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 increasingboth 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 hours11. It is reasonable to assume that in these cases capacity is not
11The 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.
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 capac-ity 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 (6.2) with respect to demand. Marginal social costs can be split into three components:
∂T C(F, S, Q)
∂Q =aF−1+βt+ϕQ(F S)−1βt
| {z }
marginal user cost
+
+aQ∂F−1
∂Q
| {z }
i.w.t.ext.
+ϕ Q F Sβt
| {z }
d.cr.ext.
+ϕ∂(F S)−1
∂Q βtQ2
| {z }
i.cr.ext.
+
+v∂F
∂Qt+∂F
∂QtwSδ+F twδSδ−1∂S
∂Q
| {z }
marginal operational cost
.
(6.16)
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 demand12. We expect that both indirect externality compononets have a negative sign, i.e.
capacity adjustment has a positive effect on both the headway and the average in-vehicle area per passenger. Finally, the marginal passenger induces incremental operational costs
12For the sake of simplicity we did not indicate that capacity is optimised in this cost function, so that F=F(Q, S) andS=S(Q, F).
too. Capacity adjustment affects in this case both operational cost elements in equation (6.2).
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)
∂Q =aF−1+βt+ϕQ(F Sm)−1βt
| {z }
marginal user cost
+
+aQ∂F−1(Q|Sm)
∂Q
| {z }
i.w.t.ext.
+ϕ Q F Smβt
| {z }
d.cr.ext.
+ϕ∂F−1(Q|Sm)
∂Q
Q SmβtQ
| {z }
i.cr.ext.
+
+v∂F(Q|Sm)
∂Q t+ ∂F
∂QtwSmδ + 0
| {z }
marginal operational cost
.
(6.17)
Due to the fact that vehicle size is now exogenous, we can identify two differences compared to the unconstrained case and equation (6.16): the third component of the marginal opera-tional 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 is fixed withFm.
∂T C(S, Q|Fm)
∂Q =aFm−1+βt+ϕQ(FmS)−1βt
| {z }
marginal user cost
+
+ 0
|{z}
i.w.t.ext.
+ϕ Q FmSβt
| {z }
d.cr.ext.
+ϕ∂S−1(Q|Fm)
∂Q
Q Fm
βtQ
| {z }
i.cr.ext.
+
+ 0 + 0 +FmtwδSδ−1∂S(Q|Fm)
∂Q
| {z }
marginal operational cost
.
(6.18)
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 adjust-ment 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)
∂Q =aFm−1+βt+ϕQ(FmSm)−1βt
| {z }
marginal user cost
+
+ 0
|{z}
i.w.t.ext.
+ϕQ(FmSm)−1βt
| {z }
d.cr.ext.
+ 0
|{z}
i.cr.ext.
+ 0
|{z}
marginal operational cost
. (6.19)
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 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 8trains/hour and 800m2/train in scenario S1, and 16 trains/hour and 400m2/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 6.4 depicts the results of the numerical optimisation of equation (6.2) with con-strained 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 thiscompensation 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 (6.16)-(6.19) in Figure 6.5.
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 to a minor reduction in crowding.
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051015
S1 | Optimal frequency
Demand (pass/h)
F (1/h)
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051015
S2 | Optimal frequency
Demand (pass/h)
F (1/h)
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0200400600800
S1 | Optimal vehicle size
Demand (pass/h) S (m2 )
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0200400600800
S2 | Optimal vehicle size
Demand (pass/h) S (m2 )
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1.01.11.21.31.41.51.6
S1 | Optimal occupancy rate
Demand (pass/h) φ (pass m2 )
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1.01.11.21.31.41.51.6
S2 | Optimal occupancy rate
Demand (pass/h) φ (pass m2)
Figure 6.4: Optimal capacity under infrasturctural constraints. Frequency and vehicle size constraints are 8trains/hour and 800m2/trainin scenario S1, and 16trains/hour and 400
m2/trainin scenario S2. First-best optima are depicted with dashed lines
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−3−2−10123
S1 | Marginal crowding externalities
Demand (pass/h)
monetary units
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−3−2−10123
S2 | Marginal crowding externalities
Demand (pass/h)
monetary units
0 2000 4000 6000 8000 10000
−3−2−10123
S1 | Marginal user externalities
Demand (pass/h)
monetary units
0 2000 4000 6000 8000 10000
−3−2−10123
S2 | Marginal user externalities
Demand (pass/h)
monetary units
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−20246
S1 | Marginal social costs
Demand (pass/h)
monetary units
0 2000 4000 6000 8000 10000
−20246
S2 | Marginal social costs
Demand (pass/h)
monetary units
Direct crowding externality
Indirect crowding externality Net marginal external crowding cost
Direct crowding externality
Indirect crowding externality Net marginal external crowding cost
Net crowding externality
Net marginal external user cost Waiting time externality
Net crowding externality
Net marginal external user cost Waiting time externality
Marginal operational cost
Net marginal external user cost Net marginal social cost
Marginal operational cost
Net marginal external user cost Net marginal social cost
Figure 6.5: Marginal social costs and its components under infrastructural constraints. Frequency and vehicle size constraints are 8trains/hourand 800 m2/trainin scenario S1, and 16trains/hour
and 400m2/trainin scenario S2
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 6.5 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, ifF 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 6.5 compares the net marginal user externality (dashed line) with marginal operational costs. With the current simulation parameters marginal 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 exter-nalities 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.