will no longer coincide. Although at first glance Lave’s approach seems to offer a promising way of dealing with this issue, it is my opinion that the analysis in Section 3 of Lave (1994) is not entirely satisfactory. *As there is no reason to have any assumption on the specific curvature of the demand and the cost curves (as long as the former is negatively sloped and the latter two are positively sloped and consistent), there is also no reason for assuming any specific curvature of these MNPB, curves (convexity, concavity, linearity)-as long as they are downward sloping and have equal derivatives (for each x), they can be generated by some possible set of demand and cost curves (note that the curves in the right panel of Fig. 1 are consistent with the curves in the left panel of Fig. 1).
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3. COPING WITH A HETEROGENEOUS
GROUP OF ROAD USERS
The problem I have with the way in which Lave (1994) copes with heterogeneity among individual road users actually starts in Section 2, where Lave asserts that the demand curves slope “... downward for two reasons: (a) The value of time differs across individuals; and (b) given the diversity of origins and destinations, some drivers have better alternatives than others” (Lave, 1994, p. 84). In the first place, the first of these two reasons is not treated very carefully when Lave continues that drivers with a high value of time are to be found at the left-hand side of the curve, and drivers with a low value of time at the right-hand side of the curve. I will argue below that, although this may be the case for curves associated with high speeds, the reverse may actually hold for curves associated with low speeds. Related to this, it seems to me that Lave leaves out a couple of important reasons for differences in the individuals’ willingness to pay for making a trip at a certain moment. The most important of these are: (a) the purpose of the trip; (b) the individual marginal utility of money (as related to income and/or purchasing power); and (c) the possibility of rescheduling the trip. These issues become crucial when the analysis focuses on the political feasibility of a road pricing scheme given heterogeneity among individuals, which is actually the central topic of Lave’s article. It is my opinion that a one-dimensional diagrammatic approach is simply not capable of dealing with the complexities introduced when allowing for heterogeneity among road users. Figure 2 represents Fig. 4 as presented in Lave (1994). The idea behind this diagram is to ignore the allocation of the tax revenues, and to focus on individual changes in surpluses after the introduction of road pricing as the driving force behind the ‘political response’ of the road users involved. The bold curves give the marginal net private benefit at two speed levels: 25 mph and 55 mph respectively. The former is the ‘zero-road-price’ market outcome (readily giving individual surpluses of road use), and the latter is associated with a road price P2 (so that the individual surplus can be found by subtracting the road price PZ from the net private benefit). According to Lave (1994), the introduction of a road price PZ, necessary to accomplish an increase in speed from 25 mph up to 55 mph, may be difficult to realize from the perspective of political feasibility. Enumerating the winners and losers, “... all drivers to the right of Ed may perceive that they are worse off after the fee is imposed, and their political actions will depend on these perceptions” (Lave, 1994, p. 87). Ed, then, is that particular driver for whom the surplus (labelled a) is
Road price
I
b
JLU
ED
Fig. 2. Congestion pricing: who wins, who whines?
