Impact of Congestion Charge and Minimum Wage on TNCs: A Case Study for San Francisco
IImpact of Congestion Charge and Minimum Wage on TNCs:A Case Study for San Francisco
Sen Li a , Kameshwar Poolla b , Pravin Varaiya b a Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology b Department of Electrical Engineering and Computer Science, The University of California, Berkeley
Abstract
This paper describes the impact on transportation network companies (TNCs) of the imposition of acongestion charge and a driver minimum wage. The impact is assessed using a market equilibrium modelto calculate the changes in the number of passenger trips and trip fare, number of drivers employed, theTNC platform profit, the number of TNC vehicles, and city revenue. Two types of charges are considered:(a) a charge per TNC trip similar to an excise tax, and (b) a charge per vehicle operating hour (whetheror not it has a passenger) similar to a road tax. Both charges reduce the number of TNC trips, but thisreduction is limited by the wage floor, and the number of TNC vehicles reduced is not significant. Thetime-based charge is preferable to the trip-based charge since, by penalizing idle vehicle time, the formerincreases vehicle occupancy. In a case study for San Francisco, the time-based charge is found to be Paretosuperior to the trip-based charge as it yields higher passenger surplus, higher platform profits, and highertax revenue for the city.
Keywords:
TNC, ride-sourcing, congestion charge, wage floor, regulatory policy.
1. Introduction
Transportation network companies (TNCs) like Uber, Lyft and Didi, have dramatically changed urbantransportation. While the emergence of TNC significantly benefits passengers and drivers, it also bringsnegative externalities that have to be addressed by regulatory intervention. In recent years, this concernhas prompted several cities to take actions to regulate TNCs [1, 2, 3, 4]. Despite numerous works onthe operation and management strategies of TNC platforms, only a handful of works have consideredthe mathematical model for policy analysis on the ride-hailing market. This paper aims to formulate aneconomic equilibrium model to evaluate the impacts of various regulations on the TNC economy.
Background and Motivation
TNCs are disrupting the urban transportation systems. On the one hand, they offer on-demand ride servicesat prices that many riders can afford. On the other hand, they create numerous job opportunities for driversworking as independent contractors. These favorable demand and supply factors led to the TNC’s explosivegrowth. However, the resulting growth has raised two public concerns in large metropolitan areas. The firstis due to increased traffic congestion. In New York City, Uber, Lyft, Juno and Via together dispatch nearly600,000 rides per day, involving about 80,000 vehicles. Schaller [5] estimates that from 2013 to 2017 TNCtrips in NYC increased by 15%, traffic speed dropped by 15%, VMT increased by 36%, and the number
Email addresses: [email protected] (Sen Li), [email protected] (Kameshwar Poolla), [email protected] (PravinVaraiya)
Preprint submitted to Elsevier February 2, 2021 a r X i v : . [ ec on . E M ] F e b f TNC vehicles increased by 59%. He suggested regulation to reduce TNC vehicles deadhead time (whenvehicles are carrying no passengers) in order to limit congestion. Two reports [6, 7] by the San FranciscoCounty Transportation Authority identified TNC impact on traffic congestion and estimated that TNCsaccount for approximately 50 percent of the increase in congestion in San Francisco between 2010 and 2016.More recently, Uber and Lyft commissioned Fehr & Peer to estimate the TNC share of VMT in six USMetropolitan Regions, Boston, Chicago, Los Angeles, Seattle, San Francisco and Washington. Their report[8] concludes that Uber and Lyft have a nontrivial impact in core urban areas such as San Francisco County,where they account for 12.8% of total VMT.The second concern is provoked by the very low earnings of TNC drivers. The success of the on-demandride-hailing business relies on short passenger waiting times that require a large pool of available but idleTNC drivers. This pushes down driver wages. Parrott and Reich [9] revealed that the majority of for-hirevehicle drivers in NYC work full-time. They found that the median driver earnings declined almost $ $ $ Research Problem and Contribution
This paper presents a study calculating the impact on TNCs of the joint imposition of a congestion chargeand a driver minimum wage. The impact is formulated within a framework comprised of a queuing theoreticmodel of the arrivals of passengers and drivers, a general equilibrium model that predicts market prices,passenger demand and driver supply, and a profit maximizing model of the TNC platform decisions. Thisframework enables the assessment of the impact in terms of changes in ride prices, passenger waiting time,driver wage, numbers of passengers and drivers, vehicle occupancy rate, platform rent, and city tax revenue.The key conclusions of this study are:• The congestion charge does not significantly affect TNC ridership. It does not directly curb trafficcongestion by reducing the number of TNC vehicles on the road. This is because the impact of thesurcharge is mitigated by the wage floor on TNC drivers.2 The time-based congestion charge is preferable to the trip-based charge because the former penalizesidle vehicle hours, thereby increasing vehicle occupancy (we use the terms congestion charge andtax interchangeably.) Furthermore, the increased occupancy generates a surplus that offers a Paretoimprovement in a certain regime, bringing higher consumer surplus, higher platform profit and highertax revenue for the city.• The case study for San Francisco employs a model whose parameters are calibrated to match reportedSan Francisco TNC data, and the model is used to predict the likely effect of regulatory policies onthe San Francisco TNC market.• Through numerical simulation, we show that the tax burden mainly falls on the ride-hailing platformas opposed to passengers and drivers. Under a trip-based tax of $2/trip (with average trip fare of$11.6), passenger travel cost increases by 0.6%, driver wage remains unchanged, while the platformprofit is reduced by 59.5%. Under a time-based tax in the regime of practical interest, both passengersand drivers are unaffected, while the platform assumes all of the tax burden.
2. Related Works
There is an extensive literature on app-based ride-hailing platforms. Many studies investigated the platformpricing strategy under various interacting factors. Zha et al [13] developed an aggregate model to capturethe interactions among passengers, drivers and the platform, and found that the first-best solution isnot sustainable when the matching function exhibits increasing returns to scale and the cost function ofthe platform is subject to economies of scale. Bai et al [14] considered an on-demand service platformusing earning-sensitive independent providers with heterogeneous reservation price, and concluded that itis optimal to charge a higher price when demand increases, and that the platform should offer a higherpayout ratio as demand increases, capacity decreases, or customers become more sensitive to waiting time.Taylor [15] examined how delay sensitivity and agent independence affect the platform’s optimal price andwage and identified the complexity caused by uncertainty in customer valuation. Hu and Zhou [16] studiedthe commission setting of the ride-sourcing platform and showed that an optimal fixed-commission contractcan achieve at least 75% of the optimal profit when there is no pre-committed relationship between priceand wage.Platform pricing has also been studied with temporal and spatial considerations. From the temporal aspect,Cachon et al [17] showed that surge pricing can significantly increase platform profit relative to contractsthat have a fixed price or fixed wage, and that all stakeholders can benefit from the use of surge pricing ona platform with driver self-scheduling capacity. Castillo et al [18] showed that surge pricing can avoid caseswhere vehicles are sent on a wild goose chase to pick up distant customers, wasting driver time and reducingearnings. Zha et al [19] investigated the impact of surge pricing using a bi-level programming framework,and showed that compared to static pricing, the platform and drivers are found generally to enjoy higherrevenue while customers may be made worse off during highly surged periods. Banerjee et al [20] developeda queuing theoretic model to study the optimal (profit-maximizing) pricing of ride-sharing platforms. Theyshow that the performance of a dynamic price (in terms of revenue and throughput) does not exceed that ofa static price, but it is more robust to fluctuations of model parameters. From the spatial aspect, Bimpikiset al [21] considered the price discrimination of a ride-sourcing platform over a transportation network andestablished that profits and consumer surplus at the equilibrium corresponding to the platform’s optimalpricing are maximized when the demand pattern is “balanced” across the network’s locations. Guda andSubramanian [22] studied the spatial pricing of a ride-sourcing platform over a transportation network andshowed that surge pricing can be useful even in zones where supply exceeds demand. Zha et al [23] developeda model to investigate the effects of spatial pricing on ride-sourcing markets and found that the platformmay resort to relatively higher price to avoid an inefficient supply state if spatial price differentiation is3ot allowed. In addition to platform pricing, studies