Jane is buying a plane ticket to Miami, Florida, to celebrate New Years with some friends....

  

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Jane is buying a plane ticket to Miami, Florida, to celebrate New Years with some friends. For her return ticket on Jan 2, her ideal flight leaves at 11:00am, but costs $35 more than a flight at 8:00am that would require her to wake up three hours earlier than normal (for the 11:00 flight, she could wake at her normal time). On an average day, waking up three hours earlier than normal would cause her disutility worth $20, but on a day when she is already sleep-deprived, the disutility is twice as high: she would pay $40 to avoid it. (These disutilities are defined relative to waking at her normal time, which we normalize to 0). Jane knows that she will be sleep-deprived after a few days on vacation with her friends, but she also has projection bias. Projection bias: If she is currently in state st, and will be in state s, on future day 7, Jane perceives her utility of consuming c, on that future day to be: au(cr, st) + (1 - a)u(c7, ST). (c represents, for example, consuming an early wake-up time on day 7.) Suppose that Jane is an exponential discounter with d = 1, so even though the disutility on day 7 is several periods away when she buys the ticket, she does not discount it when making the purchase decision. (a) (5 pts) What are the two relevant states here? (In words, not notation.) (b) (5 pts) Let's write down the utility for waking up at the two different times in one of these states, as: U(11am, S₁) = -35 U (8am, S₁) = -40 Notice that -35 represent the higher price of the 11:00 ticket. You could think about both utility functions also including -p, the price of the 8:00 ticket, but this is just going to drop out when the two utilities are compared. (Since she is going to buy a flight regardless, the only portion of the price that matters for her choice is the difference in price between the two.) Write down the utility functions for the other state, and also denote which of the states you named in part (a) is S₁ and which is S2. U(11am, S₂): U (8am, S₂) = S₁ S₂ = (c) (5 pts) Suppose Jane is in her usual state when she buys her ticket, but has complete projection bias (a = 1). Would she buy the 11:00 ticket or the 8:00 ticket? (d) (5 pts) For what range of values of a does Jane buy the 11:00 ticket, assuming she is in a normal state when she buys the ticket? Hint: Find the value of a that makes her indifferent between the two tickets. (e) (5 pts) Suppose that Jane bought the 8:00 ticket, but when the day arrives, she really wants to sleep longer. If she could pay the $35 fare difference now, to swap her ticket for the 11:00 flight, would she do so? (f) (5 pts) Suppose Jane would have to pay a $25 change fee in addition to the fare difference. Would she do so? Jane is buying a plane ticket to Miami, Florida, to celebrate New Years with some friends. For her return ticket on Jan 2, her ideal flight leaves at 11:00am, but costs $35 more than a flight at 8:00am that would require her to wake up three hours earlier than normal (for the 11:00 flight, she could wake at her normal time). On an average day, waking up three hours earlier than normal would cause her disutility worth $20, but on a day when she is already sleep-deprived, the disutility is twice as high: she would pay $40 to avoid it. (These disutilities are defined relative to waking at her normal time, which we normalize to 0). Jane knows that she will be sleep-deprived after a few days on vacation with her friends, but she also has projection bias. Projection bias: If she is currently in state st, and will be in state s, on future day 7, Jane perceives her utility of consuming c, on that future day to be: au(cr, st) + (1 - a)u(c7, ST). (c represents, for example, consuming an early wake-up time on day 7.) Suppose that Jane is an exponential discounter with d = 1, so even though the disutility on day 7 is several periods away when she buys the ticket, she does not discount it when making the purchase decision. (a) (5 pts) What are the two relevant states here? (In words, not notation.) (b) (5 pts) Let's write down the utility for waking up at the two different times in one of these states, as: U(11am, S₁) = -35 U (8am, S₁) = -40 Notice that -35 represent the higher price of the 11:00 ticket. You could think about both utility functions also including -p, the price of the 8:00 ticket, but this is just going to drop out when the two utilities are compared. (Since she is going to buy a flight regardless, the only portion of the price that matters for her choice is the difference in price between the two.) Write down the utility functions for the other state, and also denote which of the states you named in part (a) is S₁ and which is S2. U(11am, S₂): U (8am, S₂) = S₁ S₂ = (c) (5 pts) Suppose Jane is in her usual state when she buys her ticket, but has complete projection bias (a = 1). Would she buy the 11:00 ticket or the 8:00 ticket? (d) (5 pts) For what range of values of a does Jane buy the 11:00 ticket, assuming she is in a normal state when she buys the ticket? Hint: Find the value of a that makes her indifferent between the two tickets. (e) (5 pts) Suppose that Jane bought the 8:00 ticket, but when the day arrives, she really wants to sleep longer. If she could pay the $35 fare difference now, to swap her ticket for the 11:00 flight, would she do so? (f) (5 pts) Suppose Jane would have to pay a $25 change fee in addition to the fare difference. Would she do so?