3. A government procurement officer is trying to decide how many doses of a new coronavirus vaccine to order. This decision will depend on the effectiveness of the vaccine, which will be determined by clinical trials conducted by a scientific advisor. You may assume that the effectiveness of the vaccine is given by a random variable , uniformly distributed on the interval [Eo, Eo + 1]. The scientific advisor believes the utility of quantity Q (measured in millions of doses) is UA(QE) = 1 + EQ - Q; if perfectly informed about effectiveness, the procurement officer would value Qat UG (QE) = 1 + (8 + B)Q- Q where is a non-negative constant. After the trials, the government officer asks the scientific advisor to report on the vaccine's effectiveness and purchases the quantity that maximises the procurement officer's expected utility. The scientific advisor is not paid for their efforts, but seeks to maximise UA. (a) How would you set up this problem? Can the scientific advisor be sure that the government will purchase the optimal quantity (according to the scientific advisor's preferences)? If so, how? If not, why not? How does your answer depend on the size of B? (15 marks) (b) Suppose that the minimum effectiveness is Eo = 25%, and that B = 5%. Find the 'babbing equilibrium' for this situation - how much will the government order and what expected utilities will the two parties get? (7 marks) (c) Now construct a two-part equilibrium - depending on the advice they receive the government will place either a small order QS or a large order Q. At what reported level of effectiveness will the government switch its order size, and what are the values of Qs and Q? (10 marks) (d) How would you find the most efficient equilibrium (you do not have to compute it explicitly, but should say how it could be identified)? (10 marks)