Price Beliefs and Experience:
Do Consumers’ Beliefs Converge to Empirical
Distributions with Repeated Purchases?
Preliminary and Incomplete
Brett Matsumoto† and Forrest Spence‡
April 25, 2013
Abstract
We use data on subjective beliefs about price distributions to investigate two questions of interest. First, when consumers face price uncertainty, to what extent do their
beliefs about the distribution of prices reflect the actual empirical distribution? Second, if consumers have biased or highly variable beliefs, do these beliefs converge to
the empirical distribution as consumers repeatedly participate within a market (i.e. are
consumers learning about this distribution through experience)? Preliminary results
suggest that price beliefs are biased upwards for inexperienced consumers, but that
these beliefs converge to the empirical distribution as consumers acquire experience.
1
Introduction
For a wide range of goods and services, prices can vary considerably both across and within
retailers (e.g., online marketplaces). In these environments, consumers make decisions to
price a particular good or service based on beliefs about the distribution of prices they face.
†
‡
UNC-Chapel Hill
UNC-Chapel Hill
1
This paper uses data on subjective beliefs to investigate the extent to which a consumer’s
belief about a price distribution reflects the observed empirical price distribution and whether
this belief converges to the empirical distribution as a consumer repeatedly participates
within a market.
There is a growing literature focusing on the development and estimation of structural
models of consumer search (De los Santos et al., 2012b; Hortaçsu and Syverson, 2004). These
models have been used to explain observed price dispersion for homogeneous goods (Hong and
Shum, 2006) and to recover unbiased demand estimates in markets where price uncertainty is
important (Koulayev, 2012; Moraga-González et al., 2009). A critical assumption used in the
previously cited studies is that consumers have Rational Expectations (i.e. consumers face
price uncertainty, but know the empirical distribution of prices). However, if consumers have
biased beliefs about the empirical distribution of prices, this will lead to biased search costs.
In particular, if consumers’ beliefs about prices are biased upward, but are assumed to have
rational expectations, search cost estimates will also be biased upward (low levels of search
can be explained by either high search costs or low expected benefits from search). The
magnitude of estimated search costs in recent empirical studies suggests that it is important
to test the appropriateness of this assumption.
Some recent research has focused on understanding how consumers learn about empirical
price distributions within purchasing decisions (De los Santos et al., 2012a; Koulayev, 2009,
2013)1 . In these models, consumers acquire information about the distribution of prices
as they repeatedly search, prior to making a final purchasing decision. We instead focus
on learning across purchasing decisions; in particular we are examining the hypothesis that
more experienced consumers have acquired information about the empirical price distribution
through repeated participation.
This hypothesis is supported by work done in the labor literature using subjective beliefs about future earnings (Arcidiacono et al., 2012; Stinebrickner and Stinebrickner, 2011;
Wiswall and Zafar, 2012). These studies are relevant to our paper for two reasons. First,
Wiswall and Zafar (2012) show that college students’ beliefs about future earnings are more
consistent with actual earnings distributions as they proceed through school. Second, these
studies show that using students’ subjective beliefs can lead to significantly different estimates than under the assumption of Rational Expectations.
1
Need to check the assumptions these papers make on consumers’ priors to see if they can say if priors
are biased.
2
We approach this line of inquiry by collecting data on subjective beliefs about the distribution of prices for textbooks from online retailers. We provide seven hundred undergraduate
students with multiple textbook purchasing scenarios and elicit their beliefs about prices.
This market provides an ideal environment to examine consumers’ beliefs about price distributions for a few notable reasons. The first reason is that there is significant variation
in price across online retailers for what are essentially homogeneous goods; this generates a
large amount of price uncertainty for consumers which leads to the benefit of search. The
second reason is that consumers in this market face discrete, structured intervals in which
they face a round of textbook purchasing decisions. This structure allows us to observe
an informative measure of a consumers’ experience in the market (i.e. semesters enrolled).
Finally, anecdotal evidence suggests that search behavior within this market is significantly
different for consumers across experience levels.
The following section provides theoretical motivation for this project and expands on our
goals. Section 3 describes the data and Section 4 presents preliminary results.
2
Theoretical Motivation
We use the following simple model of consumer search to motivate the empirical section
of this paper. Individuals can purchase a given product from two sources. The price of the
product at each source is unknown by the individual. One of the sources has a higher search
cost, which is defined as the cost associated with determining the price of the product at that
location. Assume for simplicity that the search cost is zero for the low search cost source.
