E100 Microeconomics 2020-21
Problem set for Easter Term revision
Question 1
A profit maximising monopolist is selling a single item online. There are N consumers
willing to buy the product, and each consumer (i) is prepared to pay at most (vi) for
the item, where vi is a random variable drawn from a uniform distribution over the unit
interval [0,1]. The monopolist proposes to sell the item using an English (ascending bid)
auction.
a. Calculate the Expected Revenue for the seller from the auction.
b. The monopolist asks for advice about whether or not they could adopt an alternative
selling mechanism that would increase their expected revenue from the sale of the
item. Would you advise using an alternative selling mechanism? If so, describe an
optimal selling mechanism the monopolist should adopt, and calculate the expected
revenue using this mechanism. If not, carefully explain why not.
c. How would your answer to b. vary if the monopolist was unsure of the actual
number of potential buyers (N)?
Question 2
a. A club needs to decide on the membership fee at its annual members’ meeting.
The club uses the majority rule to make decisions and any member can make a
proposal to be considered at the meeting. The members have different views on the
appropriate membership fee. What will the meeting decide? Explain in detail your
reasoning.
b. Suppose that the members’ meeting in addition to the membership fee also needs
to decide on how the total fee income should be allocated between two different ac-
tivities. The members have different views on what the appropriate membership fee
and allocation is. What will the meeting decide? Explain in detail your reasoning.
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Question 3
Consider a world with N = 100 countries. Each country i ∈ {1, ..., 100} must choose a
level of abatement qi ∈ {0, 1}. The public benefit to each country of qi = 1 is b = 8 and
the private cost to each country of qi = 1 is c = 100. The costs and benefits of qi = 0
are both 0. Countries are considering signing a standard International Environmental
Agreement (IEA) as covered in the lectures.
(a) What is the Nash equilibrium outcome without an IEA and with an IEA?
(b) Suppose now that signatories to an IEA can use trade sanctions on non-signatories.
These impact country payoffs as follows: for an IEA with k signatories, trade
sanctions benefit each signatory by τ(N−k) and impose costs on each non-signatory
of τk. What values of τ can sustain full cooperation in equilibrium?
(c) How do your answers from parts (a) and (b) relate to the conclusions of Nordhaus
(2015) Climate Clubs: Overcoming Free-riding in International Climate Policy?
Question 4
Consider a society inhabited by a continuum of citizens and normalize the size of the
population to 1. Suppose that the preferences of agent i over a public good y and a
private good c are
wi = ci + ln y. (1)
Individuals are endowed with different levels of wealth, denoted ei. The wealth distri-
bution is uniform and defined on the interval [e, e], with e > e > 1. Public goods are
produced from private goods according to the technology
y = aQ, (2)
where Q is the total input of private goods and a > 0 is the productivity of public
production. The production of public goods is financed via a tax, t, on the wealth of each
individual and so, individual i′s budget constraint is
ci ≤ ei(1− t). (3)
(a) What is individual i’s preferred choice of the public good and the tax rate? Are
the conditions for the Median Voter Theorem satisfied? Explain.
(b) Under majority rule and universal franchise, what is the selected policy?
(c) Explain what happens to the supply of public goods and to the tax rate if
(i) The productivity of government production a increases.
(ii) Average wealth increases.
(d) Suppose the franchise is restricted in such a way as to exclude some poor individuals.
Assume that only individuals with wealth above ê ∈ (e, e) are allowed to vote. How
does that affect the supply of public goods and the tax rate? Can an extension of
the franchise or an increase in wealth explain an increase in the size of the public
sector? Explain.
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