Quantitative Methods: Decision Making Under Uncertainty

Round numeric responses to two decimal places

i) When in doubt, round numeric responses to upto two decimal places
ii) It is okay to discuss the course material with your peers, but please solve all the questions independently and write all your responses in your own words.On the coversheet, please write and sign the following honor pledge: “I pledge on my honor that I have not given or received any unauthorized assistance on this problem set.”
iii) Late assignments would not be accepted.

Problems:
1) Jack, a struggling artist and die-hard romantic, dreams to make it to the East Coast of the USA from Southampton (England), and is planning to buy a trans-Atlantic ship-ticket. The year is 1912, and there are two ships plying that route: The Mariner and the Titanic. He consults a local fortune teller who foretells him the following information: The journey by The Mariner has a 50% chance of being worth a net $80 to Jack (because too much sea-food served during the travel, and Jack hates sea-food) and a 50% chance of being worth a net $200 (because the voyage and food served is really good). The journey by The Titanic has a 50% chance of being worth a net $0 (because the ship hits an iceberg!) to Jack and a 50% chance of being worth a net $280 (because the ship avoids icebergs and Jack gets to meet the love of his life!).

Suppose Jack’s preferences are given by U (w) = ?w. Show that the two ship-tickets have an equal expected value but different variances in terms of their net worth to Jack. Show that Jack prefers to buy a ticket for The Mariner because his expected utility is higher with it.
2. i) Suppose that Mary’s utility function is U (W ) = W 1
3 where W is wealth.
Is she risk averse? Suppose that Mary has initial wealth of $125,000. What is her certainty equivalent for the lottery that offers a 50% probability of raising her wealth by $35,000 and a 50% probability of lowering her wealth by $35,000? What is the risk-premium of that lottery?
2.ii) Suppose, a businessman has a utility function of u(w) = ?w, and an initial holding of $10,000. Consider a risky project (buying a mortgage pool). When successful, this project reaps a great outcome (a net gain of $6900 in 20% of cases when a bunch of mortgage loans are repaid with interest) and otherwise results in a net loss (a net loss of $1500 in 80% of cases, when multiple home owner defaults happen on the mortgaged loans). Show that the businessman would not undertake this project, but would be willing to buy a 1/100th share of the project at a price of $1.
3) Burkhard and Anujit are neighbors in Davis. Burkhard is a full professor,whereas Anujit is a lowly Assistant Professor who has recently joined the same University. Each owns a car valued at $12000. Neither has comprehensive insurance (which covers losses due to theft). Burkhard’s wealth, including the value of his car, is $100,000. Anujit’s wealth, including the value of his car, is $60,000.
Burkhard and Anujit have identical utility of wealth functions, U (W ) = W 0.5.

Burkhard and Anujit can park their cars on the street or in a rented garage. In Davis, the probability that a street-parked car is stolen is 20% over a one-year period. A garage-parked car will not be stolen.What is the largest amount that Burkhard and Anujit are willing to pay for a garage? What are their Arrow-Pratt measures of risk aversion computed at their respective wealth levels? Is the person with the high measure of risk aversion willing to pay more? (hint: Buying garage rights is equivalent to buying
insurance, solve this like an insurance problem).

4) States and provinces in North America have long used lotteries to raise money for public expenditures. You can see the whole host of state lotteries here. The revenues from lotteries are invested in education, park services, veterans and senior citizens. This question would introduce you to a new financial innovation called prize-linked savings account, that might use people’s bias towards buying lotteries to help fix USA’s very low saving’s rate ”problem”.
Let us start by introducing the roots of the problem. What was USA’s sav-ings rate in Jan 2020, right before the pandemic hit? You can find it here: Do you think it was high or low compared to recent history (last 50 years)?

Why are savings important? (try to come up with 2 reasons of your own) Finally read the article here. How is the program called SaveYourRefund planning to help low-income working families save more? If this gets you interested, you might also want to listen to the longer podcast.