Unformatted text preview: Econ 330 Spring 2010 Prof. Sydnor Problem Set 2 Solutions 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. The ganansiya box was basically like a piggy‐bank that you could keep at your house. It was a locked box that the person with the account did not have a key for. It was designed to help people save, by giving them a place to put their money that they could not then get into later when they were tempted by some savings. That is easier and more convenient than going to the bank with deposits every day. At the same time, they tried to collect this money and get it into the bank regularly, because with enough temptation in there (i.e., with enough money sitting in the box) the danger is that your current self will find a sledge‐hammer or axe to undo the good intentions of your past self. 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. If you want to save for retirement, you have to start sometime. If we do a large survey and most people tell us that they are saving less than they think they need to, it means that most people are thinking they will save more in the future. The only way that could be true with exponential discounting (where everyone is time consistent) is if most of the people we asked were very young and not earning money or if it happened to be a randomly bad time in the economy when most people had to temporarily stop saving, but anticipated saving more later. With hyperbolic discounting, however, it is possible for both naives and sophisticates to put off saving (or more generally under save) each period because of their self‐control problems. People like that want to save more in the future, yet may never actually do it. Surveys of savings rates and attitudes suggest that most Americans have some level of time inconsistency in their preferences. 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.] Econ 330 Spring 2010 Prof. Sydnor 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 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. Let’s ask what BoJangles wants to do from the perspective of both Period 1 and Period 2 (the two periods in which he has a choice). If he does it in Period 1: U1 = ‐10 If he does it in Period 2: U1 = .25*(‐20) = ‐5 If he does it in Period 3: U1 = .25*.25*(‐30) = ‐1.87 So from the perspective of Period 1, BoJangles thinks it is best to do it in Period 3. Now let’s ask about Period 2 Utility: If he does it in Period 2: U2 = ‐ 20 If he does it in Period 3: U2 = .25(‐30) = ‐7.5 So from the perspective of Period 2, he also wants to do it in Period 3. BoJangles will do it in Period 3. All selves agree on that plan. We should not be surprised by that, because BoJangles is time consistent – he is simply an exponential discounter. He clearly looks impatient, because he is willing to wait until the end to do the project when it is much more costly to do so. That is driven by the fact that he has a low discount factor, which makes him very impatient. But he is not time inconsistent (and no exponential discounter is), because his ranking over the options does not change over the periods. 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 Econ 330 Spring 2010 Prof. Sydnor 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. Let’s look at some inequalities: Period 1 BoJangles prefers doing it in Period 1 vs. Period 2 if: ‐10 > δ(‐20) which means δ > ½ Period 1 BoJangles prefers doing it in Period 1 vs Period 3 if: ‐10 > δ2(‐30) which means δ > .58 Period 1 BoJangles prefers doing it in Period 2 vs Period 3 if: ‐20δ > δ2(‐30) which mean δ > 2/3 Period 2 BoJangles prefers doing it in Period 2 vs. Period 3 if: ‐20 > δ(‐30) which means δ > 2/3 So if δ > .58, then Period 1 prefers doing it in Period 1 over both Period 2 and Period 3 and will do it then. If δ < .58, then both Period 1 and Period 2 want to see it get done in Period 3, so it will get done in Period 3. 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. Without commitment, BoJangles will do the project in Period 3. Period 0 Self, thinks (just like his other selves) that the best idea is to do it in Period 3. To see this, note: Period 0 Utility from doing it in Period 1 = .25(‐10) = ‐2.5 Period 0 Utility from doing it in Period 2 = .25*.25*(‐20) = ‐1.25 Period 0 Utility from doing it in Period 3 = .25*.25*.25(‐30) = ‐0.47 Since he is already going to do it in the period that Period 0 likes best, of course Period 0 would not pay any utility cost (negative little u0) to commit to the plan. This is always true for time consistent exponential discounters – they do not pay for commitment. 