# Price dispersion across online platforms: Evidence from hotel room prices in
London (UK)††thanks: The authors are grateful to Frank Verboven, Jan De
Loecker, Jo Van Biesebroeck, Luis Vasconcelos, for invaluable help, advice,
and encouragement. We sincerely thank the editor and the two anonymous
referees for their detailed comments which led to significant improvements in
the content and exposition of the paper. We also sincerely thank the seminar
participants at KU Leuven and at the University of Essex for their valuable
comments and helpful suggestions. Lastly we thank Sunhyung Lee for helpful
discussions.

Debashrita Mohapatra Department of Agriculture and Resource Economics,
University of Connecticut, Storrs, CT, USA E-mail:
<EMAIL_ADDRESS>Debi Prasad Mohapatra Department of Resource
Economics, University of Massachusetts Amherst, Amherst, MA, USA E-mail:
<EMAIL_ADDRESS>Ram Sewak Dubey Economics Department, Montclair State
University, Montclair, NJ, USA 07043; E-mail<EMAIL_ADDRESS>(Corresponding Author)

###### Abstract

This paper studies the widespread price dispersion of homogeneous products
across different online platforms, even when consumers can easily access price
information from comparison websites. We collect data for the 200 most popular
hotels in London (UK) and document that prices vary widely across booking
sites while making reservations for a hotel room. Additionally, we find that
prices listed across different platforms tend to converge as the booking date
gets closer to the date of stay. However, the price dispersion persists until
the date of stay, implying that the “law of one price” does not hold. We
present a simple theoretical model to explain this and show that in the
presence of aggregate demand uncertainty and capacity constraints, price
dispersion could exist even when products are homogeneous, consumers are
homogeneous, all agents have perfect information about the market structure,
and consumers face no search costs to acquire information about the products.
Our theoretical intuition and robust empirical evidence provide additional
insights into price dispersion across online platforms in different
institutional settings. Our study complements the existing literature that
relies on consumer search costs to explain the price dispersion phenomenon.

Key Words: Price Dispersion, Online Platforms, Law of one price, Hotel Booking
JEL Codes: D40, D81, L83, L86.

##  1 Introduction

Price dispersion is a widely observed phenomenon in offline and online
markets, where sellers offer different prices for similar products.111See Baye
et al. (2006a), Baye et al. (2006b), Escobari (2012), Brynjolfsson and Smith
(2000) for some of the evidences available in recent literature. Traditional
literature has explained the existence of price dispersion for homogeneous
products through imperfect information among consumers and sellers (Salop and
Stiglitz (1982), Burdett and Judd (1983)). To quote Sorensen (2000), “
_Generally speaking, price dispersion will arise when there is a positive (but
uncertain) probability that a randomly chosen customer knows only one price.
Thus, even in markets with symmetric firms selling homogeneous products,
prices may differ in equilibrium if consumers must incur search costs to
obtain price information._ ”

However, with the growth of technology and easy accessibility to internet
services, gathering information about prices and product characteristics has
become increasingly more accessible. In the new age of the data revolution,
the marginal costs of acquiring and transmitting information have consistently
fallen. With a single click, consumers are now able to obtain information
about prices charged by different sellers for products that range from
computer hardware and software (_Shopper.com_) to mortgages
(_Mortagequotes.com_) and airlines (_farecompare.com_ , _kayak.com_). In the
hotel industry, a consumer can access multiple platforms while booking a room.
Different booking websites (e.g., Expedia.com, Booking.com, Hotels.com, etc.)
post prices and availability for a given type of room at a given hotel for a
specific date of stay. However, a consumer can also use a price comparison
website (such as _Skyscanner.com_ , _Trivago.com_), where she can check all
the prices (and other room-related details) offered by different booking
websites, compare them and book the hotel room using the platform that
provides the lowest price. Effectively, the price comparison website works as
an information exchange as it collates information from different booking
websites and displays the combined information on prices (and other details
for a hotel room) posted by different platforms as its search results.
However, even though price comparison websites have been steadily increasing
in popularity among consumers (Bodur, Klein, and Arora (2015)), it is puzzling
that price dispersion remains persistent across different markets selling
homogeneous products.

In this paper, we focus on the hotel industry and investigate the dispersion
of prices listed by booking platforms on a comparison website while booking a
given type of room in a specific hotel for a given booking date. We provide
robust empirical evidence for two findings: first, we document the existence
of widespread price dispersion among homogeneous products in the hotel
industry. In particular, we show that for a “given date of stay for a specific
room type in a particular hotel”, prices across platforms vary widely for any
booking date. Second, we focus on the evolution of price dispersion across
booking platforms as the booking date is closer to the date of stay. We find
that although the price dispersion persists, i.e., the law of one price does
not hold, prices tend to converge (hence, dispersion goes down) as the booking
date approaches the date of stay.

We do so by collecting detailed data from a price comparison website
_Skyscanner.com_ for hotel rooms for the 200 most popular hotels in London,
United Kingdom. To capture the existence of price dispersion across different
platforms, we collect (through web scraping) prices posted for a specific date
of stay across different booking platforms for a specified room type in a
hotel. To ensure that the products are _homogeneous_ , we collect this
information for a one-night stay (specified for two guests, 1 room) on a given
date of booking. We collected the price information for seven different dates
of stay in November 2017 for all 200 popular hotels in London (UK). For each
date of stay, we collected the price information on fifteen different booking
dates. Therefore, each price information in our dataset corresponds to a _date
of stay–hotel id–room type_ combination on a given website for a specific
booking date.

To document the existence of price dispersion, we compute the coefficient of
variation of prices posted across booking sites for each _date of stay–hotel
id–room type_ combination for a given booking date.222The coefficient of
variation is given by the ratio of the standard deviation to the average price
and captures the extent of variability about the average price. Our results
show that the coefficient of variation varies widely in our sample, providing
strong evidence of price dispersion among booking websites for hotel rooms. On
average, prices vary by 9% around the mean hotel room prices. To further take
care of any potential heterogeneity that may exist between booking websites
(such as website reputation and user-friendliness of the booking site), we
resort to regression analysis where we take into account the potential price
variations due to time-invariant website characteristics, in addition to the
hotel identity, room type, date and time of booking, and date of stay. We show
that extensive price dispersion still exists across different booking websites
even after controlling for a wide range of observable covariates.

We then investigate how average price and price dispersion evolve as the date
of booking approaches the date of stay. We find that the average hotel room
price increases as we approach the date of stay. Additionally, as the booking
date gets closer to the date of stay, prices listed across different platforms
tend to converge, leading to a drop in the dispersion. Nevertheless, the price
dispersion persists until the date of stay, implying that the law of one price
does not hold.

Given the empirical evidence, we then investigate why price dispersion may
exist for homogeneous products in our institutional setting. In particular, we
ask: “in markets with symmetric firms selling homogeneous products, can prices
differ in equilibrium if consumers incur zero search costs to obtain price
information?” We present a simplified model of the market to show that in the
presence of aggregate demand uncertainty and capacity constraints, price
dispersion could exist as an equilibrium outcome of the pricing game even when
products are homogeneous, consumers are homogeneous, all agents have perfect
information about the market structure, and consumers face no search costs to
acquire information about the products. Uncertain demand is a common feature
of the hotel booking industry, especially in big cities such as London, where
the flow of customers to a specific hotel is often difficult to predict.
Similarly, online booking platforms in the hotel industry face capacity
constraints while making reservations as the hotels have only a limited number
of rooms open for reservation. Additionally, a booking platform often has
access to only a subset of rooms from a given hotel, adding to the capacity
constraint within a booking website. Therefore, by showing the existence of
price dispersion under such institutional settings but with zero search costs,
our results provide additional insights into the existence of price
dispersion. In doing so, we complement the existing literature that relies on
consumer search costs to explain the price dispersion in equilibrium.

