CHAPTER 19:
Externalities and Public Goods
The problems in this chapter illustrate how externalities in consumption or production can affect
the optimal allocation of resources and, in some cases, describe the remedial action that may be
appropriate. Many of the problems have specific, numerical solutions, but a few (Problems 19.4
and 19.5) are essay-type questions that require extended discussion and, perhaps, some
independent research. Because the problems in the chapter are intended to be illustrative of the
basic concepts introduced, many of the simpler ones may not do full justice to the specific
situation being described. One particular conceptual shortcoming that characterizes most of the
problems is that they do not incorporate any behavioral theory of government—that is, they
implicitly assume that governments will undertake the efficient solution (i.e., a Pigovian tax)
when it is called for. In discussion, students might be asked whether that is a reasonable
assumption and how the theory might be modified to take actual government incentives into
account.
Comments on Problems
19.1 This problem provides an example of a Pigovian tax on output. Instructors may wish to
supplement this with a discussion of alternative ways to bring about the socially optimal
reduction in output.
19.2 This problem provides a simple example of the externalities involved in the use of a
common resource. The allocational problem arises because average (rather than
marginal) productivities are equated on the two lakes. Although an optimal taxation
approach is examined in the problem, students might be asked to investigate whether
poses a nice introduction to discussing “compulsory unitization” rules for oil fields and,
more generally, for discussing issues in the market’s allocation of energy resources.
19.4 This is a descriptive problem involving externalities, now in relation to product liability
19.5 This problem is an illustration of the second-best principle to the externality issue. It
shows that the ability of a Pigovian tax to improve matters depends on the specific way in
19.6 This is an algebraic public-goods problem in which students are asked to sum demand
curves vertically rather than horizontally.
19.7 Another public-goods problem. In this case, the formulation is more general than in
Problem 19.6 because there are assumed to be two goods and many (identical)
individuals. The problem is fairly easy if students begin by developing an expression for
the RPT and for the MRS for each individual and then apply Equation 19.40.
Analytical Problems
19.8 More on Lindahl equilibrium. This problem asks students to generalize the discussions
of Nash and Lindahl equilibria in public goods demand to
n
individuals. In general,
inefficiencies are greater with
n
individuals than with only two.
19.9 Taxing pollution. This problem primarily focuses on the idea that a Pigovian tax must
tax the externality, not just the output of the externality-creating firm.
19.10 Vote trading. This problem shows that voluntary trading of votes may still be unable to
yield a sensible social welfare equilibrium.
19.11 Public choice of unemployment benefits. A simple problem focusing on an individual’s
choice for the parameters of an unemployment compensation policy.
19.12 Probabilistic voting. This problem shows how to introduce continuity into voting
decisions by specifying a probability of voting function. This process is similar to
developing the mixed strategy concept in game theory.
Solutions
19.1 a. Given
0.4MC q
and
20.p
Setting
p MC
implies
20 0.4 ,q
in turn
implying
50.q
0.5 .SMC q
p SMC
20 0.5 ,q
gives
10 0.5 5,
x
l =
implying
10.
xy
ll
Substituting,
50 0.5(100) 100 100.
T
F
2
5 0.5 100,
Txx
= l l +
F
5 0,
Txx
dF dl l
implying
5,
x
l
15,
y
l
and
112.5.
T
F
c. In part (a),
50,
x
F
so the average catch = 50/10 = 5.
In part (b),
37.5,
x
F
so the average catch = 37.5/5 = 7.5.
Thus, the license fee on Lake X = 2.5.
d. The arrival of a new fisher on Lake X imposes an externality on the fishers
already there in terms of a reduced average catch. Lake X is treated as common
property here. If the lake were private property, its owner would choose
x
l
to
maximize the total catch less the opportunity cost of each fisher (the 5 fish he/she
can catch on Lake Y). So the problem is to maximize
5,
xx
Fl
which yields
5,
x
l
as in the optimal allocation.
19.3 Given
1,000 per well.AC MC
a. Produce where revenue per well equals
1,000 10 5,000 10 .qn
Solving,
400.n
There is an externality here because drilling another well reduces output
in all wells.
19.5
of
t
would cause the monopoly to produce output QR which is below the optimal level.
Since a tax will always cause such an output restriction, the tax may improve matters
only if the optimal output is less than QM, and even then, in many cases it will not.
19.6 a. To find the total demand for mosquito control, demand curves must be summed
19.7 a. In the general equilibrium model of Chapter 13, we saw that the perfectly
competitive equilibrium involved a tangency between the production possibility
frontier and individual’s indifference curve. In other words,
.
i
RPT MRS
To find
,RPT
the production possibility frontier can be rewritten as the implicit function
22
( , ) 100 5,000 0.f x y x y
valuated at the shadow price
.
b. With a tax
t
on output, the profit function for each firm would be
( ) ( ).
pf l wl tf l
( ) ( )
i i i i i i
pf l wl tg l
The first-order condition from profit maximization is
0.
iii
pf w tg
l
19.10 Vote trading
alternative,” say D.
19.11 Public choice of unemployment benefits
a. So long as this utility function exhibits diminishing marginal utility of income,
.yy
a. Candidate A will choose a set of
A
i
so as to maximize the expected probability of
his/her election. Thus, he/she will want to maximize
11
( ) ( ) .
nn AB
i i i i i
ii
f U U
Similarly, candidate B will choose to maximize his/her expected votes:
(1 ) ( ) ( ) .
nn
AB
i i i i i
n f U U
U
c. For each candidate, the optimal platform satisfies the condition that
i
fU
is a
U