Partial peak-load pricing of a transportation bottleneck with homogeneous and heterogeneous values of time☆
Introduction
Vickrey (1969) and many subsequent authors have argued that the first-best solution for a transportation bottleneck (such as a bridge) during the morning rush hour is to impose a first-best peak-load toll that varies continuously over time, reaching its maximum value at the most desirable crossing time, and thereby eliminating the wasteful queueing that exists in the absence of this toll.1 However, first-best peak-load tolls of this sort are never observed in the real world (see Lindsey et al., 2012), even though they are technologically feasible (see de Palma and Lindsey, 2011).2 The reason that first-best peak-load tolls are not used, in spite of the large efficiency gains that would be realized (see the numerical examples of Vickrey, 1969, Arnott et al., 1990a, and other authors), is presumably mainly political.3
Peak-load tolls that are used in the real world typically consist of a variation of a “step toll” that is constant over the off-peak hours, and a somewhat higher toll that is constant over the peak hours.4 Tolls of this sort can achieve a moderate efficiency gain by reducing the number of peak-period users, and thus reducing queueing to some extent, assuming that peak-period demand is not completely inelastic. However, queueing is reduced only to the extent that peak-period users are reduced, and not in the more fundamental way that the first-best toll completely eliminates queueing, by giving users an incentive to change their departure times.
Arnott et al. (1990a) presented the first analysis of a “step toll” that is positive and constant within an interval of time during the middle of the peak period, and zero (although it could be a lower constant value) at all other times, including less central times during the peak period.5 Thus it is applied wholly within the peak period, unlike the step toll described above, which applies different toll levels to the peak period and the off-peak period.6 This is (I believe) used less than the type of step toll mentioned above, but it is sometimes used in the real world, and sometimes with more than one step (see Lindsey et al., 2012). Under certain assumptions about consumer behavior, Arnott et al. (1990a) show, with a fixed number of peak-period users, that a step toll with one step is optimally set at half of the maximum value of the first-best toll, and can achieve 54.2% of the efficiency gain of the first-best toll.7 However, under more realistic assumptions about consumer behavior, a step toll with one step, which (once again) is optimally set at half of the maximum value of the first-best toll, achieves only 42.8% of the efficiency gain of the first-best toll, based on the “braking model” of Lindsey et al. (2012), discussed in my Section 12.
In this paper, I consider continuously time-varying tolls that have a much lower maximum value than the first-best peak-load toll, but which still achieve a large fraction of the efficiency gain of the first-best peak-load toll. The reason I consider these tolls, which I refer to as “partial peak-load tolls”, is that it is possible that the main political objection to a first-best peak-load toll is its high maximum value, rather than the fact that it varies continuously with time, so that a step toll may not be the only politically acceptable option. My partial peak-load tolls might possibly be as politically acceptable as a step toll that has a single step during the peak period, and they can be more efficient, relative to their maximum values, mostly because they are more effective at eliminating wasteful queueing. Also, they avoid the difficulties with consumer behavior resulting from a step toll that are identified by various authors and discussed in Section 12.
Sections 2 Assumptions and equilibrium conditions, 3 Queueing equilibrium with no pricing, 4 First-best peak-load pricing, 5 The first method of partial peak-load pricing, 6 The second method of partial peak-load pricing, 7 The second-best optimality of the second method of partial peak-load pricing assume that consumers are homogeneous. Section 2 presents the basic assumptions and equilibrium conditions of the model. A single route, possibly with a bridge, is subject to pure bottleneck-and-queue congestion. I assume that a fixed number of drivers must get to work, so that overall demand is completely inelastic. All drivers have the same preferred time of crossing the bottleneck, which is designated as time 0.8 Each driver has the same disutility per minute from waiting in a queue. Each driver has the same “schedule delay cost” function, with three parameters, of crossing the bottleneck before or after time 0 and thus being early or late to work. If the parameter , then the schedule delay cost function is piecewise linear, as in Arnott et al. (1990a, 1993) and most other papers in the literature. All of the figures assume this case. In Sections 2 Assumptions and equilibrium conditions, 3 Queueing equilibrium with no pricing, 4 First-best peak-load pricing, 5 The first method of partial peak-load pricing, 6 The second method of partial peak-load pricing, 7 The second-best optimality of the second method of partial peak-load pricing (but not in Sections 8 - 12) I also consider the case that , which means that the schedule delay cost function is strictly convex, since some authors in the literature consider this case. Drivers choose crossing times rationally to minimize private costs, including queueing costs, schedule delay costs, and toll costs.
