A New Disneyland Dilemma: Seasonal Pricing for Emotional Mickey Mouse Fans∗ Cristina Nistor†

Matthew Selove‡

November 16, 2016

Abstract Fans are disappointed when an entertainment venue is sold out, but they also consider it unfair to pay a higher price than others do. We develop a model in which a firm maximizes profits given these emotional aspects of customer behavior. If customers are “naive” about how unfairness and disappointment can reduce future consumption utility, then the firm must either set a high price during all periods or exit the market during peak demand to avoid antagonizing customers. On the other hand, if “sophisticated” customers anticipate how unfairness and disappointment can reduce future consumption utility, then the firm stays active in the market during peak demand and sets a low price during off-peak demand. We derive conditions in which profits are higher, and average customer utility is also higher, if customers are sophisticated rather than naive.

∗

Our paper’s title refers to Oi (1971), “A Disneyland Dilemma: Two-Part Tariffs for a Mickey Mouse Monopoly.” Our model abstracts away from two-part tariffs and assumes, in each period, the firm charges only a lump sum admission fee; in fact, Disneyland’s current policy is that rides are free conditional on paying for admission. † Argyros School of Business and Economics, Chapman University, email: [email protected] ‡ Argyros School of Business and Economics, Chapman University, email: [email protected]

1

Introduction

Customers who attempt to buy a consumer packaged good that is out of stock are less likely to buy the seller’s products in the future (Fitzsimons 2000; Anderson et al. 2006; Jing and Lewis 2011). Managers of theme parks, sports teams, and popular musicians also worry that disappointed fans, who make unsuccessful purchase attempts, become less profitable customers in the future. Entertainment firms depend on emotional connections with fans to generate profits, and fans who unsuccessfully attempt to purchase access to a sold-out venue often develop negative emotional associations with the seller’s brand (News12 2009; Fritz 2015; DailyMail 2016). The challenge these firms face is some times of year, or some events, are so popular that demand will exceed capacity unless firms set very high prices. On the other hand, some customers consider it unfair for firms to increase prices during peak demand (Kahneman et al. 1986; Bolton et al. 2003). Many entertainment firms worry about antagonizing their fans if they set prices that vary across events or across seasons. For example, National Basketball Association (NBA) commissioner David Stern stated in an interview (in the year 2003) that the league had not adopted “variable pricing,” which involves a team setting higher ticket prices for more popular games than for less popular games, because doing so “raises questions about the fairness of your pricing” (Lefton and Lombardo 2003).1 Thus, entertainment firms often face a choice between maintaining stable prices (and disappointing some customers who are unable to purchase during peak demand) or setting higher prices during peak demand (which some customers consider unfair). Either pricing strategy is likely to antagonize some of a firm’s loyal customers. Disneyland represents an example of this dilemma. The theme park has recently had to close their gates to new customers after reaching capacity during winter break (Martin 2014; Munarriz 2015) and during special events such as the first day of the 1

Most professional sports teams have traditionally set ticket prices that are constant across games (Drayer et al. 2012). However, some sports teams have recently started charging higher prices for tickets to more popular games, and there is ongoing debate and concern within the industry about the extent to which fans consider such price practices unfair (Drayer et al. 2012; Shapiro et al. 2016).

2

park’s sixtieth anniversary celebration (Pimentel and Yee 2015; Niles 2015). Imagine the disappointment of a family who decides to spend Christmas Day at Disneyland, but when they arrive at the theme park gates, an employee in the ticket booth tells them they cannot enter because the park it is at capacity.2 Concerned that such experiences are generating unhappy memories of the theme park that aspires to be “The Happiest Place on Earth,” Disneyland recently changed from a policy in which ticket prices stayed the same all year to a new policy in which prices of one-day tickets vary, with higher prices during times of year with high demand (Fritz 2015, 2016). However, such policy changes antagonize some Disneyland fans, who react with “outrage” on social media to price changes they perceive as unfair (Pimentel 2015). In an article praising Disneyland’s new seasonal price policy, one pricing consultant acknowledged that some people consider the theme park’s decision to set higher prices during peak demand to constitute price “gouging,” and noted that “managers are often wary of raising prices during times of high demand because they fear a consumer backlash” (Mohammed 2015).3 Given that The Walt Disney Company’s theme parks, including Disneyland and other parks throughout the world, generate revenues of over $11 billion per year and profits of over $2 billion per year, setting an optimal price policy that accounts for these emotional reactions by customers is an important priority for senior management of the company (Fritz 2015). This paper develops a formal game theoretic model to explore this type of pricing problem. Although it is not intended to reflect all the details of any particular company, our model is consistent with several aspects of Disneyland’s theme park business, including seasonal demand fluctuations with a capacity constraint, repeat customers with utility functions that can change in response to perceived unfairness or disappointment, and a firm that sets prices to maximize profits over multiple seasons, 2 If they had already bought tickets online, they could either wait for the gates to open again (which might take several hours) or return on a different day to use their tickets. 3 We have interviewed Disneyland executives who told us their theme park has been considering seasonal ticket price variation for several years but had not previously adopted such a price policy because of potential fairness concerns.

3

with equilibrium price outcomes that can involve either stable prices or prices that vary by season. In our model, if all customers were rational, the firm would set a high price when demand is high, and a low price when demand is low.4 However, we assume some customers react to the firm’s strategy in an emotional manner. In particular, some customers exhibit fairness concerns, some exhibit disappointment aversion, and some exhibit both of these effects. If a customer with fairness concerns pays a high price in one period and then observes a lower price in another period, this customer will experience reduced utility from consuming the product in the latter period and subsequent periods. Similarly, if a customer with disappointment aversion attempts to purchase the product in a given period but is denied the opportunity to do so because of the capacity constraint, this customer will experience reduced utility from consuming the product in subsequent periods. We first analyze the model under the assumption that “naive” behavioral customers do not anticipate how unfairness and disappointment can reduce their future consumption utility, either because they lack self-awareness about their own emotional reactions or because they have incorrect (but rationalizable) beliefs about the strategies of other customers. If naive customers have strong fairness concerns and strong disappointment aversion, the firm either sets a high price during all periods, or it temporarily exits the market during peak demand, in order to avoid causing customers to experience feelings of unfairness or disappointment.5 We next analyze the model under the assumption that “sophisticated” behavioral 4

We refer to “high” and “low” prices in relative terms. Given Disneyland’s premium position among theme parks, their profit-maximizing price during off-peak times of year, which we call a “low” price, could still be quite high. 5 In our model, the firm can temporarily exit by setting a very high price for the period. Exiting the market during peak demand might seem like an unusual strategy, but some real world firms do follow this strategy. For example, on the day after Thanksgiving, a popular shopping day known as “Black Friday,” many American retailers face a capacity constraint in terms of physical retail space and inventory. As a result, many disappointed shoppers are unable to buy all the products they want (Seufert 2015). Rather than disappointing customers with stock-outs or angering customers with high prices, Recreational Equipment Inc. (REI) remains closed on Black Friday. The outdoor clothing and equipment retailer encourages customers to “spend Black Friday outdoors instead of in a stuffy, crowded mall.”(Johnson 2015)

