Few questions on microeconomics, provide solutions for these below.
(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.There are three periods, t = 0, 1,2. In t = 1 Mary maximizes her utility over leisure and consumption given the following function: UI(M, CI) = NICE subject to the following budget constraint: Ci + wiN = 24w1 where wi = 10. Note the price of the consumption good is assumed to be one in all periods. After she has made this decision, in t = 2 she maximizes this utility function: U2(N2, C2) = Nici subject to the following budget constraint: C2 + w2 N2 = 24w2 where w2 = 20. (a) (6 points) For t = 1, 2 calculate Mary's choice of leisure and consumption in each period. (b) (6 points ) For t = 1, provide economic intuition for the income and substitution effects of a wage increase on leisure. Can you say anything about the relative magnitudes of these income and substitution effects? (c) (7 points) Go back to your solution in part (a). If the interest rate is 10% per period, what is the present value of her consumption in t = 0? Please use 0.9 and 0.8 as approximations for 1/(1.1) and 1/(1.1) respectively. (d) (7 points) Mary now has the option of obtaining additional job training in t = 0 at an investment cost of $200. As a result, her wage rate increases in t = 1 to wj = 20 and in t = 2 to w2 = 30. Calculate the net present value of this investment on consumption. Consider only the value of consumption (and not the value of leisure). (e) (7 points ) For more general utility functions, when will the net present value of the investment on consumption from part (d) likely be negative? Use income and substi tution effects in your explanation. (f) (7 points) Does Mary have a Laffer curve for income taxes (as opposed to consumption taxes)
