The Importance of Price Beliefs in Consumer Search

Abstract

A consumer’s decision to engage in search depends on the beliefs the consumer has about an unknown product characteristic (e.g., price). Because beliefs are rarely observed, researchers typically assume that consumers have rational expectations or update beliefs consistent with Bayesian updating. These assumptions are restrictive and do not enable the researcher or the retailer to price discriminate among consumers on the basis of heterogeneity in beliefs. The authors use Monte Carlo experiments to show how these assumptions affect estimates of search cost. Next, they design an incentive-aligned online study in which participants search over the price of a homogeneous good, and the authors elicit distributions of price beliefs before and after each search. Drawing on data collected from a nationally representative panel, they find substantial heterogeneity in prior price beliefs, such that participants update their beliefs in response to search outcomes but deviate from Bayesian updating in that they underreact to new information. Importantly, the authors show that (1) assuming Bayesian updating does not significantly bias search cost estimates at the aggregate level if the researcher accounts for heterogeneous prior beliefs, (2) eliciting heterogeneity in prior expected prices is much more important than eliciting heterogeneity in prior price uncertainty, and (3) a retailer can increase profits through third-degree price discrimination by recognizing the heterogeneity in prior beliefs.

Keywords

Bayesian updating consumer search price beliefs rational expectations search costs sequential search

Searching over prices is ubiquitous in online and offline product markets such as hotels, grocery stores, automobiles, and others, thus making it imperative for researchers and practitioners to understand the underlying primitives that govern consumers’ search process. Consumers typically trade off the cost of searching (hereinafter referred to as “search cost”) with the expected benefit from search, which is computed using consumers’ prior beliefs about expected prices, the uncertainty in these beliefs, and how consumers update their beliefs in response to searched prices (Baye, Morgan, and Scholten 2006; Stigler 1961). Because beliefs are typically not observed, researchers (1) routinely assume that consumers have rational expectations or update their beliefs consistent with Bayesian updating and (2) infer search cost on the basis of these assumptions and observed search patterns (e.g., Mehta, Rajiv, and Srinivasan 2003; Hong and Shum 2006; Koulayev 2013). However, little is known about the implications of these assumptions on search cost estimates and the extent to which researchers and practitioners might benefit from relaxing these assumptions. In this study, we elicit the distribution of price beliefs in a sequential search paradigm to examine the implications of assuming rational expectations and Bayesian updating, and we quantify the relative importance of accounting for heterogeneity in expected prices versus prior uncertainty in prices. An assumption of rational expectations also implies that consumers are homogeneous in their beliefs, which rules out an opportunity for retailers to price discriminate on the basis of belief heterogeneity. Our investigation thus not only informs researchers on the drivers of search cost estimates but also quantifies the benefit to a retailer of engaging in price discrimination on the basis of heterogeneous prior beliefs.

Accounting for beliefs is central to the literature on consumer search as well as to the economics and marketing literature on dynamic discrete choice, consumer learning, risk, and insurance and investments (see, e.g., Erdem and Keane 1996; Kahneman and Tversky 1979; Rust 1987; Shin, Misra, and Horsky 2012). Explicitly eliciting heterogeneous beliefs across consumers and over time for the same consumer in a sequential search framework alleviates the need to assume rational expectations (i.e., consumer beliefs coincide with the true price distribution) and Bayesian updating.¹ Our goal in this study is not to critique these assumptions, which we believe are necessary in the absence of belief information; rather, we study the relative importance of different assumptions and provide guidance to researchers and managers on how to relax some of these assumptions with minimal data requirements and how to managerially benefit from measuring price beliefs.

We conduct our investigation by first running Monte Carlo simulations to understand how assuming rational expectations and Bayesian updating affect both an individual consumer’s search cost estimates and the estimated distribution of search cost. We then apply a sequential search model with heterogeneous beliefs to data collected from an incentive-aligned online survey among representative consumers who search over price for a homogeneous home appliance: a KitchenAid mixer. We elicit prior price beliefs from each participant and then exogenously generate search scenarios, each with a different mixer price. After each search, we again elicit the (posterior) price beliefs and allow participants to decide between searching more versus purchasing at the lowest observed price (i.e., the case of perfect recall). Unlike previous research, which directly elicits expectations (e.g., Delavande 2008; Erdem et al. 2005), we elicit belief quantiles, which enables us to account for price uncertainty in addition to the expected prices. In our study, we incentive-align both belief elicitation and search tasks by utilizing the binarized scoring rule (Hossain and Okui 2013) and by introducing monetary incentives to participants’ trade-offs between search and purchase, respectively.

We find substantial heterogeneity across participants in not only the prior expected price but also the prior uncertainty around this price. Furthermore, participants update the prior expected price in response to search outcomes, and the uncertainty associated with this price decreases as participants search more. Given the elicited prior and posterior beliefs, we find that assuming either rational expectations or Bayesian updating with informative priors (i.e., the expected value of the realizations from prior beliefs coincides with the true price distribution) leads to overestimation of search costs, on average. Participants in our survey deviate from Bayesian updating in that they attach more weight to the prior beliefs and underreact to new information. However, after we account for heterogeneous prior beliefs (instead of informative priors), the extent of deviation from Bayesian updating does not significantly bias the estimated distribution of search costs.

A participant’s prior price belief is characterized by both the expected price and price uncertainty. Whereas eliciting the expected price is straightforward, estimating price uncertainty is more demanding because it requires eliciting additional information on the variance in price beliefs. Similar to assuming Bayesian updating, not accounting for individual heterogeneity in prior price uncertainty biases each participant’s estimate of search cost; however, these biases at the individual level do not translate to a bias in the distribution of search costs at the aggregate level. By contrast, ignoring individual heterogeneity in prior expected prices biases not only an individual’s search cost estimate but also the estimated distribution of search costs, such that the median search cost from ignoring heterogeneity in prior expected prices is 51% higher than the true median search cost (estimated from elicited beliefs). These results are consistent with our finding that a 1% increase in prior expected price (prior uncertainty) decreases (increases) search cost by 12% (3%), respectively. Together, these findings suggest that if researchers are only interested in the distribution of search costs, eliciting only individual-level expected prices should suffice.

Our findings also have important implications for retailers that can potentially benefit from price discrimination by taking advantage of heterogeneity in prior beliefs. Using our estimates, we show that third-degree price discrimination (i.e., price segmentation) based on beliefs, if observed, results in a 6.4% increase in profits. However, retailers do not typically observe prior beliefs and can, at best, collect beliefs from only a sample of consumers. We thus run a discriminant analysis to predict belief-based segment membership and allocate participants to different segments, using observable demographic and attitudinal variables. Compared with the base case, setting different prices to segments from the discriminant analysis increases profits by 4.0%. These results are promising in that a retailer can capture 63% of the ideal increase in profit (a scenario in which consumers’ price beliefs are known) by eliciting beliefs from a sample of consumers and engaging in third-degree price discrimination based on observable demographics of all consumers. We note that our analysis relies only on heterogeneity in price beliefs and is independent of a participant’s willingness to pay (WTP), which has been the basis of price discrimination in the extant literature.

Our article makes several contributions to the marketing and economics literature. First, we quantify how assumptions of rational expectations and Bayesian updating influence search cost estimates. Our findings provide evidence that Bayesian updating is a safer assumption to make even when consumers underreact to new information, granted that researchers take heterogeneous prior beliefs into account. Second, accounting for the entire distribution of beliefs enables us to quantify the relative importance of expected prices and price uncertainty in accurately estimating search costs. The analysis shows that if the goal is to estimate the distribution of search costs, then provided that consumers are somewhat knowledgeable about the price dispersion in the market, researchers only need to elicit expected prices without having to worry about the consumers’ price uncertainty. Our finding—that ignoring heterogeneity in prior expected prices results in overestimating search costs (for commonly assumed belief distributions)—also extends the previous literature by providing an alternate explanation for overestimation of search costs. Finally, we quantify the increase in a retailer’s profit from engaging in third-degree price discrimination that is based on heterogeneity in prior beliefs and observable consumer characteristics, independent of heterogeneity in WTP. We believe this is the first study to quantify the managerial impact of assumptions about prior beliefs and belief updating in the context of sequential search.

The rest of the article is organized as follows. The next section provides an overview of the search literature. Next, we outline a generalized search model to account for heterogeneous price beliefs and report results based on Monte Carlo experiments. We outline the details of our experimental design in the following section, then describe the data and discuss the search cost estimates. We present results on the implications of assuming Bayesian updating and the relative importance of expected prices versus uncertainty in prices. We then discuss the managerial implications of our findings for retailer pricing. Finally, we conclude and suggest directions for future research.