4

customers anticipate how paying an unfair price or being disappointed will affect their future consumption utility, and customers and the firm adjust their strategies accordingly. In this case, when fairness concerns and disappointment aversion are both strong, two alternative outcomes can arise in equilibrium. If disappointment aversion is particularly strong, the firm sets a high price during high demand, and sophisticated customers with fairness concerns do not purchase during this period (even if their valuation exceeds the price), allowing the firm to reduce its prices during subsequent periods without antagonizing customers. If fairness concerns are particularly strong, the firm sets low prices during all periods, and sophisticated customers with disappointment aversion randomize when deciding whether to attempt to purchase during peak demand, and some (but not all) of these customers end up being disappointed (denied a chance to purchase). If there are sufficiently many periods with low demand, then the firm earns higher profits if customers are sophisticated rather than naive. Because sophisticated consumers avoid purchasing in situations that might lead to negative emotional experiences, the firm can stay active in the market during peak demand (with either a high price or a low price) and then set a low price during low-demand periods, while antagonizing fewer customers than would occur if all customers naively attempted to purchase during peak demand. Customers also experience higher average utility in the equilibrium with sophisticated customers and low prices than in any possible equilibrium with naive customers. One interpretation of our model is that naive behavior results from a coordination failure in which customers make too many purchase attempts, because they have rationalizable, but incorrect, beliefs about the strategies of other customers. For example, if the firm sets a low price during peak demand, then under some conditions, it is rationalizable for all customers to attempt to purchase based on an incorrect belief that their purchase attempt is certain to succeed because many other customers will

5

not attempt to purchase.6 Previous research has shown that pre-game communication helps solve such coordination problems (Cooper et al. 1992; Rabin 1994). In the context of our model, if the firm or a third party provided customers with accurate information about other customers’ strategies in the form of demand forecasts, for example, such information could potentially increase profits and customer welfare by discouraging purchase attempts that are likely to generate negative emotions. Though Disneyland has a policy of not reporting daily attendance numbers or projections of future daily attendance (Martin 2015), several third-party websites and smartphone applications do provide such information (Tully 2012). Our results imply Disneyland might benefit from the existence of these third-party sources of attendance projections.

2

Literature Review

Many previous theoretical and experimental papers have studied fairness concerns. Experiments have shown that people may reject offers they perceive as unfair even if doing so reduces their own monetary payoff (Hoffman et al. 1994; Camerer and Thaler 1995; Zwick and Chen 1999). Theoretical literature has developed models in which unfairness either directly reduces a player’s utility (Fehr and Schmidt 1999; Bolton and Ockenfels 2000; Cui et al. 2007; Guo 2015; Li and Jain 2016); or causes a player to want to punish those who have treated him unfairly (Rabin 1993; Rotemberg 2005). Our paper is agnostic between these two modeling approaches. We assume customers with fairness concerns who believe they have paid an unfair price reduce their future consumption of the firm’s product, which could occur either because the customer desires the product less frequently or because the customer wants to punish the firm. Particularly relevant to our model is a field experiment showing that, when customers buy a product and later observe the same product being sold at a lower price, some of these customers become angry at the firm and reduce their future purchases of the 6

Strategies are rationalizable if they survive iterative elimination of strictly dominated strategies (Fudenberg and Tirole 1991).

6

firm’s products (Anderson and Simester 2010). Similarly, in our model, customers with fairness concerns who buy at a high price and later observe a lower price reduce their purchase frequency. Previous literature has also studied disappointment aversion. Experiments have shown that participants choosing among lotteries deviate from rational choice axioms in a manner that is consistent with assigning an additional negative weight to outcomes that are worse than the expected outcome (Loomes and Sugden 1986, 1987). Theoretical literature has developed models in which players try to avoid such disappointing outcomes (Bell 1985; Gul 1991; Liu and Shum 2013). Because our model focuses instead on how disappointment affects subsequent purchase decisions, our paper is more closely related to various lab experiments showing that disappointment reduces subsequent product valuations. Product performance that falls short of expectations has a greater absolute impact on participants’ reported happiness with their chosen product than performance that exceeds expectations (Inman et al. 1997). If participants are endowed with a product and induced to feel disappointment or other negative emotions through an unrelated task, they associate their negative emotions with the product and are willing to sell it at a lower price than respondents in a control condition (Lerner et al. 2004; Martinez et al. 2011). Participants are, on average, less likely to purchase again from a simulated store after attempting to buy an out-of-stock product, although there is a segment of participants (who score high on a personality trait called “reactance”) who desire a product more after it is out of stock (Moore and Fitzsimons 2014). In our model, customers with disappointment aversion who make a failed purchase attempt experience disappointment, and this negative emotion results in less frequent consumption by this group of disappointed customers. Our paper is part of the theoretical literature in behavioral industrial organization, which studies how profit-maximizing firms react to customers who deviate from standard assumptions of rational choice models (Gabaix and Laibson 2006; Orhun 2009; Guo and Zhang 2012). Particularly relevant to our paper are models of self7

control, which have introduced a distinction between “naive” behavioral types who are unaware of their deviations from rationality and “sophisticated” behavioral types who are self-aware about these deviations (O’Donoghue and Rabin 2001; DellaVigna and Malmendier 2004). Similarly, we allow customers to be either naive (unaware) or sophisticated (aware) about how unfairness and disappointment can affect their future consumption utility. Our paper is also related to literature on dynamic pricing with capacity constraints (Xie and Shugan 2001; Su 2010; Nasiry and Popescu 2012). Our contribution is to analyze the interaction of fairness concerns with disappointment aversion in such a model. Liu and Shum (2013) also incorporate disappointment aversion into a multiperiod pricing model with capacity constraints, and derive conditions in which the firm sets either constant prices or prices that vary across periods. However, their model does not involve fairness concerns or repeat purchases. As a result, their paper’s focus and results are quite different than ours. For example, in their model, the firm tries to induce each customer to purchase one time at the highest possible price; whereas in our model, the firm tries to prevent customers from experiencing negative emotions so that each customer purchases multiple times. As far as we know, our paper is the first theoretical analysis of the interaction between fairness concerns and disappointment aversion. In our model, the firm’s desire to treat customers fairly its desire to avoid disappointing customers are in conflict, and we study how the firm resolves this conflict under different conditions.

3

Model

We first present a baseline version of the model with rational customers. We then incorporate fairness concerns and disappointment aversion into the model, and show how these effects change the firm’s optimal pricing strategy.

8

3.1

Baseline Model with Rational Customers

Assume a monopolist can sell a product in T + 1 time periods, consisting of one high-demand period followed by T low-demand periods, where T ≥ 1. During the high-demand period, potential customers consist of a mass DH of high types and a mass DL of low types. During each low-demand period, a fraction α of each customer type is randomly selected to be available in the market, where 0 < α < 1, so that there are αDH high types and αDL low types. Starting with the high-demand period allows us to explore (in the following sections of the paper) how the company’s management of the initial demand spike affects customers’ emotions and decisions in subsequent periods. Several components of our model, including the choice of which customers are available in each period, involve randomization to select a subset from a set of customers. We assume these random draws occur in such a way that they are independent across customers and across time periods. This assumption implies, for example, if two particular customers are available in the market in one period, knowing one of them is available in the next period provides no information about whether the other is also available in the next period. This independence assumption is not technically essential for the insights of our paper to hold, but it simplifies the computation and analysis of the model.7 The product valuations of the customers who are available in a given period are VH for high types and VL for low types, where 0 < VL < VH . Each customer can purchase during multiple periods, and we assume in this baseline version of the model that a customer’s purchase decision in a given period does not directly affect his utility from purchasing in subsequent periods. Each customer’s objective is to maximize total utility over all T + 1 periods. 7

If we have a mass M of customers and need to choose a subset with mass N , imagine the original set of customers are uniformly distributed around a unit circle. An interval of length N/M would be selected, with the starting point of the interval drawn from a uniform distribution over the entire circle. We assume that a particular customer’s position on the circle is independent across draws, so that there is no correlation between the draws for any pair of customers, and no correlation between the customer’s position on the circle for a particular draw and any other attribute of the customer.

