CSIS Discussion Paper No. 135

Economic analysis of tariff integration in public
transport

Takaaki Takahashi
Center for Spatial Information Science, University of Tokyo, 5-1-5, Kashiwa-no-ha, Kashiwa,
Chiba 277-8568, Japan

November 2014

Abstract
The attempts to integrate the tariffs of public transport operated by various institutions are becoming more and more widespread. Their main purpose is a reduction of distortion in consumers’
choices, which arises when a more efficient route costs more. In this paper, we examine this effectof tariff integration based on a simple model with users and operators of public transport. It isshown that under certain conditions, there exists a common fare that makes both of them betteroff given redistribution of incomes among users and that of revenues among operators.
Keywords: common fare; distortion in consumers’ choices; distribution of wage; economy ofscale; heterogenous consumers
JEL Classification Numbers: R4
E-mail address: takaaki-t@csis.u-tokyo.ac.jp
This research is partly supported by the Grants in Aid for Research by the Ministry of Education, Science and Culture in
Japan (No. 25285070).

1 Introduction
In many countries, especially in developed ones, the attempts to integrate public transport are becoming more and more widespread (see the report commissioned by the European Commission,
NEA (2003), among others). One reason is the growing concerns for environmental issues: theintegration is expected to improve the accessibility to public transport and thereby to induce theshift from automobile to it.
It is common to distinguish three fields of integration (see Abrate et al. (2009)): informative
integration, which provides users an easier access to information on the total networks, timetables and fares; physical integration, which improves the infrastructure necessary to use the servicesof different operators or by different modes; and tariff integration, which is a topic of this paper.
The most simple type of tariff integration takes the form that different operators adopt the same
pricing system: several bus companies charge the same fare for a single ride, or several train
companies charge the same fare for a travel of a given distance, for instance. In broader integration, this same pricing system applies even to the multi-leg services involving more than one
operators. That is, the fare for a travel with a given distance is the same whether one uses only
the services of one operator or mixes the services of different operators. What we observe mostoften is this broader type of integration combined with a flat fare. In most of European big cities,
for example, they sell the tickets valid for the unlimited number of rides on subways, trams andbuses, possibly run by different operators, within a predetermined time period, say 1 hour or 1
day.
One of the immediate benefits of tariff integration is the reduction of transaction costs (forreviews, see White (1981), Carbajo (1988) and Gilbert and Jalilian (1991)). In many cases, tariff
integration has been realized with the help of a brand-new fare collection system often accompanied by electronic cards with or without IC chips, which greatly facilitates the usability of public
transport. This improvement results in the increase in its ridership, as has been recorded in asizable body of literature.1
However, another benefit is no less important although it has seldom attracted the interestsof researchers: it is the removal or alleviation of a distortion in users’ choices. Users of public
transport decide their travel routes taking into account various factors such as travel time and afare, most importantly, and the number of transfers and the degree of congestion, less importantly.
If tariff is not integrated, a user may choose a less expensive route even though there are otherroutes that are better in terms of the other factors. If it is integrated, she will pay the same amountof money regardless of the routes to take, and consequently, choose her best route. In this way, thetariff integration, inducing users to commit themselves to take the most efficient routes, can get
rid of a distortion in their choices. It is noteworthy that this distortion is much more serious in big
1 See Taylor and Carter (1998), Lee (1999), Hirsch et al.

(2000), Giuliano et al. (2000), and Ungemah et al. (2006) forthe cases in the United States; Dargay and Pekkarinen (1997) in Finland; FitzRoy and Smith (1988, 1999) in Germany and
Switzerland; Matas (2004) n Spain; Abrate et al. (2009) in Italy; and Sharaby and Shiftan (2012) in Israel.

