problem1.  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.

i) 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 ε, 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 ε).  
 
ii) Show that this background risk reduces the demand for risky assets. 
 
problem2. 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 θ₁) is 30%. The expected revenue is €18 million if there are no injuries (state θ₂) and €3 million if there is an injury (state θ₁). Assume that the club is risk neutral and wants to maximize the expected profit.
 
i) 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 (β₁), suggesting that his knee is not strong enough to endure the season, or positive (β₂), suggesting that his knee is fine. The probability that the medical report is negative when the knee is really bad, P(β₁¦θ₁), is 80%, the probability that the report is positive when the knee is really good, P(β₂¦θ₂), is also 80% (in other words, in each state there is a 20% chance that the doctors are wrong).

ii) Represent the decision tree when the medical examination is done. 
 
iii) Compute the joint probabilities P(β_j, θ_i), the probabilities of the signals P(β_j), and the conditional probabilities P(θ_i¦β_j). 
 
iv) What is the club’s optimal strategy? What is the expected profit?
 
v) What is the value of the doctors’ opinion?

problem3.  There are two agents, A and B. Both have preferences represented by a von Neumann-Morgenstern utility function u (c_s^j) = ln (c_s^j), where c_s^j is consumption of agent j in state s. Agents have risky endowments ω_s^j and z_s = ω_s^A + ω_s^B is the aggregate amount of resources in states. Suppose there are two possible states, 1 and 2, with probabilities π₁ and π₂.
 
i) prepare down the problem that determines the efficient allocations of consumption in this economy (indicate with λ_A and λ_B the Pareto weights of the agents).
 
Take now ω₁^A = 60, ω₂^A = 0, ω₁^B = 30, ω₂^B = 60, and π₁ = π₂ = 1/2.  
 
ii) Find the efficient allocation of consumption when the Pareto weight of B is twice the weight of A. 
 
iii) Represent it in an Edge worth box. What is the marginal rate of substitution (the slope Of agents’ indifference curves) at the optimum?
 
iv) 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 Edge worth 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?