7.2.1 Baseline model
In this section we develop a simple model as an illustration of (1) the trade-off between various consequences that variable seat supply implies, and (2) the consumption externalities when the effect of crowding on seated and standing passengers is differentiated. The main contribution of this part is the analytical treatment of seat supply and its effect on marginal external crowding costs.
Assume that the average user cost of standing and sitting are denoted by functions cst(Q, S, σ) and cse(Q, S, σ) respectively, where the disutility of crowding associated with the density of passengers depends on three main factors: the amount of passengers (Q), the available in-vehicle floor space (S), and the number of seats installed in the vehicle (σ). De-mand is elastic with respect to the generalised cost of travelling, which is composed of the fare price and the expected cost of crowding. We have to introduce here another important assumption: passengers have equal chance to find a seat and are assumed to gain perfect information about the probability of finding a seat prior to their decision on consumption.
That is, crowding appears in the generalised cost function as the average of cst and cse, weighed by the proportion of standing and seated passengers:
GC=p+Q−σ
Q cst(Q, σ;S) + σ
Qcse(Q, σ;S), (7.1)
where p denotes the fare set by the operator, and the vehicle’s interior space is fixed in the short run. We also assume that passengers always prefer sitting over standing, and demand is always higher than the number of seats supplied2. We describe demand with a downward sloping inverse demand function, d(Q). In equilibrium, inverse demand equals to the generalised user cost of travelling.
We analyse two possible economic objectives to model the operator’s optimal choice of seat supply and fare price. The operator may wish to maximise social welfare calculated as the sum of consumer’s surplus net of crowding disutilities and operational costs. Assuming a linear ridership-dependent operational cost function with slopeu, the social welfare function
2OtherwiseQ−σ takes a negative value in equation (7.1), thus resulting in a user benefit when certain seats remain unused. In the realistic but in this chapter neglected case whenQ < σ, so that all passengers find a seat, the only consumption externality may stem from frictions between seated users. Wardman and Murphy (2015) provides preliminary guidelines about the magnitude of such externalities.
to be maximised is
maxσ,p W =
Q
Z
0
d(q)dq−(Q−σ)cst(Q, σ;S)−σ cse(Q, σ;S)−uQ. (7.2)
The profit seeking operator’s objective is to maximise
maxσ,p π= (p−u)Q(σ, p), (7.3)
where one can immediately see that the choice of seat supply remains relevant through its impact on demand.
The optimal seat supply and fare price values can be obtained by solving the optimisation problems outlined in equations (7.2) and (7.3) subject to the equilibrium constraintd(q) ≡ GC. After establishing Lagrangian functions and taking first order conditions, the welfare sensitive operator’s optimal price (pW) and seat supply rule become
pW =u+ (Q−σ)∂cst
∂Q +σ ∂cse
∂Q + σ
Q(cst−cse) (7.4)
and
(Q−σ)∂cst
∂σ +σ ∂cse
∂σ =cst−cse. (7.5)
It is easy to show that the optimal price in equation (7.4) complies with the basic Pigovian externality tax. The user should pay the marginal cost imposed on the operator (u), plus the marginal external crowding cost that can be split into two elements. The first part, (Q−σ)∂c∂Qst+σ∂c∂Qse, equals to the marginal increase in personal crowding costs multiplied by the number of standing and seated passengers, reflecting that an additional consumer makes all fellow passengers’ trip slightly less comfortable. In the second part Qσ is the probability that the marginal traveller will find a seat; in this case she would enforce someone else to stand and bear higher user cost measured by the difference between cst and cse. When the marginal passenger boards the train, beside the density effect it is sure that one more traveller will have to stand – this incremental standing cost may be born by the marginal traveller herself, or someone else. The likelihood of the second case equals to the probability that the marginal passenger finds a seat.
We refer to these two effects as density and occupancy externalities, respectively. Recall
again that the optimal prices derived in this section are applicable only to the case when demand is larger than seat supply. When all passengers find a seat, in this model the density of standees as well as both crowding externalities drop to zero.
