1. Congestion Pricing Economic Models and international experience
Aya Aboudina, PhD Student
Civil Engineering Dept, University of Toronto

2. Outline Introduction Congestion Pricing Models
Motivation
Microeconomic foundations
Congestion Pricing Models
Static vs. dynamic pricing
Profit-maximizing vs. social-welfare maximizing pricing
Social-welfare maximizing pricing
Congestion pricing models: a conclusion
International Experience and Research Vision

3. Motivation
Users/consumers should pay the full cost of whatever they consume
Otherwise, they are subsidized
Therefore, they unnecessarily consume more to the detriment of all
i.e. “Tragedy of the Commons”
Garrett Hardin, journal Science in 1968
Add to Motivation:
Boosting traffic flow up to capacity
Reducing trip delay through departure time rescheduling

4. Tragedy of the Commons
A dilemma arising from the situation in which multiple individuals, acting independently and rationally consulting their own self-interest, will ultimately deplete a shared limited resource even when it is clear that it is not in anyone's long-term interest for this to happen.
Examples:
Overgrazing
Congestion
Criticized for promoting privatization
Used here to encourage “control”

5. Motivation (cont’d)
Recent research conducted at UofT eliminates both capacity expansions and extensions to public transit as policies to combat traffic congestion. On the other hand, it indicates that vkt (vehicle kilometres travelled) is quite responsive to price.
Together these findings strengthen the case for congestion pricing as a policy response to traffic congestion.
As mentioned, extensions to public transit may not be the optimum solution to combat congestion (Durantor, Turner).
Additionally, investing the pricing revenue back in transit may cause social welfare loss.
It needs a case-by-case specific study to find the optimum allocation of revenues (transit or roads or rebating taxes).

6. Microeconomic Foundations I: Demand, Cost, Revenue, and Profit
Demand (aka Marginal Benefit)
The change in the quantity purchased to the price.
Downward slopping curve
Demand
Output (Q) $
Demand: Downward slopping curve (demand increases as the price falls since more consumers can afford (more of) the good).

7. Microeconomic Foundations I: Demand, Cost, Revenue, and Profit (cont’d)
Costs
Average Cost (AC) = Total Cost (TC)/Total Production (Q)
Marginal Cost (MC): the change in total cost required to increase output by one unit = ΔTC/ ΔQ → d(TC)/d(Q) MC AC
Demand
Output (Q) $

8. Microeconomic Foundations I: Demand, Cost, Revenue, and Profit (cont’d)
Total Revenue (TR) = Price (p) * Total Production (Q)
Marginal Revenue (MR) = ΔTR/ ΔQ → d(TR)/d(Q)
Profit = TR – TC (maximized when MR = MC) MC AC
Demand MR
Output (Q) P Q $
Profit

9. Microeconomic Foundations I: Demand, Cost, Revenue, and Profit (cont’d)
In Transportation
AC = Monetary Expenses + VOT * Travel Time
The AC increases with the level of road use. This implies that the MC exceeds the AC. This is because the MC includes both the cost incurred by the traveler himself (AC) and the additional cost (s)he imposes on all other travelers. This additional cost is known as the marginal external congestion cost (mecc).
Monetary expenses: for ex. fuel consumption and maintenance
Travel Time = free-flow travel time + excess delay
Level of Road Use: traffic flow (V) in static models, or total number of travelers over the peak (Q) in dynamic models
The physical meaning of the difference between the AC and the MC will be illustrated (through a numerical example) later in this presentation. MC AC
Flow $ V
mecc

