Keywords

JEL Classifications

Models of Intertemporal Choice

Most choices require decision-makers to trade-off costs and benefits at different points in time. Decisions with consequences in multiple time periods are referred to as intertemporal choices. Decisions about savings, work effort, education, nutrition, exercise, and health care are all intertemporal choices.

The theory of discounted utility is the most widely used framework for analysing intertemporal choices. This framework has been used to describe actual behaviour (positive economics) and it has been used to prescribe socially optimal behaviour (normative economics).

Descriptive discounting models capture the property that most economic agents prefer current rewards to delayed rewards of similar magnitude. Such time preferences have been ascribed to a combination of mortality effects, impatience effects, and salience effects. However, mortality effects alone cannot explain time preferences, since mortality rates for young and middle-aged adults are at least 100 times too small to generate observed discounting patterns.

Normative intertemporal choice models divide into two approaches. The first approach accepts discounting as a valid normative construct, using revealed preference as a guiding principle. The second approach asserts that discounting is a normative mistake (except for a minor adjustment for mortality discounting). The second approach adopts zero discounting (or near-zero discounting) as the normative benchmark.

The most widely used discounting model assumes that total utility can be decomposed into a weighted sum – or weighted integral – of utility flows in each period of time (Ramsey 1928):

$$ {U}_t=\sum_{\tau =0}^{T-t}D\left(\tau \right)\cdot {u}_{t+\tau }. $$

In this representation: Ut is total utility from the perspective of the current period, t; T is the last period of life (which could be infinity for an intergenerational model); ut + τ is flow utility in period t + τ (ut + τ is sometimes referred to as felicity or as instantaneous utility); and D(τ) is the discount function. If delaying a reward reduces its value, then the discount function weakly declines as the delay, τ, increases:

$$ {D}^{\prime}\left(\tau \right)\le 0. $$

Economists normalize D(0) to 1. Economists assume that increasing felicity, ut + τ, weakly increases total utility, Ut. Combining all of these assumptions implies,

$$ 1=D(0)\ge D\left(\tau \right)\ge D\left({\tau}^{\prime}\right)\ge 0, $$

where 0 < τ < τ’.

Time preferences are often summarized by the rate at which the discount function declines, ρ(τ). For differentiable discount functions, the discount rate is defined as

$$ \rho \left(\tau \right)\equiv -\frac{D^{\prime}\left(\tau \right)}{D\left(\tau \right)}. $$

(See Laibson 2003, for the formulae for non-differentiable discount functions.) The higher the discount rate the greater the preference for immediate rewards over delayed rewards.

The discount factor is the inverse of the continuously compounded discount rate.

ρ(τ). So the discount factor is defined as

$$ f\left(\tau \right)=\underset{\Delta \to 0}{\lim }{\left(\frac{1}{1+\rho \left(\tau \right)\Delta}\right)}^{1/\Delta}={e}^{-\rho}\left(\tau \right). $$

The lower the discount factor the greater the preference for immediate rewards over delayed rewards.

The most commonly used discount function is the exponential discount function:

$$ D\left(\tau \right)={\delta}^{\tau }, $$

with 0 < δ < 1. For the exponential discount function, the discount rate is independent of the horizon, τ. Specifically, the discount rate is − ln(δ) and the discount factor is δ. Figure 1.

Intertemporal Choice, Fig. 1
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Three calibrated discount functions

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The exponential discount function also has the property of dynamic consistency: preferences held at one point in time do not change with the passage of time (unless new information arrives). For example, consider the following investment opportunity: pay a utility cost of C at date t = 2 to reap a utility benefit of B at date t = 3. Suppose that this project is viewed from date t = 1 and judged to be worth pursuing. Hence, − δC + δ2B> 0. Imagine that a period of time passes, and the agent reconsiders the project from the perspective of date t = 2. Now the project is still worth pursuing, since − C + δB> 0. To prove that this is true, note that the new expression is equal to the old expression multiplied by 1/δ. Hence, the t = 1 preference to complete the project is preserved at date t = 2. The exponential discount function is the only discount function that generates dynamically consistent preferences.

