JOURNAL OF ADVANCED TRANSPORTATION
J. Adv. Transp. 2013
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/atr.1253

Incentive subsidy scheme design with elastic transport demand
Wenqian Zou and Shoshi Mizokami*
Graduate School of Science and Technology, Kumamoto University, Kumamoto, JapanSUMMARY
Huge public transport subsidies caused by deﬁcits have become a heavy ﬁnancial burden on some local
governments due to the decline of bus passenger numbers. It is essential to apply the performance-based
contract to bus services considering maximization of social welfare. This paper constructs an incentive subsidy
contract considering the decision-making powers of the service level and calculating the proper frequency elasticity aiming at two problems of performance-based contracts. Meanwhile, we consider a role of bus operators
ignored by most researchers. Under the scheme, the decision-making power of the service level is discussed
based on ﬁve assumptions, and meanwhile, bus operators are motivated to reduce cost and improve service levelin the scheme. The case of the bus service of Arao city indicates that the optimal frequency equals to zero whenbus operators decide frequency. If bus operators determine efforts, the optimal effort also equals to zero with thegoal of maximizing the proﬁt. Also, bus operators can play their roles in lessening cost and improving service
level to help bus operators and the local government achieve a win-win situation, which maximizes the social
beneﬁt in this subsidy scheme when all factors are decided by the government. Copyright © 2013 John Wiley &Sons, Ltd.
KEY WORDS: bus services; incentive subsidy scheme; Laffont-Tirole’s model; frequency elasticity

1. INTRODUCTION
External costs are incurred in urban areas that experience substantial levels of trafﬁc congestion due
to the increasing number of private cars. The existence of these external costs is reﬂected in governments around the world seeking more sustainable means of meeting passenger transport requirements, including support for public transport because of its capacity to meet social obligations andto reduce the external costs of private car use. Therefore, transit subsidies are necessary, given
substantial economies of scale, in order to permit services to be set at a level that will result inreasonably efﬁcient use of the public transport. Also, it is regarded as important to provide service
for those who are unable to drive, in addition to helping cities ﬁnd some relief from urban congestion
and reducing fuel consumption.
Although transit subsidies are necessary to lower the price of public transport and diminish discrimination, they have long been controversial. Pucher et al. [1] noted that costs and subsidies are jointly
determined, as higher costs elicit more subsidies, which in turn induce higher costs.
As a mechanism with a major objective of containing the cost to government of service provision,competitive tendering (CT) was proposed and applied. The CT contract attracted signiﬁcant interest
because of its attempt to apply responsibilities and incentives to contract operators to reduce cost.
The three kinds of incentive contracts are gross-cost, net-cost, and cost-plus contracts. The gross-cost
contract introduced in London is payment to the operator of a speciﬁed sum of provision of thespeciﬁed service for a speciﬁed period, with all revenue collected being for the account of thegovernment [2]. Under the net-cost contract, instituted to Hong Kong bus companies, revenues
must cover cost and guarantee proﬁt. Worcman [3] concluded that contract payments of*Correspondence to: Shoshi Mizokami, Graduate School of Science and Technology, Kumamoto University,
Kumamoto, Japan. E-mail: smizo@gpo.kumamoto-u.ac.jp

Copyright © 2013 John Wiley & Sons, Ltd.

W. ZOU AND S. MIZOKAMI

Curitiba bus services are calculated from an analysis of each operator’s cost structure, and proﬁt
margins are set at 10% of turnover (cost-plus contract).
The success of the aforementioned CT regimes is measured by cost reduction. The regimes are also
used to force bus companies to improve service level. Gargett and Wallis [4] and Radbone [5]introduced the Adelaide competitive tendering and contracting that was initiated by the governmentof South Australia in 1994. Under this, a ﬁxed sum, which was the basis of the tender price bid anda patronage-related amount, which was calculated according to the change in patronage from the baseyear, comprised the contract payment.
However, there are some criticisms of competitive tendering. Hensher and Stanley [6] pointed out
weakness of competitive tendering. Although competitive tendering has lead to some noticeable gains
in cost efﬁciency, the sizable ﬁnancial gains are once-off, and subsequent retendering has delivered
fewer ﬁnancial beneﬁts. Another reason is that CT is focused on individual contracts with no mechanism to ensure that the incentive payment support sums to the optimal subsidy commitment across a
broader geographic area. Consequently, researchers have looked for alternative ways for the appealof active competition. Performance-based quality contracts align with this view through incentive
payments and benchmarked best practice costs.
Hensher and Stanley [6] consummated the concept of performance-based contracts (PBCs). They
proposed that the PBC framework takes into account the commercial angle of operators and theobligations of government to ensure that subsidy support is spent in a way that maximizes the net
beneﬁts of society. A PBC was proposed that combines payment for delivering a minimum level ofservice and established the optimum subsidy on the basis of the system-wide maximization of social
surplus [7]. The reward system can both meet government community service obligations and realize
an incentive regime that rewards operators for passengers increases (above minimum level of service
patronage levels).
Gonzalez-Dıaz and Montoro-Sanchez [8] concluded some lessons from incentive theory promoting quality in bus transport. Most performance-based contracts regulate the service level as the performance indicator; when bus operators improved the service level, public transport demand may likely be inﬂuenced.
There are two questions regarding performance-based contracts. The ﬁrst is which part will
determine the service level. In most contracts, the service level is usually determined by the bus
operators. The reason is that a public transport operator usually has the best and most detailed
knowledge of the market they serve and also better knowledge of the cost structure of public
transport services to assess the cost of increasing or decreasing frequencies and so forth. However,
the objective of the bus operator is maximizing proﬁt, and political considerations have weighedheavily in the determination of a level barely sufﬁcient to maintain a tolerable service, with little
attempt at a rational determination based on costs and beneﬁts. The second question is that ofelasticity between the service level and the demand. Bus service demand generally rises whenbus service quality achieves a viable and reasonable standard. There are many important aspects
to be considered in increasing bus service demand, including the changes of bus service elements
and characteristics of ridership factors. Suwardo et al. [9] concluded that frequency change is likely
a more preferable recommendation than change in capacity because of the opportunity of raisingload factor in the future. Also, in some of the researches of PBC, frequency was studied as an index
of performance. However, there is less incentive theory research on how to get the elasticity, which
they employed in their papers.
Aiming these two questions, the main purpose of this paper is to propose an optimal performancebased incentive subsidy scheme, in which the decision-making power regarding service level isdiscussed through the proper frequency elasticity value. Under this scheme, the bus operator is motivated not only to attract more passengers through improving service level, but also to reduce deﬁcits toobtain the premium, a point which is ignored in existing performance-based contracts.
The structure of this paper is organized as follows: Section 2 provides an incentive subsidy
scheme based on Laffont-Tirole’s model, and ﬁve solutions aiming at the decision-making powerof the service level are proposed; Section 3 contains results and analysis of the situation of deﬁcit
and social welfare under the ﬁve solutions through investigation of the public transport of Arao inJapan as the research object; ﬁnally, Section 4 presents conclusions and the feasibility of introducing
the incentive scheme.
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

