OPTIMUM ROAD PRICING UNDER CONGESTED CONDITIONS
Abstract:
This paper looks at the question “what is the optimal congestion price?” Many published
papers attempt to calculate the congestion price by estimating the short-run marginal cost
and hence the ‘congestion externality’. This paper asks how that relates to the marketclearing price and shows that the optimum price tends towards the market clearing price1
when congestion is severe. Although the short run marginal cost is the “correct” measure, it
is impossible to calculate in practice as it depends on knowing the shape of the
speed/density curve close to capacity and the value of time for just those motorists who
continue to travel.
The purpose of this paper is to re-assert the importance of matching supply and demand as
a basis for pricing policy and to demonstrate that this leads to a rule that is both pragmatic
and defensible.
Key words: Congestion, Pricing, Demand, Externalities
1
INTRODUCTION
This paper looks at the question “what is the optimal congestion price?” We are hesitant to
venture into a subject which has already been the subject of considerable scholarship. (See
for example Dupuit, 1844; Walters, 1961; Vickrey, 1969; Newbery, 1990; Verhoef,
1999a). However it appears to us that the debate is in danger of becoming so esoteric that
the basic principles may have been overlooked.
Most papers appear to make at least one of four fundamental errors:
1

Treating the value of time as constant across the population. Using a demand curve
or a diversion curve implicitly assumes that values of time differ2, but many ex post
evaluations value the time savings of those who continue to pay at the same rate as
those who are “tolled off”. It is not surprising that these studies find limited benefits
– the surprise is that they find benefits at all

Treating flow as the exogenous variable3. For low levels of traffic all demand can
be met, but at higher traffic levels flow is constrained. Flow is an endogenous
variable. It is possible to draw a graph showing speed, density and flow and to
relate each to each other (Haight 1963) but that is irrelevant. The mistake it leads to
is calculating the externality by differentiating with respect to the flow. This results
in the preposterous conclusion that if a road is operating at capacity, the externality
is infinite. We define the externality as those costs consequent on a decision that are
not taken into account by the decision maker. The decision in this case is to travel
or not. This results in a unit change in demand, but may have no impact on flow.

Treating the cost curve as the supply curve and averring that there is equilibrium
when demand equals supply. While the cost curve will be coincident with the
supply curve under perfect competition, there is no reason to expect this to be the
the price(s) at which the demand for road space during each time interval equals the road space available.
2
It might be argued that a diversion curve could result from benefits being different while the value of time is
constant across the population. We show later that the benefits and the value of time must be related.
3
There is a long-running controversy in the literature on whether it makes sense to define supply and demand
for trips in terms of flow. We are absolutely on the side that says that it doesn’t.
1
case for a monopoly or oligopoly. The problem we are addressing arises because at
the point where the demand and the marginal private cost curve cross, the demand
exceeds the capacity of the road (ie demand exceeds supply). Some authors regard
the situation under hypercongestion as an equilibrium by regarding the time spent
queuing as part of the generalised cost of travel. This is again confusing cause and
effect. On this definition, the soviet union was a market economy4.

Assuming a fixed demand. The purpose of road pricing is to change behaviour. If
we assume behaviour is fixed it is not surprising that we see some strange
consequences.
Most published papers attempt to calculate the congestion price by estimating the short-run
marginal cost and hence the ‘congestion externality’ (following Pigou, 1920). There
appears to be an assumption that this will result in demand being constrained to capacity,
but no proof that this is the case5. In this paper, we investigate the relationship between the
concept of matching supply and demand as a basis for pricing policy and the maxim to
price at social marginal cost. Unlike some authors, we are interested in the situation where
“hypercongestion” occurs, or would occur in the absence of road pricing6.
2
BASIC CONCEPTS
Road vehicles interact with one another. Addition of one more vehicle to a traffic stream
has a tiny but quantifiable effect on other vehicles, which are slowed by a small amount.
This economic externality is of little significance when traffic is sparse. When traffic is
dense, the effect of one more vehicle is bigger, although still tiny, but the additional vehicle
now affects a large number of other vehicles. The aggregate externality (summed over all
vehicles) can be large.