9

The firm faces a capacity constraint, so that in each period it can serve at most K customers. All costs are normalized to zero. In each time period t, the firm sets price Pt , and potential customers decide whether to attempt to buy at this price. (Assume customers attempt to buy if their expected utility from doing so is weakly greater than zero.) If there is excess demand, that is, the number of customers who attempt to buy exceeds capacity, then K of the customers are randomly selected to buy the product, with all of the customers who attempt to buy having equal chances of being chosen. During the high-demand period, the firm has capacity to serve all high types, but does not have capacity to serve all of both types: Assumption 1. DH < K < DH + DL During the low-demand period, the firm has capacity to serve all of both types: Assumption 2. α(DH + DL ) < K We also make the following assumption about product valuations: Assumption 3. VL K < VH DH < VL (DH + DL ) This assumption implies that it is more profitable to serve only the high types than to serve a mix of both types when the capacity constraint binds (during the high-demand period); however, it is more profitable to serve both types when the firm has capacity to do so (during the low-demand periods). The firm’s objective is to maximize total profits over all T + 1 periods. To summarize, in each period t, the game timing is as follows: 1. If t = 1, all customers are available in the market. If t ≥ 2, a fraction α of customers are randomly selected to be available in the market. 2. The firm sets price Pt for the period. 3. Customers simultaneously decide whether to attempt to purchase at this price. 10

4. If more than K customers attempt to purchase, then K of these customers are randomly selected to purchase, and all other customers who attempted to purchase are denied the opportunity to do so. All payoffs for the period are realized. Our solution concept is subgame perfect Nash equilibrium. The appendix proves the following result formally. Proposition 1. If customers react to prices in a rational manner (with no fairness concerns or disappointment aversion), in equilibrium, the firm sets price VH during the first period and price VL during each subsequent period. In this baseline model with rational customers who have complete information about demand in each period, setting a high price or, alternatively, denying customers a purchase opportunity during peak demand does not reduce purchases in subsequent periods. By contrast, such effects are a key feature of our model with fairness concerns and disappointment aversion, developed in the following sections of the paper.

3.2

Fairness Concerns and Disappointment Aversion

This section incorporates two behavioral phenomena into the model. We then explore how the firm’s optimal price policy is influenced by the strength of customers’ fairness concerns, the strength of their disappointment aversion, and the relative strength of these two effects. Similar to the model by Rotemberg (2005), we assume the effects of fairness concerns are a step-function of prices, and any non-zero variance in prices is sufficient to provoke these concerns.8 In particular, a fraction f of each customer type exhibits fairness concerns, where 0 ≤ f ≤ 1. If one of these customers ever purchases at a given price, and the firm subsequently offers the product at any lower price, this particular 8

One justification for this assumption is that any change in prices could motivate customers to update their beliefs about whether the firm is treating them fairly (Rotemberg 2005). The conclusion of our paper discusses other possible functional forms to represent the effects of unfairness.

11

customer becomes angry with the firm; therefore, the customer’s probability of being in the market in each period is reduced to αf , where 0 ≤ αf < α. Thus, the proportional reduction in demand by customers who buy at high price and then observe a low price is: f (α − αf ) fb ≡ α

(1)

Because of their aversion to paying a higher price than others, customers with fairness concerns also will never purchase at any price that exceeds the minimum price from all prior periods.9 A fraction d of each customer type exhibits disappointment aversion, where 0 ≤ d ≤ 1. If one of these customers ever attempts to purchase the product but is not able to do so because of the capacity constraint, this customer becomes angry with the firm, and has probability of being in the market in each subsequent period reduced to αd , where 0 ≤ αd < α.10 The proportional reduction in demand from customers who make failed purchase attempts is: d(α − αd ) db ≡ α

(2)

Recall that we have assumed all potential customers are in the market during the initial high-demand period. If we instead allowed some old customers to exit and some new customers to enter during each period, the new potential customers would not be exposed to unfairness or disappointment, because they would not have been in the market during peak demand. Thus, this change to the model would be equivalent 9

Given our assumption that the high-demand period occurs first, the firm does not increase prices, from one period to the next, in any of the equilibria we find. However, the assumption that customers with fairness concerns will not purchase at a price that is higher than any previous price simplifies our formal proofs by making it technically easier to rule out strategies in which the firm sets a price strictly less than VL during peak demand. 10 We assume these behavioral types occur independently, so a fraction f d of customers exhibits both types of behavior, and that such a customer who experiences both unfairness and disappointment has a probability of being in the market that is the minimum of αf and αd . None of our results involve both unfairness and disappointment occurring on the same equilibrium path, but these assumptions help to characterize what happens off the equilibrium path.

12

to reducing f and d. The goal of this model set-up is to capture some of the key factors involved in the interaction between fairness concerns and disappointment aversion while also maintaining analytical tractability. Any model in which fairness concerns compel a firm to set stable prices could generate periods of excess demand and therefore disappointed customers, and thus could generate a similar trade-off to the one we study. However, other approaches to modeling fairness and disappointment would affect some of the details of our results. For example, in our model, customers exhibit “self-centered inequity aversion” (Fehr and Schmidt 1999), meaning that only customers who actually pay an unfair price become angry at the firm. If we assumed instead that even customers who do not pay an unfair price can become angry because of seasonal price variation, then such price variation would lead to an even greater drop in demand during the low-demand periods than occurs in our model. Furthermore, we assume that only customers who make failed purchase attempts can become disappointed. If we assumed instead that some customers become disappointed if high prices or the firm’s market exit prevent them from making a purchase attempt during peak demand, then the firm would have no way to avoid provoking feelings of disappointment in these customers; thus, the trade-off we study holds if, at least for some customers, the act of attempting to purchase is what generates an expectation of consumption and thus the potential for disappointment. Finally, if we allowed for some low-demand periods to precede the high demand period, then customers with fairness concerns who are informed about the firm’s previous low prices may avoid purchasing during peak demand, and thus behave as in the case with sophisticated customers studied in the next section.

13

Table 1. Overview of notation DH , DL

Number of high-type and low-type type customers, respectively

VH , VL

Product valuation of high-type and low-type type customers, respectively

T

Number of low-demand periods (following one high-demand period)

Pt

Price in period t

K

Capacity constraint in each period

f

Fraction of customers with fairness concerns

d

Fraction of customers with disappointment aversion

α ∈ (0, 1)

Initial probability of a customer being available during each low-demand period

αf ∈ [0, α)

Probability of being available in each period after unfairness

αd ∈ [0, α)

Probability of being available in each period after disappointment

fb ≡

f (α−αf ) α

Reduction in demand from customers who pay a higher price than others

db ≡

d(α−αd ) α

Reduction in demand from customers with failed purchase attempts

In this section of the paper, we assume customers do not anticipate how unfairness and disappointment might reduce their subsequent consumption utility. For example, customers who attempt to buy during the first period do not account for the possibility that a failed purchase attempt could cause them to experience less frequent desire to consume the product. In the language of behavioral industrial organization literature, we assume customers are “naive” about these behavioral effects (O’Donoghue and Rabin 2001). One interpretation is naive behavior reflects customers’ inability to anticipate their own emotional reactions. Another interpretation is naive behavior occurs because customers have incorrect beliefs about other players’ strategies. For example, a customer with disappointment aversion who does not realize that many other customers will attempt to purchase in the first period may also attempt to purchase in this period, even though doing so reduces the customer’s expected utility because there is a high probability the purchase attempt will be unsuccessful. Similarly, a customer with fairness concerns may purchase at a high price in the first period based on a belief that enough other customers with fairness concerns are 14

also purchasing at this price to compel the firm not to reduce prices in the future.11 The next section of the paper presents a model extension in which customers are “sophisticated,” so that they do anticipate these behavioral effects. Given this set-up, we show that there are four possible optimal price strategies. First, the firm could follow the same strategy as in the model with rational customers, setting price VH in the first period and price VL in all subsequent periods. This strategy causes all high types who have fairness concerns to reduce their consumption frequency after the first period, resulting in total profits of:   πHL = VH DH + αT VL (1 − fb)DH + DL A second strategy is to set price VL in all periods.

