help with question 5

5. [20 points total] An agent lives for two periods, 0 and 1. In period 0, she has no uncer- tainty. But in period 1, she is either healthy with probability p, sick with probability q, 2 or dead with probability 1 - p - q. Thus, the agent's lifetime expected utility is written as " (co) + B pu (ci ) + qu (ci)] , where co, ci, and of denote consumption in period 0 and in period 1 when healthy and sick. u (c) is the utility function with u' > 0 > u", and B E (0, 1) is a discount factor. In period 0, the agent earns labor income y, consumes co, and buys bond b and annuity a. Hence her budget constraint is co +b + a =y. In period 1, the agent consumes the asset income: (1 + r) b from bond and (1 + ra) a from annuity. Therefore, her budget constraints in period 1 are given as follows. healthy: ci = b (1 + r) + a(1 +ra) sick: ci = b (1 +r) + a(1+ra) -m Note that m > 0 represents medical expenses if she is sick. We assume perfect com- petition in the annuity market. (a) [5 points] Show that b = 0. (b) [5 points] Derive the following intertemporal budget constraints. (1 + ra) cot ci = y(1+ra) (1 + ra)cot citm = y(1+ra) (c) [10 points] Prove that the optimal consumption (co, ch, c;) satisfies u' (co) = B(1 +r) P u' (ci ) + 9 u (c ). Interpret this equation