6. Lucky's wealth consists of R10 000 currently in his bank account. He must choose an amount of money, x E [0, 10 000], to invest in a risky investment, which will deliver a return of 16% with a probability of 50% and a return of – 5% with a probability of 50%. Whatever money is left in his bank account, 10 000 – x, will earn 5% interest from the bank without risk. Lucky's utility level will be determined by the natural logarithm of his wealth at the end of the year, u(w) = In (w). 6.1. Represent the lottery L that Lucky faces, using (p1 • a1, ..., pN ° an) notation. 6.2. Determine Lucky's expected wealth at the end of the year as a function of his decision x. 6.3. What value of x would a risk-neutral investor choose? Explain. 6.4. What value of x would Lucky choose if he is an expected utility maximiser? [Hint: Recall dg(x)/dx g(x) df (x) that if f(x) = In g(x), then dx 6.5. What will his expected wealth be at the end of the year, given this decision? What is the certainty equivalent of the lottery that Lucky's decision initiates? What does a comparison of these two values tell us about Lucky's preferences?