Sunk Cost Effect,Self-Control,and Contract Design

Abstract

This article examines the role of the sunk cost effect as a commitment device in mitigating the self-control problem and analyzes its implications for optimal contract design. Consumers may anticipate the effect ex ante and strategically use it to mitigate their self-control problems. Although the sunk cost effect may lead to a loss of consumption flexibility in the event of high consumption costs, it can serve as a commitment device to enforce self-control. A firm's optimal policy should balance the consumer's demand for flexibility in consumption with the demand for commitment. Under a simple fixed-fee contract, sunk costs have a nonmonotonic effect on profits for investment goods; that is, profits first decrease and then increase with the sunk cost effect. The firm can use a two-part tariff or a refundable fixed-fee contract to mitigate the sunk cost effect. This article also compares the implications of alternative psychological mechanisms underlying the sunk cost effect (regret-based vs. memory-cue-based) for contract design.

Keywords

sunk cost effect present bias contract design two-part tariff refund policy

In many markets, consumers pay a fixed up-front fee to access services and products, for instance, paying the membership fee for a health club, buying a mobile app, or paying to take an online course. The fixed fee is typically nonrefundable and independent of future usage, and therefore is a sunk cost (Dick and Lord 1998). Since it is irreversible, a sunk cost should be irrelevant to decision making. Thaler (1980, p. 47) illustrates with a succinct example of how a consumer violates this principle in decision making:
A man joins a tennis club and pays a $300 yearly membership fee. After two weeks of playing he develops a tennis elbow. He continues to play (in pain) saying “I don’t want to waste the $300!”

The standard economic argument is that the decision whether to continue playing should only depend on the marginal benefits and marginal costs of playing, and the membership fee (a sunk cost) should not matter. A sunk cost effect arises when the consumer has a disutility for changing a chosen plan of action because of a sunk cost that has already been incurred and is irreversible, even though a rational consumer should ignore such a cost and only consider the marginal benefits and marginal costs of the action. Empirical evidence shows that people make decisions conditional on the sunk cost, which leads to suboptimal decisions such as overconsumption (Ho, Png, and Reza 2018; Ho, Wu, and Zhang 2020; Just and Wansink 2011) and an escalation of commitment in investment (Camerer and Weber 1999; Staw 1981). Although it is a textbook example of inconsistent (irrational) behavior in economics (Mankiw 2011), the philosopher Robert Nozick has argued that consumers can benefit from the sunk cost effect (Nozick 1993), especially when they have an issue of underconsumption due to self-control problems. He offers the following example (Nozick 1993, p. 22):
If I think it would be good for me to see many plays or attend many concerts this year, and I know that when the evening of the performance arrives I frequently will not feel like rousing myself at that moment to go out, then I can buy tickets to many of these events in advance. … Since I will not want to waste the money I have already spent on the tickets, I will attend more performances than I would if I left the decisions about attendance to each evening.

For investment goods such as health clubs or online education, consumers often face underconsumption problems due to the lack of self-control. For example, although the consumer would prefer to use a health club at a certain frequency in a future period, when that period arrives, their actual usage frequency is lower. Although the sunk cost effect can give rise to the overconsumption problem as illustrated in Thaler (1980), it also has the potential to serve as a commitment device to countervail one's self-control problem. Nozick's example describes precisely such an effect: he anticipates that he will not incur the cost of “rousing [himself] … to go out” on the evening of the concert. However, if he buys the tickets in advance, the sunk cost effect would kick in, counteracting his cost of “rousing [himself] … to go out” as he would not want to waste what he has already spent.

Two questions arise in the presence of the sunk cost effect. First, do consumers anticipate that they will suffer from the sunk cost effect associated with an up-front fixed fee? Second, if this were to be so, what would be the optimal contract design and the resulting firm profits when facing consumers who are subject to the sunk cost effect and self-control problem?

Anecdotal and empirical evidence suggests that individuals exploit their own future sunk cost effect to exert more effort in planned tasks. Steele (1996) and Walton (2002) recount stories of individuals who buy expensive exercise machines or gym memberships, reasoning that the high cost will motivate them to exercise more in the future. Using user engagement data on Coursera, a massive open online course platform, Goli, Chintagunta, and Sriram (2022) show that users might use the sunk cost effect as a commitment device. Coursera provides free access to course content, homework, and a final grade. Customers can pay to acquire a certificate upon course completion. On the platform, because users can opt to defer payment until 24 days after the commencement of the course, consumers are always weakly better off by deferring the payment. Despite this, a significant fraction of paying users make the payment right after the start of the course, which is suggestive of the fact that consumers may be using early payment as a potential commitment device to increase their future engagement. The authors find that solely paying for the courses up front increases user engagement by 17% to 20% in the weeks following the payment.

To strategically use the sunk cost effect as a commitment device, a consumer should ex ante anticipate the future sunk cost effect when they sign the contract with a fixed fee. Prior studies have demonstrated that people experience the sunk cost effect ex post, and it is an empirical question whether they can anticipate the sunk cost effect ex ante. To examine whether consumers are able to anticipate that the sunk cost will influence their future consumption, we conducted a pilot study involving 186 online participants (see Appendix A for more details). We asked participants to imagine that they are planning to join a health club with a regular membership fee of $2,500 a year. Because of a promotion, they only need to pay a discounted price ($2,000/$1,500/$500/$100). For each of the four discounted price points, we then asked, “How often do you think you will go to the club?” We found that as the price was increased from $100 to $2,000, participants forecast that they would attend the club around three times more per month.

The model in the present article consists of a firm selling to a consumer who is subject to both the sunk cost effect and potential self-control problems. Specifically, the consumer has a regret disutility when they do not carry out the planned action after paying the up-front fixed fee (e.g., failing to attend the gym after paying the membership fee), and this captures the sunk cost effect. This is consistent with the “aversion to waste” argument by Arkes and Blumer (1985) that the sunk cost effect arises because the consumer does not want a cost incurred in the past to be wasted. The lack of self-control stems from the present bias (O’Donoghue and Rabin 1999; Strotz 1955), namely, that the consumer cannot resist immediate urges and temptation, behaving myopically in the short run, even though they would prefer otherwise in the long run. That is, the preference is time inconsistent between the short-run and long-run selves.

We first characterize the optimal fixed-fee contract for investment goods and the implications for the firm and the consumers. After paying the fixed fee, the consumer faces a stochastic consumption cost if they consume, and a regret disutility if they do not. On the one hand, the sunk cost effect of the fixed fee curtails the consumer's consumption flexibility because the consumer may end up consuming even when the realized consumption cost is higher than the benefit of consumption. As a result, the sunk cost effect reduces the consumer's willingness to pay. On the other hand, the effect can serve as a commitment device to mitigate the underconsumption due to the self-control problem, which increases the willingness to pay. The firm's pricing policy trades off this demand for flexibility with the demand for commitment against the self-control problem.

This trade-off leads to an interesting nonmonotonic effect of the sunk cost effect on the optimal fixed fee: as the degree of the sunk cost effect increases, the optimal fixed fee first decreases and then increases. When the degree of the sunk cost effect is small, any increase in the sunk costs increases the regret disutility from nonconsumption by more than the countervailing commitment benefit of mitigating the self-control problem. As a result, the optimal fixed fee decreases with the sunk cost effect. But when the degree of the sunk cost effect is large enough, it can provide sufficient commitment power to mitigate the self-control problem without causing too much inefficient overconsumption. This allows the firm to charge a higher fixed fee that balances the demand for commitment versus that for flexibility. As a result, the optimal fixed fee increases with the sunk cost effect. In contrast, for leisure goods (e.g., mobile apps for gaming), the firm's optimal fixed fee and profits always decrease with the sunk cost effect. Here, the sunk cost effect aggravates the existing concern of overconsumption problems due to the lack of self-control.

Next, we examine the profit implications of the sunk cost effect. We compare the two most commonly observed contractual forms in markets with self-control problems: fixed-fee contracts and variable pay-per-use contracts. The pay-per-use contract, which does not induce the sunk cost effect, also provides a benchmark for comparison with the fixed-fee contract. When the firm incurs a marginal operating cost each time a consumption takes place, the firm has to balance the up-front lump-sum fee and the expected consumption amount associated with the sunk cost effect. The analysis shows that the fixed-fee contract yields higher equilibrium profits than the pay-per-use contract when the firm's marginal cost is low and the consumer's self-control problem is moderate.

We then consider the more general two-part tariff contract. The pay-per-use fee in the two-part tariff can be potentially used to manage the overconsumption caused by the fixed-fee-induced sunk cost effect. Although the overall profits will obviously be higher under the more general contract, comparative statics of the firm's profits uncover negative internalities that result in firm profits decreasing with the sunk cost effect even though the pay-per-use fee in the two-part tariff can be used to counteract the overconsumption problem. We also investigate a contract with a refund policy that allows the consumer to cancel the contract and get a refund. Providing a refund can mitigate the negative impact of the sunk cost effect because it insures the consumer in the event of no consumption. However, as in the case of the two-part tariff, the presence of the sunk cost effect reduces the firm's profits.

We extend the analysis to investigate the implications of an alternative psychological mechanism of the sunk cost effect. Baliga and Ely (2011) and Hong, Huang, and Zhao (2019) argue that sunk cost can serve as a device for coping with limited memory and can be a memory/perceptual cue for the consumer regarding the importance of the associated activity. Specifically, the sunk cost only affects the likelihood of consumption and does not affect the utility. When sunk costs act as a memory cue, the optimal fixed fee increases if the extent of the sunk cost effect is sufficiently low, but decreases if the extent of the effect is sufficiently high. This is in contrast to the case of the regret-based sunk cost effect, whereby only a higher level of the sunk cost effect can increase the optimal fixed fee. We find that if the sunk cost effect is memory-cue-based (and not regret-based), offering a two-part tariff contract or a contract with a refund can achieve the first-best profits.

Related Research

The existing research has focused on empirically documenting the sunk cost effect and on explaining the underlying mechanisms of the effect. Proposed accounts include diminishing sensitivity toward loss (Thaler 1980), aversion to being wasteful (Arkes and Blumer 1985), and need to justify a prior action (Staw 1981). More recently, Baliga and Ely (2011) and Hong, Huang, and Zhao (2019) show that the sunk cost effect can endogenously arise as a cue for coping with limited memory. However, little work has addressed whether consumers anticipate the effect ex ante or the manner in which the anticipated sunk cost effect influences contract design. This article highlights consumer sophistication and awareness of the effect, and how it affects consumer choice and the firm’s optimal contract design. We capture sunk cost effect as the negative emotion related to regret. Thus, our article adds to the literature on the role of emotion in decision making and its strategic implications for the firm (e.g., Diao, Harutyunyan, and Jiang 2023; Iyer and Kuksov 2010, 2012).

This article is also related to the large literature on the demand for commitment (DellaVigna and Malmendier 2004; Jain 2009; Jain and Li 2018; Wertenbroch 1998). Given the lack of self-control, consumers may have the incentive to seek commitment devices (Bryan, Karlan, and Nelson 2010; Carrera et al. 2022) that help them counter their self-control problems. Here, we investigate the role of the sunk cost effect as a commitment device. In a related vein, Jain (2009) and Amaldoss and Harutyunyan (2022) investigate consumers’ goal-setting behavior in mitigating self-control problems. Optimal goals arise because when a consumer does not achieve a goal, the consumer suffers costs from negative emotions. Amaldoss and Harutyunyan analyze the optimal per-unit pricing of leisure goods in the presence of goal-setting behavior. Here, the sunk cost effect induces negative emotions when the consumer fails to consume. However, unlike consumers’ goal-setting behavior, the sunk cost is endogenous to firm incentives (i.e., the fixed fee that is strategically set by the firm). We mainly focus on how the fixed-fee-induced sunk cost effect influences consumer behavior and contract design.

