An Introduction to Behavioral Economics

Intertemporal choices relate to decisions involving trade-offs between costs and benefits occurring in different time periods. Governments, firms and individuals are all faced with such decisions on a frequent and ongoing basis, for example: investing in roads, schools and hospitals; building a new factory or launching a new product; buying a new car, spending on a vacation, or joining a health club. Economists have been interested in such decisions since at least the days of Adam Smith, but the current model that is generally used by most economists, as well as by governments and firms, is the Discounted Utility Model (DUM), originally proposed by Samuelson in 1937. The widespread use of the model seems strange to many practitioners in behavioral economics, in view of the fact that Samuelson himself had significant reservations regarding both the normative and descriptive aspects of the model, and that many anomalies have been observed over the last few decades.

Because the topic of intertemporal choice is so broad in scope, the structure in the previous chapters is somewhat modified. Up to this point we have started with the fundamental aspects of the standard model, observed the anomalies and then extended the model, introducing behavioral modifications. In this chapter, however, we will concentrate entirely on describing the DUM (as the relevant aspect of the standard model), and examining its implications and anomalies. Only in the next chapter will we move on to discuss alternative behavioral models. At that point we will see that various factors discussed in previous chapters are relevant in terms of formulating such models.

In this chapter therefore we begin by considering the historical origins of the DUM, before describing the essential features of the model. The discussion of origins is important, since many of the more recent behavioral developments are in many ways a reversion to much earlier work involving psychological and sociological factors. We shall also see in the next chapter that recent theories of intertemporal choice also incorporate elements of theory from evolutionary biology and neuroeconomics. After the features of the DUM are described, its many anomalies are discussed. The nature and causes of these anomalies involve a discussion of the complex concept of time preference, which is deferred until the next chapter. Finally, various policy implications of the DUM are considered.

7.2 Origins of the DUM

Although Adam Smith was the first economist to discuss the importance of intertemporal choice as far as the wealth of nations was concerned, it was Rae who essentially provided a psychological foundation for a theory of intertemporal choice.

John Rae and the desire for accumulation

In the early nineteenth century Rae (1834) identified ‘the effective desire for accumulation’ as being the key psychological factor determining a society’s decisions to save and invest, which in turn determined a country’s rate of economic productivity and growth. Rae also identified four psychological factors that either promoted or inhibited this desire, and which varied across different societies. The two promoting factors were:

1 The bequest motive – to accumulate resources for one’s descendants.

2 The propensity to exercise self-restraint – this involves the intellectual capacity to foresee probable future outcomes of decisions, and the willpower to put long-term interests ahead of short-term ones.

The two inhibiting factors described by Rae were:

1 The uncertainty of human life – there is no point in saving for the future if it is unlikely that we will have much of one. Rae summed this up by saying:

When engaged in safe occupations, and living in healthy countries, men are much more apt to be frugal, than in unhealthy, or hazardous occupations, and in climates pernicious to human life (p. 57).

2 The urge for instant gratification exacerbated by the prospect of immediate consumption – Rae again expresses this in vivid terms:

The actual presence of the immediate object of desire in the mind by exciting the attention, seems to rouse all the faculties, as it were, to fix their view on it, and leads them to a very lively conception of the enjoyments which it offers to their instant possession (p. 120).

Two different approaches

Later theorists developed two different views of time preference stemming from Rae’s work. One view took the approach that the default situation was that people weighted the present and future equally, but the discomfort of delaying gratification caused people to weigh future outcomes less heavily than present ones (Senior, 1836). The second view took the opposite approach, proposing that people generally only considered immediate utility, but the anticipation of future utility might on occasion more than offset any loss of current utility, causing them to delay gratification (William Jevons, 1888; Herbert Jevons, 1905). Both approaches emphasize the importance of current feelings, but explain variations in time preference between people in different ways. According to the first approach people vary in terms of the discomfort they experience in delaying gratification. According to the second approach variations in time preference arise because people have different abilities in anticipating the future.

Böhm-Bawerk and trade-offs

The next major development in the theory of intertemporal choice came from Böhm-Bawerk (1889). He, and later Pigou (1920), introduced the notion that people generally underestimate future wants, leading to a time preference pattern biased towards the present. It should be noted at this stage that this notion does not involve a discounting of future outcomes; instead it is the utility of these outcomes that is underestimated. This is a key distinction that will be discussed in more detail later in the chapter.

Böhm-Bawerk introduced another important innovation in the theory of intertemporal choice: such choices were seen as trade-offs in terms of allocating resources to oneself in different periods of time, similar to trade-offs in allocating resources between consuming different current goods.

Irving Fisher and indifference curve analysis

The final development before the introduction of the discounted utility model came from Irving Fisher (1930). He formalized much of the work outlined above, extending the Böhm-Bawerk framework of analysis in terms of using indifference curves to illustrate the relevant trade-offs. Current consumption was plotted on the horizontal axis and future consumption (usually for the following year) was plotted on the vertical axis. The concept of the marginal rate of substitution between current and future consumption was also applied; this depended both on time preference and diminishing marginal utility.

It should also be emphasized that Fisher discussed at length the psychological factors affecting time preference. Thus he not only took account of future wealth and risk but he also referred to the four factors described by Rae, and to foresight, which is the other side of the coin of Böhm-Bawerk’s notion of the underestimation of future wants. Fisher also stressed the importance of fashion:

This at the present time acts, on the one hand, to stimulate men to save and become millionaires, and, on the other hand, to stimulate millionaires to live in an ostentatious manner (p. 87).

