Simple vs. Optimal Congestion Pricing

Simple vs. Optimal Congestion Pricing Dev ansh Jalota Colum bia dj2757@columbia.edu Sharon Di Colum bia sharon.di@columbia.edu Adam N. Elmac h toub Colum bia adam@ieor.columbia.edu F ebruary 26, 2026 Abstract Congestion pricing has emerged as an eﬀectiv e to ol for mitigating traﬃc congestion, yet implemen t- ing welfare or rev enue-optimal dynamic tolls is often impractical. Most real-w orld congestion pricing deplo yments, including New Y ork Cit y’s recen t program, rely on signiﬁcan tly simpler, often static, tolls. This discrepancy motiv ates the question of how m uch rev enue and w elfare loss there is when real-world traﬃc systems use static rather than optimal dynamic pricing. W e address this question by analyzing the p erformance gap betw een static (simple) and dynamic (optimal) congestion pricing schemes in tw o canonical frameworks: Vic krey’s b ottleneck model with a public transit outside option and its cit y-scale extension based on the Macroscopic F undamen tal Diagram (MFD). In b oth mo dels, we ﬁrst characterize the reven ue-optimal static and dynamic tolling p olicies, whic h hav e receiv ed limited atten tion in prior w ork. In the w orst-case, reven ue-optimal static tolls ac hieve at least half of the dynamic optimal reven ue and at most t wice the minim um ac hiev able system cost across a wide range of practically relev an t parameter regimes, with stronger and more general guarantees in the b ottleneck mo del than in the MFD model. W e further corrob orate our theoretical guarantees with n umerical results based on real-world datasets from the San F rancisco Bay Area and New Y ork City , whic h demonstrate that static tolls achiev e roughly 80-90% of the dynamic optimal reven ue while incurring at most a 8-20% higher total system cost than the minimum achiev able system cost. 1 In tro duction T raﬃc congestion has surged in ma jor cities w orldwide, straining infrastructure, degrading air qualit y , and imp osing substan tial economic costs (Fleming 2019, F ernandez 2025). Against this backdrop, congestion pricing has emerged as an eﬀective tool for mitigating the ineﬃciencies of traﬃc congestion and empirical evidence from recen t large-scale deplo yments underscores its promise. F or instance, New Y ork City’s newly implemen ted congestion pricing program in January 2025 has substantially reduced vehicle volumes, with the potential to generate billions of dollars in rev enue for transit inv estments (Cook et al. 2025, Ostro vsky and Y ang 2024). Similar successes ha v e b een documented in Stockholm and Singap ore (Hu et al. 2023). While man y large-scale congestion pricing deplo ymen ts, including those in London (2003), Sto c kholm (2007), and New Y ork City (2025), ha v e emerged in the past tw o decades, their in tellectual roots date back to Pigou (1920). Under Pigou’s framework, the socially optimal toll charges each tra veler the marginal cost they imp ose on others in the netw ork. Ho wev er, implementing Pigouvian tolls in practice is challenging and often impractical, as it requires time-v arying, state-dep enden t tolls that resp ond to ev olving traﬃc conditions, p otentially at the granularit y of individual netw ork links. Suc h ﬁne-grained dynamic pricing demands sophisticated sensing, comm unication, and computation infrastructure, while placing substan tial informational and cognitive burdens on tra v elers. Consequen tly , despite its theoretical app eal, fully dynamic marginal-cost or Pigouvian tolling remains largely infeasible in real-w orld congestion pricing deploymen ts. Instead, most op erational congestion pricing systems rely on m uc h simpler, often static, tolling structures. F or instance, ﬁxed (static) c harges are levied on sp eciﬁc b ottlenec ks, bridges, or crossings, suc h as the San F rancisco (SF)–Oakland Bay Bridge in the SF Bay Area (Gonzales and Christofa 2015). Likewise, large 1 urban cordon or area-based systems, suc h as those in London and New Y ork Cit y , t ypically impose a ﬁxed (static) entry or usage fee for designated congestion zones during peak perio ds. Such static tolling schemes are app ealing as they are transparent, easy to communicate to users, and require less real-time information and computational infrastructure than theoretically optimal dynamic pricing. Y et, the simplicity of static tolling means that it may deviate substantially from idealized marginal- cost pricing, raising a central question: How much p erformanc e is sacriﬁc e d when r e al-world tr ansp ortation systems use simple static pricing inste ad of optimal dynamic c ongestion pricing? Understanding this gap is crucial for p olicymakers as cities seek congestion pricing schemes that are operationally feasible and deliv er substan tial w elfare and rev enue gains. This question is also increasingly relev ant for transportation authorities exploring more dynamic approaches, such as managed express lanes that adjust tolls based on real-time conditions (Pandey and Boyles 2018, Jang et al. 2014), or Singap ore’s ERP 2.0 system, which aims to enable real-time, link-lev el pricing across its road netw ork (of T ransp ort 2025). This w ork addresses the abov e question b y analyzing the p erformance gap in terms of rev en ue and welfare (or system cost) betw een simple static and optimal dynamic congestion pricing sc hemes in tw o canonical mo deling frameworks that capture the dominant real-w orld use cases of congestion pricing. In particular, we study (i) Vickrey’s bottleneck mo del (Vic krey 1969), a foundational and analytically tractable represen tation of p eak-p erio d congestion that captures queue formation and dissipation at a b ottlenec k, and (ii) its cit y- scale urban system extension based on the Macroscopic F undamen tal Diagram (MFD) (Daganzo 2007), which incorp orates spatially distributed origins and destinations and mo dels congestion through a state-dep endent capacit y , rather than a ﬁxed b ottleneck capacity . W e fo cus on these mo dels as they provide clean, tractable framew orks in which to isolate the core mechanisms gov erning static versus dynamic tolling performance, while also reﬂecting the structure of con temp orary deploymen ts, from b ottleneck-priced corridors such as the SF–Oakland Ba y Bridge to cit y-scale congestion pricing systems in New Y ork City and London. In studying these mo dels, we fo cus on reven ue-maximizing tolls, which hav e received limited attention in the literature that has largely emphasized welfare-optimal or system-cost-optimal tolls. Beyond ﬁlling this research gap, our fo cus on reven ue maximization is motiv ated by several practical considerations. First, rev enue generation is an explicit p olicy ob jective of many congestion pricing programs (e.g., New Y ork Cit y), where toll reven ues are earmarked for transit inv estmen ts (MT A 2025). Next, reven ue ob jectives are central to public–priv ate toll road partnerships, including tolled road netw orks op erated by ﬁrms such as T ransurban (O’Neill 2022). Finally , our numerical exp erimen ts in Section 6 demonstrate that static rev enue-optimal tolls often coincide, or p erform nearly iden tically to, system-cost-optimal static tolls across empirically relev ant parameter regimes, reinforcing our fo cus on rev en ue-optimal tolling. Ov erall, while a k ey focus of this work is to study reven ue-optimal tolling p olicies, we also ev aluate their welfare (system cost) implications, pro viding a comprehensiv e assessment of p erformance across the metrics most relev ant to p olicymakers. Our Con tributions: This work analyzes the p erformance of static versus dynamic congestion pricing in t wo canonical settings: Vickrey’s bottleneck model with a public transit outside option (e.g., a subw ay) and its MFD-based urban system extension. In doing so, a central contribution is the characterization of rev enue-optimal static and dynamic tolls in b oth mo dels. F or the b ottleneck mo del with an outside option, w e lev erage the equilibrium prop erties of Vickrey’s framework to sho w that computing rev enue-optimal static and dynamic tolls reduces to solving single-variable quadr atic pr o gr ams that admit closed-form solutions. F or urban systems, suc h closed-form results are t ypically intractable for arbitrary MFD relations; thus, we fo cus on the widely studied and empirically supp orted triangular fundamental diagram (see Section 5 for more details). W e c haracterize the reven ue-optimal dynamic toll, showing that it maintains the system at the throughput-maximizing capacity , and deriv e a closed-form expression for the rev enue generated by any static toll. Lev eraging these c haracterizations, we compare static rev en ue-optimal tolling with its optimal dynamic coun terparts on b oth rev enue and total system cost metrics (see Section 3.3 for details), which are of direct relev ance to p olicymak ers. Our results establish low er b ounds on the reven ue and upp er b ounds on the total system costs achiev able under static-reven ue optimal tolls relative to the dynamic b enchmarks. In Vickrey’s b ottlenec k mo del with an outside option, we sho w that static reven ue-optimal tolls obtain at le ast half of 2 the rev enue ac hieved by dynamic reven ue-optimal tolls across all parameter regimes and at le ast two-thir ds in many practically relev an t regimes. Static rev enue-optimal tolls do not alwa ys guaran tee constant-factor appro ximations for system cost. How ev er, when the public transit outside option is not substantially less attractiv e than driving a car, a condition that holds in many real-world settings, static reven ue-optimal tolls incur at most twic e the optimal system cost. In the MFD framework, the guarantees we obtain are w eaker (as the capacit y is state-dependent) but still robust: across a range of practically relev ant parameter regimes, static reven ue-optimal tolls still achiev e at le ast two-thir ds of the dynamic optimal reven ue and incur at most twic e the minimal system cost. Finally , we complement our theory with numerical experiments based on tw o real-w orld case studies: the SF–Oakland Ba y Bridge for the b ottlenec k m odel and New Y ork Cit y’s congestion reduction zone for the MFD framework. Our results show that static reven ue-optimal tolls ac hieve roughly 80-90% of the dynamic optimal reven ue while incurring at most a 8-20% higher total system cost than the minimum ac hiev able for practically relev ant parameter regimes, though this gap can widen when the public transit option is signiﬁcantly less attractive (i.e., more costly), consistent with theory . Moreo ver, when ev aluated through the lens of our model, w e ﬁnd that the static tolls curren tly implemen ted in practice in both case studies corresp ond to parameter regimes in whic h static reven ue-optimal tolls capture nearly all the b eneﬁts of dynamic tolling, achieving roughly 98% of the dynamic optimal rev enue and a 3% higher total system cost than the minimum ac hiev able. Beyond the attractiveness of public transit relative to driving, a k ey determinan t of the performance of static reven ue-optimal tolling is the ratio of the maxim um system throughput to the desired user arriv al rate. When this ratio is close to one, as in the Bay Bridge study , static reven ue-optimal tolls capture a larger fraction of the dynamic optimal reven ue but incur a higher m ultiplicative gap relativ e to the minim um achiev able system cost. This is in comparison to the New Y ork Cit y study , where this ratio is substantially lo wer due to its m uch higher public transit mode share. Th us, in addition to highlighting the eﬃcacy of static reven ue-optimal tolls, our numerical results illustrate a fundamen tal trade-oﬀ betw een reven ue and system cost ob jectives under static tolling. Ov erall, our results demonstrate that simple static tolls can deliver robust performance on b oth reven ue and system cost metrics, underscoring its practical eﬃcacy and shedding light on the strong empirical perfor- mance of the many static congestion pricing systems already in op eration. Moreov er, since our comparisons abstract from the substan tial informational and computational burdens that dynamic tolling places on b oth transp ortation planners and users, our theoretical b ounds should b e in terpreted as worst-case guarantees. Incorp orating realistic b eha vioral or op erational constraints w ould only further strengthen the case for static tolling in real-w orld deplo ymen t. Or ganization: The remainder of this paper is organized as follows. Section 2 reviews related literature. Section 3 in tro duces our mo del and reviews equilibrium outcomes in Vickrey’s bottleneck mo del with and without an outside option, whic h form the foundation of our analysis. Then, Section 4 derives the rev en ue- optimal static and dynamic tolling p olicies in Vic krey’s b ottlenec k mo del with an outside option and compares the performance of reven ue-optimal static tolls to its dynamic b enc hmarks. Section 5 extends these results to urban systems based on the MFD. Section 6 presents n umerical exp eriments. Section 7 concludes and pro vides directions for future work. 2 Related Literature The study of p eak-p erio d congestion originates from Vickrey’s b ottlenec k mo del (Vickrey 1969), which c har- acterizes the equilibrium departure time choices of users tra versing a b ottleneck with a ﬁxed capacity . This framew ork has served as the foundation for an extensive literature, with numerous extensions including mo dels of parallel-route choice (Arnott et al. 1990a), mo dal split (T abuchi 1993), heterogeneous prefer- ences (Newell 1987, Lindsey 2004), carp o oling (Xiao et al. 2016, Ostrovsky and Sch warz 2025), stochastic demand (Arnott et al. 1999, de Palma et al. 1983), and b ounded rationality (Mahmassani and Chang 1987), among others. While the b ottlenec k mo del and its extensions capture queuing delays as a p oint queue, they do not accoun t for h yp ercongestion, wherein system throughput declines once road density exceeds a crit- ical threshold. This limitation has motiv ated cit y-scale extensions, including bath tub mo dels of down town 3 traﬃc (Arnott 2013) and queuing formulations (F osgerau and Small 2013) in which the b ottleneck capacity is state-dep enden t and degrades under heavy congestion. Relatedly , the MFD (Daganzo 2007, Geroliminis and Daganzo 2008) provides a parsimonious represen tation of the relationship betw een system-wide v ehicle densit y and throughput. F or a comprehensive surv ey of the b ottleneck model and its extensions, see Li et al. (2020). Building on these foundations, a substantial literature has examined congestion pricing as a mechanism for mitigating traﬃc, with a fo cus on system-cost-optimal tolls that in ternalize congestion externalities (Pigou 1920). In the b ottleneck mo del, the system-cost-optimal toll is time-v arying, requiring contin uously ad- justable charges (Vic krey 1969, Arnott et al. 1990b, Newell 1987). While suc h dynamic pricing is theoretically eﬃcien t, it is op erationally complex, motiv ating the design of more implementable sc hemes, including uni- form, stepwise, and coarse time-of-day tolls (Chu 1999, Arnott et al. 1993a, Braid 1989). F or example, Laih (1994) show ed that an optimal n -step toll can eliminate at most n n +1 of the queuing delay compared to the time-v arying optimal toll. Related studies extend these ideas b eyond a single b ottleneck to parallel net works (Arnott et al. 1990a, Braid 1996). Like these works, w e compare simple static tolls with (optimal) dynamic congestion pricing. Ho w ever, we consider a mo del with an outside option (e.g., public transit), whic h substan tially alters equilibrium behavior, and ev aluate tolling p olicies under a reven ue-maximization ob jective. In this regard, our w ork connects to prior studies that incorporate an outside option or elastic demand in to Vickrey’s b ottleneck mo del (Arnott et al. 1993b, Gonzales and Daganzo 2012), and to the literature on rev enue-maximizing tolls for priv ately op erated facilities (De P alma and Lindsey 2000, de Palma and Lindsey 2002, F u et al. 2018). Existing reven ue-maximization studies in the b ottleneck framework primarily examine equilibrium outcomes arising from comp etition among ﬁrms, and w ork incorporating outside options predominan tly fo cus on devising system-cost-optimal tolling p olicies. In contrast, w e study the design of rev en ue-maximizing tolling p olicies and pro vide explicit guaran tees that quan tify the p erformance gap b et w een static and dynamic tolling. Bey ond the tolling literature rooted in Vickrey’s b ottleneck framework, our work also connects to the broader work on congestion pricing in settings where ﬁrst-b est Pigouvian pricing is infeasible due to policy , b eha vioral, or infrastructural constraints. This includes research on second-b est congestion pricing, which examines toll design when only a subset of netw ork links can b e priced (V erho ef 2002, Labb´ e et al. 1998, Larsson and P atriksson 1998, P atriksson and Rock afellar 2002, Di et al. 2016), and optimal cordon pricing, whic h uses zone-based tolls to manage congestion in urban systems (il Mun et al. 2003, Zhang and Y ang 2004). Like these works, we study simple, implementable tolling p olicies; ho w ever, unlik e the large-scale bi-lev el or mixed-in teger optimization mo dels used in these works, we deriv e closed-form optimal static and dynamic tolls in both the bottleneck mo del and its MFD extension, yielding transparent structural insights and theoretical guaran tees. Our w ork also connects to the literature on simplicity versus optimalit y in algorithm and mec hanism design (Hartline and Roughgarden 2009, Hart and Nisan 2017), including studies of static pricing in rev enue managemen t (Elmach toub and Shi 2025, Besb es et al. 2022) and online platforms (Banerjee et al. 2015), whic h sho w that simple, time-inv ariant prices can achiev e strong p erformance relativ e to their optimal dynamic pricing counterparts. In this spirit, we also show that simple static tolls can achiev e p erformance guarantees relativ e to optimal dynamic tolling. Finally , since we compare tolling policies in terms of reven ue and w elfare (or system cost), our w ork relates to the broader economics literature on the tension betw een reven ue and welfare-maximizing pricing rules. This trade-oﬀ is w ell kno wn in auction theory , most notably in the con trast b et w een welfare-maximizing Vic krey auctions (Vickrey 1961) and Myerson’s reven ue-maximizing mechanisms (Myerson 1981), and has also b een studied in settings including security games (Jalota et al. 2024), congestion games (Zhang et al. 2024), and rev enue managemen t (Chen and Gallego 2019, Zhang and Dong 2025). Con tributing to this literature, we analyze reven ue-welfare tradeoﬀs in a congestion pricing con text and quantify how muc h rev enue can b e retained and how muc h welfare is lost when moving from optimal dynamic pricing to simple static tolls. 