Related Literature

The theoretical and empirical literature typically characterizes and models search as either simultaneous (fixed sample) or sequential. For homogeneous goods, starting with Stigler (1961), simultaneous search is characterized by the consumer choosing the number of searches to make before actually searching and then choosing the searched alternative with highest utility. By contrast, McCall (1970) and Mortensen (1970) characterize search as a sequential process in which a consumer’s decision to continue searching is based on the last searched price. Lippman and McCall (1976) and Landsberger and Peled (1977) show that for a homogeneous good with a known (common) distribution of prices in the market, it is optimal to randomly search for the lowest price and stop searching when the lowest sampled price is lower than the reservation price—the price that makes the consumer indifferent between searching and stopping. Weitzman (1979) allows for heterogeneous products to show that it is optimal for consumers to sort products on the basis of reservation utility and start searching with the product with highest reservation utility.

Assuming rational expectations, or that consumers know the true price distribution, is restrictive and gives rise to a search paradigm in which consumers will never recall an alternative searched previously, provided that they have not exhausted all search outcomes. To account for this, Rothschild (1974) studies a setting where consumers update their beliefs in response to search outcomes and shows that with Dirichlet priors, the optimal search rule under sequential search with an unknown utility distribution is still based on reservation utilities. Rosenfield and Shapiro (1981) and Bikhchandani and Sharma (1996) further generalize this and prove that if the beliefs follow a distribution that satisfies certain assumptions, or if belief updating follows certain rules, then the optimal stopping rule is myopic, such that if it is optimal to stop searching given current beliefs, then it will never be optimal to search in the future even though consumers learn about the true distribution of prices.²

Researchers typically do not observe a consumer’s price beliefs (prior or posterior), which necessitates an assumption about the learning process and prior beliefs. Previous literature assumes either that consumers have rational expectations and do not learn about the price distribution (Honka 2014; Honka and Chintagunta 2017; Mehta, Rajiv, and Srinivasan 2003; Zwick et al. 2003) or that consumers update their beliefs, which can explain revisits to previously searched information (Bronnenberg, Kim, and Mela 2016; Dang, Ursu, and Chintagunta 2020). The empirical literature allowing for learning assumes that consumers engage in Bayesian updating with either normally distributed priors (Chick and Frazier 2012; Ursu, Wang, and Chintagunta 2020; Zhang, Ursu, and Erdem 2020), Dirichlet priors (Hu, Dang, and Chintagunta 2019; Koulayev 2013; Wu 2017), or Dirichlet process priors (De los Santos, Hortacsu, and Wildenbeest 2017; Häubl, Dellaert, and Donkers 2010).

While previous research assumes that consumers have an informative prior (De los Santos, Hortacsu, and Wildenbeest 2017; Koulayev 2013), such that the expected value of the realizations from the process that generates prior beliefs corresponds to the true distribution of prices, more recent research attempts to estimate prior beliefs from observational data (Hu, Dang, and Chintagunta 2019; Ursu, Wang, and Chintagunta 2020; Zhang, Ursu, and Erdem 2020). Hu, Dang, and Chintagunta (2019) use relative variation in clicking and purchase to estimate Dirichlet prior beliefs that are homogeneous across consumers. By contrast, Ursu, Wang, and Chintagunta (2020) and Zhang, Ursu, and Erdem (2020) estimate heterogeneous prior uncertainty either as latent segments or as a function of the prior consumer experience using eye-tracking data, respectively. Zhang, Ursu, and Erdem also model heterogeneous expected beliefs as a function of prior ownership. Both these works, however, focus on a setting where the consumer searches for their match value, and both assume Bayesian updating with a normal prior and posterior distribution. Our article complements this recent stream of research in that we focus on the role of, and explicitly elicit, prior and posterior beliefs in a setting where the consumer searches for the lowest price. Our primary goals, thus, are to study how assumptions such as rational expectations and Bayesian updating influence search cost estimates and to quantify the potential benefit to a retailer from recognizing heterogeneity in prior beliefs.

Finally, our article is related to Ching et al. (2020), who explore how consumers update beliefs. Similar to our research, Ching et al. also elicit prior and posterior beliefs using the binarized scoring rule (Hossain and Okui 2013) and allow for uncertainty in price beliefs. Although we also explore whether participants deviate from Bayesian updating, our article differs from Ching et al. in that our primary focus is on understanding the implications of assuming rational expectations and Bayesian updating on search cost estimates. By contrast, Ching et al. focus primarily on belief updating and do not elicit search decision, which is crucial to estimating search costs.

Sequential Search with Heterogeneous Beliefs

We focus on the sequential search model with heterogeneous prior beliefs and learning. Consider consumer $i$ searching for the lowest price of a homogeneous good across different retailers. The consumer faces a constant cost $c_{i}$ for every single search. Let $F_{i k} (p)$ denote the distribution of price beliefs the consumer has after making $k$ searches. The $i$ subscript in $F_{i k} (p)$ allows for consumers to have heterogeneous beliefs, and the $k$ subscript indexes learning about the price distribution. In the absence of learning, the beliefs will be independent of the number of searches. The special case, widely considered in the literature as when all consumers know the true distribution of prices (rational expectations), is represented by consumers having beliefs $F (p)$ (no $i$ or $k$ subscript).

Assuming that a consumer’s belief distribution satisfies the assumptions in Bikhchandani and Sharma (1996), the optimal stopping rule is myopic in nature. Let the lowest price a consumer has sampled after $k$ random searches be given by ${\bar{p}}_{i k}$ . The expected gain from searching once more (for the $k + 1 th$ time) excluding the search cost is given by

Γ_{i k + 1}^{s e q} = \int_{0}^{{\bar{p}}_{i k}} ({\bar{p}}_{i k} - p) d F_{i k} .

If the consumer searches $k$ times, then for $k$ searches to be optimal, the bounds of search cost $c_{i}$ can be specified as $c_{i} \in [Γ_{i k + 1}^{s e q}, Γ_{i k}^{s e q}]$ , which can be inferred directly from the data provided that we know the distribution of beliefs $F_{i k}$ and $F_{i k - 1}$ . Thus, in a simple setting where a consumer searches over the price of a homogeneous good, knowledge of (1) the number of searches a consumer undertook, (2) the distribution of lowest prices sampled, and (3) either prior and posterior beliefs, or prior beliefs and belief updating process, provide a lower and an upper bound on search costs. Assuming rational expectations implies that the expected benefit (Equation 1) for a consumer searching $k$ times and having searched a lowest price of ${\bar{p}}_{i k}$ is given by

Γ_{i k + 1}^{R E} = \int_{0}^{{\bar{p}}_{i k}} ({\bar{p}}_{i k} - p) d F_{0},

where $F_{0}$ is the prior belief distribution of the consumer, which is assumed to be homogeneous across consumers (no $i$ subscript) and equal to the true price distribution. If, instead, we assume that the consumer engages in Bayesian updating such that the prior beliefs follow a Dirichlet process with a base distribution $F_{i 0}$ , then the prior distribution $D$ is given by $D \sim D P (W, F_{i 0})$ where $W$ is the concentration parameter of the Dirichlet process. The Dirichlet process is a generalization of the Dirichlet distribution. Whereas the Dirichlet distribution is the conjugate prior for a multinomial distribution, the Dirichlet process is the conjugate prior for infinite, nonparametric discrete distributions. The Dirichlet process draws distributions around the base distribution, which is its expected value. Although the base distribution $F_{i 0}$ is continuous, the distributions $D$ (i.e., its realizations) drawn from the Dirichlet process are discrete, made up of countably infinite number of point masses (Ferguson 1973). The concentration parameter, which is a positive real value, dictates the extent of discreteness; as the concentration parameter goes to infinity, realizations of the Dirichlet process become continuous. The posterior distribution after $k + 1$ searches also follows a Dirichlet process such that

D \sim D P (W + k + 1, \frac{W}{W + k + 1} F_{i 0} + \frac{k + 1}{W + k + 1} {\hat{F}}_{i 0}),

where ${\hat{F}}_{i 0}$ is the empirical distribution of observed prices. As De los Santos, Hortacsu, and Wildenbeest (2017) show, the expected benefit from searching $k + 1$ times (lower bound on search for a consumer searching $k$ times) is given by

Γ_{i k + 1}^{B U} = \frac{W}{W + k + 1} \int_{0}^{{\bar{p}}_{i k}} ({\bar{p}}_{i k} - p) d F_{i 0} .

Thus, similar to Equation 1, conditional on assuming Bayesian learning with Dirichlet process and initial prior beliefs, researchers can infer bounds on search costs. The specification of the Dirichlet process allows for a continuous prior base distribution of prices (Bikhchandani and Sharma 1996), which is in line with our belief elicitation procedure (discussed in the next section). Yet its realizations are discrete, better capturing price distributions typically observed in the marketplace.

