Efficient Fare and Value Capture
A Unified Theoretic Approach
YING, Jiang Qian
Faculty of Regional Studies
Gifu University
Yanagido 1-1 Gifu 501, Japan
Tel: +81-58-2933306 Fax: +81-58-2933118
Email: ying@cc.gifu-u.ac.jp
Abstract
Due to the existence of economies of scale in public transit systems, efficient
marginal cost pricing brings about a deficit to the supplier. Based on an
examination of the composition of the travel demand for public transit, we study
the economic implications of average cost pricing and marginal cost pricing with
various demand compositions. As a result, the bundling of passengers with third
party beneficiaries of public transit is suggested as a potential method for
improving the fiscal performance of transit systems. This serves as a rigorous
economic theoretic foundation for marketing techniques for increasing public
transit ridership.
1. Introduction
There are two main causes of fiscal problems in public transit systems: the
first is that a large part of the benefit of public transit is accrued to third parties
other than the passengers and the transit supplier, e.g., businesses, retailers and
owners of properties in the proximity of public transport facilities, which is
difficult to be sufficiently captured by the supplier; the second is that there exist
economies of scale in public transit systems due to which efficient marginal cost
pricing brings about a deficit to the supplier. Therefore, any public policy that
aims to improve the performance of public transport systems and to alleviate the
fiscal stress in transport investment and operation, should effectively increase the
use of public transit and should have a mechanism to collect the lost benefit back
from the third party beneficiaries. Indeed, these two approaches have been
adopted in practice and have been a theme of extensive research. See
Transportation Research Board (1999 a, b), Taylor et al. (2002), and Nuworsoo
(2004) for the methods for increasing the use and hence revenues of public transit,
and Smith and Gihring (2006) for a review of the literature on benefit capture
issues.
The purpose of this paper is to establish a unified economic theoretic
foundation for these two approaches. In this paper, the composition of the travel
demand for public transport systems is examined. The relationship between
average cost pricing and marginal cost pricing with various demand compositions
will be studied. Based on this, efficient pricing schemes will be examined. It will
be shown that the conventional efficiency-oriented marginal cost pricing based
fare scheme lose benefit simply because transit passengers do not belong to a
single organization. Thus as a complementary of conventional hostile instrument
of value capturing, fare programs aimed at the organizing of passenger groups
may serve as a friendly value capturing instrument.
In Section 2, the theoretical and practical issues regarding economies of scale
and value capture problems in public transport will be reviewed. In Section 3, the
composition of travel demand on public transport is analyzed. Economic
implications of average cost pricing and marginal cost pricing are examined. In
2
Section 4, we discuss how these theoretical results are related to practical
instruments for improving the fiscal performance of public transit.
2. Economies of Scale and Value Capture in Public Transport
In this section we review the formulations of marginal cost pricing and
average cost pricing and their economic implications in a public transit system
with economies of scale. We then briefly review the theory and practice of value
capture, and marketing techniques for promoting transit use.
2.1. Economies of Scale, Average Cost Pricing and Marginal Cost Pricing
In this paper, the cost of a public transit system refers to the cost of building
the transit system and providing transit services. Therefore, average cost denotes
the total supply cost divided by the number of passengers, marginal cost denotes
the extra cost for providing travel service to a marginal passenger.
Due to the fact that the fixed costs invested in public transit are usually very
high, the supply of public transit services exhibits significant economies of scale
(Savage 1997). In this case, average cost pricing in general does not yield the
optimal level of patronage of the system. Let p be transit fare level, q be the transit
demand. Let D(p) denote the demand function and D 1 (q) its inverse. Let MC(q)
and AC(q) denote the marginal cost function and the average cost function,
respectively. As shown in Figure 1, let E be the intersection of marginal cost curve
MC(q) with inverse demand curve D 1 (q) , F the intersection of the average cost
curve AC(q) with D 1 (q) , and G the point on MC corresponding to the demand at
point F. Under the average cost pricing program, the demand is smaller than the
optimum level and a deadweight loss of surplus equal to the area EFG occurs.
On the other hand, if the marginal cost pricing scheme is enforced, then the
patronage will achieve an optimal level (point E), but it will be accompanied by a
fiscal deficit of the transit supplier because the average cost at the demand level of
E is greater than the marginal cost (Hotelling 1938).
2.2. Who Benefits from Marginal Cost Pricing?
3
Although marginal cost pricing yields the most efficient use of the transit
system, it brings about a deficit. This also has negative impact on social equity
because some tax payers who do not use the public transit and do not benefit from
the transit system also have to pay for the deficit through taxation.
According to Vickrey (2001), the total cost for urban public transit investment
and operation differs from the revenue collected based on marginal cost pricing by
an amount which is capitalized into “site-values” in the relevant area. Vickrey
proposed that “the subsidy should be covered by a tax on site values--the value of
urban locations.”
2.3 Conventional Value Capturing Schemes
The above logic justifies the proposal of collecting fees from the beneficiaries
(Vickrey 2001, Batt 2001, Smith and Gihring 2006). In reality, part of such benefit
is actually captured through taxation, including property tax, corporate tax, sales
tax, and employment tax. Excess charges to property owners and companies in the
form of benefit assessment have also been conducted. See Stopher (1993) for the
case of Los Angeles.
However, the effect of hostile benefit assessment has its limitations. This is
because in reality there are unknown and uncertain factors which make the precise
evaluation of the benefit accrued to the properties in the proximity of public
transit facilities (e.g., rail stations, etc.) very difficult. In fact, it has been observed
that public transit facilities do not raise property values in a uniform manner, and
that there are also cases where such facilities even decrease the values of
properties in their proximity (Cervro and Duncan 2002). All these factors prevent
the smooth implementation of benefit assessment schemes (Stopher 1993).
An effective measure for value capturing is joint development or development
by the transit agency in the relevant areas (Hayashi 1989). However, this method
requires that the transit agency has control of the land near transit lines.
2.4. Increasing Revenues by Marketing Methodology
4
In cases where the marginal cost is low, the critical method for reducing the
deficit is to promote transit ridership and increase revenues. Key factors for
promoting transit ridership include service and amenity improvements, marketing,
partnerships and community collaborations (Taylor et al. 2002). In particular, the
combinations of the price differentiation technique with the cooperation with
businesses, universities, residential communities, are well established practical
methods. Typical methods are various discount pass programs. For instance, with
employer-based pass programs, companies purchase transit passes for employees
or subsidize their purchase. It is reported that such programs are very effective in
both promoting transit ridership and in increasing transit fare revenues (Taylor et
al. 2002 and Nuworsoo 2004).
2.5. Relationship between Economies of Scale and Value Capture
As reviewed above, while in theory the need for value capture is credited
primarily to the economies of scale of public transit systems, in practice the
techniques for value capture are not related to economies of scale, and the
techniques for transit ridership promotion are not related to value capture. This
might be accounted for by the fact that the mechanism of how the benefit lost due
to marginal cost pricing is accrued to third parties has not been explicitly
investigated in the literature. In the following section this mechanism will be
explored based on an examination of the compositions of the demand for public
transit.
3. Transport Demand Curve and Pricing Principles
With very few exceptions where people simply enjoy “trips” either by using
public transit or by car, the travel demands of most people are derivatives of other
activities: commutation, shopping, excursion, tourism, and so on. According to the
purpose of their travel, passengers can be grouped by the entities accommodating
their activities at their destination or origin (Figure 2). For example, commutation
can be seen as an input factor into a production sector. In some countries,
commuting travel costs are actually paid by the employers. For instance, about
5
91% of Japanese companies pay the commutation costs for their employees
(Japan Ministry of Health, Labor and Welfare 2005). This means that subsidies to
public transit use are eventually accrued to employers and other third party
beneficiaries. This then raises the question: instead of charging the marginal cost
based fares and managing to capture the lost values by other hostile measures,
why not charge passengers the average costs, if the transit fares are eventually
owed by the potential beneficiary third parties? We shall see that the answer to
this question depends on the composition of the travel demand.
We will develop our theoretic analysis with the following two assumptions.
First, there are no congestion externalities in the public transit. This means that the
marginal cost for providing transit service to a marginal passenger is equal to the
marginal social cost of the whole transit system. Second, externalities of
alternative transport mode (auto transport, in most cases) are efficiently priced.
The primary benefits of a public transit system, including the reduction of road
congestion and the improvement of safety and environment, all arise from the
shift of auto travel to transit travel. Therefore the second assumption means that
the benefits of the public transit can be measured from the demand functions for
the transit system.
Under these assumptions, it will be shown that marginal cost pricing creates a
gap between the benefit of a group of passengers and their total cost, which is
exactly the value lost to the third party beneficiaries and is to be captured by
applying innovative group pricing methods advocated in the this paper.
3.1 Marginal Cost Pricing when Passengers Belong to a Single Group
Suppose that public transit passengers are solely employees of a single firm
and that it pays for the fares for its employees who commute by public transit. The
sum of fares is a function TF(q) in the number q of transit commuters. The
employees may have alternative modes of commuting, which may be costly. For
example, the firm has to provide parking facilities for employees who commute
by car. Suppose that by having q employees commuting by the transit system, the
firm receives a benefit B(q). The demand q is determined by the following
6
condition.
q  arg max q {B(q)  TF (q)} .
(1)
This requires that B(q)  TF (q) . Obviously, the curve depicted by the marginal
benefit function B(q ) is exactly the inverse travel demand curve for the group of
passengers affiliated to the firm, i.e., D 1 (q)  B(q) . Suppose that B(q) is
concave, so that the marginal benefit B(q ) is decreasing.
In this case, if the fare is set equal to the average cost of the transit system, then
TF (q)  (qAC (q))  TC (q)  MC (q) .
(2)
The net benefit function