4 3 Mo del and Background: Bottlenec k with Outside Option This section presen ts Vickrey’s bottleneck model with an outside option (Section 3.1), reviews its untolled equilibrium outcome established in Gonzales and Daganzo (2012) (Section 3.2), and deﬁnes the reven ue and system cost metrics to ev aluate the tolling p olicies w e study (Section 3.3). 3.1 Setup W e study Vic krey’s bottleneck mo del (Vickrey 1969), a canonical represen tation of p eak-p erio d congestion in capacit y-constrained facilities such as bridges and tunnels, augmen ted with a public transit outside option. In this framew ork, a mass of Λ users makes morning comm ute trips using either (i) a car, whic h requires tra versing a b ottleneck of capacity (service rate) µ , or (ii) a public transit alternative (e.g., a sub wa y). As in Vic krey (1969), users’ desired b ottlenec k crossing times are uniformly distributed ov er the time in terv al [ t 1 , t 2 ] (e.g., morning rush). In the absence of the b ottleneck, users would arrive at their destination at their desired times, with a corresp onding desired arriv al rate λ = Λ t 2 − t 1 . Unlike stochastic queuing models, e.g., based on P oisson processes, w e clarify that arriv als and service are deterministic. In addition to b ottlenec k congestion that may induce deviations from users’ desired bottleneck crossing times and inﬂuence mo de c hoices, users’ tra v el decisions can also be inﬂuenced via a (possibly time-v arying) toll ˜ τ ( t ) on cars crossing the b ottleneck at time t . The k ey distinction b et ween car and public transit is how they inﬂuence users’ tra v el costs. F or car users, the tra v el cost dep ends on their desired bottleneck crossing time t ∗ , the prev ailing level of congestion, and tolls. Normalizing the queuing delay at the b ottleneck to zero under free-ﬂo w conditions (i.e., in the absence of congestion), the cost of a car trip comprises four comp onen ts: (i) a congestion indep enden t free-ﬂo w generalized cost ˜ z C , (ii) queuing (or waiting) delay w ( t ) to tra verse the b ottleneck due to congestion, (iii) sc hedule delay | t ∗ − t | , denoting the deviation b etw een a user’s desired and actual bottleneck crossing time, and (iv) the toll ˜ τ ( t ). Let c W denote the p enalty for incurring a unit of waiting time delay , and c e and c L b e the sc hedule delay p enalties for arriving early or late, resp ectively , which is assumed to be identical across users (Vickrey 1969, Ostrovsky and Sch w arz 2025), with 0 < c e < c W and c L > 0, as is standard in the literature (Vic krey 1969). Then, the total trav el cost for a car user with a desired b ottleneck crossing time t ∗ who crosses the b ottlenec k at time t is: ˜ c ( t, t ∗ ) = ˜ z C + c W w ( t ) + c e ( t ∗ − t ) + + c L ( t − t ∗ ) + + ˜ τ ( t ) . (1) Unlik e car users, w e assume that transit users incur a ﬁxed generalized cost ˜ z T , as is standard in the literature (Gonzales and Daganzo 2012). This sp eciﬁcation is reﬂective of settings in whic h the public transit system charges a ﬁxed fare, op erates at ﬁxed headw a ys, and is segregated from the road net w ork, features t ypical of metro or subw ay systems (e.g., New Y ork City’s subw a y), and thus does not exp erience congestion dela ys. While richer mo dels could also allo w the transit cost to dep end on ridership (e.g., to capture crowding or discomfort eﬀects), a ﬁxed transit cost enables analytical tractability , allo wing us to isolate the core mechanisms through whic h static and dynamic pricing inﬂuence user b ehavior and system outcomes. Nevertheless, relaxing this assumption to incorp orate crowding eﬀects or other dep endencies of the transit cost on ridership is a v aluable direction for future work. In the remainder of this work, for analytical clarity , we normalize user costs by the waiting time p enalty c W . Then, the normalized transit cost is z T = ˜ z T c W , and letting z C = ˜ z C c W , τ ( t ) = ˜ τ ( t ) c W , c e c W = e , and c L c W = L , the normalized cost for a car user from Equation (1) is giv en by: c ( t, t ∗ ) = z C + w ( t ) + e ( t ∗ − t ) + + L ( t − t ∗ ) + + τ ( t ) . (2) 3.2 Equilibria in the Untolled Bottlenec k Mo del Under the setup and user costs deﬁned in the previous section, an e quilibrium arises when all users minimize tra vel costs by choosing a mo de (car or transit) and, if trav eling by car, a b ottlenec k crossing time, resulting in a pattern of departure times, mo de c hoices, and induced congestion eﬀects under which no user has an incen tive to deviate (Vic krey 1969, Hendrickson and Kocur 1981). Giv en cost-minimizing user b ehavior, w e no w review well-established equilibrium characterization results of the b ottleneck mo del in the untolled 5 setting ( τ ( t ) = 0 for all t ), b oth without and with a public transit outside option, whic h forms the foundation for our analysis of static and dynamic tolling sc hemes in Sections 4 and 5. Bottlene ck without Outside Option: T o build intuition, w e ﬁrst consider the setting without an outside option, in whic h all users tra vel by car and choose only their b ottleneck crossing time t . Sp eciﬁcally , a user with a desired crossing time t ∗ selects a time t to pass the b ottleneck to minimize their tra vel cost c ( t, t ∗ ) in Equation (2). If the b ottlenec k service rate is at least the desired arriv al rate, i.e., µ ≥ λ , no queue forms at equilibrium and all users cross the b ottleneck at their desired times, incurring a cost z C . When µ < λ , as is t ypical during morning and evening rush p erio ds, the b ottleneck cannot serve all Λ users during the in terv al [ t 1 , t 2 ] and congestion dela ys arise. In this congested regime, users incur waiting and sc hedule delay costs, and the Λ users are served at the b ottlenec k capacity µ ov er an interv al [ t ( N o ) A , t ( N o ) D ], where t ( N o ) D − t ( N o ) A = Λ µ > t 2 − t 1 . Here, the sup erscript No denotes the setting with no outside option. Over this interv al, the equilibrium waiting-time function w ( t ) is determined by the ﬁrst-order optimality condition for cost-minimizing users, c ′ ( t, t ∗ ) = 0, under whic h no user can reduce their trav el cost through a marginal adjustmen t in their bottleneck crossing time. This condition implies that the slop e of the waiting-time function satisﬁes w ′ ( t ) = e for early arriv als and w ′ ( t ) = − L for late arriv als, which is indep endent of users’ desired crossing times t ∗ , thus characterizing the equilibrium condition for all users. T o characterize the waiting time proﬁle from this diﬀerential equation, note that the user with the maxim um waiting time at equilibrium must cross the b ottlenec k at their desired time; otherwise, a marginal shift in their departure time would reduce sc hedule delay without increasing w aiting time, th us low ering their trav el cost. Let t ∗ = ˜ t denote this user’s desired crossing time. Then, the w aiting time proﬁle is triangular with w ′ ( t ) = e for t < ˜ t and w ′ ( t ) = − L for t > ˜ t , as sho wn in Figure 1 (left). The critical user with a desired crossing time ˜ t incurs the maxim um tra v el cost, and experiences the maximum waiting time T C = Λ eL µ ( e + L ) . F or more details on b ottlenec k equilibria and the resulting equilibrium bottleneck arriv al and departure (crossing) time proﬁles of users, see Vic krey (1969), Hendrickson and Ko cur (1981) and Figure 7 in App endix A.1. Figure 1: Equilibrium w aiting time p roﬁles in the bottleneck mo del without an outside option (left) and with a public transit outside option in a mixed-mo de equilibrium where b oth car and transit are used (right) in the setting when µ < λ . The slop es e and L denote the normalized schedule-delay p enalties for ea rly and late arrivals, resp ectively , and [ t 1 , t 2 ] rep resents users’ desired b ottleneck crossing times. Bottlenec k with Outside Option: When a public transit outside option is av ailable, users choose not only when to tra vel but also whether to trav el by car or transit. While car users ma y incur congestion delays, transit users (e.g., subw ay riders) face a ﬁxed cost z T and can arrive at their destination at their desired time; hence, they do not face a departure-time decision. Then, equilibrium mode choice and, for car users, departure-time decisions are determined b y the relativ e magnitudes of the costs z T , z C , and the maximum w aiting time T C in the car-only b ottleneck. This yields three parameter regimes, corresp onding to diﬀerent equilibrium mo de-use patterns, summarized below. Case 1 ( z T < z C ): All users take transit, as it is strictly preferred to a car trip at free-ﬂo w. Case 2 ( z T ≥ z C + T C ): T ransit is never cost-eﬀective relative to car trav el, even for the user facing the highest trav el cost with a w ait time of T C in the car-only b ottleneck. Th us, all users tra vel by car, and the equilibrium coincides with the b ottleneck mo del without an outside option. Case 3 ( z T ∈ [ z C , z C + T C ] ): In this intermediate regime, a mixed-mo de equilibrium arises in which b oth car and transit are used, as characterized by the following prop osition from Gonzales and Daganzo (2012). 6 Prop osition 1 (Two-Mode Equilibrium (Gonzales and Daganzo 2012)) . Supp ose users c an cho ose b etwe en two mo des, tr aveling by c ar thr ough a b ottlene ck with a fr e e-ﬂow c ost z C , and using public tr ansit with a ﬁxe d c ost z T , wher e z T ≥ z C . L etting z T − z C < T C and assuming users p ass the b ottlene ck in or der of their desir e d b ottlene ck dep artur e times, ther e exists an e quilibrium such that (se e right of Figur e 1): 1. The numb er of e arly c ar users N e = µ ( z T − z C ) e , who tr avel at the start of the rush b etwe en [ t A , t B ] . The numb er of late c ar users N L = µ ( z T − z C ) L , who tr avel at the end of the rush b etwe en [ t C , t D ] . 2. The numb er of on time c ar users and tr ansit riders N o ( · ) , N T ( · ) ar e strictly de cr e asing functions of the c ost diﬀer enc e z T − z C and they tr avel in the midd le of the rush b etwe en [ t B , t C ] . Prop osition 1 characterizes the equilibrium structure in the regime z T ∈ [ z C , z C + T C ], which is arguably the most practically relev ant case: while public transit is typically less attractive than driving under free-ﬂo w conditions, it is often not suﬃciently worse for all users to trav el b y car, so that b oth mo des are used at equilibrium. In this regime, Prop osition 1 shows that the w aiting time proﬁle for car users is trap ezoidal (see righ t of Figure 1), whic h arises as the queuing dela ys at the car bottleneck can nev er exceed z T − z C , since any such user would prefer to switch to transit. W e also note that the interv al [ t A , t D ] in Prop osition 1 need not coincide with, and is generally a subset of, the corresponding interv al [ t (No) A , t (No) D ] in the car-only b ottlenec k. F or more details on equilibria in the b ottlenec k mo del with an outside option and the resulting equilibrium b ottleneck arriv al and departure (crossing) time proﬁles of car users, see Gonzales and Daganzo (2012) and Figure 8 in Appendix A.1. 3.3 P erformance Metrics to Ev aluate Equilibria Induced b y T olling P olicies Ha ving presented the untolled equilibrium outcomes in the b ottlenec k mo del with and without an outside option, we now presen t the metrics to assess the equilibria induced b y a (possibly time-v arying) tolling policy τ ( · ). W e fo cus on reven ue and total system cost metrics, b oth of which w e normalize b y the w aiting time p enalt y c W . The reven ue of a tolling p olicy , denoted R ( τ ( · )), is deﬁned as the total toll pa yments collected from car users at the resulting equilibrium. The total system cost, denoted S C ( τ ( · )), measures the aggregate tra vel burden b orne b y users at equilibrium and is deﬁned as the sum of w aiting time costs, schedule dela y costs, and generalized costs (i.e., z T or z C ) asso ciated with car or transit use. Equiv alen tly , the total system cost is the sum of users’ total tra vel costs at equilibrium minus their toll paymen ts. Consistent with standard practice in congestion pricing (Vickrey 1969, Gonzales and Daganzo 2012) and e conomics (Vickrey 1961) that excludes tolls (or prices) when deﬁning system cost or w elfare, toll paymen ts are excluded from total system cost, as they represent transfers b etw een users and the planner. F or formal mathematical deﬁnitions of b oth metrics, see Appendix B. 4 Static vs. Dynamic T olling in the Bottlenec k Mo del This section b egins our p erformance comparison betw een static reven ue-optimal tolling and its rev enue and system-cost-optimal dynamic tolling counterparts in the b ottleneck mo del with an outside option. In this setting, w e ﬁrst deriv e closed-form expressions for reven ue-optimal static and dynamic tolls in Sections 4.1 and 4.2, resp ectiv ely . Leveraging these characterizations, we then quantify the p erformance gap b etw een static reven ue-optimal tolls and its dynamic optimal b enc hmarks by establishing lo wer b ounds on the fraction of optimal rev enue attained and multiplicativ e upp er bounds on total system cost in Sections 4.3 and 4.4. W e fo cus on the non-trivial and empirically relev an t regimes when µ < λ , i.e., the congested regime when the bottleneck service rate is strictly b elow the desired bottleneck departure rate, and z T ≥ z C , i.e., car tra vel under free-ﬂow is preferred to transit. Note that when µ ≥ λ , congestion do es not arise and the static and dynamic optimal tolls on b oth reven ue and system cost metrics coincide, corresp onding to a uniform toll of max { z T − z C , 0 } . Moreo ver, when z T < z C , all users strictly prefer transit under any tolling policy , yielding no rev enue and a total system cost of z T Λ. 7 4.1 Static Rev enue-Optimal T olls This section c haracterizes the reven ue-optimal static toll τ in the b ottleneck mo del with an outside option, where the toll is constant o ver time (i.e., τ ( t ) = τ for all t ). Under a static toll τ , let N T ( τ ) denote the n um b er of users that choose transit at equilibrium. The toll reven ue is given by R ( τ ) = τ (Λ − N T ( τ )). Prop osition 2 deriv es an expression for the rev enue-optimal static toll τ ∗ s , showing that it dep ends on the magnitude of the cost diﬀerence b etw een transit and car trav el at free-ﬂow, giv en by z T − z C . When this diﬀerence is b elo w a threshold, reven ue is maximized by setting the toll at its highest feasible lev el, z T − z C , b eyond whic h all users w ould switc h to transit. In con trast, when this diﬀerence is larger than that threshold, the rev enue-optimal static toll is strictly lo wer and dep ends on the remaining model parameters, as c haracterized in Prop osition 2. Prop osition 2 (Reven ue-Optimal Static T olls) . Supp ose µ < λ and users cho ose b etwe en two mo des, tr aveling by c ar thr ough a b ottlene ck with a fr e e ﬂow c ost of z C , and using a public tr ansit alternative with a ﬁxe d c ost of z T , wher e z T ≥ z C . Then, the r evenue-optimal static tol l is given by: τ ∗ = ( max n z T − z C 2 + Λ eL 2( λ − µ )( e + L ) , z T − z C − Λ eL µ ( e + L ) o , if z T − z C ≥ Λ eL ( λ − µ )( e + L ) z T − z C , if 0 ≤ z T − z C < Λ eL ( λ − µ )( e + L ) . (3) Pr o of. First note that a static toll τ ≥ 0 must be such that z T − z C − τ ≤ T C . Note that if z T − z C − τ > T C , then τ can b e increased to z T − z C − T C without changing the num b er of car users, resulting in a higher rev enue. Next, for any τ ≥ max { 0 , z T − z C − T C } , note that the equilibrium is as sp eciﬁed in Proposition 1 other than that the maximum waiting time is ¯ w = z T − z C − τ . Next, we derive an expression for the num ber of users trav eling by transit under the toll τ . By the linearit y of the w aiting time proﬁle, all car users that pass the bottleneck either early or late (i.e., betw een [ t A , t B ] and [ t C , t D ], resp ectiv ely , on the righ t of Figure 1) corresp ond to a ¯ w T C fraction of all users. The remaining 1 − ¯ w T C fraction arrive exactly on time (either using transit or car), where the total on-time car users is N o ( τ ) = (1 − ¯ w T C )( t 2 − t 1 ) µ = (1 − ¯ w T C ) Λ λ µ and transit users is N T ( τ ) = (1 − ¯ w T C )Λ  1 − µ λ  . W e then obtain the following expression for the reven ue as a function of τ : R ( τ ) = τ [Λ − N T ( τ )] = τ h Λ −  1 − z T − z C − τ T C  Λ  1 − µ λ  i ( a ) = µτ h Λ λ + ( z T − z C − τ )( e + L ) eL  1 − µ λ  i , where (a) follows as T C = Λ eL µ ( e + L ) . Con- sequen tly , the static reven ue optimization problem is: max τ ∈ R R ( τ ) := µτ  Λ λ + ( z T − z C − τ )( e + L ) eL  1 − µ λ   s.t. max { 0 , z T − z C − T C } ≤ τ ≤ z T − z C . (4) T aking the ﬁrst-order condition of this single dimensional quadratic optimization problem, we can sho w that its optimal solution corresp onds to the static tolls in the statemen t of the prop osition. Prop osition 2 characterizes the reven ue-optimal static toll by casting the static rev en ue optimization problem as a single-v ariable quadratic program, where the optimal toll dep ends on the magnitude of z T − z C relativ e to the threshold Λ eL ( λ − µ )( e + L ) . A key step in proving this result requires extending the equilibrium analysis in Prop osition 1 (Gonzales and Daganzo 2012), whic h characterizes monotonicity prop erties of the n umber of users choosing transit at equilibrium, to obtain an explicit expression for this quantit y . This explicit characterization enables closed-form expressions for b oth reven ue and total system cost under the optimal static toll across parameter regime s, whic h underpins our p erformance comparison b etw een static and dynamic tolling in Sections 4.3 and 4.4. 