1cities in Japan than in the counterparts in Europe and the United States. In the latter cities, urban
transit is usually run by a public sector and therefore, the problem of integration mainly concerns
the one between different modes of transport such as subways and commuting trains, subway
and buses, and trams and buses. In big Japanese cities, contrastingly, urban transit services areprovided by a number of private companies. Mass transit in Tokyo metropolitan area is, forinstance, operated by the JR East, part of the former national railway company, nine “big” private
commuting rail companies, the Tokyo metropolitan government, and more. In such cities, it is
not rare that users take less efficient routes to save transport costs and thus, there is a great needfor tariff integration.
The first aim of this paper is to analyze this distortion in a rigorous manner. For that purpose,
we construct a simple model with the consumers who use transport services and the two transport
firms that provide them. The consumers are heterogenous in terms of their income levels. Eachtransport firm possesses its own transport route and offers the consumers a pair of travel timeand a fare for its service. The consumers decide which firm’s service to use comparing the travel
times and fares.
In Tokyo, what is more, even the subway lines are operated by two institutions, the Tokyo
Metro, a private company, and the Toei, the sector of the Tokyo metropolitan government incharge of its transport network. To integrate the tariff schemes of the two operators, policy makers have been insisting on their merger in recent years. Although its principal aim is allegedly
the reduction of the distortion in users’ choices, the merger has far-reaching effects on a local
economy: it will affect the financial conditions of respective operators, alter the competitive environment surrounding the railroad companies in Tokyo, and even change the spatial structureof the city in a long run, to name a few. In order to justify the measure, therefore, one has to
scrutinize its costs and benefits in a broad array of its aspects. Moreover, even if it is justifiable on
economic grounds, the diversity of the effects entails conflicts among individuals and organizations concerned, which can politically hinder the adoption of the measure. For these reasons, themerger does not seem to be a realistic solution to the problem.
Alternatively, this paper discusses a more basic policy, the tariff integration combined withthe redistribution of revenues among mass transit operators. The second aim of this paper is to
study the effects of this policy. More specifically, we show, using the model explained earlier,that under certain conditions, there exists a fare scheme common to the two transport firms thatmakes both the consumers and firms better off with some redistributions. In other words, wedemonstrate the existence of a fare that satisfies the following two requirements: first, it yields a
sufficient amount of total revenue that both of the two operators continue to record a surplus aftersome redistribution of their revenues; and second, consumers’ surplus increases as a result of theintroduction of the common fare.
Most of the studies on tariff integration are the empirical works that estimate the increase in
the ridership of public transport, as has been referred to earlier. Two exceptions are worth men-2

tioning. First, Cassone and Marchese (2005) compare the welfare impacts of tariff integration under monopoly pricing and under the benevolent regulation through Ramsey pricing to unveil the
distortion caused by a monopolistic behavior. Second, Marchese (2006) examines through a nonlinear pricing approach how consumers’ surplus is extracted when tariff is integrated. However,she does not discuss the effects of tariff integration on individual transport firms and consumers.
The rest of the paper consists of six sections. In the next section, we briefly come back to theexample of Tokyo’s subway system to casually illustrate how severe the problem of the distortion
can be. Section 3 presents a basic model. In Section 4, we examine the effects of tariff integrationon the welfare of consumers. A case with of a specific functional form is analyzed in Section 5. In
Section 6, the effects on the total welfare, namely, the welfare of consumers and transport firms,
are discussed. The last section concludes.

2

An introductory case study of the Tokyo subway system

In this section, we view the subway system in Tokyo in a little more detail to get a rough imageof the severity of the problem. The purpose is not to present a rigorous empirical analysis but toprovide a casual observation to enhance readers’ motivation.
As has been mentioned, the Tokyo’s subway system is operated by two institutions, the Tokyo
Metro and the Toei. Among its 13 subway lines, the Tokyo Metro operates 9, which total 195.1
km in length and carry 6.44 million passengers daily in 2012. The Toei operates 4 lines with total
length and daily passengers being 109 km and 2.46 million, respectively, in 2013. The tariff is not
integrated. For one thing, the fare schemes are different. Both operators set fares depending on
travel distances and the Tokyo Metro charges a lower fare than the Toei for any given distance.
Furthermore, if a user wants to take a route consisting of two or more than two legs, some ofwhich are operated by the Tokyo Metro and the rest are operated by Toei, she basically needs
to pay the sum of the fares payable to respective operators.2 Therefore, such a route becomes
relatively more expensive compared to the routes consisting of the legs all of which are run by thesame operator. On these two accounts, consumers are often enticed to use a less expensive but
less efficient route.
Now, let us take a look at travel times and fares between a pair of stations. Because there
are too many subway stations in Tokyo (183 stations) to examine travel times and fares for all
possible routes, we focus on a subset of the 29 stations that are served by both the Tokyo Metroand the Toei.3 One reason for picking such stations is that there are at least two routes between
these stations. For them, 812 (29 multiplied by 28) origin-destination (OD) pairs can be identified.
2 In fact, some transfer discount is applicable provided that certain conditions are met.
3 Even if a pair of stations have different names, we regard them as one station if the two operators do so in offeringinter-operator transfer discounts. Furthermore, 2 stations, Megro and Shirokane-Dai, are, although served by both operators, not included in the 29 stations, because they run trains on the same tracks between Megro station and ShirokaneTakanawa station through Shirokane-Dai station.