As traffic flow increases, travel speed reduces. As traffic becomes congested, speed falls
off at an increasing rate. Speed cannot keep falling monotonically to zero because at zero
speed there is zero flow.7 Hence there must be a point, call it ‘capacity’, at which the flow
rate is a maximum. If flow is a maximum at capacity, then at capacity, the incremental
vehicle slows the traffic stream by an amount that leaves the flow rate unchanged.
What happens beyond capacity? If we plot speed against the actual vehicle flow rate, a
level of flow exceeding capacity cannot be represented analytically. Yet, operating at
capacity is no barrier to more vehicles attempting to use the road. When more vehicles
attempt to use the road, speed continues to fall. What is more, the flow rate falls. Vehicles
travel increasingly close to each other, increasingly slowly.8 A speed-flow relationship is
4
If you think of regular commuters adjusting their start times to ensure that they arrive at their destination
within a particular time range, you might describe this as an equilibrium. This is not what we mean by supply
equalling demand – rather it is a response to market failure.
5
Of course it will be if you treat flow as if it were demand. Flow can never exceed capacity.
Some authors claim hypercongestion is a temporary phenomenon and don’t study it, or even claim it doesn’t
exist. This seems to assume away the problem we are trying to solve. Since the speed does not fall
significantly until close to capacity, the optimal congestion charge for a road operating below capacity is near
zero. This is a non-problem.
6
7
Mathematically there is another possibility, which is that the speed falls asymptotically to a limiting value, but
this is contrary to observation.
8
If this situation were to continue indefinitely, queues of traffic would increase without limit. Since peak periods
are of finite duration, this does not happen.
2
not very useful when demand exceeds the road capacity. It is better to use demand, not
flow, as the independent variable.
Demand for what? Hills and Gray (1999) adopt vehicle trips as the basic unit of demand.
We will call this aggregate demand. These trips will have a preferred arrival time and
associated travel time and schedule delay costs. We are particularly interested in the
situation where not all trips can be accommodated at the preferred travel time, so that some
vehicles arrive at their destination early or late and thus incur schedule delay costs. Drivers
will select their departure time in order to minimise their expected total travel costs.
Now consider a time interval of some (arbitrary) length during this period. Then the
number of vehicles wishing to start their journey during this time interval is the interval
demand (referred to as “demand” unless it is not clear from the context). This demand is
influenced by the expected travel time and schedule delay costs, plus tolls etc if any. The
“flow” is the actual number of trips that commence in that time interval, while the
“capacity” is the maximum achievable flow. All three measures are in units of vehicles per
hour. An increase in aggregate demand would result in the demand in each time interval
increasing. This may be accommodated by an increase in flow if demand in the interval is
less than capacity and/or by some trips being displaced to later or earlier time intervals.
Many practitioners refer to “speed-flow” or “volume-delay” curves as if the independent
variable was the flow (volume). The curves commonly used for transport modelling are
generally speed (or time) vs demand curves. This is a common source of confusion. Hills
and Gray (1999) overcomes this by referring to speed flow curves as “performance curves”
and time vs demand curves as “supply and demand curves”.
DIGRESSION: The notion of capacity is ill-defined. There is no identifiable point at which an
increasing traffic flow reaches a limit, viz the road’s capacity. It was once thought that
1800veh/h was the capacity of a single lane. That increased to 2000veh/h. But
supersaturated flow rates of 2700veh/h per lane have been observed. Such supersaturated
traffic flows are common on freeways with ‘ramp metering’, which injects vehicles into
gaps in the approaching traffic stream. Supersaturated flows are unstable. A perturbation
such as one vehicle braking a little too hard can surprise the following driver who has to
brake harder still. This set ups a shock wave that causes all following vehicles to come to a
standstill before carrying on, an effect which persists throughout the remainder of the peak
period. Vehicles come to a stop at some point on the road for no apparent reason (that
reason having long gone). After the perturbation, the traffic flow cannot recover to its
former high rate (until the next peak period).