INCENTIVE SUBSIDY SCHEME DESIGN

2. INCENTIVE SUBSIDY SCHEME MODEL
In the present subsidy scheme in Japan, the government supplies enough subsidies to make up for thegradually increasing deﬁcit of bus operators, and there is not any stimulation for bus operators to
reduce deﬁcit. The local government has to pay the same amount of subsidies as the deﬁcit statedby bus operators as shown on the left side of Figure 1.
Zou and Mizokami [10] proposed a new incentive subsidy scheme as represented on the right side of
Figure 1. In the scheme, the local government will give a premium to the bus operator if it successfully
makes efforts to reduce deﬁcit. The result is that bus operators get their premium and the local
government lessens subsidies, a win-win situation. The goal of bus operators is maximizing proﬁt
U = t  ψ(d), whereas the objective of the government is maximizing social welfare, that is, maximizing the sum of passenger surplus beneﬁt S  (1 + λ)( β  d + t) and the excess proﬁt U = t  ψ (d).
Max S  ð1 þ λÞ½ β  d þ ψ ðd Þ  λðt  ψ ðdÞÞt;ds:t:t  ψ ðdÞ≥0(2:1)

Where, S =passenger beneﬁt; λ = shadow cost of public fund; ψ(d) = disutility of effort d; t = netmonetary premium; β = deﬁcit without effort to reduce cost; and d = bus operator’s effort.
Under complete information, the government knows the deﬁcit β and observes the bus operator’seffort d. The maximum social welfare is determined by the reduced cost d* and the excess proﬁt U*.
When the social welfare is maximized, U* = 0.
However, the service level was assumed to be the same as the current situation in Equation (2.1).
Actually, one requirement of the incentive contract is improvement in the service level [11]. In orderto analyze how the change in service level affects passengers and subsidies, this article improves on the
aforementioned model, introducing frequency as the service level variable, and builds an incentive
subsidy model with elastic transport demand.
2.1. Model description
The objective of the model is to maximize social welfare, that is, to maximize of the sum of passenger
surplus beneﬁt and excess proﬁt, as Equation (2.2) shown.
Max

SB ¼ ∑ ðU h þ UBh Þ
h

(2:2)

Where, SB = the social welfare; Uh = the excess proﬁt of line h; and UBh = the surplus beneﬁt ofpassengers of line h.
The excess proﬁt Uh is the difference between the premium th and the disutility ψ(dh, fh), as Equation (2.3)
shown.

Figure 1. Mechanism design of incentive subsidy scheme model.
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

W. ZOU AND S. MIZOKAMI

U h ¼ t h  ψ ðd h ; f h Þ(2:3)

Where, th = the premium that rewards bus operators for improving service level and reducing cost ofline h.ψ(dh, fh) = the disutility of bus operators because of their efforts to improve the service andreduce cost of line h. When bus operators take efforts, disutility of efforts will occur. The ψ(dh, fh)
is the increasing function of dh, which means the more is the disutility of efforts, the higher is the
reduced cost.
Meanwhile, the bus operator is constrained by individual rationality. The excess proﬁt should not be
negative, as Equation (2.4) shown.
U h ¼ t h  ψ ðdh ; f h Þ≥0(2:4)

The subsidies and premium supplied to the bus operator are gathered from both taxes and frompassengers. Therefore, the surplus beneﬁt UBh is the difference between the beneﬁt that the public
transport brings to passengers and the paid. Due to the existence of social cost coefﬁcient, UBh isexpressed as Equation (2.5).
UBh ¼ Sðf h Þ  ð1 þ λÞfC ðd h ; f h Þ þ t h g(2:5)

