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by
William Joseph Brennan
2005
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The Dissertation Committee for William J oseph Brennancertifies that this is the approved version of the following dissertation:
Airline Pricing and Capacity Behavior
Committee:
____________________________________ Maxwell Stinchcombe, Co-Supervisor
____________________________________ R. Preston McAfee, Co-Supervisor
____________________________________ Dale O. Stahl
____________________________________ David Sibley
____________________________________
Peter Wilcoxen
____________________________________ Vijay Mahajan
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Airline Pricing and Capacity Behavior
by
William Joseph Brennan, B.S.; M.S.
Dissertation
Presented to the Faculty of the Graduate School of
the University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
August, 2005
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Dedication:
I dedicate this dissertation to my parents William Joseph Brennan, Jr. and
Kathleen Mary Brennan who have supported me in every way toward my completion of
the Ph.D. They have always encouraged me to do my best and this work is to their
support and credit. I would also like to dedicate this dissertation to my sister Kaela
Brennan and brothers Patrick and Jim Brennan. They have also been very supportive in
writing this dissertation.
I’d like to thank my grandmother Kathleen Dillon and Godmother Maggi Duncan
for all of their encouragement. I’d like to thank Alice Chakkalakal for all of her support.
I’d like to thank my friends including Mike Fell, Mark Daeges, Pat Dempsey, Darren and
Michaela Cook, Fr. Eric Schimmel, Phil and Megan Tomsik, Brian and Marita Connor,
Christina Frank, Mary Finnegan, Doug Maurer, Risa Kumazawa (who encouraged me to
“just slap [the pages] together”), Melissa Halac, Dan and Carol Pier, Paul LaBarre, Sam
and Kathleen Rauch, Hal and Nan Kuehn, the Franco family, and Dan and Lisa Gaynor.
All of them plus many more unmentioned friends encouraged me and listened to many a
conversation about the process and about life in general. I’d like to thank Dr. Hudson
Hsieh, Dr. Lilly Stoller, and Sandy Trandahl for their support. Finally I’d like to thank
my uncle Gerry Dillon for his support. At one point he and I were about the same point
in different Ph.D. programs but duty to his country has prevented him from reaching this
stage at this point in his life.
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Acknowledgements:
I would like to thank my Co-Chair, Dr. R. Preston McAfee, for all of his
comments, suggestions, and feedback on this dissertation. I am greatly honored to have
had his services. I would also like to thank my other Co-Chair, Dr. Max Stinchcombe,
for all of his comments and suggestions. I am also truly honored to have had his services.
He too has been very helpful, especially near the end of the process. I would like to
thank Dr. Pete Wilcoxen for all of his comments, suggestions, and support on the
dissertation and his help in developing the C computer program that allowed me to view
the airline data in Chapter 1. I would like to thank Dr. Dale Stahl, Dr. David Sibley, and
Dr.Vijay Mahajan for all of their comments and suggestions as committee members. I
would like to thank Dr. Doug Dacy for attending my oral defense and providing
intriguing questions and support during the defense and during the University of Texas
Job Placement Seminar. I would also like to thank Dr. Dan Slesnick, Dr. Vince Geraci,
Damien Eldridge, and the University of Texas Job Placement Seminar for useful
comments. I would like to thank Dr. Elsie L. Echeverri-Carroll, Dr. Louise Wolitz, Dr.
Darryl Young, Ruhai Wu, Shane Carbonneau, Abe Dunn, Brett Wendling, Lisa Dickson,
and Rich Prisinzano for all of their useful comments and discussions. Finally, I would
like to thank Vivian Goldman-Leffler for all of her support on the administrative and
personal level. Truly she is one great Graduate Coordinator to have on your side. Any
errors are strictly my own.
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Airline Pricing and Capacity Behavior
Publication No. _____________
William Joseph Brennan, Ph.D.The University of Texas at Austin, 2005
Supervisors: R. Preston McAfeeMaxwell Stinchcombe
Standard models of price dispersion (Butters, Varian, Burdett & Judd) give some
explanation of how equilibria of firms selling the same product at different prices occur by
providing consumers with dispersed information or loyalty but a common reservation price. This
model extends these models by having business travelers paying more than tourists. There is a
continuum of prices broken by a gap right above the monopoly price of the tourists. In this
region, expected firm profits are lower, as firms do not make up for the discrete loss of leisure
travelers. The model is compared to Data Bank 1A – a ten-percent random sample of airline
ticket itinerary. Individual airline routes are shown to have up to several peaks in the estimated
price kernel density. This is where the theory matches the data: airline fares cluster around
prices.
Kreps and Scheinkman generate a limited capacity model that generates price
randomization when two firms have unequal capacities. The large capacity firm has the
ability to set prices lower than the smaller firm. Butters, Varian, Burdett & Judd develop
price dispersion models as described before. A symmetric two-firm model is developed
that has both features, limited capacity and limited price information, thus unifying two
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disparate literatures. The lower price firm sells out to capacity, while the higher price
firm receives the leftover customers from the sold out firm plus the loyal customers who
did not see the lower priced firm. This leads to price randomization. Limited
information and capacity are thus identical in economic effects.
There is a variety of sizes of loyal customers within the airline, hotel, and car
rental companies. A price randomization model is created with one large firm of loyal
customers and many smaller firms having the same size of loyal customers. Firms
randomize by charging the monopoly price for the loyal customers and discounting to
obtain a group of shoppers seeing the lowest price of all the firms. The largest firm has
an atom at the highest price in the distribution. The smaller firms compete for shoppers
at all prices in the distribution and have no atom.
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Table of Contents
0. INTRODUCTION……………………………………………………………….1
0.1 Business and Leisure Travelers…………………………………………...1
0.2 Frequent Flier Programs………………………………………………..…2
0.3 Differing Sizes of Frequent Flier Programs……………………………….5
0.4 The Elite Frequent Flier…………………………………………………...7
0.5 Yield Management and Capacity Limitations…………………………….8
0.6 The Rise of Low-Cost Carriers and Falling Average Ticket Prices…….12
0.7 Price Dispersion………………………………………………………….14
0.8 Summary of Developments Since Deregulation………………………....18
0.9 Summary of Dissertation………………………………………………...19
0.10 Future Research……………………………………………………….....25
1. “PRICE DISPERSION WITH DIFFERING CONSUMER VALUES” ……..26
1.1 Introduction………………………………………………………………26
1.2 Literature Review…………………………………………………...……30
1.3 Motivation……………………………………………………………..…36
1.4 The Model………………………………………………………………..44
1.5 Comparative Statics……………………………………………………...75
1.6 Explaining the Graphs: Matching the Comparative Statics to Airline
Data……………………………………………………………………..134
1.6a Discussing the Computer Generated Graphs…………………...134
1.6b Matching the Computer Graphs to the Airline Data……………1371.7 Analysis and Conclusion………………………………………………..143
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2. “CAPACITY AND RANDOM PRICES”...........................................................149
2.1 Introduction……………………………………………………………..149
2.2 The Symmetric Two-Firm Model………………………………………152
2.3 Generalizing to the Case Where All But One of the Firms are Sold to
Capacity………………………………………………………………...158
2.4 Generalizing to the Asymmetric Model………………………………...163
2.5 Building Capacity First and then Setting Prices………………………..176
2.6 Conclusion………………………………………………………….......183
3. “EQUILIBRIUM PRICE RANDOMIZATION WITH ASYMMETRIC
CONSUMER LOYALTY”…………………………………………………..…185
3.1 Introduction……………………………………………………………..185
3.2 The Model with Two Firms…………………………………………….187
3.3 Comparative Statics with the Two Firm Model………………………...195
3.4 The Model with n Firms – 1 Large and (n-1) Similar Small Firms…….258
3.5 Comparative Statics with the Modified n-Firm Model…………………264
3.6 Conclusion………………………………………………………...……337
APPENDIX PDF, 3 VAL UATION CASE, AND DATA GRAPHS……….....……342
BIBLIOGRAPHY……………………………………………………………………..363
VITA…………………………………………………………………………………...368
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Introduction
0-1. Business and Leisure Travelers
Since deregulation, the airline industry has been a dynamic and exciting industry
prone to many changes. Several major trends developed. The first is that airlines found
that not all travelers are the same. Some traveled more frequently than others and were
willing to pay much more for their tickets. Airlines found that in a post deregulation
environment there are business travelers and leisure travelers.
Business travelers are those willing to pay top dollar for flights and making repeat
business. Airlines depend on this smaller group of travelers to make up a substantial
proportion of their revenue. There are a variety of estimates of how big the overall group
of business traveler group is to the airlines. During the expansion of the late 1990’s,
United Airlines had 9% of its premium business travelers paying top fares generating
46% of the entire company’s revenue! (Brannigan, Carey, McCartney) The revenue
produced today by this group of travelers is much lower today. Southwest Airlines
estimates that thirty – five percent of its passengers pay full fare (Koenig). The National
Business Travel Association estimates that one third of travelers deemed to be business
account for one half of airline revenue (Bjorhus). The American Transport Association
more liberally1
estimates that 45% - 55% of airline passengers that travel for “business
purposes” comprise 60 to 70 percent of an airline’s revenue (Fedor).
1 Some travelers combine business with leisure so this estimate could be high.
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Deregulation resulted in the continued rise of the leisure class. Leisure travelers
generally search vigorously for lower fares as they are not as willing to pay as high fares
as business travelers. They generally do not make as many repeat trips as business
travelers. There are generally more leisure travelers than business travelers, even though
business travelers are the ones making the trips much more frequently. From the
estimates above there are anywhere from one half to two thirds leisure travelers on an
average flight.
0-2. Frequent Flier Programs
From an airline’s perspective, business travelers are their best customers because
they are willing to pay higher fare and they travel more frequently. Airlines want the
business travelers (and the leisure travelers if they continue to make repeat trips) to keep
making repeat transactions with their company. The frequent flier program was first
developed by American Airlines in 1981, and soon copied by other airlines, to reward
travelers that made frequent trips with a particular airline (Bailey, Graham, and Kaplan,
p. 60). Travelers could receive perks such as early check-ins, separate reservation
assistance, upgrades to first class, free flights to regular destinations the airline serves,
and free flights to exotic destinations that the airline served – such as the Caribbean,
Europe, or Hawaii. Each of these perks depends on the number of frequent flier “miles”
that a traveler earned with an airline. This, in turn, depended on a basic level how often a
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traveler traveled with an airline, how far within an airline2
a traveler travels, what fare a
traveler pays, and (today) what recognized status3 the traveler has with the airline. The
more miles a traveler obtains, the more choices the traveler has with which award can be
redeemed with the miles.
The frequent flier programs help create a loyal base of travelers to individual
airlines. Business travelers who split their business between several airlines lose out on
the accrual of benefits that placing their business with one airline could provide. The
incentive in the frequent flier programs is strong for travelers to be loyal to a particular
airline or two. This, in turn causes travelers in frequent flier programs, especially
business travelers, to be less price sensitive to other airlines. This can be seen with the
following example. Suppose that a business traveler short 4,000 miles of the 125,000
award4 level for a free premium first class ticket in the peak season between the United
States and Europe (and desires the award for a planned vacation) and is purchasing a
roundtrip ticket trip between New York and Los Angeles (a distance that would put him
or her over the award limit). How much cheaper must the ticket be on a competing
airline that the traveler has no ties for the business traveler to switch or even give
considerable consideration, especially if the business traveler is not footing the bill?
2 Southwest Airline’s Rapid Rewards Program does not depend on actual miles flown but on segmentsflown. Thus an Austin – Dallas Love segment counts as much toward a Rapid Reward credit as a Phoenix – Baltimore segment. Most other airlines have a minimum floor of 500 miles that can be earned by asegment on their frequent flier programs.3 Most of the legacy carriers have established in the last decade programs that reward travelers that makeheavy travel within the past year. There are generally three levels based on the number of segmentstraveled or the number of actual miles traveled. The rewards for obtaining one of these levels areassociated with frequent fliers with very high mileage such as early check-ins, separate reservationassistance, etc.4 Using American Airlines Advantage Reward Program found at www.aa.com.
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Leisure travelers can be softened in a similar way to a lesser extent by frequent
flier programs. Whereas leisure travelers may not travel as much as business travelers,
they can accumulate frequent flier miles through promotions such as credit card points,
auto rentals, and hotel programs. According to Randy Petersen, Publisher/Editor Inside
Flyer magazine, 40% of all frequent flier miles are accumulated by not flying
(http://www.webflyer.com). The fastest growing segment of the population with
frequent flier miles is those that are “`mileage consumers’ not frequent fliers” (Ibid).
With frequent flier miles being accumulated by a class of people not traveling very much,
the occasional flight by these consumers is more likely to be done on the airline that they
have miles, especially if they are close to a free ticket. With the introduction of frequent
flier programs, a class of travelers loyal to a particular airline has been created since
deregulation.
Most carriers evolved into hub and spoke operations in which one city is the hub
where most flights originate and depart and the other cities in the network feed into the
hub. Passengers traveling throughout the network make connections throughout the hub
thus allowing an airline to serve many markets with one or fewer connections. Airlines
competed over the size of their networks and often the large airlines had more than more
than one hub in their network to broaden their scope. Combined with frequent flier
programs, airlines could use the breath of their network to win approval of the repeat
frequent flier, who needed to use the services of the network often. This combination
especially works well in hub cities, where airlines offer their best service – nonstop
service – several times a day to many destinations. Borenstein (1989) found evidence
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that fares are higher in hub cities than non-hub cities at a higher percentile of fares.
Airlines could offer the expansive network ready for business use when travelers are
paying top fares and then offer the reach of that same network to exotic locations at the
time frequent flier miles are redeemed for free trips. “ 'You can fly a million miles on
Frontier [Airlines], but it will never get you to Paris,' said United [Airlines] spokesman
Jason Schechter. 'It's apples versus oranges. We have an unmatched global route. We
have first class'” (Aguilar p. C 01).
0-3. Differing Sizes of Frequent Flier Programs
The sheer size of United Airlines versus Frontier Airlines is not only matched in
annual revenue, network size, but in the size of frequent flier programs. United boasts of
approximately 43 million frequent flier members while Frontier Airlines reports over one
million frequent flier members (http://www.webflyer.com). There is a variety in the
sizes of frequent flier programs. Table 1 shows this difference as it lists actual
enrollment of the frequent flier programs5. The information comes from
http://www.webflyer.com:
5 webflier.com also lists hotel frequent stay membership totals and American Express MembershipRewards totals.
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Table 1: Size of Frequent Flier Programs – US, Canada, Mexico
Airline Members Data Reported as of:
AeroMexico 1,600,000 6/03
Air Canada 6,000,000 11/03
Alaska Airlines 3,700,000 8/03
Aloha Airlines 300,000 12/03
America West 4,100,000 8/03
American Airlines 48,000,000 1/05
Continental Airlines 19,000,000 4/01
Delta Airlines 35,000,000 9/04
Frontier Airlines 1,000,000 9/03
Hawaii Airlines 880,000 8/03
Mexicana Airlines 825,000 8/03
Midwest Airlines 1,654,000 3/04
Northwest Airlines 25,000,000 8/03
United Airlines 43,000,000 12/03
US Airways 21,200,000 4/01
The list above only shows enrollment numbers. Randy Petersen, Publisher/Editor
Inside Flyer magazine, estimates that only 27 – 28% of enrolled frequent fliers are
actually active (http://www.webflyer.com). Furthermore, fliers do enroll in more than
one program. Thus the actual enrollment numbers do not tell the entire story of the
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differing sizes of frequent flier bases for each airline. If information was available, the
totals of frequent fliers could be broken down into metropolitan areas. Though American
Airlines has a higher frequent flier membership total than Northwest Airlines, the total
Northwest Airlines WorldPerks members in Detroit, Michigan is much higher than
American Airlines AAdvantage members because Northwest Airlines has a hub in
Detroit. Frontier Airlines may be closer in size to United Airlines in the Denver
metropolitan area in terms of frequent flier memberships, where both airlines have hubs.
0-4. The Elite Frequent Flier
The frequent travelers of the frequent fliers tend to bring airlines a significant
portion of their revenue. Airlines reward this group of travelers with extra miles,
frequent upgrades to first class, priority calling to a 1-800 line, and priority boarding.
Usually, airlines have a tier class with higher tiers being for higher travel with the airline.
Travel can be defined as within the past year or lifetime travel with the airline. American
Airlines, for instance, has Gold, Platinum, and Executive Platinum for its best customers.
(www.aa.com) These travelers contribute more to the airline revenue wise than the
average frequent flier. Peterson estimates that United Airlines has approximately
800,000 total flier members in their three elite classes (Aguiar, p C01). Peterson
estimates that there are approximately 307,000 total fliers that have over a million miles
with a particular airline with 250,000 combined with American, Delta, and United
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Airlines (Stoller, 8B)6. For American Airlines, a traveler earning one million miles with
the airline earns a lifetime Gold Membership. Whether measured by cumulative activity
traveled for one airline or by cumulative activity in the past year for one airline, this class
of frequent travelers may be another way to measure the strength and reach of airlines’
frequent flier programs.
0-5. Yield Management and Capacity Limitations
Another major trend that continued after deregulation was the continued rise of
yield management systems. Airlines started to inventory their seats in a computerized
inventory system well before deregulation first to provide real time information to airline
employees and travel agents making reservations for passengers. Seats were
compartmentalized in to various categories reflecting the different types of rules
accompanied each tariff or fare. During the period of regulation, each tariff or fare was
controlled by the Civil Aeronautics Board in Washington, DC. Sales and promotions
were present in the regulation era. However, the present system of yield management
still had to wait, as it was cumbersome to apply to Washington to have approval for fares
codes changed on routes prior to deregulation. Routes that had fares changed were met
or even challenged by competing airlines (Bailey, Graham, and Kaplan p. 16).
After deregulation, the tools were set in place for the growth in price dispersion.
Airlines kept the fare class system. New classes of discount class tariffs appeared for the
6 According to Peterson at webflier.com, the person with the highest unclaimed frequent flier mileage totalhas 23 million miles!
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leisure traveler offering great savings off the standard coach fare (Bailey, Graham, and
Kaplan, p. 47). Since these cheaper seats were not as profitable as a standard coach seat
at a high price, airlines found ways to control how they sold discount seats. Airlines use
price discrimination tactics in tariffs in how many days in advance the passenger is
purchasing the ticket in advance or whether the passenger is staying over on a Saturday
night to determine if a discount tariff applies (ibid). More subtly, airlines set a limit on
the number of seats that can be sold at the discount fare on a flight basis (ibid). Once that
discount seat capacity limit is reached on a flight passengers have to pay the next higher
fare or choose another flight with the same discount seats. To keep track of the explosion
of new tariff classes, rules, and inventory on each class on routes across their networks,
airlines needed the services of sophisticated computerized reservation systems.
Airlines use the computerized reservation systems to constantly readjust their
inventory of seats to maximize their possible revenue. The entire process developed was
called yield management systems that maximized revenue from every airline seat given
market conditions (Sloane). These conditions included costs to the airlines, prices that
competitor airlines are charging, historical trends of business and leisure travelers, the
overall economy performance, and supply and demand conditions within a given subclass
of seats, flight, and route (ibid). Airlines usually change flights within their system
slowly so supply of overall seats on a flight may be fixed but demand each day on a
particular flight can be quite variable (Schmitt and Williams). Airlines use several years
worth of data on flights to predict passenger traffic to forecast demand on a route
(Schmitt and Williams). Subcategories of seats with separate tariffs are initially
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distributed across each flight as it is initially entered into the system almost a year prior to
departure (Schmitt). Airlines leave as many seats as predicted to business traveler, who
are willing to pay much more than the leisure traveler, while selling seats to the leisure
traveler (ibid).
As demand conditions change between leisure and business travelers, airlines
change inventories of seats between classes of seats on a flight. Suppose more business
travelers make more early bookings on the American Airlines Dallas – Tulsa 8:00 P.M.
flight than expected. Then American will reduce the number of discount seats available
for booking and increase the number of higher priced business classes of seats. The
adjustment could be weeks out or within hours of departure (Schmitt). This adjustment
process is generally invisible to the consumer.
The net result of yield management is an increase in revenues for the airlines and
an increase in load factors (ibid). Every day airlines face a problem of perishing
inventory, namely seats that go empty when flights take off. As of now, major airlines
generally keep a fixed schedule with minor adjustments over time with equipment and
flights for a set period of time. With set capacity levels to markets, airlines face the
problem of uncertain demand to their markets, sometimes not enough capacity,
sometimes too much capacity. Complicating the uncertainty of demand is the uncertainty
of the mix of travelers between leisure and business travelers. Yield management helps
airlines maximize revenues given the capacity they have on the route and minimize
empty seats.
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Due to yield management, airlines offered discount seats in limited capacity.
Travelers are conditioned to shop around for the lowest fare possible. If one airline sells
out in discount seats on a flight or quotes a higher price because it does not offer cheap
seats on that flight, customers will shop for another flight for the same carrier that offers
lower prices. If a carrier’s flight selection is sold out of the lowest fares, those not overly
loyal to a particular airline will go to other airlines to obtain cheap seats. Thus airlines
that do no sell out to capacity have a residual demand from other airlines that sold out
their inventory.
This system of offering discount seats is constantly in flux. Since the start of
deregulation, airlines began experimenting with changing fares or fare classes constantly
or offering all out fare wars. In the late 1980’s United Airlines made 30,000 fare changes
daily (Hamilton, H1). Furthermore, United monitored advanced ticket sales on 120,000
of its total flights in the computer system and changed the fare class distribution of seats
on 15,000 flights daily. (ibid). Changes in capacity were responsible for most of
United’s daily fare changes (ibid). If passenger totals increased or a competitor reduced
capacity, then fares increased at United (ibid).
Airlines sometimes run system-wide sales. Sometimes these sales are promoted
by the airlines in response to system-wide bookings being lower than capacity. Airlines
will put a certain percentage of seats on discount in their whole system for some time.
Sometimes sales are airlines acting in retaliation to another airline for some action.
American Airlines, for instance, responded to Northwest undercutting its simplified fare
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structure in 1992 with a 50% off all discount coach seats and moved the 14 day in
advance purchase requirement to seven days in advance (Salpukas, p. D1).
The process of putting seats on discount has become refined that airlines are
targeting specific markets or specific period of time without going to a broad price war.
American Airlines, for instance, put seats on discount for Mother’s Day weekend 2005
(American Airlines Press Release 4/27/05). The Mother’s Day Sale did not lead to a
broad fare war. Airlines have developed weekly specials broadcasted by email that puts
certain routes on sale. Southwest Airlines has taken the process further with its DING!
program, offering specials to participating computer users for a few hours (Southwest
Airlines Press Release: February 28, 1995). By limiting the period of discount to a short
period of time, airlines are avoiding all-out price wars on all of their discount seats.
0-6. The Rise of Low-Cost Carriers and Falling Average Ticket Prices
There has been a rise of the low cost carrier since deregulation. Southwest
Airlines has grown to one of the top domestic passenger airlines from a small regional
airline at the beginning of deregulation. American Trans Air, Independence Air, Jet
Blue, Frontier, America West, Air Tran, and Alaska are all now considered low cost
carriers. All of them except for Alaska have been created since deregulation. Other low-
cost carriers such as Midway Airlines, People’s Express, and Piedmont Airlines either
were merged or liquidated. Even the majors or legacy carriers are copying the low cost
airlines as Ted and Song are separate low-cost airlines within United and Delta
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respectively. Low cost carriers’ market share jumped from 20% in 2000 to slightly over
30% in 2004 (Walsh, p. 1B). More impressively, the percentage of seats in markets
longer than 2,000 miles in the contiguous United States served by low cost carriers has
jumped from 13% in 1999 to 37% in 2004 (Maynard, Section3, p7).
The overall effect of the introduction of the yield management system and low-
cost carriers was that average airline fares fell as a result of deregulation as airlines
competed for the leisure traveler. The Airline Transport Association reports that
domestic passenger yield per passenger mile fell 48.8 percent between 1978 and 2003,
adjusting for the Consumer Price Index (Air Transport Association 2004 Economic
Report p. 11). The General Accounting Office reports that between 1979 and 1994 fares
per passenger mile, adjusted for inflation, fell 9% for small community airports, 11% for
medium sized community airports, and 8% for large community airports (GAO 4/25/96
p. 5). The Airline Transport Association reports that nominal total (including
international travel) airline revenue per passenger mile fell 10.6% from 13.13 cents per
passenger mile in 1993 to 11.74 cents per passenger mile in 2003 (Air Transport
Association 2004 Economic Report p. 7). This acceleration of the erosion of airfares
between 1993 and 2003 translates to an average loss of $13.90 – from $131.30 to $117.40
- for a one- way flight of 1000 miles. Even with the complicated yield management
system available, there has been an explosion in discount seats for air travel.
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0-7. Price Dispersion
Despite the decline in average ticket prices, there has been a rise of price
dispersion within the industry. Airlines discounted for the leisure travelers but raised
prices for business fares. Business travelers paid seventy percent higher fares in 2000
than in 1980 (Hamburger)7. Leisure fares fell in the same period as business fares rose
(Hamburger)8. eCLIPSE advisors, a subsidiary of American Express Business Travel,
estimates that the typical business fare (refundable, three days in advance) on routes it
monitors is $402 one-way in 2004 (American Express Business Travel Monitor Press
Release 11/22/04). The same subsidiary estimates that average fares paid by its
customers traveling on routes monitored by the same BTM group are at a lower $217
(Ibid)9. These business fares are much higher than the average one-way fares listed by
the American Transport Association. Yield management systems take advantage of this
wide spread of willingness to pay among consumers with many different fares.
Price dispersion in the airline industry can be seen by looking at the second
quarter of 1995 Data Bank 1a, which is a 10% random sample of airline tickets. The
following chart shows adjusted one-way fare differences on selected routes (both
7 Some of that gain between 1980 and 200 has eroded. In a SABRE study business fares (average roundtripfare purchased three days from departure) fell 12% between May 2001 and May 2003 (Reed 1A). With theintroduction of SimpliFares by Delta in January 2005 that cuts the maximum one-way fare to $499 ondomestic routes, the trend of lower business fares continues well into 2005 (http://www.delta.com). In July2005, Delta raised (matched by other legacy carriers) the maximum one-way fare to $599 due to higher oil prices.8 James Higgins, airline analyst at Credit Suisse First Boston reports that leisure fares have been creepingup between 2003 and 2004 due to discount carriers raising prices (Grantham p. 1F). This trend hascontinued today with the legacy carriers raising fares on a couple of times this year due to high oil prices.9 These average business fares include full fares, cheaper nonrefundable fares, and negotiated corporatediscounts (American Express Business Travel Monitor Press Release 11/22/04).
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directions included) between the 25%ile and 75%ile plus the mean, median, and standard
deviation. Itineraries are either one-way or closed loop round – trips. In the case of a
closed-loop round-trip, the fare is divided by two to get the one-way fare.
Table 2: Route Price Dispersion
Route
(Both
Directions)
# Pax
in 2nd
Qtr ‘95
25%ile Median Mean 75%ile Difference
75%ile –
25%ile
Standard
Deviation
Std.
Dev./
Mean
LAXEWR 171,420 $153 $192 $254.94 $253 $100 $233.34 0.904
OMALGA 9,800 $117 $167.25 $213.55 $300.50 $183.50 $145.94 0.683
MCOSEA 32,990 $152.50 $196 $211.20 $247 $94.50 $169.29 0.802
LASPHX 227,500 $37 $40 $50.06 $67 $30 $20.83 0.416
SFOORD 171,570 $130.50 $190.50 $278.47 $381 $250.50 $258.88 0.930
RDUSAN 8,480 $150.50 $193 $239.04 $309.25 $158.75 $208.70 0.873
AUSBOI 2,320 $135 $143.50 $173.52 $223.50 $88.50 $84.04 0.484
FARMSP 9,300 $95.50 $111.50 $112.13 $130.50 $35 $37.33 0.333
LANRNO 720 $145 $176.50 $179.36 $240 $95 $128.35 0.716
LAXDFW 103,710 $142 $189.50 $259.26 $423 $281 $183.07 0.706
PHLDEN 50,420 $138 $163.50 $242.12 $304.50 $166.50 $208.43 0.861
RICGEG 630 $175 $210.50 $217.68 $240 $65 $100.88 0.463
BOSMIA 73,040 $116.50 $151 $161.71 $177.50 $61 $114.68 0.709
TPALGA 88,420 $113 $131 $152.22 $181.50 $68.50 $89.10 0.585
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DTWPIT 20,070 $111.50 $286 $224.64 $316.50 $205 $105.51 0.470
PVDORF 8,130 $88.50 $102 $136.70 $157 $68.50 $75.77 0.554
PWMLAX 3,920 $147 $221 $221.26 $266 $119 $178.43 0.806
ATLDCA 105,150 $90.50 $207.50 $209.17 $305.50 $215 $118.21 0.565
AMADCA 1,150 $188.50 $208.50 $261.33 $290.50 $102 $155.29 0.594
LAXORD 197,550 $141.50 $184.50 $256.80 $299.50 $158 $239.74 0.934
PHXBOS 59,640 $147.50 $185.50 $217.12 $257 $109.50 $167.31 0.771
IAHLGA 57,310 $175.50 $287 $379.74 $650.50 $475 $242.54 0.634
BILLAX 3,280 $144.50 $161 $189.79 $231 $86.50 $98.18 0.517
SAVANC10
210 $0 $209.50 $235.71 $440 $440 $269.81 1.145
STLSEA 35,280 $119 $144.50 $161 $210 $91 $106.07 0.659
FSDCLT 500 $139 $161 $196 $259 $120 $142.50 0.727
MCISJC 8,990 $117.50 $143 $163.69 $223.50 $106 $116.75 0.713
HNLORD11 20,870 $0 $306 $283.19 $387.50 $387.50 $245.37 0.866
IADSFO 124,810 $165.50 $224 $305.79 $317 $151.50 $279.38 0.914
ANCFAI 44,660 $69 $98 $90 $99 $30 $24.18 0.269
Averages 54,728 $121.52 $179.49 $209.24 $272.93 $151.41 $151.60 0.687
Weighted
Average
$117.32 $165.31 $210.89 $267.35 $150.03 $168.65 0.721
10 Delta is the only carrier reporting passengers between these two cities. Frequent fliers make up almosthalf of the 210 travelers. If frequent fliers are eliminated, the 50%ile on fares paid jumps to $440.11 The 25%ile, median, average, and 75%ile fares rise to $293.50, $334, $393.21, $416.50 respectively if frequent fliers and non-revenue passengers are eliminated from the sample.
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Table 2 shows that there is dispersion in all routes. The minimum standard
deviation in fares is $20.83 on the Phoenix, AZ to Las Vegas, NV route. However, the
average fare on that route is extremely low at $50.06. Thus the standard
deviation/average on that route is not the lowest in the sample. Some routes have
extreme price dispersion. For instance, the Houston Intercontinental, TX – New York,
NY LaGuardia airport has a standard deviation of $242.54 and a whopping $475
difference between fares at the 25th percentile level and 75th percentile level! When
measured by standard deviation/average price, the Houston-New York route is
approximately 50% higher than the Phoenix – Las Vegas route.
A simple regression with passengers being the dependent variable and median,
standard deviation, distance, dummy for low cost carrier, and number of nonstop carriers
being the independent variables produces the following result:
Table 3 A Simple Regression with Passengers as Dependent Variable
R Squared =0.7193
Adj. R Squared =0.6608
F(6,23) = 12.30 Prob > F =0.0000
Independent var. Coefficient Standard error t Prob > |t|
Median - 368.3146 181.9153 -2.02 0.054
Standard dev. 711.7568 171.0115 4.16 0.000
Distance - 23.04458 11.77245 1.96 0.062
Low Cost 34708.72 21566.98 1.61 0.121
# NonstopCarriers
33951.22 5983.681 5.67 0.000
constant 5124.998 30006.02 0.17 0.866
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Obeying the law of demand, the coefficient of median is negative.12
. Interesting is that
standard deviation is positive and significant. Larger markets generally have more
variation in prices by the measure of standard deviation. The number of nonstop carriers
is extremely significant and indicates a large portion of the variation in passengers. A
better regression might include the Herfindahl index and adjust for the frequent flier/non-
revenue passengers.
0-8. Summary of Developments Since Deregulation
There have been several trends since deregulation. There has been the rise of two
different types of travelers – business and leisure travelers. Average fares paid by these
business travelers are much higher than leisure travelers. There is a class of business
travelers that travel much more than the average public. The development of frequent
flier programs marks a way for airlines to reward those contributing the most revenue.
Not all airlines are the same in terms of frequent fliers. As has been seen, American
Airlines’ frequent flier program today is almost fifty times the size of smaller Frontier
Airlines. (http://www.webflyer.com/company/press_room/facts_and_stats).
With the difference of business travelers’ willingness to pay for flights compared
to leisure travelers, airlines have developed sophisticated yield management systems that
limit the capacity of discount seats in favor of last minute business travelers. Once
12 If mean is used instead of median, the regression is not as robust as median. Adding 25ile and 75ile asindependent variables lowers the adjusted R squared and F. Using 75ile minus 25ile raises the adjusted R squared but lowers F. In this regression of adding 75ile minus 25ile, median becomes insignificant.Standard deviation/mean is insignificant.
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discount seats sell out, higher fares result. Airlines predict using past data, current
economic and market conditions, what seating capacities should be for the various classes
of travelers. Should conditions change or not go as predicted, airlines make adjustments
to the system.
The rise of yield management systems and low cost carriers has resulted in two
trends: decreasing overall average fares and rise of price dispersion. Average fares have
fallen since deregulation. The American Transport Association reports average domestic
revenue per passenger mile has fallen almost 49% since 1978, adjusting for CPI inflation
(American Transport Association 2004 Economic Report p. 11). The ATA reports that
average fares on a one-way 1000 mile flight are $117.40 in 2003 (Ibid). The average fare
misses the massive price dispersion inside the industry. Typical three day refundable
fares are four times that amount in 2004 (American Express Business Travel Monitor
Press Release 11/22/04). The average weighted standard deviation of a sample of thirty
markets from the second quarter 1995 is $168.65. The weighted standard deviation
divided by the weighted mean results in a ratio of 0.721. Thus one can conclude that
there is price dispersion within the industry.
0-9. Summary of Dissertation
This dissertation models some of these trends within the airline industry since
deregulation. Chapter one models price dispersion in the industry with differing values
of business and leisure travelers. Airlines are the same in size of loyal customers, who
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see only one price. Each airline has the same amount of loyal travelers, who will pay any
price at hand up to their reservation value. Airlines compete for a group of shoppers,
who search all the airlines for the lowest price. Airlines can choose to serve the business
market, which have both loyal customers and shoppers. Airlines can serve the leisure
market, which also have a different amount of loyal customers and shoppers. In either
market, firms randomize over a range of prices as there is no pure price strategy.
Under certain conditions of profits not being too big in the leisure market and
profits not being too big serving only the loyal business travelers, airlines will choose to
serve both markets. Firms randomize in their prices over an interval of prices large
enough to serve both markets. In covering both markets, there is a gap in prices
immediately above the reservation price of the leisure traveler, where airlines do not
randomize. In this region, profits are less than region of prices immediately lower than
the monopoly price of business travelers and monopoly price of leisure travelers. Thus
there are two regions where fares cluster. The model can be extended to the case where
there are more than two consumer valuations. A cumulate distribution function and
probability density function are generated with three consumer valuations, resulting in
two gaps and three regions where fares cluster.
The model is compared to the 1995 Origin and Destination Data Bank 1a second
quarter. Some similarities are revealed. Using the Epanechnikov method of kernel
density estimation, several airline markets fares are graphed. Similar in each of these
markets is fares cluster around one or several prices. This clustering of prices matches
what the theory predicts.
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Chapter two of the dissertation models price dispersion and capacity limitations
found in the airline market. Two firms modeled sell to customers that see none, one, or
both prices. The customers that see one price are loyal customers. Those that see both
firm’s prices are shoppers. Firms have a capacity limitation. Each firm cannot serve the
entire market of its loyal customers plus the shoppers.
When the price of one firm is lower than the other, the lower priced firm sells out
to capacity. The higher priced firm sells to its loyal customers plus the left-over shoppers
that cannot purchase from the lower priced firm. Thus the higher price firm receives an
extra bonus of customers that could not buy from the lower priced firm. Assuming that
there is enough capacity in the marketplace for the higher priced firm to handle its loyal
customers plus the left-over shoppers that could not purchase from the lower priced firm,
firms randomize between socking their own loyal customers plus the spillover customers
at the monopoly price and discounting to win over the shoppers and sell out to capacity.
The lowest price in the distribution is profits at the monopoly price of the higher priced
firm divided by capacity.
The model can be extended to the case of w firms where w-1 sell out to capacity.
The lowest price in the distribution still is monopoly profits of the highest priced firm
divided by capacity. Again this case assumes that the highest price firm has enough
capacity to handle the left-over consumers from the other firms.
In the case of asymmetric capacity, there is not an equal lowest price in the
distribution. If total capacity is small, the smaller firm has the smallest lowest price and
thus has the atom at the highest price. The smaller firm has more slack to undercut the
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higher capacity firm. If total capacity is high, then the atom shifts to the larger firm as it
has more slack in its capacity to undercut the smaller firm. In both cases, it is assumed
that the smaller firm has enough capacity to handle its loyal customers plus the left-over
customers of the high capacity firm when the low capacity firm has a lower price.
When the two firms play a two stage game that involves setting capacities first
and then announcing prices, the optimum equilibrium is a symmetric one as the reaction
curves of both firms cross each other at the symmetric point. With a cost of building
capital and profits being lower with more capacity, the two firms will build the smallest
possible capacity. Each firm sets the price to the monopoly price.
Chapter three models price dispersion and differing level of loyalty of customers.
The first half of the paper has two firms: one larger in the amount of loyal customers that
it serves than the other. The second half of the paper has many firms: one firm that is
larger in loyal customers than the other same sized firms. The setup is the same in each
half of the paper. The loyal customers of each firm pay whatever price is at hand. Each
firm has the incentive to charge their group of loyal customers the monopoly price. The
largest firm has more of them so when the price is at the monopoly price, the largest firm
earns more profits than the smaller firm(s).
However, there is a group of non-loyal shoppers that buy from the lowest price
firm. If there are enough of these shoppers, all firm lower their prices to try to win the
shoppers. Firms price over an interval as there is no pure strategy on a price that works
with these two different groups of customers.
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The interval of randomization of prices for the largest firm is not the same as the
interval of prices for the smallest firm(s). The largest firm has a lower minimum price
than the smaller firm(s). This implies that the largest firm has an atom in its distribution
at the monopoly price. With the atom, the largest firm does not compete as aggressively
and thus all firms have the same interval of prices.
Increasing the group of loyal customers for the largest firm softens competition
for all firms since the largest firm places more weight on the monopoly price. Increasing
the group of loyal customers for the smaller firms has no effect on the smaller firm’s
probability distribution as they compete for shoppers at all prices in their distribution.
Profits for the smaller firms, however, increase when the loyal customers for the smaller
firms increase. Increasing the loyal customers for the smaller firm(s) lowers the atom for
the largest firm and increases the discounting of the largest firm. Increasing the customer
group of shoppers causes all firms to be more aggressive in discounting. The lowest
price falls and cumulative probability weight increases at every price below the
monopoly price. The atom for the largest firm decreases.
If the loyal group of customers for the largest firm is increased at the same time
the loyal customers for the smallest firms are decreased, then all firms discount less and
more weight is placed on the monopoly price for the largest firm. If the loyal group of
customers for the largest firm is increased at the same time the shoppers are decreased,
then the smaller firm(s) discount less. The larger firm generally discounts less, especially
if the number of firms grows larger. If the loyal group of customers for the smallest firms
is increased at the expense of the shoppers, then the smaller firms discount less, because
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there are less shoppers. Under this scenario, the largest firm discounts more at lower
prices and discounts less at higher prices.
The cumulative probability of the minimum price is created and applied to the
model. Increasing the loyal customers for the largest firm decreases the cumulative
probability of the minimum price. Increasing the loyal customers for the smaller firms
increases the cumulative probability of the minimum price at all prices except at the
monopoly price. At the monopoly price, the minimum price statistic is zero, thus
showing that the atom has to be with the largest firm. Increasing the shoppers generally
increases the cumulative probability of the minimum price. Increasing the number of
firms lowers the cumulative probability of the minimum price. Increasing the amount of
loyal customers for the largest firm, while decreasing the loyal customers for the smaller
firms, lowers the cumulative probability of the minimum price. Increasing the amount of
loyal customers for the largest firm, while decreasing the shoppers, generally lowers the
cumulative probability of the minimum price. Increasing the amount of loyal shoppers
for the smaller firms while decreasing the group of shoppers lowers the cumulative
probability of the minimum price as the number of firms grows larger.
These three chapter, thus begin modeling some of the characteristics of the airline
industry since deregulation. All three chapters have price dispersion between firms. As
airlines have complicated yield management strategies to maximize revenue, these three
chapters model airline pricing behavior as a mixed strategy over an interval of prices. All
three chapters model loyal customers. Frequent fliers are important to airlines’ business
strategy. Modeling the strategy airlines use to keep frequent fliers paying high fares
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takes a variety of approaches in the dissertation. Chapter one separates loyal customers
as business and leisure travelers while chapter three models different sizes of loyal
customers. Chapter two is unique in that it considers capacity limitations that airlines
face. Loyal travelers, plus the residual customers that are rationed from buying from the
lower priced firm, purchase from the higher priced firm.
0-10. Future Research
Future research could include capacity limitations with differing valuations of
customers. Airlines many times sell out their lower price seats and then leave the highest
seats left for the business traveler. Differing costs of the airline industry could be
modeled. How would the addition of a lower cost carrier affect price distribution?
Extending these models into multiple periods might allow more flexibility in modeling
frequent fliers. Airlines sometimes allow twenty five percent of seats on certain routes to
be award and non-revenue passengers. Finally, network and multi-market effects can be
studied. All three chapters focus on one particular route. What happens if there is more
than one route? Thus like the ever changing airline industry, there is an everlasting
supply of future models that can be used to further explain the industry.
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1: “Price Dispersion with Differing Consumer Valuations”
1-1. Introduction
In the last few years questions have arisen on how the growing influence of the
internet and electronic commerce will ultimately impact firms’ pricing power. Early
analysis in the press focused on the Bertrand perspective, describing the very diminished
pricing power. Most predictions were that old economy firms with established market
power (e.g., “brick and mortar” companies) that did not adjust quickly to this new
medium of commerce were going to be undercut, and that the only firms gaining market
share would be the lowest cost firms, those had adjusted to the new cyberspace world by
selling at razor thin margins.
These predictions are no longer found in the mainstream business press. Brick
and mortar companies that were slow to enter cyberspace are not generally moving
toward bankruptcy as many had initially predicted. Many have entered the world of
cyberspace, but still conduct a significant portion of their business away from cyberspace,
while almost all of the high flying internet companies’ stock valuations have crashed.
The clear-cut predictions of how industry was going to be shaped have either been
discredited or extremely clouded by recent market events. What then does the rise of the
internet mean for markets?
The answer may lie in markets that have already had a history with electronic
commerce. Those industries that had already been subjected to electronic price
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competition have only continued to evolve with the spreading of cyberspace. How the
internet affects these industries may be a preview of how others may be affected in the
future. One such industry is the airline industry, which has had to contend with the
emergence of internet ticket sales.
In October 1999, the Wall Street Journal ran a front page article discussing the
impact of the electronic commerce on airline pricing. In this article, Wharton Business
School Professor Eric Clemons argues that the online travel agents help prevent a "full-
scale price war for airline tickets (Clemons, 1999)." Further, he states that
"Economists were offended by this [result]. They said it can't be stable - the
Internet must provide perfect competition. … [However,] it isn't [a perfectly
competitive environment]. If a product is complex enough, sellers can avoid
competition by serving different customers."
The offended economists referred by Clemons generally assume, in the Jevons
tradition, that there is one price that clears markets. Most economic models are built
under this assumption. However, there are economists who have shown how the world
outside of one clearing price can exist, well before the Internet became mainstream.13
In
fact, three patterns suggesting equilibria beyond one price have become more
widespread: promotions, coupons, and major price swings.
13 For example, Stigler (1961) reports of an earlier study about the identical make and model of Cheverolets, which have a standard deviation of $42 on a average price of $2436 in February 1959, evenafter assuming an "average amount of 'higgling.'" This works out to be plus or minus 3.4% for twostandard deviations.
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First, as the importance of promotions grew, major retailers such as Albertsons,
Wal-Mart, and Target continue today to promote sale information to their consumers by
the mail or newspaper. Flyers advertise different items that are on sale each week.
Today, major airlines, such as Southwest Airlines, email their customers about the
markets that have reduced prices that week. Second, the distribution of coupons14 have
become ubiquitous: Sunday newspapers have their own coupon section (or two).
Booklets are distributed around campuses, stores, and malls. Packaged products have
their own coupons. Stores have kiosks generating in store coupons. Finally, consumers
can print coupons off the internet. Their use may be advertising toward a new product or
to get consumers to buy more of a certain product. Sometimes coupons try to entice
consumers to change their cross-brand purchasing habits such as what brand of
hamburger buns they purchase with the hamburger meat (Nazareno 2003). Coupons
requiring two or three different items purchased before the coupon face savings is passed
on to the consumer are more common today. Department stores have coupons that have
consumers choose what item receives the percentage discount.
Finally, some markets have major price swings. Retailers may offer an all out
sale of up to 40% - 50% off instead of a controlled weekly promotion. Department stores
are more likely to practice in this behavior than the retailers such as Wal-Mart. Airlines
also engage in this sort of behavior. Large fare wars often develop between the large
carriers that end with ever changing deadlines when the sale will end. Sometimes an
individual airline plays guerrilla tactics: offering large price discounts within markets
14 Though the distribution of coupons have remained constant the past five years, the redemption of coupons has declined the last five and ten years. (Nazareno 8H)
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1-2. Literature Review
The ‘law of one price’ and its competitive equilibrium foundations date to Jevons
(1871). Starting with Diamond (1971) there have been a large number of search-theoretic
explanations of non-competitive pricing. Explanations have differed both in
assumptions- differing search costs of consumers, differing abilities to price discriminate
consumer types, and different information seen by consumers – and in conclusions –
convergence to monopoly price, divergence to a few prices, and dispersion over a price
interval.
Diamond (1971) gave the first search theoretic explanation of convergence to
non-competitive prices. Diamond emphasized in his paper that adjustment processes
should be based on real influences in the economy rather than being an arbitrary process
designed to yield the competitive equilibrium. He develops a dynamic model that had
prices converge to the monopoly price due to the cost of search.
Diamond assumes that the mfirms symmetrically serve a market for a discrete
good every period; each having 1/mfraction of the certain market demand curve Xt(p).
Diamond uses the cost of search in two ways. First, he demonstrates how the consumer
choke price qt+1 for the good one period from now would be greater than the choke price
today qt due to the cost of search. Using this individual consumer behavior, he shows
that the minimum price tp in the market that allows everyone to buy the product (or the
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minimum of all consumer choke prices) in a particular period must be nondecreasing
over time. When it is profitable ( tp < p*, the optimal monopoly price) firms will price
between the minimum full participation price tp and the price tp + c, where consumers
will find it worthwhile to switch to another firm. Given that tp is nondecreasing over
time, the price charged in the market pt converge to the monopoly price p*.
Rothschild (1973) echoed Diamond in challenging economists to develop models
that have endogenous causes for price dispersion. From this challenge, there became
three different schools of thought how prices can vary. Firms with pricing power can
employ price discrimination to distinguish between the varying types of consumers.
Using observable characteristics or choices that consumers have made themselves from a
list or menu, firms distinguish between the various types of purchasers. The dispersion of
prices comes from consumers in each category paying different prices. Examples of
price discrimination include separating consumer types by age or coupons.
Two main issues focused by the price discrimination literature has focused on is
how price discrimination affects welfare and the role of arbitrage. Does welfare improve
or decline with price discrimination compared with uniform pricing? For instance,
Schmalensee (1981) and Varian (1985) set conditions on output where price
discrimination will enhance or at least not harm welfare. Nahata, Krzysztof, and
Ostaszewski (1990) tackle the welfare question in a different angle. They show that that
prices may increase or decrease within submarkets due to third degree price
discrimination but profits will increase unilaterally.
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Beside welfare issues, price discrimination has focused on the issue of arbitrage.
On the global level, will another firm go into business into a market where firms are price
discriminating? By buying at the lower price and then selling it at a slightly higher price
to the segmented consumers paying higher prices the firm helps break down the effects of
price discrimination. The issue of arbitrage can also be modeled on a personal level.
Firms offer a menu of prices for consumers to choose. Seeing this menu, an individual
consumer may misrepresent his or her type or change behavior to get a lower price.
Taking this consumer arbitrage into account, firms create the right incentives for
consumers. Firms maximize profits subject to individual rationality constraints –
ensuring that all types of consumers participates - and individual compatibly constraints –
making sure each offered menu only appeals to those right types. The bundles offered to
the consumers are set by this method so that the incentive for a type to choose a price
offering not directed to that type is minimized.
Other factors can explain price dispersion besides price discrimination on the
basis of observable characteristics. In these dispersion models, firms “offer” a random
price and it is this randomness that is the source of price dispersion. Rather than offering
a single price for sure, firms engage in mixed strategy behavior.
Mixed strategy behavior by firms may produce higher profits than pure strategy
behavior when consumers have positive search costs or when consumers do not have all
the relevant information. Mixed strategy behavior also could provide the only solution in
many of these games as pure strategy behavior cannot rest on any price due to the
incentives facing each firm. Firms could undercut each other at a particular price to gain
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more market share only to be further undercut at lower prices. This process continues to
marginal cost where profits are then higher at the monopoly price. The other two classes
of models explaining price dispersion fall under the mixed strategy equilibria.
The second category of models explaining price dispersion arise from search costs
of consumers. A typical example is Robert and Stahl (1993). They assume advertising
reaches some consumers but not others. The others must expend effort searching to find
the prices. In their model, when search costs increase, some consumers might find it too
costly to search for the best available price. Firms fearing the potential of losing their
base consumers randomly offer some discounts to induce people to shop at their store.
Firms advertise more heavily those discounts to consumers. To make up for these
discounts, firms raise the highest prices that the uninformed shoppers pay. Consumers
become more informed about the potential discounts as the probability of being
completely uninformed about any store’s offer falls. Reservation prices of consumers,
where the expected gain and cost of search are equal, rise with the increased dispersion of
prices and cost of search. When reservation prices rise to the point of consumer valuation
of the object or service, consumers start ending their search at the point where reservation
price equals valuation. Salop (1977), Salop and Stiglitz (1977), Carlson and McAfee
(1983), Rob (1984), and Stahl (1989) are other examples of this class of models.
The third class of models explaining price dispersion focuses on the ex post
information received by consumers rather than the search process. A complicated selling
strategy coupled with the incomplete ability of consumers to see all relevant information
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allows consumers to view firm pricing as random. Varian (1980) emphasized this point
in his model that consumers cannot learn the pricing behavior of firms over time.
Of this final approach to explaining price dispersion by deliberate randomization
in the years after Rothschild's challenge, three models - Butters (1977), Varian (1980),
and Burdett and Judd (1983) - emphasize the ex post information received by consumers.
Underlying each of these three models is the incomplete ability of consumers to see the
prices of other competitors. Some consumers were informed of the lowest price in the
marketplace and some consumers are less than fully informed, shopping only at one firm.
Dispersion arises in these models as firms balance between capturing monopoly profits
from consumers shopping from one firm and capturing the entire group of consumers that
search for the lowest price in the marketplace. The number of ads that consumers receive
in the mail drives Butters' results. Varian outright assumes the heterogeneity in
consumers by splitting his consumers into informed and uninformed types. Burdett and
Judd divides the informed and uninformed by allowing the probability of consumers
seeing more than one price to between zero and one.
Butters (1977) developed a model in which firms send out ads at a fixed cost per
ad (i.e. mailing costs) notifying consumers which store is running the special. A
consumer may or may not receive ads from firms. If a consumer receive ads in the mail,
she will purchase from the lowest price ad. Sellers know the limit prices of consumers
and the distribution of prices that other sellers offer but do not know exactly how many
ads that a consumer will receive. In choosing the number of ads to sell out, sellers will
weigh the expected benefit of sending one more ad out against the expected cost of
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printing the ad. The price randomization results from two forces in the model: First,
sellers will not price at marginal cost to capture the entire market since they will lose
money after factoring in the fixed cost per ad. Second, sellers cannot concentrate
advertising above this price since another firm can undercut the targeted price by epsilon
and capture the entire market. Thus sellers will have to randomize in equilibrium.