The reason for underlining the fact that these early pre-DUM developments in the theory of intertemporal choice all took different psychological factors into account is that the DUM itself swept away the explicit consideration of these psychological factors, condensing them into a single construct: the discount rate. The implications of this ‘revolution’ need to be discussed at some length in the remainder of this chapter.

Samuelson and the DUM

Samuelson introduced the DUM in 1937 in a short article modestly titled: ‘A note on measurement of utility’. Apart from commenting that the comparison of intertemporal trade-offs required a cardinal as opposed to ordinal measure of utility, it extended Fisher’s indifference curve analysis, essentially limited to the comparison of two periods, to multi-period situations. However, as noted above, it combined all the psychological factors involved in time preference into the single parameter of the discount rate. The nature of the model can be best described in mathematical terms. It specifies an intertemporal utility function U_t(c_t,...,c_T) which describes the utility at time t of the consumption profile (c_t, c_t₊₁, c_t₊₂,…) starting in period t and continuing to period T. The model incorporates the general axioms of the standard model described in previous chapters regarding the completeness, transitivity and independence principles. The DUM then goes on to assume that a person’s intertemporal utility function can be described by the following specific functional form:

This is essentially a similar expression to (1.1), where the second component is expanded. The term u(c_t_+k) can be interpreted as the person’s instantaneous utility function, meaning their perceived wellbeing in period t + k. The term D(k) refers to the person’s discount function, meaning the relative weight that the person attaches in time period t to their wellbeing in period t + k. Finally, the term ρ refers to the person’s discount rate, meaning the rate at which they discount expected future utilities. This term therefore combines all the various psychological factors involved in time preference that were discussed earlier.

It will aid an understanding of the workings of the model, as well as the implications and anomalies discussed later, if a simplified example is examined at this stage. Let us take the following consumption profile, measured in thousands of dollars per year over a period of the next three years: (20, 20, 20). We will also assume for simplicity that these consecutive equal amounts of consumption yield equal amounts of utility. It should be noted that this assumption violates some of the behavioral factors considered in earlier chapters, such as the effects of reference points, habit formation and the desire for increasing consumption profiles (the ‘happiness treadmill’). These complicating factors will be discussed later in terms of their implications for the DUM.

Since the measurement of utility is in arbitrary units of ‘utils’, we can consider the utility profile as consisting of the terms (20, 20, 20). It is assumed that the model is discrete and that the utilities are all received at points in time at the end of each period, rather than being continuous flows throughout each period. This may appear unrealistic but the DUM can be modified to transform it into a continuous-time model; the utility function becomes an integral of a negative exponential function. For simplicity we will use the discrete form of the model. If the consumer discounts future utility at the rate of 10% per year the current utility of the consumption and utility profiles can now be calculated as follows:

(7.1)

This calculation illustrates an important feature of the DUM: it closely resembles the compound interest formula used in calculating net present values. This analogue with the common technique for evaluating financial investments has been largely responsible for the rapid assimilation of the model by economists, particularly after Koopmans (1960) showed that the model could be derived from a set of axioms or basic principles that are superficially plausible. The model also gained normative status, as it was shown that an exponential discounting function was consistent with rationality, in the sense that it resulted in dynamically consistent choices. Other economists, including Lancaster (1963), Fishburn (1970) and Meyer (1976), have since provided alternative axiom systems for the DUM, further increasing its perceived legitimacy and popularity.

7.3 Features of the DUM

There are a number of features of the DUM that now need to be examined, since these relate to the implicit psychological assumptions underlying the model. These features have been analyzed and their validity critically reviewed by Frederick, Loewenstein and O’Donoghue (2002), and the following discussion draws extensively from that review.

Integration of new alternatives with existing plans

It is a common assumption in most decision or choice situations relating to the standard model that people evaluate new alternatives by integrating them with existing plans. This means that if a person is offered a prospect A (say, investing $10,000 now to gain $15,000 in three years), the effects of this prospect on the person’s whole consumption profile must be considered. Thus, if the person currently has the consumption profile (c_t,...,c_T), then they must estimate their new consumption profile if they were to accept prospect A, for example (c′_t,...,c′_T). Prospect A would then be accepted if U_t(c′_t,...,c′_T) > U_t(c_t,...,c_T).

Although this approach may seem logical from a normative point of view, meaning that people should want to integrate new alternatives in order to maximize welfare, we have already seen in previous chapters that this places unrealistic demands on people’s mental capacities. People may not have well-formed consumption profiles relating to all future periods, and they may be unable or unwilling to reformulate such plans every time that new prospects are encountered. The next section relating to anomalies in the DUM will examine empirical evidence regarding people’s behavior in this regard.

Utility independence

It is assumed in the DUM that it is simply the sum of all the discounted future utilities, for example, the value of 49.74 in (7.1), which is relevant in terms of making intertemporal choices. This ignores the possibility that the distribution of utilities over time may be relevant. As we shall see, people may prefer a flat or rising utility profile to a falling one, or a pattern of dispersion of utilities rather than a concentrated pattern. This assumption is also related to the next one.

Consumption independence

It is assumed in the DUM that a person’s welfare in any time period is independent of consumption in any other period. This means that preferences over consumption profiles are not affected by the nature of consumption in periods in which consumption is identical in the two profiles. It is, therefore, analogous to the independence axiom in EUT. As Frederick, Loewenstein and O’Donoghue (2002) state:

Consumption independence says that one’s preference between an Italian and Thai restaurant tonight should not depend on whether one had Italian last night nor whether one expects to have it tomorrow (p. 357).