Our study makes several contributions. First, we contribute by documenting
widespread and persistent price dispersion among homogeneous products in the
context of the hotel booking industry, where consumers can easily access price
comparison websites. We show that although the law of one price does not hold,
the prices tend to converge as we approach the date of stay. We highlight a
unique setting, collect novel and detailed data from different hotel booking
websites, and report robust evidence of price dispersion and convergence
across different booking platforms for hotels in London. Second, our paper
contributes to the growing literature on the study of price comparison
websites by highlighting the role of such platforms in gathering information
and potentially reducing consumers’ search costs while making online
purchases. Finally, we develop a simplified model to highlight the fundamental
economic forces (in particular, demand uncertainty and capacity constraints)
that may lead to price dispersion as an equilibrium outcome even when the
consumers can acquire price information at zero search cost.

The rest of the paper is organized as under. In Section 2, we provide a review
of the recent literature on price dispersion in online markets. Then we
describe the details of the dataset collected from _Skyscanner.com_ , and
document the empirical evidence of price dispersion in Section 3. Section 4
describes the theoretical model, showing the existence of price dispersion
even under zero search costs. Section 5 documents the evidence of price
convergence in the booking prices as we near the date of stay. We conclude in
Section 6.

##  2 Literature review

Our paper is related to the extensive literature on online and offline markets
that documents price dispersion for homogeneous products across sellers. The
idea of “Law of one price” has been widely challenged in several theoretical
and empirical studies in the existing literature. Starting with the seminal
work by Stigler (1961), several economists have developed various theoretical
models based on information asymmetries, search frictions, price volatility,
and demand uncertainty, among others, to explain this phenomenon and have
documented the existence of price dispersion in offline markets (Goldberg and
Verboven (2005)). With the increased adoption of online transactions, although
consumers could access broader markets at relatively low cost, patterns of
price dispersion among homogeneous products remain a widely reported
phenomenon. There are various reasons, however, to expect price dispersion to
be lower on the internet platforms, as search costs are typically lower, and
the websites offer significantly easier entry to the new sellers. With the
advent of price comparison websites that collates information from different
platforms and provides it to the consumers in one place, we expect the
dispersion of prices for homogeneous product to reduce even further.

It is pertinent to discuss empirical evidence in some of the prominent recent
papers. Bailey (1998) compared prices of 125 books, 108 music CDs, and 104
software titles in 1996 and 1997 through 52 internet and traditional outlets
and found that price dispersion in online markets is as prevalent as in
offline retailers. Also, he observed that prices on the internet are higher
than on the offline channels. These findings contradict the theoretical
predictions that expect a decline in price dispersion in a comparatively
frictionless market on the internet. Similarly, Baylis and Perloff (2002) has
documented that the pricing pattern observed in the market for Olympus C-2000Z
digital camera (data for September 24- December 19, 1999, fourteen weeks
period) and Hewlett-Packard 6300 flatbed scanner (data for October 7- December
19, 1999, eleven weeks period) could be explained via price dispersion.
Brynjolfsson and Smith (2000) compared prices for a matched set of 20 books
and 20 CDs from 41 online and offline outlets between February 1998 and May
1999. They found that internet retailer prices differ by an average of $33\%$
for books and $25\%$ for CDs. However, by weighing the prices by proxies for
market share, the dispersion is lower in internet channels than in
conventional channels, reflecting the dominance of certain heavily branded
retailers. They figure out that, even if there is lower friction in many
dimensions of internet competition, branding, awareness, and trust remain
essential sources of heterogeneity among Internet retailers. Orlov (2011)
studied the influence of the internet on price dispersion by taking empirical
evidence from the airline industry. This study shows that an increase in
internet penetration reduces the average price and leads to higher intra-firm
price dispersion. Möller and Watanabe (2016) did an empirical analysis to
study the effect of price discrimination in airline markets and showed the
positive effect of competition on price dispersion when demand uncertainty is
high, or product differentiation is low. On the contrary, the effect is
negative when preferences are sufficiently certain, and products are
heterogeneous.333See Silva et al. (2019)\- a study of price dispersion over
time and across geographical location, Chakrabarty and Kutlu (2014)\- a study
of price dispersion and inter-firm, inter-flight and frequency competition,
Xing (2010)\- a study of price dispersion for online branches of Multi-Channel
Retailers and online-only retailers for recent references, Choi et al.
(2019)\- a study of price dispersion for gasoline in 25 regions of Seoul,
Korea in response to asymmetric information between retailers and consumers.

To explain those empirical patterns, existing literature has either relied on
heterogeneity or search costs or both (Arnold et al. (2011), Sorensen (2000),
Ellison and Ellison (2009), Ellison and Wolitzky (2012), Ellison and Ellison
(2018), Honka et al. (2019)) to explain the existence of price dispersion in
equilibrium. With the wider adoption of price comparison websites among
consumers, several theoretical and empirical research has been done in the
recent past to understand the price dispersion that persists even after the
growth of the internet and the popularity of price comparison websites. The
study by Baye and Morgan (2001) focuses on the strategic incentive of a price
comparison website known as “Information gatekeeper” and finds that it
positively affects the consumers who enjoy free access to the sites. Access to
information leads to competitive pricing and reduces the degree of price
dispersion over time. A number of papers on clearing house models (for
example, Arnold et al. (2011); Arnold and Zhang (2014); Baye and Morgan
(2001); Baye et al. (2004); Chioveanu (2008); Moraga-González et al. (2021);
Rosenthal (1980); Salop and Stiglitz (1977); Varian (1980) have rationalized
price dispersion in homogeneous goods markets as an equilibrium outcome
through costly information acquisition leading to information asymmetry among
consumers. Indeed, price dispersion has persisted despite technological
advances such as the internet, extensive advertisement, and comparison sites.
Ronayne et al. (2015) has investigated the impact of introducing a price
comparison website to a market for a homogeneous good, where the sites charge
firms for sale. They predict that a single price comparison website increases
prices for all consumers, whereas multiple price comparison websites would
lead to a profitable equilibrium. The paper obtains price dispersion even in
the presence of price comparison websites endogenously in equilibrium, which
comes from the fact that some consumers have incomplete information.
Similarly, Bodur et al. (2015) reports the effect of price comparison websites
on online and offline prices.

Our study complements this literature by empirically documenting widespread
price dispersion in the hotel booking industry and provides a simple
theoretical foundation where price dispersion can exist in equilibrium in
markets with symmetric firms selling homogeneous products even when consumers
incur zero search costs to obtain price information.

##  3 Existence of Price Dispersion: empirical evidence

For our analysis, we define a product as a “date of stay–hotel id–room type”
combination. Depending on when a consumer books a hotel room for a given date
of stay, the price of a product might vary. For a specific date of stay,
different booking platforms (e.g., Expedia.com, Booking.com, Hotels.com, etc.)
post prices and availability for a given type of room in a given hotel.
Therefore, prices can differ across platforms even for the same booking date.
In our analysis, we focus on the price dispersion across platforms for a given
date of booking. To document the price dispersion of products across different
platforms, we collected detailed data from a price comparison website
_Skyscanner.com_ for hotel rooms for the 200 most popular hotels in London,
United Kingdom.444We used the web services provided by data extraction
platform _grepsr.com_ to scrape the data for this study. The price comparison
website _Skyscanner.com_ works as an information exchange. For specific dates
of stay, it collates information from different booking websites. It displays
the combined price information (and other hotel room details) posted by
different platforms as its search results. Hence, it provides the consumer
with all relevant information in one place, saves the consumer any search
costs of visiting various websites separately, and helps her to find the room
at the lowest available price.

Figure 1: Price Comparison Website

  * Notes: This figure illustrates how data is organized on price comparison websites. On the left, one can see the search criteria and what a typical search displays on the Skyscanner website. Web page 1 presents hotel results from 1 to 25, web page 2 shows hotels 26 to 50, and so forth. If one clicks on Hotel 1, a new page opens with room prices for that hotel on various websites (shown on the right panel of Figure 1).

Figure 1 illustrates how data is organized on price comparison websites. As
the left panel of the figure shows, when a user provides her travel details
(like date of stay and number of guests), the price comparison website
displays a list of hotels on different pages (also called web pages). Each web
page typically lists around 20-25 hotels and their prices. During our sample
period, a hotel search for London provided at most 14 such web pages. Clicking
on a hotel link reveals its room prices on various websites. For instance, as
the bottom left panel of figure 1 shows, web page 1 lists prices for hotels
numbered 1 to 25, and web page 2 lists prices for hotels 26 to 50. Clicking on
Hotel 1 shows its prices on different websites (right panel of figure 1).

