Question 2 (23 marks total)

Mr. Scrooge did not trust the banks. He put only $1 in his bank account but stored all his other wealth, $100,000, in cash under his mattress. He has a von Neumann-Morgenstern utility over wealth (all the money he has), given by u(w) = lnw. Due to a recent pandemic, people stop using cash. Fearing that his cash might soon be useless, Mr. Scrooge decides to take all his cash to the bank. You can round off any utility number in your answer to the nearest 4 decimal places.

(a) (3 marks) Suppose there is a 20% probability that Mr. Scrooge is robbed on his way to the bank, in which case he will lose all the cash he is transporting. Calculate Mr. Scrooge's expected utility.

(b) (5 marks) Suppose instead of carrying all the cash in one-go, Mr. Scrooge can split his cash into two halves and make two trips to the bank. The probability of being robbed on each trip is still 20% and is independent across trips. If he is robbed on a trip, he loses all the cash he is carrying on that trip, but not the cash on the other trip (unless he also gets robbed on the other trip). Calculate Mr. Scrooge's expected utility if he makes two trips.

(c) (3 marks) Would Mr. Scrooge make one trip or two trips? Give an intuitive explanation to your answer.

(d) (4 marks) For this part of the question, suppose Mr. Scrooge can only make one trip to the bank. A transport company, Marley Inc., offers to transport Mr. Scrooge's cash to the bank for him. Marley Inc. would guarantee that Mr. Scrooge's cash arrives safely at the bank, but charges a fee in return. Find the highest amount of fee that Marley Inc. can charge Mr. Scrooge (so that Mr. Scrooge is still willing to use its service).

(e) (8 marks) Now suppose Mr. Scrooge has no access to Marley Inc., but that he can make as many trips to the bank as he wishes (the number of trips must be an integer, though), carrying equal portions of his cash each trip. As in part (b), the probability of being robbed on each trip is 20% and is independent across trips. Nonetheless, each trip incurs a disutility of 0.2 utils to Mr. Scrooge. How many trips will Mr. Scrooge make?