also touch upon driver supply [24], [25], platformoperations [26], [27], platform competition [28], [29], and regulations [30], [31], [32], [33]. Please see [34] fora comprehensive literature review.Road pricing has attracted substantial research attention for decades. The idea was initially proposed byPigou [35], which inspired several seminal works including Vickery [36], Walters [37] and Beckmann [38].Since then, various taxing schemes have been proposed in the literature, including charge based on cordon-crossing, distance traveled, time spent traveling, or time spent in congestion [39]. For instance, Zhangand Yang [40] investigate the cordon-based second-best congestion pricing problem on road networks thatjointly consider toll levels and toll locations. Yang et al [41] study road pricing for effective congestioncontrol without knowing the link travel time and travel demand. Liu and Li [42] derive a time-varying tollcombined with a flat ride-sharing price to nudge morning travelers to depart in off-peak hours. Despitethis large literature in transportation economics, the research on congestion charges for TNCs is relativelyscarce. A TNC congestion charge is distinctive since it involves decisions of the profit-maximizing platformand the passengers and drivers in the two-sided ride-hailing market. Li et al [30] proposed a marketequilibrium model to evaluate the impact of various regulatory policies and analyzed the incidence of aTNC tax on passengers, drivers, and the TNC platform. Schaller [43] conducted an in-depth analysis ofhow to apply pricing to new mobility services, and recommended that a surcharge on taxi/for-hire tripsin central Manhattan be applied as an hourly charge. Recent work of Vignon and Yin [33] investigatedthe performance of various regulation policies on ride-sourcing platforms with congestion externality andproduct differentiation taken into account. They compared a uniform toll that treats all vehicles identicallywith a differentiated toll that treats idle vehicles, solo rides and pooled rides differently, and showed thata differentiated toll offers little advantage over a uniform one.Only a handful of studies considered wage regulation of TNCs. Gurvich [44] studied the platform’s profitmaximizing wage level for self-scheduling drivers, and showed that under a minimum wage, the platformlimits agent flexibility by restricting the number of agents that can work during some time intervals. Parrottand Reich [9] utilized administrative data of New York City and showed by simulation that the proposedminimum wage standard in New York City will increase driver wage by 22.5 percent while hurting passengersby slightly increasing ride fare and waiting time. Li et al. [30] and Benjaafar et al. [31] developed marketequilibrium models to show that wage regulations on TNC will benefit both passengers and drivers, becausewage regulation curbs TNC labor market power [30]. Zhang and Nie [45] proposed a market equilibriummodel for ride-sourcing platforms that offers a mix of solo and pooled rides. They showed that a wagefloor on TNC drivers will force the platform to hire more drivers, which will reduce the appeal of collectivemodes and the supply efficiency and is likely to worsen traffic congestion.This paper differs from the aforementioned works in that we explore the joint impact of congestion chargeand driver minimum wage on the TNC market. We are the first to point out that distinct regulatorypolicies on TNCs do interfere with each other when they are jointly implemented, which may producesurprising market outcomes that deviate from the expectation of the policy maker. We are also the first toestablish models that compare the trip-based congestion charge and the time-based congestion charge andidentify the superiority of time-based congestion in certain regimes of practical interest. These results willprovide valuable insights for city planners who are considering implementing (e.g., San Francisco), or havealready implemented (e.g., NYC and Seattle) a congestion charge and a minimum wage to address TNCexternalities.
3. Market Equilibrium Model
We consider a transportation system comprised of a city council, a TNC platform, and a group of passengersand drivers. The city council approves legislation (e.g., cap on the total number of vehicles, minimum wage4or TNC drivers, congestion charge on TNC trips) to regulate the operations of the TNC platform. Theplatform sets fares and wages and hires drivers to maximize its profit under these regulations. The pricingdecisions affect the choices of passengers and drivers, and these choices collectively determine the platform’sprofit. We will describe a market equilibrium model to capture the decisions of passengers, drivers, and theTNC platform. The model will be used to investigate how TNC market outcomes are affected by regulation.
The TNC platform matches randomly arriving passengers to idle TNC drivers. Upon arrival, each passengerjoins a queue and waits until she or he is matched to an idle driver . This matching is modeled as acontinuous-time queuing process, in which each passenger defines a “job” and each driver is a “server”. Theserver is “idle” if the vehicle is not occupied, and it is “busy” if a passenger is on board or if the vehicle isdispatched and on its way to pick up a passenger. Assume that passenger arrivals form a Poisson processwith rate λ > , and denote N as the total number of TNC drivers. This matching process forms a M/G/Nqueue, and the expected number of idle servers (vehicles) is N I = N − λ/µ , with µ being the service rate( /µ is the amount of time a passenger occupies a vehicle on average). We assume that N > λ/µ . Giventhe model parameters, the average waiting time for the M/G/N queue can be derived approximately in ananalytical form. We comment that this is the ride confirmation time, which represents the time elapsedafter the ride is requested and before the ride is confirmed. It differs from the pickup time (from rideconfirmation to pickup), which will be treated below.
The total travel cost of the TNC passenger consists of the waiting time for pickup, the travel time duringthe trip, and the monetary payment for the ride service. We refer to this total travel cost as the “generalizedcost” and define it as the weighted sum of waiting time, travel time, and trip fare. It may differ for distinctpassengers due to the randomness in trip length, trip duration, and the matching process of the TNCplatform. Since we primarily focus on aggregate market outcomes, we define the average generalized costas: c = αt w + βt + p f , (1)where t w is the average waiting time, t is the average trip duration (in minutes), and p f is the average priceof a TNC ride. The parameters α and β specify the passenger trade-off between time and money. Note that α is generally larger than β since empirical study suggests that the value of time while waiting is larger thanthe value of time while traveling in the vehicle. It is important to emphasize that we do not need to assumethat all passengers have the same travel cost. The heterogeneity in passengers is irrelevant as we focus onthe aggregate market outcome, which typically depends on the average cost c . A widely-studied example isthe logit choice model, where the total number of agents choosing a particular mode only depends on theaverage cost of each mode. In this spirit, we define a demand function that determines the arrival rate ofTNC passengers as a function of the average generalized cost: λ = λ F p ( c ) , (2)where λ is the arrival rate of potential passengers (total travel demand in the city), and F p ( · ) is theproportion of potential passengers who choose a TNC ride. We assume that F p ( · ) is a strictly decreasing andcontinuously differentiable function so that a higher TNC travel cost c will lead to fewer TNC passengers.The logit model is a special case of (2). For simplicity, we do not consider the case of multiple passengers sharing the same vehicle. t w intimately interacts with other endogenous decision variables λ and N .To delineate this relation, we divide a TNC ride into three time periods: (1) from ride being requestedto the ride being confirmed, (2) from the ride being confirmed to passenger pickup, (3) from passengerpickup to drop-off. Let t m , t p , and t represent the length of these three periods, respectively, then we have t w = t m + t p , and t as the average trip distance L divided by traffic speed v , i.e., t = L/v . Since theplatform immediately matches each newly arrived passenger to the nearest idle vehicle, t m is the averagewaiting time in the queue, and t p depends on the traffic speed v and the distance of the passenger to thenearest idle vehicle, which further depends on the number of idle vehicles N I . Therefore, we write t p as afunction of N I and v , i.e., t p ( N I , v ) . The following assumption is imposed on t p ( · ) : Assumption 1. t p ( N I , v ) is twice differentiable with respect to N I and v. It is decreasing and strictlyconvex with respect to N I , and it is decreasing with respect to traffic speed v. Assumption 1 requires that the pickup time decreases with respect to the number of idle vehicles and thetraffic speed. We suppose traffic speed v ( N ) is a function of the total number N of vehicles and imposethe following assumption on v ( · ) : Assumption 2. v ( N ) is decreasing and continuously differentiable with respect to N . Using data of San Francisco and New York City for the M/G/N queue, we find that the ride confirmationtime t m is very short, i.e., less than 1 seconds. This is negligible compared to the pickup time t p , which istypically