We take several steps while collecting the data to focus on the price
dispersion of _homogeneous_ products across different platforms. First, we
only consider prices for a one-night stay in a hotel room (specified for two
guests, 1 room). The different dates of stay belong to the second week of
November, such as November 7th check-in–November 8th check-out; November 8th
check-in–November 9th check-out, and so on until November 13th check-
in–November 14th check-out. Second, we collect detailed information about the
website-hotel room-specific characteristics (whether the room is double or
single).555A hotel might offer more room types (such as a suite or a family
room) that may accommodate more than two guests. We did not collect
information for those rooms in order to keep the set of products homogeneous.
Finally, for a given date of stay, we collect price data for all hotels on a
specific date of booking at a particular time (for example, say for date-of-
stay November 7th, we collect price data on October 24th, 2017, where October
24th is the date of booking). Day 1 of data collection starts on October 24th,
2017, at 10:30 am EST. On the first day of the data collection, we searched
for all hotels to stay in London and sorted them by popularity. For our
analysis, we considered the top 200 hotels ranked by popularity by the
Skyscanner website on day 1. In other words, the 200 hotels in our sample were
ranked as the 200 most popular hotels in London as selected by Skyscanner’s
algorithm on October 24th, 2017, at 10:30 a.m. EST, the first day of our data
collection.666Skyscanner uses consumer reviews, hotel star ratings (2-star,
3-star, 4-star, and 5-star), and other details from booking and text reviews
to rank the hotels by popularity. We collected information for these 200
hotels in parallel every day at 10:30 a.m. EST until November 9, 2017.

For each date of stay and the booking date, our search listed at most 14 web
pages of results. We collected the information on the 200 hotels across the
entire set of search results. Note that a hotel won’t appear in the search
results if it is fully booked. If any of the 200 hotels do not appear on the
search results, it implies that they’re booked for that date of stay. Our
final dataset from London contains 179,234 observations. Each observation in
our dataset corresponds to a price listed on a website for a _date of
stay–hotel id–room type_ combination on a given booking date. To further
verify whether the prices posted on the price comparison website are indeed
the actual price to be paid by the consumer, we take a random sample of 1000
observations from the dataset. For each observation, we verify the price on
the payment page for each posted website for the hotel room. We have confirmed
that the prices posted on Skyscanner.com are the actual price to be paid if
the consumer clicks on the corresponding website and makes the purchase.

Table 1: Summary statistics for prices across websites for hotels in London Variables | Mean | Std.Dev. | Min | Max
---|---|---|---|---
Posted Price (in £)11footnotemark: 1 | 219 | 149 | 36 | 998
No. of websites22footnotemark: 2 | 9 | 3 | 1 | 19
Range (in £)33footnotemark: 3 | 55 | 75 | 0 | 986
Coefficient of Variation | 0.09 | 0.18 | 0 | 0.82
11footnotemark: 1 Posted Price refers to the price posted by different website
22footnotemark: 2 No. of websites refers to the number of websites posting
price for
the same hotel room in skyscanner.com
33footnotemark: 3 Range refers to the gap between the highest and lowest
prices
posted for a hotel room across websites

Table 1 documents summary statistics from our data. The average prices posted
across websites varies between £36 and £998. This suggests that our dataset
covers a wide range of hotels, both cheap, low-quality, and high-quality,
expensive hotels. We observe that consumers can access posted prices from 9
different websites on average for each hotel room. The number of websites for
posted prices varies between 1 to 19. To capture the price dispersion on a
given booking date across websites for a specific “date of stay–hotel id–room
type” combination, we compute the price range as well as the coefficient of
variation. For a specific date of stay, on a given date of booking, the price
range refers to the difference between the maximum posted price and the
minimum posted price across websites for a hotel room. Our results show on
average, the maximum price exceeds the minimum price by £55.

To compute the coefficient of variation on a given date of booking, we
determine the average price across websites and the standard deviation of
prices across websites for a “date of stay–hotel id–room type” combination.
The coefficient of variation is given by the ratio of the standard deviation
to the average price and captures the extent of variability in relation to the
average price. If there is no price variation across websites, the standard
deviation is 0, and hence the coefficient of variation is given by 0. A higher
coefficient of variation reflects higher price dispersion across websites. As
documented in table 1, the average coefficient of variation is 0.09,
suggesting that the relative variation of prices across websites is around 9%
of the average price.

Figure 2: Scatter plot of Coefficient of Variation

Note: Here we have plotted the average prices for a “date of stay–date of
booking–hotel id–room type” combination in the x-axis, and the coefficient of
variation of prices in the y-axis.

As Figure 2 shows, the coefficient of variation varies from 0 to about 0.8.
This indicates significant price differences among various websites when
looking at a particular hotel room, date of stay, and booking date. In
appendix Section B, we plot the standard deviation with respect to prices. We
observe wide variation in the standard deviation values further providing
evidence of price dispersion.

Note that, although we have considered the hotel identity, the room type, date
and time of booking, and date of stay, the differences in the websites
displaying prices may lead to heterogeneity across products and may explain
the price dispersion. To further address this concern, we run the following
regression for each “date of stay–date of booking” combination.

$\displaystyle\underset{d_{s},d_{b},h,r,w}{\text{Price}}=$
$\displaystyle\beta_{0}+\beta_{1}\left(\text{Hotel dummy}\times\text{Hotel
room type dummy}\right)$ (1) $\displaystyle+\beta_{2}(\text{Website
dummy})+\varepsilon_{d_{s},d_{b},h,r,w}$

where $d_{s}$ stands for date of stay, $d_{b}$ for date of booking, $h$ for
hotel id, $r$ for room type, and $w$ for the website. We run 106 separate
regressions for each “date of stay–date of booking” combination.777Note that,
for the date of stay Nov 7, 2017, we have data from 14 dates of booking prior
to the date of stay. Similarly, for the date of stay Nov 13, 2017, we have
data from 17 dates of booking prior to the date of stay. For all other 5
dates-of-stay, we have data from 15 days prior to the date of stay. Therefore,
we have 106 regressions in total. In each regression, we consider the prices
of a specific room in a given hotel as we control for $\left(\text{Hotel
dummy}\times\text{Hotel room type dummy}\right)$ in the regression.
Additionally, we control for website dummies to control for any differences
between the websites. For example, some websites may have a loyal consumer
base, which may be the preferred website for that set of consumers. If the
differences in websites can explain all price heterogeneity for a specific
room in a given hotel, then given the regression specification, we should be
able to explain the data perfectly. In particular, if website heterogeneity
leads to price dispersion, controlling for website dummies should result in an
$R^{2}$ close to 1 for most of the regressions.

Figure 3: Evidence of price dispersion across websites

Note: Here we have plotted the histogram of R-square for different “date of
stay–date of booking” combinations as specified in equation 1. .

In figure 3, we plot the histogram of $R^{2}$ values from 106 different
regressions. Our results show that, only in 33% of the cases, the $R^{2}$ is
above 90%. In 55% of the cases, the estimated $R^{2}$ is less than 50%
suggesting that even controlling for heterogeneity across websites, for a
given hotel room for a specific “date of stay–date of booking” combination,
there exists substantial variation between prices posted across websites.

Our empirical finding of price dispersion in the case of homogeneous products
leads to the following question: “In markets with symmetric firms selling
homogeneous products, can prices differ in equilibrium if consumers incur zero
search costs to obtain price information?”. In the Section 4, we present a
simplified model of the market to show that in the presence of _aggregate
demand uncertainty_ and _capacity constraints_ , price dispersion could exist
as an equilibrium outcome of the pricing game even when products are
homogeneous, consumers are homogeneous, all agents have perfect information
about the market structure, and consumers face _zero_ search costs to acquire
information about the products. Our model complements the existing literature
that relies on consumer search costs to explain the price dispersion in
equilibrium.

