Taxi Travel Should Be Subsidized

Ž .JOURNAL OF URBAN ECONOMICS 40, 316]333 1996ARTICLE NO. 0035

Taxi Travel Should Be SubsidizedU

RICHARD ARNOTT

Department of Economics, Boston College, Chestnut Hill, Massachusetts 02167

Received May 16, 1994; revised July 12, 1995

In a first-best environment, taxi travel should be subsidized. The result derivesfrom economies of density}doubling trips and taxis reduces waiting time. Thesubsidy should cover the shadow cost of taxis’ idle time, evaluated at the optimum.The paper provides a proof of the result for dispatch taxis and then discusses thepracticality of its implementation. Q 1996 Academic Press, Inc.

In a first-best environment taxi travel should be subsidized. The resultderives from the technology of waiting time. Double the number of alllengths of taxi trips and simultaneously double the number of taxis. Doingthis doubles the density of vacant taxis and so decreases average waitingtime. This is the source of increasing returns to scale which would causethe taxi industry to operate at a loss with marginal cost pricing. The degreeof subsidization may be established by another argument. A taxi travelershould pay for the marginal social cost of a trip, which equals the shadowcost of the occupied time of her taxi. Summing over trips gives that farerevenue should cover the shadow value of taxis’ occupied time. Thesubsidy should therefore cover the shadow value of taxis’ äcant time atthe optimum.

w xThis result was originally derived by Douglas 2 , who made reference tow xMohring 6 in which an analogous result was obtained for bus travel.

* I thank two former students at Queen’s University, Simon Anderson and Ian Laxer, forstimulating my interest in the economics of taxis, Marvin Kraus for constructive criticisms onan earlier draft, Ece Yolas for research assistance using MATHEMATICA, the editor andtwo referees for helpful comments, and many taxi drivers for informative discussions.

316

0094-1190r96 $18.00Copyright Q 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

SUBSIDIZED TAXI TRAVEL 317

Douglas’ model of cruising taxis is aggregative and is summarized by

P PP s P Q, W , - 0, - 0 iŽ . Ž .

Q W

WW s W V , - 0 iiŽ . Ž .

V

TC s m Q q V , iiiŽ . Ž .

where Q is occupied taxi-hours, V vacant taxi-hours, P price per occupiedtaxi-hour, W expected waiting time, TC total costs, and m cost per

Ž .taxi-hour. Equation i states that the marginal willingness-to-pay is in-Ž .versely proportional to occupied taxi-hours and expected waiting time; ii

states that expected waiting time is inversely proportional to vacant taxi-Ž .hours; and iii states that total costs equal total taxi-hours times a

constant cost per taxi-hour. The first-best optimum is derived by maximiz-ing social surplus with respect to Q and V:

Q X Xmax SS s P Q , W V dQ y m Q q V ivŽ . Ž . Ž .Ž .HQ, V 0

Q: P Q, W V y m s 0 vŽ . Ž .Ž . P QX , W V W VŽ . Ž .Ž .Q XV : dQ y m s 0. viŽ .H

W V0

Social surplus equals social benefit minus social cost. Occupied taxi-hoursshould be such that marginal willingness-to-pay equals marginal cost. Andvacant taxi-hours should be such that the marginal benefit from a vacant

Ž .taxi-hour which stems from reduced waiting time equals marginal cost.Ž .From v alone it follows that taxi revenues should cover only the cost of

occupied taxi time so that, in the aggregate, taxi operation makes a lossequal to the cost of vacant taxi time.

Subsequent papers on the economics of taxis have adopted Douglas’general analytical framework. In contrast to Douglas’ highly aggregative

w xmodel of cruising taxis, Manski and Wright 5 provided a specific ‘‘struct-ural’’ model of a taxi stand. Arrivals at the taxi stand occur according to a

Ž .Poisson process with the arrival rate linear in fare per unit occupied timeand in waiting time. The single queue has parallel servers, exponentiallydistributed services times, first-in first-out service order, no balking andunlimited queuing capacity. As in Douglas, the cost per unit time forvacant and occupied taxis is the same and independent of scale. Beesley

w xand Glaister 1 considered a cruising taxi model that is essentially thesame as Douglas’, but particularizes it somewhat by assuming that ex-

RICHARD ARNOTT318

pected waiting time is inversely proportional to the number of vacantŽ Ž . .taxi-hours W V s grV, with g a constant . This assumption is based on

the argument that doubling the number of vacant cruising taxis shouldw xhalve expected waiting time. Frankena and Pautler 3 adopted Douglas’

model of cruising taxis without modification. All these papers focused ontaxi regulation to achieve a second best with no subsidization, treating thefirst best only in passing as an unattainable ideal.

This paper differs from the previous literature in three principal re-spects. First, it focuses on the first best, and considers at some length theincentive and monitoring problems associated with trying to decentralizethe first best. Second, reflecting the author’s distrust of aggregative model-ing, it presents a structural model which attempts to treat explicitly thetechnological and informational aspects of the problem. And third, itconsiders dispatch rather than cruising taxi service.1

Section 1 presents the model. Section 2 derives the social optimum inthe model, while section 3 considers its decentralization. Section 4 exam-ines an example. Section 5 presents some concluding comments.