Where, S( fh) = the surplus of passengers of line h;λ = shadow cost of public fund; C(dh, fh) = thedeﬁcit of line h after efforts taken to reduce cost and improve service.
Eventually, the social beneﬁt SB can be written as Equation (2.6) shown.
Max SB ¼ ∑ SBh ¼ ∑ðU h þ UBh Þhh¼ ∑fSðf h Þ  ð1 þ λÞðC ðdh ; f h Þ þ t h Þ þ U h g
h

(2:6)

¼ ∑fSðf h Þ  ð1 þ λÞðC ðdh ; f h Þ þ ψ ðd h ; f h ÞÞ  λU h gh2.2. Maximizing social beneﬁt SBh
This study solves the social beneﬁt maximization problem based on the Stackelberg model [12]. The
government is the leader whose object is maximizing the social welfare. The bus operator is thefollower whose object is maximizing proﬁt. The government must know ex ante that the bus operator
observes his actions. In the model, the premium th、reduced cost dh and the frequency fh is variable. The
premium th, determined by the government, is supplied by the government to reward good performance ofthe operator. As we said before, there is dispute about the decision-making power of the service level. In
order to solve the problem, we make several assumptions about the decision-making power of dh and fh
shown as Table I. According to assumptions, there are ﬁve possible choices; we will respectively calculate
the maximal social beneﬁt of each case to get the right decision of the service level.
(1) Case 0
Under Case 0, the existing service level continues unchanged, whereas the reduced cost dh and thepremium th are determined by the government. nWhen the social
o beneﬁt is maximal, the excess proﬁt 
dC ðd h Þdψ ðdh Þequals to 0. t h ¼ ψ d h . And d h ﬁts ð1 þ λÞ ddh þ ddh ¼ 0.
(2) Case 1
Under Case 1, th is determined by the government, whereas dh and fh are determined by bus
operators. As we said, ψ(dh, fh) is the decreasing function of dh, but we do not know its relationshipwith fh. Therefore, the model is solved based on two hypotheses.
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

INCENTIVE SUBSIDY SCHEME DESIGN

Table I. Decision-making power of variables and the optimal solution.
Case 1Case 0Governmentt hGovernment
dh

th
dh
fh

∂ψ ðd h ;f h Þ≤0∂f h∂ψ ðd h ;f h Þ≥0∂f h

Governmentt hOperatordh ¼ 0Operatorf hGovernmentt h ¼ 0
Operatordh ¼ 0Operatorf h ¼ 0

Case 3
Case 2∂ψ ðd h ;f h Þ≤0∂f h∂ψ ðd h ;f h Þ≥0∂f hCase 4Governmentt hOperatord h ¼ 0
Governmentf hGovernmentthGovernmentdhOperatorf hGovernmentt h ¼ 0
Governmentdh ¼ 0Operatorf h ¼ 0

Governmentt hGovernmentdhGovernmentf hIf ψ(dh, fh) is the decreasing function of fh, the ﬁrst optimal order condition that makes the excess proﬁt
maximum is d h ¼ 0. When the optimal frequency f h satisﬁes t h ¼ ψ 0; f h , the social beneﬁt is maximal.
If ψ(dh,fh) is the increasing function of fh, the ﬁrst optimal order condition that makes the excess
proﬁt maximum is d h ¼ 0 f h ¼ 0. When t h ¼ ψ ð0; 0Þ ¼ 0, the social beneﬁt is maximal.
(3) Case 2
Under Case 2, th and fh are determined by the government, whereas dh is determined by the bus operator.
When the ﬁrst optimal order is d h ¼ 0, the excess proﬁt of the bus operator is maximal.
The optimal solution of social beneﬁt ist h ¼ ψ 0; f h :Andf hsatisﬁes

dSð f h Þ∂C ð0; f h Þ-ð1 þ λÞ ∂ψð∂f0; f h Þ ¼ 0.
df h  ð1 þ λÞ∂f hh(4) Case 3
Under Case 3, th and dh are determined by the government, whereas fh is determined by the bus
operator. Because the relationship between ψ(dh, fh) and fh is unknown, the model is solved basedon the same hypotheses.
If ψ(dh,fh) is the decreasing function of fh, the ﬁrst order condition that makes the excess proﬁt
maximum is f h ¼ f h ðd h Þ.
And when t h ¼ ψ dh ; f h , the social beneﬁt is maximal. The optimal frequency d h satisﬁesno; f h ðd h ÞÞ
¼ 0.
ð1 þ λÞ dCðdhdd;f hh ðdh ÞÞ þ dψðdhddhIf ψ(dh, fh) is the increasing function of fh, the ﬁrst order condition that makes the excess proﬁt
maximum is dh ¼ 0 f h ¼ 0. When t h ¼ ψ ð0; 0Þ ¼ 0, the social beneﬁt is maximal.
(5) Case 4
Under Case 4, th, dh, and fh are determined  by the government.
The optimal ﬁrst condition is t h ¼ ψ d h ; f h , dh ¼ dðf h Þ . The optimal frequency f h satisﬁesnodSðf h Þ
dC ðd ðf h Þ;f h Þdψ ðdðf h Þ;f h Þ