Varian (1980) develops a monopolistically competitive model to describe price
dispersion by firms. Costs by firms are assumed to be identical. Firms set their prices by
the week. Customers are divided between the informed and the uninformed types. Those
consumers that are informed will know what every firm is charging in the marketplace
and shop at the firm with the lowest price. The uninformed consumers will randomly
choose one store and pay whatever the price that the store is charging. Varian assumes
that the number of uninformed consumers is small enough that firms will have to also
seek the business of informed consumers. Like Butters, Varian shows that there cannot
be a mass point at any other price lower than the monopoly price because there can be
another firm that can capture the entire market of informed consumers by just pricing
epsilon lower than the concentrated price. In a specific case that the cost function has a
fixed cost and no marginal cost, Varian finds that distribution of prices will be
concentrated at the endpoints of the cost distribution: at the marginal cost and at the
monopoly price. Thus, the price density function of the fixed and no variable cost case
would suggest a pattern of firms occasionally running sales to get a portion of the
informed consumers.
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Burdett and Judd (1983) generate equilibria with price dispersion through the
search patterns of consumers. Firms produce a singular good or service at the same
marginal cost and select the price that they offer consumers. Consumers have the same
reservation prices, search costs and same methodology of searching. In a sequential
search, consumers obtain a number of price quotes at a time. A price dispersion
equilibrium occurs in the sequential search when the probability of seeing only one price
is less than one. The same price dispersion result holds if the search becomes noisy -
consumers do not know how many price quotes they will see within a given search.
Burdett and Judd also find a case of price dispersion in a more constrained nonsequential
search. In this type of all or nothing type of search, consumers initially have to decide
how many price quotes to receive before any can be observed. A price dispersion
equilibrium exists when the probability of a consumer seeing only one price is less than
one but the probability of seeing one or two prices is exactly one.
1-3. Motivation
Butters (1977), Varian (1980), and Burdett and Judd (1983) in this third class of
models have started how firms cater to consumers having different ex post information.
However, a limiting point in these models is that consumers are assumed to have the
same valuations for the good or service. This paper will explicitly explore how firms will
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attempt to disperse prices with differing consumer valuations and information levels.16
I
will develop a game-theoretic model factoring both a difference of consumer valuation
and amount of information received by consumers. As this model is developed, the
airline industry provides an excellent motivation developing the model. Certain business
travelers may be willing to pay upwards of several times the value of the lowest sale
prices bought by leisure travelers. Yet, both types of consumers could be sitting next to
each other in coach. This wide valuation of airline service by consumers is also
accompanied by rapid price swings within the industry.
Figures 28 – 35 are airline one way route price data taken from a ten percent
random sample of Data Bank 1a in the second quarter of 1995. Round-trip tickets are
decomposed to one-way fares for ease of comparison. The figures are drawn using a
kernel density with the Epanechnikov method with the optimal bandwith. Notice that
there are multiple modes in most of the figures. Figure 30, for instance, shows four
major modes on the one – way ticket prices between Washington Ronald Reagan
National Airport and New York LaGuardia Airport. Minneapolis-St. Paul International
Airport to Chicago O’Hare International Airport also shows four modes in Figure 29.
Sometimes the density approaches zero between modes as Figure 32a shows on the
16 Salop (1977) developed a model factoring price discrimination and dispersion for a monopolist.
However, consumers were only differentiated in their costs of search and not of their underlying valuationsfor the good. Rob (1984) develops an elaborate multi-firm price dispersion model with different consumer search costs, but again there is not an independence between the cost of searching and private valuation.Wiesmeth (1982) develops a model closest to the objective of this model. He further develops upon Salop
by generalizing the monopolist model to include a joint distributional function ),( vcg spanning over all
consumers. In his paper c is the consumer cost to search and vis consumer valuation. Wiesmeth finds
conditions in his model generating price dispersion, but again these results only hold for the monopolist.
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Minneapolis-St. Paul International Airport to Atlanta’s Hartsfield International Airport
route.
The previous literature cannot explain the multiple modes nor the zero density in
the airline price data. Varian (1980), for instance, can generate at most two modes,
where the lowest mode is the instances when prices are on sale and a high mode, where
prices are at a monopoly price. He, however, cannot generate places in his distribution
where the incidences of prices are equal to zero. This counters what is seen in Figure 30.
He assumes that all consumers have the same valuation. This, however, does not seem to
fit completely the multiple modes that are observed on some of the routes in the airline
data, nor the experience that all travelers are alike. Business travelers are willing to pay
up to several times the fare that leisure travelers pay.
The model developed here incorporates a multiple valuation of consumers. Two
valuations are used – leisure travelers and business travelers. This simplification can
easily be extended to multiple valuations to give a result of multiple modes in the price
density. Important in the model developed here is that there is an interval where prices
are zero, thus matching what is observed in the airline price data. Figure 22, for instance,
shows this gap quite clearly in the price density generated by the model.
Other explanations, price discrimination and product differentiation, could seem
as the dominant explanations behind why there is dispersion behind airline fares.
Advance purchase requirements and Saturday night stay requirements have been well
documented as airlines use price discrimination devices to get different consumers to pay
different prices. However, a price discrimination story does not fully explain the pricing
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behavior employed by airlines. The Saturday night requirement is being weakened as
Delta Airlines announced in January 2005 its SimpliFares that lowered the maximum
ticket price and eliminated Saturday night stay requirements to obtain the lower fares
(Delta Airlines Press Release Jan 4, 2005). Furthermore, advance purchase requirements
are secondary to airline price randomization. Sometimes during sales, airlines place 7
day in advance fares cheaper than the 14 or 21 day in advance fares.171819
This makes a
strict price discrimination story by advance purchases difficult to justify.
Figures 31a -31k and Figures 32a -32f further decompose market fares into coach
Y and coach discount YD and break down the overall market shown in other diagrams to
individual carriers’ fare distribution for the New York City (3 airports – LaGuardia,
Newark, and JFK) to Orlando routes and Minneapolis-St. Paul Airport to Atlanta route.
For most carriers, coach Y fares are walk-up fares and coach discount YD fares are fares
with advance purchase. Coach Y fares may also be refundable and changeable whereas
coach discount YD fares may have restrictions on refunds and changes once purchased.
Price discrimination between the coach Y and coach discount YD might be the initial
explanation behind the difference of fares, as Y fares are generally higher than YD fares.
However, further analysis of each fare class, show price dispersion with multiple humps.
17 For instance, Independence Air announced a sale on August 2, 2005 for nonstop destinations out of Washington, DC requiring up to a seven day advance purchase for travel between August 9, 2005 andDecember 14, 2005. (http://www.flyi.com/company/pressarchive/defaul.aspx).18 On July 25, 2005 Delta Airlines announced a fall sale that had sale fares with a ten day in advance purchase requirement for the travel period between August 3, 2005 and November 16, 2005.(http://news.delta.com/news.index.cfm)19 Southwest Airlines has internet fare sales that require 7 or 14 day advance purchase. On July 5, 2005Southwest announced a 14 day advance purchase internet fare sale for numerous regions in the UnitedStates effective for travel from August 18 through October 28, 2005.(http://www.southwest.com/about_swa/press/prindex.html)
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For instance, Figure 31d shows that there is another peak for one-way fares above
$200 for Delta Airlines for the coach discount YD ticket besides the major peak of $130
one-way. Kiwi Airlines in Figure 31e has more of a pronounced second hump at fares
just above $200 one – way for its coach discount tickets besides its major peak at $130
one-way. Trans World Airlines has two peaks at $50 and $110 one way for the coach
discount tickets YD. American Trans Air has two peaks at $67 and $76 one way for the
same coach discount YD tickets. Figures 32c and 32d show multiple peaks for
Northwest and Delta Airlines respectively for coach discount YD tickets. Thus there is
price dispersion with clustering of fares controlling for price discrimination by using only
discounted coach YD tickets that have advance purchase requirements and fee – based
refund and change restrictions.
Interestingly, there is some clustering of fares into multiple peaks for the next
fare class higher than the discounted tickets. This occurs on the full coach Y ticket class
in the New York City to Orlando route. Continental Airlines, for instance in Figure 31i,
has price dispersion with multiple peaks on its full coach fare Y tickets. There is a major
peak at above $200 one-way and two higher peaks at $250 and $300 one-way.
Continental Airlines also a smaller peak at $125 one way for its full coach Y fare which
is about the same price as the single peak for coach discount YD ticket for the airline in
Figure 31c. Delta Airlines has two close peaks for its coach Y fares at $275 and $300
one-way but also has a smaller peak at $175 one-way in Figure 31j. US Airways has a
major peak at $310 for its full coach Y fare and a smaller peak at $270 one-way in Figure
31k. Thus Figures 31i – 31k show that there is price dispersion with clustering of fares
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even at the more expensive full coach Y fare. Price discrimination cannot explain the
dispersion between the fare categories between coach discount YD and coach Y fares.
In fact, the price discrimination story is countered by there not being a neat
separation of various types of consumers paying separate fares. Despite all of the
intricate number of rules accompanying each type of fare, there are always clever
consumers and travel agents working around restrictions to pay lower fares.
Unanticipated consumer actions may result from tariff restrictions.20 Furthermore,
airlines, like many other firms, do not always have the luxury of cleanly differentiating
their products to separate markets. Generally, airlines have little flexibility to
differentiate their products, as it hard to do much with coach seats closely spaced together
in a compact aluminum tube 33,000 feet above the ground. Airlines constantly adjusting
inventory, tariff rules, fares, and enforcement is evidence against a neat separation
dividing consumer types.
Further compounding the difficulty of neatly separating different consumer types,
airlines face the unique problem of keeping enough of the highest valuation consumers
paying the top price. The major US airlines (the discount carriers such as Southwest
Airlines have less of this problem) greatly depend on these consumers paying top fares.
A sizeable portion of their revenue is generated from this small group of high revenue
20 For instance, to beat the short-notice high fare, business consumers traveling frequently on a route (egMSP – DFW) can purchase an initial one-way ticket (MSP – DFW) and then round trip tickets on thereverse itinerary (DFW – MSP), using the outbound portion of the ticket to return home and changing thereturn ticket of the roundtrip (MSP – DFW) for the smaller change fee once business dictates another triprather than purchasing a new roundtrip (MSP – DFW) and incurring the large fare resulting from a purchase of a short-notice ticket. Airlines have cracked down on this practice, but they cannot do anythingto travelers that spread this tactic between airlines (such as purchase the initial one-way ticket on NorthwestAirlines between MSP and DFW and American Airlines roundtrip between DFW and MSP).
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passengers paying up to several times the amount of the lowest discounted fare. If
enough of these customers can easily figure out how to beat the system, then tremendous
savings will be reaped at the expense of the major airlines. Airlines have to be creative to
keep as much of this group as possible paying the top fare. How airlines operate in this
messy environment of serving so many different types of consumers can help provide a
motivation behind developing a price dispersion model with consumers with distinctly
different valuations.
As a result of these current airline practices, consumers cannot be sure what fare
to expect when they purchase a ticket. There is not a great way that consumers can
adjust the timing of their purchase to receive a particular fare. On one hand, certain
markets could have a limited number of discount seats sell out way in advance of the
departure date. Predicting when this event occurs is impossible; it may be even difficult
to find out what the lowest price is in the marketplace.21
However, purchasing a ticket
way in advance of the travel date does not guarantee consumers the lowest fare. Airlines
may place inventory on sale at any time, even just a short time period before upcoming
departing flights. Predicting when the next sale, let alone what routes will be covered by
fare sales is nearly impossible. Furthermore, airlines rapidly change their capacity
controlled fare class seats for posted fares very frequently. Consumers and travel agents
have very little information on immediate airline pricing objectives on particular flights,
days, cities, and/or networks.
21 All airlines have agreed in their 1999 compromise with the US Government is to display the lowestcurrent fare in a market. The fare itself, capacity assigned to the fare, or tariff rules could rapidly change.
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Besides the airline industry, the rise of the internet and electronic commerce may
also add other examples where price dispersion may coexist with different buyer
valuations. The rise of commerce through these new means has only accelerated the
ability of firms to offer varying prices. Without employing direct price discrimination
methods, firms may be able to price to different consumer types. Periodic weekly fliers
packing cutout coupons and full newspaper advertisements of upcoming sales have given
way to online auctions, email by companies listing items on sale, and computerized
search engines surveying multiple company sites for the lowest prices possible.
It is no surprise that US airlines are among the first to integrate electronic
commerce within their pricing strategies. During the past few years many of the major
US airlines have developed a program of emailing weekly Internet specials to consumers
registered at their websites. Some are even going further: United Airlines announced a
campaign to publish twenty different "DailE-Fare[s]" each day that started on October
20th , 1999 (United Airlines Press Releases - October 20, 1999). Southwest Airlines
released this year their DING! program linking users’ computers to their system to give
quicker access to lower fares. Sometimes these fares are only valid for a few hours
(Southwest Airlines Press Release February 28, 2005).
This model will allow for multiple firms to compete consumers with differing
valuations and informational gathering abilities. Given this heterogeneity, will firms
randomize prices and how much will prices fluctuate? Will a firm find it more
worthwhile to specialize to only one end of consumers? Is it possible for a firm to find a
strategy of selling the same product simultaneously to both ends of the market? By
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addressing these questions, this model will add the case of heterogeneous consumer
valuations to the price randomization literature. This will give another structured
explanation about the airline industry and possibly offer a glimpse of how electronic
commerce in some markets may evolve.
1-4. The Model
Assume that there are n firms: 1 through firm n. Firm i sells its product at price
pi. Consumers shop for one indivisible unit, such as an airline ticket. Assume that all
firms in the market have the same marginal cost c of providing service. We will simplify
the various types of consumers down to two: high valuation business types H and low
valuation leisure travelers L. Let θ be the proportion of High types, 1 - θ be the
proportion of low types.
Consumers are split into the informed and uninformed types. Of the θ proportion
of H type consumers, let α be the proportion of H consumers that see the prices of only
one firm. Keeping with our airline example, these uninformed types could also be
thought as inflexible or loyal travelers, only willing to purchase from one particular
airline. Thus the total proportion of uninformed high type consumers is αθ. Assume
initially that these α uninformed high type consumers are randomly dispersed between
the n firms. Thus θα/n be the portion of high type consumers willing to buy from only
one firm. The proportion of H consumers that see all prices is 1 - α. If a firm prices
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lowest among the n firms, it will capture these 1 - α informed high type consumers plus
its share of the high type consumers α/n.
Of (1 - θ) low type consumers, let β be the proportion of consumers that see only
one price. Again assume that these β consumers are split evenly between each firm.
Thus (1 - θ)β/n consumers see prices by all firms and the proportion β/n see only one
firm's price. The firm that prices the lowest will capture (1 - θ )(1 - β) + (1 - θ)β/n of the
low types. The first term is the low types seeing both prices whereas the second term is
the share of the low type seeing only one price. The following chart summarizes the
sales to high and low types, ignoring ties which we will show do not arise in equilibrium.
Let H and L be the prices that are the maximum reservation valuation of the high and low
types.
Table 4 Sales to High and Low Types:
Sales to High Types Sales to Low Types
pi lowest of all firm prices( ) ip
n ⎥⎦
⎤⎢⎣
⎡ +−θα
α θ 1 if Hpi ≤ ( )( )( )
ipn ⎥⎦
⎤⎢⎣
⎡ −+−−
β θ β θ
111
if Lpi ≤
pi not lowest of all firm
prices
ipn
θα
( )ip
n
β θ −1
If the firm is pricing the lowest among the low types, it will also capture the high types:
θ(1 - α) informed and θα/n uninformed consumers, or the upper left corner of the table.
Pricing the lowest among all firms captures the top two boxes of the above chart. After
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some algebra, the highest proportion T of consumers a firm can capture when pricing the
lowest is:
T = ( ) ( ) ⎟ ⎠ ⎞⎜
⎝ ⎛ −−−+⎟
⎠ ⎞⎜
⎝ ⎛ −−
nn
nn β θ α θ 11)1(11 (1)
Call the total proportion of informed types a firm can possibly capture
R = ( ) ( )( ) β θ α θ −−+− 111 (2)
Like the Varian (1980) model of sales, firms randomize between socking
consumers seeing only their price and capturing all of the consumers seeing what all
firms charge. However, the strategy becomes more complicated since the firms face
consumers with two different valuations. A firm may choose to sell only to the high
types of consumers if it charges a price above the highest price that the low valuation
consumers are willing to pay. A firm could also set its strategy of extracting monopoly
profits on low types that see only one price, losing some revenues on the high types, and
possibly capturing the informed high types if the other firm prices higher. Finally, a firm
could capture all of the informed consumers by a low price, at the expense of monopoly
profits on the uninformed high and low types. With these four different consumer types:
informed high, uninformed high, informed low, and uninformed low, we will look for
symmetric equilibrium in our model.
We will first clarify the range of prices a firm will charge. First, a firm will not
charge higher than H, the highest possible consumer valuation, this is dominated. Since
consumers are purchasing one discrete unit, H is the monopoly price a firm could charge
the high types only. Likewise, L is the monopoly price that a firm could charge its low
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types. Should a firm price above L, it will only capture the business of the high types.
Let λ be the minimum price that a firm would possibly charge. This minimum price λ is
greater than or equal to the marginal cost c, since at any price p < c, is also dominated.
In a symmetric equilibrium let F(p) be the cumulative probability distribution of
[ ]Hp ,λ ∈ . There are three important features of F(p) on the interval between these two
prices. First, if p < H, then F(p) < 1. If not, and , ,1)( 00 HppF <= a firm could raise its
price to H and increase consumer profits. This assumes that profits at H from a pool of
uninformed high valued customers are bounded below by profits at L from a pool of
uninformed high and low valued consumers. This also assumes profits at H from a group
of high valued uninformed consumers are bounded above by profits at L from a larger
pool of total high and low valued informed and high and low valued uninformed
consumers. Both of these statements are proven as Proposition 1. Second, there is no
mass point and therefore no ties on [ ]Hp ,λ ∈ . Suppose there is a mass point of F(p) in
the interval (λ,H). Then, another firm could make a discrete jump in profits by pricing ε
below this mass point. The firm doing this captures the entire group of informed
consumers that were paying the price at the mass point for only a small loss of ε below
the mass point price. Because the incentive is too great to price below the mass point
price, no mass point survives. Finally, F(p) will be strictly increasing except over an
interval where firms are switching over from collecting monopoly profits from the low
types and selling only to the high types. Proposition 9h shows that is a minimum gap
where F(p) is flat, assuming low and high valuations are sufficiently separated.
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A firm can target a number of consumer types. A firm may decide only to capture
the business of its share of the H type uninformed consumers. At the monopoly prices H,
the expected profits per consumer are:
πMH
= πH= θα/n ( H – c ) (3)
If a firm decides to capture more than uninformed H types, then the firm's price will have
to be lowest among all firms to capture all of the informed consumers with a valuation
greater than or equal to the price. Given that the probability of any firm having the
lowest price in the market is less than one, expected per consumer profits πemust equal
per consumer profits πMH at the price H. Unlike the Varian model, the number of
informed consumers a firm captures depends on whether the price charged is above or
below the highest valuation of each consumer group. When the price is below H, the
expected profit per consumer is:
πe =
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
≤−−+−−+−+=
>−−−+=−
−
(5) pif c)-)]}(1)(1()1[())(1(/)1(/{
(4) pif )]()1())(1(/[
1
1
LppFnn
LcppFn
iin
il
iin
ih
θ β θ α β θ θα π
θ α θα π
The first terms of equation (2) are the expected profits from the H consumers that only
see one price from the firms. With probability one this gives per consumer profit of
[θα/n] ( pi -c). The second term is the expected profits from the high types that see all n
prices. With probability (1 – F( pi ))n-1 the other firms -i will all have higher prices. In
this contingency, firm i will capture the profits of the informed H type consumers. Low
type consumers do not matter in equation (4) because the price is above the valuation of
the low type consumers. If the price pi charged is less than or equal to L, then low type
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consumers will matter. In this scenario, the first and second terms in equation (5) are the
expected profits from the high and low type consumers that see only that firm's price.
The second term in equation (5), (1 - θ) β/n, is the low types that see only one price. The
third term in equation (5) is the expected per consumer profits that a firm can expect from
the informed consumers. With probability (1 – F( pi ))n-1 the other firms -i will have
higher prices, thus allowing firm i to get the business of both informed high types θ (1 -
α) and the informed low types ( 1 - θ ) (1 - β ). When the price exactly equals L, the
firm will still get the business from the uninformed high types, informed high types,
uninformed low types, and informed low types with probability (1 – F(L))n-1
. The
expected per firm profit when charging L is:
πl(L) = { θα/n + (1 - θ) β/n + (1 – F(L))n-1 [(1 - α) θ+ ( 1 - θ ) (1 - β)]}( L – c) (6)
Let:
πML = πL = [θα/n +(1 - θ) β/n] (L-c) (7)
be the monopoly profits from charging L, assuming that there are no high types.
The profit equation for a firm is:
( )
( ) ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
≤≤
>≥=
(5)equationL pif
(4)equationL pHif
λ π
π π
p
pl
h
In equilibrium, all firms will randomize over prices so that expected profits are constant
at any price with positive density:
πMH = πh(pi)
= πl(pi) = θα/n ( H - c) (8)
The discrete jump of customers that are willing to buy at L creates a discontinuity
in the price distribution F(pi) of firms. As the price moves from L to L +g, a firm loses a
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discrete number of consumers – the uninformed low types while gaining an ε increase in
revenue. The change of expected profits per consumer moving from L to L + g is:
( )( ) ( ) ( ) ( ) ( )cLLFRnn
cgLgLFn
nn −⎥⎦⎤
⎢⎣⎡ −+−+−−+⎥⎦
⎤⎢⎣⎡ +−−+ −− 11
)(11
)(11β θ θα
α θ θα (9)
In a price dispersion equilibrium, this expected difference in profits moving from L to
L+g must equal zero.
( ) ( )( ) ( ) ( )( )( ) ( ) 0)(111)(11
1 11 =−−−−−+−−+−−
− −−cLLFgLFg
n
cL
n
g nn β θ α θ
β θ θα (10)
For small price increases above L, equation (9) will be negative since the benefit gained
on increasing the price by ε is greatly overshadowed by the discrete loss of revenue from
low type consumers seeing one price: (1 - θ)β/n (L - c). Any markup price over L must
be large enough to counter the discrete loss of L type consumers. Let M be the lowest
price that is greater than L where expected profits are the same. We know that the
marginal probability f(pi) equals zero in the region between L and M, or the interval
(L,M) due to the lower expected profits. Given that f(pi)=0 in this region and that there
are no mass points on the pricing distribution, F(L) = F(M). Thus, the solution to
equation (10) is well defined and simple.
This paper studies the properties of symmetric mixed strategy pricing equilibrium
in which all firms randomize according to a cumulative distribution function
demonstrated in Figure 1. In the case of equilibrium existing with this type of price
dispersion, there will be three endogenous variables that will be solved in this model:
F(ÿ), g, and λ. The exogenous variables are L, H, n, α, β, and θ. Solving for the
cumulative price distribution F(ÿ) requires accounting for the flat region between L and
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M. Solving for F(ÿ) is easier if we separate out the flat region and solve for the two
regions on either side of the flat piece of the cumulative distribution. Let the overall
probability distribution of prices F(ÿ) be broken into two pieces: FH(ÿ) and FL(ÿ). Let
FH(ÿ) be the portion of the price distribution when M≤ pi ≤H and FL(ÿ) be the portion of the
price distribution when λ≤ pi≤L. g is the flat region of prices between FH(ÿ) and FL(ÿ).
Price
F(p)
c
→)( pFL
)( pFH←
••••••
•MH
••••••
•L
•••••••••••••••••• g
↓
0
Figure 1
ô
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Table 5 Properties of F(p):
Prices Value of F(ÿ) Properties of F(ÿ)
P = l FL(l) = F (l) = 0 ----
P œ (l,L]FL(p) =
( )1/1
)(
)(1)(1
−
⎥⎦
⎤⎢⎣
⎡
−−−−−
−n
i
ii
cpnR
cppH β θ θα
Increasing
P œ (L,M) f(p) = 0 Constant Fig. 22 and 23
P œ [M,H)FH(p) =
))(1(
)(1
1/1 −
⎥⎦
⎤⎢⎣
⎡
−−−
−n
i
i
cpn
pH
α
α
Increasing
P = H FH(H) = F(H) = 1 ----
The specific values of the cumulative distribution functions in Figure 1 are given in Table
6.
Table 6:
(Prop 2) FL(pi)= ( )1/1
)()(1)(1
−
⎥⎦⎤⎢
⎣⎡
−−−−−−
n
i
ii
cpnRcppH β θ θα (15)
(Prop 3) FH(pi) =))(1(
)(1
1/1 −
⎥⎦
⎤⎢⎣
⎡
−−−
−n
i
i
cpn
pH
α
α (17)
(Prop 4) λ =( )
( ) ( )c
n
n
n
n
cHn +
⎟ ⎠
⎞⎜⎝
⎛ −−−+⎟
⎠
⎞⎜⎝
⎛ −−
−
β θ
α θ
αθ
11)1(
11
(19)
(Prop 5) g =
( )
( ))(
)())(1(
)1(
)())(1(
)1)(1(
cL
cLn
cHn
cLn
cHn −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
+⎥⎦
⎤⎢⎣
⎡ −−
−−−
−⎥⎦
⎤⎢⎣
⎡−−−
β α θ θ θα α θ
β α θ θ θα β θ
(22)
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Proposition 1: The cumulative distribution function described in Table 6 is a symmetric
equilibrium iff
a) α or β is greater than 0
b) α or β is less than 1
c) [θα/n + (1 - θ) β/n](L - c) § θα/n(H - c)
d) θα/n(H - c) § T(L - c)
e) {θα/n + (1 - θ) β/n +[θ (1 - α)+ (1 - θ)(1 - β)][1 - F(p)]n-1
}(p - c) =
θα/n(H - c) p œ [l, H]
Figure 2 below summarizes Proposition 1. On the horizontal axis is L – c and on
the vertical axis is H – c. Two conditions spelled forth from proposition one are graphed
– the third condition stating that the uninformed monopoly profits at H must be equal or
greater than the uninformed monopoly profits at L is the right hand diagonal line. To the
left of this line Proposition 1c is satisfied. The bottom triangle third region is where
Proposition (1c) does not hold. Here is where there is an equilibrium for low types only.
Likewise the left-hand diagonal line extending from the origin is the condition for
Proposition (1d). This part of Proposition 1 spelling forth the condition that the
uninformed monopoly profits at the monopoly price H must be less than the total
consumer profits at the price L is satisfied below the left diagonal line. Above the left
diagonal line Proposition (1d) fails. Here there is an equilibrium consisting only of the
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high types. Between the two diagonal lines proposition one is satisfied and a price
dispersion equilibrium with two different consumer types exists.
Figure 2
Proof (1e):
Suppose
{θα/n + (1 - θ) β/n +[θ (1 - α)+ (1 - θ)(1 - β)][1 - F(p)]n-1 }(p - c) > θα/n(H - c)
and the symmetric equilibrium described in Table 6 holds. Then firms will find it
profitable to randomize at prices higher than H. This violates the symmetric equilibrium
described in Table 6. Suppose
( )( )cL
n
n
n
n
n
cH −
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧⎥⎦
⎤⎢⎣
⎡ −−−+⎥⎦
⎤⎢⎣
⎡ −−
≤−θα
β θ α θ 1
111
1
( )( )cLcH −⎥⎦
⎤⎢⎣
⎡ −+≥−
θα
β θ 11
L-c
H-c
Price disp.with 2
cons.valuations
Conditionsfor Prop 1dfails –HighType OnlyEquilibriu
Conditions for Prop 1cfails –
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{θα/n + (1 - θ) β/n +[θ (1 - α)+ (1 - θ)(1 - β)][1 - F(p)]n-1
}(p - c) < θα/n(H – c)
and the symmetric equilibrium described in Table 6 holds. Then firms will find it not
worthwhile to randomize and price only at H. This violates the symmetric equilibrium
described in Table 6.
Proof (1a):
Suppose that α and β equal 0 and the symmetric equilibrium holds in Table 6.
Then from Proposition (1e):
{θα/n + (1 - θ) β/n +[θ (1 - α)+ (1 - θ)(1 - β)][1 - F(p)]n-1 }(p - c) = θα/n(H - c)
[1 - F(p)]n-1 }(p - c) = 0
There will be no randomization violating the equilibrium described in Table 6.
Proof (1b):
Suppose that that α and β equal 1 and the symmetric equilibrium holds in Table 6.
Then from Proposition (1e):
{θ(1)/n + (1 - θ) (1)/n +[θ (1 - 1)+ (1 - θ)(1 - 1)][1 - F(p)]n-1 }(L - c) =
θ(1)/n(H - c)
{θ/n + (1 - θ)/n +[0][1 - F(p)]n-1
]}(L - c) = θ/n(H - c)
[θ/n + (1 - θ)/n ](L - c) = θ/n(H - c)
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Thus there is no randomization violating the equilibrium described in Table 6.
Proof (1c):
Proposition (1c) deals with the case when it may be profitable for firms to sell
only to the low types. In this case, the loss of low type consumers is too large for a firm
to make up with higher prices. Suppose that there is a symmetric equilibrium described
in Table 6 but
[θα/n + (1 - θ) β/n](L - c) > θα/n(H - c). This could happen if there is a small difference
between L and H, or if there are much smaller numbers of high type uninformed
consumers than low type uninformed consumers. Thus it will not be profitable for a firm
to go after the high types. In a symmetric equilibrium, this implies that M > H.
Replacing L + g with H in equation (9) and remembering F(H) = 1 gives the highest
spread between L and M:
θα/n(H - c) -[θα/n +(1 - θ)β/n + R(1 - F(L))n-1](L - c) = 0 (11)
Notice in equation 11 that F(L) has to be less than one since F(H) = 1. If F(L) = 1 then
R(1 - F(L))n-1](L - c) will drop out since (1 - F(L)) = 0. In this case, firms will not
randomize to the high types and thus M > H. When M > H, expected profits at H will be
lower than expected profits at L. Firms will not find it worthwhile to price higher than L.
Thus M < H and for a firm to go after the high types:
θα/n ( H – c ) > [θα/n + (1 - θ) β/n](L - c) (12)
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Proof (1d):
This next part of the proposition deals with the case when it may be profitable to
sell only to the high types. A firm focusing only on high types will get expected profits of
θα/n(H - c). The most consumers a firm could get at the price L is
[θα/n + θ(1 - α) + (1- θ)β/n + (1 - θ)(1 - β)] or T. When firms find it profitable only to
focus on the high types, then it must be the case that the addition of θ(1 - α) + (1- θ)β/n +
(1 - θ)(1 - β) consumers from discounting does not make up for the lost revenue from the
uninformed high types. Expected profits will fall by selling also to the low types.
Suppose that there is a symmetric equilibrium described in Table 6 but
θα/n(H - c) > T(L - c). Thus it will be profitable for a firm to only go after the high
types. In a symmetric equilibrium, this implies that M § L. Setting equation (9) to
equal zero, moving from L to L + g in a symmetric equilibrium described in Table 6 must
result in zero profits:
( )( ) ( )( )
( ) ( ) 0)(11
)(1111 =−⎥⎦
⎤⎢⎣
⎡−+
−+−−+⎥⎦
⎤⎢⎣
⎡+−−+ −−
cLLFRnn
cgLgLFn
nn β θ θα α θ
θα (9a)
Suppose L = M and g = 0. Plugging into equation (9a):
( )( ) ( ) ( ) ( ) ( )( )[ ]( ) ( ) 0)(1111/1/)(11/ 11 =−−−−+−+−+−−−−+ −− cLLFnncLLFn nn β θ α θ β θ θα α θ θα
( )( ) ( ) ( ) ( )( )[ ]( ) 0)(1111/1)(1111 =−−−+−+−−−− −− nn
LFnLF β θ α θ β θ α θ
( ) ( )( )[ ]( ) 0)(111/11 ≠−−−+−− −n
LFn β θ β θ
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Thus M § L cannot hold. From equation (9a), there is a minimum for which ε can be to
make the equation hold. This minimum g will be found in a later proposition.
Proof Symmetric Equilibrium:
Suppose Propositions (1a) – (1e) hold but there is no symmetric equilibrium as
described in Table 6.
A. Proposition (1e) can be used to find FL(p):
( )( )
( ) ( ) ( )( )[ ] ( )cppFnn
cHn
in
i −⎭⎬⎫
⎩⎨⎧ −−+−−+
−+=− −
β θ α θ β θ θα θα
111)(11 1 (14)
Again, we replace F(ÿ) with FL(ÿ), use our definition for R (total proportion of informed
types) and combine terms:
( )( )
( ) ( )cpRpFnn
cHn
in
i −⎭⎬⎫
⎩⎨⎧
−+−
+=− −1)(1
1 β θ θα θα
( ) ( ) ( )( )
( )cpn
pHn
cpRpF iiin
iL −−
−−=−− − β θ θα 1)(1
1
( ) 1/1
)(
)(1
)(
1)(
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−
−−−=
n
i
ii
iLcpR
cpn
pHnpF
β θ θα
(15)
The ratio is similar to FH(ÿ) except that the low types now need to be included in the
analysis. The first term numerator still describes the decrease profits that a firm earns
from the uninformed high types when it prices far below H - when pi is in the [λ,L]
interval. However, by pricing low, the firm obtains the low valuation consumers. The
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second term in the numerator is the expected revenue obtained from the uninformed low
types. This term is negative because the firm will gain the business of these consumers
when it prices on the [λ,L] interval. Thus the numerator describes the net change in
revenue from pricing below H. The denominator is the total revenue obtained from all
informed consumers R: both low and high types.
B. Proposition (1e) can be used to find FH(p):
The β terms drop out of Proposition (1e) since the price is above L:
( ) ( )( ) ( )cppFn
cHn
in
i −⎥⎦
⎤⎢⎣
⎡ −−+=− −1)(11 α θ
θα θα (16)
Replacing F(ÿ) with FH(ÿ). and isolating the FH(ÿ) term we have:
( )( )[ ]( ) ( ) ( )[ ]cpcHn
cppF iin
i −−−=−−− − θα α θ
1)(11
))(1(
)(
1 )(
))(1(
)( )](1[
1/1
1
−
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−=
−−−=−
n
i
i
iH
i
in
iH
cp
pHnpF
cp
pHnpF
α
α
α
α
(17)
The ratio describing FH(ÿ) is quite straightforward: In the second term the numerator
describes the revenue lost when a firm prices below the monopoly price. The
denominator is the total market profit that can be earned from the informed consumers at
the price pi. This ratio and entire expression will equal zero and one respectively when pi
= H, thus matching the earlier statement that the highest price a firm will charge is H.
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C. l can be found by setting FL(p) =0:
Plugging in for FL(λ):
( ) 1/1
))](1)(1()1([
)(1
)(
1
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−+−
−−
−−−
n
c
cn
Hn
λ θ β α θ
λ β θ
λ θα
= 0 (18)
Remember R = [θ(1 - α) + (1 - θ)(1 - β)] or the total proportion of informed consumers.
Plugging R into equation (18a) and simplifying:
( ) ( )( )[ ] ( ){ }( ) ( )cHcn
cHcccnR
cnR
cH
−=−−++−−+−
−=−+−−+−
=−
−−−−
θα λ β θ θα β θ α θ
θα λ θα λ β θ λ
λ
λ β θ λ θα
1111
)()()()1()(
1)(
)()1()(
( )( ) ( )( )[ ] ( )
cn
cH+
−++−−+−
−=
β θ θα β θ α θ
αθ λ
1111(19a)
=( )
cnR
cH+
−++−
β θ θα
αθ
1
)((19b)
( )
( ) ( )( )[ ]( )
c
nn
cHn +
−++−−+−
−=
β θ θα β θ α θ
αθ
λ 1
111
( )
( )c
n
n
n
n
cHn +
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−−+⎟
⎠
⎞⎜⎝
⎛ −−
−=
111
11 β θ α θ
αθ
λ (19)
= cnT
cH+− )(αθ
(20)
= c))1n(n()1n)([(
)cH(+
β−−+θ−α−β
−αθ(21)
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= c])1([(])1(1[n
)cH(+
θα+θ−β+θα−θ−β−−αθ
(21a)
Equations (19a), (19b), (20), (21) and (21a) are modified versions of (19) that make
comparative statics calculations easier.
D. g can be found by plugging in L +g and L for the price in Proposition (1e)
and then subtracting the two equations:
( ) ( )[ ][ ] ( )
( )n
cH
cgLgLFcgLn
n −
=−++−−+−+− θα
α θ
θα 1
)(11
( )( )
( ) ( ) ( )( )[ ][ ] ( )( )n
cHcLLFcL
ncL
n
n −=−−−−+−+−
−−− − θα
β θ α θ β θ θα 1
)(11111
( )( ) ( )[ ] ( )( )[ ] ( ) 0)(111)(11
1 11 =−−−−−+−−+−−
− −−cLLFggLFcL
ng
n
nn β θ α θ
β θ θα
Adding and subtracting ( )[ ] ( )cLLFn −−− −1
)(11 α θ to both sides gives:
( ) ( ) ( )[ ] ( )[ ][ ] ( ) 0)(11)(111 11 =−−−−++−−+−−− −− cLLFRggLFcLn
gn
nn α θ α θ β θ θα
Simplifying we get:
[ ] [ ][ ] )()(1)1()1(
)(1)1(11
cLLFRn
ggLFn
g nL
nH −⎥⎦
⎤⎢⎣
⎡ −−−+−
=+−−+ −−α θ
β θ α θ
θα
[ ] )()(
)()1()()1(
)1(
)(
)(cL
cLnR
cLLHR
ncgL
cH
n
g−⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
−−−−−
−−+−
=⎥⎦
⎤⎢⎣
⎡
−+− β θ θα
α θ β θ θα
[ ] ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ −−−+⎥⎦
⎤⎢⎣
⎡ −−−=⎥
⎦
⎤⎢⎣
⎡
−+−
nR
cL
nR
LH
cgL
cH
n
g )()1)(1()()1)(1(
)(
)( β θ α θ θα β θ
θα
[ ] ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ −−−+⎥⎦
⎤⎢⎣
⎡ −−−=⎥
⎦
⎤⎢⎣
⎡
−+−
nR
cL
nR
LH
cgL
cH
n
g )()1)(1()()1)(1(
)(
)( β θ α θ θα β θ
θα
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⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−⎥⎦
⎤⎢⎣
⎡ −−−−−
=⎥⎦
⎤⎢⎣
⎡
−+−
R
cLn
cgL
cH
n
gH )(
)()1()1)(1(
)(
)(
β α θ θ π β θ
θα
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎦
⎤⎢⎣
⎡ −−−−−
=⎥⎦
⎤⎢⎣
⎡ −+
)()(
)1()1)(1(
)(
cLn
R
g
cgL
HH β α
θ θ π β θ π
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥
⎦
⎤⎢
⎣
⎡ −−−−−
−⎥⎦
⎤⎢⎣
⎡ −−+−−−
=⎥⎦
⎤⎢⎣
⎡ −
)()(
)1()1)(1(
)()(
)1()1)(1()(
cL
n
cLn
R
g
cL
H
HH
β α θ θ π β θ
β α θ θ π β θ π
g =)(
)())(1(
)1(
)())(1(
)1)(1(
cL
cLn
cLn
H
H
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
+−
−−−
−−−
β α θ θ π α θ
β α θ θ π β θ (22)
g = )()1(
)1)(1(cL
g
g
termH
termH
−⎥⎦
⎤⎢⎣
⎡
+−
−−−π α θ
π β θ (22a)
g =
( ) ( )( )[ ][ ] ( )
( )[ ] 1
1
)(11
)(111
−
−
−−+
−−−−+−−
nL
nL
LFn
cLLFR
n
cL
α θ θα
α θ β θ
(22b)
M = L)cL(
)cL(n
))(1()1(
)cL(n
))(1()1)(1(
H
H
+−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−β−αθ−θ
+πα−θ
−β−αθ−θ
−πβ−θ−(23)
The gap g between L and M depends on a ratio between the informed low types and the
informed high types. This ratio is adjusted by a constant called
gterm= )())(1(
cLn
−−− β α θ θ
. This constant gterm is subtracted from the numerator and
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63
added to the denominator. gterm adjusts for the case that the uninformed high and low
types are not the same. In the case that the proportion of uninformed low and high types
is the same, then the constant equals zero. gterm will also become smaller, the more
uneven the proportion of low and high types. The overall ratio g is large when there are a
large number of uninformed types. A firm will have to price substantially higher than L
to make up in expected revenue from these lost low types.
Notice that from parts A – D the solutions match the symmetric equilibrium
described in Table 6. Thus Propositions (1a) – (1e) give the symmetric solution
described in Table 6.
QED
Proposition 2: An equilibrium with only high types exists iff:
a) 0 < α < 1
b) [θα/n + (1 - θ) β/n](L - c) < θα/n(H - c)
c) θα/n(H - c) > T(L - c)
d) {α/n +[ (1 - α)][1 - F(p)]n-1 }(p - c) = α/n(H - c) p œ [lH, H]
Proof (2c):
Suppose that θα/n(H - c) < T(L - c) and the univariate equilibrium with high types holds.
Then firms find it profitable to randomize to the low types. This violates the univariate
equilibrium with only high types.
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Proof (2d):
Suppose {α/n +[ (1 - α)][1 - F(p)]n-1 }(p - c) > α/n(H - c)
and the univariate equilibrium with high types hold. Then firms will find it profitable to
randomize at prices higher than H. This violates the univariate equilibrium with high
types.
Suppose {α/n +[ (1 - α)][1 - F(p)]n-1 }(p - c) < α/n(H – c)
and the univariate equilibrium with high types only holds. Then firms will find it not
worthwhile to randomize and price only at H.
Proof (2b):
Suppose that [θα/n + (1 - θ) β/n](L - c) > θα/n(H - c) and the univariate
equilibrium with high types holds. Then firms will find it more profitable to set the
monopoly price at L. This violates the univariate high equilibrium.
Suppose that [θα/n + (1 - θ) β/n](L - c) = θα/n(H - c) and the univariate high
equilibrium holds. Then from Proposition (2c):
θα/n(H - c) > T(L - c)
[θα/n + (1 - θ) β/n](L - c) > [θα/n + (1 - θ) β/n +θ (1 - α) +(1 - θ) (1 - β)](L - c)
This cannot occur.
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Proof (2a):
Suppose that α = 0 and the univariate equilibrium with high types holds. Then
Proposition (2b) can be written as:
[0 + (1 - θ) β/n](L - c) < 0
This is violated.
Suppose that α = 1 and the univariate equilibrium with high types holds. Then
Proposition (2d) can be written as:
{θα/n +[0][1 - F(p)]n-1 }(p - c) = θα/n(H - c)
Then the univariate high equilibrium is violated since there is no F(p).
Proof Equilibrium with only High Types:
Suppose Proposition (2a) – (2d) hold but there is no univariate equilibrium with high
types only.
A. F(p) can be solved from Proposition 2d:
{α/n +[ (1 - α)][1 - F(p)]n-1 }(p - c) = α/n(H - c)
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))(1(
)(1 )(
))(1(
)(
)](1[
1/1
1
−
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−−
−−=
−−
−=−
n
i
i
i
i
in
i
cp
pHnpF
cp
pHnpF
α
α
α
α
Notice that F(p) is the same as FH(p) in Proposition 1. It is also the same F(p) as in
Varian (1980).
B. The lowest pricelH can be solved by setting F(p) =0.
1/1
))(1(
)(
1 0
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−=
n
H
H
c
Hn
λ α
λ α
⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢
⎢
⎣
⎡
−−
−=
))(1(
)(
1c
Hn
H
H
λ α
λ α
( ) )(1
1
)( ))(1(
cHn
cn
n
Hn
c
H
HH
−=−⎟ ⎠
⎞⎜⎝
⎛ −−
−=−−
α λ α
λ α
λ α
( )( )c
nn
cH
c
n
n
cH
n
H
H
+−−−
=
+⎟ ⎠
⎞⎜⎝
⎛ −−
−
=
α
α λ
α
α
λ
1
)(
11
)(
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This is the same lowest price as Varian (1980).
QED
Proposition 3: An equilibrium with only low types exists iff:
a) α or β is greater than 0
b) α or β is less than 1
c) [θα/n + (1 - θ) β/n](L - c) > θα/n(H - c)
d) θα/n(H - c) < T(L - c)
e) {θα/n + (1 - θ) β/n +[θ (1 - α)+ (1 - θ)(1 - β)][1 - F(p)]n-1
}(p - c) =
[θα/n + (1 - θ) β/n](L - c) p œ [lL, L]
Proof (3c):
Suppose [θα/n + (1 - θ) β/n](L - c) < θα/n(H - c) and the univariate low
equilibrium holds. Then firms will find it worthwhile to set the monopoly price at H.
Proof (3d):
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Suppose θα/n(H - c) ¥ T(L - c) and the univariate low equilibrium holds. Then from
Proposition (1d) firms will not randomize to the low types and only go after the high
types.
Proof (3e):
Suppose
{θα/n + (1 - θ) β/n +[θ (1 - α)+ (1 - θ)(1 - β)][1 - F(p)]n-1 }(p - c) >
[θα/n + (1 - θ) β/n](L - c)
and the univariate low equilibrium holds. Then firms will find it profitable to randomize
at prices higher than L. This violates the univariate low equilibrium.
Suppose {θα/n + (1 - θ) β/n +[θ (1 - α)+ (1 - θ)(1 - β)][1 - F(p)]n-1
}(p - c) <
[θα/n + (1 - θ) β/n](L - c)
and the univariate equilibrium with low types holds. Then firms will not find it
worthwhile to randomize and price at L.
Proof (3a):
Suppose α and β equal 0 and the univariate low equilibrium holds. Then Proposition (3c)
can be written as:
[θ(0)/n + (1 - θ) (0)/n](L - c) > θ(0)/n(H - c)
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This is violated as both sides equal 0.
Proof (3b):
Suppose α and β equal 1 and the univariate low equilibrium holds. Then Proposition (3e)
can be written as:
{1 /n +[θ (0)+ (1 - θ)(0)][1 - F(p)]n-1
}(p - c) = [1/n](L - c)
The univariate low equilibrium is violated as firms will not randomize.
Proof Equilibrium with only Low Types:
Suppose that Propositions (3a) – (3e) hold but there is no univariate low equilibrium.
A. F(p) can be found from Proposition (3e):
( )( )
( )( ) ( ) ( )( )[ ] ( )cppF
nncL
nni
ni −
⎭⎬⎫
⎩⎨⎧
−−+−−+−
+=−⎥⎦
⎤⎢⎣
⎡ −+ −
β θ α θ β θ θα β θ θα
111)(111 1
( )( )
( )( ) ( )cpRpF
nncL
nni
ni −
⎭⎬⎫
⎩⎨⎧
−+−
+=−⎥⎦
⎤⎢⎣
⎡ −+ −1
)(111 β θ θα β θ θα
( ) ( )( )
( )iin
i pL
nn
cpRpF −⎥
⎦
⎤⎢
⎣
⎡ −+=−− − β θ θα 1
)(11
( )1/1
)(
)(1
1)(
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−⎥⎦
⎤⎢⎣
⎡ −+
−=
n
i
i
icpR
pLnn
pF
β θ θα
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B. The lowest pricelL can be found by setting F(p) =0.
( )1/1
)(
)(1
10
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−⎥⎦
⎤⎢⎣
⎡ −+
−=
n
L
L
cR
L
nnλ
λ β θ θα
( )
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−⎥⎦
⎤⎢⎣
⎡ −+
=)(
)(1
1cR
Lnn
L
L
λ
λ β θ θα
( ) ( )( )[ ] ( ) )(1)(111 LL Lnn
c λ β θ θα λ β θ α θ −⎥⎦⎤⎢⎣
⎡ −+=−−−+−
( )( )
( ) ( ))(
1)(
111
11 cL
nnc
n
n
n
nL −⎥⎦
⎤⎢⎣
⎡ −+=−⎥
⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−−+⎟
⎠
⎞⎜⎝
⎛ −−
β θ θα λ
β θ
α θ
( )
( )( )
( )c
n
n
n
n
cLnn
L +
⎥
⎦
⎤⎢
⎣
⎡⎟
⎠
⎞⎜
⎝
⎛ −−−+⎟
⎠
⎞⎜
⎝
⎛ −−
−⎥⎦
⎤⎢⎣
⎡ −+
= β
θ α
θ
β θ θα
λ 1
111
1
)(1
QED
Figures 3 – 21 are created from the model using Mathematica in a cumulative
distribution format. The solid line is the graph of entire function F(ÿ). The right-hand
graph is the graph of FH(ÿ) whereas the left-hand graph is FL(ÿ). The dashed portions of
the graph is the portions of FH(ÿ) and FL(ÿ) that are not part of F(ÿ). Below are three
graphs (Figures 3, 4, and 5) showing when Proposition 1 holds and does not hold. Figure
3 is a case where Proposition 1 holds and there is an equilibrium as described in Table III.
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Figure 4 is where Proposition (1c) fails and is a univariate low equilibrium. Figure 5 is
where Proposition (1d) fails and is a univariate high equilibrium.
200 400 600 800 1000 Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 3
Figure 3: α = 0.4 θ = 0.3 H = 1000 c = 50 λ = 101.82
β = 0.4 n = 3 L = 200 M = 550
Figure 3 is an example of price dispersion equilibrium with two different
customer types and three firms. The heavy solid line is the cumulative distribution
function of price for each firm. As will be shown, both proposition one conditions hold
for Figure 3. For the first part of Proposition 1, condition 12 must hold. Profits of the
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uninformed types at price H must be at least as great as profits of the uninformed types at
L. Repeating condition 12:
θα/n ( H – c ) ≥ [θα/n + (1 - θ) β/n](L - c) (12)
( )( )( ) ( )( ) ( )( )[ ]( )502003/4.07.03/4.03.03/5010004.03.0 −+≥−
( )( ) ( ) ( )[ ]( )5028.012.050100004.0 +≥−
( )( ) [ ]( )504.95004.0 ≥
2038 ≥
Thus Proposition (1c) holds for Figure 3. Figure 3 with the price dispersion with two
different consumer types passes Proposition (1d). Repeating condition 13:
[θα/n + θ(1 - α) + (1- θ)β/n + (1 - θ)(1 - β)] (L – c) ≥ θα/n (H - c) (13)
( )( ) ( )( ) ( )( ) ( )( )[ ]( ) ( )( )( ) 3/5010004.03.0502006.07.03/4.07.06.03.03/4.03.0 −≥−+++
( ) ( ) ( ) ( )[ ]( ) ( )( )95004.015042.0093333.018.004.0 ≥+++
38110 ≥
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200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 4
Figure 4: α = 0.2 θ = 0.3 H = 1000 c = 50
β = 0.6 n = 3 L = 200
In Figure 4 the solid line is only at the left hand cumulative distribution function. This is
not an equilibrium of firms randomizing to two different consumer types. Instead, this is
an equilibrium with low types only. Firms find it profitable only to randomize below L.
Proposition (1c) fails for Figure 4. Repeating condition 12:
θα/n ( H – c ) ≥ [θα/n + (1 - θ) β/n](L - c) (12)
( )( )( ) ( )( ) ( )( )[ ]( )502003/6.07.03/2.03.03/5010002.03.0 −+≥−
( )( ) ( ) ( )[ ]( )5042.006.095002.0 +≥
The above equation does not hold since
2419 <
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Thus condition 12 fails when the values corresponding to Figure 4 are used. There is no
need for firms to randomize to exclusively to go after the high types only. There are
more profits to be made by going after the lower types only.
200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 5
Figure 5: α = 0.9 θ = 0.3 H = 1000 c = 50
β = 0.9 n = 3 L = 200
In Figure 5 firms randomize only at the solid line. They find it profitable only to
serve the high types; it is not profitable to go after the low types. Again, this is not an
equilibrium of firms randomizing to two different consumer types. Proposition (1b) fails
for this equilibrium. Repeating condition 13:
[θα/n + θ(1 - α) + (1- θ)β/n + (1 - θ)(1 - β)] (L – c) ≥ θα/n (H - c) (13)
( )( ) ( )( ) ( )( ) ( )( )[ ]( ) ( )( )( ) 3/5010009.03.0502001.07.03/9.07.01.03.03/9.03.0 −≥−+++
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( ) ( ) ( ) ( )[ ]( ) ( )( )95009.015007.021.003.009.0 ≥+++
( )( ) ( )( )95009.01504.0 ≥
The above equation does not hold since
5.8560 <
Thus condition 13 fails when the values corresponding to Figure 5 are used. In this case,
Proposition (1b) fails. The three firms find it profitable only to randomize to the high
types.
