National Tax Journal, Vol. 44, no. 4,
(December, 1991), pp. 529-32
PRICE
DISCRIMINATION
STEPHEN
SHMANSKE*
XISTING models of the economics of
E congestion generally reach the same
result: prices should cover all marginal
production costs for use of a crowded facility as well as an extra premium for costs
imposed on others using the crowded facility.' In formal models the congestion cost
has typically been related to the number
of people using or desiring to use the facility. There is a hidden assumption
in this
approach that all users are equally harined
by congestion.
This assumption
allows a
substantial
simplification
of the model.
Implicitly, all users impose the same costs,
bear the same costs, and pay the same
price. These results correspond nicely to
results
in economic models with no
congestion. However, once differences in
congestion costs are recognized,
the theoretical optimality
of uniform pricing
across individuals
no longer holds.' Generally, it is unrecognized
that congestioninternalizing
pricing does not lead to uniform pricing. This note illustrates
a case
in which congestion results in a waiting
queue for system access. In this setting,
price discriminatory
user fees both increase the revenue earned by the system
and reduce congestion costs bom6 by users.
Practically,
congestion
may take the
form of decreased benefits to each user of
the crowded facility. Alternatively,
it may
mean an extra cost has to be incurred in
order to achieve the same benefit the user
could have achieved in the absence of
congestion. This extra cost may be a time
cost if it takes longer to use a crowded facility. Extra time may be spent either actually using the facility or in queuing to
gain access to it. The possibility of queuing
presents a nexus between the economics
of congestion and the economics of rationing by waiting. 3
An example helps to highlight this connection. Consider a toll bridge. When not
crowded, the bridge is crossable at the
speed limit in a given amount of time.
During rush hour, however, slowed egress
*California State University, Hayward CA 94542.
529
L
AND CONGESTION
from the bridge may back up traffic onto
the bridge itself, slowing drivers and
causing delays. As an alternative
to a
backed-up bridge, entrance to the bridge
could be slowed through the use of a toll
booth and metering devices, thus causing
the waiting to take place at the toll booth
instead of on the bridge itself. Since optimal congestion-intemalizing
pricing does
not in general reduce congestion to zero,
the queues at toll booths remain and the
economics of queuing is relevant.
The usual assumption
of equal waiting
costs is unrealistic
in this setting. V&ile
a first-come-first-served
system seems
fairest when all people waiting are identical, an enlightened
policy when people
are different suggests that those least
willing and able to wait be allowed to go
first. Indeed, except for overwhelming
transactions
costs and logistic difficulties,
opportunities
for exchanges
would arise
between patient drivers at the head of the
line and those in a great hurry at the line's
end. Fortunately,
there ir, an indirect
method of making this exchange-price
discrimination.
A high-priced lane at the
toll bridge, for instance, would attract
those in the biggest hurry. The wait for
everyone else in the lower-priced
lanes
would be only marginally
longer, while
the total congestion cost will diminish due
to the more rapid processing of those with
the highest waiting costs.
This note demonstrates
how different
prices, where higher prices are linked to
shorter waits, will cause a self-selection
among consumers that can potentially
increase the efficiency of the waiting system. Note that the gains come essentially
from lowering the total waiting costs by
altering the placement of consumers in the
waiting queue for a single facility, as opposed to either the reduction of conges
tion by raising all prices to limit the
quantity demanded or the establishment
of a second facility with different characteristics
and different prices as in the
segregated
clubs models.
National Tax Journal, Vol. 44, no. 4,
(December, 1991), pp. 529-32
530
NATIONAL TAX JOURNAL
A Simple Model
The simplest framework within which
to examine such a pricing system has two
lanes and three people of two different
types. In each lane the service per customer takes one unit of time. There are
two "type A!'individuals with waiting costs
of w per unit of time, and one "type B"
individual with waiting cost of w + k per
unit of time with w > 0, k > 0. The three
customers
arrive simultaneously
but in
random
order (there are three distinct
patterns).
With the same price, P, on each toll, the
first two customers
to arrive start to get
served immediately
and the third must
wait for one unit of time before obtaining
service. The expected
cost for an individual of type i, E(Ci), is the sum of the price
paid and the total expected waiting
cost.
Therefore: E(CA)
= P + (4/3)w, E(CB) =
P + (4/3) (w + k). (Note that the time
spent being served is also costly.) The total amount
of money collected is 3P.
I will now prove the following proposition:
Proposition 1: For the setting described
above there exists a set of discriminatory
prices that simultaneously:
(a) lowers the
expected cost to both type A and type B
individuals;
and (b) maintains
the toll
collections.
