Taxi Travel Should Be Subsidized

JOURNAL OF URBAN ECONOMICS ARTICLE NO.

40, 316]333Ž1996.

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 derives from economies of density}doubling trips and taxis reduces waiting time. The subsidy 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 the practicality of its implementation. Q 1996 Academic Press, Inc.

In a first-best environment taxi travel should be subsidized. The result derives from the technology of waiting time. Double the number of all lengths of taxi trips and simultaneously double the number of taxis. Doing this doubles the density of vacant taxis and so decreases average waiting time. This is the source of increasing returns to scale which would cause the taxi industry to operate at a loss with marginal cost pricing. The degree of subsidization may be established by another argument. A taxi traveler should pay for the marginal social cost of a trip, which equals the shadow cost of the occupied time of her taxi. Summing over trips gives that fare revenue should cover the shadow value of taxis’ occupied time. The subsidy should therefore cover the shadow value of taxis’ ¨ acant time at the optimum. This result was originally derived by Douglas w2x, who made reference to Mohring w6x in which an analogous result was obtained for bus travel.

* I thank two former students at Queen’s University, Simon Anderson and Ian Laxer, for stimulating my interest in the economics of taxis, Marvin Kraus for constructive criticisms on an earlier draft, Ece Yolas for research assistance using MATHEMATICA, the editor and two referees for helpful comments, and many taxi drivers for informative discussions. 316 0094-1190r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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SUBSIDIZED TAXI TRAVEL

Douglas’ model of cruising taxis is aggregative and is summarized by P s P Ž Q, W . , W s WŽV . ,

­P ­Q

­P

- 0,

­W ­V

­W

-0

-0

TC s m Ž Q q V . ,

Ž i. Ž ii . Ž iii .

where Q is occupied taxi-hours, V vacant taxi-hours, P price per occupied taxi-hour, W expected waiting time, TC total costs, and m cost per taxi-hour. Equation Ži. states that the marginal willingness-to-pay is inversely proportional to occupied taxi-hours and expected waiting time; Žii. states that expected waiting time is inversely proportional to vacant taxihours; and Žiii. states that total costs equal total taxi-hours times a constant cost per taxi-hour. The first-best optimum is derived by maximizing social surplus with respect to Q and V: max SS s Q, V

Q

H0

P Ž QX , W Ž V . . dQX y m Ž Q q V .

Q: P Ž Q, W Ž V . . y m s 0 V:

Q

H0

Ž iv. Ž v.

X

­PŽ Q , WŽV .. ­WŽV . ­W

­V

dQX y m s 0.

Ž vi.

Social surplus equals social benefit minus social cost. Occupied taxi-hours should be such that marginal willingness-to-pay equals marginal cost. And vacant taxi-hours should be such that the marginal benefit from a vacant taxi-hour Žwhich stems from reduced waiting time. equals marginal cost. From Žv. alone it follows that taxi revenues should cover only the cost of occupied taxi time so that, in the aggregate, taxi operation makes a loss equal 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 model of cruising taxis, Manski and Wright w5x provided a specific ‘‘structural’’ model of a taxi stand. Arrivals at the taxi stand occur according to a Poisson process with the arrival rate linear in fare per unit Žoccupied. time and in waiting time. The single queue has parallel servers, exponentially distributed services times, first-in first-out service order, no balking and unlimited queuing capacity. As in Douglas, the cost per unit time for vacant and occupied taxis is the same and independent of scale. Beesley and Glaister w1x considered a cruising taxi model that is essentially the same as Douglas’, but particularizes it somewhat by assuming that ex-

