Crowding and the Marginal Cost of Travelling
Under Second-Best Capacity Provision
Daniel Hörcher∗ and Daniel J. Graham
Railway and Transport Strategy Centre
Department of Civil Engineering
Imperial College London
Junior research paper submitted for
the ITEA Annual Conference, 2016
14 February 2016
Abstract
The classic economic theory of capacity optimisation in public transport
suggests that the welfare maximising frequency and vehicle size increase with
demand, and therefore the optimal occupancy rate is independent of demand,
crowding is internalised through capacity adjustment. On the other hand, empirical studies show that the crowding externality does contribute significantly to
the social cost of public transport usage in large metropolitan areas. This paper
presents a theoretical framework that explains why rational second-best capacity
provision may lead to a wide range of demand dependent crowding levels under
economies of vehicle size, infrastructure constraints and demand fluctuations.
We derive the marginal external waiting time, crowding and operational costs
of travelling for second-best scenarios, and explore the resulting subsidy rates.
Thus, we take an important step towards the full understanding of optimal demand and crowding dependent pricing in public transport.
∗
PhD Student and Corresponding author – Electronic address: [email protected]
1
Introduction
Crowding is a special term for passenger congestion in public transport. Recent empirical works summarised by Wardman and Whelan (2011) and Li and Hensher (2011)
show that the user cost of crowding disutilites as a function of passenger density behaves in a similar way as travel time costs in road transport in function of traffic
density. In other words, it is apparent that crowding is also a consumption externality.
As a consequence, the well-known Pigouvian theory of externality pricing should be
applied to internalise the crowding externality. Tranforming decades of research on
road congestion pricing into public transport applications became an appealing topic
for transport economists.
However, there is a huge obstacle hindering the direct application of road congestion
pricing theory in public transport. This is Herbert Mohring’s square root principle
(Mohring, 1972, 1976) which suggests that the optimal capacity of a public transport
service depends on demand. In an extensions of the Mohring model Jara-Dı́az and
Gschwender (2003) derived that crowding does not emerge at all, the optimal occupancy
rate is independent of demand. According to this result the marginal passenger does not
impose any crowding externality on fellow travellers, the operator internalises crowding
through frequency, and there is no reason to introduce any kind of crowding pricing
whatsoever.
The aim of this paper is to establish a theoretical framework which does not contradict with the existing literature of capacity optimisation, but in which the emergence
of crowding can be explained under rational capacity setting regimes. We believe that
crowding in public transport is a result of second-best capacity optimisation decisions
subject to operational constraints. We focus on two such scenarios.
First, in many cases the maximum frequency and vehicle size that the operator
can set is limited by the available infrastructure. One may argue that it is easier to
adjust frequency and vehicle size to demand than the capacity of a road, but it is hardly
convincing that the throughput of rail infrastructure, for example, is more flexible than
highway capacity. Therefore, similarly to most of the road congestion pricing studies,
the upper limit of public transport capacity cannot be varied in the short run. On the
other hand, it is not necessary that the frequency and vehicle size limits are reached
in the same time as demand grows. The analysis presented in Section 4 pays special
attention to the intermediate stages when one of the two variables cannot be increased
any more, but the operator can still react to variations in demand by adjusting the
other.
2
Second, another usual explanation for capacity shortages stems from demand fluctuations: the optimal capacity in peak hours would induce wasteful operations off-peak,
if capacity cannot be adjusted to the actual level of demand, and the optimal off-peak
capacity is insufficient to serve peak demand. Such demand fluctuations may arise in
temporal, spatial as well as directional terms in transport systems. We investigate in
Section 5 how the magnitude of demand imbalances affect the optimal second-best capacity, and derive the marginal external crowding, waiting time and operational costs
of peak and off-peak trips.
In each second-best scenario we investigate the following research questions:
1. What is the second-best optimal frequency, vehicle size, and the resulting occupancy rate?
2. What is the optimal capacity adjustment rate as demand grows?
3. What indirect externaties does the marignal trip induce through optimal capacity
adjustment?
4. How do direct and indirect marginal external user and operation costs relate to
each other? Does the resulting optimal fare justify subsidisation? Is the optimal
fare positive?
This paper links the gap between the classic capacity management literature and
research efforts towards crowding-dependent public transport pricing. We identified a
number potential extensions for the models presented below. These include the joint
optimisation of infrastructure, vehicle size and frequency, and modelling the ability to
partially adjust capacity in fluctuating demand. These ideas are summarised in Section
6, together with the main contributions of this paper.
2
Literature review
The mainstream economic literature of capacity optimisation is based on the gamesetting model of Mohring (1972, 1976), in which he investigated the balance between
the operational and user cost implicatons of service frequency. In essence, waiting
time for the average passenger declines as the service provider increases frequency,
while this capacity expansion will most likely induce incremental operational costs.
Assuming that passengers are not familiar with the timetable and therefore they arrive
randomly to the bus stop, the average waiting time becomes half of the headway
between consecutive services, which equals to half of the reciprocal of frequency (F ).
3
Mohring assumed inelastic demand (Q) and expressed the capacity management
decision as a cost minimisation problem. In the social cost function to be minimised
by the optimal frequency the waiting time component is proportional to QF −1 , while
operational costs grow with neutral scale economies in F . Therefore the marginal
operational cost is constant and the first order condition with respect to frequency
always contains an element proportional to −QF −2 , from which the optimal frequency
becomes a function of the square root of demand – this is often referred to as Mohring’s
square root principle. Due to this fundamental relationship between waiting time costs
and frequency, all extensions of the Mohring model, that kept the assumption of random
arrivals to the bus stop and neutral scale economies in operational costs with respect
to frequency, found that the square root principle holds.
As Small and Verhoef (2007) discuss in details, the Mohring-type capacity adjustment has an important implication for the marginal cost of travelling and the optimal fare. As the marginal consumer induces an incremental reduction in headways,
marginal operational costs are somewhat compensated by a reduction in the waiting
time cost that fellow passengers bear. In Mohring’s simple specification the two effects
have exactly the same magnitude, so the marginal social cost and thus the optimal fare
equals to zero. The positive waiting time externality became an important argument
supporting the subsidisation of public transport.
Mohring’s basic model has been extended in multiple directions by capacity optimisation studies. Jara-Dı́az and Gschwender (2003) provides a comprehensive review
of the evolution of models until 2003. Most of these contributions delt with bus operations and kept the methodolical framework of (1) assuming inelastic demand, (2)
constructing a social cost function, and (3) minimising it with respect to the optimal
frequency and other supply-side variables.
