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  1. Tradable mobility permits in a monocentric city with pre-existing labor taxation
    a general equilibrium perspective
    Autor*in: Candia, Diego; Verhoef, Erik T.
    Erschienen: [2021]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    This paper considers tradable mobility permit schemes in a monocentric city with a distorting labor tax. Three schemes are analyzed, that differ by the (spatial) allocation of permits to households. Numerical results show that the scheme with permits... mehr

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    This paper considers tradable mobility permit schemes in a monocentric city with a distorting labor tax. Three schemes are analyzed, that differ by the (spatial) allocation of permits to households. Numerical results show that the scheme with permits allocated in proportion to labor supply reaches about 99.9% of the first-best welfare, being the only welfare-increasing scheme for higher levels of labor tax, and being more efficient than the second-best tax for levels of the labor tax of 20% or higher. This is due to the welfare gains of incentivizing labor supply, which has an effect similar to the revenue recycling effect of conventional road pricing, but that can be better optimized by the social planner by exploiting spatial differentiation. When permits are allocated on the basis of households' residential distance from the CBD, a spatially differentiated component leads to higher welfare levels than with the third scheme considered, which is the allocation in proportion to households' kilometers traveled.

     
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    Sprache: Englisch
    Medientyp: Buch (Monographie)
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    Weitere Identifier:
    hdl: 10419/248760
    Schriftenreihe: Array ; TI 2021, 076
    Schlagworte: Tradable permits; Congestion pricing; Monocentric city; Revenue recycling; Spatial general equilibrium
    Umfang: 1 Online-Ressource (circa 31 Seiten), Illustrationen
  2. Business models for interoperable mobility services
    Autor*in: Berg, Vincent A. C. van den; Meurs, Hendrik J.; Verhoef, Erik T.
    Erschienen: [2020]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    Travelers often combine transport services from different firms to form trip chains: e.g. first train and then a bus. Integration of different forms of public and private transport into a single service is gaining attention with the concept of... mehr

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    Travelers often combine transport services from different firms to form trip chains: e.g. first train and then a bus. Integration of different forms of public and private transport into a single service is gaining attention with the concept of Mobility as a Service (MaaS). Usually, the focus is on such things as ease of use, and shifting demand away from the car. We solely focus on the effects on behaviour and welfare via the market structure of transport. In particular, we analyse three archetype ways in which MaaS could be operationalized: Integrator, Platform, and Intermediary. We find that these models differ strongly in how consumers and firms are affected. The Integrator seems best for consumers and social welfare. It always leads to lower prices than Free Competition without Maas and therefore benefits consumers; transport firm profits can be lower or higher. The Platform tends to lead to an outcome that is relatively close to Free Competition without Maas: prices can be higher or lower, while transport firm profits are lower. Finally, the Intermediary tends to lead to much higher prices. Regulation of the price that the MaaS firm has to pay may further lower prices, but compared to the Integrator the difference is often small. So, even without price regulation, MaaS supply can already benefit consumers by increasing competition and removing serial marginalization, even before we consider other benefits of MaaS such as information provision, ease of use and a demand shift towards public transport.

     
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    Weitere Identifier:
    hdl: 10419/229671
    Schriftenreihe: Array ; TI 2020, 051
    Umfang: 1 Online-Ressource (circa 34 Seiten), Illustrationen
  3. Will urban air mobility fly?
    the efficiency and distributional impacts of UAM in different urban spatial structures
    Autor*in: Straubinger, Anna; Verhoef, Erik T.; Groot, Henri L. F. de
    Erschienen: [2021]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    Recent technological developments open up possibilities for introducing a vast number of novel mobility concepts in urban environments. One of these new concepts is urban air mobility (UAM). It makes use of passenger drones for on-demand transport in... mehr

