Staged experiments must always be viewed with caution, especially when
the subjects are shown to be somewhat confused by the experiments. None-
theless, the overwhelming weight of the free-rider experiments is that people
are willing to behave cooperatively even when it is clearly in their interests to
behave selWshly. And, as Andreoni points out, the real world is likely to be
more conducive to acts of kindness than these experimental settings are.
These Wndings are somewhat encouraging for the mainstream normative
public sector theory.
AGGREGATE EXTERNALITIES
Thus far we have considered two kinds of externalities that are likely to cause
severe practical problems for the government: (1) individualized externalities
arising from exclusive activities for which the identity of each individual
consumer matters and (2) nonexclusive goods. Fortunately, a number of
important externalities have a special form that is much more amenable to
corrective public policy action
Consider the example of highway congestion. An additional car on a
congested highway generates an external diseconomy to anyone driving on
the highway because it adds to the total number of vehicles on the road, to the
total amount of congestion. But no one cares who is actually driving the
additional car. This is an example of an aggregate externality, meaning that
the external eV ect depends only upon the aggregate level of some exclusive
economic activity. The identity of the individuals within the aggregate is
irrelevant.
To formalize the idea of an aggregate externality, let X
ik
person i's
driving on a particular highway, good k. Write:
C C
P
H
i1
X
ik
C(X
k
),
qC
qX
ik
qC
qX
k
all i 1, ...,H (6:46)
where:
C congestion on the highway.
X
k
aggregate number of cars on the highway at any given time
(assuming one person per car).
The condition qC/qX
ik
qC/qX
k
implies that a decision by anyone to drive
on the highway has an identical marginal eVect on total congestion.
If consumers only care about the aggregate level of congestion, then they
each have a utility function of the form:
U
h
U*
h
X
hn
;X
hk
; C(X
k
)U
h
X
hn
;X
hk
;
P
H
i1
X
ik
(6:47)
6. CONSUMPTION EXTERNALITIES 193
where:
X
hn
good (factor) n consumed (supplied) by person h,
n 1, ...,k1, k 1, ..., N, each assumed to be a pure private
good (factor):
X
hk
use of the highway by person h.
CandX
k
, as above.
U
h
( ) has the following properties:
qU*
h
qX
ik
qU
h
qX
k
, for i 6 h(6:48)
qU*
h
qX
ik
qU
h
qX
hk
qU
h
qX
k
, for i h(6: 49)
If anyone but person h uses the road, his utility is aVected simply because
aggregate road use increases, thus increasing congestion. When person h uses
the road, however, there are two distinct eVects. On the one hand, person h
has some private reason for choosing to drive on the road that is totally
unrelated to the congestion problem. On the other hand, he is adding to the
congestion exactly as any other driver would and with the same consequences
for his utility. He may or may not consciously understand that his choice to
drive on the road necessarily contributes to the congestion and thereby lowers
his utility (a point we will return to later), but he certainly views his own use of
the road diVerently from anyone else's use of the road. This is why the
derivative of U
h
with respect to X
hk
has two separate terms. A private-use
term and a congestion term. Note, Wnally, that congestion must be a function
of aggregate road use and not a general function of individual road use such
as C* C*(X
1k
, ...,X
ik
, ...,X
Hk
). With this more general formulation,
qU*
h
/qU
ik
6 qU*
h
/qX
jk
for i 6 j, and we are back in a situation of individu-
alized externalities in which the identity of the individual consumer matters.
Congestion is not the only important example of an aggregate externality
by any means. Virtually all pollution externalities aVecting consumers,
whether caused by other consumers or by producers, can be thought of as
aggregate externalities arising from exclusive economic activities. Smog,
airport noise, and industrial air and water pollution usually exhibit this
property.
The Pigovian Tax
Aggregate externalities are far more amenable to government policy than are
the individualized externalities or nonexclusive goods. The government need
not design a set of H taxes, one speciWc to each individual. They can be
corrected by a single tax levied on the externality-causing activity. The single
194 AGGREGATE EXTERNALITIES
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