Mohring (1972), for example, presents a more complicated model of bus operations
too, where the number of stops along an isolated bus line is also a decision variable.
Bus stop density affects the time that passengers spend with walking to the nearest
bus stop as well as the probability that more than zero passenger appears at a bus
stop, so that the bus cannot skip the stop.
Jansson (1980) dropped the idea of endogenous stop density in order to make the
frequency optimisation analytically tractable. However, he kept the cycle time of buses
endogenous, assuming that dwell times depend on the number of boarding and alighting passengers per stop. In other words, he used the fact that frequency has an indirect
impact on operational costs and passengers’ in-vehicle travel time costs through cycle
time. Endogenous cycle time remained an important component of capacity models
4
focusing primarily on bus operations, e.g. Jara-Dı́az and Gschwender (2003), Jara-Dı́az
and Gschwender (2009), Tirachini et al. (2010) and Tirachini (2014). Note, however,
that travel times of rail services are generally much less sensitive to the number of
boarding and alighting passengers than buses with a front door boarding policy. Assuming endogenous train size, dwell times are definitely not linear in the number of
boardings, because the number of doors may increase with demand. In this study we
focus on rail services.
The first capacity model that incorporated vehicle size as a decision variable was
again published by Jansson (1980). He considered that the average operational cost
of a bus depends on its size. However, the model’s vehicle size setting objective was
simply to keep the occupancy rate within an exogenous occupancy rate constraint,
without considering the user cost of crowding. Oldfield and Bly (1988) assumed that
the main impact of high vehicle occupancy on users is that it increases the probability
that passengers cannot board the first vehicle, thus lengthening the expected waiting
time. Oldfield and Bly (1988) were also innovative in the sense that they applied a
demand function with elasticity with respect to generalised user costs – although this
improvement did not imply fundamental differences in the resulting optimal frequency.
During the 1990s the rapid evolution of discrete choice modelling techniques provided new insights and quantitative measurement results about the user cost of crowding. The main contributions in this area were reviewed by Wardman and Whelan
(2011) and Li and Hensher (2011). An important methodological novelty was the appearance of crowding multiplier functions, i.e. that the cost of crowding was expressed
in terms of the equivalent incremental travel time cost. Following this logic Jara-Dı́az
and Gschwender (2003) extended Jansson (1980) with a crowding dependent linear
value of time function. They found that the optimal occupancy rate is independent of
demand. We may attribute this intuitively suspicious result to the fact that operational
costs are directly, while the user cost of crowding is inversely proportional to demand.
The constant occupancy rate result of Jara-Dı́az and Gschwender (2003) implies that
crowding is internalised by the welfare maximising operator – a conclusion that might
not hold under economies of vehicle size or second-best operational conditions, as we
discuss later in the paper.
There is another branch in the capacity related literature, hallmarked by Piet
Rietveld and co-authors, that deserves attention. Their models went less into details in specifying operational characteristics affecting cycle time, but introduced scale
economies in operational costs. This difference can be explained by a shift in focus from
bus services to railway operations. Rietveld et al. (2002) examined whether Mohring’s
5
square root principle holds when demand is elastic with respect to the fare, travel time
and waiting time with constant elasticity, and scale economies exist in operational and
environmental costs with respect to vehicle size. Unfortunately, they did not account
for crowding costs in this paper, assuming that the operator intends to achieve 100%
occupancy rate in capacity provision1 .
Both Rietveld et al. (2002) and Rietveld (2002) raised the issue of interrelations
between peak and off-peak capacity. Earlier on, Oldfield and Bly (1988) also analysed
the impact of joint capital and labour costs between peak and off-peak periods, and
expressed the optimal size of bus fleet as a function of demand imbalances. Rietveld
(2002) simplified the assumption on the operator’s supply strategy: the new hypothesis
was that capacity is adjusted just to meet peak demand at maximum occupancy, and
then reduced in low demand periods as much as possible, but never to the actual level
of demand due to the high operational costs of capacity reduction (e.g. decoupling and
storing vehicles, etc.). He concluded that the marginal operational and environmental
cost of an off-peak trip is basically zero, while in the rush hours the marginal burden
is very high, because the incremental capacity remains in operation throughout the
entire day.
Rietveld and Roson (2002) and Rietveld and van Woudenberg (2007) investigated
directional and spatial demand fluctuations and the implied second-best behaviour in
pricing and capacity provision. Rietveld and Roson (2002) is special in the sense that
they modelled monopolistic behaviour with a profit maximising objective – they found
that even with this objective price differenitation between directions can be befitial
from a social welfare perspective. Rietveld and van Woudenberg (2007) compared the
welfare loss that uniformed supply-side variables cause in fluctuating demand and revealed that differentiated service frequency would provide significantly more benefits
for society than dynamic pricing or adjustable vehicle size. Even though these contributions provide relevant policy conclusions for public transport operators, they neglect
the external cost of crowding disutilities, thus leaving a gap in capacity and pricing
research. Chapter 5 investigates a simple framework to analyse the effect of demand
imbalances on the second-best choice of frequency and vehicle size, and the resulting
peak and off-peak occupancy rate.
1
They observed though using data on a sample of train services operated by the Dutch Railways
that this rule may not apply in reality, and urged that ’future theoretical work in this field [...] should
pay explicit attention to various reasons why occupancy rates may be systematically below 100% ’.
6
3
Baseline model: Crowding and scale economies
The basis of our analysis complies with the mainstream literature of capacity optimisation: we investigate the interactions between user-born and operational costs assuming
inelastic demand. The goal here is to develop a social cost function that captures the
main characteristics of railway operations: we attach importance to density economies
in vehicle size and the fact that capacity shortages cause crowding and inconvenience
for passengers. Therefore we merge the operational cost specification of Rietveld et al.