3
3.1
PRELIMINARY RESULT
Simple Model
We assume a one kilometre section of road operating under congested conditions (level of service
“D” or worse). In these conditions all traffic moves at approximately the same speed.
Let speed be v, in km/h, flow be F, in vehicle/h, and density be D, in vehicles/km.
Speed is a function of the density ie v = f(D)
Then the time to traverse a section of road t = 1/v = 1/f(D)
The cost to the ith user is μi t = μi/f(D) +vehicle operating cost
Ignoring the variation in vehicle operating costs the externality is
E = Σ μi . dt/dD = μiD.(-f’(D)/f(D)2) where μ is the average value of time
3
The flow F =f(D). D
If we introduce the concept of capacity as the demand at which flow is a maximum, then at
capacity:
dF/dD = f(D) + D.f’(D) = 0
ie at maximum flow, f(D) = - D.f’(D)
Thus at maximum flow, E = μ D.(-f’(D)/f(D) 2 = μi/f(D) = μ t
In other words, at capacity, the additional cost imposed by the marginal vehicle is exactly equal to
the travel time valued at the average value of time.
3.2
Illustration
Early work (e.g. Greenshields (1935), Almond, 1965, Homberger, 1982) on the
relationship between speed and traffic flow assumed that speed decreased linearly with
increasing density of vehicles. ie that the function is of the form v=v0 – k*D
Then F = v*D = (v0v - v2)/k (a parabolic speed flow curve)
or F = v0D –k D2
F is maximum when dF/dD = v0 – 2k D =0
ie when D= v0/2k
Thus capacity = Fmax =(v0 – v0/2).v0/2k = v02/4k
Travel time (in hours) for one kilometre is
1/v = 1/(v0 - kD).
Differentiating gives the marginal cost when the density is increased by one veh/km,
namely, k/(v0 - kD)2 hours per vehicle for one kilometre of travel. Since D vehicles are
affected, the congestion externality is Dk/(v0 - kD)2.
Table 1 displays these quantities with v0=50 and k=5/8. Flow reaches a maximum of 1000
veh/h at a density of 40 veh/km. At this point, the externality cost and the travel time are
equal, at 2.40 minutes.
The speed-flow curve has the property that when the density exceeds 40 vehicles/km, both
the speed and the volume drop. This is a consequence of assuming that there is a maximum
flow, and results in the phenomenon of the “backward-bending speed-flow curve”
Table 1. Parabolic Speed-Flow Curve
Density
(veh/km)
Speed (km/h)
5
10
20
30
40
50
60
70
75
Flow
(veh/h)
46.9
43.8
37.5
31.3
25.0
18.8
12.5
6.3
3.1
234
438
750
938
1000
938
750
438
234
4
Externality
(min/km)
0.09
0.20
0.53
1.15
2.40
5.33
14.4
67.2
288.0
Travel Time
(min/km)
1.28
1.37
1.60
1.92
2.40
3.20
4.80
9.60
19.20
Figure 1. Typical Speed/Volume Relationship
This backward-bending phenomenon is illustrated in Figure 1, for data observed on the
northern motorway in Auckland, New Zealand. Note that rather than truly parabolic, the
speed actually only declines slowly until the flow breakdown zone.
The backward-bending phenomenon arises because flow is plotted as the dependent
variable. In fact, the flow is not independent. The true independent variable is the demand. In
this analysis we use density as a proxy for demand. This is equivalent to assuming that all
vehicles wishing to travel enter the link and queuing occurs on the road9.
3.3
Benefit Maximisation
We now introduce the concept of user benefit. For simplicity of exposition, suppose
congestion is the only externality. This simplification can be removed later. We assume all
vehicles travel at the same speed t. While this is not true in general, it will be true under the
congested conditions that interest us.
If there are no other externalities, then the net benefit to society will equal the sum of the
net benefits to vehicle users.
Net Benefit per hour
Then
= D.(B - t)* v
= B.F - D
where B is average benefit per user
dN/dD = B.dF/dD + F.dB/dD - 1
Verhoef has shown that density is not an ideal proxy for demand. I don’t have this article, so am not clear
whether this has any real impact on my conclusions. I suspect not.