The extant literature in learning and dynamic discrete choice (see, e.g., Erdem and Keane 1996) as well as search (Ursu, Wang, and Chintagunta 2020; Zhang, Ursu, and Erdem 2020) models Bayesian learning with normally distributed prior beliefs. We, however, focus on the Dirichlet process priors for the following reasons. First, the Dirichlet process is more flexible than the normal distribution in that although it allows consumers to express their prior beliefs as a continuous distribution, the realizations from the posterior are discrete. This is especially important given that consumers typically conduct only a small number of searches (Bronnenberg, Kim, and Mela 2016; De los Santos, Hortacsu, and Wildenbeest 2012; Honka 2014). Second, under the Dirichlet process priors, the expected benefit from search depends only on the empirical distribution of lowest prices (Equation 4). By contrast, when searching for the lowest price, normally distributed priors imply that the expected benefit from search is a function of the empirical distribution of both lowest prices and all searched prices. Because, in our setting, consumers search for the lowest price, it is reasonable to assume that the expected benefit from search does not depend on the magnitude of the searched price if it is higher than the lowest sampled price so far. Having said this, in other applications where consumers learn about the product quality or match value of each alternative in the choice set, the empirical distribution of all searched outcomes matter, which could make normal priors a more suitable assumption. For the simulations and the results based on our experimental data, we perform robustness checks on the assumption of the Dirichlet process versus the normal updating process and find qualitatively similar results.

Relative Importance of Bayesian Updating and Prior Beliefs

We next run Monte Carlo simulations to provide intuition into how assuming rational expectations and Bayesian updating affects search cost estimates when the prior beliefs and the belief updating process deviate from these assumptions, respectively. Deviation from rational expectations can be measured on the basis of the degree to which the prior belief ( $F_{i 0}$ ) differs from the true price distribution. To measure the extent of deviation from Bayesian updating, we let the posterior distribution of beliefs follow a Dirichlet process, such that

D \sim D P [W + θ (k + 1), \frac{W}{W + θ (k + 1)} F_{i 0} + \frac{θ (k + 1)}{W + θ (k + 1)} {\hat{F}}_{i 0}],

where $θ$ measures the extent of updating. For $θ = 0$ , the posterior distribution is the same as the prior distribution (i.e., no learning), and for $θ = 1$ , the posterior distribution is the same as that from Bayesian updating (Equation 3). Furthermore, $θ < 1$ ( $θ > 1$ ) implies that consumers underreact (overreact) to new information. The expected benefit from search for the $k + 1 th$ time based on this generalization is given by

Γ_{i k + 1}^{G} = \frac{W}{W + θ (k + 1)} \int_{0}^{{\bar{p}}_{i k}} ({\bar{p}}_{i k} - p) d F_{i 0} .

If the consumer searches $k$ times, then Equation 6 provides the lower bound on search cost. It is straightforward to see that for $θ = 0$ (and assuming homogeneous prior beliefs), the expected benefit from search is the same as that in Equation 2, and for $θ = 1$ , the expected benefit is the same as that in Equation 4.

We first consider a representative consumer and simulate the number of searches this consumer makes on the basis of different values of prior expected price, prior uncertainty, or the extent of deviation from Bayesian updating with Dirichlet process priors.³ The Appendix provides more details about the simulation. Figure 1, Panel A, plots the difference between search costs estimated assuming either rational expectations (dashed line; $θ = 0$ ) or Bayesian updating with informative priors (solid line; $θ = 1$ ) and the true search costs for varying values of prior expected price (i.e., the error in search cost). We assume that the consumer’s prior uncertainty coincides with the standard deviation of the true price distribution and that they engage in Bayesian updating, so any differences in search cost can be attributed to differences in prior expected price and the true average price.

Figure 1.

Monte Carlo simulations based on one representative consumer.

If the consumer’s prior expected price is lower (higher) than the true average price (vertical line), then imposing either rational expectations (dashed line) or Bayesian updating with informative priors (solid line) generally results in underestimating (overestimating) search costs. This pattern is driven by the fact that if the consumer’s prior beliefs are higher than the true price distribution, then the consumer will search less, all else being equal. Therefore, assuming rational expectations or informative priors, we would infer higher search cost to rationalize the small number of searches. The error in search cost from not accounting for the prior expected price can be substantial given that we used a search cost of $10 to simulate the data. Furthermore, search cost estimates assuming Bayesian updating with informative priors are lower than those assuming rational expectations. We provide an analytical proof based on the Dirichlet process updating to support these results in Web Appendix A. We also find similar results based on simulations with normally distributed beliefs and updating (Web Appendix B).

Figure 1, Panel B, plots the error in search cost as a function of prior uncertainty, assuming that the consumer engages in Bayesian updating and their prior expected price is the same as the average true price. Assuming rational expectations or Bayesian updating with informative priors results in overestimation (underestimation) of search costs if the prior uncertainty is below (above) the standard deviation of the true price distribution. This pattern is driven by the fact that if the prior uncertainty is low, then the perceived benefit from searching is low and the consumer engages in less search, all else being equal. Assuming rational expectations or informative priors (i.e., uncertainty is higher than what the consumer perceives) therefore results in inferring higher search cost to rationalize the small number of searches. Importantly, we find that the error in search costs from assuming rational expectations or informative priors is driven more by prior expected prices than by prior price uncertainty.⁴ As we show in Web Appendix A, the relatively small impact of ignoring prior uncertainty is driven by the finding that the difference between prior uncertainty and true standard deviation is relatively small compared with the true average price. In situations where participants have little or no knowledge of the price dispersion in the market (i.e., the difference between prior uncertainty and the true standard deviation is high compared with the mean of the true price distribution), the error in search costs from assuming rational expectations or informative priors is also likely to be driven by the prior price uncertainty.⁵

To explore the impact of assuming Bayesian updating when the consumer either under- or overreacts to searched prices, in Figure 1, Panel C, we plot the error in search cost from assuming Bayesian updating ( $θ = 1$ ) with Dirichlet process priors (solid line) when the consumer deviates from Bayesian updating ( $θ \neq 1$ ). In this scenario, we hold the prior beliefs fixed and equal to the true price distribution so any changes in search cost can be attributed only to the extent to which the consumer deviates from Bayesian updating. Assuming Bayesian updating results in underestimating (overestimating) search costs if the consumer underreacts (overreacts) to new information (i.e., $θ < 1$ [ $θ >$ 1]). This is driven by the fact that the derivative of the bound on search cost in Equation 6 with respect to $θ$ is negative (i.e., if consumers underreact to new information [ $θ < 1$ ], then imposing Bayesian updating [ $θ = 1$ ] will result in lower search costs). Importantly, the possible error in search cost from assuming Bayesian updating is much smaller in magnitude than the possible error from not accounting for the prior expected price. We show in Web Appendix A that this finding can be attributed to differences in the elasticity of search cost with respect to the prior expected price and $θ$ . The dashed line in Figure 1, Panel C, shows the error in search costs from assuming Bayesian updating when the consumer has normally distributed prior beliefs. The similarity between the two curves suggests that our qualitative findings on the importance of assuming Bayesian updating do not hinge on the assumed updating process.

Heterogeneity in Prior Beliefs

The simulations based on a representative consumer show that accounting for prior beliefs is much more important than accounting for the true updating process. Furthermore, the error in search cost that results from not accounting for prior expected price is much larger than the error from not accounting for prior price uncertainty, provided that the difference in prior price dispersion and true standard deviation is small relative to the true average price. Consumers, however, are likely to have heterogeneous prior beliefs (for both expected prices and uncertainty), which influence the distribution of search costs. From a researcher’s perspective, accounting for or eliciting heterogeneity in both prior expected prices and prior price uncertainty may not always be feasible. Thus, we explore the implications of ignoring heterogeneity in either prior expected price or prior price uncertainty on the distribution of search costs.

Panel A (Panel B) of Figure 2 plots the average error in search cost (across consumers) from ignoring heterogeneity in prior expected prices (prior uncertainty) for varying degrees of heterogeneity in prior expected prices (prior uncertainty).⁶ While the average error in search cost that results from ignoring heterogeneity in prior expected prices increases with the degree of heterogeneity in prior expected prices, we find minimal effect of ignoring heterogeneity in individual-level prior uncertainty on the search cost estimates. Panel A also plots the error in search costs for different values of the average of prior expected prices compared with the true average price (i.e., average bias in prior beliefs). Notably, the marginal effect on search cost from ignoring heterogeneity in prior expected prices is similar regardless of the extent of bias in prior beliefs (i.e., the degree to which the consumers’ beliefs are wrong, on average).

Figure 2.

Monte Carlo simulations based on heterogeneous consumers.

The differences in the impact of ignoring heterogeneity in prior expected prices and prior uncertainty are driven by the error in search cost at the individual level. Specifically, as Figure 1, Panel A, shows, the error in search cost is convex in prior expected price, which implies that the magnitude of the error in search cost depends on whether the prior expected price is higher or lower than the true average price (holding fixed the absolute difference between the prior expected price and the true average price). Thus, for a symmetric distribution of prior expected price, as the heterogeneity in prior expected price increases (holding average of prior expected price fixed), the positive error in search cost (overestimation) from ignoring this heterogeneity will outweigh the negative error in search cost (underestimation); the combined effect of which is reflected in Figure 2, Panel A. The nonlinearity of search cost error in prior expected price (Figure 1, Panel A) stems from the calculation of expected benefit and exists even for a uniform distribution with a linear cumulative distribution function (CDF). Finally, we note that ignoring heterogeneity in prior expected prices could also result in underestimating search costs if the distribution of prior expected prices is skewed to the right.