(3) This strategy avoids

antagonizing customers with fairness concerns, but it does cause some customers with disappointment aversion to reduce their consumption frequency after the first period (in which they are denied a opportunity to purchase because of the capacity constraint), resulting in total profits of:   b H + DL ) + dK b πLL = VL K + αT VL (1 − d)(D

(4)

A third strategy is to set price VH in all periods. This strategy avoids provoking either fairness concerns or disappointment aversion, resulting in total profits of: πHH = VH DH + αT VH DH

(5)

A fourth strategy is to set a price greater than VH in the first period, effectively exiting the market for this period, and then set a price of VL in all subsequent periods. This strategy also avoids provoking either fairness concerns or disappointment 11

In fact, if enough customers with fairness concerns purchase in the first period, the firm will not reduce its price in subsequent periods, and so the beliefs of customers would be confirmed. Thus, in some cases, naive customer behavior is equivalent to selecting a Pareto-dominated equilibrium in which prices are always high.

15

aversion, resulting in total profits of: πEL = αT VL (DH + DL )

(6)

The appendix proves that, in equilibrium, the firm chooses whichever of these four strategies results in the highest profits. In particular, we show that this equilibrium is subgame perfect. For example, under conditions such that πH,H is the highest of the four profit levels, the firm’s desire not to antagonize customers with fairness concerns prevents it from cutting prices in the later periods of the game, so that the firm’s pricing strategy is dynamically consistent. Proposition 2. With naive behavioral customers, the equilibrium outcome depends on the relative values of πHL , πLL , πHH , and πEL . If πHL is the greatest of these four values, the firm sets price VH in period 1 and price VL in all subsequent periods. If πLL is the greatest, the firm sets price VL in all periods. If πHH is the greatest, the firm sets price VH in all periods. If πEL is the greatest, the firm sets a price greater than VH in period 1 (and no customers purchase in this period) and sets price VL in all subsequent periods. Figures 1 and 2 illustrate equilibrium prices, based on Proposition 2, as a function of the strength of fairness concerns and the strength of disappointment aversion. Figure 1 assumes a relatively small number of low-demand periods (T = 5), whereas Figure 2 assumes a larger number of low-demand periods (T = 15).

16

Figure 1. Equilibrium Prices with Few Low-Demand Periods (T = 5)12

Both figures show that, if fairness concerns are weak, then the firm follows the same strategy as in the rational model, setting a high price in the first period and low prices in all subsequent periods. Proposition 2 implies that this high-then-low price strategy is optimal whenever fb is sufficiently close to zero. Both figures also show that, if fairness concerns are strong but disappointment aversion is weak, then the firm sets low prices in all periods. Proposition 2 implies, if fb is sufficiently large and db is sufficiently small, the firm sets the same price in all periods to avoid provoking fairness concerns. Under such conditions, always setting low prices is guaranteed to be optimal if there are sufficiently many periods with low demand.13

12

The other parameter values used in this figure are: DH = 1, DL = 1, K = 1.4, α = 0.6, VH = 1.8, and VL = 1.1. 13 Either always-low prices or always-high prices could be optimal if there are few periods with low demand, depending on the other parameter values involved.

17

Figure 2. Equilibrium Prices with Many Low-Demand Periods (T = 15)14

Finally, Proposition 2 implies, if fairness concerns and disappointment aversion are both strong, the firm has two possible strategies to avoid antagonizing customers. If there are sufficiently few periods with low demand (as in Figure 1), the firm sets high prices in all periods. If there are sufficiently many periods with low demand (as in Figure 2), the firm exits the market in the first period, and then sets low prices in all subsequent periods. The latter strategy (temporarily exiting the market and then setting low prices) generates no profits in the first period, but it generates the greatest possible profits in all subsequent periods.

3.3

Sophisticated Behavioral Customers

We now assume customers are sophisticated in the sense that they anticipate the potential lost utility that could result from unfairness or disappointment. Customers maximize their total expected utility over all T +1 periods, accounting for such effects. To help focus our analysis, this section places three additional restrictions on the 14

The other parameter values used in this figure are the same as in Figure 1.

18

model’s parameter values. The first two restrictions ensure there are enough periods with low demand to influence how sophisticated customers with fairness concerns and disappointment aversion behave in the initial high-demand period. The third restriction ensures the firm has sufficient capacity to serve all customers who do not have disappointment aversion and still retain some capacity for those who do have disappointment aversion. Following is a more detailed explanation of each of these assumptions. Sophisticated customers with fairness concerns may avoid purchasing, even if their valuation exceeds the current price, if they expect the firm to reduce prices in the future. The following assumption ensures that a sophisticated high type customer with fairness concerns will not purchase in the first period at any price P1 > VL if this customer expects the firm to set price Pt = VL in all subsequent periods: Assumption 4. (α − αf )T > 1 Note this assumption is guaranteed to hold if there are sufficiently many periods with low demand (T is sufficiently large). Sophisticated customers with disappointment aversion may not attempt to purchase, even if their valuation exceeds the price, if they believe there is a high enough probability they would be denied the opportunity to purchase. The next assumption ensures capacity is low enough that there cannot be an equilibrium in which all customers with disappointment aversion attempt to purchase in the first period at price P1 = VL if they expect the firm to set price Pt = VL in all subsequent periods: Assumption 5. (α − αd )T K < DH + DL 1 + (α − αd )T Note that this assumption also is guaranteed to hold if there are sufficiently many periods with low demand (T is sufficiently large), because Assumption 1 implies the 19

term on the left of this inequality is strictly less than one. The following assumption ensures that at least some of the customers with disappointment aversion will attempt to purchase in the first period at price P1 = VL if they expect the firm to set price Pt = VL in all subsequent periods: Assumption 6. (1 − d)(DH + DL ) < K Note this assumption, which implies the number of customers who do not have disappointment aversion is less than the firm’s capacity constraint, is guaranteed to hold if there are sufficiently many customers with disappointment aversion (d is sufficiently large). Table 2. Assumptions in model extension with sophisticated customers15 DH < K < DH + DL α(DH + DL ) < K VL K < VH DH < VL (DH + DL ) (α − αf )T > 1 K DH +DL

<

(α−αd )T 1+(α−αd )T

(1 − d)(DH + DL ) < K

Capacity constraint binds during high-demand period Capacity constraint does not bind during low-demand periods Profits are greatest from serving both types at price VL , when possible Reduced consumption from unfairness exceeds peak-demand consumption Capacity small enough that not all customers attempt to purchase Capacity large enough for all customers without disappointment aversion

Given these assumptions, we show that one possible equilibrium outcome is the following: the firm sets price VH in the first period; only the high types who do not have fairness concerns purchase in the first period; and then the firm sets price VL and sells to all available customers in all subsequent periods.16 This outcome generates 15

As a numerical example, all assumptions are satisfied for: DH = 1, DL = 1, K = 1.4, α = 0.6, αf = 0.1, αd = 0.1, VH = 1.8, VL = 1.1, d ≥ 0.35, and T ≥ 5. 16 If VH DH > VL [(1 − fb)DH + DL ], then the subgame that starts with the firm setting P1 = VH has another equilibrium in which all high types buy in the first period, and the firm always sets price VH and sells to all available high types in subsequent periods. Intuitively, if all high types buy at price VH in the first period, fairness concerns can prevent the firm from reducing its price in subsequent periods. However, if T is large enough that f VH DH < αT [VL (DH + DL ) − VH DH ], the firm earns higher profits in the equilibrium discussed in the body of the paper (which has a high price in the first period and low prices in all subsequent periods) than in this alternative equilibrium (with high prices in all periods), and so this alternative equilibrium is Pareto-dominated.

