Preliminary Examination
Public Economics
Fall 2009
University of Minnesota
Department of Economics
Ph.D. Preliminary Examination
Public Economics
Instructions: Answer THREE (3) of the four questions. Make your answers
as neat and concise as possible. If you need to make additional assumptions
when answering the questions then make sure you note them.
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Preliminary Examination
Public Economics
Fall 2009
Question 1
Providing Public Goods
Consider the following public goods problem. The economy has n agents. Each
agent’s preferences are given by c + δv, where c denotes consumption of a private good, v
denotes utility from consumption of the public good if it is provided, and the parameter
δ equals 1 if the public good is provided and 0 if the public good is not provided. The
parameter v is common knowledge. Each agent i = 1, ..., n is endowed with mi units of
the private good. This endowment is private information and is drawn from a uniform
distribution with support [0, m̄]. The cost of providing the public good is kn where the
parameter k < v and k < E(m).
A social planner desires to maximize ex-ante expected utility of the agents by designing a mechanism which prescribes when the public good should be provided. No agent
can be forced to contribute more than the endowment of the private good that agent
possesses.
1. Set up the social planner’s problem.
2. Analyze what happens to the solution of the social planners’s problem as n converges
to infinity. How does your answer to this question change if k > v.
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Preliminary Examination
Public Economics
Fall 2009
Question 2
Consider the following two period economy. In each period, the economy is populated by a single household and a single government. The same government lives in both
periods, but the second period has a different household than the first period. In each
period, the household has a unit endowment of the single divisible consumption good and
√
the household can eat this endowment, or invest it in a technology f (x) = 2 x, where
x ∈ [0, 1] is the amount the household invests and f (x) is the amount of the consumption
good the technology puts out.
The government is of two possible types. With probability ρ0 ∈ (0, 1), the government is a non-strategic robot which cannot confiscate. With probability 1 − ρ0 , the
government is strategic and can either confiscate (take all output from the household) or
not confiscate (take no output from the household).
The portion, (1 − x), of the household’s endowment that the household doesn’t
invest is not subject to confiscation by the government. The output of the technology,
f (x), is subject to confiscation. Thus a household which invests x consumes (1 − x) if
output is confiscated and (1 − x) + f (x) if output is not confiscated. When a government
confiscates, it consumes f (x). When it does not confiscate, it consumes nothing. The
household in each period and the strategic government care only about consumption and
are risk neutral, and the strategic government does not discount.
(a) Characterize the second period taking as a parameter, ρ1 , the probability that
the government is non-strategic at the beginning of the second period. What is a
strategic government’s equilibrium second period payoff, V (ρ1 )?
(b) Is it possible for their to be an equilibrium where the strategic government confiscates with certainty in the first period? Show why or why not.
(c) Is it possible for their to be an equilibrium where the strategic government does not
confiscates with certainty in the first period? Show why or why not.
(d) Is it possible for their to be an equilibrium where the strategic government confiscates with probability π ∈ (0, 1) in the first period? If so, characterize.
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Preliminary Examination
Public Economics
Fall 2009
Question 3
Implementing Efficient Allocations
Consider the following two period insurance incentive problem. The economy has a
continuum of agents in the unit interval and lasts for two periods, denoted by t = 0, 1. All
agents are endowed with y0 units of a single consumption good in period 0. The technology
for producing the consumption good in period 1 is
y1i = θi li
where yi denotes output by agent i ε [0, 1], θi denotes agent i s productivity, and li denotes
labor input by agent i. Let θi denote the type of agent i. This type can take on two values
θi = 0 and θi = 1. The fraction of the population of type 0 is π. An agent’s type is realized
at the beginning of period 1 and is private information to each agent. Goods can be stored
over time at an exogenous gross interest rate R. Each agent believes the probability of
being of type 0 is π. Each agent’s preferences over deterministic allocations is given by
u(c0i ) + β{u(c1i ) − v(li )}
where u denotes the utility function over consumption, β is the discount factor, and v is
the disutility function over labor. Agents maximize expected utility.
1. Define an incentive-feasible allocation.
2. Assume that social welfare is given by the integral over each agent’s utility. Set up
the planner’s problem of solving for an incentive-feasible allocation which maximizes
social welfare. What are the first-order conditions that the solution to this problem
must satisfy?
3. Consider a government which is restricted to using linear taxes on savings and
income, and lump-sum transfers. Can this government implement the solution in
part 2? Explain why or why not. If not, what additional instruments would help?
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Preliminary Examination
Public Economics
Fall 2009
Question 4
Consider a static world with a unit continuum of agents. There are two goods:
Apples and Bananas. Suppose type 1’s have an endowment of 1 apple and 2 bananas and
preferences represented by the utility function u1 (ca , cb ) = 2 log(ca ) + 2 log(cb ). Suppose
type 2’s have an endowment of 3 apples and 2 bananas and preferences represented by the
utility function u2 (ca , cb ) = ca + cb . There are exactly
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2
of each type.
(a) If endowments are observable, solve for the ex-ante social optimum.
(b) Now suppose that type is private. In particular, type 2 agents can claim to be type
1 agents by secretly hiding apples. Is the ex-ante social optimum from the previous
question incentive compatible? If not, show why not.
(c) Solve for the incentive feasible ex-ante optimum when type 2’s can claim to be type
1’s. Devise a tax and transfer scheme to implement this.
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