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RICHARD ARNOTT

pected waiting time is inversely proportional to the number of vacant taxi-hours ŽW Ž V . s grV, with g a constant.. This assumption is based on the argument that doubling the number of vacant cruising taxis should halve expected waiting time. Frankena and Pautler w3x adopted Douglas’ model of cruising taxis without modification. All these papers focused on taxi regulation to achieve a second best with no subsidization, treating the first best only in passing as an unattainable ideal. This paper differs from the previous literature in three principal respects. First, it focuses on the first best, and considers at some length the incentive and monitoring problems associated with trying to decentralize the first best. Second, reflecting the author’s distrust of aggregative modeling, it presents a structural model which attempts to treat explicitly the technological and informational aspects of the problem. And third, it considers dispatch rather than cruising taxi service.1 Section 1 presents the model. Section 2 derives the social optimum in the model, while section 3 considers its decentralization. Section 4 examines 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 uniformly distributed over space. When at home, each identical resident randomly receives trip opportunities according to an exogenous Poisson process. A trip opportunity specifies a random dollar benefit if she travels to a specific destination. The destination of trip opportunities is random and uniform over space. Taxi is the only mode of travel. If a resident decides to take a trip, she telephones for a taxi, waits until it arrives, and travels by taxi to the destination. For her return, she again telephones for a taxi to take her home. Taxi speed is constant Žno traffic congestion., and travel distance is crow-line or Euclidean distance. Cruising taxis are distinguished according to whether they are occupied or vacant. Under a dispatch system, however, a vacant taxi is engaged if it is en route to pick up a passenger; thus, the relevant distinction is between engaged and idle taxis. The planner’s objective is to maximize social surplus per resident. The planner is omniscient}he knows all the trip opportunities received by According to Frankena and Pautler Žp. 130., ‘‘wIn the United Statesx cruising is important only in the central areas of a few large, dense cities including New York, Chicago, and Washington, D.C. With these exceptions, in large cities radio-dispatch typically accounts for 60]75 percent of taxi trips, cab stands for 15]30 percent, and contracts for 10]20 percent. In small cities, radio-dispatch accounts for over 85 percent of taxi trips.’’ 1

SUBSIDIZED TAXI TRAVEL

319

residents, as well as the location of all taxis. But he has limited computational powers. As a result, instead of continuously solving a stochastic scheduling problem, he adopts a simple scheduling rule: Send the closest idle taxi to transport any resident who has recei¨ ed a trip opportunity for which the social benefit exceeds the ‘‘expected’’ social cost. Expected social cost is computed ignoring the actual location of taxis. Two comments are in order concerning this heuristic rule. First, expected cost rather than actual cost is chosen since it is the decentralized economy that is of practical interest, and in the decentralized economy a prospective trip taker does not know the location of taxis and hence bases her trip decision on expected rather than 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 traveler than any idle taxi.3 The planner also chooses the number of taxis. The analysis is conducted for a unit of time and in terms of average values.4 The following notation is employed.

G x u T uT nŽ x . GŽ n .

t t Ž x, uT . y k c z

population density trip length proportion of taxi time that is idle Žnot engaged. density of taxis density of idle taxis number of trips by a resident to a unit area of destinations a distance x away from home gross social benefit from n trips to a unit area of destinations Žwith GX ) 0, GY - 0, GX Ž0. s `. 5 pick-up and drop-off time on a one-way trip, a constant average trip time for a return trip to x, including waiting, pick-up, and drop-off time, ­ t Ž?.r­ u T - 0 opportunity cost of a taxi driver’s time taxi operating costs per unit idle time additional taxi operating costs per unit engaged time opportunity cost of a resident’s time

2 The prospective trip taker may, however, have some knowledge. Specifically, when she calls for a taxi, the dispatcher may tell her how long the expected wait is. She may then cancel the 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 and which 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 would not alter the qualitative results, but would add notational complexity. 5 X G Ž0. s ` is assumed simply to rule out nuisance corner solutions.

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RICHARD ARNOTT

2. SOCIAL OPTIMUM Ž 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 from trips to distances between x and x q dx away from home is GŽ nŽ x ..2p x dx, and her total gross benefits are H0` GŽ nŽ x ..2p x dx. Note that this specification assumes for simplification that the benefit from trips of length x is independent of the number of trips taken of other lengths. Social surplus per resident equals these gross benefits, less: Ži. the opportunity cost of the resident’s trip time Žwhich includes the time waiting for a taxi., H0` znŽ x . t Ž x, uT .2p x dx; Žii. the opportunity cost of taxi drivers’ time per resident, y ŽTrG .; and Žiii. taxi operating costs per resident, Ž kTrG . q Ž cTrG .Ž1 y u..6 Thus, SS s

`

`

H0 G Ž n Ž x . . 2p x dx y H0 znŽ x . t Ž x, uT . 2p x dx y

T G

Ž y q k q c Ž 1 y u. . .

Ž 1.

In a stationary state, the proportion of time a resident spends on taxi trips equals the proportion of time a taxi is engaged times the ratio of the density of taxis to the density of residents, i.e., T Ž 1 y u. G

s

`

H0 n Ž x . t Ž x, uT . 2p x dx.

Ž 2.

The planner chooses T, u, and ² nŽ x .: to maximize Ž1. s.t. Ž2.. Letting l denote the shadow price on the constraint Žthe shadow value of a unit of engaged taxi time., the first-order conditions are 1

`

H0 znut 2p x dx y G Ž y q k q c Ž 1 y u . .

