Covid-19 and optimal urban transport policy.

Covid-19 has important implications for public transport operations. Increased teleworking and the perceived infection risk on public transport vehicles have drastically reduced demand in many cities. At the same time, physical distancing has effectively reduced available peak-period public transport capacity. In this paper, we use a simple model to study the effect of these changes on second-best optimal pricing and frequency provision, assuming that car use is underpriced. A numerical application reflecting the public transport situation in Brussel is provided. Results include the following. First, more telework and the increased perceived infection risk have opposite effects on the fare, so that it may be optimal not to change the fare at all. Optimal frequency is likely to decline. Second, holding the fare and frequency constant at their pre-Covid second-best optimal values, more telework reduces the public transport deficit if car use is underpriced. Third, extending the model to allow for passengers with different vulnerability towards Covid-19, allowing fare and frequency differentiation implies that vulnerable users will face higher fares only if their risk perception is sufficiently higher than that of the non-vulnerable, and car use is not too much underpriced. Occupancy rates will be lower for the vulnerable passengers. Fourth, the numerical results for Brussels show that telework and a high perceived infection risk for workers may yield a welfare optimum whereby commuters do almost not use public transport. Offering a low frequency suffices to deal with the captive demand by school children and students. Lastly, reserved capacity for the vulnerable users and stimuli for walking and biking to school may be useful policies to deal with the crowding risk.

Introduction

The Covid-19 crisis has drastically affected individuals’ way of life around the world. Moreover, it has implications in all major sectors of the economy. In this paper, we focus on the effects of the virus for public transport (PT) in urban areas. The virus has important effects on both the demand and supply sides of the market. On the demand side, some passengers may be reluctant to use public transport because the perceived infection risk is higher (both the in-vehicle risks and the risk of contacts in stations) in comparison with a trip by car; others may find public transport less attractive because of the requirement to wear a mouth mask. Moreover, the large shift in teleworking has reduced the total demand for commuting trips, including trips by public transport. On the supply side, social distancing has affected the real capacity of public transport vehicles, and operators have faced extra costs trying to keep public transport use as safe as possible.

In many countries the total number of transport trips went down due to telework, but on top of that the market share of public transport dropped drastically1 . In this paper, we study the effect of teleworking and the high perceived infection risk in public transport vehicles on the socially optimal pricing and frequency of public transport trips. We look at a second-best equilibrium when road congestion cannot be priced. We first present a simple graphical analysis, using linear cost and demand functions that allows us to develop some intuition for the effect of the virus on the socially optimal fare when frequency cannot be changed in the short-run (and assuming a homogeneous population). We then study more formally the role of teleworking, the perceived infection risk and the restricted capacity of public transport vehicles on fares and frequencies.

The pandemic implies an objective risk of transmitting the infection to others on public transport vehicles. Equally important is that it also increases the ‘perceived’ infection risk by passengers. A comprehensive model to study the effect of the Covid-crisis on fares and frequency would require information on both risk concepts. Unfortunately, there is insufficient information on the precise relation between public transport use and the objective infection risk (see Galmiche et al., 2021 and the literature survey below). One of the problems in isolating the objective infection transmission in public transport vehicles is to distinguish it from the infection transmission at other places prior to entering the vehicles. In this paper we therefore base our policies on the perceived risk by potential passengers. This is what affects their behavior and determines their ultimate demand for public transport.

The model we use is a traditional model of second-best public transport decision-making (early versions of such models are, among many others, Mohring (1972), Jansson (1983), and Frankena, 1981, Frankena, 1983) that captures the interaction with the competing car mode and allows to introduce telework and infection risk in a straightforward way. Moreover, the model allows for different types of passengers that may differ in their vulnerability to a Covid-19 infection (for example, younger versus older passengers). Our paper focuses on second-best socially optimal policies and only touches sporadically on political support towards implementation. We numerically illustrate the model using data for the city of Brussels.

Our findings include the following. First, an increase in telework reduces the demand for both transport modes. If the public transport firm could flexibly adapt its policies, the efficient second-best response would be to reduce both the fare and the frequency, mainly because there is less congestion on the road and less crowding on public transport vehicles. This finding has important implications also for the long run, because it is quite likely that some telework will remain even when the pandemic is over (Hensher, Beck and Wei 2021). Second, the increase in the external crowding cost due to the (perceived) infection risk reduces public transport demand and increases car use. We show that it plausibly increases the optimal fare; the effect on optimal frequency depends on the relative importance of higher frequency on demand. Noting that telework and the infection risk have opposite effects on the fare implies that in practice it may be optimal for the firm not to change the fare at all. Third, assuming that car use is underpriced, holding the fare and frequency constant at their pre-Covid second-best optimal values implies that more telework reduces the public transport deficit. The effects of a higher perception risk on the deficit are ambiguous in general. However, if car use is strongly underpriced, a higher risk perception reduces the deficit. Fourth, extending the model to allow for passengers with different vulnerability towards Covid-19, we show that allowing fare and frequency differentiation implies that lower occupancy rates per vehicle for the vulnerable are socially optimal. They will face higher fares only if their risk perception is sufficiently higher than that of the non-vulnerable and car use is not too much underpriced2 . Fifth, more restrictive social distancing on wagons for the vulnerable raises the fare for vulnerable passengers, and it implies that more wagons are reserved for the vulnerable.

The application to Brussels yields additional insights. The model considers the peak period and allows commuters to choose between car and public transport use; however, students and school children can only use public transport. In the baseline, a very low fare and a high frequency are justified to address un-priced car congestion. Whenever telework reduces the demand for trips by at least 25% and the perceived crowding externality becomes higher due to Covid19, we find that from a welfare perspective the best option is to decrease the frequency. We do find that this outcome may imply that most work trips switch to car use, an outcome that may be difficult to accept by policymakers. However, it performs much better from a welfare viewpoint than continuing to offer high frequency and keeping a fair share of public transport trips to work. As telework may continue to be important in the future, even after the Covid pandemic, car congestion in the peak period will also be less important and one may opt for an equilibrium with a much smaller frequency of public transport. Lastly, the numerical exercise suggests that reserving part of the available capacity for vulnerable users and stimuli for walking and biking to school may be useful policies to deal with the infection risk due to crowding.

The structure of the paper is the following. In Section 2 we start with a brief literature review. In the next section we present a graphical analysis showing what the main implications of the virus are for the socially optimal fare, holding frequencies fixed. We then develop a formal model in Section 4 to study the effect of telework and the infection risk on second-best optimal fare and frequency, given underpriced car use. In Section 5 the model is extended to allow for passengers that differ in vulnerability with respect to the virus and we ask whether and under what conditions discriminatory policies are warranted. A numerical model, calibrated using data for the city of Brussels, is used in Section 6 to illustrate the effects of a number of Covid19-policies. A final section concludes.

Literature review

This paper introduces telework, the perceived infection risk of public transport passengers, and the capacity restrictions implied by Covid19-related policy intervention into a standard model of public transport decision-making. We therefore need information on the implications of the pandemic for public transport demand and supply, and we borrow ideas from the literature discussing optimal fare and frequency choice by public transport firms.