B(q)  TF (q)  ( D1 (q)  MC(q))dq
(3)
is equivalent to the social surplus (consumer surplus plus supplier surplus) up to
an integral constant. Therefore we have the following proposition:
Proposition I. If the firm pays for the total transit fares for its employees, then the
average cost pricing scheme imposed on individual passenger results in the
optimal state (where the use of the transit system is most efficient) as if
marginal cost pricing is imposed on the market demand of transit ridership.
If the transit agency does not negotiate with the firm, and simply sets the fare
equal to the marginal cost MC(q) based on revealed market demand q, which is
paid directly by the passengers, then the firm saves an amount TC(q)-qMC(q)
which is exactly the benefit accrued to the firm and is exactly the value to be
captured. If the transit agency negotiates with the firm and charge the total cost,
then it will break even and still realize the efficient use of the transit system. This
yields the same result as the conventional method of setting marginal cost pricing
to individual passengers and collect an amount equal to the deficit back from the
firm.
3.2. The Case of Multiple Passenger Groups—Average Cost Pricing
Suppose that there are m firms each with an identical collective transit travel
7
benefit function
Bi ( qi ) , i 1 , m; 
, j 1
m
qj . q
The travel cost shared by firm i is
TCi 
qi
q
TC (q) , i  1,m .
(4)
With the other firms choices fixed, implying that q0   j i q j is given fixed, the
choice of firm i is determined by
dBi TCi
.

