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 state s. Suppose there are two possible states, 1 and 2, with probabilities Π₁ and Π₂.

a) Write 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).

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?