ð
1
þ
λ
Þ
þ
¼0
df
df
df
h

h

h

From the aforementioned calculation, we can see that when fh is determined by the bus operator under
Case 1 and Case 3, the model is solved based on two situations because of the relationship between
ψ(dh, fh) and fh. When ψ(dh, fh) is the increasing function of fh, the bus service is terminated f h ¼ 0. Thismeans operators will stop the bus service. When ψ(dh, fh) is the decreasing function of fh, the optimal frequency f h can be obtained through the formulation. When dh is determined by bus operators under Case 1
and Case 2, d h ¼ 0. This means the best behavior of the bus operator is to take no efforts to reduce cost.
3. APPLICATION OF THE INCENTIVE SUBSIDY MODEL BASED ON ELASTIC DEMAND
3.1. The bus services situation of Arao city
This paper aims at the bus services in Arao city as the research project. Arao is a city located in
Kumamoto, Japan. As of 2011, the city has an estimated population of 56 144, with the density of
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

W. ZOU AND S. MIZOKAMI

982.40 persons per km2. At present, the public transport of Arao city is operated by Kyushu Sanko Bus
Co., Ltd and Nishi-Nippon Railroad Co.,Ltd (NNR). Figure 2 shows the trend of the passengers andthe subsidy from 2003 to 2009.
Due to the growing popularity of private vehicles, the operation situation has sharply deteriorated.
In order to avoid the lessening of service levels bus operators—and thus to avoid further deﬁcit—thelocal government supplies sufﬁcient subsidies. These subsidies reached nearly ¥160m in 2003, causing a large ﬁnancial burden on the government. In order to relieve the burden, transit privatization
began in 2004. Savage [13] discussed the effects of privatization and reported that operating costs
and subsidization had decreased after privatization. However, he mentioned that demand had
declined as a result of service changes and that services had been concentrated to the most popularroutes. The transit privatization in Arao also faced these problems, and it resulted that the amount ofsubsidies fell from ¥154m to ¥55m and in passenger number by about 35% in the same year. The
object of the bus operator is to maximize proﬁt through controlling costs [14] and cutting bus lines
with large deﬁcits. Bus service was considered by the bus operator as a business and not as a merit
good, and the situation continued to deteriorate in 2005. The government began to increase subsidies
from 2006 in order to attract more passengers. However, the number of passengers continued todecrease. Thus, transit privatization in Arao reduced subsidies at the expense of a decrease indemand. Application of a subsidy mechanism considering passenger demand is thus crucial forbus services in Arao city.
3.2. Parameter calibration
The bus service is operated by two companies in Arao city. Kyushu Sanko Bus Co., Ltd owns 20 bus
lines, whereas NNR has two bus lines. The bus lines of Kyushu Sanko Bus Co., Ltd cover the whole
city, whereas bus lines of NNR are running in the suburbs with very low frequencies. Therefore, thepaper takes bus lines of Kyushu Sanko Bus Co., Ltd as the research object. First, we need to conﬁrm
C( fh,dh),Dij( fh), S( fh)and ψ(dh, fh) that are needed to get the social welfare.
3.2.1. The deﬁcit C(fh,dh) of line h
The deﬁcit of line h is the difference among the cost of line h before effort, the revenue of line h and the
reduced cost due to efforts. The formulation is as Equation (3.1) shown.
C ðd h ; f h Þ ¼ Pð f h Þ  ∑ Dij ð f h ÞchðijÞ  d h 3652f h L h

(3:1)

i; j

Where P( fh ) = the cost of line h before effort; Dij( fh) = estimated demand between stop ij of line hafter effort; ch(ij) = the ticket price between stop ij of line h; dh = the reduced cost per mile of line h; fh =the frequency per day of line h; and Lh = the length of line h.
P(fh) is related to the revenue kilometer of line h. Regression analysis is carried out to calculate P( fh)through using 20 bus lines data. The result is shown in Table II. The revenue kilometer is the functionof fh, and is expressed as 365 × 2 × fh × Lh.

Figure 2. The current situation of bus services in Arao.
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

INCENTIVE SUBSIDY SCHEME DESIGN

Table II. The calibration results.
Variant

Parameter

Constant
Revenue kilometer
Relevance

5.58 × 10
191.4
0.91

t-value
5

0.95
13.6

3.2.2. Passengers between stop i and stop j of line h after effort Dij(fh)If the frequency changes, the demand of passengers must change with it. The new demand is estimatedbased on the current passengers between stop i and stop j as Equation (3.2) shown. The data of DBhðijÞ isfrom the origin–destination survey that was carried out to passengers in 2010.
Dij ð f h Þ ¼ DBhðijÞ þ εDBhðijÞ
f Bh

!



 f h  f Bh

(3:2)

Where, DBhðijÞ = passengers of line h between stop ij; ε = frequency elasticity; f Bh ¼ the frequency
before; and fh =the frequency after changing the service level.
. There has been much research on frequency elasticity. However, there are fewer studies on how to
calculate the elasticity. Most of the elasticity value was obtained from data statistics [15]. This paper
built a model to calculate frequency elasticity while considering the trend of public transport and otherfactors. Effective factors considered by this paper include bus services, population level and number ofprivate cars. Factors for bus services include frequency and the inherent trend of bus services. We also
consider inﬂuences that the adding or cancelling of lines may have on demand. The local government
respectively reorganized the bus network in 2008 and 2009. Some bus lines were added and somecanceled in each reorganization of the network, and these changed bus lines may affect demand in
the subsequent year. The factors of the previous year may inﬂuence the demand in the subsequent year,
as Equation (3.3) shown. The affecting factors are as presented in Equation (3.4).