dqi
qi
(5)
The following formula can be derived
dBi TCi
q

 AC (q)  i MC(q)  AC (q)  .
dqi
qi
q
(6)
dBi
q
 AC (q)  i MC(q)  AC (q) means that the supply price curve confronted by
dqi
q
group i is p  AC (q) 
qi
MC(q)  AC (q)  , rather than the “average cost” AC(q).
q
This price curve lies in between the AC and the MC curves (Figure 4). When qi q
is small, that is, when the size of each passenger group associated with a firm is
small, average cost pricing yields the same results as conventional average cost
pricing.
Let Di ( p ) be the demand function of group i, that is, the inverse function of
Bi / qi , that is, Di (dBi dqi )  qi . Then the equilibrium demands are given by
dBi
 pi , qi  Di ( pi ), i  1,
dqi
pi  AC (q ) 
, m;
qi
 MC (q)  AC (q)  , i  1,
q
, m;
(7)
 D ( p )  q.
i
i
i
When scale economies exist, MC(q)  AC (q) , a small-size group faces a large
deviation of price from marginal cost, yielding insufficient use of the system. If
there are a lot of passenger groups represented by potential “third party”
8
beneficiaries who are mutually irrelevant, average cost pricing results in a loss of
efficiency. This serves as a partial answer to our earlier question “why not charge
the passengers the average costs, if the transit fares are eventually owed by the
potential third party beneficiaries?”
Suppose that two distinct groups i and j jointly determine their use of transit.
Their aggregated demand will be determined so that their aggregated benefit is
maximized. By Proposition II to be established in Section 3.3, it can be seen that
their common price curve will be
pi  p j  AC (q) 
qi  q j
q
MC(q)  AC(q) .
(8)
This is closer to the marginal cost curve than each of the price curves of individual
groups was. This implies that under the average cost pricing scheme, the grouping
of small groups into large ones will increase transit use, because larger groups will
face with a lower price curve. A deeper exploration on the effect of grouping will
be undertaken in Section 3.4.
3.3 The Case of Multiple Passenger Groups—Optimal State
In the case where there are multiple passenger groups, if there exists some
mechanism of cooperation among them, the optimal state can still be realized
by
average cost pricing. This can be shown as follows:
Let W  i 1 Bi TC (q) be the total benefit of all the groups minus total transport
m
cost. If there is a mechanism for redistributing benefit among the groups, then the
total travel demand is determined by the condition that W  i 1 Bi TC (q) is
m
maximized. Consider the following maximum total net benefit function
W (q)  max