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    Recent technological developments open up possibilities for introducing a vast number of novel mobility concepts in urban environments. One of these new concepts is urban air mobility (UAM). It makes use of passenger drones for on-demand transport in urban settings, promising high travel speeds for those willing and able to pay. This research aims to answer the question how benefits from UAM will be distributed, taking into account the spatial dimension and the differential impacts on low- and high-skilled households. We develop a framework that can more generally be used to assess the welfare impacts resulting from the introduction of novel transport modes. The development of an urban spatial computable general equilibrium model building on the polycentric modelling tradition developed by Anas and coauthors allows for an analysis of mutually dependent effects on the land, labour and product markets, triggered by changes on the transport market. Allowing for an endogenous spatial structure through the introduction of agglomeration effects and an amenity-based approach, the framework investigates the relevance of the initial spatial structure for the impact of the introduction of UAM. Incorporating different skill levels of households allows to assess location choice and travel behaviour for households with different characteristics. A numerical simulation of the model shows that the different initial spatial structures impose comparable welfare changes. Variations in UAM features like marginal cost, prices, land demand for infrastructure, vertical travel speed and access and egress times have a (much) more decisive impact on modal choice and welfare effects than the initial urban structure. Simulations show that considering households of different skill levels brings additional insights, as welfare effects of UAM introduction strongly differ between groups and sometimes even go in opposing directions.

     
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    hdl: 10419/237754
    Schriftenreihe: Array ; TI 2021, 021
    Schlagworte: Urban air mobility; spatial equilibrium; welfare effects; agglomeration effects
    Umfang: 1 Online-Ressource (circa 41 Seiten), Illustrationen
  4. Pcoins for parking
    a field experiment with tradable mobility permits
    Autor*in: Brands, Devi; Verhoef, Erik T.; Knockaert, Jasper
    Erschienen: [2021]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    With congestion being one of the most important externalities in transportation, it remains important to investigate effective and politically feasible solutions for it. We have conducted an 8-week experiment with tradable mobility permits,... mehr

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    With congestion being one of the most important externalities in transportation, it remains important to investigate effective and politically feasible solutions for it. We have conducted an 8-week experiment with tradable mobility permits, specifically applied to the use of parking facilities at a Dutch employer. We combine actual mobility behaviour with trading behaviour and survey responses of participants and non-participating employees of the same company. We have analysed the choice to participate in a voluntary experiment, and the behavioural response to tradable permits. Our results provide suggestive evidence that active participants do adjust their behaviour as intended. Furthermore, participation takes less time than people anticipate, and permits are viewed as a fairer and better functioning alternative to paid parking.

     
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    hdl: 10419/237762
    Schriftenreihe: Array ; TI 2021, 029
    Schlagworte: Parken; Genehmigung; Handelbare Güter; Feldforschung; Niederlande
    Umfang: 1 Online-Ressource (circa 22 Seiten), Illustrationen
  5. Autonomous cars and dynamic bottleneck congestion revisited
    how in-vehicle activities determine aggregate travel patterns
    Autor*in: Yu, Xiaojuan; Berg, Vincent A. C. van den; Verhoef, Erik T.
    Erschienen: September 20, 2019
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    We investigate the impacts of in-vehicle activities of commuters in the autonomous car on aggregate travel patterns. We allow for an autonomous car to affect the utility difference between being at home and being in the vehicle differently than the... mehr

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    We investigate the impacts of in-vehicle activities of commuters in the autonomous car on aggregate travel patterns. We allow for an autonomous car to affect the utility difference between being at home and being in the vehicle differently than the utility difference between being at work and being in the vehicle, compared to the differences experienced with a normal car. This affects the relative importance of values of travel delays, schedule delays early, and schedule delays late. Hence multiple possible changes in travel patterns may occur when autonomous cars become available. Switching to an autonomous vehicle may impose a net negative or positive externality, by raising the marginal external cost of autonomous cars themselves while lowering that of normal cars. We examine three provision regimes: marginal cost pricing, second-best pricing and profit-maximizing pricing by a private monopoly. The second-best mark-up (over marginal cost) rises with the price sensitivity, due to the increasing marginal external cost. Surprisingly, for the monopoly, mark-up may rise or fall with the price sensitivity, depending on the relative strength of the externality and of market power, where the former tends to raise it, and the latter tends to reduce it. Furthermore, the difference of the mark-up between private monopoly and second-best public provision falls as the demand becomes more price-sensitive.