Let us now turn to the seat supply rule in equation (7.5). The expression on the left hand side equals to the additional disutility that the marginal seat imposes on standing and seated passengers via the reduction in standing area and the increase in standing density. On the right hand sidecst−cse is the benefit of a standing passenger who will occupy the marginal seat. In other words, equation (7.5) tells that the number of seats should be increased until the benefit delivered to an additional seated passenger becomes equal to the marginal social cost of reducing the standing area. The seat supply rule is only valid when the service is optimally priced: Q can be held constant because the envelope theorem assures that the effect of a marginal change in seat supply on demand has no welfare implication3.
pπ =u+ (Q−σ)∂cst
∂Q +σ ∂cse
∂Q + σ
Q(cst−cse)−Q ∂d
∂Q (7.6)
From equation (7.3) and the equilibrium constraint, we derive in equation (7.6) the fare that the monopolist applies to maximise its profits. It is not surprising that the profit maximising price also internalises the marginal external crowding costs entirely. In fact, the only difference between equations (7.4) and (7.6) is the monopoly mark-up determined by the generalised cost elasticity of demand4. Solving the constrained profit maximisation problem for the optimal seat supply rule we get exactly the same expression as for the welfare oriented case in equation (7.5). This result, however, does not mean that the monopolist supplies the same number of seats and the same level of crowding will emerge. Aspπ is higher than the socially optimal fare, the monopolist faces lower demand and less crowding. Thus, ceteris paribus, the private operator’s vehicle is expected to be equipped with more seats.
7.2.2 Seat supply under first-best conditions
In the rest of Section 7.2, we take a closer look at the optimal supply of seats. What the supply rule in equation (7.5) reveals is just the first order condition of optimality, from which the impact of major determinants of the actual value ofσ may not be understood directly.
3This reasoning is equivalent to the optimal investment rule discussed by Small and Verhoef (2007, Page 164), and the frequency and vehicle size rules derived in equation (6.5) and (6.6).
4By rewriting equation (7.6) we get the usual expression for the profit mark-up: pπ−mscpπ =−1
GC, where GC is the generalised price elasticity of demand andmscis the marginal social cost of a trip in crowding.
After a brief analysis of the first-best seat supply under exogenous capacity, we spend more time with two extensions of the model that brings the investigation closer to real-world applications. First, we explore the case of simultaneously optimised frequency and vehicle size. The crucial question here is whether reducing seat supply or increasing frequency (or both) is the optimal reaction to any growth in demand. Then, Section 7.2.3 puts the choice of seat supply into the context of fluctuating demand, and derives second-best policies in the back-haul problem.
Exogenous capacity
Let us investigate in a simple simulation exercise the pattern of optimal seat supply in function of hourly ridership. For the sake of simplicity, we choose a social cost minimisation approach.
This may obscure some features of elastic demand, but the results below should hold in the equilibrium state of an endogenous demand setting as well. The optimal seat supply can be derived by solving the following minimisation problem:
minσ T C=Cop+Q·cu, (7.7)
where the operational costs are simply linear in frequency, Cop =vF. The generalised user cost expression has three main components: waiting time proportional to half of the headway F−1, in-vehicle travel time (βt), and crowding costs that enter the function as a travel time multiplier.
cu= 0.5α F−1+βt
1 +max
0,1−F σ Q−1
·βp+max
0,QF−1−σ S−σas
·βc
(7.8) The notation is consistent with Chapters 3 and 6, and summarised in Table 7.1. Note that in user cost expression (7.8) there are two non-negative factors. The first captures the probability of standing, given that F σ Q−1 is the ratio of the available seats and ridership per hour, i.e. the probability of sitting. This may turn into negative if F σ > Q, but we deliberately exclude this option by setting the minimum of the factor to zero. Then, the probability of standing is multiplied with βp, the standing multiplier estimated in Chapter 5. The second fraction of equation (7.8) is the density of standing passengers, which is the ratio of the number of standing passengers per train and the available standing area. The latter comes as the difference between the in-vehicle area (S), and the space occupied by seats and seated passengers. Again, we prevent the standing density to turn into negative