(2002) with the crowding multiplier approach of Jara-Dı́az and Gschwender (2003),
and define the following objective function:
min T C(F, S, Q) = aF −1 Q +
| {z }
F,S
waiting time
βtQ
|{z}
in−veh. time
+ ϕQ(F S)−1 βtQ + vF
t +{zwS δ F }t , (1)
|
{z
} |
crowding
operational cost
where a is half of the value of waiting time with the usual assumption that passengers
arrive randomly to the station, so that the expected waiting time is half of the headway
(F −1 ). Furthermore, β is the value of uncrowded in-vehicle travel time, and the last
component of user costs is a crowding-dependent linear travel time multiplier function
that reflects the invonvenience of crowding. Here we express the occupancy rate of
vehicles with the ratio of demand and capacity, Q(F S)−1 . The vehicle size variable
can be interpreted as the number of seats in a long distance vehicle or the available
useful floor area of an urban transit vehicle. The user cost of a unit of in-vehicle travel
time is linear in the occupancy rate with slope ϕ. In case of railway operations it can
be assumed that the travel time (t) is independent of the operator’s capacity decision,
because door capacity is increased proportionally with vehicle size. Alternatively, one
may assume that ϕ in the multiplier function takes account of both crowding disutilities
and the impact of excess demand on dwell times.
Operational costs are modelled as the product of total vehicle-hours supplied (tF )
and the unit cost of a vehicle-hour, wS δ , where δ is the vehicle size elasticity of operational costs. Finally, we add a purely frequncy dependent component to the objective
function to reflect driver costs, the price of train paths supplied by the infrastructure
manager, or the operational cost of a locomotive, when applicable. Table 1 summarises
the notation used throughout this paper.
∂T C
δ−1
= − ϕβtF −1 S −2 Q2 + wδS
F t = 0,
|
{z
} | {z }
∂S
crowding
operations
7
(2)
∂T C
= − aF −2 Q − ϕβtS −1 F −2 Q2 + |vt +{zwS δ}t = 0,
| {z } |
{z
}
∂F
waiting
(3)
operations
crowding
The optimal capacity variables satisfy the first order conditions provided in equations (2) and (3). That is, any increase in vehicle size reduces the cost of crowding and
induces additional operational costs, while the marginal frequency relieves waiting as
well as crowding costs at the expense of the operator’s budget. Figure 1 depicts the
results of a numerical simulation2 of the optimal first-best capacity values.
Optimal frequency
Optimal occupancy rate
2000
4000
6000
Demand (pass/h)
8000
10000
1.25
1.20
1.10
1.05
100
0
1.15
φ (pass m2)
500
S (m2)
300
8
4
6
F (1/h)
10
12
700
1.30
Optimal vehicle size
0
2000
4000
6000
Demand (pass/h)
8000
10000
0
2000
4000
6000
8000
10000
Demand (pass/h)
Figure 1: First-best frequency, vehicle size and occupancy rate under economies of vehicle
size in operational costs
Both the optimal frequency and vehicle size are less than proportional to demand.
The elasticy of frequency with respect to demand ranges between 0.47 and 0.38 as
demand grows from zero to ten thousand, which is lower than what Mohring’s square
root principle suggests (0.5). As opposed to Jara-Dı́az and Gschwender (2003), the
elasticities of the optimal F and S with respect to demand add up to more than one.
This implies that the optimal occupancy rate depends on demand. As the presence of
increasing returns to vehicle size suggests, high demand allows the operator to reduce
the average cost of capacity provision and ease crowding under first-best conditions.
This is a robust result that applies for any reasonable parameter values as long as
δ < 1.
2
Based on Rietveld et al. (2002) and common intuition we chose the following parameters for the
simulations throughout this paper: a = 15; β = 20, t = 1, ϕ = 0.1, v = 500, w = 10, δ = 0.8.
Description and measurement units are provided in Table 1. Of course, these values may differ
significantly between railways, so the primal goal of this simulation is just to illustrate the mechanics
of the model.
8
We discussed in Section 2 that if the optimal frequency grows with demand, then
the marginal trip has a beneficial impact on the average waiting time, which can be
considered as an indirect positive externality. The fact that under density economies
the optimal vehicle size also falls with demand implies that the marginal trip has a
similar positive impact on fellow passengers’ wellbeing through a reduction in crowding. This provides another justification for the subsidisation of rail services if perfect
capacity adjustment is possible.
Table 1: Notation and simulation value of frequently used variables
4
Symbol
Description
Dimension
Q
F
S
t
a
β
φ
ϕ
v
w
δ
ω
Demand
Service frequency
Vehicle size
In-vehicle travel time
Half of the value of waiting time
Value of uncrowded in-vehicle time
Occupancy rate, φ = Q(F S)−1
Crowding multiplier parameter
Fixed operational cost per train hour
Variable operational cost per hour per m2
Elasticity of operational costs w.r.t. vehicle size
Demand imbalance factor, ω = Q2 /Q1
pass/h
1/h
m2
h
$/h
$/h
pass/m2
(pass/m2 )−1
$/h
$/(m2 · h)
–
–
Value
1
15
20
0.1
500
10
0.8
Infrastructure constraints
The key precondition of Mohring-type capacity models is perfect capacity adjustment.
That is, we have to assume that the operator is able to react to growing demand
by increasing both frequency and vehicle size. Although in an off-peak situation, for
example, this assumption is not threatened, in many cases at least one of these variables
is already set at the highest value that the infrasturcture allows. Frequency may be
constrained by the signalling system and other safety regulations, while the maximum
train size is usally limited by the shortest platform length and the clearance at bridges
or tunnels. On densely used, aging metro systems it is not unusual that both the
frequency and the vehicle size reach their respective maxima during rush hours3 . It
3
The classic example is the deep-level Tube network of London where the tunnel diameter defined
more than a century ago is likely to be sub-optimal nowadays.
9
is reasonable to assume that in these cases capacity is not adjustable any more in the
short run, and therefore the indirect positive externalities, that the marginal trip could
induce through capacity expansion, disappear.
The purpose of this section is to show how capacity constraints affect the interplay
between marginal operational and user-borne costs. We pay particular attention to
intermediate stages where only one of the two capacity constraints become active, so
that the operator is still able to internalise crowding costs through the other. These
are certainly not unrealistic scenarios, especially if the infrastructure constraints are
exogenous to the current operations.
Based on these considerations one can distinguish four states of operations: perfect
capacity adjustment, fixed vehicle size with flexible frequency, constrained frequency
with variable vehicle size, and totally fixed capacity. We derive the marginal cost of
travelling for each of these cases
Unconstrained capacity
In case both capacity variables are adjustable to varying demand condtions, the marginal
cost of a trip is simply the partial derivative of equation (1) with respect to demand.
Marginal social costs can be split into three components:
∂T C(F, S, Q)
= aF −1 + βt + ϕQ(F S)−1 βt +
|
{z
}
∂Q
marginal user cost
−1
Q
∂F
∂(F S)−1
+ϕ
βt + ϕ
βtQ2 +
+ aQ
∂Q
FS
∂Q
| {z } | {z } |
{z
}
i.w.t.ext.
d.cr.ext.