     
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    Weitere Identifier:
    hdl: 10419/205357
    Schriftenreihe: Array ; TI 2019, 067
    Umfang: 1 Online-Ressource (circa 38 Seiten), Illustrationen
  6. Autonomous cars and activitybased bottleneck model
    how do in-vehicle activities determine aggregate travel patterns?
    Autor*in: Yu, Xiaojuan; Berg, Vincent A. C. van den; Verhoef, Erik T.
    Erschienen: [2022]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    When traveling in an autonomous car, the travel time can be used for performing activities other than driving. This paper distinguishes users' work-related and home-related activities in autonomous cars and proposes an activity-based bottleneck model... mehr

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    When traveling in an autonomous car, the travel time can be used for performing activities other than driving. This paper distinguishes users' work-related and home-related activities in autonomous cars and proposes an activity-based bottleneck model to investigate travelers' behavior in the morning commute, shedding light on how the scope to undertake in-vehicle activities affects travelers' trip-timing preferences and decisions, and therewith social welfare. These welfare effects can be expected to depend on the optimality of both the market for trips, and the market for vehicles. We therefore consider different supply regimes for automobiles, and un-priced congestion versus queue-eliminating road pricing. We reveal analytically the relationship between users' various in-vehicle activities and trip timing choices by autonomous and normal car users. Three supply regimes for autonomous cars are investigated: welfare-maximizing public supply, competitive marginal cost supply, and profit-maximizing private supply. Pricing rules under different supply regimes are compared analytically, and the relative efficiencies in terms of the welfare gains are compared numerically. Results show that travelers' in-vehicle activity choices have significant impacts on the travel patterns, congestion externality, supply decisions and the associated welfare effects.

     
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    hdl: 10419/263924
    Schriftenreihe: Array ; TI 2022, 004
    Schlagworte: Activity based modelling; Autonomous cars; Bottleneck model; Private vs public supply; Traffic congestion
    Umfang: 1 Online-Ressource (circa 55 Seiten), Illustrationen
  7. Will all autonomous cars cooperate?
    brands’ strategic interactions under dynamic congestion
    Autor*in: Yu, Xiaojuan; Berg, Vincent A. C. van den; Verhoef, Erik T.; Li, Zhi-chun
    Erschienen: [2022]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    Autonomous cars allow safe driving with a smaller headway than that required for normal human-driven cars, thereby potentially improving road capacity. To attain this capacity benefit, cooperation among autonomous cars is vital. However, the future... mehr

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    Autonomous cars allow safe driving with a smaller headway than that required for normal human-driven cars, thereby potentially improving road capacity. To attain this capacity benefit, cooperation among autonomous cars is vital. However, the future market may have multiple car brands and the incentive for them to cooperate is unknown. This paper investigates competition and cooperation between multiple car brands, which may offer both autonomous and normal cars. In particular, we develop a two-stage game theoretic model to investigate EUDQGV¶strategic interactions and evaluate, from both policy and organizational perspectives, the implications of their cooperation incentives and pricing competition. We compare four market structures: duopoly competition, perfect competition, a public welfare-maximizing monopoly, and a private profit-maximizing monopoly. Various parameters are evaluated, including factors such as price elasticity, capacity effects, and cooperation cost. This evaluation provides policy insights into actions that could be considered by regulators and organizations for the operation of autonomous cars.

     
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    hdl: 10419/263923
    Auflage/Ausgabe: Revision: July 2022
    Schriftenreihe: Array ; TI 2022, 003
    Schlagworte: Autonomous cars; Cooperation strategy; Duopoly competition; Game theory; Regulatory policy
    Umfang: 1 Online-Ressource (circa 43 Seiten), Illustrationen
  8. Business models for mobility as an service
    Autor*in: Berg, Vincent A. C. van den; Meurs, Hendrik J.; Verhoef, Erik T.
    Erschienen: February 2022
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    Travellers often combine transport services from different firms to form trip chains: e.g. first taking a train and then a bus. Integration of different forms of public and private transport into a single service is gaining attention with the concept... mehr

