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Problem Set 1

Behavioral Finance and Economics

2022

1. Roger can go to a ﬁlm festival at which a diﬀerent movie will be screened in each week, t = 1, 2, 3. The movies get progressively better, so the instantaneous utility that Roger gets from watching a movie in week t is

u1 (movie) = 5

u2 (movie) = 8

u3 (movie) = 13.

Roger’s instantaneous utility is ut (no movie) = 0 in a week when he doesn’t watch a movie. Assume that Roger is a hyperbolic discounter with 8 = and 6 = 1.

(a) Suppose that in period t = 0, Roger receives one movie ticket, which he can use to go to one movie during the festival (or he could go to none at all).

i. If Roger is naive, in which week will he plan to go when he gets the ticket? In which week will he end up going to a movie?

ii. If Roger is sophisticated, when will he go to a movie?

(b) Suppose that in period t = 0, Roger instead receives two movie tickets, which he can use to go to two movies during the festival (or he could go to one or none).

i. If Roger is naive, which movie week will he plan to miss when he gets the tickets? Which movie week will he end up missing?

ii. If Roger is sophisticated, which movie week will he miss?

2. There are three little pigs, Wilbur, Babe, and Piglet. Each of the three pigs lives for ﬁve periods, 1, 2, 3, 4, and 5, and is a hyperbolic discounter. In period 1, Wilbur and Babe have to build a house of either bricks or straw. Building a house of straw has no instantaneous utility cost (it takes no eﬀort), but building a house of bricks has an instantaneous utility cost of 3 (it requires some eﬀort).

Every day the three little pigs get utility from rolling around in the mud near the house they live in. This includes the ﬁrst period, in which they roll around in the mud once their houses have been built. But how much utility they get depends on whether they are safe from the big bad wolf, and this in turn depends on what their house is made of. If a pig lives in a brick house, he gets an instantaneous utility of 5 each period from rolling around in the mud. But if a pig lives in a straw house, he knows that the big bad wolf could come and huﬀ and puﬀ and blow his house down, and so he only gets instantaneous utility of 4 each period from rolling around in the mud.

Food for thought: Why doesn’t it matter whether or not the three little pigs are sophisti- cated or naive in this problem?

(a) Wilbur has a long-run discount factor 6 = 0.9 and a short-run discount factor 8 = 0.9. Which type of house will he build in the ﬁrst period? Show your work.

(b) Babe instead has 6 = 0.9 and 8 = 0.5. Which type of house will he build in the ﬁrst period? Show your work.

(c) Piglet has a cold and cannot build a house in the ﬁrst period, so Mama Pig lets him stay in her brick house for the ﬁrst period and roll around in the mud near her house. Hence, Piglet must wait to build his house in the second period. Like Babe, he has 6 = 0.9 and 8 = 0.5. From the perspective of period 1, which type of house would he like to build in the second period? Show your work. Compare this preference to what he would have built in the ﬁrst period (if he had not had a cold) and explain intuitively.

(d) Without commitment, which type of house will Piglet end up building in the second period? Show your work.

(e) Suppose that Mama Pig oﬀers to give Piglet a housewarming gift of acorns in the second period, which gives Piglet instantaneous utility of z > 0. However, Piglet will only receive the acorn gift if he sticks to his period-1 plan. What is the minimum acorn gift that Mama Pig must oﬀer to induce Piglet to stick to his period-1 plan?

(f) Suppose that it takes time for the acorn gift to be shipped from the acorn store to Piglet’s new house. So Piglet would receive Mama Pig’s gift for sticking to his period-1 plan in the fourth period, rather than the second period. Explain qualitatively how should Mama Pig adjust the size of her gift due to the shipping delay, if at all, and why.

3. For each of the ﬁndings in parts (a) and (b), argue how it is or is not consistent with naive hyperbolic discounting (no more than 5 sentences), and argue how it is or is not consistent with sophisticated hyperbolic discounting (no more than 5 sentences). In each of the answers, you should address the following questions: (i) what does each type of person predict she will do in the future, as a function of what she does now?; (ii) what does each type of person actually do at the relevant future time?; and (iii) how can each type of person’s behavior as you described above generate the observed phenomenon? Note that there may be other explanations to these phenomena, but you should only explain it with hyperbolic discounting.

(a) In a 2003 study, students were oﬀered mail-in rebates on movie tickets. They could choose between paying ✩11 for two tickets to a local movie theater, or ✩13 with a rebate oﬀer of ✩6. Students were given either one, seven or 21 days to redeem the oﬀers, which involved mailing in the original receipt and a completed rebate form before the respective expiration date.

A total of 184 out of 1,233 students purchased the rebate oﬀers, of which 67% followed through and redeemed them. The 21-day rebate was most popular, chosen by 20% of the 410 students to whom it was oﬀered, compared with 12% of the 412 students oﬀered the 7-day rebate and 13% of 411 students oﬀered the one-day rebate. The redemption rate decreased with the length of the redemption period: 58% of those who bought the 21-day rebates redeemed them, compared with 70% of those who opted for the seven-day rebates and 77% of those who chose the one-day oﬀer.

(b) In a 2002 study, students were recruited to proofread three essays within a 21-day period. There were two groups of participants. Students in the ﬁrst, “end-deadline,” group just had to turn in the three essays by the end of the 21 days. Students in the second, “own-deadline,” group could select any deadline within the 21 days for each of the essays (including choosing the last day for all three essays). Students were paid 10 cents per correctly spotted mistake, and were penalized ✩1 per day for turning in a corrected essay past the deadline, independently of whether that deadline was set by themselves or imposed by the researchers. Turning in an essay earlier than the relevant deadline carried no rewards, since all work was assessed and payments were made at the end of the 21-day period.

A vast majority of students in the own-deadline group set at least one of their deadlines to be before the ﬁnal possible date. In addition, students in the own-deadline group spotted more mistakes and lost less money on being late than students in the end-deadline group.