Q
The demand curve under road
pricing
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the same before and after the introduction of the road price P2. For drivers to the left of Ed, the surplus has increased, whereas the opposite holds for drivers to the right of Ed. However, the critical implicit assumption Lave makes is that the individual road users remain on their original ‘position’ on the horizontal axis after the road price is imposed. This is only true if there were a perfect rank correlation between the drivers’ individual values of time and their marginal utilities of income (or ‘values of money’), which needs certainly not be the case. Rather, when allowing for heterogeneity among road users where the value of time and the marginal utility of income are allowed to differ across individuals, one would expect the former to increase, and the latter to decrease with higher incomes. It may for instance well be the case that the reason that drivers to the right of Ed have a relatively low willingness to pay for making a 25 mph trip is their high value of time. However, when road pricing is imposed and the speed increases up to 55 mph, they may find it attractive to make the trip. If their high value of time is accompanied with a relatively low value of money (which is a likely combination for high income groups), they may actually have a relatively high willingness to pay for making a 55 mph trip. Taking an extreme example, suppose Sue is a yuppie-like business woman. In that case, she may actually shift to the outer left position on the horizontal axis after road pricing is imposed (taking over Jim’s initial position), simply because she feels she can afford little time but a lot of money for using the road considered. Taking the other extreme, suppose Jim is a retired man, with plenty of time but with a modest budget. In the most extreme case, he does not care at all about speeding up (he has a zero value of time), and may find himself not willing to pay the road price of P *. Poor Jim will then be priced off the roadalthough Lave (1994) predicts him to be the absolute winner of a road pricing scheme. More generally, the shift in the marginal net private benefit curve does not convey all information on the political feasibility of the proposed road pricing scheme. If the drivers between Ed and Sue have a high value of time and a low marginal utility of income, they may all win as a result of the scheme. For convenience, in Fig. 2 it is assumed that the members of this group all shift to the left along the horizontal axis, but retain their relative positions. In that case, the vertical distance between the bold 55 mph marginal net private benefit curve and the dashed curve (giving their original surplus at 25 mph) represents the increase in their surpluses. On the other hand, the drivers between Jim and Ed with a low value of time and a high value of money may all be priced off the road (note that, if all of these drivers had a zero marginal utility of time, the 55 mph marginal net private benefit curve should actually have a relatively flat segment between Ed’s and Jim’s original willingness to pay). Clearly, contrary to Lave’s prediction, a referendum among existing and potential road users would in this case result in implementation of the proposed road pricing scheme. However, it may be noted that the intensity of welfare change is here much more severe for those priced off the road, than for those who will remain using the road. In conclusion, the type of data needed to predict the political feasibility of road pricing is even more complicated than Lave (1994) asserts. His concern with “the right-hand end of the demand curve” (Lave, 1994, p. 87) is not sufficient, as the distribution of individual road users along the horizontal axis will generally not remain the same after the introduction of a road pricing scheme. Rather, when predicting something like the political feasibility of a certain road pricing scheme (let alone the optimal level of such a fee), one would have to know: (a) the effect of a certain road price on road usage and speed; (b) individual marginal utilities of income; (c) individual values of time; and (d) (the valuation of) the options open to those who decide to quit. Only then can the winners and losers, as well as the intensity of their welfare change, properly be predicted. The next question, concerning the extent to which these effects on individual surpluses will subsequently translate into political actions, is therewith still unanswered. The problem of the political feasibility of road pricing gains in complexity because of the interactions between the elements (a)-(d) mentioned above. Still, it is crucial that these factors should be studied over the whole population of (potential) road users; not merely
Erik Verhoef
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for those on the right-hand end of the demand curve. From a theoretical perspective [and given the elements (a)-(d) mentioned above and their interactions], the only conclusion one can generally draw is that, if the possible allocation of the tax revenues is indeed not taken into account, a road pricing scheme is likely to receive more public support, the less homogeneous the population of road users in terms of valuations of time and money (compare Fig. 1, where it was shown that, with identical road users, everybody is worse off at any positive road price). However, it seems to me that the potential allocation of tax revenues should not be entirely excluded from the analysis. First, this would imply that the economist would more or less voluntary let go of the whole concept of Pareto optimality-which is the very economic reason for implementing congestion pricing in the first place. Secondly, it may be worthwhile to consider the possibility of somehow convincing the public of the close relationship that exists between the revenues of optimal road pricing and the cost of the optimal level of infrastructure supply. 4.
