Economics of Information and Uncertainty Problem Set, Microeconomics

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1. Consider a world with two assets: a riskless asset paying a zero interest rate, and a
risky asset whose return r can take values +10% or –8% with equal probability.
An individual has preferences represented by the utility function u(x) = ln x and an
initial wealth w0 = 10.
a) Solve the portfolio choice problem of the agent. What is the optimal amount z* of
risky assets?
Assume now that the agent also faces an exogenous additive background risk e, with a
distribution independent of r, that can take values +4 or –4 with equal probability
(additive means that the agent’s final wealth x is given by the portfolio of assets plus e).
b) Show that this background risk reduces the demand for risky assets.
(Note: I am not asking for the value of the new solution z**, just to show that it must be
smaller than the previous z*)
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2. The Barcelona Football Club is considering the signing of a player of international
fame. The problem is that the player has a reputation for having a weak knee. The
probability that the club assigns to the event that the player is injured during the season
(state ?1) is 30%. The expected revenue is €18 million if there are no injuries (state ?2)
and €3 million if there is an injury (state ?1). Assume that the club is risk neutral and
wants to maximize the expected profit.
a) If the cost of the contract is €15 million, what is the club’s optimal decision? What is
the expected profit?
We now consider the possibility that the club, before making its decision, can have the
player pass a medical examination. Doctors can issue a negative report (ß1), suggesting
that his knee is not strong enough to endure the season, or positive (ß2), suggesting that
his knee is fine. The probability that the medical report is negative when the knee is
really bad, P(ß1¦?1), is 80%, the probability that the report is positive when the knee is
really good, P(ß2¦?2), is also 80% (in other words, in each state there is a 20% chance
that the doctors are wrong).
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b) Represent the decision tree when the medical examination is done.
c) Compute the joint probabilities P(ßj, ?i), the probabilities of the signals P(ßj), and the
conditional probabilities P(?i¦ßj).
d) What is the club’s optimal strategy? What is the expected profit?
e) What is the value of the doctors’ opinion?
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3. There are two agents, A and B. Both have preferences represented by a von
Neumann-Morgenstern utility function u(cs
j) = ln (cs
j), where cs
j is consumption of agent
j in state s. Agents have risky endowments ?s
j and zs = ?s
A + ?s
B is the aggregate
amount of resources in state s. Suppose there are two possible states, 1 and 2, with
probabilities p1 and p2.
a) Write down the problem that determines the efficient allocations of consumption in
this economy (indicate with ?A and ?B the Pareto weights of the agents).
Take now ?1
A = 60, ?2
A = 0, ?1
B = 30, ?2
B = 60, and p1 = p2 = 1/2.
b) Find the efficient allocation of consumption when the Pareto weight of B is twice the
weight of A.
c) Represent it in an Edgeworth box. What is the marginal rate of substitution (the slope
of agents’ indifference curves) at the optimum?
d) Why should agent A’s consumption be lower than B’s even when her income is
higher than B’s?
e) Suppose that prior to the realization of the uncertainty, agents can trade (buy or sell)
two types of securities: asset 1 that promises a payment of 1 unit of resources if state 1
is realized (0 in state 2); asset 2 that promises a payment of 1 unit of resources if state 2
is realized (0 in state 1). Suppose further that the price of asset 1 is 2/3, the price of asset
2 is 1; agents decide how much to buy or sell of each asset taking their prices as given
(think of there being many agents like A and B, no one has market power, so we have
perfectly competitive markets). Look at the Edgeworth box. How much of each asset do
you think agent A would buy or sell? What will agent B want to do? Would they
manage to implement an efficient risk sharing?

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