Dht ¼ A xhðt1Þ Dhðt1Þ

(3:3)

Where, Dht = the number of passengers of line h in t year; A(xh(t  1)) = the index that affecting number of passengers of line h in t  1 year; Nh(t  1) = the number of passengers of line h in t  1 year.

ln A xiðt1Þ ¼ μp ln xpðt1Þ þ μcar ln xcarðt1Þ þμbus þ ηbus δiðt1Þ þ εbus f iðt1Þ ln N iðt1Þ

(3:4)

Where, μp = the inﬂuence coefﬁcient of population; xp(t  1) = the population in t  1 year; μcar = theinﬂuence coefﬁcient of car; xcar(t  1) = the number of private cars in t  1 year; μbus = the inherent
inﬂuence coefﬁcient of bus; ηbus = the inﬂuence coefﬁcient of cancelling or adding lines;δh(t  1) =passengers of line h that were inﬂuenced by cancelling or adding lines in t  1 year; εbus = the elasticity
of frequency; fh(t  1) = the changed frequency of line h in t  1 year.
Utilizing the public transport data, the size of population, and the number of private cars from 2005
to 2010, the SPSS (IBM, Armonk City, NY, USA) was employed to calibrate parameters with 95%
conﬁdence level. Results are shown in Table III.
The variant of population was excluded as a calibration parameter by the software, perhaps because
the population size has less inﬂuence on the demand because of low population growth rate. In 2005,
the population of Arao was 56 420 and the growth rate 0.3%. There is scarcely population change in
Arao. Therefore, the parameter of size of population was excluded by the software.
The parameter of frequency is 0.349. Currie and Wallis [16] concluded that the average elasticity is0.35. Paulley et al. [17] calculated the frequency elasticity of bus demand as approximately 0.4 in the
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

W. ZOU AND S. MIZOKAMI

Table III. The estimation results.
Variant

Parameter

Signiﬁcant values

Constant
Inﬂuence coefﬁcient of car
Inherent coefﬁcient of bus
Coefﬁcient of changed lines
Elasticity of frequency

1.12
0.331
0.064
7.50*10-6
0.349

0.037
0.019
0.020
0.001
0.001

short run. Therefore, the frequency elasticity of Arao is reasonable. The value of elasticity is positive,
and that means the passenger demand is the increase function of the frequency.
3.2.3. The surplus S(fh) of passengers of line h
The generalized cost borne by passengers is deﬁned as the sum of ticket price and the cost of waiting
and trip times. When bus operators increase frequency, passenger generalized cost will decrease fromgB to g due to the reduction of the waiting time, as shown in Figure 3. As such, the demand will
increase from DB to D. The area gBBAg that is derived from a reduction in the generalized cost andthe rise in demand is the passenger surplus measured as Equation (3.5).
ono
1 nSð f h Þ ¼ ∑ DBhðijÞ þ Dij ð f h Þ  gBhðijÞ ghðijÞ
2 ij

o
1 n
6013
¼ ∑ DBhðijÞ þ Dij ð f h Þ  gBhðijÞ  chðijÞ þ ωTimehðijÞ þ ω
2 ij
2*f h

ð3:5Þ

Where, DBhðijÞ ¼ the demand between stop ij of line h before changing the frequency; Dij(fh) = thedemand between stop ij of line h after changing the frequency; gBhðijÞ ¼ the generalized cost between
stop ij of line h before; gh(ij) = the generalized cost between stop ij of line h after; ch(ij) = the ticket pricebetween stop ij of line h; ω = the time value of passengers (ω is set 24.94 (yen/min) based on theJapanese national standard); and Timeh(ij) = the trip time between stop ij of line h.
3.2.4. Disutility of efforts ψ(dh,fh)
When the bus operator takes efforts to increase frequency or reduce cost, these efforts would incur thecost ψ(dh, fh). A reasonable subsidy mechanism applied to public transport operators in Kyushu, Japanis used to calculate the disutility. The mechanism, which acts as an incentive to promote bus operators

Figure 3. Composite demand function.
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J. Adv. Transp. 2013
DOI: 10.1002/atr

INCENTIVE SUBSIDY SCHEME DESIGN

to improve the service level while controlling cost, includes two conditions. One is the cost of the bus
operator should be below the average cost of bus operators in the area; the other is the operating
situation should be improved compared with the previous year by either reducing cost or increasing
revenues. When both conditions are satisﬁed, the bus operator can obtain the subsidies.
A questionnaire survey was carried out to bus operators that had obtained the subsidy to gather data.
The content of the questionnaire is revealed as Table IV. The data was used to calibrate the function of
ψ(dh, fh) deﬁned as Equation (3.6).
ψ ðd h ; f h Þ ¼ expðα0 þ α1 gap þ α2 dh þ α3 3652Lh f h Þ(3:6)

Where, gap= the difference between the average operating cost of all bus operators and the operating cost of one bus operator, that is, a kind of yardstick cost.
Table V presents the calibration results. The parameter of the revenue kilometer is positive inTable V. That means ψ(dh, fh) is the increasing function of fh. When the bus operator decides fh andψ(dh, fh) is the increasing function of fh under Case 1 and Case 3, the optimal frequency f h is zero. Thatmeans the bus operator stops supplying the bus service under this situation, and this is unreasonable.
Therefore, the two situations will not be discussed during the following analysis.

3.3. Result analysis
This paper takes the bus services of Arao city as an example to verify the incentive subsidy model on
the basis of elastic demand. Results are shown as Table VI.

