1. (40 points) Consider a decision maker (DM) that has preferences over lotteries, with Z C R+ + being the set of monetary prizes'. Let v : Z - R; then the DM assigns the number v(z) to each amount of money z E Z. The function v is continuous and increasing, and strictly positive. Define U : 2(Z) - R by U(p) = a max P(z) v(2) | z EZ, where o is any real number different from zero. Let _ denote the preference relation on 2 (Z) that is represented by U. (a) (20 points) Does the DM's preferences over lotteries ~ admit a Expected Utility representation? (b) (20 points) The notion of risk aversion is independent of Expected Utility Theory, so irrespective of your answer to the previous item, we can define risk aversion as follows.? A DM is risk averse if she always prefers the expected value of any lottery with certainty to the lottery itself, that is, for any p E 2(Z), [E(p)] _ p. Characterize decision makers with this type of preference that are risk averse for any o and v