Consider an economy in which there are two dates, t = 0 and t = 1,...

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Consider an economy in which there are two dates, t = 0 and t = 1, and two possible states i = 1 and i = 2, at t = 1, which occur with equal probability,so that 1 = 2 = 1/2. Suppose, as well, that there are two types of investors, j = 1 and j = 2, both of whom have logarithmic Bernoulli utility functions, so that in particular the representative investor of type j has expected utility In c + BE[In č;], (1) where the discount factor = 0.9, c denotes this investors consumption at t = 0, and c, is the investor's consumption at t = 1. In state i = 1, c, = c) and in state i = 2, č, = c². Suppose that the aggregate endowment is wo = 30 at t = 0, w₁ i = 1 at t = 1, and w₂= 60 in state i = 2 at t = 1. = 30 in state A social planner in the economy maximizes a weighted sum of investors' ex- pected utility: (In c + BE[In či]) + (1 - 0) (In c₂ + BE[In č₂]). (2) (a) Write down the Lagrangian for the social planners problem: choose c and c to maximize the weighted sum of utilities subject to the resource constraints. (b) Write down the first-order conditions for the optimal choices of c and c (c) Suppose that the social planner attaches a weight of = 2/3 to the type j = 1 investors expected utility that is twice as large as the weight 1-0 = 1/3 given to the type j = 2 investors expected utility. Under this assumption, what are the numerical values of the optimal choices of cand c. Consider an economy in which there are two dates, t = 0 and t = 1, and two possible states i = 1 and i = 2, at t = 1, which occur with equal probability,so that 1 = 2 = 1/2. Suppose, as well, that there are two types of investors, j = 1 and j = 2, both of whom have logarithmic Bernoulli utility functions, so that in particular the representative investor of type j has expected utility In c + BE[In č;], (1) where the discount factor = 0.9, c denotes this investors consumption at t = 0, and c, is the investor's consumption at t = 1. In state i = 1, c, = c) and in state i = 2, č, = c². Suppose that the aggregate endowment is wo = 30 at t = 0, w₁ i = 1 at t = 1, and w₂= 60 in state i = 2 at t = 1. = 30 in state A social planner in the economy maximizes a weighted sum of investors' ex- pected utility: (In c + BE[In či]) + (1 - 0) (In c₂ + BE[In č₂]). (2) (a) Write down the Lagrangian for the social planners problem: choose c and c to maximize the weighted sum of utilities subject to the resource constraints. (b) Write down the first-order conditions for the optimal choices of c and c (c) Suppose that the social planner attaches a weight of = 2/3 to the type j = 1 investors expected utility that is twice as large as the weight 1-0 = 1/3 given to the type j = 2 investors expected utility. Under this assumption, what are the numerical values of the optimal choices of cand c.

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