Derive the equilibrium of the economy Arrow Debreu Economy

Consider a competitive financial economy with two securities and two dates, i.e. date 0 and date 1. There is only one commodity (consumption good), and consumption takes place only at date 1, while there is no production in the economy. There are two consumers in the economy, indexed by h = 1, 2, and they receive an endowment of the consumption good at date 1. However, there are two states at date 1, indexed by s = 1, 2. Let ωhs denote consumer h’s endowment of the consumption in state s. Assume then that the endowments are specified as follows: (ω11, ω12) = (9, 3), (ω21, ω22) = (1, 9). The two securities have the following return structures (they are in terms of the consumption good): r1 = (2, 1), r2 = (0, 1), where rk = (rk1, rk2) and rks denotes the return of security k in state s, for k = 1, 2 and s = 1, 2. State 1 takes place with probability 1/5 and state 2 with probability 4/5, and that the consumers know the probabilities. Both consumers are expected utility maximisers with an identical von Neumann-Morgenstern utility function: uh(xhs) = ln(4xhs)2, where xhs is the consumption of consumer h in state s. The timing of the transactions is as follows: 1. At date 0, markets for the two securities open. 2. At date 1, the spot market for the consumption good opens. With this setup, answer the following questions. a) Show that each consumer h’s preference can be represented by the following expected utility function: Uh = ln xh1 + 4 ln xh2. b) Derive the equilibrium of the economy.