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\title{%
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Search, Price Dispersion, and Local Competition: Estimating Heterogeneous
Search Cost in the Retail Gasoline Markets\thanks{We have benefited from
discussions with Maqbool Dada, Babur De Los Santos, Emin M. Dinlersoz,
Elisabeth Honka, Han Hong, Ali Horta\c{c}su, Sergei Koulayev, Jos\'{e} Luis
Moraga-Gonz\'{a}lez, Aviv Nevo, Zsolt S\'{a}ndor, Stephan Seiler, Chad
Syverson, Matthijs Wildenbeest, Xiaoyong Zheng, and seminar participants at
North Carolina State University, Kyoto University, Johns Hopkins University,
University of Tokyo, Center for Economic Studies at US Census Bureau, 2013
IIOC, and the 5th workshop on consumer search and switching costs. We thank
Reid Baughman, Autumn Chen, Yajing Jiang, Sam Larson, Jianhui Li, and Karry Lu
for research assistance. The retail gasoline price data for this project were
generously provided by Mariano Tappata.}%
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\author{Mitsukuni Nishida\thanks{The Johns Hopkins Carey Business School, 100
International Drive Baltimore, MD 21202. Email: nishida@jhu.edu.}
\and Marc Remer\thanks{Economic Analysis Group, U.S. Department of Justice. Email:
marc.remer@usdoj.gov. The views expressed in this article are entirely those
of the author and are not purported to reflect those of the U.S. Department of
Justice. }}
\date{May 24, 2014}
\maketitle

\begin{abstract}%
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Information frictions play a key role in a wide array of economic environments
and are frequently incorporated into formal models as search costs. Yet, as
search costs are typically unobserved, little empirical work investigates the
determinants of the distribution of consumer search costs and the implications
for policy. This paper explores the sources of heterogeneity in consumer
search costs and how this heterogeneity and market structure shape firms'
equilibrium pricing and consumers' search behavior in retail gasoline markets.
We estimate the distribution of consumer search costs using price data for a
large number of geographically isolated markets across the United States. The
results demonstrate that the distribution of consumer search costs varies
significantly across geographic markets and that market and population
characteristics, such as household income, explain some of the variation.
Policy counterfactuals suggest that the shape of the consumer search cost
distribution has important implications for both government policy and firms'
strategic pricing behavior. The experiments reveal that (1) the search cost
distribution needs to be sufficiently heterogeneous to generate equilibrium
price dispersion, and (2) the market-level expected price paid decreases in
the number of firms, but consumers with high search costs may be worse off
from an increased number of firms.

\noindent\textbf{JEL Classification Numbers}: D4, D8, L8

\noindent\textbf{Keywords}: Search Costs, Price Dispersion, Retail Gasoline,
Market Structure.





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\end{abstract}

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\section{Introduction}

Information frictions play a key role in explaining many aspects of economic
activity. For instance, a robust body of economic research has identified and
explained the existence of price dispersion in both homogeneous and
differentiated product markets as a consequence of consumer search costs.
Since Stigler's (1961) seminal article, a number of influential theoretical
papers, such as Varian (1980), Burdett and Judd (1983), and Stahl (1989),
demonstrate that information frictions resulting from consumer search costs
can lead to competing firms setting different prices for homogeneous
goods.\footnote{See Baye, Morgan, and Scholten (2006) for a broad review of
the consumer search and price dispersion literature.} Search costs have also
played an important role in characterizing labor and monetary
markets.\footnote{For reviews of search-theoretic models in labor economics,
see Rogerson, Shimer, and Wright (2005) and Eckstein and Van den Berg (2007).
For reviews in monetary economics, see Rupert, Schindler, Shevchenko, and
Wright (2000).}

Although search costs are an important component of many theoretical models,
we know very little about what determines consumers' search cost distributions
and its implications for policy and pricing as search costs are typically
unobserved. For instance, there exists little empirical work that documents
how and why consumer search costs vary across geographic markets. This gap in
the literature is unfortunate because measuring and understanding the source
of variation in search costs can benefit government policy and help understand
firms' pricing strategy, which critically depends upon both market structure
and the distribution of consumer search costs in a market. For instance, as we
later show, a policy or technological improvement that lowers the average cost
of search, but also reduces the variance of the search cost distribution, can
lead to higher equilibrium prices.

This paper fills this gap by exploring the determinants of consumer search
costs and their role in shaping equilibrium pricing and search behavior. To do
so, we first structurally estimate the search cost distributions for each of
many retail gasoline geographic markets. We document that search costs vary
considerably both within and across markets. As detailed below, the shape of
the consumer search cost distribution has important implications for both
government policy and firms' strategic pricing behavior. In our counterfactual
policy experiments, we find that (1) the search cost distribution needs to be
sufficiently heterogeneous to generate equilibrium price dispersion, and (2)
the market-level expected price paid decreases in the number of firms, but
consumers with high search costs may be worse off from an increased number of
firms.\footnote{We use the terms \textquotedblleft gas
station\textquotedblright\ and \textquotedblleft firm\textquotedblright%
\ interchangeably.}

We proceed in two steps. First, the extent to which consumers' search costs
vary across geographic markets is explored. Instead of relying on indirect
measures of search behavior, such as internet usage for searching for online
insurance products (Brown and Goolsbee 2002), we leverage the non-sequential
search model developed in Burdett and Judd (1983) to directly recover the
consumer search cost distribution that rationalizes observed gasoline prices
as an equilibrium outcome generated by gas stations pricing to consumers with
heterogenous search costs (Hong and Shum 2006; Moraga-Gonz\'{a}lez and
Wildenbeest 2008; Wildenbeest 2011). To obtain multiple cross-sectional
observations, which we need to examine the heterogeneity of search costs
across markets, we define geographically isolated markets in the spirit of
Bresnahan and Reiss (1991). Facilitated by daily gasoline prices for many
geographically diverse local markets in the United States, we estimate the
distribution of search costs for each of these markets. We establish that both
the mean and variance of the search cost distributions vary considerably
across geographic markets. By relating the variation in the distribution of
search costs across markets to variation in market characteristics, we find
that the search cost distribution is closely related to the distribution of
household income; markets with a higher earning population are characterized
by higher search costs. Furthermore, markets with more dispersed household
income have more dispersed search costs. These results suggest that consumers'
search costs are, in part, driven by opportunity costs. Meanwhile, we do not
find a relationship between search costs and other potentially informative
population characteristics such as the age, education, and the mean distance
among stations.

Second, using the estimated structural parameters, we conduct policy
experiments to investigate the effect of heterogeneity in consumer search
costs on equilibrium prices and consumer welfare. We run two experiments,
which suggest that the shape of the consumer search cost distribution has
important implications for both government policy and firms' strategic pricing
behavior. The first experiment studies how two exogenous changes in the search
cost distribution, in the sense of first-order stochastic dominance and
second-order stochastic dominance, respectively, changes price equilibria. As
Armstrong (2008) notes, competition policy affecting consumer search costs has
an ambiguous affect on the price paid by \textquotedblleft
searchers\textquotedblright\ and \textquotedblleft
non-searchers\textquotedblright. We find that a decrease in search costs such
that the new search cost distribution is first-order stochastically dominated
by the original distribution leads to a decrease in the expected price paid
for all consumers; however, the paid search costs decrease only for people
with low search costs, whereas the paid search costs increase for people with
median search costs. The total expenditure decreases for nearly all consumers,
and the benefit is larger for consumers with smaller search costs. For the
second-order stochastically dominant change, we confirm that heterogeneity and
not the level of expected search costs is the key to generating equilibrium
price dispersion. We find that making search costs more homogeneous such that
the new search costs distribution second-order stochastically dominate the
original distribution may lead to higher total expenditure in terms of prices
and paid search costs. If the distribution of search costs become sufficiently
homogeneous (although the distribution need not be degenerate), all firms set
the monopoly price. Overall, our findings highlight that competition policy
should incorporate search cost distributions to fully capture the effect on
prices and consumer surplus.

