Your friend needs to write a massive term paper over the next three weeks, but the NBA playoffs are heating up. If she concentrates, she can write the term paper in a week. This week (t = 1), she can write the paper instead of watching the playoffs and miss the conference semifinals at an immediate cost equal to 4. Next week (t = 2), she can write the paper and miss the conference finals, which has an immediate cost of 6. If she waits until t = 3, she misses the NBA finals, and incurs an immediate cost of 9.
(a) Suppose your friend if time-inconsistent and naive, with β =1/2 and δ = 1. When will she write the paper?
(b) Suppose instead she is sophisticated about her time-inconsistency, with β =1/2 and δ = 1. When will she write the paper?
(c) You are an entrepreneurial friend, and you decide to make some money off of the situation. You offer her a commitment device, whereby she can pay you some amount of money, and in exchange, you can disconnect her TV and prevent her from watching basketball for one week (specified in advance). If she is naive, how much will she be willing to pay you at t = 1? [Assume all costs mentioned in the problem map one-to-one into dollars.]
(d) If she is sophisticated, how much will she be willing to pay you at t = 1?