Proof.- It is sufficient
to show that the
following set of prices, with 0 < e < min
(w/3, k/3) satisfies (a) and (b):
P, = P - (1/3)w - e
P2 = P + (2/3)w
sider the case where arrivals are in the
order B, A, A. The type B customer arrives first and goes to lane one. The first
type A person would rather wait for lane
one (costing P, + 2w), while the second
type A person will go through lane two
(costing P2 + w) instead of also waiting
for lane one (costing P, + 3w), because
P, + 2w < P2'+ w < P, + 3w, as substituting from the definitions of P, and P2
verifies.
Proposition
lb is now established.
To
compare expected costs, note that for a type
A individual,
if he arrives first (probability = 1/3), he immediately
goes through
lane 1. If he arrives second (probability
=
1/3), he waits on line on lane 1. If he arrives third (probability
= 1/3), he waits
on line 1 if there is no one else in line
(conditional
probability
= 1/2) and goes
through
lane two if there is already a line
for lane 1 (conditional
probability
= 1/2).
Thus the expected
cost is:
E(CA)
= (1/3)(Pl
+ 2e.
Queuing decision (lane)
1,1,2
1,2,1
1,1,2
To illustrate
the type of calculation
necessary
to reach these conclusions,
con-
+ w) + (1/3)(PI
+ (1/3)[(1/2)(P,
P + (4/3)w
E(CB)
+ 2w) + (1/2)(P2 + W)l
= (1/3)(Pi
+ (2/3)(P2
+ 2w)
- (e/2).
This is less than calculated
form pricing.
For the type B individual,
cost is:
above
for uni-
the expected
+ w + k)
+ w + k)
= P + (4/3)w
Notice first that 2P, + P2 = 3P; therefore, if exactly two people go through
the
low-price
lane, revenues
will be preserved. It is straightforward
to show that
the three patterns
of arrival
lead to the
related queuing
decisions.
Arrival order (type)
A, A, B
A, B, A
B, A, A
[Vol. XLIV
+ k + e
which is less than
long as e < k/3.
for uniform
pricing
as
Q.E.D.
Discussion
First, the micro analysis
of the queuing
discipline
indicates
that this model is not
an example
of segregated
clubs. Indeed,
each type A consumer uses the high-priced
lane one-sixth
of the time. The relatively
impatient
type B person uses the highpriced lane two-thirds
of the time."
Second, to the extent
that wages and
National Tax Journal, Vol. 44, no. 4,
(December, 1991), pp. 529-32
No. 4, Part 21
PRICE DISCRIMINATION
incomes are correlated to the value of time,
the toll collection scheme represents a
progressive tax because the type B person
pays the higher toll more often.
Third, if k = 0 so that there is no difference between type A and type B, then
no e can be found that satisfies the conditions in Proposition 1. It is not surprising that price discrimination will not help
when people are identical.
Fourth, if the type A person has no cost
of waiting (w = 0), then no positive e satisfies the conditions in Proposition 1.
However, further analysis reveals that the
following set of prices lowers the expected
cost for both types while keeping toll revenues constant in an expected value sense:
P, = P - e; P2 = P + (7/2)e,
for 0 < e < k/6. The proof will be left to
the interested reader. Revenues will equal
3P only in an expected value sense, because some of the time the high-priced lane
will 90 unused.
Fifth, since the magnitude of the original uniform price is left unspecified, it
should be clear that the policy of nonuniform tolls is a complement to, rather than
a substitute for, a general increase in the
level of tolls. That is, in places where dramatic increases in tolls have been instituted or are being considered
as
congestion-relieving
policies, there are still
further gains to be captured with discriminatory tolls. Furthermore,
discriminatory tolls can also complement timevarying tolls unless the time-varying tolls
eliminate waiting completely.
Sixth, simple arithmetic exercises will
also reveal sets of prices that raise toll
collections and reduce the expected cost of
a type A (type B) individual while holding the expected waiting cost of the type
B (type A) person constant. Indeed, by inspection of the system described in the
paper it is clear that small increases to
both P, and P2 will cause toll collections
to rise above 3P and still leave all consumers better off.
Seventh, in this last case the commuters are better off even before consideration of what will be done with the added
531
revenue. Individuals
will benefit even
more as long as the additional revenues
are not squandered. The additional toll
collections could go toward deficit reduction, programs for the homeless, AIDS research, mass transit subsidies, education,
or reductions in other taxes.
Eighth, since it is not often that an
economist gets to propose such a win/win/
win policy change, I would like to offer an
explanation as to why the possibility has
gone previously unnoticed. Most tolls were
originally put in place not as congestionrelieving devices, but rather as costrecouping mechanisms. Indeed, there are
even some tolls that were supposed to revert to zero when the capital costs were
completely repaid. If congestion is not an
issue, then Pareto-efficient
distribution
does require a uniform toll (perhaps zero)
to make each consumer face the sam cost
at the margin. Further, as mentioned
above, even if there is congestion, uniform tolls are still called for if all individuals incur the same time cost. Once
differences in the value of time are recognized, however, uniform tolls should be
scrapped when the facility is congested.
