Consumer-Driven Class Pricing

1. Introduction

Advancements in information technologies and artificial intelligence (AI)-based pricing algorithms enable firms to set prices that closely reflect product costs or values. Yet, the prices of individual products do not often align with these factors. One prominent example is class pricing, a practice in which a number of differentiated products are priced with only a few price points. Wernerfelt (2008) describes this practice as dividing “a large set of goods or services into [a smaller number of] classes, and assign[ing] a single price to any element of a class” (Wernerfelt 2008, p. 755). Table 1 illustrates class pricing at a bar where the eight beers on tap carry only three prices, despite the self-evident differences in their costs or values.

Table 1. Class Pricing of Draft Beer

This pattern of strikingly fewer assigned prices than the assortment size is frequently observed across various product categories. Table 2 shows class-pricing patterns in supermarket assortments of fast-moving consumer packaged goods. Additional examples cited by Wernerfelt (2008) include books, pizzas, dollar stores, auto rentals, college tuition, and all-inclusive resorts.

Table 2. Class Pricing (from Zhang and Krishna 2007)

Despite its prevalence, class pricing has been studied rigorously in the extant literature in just a single paper. Wernerfelt (2008) proposes the costly process of setting prices (i.e., gathering information about cost and demand, analyzing the data, bargaining costs, posting prices, etc.) as a firm-side explanation. The pricing costs induce firms to pass up some opportunities for assigning fine-grained prices.

This research aims to add to Wernerfelt (2008) by providing a different rationale for class pricing based on the consumer perspective. Drawing from prospect theory (Kahneman and Tversky 1979), I assume that consumers exhibit reference dependency and loss aversion in their preferences. Specifically, consumers not only evaluate their economic payoffs from consuming a product, but also, they compare their outcomes relative to some reference point and experience reference gains or losses from these comparisons. As is well known, their sense of loss looms larger than the equivalent-sized gain, a phenomenon referred to as “loss aversion.” I develop an analytical model showing that class pricing emerges as an optimal strategy for firms to engage reference-dependent and loss-averse consumers.

I build on the seminal expectations-based prospect theory developed by Kőszegi and Rabin (2006), which posits that consumers’ reference points are their rational expectations about future consumption outcomes. This framework enables endogenous determination of reference points by aligning their optimal behaviors conditional on their rational expectations. Despite the elegance and significance of Kőszegi and Rabin (2006), its application to marketing strategies has been quite modest because the challenges of applying Kőszegi and Rabin (2006) stem from the self-fulfilling nature of consumers’ expectations to identify consumers’ optimal behaviors. Consequently, much of the existing economic literature (Heidhues and Kőszegi 2008, 2014; Karle and Peitz 2014) has focused solely on single-product firms.

My research overcomes these obstacles and advances Kőszegi and Rabin (2006) to a multiproduct context. I establish a model wherein a representative firm sets prices for its differentiated products and sells to a continuum of consumers with heterogeneous tastes. Consumers are ex ante uncertain about their ideal tastes and hold prior beliefs regarding these tastes. This uncertainty is resolved just prior to purchase, at which point each consumer privately observes the realization.

Central to my analysis of consumers’ optimal behaviors is the concept of state-contingent consumption plans, which outline consumers’ planned product choices for every possible realization of their ideal tastes. These plans generate distributions of future possible consumption outcomes along both price dimension and taste dimensions. In accordance with Kőszegi and Rabin (2006), these distributions function as consumers’ reference distributions. Upon taste realization, consumers compare consumption outcomes from each product choice against the reference distributions to determine reference gain/loss utilities, and they select ex post optimal product(s) to maximize their total utilities. My solution employs credible consumption plans, ensuring that consumers’ ex post optimal product choices after taste realization align with their ex ante consumption plans. I characterize the structure and uniqueness of credible consumption plans and identify conditions for class pricing to be optimal. I further characterize how the optimal numbers of classes, grouping patterns, and class sizes can be determined and how loss aversion can affect class size in a more general model.

My findings reveal a core driver of class pricing centering around demand-side effects rooted in consumers’ loss aversion. At its core, although cost differences motivate unequal prices to expand demand for a low-cost product, consumers asymmetrically evaluate the induced price difference as weighted reference price gain for the low-cost (cheaper) product but weighted reference price loss for its high-cost (expensive) alternative. Because of loss aversion in prices, this price difference reduces consumers’ utility for the expensive product more than it can increase consumers’ utility for the cheaper one. Therefore, when a monopolist fully extracts indifferent consumers’ value, the demand expansion for the low-cost product reduces its own willingness to pay (WTP) more significantly than it can elevate the WTP for the high-cost alternative. In case of very strong loss aversion, WTP for both products may decline. Therefore, this asymmetry induced by loss aversion in prices can reduce total profit from selling multiple products, incentivizing the monopolist to adopt class pricing.

More specifically, an increase in a cheaper product’s demand dilutes its alignment with indifferent consumers’ taste and decreases its product fit, generating increased reference loss in the taste dimension. To make this product attractive to the outside option, the increased loss in the taste dimension needs to be counterbalanced by an increased reference gain in the price dimension (namely, a reduced WTP for the cheaper product). Conversely, this demand shift simultaneously enhances the fit of its expensive alternative and thereby, reduces reference loss for buying the expensive alternative in the taste dimension. As the fit of the outside option is unchanged, this demand shift increases the expensive product’s marginal taste advantage over the cheaper one up to twice as much as it decreases the cheaper product’s marginal taste advantage over the outside option.

