When are anonymous congestion charges consistent with marginal cost pricing?

Journal of Public Economics 67 (1998) 45–64

When are anonymous congestion charges consistent withmarginal cost pricing?

*Richard Arnott, Marvin KrausDepartment of Economics, Boston College, Chestnut Hill, MA 02167, USA

Received 1 May 1995; received in revised form 11 April 1997

Abstract

There are constraints on pricing congestible facilities. First, if heterogeneous users areobservationally indistinguishable, then congestion charges must be anonymous. Second, thetime variation of congestion charges may be constrained. Do these constraints underminethe feasibility of marginal cost pricing, and hence the applicability of the first-best theory ofcongestible facilities? For an important class of congestible facilities, we show that ifheterogeneous users behave identically when using the facility and if the time variation ofcongestion charges is unconstrained, then marginal cost pricing is feasible with anonymouscongestion charges. If, however, the time variation of congestion charges is constrained,optimal pricing with anonymous congestion charges entails Ramsey pricing. 1998Elsevier Science S.A.

Keywords: Clubs; Congestion; Congestion pricing; Externalities

JEL classification: H2

1. Introduction

It has long been recognized that the theory of clubs, the economic theory ofurban transportation, and the theory of congestible facilities have many features in

1common (e.g., Berglas and Pines (1981)). Some would argue that each treats a

*Corresponding author. Tel.: 11 617 552 3692; fax: 11 617 552 2308.1The theory of local public goods is similar but has distinctively different features. The local public

good is provided only to those who own or rent property in the jurisdiction, and the congestion occurswith respect to land. In what follows, we ignore local public goods.

0047-2727/98/$19.00 1998 Elsevier Science S.A. All rights reserved.PII S0047-2727( 97 )00059-5

46 R. Arnott, M. Kraus / Journal of Public Economics 67 (1998) 45 –64

different subset of models within a common general class of models of congestiblefacilities. Others would argue that the theories are essentially identical, but haveevolved to a large extent independently due to the different policy issues andliteratures from which they derive their inspiration. Whatever the extent of overlap,the theories continue to develop to a large extent independently.

In recent years, a new class of dynamic models has been developed in theeconomic theory of urban transportation (Arnott et al. (1993), provides a goodintroduction). Previous models of congestible facilities deal with the peak-loadproblem by dividing up the period of use (typically a day) into intervals, and thenapplying static analysis to each interval separately (though they do take intoaccount cross-price effects between intervals). The new class of dynamic modelsinnovates in two distinct ways. First, it incorporates state-variable congestioneffects which take into account that the current characteristics of congestiondepend not only on the current number of individuals entering the facility but alsoon the time pattern of entrance earlier in the day (put alternatively, congestion mayspill over from one time interval to the next). The most obvious example is aqueue (indeed, the first of the new class of dynamic models, due to Vickrey(1969), was of a queue of cars behind a bottleneck in the morning rush hour). Butthere are many others: the quality of a seat in a movie theatre, occupancy of aparking lot, voltage fluctuations after an interval of intense electrical use, and cold,low-pressure nine o’clock showers are just a few examples. Second, it explicitlymodels individuals’ time-of-use decisions, which take into account the tradeoffbetween using the facility at a more convenient time versus at a less congestedtime, as well as possibly time-varying user charges. This additional structure leadsto new economic insights and in some circumstances facilitates solving forequilibrium.

This paper has two objectives, one more general, one more specific. The moregeneral objective is to illustrate that the new class of dynamic transportationmodels can be adapted to a class of congestible facilities. This is not surprisingsince individuals always care about time of use to some extent and all congestionexhibits some history dependence. For some problems, the adaptation requiresonly a relabelling of variables; for others, the new class of models needs to besubstantially modified prior to application. The more specific objective is toanswer the question posed in the paper’s title. With virtually any congestiblefacility, there are constraints on congestion pricing. For one thing, congestioncharges can be differentiated across heterogeneous users only on the basis ofobservable differences. For another, there may be constraints on the time variationof congestion charges. Do these constraints undermine the feasibility of marginal(social) cost pricing? Put differently, does the first-best theory of congestiblefacilities (which includes the well-known result that, for a congestible facilitywhich exhibits constant long run average cost, the revenue from the optimalcongestion charge exactly covers capacity costs when the facility’s capacity isoptimal) continue to apply in the presence of these constraints? We answer thesequestions for congestible facilities with a particular set of characteristics.

R. Arnott, M. Kraus / Journal of Public Economics 67 (1998) 45 –64 47

The paper is organized as follows. In Section 2, we review what light theexisting literature on dynamic transportation models throws on these questions.The existing literature does provide answers, but only for the bottleneck model ofrush-hour auto congestion. It is not clear a priori whether the results are specific tothat model, generalize to only other dynamic transportation models, or generalizeto some other or perhaps all other models of congestible facilities. To investigatethis, in Section 3 we present a dynamic model of congestion that is considerablymore general than the bottleneck model. We also discuss the scope of itsapplication. To do this, we develop a typology of congestible facilities which mayprove useful in future work on congestion. Section 4 examines whether anonymityin congestion pricing and constraints on the time variation of congestion chargesundermine marginal cost pricing for this more general dynamic model ofcongestion, and contains the central analysis and results of the paper. The analysisgives rise naturally to two alternative definitions of the congestion externality in adynamic context, and we investigate the relationship between them. In the courseof the analysis, we generalize existing theorems on the extent to which self-financing occurs under optimal congestion charges and capacity. Section 5presents some concluding observations.