This article is related to the research on price tariff choice biases (e.g., Lambrecht and Skiera 2006; Miravete 2003) and usage behavior in telecommunication markets (e.g., Danaher 2002; Iyengar et al. 2011). Here, we focus on the markets in which consumers are concerned about self-control problems. We show that in these markets, the sunk cost effect can be a potential cause of tariff-choice biases. Because the sunk cost effect increases (decreases) the willingness to pay for the fixed fee in investment (leisure) goods markets, our framework offers an alternative viewpoint regarding two types of tariff-choice biases found in empirical studies: a flat-rate bias in investment goods markets (e.g., health club attendance) and a pay-per-use bias in leisure goods markets (e.g., digital content consumption). Iyengar et al. (2011) empirically investigate the role of the access fee in a two-part tariff on usage. Their results support the combination of the notion of mental depreciation and the reference price effect (Heath and Fennema 1996). Consumers tend to mentally depreciate the sunk cost to the future per-use fee, so that they can align the benefits and costs when they evaluate the entire transaction experience. The mental depreciation process shares similar underlying psychological features with the sunk cost effect in that consumers try to make the sunk cost worthwhile.

This article is complementary to the work of Zhang (2015) and Jain and Chen (2022), who also investigate the interaction of the sunk cost effect and time inconsistency and its implication for pricing. Zhang analyzes the sunk cost effect as a cue that increases the consumption probability and focuses on pricing for investment and leisure goods. Jain and Chen study firm pricing in a market in which heterogeneous consumers have different valuations of a durable product. They look at how pricing and profits are influenced by the self-control problem and the sunk cost effect. Our focus is on examining the optimal contractual design and explaining commonly observed contracts, namely, the fixed-fee contract (which induces the sunk cost effect) and pay-per-use contract (which does not) to understand the trade-offs facing the firms. We also study the role of a two-part tariff and refund policy under the sunk cost effect and show how different psychological foundations of the sunk cost effect have different implications for contract design.

The rest of the article proceeds as follows: We first set up the model and present the main insights on how the optimal fixed fee is driven by the interaction between sunk cost effect and self-control and discuss the profit implications of the sunk cost effect under the fixed-fee, the pay-per-use, and the two-part tariff contract. We then explore the implication of the sunk cost effect for the refund policy as a possible extension of the main contract design. Before concluding, we investigate the implication of the memory cue as an alternative psychological mechanism underlying the sunk cost effect.

Model Setup

The market consists of a firm selling to a consumer. The firm sells an investment good whose consumption incurs an immediate cost but results in a long-run benefit. Examples of such goods include going to a health club, visiting a dentist, and taking a professional course, among others. The consumer exhibits two behavioral biases: present bias and sunk cost effect.

The present bias, which refers to the consumer’s tendency to behave overly impatiently with regard to immediate rewards or costs, represents the self-control problem. In the literature, the present bias is modeled using the quasi-hyperbolic β − δ preferences (Laibson 1997; O’Donoghue and Rabin 1999; Phelps and Pollak 1968; Strotz 1955). Specifically, the β − δ model allows for the standard discounting with δ and captures the present-biased preference with β as follows. At any period t, benefits and costs for t + 1, t + 2, and so on are discounted with βδ, βδ², and so on with 0 ≤ β ≤ 1. The discount factor β captures “present biasedness”: the discounting between the present period and the next period is βδ, while the discounting between any two consecutive periods in the future is δ. Since βδ ≤ δ, the consumer is more impatient when considering the trade-offs involving immediate rewards/costs than when considering the trade-offs involving delayed rewards/costs. This discrepancy between the immediate and the future discounting gives rise to a time-inconsistent preference. The smaller β is, the more significant the time inconsistency of preferences is. This time inconsistency of preferences results in a potential conflict between the current preferences and the preferences that will be held in the future, and generates the issue of self-control. The quasi-hyperbolic discounting allows for both a standard time-consistent preference (β = 1) and a present-biased preference (β < 1).

The sunk cost effect is captured by a disutility associated with the sunk cost if the consumer does not consume, which captures the negative feelings analogous to a psychological “regret effect.” This modeling choice is consistent with the leading explanation that the sunk cost effect can arise as a result of the aversion to being wasteful (Arkes and Blumer 1985). In the “Extensions” section, we analyze a memory-cue-based model and compare it with the regret-based setup.

In the main model, we consider full consumer sophistication and rational expectations of both the sunk cost effect and time inconsistency. In particular, we expect that the consumer anticipates the sunk cost effect, and our survey study (Appendix A) shows that consumers do. Also, the consumer is sophisticated in anticipating the self-control problem. So, anticipating the self-control problem, the consumer chooses whether to take the contract offered by the firm with the hope that when the time to consume comes, the sunk cost effect induced by the up-front fee will overcome their present-bias-induced procrastination, thereby facilitating consumption.

Consider an interaction between a consumer and a firm (e.g., a health club) that has a marginal operating cost of a. Following DellaVigna and Malmendier (2004), we set up a two-period model with the following timing (see Figure 1). In period t = 0, the firm offers a contract with an up-front fixed fee of L and nothing to pay at the time of consumption. If the consumer rejects the offer, the firm receives zero profit, and the consumer gets a reservation utility $\underline{u}$ . If the consumer signs the contract, they pay the fixed fee of L.

Figure 1.
Timing of the Game.

In period t = 1, the consumer observes the private consumption cost c. The cost is stochastic with a range $0 < c \leq \bar{c}$ and follows the distribution F(·), which is smooth, continuous, and differentiable. For example, the cost of attending the gym is related to some random factors such as the weather, the consumer's physical condition of the day, and so on. If the consumer decides to proceed with consumption, the consumption cost realization c is incurred and the consumer receives a benefit $0 < b \leq \bar{c}$ subsequently in period t = 2. If the consumer decides not to proceed with consumption, a sunk-cost-induced disutility −γL is experienced.¹ Note that L is the magnitude of the sunk cost and γ is the parameter that captures the magnitude of sensitivity to the sunk cost.

We derive the expected utility of the consumer as follows. Suppose the consumer only exhibits time inconsistency. At t = 0, the consumer thinks they will consume the product (say, go to the gym) at t = 1 if the net discounted benefit is positive, that is, δβ(δb − c) ≥ 0, which implies c ≤ δb. At t = 1, the consumer will actually consume if βδb − c ≥ 0 or c ≤ βδb. If β < 1, for the consumers with βδb < c < δb, consumption is desirable at t = 0 but undesirable at t = 1, a change in preference due to time inconsistency. Given the distribution of the stochastic cost c, the probability that the consumer will consume is F(βδb). Note that the time-consistent consumer has a higher likelihood of consuming because F(δb) > F(βδb) if β < 1. If the consumer exhibits the sunk cost effect, the consumer chooses to consume the good if βδb − c ≥ −γL or c ≤ βδb + γL. As a result, the probability of consumption is F(βδb + γL). Putting all of this together, the consumer's expected utility from signing the contract is
$\begin{aligned} E (U_{t = 0}) = δ β \cdot \\ (- L + \int_{0}^{β δ b + γ L} (δ b - c) dF (c) + \int_{β δ b + γ L}^{\bar{c}} (- γ L) dF (c)) . \end{aligned}$
(1)
As shown in Equation 1, the sunk cost effect has two impacts on the consumer's expected utility. First, the sunk cost effect increases the probability of consumption. Recall that without the sunk cost effect, the probability of consumption is F(βδb), whereas in the presence of the sunk cost effect, the probability becomes F(βδb + γL) > F(βδb). The probability of consumption is affected by both the sunk cost effect γ and time inconsistency governed by β. Second, the sunk cost effect directly affects the consumer's utility. Due to the randomness of the consumption cost, there is some likelihood that the consumer will experience disutility −γL. Similar to previous research (DellaVigna and Malmendier 2004; Jain 2009; O’Donoghue and Rabin 1999), we assume δ = 1 without loss of generality of the insights.

We begin by demonstrating the impact of the sunk cost effect in the model. Consider the first-order partial derivative of E(U_t=0) with respect to γ:
$\frac{\partial E (U_{t = 0})}{\partial γ} = β [bLf (β b + γ L) - (β b + γ L) Lf (β b + γ L) + γ L^{2} f (β b + γ L) - \int_{β b + γ L}^{\bar{c}} L dF (c)] .$
(2)
As can be seen, Equation 2 has four terms, which represent the decomposition of the sunk cost effect on consumer expected utility. The first two terms capture the commitment effect, the impact of the sunk cost effect on the expected utility when the consumer consumes. The first term is positive, which is related to the increase in the probability of experiencing the benefit. The second term is negative, which is related to the increased chance of experiencing the cost c. The last two terms capture the impact of the sunk cost effect on the expected regret disutility when the consumer fails to consume. The third term is positive, which shows that the increase of the sunk cost effect reduces the likelihood of experiencing regret disutility by increasing the likelihood of consuming. The fourth term is negative, which captures the increased disutility associated with not consuming. Rearranging Equation 2 gives
$\frac{\partial E (U_{t = 0})}{\partial γ} = β [\underset{Commitment effect (+)}{\underset{⏟}{(1 - β) bLf (β b + γ L)}} - \underset{Regret effect (-)}{\underset{⏟}{\int_{β b + γ L}^{\bar{c}} L dF (c)}}] .$
(3)
Equation 3 shows that the impact of the sunk cost effect on consumer expected utility is a combination of two effects: The first is a nonnegative commitment effect. It shows that when the consumer is time inconsistent (β < 1), the commitment effect can have a positive impact on the expected utility because it drives more consumption to mitigate the self-control problem. The second effect is related to regret when the consumer does not consume. Both effects depend on the consumer's degree of time inconsistency.

Lemma 1:
If the consumer is time consistent (i.e., β = 1), the sunk cost effect has a negative impact on the consumer's expected utility (i.e., $\frac{\partial E (U_{t = 0})}{\partial γ} < 0$ ).

If the consumer is time inconsistent (i.e., β < 1) and F(·) is weakly convex, the consumer's expected utility first decreases and then weakly increases with the sunk cost effect γ.

The proof is in Appendix B. If the consumer is time consistent (i.e., β = 1), a larger sunk cost effect always results in a lower expected utility. This is because a time-consistent consumer does not need a commitment device, but the sunk cost effect generates regret if the consumer does not consume. As the degree of time inconsistency increases, the benefit of the commitment effect looms larger since the sunk cost effect mitigates the self-control problem and motivates the consumer to consume. However, a higher degree of time inconsistency can also have a countervailing effect by causing the consumer to fail to consume, resulting in regret disutility if the consumer has already paid L. The overall impact of the sunk cost effect on consumer utility depends on which of these two effects dominates. When the sunk cost effect is small, the benefit from the commitment effect is small whereas the regret effect is large. As a result, the consumer's expected utility decreases with the sunk cost effect. As the sunk cost effect further increases, the commitment effect starts to dominate the regret effect, and the consumer's expected utility increases with the sunk cost effect.² When setting the optimal fixed fee, the firm has to consider the trade-off between the commitment effect and the regret effect, both of which depend on the degree of the sunk cost effect.