T:y

2

ql

ž

Ž 1 y u. G

y

`

H0 nut 2p x dx 2

/

s0

Ž 3a .

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

321

SUBSIDIZED TAXI TRAVEL `

H0 znTt 2p x dx q

u: y

2

Tc G

ql y

ž

T G

y

`

H0 nTt 2p x dx 2

n Ž x . : GX y zt y l t s 0,

/

s0

Ž 3b . Ž 3c .

where arguments of functions have been suppressed to simplify notation and t 2 ' Ž ­ t Ž x, uT ..r­ u T. Equation Ž3c. is a marginal benefit equals marginal cost condition. GX Ž nŽ x .. is the gross social benefit from a marginal trip to x, zt Ž x, uT . is the shadow value of the resident’s time on the trip, and l t Ž x, uT . is the shadow taxi fare. Multiplying Ž3b. by urT and subtracting the resulting equation from Ž3a. gives

l s y q k q c,

Ž 4.

which states that the shadow value of a unit of engaged taxi time is y q k q c. Integrating the shadow taxi fare over all a resident’s trips yields Rˆ s

`

H0 l n Ž x . t Ž x, uT . 2p x dx s

lT Ž 1 y u .

Ž using Ž 2 . . , Ž 5a .

G

so that RU s GRˆU s Ž y q k q c . EU

Ž using Ž 4 . . ,

Ž 5b .

where RU is shadow taxi revenue per unit area, EU is engaged taxi time per unit area, and U denotes evaluation at the optimum. Equation Ž5b. states that at the social optimum, shadow taxi fare revenue equals the shadow value of engaged taxi time. Thus, at the social optimum, the shadow loss of the taxi industry Žper unit area. equals the shadow value of idle taxi time Žper unit area., i.e., LU s Ž y q k . T U uU ,

Ž 6.

so that the shadow loss per taxi equals lU s Ž y q k . uU . Under the particular assumptions of the model,7 t Ž x, uT . s 2

ž

x

n

qtq

1 2 n'uT

/

.

Ž 7.

7 The expected distance of the closest idle taxi is Žsee Appendix 1 for derivation. H0` 2p uTx 2 ŽexpŽyp x 2 uT .. dx s 1rŽ2'uT . Že.g., with unit idle taxi density, the expected distance is 1r2., so that the expected waiting time is 1rŽ2 n'uT .. The expected travel time on a one-way trip equals this plus the pick-up and drop-off time plus the trip travel time. To account for the return trip, multiply by 2. It has been assumed that travel distance equals Euclidean distance. If, instead, the city has a grid network so that travel distance equals Manhattan distance, the only part of the analysis which changes is that t Ž x, uT . s 2Ž xrn q t q pr2 rŽ2 n'uT .. replaces Ž7..

'

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RICHARD ARNOTT

It is interesting to note that, under a dispatch system, expected waiting time 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 the density of vacant taxis Žrecall 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 with relatively low population densities, while cruising taxis predominate in the central 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 well as the number of taxis, and lets residents decide which trips to take.8 If a resident decides to take a trip, she calls the single taxi company and the nearest idle taxi is dispatched to pick her up. A resident will accept a trip if its gross benefit exceeds the expected fare plus the opportunity cost of her expected trip travel time. Thus, nŽ x . is determined by the condition GX Ž n Ž x . . y zt Ž x, uT . s ft Ž x, uT . .

Ž 8.

Comparing Ž8. and Ž3c. indicates that a necessary condition for decentralization of the social optimum is that, for all trips, the regulated fare pŽ x . s ft Ž x, uT . equal the corresponding taxi fare at the optimum, pU Ž x . s lU t Ž x, uU T U ., which requires that f s lU . Note that, since the engaged taxi 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 a component that is independent of trip distance plus a component linear in trip distance. If this decentralization procedure ‘‘works,’’ then taxi fare revenue equals the social cost of engaged taxi time. To break even, the taxi industry should receive a subsidy equal to the cost of idle taxi time. This decentralization procedure works if, with T U and pŽ x . s U l t Ž x, uT U ., the market generates u s uU . The market u is a solution to Ž2., subject to nŽ x . satisfying GX Ž nŽ x .. y zt Ž x, uT U . s lU t Ž x, uT U . and T s T U , i.e., T U Ž 1 y u. G

s

`

Xy1

H0 G

Ž Ž lU q z . t Ž x, uT U . . t Ž x, uT U . 2p x dx.

