advise individual consumers to consume solely to enjoy the pizza and forget about how much they paid. Once the pizza is purchased, there is no longer a reason to consider the costs involved. In general, every project should be evaluated based on its future costs and future contribution to utility or profit rather than on any prior considerations.

History and Notes

It is difficult to pin down exactly when the sunk cost fallacy was discovered or named. Economists generally recognized the irrelevance of

sunk costs before 1900, and the topic was covered briefly in the classic text Economics by Paul Samuelson and William Nordhaus in 1948.

Nineteenth-century works on agriculture and railroad management make mention of the concept of sunk costs and often argue the concept as if the sunk cost fallacy were pervasive, though it was not yet named.

Rational Explanations for the Sunk Cost Fallacy

As alluded to previously, there are some reasons sunk costs might truly matter in future decisions. As with examples in politics, discontinuing a project often induces a future cost by publicly signaling a failure. A business might negatively influence future investment, or a politician might negatively influence public opinion of her ability, with such a public signal. Further, sometimes ending a project involves substantial fixed costs in terms of disposal of equipment or waste. In this case, marginal costs of production can exceed marginal benefits at the optimum given the fixed costs that may be incurred if production were stopped short.

In consumer choice, we have already noted that sunk costs can influence choice if they create an income effect (as displayed in Figure 2.3). In this case, higher sunk costs reduce the amount of wealth that can be allocated, leading to substitution between goods as consumers move along their income expansion path (or in this case, down toward the x1 axis). Agnar Sandmo provides an argument as to why fixed costs can also influence the production of a firm. If the firm must make decisions about the amount of production before they know what price they will receive for their output, they might wish to reduce production as a means of reducing risk. This will occur if the firm maximizes the expected utility of profit rather than expected profit, topics that are covered in Chapter 9. In this model, aversion to risk is represented by the shape of a utility of profit function, uπ. Suppose π = py − cy − FC, where p is the random price, y is the output chosen by the firm, cy is the variable cost of production, and FC is the fixed cost. In this case, the fixed cost shifts the distribution of profits and thus alters the shape of the utility curve at any particular price. The firm will alter production in response to changes in fixed costs because the change in the shape of the utility function at any given price necessarily alters their preferences regarding risk.

Finally, people might derive some (fixed) added joy from finishing a project that is unaccounted for in the marginal calculations. Thus, the marathon runner might obtain

positive enjoyment from running the first 13 miles and negative marginal enjoyment for the next 13. In fact, the overall enjoyment for the race may be negative between mile 20 and mile 26 if the race is not finished. But if the joy of crossing the finish line itself is enough to wipe out all negative utility realized between mile 13 and mile 26, it is rational to continue running until the end. In this case, even if a project begins to be a loser, a management team might consider the joy of completing the project more important than the monetary loss incurred. Such a decision may be rational, but it is far from the profit- maximizing assumption that is most often used in economic theory.

Transaction Utility and Flat-Rate Bias

A phenomenon that is related to the sunk cost fallacy is flat-rate bias. Many services can be purchased on a per-use basis or with a fixed fee for access. For example, a consumer can buy issues of a magazine at the newsstand or purchase a subscription to that magazine. A subscription usually offers a substantial discount over the newsstand price, but it would only truly be worth the cost if the subscriber reads the magazines she has ordered. If the subscription reduces the price per magazine by 50%, that may be a good deal, but only if you read at least half of the magazines that are delivered to your door. Similarly, a monthly bus pass often provides a substantial discount over paying for individual trips if the rider makes enough trips; but a consumer would need to determine that her level of ridership would lead to a discount rather than an added expense. The flat-rate bias occurs when consumers choose to use the fixed-fee option when they would have been better off choosing the per-use option. For example, telephone services can be purchased by the minute using a pay-as-you-go plan or can be based on a monthly fee for unlimited access. Suppose the consumer can choose either a fixed rate p0 for unlimited consumption of a good or a linear price p1 per unit consumed. Then, we can modify the consumer problem

subject to the budget constraint

where δ is equal to 1 if the fixed-price plan is chosen and 0 if the linear-pricing plan is chosen, x1 is the number of minutes used, and x2 is consumption of other goods. The notation in equation 2.6 indicates that the consumer must choose δ, which can either equal 1 or 0, in addition to x1 and x2, to maximize her utility of consumption.