3

Assuming that a travel starts at the noon on the September 1st in 2014, we look up the travel
times and fares required for the corresponding 812 travel patterns and find the fastest route andthe least expensive route for each pattern.4,5The first four columns of Table 1 summarize the basic statistics of the travel times and faresof such routes for the 812 OD pairs. They indicate that the fastest route is faster than the least
expensive route by 4.42 minutes on average, which amounts to 25.9% of the average travel timeof the fastest route. The least expensive route is, furthermore, less expensive than the fastest routeby 25.15 yen on average, which equals 13.4% of the average fare of the least expensive route. In
an extreme case in which users always choose the least expensive routes, therefore, the travel
time will decline by 25.9% (4.42 minutes) on average if tariff is integrated. However, the least
expensive routes coincide the fastest routes for 353 OD pairs. For the remaining 459 OD pairs,
which we call ”irregular” OD pairs, the least expensive routes are not the fastest ones.6 When
only the irregular OD pairs concern, the time saving effect becomes much greater. The last fourcolumns of the table show figures for the irregular OD pairs. The fastest route is now faster thanthe least expensive route by 7.81 minutes (43.22% of the average travel time of the fastest route)while the least expensive route is now less expensive than the fastest route by 44.25 yen (22.93%of the average fare of the least expensive route). If users always choose the least expensive routes,
therefore, tariff integration will reduce the travel time by 43.22% for the irregular OD pairs.

4 There are a few travel patterns for which either the fastest route or the least expensive route, or both of them include

the legs served by the other train companies like the JR East. For the sake of simplicity, we disregard such routes andconcentrate on the routes along which every leg is served by either of the two subway operators.
5 Travel time includes the one to wait for trains’ arrivals and, if any, the one to change trains.
6 We say that these pairs are “irregular” by the following reason. In the Tokyo subway system, like many others
elsewhere, there is no faster train service with an additional fee (with only one exception). Furthermore, the speeds oftrains are not much different among lines. If the fare schemes were the same for the two operators, therefore, a travelwith a longer distance would need more money and more time. Consequently, in this “regular” situation, fare would
monotonically increase with travel time.

4

5

17.08

17

7.83

averagemedianstandard deviation

42.17

200

212.76

320

170

fare (yen)

812

9.58

21

21.50

54

1

time (min.)

19.94

180

187.61

240

170

fare (yen)the least expensive route

7.87

18

18.07

43

2

39.49

220

237.21

320

180

fare (yen)

459

8.60

25

25.88

54

4

time (min.)

21.16

200

192.96240

170

fare (yen)the least expensive route

only the irregular OD pairs
the fastest route
time (min.)a All the figures are manually derived from an internet site, Jorudan, by the author (http://www.jorudan.co.jp/).

number of samples

43

1

maximum

minimumtime (min.)the fastest route

all the OD pairs

Table 1: Travel times and fares for the selected OD pairs in the Tokyo subway systema

Table 2: Number of the irregular OD pairs in the Tokyo subway systema
difference in fare (yen)

sum

10

20

30

40

50

60

70

80

90

100

110

120

1-5

31

56

11

0

26

0

11

12

0

5

37

2

191

difference

6-10

34

38

11

0

5

4

11

23

0

2

16

0

144

in

11-15

10

26

13

1

8

0

5

11

0

0

2

0

78

time

16-20

13

14

3

0

0

0

0

6

0

0

1

0

37

(min.)

21-27

4

6

0

0

0

0

0

1

0

0

0

0

11

92

140

38

1

39

4

27

53

0

7

56

2

459

sum
a All

the figures are computed from the data manually derived from an internet site, Jorudan, by the author(http://www.jorudan.co.jp/).

To investigate these points in more detail, we, focusing on the irregular OD pairs, compute thedifference in travel time and fare between the fastest route and the least expensive route for eachpair. The result is summarized in Table 2. It says, for instance, that for 31 OD pairs, the fastest
route is faster than the least expensive route by 1 to 5 minutes but more expensive by 10 yen. We
can see that the difference in travel time is large for a considerable number of pairs. Indeed, 270
pairs exhibit the time difference of more than 5 minutes, 126 pairs more than 10 minutes, and 48
pairs more than 15 minutes. Moreover, the difference in fare is also quite large for not small partof such pairs. Among the 270 pairs with more-than-5-minute time difference, for example, 95
exhibit the fare difference of 50 yen or larger. True, many consumers will probably use the fastest
route if it is not much more expensive; but otherwise, they will choose a less expensive route. For
the 95 pairs, thus, it is plausible that consumers do not take the fastest route to save 50 yen or
larger, although it reduces travel time by more than 5 minutes.