2. The existing literature

A toll will be said to have the form V( ? ) if it is a member of the family oftime-dependent toll functions t(t) 5 V(t) 1 k, where k, a constant, is the toll level.Arnott et al. (1993) demonstrated, in the context of Vickrey’s bottleneck model ofmorning rush-hour auto congestion (Vickrey (1969)), that with identical commu-ters, marginal cost pricing can be achieved independently of the form of the toll.Two points, A and B, are considered where commuters live and work, respective-ly. A and B are connected by a single road which has a single bottleneck withcapacity s – only s commuters can pass through the bottleneck per unit time. If thearrival rate at the bottleneck exceeds s, a queue develops. Commuters have a

*common work start time t and incur a schedule delay cost for arriving at work*before or after t . User cost equals schedule delay cost plus queuing time cost, and

trip price equals user cost plus the toll. An individual decides when to leave hometo minimize trip price. The crucial point is that in equilibrium the trip price mustbe constant over the departure interval and at least as high outside this interval.Otherwise, some individuals would change their departure times. For any numberof daily trips N, solving for equilibrium entails solving for a time pattern ofdepartures, which determines a time pattern of queuing time cost and of scheduledelay cost, such that the equilibrium trip-price condition holds. Equilibrium isunique, so that total user costs are determined as a function of the number of dailytrips and the form of the toll; i.e. TC 5 TC(N,V ). Marginal social cost is thendetermined as MSC(N,V ) 5 ≠TC(N,V ) /≠N. Thus, marginal social cost, too, isconstant over the equilibrium departure interval. Now consider a change in the toll


level, k. For any given value of N, this has no effect on the equilibrium departurepattern and thus on marginal social cost. However, the equilibrium trip price goesthrough the same change as k. Thus, regardless of the form of the toll, k can be setso that price equals marginal social cost.

Arnott and Kraus (1995) extended the bottleneck analysis of Arnott et al. (1993)to treat heterogeneous commuters who behave the same way in traffic. Commutersare assumed to behave the same way in traffic but to differ from one another inobservationally indistinguishable ways such as the shadow value of time or workstart time, which precludes the possibility of nonanonymous tolling. The questionis whether the anonymity of tolling in this situation renders marginal cost pricinginfeasible and whether the form of the toll matters for this (for example, doesconstraining the toll to be uniform over the rush hour affect the answer?). Theargument with two commuter types (indexed by i51,2) goes as follows. Givenany anonymous toll function t(t) 5 V(t) 1 k, total user costs in equilibrium can bedetermined as a function of the number of daily trips of each type, from which thecorresponding marginal social cost functions, MSC (N ,N ,V ), can be determined.i 1 2

Also, trip price for each type must be constant over its equilibrium departureinterval and at least as high at other times. Now suppose that k is set so that theprice of a type 1 trip equals its marginal social cost, implying that t(t) equals thecongestion externality of a type 1 trip for all t in the 1’s departure interval. Theissue is whether this results in the price of a type 2 trip equalling its marginal

˜social cost. If the two types have a common departure time t, then this is˜ ˜equivalent to their imposing the same congestion externality at t, since t(t ) must

˜equal each type’s congestion externality at t. Arnott and Kraus (1995) show thatthis condition is met with the optimal time-varying toll, which has both V( ? ) andk optimal, but not generally otherwise. The reason is that with the optimaltime-varying toll, the congestion externality is anonymous – at any departure time,it is the same for a 2 as for a 1 – but with other toll forms the congestionexternality is generally nonanonymous. The results can be generalized to anarbitrary number of commuter types. Thus, in the bottleneck model with heteroge-neous commuters who behave the same way in traffic, marginal cost pricing ispossible with the optimal anonymous time-varying toll, but not generally withanonymous tolls of other forms.

The possibility of marginal cost pricing in these circumstances therefore turnson whether or not the congestion externality is anonymous. How can twocommuters who depart at the same time and who behave the same way in trafficgenerate different congestion externalities? A simple illustration comes from thepreceding bottleneck model when there is no toll.

* *Assume that work start time is t for both types, that arrivals after t areprohibited, and that the user cost functions are c (t) 5 a (queuing time (t)) 1i i

b(time early (t)), where t is departure time, a , which is the only way the typesi

differ, is the shadow cost of queuing time, and b is the shadow cost of time early.Assume that a . a . b – 1’s dislike being stuck in traffic more than 2’s, and1 2


both types prefer time early at work to time spent in traffic. With no toll, trip priceequals user cost. Thus, in equilibrium all commuters of a given type will incur thesame user cost. Since commuters who depart later arrive less early, they mustincur greater queuing costs and hence face a longer queue than those who departearlier. The 1’s dislike queuing more, so they depart earlier when the queue isshorter. In Fig. 1, abcd and ad are respectively the equilibrium cumulative

˜ ˜departure and arrival schedules; 1’s depart over [t ,t ], 2’s over [t,t ]. To satisfy theo f

equal-trip-price conditions, the queue (given by the vertical distance between thecumulative departure and arrival schedules) grows more rapidly over the 2’sdeparture interval than over the 1’s. Since time early cost increases linearly withtime early at the same rate for both types, queuing cost must decrease linearly withtime early at the same rate for both types. But since a . a , queuing time – and1 2

therefore queue length – must grow faster over the 2’s departure interval than overthe 1’s.

˜Now consider removing one type 1 commuter at t and adding a type 2˜commuter at t. This causes the equilibrium cumulative departure schedule to

change to ab9c9d, implying that all other type 2’s face a longer queue. Otherwise,the equilibrium is unchanged. This implies that a type 2 commuter imposes a

˜larger congestion externality at t than a type 1 commuter. Thus, two commuterswho depart at the same time and who behave the same way in traffic can generatedifferent congestion externalities because they cause the queue to evolve different-ly in order to satisfy the equal-trip-price conditions.

With the optimal time-varying toll, there is no queue. The source of the

Fig. 1. The no-toll equilibrium in the bottleneck model. Notes: ad has a slope of s. ab and bc haveslopes of a s /(a 2 b ) and a s /(a 2 b ), respectively.1 1 2 2


difference in the two types’ congestion externality disappears, making thecongestion externality anonymous.