around 3-5 minutes. Therefore we ignore t m and express the total waiting time t w as t w = t p ( N I , v ) . (3)The number of idle vehicles N I depends on λ and N , whereas the average traffic speed v depends on N . In the TNC market, drivers can decide whether to remain subscribed to the TNC platform depending onthe long-term average earnings offered by the platform. The average hourly wage of drivers depends on theride fare of the TNC trip, the commission rate set by the platform, and the occupancy rate of the vehicles.It can be described as: w = λp d N , (4)where p d is the average per-trip payment to drivers. The driver payment p d differs from the passengertrip fare p f . The difference p f − p d is kept by the platform as profit. Therefore, the commission rate ofthe platform (typically 25%-40%) can be written as ( p f − p d ) /p f . The average hourly wage (4) is just thetotal platform payment to all drivers λp d divided by the total number of drivers N . Each driver may havean hourly earning that differs from the earning of others due to the randomness in work schedule, driverlocation, and repositioning strategy. However, as we primarily focus on the aggregated market outcome,the heteregeneity in driver earnings is irrelevant as far as the aggregate market outcome (e.g.,total numberof TNC passengers or drivers) only depends on the average hourly earning over all TNC drivers. Note thatthis is the case for the well-established logit choice model. More generally, we define a supply function thatdetermines the total number of TNC drivers as a function of the average hourly wage: N = N F d ( w ) , (5)where N is the number of potential drivers (all drivers seeking a job), and F d ( w ) is a strictly increasingand continuously differentiable function that gives the proportion of drivers willing to join TNC. Note thatthe logit model is a special case of (5) The waiting time can be significantly larger in rush hours. In this case, one can add t m to t w as the waiting time in thequeue. We believe that this will not affect our conclusion, but we neglect this term in this paper for analytic tractability. .4. Platform decisions in absence of regulation The TNC platform determines the ride prices and the driver payment to gauge passengers and driversto maximize its profit. In each time period, the platform revenue is the total ride fares received frompassengers, i.e., λp f , and the platform cost is the total payment made to the drivers, i.e., λp d . The profitof the platform can be thus written as the difference between the revenue and the cost max p f ,p d λ ( p f − p d ) (6) λ = λ F p ( αt p + βt + p f ) (7a) N = N F d (cid:18) λp d N (cid:19) (7b)where (7a) is the demand function and (7b) is the supply function. Note that t p depends on λ and N , and t depends on the traffic speed which is a function of N . The overall problem not only involves p f and p d as decision variables, but also involves N , λ , t p , v and t as endogenous variables. The optimal solution to(6) represents the platform’s profit-maximizing pricing decision in absence of the regulatory intervention.The profit maximization problem (6) is a constrained optimization which can be solved by various gradient-based algorithms [46]. However, since the problem is non-concave with respect to p d and p f , it is difficultto assert whether the obtained solution is globally optimal. To address this concern, we apply a change ofvariable and treat λ and N as the new decision variables. More specifically, given λ and N , we can use(7a)-(7b) to uniquely determine p f and p d as follows: p f = F − p (cid:18) λλ (cid:19) − αt p ( N I , v ) − βp (8a) p d = Nλ F − d (cid:18) NN (cid:19) (8b)where (8a) is derived from (7a), and (8b) is derived from (7b). Note that the right-hand sides of (8a)and (8b) are both functions of λ and N . By plugging (8a) and (8b) into (6), we can transform the profitmaximization problem (6) into the following unconstrained optimization: max λ,N λ (cid:18) F − p (cid:18) λλ (cid:19) − αt p ( N I , v ) − βp (cid:19) − N F − d (cid:18) NN (cid:19) (9)where λ and N are decision variables. Clearly, (9) is equivalent to (6). We note that although (9) isnon-concave with respect to λ and N , under certain mild conditions, it is concave with respect to λ forfixed N . We formally summarize this result as the following proposition: Proposition 1.
Assume the demand function F p ( · ) is a logit model represented as: λ = λ e − (cid:15)c e − (cid:15)c + e − (cid:15)c , (10) where (cid:15) > and c are parameters. Further assume that given v , the waiting time function t p ( N I , v ) isconvex with respect to N I , then we have the following results:(1) the profit maximization problem (9) is concave with respect to λ under a fixed N ,(2) Given N , there exists a unique λ that maximizes the platform profit (9) The proof can be found in Appendix A. Proposition 1 suggests that for any fixed N , we can efficiently derivethe unique optimal λ that maximizes the profit by solving a concave program. This result is based on a fewmild assumptions: (a) the logit model (10) is used for studying customer discrete choice, (b) the convexity7f t p ( · ) simply requires that the marginal benefit of adding extra idle vehicles in reducing passenger waitingtime decreases with respect to N I , which is consistent with intuition. Based on this result, we can obtainthe optimal combination of ( λ, N ) by enumerating over N . This provides the globally optimal solution to(9). Remark 1.
Many works study the spatial and temporal aspects of the TNC market. These aspects areneglected in our model since we primarily focus on the evaluation of regulatory policies (e.g., minimumwage) that are imposed on a uniform basis regardless of the time of the day or the location of the driver.This makes it legitimate to consider the impact of these policies at the aggregate scale, which suffices toprovide valuable insights for city planners to assess their policies. A spatial-temporal analysis is necessaryif policy makers further consider fine-tuning these policies so that they differentiate trips at different timeinstances or different locations. This is left for future work.3.5. Modeling regulation policies
Regulation policies, such as congestion charge and driver minimum wage, modify the incentives of passengersand drivers and affect the pricing decision of the TNC platform. To capture this effect, we formulate theplatform pricing problem under the minimum wage, trip-based congestion charge, and time-based congestioncharge.
Minimum wage:
To capture the impact of a driver wage floor w , we impose the constraint that requiresthe driver hourly earning to be greater than w . The optimal pricing problem under minimum wageregulation can be formulated as: max p f ,p d ,N λ ( p f − p d ) (11) λ = λ F p ( αt p + βt + p f ) (12a) N ≤ N F d (cid:18) λp d N (cid:19) (12b) λp d N ≥ w (12c)where constraint (12c) captures the wage floor on TNC driver earnings. Note that we relax the equalityconstraint (7b) to inequality constraint (12b). This permits the TNC platform to hire a subset of driverswho are willing to work for TNC in case the minimum wage is set so high that it is unprofitable for theplatform to hire all the willing drivers in the market. Remark 2.
Note that the minimum wage constraint (12c) places a lower bound on the average driverwage w . Since the hourly wage may differ from one driver to another, when (12c) is satisfied, it does notnecessarily mean that all drivers earn at least the minimum wage. Instead, it only indicates that driverscan earn more than the minimum wage on average. We emphasize that this formulation is consistent withthe practice: the minimum wage for TNC drivers in New York City and Seattle are both implemented on aplatform-wide average basis, instead of an individual driver basis [2], [47]. Trip-based congestion charge:
Many existing congestion charge schemes are trip-based (e.g., New YorkCity, Seattle, Chicago). The trip-based congestion charge assesses an extra fee of p t on each TNC trip inthe congestion area. When a congestion charge p t and a minimum wage w are imposed concurrently, theoptimal pricing problem can be formulated as follows: max p f ,p d ,N λ ( p f − p d ) (13)8 λ = λ F p ( αt p + βt + p f + p t ) (14a) N ≤ N F d (cid:18) λp d N (cid:19) (14b) λp d N ≥ w (14c)where the per-trip congestion charge p t is incorporated into the passenger travel cost within the demandfunction (14a). Another way to formulate the congestion charge is by adding it to the cost of the platform.This is easier to implement as it only requires the platform to transfer the accumulated congestion chargeof all trips within certain period to the city. In this case, the optimal pricing problem can be written as: max p f ,p d ,N λ ( p f − p d ) − λp t (15) λ = λ F p ( αt p + βt + p f ) (16a) N ≤ N F d (cid:18) λp d N (cid:19) (16b) λp d N ≥ w (16c)where p t is incorporated into the profit of the platform instead of the travel cost of the passengers.Economists find that whether a tax is levied on the buyer or seller of the good does not matter becausethey always share the tax burden based on their elasticities [48, Chap. 16]. This principle also applies here: Proposition 2.
Let ( p ∗ f , p ∗ d , N ∗ , λ ∗ ) and ( p ∗∗ f , p ∗∗ d , N ∗∗ , λ ∗∗ ) denote the optimal solutions to (13) and (15),respectively, then we have p ∗ f + p t = p ∗∗ f , p ∗ d = p ∗∗ d , λ ∗ = λ ∗∗ , and N ∗ = N ∗∗ . Proposition 2 states that the two formulations of trip-based congestion charge, i.e., (13) and (15), lead tothe same market outcome. The proof is omitted since it can be simply derived by a change of variable.