Section 3 presents the no-toll solution. A queue forms that is longest at time 0, as is very familiar in the literature. Section 4 presents the first-best solution, which is also very familiar in the literature. A first-best peak-load toll, which reaches its largest value at time 0, eliminates the wasteful queueing and restores efficiency.
The first partial peak-load toll, presented in Section 5, is very simple to describe and analyze. It is just a fraction of the first-best peak-load toll of Section 4 at each value of time, , and it results in a fraction of the queueing of Section 3 at each value of . If , its maximum value is 50% of the maximum value of the first-best peak-load toll, and it results in 50% of the efficiency gain of the first-best peak-load toll.
The second partial peak-load toll is presented in Section 6. It is also simple to describe and analyze. At each instant of time, it is the minimum of the first-best toll and a fraction of the maximum value of the first-best peak-load toll. In the case where the maximum value of this toll is 50% of the maximum value of the first-best peak-load toll (), the efficiency gain is 75% of that of the first-best peak-load toll (if ).
Section 7 demonstrates that the second type of partial peak-load toll has a second-best optimality property. The partial peak-load toll that maximizes the efficiency gain subject to a maximum value of the toll is of the type described in Section 6.
The various tolls in Sections 4 First-best peak-load pricing, 5 The first method of partial peak-load pricing, 6 The second method of partial peak-load pricing, 7 The second-best optimality of the second method of partial peak-load pricing leave the private costs of all consumers unchanged, and they generate efficiency gains equal to the toll revenues collected, but the high maximum value of the first-best peak load toll of Section 4 might nevertheless make it politically unacceptable. However, it is useful to have a model in which there is a purely economic argument, based on the private costs of consumers, that the first-best peak-load might be politically unacceptable. This leads to the analysis of Sections 8 Queueing equilibrium with no toll and heterogeneous values of time, 9 The first-best peak-load toll with heterogeneous values of time, 10 A partial peak-load toll with heterogeneous values of time.
Sections 8 Queueing equilibrium with no toll and heterogeneous values of time, 9 The first-best peak-load toll with heterogeneous values of time, 10 A partial peak-load toll with heterogeneous values of time consider a model with low-income (type 1) consumers with low time values of both waiting and schedule delay, high-income (type 2) consumers with high time values of both waiting and schedule delay, and a small number of “type 3” consumers who have a low value of waiting time but a very high value of schedule delay time and thus must choose to cross the bottleneck at time 0 (the most desirable time). Section 8 examines the no-toll queueing equilibrium. Section 9 considers the first-best peak-load toll that eliminates all queueing. Section 10 considers a partial peak-load toll that is very similar to the one examined in Sections 6 The second method of partial peak-load pricing, 7 The second-best optimality of the second method of partial peak-load pricing.
One representative case based on the analysis of Sections 8 Queueing equilibrium with no toll and heterogeneous values of time, 9 The first-best peak-load toll with heterogeneous values of time, 10 A partial peak-load toll with heterogeneous values of time is summarized near the end of Section 10. The partial peak-load toll has a maximum value that is 1/3 of the maximum value of the first-best peak-load toll, but it collects 3/5 of the toll revenues of the first-best toll and achieves 5/7 of its efficiency gain.
The first-best toll of Section 9 and the partial peak-load toll of Section 10 both leave the private costs of type 1 consumers unchanged, and both decrease the private costs of type 2 consumers. However, the partial peak-load toll might be more politically acceptable since it leaves the private costs of type 3 consumers unchanged, whereas the first-best toll increases them significantly.
In Section 11, I consider how the results change, if there are different numbers of the three types of consumers, or if the ratios of two important time cost parameters (one for waiting time, the other for schedule delay) are not the same for consumers of types 1 and 2. In all of the cases except one, the partial peak-load toll has the significant political advantage of either leaving the private costs of type 3 consumers unchanged at the no-toll value (as in Sections 8 Queueing equilibrium with no toll and heterogeneous values of time, 9 The first-best peak-load toll with heterogeneous values of time, 10 A partial peak-load toll with heterogeneous values of time), or increasing them by a small fraction as much as the first-best peak-load toll.
Section 12 illustrates the results using a numerical example that extends the one in Arnott et al. (1990a). It also considers step tolls, considered by some authors, and sometimes used in the real world. Section 13 makes concluding remarks.
Section snippets
Assumptions and equilibrium conditions
The assumptions of the model can be outlined. A fixed number () of identical consumers must get to work by a route which has a bottleneck section with capacity . The route is subject to pure bottleneck-and-queue congestion, defined as follows. If the traffic flow on the route is less than , then no congestion occurs. However, if the flow of traffic attempting to use the route exceeds its capacity, then actual traffic flow equals and cars must wait in a queue before crossing the bottleneck
Queueing equilibrium with no pricing
Suppose that there is no toll, which is a common situation in the real world. From (9) with ,
Substituting (14a) into (12), using (1a), (1b), (14a) and (7), and then using (8), it is found thatwhere is defined by (11). From (10), (13) and (15), and then using (8) and (11),
This can also be derived by arguing that are , and using (1a), (1b), (8) and (11).