10. Microeconomic Foundations I: Demand, Cost, Revenue, and Profit (cont’d)
Numerical Example
When the flow entering some bridge is 10,999 veh/hr, it takes min to cross bridge. Whereas it takes 1.09 min when the flow is 11,000 veh/hr. Then we have the following cost values at flow = 11,000:
TC = 11,000 *(1.09 min * VOT)
AC = TC/Flow = 1.09 min * VOT
MC = TC (11,000) –TC (10,999)
= 11,000*(1.09 min*VOT) –10,999 *(1.07 min*VOT)
= (1*1.09min*VOT) +10,999 (( )min*VOT)
= AC + externality cost MC AC
Flow
Travel Time
externality = 10,999* 0.02 min
1.09
Notes:
The relation obtained between MC and AVC indicates that the extra cost added when a new vehicle enters the HYWY is composed of two parts: the variable private cost incurred by the new vehicle itself in addition to an extra cost (due to the increase in travel time caused by the new vehicle) incurred by the existing vehicles on the HYWY.
As flow increases, the gap between MC and AVC (i.e., the externality cost) increases. Because:
1. The no. of existing vehicles (that are badly affected by entrance of new vehicle) is high
2. The difference in travel time increases (since AVC curve is highly deteriorating as flow increases)
1.07
10,999
11,000

11. Microeconomic Foundations II: Consumer ,Producer, and Social Surpluses
Consumer Surplus (CS) = Total Benefit – Amount Paid
Producer Surplus (PS) = Total Revenue – Total Cost = Profit
Social Surplus (SS) = CS + PS = Total Benefit – Total Cost P Q
Consumer Surplus
Producer Surplus
Demand $
Quantity MC
Social Surplus
When computing the above values, we should consider the Total Cost, not just the Total Variable Cost (which is the area under the MC curve). However, the fixed costs do not vary with the Quantity produced and hence will not affect the computation of Quantity that maximizes any surplus (consumer, producer, or social).
Note: the Total Benefit is the area under the Marginal Benefit (Demand) curve

12. Microeconomic Foundations III: Engineers vs
Microeconomic Foundations III: Engineers vs. Economists Definition of Congestion
Economists: system is congested if the performance of the system (e.g. travel time) rises with the intensity of use (e.g. flow levels).
Traffic engineers: system is congested when traffic density exceeds the critical density, resulting in traffic breakdown.
Economists
Traffic Engineers
Capacity
Critical density
Density (veh/km)
Flow (veh/hr)
Breakdown AC p
Cost
(1/speed)
Flow
(veh/hr)
Capacity (max flow)
Economists
Traffic
Engineers
Critical density: is the density corresponding to capacity (maximum flow) as shown on Figure.

13. Microeconomic Foundations III: Engineers vs
Microeconomic Foundations III: Engineers vs. Economists Definition of Congestion
“Congestion” for traffic engineers is termed “hyper- congestion” for economists.
Hyper-congestion causes a significant drop in capacity (at critical density).
Eliminating hyper-congestion allows the sustenance of the original capacity.
Critical density
Density (veh/km)
Flow (veh/hr)
Uncongested capacity
Congested capacity
Breakdown
Normal- Congestion
Hyper- Congestion
For instance, if capacity is 2200 veh/hr/lane on a freeway, this capacity drops to about 1700~1800 veh/hr/lane under hyper-congestion, causing some 25% or so loss in capacity at rush hours, the time we need capacity most!

14. Outline Introduction Congestion Pricing Models
Motivation
Microeconomic foundations
Congestion Pricing Models
Static vs. dynamic pricing
Profit-maximizing vs. social-welfare maximizing pricing
Social-welfare maximizing pricing
Congestion pricing models: a conclusion
International Experience and Research Vision

15. Static vs. Dynamic Pricing
Congestion Pricing
Dynamic
Static
Open Loop
Vary according to fixed schedule (based on typical conditions per time of day)
Closed Loop
vary depending on time of day and level of congestion (based on real-time system state)
This categorization determines the nature of tolls (not how to set toll values)
We can further classify each of the tolling strategies (that we will consider later in this presentation) according to the above categorization.
For ex:
First-best and second-best static ->static
First-best (multiple time periods) -> quasi dynamic open-loop
First-best and second-best dynamic -> dynamic open-loop
Mahmassani -> dynamic closed-loop
Reactive
Predictive