An ideal description of the implications of the pandemic for public transport demand and operating costs would have at least three components. It requires a model of the mechanism by which public transport use contributes to additional infections, and it requires a behavioral model that explains how passengers’ risk perception (as opposed to the ‘objective’ risk) is determined. Moreover, a model is needed that describes how public transport users react to specific policy interventions.

First, although there is continuous progress on the relative importance of infection mechanisms in the public health literature (see Chu et al., 2020, Leung et al., 2020, CDC, 2021, and the literature cited in the latter survey paper), we still have only a vague idea on the role of public transport in the infection transmission. Several studies did focus on the link between public transport and the spreading of infections (see, for example, Cooley et al., 2011, Goscé and Johansson, 2017), but the specific evidence for Covid19 is scarce. One of the main problems is to know who is infected when they enter the public transport system, and where transmission exactly takes place: inside the system (stations, vehicles) or at the origin or destination. The few empirical studies on the contribution of public transport to Covid19-infections offer mixed evidence. Some case studies suggest a positive contribution. These include studies on infections on trains and buses in China (Hu et al., 2020, Shen et al., 2020) and in the New York metro (Harris 2020). More recently, however, Galmiche et al. (2021) find in a nation-wide online case-control study on the French population that it is difficult to attribute infections specifically to the use of public transport.

Second, how public transport users react to changes in small and unknown probabilities to be infected with Covid 19 is also largely unknown. Small objective probabilities may be overestimated and the attitude to the “public good” or the follow-up of protective measures also varies widely among population groups. Lastly, little information is available on the reaction of public transport users to government information and operator recommendations to wear masks and to adhere to the physical distancing rules (see Chu et al., 2020, Dzisi and Dei, 2020).

There is overwhelming evidence that demand and market share has declined drastically for public transport systems around the world. Evidence is available from the Google Mobility Data and from private mobility firms such as Moovit. Moreover, recent papers document the decline in detail for specific countries (see Aloi et al., 2020, Jenelius and Cebecauer, 2020, Arrelana et al., 2020). The decline was partly due to government policy in response to the pandemic. For example, many governments strongly encouraged telework, required wearing masks on public transport vehicles and to observe social distance, both when boarding and in the vehicles. But consistent with the observation above, partly the demand reduction was also due to the perception of potential passengers that public transport implied a higher risk of infection than the use of other modes. This happened despite the safety measures taken by many governments on the supply side (social distancing rules, etc.) that reduced the capacity available and lowered the infection risk.

The main challenges of Covid-19 for public transport from an economic point of view are easily summarized (also see Tirachini and Cats 2020). First, what is the effect of teleworking on the demand side of the public transport market? Second, how large should the capacity of public transport systems be to allow passengers to make trips under “perceived” safe conditions. These include the role of physical distancing and of mouth masks, disinfection, contact tracing and enforcement. Third, how to regulate access to public transport: do we count on fare policies or on permit or reservation systems? Lastly, how to manage the short-run financial implications of the crisis for the public transport sector, and what are the implications for the longer term. In this paper, we explicitly deal with several of these issues.

Our paper builds upon the extensive literature on optimal pricing and supply characteristics of public transport firms, initiated by Mohring (1972). He studied socially optimal fare and frequency policies, emphasizing the existence of economies of density in scheduled public transport services. He showed that constant returns to scale in operational costs and randomly arriving passengers at bus stations imply that the marginal cost of frequency adjustment is compensated by the marginal benefit of waiting time reduction. Marginal cost pricing leads to a zero fare, and operating costs have to be fully recovered by subsidies.

The model has been extended to capture, among others, the role of crowding, second-best considerations due to underpriced car use, the implications for government subsidies, and how to optimize additional decision variables (including the number of vehicles, the size of vehicles, and the number of stops). Crowding implies that additional passengers impose an external cost on other users (see, among many others, Jansson, 1980, Jara-Díaz and Gschwender, 2003, Haywood and Koning, 2015) so that, in very crowded public transport systems, welfare-optimal fares may be positive and quite large. If car use is underpriced due to absence of congestion tolls, low public transport fares can be defended as a standard second-best policy, reducing the social costs of congestion (see, for example, Parry and Small, 2009, Anderson, 2014, De Borger and Proost, 2015). Unsurprisingly, studies that jointly optimize prices for road and public transport use in the presence of road congestion and crowding in public transport find quite high optimal public transport fares (see, for example, Basso and Silva, 2014, De Palma et al., 2015). Subsidies to public transport are socially costly because they have to be financed by distortionary taxes. The implications of optimal fares and frequencies for public transport subsidies have been scrutinized by Frankena, 1981, Frankena, 1983, De Borger and Wouters, 1998, and Parry and Small (2009), among others. The efficient public transport subsidy strongly depends on the cross-price elasticities between car and public transport use, whether or not car use is correctly priced, and the cost of public funds (Proost and Van Dender 2008). Moreover, distributional motives may lead to low fare policies that require large subsidies.

Given the huge uncertainties involved in the role of public transport on the transmission of infections and the response of passengers to the risk of infections and policies to reduce these risks, we introduce telework, the perceived infection risk and capacity restrictions due to policy interventions in the simplest possible way into a traditional public transport model. In addition, as there is no clear evidence on the contribution of public transport use to the pandemic, we did not include the social cost of additional infections to the social welfare function, see below.

Covid-19 and public transport: A simple graphical approach

In this section, we use a simple graphical approach to start the discussion of the effect of Covid-19 for optimal public transport policies. We start from a baseline pre-Covid second-best equilibrium where the relevant public transport policy variables (fare, frequency) have been optimized. Then we introduce an unexpected Covid-19 shock and analyze how this affects the public transport system and how optimal transport policy needs to be adapted. Of course, our simplified approach needs to be extended and completed by an analytical and numerical model in the next sections.

We use a comparative statics framework, and we focus on the peak period only. In the initial pre-Corona equilibrium, there is no congestion toll on road use, and the fare and the frequency in public transport are second-best optimal.

Consider a homogeneous corridor where all passengers want to move from an Origin to a Destination. Fig. 1 represents the demand for road transport and the demand for public transport in the peak period as a function of their generalized cost, holding the total number of trips constant. The horizontal distance measured from the left represents car trips, the distance measured from the right represents public transport trips. Both modes are assumed to be perfect substitutes, so that, in equilibrium, the generalized prices are equal. For car use the generalized cost consists of the money cost plus the time cost. The average user cost (ACroad) is increasing with car use and, since we use linear functions for simplicity, the marginal social cost (MSCroad) has twice the slope of the ACroad-function. For public transport, the average user cost (AC PT) is the sum of the (second-best optimal) fare, the access cost, the cost of waiting time and, lastly, the in-vehicle cost. The latter is linearly increasing in the number of passengers because of crowding discomfort. As the crowding discomfort is an externality, the slope of the MSC of PT (not shown on Fig. 1) is larger than the AC PT curve. In allocation A, we represent a second best equilibrium: because there is no road pricing, the public transport fare is set below its optimal level.

Fig. 1

Effect of Telework and Covid-19 on the transport market.