There are two possible solutions to this problem. If the consumer chooses the fixedprice plan, she will consume good 1 until she reaches her bliss point as in Figure 2.3. Otherwise, she will consume until she reaches the budget constraint, as in the standard consumer model presented in Chapter 1. Importantly, two potential behaviors are ruled out by the rational model. The two choices have equal cost at the level of consumption defined by p0 = p1x1, for x1 = p0p1. If one consumes more than x1, it will always be cheaper to use the fixed-price plan. Thus, no one should be observed to consume more

than x1 under the linear-pricing plan. Alternatively, below x1 it is always cheaper to use the linear-pricing plan. Thus, no one should be observed to consume less than x1 under the fee-for-unlimited-service plan.

In actuality this does not appear to be the case. Rather, in most cases, it appears that consumers prefer flat-rate pricing even if their usage will not justify it.

EXAMPLE 2.4 Telecommunications

Telephone and cell phone plans are often offered in both flat-rate and pay-as-you-go options. In the late 1980s, Southwestern Bell Telephone introduced extended area service (EAS), which offered unlimited calling to the Dallas area for a set of customers in a nearby community who would normally have to pay relatively high per-minute longdistance fees. The cost of EAS was $19.85 per month. Donald Kridel, Dale Lehman, and Dennis Weisman examined a sample of 2,200 EAS customers and found that only 24% of the customers had placed enough calls to the Dallas calling area to exceed the $19.85 cost of service had they instead been charged the per-minute fee. Thus, 76% had chosen the flat rate when they should have chosen a linear-pricing option, displaying the flat-rate bias. Similar results (though perhaps not as strong) were found when examining behavior when selecting more general long-distance usage. Thus the telephone company might have been able to increase their profits by inducing customers to pay for services that 76% never used. Analogously, consumers today need to be wary of relatively expensive flat fees for unlimited phone, text, or data plans. Many of us may be lining the pockets of the phone companies without enjoying any greater benefits.

EXAMPLE 2.5 Gym Memberships

Another context in which the consumer can choose between flat-rate and linear pricing is in attending a gym. Members usually pay a monthly fee that allows them to attend the gym whenever they like. Alternatively, some gyms offer a fee for use or a pass that is good for a small number of uses. Gyms regularly lament that a large number of customers join the gym when they resolve to finally get in shape but, soon after, their resolve fades and they stop coming. Stefano Della Vigna and Ulrike Malmendier analyzed the membership decisions and attendance records for close to 8,000 gym members over a three-year period. These members could pay a monthly fee of $70 or could purchase a 10-visit pass for $100. The average gym member attended the gym just 4.3 times per month. At that rate, a person could use a 10-visit pass for an entire year and pay just $600. Instead, using the monthly membership option costs $840 per year. Over the course of membership, an average person pays about $600 more than necessary ($1,400 total) for the level of gym attendance, making the flat-rate bias rather costly.

Procedural Explanations for Flat-Rate Bias

The notion of transaction utility seems to suggest that using a flat rate would encourage the consumer to use more of the product in order to reduce the average price per unit of consumption. Though consumers displaying the flat-rate bias are clearly not getting a good deal, this thought process might still take place. Nonetheless, before purchasing, consumers might think very differently about transaction utility and their future potential use of a consumption good. Several motivations for the flat-rate bias have been proposed. The most central to the topic of this chapter are based on the notion of transaction utility.

A consumer considering a gym membership might believe that $10 for a single visit to the gym sounds like a high average cost resulting in a low level of transaction utility. Alternatively, paying $70 for the opportunity to go to the gym as many times as you like for a month might sound like a much better deal. The consumer might either neglect to consider, or fail to accurately project, the number of times she will attend. The monthly option is not stated in a way that allows the consumer to easily compare the average cost. Thus instead of choosing the minimum cost option for the level of use the consumer is intent upon, she might simply consider the transaction utility associated with each option, where perception of transaction utility might depend heavily on the way the problem is stated.