Time-based congestion charge:
Distinct from the trip-based congestion charge, the time-based chargeis levied on TNC vehicles based on vehicle hours instead of trip volumes. The key difference between thetwo congestion charge schemes is that time-based congestion charge not only penalizes TNC trips, butalso penalizes idle TNC hours and thus incentivizes the platform to increase vehicle utilization. When thetime-based congestion charge p h and a minimum wage w are concurrently levied on TNC drivers, we havethe following formulation: max p f ,p d ,N λ ( p f − p d ) (17) λ = λ F p ( αt p + βt + p f + p t ) (18a) N ≤ N F d (cid:18) λp d N − p h (cid:19) (18b) λp d N − p h ≥ w (18c)When the time-based congestion charge is levied on the TNC platform, we have the following formulation: max p f ,p d ,N λ ( p f − p d ) − N p h (19) λ = λ F p ( αt p + βt + p f ) (20a) N ≤ N F d (cid:18) λp d N (cid:19) (20b) λp d N ≥ w (20c)Similar to the trip-based congestion charge, these two forms of formulations are equivalent.9 roposition 3. Let (¯ p d , ¯ p d , ¯ N , ¯ λ ) and (˜ p f , ˜ p d , ˜ N , ˜ λ ) denote the optimal solutions to (17) and (19), respec-tively, then we have ¯ p f = ˜ p f , ¯ λ ¯ p d ¯ N − p h = ˜ λ ˜ p d ˜ N , ¯ λ = ˜ λ, and ¯ N = ˜ N . Proposition 3 states that the two formulations of time-based congestion charge, i.e., (17) and (19), lead tothe same market outcome. The proof is omitted since it is similar to that of Proposition 2.
4. Profit maximization under trip-based congestion charge
This section analyzes the joint impact of a trip-based congestion charge and a minimum wage for TNCdrivers. We consider a platform that determines the ride fare p f and the per-trip driver payment p d tomaximize its profit λ ( p f − p d ) under the trip-based congestion charge p t and a minimum wage w . Theoptimal pricing problem can be formulated as (13) or (15). For sake of exposition, we will start with arealistic numerical example for San Francisco. The numerical example will be complemented by a theoreticalanalysis presented later that shows the insights derived from the numerical example can be generalized. We investigate the impact of the proposed regulations via a case study for San Francisco (followed bytheoretical analysis in the next subsection). Assume that passengers choose their transport mode based onthe total travel cost. We use a logit model so the demand function for TNC rides is λ = λ e − (cid:15)c e − (cid:15)c + e − (cid:15)c , (21)where c is the total travel cost of a TNC trip, and (cid:15) > and c are parameters. Similarly, drivers chooseto work for the TNC depending on its wage. Under a logit model, the supply function is N = N e σw e σw + e σw , (22)where σ is a parameter. We note that (21) is a special case of the general demand function (2), and (22)is a special case of the general supply function (5).Passenger pickup time t p follows the “square root law” established in [49] and [30]: t p ( N I , v ) = Mv (cid:112) N − λ/µ , (23)where the constant M depends on the travel times in the city. The square root law establishes that theaverage pickup time is inversely proportional to the square root of the number of idle vehicles in the city, ( N − λ/µ ) . The intuition behind (23) is straightforward. Suppose all idle vehicles are uniformly distributedthroughout the city, then the distance between any two nearby idle vehicles is inversely proportional to thesquare root of the total number of idle vehicles. This distance is proportional to that between the passengerand the closest idle vehicle, which determines the pickup time. A justification of the square root law canbe found in [30].The average traffic speed v is a function of the total traffic. Using Greenshield model [50] gives the linearspeed-density relation , v = v − κ ( N + N b ) , (24) Since TNC vehicles only account for a small percentage of the overall traffic, the Greenshield model can be regarded as alinear approximation in a small neighborhood of a nonlinear speed-density function. N b is the background traffic , N is the number of TNC vehicles, and v and κ are model parameters.Assuming that N b is constant, (24) is equivalent to v = v f − κN. (25)In summary, the model parameters are Θ = { λ , N , M, L, v f , κ, α, (cid:15), c , σ, w } . In the numerical study we set the parameters values so that the optimal solution to (6) matches the realdata of San Francisco city. The values of these model parameters are summarized below: λ = 1049 / min , N = 10000 , M = 41 . , L = 2 . mile , v f = 15 mph , κ = 0 . ,α = 2 . , (cid:15) = 0 . , c = 31 . , σ = 0 . , w = $31 . / hour . For the data source (from San Francisco) and justification of these parameter values, please refer to Ap-pendix B.We solve the profit maximizing problem (13) for different values of congestion charge p t under a fixed wagefloor w , and plot all the variables as a function of p t . The minimum wage of TNC drivers in San Franciscois set in a way similar to that in NYC. Under current NYC regulations, the TNC driver minimum wage is $25 . /hour, which is equivalent to the $15 /hour minimum wage of NYC after deducting vehicle expensessuch as insurance, maintenance and taxes. Since the hourly minimum wage of San Francisco is $0 . higherthan in NYC, we set w = $25 .
76 + $0 .
59 = $26 . /hour to compensate for this difference. Figure 1- Figure 3 show the number of drivers, passenger arrival rate, and the occupancy rate of TNCvehicles as a function of the congestion charge p t when the minimum wage is set at w = $26 . /hour.Figure 4 shows the per-trip ride fare p f and the driver payment p d . Figure 5-Figure 6 show the passengerpickup time and travel cost. Figure 7 shows the driver wage (which equals the minimum wage). Figure 8and Figure 9 show the platform profit and city’s tax revenue under different values of p t , respectively.Clearly, the optimal solution as a function of p t has two distinct regimes:• when p t ≤ $2 . /trip, the number of drivers remains constant, while the number of passengers reduces;vehicle occupancy drops, passenger pickup time decreases, ride fare increases, and the passenger totaltravel cost increases. At the same time, driver wage remains constant and equals the minimum wage,platform profit reduces, and the tax revenue increases.• when p t > $2 . /trip, both the passenger arrival rate and number of TNC drivers reduce sharply;vehicle occupancy reduces, ride fare and pickup time increase, and the total travel cost increases. Thedriver wage remains constant and equals the minimum wage, while the platform revenue declines, andthe tax revenue increases.This is a surprising result: the number of drivers is unaffected by the congestion charge p t when p t ≤ $2 . /trip. It is in contrast with the case when there is only a congestion charge and no minimum wage TNC trips may substitute taxis or private vehicles. This may introduce coupling between the TNC demand and thebackground traffic N b . For simplicity, we neglect this substitution effect and assume N b is exogenous. We leave it for futurework to investigate how the coupling between λ and N b affect the conclusion of this paper.
11 1 2 33.2K3.6K4K per-trip tax N u m b e r o f d r i v e r s Figure 1: Number of drivers under dif-ferent trip-based congestion charge. per-trip tax P a ss e n g e r a rr i v a l Figure 2: Passenger arrivals under dif-ferent trip-based congestion charge. . . . . . per-trip tax o cc up a n c y Figure 3: Occupancy rate under differ-ent trip-based congestion charge. per-trip tax p r i ce / p a y m e n t p f p d Figure 4: Per-trip ride price and driverpayment under different trip-based con-gestion charge. . . . per-trip tax p i c k up t i m e Figure 5: Passenger pickup time in min-utes under different trip-based conges-tion charge. . . per-trip tax p a ss e n g e r c o s t Figure 6: Passenger travel cost in $ pertrip under different trip-based conges-tion charge. per-trip tax d r i v e rr e v e nu e Figure 7: Per-hour driver wage underdifferent trip-based congestion charge. · per-trip tax p l a t f o r m r e v e nu e Figure 8: Per-hour platform profit underdifferent trip-based congestion charge. · per-trip tax t a x r e v e nu e Figure 9: Per-hour tax revenue underdifferent trip-based congestion charge. (see [30]). Therefore, this set of result indicates that the effect of a congestion charge on congestion reliefis mitigated by the wage floor on TNC drivers. In certain regimes, the congestion charge cannot directlycurb traffic congestion by reducing the number of TNC vehicles.The reason behind this surprising result is rooted in the platform’s power in the labor market. The platformis a monopoly in the labor market and sets driver wages. When there is no regulation (i.e., p t = 0 and w = 0 ), the platform hires fewer drivers to maximize its profit compared to a competitive labor marketwhere the TNC faces the competitive driver wage. In a certain regime, the minimum wage squeezes theplatform’s market power and induces it to hire more drivers [30]. This indicates that the marginal profitof hiring additional drivers under the minimum wage regulation is positive. When the congestion charge isinsignificant, this marginal profit reduces but remains positive, and thus the platform still hires all driversavailable in the labor market. The number of drivers is upper bounded by N ≤ N F d ( w ) . Therefore, in thefirst regime, N remains constant and satisfies N = N F d ( w ) . If the congestion charge is further increased,the marginal profit of hiring an additional driver reduces to zero, and the system enters the second regime. Remark 3.