Main Results

The Fixed-Fee Contract

Let us consider the firm's decision problem of finding an optimal L to maximize profits. If the consumer signs the contract, the firm gains expected profits E[Π_t=0] from charging an up-front fixed fee L. If the consumer consumes, the firm incurs a marginal cost a. The firm maximizes its profits subject to the consumer's and its own participation constraints. The firm solves the following decision problem:
${max}_{L} (L + \int_{0}^{β b + γ L} (- a) dF (c)),$
(4)
subject to the consumer's participation constraint:
$β (- L + \int_{0}^{β b + γ L} (b - c) dF (c) + \int_{β b + γ L}^{\bar{c}} (- γ L) dF (c)) \geq β \underline{u},$
and firm's participation constraint:
$L + \int_{0}^{β b + γ L} (- a) dF (c) \geq 0.$
Without loss of generality, we assume $\underline{u} = 0$ . To obtain a closed-form solution for the optimal L, we assume c ∼ U[0,1]. Throughout this analysis, we assume that the marginal cost a is sufficiently low (i.e., $a \leq \bar{a}$ ) and the benefit is sufficiently high (i.e., 3/4 ≤ b < 1) so that the firm always gets nonnegative expected profits over the full range of parameters and the participation constraint of the firm and the consumer is satisfied.³ We also restrict 0 ≤ γ ≤ 1 because the regret of paying γL cannot exceed L.⁴ By solving the optimal fixed fee, we have
$L * = {\begin{matrix} \frac{β (2 - β) b^{2}}{(1 + γ - γ b) + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}} & if 0 \leq γ \leq {\bar{γ}}_{1} . \\ b - \frac{1}{2} & if γ > {\bar{γ}}_{1} . \end{matrix}$
(5)
Proposition 1:

If the degree of time inconsistency is sufficiently large (i.e., $0 \leq β \leq {\bar{β}}_{1}$ ), the optimal fixed fee decreases with the sunk cost effect for $γ \leq {\bar{γ}}_{1}$ .

If the degree of time inconsistency is sufficiently small (i.e., ${\bar{β}}_{1} < β < 1$ ), the optimal fixed fee first decreases when $0 < γ < {\bar{γ}}_{2}$ and then increases with the sunk cost effect when ${\bar{γ}}_{2} \leq γ \leq {\bar{γ}}_{1}$ .

If the degree of the sunk cost effect is larger than the threshold ${\bar{γ}}_{1}$ , the optimal fixed fee L* is a constant that induces the consumer to consume with probability 1.

Here ${\bar{γ}}_{1} = min {\frac{2 (1 - β b)}{2 b - 1}, 1}$ , ${\bar{γ}}_{2} = \frac{2 (1 - b)}{2 b - 1 - b^{2} {(1 - β)}^{2}}$ , and ${\bar{β}}_{1} = 1 - \frac{\sqrt{4 b - 3}}{b}$ . The proof can be found in Appendix C.⁵

If the consumer has a severe time-inconsistency problem (i.e., $0 \leq β \leq {\bar{β}}_{1}$ ), the regret effect would dominate the commitment effect because the consumer has a high chance of failing to consume due to time inconsistency. The higher the degree of the sunk cost effect, the larger the regret disutility the consumer will experience, which reduces the willingness to pay.

If the consumer's time inconsistency is not too severe ( ${\bar{β}}_{1} < β < 1$ ), the optimal fixed fee and the sunk cost effect have a U-shaped relationship (see Figure 2). When the sunk cost effect is small, the consumer has a relatively large chance of experiencing the disutility −γL. In addition, although the commitment effect can drive more consumption, the positive commitment effect is not sufficiently large to compensate for increased regret disutility. As a result, the willingness to pay and the optimal fixed fee decrease with the sunk cost effect. As the sunk cost effect increases, the consumer will be more likely to consume, resulting in a higher chance of getting the benefit from consuming the investment goods and a lower chance of experiencing regret. Thus, the commitment effect can dominate the regret effect. As a result, the willingness to pay and the optimal fixed fee increase with the sunk cost effect. Note that the presence of the sunk cost effect (γ > 0) can lead to a higher or lower optimal fixed fee compared with the case in which the sunk cost effect is absent (γ = 0). The presence of the sunk cost effect induces a higher optimal fixed fee when the sunk cost effect is sufficiently high and the consumer's time inconsistency is small.⁶

Figure 2.
Sunk Cost Effect and Optimal Fixed Fee.

When the sunk cost effect is sufficiently large (i.e., $γ > {\bar{γ}}_{1}$ ), the consumer will consume with probability 1. In such a case, the firm charges a fixed fee that equals the expected benefits of the consumer who consumes irrespective of the consumption cost c; that is, L* = b − $\frac{1}{2}$ .

The results have interesting implications. For fully time-consistent consumers and for consumers with severe time-inconsistency problems, the fixed-fee-induced sunk cost effect leads to a lower equilibrium fixed fee. However, for the consumer whose time inconsistency is small, a sufficiently large sunk cost effect associated with the fixed-fee contract can increase the willingness to pay for the contract. Recent empirical studies have found estimates of the present-bias parameter β ≈ .9 for unpleasant tasks (Augenblick, Niederle, and Sprenger 2015; Imai, Rutter, and Camerer 2021), which suggests that consumer time inconsistency is small in such a market. Thus, the analysis can provide a possible rationalization of the fixed-fee bias in DellaVigna and Malmendier (2006). DellaVigna and Malmendier find that the consumers pay more than $70 for monthly membership and on average attend the gym 4.8 times a month, resulting in $17 per visit. However, the gym also offers a pay-per-use contract that costs only $10 per visit. In the work of DellaVigna and Malmendier, this is explained by consumers’ overconfidence about their self-control. The alternative explanation is that if the consumer is aware of their self-control problem and the sunk cost effect, choosing the fixed-fee contract with an up-front fixed fee induces commitment to greater future usage. Thus, the flat-rate bias can arise from sophisticated consumers with rational expectations about their future behavior.

Implications of the sunk cost effect for firm profits: investment goods

We next examine the implications of the sunk cost effect for firm profits in the investment goods market. If the firm charges the fixed fee L* in equilibrium, the profit is $Π (L ) = L + \int_{0}^{β b + γ L } (- a) dF (c)$ . The sunk cost effect influences the firm's profit in two ways. First, as shown in Equation 5, the sunk cost effect has a nonmonotonic impact on the revenue: fixed fee L. The sunk cost effect can increase the optimal fixed fee by increasing the willingness to pay when it serves as a commitment device in counteracting the self-control problem. Second, the sunk cost effect increases the consumption probability F(βb + γL), and hence the probability of incurring the marginal cost a. The first-order partial derivative of Π(L) with respective to γ yields
$\frac{\partial Π (L )}{\partial γ} = \frac{\partial L }{\partial γ} - a \cdot \frac{\partial F (β b + γ L )}{\partial γ} .$
As can be seen, the first part is about the impact of the sunk cost effect on revenue, and the second part is about the impact on expected marginal cost. The overall impact of the sunk cost effect on the profits is summarized in the following proposition:
Proposition 2:

If the marginal cost a = 0 and the consumer's time inconsistency is sufficiently small ( ${\bar{β}}_{1} < β < 1$ ), the profits decrease with the sunk cost effect when the sunk cost effect is small, and increase with the sunk cost effect when the effect is large.

If the marginal cost is sufficiently high ( $a > {\bar{a}}_{1}$ ),⁷ the profits decrease with the sunk cost effect.

The proof of Proposition 2 can be found in Appendix D. If the marginal cost a = 0, the profits are equal to the fixed fee and $\frac{\partial Π (L )}{\partial γ} = \frac{\partial L }{\partial γ}$ . As shown in Proposition 1, the profits would first decrease and then increase with the sunk cost effect. The firm can benefit from the sunk cost effect only when the sunk cost effect is sufficiently large. If the marginal cost a is sufficiently high, then the profits decrease with the degree of the sunk cost effect. This is because even though the sunk cost effect can potentially increase the revenue L, the consumption probability and the firm's expected cost also increase with the sunk cost effect. As the degree of the sunk cost effect increases, the effect on expected cost dominates the effect on revenue. In general, can the presence of the sunk cost effect (γ > 0) increase the firm's profits? We can show that when the marginal cost is low enough and the consumer's time-inconsistency problem is not too severe, the sunk cost effect increases profits when the effect is sufficiently large.⁸

Implications of the sunk cost effect for firm profits: leisure goods

Consumption of leisure goods such as gaming involves immediate benefits and delayed costs. The time-inconsistent consumer is concerned about the overconsumption problem associated with the immediate benefits. In the presence of a fixed fee, the sunk cost effect would further exacerbate the overconsumption problem. As a result, a higher degree of the sunk cost effect would lead to a lower expected utility given the fixed fee, and therefore a lower willingness to pay. We summarize the results for the leisure goods market in the following corollary (the proof is in Appendix E):
Corollary 1:
In the leisure goods market, the firm's profits weakly decrease with γ regardless of the marginal cost a.

Comparison with the Pay-Per-Use Contract

Alternatively, the firm can use a variable pay-per-use contract that does not entail sunk costs and does not induce the sunk cost effect. Therefore, it is a contractual benchmark to compare the sunk cost effect created by the fixed-fee contract. Suppose the firm offers a pay-per-use contract with a price p > 0, which the consumer only pays when they consume. As there is no sunk cost effect, the consumption decision is determined only by the benefit and the cost of the consumption. The firm's optimization problem is
$max_{p} (\int_{0}^{β b - p} (p - a) dF (c)),$
(6)
subject to
$E [U_{t = 0}] = β (\int_{0}^{β b - p} (b - p - c) dF (c)) \geq 0.$
Solving Equation 6 gives
$p * = \frac{1}{2} (a + β b) .$
(7)
The firm's profits from the pay-per-use contract are $\frac{1}{4}$ (βb − a)², which does not depend on the sunk cost effect. The following proposition compares the profitability between fixed-fee and pay-per-use contracts.
Proposition 3:

When the marginal cost a is sufficiently low (i.e., $a < {\bar{a}}_{2}$ ) and the time inconsistency is moderate (i.e., ${\bar{β}}_{2} \leq β \leq {\bar{β}}_{3}$ ), the fixed-fee contract is always more profitable than the pay-per-use contract, irrespective of the degree of the sunk cost effect.

If the marginal cost a is higher (i.e., ${\bar{a}}_{2} \leq a \leq \min {{\bar{a}}_{3}, \bar{a}}$ ) and the consumer is sufficiently time consistent ( ${\bar{β}}_{4} \leq β < 1$ ), the fixed-fee contract is more profitable when the sunk cost effect is low (i.e., $γ < {\bar{γ}}_{3}$ ); when the sunk cost effect is large (i.e., $γ > {\bar{γ}}_{3}$ ), the pay-per-use contract is more profitable.⁹

The proof of Proposition 3 can be found in Appendix F. Under the pay-per-use contract, the consumer has to incur both a consumption cost and a per-use fee, which reduces the probability of consumption, whereas under the fixed-fee contract, the consumer has a higher probability of consumption even without the sunk cost effect. When the marginal cost is low, the firm can use the fixed-fee contract to extract all the consumer surplus, and it results in a higher profit level than the pay-per-use contract.

When the marginal cost is higher, the fixed-fee-induced sunk cost effect leads to two negative impacts. First, the sunk cost effect can increase the probability of consumption and therefore the expected marginal cost for the firm. Second, the consumer who is under a fixed-fee contract faces a potential overconsumption problem if the realized consumption cost is high. These two impacts reduce the profitability of the fixed-fee contract. In contrast, the pay-per-use contract provides the consumer with more flexibility in consumption. The consumer only needs to pay when the benefit is larger than the realized consumption cost. As a result, the pay-per-use contract can yield higher profits.

The results have some testable implications about which contract the firm would prefer based on the marginal operating costs. For example, in some industries with relatively low marginal costs, such as health clubs, it is beneficial for the firm to offer a fixed-fee contract. In DellaVigna and Malmendier's (2004) telephone survey of 67 health clubs in the metropolitan Boston area, the researchers asked which contracts were available at each health club. Although both the fixed-fee and pay-per-use contracts are feasible, a majority of the clubs initially offered a fixed-fee contract. Only 2 of 67 clubs initially mentioned a pay-per-use contract. In other industries with relatively high marginal costs, such as personal coaching and dental care, the pay-per-use contract seems more prevalent. For example, tutors in leading marketplaces for private tutors such as Apprentus and TakeLessons adopt pay-per-teaching-session contracts.