1-5. Comparative Statics
Given these four endogenous solutions – FH(p), FL(p), l, and g - how does
varying the exogenous parameters - α, β, θ, L, H and n affect the four endogenous
solutions? Most comparative statics are quite intuitive. For a few comparative statics, it
is easier to break α and β ratios into their raw consumer numbers. Let t = total amount of
consumers in the marketplace. Let UH = total amount of uninformed high consumers.
The proportion of uninformed high consumerst
UH=α . Let UL = total amount of
uninformed low consumers. The proportion of uninformed low type consumers t
UL= β .
Table 7 below defines the various exogenous and endogenous variables. Table 8 lists
where comparative statics of the variables can be found in the following propositions.
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Some of the more important comparative statics include ∑FL(p)/ ∑q , ∑FL(p)/ ∑H,
∑FH(p)/ ∑H, ∑l/ ∑H, ∑FL(p)/ ∑c, and ∑FH(p)/ ∑c. The comparative statics
∑FL(p)/ ∑q , ∑FH(p)/ ∑c, ∑FL(p)/ ∑c can be seen by comparing two routes in the Appendix:
Figure 26 Los Angeles to Honolulu and Figure 27 Minneapolis to Chicago O’Hare.
Figure 26 appears to have more leisure travelers than business travelers as the density is
rather large around low prices. Figure 27: Minneapolis to Chicago O’Hare has more
business travelers so the peak density appears to be higher in price than Los Angeles to
Honolulu. When further considering the marginal cost of traveling from Los Angeles to
Honolulu is much higher than the marginal cost of traveling from Minneapolis to
Chicago O’Hare due to the large difference in air distance in the two routes, the fares
adjusted for distance are even lower for Figure 26 and higher for Figure 27.
Two routes that are more equidistant in terms of air mileage are Figure 28
Washington National – New York LaGuardia and Figure 31 Los Angeles to Las Vegas.
The fares in the Washington – New York route are higher than the fares in the Los
Angeles to Las Vegas. Specifically the maximum fares in the Washington – New York
route appear higher than the Los Angeles to Las Vegas route. There are modes well
above $150 in the Washington – New York route whereas the highest mode in the Los
Angeles – Las Vegas route is below $80. There appears to be less density between $0
and $50 in the Washington – New York route than the Los Angeles - Las Vegas route.
These two routes give some credence to the comparative statics ∑FL(p)/ ∑H, ∑FH(p)/ ∑H,
and ∑l/ ∑H. The density reaches higher peaks for a lower maximum price in the Los
Angeles – Las Vegas route than for the New York – Washington route, as comparative
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statics predicts. The minimum set of prices appear to be lower in the Los Angeles – Las
Vegas route than the Washington – New York route, just as comparative statics predicts.
Table 7: Variable Definitions
Variable Exogenous/
Endogenous
Definition
F(p) Endogenous Cumulative price distribution function of each firm
FH(p) Endogenous Upper portion of cumulative price distribution for each firm
spanning where firms sell to high type consumers only
FL(p) Endogenous Lower portion of cumulative price distribution for each firm
spanning where firms sell to both high and low type
consumers
g Endogenous Flat portion of cumulative price distribution above the
monopoly low price for each firm where it is unprofitable for
firms to randomize to consumers
l Endogenous Lowest price spanning each firm’s cumulative price
distribution function
α ExogenousProportion of uninformed high type consumers
t
UH=α
β ExogenousProportion of uninformed low type consumers
t
UL= β
θ Exogenous Proportion of high type consumers
H Exogenous Monopoly price of high types
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L Exogenous Monopoly price of low types
c Exogenous Marginal cost
n Exogenous Number of firms
t Exogenous Total number of consumers
UH Exogenous Total number of uninformed high type consumers
(comparative statics not shown in paper)
UL Exogenous Total number of uninformed low type consumers
(comparative statics not shown in paper)
Variable Exogenous/
Endogenous
Definition
Below is Table 8 that gives a consolidated place to preview results of this section
of comparative statics. The entry in the first row and first column is read as ∂FH (ÿ)/∂α.
The comparative static ∂FH (ÿ)/∂α is less than zero and the actual proof can be found in
Lemma (1b) after Proposition 6.
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Table 8: Comparative Statics Preview
∂FH (ÿ) ∂FL(ÿ) ∂l ∂g
∂α Lemma (1b) after
Prop 6
< 0
Prop (7b)
< 0
Prop (8d)
> 0
Prop (9b)
> 0 if
( )[ ] (1
)( 22 cLcH −⎥
⎦
⎤⎢⎣
⎡
−−
+>− β
β α θ β θα
< 0 if
( )[ ] (1
)( 2
2 cLcH −⎥⎦
⎤⎢⎣
⎡
−−
+<− β
β α θ β θα
∂β N/A Prop (7d)
> 0 or < 0
Prop (8a)
> 0
Prop (9c)
> 0 if θα( H - c) < (L - c)
< 0 if θα( H - c) ≥ (L - c)
∂θ N/A Prop (7c)
< 0
Prop (8b)
> 0
Prop (9a)
< 0
∂H Lemma (1a) after
Prop 6
< 0
Prop (7a)
< 0
Prop (8c)
> 0
Prop (9e)
> 0 if α > β
< 0 if α < β
∂L N/A N/A N/A Prop (9d)
> 0
∂c Lemma (1c) after
Prop 6 < 0
Prop (7e)
< 0
Prop (8e)
> 0
Prop (9g)
< 0
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∂n Prop (6a)
< 0 except at p =
H
Prop (6b)
< 0
Prop (6d)
< 0
Prop (6c)
= 0
∂t Prop (4a)
> 0
Prop (4b)
> 0
Prop (4c)
< 0
Prop (9f)
> 0 if UH > UL
< 0 if UH < UL
Proposition 4: For all n, if α and β approach zero and the conditions of
Proposition 1 hold, the cumulative price density shifts monotonically toward the lowest
price and the lowest price gets lower toward marginal cost. In other words let F(ÿ | α, β)
be the equilibrium as described in Table 3. F(ÿ | α, β) Ø dc, where dc is a point mass on
marginal cost. The market becomes comprised of firms very occasionally raising prices
to a moderate or high level and the rest of the time of firms charging very, very close to
marginal cost.
Proof:
Proving this proposition requires establishing these points:
a) ∂FH (ÿ)/∂t > 0
b) ∂FL (ÿ)/∂t > 0
c) ∂λ/∂t < 0
d) " x > c, lim (α, β) = 0 ö F(x | α, β) = 1
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Proof (4a):
FH(pi) =))(1(
)(1
1/1 −
⎥
⎦
⎤⎢
⎣
⎡
−−−
−n
i
i
cpn
pH
α
α (17)
Rearranging FH(∏) to put in t and UH for α:
FH(pi) =))(1(
)(
1
1/1 −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−
n
iH
iH
cpt
Un
pHt
U
FH(pi) =
))((
)(1
1/1 −
⎥
⎦
⎤⎢
⎣
⎡
−−
−−
n
iH
iH
cpUtn
pHU
∂FH /∂t =222
1
2
)()(
)()(
))((
)(
1
1
cpUtn
cpnpHU
cpUtn
pHU
n H
Hn
n
H
H
−−
−−−⎥⎦
⎤⎢⎣
⎡
−−−
−−
−
−
= 0)()(
)(
))((
)(
1
12
1
2
>−−
−⎥⎦
⎤⎢⎣
⎡
−−−
−
−
−
cpUtn
pHU
cpUtn
pHU
n H
Hn
n
H
H
With the positive relationship between t and FH, increasing t to infinity causes FH
to increase. Since t and α have a negative relationship byt
UH=α , increasing t to
infinity when UH is held constant causes α to fall to near zero. Thus α and FH have a
negative relationship when t is increased and UH is held constant. Thus increasing t to
infinity holding UH constant or lowering α to zero causes FH to increase for prices along
its interval. Almost no consumer pays the full monopoly price as all consumers receive
some sort of price break on their purchase.
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Proof (4b):
( ) )(
)(1)(1)(
1/1 −
⎥⎦
⎤⎢⎣
⎡−
−−−−−=n
i
iiiL
cpnR
cppHpF
β θ θα (15)
Rearranging FL(ÿ) to put in t and UH for α and t and UL for β:
( )
( )
)(111
)(1)(
1)(
1/1 −
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−+⎟
⎠
⎞⎜⎝
⎛ −
−−−−−=
n
iLH
iL
iH
iL
cpt
U
t
Un
cpt
UpH
t
U
pF
θ θ
θ θ
( )( ) ( )( )[ ]
)(1
)(1)(1)(
1/1 −
⎥⎦
⎤⎢⎣
⎡
−−−+−−−−−
−=n
iLH
iLiHiL
cpUtUtn
cpUpHUpF
θ θ
θ θ
∂FL(pi)/∂t = [ ]( )[ ] ( )( )( )[ ]
( ) ( )( )[ ] ⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−
−−+−−−−−−−
−
−
2221
2
)(1
1)(1)(
1
1
cpUtUtn
cpncpUpHULfract
niLH
iiLiHn
n
θ θ
θ θ θ θ
∂FL(pi)/∂t = [ ]( )
( ) ( )( )[ ]0
)(1
)(1)(
1
12
1
2
>⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−
−−−−−
−
−
cpUtUtn
cpUpHULfract
niLH
iLiHn
n
θ θ
θ θ
With the positive relationship between t and FL, increasing t to infinity causes FL
to increase. Since t and α have a negative relationship byt
UH=α and t and β have a
negative relationship byt
UL= β , increasing t to infinity when UH and UL are held
constant causes α and β to fall to near zero. Thus FL and α and β jointly have a negative
relationship when t is increased and UH and UL are held constant. (In a future proposition
β and FL are shown not to have a sole negative relationship. Thus the joint relationship is
needed with α and β.) Thus increasing t to infinity holding UH and UL constant (keeping
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proposition one satisfied) or lowering α and β to zero causes FL to increase for prices
along its interval. Firms are often running sales to capture the informed consumers in the
marketplace.
Proof (4c):
λ =( )
( ) ( )c
n
n
n
n
cHn +
⎟ ⎠
⎞⎜⎝
⎛ −−−+⎟
⎠
⎞⎜⎝
⎛ −−
−
β θ
α θ
αθ
11)1(
11
(19)
Rearranging λ to put in t and UH for α and t and UL for β:
λ =( )
( ) ( )c
nt
Un
nt
Un
cHtn
U
LH
H
+⎟ ⎠
⎞⎜⎝
⎛ −−−+⎟
⎠
⎞⎜⎝
⎛ −−
−
11)1(
11 θ θ
θ
λ =( )
( ) ( )c
n
Unt
n
Untn
cHU
LH
H +
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−−+⎟
⎠
⎞⎜⎝
⎛ −−
−
1)1(
1θ θ
θ
∂λ/∂t =
( ) ( )( )[ ]
( ) ( )2
2 1)1(
1
1
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−−+⎟
⎠
⎞⎜⎝
⎛ −−
−+−−
n
Unt
n
Untn
ncHU
LH
H
θ θ
θ θ θ
∂λ/∂t =( )
( ) ( )0
1)1(
12<
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−−+⎟
⎠
⎞⎜⎝
⎛ −−
−−
n
Unt
n
Untn
cHU
LH
H
θ θ
θ
With a negative relationship between λ and t, increasing t to infinity causes λ to
fall near marginal cost c. Since t and α have a negative relationship byt
UH=α and t
and β have a negative relationship byt
UL= β , increasing t to infinity when UH and UL
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are held constant causes α and β to fall to near zero. As will be shown in a future
proposition both ∂λ/∂α and ∂λ/∂β are greater than zero. Thus when α and β to fall to
near zero, λ falls near marginal cost.
Proof (4d):
Proposition (4a) – (4c) show that as α and β approach zero, FH(ÿ), FL(ÿ) are
monotonically increasing while l is monotonically decreasing to marginal cost. This
part of the proposition, shows that any price x above marginal cost has F(x | α, β) = 1.
We first know that for any price x < c, F(x | α, β) = 0 as firms will not suffer losses. For
the upper portion of the distribution:
))(1(
)(
1 )(
1/1 −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−=
n
Hcx
xHnxFα
α
When α ö 0, the (1 - α) term in the denominator approaches 1 while the numerator
approaches 0. Thus the fraction approaches 0 and FH(x | α) ö 1. For the lower part of
the distribution:
( )
( ) ( )( )[ ]
1/1
)(111
)(1
)(
1 )(
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−+−
−−
−−−=
n
Lcx
cxn
xHnxF
β θ α θ
β θ θα
When α,β ö 0, both (1 - α) and (1 - β) approach 1 in the denominator resulting in the
denominator approaching (x – c). The numerator approaches 0 as both terms approach 0.
Thus the fraction approaches 0 for all x > c and thus FL(x | α, β) ö 1.
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With FH(x | α) ö 1 and FL(x | α, β) ö 1 when α,β ö 0, F(x | α, β)
approaches a point mass. This point mass dc is a point mass approaching marginal cost as
Proposition (4c) shows the lowest price approaching marginal cost monotonically. Since
F(x | α, β) increases monotonically and the lowest price l decreases monotonically to
marginal cost as α,β ö 0, F(x | α, β) monotonically approaches dc.
Figure 6 below is an example of an equilibrium of α and β approaching zero with
proposition one holding. Notice that most of the weight of the cumulative distribution is
located near λ = 50.86, which is extremely near marginal cost c = 50. The cumulative
distribution function does not really flatten out at the parameter values α = 0.009 and β =
0.004 until after F(p) reaches 0.8. Epsilon is calculated at 265 and this is at height of
.938. Thus from M = 465 to H = 1000, the cumulative distribution function is quite flat.
Firms generally run extreme sales most of the time as most customers are extremely
informed. Occasionally, firms raise their prices anywhere from moderate to high levels
to capture profits from the few uninformed customers. As α and β get closer to zero this
occurs less and instead Bertrand pricing occurs more often.
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200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 6
Figure 6: α = 0.009 θ = 0.3 H = 1000 c = 50 λ = 50.86
β = 0.004 n = 3 L = 200 M = 465.87
QED
Proposition 5: If α and β approach one and the conditions of Proposition 1 holds,
dispersion in the marketplace decreases. The cumulative distribution function shifts its
weight towards the monopoly prices H and L. λ is positioned right below L and M is
positioned just below H. The probability density monotonically approaches a bimodal
distribution at L and H though it never becomes one.
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Proof:
From Proposition 4:
1. ∂FH (p)/∂t > 0 implies 0/)( <ΔΔ α pFH
2. ∂FL (p)/∂t > 0 implies 0/)( <ΔΔ α pFL and 0/)( <ΔΔ β pFL
3. ∂λ/∂t < 0 implies 0/ >ΔΔ α λ and 0/ >ΔΔ β λ
The first two points indicate that FH(p) and FL(p) are monotonically increasing as
α,β ö 1. The third point indicates that λ is monotonically increasing as α,β ö 1.
Now it needs to be shown that
a) 0),|(1,
→→
β α β α
xFLim L for x < L.
b) HgLLim =+→1, β α
c) 1),|(1,
→→
β α β α
xFLim H for x > M
d) 1),|(1,
=→
β α β α
HFLim H
e) 1
1
1,
11),|(
−
→ ⎥⎦
⎤⎢⎣
⎡−=n
Ln
LFLim β α β α
Proof (5a):
For prices x less than l, FL(x | α, β) = 0.
( )
( ) ( )( )[ ]( )
c
nn
cHn +
−++−−+−
−=
β θ θα β θ α θ
αθ
λ 1
111
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As α,β ö 1, the bracketed expression in the denominator becomes zero. l can be
rewritten as
( )( ) ccHc
n
cHn +−=+
−= θ
θ
λ 1
. Proposition 1 says the following inequalities must hold:
[θα/n + (1 - θ) β/n](L - c) § θα/n(H - c) § [θα/n + (1 - θ) β/n + θ(1-α) +(1 - θ) (1-β)](L -
c)
Taking the limit α,β ö 1, the inequality reduces to:
(1/n)(L - c) § θ/n(H - c) § (1/n)(L - c)
(L - c) § θ(H - c) § (L - c)
Thus l ö L when α,β ö 1. All prices x, below L have FL(x | α, β) = 0.
Proof (5b):
g =
( ) ( ) ( )[ ][ ] ( )
( )[ ] 1
1
)(11
)(111
−
−
−−+
−−−−+−−
nL
nL
LFn
cLLFRn
cL
α θ θα
α θ β θ
(22b)
The right hand terms of the numerator and denominator become 0. Thus g can be
rewritten as
g =
( )( )
n
n
cL
θ
θ −−1
g =( )( )
θ
θ cL −−1
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Equation 11 gives the maximum gap between L and L+ g = H:
θα/n(H - c) -[θα/n +(1 - θ)β/n + R(1 - F(L))n-1](L - c) = 0 (11)
Taking the limit as α,β ö 1:
θ/n(H - c) – (1/n )(L - c) = 0
θ(H - c) – (L - c) = 0
Plugging in L + g – c for H - c verifies that M = H:
( )( )( )
( )( ) ( ) 01
01
=−−−−−+=
=−−⎥⎦
⎤⎢⎣
⎡ −−−
+
cLccLL
cLccL
L
θ θ θ
θ
θ θ
( ) ( ) 0=−−− cLcL
Proof (5c):
From Proposition (2), the upper portion FH(p) minimum:
c
n
ncHn
H +⎟ ⎠
⎞⎜⎝
⎛ −−
−=α
α
λ 1
1
)(
Taking the limit as α ö 1:
( )( ) HccHc
nn
cHc
n
n
cHn
H =+−=+−−
−=+
⎟ ⎠
⎞⎜⎝
⎛ −−
−=
→ 1)1(
11
)(1
lim1λ
α
Proof (5d):
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))(1(
)(
1 )(
1/1 −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−=
n
i
i
iHcp
pHnpFα
α
( )
))(1(
)(
1 )(lim
1/1
1
−
→
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−−−=
n
HcH
cHcHnHF
α
α
α
( )
))(1())(1(
)(1 )(lim
1/1
1
−
→⎥⎦
⎤⎢⎣
⎡
−−−
−−−
−−=
n
HcHn
cH
cHn
cHHF
α
α
α
α
α
)1()1(1 )(lim
1/1
1
−
→ ⎥⎦⎤⎢
⎣⎡ −−−−=
n
Hnn
HFα
α α
α α
1 )(lim1
=→
HFHα
Proof (5e):
( ) 1/1
)(
)(1
)(
1 )(
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−
−−
−=
n
i
ii
iLcpR
cp
n
pH
npF
β θ θα
( ) 1/1
1, )(
)(1
)(
1 )(lim
−
→
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−
−−−=
n
LcLR
cLn
LHnLF
β θ θα
β α
( ) ( )1/1
1,
)(
)(1)(1 )(lim
−
→⎥
⎦
⎤⎢
⎣
⎡
−
−−−−−−−=
n
L
cLnR
cLcLcHLF
β θ θα θα
β α
( )( ) ( )
1/1
1, )(
)(1
1 )(lim
−
→
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−−−−−
−=
n
LcLnR
cLcLcL
LF β θ θα
θ θα
β α
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( ) ( )( ) ( )( )[ ]
1/1
1, 111
111 )(lim
−
→ ⎥⎦
⎤⎢⎣
⎡
−−+−−−−
−=n
Ln
LF β θ α θ
θ β θ α
β α
Applying L’Hôpital’s Rule by taking ∑/∑b of the numerator and denominator:
( )( )[ ]
1/1
1, 1
11 )(lim
−
→ ⎥⎦
⎤⎢⎣
⎡
−−−−
−=n
Ln
LFθ
θ
β α
1/1
1,
11 )(lim
−
→ ⎥⎦
⎤⎢⎣
⎡−=
n
Ln
LF β α
Since α and β are getting larger, FH(ÿ) and FL (ÿ) are getting smaller for most prices.
The lowest price λ is getting larger as firms are not competing intensely for the shopper.
Figures 7 and 8 provide an example of near limit cases. These are examples of equilibria
where firms randomize between two different types of consumers. Generally, the
window for these type of equilibria is quite small, with only the whole number L = 334
working for the given parameters in Figure 8. With these type of equilibria with two
different type of consumers, there is not much price dispersion. Prices bounce between a
small interval below and including the highest price H and an interval right below and
including the monopoly price for the low types L. The lowest price λ is just below L
while M is just below H. As α and β approach one, the distribution in a two consumer
equilibrium approaches a bimodal distribution at L and H.
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0 200 400 600 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1
CDF FHLFi gure 7
Figure 7: α = 0.99 θ = 0.85 H = 1000 c = 50 λ = 833.75
β = 0.99 n = 3 L = 850 M = 991.18
200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 8
Figure 8: α = 0.99 θ = 0.3 H = 1000 c = 50 λ = 326.62
β = 0.99 n = 3 L = 334 M = 996.67
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QED
Proposition 6: For all α and β such that the conditions of Proposition 1 holds, if n
approaches infinity, the cumulative price density monotonically shifts toward the highest
price and the lowest price gets lower as more firms enter the market. In other words, F(ÿ)
ö dH, where dH is a point mass on the monopoly high price. The market is comprised
of the very occasional extreme sale of consumers paying the lowest price and the rest of
the time of everyone paying the monopoly high price.
Proof:
This proposition deals with the case when there is extreme dispersion in the market and
what happens if more firms enter the market.
Proving this proposition requires establishing these four points:
(a) ∂FH (p)/∂n < 0 (except at H).
(b) ∂FL (p)/∂n < 0.
(c) ∂g/∂n = 0
(d) ∂λ/∂n < 0
(e) 0)|( →∞→nxFLim
nfor x < H
(f) 1)|( =∞→
nHFLim Hn
Proof (6a):
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Calculating ∂FH/∂n requires the trick of taking ln() of both sides of equation (17):
FH(pi) = ))(1(
)(
1
1/1 −
⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢
⎢
⎣
⎡
−−
−
−
n
i
i
cp
pHnα
α
(17)
FH(pi)-1 =))(1(
)(
1/1 −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−
n
i
i
cp
pHnα
α
ln(1-FH) =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−
− ))(1(
)(
ln
1
1
cp
pHn
n i
i
α
α
Taking ∂/∂n on both sides will allow us to calculate ∂FH/∂n.
( )H
H
F
nF
−−∂∂
1
/=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
−
−−−
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−− −
2
2 1
))(1(
)(
)(
))(1(
1
1
))(1(
)(
ln)1(ncp
pH
pHn
cp
ncp
pHnn
i
i
i
i
i
i
α
α
α
α
α
α
= ⎥⎦
⎤⎢⎣
⎡
−+⎥
⎦
⎤⎢⎣
⎡
−−
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−
22 )1(
1
)1(
1
)1(
1))(1(
)(ln
nnnn
cp
pHn
i
i
α
α
=( )
⎥⎦
⎤⎢⎣
⎡
−
+−−+
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−
nn
nn
n
cp
pHn
i
i
22 )1(
1
)1(
1))(1(
)(
lnα
α
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=2)1(
1
))(1(
)(
ln
−
⎥⎦
⎤⎢⎣
⎡ −−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−
n
n
n
cp
pHn
i
i
α
α
∂FH/∂n =⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ −+⎥
⎦
⎤⎢⎣
⎡
−−−
−−
n
n
cpn
pH
nF
i
iH
1
))(1(
)(ln
)1(
1)1(
2 α
α (24)
The value of the right hand expression depends on the value of n. The sum of
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ −+⎥
⎦
⎤⎢⎣
⎡
−−−
n
n
cpn
pH
i
i 1
))(1(
)(ln
α
α (24a)
will determine whether ∂FH/∂n is greater or less than zero. The first term, or the natural
log term, will be less than zero since the fraction inside the expression is less than one.
How close to zero is this fraction inside the natural log expression determines the sign of
the ∂FH/∂n. As n → ∞, the left hand term in (24a) becomes much less than negative one
because the term in square brackets that is taken the natural log of approaches zero. The
right hand term of equation (24a) approaches one but this is not enough to counterbalance
a large negative term. Equation (24a) then is negative and thus equation (24) or ∂FH/∂n <
0. The exception is that for prices → H, FH(ÿ) become steeper, looking more like a spike
at H, with an increase in n. Here most customers are more likely to be paying the
monopoly price.
Proof (6b):
To assist in reducing the sprawling symbols, FL will be abbreviated:
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FL=)c p(nR
)c p()1() pH(1
1n/1
i
ii
−
⎥⎦
⎤⎢⎣
⎡
−
−βθ−−−θα−
= 1 - (Lfract)1/n-1 = 1 -1n
1
LfractD
LfractN −
⎥⎦
⎤⎢⎣
⎡(25)
Like ∂FH/∂n, calculating ∂FL/∂n requires the trick of taking ln() of both sides of equation
(15):
1/1
)(
)]()1()([1)(
−
⎥⎦
⎤⎢⎣
⎡
−−−−−
−=n
i
iiiL
cpnR
cppHpF
β θ θα
ln(1-FL) =1n
1
i
ii
)c p(nR
)c p()1() pH(ln
1n
1 −
⎥⎦
⎤⎢⎣
⎡
−
−βθ−−−θα
−
Taking ∂/∂n on both sides will allow us to calculate ∂FL/∂n.
( )L
L
F
nF
−−∂∂
1
/= [ ] ⎥
⎦
⎤⎢⎣
⎡
−−−
−−−
−−−− −
2
2 1
))(1(
)(
)(/
))(1(
1
1ln)1(
ncp
pH
pHn
cp
nLfractn
i
i
i
i
α
α
α
α
= [ ][ ] ⎥⎦⎤⎢
⎣⎡
−+⎥
⎦⎤⎢
⎣⎡
−−
−−−
22 )1(1
)1(1
)1(1ln
nnnnLfract
=[ ]
⎥⎦
⎤⎢⎣
⎡
−
+−−+⎥
⎦
⎤⎢⎣
⎡
−
−−
nn
nn
n
Lfract22 )1(
)1(
)1(
1ln
=
[ ]
2)1(
1ln
−
⎥⎦
⎤⎢⎣
⎡ −−−
n
n
nLfract
∂FL/∂n =⎥⎥⎦
⎤
⎢⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ −+⎥
⎦
⎤⎢⎣
⎡
−−−−−
−−
n
n
cpnR
cppH
nF
i
iiL
1
)(
)()1()(ln
)1(
1)1(
2
β θ θα (26)
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The value of the right hand expression depends on the value of natural log function. The
analysis is identical to calculating ∂FH/∂n. The sum of
⎥⎦
⎤⎢⎣
⎡⎥⎦⎤
⎢⎣⎡ −+⎥
⎦
⎤⎢⎣
⎡−
−βθ−−−θαn
1n
)c p(nR
)c p()1() pH(ln
i
ii (26a)
will determine whether ∂FL/∂n is greater or less than zero.
As n → ∞ (assuming Proposition one is satisfied), ∂FL/∂n < 0 iff the natural log
term is less than negative one. When this happens, equation (26a) will be satisfied.
Examining the square bracket term in the natural log function (left-hand term) more
closely,[ ] ⎥
⎦
⎤⎢⎣
⎡
−−−+−−−−−
)()1)(1()1(
)()1()(
cpn
cppH
i
ii
β θ α θ
β θ θα when n gets very large the denominator gets
very large. The entire square bracket term approaches zero and the natural log term
becomes a large negative term. The right hand in equation (26a) approaches one, but is
not able to offset such a large negative number. Thus ∂FL/∂n < 0.
Proof (6c):
g = )(
)())(1(
)1(
)())(1(
)1)(1(
cL
cLn
cLn
H
H
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
+−
−−−
−−−
β α θ θ π α θ
β α θ θ π β θ
(22)
Dividing the numerator and denominator by n in equation (22) gives
g = )())()(1()()1(
))()(1()()1)(1(cL
cLcH
cLcH−⎥
⎦
⎤⎢⎣
⎡−−−+−−
−−−−−−− β α θ θ θα α θ
β α θ θ θα β θ (22b)
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Taking the derivative of g with respect to n gives a result of zero. Given that both FH and
FL are shifting downward the flat region g remains constant with a change in the number
of firms.
Proof (6d):
∂λ/∂n =2])1())1(1(n[
])1(1)[cH(
θα+θ−β+θα−θ−β−
θα−θ−β−−θα−< 0 (23)
The increase in the number of firms causes firms to increasingly compete for the
shoppers. However, their exponential ( ) 1)(1
−− npF odds of winning the shoppers falls as n
approaches infinity much faster than the fall in expected profits of receivingn
1of their
uninformed profits. Firms compete vigorously for the shoppers as the lowest price λ falls
when n increases to infinity. However, not much weight is placed on prices around λ
because the odds of winning against the other n – 1 firms is not so good.
Proof (6e):
( ) 1/1
)(
)(1
)(
1 )(
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−
−−−=
n
i
ii
iLcpR
cpn
pHnpF
β θ θα
( )
1/1
)()(1)(1 )|(lim
−
∞→⎥⎦⎤⎢
⎣⎡
−−−−−−=
n
Ln cxnR
cxxHnxF β θ θα
There are two competing factors: the n in the denominator of the fraction inside
the brackets and the 1/(n-1) in the exponent. The fraction inside the brackets approaches
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0 as n gets larger but the ( )thn 1− root in the exponent dominates and
( )1/1
)(
)(1)(−
⎥
⎦
⎤⎢
⎣
⎡
−
−−−−n
cxnR
cxxH β θ θα approaches one. Thus Lx 0 )|(lim ≤∀=
∞→nxFL
n.
))(1(
)(
1 )(
1/1 −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−=
n
i
i
iHcp
pHnpFα
α
))(1(
)(1 )|(lim
1/1 −
∞→ ⎥⎦
⎤⎢⎣
⎡
−−−
−=n
Hn cxn
xHnxF
α
α for x < H.
Again there are two competing factors: the n in the denominator of the fraction
inside the brackets and the 1/(n-1) in the exponent. Again, the fraction inside the
brackets approaches 0 as n gets larger, but the ( )thn 1− root in the exponent dominates
and))(1(
)(1/1 −
⎥⎦
⎤⎢⎣
⎡
−−−
n
cxn
xH
α
α approaches one. Thus Hx 0 )|(lim <≤∀=
∞→MnxFL
n
Proof (6f):
( )
))(1(
)(1 )(lim
1/1 −
∞→⎥⎦
⎤⎢⎣
⎡
−−−−−
−=n
Hn cHn
cHcHHF
α
α α
)1()1(1 )(lim
1/1 −
∞→ ⎥⎦
⎤⎢⎣
⎡
−−
−−=
n
Hn nn
HFα
α
α
α
101 )(lim =−=∞→
HFH
n
Figure 9 is an example of an equilibrium of two different types of consumers
where n is allowed to get large. Figure 9 has the same parameter values as Figure 1
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100
except for the larger value of n. Figure 9 is blown up on the vertical axis to show F(p) is
extremely small for all prices except for those right around H. This is as predicted by the
comparative statics. As predicted, λ has moved very low – right around marginal cost.
M, and therefore ε, have the same values in figures 1 and 9, just as the model predicted.
Thus, firms mostly price at H but occasionally run extreme sales on the small chance they
have winning the informed consumers. As n gets larger, F(p) only gets smaller for most
prices, the exception being prices immediately less than H. Also λ inches closer to
marginal cost.
200 400 600 800 1000Pri ce
0. 05
0. 1
0. 15
0. 2
0. 25CDF FHL Fi gure 9
Figure 9: α = 0.4 θ = 0.3 H = 1000 c = 50 λ = 51.89
β = 0.4 n = 100 L = 200 M = 550
QED
This next lemma has been already found in previous literature:
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101
Lemma 1: (a) ∂FH /∂H < 0
(b) ∂FH /∂α < 0
(c) ∂FH /∂c < 0
Proof:
The endogenous parameter FH (ÿ) can only be affected by four parameters: H, α, c and n.
The case of n was dealt in Proposition 6. Varying L, θ, and β will have no effect on the
high portion of the cumulative probability distribution FH(ÿ).
Proof Lemma 1a:
)c p)(1(
) pH(n/1) p(F
1n/1
i
iiH
−
⎥⎦
⎤⎢⎣
⎡
−α−−α
−=
Obviously, ∂FH (pi)/∂H < 0 since
∂FH (pi)/∂H =)c p)(α1(
n/α
)c p)(α1(
) pH(n/α
1n
1 1n
n2
−−⎥⎦
⎤⎢⎣
⎡
−−
−
−−
−
−
< 0 (24)
See Proposition 7a for graphs.
Proof Lemma 1b:
∂FH (pi)/∂α
=22
1n
n2
)c p()α1(
)c p)( pH(n/α)c p)(α1)( pH(n/1
)c p)(α1(
) pH(n/α
1n
1
−−
−−−−−−−⎥⎦
⎤⎢⎣
⎡
−−−
−−
−
−
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102
=)c p()α1(
]α)α1)[( pH(n/1
)c p)(α1(
) pH(n/α
1n
12
1n
n2
−−
+−−⎥⎦
⎤⎢⎣
⎡
−−−
−−
−
−
=)c p()α1(
) pH(n/1
)c p)(α1(
) pH(n/α
1n
12
1n
n2
−−
−⎥⎦
⎤⎢⎣
⎡
−−−
−−
−−
< 0 (25)
See Proposition 7b for graphs.
Proof Lemma 1c:
∂FH (pi)/∂c = ⎥⎦
⎤⎢⎣
⎡
−−−−+
⎥⎦
⎤⎢⎣
⎡
−−−
−−
−
−
222
1
2
)()1(
)()1(0
))(1(
)(/
1
1
cpn
pHn
cp
pHn
n
n
n
α
α α
α
α
= 0)()1(
)(
))(1(
)(/
1
1222
1
2
<⎥⎦
⎤⎢⎣
⎡
−−−
⎥⎦
⎤⎢⎣
⎡
−−−
−−
−
−
cpn
pH
cp
pHn
n
n
n
α
α
α
α
See Proposition 7e for graphs.
QED
Proposition 7:(a) ∂FL(pi)/∂H < 0
(b) ∂FL(pi)/∂α < 0
(c) ∂FL(pi)/∂θ < 0
(d):
(1) When L is low and ( )( )
n
cLcH
n
−≥−
θα , ∂FL(pi)/∂β < 0
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103
(2) In the case of a higher L, if ( )( )
n
cLcH
n
−<−
θα , ∂FL(pi)/∂β
> 0 as pi → L and ∂FL(pi)/∂β < 0 as pi → λ
(e) ∂FL(pi)/∂c < 0
Proof (7a):
Calculating ∂FL/∂H is fairly basic:
( )
)(
)(1)(
1)(
1/1 −
⎥
⎥⎥⎥
⎦
⎤
⎢
⎢⎢⎢
⎣
⎡
−
−−−−−=
n
i
ii
iLcpR
cpn
pHnpF
β θ
θα
(15)
∂FL(pi)/∂H
=⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−αθ⎥⎦
⎤⎢⎣
⎡
−
−βθ−−−θα
−− −
−
2i
2
i1n
n2
i
ii
)c p(R
)c p(R )n/(
)c p(R
)c p)(n/)(1() pH(n/
1n
1< 0 (27)
Figures 10 and 11 provide an example of what happens if H is decreased, holding all
other exogenous variables constant. Notice that all of F(ÿ) increases when H decreases
from 1400 to 700. Both FH(ÿ) and FL(ÿ) rise with a fall in H.
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200 400 600 800 1000 1200 1400Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 10
Figure 10: α = 0.4 θ = 0.3 H = 1400 c = 50 λ = 123.636
β = 0.4 n = 3 L = 200 M = 550
100 200 300 400 500 600 700 Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 11
Figure 11: α = 0.4 θ = 0.3 H = 700 c = 50 λ = 85.4545
β = 0.4 n = 3 L = 200 M = 550
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The next three comparative static calculations are more lengthy.
Proof (7b):
( )
)(
)(1)(
1)(
1/1 −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−−−−=
n
i
ii
iLcpR
cpn
pHnpF
β θ
θα
(15)
∂FL/∂α= [ ]⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
−
−βθ−−−α−θ+−−θ
−−
−
−
2i
2
iiiii1n
n2
)c p(nR
)c p()1() pH()[c p()c p(R ) pH([Lfract
1n
1
=⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−βθ−θ−−θα+−θ⎥⎦
⎤⎢⎣
⎡
−
−βθ−−−θα
−− −
−
)c p(nR
)c p()1() pH(R ) pH([
)c p(R
)c p(n/)1() pH(n/
1n
1
i2
iii1n
n2
i
ii
= [ ]⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−βθ−θ−α+θ−β−+θα−−θ
−−
−
−
)c p(nR
)c p()1(])1)(1()1)[( pH([Lfract
1n
1
i2
ii1n
n2
= [ ]⎥⎥⎦
⎤
⎢⎢⎣
⎡
−θ
−θ−θβ+θ−β−+−−θ−α+θ−β−+
−−
−
−
)c p(R )/n(
)c p)](1()1)(1(1[)cH)](1()1)(1(1[Lfract
1n
1
i2
i1n
n2
∂FL/∂α < 0 iff the numerator on the right-most fraction is greater than zero, or
.0)c p)](θ1(θβ)θ1)(β1(1[)cH)](θ1(α)θ1)(β1(1[ i >−−+−−+−−−+−−+
This condition can be rewritten as:
)c p()cH()]θ1(θβ)θ1)(β1(1[)]θ1(α)θ1)(β1(1[ i −>−−+−−+ −+−−+ (a)
We know from proposition (1c) that firms will sell to both low and high type consumers
rather than just to low types if:
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θα/n ( H – c ) > [θα/n + (1 - θ) β/n](L - c) (12)
Rearranging the above equation to make an easier comparison with condition (a):
)cH(β)θ1(θα
θα−⎥
⎦
⎤⎢⎣
⎡−+
> (L – c) (28)
If the left side of (a) can be shown to be greater than the left side of equation (28)
)cH()]θ1(θβ)θ1)(β1(1[
)]θ1(α)θ1)(β1(1[−⎥
⎦
⎤⎢⎣
⎡
−+−−+−+−−+
> )cH(β)θ1(θα
θα−⎥
⎦
⎤⎢⎣
⎡
−+(b)
then we will have shown that condition (a) holds by taking advantage of transitivity:
)cH()]θ1(θβ)θ1)(β1(1[
)]θ1(α)θ1)(β1(1[−⎥
⎦
⎤⎢⎣
⎡
−+−−+−+−−+
> )cH(β)θ1(θα
θα−⎥
⎦
⎤⎢⎣
⎡
−+> (L – c) > (pi –c) (c)
The last relation (L - c) > (pi –c) holds since L ≥ pi in the domain of FL(pi).
Using a ratio test: setting ⎥⎦
⎤⎢⎣
⎡
−+−−+−+−−+
)]θ1(θβ)θ1)(β1(1[
)]θ1(α)θ1)(β1(1[= a/b and ⎥
⎦
⎤⎢⎣
⎡
−+ β)θ1(θα
θα= c/d
ad = θα + θα(1-θ)(1 - β)+(1- θ)β + β(1-β)(1 - θ)2 +α2θ(1-θ)+αβ(1-θ)2 >
θα +θα(1-β)(1- θ) + αβθ2(1-θ) = bc
Subtracting the right-hand side so that bc = 0:
ad = (1- θ)β + (1- θ)β(1- β)(1- θ) - β(1 - θ)αθ2 + α2θ(1 - θ) + αβ(1 - θ)2 > 0 = bc
Thus ad = (1-θ)β[1 + (1 + α - β)(1 - θ) - α θ2] + α2θ(1 - θ) > 0 = bc
Thus condition (c) holds implying ∂FL/∂α < 0.
Figures 12 and 13 provide an example of changing α and holding all other
exogenous variables constant. The effect of decreasing α from Figure 12 to Figure 13 is
an increase in F(ÿ). Both FH(ÿ) and FL(ÿ) rise with a fall in α.
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200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 12
Figure 12: α = 0.9 θ = 0.3 H = 1000 c = 50 λ = 175.735
β = 0.3 n = 3 L = 200 M = 802.284
200 400 600 800 1000 Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 13
Figure 13: α = 0.2 θ = 0.3 H = 1000 c = 50 λ = 73.1707
β = 0.3 n = 3 L = 200 M = 642. 735
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Proof (7c):
( )
)(
)(1)(
1)(
1/1 −
⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢
⎢
⎣
⎡
−
−−−−
−=
n
i
ii
iL cpR
cpn
pHn
pF
β θ
θα
∂FL(pi)/∂θ
[ ]
[ ]⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−βθ−−−θαα−β−−β+−α
−−
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−β−−α−−−−β+−α
−−
=
−
−
−
−
)c p(nR
)c p()1() pH()[((F)]c p() pH([Lfract
1n
1
)c p(nR
]LfractN*n)[c p)](1()1[()c p(F)]c p() pH([Lfract
1n
1
i2
iiii1n
n2
2i
2
iiii1n
n2
= [ ]⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−θ−α−β+β+α−βθ−−α
−−
−
−
)c p(nR
)c p)](1)((R [)](R )[ pH([Lfract
1n
1
i2
ii1n
n2
= [ ]⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−α−β+β−−α
−−
−
−
)c p(nR
)c p)(1()1)( pH([Lfract
1n
1
i2
ii1n
n2
= [ ]
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−α−β−β−α−β−−α
−−
−
−
)c p(nR
)c p)](1()1([)1)(cH([Lfract
1n
1
i
2
i1n
n2
= [ ]⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−β−α−−β−α
−−
−
−
)c p(R 2
)c p)(()cH)(1(Lfract
1n
1
i2
i1n
n2
(29)
∂FL/∂θ < 0 if the numerator of the right fraction is greater than zero. Rearranging this, we
obtain condition (d):
)cH(βα
)β1(α
−⎥⎦
⎤
⎢⎣
⎡
−
−
> (pi –c) (d)
Again, we use equation twenty – eight to make easier comparisons with (d):
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)cH(β)θ1(θα
θα−⎥
⎦
⎤⎢⎣
⎡
−+> (L – c) (28)
Iff (d) holds, then the left hand side of (d) should be greater than the left-hand side of (28)
or condition (e):
)cH(βα
)β1(α−⎥
⎦
⎤⎢⎣
⎡
−−
> )cH(β)θ1(θα
θα−⎥
⎦
⎤⎢⎣
⎡
−+(e)
then we will have shown that condition (d) holds by taking advantage of transitivity:
)cH(
)]θ1(θβ)θ1)(β1(1[
)]θ1(α)θ1)(β1(1[−⎥
⎦
⎤⎢⎣
⎡
−+−−+
−+−−+> )cH(
β)θ1(θα
θα−⎥
⎦
⎤⎢⎣
⎡
−+
> (L – c) > (pi –c) (f)
Again, using a ratio test: letting ⎥⎦
⎤⎢⎣
⎡
−
−
βα
)β1(α= a/b and ⎥
⎦
⎤⎢⎣
⎡
−+ β)θ1(θα
θα= c/d
ad = α(1 - β)θα + β(1 - θ)α(1 - β) > θα(α - β) = bc
Subtracting the right hand side to allow bc to equal zero:
ad = α2θ - α2βθ + β( 1 - θ)α( 1 - β) - α2θ + αθβ > 0 = bc
= - α2βθ + βα - αθβ - β2α + β2αθ + αθβ
= βα [ -αθ + 1 - β(1-θ) ] > 0
since β(1 - θ) - αθ < 1
Thus conditions (d) holds implying ∂FL/∂θ < 0.
Figures 14 and 15 provide an example of changing θ and holding all other
exogenous variables constant. The effect of decreasing θ from Figure 14 to Figure 15 is a
decrease in FL(ÿ) .
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Since FH(ÿ) does not depend on θ, there is no change in FH(ÿ) when θ changes. Another
way to describe the effect of changing θ on FL(ÿ) is that a high θ implies that there is a
small distance between FL(ÿ) and FH(ÿ) as Figure 14 demonstrates. When θ decreases,
there is less high consumer types and more low consumer types. Firms discount more to
the low types and FL(ÿ) is higher at low types, or is further away from FH(ÿ) as Figure 15
demonstrates.
200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 14
Figure 14: α = 0.4 θ = 0.8 H = 1000 c = 50 λ = 188.182
β = 0.4 n = 3 L = 200 M = 237.5
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200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 15
Figure 15: α = 0.4 θ = 0.3 H = 1000 c = 50 λ = 101.818
β = 0.4 n = 3 L = 200 M = 550
Proof (7d):
( )
)(
)(1)(1)(
1/1 −
⎥⎦
⎤⎢⎣
⎡
−
−−−−−=
n
i
iiiL
cpnR
cppHpF
β θ θα
Calculating ∂FL/∂β gives the least straightforward answer in Proposition 7.
∂FL(pi)/∂β
= [ ] ⎥⎦
⎤⎢⎣
⎡
−
−−−−−−+−−−−−
−
−
222
2
1
2
)(
)]()1()()[)(1()()1([
1
1
cpRn
cppHcpncpnRLfract
n i
iiiin
n β θ αθ θ θ
[ ] ⎥⎦
⎤
⎢⎣
⎡
−
−−−−−−−+−−−
−
−
= −
−
)(
)()1()()()[1()()1([
1
1
21
2
cpnR
cpcpcHcpR
Lfractn i
iii
n
n β θ αθ αθ θ θ
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[ ]⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−βθ−+αθ+θ−β−+θα−θ−−−αθθ−
−−
= −
−
)c p(nR
)c p]()1()1)(1()1)[(1()cH()1(Lfract
1n
1
i2
i1n
n2
[ ] ⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
−θ−+θθ−−−αθθ−
−
−
= −
−
)c p(nR
)c p)](1()[1()cH()1(
Lfract1n
1
i2
i
1n
n2
[ ] [ ])c p()cH()c p(nR
)1(Lfract
1n
1i
i2
1n
n2
−−−αθ⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
θ−
−−
= −
−(30)
∂FL/∂β < 0 if the third bracketed term [θα(H – c )-(pi –c) ] in equation (30) is greater than
zero. Writing this as condition (g) we have:
θα(H – c) > (pi –c) (g)
Again, we use equation 28 to make easier comparisons with (d):
)cH(β)θ1(θα
θα−⎥
⎦
⎤⎢⎣
⎡
−+> (L – c) (28)
Unlike the previous two derivatives, we cannot say that αθ(H - c) is greater than (L - c),
since the denominator in the second term of )cH(
β)θ1(θα
θα−
⎥⎦
⎤
⎢⎣
⎡
−+is less than one. We
do know that θα(H - c) is really monopoly profits of each firm at H since a n can be
factored out from the denominator of equation (30):
[ ] ⎥⎦
⎤⎢⎣
⎡ −−
−⎥⎦
⎤⎢⎣
⎡
−−
−−
= −
−
n
cp
n
cH
cpRLfract
ni
i
n
n )()(
)(
)1(
1
12
1
2 αθ θ (30a)
With a small L that θα( H - c) > (L – c). If θα( H - c) is greater than (L - c), then ∂FL/∂β
< 0 everywhere on [λ,L]. This constitutes cases where the fraction of uninformed high
types is large. The opposite case is when the uninformed high types are not large. If θα(
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H - c) is less than (L - c) but greater than (λ -c), then ∂FL/∂β = 0 for some pi on the
interval [λ,L], where θα(H - c) + t = pi- c. In this case, FL(ÿ) rotates counterclockwise as
β is increased. Prices near L have ∂FL/∂β > 0 while prices near λ have ∂FL/∂β < 0.
Figures 16 through 19 provide examples of how FL(ÿ) changes with β. Figures 16
and 17 provide a case where ∂FL/∂β < 0 everywhere on [λ,L]. In these two diagrams
θα( H - c) = 152 is greater than (L - c) = 150. The two endpoints of FL(ÿ) in Figures 16
and 17 match the case of ∂FL/∂β < 0 everywhere on [λ,L]. λ in Figure 17 is lower than λ
in Figure 16 and FL(L) is lower in Figure 16 than FL(L) in Figure 17. This corresponds
with FL(ÿ) being higher in Figure 17 with a lower β = 0.2 than Figure 16 where β = 0.6.
Figures 18 and 19 demonstrate the case where prices near L (and for the most part of
FL(ÿ) ) have ∂FL/∂β > 0 while prices near λ have ∂FL/∂β < 0. In these diagrams θα( H -
c) = 114 is less than (L - c) = 150. FL(ÿ) does a slight rotation to the counterclockwise
direction. In Figure 18 with a higher β, λ and FL(L) are both higher than λ and FL(L)
respectively in Figure 19. This setup means that if the two FL(ÿ)’s from Figures 18 and 19
are drawn on the same diagram, the two would cross. ∂FL/∂β is slightly positive at prices
near L and negative at all other prices.
The four diagrams also demonstrate that changing β alone does not produce large
changes in FL(ÿ). Visual inspection can verify this result between Figures 16 and 17 and
between Figures 18 and 19. As an added check, λ does not vary widely, especially
between Figures 18 and 19. FL(L) does not vary widely, especially when considering
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Figures 16 and 17. Thus the indeterminate result of ∂FL/∂β is not as important since the
actual variance in FL(ÿ) from β is small.
200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 16
Figure 16: α = 0.8 θ = 0.2 H = 1000 c = 50 λ = 138.372
β = 0.6 n = 3 L = 200 F(L)= .412055 M = 804.412
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200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 17
Figure 17: α = 0.8 θ = 0.2 H = 1000 c = 50 λ = 114.407
β = 0.2 n = 3 L = 200 F(L)= .417017 M = 807.031
200 400 600 800 1000 Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 18
Figure 18: α = 0.3 θ = 0.4 H = 1000 c = 50 λ = 100
β = 0.4 n = 3 L = 200 F(L)= .543565 M = 436.441
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200 400 600 800 1000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 19
Figure 19: α = 0.3 θ = 0.4 H = 1000 c = 50 λ = 92.2222
β = 0.05 n = 3 L = 200 F(L)= .510903 M = 405.205
Proof (7e):
( ) )(
)(1)(1)(
1/1 −
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−−−=
n
i
ii
iLcpR
cpnpHnpF
β
θ
θα
∂FL(pi)/∂c =
( ) ( ) ( ) ( ) ( ) ( )[ ]( )⎥⎦
⎤⎢⎣
⎡
−−−−−−−−−
⎥⎦
⎤⎢⎣
⎡
−−−−−
−−
−
−
222
1
2
)(
11
)(
)(1)(
1
1
cpRn
nRcppHcpnR
cpnR
cppH
n i
n
n
i
ii β θ θα β θ β θ θα
=( ) ( )
0)(
)(
)(1)(
1
12
1
2
<⎥⎦
⎤⎢⎣
⎡
−
−⎥⎦
⎤⎢⎣
⎡
−−−−−
−−
−
−
cpnR
pH
cpnR
cppH
n i
n
n
i
ii θα β θ θα
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Figures 20 and 21 provide an example of changing c and holding all other exogenous
variables constant. The effect of decreasing c from Figure 20 to Figure 21 is an increase
in F(ÿ). Both FH(ÿ) and FL(ÿ) rise with a fall in c.
200 400 600 800 1000 1200Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 20
Figure 20: α = 0.3 θ = 0.4 H = 1200 c = 300 λ = 340
β = 0.05 n = 3 L = 400 M = 553.311 g = 153.311
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200 400 600 800 1000 1200Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 21
Figure 21: α = 0.3 θ = 0.4 H = 1200 c = 50 λ =101.111
β = 0.05 n = 3 L = 400 M = 738.38 g = 338.38QED
Proposition 8: (a) ∂λ/∂β > 0
(b) ∂λ/∂θ > 0
(c) ∂λ/∂H > 0
(d) ∂λ/∂α > 0
(e) ∂λ/∂c > 0
Proposition 8 is the comparative statics dealing with the lowest price l. All of the
comparative statics are positive.
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Proof (8a):
Comparative statics calculations on λ are straightforward:
∂λ/∂β =2])1n(n))(1n([
)]1n()1n()[cH(
β−−+α−β−θ−−−θ−θα−
=2])1n(n))(1n([
)1)(1n)(cH(
β−−+α−β−θ
θ−−−θα> 0 (31)
Figures 16 – 19 provide two examples that demonstrate that ∂λ/∂β > 0. When β
falls from 0.6 in Figure 16 to 0.2 in Figure 17, λ falls from 138.372 in Figure 16 to
114.407 in Figure 17. The same trend occurs in Figures 18 and 19. When β falls from
0.3 in Figure 18 to 0.05 in Figure 19, λ falls from 100 in Figure 18 to 92.2222 in Figure
19.