1. THE MODEL

The model is illustrative rather than general.Consider a spatially and temporally homogeneous two-dimensional city

which extends infinitely far in every direction. Thus, residences are uni-formly distributed over space. When at home, each identical residentrandomly receives trip opportunities according to an exogenous Poissonprocess. A trip opportunity specifies a random dollar benefit if she travelsto a specific destination. The destination of trip opportunities is randomand uniform over space. Taxi is the only mode of travel. If a residentdecides to take a trip, she telephones for a taxi, waits until it arrives, andtravels by taxi to the destination. For her return, she again telephones for

Ž .a taxi to take her home. Taxi speed is constant no traffic congestion , andtravel distance is crow-line or Euclidean distance. Cruising taxis aredistinguished according to whether they are occupied or vacant. Under adispatch system, however, a vacant taxi is engaged if it is en route to pickup a passenger; thus, the relevant distinction is between engaged and idletaxis.

The planner’s objective is to maximize social surplus per resident. Theplanner is omniscient}he knows all the trip opportunities received by

1 Ž . w xAccording to Frankena and Pautler p. 130 , ‘‘ In the United States cruising is importantonly in the central areas of a few large, dense cities including New York, Chicago, andWashington, D.C. With these exceptions, in large cities radio-dispatch typically accounts for60]75 percent of taxi trips, cab stands for 15]30 percent, and contracts for 10]20 percent. Insmall cities, radio-dispatch accounts for over 85 percent of taxi trips.’’


residents, as well as the location of all taxis. But he has limited computa-tional powers. As a result, instead of continuously solving a stochasticscheduling problem, he adopts a simple scheduling rule: Send the closestidle taxi to transport any resident who has recei ed a trip opportunity for whichthe social benefit exceeds the ‘‘expected’’ social cost. Expected social cost iscomputed ignoring the actual location of taxis. Two comments are in orderconcerning this heuristic rule. First, expected cost rather than actual cost ischosen since it is the decentralized economy that is of practical interest,and in the decentralized economy a prospective trip taker does not knowthe location of taxis and hence bases her trip decision on expected ratherthan actual waiting time.2 Second, this rule is inefficient in another respect}an engaged taxi may be closer, in terms of time, to a waiting travelerthan any idle taxi.3 The planner also chooses the number of taxis. Theanalysis is conducted for a unit of time and in terms of average values.4

The following notation is employed.

G population densityx trip length

Ž .u proportion of taxi time that is idle not engagedT density of taxisuT density of idle taxisŽ .n x number of trips by a resident to a unit area of destina-

tions a distance x away from homeŽ .G n gross social benefit from n trips to a unit area of destina-

Ž X Y XŽ . .5tions with G ) 0, G - 0, G 0 s `t pick-up and drop-off time on a one-way trip, a constantŽ .t x, uT average trip time for a return trip to x, including waiting,

Ž .pick-up, and drop-off time, t ? r u T - 0y opportunity cost of a taxi driver’s timek taxi operating costs per unit idle timec additional taxi operating costs per unit engaged timez opportunity cost of a resident’s time

2 The prospective trip taker may, however, have some knowledge. Specifically, when shecalls for a taxi, the dispatcher may tell her how long the expected wait is. She may then cancelthe trip if the expected wait is too long. This complication is ignored.

3 If the heuristic rule is modified to: ‘‘Send the taxi which is closest in terms of time andwhich is not already committed to a subsequent trip . . . ,’’ the qualitative results are unaltered.

4 We ignore the aggregate stochasticity generated by the Poisson process. Treating it wouldnot alter the qualitative results, but would add notational complexity.

5 XŽ .G 0 s ` is assumed simply to rule out nuisance corner solutions.

RICHARD ARNOTT320

2. SOCIAL OPTIMUMŽ .The resident takes n x trips to a unit area at a distance x away. The

area between x and x q dx away is 2p x dx. Thus, her gross benefit fromŽ Ž ..trips to distances between x and x q dx away from home is G n x 2p x dx,

` Ž Ž ..and her total gross benefits are H G n x 2p x dx. Note that this specifica-0tion assumes for simplification that the benefit from trips of length x isindependent of the number of trips taken of other lengths. Social surplus

Ž .per resident equals these gross benefits, less: i the opportunity cost of theŽ .resident’s trip time which includes the time waiting for a taxi ,

` Ž . Ž . Ž .H zn x t x, uT 2p x dx; ii the opportunity cost of taxi drivers’ time per0Ž . Ž . Ž .resident, y TrG ; and iii taxi operating costs per resident, kTrG q

Ž .Ž . 6cTrG 1 y u . Thus,

` `

SS s G n x 2p x dx y zn x t x , uT 2p x dxŽ . Ž . Ž .Ž .H H0 0

Ty y q k q c 1 y u . 1Ž . Ž .Ž .

