The Economics of Crowding in Public Transport

Andre de Palmaa Robin Lindseyb

Guillaume Monchamberta,c,1

aEcole Normale Superieure de Cachan

bUniversity of British Columbia

cKULeuven

Railway Operations Research Seminar: ”Put Passengers first”

KUL - May 3rd, 2016

1guillaume.monchambert@kuleuven.be

Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Motivation

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Motivation

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Modal split - Intuition 1/4Generalized cost• is the sum of the monetary and non-monetary costs of a

journey (travel time, waiting time...),• may depend on the patronage.

Figure 1: Generalized cost of using road as a function of the number ofroad users

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Modal split - Intuition 1/4Generalized cost• is the sum of the monetary and non-monetary costs of a

journey (travel time, waiting time...),• may depend on the patronage.

Figure 1: Generalized cost of using road as a function of the number ofroad users

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Modal split - Intuition 2/4The equilibrium on a network is reached when the generalized costis the same for all modes (Wardrop equilibrium).

Figure 2: Bi-modal equilibrium with economies of scale in publictransport (Mohring effect)

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Modal split - Intuition 2/4The equilibrium on a network is reached when the generalized costis the same for all modes (Wardrop equilibrium).

Figure 2: Bi-modal equilibrium with economies of scale in publictransport (Mohring effect)

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Modal split - Intuition 3/4

Figure 3: Bi-modal equilibrium without economies of scale in publictransport

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Modal split - Intuition 4/4

Figure 4: Bi-modal equilibrium with diseconomies of scale in publictransport (Crowding)

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Characteristics of public transport crowding

Crowding in public transport have characteristics which are similarto congestion (traffic jam) on road:

• a negative externality,

• caused by an excess of demand,

• which increases with the demand,

• which degrades the individual experience of travel,

• and which raises the generalized travel cost.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

In the literature

• There is an economic theory of road congestion (mainlythrough the bottleneck model : Vickrey, 1969; Arnott et al.,1990, 1993),

• but few works on crowding in public transport.

Almost all works study competition between road and publictransport including congestion on road but not on public transport(Tabuchi, 1993 ; Danielis et Marcucci, 2002 ; Mirabel et Reymond,2011 ; Gonzales et Daganzo, 2013 ; Tian et al., 2013...).

=⇒ Fill this gap in the literature.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Goals of this work

• Develop a dynamic model of public transit usage that includescrowding (PTM model).

• Derive:- User equilibrium- System (i.e., social) optimum- Optimal fares: uniform and time-varying- Optimal service

• Timetable, no. trains, train capacity• Dependence on fare regime

• Comparisons with bottleneck model of road traffic congestion(the most widely adopted road congestion model in theeconomic literature).

• Application to a segment of RER A transit line in the ParisRegion.

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Outline

1. Literature review

2. Hypothesis of the model

3. PTM Model

4. Capacity adjustment

5. Application to RER A

6. Conclusions

Outline

1. Literature review1.1 Empirical microeconomic evidences1.2 Empirical macroeconomic valuations

2. Hypothesis of the model

3. PTM Model

4. Capacity adjustment

5. Application to RER A

6. Conclusions

Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Empirical microeconomic evidences

Mircoeconomic valuations of crowding cost:

• Fixed penalties, about 2,5$US/trip (Pepper et al., 2003 ;Hensher et al., 2011)

• Time multipliers

- Whelan and Crockett, 2009 (next slide).

- Wardman and Whelan, 2011 : standing = 2,32 ; seating =1,32.

- Kroes and al., 2013 :• Seating when all seats are occupied = 1,1.• Standing when the vehicle is full = 1,6.

- Haywood and Koning, 2015 (next slide).

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Figure 5: Time multipliers as a function of the in-vehicle density - Source:Haywood and Koning, 2015.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Table 1: Times multipliers as a function of the trip characteristics -Source : Whelan and Crockett, 2009.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Empirical macroeconomic evidences

Valuations of crowding cost at a city scale:

• Prud’homme et al., 2012 : From 2002 to 2007, an 8%increase in ridership on the Paris subway system caused awelfare loss due to extra crowding of at least e75M/year.

• Veicht et al., 2013 : In 2011, welfare loss of over-crowding inMelbourne: $280M.

• de Palma, Monchambert and Picard, 2015 : crowding cost inParis Region public transport reaches from 11 to 15 millionseuros per working day.