By contrast, for reasonable values of prior uncertainty, the error in search cost is almost linearly decreasing in prior uncertainty (Figure 1, Panel B). This implies that for a symmetric distribution of prior uncertainty, the average error in search cost (at the aggregate level) from ignoring heterogeneity in prior uncertainty does not depend on the degree of heterogeneity in prior uncertainty. Ignoring this heterogeneity, however, does affect an individual consumer’s search cost estimate, as is evident in the Figure 2, Panel D, which plots the average of the absolute error in search costs. Taking the average over absolute value of the error provides a measure of the error at the individual consumer level. We find a similar, but much larger in magnitude, effect of ignoring heterogeneity in prior expected prices on search costs (Panel C).

In summary, the Monte Carlo simulations based on a representative consumer and heterogeneous consumers provide us a preliminary understanding of how assumptions such as rational expectations and Bayesian updating potentially influence search cost estimates at the individual and aggregate level, respectively. However, to derive consumers’ true search costs, we need to know both prior and posterior beliefs after each search. In the simulations, we alleviate this issue by assuming the process through which the consumer updates beliefs (Equation 5), which may not necessarily be true in practice. Next, we conduct an incentive-aligned experiment in which we elicit both prior and posterior price beliefs directly from consumers. The experimental data enable us to quantify the degree of heterogeneity in beliefs across consumers and accurately determine the bias in search cost estimates from routinely made assumptions. We also use these data to test whether consumers update beliefs consistent with Bayesian updating, without assuming that the true data-generating process follows Equation 5. Finally, we use the experimental data to quantify the potential benefit to a retailer from engaging in third-degree price discrimination on the basis of heterogeneity in prior beliefs—an undertaking that is not possible using Monte Carlo simulations.

Experimental Design

We designed an online incentive-aligned experiment in which we asked participants to search for prices of a KitchenAid Artisan 5-Quart mixer. We conducted our experiment among participants who were recruited from a nationally representative commercial online panel of Lightspeed Research. We controlled for age, gender, and income distributions to ensure that participants were reflective of the general U.S. population.

Quality control procedures

To further ensure the qualification of participants, we screened them such that they (1) have purchased or have been actively involved in decision making regarding kitchen appliances and (2) have a basic understanding of probability. We asked them three questions pertaining to probability and distributions (Web Appendix C) and they had to answer two of them correctly to proceed. After participants passed the screening questions, we collected information about their usage of and interest in purchasing the KitchenAid mixer, their cooking behavior, and their knowledge about prices of the KitchenAid mixer and appliances in general. Next, we provided participants an opportunity to familiarize themselves with belief elicitation tasks using a Nutri Ninja blender as the focal product. Participants performed a prior belief elicitation task and a posterior belief elicitation task after seeing a retail scenario in which a seller was selling the Nutri Ninja blender for $150. In our setting, a retail scenario corresponds to a search task, and moving forward, we use the words “retail scenario” and “search task” interchangeably. We describe these elicitation tasks in detail next. We concluded the practice session by showing how much money the participants would have made from the task, though we did not actually pay them for the practice tasks. Participants continued to the main study only if they felt comfortable performing the belief elicitation tasks.

Belief elicitation and search tasks

Once the participants agreed to continue, we asked them to imagine that they were in the market for a KitchenAid mixer, which was sold online (e.g., Amazon, eBay) and in local stores. We asked them to assume that there were 100 sellers from which they could potentially purchase the mixer and that these sellers may charge different prices. These prices include all applicable taxes and handling/shipping charges, and the warranty is provided by the manufacturer. Figure 3 illustrates the experimental flow. We began by eliciting participants’ prior beliefs about prices of the mixer. Because a consumer’s decision to search hinges on not only the expected price but also the uncertainty in this price, we follow Manski (2004) and the elicitation routine outlined in previous research (Dominitz and Manski 1996, 1997, 2004, 2005) to account for the entire distribution of price beliefs. This enables us to examine the role of uncertainty in price beliefs separately from that of expected prices.⁷ We note, however, that if a researcher is interested in only expected prices, then the elicitation procedure can be made considerably simple, as previously incorporated in Erdem et al. (2005) and Delavande (2008). To ensure that participants are incentivized to reveal their true beliefs, we use the binarized scoring rule proposed by Hossain and Okui (2013), which makes belief elicitation incentive-aligned irrespective of a participant’s risk preferences.

Figure 3.
Survey design (flowchart).

For prior beliefs, we first asked the participants what they thought were the minimum and maximum prices of the KitchenAid mixer to establish the belief bounds (Dominitz and Manski 1997; Oakley and O’Hagan 2007). With the price range obtained from each participant, we showed them four custom-designed price points ( $p_{d}^{}$ ; $d \in [1, 4]$ ) that were spread out evenly as follows: the minimum price plus the price range multiplied by .2, .4, .6, and .8.⁸ We then elicited a quantile (i.e., a value from the CDF) associated with each price point by asking for the number of sellers (out of 100) the participants thought would have a price below each of the price thresholds (the quantile scale is also shown in Web Appendix C). We designed our quantile elicitation task to resemble those used by Garthwaite, Kadane, and O’Hagan (2005) and Manski (2004).⁹ Next, we presented participants with a retail scenario where they could potentially purchase the mixer and then asked them to report what they believed to be the median price (Dominitz and Manski 1996). Each participant was shown four retail scenarios. We created 20 blocks of prices for each of the four scenarios and randomly drew a block of prices to show to each participant in random order.¹⁰

After the price exposure, we elicited the posterior (i.e., updated) price belief using a new set of custom-designed price points given by $m e d i a n \pm κ [(p_{H} - p_{L}) / 4]$ , where $m e d i a n$ is the median price elicited, $p_{H} - p_{L}$ is the price range calculated using the lowest and highest price elicited previously, and $κ$ takes on values .5 and 1.5. As before, participants provided us the quantiles associated with each of these price points. Note that the price range for the prior and subsequent belief elicitations is held constant to ensure that any changes in participants’ price uncertainty are not driven by the spread of the price points. After the posterior belief elicitation task, we asked participants whether they would be interested in purchasing the mixer or would continue to search. We allow for search with recall (i.e., participants can purchase the KitchenAid mixer at the lowest price they have observed once they decide to purchase). After each retail scenario, we elicited posterior beliefs and provided an opportunity for participants to purchase. Eliciting beliefs after every search task enables us to explore the implications of assuming Bayesian updating.

If a participant decided to purchase before observing all four retail scenarios, we continued to show them the remaining retail scenarios and elicited posterior beliefs after each retail scenario, but without incurring search costs (to be described in the next section). This ensured that participants did not stop searching early to avoid completing belief elicitation tasks and also that the payoff from the belief elicitation tasks did not depend on when participants stopped searching. In addition, in such a case, we did not ask the participant about their purchase decision in any of the remaining retail scenarios. Notably, the price a participant saw in each retail scenario was independent of the participant’s purchase decision. All participants were told that they would see four search scenarios and that their payoff from the experiment would be based on solely the lowest price at which they purchase and their performance on belief elicitation tasks.¹¹ We concluded our survey by collecting additional information about the participants’ knowledge about the KitchenAid’s prices after the survey, attitude toward spending money on kitchen appliances, and demographics.

Two-part incentive-aligned design

We designed both belief elicitation and search tasks to be incentive-aligned. For the belief elicitation tasks, we informed the participants that, in addition to the base compensation (as reward points), they could earn a monetary reward based on how accurate they were in estimating the number of sellers in the market that offered the mixer below a certain price point. We researched the market and found that prices of a KitchenAid mixer follow a normal distribution, with a mean and standard deviation of $325 and $37, respectively.¹² For each designed price point $p_{d}^{}$ , we calculated the probability ( $q_{t r u e} (p_{d}^{})$ ) of observing a price below the design point based on the true price distribution and derived a measure of loss as the square root of the absolute difference between the self-reported quantile (probability) and its associated true quantile value ( $l o s s = \sqrt{| q_{s e l f} (p^{}) - q_{t r u e} (p^{}) |}$ ).¹³ Finally, we drew a random number $r$ from a Uniform(0,1) distribution. If $l o s s < r$ , the participant earned $.25 for that design point. Given four price points in each elicitation task, participants could earn up to $1 per belief elicitation. This scoring rule is consistent with the binarized scoring rule (Hossain and Okui 2013), which provides correct beliefs about a random variable independent of the participant’s risk preferences and the expected utility hypothesis.

