Naïve vs. Sophisticated Hyperbolic Discounting: A Behavioral Economics Example

1. A national space agency has developed a test to identify people with the ability to be astronauts. Suppose that one out of every 20,000 people has the ability to be an astronaut and that the test can identify people who have the ability to be an astronaut with 95% accuracy (i.e., 95% of people who pass the test have the ability to be an astronaut and 95% of people who fail the test do not have the ability to be an astronaut). Edwina passes the test and claims, I can be an astronaut!

1. Let Adenote that Edwina has the ability to be an astronaut and P denote that she passed the test. Given that she passed the test, what is the probability that Edwina has the ability to be an astronaut, Pr(A|P)?            [10 points]
   
   
2. In simple terms, explain why it is unlikely that Edwina has the ability to be an astronaut. [6 points]

2. Suppose you are planning a bird watching trip in April and are choosing between three potential locations: T?wharanui, Tahuna Torea, and Motuora Island. The number of birds that you will see at each location will depend on weather conditions, but you have no way to predict weather conditions so far in the future. If you go to T?wharanui, you will see 7 birds if it is raining and 5 birds if it is not raining. If you go to Tahuna Torea, you will see 14 birds if it is raining and 2 birds if it is not raining. If you go to Motuora Island, you will see 12 birds if it is raining and 3 birds if it is not raining.

1. According to the maximin criterion, where should you go bird watching? [2 points]
   
   
2. According to the maximax criterion, where should you go bird watching? [2 points]
   
   
3. According to the minimax-risk criterion, where should you go bird watching? Make sure you write out the risk-payoff matrix for this criterion. [5 points]
   
   
4. Suppose that you now have access to a long-range weather forecast. The probability that it will rain on the day of the bird watching trip in 0.25 and the probability that it will not rain is 0.75. What is the expected number of birds that you will see at each location? Where should you go bird watching if you want to maximize the expected number of birds that you see? [4 points]

3. Nicole has $1,000 and is considering whether to buy bonds or shares. If she buys bonds, she will end up with $1,015 with certainty. If she buys shares, there is a 2/5 chance that she will end up with $750 and a 3/5 chance that she will end up with $1,200. Nicole’s utility function is u(x)=?x , where is the amount of money she has.

1. What is the expected value of investing in bonds? What is the expected value of investing in shares? If Nicole wants to maximize the expected value of her investment, should she invest in bonds or shares? [3 points]
2. What is Nicole’s expected utility from investing in bonds? What is Nicole’s expected utility from investing in shares? If Nicole wants to maximize the expected utility of her investment, should she invest in bonds or shares? [3 points]
3. What amount would Nicole need from investing in bonds (that provide a certain pay out) for her to be indifferent between investing in bonds and shares? [3 points]
4. Bill claims that, as Nicole is risk averse, she should always buy (risk-free) bonds rather than (risky) shares. Is Bill correct, why or why not? [4 points]

4. Tama is renting a car for 24 hours. The rental fee, which he has already paid, includes a ‘Basic’ insurance policy. Under this policy, Tama will have to pay the first $3,600 if he is in a car accident (and the rest will be covered by the insurance policy). Any accident, even a minor one, will result in at least $3,600 of repair costs. When picking up the rental car, Tama is asked if he wants to upgrade to a ‘Full’ insurance policy at a cost of $36. If he purchases the Full insurance policy, he will not have to pay anything if he has an accident. Based on his past driving record, the probability that Tama will have an accident in any 24 hour period is 0.01. Tama’s value function is v(x) =?x/2 for gains and v(x)=-2?x for losses, where  is the change in the amount of money he has.

1. If Tama’s reference point is that he will not have an accident, what is the value to him of purchasing Full insurance? What is the value to Tama of not purchasing Full insurance? Should he purchase Full insurance? [4 points]
2. If Tama’s reference point is that he will have an accident, what is the value to him of purchasing Full insurance? What is the value to Tama of not purchasing Full insurance? Should he purchase Full insurance? In your answers, assume that Tama always integrates when evaluating options that include two or more changes. [4 points]
3. With reference to your answers above, will requiring customers to pay a $3,600 deposit if they do not purchase Full insurance and asking them if they want to ‘save’ their deposit by upgrading to Full insurance’ make it or more or less likely that they will purchase full insurance?      [6 points]

5. Paula supports the Wildcats in a national sports league. If the Wildcats win the league, a bookmaker will pay $700 for every dollar bet on the Wildcats achieving that outcome. Using a computer model, Paula estimates that the Wildcats’ chances of winning the league are 1/1200. Paula is considering betting $10 on the Wildcats to win the league.

1. What is the expected value of the bet? [2 points]
2. If Paula places the bet, is her behaviour consistent with prospect theory? Why or why not? [7 points]

6. Joe and Sam wish to complete the Auckland marathon. They each have the following two options: (a) training for the marathon at time 1 (utility=0) and completing the marathon at time 2 (utility=16); and (b) going to the beach at time 1 (utility=8) and not completing the marathon at time 2 (utility=0).

4. Joediscounts the future exponentially. His ? =3/4.

1. From the point of view of time 0: What is his utility of a? What is his utility of b?               [4 points]
2. From the point of view of time 1: What is his utility of a? What is his utility of b? [4 points]

1. Sam discounts the future hyperbolically. His ?=1/4 and his ? = 1.

1. From the point of view of time 0: What is his utility of a? What is his utility of b? [4 points]
2. From the point of view of time 1: What is his utility of a? What is his utility of b? [4 points]

1. Who will not complete the marathon? Why? [2 points]
2. Who will complete the marathon? Why? [2 points]

7. Describe a situation when your behaviour was consistent with naïve hyperbolic discounting and outline how acting like a sophisticated hyperbolic discounter may have changed the outcome you experienced. If your decisions have never been consistent with hyperbolic discounting, describe how you evaluate options that provide rewards in different time periods. You may not use the runaway alarm clock example discussed in class (even if you own such an alarm clock). Word limit: 250 words.