4.2 Dynamic Rev enue-Optimal T olls W e now c haracterize the reven ue-optimal dynamic tolling p olicy τ ∗ d ( · ). Theorem 1 establishes that the rev enue-optimal dynamic toll has the trapezoidal structure shown in Figure 2, where the tolling p olicy is constan t and equal to z T − z C o ver the middle of the rush, i.e., ov er the interv al [ t ∗ B , t ∗ C ], and decreases linearly on either side of this in terv al. The corresp onding equilibrium arriv al and departure (crossing) time proﬁles of users at the bottleneck under this tolling p olicy are as depicted in Figure 9 in App endix A.2. 8 Theorem 1 (Rev en ue-Optimal Dynamic T olls) . Supp ose µ < λ and users cho ose b etwe en tr aveling by c ar thr ough a b ottlene ck with a fr e e ﬂow c ost z C and using public tr ansit with a ﬁxe d c ost z T , wher e z T ≥ z C . Then, the r evenue-optimal dynamic tol ling p olicy τ ∗ d ( · ) is such that (se e Figur e 2): 1. The tol l is c onstant and ﬁxe d at τ ∗ d ( t ) = z T − z C for al l t ∈ [ t ∗ B , t ∗ C ] during the midd le of the rush. Mor e over, the time interval [ t ∗ B , t ∗ C ] over which the tol l r emains c onstant c orr esp onds to a fr action f ∗ = max { 1 − ( z T − z C ) µ ( e + L ) Λ eL  1 − µ λ  , 0 } of the interval [ t 1 , t 2 ] , i.e., t ∗ C − t ∗ B t 2 − t 1 = f ∗ . 2. F or al l t ∈ [ t ∗ A , t ∗ B ] , τ ∗ ( t ) = z T − z C − e ( t ∗ B − t ) , and for al l t ∈ [ t ∗ C , t ∗ D ] , τ ∗ ( t ) = z T − z C − L ( t − t ∗ C ) , wher e t ∗ B − t ∗ A = L e + L (1 − f ∗ )Λ µ , and t ∗ D − t ∗ C = e e + L (1 − f ∗ )Λ µ . Mor e over, under this tol ling p olicy, the optimal r evenue is given by: R ∗ = ( ( z T − z C )Λ µ λ + ( z T − z C ) 2 µ ( e + L ) 2 eL  1 − µ λ  2 , if z T − z C ≤ Λ eL ( λ − µ )( e + L ) λ µ , ( z T − z C )Λ − Λ 2 2 µ eL e + L , if z T − z C > Λ eL ( λ − µ )( e + L ) λ µ . (5) Figure 2: Depiction of the revenue optimal dynamic tolling p olicy . Here, [ t ∗ A , t ∗ D ] denotes the equilibrium interval over which users pass the b ottleneck under the dynamic revenue-optimal tolling p olicy τ ∗ d ( · ) , analogous to the no-toll equilibrium interval [ t A , t D ] sho wn in the right of Figure 1. The sub-interval [ t ∗ B , t ∗ C ] co rresp onds to the middle of the rush, during which the toll is constant and equal to z T − z C . Pr o of. Under the cost function in Equation (2), a necessary condition at equilibrium is that all users minimize tra vel costs when using a car, i.e., the follo wing condition must hold: c ′ ( t, t ∗ ) = 0, w hic h implies that w ′ ( t ) + τ ′ ( t ) = e for early arriv als and w ′ ( t ) + τ ′ ( t ) = − L for late arriv als. Moreo ver, at equilibrium, w ( t ) + τ ( t ) ≤ z T − z C for all t , as otherwise users would switch to transit. Th us, the sum of the equilibrium w aiting and toll costs tak e the form depicted in Figure 3, where the exact times t ′ A , t ′ B , t ′ C , t ′ D sho wn in the ﬁgure are determined endogenously by the tolling policy . Figure 3: Sum of the waiting plus toll costs at equilibrium. The time points t ′ A , t ′ B , t ′ C , t ′ D a re determined endogenously b y the tolling p olicy and ma y , in general, diﬀer from the corresponding time p oints sho wn in Figure 1 (right) and Figure 2, which correspond to the no-toll and revenue-optimal dynamic toll settings, resp ectively . Lev eraging the structure of the sum of the w aiting and toll costs at equilibrium in Figure 3, w e proceed b y showing that in analyzing dynamic reven ue-optimal p olicies, it suﬃces to restrict atten tion to tolling sc hemes under which the waiting time satisﬁes w ( t ) = 0 for all p erio ds t . T o see this, for any p olicy τ ( · ) with asso ciated p eriods t ′ A , t ′ B , t ′ C , t ′ D as depicted in Figure 3, we construct an alternate p olicy ˜ τ ( · ), where ˜ τ ( t ) = w ( t ) + τ ( t ) for all t . Then, we hav e R ( τ ( · )) = R t ′ D t ′ A µτ ( t ) dt ≤ R t ′ D t ′ A µ ˜ τ ( t ) dt = R ( ˜ τ ( · )) , where the inequalit y follows as ˜ τ ( t ) ≥ τ ( t ) for all t . 9 Th us, fo cusing on tolling p olicies ˜ τ ( · ) where the waiting time w ( t ) = 0 for all t , w e ha v e: R ( ˜ τ ( · )) = Z t ′ D t ′ A µ ˜ τ ( t ) dt = Z t ′ B t ′ A µ ˜ τ ( t ) dt + Z t ′ C t ′ B µ ˜ τ ( t ) dt + Z t ′ D t ′ C µ ˜ τ ( t ) dt, = µ  Z t ′ B t ′ A z T − z C − L ( t ′ B − t ) dt + Z t ′ C t ′ B ( z T − z C ) dt + Z t ′ D t ′ C z T − z C − L ( t − t ′ C ) dt  , = µ  ( z T − z C )( t ′ D − t ′ A ) − e ( t ′ B − t ′ A ) 2 2 − L ( t ′ D − t ′ C ) 2 2  . Let f be the fraction of time in the horizontal p ortion of the waiting time curv e, i.e., t ′ C − t ′ B t 2 − t 1 = f . Then, t ′ C − t ′ B = f ( t 2 − t 1 ) = f Λ λ . Since the remaining (1 − f ) fraction of users all use a car, it follo ws that µ ( t ′ B − t ′ A + t ′ D − t ′ C ) = (1 − f )Λ. Th us, the abov e rev en ue expression reduces to: R ( ˜ τ ( · )) = ( z T − z C )  f Λ µ λ + (1 − f )Λ  − eµ ( t ′ B − t ′ A ) 2 2 − Lµ ( t ′ D − t ′ C ) 2 2 . Next, we kno w that t ′ B − t 1 + t 2 − t ′ C = (1 − f ) Λ λ , where λ ( t ′ B − t 1 ) = µ ( t ′ B − t ′ A ) and λ ( t 2 − t ′ C ) = µ ( t ′ D − t ′ C ). Then, it is straigh tforw ard to see that to maximize eµ ( t ′ B − t ′ A ) 2 2 + Lµ ( t ′ D − t ′ C ) 2 2 , it must b e that t ′ B − t ′ A = L e + L (1 − f )Λ µ and t ′ D − t ′ C = e e + L (1 − f )Λ µ . W e thus obtain the follo wing expression for the reven ue, whic h, with a slight abuse of notation, we re-express as a function of the fraction f : R ( f ) = ( z T − z C )  f Λ µ λ + (1 − f )Λ  − Λ 2 2 µ eL e + L (1 − f ) 2 . (6) Giv en this relation for the rev en ue, we hav e the following reven ue optimization problem: max f ∈ [0 , 1] R ( f ) = ( z T − z C )  f Λ µ λ + (1 − f )Λ  − Λ 2 2 µ eL e + L (1 − f ) 2 s.t. 1 − min  z T − z C T C , 1  ≤ f ≤ 1 . (7) W e hav e thus reduced the dynamic rev en ue optimization problem from one of optimizing o ver a set of tolling functions to optimizing o ver a single dimensional v ariable f . Note here that when z T − z C ≤ T C = Λ eL µ ( e + L ) , the low er b ound constraint reduces to f ≥ 1 − z T − z C T C to ensure that the tolls are non-negative at all p erio ds. Then, taking the ab ov e problem’s ﬁrst order condition, we obtain f ∗ = max n 1 − ( z T − z C ) µ ( e + L ) Λ eL  1 − µ λ  , 0 o . Finally , substituting f ∗ in the rev enue expression in Equation (6), we obtain the expression for the optimal rev enue in the theorem statement. Theorem 1 c haracterizes the rev enue-optimal dynamic toll in the bottleneck mo del with an outside option. T o achiev e this result, we reduced the reven ue maximization problem, which requires optimizing ov er tolling functions, to a single-v ariable quadratic program. T o our knowledge, this is the ﬁrst analysis of reven ue maximization in the b ottleneck mo del with an outside option. The resulting reven ue-optimal policy closely resem bles the system-cost-optimal dynamic toll in Gonzales and Daganzo (2012), which also has a trapezoidal form and is, in particular, identical to the equilibrium waiting time proﬁle shown on the right of Figure 1 and eliminates congestion dela ys at equilibrium. Despite this similarity , the tw o policies diﬀer in k ey w ays. First, by comparing the functions in Figure 1 (right) and Figure 2, the reven ue-optimal p olicy can b e viewed as a v ertical shift of the system-cost-optimal tolling proﬁle. Additionally , the duration of the ﬂat segmen t, where the toll equals z T − z C , is chosen to maximize reven ue and thus generally diﬀers from that under the system-cost-optimal p olicy . 4.3 Rev en ue Comparison for Static vs. Dynamic Reven ue-Optimal T olls Lev eraging the c haracterizations of the static and dynamic reven ue-optimal tolls, w e compare their perfor- mance by deriving lo w er b ounds on the fraction of the optimal dynamic reven ue attained under the static rev enue-optimal p olicy . Theorem 2 shows that the reven ue gap b et w een the tw o policies dep ends on the magnitude of z T − z C , the diﬀerence betw een the cost of transit and car tra v el at free-ﬂo w. When z T − z C lies b elo w a lo w er threshold or ab o v e an upp er threshold, the static rev en ue-optimal toll ac hieves at least tw o- thirds of the optimal dynamic rev en ue. In the in termediate regime, this reven ue fraction is b ounded below 10 b y one-half, implying that static rev enue-optimal tolls ac hieve at least half of the dynamic optimal rev enue across all parameter regimes. Sharp er, regime-speciﬁc bounds are provided in the theorem statemen t. Theorem 2 (Reven ue Ratio: Static vs. Dynamic Reven ue-Optimal T olls) . Supp ose z T ≥ z C and µ < λ . Mor e over, let τ ∗ s b e the static r evenue-optimal tol l, τ ∗ d ( · ) b e the dynamic r evenue-optimal p olicy, and let s = Λ eL ( z T − z C )( λ − µ )( e + L ) ≥ 0 . Then, the r atio of the r evenues of the two p olicies satisﬁes: R ( τ ∗ s ) R ( τ ∗ d ( · )) ≥        2 3 − µ λ , if 0 ≤ z T − z C < Λ eL ( λ − µ )( e + L ) min n 2+ s 4 , 1 2(1 − µ λ ) o , if z T − z C ∈ h Λ eL ( λ − µ )( e + L ) , Λ eL ( e + L )  1 λ − µ + 2 µ i 2 3 , if z T − z C > Λ eL ( e + L )  1 λ − µ + 2 µ  . W e establish this result b y b ounding the ratio of the rev enue of the static reven ue-optimal toll to the dynamic optimal reven ue in Equation (5) in the three regimes for z T − z C in the theorem statement, eac h cor- resp onding to a distinct static rev en ue-optimal toll characterized in Proposition 2. F or a pro of of Theorem 2, see App endix D.1. Theorem 2 establishes that reven ue-optimal static tolls achiev e a constant fraction of the optimal dynamic reven ue, at least one-half in some parameter regimes and at least tw o-thirds in others, highligh ting the eﬃcacy of simple static tolls. 4.4 System Cost Bounds of Rev en ue-Optimal T olling P olicies W e now compare the total system cost under static and dynamic reven ue-optimal tolls to that under the dynamic system-cost-optimal p olicy . Unlike the reven ue comparisons in the previous section, constan t- factor system cost guarantees do not hold across all parameter regimes (see Prop osition 3 at the end of this section). Nevertheless, in the regime where b oth car and transit are used at the untolled equilibrium, i.e., when z T − z C ≤ T C = Λ eL µ ( e + L ) , a condition characteristic of cities with viable public transit and holds for real systems in our numerical exp eriments in Section 6, w e establish a constant-factor guarantee. Speciﬁcally , in this regime, both static and dynamic rev enue-optimal tolls incur at most twice the minimum ac hiev able system cost. Theorem 3 (System Cost Comparison) . Supp ose µ < λ , and let τ ∗ s b e the static r evenue-optimal tol l, τ ∗ d ( · ) b e the dynamic r evenue-optimal tol l, and S C ∗ b e the minimum achievable system c ost. If z T − z C ≤ Λ eL µ ( e + L ) , then S C ( τ ∗ d ( · )) ≤ 2 S C ∗ and S C ( τ ∗ s ) ≤ 2 S C ∗ . Pr o of (Sketch). T o prov e this claim, we ﬁrst leverage the characterizations of the rev en ue-optimal static and dynamic tolling p olicies in Prop osition 2 and Theorem 1, resp ectively , to derive expressions for their corresp onding total system costs at equilibrium. In particular, for b oth p olicies, w e compute the total system cost by , summing across all users, the following four terms: (i) the cost of using transit, (ii) the free-ﬂow cost of using a car, (iii) sc hedule delay costs, and (iv) w aiting time dela ys, where, recall from the pro of of Theorem 1 that there are no waiting time dela ys under the dynamic reven ue-optimal p olicy . Summing these relations, w e then prov e that the total system cost under b oth p olicies remains upp er b ounded by z C Λ µ λ + z T  1 − µ λ  Λ. Then, comparing this upp er bound to the minimum ac hiev able total system cost deriv ed in Gonzales and Daganzo (2012), combined with the fact that z T − z C ≤ Λ eL µ ( e + L ) , we obtain our desired b ounds. F or a complete pro of of Theorem 3, see App endix D.2. Theorem 3 shows that when transit provides a viable outside option, i.e., when z T − z C ≤ Λ eL ( λ − µ )( e + L ) , rev en ue-optimal tolls achiev e total system costs within a mo dest factor of the system-cost-optimal benchmark. T ogether, Theorems 2 and 3 sho w that static reven ue- optimal tolling simultaneously achiev es strong reven ue and system cost guarantees b y capturing at least a constan t fraction of the optimal dynamic rev enue while incurring at most twice the minimum achiev able system cost (when z T − z C ≤ Λ eL µ ( e + L ) ). Since our comparisons abstract from the substantial informational and computational burdens that dynamic tolling places on traﬃc planners and users, these b ounds should b e in terpreted as w orst-case relative guaran tees. Incorp orating realistic b ehavioral or op erational constraints w ould likely strengthen these bounds, further reinforcing the case for static tolling in practical applications. 