CONCLUSION
Although Lave (1994) presents an intriguing way of analyzing the political feasibility of road pricing, his approach is unfortunately somewhere halfway between what we already knew and what we would like to know. In Section 2, I showed that in a simple model where road users are identical in terms of private costs and values of time, Lave’s approach is inferior to the traditional approach for two main reasons: (a) the interdependence between the private cost and the congestion externality is completely clouded, as information on the former is hidden along the horizontal axis, whereas the latter is modelled as a negative external benefit rather than as a positive external cost; and (b) in the traditional approach it is much easier to show the optimal road price and the welfare gain resulting from it. When switching to a more realistic setting, where differences across individuals are allowed for [the type of situation that Lave (1994) actually wants to deal with], a more complex approach is needed, which should at least allow for different marginal utilities of income in addition to differences in valuation of time. Lave’s concern with “the right-hand end of the demand curve” (Lave, 1994, p. 87) is not sufficient, as the distribution of individual road users along the horizontal axis will generally not remain the same after introduction of a road pricing scheme. Rather, when predicting something like the political feasibility of a certain road pricing scheme (let alone the optimal level of such a fee), one would have to know: (a) the effect of a certain road price on speed; (b) individual marginal utilities of income; (c) individual values of time; and (d) (the valuation of) the options open to those who decide to quit. Only then can the winners and losers, as well as the intensity of their welfare change, properly be predicted. Acknowledgements-1 would like to thank Richard Emmerink, Henk van Gent and Piet Rietveld for valuable contributions to this comment.
REFERENCE Lave C. (1994) The demand curve under road pricing and the problem of political feasibility. Trunspn Rex 28A, 83-91.
AUTHOR’S REPLY Verhoef has two main points. The first is incorrect, the second is valid but overstated. For his first point, Verhoef goes to some trouble to show he can convert most of my diagram into the conventional one. But the ability to manipulate one diagram into another is not grounds for saying they are the same. The focus of the conventional diagram is on the difference between private and social cost and the deviation from optimality that results when drivers pursue their individual interest. It does this very well, but its very ability to focus on externalities displaces a view of other issues. Although the conventional diagram might be used to examine why road users oppose road tolls, researchers have not used it in that fashion. Perhaps different questions require different approaches. Moreover, the conventional diagram cannot consider changes in consumer welfare when different consumers have different values
The demand curve under road pricing
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of time. My diagram can do this. Indeed all the comments that Verhoef makes in the second half of his comment can only be made in the context of my diagram, not the conventional one. It has been a commonplace event for transportation economists to put the conventional diagram on the board, note the self-evident optimality of pricing solutions, and then sit down waiting for the world to adopt this obviously correct solution. Well, we have been waiting for 70 years now, and it’s worth asking what are the facets of the problem that we have been missing? Why is the world reluctant to do the obvious? My diagram concentrates on that question: why don’t people accept the pricing solution? It does this by explicitly focusing on the winners and losers: who gets pushed off by the toll, and who remains-and it makes it easy to see that even those who remain can be worse off than they were before the toll. Finally, my diagram makes explicit the problems that confront a new toll road that parallels an existing congested highway. Verhoef s second point is correct, but overstated. It is possible that a congestion toll will cause a difference in the ranking of individuals willingness to pay: their relative positions on horizontal axis may change when given the opportunity to drive faster. In most situations I would expect these shifts to be small. In what Verhoef himself describes as “an extreme example” (p.7) the shift is quite large. Verhoef has made a valuable contribution by clarifying the implicit assumptions behind my diagram, but my main points are unaltered: I showed that the political feasibility of congestion pricing depended upon the ratio of winners to losers, and that even many users who remained on the road would still be losers. Verhoef does not disagree. I showed that public reaction to the newly built toll roads could not be used to estimate public reaction to a situation where we add a toll to an existing road-because the users of the new road come from the wrong end of the demand curve. Verhoef goes even farther, saying that in some cases the users may come from even more diverse segments of the demand curve. I showed that the problem of estimating winners and losers is too complex to be solved by analytical methods, that it will require data from actual experiences on the actual road link. Verhoef agrees. Finally, I showed that some of the new toll roads will incur public opposition after they are opened. They must be operated far below their capacity in order to keep congestion levels high on the parallel free road. The toll road is selling a speed differential-if it admits too many people it will decongest the free-road and lose its speed advantage. Thus almost all drivers will still be stuck on the congested free road, viewing the essentially empty toll road that is the solution to their frustration. Verhoef does not comment on this result.
n(A)
29-6-F
Pergemon
THE DEMAND CURVE UNDER ROAD PRICING AND THE PROBLEM OF POLITICAL FEASIBILITY: A COMMENT ERIK VERHOEF* Department of Spatial Economics, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands (Received I December 1994)
Abstract-This paper contains a comment on an article by Professor Lave, recently published in this journal (Trunspn Res. UIA, 83-91, 1994). Lave’s approach towards analyzing the political feasibility of road pricing is challenged on several grounds. In a simple setting, where individual road users are identical in terms of private cost of driving and valuation of time, Lave’s approach is seen to be, although less clear, in essence equivalent to the traditional textbook analysis. In a more complex setting, where differences across individuals are allowed for, his approach is seen to suffer from lacking recognition of differences in individual marginal utilities of income.