The Two-Part Tariff Contract

The fixed-fee and pay-per-use contracts are special cases of a two-part tariff contract consisting of the lump-sum fixed fee L and the per-use fee p. We next analyze the implication of the sunk cost effect for profits when the firm adopts a two-part tariff contract. The firm solves the following problem:
$max_{L, p} (L + \int_{0}^{β b - p + γ L} (p - a) dF (c)),$
(8)
subject to the consumer's participation constraint:
$β (- L + \int_{0}^{β b - p + γ L} (b - p - c) dF (c) + \int_{β b - p + γ L}^{1} (- γ L) dF (c)) \geq 0.$

No sunk cost effect (γ = 0)

In the case when the consumer does not exhibit the sunk cost effect but only exhibits time inconsistency, that is, γ = 0, the equilibrium profits are $\frac{1}{2}$ (b − a)². The optimal two-part tariff consists of L* = $\frac{1}{2}$ (b − a)(b(3 − 2β) − a) and p* = a − (1 − β)b.

Under the two-part tariff, the firm can lower p to attenuate the self-control problem. Specifically, to address the underconsumption problem due to the lack of self-control, the firm may even be willing to pay the consumer to participate (p* < 0) when the consumer is sufficiently time inconsistent (β < 1 − a/b), because the fixed fee can be used to extract surplus. A lower per-use fee can induce the time-inconsistent consumer to consume more, and hence the firm reduces p and increases L as the time inconsistency increases. In the equilibrium, the probability of consumption is F(b − a) = b − a, and the firm's profits are $\frac{1}{2}$ (b − a)², and both are independent of the degree of time inconsistency. Note that since the maximum social surplus is $\frac{1}{2}$ (b − a)², when γ = 0, the firm can extract all the surplus and achieve the first-best profits (see Appendix G).

Sunk cost effect (γ > 0)

Next, we analyze the firm's profits when the consumer also exhibits the sunk cost effect. The firm solves the profit-maximization problem shown in Equation 8. We find that the presence of the sunk cost effect in general has a negative impact on the firm's profits:

Proposition 4: Under a two-part tariff, the firm's profits decrease with the sunk cost effect γ.

The proof is shown in Appendix G. As we just showed in the case where γ = 0, the two-part tariff contract can fully address the self-control problem and achieve the first-best profit level. In general, under a two-part tariff, as γ increases, the sunk cost effect gives rise to overconsumption problems due to the commitment effect and the regret disutility if the consumer fails to consume. The firm may try to mitigate the overconsumption problem by decreasing L and increasing p, but reducing the consumption probability leads to a higher chance of realizing the regret disutility. Conversely, the firm can mitigate the regret disutility by increasing the consumption probability (e.g., by increasing L and decreasing p), but that inevitably leads to overconsumption problems. As a result, the two-part tariff cannot fully address the negative impact caused by the sunk cost effect, and the increase in the sunk cost effect reduces the firm's profits.

Although it should be obvious that the profit level can be higher under the more general two-part tariff than with either the fixed fee or pay-per-use fee, the two-part tariff still does not generate the first-best outcome. The sunk cost effect will generate a net loss in social surplus because when the consumer fails to consume, the total surplus will be reduced by the regret −γL, which prevents the firm from achieving the first-best profits. Only when the consumer chooses to consume with probability 1, the loss in social surplus −γL will not occur. However, if the consumer chooses to consume with probability 1, there is a probability of (1 − b + a) that the sum of the consumption cost and the marginal cost c + a is higher than the benefit b. Put differently, when the consumer consumes with a probability of 1, the social welfare is W = b − a − $\frac{1}{2}$ , is also strictly lower than the first-best outcome $\frac{1}{2}$ (b − a)² if b < 1 and a > 0. Therefore, only in the knife edge case when b = 1 and a = 0, the firm can possibly achieve the first-best profits $\frac{1}{2}$ (b − a)². In such a case, the consumer always consumes, and the loss in social surplus −γL never occurs (Table 1).

Table 1.
The Impact of the Sunk Cost Effect on Pricing and Profits.

Contract Type Severity of Self-Control Problem

Mild (i.e., High β) Severe (i.e., Low β)

Fixed-fee contract (L) Pricing L first decreases and then weakly increases with γ L* decreases with γ

Profits The sunk cost effect increases profits when γ is high and a is low The sunk cost reduces profits

Pay-per-use contract (p) Pricing p is independent of γ

Profits The sunk cost effect has no effect on profits; when γ and a are high, the pay-per-use contract is more profitable than the fixed-fee contract

Two-part-tariff contract (L, p) Pricing L* and p* have a nonmonotonic relationship with γ (see Web Appendix A)

Profits The sunk cost effect reduces profits

Notes: β is the degree of time consistency; γ represents the degree of the sunk cost effect; and L* and p* represent the optimal fixed fee and per-use fee, respectively.

Extensions

Refund Policy

In the price contract, the firm can offer a refund f (f ≤ L) if the consumer decides not to consume. The refund can serve as an instrument to influence the magnitude of the sunk cost effect the consumer commits to. Similar to the previous setting, when the consumer chooses to consume at t = 1, they get βb − c; otherwise, they get −γL. Given the refund policy, the firm refunds f if the consumer decides not to consume and cancels the contract. In this case, the consumer’s utility is −γ(L − f) + f.

Assume that the consumer will always get a positive refund if they decide not to consume. Therefore, the consumer chooses to consume if βb − c > −γ(L − f) + f. The probability of consumption is F(βb + γ(L − f) − f). The refund f reduces the probability of consumption via two channels: first, it reduces the sunk cost to L − f and thus reduces the sunk cost effect; second, since the consumer can get f by canceling the contract, it creates an incentive for the consumer not to consume ex post.

The firm's problem is
$ma x_{L, f} (L + \int_{0}^{β b + γ (L - f) - f} (- a) dF (c) + \int_{β b + γ (L - f) - f}^{1} (- f) dF (c)),$
(9)
subject to
$\begin{aligned} β (- L + \int_{0}^{β b + γ (L - f) - f} (b - c) dF (c) \\ + \int_{β b + γ (L - f) - f}^{1} (- γ (L - f) + f) dF (c)) \geq 0. \end{aligned}$
If we consider the case when the consumer does not exhibit the sunk cost effect but only time inconsistency, that is, γ = 0, then analogous to the two-part tariff case, by setting the optimal fixed fee and refund, the equilibrium profits are $\frac{1}{2}$ (b − a)², which is at the first-best profit level. The optimal fixed fee and refund are L* = $\frac{1}{2}$ (b − a)(b + a) + (a − b + βb)(1 + a − b) and f* = a − (1 − β)b. Because a positive refund insures the consumer in the event of no consumption, it can exacerbate the underconsumption problem due to time inconsistency. Therefore, the optimal refund decreases with the severity of the time-inconsistency problem. When the marginal cost is small and the self-control problem is severe, the optimal refund may be negative, which can be interpreted as an inactivity or dormancy fee.¹⁰ Next, we examine the implications of the sunk cost effect for profits.

Proposition 5: Under the contract consisting of a fixed fee and a refund, the presence of the sunk cost effect reduces profits.

The proof is in Web Appendix B.

The role of the refund is analogous to the pay-per-use fee in two-part tariff pricing because both can discourage consumption. The difference is that the per-use fee penalizes consumption, whereas the refund “rewards” the lack of consumption. Note that it is never optimal for the firm to offer L = f. This is because if a full refund is provided, the consumer will consume if and only if there is a positive surplus, that is, βb > c; thus, the firm has incentives to decrease the refund.

In the presence of the sunk cost effect, if the consumer fails to consume, the total social surplus will be reduced by −γ(L − f), which prevents the firm from achieving the first-best outcome. Since it is never optimal for the firm to offer L = f, the net loss always exists unless the consumer chooses to consume with probability 1. As the social welfare should be higher than the social cost, the sum of the consumption cost and the marginal cost c + a is lower than the benefit b (i.e., c < b − a); otherwise, the net social surplus is negative. However, if the consumer chooses to consume with probability 1 when b < 1 or a > 0, there is a probability of 1 − b + a that consumption leads to a negative net social surplus. Therefore, the first-best outcome can never be achieved unless b = 1 and a = 0.

Sunk Cost Effect as a Memory Cue

Consumers develop strategies to cope with limited memories (Baliga and Ely 2011; Chen, Iyer, and Pazgal 2010; Guo 2023; Hong, Huang, and Zhao 2019). In this section, we examine an alternate account of the sunk cost effect based on limited memory (Baliga and Ely 2011; Hong, Huang, and Zhao 2019). Consistent with this account, the sunk cost effect can be modeled as a memory/perceptual cue for the consumer regarding the importance of the activity associated with the sunk cost. That is, the sunk cost only affects the likelihood of consumption but does not yield disutility. The strength of the cue depends on both how much the consumer paid (L) and the degree of recall salience. Here, we use s ( $s \in [0, 1]$ ) to capture the degree of the recall salience. As a result, the consumer's expected utility from signing the fixed-fee contract is
$E [U_{t = 0}] = β (- L + \int_{0}^{β b + sL} (b - c) dF (c)) .$
(10)
Note that the sunk cost effect is incorporated only in the probability of consumption, F(βb + sL), which is larger than the probability without sunk cost, F(βb). The impact of the sunk cost effect s on expected utility is as follows:
$\frac{\partial E [U_{t = 0}]}{\partial s} = β L [(1 - β) b - sL] f (β b + sL) .$
(11)
As can be seen from Equation 11, if the consumer is time consistent (β = 1), then $\frac{\partial E [U_{t = 0}]}{\partial s} < 0$ , that is, the consumer's expected utility decreases with a higher degree of sunk cost effect. This result is consistent with that in the main model shown in Lemma 1. However, the mechanism of the negative effect of the sunk cost effect is different. In the main regret-based sunk cost effect model, the negative impact comes from two sources: (1) the disutility (e.g., regret) when the consumer fails to consume (e.g., attend the gym) due to high realized cost; and (2) the overconsumption problem associated with the sunk cost effect. Under the memory-cue account, the negative impact stems only from the overconsumption problem. When the consumer is time inconsistent (β < 1), the consumer's utility is not monotonic in the degree of sunk cost effect (s); specifically, when the consumer is sufficiently time inconsistent, that is, $β < 1 - \frac{sL}{b}$ , then the consumer utility increases in s (i.e., $\frac{\partial E [U_{t = 0}]}{\partial s} > 0$ ). If we assume c ∼ U[0,1], then we have the following proposition:
Proposition 6:

If the degree of time inconsistency is sufficiently large (i.e., $β \leq 1 - \frac{b}{2}$ ), the optimal fixed fee increases with s.

If the degree of the time inconsistency is sufficiently small (i.e., $1 - \frac{b}{2} < β < 1$ ), the optimal fixed fee first increases when $s < {\bar{s}}_{2}$ and then decreases with s when ${\bar{s}}_{2} \leq s \leq {\bar{s}}_{1}$ .

If s is sufficiently large (i.e., $s > {\bar{s}}_{1}$ ), the optimal fixed fee is a constant L* = b − $\frac{1}{2}$ , which induces the consumer to consume with probability 1.

The proof can be found in Web Appendix C.

Here ${\bar{s}}_{1} = \min {1, \frac{2 (1 - β b)}{2 b - 1}}$ and ${\bar{s}}_{2} = \frac{2 - 2 β}{b}$ . When the consumer's time inconsistency is high, the higher degree of sunk cost effect will result in a higher chance of consumption, canceling out the negative impact of the self-control problem. Thus, the consumer's willingness to pay the optimal fixed fee increases with the degree of the sunk cost effect. However, if the degree of the time inconsistency is small, although a small degree of the sunk cost effect can increase profits, a higher degree of the effect will drive the consumer to consume under a greater set of undesirable circumstances (i.e., c > b), and this results in a lower willingness to pay the fixed fee.