The obvious question is to what extent the results from the traffic bottleneckmodel generalize to other models of congestible facilities. We address this issue inthe next two sections. We show that the results do indeed generalize to thecongestible facilities described by our model. With users who behave identicallywhen using a congestible facility but differ in other, unobservable ways, thetime-varying congestion externality is anonymous with the efficient time pattern ofutilization. Setting an anonymous congestion charge equal at all points in time tothe corresponding congestion externality decentralizes the efficient time pattern ofutilization. Under these circumstances, anonymous congestion charges are con-sistent with marginal cost pricing. When, however, the form of the toll isconstrained, the time pattern of utilization is inefficient, the congestion externalityis generally nonanonymous, and anonymous congestion charges are inconsistentwith marginal cost pricing.

3. A more general dynamic model of congestion

3.1. The model

A congestible facility can be used by G types of consumers. The number ofdaily utilizations by consumers of type i (i 5 1, . . . ,G) that begin by time of day tis denoted by N (t). Let t and t denote the earliest and latest times of day,i o f

respectively, that a utilization by any consumer begins. Then, for all i 5 1, . . . ,G,the number of daily utilizations by type i consumers is given by N (t ).i f

For a utilization beginning at time t, a consumer incurs a cost (excluding thecongestion charge) given by c (t,n(t),N(t),x(t),s), where i is the consumer’s type;i

2~ ~N(t) 5 (N (t), . . . ,N (t)); n(t) 5 (n (t), . . . ,n (t)) 5 (N (t), . . . ,N (t)); x(t) 51 G 1 G 1 G

(x (t), . . . ,x (t)), a vector of state variables at time t; and s 5 (s , . . . ,s ), a vector1 J 1 K

of variables which characterizes the various dimensions of the facility’s capacity.The state equations for x , . . . ,x are given by1 J

~x (t) 5 g (t,n(t),N(t),x(t),s), j 5 1, . . . ,J. (1)j j

x(t ) is given, while x(t ) is free. Capacity costs are given by h(s).o f

Note that different consumer types may differ in terms of their cost functions orhow their usage directly affects other consumers’ costs or the state of the

2If type i consumers begin utilizations only over some subinterval of [t ,t ], then N ( ? ) may have ao f i

kink at either the beginning or the end of the subinterval. It is straightforward to show that one canallow for a finite number of times for which a kink exists along the optimal time path of one or more ofthe N without affecting the results.i


congestible facility. Later, we shall particularize the treatment of congestion toconsider types that are observationally indistinguishable and whose usage affectsother users’ costs and the state of the facility in the same way.

A type i utilization beginning at time t is subject to a congestion charge t (t),i

which for the moment can differ across consumer types. The price of theutilization is given by

p (t) 5 c (t,n(t),N(t),x(t),s) 1 t (t). (2)i i i

Consumers choose when to begin utilizations to minimize price. Let t9 be a time* *chosen by a type i consumer to begin a utilization. Then p (t9) 5 p , where p isi i i

the minimum of p (t) over all t.i

The number of daily utilizations by type i consumers, N (t ), is given by thei f

*demand relationship N (t ) 5 D ( p ). This assumes that the benefit from ai f i i

utilization is independent of time of use. Put differently, utilizations at differenttimes are perfect substitutes in demand. Preferences regarding time of use are

*captured in the cost function. Denoting the inverse of D by P , p 5 P (N (t )).i i i i i f

Let T denote the set of times at which type i consumers begin utilizations. Thei

economy’s equilibrium conditions are that, for all i 5 1, . . . ,G,

p (t) 5 P (N (t )), t [ T , (3a)i i i f i

p (t) $ P (N (t )), t [⁄ T . (3b)i i i f i

33.2. Scope of application

Congestible facilities differ considerably from one another in their demandcharacteristics, in their congestion technologies, and in the allocation schemesutilized. There are no doubt many possible classification schemes. The one weadopt is designed to identify what types of congestible facilities are reasonablywell described by our model.

3.2.1. Demand characteristicsDecisions faced by consumers. Some congestible facilities are visited only once,

others multiple times per day by the same consumer. An individual typically visitsan art gallery at most once in a day. But she may utilize the photocopy machine atwork several times in a day.

For some facilities, the severity of congestion depends primarily on the numberof individuals using the facility (e.g., public transportation). For others, it dependsas well on decisions consumers make concerning their use of the facility. The mostpervasive of these is duration of use – length of visit to a swimming pool or lengthof a telephone call at a telephone booth. Other examples are the rate at which an

3This section draws on Lindsey (1990).


individual consumes electricity, which can vary over the course of a day,roughhousing in a swimming pool, and standing close to paintings in a gallery.

3.2.2. Congestion technologiesNature of congestion. Congestion may manifest itself as delay, degradation in

quality, or denial of service. A line at a store check-out counter entails delay; acrowded swimming pool entails degradation in quality; and a sold-out concertentails some consumers being denied entry.

Service discipline. Perhaps the most common service discipline is FIFO (first-in,first-out). The most obvious example of this is queuing; another is service in ashop. In other cases, the service discipline is random access. A good example ofthis is long-distance telephone service. In still others, such as the assignment of ataxi under a dispatch system, the service discipline is intermediate to these twoextremes. The service discipline may also be endogenous in that it may depend onconsumer behavior. Push-and-shove is one example, overtaking another.

Stock or flow capacity. A toll booth or ticket window has a flow capacity, atelephone line or swimming pool a stock capacity.

Single or multiple congestible elements of capacity. Most congestible facilitieshave multiple congestible elements of capacity. For example, attending a profes-sional baseball game involves queues at the turnstiles, congestion in the ballpark,and delays in exiting. And in driving to work, each segment of the route is adifferent congestible element of capacity.

3.2.3. Allocation schemesThe three main types of allocation schemes are free access, pricing and

rationing. And pricing may be standardized or classificatory, with higher classes4getting preferential treatment of some sort in their use of the facility.