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Outline

1. Literature review

2. Hypothesis of the model2.1 Bottleneck theoretical framework2.2 Specific to public transport2.3 Crowding cost function

3. PTM Model

4. Capacity adjustment

5. Application to RER A

6. Conclusions

Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Bottleneck theoretical framework

Hypothesis from Arnott et al., 1990, 1993 :

• N identical individuals commute from A to B every day.

• All individuals have the same preferred arrival time, t∗.

• The strategic variable is the departure time, t : individualstrade-off between schedule delay cost and congestion cost(extra travel time on road and crowding for public transport).

• Individuals incur a schedule delay cost if they do not arrive att∗, denoted δ (t)

δ (t∗) = 0,

{δ′ (t) < 0 si t < t∗

δ′ (t) > 0 si t > t∗.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Specific to public transport

• A PT line connects A and B with no intermediate stops.

• Travel time is fixed (and normalized to zero).

• The public transport authority operates m trains, indexed bydeparture time k = 1, ...,m (train m leaves last).

• Train k leaves (and arrives) at tk.

• Users know the timetable.

• Departure time decision is discrete.

• There is no upper limit to train capacity.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Crowding cost function

Linear form:g (n) = λ

n

s

where

• n is the number of users in the same train,

• s > 0 is a measure of the train capacity (might be m2),

• and λ is a scale parameter.

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Outline

1. Literature review

2. Hypothesis of the model

3. PTM Model3.1 Equilibrium - graphical intuition3.2 Inefficiency of equilibrium3.3 Social optimum

4. Capacity adjustment

5. Application to RER A

6. Conclusions

Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Equilibrium - graphical intuition

t7t6t∗ = t5t4t3t2t1 t

Arrival time of PT

User private cost

SDC

Scheduling cost

ce

Crowding cost

Figure 6: Equilibrium distribution of departure times when m = 7.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Equilibrium - graphical intuition

t7t6t∗ = t5t4t3t2t1 t

Arrival time of PT

User private cost

SDC

Scheduling cost

ce

Crowding cost

Figure 6: Equilibrium distribution of departure times when m = 7.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Equilibrium - graphical intuition

t7t6t∗ = t5t4t3t2t1 t

Arrival time of PT

User private cost

SDC

Scheduling cost

ce

Crowding cost

Figure 6: Equilibrium distribution of departure times when m = 7.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Inefficiency of equilibrium

In equilibrium, the pattern of departure times is sub-optimal (themarginal social cost of a trip is higher than the private cost).

Crowding is a negative externality. It implies two distorsions:

• There are too many users using the facility (if the demand iselastic).

• Ridership is not optimally distributed over trains.(Presumption: riders are too concentrated on ”timely” trains).

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Social optimum - Graphical intuition

Figure 7: Equilibrium cost and pattern of departure times

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Social optimum - Graphical intuition

Figure 8: Optimal private and marginal social costs

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Social optimum - Graphical intuition

Figure 9: Equilibrium and optimal patterns of departure times

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Social optimum - Main result

The social optimum can be decentralized with a time-dependentfare:

τ ok = nokg′ (nok) .

Users are made worse of by the fare (proof: at least one train ismore heavily loaded in the social optimum than in the equilibrium).

In the bottleneck model, users are equally well off.

The welfare gain from optimal pricing is independent of N , thepatronage.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

The Parisian subway case

Figure 10: Daily distribution of trips in the Parisian subway during awinter workday - Source: RATP 2008.

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Outline

1. Literature review

2. Hypothesis of the model

3. PTM Model

4. Capacity adjustment4.1 Three fare regimes4.2 Results

5. Application to RER A

6. Conclusions

Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Three fare regimes

We maximize the social surplus with respect to the number oftrains operated, m and the train capacity, s, under three pricingregimes:

1. No pricing (free regime) n,

2. Optimal flat pricing (uniform regime) u,

3. Optimal time-dependent pricing (optimal regime) o.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Results

Free regime:

• With underpriced usage, latent demand dilutes benefit ofcapacity expansion.

Uniform regime:

• No welfare loss from latent demand.

Optimal regime:

• Generation of variable extra revenue effectively reducesmarginal cost of expanding s or m. Expanding capacity ismore profitable in the optimal regime.