11 The constant-factor guaran tees in Theorem 3 rely on the cost diﬀerence b et w een transit and car trav el at free ﬂow b eing b ounded by Λ eL µ ( e + L ) , ensuring b oth mo des are used at equilibrium (see Section 3.2). When this condition fails, constan t-factor system cost ratios generally do not hold. Prop osition 3 (Un bounded System Cost Ratios) . L et τ ∗ s b e the static r evenue-optimal tol l and S C ∗ b e the minimum achievable total system c ost. F urther, supp ose that z T − z C > Λ eL µ ( e + L ) , with z C = 0 and z T > Λ eL ( e + L )( λ − µ )  2 λ − µ µ  . Then, S C ( τ ∗ s ) =  1 + 1 1 − µ λ  S C ∗ , which is unb ounde d when µ λ → 1 . F or a pro of of Prop osition 3, see App endix D.4. This result also extends when comparing the system cost of dynamic reven ue-optimal tolls to the minimum ac hiev able system cost. The main driver behind these un b ounded system cost ratios is that when z T − z C > Λ eL µ ( e + L ) , transit is suﬃciently unattractiv e compared to car tra v el that rev enue-optimal tolls scale with z T − z C to extract maximal reven ue. In contrast, system-cost- optimal tolls are upp er b ounded by Λ eL µ ( e + L ) and are calibrated to eliminate congestion rather than exploit users’ willingness to pay . Th us, when z T − z C > Λ eL µ ( e + L ) , the reven ue and system-cost-optimal p olicies op erate on fundamen tally diﬀerent scales, precluding constant factor system cost guaran tees. Despite this un b oundedness result, Theorem 3 demonstrates the robustness of reven ue-optimal tolling even on the system cost metric in regimes where b oth car and public transit are use d at the untolled equilibrium, an empirically relev ant regime that characterizes many transp ortation systems with viable public transit alternativ es. 5 Rev enue-Optimal T olling in Urban Systems The previous section analyzed static and dynamic rev enue-optimal tolling in the bottleneck model with an outside option, where the b ottlenec k represents a single capacity-constrained facility (e.g., bridge or tunnel) and has a ﬁxed capacit y independent of the n umber of v ehicles in the system. W e no w extend this analysis to cit y-scale urban traﬃc systems, where congestion emerges from in teractions across an en tire urban area rather than at a single facility . In such settings, the outﬂow capacity (or service rate) dep ends on the n umber of vehicles in the urban system and decreases once vehicle accumulation exceeds a critical threshold, e.g., due to queue spill back across intersections. W e mo del these system-wide congestion dynamics using the Macroscopic F undamen tal Diagram (MFD), whic h provides a parsimonious yet empirically grounded c haracterization of aggregate traﬃc behavior in urban systems (Daganzo 2007). While the MFD is commonly expressed as a relationship betw een a v erage system-wide v ehicle densit y and aggregate ﬂow, when the av erage trip distance D is constant across users, it can b e in terpreted as describing a state-dep endent service rate as a function of total vehicle accumulation (Gonzales and Daganzo 2012). W e focus on a w ell-established class of MFDs satisfying t wo standard properties: • Prop erty 1: There exists a critical vehicle accumulation level n c b ey ond which additional v ehicles reduce total outﬂo w due to spill-bac k and gridlo c k eﬀects. • Prop erty 2: F or vehicle accumulations up to n c , the throughput is linear, corresp onding to free-ﬂow conditions, achieving its maximum throughput µ f at n = n c . Figure 4 illustrates the canonical triangular MFD satisfying these prop erties. F or vehicle accum ulations b elo w the threshold n c , the system op erates at free-ﬂow with no queuing delays. Once accumulation exceeds n c , congestion sets in, reducing the eﬀectiv e system capacit y , which drops to zero at the jam accum ulation lev el n j . Vic krey’s b ottlenec k mo del can b e interpreted as a sp ecial case of suc h MFD relations, with a capacit y plateau rather than a drop in the congested regime (see App endix C). F or urban systems gov erned by MFDs satisfying th e ab o ve prop erties, in Section 5.1, we ﬁrst c haracterize the dynamic reven ue-optimal p olicy and show that it maintains the system at the throughput-maximizing capacit y µ f ac hieved at n c . Then, in Section 5.2, w e derive a closed-form expression for the rev enue generated b y any static toll for triangular MFDs (see Figure 4). Finally , while static reven ue-optimal tolls generally ac hieve weak er guarantees than in the bottleneck setting (as the system service rate is state-dependent), in Section 5.3, we show that, across a broad and practically relev ant range of parameters, static rev en ue-optimal 12 tolls attain at least t wo-thirds of the dynamic optimal reven ue while incurring at most twice the minimum attainable system cost. Figure 4: Depiction of the triangular MFD relating the system outﬂow capacity to total vehicle accumulation. The slop e of the segment connecting (0 , 0) to any point on the curve determines the average system speed, equal to average trip distance D times the slop e. The MFD consists of an uncongested regime in which the system operates at free-ﬂo w with a sp eed v f up to the critical threshold n c . Beyond n c , the system enters a congested regime in which sp eeds decline, eventually dropping to zero at the jam accumulation level n j . 5.1 Rev en ue-Optimal Dynamic T olling This section characterizes the reven ue-optimal dynamic tolling p olicy for urban systems gov erned b y MFDs satisfying Prop erties 1 and 2, such as the triangular MFD in Figure 4, and sho ws that it main tains the system at the critical vehicle accumulation level that maximizes system throughput. In pro ving this result, akin to Section 4, we fo cus on the setting when the maximum ac hiev able system throughput µ f is strictly b elo w the desired arriv al rate λ , i.e., µ f < λ . F or a depiction of the arriv al and departure time proﬁles of users in the urban system under the reven ue-maximizing tolling policy , see Figure 10 in App endix A.2. Theorem 4 (Rev enue-Optimal Dynamic T olls under MFD) . Supp ose µ f < λ and users cho ose b etwe en two mo des, tr aveling by c ar thr ough an urb an system char acterize d by an MFD satisfying pr op erties 1 and 2, and using public tr ansit with a ﬁxe d c ost z T , wher e z T ≥ z C . Then, under the r evenue-optimal dynamic tol ling p olicy, the system always op er ates at the critic al vehicle ac cumulation level n c c orr esp onding to a thr oughput-maximizing c ap acity µ f . Pr o of (Sketch). First, as with the b ottleneck mo del, a necessary equilibrium condition is that w ′ ( t ) + τ ′ ( t ) = e for early arriv als and w ′ ( t ) + τ ′ ( t ) = − L for late arriv als, and that w ( t ) + τ ( t ) ≤ z T − z C for all t , i.e., the sum of the waiting and toll costs are as depicted in Figure 3. Here, the times t ′ A , t ′ B , t ′ C , t ′ D sho wn in the ﬁgure are determined endogenously by the tolling p olicy and the waiting time dep ends on the num ber of users n ( t ) in the system at time t . Under this equilibrium condition, we compare t wo p olicies: (i) a candidate rev enue-optimal p olicy τ ∗ ( · ) with asso ciated times t ∗ A , t ∗ B , t ∗ C , t ∗ D in Figure 3, and (ii) a p olicy ˜ τ ( · ), with asso ciated times ˜ t A , ˜ t B , ˜ t C , ˜ t D , under which the system operates at the capacity µ f with no waiting delays, where ˜ t B = t ∗ B and ˜ t C = t ∗ C , such that ˜ τ ( t ) = z T − z C for the p erio d [ t ∗ B , t ∗ C ]. Finally , using Prop erties 1 and 2, we show R ( ˜ τ ( · )) ≥ R ( τ ∗ ( · )). F or a complete pro of of Theorem 4, see App endix D.3. The challenge in establishing this result and, in particular, proving R ( ˜ τ ( · )) ≥ R ( τ ∗ ( · )) lies in handling a key trade-oﬀ under a reven ue-maximization ob jective. Sp eciﬁcally , we compare the rev en ue of a candidate optimal policy τ ∗ ( · ) that op erates o v er a longer interv al [ t ∗ A , t ∗ D ] ov er which b oth throughput and tolls may be low er, with that of an alternative policy ˜ τ ( · ) that enforces op eration at the throughput-maximizing lev el µ f b y c harging (weakly) higher tolls ov er a shorter interv al [ ˜ t A , ˜ t D ]. Apriori, it is not obvious which eﬀect dominates: higher tolls ov er a shorter in terv al or low er tolls ov er a longer interv al. Our analysis shows that despite this apparent tradeoﬀ b etw een toll intensit y and duration, rev en ue is maximized b y maintaining the system at the throughput-maximizing op erating p oint. W e note that this trade-oﬀ do es not arise in the b ottleneck setting, where throughput is ﬁxed at the bottleneck capacity; in con trast, under the MFD, throughput is endogenous and dep ends on v ehicle accumulation in the system, making the analysis fundamen tally more in volv ed. Theorem 4 establishes that the dynamic reven ue-optimal toll eliminates queuing delays and maintains the system at the throughput-maximizing capacity µ f , a desirable prop erty in real-world traﬃc systems. 13 Consequen tly , as immediate corollaries of Theorems 1 and 4, the dynamic reven ue-optimal p olicy is akin to the tolling p olicy depicted in Figure 2 (as it sets waiting times p oin t-wise to zero) and the reven ue and system cost expressions under this p olicy coincide with those in the b ottlenec k setting in Section 4, with the b ottlenec k capacity µ replaced b y µ f . Since the dynamic reven ue-optimal toll eliminates queuing delays and sustains an equilibrium at the throughput-maximizing p oin t of the MFD, it closely resem bles the system-cost-optimal p olicy in Geroliminis and Levinson (2009), which also maintains the system at the critical accumulation n c . Nevertheless, the tw o p olicies diﬀer in imp ortant resp ects, for the same reasons discussed in Section 4.2. More broadly , Theorem 4 can be interpreted through the lens of standard rev enue-maximization trade- oﬀs. In man y settings, tolls or prices directly aﬀect the num b er of users serv ed, and the optimal reven ue is ac hiev ed b y balancing higher paymen ts p er user against reduced demand. In the setting studied here, Theorem 4 sho ws that this balance is achiev ed at the b oundary of the feasible op erating region, with the rev enue-optimal p olicy main taining the system at the critical accum ulation n c , rather than at higher accu- m ulation levels on the congested branc h of the MFD. 5.2 Static T olling under T riangular MFD This section characterizes the rev enue under static tolls for urban systems gov erned by a triangular MFD (see Figure 4). Unlik e the dynamic tolling analysis in the previous section, which applies to general MFDs satisfying Properties 1 and 2, static tolling can induce op eration on the congested branch of the MFD and th us requires additional structure for analytical tractabilit y . Thus, fo cusing on triangular MFDs, w e derive an expression for the reven ue as a function of any static toll τ . Theorem 5 (Reven ue under Static T olls in MFD F ramew ork) . Supp ose µ f < λ and users cho ose b etwe en two mo des, tr aveling by c ar thr ough an urb an system char acterize d by a triangular MFD (se e Figur e 4), and using public tr ansit with a ﬁxe d c ost z T , wher e z T ≥ z C and al l trips ar e of a ﬁxe d distanc e D . Mor e over, let τ ≥ 0 b e the minimum tol l at which ther e is some user that is indiﬀer ent b etwe en using c ar and tr ansit. Then, the r evenue under any static tol l τ ∈ [ τ , z T − z C ] is given by: R ( τ ) = τ   Λ λ n j n j µ f + z T − z C − τ + n j e + L eL ln  1 + ( z T − z C − τ ) µ f n j    1 − n j n j µ f + z T − z C − τ λ     . (8) Pr o of. W e pro ve this claim in tw o steps. First, under a triangular MFD, w e characterize the equilibrium system throughput µ ( n ( t )), which is time-v arying and dep ends on the v ehicle accumulation in the system, unlik e in the b ottleneck setting, under a static toll τ . Using this relation for the system throughput, we then deriv e the rev en ue expression in the statement of the theorem. T o establish a relation for µ ( n ( t )), ﬁrst note that the static toll τ ≥ 0 must be such that there is some user with a cost of z T at equilibrium. If not, all users would tak e the car at equilibrium and thus the toll can b e increased without reducing car ridership or reven ues. Thus, let τ ≥ 0 b e the minimum toll at whic h there is some user who is indiﬀerent b etw een using car and transit. Next, recall that under a static toll τ ≥ τ , the equilibrium waiting time is akin to the righ t of Figure 1, with the time p oin ts t A , t B , t C , t D determined endogenously b y the tolling p olicy and where the p eak w aiting time is z T − z C − τ during the in terv al [ t B , t C ]. Moreov er, note that in the in terv al [ t A , t B ], the waiting time is given b y z T − z C − τ − e ∆ where ∆ ∈ [0 , t B − t A ]. F urthermore, in the interv al [ t C , t D ], the w aiting time is given by z T − z C − τ − L ∆ where ∆ ∈ [0 , t D − t C ]. Next, let v f b e the free-ﬂow sp eed in the uncongested part of the MFD, so free-ﬂow trav el time is D v f . Moreov er, letting v ( n ( t )) b e the system sp eed at time t , the corresp onding trav el time with v ehicle accum ulation n ( t ) is D v ( n ( t )) . Here, v ( n ( t )) = D µ ( n ( t )) n ( t ) , since the system sp eed equals to the trip distance D times the slop e from the origin to any p oint on the triangular MFD in Figure 4. Consequently , with slight abuse of notation, we hav e the following relation for the waiting time: w ( n ( t )) = D v ( n ( t )) − D v f = n ( t ) µ ( n ( t )) − n c µ f . Moreo ver, by our triangular MFD assumption, µ f µ ( n ( t )) = n j − n c n j − n ( t ) , whic h implies n ( t ) = n j − µ ( n ( t )) µ f ( n j − n c ). 14 Substituting this relation in the ab ov e w aiting time equation and rearranging, w e obtain that: µ ( n ( t )) = n j n j µ f + w ( n ( t )) . Let µ τ b e the ﬁxed system throughput during the p erio d [ t B , t C ] when the w aiting time function is ﬂat and at its p eak under the static toll τ . Then, using the ab ov e relation for the system throughput as a function of t , w e hav e the follo wing expression for the reven ue given a ﬁxed toll τ : R ( τ ) = τ Z t D t A µ ( n ( t )) dt = τ  Z t B t A µ ( n ( t )) dt + Z t C t B µ ( n ( t )) dt + Z t D t C µ ( n ( t )) dt  , ( a ) = τ  Z t B − t A 0 n j n j µ f + ( z T − z C − τ − ∆ e ) d ∆ + Z t D − t C 0 n j n j µ f + ( z T − z C − τ − ∆ L ) d ∆ + µ τ ( t C − t B )  , ( b ) = τ  n j e ln  1 + ( z T − z C − τ ) µ f n j  + n j L ln  1 + ( z T − z C − τ ) µ f n j  + n j n j µ f + z T − z C − τ ( t C − t B )  , = τ  n j e + L eL ln  1 + ( z T − z C − τ ) µ f n j  + n j n j µ f + z T − z C − τ ( t C − t B )  , where (a) follo ws b y the v ariable transformation ∆ = t B − t for the in tegral betw een [ t A , t B ] and the v ariable transformation ∆ = t − t C for the in tegral betw een [ t C , t D ], and (b) follo ws b y ev aluating the in tegral and noting that the w aiting time at t A and t D is zero. Next, noting that R t B t A µ ( n ( t )) dt = λ ( t B − t 1 ) and R t D t C µ ( n ( t )) dt = λ ( t 2 − t C ), we ha ve: t C − t B = Λ λ − 1 λ h R t B t A µ ( n ( t )) dt + R t D t C µ ( n ( t )) dt i = Λ λ − 1 λ n j e + L eL ln  1 + ( z T − z C − τ ) µ f n j  . Substituting this into the abov e relation for the rev enue and simplifying, we obtain our desired result. While Theorem 5 yields a closed-form expression for the reven ue under a static toll τ , its complexity makes it challenging to characterize the reven ue-optimal static toll in closed form, in contrast to the b ottleneck setting in Section 4.1. This diﬃcult y arises since, unlik e dynamic reven ue-optimal tolls, static reven ue- optimal tolling can, in general, induce operation on the congested branch of the MFD, resulting in congestion dela ys. That said, since the reven ue function in Equation (8) is Lipschitz contin uous ov er τ ∈ [0 , z T − z C ], an appro ximately optimal toll can b e computed b y discretizing this in terv al and selecting the toll with the highest reven ue from this set. 5.3 Static vs. Dynamic T olling under T riangular MFD Due to the state-dep endent capacities under the MFD framework, static tolls generally admit weak er p erfor- mance guaran tees than in the bottleneck mo del. F or instance, while static reven ue-optimal tolls ac hieve at least half of the dynamic optimal reven ue in the b ottleneck mo del (Theorem 2), our numerical exp eriments in Section 6 show that this reven ue ratio ma y drop b elow half for urban systems gov erned b y a triangular MFD. Nevertheless, for such systems, we show that there exists a toll that, ov er a broad and practically relev ant range of parameters, achiev es at least tw o-thirds of the dynamic optimal reven ue and incurs at most t wice the minim um attainable system cost. W e consider the regime in which the cost diﬀerence betw een transit and the free-ﬂow cost of using a car satisﬁes z T − z C ≤ Λ eL ( λ − µ f )( e + L ) , an empirically relev ant condition for urban systems where public transit pro vides a viable alternativ e to car tra vel, as supported by our n umerical experiments in Section 6. In this regime, the toll τ = z T − z C , the reven ue-optimal static toll in the b ottleneck setting (see Prop osition 2), main tains the system at the throughput-maximizing capacity µ f with no congestion delays, as any dela ys w ould induce car users to switc h to transit. As a result, b oth the reven ue expression in Equation (8) and the corresp onding system cost reduce to their b ottleneck counterparts, with the capacit y µ replaced by µ f . Then, following the proofs of Theorem 2 and 3, if z T − z C ≤ Λ eL ( λ − µ f )( e + L ) , the static toll τ = z T − z C (i) ac hieves at least a 2 3 − µ f λ fraction of the optimal dynamic reven ue and (ii) incurs at most twice the minim um ac hiev able system c ost. While the ab ov e guarantees do not extend to the regime in whic h z T − z C > Λ eL ( λ − µ f )( e + L ) , they nonetheless highligh t that static tolling can achiev e robust performance across b oth rev en ue and system cost metrics o ver 15 a broad and practically relev ant range of parameters, even for urban systems gov erned b y a triangular MFD. That said, owing to the complexit y of the reven ue expression under static tolls in Equation (8), a general c haracterization of the p erformance gap b etw een static reven ue-optimal tolling and its dynamic b enchmarks is considerably more c hallenging than in the bottleneck setting. Therefore, we pro vide a more comprehensive assessmen t of this p erformance gap through numerical exp eriments in the next section. 