(4)
i.cr.ext.
∂F
∂F
∂S
+v
t+
twS δ + F twδS δ−1
.
∂Q
∂Q
∂Q
|
{z
}
marginal operational cost
Fist, the marginal user will of course have to bear the cost of waiting time, travel
time and in-vehicle crowding. Second, she imposes externalities on fellow passengers:
a direct crowding externality which is proportional to the in-vehicle area that she occupies, and indirect waiting time (i.w.t.ext.) and crowding (i.cr.ext.) effects resulting
from the fact that the operator adjusts the frequency and in-vehicle area according
to the marginal increase in demand4 . We expect that both indirect externality compononets have a negative sign, i.e. capacity adjustment has a positive effect on both
4
For the sake of simplicity we did not indicate that capacity is optimised in this cost function, so
that F = F (Q, S) and S = S(Q, F ).
10
the headway and the average in-vehicle area per passenger. Finally, the marginal passenger induces incremental operational costs too. Capacity adjustment affects in this
case both operational cost elements in equation (1).
Note, that the direct crowding externality equals to the average crowding cost in
this model, which would not obviously hold if standing and seated travelling and the
respective user costs were differentiated. In that case the ratio of personal and external
crowding costs would depend on the probability that the marginal user finds a seat.
Fixed vehicle size, unconstrained frequency
Let us now investigate the case when vehicle size cannot be increased any more, but
the operator is still able to adjust the frequency, and thus partly or fully internalise
the marginal crowding impact of a trip. Given that vehicle size is limited in Sm , the
marginal social cost becomes
∂T C(F, Q|Sm )
= aF −1 + βt + ϕQ(F Sm )−1 βt +
|
{z
}
∂Q
+ aQ
|
marginal user cost
∂F −1 (Q|Sm )
∂Q
{z
i.w.t.ext.
∂F −1 (Q|Sm ) Q
Q
βt + ϕ
βtQ +
+ϕ
FS
∂Q
Sm
(5)
} | {zm } |
{z
}
d.cr.ext.
i.cr.ext.
∂F
∂F (Q|Sm )
δ
t+
twSm
+ 0.
+v
∂Q
∂Q
|
{z
}
marginal operational cost
Due to the fact that vehicle size is now exogenous, we can identify two differences
compared to the unconstrained case and equation (4): the third component of the
marginal operational cost disappeared, and the indirect crowding externality is now
limited to the in-vehicle capacity expansion resulting from the increase in the optimal
frequency. However, we expect that the indirect capacity externalities still have a
positive sign. Moreover, as subsequent simulation results will show, it is likely that
the operator will increase frequency in a faster rate to compensate for its inability of
adjust vehicle size: ∂F (Q|Sm )/∂Q > ∂F (Q, S)/∂Q.
Fixed frequency, unconstrained vehicle size
It may also be the case that the optimal frequency reaches its infrastructure constraint
earlier than the optimal vehicle size, so that the operator’s only option to internalise
crowding is to adjust the capacity of trains. We denote the value at which frequency
11
is fixed with Fm .
∂T C(S, Q|Fm )
= aFm−1 + βt + ϕQ(Fm S)−1 βt +
|
{z
}
∂Q
marginal user cost
∂S −1 (Q|Fm ) Q
Q
+ |{z}
0 +ϕ
βt + ϕ
βtQ +
F S
∂Q
Fm
{z } |
| m
i.w.t.ext.
{z
}
d.cr.ext.
(6)
i.cr.ext.
∂S(Q|Fm )
.
+ 0 + 0 + Fm twδS δ−1
∂Q
{z
}
|
marginal operational cost
The most obvious consequence of constrained frequency is the absence of indirect waiting time externalities. On the other hand, the incremental burden of crowding is
still partly or fully compensated by vehicle size adjustment. In the marginal operational cost expression all components that depend on the elasticity of frequency
disappeared, but vehicle size adjustment still implies some cost for the operator.
We expect though that the optimal vehicle size now increases with a higher rate,
∂S(Q|Fm )/∂Q > ∂S(Q, F )/∂Q, so that we cannot declare with certainty that either
the marginal external or operational costs are lower than in the fully unconstrained
case.
Fixed frequency, fixed vehicle size
In the most extreme case both capacity variables are exogenous due to limitations
in the available technology or infrastructure. Thus, the marginal social cost function
simplifies to
∂T C(Q|Fm , Sm )
= aFm−1 + βt + ϕQ(Fm Sm )−1 βt +
|
{z
}
∂Q
marginal user cost
+ |{z}
0 + ϕQ(Fm Sm )−1 βt + |{z}
0 +
|
{z
}
i.w.t.ext.
d.cr.ext.
i.cr.ext.
(7)
0
|{z}
.
marginal operational cost
As it was expected, all marginal external and operational costs related to capacity
expansion disappears, and the only externality component that prevails is the direct
crowding externality that the marginal consumer imposes on fellow passengers. A
straightforward policy conclusion of this state is that the optimal fare for the public
transport service equals to the pure marginal external crowding cost. Is this optimal
fare higher or lower than in the earlier cases? It depends on the relative magnitude of
12
indirect external and operational costs of capacity expansion. We investigate the transition between the operational states introduced above with
two hypothesised scenarios. The difference between the two scenarios is whether the
frequency or the vehicle size limit is reached first as demand grows. Frequency and
vehicle size constraints are 8 trains/hour and 800 m2 /train in scenario S1, and 16
trains/hour and 400 m2 /train in scenario S2. Thus, the overall capacity is maximised
in 6400 square metres of in-vehicle area per hour in both scenarios, allowing us to
compare the two transition regimes.
Figure 2 depicts the results of the numerical optimisation of equation (1) with
constrained capacity variables according to scenarios S1 (left column) and S2 (right
column). The unconstrained optima are shown by the dashed lines in all graphs. As
expected, both the second-best optimal frequency and vehicle size are higher in the
intermediate stage in order to compensate for the constraint in the other variable, and
internalise crowding and waiting time in a faster rate compared to the first-best case.
Note that this compensation is much more effective in S1 where the vehicle size can be
adjusted. The resulting second-best occupancy rate is even lower than the first-best
optimum. This is not the case in S2, where even though the second-best frequency is
higher than its uncontrained value, the resulting occupancy rate increases as soon as
the vehicle size constraint becomes active. We explain this result with the presence
of density economies in vehicle size provision, which has an even stronger effect when
increasing train length is the only way to reduce crowding costs.