→ Congestion pricing and capacity expansion are complementary.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Results

Free regime:

• With underpriced usage, latent demand dilutes benefit ofcapacity expansion.

Uniform regime:

• No welfare loss from latent demand.

Optimal regime:

• Generation of variable extra revenue effectively reducesmarginal cost of expanding s or m. Expanding capacity ismore profitable in the optimal regime.

→ Congestion pricing and capacity expansion are complementary.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Results

Free regime:

• With underpriced usage, latent demand dilutes benefit ofcapacity expansion.

Uniform regime:

• No welfare loss from latent demand.

Optimal regime:

• Generation of variable extra revenue effectively reducesmarginal cost of expanding s or m. Expanding capacity ismore profitable in the optimal regime.

→ Congestion pricing and capacity expansion are complementary.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Results

Free regime:

• With underpriced usage, latent demand dilutes benefit ofcapacity expansion.

Uniform regime:

• No welfare loss from latent demand.

Optimal regime:

• Generation of variable extra revenue effectively reducesmarginal cost of expanding s or m. Expanding capacity ismore profitable in the optimal regime.

→ Congestion pricing and capacity expansion are complementary.

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Outline

1. Literature review

2. Hypothesis of the model

3. PTM Model

4. Capacity adjustment

5. Application to RER A5.1 RER A, some facts5.2 Data5.3 Results5.4 Short term vs long term5.5 Many to one network

6. Conclusions

Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

RER A, some facts

Figure 11: RER A map

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

RER A, some facts

RER A, it is:

• More than 300 millions trips per year.

• More than one million trips per working day.

• 60% concentrated on the central segment.

La Defense station:

• Every working day, 32 600 users stop at La Defense stationbetween 8:25am and 9:25am.

• Average travel time in RER of these users: 40 minutes.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Data

Calibration of the model to describe RER A - heading to LaDefense station during the morning peak hour:

• K (m, s) = (ν0 + ν1s)m+ ν2s (Kraus and Yoshida, 2002)

• β = 7, 4e/h, γ = 17, 2e/h, λ = 4, 4e/user

• h = 2, 5 min

• Nu = 32 600, mu∗ = 24 et su∗ = 1 733

• Price-elasticity of the demand -1/3

• Cost recovery ratio = 5/6

−→ N0 = 69 003 potential users, ν0 = 936.7 e/train, ν1 = 0.1344e/train/user, and ν2 = 61.63e· usager.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Results

Free Uniform Optimal

Number of trains 25.26 24 26.70Trains capacity (m2) 1,762 1,733 1,710Patronage 37,173 32,600 32,907

(-12%) (-11%)Ave. scheduling cost (e) 2.1 1.9 2.4Ave. crowding cost (e) 4.3 4.1 3.4Ave. fare (e) - 3.5 3.4Private cost (e) 6.4 9.5 9.2Priv. cost - ave. fare (e) 6.4 6 5.8Social surplus (ke) 1,735 1,743 1,749Gain/user (e) - 0.27 0.45

Table 2: Individual costs, fares and surplus in three pricing regimes onRER A during morning peak.

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Short term vs long term

Regime change n→ u u→ o n→ o

Long term gains +8 714 e +6 076 e +14 788 e(Flexible supply)

Short term gains +8 336 e +5 273 e +14 589 e(Fixed supply)

Difference ST → LT +4,5% +15,2% +1,2%

Table 3: Social gains due to a change in the pricing regime at long andshort terms (n = free, u = uniform and o = optimal).

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Many to one network

Figure 12: Departure pattern from three departure stations in equilibrium

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Many to one network

Figure 13: Application to the West branch of RER A: from SaintGermain-en-Laye to La Defense

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Many to one network

Figure 14: Distribution of equilibrium arrival times at La Defense

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Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Many to one network

Figure 15: Distribution of equilibrium and optimal arrival times at LaDefense

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Outline

1. Literature review

2. Hypothesis of the model

3. PTM Model

4. Capacity adjustment

5. Application to RER A

6. Conclusions

Introduction Literature review Hypothesis PTM Model Capacity Application to RER A

Conclusions

1. A general tractable model of crowding in public transport.

2. Microeconomic analysis of this externality.

3. Dynamics of crowding differ from the dynamics of roadcongestion. Crowding pricing seems less profitable.

4. However, the revenue generated can be used to improvecapacity and service quality.

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Thank you!

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