6 Numerical Exp erimen ts This section presents numerical exp eriments comparing the eﬃcacy of static reven ue-optimal tolling against its dynamic reven ue and system-cost-optimal counterparts based on tw o real-world application cases of congestion pricing: (i) the SF–Oakland Bay Bridge for the b ottleneck mo del and (ii) New Y ork City’s congestion reduction zone (CRZ) for the MFD framework. Our results, which are grounded in practical datasets, v alidate our theoretical guaran tees and demonstrate that simple static tolling p olicies can deliver robust p erformance on both rev enue and system cost metrics, underscoring their practical eﬀectiv eness and shedding light on the strong empirical p erformance of the many static congestion pricing systems already in op eration. In the following, w e pro vide an o verview of our exp erimental setup (Section 6.1) and present results comparing the relative eﬃcacy of static and dynamic tolling for b oth application cases (Section 6.2). F or complete details on our data sources and mo del calibration pro cedure for b oth case studies, see App endix E. The co de to generate the results are av ailable at the following Link. 6.1 Ov erview of Exp erimental Setup F or b oth the SF-Oakland Bay Bridge and New Y ork City (NYC) case studies, we fo cus on the weekda y morning commuting p eriod b etw een 5-10 AM, when congestion is the most pronounced and a ma jorit y of cars are sub ject to a ﬂat, time-invariant toll of $ 8.50 for westbound trips on the Bay Bridge and $ 9 for entry in to NYC’s Congestion Relief Zone (CRZ). The use of simple static tolls in b oth settings makes them natural empirical settings for ev aluating the p erformance of static tolling relativ e to its dynamic benchmarks. In eac h case, we consider a setting where users c ho ose betw een trav eling b y car or using a lo cal public transit alternativ e, Bay Area Rapid T ransit (BAR T) in the Bay Bridge corridor or the NYC subw a y system, which w e mo del as the outside option. T o calibrate the parameters of the bottleneck mo del with an outside option for the Ba y Bridge study and the MFD framework for the NYC study , we draw on a broad set of data sources. These include BAR T and NYC sub wa y ridership data (Ba y Area Rapid T ransit (BAR T) 2026, State of New Y ork 2025a), v ehicular demand and ﬂow data (suc h as hourly traﬃc counts for the Ba y Bridge and C RZ entry coun ts for NYC) (California Department of T ransportation 2024, State of New Y ork 2025b), taxi and for-hire v ehicle trip records in NYC (New Y ork City T axi and Limousine Commission 2025), and Op enStreetMap data (Op enStreetMap contributors 2025) to compute total roadwa y length within the CRZ for calibrating the parameters of the triangular MFD. W e combine these data sources with standard estimates of the v alue of time and sc hedule delay p enalties from the literature. A key quantit y w e v ary in our experiments is the cost diﬀerence b etw een transit and car tra v el at free-ﬂo w, z T − z C , which plays a central role in our theoretical results in Sections 4 and 5. T o do so, while we ﬁx z C to reﬂect the sum of the costs of parking and the a verage free-ﬂo w trav el time for morning commutes in each setting, w e v ary z T through a single parameter: a disc omfort multiplier η . Sp eciﬁcally , in our exp eriments, w e mo del z T as the sum of the transit fare and a time-based cost comp osed of walking to and from stations, w aiting time on platforms, and in-v ehicle BAR T or subw ay trav el time, with these time comp onents scaled b y η to reﬂect the empirically observed fact that time sp ent using transit is p erceived as more onerous than time sp ent driving (W ardman 2012). Because there is no single, well-established v alue for the discomfort multiplier and it can v ary substan- tially across settings, w e conduct a sensitivity analysis by v arying η ∈ [1 . 5 , 18], with v alues in the range [1 . 5 , 5] consistent with empirical estimates (W ardman 2012). Exploring a broader range allows us to capture heterogeneit y in user preferences and assess ho w tec hnological or infrastructural c hanges, suc h as impro ve- men ts in transit quality or the increased attractiveness of car trav el due to autonomous v ehicles (Ostro vsky 16 and Sch warz 2025), aﬀect the cost diﬀeren tial z T − z C . In addition, v arying η ov er this wider range enables us to capture all the regimes for z T − z C c haracterized in our theoretical results. F or further details on our data sources and mo del calibration pro cedure, see App endix E.1. Moreov er, w e summarize the parameter v alues used in our exp eriments for the Bay Bridge and NYC case studies in T ables 2 (App endix E.3) and 3 (Appendix E.6), resp ectively . 6.2 Results This section compares the p erformance of static and dynamic tolls on b oth reven ue and system cost metrics for the Ba y Bridge and NYC case studies. T o this end, Figures 5 and 6 plot the reven ue and system cost ratios for a set of static and dynamic tolling p olicies, normalized by the corresp onding dynamic reven ue-optimal and system-cost-optimal b enchmarks, as the discomfort multiplier η is v aried for the Bay Bridge and NYC settings, resp ectiv ely . W e denote the static rev en ue-optimal and system-cost-optimal policies by Static-RO and Static-SO , and their dynamic coun terparts by Dynamic-RO and Dynamic-SO . F or the exp eriments, the static and dynamic reven ue-optimal p olicies are computed using the deriv ations in Sections 4 and 5. The dynamic system-cost-optimal p olicy is com puted using established results from the literature (Gonzales and Daganzo 2012, Geroliminis and Levinson 2009), while the static system-cost-optimal p olicy under the b ottlenec k and MFD mo dels are computed based on the deriv ations in App endices F.1 and F.2. Our results show that for practically relev ant v alues of the discomfort multiplier (i.e., η ∈ [1 . 5 , 5]), static rev enue-optimal tolls incur at most a 10% rev enue loss in the Ba y Bridge study and 20% in the NYC study compared to the dynamic reven ue-optimal policy . As public transit b ecomes substan tially less attractiv e ( η >> 5), this gap widens to around 15% in the Bay Bridge study and can exceed 50% in NYC, consistent with our theory . While static reven ue-optimal tolling achiev es b etter reven ue ratios in the Bay Bridge study , it p erforms w orse on the system cost metric. Speciﬁcally , in the NYC study , static rev enue-optimal tolls incur at most an 8% increase relative to the minim um ac hiev able system cost, whereas, in the Bay Bridge study , the system cost increase reaches 25% for η ∈ [1 . 5 , 5]. F urthermore, when η ≫ 5, the system cost ratio in the Bay Bridge study can even exceed tw o, incurring more than twice the optimal cost. Thus, while static rev enue-optimal tolls in NYC capture a smaller fraction of dynamic optimal rev enue, they ac hieve a system cost m uc h closer to the dynamic optimum compared to the Ba y Bridge study , highligh ting a core trade-oﬀ b et w een the rev en ue and system cost ob jectives under static tolls. Additionally , our results for the Bay Bridge study in Figure 5 highlight a non-monotonic relationship b et w een the discomfort multiplier and the p erformance gap b etw een static and dynamic p olicies. Static rev enue-optimal tolls p erform well when transit is only slightly less desirable than driving (i.e., low v alues of η ), their p erformance deteriorates as transit b ecomes mo derately w orse corresp onding to intermediate v alues of η , and then improv es somewhat again when transit b ecomes highly unattractive (i.e., v ery large v alues of η ). This non-monotonic b eha vior is consistent with our theoretical b ounds on the reven ue ratio b et w een static and dynamic reven ue-optimal p olicies in Theorem 2. Moreo v er, this non-monotonic b ehavior con trasts the strictly monotonic pattern observed in the NYC study under the MFD framework in Figure 6, where the p erformance gap of static rev enue optimal tolls relativ e to its dynamic benchmarks worsens as η increases. Figure 5 further sho ws that the t wo static tolling p olicies (Static-R O and Static-SO) exhibit near iden tical p erformance across most v alues of the discomfort m ultiplier η . Notably , in the practically relev ant range η ∈ [1 . 5 , 5], the tw o policies coincide, with a sligh t div ergence b et w een these p olicies at in termediate v alues of η . A similar pattern holds for the NYC study under the MFD framew ork, where the static system-cost- optimal and reven ue-optimal tolls coincide for all discomfort multipliers sho wn in Figure 6; hence, we plot only the latter for visual clarity . Overall, these results suggest that if a central planner is required to set static tolls, as is common in practice, both rev enue and system cost ob jectiv es can be ac hiev ed sim ultaneously particularly ov er practically relev ant parameter ranges of the discomfort m ultiplier. Bey ond the discomfort multiplier, a key driver of performance is the ratio of the maximum system throughput to the desired user arriv al rate ( µ λ for the b ottleneck mo del and µ f λ for the MFD framework). When this ratio is m uch low er than one, as in NYC, where it is around 0 . 2 due to its high public transit mo de share, the dynamic reven ue and system-cost-optimal tolls nearly coincide, and static tolls ac hieve lo wer 17 rev enues and higher system costs compared to b oth dynamic b enc hmarks. Despite this Pareto dominance of dynamic tolling across b oth metrics in the transit-in tensive NYC study , the performance losses of static tolling remain small for most practically relev an t v alues of η ∈ [1 . 5 , 5]. In contrast, when the ratio of the maxim um system throughput to the desired user arriv al rate is closer to one, as in the Ba y Bridge morning p eak where it is roughly 0 . 7, the tw o dynamic tolling p olicies diﬀer considerably . In this case, while dynamic rev enue-optimal tolls Pareto dominate b oth static p olicies by achieving higher reven ues and low er system costs for all η , such a P areto dominance do es not hold for dynamic system-cost-optimal tolls, whic h can even generate up to 20% less rev en ue than both static p olicies for practically relev an t v alues of η . Finally , in b oth case studies, the static and dynamic rev en ue-optimal tolls align closely with the currently implemen ted tolls on the Ba y Bridge and in NYC’s CRZ for most practical ranges of the discomfort multiplier η . Notably , the static reven ue-optimal toll equals the $ 8.50 Bay Bridge toll at appro ximately η = 2 . 1 and the $ 9 CRZ toll at η = 1 . 7. This low er v alue of the discomfort multiplier in NYC is reﬂective of its robust public transit system, which con trasts the dominance of car trav el in the Ba y Bridge corridor. At these v alues of η , we ﬁnd that static tolls capture nearly all of the b eneﬁts of dynamic tolling, ac hieving roughly 98% of dynamic optimal reven ue and increasing total system cost by only 3% relative to the minimum ac hiev able system cost in b oth settings. Ov erall, our results highligh t the merits of static reven ue optimal tolling policies relative to their dynamic optimal tolling coun terparts on b oth reven ue and total system cost metrics. 5 10 15 0 . 8 1 1 . 2 Discomfort Multiplier ( η ) Reven ue Ratio Practical Range ( η ∈ [1 . 5 , 5]) Static-RO Static-SO Dynamic-SO 5 10 15 1 1 . 5 2 Discomfort Multiplier ( η ) System Cost Ratio Static-RO Static-SO Dynamic-RO Figure 5: Depiction of the revenue (left) and system cost (right) ratios of a set of static and dynamic tolling p olicies, no rmalized by the corresponding dynamic revenue-optimal and system-cost-optimal b enchmarks, as the discomfort multi- plier η is varied for the Bay Bridge case study . 5 10 15 0 . 4 0 . 6 0 . 8 1 Discomfort Multiplier ( η ) Reven ue Ratio Practical Range ( η ∈ [1 . 5 , 5]) Static-RO Dynamic-SO 5 10 15 1 1 . 2 1 . 4 1 . 6 1 . 8 Discomfort Multiplier ( η ) System Cost Ratio Static-RO Dynamic-RO Figure 6: Depiction of the revenue (left) and system cost (right) ratios of a set of static and dynamic tolling p olicies, no rmalized by the corresponding dynamic revenue-optimal and system-cost-optimal b enchmarks, as the discomfort multi- plier η is varied for the New Y ork City case study . 7 Conclusion and F uture W ork This w ork analyzed the p erformance gap b et w een simple static and optimal dynamic congestion pricing sc hemes in tw o canonical mo dels capturing the dominant real-world use cases of congestion pricing. In b oth mo dels, we deriv ed, in closed form, the reven ue-optimal static and dynamic tolling p olicies and show ed that static reven ue-optimal tolls deliver robust p erformance, achieving constant-factor approximations, on 18 b oth reven ue and system cost metrics for a wide range of practically relev an t parameter regimes. W e further v alidated our theory with experiments based on t w o real-world congestion pricing case studies. Ov erall, our results demonstrate the practical eﬃcacy of simple static tolls and help shed light on the strong empirical performance of the man y static congestion pricing systems already in op eration. Moreov er, since our work abstracts from the substan tial informational and computational burdens dynamic tolling places on transp ortation planners and users, our results can b e interpreted as worst-case guarantees. Incorp orating realistic b ehavioral or op erational constrain ts w ould only strengthen the case for static tolling. There are several future research directions. First, it w ould b e v aluable to study the p erformance gap b et w een static and dynamic tolling under standard extensions of the b ottleneck mo del, including heteroge- neous v alues of time, sto chastic arriv als and utilities, non-uniform desired b ottleneck departure distributions, and congestion-dep endent outside options whose cost v aries with utilization. It would also b e w orthwhile to examine tolling p olicies for MFDs, suc h as Greenshield’s relation (Greenshields 1947), that do not satisfy Prop erties 1 and 2 or are non-triangular in the case of static tolling. Finally , there is scop e to study ob- jectiv es b eyond reven ue and system cost, and diﬀerentiated tolling sc hemes (e.g., carpo ol or taxi discounts) that b etter reﬂect real-world congestion pricing structures. Ac knowledgmen ts This work was supp orted b y the Data Science Institute Postdoctoral F ello wship at Columbia Univ ersit y . 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Subw ay frequencies: Left in the dark. https://comptroller.nyc.gov/wp- content/ uploads/documents/Subway- Frequencies_Left- in- the- Dark.pdf , 2023. Accessed: 2026-01-10. StreetEasy. Nyc sub w ay access: Neighborho o ds with the b est and worst access. https://streeteasy.com/blog/ nyc- subway- neighborhoods- best- worst- access/ , 2018. Accessed: 2026-01-10. A Arriv al and Departure Time Proﬁles A.1 Equilibrium Arriv al and Departure Time Proﬁles In the following, we depict the equilibrium arriv al and departure time proﬁles in the b ottleneck mo del without an outside option (Figure 7) and with a public transit outside option in a mixed-mo de equilibrium where b oth car and transit are used (Figure 8) in the setting when µ < λ . F or more details on these equilibrium arriv al and departure time proﬁles, see Gonzales and Daganzo (2012). Note that the waiting time proﬁles depicted in Figure 1 for car users is given by w ( t ) = D ( t ) − A ( t ) for the b ottleneck mo del without an outside option and is given by w ( t ) = D C ( t ) − A C ( t ) for the b ottlenec k mo del with a public transit outside option. W e note that in the b ottlenec k mo del, the arriv al time repres en ts the time at which users arrive at the p oint queue depicted in Figure 11 and the departure or cross time is the time at which they exit the p oint queue. A.2 Arriv al and Departure Time Proﬁles under Dynamic Rev en ue Optimal T oll Figure 9 depicts the equilibrium b ottleneck arriv al and departure (crossing) time proﬁles in the bottleneck mo del with a public transit outside option under the dynamic reven ue-optimal tolling sc heme depicted in Figure 2 in the setting when µ < λ and z T ≥ z C . Since there are no waiting delays under the dynamic rev enue optimal tolling p olicy , the departure and arriv al time proﬁles coincide, i.e., w ( t ) = D C ( t ) − A C ( t ) = 0 for all t . Figure 10 depicts the equilibrium arriv al and departure time proﬁles in an urban system gov erned by an MFD satisfying Prop erties 1 and 2 (see Section 5) under the dynamic reven ue optimal tolling scheme in the 23 Figure 7: Bottleneck arrival and departure (crossing) time proﬁles in the b ottleneck mo del without an outside option in the setting when µ < λ . Here, e and L denote the normalized schedule delay p enalties for early and late arrivals, resp ectively , and [ t 1 , t 2 ] represents users’ desired b ottleneck crossing times. Mo reover, A ( t ) represents the arrival time distribution of the users at the b ottleneck and D ( t ) represents the numb er of car users that cross the b ottleneck at their desired time. The curve in blue represents the desired b ottleneck cross time of the users. Figure 8: Bottleneck arrival and depa rture (crossing) time proﬁles for ca r users in the b ottleneck mo del with a public transit outside option in a mixed-mo de equilib rium where both car and transit a re used in the setting when µ < λ and z T ≥ z C . Here, e and L denote the normalized schedule dela y penalties for early and late arrivals, respectively , and [ t 1 , t 2 ] represents users’ desired b ottleneck crossing times. A C ( t ) represents the a rrival time distribution of car users at the b ottleneck and D C ( t ) rep resents the departure or crossing time of car users from the bottleneck. Moreover, N T ( z T − z C ) rep resents the numb er of transit users and N