Another consequence to be attributed to density economies is that in scenario S1
the full capacity constraint is reached earlier than in the second regime, i.e. the transition period is shorter, although in both cases the ultimate hourly capacity is the same.
The fact that the second-best occupancy rate is downward sloping in S1 and upward
sloping in S2 will have an important consequence on the sign of marginal indirect
crowding costs. In order to further investigate this and the evolution of other externality components, we derived the marginal frequency and vehicle size curves from the
numerical results and visualised equations (4)-(7) in Figure 3.
Let us focus on crowding-related externalities in the first row of the figure. The
direct and indirect crowding externalities are very similar in magnitude in the unconstrained state, which means that the operator is able to internalise the impact that
an additional passenger would have of fellow travellers. In fact, due to the presence
of scale economies, the indirect effect is slightly stronger, as capacity adjustment leads
13
2000
4000
6000
8000
10000
0
4000
6000
8000
S1 | Optimal vehicle size
S2 | Optimal vehicle size
10000
600
400
0
0
200
400
S (m2)
600
800
Demand (pass/h)
200
2000
4000
6000
8000
10000
0
2000
4000
6000
8000
Demand (pass/h)
S1 | Optimal occupancy rate
S2 | Optimal occupancy rate
10000
1.5
1.4
1.3
1.1
1.0
1.0
1.1
1.2
1.3
φ (pass m2)
1.4
1.5
1.6
Demand (pass/h)
1.6
0
1.2
S (m2)
2000
Demand (pass/h)
800
0
φ (pass m2)
10
0
5
F (1/h)
10
0
5
F (1/h)
15
S2 | Optimal frequency
15
S1 | Optimal frequency
0
2000
4000
6000
8000
10000
0
Demand (pass/h)
2000
4000
6000
8000
10000
Demand (pass/h)
Figure 2: Optimal capacity under infrasturctural constraints. Frequency and vehicle size
constraints are 8 trains/hour and 800 m2 /train in scenario S1, and 16 trains/hour and
400 m2 /train in scenario S2. First-best optima depicted with dashed lines
14
S2 | Marginal crowding externalities
1
0
monetary units
−2
4000
6000
8000
10000
0
4000
6000
8000
S1 | Marginal user externalities
S2 | Marginal user externalities
10000
2
1
0
monetary units
Net crowding
externality
Waiting time
externality
−2
−2
Waiting time
externality
−1
2
3
Demand (pass/h)
Net crowding
externality
Net marginal
external user cost
−3
−3
Net marginal
external user cost
2000
4000
6000
8000
10000
0
2000
4000
6000
8000
Demand (pass/h)
S1 | Marginal social costs
S2 | Marginal social costs
10000
6
Demand (pass/h)
6
0
Net marginal
external user cost
0
2000
4000
2
Net marginal
social cost
−2
−2
0
Net marginal
social cost
0
monetary units
2
4
Marginal
operational cost
4
Marginal
operational cost
monetary units
2000
Demand (pass/h)
1
0
Indirect crowding
externality
−3
2000
3
0
monetary units
Net marginal external
crowding cost
−1
1
0
−1
Indirect crowding
externality
−3
−2
monetary units
Net marginal external
crowding cost
−1
Direct crowding
externality
2
3
Direct crowding
externality
2
3
S1 | Marginal crowding externalities
6000
8000
10000
Net marginal
external user cost
0
Demand (pass/h)
2000
4000
6000
8000
10000
Demand (pass/h)
Figure 3: Marginal social costs and its components under infrastructural constraints.
Frequency and vehicle size constraints are 8 trains/hour and 800 m2 /train in scenario S1,
and 16 trains/hour and 400 m2 /train in scenario S2
15
to a minor reduction in crowding. This beneficial net impact further increases when
the frequency constraint becomes active. However, in scenario S2 the indirect externality drops and the net crowding effect becomes negative (positive when expressed
as a cost) as soon as only frequency can be adjusted. As one may expect, when both
capacity constraints are active, there is no indirect crowding relief any more and the
direct crowding externality becomes dominant.
In the second row of Figure 3 we aggregeted the two marginal crowding cost components (dashed line) and compared it with the magnitude of waiting time externalities
(see the thin, solid line). It is clear that in the unconstrained stage the waiting time
effect is more important than the net crowding externality, and both have a positive
impact on fellow passengers. This justifies the assumption of Mohring and other early
contributors of the capacity management literature that the focus of basic first best
models should be on waiting time costs instead of crowding. Above the frequency
limit in S1, however, the indirect waiting time externality drops to zero. In the intermediate phase of scenario S2, crowding and waiting time externalities have different
signs, so the sign of the net marginal external user cost becomes ambiguous. With
the current simulation parameters the aggregate marginal user externality (thick line)
remains positive (negative, when expressed as a cost). Obviously, as soon as there is
neither frequency nor vehicle size adjustment, the only the direct crowding externality
prevails with strong negative impact on other users.
The sign of marginal external costs has a crucial impact on the optimal pricing
of public transport services. As long as it is negative, so that the user cost for the
average fellow passenger decreases on the margin, subsidisation of the service, i.e. a
fare below marginal operation cost, is justified. It is clear from the simulation model
that in the unconstrained stages the service should be subsidised, while in the fully
constrained stage the fare should be above the zero profit level. The optimal subsidy
in the intermediate stage, however, depends significantly on which capacity variable’s
constraint is reached first. When only vehicle size can be adjusted, the subsidy is
proportional to the magnitude of vehicle size economies. By contrast, if F is the only
decision variable, the need for subsidy depends on the relative value of waiting time
and crowding cost parameters: crowding works against the subsidy, while waiting time
externalities supports it.
Now we turn to another frequently raised policy question: should public transport
be completely free, as Small and Verhoef (2007) derived from Mohring (1972, 1976)?
The third row of Figure 3 compares the net marginal user externality (dashed line)
with marginal operational costs. With the current simulation parameters marginal
16
operational costs are appreciably higher. When frequency cannot be increased any
more, an important marginal operational cost component drops zero, but in the same
time the indirect waiting time externalities also disappear, so there is no significant
change on the aggregate level. The minimum point of the net marginal social cost
curve is always at the demand level where the vehicle size constraint becomes active.
Our simulation results suggest that despite the presence of indirect scale economies in
user costs the optimal fare is positive, assuming marginal social cost pricing.