20

profits of: π bHL = (1 − f )VH DH + αT VL (DH + DL )

(7)

In the equilibrium described above, the firm does not have to worry about antagonizing customers with fairness concerns by setting low prices in periods t > 1, because sophisticated customers anticipate their potential feelings of unfairness and therefore do not purchase at the high price set in the first period if they have fairness concerns. We show that another possible equilibrium outcome is the following: the firm sets price VL in all periods; customers with disappointment aversion randomize when deciding whether to attempt to purchase in the first period, and some of these customers are denied the opportunity to purchase and therefore exit the market. In this case, customers with disappointment aversion randomize in such a way that the profits they generate from successful purchases in the first period equals the foregone profits that result from some of these customers being disappointed and exiting the market in subsequent periods. Therefore, total profits equal the profits generated in the first period by customers do who not have disappointment aversion plus the profits that would arise from selling to all available customers (including those with and without disappointment aversion) in subsequent periods:17 

π bLL

 (α − αd )T = (1 − d)VL (DH + DL ) + αT VL (DH + DL ) 1 + (α − αd )T

(8)

In the equation above, the term in square brackets represents the equilibrium success probability for purchase attempts in the first period. As the purchase frequency reduction for disappointed customers (α−αd ) grows in absolute value, fewer customers 17

Such randomization can also occur with more than two valuation types. If fairness concerns are strong enough that equilibrium prices are constant, then all customers with disappointment aversion and a valuation greater than the equilibrium price randomize when faced with the same success probability. Intuitively, as a customer’s product valuation increases, the increase in expected utility from potentially purchasing in the first period exactly equals the decrease in expected utility because of potential disappointment.

21

with disappointment aversion attempt to purchase in the first period, so the success probability increases. Thus, with sophisticated customers, profits decrease as the number of customers with disappointment aversion grows, but profits increase as the reduction in purchase frequency that results from disappointment grows. The firm benefits from such strong disappointment aversion, because it discourages customers with disappointment aversion from attempting to purchase in the first place. The appendix formally proves all of the results in this section, which are summarized on the following proposition. Proposition 3. With sophisticated behavioral customers, there is an equilibrium in which the following occurs. If π bHL > π bLL , the firm sets price VH in the first period, only high types who do not have fairness concerns purchase in the first period, and the firm then sets price VL and sells to all available customers in subsequent periods. If π bLL > π bHL , the firm sets price VL in all periods, customers with disappointment aversion randomize when they decide whether to attempt to purchase in the first period, and some of these customers are disappointed. Note that the equilibrium outcome depends on the relative values of f and d. When d is relatively large, the firm sets a high price in the first period to avoid disappointing any customers. When f is relatively large, the firm sets a low price in the first period to induce customers with fairness concerns to purchase. In both cases, the firm sets low prices in all subsequent periods. Thus, in this equilibrium with sophisticated customers, there are no conditions in which the firm exits the market or always sets high prices. If there are sufficiently many periods with low demand, then the equilibrium with sophisticated customers generates greater profits than the equilibrium with naive customers. With naive customers, the firm has to exit the market during the high-demand period if it wants to avoid provoking either fairness or disappointment 22

concerns and also wants to set a low price during the low-demand periods. By contrast, with sophisticated customers, the firm can remain active in the market during the high-demand period (with either a high or low price) and still set a low price during the low-demand periods, and sophisticated customers will adjust their strategies accordingly during the high-demand period to help avoid future feelings of unfairness or disappointment. The appendix formally proves the following corollary to Propositions 2 and 3. Corollary 1. The firm earns strictly greater profits in the equilibrium with sophisticated customers than in the equilibrium with naive customers if T is large enough that both of the following hold:

f VH DH < αT VL fbDH

(9)

f VH DH < αT [VL (DH + DL ) − VH DH ]

(10)

The left side of each condition of this corollary (the term f VH DH ) represents the profits the firm can generate in the first period, if customers are naive, by selling at a high price to customers with fairness concerns. The right side of the first condition represents the long-term reduction in profits resulting from less frequent purchases by these customers if they pay an unfair price. The right side of the second condition represents the long-term reduction in profits from keeping prices high to avoid provoking fairness concerns of these customers. If T is sufficiently large, each of these potential costs outweighs the benefit of selling, in the first period, to naive customers with fairness concerns. Under such conditions, the firm is guaranteed to be better off with at least one possible price strategy and sophisticated customers than with any possible price strategy and naive customers. Furthermore, the sophisticated-customer equilibrium in which the firm always sets 23

low prices generates greater average customer utility than any of the price equilibria with naive customers. Because some sophisticated customers with disappointment aversion do not attempt to purchase when the firm sets a low price during peak demand, fewer customers are disappointed, which results in more purchases and greater customer surplus in subsequent periods, relative to the case with naive customers and low prices. The appendix formally proves the following corollary to Propositions 2 and 3. Corollary 2. Average customer utility is strictly greater in the equilibrium with sophisticated customers than in the equilibrium with naive customers if π bLL > π bHL . This corollary applies to parameter values for which the firm always sets low prices given sophisticated customers. On the other hand, there exist parameter values for which the firm always sets low prices given naive customers, but uses a high-then-low price strategy given sophisticated customers. In such cases, customer surplus could be higher if customers are naive rather than sophisticated.

4

Conclusion

This paper studies an important dilemma for firms that have an emotional customer base, a capacity constraint, and seasonal demand fluctuations. Customers’ sense of fairness compels such firms not to vary prices over time. On the other hand, customers may become disappointed if they are denied the opportunity to purchase during peak demand. If customers are naive about how unfairness and disappointment can reduce future consumption utility, then in order to avoid provoking these emotions, the firm must either set a high price at all times or exit the market during peak demand. By contrast, if customers are sophisticated about anticipating the effects of unfairness 24

and disappointment, two other strategies are available to the firm. First, it can set a high price during peak demand, during which time sophisticated customers with fairness concerns will avoid purchasing; thus, the firm can later lower its price without antagonizing these customers. Alternatively, the firm can set a low price at all times, and sophisticated customers with disappointment aversion will limit their total purchase attempts during peak demand, for example, with each of these customers randomizing over the decision of whether to attempt to purchase. Future research could extend our model to include features that are not in our current model but that apply to some real world firms. For example, the firm in our model has an exogenously fixed capacity constraint, which is a reasonably accurate assumption for many entertainment firms. Disneyland has limited room to expand because of its location in the city of Anaheim (Vaux 2010). Similarly, the Boston Red Sox baseball team cannot, from a structural engineering perspective, add many new seats to their current stadium; they would need to build an entirely new stadium to increase their seating capacity (Charlotin 2010). However, some firms can expand capacity quickly. When ridesharing services such as Uber and Lyft increase prices during peak demand, it encourages more drivers to become active during peak demand. Such price increases anger some customers, but the resulting supply increase helps ensure sufficient capacity of vehicles (Kosoff 2015). Future research could incorporate endogenous capacity into our model to allow for this additional benefit of high prices. Our model does not include transaction costs or travel costs that arise from attempting to purchase. Our model also does not include congestion costs, that is, the negative externality that arises from overcrowding. One could potentially modify our model to allow such costs to contribute to the negative emotions that reduce future demand. For example, customers who travel a long distance in an unsuccessful

25

attempt to visit a theme park might become particularly angry with the firm, whereas local customers who travel a shorter distance might not be so angry. Future research could develop information-based models that study a similar managerial problem as our emotion-based model. For example, a customer who is unable to access a car using Uber during New Year’s Eve might use the service less frequently in the future, either because they are disappointed and angry at the firm (as in our model), or because they no longer believe the service is reliable. In the latter case, setting a high price during peak demand could help ensure that customers are able to purchase and therefore that they infer the service is reliable. Future research could also incorporate more complex representations of unfairness and disappointment into our model. For example, fans might not become angry if a sports team charges slightly higher ticket prices for a game against a major rival, but they might become angry if this price premium is too large. As a result, the firm might set prices that vary across events, but with smaller variance than in a rational model. The problem of potentially disappointing customers would still arise, as in our model, if this reduced variance in prices caused excess demand during some events.