In the equilibrium after the Covid19-shock, we assume that a given share of the population is infected. The fear of being infected increases potential passengers’ aversion to crowding, leading to higher discomfort and to a decrease in the modal share of public transport. Moreover, the Covid-crisis has strongly increased telework, reducing the total number of peak-period trips.

Keeping the capacity, the frequency and the fare of public transport constant, the Covid-19 shock implies two effects in Fig. 1:

Telework reduces the total number of trips by the amount A’P’: this reduction is distributed over road users and public transport users. It will also reduce the market share of public transport when the slope of the average cost function of road use is higher than that of public transport use. The decline in road use implies less congestion.

The higher perceived infection risk for Covid-19 in public transport translates into a larger discomfort for users: the average cost of public transport AC PT rotates upwards. This implies that, at unchanged fares and frequencies, users will substitute part of their public transport trips by road trips, and road congestion will increase somewhat: many people prefer car use above a crowded public transport vehicle. We end up with a lower market share for public transport and a level of car use (OA’’) that is higher than when there would only be telework (OA’).

Of course, telework and the fear for a Covid infection will have second-order effects on the total number of trips not captured in Fig. 1. Telework will decrease the generalized cost of trips because road congestion and crowding in public transport will decrease. This leads to a small compensation of the downward demand shock. The perceived Covid infection risk will increase the cost of public transport use and, by modal substitution to road use, it will also increase the generalized cost of all trips. This may re-enforce the reduction of the total number of trips induced by telework.

In Fig. 1, we kept the frequency and the fare of public transport constant at the initial second-best equilibrium but, of course, both the optimal fare and the optimal frequency may need to be adapted, although a priori it is not clear in which direction3 . To study the effects of Covid-induced telework and the increased costs of crowding, a more formal analysis is warranted, to which we turn in the next section.

The effect of infection risk and telework on public transport policies: A basic model

In this section, we use a simplified version of the model discussed in De Borger and Proost (2015) to study the effect of more telework and the effect of a higher perceived risk by public transport passengers on optimal transport policies. The model describes behavior in the peak period. It is assumed that people can commute by car or by public transport. The road is congestible, and a toll may be in place for car use. Public transport is subject to crowding; when there is a perceived risk of infection, the perceived cost of crowding increases.

The model

To keep the analytics tractable without affecting the main insights, we make a number of strong assumptions. First, we assume that car and public transport use are perfect substitutes: peak-period transport users prefer the mode with the lowest generalized cost (including the time cost as well as the monetary cost and, in case of public transport, waiting time and crowding discomfort costs). This much simplifies the derivations and, although the technical details will differ, it does not affect the qualitative insights derived from the model (such as the signs of the various effects of telework and infection risk derived below). Second, as mentioned, we study the peak period. Adding an off-peak component would make the model more complete, but the additional complexity would not add relevant new insights4 . Moreover, as in many cities and regions peak-period frequency of public transport service is quite high, we assume that timetables are not guiding users’ decisions5 .

Total demand for peak-period trips is assumed to be linear. Denoting the generalized price by P, the inverse demand function relates the generalized price to the total number of trips. We denote public transport trips by Y > 0, car trips are denoted X > 0. The inverse demand function is given as.

We assume that parameters a and b are positive; moreover, P(X + Y) > 0 for X + Y > 0.

The generalized user costs of public transport and car use are formulated as, respectively:

The generalized cost of public transport consists of the fare (denoted p), the cost of waiting time (f is the frequency, is half the value of waiting time), and the cost of crowding ( is the crowding parameter). The generalized cost of car use is a fixed monetary user cost inclusive of a possible toll plus the congestion cost; the slope of the congestion function is v.

Given our assumptions a user equilibrium requires:

Using the demand and generalized costs defined before, and solving the equilibrium conditions gives the reduced-form demand functions, relating demand for public transport and for car use to the public transport policy variables (p, f) and to exogenous parameters. We find:

Where

We then have the following fare and frequency effects on the demand for public transport and car use:

A higher fare reduces public transport demand and raises car use. Higher frequency raises public transport demand and reduces car use.

We determine the fare and frequency that maximizes the following social welfare function:

The first three terms together define net consumer surplus (gross consumer surplus minus de generalized costs) of public and private transport trips. The term between square brackets captures the net revenues of the public transport operator. It equals the fare revenues minus the operating costs of providing the service. The operating costs are specified in the simplest possible manner: the cost is linear in its two main determinants, the frequency and the number of passengers. Each extra bus trip offered by the public transport firm implies an extra frequency cost c, assumed to be constant. Moreover, there is a small marginal cost c of carrying extra passengers. The final term in the social welfare function consists of the road toll revenues (we normalize other monetary costs of car use to zero – so that can be directly interpreted as the toll -- without affecting the qualitative results).

Using equality between generalized price and generalized cost in equilibrium, the first-order conditions can be written as.

Rearranging yields the second-best socially optimal fare and frequency, denoted p* and f*, respectively:

The socially optimal fare equals the marginal social cost corrected for suboptimal pricing of car use. The frequency rule is the standard square root expression familiar from the public transport literature surveyed in the introduction. Substituting the reduced-form demand functions (see (5)-(6) above) into (10), one obtains two equations in p* and f* that depend on parameters only. Closed-form solutions are typically not possible, but the equations can be numerically solved for the optimal fare and frequency.

Note that this second-best optimum, assuming underpriced car use, necessarily produces a public transport deficit. To see this, note that operating profit for the public transport firm is:

Using the first-order conditions (9) in this expression, we have:

Working out we find:

If car use is priced below marginal external cost this expression is necessarily negative, so that a deficit occurs:

The effect of telework and infection risk on fare and frequency

We are interested in the effect of exogenous changes in telework and a higher perceived infection risk on the policy variables of the public transport sector. We consider both in turn.

First, we view telework as an exogenous policy of the government in response to the pandemic; the explicit purpose is to reduce social contacts at the workplace and on the journey-to-work, mainly contacts in public transport vehicles and stations. Within the framework of our model, exogenously imposed telework boils down to an exogenous downward shift in demand. More precisely, we interpret more telework as a downward shift in the intercept, the parameter a, of the inverse demand function. Importantly, as the productivity effects of telework are highly uncertain (see Bloom, 2020, Behrens et al., 2021 for details) we do not capture them in our welfare function.

Second, Covid19 has also given rise to a higher perceived infection risk for public transport users. The most natural way to introduce this in the model is via an increase in the crowding parameter: apart from the standard crowding cost (more users reduce the available space per passenger and therefore the utility associated with the trip) the pandemic implies an additional external cost component of crowding: having more passengers on bus or train increases the perceived risk of being infected.6 Of course, this perceived risk depends on a variety of operating practices introduced in public transport, including the requirement to wear a mouth-mask and to keep a minimum distance from other passengers, extra ventilation in vehicles, etc. We return to the role of protective measures below.