However, loss aversion in the price dimension constrains the monopolist’s ability to fully capitalize on the expensive product’s increased taste advantage. The price difference between products is asymmetrically weighted as a partial reference gain for the cheaper product but a partial reference loss for the expensive one. Consequently, choosing the expensive product yields both a realized partial loss and a foregone partial gain for the cheaper one. This dual cost results in a price disadvantage for the expensive product scaled by a composite weight that captures both the incurred partial loss and the foregone partial gain. Because of loss aversion, this weight exceeds the complete gain associated with the cheaper product’s price disadvantage over the outside option. Therefore, under full value extraction, this asymmetry caps the permissible marginal increase in the price difference to be less than twice the magnitude of the marginal reduction in the cheaper product’s WTP. It follows that even if the expensive product’s WTP rises when loss aversion is not too strong, this increase is insufficient to offset the larger price reduction of the cheaper product. Under severe loss aversion, the price difference becomes even less responsive than the price reduction of the cheaper product, and the WTP for both products declines. In both scenarios, this demand shift results in a net decrease in total revenue.

The core mechanism, which is built on asymmetric consumer responses to price expectations driven by loss aversion, is both theoretically consistent and well supported by extensive empirical evidence. Asymmetric reference price effects have been robustly documented across many frequent purchase categories in supermarket and retailing settings, including coffee (Kalwani et al. 1990, Krishnamurthi et al. 1992), yogurt (Mayhew and Winer 1992), eggs (Putler 1992), refrigerated orange juice (Hardie et al. 1993), and sweetened and unsweetened drinks (Kalyanaram and Little 1994) as well as ketchup, peanut butter, and tuna (Erdem et al. 2001). Moreover, loss aversion in pricing has been observed in diverse contexts, such as tourism (Nicolau 2008), restaurants (Morgan 2008), telecommunications (Bidwell et al. 1995), and energy use (Adeyemi and Hunt 2007, Ryan and Plourde 2007). It is noteworthy that many of these product categories and contexts are precisely those where class pricing is commonly employed. This consonance lends suggestive support to the underlying mechanism proposed in my model.

My research contributes to the literature in several ways. First, this research supplements the Wernerfelt (2008) friction cost explanation for class pricing with my consumer-driven explanation based on reference dependency and loss aversion. Second, I extend Kőszegi and Rabin (2006) to multiproduct settings, enriching the current economic literature that applies Kőszegi and Rabin (2006) in consumer purchase contexts (Heidhues and Kőszegi 2008, 2014; Karle and Peitz 2014). Lastly, my work contributes to the growing marketing literature that explores marketing strategies with loss aversion (Ho and Zhang 2008, Orhun 2009, Kuksov and Wang 2014, Amaldoss and He 2018, Zhang and Li 2021). Although much existing work employs fixed and context-specific reference points, by drawing on Kőszegi and Rabin (2006), my work employs stochastic and endogenous reference points to sidestep context-specific references, and thus, it adds to the limited formal modeling of applying Kőszegi and Rabin (2006) to marketing strategies.

From a managerial perspective, my findings provide valuable insights into the necessity and desirability of class pricing. My results suggest that firms can benefit from implementing class pricing across various product categories when consumers exhibit strong loss aversion in pricing. Of immediate practical utility are my results about the number, the size, and the grouping patterns of pricing classes in implementing class pricing and about profit and consumer surplus implications with class pricing. Furthermore, I argue that although big data and emerging technologies enable firms to assign different prices to different products and the implementation of high-frequency AI pricing algorithms can exacerbate the magnitude of price differentials among products, doing so may not necessarily be more profitable. The central role of consumers with loss aversion motivates firms to adopt fewer rather than more prices.

The remainder of the paper is organized as follows. Section 2 reviews related literature. Section 3 sets up the model and introduces the firm’s problem. Section 4 characterizes the existence and properties of consumers’ personal equilibria (PE), and it analyzes marginal consumers’ behavior and the firm’s optimal strategies in a two-product analysis. I identify conditions on the optimality of class prices. In Section 5, I generalize my model results in an N-product case on how number of pricing classes and class sizes are determined. Section 6 concludes with a discussion of academic contributions and managerial implications of the results as well as directions for future work.

2. Related Literature

Despite the literature on class pricing per se being limited to Wernerfelt (2008), there is a stream of work on a uniform price for all products. Softened producer competition is identified as the primary motivation, with competing firms averting a prisoner’s dilemma through a credible commitment to uniform pricing (Holmes 1989, Corts 1998, Chen and Cui 2013). Additional explanations encompass firm-side friction costs and regulatory constraints on segmented prices as reviewed by Einav and Orbach (2007). As uniform pricing can be considered a restrictive form of class pricing, my rationale for class pricing can also be applied to uniform pricing. Consequently, this paper supplements existing explanations for uniform pricing by offering an alternative explanation that is applicable in monopolistic scenarios, driven by customer behaviors, and valid in the absence of regulatory constraints.

