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The Impact of Service Refusal to the Supply–DemandEquilibrium in the Taxicab Market
Dali Wei1 & Changwei Yuan2& Hongchao Liu3
&
Dayong Wu3& Wesley Kumfer3
Published online: 16 May 2016# Springer Science+Business Media New York 2016
Abstract Service refusal is a significant problem in the taxicab market, especially indeveloping countries where policies and regulations have not been well developedagainst this unpleasant phenomenon. Understanding the disturbance of service refusalto the demand–supply equilibrium is essential for the governing authorities to developeffective pricing policies and regulations to tackle the issue. This paper proposes asigmoid function that depicts the service refusal behavior due to the lower-than-expected profit. Integrated with this service refusal function, the interrelation betweenthe fleet size, fare, and passenger demand is well evaluated at an aggregate level. Thesocial optimum and maximum profit solutions are examined with consideration of thepresence of service refusal. It is found that in a market with a non-negligible refusalproblem, raising fares at a certain level would drive passenger demand as the benefit
Netw Spat Econ (2017) 17:225–253DOI 10.1007/s11067-016-9324-z
* Changwei [email protected]
Dali [email protected]
Hongchao [email protected]
Dayong [email protected]
Wesley [email protected]
1 Partners for Advanced Transportation Technologies, University of California Berkeley, 1357 S46th St, Richmond, CA 94804, USA
2 School of Economy and Management, Chang’an University, 161 Chang’an Middle Road, Yanta,Xi’an, Shaanxi 710064, China
3 Civil and Environmental Department, Texas Tech University, 10th and Akron, Lubbock,TX 79409, USA
from relieved service refusal outweighs the negative impact of the markup itself. This iscontrary to the common understanding that raising fares will always reduce thedemand. The maximum profit and maximum social welfare achievable will drop ifthe service refusal becomes severer due to higher expected profits. However, at thesocial optimum the profit could be positive in the presence of service refusal. Theseproperties of the model are demonstrated by a numerical study.
Keywords Servicerefusal .Expectedprofit .Demand.Socialoptimum.Maximumprofit
1 Introduction
Taxicab is an active mode of transportation whose flexible service is irreplaceable inmetropolitan areas. In Beijing, for example, there are about 67,000 taxicabs servingnearly 6 million rides a day; on average, each taxi vehicle runs approximately 400 km aday (Beijing Transportation Research Center 2011; China Statistics Bureau 2012; Zhaoet al. 2015). While the taxi market in Beijing is under strict fare and entry regulations,refusal to service is not uncommon. A recent survey in Beijing found as many as nearly10,000 taxi vehicles were at rest during the afternoon peak hours, evading service incongested hours (Beijing Municipal Commission of Transport 2013; Cao et al. 2006;Institute for Energy and Environmental Research Heidelberg 2008).
There are many reasons a driver would escape from serving during peak hours. Oneof the most obvious is the lower-than-expected profit. The drivers often complained theprofit is too low and the level of stress is high when driving in congested traffic. As aresult, the drivers would rather suspend the service and take rests. Studies to date havemostly focused on well-regulated markets where service refusal is not present or at aminimum with negligible impacts (Cairns and Liston-Heyes 1996; De Vany 1975;Douglas 1972; Pachon and Johansen 1989). In a regulated market, once the supply (i.e.,the fleet size) has been determined, it is usually considered constant, and the number oftaxicabs in service is independent of the taxi fare. Economic models, therefore, havefocused on developing taxi fare structures and fleet size that maximize a selectedobjective, such as the profits and social welfare (Chang and Chu 2009; Kim andHwang 2008; Salanova et al. 2011). As most of the economic models lack consider-ation of the traffic impacts, Yang et al. (2005b) introduced distance-based and delay-based taxi fare models that take into account the effects of traffic congestion on the taximarket (Yang et al. 2010a; Yang et al. 2005b).
As in many other systems, equilibrium can also be found in a taxicab systembetween the supply and demand (Hasan et al. 2007; Kitthamkesorn et al. 2013;Rouhani and Gao 2015; Ben-Akiva et al. 2015). The supply–demand equilibriumand the relation between the fleet size, elastic demand, and taxi fare have been wellanalyzed in literature such as Arnott (1996), Cairns and Liston-Heyes (1996), andWong et al. (2001b). The road network, the origin–destination based demand and thesearch friction were modeled by Yang and Wong (1998), Yang et al. (2002) and Yanget al. (2010b). Taking into account the spatial structure of the taxi market, their modelsare capable of quantifying a number of system performance measures such as utiliza-tion rate and level of service quality for different zones. Recently, Yang et al. (2010b)further improved the model by considering the bilateral waiting and search behaviors,
226 D. Wei et al.
and more realistic waiting time functions were developed. Notably, to consider thevariations in both demand and supply, Yang et al. (2005a) proposed a multi-perioddynamic model. The service intensity was treated as an endogenous variable, whichwas determined through finding the equilibrium state in which drivers cannot alter theirwork schedules to increase their profits. Because of the competition, the net profits ofdrivers in service and out of service are both zero in the period when the number oftaxis in service is less than the fleet size. It is theoretically sound but hard to justify inreal world situation, where the profit for drivers in service is obviously positiveotherwise they will not choose to serve. The most important reason why the driverevade service during the peak hour is that the net profit per ride is less than theexpected, which depends on drivers’ own judgments of both monetary factors (e.g.,expected income per day) and non-monetary factors (e.g., fatigue, stress).
A common assumption of previous studies is that the market is well regulated andlittle attention has been paid to the issue of service refusal as well as its impact to thesupply–demand equilibrium. Refusal, however, may affect the demand–supply relationin many ways (von Massow and Canbolat 2010; Flötteröd et al. 2012; Watling andCantarella 2015). On one hand, service refusal decreases the actual size of the fleet inservice, leading to larger waiting time. On the other hand, the degraded quality ofservice in terms of waiting time affects the demand side by reducing the number ofpassengers willing to take the service. Of course, this idling-type service refusal islegitimate; and therefore, transportation agencies cannot apply punitive policies orcoercive measures to eliminate this phenomenon. Instead, their focus has been directedto motivate and stimulate taxi drivers back to service by regulating the market throughfare adjustment. Understanding the underlying disturbance of this unpleasant phenom-enon to the demand–supply equilibrium is essential to such policy development.
