time in the gym), x12 < 0. At this point, the member can choose to close the account and cease attending. If instead, the fair price for attending 12 times was much less than 0.316 times the price of a six-month membership, the member could continue to attend for the potential gain in transaction utility. Note that not attending would result in va− 0.002p + vt− 0.316p + pr11. This is less than the value of additional attendance if x12 is small in absolute value relative to the change in the fair price pr12 − pr11. In this case, the member is reluctant to close the account and could continue to consume simply for the increase in transaction utility, thus demonstrating the sunk cost fallacy.

Reference Points and Indifference Curves

Up until now, we have represented all indifference curves using the familiar concave shapes found commonly in the economics literature (e.g., those depicted in Figures 3.4 and 3.5). However, prospect theory suggests that this shape does not hold universally. This topic is covered more thoroughly in Chapter 4, but for completeness in presenting the theory of mental accounting, it is important to have some understanding of how reference points affect indifference curves.

We have commonly written the prospect theory value function in terms of monetary outcomes and a reference point as a monetary amount. This type of analysis suffices when considering a single consumption activity or good. However, if we are considering multiple consumption activities, it may be that the consumer faces a reference point that includes a consumption level of each activity. Thus, someone who consumes two eggs and a piece of toast every morning for breakfast might consider two eggs and two pieces of toast a gain. Additionally he might consider three eggs and one piece of toast a gain. Further, reducing the number of eggs or eliminating the toast may be considered a loss. But suppose we considered consuming just one egg and two pieces of toast. This is a loss in the number of eggs but a gain in the number of pieces of toast. If the person is loss averse, consuming less than the reference point in any dimension reduces utility by much more than increasing consumption above the reference point in the same dimension.

This has implications for the shape of indifference curves. If I lose one egg (i.e., consume one less than the reference amount) this has a sharply negative impact on utility relative to gaining eggs. To compensate this loss and place me back on my indifference curve, I must be given more toast. But, gaining toast has a small impact on my utility relative to losing toast. Thus, I need to be compensated with a lot more toast for a loss of one egg than I would be willing to give up in order to gain another egg (beyond the reference level). This implies a kink in indifference curves around the reference point, as depicted in Figure 3.8.

In Figure 3.8, the reference amount of good 1 is given by xr1 and the reference amount of good 2 is given by xr2. Anything to the southwest of xr1, xr2 is considered a loss in both domains, and thus utility drops quickly in this direction. Anything to the northeast of xr1, xr2 is considered a gain in both domains, and thus utility increases more slowly in this direction. Alternatively, anything to the southeast of the reference point is a gain (slow ascent) in terms of good 1 but a loss (rapid decline) in terms of good 2. The indifference curve must be relatively flat in this region so that larger quantities of good 1 compensate for losses in good 2. Anything to the northwest of the reference point is a gain (slow ascent) in terms of good 2 but a loss (rapid decline) in terms of good 1. The

x2

x2r

v(x1, x2 ǀ x1r , x2r) = k

indifference curve must be relatively steep in this quadrant so that large quantities of good 2 compensate for losses in good 1. The fact that these slopes change abruptly at the reference point leads to the kink in the indifference curve at the reference point.

Kinks in curves cause problems with the conditions for maximizing utility subject to a budget constraint. In particular, there may not be any point on some indifference curves that have the same slope as the budget constraint. No point of tangency exists for some indifference curves. If the optimum occurs on a curve without a tangency point, the conditions for optimization are rather different from the standard conditions. This is the case presented in Figure 3.8. Here, the slope of the budget constraint is between the slope of the upper portion of the indifference curve as you approach the reference point and the slope of the lower portion of the indifference curve as you approach the reference point. If the kink of an indifference curve lies on the budget constraint and satisfies this condition, then the consumption bundle depicted at the point of the kink is optimal. Chapter 4 discusses more of what happens in the southwest and northeast quadrants, as well as the shape of indifference curves that do not pass directly through the reference point.

Rational Choice, Temptation and Gifts versus Cash

Rational choice models have a difficult time explaining the notion of temptation. Inherent in the problem of temptation is the idea that consumers want something but don’t think they should have that thing. Traditional economics uses the utility function to capture both what consumers want and what they think they should have, eliminating the possibility of temptation. One way that has been proposed to model cases in which a decision maker feels temptation is to differentiate between the short-term and long-term impacts of items. Thus, a good can generate an immediate positive utility (say, the taste of a particularly desirable dessert), but a negative long-term impact on utility (e.g., additional unattractive pounds). Then, the consumer problem could be written as

where u is the utility of consumption in any period given the state of health, ht, x is the amount of cake the consumer can choose to eat in the first period, z is the level of consumption that occurs in the future (which is taken as given in equation 3.9), δ is the time discounting applied to future utility, h0 is the initial state of health, and h1x is the state of health in the future as a function of cake eaten now. Presumably, increasing the consumption of cake decreases health in the future, thus decreasing utility in the future.