According to this classification scheme, the congestible facilities covered by ourmodel: (1) involve only two decisions on the part of a consumer – whether to visitthe facility once or not at all and, if once, when to begin the utilization; (2) allowcongestion to manifest itself as delay or degradation of quality, but not denial ofservice; (3) have FIFO service disciplines; (4) involve flow but not stock capacity;(5) can have multiple congestible elements of capacity; and (6) have standardizedpricing. Generalizing the model to allow for stock capacity is quite

4Club theory, the theory of congestible facilities, and much of the economic theory of transportationapply the same generic black-box analysis to all congestible facilities. It should be evident from theabove classification scheme, however, that at the micro level congestible facilities may differsignificantly from one another. It may be, as the traditional theories assume, that all congestiblefacilities have a common reduced-form representation. But this is certainly not obvious, particularlywhen distortions are present. Thus, we favor the use of more micro models of congestible facilities –but that is not the theme of this paper.


5straightforward. So too is augmenting the model to allow for differentiated pricing6according to the quality or speed of service provided. But characteristics (1) and

(3) are essential in our treatment and restrictive. As for the first of these, the firstbest requires marginal cost pricing or efficient regulation on all margins of choice.The model could be extended to treat other margins of choice that can be priced(such as visit length) or regulated (a barrier can be put around paintings in agallery to prevent visitors from blocking others’ view). But if there are anysignificant margins of choice that cannot be efficiently priced or regulated, theproblem immediately becomes second-best. Characteristic (3) is essential becauseof our state system formulation, under which the congestion that an individualexperiences depends on utilization of the facility only through the number ofindividuals initiating utilizations at the same time and the time pattern ofutilizations initiated earlier in the day, as captured by state variables. With otherservice disciplines, the congestion that an individual experiences depends as wellon utilizations initiated after her own. Examples of this are random access

7long-distance telephone service and overtaking on a hiking trail.How extensive then is the scope of application of our model? Its feature that

consumers decide only on participation and time of entry makes it inappropriatefor facilities for which an important decision variable is visit length (e.g., parkinggarages and many recreational facilities) and excludes electricity, whose consump-tion by the typical user varies throughout the day. Its FIFO queue disciplineexcludes regular long-distance telephone service. Its assumption that the conges-tion that an individual experiences does not depend on utilizations that begin afterhers renders it inapplicable to many recreational facilities for which the visit lengthis extended, such as a swimming pool or a network of hiking or ski trails. And itsassumption of standardized pricing rules out its application to ticket classes inairline travel, for example, though as noted above extension to treat differentiatedpricing for differentiated service is straightforward. Apart from these situations,applicability of the model is more a matter of degree than of kind.

In the context of auto traffic, the model is applicable to congestion on a link(unless overtaking is important) but not on a network where early-departingexurban commuters are impeded by later-departing suburban commuters. A golf

5With stock capacity, there may be times in the day when it is optimal for individuals to enter thefacility in masses. This means allowing for points of discontinuity in the functions N ( ? ). This in turni

requires the integral in Eq. (4) to be written instead as a sum of integrals over subintervals defined bythe points of discontinuity and that a term be added to Eq. (4) to capture the user costs of individualswho are part of entry masses.

6An example of the former is classes of concert tickets, and of the latter a freeway with regular andexpress lanes.

7We conjecture that our results extend to the case in which a consumer’s cost depends on the entiretime path of usage, both before and after she enters the facility. But for reasons of tractability,investigating this will require that we switch to a discrete time formulation.


course without a reservation system is an example of a recreational facility whichaccords well with the model. Special exhibitions at museums and art galleries fitthe model well. So too do concerts and movie theatres without reservations andpriority pricing, for which early arrival means better seating. And the modelextended to treat variable visit length applies to the batch processing of short jobswithout priority service and to 900 numbers for which calls are taken in order.

Thus, while our model is certainly not generally applicable, it does encompassan important class of facilities. The value of our model should not, however, bejudged primarily on the basis of its scope of application. From the precedingdiscussion, it should be evident that congestible facilities are qualitativelyheterogeneous in many dimensions. Any general model would therefore becomplex and cumbersome. A major contribution of our paper is methodological –to illustrate how equilibrium in time-of-use decisions and the time pattern ofcongestion can be modelled for a class of congestible facilities, and how thequalitative characteristics of equilibrium depend on the form of pricing and theanonymity of users. Many features of the method should generalize to the analysisof congestible facilities not covered by our model.

4. Analysis

4.1. The social optimum

N (t )i fLet the benefits to type i consumers be given by e P (q)dq. Society’s0 i

problem can then be stated as

N (t ) ti f fG Gmax O E P (q)dq 2 h(s) 2EOn (t)c (t,n(t),N(t),x(t),s)dt, (4)i i is,t ,t ,n( ? )o f i51 i51

t0 o

s.t.

~N (t) 5 n (t), i 5 1, . . . ,G, (5)i i

~x (t) 5 g (t,n(t),N(t),x(t),s), j 5 1, . . . ,J, (6)j j

with initial conditions

N (t ) 5 0, i 5 1, . . . ,G, (7)i o

ox (t ) 5 x , j 5 1, . . . ,J, (8)j o j

o ofor given x , . . . ,x , and inequality constraints1 J

n (t) $ 0, i 5 1, . . . ,G. (9)i


Eqs. (4)–(9) define an optimal control problem with state variables N , . . . ,N and1 G

x , . . . ,x and control variables n , . . . ,n . Assume that c ( ? ), . . . ,c ( ? ),1 J 1 G 1 G

g ( ? ), . . . ,g ( ? ) and h(?) have continuous first derivatives, and define the1 J

HamiltonianG G

H(t,n(t),N(t),x(t),m(t),l(t),s) 5 2On (t)c (t,n(t),N(t),x(t),s) 1Om (t)n (t)i i i ii51 i51

