Please answer this question. The total point is worth 100 points. ? Please use word processing softwares (e.g., Word, LaTeX). A handwritten one will not be accepted. If you do not follow this rule, your score will be halved. ? If you submit late, your score will be deducted based on how much time you do so. ? Write down in English. ? Please read the question carefully and answer questions very clearly. ? You must complete the exam individually. If I catch you working with your friends, score for both you and your friends will be 0.

Question 2 (30 Points) Consider a two-period consumption-savings problem under uncertainty where a consumer has preferences: logCo + Eo [BlogCi], where Eo is the expectation formed as of date 0. Income at date 0 is known, but income, consumption, and asset payoffs in date 1 are random variables. Suppose the consumer can buy or sell an asset at price P in date 0 which yields a stochastic payoff X in the next period. She gets the exogenous, possibly stochastic income y, each period. If 0 are her purchases of the asset at date 0, she thus faces the budget constraints: Co+0 = yo, CI = yI+OX. Answer following questions. (a) (10 Points) Find the optimality condition (Euler equation) for the choice of asset holdings 0 (or equivalently the choice of consumption) (Warning: Your score will be lower if you derive Euler equation incorrectly). (b) (5 Points) Now suppose that there are many such identical consumers, and that the asset is in zero net supply, so in equilibrium C, = y,. Moreover suppose that yo is known and there are two states of the world in period 1 which determine (y1, X) . With probability 0.5, y1 = (1 -A) yo and X = X1, and with probability 0.5 and y1 = (1 + A)yo and X = X2, where A > 0 is a constant and X1, X2 are specified payoffs in each state. Now find an expression for the equilibrium asset price P. (c) (5 Points) Consider three different assets which differ in their payoffs: asset A has X1 = X2 = 1, asset B has X1 = 1 - A and X2 = 1 + A, and asset C has X1 = 1 + A and X2 = 1 -A. Find and rank the prices of these three assets. (d) (10 Points) Interpret your answer to (c) by using the following formula: E[YZ] = E[Y]E [Z] - COV (Y,Z), where Y, Z are random variables