PacBaby just found a $100 bill—it is the only thing she owns. Ghosts are nice enough not to kill PacBaby, but when they find PacBaby they will steal all her money. The probability of the ghosts finding PacBaby is 20%. PacBaby’s utility function is U(x) = log(1 + x) (this is the natural logarithm, i.e., log e^x = x), where x is the total monetary value she owns. When PacBaby gets to keep the $100 (ghosts don’t find her) her utility is U($100) = log(101). When PacBaby loses the $100 (per the ghosts taking it from her) her utility is U($0) = log(1 + 0) = 0. 

a. What is the expected utility for PacBaby?

b. Pacgressive offers theft insurance: if PacBaby pays an insurance premium of $30, then they will reimburse PacBaby $70 if the ghosts steal all her money (after paying $30 in insurance, she would only have $70 left). What is the expected utility for PacBaby if she takes insurance? For PacBaby to maximize her expected utility should she take this insurance? 

c. In the above scenario, what is the expected monetary value of selling the insurance from Pacgressive’s point of view?