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    Travellers often combine transport services from different firms to form trip chains: e.g. first taking a train and then a bus. Integration of different forms of public and private transport into a single service is gaining attention with the concept of Mobility as a Service (MaaS). Usually the attention focuses on such things as ease of use for travellers and shifting demand away from the car. We focus on the effects of MaaS on behaviour and welfare via the market structure of transportation. In particular, we analyse three archetypical ways in which MaaS could be operationalised: Integrator, Platform, and Intermediary. We find that these models differ strongly in how consumers and firms are affected by the availability of MaaS technologies. The Integrator model seems best for consumers and social welfare. It always leads to lower prices than free competition without MaaS and therefore benefits consumers; transport firm profits can be lower or higher. The Platform model tends to lead to an outcome that is relatively close to free competition without MaaS: prices can be higher or lower, while transport firm profits are lower. Finally, the Intermediary model tends to lead to much higher prices. Regulation of the price that the MaaS firm has to pay may lower prices, but, compared to the Integrator model, the change is often small. So, even without price regulation, MaaS supply can benefit consumers by increasing competition and removing serial marginalisation, even before we consider other benefits of MaaS.

     
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    hdl: 10419/263922
    Auflage/Ausgabe: Revision
    Schriftenreihe: Array ; TI 2022, 002
    Schlagworte: MaaS; market structure; platform; intermediary; integrator; regulation
    Umfang: 1 Online-Ressource (circa 46 Seiten), Illustrationen
  9. Emergent hypercongestion in Vickrey bottleneck networks
    Autor*in: Frascaria, Dario; Olver, Neil; Verhoef, Erik T.
    Erschienen: [2020]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    Hypercongestion - the phenomenon that higher traffic densities can reduce throughput - is well understood at the link level, but has also been observed in a macroscopic form at the level of traffic networks; for instance, in morning rush-hour traffic... mehr

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    Hypercongestion - the phenomenon that higher traffic densities can reduce throughput - is well understood at the link level, but has also been observed in a macroscopic form at the level of traffic networks; for instance, in morning rush-hour traffic into a downtown core. In this paper, we show that macroscopic hypercongestion can occur as a purely emergent effect of dynamic equilibrium behaviour on a network, even if the underlying link dynamics (we consider Vickrey bottlenecks with spaceless vertical queues) do not exhibit hypercongestion.

     
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    hdl: 10419/220039
    Schriftenreihe: Array ; TI 2020, 002
    Umfang: 1 Online-Ressource (circa 24 Seiten), Illustrationen
  10. Leaving the tub
    the nature and dynamics of hypercongestion in a bathtub model with a restricted downstream exit
    Autor*in: Bao, Yue; Verhoef, Erik T.; Koster, Paul
    Erschienen: [2020]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    Hypercongestion is the situation where a certain traffic flow occurs at a combination of low speed and high density, and a more favorable combination of these could produce the same flow. The macroscopic fundamental diagram (MFD) allows for such... mehr

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    Hypercongestion is the situation where a certain traffic flow occurs at a combination of low speed and high density, and a more favorable combination of these could produce the same flow. The macroscopic fundamental diagram (MFD) allows for such hypercongestion, but does not explicitly describe the dynamic process leading up to hypercongestion. Earlier studies of hypercongestion on single links have, however, confirms that such dynamic processes are important to consider. The bathtub model is one class of model that can be used to investigate how hypercongestion can arise in urban areas, when drivers can choose their departure times. This paper investigates equilibrium outcomes and user costs under the realistic assumption that there is finite capacity to exit the bathtub, without which it would be hard to explain why hypercongestion would not dissolve through shockwaves originating from the bathtub exit. We find that when the exit capacity of the bathtub is lower than the equilibrium exit flow from the bathtub, no additional inefficiencies arise due to hypercongestion. However, when the exit capacity is higher than the equilibrium exit flows from the bathtub inefficiencies do occur. The implication of this result is that a restricted downstream capacity regulating hypercongestion may not be necessary because time gains in the bathtub are lost in the queue at the exit.