Table IV. The questionnaire survey content.
Characteristics of bus lines

Name of the bus line
Starting/ terminal stop
Length of the bus line
Frequency (workday, day off)
Revenue, cost
Average operating cost in the area (yen/km)
The change of the operating cost compared with that in last year (yen/km)
The change of revenues compared with that in last year (yen/km)
The maintaining lines reasonable subsidies

Operating situation

Table V. The calibrating results.
Variant

Parametert-valueConstant
Difference
Reduced deﬁcit
Revenue kilometer
Relevance

10.82538
0.007861
0.049383
8.71E-06
0.97

73.19899
12.40475
6.718688
15.49588

Table VI. The calculating results
D(f)(thousandd* (million*
f (runs/day) persons/year) yen/year)
Current
Case 0Case 2
Case 4

62.5
62.5
143.9
156.4

382.3
382.3
505.8
520.6

0
26.2
0
70.3

Copyright © 2013 John Wiley & Sons, Ltd.

C (million t* (million
yen/year) yen/year)
72.6
46.4
176.4
147.0

0
13.3
3.2
31.1

C + t*S(f)SB(f)(million (million (million
yen/year) yen/year) yen/year)
72.6
59.7
179.6
178.1

0
0
1299.81484.00.0
13.5
1111.11329.7

J. Adv. Transp. 2013
DOI: 10.1002/atr

W. ZOU AND S. MIZOKAMI

3.3.1. The optimal frequncy f hFigure 4 shows the optimal frequency of the bus network under different cases. The frequency stays
constant and is 62.5 runs/day under Case 0, whereas the optimal frequency under Case 2 and Case 4 is
sharply increased. The number of runs is 143.9 per day under Case 2, 2.3 times that of the current
frequency; the number is 156.4 per day under Case 4, 2.5 times. The frequency under the Case 4 is higher
than that under Case 2.
3.3.2. Reduced deﬁcit dhAt present, the average operating cost per kilometer in Arao city is ¥187. The average operating costis reduced by ¥32/km under Case 0, whereas it is reduced by ¥43/km under Case 4. The situation of
reduced cost of bus lines under Case 0 and Case 4 is presented as Figure 5. The ﬂuctuant range ofreduced cost of each bus line is ¥13–48 under Case 0, whereas the ﬂuctuant range under Case 4 is
¥28–50. The reduced cost is relevant with the revenue kilometer. Taking the Case 0 as an example,
it is seen that the reduced cost is more when the revenue kilometer is higher.
Figure 6 shows the relationship between the frequency and the reduced cost. As stated earlier, thefrequency under Case 4 is higher than that under Case 2. Through Figure 6, it is seen the frequencyof the line under Case 4 is higher than the frequency of same line under Case 2 when the reduced costof the bus line is larger. That means the bus operator would prefer to improve the service of the bus linewith larger cost reduction. In such a case, the bus operator can obtain a higher premium with less cost.
3.3.3. The premium t h
The premium under different cases is presented in Table VI. The premium under Case 0 is ¥13.3m per
year due to the reduced cost of 26.2 million. Under Case 2, the bus operator gets ¥3.2m because of theincrease of frequency; the premium under Case 4 is ¥31.1m, which is 2.3 times the premium under
Case 0. The reason is that the frequency under Case 4 has a sharp augment, reaching 156.4 runs aday, and meanwhile the bus operator takes efforts to reduce costs to ¥70.3m. In conclusion, when

Figure 4. The frequency under cases.

Figure 5. Optimal reduced cost of each bus line.
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

INCENTIVE SUBSIDY SCHEME DESIGN

Figure 6. Relationship between the frequency and the reduced cost.

the bus operator takes more efforts to increase the frequency and reduce cost, it will obtain a higher
premium from the government as a reward.
3.3.4. The deﬁcit Ch and total subsidies
Figure 7 shows a comparison of deﬁcit and subsidies. Total deﬁcit of bus services in Arao city is¥72.6m. When the bus operator takes efforts to reduce deﬁcit to ¥46.4m under Case 0, the total amountof subsidies declines to ¥59.7m, which is ¥12.9m lower compared with the current total subsidies. The
bus operator increases frequency to attract more passengers and get more revenues under Case 2 andCase 4, but it pays more cost. Therefore, deﬁcit and total subsidies have a signiﬁcant increase under thetwo situations. The deﬁcits rise to ¥176.4m, and the total subsidies are ¥179.6m under Case 2. Because
the bus operator not only increases the frequency but also takes efforts to reduce cost under Case 4, thebus operator supplies more service with fewer deﬁcit compared with that under Case 2. The deﬁcit is¥147m, and the total subsidies are ¥178.1m under Case 4.
3.3.5. The passenger surplus S(fh)Under Case 0, the bus operator keeps the frequency constant, and there is no passenger surplus. The
passenger surplus beneﬁt increases to ¥1299.8m under Case 2 due to the augment of the frequency,
whereas the increase of passenger surplus beneﬁt is ¥1484.0m under Case 4. The surplus under Case 4
is larger than Case 2 because of the higher frequency and less subsidies.
3.3.6. The social welfare SB
The social welfare under Case 0 is ¥13.5m because the bus operator takes efforts to reduce deﬁcit, so
the amount of subsidies eventually decreased. Under Case 2, the social welfare reaches ¥1111.1m
because the bus operator takes measures to increase frequency to about 2.3 times that under the current
situation. Improving frequency increases the number of passengers, and subsidies thus increase. It isfound that the effect to passengers caused by increasing frequency is more signiﬁcant compared withthe effect on subsidies; as such, social welfare has a positive relationship with frequency.