G

In a stationary state, the proportion of time a resident spends on taxitrips equals the proportion of time a taxi is engaged times the ratio of thedensity of taxis to the density of residents, i.e.,

`T 1 y uŽ .s n x t x , uT 2p x dx. 2Ž . Ž . Ž .H

G 0

² Ž .: Ž . Ž .The planner chooses T , u, and n x to maximize 1 s.t. 2 . Letting lŽdenote the shadow price on the constraint the shadow value of a unit of

.engaged taxi time , the first-order conditions are

` 1T : y znut 2p x dx y y q k q c 1 y uŽ .Ž .H 2 G0

`1 y uŽ .ql y nut 2p x dx s 0 3aŽ .H 2ž /G 0

6 This specification of costs ignores two important considerations. First, the model is moststraightforwardly interpreted as assuming that a taxi stays on the road all the time. But theutilization rate is an important margin. Since neither full nor zero utilization is cost-minimiz-ing, there is presumably an optimal utilization rate, which owners would choose with first-bestpricing. Thus, a more sophisticated interpretation is that costs are measured at the optimalutilization rate for which marginal cost equals average cost. Second, dispatching costs are notexplicitly considered. On the one hand, there are fixed costs to dispatching; on the other, thecomplexity of the dispatching operation tends to rise exponentially with the aggregatefrequency of trips. Since dispatching is becoming computerized, it is not unreasonable tosuppose that these effects more or less balance out.


` `Tc Tu: y znTt 2p x dx q q l y y nTt 2p x dx s 0 3bŽ .H H2 2ž /G G0 0

n x : GX y zt y lt s 0, 3cŽ . Ž .where arguments of functions have been suppressed to simplify notation

Ž Ž .. Ž .and t ' t x, uT r u T. Equation 3c is a marginal benefit equals2XŽ Ž ..marginal cost condition. G n x is the gross social benefit from a marginal

Ž .trip to x, zt x, uT is the shadow value of the resident’s time on the trip,Ž .and lt x, uT is the shadow taxi fare.

Ž .Multiplying 3b by urT and subtracting the resulting equation fromŽ .3a gives

l s y q k q c, 4Ž .which states that the shadow value of a unit of engaged taxi time isy q k q c.

Integrating the shadow taxi fare over all a resident’s trips yields

` lT 1 y uŽ .R s ln x t x , uT 2p x dx s using 2 , 5aŽ . Ž . Ž . Ž .Ž .H

G0

so thatU U UR s GR s y q k q c E using 4 , 5bŽ . Ž . Ž .Ž .

where RU is shadow taxi revenue per unit area, EU is engaged taxi timeU Ž .per unit area, and denotes evaluation at the optimum. Equation 5b

states that at the social optimum, shadow taxi fare revenue equals theshadow value of engaged taxi time. Thus, at the social optimum, the

Ž .shadow loss of the taxi industry per unit area equals the shadow value ofŽ .idle taxi time per unit area , i.e.,

LU s y q k TU uU , 6Ž . Ž .U Ž . Uso that the shadow loss per taxi equals l s y q k u .

Under the particular assumptions of the model,7

x 1t x , uT s 2 q t q . 7Ž . Ž .ž /'n 2n uT

7 Ž . ` 2The expected distance of the closest idle taxi is see Appendix 1 for derivation H 2p uTx02 'Ž Ž .. Ž . Žexp yp x uT dx s 1r 2 uT e.g., with unit idle taxi density, the expected distance is

'. Ž .1r2 , so that the expected waiting time is 1r 2n uT . The expected travel time on a one-waytrip equals this plus the pick-up and drop-off time plus the trip travel time. To account for thereturn trip, multiply by 2.

It has been assumed that travel distance equals Euclidean distance. If, instead, the city hasa grid network so that travel distance equals Manhattan distance, the only part of the analysis

'Ž . Ž Ž .. Ž .which changes is that t x, uT s 2 xrn q t q pr2 r 2n uT replaces 7 .'

RICHARD ARNOTT322

It is interesting to note that, under a dispatch system, expected waitingtime is inversely proportional to the square root of the density of idle taxis,while under a cruising taxi system, waiting time is inversely proportional to

Žthe density of vacant taxis recall that this was the assumption made by.Beesley and Glaister in their model of cruising taxis . This explains why

dispatch systems are relied on more heavily in smaller communities withrelatively low population densities, while cruising taxis predominate in thecentral cities of large metropolitan areas.

3. DECENTRALIZATION OF THE SOCIAL OPTIMUM

Now consider partial decentralization of the optimum. In particular,suppose that the planner chooses the fare per unit engaged time, f , as wellas the number of taxis, and lets residents decide which trips to take.8 If aresident decides to take a trip, she calls the single taxi company and thenearest idle taxi is dispatched to pick her up. A resident will accept a trip ifits gross benefit exceeds the expected fare plus the opportunity cost of her

Ž .expected trip travel time. Thus, n x is determined by the condition

GX n x y zt x , uT s ft x , uT . 8Ž . Ž . Ž . Ž .Ž .

