Hrlp in giving correct answers
(i) (For Grade) There are n individuals. Each individual i has constant absolute risk aversion my >0 and an asset that pays X, where (X1,.... X,) ~ N ((#1. ... . An), E)- (a) What are the optimal risk sharing contracts? What is the vector of payoffs from an optimal risk-sharing contract? Characterize the set of the vectors of certainty equivalents from optimal risk sharing contracts. (b) Answer (a) for a1 = = 0n, #1 = .. = and p P ... p How much the society as a whole are willing to pay all of these assets? Assuming that they write a symmetric contract, what is the preference relation of an individual on (o', p) pairs? Briefly discuss. (ii) Exercise 2.1 in lecture notes. (iii) Consider the set of lotteries (pr, py, p=) on the set of outcomes (x. y, z} where pe, Py, and p: are the probabilities of r, y, and z, respectively. (a) For each (partial) preference below, determine whether it is consistent with expected utility maximization. (If yes, find a utility function; if no, show that it cannot come from an expected utility maximizer.) i. (0, 1,0) > (1/8, 6/8, 1/8) and (7/8, 0, 1/8) > (6/8, 1/8, 1/8) ii. (1/4, 1/4, 1/2) > (3/4, 0, 1/4) > (5/6, 1/6,0) > (1/2, 1/3, 1/6) (b) For each family of indifference curves below, determine whether it is consistent with expected utility maximization. (If yes, find a utility function; if no, show that it cannot come from an expected utility maximizer.) i. Py = c - 2px (where c varies) ii. py = c(px + 1) (where c varies) ili. Py = c - 2vps (where c varies) (c) Find a complete and transitive preference relation on the above lotteries that satisfies the independence axiom but cannot have an expected utility representation. (iv) Alice has M dollars and has a constant absolute risk aversion a (i.e. u (1) = -e-or) for some a > 0. With some probability * e (0, 1) she may get sick, in which case she would need to spend & dollars on her health. There is a health-insurance policy that fully covers her health care expenses in case of sickness and costs P to her. (If she buys the policy, she needs to pay P regardless of whether she gets sick.) (a) Find the set of prices P that she is willing to pay for the policy. How does the maximum price P she is willing to pay varies with the parameters M, L, a, and *? (b) Suppose now that there is a test te {-1, +1) that she can take before she makes her decision on buying the insurance policy. If she takes the test and the test t is positive, her posterior probability of getting sick jumps to a* > > and if the test is negative, then her posterior probability of getting sick becomes 0. What is the maximum price c she is willing to pay in order to take the test? (Take P & P.)1. (For Grade) There are finitely many states s 0. Derive the demand of the decision maker for these securities as a function of the price vector p = (Ps) SES . 2. Ann is an expected utility maximizer, but she does not know her preferences, which she can learn by costly contemplation. To model this situation, take S = [0, 1], and let Z C R be a finite set of consequences with at leat two elements. Assume that Ann's von Neumann utility function is ZZA z = (2)n and her belief on S is represented by uniform distribution. For any n and some fixed c > 0, by spending en utils, Ann can obtain a partition Pn = {[0, 1/2"], (1/2", 2/2") , .... (k/2" , (k + 1) /2") ,..., [(2" - 1) /2", 1]} and observe the cell In (s) , g if Ann may end up choosing f when the true state happens to be s. Check which of the postulates P1-P5 of Savage is satisfied by _, for any fixed s. 3. Under the assumptions P1-P5, prove or disprove the following statements. (a) For any partition A1, ..., An of S, and for any acts f, ge F, If > g given Ax for all Ax] = f _ g. (b) If All Bi, Azz B2, and Aj n Az = 0, then A, U A,_ B, UBz. (c) For any given event D, define "_ given D" by A> B given D iff An DEBn D. The relation > given D is a qualitative probability.2. Consider the set of lotteries (Pr: Py; p=) on the set of outcomes {r, y, z} where Pr, Py; and p, are the probabilities of r, y, and z, respectively. (a) For each (partial) preference below, determine whether it is consistent with ex- pected utility maximization. (If yes, find a utility function; if so, show that it cannot come from an expected utility maximizer.) 1. (0, 1, 0) > (1/8, 6/8, 1/8) and (7/8, 0, 1/8) > (6/8, 1/8, 1/8) 2. (1/4, 1/4, 1/2) > (3/4, 0, 1/4) > (5/6, 1/6, 0) > (1/2, 1/3, 1/6) (b) For each family of indifference curves below, determine whether it is consistent with expected utility maximization. (If yes, find a utility function; if so, show that it cannot come from an expected utility maximizer.) 1. py = c - 2pr (where c varies) 2. Py = c(pr + 1) (where c varies) 3. Py = c - 2vp, (where c varies) (c) Find a complete and transitive preference relation on the above lotteries that sat- isfies the independence axiom but cannot have an expected utility representation. 3. Under the assumptions P1-P5, prove or disprove the following statements. (a) If All Bi, Azz B2, and Aj n A, = 0, then Aj U Azz B, U B2. (b) For any given event D, define "_ given D" by A> B given Diff An DEBAD. The relation > given D is a qualitative probability. (c) For any two partitions (A1. . .., A,) and (B1, .... B.) of S with A ... ~ An and B1~ . .. ~ Bn, we must have A| ~ B1