Ž 9.

Unfortunately, while u s uU solves Ž9., it may not be the only economic solution. Whether Ž9. has a unique solution depends on the form of the 8 One can posit alternative decentralization procedures. For example, the planner could choose the fare schedule, pŽ x ., rather than the fare per unit engaged time.

SUBSIDIZED TAXI TRAVEL

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demand function. If there is more than one ‘‘stable’’ 9 solution, then the economy could end up at u - uU . The reason this could arise is analogous to that for a stable, hypercongested equilibrium in traffic flow theory. With an equilibrium u - uU , taxi drivers take more time on average to collect their passengers, and may take so much more time that, even though trip demand is reduced, engaged taxi time increases to satisfy Ž9.. In what follows, unless stated otherwise, it is assumed that when there are multiple equilibria, the market settles in the Pareto efficient equilibrium. In the next section an example will be presented with this decentralization mechanism for the special case where the benefit from a trip is fixed. The above decentralization procedure was only partial because the planner controls the size of the taxi fleet and sets the fare per unit engaged time, and taxi drivers respond obediently to the planner’s instructions. How much further can decentralization be taken? As will become apparent, a satisfactory answer to this question requires the solution of a complicated mechanism design problem that is beyond the scope of this paper.10 The aim of the rest of this section is more modest}to argue that it should be possible to design a mechanism, albeit somewhat imperfect, that subsidizes taxi travel. The focus is on a particular mechanism in which the fare schedule is regulated and set at the first-best level, drivers are paid their opportunity wage, and owners receive all fare revenue, incur all costs associated with their taxis’ operation, and together receive a lump-sum subsidy from the government. There has been some discussion in the literature of the decentralized determination of taxi fares with cruising cabs. The literature has considered the situation where taxi drivers negotiate the fare with each passenger. There is no fare structure equilibrium. Suppose the contrary and that the equilibrium fare for a particular trip is p. A cruising taxi driver who stops for a prospective passenger has an incentive to quote a fare above p since he knows that the passenger is willing to pay a premium not to wait for the next vacant taxi that passes by. In this situation, taxi drivers would bargain with their prospective passengers. Such fare negotiations would be costly, in terms not only of the prospective passenger’s and driver’s stress and time but also of the congestion caused by the stationary taxi. It is more reasonable to assume that taxi drivers would choose to or be required to commit themselves to a posted fare schedule. With heterogeneity in passengers’ reservation fares, one might observe a distribution of fares for a particular trip. Taxis that charged a higher fare would be refused more 9 Characterization of out-of-equilibrium behavior is somewhat problematical in this context. 10 See Laffont and Tirole w4x for a very good introduction to the mechanism design literature.

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often and therefore vacant a higher proportion of the time. Average waiting time under such a system would be inefficiently high. It would also be inconvenient and disruptive for passengers to find out taxis’ fares. In any event, there are persuasive reasons for the regulation of cruising taxi fares. The situation with dispatch taxis is somewhat different. First, search costs are significantly lower than for cruising taxis since only a telephone call is required to find out the fare and perhaps expected waiting time. Second, waiting time is a potentially contractible element of service. And third, economies of density with respect to waiting time are present at the level of the firm. What fare structure would arise in the absence of regulation? Different taxi companies might offer different fare schedulerwaiting time packages, among which individuals would sort themselves out on the basis of the shadow value of waiting time. Alternatively, a particular taxi cab company might offer a menu of fare schedulerwaiting time packages, akin to the priority pricing systems that used to be employed on computers under batch processing. An obvious difficulty is that firms would have an incentive to understate expected waiting time or to renege on guarantees of maximum waiting time. Truth in advertising could be monitored and enforced, though doing so would be costly and cumbersome. Reputation and repeat usage would also provide some discipline. Given current technology, however, it seems most reasonable to assume that expected waiting time and actual waiting time are non-contractible, though customers would consider their previous experiences with waiting time in deciding which taxi company to call up. Despite economies of density with respect to waiting time at the level of the firm, there are typically many taxi companies with overlapping service areas. The main reason for this is probably that, beyond some scale of operation, there are diseconomies in dispatching, which along with the economies of density, result in a Žprivately. cost-minimizing scale. The diseconomies of scale in dispatching result from problems of coordination with more than one dispatcher, as well as limits on the scheduling capabilities of the individual dispatcher. Another consideration is that most taxi journeys are local. Then one would expect an equilibrium in which firms operate at close to minimum Žprivate. cost and have localized through overlapping service areas, with pricing similar to that obtained in models of spatial monopolistic competition with delivered pricing}whereby the markup over cost on trips originating close to the center of a firm’s service area is higher than those originating near its boundary.11 The central 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 taxi industry is that computerized dispatching should result in a significant increase in firm size.