16. Profit-Maximizing vs. Social-welfare Maximizing PricingQ*
Consumer Surplus
Producer Surplus
Demand $
Output MC
Social Welfare maximized
Social-Welfare Maximizing Price Pm Qm
Consumer Surplus shrinks
Monopolist maximizes its Producer Surplus
Demand MR $
Output MC
Profit Maximizing (Monopoly) Price
Dead weight loss
But total social welfare declines by yellow area- “dead weight loss”
They differ in the objective function to be maximized while formulating pricing
Social-Welfare maximizing pricing: each economic agent faces a price equal to its activity’s social marginal cost
Profit-maximizing price: the price is determined by the demand curve
Producer Surplus maximized
Set to maximize the social welfare.
Achieved when the “Demand” equals the MC.
Set to maximize profits (PS).
It is the price consistent with the output where the MR equals the MC.

17. Social-Welfare Maximizing Pricing
First-best pricing
Second-best pricing
Set to maximize the social welfare.
No constraints on congestion prices.
It is often impossible!
It optimizes welfare given some constraints on policies (e.g. the inability to price all links on a network and the inability to distinguish between classes of users).
Rules are more complex.
Relative efficiency less than one.
Static congestion
Dynamic congestion

18. First-Best Pricing with Static Congestion
This is the conventional diagram of optimal congestion pricing.
The un-priced equilibrium occurs at the intersection of demand and AC curves; it involves traffic flow V0 and cost c0.
The optimal flow V1 occurs at the intersection of demand and MC; it can be achieved by imposing a toll τ (marginal-cost pricing).
The gain in the social surplus is depicted by the shaded triangle, which gives the difference between social cost saved (under MC curve) and benefit forgone
(under demand curve) when
reducing traffic flow from
V0 to V1.
Flow (V) $ AC
Toll Revenue MC
Social Subsidy
MC1
AC1
τ = mecc V1 V0
Demand

19. First-Best Pricing with Static Congestion
Static models limitations:
Assumes static demand and cost curves for each congested link and time period.
Appropriate when traffic conditions do not change too quickly or when it is sufficient to focus attention on average traffic levels over extended periods.
Cannot explain whether or not hyper-congestion will occur in equilibrium (i.e., whenever demand intersects the AC curve in its hyper-congested segment).

20. First-Best Pricing from a Dynamic Perspective
The basic bottleneck model
Alternative dynamic congestion technologies
Chu 1995
Verhoef 2003

21. The Basic Bottleneck Model
The most widely used conceptual model of dynamic congestion pricing
Assumptions:
No delay if inflow is below capacity
Queue exit rate equals capacity (when a queue exists)
Single desired queue exit time t* (for all users)
Total demand for passages Q is inelastic
Two costs in un-priced equilibrium:
Travel delay cost cT(t)
Schedule delay cost cs(t) (early and late arrival costs)
Queue forms
Time
Cumulative queue exits (slope: Vk)
Cumulative queue entries
Queue disperses
Number of vehicles tq
tq’ t* tq
tq’
Exit time (t) t*
cs(t)
cT(t)
c(t)
Average cost

22. The Basic Bottleneck Model (cont’d)
The optimum toll τ(t):
Replicates travel delay costs (it has a triangular shape).
Affects the patterns of entries (queue entry rate will equal capacity) (flattening the peak).
Results in the same pattern of schedule-delay cost.
Produces zero travel delay costs.
The main source of efficiency gains from optimal dynamic pricing is the rescheduling of departure times from the trip origin
This illustrates that the reallocation of departure times may be a greater source of benefit from time-of-day pricing than the reduction in total trips tq
tq’
Exit time (t) t*
cs(t)
τ(t)
c(t)
Average cost