The prices of the product at the different locations in a given period are draws from some
distribution with a cumulative density function given by F (p). The individual knows the
price from one of the locations, and he can either buy the product from that location or pay
some cost, c, to search and discover the price at the other location. If the individual decides
to search, then he can purchase the product from either location.
Let u(p) denote the utility an individual receives from the product. Utility is assumed to
only be a function of the price paid, with u0 (p) < 0. The decision rule for the static search
problem is given by Equation 1. An individual chooses to search if,
Z
pLC
u(p) − u(pLC ) dF (p) > c
0
3
(1)
where pLC denotes the price from the low cost source which is known at the time of the
search decision. The LHS of Equation (1) is the expected benefit of search and the RHS is
the cost of search.
The structural consumer search literature attempts to recover the parameters that characterize the distribution of individuals’ search costs. Estimating the structural search model,
however, requires assumptions regarding the individual’s beliefs as well as a parametric assumption on F (p)2 . A common assumption regarding the individual’s beliefs is to assume
Rational Expectations. In other words, the individual is assumed to know the parameters of
the distribution of p. The distribution of p also needs to be specified by the econometrician
(i.e., Normal, Log-Normal, Gamma, Weibull, etc.). The estimated search cost parameters
are highly sensitive to these assumptions. The unique features of our data allow us to empirically test both the assumption of Rational Expectations and the parametric form of the
distribution of beliefs.
An alternative to Rational Expectations is to allow individuals to have potentially biased
beliefs. When individuals search and observe a price draw, they can use this information to
update their beliefs according to a learning process (e.g., Bayesian). Let γ denote the true
parameters of the price distribution. Individuals with no experience in the market believe
that the price distribution parameters are γ̃. An implication of the learning process is that
γ̃ → γ as the number of signals (or observed price draws resulting from the search decisions)
goes to infinity. In the empirical section we will test the following hypotheses:
1. Do individuals have unbiased beliefs? I.e., does E[p|γ] = E[p|γ̃]?
2. Are individuals learning about the distribution of prices? I.e., is |γ̃ − γ| decreasing in
experience3 ?
Finally, we will test the distributional assumption for the price distribution. The questionnaire is designed to elicit information about the shape of individual’s beliefs about the
price distribution.
2
Alternatively, some studies have non-parametrically identified a distribution of search costs (Hortaçsu
and Syverson, 2004; Hong and Shum, 2006)
3
Note that unbiased beliefs does not imply no learning. Also learning does not imply biased beliefs.
4
3
Data
Data on beliefs about price distributions were collected through online questionnaires sent
to students at the University of North Carolina. The online questionnaires elicit information
on respondents’ previous textbook purchasing behavior, their perceived costs from searching
in the online market, and then presented respondents with hypothetical textbook purchasing
scenarios. The data used in this paper is primarily drawn from the responses to these
textbook purchasing scenarios. These data are supplemented with data scraped from online
retailers for a large number of textbooks.
Before providing a summary of both datasets, we will provide more detailed information
about the textbook purchasing scenarios.
3.1
Textbook Purchasing Scenarios
Each questionnaire respondent was provided with three hypothetical textbook purchasing
scenarios. For each respondent, the three scenarios were randomly assigned from twelve
potential scenarios. The following figure is a screenshot from the information provided in
one particular scenario.
For each scenario, respondents were randomly assigned to a full information case (title,
author, publisher, picture, etc.) or a limited information case. As opposed to the full
information case, as seen in the figure, the limited information case only provided information
on the title, author, and course.
After being presented with information about the scenario, respondents were then provided with the (actual) price of a new copy of the textbook from the campus bookstore and
asked to give their expectations about the lowest price they would find for this textbook if
they searched one online retailer.
5
6
In order to gauge the variance in individuals’ beliefs, we then asked respondents for the
probability that the price realized after search would be less than X% or greater than Y%
of their reported expected price. For example, in the figure above, the new price of the
textbook at the campus bookstore for the Fall 2012 semester was $87.00. If the respondent
reported that her expectation of the lowest price from one online retailer was $50.00, then
the following questions would ask her the probability that price would be less than $45.00
and greater than $55.00. In practice, X was randomly drawn from {.85, .90, .95} and Y was
randomly drawn from {1.05, 1.10, 1.15}. This set of questions was then repeated for used
and rental alternatives.