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. Period 1 utility of doing it in Period 1: U1 = ‐10 Period 1 utility of doing it in Period 2: U2 = .45(‐20) = ‐9 Period 1 utility of doing it in Period 3: U3 = .45(‐30) = ‐13.5 Econ 330 Spring 2010 Prof. Sydnor So Period 1, in order from favorite to least, likes doing it in Period 2, then Period 1, then Period 3. e. From the perspective of his Period 2 self, in what period would Mr. BoJangles like to do the project? Period 2 self utility from doing it in Period 2 is: U2 = ‐20 Period 2 self utility from doing it in Period 3 is: U2 = .45(‐30) = ‐13.5 So Period 2 self prefers to wait to Period 3. f. Explain whether you think BoJangles has a time inconsistency problem. BoJangles clearly has a time inconsistency problem. His preferred plan (ranking of options) changes as he goes from Period 1 to Period 2. 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. If Bojangles is naïve, his Period 1 self wrongly believes that his Period 2 self has the same ranking of options. So his Period 1 self will anticipate that his Period 2 self will do the project, and as such Period 1 will not do it. Once Period 2 comes, however, BoJangles wants to do it in Period 3 instead and will wait until then. 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. Period 1 BoJangles would like to wait and do the project in Period 2. However, being aware of his future preferences, he knows that if he waits, his Period 2 self will then wait again and the project won’t get done until Period 3. Anticipating that, BoJangles has to decide between doing it right away or knowing that it won’t get done until Period 3. BoJangles thinks it is better to do it right away (by which I mean he gets higher utility U1) than to wait until Period 3, and thus gets the project done right away. 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. Commitment problems are potentially more interesting for time inconsistent people, because they may want to get themselves to change behavior. It turns out, however, that in this example, there is still no demand for commitment. To see this, let’s consider two cases. First, consider the naïve BoJangles at Time 0. In Period 0, BoJangles wants himself to do the project in Period 1 (you should be able to verify). Now we know that the naïve BoJangles won’t do it actually until Period 3. So Period 0 could benefit in utility by committing and getting it done. How much could he benefit? Well U0 = .45*(‐10) = ‐4.5 from doing it in Period 1 and U0 = .45(‐30) = ‐13.5 from doing it in Period 3. So there would be an improvement of utility of 9 by committing. However, and unfortunately from BoJangles, when he is naïve, he does not realize that he won’t do it until Period 3. His Period 0 self wrongly thinks that his Period 1 self will get the task done. So he does not perceive a need for commitment and won’t pay. The Period 0 sophisticate is a little different. He doesn’t pay for commitment because he knows that his Period 1 self is going to get it done. Now, if Period 0 had a choice between committing his Period 1 self or giving his Period 1 self the chance to commit Econ 330 Spring 2010 Prof. Sydnor his Period 2 self, then Period 0 self might want commitment – but that’s a different question. 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. 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? Both types of investors would immediately switch to Option B. There is no cost to doing so and no benefit from waiting. So there is no scope for time inconsistency – everyone can recognize the better deal and do it. It is important to recognize this. The hyperbolic discounting model does not mean people are always inconsistent, crazy, or behave Econ 330 Spring 2010 Prof. Sydnor surprisingly – they only do nutty things when there is a tension between current desires and future costs/benefits. Mathematically the discounted utility in Period 1 from staying with investment A forever is 50, while it is ½(107) = 53.5 if she switches right away to Option B. By waiting one period, she simply decreases the amount she has at retirement from 107 to 106, and as such decreases the discounted utility at Period 0 to 53. So switching right away is the best option from Period 0 perspective, and since that is true, there is no temptation to procrastinate and thus no difference in behavior between a sophisticate and a naïf. 