5
Fluctuating demand with joint costs
Section 4 showed that exogenous infrastructure constraints can explain the emergence
of crowding even if the operator follows a welfare maximising objective in setting
service capacity. This section models another potential source of crowding: demand
fluctuations. Demand for public transport services varies by time, location as well
as direction, and operators are not obviously able to adjust capacity to the first-best
optimum of each journey leg. Thus, supply interdependencies arise between different
markets served with the same capacity, i.e. demand on one market may affect the
capacity supplied on another. Ultimately, this implies that the marginal social cost of
a trip becomes dependent on operational and user costs in seemingly unrelated markets.
In this paper we model the simplest form of demand fluctuations: a bidirectional
train service subject to unbalanced inelastic demand on the two markets. We denote
demand on the busy direction with Q1 , while Q2 is passenger volume in the opposite
direction, Q2 = ωQ1 . This setting is often referred to as the backhaul problem (Demirel
et al., 2010, Rietveld and Roson, 2002). The main goal of this analysis is to investigate
the impact of the magnitude of demand imbalances on supply decisions and the implied
marginal costs. Although in our case 1 ≥ ω ≥ 0 is simply the ratio of the two demand
levels, for most of our conclusions it can be interpreted as the spread of willingness to
pay on markets served with the same capacity.
We keep equation (1) as the engine of the model, and express the function of social
costs as
min T C(F, S, Q1 , Q2 ) = aF −1 (Q1 + Q2 ) + βt(Q1 + Q2 ) + ϕ
|
{z
} |
{z
} |
F,S
waiting time
in−veh. time
δ
+ |vF 2t +{zwS F 2t} .
operatonial cost
17
Q21 + Q22
βt
F{z
S
}
crowding
(8)
As before, travel time is assumed to exogenous, but now counted twice in the
operational cost function. Note that waiting time costs are proportional simply to the
sum of passenger numbers in the two directions, while crowding costs depend on the
sum of squared demand levels. Later on we will see that this implies that the secondbest capacity is skewed towards the first-best optimum of the busy direction, which
would not be the case if only waiting time and operational costs were considered. The
numerical solution of equation (8) for Q1 = 5000 pass/h, and the implied second-best
occupancy rates are plotted in Figure 4.
The U-shaped optimal vehicle size curve may be surprising for the first sight. It
suggests that the same train length should be applied when demand equals on the
two directions and when there are no passengers at all on the back-haul5 . Numerical
tests have shown that this feature, and even the minimum point of the curve, are
unaffected by the strength of density economies. Therefore we can investigate this
property analytically assuming that δ = 1. Given this assumption the social cost
minimising first order conditions are
∂T C
= −ϕ(Q21 + Q22 )F −1 S −2 βt + F w2t = 0,
∂S
(9)
and
∂T C
= −a(Q1 + Q2 )F −2 − ϕ(Q21 + Q22 )F −2 S −1 βt + v2t + wS2t = 0.
∂F
Equation (9) provides an implicit function for the optimal vehicle size,
r
ϕ(Q21 + Q22 )β
S=
,
2wF 2
(10)
(11)
which, after inserting the optimal F from equation (10) and using that Q1 + Q2 =
Q1 (1 + ω) and Q21 + Q22 = Q21 (1 + ω 2 ), simplifies to
s
ϕvβtQ1 (1 + ω 2 )
.
(12)
S(ω) =
aw(1 + ω)
Indeed, ω = 1 and ω = 0 give the same optimal S to this function, and the minimum
√
second-best vehicle size is always at the critical value of ω ∗ = 2 − 1.
5
The optimal frequency, however, is lower in the second case, so total capacity does depend on the
back-haul demand.
18
Optimal frequency
450
S (m2)
440
8.5
7.0
420
7.5
430
8.0
F (1/h)
9.0
460
9.5
470
Optimal vehicle size
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
Demand imbalance, ω
Occupancy rate
Average occupancy rate
1.5
Demand imbalance, ω
(pass m 2)
1.0
Busy
direction
per passenger
per train
0.0
0.0
0.5
Calm
direction
0.0
1.0
0.8
0.5
(pass m 2)
1.5
1.0
1.0
0.8
0.6
0.4
0.2
0.0
1.0
Demand imbalance, ω
0.8
0.6
0.4
0.2
0.0
Demand imbalance, ω
Figure 4: Second-best optimum of vehicle size and frequency provision as a function of the
magnitude of demand imbalance (ω). Demand in the busy direction is fixed at
Q1 = 5000 pass/h.
Let us return to Figure 4 and investigate the resulting second-best occupancy on the
two markets. Unsurprisingly, crowding decreases in the calm direction, simply because
passengers disappear as we move towards strong demand imbalances. Even though
vehicle size recovers at very low off-peak demand, the second-best frequency decreases
steadily, and therefore crowding on the busy direction grows with a decreasing rate.
The average occupancy rate on the two services slightly decrease with ω. However, as
Rietveld (2002) pointed out correctly, this ’operational average’ is not what passengers
experience in reality. The average crowding density per passenger, measured as
φ̄p =
Q1 φ1 + Q2 φ2
Q1 + Q2
(13)
increases with the difference in demand between the two markets. Indeed, when the
19
Marginal vehicle size
Marginal capacity
0.2
0.0
−0.06
1.0
0.8
0.6
0.4
Demand imbalance, ω
0.2
0.0
0.4
(m2/h)
m2
−0.02
6e−04
4e−04
(1/h)
0.02
0.6
8e−04
0.06
0.8
Marginal frequency
1.0
0.8
0.6
0.4
Demand imbalance, ω
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Demand imbalance, ω
Figure 5: Change in the optimal frequency, vehicle size and overall capacity after a
marginal increase in peak (solid lines) and off-peak (dashed lines) demand, as a function of
ω. Demand in the busy direction is fixed at Q1 = 5000 pass/h.
off-peak train is empty, i.e. ω = 0, the average occupancy rate per train is half of what
passengers experience.
Figure 5 shows how an additional peak or off-peak passenger would affect the optimal capacity variables depending on the magnitude of demand imbalances. Interestingly, the impact of back-haul trips on the optimal frequency is always greater than
in the busy direction, and the higher the difference between the two directions, the
faster the optimal F reacts to demand. By contrast, the marginal off-peak trip has
less influence on the second-best vehicle size. Below the critical ω ∗ that we identified
above, what an incremental trip in the calm direction induces is a reduction in vehicle
size (alongside with a massive increase in frequency). A potential explanation for this
seemingly counter-intuitive result is that in this case the frequency adjustment provides large waiting time savings in both directions, while the direct crowding impact of
the marginal traveller is negligible when she boards the almost empty back-haul train.