Figure 7. Deﬁcit and total subsidies under different cases.
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

W. ZOU AND S. MIZOKAMI

Social welfare under Case 4 is the highest because of the bus operator not only taking efforts to
reduce cost but also taking measures to increase frequency.

3.3.7. fVSD(f) and fVSB
Figure 8 shows the relationship between the frequency and the demand under Case 4. The current
frequency is 62.5 runs/day, whereas the frequency under Case 4 is 156.4 runs/way. This paper takes
10 frequency points between 62.5 runs/day and 200 runs/day to do the sensitivity analysis. It is seenthat the demand increases as the frequency rising. The increasing range gets smaller when thefrequency exceeds the optimal frequency 156.4 runs/day. The reason is that the demand may besaturated as the frequency increases.
Figure 9 shows the relationship between the frequency and the social welfare under Case 4 throughtaking 10 frequency points. The frequency of the ﬁrst point is the current level, and the social welfare iscaused by the reduced subsidies. The social welfare of the other 9 points is caused by the improved
service frequency and the reduced subsidies. There is a positive relationship between the frequency
and the social welfare. However, it is seen that the social welfare of points 9 and 10 increases very slow
when the frequency is higher than the optimal frequency 156.4 runs/day. The reason is that the demandof the two points increases fewer shown as Figure 8 while there is still signiﬁcant growth of subsidies
due to the increasing frequency.
Above all, the relationship between the frequency and the demand and the relationship between thefrequency and the social welfare are positive. However, the increase trend of the demand and the social
welfare of points 9 and 10 is not signiﬁcant when the frequency is higher than the optimal frequency
156.4 runs/day. This means the larger frequency is not better, and it should be considered with thelocal public transport demand.

Figure 8. The relationship between frequency and demand.

Figure 9. The relationship between frequency and social welfare.
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

INCENTIVE SUBSIDY SCHEME DESIGN

4. CONCLUSION
Public transport subsidies caused by deﬁcit have become a huge ﬁnancial burden of the local
governments in Japan. Most governments support the loss-making subsidy scheme. Meanwhile,
passengers are decreasing year by year because of the popularity of private cars and decline of population.
There is no motivation to encourage bus operators to improve the service level to attract more passengers
under the present subsidy scheme. Therefore, many researchers propose reformation of public transport
contracts. During the development process of the contract, competitive tendering contracts and performance-based contracts are respectively put forward. The performance-based contract suggested that thesubsidy mechanism should be related to the service level of bus operators. However, all the mechanisms
ignore the role of bus operators in lessening subsidies. Therefore, it is essential to design a new kind of
subsidy scheme, which will motivate bus operators not only to improve service level to attract morepassengers, but also to take efforts to reduce cost. The reward of efforts is the premium based on the
improved service level and the reduced costs supplied by the government.
Referring Zou and Mizokami [10], this paper proposed an incentive subsidy model based on elastic
demand. Under the scheme, two questions of the PBC are considered. One is the decision-making
power of the service level; the other is how to measure the elasticity of the service level. Referring
Suwardo et al. [9], this paper uses frequency as the index of service level and calculates its elasticity
while considering intrinsic trends of public transport and other factors. Meanwhile, decision-making
power is respectively assumed to yield the maximizing social welfare based on elastic demand. It is
proved that the model can help local governments and bus operators reach a win-win solution by taking public transport in Arao as the research object.
The paper gives ﬁve cases based on different powers of decisions of the service level. Premium andeffort are determined by the government with frequency invariant under Case 0. The premium isdetermined by the government, whereas the effort to reduced cost and the frequency are determinedby bus operators under Case 1. In Case 2, the premium and the frequency are determined by thegovernment, whereas the effort is determined by bus operators. In the Case 3, besides the frequency,
other two factors are determined by the government. All factors are determined by the government
under Case 4. Case 1 and Case 3 are eliminated because of inconsistent with the actual through taking
public transport in Arao as research object to calculate.
The current deﬁcit is ¥72.6m. Under Case 0, the bus operator takes efforts to reduce costs ¥26.2m.
The government not only makes up deﬁcit of ¥46.4m, but also supplies a premium of ¥13.3m. Total
subsidies are ¥59.7m, ¥12.9m less than the current subsidies. The number of passengers does notchange because of frequency invariant. The social welfare increases ¥13.5m.
The optimal condition under Case 2 is d h ¼ 0, meaning the bus operator would make no efforts to
reduce cost. The optimal frequency is 143.9 runs/day, 2.3 times of the current frequency. Due to theincreasing frequency, deﬁcit of the bus operator sharply increases to ¥176.4m. The bus operator also
gets ¥3.2m premium because of the improved service. The total subsidy supplied by the governmentis ¥179.6m, an increase of ¥107m from current deﬁcit. The number of passengers also rises 1.3 times,
and the passenger surplus is ¥1299.8m. The social welfare is ¥1111.1m.
The optimal frequency is 156.4 runs/day, 2.5 times the current frequency under Case 4. The bus
operator takes efforts to reduce cost by ¥70.3m, and the deﬁcit after efforts is ¥147.0m. The degreeof efforts is positively related to the revenue kilometer. The government supplies subsidies to makeup the deﬁcit, and pays a ¥31.1m premium at the same time. Total expenses reach ¥178.1m, a¥105.5m increase from current subsidies. The number of passengers also rises more than 1.5 times,
and the passenger surplus is ¥1484.0m. The social welfare is ¥1329.7m.
It is seen that the passenger surplus and the social welfare are both greatly improved under Cases 2 and 4,
when the frequency is determined by the government. In Case 2, the bus operator determines the effort, andthe optimal choice for it is taking no effort to reduce cost. Therefore, the total subsidy under Case 4 islower than that under Case 2, whereas the frequency is higher under Case 4 compared with that under
Case 2. The reason is that all parameters are determined by the government under Case 4. As statedearlier, the government will make decisions considering maximization of social welfare. Therefore, thesocial welfare is maximizing under Case 4. Under this scheme, the bus operator is encouraged to takeefforts to reduce cost and improve service level. Meanwhile, social welfare can be maximized.
Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
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W. ZOU AND S. MIZOKAMI