This study was developed to investigate the topics such as the interactions betweenthe service refusal behavior and the demand–supply equilibrium, as well as its impactto the optimal social welfare and maximum profit of cruising taxi services. The farestructure currently in use in Beijing, China is adopted in this research, and a function ofservice refusal is introduced into the system to analyze its impact. The static effects ofthe regulating variables such as the fare, the fleet size, and the expected profit areintegrated into a mathematic model to investigate the changes in waiting time andpassenger demand as related to service refusal.
A definitional approach is provided to describe the relation between the regulatingvariables, the service refusal phenomena, and the variables of the taxi market. As theaggregated effects of the regulating variables and the service refusal exhibit a differentequilibrium mechanism, the solutions for the maximum social welfare and maximumprofit are investigated, and comparison was made between the two conditions, eitherwith or without the service refusal phenomenon. A case study is provided using the taxifare structure currently in use in Beijing, China to demonstrate the property ofthe model.
The paper is structured as follows: In section 2, we introduce the variables and theservice refusal function. In section 3, the static effects of the fare, the fleet size and themean expected profit are investigated. In section 4, we examine the maximized socialwelfare solution. Section 5 addresses the maximized profit solution with service refusal.Beijing’s case study is provided in Section 6 to highlight the major theoretical findings.Conclusions and recommendations are given in Section 7.
The Impact of Service Refusal to the Supply–demand Equilibrium 227
2 Fare Structure and Service Refusal Function
2.1 Variable Definitions and Fare Structure
Major variables used in the paper are defined in Table 1.A realistic fare structure commonly adopted in China is specified as:
P ¼ ps þ plLþ pttc ð1Þ
which consists of an initial flag-drop charge of ps, the charge based on the ridelength plL and the congestion based charge of pttc. tc is the length of the period whenthe taxi is traveling at a speed less than a specified threshold of u0 indicating acongested traffic condition. For example, u0 is set to be 12 km/hour in Beijing. tc isthen determined by the following equation:
tc ¼ σL=uc ð2Þ
where the parameter of σ accounts for the proportion of congested road segments, forwhich the traveling speed is less than u0. uc denotes the average traveling speed onthese congested road segments.
Table 1 Variable notations and definitions
Variable notations Definition
P Fare per ride (CNY)
ps Initial flag-drop price (CNY)
pt Congestion charge per hour (CNY/hour)
pl Charge per kilometer (CNY/km)
L Average length of each ride (km)
uc Average traveling speed in congested network (km/hour)
uf Free-flow speed (km/hour)
T Trip time per ride (hour)
C Cost per hour of each taxicab in service (CNY)
Co Cost per hour of each taxicab out of service (CNY)
Q Demand per hour (trips/hour)
W Average waiting time (hour)
u0 Speed threshold below which the congestion based charge is activated (km/hour)
tc Period when the taxi is traveling under the threshold of u0 (hour)
σ Proportion of congested road segments
N Fleet size
Nv Number of vacant taxis
r Service refusal rate
The Renminbi (code: CNY) is the currency of the People’s Republic of China (PRC), whose principal unit isYuan
228 D. Wei et al.
2.2 The Function of Service Refusal and its Properties
Taxi drivers’ willingness to serve during the peak hours relies on the expected profitgained from each ride. If the profit per ride is less than the expected, taxi drivers aremore inclined to refuse service. To depict this behavior, a sigmoid function of r(P)dependent on the fare per ride is proposed as:
r Pð Þ ¼ 1−1
1þ e− P−CT−sð Þ=μ whereμ > 0 ð3Þ
where s and μ are parameters to be explained later. C is the cost per hour ofeach taxicab in service and and T is the trip time per ride (hour). r(P), boundedbetween 0 and 1, describes the tendency of a taxi driver to refuse to serve. Ifr(P) approaches to 0, the taxi driver would be willing to serve. In that case, itis very unlikely for a taxi driver to exhibit the refusal behavior. However, inthe opposite, r(P) approaching to 1 implies that the taxi driver is very unsat-isfied with the profit and more inclined to refuse to serve the passengers.Assuming the homogeneous behavior among taxi drivers, r(P) also representsthe service refusal rate in the entire fleet size.
Now let’s explore the properties of this function, as well as the physical meanings ofthe parameter s and μ. To examine how the service refusal rate changes with respect tothe fare, the partial derivative of r(P) with respect to P is:
δ≡∂r∂P
¼ −1−rð Þ2μ
e−P−CT−s
μ ð4Þ
Obviously, ∂r/∂P is negative, which indicates that a higher value of fare per ridewould result in a lower rate of service refusal as the profit gained per ride increases withthe fare per ride.
The parameter of s actually implies the mean expected profit per ride since theservice refusal rate equals 0.5 if the profit per ride (P−CT) equals s. A larger valueof s would result in a higher service refusal rate given the same fare per ride. Thiscan be easily verified by the partial derivative of r with respect to s, which is in theform of:
γ≡∂r∂s
¼ 1−rð Þ2μ
e− P−CT−sð Þ=μ ð5Þ
since μ is positive as defined in Eq. (3), γ is always greater than zero, which impliesa positive correlation between the service refusal rate and the mean expected profit s.This is illustrated in Fig. 1, where service refusal curves with respect to the fare per rideshift to the right given a higher mean expected profit of s.
The parameter μ indicates the smoothness of the service refusal rate withrespect to the fare per ride as shown in Fig. 2. It can be viewed as thesensitivity of the service refusal behavior with respect to the fare per ride.Taxi drivers are more sensitive to the fare per ride with smaller values of μ,
The Impact of Service Refusal to the Supply–demand Equilibrium 229
which result implies that when the profit per ride is close to the driver’sexpectation, a small change in the fare would give rise to a large increase ordecrease in service refusal rate and the size of the fleet in service.
