Choosing When to Do It

With many activities, such as studying for an exam or taking a vacation, comes the decision of when it should be done. Pleasurable activities, like taking a vacation, can lead us to an immediate reward but a long-term cost. Other activities, like cleaning a bathroom, give us an upfront cost but a longer-term reward. As experience informs us, people might want to postpone costs and speed up rewards. Thus, we might tend to do pleasurable things with long-term costs earlier than those with near-term costs and long-term benefits.

Ted O’Donoghue and Matthew Rabin propose a very simple model of such decisions of when to complete an activity. Suppose the activity can only be performed once, and the decision maker must decide on the period in which to do it. The activity can be

where vt ≥ 0 is the reward that is realized if the task is completed in period t. Thus, if the task is completed in period 3, the decision maker will receive v3. Similarly, the costs, in terms of utility lost, are given by c c1, , cT, where ct ≥ 0 is the cost of the task if completed in period t. However, the rewards and costs are not necessarily realized in the same period in which the project is completed. We will consider the decision maker to be a quasi-hyperbolic discounter.

Now let us compare the behavior of those who are naïve (naïfs) and those who are sophisticated (sophisticates) regarding their time inconsistency, with those who display no time inconsistency. We assume that time-consistent preferences in this case are represented by β = δ, and that in other cases, β takes on some smaller value representing a misperception (an underestimate) of the discounting of future utility. This makes a very specific assumption about the nature of the irrationality of both naïfs and sophisticates. In particular, it assumes that irrationality derives from a discount factor of the first future period being too low. Because we cannot observe preferences, but only actions, this is truly a philosophical assumption. Alternatively, for example, it could be the case that people do not discount all future periods heavily enough (δ could be lowered until it equals β). Intuitively, this makes less sense, but there is no behavioral evidence that could point us to one conclusion versus the other (or any of the other possibilities that could be compared to a case of β = δ).

Given this assumption about the true realized utility, O’Donoghue and Rabin note two requirements for behavior stemming from the time-consistent model. No rational decision maker should violate either of these properties given these assumptions about underlying preferences.

PROPERTY 13.1: DOMINANCE A decision maker obeys dominance if whenever there exists some period t with vt > 0 and ct = 0, the person will not choose to complete the task in any period t′ with ct′ > 0 and vt′ = 0.

Dominance simply says that if decision makers can complete a task in some period t for some positive benefit and no cost, then they will never choose to complete the same task in another period for which there is no benefit but some cost.

u = 8β or doing it in period 3 and receiving − βδ2. In this case, the naïf would always choose to complete the task now and obtain positive utility. Removing the unselected second-period option creates a preference reversal and thus a violation of the independence of irrelevant alternatives.

Sophisticates may also violate dominance if they fear that their future self will choose the dominated option. Suppose that the sophisticate faces v = 0, 9, 1, and c = 6, 16, 0, with kv = 0, and kc = 1. We can solve for the sophisticate’s decision using backward induction. In the second period, the sophisticate either chooses to complete the task at that point and receive u = 9 − 16β = 2, or chooses to postpone the task to the last period and receive u = β = 12. In this case, the sophisticate would prefer to complete the task in the second period than in the last. In the first period, the sophisticate must then decide whether to complete the task then and receive u = − β6 = − 3, or postpone until the second period and receive u = β9 − δ16 = − 3.5. In this case, the sophisticate will choose to complete the task in the first period, despite this outcome being dominated by completing the task in the last period. Sophisticates do this because they are afraid they will choose to complete the task in the second period given the chance, which is perceived in the first period to be much worse than completing the task in the first period. Now suppose we again remove the option of completing the task in the second period. In this case, the sophisticate in the first period must choose between doing it now and receiving u = − 6β = − 3, or completing it in the last period and receiving u = δβ = 12. In this case, the sophisticate would choose to complete the task in the last period. Again, removing an unselected option changed the decision, which is a violation of the independence of irrelevant alternatives.

From this point on, we will assume δ = 1. Because the decision maker does not discount tradeoffs in utility between two future periods, we can treat utility changes in future periods as if they all occur in the same period. For example, if someone is deciding whether to complete the activity in period t, which would produce a reward in period t + 3, then the decision maker values this reward as u = βδ2vt = βvt. Alternatively, if the reward would be given in period t + 27, the decision maker would value this reward as u = βδ26vt = βvt. This simple form of discounting allows us to more easily illustrate the intuition of the quasi-hyperbolic discounting model.

Suppose there are two types of activities: immediate cost and immediate reward. In immediate-cost activities, costs are realized in the period in which the activity is completed, but the rewards are received in future periods, kv > 0, kc = 0. Alternatively, in immediate-reward activities the rewards are received in the period in which the activity is completed, and the costs are paid in future periods, kv = 0, kc > 0.