1. INTRODUCTION
In a recent article in this journal, Lave presents what he claims to be “a new way to depict the demand for travel on a priced road” (Lave, 1994, p. 83). Using this approach, an analysis is made of the political feasibility of road pricing schemes, essentially focusing on the number of ‘winners’ and ‘losers’ of such a scheme, and the intensity of welfare change for individual members in both groups. Although Lave’s article is intriguing and points out some very important topics, it may be challenged on several grounds. In Section 2, I will demonstrate that Lave’s approach is not at all that much different from traditional textbook analyses of road pricing. Then, in Section 3, I will argue that the potentially strong side of Lave’s way of analysis, being the explicit recognition of differences in individual preferences of time and money, is unfortunately not dealt with in an entirely satisfactory way in his paper. Section 4 concludes. 2. LAVE’S VERSUS THE TEXTBOOK
ANALYSIS OF ROAD PRICING
Whereas traditional diagrammatic representations of the problem of road pricing make use of one single demand curve, a rising marginal private cost curve and a rising marginal social cost curve for road use, Lave (1994) presents an analysis in which the latter two are left out. Instead, a family of demand curves is used, each of which represents what may be called the ‘net’ willingness to pay for using the road considered at a particular speed. The prefix ‘net’ indicates that the horizontal axis of Lave’s diagram actually corresponds to the private cost of road use at a certain speed, so that the different demand curves give the ‘willingness to pay a user fee’ for different speeds. Apparently, Lave interprets the congestion externalityt as a negative external benefit rather than as a positive external cost. At first glance, there is of course not much that can be said against such an approach; however, the question is whether it really yields important new insights. In this section, I will argue that it does not, at least not in the simplest case where all car drivers are iden*The author is affiliated to the Tinbergen Institute in Amsterdam and participates in the VSB Foundation sponsored research project “Transport and Environment”. tcongestion costs may include time costs as well as any other cost component negatively related to speed. Like Lave (1994), I will merely consider the time component in this comment. 459
Erik Verhoef
460
$
I
I
Q, Q,
MPB
Q
Fig. 1. Equivalence between the traditional diagrammatic representation of congestion (left) and Lave’s approach N&t).
tical as far as their private cost and value of time is concerned, and where they only differ with respect to the willingness to pay to make a trip. A discussion of the case where individual valuations of time differ among road users is postponed to Section 3. Figure 1 compares the traditional approach to Lave’s representation. The left panel shows the traditional diagrammatic representation of congestion, where the MPB (marginal private benefit) curve gives the demand, and the MPC gives the marginal private cost, which are rising because of congestion. The market outcome will be at Qt. As the identical individual road users perceive the average social cost (ASC) as their marginal private cost, the marginal social cost (MSC) will be higher than MPC; the vertical distance between these two represents the external congestion cost. The socially optimal level of road use is at Q2, where marginal benefits are equal to marginal social cost and the welfare loss represented by triangle bzc is avoided. The optimal road price therefore amounts to r*. Clearly, with road users identical in terms of costs and values of time, everyone is worse off if the tax revenues are not somehow redistributed: drivers between Q2 and Q, are priced off the road and have to choose an option that is inferior to them (otherwise they would not have been on the road before road pricing was implemented), and the QZ drivers that remain on the road see their individual surplus decline by f - a, which can be broken down into a cost advantage of a - g due to decreased congestion on the one hand, and the road price f - g they have to pay in exchange for this on the other. The right panel of Fig. 1. shows Lave’s representation of the same problem (see Lave, 1994, for more details). Whereas in the left panel the MPB was solely related to benefits associated with making the trip, the marginal net private benefit (MNPB) curves depicted here already contain a discount depending on the private cost of making the trip at the speed given in the subscripts. Therefore, for a certain speed X, MNPB, in the right panel is equal to MPB-MPC(Q,) in the left panel, where QX denotes the usage Q consistent with speed X, and MPC(Q,) the marginal private cost at this speed. Both represent the individual surplus of making a trip under given conditions (that is: at a certain speed), and are therefore equal to the marginal ‘willingness to pay a user fee’. As said before, time costs are one of the elements in MPC, and therefore different MNPB curves are drawn for different speeds (we assume the market outcome Qt to correspond to a speed of 45 mph, and the optimal outcome Qz to correspond to a speed of 50 mph).* *By connecting points such as b and c in the right panel, the ‘general’ MNPB curve can be found, which is equivalent to the vertical distance between MPB.and MPC in the left panel. This curve [which Lave (1994) calls the ‘composite demand curve] gives the equilibrium levels of road usage for each possible road price. In the sketched case, this curve (not included in the diagram) would be concavely shaped.