For the search tasks, we determine a participant’s payoff on the basis of a trade-off between searching once more at a cost of $1 versus the potential benefit from finding a price lower than the currently available best price. Participants were endowed with $3 at the beginning of the study, and barring the first search (which is free), for each incremental search participants incurred $1. Each incremental search also had a potential benefit wherein a $50 reduction in price (compared with the currently available best price) results in a reward of $1. This induces a trade-off between searching more by paying $1 to get a possibly lower price versus purchasing at the current lowest price and saving $1 in search costs, which makes the search task (decision to purchase) incentive-aligned. Once participants decided to purchase, we showed them the remaining search scenarios (if any) without them incurring any additional search costs. As we have mentioned, this ensures that the payoff from belief elicitation tasks does not depend on the number of searches participants chose to engage in before purchasing.

Data Description

Three hundred participants completed the survey, out of which we dropped 19 participants who provided invariant responses, thus resulting in a sample of 281 participants. Table 1, Panels A and B, report participants’ familiarity with prices and purchase decisions for kitchen appliances as well as the KitchenAid mixer. Before taking the survey, over 80% of the participants stated that they were familiar with the average prices (and possibly variation) of kitchen appliances, and over 90% of participants have purchased a kitchen appliance in the past. Participants in the study cooked on average four times a week, baked once a week, expressed interest in purchasing a stand mixer, and were somewhat knowledgeable about the prices of a KitchenAid mixer. In addition, the participants agreed that price was an important factor in their purchase decision and that they searched extensively before buying a kitchen appliance. Finally, 88% of the participants found the survey questions to be clear and 82% found the prices shown to be realistic. Thus, the sampled participants are representative of the U.S. population, are relevant for this research, and understood the survey questions. Web Appendix D reports detailed information about the sample.¹⁴

Table 1.
Survey Summary: Pre- and Postsurvey Familiarity and Purchase Behavior for Home Appliances and KitchenAid Mixer.

N Percentage

A: Presurvey Measures

Current familiarity with the kitchen appliance prices

Average price and variation 103 37%

Average price but not variation 134 48%

Do not have a good sense 44 16%

Have purchased kitchen appliance in the past 256 91%

Own stand mixer 158 56%

How often use stand mixer (per month) 1.98

How often cook at home (per week) 4.41

How often bake at home (per week) 1.07

Interest in purchasing mixer in near future (five-point scale) 3.11

Knowledge about prices of KitchenAid mixer (seven-point scale) 4.12

B: Postsurvey Measures

Extent of agreement (five-point scale)

Purchase most home appliances online 2.74

Like to see the appliance in person before purchasing 4.06

Extensively search for the best price before purchasing 4.27

Purchase once I find the appliance with all the features I like 3.89

Stores are unable to match prices offered by websites 3.07

Can purchase home appliances at lower prices online 3.55

Great kitchen appliance is worth paying a lot of money for 3.48

Less willing to buy new appliances if they have a high price 3.76

Price of buying new appliances is important to me 4.20

Don’t mind spending a lot to buy a new appliance 2.69

Involvement in purchasing home appliances N Percentage

I decide 158 56%

My spouse/partner and I together decide 111 40%

My spouse/partner decides 7 2%

Someone else decides 5 2%

Familiarity with features of KitchenAid mixer (seven-point scale) 5.2

Knowledge about prices of KitchenAid mixer (seven-point scale) 5.87

Clear and realistic? N Percentage

Survey questions were clear 247 88%

Price thresholds were realistic 230 82%

Notes:* The table summarizes participants’ familiarity and purchase intentions for home appliances and KitchenAid mixer both before and after belief elicitation and search tasks.

In the survey, each participant provided five belief distributions (one before and one after each of the four search scenarios), each of which include four probabilities at different price thresholds. The Figure 4, Panel A, plots, by price threshold, the distribution of the average elicited probability across participants. Not only do we find significant variation in elicited probabilities across participants, but the S-shaped pattern in probabilities across quantiles also resembles the CDF of a normal distribution. We find a similar S-shaped pattern in almost 95% of all belief elicitations at the individual participant-search task level. Figure 4, Panel B, plots, by price threshold, the distribution of uncertainty (standard deviation) of elicited probabilities across participants. Values greater than zero indicate variation in responses within a participant to the same price threshold across belief elicitation tasks. We observe a sizable within-participant variation in responses across all price quantiles, which is crucial to studying whether participants update beliefs in response to search outcomes.

Figure 4.
Variation in participant’s responses to price thresholds, price changes and search decisions.

Figure 4, Panel C, shows the distribution of number of searches (choices) across participants. The last bar indicates the number of participants who chose to continue searching after the four search tasks. Overall, we find variation across participants in when they decide to stop searching. This, combined with the variation in price beliefs, allows for estimation of search costs. To understand whether participants respond to search outcomes, in Figure 4, Panel D, we show a scatter plot between the change in the searched price and the change in the elicited median price between consecutive search tasks. The positive relationship provides preliminary evidence that consumers update price beliefs in response to changes in observed prices. Such a pattern is evidence of participants learning about the price distribution, which is consistent with our focus on sequential search (McCall 1970; Rothschild 1974).

Belief estimation

We follow Manski (2004) to estimate the belief distribution using the elicited quantiles. For expositional convenience, we omit the participant-specific subscript $i$ but note that each participant saw different thresholds in line with their reported upper and lower bounds or median prices. For each price threshold (design point) $p_{d}^{}$ , let $f_{d}$ denote the probability that the price is below this threshold (i.e., $f_{d} = Pr (p \leq p_{d}^{})$ ). We thus have four points $f_{d}$ on the CDF describing the participant’s subjective beliefs about prices, and we use these to recover the distribution of price $F (p)$ using the approach outlined in Dominitz and Manski (1996).

The estimation routine we have outlined requires assuming a belief distribution. In this study, we limit our attention to unimodal distributions.¹⁵ We first estimate the belief distribution subject to the assumption that beliefs follow a skewed normal distribution. The skewed normal distribution has a density function given by $g (x) = (2 / ω) ϕ [(p - ζ) / ω] Φ \{α [(p - ζ) / ω]\}$ , where $ϕ (.)$ and $Φ (.)$ are the density and CDFs of a standard normal distribution, respectively; $ζ$ and $ω$ are the location and scale parameters, respectively; and $α$ measures the skewness of the distribution. The normal distribution is a special case of the skewed normal distribution with $α = 0$ . Across all participants and belief elicitations, over 65% of estimated distributions have a value of $α$ between −2 and 2, which points to almost symmetric belief distributions. Furthermore, none of the estimated $α$ s are significantly different from 0. This supports the assumption that beliefs follow a normal base distribution, which has also been assumed in previous literature (see, e.g., Dominitz and Manski 1996).¹⁶ Let $(μ, σ)$ denote the mean and standard deviation of the normally distributed belief distribution. The estimated parameters minimize the sum of the squared errors between the price thresholds and the predicted prices given the elicited quantiles:

$(\hat{μ}, \hat{σ}) = \underset{μ, σ}{argmin} \sum_{d} {[p_{d}^{} - Φ^{- 1} (f_{d}; μ, σ)]}^{2},$
7

where $\hat{μ}$ and $\hat{σ}$ are the estimated mean and standard deviation of the participant’s belief distribution.

Belief Heterogeneity and Updating

Figure 5, Panel A, plots the distribution (across participants) of prior expected prices (dotted density) and the true price distribution (dashed density). The prior distribution is elicited before providing participants with any price information about the KitchenAid mixer. Overall, there exists substantial heterogeneity in expected prices across participants, and they underestimate the true prices of the KitchenAid mixer in the absence of any price information. The solid density shows the distribution of estimated means of the posterior distribution obtained at the end of all four search scenarios. Compared with the prior distribution (dotted density), the posterior distribution is closer to the true distribution (dashed density) and also has a smaller standard deviation, which provides preliminary evidence that participants learn about the true price distribution. We also see evidence of learning in Figure 5, Panel B, which shows a box plot of the expected prices across participants by search task. Participants update their beliefs quickly in response to the outcome of the first search, and the distribution of beliefs across tasks is very close to the average market price of $325 (dashed line). In fact, the average expected price across participants after the first search is slightly higher than the average of the true price distribution.

Figure 5.
Participants’ belief distributions.

Figure 6, Panel A, shows the distribution of the uncertainty (standard deviation) in prior expected prices and the true standard deviation (dashed line). Again, we find that participants differ in their prior price uncertainty for the mixer with majority of the participants overestimating the degree of price dispersion. Nonetheless, for majority of the participants, the difference between the prior uncertainty and the standard deviation of the true price distribution is small relative to the average market price. Finally, Figure 6, Panel B, shows that participants’ uncertainty somewhat decreases as they observe more search outcomes (i.e., participants seem to update their uncertainty meaningfully in response to search outcomes).

Figure 6.
Participants’ belief distributions (standard deviation).