J

1Ol (t)g (t,n(t),N(t),x(t),s), (10)j jj51

where m (t) and l (t) are the costate variables corresponding to Eqs. (5) and (6),i j

respectively. The first-order conditions, in addition to Eqs. (5)–(9), are

≠g≠c≠H ji] ] ]5 2 c 2On 1 m 1Ol # 0, i9 5 1, . . . ,G, (11)i 9 i i 9 j≠n ≠n ≠ni 9 i 9 i 9i j

≠g≠c≠H ji] ] ]n 5 n 2 c 2On 1 m 1Ol 5 0, i9 5 1, . . . ,G, (12)S Di 9 i 9 i 9 i i 9 j≠n ≠n ≠ni 9 i 9 i 9i j

≠g≠c≠H ji]] ]] ]]~m 5 2 5On 2Ol , i9 5 1, . . . ,G, (13)i 9 i j≠N ≠N ≠Ni 9 i 9 i 9i j

≠g≠c≠H ji~ ] ] ]l 5 2 5On 2Ol , j9 5 1, . . . ,J, (14)j 9 i j≠x ≠x ≠xj 9 j 9 j 9i j

P (N (t )) 5 m (t ), i 5 1, . . . ,G, (15)i i f i f

l (t ) 5 0, j 5 1, . . . ,J, (16)j f

H(t ,n(t ),N(t ),x(t ),m(t ),l(t ),s) 5 0, (17)f f f f f f

H(t ,n(t ),N(t ),x(t ),m(t ),l(t ),s) 5 0, (18)o o o o o o

t f

≠g≠c≠h ji] ] ]2 2E On 2Ol dt 5 0, k 5 1, . . . ,K. (19)S Di j≠s ≠s ≠sk k ki j

to

4.2. Decentralization of the optimum

There are two distinct ways to identify the decentralizing congestion charges,depending on how congestion externalities are conceptualized. The first approachis based on the left-hand side of Eq. (11). At any time t, this expression equals themarginal net social benefit of a type i9 utilization beginning at t, conditional on thevalues of state variables at t, but allowing the controls to optimally readjust attimes greater than t (see, e.g., Dorfman (1969)). Because of the conditioning on


state variables at t, we term this the ceteris paribus marginal net social benefit. Inthe absence of a congestion charge, a type i9 utilization beginning at t has amarginal net private benefit given by P (N (t )) 2 c (t). Subtracting this from thei 9 i 9 f i 9

ceteris paribus marginal net social benefit and reversing the sign gives the ceterisparibus congestion externality. Denoting the ceteris paribus congestion externalityby n (t),i 9

≠g≠c ji] ]n (t) 5On 2 m 2Ol 1 P (N (t )). (20)i 9 i i 9 j i 9 i 9 f≠n ≠ni 9 i 9i j

We now show that this is the optimal congestion charge.

Proposition 1. An optimum to Eqs. (4)–(9), conditional on s, can be decentralizedwith the ceteris paribus congestion charge t (t) 5 n (t), where n (t) is evaluatedi 9 i 9 i 9

at the conditional optimum.

Proof. See Appendix A.

Remark 1. In the special case of static congestion, Eq. (20) reduces to the classiccongestion externality

≠ci]n (t) 5On . (21)i 9 i ≠ni 9i

Congestion is static when the only state variables of the model are N , . . . ,N and1 G

these do not appear as arguments of the cost functions c ( ? ). Then the third termi

~on the right-hand side of Eq. (20) can be dropped, and Eq. (13) reduces to m 5 0,i 9

implying that m in Eq. (20) is equal to m (t ). The result then follows from Eq.i 9 i 9 f

(15).

The second approach is based on the equivalence between the optimal controlproblem defined by Eqs. (4)–(9) and a two-stage optimization problem in whichN (t ), . . . ,N (t ) are given at the first stage and optimized at the second stage. The1 f G f

problem at the first stage is to minimize the last term in Eq. (4), subject to Eqs.(5)–(9) and given values for N (t ), . . . ,N (t ) and s. The minimum value function1 f G f

for this problem is the facility’s short run cost function, which we denote byC(N (t ), . . . ,N (t ),s). At the second stage, values for N (t ), . . . ,N (t ) and s are1 f G f 1 f G f

determined to maximize net social benefits, written asN (t )i f

GO E P (q)dq 2 h(s) 2 C(N (t ), . . . ,N (t ),s). (22)i 1 f G fi51

0

For later use, we note that this requires

P (N (t )) 2 ≠C /≠N (t ) 5 0, i 5 1, . . . ,G. (23)i i f i f


Starting from a solution to the stage one problem, what is the marginal cost of atype i9 utilization beginning at time t when the utilization times of otherconsumers are adjusted, if necessary, to minimize costs? If t [ T , this mutatisi 9

mutandis marginal cost must equal ≠C /≠N (t ), while if t [⁄ T , it cannot be lessi 9 f i 9

than this value. The mutatis mutandis congestion externality is simply thedifference between the mutatis mutandis marginal cost and the marginal user’sown cost, c (t). Denoting the mutatis mutandis congestion externality by w (t),i 9 i 9

w (t) 5 ≠C /≠N (t ) 2 c (t) for t [ T , (24a)i 9 i 9 f i 9 i 9

w (t) $ ≠C /≠N (t ) 2 c (t) for t [⁄ T . (24b)i 9 i 9 f i 9 i 9

The relationship between w (t) and n (t) is given by:i 9 i 9

Lemma 1. At an optimum to Eqs. (4)–(9), conditional on s, the ceteris paribuscongestion externality n (t) is equal to the mutatis mutandis congestion externalityi 9

8w (t) for all t [ T .i 9 i 9


It can then easily be shown (see Appendix A for the proof) that

Proposition 2. An optimum to Eqs. (4)–(9), conditional on s, can be decentralizedwith the mutatis mutandis congestion charge t (t) 5 w (t), where w (t) isi 9 i 9 i 9

evaluated at the conditional optimum.