This is closely related to the notion of framing, which is covered more thoroughly in later chapters. The wording or phrasing of a choice can have serious implications for how consumers perceive the tradeoffs. In this case, consumers may be influenced by the comparison of potential (though not actual) use between a single trip to the gym versus a month’s worth of attendance. The decision may be written by modifying equations 2.6 and 2.7 as follows:

subject to the budget constraint

where z0 and z1 are the anticipated transaction utility associated with fixed and linearpricing options, respectively. For the purpose of making a managerial decision, it would be important to model the factors that can affect the transaction utility under each option.

Flat-rate pricing can allow the consumer to disassociate the payment from the actual consumption. Drazen Prelec and George Loewenstein refer to this as the effect of payment decoupling. When we pay for a good at the time of consumption, we link the price and consumption directly to each other. When payment and consumption are separated substantially in time, price and consumption might not be as closely linked. For example, some consumers might save substantial money by using public transportation or taxis rather than owning a car. However, when one already owns a car, one has the luxury of considering a trivial trip to the convenience store for a forgotten toiletry to

be a “free” trip—or at least this trip to the store comes at zero marginal cost. On the other hand, the person without the car would be confronted by the cost of this simple trip, potentially paying $2.50 for a trip to the store to purchase a $2-tube of toothpaste. The potential negative transaction utility from trivial uses such as this can lead to choosing the fixed-price option even if the linear price produces a lower cost. Payment decoupling may be of particular importance in examining credit card behavior, which can allow the consumer to ignore the direct cost of some items both because of the time delay in receiving the bill and because the bill will include potentially dozens of purchases all lumped together under a single total amount due.

A consumer might also simply have a distaste for linear pricing. Using a service, like a cell phone, on a pay-as-you-go plan requires the rational consumer to evaluate before each use (or each minute of use) whether the use really justifies the cost. For example, Verizon Wireless currently offers a pay-as-you-go plan that charges you $3.99 for your first call on any day. Before you make your first call of the day, you must determine that it will be a relatively valuable call. Alternatively, with a flat-rate price, you would not need to evaluate this because the call will not add to the bill on the margin. Thus, even if you do not use the service enough to justify the flat rate, you might use the flat rate to eliminate the nagging feeling that can accompany a linear-pricing plan. Such an effect could be modeled by introducing an added utility cost to the consumer for consuming under linear pricing, replacing the utility function with Ux1, x2 − 1 − δkx1. Here k is the marginal cognitive cost of using the service under a linear price. If this cost is great enough, it can lead the individual to choose flat-rate pricing despite a monetary cost disadvantage.

A final potential motivation is the notion that the consumer might have problems with self-control. This topic is covered more thoroughly in later chapters. In essence, if you believe that going to the gym is good for you, you might worry that a marginal fee for service could discourage future use, whereas a fixed fee for service can introduce, through transaction utility, an incentive for higher levels of use. In this case, you might try to induce gym attendance by purchasing the monthly membership, so that you will feel guilty for not attending because it keeps your average price for use high and your transaction utility low. Our data on gym usage may simply be the evidence of how ineffective this strategy is.

A manager who is considering offering a flat rate for services could potentially benefit from the flat-rate bias. If consumers have a bias for purchasing using a flat-rate pricing plan even when their use does not justify it, there is the potential to increase profits from those who do not increase use above the threshold where the flat rate is actually cheaper. However, there is also the potential of increased use by those who would pay more under the linear-pricing plan. Hence, this is not always a profit-maximizing strategy. The actual benefit to the firm depends on how sensitive consumers are to the price of the flat-rate program and on how many adjust their consumption above the break-even level of consumption.

From a normative perspective, some consumers could be better off by consuming the same amount under the linear-pricing plan. However, to determine which pricing plan is better requires the effort of examining one’s usage and comparing the cost under different plans. One should at times analyze the use of flat-rate services (such as cell phones) and determine if cheaper pricing alternatives exist given one’s own historical usage behavior.