The second experiment analyzes how an increase in the number of firms affects
the equilibrium price distribution. Not surprisingly, we find that the minimum
market price decreases in the number of gas stations. More interestingly, the
experiment illustrates that increasing the number of stations in a market
initially decreases the expected price, but as the number of firms increases
beyond four the expected price increases, which stands in contrast to the
predictions of the standard Cournot and differentiated Bertrand models.
Expected price paid, on the other hand, declines as the number of gas stations
increases and attains its minimum at $13$ stations.\footnote{The expected
market price is the expected value of the price cumulative distribution
function (i.e. the expected price from a random draw in a given market. The
expected price paid, on the other hand, is the expected minimum price among a
consumer's set of price quotes. In other words, the price paid factors in what
the consumer expects to actually pay, which depends upon the consumer's cost
of search and the number of searches. See Section 4 for the formal definition
of the expected price paid.} We also confirm a non-monotonic relationship
between one measure of price dispersion and the number of firms; the standard
deviation of prices has an inverse u-shape and attains its maximum in a market
with around $18$ stations. Finally, we observe that a change in market
structure differentially impacts people with different search costs. For
example, when the number of stations increases from five to six, consumers in
the $10$th percentile of the search cost distribution decrease their total
expenditures, whereas consumers in $75$th percentile increase total expenditures.

This paper is a continuation of a recent strand of research in industrial
organization that uses structural assumptions to estimate consumer search
costs from price data. Horta\c{c}su and Syverson (2004) use data on S\&P 500
index funds to estimate search costs; the econometric framework in that
article allows for horizontal product differentiation but requires both price
and quantity data - the latter of which is often difficult to obtain. Hong and
Shum (2006) are the first to demonstrate how, in homogeneous goods markets,
firms' profit-maximizing conditions can be used as moment restrictions in an
empirical likelihood estimation routine to back-out the consumer search cost
distribution from only price data. Moraga-Gonz\'{a}lez and Wildenbeest (2008)
extend Hong and Shum (2006) through the maximum likelihood estimation approach
and achieve more favorable convergence properties. Wildenbeest (2011) builds
on Hong and Shum (2006) to include vertical product differentiation to
estimate the distribution of search costs using price data from four grocery
stores in the UK. Using the methods developed in Hong and Shum (2006),
Moraga-Gonz\'{a}lez and Wildenbeest (2008), and Wildenbeest (2011), we
estimate the parameters of the search cost distribution that justify the
observed regular gasoline price distributions. Unlike such previous research
that estimates consumer search costs for a single market, however, this paper
uncovers the distribution of consumer search costs for $354$ local markets to
investigate the heterogeneity of search costs across markets. This paper is
related to a recent study by Moraga-Gonz\'{a}lez, S\'{a}ndor, and Wildenbeest
(2013b), which use price observations from multiple markets to achieve, via
semi-nonparametric estimation, a more precise search cost distribution that is
common to all product markets. Our paper, by contrast, estimates the search
costs market by market to document the heterogeneity of search costs across
geographical markets.

This paper is related to the literature on price dispersion and consumer
search in the retail gasoline market. A number of studies, such as Marvel
(1976), Lewis (2008), Chandra and Tappata (2011), Pennerstorfer,
Schmidt-Dengler, Schutz, Weiss, and Yontcheva (2014) have identified patterns
of temporal and cross-sectional price dispersion in retail gasoline markets
that are consistent with models of costly consumer search.\footnote{For
example, using a superset of the data used in this study, Chandra and Tappata
(2011) find that the price ranking of firms varies less for more closely
located firms and that price dispersion increases in the number of firms in a
market. Lewis (2008) uses weekly price data for stations in the San Diego area
and similarly finds that patterns of price dispersion are consistent with a
model of consumer search. Lewis and Marvel (2011) use website traffic data for
gasoline price comparison sites to characterize the patterns of consumer
search on the internet. Barron, Taylor, and Umbeck (2004) use a large
cross-section of station-level price data in four large US cities and find
patterns of price dispersion consistent with some models of consumer search.
More broadly, this paper is also related to several studies on price
dispersion in the retail gasoline markets. See, for example, Hosken, McMillan,
and Taylor (2008), and Lach and Moraga-Gonz\'{a}lez (2012). For recent
empirical work on retail gasoline markets, see Eckert (2013) and references
therein.} The reduced-form approach of studying the relationship between price
dispersion and market characteristics is also conducted in other product
markets.\footnote{{See, for example, Baye, Morgan, and Scholten, 1994
(consumer electronics), Sorensen, 2000 (prescription drugs), Lach, 2002
(grocery stores), Brown and Goolsbee, 2002 (life insurance), and Vukina and
Zheng, 2010 (live hog).}} Although the analysis in these studies is carefully
executed, because search costs are not directly observed, the evidence has
been limited to reduced-form testing of the comparative static relationships
implied by a particular theoretical model. By contrast, by directly estimating
the search cost distributions that rationalize the data, we push the
literature forward by quantifying the effect consumer search cost
heterogeneity has on changes in market structure and policies that effect
consumer search.

Finally, this paper is related to the extensive literature, both theoretical
and empirical, on how market structure affects the equilibrium price
distribution in the presence of market frictions and consumer search
(Rosenthal 1980; Varian 1980; Stiglitz 1987; Stahl 1994; Janssen and
Moraga-Gonz\'{a}lez 2004; Moraga-Gonz\'{a}lez, S\'{a}ndor, Wildenbeest 2010
and 2013a; Lach and Moraga-Gonz\'{a}lez 2012). Our work differs from those
papers in that we employ the estimated structural parameters of the model to
quantify the effects of changes in market structure on the prices and paid
search costs for consumers with different levels of search costs.

Our paper proceeds as follows. Section 2 describes the data, how we choose
markets within which to perform the estimation, and reduced-form analysis on
price dispersion. Section 3 details the empirical model and estimation
results. Section 4 conducts the counterfactual experiments. Section 5 concludes.

\section{The Data}

\subsection{Price Data}

The analysis in this paper benefits from a large panel data set of daily
gasoline prices. The data originate from the Oil Price Information Service
(OPIS), which obtains data either directly from gas stations or indirectly
from credit card transactions.\footnote{OPIS's website states that their data
originates from \textquotedblleft exclusive relationships with credit card
companies, direct feeds, and other survey methods.\textquotedblright} OPIS's
data have frequently been relied upon in academic studies of the retail
gasoline industry (e.g. Lewis and Noel 2011; Taylor, Kreisle, and Zimmerman
2010; and Chandra and Tappata 2011).

The data cover daily station prices from January 4th, 2006 through May 16th,
2007 for stations in California, Florida, New Jersey, and Texas, which amounts
to more than $20,000$ stations. This data set was previously utilized in
Chandra and Tappata (2011).\footnote{We refer interested readers to Chandra
and Tappata (2011) for a more detailed data description.}

\subsection{Location Data and Selecting Isolated Markets}

In the retail gasoline markets, competition among stations is highly localized
(Eckert 2013). The location of firms plays an important role in our analysis,
both in the selection of markets within which to estimate search costs and in
constructing control variables for subsequent regressions. To pinpoint the
geographic location of each firm, street addresses were converted to longitude
and latitude coordinates using ArcGIS and then cross-referenced with
coordinates outputted from Yahoo maps.