We next introduce the assumption(A.1) For all i 5 1, . . . ,G, c ( ? ) is of the form c (t,n (t) 1 ? ? ? 1 n (t),N (t) 1 ?i i 1 G 1

? ? 1 N (t),x(t),s), while for all j 5 1, . . . ,J, g ( ? ) takes the form g (t,n (t) 1 ? ? ?G j j 1

1 n (t),N (t) 1 ? ? ? 1 N (t),x(t),s).G 1 G

8The explanation for Lemma 1 is that ≠C /≠N (t ), which gives the mutatis mutandis marginal cost ofi 9 f

a type i9 utilization beginning at any time t [ T , is unaffected by being conditioned on the values thati 9

the state variables take on at an arbitrary time t9 in an initial optimum to the stage one problem.Demonstrating this result is a straightforward application of the Envelope Theorem. Consider a variantof the stage one problem in which N and x are given at some arbitrary time t9, and write the problem’s

˜ ˜minimum value function as C(N(t ),s,N(t9),x(t9)). Consider the minimum of C over N(t9) and x(t9), andf˜* * * *assume that this occurs at N(t9) 5 N (t9) and x(t9) 5 x (t9). Then C(N(t ),s) 5 C(N(t ),s,N (t9),x (t9))f f

and

˜ ˜* * * *≠C(N(t ),s) ≠C(N(t ),s,N (t9),x (t9)) ≠C(N(t ),s,N (t9),x (t9)) ≠N (t9)f f f i]]] ]]]]]] ]]]]]]]]5 1O

≠N (t ) ≠N (t ) ≠N (t9) ≠N (t )ii 9 f i 9 f i i 9 f˜ ≠x (t9)* *≠C(N(t ),s,N (t9),x (t9)) jf]]]]]]]]1O . (25)

≠x (t9) ≠N (t )j j i 9 f

The stated result then follows from the fact that the last two terms in Eq. (25) vanish.


(A.1) provides a formal characterization of consumers behaving identically9when using the congestible facility.

Our key result is:

Lemma 2. Suppose that (A.1) holds. Then at an optimum to Eqs. (4)–(9),conditional on s, the ceteris paribus congestion externality n (t) is the same for alli 9

consumer types i9.

Proof. From Eqs. (20) and (15) we have that at any time t9

≠g≠c ji] ]n (t9) 5On 2 m 2Ol 1 m (t )i 9 i i 9 j i 9 f≠n ≠ni 9 i 9i j

t f

≠g≠c ji] ] ~5On 2Ol 1Em dt. (26)i j i 9≠n ≠ni 9 i 9i j

t9

From Eqs. (26) and (13)

t f

≠g ≠g≠c ≠cj ji i] ] ]] ]]n (t9) 5On 2Ol 1E On 2Ol dt. (27)S Di 9 i j i j≠n ≠n ≠N ≠Ni 9 i 9 i 9 i 9i j i j

t9

But under (A.1) each term on the right-hand side of Eq. (27) is independent of i9.The intuition for Lemma 2 goes as follows. It is a well-known result of optimal

control theory that at an optimum to Eqs. (4)–(9), conditional on s, the shadowvalue of any state variable (which allows for optimal readjustment of subsequentcontrols) is equal to the corresponding constrained shadow value, which does not

10permit subsequent controls to readjust. It follows immediately that the same istrue of n (t). At some time t, then, consider a marginal increase in n (t), withi 9 i 9

n , . . . ,n held fixed at times greater than t. Regardless of the specific choice of i9,1 G

the new time paths followed by o n and o N are the same. Together with (A.1),i i i i

9Club models make no mention of time of use. They can be interpreted as truly static or asemploying a reduced-form specification which implicitly treats time of use. A distinction is made inclub theory between anonymous and nonanonymous crowding (see Scotchmer (1994)), based onwhether or not consumers enter in the same way in the club cost function. According to the formerinterpretation of club models, which is standard, anonymous crowding occurs when consumers behavethe same way when using the facility. According to the latter, more sophisticated interpretation,however, whether crowding is anonymous depends on the form of pricing used, but in general requiresthat consumers not only behave the same way when using the facility but also have the same valuationsof time and tastes over quality attributes of the facility. Because of this ambiguity, we have avoided theterms.

10Allowing the controls to adjust introduces additional terms which vanish when evaluated at theinitial optimum.


this implies that the same is true for the new time path followed by x and hencec , . . . ,c . The congestion externality is therefore independent of i9.1 G

Together with Proposition 1, Lemma 2 implies the main result of the paper:

Proposition 3. Suppose that (A.1) holds. Then an optimum to Eqs. (4)–(9),conditional on s, can be decentralized with anonymous congestion charges.

Proof. Proposition 3 is a direct implication of Proposition 1 and Lemma 2.

When constraints exist on the time variation of the congestion charge, it isgenerally not possible to establish marginal cost prices for all types. Determiningthe optimal congestion charge (subject to the constraints on its time variation) isthen an exercise in the theory of the second best. The optimal congestion chargewill entail Ramsey pricing (see Braeutigam (1989), for a survey of Ramseypricing), under which the absolute value of the deviation from marginal cost will‘on average’ be higher for types with less elastic demand.

4.3. Self-financing results

In this section, we develop two self-financing results analogous to thosedeveloped by Mohring and Harwitz (1962) and Strotz (1965) for the traditionalmodel of a congestible facility. Both depend on the assumption

(A.2) c ( ? ), . . . ,c ( ? ) and g ( ? ), . . . ,g ( ? ) are homogeneous of degree zero in1 G 1 J

n(t), N(t) and s.For (A.2) to hold, the state variables x , . . . ,x must be scale-invariant. In the1 J

case of a queue, the appropriate state variable is queuing time rather than queuelength.