Consider a task with immediate costs. In this case, in any period t in which a timeconsistent decision maker completes the task, the utility of doing so will be vt − ct (recall that for time-consistent decision makers β = δ = 1). For example, consider someone who must go in for a thorough (and uncomfortable) medical exam sometime in the next four days. Completing this medical exam will allow him to take part in a high-adventure activity that should be enormous fun several weeks later. The next four days, however, might each provide different costs in terms of opportunities forgone. For example, a close friend has invited the decision maker to an amusement park on day 4, and a gettogether with family is planned for day 2. We may represent the costs of these four days by c = 7, 8, 9, 10. However, no matter when the decision maker completes the medical

exam, the benefit will be the same, v = v, v, v, v, realized one week from now. A timeconsistent person will thus choose when to have the medical exam, t, so as to maximize the net benefit of the action. The problem becomes

Thus, in this example, the time-consistent person will choose to complete the medical examination on the first day. If he were only allowed to choose from the last three days, he would choose the first day in which he was allowed to undergo the exam.

Now suppose that the decision maker is a naïf. Then all future utility would be discounted by β. In this case, the person considering when to go in for the exam from the perspective of the first day solves

In this case, the benefit always arrives in the future and is thus discounted. However, taking the exam on the first day will not be discounted, but it will be on all other possible days. Thus, on the first day, the naïve person puts off the exam until the second day, so long as 8β < 7. However, he will plan to complete the task on the second day because 8β < 9β < 10β. The next day, he would decide to postpone again as long as 9β < 8. The naïve decision maker would procrastinate again on the third day if 10β < 9 but would then be forced to complete the exam on the fourth day. Procrastinating to the very last day requires only that β < 0.875.

PROPOSITION 13.1 A naïf always completes an immediate-cost task either at the same time as or later than a time-consistent decision maker does.

Proof

A time-consistent decision maker will choose t such that vt − ct > vt′ − ct′ for any t′ t. Consider some time t′ < t. At t′ naïfs perceive the utility of completing the task at time t′ to be βvt′ − ct′, and their perception of completing the task at time t would be βvt − ct. Because vt − ct > vt′ − ct′, and 0 < β < 1, it must be that βvt′ − ct′ < βvt − ct. Thus, naïfs at least put off the activity until the same time as the time-consistent decision maker. ▪

Consider alternatively a time t′ > t. At time t, the naïf must now decide whether to complete the task at time t with perceived utility βvt − ct, or to complete it at t′ with perceived utility βvt′ − ct′. If it is the case that βct′ < ct, then the naïf will decide to postpone the activity beyond t and thus choose to complete the task later than the timeconsistent decision maker would.

Consider instead how a sophisticate would respond to an immediate-cost problem. Consider again the immediate-cost example above with c = 7, 8, 9, 10, and v = v, v, v, v. In this case, we can solve for the sophisticate’s decision using backward induction. On the fourth day, given he had not yet gone for the exam, the sophisticate would be forced to go for the examination, receiving u = β v − 10. On the third day, given he had not yet gone for the exam, he must decide between going now and receiving u = β v − 9, or going the following day and receiving u = βv − 10. If 10β < 9, then the

sophisticate would in this case choose to go on the last day rather than the third. Let us suppose this is the case. Now consider the decision on the second day. The sophisticate now decides between doing it on the second day and receiving u = β v − 8, or doing it on the last day (because the third day has been eliminated) and receiving u = βv − 10. The sophisticate will choose the last day only if 10β < 8. If this were the case, the sophisticate in the first day would choose between doing it the first day and receiving u = β v − 7, or doing it on the last day and receiving u = βv − 10. Again, the sophisticate would only postpone the exam if 10β < 7. Thus, while the naïf puts off the exam to the last day if β < 0.875, the sophisticate would only ever procrastinate to the last day if β < 0.7.

PROPOSITION 13.2 A sophisticate always completes an immediate-cost task either at the same time as or before a naïf would.

Proof

Suppose a naïf eventually performs an immediate-cost task at time t. Then βvt − ct > βvt′ − ct′ for all t′ > t. If this is the case, a sophisticate would also never choose to postpone beyond time t. ▪

Note that the sophisticate might do it sooner than the naïf. Suppose the naïf completes the task at time t, then the sophisticate will complete the task sooner if for some t′, t′′ with t′ < t″ < t

The first inequality, equation 13.57, implies that at time t′ the naïf will postpone, potentially believing he will take action at time t′′ (or some better time). The second inequality tells us that at time t′′ the naïf would prefer to postpone to time t. The last inequality tells us that at time t′ the sophisticate will recognize that postponing to time t will make him worse off. In this case, the sophisticate would not postpone past t′ because he recognizes that he would never choose to engage in the activity at time t′′, even though it looks attractive at time t′.

