(b) A consumer has a utility function over wealth given by U (w) = In(w) . He is offered an opportunity to bet on a flip of a coin that has probability p of coming up heads. If he bets x dollars he will have w + x if heads comes up and w - x if tails comes up. Solve for the optimal x as a function of p . What is the optimal choice if p = 0.5 ? (10 points)