Proof (8b):
∂λ/∂θ =2])1n(n))(1n([
))(1n)(cH(])1n(n))(1n()[cH(β−−+α−β−θ
α−β−−θα−β−−+α−β−θ−α
=2])1n(n))(1n([
)])(1n()1n(n))(1n()[cH(
β−−+α−β−θ
α−β−θ−β−−+α−β−θ−α
=2])1n(n))(1n([
])1n(n)[cH(
β−−+α−β−θ
β−−−α> 0 (32)
Figures 14 and 15 provide an example that demonstrates that ∂λ/∂θ > 0. When θ
falls from 0.8 in Figure 14 to 0.3 in Figure 15, λ falls from 188.182 in Figure 14 to
101.818 in Figure 15.
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Proof (8c):
∂λ/∂H =θα+θ−β+θα−θ−β−
θα)1(])1(1[n
> 0 (34)
Figures 10 and 11 provide an example that demonstrates that ∂λ/∂H > 0. When H
falls from 1400 in Figure 10 to 700 in Figure 11, λ falls from 123.636 in Figure 10 to
85.4545 in Figure 11.
Proof (8d):
∂λ/∂α =2])1n(n))(1n([
))1n()(cH(])1n(n))(1n()[cH(
β−−+α−β−θ
−θ−−θα−β−−+α−β−θ−θ
=2])1n(n))(1n([
)]1n()1n(n))(1n()[cH(
β−−+α−β−θ
−θα+β−−+α−β−θ−θ
=2])1n(n))(1n([
])1n(n))(1n()[cH(
β−−+α−β−θ
β−−+α+α−β−θ−θ
=2])1n(n))(1n([
])1n(n)1n()[cH(
β−−+α−β−θ
β−−+β−θ−θ
=2])1n(n))(1n([
)]1)(1n(n)[cH(
β−−+α−β−θ
θ−−β−−θ> 0 (35)
Figures 12 and 13 provide an example that demonstrates that ∂λ/∂α > 0. When α
falls from 0.9 in Figure 12 to 0.2 in Figure 13, λ falls from 175.735 in Figure 12 to
73.1707 in Figure 13.
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Proof (8e):
λ = cnT
cH+
− )(αθ (20)
∂λ/∂c = 1+−nT
αθ
=( )[ ] ( )[ ]
1111
++−+−−−
−θα β θ β θ θα
θα
n(36)
The denominator in equation (36) is always greater than the absolute value of the
numerator. Thus the left fraction is always between 0 and -1. Thus ∂λ/∂c > 0. Figures
20 and 21 provide an example that demonstrates that ∂λ/∂c > 0. When c falls from 300 in
Figure 20 to 50 in Figure 21, λ falls from 340 in Figure 20 to 101.111 in Figure 21.
Proposition 9: (a) ∂g/∂θ < 0
(b) ∂g/∂α < 0 if ( )[ ] )(1
)( 22 cLcH −⎥
⎦
⎤⎢
⎣
⎡
−−
+<− β
β α θ β θα
∂g/∂α > 0 if ( )[ ] )(1
)( 22 cLcH −⎥
⎦
⎤⎢⎣
⎡
−−
+>− β
β α θ β θα
(c): ∂g/∂β < 0 if θα( H - c) ≥ (L - c)
∂g/∂β > 0 if θα( H - c) < (L - c)
(d) ∂g/∂L > 0
(e) ∂g/∂H > 0 if α > β
∂g/∂H < 0 if α < β
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(f) ∂g/∂t> 0 if UH > UL.
∂g/∂t< 0 if UH < UL.
(g) ∂g/∂c< 0
(h) Min g=( ) ( )[ ] ( ) )(1
)(1
)()(1112 Lf n
LF
Lf LFnnn −
−+
−−− −α
α
Proof:
When finding the comparative statics for g, it is important to keep in mind some
background information. First, g measures the gap between the two cumulative
distribution function FH and FL. As each of these exogenous parameters change, one or
both of the cumulative distribution functions will change. Furthermore L is fixed for all
comparative statics calculations (except obviously when finding ∂ε/∂L). Thus as FL
changes, FL(L) changes as well: increasing as FL(ÿ) increases and decreasing as FL(ÿ)
decreases. Since FH(M) = FL(L), the gap g can be thought as a line connecting FH(M)
and FL(L) sliding up and down both cumulative distribution functions.
With so much going on in finding the comparative statics for g, equation (22)
will be rewritten with abbreviated terms gterm, gD, and g N to reduce the sprawling of some
terms.
M = L)cL(
)cL(n
))(1()1(
)cL(
n
))(1()1)(1(
H
H
+−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−β−αθ−θ
+πα−θ
−β−αθ−θ
−πβ−θ−(23)
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g = )(
)())(1(
)1(
)())(1(
)1)(1(
cL
cLn
cLn
H
H
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
+−
−−−
−−−
β α θ θ π α θ
β α θ θ π β θ
(22)
g = )()1(
)1)(1(cL
g
g
termH
termH
−⎥⎦
⎤⎢⎣
⎡
+−−−−
π α θ
π β θ (22a)
g = )( cLg
g
D
N −⎥⎦
⎤⎢⎣
⎡(22b)
Proof (9a):
∂g/∂θ =
[ ] [ ])(
))()(21()1(2))()(21()(/)21)(1(2
cLg
gcLgcLcHn
D
NH
D −⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−−+−−−−−+−−− β α θ π α α β θ α θ β
Expanding the left term of the numerator:
[ ] DgcLcHn ))()(21()(/)21)(1( −−−+−−− α β θ α θ β
[ ][ ] [ ][ ]termH
termH gcLgcHn +−−−−−+−−−− π α θ β α θ π α θ α θ β )1())()(21()1()(/)21)(1(
Expanding the right term (carrying the negative sign):
[ ] DH gcL ))()(21()1(2 −−−+−− β α θ π α
[ ][ ] [ ][ ]termH
termHH gcLg −−−−−−−−−−−− π β θ β α θ π β θ π α )1)(1())()(21()1)(1()1(2
After a couple of cancellations, the first term reduces to:
[ ] [ ]HH
cLcHn π α θ β α θ π α θ α θ β )1())()(21()1()(/)21)(1( −−−−−−−−−
The second term reduces to:
[ ][ ]termHH g−−−−− π β θ π α )1)(1()1(2
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Combining the two terms in the numerator:
HH cL π α θ β α π α β )1())(())(1)(1( 2 −−−+−−−
[ ]))(()()1()1(/ cLcHn H −−−−−−− α β α β π α θ
∂g/∂θ < 0
iff –(1 - β)α(H –c) – (β - α)(L - c) < 0
iff )()(
)1(cH −⎥
⎦
⎤⎢⎣
⎡−
− β α
α β > (L - c) (37)
We know from equation (28) that
)cH(β)θ1(θα
θα−⎥
⎦
⎤⎢⎣
⎡
−+> (L – c) (28)
Equation (36) holds iff
⎥⎦
⎤⎢⎣
⎡
−−
)(
)1(
β α
α β > ⎥
⎦
⎤⎢⎣
⎡
−+ β θ θα
θα
)1(
iff (1 - β)αθα+(1 - θ)(1 - β)αβ > θα(α - β)
iff θα2 - αθαβ +αβ - θαβ + θαβ2 - αβ2 > θα2 - θαβ
iff - αθαβ +αβ + θαβ2 - αβ2 > 0
iff - αθ +1 + θβ - β > 0
Since this last expression holds, equation (36) holds and thus ∂g/∂θ < 0.
Figures 14 and 15 provide an example that demonstrates that ∂g/∂θ < 0. When θ
falls from 0.8 in Figure 14 to 0.3 in Figure 15, g rises from 37.5 in Figure 14 to 350 in
Figure 15.
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Proof (9b):
∂g/∂α=
[ ] [ ] )()(/)1()(/2)(/)1()(/)1)(1(
2
2
cLg
gcLncHngcLncHn
D
NH
D −⎥⎥⎦
⎤⎢⎢⎣
⎡ −−+−+−−−−−−−− θ θ θ θπ θ θ θ θ β
Again expanding the left term of the numerator:
[ ][ ] [ ][ ]termH
termH gcLngcHn +−−−−+−−−− π α θ θ θ π α θ θ θ β )1()(/)1()1()(/)1)(1(
Expanding the right term of the numerator (carrying the negative sign):
[ ][ ] [ ][ ]termH
termHH gcLngcHn −−−−−−−−−−−− π β θ θ θ π β θ θπ θ )1)(1()(/)1()1)(1(2)(/2
After cancellations, the left side of the numerator reduces to:
[ ][ ] [ ][ ]Hterm
H cLngcHn π α θ θ θ π α θ θ θ β )1()(/)1()1()(/)1)(1( −−−−+−−−−
The right side of the numerator becomes:
[ ][ ] [ ][ ]Hterm
HH cLngcHn π β θ θ θ π β θ θπ θ )1)(1()(/)1()1)(1(2)(/2 −−−−−−−−−−−
Combining both sides of the numerator yields:
( ) [ ] [ ])(/)1()(/2)(/)1)(1()1)(1( 22cLnRgcHncHn term
HH −−−−+−−−−+⎥⎦⎤
⎢⎣⎡ −− θ θ θ θπ θ θ β π θ θ β
Further simplification yields:
( )[ ] [ ] [ ][ ])())((2)1)(1()1())(1( 2
2
2
cLRcLcHn
cH−−−−−+−−+−−
−−α β α θα θ θ β θα β
θ θ
( )[ ] [ ][ ])()()1())(1( 222
2
2
cLcHn
cH−−−+−+−−
−− β α θ β β θα β
θ θ
( )[ ] ⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡
−
−+−−
−−−)(
1
)()1)()(1( 22
2
2
cLcHn
cH
β
β α θ β θα
β θ θ
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Thus,
∂g/∂α=
( )[ ]( )
)(
)(1
)()1)()(1(
22
22
cLgn
cLcHcH
D−
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡−⎥
⎦
⎤⎢
⎣
⎡
−
−+−−−−−
α
β α
β α θ
α
β θα β θ θ
(38)
Equation (38) is positive more often than it is negative. Theα
β term in the right –
hand expression is generally driving the size of the expression in front of (L – c). When
α > β or when α is slightly less than β, equation (38) is positive.α
β is less than one or
slightly greater than one, which is enough to keep( )
)(1
)( 2
cL −⎥⎦
⎤⎢⎣
⎡
−−
+ β α
β α θ
α
β less than
( )[ ]cH −θα . When β > > α, the expression is negative asα
β is large. Proposition (7b)
and Lemma (2a) demonstrate that ∂FL/∂α and ∂FH/∂α have the same sign: both being
negative. ∂g/∂α attempts to find which portion of the distribution flattens out faster with
a change in α. A high alpha causes both cumulative functions to be steeper – especially
FH. Lowering α from a high value will flatten FH faster (since it was very steep to begin
with) and thus lower the gap between the two functions at p = L.
Proof (9c):
∂g/∂β =[ ] [ ]
)()(/)1()(/)1()1(
2cL
g
gcLngcLn
D
NDH
−⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−−−−−+−− θ θ θ θ π θ
Expanding the left term of the numerator:
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[ ][ ] [ ][ ]termH
termHH gcLng +−−−++−−− π α θ θ θ π α θ π θ )1()(/)1()1()1(
Expanding the right term of the numerator (carrying the negative sign):
[ ][ ]termH gcLn −−−−−−− π β θ θ θ )1)(1()(/)1(
After cancellations and some rearranging, the numerator becomes:
= [ ][ ] [ ][ ]Hterm
HH RcLng π θ θ π α θ π θ )(/)1()1()1( −−++−−−
= )(//))(1()()1()1( cLnRncLHH −+−−−−−−− θ θ θ β α π α θ π θ
= ncLHH /))(1()1()1( −−+−−− α π α θπ θ
[ ])()()1)(1(
cLcHn
H
−+−−−−
θα θπ α θ
Putting the numerator back into the entire fraction gives:
∂g/∂β =[ ]
)()()()1)(1(
2
cLn
cLcH
D
H
−⎥⎦
⎤⎢⎣
⎡ −+−−−−
ε
θα θπ α θ (39)
Like proposition (7d), this equation will be positive if (L – c) > θα( H – c) and
negative otherwise. Again, FH(ÿ) does not depend on β and thus does not change with β.
If θα(H - c) < (L- c), then an increase in β will cause FL(ÿ) to pivot counterclockwise
around some price on the [λ.L) interval. FL(L) will increase and the gap g between L and
M will increase. If θα(H - c) > (L - c) then an increase in β will cause FL to decrease
everywhere in [λ,L] and thus g will decrease.
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Another way to see this result is remembering that β is the proportion of
uninformed low type consumers. A firm will have to increase its price by a greater g
from L to M to counter the discrete loss of these consumers not willing to pay more than
L. If (L – c) is less than θα(H – c), then the loss of the low type consumers is not as bad
for firms because there is a large enough proportion of uninformed high type consumers
that firms can rely upon. However, if there is not enough of these uninformed high types
(ie θα(H – c) < (L – c)) then the loss of the low types is important to firms and g will
have to increase as β increases.
Proof (9d):
This comparative static tells about the slope of the two cumulative probability
distribution functions. Unlike the other parts in proposition (9), both cumulative
probability distribution functions remain constant when changing L. If FH is steeper than
FL, then increasing L will narrow g. If FL is steeper than FH then increasing L will widen
g.
∂g/∂L =[ ] [ ]
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−+
2
D
NtermDterm
D
N
g
gggg
g
g
=⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−2
)(
D
DNtermDN
g
ggggg
= [ ]
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−+−−2
)1)(1()1(
D
termDN
g
ggg β θ α θ
=⎥⎥⎦
⎤
⎢⎢⎣
⎡ −2
D
termDN
g
Rggg(40)
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The first term g NgD of the numerator is positive. The term is generally large unless α is
extremely small. If α is very small, gD in the left – hand term becomes very small and ε N
also becomes small. Thus the left-hand term g NgD in the numerator of equation (40) is
very small with α being small. However, with α being small, the second term in the
numerator becomes positive so the numerator remains positive. A larger α means that
the g NgD term in the numerator becomes large and can handle the subtraction of Rgterm.
Thus, the numerator remains positive in all cases of α, β, θ, L, and H such that
Proposition 1 holds. Thus ∂g/∂L > 0.
Proof (9e):
∂g/∂H =[ ] [ ]
)(/)1(/)1)(1(
2cL
g
gngn
D
ND −⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−−− θα α θ θα β θ
Expanding the left term of the numerator:
[ ][ ]termH gn +−−− π α θ θα θ β )1(/)1)(1(
Expanding the right side of the numerator:
[ ][ ]termH gn −−−−− π β θ θα α θ )1)(1(/)1(
Recombining the two sides of the numerator:
[ ][ ]nngterm /)1(/)1)(1( θα α θ θα θ β −+−−
[ ][ ])1()1)(1(/ θ θ β θα −+−−ngterm
⎥⎦
⎤⎢⎣
⎡ −−−2
2 ))()(1(
n
cLR β α θ αθ
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130
Plugging the simplified numerator back into the original calculation:
∂g/∂H = 2
22
2
)())(1(
cLn
R
D
−⎥⎥
⎦
⎤
⎢⎢
⎣
⎡ −−
ε
β α θ αθ (41)
If α > β, then ∂g/∂H > 0. The g jump from FL(L) to FH(M) will be smaller as the
proportion of uninformed high types falls. Less uninformed higher type consumers will
counter the discrete loss of low types at prices above L.
Proof (9f):
g = )(
)())(1(
)1(
)())(1(
)1)(1(
cL
cLn
cLn
H
H
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
+−
−−−
−−−
β α θ θ π α θ
β α θ θ π β θ
(22)
g =( )
( ))(
)())(1(
)1(
)())(1(
)1)(1(
cL
cLnt
UUcH
nt
U
t
U
cLnt
UUcH
nt
U
t
U
LHHH
LHHL
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−
+−−
−−−
−−−−
θ θ θ θ
θ θ θ θ
g =( )
( ))(
))()(1()1(
))()(1()1)(1(
cL
cLUUcHUt
U
cLUUcHUt
U
LHHH
LHHL
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−+−−
−−−−−−−
θ θ θ θ
θ θ θ θ
g =( )
( ))(
))()(1()(
))()(1())(1(cL
cLUUtcHUUt
cLUUtcHUUt
LHHH
LHHL −⎥⎦
⎤⎢⎣
⎡
−−−+−−−−−−−−−
θ θ θ θ
θ θ θ θ
∂g/∂t=
( ) ( )[ ] ( )[ ]( )[ ]
)())()(1()(
))()(1()())()(1(12
cLcLUUtcHUUt
cLUUtcHUUtcLUUcHU
LHHH
LHHHLHH −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−+−−−−−−−−
θ θ θ θ
θ θ θ θ θ θ θ θ
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131
( )[ ] ( )[ ]( )[ ]
)())()(1()(
))()(1())()(1())(1(2
2
cLcLUUtcHUUt
cLUUcHUcLUUtcHUUt
LHHH
LHHLHHL −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−+−−−−−−−−−
θ θ θ θ
θ θ θ θ θ θ θ
=
( ) ( ) ( )
( )[ ])(
))()(1()(
)()()1()(122
22223
cLcLUUtcHUUt
cLcHUUtUcHUtU
LHHH
LHHHH −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−−+−−−
θ θ θ
θ θ θ θ
( ) ( ) ( ) ( )( )[ ]
)())()(1()(
)(1)())((122
22223
cLcLUUtcHUUt
cLUUtcLcHUUtUU
LHHH
LHHHLH −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−−−−−−−−+
θ θ θ
θ θ θ θ
( ) ( )( )[ ]
)())()(1()(
)()()()1()()1(22
22223
cLcLUUtcHUUt
cLcHUUUUtcHUUt
LHHH
LHHLHL −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−−−−−−−−+
θ θ θ
θ θ θ θ
( ) ( )( ) ( ) ( )( )[ ]
)())()(1()(
)(1)(12
22223
cLcLUUtcHUUt
cLUUtcLcHUUUt
LHHH
LHHLH −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−+−−−−+
θ θ θ
θ θ θ θ
=
( ) ( ) ( )
( )[ ])(
))()(1()(
)())((1)()()1(2
2
cLcLUUtcHUUt
cLcHUUtUUcLcHUUtU
LHHH
HHLHLHH −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−−−−−−−−
θ θ
θ θ θ
( ) ( ) ( )( )
( )[ ])(
))()(1()(
)(1)()()()1(22
2
cLcLUUtcHUUt
cLcHUUUtcLcHUUUUt
LHHH
HLHLHHL −⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−−+−−−−−−+
θ θ θ
θ θ θ
=( )[ ]
( ) ( ) 2
2)()(1
))()(1()(
)1(cLcHUU
cLUUtcHUUt
tUtULH
LHHH
HH −−−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
+−θ
θ θ
θ θ
( )[ ]( ) ( ) 2
22)()(1
))()(1()(
)())(1(cLcHUU
cLUUtcHUUt
UUtUUtLH
LHHH
HHHL −−−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−−−−+ θ
θ θ θ
θ θ
=( )[ ]
( ) ( ) 2
2)()(1
))()(1()(
)(cLcHUU
cLUUtcHUUt
UUttULH
LHHH
HLH −−−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−
−−θ
θ θ
=( )[ ]
( ) ( ) 2
22)()(1
))()(1()(cLcHUU
cLUUtcHUUt
UULH
LHHH
HL −−−−⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−+−−θ
θ θ θ (42)
∂g/∂t> 0 if UH > UL.
∂g/∂t< 0 if UH < UL.
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132
Proof (9g):
g = )()(
))(1()1(
)())(1(
)1)(1(
cLcL
n
cLn
H
H
−⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢
⎢
⎣
⎡
−−−+−
−−−
−−−
β α θ θ π α θ
β α θ θ π β θ
(22)
=( )
( ))(
))()(1()1(
))()(1()1)(1(cL
cLcH
cLcH−⎥
⎦
⎤⎢⎣
⎡
−−−+−−−−−−−−−
β α θ θ θα α θ
β α θ θ θα β θ
= )( cLg
g
denr
numr −⎥⎦
⎤⎢⎣
⎡
∂g/∂c =
[ ] [ ]( )[ ] ⎥
⎦
⎤⎢⎣
⎡−−⎥
⎦
⎤⎢⎣
⎡
−−−+−−
−−−−−−−−+−−−
denr
numrnumrdenr
g
gcL
cLcH
gg)(
))()(1()1(
))(1()1())(1()1)(1(2
β α θ θ θα α θ
β α θ θ θα α θ β α θ θ θα β θ
= [ ] ( )[ ]( )[ ] ⎥
⎦
⎤⎢⎣
⎡−−⎥
⎦
⎤⎢⎣
⎡
−−−+−−
−−−+−−−−+−−−
denr
numr
g
gcL
cLcH
cLcH)(
))()(1()1(
))()(1()1())(1()1)(1(2
β α θ θ θα α θ
β α θ θ θα α θ β α θ θ θα β θ
[ ] ( )[ ]( )[ ]
)())()(1()1(
))()(1()1)(1())(1()1(2
cLcLcH
cLcH−⎥
⎦
⎤⎢⎣
⎡
−−−+−−
−−−−−−−−−−−−−+
β α θ θ θα α θ
β α θ θ θα β θ β α θ θ θα α θ
= ( ) ( )[ ]( )[ ] ( )[ ]
)(1/))()(1()1(
))()(1()1()1()()1()1(2
cLcLcH
cLcHcH−⎥
⎦
⎤⎢⎣
⎡
−−−−+−−
−−−−−−−−+−−−−
θ θ β α θ θ θα α θ
β α θ αθ β θα α θ β α θα α αθ β
[ ]( )[ ] ( )[ ] ⎥
⎦
⎤⎢⎣
⎡−−⎥
⎦
⎤⎢⎣
⎡
−−−−+−−
−−−−+
denr
numr
g
gcL
cLcH
cL)(
1/))()(1()1(
))()(1()(2
θ θ β α θ θ θα α θ
β α θ θ β α
( ) ( )[ ]( )[ ] ( )[ ]
)(1/))()(1()1(
)1)(1)(()1()1(2
cLcLcH
cHcH−⎥
⎦
⎤⎢⎣
⎡
−−−−+−−
−−−−+−−−+
θ θ β α θ θ θα α θ
θα β θ β α θα β θα α
[ ]( )[ ] ( )[ ]
)(1/))()(1()1(
))()(1()())(()1(2
cLcLcH
cLcL−⎥
⎦
⎤⎢⎣
⎡
−−−−+−−
−−−−−−−−−+
θ θ β α θ θ θα α θ
β α θ θ β α β α θαθ α
=( )[ ]
( )[ ]( ) ( )[ ] ⎥
⎦
⎤⎢⎣
⎡−−−⎥
⎦
⎤⎢⎣
⎡
−−−+−−
−−−−+−−−
denr
numr
g
gcL
cLcH
LHLHθ θ
β α θ θ θα α θ
β β α α θ θ α β α α θ 1
))()(1()1(
))(1)((1))(1)((2
2
(43)
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Equation (43) is negative, evaluated at various parameters of a, b,q , and c. If the
expression is simplified further, the right hand term of ⎥⎦
⎤⎢⎣
⎡
denr
numr
g
gsimplified produces a
term ( )( )2
11 Hπ α β −−− in the numerator that appears to dominate all other terms. Thus
∂g/∂c < 0. Figures 20 and 21 provide an example that demonstrates that ∂g/∂c < 0.
When c falls from 300 in Figure 20 to 50 in Figure 21, ε rises from 153.311 in Figure 20
to 338.38 in Figure 21.
Proof (9h):
Repeating Equation (10): In a price dispersion equilibrium, the expected difference in
profits moving from L to L+g must equal zero.
( ) ( )( ) ( ) ( )( )( ) ( ) 0)(111)(11
1 11 =−−−−−+−−+−−
− −−cLLFgLFg
n
cL
n
g nn β θ α θ
β θ θα (10)
Minimizing this with respect to g gives the minimum possible g for a given set of
parameter values. Notice that g cannot be negative because Equation (10) would not
hold. (F(L+g) = F(L))
Minimizing equation (10) with respect to g:
( ) ( )( )[ ] ( )( )[ ] ( ) 0)(111)(11
1min
11 =−−−−−+−−+−−
− −−cLLFgLFg
n
cL
n
g nn
g β θ α θ
β θ θα
( )[ ] ( )( )[ ] 0)()(111)(11:
21
=++−−−−+−−+
−−
gLf gLFngLFnFOC
nn
α θε α θ
θα
( )( )[ ] ( )[ ] 12
min )(11)()(111−− +−−+=++−−− nn
gLFn
gLf gLFng α θ θα
α θ
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134
( ) ( )[ ][ ]
( ) )(1
)(1
)()(1112min
gLf n
gLF
gLf gLFnng
n +−+−
+++−−−
= −α
α
( ) ( )[ ][ ]
( ) )(1
)(1
)()(111 2min Lf n
LF
Lf LFnng n −
−
+−−−= −α
α
(44)
( )( )( )
( ) ( ) ( )( )
( )( )
⎥⎦
⎤⎢⎣
⎡
−
−
⎥⎦
⎤⎢⎣
⎡
−−−−−
+
⎥⎦
⎤⎢⎣
⎡
−
−−
=
22
min
1
1cLnR
cH
cLnR
cLLH
cLnR
cHn
gθα
β θ θα
θα α
α
( )( )
( )( ) ( ) ( )
( )
( ) ( )( )
⎥⎦⎤⎢
⎣⎡
−−−
⎥⎦
⎤⎢⎣
⎡
−−−−−
−+−−
=
2
min
1
11
cLnRcHn
cLnR
cLLHn
cLnR
cLnR
g
θα α
β θ θα α
α
( ) ( ) ( ) ( ) ( )[ ]( ) ( )
( )cLcH
cLLHcLRg −⎥
⎦
⎤⎢⎣
⎡
−−−−−−−+−
=θα α
β θ θα α α
1
11min
( ) ( ) ( )( )( )( ) ( )
( )cLcH
cLcHg −⎥
⎦
⎤⎢⎣
⎡
−−−−−+−−
=θα α
θ β α θα α
1
11min
( )( )( )( )
( ) ( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−
−−−+−= cH
cLcLg θα α
θ β α
1
12
min (45)
1-6. Explaining the Graphs: Matching the Comparative Statics to Airline Data
1-6a. Discussing the Computer Generated Graphs
From Figures 3 - 21, there can be six conclusions that can be drawn. First, entry
by firms changes the price distribution. The lowest price gets lower while more and more
weight gets placed on the highest price. Figure 9 is an example of many firms in the
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135
market. In that diagram, the spike in the cumulative distribution at H starts when F(p)
equals about .10.
A second conclusion that can be drawn is that shifting the proportion of
uninformed consumers to the extremes of zero and one causes the cumulative distribution
to collapse on certain prices. When α and β collapse to near zero as in Figure 6, the
cumulative distribution function nearly collapses to a point mass near marginal cost.
Firms price at this low price almost always with a very occasional price markup to take
advantage of whatever few uninformed consumers there are. When α and β move to near
one as in Figures 7 and 8, the cumulative distribution function moves to a near bimodal
distribution function at the monopoly high price and monopoly low price. Firms
essentially randomize between these two prices.
A third conclusion is that the two distributions FL(ÿ) and FH(ÿ) become closer
together as θ gets close to one. Figure 14 shows a case where θ is eighty percent. g is
small and only a small portion of the weight is put on the distribution below L. Figure 15
shows the proportion of business travelers θ falling to 0.3. g increases and the distance
between the two functions FL(ÿ) and FH(ÿ) increases. As can be seen from the two
diagrams, the proportion of high types θ is a big driver where FL(ÿ) is drawn.
The conditions of Proposition 1 which yield a dispersed equilibrium may be
violated for some airline markets. Figure 5 shows a case of an equilibrium where there is
only one group of customers. In this case it is the high customers only as the solid line is
on the FH(ÿ) portion of the graph only. Here it does not make sense for firms to discount
below L. With α so close to one, firms do not have to discount much since there are so
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few informed customers. Figure 4 is another case of an equilibrium where there is only
one type of customer being served by firms. This time, however, it is the low type of
customers. There are too few uninformed high types with α equal to 0.2 and too few
high types in general with θ being 0.3 for firms to place any weight on prices above L
and lose the business of leisure customers. Here the proportion of uninformed low types
is higher as β equals 0.6.
The informed low types are important in determining an equilibrium with two
types. In Figure 12, β is 0.3 and α is 0.9. Despite this large α, there is still an
equilibrium with two different types. Firms run extreme sales occasionally to capture the
group of informed low types. There is still not much weight on prices below L as the
cumulative density at L is rather low. Even though the uninformed high types is high,
there still is an equilibrium with two differing types of consumers because there are more
informed low types 1- β and less uninformed low types β.
However, the shape of the cumulative distribution function is determined by the
proportion of uninformed high types α and proportion of high types θ. Figures 18 and 19
and a lesser extent Figures 16 and 17 demonstrate that FL(ÿ) does not change much with a
change in the uninformed low types β, holding all other variables constant.
Figures 24 and 25 are probability density function graphs of the model. Figure 24
is the pdf from Figure 15’s cumulative distribution parameters. Figure 25 has new
parameters not encountered in Figures 3 – 19. In both of these diagrams there is a spikes
in the probability density function near the lowest price l and near the highest price H.
The spike in the pdf at the lowest price is thicker than the highest price. The region
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above M in both diagrams has some probability mass, but not a gigantic spike. There is a
region between L and M in both diagrams where probability density is zero. One can
conclude from these diagrams, that the fares cluster around certain regions, especially for
the lower prices.
This holds up in a three – valuation case as Figures 26 and 27 indicate. In these
diagrams, there are two regions where there is no randomization by individual firms, as it
is less profitable to randomize in these regions. The cumulative distribution rises
between l and L1, M1 and L2, and M2 and H in Figure 26. The probability density has
three areas of clustering, two below $600 and one above $1700. The lowest area of
clustering below $280 has the highest peak of the lower two area that have density.
1-6b. Matching the Computer Graphs to the Airline Data
Figures 28 through 35 are kernel density estimates taken from the second quarter
1995 airline passenger Data Bank 1a. Data Bank 1a is a ten percent random sample of all
passenger itineraries within the US and international. Among some of the variables
included in the data set is the itineraries that passengers take, the total fare paid for the
itinerary, the nonstop segments a passenger takes on the itinerary, the distance of each
segment, the airline carrying on the segment, type of seat (first class, business, or coach)
on each segment, and the number of passengers taking this exact same itinerary and
paying the same fare. For instance, one entry in the data set might be 5 passengers
paying $324 for an Austin – Providence roundtrip trip on American Airlines connecting
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in Chicago O’Hare each way with each of the four segments being coach. For the
purposes of these markets, the data bank is cleaned up of international flights and
itinerary beyond a nonstop flight (with the exception of Austin – Providence where there
are no carriers offering nonstop service). The markets chosen are not designed to be a
representative selection of the US market, but provide enough of a contrast that some of
the varying features of the US airline market can be exploited for study. Below each
kernel estimate is a market share breakdown of the various carriers that serve the market.
The tallies beside each carrier need to be multiplied by ten to get actual passenger totals
since this is a ten percent sample.
Figure 28 is an example of a primarily leisure market – Los Angeles to Honolulu.
There is a spike of passengers that are paying zero dollars – as lots of people use frequent
flier awards on this segment. Then there is a large single peak somewhere at two
hundred dollars one way and then very small bumps at extremely high fares. These fares
are several times the fare of the most frequently paid leisure fare. Most people pay the
discounted fare and few passengers actually pay the posted business fare. This market
has seven nonstop carriers serving the route and the share between each carrier generally
falls between ten and eighteen percent.
There are markets that have a strong business customer presence. Figure 29 is an
example of this. Here there is a large proportion of business types or q . The Minneapolis
– Chicago O’Hare market in 1995 was known as one of the top revenue grossing yield
per mile routes in the United States. The three carriers United, American, and Northwest
Airlines all offered business travelers numerous flight options for them to choose to fly
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the route. There is a narrow, but sharp spike around $300 as lots of business travelers
paid that fare one way between the two cities. Then there are two smaller spikes in the
probability density functions $100 and $170 for more of the leisure travelers.
Figure 32a is another example of a strong business market. This figure shows a
case of two large carriers competing between their hubs: Minneapolis and Atlanta. There
is also a solid mix of leisure travelers, judging by the two separate humps for each type of
consumer. One hump, indicating the leisure travelers, has its peak in the $180 range.
The other hump is at $420 one way and this hump has the same height and nearly the
same width as the one in the $180 range.
Breaking down the Minneapolis to Atlanta market into discounted coach YD and
full coach Y fares reveals that there is still price dispersion with multiple peaks at the
discounted coach YD level. The overall coach discount YD market fare distribution in
Figure 32b shows two large peaks. Northwest Airlines shows two peaks in the discount
coach YD tickets in Figure 32c. Delta Airlines shows four local peaks in the discounted
coach YD one-way fares. The largest two peaks for Delta are at $175 and $210 one-way.
Unlike the New York City – Orlando market, Delta’s full coach Y one-way fares in
Figure 32e shows very little price dispersion. There is one narrow peak at $425 one –
way with a density height that is about 10 times the height of the Delta coach discount
YD fares. Table 33 reveals that Delta sold 23.9% of its fares on the Minneapolis –
Atlanta route as full coach tickets, almost four times as much as the New York City to
Orlando market. Northwest Airlines only sold 1.5% of its tickets on the Minneapolis –
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Atlanta route as full coach Y tickets. Most of Northwest Y tickets are priced just above
$0. These are most likely free tickets redeemed by frequent fliers.
Figure 35 is the only case of a market where there is no nonstop service provided
by airlines between the two cities. The market Austin to Providence is much smaller than
the rest of the markets that have been studied, but is a good market to look at because
there are several ways that a passenger can connect between the two cities. A passenger
can make single plane connections on American through Chicago, Delta through
Cincinnati or Atlanta, or United through Chicago. This market shows that there are two
different types of customers in the market. The mode in the fares is $200 one way but
there is another sizeable spike at $580 one way. This $580 has been the highest fares that
spikes of all the city pairs chosen in Figures 28 through 35. American Airlines has the
dominant market position in the market with over 40% market share.
There are markets that fall in the middle of these two extremes where there are
leisure travelers and a smaller presence of business travelers. Figure 31a shows a market
of mostly leisure travelers between New York City and Orlando, Florida. Here the
proportion of customers with a reservation value of L, (1- q ), might be high. There is a
spike in passengers paying $150 one way. This might be because there are so many
leisure passengers on the route or that there is the presence of low cost carrier Kiwi
Airlines or American Trans Air on the route that causes other carriers on the route to
match their lower fares. After the large spike at $150, there is only a couple of smaller
spikes at $200 and $300 one way, suggesting that there still are relatively few business
travelers on the route that are paying higher fares.
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However, if the overall market is broken down into discounted coach YD fares
and coach Y fares, there is still dispersion within each fare class. Some airlines, like
Delta, Kiwi, TWA, and American Trans Air have multiple peaks at the discounted coach
YD fares. Other airlines, such as Continental, Delta, and US Airways have multiple
peaks at the full Y coach fare. Figure 31a shows the massive number of leisure tickets
that are sold on the route. However, Table 32 indicates that the three largest carriers on
the route sold many full coach Y tickets. Continental, Delta, and US Airways sold
13.3%, 6.8%, and 6.3% of their total tickets on the New York City to Orlando flights
respectively as full coach Y tickets.
Figure 34 is another market that looks quite similar to New York City to Orlando.
In this case, the market is New York City to Chicago’s two airports. Like New York City
to Orlando, there is more leisure travelers and thus a spike in fares paid around $100.
There are smaller spikes at the $220 and $420 range, indicating areas that business
customers are paying up to several times the leisure fare in this market. New York City
to Orlando did not have a spike as upward as $400, thus indicating that New York City to
Chicago may have a stronger business presence than the New York City to Orlando.
Finally, there are markets dominated by low cost carriers. Figure 33 is an
example of a market dominated low cost Southwest Airlines. Notice the several clusters
of fares. There are more peaks in this market than some of the other airline markets.
Southwest has over a sixty percent market share on the route with America West and
United Airlines having market shares only in the lower teens. Fares in this market are
substantially less compared to the other short hauled route Minneapolis – Chicago
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O’Hare. The air mileage between Los Angeles and Las Vegas is only 232 miles while
the miles between Minneapolis and Chicago O’Hare is 328 miles. The highest spike in
fares between Los Angeles and Las Vegas are approximately one fourth as large at $75
one way compared with Minneapolis to Chicago O’Hare. The other spike in fares occurs
at $0 and at $40 one way, substantially below the one-way fares in the Minneapolis to
Chicago O’Hare market. There is a small hump in the $140 area, suggesting that if
Southwest Airlines was not in the Los Angeles – Las Vegas market, other airlines might
be charging their passengers this fare or higher. Passenger loads are quite strong in this
market as there are five times as many passengers as the Minneapolis – Chicago O’Hare
market.
Figures 28 through 35 provide evidence that there is dispersion in fares that
consumers pay and that fares do cluster around certain prices in particular markets. This
clustering phenomenon is a continuation from Figures 24, 25, and 27, the pdf diagrams.
In some markets, there is a resemblance to the earlier figures in that there are two spikes
in the data, such as in the Minneapolis – Atlanta or Austin – Providence markets. Each
of those spikes could be thought of as the different types of consumers – business or
leisure customers that the airlines are targeting with the dip between the spikes as the gap
where firms find it unprofitable to target consumers. Even, in the cases where there are
more than two spikes in the fare data, the theory could still apply as there could be more
than two types of customers in the data. Figure 27 is an pdf example of 3 valuations.
Figures 31b – 31k and Figures 32b - Figure 32f control for price discrimination between
full coach Y fares and discount coach YD fares. Price dispersion with multiple peaks
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occur in the YD fares and sometimes occur in the full coach Y fares. Thus the data might
provide some evidence that there is heterogeneity of consumer types within dispersed
airline prices.
1-7. Analysis and Conclusion
Previous price randomization models have shown that firms can deliberately vary
prices as a means to partition customers with differing information. Those with more
information of prices in the marketplace will on average pay a lower price than those
with limited information. Striking in the models of Butters (1977), Varian (1980), and
Burdett and Judd (1983) is that consumers differ only in the amount of information
received in the marketplace; the willingness to pay of all consumers in this class of
models is the same. Given the experience of certain markets – especially the airline
market – this assumption seems quite unrealistic.
Besides heterogeneity in information, this model allows heterogeneity in
consumer valuation. The model finds that under the right mix of proportion of consumer
types and valuations, that there will exist a price randomization equilibrium where firms
sell to all consumers. The presence of the gap, g, is a good indication of how well price
randomization works selling to consumers with different values. First, if g is greater than
the spread between any price L in the support of FL and H, then price randomization
between consumers with different valuations is not a feasible equilibrium. In this case,
firms will serve the low end of the market through price dispersion rather than serving
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both groups of consumers. Charging the high types higher prices does not completely
make up for the discrete loss of low types. On the other hand, too many uninformed high
types may make steep discounting unprofitable for firms. In this case g is negative. The
extra revenue generated selling to the low types will not outweigh the revenue lost by
discounting to the uninformed high types. Only g values in the middle will be associated
with equilibria with two different consumer types.
The comparative statics section suggests how price dispersion adjusts to changing
consumer characteristics. Sometimes there will be a consistent change in the entire
distribution of prices. Increasing both H and α lowers the value of both FH and FL at any
given price and concentrates the weight of the distribution functions at the high end.
With higher willingness to pay of the business travelers or proportion of loyal business
travelers, firms will discount less as they can make greater revenue from this smaller
group of business travelers.
Changing other consumer characteristics only affects one part of the price
distribution. β and θ have no effect on FH. Increasing θ decreases FL. θ has a big
impact on where FL is placed. Increasing β could either increase or decrease FL
depending on the proportion of uninformed high types θα and the spread between H and
L. A large spread between H and L or high θα will result in ∂FL/∂β being negative. For
the parameter values used in this paper, FL does not change much with a change in β.
Proposition 9 provides good perspective of the how the two cumulative
probability distribution functions move when an exogenous variable changes. As each
cumulative distribution function changes, the interval g connecting FL(L) and FH(M)
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changes as well. Sometimes both functions change in a way that makes predicting g
simple – such as when θ changes. As θ, the proportion of high types increases, the lower
probability distribution function FL decreases - moving closer to a stationary FH thus
reducing g. Changing α - the proportion of uninformed high types – is not as simple.
Here the proportion of uninformed low types β determines how the two functions move
together when α changes. A high β generally implies that ∂g/∂α will generally be
positive.
One important result is the case where the uninformed consumers approach one or
zero. In the case where the uninformed α and β consumers approach zero, the
cumulative distribution collapses down to the lowest price which is near marginal cost.
Near Bertrand pricing becomes the norm with very occasional price jumps to take in
account the few uninformed consumers. In the case where the uninformed α and β
consumers approach one, there is a spike in the cumulative distribution function around
the monopoly high price for consumers and monopoly low price for consumers. The
probability distribution for prices becomes very close to a bimodal one at the monopoly
high and low prices.
Another important result in the model is in the area of increasing the number of
firms and the resulting price distribution. This result continues some of the interesting
result from previous price randomization models. Varian (1980), for instance, gives a
clear answer about what happens to the lowest price when the number of firms increases.
In Varian’s model, the lowest price decreases but weight around the highest price
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increases when the number of firms increases. This model predicts the same result as
Varian (1980).
The model developed allows some flexibility. Two counterintuitive results can
occur under limited conditions. First, price randomization can occur even in cases not
representing the usual demand case where there is a larger proportion of high types θ than
low types (1 – θ). In this type of equilibrium, the two probability distributions FH and FL
are close together. g is small as the markup above L does not need to be very large to
make up for the discrete loss of the low types. A small α of uninformed high types also
helps make such equilibria possible.
Price randomization can also occur in cases where α is less than β. Normally we
would expect α to be larger than β as there could be some correlation between value and
the tendency to shop around. In this case, the lower cumulative distribution function FL
will become steeper. Under this case, ∂g/∂H changes signs to be negative.
There are a few main lessons learned from the graphs. First, the informed low
types are important in determining equilibria with two different consumer types. Figure
12 provided an example of equilibrium with two different consumer types occurring
despite the uninformed high types being so close to one. However, the shape of the
distribution functions depend on the proportion of the uninformed high types. Figures 18
and 19, as well to a lesser extent Figures 16 – 17 show that the shape of the FL(ÿ) curve
does not change dramatically as β is changed, holding all other variables constant.
Another conclusion that can be drawn is that equilibria do exist even when the proportion
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of uninformed business and leisure consumers are quite low. This is important in
showing that this model is relevant even when loyalty by consumers is very low or when
search costs are low. Figure 19 provides an example of this phenomenon. A third result
is that there are equilibria with two different types of customers in cases when firms
would appear to not have an incentive to run sales because the proportion of business
travelers who are uninformed is high. These equilibria need the proportion of low types
who are informed to be high and the overall proportion of business travelers to be
relatively low. As Figure 12 shows, firms mostly randomize prices around the monopoly
price of the business type, but occasionally run extreme sales to capture the informed low
types. Finally, as the pdf graphs indicate in Figures 24 – 25, fares in this model cluster
around certain prices. This holds in the two valuation model, as well as a three valuation
model depicted in Figures 26 and 27.
Finally, the ten percent airline sample data shows that fares cluster around certain
prices, indicating a place where the theory matches the data. Sometimes they cluster only
around a leisure price – such as in the Los Angeles to Honolulu market. Other times they
mainly cluster around a business price – such as in the Minneapolis to Chicago O’Hare
market. Sometimes they cluster in more than one place – sometimes almost
symmetrically as in the Minneapolis to Atlanta market or non- symmetrically as in the
Austin – Providence market. Sometimes there are several modes as there is in the
Washington National to New York LaGuardia market. This clustering of fares around
certain prices might suggest that there are regions in the distribution of prices that airlines
find it profitable to target customers while regions where there are dips are places in the
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distribution of prices where it is not profitable to target consumers. This would be a
place where the theory matches the data.
Even with price discrimination providing an explanation behind differences
between airline fares, the price dispersion story with different consumer types still applies
in the face of price discrimination. This clustering of fares observed in overall markets
occurs even controlling for price discrimination between fare types coach Y and coach
discount YD. As the Minneapolis - Atlanta and New York City - Orlando markets
indicate, there is fare clustering at the YD level for many individual airlines. Clustering
occurs for the full coach Y fare for the largest carriers in terms of market passenger share
- Continental, Delta, and US Airways - on the New York - Orlando market. Price
discrimination cannot explain these clustering of fares on the subclasses of coach fares.
This model begins an important step in the direction of including valuation as part
of the analysis behind price randomization. Future research can now include linking up
the literature on price discrimination and price randomization since consumers in both
types of models have heterogeneous valuations. Adding asymmetry in the model also is
good area for future research. How does dispersion change if one firm has more of the
uninformed types or if another firm has lower costs than the existing firms? Could there
be a cost to keep a certain group of customers loyal, or not willing to compare with other
firms? Does entry by new firms become deterred with firms with large shares of loyal
high value customers? Answering these questions could provide further predictions on
how electronic commerce will continue to shape markets.
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2:“Capacity and Random Prices”
2-1. Introduction
Airlines are characterized by rapidly changing prices and mobile but limited
capacity. Prices are notorious for changing frequently and at the last minute. Price
dispersion is large. Firms offering heterogeneous prices for the same product, based on
imperfect consumer information, have been modeled with such price dispersion models
such as Butters (1977), Varian (1980), and Burdett and Judd (1983). Consumers with the
same valuation of the good are segmented on the basis of the amount of information that
they receive about the price from each of the firms in the market. The more information
they receive, the greater their ability to shop between offers, and thus the lower price on
average they pay for the product or service. Those shoppers with little information do not
have this luxury of comparing between firms so they go with the offer that they receive.
These models are formally identical to models in which some consumers are “locked-in”
not by loyalty but by preference, and others are indifferent between suppliers.
Understanding the difference in the information that shoppers are facing, firms
face a tradeoff between capturing these “shoppers” (consumers with more information)
by pricing lower versus pricing higher and getting more revenue from the consumers with
little information in the marketplace. The nature of this tradeoff induces randomization
in equilibrium. The reason is that a pure strategy is easily exploited by rivals, by a price
just below the pure strategy price. In such a circumstance, the firm is better off with a
monopoly price on the loyal customers. The overall randomization of pricing depends on
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the distribution of consumer information, the number of firms competing in the
marketplace, and the profits of the firm in serving customers with few options.
Randomization works as a best response in this type of situation because firms can
balance the risk of going for the group of informed consumers against the steady revenue
of the less informed shoppers. Standard models provide a satisfactory account of the
behavior of airlines with regard to their prices – randomization induces a dispersion that
is observed.
Besides randomization of prices, limited capacity is important within the airline
industry. Schedules are set ahead of time, with many flights departing either with excess
demand or with many excess seats that are left empty at the time of departure. Since it is
costly to shift around schedules more than a few times a year, airlines have to price
within the confines of a fixed seating allotment on routes by the flight, day, week, and
month. This problem of capacity is not unique to the airline industry, as hotels, stadiums,
heavy manufacturing (whose lines cannot be flexible to be changed into another product
quickly) industries all grapple with the same issues of pricing within the same capacity
within the same period of time.
Kreps and Scheinkman (1983) provided a two-stage model that allowed two firms
to build capacities in the first period and then announce prices in the second period. If
each firm built capacities in the first period below the best response function of the other
firm’s reaction curve, then both firms announced Cournot prices with probability one in
the second period, provided the consumers are allocated using the efficient rationing rule.
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However, if one firm has a larger capacity than the others, but the sum of the
capacities doesn’t clear the market at a price of zero, price randomization will result.
(This is the mixed strategy solution to the classic Bertrand Paradox; see Baye and
Morgan, 1997.) The smaller firm will have a higher probability density at each price than
the larger firm until the cumulative density function equals one and will earn a smaller
portion of profits due to the lower capacity size. Yet the large firm benefits as well since
part of the time it will place an atom on the highest price in the probability distribution
and earns more profits due to its larger capacity size.. The big assumption in this price
randomization model is that the smaller firm did not build enough capacity to have the
option of forcing prices to zero (assuming zero marginal cost of production), regardless
of what action the larger firm takes.
Both information and capacity models lead to a similar result of price
randomization. I will show that the similar outcomes of these two distinct models is not
an accident. In the equilibrium formula for the pricing distributions, there is a
multiplicative term involving capacity or the amount of information that consumers are
receiving. This term is important in the actual calculation of both the price distribution
and ultimately the profits that firms will earn in each of the models. Given that these type
of models are similar in how they solve for random prices, how does a model that
encompasses both kinds of features – limited information of consumers and limited
capacity of firms – work? What does pricing look like in that type of model? Would that
validate these two different types of models or would something different be said about
price randomization? This chapter answers that question by synthesizing the two models.
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2-2. The Symmetric Two-Firm Model
Let there be initially two firms within a marketplace of size npotential customers,
each of whom values the product at H. Each firm serves three different types of
customers. Let α2 be the proportion of informed customers that see the prices of both
firms. Let α1 be the proportion of customers that only see one firm’s price, called loyal or
uninformed customers. These consumers are relatively uninformed as they cannot see
the entire market place. Let α0 be the proportion of customers that see the prices of
neither firm and hence do not buy. The table below shows all three classes of consumers,
with α1nconsumers split evenly between the two firms:
Table 9: Classes of Consumers for Two-Firm Symmetric Model
Sees Firm 2’s Price: Does Not See Firm 2’sPrice:
Sees Firm 1’s Price: α2n α1n/2
Does Not See Firm 1’sPrice:
α1n/2 α0n
Let each firm be endowed with capacity to serve kconsumers. Let k< α2n +
α1n/2. Set the marginal cost c of serving consumers to zero. Suppose the price of firm 1
is less than firm 2. Then firm 1 sells to
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kn
nn
n
kq =⎥⎦
⎤⎢⎣
⎡ +⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+=
2
2
12
12
1α
α α
α
(1)
consumers. The expression
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+2
12
nn
k
α α
is the share of customers that firm 1, with its
lower price, can serve. The share is applied equally to both searchers and non-searchers,
and not surprisingly, the total number of customers is the capacity k. Firm 2 sells to
[ ]nn
n
knq 2
12
12
2
12
α α
α
α
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+−+= (2)
consumers. The first term21nα
represents those customers only seeing firm 2’s price.
The second term of equation two represents those customers who saw both prices that
couldn’t buy from firm 1, and are rationed to firm 2. When firm 2 has the lower price,
the outcome is analogous.
Loyal customers, who represent α1, play a very important role. Their role is
analogous is similar to the role they play in the price dispersion models of Butters (1977)
and Varian (1980). As an added feature, however, is the second term of the higher priced
firm – in our example being firm 2. This term [ ]nn
n
k2
12
2
1 α α
α ⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+− represents spillover
informed customers that the lower priced firm could not serve because it was capacity
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constrained. This term is not present in the price dispersion models. The effect of this
term gives the higher priced firm less of a penalty for not winning the informed
consumers and instead adds to the loyal customers. The more that the lower price firm is
constrained from meeting the market demand, the better advantage this is to the higher
priced firm. As will be shown, this setup will give rise to price randomization leading to
an interaction of capacity and information constraints.
Let nn
n
knk 2
1
2
1
2
12
α α
α
α
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+
−+> (3)
Inequality (3) insures that the higher-priced firm is not capacity constrained. Simplifying
expression (3):
kn
n
nkn
n<
+−+
2
2 12
22
1
α α
α α
α
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
++<+
2
12 1
2
22
1
nn
nkn
n
α α
α α
α
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+
+<+
2
22
2 12
12
21
nn
nn
knn
α α
α α
α α
kn
n
nn
<+
⎟ ⎠
⎞⎜⎝
⎛ +
22
2
12
21
2
α α
α α
(3a)
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There are no pure strategy equilibria for each firm. Suppose that firm 1 plays a
pure strategy of a monopoly price. Then firm 2 can set its price at slightly lower and
receive the profits of all the informed consumers. Likewise, if firm 1 places probability
one on any price within the support then firm 2 can undercut by a very small amount and
receive the informed consumers and sell all its capacity. Once the price of firm 1 is
sufficiently low, firm 2’s best response is the monopoly price, rendering firm 1’s price
too low to be optimal.
Suppose firm j follows randomized strategy with cumulative distributive function
F j. We seek a symmetric mixed strategy equilibrium. The profits πi of firm i not equal to
j given price pare:
[ ] .