To solve for the effect of more telework (captured by a reduction in a) and increasing perceived infection risk (captured by an increase in ) on the optimal fare and frequency, first note that the effects of the relevant parameters on the demand for public transport and car use are obtained by differentiating the reduced-form demand functions given in (5) above. We find:

To arrive at the final result in (13) we used the equilibrium conditions (4). Unsurprisingly, an exogenous increase in teleworking (a decline in a) reduces the demand for both modes of transport. A higher Covid risk reduces public transport demand, while it increases car use.

At this point it is useful to note that, as mentioned above, captures both the standard crowding cost as well as the perceived infection risk. A plausible hypothesis is that the latter increases in the infection rate in the population, as communicated by the authorities, and that it can be reduced by imposing protective measures. A convenient specification could be the following7 :

The perceived cost of crowding increases with the infection rate i in the population and it is negatively affected by the impact of a set of protective measures summarized via the function g(.). Protective measures could include wearing mouth masks m, ventilation on vehicles v and restricted seating r. The parameter is the ‘standard’ crowding parameter, absent the pandemic. Unfortunately, the literature has not yet produced reliable estimates of the role of the infection risk and of the protective measures taken on passengers’ perceived crowding cost. We will therefore focus on the effect of increases in on demand and on optimal fares and frequencies, but keeping in mind that these increases are determined by the severity of the pandemic and the protective policies of the governemnt and the public transport firm8 .

Totally differentiating the first-order conditions of the welfare optimization problem and using the demand effects just derived, we show in Appendix that, in general, the effects of telework and infection risk on the optimal fare and frequency are all ambiguous. This is not surprising. For example, telework reduces the demand for public transport as well as car use. The first-order condition for the optimal fare suggests that less public transport reduces the optimal fare, while less car traffic (and hence less congestion and a smaller distortion on the car market) raises the optimal fare. Moreover, telework affects frequency which has itself an effect on the optimal fare. The lower demand for public transport suggests reducing frequency, but doing so may raise infection risk which implies higher optimal frequency.

Although the effects of telework and infection risk are ambiguous in general, in Appendix A we show that the first-order effects (more precisely, the effect on the fare at given frequency, and the effect on frequency at given fare) are as follows9 :

Since telework is interpreted as a decline in ‘a’, these first-order effects suggest that telework reduces both the optimal fare and the optimal frequency. Telework reduces both public transport demand (and this is an argument to lower the fare and the frequency) and the demand for car trips (this reduces congestion and raises the optimal public transport fare, a standard second-best argument). Clearly, the first effect on the fare dominates the second.

A higher infection risk raises the optimal fare, and it can raise or reduce the optimal frequency. The reason is that offering more frequency offers safer travel, holding the number of passengers fixed; however, it also increases demand, which reduces safety. As shown in Appendix A, a higher risk of infection will increase the optimal frequency if the infection risk only mildly reduces demand. If demand strongly declines, a higher infection risk reduces the optimal frequency.

Proposition 1. The effects of telework and the perceived infection risk in the peak period

Telework reduces the demand for both transport modes. A higher perceived infection risk reduces public transport demand and increases car use.

The effect of telework on the socially optimal fare and frequency is ambiguous in general. Plausibly, however, it reduces both the fare and the frequency.

A higher perceived infection risk plausibly increases the optimal fare. If the perceived risk of being infected with Covid-19 strongly reduces demand then the optimal frequency declines.

These findings have a few simple practical policy implications. First, more telework reduces the optimal fare, the increase in perceived crowding costs increases the optimal fare. As these two effects have opposite signs the joint effect on fares may be very small, and the associated welfare gain may be limited. Given the cost of adapting fares, it may be optimal not to change fares in response to the Covid pandemic at all. Second, under plausible conditions, is does make sense to offer lower frequency in response to the joint increase in telework and infection risk. For the post-pandemic period (when it is hoped that the infection risk disappears) it will be crucial to know to what extent telework is there to stay (see Hensher et al., 2021, Behrens et al., 2021 on this issue). Unless commuting returns to its pre-Covid19 level, reducing frequency might well be a desirable option. This conclusion is corroborated by the empirical observation that telework is easier to maintain in the employment zones of large urban areas, where public transport has typically a fairly large market share.

Implications of telework and infection risk for the public transport deficit

Policymakers are concerned about the implications of telework and the infection risk on the public transport deficit. We can make the following observations.

First, if the firm does not adapt its fare-frequency policy in response to the changes in telework and perceived infection risk, then the implications for the deficit depend on whether or not the fare p exceeds the marginal cost of an extra passenger. This follows directly from the definition of profit,. If the fare exceeds marginal cost, profit will decline when telework increases, because demand declines; the same holds for an increase in infection risk. If the fare falls short of the marginal cost, then telework and infection risk raise profit (reduce the deficit).

Second, the previous general statement obviously also applies at the pre-Covid second-best optimum. Suppose the pre-Covid fare and frequency policy was second-best optimal and assume the public transport firm holds the fare and the frequency constant when telework increases. For example, it may be politically as well as commercially unattractive to adapt fare and frequency during the Covid-crisis. In that case, more telework reduces the deficit. To see this, differentiating (11) with respect to a gives.

The term between brackets is negative when car use is underpriced, and we showed before that. As a consequence, we have More telework (a reduction in a) raises profit or, more precisely, it reduces the deficit. Similarly, the effect of a higher perceived infection risk can be written as.

The sign of this expression is ambiguous. If the toll is small then – using earlier results (13) – we see that the effect of Covid-19 is to reduce the deficit.

Third, suppose that the firm has implemented the socially optimal fare-frequency policy prior to changes in telework and infection risk, but now assume that the firm optimally adapts the frequency downward while keeping the fare fixed. In Appendix A we then show that, assuming the frequency decline itself does not much affect car use, more telework again unambiguously reduces the deficit. To properly interpret this finding, note that we assume throughout that all costs are real opportunity costs. In the short term and during the pandemic, the opportunity cost of labor and equipment is clearly lower than their book values; reducing the deficit by laying off workers and selling equipment is then probably not an ideal policy. However, in the longer term, lower demand for public transport due to telework (and possibly still some fear of using public transport) implies that a smaller frequency is needed. As after the pandemic nominal costs are probably closer to the real opportunity costs, the message is then not to stick to high frequencies and a large deficit.

Proposition 2. The effects of telework and the perceived infection risk on the peak-period public transport deficit.

Holding the fare and frequency constant at their pre-Covid optimal values, more telework reduces the public transport deficit. The effects of a higher perception risk is ambiguous. If car use is strongly underpriced, higher risk perception reduces the deficit.

If the firm optimally adapts frequency but holds the fare fixed, more telework again plausibly reduces the deficit.

Distinguishing vulnerable and less vulnerable passengers

It is straightforward to extend the model for two types of commuters that may differ in vulnerability with respect to Covid-19. Let us assume the morning commute consists of workers and students. Assume students are less vulnerable to the illness (call them the non-vulnerable, indexed N) than workers (the vulnerable, indexed V). We first assume that all passengers use the same public transport vehicles. Next we allow different vehicles to be used for the two types of passengers.

Extending the model: Two types of passengers

Suppose students have no access to a car; they either go by public transport, or they walk or bike to school. The workers (the vulnerable) have the option to go by car or public transport. The two types of public transport users have different generalized costs of public transport use:

Differentiating the total generalized cost of all public transport trips,, with respect to we find that the marginal external cost of crowding due to an extra passenger is independent of the passenger type; in both cases, it equals.