In the structural model, prices are generated by an equilibrium in which a
defined set of gas stations all compete against each other for the same set of
customers. Markets in the data must be carefully defined to be consistent with
this assumption. Typically, the literature on retail gasoline markets defines
each firm in the data to be at the center of a market of a specified
radius.\footnote{For example, see Hastings (2004), Barron, Taylor, and Umbeck
(2004), Lewis (2008), and Remer (2013).} A potential difficulty and source of
estimation bias inherent in this market definition is overlapping markets; two
firms within a specified distance that compete against each other may not
share the same set of total competitors.\footnote{For example, suppose three
firms, A, B, and C, are located one mile apart in sequence along a road. Using
a cutoff distance of $1.5$ miles for the radius, we obtain three markets, A
and B, A, B, and C, and B and C. It would, however, be inappropriate to treat
market outcomes from these three markets as three independent observations to
draw inferences on each of those markets because firm A and C's pricing is
dependent through firm B's pricing.}

To circumvent this problem, the analysis focuses on what we define to be
\textquotedblleft isolated\textquotedblright\ markets in the spirit of
Bresnahan and Reiss's (1991) geographic market definition. We use two strict
criteria. First, let $J$ be a set of firms and let $d(i,j)$ be the Euclidean
distance between any two firms $i,j\in J$. Then, $J$ is an isolated market if
for all $i,j\in J$, $d(i,j)\leq X$, and for all $k\notin J$ and $i\in J$
$d(i,k)>X$. For the analysis presented below, $X$ is set be $1.5$ miles; thus
an isolated market is a set of firms all within $1.5$ mile of each other and
no other competitor is within $1.5$ mile of any firm in the market. The
maximum distance between firms in a market is chosen to be $1.5$ miles, which
is consistent with previous studies such as Barron, Taylor, and Umbeck (2004),
Hosken, McMillan, and Taylor (2008), and Lewis (2008). This market definition
ensures that the observed prices in a market are not influenced by competition
with unobserved competitors.

To analyze the relationship between search costs and census tract level
characteristics, isolated markets are further restricted to only include
markets where all gas stations are located within a single census tract.
Figure 1 depicts a map that contains such an isolated markets with multiple
firms and a single census tract (\textquotedblleft Market 1\textquotedblright%
). Using this market definition, we estimate the model using $11,736$ price
observations from $1,127$ stations in $354$ isolated (and single census tract)
markets. Figure 2 shows the distribution of those markets by market structure.
Although our market definition still contains some potential issues, such as
people may purchase gasoline not only where they live, but also where they
commute for work, the requirements for the market definition in this study are
more stringent than those used by most of the existing literature.\footnote{A
notable exception is Houde (2012), who focuses on the Quebec City gasoline
market and takes commuting routes into account.}

\subsection{Cost data}

The structural model that we estimate requires only price data, and reconciles
the observed price dispersion as a consequence of vertical product
differentiation and heterogeneity in consumers' cost of search. In our data,
however, prices may vary over time in response to changes in the wholesale
cost of gasoline. To minimize the effect of marginal cost changes on the
observed price dispersion, the model is estimated using the 30 day window in
the data where the variation in wholesale price of gasoline is minimized. As
the price at which each retailer purchased their product is privately
negotiated and unavailable, we utilize the price of wholesale gasoline traded
on the New York Mercantile Exchange (NYMEX), which is likely to be highly
correlated with actual marginal costs. These data are commonly employed as a
measure of marginal cost in studies of the retail gasoline
industry.\footnote{See, for example, Borenstein, Cameron, and Gilbert (1997),
Velinda (2008), and Lewis (2011).} We estimate the model using price data from
March 3rd through April 1st, 2006, where the standard deviation of the daily
price of wholesale unleaded fuel shipped from the NY Harbor was 6.3 cents per
gallon.\footnote{Note also that there are brief windows in the data set where
price data are missing. These days were a priori excluded as potential times
over which to perform the estimation.}

\subsection{Descriptive Evidence of Price Dispersion}

Both survey evidence and the economic literature demonstrate that consumer
search plays an important role in the retail gasoline industry. The National
Association for Convenience and Fuel Retailing (NACS) has been surveying
gasoline consumers since 2007 and has consistently found that price is
overwhelmingly the most important factor in buying gasoline.\footnote{See
http://www.nacsonline.com/YourBusiness/FuelsReports/GasPrices\_2013/Pages/Consumers-React-to-Gas-Prices.aspx.}
They also find that about $68\%$ of people would drive five minutes out of
their way to save 5 cents per gallon, but only $36\%$ of people would drive
ten minutes to save the same amount. Similar results throughout the survey
suggest that (i) a large fraction of people search to save money on gasoline
expenditures and (ii) the intensity with which customers search varies.

We run OLS regressions to document the relationship between measures of price
dispersion and market characteristics. The census-level data used as
explanatory variables is taken from the 2006-2010 American Community Survey
(ACS), which is an ongoing survey conducted under the auspices of the US
Census Bureau. Mean income and age are taken directly from the survey, while
mean years of education are calculated by taking a weighted average of the
proportion of people in a census tract who have reached a particular
educational attainment. For each census tract, the ACS reports the number of
households that fall within a particular income bracket; by assuming that
average household income within a bracket is the mid-point of the bracket, we
calculate the standard deviation of income. The mean distance between stations
in a market is constructed by measuring the mean Euclidean distance between
all stations in a market. Table A1 in Appendix presents basic summary
statistics of the variables from ACS for $354$ single-census isolated markets
in our data.

Table 1 documents general price statistics for $354$ isolated markets. As the
number of stations increases the sample range monotonically increases, whereas
there is no uniform relationship between the the sample standard deviation and
the number of firms. This pattern suggests that different measures of price
dispersion correlate differently with market structure, which may limit the
ability of reduced-form analysis to infer the link between price dispersion
and search costs.

Table A2 in Appendix shows the results of regressing two measures of price
dispersion on the market characteristic variables. We define sample range as
the difference between the maximum and minimum price for a given market. We
find a robust relationship between price dispersion and some market
characteristics, such as income and the number of firms in a market, but not
all results are robust to how price dispersion is measured. For example, mean
household income is always positively and significantly correlated with price
dispersion with at least $95\%$ confidence. On the other hand, the standard
deviation of household income has a significant negative relationship with
price dispersion (at the $90\%$ confidence level) only when specified in logs,
and otherwise has no significant link. Consistent with Table 1, the number of
firms in the market has a positive and significant relationship with the
market price range, but is not found to have any relationship with the
standard deviation of prices. This result highlights a benefit of using a
structural model to study the relationship between price dispersion and search
costs; the relationship can be directly estimated and is not tied to a
particular measure of price dispersion.

\section{Model Estimation}

\subsection{Empirical Model of Search and Price Equilibrium}

This subsection introduces a non-sequential search model based on Burdett and
Judd (1983) and explains how the model can be used to estimate the
distribution of consumer search costs.\footnote{We assume that consumers
search non-sequentially in retail gasoline markets. See De Los Santos,
Horta\c{c}su, and Wildenbeest (2012) and Honka and Chintagunta (2013) for
discussions and empirical testing of these two strategies.} Our model
presentation borrows from Hong and Shum (2006) and Moraga-Gonz\'{a}lez and
Wildenbeest (2008).