In summary, we find that (1) there exists substantial heterogeneity in beliefs across participants, (2) participants initially underestimate the market prices in the absence of any price information, and (3) participants learn quickly and their updated beliefs, even after the first search, are in line with the true price distribution. Next, we use the estimated beliefs and observed choices to compute the distribution of search costs. We note that even though participants underestimate market prices initially, the fact that they quickly learn about the prices justifies our investigation that compares search cost estimates under elicited beliefs with those under the rational expectations and Bayesian learning assumptions.

Search Cost Estimates

In the survey, the first search is free for the participants, so we do not include the first search (i.e., the prior belief before the first search does not influence our results) in our estimation of search costs. Furthermore, for participants who search once (or continue to search after four tasks), we can infer only a lower (upper) bound on search costs, respectively. We first compute bounds (in survey dollars) on search costs based on Equation 1 and, in Figure 7, plot the CDF corresponding to these bounds based on elicited beliefs, subject to either rational expectations or Bayesian updating with informative priors assumptions (Panel A).¹⁷

Figure 7.
CDF of bounds on search costs.

For the majority of the participants, the bounds on search cost estimated using elicited beliefs are lower than those estimated subject to assuming either rational expectations or Bayesian updating with informative priors. Furthermore, consistent with the simulation results, we find that the distribution of search costs when rational expectations are assumed exhibits first-order stochastic dominance over the distribution when assuming Bayesian updating with informative priors. These results are also reflected in Table 2, Panels A–C, in which we present the median and standard deviation of the estimated search costs (we report these due to the skewed nature of the distribution). To derive search costs, we take the average of the bounds shown in Figure 7, Panels A and B, but focus only on 144 participants who search more than once and fewer than five times, for whom we have both an upper and a lower bound on search cost.

Table 2.
Search Costs Under Sequential Search.

Median ($) SD ($)

A: Elicited Beliefs

Elicited belief distribution 7.47 (1.60) 28.96 (2.84)

Rational expectations 15.39 (.72) 11.80 (1.41)

Bayesian updating (informative prior) 11.02 (.57) 9.63 (1.24)

B: Bayesian Updating with Elicited Priors

Dirichlet process prior 6.82 (1.41) 23.97 (2.11)

Normal distribution 4.95 (1.11) 17.87 (2.11)

C: Importance of Prior Beliefs

Elicited means but homogeneous (true) uncertainty 6.11 (1.31) 23.24 (2.19)

Elicited uncertainty but homogeneous (true) means 11.31 (.84) 11.51 (1.09)

Notes:* Panel A reports the median and standard deviation of the distribution of search costs estimated under different assumptions about prior beliefs and belief updating. We calculate standard errors (reported in parentheses) through bootstrapping by drawing 1,000 different samples with replacement. Panel B reports the median and standard deviation of the distribution of search costs estimated subject to Bayesian updating assumptions and elicited prior beliefs but for different distributional assumptions. Panel C reports the median and standard deviation of the distribution of search costs estimated by constraining all participants to have either the same prior uncertainty or the same prior expected prices. We assume that participants have Dirichlet process priors and update beliefs consistent with Bayesian updating.

Drawing on the elicited prior and posterior beliefs, we estimate a median search cost of $7.47 and a standard deviation of $28.96 (top row of Table 2; solid line in Figure 8). The second row of Table 2 reports search cost estimates assuming rational expectations (i.e., participants know the true distribution of prices and do not learn). We estimate a median search cost of $15.39 with a standard deviation of $11.80 (dotted line in Figure 8). Such overestimation of search costs is consistent with the simulation results (Figure 2) because the rational expectations assumption ignores heterogeneity in prior beliefs.¹⁸ Instead of assuming no learning, if we enable participants to update beliefs in a Bayesian manner with informative priors, we estimate a lower (compared with rational expectations) median search cost of $11.02 and a standard deviation of $9.63 (third row of Table 2; dash-dotted line in Figure 8). Compared with the search cost estimates based on the elicited beliefs (a median of $7.47), these estimates differ because they are still restricted by the assumption about the updating mechanism, average prior beliefs, and the homogeneity in prior beliefs across participants. Given the similarity of these estimates to those based on a rational expectations assumption, the differences in search cost estimates from assuming Bayesian updating with informative priors appear to stem from the fact that imposing informative priors ignores heterogeneity in prior expected prices across participants. We test this theory further in the following section.

Figure 8.
Distribution of search costs.

Collectively, our results show that both assumptions—rational expectations and Bayesian updating with informative priors, both of which are routinely made in the literature—lead to overestimation of search costs. Although allowing for learning mitigates the magnitude of bias, the estimates based on the Bayesian updating with informative priors assumption are still significantly different from those based on elicited beliefs. These differences in estimated search costs have important implications for how consumers search and the retailer’s market share in response to small changes in search costs, as we show in Web Appendix F.

Compared with previous literature, in which high search costs can be attributed to whether individuals make decisions on the basis of marginal or total returns (Kogut 1990; Sonnemans 1998), observed partial or incomplete search history (Bronnenberg, Kim, and Mela 2016), assumed search method (Honka and Chintagunta 2017), heuristic rules (Camerer 1995; Zwick et al. 2003), or how paid search (Blake, Nosko, and Tadelis 2015) or recommendations (Dellaert and Häubl 2012) are accounted for, we show that ignoring heterogeneity in prior beliefs can result in overestimating search costs even in the absence of paid search or recommendations and when the complete search history and search method are known. Our results, thus, lend support to the more recent literature in marketing that estimates and allows for heterogeneity in prior beliefs (Ursu, Wang, and Chintagunta 2020; Zhang, Ursu, and Erdem 2020).

In the survey, participants incur $1 for every incremental search and receive $1 for every $50 reduction in the lowest searched price. Thus, assuming fungibility of money, the survey implicitly imposes a search cost corresponding to 50 survey dollars, which is substantially higher than the search cost estimates. The difference between estimated and imposed search cost cannot be attributed to participants misunderstanding the tasks, given their responses in Table 1, Panel B. We believe this difference can be attributed to the “house money effect” reported in Thaler and Johnson (1990), in which participants are more risk seeking in the presence of prior gain. The $3 we endowed the participants with at the beginning is essentially a prior gain (house money) and is treated differently than the amount earned from a lower searched price. While the impact of this prior gain on search cost estimate exists regardless of the prior belief specification, understanding its consequences is important, and we defer this to future research. The estimate of search cost might also reflect the cost of time and effort a participant incurs in completing the survey. While we acknowledge the difference in search cost estimates, it is important to note that this difference does not affect our ability to compare search cost estimates under different assumptions about prior beliefs and belief updating. We conduct robustness checks to our modeling assumptions in Web Appendix G and do not find any qualitative differences in search cost estimates.

Bayesian Updating and Prior Beliefs

The previous section’s analysis, while insightful, does not enable us to pin down the resulting bias from assuming Bayesian updating. Next, we use the data to explore the extent to which participants deviate from Bayesian updating and its implications for search costs. We also quantify the extent of bias in search cost from ignoring heterogeneity in prior expected prices and test whether the impact (or lack thereof) of ignoring heterogeneity in prior uncertainty conforms to the simulation results.

Implications of Assuming Bayesian Updating

Do participants engage in Bayesian updating?

We first use the elicited prior beliefs and updated (posterior) beliefs to examine whether participants update their beliefs consistent with Bayesian updating. We use the test proposed in Epstein (2006) and Epstein, Noor, and Sandroni (2010) to quantify departure from Bayesian updating. Specifically, for participant $i$ , given an expected price $μ_{i k}$ after $k$ searches (i.e., prior), we calculate the updated expected price, $B U (μ_{i k}; p_{i, k + 1})$ , after the $k + 1 th$ search assuming that the posterior beliefs follow a Dirichlet process (Equation 3) and that the participant engages in Bayesian updating. We then run the following regression:

$μ_{i, k + 1} = κ B U (μ_{i k}; p_{i, k + 1}) + (1 - κ) μ_{i k} + ∊_{i k},$
8

where $μ_{i, k + 1}$ is the elicited expected price after $k + 1$ searches (i.e., posterior) and $κ \geq 0$ . The case $κ = 1$ nests Bayesian updating. $κ < 1$ implies that participants underreact to new information, and $κ > 1$ implies participants overreact because they attach too little weight to the prior. $κ$ , thus, measures the extent to which participants engage in Bayesian updating. Pooling data across all participants and search tasks, we estimate $κ = .686$ , with a standard error of $.046$ . Thus, we find evidence that participants’ belief updating is inconsistent with Bayesian updating, and they underreact to searched prices.¹⁹

Bias from assuming Bayesian updating

The Monte Carlo simulations show that assuming Bayesian updating has a relatively small impact on search cost estimates. Results in the previous section show that assuming Bayesian updating with informative priors reduces the error in search cost estimates (relative to assuming rational expectations). Informative priors, however, imply homogeneity in prior beliefs. Next, we isolate the impact of Bayesian updating from assuming informative priors and explore how assuming Bayesian updating influences search cost estimates after one accounts for heterogeneity in prior beliefs. Specifically, we estimate search costs assuming that participants engage in Bayesian updating with prior beliefs (after the first free search) as elicited.