As before, the cost of car use depends on the number of vulnerable users, viz. workers driving to work by car:

The inverse demand for public transport trips by the non-vulnerable students is given by:

As before, we assume car and public transport use are perfect substitutes for the workers, so that inverse demand reads:

Note that -- to save on notation but without implications for the main results -- we assume the same intercept but a different slope for the inverse demand functions of the two types.

Equilibrium now requires the following three conditions:

Solving this system yields the reduced-form demand functions; they are complex functions of the parameters. However, it is easy to show that:

Here. Note that a higher fare unambiguously reduces the demand for public transport by the non-vulnerable; it also unambiguously reduce total public transport demand. However, the sign of the effects on public transport and road use by the vulnerable are ambiguous in general. Provided we have the expected signs: public transport demand declines and road use increases.

To find the optimal fare and frequency we solve.

The first-order condition for the fare p is:

Rearranging we find, using earlier expressions:

The fare equals the marginal production cost plus the marginal external cost of an extra passenger, corrected for the inefficient pricing of road use. Assuming that car use is underpriced and that a higher public transport fare increases road use the optimal fare is less than the marginal social cost of an extra passenger.

The optimal frequency is.

The frequency rule takes account of infection risk for both passenger types.

Allowing discrimination between vulnerable and non-vulnerable users

To study the effect of differences in vulnerability on fares and frequency, we consider the possibility of discriminating between types. Note that fare discrimination between different types of passengers based on differences in vulnerability does not make much sense, except when different vehicles can be reserved for the different types. The intuitive reason is that, as shown above, the marginal external crowding cost each type imposes on others is equal for both types. Consequently, in first-best fares will be equal for all passengers. Of course, in second-best with underpriced road use a small fare difference between the vulnerable and non-vulnerable will result due to the different response of car users to the two fares, but not due to differences in vulnerability itself.

Fare differentiation does make sense if one can separate the two types by offering each type different vehicles. The idea of separating vulnerable and non-vulnerable people is advocated as a general principle in the Covid-literature by several authors (see, for example, Acemoglu et al., 2020, Gollier, 2020). Fare differentiation may be more controversial in some countries, although crude applications of the overall principle are available10 .

Suppose that the public transport firm operates separate busses for the two types, and assume passenger type can easily be established. Denote the frequencies of the buses reserved for types V, N by. Since passengers are now confronted (making abstraction of interaction at stations…) only with their own types, generalized costs are:

Equilibrium requires:

We immediately find the price and frequency effects of demand:

Here. All price and frequency effects are as expected.

We determine optimal policies by solving.

First-order conditions for fares and frequencies lead to the following rules:

Whether the vulnerable pay more than the non-vulnerable is a priori unclear. On the one hand, at given demand and frequency their marginal crowding cost is higher, raising the optimal fare. On the other hand, one easily shows that the optimal occupancy rate is declining in the infection risk:

Moreover, competition with car use implies a lower public transport fare if car use is underpriced. Therefore, the vulnerable will face higher fares than the non-vulnerable only if the difference in infection risk is sufficiently large11 .

Proposition 3. Price and frequency discrimination in the peak period.

Fare differences between different types of passengers based on differences in vulnerability are unwarranted unless for each type separate vehicles can be used.

Allowing fare and frequency differentiation implies lower occupancy rates per vehicle for the vulnerable. They will face higher fares only if their risk perception is sufficiently higher than that of the non-vulnerable and car use is not too much underpriced.

Covid infection risk and the optimal composition of trains

Consider a similar but slightly different aspect of the implications of Covid-19 for public transport, which is potentially important mainly for rail and metro service. Specifically, in this section we focus on a given peak period train or metro trip and consider the problem of reserving train wagons to different passenger types12 . The idea is again to separate vulnerable from non-vulnerable passengers by offering different wagons at different prices for the same trip. Vulnerable passengers are then protected in two ways. By using different wagons, they are protected from the interaction with non-vulnerable passengers. Moreover, the wagons reserved for vulnerable passengers restrict the number of passengers per wagon.

There are two ways to make sure passengers use separate wagons according to their type. First, if the distinction between types is based on observable characteristics (for example, if the vulnerable are defined as passengers older than 65, or the non-vulnerable are defined as students) the separation can be implemented at low cost by a simple passport control. Second, if the distinction is based on unobservable characteristics or on passengers’ own perception, appropriate price discrimination can induce a separating equilibrium whereby neither type has an incentive to use a vehicle reserved for the other type.

In an equilibrium where passengers use wagons according to their type, the generalized costs for the vulnerable and non-vulnerable are given by, respectively:

Here the y’s are the number of passengers of a given type on a given peak-period trip. The Q’s are the number of wagons allocated to the two types.

We assume the decision-maker decides on the number of wagons to reserve for the vulnerable and non-vulnerable users and the fares to be charged for admission to each wagon type. It faces two constraints. In the peak period there is a limited number of wagons available, determined by the overall capacity of available rolling stock. Moreover, the wagons reserved for the vulnerable have limited capacity due to stricter social distancing rules than the vehicles reserved for the non-vulnerable. The problem is to:

The first constraint is a capacity restriction on the available wagons. The second restricts the occupation rate on the wagons for the vulnerable due to social distancing. The r’s are the costs the firm incurs for operating wagons of each type.

First consider the solution when separating the two types can be based on observable indicators. We find, under the reasonable assuming that both the overall capacity constraint and the occupancy restriction for the wagons reserved for the vulnerable are binding at the optimum:

The optimal fare for use of the wagon reserved for the non-vulnerable is the marginal social cost, consisting of the production cost c plus the marginal external inconvenience cost. The fare for the vulnerable has an extra component because of the occupation restriction due to social distancing. Whether the fare is higher or lower for the vulnerable or the non-vulnerable is a priori unclear: vulnerable passengers can be assumed to perceive higher inconvenience at the margin () and they are charged for the capacity constraint due to social distancing, but the occupation rate of their wagons is lower.

Looking at the optimal allocation of the available peak period wagons, we note that the overall capacity constraint restricts the number of both wagon types. More importantly, the restriction on the occupation rate of the wagons reserved for the vulnerable raises the number of wagons allocated to the vulnerable passengers. Whether more than half of the wagons is reserved for the vulnerable depends on the number of vulnerable versus non-vulnerable passengers, and on the severity of the social distancing rules implemented in the wagons reserved for the vulnerable.

Of course, price discrimination is not necessary. The solution to the following problem gives an optimal equilibrium without price discrimination:

We find

where.

The optimal fare is the marginal production cost plus a weighted sum of the marginal external inconvenience costs, including the correction for the constraint on the wagon capacity for the vulnerable travelers. Assuming that T > 0, the larger is T the larger will be the fraction of wagons reserved for the vulnerable.

Finally, if the separation of vulnerable from non-vulnerable passengers is not based on observable characteristics but passengers self-select the wagons to enter, fares and capacity allocation should be such that neither type has an incentive to use a wagon reserved for the other type. This requires.