9
5
Since dN/dD > 0 when D is small and dB/dD <0, this says that the net benefit is maximum
before flow is maximum. In other words for benefit maximization, demand must be kept
within the capacity of the road.
Introduce a price p such that flow is at a maximum. Now consider an increase in price of
dp that results in a reduction in demand of dD and a consequential reduction in time of dt.
Then the benefit will change by μD.dt – p.dD
This is an increase if p<μD.dt/dD
Thus the price should increase if p < E (or if p <μt at capacity)
The optimum price is (as expected) p = E.
3.4
Revenue Maximisation
As an aside, we can also show that revenue is maximized when
or
D.(dp + μdt) = p dD
p = D.dp/dD + μD.dt/dD
define the elasticity e as e= -dD/dp * p/D
Then p = - p/e + μD.dt/dD
The revenue maximizing price is thus
p= (E /(1+1/e)
where E is the externality
Hence if e is large (demand is price sensitive) p will be close to the optimum. If e is small
(demand is price insensitive) p will be significantly higher than optimum.
3.5
Relationship between value of travel time savings and benefits
Let the occupiers of vehicle i have a value of time μi and a benefit Bi
Assume that there is a price p that ensures that the road operates at capacity
Then p = μt = μdt/dD
(μ = Σ μi/ n)
But the benefit to the marginal user Bm = p + μmt
= μt + μmt
In other words Bi and μi must be related.
Now assume an exogenous increase in demand (a new subdivision opens) but the benefit to
former users is unchanged. The price must increase to keep demand to the same level. The
former marginal user is “tolled off” and there is a new marginal user. The travel time and
number of users is the same. For the new marginal user n:
Bn = p + μnt = μ’t + μnt
(p’ = μ’t
where p’> p and μ’ > μ)
The average value of time must be higher.
[I think we can take this further …]
6
4
GRAPHICAL ILLUSTRATION
We now present a simplified graphical
representation of the problem. This is not intended
to be rigorous, but to provide an intuitive
illustration.10
P
cost
Fig 2 shows the demand for road space and the
private user cost of travel. In the absence of road
charges, the user cost is made up of time and
vehicle operating costs. If the road is operating at
less than capacity, an equilibrium is reached.
What happens if the demand for road space
exceeds the space that can be provided? Fig 3
shows the situation where supply of road space is
fixed (independent of price). Drawing the supply
curve this way may be controversial, but note that
Q is demand not flow. There is no reason why
demand should not exceed the quantity offered.
However roads differ from simple goods in that
people continue to attempt to consume (and
therefore impact on other consumers) more than
the amount supplied (ie the capacity) The point
where the cost and demand curves cross is no
longer an equilibrium.
Because each user increases the travel time for all
other users, the social cost of travel exceeds the cost
perceived by the individual. Following Pigou, we
assert that the benefit will be maximised if the user
is charged the marginal social cost Fig 4.
Equilibrium occurs where the social cost curve
crosses the demand curve. (these are sometimes
misleadingly labelled “average cost” and “marginal
cost”)
But what if following this rule results in demand
exceeding the supply as appears to be happening in Fig
5? Increasing the price from p0 to p1 would enable
more vehicles to pass at higher speeds. p0 therefore
cannot be the optimum price. In fact since at p1 the
externality equals the value of the travel time, and at p0
it must be greater than the travel time, p0 must be
greater than p1. In other words demand at the point
where the demand and marginal cost curves cross must
be less than capacity. The figure as drawn must be
incorrect.
demand
Q
P
cost
demand
supply
Q
P
social cost
private cost
Popt
Q
P
social cost
P1
Po
private cost
Q
10
I only put these in to illustrate some of the points, I know using graphs can be misleading as you can draw
things that are impossible in practice. If I do use them I must label the figures.
7
social cost
P
We conclude that charging a price equal to the
externality (ie ensuring that the user pays the
marginal social cost) will result in the demand
being less than the capacity of the road as
shown in fig 6.