SUBSIDIZED TAXI TRAVEL

325

with the friction of space, would give firms market power, which they would exploit by pricing above marginal cost. This ‘‘model’’ provides an argument in favor of regulated fares. Thus, in what follows, it will be assumed that taxi fares are fixed by regulation at the first-best level. The question then arises concerning how the subsidy should be paid to the dispatch taxi industry, taking into account the margins of choice of taxi owners and drivers. Consider drivers first. Incentives should be structured in such a way that drivers, acting in their self-interest, make socially efficient decisions. A driver faces at least seven decisions: whether to become a taxi driver, how many hours to drive each day, whether to pick up 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 with the right degree of safety}whether to charge the correct fare, and whether to behave with courtesy. Suppose that a representative riskneutral taxi driver bases these decisions on financial considerations alone, and that, if he is indifferent concerning how to behave, will act in the socially efficient manner. It is easy to posit payment mechanisms that distort his decisions. Suppose, for example, that the taxi company pays for the driver’s gas, insurance, and repairs, and pays the taxi driver a proportion Žpossibly greater than 100%. of the fare revenue he collects such that on average he earns his opportunity wage. Then the taxi driver has an incentive to maximize fare revenue per unit time. This encourages him to drive too fast and with insufficient safety, overcharge where possible, drive circuitously, skip unprofitable Žbecause travel time to pick up the passenger is high, or because the passenger’s destination is likely a long distance from the closest subsequent fare. trips, and steal fares Žwhere 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 the marginal 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 be indifferent concerning all other margins. Thus, given the assumptions, this mechanism is efficient. One might reasonably object to the assumption that, when a driver is indifferent as to how to behave, he will behave in the socially efficient manner. This objection can be at least partially countered by augmenting the model to include tips. Tipping is a strange phenomenon from an economist’s perspective, at least for non-repeat interactions, since apart from the fear of being cursed at and the warm glow from being generous, 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, drivers could receive part of their remuneration in the form of wages and the rest via profit-sharing.

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any event, assume that on average the tip is based on quality of service. Since providing quality service entails little cost, with the mechanism under consideration a driver would have an incentive to provide quality service which would, largely though not completely, result in his making socially efficient decisions. The exceptions include speeding to get a passenger to an appointment on time, skipping assigned fares when the expected tip falls short of the expected opportunity tip, and stealing fares. These abuses could be checked by the taxi companies themselves. This mechanism has two potentially serious flaws. First, drivers would have an incentive to pocket fare revenue. A driver could do this by negotiating a fare with the passenger that is lower than the regulated fare, on condition that the meter be turned off, and telling the dispatcher that the passenger did not show up. Second, drivers would have an incentive to pad their hours, claiming that they were available but idle when they were in fact shopping, having a donut and coffee, etc. Appropriate, cost-effective technology to check such abuses is almost certainly available, though drivers would no doubt resist its introduction. Since taxi owners would have an incentive to monitor their own drivers, the government could stay out of the business of monitoring driver behavior, except for extreme abuses. Where dispatch taxis spend some time at taxi stands and some time cruising, the incentive and monitoring systems would have to be designed so that they are suitable for all modes of service, or else taxi companies would have to specialize according to type of service, which would result in giving up some benefits from economies of density. Now consider the taxi owner Žor firm.. He has essentially two decisions }how many taxis to operate, and at what rate to utilize them, including whether to keep them on the road at all. How he would behave depends crucially on how the subsidy is paid. Suppose that the subsidy is paid to each taxi, without restriction. Then there is an obvious incentive for every car owner to register his car as a taxi, get the subsidy, and continue using the car for personal use. Suppose alternatively that the subsidy is per taxi-hour of operation. Taxi owners would then have an incentive to overstate their hours of operation. Taxi owners and drivers would have an incentive to collude to overstate the hours driven. And if the subsidy per hour exceeds the hourly wage, taxi owners would have an incentive to hire and pay family members and friends who would not drive the taxis. One way around these problems would be for the government to pay the taxi industry a lump sum, have the taxi association divvy up the lump sum between companies on the basis of hours driven, and have the companies police one another. This might be exploited as a means of preventing entry, and could invite gangsterism. But it would also provide an incentive

