Assume you have the following von Neumann-Morgenstern utility function: U(w) = 5000 − (50 − w 1000) 2 where w denotes wealth. 

a) Work out the utility maximization problem and illustrate the answer in a diagram.

b) Briefly explain what Arrow-Pratt coefficients of absolute risk aversion (ARA) and relative risk aversion (RRA) seek to measure. Find out and interpret their values for the above utility function. 

c) The current average diesel price is about 145 pence per litre. Your friend firmly believes that the average diesel price will shoot over 160 pence per litre for the Christmas period due to supply and logistics problems while you think, with the mitigation policies from the government, that there is a 60% probability that it will remain below 160 pence per litre. The two of you decide to bet on the outcome with x pounds: if you win, your friend pays you x pounds and vice versa. Your current wealth is 5,000 pounds which is also the maximum amount you can bet. As an expected utility maximiser, should you bet, and why or why not? If you do bet, what is the optimal amount that you should bet to maximize your expected utility?