Assuming the Dirichlet process prior, we estimate a median search cost of $6.82 and a standard deviation of $23.97 (Table 2, Panel B; dashed line in Figure 8), which, while lower, is not significantly different from the search cost estimates based on elicited beliefs (Table 2, Panel A). This is also evident in Figure 7, Panel B, which plots the CDF of the lower and upper bounds on search costs based on elicited beliefs and subject to assuming Bayesian updating. Importantly, these estimates are much lower than those based on assuming either rational expectations or Bayesian updating with informative prior, which indicates the importance of accounting for heterogeneous prior beliefs. Thus, after one accounts for heterogeneous prior beliefs, assuming Bayesian updating does not significantly affect the distribution of search costs even when participants underreact to new information. These findings not only validate the use of the Bayesian updating assumption in the search literature but also alleviate the need to elicit updated beliefs after each search occasion.

The second row in Table 2, Panel B, reports estimates based on the assumption that participants engage in Bayesian updating but with normally distributed prior and posterior beliefs with known variance of the true distribution. We estimate a median search cost of $4.95, which is not significantly different from the median estimates based on either elicited beliefs (the first row) or elicited priors with Dirichlet process updating (the fourth row). The estimated standard deviation of $17.87, however, is lower than the estimates in the first and fourth rows. Importantly, the qualitative differences in search costs from accounting for heterogeneous priors versus not (assuming rational expectations and Bayesian updating with informative priors) are the same regardless of whether the updating process follows Dirichlet process priors or normally distributed priors. Understanding which of these processes better characterizes how consumers update beliefs is an interesting topic that we defer to future research.

Relative Importance of Prior Uncertainty

Given the importance of accounting for prior beliefs, and the observed heterogeneity in prior beliefs, we next quantify the implications of this heterogeneity on search cost estimates. We first explore the extent to which differences in prior beliefs across participants influence their search cost estimates. To do so, we regress the participant-specific search cost estimated with elicited prior beliefs and assuming Bayesian updating (first row of Table 2, Panel B) on the prior expected price and prior uncertainty. Assuming Bayesian updating standardizes the learning process across participants, thereby enabling us to focus on the marginal effect of changes in prior beliefs. We estimate that a 1% increase in prior expected price lowers search cost by 12%, whereas a 1% increase in prior uncertainty increases search cost by 3%. Thus, in line with the simulation results, at the individual participant level, changes in prior expected price affect search cost estimates more than changes in prior price uncertainty.

Transitioning to inference about search costs at the aggregate level, we estimate search costs subject to two different belief specifications. In the first specification, we allow participants to have prior expected prices as elicited but assume that the prior uncertainty coincides with the standard deviation of the true price distribution. In the second specification, we allow participants to have prior uncertainty as elicited, but the prior expected prices are assumed to coincide with the true average price. In either case, we assume that participants update beliefs consistent with Bayesian updating. In line with simulation results (Figure 2), ignoring participant-level heterogeneity in prior expected prices results in overestimating search costs, with a median search cost of $11.31 and standard deviation of $11.51 (Table 2, Panel C). The median search cost from ignoring heterogeneity in prior expected prices is over 50% higher than the true median search cost. By contrast, ignoring heterogeneity in prior uncertainty results in median search cost of $6.11 and a standard deviation of $23.24, which is not significantly different from that based on elicited prior beliefs. This result is driven by the fact that the difference between prior uncertainty and the true standard deviation is small for majority of the participants (Figure 6, Panel A).

The experimental data thus not only confirm that ignoring heterogeneity in prior expected prices can significantly bias the estimates of search cost but also show that a researcher may not need to elicit heterogeneity in prior price uncertainty if participants are somewhat knowledgeable about the price dispersion in the market. This is especially important because eliciting prior uncertainty may require eliciting price quantiles, which is burdensome for participants. By contrast, expected prices are more easily understood and simpler to elicit. Together, this analysis highlights the value of augmenting transactions data with elicited belief information for researchers and retailers. The key takeaway is that if the primary objective is to infer the distribution of search costs, researchers and retailers can choose to not account for uncertainty at the individual level. That is, they can simply elicit expected prices (or outcomes), following Erdem et al. (2005) and Delavande (2008), respectively. If, however, the objective is to target customers on the basis of individual-level estimates of search costs, then it is also important to account for individual-level heterogeneity in prior price uncertainty.

Belief Heterogeneity and Price Discrimination

Much of the emphasis in the price discrimination literature has been on utilizing differences in consumer’s WTP, patience when making intertemporal choices, bargaining power when negotiating, and so on to charge different prices to different consumers. In the context of search, researchers can quantify the benefit from price discrimination by relying on differences in search costs (in addition to differences in WTP) across consumers. These search costs, however, are typically estimated subject to the assumption that all participants have the same prior beliefs. We quantify the benefit to a retailer from price discrimination based on prior beliefs, independent of that based on WTP. Specifically, we study how a retailer can benefit from third-degree price discrimination if it can elicit prior beliefs from a small sample of consumers and correlate these beliefs with observable demographics.²⁰ Our primary objective in conducting the pricing analysis is thus to show that utilizing belief heterogeneity can lead to an economically meaningful increase in profits, as opposed to prescribing how to implement price discrimination for a specific product such as the KitchenAid mixer.

The analysis assumes that each participant has a WTP high enough that they will purchase the mixer from either the focal retailer or some other retailer. In assuming so, we abstract away from the impact of WTP on price discrimination. Because all participants always purchase the mixer, we estimate a lower bound on the increase in profits from price discrimination. If participants differ in their WTP, the retailer can additionally benefit from price discrimination based on WTP, where the incremental benefit may depend on the correlation between prior beliefs and WTP. We consider a setting where participants first search from the focal retailer and, based on the searched price, their updated beliefs (assuming Bayesian updating), and search costs, decide whether to conduct additional searches. A participant purchases from the focal retailer if they decide not to search any further. However, if the participant searches additional retailers, they purchase from the focal retailer only if it has the lowest price. To compute the focal retailer’s profit, we first solve for the optimal price of the focal retailer, taking into account participants’ prior beliefs. Next, we take the optimal price of the focal retailer and participants’ updated price beliefs as given and compute the optimal price set by a retailer who is searched second. We repeat this exercise for two more retailers in the sequence in which they are searched.

We first compute the focal retailer’s optimal price and implied profits assuming that consumers have informative priors and that they engage in Bayesian updating. This scenario ignores prior beliefs and provides a benchmark for comparison of profits drawing on other scenarios. Next, we segment participants into three clusters on the basis of their elicited prior beliefs and compute profits in line with these elicited beliefs and the assumption that participants engage in Bayesian updating.²¹ In doing so, we allow the retailer to charge different profit-maximizing prices to each segment. Because beliefs are typically not observed, consumers cannot be classified into different segments on the basis of prior beliefs. This scenario thus provides a theoretical upper bound on the profits the retailer can achieve if it had perfect information. Finally, we run a discriminant analysis to allocate consumers to different belief-based segments using observable demographics and attitudinal variables. We again compute the retailer’s optimal profit while allowing for segment-specific prices, where the segments are derived from the discriminant analysis. A retailer can elicit beliefs from a small subset of consumers and classify them into different segments on the basis of these elicited beliefs. The discriminant analysis, then, enables the retailer to use demographic variables, which are observed for all consumers, to allocate consumers to the belief-based segments. This scenario is readily implementable and presents a more realistic scenario for managerial implications.

Table 3 summarizes the means of the prior beliefs, and we use these to divide participants into three segments for the pricing analysis. Overall, we find significant differences across segments in their prior beliefs pointing to possible gains from price discrimination. Figure 9 shows the optimal profits under each of the three pricing scenarios. As with the base case of Bayesian updating with informative priors, segmenting participants into three groups based on prior beliefs and optimally pricing to each segment differently results in a 6.4% increase in profits. However, optimally pricing to segments on the basis of discriminant analysis increases the retailer’s profit by 4.0%. Both differences are statistically significant at the 95% confidence level. Thus, the retailer is able to extract approximately 63% of the ideal profit (based on prior beliefs) by allocating participants to different segments on the basis of the correlation between belief-based segments and observable demographics. This not only highlights the importance of recognizing heterogeneity in prior beliefs but also shows how a retailer can benefit from third-degree price discrimination where the consumer segments correlate with prior beliefs, independent of the distribution of WTP.

Figure 9.
Optimal profits.

Table 3.
Segments Based on Prior Beliefs.

Aggregate Segment 1 Segment 2 Segment 3

N 144 26 72 46

Prior mean 334.06 (5.35) 239.50 (7.32) 325.86 (2.77) 400.35 (6.09)

Prior SD 42.05 (1.96) 53.24 (6.09) 37.23 (2.21) 43.26 (3.47)

Segment-specific average is significantly different from the overall average at the 95% confidence level.