3 Impact of the Service Refusal on the Equilibrium in Taxicab Market
With the service refusal function defined in Section 2, this section investigates itsimpact on the interaction between the demand and the supply. Particularly, the analysiswill focus on how the demand, the customer waiting time, and the consumer
0 10 20 30 40 50 60 70 80 900
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fare per ride (CNY)
Ser
vice
ref
usal
rat
e
-20 -10 0 10 20 30 40 50 60 70
Profit per ride (CNY)
=5=10=15=20=25
S=20C=40T=0.7
Fig. 2 Service refusal rate with respect to the congestion price and the parameter of μ
0 10 20 30 40 50 60 70 80 90
Fare per ride (CNY)
-20 -10 0 10 20 30 40 50 60 70
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Profit per ride (CNY)S
ervi
ce r
efus
al r
ate
s=5s=10s=15s=20s=25
=10C=40T=0.7
Fig. 1 Service refusal rate with respect to the congestion price and the parameter of s
230 D. Wei et al.
surplus change with respect to the fare per ride, the fleet size and the meanexpected profit.
3.1 Basic Assumptions
Let’s assume the demand of Q (trip/hour) is in the form of:
Q ¼ f P;W ; Tð Þ ¼ ~Qe−α PþκWþτTð Þ ð6Þ
where α, κ, τ and ~Q are positive parameters (Wong et al. 2001a; Yang et al. 2000). ~Q isthe potential customer demand. κ and τ are the monetary values of unit waitingtime and in-vehicle travel time, respectively. α is a scaling parameter, whichimplies the sensitivity of demand to the full trip price. Eq. (6) indicates thedemand would decrease with a higher fare per ride, a longer waiting time and alonger travel time.
The waiting time W, as a measure of the service quality, depends on the number ofvacant taxis Nv and can be derived as (Douglas 1972):
W ¼ ANv
ð7Þ
where A represents presenting the number of vehicle-hours needed to cover the streetnetwork. Nv is the number of vacant taxis. This relation was theoretically developed byDouglas (1972). The demand Q, and the travel time T are related by:
Nv ¼ 1−rð ÞN−QT ð9Þ
where (1− r)N is the number of taxis in service. The travel time of T can becalculated by:
T ¼ Lσuc
þ L 1−σð Þuf
ð9Þ
As shown from Eq. (6) to Eq. (9), the demand without service refusal is a specialcase when the mean expected profit tends to be negative infinity and the service refusalrate tends to be one.
3.2 Effects of the Service Refusal on Demand
To facilitate the discussion of the equilibrium properties, let fP be the partial derivativeof Q with respect to P, where W is treated as an independent variable. fW is denoted asthe partial derivative of Q with respect toW. From the definition of Q in Eq. (6), fP andfW are both negative.
Let’s first look at the impact on the demand Q of three variables of interest,specifically the fare per ride P, the fleet size N, and the mean expected profit s.
The Impact of Service Refusal to the Supply–demand Equilibrium 231
From Eqs. (3) to (9), the partial derivative of Q with respect to P can bederived as:
QP≡∂Q∂P
¼ f P þ f W∂W∂Nv
−N∂r∂P
−QPT� �
ð10Þ
By letting wNv denote ∂W/∂Nv, and reorganizing, we have:
QP ¼ f P− f WwNvNδ1þ f WwNvT
ð11Þ
Eq. (11) shows an aggregate effect of the fare per ride on the demand. The first termof fP in the numerator is negative, which represents the decreased demand if the fare perride rises by one unit. But this is compensated by a positive term of − f WwNvNδ(δ<0, fW<0 and wNv < 0). This term implies an increase in demand due to thedecreased waiting time, which is attributed to the decreased service refusal rate andthe increased number of taxis in service.
Therefore, the sign of QP is undetermined. Increasing the price at acertain level could drive the demand. In that case, customers are willing toaccept the markup, since the benefit from decreased service refusal rate ismore significant compared with the markup itself. This is fundamentallydifferent from the market without service refusal, where QP would besimplified from Eq. (11) to f P= 1þ f WwNvTð Þ, which is apparently negative.That implies the demand always decreases as the fare rises in the marketwithout service refusal.
The impact of the fare per ride on the demand also implies that the trip cost,which is the sum of the full trip price, the cost of the waiting time and the costof the travel time, would not necessarily increase given a higher fare per ride.The benefit from the decreased waiting time due to the increased number ofoperating taxis could outweigh the increase in the fare. It is interesting to notethat the service refusal behavior is only reflected through δ, the derivative ofservice refusal rate with respect to the fare per ride. That means if the servicerefusal rate is constant or irrelevant to the fare (δ= 0), the impact on thedemand would be the same as in the market without service refusal.1
Now let’s examine the impact of the fleet size on the demand with respect to servicerefusal. The partial derivative of the demand with respect to the fleet size N can bederived similarly as:
QN ≡∂Q∂N
¼ f WwNv 1−rð Þ1þ f WwNvT
ð12Þ
which is obviously positive and implies that an increase in fleet size wouldresult in an increase in the demand. Without the service refusal, it would be
1 If the service refusal rate is irrelevant of the fare per ride, the number of operating taxis is independent of thefare and the benefit from the increased taxis in service would vanish.
232 D. Wei et al.
simplified to f WwNv= 1þ f WwNvTð Þ, which is also positive. Hence, the effectsof the fleet size for a market, either with or without the service refusal, havethe same trend. But the service refusal does shrink the impact of the fleet sizewith the coefficient of (1 − r) between 0 and 1. This is easy to understandsince only a proportion (1 − r) of the change in the fleet size would actuallyaffect the market.
The expected mean profit is directly related to the service refusal rate and the level ofseverity of service refusal in a market. The partial derivative of the demand with respectto the mean expected profit can be derived similarly as:
Qs ≡∂Q∂s
¼ − γN f WwNv
1þ f WwNvTð13Þ
Qs is negative since f WwNv and γ are both positive. This implies that the demanddecreases if taxi drivers expect a higher profit per ride. Examination of the term of γN f WwNv in Eq. (13) shows that the decrease in demand is essentially attributed fromthe higher service refusal rate implied by γ, the decreased number of taxis in service δNand finally the increased waiting time of γN f WwNv .