If the above-described solution satisfies these constraints we have a separating equilibrium. Note that, since the number of passengers in the wagons for the non-vulnerable will be higher than for the vulnerable, satisfying the constraints implies that the vulnerable will face a higher fare in a separating equilibrium. We will necessarily have.

Proposition 4. Optimal allocation of peak-period capacity for the vulnerable versus non-vulnerable.

More restrictive social distancing on wagons for the vulnerable raises the fare for vulnerable passengers, and it implies that more wagons are reserved for the vulnerable.

A separating equilibrium implies that fares for trips in the wagons intended for the vulnerable are sufficiently higher than fares in the wagons for the non-vulnerable to prevent non-vulnerable passengers to use wagons reserved for the vulnerable.

Application to public transport in Brussels

In this section, we use data for Brussels, a city of 1 million inhabitants, to illustrate some of the implications of Covid-19 for public transport. As explained in more detail below, the numerical model is not a literal translation of the theory presented above. It is calibrated to reflect the local transport situation in Brussels prior to the Corona-crisis, where public transport consisted of a mixture of 75% metro trips and 25% bus trips13 . We impose other simplifying assumptions as well. We (realistically) restrict frequencies to integer values. Moreover, the application takes into account the institutional setting; this implies, among others, that most peak-period public transport passengers have subscriptions that are financed by employers and/or the government.

Description of the case study

We distinguish between peak period trips to school or to the university (schoolchildren, students), and trips to work or for other purposes. For school trips, we assume there is no modal choice between public transport and car use; children can only forego the trip, or they can walk or bike to their destination. For the other (mainly work) trips we do allow the choice between road use and public transport use; the use of other modes is not explicitly represented. Car and public transport are considered to be perfect substitutes: their user costs are equal in equilibrium.

The public transport data come from the local public transport firm. Information on car use was provided by the Brussels Regional government. We represent the relative trip levels in the peak for a typical day in the week in pre-Corona times in Table 1 . We normalize the total number of trips at 100 and use rounded figures; our main interest is in showing the broad effects of telework and Covid-19 on fares, frequencies etc. Based on the information provided, about 40% of public transport trips in the peak period are school trips. For the journey to work to Brussels, slightly less than 30% of all trips are by public transport, the remainder are car trips. The total trip volume for non-school trips (car and public transport) amounts to 210.

Table 1

Trip distribution in the pre-Corona equilibrium.

Relative number of peak trips/day	Substitutable trips (road, public transport): non-school (mainly work) trips		Non-substitutabletrips: school trips	Total public transport
	Road	Public transport	Public transport
Pre-Covid	150	60	40	100

We will consider school children as less vulnerable than workers commuting to work. In our analysis we will therefore let the aversion to crowding (due to the perceived risk of infection because of the Corona-pandemic) increase only for trips to work, and not for school trips, see below.

Finally, we assume homogeneous adaptation to telework. As it is exogenously imposed for most sectors by the government, and given the radial transport network of Brussels, we assume that telework does not affect the relative importance of the different origins and destinations.

Model calibration

The calibration of the demand and generalized cost functions is explained in detail in Appendix B. We calibrate the demand functions for substitutable work trips as linear functions with price elasticity −0.4. We further assumed a value of time for substitutable work trips of 10€/hour and 5€/hour for non-substitutable school trips. As there is no toll in the baseline, the external costs are not charged for road use, so that the generalized price is the user cost that consists of money costs (fuel, wear and tear) and time costs. For public transport, the average cost consists of the fare, the time costs associated with walking and waiting at the metro station or bus stop, and the in-vehicle time that includes crowding discomfort.

We interpret the baseline equilibrium as a second-best equilibrium. The second-best fare (see the theoretical section) implies charging for the external cost of crowding, but it also implies a subsidy to correct for the absence of a road toll on car use. This gives (see Appendix B) a second-best fare in the baseline very close to zero. Note that this is entirely consistent with the observed fares. Most public transport users in Brussels have a monthly or annual pass that in fact makes the marginal price of an extra trip zero. Moreover, the cost of the pass is in the huge majority of cases paid by the government (for school trips) or by the employer. As a reasonable approximation, we therefore set the second-best baseline fare equal to zero. The implied subsidy to public transport is a second-best subsidy shifting users from road to public transport. For work trips, the subsidy is often justified to correct for the absence of road pricing. For school trips, the zero fare in the baseline is often motivated by income distribution concerns. Note that the zero fare remains second-best because it leads to a combined (road and public transport) equilibrium trip cost that is too low compared to the first-best optimum.

Based on the information provided in Appendix B, we give in Table 2 the composition of the generalized costs of car and public transport use in the baseline equilibrium. Fig. 2 represents the baseline (pre-Covid) equilibrium graphically.

Table 2

Composition of generalized costs in the baseline equilibrium.

Car use	AverageCost (€/trip)	Public transport (non-school)	AverageCost(€/trip)	Public transport(school)	AverageCost(€/trip)
Vehicle cost	0.66	Walk	1.14	Walk	0.57
Min time cost	1.83	Wait time	0.5	Wait time	0.25
Congestion	1.5	In-vehicle	1	In-vehicle	0.5
		Crowding	1.36	Crowding	0.68
Toll	0	Fare	0	Fare	0
Total generalized cost	4	Total generalized cost	4	Total generalized cost	2

Fig. 2

The baseline equilibrium for Brussels.

Model simulations and results

We test three different policies. We first study the effects of Covid19 (changes in telework and perceived infection risk) on ridership of public transport and its implications for the optimal frequency. Next we look into the potential effect of reserving capacity for the more vulnerable. Finally, we consider the effects of policies that promote the use of other transport modes for trips to school.

Note that we do not investigate variations in the public transport fare. The reason is twofold. First, as argued in the theoretical sections, telework and a high perceived infection risk have opposite effects on the optimal fare, with the joint effect likely to be very small. Second, fare adaptations are politically difficult in the short run. We therefore prefer to simulate some of the policies that are potentially easier to implement.

Studying the effects on ridership, frequency and the deficit

We simulate results for 3 cases of lower demand for work trips (telework) and for trips to school: demand is reduced by 25%, 50% and 75%, respectively. In each case, the demand function for work trips is recalibrated so as to reproduce a decrease of trips by the relevant percentage, keeping the same slope at baseline fare and frequency conditions. We also consider three levels for the crowding aversion factor, capturing different levels of the perceived infection risk. The index in the baseline is set at 100, and increases by 50% and by 200% are implemented in the model.

Results are summarized in Table 3 . Two preliminary remarks are useful to properly interpret the figures in the table. First, note that we restrict, unlike in the theory, frequency to integer figures. We therefore report simulation results for different values of frequency, identifying the optimal integer value by evaluating the welfare index. Second, although the meaning of the other columns is self-explanatory, this may not be the case for the deficit reported in the final column. We calculated the deficit taking into account the institutional setting in which public transport operates. The costs include the cost of offering frequency (c in the notation of our theoretical model) and the variable cost of carrying passengers (c). The revenues are computed in function of the current institutional system. Employers pay for a public transport subscription of their employees, so that employees travel for free. Employers pay the cost of frequency and the variable passenger cost in proportion to their use (which was 60% in our baseline). For school trips, children and students benefit from a subscription at a very low price. For practical purposes, it is fair to say that the government pays for all school trips. To arrive at the deficit figures reported in the final column of Table 3, we first calculated the deficit in absolute terms. For example, in the baseline, the deficit amounted to 40% of variable costs. We then normalized the absolute baseline deficit to equal 100.