P1
Po
private cost
This confirms the approach of most authors,
who assume that this will be the case.
Q
P
But what happens if there is an exogenous
increase in the demand for travel (eg a new
subdivision is opened. This will result in a shift
right of the demand curve. This would appear to
push the ‘equilibrium’ point to the right so that
demand exceeds capacity (Fig 7). We know this
cannot happen
What must happen is that the cost curves move
anticlockwise as shown in Fig 8. Why?
Because the cost curve is the travel time
valued at the average value of time of those
who remain travelling. As demand increases, a
smaller proportion of the potential demand can
travel and the average of their time values will
consequently be higher. So both the private
cost and the social cost are higher.
social cost
private cost
Q1
Q2
Q
SC2
SC1
P
PC2
PC1
P2
P1
D2
D2
At p1 the demand will be greater than it was at
p0, but it still will be less than C, the capacity.
We conclude that as demand increases, flow will increase asymptotically towards a value of C.
5
OPTIMUM PRICING RULE
We can now state our optimum pricing rule.
Theoretical Rule
We have confirmed the validity of the conventional congestion pricing rule, which is to set the price
equal to the marginal social cost. What we have done is shown that this results in a price that is higher
than the market clearing price, but that the optimum price tends towards the market clearing price as
demand increases. We note that this relationship between the optimum price and the market clearing
price implies that the value of travel time is not constant.
Practical Rule
Applying the theoretical rule requires knowledge of
 The shape of the cost curve (and thus the marginal cost curve) close to capacity
 The average value of travel time savings for those who actually travel, noting that
this will depend on the ratio of satisfied to potential demand.
8
Neither of these can be determined with any certainty.
Fig 9, taken from the
Highway
Capacity
Manual, shows sample
speed flow (not demand)
curves
estimated
in
practice. These show the
relationship between the
flow rate and the average
speed for different free
flow speeds (these would
reflect
different
road
conditions,
roadside
hazards, etc). The most
notable feature is that they
are almost flat until the
last few hundred vehicles. Beyond the dotted line, the flow breaks down and the
relationship is indeterminate. This suggests that rather than the gently sloping curves of our
examples, real marginal cost curves are flat until close to capacity and then turn up sharply.
Which in turn implies that the optimum price will be close to the market clearing price.
The market clearing price is (again in theory) much easier to determine. The price can be
adjusted in small increments up or down to find the price that maximises the flow.
Unfortunately flows are unstable close to capacity, and small perturbations in the flow can
cause the flow to collapse. It is thus better to price a little higher – the aim is the maximum
consistently sustainable flow.
Fig 9 shows five zones labelled LOS (level of service) A to E which depend on the density
of the traffic. LOS C represents a range in which the influence of traffic density on
operations first becomes marked. The ability to manoeuvre within the traffic stream is
affected by the presence of other vehicles and average travel speeds show some reduction.
LOS D represents a range in which ability to maneuver is severely restricted and speeds are
curtailed because of the traffic volume. Only minor disruptions can be absorbed without the
formation of extensive queues and the deterioration of service to LOS E and LOS F. LOS E
represents operations at or near capacity and is quite unstable. At LOS E, vehicles are
operating with the minimum spacing at which uniform flow can be maintained. Thus
disruptions cannot be damped or readily dissipated, and most disruptions will cause queues
to form and service to deteriorate to LOS F. Speeds are highly variable and unpredictable.
LOS F (not shown) represents forced or breakdown flow (the backward section of Fig 1).
The practical prescription is thus to price to ensure a LOS of somewhere between C and D.
Pricing at this level would appear likely to be both economically and technically efficient.
6
CONCLUSION
We have confirmed the validity of the standard prescription to price at social marginal cost, but
demonstrated that calculating such a price would be a practical impossibility. On the other hand
pricing to ensure that the level of service (LOS) is between C and D (as defined in the Highway
Capacity Manual) provides an operationally implementable rule that will ensure both economic
optimality and technical efficiency.
9
REFERENCES
To come
10