We would like to clarify that the aforementioned result relies on the assumption that the TNC latform has market power that can influence the driver wage , but does not rely on the assumption thatTNC is a monopolistic wage-setter. To validate this, we considered the duopolisitic setting, where twosymmetric TNCs compete against each other on both passenger and driver side to maximize their ownprofits. The numerical study reveals that the number of drivers and number of passengers at the Nashequilibrium demonstrate the same properties as shown in Figure 1 and Figure 2, respectively. We believethat this can be further extended to the case of more than two competing TNCs. Figures 6-8 show that the tax burden primarily falls on the ride-hailing platform as opposed to passengersand drivers. As the trip-based charge increases, the passenger cost increases slightly, the driver wageremains unchanged, while platform profit reduces significantly. In particular, under a trip-based tax of$2/trip, passenger cost increases by 0.6%, driver wage remains constant, and platform profit declines by59.5%. This is because drivers are protected by the minimum wage, and the passenger’s price elasticity isrelatively high (Figure 10) so that the platform has to refrain from significantly increasing the ride fare.0 1 2 3 per-trip tax E l a s t i c i t y driverpassenger Figure 10: Absolute value of passenger price elasticity and driver wage elasticity under different trip-based tax .We show that the result reported in Figure 1-Figure 9 (including number of drivers, number of passen-gers, platform revenue, and tax revenue) is robust for a large range of model parameters. For notationconvenience, let ˜ w be the optimal driver wage set by the platform in the absence of any regulation (i.e., p t = w = 0 ), and denote by N ∗ t ( p t ) the optimal number of drivers to (13) under a fixed wage floor, whichdepends on p t . We then have the following result. Theorem 1.
Assume that (13) has a unique solution. For any model parameters λ , N and α , any strictlydecreasing function F p ( c ) , any strictly increasing function F d ( w ) , any pickup time function t p that satisfiesAssumption 1, and any speed-density relation v ( N ) that satisfies Assumption 2, there exists w > ˜ w, suchthat for any ˜ w < w < w , there exists ¯ p t > , so that ∂N ∗ t /∂p t = 0 for p t ∈ (0 , ¯ p t ) . The proof of Theorem 1 is can be found in Appendix C. It states that for any wage floor in an appropriaterange, there is always a regime in which the congestion charge does not affect the number of TNC vehicles ordrivers. In this case, the congestion charge will not directly curb the congestion by reducing the number ofTNC vehicle on the city’s streets. Instead, it can only indirectly mitigate the traffic congestion by collectingtaxes to subsidize public transit to attract passengers. Note that ˜ w and w can be calculated numerically,and ¯ p t depends on the wage floor w . For the case of San Francisco, we calculate that ˜ w = $21 . /hour, w = $29 . /hour, and ¯ p t = $2 . /trip when w = $26 . /hour. In a competitive labor market where driver wage is given, the conclusions of this numerical study no longer hold. The passenger price elasticity is ∂λ∂p f p f λ . We calculate ∂λ∂p f assuming that the waiting time t w is fixed under different p t .This is a reasonable approximation since t w does not change significantly under distinct p t (Figure 5). N u m b e r o f d r i v e r s Figure 11: Number of drivers under dif-ferent time-based congestion surcharge. time-based tax P a ss e n g e r a rr i v a l Figure 12: Passengers arrival rate underdifferent time-based congestion charge. . . . time-based tax O cc up a n c y Figure 13: Occupancy rate under differ-ent time-based congestion charge. time-based tax p r i ce / p a y m e n t p f p d Figure 14: Per-trip ride price and driverpayment under different time-based con-gestion charge. . . . time-based tax p i c k up t i m e Figure 15: Passenger pickup time inminutes under different time-based con-gestion charge. . . time-based tax p a ss e n g e r c o s t Figure 16: Passenger travel cost in $under different time-based congestioncharge. time-based tax d r i v e rr e v e nu e Figure 17: Per-hour driver wage underdifferent time-based congestion charge. · time-based tax p l a t f o r m r e v e nu e Figure 18: Per-hour TNC profit underdifferent time-based congestion charge. · time-based tax t a x r e v e nu e Figure 19: Per-hour tax revenue underdifferent time-based congestion charge.
5. Profit maximization under time-based congestion surcharge
This section considers the profit maximization problem under a wage floor and a time-based congestioncharge. Under the time-based charge, each vehicle is penalized based on the total time it stays activeon the platform (whether there is a passenger on board or not). Let p h denote the per-vehicle per-unit-time congestion charge. The total charge (per unit time) is N p h , and the profit maximization problemis cast as (17). For sake of exposition, we will first present a numerical example for San Francisco. Theinsights derived from the numerical study will be examined by theoretical analysis later to demonstrate itsindependence on model parameters.In the numerical study, we will solve the profit maximization problem (17) for different time-based conges-tion charge p h under a fixed wage floor w = $26 . /hour. The model parameters of (17) are the same asthose in Section 4.1.Figure 11 - Figure 13 display the number of drivers, passenger arrival rates, and vehicle occupancy as afunction of the time-based congestion charge. Figure 14 shows the ride fare and per-trip driver payment.Figure 15 and Figure 16 show the passenger pickup time and total travel cost. Figure 17 shows the driver14age. Figure 18 and Figure 19 present the platform profit and tax revenue, respectively. Clearly, the plotsin Figure 11-19 have two distinct regimes:• when p h ≤ $6 . /hour the number of TNC drivers and the passenger arrival rate remain constant.So do the occupancy rate, ride fare, per-trip driver payment, pickup time, passenger travel cost anddriver wage. The platform revenue reduces linearly, and the tax revenue also increases linearly.• when p h > $6 . /hour the numbers of drivers and passengers decline. Vehicle occupancy, ride fare ( p f )and per-trip driver payment ( p d ) also decline. The pickup time and passenger travel cost increase.The driver wage is constant and equals the minimum wage. The platform profit reduces and the taxrevenue increases.Simulation results suggest that the time-based congestion charge does not affect the number of TNC vehiclesunless the charge is greater than $6 . /hour. In that case the effect of the congestion charge on congestionrelief is mitigated by the minimum wage on TNC drivers. This observation is consistent with the resultsin Section 4.2 and for the same reason. However, in contrast with the trip-based charge, the time-basedcharge does not affect passenger arrivals (Figure 12). This indicates that the time-based charge leads to adirect money transfer from the platform to the city in the first regime without affecting the passengers ordrivers. This is evidenced by the linear curves in the first regime of Figure 18-19.The quantitative results in Figure 11-Figure 19 are robust with respect to the variation of model parameters.Formally, denote by N ∗ h ( p h ) and λ ∗ h ( p h ) the optimal number of drivers and passenger arrival rates to (17)under a fixed wage floor. We have the following result. Theorem 2.
Assume that (17) has a unique solution. For any model parameters λ , N , and α , any strictlydecreasing function F p ( c ) , any strictly increasing function F d ( w ) , any pickup time function t p that satisfiesAssumption 1, and any speed-density relation v ( N ) that satisfies Assumption 2, there exists w > ˜ w, suchthat for any ˜ w < w < w , there exists ¯ p h > , so that ∂N ∗ h /∂p h = 0 and ∂λ ∗ h /∂p h = 0 for ∀ p h ∈ (0 , ¯ p h ) . The proof of Theorem 2 can be found in Appendix D. Theorem 2 states that there exists a regime underwhich both the number of TNC drivers and the passenger arrival rates are unaffected by the congestioncharge. This indicates that the ride fare, driver wage and passenger cost remain constant in this regimeand the congestion charge is entirely imposed on the platform through a direct money transfer from theplatform to the city. In this scheme, congestion charge will not directly curb the congestion by reducingtraffic in the city. Instead, it can only indirectly mitigate traffic congestion by collecting taxes to subsidizepublic transit.