Table 3

The effect of increases in telework and in the perceived infection risk: simulation results for Brussels (the bold columns represent simulation results, other columns are assumptions).

n	Relative demand(telework)	Relative Crowdingaversionparameter	PT-tripswork	PT-tripsschool	FarePT work	Freq-uency	Gen.PriceCar trip	Welfare	Crowdingindex	DeficitIn absoluteTerms
1(baseline)	1	100	60	40	0	10	4	891	100	100
2	1	150	36	40	0	10	4,2	860	76	129
3	1	150	50	40	0	12	4,1	850	74	131
4	1	150	13	40	0	7	4,4	863	75	131
5	0,75	100	13	30	0	5	4,0	507	72	111
6	0,75	100	0	30	0	3	4,1	517	100	75
7	0,75	150	28	30	0	10	3,8	472	58	125
8	0,75	150	0	30	0	3	4,1	517	100	75
9	0,75	200	0	30	0	3	4,1	517	100	75
10	0,5	100	6	20	0	6	3,6	222	44	109
11	0,5	100	0	20	0	3	3,6	250	67	73
12	0,5	150	20	20	0	10	3,5	178	40	147
13	0,5	150	0	20	0	2	3,6	252	100	50
14	0,5	200	0	20	0	3	3,6	252	100	50
15	0,25	150	12	10	0	10	3,1	−24	22	107
16	0,25	150	0	10	0	2	3,2	78	50	48

The first line in Table 3 represents the baseline. In the next three lines we keep the same total demand for work and school trips and investigate the effect of a higher perceived infection risk on the optimal frequency. We specifically increase the discomfort factor for work trips by 50%. The result in line 2 suggests that, keeping frequency at its baseline value of 10 services per hour leads, as expected, to a strong decline in the use of public transport for work trips (36 instead of 60). Crowding declines, whereas congestion on the road increases: the generalized cost for car use increases from 4 to 4,2 €/trip. The deficit increases because of the decrease in ridership by employees. To understand this, remember that the fare for the employees is zero because the employers pay for the subscription; this covers the variable cost per passenger and the costs of frequency. We assume that the employers reduce their contribution to the public transport costs in proportion to the reduction in ridership. Of course, if it were the government (and not the employers) who pay for the subsidy and continues to pay for it, then the public transport operator would see a reduction of the deficit, because he saves part of the variable costs of passengers and of frequency (see Proposition 2).

One may be tempted to address the higher discomfort in public transport use and the increased road congestion by increasing the frequency. This is illustrated in line 3, where an increase of the frequency from 10 to 12 services per hour is able to attract more public transport users for the trip to work than in line 2, and to decrease the level of car congestion. However, augmenting frequency is welfare-reducing (welfare declines from 860 to 854); moreover, it raises the deficit. It turns out that, in line with the theory (see Proposition 1), the optimal response to a higher perceived infection risk is to reduce frequency to 7 services per hour. This is illustrated in line 4; welfare increases to 863. Clearly, the effect of the higher perceived infection risk is to strongly reduce demand, allowing a lower frequency to be offered.

Lines 5 to 14 in Table 3 consider the effect of demand reductions due to the government’s effort (or even requirement) to increase telework. First we discuss lines 10 to 14, where we look at the implications of a 50% demand reduction relative to the baseline. Interestingly, we find (see the comparison of lines 10 and 11) that, if there were no increase in crowding discomfort, it is optimal not to have any work trips by public transport at all. Line 10 (a frequency of 6 per hour and an index of 6 trips to work) generates lower welfare than the optimum given in line 11 (a frequency of 3 per hour but no more public transport work trips). The main reason is that telework reduces the demand for road use and leads to a decline in congestion; addressing congestion by encouraging public transport use for work trips is then costly in terms of the extra frequency offered; it is not efficient. The lower frequency in line 11 decreases the deficit (73 instead of 109). All this under the assumption one can save the costs of frequency, which may be economically and politically difficult in the short term as the opportunity cost of labor and equipment is low in the short term.

In lines 12 to 14 we add a higher Covid-inspired discomfort for public transport trips to work to the 50% demand reduction due to telework. Comparing lines 12 and 13 we see that workers avoid public transport (their demand drops to zero), and we find the optimal frequency dropping to 2 (line 13); this leads to a higher welfare level than keeping the baseline frequency of 10 and keeping some commuters to use public transport (line 12). Offering a frequency of 2 per hour is optimal; it serves to accommodate the demand for school trips. For an even higher Covid-inspired discomfort, one has similar results (line 14).

It is argued that once the Covid pandemic is over, telework will be more common than in pre-Covid times, but less pronounced than during the pandemic. To simulate this case, we look at a demand reduction of only 25%, see lines 5–9. If the perceived infection risk drops to its baseline value, there are two options for public transport. This is illustrated in lines 5 and 6. One option is to have an internal solution with some work trips going by public transport, see line 5. The frequency is kept relatively high at 5 per hour (but much below the baseline value of 10). The alternative is a corner solution where public transport is only used for trips to school (and an optimal frequency of 3). We find that the equilibrium with low frequency is welfare optimal. The main reason is again that road congestion has decreased so that the main motivation to have low fares and high frequencies disappears. Combining high levels of telework with higher perceived infection risk (lines 7 to 9) shows that, as before, it becomes optimal to have no more work trips by public transport and a low frequency at 3 vehicles per hour. All this holds, of course, assuming one can save the full costs of high frequencies.

The main (and politically) controversial message from these results is that, if more telework becomes the norm and the perceived infection risk does not return to pre-Covid times, from a welfare point of view it becomes difficult to justify substantial public transport use for commuting trips.

Reserving capacity for the more vulnerable

In Section 5.3, it was shown that reserving a larger part of the capacity for the vulnerable may be a useful policy. We illustrate this idea by showing the effects of such a reservation system in Table 4 . In practice, reserving capacity for the vulnerable should not be too difficult to organize. For example, it can be implemented by asking the users of the reserved spaces to show their subscriptions. Another option is to use a small entrance fee for the reserved spaces.

Table 4

Effects of reserving public transport capacity for the vulnerable (the bold columns represent simulation results, other columns are assumptions).

n°	Relativedemand	Frequency	Relative crowding aversion	ShareCapacityVulnerable	ShareCapacityNon-Vulnerable	ACCar andPT	Trips to work by car	Trips to workby PT
1	0,5	10	150%	Full sharing		3,46	96	20
2	0,5	10	150%	0,7	0,3	3,39	90	26
3	0,5	10	150%	0,8	0,2	3,37	88	28
4	0,5	10	150%	0,9	0,1	3,34	85	31

In line 1, we start from “full sharing” of capacity among the vulnerable and the non-vulnerable. We assume 50% telework and a relative crowding parameter of 50% more than the baseline, and we set the frequency at 10. This line corresponds to line 12 in Table 3. In this full sharing case, there are 20 trips to work (and 20 trips to school). As trips to work are more vulnerable to crowding, we then simulate what happens when one reserves more than 50% of the capacity of each train for the trips to work. This will have two beneficial effects: the vulnerable suffer less from crowding, and the average trip cost to work declines because more work trips will take public transport.