SUBSIDIZED TAXI TRAVEL

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for taxi companies to introduce procedures to facilitate monitoring abuses.13 A final issue is whether the government should restrict the number of taxis or permit free entry. With the mechanism under consideration, free entry with certification of drivers and regulation of taxis would seem best, since this would permit flexibility in the face of predictable and unpredictable Že.g., a dry winter, transit strikes. fluctuations in demand. The particular features of the mechanism discussed above are of secondary importance. Detailed analysis that costs out the monitoring technologies and quantifies the deadweight losses due to distortion might well find that a significantly different scheme is optimal. What is important are two general points that the above discussion illustrates. First, design of a mechanism for the effective provision of taxi services should pay considerable attention to the behavioral responses of both taxi drivers and taxi owners, as well as to the availability of technologies to counter perverse incentives. Second, while attainment of the full first best is most likely infeasible given current technology, it should be possible to design a mechanism that subsidizes taxi travel and that improves considerably on current systems. 4. AN EXAMPLE The example is constructed with two considerations in mind. First, the form of trip demand assumed is deliberately simple in order to illustrate that, even in the simplest situations, multiple equilibria may arise. Second, observable parameters are given realistic values, while the values of unobservable parameters are determined by calibration to a plausible base case. It is assumed that all trip opportunities provide a benefit b . Since the marginal cost of a trip to x, including the opportunity cost of the resident’s time, is Ž lU q z . t Ž x, uT . s 2Ž y q k q c q z .ŽŽ xrn . q t q 1rŽ2 un .., where u ' 'uT , the planner will dispatch taxis for all trip opportunities up to a distance x such that Ž lU q z . t Ž x, uT . s b , and will refuse all trip opportunities beyond this distance. The following parameter values are assumed: y s 10 ks1 cs5 z s 20

t s .05 n s 12 G s 10000

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

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RICHARD ARNOTT

where units are dollars, hours, and miles. The opportunity cost of a taxi driver’s time Ž y . is $10rhour; the operating costs of an idle taxi Ž k . are $1rhour; the additional operating costs of an occupied taxi Ž c . are $5rhour; the opportunity cost of a resident’s time Ž z . is $20rhour; pick-up and drop-off time on a one-way trip Žt . is .05 hours; taxi speed Ž n . is 12 m.p.h.; and population density is 10,000 residents per square mile. That leaves two parameters, n and b , which are chosen to be consistent with a base case social optimum in which T s 2.4384, u s 0.20095, and x s 5; taxi density ŽT . is approximately 2.5 per square mile; taxis are idle Ž u. about a fifth of the time; all trip opportunities up to a distance Ž x . of 5 miles are accepted and all trip opportunities beyond that distance are rejected. Expected waiting time in this optimum is 1rŽ2 n'uT . f 0.05952, about 3 12 minutes. From the optimum condition for x, given in the previous paragraph, b f 37.89, and from the stationary-state-condition Ž2., nŽ x . s n f 3.2025 = 10y6 . Thus, the trip benefit is close to $38, while the value of n implies that a resident takes a trip on average every 3976 Žprecisely, 1rŽ np x 2 .. hours. It is straightforward, though tedious, to check that the above ‘‘solution’’ is indeed the social optimum for the specified parameters; the details of the analysis are given in Appendix 3. Holding the number of taxis and the taxi fare per unit engaged time at their first-best optimal levels, the full set of equilibria can be solved for 14 ; the details of the procedure are given in Appendix 2. As noted in Section 3, there is always an equilibrium corresponding 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. The results are plotted in Fig. 1. The lower locus corresponds to the equilibria that coincide with the social optima. The upper locus corresponds to Pareto-dominated equilibria. Three points bear mention. First, for each value of G there are two equilibria. Second, there is the issue of equilibrium selection. On the basis of a plausible adjustment mechanism, the lower equilibrium is analogous to a stable, congested equilibrium in traffic flow theory, and the upper equilibrium to an unstable, hypercongested equilibrium. Unstable equilibria typically have perverse comparative static properties. This is true of the upper equilibrium in Fig. 1, for which waiting time increases with an increase in population density. Thus, it appears that the upper equilibrium can be excluded on stability grounds. If waiting time is initially at a high level Žabove that corresponding to the unstable equilibrium. it will con14 MATHEMATICA was used for all the calculations and for the graphing. MATHEMATICA had to be ‘‘helped’’ to find all the equilibria. The equilibrium conditions were reduced to a single quintic equation in u . For the range of parameter values considered, there were two real and two imaginary solutions, and a singularity.