Generally, in the market with service refusal, an increase in the fare would notnecessarily decrease the demand and increase the trip cost. Those changes actuallydepend on the interaction between the benefit from increased number of taxis in serviceand the negative impact of the increased price itself. This is the fundamental differencecompared to the market with taxi drivers’ full compliance. Also, the service refusalwould weaken the regulating impact of the fleet size on the demand. Last but not least,severe service refusal brought by the high expected profits per ride would hurt thedemand due to the long waiting time.
3.3 Effects of the Service Refusal on Waiting Time
The average customer waiting time, as a measure of the service quality, is investigatedin this section. Using Eqs. (3) to (9), the partial derivative of W with respect to P is:
WP ≡∂W∂P
¼ −wNv
Nδ þ f PT1þ f WwNvT
ð14Þ
which is negative given δ<0, fP<0 and wNv < 0. The first term of Nδ implies therewill be more vacant taxis are available due to the increased fare per ride and the lowerservice refusal rate. The second term of fPT implies the decreased number of occupiedtaxi-hours due to the lower demand brought by the increased fare. The combinedeffects from these two terms would result in a decrease in waiting time as the fare rises.
Without considering the service refusal, WP would only include the second term inthe numerator, while the benefit from the increased number of taxis in service vanishes.The impact of the fare on the waiting time is actually enhanced due to the additionalimpact from the increased number of taxis in service as the fare rises. Similar to Eq.(11), the service refusal is only reflected through the derivative δ, which indicates thatthe impact of service refusal on the waiting time would vanish if the service refusal andthe fleet size actually in service is irrelevant to the fare per ride.
The Impact of Service Refusal to the Supply–demand Equilibrium 233
Similarly, we obtain the partial derivative of the waiting time with respect to the fleetsize N as:
WN ≡∂W∂N
¼ wNv
1−r1þ f WwNvT
ð15Þ
which is negative, indicating the waiting time would decrease given a larger fleetsize. For the market without service refusal (r=0), Eq. (15) would be simplified towNv= 1þ f WwNvTð Þ. Obviously, the impact of the fleet size in presence of servicerefusal is weakened by a coefficient of (1− r) since only a proportion of the fleet isactually in service.
Now let’s examine how the mean expected profit affects the waiting time. Thepartial derivative is:
Ws ≡∂W∂s
¼ wNv
− γN1þ f WwNvT
ð16Þ
Which is positive, implying that the waiting time becomes longer as the expectedprofit rises. This is due to the higher service refusal rate and fewer taxis in service. Thisresult indicates that the service refusal would degrade the service quality and lead tolonger customer waiting time.
In contrast to the demand, the effects of the regulating variables on thewaiting time are consistent for both markets with and without service refusal.The waiting time rises as the fare decreases and the fleet size increases.However, the magnitude of the impact is different. Compared with the marketwithout service refusal, the impact of the fare per ride on the waiting time isenhanced since the service refusal rate imposes additional influence on thenumber of vacant vehicles. Meanwhile, the impact of fleet size is weakenedas only a proportion of the fleet is actually in service. Moreover, the servicerefusal would prolong the customer waiting time as the mean expected profitrises.
3.4 Effects of the Service Refusal on Consumer Surplus
The consumer surplus is an economic measure of consumer satisfaction, whichis calculated by analyzing the difference between what consumers are willingto pay for the service relative to its market price (Flores-Guri 2003). As thewaiting time enters the demand curve, the consumer surplus is obtained byintegrating under a demand curve in which the waiting time is held fixedwhile the fare varies (Cairns and Liston-Heyes 1996). The demand Q isrewritten as the function of q(ω) = q(P+κW+ τT), and the consumer surplusis obtained as:
Sc ωð Þ ¼Z ∞
ωq xð Þdx ð17Þ
234 D. Wei et al.
The partial derivative of Sc with respect to the fare can be derived as:
Sc;P ≡∂Sc∂P
¼ −Q∂ω∂P
ð18Þ
Given that
QP ¼ f ω∂ω∂P
¼ −αQ∂ω∂P
ð19Þ
Eq. (18) can be written as:
Sc;P ¼ −QQP
f ω¼ QP
αð20Þ
where fω is the derivative of f with respective to ω.Since the sign of QP is undetermined from Eq. (11), the consumer surplus does not
necessarily decrease if the fare rises. Hence, in a market with service refusal, the fare atwhich the customers can obtain the maximum surplus is not necessarily at zero. Instead,the price at the optimal consumer surplus would also maximize the demand becauseQP=0 also makes Sc,P=0. In contrast, in a market without service refusal or with aconstant service refusal rate where δ=0, the consumer surplus would decrease as farerises and the fare at the optimal consumer surplus is always zero.
Similarly, the impact on consumer surplus regarding the change of fleet size N canbe examined by deriving the following partial derivative:
Sc;N ¼ −Q∂ω∂N
¼ −QQN
f ω¼ QN
αð21Þ
Apparently Sc,N is positive, meaning the consumer surplus would increase as morevacant taxis are available and the waiting time is shortened. However, the service refusalbehavior would weaken the effects of fleet size in the same way it weakens the demand,since only a proportion of the change in the fleet size actually affects the market.
Now let’s check how the expected profit would impact the consumer surplus. Thepartial derivative of Sc with respect to s is:
Sc;s ¼ −Q∂ω∂s
¼ −QQs
f ω¼ Qs
αð22Þ
Sc,N is negative as it has the same sign as Qs. Obviously, it states that the consumersurplus would shrink if taxi drivers expect higher profits and exhibit a higher servicerefusal rate.
3.5 Summary of the Static Effects of Relative Variables
The above analysis examines the static effects on the demand, waiting time andconsumer surplus regarding the fare, the fleet size, and the expected profit per ride,
The Impact of Service Refusal to the Supply–demand Equilibrium 235
where the service refusal phenomena has been explicitly considered by integrating asigmoid refusal function. The results are summarized in Table 2.
In summary, the impacts exhibit the following unique features in comparison withthe market without service refusal.
The most significant characteristic is that both the demand and the consumersurplus do not monotonously decrease function with respect to the fare. Theinteraction between the opposite impacts brought by the increased fare and theshortened waiting time determines whether or not the demand and consumer surpluswould increase. Counter-intuitively, this result implies that raising the fare to adecent level could actually drive the demand when the benefit of the shorter waitingtime due to the decreased service refusal rate outweighs the negative impact fromthe markup. This is different from the market without service refusal, where thedemand decreases and the maximum demand is always achieved when the fareequals zero Table 3.