Two-Part Tariff and Refund Policy

Note that when the consumer does not exhibit the sunk cost effect, the two-part tariff and the refund policy generate the same equilibrium profit, that is, $\frac{1}{2}$ (b − a)². Interestingly, if the sunk cost effect is memory-cue-based and does not generate disutility, we can show that the potential negative impact of the effect is eliminated in the equilibrium if the firm uses a two-part tariff or a refund policy. That is, the firm can achieve the first-best profit level using the two-part tariff or the refund policy (see Web Appendix D). This result is driven by the fact that by appropriately setting the fixed fee and per-use fee in the two-part tariff (or the refund in the refund policy), the sunk cost effect and time inconsistency counterbalance each other in equilibrium. Under a two-part tariff, the optimal fixed fee L* decreases in s. Lowering the fixed fee will give the consumer more flexibility to choose whether to consume in the future, and this mitigates the overconsumption problem due to the sunk cost effect. Meanwhile, the per-use price increases in s since a higher degree of sunk cost effect makes the consumer likely to consume even with a higher per-use fee. Under a refund policy, as the sunk cost effect increases, the firm will increase its refund to mitigate consumers’ overconsumption; as a result, the firm will also set a higher up-front fixed fee to ensure its profits.

Hence, if the sunk cost effect serves only as a memory cue, the negative impact of this bias can be internalized in the two-part tariff contract or the refund policy, whereas that is not the case when the sunk cost effect stems from regret disutility. The contrast between the two psychological accounts helps disentangle the effect on consumption probability from the direct effect on utility, which has different implications on firm profits. The results show that it is important for the firm to understand the psychological mechanism underlying the sunk cost effect.

Discussion and Conclusion

The sunk cost effect is a canonical example of behaviors that are inconsistent with standard economic theory predictions. It can result in overconsumption, escalation of commitment to investments, and insufficient adaptation to new situations. In Nozick’s (1993, p. 22) words, the sunk cost effect “can be rationally utilized to check and overcome another behavioral bias (the self-control problem).” This article analyzes the role of the sunk cost effect as a self-commitment device and its implication for optimal contract design. For investment goods, a firm balances the demand for flexibility in response to the sunk cost effect and the demand for a commitment to counter the self-control problem when designing an optimal price contract. Under a fixed-fee contract for investment goods, the firm's profits can increase or decrease with the effect in the investment goods market, depending on the degree of the effect. The firm can use a two-part tariff or a refundable fixed-fee contract to mitigate the sunk cost effect. We show that under a two-part tariff or a refund policy, the sunk cost effect reduces profits, and the adverse impact on profits arises when the sunk cost effect is regret-based but not memory-cue-based.

Managerial Implications

In our main analysis of the fixed-fee contract in investment goods markets, we show that the optimal fixed fee can increase with the degree of the sunk cost effect. Interestingly, in leisure goods markets (e.g., games, smoking) where the consumption involves immediate benefits and delayed cost, the optimal fixed fee decreases with the degree of the sunk cost effect. This is because the sunk cost effect exacerbates the overconsumption problems, and therefore reduces the consumer's willingness to pay. Our analysis can shed light on mobile app pricing. For apps associated with investment goods consumption (e.g., education apps), the fixed fee can serve as a commitment device for the consumers, and hence the firm can charge a higher fixed fee. For the apps associated with leisure goods consumption (e.g., game apps), consumers who lack self-control have a concern about overconsumption problems, and the fixed-fee-induced sunk cost effect aggravates the concern. Therefore, the consumer has a lower willingness to incur the fixed fee. The apps’ pricing strategy on the Apple App Store appears consistent with our model prediction. Among the 119,174 apps in the education genre, associated with investment goods consumption, about 15% charge a fixed fee. The average fixed fee charged is $6.70. However, among the 193,749 apps in the games genre, associated with leisure goods consumption, only about 8% charge a fixed fee. The average fixed fee charged is $3.40. The difference in fixed fees may be due to the consumers’ concern for the sunk cost effect and the self-control problem.

We demonstrate that regret-based and memory-cue-based sunk cost effects can have different implications for firm profits and contract design. Although a two-part tariff can achieve the first-best profit level when the sunk cost effect is memory-cue-based, there is a deadweight loss of surplus, and profits will be below the first-best level when the effect is regret-based. Framing the sunk cost as a memory cue in marketing communications can help firms increase the available surplus. Certain consumer characteristics might be associated with the types of sunk cost effect they suffer. For example, women are more susceptible to regret emotion than men (Li et al. 2018), and this again may have implications for the nature of marketing communications. Firms can conduct marketing research to identify consumer segments or product markets that are more likely to involve one or both types of sunk cost effect.

In the presence of consumer heterogeneity in sensitivity to sunk costs, it is common to observe firms offering consumers a menu consisting of both fixed-fee and pay-per-use contracts. Interestingly, it is never optimal for the firm to offer the fixed-fee contract to the sunk-cost-sensitive consumer nor to offer the pay-per-use contract to the consumer who is not affected by the sunk cost (see analysis and discussion in Web Appendix E). It is interesting that in a separating equilibrium, it is optimal for the firm to design a menu such that the consumer who is prone to the sunk cost effect chooses the pay-per-use contract, whereas the consumer who does not suffer from the sunk cost effect chooses the fixed-fee contract. The screening contract menu is more profitable than a single fixed-fee contract and than a single pay-per-use contract when the difference in the degree of the sunk cost effect is large; otherwise, when the difference is small, the two contracts can cannibalize each other, resulting in lower profits.

Future Research Directions

Some interesting aspects of the problem may be pursued in future work. First, it would be useful to consider repeated interactions between the firm and the consumer, and the role of the sunk cost effect in customer retention. In our main model, although the sunk cost effect can mitigate the self-control problem, the consumer might overconsume to avoid regret or might experience regret if they do not consume. In either case, the fixed-fee-induced sunk cost effect reduces consumer welfare and the willingness to continue the contract with the firm. The sunk cost effect might help explain the empirical results that the users are more likely to churn if the contract includes a fixed fee (e.g., Danaher 2002; Iyengar et al. 2011; Lambrecht and Skiera 2006). To address this issue, the firm could offer a voucher or allow the consumer to carry over the service to a future period if the consumer decides not to consume in the current period. These tactics may potentially mitigate the regret associated with the sunk cost effect. How the firm should design a contract involving repeated interactions warrants formal analysis.

It may also be interesting to consider the quantity decisions of individual consumers. The presence of the sunk cost effect may have implications for pricing structures such as the three-part tariff in telecommunication services (Ascarza, Lambrecht, and Vilcassim 2012) and the bucket pricing discrimination in online rental services (Sun, Li, and Sun 2015). Both pricing schemes include an up-front fixed fee and a quota for free usage of the service. When the consumption quantity hits the quota, the consumer faces a higher marginal price under a three-part tariff or a cap under bucket pricing discrimination. Although the up-front fixed fee can result in a high tendency to consume more, the firms may set an optimal quota to mitigate the overconsumption problem due to the sunk cost effect.

Supplemental Material

sj-pdf-1-mrj-10.1177_00222437231196824 - Supplemental material for Sunk Cost Effect, Self-Control, and Contract Design

Supplemental material, sj-pdf-1-mrj-10.1177_00222437231196824 for Sunk Cost Effect, Self-Control, and Contract Design by Xing Zhang, Ganesh Iyer, Xiaoyan Xu and Juin Kuan Chong in Journal of Marketing Research

Contract Type	Severity of Self-Control Problem
Fixed-fee contract (L*)	Pricing	L* first decreases and then weakly increases with γ	L* decreases with γ
	Profits	The sunk cost effect increases profits when γ is high and a is low	The sunk cost reduces profits
Pay-per-use contract (p*)	Pricing	p* is independent of γ
Profits	The sunk cost effect has no effect on profits; when γ and a are high, the pay-per-use contract is more profitable than the fixed-fee contract
Two-part-tariff contract (L, p)	Pricing	L* and p* have a nonmonotonic relationship with γ (see Web Appendix A)
Profits	The sunk cost effect reduces profits

Footnotes

Appendix A: Consumer’s Anticipated Sunk Cost Effect: Pilot Study

We conducted a pilot study to investigate whether people anticipate that their future consumption will be affected by the sunk cost. The replication package can be found at https://www.openicpsr.org/openicpsr/project/192762/version/V1/view.

Appendix B: Proof of Lemma 1

In this proof, we first examine the general setting whereby the cost c incurred by the consumer follows a distribution function F(c), and F(c) is smooth, continuous, and differentiable. Furthermore, the effect of sunk cost is captured by a general functional form g(γ, L), which satisfies g(γ, L) = 0 when γ = 0 or L = 0, $\frac{\partial g (γ, L)}{\partial γ} > 0$ , and $\frac{\partial g (γ, L)}{\partial L} > 0$ . By denoting the consumer’s expected utility when signing a fixed-fee contract as E(U_t=0), we have (\rm A1)

\begin{aligned} E (U_{t = 0}) = β · \\ (- L + \int_{0}^{β b + g (γ, L)} (b - c) dF (c) + \int_{β b + g (γ, L)}^{\bar{c}} (- g (γ, L)) dF (c)) . \end{aligned}

By using the Leibniz rule, we take the first-order derivative of E(U_t=0) with respect to γ: (\rm A2)

\frac{\partial E (U_{t = 0})}{\partial γ} = β [(1 - β) bf (β b + g (γ, L)) \frac{\partial g (γ, L)}{\partial γ} - \int_{β b + g (γ, L)}^{\bar{c}} \frac{\partial g (γ, L)}{\partial γ} d F (c)] .

When g(γ, L) = γL, Equation A2 reduces to

\frac{\partial E (U_{t = 0})}{\partial γ} = β [(1 - β) bLf (β b + γ L) - \int_{β b + γ L}^{\bar{c}} L dF (c)]

When β = 1, $\frac{\partial E (U_{t = 0})}{\partial γ} = - \int_{β b + γ L}^{\bar{c}} L dF (c) < 0$ , and the expected utility decreases with γ.

In this proof, we show that when the consumer is time inconsistent (β < 1) and the distribution function F(·) is weakly convex, the consumer’s expected utility E(U_t=0) first decreases and then weakly increases with γ. Note from the preceding analysis that $\frac{\partial E (U_{t = 0})}{\partial γ} |_{γ = 0} = β [(1 - β) bLf (β b) - \int_{β b}^{\bar{c}} L dF (c)]$ , so when F(·) is weakly convex and the density function f(·) is nondecreasing, $\frac{\partial E (U_{t = 0})}{\partial γ} |_{γ = 0} = β [(1 - β) bLf (β b)$ $- \int_{β b}^{\bar{c}}$ $L dF (c)] \leq (1 - β) bLf (β b) - (\bar{c} - β b) Lf (β b) = (b - \bar{c}) Lf (β b) < 0$ when $b < \bar{c}$ . Also, $\frac{\partial E (U_{t = 0})}{\partial γ} |_{γ L + β b = \bar{c}} = β L (1 - β) bf (\bar{c}) > 0$ when β < 1. Last, by taking the second-order derivative of E(U_t=0) with respect to γ, we have $\frac{\partial^{2} E (U_{t = 0})}{\partial γ^{2}} = β L^{2} [(1 - β) b f^{'} (β b + γ L) + f (β b + γ L)] > 0$ since f(·) is nondecreasing. In sum, we have shown that when β < 1, $b < \bar{c}$ , and F(·) is weakly convex, then $\frac{\partial E (U_{t = 0})}{\partial γ} |_{γ = 0} < 0$ , $\frac{\partial E (U_{t = 0})}{\partial γ} |_{γ L + β b = \bar{c}} > 0$ , and $\frac{\partial^{2} E (U_{t = 0})}{\partial γ^{2}} > 0$ , indicating that the consumer’s expected utility first decreases and then weakly increases with γ.

