1.There are three states, s = 1, 2, 3, and three agents. Agents have expected utility functions with the same von Neumann-Morgenstern utility v(y) = ln(y), and probabilities ¹₃ for each state. There is no consumption at date 0. Date-1 endowments are as follows: agent 1 gets L units of the good in state 1 and zero in states 2 and 3, agent 2 gets M units in state 2 and zero in states 1 and 3, and agent 3 gets H units in state 3 and zero in states 1 and 2. Assume that H > M > L > 0.

(i)What can you say about Pareto optimal consumption allocations in this en-vironment, in particular, about the comparison of consumption across states? If you use a general result about Pareto optimal allocations, state that result clearly and make sure that it does apply to this environment.

(ii)Consider complete security markets. In an equilibrium in security markets, does agent 1 consume more in state 1 (where he gets the larger endowment) than in states 2 and 3? Does agent 2 consume more in state 2 than in states 1 and 3? Justify your answer. .

(iii)Let the market portfolio of securities be such that its payo equals the aggregate endowment. Show that the expected return on the market portfolio in an