Some questions on microeconomics, try and provide solutions for the same

3. Bergson becomes a benevolent dictator. He has a subjects i = 1,... .n with CARA utilities w1...., un, respectively. (Write o, for the absolute risk aversion of i.) The total wealth in the society, Y, is a function of an unknown state w and is normally distributed with mean / and variance of. Bergson can choose any allocation r = (21, ... .2,) such that an (@) + ... +2, (w) I from her savings so that her wealth at t + 1 is my+1 = " (wt - 2,) if her wealth at t is wy and she consumes r, at t. (b) Find a sophisticated-optimal consumption strategy for her in which the self at any given date s consumes yes. Compute the constant y and briefly verify that this is indeed a subgame-perfect equilibrium of the multi-agent game. (c) For 8 1 from her savings so that her wealth at t + 1 is wy+1 = " (14 - It) if her wealth at t is wy and she consumes r, at t. (b) Find a sophisticated-optimal consumption strategy for her in which the self at any given date s consumes yu's. Compute the constant y and briefly verify that this is indeed a subgame-perfect equilibrium of the multi-agent game. (c) For #