The greatest challenge to the normative account – and one of the most important behavioural economic discoveries – has come from the observation that the discount rate is not constant but seems to decrease with time. Numerous studies have shown this to be the case (e.g. Benzion et al., 1989; Chapman, 1996; Chapman and Elstein, 1995; Kirby, 1997). In a simple demonstration, Thaler (1981) asked subjects to specify the amount of money they would require in one month, one year or ten years, to make them indifferent between that option and receiving $15
now. Their median responses implied an average annual discount rate of 19% over a ten year horizon, 120% over a one year horizon and 345% over a one month horizon. Similar observations have been demonstrated in non-monetary domains such as health and in credit markets (Chapman 1996; Chapman et al., 2001;
Chapman and Elstein 1995; Pender, 1996; Redelmeier and Heller, 1993). This pattern also emerges from a meta-analysis of discount rates computed from studies using different time spans (Frederick et al., 2002).
Moreover, when mathematical functions are fit to such data, a multitude of studies have demonstrated that hyperbolic or quasi-hyperbolic discount functions provide a superior fit compared to exponential functions in both humans and animals, for monetary and other forms of delayed reward and punishment (e.g.
Estle et al., 2006; Frederick et al., 2002; Green and Myerson, 2004; Green et al., 1994a, 1994b; Green et al., 1997, 1999a, 1999b; Ho et al., 1999; Kirby, 1997; Kirby and Marakovic, 1995; Kirby and Santiesteban, 2003; Kirby et al., 1999; Myerson and Green, 1995; Ostaszewski et al., 1998; Rachlin et al., 1991; Richards et al., 1997a,b;
Simpson and Vuchinich, 2000 – and most of the studies reviewed below in the pharmacology, neurobiology and psychiatry of intertemporal choice). The standard and most widely used functional form for hyperbolic discounting in the behavioural literature, was proposed by Mazur (1987) and based on earlier work by Ainslie and Herrnstein (e.g. Ainslie, 1975; Ainslie and Herrnstein, 1981;
Herrnstein, 1981). Using the same terminology as the exponential discounting model, the discounted value of a delayed reward or punishment is calculated as follows:
Here delay is represented by d. It is important to note that other functional forms which capture decreasing rates of discounting have also been proposed (see
Chapter 3, Frederick et al., 2002; Green and Myerson, 2004; Loewenstein and Prelec, 1992; Phelps and Pollak, 1968). Therefore, unlike exponential discounting, where the reward is devalued at a constant proportion per unit time, in hyperbolic discounting the reward looses a gradually smaller proportion of its value per increasing unit time – so it will lose a large proportion of its value in the initial stages of the delay and less throughout the later stages (see Figure 1).
One interesting prediction that emerges from hyperbolic (but not exponential) models is that preference in intertemporal choice should be observed to reverse depending on the time that the choice is made. Thus a person may prefer $1 today to $1.50 tomorrow but prefer $1.50 in 51 days to $1 in 50 days (Figure 2). Such
‘preference reversal’ is a reliable experimental finding in both humans and animals (Ainslie and Haendel, 1983; Ainslie and Haslam, 1992; Ainslie and Herrnstein, 1981; Bradshaw and Szabadi, 1992; Green and Estle, 2003; Green et al., 1981, 1994;
Herrnstein, 1981; Kirby and Herrnstein, 1995; Mazur, 1987; Millar and Navarick, 1984; Rachlin, 1974; Rachlin and Green, 1972; Rachlin and Raineri, 1992; Rodriguez and Logue, 1988; Solnick et al., 1980). Green et al. (1994) for example, asked subjects whether they would prefer $20 now or $50 in 1 month. In this case, most respondents said they preferred the $20 option. They then added a constant delay to each option, in increasing increments – increasing the delay to the first option while keeping the delay between the two options constant. Thus, subjects had to subsequently choose between $20 in six months and $50 in seven months, $20 in one year and $50 in one year and one month, and so on. As the delay to the first option increased, subjects increasingly switched their preference to the larger-delayed option, such that most participants preferred $50 in one year and one month to $20 in one year. In an analogous experiment (Green et al., 1981) pigeons were given a choice between a smaller-sooner pellet of food and a larger-later pellet. To make the choice they had to peck one of two response keys. They found
that if the choice was presented two seconds before the outcome of either option was initiated, the pigeons most often opted for the smaller-sooner option, whereas if the choice outcome occurred after 28 seconds, they opted for the larger-later option. Holt et al., (2008) also observed preference reversals in the loss domain.
Preference reversals are consistent with most functional forms of discounting where the discount rate decreases over time. They occur because the subjective value of the larger-later reward decreases more slowly, as it becomes more delayed, than the subjective value of the smaller-sooner reward. In exponential discounting such a scenario is not possible since the rate of discounting is constant across all time periods. Introspection and simple anecdotal observation of human behaviour confirms the existence of preference reversals. Take the classic example of a smoker or dieter who says that from now on they intend to quit smoking, or refrain from eating highly calorific foods. When the decision is made at time t1 they are stating their decision intention about a future choice between a larger-later (quitting smoking, health and financial benefits etc.) and a smaller-sooner (enjoying the next cigarette) option. Their statement at t1 indicates that they value the larger-later option as the greater option (higher utility). Yet when the decision gets closer they often succumb and instead choose the smaller-sooner option, indicating (assuming they have a decision-making system which is based on their value systems) that they now value the smaller-sooner option as the greater – thus constituting a preference reversal. It is during the brief period, close to the possible receipt of the smaller-sooner option that values can cross and a lifetime’s resolve can be overcome by a moment’s weakness (Figure 2).
This key feature of hyperbolic discounting has led to its being described as irrational (e.g. Ainslie, 1992, 2001; Olson and Bailey, 1981; however see Becker and Murphy, 1988 for a rational and exponential take on preference reversals). As Strotz (1955) argues, if we make plans for future consumption, we should stick to
them unless we have a good reason to do otherwise. Moreover, dynamically inconsistent choices may be seen as a violation of the independence axiom (see above).
Figure 1. Temporal discounting. Rewards lose value with increasing delay. In the exponential model (black line) rewards lose a constant proportion of their value per unit time. Hyperbolic discounting (red line) implies that the reward loses a decreasing proportion of its value per unit time and is characterized by a steeper loss in the initial phases of the delay and a shallower loss in later phases.
Figure 2. Dynamically inconsistent choice. Hyperbolic but not exponential discounting can lead to preference reversals. When the agent makes a decision between a smaller-sooner reward and a larger-later reward when both options are far away, the larger-later option may be valued more (e.g. a smoker says he intends to quit rather than smoke the next cigarette). As time approaches the sooner option, its value may increase above that of the larger-later option, leading to a preference reversal.