O ( z T − z C ) represents the numb er of car users that cross the bottleneck at their desired time. The curve in blue represents the desired b ottleneck cross time of the users. Figure 9: Bottleneck arrival and departure (crossing) time proﬁles for car users under the dynamic revenue-optimal toll in the b ottleneck mo del with a public transit outside option in the setting when µ < λ and z T ≥ z C . Here, [ t 1 , t 2 ] rep resents users’ desired b ottleneck crossing times and the curve in blue represents the desired b ottleneck cross time of the users. Since w aiting or queuing dela ys are eliminated under the dynamic revenue-optimal toll, b oth a rrival and departure time p roﬁles fo r car users coincide, i.e., D C ( t ) = A C ( t ) . Moreover, N ∗ T rep resents the number of transit users and N ∗ O rep resents the numb er of car users that cross the b ottleneck at their desired time. setting when µ f < λ and z T ≥ z C . Since there are no waiting dela ys under the dynamic reven ue optimal tolling p olicy , the departure and arriv al time proﬁles are oﬀset by a constant free-ﬂo w trav el time of t f = D v f , whic h corresp onds to the time taken to tra vel through the urban system if all users trav erse a ﬁxed distance D . Note that the b ottlenec k model do es not inherently include a notion of distance—hence the arriv al and 24 departure curv es coincide in that setting. How ev er, distance can b e incorp orated by modeling the b ottleneck as comprising an uncongested physical segmen t of length D follow ed by a p oin t queue (see Figure 11). In this case, the arriv al and departure time proﬁles in the b ottlenec k framework will lo ok akin to that depicted in Figure 10. Figure 10: Arrival and depa rture time proﬁles for car users in an urban system under the dynamic revenue-optimal toll in the MFD mo del in the setting when µ < λ and z T ≥ z C . Here, the curve in blue represents users’ desired departure time p roﬁles from the urban system (or arrival times at the destination). Since waiting or queuing delays are eliminated under the dynamic revenue-optimal toll, both a rrival and depa rture time p roﬁles for car users a re oﬀset b y a constant free-ﬂow travel time of t f = D v f , which corresponds to the time taken to travel through the urban system if all users traverse a ﬁxed distance D . Moreover, N ∗ T rep resents the number of transit users and N ∗ O rep resents the number of car users that cross the urban system at their desired time. B F ormal Deﬁnitions of Reven ue and T otal System Cost Metrics W e ev aluate tolling p olicies along tw o p erformance metrics of direct relev ance to a cen tral planner: (i) cum ulative toll rev enue and (ii) total system cost, deﬁned b elow. R evenue: The cumulativ e toll reven ue is deﬁned as the total toll pa yments collected from all users at the equilibrium induced b y a tolling p olicy τ ( · ). Under suc h a p olicy , let t τ ( · ) ( t ∗ ) denote the time at which a user with a desired b ottleneck departure time t ∗ passes the bottleneck under τ ( · ) at equilibrium, and let x τ ( · ) C ( t ∗ ) ∈ [0 , 1] denote the equilibrium fraction of users of t yp e t ∗ who trav el by car (with the remaining fraction trav eling by transit). Then, the cumulativ e toll rev enue under τ ( · ), normalized b y c W , is giv en b y: R ( τ ( · )) := Z t 2 t 1 λ · x τ ( · ) C ( t ∗ ) · τ ( t τ ( · ) ( t ∗ )) dt ∗ . While the ab ov e expression for the reven ue is in terms of users’ desired bottleneck departure times t ∗ , it is often more conv enient to express the ab ov e in tegral in the time at whic h users actually pass the b ottleneck. Accordingly , let [ t τ ( · ) A , t τ ( · ) D ] denote the equilibrium in terv al o ver whic h users pass the b ottleneck under the tolling p olicy τ ( · ), analogous to the no-toll equilibrium interv al [ t A , t D ] shown in the righ t of Figure 1. Then, in the case z T ≥ z C , the abov e rev en ue expression simpliﬁes to: R ( τ ( · )) = Z t τ ( · ) D t τ ( · ) A min { µ, λ } τ ( t ) dt as the rate at whic h users can b e pro cessed at the b ottleneck is the minimum of the user arriv al rate and the b ottlenec k service rate. Note that when z T < z C , all users strictly prefer transit, so x τ ( · ) C ( t ∗ ) = 0 for all t ∗ and no toll rev en ue is generated. In the remainder of this work, for brevit y of notation, we drop the sup erscript τ ( · ) in the notation for the equilibrium interv al [ t τ ( · ) A , t τ ( · ) D ] when it is clear from con text. F urthermore, in studying equilibrium outcomes, when users are indiﬀerent b etw een car and transit, without loss of generalit y , we break ties in fa v or of higher reven ue outcomes. 25 T otal System Cost: W e deﬁne the total system cost as the sum (across all users) of waiting costs, schedule dela y costs, and generalized trav el costs asso ciated with car or transit use at the equilibrium induced b y a tolling policy τ ( · ). Equiv alen tly , the total system cost is the sum of users’ total trav el costs at equilibrium minus their toll paymen ts, and is given by: S C ( τ ( · )) = Z t 2 t 1 λ h x τ ( · ) C ( t ∗ )  c  t τ ( · ) ( t ∗ ) , t ∗  − τ  t τ ( · ) ( t ∗ )   +  1 − x τ ( · ) C ( t ∗ )  z T i dt ∗ , where c ( t, t ⋆ ) denotes the trav el cost (in time units) for a car user from Equation (2). The ab o v e expression for the total system cost subtracts toll pa yments from the user costs, since tolls represent transfers b etw een users and the central planner, consisten t with standard practice in congestion pricing (Vic krey 1969, Gonzales and Daganzo 2012) and economics (Vic krey 1961) that excludes tolls (or prices) when deﬁning system cost or welfare metrics. C Conceptual Relationship Betw een Bottlenec k and MFD Mo dels This section describes a high-level conceptual connection betw een the b ottleneck and MFD models, illus- trating ho w Vickrey’s b ottleneck model can b e interpreted as a special case of the MFD mo del. Figure 11 depicts a bottleneck system as an uncongestible ph ysical section follo w ed by a downstream p oin t queue, and shows the resulting MFD-like ﬂow–accum ulation relationship. Akin to the triangular MFD in Figure 4, throughput in the b ottleneck setting increases linearly with vehicle accumulation up to the critical threshold n c . Beyond n c , the throughput remains at the bottleneck capacity µ rather than declining in the congested regime as in the case of the triangular MFD. Figure 11: Bottleneck system with a point queue and its implied MFD-like relation. The b ottleneck system can b e rep resented via an uncongestible physical section (e.g., a bridge of length D ) follow ed by a downstream p oint queue that o ccupies no physical space and captures congestion and queuing dela ys in the b ottleneck. The physical section can hold at most n c vehicles and any additional vehicles join the p oint queue. In this system, when the total vehicle accumulation (i.e., the sum of the vehicles in the physical section and the point queue) satisﬁes n ≤ n c , the system op erates in free ﬂow with a sp eed v f . Beyond this accumulation threshold n c , the system throughput remains at the b ottleneck capacit y µ and vehicles incur queuing delays. Note that the slop e of the segment connecting (0 , 0) to any p oint on the curve determines the average system sp eed, equal to trip distance D times the slope, which declines when the vehicle accumulation exceeds n c . 26 D Pro ofs D.1 Pro of of Theorem 2 W e establish this result by analyzing three regimes for z T − z C , each corresp onding to a distinct rev enue expression under the static reven ue-optimal toll c haracterized in Prop osition 2: R ( τ ∗ s ) =        ( z T − z C )Λ µ λ , if 0 ≤ z T − z C < Λ eL ( λ − µ )( e + L ) µ eL 4 ( e + L ) ( 1 − µ λ )  Λ λ + e + L eL  1 − µ λ  ( z T − z C )  2 , if z T − z C ∈ h Λ eL ( λ − µ )( e + L ) , Λ eL ( e + L )  1 λ − µ + 2 µ i ( z T − z C )Λ − Λ 2 eL µ ( e + L ) , if z T − z C > Λ eL ( e + L )  1 λ − µ + 2 µ  . Case i  z T − z C < Λ eL ( λ − µ )( e + L )  : In this case, note that z T − z C < Λ eL ( λ − µ )( e + L ) λ µ as λ > µ ; hence, from Theorem 1, the rev enue of the reven ue-optimal dynamic toll is ( z T − z C )Λ µ λ + ( z T − z C ) 2 µ ( e + L ) 2 eL  1 − µ λ  2 . Moreo ver, let s ≥ 1 b e a constan t such that s ( z T − z C ) = Λ eL ( λ − µ )( e + L ) . Then, w e hav e: R ( τ ∗ d ( · )) = ( z T − z C )Λ µ λ + ( z T − z C ) 2 µ ( e + L ) 2 eL  1 − µ λ  2 ( a ) = R ( τ ∗ s ) + ( z T − z C ) 2 µ ( e + L ) 2 eL  1 − µ λ  2 , ( b ) = R ( τ ∗ s ) + ( z T − z C ) Λ eL s ( λ − µ )( e + L ) µ ( e + L ) 2 eL  λ − µ λ  2 = R ( τ ∗ s ) + 1 2 s ( z T − z C )Λ µ λ  1 − µ λ  , = R ( τ ∗ s ) + R ( τ ∗ s ) 1 2 s  1 − µ λ  = R ( τ ∗ s )  1 + 1 2 s  1 − µ λ   where (a) follo ws as R ( τ ∗ s ) = ( z T − z C )Λ µ λ and (b) follo ws as s ( z T − z C ) = Λ eL ( λ − µ )( e + L ) . Thus, the optimal static toll ac hieves at least 2 s 2 s +1 − µ λ ≥ 2 3 − µ λ fraction of the optimal dynamic rev en ue. Case ii  z T − z C ∈ h Λ eL ( λ − µ )( e + L ) , Λ eL ( e + L )  1 λ − µ + 2 µ i : In this case, letting s ≤ 1 b e a constant such that s ( z T − z C ) = Λ eL ( λ − µ )( e + L ) , we hav e R ( τ ∗ s ) = µeL 4( e + L )  1 − µ λ   Λ λ + e + L eL  1 − µ λ  ( z T − z C )  2 , ( a ) = Λ s 4 µ λ ( z T − z C ) + Λ 2 µ λ ( z T − z C ) + ( z T − z C ) 2 µ ( e + L ) 4 eL  1 − µ λ  , ≥ min  1 2 + s 4 , 1 2(1 − µ λ )   Λ µ λ ( z T − z C ) + ( z T − z C ) 2 µ ( e + L ) 2 eL  1 − µ λ  2  ( b ) ≥ min  2 + s 4 , 1 2(1 − µ λ )  R ( τ ∗ d ( · )) , where (a) follows as s ( z T − z C ) = Λ eL ( λ − µ )( e + L ) and (b) follows as the term in the square brac kets is the unconstrained optimal ob jectiv e of the dynamic reven ue optimization Problem (7). Th us, optimal static tolls achiev e at least min n 2+ s 4 , 1 2(1 − µ λ ) o fraction of the optimal dynamic rev en ue. Case iii  z T − z C > Λ eL ( e + L )  1 λ − µ + 2 µ  : In this case, τ ∗ s = z T − z C − Λ eL µ ( e + L ) , which implies: z T − z C − Λ eL µ ( e + L ) > z T − z C 2 + Λ eL 2( λ − µ )( e + L ) , = ⇒ z T − z C > Λ eL ( λ − µ )( e + L ) + 2 Λ eL µ ( e + L ) = Λ eL e + L  1 λ − µ + 2 µ  > Λ eL e + L  1 λ − µ + 1 µ  = Λ eL ( λ − µ )( e + L ) λ µ 27 When z T − z C > Λ eL ( λ − µ )( e + L ) λ µ , the dynamic optimal reven ue is giv en b y ( z T − z C )Λ − Λ 2 2 µ eL e + L . Then: R ( τ ∗ s ) R ∗ d = ( z T − z C )Λ − Λ 2 µ eL e + L ( z T − z C )Λ − Λ 2 2 µ eL e + L = 1 − Λ 2 2 µ eL e + L ( z T − z C )Λ − Λ 2 2 µ eL e + L = 1 − 1 ( z T − z C )Λ Λ 2 2 µ eL e + L − 1 = 1 − 1 ( z T − z C )( e + L ) Λ eL (2 µ ) − 1 ( a ) ≥ 1 − 1  1 λ − µ + 2 µ  (2 µ ) − 1 = 1 − 1 2(2 λ − µ ) λ − µ − 1 = 1 − λ − µ 3 λ − µ ≥ 1 − λ − µ 3( λ − µ ) = 2 3 where (a) follo ws as z T − z C > Λ eL ( λ − µ )( e + L ) λ µ . This establishes our claim. D.2 Pro of of Theorem 3 T o prov e this claim, w e ﬁrst derive expressions for the total system cost under the static and dynamic rev enue-optimal tolling policies as w ell as the minimum achiev able system cost S C ∗ . T otal System Cost of Dynamic R evenue-Optimal T ol ls: Under the dynamic rev en ue-optimal policy , recall from the pro of of Theorem 1 that there are no queuing delays. Letting, N e b e the n um b er of users that pass the b ottlenec k early (i.e., b et ween [ t ∗ A , t ∗ B ] in Figure 2) and N L b e the n um b er of users that pass the b ottlenec k late (i.e., b etw een [ t ∗ C , t ∗ D ] in Figure 2), the total system cost under the dynamic reven ue-optimal p olicy for the fraction f ∗ deﬁned in Theorem 1 is giv en by: S C ( τ ∗ d ( · )) = z T f ∗ Λ  1 − µ λ  + z C  f ∗ Λ µ λ + (1 − f ∗ )Λ  + eN e t 1 − t ∗ A 2 + LN L t ∗ D − t 2 2 , whic h corresp onds to the sum of (i) the cost of using transit, where f ∗ Λ  1 − µ λ  users av ail transit, (ii) the free-ﬂo w cost of using a car, where f ∗ Λ µ λ + (1 − f ∗ )Λ users av ail a car, and (iii) sc hedule delay costs. Then, noting that N e = L e + L (1 − f ∗ )Λ, N L = e e + L (1 − f ∗ )Λ, and substituting the corresp onding relations for t 1 − t ∗ A and t ∗ D − t 2 based on the analysis in the proof of Theorem 1, w e obtain the following relation for the total system cost of the reven ue-optimal dynamic tolling p olicy: S C ( τ ∗ d ( · )) = z T f ∗ Λ  1 − µ λ  + z C  f ∗ Λ µ λ + (1 − f ∗ )Λ  + Λ 2 eL 2 µ ( e + L ) (1 − f ∗ ) 2  1 − µ λ  . In the regime z T − z C ≤ Λ eL µ ( e + L ) , w e hav e z T − z C ≤ Λ eL µ ( e + L ) λ λ − µ , and therefore Theorem 1 implies that f ∗ = 1 − ( z T − z C ) µ ( e + L ) Λ eL  1 − µ λ  , yielding the follo wing expression for the total system cost: S C ( τ ∗ d ( · )) = z C Λ µ λ + z T  1 − µ λ  Λ − ( z T − z C ) 2 µ ( e + L ) 2 eL  1 − µ λ  3 . (9) T otal System Cost of Static R evenue-Optimal T ol ls: F or a static toll τ , w e show that the total system cost as a function of the toll τ is giv en b y (see Appendix F.1 for a deriv ation): S C ( τ ) = z T " 1 − z T − z C − τ Λ eL µ ( e + L ) # Λ  1 − µ λ  + z C " µ ( z T − z C − τ )  1 e + 1 L  + 1 − z T − z C − τ Λ eL µ ( e + L ) ! Λ µ λ # (10) + µ ( z T − z C − τ ) 2 2  1 − µ λ   1 e + 1 L  + ( z T − z C − τ ) 1 − z T − z C − τ Λ eL µ ( e + L ) ! Λ µ λ + µ ( z T − z C − τ ) 2  1 e + 1 L  ! , whic h corresponds to the sum of the (i) cost of using transit, (ii) the free-ﬂo w cost of using a car, (iii) schedule dela y costs, and (iv) waiting time dela ys. F rom this expression, in the regime z T − z C ≤ Λ eL µ ( e + L ) , w e deriv e the total system cost under the t wo feasible static reven ue-optimal tolls in Equation (3): (i) τ ∗ s = z T − z C and (ii) τ ∗ s = z T − z C 2 + Λ eL 2( λ − µ )( e + L ) . Sp eciﬁcally , we obtain: S C ( τ ∗ s ) = ( z C Λ µ λ + z T  1 − µ λ  Λ , if τ ∗ s = z T − z C z C Λ µ λ + z T  1 − µ λ  Λ − µ (2 λ − µ ) 8 λeL ( e + L )( λ − µ ) 2 (( z T − z C )( e + L )( λ − µ ) − Λ eL ) 2 , if τ ∗ s = z T − z C 2 + Λ eL 2( λ − µ ) ( e + L ) 28 W e note that the third candidate rev enue-optimal toll, τ ∗ s = ( z T − z C ) − Λ eL µ ( e + L ) (see Prop osition 2), is non-p ositiv e in the parameter regime of interest and th us is not relev an t for our analysis here. Optimal System Cost: F rom Gonzales and Daganzo (2012), the minimum achiev able system cost is S C ∗ = z C Λ +  1 − µ λ  Λ( z T − z C ) −  1 − µ λ  µ ( e + L ) 2 eL ( z T − z C ) 2 in the regime when 0 < z T − z C ≤ Λ eL µ ( e + L ) . System Cost Comp arisons: F rom the ab ov e deriv ed relations for the total system cost under static and dynamic rev enue-optimal tolls, it follows that S C ( τ ∗ s ) , S C ( τ ∗ d ( · )) ≤ z C Λ µ λ + z T  1 − µ λ  Λ. Then, we obtain the following relation for the ratio of the system costs under these tw o p olicies (which w e present here for the static rev enue-optimal p olicy): S C ( τ ∗ s ) S C ∗ ≤ z C Λ µ λ + z T  1 − µ λ  Λ z C Λ +  1 − µ λ  Λ( z T − z C ) −  1 − µ λ  µ ( e + L ) 2 eL ( z T − z C ) 2 , = 1 +  1 − µ λ  µ ( e + L ) 2 eL ( z T − z C ) 2 z T Λ − ( z T − z C ) µ λ Λ − ( z T − z C ) Λ 2  1 − µ λ  , ( b ) ≤ 1 + ( z T − z C ) Λ 2  1 − µ λ  z T Λ − ( z T − z C ) µ λ Λ − ( z T − z C ) Λ 2  1 − µ λ  , ( c ) ≤ 1 + ( z T − z C ) 2  1 − µ λ  ( z T − z C ) 2  1 − µ λ  , = 2 , where (b) follows as z T − z C < Λ eL µ ( e + L ) , and (c) follo ws b y subtracting z C Λ in the denominator and simplifying. These inequalities establish our desired system cost ratio for static reven ue-optimal tolling. Note that the same set of inequalities applies for the dynamic rev enue-optimal tolling p olicy , as its total system cost also satisﬁes S C ( τ ∗ s ) , S C ( τ ∗ d ( · )) ≤ z C Λ µ λ + z T  1 − µ λ  Λ. This establishes our claim. D.3 Pro of of Theorem 4 First, as with the b ottlenec k mo del, a necessary equilibrium condition is that w ′ ( t ) + τ ′ ( t ) = e for early arriv als and w ′ ( t ) + τ ′ ( t ) = − L for late arriv als, and that w ( t ) + τ ( t ) ≤ z T − z C , i.e., the sum of the w aiting and toll costs are as depicted in Figure 3. Note here that the waiting time w ( t ) can b e expressed as a function of the n um b er of users n ( t ) in the system at time t , i.e., w ( t ) = w ( n ( t )), and the exact times t ′ A , t ′ B , t ′ C , t ′ D sho wn in Figure 3 are determined endogenously b y the tolling p olicy . W e no w show that ev en for urban systems go verned by an MFD, the dynamic reven ue-optimal tolls correspond to operating on the throughput-maximizing p oint of the MFD where the w aiting times are set point-wise to zero. T o establish this claim, we consider tw o tolling p olicies corresp onding to p ossibly diﬀerent v alues of the endogenously determined time p oints t ′ A , t ′ B , t ′ C , t ′ D in Figure 3. First, consider the dynamic rev en ue- optimal p olicy τ ∗ ( · ) with associated times t ∗ A , t ∗ B , t ∗ C , t ∗ D in Figure 3. Moreo ver, let f be the fraction of time corresp onding to the horizon tal p ortion of the curve in Figure 3 under the optimal dynamic tolling p olicy . Additionally , let f 1 b e the fraction of users that seek to arrive in the in terv al [ t ∗ A , t 1 ] and f 2 b e the fraction of users that seek to arriv e in the interv al [ t ∗ C , t 2 ], where f 1 + f 2 = 1 − f . W e deﬁne the second tolling p olicy as ˜ τ ( · ), with no w aiting time dela ys (i.e., w ( t ) = 0 for all t and the system op erates at maximum throughput µ f ) and ˜ τ ( t ) = z T − z C for the p erio d [ t ∗ B , t ∗ C ]. In this case, we deﬁne the associated times