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). Now there is a potential tension – there is an immediate cost that results in a future benefit. This is a classic situation where time inconsistency and the distinction between naifs and sophisticates matters. Let’s think about Period 1 Self, who might first consider switching right away to Investment C. Doing so results in an immediate utility cost (in period 1) of 11, but leads to having 170 at retirement in period 8. Let’s see the discounted utility of this switch from the Period 1 perspective: Switch to C in Period 1: That discounted utility of 74 is higher than staying in option A, which has discounted utility of only 50. But now the person has to consider the option of waiting one period to do the switch. Intuitively, this may be attractive, since you delay experiencing the cost of switching into the future and you don’t weight the future soheavily. Let’s think about Period 1’s utility from switching in Period 2: . So we see that the Period 1 self would like to wait one day and switch in Period 2 rather than Period 1. Intuitively what is going on is that they are pushing back the pain of switching for one period so that the ‐11 cost is now discounted down to ‐5.5 from the Period 1 perspective. That reduction in the perceived cost outweighs the reduction in the discounted benefits of the trust fund balance you get to enjoy when you retire. (Delaying the cost of 11 by 1 period and having it therefore discounted by ½ is worth the discounted reduction in benefits, which is ‐5 (the 10 less, discounted by ½)). Now a naïve person will therefore want to procrastinate 1 day and believing that she will be patient in the future, thinks she will follow through with that plan. However, when the next Econ 330 Spring 2010 Prof. Sydnor day comes, this same logic repeats itself. It again seems better to push the cost of switching back 1 day. And so the naïve investor ends up never switching to Option C. Notice that the naïve person is actually better off only having the option of the low‐return option (B) that does not require a cost of switching than being given the option with a higher return. 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. Here it is useful to make some tables that show the discounted utility from each Period’s perspective of the possibility of switching to the different investments in different periods. First note that the discounted utility to all of the relevant decision‐making periods (Period 1 through 7) of staying in Option A is simply 50. What are the discounted utility of switches to Investment B? Self T = 1 T = 2 T =3 T = 4 T =5 T= 6 T = 7 Period 1 (U1) Period 2 (U2) Period 3 (U3) Period 4 (U4) Period 5 (U5) Period 6 (U6) Period 7 (U7) Never switch 53.5 53 52.5 52 51.5 51 50.5 50 ‐‐ 53 52.5 52 51.5 51 50.5 50 ‐‐ ‐‐ 52.5 52 51.5 51 50.5 50 ‐‐ ‐‐ ‐‐ 52 51.5 51 50.5 50 ‐‐ ‐‐ ‐‐ ‐‐ 51.5 51 50.5 50 ‐‐ ‐‐ ‐‐ ‐‐ ‐‐ 51 50.5 50 ‐‐ ‐‐ ‐‐ ‐‐ ‐‐ ‐‐ 50.5 50 A few things to note here in the table. First, I have used dashes to blank out each self’s discounted utility from switching in a prior period. We could fill these in (and would simply match the rows above), but I have left them blank to highlight that they are not relevant to the decision‐making. You only have the choice to switch once and my feeling as the Period 2 Self about how good it was to switch in Period 1 is totally irrelevant – I can’t go back in history and change my action. The second thing to notice is that when we are considering Investment B, each row looks the same. That is, the person looks time consistent. That is because there is no tension in this investment between current attitudes and future costs/benefits. Econ 330 Spring 2010 Prof. Sydnor Now instead let’s consider the discounted utility of switches to Investment C. Self T = 1 T = 2 T =3 T = 4 T =5 T= 6 Period 1 (U1) Period 2 (U2) Period 3 (U3) Period 4 (U4) Period 5 (U5) Period 6 (U6) Period 7 (U7) T = 7 Never switch 74 74.5 69.5 64.5 59.5 54.5 49.5 50 ‐‐ 69 69.5 64.5 59.5 54.5 49.5 50 ‐‐ ‐‐ 64 64.5 59.5 54.5 49.5 50 ‐‐ ‐‐ ‐‐ 59 59.5 54.5 49.5 50 ‐‐ ‐‐ ‐‐ ‐‐ 54 54.5 49.5 50 ‐‐ ‐‐ ‐‐ ‐‐ ‐‐ 49 49.5 50 ‐‐ ‐‐ ‐‐ ‐‐ ‐‐ ‐‐ 44 50 Let’s stare at this for a second. Note that all selves think that switching in Period 7 is worse than never switching (compare column 7 to the never‐switch column). That’s because the cost of switching (‐11) outweighs the benefit of being invested for only 1 period. All selves, with the exception of Self 6, think, however, that switching in Period 6 is better than never switching. Now the cost of 11 is less than 2 periods of the investment (a total of 20). However, the Period 6 self disagrees. The Period 6 self thinks that the cost of 11 outweighs the discounted cost of the additional 20, which because it comes in the future is only worth 10 to him. The previous selves were also discounting, but for them both the cost of 11 to be paid in period 6 and the extra 20 that came in period 8 were both discounted by ½ and 10 is higher than 5.5. Note also that we can see in the table the inconsistency we discussed above. Period 1 thinks that switching in period 2 is the best thing to do. Note that period 1, however, thinks that switching in period 1 is better than waiting until period 3 to switch. Intuitively, period 1 would like to push the cost of switching so that it is “in the future”. However, once considering costs and benefits that all occur “in the future”, Period 1 thinks that delaying a period is just wasteful. Now this is true for each period. So again in Period 2, the