The waiting time experiences a similar decreasing trend as the fare rises in bothmarkets with and without service refusal. The difference lies in the fact that the servicerefusal has an additional effect in terms of the increased taxis available due to thedecreased service refusal rate. Hence, the waiting time would decrease more rapidlycompared to the market without service refusal as the fare rises.
With respect to the fleet size, the service refusal would weaken its impact on themarket in terms of the demand, the waiting time and the consumer surplus by acoefficient of (1− r) since only a proportion of the changes in the fleet size wouldactually be in service and have an effect on the market.
Lastly, if taxi drivers expect higher profits per ride, the service refusal rate wouldrise. That would result in severer service refusal behavior in the taxicab market, andlead to a contraction in both demand and the consumer surplus. The waiting time wouldalso increase since fewer taxis are in service.
Table 2 Comparative static effects in markets with and without service refusal
With service refusal Without service refusal
With respect to fare per ride P
Demandf p− f wwNv Nδ
1þ f 2w0 T
þ=−ð Þ f p1þ f 2w
0 T−ð Þ
Waiting time −wNv
Nδþ f PT1þ f WwNv T
−ð Þ wNv
f pT1þ f wwNv T
−ð ÞConsumer surplus α
f p− f wwNv Nδ
1þ f ww0 T
þ=−ð Þ αf p
1þ f ww0 T
−ð ÞWith respect to fleet size N
Demand wNv
Nδ− f pT1þ f wwNv T
þð Þ wNv
− f pT1þ f wwNv T
þð ÞWaiting time wNv
1−r1þ f wwNv T
−ð Þ wNv1
1þ f wwNv T−ð Þ
Consumer surplus αwNv
Nδ− f pT1þ f wwNv T
þð Þ αwNv
− f 1T1þ f wwNv T
þð ÞWith respect to mean expected profit s
Demand −γN f wwNv1þ f wwNv T
(−) 0
Waiting time wNv−γN
1þ f wwNv T(+) 0
Consumer surplus −α γN f wwNv1þ f wwNv T
(−) 0
(+), (−) and (+/−) means the derivative is positive, negative or undetermined, respectively
236 D. Wei et al.
4 Optimal Social Welfare Solution
As discussed in the previous section, the service refusal behavior imposes differentimpacts to the taxicab market. This section explores the conditions of optimal socialwelfare with service refusal regarding the two regulating variables: the fare andthe fleet size.
The social welfare comprises both the consumer surplus of Eq. (17) and thetaxicab profits. It’s important to note that the costs associated with the taxioperation normally consist of the gas cost and the license fee. In China, taxidrivers normally have to pay a certain amount of the license fee per month totheir companies. Hence, the profit is the income minus the gas cost and thelicense fee for operating taxis, while the taxi drivers out of service could face aloss of the license fee.
The profit of taxis is formulated as:
I ¼ PQ− 1−rð ÞCN−rC0N ð23Þ
where C0 accounts for the license fee per hour for taxis out of service. C is the cost ofoperating taxis including both the license fee C0 and the gas cost.
With Eq. (23), the social welfare is specified as:
S P; Nð Þ ¼ Sc ωð Þ þ PQ− 1−rð ÞCN−rC0N ð24Þ
where Sc (ω) is defined in Eq. (17).
Table 3 Values of parameters
Parameters Values
Initial flag-drop price ps 6.1 CNY
Charge per kilometer (km) pl 2.3 CNY
Charge per waiting wait pt 55 CNY per hour
Average trip length L 8 km
Average traveling speed in congested network (km/hour) 10 km per hour
Free flow speed uf 40 km per hour
Cost per hour C for taxis in service 40 CNY
Cost per hour Co of taxis out of service 27.5 CNY
Potential demand Q 1, 100, 000
Proportion of congested segment σ 0.6
Value of waiting time κ 60 CNY/hour
Value of trip time τ 35 CNY/hour
Customer’s choice behavior α 0.03
Waiting time parameter A 1,000 vehicle hour
Driver diversity parameter μ 10
The Impact of Service Refusal to the Supply–demand Equilibrium 237
The partial derivative of S regarding P is:
∂S∂P
¼ −QQP
f ωþ Qþ PQP þ δCN−δC0N ð25Þ
The partial derivative of S regarding N is:
∂S∂N
¼ −QQN
f ωþ Qþ PQN− 1−rð ÞC−rC0 ð26Þ
The necessary condition of the maximum social welfare is ∂S/∂P=0 and ∂S/∂N=0.Using Eq. (11) and Eq. (20), Eq. (25) can be written as:
P ¼ −QwNvκT−δNC−C0ð Þ þ QκwNv
Qpð27Þ
Similarly, Eq. (26) can be written as:
P ¼ −QwNvκT þ 1−rð Þ C−C0ð Þ þ QκwNv
QNþ C0
QNð28Þ
Combining Eq. (27) and Eq. (28), we obtain:
C−C0ð Þ þ QκwNv ¼−C0QP
δNQN þ 1−rð ÞQPð29Þ
Taking Eq. (30) into Eq. (28), we have:
P ¼ C−C0ð ÞT þ C0 QPT þ δNð ÞδNQN þ 1−rð ÞQP
ð30Þ
Eq. (11) of Qp and Eq. (12) of QN, Eq. (31) can be further simplified to:
P ¼ C−C0ð ÞT þ C0 1þ f WwNvTð Þ QPT þ δNð Þ1−rð Þ f P
¼ C−C0ð Þ T þ C0
1−rT þ δN
f P
� �
¼ C T þ C0r
1−rT þ δN
1−rð Þ f P
� � ð31Þ
Eq. (31) is the central result of the optimal social welfare condition withservice refusal behavior. The first term of CT is the operation costs for theoperating taxis. Without the service refusal behavior (r= 0 and δ= 0), the farewould equal to CT. That means the fare per ride just covers the operation costsduring the trip and taxis are operated at a loss equal to the cost of vacant taxi-hours (Yang et al. 2005b). This result is exactly the same as that derived by
238 D. Wei et al.
Arnott (1996). Also, if the license fee cost C0 is ignorable, the fare equation issimplified to CT. That implies if there is no loss to the taxi drivers out ofservice, the fare at the optimal social welfare would also equal that in a marketwithout service refusal.