26

References Anderson, E. T., G. J. Fitzsimons, and D. Simester (2006). Measuring and mitigating the costs of stockouts. Management Science 52 (11), 1751–1763. Anderson, E. T. and D. I. Simester (2010). Price stickiness and customer antagonism. The Quarterly Journal of Economics 125 (2), 729–765. Bell, D. E. (1985). Disappointment in decision making under uncertainty. Operations Research 33 (1), 1–27. Bolton, G. E. and A. Ockenfels (2000). ERC: A theory of equity, reciprocity, and competition. The American Economic Review 90 (1), 166–193. Bolton, L. E., L. Warlop, and J. W. Alba (2003). Consumer perceptions of price (un)fairness. Journal of Consumer Research 29 (4), 474–491. Camerer, C. F. and R. H. Thaler (1995, June). Anomalies: Ultimatums, dictators and manners. Journal of Economic Perspectives 9 (2), 209–219. Charlotin, R. (2010). Boston Red Sox need a new Fenway Park. Bleacher Report. Cooper, R., D. V. DeJong, R. Forsythe, and T. W. Ross (1992). Communication in coordination games. The Quarterly Journal of Economics 107 (2), 739–771. Cui, T. H., J. S. Raju, and Z. J. Zhang (2007). Fairness and channel coordination. Management Science 53 (8), 1303–1314. DailyMail (2016). Thom Yorke shares Radiohead fans’ anger over UK tour ticket sales. Daily Mail . DellaVigna, S. and U. Malmendier (2004). Contract design and self-control: Theory and evidence. The Quarterly Journal of Economics 119 (2), 353–402. Drayer, J., S. L. Shapiro, and S. Lee (2012). Dynamic ticket pricing in sport: An agenda for research and practice. Sport Marketing Quarterly 21 (3), 184–194. Fehr, E. and K. M. Schmidt (1999). A theory of fairness, competition, and cooperation. The Quarterly Journal of Economics 114 (3), 817–868. Fitzsimons, G. J. (2000). Consumer response to stockouts. Journal of Consumer Research 27 (2), 249–266. Fritz, B. (2015). Disney parks consider off-peak prices. Wall Street Journal . Fritz, B. (2016). Disney rolls out seasonal pricing for one-day park tickets. Wall Street Journal . Fudenberg, D. and J. Tirole (1991). Game Theory. Cambridge, MA: MIT Press. 27

Gabaix, X. and D. Laibson (2006). Shrouded attributes, consumer myopia, and information suppression in competitive markets. The Quarterly Journal of Economics 121 (2), 505–540. Gul, F. (1991). A theory of disappointment aversion. Econometrica 59 (3), 667–686. Guo, L. (2015). Inequity aversion and fair selling. search 52 (1), 77–89.

Journal of Marketing Re-

Guo, L. and J. Zhang (2012). Consumer deliberation and product line design. Marketing Science 31 (6), 995–1007. Hoffman, E., K. McCabe, K. Shachat, and V. Smith (1994). Preferences, property rights, and anonymity in bargaining games. Games and Economic Behavior 7 (3), 346 – 380. Inman, J. J., J. S. Dyer, and J. Jia (1997). A generalized utility model of disappointment and regret effects on post-choice valuation. Marketing Science 16 (2), 97–111. Jing, X. and M. Lewis (2011). Stockouts in online retailing. Journal of Marketing Research (JMR) 48 (2), 342 – 354. Johnson, H. (2015). REI to customers: We’ll be closed on Black Friday, go outside. The Simple Dollar . Kahneman, D., J. L. Knetsch, and R. Thaler (1986). Fairness as a constraint on profit seeking: Entitlements in the market. The American Economic Review 76 (4), 728– 741. Kosoff, M. (2015). Don’t complain about Uber’s surge pricing tonight. Business Insider . Lefton, T. and J. Lombardo (2003). Stern’s NBA shows its transition game. Street and Smith’s Sports Business Journal . Lerner, J. S., D. A. Small, and G. Loewenstein (2004). Heart strings and purse strings: Carryover effects of emotions on economic decisions. Psychological Science 15 (5), 337–341. Li, K. J. and S. Jain (2016). Behavior-based pricing: An analysis of the impact of peer-induced fairness. Management Science (forthcoming). Liu, Q. and S. Shum (2013). Pricing and capacity rationing with customer disappointment aversion. Production and Operations Management 22 (5), 1269–1286. Loomes, G. and R. Sugden (1986). Disappointment and dynamic consistency in choice under uncertainty. The Review of Economic Studies 53 (2), 271–282. 28

Loomes, G. and R. Sugden (1987). Testing for regret and disappointment in choice under uncertainty. The Economic Journal 97, 118–129. Martin, H. (2014). Disney parks on both coasts close temporarily on Christmas. Los Angeles Times. Martin, H. (2015). Disneyland prepares for crush of visitors during 60th anniversary celebration. Los Angeles Times. Martinez, L. F., M. Zeelenberg, and J. B. Rijsman (2011). Regret, disappointment and the endowment effect. Journal of Economic Psychology 32 (6), 962–968. Mohammed, R. (2015). Of course disney should use surge pricing at its theme parks. Harvard Business Reviewl . Moore, S. G. and G. J. Fitzsimons (2014). Yes, we have no bananas: Consumer responses to restoration of freedom. Journal of Consumer Psychology 24 (4), 541– 548. Munarriz, R. (2015). Sorry, folks, park’s closed – another Disney Christmas. The Motley Fool . Nasiry, J. and I. Popescu (2012). Advance selling when consumers regret. Management Science 58 (6), 1160–1177. News12 (2009). Yankees fans angry after tickets sell out. News 12 Brooklyn. Niles, R. (2015). Post-mortem: Three things Disneyland could have done better at the 24-hour party. Theme Park Insider . O’Donoghue, T. and M. Rabin (2001). Choice and procrastination. The Quarterly Journal of Economics 116 (1), 121–160. Oi, W. Y. (1971). A Disneyland Dilemma: Two-part tariffs for a Mickey Mouse monopoly. The Quarterly Journal of Economics 85 (1), 77–96. Orhun, A. Y. (2009). Optimal product line design when consumers exhibit choice set-dependent preferences. Marketing Science 28 (5), 868–886. Pimentel, J. (2015). Disneyland raises annual pass prices, introduces $1,000 pass, and discontinues Premium pass. The Orange Country Register . Pimentel, J. and C. Yee (2015). Disneyland re-opens its gates after shutting them to control overflowing crowd. The Orange County Register . Rabin, M. (1993). Incorporating fairness into game theory and economics. The American Economic Review 83 (5), 1281–1302.