Ž . Ž .Comparing 8 and 3c indicates that a necessary condition for decentral-ization of the social optimum is that, for all trips, the regulated fareŽ . Ž . U Ž .p x s ft x, uT equal the corresponding taxi fare at the optimum, p x

U Ž U U . Us l t x, u T , which requires that f s l . Note that, since the engagedtaxi time on a trip equals the waiting time, and pick-up and drop-off time,in addition to the actual travel time on the trip, the fare contains acomponent that is independent of trip distance plus a component linear intrip distance. If this decentralization procedure ‘‘works,’’ then taxi farerevenue equals the social cost of engaged taxi time. To break even, the taxiindustry should receive a subsidy equal to the cost of idle taxi time.

U Ž .This decentralization procedure works if, with T and p x sU Ž U . Ul t x, uT , the market generates u s u . The market u is a solution toŽ . Ž . XŽ Ž .. Ž U . U Ž U .2 , subject to n x satisfying G n x y zt x, uT s l t x, uT andT s TU , i.e.,

U`T 1 y uŽ .

Xy1 U U Us G l q z t x , uT t x , uT 2p x dx. 9Ž . Ž . Ž . Ž .Ž .HG 0

U Ž .Unfortunately, while u s u solves 9 , it may not be the only economicŽ .solution. Whether 9 has a unique solution depends on the form of the

8 One can posit alternative decentralization procedures. For example, the planner couldŽ .choose the fare schedule, p x , rather than the fare per unit engaged time.


demand function. If there is more than one ‘‘stable’’9 solution, then theeconomy could end up at u - uU. The reason this could arise is analogousto that for a stable, hypercongested equilibrium in traffic flow theory. Withan equilibrium u - uU , taxi drivers take more time on average to collecttheir passengers, and may take so much more time that, even though trip

Ž .demand is reduced, engaged taxi time increases to satisfy 9 . In whatfollows, unless stated otherwise, it is assumed that when there are multipleequilibria, the market settles in the Pareto efficient equilibrium.

In the next section an example will be presented with this decentraliza-tion mechanism for the special case where the benefit from a trip is fixed.

The above decentralization procedure was only partial because theplanner controls the size of the taxi fleet and sets the fare per unitengaged time, and taxi drivers respond obediently to the planner’s instruc-tions. How much further can decentralization be taken? As will becomeapparent, a satisfactory answer to this question requires the solution of acomplicated mechanism design problem that is beyond the scope of thispaper.10 The aim of the rest of this section is more modest}to argue thatit should be possible to design a mechanism, albeit somewhat imperfect,that subsidizes taxi travel. The focus is on a particular mechanism in whichthe fare schedule is regulated and set at the first-best level, drivers arepaid their opportunity wage, and owners receive all fare revenue, incur allcosts associated with their taxis’ operation, and together receive a lump-sumsubsidy from the government.

There has been some discussion in the literature of the decentralizeddetermination of taxi fares with cruising cabs. The literature has consid-ered the situation where taxi drivers negotiate the fare with each passen-ger. There is no fare structure equilibrium. Suppose the contrary and thatthe equilibrium fare for a particular trip is p. A cruising taxi driver whostops for a prospective passenger has an incentive to quote a fare above psince he knows that the passenger is willing to pay a premium not to waitfor the next vacant taxi that passes by. In this situation, taxi drivers wouldbargain with their prospective passengers. Such fare negotiations would becostly, in terms not only of the prospective passenger’s and driver’s stressand time but also of the congestion caused by the stationary taxi. It is morereasonable to assume that taxi drivers would choose to or be required tocommit themselves to a posted fare schedule. With heterogeneity inpassengers’ reservation fares, one might observe a distribution of fares fora particular trip. Taxis that charged a higher fare would be refused more

9 Characterization of out-of-equilibrium behavior is somewhat problematical in this con-text.

10 w xSee Laffont and Tirole 4 for a very good introduction to the mechanism designliterature.

RICHARD ARNOTT324

often and therefore vacant a higher proportion of the time. Averagewaiting time under such a system would be inefficiently high. It would alsobe inconvenient and disruptive for passengers to find out taxis’ fares. Inany event, there are persuasive reasons for the regulation of cruising taxifares.

The situation with dispatch taxis is somewhat different. First, searchcosts are significantly lower than for cruising taxis since only a telephonecall is required to find out the fare and perhaps expected waiting time.Second, waiting time is a potentially contractible element of service. Andthird, economies of density with respect to waiting time are present at thelevel of the firm. What fare structure would arise in the absence ofregulation? Different taxi companies might offer different fareschedulerwaiting time packages, among which individuals would sortthemselves out on the basis of the shadow value of waiting time. Alterna-tively, a particular taxi cab company might offer a menu of fareschedulerwaiting time packages, akin to the priority pricing systems thatused to be employed on computers under batch processing. An obviousdifficulty is that firms would have an incentive to understate expectedwaiting time or to renege on guarantees of maximum waiting time. Truthin advertising could be monitored and enforced, though doing so would becostly and cumbersome. Reputation and repeat usage would also providesome discipline. Given current technology, however, it seems most reason-able to assume that expected waiting time and actual waiting time arenon-contractible, though customers would consider their previous experi-ences with waiting time in deciding which taxi company to call up.