Despite its many appealing properties, the exponential discount function fails to match several empirical regularities. Most importantly, a large body of research has found that measured discount functions decline at a higher rate in the short run than in the long run. In other words, people appear to be more impatient when they make short-run trade-offs – today vs. tomorrow – than when they make long-run trade-offs – day 100 vs. day 101. This property has led psychologists (Herrnstein 1961; Ainslie 1992; Loewenstein and Prelec 1992) to adopt discount functions in the family of generalized hyperbolas:

$$ D\left(\tau \right)={\left(1+\alpha \tau \right)}^{-\gamma /\alpha }. $$

Such discount functions have the property that the discount rate is higher in the short run than in the long run. Particular attention has been paid to the case in which γ = α, implying that D(τ)= (1 + ατ)−1.

Starting with Strotz (1956), economists have also studied alternatives to exponential discount functions. The majority of economic research has studied the quasi-hyperbolic discount function, which is usually defined in discrete time:

$$ D\left(\tau \right)=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill \kern-2em \mathrm{if}\;\tau =0\hfill \\ {}\hfill \beta \cdot {\delta}^{\tau}\hfill & \hfill \kern1.5em \mathrm{if}\;\tau =1,2,3,\dots \hfill \end{array}\right\}. $$

This discount function was first used by Phelps and Pollak (1968) to study intergenerational discounting. Laibson (1997) subsequently applied this discount function to intra-personal decision problems. When 0 < β < 1 and 0 < δ < 1 the quasi-hyperbolic discount function has a high short-run discount rate and a relatively low long-run discount rate. The quasi-hyperbolic discount function nests the exponential discount function as a special case (β = 1). Quasi-hyperbolic time preferences are also referred to as ‘present-biased’ and ‘quasi-geometric’.

Like other non-exponential discount functions, the quasi-hyperbolic discount function implies that intertemporal preferences are not dynamically consistent. In other words, the passage of time may change an agent’s preferences, implying that preferences are dynamically inconsistent. To illustrate this phenomenon, consider an investment project with a cost of 6 at date t = 2 and a delayed benefit of 8 at date t = 3. If β = 1/2 and δ = 1 (see Akerlof 1991), this investment is desirable from the perspective of date t = 1. The discounted value is positive:

$$ \beta \left(-6+8\right)=\frac{1}{2}\left(-6+8\right)=1. $$

However, the project is undesirable from the perspective of date 2. Judging the project from the t = 2 perspective, the discounted value is negative:

$$ -6+\beta (8)=-6+\frac{1}{2}(8)=-2. $$

This is an example of a preference reversal. At date t = 1 the agent prefers to do the project at t = 2. At date t = 2 the agent prefers not to do the project. If economic agents foresee such preference reversals they are said to be sophisticated and if they do not foresee such preference reversals they are said to be naive (Strotz 1956). O’Donoghue and Rabin (2001) propose a generalized formulation in which agents are partially naive: the agents have an imperfect ability to anticipate their preference reversals.

Many different microfoundations have been proposed to explain the preference patterns captured by the hyperbolic and quasi-hyperbolic discount functions. The most prominent examples include temptation models and dual-brain neuroeconomic models (Bernheim and Rangel 2004; Gul and Pesendorfer 2001; McClure et al. 2004; Thaler and Shefrin 1981). However, both the properties and mechanisms of time preferences remain in dispute.

Individual Differences in Measured Discount Rates

Numerous methods have been used to measure discount functions. The most common technique poses a series of questions, each of which asks the subject to choose between a sooner, smaller reward and a later, larger reward. Usually the sooner, smaller reward is an immediate reward. The sooner and later rewards are denominated in the same goods, typically amounts of money or other items of value. For example: ‘Would you rather have $69 today, or $85 in 91 days?’ The subject’s discount rate is inferred by fitting one or more of the discount functions described in the previous section to the subject choices. Most studies assume that the utility function is linear in consumption. Most studies also assume no intertemporal fungibility – the reward is assumed to be consumed the moment it is received. Many factors may confound the analysis in such studies, leading numerous researchers to express scepticism about the conclusions generated by laboratory studies. Table 1 provides a summary of such critiques.