The demand curve under road pricing
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At a higher speed, one is prepared to pay a higher fee, hence the positions of the curves. However, the (relative) positions of the MNPB curves are of course by no means coincidental. The vertical distance represents the private valuation of speeding up from 45 mph to 50 mph, and it should make no difference whether one interprets this as a decline in private costs (as in the left panel), or as an increase in private benefits (as in the right panel). In other words, given the assumption of road users being identical in terms of costs and values of time, the vertical distance d - e in the right panel should be equal to a - g in the left panel.* Given the equivalence of the two approaches sketched above, it should be no surprise that welfare measures as discussed in Lave (1994) can easily be found in the left panel as well. That is, the private users’ surplus associated with unpriced road use at 45 mph (abe), or at 50 mph with the associated necessary road price r* (fed) appear in both diagrams. However, finding the optimal road price is a bit harder in the right panel. As a matter of fact, it is remarkable that Lave (1994) does not mention optimality at all when discussing his approach, but rather sort of postulates the regulator’s wish to decrease congestion and increase speed. This is illustrated by phrases like: ‘Suppose we want to decrease congestion enough to increase the speed to 40 mph” (Lave, 1994, p. 84). Still, the optimal level of road use can be found in the right panel by searching that particular speed for which the sum of the users’ surplus and the tax revenues (areas such as ghcd) is maximized. Likewise, the welfare loss bzc cannot easily be found in the right-hand panel, since in that approach it is actually given by ghcd - abe. Finally, Lave suggests that one might construct a “composite demand curve that incorporated the three-way relationship between price, usage and speed” (Lave, 1994, p. 85) by connecting points such as b and c in the right panel (see also footnote * on p. 2). It may now be clear that the need for this ‘composite’ demand curve follows directly from the fact that, for each different speed level, the horizontal axis in the right panel corresponds to a different level of private costs. Consequently, whereas in the left panel it is obvious that a restriction of road use from Qi to Q2 requires a road price of f-g rather than f-a, Lave needs two paragraphs to convince us on that (Lave, 1994, pp. 84-85). Although the two diagrammatic representations may be formally equivalent, in my opinion the traditional one deserves preference. A main reason is that the interdependence between the private cost and the congestion externality is completely clouded in Lave’s diagram, as information on the former is hidden along the horizontal axis, whereas the latter is modelled as a negative external benefit rather than as a positive external cost. Apart from that, in the traditional approach it is much easier to show the optimal road price and the welfare gain resulting from it. On the contrary, in Lave (1994) the question of optimality tends to be neglected. Therefore, a diagram as Fig. 2 in Lave (1994) does not really say much, as the question of political feasibility of road pricing is likely to become more pregnant the closer one gets to its optimal level. It may well be that Qi and Q2 mentioned there are both way above the optimal level of road use. In that case, the conclusion would be that for very modest, far below optimal levels of road pricing, it may be easy to find social support. However, in this section I had to make a very restrictive assumption, namely that the private cost of driving and the value of time are equal for all road users. As Lave rightly points out, this need certainly not be the case. It is indeed difficult to see how this phenomenon could be captured in the traditional approach, as in that case the average social cost and marginal private cost will no longer coincide. Although at first glance Lave’s approach seems to offer a promising way of dealing with this issue, it is my opinion that the analysis in Section 3 of Lave (1994) is not entirely satisfactory. *As there is no reason to have any assumption on the specific curvature of the demand and the cost curves (as long as the former is negatively sloped and the latter two are positively sloped and consistent), there is also no reason for assuming any specific curvature of these MNPB, curves (convexity, concavity, linearity)-as long as they are downward sloping and have equal derivatives (for each x), they can be generated by some possible set of demand and cost curves (note that the curves in the right panel of Fig. 1 are consistent with the curves in the left panel of Fig. 1).
