A customer of an insurance company is an EU maximizer with a total wealth of 100. She/he is facing a random loss ξ for which P(ξ = 0) = 0.9, P(ξ = 50) = 0.05, P(ξ = 100) = 0.05.

(a) Let the utility function of the customer be u(x) = x−0.005x² for 0 ≤ x ≤ 100. Graph it. Why do we consider this range of x’s? Is the customer a risk averter?

(b) What would you say in the case u(x) = x+0.005x²?

(c) For the case 16a, find the maximal premium the customer would be willing to pay. Is the premium you found greater or less than E{X}? Might you predict it from the very beginning?

(d) Solve Exercise 16c for u(x)=200x−x²+349. (Advice: Look at this function carefully before starting calculations.)