We first consider a set of gas stations selling a homogeneous product to a
continuum of consumers that are identical except for their search costs that
are unobservable to the firm.\footnote{Wildenbeest (2011) demonstrates that
specifying the model in terms of utility, rather than price, allows for
vertical product differentiation to be incorporated into the model, and
controlled for in the estimation by simply adding a fixed-effect. While our
empirical model accounts for vertical product differentiation, for
expositional purposes in this section, we discuss a homogeneous product market
where the price $p$ is the strategic variable. All of the subsequent
discussions go through if we replace price with utility.} Firms, however, do
know the distribution of consumer search costs. Each firm in the market
simultaneously chooses its price, $p$; the cumulative equilibrium price
distribution is denoted as $F_{p}$, where \textit{\b{p}} and $\bar{p}$ are the
lower and upper bound, respectively, of the support of $F_{p}$. Firms have
constant and identical marginal costs of production, denoted as $r$. In
equilibrium, firms play mixed strategies and therefore vary their prices over
time.\footnote{Several empirical works document that the pricing of retail
gasoline stations is consistent with a mixed strategy; see Lewis (2008),
Hosken, McMillan and Taylor (2008), Lach and Moraga-Gonz\'{a}lez (2009), and
Chandra and Tappata (2011).}\footnote{For expositional simplicity, we present
the model by assuming products are homogeneous. Our empirical model extends
this framework by assuming that firms play mixed strategies in utilities,
which consist of price and quality of a product (Wildenbeest 2011). See the
next subsection for details.}

Consumers draw an i.i.d. search cost, $c\geq0$, from the cumulative search
cost distribution, $F_{c}$. All consumers have an inelastic demand for a unit
of gasoline. Consumers know the distribution of market prices,\thinspace
$F_{p}(p)$, but they do not know individual firms' prices. Consumers receive
one free price quote and must pay a cost, $c$, for each additional quote.
Consumers learn the realization of prices after deciding how many additional
quotes to obtain, including no additional quote, which implies the consumer
goes with the free quote. With a sample of $l(\geq1)$ gas prices from
$l$\ stations, each consumer purchases one unit from the lowest-priced gas
station in their sample. A consumer's problem is to minimize the total
expected expenditure of purchasing a product by choosing the number of gas
stations to search, $l-1$, where
\[
l=\arg\min_{l\geq1}c\cdot(l-1)+\int_{\text{\textit{\b{p}}}}^{\bar{p}}l\cdot
p(1-F_{p}(p))^{l-1}f(p)dp.
\]
The first term, $c(l-1),$ is the total costs of search.\footnote{We subtract
$1$\ because the first quote is free.} The second term is the expected price
paid for the product when a consumer has $l$ quotes from $l$\ stations.
$(1-F_{p}(p))^{l-1}$ is the probability that all other stations charge a price
higher than $p$. By searching $i+1$ rather than $i$ stores, a consumer obtains
an expected marginal savings, which is denoted as $\Delta_{i}\equiv
Ep_{1:i}-Ep_{1:i+1}$, $i=1,2,...$ Here, $p_{1:i}$ represents the minimum price
when a consumer takes $i$ draws from $F_{p}.$ Accordingly, a consumer with
search cost $c$\ will sample $i$ stores when $\Delta_{i-1}>c>\Delta_{i}$. A
consumer with $c=\Delta_{i}$ is indifferent between searching $i$ stores and
$i+1$ stores. The proportion of consumers with $i$ price quotes $q_{i}$ will
be $q_{1}\equiv1-F_{c}(\Delta_{1})$ and $q_{i}\equiv F_{c}(\Delta_{i-1}%
)-F_{c}(\Delta_{i})$ for $i\geq2$.

Firms maximize profits by choosing a symmetric, mixed-pricing strategy $F_{p}$
for all $p\in\lbrack$\textit{\b{p}}$,\bar{p}]$. Based on the consumer
behavior, each firm's total profit can be denoted as $\Pi(p)=(p-r)[\sum
_{i=1}^{N}q_{i}\cdot\frac{i}{N}(1-F_{p}(p))^{i-1}]$. The firms'
profit-maximizing mixed strategies imply a condition that each firm is
indifferent between charging the monopoly price $\bar{p}$ and any other price
$p\in\lbrack$\textit{\b{p}}$,\bar{p}],$ namely%
\begin{equation}
\frac{(\bar{p}-r)\tilde{q}_{1}}{N}=(p-r)[\sum_{i=1}^{N}\tilde{q}_{i}\cdot
\frac{i}{N}(1-F_{p}(p))^{i-1}], \label{eq_condition}%
\end{equation}
where $r$ is the common marginal cost of selling gasoline for each gas
station. In words, firms face a trade-off between setting a high price and
selling to less informed customers or setting a low price and capturing more
informed customers. In equilibrium, firms are indifferent between all points
in the price distribution and therefore maximize profits by playing a mixed
strategy over the price distribution. Solving equation (\ref{eq_condition})
for price allows us to represent price as the inverse function%
\begin{equation}
p(z)=\frac{\tilde{q}_{1}(\bar{p}-r)}{\sum_{i=1}^{N}iq_{i}(1-z)^{i-1}}+r,
\label{inverse_p}%
\end{equation}
where $z=F_{p}(p)$.

\subsection{First Stage: Nonparametric Estimation of Search Cost at the Market
Level}

Our first stage estimation strategy is based on Hong and Shum (2006) and the
extensions developed in Moraga-Gonz\'{a}lez and Wildenbeest (2008) and
Wildenbeest (2011). The Hong and Shum (2006) framework allows the distribution
of consumer search costs to be recovered using price data alone by
rationalizing the observed price dispersion as a consequence of search costs.
We apply this methodology to each geographically isolated retail gasoline
market to document the heterogeneity of search costs across markets.

To identify the model parameters, we use the equilibrium condition specified
in equation (\ref{eq_condition}), that in a mixed strategy equilibrium,
expected profits are the same for all prices in the support of the equilibrium
price distribution. We conduct nonparametric estimation using this optimality condition.

We conduct the maximum likelihood technique developed by Moraga-Gonz\'{a}lez
and Wildenbeest (2008). Denoting the number of gas stations in a market by $N$
and the number of price observations in that market by $M$, we employ the MLE
estimation strategy to obtain the estimated model parameter $\hat{\theta
}_{MLE}=\{\hat{q}_{i}\}_{i=1}^{N-1}$ such that:
\[
\hat{\theta}_{MLE}=\arg\max_{\{q_{i}\}_{i=1}^{N-1}}%
%TCIMACRO{\tsum \limits_{l=2}^{M-1}}%
%BeginExpansion
{\textstyle\sum\limits_{l=2}^{M-1}}
%EndExpansion
\log f_{p}(p_{l};q_{1},..,q_{N}),
\]
where $f_{p}$ is the density for $F_{p}$ and $F_{p}(p_{l})$ solves equation
(\ref{eq_condition}).\footnote{Appendix provides the representation of $f_{p}%
$\ in $F_{p},q_{i},p,$ and $r$.} The maximum likelihood routine yields
estimates of a non-parametric search cost CDF represented by a combination of
points $\{q_{i},\Delta_{i}\}$.