It should be noted that neither Samuelson nor Koopmans proposed that this assumption had either normative or descriptive validity. Indeed we will review a number of empirical anomalies in the remainder of this chapter and the next.

Stationary instantaneous utility

The DUM generally assumes that the instantaneous utility function is constant over time, meaning that the same activity yields the same utility in the future, viewed from the future, as now. For example, if activity A is expected to yield 20 utils in three years time, the current utility of A, which can be written as u⁰(A), will be discounted to about 15 utils (at a 10% discount rate). However, it is assumed that the utility of A in three years time, u³(A), will still be 20 utils. This implies that people’s preferences do not change over time, an obviously unrealistic assumption.

Evidence suggests that people tend to exaggerate the degree to which their future preferences will resemble their current ones, a phenomenon referred to as projection bias (Loewenstein, O’Donoghue and Rabin, 2003). Thus they may expect to like the music of Oasis in 20 years because they like it now, but then find in 20 years time that they can’t stand it.

Stationary discounting

It is assumed in the DUM that people use the same discount rate over their lifespan. In contrast, there is considerable evidence that discounting rates vary according to age. Mischel and Metzner (1962) have found that willingness to delay gratification increases with age, implying that older people have a lower discount rate. However, the relationship between discounting and age appears to be a complex one. In an experimental study of respondents between the ages of 19 and 89 it was found that older people discount more than younger ones, and that middle-aged people discount less than either group (Read and Read, 2004). Similar results were found by Harrison, Lau and Williams (2002). It would appear, therefore, that young people are more impatient than their parents, but discount less than their grandparents, whose future involves a greater degree of uncertainty.

Constant discounting

The DUM assumes that at any period of time the same discount rate is applied to all future periods. In mathematical terms this means that, given the discount function

D(k) = (1 / 1+ ρ)^k

at time period t the same per-period discount rate ρ is applied to all periods in the future. This condition ensures time-consistent preferences; since the standard model views time-consistent preferences as being rational, and inconsistent preferences are usually seen as irrational, this feature of the DUM is often perceived to be legitimate. However, there is plentiful empirical evidence that discount rates are not constant over time, but rather tend to decline. Inconsistent time preferences have been widely observed. This evidence, and alternative models of accommodating it, is reviewed in the next chapter.

It should also be noted at this stage that constant discounting is not a sufficient condition for consistent time preferences to exist. Stationary discounting is also necessary. If discounting is not stationary, then in the following time period t + 1 one may observe the different, but still constant, discount function

D_{t +}₁(k) = (1 / 1+ ρ′)^k

where ρ′ ≠ ρ′. For example, the evidence in the Read and Read study mentioned above suggests that, as people enter middle age, ρ′ < ρ. In this situation discounting is constant in both periods t and t + 1, but the discount rate changes with the passage of time. The consequence of this is that once again time preferences will be inconsistent.

Independence of discounting from consumption

Another assumption of the DUM is that all forms of consumption are discounted at the same rate. Without this assumption it is impossible to reduce the discount rate to a single parameter or to talk about a uniform time preference. As Frederick, Loewenstein and O’Donoghue (2002) put it:

We would need to label time preference according to the object being delayed – ‘banana time preference’, ‘vacation time preference’, and so on (p. 358).

There is indeed evidence that different products are discounted at different rates, and even that different attributes of products are discounted differently. For example, Soman (2004) has found that perceived effort and perceived price are discounted differently. Do-it-yourself products involve effort in terms of consumption and appear more attractive when the purchase is planned for some time in the future. When purchase is immediate such products appear less attractive, implying that perceived effort is discounted more heavily than price. Once again the consequence is preferences that are inconsistent over time.

Diminishing marginal utility and positive time preference

Although these two features are not essential elements of the DUM, in practice most analyses involving intertemporal choice assume that both conditions exist. In actuality both elements predate the DUM, since Fisher (1930) emphasized the importance of each of them in his indifference analysis approach. Indifference curves generally require the assumption of diminishing marginal utility or concave utility functions. The implication of this condition is that it might cause people to delay consumption until later time periods. For example, when we eat a delicious and generously portioned meal we may prefer to leave some of it for the next day, or, if at a restaurant, take the remaining portion home in a doggy bag.

It should be noted that this effect of diminishing marginal utility operates in the opposite direction to the normal direction of time preference. Normally time preference is positive, meaning that people apply a positive discount rate to future utilities. For this reason many economists have not been happy with Fisher’s approach, on the grounds that it confounds the effect of diminishing marginal utility with ‘pure’ time preference. However, we shall see in a later section that it is very difficult to define the term ‘pure’ time preference, since there are in reality a number of confounding factors, not just the phenomenon of diminishing marginal utility.

Evolutionary biology and the DUM

Although the economic model of the DUM was developed without any reference to evolutionary biology, it is worthwhile considering evolutionary aspects at this stage, first modeled by Rogers (1994). Evolutionary theory suggests a sound reason why all animals should discount future outcomes. To consider this we need to recall that biological or inclusive fitness operates at the level of the gene rather than the individual. Mothers prefer consuming now, to increase their own offspring, to saving and passing an equal amount of consumption to their daughters later, to increase the daughters’ offspring. This is because only half of one’s genes are passed on to one’s offspring, so a mother is more closely related to her daughters than to her granddaughters (twice as closely, in fact). This is the biological basis for impatience or discounting the future. However, if the return for saving is high enough, passing on consumption to the future would pay, since the increase in the number of grandchildren would more than compensate for the fact that they were less closely related. Rogers (1994) estimated the rate of time preference based on this simple evolutionary model to be about 2% per year. This estimate and the methodology involved have been questioned by other studies, but a discussion of this aspect is best left until the next chapter.