##  4 A Model to Explain Price Dispersion

Can price dispersion exist for homogeneous products when all consumers can
acquire information about the availability and prices of the products at zero
search cost? Note that, in our setting, a price comparison website effectively
acts as an information exchange; it collates all relevant information and
provides it to consumers at zero search cost. In this section, we present a
simplified model of the market to show the existence of price dispersion under
this setting. It is shown that price dispersion could exist even when products
are homogeneous, consumers are homogeneous, all agents have perfect
information about the market structure, and consumers face no search costs to
acquire information about the products.

The model specification follows the approach in Arnold and Saliba (2011).
There are two firms labeled $i=\\{1,2\\}$ producing identical goods with same
marginal cost $c>0$ and zero fixed cost. Each firm is an expected profit
maximizer. Thus firm $i$ earns a profit per unit of $P_{i}-c$ by selling the
good at the price of $P_{i}$. The production capacity of each firm is
constrained by a maximum of one unit of the good.

The consumers are assumed to have identical reservation value $v$, with $v>c$
to avoid trivial outcomes. There is aggregate demand uncertainty (common to
each firm) in this market with the following feature. There are two possible
realizations of aggregate demand - high and low. In the high demand state
(which occurs with probability $0<\alpha<1$, there are two active consumers in
the market, whereas, in the low demand state (with probability $1-\alpha$),
there is one active consumer. Both the probability of occurrence of each state
and the demand quantity is common knowledge between the two firms. The firm’s
decision problem is to set the price $P_{i}$, which they are required to
complete simultaneously and non-cooperatively before the realization of the
high/low demand state.888In this model, the only factors contributing to price
dispersion are capacity constraint and demand uncertainty. In this model
specification, it is straightforward to prove that the equilibrium price set
by each firm $P_{i}$ will be restricted to the interval $[c,v]$; the details
are left for the reader as an easy exercise. The proposition 1 below shows
that there exists no symmetric equilibrium in pure strategy.

######  Proposition 1.

There is no symmetric pure strategy Nash equilibrium in the market described
above.

###### Proof.

Please see the proof of Proposition 1 in appendix section A. ∎

Proposition 1 establishes the existence of price dispersion in equilibrium
under the presence of demand uncertainty and capacity constraints. Next, we
will show that, relaxing either the assumption of demand uncertainty or the
assumption of capacity constraint would support a pure strategy symmetric Nash
equilibrium.

######  Proposition 2.

Under demand uncertainty and no capacity constraint, if consumers are
homogeneous, there exists a symmetric pure strategy Nash equilibrium.

###### Proof.

Please see the proof of Proposition 2 in appendix section A. ∎

In the next proposition, we show that symmetric equilibria exist (one for each
state) when the firms are able to learn the state of the market demand (high
or low) but continue to face capacity constraint.

######  Proposition 3.

Under no demand uncertainty, but with capacity constraint, there exists a
symmetric pure strategy Nash equilibrium.

###### Proof.

Please see the proof of Proposition 3 in appendix section A. ∎

Propositions 2 and 3 demonstrate that relaxing either demand uncertainty or
capacity constraint would support a pure strategy Nash equilibrium in the
pricing game where both firms charge identical prices leading to no price
dispersion. In proposition 4, we characterize a

######  Proposition 4.

There exists a symmetric mixed strategy Nash equilibrium in the market
described above.

###### Proof.

Please see the proof of Proposition 4 and characterization of the symmetric
mixed strategy Nash Equilibrium in the appendix section A. ∎

Our characterization of the symmetric mixed strategy Nash Equilibrium shows
that each seller will charge a price in the interval specified below under the
cdf as described here, i.e.,

$P_{i}\in[\alpha
v+(1-\alpha)c,v],\;\text{and}\;F(P_{i})=\frac{(P_{i}-c)-\alpha(v-c)}{(P_{i}-c)(1-\alpha)},\;i=1,2,$
(2)

is the symmetric mixed strategy Nash equilibrium of this game. Note that, as
uncertainty is resolved where the high demand state occurs with probability
one, i.e., $\alpha\rightarrow 1$, $P_{i}\rightarrow v$.

##  5 Evolution of Price Dispersion: empirical evidence

This section investigates how average price and price dispersion evolve as the
date of booking approaches the date of stay.

Figure 4: Mean Price up to the date of booking

To track how the average price evolves as the booking date gets closer to the
date of stay, we calculated the average price across all hotels, room types,
websites, and stay dates based on a specific number of days before the stay
date. This average price was then plotted against the number of days before
the stay date in Figure 4. Note that we gathered hotel price data for various
stay dates and room types. For each date and room type, we collected prices
from different websites for the 15 days leading up to the stay date. As we
collected data for the 15 days preceding the stay date, there are 15 average
values displayed on the graph.

As figure 4 indicates, the average hotel room price increases as we approach
the check-in date (i.e., the x-axis values are closer to zero). This may
result from hotels charging higher prices as they approach the capacity
constraint. It may also indicate that price-insensitive business consumers
book hotel rooms close to the booking date, and hence hotels may charge a
higher price from those consumers effectively raising the average price. Next,
we investigate if this implies higher (or lower) price dispersion while
approaching the check-in date.

We conduct a set of regression analyses to provide evidence of price
convergence across websites as we approach the date of stay. First, we run
one-day ahead regression up to $14$ days before the date of stay and plot the
coefficients and corresponding confidence intervals in figure 5. The $k^{th}$
dot in the diagram refers to the coefficient of $CV_{t-k}$ in a regression
where the dependent variable is $CV_{t-k+1}$ and the independent variable is
$CV_{t-k}$. For example, the first dot in figure 5 plots the coefficient where
we regress the CV of prices for a given hotel on date $t$ on the CV of prices
for the same hotel on date $(t-1)$. The second dot corresponds to the
coefficient of $CV_{t-2}$ when the dependent variable is $CV_{t-1}$.

Figure 5: Evolution of Price Dispersion (CV) up to the date of booking

Our results from figure 5 suggests that all of the coefficient estimates are
less than $1$ and almost all the confidence intervals of the coefficient
estimates exclude $1$ with $95\%$ probability. This suggests that the CV on
any day is less than the corresponding CV on the previous day as the
coefficient is always less than $1$. Additionally, the figure suggests that
the coefficient estimates for the last 6 days decline at an increasing rate.
For example, moving from day $(t-6)$ to $(t-5)$, CV decreases by around $10\%$
where as moving from day $(t-5)$ to $(t-4)$, it decreases by more than $20\%$.
Our supporting model shows that price dispersion exists in the equilibrium
under demand uncertainty and capacity constraint. In addition to this, current
literature suggests that consumer search costs may also contribute to price
dispersion in equilibrium. Our results are consistent with both of these
explanations. Our estimates show that in consistency with search theory, price
dispersion persists over time, never converges to zero, and hence the law of
one price does not hold (similar to Baye and Morgan (2001) and Escobari
(2012)).999Note that if the law of one price holds on the date of stay, then
$CV_{t}$ should be close to zero. This implies that the regression coefficient
of $CV_{t}$ on $CV_{t-1}$ would be very close to zero. As figure 5 shows, the
confidence interval for the regression coefficient of $CV_{t}$ on $CV_{t-1}$
is bounded away from zero and is close to 1, suggesting that price dispersion
persists. However, the decrease in the price dispersion over time is
consistent with the hypothesis that since websites learn about the demand
characteristics, the demand uncertainty gets resolved. This leads to a lower
price dispersion as we get closer to the date of stay.

To provide further evidence, we regress the coefficient of variation (CV) at
date $t$ for a given hotel on the corresponding CV for the same hotel from
date $t-1$ while controlling for a set of hotel characteristics and several
fixed effects. The regression specification is given by:

$CV_{t}=\alpha CV_{t-1}+\beta_{w}X_{w}+\beta_{h}X_{h}+\epsilon$ (3)

Our key parameter of interest is $\alpha$ which captures the evolution of CV
after one day lag. $X_{w}$ denotes the website-specific characteristics, and
$X_{h}$ denotes the hotel-specific characteristics. The controls include the
number of websites that list the hotel price, the webpage number on which the
hotel is displayed on Skyscanner, a dummy for the last three days before the
date of stay, number of reviews posted on websites for a hotel, hotel star
rating (0 stars to 5 stars), and ratings by consumers (varies continuously
from 1 to 10). We include fixed effects for hotels, date of stay, hotel
quality, and “hotel-date of stay-room type” combinations in different
specifications. Based on the ratings on the Skyscanner website, the hotels are
grouped into different categories: Excellent, Very Good, Good, Average, and
Satisfactory. The quality fixed effects are based on this categorical
variable. We report the results in table 2. We present estimation results from
seven different specifications.

