Problem Set 2

Problem Set 2 - Econ 330 Spring 2010 Prof. Sydnor Problem...

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Unformatted text preview: Econ 330 Spring 2010 Prof. Sydnor Problem Set 2 Due Wed, March 24th Problem 1: Look at the paper by Ashraf, Karlan and Yin (QJE 2006) and figure out what the “ganansiya box” was. Explain briefly how the box fit in with the SEED program and discuss briefly how it relates to the concepts of time inconsistency and self‐control we have discussed in class. Note: you do not need to read this whole article, just figure out what the box was about. Problem 2: In a survey, households were asked (i) what percentage of your annual household income do you think you should save for retirement? and (ii) what percentage of your annual household income are you now saving for retirement? The median difference between the answers to questions (i) and (ii) was 10% of income and the mean difference was 11.1 % of income. Explain why this pattern is hard to explain with exponential discounting and why it is consistent with hyperbolic discounting. Problem 3: Mr. BoJangles has to get an important, but unpleasant, project done at work. He will be given the information he needs to complete the project in week 1 and can choose to work on it in either week 1, week 2, or week 3. For simplicity, we will assume the task is either fully completed or not done at all in one of these weeks. We will also simplify and assume that the project is so important that he will definitely do it in one of the periods. [Note, we could formalize that by adding a future period with a very large negative utility from not having the project done, but that’s just complicating things and not adding anything interesting to the model.] Imagine initially that Mr. Bojangels’ preferences are as follows: U1 = u1 + δu2 + δ2u3 U2 = u2 + δu3 U3 = u3 Where as usual, the little u are the instantaneous utility in that period, while the big U are the overall utility for the “self” that makes a decision in a given period. We normalize the instantaneous utilities to be zero in any week that BoJangles does not work on the project. His disutility from doing the project is increasing the longer he waits to do it. Specifically, if he does the project in week t, the disutility is: u1 = ‐ 10 u2 = ‐ 20 u3 = ‐30 Econ 330 Spring 2010 Prof. Sydnor a. If δ = .25 (i.e., 1/4th), in what period will BoJangles complete the project? Explain why we would say that BoJangles seems very impatient but does not have a time inconsistency problem. b. Calculate the ranges of δ for which BoJangles will do the project in Period 1 and the range for which he will do it in Period 3. Explain briefly what it is about the structure of this particular problem that means there is no δ for which BoJangles would do the task in Period 2. [Hint: it has to do with the particular levels of disutility across periods.] Recognize internally that this is not a general phenomenon and that if I wanted to, I could have come up with payoffs such that BoJangles might do the project in Period 2. c. Again assuming that δ = .25, imagine the Period 0 Mr. Bojangles, who has learned about the project, but cannot do the project in that period. Period 0 has the preferences we would expect, namely U0 = u0 + δu1 + δ2u2 + δ3u3. Calculate the utility cost (i.e., amount of negative utility) that the Period 0 self would be willing to pay to commit his Period 1 self to doing the project. Note that if Period 0 self does not pay for commitment, then whatever you found in part a will be what the pattern of behavior will be. Now assume that BoJangles actually has quasi‐hyperbolic preferences with β = .45 and δ = 1. So that his utility is given by: U1 = u1 + βδu2 + βδ2u3 U2 = u2 + βδu3 U3 = u3 d. From the perspective of his Period 1 self, in what period would Mr. BoJangles like to do the project? Rank his preferences from the Period‐1 perspective between the 3 weeks that he could do the project. e. From the perspective of his Period 2 self, in what period would Mr. BoJangles like to do the project? f. Explain whether you think BoJangles has a time inconsistency problem. g. Explain what you expect to happen (i.e., when will the project get done) and Mr. Bojangles’ thought process if his is completely naïve about his present bias. h. Explain what you expect to happen (i.e., when will the project get done) and Mr. Bojangles’ thought process if his is completely sophisticated about his present bias. i. Now imagine the Period 0 BoJangles who has learned about the project, but cannot complete the project yet. Period 0 has the preferences we would expect, namely U0 = u0 + βδu1 + βδ2u2 + βδ3u3. Calculate the utility cost that the Period 0 self would be willing to pay to commit his Period 1 self to doing the project. Problem 4: This problem is a very stylized model of making decisions about where to invest money. It obviously does not capture all of the features of that real‐life situation, but it captures a lot of what might be important. The question here is how hyperbolic discounting affects the ability to invest money in ways that earn high returns. This is a challenging question and may take you some time to work through. It is clearly way too long/tough for an in‐class midterm, however, it is good practice for thinking about the hyperbolic discounting model. Econ 330 Spring 2010 Prof. Sydnor An investor lives for eight periods, t = 1, 2, 3, 4, 5, 6, 7, 8. At the beginning of period 8, she retires and consumes whatever money she has made up to that point. Her instantaneous utility for retirement wealth u8(w) = w, and she does not consume in periods 1 through 7. So the idea here is that all we care about is how much money there will be for retirement. At the beginning of period 1 she starts off with $100 to invest for retirement. You can think of this as money she inherited in a trust fund and cannot spend until retirement. The money is currently sitting in a checking account, Investment A, in which it is earning no interest. There are two other potential investments out there. Investment B earns a return of $1 in every period in which the person’s wealth is invested in it. She can costlessly transfer her wealth from Investment A to Investment B. The other option is Investment C. This investment earns a return of $10 every period that the money is invested in it, but it takes a one‐time immediate effort cost of $11 to transfer wealth from investment A to C. For example, the high‐return alternative investment might take some effort to find. Suppose that wealth cannot be transferred between Investment B and C, and that once the investor transfers her wealth, she transfers all of it. The investor is a hyperbolic discounter with β = ½ and δ = 1. She cannot commit her future behavior. She decides at the beginning of each period, including period 1, whether to transfer her money out of Investment A, and if so, which alternative investment to use. If she transfers her money at the beginning of period t, it will earn the return of the new investment in period t. Remember, however, that the enjoyment of that return does not happen until Period 8, in which she retires and spends the money. If she transfers to Investment C, she incurs the cost of $11 in the period in which she transfers the money and enjoys the returns from Investment C in Period 8 when she retires. Note also that she retires at the beginning of Period 8 and does not reap returns from investments in Period 8. a) Suppose the investor only has access to investments A and B. What does a naïve investor do? What does a sophisticated investor do? b) Suppose the investor only has access to Investments A and C. What does a naïve investor do? Explain intuitively (i.e., you don’t need a lot of math here). c) Suppose the investor has access to all three investment options. Show that a naïve investor waits until period 6, and then transfers her wealth into investment B. Explain intuitively why the investor waits so long to move money into a superior investment into which she could costlessly have moved her wealth all along. d) Argue that a sophisticated investor would never wait in this situation. Here I want you to compare to part (c) in particular, and argue the intuition. I am not asking you to solve for what exactly the sophisticated investor would do – that’s a harder prediction to make. e) For extra credit: Give a solution to what the sophisticated investor would do. ...
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