My work builds upon the expectations-based prospect theory developed by Kőszegi and Rabin (2006), which posits that consumers’ reference points are their rational expectations about future consumption outcomes. Extensive empirical evidence from both laboratories and the field has demonstrated the predictive value of Kőszegi and Rabin (2006). List (2003) shows that the endowment effect vanishes among experienced traders who expect to trade. Crawford and Meng (2011) find that drivers with expectations about wage and working hours stop working upon reaching anticipated target hours. Ericson and Fuster (2011) validate a laboratory experiment in which individuals anticipating a higher probability of trading engage in more trading. Abeler et al. (2011) reveal consistent correlation between expectations and individuals’ effort behaviors.

In the context of pricing that adopts the framework of Kőszegi and Rabin (2006), Heidhues and Kőszegi (2008) examines how single-product firms with varying costs compete for loss-averse consumers using an identical price. Their intuition is that consumer-determined reference points lead to a kink in the demand curve; therefore, increasing a price above consumers’ expected price results in a substantial loss of demand to competitors, and decreasing a price below cannot attract sufficient demand from competitors. The issue of kinked demand in my paper does not exist because the endogeneity in a firm’s determination of reference points smooths out the demand responsiveness. Moreover, my focus on a monopolistic setting with complete market coverage eliminates demand stealing effects induced by competition.¹ Instead, my mechanism centers on the asymmetric effects of loss aversion on consumers’ WTP for different products—effects that are absent from Heidhues and Kőszegi (2008). Karle and Peitz (2014) also examines firms’ competition with loss-averse consumers similar to Heidhues and Kőszegi (2008) but in the context where firms can determine reference prices. Interestingly, in contrast to the focal pricing discovered in Heidhues and Kőszegi (2008), they find that loss aversion exacerbates price differences between competing firms. Heidhues and Kőszegi (2014) study a monopolist’s high-low pricing strategies with a loss-averse consumer in a single-product context. Whereas both papers emphasize how loss aversion contributes to price disparities or variations in a setting where firms can determine reference prices, my paper shows the formation of class pricing by a multiproduct monopolist in such a setting.

My research contributes to the literature examining firm strategies for loss-averse consumers/agents, which has addressed channel contracting (Ho and Zhang 2008), produce line design (Orhun 2009), dynamic price competition (Kuksov and Wang 2014), differentiated price competition (Amaldoss and He 2018), and quality disclosure (Zhang and Li 2021). This paper contributes to this stream of literature both on the methodology side and with managerial insights. On the methodology side, as the canonical prospect theory by Kahneman and Tversky (1979) provides limited guidance on determining reference points, many of these studies employ context-specific reference points. By employing Kőszegi and Rabin (2006), a theory with demonstrated portability across different contexts, this paper contributes to this body of literature by sidestepping context-specific reference points and expanding the applications of Kőszegi and Rabin (2006) into marketing strategies. On the managerial insight front, although loss aversion typically presents a negative externality to consumers’ utility functions, its impacts can vary across different contexts.² For example, Orhun (2009) shows that loss aversion can lead to a compressed product line, offering consumers similar products. In contrast, Amaldoss and He (2018) demonstrate that loss aversion may enhance product variety depending on consumer valuations. This research contributes to this body of literature by studying optimal pricing strategies with multiple products in monopolistic environment and identifying conditions for class pricing despite cost variances.

3. The Model

In this section, I set up a model to study a representative firm’s pricing decisions that incorporate consumers’ endogenous reference-dependent and loss-averse preferences. Consider a firm selling an assortment of $N$ ($N\ge 2$) products in a product category. These horizontally differentiated products³ are located on a Hotelling line from zero to $(N-1)m$ with an equal distance from each other, and $m$ denotes the equidistance between two neighboring products (Hotelling 1929). Thus, the location of product $i$ is $a_i=\left(i-1\right)m$, $i=\mathrm{1,2},\dots ,N$. I consider the case where these products differ in their marginal costs $c_i$ with $c_1<c_2<\dots <c_N$ so that the firm has incentive to charge different prices for different products.⁴

3.1. Consumer Behavior

A mass $(N-1)$ of consumers have tastes for their ideal product denoted as $x$, uniformly distributed on the same Hotelling line. Each consumer buys at most one product and evaluates their consumption (economic) outcomes along two dimensions: taste dimension and price dimension defined as $\delta =(t,p)$. Therefore, ${\pi }_{ti}^j$ and ${\pi }_{pi}^j$ are denoted as consumption outcomes in the taste and price dimension, respectively, for consumers located in the $j$th interval of the Hotelling line (or correspondingly, for $x\in [a_j,a_{j+1}]$) when purchasing product $i$. In the taste dimension, a consumer incurs linear transportation costs to buy product $i$ for traveling from $x$ to $a_i$, and ${\pi }_{ti}^j=v-t|x-a_i|$, where $v$ is consumers’ intrinsic value for all of the products and is the same for all of the consumers, $t|x-a_i|$ is the mismatch taste cost (transportation cost) between their ideal product $x$ and the actual product $i$, and $t>0$ measures the unit transportation cost. In the price dimension, consumers pay $p_i$ for product $i$, so ${\pi }_{pi}^j=-p_i$.