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3. COPING WITH A HETEROGENEOUS
GROUP OF ROAD USERS
The problem I have with the way in which Lave (1994) copes with heterogeneity among individual road users actually starts in Section 2, where Lave asserts that the demand curves slope “... downward for two reasons: (a) The value of time differs across individuals; and (b) given the diversity of origins and destinations, some drivers have better alternatives than others” (Lave, 1994, p. 84). In the first place, the first of these two reasons is not treated very carefully when Lave continues that drivers with a high value of time are to be found at the left-hand side of the curve, and drivers with a low value of time at the right-hand side of the curve. I will argue below that, although this may be the case for curves associated with high speeds, the reverse may actually hold for curves associated with low speeds. Related to this, it seems to me that Lave leaves out a couple of important reasons for differences in the individuals’ willingness to pay for making a trip at a certain moment. The most important of these are: (a) the purpose of the trip; (b) the individual marginal utility of money (as related to income and/or purchasing power); and (c) the possibility of rescheduling the trip. These issues become crucial when the analysis focuses on the political feasibility of a road pricing scheme given heterogeneity among individuals, which is actually the central topic of Lave’s article. It is my opinion that a one-dimensional diagrammatic approach is simply not capable of dealing with the complexities introduced when allowing for heterogeneity among road users. Figure 2 represents Fig. 4 as presented in Lave (1994). The idea behind this diagram is to ignore the allocation of the tax revenues, and to focus on individual changes in surpluses after the introduction of road pricing as the driving force behind the ‘political response’ of the road users involved. The bold curves give the marginal net private benefit at two speed levels: 25 mph and 55 mph respectively. The former is the ‘zero-road-price’ market outcome (readily giving individual surpluses of road use), and the latter is associated with a road price P2 (so that the individual surplus can be found by subtracting the road price PZ from the net private benefit). According to Lave (1994), the introduction of a road price PZ, necessary to accomplish an increase in speed from 25 mph up to 55 mph, may be difficult to realize from the perspective of political feasibility. Enumerating the winners and losers, “... all drivers to the right of Ed may perceive that they are worse off after the fee is imposed, and their political actions will depend on these perceptions” (Lave, 1994, p. 87). Ed, then, is that particular driver for whom the surplus (labelled a) is
Road price
I
b
JLU
ED
Fig. 2. Congestion pricing: who wins, who whines?
