(a) Expected Utility Theory implies that a risk averse agent will always buy some amount of insurance". Discuss this statement.

(b) A consumer has a utility function over wealth given by U w w ( ) ln( ) = . 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 ?