Therefore, the total consumption payoff is ${\pi }_i^j={\pi }_{ti}^j+{\pi }_{pi}^j=v-t\left|x-a_i\right|-p_i$. To best investigate consumers’ switch between products, I focus on the case when $v$ is sufficiently high so that all of the consumers find it better to buy one product than buy nothing.

3.1.1. Ex Ante Taste Variability.

The literature has long demonstrated that consumers’ tastes may vary across by different settings and contexts, which is evident by their variety-seeking and brand-switching behaviors (McAlister and Pessemier 1982, Givon 1984). I incorporate this idea of taste variability and assume that ex ante consumers located in $[a_j,a_{j+1}]$ ($j=\mathrm{1,2},\dots ,N-1$) hold a prior that their ideal tastes are uniformly distributed over this region (equivalently $x\sim U[a_j,a_{j+1}]$).⁵ With $j$ ranging from $1$ to $N-1$, consumers located in different intervals have different priors; thus, this setup incorporates heterogeneity in consumers’ priors.

Just prior to the purchase decision, every consumer privately observes his realized taste $\tilde{x}\in [a_j,a_{j+1}]$, and such uncertainty in taste is resolved.

3.2. Credible Plans/Personal Equilibria (PE)

Consumers evaluate not only their consumption outcomes, but also, they compare their consumption outcomes with reference points. Per Kőszegi and Rabin (2006), consumers’ rational expectations held ex ante about outcomes serve as references. In the context of multiple products, I incorporate consumers’ rational expectations as generated by consumers’ credible state-contingent consumption plans, which are defined as planned product choices that every consumer forms ex ante for every possible state in their prior belief distribution. A state-contingent plan is credible when a consumer’s ex post optimal product choice (after uncertainty is resolved) aligns with their ex ante planned choice for every possible state. If such a credible plan exists, we say a personal equilibrium (PE) exists. Figure 1 illustrates a closed-loop process of determining credible plans (PE).

Specifically, a consumer with a prior belief about the taste $x\sim U{[a}_j,a_{j+1}]$, where $j=\mathrm{1,2},\dots ,N-1$, forms a taste-contingent consumption plan $M_j$, which outlines their planned product choice(s) from the $N$ available products for every possible taste (equivalently, $M_j(x)\subset \{\mathrm{1,2},\dots ,N\}$ for every $x\in {[a}_j,a_{j+1}]$). This plan generates two probability distributions of anticipated future consumption outcomes in the price dimension and the taste dimension denoted as ${\mathrm{\Gamma }}_p^j$ and ${\mathrm{\Gamma }}_t^j$, respectively, which will serve as reference distributions.

When the uncertainty in their taste is resolved, the consumer privately observes a realized $\tilde{x}$, and the consumer’s ex post consumption outcomes from buying product $i$ are ${\pi }_{pi}^j\left(\tilde{x}\right)=-p_i$ in the price dimension and ${\pi }_{ti}^j\left(\tilde{x}\right)=v-t|\tilde{x}-a_i|$ in the taste dimension.

To formally model reference-dependent and loss-averse preferences, in demension $\delta$, for a consumption outcome ${\pi }_{\delta i}^j\left(\tilde{x}\right)$ and a reference point $r_{\delta }^j$ in its distribution ${\mathrm{\Gamma }}_{\delta }^j$, a weighted gain-loss utility $g_{\delta }^j\left({\pi }_{\delta i}^j\left(\tilde{x}\right)|{\mathrm{\Gamma }}_{\delta }^j\left(r_{\delta }^j\right)\right)$ is given by

Here, $0<\eta \le 1$ is the weight given to the gain-loss utility as compared with consumption utility, and the loss aversion coefficient $\lambda$ is the additional weight attached to reference losses relative to reference gains. The reference gains are associated with a slope for $\eta$, whereas reference losses are associated with a slope of $\eta \lambda (\lambda >1)$. Loss aversion is captured with $\lambda >1$ such that losses loom larger than the same-sized gains.

Notably, the ex post consumption outcome (${\pi }_{\delta i}^j$) is deterministic, whereas the reference distribution ${\mathrm{\Gamma }}_{\delta }^j$ is stochastic. Per Kőszegi and Rabin (2006), when the consumer exhibits a reference-dependent preference, they compare ${\pi }_{\delta i}^j$ with every possible reference point within ${\mathrm{\Gamma }}_{\delta }^j$, with each comparison weighted by the occurrence probability of that reference point. Thus, $g_{\delta }^j\left({\pi }_{\delta i}^j|{\mathrm{\Gamma }}_{\delta }^j\right)$ is a weighted gain-loss utility based on the probability density of the reference distribution.

I consider that both consumption outcomes and reference gain-loss utilities are additive across dimensions, so the ex post total utility for a consumer with $\tilde{x}\in [a_j,a_{j+1}]$ to buy product $i$ is

Accordingly, with the realized $\tilde{x}$, the consumer’s optimal product choice $i^{*}$ is determined by $i^{*}={\text{argmax}}_{i=\mathrm{1,2},\dots ,N}{u}_i^j\left(\tilde{x}\right)$, the products that give them the highest ex post total utility among the $N$ available products.