As Eq. (31) shows, with the impact of service refusal, the fare at theoptimal social welfare covers an additional term regarding the cost for idletaxis. Since δ < 0 and fP< 0, this term implies an extra charge, which compen-sates the costs of idle taxis. Hence, the fare per ride at social optimum ishigher due to the service refusal behavior. This brought a possibility that theprofit of the taxi firms may not be negative at the social optimum in a marketwith service refusal. These properties will be verified and discussed in thenumerical study.
5 Maximum Profit Solution
In this section of the research, the focus was to regulate the fleet size and the fare in amonopoly taxi market to obtain the maximum profit. The comparison to the marketwithout service refusal behavior is discussed.
The profit of a taxi firm is given by:
I P;Nð Þ ¼ PQ− 1−rð ÞCN−rC0N ð32Þ
with the costs of both operating taxis and taxis out of service taken intoaccount. To maximize the profit, the first-order condition has to be satis-fied, which is the partial derivatives of I with respect to P and N shouldequal to zero.
The partial derivative of the profit with respect to P is:
∂I∂P
¼ Qþ QPP þ CNδ−CONδ ¼ 0 ð33Þ
and further written as:
P ¼ CNδ−CONδ−QQP
ð34Þ
The partial derivative of the profit with respect to N is:
∂I∂N
¼ QNP−C 1−rð Þ−COr ¼ 0 ð35Þ
which can be written in the similar form to Eq. (34):
P ¼ C 1−rð Þ þ COrQN
ð36Þ
The Impact of Service Refusal to the Supply–demand Equilibrium 239
Combining Eq. (34) and Eq. (36), one can obtain:
C 1−rð Þ þ COrQN
¼ CNδ−CONδ−QQP
ð37Þ
Taking QN from Eq. (11) and QP from Eq. (12), we have:
rCo þ C 1−rð Þ ¼ f wwNv CoNδ−Q 1−rð Þ½ �f p
ð38Þ
Now taking QN into Eq. (36), one obtains:
P ¼ C 1−rð Þ þ COr½ � 1þ f wwNvTf wwNv 1−rð Þ
¼ C 1þ f wwNvTð Þf wwNv
þ COr 1þ f wwNvTð Þf wwNv 1−rð Þ
¼ CT þ COr1−rð Þ T þ 1
f wwNv 1−rð Þ rCO þ C 1−rð Þ½ �
ð39Þ
Applying Eq. (38) to Eq. (39), we have:
P ¼ CT þ COr1−rð Þ T þ CoNδ
1−rð Þ f p−Qf p
ð40Þ
By comparing the fare at the social optimum, the charge per ride at the maximumprofit is apparently higher than the fare at the social optimum by an additional amountof Q/fp, which represents the consumer’s marginal net willingness-to-pay for a ride. Ifignoring the service refusal (r=0, δ=0), the consumer would be charged at a lower fareof CT+ Q/fp, which is the same result derived in Yang et al. (2005b). However, with thepresence of the service refusal, the fare does need to be raised in order to achievemaximum profit.
6 A Numerical Study and Discussion
To demonstrate the analytical analysis obtained so far and further explorenumerical properties of the market equilibrium with service refusal, a casestudy is presented and discussed in this section. The basic parameters, suchas the average trip distance, operation cost, license fee, congested speed andfree flow speed were directly obtained and derived from the Annual Reportof Beijing Transportation (Beijing Transportation Research Center 2011),while the other parameters such as the sensitivity parameter of α, value ofwaiting time and value of travel time refer to travel surveys and studies ofBeijing (Beijing Municipal Commission of Transport 2013; Jia et al. 2007;Liu et al. 2008). The values of the parameters are listed in Table 3.
240 D. Wei et al.
6.1 Impact on Demand
Figure 3 illustrates the demand curves with respect to fare per ride in markets with andwithout service refusal. The solid line clearly demonstrates that the demand decreasesas fare rises and the maximum demand is achieved when the fare equals zero if theservice refusal is ignored.
This result is, however, essentially different from demand curves in markets withservice refusal, as illustrated in Fig. 3. The demand curve exhibits a convex shape as thefare rises. The maximum demand is achieved when the fare is set to a certain thresholdthat makes QP equal zero. If the fare is below this threshold, raising the price does nothurt the demand. Instead, the benefit brought by the shortened waiting time due toincreased operating taxis outweighs the negative impact of the raised price, whichwould boost the demand. This stage corresponds to the ascent phase of the curve.However, when the fare is higher than the threshold, raising the fare would hurt thedemand as the benefit from the increased number of taxis in service is less significantthan the impact of the markup. As the fare rises, the demand curves in markets with andwithout service refusal eventually overlap when the service refusal rate tends to zero.
Comparison of the demand curves between markets with and without service refusalshows that, given the same fare and the same fleet size, the demand is apparently lowerwith service refusal. This result implies that solely regulating the fare, either bydecreasing or increasing, cannot eliminate the negative impact on the demand if thedesired demand is beyond the maximum demand attainable in the market with servicerefusal.
Furthermore, if the taxi drivers expect greater profit per ride, the service refusal raterises, which further shrinks the demand as seen in Fig. 3. Meanwhile, the threshold farethat maximizes the demand also rises, since the peak of the curve shifts to the right. Themaximum demand decreases in the market with severer service refusal. These obser-vations are consistent with and further extend our analytical discussion in Section 3.
Fig. 3 Demand curves with respect to fare per ride in market with and without service refusal
The Impact of Service Refusal to the Supply–demand Equilibrium 241
Figure 4 plots demand curves with respect to the fleet size. Obviously, the demandgrows as the fleet size increases in both markets with and without service refusal.However, the service refusal renders this growing at a slower rate. This coincides withour analysis in Section 3, and the reason is that only a proportion (1− r) of the increasein the fleet size has an effect on the market. As the expected profit and the servicerefusal rate rise, the growing rate would be further decreased. This implies that toachieve the same demand while keeping the fare unchanged, the fleet size has to beincreased to compensate for taxis out of service.