m
B (qi )  TC (q) subject to
i 1 i

m
q
i 1 i
q.
(9)
The Lagrangean for the maximization problem is
L

m
i 1
Bi (qi )  TC (q)   (

m
q
i 1 i
9
 q) .
(10)
The conditions L qi  0 imply that dBi dqi   , i  1,
,m .
By the Envelop Theorem (Takayama 1985, p.137, and Varian 1992, p. 490) we
have
W q  L q   MC (q)   .
(11)
W q  dBi dqi  MC (q) , i  1,, m .
(12)
Therefore we have
The optimal q is determined by the following equations
W q  dBi dqi  MC (q)  0, i=1,…,m,
or dBi dqi  MC (q), i=1,…,m

m
q
i 1 i
and
(13)
q.
Note that dBi dqi is actually the inverse demand function of the individual
passenger group i . Therefore we have the following proposition.
Proposition II. If total cost is shared by the whole passenger groups and the union
of the groups maximizes the sum of their net benefits, then the pricing system
just yields the same result for each passenger group as marginal cost pricing
schemes are applied to individual groups. In other words, the optimal state of
the use of the transit can be realized by applying a pricing scheme satisfying
dBi dqi  MC (q), i=1,…,m.
The optimal state can actually be achieved by the following incremental
pricing mechanism. The incremental pricing scheme charges group i the cost
which could be avoided had it not used the transit system:
Ci  TC (q)  TC (q  qi ) .
(14)
If each group independently decides its amount qi of use of the system, then qi
is determined by
dBi dqi  Ci qi  TC (q  qi ) qi  MC (q) .
(15)
Proposition III. If the passengers are divided into multiple groups, with perfect
cooperation within each group but without any cooperation among the groups,
10
then the optimal state can be achieved by imposing separate incremental cost
pricing schemes on each of the groups.
A special case is that each group consists of one passenger. In this case
incremental cost pricing is exactly the conventional marginal cost pricing. Another
extreme case is the single group case where average cost pricing results in the
most efficient use of the transit system.
However, as expected by the incremental pricing scheme, if scale economies
exist and there are multiple groups, then the transit system runs into deficit. This
can be shown as follows:
As the transit exhibits economies of scale, the average cost is a decreasing
function. We have
AC (q  qi ) 