I also establish the consumer’s participation constraint, which ensures that they find it better to buy something than to buy nothing. If the consumer chooses to buy nothing, then they get their initial endowment normalized to zero in both dimensions (equivalently, $({\pi }_{p0}^j,{\pi }_{t0}^j)=\left(\mathrm{0,0}\right)$). Thus, the ex post total utility of not buying, $u_0^j$, is given by

Consumers can form any consumption plan as desired, but only credible plans are rationally expected to survive at the purchase stage. The key step in determining PE is to eliminate implausible plans and retain only credible plans in which consumers’ planned product choice generates higher ex post total utility than all other alternatives, including the option of not buying.

Formally, $M_j\left(x\right)$ is a PE for a consumer with a prior belief $x\sim U{[a}_j,a_{j+1}]$ if for every $x\in [a_j,a_{j+1}]$ ($j=\mathrm{1,2},\dots ,N-1$), we have the following.

• i. Optimality and internal consistency. $\begin{array}{c}{M}_j\left(x\right)={\mathrm{\text{argmax}}}_{i=\mathrm{1,2},\dots ,N}{u}_i^j\left(x|M_j\right)={\pi }_{pi}^j\left(x\right)+{\pi }_{ti}^j\left(x\right)+g_p^j\left({\pi }_{pi}^j\left(x\right){|\mathrm{\Gamma }}_{p|M_j}^j\right)+g_t^j\left({\pi }_{ti}^j\left(x\right)\right|{\mathrm{\Gamma }}_{t|M_j}^j)\end{array}$, where ${\mathrm{\Gamma }}_{p|M_j}^j$ and ${\mathrm{\Gamma }}_{t|M_j}^j$ are distributions of anticipated outcomes in price and taste dimensions, respectively, generated by $M_j$.

  ii. Participation constraints. $u_{M_j\left(x\right)}^j\left(x\right)\ge u_0^j$.

3.3. Firm’s Problem

The firm is a standard profit-maximizing monopolist that knows the distribution of consumers’ stochastic tastes but does not observe their realizations. The firm chooses $N$ prices, $p_1,p_2,\dots ,p_N$ (these $N$ prices can be the same or different from each other), to maximize its expected profit. I assume that prices weakly increase in costs; that is, a price assigned to a higher-cost product is no lower than a price assigned to a lower-cost product (equivalently, $p_1\le p_2\le \dots \le p_N$ for products with costs $c_1<c_2<\dots <c_N$).⁶

3.3.1. Consumer Demand.

With credible plans, every consumer follows through their plan. The consumer demand of product $i$ can be generalized and written as ${\sum }_{j=1}^{N-1}\left[{\int }_{a_j}^{a_{j+1}}\right(\mathbb{I}\left(M_j\left(x\right)=i\right)/m\left)dx\right]$, where $\mathbb{I}\left(M_j\left(x\right)=i\right)=1$ if $M_j\left(x\right)=i$ and zero if otherwise. Here, ${\int }_{a_j}^{a_{j+1}}\left(\mathbb{I}\left(M_j\left(x\right)=i\right)/m\right)dx$ is demand of product $i$ from consumers located in ${[a}_j,a_{j+1}]$. Given that there are $N-1$ different intervals, the total demand of product $i$ can be aggregated from the demand from all of the intervals.

Therefore, the firm’s optimization problem is given by

subject to $p_1\le p_2\le \dots \le p_N$, and the constraints derived from PE conditions.

The timing of events is given as follows.

Stage 1. The firm announces (or makes commitment on) prices for all $N$ products.

Stage 2. For any $j=\mathrm{1,2},\dots ,N-1$, consumers located in ${[a}_j,a_{j+1}]$ form an ex ante taste-contingent consumption plan $M_j(x)$ for all $x\in {[a}_j,a_{j+1}]$, based on which they derive expectations of future possible consumption outcomes, ${\mathrm{\Gamma }}_{p|M_j}^j$ and ${\mathrm{\Gamma }}_{t|M_j}^j$ that serve as their reference distributions.⁷

Stage 3. Consumers privately observe their realized tastes $\tilde{x}$ (i.e., uncertainty is resolved for all of the consumers).

Stage 4. Consumers decide which product to buy (PE conditions guarantee that their choices are consistent with their ex ante state-contingent consumption plans).

4. Two-Product Analysis

To best understand the insights of class pricing, I start with a scenario involving two products (equivalently, $N=2$). Consequently, there is only one interval between products with $a_1=0$ and $a_2=m$, so we have $j=1$. Ex ante consumers form a state-contingent consumption plan $M_1(x)$ for all $x\in \left[0,m\right]$, which generates reference distributions ${\mathrm{\Gamma }}_p^1$ and ${\mathrm{\Gamma }}_t^1$.

4.1. Characterization of PE

To characterize the structure of a credible plan $M_1(x)$, for a realized taste $\tilde{x}$, I denote $S_{i^{*}}^{1*}=\{\tilde{x}|i^{*}={\text{argmax}}_{i=\mathrm{0,1},2}{u}_i^1\left(\tilde{x}|M_1\right)\}$ as the set of ex post realized tastes with which consumers find it optimal to buy product $i^{*}$ over other alternatives. I first explore several properties about consumers’ ex post total utilities from buying products 1 and 2 ($u_1^1\left(\tilde{x}\right)$ and $u_2^1\left(\tilde{x}\right)$, resepctively, as defined by Equation (1)) and the sets $S_1^{1*}$ and $S_2^{1*}$. These properties hold true for any ${\mathrm{\Gamma }}_p^1$ and ${\mathrm{\Gamma }}_t^1$.