Despite economies of density with respect to waiting time at the level ofthe firm, there are typically many taxi companies with overlapping serviceareas. The main reason for this is probably that, beyond some scale ofoperation, there are diseconomies in dispatching, which along with the

Ž .economies of density, result in a privately cost-minimizing scale. Thediseconomies of scale in dispatching result from problems of coordinationwith more than one dispatcher, as well as limits on the schedulingcapabilities of the individual dispatcher. Another consideration is thatmost taxi journeys are local. Then one would expect an equilibrium in

Ž .which firms operate at close to minimum private cost and have localizedthrough overlapping service areas, with pricing similar to that obtained inmodels of spatial monopolistic competition with delivered pricing}where-by the markup over cost on trips originating close to the center of a firm’sservice area is higher than those originating near its boundary.11 Thecentral point is that economies of density at the level of the firm, along

11 An interesting implication of this model of the market structure of the dispatch taxiindustry is that computerized dispatching should result in a significant increase in firm size.


with the friction of space, would give firms market power, which theywould exploit by pricing above marginal cost. This ‘‘model’’ provides anargument in favor of regulated fares. Thus, in what follows, it will beassumed that taxi fares are fixed by regulation at the first-best level.

The question then arises concerning how the subsidy should be paid tothe dispatch taxi industry, taking into account the margins of choice of taxiowners and drivers. Consider drivers first. Incentives should be structuredin such a way that drivers, acting in their self-interest, make sociallyefficient decisions. A driver faces at least seven decisions: whether tobecome a taxi driver, how many hours to drive each day, whether to pickup all his assigned fares and only them, whether to take the direct route,whether to drive in a socially efficient way}at the right speed and withthe right degree of safety}whether to charge the correct fare, andwhether to behave with courtesy. Suppose that a representative risk-neutral taxi driver bases these decisions on financial considerations alone,and that, if he is indifferent concerning how to behave, will act in thesocially efficient manner. It is easy to posit payment mechanisms thatdistort his decisions. Suppose, for example, that the taxi company pays forthe driver’s gas, insurance, and repairs, and pays the taxi driver a propor-

Ž .tion possibly greater than 100% of the fare revenue he collects such thaton average he earns his opportunity wage. Then the taxi driver has anincentive to maximize fare revenue per unit time. This encourages him todrive too fast and with insufficient safety, overcharge where possible, drive

Žcircuitously, skip unprofitable because travel time to pick up the passen-ger is high, or because the passenger’s destination is likely a long distance

. Žfrom the closest subsequent fare trips, and steal fares where a driver.finds himself closer to a waiting passenger than the assigned taxi }all

problems in the current system.Consider the following mechanism: The taxi company pays all a driver’s

expenses, as well as an hourly wage equal to the opportunity wage of themarginal driver, and the driver gives the company all his fare revenue.12

Then drivers will choose to drive the optimal number of hours, and will beindifferent concerning all other margins. Thus, given the assumptions, thismechanism is efficient. One might reasonably object to the assumptionthat, when a driver is indifferent as to how to behave, he will behave in thesocially efficient manner. This objection can be at least partially counteredby augmenting the model to include tips. Tipping is a strange phenomenonfrom an economist’s perspective, at least for non-repeat interactions, sinceapart from the fear of being cursed at and the warm glow from beinggenerous, the passenger has no incentive to tip at the end of a journey. In

12 Under this mechanism, owners would bear all the risk. If this is undesirable, driverscould receive part of their remuneration in the form of wages and the rest via profit-sharing.

RICHARD ARNOTT326

any event, assume that on average the tip is based on quality of service.Since providing quality service entails little cost, with the mechanism underconsideration a driver would have an incentive to provide quality servicewhich would, largely though not completely, result in his making sociallyefficient decisions. The exceptions include speeding to get a passenger toan appointment on time, skipping assigned fares when the expected tipfalls short of the expected opportunity tip, and stealing fares. These abusescould be checked by the taxi companies themselves.

This mechanism has two potentially serious flaws. First, drivers wouldhave an incentive to pocket fare revenue. A driver could do this bynegotiating a fare with the passenger that is lower than the regulated fare,on condition that the meter be turned off, and telling the dispatcher thatthe passenger did not show up. Second, drivers would have an incentive topad their hours, claiming that they were available but idle when they werein fact shopping, having a donut and coffee, etc. Appropriate, cost-effec-tive technology to check such abuses is almost certainly available, thoughdrivers would no doubt resist its introduction. Since taxi owners wouldhave an incentive to monitor their own drivers, the government could stayout of the business of monitoring driver behavior, except for extremeabuses.

Where dispatch taxis spend some time at taxi stands and some timecruising, the incentive and monitoring systems would have to be designedso that they are suitable for all modes of service, or else taxi companieswould have to specialize according to type of service, which would result ingiving up some benefits from economies of density.