23. Alternative Dynamic Congestion Technologies: Chu 1995
Chu extends the same basic logic of (marginal-cost) toll to a dynamic setting in which traffic flow varies with time.
The optimal time-varying toll is a dynamic generalization of the standard toll for static congestion.
Hyper-congestion is absent in this model and both inflow and outflow from the road are assumed to have equal sub-critical value.
Chu investigates optimal pricing for a road that is characterized by no-propagation flow congestion (no hyper-congestion or spreading of the congestion upstream) in which the travel time depends only on the flow at the road's entrance (or exit) at the time the trip is started (or completed) (as opposed to bottleneck where tt depends on inflow + enter time)
Flow $ AC
Toll Revenue MC
MC1
AC1
τ = mecc V1 V0
Demand

24. Alternative Dynamic Congestion Technologies: Verhoef 2003
It uses the car-following model to investigate optimal time-varying tolls on a road with a sudden reduction in the number of lanes by half (which produces hyper- congestion in the un-priced equilibrium).
The study provides a time- and location-specific version of Chu's toll; it extends the basic “marginal-cost" toll to more realistic cases where congestion varies both temporally and spatially in a continuous manner.
Verhoef’s model uses the car-following model to investigate optimal time-varying tolls on a road with a sudden reduction in the number of lanes by half, which produces hyper-congestion in the un-priced equilibrium, in the form of queue immediately upstream of the bottleneck.
Copying the triangular toll schedule from the basic bottleneck model, to eliminate all travel delays, is a disastrous policy in Verhoef’s simulations, as it attempts to create a road system running at free-flow speed in which case pricing will be excessive as it produces a welfare loss.
Verhoef’s pricing model eliminates hyper-congestion but it does not fully eliminate travel delays (delay in this model is defined relative to free-flow speed and not capacity speed).

25. Congestion Pricing: a Conclusion
Two types of models may be considered to set socially optimal prices, namely, static and dynamic methods.
Static pricing models:
Assume static demand and cost curves and hence come out with static (time-independent) tolls.
The main benefit from static pricing is to force vehicles to pay their full congestion externalities to the road (spatial distribution).
Dynamic pricing models
Take into account the variations of demand (arrival rates) with time (how peak demand evolves then subsides); accordingly they produce dynamic (time-varying) tolls.
A main source of efficiency gains from optimal pricing would be the rescheduling of departure times (temporal distribution) from the trip origin, which can produce significant benefits going beyond those produced by static tolls.

26. Outline Introduction Congestion Pricing Models
Motivation
Microeconomic foundations
Congestion Pricing Models
Static vs. dynamic pricing
Profit-maximizing vs. social-welfare maximizing pricing
Social-welfare maximizing pricing
Congestion pricing models: a conclusion
International Experience and Research Vision

27. International Experience
USA, UK, France, Norway, Sweden, Germany, Switzerland, Singapore, and Australia have implemented major road pricing projects.

28. London Congestion Pricing
In service since 2003.
The first congestion pricing program in a major European city.
£10 daily cordon fee (flat price) for driving in “Central London Congestion Pricing Zone” during weekdays (from 7am to 6pm) (one time per chargeable day).
Bus and taxi service improved.
Accidents and air pollution declined in city center.
After 1 year of cordon tolls and during charging:
Traffic circulating within the zone decreased by 15%.
Traffic entering the zone decreased by 18%.
Congestion (measured as the actual minus the free-flow travel time per km) decreased by 30% within the zone.
LEZ: to encourage the most polluting heavy diesel vehicles driving in the Capital to become cleaner. The LEZ covers most of Greater London. To drive within it without paying a daily charge these vehicles must meet certain emissions standards that limit the amount of particulate matter (a type of pollution) coming from their exhausts.
note this was an original sign, the hours shown are no longer accurate

29. London Congestion Pricing (cont’d)
The Central London Congestion Pricing Zone

30. I-15 HOT Lanes, San Diego, CA
First significant value pricing project.
Implemented in 1996 along the 13-km HOV section of I-15 in San Diego.
Convert HOV to HOT; solo drivers pay a monthly pass ($50 to $70) to use HOV during peak periods.
In 1998, automated and dynamic pricing scheme.