Intertemporal Choice, Table 1 Potential confounds that may arise in attempts to measure discount rates in laboratory studies
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Discount functions may also be inferred from field behaviour, such as consumption, savings, asset allocation, and voluntary adoption of forced-savings technologies (Angeletos et al. 2001; Shapiro 2005; Ashraf et al. 2006). However, field studies are also vulnerable to methodological critiques. There is currently no methodological gold standard for measuring discount functions.

Existing attempts to measure discount functions have reached seemingly conflicting conclusions (Frederick, Lowenstein and O’Donoghue, Frederick et al. 2003). However, the fact that different methods and samples yield different estimates does not rule out consistent individual differences. Dozens of empirical studies have explored the relationship between individuals’ estimated discount rates and a variety of behaviours and traits. A significant subset of this literature has focused on delay discounting and behaviour in clinical populations, most notably drug users, gamblers, and those with other impulsivity-linked psychiatric disorders (see Reynolds 2006, for a review). Other work has explored the relationship between discounting and traits such as age and cognitive ability. Table 2 summarizes representative studies.

Intertemporal Choice, Table 2 Representative empirical studies linking estimated discount rates for monetary rewards to various individual behaviours and traits
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Smoking

A number of investigations have explored the relationship between cigarette smoking and discounting, together providing strong evidence that cigarette smoking is associated with higher discount rates (Baker et al. 2003; Bickel et al. 1999; Kirby and Petry 2004; Mitchell 1999; Ohmura et al. 2005; Reynolds et al. 2004).

Excessive Alcohol Consumption

While the association with alcoholism has received relatively little attention, the available data suggest that problematic drinking is associated with higher discount rates. Heavy drinkers have higher discount rates than controls (Vuchinich and Simpson 1998), active alcoholics discount rewards more than abstinent alcoholics, who in turn discount at higher rates than controls (Petry 2001a), and detoxified alcohol-dependents have higher discount rates than controls (Bjork et al. 2004).

Illicit Drug Use

Recent studies document a positive association between discount rates and drug use for a variety of illicit drugs, most notably cocaine, crack-cocaine, heroin and amphetamines (Petry 2003; Coffey et al. 2003; Bretteville-Jensen 1999; Kirby and Petry 2004).

Gambling

Pathological gamblers have higher discount rates than controls, both in the laboratory (Petry 2001b) and in a more natural setting (Dixon et al. 2003), and among a population of gambling and non-gambling substance abusers (Petry and Casarella 1999). Moreover, Alessi and Petry (2003) report a significant, positive relationship between a gambling severity measure and the discount rate within a sample of problem gamblers. Petry (2001b) finds that gambling frequency during the previous 3 months correlates positively with discount rate.

Age

Patience appears to increase across the lifespan, with the young showing markedly less patience than middle-aged and older adults (Green et al. 1994; Green et al. 1996; Green et al. 1999). Read and Read (2004) report that older adults (mean age = 75) are the most patient age group when delay horizons are only 1 year. However, this study also finds that older adults are the least patient group when delay horizons are from three to ten years. This reversal probably reflects the fact that 75-year-olds face significant mortality/disability risk at horizons of three to ten years.

Cognitive Ability

Kirby et al. (2005) report that discount rates are correlated negatively with grade point average in two college samples. Benjamin et al. (2006) find an inverse relationship between individual discount rates and standardized (mathematics) test scores for Chilean high school students. Silva and Gross (2004) show that students scoring in the top third of their introductory psychology course have lower discount rates than those scoring in the middle and lower thirds. Frederick (2005) shows that participants scoring high on a ‘cognitive reflection’ problem-solving task demonstrate more patient intertemporal choices (for a variety of rewards) than those scoring low. Finally, in a sample of smokers, Jaroni et al. (2004) report that participants who did not attend college had higher discount rates than those attending at least some college.

All of these empirical regularities are consistent with the neuroeconomic hypothesis that prefrontal cortex is essential for patient (forward-looking) decision-making (McClure et al. 2004). This area of the brain is slow to mature, is critical for general cognitive ability (Chabris 2007), and is often found to be dysfunctional in addictive and other psychiatric disorders.

More research is required to clarify the cognitive and neurobiological bases of intertemporal preferences. Future research should evaluate the usefulness of measured discount functions in predicting real-world economic decisions (Ashraf et al. 2006). Finally, ongoing research should improve the available methods for measuring intertemporal preferences.

See Also