The traditional model has the consumer selecting an amount x* that maximizes intertemporal utility, balancing current utility against future utility. This clearly represents the notion that the diner desires the cake now but must also dread the impact it could have on his future utility. However, problems of temptation also often involve regret. Thus, on a regular basis, a diner might eat so much cake that afterwards he regrets his actions and believes he should have shown more restraint. This sort of regret suggests that either the individual decision maker did not perceive the problem correctly, or he did not have complete control of his actions. Rational models do not account for such systematic regret. Although the consumer in the future might desire better health, the consumer who behaves according to equation 3.9 must acknowledge that he made the correct choice.

Gift givers, on the other hand, are not always so accurate. Consider, for example, that your grandmother sends you a sweater she purchases for $75. The sweater is nice, but had you been given the $75, you would have purchased a new MP3 player instead. In this case, you must prefer the MP3 player to the sweater. Further, you would have been better off if your grandmother had given you the $75 directly. Under the assumptions of rationality, so long as consumers are aware of all the possible options they could spend their money on, they will always be at least as well off receiving cash as receiving a gift that cost the same amount. But if money always makes people better off, why would one give anything else?

Budgets, Accounts, Temptation, and Gifts

Because budgets are treated as nonfungible, consumers can use accounts as a means of limiting temptation. Consumers often view their checking account as being much more easily accessed than their savings account. In essence, they place in this account money that they are comfortable spending. Alternatively, they place money in their savings account partially to limit the temptation to use it. Hersh Shefrin and Richard Thaler propose that consumers classify each physical monetary account into one of three categories: current income, current wealth, and future wealth. Current income consists of accounts intended to be spent in the immediate term. Current wealth consists of money accumulated to purchase items too expensive for paycheck-to-paycheck purchases. Finally, future wealth is money that is intended for future consumption, such as retirement savings.

Corresponding to the three different orientations of these accounts, each type provides a different level of temptation to spend. Money in the current income account is very tempting because it is intended to spend in the near term. Money in the current wealth account is less tempting, and one needs to find some substantial justification for spending from this account. Finally, future wealth accounts may be treated as nearly untouchable. People might place money into these accounts to restrict their temptation to spend. Thus,

one might want a portion of one’s paycheck to be placed in each of these three accounts to ensure that one doesn’t spend the entire paycheck. Shefrin and Thaler suggest that the propensity to spend from each account differs. People code income into differing categories based on their intention to spend it, and they place the money in the appropriate account.

The theory predicts that people are able, to some extent, to overcome temptation by viewing different accounts as nonfungible and setting the amount in the more-tempting accounts artificially lower than the amount they are tempted to spend. Similar behavior is possible in limiting specific consumption temptations. Richard Thaler gives the example of a couple who are tempted to purchase and consume expensive wine. To limit their expenditures on and consumption of expensive wine, they might limit themselves to purchasing only bottles of wine that cost less than $20. They set their budget as an artificial mechanism to prevent giving in to temptation. This artificial rule might not always lead to optimal behavior. It might save them from overspending, but it might prevent them from consuming more-expensive wine on an occasion when such a purchase may be justified by the utility it would yield. Thaler points out that this budget constraint necessarily implies that the couple might, on such an occasion, be made better off by receiving a gift of a $50 bottle of wine than they would by receiving $50 in cash. The $50 in cash would be artificially budgeted away from the wine that would make them better off. Alternatively, a gift of wine would not fall under such a restriction and would be consumed, making the person better off than the equivalent cash. Thus, if a budget is set arbitrarily too low owing to an aversion to some temptation, the person might at times be made better off by a gift. This increase in well-being by receiving a gift rather than cash can only take place if the budget is set to arbitrarily eliminate the optimal consumption bundle. If the couple truly preferred $20 bottles of wine to $50 bottles of wine, a gift of $50 would clearly be preferred.

EXAMPLE 3.6 Limiting Temptation

Some consumer items are considered tempting or sinful if they are pleasurable to consume but cause negative long-term effects. Other items are not considered tempting but rather are considered virtuous because they have very positive long-term benefits relative to the immediate consumption experience. Some people use consumption budgets to limit the temptation from sinful items. A consumption budget is a limit on how much one will consume under any circumstance. If people use strict budgets to limit consumption quantities of tempting items, then we would expect people to be less prone to purchasing more of the tempting item if offered a larger quantity at a per-unit discount. When virtuous items (carrots) are buy one get one free, people are likely to take advantage of the offer and the transaction utility that comes with it. Alternatively, when a tempting item (cheesecake) is offered on special as buy one get one free, people might not buy because they have set a limit on their consumption of the item.

Klaus Wertenbroch set out to test this theory performing an experiment. Subjects were presented six-ounce bags of potato chips and told they could purchase one bag or could purchase three bags at a discount. The amount of the discount was varied among subjects. Some of the subjects were told the potato chips were 75 percent fat free, framing the potato chips as being relatively healthy. Others were told the chips were