From (10),
First-best peak-load pricing
The first-best solution is attained by using the following first-best peak-load toll, chosen so that it eliminates all of the queueing (which is a pure social waste).From (17a), (17b), . Also, from (1a), (1b) and (7), .
From (9), (12) and (17a), (17b), it is seen thatso that the queue as a function of time, and aggregate queueing costs, are both 0, as desired. From (10), (13) and (19), and then using (8) and (11),
The first method of partial peak-load pricing
Consider the following partial peak-load toll, which is just a fraction of the first-best peak-load toll (see (17a), (17b)) at each value of .
From (21), . From (1) and (7), . This toll, or that of Section 6, is a possible example of the time-varying component of one of the tolls that can be chosen by the private owners of two substitute bottleneck routes in de Palma and Lindsey (2002).17
The second method of partial peak-load pricing
I now consider a second type of continuous partial peak-load toll, given by
The parameter is the maximum value of the toll, , as a fraction of the maximum value of the first-best peak-load toll of Section 4.
Consider the toll graphically. Recall that . From (25), . Also, from (1) and (7), . In Fig. 2, which assumes that , and (with points A, B,
The second-best optimality of the second method of partial peak-load pricing
The second method of partial peak-load pricing that was examined in Section 6 has a second-best optimality property. In particular, it is the partial peak-load toll that maximizes the efficiency gain subject to a maximum value of the toll.
First note that with the first-best peak-load toll of Section 4 and all of the partial peak-load tolls of Sections 5 The first method of partial peak-load pricing, 6 The second method of partial peak-load pricing, the following statement holds true. The
Queueing equilibrium with no toll and heterogeneous values of time
It is possible to extend my analysis to consider a distribution of consumers who have different queueing time costs (different values of ) and/or or different schedule delay cost functions (different values of and/or ). Papers that have considered this (but not with the partial peak-load tolls of this paper) include Vickrey (1973), Cohen (1987), Arnott et al. (1988, 1994), and van den Berg and Verhoef (2011a, 2011b).
In Sections 8 Queueing equilibrium with no toll and heterogeneous values of
The first-best peak-load toll with heterogeneous values of time
The first-best peak-load toll is chosen so that it completely eliminates all of the queueing (since queueing is a pure social waste). Thus must be chosen so that
The queue, , is zero and thus runs along the horizontal axis in Fig. 3.
I will also assume that
Since I want to be a continuous function of time, it follows that
With the first-best toll, the indeterminacy in the matching of type 1 and type 2
A partial peak-load toll with heterogeneous values of time
In this section, I consider the following partial peak-load toll, which is similar to the second type of partial peak-load toll considered earlier in the paper, in Sections 6 The second method of partial peak-load pricing, 7 The second-best optimality of the second method of partial peak-load pricing. I assume that (42) and (43) are satisfied, and that
Recall that (32) relates and for
Results for heterogeneous values of time with other assumptions
In this section, I consider how the results of Sections 8 Queueing equilibrium with no toll and heterogeneous values of time, 9 The first-best peak-load toll with heterogeneous values of time, 10 A partial peak-load toll with heterogeneous values of time change, if there are different numbers of the three types of consumers, or if the ratios of two important time cost parameters are not the same for consumers of types 1 and 2, as was assumed by (32). This section is substantially rewritten and
A numerical example and step tolls
In this section, I illustrate the results of Sections 3 Queueing equilibrium with no pricing, 4 First-best peak-load pricing, 5 The first method of partial peak-load pricing, 6 The second method of partial peak-load pricing, 7 The second-best optimality of the second method of partial peak-load pricing, 8 Queueing equilibrium with no toll and heterogeneous values of time, 9 The first-best peak-load toll with heterogeneous values of time, 10 A partial peak-load toll with heterogeneous values of
Concluding remarks
This section first summarizes the advantages of partial peak-load tolls based on the results in this paper. It then considers possible extensions of the analysis.
Sections 2 Assumptions and equilibrium conditions, 3 Queueing equilibrium with no pricing, 4 First-best peak-load pricing, 5 The first method of partial peak-load pricing, 6 The second method of partial peak-load pricing, 7 The second-best optimality of the second method of partial peak-load pricing assume homogeneous consumers. The
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I thank two anonymous referees for very helpful comments and suggestions, and Julia Braid and Liang Hu for help with the figures, table and equation editor.