Next, we further show that the preceding results hold when the sunk cost effect follows a general functional form g(γ, L) when $\frac{\partial^{2} g (γ, L)}{\partial γ^{2}} = 0$ . When γ = 0, $\frac{\partial E (U_{t = 0})}{\partial γ} |_{γ = 0} = β \frac{\partial g (γ, L)}{\partial γ}$ $[(1 - β) bLf (β b) - \int_{β b}^{\bar{c}} 1 dF (c)]$ . Similar to the previous reasoning, $\frac{\partial E (U_{t = 0})}{\partial γ} |_{γ = 0} < 0$ when F(·) is weakly convex. When $γ L + β b = \bar{c}$ , $\frac{\partial E (U_{t = 0})}{\partial γ} |_{γ L + β b = \bar{c}} = β \frac{\partial g (γ, L)}{\partial γ} (1 - β) bf (\bar{c}) > 0$ holds when β < 1. By taking the second-order derivative of E(U_t=0) with respect to γ, we have $\frac{\partial^{2} E (U_{t = 0})}{\partial γ^{2}} = β \frac{\partial^{2} g (γ, L)}{\partial γ^{2}}$ $[(1 - β)$ $bf (β b + g (γ, L)) - \int_{β b + g (γ, L))}^{\bar{c}} 1 dF (c)] + β [\frac{\partial g (γ, L)}{\partial γ}]^{2} [(1 - β) b$ $f^{'} (β b + g (γ, L)) +$ $f (β b + g (γ, L))] = β [\frac{\partial g (γ, L)}{\partial γ}]^{2} [(1 - β) b f^{'} (β b + g$ $(γ,$ $L)) + f (β b + g (γ, L))] > 0$ when $\frac{\partial^{2} g (γ, L)}{\partial γ^{2}} = 0$ .

Appendix C: Proof of Proposition 1

The firm solves the profit maximization problem described in Equation 4. By rewriting the consumer’s participation constraint in Equation 4 as $\frac{γ^{2} L^{2}}{2} - L (1 + γ (1 - b)) + β b^{2} -$ $\frac{β^{2} b^{2}}{2} \geq 0$ , we find that this condition is satisfied when $L \geq$ $\frac{(1 + γ - γ b) + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}{γ^{2}}$ , or $L \leq \frac{(1 + γ - γ b) - \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}{γ^{2}}$ . However, note that when $L \geq \frac{(1 + γ - γ b) + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}{γ^{2}}$ , $γ L + β b > \frac{1}{γ} + β b > 1$ , the boundary condition βb + γL ≤ 1 does not hold. In our analysis, we focus on the solution when $L \leq \frac{(1 + γ - γ b) - \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}{γ^{2}}$ . In equilibrium, the consumer participation constraint must be binding to 0, since otherwise, the firm can increase its fixed fee to extract more surplus from the consumer. By solving $\frac{γ^{2} L^{2}}{2} -$ $L (1 + γ (1 - b)) + β b^{2} - \frac{β^{2} b^{2}}{2} = 0$ , we have the following:

If γ = 0, then the optimal fixed fee is $L * = β b^{2} - \frac{β^{2} b^{2}}{2}$ .

If γ > 0, the optimal fixed fee is (\rm A3)

L * = \frac{(1 + γ - γ b) - \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}{γ^{2}} .

Equivalently, $L * = \frac{β (2 - β) b^{2}}{(1 + γ - γ b) + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}$ .

Proof of 1. When 0 < γ ≤ 1 and β < 1, from the implicit function derivative, we have $\frac{\partial L *}{\partial γ} = \frac{L * (1 - b - γ L *)}{- 1 - γ (1 - b - γ L *)} \leq 0$ if and only if γL* ≤ (1 − b). By plugging L* into Equation A3, γL* ≤ (1 − b) can be reduced to (1 + γ − γb)² − γ²(2 − β)βb² ≥ 1, and the condition can be simplified as γ[(1 − b)² − (2 − β)βb²] + 2(1 − b) ≥ 0. Note that this condition is linear in γ, and this condition can be satisfied if and only if it is satisfied when γ = 0 and γ = 1. When γ = 0, the condition is satisfied automatically since 2(1 − b) > 0; when γ = 1, the condition is satisfied when (1 − b)² − (2 − β)βb² + 2(1 − b) > 0, which can be reduced to $β \leq {\bar{β}}_{1} = 1 - \frac{\sqrt{4 b - 3}}{b}$ . In sum, we have shown that when $β \leq {\bar{β}}_{1}$ , L* is decreasing in γ.

Proof of 2 and 3. Note from the previous proof (1) that $\frac{\partial L *}{\partial γ} = \frac{L * (1 - b - γ L *)}{- 1 - γ (1 - b - γ L *)} \leq 0$ if and only if γ[(1 − b)² − (2 − β)βb²] + 2(1 − b) ≥ 0. When $\frac{3}{4}$ < b and ${\bar{β}}_{1} < β$ , $\frac{\partial L *}{\partial γ} \leq 0$ holds if $γ \leq {\bar{γ}}_{2} = \frac{1 - b}{[(2 - β) β b^{2} - {(1 - b)}^{2}]}$ ; otherwise, $\frac{\partial L *}{\partial γ} > 0$ if $γ > {\bar{γ}}_{2}$ . Note that the boundary condition γL* + βb ≤ 1 is equivalent to γβb + γ²L* ≤ γ. By plugging L* into Equation A3, the condition can be reduced to $(1 - γ b + γ β b) \leq \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}$ . By solving this inequality, we get 2(1 − βb) ≥ γ(2b − 1), and the boundary condition is satisfied when $γ \leq \frac{2 (1 - β b)}{2 b - 1}$ . As a result, when $\frac{3}{4}$ < b and ${\bar{β}}_{1} < β < 1$ , $\frac{\partial L *}{\partial γ} \leq 0$ when $γ \leq {\bar{γ}}_{2}$ , and $\frac{\partial L *}{\partial γ} > 0$ when ${\bar{γ}}_{2} < γ \leq \frac{2 (1 - β b)}{2 b - 1}$ . When $1 \geq γ > \frac{2 (1 - β b)}{2 b - 1}$ , γL* + βb ≥ 1, the consumer will consume with probability 1, and the consumer’s expected utility is E(U_t=0) = −L + b − $\frac{1}{2}$ , and L* = b − $\frac{1}{2}$ .

Last, we show that our results here can be extended to a more general functional form of sunk cost effect g(γ, L) when $0 < \frac{\partial g}{\partial L} < 1$ . When the sunk cost effect is g(γ, L), the profit maximization problem in Equation 4 can be written as max_LΠ = (L − ag(γ, L) − aβb) subject to $\frac{1}{2}$ g(γ, L)² − (1 − b)g(γ, L) − L + $\frac{1}{2}$ β(2 − β)b² ≥ 0. Note that the profit Π increases in L since $\frac{\partial Π}{\partial L} = 1 - a \frac{\partial g}{\partial L} \geq 1 - a > 0$ . The firm’s profit is maximized when the consumer’s participation condition is binding to 0 such that $\frac{1}{2}$ g(γ, L)² − (1 − b)g(γ, L) − L + $\frac{1}{2}$ β(2 − β)b² = 0. By using the implicit derivative, we take the first-order partial derivative of the consumer’s participation condition with respect to γ, and we have $g (γ, L) (\frac{\partial g}{\partial γ} + \frac{\partial g}{\partial L} \frac{\partial L}{\partial γ}) - (1 - b) (\frac{\partial g}{\partial γ} + \frac{\partial g}{\partial L} \frac{\partial L}{\partial γ}) - \frac{\partial L}{\partial γ} = 0$ , which leads to $\frac{\partial L}{\partial γ} = \frac{- \frac{\partial g}{\partial L} (1 - b - g (γ, L))}{1 + \frac{\partial g}{\partial L} (1 - b - g (γ, L))}$ . Note that when $0 < \frac{\partial g}{\partial L} < 1$ , g(γ, L) < L and the denominator $1 + \frac{\partial g}{\partial L} (1 - b - g (γ, L))$ is always positive since $1 + \frac{\partial g}{\partial L} (1 - b - g (γ, L)) > 1 - | 1 - b - g (γ, L) | = \min {b + g (γ, L), 2 - b - g (γ, L)} > 0$ and g(γ, L) ≤ L < b. Then $\frac{\partial L}{\partial γ} > 0$ if and only if (1 − b − g(γ, L)) < 0, which is equivalent to g(γ, L) > 1 − b. By rewriting the consumer’s participation condition as L = [g(γ, L) − (1 − b)]² + (− $\frac{1}{2}$ β²b² + βb² − $\frac{1}{2}$ (1 − b)²), we find that when $β < 1 - \frac{\sqrt{2 b - 1}}{b}$ , (− $\frac{1}{2}$ β²b² + βb² − $\frac{1}{2}$ (1 − b)²) < 0, indicating that [g(γ, L) − (1 − b)]² is always positive for the full range of parameters since L > 0. Note that g|_γ=0 = 0, due to the continuity of g(γ, L), g(γ, L) − (1 − b) < 0 holds when $β < 1 - \frac{\sqrt{2 b - 1}}{b}$ , and as a result $\frac{\partial L}{\partial γ} < 0$ . Conversely, when $β \geq 1 - \frac{\sqrt{2 b - 1}}{b}$ , $\frac{\partial L}{\partial γ} \leq 0$ if g(γ, L) − (1 − b) ≤ 0, while $\frac{\partial L}{\partial γ} > 0$ if g(γ, L) > (1 − b). In sum, we have shown that when the sunk cost effect follows a more general format g(γ, L), which satisfies g(γ, L) = g(0, L) = g(γ, 0) = 0, $0 < \frac{\partial g}{\partial L} < 1$ , and $\frac{\partial g}{\partial γ} > 0$ , the optimal fixed fee L* decreases with γ when $β < 1 - \frac{\sqrt{2 b - 1}}{b}$ ; otherwise if $β \geq 1 - \frac{\sqrt{2 b - 1}}{b}$ , L* first decreases with γ when g(γ, L) ≤ (1 − b) and then increases with γ when g(γ, L) > (1 − b).

Appendix D: Proof of Proposition 2

Following the analysis in the proof of Proposition $1$ , we can get this result immediately.

