3 Assume that an individual owns a car whose value is 10000. The car can be stolen with probability equal to 5% if he is careful, and with probability 25%, if he is careless. His Bernoulli utility inction is u(x,e)= ln(x+l)e/ 10 where x equals the value of the car less the insurance premium( if the car is not stolen), and 1: equals the value of the compensation payed by the insurance company minus the insurance premium (if the car is stolen), e=0 if the individual is careless, and e=l if he is careful. 3.1 Show that, if there were no insurance companies, the individual would choose to be careful. 3.2 Assume that there is a large number of insurance companies that operate in a perfectly competitive market. Assume that insurance companies are willing to accept any insurance contract (y,p) that generates a non-negative expected profit (p represents the insurance premium and y represents the compensation payed by the insurance company, if the car is stolen). Show that if the level of care is observable, the individual chooses to be careful and to buy Jll insurance (he gets the same utility level in both states). 3.3 Specify the optimal contract when effort is not observable. What is the expected utility in this case? Reference: Exercise based on Section 8.5 "Insurance games | and II" of the book "GAMES AND INFORMATION, FOURTH EDITION An Introduction to Game Theory" by Eric Rasmusen