Ž .Now consider the taxi owner or firm . He has essentially two decisions}how many taxis to operate, and at what rate to utilize them, includingwhether to keep them on the road at all. How he would behave dependscrucially on how the subsidy is paid. Suppose that the subsidy is paid toeach taxi, without restriction. Then there is an obvious incentive for everycar owner to register his car as a taxi, get the subsidy, and continue usingthe car for personal use. Suppose alternatively that the subsidy is pertaxi-hour of operation. Taxi owners would then have an incentive tooverstate their hours of operation. Taxi owners and drivers would have anincentive to collude to overstate the hours driven. And if the subsidy perhour exceeds the hourly wage, taxi owners would have an incentive to hireand pay family members and friends who would not drive the taxis. Oneway around these problems would be for the government to pay the taxiindustry a lump sum, have the taxi association divvy up the lump sumbetween companies on the basis of hours driven, and have the companiespolice one another. This might be exploited as a means of preventingentry, and could invite gangsterism. But it would also provide an incentive


for taxi companies to introduce procedures to facilitate monitoringabuses.13

A final issue is whether the government should restrict the number oftaxis or permit free entry. With the mechanism under consideration, freeentry with certification of drivers and regulation of taxis would seem best,since this would permit flexibility in the face of predictable and unpre-

Ž .dictable e.g., a dry winter, transit strikes fluctuations in demand.The particular features of the mechanism discussed above are of sec-

ondary importance. Detailed analysis that costs out the monitoring tech-nologies and quantifies the deadweight losses due to distortion might wellfind that a significantly different scheme is optimal. What is important aretwo general points that the above discussion illustrates. First, design of amechanism for the effective provision of taxi services should pay consider-able attention to the behavioral responses of both taxi drivers and taxiowners, as well as to the availability of technologies to counter perverseincentives. Second, while attainment of the full first best is most likelyinfeasible given current technology, it should be possible to design amechanism that subsidizes taxi travel and that improves considerably oncurrent systems.

4. AN EXAMPLE

The example is constructed with two considerations in mind. First, theform of trip demand assumed is deliberately simple in order to illustratethat, even in the simplest situations, multiple equilibria may arise. Second,observable parameters are given realistic values, while the values ofunobservable parameters are determined by calibration to a plausible basecase.

It is assumed that all trip opportunities provide a benefit b. Since themarginal cost of a trip to x, including the opportunity cost of the resident’s

Ž U . Ž . Ž .ŽŽ . Ž ..time, is l q z t x, uT s 2 y q k q c q z xrn q t q 1r 2un ,'where u ' uT , the planner will dispatch taxis for all trip opportunities up

UŽ . Ž .to a distance x such that l q z t x, uT s b , and will refuse all tripopportunities beyond this distance.

The following parameter values are assumed:

y s 10 t s .05k s 1 n s 12c s 5 G s 10000z s 20

13 One way would be to make computerized dispatching compulsory and to require thatrecords be kept and cross-referenced with computerized fare recording.

RICHARD ARNOTT328

where units are dollars, hours, and miles. The opportunity cost of a taxiŽ . Ž .driver’s time y is $10rhour; the operating costs of an idle taxi k are

Ž .$1rhour; the additional operating costs of an occupied taxi c are $5rhour;Ž .the opportunity cost of a resident’s time z is $20rhour; pick-up and

Ž . Ž .drop-off time on a one-way trip t is .05 hours; taxi speed n is 12 m.p.h.;and population density is 10,000 residents per square mile. That leaves twoparameters, n and b , which are chosen to be consistent with a base casesocial optimum in which T s 2.4384, u s 0.20095, and x s 5; taxi densityŽ . Ž .T is approximately 2.5 per square mile; taxis are idle u about a fifth of

Ž .the time; all trip opportunities up to a distance x of 5 miles are acceptedand all trip opportunities beyond that distance are rejected. Expected

1'Ž .waiting time in this optimum is 1r 2n uT f 0.05952, about 3 minutes.2

From the optimum condition for x, given in the previous paragraph,Ž . Ž .b f 37.89, and from the stationary-state-condition 2 , n x s n f 3.2025

= 10y6. Thus, the trip benefit is close to $38, while the value of n implies2Ž Ž ..that a resident takes a trip on average every 3976 precisely, 1r np x

hours.It is straightforward, though tedious, to check that the above ‘‘solution’’

is indeed the social optimum for the specified parameters; the details ofthe analysis are given in Appendix 3. Holding the number of taxis and thetaxi fare per unit engaged time at their first-best optimal levels, the full setof equilibria can be solved for14 ; the details of the procedure are given inAppendix 2. As noted in Section 3, there is always an equilibrium corre-sponding to the social optimum, but there may be other equilibria as well.This procedure is repeated for values of G ranging from 500 to 50000. Theresults are plotted in Fig. 1. The lower locus corresponds to the equilibriathat coincide with the social optima. The upper locus corresponds toPareto-dominated equilibria.