6.2 Impact on Waiting Time
The passenger waiting time both with and without service refusal is illustrated in Figs. 5and 6. Apparently waiting time decreases as the fare rises, which is the same regardlessof service refusal. However, the waiting time decreases at a faster rate in the presence ofservice refusal. Based on the analysis in Section 3, this is because both the shrinkingdemand and the increased vacant taxis contribute to the decrease in the waiting time.Given the same price and the fleet size, service refusal deteriorates the service qualitysince the waiting time is much longer. As the fare rises, the waiting times in the taxicabmarkets with and without service refusal would overlap when the service refusal ratetends to zero.
Furthermore, greater expected profits and higher service refusal rates makethe decreasing curve steepen, implying a longer waiting time. This result can beexplained by the positive sign of Ws. To achieve the same service quality (thesame waiting time) as that in a market absence of service refusal with the samefleet size, the fare has to be raised. This markup has to be higher in the marketwith severer service refusal.
The impact of the fleet size on the waiting time is illustrated in Fig. 6. Similarly, thewaiting time in the market either with or without service refusal exhibits the samedecreasing trend as the fleet size increases. But in the presence of the service refusal, the
Fig. 4 Demand curves with respect to fleet size in market with and without service refusal
242 D. Wei et al.
waiting time is longer. These observations are consistent with the derivation anddiscussion of WP, WN and Ws.
6.3 Impact on Consumer Surplus
Effects of the service refusal on the consumer surplus are demonstrated in Figs. 7 and 8in terms of the fare and the fleet size. Comparison of Fig. 8 with Fig. 4 clearly showsthat curves in both figures exhibit exactly the same trends. This can be easily verifiedfrom Eq. (20).
Hence, the features exhibited in Figs. 7 and 8 can be explained in a similar way asthe characteristics of the demand. The consumer surplus in the presence of servicerefusal shows a convex form as the fare rises. But the consumer surplus curve willeventually overlap with the curve in the market without service refusal when the servicerefusal rate tends to zero. When the fare is relatively lower, raising the fare couldincrease the consumer surplus, because the benefit to consumers from the decrease inwaiting time due to the lowered refusal rate could be greater than the negative effect
Fig. 5 Waiting time with respect to fare in market with and without service refusal
Fig. 6 Waiting time with respect to fleet size in market with and without service refusal
The Impact of Service Refusal to the Supply–demand Equilibrium 243
from the markup itself. As the expected profit rises, the demand curves shift to the rightand the maximum demand achievable also decreases. Increasing the fleet size wouldraise the consumer surplus. But with service refusal phenomena, the impact would beweakened.
6.4 Social Optimum
The social optimums for market with and without service refusal are illustrated inFigs. 9, 10, 11, and 12. First, service refusal would apparently lower the maximumsocial welfare achievable in the market. The maximum social welfare ignoringthe service refusal is 8.7 million CNY, as seen in Fig. 9. However, in thepresence of the service refusal when the expected profit is set to be 30 CNY,the maximum social welfare would be lowered to 4.9 million CNY. A higherexpected profit results in severer service refusal, which further decreases themaximum social welfare.
Fig. 7 Consumer surplus with respect to the fare in market with and without service refusal
Fig. 8 Consumer surplus with respect to the fleet size in market with and without service refusal
244 D. Wei et al.
Secondly, consistent with our analysis in Section 4, the fare should be raised toachieve the social optimum accounting for the service refusal. As seen in Fig. 9, theoptimal fare is 22 CNY for markets without service refusal, which just covers the costfor the ride time.
However, optimal fare needs to be raised to 42 CNY in the presence of servicerefusal when the expected profit is 5 CNY. The fare at the social optimum keepsgrowing as the expected profit rises. When the expected profit is 30 CNY per ride, thefare has to be raised to 62 CNY.
Fig. 9 Social welfare contour in market without service refusal
Fig. 10 Social welfare contour in market with service refusal (s = −5)
The Impact of Service Refusal to the Supply–demand Equilibrium 245
Even so, the maximum social welfare is still much lower compared with the marketin the absence of service refusal.
In contrast to the increase in the fare, the fleet size at the social optimum decreases asthe service refusal becomes severer. The fleet size is controlled at 180,000 to achievethe social optimum if ignoring the service refusal (Fig. 13). But this size has to belimited to 90,000 to account for service refusal behavior when the mean expected profitis set to be 30 CNY per ride.
Moreover, comparison of Fig. 9, 10, 11, and 12 shows that the contour tends to shiftupward as the service refusal becomes severer. This trend implies that to maintain the
Fig. 11 Social welfare contour in market with service refusal (s = 5)
Fig. 12 Social welfare contour in market with service refusal (s = 30)
246 D. Wei et al.
same amount of social welfare as that in the market without service refusal, the fare hasto be raised if keeping the fleet size stable.
6.5 Maximum Profit
The properties of the maximum profit are examined in this section. Figures 14, 15, and16 present the contour of the profit in markets with and without service refusal.Important characteristics of the effects of service refusal are identified.
Consistent with the analysis in Section 5, in a market with service refusal the fareshould be raised to achieve the maximum profit comparing the market with drivers’ fullcompliance.
From Fig. 13, the fare at the maximum profit is 58 CNY per ride, but it rises to 70CNY in presence of service refusal when the mean expected profit is 30 CNY.However, the fleet size of the optimal profit does not show a significant change inthe market either with or without service refusal. The fleet size for the case study wasaround 60,000. Furthermore, the maximum profit achievable decreases as the meanexpected profit rises and service refusal become severer. The maximum profit in themarket without service refusal cannot be achieved by regulating fare and fleet size ifconsidering the service refusal. This can be viewed as an interesting faster-is-slowerphenomena as found in many dynamic systems.
(Helbing and Mazloumian 2013; Parisi and Dorso 2007; Wei and Liu 2013; Weiet al. 2015a; Wei et al. 2015b). This phenomena implies that if drivers expect higherprofits and refuse to serve when the profit cannot match the expected, the maximumprofit they could achieve would actually decrease.