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329

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

tinue to grow without bound; if initial waiting time is at a level below that corresponding to the unstable equilibrium, waiting time will converge to the stable equilibrium level. The third point is that, at the social optimum, which coincides with the lower equilibrium, waiting time falls with population density. This is a manifestation of the economies of density in taxi travel on which this paper has focused. 5. CONCLUDING COMMENTS This paper focused on a point that the previous literature has noted only in passing. Double the number of taxis and double the number of trips, and waiting time falls. Because of these economies of density, first-best taxi pricing entails operation at a loss; more specifically, the optimum subsidy covers the shadow cost of taxi idle time at the optimum. The paper made three principal contributions. The first was to consider dispatch taxi service, in contrast to the previous literature which has concentrated on cruising taxis. The second was to point out that, even with first-best pricing-cum-subsidization, attainment of the social optimum is not guaranteed. The third point was that first-best pricing should not be quickly dismissed, as has been done in the previous literature, since it should be possible to design incentive-cum-monitoring mechanisms that are consistent with subsidization of taxi travel and come close to achieving the first best. The paper has ignored an important aspect of the ‘‘taxicab problem’’}traffic congestion. Subsidizing taxi travel may divert passengers

330

RICHARD ARNOTT

away from mass transit, exacerbating underpriced and hence excessive auto congestion. But probably of greater quantitative importance, such subsidization would encourage people to use taxis instead of their own cars for business trips during working hours, for commuting, and for other trip purposes. Thus, consideration of congestion is likely to strengthen the case for the subsidization of taxi travel. This issue, as well as the design of practicable subsidy mechanisms, are promising topics for future research. Among experts in transportation economics, there is a broad consensus that urban auto travel is excessive but at the same time that substantially expanding mass transit and subsidizing it sufficiently to induce a significant number of car travelers to use it is prohibitively expensive ŽSmall w7x.. Taxi service provides many of the advantages of the automobile}flexibility, privacy, and convenience}without significant capital costs. Providing taxi travel at its shadow price might therefore contribute significantly to solving the urban transportation problem. APPENDIX 1 Deri¨ ation of Eq. Ž7. This appendix provides a heuristic derivation of Eq. Ž7.. We wish to solve for average waiting time, which equals expected distance to the closest idle cab divided by velocity. The expected distance to the closest idle cab is H0x xg Ž x . dx, where g Ž x . is the probability density function of distance to the closest idle cab. To solve for g Ž x ., proceed as follows. Let P Ž K, x . be the probability that there are K idle taxis within a distance x. This equals Žthe 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 and x . plus Žthe 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,. 2

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

q P Ž K y 2, x y dx . Ž 2p xuT dx . q ??? . Employing a Taylor series expansion of the terms P Ž?, x y dx . around x, and collecting terms in dx yields

­PŽ K, x. ­x

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

Ž A1.1.

331

SUBSIDIZED TAXI TRAVEL

For K s 0, this reduces to

­ P Ž 0, x . ­x

s y2p xuTP Ž 0, x . ,

Ž A1.2.

since P Žy1, x . s 0. Make the transformation of variables z s p x 2 uT, and define PˆŽ K, z Ž x .. s P Ž K, x .. Then ­ PˆŽ0, z .r­z s yPˆŽ0, z ., so that PˆŽ0, z . s c 0 eyz . Since PˆŽ0, 0. s 1, c 0 s 1. Thus, P Ž 0 , x . s eyp x

2

uT

.

Ž A1.3.

Now, the probability that the closest idle taxi is between x and x q dx is the probability that there are no idle taxis within a distance x, P Ž0, x ., times the probability that there is an idle taxi between x and x q dx, 2 2p xuT dx. Thus, g Ž x . s 2p xuT eyp x uT and `

`

H0 xg Ž x . dx s H0 2p x

2

uTey p x

2

uT

dx.

Ž A1.4.

From the integral tables, the value of this integral is 1rŽ2'uT .. APPENDIX 2 Equation Ž9. for the Numerical Example This appendix derives the explicit equation corresponding to Ž9. for the special case where the benefit from a trip is fixed at b , the Poisson arrival rate of trip opportunities from each unit area is n, and the fare per unit engaged time is set at the first-best level. Under these assumptions, a resident will accept all trip opportunities up to the distance x Ž uT U . which solves b s Ž y q k q c q z . t Ž x Ž uT U ., uT U . and none beyond.Using Ž7., x Ž uT U . s

bn 2Ž y q k q c q z .

y nt y

1 2'uT U

.