Finally, we control for potential vertical differentiation by gas stations
within a market using Wildenbeest (2011). This method extends Hong and Shum
(2006) such that firms play mixed strategies in consumer utility, where
$u_{j}=\delta_{j}-p_{j}$ and $\delta_{j}$ is the value consumers realize from
purchasing one unit of the good from station $j$. To implement this model, we
parameterize the price at firm $j$ at time $t$, $p_{jt}$, as%

\begin{equation}
p_{jt}=\alpha+\delta_{j}+\epsilon_{jt}, \label{fe}%
\end{equation}
and perform the fixed effects regression specified by equation (\ref{fe}) for
a given market to recover $\hat{\delta}_{j}.$We then construct the utility
from station $j$\ by setting $\hat{u}_{jt}=$ $p_{jt}-\hat{\delta}_{j}$. By
assuming that any systematic differences in quality across gas stations are
attributed to the differences in prices across those stations, the model
reduces to a game in which firms are symmetric in their strategies to
randomize its utility.

\subparagraph{Estimation results.}

We estimate the search cost CDF (i.e., a combination of points $\{q_{i}%
,\Delta_{i}\})$ for each of $354$\ markets. Table 2 presents the statistics
that summarize estimated $\hat{F}_{c}(\hat{\Delta}_{i})$, $\hat{\Delta}_{i}$
,and the marginal cost from all those markets. This table demonstrates a large
amount of variation in the estimated search costs across markets, as shown in
the standard deviation and the range between minimum and maximum of $\hat
{q}_{1}$ and $\hat{\Delta}_{1}$. To illustrate how estimated search cost
distributions vary across markets, we randomly pick five markets that have
three stations in Texas. Figure 3a plots $(\Delta_{1},F_{c}(\Delta_{1}))$ and
$(\Delta_{2},F_{c}(\Delta_{2}))$ for each market.\footnote{We have two
combinations because consumers can search at most two stations.} We have
$\Delta_{1}$ and $\Delta_{2}$ on the horizontal axis and $F_{c}(\Delta_{1})$
and $F_{c}(\Delta_{2})$ on the vertical axis. The figure displays a
considerable variation in the distribution across those five markets. For
instance, Market 1 has the fraction of nonsearchers ($q_{1}=1-F_{c}(\Delta
_{1})$) as $0.0251$, and the gains from search for the first search is
$\$0.025$ ($=\Delta_{1}$) per gallon of regular unleaded gasoline. Market 5,
on the other hand, has the fraction of nonsearchers as $0.561$, and the gains
from search for the first search is $\$0.056$ ($=\Delta_{1}$) per gallon.

Because we normalize the minimum price in each market to $1$, the price cost
margin is on average $\$0.267$ ($=\$1-\$0.733$) per gallon.\footnote{A
different normalization would yield the same price cost margin of $\$0.267$
because what we exploit in the estimation is the variation in prices across
time after taking the average differences in prices across firms. Similarly,
the following analysis does not change quantitiatively if we choose a
different minimum price.} The equilibrium price distribution from the
estimated model approximates the empirical distribution of prices in most
markets. For instance, Figure 3b presents the empirical and estimated price
distribution from a market in Texas.

In the next subsection, we seek to explain this search cost heterogeneity
across markets by using market characteristics.

\subsection{Second Stage: Estimation of Parametric Search Cost Distribution
that Allows for Variation across Markets}

This subsection sheds light on the underlying source of heterogeneity in
search costs, which we documented in the previous subsection. Our aim is to
determine the extent to which (i) the distribution of search costs varies
across markets and (ii) this variation can be explained by observable market
and population characteristics. To achieve this goal, we take estimated points
of the search cost CDFs across all markets, which are depicted in Figure A1 in
Appendix, and use non-linear least squares regression to fit a parametric
distribution. In the regression, we let the mean and variance of the
distribution depend on market-level characteristics, which allows us to
quantify the influence of these market characteristics on the distribution of
search costs.

Table 3 presents the results of using non-linear least-squares to fit a
lognormal (columns 1 through 3) CDF to the data points in all markets. We
regard column 1 as the baseline specification, and columns 2 and 3 examine
whether the results are robust to the exclusion of imprecisely estimated parameters.

Of first note is that across all specifications the mean income in a market is
precisely estimated at the $5$\% level to positively affect the mean of the
consumer search cost distribution.\footnote{The result is robust to the
inclusion of the distance to the nearest highway exits.} This result is
intuitive in two respects; income is positively correlated with the
opportunity cost of time and people with higher income have a lower marginal
utility of wealth (and therefore gain less utility from saving money on
gasoline). Both of these effects imply higher search costs. In terms of
magnitude, a $1$\% increase in household income increases the expected search
costs by $\$0.039$ per gallon.\footnote{The results are robust to a different
parametric distribution (column 4, normal CDF) and different measures of
household income (median and median absolute deviation).}

We also find that the standard deviation of household income within a market
positively affects the standard deviation of the search costs in a market.
Thus, when household income becomes more dispersed within a market, the search
cost distribution also becomes more dispersed. This finding further reinforces
the link between income and the cost of search. Another robust pattern that
emerges from these four specifications is that after controlling for income,
higher mean age implies a lower mean of the search cost distribution at the
$10\%$ level of statistical significance. This result may reflect that retired
people, who tend to be older than the average population, have a smaller
opportunity cost of time, although we cannot rule out other explanations.
Other variables, such as mean years of education and mean distance across
stations, are not found to significantly affect the search cost distribution.
The unconditional expectation of the search costs ($=\int c$ $dF_{c})$ fitted
on the lognormal distribution for a typical market with mean household income,
mean education, mean age, and mean distance among stations is $\$0.287$ per
gallon.\footnote{The next section observes that due to the presence of
non-searchers who pay zero\ search costs, the expected search costs paid
is\ significantly smaller than this unconditional expectation of search
costs.}

Based on the estimated search cost distributions, the next section explores
how heterogeneity of search costs affect price equilibrium by conducting
policy experiments.

\section{Policy Counterfactuals}

This section adopts the parameter estimates of the second-stage search cost
distribution to perform two sets of \textquotedblleft
what-if\textquotedblright\ experiments that quantify the influence of search
costs on equilibrium prices in the retail gasoline markets. Each experiment
compares a \textquotedblleft baseline\textquotedblright\ CDF, which is
calibrated to fit the parameter estimates in the previous section, to a
\textquotedblleft hypothetical\textquotedblright\ CDF, which alters the
baseline CDF to reflect the impact of some policy. To compute the mean and
standard deviation of the baseline lognormal search cost distribution, we use
the structurally estimated model in the second stage regression and the
average value across markets of the right-hand-side variables in the second
stage regression.

To perform each policy experiment, we solve for the equilibrium price
distributions that correspond to the baseline and hypothetical CDFs (policy 1)
or number of stations (policy 2).\footnote{To solve for the price
distributions, we first compute the marginal gains from the $i$th search,
$\Delta_{i}$, based on the appropriate search cost CDF, an initial guess of
$q_{i}$ , the marginal costs, and the maximum willingness to pay. This first
step yields $\tilde{q}_{i}$\thinspace\ a vector denoting the proportion of
consumers with $i$ price quotes, implied by the model. We then iterate the
process until the initial starting values of $q_{i}$\ converge to a particular
$\tilde{q}_{i}$. Based on this $\tilde{q}_{i}$, we generate $100$ prices
according to the price inverse function in equation (\ref{inverse_p}). The
results are robust to the number of generated prices.} We assume the
market-level characteristics, marginal costs, and the maximum willingness to
pay do not change before and after the experiment. We normalize consumers'
maximum willingness to pay to be $\$2.779$ per gallon, which is the highest
price in the data.\footnote{Normalizing the maximum willingness to pay to a
certain constant is for the purpose of presention and does not affect
quantitatively the following results; an alternative normalization with a
different maximum willingness to pay, say $\$4.000$, simply shifts the
equilibrium price distribution by \$$1.221$ ($=\$4.000-\$2.779$).} In the
data, the difference between the maximum willingness to pay and marginal costs
is on average $\$0.472$, and therefore, in this section, assume that the
marginal costs are $\$2.307$ ($=\$2.779-\$0.472$) per gallon.