2
2)()(1
21
22
1 p
nn
nkn
npFkppF j ji
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−++−=
α α
α α
α π (4)
The first term is the scenario that firm i is the lowest priced firm, which happens with
probability 1-F j(p). In this case the firm earns kpdollars. With probability F j(p) firm i
will be the high priced firm. In that situation firm i will earn revenue from its loyal
customers (first term) and revenue from the spillover customers from the lower priced
firm j (last two terms), as noted above.
Plugging in the monopoly price H, or the highest price in the distribution, leads to
calculation of profits at the high price:
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H
nn
nkn
nH
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−+=
21
22
1
2
2)(
α α
α α
α π (5)
Setting the two distributions equal (for a symmetric equilibrium) and solving for the
distribution F(p) is straightforward and all that is required is taking the profits at the
monopoly price H and setting them equal to the profits at any other price p. Doing so
yields:
[ ] pn
n nknnpFkppFHn
n nknn
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−++−=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
21
22
1
21
22
1
2
2)()(1
2
2α
α α α α α
α α α α
Simplifying leads to:
p
nn
nkn
npFkppFkpH
nn
nkn
n
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−++−=−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
21
22
1
21
22
1
2
2)()(
2
2α
α
α α
α
α α
α α
α
p
nn
nkn
npk
H
nn
nkn
npk
pF
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−+−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−+−
=
21
22
1
21
22
1
2
2
2
2
)(
α α
α α
α
α α
α α
α
(6)
Here F(p) is a ratio of the difference of revenue sold at some price p at capacity minus
revenue sold at the monopoly price (assuming the firm is the high price firm) over
revenue sold at the same price at capacity minus revenue at p assuming that the firm is
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the high priced firm. Since the right hand term of the denominator will be smaller, than
the right hand term of the numerator, F(p) will be between zero and one.
Solving for the lowest price λ in the distribution is also straightforward. Setting
equation (5) to zero and multiplying both sides by the denominator:
0
2
22
1
22
1 =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+− H
nn
nkn
nk
α α
α α
α λ
λ =k
H
nn
nkn
n
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
−+
2
1
2
2
1
2
2α
α
α α
α
(7)
λ could have also been derived by solving F(λ)=0 in equation (4). λ has a simple
interpretation: monopoly profits divided by capacity. The higher capacity, the lower λ
becomes. When capacity is at highest, or each firm having enough capacity to serve the
marketplace nn
k 21
2α
α += , λ is at its lowest at H
nn
n
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+ 21
1
2
2
α α
α
. When each firm’s
capacity is at its lowest, which is just able to serve the loyal customers plus the left over
shoppers rationed to the higher priced firm,
22
2
12
2
12
nn
nn
kα
α
α α
+
⎟ ⎠
⎞⎜⎝
⎛ +
= , the lowest price equals
H.
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The rhetorical questions posed in the introduction can now be answered: the
capacity and information limitations both enter into the distribution calculations in a
semi-multiplicative fashion. The model subsumes both imperfect consumer information
and capacity constraints. In fact, when α2 = 1, the model specializes to a limited capacity
pricing game. In this game, all consumers see both prices, but consumers may not be
able to be served by the low-priced firm because of capacity limitations. If capacity is
not too large (n>2k), the unique symmetric equilibrium involves randomization. With α2
< 1, the model become richer. Now not all consumers see all prices. Firms have loyal
customers (those seeing only their prices) and also serve either their rival’s leftovers (if
the firm is a high-priced firm) or face a binding capacity constraint (if it is the low-priced
firm). Price randomization is necessary to obtain equilibrium.
2-3. Generalizing to the Case Where All But One of the Firms are Sold to
Capacity
The above model generalizes to the case when all but one firm sells to capacity.
Price randomization will still occur as a result and a closed form solution can be found
for the distribution of prices. To see the generalization, a three firm model will be
explored before the generalized model is given.
Suppose there are three firms. Let α3 be the proportion of consumers that see all
three prices. Likewise, let α2 and α1 be the proportion of consumers that see two and one
prices. Finally, let α0 be the proportion of consumers that do not see any firm’s prices.
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Thus α3nwill be the total number of customers that will see the prices of all three firms.
Table 10 below show all four classes of customers:
Table 10: Customer Types for Three-Firm Symmetric Model
Sees Firm 3’s Price:(all 4 cells) Sees Firm 2’s Price: Does Not See Firm 2’s
Price:
Sees Firm 1’s Price: n3α
A⎟⎟ ⎠
⎞⎜⎜⎝
⎛
2
32nα
BDoes Not See Firm 1’sPrice:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
2
32nα
C
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
1
31nα
D
Does Not See Firm 3’sPrice:(all 4 cells)
Sees Firm 2’s Price: Does Not See Firm 2’sPrice:
Sees Firm 1’s Price:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ 2
3
2nα
E
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ 1
3
1nα
F
Does Not See Firm 1’sPrice:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ 1
31nα
G
n0α
H
The group of α2nand α1nconsumers are broken up into thirds within the cells because
there are three possible combinations that a consumers could see one or two firm’s prices.
Let each firm be again endowed with capacity to serve k consumers. Let the sum
of the upper box, or size of the total market, be greater than 2kbut less than 3kor
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knn
nk 233
23 12
3 >++>α α
α . The size of the total market is33
2 123
nnn
α α α ++ because
that is the number of consumers that see at least one price. Assume without loss of
generality that the price of firm 1 is less than firm 2 and that the price of firm 2 is less
than the price of firm three. Firm 1 sells:
⎥⎦
⎤⎢⎣
⎡+++
++ 333
33
2122
312
3
nnnn
nnn
k α α α α
α α α
(8)
or to kconsumers. The left-hand fraction is capacity of firm 1 divided by total market
size (since it is constrained) multiplied by what the firm does sell – being cells A, B, E,
and F where firm 1’s price is posted. Cell A is those consumers that see all three prices
while cells B and E are cases where consumers only see a limited part of the marketplace-
two prices. Cell F is firm 1’s loyal consumers that only see firm 1’s price.
Firm 2 sells ⎥⎦
⎤⎢⎣
⎡+++
++ 333
33
22
312
123
nn
nn
nnn
k α α
α α
α α α
(9)
or to kconsumers. Like equation (8) the left hand fraction is capacity of firm 2 divided
by total market size while the right hand side term is what firm 2 actually sells. What
firm 2 actually sells is broken into two parts: the first two terms from cells C and G are
what firm 2 sells from consumers not seeing the price of firm 1. Cell C is those
consumers that see that firm 2’s price is lower than firm three’s price. Cell G is firm 2’s
loyal consumers that only see firm 2’s price. The other portion of what firm 2 sells is left
over from what firm 1 could not sell due to its capacity constraint. Cells A and E are
included in what firm 2 sells because consumers can see a price of firm 2.
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Firm three sells to ⎟ ⎠
⎞⎜⎝
⎛ ++
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
++−+
33
33
2
21
3
223
123
1 nnn
nnn
kn α α α
α α α
α (10)
consumers. The first term, or cell D, is firm three’s loyal consumers. Let R or remainder
designate the remainder of consumers that could not be served by firms one or two. Let
R be the expansive right hand term. Equation (10) now becomes: Rn
+3
1α . Within R,
the left hand bracket term is the proportion of customers seeing lower price not served
within the marketplace. The right hand bracket is the actual consumers that could not be
served by the other two firms that see firm three’s prices. This includes cells A, B, and
C. The right-hand bracket is multiplied by the left hand bracket because only a portion of
consumers were not served by the first two firms.
Expected profits for a firm are:
π = ppFpF jkpFR
n
j
j j
⎥⎦
⎤
⎢⎣
⎡
−⎟⎟ ⎠
⎞
⎜⎜⎝
⎛
+⎥⎦
⎤
⎢⎣
⎡
+ ∑=
−2
1
221
)())(1(
2
)(3
α
(11)
At the highest price in the support, expected profits π(H) are HRn
⎥⎦
⎤⎢⎣
⎡+
3
1α . By setting π
= π(H), a symmetric equilibrium price distribution F(p) can be found. Doing so yields:
ppFkppFpFkppFRn
HRn 2211 ))(1()())(1(2)(
33−+−+⎥⎦
⎤⎢⎣
⎡+=⎥⎦
⎤⎢⎣
⎡+
α α
Further simplification leads to:
ppkFppkFkppkFppkFppFRn
HRn 22211 )()(2)(2)(2)(
33+−+−+⎥⎦
⎤⎢⎣
⎡+=⎥⎦
⎤⎢⎣
⎡+
α α
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ppkFppFRn
kpHRn 2211 )()(
33−⎥⎦
⎤⎢⎣
⎡+=−⎥⎦
⎤⎢⎣
⎡+
α α
F(p) =
pRn
kp
HRnkp
⎥⎦
⎤⎢⎣
⎡+−
⎥⎦⎤⎢⎣
⎡ +−
3
3
1
1
α
α
(12)
The solution F(p) in the three firm model with two firms at capacity looks quite similar to
the two firm case. The left terms in the numerator and denominator are capacity at a
given price, just like the two-firm model. The right terms are loyal customers plus the
remainder that cannot be served by the lowest priced firms.
Solving for the lowest price in the distribution λ in the three firm model is done
by setting F(p) = 0 and then simplifying. Setting F(p) = 0 and squaring both sides yields:
HRn
k ⎥⎦
⎤⎢⎣
⎡+−=
30 1α
λ
or
λ =k
HRn
⎥⎦
⎤⎢⎣
⎡+
3
1α
(13)
Again, λ is simply monopoly profits at the high price in the distribution divided by
capacity of a firm. The value λ has the same interpretation in this three firm case as the
two firm case.
Many features within the two-firm model were passed to the three firm model
with two firms selling to capacity. The distribution has a familiar kp term in the
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numerator and denominator and the loyal plus remainder profits are also in the numerator
and denominator.
This pattern works for a higher number of firms in continued special cases, with
the common feature that all firms save one sell to capacity. The pattern can be
generalized for a model with wfirms and w-1 selling up to capacity. F(p) =
1
1
1
1 −
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ +−
⎥⎦
⎤⎢⎣
⎡ +−w
pRw
nkp
HRw
nkp
α
α
andk
HRw
n⎥⎦
⎤⎢⎣
⎡ +=
1α
λ , where
.)1(
11
1
1
1
)1(1R
2
1
2
1
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−−=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
−
⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
−
−−= ∑
∑∑
∑ =
=
=
=
w
i
iw
i
i
w
i
iw
i
iw
ni
w
ni
kw
i
w
n
i
w
i
w
n
i
w
kw α
α
α
α
.
2-4. Generalizing to the Asymmetric Model:
The original two-firm model described above can be extended to the case when
the firms are not of equal size. Suppose there is the same split of customers – those that
see two prices α2, those loyal customers seeing only one firm’s prices α1 (split evenly
between the two firms), and those customers not seeing either firms prices α0. Suppose
that firm 1 has built capacity greater than firm 2 or k1 > k2. Assume that the larger firm
(in this case k1) has capacity less than the total market size α2n+ α1n/2 so that the smaller
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firm takes up some of the remaining surplus customers. Assume that the smaller firm (in
this case k2) has capacity to take up the excess surplus customers or
221
2
11
2
12
knn
n
kn<
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+−+ α
α α
α .
Suppose that firm 1 has its price p1 lower than firm 2’s price p2. Then firm 1 sells
to ⎥⎦
⎤⎢⎣
⎡+
+ 2
2
12
12
1 nn
nn
k α α
α α
or k1 consumers. Firm 2 sells to the customers seeing its own
price plus those remaining customers that could not purchase from firm 1, which is
nn
n
kn2
12
11
2
12
α α
α
α
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+−+ . Now suppose that firm 2 has a lower price than firm 1. Firm
1 sells to its loyal customers plus the residual customers that could not be served by firm
2: nn
n
kn2
12
21
2
12
α α
α
α
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+−+ . Firm 2 sells to ⎥⎦
⎤⎢⎣
⎡+
+ 2
2
12
12
2 nn
nn
k α α
α α
or k 2 customers.
With these four cases, profits for the two firms can be expressed. For the larger firm,
profits are
[ ] pnn
nkn
npFpkpF
⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢
⎢
⎣
⎡
+−++−=
2
2)()(11
2
222
12121 α
α
α α
α π (14)
For the smaller firm, profits are
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[ ] pn
n
nkn
npFpkpF
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−++−=
2
2)()(1
12
212
11212 α
α
α α
α π (15)
Initially solving for the distributions, revenue at the monopoly price is set equal to
revenue to revenue at any other price within the support of prices. For firm 1 the profit
equations are:
[ ] pn
n
nkn
npFpkpFH
nn
nkn
n
⎥
⎥⎥⎥
⎦
⎤
⎢
⎢⎢⎢
⎣
⎡
+−++−=
⎥
⎥⎥⎥
⎦
⎤
⎢
⎢⎢⎢
⎣
⎡
+−+
2
2)()(1
2
2 1
2
222
1212
1
2
222
1
α α
α α
α
α α
α α
α
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+=−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+ 1
12
222
121
12
222
1
2
2)(
2
2pkp
nn
nkn
npFpkH
nn
nkn
n
α α
α α
α
α α
α α
α
11
2
222
1
11
2
222
1
2
2
2
2
2
)(
pkpn
n
nkn
n
pkHn
n
nkn
n
pF
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
=
α α
α α
α
α α
α α
α
(16)
Solving for F1(p) can be done in a similar fashion.
21
2
212
1
21
2
212
1
1
2
2
2
2
)(
pkpn
n
nkn
n
pkHn
n
nkn
n
pF
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
=
α α
α α
α
α α
α α
α
(17)
The lowest price in the distribution is solved by setting Fi(p) = 0.
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166
2
12
212
1
1
2
2
k
Hn
n
nkn
n
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
=
α α
α α
α
λ
1
12
222
1
2
2
2
k
Hn
n
nkn
n
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
=
α α
α α
α
λ
One of the noticeable things about λi in equations (16) and (17) is that they are
not equal in the proposed solution. Price randomization will not occur with λi not being
equal – the firm with the lower λ is pricing unnecessarily low. Thus, there needs to be
adjustment in the solution to induce the support of prices of the two firms to coincide,
and the only candidate is a mass point or atom in the distribution of prices. Moreover, for
an atom to exist, it must occur at H, because otherwise, were the atom at p, the rival
would never price in a small interval [p,p+ε). The firm with the lower λ solution has the
atom in its distribution. Firm 2 will have the atom in its distribution iff λ2 < λ1. This can
be seen as follows.
H
k
knnn
H
k
knnn
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟ ⎠
⎞⎜⎝
⎛ +
−++>
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟ ⎠
⎞⎜⎝
⎛ +
−++
21
1
222
221
21
21
2
212
221
21
2
4
2
4
α α
α α α α α
α α
α α α α α
. After simplifying and
rearranging, this condition becomes
⎥⎦
⎤⎢⎣
⎡−++⎟
⎠
⎞⎜⎝
⎛ +>⎥
⎦
⎤⎢⎣
⎡−++⎟
⎠
⎞⎜⎝
⎛ + 22
2
221
2
12
1221
2
221
2
12
11
4242α α α α
α α
α α α α α
α α
α knn
nkknn
nk
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or
022242
22242
2212
221
2
2
221
221
2
12
12
2212
121
1
2
221
121
2
12
11
>⎟ ⎠
⎞⎜⎝
⎛ +−⎟
⎠
⎞⎜⎝
⎛ ++⎟
⎠
⎞⎜⎝
⎛ ++⎟
⎠
⎞⎜⎝
⎛ +
−⎟
⎠
⎞⎜
⎝
⎛ +−⎟
⎠
⎞⎜
⎝
⎛ ++⎟
⎠
⎞⎜
⎝
⎛ ++⎟
⎠
⎞⎜
⎝
⎛ +
α α α
α α
α α α
α α α
α α
α α α
α α
α α α
α α α
α α
knknkn
k
knknkn
k
which reduces to
( ) ( ) 042
221
2
221
2
12
121 >⎥
⎦
⎤⎢⎣
⎡+−++⎟
⎠
⎞⎜⎝
⎛ +− α α α α
α α
α kknn
nkk
( ) ( ) 02
2221
2
21
21
21 >
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+−⎟ ⎠ ⎞⎜
⎝ ⎛ +
⎟ ⎠
⎞⎜⎝
⎛ +− α
α α
α α
kkn
nn
kk
Now if k 1 + k 2 <n
nn
2
2
21
2
α
α α
⎟ ⎠
⎞⎜⎝
⎛ +
(18)
firm two has the atom in its distribution. Adjustments will need to be made to probability
distributions to account for firm two having the atom in its distribution.
Let R 2 be the leftover customers from firm 2 selling at a price lower than firm 1.
2
12
2222 n
n
nknR
α α
α α
+−= . Similarly, let R 1 be the remainder customers from firm 1
selling at a price lower than firm 2, which is
21
2
2121 n
n
nknRα
α
α α
+−= . Since firm 2 has
the atom at H in its probability distribution, the equations showing profits are the same
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168
throughout are modified to take in account the atom. From equation (14), since firm 1
has no atom, equality of profits for firm 1 gives a constant
[ ] pRnpFpkpF ⎥⎦⎤
⎢⎣⎡ ++−= 1
11212
2)()(1 α π .
This also gives the identical λ=λ1. The solution for firm 1, using (15), gives
HRn
k ⎥⎦
⎤⎢⎣
⎡+= 1
12
2
α λ (19)
is derived from setting the profits of firm 2 from the lowest price in the distribution
against the profits of firm 2 from the highest price H in the distribution. The distribution
of firm 1 does not enter the equation because there is not an atom in firm 1’s probability
distribution. At the lowest price in the distribution, firm 2’s revenue is λk2. With
probability one firm 1 does not have the lowest price at λ, thus firm 2 is selling out to its
capacity. At the highest price in the distribution, firm 2’s revenue is HRn
⎥
⎦
⎤⎢
⎣
⎡ + 11
2
α .
With positive probability, firm 2 relies upon its loyal customers and the leftover
customers from firm 1 since it has higher prices.
The second equation
[ ] HRn
HFHkHFk ⎥⎦
⎤⎢⎣
⎡++−= −−
21
21212
)()(1α
λ (20)
pits the profits from the lowest and highest prices in firm 1’s profit equation. The
difference is that firm 2 has an atom in its distribution. The low end remains the same.
With probability one firm 1 has the lowest price at price λ. Revenue is λk1. The high
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end of the distribution is different from the case above. It is not probability one when
firm 1 reaches its highest point in its distribution that it has the highest price. That is
what the 12 )(1 HkHF −− term captures. With probability )(1 2 HF −− firm one
receives monopoly profits by pricing just below the atom of firm two. With probability
)(2 HF−
firm two has the lower price at the price just below H and thus firm one
receives expected profits of HRn
HF ⎥⎦
⎤⎢⎣
⎡ +−2
12
2)(α
from its loyal customers and
customers that firm two could not serve.
λ and F2-(H) can be solved for by these two equations.
2
11
2
k
HRn
⎥⎦
⎤⎢⎣
⎡ +=
α
λ (21)
[ ] HRn
HFHkHFk
kHR
n⎥⎦
⎤⎢⎣
⎡ ++−=⎥⎦
⎤⎢⎣
⎡ + −−2
1212
2
11
1
2)()(1
2
α α
HRnHFHkHFHkk
kHRn⎥⎦⎤
⎢⎣⎡ ++−=−⎥⎦
⎤⎢⎣⎡ + −−
21
2121
2
11
1
2)()(
2
α α
121
1
2
11
1
2
2
2)(
HkHRn
Hkk
kHR
n
HF
−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦
⎤⎢⎣
⎡ +=−
α
α
(22)
121
1
2
11
1
2
2
21)(1
HkHRn
HkkkHRn
HF
−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦⎤⎢⎣
⎡ +−=− −
α
α
(22a)
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Checking that the atom )(1 2 HF−− is between zero and one gives two more
conditions:
1)(10 2 ≤−≤ −HF
0)(1 2 ≥≥ −HF
1
2
20
121
1
2
11
1
≤−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦
⎤⎢⎣
⎡ +≤
HkHRn
Hkk
kHR
n
α
α
(23)
The left condition in equation (21) reveals a relationship between k 1 and k 2:
2
11
11
2 k
kHR
nHk ⎥⎦
⎤⎢⎣
⎡ +≥α
The sign switches because the denominator is negative.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−+≥
nn
nkn
nk
21
21212
2
2α
α
α α
α (24)
Equation (24) is the same as what we assumed before about the smaller firm having
enough capacity to handle the left-over consumers from the high capacity firm. The right
condition in equation (21) is the same condition as equation (18):
1211
2
11122 HkHRn
Hkk
kHR
n
−⎥⎦
⎤⎢⎣
⎡+≥−⎥⎦
⎤⎢⎣
⎡+
α α
The sign switches because the denominator is negative.
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171
221
111
22kR
nkR
n⎥⎦
⎤⎢⎣
⎡ +≥⎥⎦
⎤⎢⎣
⎡ +α α
( ) ( )2
2
2
1
21
2212
1
2
2kk
nn
nkkn
n−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+≥−⎟
⎠
⎞⎜⎝
⎛ +
α α
α α
α
( )21
21
22
1
2
2kk
nn
nn
n+
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+≥⎟
⎠
⎞⎜⎝
⎛ +
α α
α α
α
( )21
2
2
21
2kk
n
nn
+≥⎟ ⎠
⎞⎜⎝
⎛ +
α
α α
(18)
F1(p) and F2(p) can be solved by plugging in λ into equations (14) and (15) and
solving. [ ] pRn
pFpkpFk
kHR
n⎥⎦
⎤⎢⎣
⎡++−=⎥⎦
⎤⎢⎣
⎡+ 1
1121
2
21
1
2)()(1
2
α α
pRn
pFpkpFpkHRn
⎥⎦
⎤⎢⎣
⎡ ++−=−⎥⎦
⎤⎢⎣
⎡ + 11
121211
2)()(
2
α α
211
211
1
2
2)(
pkpRn
pkHRn
pF
−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦
⎤⎢⎣
⎡+
=α
α
(25)
This is the same result as before in equation (17). F2(p) will not be the same as equation
(16) with the atom.
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[ ] pRn
pFpkpFk
kHR
n⎥⎦
⎤⎢⎣
⎡ ++−=⎥⎦
⎤⎢⎣
⎡ + 21
212
2
11
1
2)()(1
2
α α
pRnpFpkpFpkkkHRn
⎥⎦⎤⎢⎣
⎡ ++−=−⎥⎦⎤⎢⎣
⎡ + 21
2121
2
11
1
2)()(
2α α
121
1
2
1`
1
2
2
2)(
pkpRn
pkk
kHR
n
pF
−⎥⎦
⎤⎢⎣
⎡+
−⎥⎦
⎤⎢⎣
⎡+
=α
α
(26)
If equation (18) is violated, then firm one has the atom. This means that the
combined capacity is higher thann
nn
2
2
21
2
α
α α
⎟ ⎠
⎞⎜⎝
⎛ +
. Since firm 1 has the atom at H in its
probability distribution, the equations showing profits are the same throughout are again
modified to take in account the atom. From equation (14), since firm 2 has no atom,
equality of profits for firm 1 gives a constant
[ ] pRn
pFpkpF ⎥⎦
⎤⎢⎣
⎡ ++−= 21
21212
)()(1α
π .
This also gives the identical λ=λ2. The solution for firm 2, using (13), gives
HRn
k ⎥⎦
⎤⎢⎣
⎡ += 21
12
α λ (27)
is derived from setting the profits of firm 1 from the lowest price in the distribution
against the profits of firm 1 from the highest price H in the distribution. The distribution
of firm 2 does not enter the equation because there is now not an atom in firm 2’s
probability distribution. At the lowest price in the distribution, firm 1’s revenue is λk1.
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173
With probability one firm 2 does not have the lowest price at λ, thus firm 1 selling out to
its capacity. At the highest price in the distribution, firm 1’s revenue is HRn
⎥
⎦
⎤⎢
⎣
⎡+ 2
1
2
α .
With positive probability, firm 1 relies upon its loyal customers and the leftover
customers from firm 2 since it has higher prices.
The second equation
[ ] HRn
HFHkHFk ⎥⎦⎤
⎢⎣⎡ ++−= −−
11
12122
)()(1α
λ (28)
pits the profits from the lowest and highest prices in firm 2’s distribution. The difference
is that firm 1 now has an atom in its distribution. The low end remains the same. With
probability one, firm 2 has the lowest price at price λ. Revenue is λk2. The high end of
the distribution is different from the case above. It is not probability one when firm 2
reaches its highest point in its distribution that it has the highest price. That is what the
21 )(1 HkHF
−
− term captures. With probability )(1 1 HF
−
− firm two receives
monopoly profits by pricing just below the atom of firm one. With probability
)(1 HF−
firm one has the lower price at the price just below H and thus firm two receives
expected profits of HRn
HF ⎥⎦
⎤⎢⎣
⎡ +−1
11
2)(α
from its loyal customers and customers that
firm one could not serve.
λ and F1-(H) can be solved for by these two equations.
1
21
2
k
HRn
⎥⎦
⎤⎢⎣
⎡ +=
α
λ (29)
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[ ] HRn
HFHkHFk
kHR
n⎥⎦
⎤⎢⎣
⎡ ++−=⎥⎦
⎤⎢⎣
⎡ + −−1
1121
1
22
1
2)()(1
2
α α
HRnHFHkHFHkkkHRn ⎥⎦⎤⎢⎣⎡ ++−=−⎥⎦⎤⎢⎣⎡ + −− 111212
1221
2)()(
2α α
211
21
22
1
1
2
2)(
HkHRn
Hkk
kHR
n
HF
−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦
⎤⎢⎣
⎡ +=−
α
α
(30)
211
2
1
22
1
1
2
2
1)(1HkHR
n
Hkk
kHR
n
HF−⎥⎦
⎤⎢⎣⎡ +
−⎥⎦
⎤⎢⎣
⎡ +
−=−−
α
α
(30a)
Checking that the atom is between zero and one:
1
2
210
211
2
1
22
1
≤−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦
⎤⎢⎣
⎡ +−≤
HkHRn
Hkk
kHR
n
α
α
(31)
1
2
20
211
2
1
22
1
≤−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦
⎤⎢⎣
⎡ +≤
HkHRn
Hkk
kHR
n
α
α
Taking the right condition first:
211
2
1
22
1
22HkHR
nHk
k
kHR
n−⎥⎦
⎤⎢⎣
⎡ +≥−⎥⎦
⎤⎢⎣
⎡ +α α
The sign reverses because the denominator is negative.
111
221
22kR
nkR
n⎥⎦
⎤⎢⎣
⎡ +≥⎥⎦
⎤⎢⎣
⎡ +α α
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( ) ( )21212
2
2
1
21
2
2
2
kknn
kk
nn
n−⎟
⎠
⎞⎜⎝
⎛ +≥−⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+α
α
α α
α
( )n
nn
kk2
2
21
21
2
α
α α
⎟ ⎠
⎞⎜⎝
⎛ +≥+ . This is the reverse of condition (18) when combined
capacities are large. The left condition can be reduced to:
1
22
12
2 k
kHR
nHk ⎥⎦
⎤⎢⎣
⎡ +≥α
(the denominator is negative so the sign changes)
nn
nkn
nk
21
222
11
2
2α
α
α α
α
+−+≥ (32)
Thus the largest firm has excess capacity.
F2(p) and F1(p) can be solved by plugging in λ into equations (14) and (15) and
solving. [ ] pRn
pFpkpF
k
kHR
n
⎥⎦
⎤
⎢⎣
⎡ ++−=
⎥⎦
⎤
⎢⎣
⎡ + 21
2121
12
1
2
)()(1
2
α α
pRn
pFpkpFpkHRn
⎥⎦
⎤⎢⎣
⎡++−=−⎥⎦
⎤⎢⎣
⎡+ 2
121212
1
2)()(
2
α α
121
121
2
2
2)(
pkpRn
pkHRn
pF
−⎥⎦
⎤⎢⎣
⎡+
−⎥⎦
⎤⎢⎣
⎡+
=α
α
(33)
This is the same result as before in equation (16). F1(p) will not be the same as equation
(17) with the atom.
[ ] pRn
pFpkpFk
kHR
n⎥⎦
⎤⎢⎣
⎡ ++−=⎥⎦
⎤⎢⎣
⎡ + 11
1211
22
1
2)()(1
2
α α
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pRn
pFpkpFpkk
kHR
n⎥⎦
⎤⎢⎣
⎡ ++−=−⎥⎦
⎤⎢⎣
⎡ + 11
12121
22
1
2)()(
2
α α
211
21221
1
2
2)(
pkpRn
pkkkHRn
pF
−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦⎤⎢⎣⎡ +=
α
α
(34)
2-5. Building Capacity First and then Setting Prices
As another exercise, both firms can be modeled to build capacity in stage one and
then sell to customers in stage two, just as in Kreps and Scheinkman (1983). Given the
two probability distributions, they can be plugged into equations (14) and (15) to
determine profits as a function of capacity and proportion of consumers that see one or
two prices. With that information, reaction curves can be drawn for each firm and a
solution to the two – stage asymmetric game might be obtained. Plugging22
in F1(p) -
obtained from the case the combined capacities for firm one and two are smaller than
n
nn
2
2
21
2
α
α α
⎟ ⎠
⎞⎜⎝
⎛ +
- into the profit equation (15):
22 This is the smaller capacity case where the smaller firm has the atom. Profits depend negatively oncapacity so using the smaller total capacity results in greater profits. If the larger capacity case is used, thereaction curves cross at a symmetric point and give the same symmetric profit equations as the low capacity
case. Since it will be shown that symmetric profits i j ckHRn
−⎥⎦
⎤⎢⎣
⎡ +21α
increase with decreasing
capacity, the low capacity equations eventually take hold.
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pRn
pkpR
n
pkHRn
pk
pkpR
n
pkHRn
⎥⎦
⎤⎢⎣
⎡ +
⎥
⎥⎥⎥
⎦
⎤
⎢
⎢⎢⎢
⎣
⎡
−⎥⎦
⎤
⎢⎣
⎡
+
−⎥⎦
⎤⎢⎣
⎡ ++
⎥
⎥⎥⎥
⎦
⎤
⎢
⎢⎢⎢
⎣
⎡
−⎥⎦
⎤
⎢⎣
⎡
+
−⎥⎦
⎤⎢⎣
⎡ +−= 1
1
21
1
211
2
21
1
211
22
2
2
2
21
α
α
α
α
α
π
⎥⎦
⎤⎢⎣
⎡−⎥⎦
⎤⎢⎣
⎡ +
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎦
⎤⎢⎣
⎡ +
−⎥⎦
⎤⎢⎣
⎡+
+= 211
211
211
22
2
2pkpR
n
pkpRn
pkHRn
pkα
α
α
221
1
2 pkpkHR
n
+−⎥⎦
⎤
⎢⎣
⎡
+=
α
HRn
⎥⎦
⎤⎢⎣
⎡ += 11
2
α (35)
Plugging in F2(p), obtained from the smaller capacity case, into the profit equation (14):
pR
n
pkpRn
pkk
kHR
n
pkpkpR
n
pkk
kHR
n
⎥⎦
⎤
⎢⎣
⎡
+⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎦⎤
⎢⎣⎡ +
−⎥⎦
⎤⎢⎣
⎡ +
+⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎦⎤
⎢⎣⎡ +
−⎥⎦
⎤⎢⎣
⎡ +
−= 2
1
121
1
2
11
1
1
121
1
2
11
1
1 2
2
2
2
2
1
α
α
α
α
α
π
⎥⎦
⎤⎢⎣
⎡−⎥⎦
⎤⎢⎣
⎡ +
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−⎥⎦
⎤⎢⎣
⎡+
−⎥⎦
⎤⎢⎣
⎡ ++= 12
1
121
1
2
11
1
12
2
2pkpR
n
pkpRn
pkk
kHR
n
pkα
α
α
1
2
11
11
2pk
k
kHR
npk −⎥⎦
⎤⎢⎣
⎡++=
α
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2
11
1
2 k
kHR
n⎥⎦
⎤⎢⎣
⎡ +=α
(36)
These relationships can be summarized by indexing firms with i ≠ j and adding a constant
cost cof obtaining capital k:
⎪⎪⎩
⎪⎪⎨
⎧
>−⎥⎦
⎤⎢⎣
⎡ +
<−⎥⎦
⎤⎢⎣
⎡ +=
jii j
ii
ii j
i
kkif ckk
kHR
n
jkkif ckHRn
2
2
1
1
α
α
π (37)
Profits are smaller for the lower capacity firm and higher for the larger firm. Profits are
exactly the same at HRn
i ⎥⎦⎤
⎢⎣⎡ +
21α if both firms have the same capacity. This feature will
be important in determining the solution to the two-stage asymmetric problem.
When capacity is below the other firm’s capacity, or ki< k j, profits are
i
j
ckHnn
nk
n
n
−⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢
⎢
⎣
⎡
+−+2
2 12
2
21
α α
α
α
α . This expression is decreasing in ki. When
capacity is above the other firm capacity, or ki > k j, profits
i
j
ii ckHk
k
nn
nkn
n−
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−+
2
2 12
22
1
α α
α α
α are either increasing or decreasing in ki,
depending on k j Thus in any pure strategy equilibrium, the firms have identical capacity.
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Capital for both firms can vary between
2
2
2
12
2
21
nn
nn
α α
α α
+
⎟ ⎠ ⎞
⎜⎝ ⎛ +
on the low end and taking
the market nn
21
2α
α + on the high end. When capacity of one firm is below the other,
increasing capacity results in increasing profits. Thus for capacity below the other firm’s
capacity, there is a positive correlation in the two firms’ capacities. When capacity is
above the other, decreasing capacity results in increasing profits. Thus for capacity
above the other firm’s capacity, there is a negative correlation in the two firms’ capacities
and the reaction curves take on a different slope. The two reaction curves cross each
other when the capacities are exactly the same or right at the crossing of the two curves.
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0. 2 0. 4 0. 6 0. 8 1k f i r m i
0. 2
0. 4
0. 6
0. 8
1
k f i r m j Fi gure 22
a1=.5 a2=.5 n=1 H=1 c=.5 k i*= k j*=.45 p*=.225 market=.75
p1(a1=.5, a2=.5, n=1, H=1, c=.5)outside lines= 0.10
p2(a1=.5, a2=.5, n=1, H=1, c=.5)inside lines= 0.225
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0. 2 0. 4 0. 6 0. 8 1k f i r m i
0. 2
0. 4
0. 6
0. 8
1
k f i rm j Fi gure 23
Figure 22 is one of many solutions to the problem of building capacity first and
setting prices second23 24. Notice that there is a cost of capital that is half the monopoly
price. Maximum profits occur for each firm at the inside reaction curves where capital is
23 The constraint of the atom in equations (24) and (18) are not drawn in Figure 22 so that the reactioncurves can be emphasized. If that constraint from equation (24) is drawn in, there would be two straightlines extending from the optimal symmetric solution to the outermost points at (1.125,0) and (0,1.125). Allreaction curves above those two rays will satisfy equation (24). Equation (18) is a line connecting the points (1.125,0) and (0,1.125). All reaction curves below this line satisfy equation (18). Thus satisfyingequations (24) and (18) means being inside a triangle whose vertices are the optimal symmetric solution =(0.45,0.45), (1.125,0) and (0,1.125).24 If the constraint that capital is less than the market a1n/2+a2n is imposed with the atom constraints, theresult is a pentagon. The vertices are (0.45,0.45), (0.75,0.25), (0.75,0.375), (0.375,0.75), (0.25,0.75). Allreaction curves inside the pentagon satisfy the three constraints. Figure 23 shows the pentagon.
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the same for both firms: k i= k j = .45. This level of capital for each firm is the lowest level
of capital that can be served, if the other firm is to serve the remainder of the customers
seeing two prices. This level of capital is
22
2
12
2
21
nn
nn
α α
α α
+
⎟ ⎠ ⎞⎜
⎝ ⎛ +
. When capital is at .45 and
assuming an efficient rationing rule of serving customers, the lower priced firm will serve
.45 of customers and the higher priced firm serves .45 of customers. There will be one
price charged- the monopoly price- as the lowest price λ =
k
H
nn
nkn
n
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
−+
2
1
2
2
1
2
2α
α
α α
α
equals .45H/.45 = H. Profits are .45H - .45c= .45 -
.225 = .225.
There are many such diagrams that could be drawn, but the general solution boils
down to the same idea. Many reaction curves crossings representing a continuum of
distinct profit levels could be drawn into a diagram but for simplicity imagine only the
solution curves are drawn in. Each crossing at a level of capital represents a different
level of profit, and thus the solutions are qualitatively distinct. Due to the cost of capital,
profits increase as capital usage decreases. Profits thus reach their maximum at the
lowest point of capacity due to the cost of capital. This is the solution to the asymmetric
problem: a symmetric solution at the lowest candidate level of capital, which is
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2
2
2
12
2
21
nn
nn
α α
α α
+
⎟ ⎠ ⎞
⎜⎝ ⎛ +
, where both firms are necessarily become constrained and the monopoly
price H is charged.
2-6. Conclusion:
Price randomization could be explained in the literature by either a limited
capacity story or a limited information story. As part of each of these classes of models,
there is a multiplicative term that enters the model that involves capacity or the amount of
information that consumers are receiving. A two-firm symmetric model has been
developed that takes in account both elements – capacity limitations and information
limitations – that leads to price randomization. Adding both features together in the
model enhances the ability of the higher priced firm to capture more customers than
either model would produce alone – the remainder of the customers that the other firm
could not serve due to capacity limitations plus the loyal customers that see only one
price. The result of this type of model is that there is a semi-multiplicative term of
capacity and information within the distribution calculation.
There are a number of extensions with this model. The model can also be
simplified down to a straight limited capacity model by allowing all consumers to be
informed about both firms prices or setting α2 = 1. The model can be generalized to a
closed solution where there more than two firms so long as all firms sell out to capacity
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except for one firm. Finally, the two-firm model can be carried over into asymmetric
situations where capacity of one firm is greater than the other firm. In that case and if
total capacity of both firms is not too large, (ie smaller thann
nn
2
2
21
2
α
α α
⎟ ⎠ ⎞⎜
⎝ ⎛ +
) there will
be an atom in the probability distribution of the smaller firm at the highest price. If total
capacity of both firms is larger thann
nn
2
2
21
2
α
α α
⎟ ⎠
⎞⎜⎝
⎛ +
and smaller than nn 21 2α α + , then
the largest firm has the atom in its pricing distribution. If the firms play a Kreps and
Scheinkman two-stage game of building capacity first before selling to customers, there
will be a symmetric solution to the game at the lowest possible level of capacity.
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3: Equilibrium Price Randomization with Asymmetric Customer Loyalty
3-1. Introduction
Loyal or business travelers are important to an airline’s business. These travelers
are generally a small portion of the overall customer base yet make a larger portion of the
annual airline trips than the average public. These customers generate large revenue for
airlines as they take more trips and generally pay higher fares than the average fare paid
by the public. Airlines go to great length to court these travelers with their frequent flier
programs rewarding repeat business with these travelers. With frequent flier programs
and airlines having a presence in important hub cities, frequent travelers are often loyal
with their business to one airline. The size of the loyal customer base can help determine
an airline’s fortune.
Leisure or not so loyal travelers are another group of customers that an airline
serves. These customers search from airline to airline for the lowest price. These
travelers do not make as many trips as the loyal travelers and are generally not as
important to an airline’s business. Airlines will offer these customers system-wide sales,
weekly email specials, coupons, and last-minute specials to entice these travelers to book
with them. Customers will then face prices that fluctuate.
These two qualitative features of airline travel – a small group of loyal travelers
and fluctuating prices fit as great examples from the literature of incomplete information
and random pricing. Varian (1980), for instance, uses a model of sales where there is a
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group of uninformed customers on price and informed group of customers on price.
Varian assumes in his model that firms have the same proportion of uninformed, or
customers seeing only one price, divided evenly between firms. Firms will want to
charge these customers the monopoly price. This group of customers that see only one
price is not large enough that firms could concentrate selling to these customers. The
other group of customers, those that can see all prices, shop at the firm that offers that
offers the lowest price. To win these customers, firms will have to offer the lowest price.
Firms can only offer one price, so they have problem – how do they serve both groups of
customers? Firms randomize their prices in an interval to capture as much expected
revenue from the uninformed types and expected revenue from the informed types.
Varian assumes that the proportion of loyal customers is the same for each firm.
This is not a realistic assumption in today’s economy with so many different sized firms.
If this assumption is relaxed, how do the results change? What will the probability
distribution for the firms look like? Will there still be sales? How will the size of the
firm influence the results?
As the model will show firms still randomize over an interval of prices. Unlike
Varian (1980), the largest firm now has an atom of probability at the monopoly price.
Increasing the largest firm’s loyal customers causes all firms to discount less. Increasing
the smaller firm(s)’ loyal customers causes only the largest firm to discount more.
Increasing the shoppers causes the smaller firm(s) to discount more; the largest firm has a
more complicated reaction. Increasing the number of firms causes all firms to discount
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less. A statistic showing the probability of at least one firm having the lowest price can
be created.
3-2. The Model with Two Firms
Let there be two firms initially. Let firm 1 be the firm with the larger proportion
of loyal customers and firm 2 be the firm with a lower proportion of loyal customers. Let
α1 be the proportion of customers loyal to firm one. Let α2 be proportion of customers
loyal to firm two. Thus α1 is greater than α2. Let αs be the shoppers that look for the best
prices between both firms. Let R(p) be revenue at price p, equal to (p-c) * q(p) where
q(p) is quantity at price p. Assume that R(p) is increasing in p. Let pm be equal to the
monopoly price. For a price randomization equilibrium to occur, the profits at the highest
price in the distribution, or monopoly price, must equal the profits at the lowest price in
the distribution or L and every price in between the two.
Let Fi(p) i=1,2 be the cumulative probability distribution functions that each firm
has in charging prices between the monopoly price and the lowest price in the distribution
L. The cumulative distribution function for each firm is less than one and is continuous
at all prices except possibly the monopoly price for one of the firms. In the range of
prices that these cumulative distribution functions are continuous, there is no region of
ties or some firm will price epsilon below the tie and capture the entire market. Other
than the case where one firm has a mass point at p = pm, there are no other mass points in
the region between L and pm
for the same reasoning as the ties.
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The profits for each firm can be modeled. For firm one this is:
( ) )()()()())(()(1)( 12121 pqcppFpqcppFpR sm −+−+−= α α α α
With probability ( ))(1 2 pF− firm two has the higher price. Firm one will earn revenue
from the shoppers plus its own loyal customers. With probability )(2 pF firm one has the
higher price and will only sell to its loyal customers a1. Total revenue from randomizing
equals the monopoly revenue from the loyal customers of firm one.
For firm two the profit equation is
( ) )()()()())(()(1)( 21212 pqcppFpqcppFpR s
m
−+−+−= α α α α
With probability ( ))(1 1 pF− firm one has the higher price. Firm two will earn revenue
from the shoppers plus its own loyal customers. With probability )(1 pF firm two has the
higher price and will only sell to its loyal customers a2. Total revenue from randomizing
equals the monopoly revenue from the loyal customers of firm two.
Solving for each cumulative distribution function:
[ ])()()()()()()( 11211 pRpRpFpRpR sm
s α α α α α α ++−=−+
)(
)()()()( 11
2pR
pRpRpF
s
ms
α
α α α −+=
)(
)()()()( 22
1pR
pRpRpF
s
ms
α
α α α −+=
Solving for the lowest price in the distribution L1 and L2:
)(
)()()(0
2
121
LR
pRLR
s
ms
α
α α α −+=
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s
s
s
m
LR
pR
α
α α
α
α )(
)(
)( 1
2
1 +=
)()()(
1
12
ss
m
s pRLRα α α
α α +=
cLq
pRL
s
m
++
=)()(
)(
21
12
α α
α
)(
)()(
1
12
s
mpRLR
α α
α
+=
)(
)()(
2
21
s
mpRLR
α α
α
+=
Notice that R(L2) and R(L1) are not equal. ( ) ( )12 LRLR > Decomposing R(L2) and R(L1):
( ) ( )12 LRLR >
( ) ( ) ( ) ( )1122 LqcLLqcL −>−
( )( )
( ) ccLLq
LqL +−> 1
2
12
Since R(p) is increasing in p, 12 LL > . Thus the minimum in the support is not equal.
The firm with the lower price L1, which is firm one, will have the atom. The lowest price
will need to be reset so that it is the same for both firms and the model reset to take in
account the atom for firm one.
The larger firm places a positive probability mass on the upper point in the price
distribution that the other firm does not match. This mass point is at the upper portion of
the distribution pm where the first firm makes monopoly profits from its uninformed
customers α1. The second firm cannot price at this monopoly price but prices at epsilon
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below this price. In doing so, it captures with one minus the probability of the atom the
shoppers αs that firm one does not capture at the monopoly price pm. Let F-( pm) be the
limit of p→ pm.
The profits of each firm can be rewritten.
For firm one this is:
)()())(( 11
ms pRLqcL α α α =−+ (1)
For firm two this is:
( ) )()(1()())(( 22
mm
ss pRpFLqcL
−
−+=−+ α α α α (2)
Firm one has the mass point. Thus it places positive probability on pm
and
charges its uninformed customers that price. When firm one is charging the lowest price
in the distribution, it captures the shoppers and still keeps its loyal customers. However,
it has to charge the loyal customers a lower price of L. Firm two prices from L on the
lower end to just below pm on the upper end. Like firm one, when it prices at L, firm two
receives its loyal customers α2 and the shoppers αs. When firm two prices at the upper
end of the price distribution, it prices just below the monopoly price. This firm
essentially receives monopoly revenue from its loyal customers α2 and with probability
(1- F-( pm)) essentially monopoly revenue from shoppers αs.
Equation one can be rearranged to solved for L:
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α (3)
Equation three states that the lowest price in the distribution equals marginal cost plus the
fraction of total expected monopoly revenue per customer from firm one divided by the
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total expected quantity sold per customer from firm one at the lowest price. Since the
proportion of customers and quantity sold at the monopoly price is lower, the numerator
will be lower than the denominator thus making L less than pm
.
Theorem 1: The atom for firm one at p = pm
is ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
Proof:
Using equation (3), equation (2) can be rearranged to solve for 1 - F-( pm):
( )( ) )()(1()(
)(
)()( 2
1
12
mms
s
m
s pRpFLqccLq
pR −−+=⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −+
++ α α
α α
α α α
( ))(1(
)(
)()( 2
1
12
msm
s
m
s pFpR
pR −−=−⎟⎟ ⎠
⎞⎜⎜⎝
⎛
++ α α
α α
α α α
( ) sss
smpFα
α
α α α
α α α 2
1
21 )(1)( −⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
+−=−
( ) ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−−+−=−
ss
ssmpFα α α
α α α α α α α α
1
2211211)(
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−−=−
s
mpFα α
α α
1
211)( (4)
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1 (4a)
F-( pm) is between zero and one because first the numerator is between zero and one as α1
is larger than α2. Second the numerator (α1-α2) is less than the denominator (α1+αs). F-(
pm) has an interesting interpretation. If both firms are equal in size F-( pm) equals one and
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the atom equals zero. As the size difference grows between the two firms (holding the
shoppers constant), F-( pm) grows smaller and the atom 1- F-( pm) grows larger as firm
one places more weight on the monopoly price pm. If α1 grows larger, F-( pm) grows
smaller and the atom 1- F-( pm) grows larger. If αs grows larger, F-( pm) grows larger and
the atom 1- F-( p
m) grows smaller.
QED
The cumulative probability function for each firm F1 and F2 can be solved by
setting profits at any price in the distribution to the profits at the highest price in the
distribution. For firm one this is:
( )[ ] )()()()(1 121
ms pRpqcppF α α α =−−+ (5)
Equation (5) is like equation (1) in that p is equal to L and the 1- F 2(L) term becomes one
as F2(L) is equal to zero at p = L. The profit equation for firm two at any price is:
( )[ ] ( ) )()(1()()()(1 212
mmss pRpFpqcppF −−+=−−+ α α α α (6)
Theorem 2: The cumulative distribution for firm two in the two firm case
is:)(
)()(1)( 11
2pR
pRpRpF
s
m
α
α α −−=
Proof:
F2(p) can be solved by rearranging equation (5):
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( ))(
)()(1 1
21pR
pRpF
m
s
α α α =−+
ss
m
pRpRpF
α α
α α 11
2)()(1)( +−= (7)
)(
)()(1)( 11
2pR
pRpRpF
s
m
α
α α −−= (7a)
Notice that F2(p) does not depend on α2 or firm two’s share of loyal customers. The
share of firm one’s loyal customers α1 and shoppers between both firms αs are important
in determining F2(p). The fraction is the difference between monopoly profits for firm
one and profits for firm one’s uninformed types at a lower price divided by the revenue
by the group of shoppers at that lower price. When p = pm , F2(p) equals one. When p
=( )
cLq
pRL
s
m
++
=)(
)(
1
1
α α
α ,F2(p) = 0.
QED
Theorem 3: The cumulative distribution function for firm one in the two firm case is
)()(
)()()()(1)(
1
12211
pR
pRpRpF
ss
sm
s
α α α
α α α α α α
+
+−+−=
Proof:
F1(p) can be solved similarly by rearranging equation six and plugging in for
F-( pm):
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( )[ ] )(11()()()(11
21212
m
s
ss pRpqcppF ⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+−
−−+=−−+α α
α α α α α α
( ) 2
1
2121
)(
)()(1 α α α
α α α α α −⎟
⎟ ⎠ ⎞
⎜⎜⎝ ⎛
⎟⎟ ⎠ ⎞
⎜⎜⎝ ⎛
+−+=−
pRpRpF
m
s
ss
s
m
ss pR
pRpF
α
α
α α
α α
α
α 2
1
2121
)(
)(1)( +⎟
⎟ ⎠
⎞⎜⎜⎝
⎛ ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−+−=
s
m
ss
s
ss
s
pR
pRpF
α
α
α α α
α α α
α α α
α α α 2
1
21
1
121
)(
)(
)(
)(
)(
)(1)( +⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
−+
+
+−=
s
m
ss
s
pR
pRpF
α
α
α α α
α α α 2
1
211
)(
)(
)(
)(1)( +⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
+−= (8)
)()(
)()()()(1)(
1
12211
pR
pRpRpF
ss
sm
s
α α α
α α α α α α
+
+−+−= (8a)
F1(p) depends on the loyal customer shares of both firms one and two: α1,α2 and the
shoppers αs. F1(p) involves the revenue of firm one at the monopoly price, revenue of
firm two at prices less than the monopoly price, and the revenue of shoppers at that same
price less than the monopoly price adjusted for the sum of shares of firm one and the
shoppers. At p= pm
F1(p) = 1.
At p ↑ pm ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−−=−
s
mpFα α
α α
1
211)( .
At F1(p) = 0, p =( )
cLq
pRLs
m
++
=)(
)(
1
1
α α α .
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If firm two’s loyal customers go away so that it is competing only for the shoppers, the
atom for firm one also interestingly still remains and gets larger. Firm two’s distribution
does not change.
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+=− −
s
mpFα α
α
1
1*)(1
)()(
)()(0)()0(1)(
1
111
pR
pRpRpF
ss
sm
s
α α α
α α α α
+
+−+−=
)()(
)(1)(
1
11
pR
pRpF
ss
ms
α α α
α α
+
−=
)()(
)(1)(
1
1*
1pR
pRpF
s
m
α α
α
+−=
3-3. Comparative Statics with the Two Firm Model
The standard comparative statics can be asked in which the share of customer
loyal groups is shoppers are changed and the effects on the cumulative distribution
function of each firm is measured. This approach gives several results. Also, the
question of what happens to each firm’s cumulative probability distribution when a small
amount of customers is equally taken away from one customer group and added to
another customer group. Table 11 lists the different lemmas resulting from the
comparative statics in the two firm model. The upper left-hand most entry reads
∑F1(ÿ)/∑q < 0, which is found in Lemma 1. The entry in the first column and sixth row
reads ∑F1(ÿ)/∑α1 - ∑F1(ÿ)/∑α2 < 0, which is found in Lemma 19.