3 Basic framework
3.1

Geography and transport services

Consider a city with four regions, W (West), CW (Central West), CE (Central East) and E (East),
each of which is considered a point. Regions W and E are suburbs and regions CW and CE arecity centers. In the city, there are consumers whose places of residence and workplaces are given.
The workplaces are confined to either of the two central regions, region CW or region CE. Unlessconsumers work at the region where they live, they commute. The commutes are inelastic: eachconsumer uses a constant amount, say 1 unit, of a transport service irrespective of its price (i.e.,fare).
6

Three patterns of commutes are identified. The first is an “inner-city” commute between regions CW and CE. It occurs when consumers who live in one of the two central regions work at
the other central region. Taking a unit appropriately, we normalize the mass of those inner-citycommuters at 1. The second pattern is “suburban” commutes between regions W and CW andbetween regions E and CE: consumers who live in a suburban region commute to its nearby central region. The last pattern is “city-wide” commutes between regions W and CE and between
regions E and CW. Consumers commute from a suburban region to the central region locatedfarther away from their residence than the other central region. The masses of the suburban commuters and city-wide commuters are fixed.
Two firms, say, railway companies, provide transport services for consumers’ commutes. In
parallel with commutes and commuters, the services are classified into three groups according
to the origins and destinations of carriages; the services to carry consumers between the twocentral regions, those between a suburban region and the central region closer to it, and thosebetween a suburban region and the central region farther away from it. These groups of services
are referred to as inner-city services, suburban services and city-wide services, respectively. Twoassumptions are imposed. First, city-wide commuters do not jointly consume an inner-city service
and a suburban service.7 It is because the city-wide services are, probably with the operationof direct trains, speedy and convenient enough to attract all of them. Second, without loss ofgenerality, we assume that firm 1’s inner-city service takes longer time than firm 2’s. Let ti be thetravel time involved when an inner-city commuter uses firm i’s service (i ∈ {1, 2}). Then, ourassumption is that t1 > t2 .
Firm 1 owns a transport route from region W to region CE via region CW while firm 2 does aroute from region E to region CW via region CE (see Figure 1). Thus, firm 1 supplies the innercity transport service between region CW and CE, the suburban service between regions W and
CW, and the city-wide service between regions W and CE, whereas firm 2 supplies the inner-citytransport service between region E and CE, the suburban service between regions E and W, andthe city-wide service between regions E and CW.
Figure 1: Transport routes
By construction, inner-city commuters have an access to both firms’ inner-city services. They
determine which firm’s service to use comparing the travel times and fares of the services of thetwo firms. We denote the mass of the inner-city commuters who use firm i’s service by Di with
D1 + D2 = 1 (i ∈ {1, 2}). In contrast, suburban commuters and city-wide commuters use only
the service of either firm. First, suburban services are provided by only one firm. Therefore,
the suburban commuters who commute between regions W and CW (regions E and CE, resp.)
7 In addition, there is a theoretical possibility that an inner-city commuter or a suburban commuter buys a city-wide

transport service and uses only part of it. To avoid this possibility, it is assumed that the fares of city-wide services are
sufficiently higher than those of inner-city services and suburban services.

7

can use only firm 1’s (firm 2’s, resp.) service. Second, as has mentioned earlier, it is supposedthat city-wide commuters do not jointly consume an inner-city service and a suburban service.
Consequently, the city-wide commuter who commutes between regions W and CE (regions E
and CW, resp.) are to use firm 1’s (firm 2’s, resp.) service. We denote the masses of the city-widecommuters who use firm 1’s service (or, commute between regions W and CE) by L̄1 and those
who use firm 2’s service (or, commute between regions E and CW) by L̄2 .

3.2

Consumers

The preferences of consumers are identical and described by a utility function, u(z, l ), where z
and l are the amounts of a composite good and leisure. Consumers are heterogenous in that theirwages are different. We assume that the wages are distributed in the interval [w, w] according toa given distribution function G (·), which is continuous. When a consumer uses for her commutea transport service that takes t units of time, her budget constraint is given byz + wl = w(l¯ − t) − p ≡ y.

(1)Here, l¯ is the available length of time and w and p are her wage rate and a fare of the transport
service. The price of the composite good is given and it is taken as a numeraire. Furthermore, yrepresents consumer’s “gross” income, which is a wage income, w(l¯ − t − l ), plus the opportunity
cost of leisure, wl. Consumers maximize u(z, l ) subject to (1). The resulting indirect utility isdenoted by v(w, y). Because the prices of composite good and leisure are fixed, travel time and afare affect the indirect utility only through the gross income. Therefore, consumer surplus can bemeasured by the gross income.