Ž A2.1.

Let a ˆ s 2rn , ˆb s 2Žt q 1rŽ2 n'utU . and ˆc s brŽ y q k q c q z ., so that t Ž x, uT U . s ax ˆ q ˆb and x Ž uT U . s nrŽ2Ž ˆc y ˆb ... Then Ž9. is T U Ž 1 y u. G

s

ˆ ˆ.. n Ž ax ˆ q ˆb . 2p x dx H0nrŽ2Ž cyb

s 2 np

ž

ax ˆ3 3

q

s npn 2 Ž ˆ cyˆ b.

ˆbx 2 2 2

ž

n r Ž2 Ž cyb ˆ ˆ..

/

0

2ˆ cqˆ b 12

/

.

332

RICHARD ARNOTT

Substituting back for ˆ c and ˆ b yields T U Ž 1 y u. G

npn 2

s

6

=

b

ž

yqkqcqz

b

ž

yqkqcqz

y2 tq

ž

2 n'uT U

1

q tq

ž

2

1

//

2 n'uT U

//

,

Ž A2.2.

which, multiplying through, contains a constant term plus terms in u1r2 , u, u 3r2 , u 2 , and u 5r2 . Thus, ŽA2.2. is a quintic equation in u1r2 . In constructing the numerical examples, the following procedure was followed. For each G: 1. Solve the social optimum problem Ždescribed in Appendix 3., which gives T U , uU , xU , and lU . 2. With the fare structure set as pŽ x . s lU t Ž x, uT U . and the number of taxis held at the first-best level, residents choose x according to ŽA2.1. and the set of equilibrium u’s are the Žreal. roots of ŽA2.2.. APPENDIX 3 The Social Optimum for the Numerical Example The social welfare optimization problem for the example is

max SS s

x, T , u

T

x

H0 2p n b dx y G Ž y q k q Ž c q z . Ž 1 y u . . qf

ž

T Ž 1 y u. G

y

H0

x

x

q tq

ž ž n

1 2 n'uT

//

4 np x dx .

/

Ž A3.1. Set 'Tu s u and evaluate the integrals:

max SS s n bp x 2 y

x, T , u

qf

ž

T G

Ž y q k q c q z. q

Tyu2 G

y

4 np x 3 3n

y

u2 G

2p nx 2

n

Ž c q z.

ž

tn q

1 2u

//

. Ž A3.2.

333

SUBSIDIZED TAXI TRAVEL

The corresponding first-order conditions are x : 2p n b x y f

ž

4 np x 2

n

q

4 np x

n

ž

tn q

1 2u

«xs T : Ž yŽ y q k q c q z . q f .

u:

2u G

Ž c q z. y f

«x s 2

ž

1 G 2u G

//

s 0,

bn 2f

y tn q

ž

1 2u

/

Ž A3.4.

s 0 « f s y q k q c q z Ž A3.5. y

p nx 2 u 2n

2 u 3n Ž y q k .

p nG Ž y q k q c q z .

/

s0

Ž using Ž A3.5. . .

Ž A3.6.

Substitute ŽA3.5. into ŽA3.4., and then the resulting expression for x into ŽA3.6.. This generates a quintic expression in u . If this expression has real roots, it has two of them, one with 3 x u ) 1, the other with 3 x u - 1. It is straightforward but tedious to show that the former corresponds to the social optimum.

REFERENCES 1. M. Beesley and S. Glaister, Information for regulating: The case of taxis, Economic Journal, 93, 594]615 Ž1983.. 2. G. W. Douglas, Price regulation and optimal service standards: The taxicab industry, Journal of Transport Economics and Policy, 116]127 Ž1972.. 3. M. W. Frankena and P. A. Pautler, Taxicab regulation: An economic analysis, Research in Law and Economics, 9, 129]165 Ž1986.. 4. J.-J. Laffont and J. Tirole, ‘‘A Theory of Incentives in Procurement and Regulation,’’ M.I.T. Press, Cambridge, MA Ž1993.. 5. C. F. Manski and J. D. Wright, Nature of equilibrium in the market for taxi services, Transportation Research Record, 619, 11]15 Ž1976.. 6. H. Mohring, Optimization and scale economies in urban bus transportation, American Economic Re¨ iew, 62, 591]604 Ž1972.. 7. K. Small, ‘‘Urban Transportation Economics,’’ Harwood Academic, London Ž1992..