Table 2: Regression results - Coefficient of Variation

| (1) | (2) | (3) | (4) | (5) | (6) | (7)
---|---|---|---|---|---|---|---
Dependent variable: CV | Model1 | Model2 | Model3 | Model4 | Model5 | Model6 | Model7
Previous day CV | 0.767*** | 0.758*** | 0.758*** | 0.757*** | 0.757*** | 0.507*** | 0.399***
| (0.00597) | (0.00598) | (0.00599) | (0.00600) | (0.00601) | (0.00845) | (0.00925)
Number of Websites |  | 0.0294*** | 0.0296*** | 0.0301*** | 0.0306*** | 0.0619*** | 0.0672***
|  | (0.00249) | (0.00257) | (0.00259) | (0.00261) | (0.00378) | (0.00406)
Page No in Skyscanner |  | -0.000851*** | -0.000809*** | -0.000764*** | -0.000798*** | -0.00239*** | -0.00356***
|  | (0.000249) | (0.000255) | (0.000256) | (0.000257) | (0.000410) | (0.000604)
Dummy for last 3 days |  | 0.00475** | 0.00470** | 0.00475** | 0.00489** | 0.00497** | 0.00494**
|  | (0.00201) | (0.00201) | (0.00201) | (0.00202) | (0.00197) | (0.00202)
No of Hotel Reviews |  |  | -0.000307 | -0.000112 | -0.000161 | 0.00499 |
|  |  | (0.00103) | (0.00105) | (0.00106) | (0.00940) |
Hotel star rating |  |  | -0.0113 | -0.0156 | -0.0151 | 0.0222 |
|  |  | (0.0108) | (0.0111) | (0.0111) | (0.358) |
Hotel Review Rating |  |  | -0.00490 | -0.0116 | -0.00889 | -0.0158 |
|  |  | (0.0129) | (0.0329) | (0.0330) | (0.213) |
Constant | 0.0171*** | -0.0305*** | -6.53e-05 | 0.00301 | -0.00434 | -0.127 | -0.0911
| (0.00102) | (0.00536) | (0.0275) | (0.0595) | (0.0601) | (0.592) | (0.151)
Observations | 10,829 | 10,829 | 10,829 | 10,829 | 10,829 | 10,829 | 10,829
R-squared | 0.604 | 0.610 | 0.610 | 0.610 | 0.610 | 0.666 | 0.700
Hotel FE | No | No | No | No | No | Yes | No
Date of Stay FE | No | No | No | No | Yes | Yes | No
Quality FE | No | No | No | Yes | Yes | Yes | No
Hotel x Date Of Stay x Room Type FE | No | No | No | No | No | No | Yes
Standard errors in parentheses |
*** p$<$0.01, ** p$<$0.05, * p$<$0.1 |

Our results show that the estimated $\alpha$ is statistically significant
irrespective of the model specification and lies between 0.40 and
0.77.101010Note that in model 7, we include hotel id-date of stay-room type
fixed effects. Hence, we can not include controls such as the number of hotel
reviews, hotel star rating, and hotel review rating, as those variables are
absorbed by these fixed effects. Estimated $\alpha$’s value is less than 1
suggests that compared to the previous day, on average, CV goes down the next
day as we approach the date of stay. The coefficient of the number of websites
is positive and significant, in line with the hypothesis that a higher number
of websites on which prices are posted leads to higher price dispersion. The
coefficient of the Page number in Skyscanner on which a hotel is displayed is
negative and significant. Hotels are sorted in terms of popularity, and a less
popular hotel is displayed on a later page in our sample. Our estimates
suggest that a less popular hotel faces lower demand and hence may have less
price dispersion. The coefficient of the dummy for the last three days before
the date of stay is positive and significant. This is consistent with the fact
that business consumers and last-minute consumers show up in the last 3 days
and are less price sensitive. Hence, price dispersion may increase due to the
presence of such consumers. The coefficient estimates the number of hotel
reviews, hotel star ratings, and ratings by consumers are not significantly
different from 0. Given that CV is a standardized value and we do not use the
variation across hotels, our results suggest that these hotel-specific
characteristics may not significantly affect the evolution of CV. To provide
robustness checks, we repeat the exercise for the logarithm of the coefficient
of variation and price range. Further note that if prices are posted just on
one website, then CV is equal to zero by definition. To make sure that such
observations do not drive our results, we exclude those observations with a
single website and run the regressions with the smaller sample. We report the
results in the Appendix Section C.

To summarize, our regression results highlight two consistent facts. First,
price dispersion (measured in different ways) converges over time as the
booking date approaches the date of stay. This is consistent with websites
learning about uncertain market demand over time. Second, however, the law of
one price does not hold, and price dispersion persists.

##  6 Conclusion

In this paper, we study the existence and persistence of price dispersion for
homogeneous products across different online platforms in spite of the wide
adoption of price comparison websites. We collect data from the 200 most
popular hotels in London (UK) and document that the price to book a hotel room
varies across booking websites. Our empirical evidence suggests that prices
posted across websites may vary widely, even for a given “date of stay–date of
booking–hotel id–room type” combination. Our regression analysis shows that
such price dispersion patterns are robust to controlling for heterogeneity
across the booking websites. We then investigate how the dispersion of prices
evolves as the date of booking approaches the date of stay. We find that as
the booking date gets closer to the date of stay, prices listed across
different platforms converge, leading to a drop in the dispersion.
Nevertheless, the price dispersion persists until the date of stay, implying
that the law of one price does not hold.

We provide a simplified theoretical model of why price dispersion may exist
for homogeneous products when all consumers can access an information exchange
(such as a price comparison website) and acquire information about the
availability and prices of the products at zero search cost. In contrast to
the traditional literature that relies on consumer search costs to show the
existence of price dispersion, we provide a model that shows that in the
presence of aggregate demand uncertainty and capacity constraints, price
dispersion could exist even when products are homogeneous, consumers are
homogeneous, all agents have perfect information about the market structure,
and consumers face no search costs to acquire information about the products.

Our empirical findings show that in consistence with search theory, price
dispersion persists over time, never converges to zero, and hence the law of
one price does not hold. However, the decrease in the price dispersion over
time is consistent with the hypothesis that since websites learn about the
demand characteristics, the demand uncertainty gets resolved. This leads to a
lower price dispersion as we get closer to the date of stay. Therefore, in the
context of the hotel industry, our results complement the existing literature
on search theory and provide additional insights into the existence and
persistence of price dispersion phenomenon.