Notes: The table shows the averages (SEs in parentheses) of the elicited prior mean and standard deviation used for dividing participants into three segments for the second stage of the pricing analysis.

Discussion and Conclusion

The extant literature in marketing and economics estimates search costs and provides pricing and market structure implications subject to assumptions about prior beliefs and the belief updating process. Researchers typically assume either that participants have rational expectations or that they update beliefs consistent with Bayesian updating but have informative priors. While we understand the need to make these assumptions, ex ante, it is not clear whether consumers know the true price distribution for less frequently purchased big-ticket items, especially given the frequent changes in market structure in different industries. In this study we elicit price beliefs and investigate how the knowledge of price beliefs informs consumer search costs, explore the implications of assuming Bayesian updating, and question the relative importance of accounting for prior expected prices versus uncertainty in prices. We correlate the segment classification based on elicited prior beliefs with observable demographics and show how a retailer can exploit belief heterogeneity to engage in third-degree price discrimination. We conduct our investigation in the context of a KitchenAid mixer, which is typically sold for around $300–$400 and is not purchased frequently by consumers. This provides an ideal setting to study the role of price beliefs and the implications of the assumptions routinely made in the search literature.

We find substantial heterogeneity in prior price beliefs and show that participants update beliefs in response to search outcomes. Participants, however, deviate from Bayesian updating in that they underreact to new information. Consistent with the Monte Carlo simulations, we find that assuming either rational expectations or Bayesian updating with informative priors significantly overestimates the distribution of search costs for majority of participants. Importantly, assuming Bayesian updating does not bias the distribution of search costs after accounting for the elicited prior beliefs, thereby validating this assumption, which is embedded in majority of the learning models. We also find that, in general, eliciting prior expected prices is more important than accounting for the prior uncertainty in prices, and that ignoring heterogeneity in prior uncertainty does not bias the estimated distribution of search costs. Finally, we use the estimates to study how a retailer can benefit from third-degree price discrimination by correlating beliefs-based segments and observable demographics and attitudes. To the best of our knowledge, this is the first article that quantifies the benefit from price discrimination based on heterogeneity in elicited prior beliefs.

These results have important implications for both academics and managers. From an academic perspective, the findings further our understanding of the role of price beliefs in informing search costs. While the direction of change in search costs with changes in prior mean, uncertainty, and the extent of belief updating can be derived, we contribute to the literature by quantifying the magnitude of this change. Our results on the implications of assuming Bayesian updating and the relative importance of eliciting prior expected prices versus uncertainty in prices are novel and relevant to academics and managers. Furthermore, this article provides intuition regarding how prior beliefs and belief updating influence search cost estimates and suggests which assumptions could potentially be relaxed or augmented with additional data. In doing so, we also propose heterogeneity in prior beliefs as an alternative explanation for why a researcher may overestimate search costs. We note that our findings are specific to the context we study; generalizing these to other product categories would be valuable, but we defer this to future research. From a managerial perspective, the pricing analysis highlights the importance of allowing for heterogeneous prior beliefs and illustrates how a retailer can benefit by correlating beliefs with observable demographics to take advantage of third-degree price discrimination.

Although we believe that our article makes an important contribution to the search literature as well as to the literature on belief elicitation and updating, we acknowledge several limitations of our study. First, under search with learning, a consumer may decide to stop searching and not purchase the product if the updated price beliefs make purchase unattractive. In our study, we do not account for the option of stopping search and not purchasing. Doing so would require us to make some parametric assumptions about the utility function to separate product and price preference from search costs. By contrast, our study design allows us to estimate the bounds on search costs in absence of any parametric assumptions. Second, as mentioned previously, we consider search over prices of a homogeneous good. Extending the analysis to heterogeneous goods is of value but would require more parametric assumptions, so we leave this to future research. Third, our survey design and the finding that consumers update beliefs are consistent with sequential search. Future research on how consumers search and whether they engage in a combination of simultaneous and sequential search (Harrison and Morgan 1990; Morgan and Manning 1985) would be valuable. We believe this is an interesting area of study and defer it to future research. Fourth, our estimates of search cost are substantially lower than the implied search cost based on the endowed money, pointing to the “house money effect.” Understanding how participants account for the initially endowed money differently from that implied by the trade-offs in the experiment is an interesting topic, which we defer to future research. Fifth, our pricing analysis focuses only on the role of belief heterogeneity and does not compare the benefit from price discrimination based on beliefs with that based on WTP. Understanding the relative importance of price discrimination that is based on belief heterogeneity can be of value to researchers and managers. Finally, in our analysis we assume that the base distribution of participants’ beliefs can be represented by a normal distribution. While this assumption is supported by the data, and we perform a robustness check on this assumption, it is possible to extend the analysis to allow for more flexible (nonparametric) price belief distributions. Accounting for such nonparametric price distributions may result in nonmyopic search, which we believe is an interesting topic for future research.

Supplemental Material

Supplemental Material, sj-pdf-1-mrj-10.1177_0022243720982979 - The Importance of Price Beliefs in Consumer Search

Supplemental Material, sj-pdf-1-mrj-10.1177_0022243720982979 for The Importance of Price Beliefs in Consumer Search by Pranav Jindal and Anocha Aribarg in Journal of Marketing Research

	N	Percentage
A: Presurvey Measures
Current familiarity with the kitchen appliance prices
Average price and variation	103	37%
Average price but not variation	134	48%
Do not have a good sense	44	16%
Have purchased kitchen appliance in the past	256	91%
Own stand mixer	158	56%
How often use stand mixer (per month)	1.98
How often cook at home (per week)	4.41
How often bake at home (per week)	1.07
Interest in purchasing mixer in near future (five-point scale)	3.11
Knowledge about prices of KitchenAid mixer (seven-point scale)	4.12
B: Postsurvey Measures
Extent of agreement (five-point scale)
Purchase most home appliances online	2.74
Like to see the appliance in person before purchasing	4.06
Extensively search for the best price before purchasing	4.27
Purchase once I find the appliance with all the features I like	3.89
Stores are unable to match prices offered by websites	3.07
Can purchase home appliances at lower prices online	3.55
Great kitchen appliance is worth paying a lot of money for	3.48
Less willing to buy new appliances if they have a high price	3.76
Price of buying new appliances is important to me	4.20
Don’t mind spending a lot to buy a new appliance	2.69
Involvement in purchasing home appliances	N	Percentage
I decide	158	56%
My spouse/partner and I together decide	111	40%
My spouse/partner decides	7	2%
Someone else decides	5	2%
Familiarity with features of KitchenAid mixer (seven-point scale)	5.2
Knowledge about prices of KitchenAid mixer (seven-point scale)	5.87
Clear and realistic?	N	Percentage
Survey questions were clear	247	88%
Price thresholds were realistic	230	82%

	Median ($)	SD ($)
A: Elicited Beliefs
Elicited belief distribution	7.47 (1.60)	28.96 (2.84)
Rational expectations	15.39 (.72)	11.80 (1.41)
Bayesian updating (informative prior)	11.02 (.57)	9.63 (1.24)
B: Bayesian Updating with Elicited Priors
Dirichlet process prior	6.82 (1.41)	23.97 (2.11)
Normal distribution	4.95 (1.11)	17.87 (2.11)
C: Importance of Prior Beliefs
Elicited means but homogeneous (true) uncertainty	6.11 (1.31)	23.24 (2.19)
Elicited uncertainty but homogeneous (true) means	11.31 (.84)	11.51 (1.09)

	Aggregate	Segment 1	Segment 2	Segment 3
Prior mean	334.06 (5.35)	239.50** (7.32)	325.86** (2.77)	400.35** (6.09)
Prior SD	42.05 (1.96)	53.24** (6.09)	37.23** (2.21)	43.26 (3.47)

Footnotes

Appendix: Simulation Details

Acknowledgments

The authors are grateful to Elisabeth Honka, Nitin Mehta, Raluca Ursu, Matthijs Wildenbeest, Rajdeep Grewal, Sriraman Venkataraman, Carl Mela, Camelia Kuhnen, Gonca Soysal, and Stephan Seiler for helpful comments and suggestions. They also benefited from the comments of seminar participants at the 2018 Marketing Science Conference at Temple University, 2018 Marketing Dynamics Conference at Southern Methodist University, UNC-Duke brown bag, 2019 UT Dallas Bass FORMS Conference, 2019 Yale Customer Insights Conference, 2019 Choice Symposium, 2019 Search and Switching Workshop at University of California at Los Angeles, Frankfurt School Marketing Research Camp, University of Houston, and University of Michigan Hosmer-Hall lunch talk. The authors thank Lillian Chen for providing excellent research assistance. All errors and omissions are the responsibility of the authors.

Associate Editor

Raghuram Iyengar

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The first author acknowledges the support of The M. W. “Dyke” Peebles, Jr. Faculty Development Fund.

ORCID iD

Pranav Jindal

Online supplement:

Notes

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