A further evolutionary factor underlying time preference and discounting is that individuals who have children earlier are then able to have more offspring in total, an evolutionary advantage, so that such individuals would come to dominate the population over time (Robson, 2002). However, as with the previous factor, the theory is complicated by the fact that the relationship between consumption and number of offspring is an indirect one. For example, it may be better for an individual to delay having offspring by consuming more now to build greater body size first.

Evolutionary theory also has something to say about discount rates in the DUM, which are assumed to be both stationary and constant. Chu, Chien and Lee (2010) propose that nonstationary discounting is supported by evolutionary theory, involving a U-shaped discounting function over time, with both younger and older people having higher discount rates than middle-aged people. This conforms to the findings of Read and Read (2004) and Harrison, Lau and Williams (2002) described earlier.

Basic evolutionary models suggest that people should have time-consistent preferences. The argument is as follows: if maximizing lifetime fitness involves a particular intertemporal trade-off between outcomes over two dates in the future, as time moves on and these dates become closer, Nature should choose current preferences so that this trade-off is unaffected (Robson, 2002). Thus, if initially we prefer to have offspring three years in the future rather than two, then in one year’s time our preference will remain the same as far as deferring having offspring is concerned, i.e. we will prefer having offspring in two year’s time rather than one. However, once again, if we consider additional factors relating to the basic model, the implications are that it may be more appropriate to use a nonconstant discounting function, such as the hyperbolic discounting model, rather than the exponential one of the DUM. These additional factors are examined in the next chapter.

7.4 Methodology

Before discussing anomalies in the DUM, it is necessary to consider the methodology used by investigators in empirical studies that measure discount rates. Many aspects of the methodology described also apply to other areas of research, but this discussion is particularly important in the current context, given the enormous variation in results reported: studies have computed discount rates varying from minus 6% (delayed rewards are preferred) to infinitely positive. An understanding of the different methods employed by researchers will in turn aid an understanding of the sources of these variations, as well as the anomalies and alternative models to the DUM described later. Some general aspects of methodology have already been discussed in Chapter 2, particularly relating to the nature of different types of empirical study; the more specific aspects that need to be explained at this stage concern the elicitation of relevant information regarding time preference, and the calculation of the discount rate.

Elicitation of information

Four experimental procedures tend to be used: choice tasks, matching tasks, rating tasks and pricing tasks.

1 Choice tasks

This is the most common experimental method for eliciting discount rates. Subjects are typically asked to choose between a smaller, more immediate reward and a larger, more distant reward. The obvious disadvantage of this technique is that it only provides a lower or upper limit to the discount rate. For example, if people prefer a sum of $100 now to $110 in one year this merely tells us that their discount rate is at least 10%. In order to yield more precise results investigators have to offer a series of many different choices, so that if people prefer $100 now, they are then asked if they prefer $120 in one year, and so on. Thus usually a bracket can be calculated for each subject. Apart from the complications and length involved, another problem with the technique is that it is subject to an anchoring effect, discussed in Chapter 3. Subjects tend to be influenced unduly by the first option offered. One way in which this problem can be countered is by using titration procedures. These present subjects with responses that are successively at opposite ends of a spectrum. For example, a first question might ask subjects to choose between $100 now and $101 in one year, while the next question asks them to choose between $100 now and $1000 in one year; then the next question might compare $100 now with $102 in a year, followed by $100 now and $500 in a year, as the procedure gradually narrows down to a central range.

2 Matching tasks

With this method subjects give open-ended responses which equate two intertemporal options. For example, they may be asked how much money they would require in one year to be equivalent to receiving $100 now. Such a technique has two advantages over the procedure of choice tasks. First, an exact discount rate can be computed from a single response, without the need for asking a series of questions. Second, there is no anchoring problem, since the responses are open-ended.

However, there are various problems that arise with the matching task procedure, which can lead to misleading results. One prominent problem is that subjects tend to use crude rules for making judgments. For example, it is quite common for people to use a ‘times n’ rule for generating future reward equivalents, meaning that they multiply the present amount by the number of years to obtain an equivalent delayed reward. Thus subjects may state that $100 now is equivalent to $500 in five years time. Such a rule does not necessarily mean that subjects are responding in ways that do not correspond to real-life behavior, but this suspicion is confirmed by a second problem, that results are highly inconsistent according to the method of matching used. When subjects are asked to state the future reward that is equivalent to a specified current reward they tend to use a much higher discount rate than when they are asked to state the current reward that is equivalent to a specified future reward. For example, Frederick and Read (2002) have found that when subjects were asked to state the amount that would be equivalent in 30 years to $100 now, the median response was $10,000, implying a discount factor over the 30-year period of 0.01. However, when subjects were asked to state the amount now that would be equivalent to receiving $100 in 30 years the median response was $50, implying a discount factor over the same period of only 0.5.

The same kind of inconsistency has been found when questions have been framed in terms of matching amount of reward and time of delay. For example, a study by Roelofsma (1994) asked one group of subjects to state the amount of money they would require to compensate for delaying the delivery of a purchased bicycle by nine months, obtaining a median response of 250 florins. However, when another group of subjects was asked how long they would be willing to delay delivery of the bicycle in return for the compensation of 250 florins, the mean response was only three weeks, implying a discount rate 12 times as high.

These highly inconsistent results obtained with matching procedures shed doubt on the reliability of the procedure in general, while simultaneously demanding some kind of psychological explanation. As with other cases of inconsistency and preference reversal one of the consequences is that normative models of behavior become problematical to develop. This aspect is discussed in more detail in the next chapter in the section related to policy implications.