## References

  * (1)
  * Arnold and Saliba (2011) Arnold, M. A., and C. Saliba. 2011\. “Asymmetric capacity constraints and equilibrium price dispersion.” Economics Letters 111 (2): 158–160.
  * Arnold and Zhang (2014) Arnold, M. A., and L. Zhang. 2014\. “The unique equilibrium in a model of sales with costly advertising.” Economics Letters 124 (3): 457–460.
  * Arnold et al. (2011) Arnold, M., C. Li, C. Saliba, and L. Zhang. 2011\. “Asymmetric market shares, advertising and pricing: Equilibrium with an information gatekeeper.” The Journal of Industrial Economics 59 (1): 63–84.
  * Bailey (1998) Bailey, J. P. 1998\. Intermediation and electronic markets: Aggregation and pricing in Internet commerce. Ph.D. dissertation, Massachusetts Institute of Technology.
  * Baye and Morgan (2001) Baye, M. R., and J. Morgan. 2001\. “Information gatekeepers on the internet and the competitiveness of homogeneous product markets.” American Economic Review 91 (3): 454–474.
  * Baye et al. (2004) Baye, M. R., J. Morgan, and P. Scholten. 2004\. “Price dispersion in the small and in the large: Evidence from an internet price comparison site.” The Journal of Industrial Economics 52 (4): 463–496.
  * Baye et al. (2006a) Baye, M. R., J. Morgan, and P. Scholten. 2006a. “Information, search, and price dispersion.” Handbook on Economics and Information Systems 1 323–375.
  * Baye et al. (2006b) Baye, M. R., J. Morgan, P. Scholten, and D. Jansen. 2006b. “Persistent price dispersion in online markets.” The New Economy & Beyond: Past Present and Future, Edward Elgar, S 122–143.
  * Baylis and Perloff (2002) Baylis, K., and J. M. Perloff. 2002\. “Price Dispersion on the Internet: Good Firms and Bad Firms.” Review of Industrial Organization 21 (3): 305–324.
  * Bodur et al. (2015) Bodur, H. O., N. M. Klein, and N. Arora. 2015\. “Online price search: Impact of price comparison sites on offline price evaluations.” Journal of Retailing 91 (1): 125–139.
  * Brynjolfsson and Smith (2000) Brynjolfsson, E., and M. D. Smith. 2000\. “Frictionless commerce? A comparison of Internet and conventional retailers.” Management Science 46 (4): 563–585.
  * Burdett and Judd (1983) Burdett, K., and K. L. Judd. 1983\. “Equilibrium price dispersion.” Econometrica 51 (4): 955–969.
  * Chakrabarty and Kutlu (2014) Chakrabarty, Durba, and Levent Kutlu. 2014\. “Competition and price dispersion in the airline markets.” Applied Economics 46 (28): 3421–3436.
  * Chioveanu (2008) Chioveanu, I. 2008\. “Advertising, brand loyalty and pricing.” Games and Economic Behavior 64 (1): 68–80.
  * Choi et al. (2019) Choi, Bongseok, Donghun Kim, and Hong Chong Cho. 2019\. “Price response, information, and asymmetry of price dispersion.” Applied Economics 51 (39): 4270–4281.
  * Ellison and Ellison (2009) Ellison, G., and S. F. Ellison. 2009\. “Search, obfuscation, and price elasticities on the internet.” Econometrica 77 (2): 427–452.
  * Ellison and Ellison (2018) Ellison, G., and S. F. Ellison. 2018\. “Search and obfuscation in a technologically changing retail environment: Some thoughts on implications and policy.” Innovation Policy and the Economy 18 (1): 1–25.
  * Ellison and Wolitzky (2012) Ellison, G., and A. Wolitzky. 2012\. “A search cost model of obfuscation.” The RAND Journal of Economics 43 (3): 417–441.
  * Escobari (2012) Escobari, D. 2012\. “Dynamic pricing, advance sales and aggregate demand learning in airlines.” The Journal of Industrial Economics 60 (4): 697–724.
  * Goldberg and Verboven (2005) Goldberg, P. K., and F. Verboven. 2005\. “Market integration and convergence to the Law of One Price: evidence from the European car market.” Journal of International Economics 65 (1): 49–73.
  * Honka et al. (2019) Honka, Elisabeth, Ali Hortaçsu, and Matthijs Wildenbeest. 2019\. “Empirical search and consideration sets.” In Handbook of the Economics of Marketing, Volume 1. 193–257, Elsevier.
  * Möller and Watanabe (2016) Möller, M., and M. Watanabe. 2016\. “Competition in the presence of individual demand uncertainty.” The RAND Journal of Economics 47 (2): 273–292.
  * Moraga-González et al. (2021) Moraga-González, José L., Z. Sándor, and M. R. Wildenbeest. 2021\. “Consumer search and prices in the automobile market.”
  * Orlov (2011) Orlov, E. 2011\. “How does the internet influence price dispersion? Evidence from the airline industry.” The Journal of Industrial Economics 59 (1): 21–37.
  * Ronayne et al. (2015) Ronayne, D. et al. 2015\. Price comparison websites. University of Warwick.
  * Rosenthal (1980) Rosenthal, R. W. 1980\. “A model in which an increase in the number of sellers leads to a higher price.” Econometrica 48 (6): 1575–1579.
  * Salop and Stiglitz (1977) Salop, S., and J. Stiglitz. 1977\. “Bargains and ripoffs: A model of monopolistically competitive price dispersion.” The Review of Economic Studies 44 (3): 493–510.
  * Salop and Stiglitz (1982) Salop, S., and J. E. Stiglitz. 1982\. “The theory of sales: A simple model of equilibrium price dispersion with identical agents.” The American Economic Review 72 (5): 1121–1130.
  * Silva et al. (2019) Silva, Dakshina G. De, Inkoo Lee, and Soon-Cheul Lee. 2019\. “Explaining time variation in geographic price dispersion.” Applied Economics 51 (42): 4629–4641.
  * Sorensen (2000) Sorensen, A. T. 2000\. “Equilibrium price dispersion in retail markets for prescription drugs.” Journal of Political Economy 108 (4): 833–850.
  * Stigler (1961) Stigler, George J. 1961\. “The economics of information.” Journal of Political Economy 69 (3): 213–225.
  * Varian (1980) Varian, H. R. 1980\. “A model of sales.” The American Economic Review 70 (4): 651–659.
  * Xing (2010) Xing, Xiaolin. 2010\. “Can price dispersion be persistent in the Internet markets?” Applied Economics 42 (15): 1927–1940.

Appendix: For On-line Publication Only

##  Appendix A Proofs

######  Proposition 1.

There is no symmetric pure strategy Nash equilibrium in the market described
above.

###### Proof.

Observe that the expected profit of firm $1$, conditional on firm 2 setting a
price $P_{2}$, when it sets its price equal to $P_{1}$, is given by,

$\mathbbm{E}\pi_{1}=(P_{1}-c)\Pr(q_{1}(P_{1},P_{2})),$ (4)

where $q_{1}(P_{1},P_{2})$ is the quantity sold by firm $1$ when the price
vector set by the two firms is $(P1,P_{2})$. We need to consider two cases,
(i) $P_{1}=P_{2}=P\in(c,v]$, and (ii) $P_{1}=P_{2}=P=c$.

  1. (i)

$P_{1}=P_{2}=P\in(c,v]$: We show there exists a profitable deviation. The
expected payoff for firm 2 is

$\mathbbm{E}\pi_{2}=\alpha(P-c)+(1-\alpha)\left[t(P-c)+(1-t)(0)\right]=[\alpha+t(1-\alpha)](P-c),$
(5)

where we have assumed that in the low demand state firm 2 is able to sell one
unit of good with probability $t\in(0,1)$.111111In the high demand state since
the aggregate quantity demanded is $q=2$, each firm is able to sell one unit
of good for sure due to capacity constraint. On the other hand, in the low
demand state since the aggregate quantity demanded is $q=1$, firm 1 is able to
sell one unit of good with a probability $t\in(0,1)$ and the firm 2 sells one
unit with probability $1-t$. We next consider firm 2 deviating from the
symmetric price $P$ to $P-\varepsilon\in(c,v)$, $\varepsilon>0$. The expected
profit at new price is

$\mathbbm{E}\pi_{2}^{\prime}=\alpha(P-c-\varepsilon)+(1-\alpha)(P-c-\varepsilon)=P-c-\varepsilon.$
(6)

Then,

$\mathbbm{E}\pi_{2}^{\prime}-\mathbbm{E}\pi_{2}=(P-c-\varepsilon)-[\alpha+t(1-\alpha)](P-c)=(1-\alpha)(1-t)(P-c)-\varepsilon>0$
(7)

for sufficiently small deviation (take for example
$0<\varepsilon<(1-\alpha)(1-t)(P-c)$). Due to symmetric firm assumption, the
argument goes analogously for Firm 1 where Firm 2 sets price at $P$.