3.3

Transport firms

To concentrate on the effects of tariff integration, we do not discuss the determination of the faresof the three types of transport services. They are considered fixed.
Let us look at the revenue of each firm. To begin with, the fares paid by inner-city commuters
to firm i amount to pi Di (i ∈ {1, 2}), where pi is the fare of an inner-city service charged by firmi (i ∈ {1, 2}). In contrast, the sum of the fares paid by suburban commuters and city-wide commuters becomes a constant, which we denote by R̄i . This is because their demands for transport
services and the fares of a suburban service and a city-wide service are fixed. Thus, we can write
firm i’s revenue as pi Di + R̄i .
Production cost of each transport firm is given by()
Ci ( Di ) = γ Di + L̄i + Fi ,i ∈ {1, 2}.

(2)γ(·) gives the variable cost to supply a transport service between the two central regions as afunction of the mass of its users. Here, we are implicitly assuming that the productions of innercity services and city-wide services are conducted jointly although their consumptions are done
8

separately. That is, the trains connecting the two central regions and those connecting a suburb
and the central region farther away from it are different. However, both use the same tracks
between the two central regions, and therefore, the incurred costs associated with that innercity part depend on the total number of the users that pass it. For firm i, they are Di inner-citycommuters and L̄i city-wide commuters. It is assumed that γ(·) is identical for the two firms andincreasing. Furthermore, Fi is the sum of fixed costs and the variable costs that are exogenous.
The exogenous variable costs are the variable costs of operating trains between a suburban region
and the central region closer to it. Firm 1, for instance, runs trains between regionsW and CW toprovide a suburban service between regions W and CW and a city-wide service between regions
W and CE. The variable costs incurred in this operation depends on the mass of their users, which
is fixed. Therefore, the variable costs are also fixed.
These discussions establish the profit of each firm:
πi = πi ( pi , Di ) ≡ pi Di + R̄i − Ci ( Di ),4i ∈ {1, 2}.

(3)

Effect of a common fare on consumers’ welfare

The two transport firms charge different fares, p01 and p02 , for inner-city transport services initially,
and then a common fare, pc , after the integration of tariff. We compare the welfare levels of consumers before and after the tariff integration. In what follows, superscripts 0 and c (for “commonfare”) refer to the variables before and after the tariff integration, respectively.
Let us begin with the economy before the tariff integration. Since t1 > t2 , all the inner-citycommuters use firm 2’s service if p01 ≥ p02 . This is not an interesting case and therefore, we focus
on the other case with p01 < p02 . Furthermore, the gross income of an inner-inner city commuter
who uses firm i’s service is denoted by y0i ≡ w(l¯ − ti ) − p0i .
Inner-city commuters use the service that yields a higher indirect utility. That is, they use{}{}> v(w, y0 ). If v(w, y0 ) = v(w, y0 ), they are indifferent
10)
the service of firm
if
v
(
w,
y
2
2
1
1
<
2
{ > }
{ > }
0
=
v(w, y02 ) if y01 = y02 ,between the services of the two firms. It is obvious that v(w, y1 )<<since the indirect utility is increasing in consumer’s income. Therefore, the consumers use the
}
{
{
}
{
}
/
> y0 , or equivalently, if w < w
0
0
0
if
y
service of firm 1<> b ≡ ( p2 − p1 ) (t1 − t2 ) , where212b > 0.8 This reflects the fact that a consumer with a lower wage attaches relatively smallerw
importance to the time costs of travel compared to a consumer with a higher wage. We call theb a low wage group and those with w > w
b a high wage
set of the inner-city commuters with w < wgroup.
Since the mass of inner-city commuters is normalized at 1, the size of the low wage group isb ), which is equal to the demand for firm 1’s inner-city service. Similarly, the demand for firm
G (w8 Since the measure of the consumers whose incomes satisfy y0 = y0 is zero, we can safely disregard them.
2
1

9

b ):
2’s inner-city service is equal to 1 − G (wb)
D10 = G (w

and

b ).
D20 = 1 − G (w(4)

Suppose that a common fare is introduced for inner-city services. We concentrate on the common fare in between the fares of the two firms before the tariff integration, that is, pc ∈ [ p01 , p02 ].
It is obvious that inner-city commuters receive higher gross incomes by using firm 2’s transport
service than firm 1’s since w(l¯ − t2 ) − pc > w(l¯ − t1 ) − pc . Therefore, all of them use firm 2’s
service, and consequently, the sizes of demands are given by
D1c = 0 andD2c = 1.