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Table 11: Comparative Statics Overview – Section 3
∑F1(ÿ) ∑F2(ÿ) ∑L ∑ ( ))(1 mpF −− ∑ ( ))(1 21 pF ∩−
∑q Lemma 1
< 0
Lemma 2
< 0
Lemma 3
> 0
Lemma 7a
= 0
Lemma 27
< 0
∑c Lemma 4
< 0
Lemma 5
< 0
Lemma 6
> 0
Lemma 7b
= 0
Lemma 28
< 0
∑α1 Lemma 8
< 0
Lemma 9
< 0
Lemma 10
> 0
Lemma 11
> 0 αs > α2
< 0 αs < α2
Lemma 29
< 0
∑α2 Lemma 12
> 0 else
= 0 p = L
Lemma 13a
= 0
Lemma 13b
= 0
Lemma 14
< 0
Lemma 30
> 0 else
= 0 p = pm
∑αs Lemma 15
> 0 and < 0
Generally > 0
Lemma 16
> 0
Lemma 17
< 0
Lemma 18
< 0
Lemma 31
> 0 and < 0
Generally > 0
∑α1 - ∑α2 Lemma 19
< 0
Lemma 22c
< 0
Lemma 23c
> 0
Lemma 25
> 0 and < 0
Lemma 34
< 0
∑α1 - ∑αs Lemma 20
> 0 and < 0
Generally < 0
Lemma 22a
< 0
Lemma 23a
> 0
Lemma 24
> 0 and < 0
Lemma 33
> 0 and < 0
Generally < 0
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∑α2 - ∑αs Lemma 21
> 0 high p
< 0 low p
Lemma 22b
< 0
Lemma 23b
> 0
Lemma 26
< 0
Lemma 32
> 0 and < 0
Generally < 0
The first lemmas deal with changing cost and quantity and their effects on the
cumulative distribution.
Lemma 1: 0)(1 <
∂
∂
q
pF
Proof:
)()(
)()()()(1)(
1
12211
pR
pRpRpF
ss
sm
s
α α α
α α α α α α
+
+−+−=
) ( )( )cppq
cppqcppqpF
ss
smm
s
−+
−+−−+−=
)()(
)()()()(1)(
1
12211
α α α
α α α α α α
Using the chain rule:
p
pq
p
F
q
p
p
F
q
F
∂∂
∂
⋅∂=
∂∂
∂
⋅∂=
∂
⋅∂ )(/
)()()( 111
( )[ ] ( )
( )
[ ] ( ) ( )[ ]( ) ( ) )()()()()(
)()()()()()(
)()()(
)()()()(
)(
)(
1221
11221
2221
2
1121
pqcppqcppq
pqcppqpRpR
pqcppq
pRpqcppq
pq
pF
smm
s
sssm
s
ss
sss
′−+−−+
+−′++−++
′−+
++−′+−−=
∂
∂
α α α α α α
α α α α α α α α α
α α α
α α α α α α
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The bracket ( )[ ])()( pqcppq +−′ is positive since it equals 0)( >′ pR . The two terms in the
numerator are positive. The denominator is negative since it contains the term 0)( <′ pq .
Thus 0)(
)(1 <∂∂
pq
pF.
QED
Lemma 2: 0)(2 <
∂
∂
q
pF
Proof:
)(
)()(1)( 11
2pR
pRpRpF
s
m
α
α α −−=
) ( )( )cppq
cppqcppqpF
s
mm
−
−−−−=
)(
)()(1)( 11
2α
α α
Again using the chain rule:
ppq
pF
qp
pF
qF
∂∂
∂⋅∂=
∂∂
∂⋅∂=
∂⋅∂ )(/)()()( 222
( )
( )[ ] ( ) ( ) ( )[ ] ( )[ ]
( ) )()(
)()()()()()()(222
111
2
pqcppq
pqcppqcppqcppqcppqpqcppq
q
F
s
smm
s
′−
+−′−−−+−+−′
=∂
⋅∂
α
α α α α α
( ) ( ) ( )[ ]
( ) )()(
)()()(
222
12
pqcppq
pqcppqcppq
q
F
s
smm
′−
+−′−
=∂
⋅∂
α
α α
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Again the bracket ( )[ ])()( pqcppq +−′ is positive since it equals 0)( >′ pR . All parts in the
numerator are positive. The denominator, however, is negative since 0)( <′ pq . Thus
0)(
)(2 <∂∂
pq
pF
QED
Lemma 3: 0>∂∂q
L
Proof:
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
( )( )
cLq
cppqL
s
mm
++
−=
)(
)(
1
1
α α
α
Lqq
L
∂∂=
∂∂
/
1
( )( )
0)()(
)(
/
1
1
22
1 >′−
+−=
∂∂=
∂∂
Lqcppq
Lq
Lqq
Lmm
s
α
α α since 0)( <′ Lq .
QED
Lemma 4: 0)(1 <
∂
∂
c
pF
Proof:
) ( )( )cppq
cppqcppqpF
ss
smm
s
−+
−+−−+−=
)()(
)()()()(1)(
1
12211
α α α
α α α α α α
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( )( )
( ) ( )[ ]
( )222
1
2
11221
1
112211
)()(
)()()()()()(
)()(
)()()()()()()(
cppq
pqcppqcppq
cppq
cppqpqpq
c
pF
ss
sssmm
s
ss
sssm
s
−+
+−+−−+−+
−+
−++++−−=
∂
∂
α α α
α α α α α α α α α
α α α
α α α α α α α α α
( )( )
( )[ ]( )222
12
121
1
121
)()(
)()()()(
)()(
)()()()(
cppq
pqcppq
cppq
cppqpq
ss
ssmm
s
ss
ssm
s
−+
+−+−
−+
−++−−=
α α α
α α α α α α
α α α
α α α α α α
)( )
0)()(
)()()()(
1
121 <−+
+−−++=
cppq
cpcppqpq
ss
mss
ms
α α α
α α α α α α
QED
Lemma 5: 0)(2 <
∂
∂
c
pF
Proof:
) ( )
( )cppq
cppqcppqpF
s
mm
−
−−−−=
)(
)()(1)( 11
2
α
α α
( ) ) ( )
( )222
11112
)(
)()()()()()()(
cppq
pqcppqcppqcppqpqpq
p
pF
s
smm
sm
−
−−−−−+−−=
∂
∂
α
α α α α α α
( ) )( )222
11
)(
)()()()(
cppq
pqcppqcppqpq
s
smm
sm
−
−−−−−=
α
α α α α
)( )
0)(
)()(222
1 <−
+−−=
cppq
cpcppqpq
s
ms
m
α
α α
QED
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201
Lemma 6: 0>∂∂c
L
Proof:
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
)( )
cLq
cppqL
s
mm
++
−=
)(
)(
1
1
α α
α
( )( )( ) ( )
0)(
)()()(
)(
)(
)(
)(
1
1
1
1
1
1 >+
++−=
+
++
+
−=
∂∂
Lq
LqLqpq
Lq
Lq
Lq
pq
c
L
s
sm
s
s
s
m
α α
α α
α α
α α
α α
α since ( )mpqLq >)(
QED
Lemma 7:
7a) ( )
0)(1
=∂
−∂ −
q
pF m
7b)
( )0
)(1
=∂
−∂ −
c
pF m
Proof:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
There is no quantity and cost in )(1 mpF −− thus the derivative of the atom with respect
to cost and quantity are zero.
QED
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202
What happens to the two cumulative distribution functions as the share of loyal
customers and shoppers is changed? Suppose the share of firm one’s loyal customers α1
increases. How does that affect both firms?
Lemma 8: 0)(
1
1 <∂
∂
α
pF
Proof:
( ) ( ) ( )
( ) 22
1
2
2112
1
1
)(
)()()()()(
pR
pRpRpRpRpF
ss
mssss
ms
α α α
α α α α α α α α α
α +
+−++−=
∂
∂
( )
( )0
)(
)()()(22
1
2
2
2
1
1 <+
+−=
∂
∂
pR
pRpRpF
ss
sm
s
α α α
α α α
α (9)
The addition of more uninformed customers for firm one causes them to discount less and
try less for the group of shoppers αs. Profits increase for firm one when the loyal
customers of firm one increase.
QED
The effect of an increase in firm one’s uninformed customer on firm two’s cumulative
distributive function is also straightforward:
Lemma 9: 0)(
1
2 <∂
∂
α
pF
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Proof:
ss
m
pR
pRpF
α α α
1
)(
)()(
1
2 +−=∂
∂
0)(
)()()(
1
2 <+
−=∂
∂pR
pRpRpF
s
m
α α (10)
When firm one’s loyal customers increase, firm two, like firm one, will concentrate less
on lower prices as the lowest price in the distribution increases. With firm one
concentrating less on lower prices and more on its monopoly price, firm two does not
have to randomize its prices quite as aggressive to get the shoppers. Thus firm two
places less weight on lower prices and more weight on higher prices.
QED
Lemma 10: 01
>∂∂α
L
Proof:
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
( )
( )0
)(
)()()()(22
1
11
1
>+
−+=
∂∂
Lq
LqpRLqpRL
s
ms
m
α α
α α α
α
QED
Lemma 11:
11a) ( )
0)(1
1
>∂
−∂ −
α
mpFwhen αs > α2
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11b) ( )
0)(1
1
=∂
−∂ −
α
mpFwhen αs = α2
11c) ( ) 0)(11
<∂−∂
−
α
m
pF when αs < α2
Proof:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
( ) ( ) ( )
( )
( )
( )21
2
21
211
1
)(1
s
s
s
smpF
α α
α α
α α
α α α α
α +
−=
+
−−+=
∂−∂ −
QED
Changing firm two’s share of loyal customers has a different effect than changing
firm one’s share of loyal customers.
Lemma 12: L pwhen0
)(
] p(L,in pricesfor 0)(
2
1
-m
2
1
==∂
∂
>∂
∂
α
α
pF
pF
Proof:
( ) sss
m
pR
pRpF
α α α α
α
α
1
)(
)()(
1
1
2
1 ++
−=∂
∂
( )
( ) )(
)()()(
1
11
2
1
pR
pRpRpF
ss
sm
α α α
α α α
α +
++−=
∂
∂(11)
At the lowest price p = L,( )s
mpRLR
α α
α
+=
1
1 )()( . Plugging this in to the above equation:
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( )( )
( )
( )s
m
ss
s
m
sm
pR
pRpR
LF
α α
α α α α
α α
α α α α
α
+
+
+++−
=∂
∂
1
11
1
111
2
1
)(
)()(
)(
0)(
1
11
2
1 =+−
=∂
∂
α α
α α
α s
LF
Thus for any price in the range (L, pm
] 0)(
2
1 >∂
∂
α
pFsince the right side of the numerator
of equation 11 becomes larger than the left hand of the numerator. Thus
( ]m
2
1
2
1
pL, pif 0)(
L pif 0)(
∈>∂
∂
==∂
∂
α
α
LF
LF
(11a)
QED
Lemma 13:
13a) 0)(
2
2 =∂
∂α
pF(12)
13b) 02
=∂∂α
L
Proof:
)(
)()(1)( 11
2
pR
pRpRpF
s
m
α
α α −−=
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
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There is no α2 in the expression for F2(p) or L so both 0)(
2
2 =∂
∂α
pFand 0
2
=∂∂α
L.
This result contrasts that of the previous results when firm one’s loyal customers
were changed. Firm two’s loyal customers have no effect in firm two’s distribution
because firm two is always competing for shoppers at any price in its distribution of
prices. Competing for shoppers at any price checks firm two’s desire to socking its
uninformed customers the monopoly price. At the highest possible price for firm two: the
limit as p approaches pm, (which is lower than the monopoly price for firm one) firm two
is competing with firm one for the shoppers even as firm two has uninformed shoppers
willing to pay that price. At prices lower than this limit price, firm two competes with
firm one for the shoppers and its uninformed consumers pays whatever price is offered.
Firm two randomizes to get shoppers and since they are in all possible prices in its
support, firm two’s own uninformed customers will have no bearing on its randomization
function.
QED
The size of firm two’s loyal customers α2 does matter in firm one’s distribution
function: 0)(
2
1 >∂
∂
α
pF. This is the opposite sign of the case of firm one’s loyal customers
on firm one’s distribution function. This sign is positive because the size of atom
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1 at p = pm
decreases when α2 increases.
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Lemma 14: ( )
0)(1
2
<∂
−∂ −
α
mpF
Proof:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
( ) ( )
( ) ( )0
1)(1
12
1
1
2
<+−
=+
+−=
∂−∂ −
ss
smpF
α α α α
α α
α
QED
Calculatings
pF
α ∂∂ )(
1 is more complicated:
Lemma 15:
15a)s
pF
α ∂
∂ )(1 > 0 if ( )
( ) ( ) )(2)(2
)(2
12
11
21
21
2pRpR
pR
ssm
s
mss
α α α α α α
α α α α α
−−−+
−−>
15b) s
pF
α ∂
∂ )(1 < 0 if ( )
( ) ( ) )(2)(2
)(2
12
11
21
21
2pRpR
pR
ssm
s
mss
α α α α α α
α α α α α
−−−+
−−<
15c)s
mpF
α ∂
∂ )(1 > 0 if 22
1
21
21
2
s
ss
α α
α α α α α
+
+< .
s
mpF
α ∂
∂ )(1 < 0 if 22
1
21
21
2
s
ss
α α
α α α α α
+
+> .
15d) s
LF
α ∂
∂ )(1 > 0 if ( )
( )s
ss
α α
α α α α
−
+<
1
12 and α1 > αs.
s
LF
α ∂
∂ )(1 < 0 if ( )
( )s
ss
α α
α α α α
−
+>
1
12 and α1 > αs. .
15e) s
LF
α ∂
∂ )(1 > 0 if ( )
( )s
ss
α α
α α α α
−
+>
1
12 and α1 < αs.
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15f) Generally Lemma 5 is positive at all prices except when αs is extremely small
and/or α2 approaches the size or α1 (and thus αs is small)
Proof:
s
pF
α ∂
∂ )(1
( ) ( ) ( ) ( )
( ) 22
1
2
12211121
)(
)()()(2)()()(
pR
pRpRpRpRpRpR
ss
s
m
ssss
m
α α α
α α α α α α α α α α α α α
+
+++−+−++−=
( ) ( )( )[ ]( ) )(
)(22
1
2
11111
pRpR
ss
mssss
α α α
α α α α α α α α α
+++++−=
( ) ( )( )[ ]
( ) )(
)(22
1
2
11212
pR
pR
ss
ssss
α α α
α α α α α α α α α
+
++−++
( ) )(
)(222
1
2
2
121
2
12
2
1
2
1
2
1
pR
pR
ss
msssss
α α α
α α α α α α α α α α α α α
+
++++−−=
( ) )(
)(222
1
2
2
221212
2
1
2
221
pR
pR
ss
sssss
α α α
α α α α α α α α α α α α α α α
+
−−−−+
+
( ) )(
)(2)(22
1
2
2
2212
2
1
2
121
2
1
pR
pRpR
ss
ssm
sss
α α α
α α α α α α α α α α α α α α
+
−−−+++= (13)
s
pF
α ∂
∂ )(1 >0 if the numerator of equation (13) is greater than zero.
Simplifying
( ) ( ) ( ) )()(2)(22
12
12
12
112m
ssssm
s pRpRpR α α α α α α α α α α α −−>−−−+
( )( ) ( ) )(2)(2
)(
21
211
21
21
2pRpR
pR
ssm
s
mss
α α α α α α
α α α α α
−−−+
−−< (13a)
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Equation (13a) switches signs because the denominator is generally negative at all
parameter combinations of α1, αs, and α2.
At p = pm:
s
mpF
α ∂
∂ )(1 >0 iff ( )
( ) ( ) )(2)(2
)(
21
211
21
21
2 mss
ms
mss
pRpR
pR
α α α α α α
α α α α α
−−−+
−−<
221
21
21
2
s
ss
α α
α α α α α
+
+< (13b)
With the aid of Mathematica, the following chart shows the various parameter
combinations of equation (13b) and the resulting sign of s
mpF
α ∂∂ )(1 . For each level of α2,
α1 is increased while αs is decreased. Most changes are by 0.10 except when αs becomes
extremely small. To save space, the down arrow ∞ and up arrow Æ are used next to the
sign of s
mpF
α ∂
∂ )(1 . When one row has a down arrow ∞ followed by the next row having
the up arrow Æ, the parameter combinations of α1 and αs between the down and up rows
have the sames
mpF
α ∂
∂ )(1 sign as the two arrow rows. For instance, one can conclude that
the sign of s
mpF
α ∂
∂ )(1 is positive for the parameter values of α1 = 0.3, αs = 0.65 and α2 =
0.05 because they would fit in between the two arrows. Rows with the one – sided arrow
follow the same logic. For instance, a row with a down arrow ∞ with a negatives
mpF
α ∂∂ )(1
sign means that α1 values greater and αs values less than those in the same row, holding
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α2 constant, haves
mpF
α ∂
∂ )(1 negative. The parameter values α1 = 0.40, αs = 0.10 and α2 =
0.40 haves
m
pFα ∂∂ )(1 being negative. The same logic works with one-sided rows with an
up arrow.
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Table 12: Signings
mpF
α ∂
∂ )(1
α1 αs α2 RHS of eq (13b)
s
m
pFα ∂∂ )(1
0.05 0.90 0.05 0.0526154 Positive
0.1 0.85 0.05 0.110239 Positive
0.2 0.75 0.05 0.236515 Positive ∞
0.8 0.15 0.05 0.172075 Positive Æ
0.9 0.05 0.05 0.0526154 Positive
0.94 0.01 0.05 0.0101052 Negative
0.10 0.80 0.10 0.110769 Positive ∞
0.80 0.10 0.10 0.110769 Positive Æ
0.85 0.05 0.10 0.0527586 Negative
0.89 0.01 0.10 0.0101111 Negative
0.20 0.60 0.20 0.24 Positive ∞
0.60 0.20 0.20 0.24 Positive Æ
0.70 0.10 0.20 0.112 Negative ∞
0.30 0.40 0.30 0.336 Positive
0.40 0.30 0.30 0.336 Positive
0.50 0.20 0.30 0.241379 Negative ∞
0.40 0.20 0.40 0.24 Negative ∞
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From the above table, one can conclude thats
mpF
α ∂
∂ )(1 is generally positive except
when as is relatively low against a2. For very high values of a2,s
mpF
α ∂
∂ )(1 is likely to be
negative because as is relatively small. For lower values of a2,s
mpF
α ∂
∂ )(1 is likely to be
positive because there are many such parameter combinations where as is not small
relative to a2. When another parameter search is done in Mathematica at the midpoint
price value of 2
mpL +(this price being different for each combination of α1, α2 and αs.),
the results do not change except for one parameter value of α1 = 0.30, αs = 0.50 and α2 =
0.20.
At p = L:
s
LF
α ∂
∂ )(1[ ] [ ]
( )
( )( )
)(
)(2)(2
1
121
2
1
12
2212
2
1
2
121
2
1
m
s
ss
m
s
ssm
sss
pR
pRpR
α α α α α α
α α
α α α α α α α α α α α α α α α
++
+−−−+++
=
[ ] [ ]( )
( )ss
s
sssss
α α α α
α α
α α α α α α α α α α α α α α α
+
+−−−+++
=1
2
1
1
12
2212
2
1
2
121
2
1 22
( )
( )2
1
2
1
1
2
2212
2
11
2
121
2
1 22
ss
ssssss
α α α α
α α α α α α α α α α α α α α α α α
+
−−−++++=
( )2
1
2
1
2
212
2
12
3
1
3
1
2
21
22
1
22
12
2
1
3
1 222
ss
ssssssss
α α α α
α α α α α α α α α α α α α α α α α α α α α α
+
−−−++++++=
( )2
1
2
1
23
13
12
2122
122
13
1
ss
sssss
α α α α
α α α α α α α α α α α α α
+
−+++++=
The numerator of this fraction is positive and therefores
LF
α ∂
∂ )(1 is positive when:
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( ) 3
1
22
1
22
1
3
1
3
1
2
12 sssss α α α α α α α α α α α α −−−−>−
Keeping consist with (13a), the inequality sign for s
LF
α ∂
∂ )(1 to be positive is reversed to
allow for the possibility that the denominator ( )3
1
2
1 α α α −s could be negative. If the
denominator is positive, then the inequality sign is greater than for s
LF
α ∂
∂ )(1 to be positive.
The denominator of the right hand fraction is negative when α1 is greater than αs and is
positive when αs is greater than α1:
2
1
2
32
1
2
1
3
1
2
1
3
1
22
1
22
1
3
12
2
α α
α α α α α
α α α
α α α α α α α α α
−
−−−=
−
−−−−<
s
sss
s
ssss
( )( )( )
( )( )1
1
11
21
2α α
α α α
α α α α
α α α α
−
+−=
−+
+−<
s
ss
ss
ss if α1 > αs (13c)
s
pF
α ∂∂ )(1 > 0 at p = L when (13c) holds true.
( )( )s
ss
α α
α α α α
−+>
1
12 if α1 < αs (13d)
s
pF
α ∂
∂ )(1 > 0 at p = L when (13d) holds true.
s
pF
α ∂∂ )(1 > 0 occurs for sure when αs > α1 or when specified in equation (13c).
Running different combinations in parameters in Mathematicaat p = L gives almost the
identical results at p = pm except for a couple of parameter values α1 = 0.30, αs = 0.50
and α2 = 0.20 and α1 = 0.40, αs = 0.20 and α2 = 0.40. Thus it is safe to conclude as has
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been done at p= pm for all prices:s
pF
α ∂
∂ )(1 is generally positive except when as is
relatively low against a2.
Another way to see this is to remember that a large amount of shoppers present
makes it less worthwhile for firm one to go after its uninformed customers at the
monopoly price and instead discount to win over the shoppers. As will be shown in
Lemma 18, the value of the atom ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1 falls as αs grows larger. In
that case of a large number of shoppers and lower number of uninformed loyal firm two
types, α1 does most of the falling as αs increases as α2 is not very large. The denominator
of the atom remains virtually unchanged with α1 falling and αs rising but the numerator
of the atom falls as α1 is falling. Here the shopper effect dominates the uninformed firm
two types effect. This changes if α2 is much larger than αs. An increase in αs
increasingly causes α2 to noticeably fall, the larger α2 is. If α2 is falling at a large enough
pace for an increase in αs, the atom ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1 at p = pm increases. This is
because the numerator increases faster than the denominator increases. If the atom
increases, then F1(p) will rise at high prices for an increase in shoppers as the decrease in
the uninformed firm two types dominates the shopper effect.
QED
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Calculatings
pF
α ∂
∂ )(2 is more straightforward:
Lemma 16: 0)(2 >∂∂s
pFα
Proof:
2
1
2
12
)(
)()(
ss
m
s pR
pRpF
α
α
α
α
α −
−−=
∂∂
0)(
)()()(2
112 >−
=∂
∂
pR
pRpRpF
s
m
s α
α α
α (14)
Firm two does not have the luxury like firm one to rely upon an atom to sock its
uninformed customers. Firm two is randomizing to get shoppers. An increase in the
shoppers causes firm two to discount more to win a potential larger prize. Thus
s
pF
α ∂
∂ )(2 > 0.
QED
Lemma 17: 0<∂∂
s
L
α
Proof:
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
( ) ( )0
)(
)(
)(
)()(2
1
1
221
1 <+
−=
+
−=
∂∂
Lq
pR
Lq
LqpRL
s
m
s
m
s α α
α
α α
α
α
QED
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Lemma 18: ( )
0)(1
<∂
−∂ −
s
mpF
α
Proof:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
( ) ( )
( )0
)(12
1
21 <+
−−=
∂−∂ −
ss
mpF
α α
α α
α
QED
Another exercise can be performed with the three different consumer types: α1,
α2, and αs. Holding one consumer type constant, what effect will increasing and
decreasing the other two consumer types in the same proportion have on the cumulative
probability distribution for each firm, the lowest price, and the atom for firm one?
Suppose that the shoppers are held constant. What effect will increasing firm one’s loyal
customers α1 at the expense of firm two’s loyal customers α2 have on F1(p)?
Lemma 19: Increasing α1 at the same rate as α2 is decreased holding αs constant
lowers F1(p) or 02
1
1
1 <∂∂−
∂∂
α α FF .
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Proof:
By lemma 81
1
α ∂∂F
<0
By lemma 122
1
α ∂
∂F>0
Thus 02
1
1
1 <∂
∂−
∂
∂
α α
FF
QED
Instead of the shoppers being held constant, suppose that firm two’s loyal or
uninformed customers α2 are held constant. What would happen to F1(p) if firm one’s
loyal customers are increased at the expense of the shoppers? Intuitively, one would
expect less discounting by firm one.
Lemma 20:
20a) 01
1
1 >∂
∂−
∂
∂
s
FF
α α when
[ ] [ ]( ))(2)(2
)(2
1
2
11
2
2
1
2
1
3
2pRpR
pR
ssm
ss
msss
α α α α α α α
α α α α α α
+++−−
++> and the
denominator is positive in the fraction.
20b) 01
1
1 <∂∂
−∂∂
s
FF
α α when
[ ] [ ]( ))(2)(2
)(2
1
2
11
2
2
1
2
1
3
2pRpR
pR
ssm
ss
msss
α α α α α α α
α α α α α α
+++−−
++< and the
denominator is positive in the fraction. If the denominator is negative, then
01
1
1 <∂∂
−∂∂
s
FF
α α .
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20c) ) )
01
1
1 >∂
∂−
∂
∂
s
mm pFpF
α α when
2
1
2
1
2
1
3
2α
α α α α α α
sss ++>
) )01
1
1 <∂
∂−
∂
∂
s
mm pFpF
α α when
2
1
2
1
2
1
3
2α
α α α α α α sss ++<
20d) ( ) ( )
01
1
1 >∂
∂−
∂
∂
s
LFLF
α α when
3
1
2
1
3
422
1
3
1
3
12
2
22
α α α α
α α α α α α α α
+−−
+++>
ss
ssss and the denominator is
positive in the fraction.( ) ( )
01
1
1 <∂
∂−
∂
∂
s
LFLF
α α when
3
1
2
1
3
422
1
3
1
3
12
2
22
α α α α
α α α α α α α α
+−−
+++<
ss
ssss and the denominator is positive in the fraction.
If the denominator is negative, then( ) ( )
.01
1
1 <∂
∂−
∂
∂
s
LFLF
α α
20e) Generally Lemma 20 is negative at all prices except when αs is extremely small
and/or α2 approaches the size or α1 (and thus αs is small)
Proof:
( )
( ) )(
)()(2
1
2
2
2
1
1
pR
pRpF
ss
sm
s
α α α
α α α
α +
+−=
∂
∂
( ) )(
)(2)(2)(2
1
2
2
2212
2
1
2
121
2
11
pR
pRpRpF
ss
ssm
sss
s α α α
α α α α α α α α α α α α α α
α +
−−−−++−=
∂
∂−
[ ] [ ]( ) )(
)(2)(2
)()(
2
1
2
22212
21
2121
21
322
1
1
1
pR
pRpR
pFpF
ss
ssm
sssss
s
α α α
α α α α α α α α α α α α α α α α α
α α
++++++++−
=∂
∂−
∂
∂
(15)
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For 0)()( 1
1
1 >∂
∂−
∂
∂
s
pFpF
α α the numerator of equation (15) must be greater than zero.
Factoring out α2 and subtracting non α2 to the other side of the greater than sign:
( ) )()(2)(22
12
132
12
112
2m
sssssm
ss pRpRpR α α α α α α α α α α α α α ++>+++−−
0)()( 1
1
1 >∂
∂−
∂
∂
s
pFpF
α α when
[ ] [ ] )(2)(2
)(2
1
2
11
2
2
1
2
1
3
2pRpR
pR
ssm
ss
msss
α α α α α α α
α α α α α α
+++−−
++> (15a)
At p = pm
, the condition in equation (15a) simplifies down to
( )2
1
2
1
2
1
3
2
α
α α α α α α sss ++
> (15b)
A small αs relative to α2 makes it more likely at prices near p = pm that
0)()( 1
1
1 >∂
∂−
∂∂
s
pFpF
α α as the numerator of the fraction in equation (15a) becomes smaller
with αs involved in each term. The reverse case holds.
At p = L, the condition in equation (15a) simplifies down to
[ ] [ ]( )
)(2)(2
)(
1
12
1
2
11
2
2
1
2
1
3
2m
s
ssm
ss
msss
pRpR
pR
α α
α α α α α α α α
α α α α α α
++++−−
++>
( )
[ ]( ) [ ] )(2)(2
)(
1
2
1
2
111
2
1
2
1
2
1
3
2 mss
msss
mssss
pRpR
pR
α α α α α α α α α α
α α α α α α α α
++++−−
+++>
[ ] [ ]2
1
2
1
3
1
2
1
32
1
2
1
3
1
22
1
422
1
3
1
3
1
2222 ssssss
ssssss
α α α α α α α α α α α α
α α α α α α α α α α α α
+++−−−−
+++++>
3
1
2
1
3
422
1
3
1
3
12
2
22
α α α α
α α α α α α α α
+−−
+++>
ss
ssss (15c)
If the denominator is negative in (15c) then for 0)()( 1
1
1 >∂
∂−
∂
∂
s
pFpF
α α
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3
1
2
1
3
422
1
3
1
3
12
2
22
α α α α
α α α α α α α α
+−−
+++<
ss
ssss (15d)
This cannot occur, so when the denominator is negative in equation (15c),
0)()( 1
1
1 <∂
∂−
∂
∂
s
pFpF
α α .
In equation (15c) it is more likely that 0)()( 1
1
1 >∂
∂−
∂∂
s
pFpF
α α as the denominator generally
is a negative number for a sufficiently large enough αs. The following chart shows the
different parameter combinations of equation (15c) and the resulting sign of
s
LFLF
α α ∂
∂−
∂
∂ )()( 1
1
1 .
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Table 13: Signings
LFLF
α α ∂
∂−
∂
∂ )()( 1
1
1
α1 αs α2 RHS of eq. (15c)
s
LFLF
α α ∂
∂
−∂
∂ )()( 1
1
1
0.05 0.90 0.05 -0.905279 Negative ∞
0.90 0.05 0.05 0.0562295 Negative Æ
0.94 0.01 0.05 0.0102174 Positive
0.10 0.80 0.10 -0.822535 Negative ∞
0.80 0.10 0.10 0.132727 Negative Æ
0.85 0.05 0.10 0.0566421 Positive ∞
0.20 0.60 0.20 -0.709091 Negative ∞
0.60 0.20 0.20 0.52 Negative Æ
0.70 0.10 0.20 0.139024 Positive ∞
0.30 0.40 0.30 -0.778947 Negative ∞
0.50 0.20 0.30 0.709091 Negative Æ
0.60 0.10 0.30 0.148276 Positive ∞
0.40 0.20 0.40 1.4 Negative*
0.50 0.10 0.40 0.39 Positive ∞
α1 αs α2 RHS of eq. (15c)
s
LFLF
α α ∂
∂−
∂
∂ )()( 1
1
1
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*Different sign for s
mm pFpF
α α ∂
∂−
∂
∂ )()( 1
1
1
An identical chart can be drawn for s
mm pFpF
α α ∂
∂−∂
∂ )()(1
1
1 with the same signs resulting
(obviously different values for equation (15b) ) from the various parameter combinations,
except for the one row that has an asterisk. One can also be drawn for the midpoint
between L and pm
(the midpoint prices change as the alpha parameters change). The
midpoint chart results for s
midpomidpo pFpF
α α ∂
∂−
∂
∂ )()( int1
1
int1are the same as the above chart.
Given the similarity at both ends of the price interval and at the midpoint, one can
conclude that the behavior of s
pFpF
α α ∂
∂−
∂
∂ )()( 1
1
1 is the same throughout the price interval,
with a few exceptions. The chart reveals thats
LFLF
α α ∂
∂−
∂
∂ )()( 1
1
1 , and therefore
s
pFpF
α α ∂
∂−
∂
∂ )()( 1
1
1 , is negative for large enough αs, relative to α2. A large pool of shoppers
causes the shopper effect of discounting to be dominated the effect of raising prices to
firm one’s loyal group of customers. If αs becomes relatively small, as when α2 becomes
large,s
LFLF
α α ∂
∂−
∂
∂ )()( 1
1
1 becomes positive. This would be the case that the interval
between L and pm is not very large. As will be shown in Lemma 23a, the lower price
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α is larger as the numerator in the fraction of L is greater with an
increase in α1 and the denominator remains unchanged with ( )sα α +1 remaining
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223
unchanged. This increase of L, and therefore shortening of a small price interval, causes
an increase in F1(p).
QED
Now suppose that firm one’s loyal shoppers α1 are held constant. Suppose firm
two’s loyal customers are increased at the same rate that the shoppers αs are decreased.
What happens to F1(p)?
Lemma 21:
21a) 01
2
1 >∂
∂−
∂
∂
s
FF
α α when
[ ] )(2)(2
)(2)(222
1
2
11
32
1
2
1
2
1
2
12
pRpR
pRpR
ssm
s
sssm
ss
α α α α α α
α α α α α α α α α α
+++−
++−+>
21b) 01
2
1 <∂
∂−∂
∂
s
FF
α α when
[ ] )(2)(2
)(2)(222
1
2
11
32
1
2
1
2
1
2
12
pRpR
pRpR
ssm
s
sssm
ss
α α α α α α
α α α α α α α α α α
+++−
++−+<
21c) ) )
01
2
1 >∂
∂−
∂
∂
s
mm pFpF
α α when
22
1
32
12
s
ss
α α
α α α α
+
−> .
) ) 01
2
1 <∂
∂−∂
∂
s
mm pFpF
α α when 221
32
12
s
ss
α α
α α α α
+
−< .
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21d) Generally( ) ( )
01
2
1 >∂
∂−
∂
∂
s
pFpF
α α holds at high prices with a few exceptions when α2
is small.
21e) ( ) ( )
01
2
1 >∂
∂−
∂
∂
s
LFLF
α α when
( )
s
ss
α α
α α α α
−
+>
1
12 and α1 > αs
21f) ( ) ( )
01
2
1 <∂
∂−
∂
∂
s
LFLF
α α when
( )
s
ss
α α
α α α α
−
+<
1
12 and α1 > αs
or αs > α1
21g) Generally( ) ( )
s
pFpF
α α ∂
∂−∂
∂ 1
2
1
is higher at high prices and( ) ( )
s
pFpF
α α ∂
∂−∂
∂ 1
2
1
is lower at
low prices, with a few exceptions
Proof:
( )( ) )(
)()()(
1
11
2
1
pR
pRpRpF
ss
sm
α α α
α α α
α +++−
=∂
∂=
( ) ( )( ) )(
)()(
1
2
111
pR
pRpR
ss
ssm
ss
α α α
α α α α α α α
++++−
( ) )(
)(2)(2)(2
1
2
2
2212
2
1
2
121
2
11
pR
pRpRpF
ss
ssm
sss
s α α α
α α α α α α α α α α α α α α
α +
−−−−++−=∂
∂−
[ ] [ ]( ) )(
)(22)(222
)()(
2
1
2
32
1
2
1
2
2212
2
1
2
121
2
1
1
2
1
pR
pRpR
pFpF
ss
sssssm
sss
s
α α α
α α α α α α α α α α α α α α α α α α α
α α
+
++++++++−
=∂
∂−
∂
∂
For 0)()( 1
2
1 >∂
∂−
∂
∂
s
pFpF
α α the numerator must be greater than zero. Factoring out α2 and
subtracting non α2 to the other side of the greater than sign:
( )[ ] [ ] )(2)(22
)(2)(2
32
1
2
1
2
1
2
1
2
1
2
112
pRpR
pRpR
sssm
ss
ssm
s
α α α α α α α α α
α α α α α α α
++−+
>+++−
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[ ] )(2)(2
)(2)(222
1
2
11
32
1
2
1
2
1
2
12
pRpR
pRpR
ssm
s
sssm
ss
α α α α α α
α α α α α α α α α α
+++−
++−+> (16)
The greater than sign holds as it is the opposite of the denominator of equation (13a).
At p = pm
equation (16) simplifies down to:
[ ] )(2)(2
)(2)(222
1
2
11
32
1
2
1
2
1
2
12 m
ssm
s
msss
mss
pRpR
pRpR
α α α α α α
α α α α α α α α α α
+++−
++−+>
22
1
32
12
s
ss
α α
α α α α
+
−> (16a)
The following table shows the various parameter combinations of α1, α2, and αs
have ons
mm pFpF
α α ∂
∂−
∂
∂ )()( 1
2
1 . Most values are positive except for a few values when α2 is
small. When αs > α2, equation (16a) always holds as the numerator is negative and thus
0)()( 1
2
1 >∂
∂−
∂
∂
s
pFpF
α α at high prices in the distribution.
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Table 14: Signings
mm pFpF
α α ∂
∂−
∂
∂ )()( 1
2
1
α1 αs α2 RHS of eq. (16a)
s
mm
pFpFα α ∂∂−∂∂ )()( 1
2
1
0.05 0.90 0.05 -0.894462 Positive ∞
0.50 0.45 0.05 0.0472376 Positive Æ
0.60 0.35 0.05 0.17228 Negative ∞
0.85 0.10 0.05 0.0972696 Negative Æ
0.90 0.05 0.05 0.0496923 Positive ∞
0.10 0.80 0.10 -0.775385 Positive ∞
0.50 0.40 0.10 0.0878049 Positive Æ
0.60 0.30 0.10 0.18 Negative
0.70 0.20 0.10 0.169811 Negative
0.80 0.10 0.10 0.0969231 Positive ∞
0.20 0.60 0.20 -0.48 Positive ∞
0.30 0.40 0.30 -0.112 Positive ∞
0.40 0.20 0.40 0.12 Positive ∞
α1 αs α2 RHS of eq. (16a)
s
mm pFpF
α α ∂
∂−
∂
∂ )()( 1
2
1
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At p = L equation (16) simplifies to:
[ ] [ ]
[ ] )()(
2)(2
)()(
2)(22
1
12
1
2
11
1
132
1
2
1
2
1
2
1
2m
s
ssm
s
m
s
sssm
ss
pRpR
pRpR
α α
α α α α α α α
α α
α α α α α α α α α α
α
++++−
+++−+
>
[ ]1
2
1
2
111
1
32
1
2
11
2
1
2
12
2)(2
2)(22
α α α α α α α α α
α α α α α α α α α α α α α
ssss
ssssss
++++−
++−++>
[ ]2
1
2
1
3
1
2
1
2
1
3
1
22
1
3
1
22
1
3
1
3
1
22
12
222
22222
ssss
sssssss
α α α α α α α α α
α α α α α α α α α α α α α α α
+++−−
++−+++>
3
1
2
1
22
1
3
1
3
12
2
α α α
α α α α α α α
+−
++>
s
sss
2
1
2
2
1
32
12
2
α α
α α α α α α
+−
++>
s
sss
( )
s
ss
α α
α α α α
−
+>
1
12 (16b)
When αs larger than α1, for 0)()( 1
2
1 >∂
∂−∂
∂
s
LFLF
α α , the greater than sign in equation (16b)
switches to a less than sign in equation (16c) since there is division by a negative number.
( )
s
ss
α α
α α α α
−
+<
1
12 (16c)
This condition (16c) does not hold since the denominator in equation (16c) is less than
zero, thus guaranteeing 0)()( 1
2
1 <∂
∂−∂
∂s
LFLFα α
. For smaller αs (and larger α1), equation
(16b) may not hold thus meaning 0)()( 1
2
1 <∂
∂−
∂
∂
s
pFpF
α α for prices immediately above L.
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Notice that the right hand side of equations (16b)/(16c) is exactly the same as equations
(13c) and (13d). The difference is that the greater than/less than signs flip between the
two equations thus guaranteeing the opposite signs of s
LFα ∂
∂ )(1 ands
LFLFα α ∂
∂−∂
∂ )()( 1
2
1 .
(At( )
cLq
pRL
s
m
++
=)(
)(
1
1
α α
α , α2 does enter into the equation, thus
s
LFLF
α α ∂
∂−
∂
∂ )()( 1
2
1 only
reflects the effect of αs.)
The following table shows the various parameter combinations of α1, α2, and αs
have ons
LFLFα α ∂
∂−∂
∂ )()( 1
2
1 . There are more negative values of s
LFLFα α ∂
∂−∂
∂ )()( 1
2
1 compared
tos
mm pFpF
α α ∂
∂−
∂
∂ )()( 1
2
1 in the previous chart. At the monopoly price p = pm,(as will be
shown in Lemma 26) the atom ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1 is smaller as the numerator and
denominator fall by the same amount. As the previous chart indicated, that lost atom
weight is going to high prices.( )
cLq
pRL
s
m
++
=)(
)(
1
1
α α
α is higher than before α2 and αs are
changed because the denominator is smaller (shown in Lemma 23b). Less weight is
being placed on lower prices as there are less shoppers available to win. The following
table shows that there are more negative outcomes for s
LFLF
α α ∂
∂−
∂
∂ )()( 1
2
1 , especially when
αs is falling from a relatively high number. These effects – a falling atom (Lemma 26),
rising lower prices, and less shoppers – cause firm one to randomize more. However
firm one randomizes more at high prices because there are less shoppers available to
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steeply discount. The exception of this behavior is when αs is small. In this case a few
parameter combinations in the previous chart become negative.
Table 15: Signings
LFLFα α ∂
∂−∂
∂ )()( 1
2
1
α1 αs α2 RHS of eq. (16b)/(16c)
s
LFLF
α α ∂
∂−
∂
∂ )()( 1
2
1
0.05 0.90 0.05 -1.00588 Negative ∞
0.90 0.05 0.05 0.0558824 Negative Æ
0.94 0.01 0.05 0.0102151 Positive ∞
0.10 0.80 0.10 -1.02857 Negative ∞
0.80 0.10 0.10 0.128571 Negative Æ
0.85 0.05 0.10 0.05625 Positive
0.20 0.60 0.20 -1.2 Negative ∞
0.60 0.20 0.20 0.4 Negative Æ
0.70 0.10 0.20 0.133333 Positive ∞
0.30 0.40 0.30 -2.8 Negative ∞
0.50 0.20 0.30 0.466667 Negative Æ
0.60 0.10 0.30 0.14 Positive ∞
0.40 0.20 0.40 0.60 Negative
0.45 0.15 0.40 0.30 Positive ∞
α1 αs α2 RHS of eq. (16b)/(16c)
s
LFLF
α α ∂
∂−
∂
∂ )()( 1
2
1
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QED
Suppose α2 is held constant and α1 is increased at the same rate that αs is
decreased. What is the effect on F2(p)?
Lemma 22:
22a) 02
1
2 <∂
∂−
∂
∂
s
FF
α α
22b) 02
2
2 <∂
∂−
∂
∂
s
FF
α α
22c) 02
2
1
2 <∂
∂−
∂
∂
α α
FF
Proof:
By lemma 9 0
)(
)()()(
1
2 <+
−=
∂
∂
pR
pRpRpF
s
m
α α
(10)
By lemma 16 0)(
)()()(2
112 >−
=∂
∂
pR
pRpRpF
s
m
s α
α α
α (14)
Thus 02
1
2 <∂
∂−
∂
∂
s
FF
α α
By lemma 13a 0)(
2
2 =∂
∂
α
pF(12)
Thus 02
2
2 <∂
∂−
∂
∂
s
FF
α α
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Thus 02
2
1
2 <∂
∂−
∂
∂
α α
FF
QED
Lemma 23:
23a) 01
>∂∂
−∂∂
s
LL
α α
23b) 02
>∂∂
−∂∂
s
LL
α α
23c) 021
>∂∂
−∂∂
α α
LL
Proof 23a:
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
From Lemma 10:
( )
( )0
)(
)()()()(22
1
11
1
>+
−+=
∂∂
Lq
LqpRLqpRL
s
ms
m
α α
α α α
α
From Lemma 17:
( )0
)(
)(2
1
1 <+
−=
∂
∂
Lq
pRL
s
m
s α α
α
α
( )
( ) 221
111
1 )(
)()()()()()(
Lq
LqpRLqpRLqpRLL
s
mms
m
s α α
α α α α
α α +
+−+=
∂∂
−∂∂
( )
( )0
)(
)()(22
1
1 >+
+=
Lq
LqpR
s
sm
α α
α α
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Proof (23b):
From Lemma 13b:
02
=∂∂α L
From Lemma 17:
( )0
)(
)(2
1
1 <+
−=
∂
∂
Lq
pRL
s
m
s α α
α
α
Thus 02
>∂∂
−∂∂
s
LL
α α
Proof (23c):
From Lemma 10:
( )
( )0
)(
)()()()(22
1
11
1
>+
−+=
∂∂
Lq
LqpRLqpRL
s
ms
m
α α
α α α
α
From Lemma 13b:
02
=∂∂α
L
Thus 021
>∂∂
−∂∂
α α
LL
QED
Lemma 24:
24a) ( ) ( )
0)(1)(1
1
>∂
−∂−
∂−∂ −−
s
mm pFpF
α α if α1+αs-2α2 > 0
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24b) ( ) ( )
0)(1)(1
1
<∂
−∂−
∂−∂ −−
s
mm pFpF
α α if α1+αs-2α2 < 0
Proof:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
From Lemma 11:
( ) ( ) ( )
( )
( )
( )21
2
21
211
1
)(1
s
s
s
smpF
α α
α α
α α
α α α α
α +
−=
+
−−+=
∂−∂ −
From Lemma 18:
( ) ( )
( )0
)(12
1
21 <+
−−=
∂−∂ −
ss
mpF
α α
α α
α
( ) ( ) ( ) ( )
( ) ( )21
21
21
212
1
2)(1)(1
s
s
s
s
s
mm pFpF
α α
α α α
α α
α α α α
α α +
−+=
+
−+−=
∂−∂
−∂
−∂ −−
QED
Lemma 25:
25a)( ) ( )
0)(1)(1
21
>∂
−∂−
∂−∂ −−
α α
mm pFpFif α1+2αs-α2 > 0
25b) ( ) ( )
0)(1)(1
21
<∂
−∂−
∂−∂ −−
α α
mm pFpFif α1+2αs-α2 < 0
Proof:
From Lemma 11:
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( ) ( ) ( )
( )
( )
( )21
2
21
211
1
)(1
s
s
s
smpF
α α
α α
α α
α α α α
α +
−=
+
−−+=
∂−∂ −
From Lemma 14:
( ) ( )
( ) ( )0
1)(1
12
1
1
2
<+−
=+
+−=
∂−∂ −
ss
smpF
α α α α
α α
α
( ) ( ) ( ) ( )
( ) ( )21
21
21
12
21
2)(1)(1
s
s
s
ssmm pFpF
α α
α α α
α α
α α α α
α α +
−+=
+
++−=
∂−∂
−∂
−∂ −−
Lemma 26: ( ) ( ) 0)(1)(12
<∂−∂−∂−∂
−−
s
mm
pFpFα α
Proof:
From Lemma 14:
( ) ( )
( ) ( )0
1)(1
12
1
1
2
<+−
=+
+−=
∂−∂ −
ss
smpF
α α α α
α α
α
From Lemma 18:
( ) ( )
( )0
)(12
1
21 <+
−−=
∂−∂ −
ss
mpF
α α
α α
α
( ) ( ) ( ) ( )
( )
( )
( )0
)(1)(12
1
2
21
211
2
<+
+−=
+
−++−=
∂−∂
−∂
−∂ −−
s
s
s
s
s
mm pFpF
α α
α α
α α
α α α α
α α
QED
Shoppers obtain the minimum price offered by the two firms. The distribution of
the minimum price is ( )( ))(1)(11 21 pFpF −−− . I now turn to the comparative statics on
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⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −−=− ∩
)()(
)()()()(
)(
)()(1)(1
1
12211121
pR
pRpR
pR
pRpRpF
ss
sm
s
s
m
α α α
α α α α α α
α
α α . Each of
these changes are presented below.
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −−=− ∩
)()(
)()()()(
)(
)()(1)(1
1
12211121
pR
pRpR
pR
pRpRpF
ss
sm
s
s
m
α α α
α α α α α α
α
α α
Lemma 27: ( )
0)(1 21 <
∂
−∂ ∩
q
pF
Proof:
Let )(1 21 pF ∩− be represented by ( ) ( )BA−1 where
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −=
)(
)()( 11
pR
pRpRA
s
m
α
α α and ⎟
⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+=
)()(
)()()()(
1
1221
pR
pRpRB
ss
sm
s
α α α
α α α α α α
( )A
q
BB
q
A
q
pF
∂∂
−∂∂
−=∂
−∂ ∩ )(1 21
( )0
0
1
0
2)(1 21 <⎟⎟ ⎠
⎞⎜⎜⎝
⎛ <
+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ <
=∂
−∂ ∩ ALemmaBy
BLemmaBy
q
pF.
QED
Lemma 28: ( )
0)(1 21 <
∂
−∂ ∩
c
pF
Proof:
Let )(1 21 pF ∩− be represented by ( ) ( )BA−1 where
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −=
)(
)()( 11
pR
pRpRA
s
m
α
α α and ⎟
⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+=
)()(
)()()()(
1
1221
pR
pRpRB
ss
sm
s
α α α
α α α α α α
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( )A
q
BB
q
A
c
pF
∂∂
−∂∂
−=∂
−∂ ∩ )(1 21
( )0 0
5
0
4)(1 21
<⎟⎟ ⎠
⎞⎜⎜⎝
⎛
<+⎟⎟ ⎠
⎞⎜⎜⎝
⎛
<=∂
−∂ ∩
A
LemmaBy
B
LemmaBy
c
pF
QED
Suppose that the proportion of firm one’s loyal customers α1 are changed. How
does this affect ( )( ) )(1)(1)(11 2121 pFpFpF ∩−=−−− ?
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −−=− ∩
)()(
)()()()(
)(
)()(1)(1
1
12211121
pR
pRpR
pR
pRpRpF
ss
sm
s
s
m
α α α
α α α α α α
α
α α
Lemma 29: ( )
0)(1
1
21 <∂
−∂ ∩
α
pF
Proof:
( )
=∂
−∂ ∩
1
21 )(1
α
pF
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∂
∂−⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
+−+−⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−
1
2
1
1221
1
111 )(
)()(
)()()()()(
)(
)()(
α α α α
α α α α α α
α α
α α pF
pR
pRpRpF
pR
pRpR
ss
sm
s
s
m
By lemma 8( )
( )0
)(
)()(2
1
2
1
1 <+
+−=
∂∂
pR
pRpF
s
ms
α α
α α
α and by lemma 9
0)(
)()(
1
2
<
+−
=∂
∂pR
pRpRF
s
m
α α .
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237
Thus each term in the above equation for ( )
1
21 )(1
α ∂
−∂ ∩ pFis negative and thus
( ) 0)(11
21 <∂−∂ ∩α
pF .
Solving for the expression for ( )
1
21 )(1
α ∂−∂ ∩ pF
:
( )
( )
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ +−−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+−+−
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−=
)(
)()(*
)()(
)()()()(
)(
)(*
)(
)()(
1
1221
2
1
211
pR
pRpR
pR
pRpR
pR
pR
pR
pRpR
s
m
ss
sm
s
ss
mss
s
m
α α α α
α α α α α α
α α α
α α α
α
α α
( ) ( )
( )
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+−
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+++−=
)()(
))(())((*
)()(
)()()()(
)(
)()()(
1
11
1
1221
222
1
122
12
pR
pRpR
pR
pRpR
pR
pRpRpR
ss
ssm
ss
sm
s
ss
mss
mss
α α α
α α α α
α α α
α α α α α α
α α α
α α α α α α α α
( ) ( )
( )
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
++−+−++−
⎟⎟
⎠
⎞⎜⎜
⎝
⎛
+
+++−=
22
1
2
22
12
22
1
2
121
2
12
2
121
222
1
12
2
12
)()(
)()(
)()(
)()())(()()()()())((
)(
)()()(
pR
pR
pR
pRpRpRpRpR
pR
pRpRpR
ss
s
ss
mss
ms
mss
ss
mss
mss
α α α
α α α
α α α
α α α α α α α α α α α α α
α α α
α α α α α α α α
( ) ( ) ( )( )
( )⎟⎟
⎠
⎞⎜⎜
⎝
⎛
+
+−
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+++++++−=
22
1
2
22
12
222
1
2
1
2
1
2
2212
2
1
2
112
)(
)()(
)(
)()(242)(2
pR
pR
pR
pRpRpR
ss
s
ss
mssss
mss
α α α
α α α
α α α
α α α α α α α α α α α α α α α α
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238
In summary, both cumulative distribution functions F1(p) and F2(p) move in the same
direction when α1 is changed. So the probability of at least one firm offering a lower
price suffers when the proportion of firm one’s loyal customers α1 is increased.
QED
Calculating the effects of changing firm two’s loyal customers α2 on the probability of at
least one firm having a lower price is also straightforward:
Lemma 30:
30a) ( )
0)(1
2
21 >∂
−∂ ∩
α
pFfor p in [L,p
m)
30b) ( )
0)(1
2
21 =∂
−∂ ∩
α
pFwhen p = pm See proof for significance.