29

Rabin, M. (1994). A model of pre-game communication. Journal of Economic Theory 63 (2), 370 – 391. Rotemberg, J. J. (2005). Customer anger at price increases, changes in the frequency of price adjustment and monetary policy. Journal of Monetary Economics 52 (4), 829 – 852. {SNB}. Seufert, D. (2015). For many shoppers, ’Black Friday’ a disappointment. New Hampshire Union Leader . Shapiro, S. L., J. Drayer, and B. Dwyer (2016). Examining consumer perceptions of demand-based ticket pricing in sport. Sport Marketing Quarterly 25 (1), 34–46. Su, X. (2010). Optimal pricing with speculators and strategic consumers. Management Science 56 (1), 25–40. Tully, S. (2012). Labor Day crowds expected at Disney parks. The Orange County Register . Vaux, R. (2010). Disney World compared to Disneyland. USA Today Travel Tips. Xie, J. and S. M. Shugan (2001). Electronic tickets, smart cards, and online prepayments: When and how to advance sell. Marketing Science 20 (3), 219–243. Zwick, R. and X. P. Chen (1999). What price fairness? Management Science 45 (6), 804–823.

A bargaining study.

Appendix: Proofs of all Propositions Proof of Proposition 1 Consider period t = 1 (the high-demand period). If the firm sets P1 = VL , demand for period 1 is K (the capacity constraint binds because Assumption 1 from the body of the paper states DH + DL > K), and profits for the period are:

VL K

(11)

If the firm sets Pt = VH , demand is DH and profits are:

VH DH 30

(12)

Because assumption (3) from the body of the paper states that VL K < VH DH , the firm earns higher profits by setting price VH and serving only the high types. Now consider each period t > 1 (the low-demand periods). If the firm sets price Pt = VL , demand for period t is α(DH + DL ), and profits for the period are:

αVL (DH + DL )

(13)

If the firm sets Pt = VH , demand is αDH and profits are:

αVH DH

(14)

Because Assumption 3 from the body of the paper states that VL (DH +DL ) > VH DH , the firm earns higher profits by setting price VL and serving all available customers. Finally, setting an price higher than VH results in zero demand; setting a price on the interval (VL , VH ) results in strictly lower profits than setting price VH ; and setting a price lower than VL results in strictly lower profits than setting price VL . Therefore, the firm would never set a price other than VL or VH . QED Proof of Proposition 2 We will first derive the firm’s optimal strategy and continuation value in each period t > 1, given the price the firm sets in period 1. Suppose the firm sets P1 > VH . Because no customers attempt to purchase at this price, neither fairness concerns nor disappointment aversion constrains the firm’s profits in the next period. Therefore, in each period t > 1, the firm can achieve the same optimal profits from the version of the model with rational customers by setting price VL , which leads to demand of α(DH + DL ). This strategy generates total profits over all periods of πEL . 31

Suppose the firm set period one price P1 = VH . All high types purchase at this price. In subsequent periods, one possible strategy is to continue to set price VH , which avoids provoking feelings of unfairness, and generates demand of αDH in each period. This strategy leads to the following profits in each period t > 1:

αVH DH

(15)

The above strategy generates total profits over all periods of πHH . Another possible strategy in periods t > 1 is to set price VL , which causes high types with fairness concerns to reduce their consumption frequency, so that demand in each period is α[(1 − fb)DH + DL ]. This outcome leads to the following profits in each period t > 1:

αVL [(1 − fb)DH + DL ]

(16)

The above strategy generates total profits over all periods of πHL . Therefore, given a first period price of VH , the firm will follow the former strategy if πHH > πHL and will follow the latter strategy if πHL > πHH and, by setting P1 = VH , the firm can achieve which ever of these profit levels is greater. This strategy also dominates setting P1 ∈ (VL , VH ), which leads to lower first period profits and the same continuation value as setting P1 = VH (the continuation value for P1 ∈ (VL , VH ) can be derived using the same set of steps outlined above). Suppose the firm set period one price P1 = VL . All customers attempt to purchase at this price, but only K of them are able to do so. In subsequent periods, one possible strategy is to set price VH . High valuation customers with fairness concerns would not purchase at this price, because they would have previously observed a lower price. b + dK/(D b Therefore, demand would be α(1 − f )DH [(1 − d) H + DL )] in each period.

32

This strategy leads to the following profits in each period t > 1 less than or equal to:  b + α(1 − f )VH DH (1 − d)

 b dK DH + DL

(17)

Another possible strategy in periods t > 1 is to set price VL , which generates demand b + dK/(D b of α(DH + DL )[(1 − d) H + DL )] in each period. This strategy leads to the following profits in each period t > 1:  b + αVL (DH + DL ) (1 − d)

 b dK DH + DL

(18)

Assumption 3 implies that the latter strategy generates higher profits. Therefore, given a first period price of VL , the firm will also set price VL in all subsequent periods, which generates total profits over all periods of πLL . This strategy also dominates setting P1 < VL , which leads to lower first period profits and the same continuation value as setting P1 = VL (the continuation value for P1 < VL can be derived using the same set of steps outlined above). We have shown that the firm can choose from first period prices that, given the optimal continuation value, lead to total profits over all periods of πEL , πLL , or the maximum of πHH and πHL . We have also shown that at least one of these four strategies generates greater profits than all other possible strategies. Therefore, the firm will choose a strategy that generates whichever of these four profit levels is greatest. QED

Proof of Proposition 3 We will first derive an equilibrium of each possible subgame that begins after the firm sets price P1 . Consider the subgame that begins if the firm sets price P1 > VH . This subgame 33

has an equilibrium in which no customers purchase in the first period, because the price exceeds their valuations. In each subsequent period, the firm can then generate the same optimal profits that would occur with rational customers by setting price Pt = VL and selling to all available customers. This strategy generates the same profits πEL as in the naive customer case. Now consider the subgame that begins if the firm sets price P1 ∈ (VL , VH ]. We will show this subgame has an equilibrium in which only high type customers who do not have fairness concerns purchase in the first period, and in all subsequent periods the firm then sets price VL and sells to all available customers. Consider the incentives of a high type customer with fairness concerns to deviate from this proposed equilibrium. For such a customer, purchasing in the first period would generate utility of VH − P1 , but would also result in a reduction of expected utility in subsequent periods of (α − αf )T (VH − VL ), because feelings of unfairness from paying a high price and then observing a lower price would prevent the customer from enjoying the product in subsequent periods. Assumption 4 guarantees such a deviation would make the customer worse off, and so customers with fairness concerns would not purchase in the first period. Because customers with fairness concerns do not purchase in the first period, and no customers are disappointed in the first period (the capacity constraint does not bind), the firm can generate the same optimal profits as in the case with rational customers by setting price VL and selling to all available customers. This strategy generates total profits of:

(1 − f )P1 DH + αT VL (DH + DL )

(19)

Note that total profits in this set of subgames are maximized if P1 = VH , which leads to total profits π bHL . Also note that π bHL > πEL . 34

Now consider the subgame in which the firm sets price P1 = VL . We will show this subgame has an equilibrium in which customers with disappointment aversion randomize when deciding whether to attempt to purchase in the first period, and the firm then sets price VL in all subsequent periods. Given the firm’s strategy, all customers who do not have disappointment aversion will purchase in the first period, but customers who do have disappointment aversion face a trade-off when deciding whether to attempt to purchase. Let p denote the probability that a purchase attempt in the first period will be successful. For a high type customer with disappointment aversion, attempting to purchase leads to expected profits in the first period of:

p(VH − VL )

(20)

On the other hand, for such a customer, the possibility that the purchase attempt will fail (and cause disappointment) implies a reduction in expected future utility of:

(1 − p)(α − αd )T (VH − VL )

(21)