Bureaucratic inertia, predictable along the
lines of public choice theory, explains why
this does not generally happen. Since the
facilities directors are not principals and
do not stand to gain from more efficient
or more profitable operations, such opportunities generally remain unnoticed.5
Ninth, there are several minor logistic
difficulties that could be mentioned. For
example, there is the cost of signs and lane
dividers, and the problem of aggressive
drivers using the high-priced lane to pass
others and then cutting back into the lowpriced lane. These problems, however, are
not insurmountable,
and are probably not
even formidable.
Indeed, these exact
problems have already been overcome, as
evidenced in the preferential High Occupancy Vehicle (HOV) lanes that exist
around the nation.' One final possibility
is that traffic safety may be affected as
drivers jockey for arrival position in the
lane of their choice. Although this influence could cut either way, I suggest reason to believe that traffic safety will improve through this policy. Assuming that
National Tax Journal, Vol. 44, no. 4,
(December, 1991), pp. 529-32
NATIONAL TAX JOURNAL
532
it is the least patient drivers that do the
most lane switching and jockeying (after
all they have the most incentive to), a
nonuniform toll system will reduce such
lane switching because the least patient
drivers will know immediately to head for
the short line in the high-priced lane.
Finally, such discriminatory
user fees
can be (and already are) applied in other
areas. The postal service charges more for
express mail. Different prices can be
charged for different levels of priority in
obtaining computer access. Reservation
fees can secure priority access to municipal golf courses, etc. The policy conclusion is clear. Price discriminating user fees
on crowded facilities with rationing by
queues can simultaneously
raise revenue
(progressively) and make consumers better off.
ENDNOTES
**I would like to thank participants of the Economics Workshop at California State University, Hayward, and three referees of this journal.
'For example, see Berglas and Pines (1981), Deserpa (1978), Dorfman (1984), Muzondo (1978), Oakland (1972), Sandler (1975), and Sandler and Tschirart (1984).
'Berglas and Pines (1981) develop a model of mixed
and segregated clubs with heterogeneous consumers.
They show that, barring insurmountable scale economies, the optimal solution consists of having several
segregated clubs with different prices, characteristics,
and sharing sizes. In their mixed clubs, however, the
standard result of uniform pricing holds. In my model
the possibility of segregated clubs is not considered,
and in the resulting mixed club different prices can
dominate uniform pricing.
'See Barzel [19741, Holt and Sherman [19821, and
Suen [19891.
4A referee pointed out that the random order of arrival is responsible for the outcome that separate clubs
are not optimal. If the commuters arrived in the same
order every day, they would use the same lanes every
day and each lane could be considered a separate club
with its own price.
51nthis regard it is indicative that I undertook this
research not as a transportation policy-making in-
[Vol. XLIV
sider, nor as a political regional planner, but as a citizen, taxpayer, and commuter. Indeed, it is the commuters and taxpayers who stand to gain and who must
lobby for such policy change, not members of the
transportation
bureaucracy. Researchers at
CALTRANS who saw my model indicated that it was
interetig
d technically correct, but acted as if its
main effect would be to make more work for them.
6A refereesuggestedthat the HOVlanesare an example of the discrimination I have in mind. Arranging for riders in order to use the HOV lane is an extra
cast which regular commuters in slower lanes do not
incur. However, since no revenue is raised, it is difficult to see how society benefits from this practice.
The hoped-for result is that two or three drivers will
carpool, thus reducing trafric. What actually happens
is that one driver picks up two strangers at a bus stop,
thus avoiding the wait at the toll while the mass
transit grid loses two fares. When granting preferential treatment for a price, society at large benefits
from the revenue raised.
REFERENCES
Barzel, Yoram, "A Theory of Rationing by Waiting,"
Journal of Law and Economics, V. 17 (Apr. 1974),
73-95.
Berglas, Eitan and David Pines, "Clubs, Local Public
Goods and Transportation Models: A Synthesis,"
Journal ofpublic Economics, V. 15 (1981),141-62.
Deserpa, Allan C., "Congestion, Pollution and Impure
Public Goods,"Public Finance, V. 33, No. 1-2 (1978),
68-83.
Dorfinan, Robert, "On Optimal Congestion," Journal
of Environmental Economics and Management, V.
11, No. 2 (June 1984), 91-106.
Holt, Charles A. Jr. and Roger Sherman, "WaitingLine Auctions," Journal of Polftkd Economy, V. 90,
No. 2 (Apr. 1982), 280-94.
Muzondo, Timothy R., "Mixed and Pure Public Goods,
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Oaldand, William, "Congestion, Public Goods and
Welfare," Journal of Public Economics, V. 1 (1972),
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No. 1 (Feb. 1975), 25-38.
Sandler, Todd and John T. Tachirart, "Mixed Clubs:
Further Observations," Journal of Public Economics, Vol. 23 (1984), 381-89.
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