Proof:
( )=
∂
−∂ ∩
2
21 )(1
α
pF
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∂∂
−⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+−⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂∂
−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −−
2
2
1
1221
2
111 )(
)()(
)()()()()(
)(
)()(
α α α α
α α α α α α
α α
α α pF
pR
pRpRpF
pR
pRpR
ss
sm
s
s
m
( ) ( ) 0*)(1)()(
)()()()(
)()()(1 1
1
1111
2
21 pFpR
pRpRpR
pRpRpFss
s
m
s
m
−−⎟⎟ ⎠ ⎞⎜⎜
⎝ ⎛ + +−⎟⎟
⎠ ⎞⎜⎜
⎝ ⎛ −−=∂−∂ ∩
α α α α α α
α α α
α
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239
1-F2(p) (the leftmost term) is negative and2
1 )(
α ∂
∂−
pFis negative thus
( )
2
21 )(1
α ∂
−∂ ∩ pFis
positive. The lone exception is at p = pm where 1-F2(p) equals zero thus making the
entire expression zero. This is significant because this states that the minimum price has
no mass point at the monopoly price. Only the largest firm has an atom at p = pm
. The
above expression can be multiplied out for the following expression:
( )2
1
2
2
1111
2
1
22
1
2
21
)()(
)()()()()()()()()(1
pR
pRpRpRpRpRpRpF
ss
sm
smm
α α α
α α α α α α α α
α +
+−+++−=
∂−∂ ∩
QED
Lemma 31:
31a) ( )
0)(1 21 >
∂−∂ ∩
s
pF
α when
( )( ) ( ) )(242)(3
)(222
12
112
1
21
21
2pRpR
pR
ssm
s
mss
α α α α α α α
α α α α α
−−−++
−−<
31b) ( )
0)(1 21 <
∂−∂ ∩
s
pF
α when
( )( ) ( ) )(242)(3
)(222
12
112
1
21
21
2pRpR
pR
ssm
s
mss
α α α α α α α
α α α α α
−−−++
−−>
31c) )
0)(1 21 >
∂
−∂ ∩
s
mpF
α when
( )( )2
12
1
21
21
22
22
ss
ss
α α α α
α α α α α
−−−
−−<
)0
)(1 21 <∂
−∂ ∩
s
mpF
α when
( )( )2
12
1
21
21
22
22
ss
ss
α α α α
α α α α α
−−−
−−>
31d) ( ) 0)(1 21 >
∂−∂ ∩
s
LF
α when ( )
1
12
2
α α
α α α α
−+−<
s
ss and a1 > as
or as > a1
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31e) ( )
0)(1 21 >
∂
−∂ ∩
s
LF
α when
( )
1
12
2
α α
α α α α
−
+−>
s
ss and a1 > as
31f) Generally( )
0)(1 21
>∂
−∂ ∩
s
pF
α except when a2 gets large or as is extremely small
Proof:
( )=
∂
−∂ ∩
s
pF
α
)(1 21
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∂
∂−⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+−+−⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∂
∂−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −−
sss
s
m
s
ss
m pF
pR
pRpRpF
pR
pRpR
α α α α
α α α α α α
α α
α α )(
)()(
)()()()()(
)(
)()(2
1
1221111
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
+−++⎟⎟
⎠
⎞⎜⎜⎝
⎛
∂
∂−⎟⎟
⎠
⎞⎜⎜⎝
⎛ −−=
sss
sm
s
ss
m
pR
pRpRpF
pR
pRpR
α α α α
α α α α α α
α α
α α 1
)()(
)()()()()(
)(
)()(
1
1221111
( )[ ])(2)()()(
)()( 2
121
2
12
1
2
11 msss
sss
m
pRpRpR
pRpRα α α α α α α
α α α α
α α ++⎟
⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=
( )[ ])(2
)()()(
)()( 2
2212
2
1
21
2
11 pR
pRpR
pRpRss
sss
m
α α α α α α α
α α α α
α α −−−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−+
[ ])()()())(()()()(
)()( 2
121212
1
2
11 pRpRpRpR
pRpRs
mss
sss
m
α α α α α α α α α α α α
α α +−++⎟
⎟ ⎠
⎞⎜⎜⎝
⎛
+
−+
Let M1 = ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−
)()()(
)()(2
1
2
11
pRpR
pRpR
sss
m
α α α α
α α .
( )=
∂−∂ ∩
s
pF
α
)(1 21
( ) )(22
121
2
11
msss pRM α α α α α α α ++= ( ) )(2
2
2212
2
11 pRM ss α α α α α α α −−−+
)()()())(( 2
121211 pRpRM sm
ss α α α α α α α α +−+++
For ( )
0)(1 21 >
∂
−∂ ∩
s
pF
α , the terms multiplied by M1 must be greater than zero since
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241
M1 > 0.
Simplifying:
( ) )(2)(2 22212
21211 pRpRM ss
ms α α α α α α α α α α −−−+
)()()())(( 2
121211 pRpRM sm
ss α α α α α α α α +−+++ ( ) )(2
1
2
11
mss pRM α α α α −−>
( ) ( ) )(2)(22
1
2
112 pRpR ssm
s α α α α α α α −−−+=
)()()()( 2
1112 pRpR sm
s α α α α α α +−++ )()( 11
2
1
2
1
mssss pRα α α α α α α α +−−−>
( ) ( ) )(2)(32
11
2
12 pRpR sm
s α α α α α α +−+= )()( 11
2
1
2
1
mssss pRα α α α α α α α +−−−>
2α =( )
( ) ( ) )(242)(3
)(222
1
2
11
2
1
2
1
2
1
pRpR
pR
ssm
s
mss
α α α α α α α
α α α α
−−−++
−−< (17)
for ( )
0)(1 21 >
∂
−∂ ∩
s
pF
α .
The sign reverses in equation (17) because the denominator is less than zero.
At p = pm
equation (17) simplifies down to equation (17a) for
( )0
)(1 21
>∂
−∂ ∩
s
mpF
α :
( )( )2
1
2
1
2
1
2
12
2
22
ss
ss
α α α α
α α α α α
−−−
−−< (17a)
The sign reverses because the denominator is negative. The distribution of the minimum
price is not very likely to have an increase in weight at higher prices when there is an
increase in shoppers.
At p = L equation (17) simplifies down to equation (17b) for ( )
0)(1 21 >
∂
−∂ ∩
s
LF
α :
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242
( )
( ) ( )( )
)(242)(3
)(22
1
121
211
21
21
21
2m
s
ssm
s
mss
pRpR
pR
α α
α α α α α α α α
α α α α α
+−−−++
−−<
( ) ( )21
21
31
21
21
21
31
31
221
221
31
224233
2222
sssss
ssss
α α α α α α α α α α α α
α α α α α α α α α
−−−++++
−−−−<
( )
1
1
22
1
321
21
2
1
3
1
31
221
31
2
2242242
α α
α α α
α α
α α α α α
α α α
α α α α α α α
−
+−=
+−
−−−=
+−
−−−<
s
ss
s
sss
s
sss (17b)
If αs > α1, then the less than sign in equation (17b) reverses to a greater than sign in (17c)
for ( )
0)(1 21 >
∂
−∂ ∩
s
LF
α since the denominator is now positive.
( )
1
12
2
α α
α α α α
−
+−>
s
ss (17c)
Since the right hand expression is negative,( )
0)(1 21 >
∂
−∂ ∩
s
LF
α will always occur when
αs > α1.
The following chart shows the different parameter combinations that affect the sign of
( )
s
LF
α ∂
−∂ ∩ )(1 21 :
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Table 16: Signing( )
s
LF
α ∂
−∂ ∩ )(1 21
α1 αs α2 RHS of eq. (17b)/(17c) ( )
s
LF
α ∂
−∂ ∩ )(1 21
0.05 0.90 0.05 -2.01176 Positive ∞
0.90 0.05 0.05 0.111765 Positive Æ
0.94 0.01 0.05 0.0204301 Negative
0.10 0.80 0.10 -2.05714 Positive ∞
0.80 0.10 0.10 0.257143 Positive Æ
0.85 0.05 0.10 0.1125 Negative ∞
0.20 0.60 0.20 -2.4 Positive ∞
0.70 0.10 0.20 0.266667 Positive Æ
0.75 0.05 0.20 0.114286 Negative ∞
0.30 0.40 0.30 -5.6 Positive ∞
0.50 0.20 0.30 0.933333 Positive Æ
0.60 0.10 0.30 0.28 Negative ∞
0.45 0.15 0.40 0.6 Positive Æ
0.50 0.10 0.40 0.3 Negative ∞
α1 αs α2 RHS of eq. (17b)/(17c) ( )
s
LF
α ∂
−∂ ∩ )(1 21
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244
The signs on)
s
mpF
α ∂
−∂ ∩ )(1 21 are similar to( )
s
LF
α ∂
−∂ ∩ )(1 21 except for a couple more
negative values when α2 becomes larger. Thus ( ) 0)(1
21 >∂
−∂∩
s
pF
α at most values of
α1,αs, and α2 – except when α2 becomes large or as becomes extremely small.
QED
The exercise of calculating the effect on the firm’s cumulative probability
distribution from holding one consumer group constant while changing the other two in
opposite directions of the same magnitude can be done with the probability of at least one
firm having the lower price.
Lemma 14:
32a)
( ) ( )0
)(1)(1 21
2
21 >∂
−∂−
∂
−∂ ∩∩
s
pFpF
α α iff
( ) ( ) ( )( ) ( ) ( ) 22
12
12
12
122
11
2321
21
21
321
221
21
2)(242)()(327)(3
)(2)()(45)(33
pRpRpRpR
pRpRpRpR
ssm
ssm
s
sssm
sssm
ss
α α α α α α α α α α α
α α α α α α α α α α α α α α α
−−−++++−−
+++−−−++>
32b)( ) ( )
0)(1)(1 21
2
21 <∂
−∂−
∂
−∂ ∩∩
s
pFpF
α α iff
( ) ( ) ( )( ) ( ) ( ) 22
12
12
12
122
11
2321
21
21
321
221
21
2)(242)()(327)(3
)(2)()(45)(33
pRpRpRpR
pRpRpRpR
ssm
ssm
s
sssm
sssm
ss
α α α α α α α α α α α
α α α α α α α α α α α α α α α
−−−++++−−
+++−−−++<
32c) Generally( ) ( )
0)(1)(1 21
2
21 >∂
−∂−
∂
−∂ ∩∩
s
pFpF
α α at prices near pm with some exceptions
when α2 is small
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32d) ( ) ( )
0)(1)(1 21
2
21 >∂
−∂−
∂
−∂ ∩∩
s
LFLF
α α when
( )
s
ss
α α
α α α α
−
+>
1
12
2and α1 is larger than αs
32e) ( ) ( )
0
)(1)(1 21
2
21
<∂
−∂−∂
−∂ ∩∩
s
LFLF
α α when αs is larger than α1
32f) Generally( ) ( )
0)(1)(1 21
2
21 <∂
−∂−
∂
−∂ ∩∩
s
LFLF
α α except when αs is small
Proof:
( )2
1
2
2
1111
2
1
22
1
2
21
)()(
)()()()()()()()()(1
pR
pRpRpRpRpRpRpF
ss
sm
smm
α α α
α α α α α α α α
α +
+−+++−=
∂
−∂ ∩
( ) ( ) ( )
( )22
1
3
22
11
22
1
3
2
111
2
12
1
2
1
)()(
)(
)()(
)()()()()(
pR
pR
pR
pRpRpRpRpR
ss
ss
ss
mss
mss
mss
α α α
α α α α
α α α
α α α α α α α α α α α α
+
+−
+
+++++−=
( ) ( )
( )22
1
3
23
1
22
1
3
1
221
3
3
1
22
1
3
1
222
1
3
1
)()(
)(2
)()(
)()(32)(
pR
pR
pR
pRpRpR
ss
sss
ss
msss
mss
α α α
α α α α α α
α α α
α α α α α α α α α α
+
−−−
+
+
+++−−=
( )=
∂−∂
− ∩
s
pF
α
)(1 21
( )[ ])(2)()()(
)()( 2
121
2
12
1
2
11 msss
sss
m
pRpRpR
pRpRα α α α α α α
α α α α
α α ++⎟
⎟ ⎠
⎞⎜⎜⎝
⎛
+
−−
( )[ ])(2
)()()(
)()( 2
2212
2
12
1
2
11 pR
pRpR
pRpRss
sss
m
α α α α α α α
α α α α
α α −−−⎟
⎟
⎠
⎞⎜⎜
⎝
⎛
+
−−
[ ])()()())(()()()(
)()( 2
121212
1
2
11 pRpRpRpR
pRpRs
mss
sss
m
α α α α α α α α α α α α
α α +−++⎟
⎟ ⎠
⎞⎜⎜⎝
⎛
+
−−
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246
( ) ( )22
1
3
2
2
3
1
22
12
2
1
3
121
)()(
)(232)(1
pR
pRpF
ss
msss
s α α α
α α α α α α α α α
α +
−−−−=
∂
−∂− ∩
( )221
3
23
12
2122
122
13
1
)()()()(32272
pRpRpR
ss
mssss
α α α α α α α α α α α α α α α
++++++
( )22
13
22212
212
31
)()(
)(242
pR
pR
ss
ss
α α α
α α α α α α α α
+
−−−+
( ) ( ) ( )
( )22
13
23
1
22
1
3
1
221
3
3
1
22
1
3
1
222
1
3
1
2
21
)()(
)(2
)()(
)()(32)()(1
pR
pR
pR
pRpRpRpF
ss
sss
ss
msss
mss
α α α
α α α α α α
α α α
α α α α α α α α α α
α
+
−−−+
+
+++−−=
∂−∂ ∩
( ) ( )=
∂−∂
−∂
−∂ ∩∩
s
pFpF
α α
)(1)(1 21
2
21
22
1
3
2
2
3
1
22
12
2
1
3
1
)()(
)(333
pR
pR
ss
msss
α α α
α α α α α α α α α
+
−−−−
( )22
13
23
13
13
12
2122
122
1
)()(
)()(34257
pR
pRpR
ss
msssss
α α α
α α α α α α α α α α α α α α
+
++++++
( )22
13
231
221
31
2212
212
31
)()(
)(2242
pR
pR
ss
sssss
α α α
α α α α α α α α α α α α α α
+
−−−−−−+ (18)
For ( ) ( )
s
pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
2
21 >0, the numerator in equation (18) must be positive.
Rearranging, this implies:
( ) ( )( ) ⎥
⎥⎦
⎤
⎢⎢⎣
⎡
−−−+
+++−−22
12
13
1
3
1
2
1
2
123
1
2
12
)(242
)()(327)(3
pR
pRpRpR
ss
mss
ms
α α α α α
α α α α α α α α α
( ) ( )( ) 23
1
22
1
3
1
31
31
221
2221
31
)(2
)()(45)(33
pR
pRpRpR
sss
msss
mss
α α α α α α
α α α α α α α α α α
+++
−−−++>
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( ) ( ) ( )( ) ( ) ( ) 22
12
13
13
12
12
123
12
1
231
221
31
31
31
221
2221
31
2
)(242)()(327)(3
)(2)()(45)(33
pRpRpRpR
pRpRpRpR
ssm
ssm
s
sssm
sssm
ss
α α α α α α α α α α α α α
α α α α α α α α α α α α α α α α
α
−−−++++−−
+++−−−++
>
( ) ( ) ( )( ) ( ) ( ) 22
12
12
12
122
11
2321
21
21
321
221
21
2)(242)()(327)(3
)(2)()(45)(33
pRpRpRpR
pRpRpRpR
ssm
ssm
s
sssm
sssm
ss
α α α α α α α α α α α
α α α α α α α α α α α α α α α
−−−++++−−
+++−−−++> (18a)
The denominator in condition (18a) is exactly the same except the opposite in sign from
the denominator in the condition found in( )
s
pF
α ∂
−∂ ∩ )(1 21 (when the numerator of the M1
term is multiplied into the equation). Thus the > found here in condition (18a) for
( ) ( )s
pFpFα α ∂−∂−∂−∂ ∩∩ )(1)(1 21
2
21 to be positive is opposite of the < sign found in the condition
for ( )
s
pF
α ∂−∂ ∩ )(1 21 to be positive.
( ) ( ) ( )( ) ( ) ( ) 22
12
12
12
122
11
2321
21
21
321
221
21
2)(242)()(327)(3
)(2)()(45)(33
pRpRpRpR
pRpRpRpR
ssm
ssm
s
sssm
sssm
ss
α α α α α α α α α α α
α α α α α α α α α α α α α α α
−−−++++−−
+++−−−++> (18a)
At p = pm, condition (18a) simplifies down to:
zero
zero>2α (18b)
This makes interpretation by normal inspection impossible. However, running
calculations by equation (18a) and letting R(p) get close to R(pm
) is a second – best
approach to finding the sign of ( ) ( )
s
mm pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
2
21 . The following chart
shows different parameter values of α2 compared to the right-hand side of equation (18a)
with differing values of α1 and αs.
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Table 17: Signing) )
s
mm pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
2
21
α1 αs R(pm) R(p)
α2 RHS of eq (18a)
( )( )
s
m
m
pF
pF
α
α
∂
−∂−
∂−∂
∩
∩
)(1
)(1
21
2
21
0.3 0.65 20 19.999 0.05 -0.0272992 Positive Æ
0.4 0.55 20 19.999 0.05 0.132652 Negative ∞
0.9 0.05 20 19.999 0.05 0.0966697 Negative Æ
0.94 0.01 20 19.999 0.05 0.0198917 Positive
0.3 0.6 20 19.999 0.10 0.0000245475 Positive Æ
0.4 0.5 20 19.999 0.10 0.157015 Negative ∞
0.8 0.1 20 19.999 0.10 0.182458 Negative Æ
0.85 0.05 20 19.999 0.10 0.0964413 Positive ∞
0.3 0.5 20 19.999 0.20 0.0540804 Positive Æ
0.4 0.4 20 19.999 0.20 0.2004 Negative ∞
0.6 0.2 20 19.999 0.20 0.285758 Negative Æ
0.7 0.1 20 19.999 0.20 0.179336 Positive ∞
0.31 0.39 20 19.999 0.30 0.120502 Positive ∞
0.41 0.19 20 19.999 0.40 0.225743 Positive ∞
α1 αs R(pm) R(p) α2 RHS of eq (18a) )
( )s
m
m
pF
pF
α
α
∂
−∂−
∂
−∂
∩
∩
)(1
)(1
21
2
21
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In most cases, at high prices near p = pm
, the above expression is generally positive with
some exceptions when α2 is small.
At p = L, there is more instances where ( ) ( )s
pFpFα α ∂
−∂−∂
−∂ ∩∩ )(1)(1 21
2
21 is positive:
( )( ) ( )( ) ( )[ ]( )( ) ( )( ) ( )[ ] 223
14
15
114
122
13
12
13
12
1
2331
241
511
41
321
231
21
221
31
2
)(2423273
)(24533
msssssss
mssssssssss
pR
pR
α α α α α α α α α α α α α α α α α
α α α α α α α α α α α α α α α α α α α α
α
−−−++++++−−
++++−−−+++
>
Simplifying the numerator for the condition at p=L:
( )
( ) ( )
2
33
1
24
1
5
1
24
1
42
1
33
1
5
1
33
1
24
1
421
331
241
331
241
51
)(
24545
363363 m
sssssssss
sssssspR
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+++−−−−−−+
+++++
α α α α α α α α α α α α α α α α α α
α α α α α α α α α α α α
( ) 2421
331
241 )(442 m
sss pRα α α α α α ++=
Simplifying the denominator for the condition at p=L:
( )( ) ( )
2
231
41
51
41
321
231
51
231
41
231
41
51
321
231
41
)(242327327
2363 m
sssssss
ssssspR
⎥⎥⎦
⎤
⎢⎢⎣
⎡
++−++++++
−−−−−−
α α α α α α α α α α α α α α α α
α α α α α α α α α α α
( ) 2321
231
41 )(4 m
sss pRα α α α α α −+
( )
s
ss
s
sss
ss
sss
α α
α α α
α α
α α α α α
α α α α
α α α α α α α
−
+=
−
++=
−
++>
1
1
221
321
21
321
41
421
331
241
2
2242242(18c)
For ( ) ( )
s
LFLF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
2
21 > 0 condition (18c) becomes( )
s
ss
α α
α α α α
−
+<
1
12
2(18d)
when αs > α1. Equation (18d) cannot occur so( ) ( )
s
LFLF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
2
21 < 0 when αs
> α1. The following chart shows α2 compared to the right hand side of equations (18c)
and (18d) when the parameters of α1, α2 and αs are varied. The signs for
( ) ( )
s
LFLF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
2
21 are the exact opposite for the signs of ( )
s
LF
α ∂
−∂ ∩ )(1 21 .
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Table 18: Signing( ) ( )
s
LFLF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
2
21
α1 αs α2 RHS of eq. (18c)/(18d) ( )
( )
s
LF
LF
α
α
∂
−∂−
∂
−∂
∩
∩
)(1
)(1
21
2
21
0.05 0.90 0.05 -2.01176 Negative ∞
0.90 0.05 0.05 0.111765 Negative Æ
0.94 0.01 0.05 0.0204301 Positive
0.10 0.80 0.10 -2.05714 Negative ∞
0.80 0.10 0.10 0.257143 Negative Æ
0.85 0.05 0.10 0.1125 Positive ∞
0.20 0.60 0.20 -2.4 Negative ∞
0.70 0.10 0.20 0.266667 Negative Æ
0.75 0.05 0.20 0.114286 Positive ∞
0.30 0.40 0.30 -5.6 Negative ∞
0.50 0.20 0.30 0.933333 Negative Æ
0.60 0.10 0.30 0.28 Positive ∞
0.45 0.15 0.40 0.6 Negative Æ
0.50 0.10 0.40 0.3 Positive ∞
α1 αs α2 RHS of eq. (18c)/(18d) ( )
( )
s
LF
LF
α
α
∂
−∂−
∂−∂
∩
∩
)(1
)(1
21
2
21
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Generally, the sign of ( ) ( )
s
LFLF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
2
21 at low prices in the distribution is
negative except when αs is small.
QED
Lemma 33:
33a)( ) ( )
s
pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 > 0 iff
( ) ( )( ) ( ) 233
12
12
122
13
12
1
3
1
22
1
3
1
23
1
22
1
3
12
)()(369)(24)()(232)(232
K pRpRpRpRpRpR
msss
mss
m
sss
m
sss
+++++−−−−−−+++>α α α α α α α α α α α
α α α α α α α α α α α α α
where ( ) 2321
21
312 )(452 pRK sss α α α α α α −−−−=
33b)( ) ( )
s
pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 < 0 iff
( ) ( )( ) ( ) 2
331
21
21
221
31
21
31
221
31
231
221
31
2)()(369)(24
)()(232)(232
K pRpRpR
pRpRpRm
sssm
ss
msss
msss
+++++−−−
−−−+++<
α α α α α α α α α α α
α α α α α α α α α α α α α
where ( ) 2321
21
312 )(452 pRK sss α α α α α α −−−−=
33c) Generally( ) ( )
s
pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 < 0 at high prices near p = pm
except when
αs is relatively small to α2
33d) ( ) ( )
s
LFLF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 > 0 when32
1
3
1
431
221
31
22
2552
ss
ssss
α α α α
α α α α α α α α
−−
+++> and
the denominator is positive
33e) ( ) ( )
s
LFLF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 < 0 when 0232
13
1 <−− ss α α α α
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33f) Generally( ) ( )
s
pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 < 0 except when αs is relatively small to α2
Proof:
( )
1
21 )(1
α ∂
−∂ ∩ pF
( ) ( ) ( )( )
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
+−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
+++++++−=
221
2
2212
2221
21
21
22212
21
2112
)(
)()(
)(
)()(242)(2
pR
pR
pR
pRpRpR
ss
s
ss
mssss
mss
α α α
α α α
α α α
α α α α α α α α α α α α α α α α
( ) ( )( )
( ) ( )( ) ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛
+
++−++
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+++−−−−=
221
3
232
2212
21
221
31
2321
32
2212
21
2221
312
21
221
)(
)(2)()(2
)(
)()(42)(22
pR
pRpRpR
pR
pRpRpR
ss
sssm
ss
ss
msss
mssss
α α α
α α α α α α α α α α α α
α α α
α α α α α α α α α α α α α α α α α α
( ) ( )22
13
22
31
2212
21
3121
)()(
)(232)(1
pR
pRpF
ss
msss
s α α α
α α α α α α α α α
α +
−−−−=
∂
−∂− ∩
( )22
13
23
12
2122
122
13
1
)()(
)()(32272
pR
pRpR
ss
mssss
α α α
α α α α α α α α α α α α
+
+++++
( )22
13
22212
212
31
)()(
)(242
pR
pR
ss
ss
α α α
α α α α α α α α
+
−−−+
( ) ( )
( )22
13
231
2212
31
2212
21
31
21
1
21
)()(
)(22342
)(1)(1
pR
pR
pFpF
ss
msssss
s
α α α
α α α α α α α α α α α α α α
α α
+
−−−−−−
=∂
−∂−
∂
−∂ ∩∩
( )22
13
31
322
31
221
2212
21
31
)()(
)()(236392
pR
pRpR
ss
mssssss
α α α
α α α α α α α α α α α α α α α α
+
+++++++
( )22
13
232
2212
212
31
)()(
)(452
pR
pR
ss
sss
α α α
α α α α α α α α α α
+
−−−−+ (19)
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For ( ) ( )
s
pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 >0 the numerator in equation (19) must be positive.
Simplifying:
( ) ( )( )
>⎥⎥⎦
⎤
⎢⎢⎣
⎡
−−−−+
++++−−−232
12
13
1
331
21
21
221
31
21
2)(452
)()(369)(24
pR
pRpRpR
sss
msss
mss
α α α α α α
α α α α α α α α α α α α
( ) ( ) )()(232)(2323
122
13
123
122
13
1 pRpRpR msss
msss α α α α α α α α α α α α −−−+++
For ( ) ( )
s
pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 >0,
( ) ( )( ) ( ) 2
331
21
21
221
31
21
31
221
31
231
221
31
2)()(369)(24
)()(232)(232
K pRpRpR
pRpRpRm
sssm
ss
msss
msss
+++++−−−
−−−+++>
α α α α α α α α α α α
α α α α α α α α α α α α α (19a)
where ( ) 2321
21
312 )(452 pRK sss α α α α α α −−−−=
At p = pm, equation (19a) reduces to:
zero
zero>2α (19b)
. Again, running calculations by equation (19b) and letting R(p) get close to
R(pm) is a second – best approach to finding the sign of
( ) ( )s
mm pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 .
The following chart shows different parameter values of α2 compared to the right-hand
side of equation (19a) with differing values of α1 and αs.
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Table 19: Signing) )
s
mm pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21
α1 αs R(pm) R(p)
α2 RHS of eq (19a)
( )( )
s
m
m
pF
pF
α
α
∂
−∂−
∂−∂
∩
∩
)(1
)(1
21
1
21
0.9 0.05 20 19.999 0.05 0.10232 Negative Æ
0.94 0.01 20 19.999 0.05 0.020105 Positive
0.85 0.05 20 19.999 0.10 0.102427 Negative Æ
0.89 0.01 20 19.999 0.10 0.0201106 Positive
0.7 0.1 20 19.999 0.20 0.208133 Negative Æ
0.75 0.05 20 19.999 0.20 0.102669 Positive ∞
0.5 0.2 20 19.999 0.30 0.394671 Negative Æ
0.6 0.1 20 19.999 0.30 0.208327 Positive ∞
0.41 0.19 20 19.999 0.40 0.364301 Positive ∞
α1 αs R(pm) R(p) α2 RHS of eq (19a) ( )
( )s
m
m
pF
pF
α
α
∂
−∂−
∂
−∂
∩
∩
)(1
)(1
21
1
21
Generally( ) ( )
s
pFpF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21 < 0 at high prices near p = pm except when αs is
relatively small to α2. Less weight is placed on the cumulative distribution function on
the minimum prices on high prices when shoppers are reduced in favor of adding loyal
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customers of firm one. This is driven by both firms one and two discounting less
intensely when shoppers are taken away in favor of firm one loyal customers.
At p = L, the numerator of the fraction of equation (19a) reduces to:
( )( ) ( )( ) 21
321
231
41
221
31
221
31 )(232)(232 m
ssssm
ssss pRpR α α α α α α α α α α α α α α α α +−−−++++
( )( ) 242
133
124
133
124
15
1
251
421
331
421
331
241
331
241
51
)(232232
)(242363242
mssssss
msssssssss
pR
pR
α α α α α α α α α α α α
α α α α α α α α α α α α α α α α α α
−−−−−−+
++++++++=
( ) 251
421
331
241 )(2552 m
ssss pRα α α α α α α α +++=
At p=L, the denominator of the fraction of equation (19a) reduces to:
)( ) )( )
( ) 2321
231
41
51
21
31
41
221
31
221
21
31
21
)(452
)(369)(24
msss
mssss
msss
pR
pRpR
α α α α α α α
α α α α α α α α α α α α α α α α
−−−−+
++++++−−−
( )( )( ) 232
123
14
15
1
241
41
321
231
321
51
231
41
241
321
231
231
41
51
321
231
41
)(452
)(369369
)(2422484
msss
msssssss
mssssssss
pR
pR
pR
α α α α α α α
α α α α α α α α α α α α α α α
α α α α α α α α α α α α α α α α α
−−−−+
++++++++
−−−−−−−−−=
( ) 241
41
321 )(2 m
sss pRα α α α α α −+−=
( )( ) 24
132
14
1
251
421
331
241
2)(2
)(2552
msss
mssss
pR
pR
α α α α α α
α α α α α α α α α
−−
+++>
321
31
431
221
31
22
2552
ss
ssss
α α α α
α α α α α α α α
−−
+++> (19c)
When the denominator is negative the condition for ( ) ( )
0)(1)(1 21
1
21 >∂
−∂−
∂
−∂ ∩∩
s
LFLF
α α
changes to
32
1
3
1
431
221
31
22
2552
ss
ssss
α α α α
α α α α α α α α
−−
+++< (19d)
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Since α2 is positive( ) ( )
0)(1)(1 21
1
21 <∂
−∂−
∂
−∂ ∩∩
s
LFLF
α α when the denominator in equations
(19c) and (19d) is negative. The following chart shows different parameter values of α2
compared to the right-hand side of equation (19c) and (19d) with differing values of α1
and αs:
Table 20: Signing( ) ( )
s
LFLF
α α ∂
−∂−
∂
−∂ ∩∩ )(1)(1 21
1
21
α1 αs α2 RHS of eq. (19c)/(19d) ( )
( )
s
LF
LF
α
α
∂−∂−
∂
−∂
∩
∩
)(1
)(1
21
1
21
0.90 0.05 0.05 0.11541 Negative Æ
0.94 0.01 0.05 0.0205423 Positive
0.85 0.05 0.10 0.116421 Negative Æ
0.89 0.01 0.10 0.0205734 Positive
0.70 0.10 0.20 0.295122 Negative Æ
0.75 0.05 0.20 0.1189 Positive ∞
0.60 0.10 0.30 0.317241 Negative Æ
0.65 0.05 0.30 0.122258 Positive ∞
0.45 0.15 0.40 0.87 Negative Æ
0.50 0.10 0.40 0.352632 Positive ∞
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Generally( ) ( )
0)(1)(1 21
1
21 <∂
−∂−
∂
−∂ ∩∩
s
LFLF
α α except for small αs relative to α2. Since this
same result held for ) )s
mm pFpF
α α ∂
−∂−∂
−∂∩∩
)(1)(121
1
21 ,( ) ( )
0)(1)(1
21
1
21 <∂
−∂−∂
−∂∩∩s
pFpF
α α
except for small αs relative to α2. QED
Lemma 34: Suppose the shoppers αs are held constant. The probability of having the
minimum price )(1 21 pF ∩− decrease if firm one’s loyal customers α1 is increased at the
same rate as firm two’s loyal customers α2 is decreased.
Proof:
By lemma 29( )
0)(1
1
21 <∂
−∂ ∩
α
pF.
By lemma 30( )
0)(1
2
21 ≥∂
−∂ ∩
α
pF.
Thus ( ) ( ) 0)(1)(1
2
21
1
21 <∂
−∂−∂
−∂ ∩∩
α α pFpF .
Loyal customers to firm one matter more to both firms in deciding how much to
randomize.
QED
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3-4. The Model with n Firms – 1 large and (n - 1) Similar Small Firms
Now suppose that there are n firms in the marketplace. Suppose that there is one
large firm who has loyal customers α1. Suppose that there are (n – 1) smaller firms, each
with loyal customers α2. Let αs still be the proportion of shoppers in the marketplace.
Let )(*)()( pqcppR −= be the revenue earned by a firm at price p. Assume again that
this is increasing in p.
If the support from the initial profit calculations reveals differing lower prices,
then one of the firms has an atom. Checking profit equations for each type of firm will
help determine whether there is an atom. For firm one profits are:
( )( ) ( ) ( )21
1
221
1
21 )()(1)( LRLFLRLFpRn
s
nm α α α α −− ++−=
Since L2 is the lowest price in the distribution, ( ) 01
2 =−nLF and ( )( ) 11
12 =− −n
LF . Thus
the above equation can be written as:
)()()( 211 LRpR sm α α α +=
Solving for R(L2):
)(
)()(
1
12
s
mpRLR
α α
α
+=
For the smaller firms two through n, profits are:
( )( ) ( ) 211212 )()(1)( α α α α LFLRLFpR sm ++−=
Since L1 is the lowest price in the distribution, ( ) 01 =LF and ( )( ) 11 1 =− LF . Thus the
smaller firms’ profit equation can be rewritten as:
)()()( 122 LRpR sm α α α +=
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Solving for R(L1):
)(
)()(
2
21
s
mpRLR
α α
α
+=
Notice that R(L1) and R(L2) are not equal. Specifically, ( ) ( )12 LRLR > . Since
R(p) is increasing in p, L2 > L1. Thus the supports of each firm do not line up. Firm one
will have the atom since the lowest price in its support is lower than firm two. The profit
equations will have to be rewritten to take into account the atom for firm one and reset so
that the lowest prices in the support of each firm’s cumulative distribution function is the
same.
Again for a price randomization equilibrium to occur, profits at the highest price p
= pm must equal profit at the lowest price p = L and all prices between the two extremes.
Again, with the asymmetric consumer groups, there will be an atom at the highest price
that firm one will place positive weight. The rest of the (n – 1) firms cannot price at this
monopoly price but prices at epsilon below this price. In doing so, they capture with one
minus the probability of the atom the shoppers αs that firm one does not capture at the
monopoly price pm
. Again let F-( p
m) be the limit of p→ p
m. The profits of each firm can
be rewritten.
Firm one’s profit equation is:
)()())(( 11
ms pRLqcL α α α =−+ (1)
Firms two through n each have the same profit equation:
( )( ) )()(1)())(( 22
mmss pRpFLqcL −−+=−+ α α α α (2)
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Firm one places a mass point at it highest point in its price distribution. Firm one socks
its uninformed consumers the monopoly price. The profits for firm one from doing so
must equal the revenue at the lowest price from its loyal customers plus the shoppers it
picks up with probability one at p = L. Firms two through n cannot sock its loyal
customers the monopoly price. However, they can price slightly below the monopoly
price, socking their loyal customers and with probability one minus the atom firm one
places on its highest price can pick up the shoppers.
Equation one can be solved for L as before:
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α (3)
The same interpretation for L still holds as before.
Theorem 1a: The atom for the modified n firm case remains ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1 .
Proof:
Equation (2) can be solved for F-( p
m):
( )( )( ) )()(1)(
)(
)()( 2
1
12
mms
s
m
s pRpFLqccLq
pR −−+=⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −+
++ α α
α α
α α α
( )( ))(1
)(
)()( 2
1
12
msm
s
m
s pFpR
pR −−=−⎟⎟ ⎠
⎞⎜⎜⎝
⎛
++ α α
α α
α α α
( ) sss
smpFα α
α α α α α α 2
1
21 )(1)( −⎟⎟ ⎠ ⎞⎜⎜
⎝ ⎛
++−=−
( ) ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−−+−=−
ss
ssmpFα α α
α α α α α α α α
1
2211211)(
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⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−−=−
s
mpFα α
α α
1
211)( (4)
⎟⎟ ⎠ ⎞⎜⎜
⎝ ⎛
+−=− −
s
mpFα α α α
1
21)(1 (4a)
(1 - F-( pm)) in the n firm case is exactly the same as the two firm (1- F -( pm)). The
proportion of firm one’s loyal customers α1, the proportion of firm two’s loyal customers
α2, and the proportion of shoppers αs affect the atom in the same way as they did in the
two firm case. The variable n does not directly affect the size of the atom, but indirectly
affects the size through α2. As n gets much larger, the α2 term gets smaller as α2
represents an individual share one of the smaller n – 1 firms. If (n – 1)α2 is unchanged,
then the size of the atom will increase when n increases. Things get more complicated if
a large increase in n and decrease in α2 also results in α1, αs, or both getting smaller. A
small increase in n coupled with no change in the size of α2 causes α1, αs or both to fall.
Again, the cumulative probability function for each firm F1 and F2 can be solved
by setting profits at any price in the distribution to the profits at the highest price in the
distribution. For firm one this is:
( ) )()()()(1 1
1
21
mn
s pRpqcppF α α α =−−+ −(20)
Notice in equation (20) on the left-hand side that the probability that all other firms have
higher prices is now ( )1
2 )(1
−
−
n
pF .
The profit equation for firm two at any price is:
( )( ) ( )( ) )()(1)()()(1)(1 2
2
212
mms
n
s pRpFpqcppFpF −− −+=−−−+ α α α α (21)
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Notice in equation (21) on the left-hand side that the probability of all other firms having
higher prices now is the probability of the larger firm having a higher price ( ))(1 1 pF−
times the probability that the (n – 2) similar smaller firms have higher
prices ( ) 2
2 )(1−− n
pF .
Theorem 4: In the modified n firm case,1
1
112
)(
)()(1)(
−
⎥⎦
⎤⎢⎣
⎡ −−=
n
s
m
pR
pRpRpF
α
α α
Proof:
F2(p) can be solved by rearranging equation (20):
( ))(
)()(1 11
21pR
pRpF
mn
s
α α α =−+ −
( )ss
mn
pR
pRpF
α
α
α
α 111
2)(
)()(1 +=− −
1
1
112
)(
)()()(1
−
⎥⎦
⎤⎢⎣
⎡ −=−
n
s
m
pR
pRpRpF
α
α α
1
1
112
)(
)()(1)(
−
⎥⎦
⎤⎢⎣
⎡ −−=
n
s
m
pR
pRpRpF
α
α α (22)
F2(p) in the n firm case looks very similar to the two firm case except now the fraction is
raised to the 1/(n - 1) power. Raising the bracketed fraction to the 1/(n – 1) power raises
the overall fraction and thus lowers F2(p) at every price. The implication of a lower F2(p)
is that smaller (n - 1) firms discount less to attempt to get the shoppers as their
marketshare and thus potential revenue from their loyal customers α2 is getting smaller
slower than the chances of winning the shoppers with higher amounts of firms in the
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marketplace. Just like the two firm case, α2 does not appear in the expression for F2(p) as
the smaller firms are randomizing for shoppers at every price that they could charge.
Theorem 5: In the modified n firm case
( ) ( )
( )1
2
111
12211
)(
)()()(
)()(1)(
−
−
⎥⎦
⎤⎢⎣
⎡ −+
+−+−=
n
n
s
m
ss
sm
s
pR
pRpRpR
pRpRpF
α
α α α α α
α α α α α α
Proof:
F1(p) can be solved by rearranging equation (21):
( ) ( )( ) )()(1)()(
)()(11)(1 2
2
1
1
1112
mms
n
n
s
m
s pRpFpRpR
pRpRpF −
−
−
−+=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ −−−−+ α α
α
α α α α
( ) )()()(
)()()(1
1
212
1
2
1112
m
s
s
n
n
s
m
s pRpRpR
pRpRpF ⎟⎟ ⎠
⎞
⎜⎜⎝
⎛
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+−
+=⎥⎥⎥⎦
⎤
⎢⎢⎢⎣
⎡
⎥⎦
⎤⎢⎣
⎡ −−+
−
−
α α
α α α α α
α α α α
( ) )()()(
)()()(1
1
212121
2
1112
m
s
sssn
n
s
m
s pRpRpR
pRpRpF ⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
−++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎦
⎤⎢⎣
⎡ −−+
−−
α α
α α α α α α α α
α
α α α α
( )( )
)(
)(
)(
)()()(1
1
211
2
1112
pR
pR
pR
pRpRpF
m
s
sn
n
s
m
s ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
+=⎥
⎦
⎤⎢⎣
⎡ −−+
−
−
α α
α α α
α
α α α α
( )( )( ) sss
ms
n
n
s
m
pR
pR
pR
pRpRpF
α
α
α α α
α α α
α
α α 2
1
211
2
111
)(
)(
)(
)()()(1 −
+
+=⎥
⎦
⎤⎢⎣
⎡ −−
−
−
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( ) ( )
( )1
2
111
12211
)(
)()()(
)()(1)(
−
−
⎥⎦
⎤⎢⎣
⎡ −+
+−+−=
n
n
s
m
ss
sm
s
pR
pRpRpR
pRpRpF
α
α α α α α
α α α α α α (23)
( )
( )1
2
11
2
1
2
111
211
)(
)()(
)(
)()()(
)()(
−
−
−
−
⎥⎦
⎤⎢⎣
⎡ −
−
⎥⎦
⎤⎢⎣
⎡ −+
+=
n
n
s
m
s
n
n
s
m
ss
ms
pR
pRpR
pR
pRpRpR
pRpF
α
α α α
α
α
α α α α α
α α α (23a)
F1(p) in equation (22) in the n – firm case is similar to F1(p) in equation (8a) in the two
firm case except for the1
2
11
)(
)()( −
−
⎥⎦
⎤
⎢⎣
⎡ − n
n
s
m
pR
pRpR
α
α α term in the denominator. This term
adjusts for the case that there are (n – 1) smaller firms competing with firm one for the
shoppers instead of just one firm. As n grows larger, this1
2
11
)(
)()( −
−
⎥⎦
⎤⎢⎣
⎡ − n
n
s
m
pR
pRpR
α
α α term
grows larger, thus making F1(p) smaller as firm one does not want to compete as much
with the (n – 1) other firms in an increasingly cutthroat competition for the shoppers.
The addition of this extra term makes it more complicated to predict the effects of
changes of consumer groups on F1(p) except for the case of α2.
3-5. Comparative Statics with the Modified n – Firm Model
As with the two firm model, one can ask how changes in consumer classes and
the number of firms affect the cumulative probability densities of the firms in the
marketplace. Table 21 shows the overview of comparative statics for section 5. Table 21
lists the different lemmas resulting from the comparative statics in the modified n- firm
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model. The upper left-hand most entry reads ∑F1(ÿ)/∑q < 0, which is found in Lemma 35.
The entry in the first column and seventh row reads ∑F1(ÿ)/∑α1 - ∑F1(ÿ)/∑α2 < 0, which is
found in Lemma 60.
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Table 21: Comparative Statics Overview – Section 5
∑F1(ÿ) ∑F2(ÿ) ∑L ∑ ( ))(1 mpF −− ∑ )(1 121pF n−∩
−
∑q Lemma 35
< 0
Lemma 36
< 0
Lemma 37
> 0
Lemma 41a
= 0
Lemma 67
< 0
∑c Lemma 38
< 0
Lemma 39
< 0
Lemma 40
> 0
Lemma 41b
= 0
Lemma 68
< 0
∑α1 Lemma 42
< 0
Lemma 43
< 0
Lemma 44
> 0
Lemma 45
> 0 αs > α2
< 0 αs < α2
Lemma 69
< 0
∑α2 Lemma 46
> 0 else
= 0 p = L
Lemma 47
= 0
Lemma 48
= 0
Lemma 49
< 0
Lemma 70
> 0 else
= 0 p = pm
∑αs Lemma 50
> 0 and < 0
Lemma 51
> 0
Lemma 52
< 0
Lemma 53
< 0
Lemma 71
> 0 and < 0
Generally > 0
∑n Lemma 54
< 0
Lemma 55
< 0
Lemma 56
= 0
Lemma 57
> 0 if (n-
1)α2
constant
Lemma 72
< 0
∑α1 - ∑α2 Lemma 60
< 0
Lemma 62b
< 0
Lemma 63c
> 0
Lemma 65
> 0 and < 0
Lemma 75
< 0
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∑α1 - ∑αs Lemma 59
> 0 and < 0
Generally
< 0
as n
increases
Lemma 61
< 0
Lemma 63a
> 0
Lemma 64
> 0 and < 0
Lemma 74
> 0 and < 0
Generally < 0
∑α2 - ∑αs Lemma 58
> 0 high p
< 0 low p
Lemma 62a
< 0
Lemma 63b
> 0
Lemma 66
< 0
Lemma 73
> 0 and < 0
∑F1(ÿ) ∑F2(ÿ) ∑L ∑ ( ))(1mpF −− ∑ )(1 121
pF n−∩−
Lemma 35: In the modified n firm model 0)(1 <
∂
∂
q
pF
Proof:
( ) ( )
( )1
2
111
12211
)(
)()()(
)()(1)(
−
−
⎥⎦
⎤⎢⎣
⎡ −+
+−+−=
n
n
s
m
ss
sm
s
pR
pRpRpR
pRpRpF
α
α α α α α
α α α α α α
( ) ( ) ( ) ( )
( ) ( )( ) ( )
( )
1
2
111
12211
)(
)()()(
)()(1)(
−
−
⎥⎦
⎤⎢⎣
⎡
−
−−−−+
−+−−+−=
n
n
s
m
ss
smm
s
cppq
cppqcppqcppq
cppqcppqpF
α
α α α α α
α α α α α α
Let ⎥⎦
⎤⎢⎣
⎡ −=
)(
)()( 11
pR
pRpRK
s
m
α
α α .
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Let D = the denominator of the fraction of F1(p) or
( )1
2
111
)(
)()()(
−
−
⎥⎦
⎤⎢⎣
⎡ −+
n
n
s
m
ss
pR
pRpRpR
α
α α α α α . Let N = the numerator of the fraction of F1(p)
or ( ) ( ) )()( 1221 pRpR sm
s α α α α α α +−+ .
( ) ( )[ ] D
N
K cppq
NpF
n
n
ss
−=−+
−=−
−1
)(
1)(
1
2
1
1
α α α
( ) ( )[ ])(
)()(2
121
pqD
Dpqcppq
q
F s
′
+−′+=
∂
∂ α α α
( ) ( )[ ])(
)()(2
1
2
1
pqD
K pqcppqN n
n
ss
′
+−′++
−
−
α α α
( ) ( )( )[ ] ( )[ ]
( )
)(
)(
)()()(
1
2)(
2
1
1
222
11
pqD
cppq
pqcppqcppq
n
ncppqN
n
s
smm
ss
′
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−
+−′−
−−
−+
+
−
−
α
α α α α α
The bracketed expression ( )[ ])()( pqcppq +−′ is positive since it equals )( pR′ , which is
assumed to be greater than zero. The numerator is positive since each of the three terms
is positive. The denominator, however, is negative since .0)( <′ pq Thus in the modified
n-firm case, 0)(1 <
∂
∂
q
pF.
QED
Lemma 36: In the modified n firm case 0)(2 <
∂
∂
q
pF
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Proof:
1
1
112
)(
)()(1)(
−
⎥⎦
⎤⎢⎣
⎡ −−=
n
s
m
pR
pRpRpF
α
α α
( ) ( )( )
1
1
112
)(
)()(1)(
−
⎥⎦
⎤⎢⎣
⎡
−
−−−−=
n
s
mm
cppq
cppqcppqpF
α
α α
( ) ( )( )
( ) ( )[ ]
( ) )()(
)()()(
)(
)()(
1
1222
11
2
112
pqcppq
pqcppqcppq
cppq
cppqcppq
nq
F
s
mms
n
n
s
mm
′−
+−′−⎥⎦
⎤⎢⎣
⎡
−
−−−
−=
∂
∂ −
−
α
α α
α
α α
Again, the bracketed expression ( )[ ])()( pqcppq +−′
is positive since it equals)( pR′,
which is assumed to be greater than zero. The left fraction or K raised to the1
2
−−
n
n
power is positive. The numerator of the right fraction is positive. The denominator of
the right fraction, however, is not positive since 0)( <′ pq is in the denominator. Thus in
the modified n-firm case 0)(2 <
∂
∂
q
pF
QED
Lemma 37: In the modified n firm model 0>∂∂q
L
Proof:
The proof is identical to Lemma 3.
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
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)( )
cLq
cppqL
s
mm
++
−=
)(
)(
1
1
α α
α
Lqq
L
∂∂=∂∂
/
1
( )( )
0)()(
)(
/
1
1
221 >
′−
+−=
∂∂=
∂∂
Lqcppq
Lq
Lqq
Lmm
s
α
α α since 0)( <′ Lq .
QED
Lemma 38: In the modified n firm model 0
)(1
<∂
∂
c
pF
Proof:
( ) ( ) ( ) ( )
( ) ( )( ) ( )
( )
1
2
111
12211
)(
)()()(
)()(1)(
−
−
⎥⎦
⎤⎢⎣
⎡
−
−−−−+
−+−−+−=
n
n
s
m
ss
smm
s
cppq
cppqcppqcppq
cppqcppqpF
α
α α α α α
α α α α α α
( ) ( )[ ] D
N
K cppq
NpF
n
n
ss
−=
−+
−=−
−1
)(
1)(
1
2
1
1
α α α
( ) ( )[ ] ( ) ( )2
1
2
112211 )()()()(
D
K cppqpqpq
q
pF n
n
sssm
s−
−
−++−+=
∂
∂ α α α α α α α α α
( ) ( ) ( ) ( )[ ] ( )2
1
2
11221 )()()(
D
K pqcppqcppq n
n
sssmm
s−
−
+−+−−+−+
α α α α α α α α α
( ) ( ) [ ]( )( )2
2221
1)()()()(
D
cppqcpcppqpqcppqN
s
m
s
m
ss
⎥⎥⎦
⎤
⎢⎢⎣
⎡
− +−−−++
α α α α α α
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( ) ( ) ( )2
1
2
121 )()(
D
cpcpK pqpq mn
n
ssm
s +−−++=
−
−
α α α α α α
( ) ( ) ( )( )
2
222
11
)(
)()()(
D
cppq
pppqpqcppqNs
msm
ss⎥⎥⎦
⎤⎢⎢⎣
⎡−
−−+
+α
α α α α α
Since ( ) 0<− mpp , each numerator term is negative. The denominator is positive. Thus
0)(1 <
∂
∂
c
pF.
QED
Lemma 39: In the modified n firm model 0)(2 <
∂
∂
c
pF
Proof:
( ) ( )( )
1
1
112
)(
)()(1)(
−
⎥⎦
⎤⎢⎣
⎡
−
−−−−=
n
s
mm
cppq
cppqcppqpF
α
α α
( ) ( )( )
( )( )222
11
2
112
)(
)()(
)(
)()(
1
1
cppq
pppqpq
cppq
cppqcppq
nq
F
s
mms
n
n
s
mm
−
−⎥⎦
⎤⎢⎣
⎡
−
−−−
−=
∂
∂ −
−
α
α α
α
α α
Again since ( ) 0<− mpp the right – hand fraction is negative. The left – hand fraction is
positive. Thus 0)(2 <
∂
∂
c
pF.
QED
Lemma 40: In the modified n firm model 0>∂∂c
L.
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Proof:
The proof is identical to that of Lemma 6.
( )c
Lq
pRLs
m
++
=)(
)(
1
1
α α
α
( )( )
cLq
cppqL
s
mm
++
−=
)(
)(
1
1
α α
α
( )( )( ) ( )
0)(
)()()(
)(
)(
)(
)(
1
1
1
1
1
1 >+
++−=
+
++
+
−=
∂∂
Lq
LqLqpq
Lq
Lq
Lq
pq
c
L
s
sm
s
s
s
m
α α
α α
α α
α α
α α
α since ( )mpqLq >)(
QED
Lemma 41: In the modified n firm model
41a) ( )
0)(1
=∂
−∂ −
q
pF m
41b)( )
0)(1
=∂
−∂ −
c
pF m
Proof:
The proof is exactly identical to Lemma 7a and 7b.
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
There is no quantity and cost in )(1 mpF −− thus the derivative of the atom with respect
to cost and quantity are zero.
QED
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273
Suppose that the proportion of loyal customers to firm one increases? How will that
affect firm one’s cumulative price distribution function F1(p)?
