USE EXCEL

It is January 2023 and Ana has just moved to Dallas. She is planning on staying here for a maximum of three years. There is a bit of uncertainty in Ana's employment: she currently has a one-year contrar (for 2023) with her employer, but the contract will have to be renewed at the beginning of each calendar year. Ana estimates that, for any given year, there is a 40% chance that the contract will not be renewed, in which case she will have to leave Dallas and move back to her hometown.

Ana quickly realizes that she will need a scooter to get around town. There are two options available for the model that she really likes. The first option is that she can purchase it from a local dealership. Currently, the retail price is $600, but the price will likely drop over the years, as the manufacturer introduces newer models. In particular, Ana believes that the price at the end of any year will definitely drop by around 22% from the value at the beginning of the year. For instance, she things the price will be 78% *$600 = $468 at the end of 2023, and 78% * $468 = $365.04 at the end of 2024, etc. If she were to purchase the scooter, Ana would simply keep it using it throughout her entire time in Dallas and take it with her whenever she moves (thus, you can assume that any resale value of the scooter is basically zero).

Her second option is to rent the scooter from a small shop for $300 per year. This rental would be on a yearly basis, i.e., she could stop renting at the end of any year and reevaluate her options (which gives her flexibility, include the alternative of purchasing). Based on her current information, she believes that the shopkeeper would not be willing to give discounts, even if newer models were to be released. So the yearly rental rate is likely to remain $300 for the foreseeable future.

Ana is trying to find a strategy that would guarantee that she has a scooter for every year of her stay in Dallas, while minimizing her expected cost.

(10 points) Build a decision tree to find the optimal strategy that minimizes Ana's expected costs. What are the resulting expected costs, and what is the optimal strategy? Clearly identify the decision nodes and the event nodes in your model, by (a) highlighting/coloring the cells, and (b) typing explicit comments on your spreadsheet. If you feel that some modeling element is ambiguous, and you need to make additional assumptions, state them explicitly in your spreadsheet and proceed.

(2 points) Under the optimal strategy that you found in 1, what is the resulting probability that Ana will spend at least $700?

(3 points) Suppose that Ana has some doubts about the yearly rental being $300. How does her decision at the beginning of 2023 (year 1) depends on this rental rate?

(5 points) Suppose that Ana has found out additional information, leading her to believe that the rental shop may adjust the rental rate once, at the end of year 1 (and then keep the same, updated rice for any future years). In particular, she believes that, with a probability of 50%, the shopkeeper may actually raise the rental rate to $330 for the second year, and with probability of 50% would drop the rate to $270. Adjust the decision tree to incorporate these changes. How would your answer in 1 change?