To analyze the effect of the policy counterfactuals on consumer and firm
well-being, we compute the expected price paid, expected paid search costs,
and the total number of searches. We calculate the total paid search costs at
the market level by integrating the product of the search costs and the number
of searches over the search cost CDF. Specifically,
\begin{align*}
E[c_{paid}]  &  =%
%TCIMACRO{\tsum \limits_{i=1}^{N}}%
%BeginExpansion
{\textstyle\sum\limits_{i=1}^{N}}
%EndExpansion
\{(i-1)%
%TCIMACRO{\dint \limits_{Q_{N+1-i}}^{Q_{_{N+2-i}}}}%
%BeginExpansion
{\displaystyle\int\limits_{Q_{N+1-i}}^{Q_{_{N+2-i}}}}
%EndExpansion
cdF_{c}\}\text{ }\\
\text{where }Q_{i}  &  =1-%
%TCIMACRO{\tsum \limits_{s=1}^{N+1-i}}%
%BeginExpansion
{\textstyle\sum\limits_{s=1}^{N+1-i}}
%EndExpansion
q_{s}\text{ if }N+1-i\geq1\text{ and }Q_{i}=1\text{ otherwise },
\end{align*}
and $N$ is the number of stations in that market. For instance, the expected
search cost paid in a market with three station is%
\[
E[c_{paid}]=2%
%TCIMACRO{\dint \limits_{0}^{q_{3}}}%
%BeginExpansion
{\displaystyle\int\limits_{0}^{q_{3}}}
%EndExpansion
cdF_{c}+%
%TCIMACRO{\dint \limits_{q_{3}}^{q_{2}+q_{3}}}%
%BeginExpansion
{\displaystyle\int\limits_{q_{3}}^{q_{2}+q_{3}}}
%EndExpansion
cdF_{c},
\]
and the first and the second term is the expected paid search costs for people
who search twice and once, respectively. Similarly, the expected price paid is
obtained by first calculating the expected price conditional on the number of
times a consumer searches, and then integrating over the number of searches:
$E[p_{paid}]=Epq_{1}+%
%TCIMACRO{\tsum \limits_{i=2}^{N}}%
%BeginExpansion
{\textstyle\sum\limits_{i=2}^{N}}
%EndExpansion
\{Ep-(%
%TCIMACRO{\tsum \limits_{s=1}^{i-1}}%
%BeginExpansion
{\textstyle\sum\limits_{s=1}^{i-1}}
%EndExpansion
\Delta_{s})\}q_{i}$. For instance, the expected price paid in a market with
three stations is
\begin{align*}
E[p_{paid}]  &  ={\small
%TCIMACRO{\TeXButton{Expected price for non searchers}{\underbrace
%{ Ep}_{\textrm{Expected price for non searchers}} *q_{1}}}%
%BeginExpansion
\underbrace{ Ep}_{\textrm{Expected price for non searchers}} *q_{1}%
%EndExpansion
+%
%TCIMACRO{\TeXButton{Expected price for people who search once}{\underbrace
%{ (Ep-\Delta_{1})}_{\textrm{Expected price for people who search once}}
%*q_{2}}}%
%BeginExpansion
\underbrace{ (Ep-\Delta_{1})}_{\textrm
{Expected price for people who search once}}
*q_{2}%
%EndExpansion
}\\
&  +{\small
%TCIMACRO{\TeXButton{Expected price for people who search twice}{\underbrace
%{ (Ep-\Delta_{1} - \Delta_{2}) }_{\textrm
%{Expected price for people who search twice}} *q_{3}}}%
%BeginExpansion
\underbrace{ (Ep-\Delta_{1} - \Delta_{2}) }_{\textrm
{Expected price for people who search twice}} *q_{3}%
%EndExpansion
.}%
\end{align*}
To demonstrate how certain policies differentially affect consumers with
different search costs, we present results for consumers in the 10th, 25th,
50th, and 75th percentiles of the search cost distribution.

\subsection{Policy Experiment 1: A Change in the Costs of Search}

We first examine an exogenous change in the cost of search, such as a
technological advance in searching for gas prices or a government policy that
publicizes gas prices on a website. Such a change would involve two aspects:
change in the mean and change in the standard deviation of a distribution, and
we discuss each aspect separately. In particular, we consider two scenarios in
which the baseline search cost CDF (i) first-order stochastically dominates
the hypothetical search cost CDF or (ii) is second-order stochastically
dominated by the hypothetical search cost CDF.

\subparagraph{Scenario (1): All consumers' cost of search decreases, and the
baseline search cost CDF first-order stochastically dominates the hypothetical
search cost distribution.}

This scenario considers a situation in which expected value of search costs
decrease (or increase) for every consumer, while keeping the variability
(i.e., standard deviation of the distribution) the same. A decrease in search
costs would represent a situation in which consumers' search effort has been
reduced due to technological developments, such as smartphones and related
apps, that affect the ease of finding gasoline price quotes. As a consequence,
the hypothetical search cost CDF ($F_{c}^{\prime}$) is first-order
stochastically dominated by the baseline search CDF, ($F_{c}$): $F_{c}%
^{\prime}(t)>F_{c}(t)$ for any $t>0$.\footnote{By contrast, an increase in
search costs would represent a situation in which search behavior becomes more
cumbersome due to an intentional price obfuscation by retailers (Ellison and
Ellison 2009) or an installation of a government policy, such as one that
prohibits the operation of price comparison gasoline websites (e.g.,
gasbuddy.com). In this scenario, the hypothetical search cost CDF
($F_{c}^{\prime}$) first-order stochastically dominates the baseline CDF
($F_{c}$).} Given the significant relationship estimated in the previous
section between search costs and household income, the counterfactual
experiments can also be interpreted as a change in household income, rather
than as a policy that directly alters the cost of search.

Table 4 presents the simulation results for a market with five firms for the
baseline specification, a $20\%$ decrease in search costs (case (1)), and a
$20\%$ increase in search costs (case (2)). To interpret the experiment as a
change in household income, cases (1) and (2) and the baseline correspond to
observationally equivalent markets except for the average household income,
which is $\$9,288$, $\$243,917$ and $\$62,270$, respectively.\footnote{This
sample range in household income roughly corresponds to the observed mean
household income's minimum and maximum (Table A1 in Appendix).} Below we
mainly discuss the difference between the baseline and the $20\%$ decrease in
search costs in case (1) in the second column in Table 4. The expected price
paid, on average, decreases by $\$0.035$ from $\$2.573$ (baseline). The
decrease in the actual price paid differs across consumers with different
search costs. For instance, consumers in the $50$th percentile experience the
largest decrease in the price paid $(-\$0.090=\$2.523-\$2.613)$, whereas
consumers in the 75th percentile experience the smallest decrease
($-\$0.025=\$2.588-\$2.613$) among the four categories of consumers ($10\%$,
$25\%$, $50\%$, and $75\%$tile).