Three points bear mention. First, for each value of G there are twoequilibria. Second, there is the issue of equilibrium selection. On the basisof a plausible adjustment mechanism, the lower equilibrium is analogousto a stable, congested equilibrium in traffic flow theory, and the upperequilibrium to an unstable, hypercongested equilibrium. Unstable equilib-ria typically have perverse comparative static properties. This is true of theupper equilibrium in Fig. 1, for which waiting time increases with anincrease in population density. Thus, it appears that the upper equilibriumcan be excluded on stability grounds. If waiting time is initially at a high

Ž .level above that corresponding to the unstable equilibrium it will con-

14 MATHEMATICA was used for all the calculations and for the graphing. MATHEMAT-ICA had to be ‘‘helped’’ to find all the equilibria. The equilibrium conditions were reduced toa single quintic equation in u . For the range of parameter values considered, there were tworeal and two imaginary solutions, and a singularity.


FIG. 1. Equilibrium waiting time as a function of population density with optimal taxidensity and optimal fare per unit engaged time.

tinue to grow without bound; if initial waiting time is at a level below thatcorresponding to the unstable equilibrium, waiting time will converge tothe stable equilibrium level. The third point is that, at the social optimum,which coincides with the lower equilibrium, waiting time falls with popula-tion density. This is a manifestation of the economies of density in taxitravel on which this paper has focused.

5. CONCLUDING COMMENTS

This paper focused on a point that the previous literature has noted onlyin passing. Double the number of taxis and double the number of trips,and waiting time falls. Because of these economies of density, first-besttaxi pricing entails operation at a loss; more specifically, the optimumsubsidy covers the shadow cost of taxi idle time at the optimum.

The paper made three principal contributions. The first was to considerdispatch taxi service, in contrast to the previous literature which hasconcentrated on cruising taxis. The second was to point out that, even withfirst-best pricing-cum-subsidization, attainment of the social optimum isnot guaranteed. The third point was that first-best pricing should not bequickly dismissed, as has been done in the previous literature, since itshould be possible to design incentive-cum-monitoring mechanisms thatare consistent with subsidization of taxi travel and come close to achievingthe first best.

The paper has ignored an important aspect of the ‘‘taxicabproblem’’}traffic congestion. Subsidizing taxi travel may divert passengers

RICHARD ARNOTT330

away from mass transit, exacerbating underpriced and hence excessiveauto congestion. But probably of greater quantitative importance, suchsubsidization would encourage people to use taxis instead of their own carsfor business trips during working hours, for commuting, and for other trippurposes. Thus, consideration of congestion is likely to strengthen the casefor the subsidization of taxi travel. This issue, as well as the design ofpracticable subsidy mechanisms, are promising topics for future research.

Among experts in transportation economics, there is a broad consensusthat urban auto travel is excessive but at the same time that substantiallyexpanding mass transit and subsidizing it sufficiently to induce a significant

Ž w x.number of car travelers to use it is prohibitively expensive Small 7 . Taxiservice provides many of the advantages of the automobile}flexibility,privacy, and convenience}without significant capital costs. Providing taxitravel at its shadow price might therefore contribute significantly to solvingthe urban transportation problem.

APPENDIX 1

Ž .Deri ation of Eq. 7Ž .This appendix provides a heuristic derivation of Eq. 7 . We wish to

solve for average waiting time, which equals expected distance to theclosest idle cab divided by velocity. The expected distance to the closest

x Ž . Ž .idle cab is H xg x dx, where g x is the probability density function of0Ž .distance to the closest idle cab. To solve for g x , proceed as follows. Let

Ž .P K, x be the probability that there are K idle taxis within a distance x.ŽThis equals the probability that there are K idle taxis within a distance

x y dx times the probability that there are no idle cabs between x y dx. Žand x plus the probability that there are K y 1 idle taxis within a

distance x y dx times the probability that there is one idle cab between.x y dx and x plus, . . . , i.e,.

2P K , x s P K , x y dx 1 y 2p xuT dx y 2p xuT dx y ???Ž . Ž . Ž .Ž .q P K y 1, x y dx 2p xuT dxŽ .

2q P K y 2, x y dx 2p xuT dx q ??? .Ž . Ž .

Ž .Employing a Taylor series expansion of the terms P ?, x y dx around x,and collecting terms in dx yields

P K , xŽ .s y2p xuTP K , x q 2p xuTP K y 1, x . A1.1Ž . Ž . Ž .

x


For K s 0, this reduces to

P 0, xŽ .s y2p xuTP 0, x , A1.2Ž . Ž .

x

Ž . 2since P y1, x s 0. Make the transformation of variables z s p x uT ,ˆ ˆ ˆŽ Ž .. Ž . Ž . Ž .and define P K, z x s P K, x . Then P 0, z rz s yP 0, z , so that

ˆ yz ˆŽ . Ž .P 0, z s c e . Since P 0, 0 s 1, c s 1. Thus,0 0

P 0 , x s eyp x 2 uT . A1.3Ž . Ž .