Results are in lines 2 to 4. Reserving progressively 70%, 80% and 90% of the capacity for work trips (lines 2,3 and 4), one attracts more commuters to public transport; this reduces the average trip costs to work for both modes. The smaller capacity available for the fixed number of trips to school leads (not shown in the table) to a small increase in average cost because crowding is less valued in school trips.

Promoting the use of other transport modes for the trips to school

The Covid-pandemic makes the vulnerable users substitute public transport trips by car trips. One of the problems is that public transport school trips take away an important part of the capacity of public transport, so that commuters face more crowding. Moreover, increasing the fares for trips to school is politically difficult. However, what can easily be done is to encourage the use of other modes (walk, (e) bike, (e) step) by subsidies and better facilities.

In Table 5 we illustrate the effects of a policy that manages to make 10 of the 20 pupils or students switch to other modes.

Table 5

Effects of exogenous changes in the modal choice for school trips.

n°	Relativedemand	Frequency	Relativecrowding	PT Trips toSchool	PT trips to work	ACRoad &PT	Welfare	Deficit
1	0,5	10	150%	20	20	3,455	178	120
2	0,5	10	150%	10	27	3,396	181	64

We start in line 1 from the case with a frequency of 10 and 50% telework and teleschool; this again corresponds to line 12 in Table 3. In line 2 we assume that 10 trips to school opt for another mode (walking, step, bike). Keeping the frequency constant, this allows to have more trips to work via public transport; these extra users value the decrease in crowding (as they are vulnerable users) leading to a decrease in the average cost AC of trips to work. The result is a small increase in welfare (181 instead of 178). There is also a decrease in the deficit because there are now more public transport trips to work and, therefore, employers contribute more to revenues.

In the simple exercise above we shifted exogenously the modal share for school trips; we kept the total number of school trips constant except for teleschool obligations. Of course also trips to work could shift to other modes than car and public transport; this would also be an efficient policy to cope with larger infection risks.

Conclusions

We study the effect of telework and perceived Covid-19 infection risk for public transport policies within the framework of a simple model to determine second-best optimal fare and frequencies for a public transport operator when the competitive mode (car use) does not pay for the full marginal external cost. An increase in telework reduces the demand for both transport modes. If the public transport firm could flexibly adapt its policies, the efficient second-best response would be to reduce both the fare and the frequency. The increase in the external crowding cost due to the perceived Corona-infection risk reduces public transport demand and increases car use. We found that under plausibly conditions the optimal fare increases; the effect on optimal frequency depends on the relative importance of higher frequency on demand. Interestingly, as telework and the infection risk have opposite effects on the fare in practice the welfare improvement of fare changes may be very small; for all practical purposes it may be optimal for the firm not to change the fare at all. Second, assuming that pre-Covid fare and frequency were second-best optimal, holding the fare and frequency constant at these values implies that more telework reduces the public transport deficit. The effect of a higher perception risk is ambiguous in general but, if car use is strongly underpriced, the higher risk perception reduces the deficit as well. Third, extending the model to allow for passengers with different vulnerability towards Covid-19, we show that fare differences between different types of passengers are unwarranted unless for each type separate vehicles can be used. Allowing fare and frequency differentiation implies that lower occupancy rates per vehicle for the vulnerable are socially optimal. They will face higher fares only if their risk perception is sufficiently higher than that of the non-vulnerable and car use is not too much underpriced. Fourth, more restrictive social distancing on wagons for the vulnerable raises the fare for vulnerable passengers, and it implies that more wagons are reserved for the vulnerable.

In the application to Brussels we assumed that trips to work can chose between car and public transport use, but school trips can only use public transport. In the baseline, a very low fare and a high frequency are justified to address un-priced car congestion. Whenever telework reduces the demand for trips by at least 25% and the perceived crowding externality becomes higher, there are two policy options. The first is to keep frequency relatively high and keep a fair share of public transport use for trips to work. The second option is a corner solution: strongly decrease the frequency, and have almost all trips to work switch to car use. The second option is difficult in the short-term but was found to be clearly superior in welfare terms; moreover, it decreases the deficit. As telework may continue to be more important in the future, even after the Covid pandemic, car congestion in the peak period is likely to be less important. One may opt for an equilibrium with a much smaller frequency of public transport. We also found that a reservation of capacity for the vulnerable users and a modal shift to soft modes could produce important efficiency gains.

Many caveats apply to the model and the numerical illustration to Brussels. First, we use perceived infection risk as basis for agents’ behavior. Of course, the objective risk of infection transmission also matters, but we do not know how the two types of risks correlate. If the objective risk is much higher than the perceived risk, the priorities in terms of fare and frequency will be very different. Second, we represent the behavior of only two categories of agents (vulnerable and less vulnerable users) that have each different options to travel. Reality offers much more diversity in types of agents and in types of transport modes available to the two types of agents. Biking, steps and simple walking saw their market share increase and this extra margin of adaptation softens the public transport versus car paradox we studied in this paper. Third, our conclusions could be different in other settings than the one considered for Brussels. For example, consider countries where car ownership is low so that for many people public transport is the only commuting option available; alternatively, suppose substitution between public transport and car use is difficult or impossible. In these cases, there is no way to escape from the infection risk by switching to car use. The restriction of transport trips via telework would then have to be much more drastic, and separation of vulnerable and non-vulnerable users could be much more important in terms of policy advice.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Table A1

Computation generalized cost public transport (work trips).

In € per PT trip in peak in baseline	Value in €	Minutes	Comments
Money price	0		Cheap subscription, so marginal price = 0
Walk time	1,44	8,64	(7,92 min/60 min) (VOT = 10 euro per hour)
Waiting time (1/(2f))	0,5	3	(1/(2f))VOT = (1/20) 10
In-vehicle time	1	6	3 km à 30 km/h × 10 euro per hour
Discomfort factor (discomfort Beta. Y/f)(in-vehicle time).VOT	1,06		Factor + 106% for in vehicle time in peak period,
TOTAL	4,00

Table A2

Computation generalized cost public transport (school trips).

In € per PT trip in peak in baseline	Value in €	In minutes	Comments
Money price	0		Cheap subscription, so marginal price = 0
Walk time	0.72	8.64	(8.64 min/60 min) 5€ VOT = 0.72
Waiting time (1/(2f))	0.25	3	0.5 (1/(2f)) VOT = 0.5 (1/10) 5 = 0.25
In Vehicle time	0.50	6	3 km à 30 km/h × VOT
Discomfort factor (discomfort Beta. Y/f)(in-vehicle time).VOT	0.53		Factor + 1.06% for in vehicle time in peak period.
TOTAL	2.00