In order to help clarify the questions within the textbook purchasing scenarios, respondents were first walked through an example where they were asked for the lowest price they
might find for a pair of jeans if they searched on retailer at the mall. This example contained
information about probabilities (e.g., that their response should be between 0 and 100) and
clarification about the nature of price uncertainty (i.e. that although their best guess might
be $20, there is some chance that the price is actually lower or greater than $20).
3.2
Online Questionnaire Data
For the Fall 2012 semester, 820 respondents completed the demographic and search cost
questions4 . The sample used in analysis is composed of 739 respondents. 52 respondents
were dropped because they had been enrolled in college greater than 10 semesters and an
additional 29 respondents were dropped for reporting nonsensical answers (e.g. reporting an
expected price of $100,000)5 . Data from an additional 702 respondents have been collected
for the Spring 2013 semester; however, these data have not been cleaned and will be omitted
from the analysis in the current version of the paper.
Table 1 displays the number of semesters enrolled for the questionnaire respondents.
The number of respondents enrolled in an odd number of semesters greatly outweighs the
number of respondents enrolled in an even number of semesters due to the nature of college
enrollment, which typically takes place in the fall.
4
979 respondents began the questionnaire, 759 completed at least one scenario, 741 completed at least
two scenarios, and 734 respondents completed the entire questionnaire.
5
In practice, this was done by removing respondents who reported expectations less than 10% or greater
than 150% of the bookstore price.
7
Table 1 - Semesters Enrolled
Semesters
Observations
1
81
2
4
3
155
4
46
5
131
6
48
7
120
8
75
9
50
10
29
Total
739
Table 2 reports respondents’ previous textbook purchasing behavior. An overwhelming
majority of respondents have purchased textbooks both at the campus bookstore and from
an online retailer. There is significant variation in how many textbooks respondents have
purchased online; 37% of our sample has purchased five textbooks or less from online retailers.
Only a quarter of respondents have rented a textbook from an online retailer, while more
than half have used a shopbot when pricing textbooks (a website that reports the lowest
price from multiple online retailers).
Table 2 - Previous Purchasing Behavior
%
Ever Purchased at Campus Bookstore
95.0
Ever Purchased Online
87.7
Purchased 1 - 5 Online
23.7
Purchased 6 - 10 Online
23.7
Purchased 11 or More Online
40.3
Ever Rented Online
25.6
Ever Used a Shopbot
54.5
8
3.3
Online Retailer Data
In order to construct an empirical distribution of prices for textbooks from online retailers,
we used a script in Perl to scrape .html files from online retailers. These data are collected
daily and span two years, twelve online retailers, and approximately 3,500 textbooks. Using
these .html files, we used a separate script in Perl to parse the lowest prices available from
the online retailers on each day. Data from November 20, 2012 to December 19, 2012 are
used in this version of the paper6 . Summary statistics on these prices will be presented
in Section 4.1, which provides reduced form evidence of differences across consumers and
convergence of beliefs.
4
Results
This section presents preliminary reduced form results using the data on reported expec-
tations. In future versions of the paper, this section will utilize data collected on the shape
of individuals’ beliefs in order to test hypotheses about parametric assumptions of beliefs.
4.1
Reduced Form Results
We first provide reduced form evidence that individuals’ beliefs do in fact differ across
experience levels and that these beliefs appear to be converging to the empirical distribution
of prices as individuals repeatedly participate within a market. First we present descriptive
statistics of individuals’ price expectations. We then proceed to compare these expectations
to the means of empirical price distributions. Finally, we consider the variance of individuals’
beliefs: presenting evidence that the dispersion of individuals’ beliefs reduces to the actual
empirical dispersion of prices as they acquire experience.
The first row of Table 3 reports the mean expectations across all individuals within our
sample normalized with respect to the bookstore price. Explicitly, on average, individuals
expect that the lowest price of a new textbook from an online retailer is 79.1% of the new
price from the bookstore. Recall that individuals are given the bookstore price when forming
6
The online questionnaire was initially distributed on November 30, 2012. In future versions of the
paper we will expand and contract the span of the data used to construct empirical price distributions as a
robustness check. The online retailers sampled in this version of the paper are Abebooks.com, Amazon.com,
Barnesandnoble.com, Chegg.com, Half.com, Textbooksrus.com.