person now wants to wait until Period 3 to switch. So let’s think about the Naïve investor. She thinks that she just has to choose what she likes best from the top rows of the two tables (since she thinks she will be consistent with these preferences in the future). She definitely thinks switching to C right away is better than switching to B right away and way better than never switching out of A. However, she thinks that procrastinating one period and switching to option C in period 2 is the best thing to do. Econ 330 Spring 2010 Prof. Sydnor Naively expecting to actually do that, she procrastinates. But when she gets to Period 2, she again looks at the world and thinks that it is best to delay one period. So she procrastinates again. This goes on and on, until she wakes up and finds herself in Period 6. In period 6 she looks at the world and still would rather wait until Period 7 to switch to option C than to switch to option C in Period 6. But now both of these options seem worse than not switching at all and both are worse than switching over to Option B. So she gives in and switches to Option B. The previous period (Period 5), she thought she would finally switch to Option C in period 6, which was the last period where it made objective sense to do it. But in Period 5, she was underestimating her level of impatience in Period 6 and didn’t realize that Period 6 would think the cost of switching to Option C was such a big deal that it was better to just switch to Option B instead. Intuitively, each period she thought it was a good idea to switch to Option C, but she wanted to just put it off a little. She did that for so long, though, that the benefits of switching got eroded and she ended up just doing the simple costless thing she could have done 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. The sophisticated person accurately perceives what her future selves would do. As such, if she knew she was going to switch to Option B in Period 6, she would definitely prefer to at the very least switch to Option B earlier. She may or may not be able to get herself to switch to Option C, but there aren’t any surprises for the Sophisticate. So she can never expect to do something that she can improve on at the beginning. e) For extra credit: Give a solution to what the sophisticated investor would do. Here we want to look at the tables again. The best way to approach these problems is to start at the end and work backward (a process known as backwards induction). If she gets to period 7 without having switched, she will decide to switch to Option B. Now go back one period. Period 6 has the following options – Switch to Investment C, Switch to Investment B, or do nothing, in which case she can anticipate switching to Option B in Period 7. Period 6’s utilities from these options are 49, 51, and 50.5 respectively. So she switches to Option B. So far nothing surprising – we had all this in Parts c and d. But now let’s think about Period 5. Period 5 has 3 relevant option: a) switch to option C, which gives utility to Period 5 of 54; b) switch to option B, which gives utility to Period 5 of 51.5; or c) do nothing, which implies that Period 6 will get to choose, which will result in switching to option B in Period 6, which from Period 5’s perspective gives a utility of 51. So Period 5 will now switch to Option C. Notice that Period 5 would still like to wait and switch to Option C in Period 6, however Period 5 knows that Period 6 won’t do it. Ok, so we switch in Period 5, right? Well… let’s keep going. Period 4 has again 3 relevant options: a) switch to Option C – utility = 59, b) switch to option B – utility = 52, or c) do nothing, leaving the decision to period 5, who will switch to Option C if Econ 330 Spring 2010 Prof. Sydnor given the chance, which from Period 4’s perspective has a utility = 59.5. So Period 4 procrastinates and let’s period 5 do the switching. And you thought we had broken the procrastination, didn’t you? Nope. Period 4 gets to procrastinate, knowing that Period 5 will do the right thing. Now think about Period 3 – just going to procrastinate like period 4, right? Nope. Take a look at it – Period 3 would of course like best to wait one period and switch to option C in period 4. But period 3 knows that period 4 will procrastinate. So period 3 has to choose to either do it now or wait two period. So Period 3 will switch to option C. Now period 2 can see that period 3 would switch to option C and thus can just wait for that to happen, which is her favorite outcome. Now period 1, knowing that period 2 would procrastinate and period 3 would switch, decides not to wait and gets it done. (Interestingly, if we add a period to this situation, we could see it getting done in the second period rather than the first.) ...
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