     
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    hdl: 10419/220040
    Schriftenreihe: Array ; TI 2020, 003
    Umfang: 1 Online-Ressource (circa 45 Seiten), Illustrationen
  11. Modelling externalities between ecological and economic systems
    a synthesis
    Autor*in: Wang, Shunli; Nijkamp, Peter; Verhoef, Erik T.
    Erschienen: 2001

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    Schriftenreihe: Discussion paper / Tinbergen Institute ; 2001,098/3
    Schlagworte: Umweltökonomik; Theorie
    Umfang: 19 S, graph. Darst
    Bemerkung(en):

    Literaturverz. S. 17 - 19
  12. Urban sustainability, agglomeration forces and the technological deus ex machina
    Autor*in: Verhoef, Erik T.; Nijkamp, Peter; Daniel, Joe
    Erschienen: 2001

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    Schriftenreihe: Discussion paper / Tinbergen Institute ; 2001,117/3
    Schlagworte: Stadtökonomik; Regionales Cluster; Agglomerationseffekt; Umweltbelastung; Räumliches Gleichgewichtsmodell; Theorie
    Umfang: 23 S, graph. Darst
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    Literaturverz. S. 23
  13. Congestition modelling
    Autor*in: Lindsey, Robin; Verhoef, Erik T.
    Erschienen: 1999

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    Schriftenreihe: Discussion paper / Tinbergen Institute ; 99-091/3
    Schlagworte: Straßenverkehr; Engpass; Theorie
    Umfang: 16 S, graph. Darst
  14. Transport investment appraisal and the environment
    Autor*in: Nijkamp, Peter; Ubbels, Barry; Verhoef, Erik T.
    Erschienen: 2002

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    Schriftenreihe: Array ; 2002-104
    Schlagworte: Infrastrukturinvestition; Umweltbelastung; Kosten-Nutzen-Analyse
    Umfang: 19 S
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  15. A structural model of traffic congestion
    endogenizing speed choice, traffic safety and time losses
    Autor*in: Verhoef, Erik T.; Rouwendal, Jan
    Erschienen: 2001

    Standort:
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    W 899 (2001.026)
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    Schriftenreihe: Discussion paper / Tinbergen Institute ; 2001,026/3
    Schlagworte: Verkehrsstau; Verkehrsaufkommen; Theorie
    Umfang: 25 S, graph. Darst
  16. Externalities in the urban economy
    Autor*in: Verhoef, Erik T.; Nijkamp, Peter
    Erschienen: 2003

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    Schriftenreihe: Array ; 2003-078
    Schlagworte: Stadtökonomik; Externer Effekt; Agglomerationseffekt
    Umfang: 24 S, graph. Darst
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  17. The economics of airport congestion pricing
    Autor*in: Pels, Eric; Verhoef, Erik T.
    Erschienen: 2003

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    Schriftenreihe: Array ; 2003-083
    Schlagworte: Flughafen; Preismanagement; Lastmanagement; Theorie
    Umfang: 27 S
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  18. Second-best pricing for imperfect substitudes in urban networks
    Autor*in: Rouwendal, Jan; Verhoef, Erik T.
    Erschienen: 2003

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    Schriftenreihe: Array ; 2003-085
    Schlagworte: Maut; Verkehrsstau; Theorie
    Umfang: 47 S, graph. Darst
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  19. Auctioning concessions for private roads
    Autor*in: Ubbels, Barry; Verhoef, Erik T.
    Erschienen: 2004

    Standort:
    ZBW - Leibniz-Informationszentrum Wirtschaft, Standort Kiel
    Signatur:
    W 899 (2004.008)
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    Sprache: Englisch
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    Schriftenreihe: Array ; 2004-008
    Schlagworte: Straßenbau; Öffentlich-private Partnerschaft; Auktion; Maut; Theorie
    Umfang: 25 S, graph. Darst
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  20. A monopolistic market for advanced traveller information systems and road use efficiency
    Autor*in: Zhang, Rong; Verhoef, Erik T.
    Erschienen: 2004

    Standort:
    ZBW - Leibniz-Informationszentrum Wirtschaft, Standort Kiel
    Signatur:
    W 899 (2004.014)
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    Schriftenreihe: Array ; 2004-014
    Schlagworte: Straßenverkehr; Information; Zahlungsbereitschaftsanalyse
    Umfang: 28 S, graph. Darst
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  21. Pricing in the stochastic bottleneck model with price-sensitive demand
    Autor*in: Liu, Qiumin; Berg, Vincent A. C. van den; Verhoef, Erik T.; Rui, Jian
    Erschienen: [2024]
    Verlag:  Tinbergen Institute, Amsterdam, The Netherlands