(5)

We denote the resulting levels of their gross income by yc ≡ w(l¯ − t2 ) − pc .
As a result of the introduction of a common fare, a consumer in the low wage group switches
the service to use from firm 1’s to firm 2’s. Thus, her gross income increases byyc − y01 = w(t1 − t2 ) − ( pc − p01 ).
This also represents the increase in her surplus, ∆s L ( pc ) (subscript L refers to the low wage group)
since consumer surplus is measured by the level of a gross income. For given w, ∆s L ( pc ) increases
with t1 − t2 , the reduction in travel time, and decreases with pc − p01 , the rise in the payment fora transport service. The sign of ∆s L ( pc ) depends on w: a consumer in the low wage group may{ > }or may not benefit from the introduction of a common fare. More specifically, ∆s L ( pc ) = 0
<
{ > }
/
c
0
=
e ≡ ( p − p1 ) (t1 − t2 ) , where w
e ∈ [0, w
b ]. Therefore, no consumer in the low wageif w
w<
group becomes worse off by the tariff integration ifew ≥ w,(6)that is, if income inequality is sufficiently small. This condition for weak Pareto-improvement ismore likely to be satisfied when t1 − t2 is larger and pc − p01 is smaller. This is because larger t1 − t2and smaller pc − p01 are associated with larger ∆s L ( pc ), as we have seen. In contrast, a consumer inthe high wage group uses firm 2’s service not only after but also before the tariff integration; andtherefore, the increase in his surplus is equal to ∆s H ( pc ) ≡ yc − y02 = p02 − pc (subscript H denotes
a consumer in the high wage group).9 As long as pc ∈ [ p01 , p02 ], ∆s H ( pc ) becomes nonnegative: noconsumer in the high wage group becomes worse off by the introduction of a common fare.
Note that the surplus of a suburban commuter and that of a city-wide commuter do notchange. Therefore, the total effect of the introduction of a common fare on the consumer surplus, denoted by ∆S( pc ), is equal to the sum of ∆s L ( pc )’s and ∆s H ( pc )’s for all the inner-city
9 The increase in a gross income is also equal to the equivalent variation (EV) and compensating variation (CV), which()()
become identical in this simple setting. For the low wage group, they are given by e w, v(w, yc ) − e w, v(w, y01 ) =()()∆s L ( pc ); and for the high wage group, they are given by e w, v(w, yc ) − e w, v(w, y02 ) = ∆s H ( pc ). Here, e(·, ·) denotes
an expenditure function.

10

commuters, that is, ∆S( pc ) =obtain∆S( pc ) = (t1 − t2 )∫ wb

cw ∆s L ( p ) dG ( w ) +

∫ wb
w

∫w

c
b ∆s H ( p ) dG ( w ). It is straightforward tow[]b ) + ( p02 − pc ) 1 − G (wb) .
w dG (w) + ( p01 − pc ) G (w(7)

The first term in the right hand side represents the time cost saving effect of a common fare. The
remaining two terms indicate the monetary cost saving effects for the low wage group and forthe high wage group, respectively. The effect is negative for the former group and positive for thelatter.
As a benchmark, we examine the common fare for which the sum of the two firms’ revenues
does not change, that is, p01 D10 + p02 D20 = pc D1c + pc D2c . (4) and (5) imply that such a fare is given[]b ) . This is the average of the two firms’ fares weighted by theb ) + p02 1 − G (wby pc∗ ≡ p01 G (wrespective masses of the users of their inner-city services.
e the lowest value of w for which no inner-city commuter becomes
For this common fare, w,
worse off, is given bye=w
b [1 − G ( wb )] .
w(8)

To get a further intuition, it is helpful to use Figure 2, which describes function G (w). Using (8),we can rewrite the weak Pareto-improvement condition (6) as
b)
1
G (w
≥ ,
b−w
b
w
w
where the left hand and right hand sides represent the slopes of lines AB and OC, respectively:
the condition is that line AB is steeper than line OC.
Figure 2: The weak Pareto-improvement conditione are ambiguous.10 This is because
The directions of the effects of changes in parameters upon w
e not only directly but also indirectly through a change in the common fare. As t1 − t2they affect w
rises (as travel time is reduced by the tariff integration more), for instance, ∆s L ( pc ) increases,b declines and, as a result, theother things being equal. However, other things are not equal: w
common fare pc∗ rises, which gives a negative impact on ∆s L ( pc ).
∫ wb( )Furthermore, it immediately follows from (7) that ∆S pc∗ = (t1 − t2 ) w w dG (w) > 0: consumers as a whole benefit from the introduction of this common fare.
Proposition 1

Consumers as a whole benefit from the tariff integration when the common fare is determined so
as to keep the total revenue of transport firms unchanged.
10 Indeed, we obtain the following results:eb ) − wGb ′ (wb)dw
1 − G (w( 0
) =0t1 − t2
d p2 − p1

and

The signs of the two derivatives are indeterminate.