Q
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pricing
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the same before and after the introduction of the road price P2. For drivers to the left of Ed, the surplus has increased, whereas the opposite holds for drivers to the right of Ed. However, the critical implicit assumption Lave makes is that the individual road users remain on their original ‘position’ on the horizontal axis after the road price is imposed. This is only true if there were a perfect rank correlation between the drivers’ individual values of time and their marginal utilities of income (or ‘values of money’), which needs certainly not be the case. Rather, when allowing for heterogeneity among road users where the value of time and the marginal utility of income are allowed to differ across individuals, one would expect the former to increase, and the latter to decrease with higher incomes. It may for instance well be the case that the reason that drivers to the right of Ed have a relatively low willingness to pay for making a 25 mph trip is their high value of time. However, when road pricing is imposed and the speed increases up to 55 mph, they may find it attractive to make the trip. If their high value of time is accompanied with a relatively low value of money (which is a likely combination for high income groups), they may actually have a relatively high willingness to pay for making a 55 mph trip. Taking an extreme example, suppose Sue is a yuppie-like business woman. In that case, she may actually shift to the outer left position on the horizontal axis after road pricing is imposed (taking over Jim’s initial position), simply because she feels she can afford little time but a lot of money for using the road considered. Taking the other extreme, suppose Jim is a retired man, with plenty of time but with a modest budget. In the most extreme case, he does not care at all about speeding up (he has a zero value of time), and may find himself not willing to pay the road price of P *. Poor Jim will then be priced off the roadalthough Lave (1994) predicts him to be the absolute winner of a road pricing scheme. More generally, the shift in the marginal net private benefit curve does not convey all information on the political feasibility of the proposed road pricing scheme. If the drivers between Ed and Sue have a high value of time and a low marginal utility of income, they may all win as a result of the scheme. For convenience, in Fig. 2 it is assumed that the members of this group all shift to the left along the horizontal axis, but retain their relative positions. In that case, the vertical distance between the bold 55 mph marginal net private benefit curve and the dashed curve (giving their original surplus at 25 mph) represents the increase in their surpluses. On the other hand, the drivers between Jim and Ed with a low value of time and a high value of money may all be priced off the road (note that, if all of these drivers had a zero marginal utility of time, the 55 mph marginal net private benefit curve should actually have a relatively flat segment between Ed’s and Jim’s original willingness to pay). Clearly, contrary to Lave’s prediction, a referendum among existing and potential road users would in this case result in implementation of the proposed road pricing scheme. However, it may be noted that the intensity of welfare change is here much more severe for those priced off the road, than for those who will remain using the road. In conclusion, the type of data needed to predict the political feasibility of road pricing is even more complicated than Lave (1994) asserts. His concern with “the right-hand end of the demand curve” (Lave, 1994, p. 87) is not sufficient, as the distribution of individual road users along the horizontal axis will generally not remain the same after the introduction of a road pricing scheme. Rather, when predicting something like the political feasibility of a certain road pricing scheme (let alone the optimal level of such a fee), one would have to know: (a) the effect of a certain road price on road usage and speed; (b) individual marginal utilities of income; (c) individual values of time; and (d) (the valuation of) the options open to those who decide to quit. Only then can the winners and losers, as well as the intensity of their welfare change, properly be predicted. The next question, concerning the extent to which these effects on individual surpluses will subsequently translate into political actions, is therewith still unanswered. The problem of the political feasibility of road pricing gains in complexity because of the interactions between the elements (a)-(d) mentioned above. Still, it is crucial that these factors should be studied over the whole population of (potential) road users; not merely
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for those on the right-hand end of the demand curve. From a theoretical perspective [and given the elements (a)-(d) mentioned above and their interactions], the only conclusion one can generally draw is that, if the possible allocation of the tax revenues is indeed not taken into account, a road pricing scheme is likely to receive more public support, the less homogeneous the population of road users in terms of valuations of time and money (compare Fig. 1, where it was shown that, with identical road users, everybody is worse off at any positive road price). However, it seems to me that the potential allocation of tax revenues should not be entirely excluded from the analysis. First, this would imply that the economist would more or less voluntary let go of the whole concept of Pareto optimality-which is the very economic reason for implementing congestion pricing in the first place. Secondly, it may be worthwhile to consider the possibility of somehow convincing the public of the close relationship that exists between the revenues of optimal road pricing and the cost of the optimal level of infrastructure supply. 4.