6. Comparison between time-based and trip-based charges
This section provides a comparison of the trip-based and time-based congestion charges. To ensure ameaningful comparison, we first set a target for the city’s tax revenue. This target can be achieved bysetting the appropriate charges. For each scheme, we find the charge that exactly attains the targeted taxrevenue and we compare the two schemes for the same target. The model parameters are consistent withprevious case studies in Section 4.1 and Section 5.Figures 20-22 compare the number of drivers, passenger arrival rate and the vehicle occupancy of the twoschemes for different targets for the city’s tax revenue. Figure 23 and Figure 24 compare the ride fare andthe pickup time for the two schemes. Figure 25 compares the platform profit under the trip-based andtime-based charges. These results reveal that for the same realized tax revenue, the time-based charge isPareto superior to the trip-based charge (as currently implemented in NYC). Under the time-based charge,15 · N u m b e r o f d r i v e r s time-based taxtrip-based tax Figure 20: Number of drivers under dif-ferent schemes of congestion surcharge. · tax revenue/hour P a ss e n g e r a rr i v a l time-based taxtrip-based tax Figure 21: Comparison of passenger ar-rival rate (per minute). · . . . tax revenue/hour O cc up a n c y time-based taxtrip-based tax Figure 22: Occupancy rate under differ-ent congestion surcharges. · . . tax revenue/hour R i d e f a r e time-based taxtrip-based tax Figure 23: Per-trip ride price under dif-ferent congestion surcharges. · . . . tax revenue/hour P i c k up t i m e time-based taxtrip-based tax Figure 24: Comparison of passengerpickup time (minute). · · tax revenue/hour P l a t f o r m r e v e nu e time-based taxtrip-based tax Figure 25: Platform revenue under dif-ferent congestion surcharges. the TNC platform earns a higher profit. For drivers, the time-based congestion charge does not affect theirsurplus in the first regime as the same number of drivers are hired at the same wage. For passengers, thetime-based charge leads to a lower ride fare but a longer waiting time. However, the time-based congestioncharge also has higher passenger arrival rate (Figure 21). Since the demand function F p ( c ) is monotonic,this implies that the total travel cost c is lower and the passenger surplus is higher under the time-basedcongestion charge.In summary, the time-based congestion charge leads to higher passenger surplus and higher platform profit(Figure 25), which benefits all participants of the transportation system. This is because the time-basedcongestion surcharge penalizes idle vehicle hours and motivates the TNC to increase the occupancy rate ofthe vehicles (see Figure 22). Based on the data for San Francisco, the surplus resulting from increased vehicleoccupancy will be distributed to all market participants, including the passengers, the TNC platform, andthe city.While the aforementioned results do not necessarily hold for all levels of targeted tax revenues, the con-clusion is indeed applicable for a large range of model parameters in the regime of practical interest.To formally present this claim, we define N ∗ t , w ∗ t , λ ∗ t , c ∗ t , P ∗ t , T r ∗ t as the optimal solution to (13) and denote N ∗ h , w ∗ h , λ ∗ h , c ∗ h , P ∗ h , T r ∗ h as the optimal solution to (17). They are respectively the optimal number of drivers,driver wage, passenger arrival rate, total travel cost, platform profit, and city tax revenue. Note that allvariables with subscript t depend on p t and w , and all variables with subscript h depend on p h and w .We suppress this dependence to simplify the notation whenever it is clear from the context. Theorem 3.
Assume that the profit optimization problems (13) and (17) both have unique solutions. As-sume that F p ( c ) and F d ( w ) satisfy the logit model as specified in (21) and (22), respectively. For any pickuptime function t p that satisfies Assumption 1, any speed-density relation v ( N ) that satisfies Assumption 2,and any model parameters Θ = { λ , N , M, L, v f , κ, α, (cid:15), c , σ, w } , there exists w > ˜ w , such that for any ˜ w ≤ w ≤ w , there exists ¯ p t so that for any trip-based congestion surcharge p t ∈ [0 , ¯ p t ] , there exists a ime-based congestion surcharge p h that offers a Pareto improvement, i.e. N ∗ h = N ∗ t , w ∗ h = w ∗ t = w , λ ∗ h > λ ∗ t , c ∗ h < c ∗ t , P ∗ h > P ∗ t , T r ∗ h > T r ∗ t The proof of Theorem 3 can be found in Appendix E. It shows that there exists a regime where a time-basedcharge offers a Pareto improvement over a trip-based one. In this regime, for any trip-based charge, onecan find an appropriate time-based charge for which the same number of drivers is hired, more passengerstake TNC rides at a lower cost, the platform earns more profit, and the city collects more tax revenues tosubsidize public transit. For the case of San Francisco, we calculate w = $29 . /hour, and ¯ p t = $2 . /tripwhen w = $26 . /hour. Remark 4.
Theorem 3 identified a regime under which a time-based congestion charge offers a Paretoimprovement. The caveat is that this regime only applies to the wage floor and congestion charge levelswithin a certain range, i.e., w ∈ [ ˜ w, w ] , p t ∈ [0 , ¯ p t ] . Outside of this range, the comparison between the twocongestion charge schemes may depend on the model parameters. However, we emphasize that it is unlikelyfor cities to impose very stringent policies that substantially raise the driver payment level (or surchargelevel), since this may drive the TNCs out of business. In practice, regulatory policies are likely to reside inor stay close to the regime identified by this paper.
7. Sensitivity Analysis
This section reports a sensitivity analysis to test the robustness of our results with respect to the modelparameters. We vary the model parameters of (17) and evaluate the impact of the time-based congestioncharge under distinct parameter values. The nominal values of the parameters are set to be the same asin Section 4.1. We perturb λ , N and α by and investigate how these perturbations affect passengers,drivers, and the TNC platform under the time-based charge.Figure 26-28 show the number of drivers, passenger arrival rate, and the platform profit as functions of thetime-based congestion charge under different λ (the nominal value is 1049). Clearly, there are two regimes.When λ increases, the TNC platform has more passengers, and therefore enjoys a higher profit. However,we note that in the first regime, the number of drivers is not affected by λ . This is because in the firstregime, both (18b) and (18c) are active, which determines N as N = N F d ( w ) .Figure 29-31 show the number of drivers, passenger arrival rate, and the platform profit as functions of thetime-based charge for different N (the nominal value is 10K). There are clearly two regimes for the threevalues of N . When N increases, the platform hires more drivers, attracts more passengers and collects ahigher profit. Platform profit is insensitive to the number of potential drivers.Figure 32-34 show the number of drivers, passenger arrival rate, and the platform profit as functions of thetime-based charge for different α (the nominal value is 2.33). When α increases, both passenger arrival rateand platform profit drop. We note that the platform profit is much more sensitive to α than it is to λ and N .
8. Conclusion
This paper describes the impact of two proposed congestion charges on TNC: (a) a charge based on vehicletrips, and (b) a charge based on vehicle hours. We used a market equilibrium model to assess the jointeffect of minimum wage with either of these two charges. Surprisingly, we find that neither charging scheme17 nu m b e r o f d r i v e r λ = 997 λ = 1049 λ = 1102 Figure 26: Number of drivers as a func-tion of p h under distinct λ . time-based tax p a ss e n g e r a rr i v a l λ = 997 λ = 1049 λ = 1102 Figure 27: Passenger arrival rate (/min)as a function of p h under distinct λ . · time-based tax p l a t f o r m p r o fi t λ = 997 λ = 1049 λ = 1102 Figure 28: Platform profit (per hour) asa function of p h under distinct λ . nu m b e r o f d r i v e r N = 9 . K N = 10 K N = 10 . K Figure 29: Number of drivers as a func-tion of p h under distinct N . time-based tax p a ss e n g e r a rr i v a l N = 9 . K N = 10 K N = 10 . K Figure 30: Passenger arrival rate (/min)as a function of p h under distinct N . · time-based tax p l a t f o r m p r o fi t N = 9 . K N = 10 K N = 10 . K Figure 31: Platform profit (per hour) asa function of p h under distinct N . , , , time-based tax nu m b e r o f d r i v e r α = 2 . α = 2 . α = 2 . Figure 32: Number of drivers as a func-tion of p h under distinct α . time-based tax p a ss e n g e r a rr i v a l α = 2 . α = 2 . α = 2 . Figure 33: Passenger arrival rate (/min)as a function of p h under distinct α . · time-based tax p l a t f o r m p r o fi t α = 2 . α = 2 . α = 2 . Figure 34: Platform profit (per hour) asa function of p h under distinct α . significantly affects the number of TNC vehicles since their effect is mitigated by the wage floor on TNCdrivers. Furthermore, we find that the time-based charge is Pareto superior compared with the trip-basedcharge that is currently imposed in New York City. Under the time-based charge, more passengers takeTNC rides at a cheaper overall travel cost, drivers remain unaffected, the platform earns a higher profit,and the city collects more tax revenue from the TNC system to subsidize public transit.The policy implication of these results are profound. First of all, our results imply that the TNC driverminimum wage mitigates the effectiveness of the congestion charge (either time-based or trip-based) inreducing the TNC traffic. Therefore, when a driver minimum wage is imposed, the city can not merelycount on the congestion charge to reduce the number of TNC vehicles on the city’s street, unless the chargeis significant and exceeds certain threshold. Second, the TNC profit is rather sensitive to regulationssuch as minimum wage and congestion charges. Based on calibrated model parameters, we showed thatthe tax burden mainly falls on the ride-hailing platform as opposed to passengers and drivers. We arguethat this effect should be taken into account in policy formulation, and an interesting research directionis to synthesize more effective policies that achieve the regulatory objective without jeopardizing the TNCbusiness model, e.g., [51]. Third, our result suggests that the time-based congestion charge is superiorto the trip-based congestion charge. While most city selects the trip-based congestion charge as a natural18andidate of its charge scheme (e.g., NYC, Chicago, Seattle), a shift to the time-based congestion chargeis not difficult to implement: the city only needs to periodically audit the operations data of the TNC andcollect the charge based on the accumulated vehicle hours on the platform.Future research directions include determining the optimal level of congestion charge that maximizes socialwelfare, extending the model to capture temporal and spatial aspect of the TNC market, and characterizingthe impact of regulatory policies on TNC competition. Acknowledgments
This research was supported by the Hong Kong Research Grant Council project HKUST26200420 andNational Science Foundation EAGER award 1839843.
