Some questions on microeconomics asked below, try and answer all of these

(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 - IN1 2 IN'. iEN iEN' (40 points) Consider the reduced normal form of the following game, in which the strategy set of Player 1 is {X, A, B}, so that the equivalent strategies X A and X B are represented by a single strategy X. X NO N 2 (a) Compute the set of rationalizable strategies. (Show your result.) (b) Compute the set correlated equilibria. (Show your result.) (c) Suppose that in addition to the type with the payoff function above, with proba- bility 0.1, Player 1 has a "crazy" type who gets 1 if he plays A and 0 otherwise. Compute the set of all sequential equilibria.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. (40 points) Consider the reduced normal form of the following game, in which the strategy set of Player 1 is {X, A, B}, so that the equivalent strategies XA and X B are represented by a single strategy X. X B a (a) Compute the set of rationalizable strategies. (Show your result.) (b) Compute the set correlated equilibria. (Show your result.) (c) Suppose that in addition to the type with the payoff function above, with proba- bility 0.1, Player 1 has a "crazy" type who gets 1 if he plays A and 0 otherwise. Compute the set of all sequential equilibria.An economy has two agents, Bill and Bob. Bill has $110, and Bob has $200. Utility of agents in this economy is characterized by the following function of income: U = u(y) = [ log(y - 60) if y 160 The minimum level of income possible in this economy is 60. Each agent is about to choose a new business venture, and has a choice between project A and project B. Neither project requires any investment up front. Project A yields revenues of 20 with probability = and revenues of -10 with probability -. Project B yields revenues of 4 with probability one-half and revenues of 5 with probability one-half. Throughout this problem, assume that fractional income is possible. (a) (5 points) Which project would each agent choose? Provide intuition for your answer. (b) (5 points ) If Bill and Bob each choose an investment project each year and receive the associated income for 20 years, will the expected gap in their incomes be larger or smaller at the end of this period than it was initially ? How does this relate to attitudes toward risk? You do not need to calculate income over 20 years, just provide intuition. (c) (10 points) Now, assume that there is a job available that provides fixed wage income. What salary would the job have to provide in order to induce Bill to take the job rather than entering a new business venture? What salary would the job have to provide in order to induce Bob to take the job? Which is higher, and why? Algebraic expressions are acceptable as answers