By observing the profit contour, a similar trend to that of the social welfare wasfound. The contour for the same amount of profit shifts upwards and skews toward theright as the mean expected profit rises. This trend implies that in order to achieve thesame amount of profit as in the market free from service refusal, the price has to be
Fig. 13 Profit contour without service refusal
The Impact of Service Refusal to the Supply–demand Equilibrium 247
raised if the fleet size remains unchanged. Similarly, the fleet size has to be enlarged if itis the only regulated variable. The social optimal price and the fleet size are alsomarked in the contour. In terms of the fleet size and the fare, the gap between themaximum profit and the optimal social welfare is shortened as the service refusalbecomes severer.
Lastly, we examined the profitability at the social optimum. Figure 17 plots thecurves of the optimal social welfare, as well as the corresponding profit and loss foroperating taxis and taxis out of service, respectively. Apparently, in the market withoutservice refusal (s→−∞), the profit is negative at the social optimum. However, in themarket in the presence of the service refusal, the profit of the taxi firms at the social
Fig. 15 Profit contour with service refusal (s = 5)
Fig. 14 Profit contour with service refusal (s = − 5)
248 D. Wei et al.
optimum could be positive. The profit of the operating vehicles increases as the servicerefusal becomes severer, and the loss of the taxis out of service also rises. However, thetotal profit increases at the social optimum as the mean expected profit rises. The trendsimplies that as the service refusal in a market becomes severer, the optimal socialwelfare decreases, but the profit at the social welfare actually increases.
6.6 Comments on the Proposed Model
It is important to note that the qualitative results derived in Sections 3,4 and 5are robust to the choices of service refusal functions. The summarized static
Fig. 17 Social welfare and profit with respect to s
Fig. 16 Profit contour with service refusal (s = 30)
The Impact of Service Refusal to the Supply–demand Equilibrium 249
effects shown in Table 2, as well as the optimal solutions in Eqs. (31) and (40),only rely on the signs of the derivatives of the service refusal function withrespect to the fare and the expected profit. As long as the function reflects thequalitative features of service refusal, the same conclusions can be expected,regardless of the specific mathematical structure. To show this, we illustrate asimple piecewise linear function to replace the Sigmoid function, which is inthe form of:
r Pð Þ ¼1
a P−CTð Þ−b0
P−CT ≤1þ ba
1þ ba
CT < P−CT <ba
P−CT ≥ba
8>>>>><>>>>>:
ð41Þ
where a (a <0) and b are parameters. The mean expected profit given thepiecewise linear function is s= (0.5 + b)/a. Substituting b with s, we have:
r Pð Þ ¼1
a P−CT−sð Þ þ 0:50
P−CT ≤0:5
aþ s
0:5
sþ s < P−CT <
0:5
a−s
P−CT ≥0:5
a−s
8>>>>><>>>>>:
ð42Þ
If 0:5s þ s < P−CT < 0:5
a −s, we have δ=a and γ=−a. Because the static effectssummarized in Table 2 only depend on the signs of δ and γ, the sameconclusions can be expected. For example, in Eq. (11), since δ= a< 0, thedecreased service refusal rate would lower the waiting time and consequentlyincrease the demand. Together with the reduction in demand due to theincreased taxi fare, the sign of Qp is still undetermined. Hence, the conclusionsdrawn in Section 3.2 are still valid. Similarly, the results regarding the maxi-mum social welfare and the maximum profit also remain valid. For example, inEq. (31), the sign of the additional term only relies on the sign of δ. Becauseδ= a< 0, the additional term is positive, which implies that the fare at the socialoptimum becomes higher in presence of service refusal behavior.
However, the proposed model assumed that the service refusal behavior is astatic function dependent on the profit per ride. It does not capture drivers’response to the market-wide profitability because the opportunity costs such asthe search time was ignored. This assumption is valid when the search time isrelatively small and the fare per ride dominates the market-wide profitability.This often occurs in peak hours when the potential customer demand is high(this is also the period when service refusal phenomena is the most severe).However, as the fare rises beyond a certain threshold, the demand will decline,resulting in a longer search time. In this case, the profitability may not increaseproportionally to the fare per ride; and the service refusal behavior cannot beproperly reflected in the proposed model.
250 D. Wei et al.
7 Conclusions
Taxicab market is a multi-actor system that involves different levels of decision-makerssuch as travelers, taxi drivers and governing agencies. In a real-world market, it iscommon that taxi drivers do not fully comply with regulating policies especially indeveloping countries.
Service refusal behavior due to the lower-than-expected profit leads to a newequilibrium mechanism of the taxi market, where the supply is not solely determinedby the regulated fleet size but also relies on the service refusal behavior. In this paper,the equilibrium properties were investigated through introducing a sigmoid function todepict the service refusal behavior. The solutions for the social optimum and maximumprofit were examined.
It was found that the demand could be boosted by raising the fare at areasonable level so that the benefit from the decreased service refusal rateoutweighs the markup itself. This fare increase would also benefit the consumersurplus and result in shorter waiting times and higher service quality. Thisresult is contrary to the common understanding that raising fare will alwaysreduce the demand and customer surplus. The ability to regulate the market viatuning the fleet size is weakened, since only a proportion of the changes in thefleet size actually affect the market.
To obtain the social optimum, both the fare and the fleet size need to beincreased. Even so, the maximum social welfare achievable cannot match thatin the market without service refusal. However, the profit at the social optimumcould be positive in the presence of the service refusal. To achieve themaximum profit, the fare has to be raised as well, but the fleet size can becontrolled at a stable size. An interesting faster-is-slower phenomenon wasfound that the maximum profit would be lowered if drivers expect higherprofits per ride and exhibit severer service refusal in the market.
It is noteworthy that the proposed model assumes service refusal behavior onlydepends on the profit per ride. The opportunity cost, i.e., the expected search time, isignored. The future work will be focused on expanding the proposed model toincorporate the market-wide profitability, where the exogenous parameter, i.e., theexpected profit per ride, will be treated endogenously as the outcome of the equilibriumof the taxicab market.
Acknowledgment The authors would like to express their appreciation to Prof. Hai Yang from Hong KongUniversity of Science and Technology for his insightful comments on an earlier version of the manuscript.This research was supported by National Natural Science Foundation of China (Project No.: 51278057).
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