An investor with assets of $10,000 has an opportunity to invest $5,000 in a venture that is equally likely to pay either $15,000 or nothing. The investor's utility function can be described by the utility function U(x) = ln(x), where x is his total wealth.

a. What should the investor do?

b. Suppose the investor places a bet with a friend before making the investment decision. The bet is for $1,000; if a fair coin lands heads up, the investor wins $1,000, but if it lands tails up, the investor pays $1,000 to his friend. Only after the bet has been resolved will the investor decide whether or not to invest in the venture. What is an appropriate strategy for the investor? If he wins the bet, should he invest? What if he loses the bet?

c. Describe a real-life situation in which an individual might find it appropriate to gamble before deciding on a course of action.