  2. (ii)

$P_{1}=P_{2}=c$: Note that the expected profit for Firm $2$ at $P=c$ is
$\mathbbm{E}\pi_{2}=0$. If Firm $2$ sets a price $c+\varepsilon\in(c,v)$ with
$\varepsilon>0$, then the expected profit is

$\mathbbm{E}\pi_{2}^{\prime}=\alpha(c+\varepsilon-c)+(1-\alpha)0=\alpha\varepsilon>0$
(8)

Note that Firm 2 sells one unit of the good in high demand state due to the
binding capacity constraint of Firm 1. Hence charging a price ($c+\epsilon$)
is a profitable deviation from setting price at $c$.

Due to symmetric firm assumption, the argument goes analogously for Firm 1
where Firm 2 sets price at $P$. ∎

######  Proposition 2.

Under demand uncertainty and no capacity constraint, if consumers are
homogeneous, there exists a symmetric pure strategy Nash equilibrium.

###### Proof.

In the absence of any capacity constraint, each firm can can cater to the
aggregate demand in both high and low demand states.

We rule out $P_{1}=P_{2}=P\in(c,v]$ as a symmetric equilibrium outcome. For
contradiction, we consider both firm charge price $P\in(c,v]$ and compute the
profit earned by Firm 2. In the high demand state, Firm 2 can sell two units
with probability $0<r<1$ and one unit with probability $0<s<1-r$. In the low
demand state, Firm 2 can sell one unit with probability $0<t<1$. If the firm
deviates to a price $P-\varepsilon\in(c,v)$, it sells two units in the high
demand state and one unit in the low demand state. At price $P$, the profit is

$\mathbbm{E}\pi_{2}=[\alpha(2r+s)+(1-\alpha)t](P-c).$ (9)

At price $P-\varepsilon$, the profit is

$\mathbbm{E}\pi^{\prime}_{2}=[\alpha(2)+(1-\alpha)1](P-c-\varepsilon)=(1+\alpha)(P-c-\varepsilon).$
(10)

There exists a profitable deviation if
$\mathbbm{E}\pi^{\prime}_{2}>\mathbbm{E}\pi_{2}$ for some $\varepsilon>0$,
i.e.,

$\mathbbm{E}\pi^{\prime}_{2}-\mathbbm{E}\pi_{2}=(1+\alpha)(P-c-\varepsilon)-[\alpha(2r+s)+(1-\alpha)t](P-c)>0.$
(11)

Simplifying, we get

$[1+\alpha-\alpha(2r+s)-(1-\alpha)t](P-c)>\varepsilon(1+\alpha).$ (12)

Or

$\left[(1-t)(1-\alpha r)+\alpha(1-r-s)+\alpha
t(1-r)\right](P-c)>\varepsilon(1+\alpha).$ (13)

Each term in the parenthesis on the left hand side is positive. Thus a
profitable deviation $\varepsilon>0$ always exists.

Next, we claim that $P_{1}=P_{2}=c$ is a pure strategy symmetric equilibrium.
Since firms offer homogeneous products, at any demand state, if firm $2$ sets
price $P=c$, it earns zero profit which would continue to be the level of
profit if deviates to a price $P_{i}>c$.121212Firm 2 ends up with zero sales
as Firm 1 does not face any capacity constraint and would be able to sell two
units in the high demand state and will sell one unit in low demand state for
sure due to its lower price. Thus there is no profitable deviation and both
firms charging same price equal to $c$ is the _unique_ symmetric pure strategy
Nash equilibrium. ∎

######  Proposition 3.

Under no demand uncertainty, but with capacity constraint, there exists a
symmetric pure strategy Nash equilibrium.

###### Proof.

In this setting, since we retain the assumption of capacity constraint, each
firms has one homogeneous product to sell. Note that, there is no demand
uncertainty, therefore, the firms are aware of the demand state while setting
the prices. We claim that $P_{1}=P_{2}=v$ is the _unique_ pure strategy
symmetric equilibrium when demand is high, and $P_{1}=P_{2}=c$ is the _unique_
pure strategy symmetric equilibrium when demand is low.

When aggregate demand is high, given that, firm $1$ sets price at $v$, firm
$2$’s payoff from setting a price $P_{2}\in[c,v]$ is given by

$(P_{2}-c)\text{Pr}(q_{2}(v,P_{2})).$ (14)

Since $\text{Pr}(\text{sale}(v,P_{2}))$ is equal to 1 under high state for any
$P_{2}\in[c,v]$, setting $P_{2}=v$ maximizes the payoff of firm 2. Hence
{$v,v$} is a pure strategy symmetric equilibrium.

At low demand state, when there are two firms and one consumer, given that
firm $1$ sets price at $c$, firm $2$ sets the same price at equilibrium giving
$0$ expected payoff to the firm. There is no profitable deviation for firm $2$
to charge more than $c$ which will lead to zero sales with probability one.
Hence $P_{1}=P_{c}=c$ is a pure strategy symmetric equilibrium in this case,
which is the equilirbium outcome in the usual Bertrand price competition
market game. ∎

######  Proposition 4.

There exists a symmetric mixed strategy Nash equilibrium in the market
described above.

###### Proof.

For any symmetric mixed strategy equilibrium in this game, the price support
$[\underline{P},\overline{P}]\subset[c,v]$ and the cdf $F(P_{i})$ is a
continuous function (i.e., it is atomless) for each of the two firms. Note
that for price $P_{1}$ of firm $1$, $P_{2}>P_{1}$ with probability
$1-F(P_{1})$. Thus the profit for the firm 1 is

$\pi_{1}(P_{1})=(P_{1}-c)\left(1-F(P_{1})\right)\left[\alpha(1)+(1-\alpha)(1)\right]+(P_{1}-c)F(P_{1})\left[\alpha(1)+(1-\alpha)(0)\right],$
(15)

which is simplified to

$\pi_{1}(P_{1})=(P_{1}-c)\left(1-F(P_{1})\right)+\alpha(P_{1}-c)F(P_{1}).$
(16)

Since $F(v)=1$, we can deteremine the profit for firm 1 at the mixed strategy
equilibrium as

$\pi^{\ast}_{1}=\pi_{1}(v)=(v-c)\left(1-F(v)\right)+\alpha(v-c)F(v)=(v-c)\left(1-1\right)+\alpha(v-c)(1)=\alpha(v-c).$
(17)

Observe that at every mixed strategy $P_{1}$, the profit of firm $1$, is given
by,

$\pi_{1}(P_{1})=(P_{1}-c)\left(1-F(P_{1})\right)+\alpha(P_{1}-c)F(P_{1})=\alpha(v-c).$
(18)

This yields the cdf as

$F(P_{1})=\frac{(P_{1}-c)-\alpha(v-c)}{(P_{1}-c)(1-\alpha)}.$ (19)

Given that the equilibrium profit for firm 1 is $\alpha(v-c)$, arbitrarily low
prices could not be chosen. In particular, $P_{1}\not<\underline{P}$ with
$\underline{P}>c$ where $\underline{P}$ is such that $F(\underline{P})=0$.

$F(\underline{P})=\frac{(\underline{P}-c)-\alpha(v-c)}{(\underline{P}-c)(1-\alpha)}=0.$
(20)

In other words,

$\underline{P}=c+\alpha(v-c)=\alpha v+(1-\alpha)c.$ (21)

and

$\overline{P}=v.$ (22)

Therefore,

$P_{i}\in[\alpha
v+(1-\alpha)c,v],\;\text{and}\;F(P_{i})=\frac{(P_{i}-c)-\alpha(v-c)}{(P_{i}-c)(1-\alpha)},\;i=1,2,$
(23)

is the symmetric mixed strategy Nash equilibrium of this game. ∎

##  Appendix B Scatter Plot of Standard Deviation

Figure 6: Scatter plot of Standard Deviation

Note: Here we have plotted the average prices for a “date of stay–date of
booking–hotel id–room type” combination in the x-axis, and the standard
deviation of prices in the y-axis.

In Figure 6, we plot the standard deviation with respect to prices. As the
figure shows, there is wide variation in the standard deviation values,
suggesting the presence of price dispersion across websites.

##  Appendix C Robustness Checks

###  C.1 Dropping Single Website Observations

Note that if prices are posted just on one website, then CV is equal to zero
by definition. To make sure that such observations do not drive our results,
we exclude those observations with a single website and run the regressions
with the smaller sample. The results are presented in table 3. Comparing the
results with table 2 suggests that our key results remain unaffected when we
drop these observations.

