Consider the relation between the magnitude of the value to be discounted and the corresponding subjective discount rate. In many cases, the subjective discount rate tends to be lower when the value to be discounted is higher. This tendency has been observed in several questionnaire surveys and economic experiments. For example, in our 2005 nationwide questionnaire survey, the mean discount rate for USD 100 that will be received in 1 year was 6.7 %, whereas the rate for USD 10,000 that will be received in 1 year was 1.16 %. Similarly, Richard Thaler, one of the founders of behavioral economics, asked undergraduate students at Oregon Uni- versity how high an interest rate they would require in order to wait for 1 year to receive varying amounts of money and found median discount rates of 139 %, 34 %, and 29 % for USD 15, USD 250, and USD 3,000, respectively (Thaler 1981).
Uri Benzion estimates a 39.9 % mean discount rate for USD 40 to be received after a 1-year delay and a 20.3 % mean discount rate for USD 5,000 to be received with the same delay (Benzion et al. 1989). This phenomenon (i.e., greater discounting rates for lower amounts) is called the “amount (or magnitude) effect.” These findings show that people can patiently wait for large gratification whereas they cannot, for small gratification.
The magnitude effect has been observed in everyday life. People are sometimes willing to pay a high interest rate in order to borrow a small amount of money, such as is often observed in the consumer loan market (this problem is also related to hyperbolic discounting, which is discussed in the next chapter). In addition, when people owe a small amount of money to friends, they often pay back more than is necessary (e.g., by paying more than their share for lunch).
2.2.2 Increasing Proportionate Sensitivity
Why are smaller amounts discounted more intensively? George Loewenstein and Drazen Prelec hypothesize that when people compare two amounts of money, they are interested not only in the relative magnitudes of the amounts but also in the difference (Loewenstein and Prelec 1992). For example, the difference between USD 100 received today and USD 110 received in 1 year is equal to just USD 10, but the difference between USD 1,000 today and USD 1,100 in 1 year is USD 100. Although there is a 10 % difference between the larger and smaller amounts in each case, people are more likely to wait 1 year to receive USD 1,100 than to receive USD 110 (i.e., they will elect to take USD 100 immediately but will wait for USD 1,100). In other words, people decide whether to wait to receive certain amounts of money based on both therateof interest and the totalamount. Yusuke Kinari and his colleagues have provided empirical support for this hypothesis,
showing that their subjects’ intertemporal choices depended on both the amount and the rate of interest. Moreover, when the amount of interest is experimentally controlled such that only reward magnitude can influence intertemporal choices, the size of the effect decreases by more than half (see Kinari et al. 2009).
Results such as Kinari et al.’s raise the following question: Why are people interested in the amount as well as the rate of interest? One hypothesis is that as reward value increases, it has a greater proportional impact on an individual’s gratification. In the context of intertemporal choices, for example, the impact on gratification of a 10 % increase in the amount of interest would be greater when the principal is USD 10,000 than when it is USD 100. Originally described by Loewenstein and Prelec (1992), this property is called increasing proportionate sensitivity(Chapman and Winquist 1998). According to this property, if a decision- maker requests a constant rate of interestin terms of gratificationor utility to accept a delay, the interest rate would be lowerin monetary termsbecause the principal is greater, as is the case of the magnitude effect. Although this explanation appears to be plausible on its surface, it merely rationalizes the magnitude effect by presupposing that people’s evaluations are more elastic for greater rewards. In economic terms, this explanation can be restated as that the reward elasticity of the value function is an increasing function of the reward. Moreover, the empirical validity of the property has not been ascertained, and it may contradict “the law of decreasing marginal utility,” which has been empirically validated. This law states that the impact of a marginal increase in reward on one’s utility decreases with an increase in the magnitude of the reward itself. Thus, it may be difficult to construct a behavioral model that can reconcile the requirements of increasing proportionate sensitivity and decreasing marginal utility.
Moreover, the increasing proportionate sensitivity of the magnitude effect is not specific to intertemporal choices. For instance, Gretchen Chapman and Jennifer Winquist have shown that the magnitude effect occurs in the domain of tipping for restaurant meals. Their subjects gave a larger percentage tip on small bills than that on large ones (Chapman and Winquist 1998). For findings such as these to be relevant, the same subjects should also display the magnitude effect in intertemporal monetary choice situations. However, the size of the effect is not significantly correlated between the two domains, suggesting that there is an additional mechanism (i.e., other than increasing proportionate sensitivity) that causes the magnitude effect.
2.2.3 Mental Fixed Costs for Waiting
Another reason for the occurrence of the magnitude effect may be that waiting entails a sort of mental fixed cost (Thaler 1981). To exercise self-control and wait for delayed gratification, people have to use mental resources, even when the waited-for reward is small. Unless the amount of interest is large enough to compensate for the mental fixed costs, “waiting” does not pay. In other words,
people will choose the immediate gratification of receiving a small amount of money today rather than the delayed gratification of receiving a larger amount in the future because the mental cost of waiting is too high. Due to the fixed-cost nature of waiting, the interest rate that compensates for the waiting costs needs to increase when the amount of delayed money is smaller.
The mental fixed-cost hypothesis is persuasive in its explanation of the magni- tude effect as considered by Thaler (1981) and Benzion et al. (1989). However, the magnitude effect has also been observed in intertemporal choices when both rewards are received with delays, but there is a longer delay to receive the large reward. For example, in our economics experiments, people display a 19 % per- sonal discount rate on average when asked to choose between USD 35 in 90 days or a certain larger amount in 97 days, whereas they exhibit a 51 % mean discount rate when the choice is between USD 10 and a larger amount. Note that the magnitude effect in this case cannot be explained by the mental fixed-cost hypothesis, because regardless of the option chosen, the decision-maker still incurs a mental fixed cost.
In other words, the fixed cost for waiting does not pose an extra burden for the decision-maker when choosing between the larger reward with a longer delay and the smaller reward with a shorter delay. Thus, the mental fixed-cost hypothesis cannot completely explain the magnitude effect.
2.2.4 Mental Accounting
Proposed by Hersh Shefrin and Richard Thaler (1988), the mental accounting hypothesis can also partially explain the magnitude effect. According to this hypothesis, people have various accounts in their mental space, and they sort money into these accounts based on the amount and source of the money. Further- more, the same objective monetary value may be managed differently in different mental accounts. For example, loose change may be sorted into a mental checking account, whereas a big sum may be sorted into a mental savings account. As a result, the costs of delaying receipt of a certain amount of money may depend on how big is the money. In the case of small coins, which are managed in a mental checking account with a 0 % interest rate, the cost of waiting is denial of the gratification that would be obtained by immediately spending the money. By contrast, a big sum of money is deposited into the mental savings account that is associated with an interest rate. Thus, the opportunity cost of waiting is the savings interest rate, which may be reasonably low. Accordingly, if spending small coins is attractive enough and hence the opportunity cost of delaying receipt of them is high, people will choose not to wait for a small amount of money even when there is a high interest rate. However, for sufficiently larger amounts of money, people will choose to wait under a reasonably low interest rate, as the occurrence of the magnitude effect indicates.