1annswer and explain

Problem 4. Driving risk and insurance (38 points). Robert is an expected-utility maximizer that likes to drive fast, so his probability of an accident is 2/3. Robert's preferences over wealth are u (w) = w.5. Suppose that Robert's initial wealth is $100. If Robert has an accident, he incurs a $51 loss. 1. What is Robert's expected utility? (5 points) 2. Now, assume there is one insurance company in existence with one policy available: Full insurance. That is to say, the insurance company charges a fixed premium and then pays Robert $51 if he gets in an accident. If the insurance company is risk neutral, what is the premium a they need to charge to break even? (5 points) 3. Compute the expected utility for Robert if he purchases the insurance at premium . Will Robert purchase the insurance? (Note: (66)" is approximately 8.1) (5 points) 4. Repeat the exercise for the case in which Robert has utility function u (w) = w. What is his utility if he does/does not purchase the insurance? Does he purchase the insurance? (Note: (66)- = 4356 and (49) = 2401) (6 points) 5. Discuss the intuition for why Robert purchases the insurance in one case, but not in another. (5 points) 6. (Harder) Consider now the general case with u (w) = w" with a > 0. Remember that Jensen's inequal- ity says Ef (x) 2 f (Ex) if and only if f concave. Use Jensen's inequality to show analytically that, in fact, Robert purchases the insurance at premium a if and only if a