in Figure 3 as ˜ t A , ˜ t B , ˜ t C , ˜ t D , where ˜ t B = t ∗ B and ˜ t C = t ∗ C . Then, letting the system throughput µ ( t ) = µ ( n ( t )) b e time-v arying and dep end on the vehicle accumu- lation in the system, the optimal reven ue is giv en by: R ∗ = Z t ∗ D t ∗ A µ ( n ( t )) τ ∗ ( t ) dt = Z t ∗ B t ∗ A µ ( n ( t )) τ ∗ ( t ) dt + Z t ∗ C t ∗ B µ ( n ( t )) τ ∗ ( t ) dt + Z t ∗ D t ∗ C µ ( n ( t )) τ ∗ ( t ) dt, 29 and the rev enue corresp onding to the second policy is giv en by: ˜ R = Z ˜ t D ˜ t A µ f ˜ τ ( t ) dt = Z t ∗ B ˜ t A µ f ˜ τ ( t ) dt + Z t ∗ C t ∗ B µ f ˜ τ ( t ) dt + Z ˜ t D t ∗ C µ f ˜ τ ( t ) dt. T o establish our claim, w e show ˜ R ≥ R ∗ b y pro ving three inequalities: (i) R t ∗ C t ∗ B µ ( n ( t )) τ ∗ ( t ) dt ≤ R t ∗ C t ∗ B µ f ˜ τ ( t ) dt , (ii) R t ∗ B t ∗ A µ ( n ( t )) τ ∗ ( t ) dt ≤ R t ∗ B ˜ t A µ f ˜ τ ( t ) dt , and (iii) R t ∗ D t ∗ C µ ( n ( t )) τ ∗ ( t ) dt ≤ R ˜ t D t ∗ C µ f ˜ τ ( t ) dt . Pro of of (i): Note that µ ( n ( t )) τ ∗ ( t ) ≤ µ f ( z T − z C ) for all t , as µ ( n ( t )) ≤ µ f and w ( n ( t )) + τ ∗ ( t ) ≤ z T − z C for all t . Then, we obtain: Z t ∗ C t ∗ B µ ( n ( t )) τ ∗ ( t ) dt ≤ Z t ∗ C t ∗ B µ f ( z T − z C ) dt = µ f ( z T − z C )( t ∗ C − t ∗ B ) = µ f ( z T − z C ) f Λ λ = µ f Z t ∗ C t ∗ B ˜ τ ( t ) dt. Pro of of (ii): First note that the reven ue under the p olicy ˜ τ ( · ) in the range [ ˜ t A , t ∗ B ] is giv en b y: Z t ∗ B ˜ t A µ f ˜ τ ( t ) dt ( a ) = Z t ∗ B − ˜ t A 0 µ f ( ˜ τ ( t ∗ B ) − e ∆) d ∆ = µ f  ˜ τ ( t ∗ B )( t ∗ B − ˜ t A ) − e ( t ∗ B − ˜ t A ) 2 2  , = µ f  ˜ τ ( t ∗ B ) − e ( t ∗ B − ˜ t A ) 2  ( t ∗ B − ˜ t A ) ( b ) =  z T − z C − e ( t ∗ B − ˜ t A ) 2  f 1 Λ , (11) where (a) follo ws from the structure of the tolling policy in Figure 3 and (b) follows as f 1 Λ = λ ( t ∗ B − t 1 ) = µ f ( t ∗ B − ˜ t A ). Then, to establish (ii), w e use a v ariable transformation ∆ = t ∗ B − t to get: Z t ∗ B t ∗ A µ ( t ) τ ∗ ( t ) dt = Z t ∗ B − t ∗ A 0 µ (∆) τ ∗ (∆) d ∆ = Z t ∗ B − t ∗ A 0 µ (∆)( z T − z C − e ∆ − w (∆)) d ∆ , ( a ) ≤ Z t ∗ B − t ∗ A 0 µ (∆)( z T − z C − e ∆) d ∆ , = Z t ∗ B − ˜ t A 0 µ (∆)( z T − z C − e ∆) d ∆ + Z t ∗ B − t ∗ A t ∗ B − ˜ t A µ (∆)( z T − z C − e ∆) d ∆ , (12) where (a) follo ws as w (∆) ≥ 0 for all ∆ ∈ [ t ∗ A , t ∗ B ]. F urthermore, for some g ∈ [0 , 1], w e deﬁne: Z t ∗ B − ˜ t A 0 µ (∆) d ∆ = g f 1 Λ , Z t ∗ B − t ∗ A t ∗ B − ˜ t A µ (∆) d ∆ = (1 − g ) f 1 Λ . W e use these expressions to upp er b ound the tw o terms in Equation (12). T o this end, note that: Z t ∗ B − t ∗ A t ∗ B − ˜ t A µ (∆)( z T − z C − e ∆) d ∆ ≤ (1 − g ) f 1 Λ( z T − z C − e ( t ∗ B − ˜ t A )) , (13) whic h follows as ∆ ≥ t ∗ B − ˜ t A for all ∆ ∈ [ t ∗ B − ˜ t A , t ∗ B − t ∗ A ]. Next, for the other term in Equation (12): Z t ∗ B − ˜ t A 0 µ (∆)( z T − z C − e ∆) d ∆ = Z t ∗ B − ˜ t A 0 µ (∆)( z T − z C ) d ∆ − e Z t ∗ B − ˜ t A 0 µ (∆)∆ d ∆ , ( a ) = g f 1 Λ( z T − z C ) − e Z t ∗ B − ˜ t A 0 µ (∆)∆ d ∆ , (14) 30 where (a) follo ws as R t ∗ B − ˜ t A 0 µ (∆) d ∆ = g f 1 Λ and z T − z C is a constan t. Next, from Equations (13) and (14), we obtain the following upp er bound on the rev en ue: Z t ∗ B t ∗ A µ ( t ) τ ∗ ( t ) dt ≤ g f 1 Λ( z T − z C ) − e Z t ∗ B − ˜ t A 0 µ (∆)∆ d ∆ + (1 − g ) f 1 Λ( z T − z C − e ( t ∗ B − ˜ t A )) , = f 1 Λ( z T − z C ) − (1 − g ) f 1 Λ e ( t ∗ B − ˜ t A ) − e Z t ∗ B − ˜ t A 0 µ (∆)∆ d ∆ . (15) No w, to bound the term R t ∗ B − ˜ t A 0 µ (∆)∆ d ∆, we deﬁne F (∆) = R ∆ 0 µ ( x ) dx . Note that F is absolutely con tinuous with F ′ (∆) = µ (∆). Consequently , using in tegration by parts, it follo ws that: Z t ∗ B − ˜ t A 0 µ (∆)∆ d ∆ = Z t ∗ B − ˜ t A 0 F ′ (∆)∆ d ∆ = [∆ F (∆)] t ∗ B − ˜ t A 0 − Z t ∗ B − ˜ t A 0 F (∆) d ∆ , = ( t ∗ B − ˜ t A ) Z t ∗ B − ˜ t A 0 µ (∆) d ∆ − Z t ∗ B − ˜ t A 0 Z ∆ 0 µ ( x ) dx ! d ∆ , = ( t ∗ B − ˜ t A ) g f 1 Λ − Z t ∗ B − ˜ t A 0 Z ∆ 0 µ ( x ) dx ! d ∆ . (16) Next, substituting Equation (16) in Equation (15), we get: Z t ∗ B t ∗ A µ ( t ) τ ∗ ( t ) dt ≤ f 1 Λ( z T − z C ) − (1 − g ) f 1 Λ e ( t ∗ B − ˜ t A ) − e  ( t ∗ B − ˜ t A ) g f 1 Λ − Z t ∗ B − ˜ t A 0 Z ∆ 0 µ ( x ) dx ! d ∆  , = f 1 Λ( z T − z C ) − f 1 Λ e ( t ∗ B − ˜ t A ) + e Z t ∗ B − ˜ t A 0 Z ∆ 0 µ ( x ) dx ! d ∆ , ( a ) ≤ f 1 Λ( z T − z C ) − f 1 Λ e ( t ∗ B − ˜ t A ) + e Z t ∗ B − ˜ t A 0 Z ∆ 0 µ f dx ! d ∆ , = f 1 Λ( z T − z C ) − f 1 Λ e ( t ∗ B − ˜ t A ) + e Z t ∗ B − ˜ t A 0 µ f ∆ d ∆ , = f 1 Λ( z T − z C ) − f 1 Λ e ( t ∗ B − ˜ t A ) + eµ f ( t ∗ B − ˜ t A ) 2 2 , ( b ) = f 1 Λ( z T − z C ) − f 1 Λ e ( t ∗ B − ˜ t A ) + ef 1 Λ t ∗ B − ˜ t A 2 , = f 1 Λ( z T − z C ) − f 1 Λ e t ∗ B − ˜ t A 2 ( c ) = Z t ∗ B ˜ t A µ f ˜ τ ( t ) dt (17) where (a) follows as µ ( x ) ≤ µ f for all x ∈ [0 , t ∗ B − ˜ t A ], (b) follows as µ f ( t ∗ B − ˜ t A ) = f 1 Λ, and (c) follo ws b y our derived relation for R t ∗ B ˜ t A µ f ˜ τ ( t ) dt in Equation (11). This establishes inequality (ii). Pro of of (iii): Using an en tirely analogous line of reasoning to that in the pro of of inequality (ii), we can sho w that R ˜ t D t ∗ C µ f ˜ τ ( t ) dt ≥ R t ∗ D t ∗ C µ ( n ( t )) τ ∗ ( t ) dt. W e omit the details here for brevity . Concluding the Proof: Com bining inequalities (i), (ii), and (iii), it is immediate that R ∗ ≤ ˜ R , i.e., for an y tolling policy τ ∗ ( · ) there exists another policy ˜ τ ( · ) that operates the system at the throughput-maximizing capacit y at all p erio ds and achiev es a weakly higher reven ue. W e note that if the throughput induced by the p olicy τ ∗ ( · ) is strictly b elo w µ f o ver any time sub-interv al, then at least one of inequalities (i)–(iii) is strict, implying that R ∗ < ˜ R , a con tradiction to the optimalit y of τ ∗ ( · ), thus establishing our claim. 31 D.4 Pro of of Prop osition 3 In the regime when z T − z C > Λ eL µ ( e + L ) , the minim um ac hiev able total system cost reduces to that in the classical b ottlenec k model without an outside option and is given b y S C ∗ = z C Λ + eL 2( e + L ) Λ 2  1 µ − 1 λ  . T o illustrate the implications of this regime, consider an instance in whic h z C = 0 (the argumen t extends directly to settings in whic h z C is a small positive constan t) and z T > Λ eL ( e + L )( λ − µ )  2 λ − µ µ  . In this setting, the optimal static toll is τ ∗ s = ( z T − z C ) − Λ eL µ ( e + L ) and its total system cost is giv en by z C Λ + Λ 2 eL 2 µ ( e + L )  1 − µ λ  + Λ 2 eL 2 µ ( e + L ) . Then, we hav e: S C ( τ ∗ s ) S C ∗ = z C Λ + Λ 2 eL 2 µ ( e + L )  1 − µ λ  + Λ 2 eL 2 µ ( e + L ) z C Λ + Λ 2 eL 2 µ ( e + L )  1 − µ λ  ( a ) = 1 + 1 1 − µ λ , where the ﬁnal equalit y follows from letting z C = 0. Consequen tly , if µ λ → 1, the system cost of the static rev enue-optimal tolling policy can b e unbounded in the regime when µ λ → 1. E Additional Details on Numerical Exp erimen ts E.1 Mo del Calibration Details This section describes our assumptions and metho dology to calibrate our model parameters for the b ottle- nec k and MFD frameworks based on the SF-Oakland Bay Bridge and New Y ork City’s CRZ case studies, resp ectiv ely . Calibration of Bottlenec k Mo del for Bay Area Case Study: W e consider w estb ound commuting trips from the East Bay in to San F rancisco, which can occur either by car via the SF–Oakland Bay Bridge or b y the Bay Area Rapid T ransit (BAR T) system, a lo cal sub wa y netw ork that runs parallel to the bridge and serv es east–west tra v el across the Bay . All westbound vehicles using the SF–Oakland Bay Bridge are sub ject to a ﬁxed (static) toll of $ 8.50 for most commuter v ehicles that do es not v ary by time of day , and empirical evidence indicates that the SF–Oakland Ba y Bridge op erates as a true b ottleneck (Gonzales and Christofa 2015). T ogether, these features make the Bay Bridge–BAR T corridor a well-suited empirical setting for studying the p erformance gap b etw een static and dynamic tolling p olicies in a b ottleneck mo del with an outside option. F or our study , we fo cus on the w eekday morning commuting p erio d b et ween 5:00 AM and 10:00 AM, during which westbound trav elers choose b etw een driving across the SF–Oakland Bay Bridge, mo deled as a b ottleneck in our framework, or using BAR T, which serv es as the outside option. T o calibrate the total n umber of users Λ seeking to trav el during this time window, we aggregate a v erage weekda y westbound trips b y car and by BAR T for August 2025. Using BAR T ridership data (Ba y Area Rapid T ransit (BAR T) 2026), w e obtain that the av erage weekda y westbound BAR T ridership b et w een 5:00 AM and 10:00 AM in August 2025 is 27,132 trips. V ehicular demand is estimated using P eMS data from a sensor (VDS 426389) lo cated immediately upstream of the Bay Bridge, which yields an a v erage ﬂow of 41,369 vehicles ov er the same p eriod. Summing these tw o comp onents giv es a total cum ulative demand of Λ = 68 , 501, which w e round to 70 , 000 users for our exp erimen ts. This corresp onds to an av erage desired arriv al rate of λ = Λ 5 = 14 , 000 users p er hour ov er the ﬁv e-hour comm uting window. Next, to calibrate the b ottleneck service rate, w e use the observed maxim um ﬂo w at detector VDS 426389 on the SF–Oakland Ba y Bridge. The p eak ﬂo w at this location is appro ximately µ = 9 , 600 vehicles p er hour, which is consistent with standard capacity ranges for a ﬁve-lane highw ay such as the SF–Oakland Bay Bridge. Under these calibrated parameters, observe that the b ottlenec k service rate during the morning comm ute is strictly low er than the desired arriv al rate (i.e., µ < λ ), implying congestion dela ys on the Bay Bridge. These dela ys will, in turn, aﬀect comm uters’ departure-time decisions and may induce mo de shifts to ward the outside option. 32 Next, w e calibrate the parameters gov erning user cost function in Equation (2) for w estb ound car trips from East Ba y to San F rancisco. W e set the v alue of waiting time to c W = $22, based on an inﬂation-adjusted estimate of the av erage v alue of tra v el time in the Ba y Area (Metrop olitan T ransp ortation Commission 2015). F or schedule dela y costs, we follo w the estimates from the seminal work of Small (Small 1982), setting the earliness parameter to e = 0 . 61 and the lateness parameter to L = 2 . 4. Moreo ver, w e mo del the free-ﬂo w cost of car tra vel, z C , in time-equiv alen t units as the sum of (i) the a verage free-ﬂow tra vel time for westbound trips from the East Ba y in to San F rancisco and (ii) the daily parking cost in San F rancisco, con verted in to time-equiv alent units through a normalization with the v alue of waiting time c W . Daily parking costs in San F rancisco are assumed to b e $ 30, consisten t with typical w eekday parking fees in the cit y . F ree-ﬂow tra vel times are calibrated using origin–destination shares and distances for westbound AM trips in to San F rancisco, yielding a w eighted av erage free-ﬂo w tra vel time of appro ximately 21 min utes. F ull details of the free-ﬂo w trav el time calibration are rep orted in App endix E.2. Then, the resulting free-ﬂow cost of car tra v el is z C = 1 . 714 hrs. Finally , w e calibrate the a verage cost z T of westbound trips from the East Ba y into San F rancisco using BAR T. Analogous to the free-ﬂo w cost of car trav el, we mo del z T as the sum of a monetary fare comp onen t and a generalized time cost. The generalized time cost is comp osed of three elements: (i) walking time to and from the BAR T station, (ii) trav el time in the BAR T, and (iii) waiting time at the station. In line with the empirical evidence that time sp en t using public transit is p erceiv ed as more onerous than time sp ent driving (W ardman 2012), w e weigh t these time comp onents by a multiplicativ e discomfort factor η when computing z T . T o estimate these comp onen ts, we use BAR T GTFS data to compute ridership v olume-weigh ted av erages for east–w est Bay trips during the morning rush, yielding an av erage fare of $ 6.14 and an av erage in-v ehicle tra vel time of approximately 32 min utes (Bay Area Rapid T ransit (BAR T) 2025a). W alking time is assumed to be 10 min utes eac h w a y , consistent with BAR T access guidelines, resulting in 20 minutes of total access and egress time (Bay Area Rapid T ransit (BAR T) 2025b). Moreo v er, the waiting time is taken to b e half of the av erage train headwa y of 20 minutes, implying an a verage w aiting time of 10 min utes for randomly arriving comm uters (Bay Area Rapid T ransit 2025). W e combine these comp onents to deﬁne z T as the sum of the fare, conv erted in to time-equiv alen t units by normalizing with the v alue of w aiting time c W , and the generalized trav el time (w alking, w aiting, and trav el time on BAR T) weigh ted by the discomfort factor. Because the discomfort multiplier associated with transit use may v ary substantially across settings, w e conduct a sensitivity analysis in our numerical exp eriments by v arying the m ultiplier o ver the range η ∈ [1 , 30]. Multiplier v alues in the range η ∈ [1 . 5 , 5] are consistent with empirical estimates in prac- tice (W ardman 2012). Exploring a broader range allows us to capture heterogeneity in trav eler preferences and to examine how technological or infrastructural changes, such as improv ements in transit quality or the increased attractiv eness of car trav el due to the proliferation of autonomous v ehicles (Ostro vsky and Sch warz 2025), aﬀect the cost diﬀerential z T − z C . In addition, v arying η ov er this wider range enables us to study the full set of regimes for z T − z C c haracterized by our theoretical results in Section 4. W e summarize the ab o v e parameter v alues for the SF-Oakland Bay Bridge case study in T able 2 in App endix E.3. Calibration of MFD F ramework for New Y ork City (NYC) Case Study: W e now describ e the calibration metho dology of our model parameters for the MFD framew ork in Section 5 based on New Y ork Cit y’s recently implemented congestion pricing program. Under this program, which began on Jan uary 5, 2025, vehicles are tolled when entering the Congestion Relief Zone (CRZ), deﬁned as the area of Manhattan south of 60th Street. While toll levels v ary b y vehicle class and other factors, most commuter v ehicles are sub ject to a ﬂat (static) toll of $ 9 to en ter the CRZ during weekda y hours b et w een 5:00 AM and 9:00 PM. These features make New Y ork City’s congestion pricing program an esp ecially comp elling empirical setting for ev aluating the performance gap b etw een static and dynamic tolling policies in city-scale systems, whic h can b e eﬀectiv ely represen ted through an MFD. F or the purp oses of our empirical calibration, we consider comm uter trips that either originate or termi- nate within the CRZ, so that at least a portion of eac h trip takes place inside the tolled region. Users may 33 c ho ose b etw een trav eling by car, incurring the congestion toll when trav eling through the CRZ, or using the lo cal sub w a y system, which is separated from the road netw ork. T o isolate the key mec hanisms of interest in this work, we fo cus on subw a y as the only outside option and thus do not explicitly mo del bus services or other transit modes in New Y ork City for our empirical study . As with the Bay Area case study , we fo cus on the weekda y morning commuting p eriod b et ween 5:00 AM and 10:00 AM. During this perio d, we estimate the cum ulative trav el demand Λ as the sum of subw ay trips and vehicle trips that o ccur at least partially within the CRZ. Averaged across weekda ys in August 2025, w e estimate 700,000 subw a y trips and 200,000 v ehicle trips during the morning p eak. Details of the demand calibration are provided in App endix E.4. Combining these comp onents yields a total calibrated demand of Λ = 900 , 000 users, corresponding to a desired arriv al rate of λ = Λ 5 = 180 , 000 users per hour. Next, we calibrate the parameters of the triangular MFD for the New Y ork City case study . T o estimate the av erage trip distance D , we use New Y ork City taxi and for-hire vehicle trip data (New Y ork City T axi and Limousine Commission 2025) and ﬁnd that the av erage trip length for vehicle trips within the CRZ is approximately 3.7 miles (5.95 km). W e round this v alue to D = 6 km and use it as the represen tativ e trip distance for all v ehicle trips in the zone. Moreov er, we set the free-ﬂo w sp eed in Manhattan to 25 mph , consisten t with posted speed limits on lo cal Manhattan streets. W e calibrate the jam density using standard traﬃc ﬂow benchmarks. Assuming an av erage v ehicle length of 5 meters and accounting for inter-v ehicle spacing under congested conditions implies appro ximately 7 meters of road space p er v ehicle, corresp onding to a jam density of about 142 v ehicles p er kilometer p er lane. W e round this num ber to set the jam density to 140 veh/km/lane, consisten t with estimates in the literature (Kno op 2021). T o extend this to the entire CRZ, we approximate the total road supply within the CRZ using OpenStreetMap data (Op enStreetMap con tributors 2025), yielding an estimate of appro ximately 1,000 lane-kilometers. This implies a maximum system-level jam accumulation level of roughly 140,000 vehicles. Given that the true jam accumulation level ma y b e low er than this v alue, w e conduct sensitivit y analyses o ver jam accumu lation levels ranging b et w een 14,000 to 140,000 v ehicles, capturing plausible v ariation in the jam density v alues in practice. Finally , to calibrate the throughput-maximizing capacit y µ f in the CRZ, we follow an analogous pro cedure to that used to calibrate the arriv al rate λ . Using this approac h, we estimate the peak achiev able ﬂo w through the CRZ to b e µ f = 45 , 000 v ehicles per hour (see App endix E.5 for details). W e ﬁnd that our results are not sensitive to mo derate v ariations in µ f , indicating that this v alue captures the k ey eﬀects relev an t for our analysis. Next, we calibrate the user cost function for car tra v el through the CRZ. T o this end, following the estimate in Cook et al. (2025), we set the v alue of waiting time to c W = $40, consistent with mean hourly w age lev els in New Y ork City (U.S. Bureau of Lab or Statistics 2025). As in the Bay Area case study , w e set the earliness parameter to e = 0 . 