For the case in which a nonanonymous congestion charge is possible, we have:

Proposition 4. Suppose that (A.2) holds and that s and t ( ? ), . . . ,t ( ? ) are1 G

unconstrained optimal. Then the ratio of receipts from congestion charges tocapacity costs is given by the elasticity of scale of the capacity cost function.


The intuition for Proposition 4 is as follows. The facility is an input in a jointproduction process in which utilizations are viewed as distinct outputs if theyeither are made by different consumer types or begin at different times. The outputlevels of the process are given by n (t), . . . ,n (t) for all t [ [t ,t ]. Suppose that a1 G o f

scale factor u which differs infinitesimally from 1 is applied to all output levelsand the facility’s capacity, s. This scales N(?) by u, so, by (A.2), x , . . . ,x and1 J

c , . . . ,c are unchanged at all times. Total user costs are therefore scaled by u.1 G

Suppose, for example, that the capacity cost function has an elasticity of scale of


1. Then capacity costs are also scaled by u, and production involves constant long11run ray average costs.

From Eqs. (3a) and (23), all outputs whose production levels are positive are12priced at long run marginal cost. With constant long run ray average costs, this

implies a total value of output equal to the total cost of output. The total value ofoutput is the sum of user costs and congestion charges, while the total cost ofoutput is the sum of user costs and capacity costs. Congestion charges and capacitycosts are therefore equal.

Now suppose that the congestion charge is constrained to be anonymous. If(A.1) holds, then by Proposition 3 this is a nonbinding constraint. It followsimmediately from Proposition 4 that

Proposition 5. Suppose that (A.1) and (A.2) hold and that s and t ( ? ), . . . ,t ( ? )1 G

are optimal subject to the constraint t ( ? ) 5 . . . 5 t ( ? ). Then the ratio of1 G

receipts from congestion charges to capacity costs is given by the elasticity ofscale of the capacity cost function.

5. Conclusion

This paper addressed the following issue: Suppose that the users of acongestible facility are heterogeneous, but behave identically when using thefacility. Users differ from one another in other, unobservable ways (such as thevalue they place on time and quality attributes of the facility), so that congestioncharges must be anonymous. Under what circumstances is marginal (social) costpricing possible?

The single-period version of the traditional model of a congestible facility,which assumes static congestion and does not explicitly account for schedule delaycosts, would give the following answer: Since each type’s user cost depends onlyon the total number of utilizations and not on their composition by type, eachutilization imposes the same congestion externality. Consequently, application ofan anonymous Pigouvian congestion charge equal to the common congestionexternality achieves marginal cost pricing. To allow for time-varying congestion,divide up the congestion period into intervals. For each user type, each interval hasits own demand function which depends on that type’s utilization prices in all ofthe intervals (reflecting cross-price effects). Since each utilization is assumed tocontribute to congestion in only one interval, the above argument for the feasibilityof marginal cost pricing applies for each interval separately.

This treatment of time-varying congestion is unsatisfactory in ignoring that

11Having capacity levels change at the same proportionate rate as output levels in the argument isjustified by the Envelope Theorem.

12When evaluated at optimal capacity levels, ≠C /≠N (t ) is long run marginal cost.i f


congestion in a time interval spills over into the subsequent interval through stockor state variables, the most obvious example being a queue. In the present paper,we examined the feasibility of marginal cost pricing for a congestible facility whenusers differ in unobservable ways, employing a model which not only accounts forsuch stock congestion effects but also explicitly models the time-of-use decision.Our model applies to congestible facilities, including clubs, which individuals useat most once a day and for which the congestion that an individual experiencesdoes not depend on utilizations that begin after hers. Our main result was that,with an anonymous congestion charge, marginal cost pricing is feasible when thetime variation of the congestion charge is unconstrained but not generallyotherwise.

What is the significance of this result? On the positive side, it indicates thatunobservable differences between individuals do not undermine the feasibility ofmarginal cost pricing. Thus, first-best rules for optimal capacity and first-bestself-financing results apply. On the negative side, it implies that when the timevariation of congestion charges is constrained, the unobservability of usercharacteristics in general renders marginal cost pricing infeasible. Not only doesthis result in inefficiency, but also calculation of the optimal congestion chargeover time (subject to the constraints on its time variation – an exercise in Ramseypricing) as well as of optimal capacity becomes considerably more difficult andinformationally demanding. Thus, our paper makes a theoretical case for flexible,time-varying congestion charges.

Acknowledgements

We are grateful to the referee and the editor for providing valuable comments.

Appendix A

Proof of Proposition 1. Suppose that t (t) 5 n (t) and that Eqs. (4)–(9) are at ani 9 i 9

optimum, conditional on s. The proof amounts to showing that Eqs. (3a) and (3b)hold.

Given any consumer type i9, first consider t [ T . This is equivalent toi 9

n (t) . 0, so from Eq. (12) we have that at t,i 9

≠g≠c ji] ]2 c 2On 1 m 1Ol 5 0. (A–1)i 9 i i 9 j≠n ≠ni 9 i 9i j

Using Eq. (A–1) in Eq. (20) along with t (t) 5 n (t) gives c 1 t (t) 5 P (N (t )),i 9 i 9 i 9 i 9 i 9 i 9 f

which together with Eq. (2) establishes Eq. (3a).


If t [⁄ T , then Eq. (11) holds at t. Starting from this relationship and carryingi 9

out the same steps as above establishes Eq. (3b).

Proof of Lemma 1. For t [ T , Eq. (A–1) holds, from which Eq. (20) can bei 9

rewritten n (t) 5 P (N (t )) 2 c (t). Using Eq. (23), this becomes n (t) 5 ≠C /i 9 i 9 i 9 f i 9 i 9

≠N (t ) 2 c (t), which is the same expression as in Eq. (24a).i 9 f i 9

Proof of Proposition 2. Suppose that t (t) 5 w (t) and that Eqs. (4)–(9) are at ani 9 i 9

optimum, conditional on s. As with Proposition 1, the proof amounts to showingthat Eqs. (3a) and (3b) hold.

