8. Mr. Smart is an investor with $15,000 to invest. He has narrowed his choice down to twin possible investments: Mutual fund Common shares in Buyme Corporation Figure 3.5 gives a decision tree for Mr. Smart's situation. Mr. Smart is risk averse. The amount of utility he derives from a payoff is Utility = 2ln(payoff) where "In" denotes natural logarithm. Because of a planned major purchase, Mr. Smart intends to sell his investment one year later. The payoffs represent the proceeds from the sale of the investment and receipt of any dividends, net of the initial investment. The probabilities on Figure 3.5 represent Mr. Smart's prior probabilities about the state of the economy (good or bad) over the coming year. Required a. Calculate Mr. Smart's expected utility for each action, and indicate which action would choose if he acted on the basis of his prior information b. Now, suppose Mr. Smart decides that he would like to obtain more informa about the state of the economy rather than simply accepting that it is just as to be good as bad. He decides to take a sample of current annual reports of corporations. Every annual report shows that its firm is doing well with increased profits over previous year. The probability that there would be such healthy profits if the state economy actually was good is 0.75. The probability of such healthy profits is only y the state of the economy actually was bad. Use Baves' theorem to calculate Mr. Smart's posterior probabilities of the high states of the economy. Will he change his decision? Note: Round your calculations to two decimal places. nual reports of major he state of the it is only 0.10 if the high and low Figure 3.5 Decision Tree for Mr. Smart's Problem Action Probability State Net Payoff Good 0.50 $8,000 Buy common shares Bad 0.50 $1,000 Invest $15,000 Good 0.50 $5,000 Buy mutual fund Bad 0.50 $2,000