S)
∂F
∂S
The marginal capacity, i.e. ∂(F
= ∂Q
S + ∂Q
F , has a similar pattern as the marginal
∂Qi
i
i
vehicle size, except that due to the strictly positive marginal frequency ω has to be
very low to let the marginal off-peak trip induce net capacity reduction.
What social costs will a peak and off-peak trip induce on the margin, given the
basic specification provided in equation (8) and the implicit second-best supply-side
behaviour discusses so far? We denote the personal (average user) cost of waiting and
in-vehicle traveling, detailed in equations (4)-(7), as ci on market i. Then the marginal
20
external user and operational cost function on a given direction becomes
M ECi =
∂T C
∂F −1
∂(F S)−1
Qi
− ci = a(Q1 + Q2 )
βt + ϕ(Q21 + Q22 )
βt
+ϕ
∂Qi
∂Qi
F{z
S }
∂Qi
|
|
|
{z
}
{z
}
i.waiting.t.ext.
d.cr.ext.
i.crowding.ext.
∂F
∂S
.
+ (v + wS δ )2t
+ F wδS δ−1 2t
∂Qi
∂Qi
{z
}
|
(14)
marginal operational cost
It is an important difference compared to the unidirectional case that now direct
crowding externalities are born by passengers travelling in the same direction only,
while indirect waiting time and crowding effects through capacity adjustment are enjoyed by all passengers. On the other hand, capacity adjustment, e.g. ∂F/∂Qi , may
now be different than in the first-best scenario, so the analytical approach cannot give
an unambiguous conclusion about the magnitude of externality components.
In Figure 6 we calculated the elements of equation (14) numerically. Figure 6(a)
depicts the value of direct (thick lines) and indirect (thin lines) marginal crowding
costs. The direct externality is proportional to the pattern of crowding density itself,
provided in Figure 4: in the busy direction it increases with ω, while in the calm market
is falls gradually to zero. The benefit that capacity adjustment provides in terms
of crowding release is very similar in magnitude, reflecting the operator’s ability to
internalise crowding externalities through crowding. The sum of the the two crowding
effects, depicted in Figure 6(b), is positive in the peak direction, which means that
the marginal trip decreases the sum of crowding related social costs. Note, however,
that this benefit goes to off-peak passengers, because Q1 decreases ω, which leads to a
reduction in off-peak crowding and an increase in peak crowding, according to Figure
4. The marginal off-peak passenger’s net crowding effect is positive above ω ∗ , and
negative below that.
Figure 6(b) compares the net crowding effect to the marginal waiting time reduction
that frequency adjustment induces. The benefit that an off-peak traveller indirectly induces is now greater than the incremental peak trip’s effect. As the marginal frequency
is always positive, the waiting time externality is also strictly positive. Interestingly,
the sum of crowding and waiting time externalities is the same in both markets, i.e.
for passengers on the back-haul the limited, or even negative crowding effects are fully
compensated by increased waiting time externalities. Overall, the marginal external
user cost is always negative in both directions. This implies that the optimal subsidy
per trip is exactly the same on the two markets.
21
As and Figure 6(c) depicts, the marginal operational cost of a trip in the calm
direction is smaller than on the busy train. Therefore the marginal social cost and
the optimal price of the off-peak trip must be lower too. With the current parameters
the optimal off-peak fare can fall below zero, if demand is strongly unbalanced. If
the environmental burden of travelling is proportional to the operational costs, then
our results are in line with the conclusion of Rietveld (2002): peak travellers are more
polluting, even if operational and environmental costs are shared by more passengers.
(b) Marginal user externalities
3
(a) Marginal crowding externalities
0.0
Net marginal external crowding cost
−0.5
monetary units
1
0
Marginal external waiting time cost
Net marginal
external user cost
−1.0
−1
Indirect crowding
externality
−3
−2
monetary units
2
Direct crowding
externality
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
Demand imbalance, ω
Demand imbalance, ω
(c) Marginal social costs
(d) Marginal social costs
0.0
4
1.0
2
1
0
1
Net marginal social cost
0
monetary units
2
Net marginal
social cost
monetary units
3
Marginal
operational
cost
−1
−1
Net marginal
external user cost
−2
−2
Net marginal
external user cost
1.0
0.8
0.6
0.4
0.2
0.0
5000
Demand imbalance, ω
10000
15000
20000
Demand, Q1=Q2
Figure 6: Marginal social cost components in the back-haul setting. Dashed lines in plots
(a) to (c) represent the off-peak market
Keep in mind that all numerical simulations in this section were conducted by
setting the inelastic demand in the busy direction to Q1 = 5000 pass/h. How do
the marginal social and external user cost curves react to a proportional increase in
demand? For the sake of simplicity what we plotted in Figure 6(d) is the marginal
22
social cost of a unidirectional trip when ω = 1, i.e. when the marginal cost is the same
in both directions. Is shows that as demand grows, the optimal fare slightly increases,
and the optimal subsidy (net marginal external user cost) diminishes.
6
Future research and conclusions
The theoretical models presented here were able to explain the presence of crowding
under optimal second-best capacity management decisions. There are a number of
potential ways to further develop these models. Section 4 revealed that infrastructure
constrains in at one of the capacity variables affects the optimal supply of the other
(e.g. constrained vehicle size implies higher second-best frequency). In this model we
assumed that the infrastructure design is exogenous, which is not obviously the case,
especially not in the planning period of a new infrastructure. In fact, public transport
supply can be modelled as parallel decisions on three levels:
1. Infrastructure capacity;
2. Vehicle capacity, i.e. frequency and vehicle size6 ;
3. Pricing.
The joint optimisation of the three dimensions leads to the undiscovered topic of cost
recovery of dedicated public transport infrastructure investments7 . In this sense our
analysis can be considered as an after-intervestment optimisation which can be relevant
on many mature rail systems facing growth in demand and unfeasible infrastructure
expansion.