61 and the lateness parameter to L = 2 . 4. Moreov er, we calibrate free-ﬂo w cost of car trav el, z C , as the generalized cost of an uncongested trip within the CRZ expressed in time-equiv alent units, com bining parking fees and the free-ﬂow tra vel time. The daily parking fee is assumed to be $ 30, consistent with an a verage parking rate of appro ximately $ 3.50 p er hour for an eight-hour w orkday in New Y ork Cit y . W e estimate the free-ﬂo w trav el time as the ratio of the av erage trip distance within the CRZ to the free-ﬂo w sp eed, giv en by D v f = 6 km 40 km/h = 0 . 15 hours. W e calibrate the subw ay cost z T for trips passing through the CRZ using the same form ulation as the Ba y Area case study . The trip fare is set to $ 3 (as of Jan uary 2026), reﬂecting the ﬂat fare structure of the New Y ork Cit y Subw a y , which applies uniformly across origins and destinations. On weekda ys, most subw a y lines op erate with headwa ys of less than ﬁve minutes, implying an av erage waiting time of approximately 2.5 minutes (New Y ork Cit y Comptroller 2023). W alking time to and from the subw a y is assumed to be 10 min utes p er access and egress leg, resulting in a total w alking time of 20 min utes (StreetEasy 2018). W e further assume that the av erage time spent trav eling on the subw a y within the CRZ is approximately 12 min utes, corresponding to roughly half the trav el time b etw een Columbus Circle (59th Street) and South F erry along the subw ay’s Red Line, under the assumption of spatially uniformly distributed trips within the CRZ. Finally , as in the Bay Area case, we v ary the discomfort multiplier o v er the range η ∈ [1 , 30] in our n umerical exp eriments to examine the implications of diﬀering relative attractiv eness of transit versus car tra vel. 34 W e summarize the ab ov e parameter v alues in the New Y ork City case study in T able 3 in App endix E.6. E.2 F ree-ﬂow T rav el Time Calibration for Ba y Area Case Study In this section, w e presen t the calibration methodology for the av erage free-ﬂow tra vel time for w estb ound trips from East Ba y in to San F rancisco. W e use origin–destination (OD) data from Metropolitan T ransp orta- tion Commission (2015), which rep orts distances and observed OD shares for ma jor East Bay sub-regions tra veling to San F rancisco, which we summarize in T able 1. The remaining OD share required to sum to 100 percent corresponds to trips originating in other East Ba y locations for which suc h data are not a v ail- able. F or the origin lo cations listed in T able 1, the free-ﬂo w trav el times are computed assuming a constan t sp eed of 50mph, consistent with the p osted sp eed limit on the SF–Oakland Ba y Bridge. These trav el time estimates are then aggregated using the reported OD shares for w estb ound AM trips to obtain a weigh ted a verage free-ﬂow trav el time of 0.35 hours (21 min utes) . This w eighted av erage provides a parsimonious and empirically grounded estimate of the free-ﬂo w trav el time for w estbound trips from East Ba y into San F rancisco, required to calibrate the parameter z C . T able 1: Calib ration of free-Flow T ravel Times for Westbound AM T rips into San Francisco. Free-ﬂo w times are computed assuming a travel sp eed of 50 mph. OD shares and distances are based on westbound AM trips rep orted in Met ropolitan T ransp ortation Commission (2015). Origin Distance (miles) F ree-Flo w Time (hours) OD Share (%) Oakland 12.6 0.252 29.8 Ric hmond 17.9 0.359 12.4 Berk eley 13.8 0.276 12.4 Ha yward 27.2 0.544 8.6 W alnut Creek 25.3 0.506 5.0 Concord 31.1 0.622 4.8 E.3 Summary of Parameter V alues for Ba y Area Case Study E.4 Calibration of T otal T ra v el Demand for New Y ork Case Study In this section, w e describ e the calibration metho dology for total trav el demand within the CRZ for the New Y ork Cit y case study . W e estimate total demand as the s um of subw ay trips and v ehicle trips, where some p ortion of the trip o ccurs within the CRZ during the weekda y morning peak b etw een 5-10 AM, av eraged o ver weekda ys in August 2025. F or subw a y demand, we classify stations according to whether they are lo cated inside or outside the CRZ and aggregate trips across three origin–destination (OD) categories: (i) trips originating within the CRZ and ending outside the CRZ, (ii) trips b oth originating and ending within the CRZ, and (iii) trips originating outside the CRZ and ending within the CRZ. T ogether, these OD categories accoun t for the v ast ma jority of sub wa y trips that pass through the CRZ. Using this approach, w e estimate an av erage of 684,306 subw a y trips trav ersing the CRZ during the 5–10 AM perio d on w eekda ys in August 2025 using the subw ay OD ridership dataset (State of New Y ork 2025a). F or our numerical expe rimen ts, we round this ﬁgure to 700,000 to conserv ativ ely account for any remaining trips not explicitly captured by the abov e procedure. F or vehicle trips, we b egin with CRZ entry data (State of New Y ork 2025b), which indicate an av erage of around 128,000 vehicle en tries into the CRZ b etw een 5-10 AM on weekda ys in August 2025, of which around 35,000 are taxi or for-hire v ehicle entries. Because these data record only v ehicle en tries in to the zone, they do not capture all v ehicle trips o ccurring within the CRZ. T o accoun t for internal trips, we additionally use New Y ork Cit y taxicab and for-hire v ehicle (FHV) trip data (New Y ork Cit y T axi and Limousine Commission 2025), including y ello w taxis, green taxis, and FHVs, to estimate the total num b er of taxi trips tra v ersing the CRZ. Aggregating across the same three origin–destination categories used for the subw ay calibration, w e ﬁnd 35 T able 2: Calib rated parameter values of the b ottleneck model with an outside option for the SF-Oakland Ba y Bridge case study . P arameter V alue Demand, A rrival R ate, and Bottlene ck Servic e R ate Λ 70 , 000 λ 14 , 000 users/hr µ 9 , 600 vehicles/hr Comp onents of User Cost of Driving c W $ 22/hr e 0 . 61 L 2 . 4 P arking fee $ 30 F ree-ﬂow trav el time 21 min z C 1 . 714 hrs T r ansit (BAR T) Par ameters to Calibr ate z T F are $ 6.14 W alking time 20 min W aiting time 10 min On-b oard BAR T trav el time 32 min Discomfort Multiplier ( η ) [1 . 5 , 18] appro ximately 50,000 taxi and FHV trips passing through the CRZ during the morning peak, compared with roughly 35,000 taxi and FHV en tries recorded in the CRZ en try data. W e therefore scale total vehicle en tries b y the corresp onding factor, yielding an adjusted estimate of approximately 128 , 000 × 50 , 000 35 , 000 ≈ 183 , 000 v ehicle trips. F or our n umerical exp eriments, we conserv ativ ely round this ﬁgure to 200,000 vehicle trips to accoun t for an y remaining trips not explicitly captured b y this pro cedure. E.5 Calibration of Throughput Maximizing Capacit y for New Y ork Case Study In this section, we describ e the calibration methodology for throughput-maximizing capacit y µ f within the CRZ for the New Y ork Cit y case study . T o estimate µ f , w e follo w a similar procedure to estimate the a v erage v ehicle trips in the CRZ in App endix E.4 and instead of lo oking at av erages, we lo ok at the maximum trav el demand. T o this end, we b egin with CRZ entry data (State of New Y ork 2025b), which indicate a maximum of approximately 137,000 vehicle entries in to the CRZ b et ween 5-10 AM on weekda ys in August 2025, of whic h about 36,400 correspond to taxi and for-hire v ehicle (FHV) entries. Because this data captures only v ehicle en tries in to the zone, w e supplemen t it with New Y ork Cit y taxicab and FHV trip data (New Y ork Cit y T axi and Limousine Commission 2025), including yello w taxis, green taxis, and FHVs, to account for trips o ccurring within the CRZ. Aggregating across the same three origin–destination categories describ ed in App endix E.4, we estimate appro ximately 54,600 taxi and FHV trips trav ersing the CRZ during the morning p eak, compared with the 36,400 taxi and FHV en tries recorded in the CRZ en try data. W e therefore scale total vehicle en tries by the corresp onding ratio, yielding an adjusted estimate of appro ximately 137 , 000 × 54 , 600 36 , 400 ≈ 205 , 000 vehicle trips. Applying the s ame conserv ativ e rounding adjustment used in App endix E.4, this implies a total of approximately 205 , 000 × 200 , 000 183 , 000 ≈ 225 , 000 vehicle trips ov er the ﬁv e-hour morning perio d, corresponding to a throughput of µ f = 45 , 000 vehicles p er hour. 36 T able 3: Calibrated pa rameter values for the MFD framewo rk in the New Y ork Cit y Case Study P arameter V alue Demand and A rrival R ates Λ 900 , 000 users λ 180 , 000 users/hr MFD Par ameters Jam density 140 veh/km/lane T otal road length in CRZ 1 , 000 lane-km Jam accumulation level ( n j ) { 14 , 000 , 42 , 000 , 70 , 000 , 140 , 000 } v ehicles Throughput-maximizing capacity µ f 45 , 000 veh/hr Av erage trip distance D 6 km Car T r avel Cost Par ameters c W $ 40/hr e 0 . 61 L 2 . 4 P arking fee $ 30 F ree-ﬂow sp eed v f 40 km/hr z C 0.9 T r ansit (Subway) Par ameters to Calibr ate z T F are $ 3 W alking time 20 min W aiting time 2 . 5 min On-b oard subw ay time 12 min Discomfort multiplier η [1 . 5 , 18] E.6 Summary of Parameter V alues for New Y ork Case Study F System Cost Optimal Static T olling F.1 Deriv ation of System Cost Optimal Static T oll for Bottleneck Mo del In the following, w e deriv e the system-cost-optimal static toll in the bottleneck mo del with an outside option. T o this end, we ﬁrst present the expression for the total system cost as a function of the static toll τ , which is the sum of the cost of using transit and the generalized cost of using the car plus scheduling and queuing dela ys. In presenting this expression, we focus on the case when τ ∈ [max { z T − z C − T C , 0 } , z T − z C ] and z T − z C ≥ 0. Note that in this regime, we hav e a mixed-mo de equilibrium as characterized in Prop osition 1. Sp eciﬁcally , letting ¯ w = z T − z C − τ , w e hav e: • N e = µ ¯ w e early arriv als • N L = µ ¯ w L late arriv als • N o = (1 − ¯ w T C ) Λ λ µ on-time arriv als via car • N T = (1 − ¯ w T C )Λ  1 − µ λ  arriv als via transit. 37 Then, we obtain the follo wing expression for the total system cost as a function of the static toll τ : S C ( τ ) = z T  Λ − µ ( e + L )( z T − z C − τ ) eL   1 − µ λ  + z C µ ( z T − z C − τ ) ( e + L ) eL + z C  Λ − µ ( e + L )( z T − z C − τ ) eL  µ λ + µ ( z T − z C − τ ) 2 2 ( e + L ) eL  1 − µ λ  + ( z T − z C − τ )  Λ − µ ( e + L )( z T − z C − τ ) eL  µ λ + µ ( z T − z C − τ ) 2 2 ( e + L ) eL . T o compute the optimal uniform toll, we ﬁrst compute the deriv ative S C ′ ( τ ) of the total system cost, which is given by: S C ′ ( τ ) = τ µ ( e + L ) eL  2 − 3 µ λ  − Λ µ λ + ( z T − z C ) µ ( e + L ) eL  2 µ λ − 1  . W e no w compute the system-cos-optimal static toll in three regimes: (i) µ λ = 2 3 , (ii) µ λ > 2 3 , and (iii) µ λ < 2 3 . Case (i): In the regime when µ λ = 2 3 , the total system cost is linear with S C ′ ( τ ) = − 2Λ 3 + 1 3 ( z T − z C ) µ ( e + L ) eL . Th us, we hav e: τ ∗ =      max { 0 , z T − z C − T C } , if z T − z C > 2Λ eL µ ( e + L ) z T − z C , if z T − z C < 2Λ eL µ ( e + L ) [max { 0 , z T − z C − T C } , z T − z C ] , if z T − z C = 2Λ eL µ ( e + L ) . (18) Note that in the regime when z T − z C > 2Λ eL µ ( e + L ) = 2 T C , it follows that max { 0 , z T − z C − T C } = z T − z C − T C . Case (ii): In the regime when µ λ > 2 3 , the total system c ost is concav e and hence the minimum m ust o ccur at one of the end-points 0 or z T − z C . Leveraging the fundamental theorem of calculus and the linearit y of the deriv ativ e S C ′ ( τ ), it follo ws that: S C ( z T − z C ) − S C (0) = Z z T − z C 0 S C ′ ( τ ) dτ = z T − z C 2 ( S C ′ (0) + S C ′ ( z T − z C )) = µ λ  ( z T − z C ) µ e + L eL − 2Λ  . Th us, we hav e: τ ∗ =      z T − z C − T C , if z T − z C > 2Λ eL µ ( e + L ) z T − z C , if z T − z C < 2Λ eL µ ( e + L ) [ z T − z C − T C , z T − z C ] , if z T − z C = 2Λ eL µ ( e + L ) . (19) Case (iii): In the regime when µ λ < 2 3 , the total system cost is conv ex and hence there is a p ossibilit y that the minimum o ccurs betw een the range (0 , z T − z C ). In this case, note that the unique unconstrained minimizer satisﬁes S C ′ ( τ ) = 0, i.e., τ = Λ µel λµ ( e + L ) + ( z T − z C )  1 − 2 µ λ  ) 2 − 3 µ λ 38 W e hav e that if Λ µel λµ ( e + L ) ≤ ( z T − z C )( 2 µ λ − 1), then τ ≤ 0, i.e., the minima o ccurs at 0, if Λ µel λµ ( e + L ) > ( z T − z C )(1 − µ λ ), then τ ≥ z T − z C . Then, we hav e: τ ∗ =          max { 0 , z T − z C − T C } , if Λ µel λµ ( e + L ) ≤ ( z T − z C )( 2 µ λ − 1) z T − z C , if Λ µel λµ ( e + L ) ≥ ( z T − z C )(1 − µ λ ) max  Λ µel λµ ( e + L ) +( z T − z C ) ( 1 − 2 µ λ ) ) 2 − 3 µ λ , z T − z C − T C  , otherwise . (20) F.2 Deriv ation of T otal System Cost Under Static T olling for T riangular MFD W e now derive the total system cost corresp onding to an y static toll τ under a triangular MFD, which consists of the following four terms: (i) cost of using transit, (ii) cost of using a car at free-ﬂow, (iii) waiting time or queuing delays when using the car, and (iv) sc hedule delays. W e now deriv e expressions for eac h of these four terms in the regime when τ ≥ τ , where τ ≥ 0 is the minimum toll at which there is some user who is indiﬀeren t betw een using car and transit. T ransit Costs: W e ha ve the follo wing expression for the cost of using transit: C T ( τ ) = z T  Λ − Z t D t A µ ( n ( t )) dt  , = z T   Λ − Λ λ n j n j µ f + z T − z C − τ − n j e + L eL ln  1 + ( z T − z C − τ ) µ f n j    1 − n j n j µ f + z T − z C − τ λ     , (21) where the equalit y follo ws from the analysis in Theorem 5. F ree-ﬂo w Car Costs: W e hav e the follo wing expression for the cost of using car: C F ( τ ) = z C Z t D t A µ ( n ( t )) dt, = z C   Λ λ n j n j µ f + z T − z C − τ + n j e + L eL ln  1 + ( z T − z C − τ ) µ f n j    1 − n j n j µ f + z T − z C − τ λ     . (22) W aiting and Queuing Delays: The total waiting and queuing dela ys are given b y the sum of the corresp onding delays for the (a) on-time car users, (b) early car users, and (l) late car users. F or the on-time car users, w e ha v e the follo wing expression for the queuing delay: C O Q ( τ ) = ( t C − t B ) µ τ ( z T − z C − τ ) , =  Λ λ − 1 λ n j e + L eL ln  1 + ( z T − z C − τ ) µ f n j  n j n j µ f + z T − z C − τ ( z T − z C − τ ) (23) 39 F or the early car users, we hav e the follo wing expression for the queuing delay: C E Q ( τ ) = Z t B t A µ ( n ( t )) w ( n ( t )) dt, = Z t B t A µ ( n ( t ))  n ( t ) µ ( n ( t )) − n c µ f  dt, = Z t B t A n j  1 − µ ( n ( t )) µ f  dt, = Z t B − t A 0 n j   1 − n j n j µ f +( z T − z C − τ − e ∆) µ f   d ∆ , = Z t B − t A 0 n j z T − z C − τ − e ∆ n j µ f + z T − z C − τ − e ∆ d ∆ . (24) Analogously , w e ha v e the follo wing expression for the queuing dela y of the late car users: C L Q ( τ ) = Z t B − t A 0 n j z T − z C − τ − L ∆ n j µ f + z T − z C − τ − L ∆ d ∆ . (25) Sc hedule Dela y: W e ﬁrst derive the total earliness delay in the system. T o this end, for a user that exits the system at t with a desired exit time t ∗ ( t ), their earliness dela y is giv en by: t ∗ ( t ) − t = t 1 + t − t A t B − t A ( t B − t 1 ) − t = ( t 1 − t A )( t B − t ) t B − t A . W e no w use this expression to derive the total earliness dela y , which is given by: C E S ( τ ) = e Z t B t A µ ( n ( t )) ( t 1 − t A )( t B − t ) t B − t A dt, = e Z t B t A n j n j µ f + ( z T − z C − τ − e ( t − t A )) ( t 1 − t A )( t B − t ) t B − t A dt, = n j e  z T − z C − τ − n j λ ln  1 + ( z T − z C − τ ) µ f n j   1 − n j µ f ( z T − z C − τ ) ln  1 + ( z T − z C − τ ) µ f n j  , where the ﬁnal equalit y is obtained by solving the integral. Analogously , the total lateness dela y is given by: C L S ( τ ) = n j L  z T − z C − τ − n j λ ln  1 + ( z T − z C − τ ) µ f n j   1 − n j µ f ( z T − z C − τ ) ln  1 + ( z T − z C − τ ) µ f n j  . Then, we hav e that the total social cost is given by: S C ( τ ) = C T ( τ ) + C F ( τ ) + C O Q ( τ ) + C E Q ( τ ) + C L Q ( τ ) + C E S ( τ ) + C L S ( τ ) . 40

Simple vs. Optimal Congestion Pricing

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