    We analyse time-varying tolling in the stochastic bottleneck model with price-sensitive demand and uncertain capacity. We find that price sensitivity and its interplay with uncertainty have important implications for the effects of tolling on travel... mehr

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    We analyse time-varying tolling in the stochastic bottleneck model with price-sensitive demand and uncertain capacity. We find that price sensitivity and its interplay with uncertainty have important implications for the effects of tolling on travel costs, welfare and consumers. We evaluate three fully time-variant tolls and a step toll that have been proposed in previous literature. We also consider a uniform toll, which affects overall demand but not trip timing decisions. The first fully time-variant toll is the "first-best" toll, which varies non-linearly over time and results in a departure rate that also varies over time. It raises the generalised price (i.e. the sum of travel cost and toll), thus lowering demand. These outcomes differ fundamentally from those found for first-best pricing in the deterministic bottleneck model. The second we call a "second-best" toll, as it aims to keep the departure rate constant over time. This is optimal without uncertainty, but it lowers welfare with uncertain capacity. Next, "third-best" tolling adds the further constraint that, besides a constant departure rate, the generalised price should stay the same as without tolling. It attains a lower welfare and higher expected travel cost than the second-best scheme, but a lower generalised price. All our tolling schemes-except for the third-best one-raise the price compared to the no-toll case. In our numerical study, when there is less uncertainty, the second-best and third-best tolls achieve welfares closer to that of the first-best toll, and the three schemes are identical without uncertainty. As the degree of uncertainty falls, the uniform and single-step tolls attain welfare gains closer to that from first-best pricing. Also, when demand becomes more price-sensitive, the uniform and single-step tolls attain outcomes closer to those in the first-best outcome. Our ADL equilibrium step toll would lower the generalised price without uncertainty but raises it in our stochastic setting.

     
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    Sprache: Englisch
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    Schriftenreihe: Array ; TI 2024, 011
    Schlagworte: stochastic bottleneck model; price-sensitive demand; time-varying toll; step toll; uncertainty
    Umfang: 1 Online-Ressource (circa 33 Seiten), Illustrationen
  22. Second-best congestion pricing in general networks
    algorithms for finding second-best optimal toll levels and toll points
    Autor*in: Verhoef, Erik T.
    Erschienen: 2000

    Standort:
    ZBW - Leibniz-Informationszentrum Wirtschaft, Standort Kiel
    Signatur:
    W 899 (2000.084)
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    Schriftenreihe: Discussion paper / Tinbergen Institute ; 2000-084/3
    Schlagworte: Maut; Second Best; Theorie
    Umfang: 25 S
  23. Traffic congestion and congestion pricing
    Autor*in: Lindsey, Robin; Verhoef, Erik T.
    Erschienen: 2000

    Standort:
    ZBW - Leibniz-Informationszentrum Wirtschaft, Standort Kiel
    Signatur:
    W 899 (2000.101)
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    Schriftenreihe: Discussion paper / Tinbergen Institute ; 2000-101/3
    Schlagworte: Verkehrsstau; Maut; Theorie
    Umfang: 27 S, graph. Darst
  24. Strategic interactions of bilateral monopoly on a private highway
    Autor*in: Wang, Judith Y. T.; Yang, Hai; Verhoef, Erik T.
    Erschienen: 2003

    Standort:
    ZBW - Leibniz-Informationszentrum Wirtschaft, Standort Kiel
    Signatur:
    W 899 (2003.038)
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    Sprache: Englisch
    Medientyp: Buch (Monographie)
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    Schriftenreihe: Array ; 2003-038
    Schlagworte: Autobahn; Straßenverkehr; Monopol; Spieltheorie
    Umfang: 44 S, graph. Darst
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  25. Speed-flow relations and cost functions for congested traffic
    theory and empirical analysis
    Autor*in: Verhoef, Erik T.
    Erschienen: 2003

    Standort:
    ZBW - Leibniz-Informationszentrum Wirtschaft, Standort Kiel
    Signatur:
    W 899 (2003.064)
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    Schriftenreihe: Array ; 2003-064
    Schlagworte: Verkehr; Engpass; Maut; Kostenfunktion; Verkehrsökonomik; Theorie
    Umfang: 21 S, graph. Darst
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