Problem Set 4: Ine¢ ciencies
1. Consider the equilibrium of the Robinson Crusoe economy with consumption taxation
that we discussed in class. Change the setting by assuming, instead, that there is no
consumption taxation and that the consumer’s labor income is taxed at the ‡at rate
of instead. As in class, assume that the equilibrium tax revenue is refunded to the
consumer.
(a) Illustrate the resulting equilibrium, its ine¢ ciency, and the deviation of the equilibrium
allocation from the Pareto e¢ cient allocation.
(b) How do your conclusions di¤er from the conclusions we drew in class for the case
of consumption taxation?
(c) What conclusions would you draw about the performance of the European economy
relative to the US economy given that both income and consumption taxation is
higher in Europe?
(d) Intuitively explain how taxation disrupts the coordinating function of the price
system and hence results in economic ine¢ ciency.
2. A government is considering whether to institute mandatory vaccination program
against an infectious disease. Absent a vaccination, an individual will catch the disease
with a chance p(x), with p0
> 0, p00
< 0, where x is the fraction of individuals who
have not been vaccinated. But getting a vaccination is a hassle, it hurts (ouch!), and
sometimes it gives you a mild disease. Individuals have linear utility functions over
wealth w, the chance p of getting the disease, and the total cost of vaccination v. That
is, u(w; p; v) = w p v. There is a continuum of individuals of measure 1 and
v U[0; 1] in the population.
(a) Characterize who gets vaccinated in a laissez-faire (competitive) equilibrium (assume
interior solutions in all cases).
(b) Characterize who gets vaccinated in the Pareto-optimal outcome, and …nd taxes
that decentralize this outcome.
(c) Rank the policies available to the government: mandatory vaccination, laissez-faire,
and an optimal tax.
2. Consider a small Vysocina town named Jihlava where too many people are named
Libor and Alena. There are altogether 10 Libors and 10 Alenas. Each Libor likes
playing a guitar and singing country music outdoors. On the other hand, Alena’s, who
generally prefer opera, despise country music. In particular, let Libor #k (k = 1; ::; 10)
derive enjoyment of monetary value 110k ln(1 + xk) from playing at the volume level
xk. The minimum volume level is zero (silence), but each Libor can really max things
up to the volume level of 1000. Jihlava is such a small town that the e¤ect of volume is
additive. For every Libor #k, there is a corresponding Alena #k (k = 1; ::; 10) who is
hurt by the amount k(x1 + ::: + x10) by the totality of singing. Suppose all the utilities
are expressed in monetary terms.
(a) How much playing do Libors do if they are unaware of Alenas’feelings?
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(b) Now assume that Alenas speak up and go to the Town Council, which imposes
an e¢ cient individual-speci…c tax on music production. What is this tax? Is it
uniform across di¤erent Libors?
(c) Now mayor Ilir gets swung into power in a major election upset, and being a true
libertarian, he instantly abolishes the tax and insists that all Libors and Alenas
cut a deal among themselves. Ilir insists that if no deal is cut, then no music is
allowed. What deal will Libors and Alenas cut? Up to how much will each Libor
be willing pay to have the deal? At least how much will each Alena need to get in
order to allow the deal? Depict your answer in a diagram of marginal costs and
bene…ts.
(d) Due to a clandestine operation of an underground country music movement, mayor
Ilir is thrown out of o¢ ce in a reactionary coup led by the pro-country mayor Olex,
who then allows country music to be played at free will. However, he allows for
the possibility of Libors striking a deal with Alenas and enforces any resulting
contract. How is the resulting total volume level and the wealth of Libors and
Alenas a¤ected?
3. Consider a two-consumer economy in which there are two types of goods: a private
good and a public good. The two consumers have identical preferences given by
u(xi; g) = xi + 4 ln g;
where xi is the consumer i’s consumption of the private good, i = 1; 2, and g is the
quantity of the public good. The total amount of the public good g is determined by
the sum of the individual contributions, i.e., g = g1 + g2. Each consumer is endowed
with 10 units of the private good that can be converted into the public good in a 1-for-1
fashion. That is, each consumer faces a budget constraint of the form
xi + gi 10:
(a) What is the Pareto-e¢ cient amount of the public good?
(b) Now suppose the public good is provided by private contributions. Also suppose
that in equilibrium, each consumer treats the contribution level of the other consumer
as given (i.e., exhibits Nash behavior). What is the equilibrium amount of
the public good and how much does each consumer contribute?
(c) Compare this equilibrium amount with the Pareto-e¢ cient amount. Do they differ?
If yes, explain why.
(d) True or false: The amount of the public good provided by private contributions
could be increased by reallocating 2 units of endowment from consumer 2 to
consumer 1.
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