Given any consumer type i9, first consider t [ T . By Lemma 1, w (t) 5 n (t),i 9 i 9 i 9

which together with Proposition 1, establishes Eq. (3a).If t [⁄ T , then from Eq. (24b), w (t) $ ≠C /≠N (t ) 2 c (t). Using Eq. (23) andi 9 i 9 i 9 f i 9

t (t) 5 w (t), this implies that t (t) $ P (N (t )) 2 c (t). Together with Eq. (2),i 9 i 9 i 9 i 9 i 9 f i 9

this establishes Eq. (3b).

Proof of Proposition 4. Expanding Eq. (12) and summing over i9,

≠g≠c ji] ]2On c 2OOn n 1On m 1OOn l 5 0. (A–2)i 9 i 9 i 9 i i 9 i 9 i 9 j≠n ≠ni 9 i 9i ji 9 i 9 i 9 i 9

Reversing the order of summation in Eq. (A–2),

≠g≠c ji] ]2On c 2OnOn 1On m 1Ol On 5 0. (A–3)i 9 i 9 i i 9 i 9 i 9 j i 9≠n ≠ni 9 i 9i ji 9 i 9 i 9 i 9

Multiplying Eq. (13) by N , summing over i9, and reversing the order ofi 9

summation,

≠g≠c ji]] ]]~ON m 2OnON 1Ol ON 5 0. (A–4)i 9 i 9 i i 9 j i 9≠N ≠Ni 9 i 9i ji 9 i 9 i 9

Adding Eqs. (A–3) and (A–4),

≠c ≠ci i] ]]~2On c 1On m 1ON m 2OnO n 1 NS Di 9 i 9 i 9 i 9 i 9 i 9 i i 9 i 9≠n ≠Ni 9 i 9ii 9 i 9 i 9 i 9

≠g ≠gj j] ]]1Ol O n 1 N 5 0. (A–5)S Dj i 9 i 9≠n ≠Ni 9 i 9j i 9

Multiplying Eq. (19) by s and summing over k,k

t f

≠g≠c≠h ji] ] ]2Os 2E OnOs 2Ol Os dt 5 0. (A–6)S Dk i k j k≠s ≠s ≠sk k kk i k j k

to

Integrating Eq. (A–5) from t to t and adding to Eq. (A–6),o f


t t tf f f

~2EOn c dt 1EOn m dt 1EON m dti 9 i 9 i 9 i 9 i 9 i 9i 9 i 9 i 9

t t to o o

t f

≠c ≠ci i] ]]2EOn O n 1 NS DFi i 9 i 9≠n ≠Ni 9 i 9i i 9

to

t f

≠g ≠g ≠g≠c ≠hj j ji] ] ]] ] ]1Os dt 1EOl O n 1 N 1Os dt 5Os .S DF GGk j i 9 i 9 k k≠s ≠n ≠N ≠s ≠sk i 9 i 9 k kk j k ki 9

to

(A–7)

By (A.2), both of the expressions that appear in square brackets in Eq. (A–7) areequal to zero, and Eq. (A–7) reduces to

t t tf f f

≠h]~2EOn c dt 1EOn m dt 1EON m dt 5Os . (A–8)i 9 i 9 i 9 i 9 i 9 i 9 k ≠skki 9 i 9 i 9

t t to o o

The third term can be rewritten,

t tf f

~ ~EON m dt 5OEN m dti 9 i 9 i 9 i 9i 9 i 9

t to o

t f

t f5O N m u 2En m dt (using integration by parts)i 9 i 9 t i 9 i 9o1 2i 9to

t f

5OP (N (t ))N (t ) 2EOn m dt (using Eqs. (7) and (15)),i 9 i 9 f i 9 f i 9 i 9i 9 i 9

to

from which Eq. (A–8) becomes

t f

≠h]OP (N (t ))N (t ) 2EOn c dt 5Os . (A–9)i 9 i 9 f i 9 f i 9 i 9 k ≠skki 9 i 9

to

The left-hand side of Eq. (A–9) is the aggregate revenue from congestion charges;the proof is completed by dividing Eq. (A–9) by h(s).


References

Arnott, R., de Palma, A., Lindsey, R., 1993. A structural model of peak-period congestion: A trafficbottleneck with elastic demand. American Economic Review 83, 161–179.

Arnott, R., Kraus, M., 1995. Financing capacity in the bottleneck model. Journal of Urban Economics38, 272–290.

Berglas, E., Pines, D., 1981. Clubs, local public goods and transportation models. Journal of PublicEconomics 15, 141–162.

Braeutigam, R.R., 1989. Optimal policies for natural monopolies. In: Schmalensee, R., Willig, R.(Eds.), Handbook of Industrial Organization, Vol. 2. North-Holland, Amsterdam, pp. 1289–1346.

Dorfman, R., 1969. An economic interpretation of optimal control theory. American Economic Review59, 817–831.

Lindsey, R., 1990. Characteristics of congestible facilities, mimeo.Mohring, H., Harwitz, M., 1962. Highway Benefits: An Analytical Framework. Northwestern

University Press, Evanston.Scotchmer, S., 1994. Public goods and the invisible hand. In: Quigley, J., Smolensky, E. (Eds.),

Modern Public Finance. Harvard University Press, Cambridge, pp. 93–119.Strotz, R.H., 1965. Urban transportation parables. In: Margolis, J. (Ed.), The Public Economy of Urban

Communities. Resources for the Future, Washington, pp. 127–169.Vickrey, W.S., 1969. Congestion theory and transport investment. American Economic Review

Proceedings 59, 251–260.

Documents

When are anonymous congestion charges consistent with marginal cost pricing?