Although the subsidy sharply increases in Case 4, the total subsidy is almost the same as the subsidy
in 2003, as Figure 2 shows. However, the number of passengers is less than the number under Case 4,
and the passenger surplus and social beneﬁt are also greatly improved. Figure 8 and Figure 9 present
that the optimal frequency 156.4 runs/day is reasonable, and we consider Case 4 to be feasible for use
in real public transport.
However, there are two questions needed to be discussed. The ﬁrst question is that the total subsidies
are still not decreased a lot under the optimal Case 4, although the number of passengers increases
compared with the current situation. It is withal high to the local government. The reason is that the
aforementioned model is a theoretical model, and the ﬁnal result is the optimum solution without
the constraint of subsidy cap. When the incentive model is applied into the public transport, thesubsidy cap is suggested to be considered by the government. Another question is the whole model
assumes complete information to obtain optimal frequency. That is a very strong assumption especiallyin the presence of a private operator. Zou and Mizokami [10] analyzed the subsidy situation underincomplete information, and the service level is assumed the same as the current in the scheme. It isfound that bus operators take fewer efforts to reduce deﬁcits when the information is more
nontransparent. More research is required on how the incentive scheme under a situation of incomplete
information affects bus service level and subsidy amount.
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Copyright © 2013 John Wiley & Sons, Ltd.

J. Adv. Transp. 2013
DOI: 10.1002/atr

INCENTIVE SUBSIDY SCHEME DESIGNAPPENDIX
These data Uh and UBh are needed to calculate the social welfare SB shown as Equation (2.2). Equation (2.3)
shows ψ(dh, fh) and th are used to get Uh, while S( fh),C( fh,dh) and th are used to get UBh shown as
Equation (2.5).
The parameter notation table explains the parameters which are used to calculate the aforementioned
data.
Parameter notation
Parameter

Description

SB
Uh
UBh
ψ(dh,fh)

The social welfare
The excess proﬁt of line h
The surplus beneﬁt of passengers of line h
The disutility of bus operators due to their efforts to improve
the service and reduce cost of line h
The surplus of passengers of line h
The deﬁcit of line h after efforts taken to reduce cost and improve service
The premium rewarding bus operators for improving service
level and reducing cost of line h
Shadow cost of public fund
The value is assumed 5% in the paper referring Laffont and Tirole [18].
The reduced cost line h
The cost of line h before effort
Estimated demand between stop ij of line h after effort
Passengers of line h between stop ijThe data is got from theorigin–destination survey done in 2010.
The frequency after
The frequency before
The ticket price between stop ij of line h
The waiting time and trip time between stop ij of line h
The time value of passengers (set 24.94 (yen/min) based on
the Japanese national standard)
The generalized cost between stop ij of line h after
The generalized cost between stop ij of line h before it iscomprised by ticket price, waiting time and trip time between stop ij of line h
The length of line hFrequency elasticity
The demand of passengers of line h in t year
The index that affecting demand of passengers of line h in t  1 year
The demand of passengers of line h in t  1 year
The inﬂuence coefﬁcient of population
The population in t  1 year
The inﬂuence coefﬁcient of car
The number of private cars in t  1 year
The inherent inﬂuence coefﬁcient of bus
The inﬂuence coefﬁcient of cancelling or adding lines
Passengers of line h that were inﬂuenced by cancelling or
adding lines in t  1 year
The changed frequency of line h in t  1 year
The difference between the average operating cost of all bus
operators and the operating cost of one bus operator the datais got through survey
The constant of the disutility
The parameter of the gap
The parameter of efforts
The parameter of the revenue kilometer

S(fh)C(dh,fh)
th
λ
dh
P(fh)
Dij(fh)DBhðijÞ
fhf Bh
ch(ij)
Timeh(ij)ωgh(ij)
gBhðijÞLhεDht
A(xh(t  1))
Nh(t  1)μp
xp(t  1)
μcar
xcar(t  1)
μbus
ηbus
δh(t  1)fh(t  1)
gap
α0
α1
α2
α3

Copyright © 2013 John Wiley & Sons, Ltd.

Unit
yen
yen
yen
yen
yen
yen
yen
—
yen/kmyen
personpersonruns/dayruns/dayyen
minyen/minyen
yenkilometer—
person—person—person—vehicle——personruns/dayyen/km————J. Adv. Transp. 2013
DOI: 10.1002/atr