3 Rating tasks

With this procedure subjects are asked to rate outcomes in different time periods in terms of either like or dislike. Thus an ordinal scale is used as opposed to a cardinal and ratio-based scale. The advantage is that this imposes less of a computational strain on subjects, but the disadvantage is that results are less well differentiated. For example, if a person rates $150 in one year as being preferable to $100 now, it is not clear if the difference in preference is large or small.

4 Pricing tasks

This procedure involves eliciting a willingness to pay (WTP) by respondents in order to receive or avoid a particular outcome in the future, expressed in either monetary or non-monetary terms (like an extra year of life or a specified amount of pain). Like rating tasks, this procedure allows the manipulation of time between subjects, since each individual may evaluate either the immediate or delayed outcome in isolation.

Methodological issues

It can be seen from Case 7.1 that even experimental studies have produced a wide variation in reported discount rates. This is because, although controls have been used, different controls have been used in different studies. To give a flavor of this we will take one of the most controlled studies, performed by Harrison, Lau and Williams (2002), and examine the principles underlying its experimental design and the issues raised. They asked a representative sample of 268 people from the Danish population the basic question: do you prefer $100 today or $100 + x tomorrow? The format of their experiment modified this question in six ways:

1 Use of choice tasks

Twenty different amounts were offered as the B option to compare with a standard A option in order to estimate narrow boundaries for an individual’s discount rate.

2 Random payments used as incentives

Subjects were simultaneously asked several questions with varying amounts of x, with one question being selected at random for actual payment after the end of the experiment. The objective was to avoid income effects that might affect the answers to later questions. This is a standard mechanism among researchers for ensuring participation, motivation and reliable responses. The main problem with this method is that it does not allow for learning effects to take place (Starmer, 2000).

3 Use of ‘front-end delay’

Both early and later options were in the future. Thus, instead of asking, for example, whether subjects preferred $100 now or $100 + x in 6 months, the investigators asked whether subjects preferred $100 in 1 month or $100 + x in 7 months. This avoids the problem, mentioned in Chapter 2, that there could be a confound between the effects of ‘pure’ time preference and transactions costs in the future. Receiving money in the future always involves some transactions cost, as it has to be collected in some way and there is a possibility of the experimenter defaulting. If both options involve future payment this effect is controlled for. This is an important control device, which has not been used in many studies. Comment on the results is given later.

4 Use of different time horizons

Four time horizons were used: 6 months, 12 months, 24 months and 36 months. Some subjects were randomly assigned to answer questions relating to a single time period while others were asked questions relating to all periods. Using both methods allows the evaluation of the effect of the subjects’ consideration of multiple time periods.

5 Information relating to the subjects’ market rate of interest is elicited

The objective here was to examine the possibility that subjects’ discount rates were censored by market rates. This means that the interest rates reported by subjects do not reflect ‘pure’ time preference, but are determined by the market rates they face. For example, if someone can borrow at 6% they might prefer $105 in one year to $100 now, since borrowing the $100 would be more expensive.

6 Provision of the annual interest rates associated with delayed payment

This facilitates comparisons between the options in the experiment and external or market options.

The format of this experimental design does not permit the testing of all the effects discussed in Section 7.5 on anomalies, for example, the ‘sign effect’ and the ‘magnitude effect’, but it does succeed in controlling for some important problems relating to misconceptions and confounds. However, there is one issue that merits some further discussion here. The authors report that:

Our results indicate that nominal discount rates are constant over the one-year to three-year horizons used in these experiments … (p. 1606)

Some commentators have concluded from this that the use of front-end delay, by eliminating the confound related to transactions cost, provides supporting evidence in favour of the constant discounting characteristic of the DUM and against hyperbolic discounting. The issue of hyperbolic discounting is discussed in the next chapter, but the merits of this claim need some clarification. Closer inspection of the study’s results indicates that the 6-month discount rate was actually 34.9%, the 12-month rate was 29.0%, and the 24-month and 36-month rates were 27.4% and 27.9%. Thus the claim that a constant discounting rate applies throughout the whole period is misleading. What would be revealing here is the use of a shorter front-end delay, of a day or a week, and an estimation of discount rates for periods shorter than 6 months. This kind of empirical evidence would facilitate the evaluation of the constant discounting claim.

Calculating discount rates

It is also a worthwhile exercise at this point to review the mathematics behind the calculation of discount rates, since different methods can be used here. The simplest method involves the standard compounding formula used in net present value and internal rate of return calculations in evaluating investment decisions. This formula can be represented below:

F = P(1 + r)ⁿ

(7.2)

where F = future value, P = present value, r = discount rate and n = number of time periods. For example, if subjects match the value of $500 in 10 years to $100 now, the method above would calculate the discount rate as follows:

500 = 100(1 + r)¹⁰

This yields a discount rate of about 17% a year.

However, most empirical studies do not use the above method of calculation because it involves discrete compounding, usually with yearly periods, as in the above example. Instead it is usually preferred to use continuous compounding. This involves using an exponential function:

F = Pe^nr

(7.3)

This formula can be manipulated to solve for the value of r as follows:

(7.4)

If the same values in the discrete example above are substituted into the exponential function, the implied discount rate is 16%. When continuous compounding is used the implied discount rates are always lower than with discrete compounding, and the difference between the two methods becomes greater as discount rates increase. For example, if the future equivalent in the above situation was $1000 instead of $100 (as would happen if a subject was using the ‘times n’ rule), then discrete compounding would yield a discount rate of 26%, while continuous compounding yields 23%.

