Sophia is a contestant on a game show and has selected the prize that lies behind door number 3. The show's host tells her that there is a 50 percent chance that there is a $10,000 diamond ring behind the door and a 50 percent chance that there is a piece of moldy bread. Before the door is opened, someone in the audience shouts, "I'll give you the option of selling me what is behind the door for $5,000 if you will pay me $3,000 for the option."
a. If Sophia only cares about the expected value of various outcomes, will she buy this option? Show me mathematically how you know. (Hint: Would she sell the ring? How about the bread?)
b. If Sophia has no starting wealth (the audience member will take an IOU for the option), and her utility of wealth is , would she benefit from taking this option? Again, show me how you know.