behave better in the future, but that future is never realized. In this case, food stamp recipients believe they will be able to restrain their appetites later in the month, but they never do. This eventually leads to severe shortages of food in the last week, as the data have documented. Often, hyperbolic discounting is misunderstood to simply embody heavy discounting of the future. Instead, it is a statement on how discounting of a pair of days in the future evolves as those days become closer to the present and how this causes recipients to change their consumption plans.

Naïve Quasi-Hyperbolic Discounting

One of the primary advantages of the exponential discounting function is the simplicity with which the exponential discount could be used in solving maximization problems. By contrast, the hyperbolic discounting model can be difficult to deal with mathematically given the functional form for the discount factor. This has led David Laibson to propose an approximation to the hyperbolic discounting function called quasi-hyper- bolic discounting.

Quasi-hyperbolic discounting separates the hyperbolic profile of time discounting into two different discount factors. These two factors represent the discount applied to utility of consumption in the second period and the discount applied to the utility of consumption for each additional period, respectively. The discount factor applied to the second period is small relative to the other discount factor, representing the notion that people discount consumption tomorrow relative to today more heavily than the day after tomorrow relative to tomorrow. In other words, any consumption in the future receives some penalty in the mind of the consumer, but trading off consumption between two different periods in the future does not face such a steep penalty. Thus, instead of the model in equation 12.2, we write

Uc1, c2, =uc1+βuc2+βδuc3+βδ2uc4+ +βδi−2uci+

where 0 < β < δ < 1, and where, if we want to consider the infinite time horizon problem, T may be . Here β represents the discount applied to utility of consumption in the second period (which also multiplies utility in all future periods), and δ represents the discount applied to utility of consumption for each additional period as we move farther into the future. In general, if β < δ the function approximates hyberbolic discounting.

Figure 12.4 displays a quasi-hyperbolic discounting function (marked in triangles) that has been selected to approximate the corresponding hyperbolic discounting function. The advantage of this form is that it closely replicates the exponential mathematical form, thus restoring the simple mathematical formulas for time-discounting problems. The solution to the utility-maximization problem for equation 12.30 again requires that discounted marginal utility is equal. However, the differential discount implies

Both of these equations are reminiscent of equation 12.18. Thus, given a functional form for the instantaneous utility function, we can find a relationship between planned consumption in one period and the next.

For example, suppose that people must maximize their utility of consumption over an infinite time horizon, given an initial endowment of wealth w. Suppose further that uc = cα, so that the marginal utility (or slope of the utility function) of consumption is

All other periods’ consumption could then be calculated from the above formula via equation 12.34.

Source: Benzion, U., A. Rapoport, and J. Yagil. ”Discount Rates Inferred from Decisions: An Experimental Study.“

Management Science 35(1989): 270–284.

Alternatively, if we modeled the same decision using a hyperbolic discount function, equation 12.36 would be replaced by

which cannot yield a closed-form solution. Thus, in many situations, economists use the quasi-hyperbolic approximation rather than the hyperbolic form. In fact, the quasihyperbolic form is much more common in practice than the hyperbolic form that it approximates.

Uri Benzion, Amnon Rapoport, and Joseph Yagil find evidence of changing discount rates over time. They asked 204 participants about their preferences between receiving bundles of money after various waiting periods. For example, one question asks participants to suppose they had just earned $200 for their labors, but after coming to pick up the money, they find their employer is temporarily out of funds. Instead, they are offered payment in six months. Participants were asked how much would need to be paid at the later time to be indifferent between receiving $200 now or the higher amount later. The amounts of money and the length of time were varied. Table 12.2 displays the estimated discount factors for each of the various scenarios similar to the scenario just described, assuming a money metric utility function.3 For each, we see a relatively large discount factor for the first six months of delay. Discounts for longer periods are much smaller, reflecting the hyperbolic nature of discounting. If a person must delay a reward for some time, longer waits are no longer thought of as so costly. Similar experiments have been run with actual money payouts, finding additional evidence that discount factors climb over time, eventually becoming stable.

3 They actually calculate average discount rate. This is the rate R such that F = P(1+R)t, where P is the present amount, and F is the future amount. The discount factor that we have discussed in this chapter is equivalent to δ = 1/ (1+R). The table displays the discount factor implied by the discount rate they report. The money metric utility function simply assumes that utility is linear in dollars.

EXAMPLE 12.2 Reading Days Are for Procrastination

It seems to have happened to all of us at one time or another. In fact, many of us have experienced it repeatedly. The big test is coming in two weeks, and we should be studying. But there is so much time before that. “I could put off studying one more day without hurting my grade,” you may tell yourself. Each day you tell yourself this, until it is so late that you truly do not have as much time to study as you probably should have. Your grade is not what you wanted or what you could have achieved. Then the next semester, despite your previous experience, you procrastinate just as before.

Hyperbolic discounting might provide one explanation for why procrastination is so prevalent. Consider a student with the following (daily) instantaneous utility function

where s is the portion of the day spent studying. Thus, one receives negative instantaneous utility from studying, with decreasing marginal pain from additional studying given by u′s = − 2s. Moreover, suppose that the grade on a particular test is measured by an index g, as a function of time studying:

Of course it is more likely that grades show declining returns to studying, but this abstraction makes our example easier to calculate, and it ensures a solution will exist. Finally, suppose that the student’s utility of receiving a grade of g is equal to g2, discounted as if received on the day the results are announced. Finally, suppose that the student discounts future utility according to a quasi-hyperbolic discount function.

Suppose further that the exam will take place in four days, and the results will be announced seven days later. The student has set a goal of studying a total of 1.05 days. For simplicity, consider that the student is choosing between two different study plans. The first plan consists of s1 =0.21, 0.21, 0.21, 0.21, 0.21, indicating that the student will study 0.21 of a day each day until the test. The second plan considers shirking today,

but making up for it later, s′ = 0.17, 0.22, 0.22, 0.22, 0.22 . These plans are not

1

binding, meaning that the student could easily change her mind tomorrow. If s is chosen,

So, for example, if β = 0.75 and δ = 0.99, then Us1 0.182. Alternatively, if s′ is chosen, then instantaneous utility in the first day will be 0.172 = − 0.0289, and instantaneous utility on each succeeding day will be 0.222 = 0.0484. Moreover, the grade will be 0.17 + 4 × 0.22 = 1.05, yielding instantaneous utility 0.525:

Choosing to procrastinate yields instantaneous utility of 0.192 = 0.0361 in the first period and 0.232 = 0.0529 in the three following periods. The grade obtained will be 0.17 + 0.19 + 3 × 0.23 = 1.05, yielding instantaneous utility of 0.525 when the grades are received, or

Thus, the student decides again to postpone the bulk of her studying.

Similar choices each day lead the student to postpone studying, planning to eventually make up the time and obtain the same grade. The next day, the student decides

whether to continue with her new plan s″ = 0.23, 0.23, 0.23 , yielding utility

3

where again the planned grade is 1.05. She again chooses to procrastinate and in the

chooses to procrastinate. In the final day, the student faces the choice to study according

The student eventually has an actual study profile of 0.17, 0.19, 0.21, 0.25, 0.16, yielding a grade of 1, less than the grade planned upon in each of the prior periods. The student’s procrastination ends up costing her almost 20 percent of the anticipated grade on the test. This is the cost of not recognizing that as the test moves closer, she will change the way she discounts between the days of the study period. For the first four days of the study period, she believes she will trade off the disutility of studying between day 4 and day 5 by δ. By the time day 4 rolls around, she instead discounts day 5 disutility of studying by β. It is the constant shifting of the β discount through time that explains the procrastination behavior.