Discount rates can also be calculated on either an average or a marginal basis. For example, subjects might be indifferent between receiving $100 now, receiving $120 in one year, and receiving $160 in five years. Using continuous compounding this implies a discount rate of 18.2% over the next year (between year 0 and year 1), and an average discount rate of 7.8% per year over the next five years (between year 0 and year 5). However, it may be useful in some situations to consider the marginal discount rate per period. In the situation above the marginal discount rate between year 1 and year 5 is 7.2% per year.

7.5 Anomalies in the DUM

Many anomalies in the DUM have already been noted, in terms of the features of the model described earlier, and the manner in which such features do not appear to be confirmed by empirical evidence. Many other anomalies arise due to factors discussed in the previous two chapters, relating to prospect theory (PT) and mental accounting. These further anomalies relate to: the ‘sign effect’; the ‘magnitude effect’; the ‘delay-speedup’ asymmetry; the ‘date/delay effect’; preference for improving sequences; and violations of independence and preference for spread. These factors are now discussed in turn.

The ‘sign effect’

This effect means that gains are discounted more than losses, as proposed by PT. For example, a study by Thaler (1981) asked subjects how much they would be willing to pay for a traffic ticket if payment could be delayed for periods of three months, a year or three years. The responses indicated that people used much lower discount rates than in situations where monetary gains were involved. This study was an experimental and hypothetical one, but in this situation it should be practical to conduct a field study, since local government authorities maintain records of fines and dates of payments. In many cases the implied discount rate used by such authorities is very large, with fines often doubling in a matter of a few weeks. The main problem with analyzing such a study would be the existence of confounding factors discussed in the previous section. In this case people may forget to pay, or delay payment in the hope of evading the fine altogether (if authorities fail to follow up on all tickets issued).

At the extreme end of the loss-discounting spectrum, there are several studies that indicate that many people prefer to incur a loss immediately rather than delay it (Mischel, Grusec and Masters, 1969; Yates and Watts, 1975; Loewenstein, 1987; Benzion, Rapoport and Yagil, 1989; MacKeigan et al., 1993; Redelmeier and Heller, 1993). This implies a zero discount rate for losses.

The ‘sign effect’ may be accounted for by a phenomenon referred to as ‘temporal loss aversion’ (Bilgin and LeBoeuf, 2010). In a series of experiments these investigators found that intervals preceding losses seem shorter than intervals preceding gains, and that this effect is driven by perceptions of the quality of the interval end point rather than by the quality of the interval itself. For example, for a person moving jobs to a new city in a couple of months’ time, if that person is not looking forward to the move, the interval may appear to be shorter than it would for the person who is looking forward to the move. If the interval is perceived to be shorter, then a smaller discount rate would be used. This temporal loss-aversion may occur because the subjective size of the effect is greater, or because losses may attract more attention than gains. There are other psychological factors related to this kind of loss-aversion. Anticipatory utility, or in the case of losses, disutility, is important. As we have seen in Chapter 3, people do not like the idea of a loss ‘hanging over’ them, and may prefer to endure the pain of the loss immediately and ‘get it over and done with’. Thus people are motivated to maximize savoring events that they look forward to and minimize the dread associated with unpleasant events in the future.

Bilgin and LeBoeuf (2010) venture an evolutionary explanation for this phenomenon: ‘when danger is imminent, it is likely more adaptive to err in the direction of exaggerating the proximity of the danger because perceiving dangers as temporally near may galvanize necessary coping resources’ (p. 528). This is particularly true for unexpected dangers, Bilgin and LeBoeuf claim, and one of their experiments found that unexpected losses loomed nearer than expected ones.

The ‘magnitude effect’

Studies that vary outcome size often find that large outcomes are discounted at a lower rate than small ones (Thaler, 1981; Ainslie and Haendel, 1983; Loewenstein, 1987; Benzion, Rapoport and Yagil, 1989; Holcomb and Nelson, 1992; Green, Fry and Myerson, 1994; Kirby, Petry and Bickel, 1999). For example, in Thaler’s study subjects were indifferent between $15 immediately and $60 in a year, $250 immediately and $350 in a year, and $3000 immediately and $4000 in a year. These matching preferences indicated discount rates of 139%, 34% and 29% respectively.

This phenomenon requires a psychological explanation, which appears to be lacking in the literature where the effect has been observed. It should be noted that the effect works in the opposite direction to the effect of diminishing marginal utility. The discount rates calculated for Thaler’s study above are based on the monetary values, not the actual utilities. If utilities were used to calculate discount rates instead of monetary values, then, assuming the law of diminishing marginal utility applies, the differences in discount rates between small and large amounts would be even greater.

The ‘delay-speedup’ asymmetry

Studies have also investigated the effect of changing the delivery time of outcomes. These changes can be framed either as delays or accelerations from some reference point in time. For example, Loewenstein (1988) has found that subjects who didn’t expect to receive a VCR for another year would pay an average of $54 to receive it immediately (a perceived gain). However, those subjects who thought they would receive the VCR immediately demanded an average of $126 to delay its receipt by a year (a perceived loss). Other studies have confirmed these findings in situations where the outcomes involved payments, i.e. negative outcomes, rather than positive ones like the delivery of a product. In these situations subjects demand more to accelerate payment (a perceived loss) than to delay it (a perceived gain).

We can note that these results are predicted by PT. Two elements of the theory are involved: reference points and loss-aversion.