11

e
e
dw
dw
).
b ( 0
= −w
d ( t1 − t2 )d p2 − p01

5 A case with a specific distribution of wages
Before proceeding to the analysis of total welfare, we examine a special case with a specific distribution of wages, which will help us intuitively understand some of the above findings.
Suppose that wages are uniformly distributed over [w, w], that is, G (w) = (w − w)/(w − w).
/e = w
b (w − wb ) (w − w), which enables us to
To begin with, it is straightforward to obtain w
rewrite (6) as
e=
w−w

[
]
b − w) wb − (w − w)
(w
w−w

≥ 0.

(9)

b − (w − w) ≥ 0 given that w
b ≥ w. In this case, therefore, (9) is always satisfied. If
If w ≤ 2w, then wb − (w − w): it is satisfied
w > 2w, instead, whether (9) is satisfied or not depends on the sign of w
if and only ifb ≥ w − w.
w

(10)

b 1),
The inverse of the left hand side of (10) is the slope of the line connecting the origin and (w,namely, line OC in Figure 2. The inverse of its right hand side is, moreover, the slope of the
distribution function, which corresponds to the slope of line AB in that figure. Consequently, theweak Pareto-improvement condition is that line AB is steeper than line OC, which is what wehave demonstrated earlier.
Furthermore, the directions of the effects of changes in parameters upon the condition areunambiguously determined in this specific example. In the relevant case with w > 2w, it is more
likely that no inner-city commuter becomes worse off by the introduction of the common fare pc∗
when p02 − p01 is larger, t1 − t2 is smaller, and w − w is smaller. This result immediately follows
from (10).
In Figure 3, the gross income, y0i , of an inner-city commuter who uses firm i’s service before
the tariff integration is described by line li0 (i ∈ {1, 2}). Lines l10 and l20 intersect each other atb ). Thus, the consumers withb Note that the former line lies above the latter for w ∈ [w, w
w = w.
b obtain a higher gross income by using firm 1’s service than firm 2’s. The opposite holds
w<w
b w̄]: the consumers with w > w
b get a higher gross income by using firm 2’s service.
for w ∈ (w,Correspondingly, firms’ revenues before the tariff integration are given by the areas of the two
shaded regions, respectively, i.e., the area of rectangular ABCD for firm 1 and that of rectangular
BEFG for firm 2.
Figure 3: The case of uniform distribution
The gross income of each consumer after the tariff integration is represented by line l c , whichis parallel to line l20 and lies above it. Its increase is measured by the vertical distance betweenlines l c and l10 for the low wage group and that between lines l c and l20 for the high wage group.
The figure describes the case where some consumers in the low wage group (the consumers withe become worse off, that is, the case where the weak Pareto-improvement
the wage lower than w)
12condition (9) is violated. The total increase of the gross income of consumers in each group isequal to the area of triangle RST minus the area of triangle PQR for the low wage group and thearea of parallelogram STUV for the high wage group. Firm 2’s revenue after the tariff integration
is equal to the area of rectangular AEJH while firm 1 receives no revenue.
In general, we cannot determine if the overall increase in gross income, namely, the area ofquadrilateral RTUV minus that of triangle PQR, exceeds the loss of total revenue, namely, the areaof rectangular FGI J subtracted by that of rectangular DCI H. However, Proposition 1 says that it
is indeed the case for pc = pc∗ . In that case, the total revenue remains constant and therefore, theloss is zero. In other words, the area of rectangular FGI J is equal to that of rectangular DCI H.
The proposition guarantees that quadrilateral RTUV is larger than triangle PQR.

6

Effect of a common fare on total welfare

In this section, we examine the change in total welfare. First of all, it is assumed that both firms
earn nonnegative profits before the tariff integration, that is, πi ( p0i , Di0 ) ≥ 0 for i ∈ {1, 2}. Next,
the change in the sum of the profits of the two firms is equal to∆Π( pc ) ≡∑i ∈{1,2}[]()πi ( pc , Dic ) − πi ( p0i , Di0 ) = pc − p01 D10 + p02 D20 − (∆C1 + ∆C2 ) ,(11)

where ∆C1 ≡ γ( L̄1 ) − γ( D10 + L̄1 ) < 0 and ∆C2 ≡ γ(1 + L̄2 ) − γ( D20 + L̄2 ) > 0 are the changes
in the costs for firm 1 and firm 2, respectively (see (2) and (3)). Note that firm 1’s cost decreases
while firm 2’s increases.
In a special case where the marginal cost is constant, the total cost remains unchanged after thetariff integration: ∆C1 + ∆C2 = 0. Therefore, (11) implies that ∆Π( pc∗ ) = 0. We have establishedthe following result.

