3. Jax is a motorcycle enthusiast, whose preference over lotteries is described by Bernoulli utility function u(w)=w. Riding a motorcycle can result in a serious accident. Assume that Jax's wealth is 100 if no accident occurs, but if an accident occurs, Jax's wealth is 36 (a drop of 64 due to the accident). The probability of accident depends on whether Jax decides to ride cautiously or aggressively. In particular, if Jax rides cautiously, then the probability of an accident 1/10. But riding cautiously is less funthe decision to ride cautiously imposes a cost, effectively reducing Jax's wealth by 36 . If Jax rides aggressively, then he enjoys himself (there is no cost), but the probability of an accident is 3/4. Thus, aggressive and cautious riding can be represented as lotteries over wealth levels: - Aggressive: w=100 with probability 1/4, and w=36 with probability 3/4. - Cautious: w=64 with probability 9/10, and w=0 with probability 1/10. (a) What is Jax's expected utility associated with the "aggressive" lottery? (b) What is Jax's expected utility associated with the "cautious" lottery? (c) Will Jax choose to be cautious or aggressive? What is his expected utility? (d) What is the probability that Jax has an accident? Suppose an insurer enters the market, and offers Jax full insurance at price p=48. In other words, if Jax buys the insurance, then his wealth will be 52 , whether he has an accident or not. If he buys the insurance, he must also decide whether to be aggressive or cautious. Let's consider the choice of whether to be cautious or aggressive if Jax buys insurance. - Insurance, Aggressive: w=52 with probability 1. - Insurance, Cautious: w=5236=16 with probability 1. (e) What is Jax's expected utility if he buys insurance and is aggressive? (f) What is Jax's expected utility if he buys insurance and is cautious? (g) Will Jax choose to be cautious or aggressive if he buys insurance? What is his expected utility? Explain your answer in a sentence or two. (h) Compare your answers to (c) and (g). Does Jax prefer not to buy insurance and ride as in part (c), or does he prefer to buy insurance and ride as in part (g)? (Hint: compare his expected utility from part (c) to his expected utility from part (g))