Preference for improving sequences

The DUM predicts that, total undiscounted utility being equal, people will prefer a declining sequence of outcomes to an increasing sequence, since later outcomes are discounted more heavily. Thus, given the two consumption profiles (50, 60, 70) and (70, 60, 50) over three consecutive time periods, the DUM predicts that people will prefer the latter to the former. In contrast, many studies have shown that people prefer improving profiles. For example, Loewenstein and Sicherman (1991) have found that, for an otherwise identical job, most subjects prefer an increasing wage profile to a declining or flat one. Hsee, Abelson and Salovey (1991) have found that an increasing salary sequence was rated as highly as a decreasing sequence that conferred a much larger monetary amount. In addition to these studies involving gains, some investigators have examined sequences of losses. Varey and Kahneman (1992) have found that subjects strongly preferred a sequence of decreasing discomfort to a sequence of increasing discomfort, even when the overall sum of discomfort over the interval was otherwise identical. Chapman (2000) investigated the responses of people to hypothetical sequences of headache pain, with the sequences being matched in terms of total pain; durations varied from one hour to as long as twenty years. For all durations, between 82% and 92% of the subjects preferred sequences of pain that were declining rather than increasing. Thus in general, for both gains and losses, people prefer an improving sequence to a sequence where outcomes are deteriorating.

These findings are in line with various effects reported in Chapter 3, for example expectations effects and anticipatory utility, and reference points. In particular it is worth recalling the colonoscopy studies of Katz, Redelmeier and Kahneman (1997), where remembered utility, which in turn may determine decision utility, may differ from real-time utility. Remembered utility, or disutility, may depend more on utility at the end of the sequence rather than at the start, as predicted by the Peak-End rule.

The ‘date/delay effect’

While all the effects described above involve objective changes and manipulations of outcomes and time periods, recent research has also indicated that people may make different intertemporal choices when logically identical situations are presented or framed in different ways. Studies by Read et al. (2005) and LeBoeuf (2006) find that people use lower discount rates in situations when time periods are described using end dates than when the same time periods are described as extents. For example, the LeBoeuf study asked subjects the following two questions (on 15 February):

1 How much money would you want to receive in 8 months to be equivalent to receiving $100 now?

2 How much money would you want to receive on 15 October to be equivalent to receiving $100 now?

Although the two questions are equivalent in logical terms, it was found that people generally demanded a larger amount in answering the first question, implying a higher discount rate when a future gain is framed in terms of extent. Similar results are reported when outcomes are expressed as losses rather than gains, and in a number of experiments when the time intervals until outcomes occur vary between two months and two years.

LeBoeuf proposes a psychological explanation for this phenomenon, utilizing aspects of both PT and mental accounting:

… When consumers consider an interval demarcated by a date, the date may be construed as relatively abstract point in time, and consumers may not even compute the interval’s length. In contrast, when that same interval is described as an extent, by definition, the amount of time is highlighted … Thus consumers may attend more to interval length when facing an extent (2006, p. 61).

If the interval length is perceived as being longer it follows that a higher discount rate will be used. This finding is supported by Ebert and Prelec (2007), who suggest that people are insufficiently sensitive to future time, and that the sensitivity they do have is easily influenced. There are various policy implications of this finding, and these are discussed in the next chapter.

Violations of independence and preference for spread

As mentioned in the previous section, the assumption of consumption independence implies that preferences between two consumption profiles should not be affected by the nature of consumption in periods in which consumption is identical in the two profiles. Again, there is evidence that suggests that consumers may respond differently in situations that are logically equivalent. For example, Loewenstein and Prelec (1993) have found that when people are given a ‘simple’ choice: (A) dinner at a fancy French restaurant next weekend or (B) dinner at the same restaurant on a weekend two weeks later, most people prefer the first option, the sooner dinner. This is predicted by the DUM, as people discount the utility of the later event more heavily. However, the investigators observed different results when subjects are offered an ‘elaborated’ choice: (C) dinner at a fancy French restaurant next weekend and dinner at home on a weekend two weeks later, or (D) dinner at home next weekend and dinner at the French restaurant on a weekend in two weeks. In this decision situation most people prefer the second option. Since the most likely event for subjects is to have dinner at home, the ‘elaborated’ options (C) and (D) amount to the same as the ‘simple’ options (A) and (B). Thus there appears to be a framing effect, causing preference reversal.

The psychological explanation here appears to be that the ‘elaborated’ options draw attention to the sequence of outcomes over time, and, as seen above, people prefer an improving sequence.

Loewenstein and Prelec (1993) also observe a preference for spreading outcomes, again a violation of independence. An experiment involved subjects hypothetically receiving two coupons for fancy restaurant dinners, and asking them when they would use them. Two different conditions were used, one involving a 2-year time limit and the other involving no time limits. Results indicated that subjects scheduled the two dinners later with the 2-year limit (8 weeks and 31 weeks) than when there were no limits (3 weeks and 13 weeks). It appears that framing the offer with a 2-year limit causes the subjects to spread their outcomes over a longer period if the 2-year horizon is longer than the implicit time horizon used by subjects who face no time limits. This is another example of an anchoring effect.

There is also evidence that people have a preference for spreading incomes as well as consumption, a phenomenon referred to as ‘income smoothing’. A natural experiment occurs in California, where about a half of the United School Districts give teachers the choice of receiving their annual salaries in 10 or 12 monthly payments. The DUM would predict that teachers should choose 10 payments and earn interest on their savings. However, in reality about 50% of the teachers choose 12 instalments, even though over a long period of time the interest foregone is considerable (Mayer and Russell, 2005). The psychological explanation, supported by a survey of the teachers, is that the 12-payment option, by spreading the receipt of incomes, facilitates self-control over spending.