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AppendixA: Proof of Proposition 1
Given N , the first-order derivative of (9) with respect to λ , denoted as ∂P∂λ , can be derived as follows: ∂P∂λ = F − p (cid:18) λλ (cid:19) − αt p ( N − λ/µ, v ) − βp + λ (cid:32) ∂F − p ∂λ + αµ ∂t p ∂N I (cid:33) (26)We note that under the assumptions of Proposition 1 , each term in the right-hand side of (26) is decreasingwith respect to λ . In particular, the first term F − p is a decreasing function of λ by the definition of F p .The second term − t p ( N − λµ, v ) is a decreasing function of λ due to the monotone property of t p . Since F − p is the logit model, we have λ ∂F − p ∂λ = − (cid:15) (cid:18) λλ − λ (cid:19) (27)which is decreasing function of λ . In addition, since t p ( N I , v ) is convex with respect to N I , the last term λ αµ ∂t p ∂N I a=is a decreasing function of λ . Therefore, the objective function (9) is a strictly concave functionof λ when N is fixed.Next, to show that there exists a unique λ that maximizes the profit, it suffices to show that there existsa unique solution to ∂P∂λ = 0 . Apparently, we have ∂P∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ =0 = ∞ and ∂P∂λ (cid:12)(cid:12)(cid:12)(cid:12) λ = Nµ = −∞ . Therefore, bycontinuity, there exists at least one λ ∈ (0 , N µ ) that satisfies ∂P∂λ = 0 . The uniqueness of the solutionfollows from the monotonicity of (26). This completes the proof. Note that this term is negative : Calibration of Model Parameters We describe how the model parameters for San Francisco are obtained. Eleven parameters are to be esti-mated,
Θ = { λ , N , M, L, v f , κ, α, (cid:15), c , σ, w } . Some of these parameters are taken directly from publishedsources, the remaining parameters are based on ‘reverse engineering’: values are selected so that the resultsof the optimization match source data.The number of TNC rides and TNC drivers are from the SFCTA report [6]. It gives hourly passenger arrivalrates and the number of TNC vehicles over an entire day. We take the rates for an‘average’ Wednesday andcalculate the average passenger arrival rate is λ ∗ = 157 . / min and the average number of TNC vehicles is N ∗ = 3000 (Figure 1 of [6]). According to [6], TNC vehicles provide approximately intra-SF vehiclestrips. This implies that the arrival rate of potential passengers is λ = λ ∗ / .
15 = 1049 / min. Assumingthat of the for-hire vehicle drivers work for TNC (the rest work for food delivery, package delivery,etc), the total number of potential drivers is N = N ∗ / . , .The average TNC trip distance in San Francisco is 2.6 miles (Table 4 of [6]), and the average arterial trafficspeed of San Francisco is about 14 mph [7]. This gives the average trip time t = 11 . min. From Lyft’spricing formula for San Francisco [52], a 2.6 mile, . min Lyft ride costs $11 . , i.e., p ∗ f = 11 . . This totalprice is comprised of a service fee of $2 . , a base fare of $2 . , a per-mile fare of $0 . and a per-minutefare of $0 . [52]. The TNC platform keeps of the service fee and of the rest, leading to anaverage of commission rate. This implies that the driver wage is w ∗ = λ ∗ p ∗ f × . /N ∗ = $21 . /hour,since of the total passenger fare is shared by all N ∗ drivers.Taking the average pickup time as 5 minutes (page 2 of [53]), we obtain M based on (23), M = vt p (cid:112) ( N − λ/µ ) = 41 . . where µ = 1 /t is the service rate. To estimate v f and κ in Greenshield’s model, we need two data pointsto fix the linear relationship (25). One data point is already available: we know that on average there are TNC vehicles and the traffic speed is v ∗ = 14 mph. Another data point can be obtained from the factthat arterial traffic speed declined from mph to mph between 2010 and 2016. Since TNC vehiclesonly contributed to part of the decrease of traffic speed. Without TNC vehicles, the traffic speed in 2016would range between 14 and 18 mph. We take v f = 15 mph, indicating that TNCs contribute to 25% of thetraffic speed reduction between 2010-2016 (TNCs may contribute more in the increase of VMT as there isa nonlinear relationship between VMT and traffic speed), which provides another data point (when N = 0 ,traffic speed is v = 15 ). We plug in these two points in (25) and obtain v f = 15 mph and κ = 0 . .Empirical study suggests that the value of travel time (VOT) for TNC customers ranges between $40 and$100 per hour [54], and the the value of time while waiting for vehicle pickup is about 1.5 to 2.5 times VOT[55, 56]. If we take VOT to be $70 /hour and assume that the value of time while waiting is 2 times thevalue of travel time, then we obtain α = 2 . .The parameters of the logit model ( (cid:15), c , σ and w ) can be uniquely determined by the aforementionedparameter values based on reverse engineering. We select (cid:15), c , σ and w so that the optimal solution tothe following unregulated profit maximization problem is consistent with the values of λ , N , M, L, v f , κ, α which we have just determined: max p f ,p d λ ( p f − p d ) (28) The total number of TNC trip is 170000/day (Table 2 of [6]). Since there are virtually no rides between 12AM-6AM, wedivide 170,000/day by 18 hours to derive 157.4 trips/min. To be consistent with passenger arrival rate, we also take the average number of vehicle hours between 6AM to 12AM asthe market volume between 12AM-6AM is trivial. λ = λ F p ( αt p + βt + p f ) (29a) N = N F d (cid:18) λp d N (cid:19) (29b)Reverse engineering yields (cid:15) = 0 . , c = 31 . , σ = 0 . , and w = $31 . /hour. C: Proof of Theorem 1
First, we note that the unregulated problem (28) is equivalent to the following: max p f ,N λp f − N w (30) (cid:26) λ = λ F p ( αt p + βt + p f ) (31a) N = N F d ( w ) (31b)where we used the definition of driver wage (4) to obtain an optimization problem over ride fare p f anddriver wage w . Let ˜ p f and ˜ N be the optimal solution to this unregulated problem (28), and let ˜ λ and ˜ w be the corresponding passenger arrival rate and driver wage.When the minimum wage is greater than the optimal wage ˜ w , i.e., w > ˜ w , the minimum wage constraint(14c) is active. In this case, the regulated profit maximization problem (13) can be reformulated as: max p f ,N λp f − N w (32) (cid:26) λ = λ F p ( αt p + βt + p f + p t ) (33a) N ≤ N F d ( w ) , (33b)where w is given exogenously . Note that (32) can be be equivalently viewed as nested maximization,where in the outer loop the platform chooses the number of drivers N and in the inner loop the platformchooses the ride fare p f to maximize the profit. To prove Theorem 1, we first consider the inner problemwhere N and w are given and the platform solves max p f λp f (34) s.t. λ = λ F p (cid:0) αt p + βt + p f + p t (cid:1) . (35)Denote the optimal value of (34) by Γ( N, p t ) , which depends on N and p t . Since (13) has a unique solutionand F d and F p are continuously differentiable, Γ( N, p t ) is a continuous function with respect to both N and p t . The regulated profit maximization problem (32) can be written as max N Γ( N, p t ) − N w (36) s.t. N ≤ N F d ( w ) . (37)Similarly, the unregulated problem (30) can be written as max N Γ( N, − N w (38) s.t. N = N F d ( w ) . (39) Compared with (31b), the supply function (33b) is an inequality since w is endogenous in (30) while w is exogenous in(32). ˜ w , when the minimum wage w = ˜ w , the optimal number of drivers for (36) and(38) are the same. Furthermore, based on (39), we have w = F − d (cid:18) NN (cid:19) . Therefore (38) can be written as max N Γ( N, − N F − d (cid:18) NN (cid:19) . (40)The first order optimality condition for (40) indicates that ∂ + Γ( ˜
