Attached are some questions below. Try and provide solutions for these.

1. Ann is pregnant. According to the tests so far, there is p = 1/3000 chance that the baby has a serious disease. There is a new test that could find out for sure whether the baby has the disease but kills the baby with probability q = 1/300. Ann also has the option of aborting the baby after taking the test. Assuming that she is an expected utility maximizer, this question asks you to help her to decide whether to take the test. Here preferences are as follows. She cares only about whether she has a baby and whether that the baby is healthy. Hence, she considers the following consequences: (A) a healthy baby; (B) a baby with the disease; (C) no baby. Her utility function is given by u (A) = 1, u (B) = v and u (C) =0, where ve (-1, 1) is known. (a) As a function of v, find whether she should take the test. (b) Suppose now that Ann will learn the test result regardless of whether the baby lives and she cares about how she would feel when she learns the test results. In addition, she now considers the following two consequences: (D) baby dies during the test and she learns that the baby was healthy; (E) baby dies during the test and she learns that the baby had the disease. (Abortion still corresponds to (C).) Assume 1 > " (E) > 0 > > > u (D). Find the condition under which she takes the test. (a) In the following pair of games, check whether the players' preferences over lotteries on the strategy profiles are identical (i.e. row player's preferences on the left to the row player's preferences on the right and column player's preferences on the left to the column player's preferences on the right). L M R L M R 2.-2 1,1 -3.7 12,-1 5.0 -3.2 1,10 0,4 0,4 5.3 3,1 3.1 -2,1 1,7 -1,-5 -1,0 5,2 1,-2 (b) Under Postulates P1-5 of Savage, let D1, D2. .... D,, be disjoint non-null events such that Di~Dy~. . .~Dn, where > and ~ are the at least as likely as and as likely as relations between events, derived from betting preferences as in the class. Given any subsets N and N' of {1, 2. ..., n}, show that UD UD - IN12 INI. iEN iEN!4. Beatrice has initial wealth of wo and suffers from quasi-hyperbolic discounting. At any date s, her utility from a consumption stream a = (ro, 21, .. .) is U (x|s) = In(z.) + 8) 8 In(1+k). where 8, 6 E (0, 1). She gets return of r > 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 #