9
their expectations. Individuals expect used prices to be 67.7% of the used price from the
bookstore. On average, individuals expect the prices available from online retailers to be
lower than bookstore prices and expect the relative savings to be greater for used options.
The second and third rows of Table 3 report normalized expectations broken down by
first year and upper level students7 . On average, upper level students expect the price of
the online alternatives to be lower than first year students. The difference is particularly
significant for used prices, with upper level students expecting the price to be roughly 7%
lower than first year students. Results from a t-test comparing the sample means are reported
in the fourth row of Table 3. The difference between groups is significant for both new and
used books.
Table 3 - Ratio of Expectations to Bookstore Prices
All Students
New
N
Used
N
0.791
2029
0.677
2021
(.164)
First Year Students
0.815
(.188)
226
(.164)
Upper Level Students
T-Test Results for
Upper Level - First Year
0.788
0.736
224
(.200)
1803
0.669
(.163)
(.187)
t = -2.3335
t = -4.7136
p = 0.020
p = 0.000
1797
The difference in expectations across types of students provides evidence that experience
plays a role in consumers’ price expectations. In Table 4 we report the same results categorized by how many online textbook purchases an individual has made in the past. The
pattern in Table 4 is striking. As individuals repeatedly participate in the online market,
their price expectations consistently fall for both new and used options.
7
First year students are defined as individuals in their first two semesters of college. Upper level students
are individuals with more than two semesters of college. The number of observations varies across New and
Used categories due to survey attrition.
10
Table 4 - Ratio of Expectations to Bookstore Prices by Previous Purchases
Number of
New
N
Used
N
0.834
256
0.734
256
Previous Purchases
None
(.164)
1-5
0.819
(.204)
479
(.155)
6 - 10
0.778
0.768
(.159)
478
(.191)
480
(.171)
More than 10
0.713
0.663
477
(.172)
814
0.645
810
(.184)
Due to the nature of the data collection, we want to control for differences in the textbook
purchasing scenarios that individuals are given. Table 5 reports results from a regression
of normalized price expectations on various measures of experience, scenario dummies, and
indicators for whether the individual has previously taken the course / been assigned the
textbook in the scenario.
The regression estimates are consistent with the mean comparisons above. First year
students and individuals who have never previously purchased a textbook have higher price
expectations than their counterparts. Column (3) provides evidence that price expectations
evolve rather slowly, as individuals with more experience consistently have lower expectations. The coefficients on indicators for whether the respondent had previously taken the
course or been assigned the textbook are consistently negative, but generally not statistically significant. Indicators for each scenario are not reported, but are generally significant
and of magnitudes between -.01 and .10, relative to the first potential textbook purchasing
scenario. These indicators capture differences in expectations due to variation in textbook
characteristics across scenarios8 .
8
Future versions of this paper will include textbook characteristics in these regressions (e.g., years since
last revision).
11
Table 5 - Regression Analysis
(1)
First Year
(2)
New
Used
0.0313***
0.068***
(0.011)
(0.014)
No Prev. Purchases
(3)
New
Used
0.049***
0.068***
(0.011)
(0.013)
1 - 5 Purchases
6 - 10 Purchases
11+ Purchases
Prev. Taken Course
Prev. Assigned Book
New
Used
-0.014
-0.022
(0.012)
(0.015)
-0.058***
-0.075***
(0.013)
(0.015)
-0.065***
-0.091***
(0.011)
(0.014)
-0.021
-0.025
-0.022
-0.029*
-0.022
-0.028
(0.016)
(0.018)
(0.016)
(0.018)
(0.016)
(0.018)
-0.023
-0.021
-0.021
-0.016
-0.021
-0.016
(0.022)
(0.024)
(0.022)
(0.023)
(0.022)
(0.024)
Although we have shown that individuals with more experience have systematically different beliefs, we have not established which group has biased beliefs relative to the actual
mean price. We now present evidence that individuals’ price expectations are converging to
the means of empirical price distributions.
The following table provides the ratio of prices of textbooks from Amazon.com relative
to the price from the campus bookstore. The first two rows report the prices of new and
used options for the full sample of books for which we have data for. On average, new
and used prices on Amazon.com are roughly 85% of their respective bookstore prices. The
median is slightly lower than this, as the distribution has a very large right tail. Note that on
certain days, the lowest price on Amazon.com for a used copy can be 33 times as expensive
as the bookstore price. These outliers are for uncommon books and are driven by pricing
algorithms used by large book retailers.