CONCLUSION
Although Lave (1994) presents an intriguing way of analyzing the political feasibility of road pricing, his approach is unfortunately somewhere halfway between what we already knew and what we would like to know. In Section 2, I showed that in a simple model where road users are identical in terms of private costs and values of time, Lave’s approach is inferior to the traditional approach for two main reasons: (a) the interdependence between the private cost and the congestion externality is completely clouded, as information on the former is hidden along the horizontal axis, whereas the latter is modelled as a negative external benefit rather than as a positive external cost; and (b) in the traditional approach it is much easier to show the optimal road price and the welfare gain resulting from it. When switching to a more realistic setting, where differences across individuals are allowed for [the type of situation that Lave (1994) actually wants to deal with], a more complex approach is needed, which should at least allow for different marginal utilities of income in addition to differences in valuation of time. Lave’s concern with “the right-hand end of the demand curve” (Lave, 1994, p. 87) is not sufficient, as the distribution of individual road users along the horizontal axis will generally not remain the same after introduction of a road pricing scheme. Rather, when predicting something like the political feasibility of a certain road pricing scheme (let alone the optimal level of such a fee), one would have to know: (a) the effect of a certain road price on speed; (b) individual marginal utilities of income; (c) individual values of time; and (d) (the valuation of) the options open to those who decide to quit. Only then can the winners and losers, as well as the intensity of their welfare change, properly be predicted. Acknowledgements-1 would like to thank Richard Emmerink, Henk van Gent and Piet Rietveld for valuable contributions to this comment.
REFERENCE Lave C. (1994) The demand curve under road pricing and the problem of political feasibility. Trunspn Rex 28A, 83-91.
AUTHOR’S REPLY Verhoef has two main points. The first is incorrect, the second is valid but overstated. For his first point, Verhoef goes to some trouble to show he can convert most of my diagram into the conventional one. But the ability to manipulate one diagram into another is not grounds for saying they are the same. The focus of the conventional diagram is on the difference between private and social cost and the deviation from optimality that results when drivers pursue their individual interest. It does this very well, but its very ability to focus on externalities displaces a view of other issues. Although the conventional diagram might be used to examine why road users oppose road tolls, researchers have not used it in that fashion. Perhaps different questions require different approaches. Moreover, the conventional diagram cannot consider changes in consumer welfare when different consumers have different values
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of time. My diagram can do this. Indeed all the comments that Verhoef makes in the second half of his comment can only be made in the context of my diagram, not the conventional one. It has been a commonplace event for transportation economists to put the conventional diagram on the board, note the self-evident optimality of pricing solutions, and then sit down waiting for the world to adopt this obviously correct solution. Well, we have been waiting for 70 years now, and it’s worth asking what are the facets of the problem that we have been missing? Why is the world reluctant to do the obvious? My diagram concentrates on that question: why don’t people accept the pricing solution? It does this by explicitly focusing on the winners and losers: who gets pushed off by the toll, and who remains-and it makes it easy to see that even those who remain can be worse off than they were before the toll. Finally, my diagram makes explicit the problems that confront a new toll road that parallels an existing congested highway. Verhoef s second point is correct, but overstated. It is possible that a congestion toll will cause a difference in the ranking of individuals willingness to pay: their relative positions on horizontal axis may change when given the opportunity to drive faster. In most situations I would expect these shifts to be small. In what Verhoef himself describes as “an extreme example” (p.7) the shift is quite large. Verhoef has made a valuable contribution by clarifying the implicit assumptions behind my diagram, but my main points are unaltered: I showed that the political feasibility of congestion pricing depended upon the ratio of winners to losers, and that even many users who remained on the road would still be losers. Verhoef does not disagree. I showed that public reaction to the newly built toll roads could not be used to estimate public reaction to a situation where we add a toll to an existing road-because the users of the new road come from the wrong end of the demand curve. Verhoef goes even farther, saying that in some cases the users may come from even more diverse segments of the demand curve. I showed that the problem of estimating winners and losers is too complex to be solved by analytical methods, that it will require data from actual experiences on the actual road link. Verhoef agrees. Finally, I showed that some of the new toll roads will incur public opposition after they are opened. They must be operated far below their capacity in order to keep congestion levels high on the parallel free road. The toll road is selling a speed differential-if it admits too many people it will decongest the free-road and lose its speed advantage. Thus almost all drivers will still be stuck on the congested free road, viewing the essentially empty toll road that is the solution to their frustration. Verhoef does not comment on this result.
n(A)
29-6-F