In this question, we will distinguish between the concepts of buying and selling gambles. If we sell a gamble, that implies we own it or have a right to it in the first place; imagine a situation where you've had a stellar performance review at work, such that you've earned a bonus. However, your boss tells you the bonus is expressed as a gamble, where you have a probability of being awarded a good bonus in the amount and a probability 1 of being awarded a bad bonus in the amount , where < and 0 < < 1. Suppose you have initial wealth and a utility function () that obeys all of our normal assumptions about rationality.

a) In this case, you have a right to the bonus gamble but wish to "sell" it by asking your boss to instead offer you a bonus with a fixed amount (i.e., you will get the same constant bonus with certainty). Provide a mathematical equation that characterizes the minimum price, , for which you would be willing to sell the gamble. In other words, if you had a functional form for (), what is the equation you would solve in order to find ? Explain.

b) Now imagine that your co-worker has gotten a bad performance review, such that they would not ordinarily be given a bonus at all. However, your employer will allow them to "buy" the same bonus gamble that you earned by paying some amount of money in exchange for getting access to the gamble. Provide a mathematical equation that characterizes the maximum price your co-worker should be willing to pay to buy the gamble. Explain.

c) Explain conceptually why both of the values from parts a) and b), and , should be greater than and less than .

d) Explain conceptually why the values and should be different, and specifically, why < .

e) For the specific case where you and your co-worker each have initial wealth = 10 and potential bonuses = 10 and = 5, with = 0.5 and () = 2 + 100, calculate the values and .