In this part, we show that when $a > \max$ ${\frac{2 (1 - β b) [(2 - β) β b^{2} + {(1 - b)}^{2}] - (1 - b) (2 b - 1)}{2 (1 - β b) (2 - 3 b) + 4 b - 2}, \frac{(1 - β) β b^{2} - (1 - b) (2 - b)}{4 - 3 b}}$ , Π(L) decreases in γ. By writing the profit function of the fixed-fee contract as Π(L) = (1 − aγ)L − aβb, we take the first-order derivative of Π(L) with respect to γ, and we have $\frac{\partial Π (L)}{\partial γ} = (1 - a γ) \frac{\partial L}{\partial γ} - aL$ . By plugging $L * = \frac{β (2 - β) b^{2}}{1 + γ - γ b + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}} = \frac{A (γ)}{B (γ)}$ into $\frac{\partial Π (L)}{\partial γ}$ , in which A(γ) = β(2 − β)b² and $B (γ) = 1 + γ$ $- γ b + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}$ , we rewrite $\frac{\partial Π (L)}{\partial γ} = (1 - a γ) \frac{\partial L}{\partial γ} - aL$ as $\frac{\partial Π (L)}{\partial γ} = - \frac{A (γ)}{B {(γ)}^{2}} [(1 - a γ) \frac{\partial B (γ)}{\partial γ} + aB (γ)]$ . Note that A(γ) > 0 and B(γ) > 0; then we show $\frac{\partial Π (L)}{\partial γ} < 0$ by showing $f (γ) = (1 - a γ) \frac{\partial B (γ)}{\partial γ} + aB (γ) > 0$ when $γ \in (0, {\bar{γ}}_{1})$ . By plugging $B (γ) = 1 + γ - γ b + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}$ and $\frac{\partial B (γ)}{\partial γ} = 1 - b + \frac{(1 - b) (1 + γ - γ b) - γ (2 - β) β b^{2}}{\sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}$ into $\frac{\partial Π (L)}{\partial γ}$ , we have $f (γ) = (1 - a γ) [1 - b +$ $\frac{(1 - b) (1 + γ - γ b) - γ (2 - β) β b^{2}}{\sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}] + a [1$ $+ γ - γ b + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}] = 1 -$ $b + a + a$ $\sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}} + (1 - a γ)$ $\frac{(1 - b) (1 + γ - γ b) - γ (2 - β) β b^{2}}{\sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}$ , and f(γ) > 0 if $a + a \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}} + (1 - a γ)$ ∙ $\frac{(1 - b) (1 + γ - γ b) - γ (2 - β) β b^{2}}{\sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}} > 0$ , which is equivalent to $a (\sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}) + a (1 + γ - γ b)$ $> γ (2 - β) β b^{2} - (1 - b) (1 + γ - γ b)$ . Since (1 + γ − γb)² − γ²(2 − β)βb² > (1 + γ − γb)² − γ²b² > (1 + γ − 2γb)², we have $\sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}} > 1 + γ - 2 γ b$ . By substituting $\sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}$ as 1 + γ − 2γb into the preceding inequality, it is sufficient to prove that f(γ) > 0 by showing a(2 + 2γ − 3γb) > γ(2 − β)βb² − (1 − b)(1 + γ − γb). Note that this inequality is linear in γ; then a(2 + 2γ − 3γβ) > γ(2 − β)βb² − (1 − b)(1 + γ − γb) holds for $γ \in (0, {\bar{γ}}_{1})$ if it holds when γ = 0, γ = 1, and $γ = \frac{2 (1 - β b)}{2 b - 1}$ . When γ = 0, this inequality reduces to 2a > −(1 − b), and it holds automatically. When $γ = \frac{2 (1 - β b)}{2 b - 1}$ , a(2 + 2γ − 3γβ) > γ(2 − β)βb² − (1 − b)(1 + γ − γb) holds when a> $\frac{2 (1 - β b) (2 - β) β b^{2} - (1 - b) (2 b - 1) + 2 (1 - β b) {(1 - b)}^{2}}{2 (1 - β b) (2 - 3 b) + 4 b - 2}$ . If γ = 1, a(2 + 2γ − 3γβ) > γ(2 − β)βb² − (1 − b)(1 + γ − γb) holds when $a > \frac{(2 - β) β b^{2} - (1 - b) (2 - b)}{4 - 3 b}$ . In sum, we have shown that when $a > {\bar{a}}_{1} = \max$ ${\frac{2 (1 - β b) (2 - β) β b^{2} - (1 - b) (2 b - 1) + 2 (1 - β b) {(1 - b)}^{2}}{2 (1 - β b) (2 - 3 b) + 4 b - 2},$ $\frac{(2 - β) β b^{2} - (1 - b) (2 - b)}{4 - 3 b}}$ , Π(L) decreases with γ.

Last, we identify the conditions under which the fixed-fee contract can be more profitable in the presence of the sunk cost effect when a > 0. By denoting the profit of a fixed-fee contract when γ = 0 as π₀, we have Π₀ =

\frac{1}{2}

β(2 − β)b² − aβb. Similarly, by denoting the profit of the fixed-fee contract when γ > 0 as Π₁, we have Π₁ = (1 − aγ)L* − aβb if

γ \leq \frac{2 (1 - β b)}{2 b - 1}

, in which

L * = \frac{β (2 - β) b^{2}}{1 + γ - γ b + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}

; otherwise, Π₁ = b − a −

\frac{1}{2}

γ \geq \frac{2 (1 - β b)}{2 b - 1}

. By comparing Π₀ and Π₁ when

γ \leq \frac{2 (1 - β b)}{2 b - 1}

, Π₁ ≥ Π₀ if and only if (1 − aγ)L* ≥

\frac{1}{2}

β(2 − β)b². By plugging

L * = \frac{β (2 - β) b^{2}}{1 + γ - γ b + \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}

into the inequality, Π₁ ≥ Π₀ holds if and only if

(1 - γ + γ b - 2 a γ) \geq \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}

. Simplifying, we have Π₁ ≥ Π₀ holds if and only if

γ \geq \frac{4 (1 + a - b)}{[4 a (1 - b + a) + (2 - β) β b^{2}]}

To check the boundary condition, we first check $\frac{4 (1 + a - b)}{[4 a (1 - b + a) + (2 - β) β b^{2}]} \leq \frac{2 (1 - β b)}{2 b - 1}$ . Note from Proposition 1 that when the profits of the firm are higher in the presence of the sunk cost effect, the optimal fixed fee L* is increasing in γ. Showing that $\frac{4 (1 + a - b)}{[4 a (1 - b + a) + (2 - β) β b^{2}]} \leq \frac{2 (1 - β b)}{2 b - 1}$ is equivalent to showing that $Π (γ = \frac{2 (1 - β b)}{2 b - 1}) = b - a - \frac{1}{2} \geq Π (γ = \frac{4 (1 + a - b)}{[4 a (1 - b + a) + (2 - β) β b^{2}]})$ $= Π_{0} = \frac{1}{2} β (2 - β) b^{2} - a β b$ . Solving this inequality, we have (1 − βb)((2 − β)b − 1) ≥ 2(1 − βb)a, which is satisfied when $β \leq 2 - \frac{1}{b}$ and a ≤ $\frac{1}{2}$ [(2 − β)b − 1]. Next, we check $\frac{4 (1 + a - b)}{[4 a (1 - b + a) + (2 - β) β b^{2}]} \leq 1$ , and it is satisfied when $β \geq 1 - \frac{\sqrt{b^{2} - 4 (1 - a) (1 - b + a)}}{b}$ . To ensure b² − 4(1 − a)(1 − b + a) ≥ 0, we have $a \leq \frac{b}{2} - \sqrt{1 - b}$ and $b \geq 2 \sqrt{2} - 2$ .

In sum, we have shown that when $a \leq \min {\frac{b}{2} - \sqrt{1 - b}, \frac{1}{2} [(2 - β) b - 1]}$ , $b \geq 2 \sqrt{2} - 2$ , and $\min {{\bar{β}}_{1}, 1 - \frac{\sqrt{b^{2} - 4 (1 - a) (1 - b + a)}}{b}} \leq β \leq 2 - \frac{1}{b}$ , the profits of the fixed-fee contract can be higher in the presence of the sunk cost effect when $\frac{4 (1 + a - b)}{[4 a (1 - b + a) + (2 - β) β b^{2}]} \leq γ$ .

Appendix E: Proof of Corollary 1

To simplify our exposition, we use the same notation as in the analysis for investment goods. If the consumer chooses to consume the goods, they receive a stochastic benefit b ∼ U[0,1] at t = 1 and incur a deterministic cost 0 ≤ c < $\frac{1}{2}$ at t = 2. Otherwise, if the cost is sufficiently high such that c ≥ $\frac{1}{2}$ , a sophisticated consumer will sign the contract if and only if they are sufficiently time consistent ( $β > 2 - \frac{1}{c}$ ). If the consumer does not consume, they will suffer from the sunk cost effect and experience a disutility −γL. From the perspective at t = 0, the consumer will consume the product (e.g., play the game) at t = 1 if b − c > −γL. As time goes by, at t = 1, they will actually consume if b − βc > −γL. Given the randomness in benefit b, the probability of consumption is 1 − F(βc − γL). The firm charges a fixed fee such that $E [U_{t = 0}] =$ $β (- L + \int_{0}^{β c - γ L} (- γ L) dF (c) + \int_{β c - γ L}^{1} (b - c) dF (c)) = γ^{2} L^{2} -$ $2 L (1 + c γ) + (1 - β c) (1 + β c - 2 c) = 0$ .

If γ = 0, then L* = $\frac{1}{2}$ (1 − βc)(1 + βc − 2c).

When γ > 0, we have $L * =$ $\frac{(1 + c γ) - \sqrt{{(1 + c γ)}^{2} - γ^{2} (1 - β c) (1 + β c - 2 c)}}{γ^{2}}$ $= \frac{(1 - β c) (1 + β c - 2 c)}{(1 + c γ) + \sqrt{{(1 + c γ)}^{2} - γ^{2} (1 - β c) (1 + β c - 2 c)}}$ . Note from the boundary condition such that 0 ≤ βc − γL < c − γL, by using the implicit derivative, we take the first-order derivative of L* with respect to γ, and we have $\frac{\partial L *}{\partial γ} = \frac{L * (c - γ L *)}{- 1 - γ (c - γ L *)} < 0$ . The profits of the firm also decrease in the degree of the sunk cost effect since $\frac{\partial Π}{\partial γ} = (1 - a γ) \frac{\partial L}{\partial γ} - aL < 0$ .

Last, we check the boundary condition such that βc − γL > 0. By plugging $L * = \frac{(1 + c γ) - \sqrt{{(1 + c γ)}^{2} - γ^{2} (1 - β c) (1 + β c - 2 c)}}{γ^{2}}$ into βc − γL > 0, the condition can be reduced to $γ < γ_{L} = \frac{2 β c}{1 - 2 c}$ . When γ ≥ γ_L, the consumer will consume with probability 1, and L* = $\frac{1}{2}$ − c.

Appendix F: Proof of Proposition 3

Following the previous analysis, we use E(L) (Π(L)) and E(p) (Π(p)) to denote the expected utility (revenue) for the consumer (firm), when the contract is based on a fixed fee (L) and a pay-per-use fee (p), respectively.

E (L) = - L + \int_{0}^{β b + γ L} (b - c) dc + \int_{β b + γ L}^{1} (- γ L) dc = \frac{1}{2} γ^{2} L^{2} - L (1 + γ - γ b) + \frac{1}{2} β (2 - β) b^{2} .

E (p) = \int_{0}^{β b - p} (b - p - c) dc = \frac{1}{2} (2 b - p - β b) (β b - p) .

Π (L) = L + \int_{0}^{β b + γ L} (- a) dF (c) = (1 - a γ) L - a β b .

Π (p) = (β b - p) (p - a) .

First, we check the consumer’s participation constraint. When the firm offers a fixed-fee contract, following the analysis in the proof of Proposition 1, we have the optimal fixed fee $L * = \frac{(1 + γ - γ b) - \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}{γ^{2}}$ . When the firm offers a pay-per-use fee contract, it is optimal for a firm to set the price as p* = $\frac{1}{2}$ (a + βb), and the consumer’s participation constraint is satisfied since $E (p *) = \frac{1}{2} (2 b - β b - \frac{β b + a}{2}) (β b - \frac{β b + a}{2}) = \frac{1}{8} (4 b - 3 β b - a) (β b - a) \geq 0$ when $a \leq \bar{a}$ . By comparing the revenue of a fixed-fee contract with that of a pay-per-use contract, we find that Π(L) ≥ Π(p) if and only if $(1 - a γ) \frac{(1 + γ - γ b) - \sqrt{{(1 + γ - γ b)}^{2} - γ^{2} (2 - β) β b^{2}}}{γ^{2}} - a β b \geq \frac{1}{4} (β b - a)^{2}$ . Simplifying, we find this condition would hold if and only if Aγ² + Bγ + C ≥ 0, in which A = a²X + 2aYZ + Z² > 0, B = 2(aZ − aX − YZ) < 0, C = (X − 2Z), X = (2 − β)βb², Y = (1 − b), and Z = $\frac{1}{4}$ (βb + a)².