Table 3: Regression results for CV after dropping observations with single
website

| (1) | (2) | (3) | (4) | (5) | (6)
---|---|---|---|---|---|---
Dependent variable: CV | Model1 | Model2 | Model3 | Model4 | Model5 | Model6
Previous day CV | 0.767*** | 0.758*** | 0.758*** | 0.757*** | 0.756*** | 0.507***
| (0.00597) | (0.00598) | (0.00600) | (0.00601) | (0.00601) | (0.00845)
Number of Websites |  | 0.0302*** | 0.0305*** | 0.0311*** | 0.0316*** | 0.0669***
|  | (0.00253) | (0.00261) | (0.00264) | (0.00266) | (0.00396)
Page No in Skyscanner |  | -0.000826*** | -0.000789*** | -0.000742*** | -0.000776*** | -0.00241***
|  | (0.000249) | (0.000255) | (0.000257) | (0.000258) | (0.000409)
Dummy for last 3 days |  | 0.00482** | 0.00478** | 0.00483** | 0.00499** | 0.00539***
|  | (0.00201) | (0.00201) | (0.00201) | (0.00202) | (0.00197)
No of Hotel Reviews |  |  | -0.000431 | -0.000229 | -0.000288 | 0.00459
|  |  | (0.00103) | (0.00105) | (0.00106) | (0.00940)
Hotel star rating |  |  | -0.0118 | -0.0162 | -0.0157 | 0.0269
|  |  | (0.0108) | (0.0111) | (0.0111) | (0.358)
Hotel Review Rating |  |  | -0.00465 | -0.0104 | -0.00776 | -0.0180
|  |  | (0.0129) | (0.0329) | (0.0330) | (0.213)
Constant | 0.0171*** | -0.0372*** | -0.00567 | -0.00455 | -0.0118 | -0.143
| (0.00102) | (0.00549) | (0.0276) | (0.0596) | (0.0601) | (0.592)
Observations | 10,817 | 10,817 | 10,817 | 10,817 | 10,817 | 10,817
R-squared | 0.604 | 0.610 | 0.610 | 0.610 | 0.610 | 0.667
Hotel FE | No | No | No | No | No | Yes
Date of Stay FE | No | No | No | No | Yes | Yes
Quality FE | No | No | No | Yes | Yes | Yes
Standard errors in parentheses
*** p$<$0.01, ** p$<$0.05, * p$<$0.1

###  C.2 Regressions with $\ln(CV)$ and $\ln(Range)$

In addition to the results presented in Section 5, here we present estimation
results from two different specifications:

$\ln(CV_{t})=\alpha\ln(CV_{t-1})+\beta_{w}X_{w}+\beta_{h}X_{h}+\epsilon$ (24)

and

$\ln(Range_{t})=\alpha\ln(Range_{t-1})+\beta_{w}X_{w}+\beta_{h}X_{h}+\epsilon$
(25)

where the range denotes the difference between the highest and lowest hotel
room prices listed across different websites for a given date of stay and date
of booking. The results documented in tables 4 and 5 show that the estimated
$\alpha$ is statistically significant and assumes a value less than 1,
irrespective of the model specification. This is consistent with the results
from Section 5 that compared to the previous day, on average, CV goes down the
next day as we approach the date of stay.

Table 4: Regression results for $\ln$ (Coefficient of Variation)

| (1) | (2) | (3) | (4) | (5) | (6)
---|---|---|---|---|---|---
Dependent variable: log(CV) | Model1 | Model2 | Model3 | Model4 | Model5 | Model6
Previous day CV (in log) | 0.791*** | 0.775*** | 0.775*** | 0.774*** | 0.773*** | 0.527***
| (0.00590) | (0.00593) | (0.00595) | (0.00596) | (0.00597) | (0.00842)
Number of Websites |  | 0.0229*** | 0.0231*** | 0.0235*** | 0.0239*** | 0.0471***
|  | (0.00152) | (0.00157) | (0.00158) | (0.00159) | (0.00229)
Page No in Skyscanner |  | -0.000632*** | -0.000603*** | -0.000568*** | -0.000589*** | -0.00176***
|  | (0.000151) | (0.000155) | (0.000156) | (0.000156) | (0.000248)
Dummy for last 3 days |  | 0.00356*** | 0.00353*** | 0.00357*** | 0.00367*** | 0.00362***
|  | (0.00122) | (0.00122) | (0.00122) | (0.00123) | (0.00119)
No of Hotel Reviews |  |  | -0.000329 | -0.000163 | -0.000200 | 0.00323
|  |  | (0.000626) | (0.000636) | (0.000642) | (0.00569)
Hotel star rating |  |  | -0.00867 | -0.0117* | -0.0114* | 0.0109
|  |  | (0.00657) | (0.00672) | (0.00673) | (0.217)
Hotel Review Rating |  |  | -0.00418 | -0.000174 | 0.00124 | -0.0116
|  |  | (0.00784) | (0.0200) | (0.0200) | (0.129)
Constant | 0.0136*** | -0.0234*** | 0.00146 | -0.0109 | -0.0148 | -0.0793
| (0.000686) | (0.00326) | (0.0167) | (0.0362) | (0.0365) | (0.358)
Observations | 10,829 | 10,829 | 10,829 | 10,829 | 10,829 | 10,829
R-squared | 0.624 | 0.632 | 0.632 | 0.632 | 0.633 | 0.688
Hotel FE | No | No | No | No | No | Yes
Date of Stay FE | No | No | No | No | Yes | Yes
Quality FE | No | No | No | Yes | Yes | Yes
Standard errors in parentheses
*** p$<$0.01, ** p$<$0.05, * p$<$0.1

Table 5: Regression results for $\ln$(Range). (Range refers to the gap between
the highest and lowest prices posted for a hotel room across websites)

| (1) | (2) | (3) | (4) | (5) | (6)
---|---|---|---|---|---|---
Dependent variable: log(Range) | Model1 | Model2 | Model3 | Model4 | Model5 | Model6
Previous day Range | 0.761*** | 0.667*** | 0.657*** | 0.656*** | 0.650*** | 0.324***
| (0.00635) | (0.00656) | (0.00669) | (0.00669) | (0.00676) | (0.00900)
Number of Websites |  | 1.018*** | 1.048*** | 1.061*** | 1.062*** | 1.591***
|  | (0.0285) | (0.0291) | (0.0294) | (0.0294) | (0.0381)
Page No in Skyscanner |  | 0.000947 | -0.00311 | -0.00248 | -0.00142 | -0.000361
|  | (0.00263) | (0.00269) | (0.00270) | (0.00271) | (0.00410)
Dummy for last 3 days |  | 0.0904*** | 0.0958*** | 0.0975*** | 0.0878*** | 0.0689***
|  | (0.0213) | (0.0212) | (0.0212) | (0.0213) | (0.0197)
No of Hotel Reviews |  |  | -0.0267** | -0.0224** | -0.0142 | 0.0229
|  |  | (0.0109) | (0.0110) | (0.0111) | (0.0940)
Hotel star rating |  |  | 0.619*** | 0.543*** | 0.539*** | -0.152
|  |  | (0.116) | (0.118) | (0.118) | (3.583)
Hotel Review Rating |  |  | 0.220 | 0.0481 | 0.157 | 2.089
|  |  | (0.136) | (0.347) | (0.347) | (2.127)
Constant | 0.683*** | -0.999*** | -2.291*** | -2.157*** | -2.519*** | -5.671
| (0.0216) | (0.0568) | (0.293) | (0.629) | (0.634) | (5.924)
Observations | 10,829 | 10,829 | 10,829 | 10,829 | 10,829 | 10,829
R-squared | 0.570 | 0.616 | 0.618 | 0.619 | 0.620 | 0.707
Hotel FE | No | No | No | No | No | Yes
Date of Stay FE | No | No | No | No | Yes | Yes
Quality FE | No | No | No | Yes | Yes | Yes
Standard errors in parentheses
*** p$<$0.01, ** p$<$0.05, * p$<$0.1