Proposition 2

When the marginal cost is constant, there is a common fare for which consumers as a whole benefit from the tariff integration and the sum of the profits of the two transport firms is nonnegative
after the integration.

In other words, we can always find a common fare that makes consumers better off and neither of
the two firms worse off with an appropriate redistribution of incomes among consumers and thatof revenues between the firms.
To study a more general case where the marginal cost is not constant, we define ϕ( x ) ≡ γ( x +D10 ) − γ( x ). Then, D10 + D20 = 1 implies that
∆C1 + ∆C2 = ϕ( D20 + L̄2 ) − ϕ( L̄1 ).

13

(12)

First, suppose that function γ(·) is concave, that is, the marginal cost is nonincreasing, in therelevant range of output. In this case, ϕ(·) is nonincreasing. When D20 + L̄2 ≥ L̄1 , therefore,ϕ( D20 + L̄2 ) ≤ ϕ( L̄1 ) and ∆C1 + ∆C2 ≤ 0 by (12). In other words, when firm 2 gets relatively larger
exogenous demand compared to firm 1, the total cost does not increase as a result of the tariff
integration. Instead, if function γ(·) is convex, ϕ(·) is nondecreasing. In this case, ∆C1 + ∆C2 ≤ 0
when D20 + L̄2 ≤ L̄1 : the total cost does not increase when firm 2 gets relatively smaller exogenous
demand. In addition, (11) implies that ∆Π( pc∗ ) ≥ 0 when ∆C1 + ∆C2 ≤ 0. This leads to thefollowing result:

Proposition 3

If the marginal cost is nonincreasing (nondecreasing, resp.) and firm 2 gets relatively larger(smaller, resp.) exogenous demand compared to firm 1, there is a common fare for which consumers as a whole benefit from the tariff integration and the sum of the profits of the two transport
firms is nonnegative after the integration.

This result is not surprising. When tariff is integrated, all the inner-city commuters use firm
2’s service, which is more efficient than firm 1’s in terms of travel time. Therefore, if there isan economy of scale (in the sense of a decreasing marginal cost), the total cost after the tariff
integration is lower when firm 2 gets a larger exogenous demand. To the contrary, if there is adiseconomy of scale, the total cost is lower when that firm gets a smaller exogenous demand.

7 Concluding remarks
One of the most important benefits of tariff integration is the removal or alleviation of the distortion in consumers’ choices, which can arise when a more efficient route costs more. The distortionis particularly serious in Japanese big cities, where urban transit services are provided by a number of different institutions and tariff is not integrated. In this paper, we have examined how
tariff integration reduces this distortion based on a simple but rigorous model. Furthermore, wehave shown that under certain conditions, there exists a common fare that makes both consumers
and transport firms better off given a redistribution of incomes among consumers and that ofrevenues among firms.
To conclude the paper, we briefly discuss the limitations of this paper. First, our results are
based on a model with a quite specific setting. It is assumed, for instance, that each consumer’s
demand for a transport service is inelastic. To give another example, the fares of a suburban
service and a city-wide service are assumed to be independent of the fare of an inner-city service. Second, we have not discussed the determination of fares. In particular, the fares prevailingbefore tariff integration are considered given. In the reality, however, they are determined byfirms themselves or authorities. For example, if the production of a transport service exhibits
14the property of decreasing cost, as widely alleged, the fares are probably regulated at averagecost. Furthermore, the analysis is focused on a special common fare, i.e., the average of respective
firms’ fares weighted by the masses of their users before the integration. The question of what
common fare is socially desirable is left unanswered. Third and finally, we have not conducteda full empirical analysis that estimates the effects of tariff integration, but have presented only
an introductory case study. One could evaluate the size of the distortion in consumers’ choices
attributed to unintegrated tariff, using detailed data on the fares of various routes and the actual
consumers’ trip behaviors, probably with a help of a discrete choice model. It might be one of
the most important agenda, although accomplishing it requires considerable efforts, particularly
when a big city like Tokyo is concerned.

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17suburban commuterscity-wide commutersWfirm 1inner-city commuters

CW

inner-city commutersCEcity-wide commutersEsuburban commuters

Figure 1: Transport routes

C

1

B

O

1
w

｝
｝

G(w)

G(w)
w–w
A
w

w

w

Figure 2: The weak Pareto-improvement conditionw

firm 2

y
V
S

lC
l20

U

l10

B
w
C

E
w

w

I

J

G

F

T
P

O
-p10

-pC
-p20

w

Q
A
D

H

R

~
w

Figure 3: The case of uniform distribution