In this proof, we show that Π(L) ≥ Π(p) always holds when $a < {\bar{a}}_{2} = \frac{1}{6} [- 2 + 2 b - β b + \sqrt{{(- 2 + 2 b - β b)}^{2} - 3 (3 β^{2} b^{2} - 4 β b^{2} + 2 {(1 - b)}^{2})}]$ and ${\bar{β}}_{2} \leq β \leq {\bar{β}}_{3}$ . By calculating the discriminant of quadratic function Aγ² + Bγ + C = 0, we have Δ = B² − 4AC ≥ 0 is equivalent to (Y + a)² ≥ (X − 2Z). In other words, Π(L) > Π(p) always holds when Δ = B² − 4AC < 0 and (Y + a)² < (X − 2Z). By plugging in X = (2 − β)βb², Y = (1 − b), and Z = $\frac{1}{4}$ (βb + a)², (Y + a)² < (X − 2Z) can be reduced to $\frac{3}{2}$ a² + a(2 − 2b + βb) + $\frac{3}{2}$ β²b² − 2βb² + (1 − b)² < 0. Solving this inequality, we have $a < {\bar{a}}_{2} = \frac{1}{3} [- 2 + 2 b - β b +$ $\sqrt{{(- 2 + 2 b - β b)}^{2} - 6 {(1 - b)}^{2} + 3 (4 - 3 β) β b^{2}}]$ . To ensure ${\bar{a}}_{2} \geq 0$ , we have ${\bar{β}}_{2} = \frac{2}{3} - \frac{1}{3 b}$ $\sqrt{4 b^{2} - 6 {(1 - b)}^{2}} \leq β \leq \frac{2}{3} + \frac{1}{3 b} \sqrt{4 b^{2} - 6 {(1 - b)}^{2}}$ . By denoting ${\bar{β}}_{3}$ as $\min {\frac{2}{3} + \frac{1}{3 b} \sqrt{4 b^{2} - 6 {(1 - b)}^{2}}, 1}$ , we have shown that when $a < {\bar{a}}_{2}$ and ${\bar{β}}_{2} \leq β \leq {\bar{β}}_{3}$ , Π(L) ≥ Π(p) always holds.

In this proof, we show that when ${\bar{a}}_{2} \leq a \leq \min {\bar{a}, {\bar{a}}_{3}}$ and ${\bar{β}}_{4} \leq β < 1$ , Π(L) > Π(p) if and only if $γ < {\bar{γ}}_{3} =$ $\frac{16 a (2 - β) β b^{2} + 4 (1 - b - a) {(β b + a)}^{2} - 2 {(β b + a)}^{2} \sqrt{4 {(1 - b + a)}^{2} - 4 (2 - β) β b^{2} + 2 {(β b + a)}^{2}}}{16 a^{2} (2 - β) β b^{2} + 8 a (1 - b) {(β b + a)}^{2} + {(β b + a)}^{4}}$ . Using the notation defined previously, we find that when $a \leq {\bar{a}}_{3} = \sqrt{(2 - β) β} b - \frac{\sqrt{2}}{2} β b$ , C = (X − 2Z) > 0 and ${\bar{γ}}_{3} = \frac{- B - \sqrt{B^{2} - 4 AC}}{2 A} > 0$ . In this case, when ${\bar{a}}_{3} \geq a \geq {\bar{a}}_{2}$ , the discriminant Δ = B² − 4AC is positive, and Aγ² + Bγ + C > 0 holds when $γ < {\bar{γ}}_{3} = \frac{- B - \sqrt{B^{2} - 4 AC}}{2 A}$ or $γ > \frac{- B + \sqrt{B^{2} - 4 AC}}{2 A}$ . However, note that when $γ > \frac{- B + \sqrt{B^{2} - 4 AC}}{2 A}$ , the boundary condition γL + βb ≤ 1 does not hold, and we only keep the solution $γ < {\bar{γ}}_{3} = \frac{- B - \sqrt{B^{2} - 4 AC}}{2 A}$ . In sum, we have shown that when ${\bar{a}}_{2} \leq a \leq \min {\bar{a}, {\bar{a}}_{3}}$ , Π(L) > Π(p) if and only if $γ < {\bar{γ}}_{3} = \frac{- B - \sqrt{B^{2} - 4 AC}}{2 A}$ . Last, to ensure the boundary condition ${\bar{γ}}_{3} \leq 1$ , we impose ${\bar{β}}_{4} = \max {\frac{3}{2 b} - 1, \frac{a}{b} + \frac{\sqrt{4 b - 4 a - 2}}{b}} \leq β < 1$ . In sum, we find that when we have shown that when ${\bar{β}}_{4} \leq β < 1$ and ${\bar{a}}_{2} \leq a \leq \min {\bar{a}, {\bar{a}}_{3}}$ , the fixed-fee contract is more profitable than the pay-per-use fee contract when $γ < {\bar{γ}}_{3}$ , whereas the pay-per-use fee is more profitable when $γ > {\bar{γ}}_{3}$ .

Appendix G: Proof of Proposition 4

In this proof, we show that under a two-part tariff, the firm’s profits decrease with the degree of sunk cost effect γ and the presence of the sunk cost effect reduces the firm’s profits. Note that in the equilibrium, the consumer participation constraint in Equation 8 is binding to 0 and $L = \int_{0}^{β b - p + γ L} (b - p - c) dF (c) + \int_{β b - p + γ L}^{1} (- γ L) dF (c)$ , since otherwise, the firm can increase its profits by setting a higher L. Note that the consumer participation constraint is equivalent to $\frac{1}{2}$ [p² − 2p(b + γL) + (b + γL)²] = $\frac{1}{2}$ [1 − (2 − β)β]b² + (1 + γ)L , so we can rewrite p as a function of L such that $p (L) = γ L + b - \sqrt{2 (1 + γ) L + {(1 - β)}^{2} b^{2}}$ . In this case, the profits of the firm in Equation 8 can be written as a function of L and p such that $Π (p, L) = \int_{0}^{β b - p + γ L} (b - c - a) dF (c)$ $+ \int_{β b - p + γ L}^{1} (- γ L) dF (c) = \frac{1}{2} (β b + γ L - p) (2 b - 2 a - β b + γ L$ $+ p) - γ L$ . By taking the partial derivative of Π(p, L) with respect to L, we have $\frac{\partial Π}{\partial L} = \frac{1}{2} (γ - \frac{\partial p}{\partial L}) (2 b - 2 a - β b + γ L$ $+ p) + \frac{1}{2} (γ + \frac{\partial p}{\partial L}) (β b + γ L - p) - γ$ . Note that in the equilibrium when L maximizes the firm’s profits, $\frac{\partial Π}{\partial L} = 0$ , and we have $γ (1 - b + a - γ L) = \frac{\partial p}{\partial L} (β b - p - b + a)$ .

Then we show that the profits of firm Π(p, L) are decreasing in γ. By taking the total derivative of Π(p, L) with respect to γ, we have $\frac{d Π}{d γ} = \frac{\partial Π}{\partial L} \frac{\partial L}{\partial γ} + \frac{\partial Π}{\partial p} \frac{\partial p}{\partial γ} = \frac{1}{2} (L + γ \frac{\partial L}{\partial γ} - \frac{\partial p}{\partial γ} - \frac{\partial p}{\partial L} \frac{\partial L}{\partial γ}) (2 b$ $- 2 a - β b + γ L + p) + \frac{1}{2} (L + γ \frac{\partial L}{\partial γ} + \frac{\partial p}{\partial γ} + \frac{\partial p}{\partial L} \frac{\partial L}{\partial γ}) (β b + γ L - p)$ $- L - γ \frac{\partial L}{\partial γ}$ . To simplify, we have $\frac{d Π}{d γ} = L (b - a + γ L - 1) + γ \frac{\partial L}{\partial γ} (b - a + γ L - 1) + \frac{\partial p}{\partial γ} (β b - b + a - p) + \frac{\partial p}{\partial L} \frac{\partial L}{\partial γ} (β b$ $- p - b$ $+ a)$ . Note that when $\frac{\partial Π}{\partial L} = 0$ , $γ (1 - b + a - γ L) = \frac{\partial p}{\partial L}$ $(β b - p - b + a)$ ; then $\frac{d Π}{d γ} = L (b - a + γ L - 1) + \frac{\partial p}{\partial γ} (β b -$ $b + a - p)$ . Next, we show $\frac{d Π}{d γ} = L (b - a + γ L - 1)$ $+ \frac{\partial p}{\partial γ} (β b - b + a - p) < 0$ .

By denoting f(L, p) = L + (βb + γL − p)(p − a) and h(L, p) = −(1 + γ)L + $\frac{1}{2}$ (βb + γL − p)(2b − p − βb + γL), the original problem in Equation 9 can be written as max_L,pf(L, p) subject to h(L, p) ≥ 0. By using the Lagrangian multiplier, we have $\frac{\partial f}{\partial L} / \frac{\partial f}{\partial p} = \frac{\partial h}{\partial L} / \frac{\partial h}{\partial p}$ . By plugging in $\frac{\partial f}{\partial L} = 1 + γ (p - a)$ , $\frac{\partial f}{\partial p} = β b + a + γ L - 2 p$ , $\frac{\partial h}{\partial L} = - 1 - γ (1 - b + p - γ L)$ , and $\frac{\partial h}{\partial p} = - b - γ L + p$ , the condition in the Lagrangian multiplier can deduce $\frac{1 + γ (p - a)}{β b + a + γ L - 2 p} = \frac{1 + γ (1 - b + p - γ L)}{b + γ L - p} = \frac{γ (- 1 - a + γ L + b)}{β b - p + a - b}$ . Next, we show (−1 − a + γL + b) < 0 by contradiction. Suppose (−1 − a + γL + b) > 0. Since 1 + γ(1 − b + p − γL) > 0 and b + γL − p > 0, from $\frac{1 + γ (1 - b + p - γ L)}{b + γ L - p} = \frac{γ (- 1 - a + γ L + b)}{β b - p + a - b}$ we can deduce βb − p + a − b > 0. Note that βb + γL − p ≤ 1, and βb − p + a − b < βb + a − b − (βb + γL − 1) ≤ 1 + a − γL − b < 0, leading to a contradiction.

Knowing that (−1 − a + γL + b) < 0, the total derivative of Π(p, L) with respect to γ can be written as $\frac{d Π}{d γ} = L (b - a + γ L - 1) + (L - \frac{L}{γ L + b - p}) \frac{γ (b - a - 1 + γ L) (γ L + b - p)}{1 + γ (1 + p - γ L - b)}$ , since $\frac{\partial p}{\partial γ} = L - \frac{L}{\sqrt{2 (1 + γ) L + {(1 - β)}^{2} b^{2}}} = L - \frac{L}{γ L + b - p}$ and [1 + γ(1 + p − γL − b)](βb − b + a − p) = γ(γL + b − a − 1)(b + γL − p) (Lagrangian multiplier $\frac{1 + γ (1 - b + p - γ L)}{b + γ L - p} = \frac{γ (- 1 - a + γ L + b)}{β b - p + a - b}$ ). To simplify, we have $\frac{d Π}{d γ} = L \frac{b - a + γ L - 1}{1 + γ (1 + p - γ L - b)} < 0$ . In sum, we have shown that the profits of the firm under a two-part tariff decrease with the degree of the sunk cost effect γ monotonically such that $\frac{d Π}{d γ} < 0$ , and as a result, the presence of the sunk cost effect reduces the firm’s profit.

Acknowledgments

The authors are grateful for the comments from the JMR review team and from Junhong Chu, Teck Ho, Fuhai Hong, Chenxi Liao, Bin Liu, Yunfeng Lu, Hang Wu, Yue Wu, Dai Yao, Songfa Zhong, and seminar participants at the University of Chinese Academy of Sciences and the CDJ Workshop at Seoul National University.

Author Contributions

The first three authors contributed equally to the article.

Coeditor

Peter Danaher

Associate Editor

Wilfred Amaldoss

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Xing Zhang

Notes

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Contract Type		Severity of Self-Control Problem
Contract Type		Mild (i.e., High β)	Severe (i.e., Low β)
Fixed-fee contract (L*)	Pricing	L* first decreases and then weakly increases with γ	L* decreases with γ
	Profits	The sunk cost effect increases profits when γ is high and a is low	The sunk cost reduces profits
Pay-per-use contract (p*)	Pricing	p* is independent of γ
Pay-per-use contract (p*)	Profits	The sunk cost effect has no effect on profits; when γ and a are high, the pay-per-use contract is more profitable than the fixed-fee contract
Two-part-tariff contract (L, p)	Pricing	L* and p* have a nonmonotonic relationship with γ (see Web Appendix A)
Two-part-tariff contract (L, p)	Profits	The sunk cost effect reduces profits