Now consider an immediate-reward activity in which costs will accrue at some later date. For example, consider a young man who has agreed to take a young woman out for an expensive date involving dinner and a show sometime in the next four days. The young man doesn’t have on hand the money to pay for the entertainment and must therefore charge it to a credit card. The particular show they go to will determine how much of a reward the young man receives. On day 1, the only show available is a rather tired musical that neither of them will enjoy much. The second through the fourth day, various newer shows are playing with varying interest. The time profile of the rewards is

given by v = 7, 8, 9, 13; however, the costs for each show is the same, c = c, c, c, c. As in the previous example, the time-consistent decision maker would solve

In this case, 13 − c clearly dominates all other choices. Thus, the time-consistent decision maker will choose to go out on the fourth night.

On the other hand, on the first day the naïf will solve

In this case, β13 − c > β9 − c > β8 − c. Thus, the naïf must decide between going the first night and receiving 7 − β c or going on the fourth night and receiving β13 − c. The naïf will plan to go out on the fourth night unless 7 > 13β, or 713 > β. Let us suppose that this is the case, so that the naïf decides to go out on the first day.

PROPOSITION 13.3 A naïf always completes an immediate-reward task either at the same time as or earlier than a time-consistent decision maker.

Proof

A time-consistent decision maker will choose t such that vt − ct > vt′ − ct′ for any t′ t. Consider some time t′ > t. At t the naïf perceives the utility of completing the task at time t′ to be βvt′ − ct′, and his perception of completing the task at time t would be vt − βct. Because vt − ct > vt′ − ct′, and 0 < β < 1, it must be that βvt′ − ct′ < vt − ct. Thus, the naïf will at least execute the activity at the same time as the time-consistent decision maker. ▪

The naïf will conduct the activity earlier than the time-consistent decision maker if for some t′ with t′ < t

Inequality 13.62 tells us that the time-consistent decision maker would prefer to complete the task at time t than at time t′. Inequality 13.63 tells us that the naïf at time t′ would prefer to complete the task at time t′ than to wait until time t. In this case, the timeconsistent decision maker will wait until t, and the naïf will perceive a greater net benefit to conducting the activity earlier.

Now consider the sophisticate facing the same problem of deciding when to go out on a previously agreed-upon date, with rewards and costs described by v = 7, 8, 9, 13 and c = c, c, c, c, and where rewards are received today but costs are delayed. Recall we had already assumed that β < 713. As before, we can solve the recursive optimization problem using backward induction. On day 4, the young man would be forced to go on

the date to meet his agreement and would receive u = 13 − βc. Thus, on the third day, the young man can either choose to go out on the date then and receive u = 9 − βc, or he can wait until the fourth day and receive u = β13 − c. So long as 913 > β, the sophisticate would choose to go out on the third day rather than the fourth day. Then, on the second day, the sophisticate can either go out that day and receive u = 8 − βc or postpone the date until the third day and receive u = β9 − c. The sophisticate would rather go out on day two than day three if 89 > β. On the first day, the sophisticate must choose to go out that day and receive u = 7 − βc or postpone until the second day and receive u = β8 − c. The sophisticate will choose to go out on the first day if 78 > β. Thus, with β > 78, the sophisticate will also choose to go out on the first day, much earlier than the timeconsistent decision maker. Note that in each round, the required β gets higher and higher for the young man to postpone rather than complete the task earlier.

PROPOSITION 13.4 A sophisticate always completes an immediate-reward task either at the same time as or earlier than a naïf does.

Proof

Suppose a naïf performs an immediate-reward task at time t. Then vt − βct > βvt′ − ct′ for all t′ > t. If this is the case, then a sophisticate would also never choose to postpone beyond time t. ▪

The sophisticate may complete the task earlier than the naïf if for some t′ and t′′ with t′ < t″ < t, where the naïf completes the task at time t,

In this case, inequality 13.64 implies that the naïf will always postpone beyond t′, potentially hoping to complete the task at t″. Inequality 13.65 shows that the naïf will postpone beyond t″. Inequality 13.66 implies that the sophisticate will never choose to complete the task at t if he could choose to complete the task at t′, recognizing that he would never actually complete the task at time t″ if given the opportunity.

If we consider the utility function of the time-consistent decision maker to be the true utility of all decision makers and the quasi-hyperbolic discount function to arise solely by misperception, then we can make some normative statements about how decision makers should behave and assess their welfare under different scenarios. For example, proposition 13.1 tells us that the naïf always postpones an immediate-cost task later than he should, leading to a reduction in realized utility. Alternatively, proposition 13.3 tells us that the naïf completes immediate-reward tasks too soon, again leading to a loss in utility. According to proposition 13.2, the sophisticate completes an immediate-cost task