Implications of anomalies

With other aspects of the standard model, violations of the model, for example preference reversals, are often perceived by subjects as being mistakes once they are pointed out. However, this is frequently not the case with the anomalies described above. A few examples will illustrate this important point:

1 The ‘sign effect’

When within-subjects studies are compared with between-subjects studies, the size of the sign effect is greater with the first type of study. Thus, when subjects are exposed to both gains and losses the disparity in discount rates is greater than when subjects are only exposed to either gains or losses but not both. If the ‘sign effect’ were regarded as a ‘mistake’ by subjects, one would expect the disparity to be smaller or non-existent in within-subjects studies where subjects could directly compare their responses to losses and gains.

2 The ‘magnitude effect’

The evidence here is similar to that relating to the ‘sign effect’. Frederick and Read (2002) have found that when subjects evaluate both small and large amounts the disparity in discount rates is greater than when they evaluate only small or large amounts. Once again this runs contrary to what one would expect if the anomaly was regarded as a ‘mistake’, since the comparison of the small and large amounts would provide an anchoring effect.

3 Preference for improving sequences

In the study by Loewenstein and Sicherman (1991) it was explained to subjects that a decreasing wage profile ($27,000, $26,000…$23,000) would enable them to consume more in every period than the corresponding increasing wage profile ($23,000, $24,000…$27,000) with the same nominal total. However, this did not affect the behavior of the subjects, who continued to prefer the increasing profile.

The significance of this behavior is that, at least in many situations, people do not regard the anomalies described as being mistakes, or errors of judgment. Actually, this should not be surprising, since at the outset the DUM was not presented as either a valid descriptive or normative model. Unfortunately, as it has gained rapid and widespread acceptance by economists and other practitioners, the DUM has covertly gained a reputation in terms of descriptive and normative validity.

Therefore, after reviewing the DUM and its shortcomings, the task is now to examine other models of intertemporal choice which have better descriptive and normative validity. This is the subject of the next chapter.

7.6 Summary

Early scholarship relating to intertemporal choice emphasized the importance of psychological factors in determining preferences.

The DUM compressed the influences of all factors affecting time preference into a single parameter, the discount rate.

The originator of the DUM, Samuelson, never intended the model to be either normative or descriptive in terms of validity.

The DUM has nine primary features: integration of new alternatives with existing plans; utility independence; consumption independence; stationary instantaneous utility; stationary discounting; constant discounting; independence of discounting from consumption; diminishing marginal utility; and positive time preference.

Empirical research may involve either field studies or experimental studies. The latter may involve either real or hypothetical rewards/losses. Either type of study may be between-subjects or within-subjects.

Experimental studies may involve choice tasks, matching tasks, pricing tasks and rating tasks. The first two techniques are most common.

Discount rates can be calculated on either a discrete or continuous compounding basis. The latter method, involving an exponential function, is more common.

There are a large number of anomalies in the DUM: the ‘sign effect’; the ‘magnitude effect’; the ‘delay-speedup’ asymmetry; preference for improving sequences; the ‘date/delay effect’; and violations of independence and preference for spread.

The anomalies in the DUM should not be regarded as mistakes or errors in judgement. Instead they imply that the model lacks both descriptive and normative validity.

7.7 Review questions

1 Explain what is meant by consumption independence, giving an example.

2 Explain the difference between stationary discounting and constant discounting.

3 Describe the method of using choice tasks to elicit information, discussing any problems and how they may be overcome.

4 Describe the method of using matching tasks, discussing any problems that tend to arise in practice.

5 Explain what is meant by ‘front-end delay’, and the advantage of using this experimental method.

6 Explain what is meant by a negative discount rate; under what circumstances might people have a negative discount rate?

7 Explain what is meant by ‘delay-speedup asymmetry’, and how it is related to PT.

8 Explain what is meant by a violation of independence in intertemporal choice, giving an example. What might cause this violation?

7.8 Applications

At this stage of our analysis of intertemporal choice we are not really in a position to examine and evaluate individual situations, since we have not yet discussed alternative behavioral models that aim to overcome the shortcomings of the DUM. Therefore the analysis of particular specific situations will be delayed until the next chapter. There is, therefore, only one application included here; this is a general overview of empirical studies measuring discount rates conducted over a 25-year period.

Case 7.1 Empirical estimates of discount rates

In a landmark article in 2002 entitled: ‘Time discounting and time preference: a critical review’, the authors Frederick, Loewenstein and O’Donoghue summarized in a table the implicit discount rates from all the studies that they could find between 1978 and 2002 where such discount rates were reported or could easily be calculated. The table lists 42 studies, and describes the type of study, the good involved, whether real or hypothetical rewards/losses were involved, the elicitation method, the time range, the annual discount rate and the discount factor. The discount factor (δ) is calculated by using the expression δ = 1/(1 + r). As we shall see in the next chapter, it can be more convenient mathematically to use the discount factor instead of the discount rate in expressing various phenomena. Their table is reproduced here (as Table 7.1).

Table 7.1 Empirical estimates of discount rates

* hypo = hypothetical

Source: Frederick, Loewenstein and O’Donoghue (2002). ‘Time discounting and time preference: a critical review’, pp. 378–9

Questions

1 What factors might account for the huge variability in the estimates of discount rates?

2 Does there appear to be any relationship between the size of discount rates and whether rewards are real or hypothetical?

3 What is the implication of a negative discount rate, and what might cause this?

4 Is there any evidence of methodological progress, in terms of a convergence of estimates as time goes on? What is the significance of any lack of progress?