Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
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
private ownership of Lake
X
would achieve the same result.
19.3 This is another example of externalities inherent in a common resource. This question
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
law. For a fairly complete analysis of many of the legal issues posed here, see S. Shavell,
Economic Analysis of Accident Law.
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
which the market is organized.
CHAPTER 19:
Externalities and Public Goods
Chapter 19: Externalities and Public Goods
215
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
Chapter 19: Externalities and Public Goods
216
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
Chapter 19: Externalities and Public Goods
217
19.4 Under caveat emptor, buyers would assume all losses. The demand curve under
Chapter 19: Externalities and Public Goods
218
19.5
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
Chapter 19: Externalities and Public Goods
219
Next, use the implicit function rule to find
i
MRS
:
b. To find the social optimum, equate
RPT
with the sum of
i
MRS
:
Next, compute the tax implementing this optimum. We want to induce
Analytical Problems
19.8 More on Lindahl equilibrium
a. The condition for efficiency is
Chapter 19: Externalities and Public Goods
220
b. In a Nash equilibrium, each person would opt for a share under which
19.9 Taxing pollution
a. The efficient allocation of resources would maximize total industry profits, given
the constraint that total pollution must equal K.
Assuming a fixed market price p and a wage w, the associated Lagrangian is
b. With a tax
t
on output, the profit function for each firm would be
c. With a tax on pollution, the profit for each firm would be
Chapter 19: Externalities and Public Goods
221
19.10 Vote trading
Suppose preferences are as follows:
b. Suppose individual 3 is very averse to A and reaches an agreement with individual
19.11 Public choice of unemployment benefits
a. So long as this utility function exhibits diminishing marginal utility of income,
19.12 Probabilistic voting
a. Candidate A will choose a set of
A
i
so as to maximize the expected probability of
b. Yes. This is a consequence of Glicksberg’s (1952) extension of Nash’s existence
c. For each candidate, the optimal platform satisfies the condition that
i
fU
is a
Chapter 19: Externalities and Public Goods
223