As ${\mathrm{\Gamma }}_p^1$ and ${\mathrm{\Gamma }}_t^1$ are independent of realized tastes, a larger value of $\tilde{x}$ shifts consumers’ ideal taste farther away from (or closer to) product 1 (or product 2), inducing a worse (or better) fit from buying product 1 (or product 2) in the taste dimension. By comparing the realized outcome with ${\mathrm{\Gamma }}_t^1$, it also generates a lower (or higher) reference gain-loss utility in the taste dimension. In contrast, a draw of $\tilde{x}$ has no impact on consumption outcomes or reference gain-loss utilities in the price dimension. Therefore, based on these monotonicity patterns in the taste dimension, Lemma 1(iii) shows that if there exists a range of realized tastes where consumers ex post choose product 1 (or product 2) over other alternatives, then in any adjacent region to the left (or right) of that range, consumers will also prefer product 1 (or product 2).

Because buying the more affordable product 1 generates both higher consumption outcome and reference gain-loss utility than buying the more expensive product 2 in the price dimension, if there exists an interior cutoff point where consumers are indifferent between products 1 and 2, the indifferent consumer must attain a better fit and reference gain-loss utility in the taste dimension by locating more closely to product 2.

When there is no difference between prices ($p_1=p_2$), comparison between prices generates no reference gain or loss in the price dimension. Anticipating equal total utilities between buying products 1 and 2 in the price dimension, the indifferent consumer must also obtain equal total utilities in the taste dimension. This can be achieved when they position themselves precisely at $m/2$, the midpoint of the Hotelling line, ensuring an equal distance from either product.

Building on Lemma 1, I formally characterize the structure and uniqueness of an interior PE as follows.

Proposition 1 establishes a subgame equilibrium on the consumer side. Intuitively, this PE partitions $[0,m]$ into two intervals, with $s_1$ as an interior cutoff point. Based on the reference distributions generated by this plan, consumers find it better to buy product 1 (or product 2) over other alternatives when the realized taste falls in $[0,s_1)$ (or $(s_1,m]$). Therefore, these preplanned product choices are indeed ex post optimal. By Lemma 1(iv), $s_1\ge m/2$, so the indifferent consumer is located more closely to product 2.

Proposition 1 also identifies conditions for the existence and uniqueness of this PE. First, I focus on interior PE, where every product has positive demand (equivalently, $s_1<m$). This happens when the price difference between the two products is not too large (condition (i) in Proposition 1); otherwise, consumers always plan to buy product 1 over product 2. Condition (ii) in Proposition 1 is a participation constraint to ensure that every consumer finds it better to buy something than buy nothing.

4.2. Indifferent Consumer’s Behavior

Before I proceed to the firm’s optimal pricing strategies, I explore the key driver of class pricing—the behavior of the consumer who is indifferent between buying product 1 and product 2. The credible plan (PE) characterized in Proposition 1 specifies that the indifferent consumer is located at $s_1$.

The credible plan generates distributions of anticipated future consumption outcomes in the price and taste dimensions, which serve as reference distributions ${\mathrm{\Gamma }}_p^1$ and ${\mathrm{\Gamma }}_t^1$, respectively. In the price dimension, this plan generates a discrete distribution with two possible future outcomes: $-p_1$ with the probability that $x$ falls in $[0,s_1)$, which is $s_1/m$, or $-p_2$ with the probability that $x$ falls in $(s_1,m]$, which is $(m-s_1)/m$. Thus, ${\mathrm{\Gamma }}_p^1=(-p_1,s_1/m;-p_2,(m-s_1)/m)$. In the taste dimension, this plan generates a dense distribution with outcomes ${\pi }_t^1(x)=\left\{\begin{array}{ll}v-tx,& \text{if} x\in [0,s_1]\\ v-t(m-x),& \text{if} x\in [s_1,m]\end{array}\right.$. Given $x\sim U[0,m]$, the probability density of ${\mathrm{\Gamma }}_t^1$ is $1/m$ for $x\in [0,m]$ and zero otherwise.

Therefore, by applying Equation (1), the indifferent consumer’s total utility from buying product 1, $u_1^1\left(s_1\right)$, is given by

The first term is the indifferent consumer’s economic consumption outcome from buying product 1. The subsequent two terms reflect the reference gain/loss in the price dimension weighted by the probability of the corresponding reference point’s occurrence. Like other consumers, because $p_1$ stands as the lowest price available in the market, the indifferent consumer experiences exclusive reference gain and zero reference loss in the price dimension when buying product 1, weighted by the probability of planning to buy product 2 ex ante.

The last two terms reflect the weighted reference gain/loss in the taste dimension. In contrast, the indifferent consumer experiences zero gain but exclusive reference loss in the taste dimension as they locate farthest from product 1 than any other consumers from their intended product choices, resulting in the worst fit.