In case of demand fluctuations this paper showed that overcrowding on the busy
market of a bidirectional route with demand imbalances can be explained by the threat
of wasted operational costs on the back-haul. The model is built on the assumption
of fixed second-best capacity for the two markets. In reality, however, operators are
able, at least partly, to adjust capacity to demand fluctuations. Joint costs between
spatially separated line sections can be reduced by decoupling cars or multiple units
at intermediate stations. In temporal demand fluctuations frequencies can be adjusted
by adding more vehicles in the peak and removing them outside of rush hours. In
some cases, at least temporarily, capacity can be differentiated directionally as well
by storing trains at the endpoint of the busy direction. But all these interventions
6
Other capacity features, such as in-vehicle seat supply can also be considered here
The road transport equivalent of this problem has been investigated extensively by Mohring and
Harwitz (1962), Newbery (1989), and Verhoef and Rouwendal (2004), among others.
7
23
induce expenditures in terms of operational, storage and capital costs. Therefore it is
a relevant question: What is the optimal rate of capacity reduction in off-peak periods?
This trade-off can be modelled with economic tools and the resulting capacity policy
would most likely distort the marginal cost values derived in Section 5 of this paper.
Another obvious limitation of our current models is the inelastic demand assumption. Rietveld et al. (2002) found that the shape of Mohring’s optimal frequency and
vehicle size functions do not change significantly under elastic demand. From a pricing
perspective, our marginal external cost results in equations (4)-(7) and (14) should be
equal to the optimal fare only in equilibrium (Jara-Dı́az and Gschwender, 2005). Suboptimal price setting leads to distortions in capacity provision under elastic demand,
as Jara-Dı́az and Gschwender (2009) showed.
From a policy perspective we can conclude the current paper with the following
statements. In the presence of vehicle size and frequency constraints due to engineering
design or regulation:
• When both capacity constraints are active, the marginal social cost of a trip is
determined predominantly by the marginal external cost of crowding, and thus
the optimal fare simplifies to crowding charging.
• When both frequency and vehicle size are adjustable, crowding is internalised
by the operator, and in the presence of economies of vehicle size the marginal
trip leads to slight reduction in crowding costs. This, together with waiting time
externalities justify subsidisation.
• If only frequency can be adjusted, the net user externality depends on the magnitude of waiting time and crowding cost parameters. If only vehicle size remains
flexible, a relatively small (smaller than 1st best) subsidy is justified, depending
on the strength of vehicle size economies.
If a fixed second-best capacity serves multiple markets with varying demand:
• Depending on the magnitude of demand imbalances, the optimal second-best
frequency decreases, while the optimal vehicle size follows a U-shaped curve.
• The marginal peak and off-peak trip induces the same marginal external user cost,
which is negative due to the indirect benefits of capacity adjustment. However,
peak trips lead to higher incremental operational costs. Therefore the optimal offpeak fare is significantly lower than in busy markets served by the same capacity.
In severe demand imbalances the off-peak fare can be negative.
• The marginal social cost of all trips slightly increases with demand, and the
optimal average subsidy decreases as demand grows.
24
The main contribution of this study from a scientific point of view is the theoretical
framework that links the classic capacity management literature built on Mohring’s
square root principle with empirical observations about the congestible nature of public transport. This framework enables researchers to investigate for example interrelations between crowding pricing and investment financing, industrial organisation and
agglomeration effects.
References
Demirel, E., Ommeren, J. V. and Rietveld, P. (2010), ‘A matching model for the
backhaul problem’, Transportation Research Part B: Methodological 44(4), 549–561.
Jansson, J. O. (1980), ‘A simple bus line model for optimisation of service frequency
and bus size’, Journal of Transport Economics and Policy pp. 53–80.
Jara-Dı́az, S. and Gschwender, A. (2003), ‘Towards a general microeconomic model for
the operation of public transport’, Transport Reviews 23(April 2013), 453–469.
Jara-Dı́az, S. R. and Gschwender, A. (2005), ‘Making pricing work in public transport
provision’, Handbook of transport strategy, policy and institutions pp. 447–459.
Jara-Dı́az, S. R. and Gschwender, A. (2009), ‘The effect of financial constraints on the
optimal design of public transport services’, Transportation 36(1), 65–75.
Li, Z. and Hensher, D. a. (2011), ‘Crowding and public transport: A review of willingness to pay evidence and its relevance in project appraisal’, Transport Policy
18(6), 880–887.
Mohring, H. (1972), ‘Optimization and Scale Economies in Urban Bus Transportation’,
The American Economic Review 62(4), 591–604.
Mohring, H. (1976), Transportation economics, Ballinger Publishing Company.
Mohring, H. and Harwitz, M. (1962), ‘Highway benefits: An analytical framework’.
Newbery, D. M. (1989), ‘Cost recovery from optimally designed roads’, Economica
pp. 165–185.
Oldfield, R. and Bly, P. (1988), ‘An analytic investigation of optimal bus size’, Transportation Research Part B: Methodological 22(5), 319–337.
25
Rietveld, P. (2002), ‘Why railway passengers are more polluting in the peak than in the
off-peak; environmental effects of capacity management by railway companies under
conditions of fluctuating demand’, Transportation Research Part D: Transport and
Environment 7(5), 347–356.
Rietveld, P., Dekkers, J. and van Woudenberg, S. (2002), ‘Choice of frequency and vehicle size in rail transport. implications for marginal external costs’, EJTIR 2(2), 95–
111.
Rietveld, P. and Roson, R. (2002), ‘Direction dependent prices in public transport: A
good idea? the back haul pricing problem for a monopolistic public transport firm’,
Transportation 29(4), 397–417.
Rietveld, P. and van Woudenberg, S. (2007), ‘Second-best decision making of railway operators: How to fix fares, frequency and vehicle size’, Journal of Transport
Economics and Policy 41, 363–385.
Small, K. A. and Verhoef, E. T. (2007), The economics of urban transportation, Routledge.
Tirachini, A. (2014), ‘The economics and engineering of bus stops: Spacing, design and
congestion’, Transportation research part A: policy and practice 59, 37–57.
Tirachini, A., Hensher, D. a. and Jara-Dı́az, S. R. (2010), ‘Comparing operator and
users costs of light rail, heavy rail and bus rapid transit over a radial public transport
network’, Research in Transportation Economics 29(1), 231–242.
Verhoef, E. T. and Rouwendal, J. (2004), ‘Pricing, capacity choice, and financing in
transportation networks’, Journal of Regional Science 44(3), 405–435.
Wardman, M. and Whelan, G. (2011), ‘Twenty Years of Rail Crowding Valuation Studies: Evidence and Lessons from British Experience’, Transport Reviews 31(March
2013), 379–398.
26