N , ∂N − ˜ w − ˜ Nf d ( ˜ w ) = 0 , (41)where f d ( w ) = ∂F d ( w ) ∂w . Since F d ( w ) is strictly increasing, we have f d ( w ) > , and so ∂ + Γ( ˜
N , ∂N − ˜ w = ˜ Nf d ( ˜ w ) > . (42)We apply (42) to the regulated case (36) and conclude that when the minimum wage satisfies w = ˜ w and the charge is zero, i.e., p f = 0 , the right derivative of the objective function (36) with respect to N is strictly positive (see equation (42)). Since the first order conditions are continuous with respect to w ,there exists w > ˜ w such that for all w ∈ [ ˜ w, w ) we have ∂ + Γ( N ∗ t , ∂N − w > , (43)where N ∗ is the corresponding optimal solution for the regulated problem. Furthermore, due to continuity,for each w ∈ [ ˜ w, w ) , there exists ¯ p t > so that for all p t ∈ [0 , ¯ p t > ∂ + Γ( N ∗ t , p t ) ∂N − w > . (44)This indicates that there exists w > ˜ w such that for w ∈ [ ˜ w, w ) , the derivative of the profit with respectto the number of drivers is strictly positive. This indicates that the platform will earn extra profit if it hiresmore drivers. Therefore, it is optimal for the platform to hire all drivers available in the market, whichgives N ∗ t = N F d ( w ) . Clearly, the optimal number of drivers does not depend on p t when w is fixed. Thiscompletes the proof. D: Proof of Theorem 2
We can first show that there exists w > ˜ w, such that for any ˜ w < w < w , there exists ¯ p h > , so that ∂N ∗ h /∂p h = 0 for p h ∈ (0 , ¯ p h ) . Therefore, we have N ∗ h ( p h ) = N F d ( w ) for ∀ p h ∈ (0 , ¯ p h ) . Hence theoptimal number of passengers will not be affected by p h when p h ∈ (0 , ¯ p h ) . This is because when we knowthat the optimal number of drivers satisfies N ∗ = N F d ( w ) , the profit maximization problem (17) can bewritten as max p f λp f (45) s.t. λ = λ F p (cid:0) αt p + βt + p f (cid:1) . (46)It is clear that p h does not affect the optimal solution to (45). This completes the proof. The proof is similar to that of Theorem 1 and is therefore omitted. : Proof of Theorem 3 To prove Theorem 3, we first show that there exists w > ˜ w , such that for any ˜ w ≤ w ≤ w , there is ¯ p t so that for any trip-based charge p t ∈ [0 , ¯ p t ] , there exists a time-based charge p h that yields the same taxrevenue, but leads to higher passenger surplus and higher platform profit, i.e., N ∗ h = N ∗ t , w ∗ h = w ∗ t = w , λ ∗ h > λ ∗ t , c ∗ h < c ∗ t , P ∗ h > P ∗ t , T r ∗ h = T r ∗ t . The result of Theorem 3 then follows by slightly increasing p h to make sure T r ∗ h > T r ∗ t without changingthe sign of other inequalities. Such p h exists due to continuity.Based on Theorem 1 and Theorem 2, there exists w > ˜ w such that for all w ∈ [ ˜ w, w ) , there is ¯ p t > and ¯ p h > so that for p t ∈ [0 , ¯ p t ) and p h ∈ [0 , ¯ p h ) we have N ∗ h ( p t ) = N ∗ h ( p h ) = N F d ( w ) . Fornotational convenience, define ˆ N ( w ) = N F d ( w ) and suppress the dependence on w to simplify thenotation whenever the context is clear. Since w ≥ ˜ w , the minimum wage constraint is active in both (13)and (17), thus w ∗ t = w ∗ h = w .To show that λ ∗ h > λ ∗ t , we first note that since N ∗ t = ˆ N , it is clear that λ ∗ t is the optimal solution to: max p f λp f (47) s.t. λ = λ F p (cid:0) αt p + βt + p f + p t (cid:1) . (48)We can regard this as an optimization problem over λ . After writing the objective function as solely afunction of λ , the first order optimality condition indicates that: F − p (cid:18) λ ∗ t λ (cid:19) − αt p (cid:18) ˆ N − λ ∗ t /µ, v (cid:19) − βt + λ ∗ t λ f p (cid:18) F − p (cid:18) λ ∗ t λ (cid:19)(cid:19) + λ ∗ t αµ ∂t p ( ˆ N − λ ∗ t /µ, v ) ∂N I − p t = 0 . (49)Similarly, since N ∗ h = ˆ N , we know that λ ∗ h is the optimal solution to max p f λp f (50) s.t. λ = λ F p (cid:0) αt p + βt + p f (cid:1) (51)The first order condition with respect to λ gives F − p (cid:18) λ ∗ h λ (cid:19) − αt p (cid:18) ˆ N − λ ∗ h /µ, v (cid:19) − βt + λ ∗ h λ f p (cid:18) F − p (cid:18) λ ∗ h λ (cid:19)(cid:19) + λ ∗ h αµ ∂t p ( ˆ N − λ ∗ h /µ, v ) ∂N I = 0 . (52)It can be verified that when F p ( c ) and F d ( w ) satisfy the logit model as specified in (21) and (22) andwhen Assumption 1 and Assumption 2 hold, every term of (49) and (52) is an increasing function of λ . Bycomparing (49) and (52) , we have λ ∗ t < λ ∗ h . By stricta monotonicity, this indicates that c ∗ t > c ∗ h .Last, we show that there exists w > ˜ w , such that for any ˜ w ≤ w ≤ w , there is ¯ p t so that for anytrip-based charge p t ∈ [0 , ¯ p t ] , there exists a time-based charge p h which ensures T r ∗ h = T r ∗ t and P ∗ h > P ∗ t .Note that T r ∗ t = λ ∗ t p ∗ t and T r ∗ h = N ∗ h p ∗ h , and that there exists ¯ p t > so that for any p ∗ t ∈ [0 , ¯ p t ] , we can µ and v depend on the number of TNC vehicles N . Since N ∗ h and N ∗ t are the same, we do not need to distinguish µ and v for these two cases. p ∗ h such that λ ∗ t p ∗ t = N ∗ h p ∗ h . After setting p ∗ h and p ∗ t so that T r ∗ t = T r ∗ h , we apply a change of variable(i.e., p f = p f + p t ) to transform (13) into: max p f ,p d ,N λ ( p f − p d ) − λp t (53) λ = λ F p ( αt p + βt + p f ) (54a) N ≤ N F d (cid:18) λp d N (cid:19) (54b) λp d N ≥ w , (54c)Then the two charging schemes (53) and (17) have the same constraints, and the optimal value to (53)satisfies: P ∗ t = λ ∗ t p ∗ f,t − w N ∗ t − λ ∗ t p t (55) = λ ∗ t p ∗ f,t − w N ∗ t − T r ∗ t , (56)where p ∗ f,t is the corresponding optimal ride fare for the trip-based congestion charge. The optimal valueto (17) satisfies P ∗ h = λ ∗ h p ∗ f,h − w N ∗ h − N ∗ h p h (57) = λ ∗ h p ∗ f,h − w N ∗ h − T r ∗ h , (58)where p ∗ f,h is the corresponding optimal ride fare for the time-based charge.Based on Theorem 2, we have λ ∗ h ( p h ) = λ ∗ h (0) and N ∗ h ( p h ) = N ∗ h (0) = for p h ∈ [0 , ¯ p h ) . Therefore, λ ∗ h ( p h ) , N ∗ h ( p h ) and p ∗ f,h ( p h ) are the optimal solutions to the profit maximization problem with minimumwage only (i.e., w > ˜ w and p h = 0 ): max p f ,N λp f − N w (59) (cid:26) λ = λ F p ( αt p + βt + p f + p t ) (60a) N ≤ N F d ( w ) . (60b)This implies that λ ∗ h p ∗ f,h − w N ∗ h > λ ∗ t p ∗ f,t − w N ∗ t . This indicates that P ∗ h > P ∗ t , which completes theproof. Otherwise, this contradicts with the fact that λ ∗ h , N ∗ h , p ∗ f h is the optimal solution to (59).is the optimal solution to (59).