We observe the policy change affects consumers with different search costs
differently for other measures as well. The expected paid search costs
decrease on average by $\$0.003$ from $\$0.017$. The decrease is the largest
for consumers in the 25th percentile ($-\$0.010$), but people in the $50$th
percentile actually increase the paid search costs by $\$0.062$ due to an
increased number of search. With respect to total consumer expenditure, which
is the sum of expected price paid and paid search costs, consumers in the
$25$th percentile gain the most ($-\$0.042$) whereas consumers in the $75$th
percentile gain the least ($-\$0.025$).

In Column 2, $q_{i}$ demonstrates that, due to a decrease in search costs, the
fraction of non-searchers decreases by $7.5$ percentage points from $56.4\%$,
and the remaining consumers who search one or more stations increase. An
implication from these results is that although search costs change across
consumers by the same percentage, there are heterogeneous effects across
people with unique search costs.

To test the sensitivity of results, we redo the simulations with a different
magnitude of change. Columns 4 and 5 in Table 4 show the results for a $50\%$
decrease in search costs (case (3)) and a $50\%$ increase in search costs
(case (4)). The effect of the change in search costs on the equilibrium price
distribution is qualitatively the same with case (1) and case (2),
respectively, with more pronounced magnitudes.

Figures 4a and 4b present the simulated search cost CDF and equilibrium price
CDF with $20\%$ changes to the expected value of the search costs,
respectively. These figures show that the case (1) price CDF is first-order
stochastically dominated by the baseline price CDF. In turn, the expected
price $E[p]$ decreases from $\$2.613$ to $\$2.588$\ per gallon when search
costs decrease by $20\%$.\ The lower bound of the price distribution drops by
$\$0.028$\ from $\$2.438$ to $\$2.410$, suggesting that the sample range
becomes wider. Conversely, when consumers' search costs increase by $20\%$ the
the baseline price CDF is first-order stochastically dominated by the
hypothetical equilibrium price CDF.\footnote{Figures A2a and A2b in Appendix
show the pattern holds for the case in which search cost mean changes by
$50\%$.}

Overall, this exercise confirms our intuition that an exogenous shifts in
search cost CDF in the FOSD sense lead to a new price equilibrium in which all
consumers benefit from lower prices, but the magnitude of the impacts, such as
the differences in total expenditure, differs across consumers with different
search costs.

\subparagraph{Scenario (2): Hypothetical search cost distribution second-order
stochastically dominates (SOSD) the baseline search cost distribution.}

This scenario considers a situation in which consumers' search costs become
less heterogeneous (but not completely homogeneous); the expected search costs
do not change, but the hypothetical search cost CDF ($F_{c}^{\prime}$) has a
lower standard deviation. More precisely, we consider a case in which
$F_{c}^{\prime}$ second-order stochastically dominates the baseline search CDF
($F_{c}$), and therefore, $\int c$ $dF_{c}^{\prime}=\int c$ $dF_{c}$ and
$\int_{0}^{x}F_{c}(t)dt\geq\int_{0}^{x}F_{c}^{\prime}(t)dt$ for all $x.$

To perform this experiment, the standard deviation parameter of the lognormal
CDF, $\sigma$, for the hypothetical distribution, $F_{c}^{\prime}$, is set to
be $15\%$ lower than the standard deviation for the baseline lognormal
distribution, $F_{c}$. We then calculate the mean parameter, $\mu$, for the
hypothetical distribution that yields the same unconditional expected search
costs, $\int c$ $dF_{c}=\int c$ $dF_{c}^{\prime}=\$0.287$ per gallon of
gasoline.\footnote{Note that we distinguish the mean parameter $\mu$ and the
expected value of the lognormal distribution, which is $e^{\mu+\frac
{\sigma^{2}}{2}}$.} Using $F_{c}$ and $F_{c}^{\prime}$\thinspace\ we simulate
gas stations' optimal pricing decisions under the baseline and hypothetical
scenarios.\footnote{We again assume the number of gas stations to be five. We
obtain similar results for market structures with different numbers of firms.}

Figures 5a and 5b present the simulated search and price CDFs when the
estimated standard deviation parameter is decreased (or increased) by $15\%$.
Prices become less dispersed when the standard deviation parameter decreases,
because as search costs become more homogeneous, firms have less incentive to
set low prices to sell to consumers with relatively low search costs. Figures
6a and 6b conduct the same experiment, but the standard deviation is changed
by $20\%.$ Figure 6b depicts a striking result -- price dispersion completely
disappears as firms maximize profits by setting the monopoly price. This
result is akin to Diamond (1971)'s monopoly equilibrium; however, it arises in
a non-sequential search environment, and all consumer search costs need not be
equal. Table A3 in Appendix presents the quantitative results. When the
standard deviation is reduced by $15\%$, consumers in the $10$th percentile of
the search cost distribution only search once and consumers in higher parts do
not search at all. When the standard deviation is reduced by $20\%$ no
consumers search. In the former case, there still exists price dispersion, as
consumers at the low end of the distribution still search and such a behavior
benefits consumers at the high end of search cost. In the latter case,
consumers in the low end of the distribution no longer search, and all firms'
profit-maximizing strategy is to set the monopoly price.

That firms may all set the monopoly price when consumer search costs are
sufficiently homogenous, but not identical, has an important policy
implication; a policy or firms' strategy that lowers both the expected value
and variance of search costs may have the unintended consequence of having all
firms increase their price to the monopoly price. The reason for this result
is that firms in our market face a trade-off between setting a low price and
selling to searchers and non-searchers and setting a high price and selling to
only people who search with low intensity. When search costs become more
homogeneous the trade-off disappears because the sales from low search costs
people becomes too small to make the expected profit the same across different
prices. To confirm this point, we run a variant of the last policy experiment;
in addition to reducing the standard deviation of search costs, we also
decrease the expected value of the search cost distribution. Figures A3a and
A3b in Appendix presents the results of decreasing the expected value of the
search cost distribution by $10\%$ and reducing the standard deviation by
$20\%$ and $22\%$, respectively. When the standard deviation is reduced by
$20\%$, we have a price dispersion unlike the previous case in Figure 6b
because it is cheaper to search due to a reduced search costs by $10\%$. When
the standard deviation is reduced by $22\%$, however, no consumers search even
when the expected value of the search costs decreases by $10\%$.

\subsection{Policy Experiment 2: A Change in Market Structure}

Now we investigate how, holding the search cost distribution constant,
different market structure affects the equilibrium price distribution and
search behavior differently. This experiment informs how firms and consumers
respond to an increase in competition via entry of firms into the retail
gasoline market.

To conduct this experiment, we assume that exogenous changes in market
structure do not affect the distribution of consumer search costs, and use the
same mean and standard deviation for the lognormal search costs distribution
we estimated in the previous section and used in the experiment above.

Figure 7 presents the simulated price distribution for hypothetical markets
with two, three, six, and twelve gas stations. Two patterns emerge from this
figure. First, the minimum price decreases as the number of stations increase,
and, as the maximum price remains fixed at\thinspace$\$2.779$, the price range
($=p_{\max}-p_{\min}$) increases in the number of stations. Second, beyond
three stations, the difference in the price distributions across markets
becomes considerably smaller. Although the minimum price declines when a
market becomes more competitive, those who do not search are worse off in less
concentrated markets with more than three firms. Indeed, Figure 7 shows that,
for prices below $\$2.48$, the simulated price CDF for a market with $12$
stations is flatter than the CDF with six stations. This pattern implies that
the lower minimum price in a market with $12$ stations will be compensated by
a steeper CDF above $\$2.48$, which means that consumers with high search
costs may face higher prices when the number of firms increase from six to twelve.