i 1

i 1
m
m
TC (q  qi )
TC (q)
 AC (q ) 
q  qi
q
TC (q  qi )   i 1
m
(16)
q  qi
TC (q)  (m  1)TC (q)
q
Ci   i 1 (TC (q )  TC (q  qi ))
m
 mTC (q )   i 1TC (q  qi )  TC (q )
m
(17)
3.4 The Effect of Grouping
As was seen above, in the case of multiple passenger groups, while the
incremental cost pricing scheme is optimal but yields a deficit, the average cost
pricing scheme has a loss of efficiency. Most realistic pricing program lies in
between these two schemes. Suppose that the transit fare is set as p0 , and that the
transit agency receives a lump sum subsidy to break even. If the lump sum
subsidy is fixed, then the transit system can be regarded as being operated with a
kind of average cost pricing scheme, where the average cost curve confronted by
passengers lies below the original true average cost curve (see Figure 5).
Suppose that at this stage two distinct groups i and j are organized into one, by
11
certain instrument, then they face with a common price curve
pi  p j  AC (q) 
qi  q j
q
MC(q)  AC(q) ,
which lies below the original price curves
pi  AC (q) 
q
qi
 MC (q)  AC (q)  and p j  AC (q)  j  MC (q)  AC (q)  .
q
q
Therefore, the unified groups can expand their demands qi  q j to improve their
benefits. This also has an effect to decrease the average cost. If the fare imposed
on the rest groups is fixed at p0 , then the transit supplier will get a surplus, which
contributes to the reduction of the deficit of the transit system.
4. Implication for Practice—Passengers Groupings
In Section 3, we have seen that:
(i)
If there is a complete cooperative mechanism among all the
passengers, then the average cost pricing scheme will yield the most
efficient state for the use of the transit system.
(ii)
If the passengers are divided into multiple groups, without any
cooperation among them, then the optimal state can be achieved by
imposing separate incremental cost pricing schemes to each of the
groups.
(iii)
The grouping of smaller passenger groups into larger ones can
simultaneously improve the efficiency of transit use and reduce
deficit.
In conclusion, the design of proper fare structures and the organization of
large groups of passengers contribute both to the efficient use of public transit and
to the reduction of deficit. The key instrument toward this purpose is the
coordination of public transit agencies with third parties which have a direct
relationship with potential passengers, such as businesses, universities, housing
developers, retailers, and so on.
There are two different situations for the organization of passengers that may
require different techniques. The first situation is that the group of potential
12
passengers (for example, employees of a firm, students of a university, or
residents in a housing complex) can be identified. The literature reveals that in
such cases conventional instruments such as discount pass programs are effective.
See, Nuworsoo 2004, Transportation Research Board 1999 (b), and Taylor et al.
2002 for scheme design and case studies of these techniques and their
effectiveness. The second is the situation where the potential passengers are
unspecified, e.g., the street or shopping mall customers. In this case, the
organization of passengers can be realized by applying advanced electronic
technologies. For instance, if a disposable or permanent card is available for the
payment of both transit fares and shop purchases, shopping customers can be
grouped using this payment system. Such transaction techniques are actually
being implemented in some countries (Smart Card Alliance, 2003)—mainly due
to user convenience. Based on our theoretical analysis, a scheme for making full
use of this coordinated payment technique in order to further promote transit
ridership is shown in Figure 6. In this scheme, a retailer buys in bulk from the
transit agency discounted “e-tokens”, i.e., tokens that can be stored electronically
on a transit-shopping payment card. The retailer may present the customer with a
transit token according to the amount purchased, in lieu of a rebate on items
purchased in the shop.
For the transit agency, this scheme has the same effect as employer-based or
university-based pass programs. On the other hand, retailers can provide their
customers with a premium at a lower cost.
Two assumptions were made for the theoretical analysis: there are no
congestion externalities in the public transit system, and externalities of
alternative mode are efficiently priced. For the practical situation in which the
main concern is the increase of revenue, congestion could not be so severe,
otherwise the pricing strategies should be accompanied with the expansion of
existing facilities. If the externalities of the alternative auto transportation mode
are not priced, the total social value of the transit system is greater than the value
enjoyed by the groups of passengers, by an amount equal to the benefit of
reduction of the negative externalities in alternative mode. This amount should not
be charged to the groups of passengers for using the transit system, and should be
13
paid for by local public finance as a subsidy. In this situation, the target of the
grouping strategies advocated in this paper is to recover the total transit system
costs except this subsidy.
5. Concluding Remarks
This paper analyzed the composition of the demand for public transit and
examined the implications of average cost pricing under various demand
compositions. As a result, the grouping of potential passengers is proven to be a
key instrument for promoting transit use and increasing revenue. The primary
contribution of this work is that it provides a rigorous economic theoretic
foundation for innovative transit fare programs that have been proven effective in
practice. Furthermore, it has also shown that based on this theoretical foundation,
novel transit fare schemes can be designed to promote transit use and increase
revenues.
It was assumed in our theoretic analysis that there are no congestion
externalities in the public transit system and that the externalities of alternative
auto transportation mode are efficiently priced. In practical situations without
these assumptions being strictly satisfied, the grouping strategies still partially
apply, as discussed in last section. Nevertheless, the development of a unified
multimodal analytic framework in which these externalities are explicitly
addressed is an important research topic to be studied in future.
Acknowledgements:
I would like to thank my colleague, Professor John Gordon Russell, who
carefully read an earlier draft of this paper and gave detailed suggestions to
improve the paper. I am also grateful to an anonymous referee whose comments
have improved the clarity of this paper.
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Report 50, National Academic Press, Washington D.C.
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16
Figure 1.
Deficit
p
F
AC(q)
E
MC(q)
G
D 1 (q)
q
17
Figure 2.
Passengers
Public
Business, University,
Transit
Developer, Retail, etc.
18
Figure 3.
p
MC(q)
D 1 (q)  B(q)
q
19
Figure 4.
p
MC(q)
TCi qi
AC(q)
))
dBi dqi
q
20
Figure 5.
Original AC(q)
p0
AC(q)
MC(q)
q
21
Figure 6.
Discounted
e-Tokens
Transit
Retailer
Agency
e-Tokens
Custome
r
Custome
r
Custome
r
22
Captions for figures.
Figure 1. Pricing schemes and demand.
Figure 2. Passengers can be grouped by the entities accommodating their
activities.
Figure 3. D 1 (q)  B(q) is the inverse demand curve.
Figure 4. In general, the curve TCi qi lies in between the AC and the MC curves.
Figure 5. With a lump sum subsidy, the transit is operated with the average cost
pricing scheme.
Figure 6. Organizing passengers through retailers.
23