31. I-15 HOT Lanes, San Diego (cont’d)
Toll levels determined from congestion level to maintain “free- flow” conditions in the HOV lane.
Tolls updated every 6 minutes ($0.5 to $4) (closed-loop regulator).
Toll level displayed on real-time sign.
48% increase in HOV volumes.
Success in congestion minimization.
Note (Kenn Small):
Toll values are most probably “reactive”, they are determined as follows:
Toll values are determined by comparing aggregated volumes (on tolled facilities) obtained from two observation intervals against volume thresholds prescribed in a look-up table. The tolls are then displayed on variable message signs.

32. 407 ETR (Express Toll Route)
Multi-lane, electronic HYWY running 69 km across the top of the GTA from HYWY 403 (in Oakville) to HYWY 48 (in Markham).
Constructed in a partnership between “Canadian Highways International Corporation” and the Province of Ontario.
Currently owned by 407-ETR International Inc.
In the figure above:
Blue parts: regular zone
Yellow parts: light zone (lower rates)

33. 407 ETR (Express Toll Route) (cont’d)
Fees at the regular zone (open-loop regulator):
Monday-Friday:
Peak period (6-7:30am, 8:30-10am, 3-4pm, 6-7pm): 22.75¢/km
Peak hours (7:30-8:30am, 4-6pm ): 22.95¢/km
Off-peak rates: 22.95¢/km
Weekends and holidays: 19.35¢/km
Speeds on HYWY 407 ~ double free HYWYs.
High level of user satisfaction.
Monopoly price!
Note: I think the pricing scheme used in HYWY 407 is open loop, i.e. it is determined a priori based on the above schedule.
About 70% of tolls are collected using electronic transponder cards that deduct charges from prepaid accounts, and 30% using a license plate photography billing system.

34. Research Vision
The purpose is to develop a comprehensive predictive real- time dynamic congestion pricing model for the GTHA.
The tolls will be set dynamically and adaptively according to location, time-of-day, distance driven (, and if possible, the level of pollution emitted from the vehicle).
The tolling system will target full cost pricing in the case of sub-critical traffic conditions and eliminating hyper-congestion and maintaining flow at the maximum capacity in case of super-critical traffic conditions.
Additionally, this research will attempt an extra step of basing the toll on the predicted (anticipated), rather than the prevailing, traffic conditions. This is to prevent traffic breakdown before it occurs.

35. Research Vision (cont’d)
To consider how our envisioned systems will be applied to:
Freeway only with HOT lanes.
Freeway corridor (e.g. Gardiner-lakeshore, where Gardiner would be tolled). 
Cordoned network, e.g. downtown Toronto.  
The concern with the first two examples is that the impact will be mostly on routing and less on departure time choice. The cordoned area will have to have an impact on departure time choice and much less on routing (I guess an impact on mode choice as well).

36. Thank You
Questions, suggestions and comments are always welcome!

37. References
The Fundamental Law of Road Congestion, Evidence from US Cities (UofT Economists)2009
CIV 1310 Infrastructure Economics lecture notes
Kenneth A. Small and Erik T. Vehoef.The Economics of Urban Transportation.
Jing Dong, Hani S. Mahmassani, Sevgi Erdoğan, and Chung-Cheng Lu. “State-Dependent Pricing for Real- Time Freeway Management: Anticipatory versus Reactive Strategies”. TRB Annual Meeting 2007 Paper #

38. Price Elasticity of Demand
Note:
When a demand curve is horizontal (or very close to), this means that a small change in price will cause a substantial change in demand, so the demand in this case is said to be “perfectly elastic”, that is demand is very sensitive to price.
For the extreme case, in which demand is vertical, any change in price doesn’t alter demand at all, so demand is said to be “perfectly inelastic” to price changes.