Also, by applying Equation (1), the indifferent consumer’s total utility from buying product 2, $u_2^1\left(s_1\right)$, is given by

In the price dimension, with $p_2$ as the highest price in the market, the indifferent consumer experiences no reference gain but exclusive reference loss from buying product 2, weighted by the probability of planning to buy product 1 ex ante.

In the taste dimension, the indifferent consumer experiences a mixture of reference gain and loss. Reference loss prevails except within the region $x\in \left[m-s_1,s_1\right]$. For tastes falling in this interval, consumers anticipate being located farther away from their intended product 1 compared with the distance between the indifferent consumer and product 2, so choosing product 2 generates better fit than from the intended product 1, generating reference gain. Note that only when $s_1=m/2$, this specific region vanishes, and the indifferent consumer experiences exclusive reference loss from buying product 2, mirroring the exclusive reference loss from buying product 1 in the taste dimension.

A unique feature of my setup is that the firm can use prices to manage consumers’ expectations and that consumers can rationally anticipate the indifferent consumer’s location $s_1$ based on prices. Given that the indifferent consumer attains equal total utilities between buying products 1 and 2 (equivalently, $u_1^1\left(s_1\right)=u_2^1\left(s_1\right)$), we have

In the price dimension, product 2 suffers a relative price disadvantage over product 1 quantified as the negative price difference amplified by a combination of partial recognized loss and partial foregone gain, $w_{p1}$. Conversely, product 2 gains a relative taste advantage over product 1 quantified as the positive taste outcome difference amplified by a reduced loss offset by partial gain, $w_{t1}$.

I denote $d_1\equiv p_2-p_1\ge 0$ as the price difference between product 2 and product 1, and derive the responsiveness of $d_1$ to $s_1$, $\partial d_1/\partial s_1$, based on Equation (5):

The derivative of $\partial d_1/\partial s_1$ is determined by two opposing effects. First, an increase in $s_1$ widens the normalized relative taste advantage of product 2 over product 1 (as captured by the first term in Equation (6)), which necessitates a larger price difference. Second, it also increases the reference price weight $w_{p1}$, which reduces the required price difference. Substituting Equation (5) into Equation (6) yields the expression of $\partial d_1/\partial s_1$ listed on the second line in Equation (6). Note that when $s_1$ is small (i.e., $s_1\to m/2$), the second term is negligible, so $\partial d_1/\partial s_1\to 2t\hspace{0.17em}{w}_{t1}/w_{p1}$ is mainly determined by the taste-to-price reference weight ratio.

When the indifferent consumer chooses not to buy (applying Equation (2), like all other consumers), their total utility of not buying, $u_0^1$, is given by

Choosing not to buy saves possible paid prices, thereby generating exclusive reference gain in the price dimension. Conversely, no buying foregoes possible intrinsic benefits from owning a product, leading to exclusive reference loss in the taste dimension.

Following participation constraints, all consumers prefer to make a purchase rather than abstain. This is ensured when even the indifferent consumer is willing to make a purchase. Equivalently, we have $u_1^1\left(s_1\right)=u_2^1\left(s_1\right)\ge u_0^1$.

4.3. The Optimality of Class Pricing

Because every consumer ex post follows through their credible ex ante plans, consumer demand is determined by the PE. Accordingly, the consumer demand for product 1 is $D_1=s_1$, and the consumer demand for product 2 is $D_2=m-s_1$. Solving Equation (5) yields an explicit expression of endogenously determined $s_1$:

Note that $s_1$ can be elegantly represented as a function of $d_1$. To ensure an interior $s_1<m$, I can further establish a constraint regarding the upper bound of $d_1$, which gives that $d_1<{\overline{d}}_1=mt\left(\eta +\lambda \eta +2\right)/\left[2\left(1+\lambda \eta \right)\right]$.

Accordingly, the firm’s optimization problem is defined as Problem 1 below .

An important aspect of a monopolistic environment is the monopolist’s unique ability to extract consumer surplus by leveraging multiple prices collectively. The following lemma establishes how the monopolist extracts consumer value in this optimization problem.

Intuitively, given that the consumer demand $s_1$ is a function of the price difference $d_1$ (Equation (8)), if the monopolist sets $p_1^{\prime }<{\overline{p}}_1$ for product 1 and $p_2^{\prime }$ for product 2, it follows that the monopolist could achieve a strictly higher profit by simultaneously raising the price for product 1 from $p_1^{\prime }$ to ${\overline{p}}_1$ and the price for product 2 from $p_2^{\prime }$ to $p_2^{\prime }+{\overline{p}}_1-p_1^{\prime }$. In this way, both prices increase without changing their products’ demands as the price difference remains at $p_2^{\prime }-p_1^{\prime }$.⁸

With a full value extraction, the monopolist sets prices at consumers’ maximum willingness to pay. Consumers’ WTP for product 1, denoted as ${\overline{p}}_1$, can be derived based on $u_1^1\left(s_1|{\overline{p}}_1\right)=u_0^1(.|{\overline{p}}_1)$. Accordingly, we have

In the price dimension, buying nothing saves all of the money, a higher economic outcome than buying product 1, generating a relative price advantage and thereby, a full gain for all of the possible states denoted as $w_{p2}=1+\eta$. In contrast, in the taste dimension, not getting a product at all yields a lower economic outcome than buying product 1, generating a relative taste disadvantage and thereby, a full loss for the whole span of possible tastes denoted as $w_{t2}=1+\eta \lambda$.

