Problem 3. General Equilibrium and Uncertainty. Consider an economy with two agents, both with the following utility function: U(c) co+E[co] - 2Var(co), where co is consumption at t = 0 and co is consumption at t are two possible states of nature at t agents endowments are given by w1 w 1, if state = 1, with probabilities occurs. There 1/4 and 2 3/4. The = (w, (w, w)) = (4, (1,5)) (w (ww)) (4, (5, 1)), where is the t = 0 endowment of agent k, (w) 0-1,2 is the t = 1 endowment of agent k in state 0. 1. Intuitively, what should a Pareto optimal allocation satisfy in this particular setup? 2. Is there more than one Pareto optimal allocation? If so, characterize the set of Pareto optimal allocations. 3. Is the following allocation Pareto optimal? Explain. w1 (4, (2,2)) w2 = (4, (4,4)). For the remainder of the problem, assume the agents both have the following utility function: U(c) = co+E[ln(c)], with the same endowments and state probabilities as before. 4. Describe the set of Pareto optimal allocations for the new utility function. [Hint: set up and solve the utility maximization problem of a benevolent social planner. Consider the three cases > 1, A = 1, and 0 < < 1.] 5. Let denote the quantity of the Arrow-Debreu security with payoff in state 0 de- manded by agent k, where qe denotes the price of the Arrow-Debreu security with payoff in state 0. In both states, the Arrow-Debreu securities are in zero net supply, i.e., z + z += 0 += 0. 6 Compute the competitive equilibrium allocation. Is it Pareto optimal? Discuss intu- itively its characteristics (the determinants of the prices of the two securities and the post-trade allocation). 6. Suppose only one Arrow-Debreu security with payoff in state 1 is traded. It is in zero net supply. Compute the competitive equilibrium allocation. Is it Pareto optimal?