These two effects exactly offset each other, so the customer is indifferent toward attempting to purchase, if the success probability is:

p∗ =

(α − αd )T 1 + (α − αd )T

(22)

Assumption 5 guarantees that, if all customers attempt to purchase, the success probability is less than p∗ , which implies customers with disappointment aversion would not want to purchase. On the other hand, Assumption 6 guarantees that, if none of the customers with disappointment aversion attempt to purchase, the success probability is 100%, which implies customers with disappointment aversion

35

would want to purchase. Therefore, the equilibrium must involve some fraction γ of customers with disappointment aversion attempting to purchase, where γ ∈ (0, 1). We focus on an equilibrium in which all customers with disappointment aversion (low and high types) randomize independently, each with equilibrium probability γ ∗ of attempting to purchase, such that γ ∗ satisfies: K   = p∗ ∗ (1 − d) + γ d (DH + DL )

(23)

Rearranging terms, we find that the equilibrium fraction customers who have disappointment aversion and also attempt to purchase is:

γ ∗d =

K − (1 − d) p∗ (DH + DL )

(24)

Recall that γ ∗ was chosen such that the success rate for purchase attempts in the first period is p∗ , which implies a fraction γ ∗ d(1 − p∗ ) of customers makes failed purchase attempts. Now consider the firm’s pricing problem in each period t > 1. If the firm sets price Pt = VH , high type customers with fairness concerns would not purchase. Therefore, the firm’s profits would be: h i (1 − f )VH DH α − γ ∗ d(1 − p∗ )(α − αd )

(25)

The firm could also set price Pt = VL , which would generate profits in each period of: h i VL (DH + DL ) α − γ ∗ d(1 − p∗ )(α − αd )

(26)

Assumption 3 implies the latter strategy generates greater profits, so the firm’s optimal strategy is to price VL in all periods t > 1, and thus the firm does not deviate from the proposed equilibrium. This outcome generates total profits for the 36

firm of: h i ∗ ∗ VL K + αT VL (DH + DL ) − T VL (DH + DL ) γ d(1 − p )(α − αd )

(27)

Noting that p∗ = (α − αd )T (1 − p∗ ), the above expression is equal to:

VL K + αT VL (DH + DL ) − VL (DH + DL )γ ∗ dp∗

(28)

Inserting the expression for γ ∗ d from (24) into the above expression, we have:

VL K + αT VL (DH + DL ) − VL K + (1 − d)VL (DH + DL )p∗

(29)

This expression is equivalent to π bLL from the body of the paper.
Now consider the subgame in which the firm sets P1 ∈ (1 − f )VL , VL . We will show this subgame has an equilibrium in which the firm always sets the same price Pt = P1 , and customers with disappointment aversion randomize when deciding whether to attempt to purchase in the first period. Similar derivations to those for the case in which the firm always sets price VL show that, if the firm always sets the same
price as P1 ∈ (1 − f )VL , VL , customers with disappointment aversion randomize in the first period, with the same probability γ ∗ of attempting to purchase and same success probability p∗ as derived above. Given that the disappointed customers reduce their purchase frequency, consider the firm’s pricing problem in each period t > 1. Continuing to set price Pt = P1 leads to profits in each period: h i ∗ ∗ P1 (DH + DL ) α − γ d(1 − p )(α − αd )

(30)

On the other hand, raising prices to Pt = VL causes customers with fairness 37

concerns not to purchase, because they will not purchase at a price higher than any previous price. This outcome generates profits in each period: h i ∗ ∗ (1 − f )VL (DH + DL ) α − γ d(1 − p )(α − αd )

(31)

Because P1 > (1 − f )VL , the firm generates greater profits by continuing to set price P1 rather than raising prices to VL . Similar derivations to those above show that always setting price P1 leads to total profits: αT P1 (DH + DL ) + (1 − d)P1 (DH + DL )p∗

(32)

Because P1 < VL , these profits are less than the value π bLL derived above. Finally, consider the subgame in which the firm sets P1 ≤ (1 − f )VL . In any such subgame, an upper bound on total profits is:

(1 − f )VL K + αT VL (DH + DL )

(33)

Given that Assumption 3 states VL K < VH DH , these profits are lower than π bHL . We have shown there is an equilibrium in which the firm can choose a first period price that generates profits of either π bHL or π bLL and in which at least one of these profit levels is greater than the profits that would arise from any other first period price. Therefore, in this equilibrium, the firm will choose the strategy that generates whichever of these two profit levels is higher. QED

Proof of Corollary 1 We will show that, for each price strategy in Proposition 2 with naive customers, the firm can generate strictly greater profits using one of the price strategies in

38

Proposition 3 with sophisticated customers. With sophisticated customers, setting a high price in the first period and a low price in subsequent periods generates profits:

π bHL = (1 − f )VH DH + αT VL (DH + DL )

(34)

With naive customers, if the firm exits the market in the first period and sets low prices in subsequent periods, it generates profits αT VL (DH + DL ). These profits are strictly less than π bHL . With naive customers, if the firm sets a high price in all periods, it generates profits VH DH + αT VH DH . The condition f VH DH < αT [VL (DH + DL ) − VH DH ] implies these profits are strictly less π bHL . With naive customers, if the firm sets a high price in the first period and a low   price in subsequent periods, it generates profits VH DH + αT VL (1 − fb)DH + DL . The condition f VH DH < αT VL fbDH implies these profits are strictly less than π bHL . Finally, with naive customers, setting a low price in all periods generates profits VL K in the first period, but leads to d(DH + DL − K) customers being disappointed and reducing their future purchase frequency. With sophisticated customers, setting a low price in all periods also generates profits VL K in the first period, and leads to fewer customers being disappointed because some customers do not attempt to purchase in the first period, and so total profits are strictly greater. QED

Proof of Corollary 2 The condition π bLL > π bHL implies, with sophisticated customers, the firm always sets low prices. We will show this equilibrium with sophisticated customers generates strictly greater customer surplus than any possible equilibrium with naive customers. Note that equilibrium utility for low-valuation customers is always zero, so we focus 39

on equilibrium utility for high-valuation customers. We first compute customer surplus with sophisticated customers and low prices in all periods. High valuation customers with disappointment aversion randomize in such a way that they are indifferent between attempting to purchase and not attempting to purchase in the first period, which implies their average utility is αT (VH − VL ). High valuation customers who do not have disappointment aversion all attempt to purchase in the first period; letting ps denote the success probability of their purchase attempts, their average utility is (ps + αT )(VH − VL ), where ps > 0. We now compute customer surplus with naive customers. If the firm sets a high price in all periods, customer surplus is zero. If the firm exits the market in the first period and sets a low price in all subsequent periods, average utility for high-valuation customers is αT (VH − VL ). If the firm sets a high price price in the first period and low prices in subsequent periods, average utility for high-valuation customers who do not have fairness concerns is αT (VH − VL ), and average utility for high-valuation customers who have fairness concerns is a strictly lower value, αf T (VH − VL ), because these customers reduce their purchase frequency as a result of paying an unfair price. Therefore, average customer utility in all three of these cases is strictly less than in the case with sophisticated customers described above. Finally, we compute customer surplus with naive customers and low prices in all periods. In this case, all customers attempt to purchase in the first period. Let pn denote the success probability of these purchase attempts. Note that pn < ps because more customers attempt to purchase and thus the success probability is lower if customers are naive. Customers without disappointment aversion experience average utility (pn + αT )(VH − VL ), which is strictly less than their average utility in the case with sophisticated customers. Customers with disappointment aversion experience   average utility pn (1 + αT ) + (1 − pn )αd T (VH − VL ). Because the success probability

40

pn is strictly less than ps , average utility for these customers is also lower than in the case with sophisticated customers. QED

41