Figure 8a further analyzes these points in detail by displaying the summary
statistics by market structure. The figure demonstrates how equilibrium price
dispersion and paid search costs change with the number of firms. Again, the
minimum price declines monotonically with the number of gas stations in a
market. Meanwhile, the unconditional expected price $E[p]=\int p$ $dF_{p}$,
the price when consumers randomly pick a gas station, attains its minimum at a
market with four stations. That the pricing in local geographic markets
becomes relatively competitive after three or four entrants is similar to the
findings in Bresnahan and Reiss (1991). Figure 8a suggests that, beyond four
stations, the expected price, $E[p]$, increases in the number of stations.
This result diverges from a canonical Cournot or differentiated Bertrand
model, which predicts that an increase in the number of firms monotonically
reduces prices.

We also calculate the expected price paid after conditioning on the number of
times a consumer searches. Figure 8a shows the expected price paid is lower
than the expected price, as people who search once or more push down the
average market price paid. Table A4 in Appendix shows the expected price paid
attains its minimum at a market with $13$ stations. We then calculate the
total expenditure for consumers by adding the expected price paid and paid
search cost. Figure 8a and Table A4 in Appendix show that the total
expenditure attains its minimum when the number of gas stations is seven, and
is smaller than with $13$ stations. This result is a natural one, as the
expected paid search costs increase monotonically with the number of
stations.\footnote{Figure 8b demonstrates that the expected paid search costs
increase with the number of stations. The fraction of people who never search
($q_{1}$) decreases with the number of stations, reflecting that the gain from
the first search increases with the number of stations in a market. On the
other hand, the fraction of \textquotedblleft shoppers\textquotedblright\ is
$32.0\%$, $23.9\%$, $9.2\%$, and $2.0\%$ for a market with two, three, six,
and $12$ stations, respectively.}

Interestingly, we observe a non-monotonic relationship between the standard
deviation of prices and the number of stations. Table A4 and Figure 8b
illustrate that the standard deviation flattens out after a market with seven
stations, and it in fact attains its maximum at a market with $17,18,$ and
$19$ stations, and thereafter decreases. Thus one of the measures of price
dispersion, the standard deviation of prices, is not monotonic in the number
of stations, which has two implications. First, this non-monotonicity
contrasts the theoretically documented monotonic relationship between the
number of stations and the gains from search $E[p-p_{\min}]$ (e.g., Chandra
and Tappata 2011), which is often used as an alternative measure of price
dispersion. Second, a linear regression of the standard deviation of prices on
the number of firms (and controls) is unlikely to be rich enough to learn the
relationship between price dispersion and search behavior.

Turning to how the change in market structure affects consumers with different
search costs, Table 5 shows consumers in the $10$th percentile benefit from
increased competition, whereas consumers in the $75$th percentile do not, and
are sometimes worse off. For example, a $10$th percentile consumer in a market
with $12$ stations (last column) pays $\$0.015$ $($=$1.5$ cents) per gallon
less than a consumer with the same search costs in a market with five stations
(baseline). On the other hand, a $75$th percentile consumer pays $\$0.003$ per
gallon more when moving from five to $12$ stations. The expected paid search
costs increase for consumers in the $10$th percentile because the number of
searches increase from four to five, whereas the expected paid search costs do
not change for consumers in the $75$th percentile because the number of search
remains zero. Overall, when the number of gas stations increase from five to
$12$, the expected total expenditure, which is a proxy for consumer utility,
does not change noticeably at the market level. The reason is that the
decrease in price paid ($-\$0.003=\$2.570-\$2.573$) is offset by the increase
in paid search costs ($\$0.003$ $=\$0.020-\$0.017$). If we look at the
distributional consequences, however, the benefits from the increased number
of stations differ by the percentile of search cost distribution. For
instance, the expected total expenditure for a consumer in the $10$th and
$75$th percentiles decreases by $\$0.005$ from $\$2.531$ and increases by
$\$0.003$ from $\$2.613$, respectively, when the number of stations changes
from five to $12$.

Overall, we find that the market-level expected price paid decreases in the
number of firms, but consumers with high search costs may be worse off from an
increased number of firms.

\section{Conclusions}

This paper documents the heterogeneity of search costs across markets and
explores how heterogeneous search costs and market structure interact to
produce equilibrium price dispersion in the U.S. retail gasoline industry.
Based on a non-sequential search model and daily price data, we recover the
distribution of search costs for a set of geographically isolated markets. We
use the estimated search costs to conduct two counterfactual policy
experiments. First, we examine the effect of FOSD and SOSD shifts in the
search cost distribution. Second, we investigate the implications of market
structure into gasoline markets. For both experiments, we examine the effect
on the equilibrium price distribution and quantify the distributional
consequences for consumers within four different percentiles of the search
cost distribution.

The paper has three major findings. First, the distribution of search costs
varies substantially across markets and some market and population
characteristics, such as household income, are able to explain some of this
variation. Second, we confirm that the search cost distribution needs to be
sufficiently heterogeneous to generate equilibrium price dispersion. Finally,
although the market-level expected price paid decreases in the number of
firms, we find that consumers with high search costs may be worse off with an
increased number of gas stations.

This study includes some simplifying assumptions, and relaxing these provide a
future research agenda. First, studying the effect of entry on pricing assumes
the change in market structure is exogenous. In the long run, however, market
structure is determined through endogenous entry and exit, which is driven by
the profitability of a market. Our work serves as a first step toward
understanding the short-run effect of market structure on pricing. Second, the
model assumes that the quantity of gasoline purchased is inelastic to the
change in prices. While in the short-run this may be true, in the long run
people may consume less gasoline or substitute from driving to other
transportation methods when the retail gasoline price increases. Third, we
assumed a lognormal search cost distribution. Although this assumption allows
us to explore hypothetical policy counterfactuals, the quantitative results in
the simulation are subject to this approximation. Finally, we abstract from
multi-station pricing under joint ownership due to data limitation. This
assumption, however, is inadequate to consider the effect of mergers for the
retail gasoline markets. Allowing for multi-station ownership is an important
topic of future research.%

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Following Moraga-Gonz\'{a}lez and Wildenbeest (2008), applying the implicit
function theorem to equation (\ref{eq_condition}) yields the density for
equilibrium price as
\[
\text{ }f_{p}(p)=\frac{%
%TCIMACRO{\tsum \limits_{i=1}^{N}}%
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{\textstyle\sum\limits_{i=1}^{N}}
%EndExpansion
iq_{i}(1-F_{p}(p))^{i-1}}{(p-r)%
%TCIMACRO{\tsum \limits_{i=1}^{N}}%
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{\textstyle\sum\limits_{i=1}^{N}}
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i(i-1)q_{i}(1-F_{p}(p))^{i-2}}\text{ ,}%
\]
and $F_{p}(p_{l})$ solves $(p_{l}-r)[%
%TCIMACRO{\tsum \limits_{i=1}^{N}}%
%BeginExpansion
{\textstyle\sum\limits_{i=1}^{N}}
%EndExpansion
\frac{iq_{i}}{N}(1-F_{p}(p_{l}))^{i-1}]=\frac{q_{1}(\bar{p}-r)}{N}$ for all
$l=2,..,M-1.$%

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\bibliographystyle{econometrica}
\bibliography{acompat,gasoline}


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\end{document}