Consequently, WTP for product 1 is derived as

This readily yields an expression of $\partial {\overline{p}}_1/\partial s_1$, the responsiveness of product 1’s WTP to its own demand $s_1$:

Note that both reference weights $w_{t2}$ and $w_{p2}$ are constants independent of $s_1$ and that $\partial {\overline{p}}_1/\partial s_1$ always stays negative. The negative relationship between ${\overline{p}}_1$ and $s_1$ occurs because the relative taste disadvantage of not buying compared with buying product 1 must be offset by its relative price advantage.

Similarly, I can also derive WTP for product 2, ${\overline{p}}_2$, based on $u_2^1\left(s_1|{\overline{p}}_2\right)-u_0^1(.|{\overline{p}}_2)=0$. Because of the complexities arising from a mix of reference gain and loss in both the price and taste dimensions between $u_2^1\left(s_1\right)$ and $u_0^1$, the expression of ${\overline{p}}_2$ is tedious. This posess a challenge in understanding the cross-price response to demand, that is, how product 2’s WTP responds to the demand of its alternative product $s_1$, captured by $\partial {\overline{p}}_2/\partial s_1$. Rather than a direct derivation, given $d_1={\overline{p}}_2-{\overline{p}}_1$ under full value extraction, I find: $\partial {\overline{p}}_2/\partial s_1=\partial {\overline{p}}_1/\partial s_1+\partial d_1/\partial s_1$. Therefore, alternatively, a more elegant way to elucidate the characteristics of $\partial {\overline{p}}_2/\partial s_1$ is to compare the relative effects between $\partial {\overline{p}}_1/\partial s_1$ and $\partial d_1/\partial s_1$. The proposition below summarizes how loss aversion constrains the responsiveness of the weakly more expensive product 2’s WTP to $s_1$.

Because $\partial {\overline{p}}_2/\partial s_1=\partial {\overline{p}}_1/\partial s_1+\partial d_1/\partial s_1$, the suppressive effect of loss aversion on the WTP for the weakly more expensive product can be decomposed into its effect on (1) the cheaper product’s price responsiveness, $\partial {\overline{p}}_1/\partial s_1$, and (2) the price difference responsiveness, $\partial d_1/\partial s_1$. The impact on $\partial {\overline{p}}_1/\partial s_1$ is direct. When comparing the outside option with product 1, higher $\lambda$ amplifies the full-loss weight in the taste dimension but leaves the full-gain weight unchanged in the price dimension. Therefore, $\partial {\overline{p}}_1/\partial s_1$ decreases linearly in $\lambda$.

Although the responsiveness of $\partial d_1/\partial s_1$ to $\lambda$ exhibits a more complicated nonmonotone pattern,⁹ it is always dominated by the responsiveness of $\partial {\overline{p}}_1/\partial s_1$, resulting in a net decrease of $\partial {\overline{p}}_2/\partial s_1$ in $\lambda$. When $s_1$ is large, both $\partial d_1/\partial s_1$ and $\partial {\overline{p}}_1/\partial s_1$ decrease in $\lambda$. When $s_1$ is small, as discussed about Equation (6), both $\partial d_1/\partial s_1$ and $\partial {\overline{p}}_1/\partial s_1$ are primarily determined by their respective taste-to-price reference weight ratios, and their responsiveness to $\lambda$ reflects the net elasticity difference between the associated taste and price weights regarding $\lambda$. Structurally, $w_{t1}$ is less sensitive to $\lambda$ than $w_{t2}$, and its loss component constitutes a smaller fraction of the total weight. This results in a lower elasticity for $w_{t1}$ than $w_{t2}$. More importantly, $w_{p1}$’s elasticity exceeds half that of $w_{t2}$. This occurs because the partial loss portion in $w_{p1}$ is no smaller than its partial gain portion, causing the loss component to be weighted more heavily than half of the full loss in $w_{t2}$. In contrast, $w_{p2}$ exhibits zero elasticity as its full gain component is independent of $\lambda$. These effects collectively imply that the net elasticity governing $\partial d_1/\partial s_1$ in $\lambda$ is less than half of that governing $\partial {\overline{p}}_1/\partial s_1$, leading to a weaker responsiveness of $\partial d_1/\partial s_1$ to $\lambda$ than $\partial {\overline{p}}_1/\partial s_1$.

To illustrate, Figure 2 compares how $\partial {\overline{p}}_1/\partial s_1$, $\partial d_1/\partial s_1$, $\partial {\overline{p}}_2/\partial s_1$, and $\partial ({\overline{p}}_1+{\overline{p}}_2)/\partial s_1$ change with $\lambda$ when $s_1=m/2$. Although $\partial d_1/\partial s_1$ increases with $\lambda$ for small $\lambda$, its rate of increase is always smaller in magnitude than the rate of decrease in $\partial {\overline{p}}_1/\partial s_1$, inducing a negative responsiveness of $\partial {\overline{p}}_2/\partial s_1$ to $\lambda$.