Now, the probability that the closest idle taxi is between x and x q dx isŽ .the probability that there are no idle taxis within a distance x, P 0, x ,

times the probability that there is an idle taxi between x and x q dx,Ž . yp x 2 uT2p xuT dx. Thus, g x s 2p xuT e and

` ` 22 yp x uTxg x dx s 2p x uTe dx. A1.4Ž . Ž .H H0 0

'Ž .From the integral tables, the value of this integral is 1r 2 uT .

APPENDIX 2

Ž .Equation 9 for the Numerical ExampleŽ .This appendix derives the explicit equation corresponding to 9 for the

special case where the benefit from a trip is fixed at b , the Poisson arrivalrate of trip opportunities from each unit area is n, and the fare per unitengaged time is set at the first-best level. Under these assumptions, a

UŽ .resident will accept all trip opportunities up to the distance x uT whichU UŽ . Ž Ž . . Ž .solves b s y q k q c q z t x uT , uT and none beyond.Using 7 ,

bn 1Ux uT s y nt y . A2.1Ž . Ž .U'2 y q k q c q zŽ . 2 uT

Uˆ 'Ž Ž . Ž .Let a s 2rn , b s 2 t q 1r 2n ut and c s br y q k q c q z , so thatˆ Û Uˆ ˆŽ . Ž . Ž Ž .. Ž .t x, uT s ax q b and x uT s nr 2 c y b . Then 9 isˆ ˆ

TU 1 y uŽ . ˆŽ Ž ..nr 2 cybˆ ˆs n ax q b 2p x dxŽ .ˆHG 0

ˆŽ Ž ..nr 2 cybˆ3 2âx bxˆs 2np qž /3 2 0

ˆ2c q bˆ22 ˆs npn c y b .Ž .ˆ ž /12

RICHARD ARNOTT332

ˆSubstituting back for c and b yieldsˆ

2U 2T 1 y u npn b 1Ž .s y 2 t q Už /ž /'G 6 y q k q c q z 2n uT

=b 1

q t q , A2.2Ž .Už /ž /'y q k q c q z 2n uT

which, multiplying through, contains a constant term plus terms in u1r2, u,3r2 2 5r2 Ž . 1r2u , u , and u . Thus, A2.2 is a quintic equation in u .In constructing the numerical examples, the following procedure was

followed. For each G:

Ž .1. Solve the social optimum problem described in Appendix 3 ,U U U Uwhich gives T , u , x , and l .

Ž . U Ž U .2. With the fare structure set as p x s l t x, uT and the num-ber of taxis held at the first-best level, residents choose x according toŽ . Ž . Ž .A2.1 and the set of equilibrium u’s are the real roots of A2.2 .

APPENDIX 3

The Social Optimum for the Numerical Example

The social welfare optimization problem for the example is

Txmax SS s 2p nb dx y y q k q c q z 1 y uŽ . Ž .Ž .H

Gx , T , u 0

T 1 y u x 1Ž . xq f y q t q 4np x dx .H ž /ž /ž /'G n 2n uT0

A3.1Ž .

'Set Tu s u and evaluate the integrals:

T u 22max SS s nbp x y y q k q c q z q c q zŽ . Ž .

G Gx , T , u

2 3 2T y u 4np x 2p nx 1q f y y tn q . A3.2Ž .ž /ž /G 3n n 2u


The corresponding first-order conditions are

24np x 4np x 1x : 2p nb x y f q tn q s 0,ž /ž /n n 2u

bn 1« x s y tn q A3.4Ž .ž /2f 2u

1T : y y q k q c q z q f s 0 « f s y q k q c q z A3.5Ž . Ž .Ž .

G

22u 2u p nxu : c q z y f y s0Ž . 2ž /G G u n

2u 3n y q kŽ .2« x s using A3.5 . A3.6Ž . Ž .Ž .

p nG y q k q c q zŽ .

Ž . Ž .Substitute A3.5 into A3.4 , and then the resulting expression for x intoŽ .A3.6 . This generates a quintic expression in u . If this expression has realroots, it has two of them, one with 3 xu ) 1, the other with 3 xu - 1. It isstraightforward but tedious to show that the former corresponds to thesocial optimum.

REFERENCES

1. M. Beesley and S. Glaister, Information for regulating: The case of taxis, EconomicŽ .Journal, 93, 594]615 1983 .

2. G. W. Douglas, Price regulation and optimal service standards: The taxicab industry,Ž .Journal of Transport Economics and Policy, 116]127 1972 .

3. M. W. Frankena and P. A. Pautler, Taxicab regulation: An economic analysis, Research inŽ .Law and Economics, 9, 129]165 1986 .

4. J.-J. Laffont and J. Tirole, ‘‘A Theory of Incentives in Procurement and Regulation,’’Ž .M.I.T. Press, Cambridge, MA 1993 .

5. C. F. Manski and J. D. Wright, Nature of equilibrium in the market for taxi services,Ž .Transportation Research Record, 619, 11]15 1976 .

6. H. Mohring, Optimization and scale economies in urban bus transportation, AmericanŽ .Economic Reïew, 62, 591]604 1972 .

Ž .7. K. Small, ‘‘Urban Transportation Economics,’’ Harwood Academic, London 1992 .

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Taxi Travel Should Be Subsidized