The third and fourth rows of Table 6 provides summary statistics for the prices of only
the textbooks which are priced greater than $40 for a new copy from the campus bookstore.
12
Relative to the full sample, the potential savings from shopping online become greater and
also less dispersed. This trend continues for textbooks priced greater than $100. On average, a textbook costing more than $100 from the campus bookstore is 33% cheaper from
Amazon.com for both new and used copies. Not surprisingly, the price of used copies are
more variable than new copies.
The final two rows provide summary statistics for the textbooks used in the hypothetical
textbook purchasing scenarios9 . On average, these prices are slightly lower than the sample
of all textbooks costing $100 or more, but the difference is not significant.
Table 6 - Amazon Prices
All Books
Price > $40
Price > $100
Scenario Books
Mean
S.D.
Min
Median
Max
N
New
0.85
(0.85)
0.08
0.77
24.79
2496
Used
0.84
(0.86)
0.04
0.77
33.09
2496
New
0.75
(0.42)
0.08
0.73
7.50
1110
Used
0.76
(0.50)
0.04
0.73
7.28
1110
New
0.67
(0.18)
0.12
0.67
1.37
481
Used
0.67
(0.29)
0.04
0.67
1.64
481
New
0.66
(0.14)
0.47
0.64
0.86
11
Used
0.63
(0.22)
0.24
0.65
0.95
11
Future versions of this paper will pool data across numerous retailers, but as 75% of respondents reported Amazon.com as the first website they would visit to search for a textbook,
the prices from Amazon.com provide a good approximation to the empirical distribution of
prices that consumers face if they only search one online retailer.
The following table compares individuals’ reported expectations with the actual mean
prices from Amazon.com of the textbooks used in the hypothetical textbook purchasing
scenarios. For both first year and upper level students, the difference between their reported
expectations and the actual means of the empirical distributions is significant. This trend
holds for new and used books.
9
Note that the total number of textbooks in the purchasing scenarios is actually 12. However, online
retailer data for one textbook is missing.
13
Table 7 - Mean Comparisons: Expectations to Actual Prices (Scenario Textbooks)
New
First Year Students
Reported
Actual
T-Test
Reported
Actual
T-Test
0.815
0.655
t = 14.6
0.736
0.626
t = 8.2
p = 0.000
(.200)
t = 34.5
0.669
p = 0.000
(.187)
(.164)
Upper Level Students
Used
0.788
0.655
(.163)
p = 0.000
0.626
t = 9.1
p = 0.000
Because respondents are forming expectations based on previous purchases for a wide
range of textbooks, we next compare their reported expectations with the mean prices of
a wider range of comparably priced textbooks. In this case, all differences are significant
except for upper level students’ expectations of used prices. We take this as evidence that
consumers expectations converging to the empirical mean. It is not surprising that this trend
is more noticeable for used books and not new books, as most purchases of textbooks from
online retailers are for used books rather than new books.
Table 8 - Mean Comparisons: Expectations to Actual Prices (> $100)
New
First Year Students
Reported
Actual
T-Test
Reported
Actual
T-Test
0.815
0.668
t = 13.4
0.736
0.675
t = 4.5
p = 0.000
(.200)
t = 31.1
0.669
p = 0.000
(.187)
(.164)
Upper Level Students
Used
0.788
(.163)
0.668
p = 0.000
0.675
t = -1.3
p = 0.206
Finally, we compare the variance in beliefs between first year and upper level students.
Recall from Table 3 that the standard deviation of expectations for new textbooks is .164 for
first year students and .163 for upper level students. A standard F-test comparing variances
results in a test statistic of .9923 and a p-value of 0.459 under the null of no difference and
the alternative that the variance of the upper level students is less than the variance of first
year students. A similar test comparing the variance in expectations for used books results
in a test statistic of .8629 and a p-value of 0.064. We take this as evidence that consumers’
beliefs are becoming more precise as they repeatedly participate within the market. That
14
being said, a standard F-test is sensitive to the assumption of normality of the population
distributions. When we use tests that relax this assumption, such as Levene’s test or the
Brown-Forsythe test, we are unable to reject the null of equality of variances for expectations
for new and used textbook prices.
15
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