Lemma 42: In the modified n firm case 0)(
1
1 <∂
∂
α
pF
Proof:
( ) ( )
( )1
2
111
12211
)(
)()()(
)()(1)(
−
−
⎥
⎦
⎤⎢
⎣
⎡ −+
+−+−=
n
n
s
m
ss
sm
s
pR
pRpRpR
pRpRpF
α
α α α α α
α α α α α α
Again, let ⎥⎦
⎤⎢⎣
⎡ −=
)(
)()( 11
pR
pRpRK
s
m
α
α α . Let D = the denominator of the fraction of F1(p) or
( )1
2
111
)(
)()()(
−
−
⎥⎦
⎤⎢⎣
⎡ −+
n
n
s
m
sspR
pRpRpR
α
α α α α α . Let N = the numerator of the fraction of F1(p)
or ( ) ( ) )()( 1221 pRpR sm
s α α α α α α +−+ .
=∂
∂
1
1 )(
α
pF
( )[ ]2
1
2
22 )()()(
D
K pRNDpRpR n
n
sm
s−
−
+−+− α α α α
( )
2
11
2
1)(
)()(
1
2)(
D
pR
pRpRK
n
npRN
s
m
n
n
ss ⎥⎦
⎤⎢⎣
⎡ −−−
+
−
−−
−
α α α α
( )[ ] ( )2
1
1
2
1221 )()()(
D
K pRpRpR n
n
ssm
s
α
α α α α α α α −
−
+−+−=
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( ) ( )[ ]2
1
1
2
11221 )()()(
D
K pRpRpR n
n
ssm
s
α
α α α α α α α α −
−
+−++
( ) ( )[ ] ( )
21
12
112211
2)()()(
D
K n
npRpRpR n
n
sssm
s
α
α α α α α α α α α −−
−−++−+
−
( ) ( )
( )01
2)(
11
121
<+
−−
+−+−=
s
ssm
s
Dn
nNpR
α α α
α α α α α α
(24)
( )( ) ( ) ( )( ) ( ) ( )( )( ) ( )s
sssm
ssm
s
Dn
npRnpRpRn
α α α
α α α α α α α α α α α α α α
+−
−+++−++−−+−=
11
11212121
1
2)(2)()(1
( )( ) ( ) ( )( ) ( )
( ) ( )( ) ( )s
ss
s
msss
mss
Dn
npR
Dn
npRpRn
α α α
α α α α α α α
α α α
α α α α α α α α α α α α α α
+−
−+++
+−
−−−−−+−−−=
11
2221
212
11
21
21212
21
2121
1
2)(2
1
2)()(1
( )( ) ( )( )( ) ( )
( )( )
( ) ( )s
ss
s
msss
Dn
pRn
Dn
pRnn
α α α
α α α α α α α
α α α
α α α α α α α α α
+−
−++
+
+−
−−−+−−−=
11
22212
21
11
212
21
2121
1
)(22
1
)(232
(24a)
Thus1
1 )(
α ∂
∂ pF<0 as firm one responds to an increase in its loyal customers α1 with placing
less weight upon lower prices. Expected profits for firm one increase.
QED
Lemma 43: In the modified n firm model 0)(
1
2 <∂
∂
α
pF
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Proof:
0
)(
)()(
)(
)()(
1
1)(1
1
1
11
1
2 <⎥⎦
⎤⎢⎣
⎡ −⎥⎦
⎤⎢⎣
⎡ −
−
−=
∂
∂−
−
pR
pRpR
pR
pRpR
n
pF
s
mn
s
m
α α
α α
α
(28)
Just like the two firm case, the smaller firms in the n firm case compete less aggressively
for shoppers when there is an increase in loyal customers for firm one α1. This is
because firm one is also competing less aggressively. Type two firms in turn place less
weight on lower prices.
QED
Lemma 44: In the modified n firm model 01
>∂∂α
L
Proof:
The proof is exactly identical to Lemma 10.
( )c
LqpRLs
m
++
=)(
)(1
1
α α α
( )
( )0
)(
)()()()(22
1
11
1
>+
−+=
∂∂
Lq
LqpRLqpRL
s
ms
m
α α
α α α
α
QED
Lemma 45: In the modified n firm model
45a) ( ) 0)(11
>∂
−∂−
α
m
pF when αs > α2
45b) ( )
0)(1
1
=∂
−∂ −
α
mpFwhen αs = α2
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45c) ( )
0)(1
1
<∂
−∂ −
α
mpFwhen αs < α2
45d) As n grows in size, it is more likely that ( ) 0)(11
>∂−∂
−
α
m
pF , assuming that αs stays
relatively the same size
Proof:
The proof is almost identical to Lemma 11a- 11c.
⎟⎟ ⎠ ⎞
⎜⎜⎝ ⎛
+−=− −
s
mpFα α
α α
1
21)(1
( ) ( ) ( )
( )
( )
( )21
2
21
211
1
)(1
s
s
s
smpF
α α
α α
α α
α α α α
α +
−=
+
−−+=
∂−∂ −
Generally as n grows larger α2 grows smaller. If αs stays near the same size then
generally)
0)(1
1
>∂
−∂ −
α
mpF.
QED
Lemma 46: In the modified n firm case
46a)2
1 )(
α ∂
∂ pF>0 for all prices in (L, p
m]
46b) 2
1 )(α ∂
∂ pF =0 when p = L
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Proof:
( ) ( )
( ) 1
2
111
12211
)()()()(
)()(1)(
−
−
⎥⎦⎤⎢
⎣⎡ −+
+−+−=
n
n
s
m
ss
sm
s
pRpRpRpR
pRpRpF
α α α α α α
α α α α α α
Changing the loyal customers of firm two α2 gives a similar effect on F1(p) in the
n – firm case as in the two firm case:
( )D
pRpRpF sm )()()( 11
2
1 α α α
α
++−=
∂∂
(25)
At p = pm
: ( ) 0)()()()( 11
2
1 >=++−=∂
∂D
pR
D
pRpRpF ms
ms
m α α α α
α
At p = L:
( )( )
0
)()()( 1
111
2
1 =+
++−
=∂
∂D
pRpRpF
m
s
sm
α α
α α α α
α
Since R(p) is increasing in p,2
1 )(
α ∂
∂ pF>0 for all prices in the (L, p
m] interval and equal to
zero at p = L. This result is not different than the two firm case as the size of the atom at
p = pm
decreases. As the size of the atom decreases, firm one discounts more at lower
prices to attract the shoppers and make up for lost monopoly profits.
QED
Lemma 47: In the modified n firm model 0)(
2
2 =∂
∂α
pF
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Proof:
1
1
112
)(
)()(1)(
−
⎥⎦
⎤⎢⎣
⎡ −−=
n
s
m
pR
pRpRpF
α
α α
( )( )
0)(
0*)()()(*0
)(
)()(
1
1)(22
11
11
1
11
2
2 =⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−⎥⎦
⎤⎢⎣
⎡ −
−−=
∂
∂−
−
pR
pRpRpR
pR
pRpR
n
pF
s
ms
n
s
m
α
α α α
α
α α
α
Changing firm two’s loyal customer base α2 has no effect on the cumulative
probability distribution F2(p) in the modified n firm case. Changing loyal customers for
the smaller firms affects profits of the smaller firms but not the cumulative distribution
function of the smaller firms. This is the same as the two firm case.
QED
Lemma 48: In the modified n firm model 02
=∂∂α
L
Proof:
The proof is identically the same as Lemma 13b.
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
There is no α2 in the expression for L so 02
=∂∂α
L
Lemma 49: In the modified n firm model( )
0)(1
2
<∂
−∂ −
α
mpF
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Proof:
The proof is identically the same as Lemma 14.
⎟⎟ ⎠ ⎞
⎜⎜⎝ ⎛
+−=− −
s
mpFα α
α α
1
21)(1
( )( )
01)(1
12
<+−
=∂
−∂ −
s
mpF
α α α
QED
Lemma 50: In the modified n firm case,
50a) s
pF
α ∂
∂ )(1 > 0 iff
( )[ ] ( ) ( ) ( )[ ] )(326432)(2
)(2
12
12
11
21
21
2pRnnnpRnn
pR
ssm
s
mss
α α α α α α α
α α α α α
+−++−++−+−−
−−<
50b) s
pF
α ∂
∂ )(1 < 0 iff
( )[ ] ( ) ( ) ( )[ ] )(326432)(2
)(2
12
12
11
21
21
2pRnnnpRnn
pR
ssm
s
mss
α α α α α α α
α α α α α
+−++−++−+−−
−−>
50c) s
mpF
α ∂
∂ −)(1 > 0 iff
( ) ( ) ( ) 2211
21
21
2325323 ss
ss
nnn α α α α
α α α α α
−+−+−
+<
As there are more firms or α2 rises for a given number of firms,s
mpF
α ∂
∂ −)(1 is more likely
to be negative
50d) At lower prices,s
LF
α ∂
∂ )(1 > 0 iff ( ) ( ) ( ) 22
11
321
21
233224
2
ss
sss
nnn α α α α
α α α α α α
−+−+−
++<
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50e) Compared tos
mpF
α ∂
∂ −)(1 , there are more possibilities for
s
LF
α ∂
∂ )(1 to be positive,
especially for when n= 3 firms. As α2 rises for a given number of firmss
LF
α ∂
∂ )(1 is more
likely to be negative
Proof:
=∂
∂
s
pF
α
)(1
[ ] ( )2
1
2
121 )(2)()(
D
K pRNDpRpR n
n
sm −
−
++−− α α α α
( )
2
2
111
1
2
1)(
)()(
1
2)(
D
pR
pRpRK
n
npRN
s
m
n
n
ss⎥⎥⎦
⎤
⎢⎢⎣
⎡ −
−−
+
−
−−
−
α
α α α α α
[ ] ( )
2
1
2
121 )()()(
D
K pRpRpR n
n
ssm −
−
+−−=
α α α α α
( ) ( )[ ]( )2
1
2
11221 )(2)()(
D
K pRpRpR n
n
ssm
s−
−
++−++
α α α α α α α α
( ) ( )[ ]( )
2
1
2
112211
2)()()(
D
K n
npRpRpR n
n
ssm
s−
−
−−
++−+−
α α α α α α α α
( ) ( )( )[ ]( )ss
mssss
D
pR
α α α
α α α α α α α α α
+
++++−
=1
11111 )(2
( ) ( )( )[ ] ( )
( )ss
sssss
Dn
nNpR
α α α
α α α α α α α α α α α
+−−
+−++−++
1
1112121
2)(2
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( )ss
msssss
D
pR
α α α
α α α α α α α α α α α α α
+
++++−−=
1
2
121
2
12
2
1
2
1
2
1 )(22
[ ] ( )( )ss
ssssss
D
n
n
NpR
α α α
α α α α α α α α α α α α α α α α α
+−−
+−−−−−++1
1
2
221212
2
1
2
2211
2
)(22
[ ] [ ] ( )
( )ss
sssm
sss
D
n
nNpRpR
α α α
α α α α α α α α α α α α α α α α
+−−
+−−−−+++=
1
1
2
2212
2
1
2
121
2
11
2)(2)(2
[ ] [ ]
( )ss
msss
msss
D
n
npRpR
α α α
α α α α α α α α α α α α α α α α
+−−
−−−−+++=
1
2
121
2
12
2
1
2
121
2
11
2)()(2
[ ] [ ]
( )ss
ssss
D
n
npRpR
α α α
α α α α α α α α α α α α α α
+−−
−−−+−−−+
1
2
2212
2
1
2
2212
2
11
2)(2)(2
( )ss
msss
D
pRn
n
n
n
n
n
n
n
α α α
α α α α α α α α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−
=1
2
2
1
2
121
2
1 )(1
2
1
21
1
22
1
21
( )ss
ss
D
pRn
n
n
n
n
n
α α α
α α α α α α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−
+ 1
2
2212
2
1 )(1
21
1
222
1
21
(26)
For equation (26) to be positive, the numerator must be positive. Simplifying:
>⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−
+⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−⎟ ⎠
⎞⎜⎝
⎛ −−
−
)(1
21
1
222
1
21
)(1
2
1
22
21
212
2
112
pRn
n
n
n
n
n
pRn
n
n
n
ss
ms
α α α α α
α α α α
)(121
121 2121 mss pR
nn
nn ⎥
⎦⎤⎢
⎣⎡ ⎟ ⎠ ⎞⎜⎝ ⎛ −−−−⎟ ⎠ ⎞⎜⎝ ⎛ −−−− α α α α
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32
11
21
21
2
)(
1
2
1
22
)(1
21
1
21
K pR
n
n
n
n
pRn
n
n
n
ms
mss
+⎥
⎦
⎤⎢
⎣
⎡⎟
⎠
⎞⎜
⎝
⎛
−
−−⎟
⎠
⎞⎜
⎝
⎛
−
−−
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−−⎟ ⎠
⎞⎜⎝
⎛ −−
−−
<
α α α
α α α α
α
where )(1
21
1
222
1
21
21
213 pR
n
n
n
n
n
nK ss ⎥
⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−= α α α α
The sign reversal comes because the denominator is negative at all parameter
combinations of α1, αs, R(pm), and R(p) except for the case when α1 is very low and αs is
very high and p is approaching L.
( )[ ] ( ) ( ) ( )[ ] )(326432)(2
)(2
12
12
11
21
21
2pRnnnpRnn
pR
ssm
s
mss
α α α α α α α
α α α α α
+−++−++−+−−
−−< (26a)
At p = pm =∂
∂ −
s
mpF
α
)(1
( )ss
msss
D
pR
n
n
n
n
n
n
n
n
α α α
α α α α α α α α α
+
⎥
⎦
⎤⎢
⎣
⎡⎟
⎠
⎞⎜
⎝
⎛
−
−−−+⎟
⎠
⎞⎜
⎝
⎛
−
−−+⎟
⎠
⎞⎜
⎝
⎛
−
−−+⎟
⎠
⎞⎜
⎝
⎛
−
−−
=1
2
2
1
2
121
2
1 )(
1
221
1
21
1
23
1
21
( )ss
ms
D
pRn
n
α α α
α α
+
⎟ ⎠
⎞⎜⎝
⎛ −−
−−+
1
22 )(
1
21
For s
mpF
α ∂
∂ −)(1 to be positive, the numerator of the above expression must be positive.
Simplifying:
>⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
− )(1
21
1
221
1
23
22112
mss pR
n
n
n
n
n
nα α α α α
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)(1
21
1
21
21
21
mss pR
n
n
n
n⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
+−+⎟ ⎠
⎞⎜⎝
⎛ −−
+− α α α α
22
11
21
21
2
1
21
1
221
1
23
1
211
21
ss
ss
n
n
n
n
n
n
n
n
n
n
α α α α
α α α α α
⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−
⎟ ⎠
⎞⎜⎝
⎛ −−
+−+⎟ ⎠
⎞⎜⎝
⎛ −−
+−<
22
11
21
21
2
1
21
1
221
1
23
1
21
1
21
ss
ss
n
n
n
n
n
n
n
n
n
n
α α α α
α α α α
α
⎟ ⎠
⎞⎜⎝
⎛ −−
++⎟ ⎠
⎞⎜⎝
⎛ −−
++⎟ ⎠
⎞⎜⎝
⎛ −−
⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−<
( ) ( ) ( ) 2211
2
1
2
12
325323 ss
ss
nnn α α α α α α α α α
−+−+−+< (26b)
The following table shows the various parameter combinations of α1, α2, n, and
αs have ons
mpF
α ∂
∂ )(1
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Table 22: Signings
mpF
α ∂
∂ )(1
α1 αs n
α2 RHS of eq (26b)
s
m
pFα ∂∂ )(1
0.05 0.85 3 0.05 0.0165944 Negative
0.1 0.8 3 0.05 0.0327273 Negative
0.2 0.7 3 0.05 0.0614634 Positive ∞
0.7 0.2 3 0.05 0.0504 Positive Æ
0.8 0.1 3 0.05 0.0254417 Negative ∞
0.25 0.35 3 0.20 0.0596591 Negative ∞
0.2 0.7 11 0.01 0.008867 Negative Æ
0.3 0.6 11 0.01 0.0113924 Positive ∞
0.5 0.4 11 0.01 0.011658 Positive Æ
0.6 0.3 11 0.01 0.00972973 Negative ∞
0.1 0.5 11 0.04 0.00470219 Negative ∞
0.2 0.7 101 0.001 0.000834382 Negative Æ
0.3 0.6 101 0.001 0.00106635 Positive ∞
0.5 0.4 101 0.001 0.00108604 Positive Æ
0.6 0.3 101 0.001 0.000906801 Negative ∞
0.1 0.5 101 0.004 0.000443918 Negative ∞
α1 αs n α2 RHS of eq (26b)
s
mpF
α ∂
∂ )(1
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Generally, this derivative at prices near p = pm
is negative as n gets larger. Also if α2
rises for a given number of firms,
s
mpF
α ∂
∂ −)(1 < 0.
The dynamics of s
pF
α ∂∂ )(1 change as prices fall toward p = L:
=∂
∂
s
LF
α
)(1
( )ss
msss
D
pRn
n
n
n
n
n
n
n
α α α
α α α α α α α α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−
1
2
2
1
2
121
2
1 )(1
2
1
21
1
22
1
21
( )( )ss
m
s
ss
D
pR
n
n
n
n
n
n
α α α α α
α α α α α α α α
++
⎥⎦
⎤⎢⎣
⎡⎟
⎠
⎞⎜
⎝
⎛
−
−−−+⎟
⎠
⎞⎜
⎝
⎛
−
−−−+⎟
⎠
⎞⎜
⎝
⎛
−
−−−
+1
1
12
2212
2
1 )(
1
21
1
222
1
21
( )2
1
2
3
1
22
12
2
1
3
1 )(1
221
1
21
1
23
1
21
ss
msss
D
pRn
n
n
n
n
n
n
n
α α α
α α α α α α α α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−
=
( )2
1
3
1
2
21
22
1
2
21 )(1
21
1
22
1
21
1
21
ss
mssss
D
pRn
n
n
n
n
n
n
n
α α α
α α α α α α α α α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−
+
( )2
1
2
2
1 )(1
2
ss
ms
D
pRn
n
α α α
α α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−
+
s
LF
α ∂
∂ )(1 >0 if the numerator of the above expression is greater than zero. Simplifying:
)(1
2
1
22
1
21
1
221
1
23
21
21
21
31
212
mssss pR
n
n
n
n
n
n
n
n
n
n⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
− α α α α α α α α α α
)(1
21
1
222
1
21
31
221
31
msss pR
n
n
n
n
n
n⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
+−+⎟ ⎠
⎞⎜⎝
⎛ −−
+−+⎟ ⎠
⎞⎜⎝
⎛ −−
+−> α α α α α α
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ssss
sss
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
α α α α α α α α α
α α α α α α
α 2
12
12
13
12
1
3
1
22
1
3
1
2
1
2
1
22
1
21
1
221
1
23
1
21
1
222
1
21
⎟ ⎠
⎞⎜⎝
⎛ −−
−⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−
⎟ ⎠
⎞⎜⎝
⎛ −−
+−+⎟ ⎠
⎞⎜⎝
⎛ −−
+−+⎟ ⎠
⎞⎜⎝
⎛ −−
+−<
2211
32
1
2
1
2
1
221
1
221
1
24
1
21
1
222
1
21
ss
sss
n
n
n
n
n
n
n
n
n
n
n
n
α α α α
α α α α α
α
⎟ ⎠
⎞⎜⎝
⎛ −−
+−+⎟ ⎠
⎞⎜⎝
⎛ −−
++⎟ ⎠
⎞⎜⎝
⎛ −−
⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−<
( ) ( ) ( ) 2211
321
21
233224
2
ss
sss
nnn α α α α
α α α α α α
−+−+−
++< (26c)
The net difference betweens
LF
α ∂
∂ )(1
ands
mpF
α ∂
∂ )(1
is that the numerator of (26c) contains
more terms than the numerator of (26b). The net result of this is there is more of a
chance thats
pF
α ∂
∂ )(1 is positive at prices near L, especially if the proportion of shoppers
αs is large and the proportion of loyal customers α1 to firm one is small. Table 23 shows
the various parameter combinations of α1, α2, n, and αs have on s
LF
α ∂
∂ )(1
Notice that there
are more positive values for s
LF
α ∂
∂ )(1 compared tos
mpF
α ∂
∂ )(1 . The largest firm is more
likely to discount at low prices when the shoppers are large than at high prices. The
probability of winning shoppers at low prices is much higher. If α2 rises for a given
number of firms, s
LF
α ∂
∂ )(1
< 0.
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Table 23: Signings
LF
α ∂
∂ )(1
α1 αs n α2 RHS of eq (26c)
s
LF
α ∂
∂ )(1
0.7 0.2 3 0.05 0.079803 Positive Æ
0.8 0.1 3 0.05 0.0361607 Negative ∞
0.25 0.35 3 0.20 0.234419 Positive
0.3 0.3 3 0.20 0.171429 Negative ∞
0.7 0.2 11 0.01 0.0110429 Positive Æ
0.8 0.1 11 0.01 0.00535714 Negative ∞
0.1 0.5 11 0.04 0.0451128 Positive
0.2 0.4 11 0.04 0.0292683 Negative ∞
0.7 0.2 101 0.001 0.0010327 Positive Æ
0.8 0.1 101 0.001 0.000506187 Negative ∞
0.1 0.5 101 0.004 0.00388853 Negative ∞
α1 αs n α2 RHS of eq (26c)
s
LF
α ∂
∂ )(1
QED
Changing the shoppers αs in the modified n firm case has the same effect on F2(p)
as the two firm case:
Lemma 51: In the modified n firm case 0)(2 >
∂
∂
s
pF
α
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Proof:
0
)(
)()(
)(
)()(
1
1)(2
11
11
1
112 >
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ +−⎥⎦
⎤⎢⎣
⎡ −
−
−=
∂
∂−
−
pR
pRpR
pR
pRpR
n
pF
s
mn
s
m
s α
α α
α
α α
α
(27)
Like the two firm case, the (n -1) smaller firms discount more when there are more
shoppers.
The discounting is not as large when the number of firms increases.
QED
Lemma 52: In the modified n firm model 0<∂∂
s
L
α
Proof:
The proof is identical to that of Lemma 17.
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
( ) ( )0
)(
)(
)(
)()(2
1
1
221
1 <+
−=
+
−=
∂∂
Lq
pR
Lq
LqpRL
s
m
s
m
s α α
α
α α
α
α
QED
Lemma 53: In the modified n firm model( )
0)(1
<∂
−∂ −
s
mpF
α
Proof:
The proof is identical to that of Lemma 18.
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⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
( ) ( )( )0)(1 2
1
21 <+−−=∂−∂
−
ss
m
pFα α α α
α QED
Varying the number of firms also affects F1(p).
Lemma 54: 0)(1 <
∂
∂
n
pF
Proof:
( ) ( )[ ]( )
( )1
2
111
2
111221
1
)(
)()()(
1
1
)(
)()(ln)()(
)(
−
−
⎥⎦
⎤⎢⎣
⎡ −+
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
−
−⎥⎦
⎤⎢⎣
⎡ −+−+
−=∂
∂
n
n
s
m
ss
s
m
sm
s
pR
pRpRpR
npR
pRpRpRpR
n
pF
α
α α α α α
α
α α α α α α α α
(28)
0)(1 <
∂
∂=
n
pF
With the increased smaller firms clamoring over the shoppers, firm one finds it harder to
win the shoppers. Assuming that the increase in firms does not take away market share
from the shoppers αs and the loyal customers for firm one α1, the individual share of type
two loyal customers α2 falls. This falling of the type two individual firm’s share of loyal
customers α2 causes the atom ⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1 for firm one at p = pm
to get
larger. Thus firm one places less weight on prices competing with the smaller firms and
instead concentrates more of its randomization at its atom at p = pm
.
QED
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Changing the number of firms also affects F2(p):
Lemma 55: 0)(2 <
∂
∂n
pF
Proof:
( )0
)(
)()(ln
)(
)()(
1
1)( 111
1
11
2
2 <⎥⎦
⎤⎢⎣
⎡ −⎥⎦
⎤⎢⎣
⎡ −
−
−−=
∂
∂ −
pR
pRpR
pR
pRpR
nn
pF
s
mn
s
m
α
α α
α
α α (29)
The probability of winning the shoppers decreases at every price for the smaller firms
when there are more of them. Instead the smaller firms focus more on maximizing
revenue from their loyal customers.
QED
Lemma 56: In the modified n firm case 0=∂∂n
L
Proof:
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
There is no n in L thus 0=∂∂n
L.
QED
Lemma 57: If ( ) 21α −n is constant,( )
0)(1
>∂
−∂ −
n
pF m
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Proof:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
If n increases when ( ) 21α −n is constant, then a2 is falling. A falling a2 implies the
numerator of the atom is increasing. Thus the atom increases. The reverse case of n
falling also holds.
QED
The same exercise in the two firm model can be performed in the modified n firm
model where one consumer group is held constant and the other two are adjusted upward
and downward by the same proportion to see the effects on the firms’ cumulative
probability distribution functions.
Lemma 58:
58a) ( ) s
pF
n
pF
α α ∂
∂
−−∂
∂ )(
1
)( 1
2
1
is positive iff
( )[ ] ( ) ( ) ( )[ ] )(326432)(2
)(2)(222
12
12
11
321
21
21
21
2pRnnnpRnn
pRpR
ssm
s
sssm
ss
α α α α α α α
α α α α α α α α α α
−+−+−+−+−
−−−++>
58b) ( )s
pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 is negative iff
( )[ ] ( ) ( ) ( )[ ] )(326432)(2
)(2)(222
1
2
1
2
11
321
21
21
21
2
pRnnnpRnn
pRpR
ss
m
s
sssm
ss
α α α α α α α
α α α α α α α α α α
−+−+−+−+−
−−−++<
58c) At high prices near p= pm, ( )s
pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 >0
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58d) At low prices near p=L, ( )s
pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 <0 so long as as is not relatively
small compared to na2.
Proof:
( ) ( ) ( )( )ss
ssm
sssm
D
pRpR
D
pRpRpF
α α α
α α α α α α α α α α
α +
+++−=
++−=
∂
∂
1
211111
2
1 )()()()()(
( ) ( )( )ss
sssm
ss
D
pRpRpF
α α α
α α α α α α α α α
α +
+++−−=
∂
∂
1
321
21
21
21
2
1 )(2)()(
( )[ ] ( ) ( ) ( )[ ]{ }( ) ( )ss
ssm
s
s
Dn
K pRnnnpRnn
pF
α α α
α α α α α α α α
α
+−
++−++−++−+−−
=∂
∂
1
42
12
12
112
1
1
)(326432)(2
)(
where )(2
1
2
14m
ss pRK α α α α +=
( )
( )[ ] ( ) ( ) ( )[ ]{ }( )ss
ssm
s
s
D
K pRnnnpRnn
pFn
α α α
α α α α α α α α
α
+
−+−++−++−+−−−
=∂
∂−−
1
4
2
1
2
1
2
112
1
)(326432)(2
)(1
( )
( ) ( )( )
( )[ ] ( ) ( ) ( )[ ]{ }( )ss
ssm
s
ss
sssm
ss
s
D
K pRnnnpRnn
D
pRpR
pFn
pF
α α α
α α α α α α α α
α α α
α α α α α α α α α
α α
+
−+−++−++−+−−−+
+
+++−−
=∂
∂−−
∂
∂
1
4
2
1
2
1
2
112
1
32
1
2
1
2
1
2
1
1
2
1
)(326432)(2
)(2)(
)(1
)(
( ) ( ) ( ) ( ){ }( )ss
ssm
s
D
K pRnnnpRnn
α α α
α α α α α α α α
+
−+−++−++−+−−−=
1
5
2
1
2
1
2
112 )(326432)(2(30)
where ( ) ( ) )(2)(2232
12
12
12
15 pRpRK sssm
ss α α α α α α α α α −−−++=
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( )s
pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 is positive when the numerator of the fraction of equation (30) is
positive. The numerator of the fraction of equation (30) is positive when the -α2 term is
bigger than –K 5.
Or:
( )[ ] ( ) ( ) ( )[ ] )(326432)(2
)(2)(222
12
12
11
321
21
21
21
2pRnnnpRnn
pRpR
ssm
s
sssm
ss
α α α α α α α
α α α α α α α α α α
−+−+−+−+−
−−−++> (30a)
The sign is kept to greater than because the denominator is generally positive, except
when α1 is low, αs is high, and p is approaching L.
At p=pm, equation (30a) reduces to
( ) ( ) ( ) 21
21
321
2326353 ss
ss
nnn α α α α
α α α α
−+−+−
−> (30b)
For high prices near p=pm, equation (30b) holds no matter how many firms are present.
All calculations in Mathematicaresult in ( )s
mm pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 > 0. Thus at high
prices, ( )s
pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 > 0.
At p=L and low prices just above that price equation (30a) reduces to:
[ ]( ) [ ]( )[ ]( ) ( ) ( ) ( )[ ] )(326432)(2
)(2)(222
12
13
112
11
31
221
311
21
21
2
mss
mss
msss
msss
pRnnnpRnn
pRpR
α α α α α α α α α α
α α α α α α α α α α α α
α
−+−+−++−+−
−−−+++
>
( ) ( ) ( ) 21
21
31
31
221
21
238453
2
ss
sss
nnn α α α α α
α α α α α α α
−+−+−
++>
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( ) ( ) ( ) 21
21
3211
238453
2
ss
sss
nnn α α α α
α α α α α α
−+−+−
++> (30c)
The story changes at low prices. There are more negative values of
( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 compared to ( )s
mm pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 . Firm one randomizes
more at high prices than at low prices under these conditions. Table 14 shows the various
parameter combinations of α1, α2, n, and αs have on ( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1 .
Generally, ( ) s
LF
n
LF
α α ∂
∂−−∂
∂ )(
1
)( 1
2
1
< 0 so long as as is not relatively small compared to
na2. In the case as is relatively small compared to na2, there is only a small range of
prices that are in randomization. As will be shown in a later proposition, L increases
under these circumstances and dominates the effect on F1(p). Thus ( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1
> 0.
Table 24 Signing ( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1
α1 αs n α2 RHS of eq (30c)( )
s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1
0.7 0.2 3 0.05 0.0809524 Negative Æ
0.8 0.1 3 0.05 0.0336806 Positive ∞
0.3 0.3 3 0.20 0.2375 Negative Æ
0.4 0.2 3 0.20 0.125 Positive ∞
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0.7 0.2 11 0.01 0.0106918 Negative Æ
0.8 0.1 11 0.01 0.00464559 Positive ∞
0.1 0.5 11 0.04 0.0551471 Negative
0.2 0.4 11 0.04 0.0393939 Positive ∞
0.6 0.3 101 0.001 0.00168108 Negative Æ
0.7 0.2 101 0.001 0.000993281 Positive ∞
0.1 0.5 101 0.004 0.00475888 Negative
0.2 0.4 101 0.004 0.00350877 Positive ∞
α1 αs n α2 RHS of eq (30c)( )
s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
2
1
QED
Lemma 59:
59a)s
pFpF
α α ∂
∂−∂
∂ )()(1
1
1 > 0 iff
( ) ( )
( ) ( ) ( )[ ] ( ) ( )[ ] )(7485)(22232
)(132
6
2
1
2
1
3
1
2
1
2
1
22
1
3
1
3
12
pRK nnpRnnn
pRnn
ssm
ss
msss
+−+−+−+−−−−
−+−+>
α α α α α α α α α
α α α α α α α
59b)s
pFpF
α α ∂
∂−
∂
∂ )()( 1
1
1 < 0 iff
( ) ( )
( ) ( ) ( )[ ] ( ) ( )[ ] )(7485)(22232
)(132
6
2
1
2
1
3
1
2
1
2
1
22
1
3
1
3
12
pRK nnpRnnn
pRnn
ss
m
ss
msss
+−+−+−+−−−−
−+−+<
α α α α α α α α α
α α α α α α α
where ( ) ( )3223
13
6 −+−= nnK s α α
59c) Generallys
pFpF
α α ∂
∂−
∂
∂ )()( 1
1
1 < 0 except when αs is very small relative to nα2
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59d) s
pFpF
α α ∂
∂−
∂
∂ )()( 1
1
1 < 0 for a greater range of parameter values as n increases
Proof:
( )( ) ( )( )( ) ( )
( )( )( ) ( )s
ss
s
msss
Dn
pRn
Dn
pRnnpF
α α α
α α α α α α α
α α α
α α α α α α α α α
α
+−
−+++
+−
−−−+−−−=
∂
∂
11
22212
21
11
212
21
2121
1
1
1
)(22
1
)(232)(
( )( ) ( )( )
( ) ( )( )( )( ) ( )ss
sss
ss
mssss
Dn
pRnDn
pRnnpF
α α α α
α α α α α α α α α α α α
α α α α α α α α α α
α
+−
−+++
+−
−−−+−−−=
∂
∂
11
3
2
2
212
2
1
11
22
12
2
1
3
1
2
21
1
1
1
)(221
)(232)(
( )[ ] ( ) ( ) ( )[ ]{ }( ) ( )ss
ssm
s
s
Dn
K pRnnnpRnn
pF
α α α
α α α α α α α α
α
+−
−+−++−++−+−−−
=∂
∂−
1
4
2
1
2
1
2
112
1
1
)(326432)(2
)(
where )(2
1
2
14m
ss pRK α α α α +=
( )[ ] ( ) ( ) ( )[ ]{ }( ) ( )
[ ]( ) ( )ss
mss
ss
ssm
s
s
Dn
pR
Dn
pRnnnpRnn
pF
α α α α
α α α α
α α α α
α α α α α α α α α
α
+−
−−+
+−
−+−+−+−+−
=∂
∂−
11
221
31
11
21
21
31
31
212
1
1
)(
1
)(326432)(2
)(
( ) ( ) ( )( ){ }( ) ( )
( ) ( )[ ]( ) ( )ss
mss
ss
sssm
ss
Dn
pRnn
Dn
pRnpRnnpF
α α α α
α α α α
α α α α
α α α α α α α α α α
α
+−
−−−−+
+−
−+++−−−−=
∂
∂
11
221
31
11
321
21
21
212
1
1
1
)(232
1
)(22)(232)(
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( ) ( ) ( )[ ]{ }
( ) ( )( ) ( ) ( ) ( )[ ]{ }
( ) ( )
( ) ( )[ ]( ) ( )ss
msss
ss
sss
ss
mss
s
Dn
pRnn
Dn
pRnnnnDn
pRnnn
pFpF
α α α α
α α α α α α
α α α α
α α α α α α α α α α α
α α α α α α
α α
+−
−−−−−+
+−
−+−+−+−+
+−
−+−−−−
=∂
∂−
∂
∂
11
22
1
3
1
3
1
11
31
321
212
11
3
1
2
1
2
12
1
1
1
1
)(132
1
)(32274851
)(22232
)()(
(31)
s
pFpF
α α ∂
∂−
∂
∂ )()( 1
1
1 is positive if the numerator in equation (31) is positive. Rearranging:
( ) ( ) ( ){ }( ) ( ) ( ) ( )[ ]{ }
( ) ( )[ ] )(132
)(3227485
)(22232
221
31
31
3
1
32
1
2
12
31
21
212
msss
sss
mss
pRnn
pRnnnn
pRnnn
−+−+
>−+−+−+−+
−+−−−−
α α α α α α
α α α α α α α
α α α α α α
( ) ( )
( ) ( ) ( )[ ] ( ) ( )[ ] )(7485)(22232
)(132
6
2
1
2
1
3
1
2
1
2
1
22
1
3
1
3
12
pRK nnpRnnn
pRnn
ssm
ss
msss
+−+−+−+−−−−
−+−+>
α α α α α α α α α
α α α α α α α
where ( ) ( )3223
13
6 −+−= nnK s α α (31a)
At p= pm, equation (31a) reduces down to
( ) ( )
( ) ( ) ( ) ( ) 321
21
31
221
31
31
22426353
132
sss
sss
nnnn
nn
α α α α α α
α α α α α α α
−+−+−+−
−+−+> (31b)
Table 25 shows the various parameter combinations of α1, α2, n, and αs have on
( )s
mm pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1 . Many values of ( )s
mm pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1 are negative and
increasingly so as the number of firms rises. With an increase in its loyal customers and
decrease in shoppers, firm one (as will be shown in a future lemma) increases the size of
its atom and decreases its randomization at high prices. With a higher number of firms,
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this trend ( ) 0)(
1)( 1
1
1 <∂
∂−−
∂
∂
s
pFn
pF
α α increases at high prices for more parameter values
due to it being more competitive to win the shoppers. Firm one will increasingly
randomize less at these higher prices with more firms because the odds of winning the
shoppers are less with more firms.
Table 25 Signing ( )s
mm pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1 and ( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1
α1 αs n α2 RHS of eq (31b) ( )s
mm
pFnpFα α ∂
∂−−∂
∂ )(1)( 1
1
1
( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1
0.7 0.2 3 0.05 0.0720231 Negative Æ
0.8 0.1 3 0.05 0.0294196 Positive ∞
0.25 0.35 3 0.20 0.227933 Negative
0.3* 0.3 3 0.20 0.18 Positive ∞
0.7 0.2 11 0.01 0.0289256 Negative Æ
0.8 0.1 11 0.01 0.00804094 Positive ∞
0.4 0.2 11 0.04 0.0456233 Negative Æ
0.5 0.1 11 0.04 0.0109967 Positive ∞
0.85 0.05 101 0.001 0.00119727 Negative Æ
0.89 0.01 101 0.001 0.0000713002 Positive
0.5 0.1 101 0.004 0.00791015 Negative Æ
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0.55 0.05 101 0.004 0.00179601 Positive ∞
α1 αs n α2 RHS of eq (31b)( )
s
mm pFn
pF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1
( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1
* negative for ( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1
At prices near p = L equation (31) can be modified to:
( ) ( ) ( )
( ) ( ) ( )[ ]( ) *613
12
12
1
122
13
13
12
)(22232
)(132
K pRnnn
pRnnm
sss
mssss
++−+−−−−
+−+−+
> α α α α α α α
α α α α α α α α
α
where ( ) ( ) ( ) ( ) )(32274854
13
122
13
16*
msss pRnnnnK −+−+−+−= α α α α α α α
( ) ( )
( ) ( ) ( )153842
32433
14
13
122
1
41
231
321
41
2−−−+−+−
−++−+>
nnn
nnn
sss
ssss
α α α α α α α
α α α α α α α α α
( ) ( )
( ) ( ) ( )153842
324333
12
12
1
4221
31
31
2−−−+−+−
−++−+>
nnn
nnn
sss
ssss
α α α α α α
α α α α α α α α (31c)
When the denominator is negative at high values αs, the greater than sign in equation
(31c) reverses to a less than sign since there is division by a negative number.
( ) ( )
( ) ( ) ( )153842
324333
12
12
1
4221
31
31
2−−−+−+−
−++−+<
nnn
nnn
sss
ssss
α α α α α α
α α α α α α α α (31d)
Table 25 above shows the various parameter combinations of α1, α2, n, and αs have on
( )s
mm pFn
pF
α α ∂∂−−
∂∂ )(
1)( 1
1
1 . The signs of ( )s
LFn
LF
α α ∂∂−−
∂∂ )(
1)( 1
1
1 for the parameter values
given are the same except for one set of values marked by an asterisk. Rather than
reprinting another table, it is easier to refer to Table 25 to check the signs of
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( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1 . Again most parameters are negative except when αs is relatively
small to nα2. As n increases, there is a greater range of parameters that support
( )s
LFn
LF
α α ∂
∂−−
∂
∂ )(1
)( 1
1
1 < 0.
QED
Lemma 60: 2
1
1
1 )()()1(
α α ∂
∂−
∂
∂−
pFpFn < 0
Proof:
By lemma 42 0)(
1
1 <∂
∂
α
pF.
By lemma 46 0)(
2
1 ≥∂
∂
α
pF.
Thus2
1
1
1 )()()1(
α α ∂
∂−
∂
∂−
pFpFn < 0.
QED
Lemma 61: 0)()( 2
1
2 <∂
∂−
∂
∂
s
pFpF
α α
Proof:
0)(
)()(
)(
)()(
1
1)(1
1
1
11
1
2 <⎥⎦
⎤⎢⎣
⎡ −⎥⎦
⎤⎢⎣
⎡ −
−−=
∂
∂−
−
pR
pRpR
pR
pRpR
n
pF
s
mn
s
m
α α
α α
α
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0)(
)()(
)(
)()(
1
1)(2
11
11
1
112 >⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −
−−=
∂
∂−
−
pR
pRpR
pR
pRpR
n
pF
s
mn
s
m
s α
α α
α
α α
α
0)(
)()(
)(
)()(
)(
)()(
1
1
)()(
2
11
2
2211
1
11
2
1
2
<⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −+
⎥⎥⎦
⎤
⎢⎢⎣
⎡ −⎥⎦
⎤⎢⎣
⎡ −
−−
=∂
∂−∂
∂
−−
pR
pRpR
pR
pRpR
pR
pRpR
n
pFpF
s
m
s
sm
sn
s
m
s
α
α α
α
α α
α
α α
α α
QED
Lemma 62:
62a) 0)()( 2
2
2 <∂
∂−
∂
∂
s
pFpF
α α
62b) 0)()(
2
2
1
2 <∂
∂−
∂
∂
α α
pFpF
Proof:
By Lemma 47 0
)(
2
2
=∂
∂
α
pF
By Lemma 51 0)(2 >
∂
∂
s
pF
α
Thus 0)()( 2
2
2 <∂
∂−
∂
∂
s
pFpF
α α
By Lemma 43 0)(
1
2 <∂
∂
α
pF
Thus 0)()(
2
2
1
2 <∂
∂−
∂
∂
α α
pFpF
QED
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302
Lemma 63:
63a) 01
>∂∂
−∂∂
s
LL
α α
63b) 02
>∂∂
−∂∂
s
LL
α α
63c) 021
>∂∂
−∂∂
α α
LL
Proof (63a):
( )c
Lq
pRL
s
m
++
=)(
)(
1
1
α α
α
From Lemma 44:
( )
( )0
)(
)()()()(22
1
11
1
>+
−+=
∂∂
Lq
LqpRLqpRL
s
ms
m
α α
α α α
α
From Lemma 52:
( )0
)(
)(2
1
1 <+
−=
∂
∂
Lq
pRL
s
m
s α α
α
α
( )
( ) 221
111
1 )(
)()()()()()(
Lq
LqpRLqpRLqpRLL
s
mms
m
s α α
α α α α
α α +
+−+=
∂∂
−∂∂
( )
( )0
)(
)()(22
1
1 >+
+=
Lq
LqpR
s
sm
α α
α α
Proof (63b):
From Lemma 48:
02
=∂∂α
L
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From Lemma 52:
( )0
)(
)(2
1
1 <+
−=
∂
∂
Lq
pRL
s
m
s α α
α
α
Thus 02
>∂∂
−∂∂
s
LL
α α
Proof (63c):
From Lemma 44:
( )
( )0
)(
)()()()(22
1
11
1
>+
−+=
∂∂
Lq
LqpRLqpRL
s
ms
m
α α
α α α
α
From Lemma 48:
02
=∂∂α
L
Thus 021
>∂∂
−∂∂
α α
LL
QED
Lemma 64:
64a) ( ) ( )
0)(1)(1
1
>∂
−∂−
∂−∂ −−
s
mm pFpF
α α if α1+αs-2α2 > 0
64b) ( ) ( )
0)(1)(1
1
<∂
−∂−
∂−∂ −−
s
mm pFpF
α α if α1+αs-2α2 < 0
Proof:
⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+
−=− −
s
mpFα α
α α
1
21)(1
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From Lemma 45:
( ) ( ) ( )
( )
( )
( )21
2
21
211
1
)(1
s
s
s
smpF
α α
α α
α α
α α α α
α +
−=
+
−−+=
∂−∂ −
From Lemma 53:
( ) ( )
( )0
)(12
1
21 <+
−−=
∂−∂ −
ss
mpF
α α
α α
α
( ) ( ) ( ) ( )
( ) ( )21
21
21
212
1
2)(1)(1
s
s
s
s
s
mm pFpF
α α
α α α
α α
α α α α
α α +
−+=
+
−+−=
∂−∂
−∂
−∂ −−
QED
Lemma 65:
65a)( ) ( )
0)(1)(1
21
>∂
−∂−
∂−∂ −−
α α
mm pFpFif α1+2αs-α2 > 0
65b) ( ) ( )
0)(1)(1
21
<∂
−∂−
∂−∂ −−
α α
mm pFpFif α1+2αs-α2 < 0
Proof:
From Lemma 45:
( ) ( ) ( )
( )
( )
( )21
2
21
211
1
)(1
s
s
s
smpF
α α
α α
α α
α α α α
α +
−=
+
−−+=
∂−∂ −
From Lemma 49:
( ) ( )( ) ( )
01)(1
12
1
1
2
<+−=
++−=
∂−∂
−
ss
sm
pFα α α α
α α α
( ) ( ) ( ) ( )
( ) ( )21
21
21
12
21
2)(1)(1
s
s
s
ssmm pFpF
α α
α α α
α α
α α α α
α α +
−+=
+
++−=
∂−∂
−∂
−∂ −−
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305
Lemma 66:( ) ( )
0)(1)(1
2
<∂
−∂−
∂−∂ −−
s
mm pFpF
α α
Proof:
From Lemma 49:
( ) ( )
( ) ( )0
1)(1
12
1
1
2
<+−
=+
+−=
∂−∂ −
ss
smpF
α α α α
α α
α
From Lemma 53:
( ) ( )( )
0)(12
1
21 <+−−=
∂−∂ −
ss
mpFα α
α α α
( ) ( ) ( ) ( )
( )
( )
( )0
)(1)(12
1
2
21
211
2
<+
+−=
+
−++−=
∂−∂
−∂
−∂ −−
s
s
s
s
s
mm pFpF
α α
α α
α α
α α α α
α α
QED
Like the two firm model, the probability of the minimum price can be created and
comparative statistics can be performed on this probability. The probability of the
minimum price is ( )( ) 1
2121)(1)(11)(1 1
−
∩−−−=− −
npFpFpF n
( ) ( )
( )
1
1
1
11
1
2
111
1221
21 )(
)()(
)(
)()()(
)()(1)(1 1
−
−
−
−∩⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ −
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ −+
+−+−=− −
n
n
s
m
n
n
s
m
ss
sm
s
pR
pRpR
pR
pRpRpR
pRpRpF n
α
α α
α
α α α α α
α α α α α α
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( ) ( )
( )
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
⎟⎟⎟
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎜⎜⎜⎜
⎝
⎛
⎥⎦⎤⎢
⎣⎡ −+
+−+−=−
−
−∩ −
)(
)()(
)()()()(
)()(1)(1 11
1
2
111
1221
21 1
pR
pRpR
pRpRpRpR
pRpRpF
s
m
n
n
s
m
ss
sm
sn
α
α α
α α α α α α
α α α α α α
The difference between )(1 121pF n−∩
− in the modified n firm model and )(1 21 pF ∩− in the
two firm model is the extra1
2
11
)(
)()( −
−
⎥⎦
⎤⎢⎣
⎡ − n
n
s
m
pR
pRpR
α
α α term in the denominator in
)(1 121pF n−∩
− . The same comparative statics exercises can be performed on this new
probability of at least one price being lower.
Lemma 67: q
pF n
∂
−∂ −∩)(1 121 < 0
Proof:
( ) ( )
( )
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ −+
+−+−=−
−
−∩ −)(
)()(
)(
)()()(
)()(1)(1 11
1
2
111
1221
21 1
pR
pRpR
pR
pRpRpR
pRpRpF
s
m
n
n
s
m
ss
sm
sn
α
α α
α
α α α α α
α α α α α α
Let( ) ( )
( ) ⎟⎟⎟
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎜⎜⎜⎜
⎝
⎛
⎥⎦⎤⎢
⎣⎡ −+
+−+=
−
−
1
2
111
1221
)()()()(
)()(
n
n
s
m
ss
sm
s
pRpRpRpR
pRpRA
α α α α α α
α α α α α α
Let ⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −=
)(
)()( 11
pR
pRpRB
s
m
α
α α
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( ) ( )BApF n −=− −∩1)(1 121
Aq
BB
q
A
q
pF n
∂∂
−∂∂
−=∂
−∂ −∩)(1 121
( )0
0
2
0
35)(1 21 <⎟⎟ ⎠
⎞⎜⎜⎝
⎛
<+⎟⎟
⎠
⎞⎜⎜⎝
⎛
<=
∂
−∂ ∩ ALemmaBy
BLemmaBy
q
pF.
QED
Lemma 68: )
c
pF n
∂
−∂ −∩)(1 121 < 0
( ) ( )
( )
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ −+
+−+−=−
−
−∩ −)(
)()(
)(
)()()(
)()(1)(1 11
1
2
111
1221
21 1
pR
pRpR
pR
pRpRpR
pRpRpF
s
m
n
n
s
m
ss
sm
sn
α
α α
α
α α α α α
α α α α α α
Again let( ) ( )
( ) ⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ −+
+−+=
−
−
1
2
111
1221
)(
)()()(
)()(
n
n
s
m
ss
sm
s
pR
pRpRpR
pRpRA
α
α α α α α
α α α α α α
Let ⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −=
)(
)()( 11
pR
pRpRB
s
m
α
α α
( ) ( )BApF n −=− −∩1)(1 121
Aq
BB
q
A
c
pF n
∂∂
−∂∂
−=∂
−∂ −∩)(1 121
( )0
0
5
0
38)(1 21 <⎟⎟ ⎠
⎞⎜⎜⎝
⎛ <
+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ <
=∂
−∂ ∩ ALemmaBy
BLemmaBy
c
pF
QED
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Suppose that firm one’s loyal customers α1 are changed. The effect on )(1 121pF n−∩
− is:
Lemma 69:1
21)(1 1
α ∂
−∂ −∩pF n
<0
Proof:
( )⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∂
∂=
∂
−∂ −∩
)(
)()()()(111
1
1
1
21 1
pR
pRpRpFpF
s
mn
α
α α
α α
( ) ( )
( )
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ −+
+−+−
−
− )(
)()(
)(
)()()(
)()(
1
2
111
1221
pR
pRpR
pR
pRpRpR
pRpR
s
m
n
n
s
m
ss
sm
s
α
α
α α α α α
α α α α α α (32)
By lemma 42 0)(
1
1 <∂
∂
α
pF.
Since both1
1 )(
α ∂
∂ pFand
1
2 )(
α ∂
∂ pFare negative,
1
21)(1 1
α ∂
−∂ −∩pF n
is negative.
Expanding and simplifying equation (32):
( )( ) ( )( )( ) ( )
( )( )( ) ( ) ⎟
⎟
⎠
⎞⎜⎜
⎝
⎛ −
+−
−+++
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
+−
−−−+−−−=
)(
)()(
1
)(22
)(
)()(
1
)(232
11
11
22212
21
11
11
212
21
2121
pR
pRpR
Dn
pRn
pR
pRpR
Dn
pRnn
s
m
s
ss
s
m
s
msss
α
α α
α α α
α α α α α α α
α
α α
α α α
α α α α α α α α α
( ) ( )( ) ( ) ( )( ) ( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛ −⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+−
+−−++−−
)(
)()(
1
)(1)(1
11
2
12112
2
1
pR
pRpR
Dn
pRnpRn
s
m
s
sm
ss
α α α α
α α α α α α α α α
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( )( ) ( )( )( ) ( )
( )( ) ( )( )[ ]
( ) ( )( )( )
( ) ( )
( )( )( ) ( )ss
ss
ss
mss
ss
msss
ss
msss
Dn
pRn
Dn
pRpRnpRDn
pRpRnn
pRDn
pRnn
α α α α
α α α α α α α α
α α α α
α α α α α α α α
α α α α
α α α α α α α α α
α α α α
α α α α α α α α α
+−
−++−
+−
−+++
+−
−++−++
+−
−−−+−−−=
11
22212
212
31
11
2212
212
31
11
312
31
2212
21
11
23
12
3
1
22
12
2
1
1
)(22
1
)()(22
)(1
)()(232
)(1
)(232
( )( )( ) ( )
( )( )
( ) ( )( )( )
( ) ( )
( )( )( ) ( ) )(1
)(12
)(1
)()(1
)(1
)()(12
)(1
)(1
11
22212
212
31
11
221
312
212
31
11
2212
212
31
11
222
1
3
12
2
12
3
1
pDRn
pRn
pDRn
pRpRn
pDRn
pRpRn
pDRn
pRn
ss
ss
ss
msss
ss
mss
ss
msss
α α α α
α α α α α α α α
α α α α
α α α α α α α α α
α α α α
α α α α α α α α
α α α α
α α α α α α α α α
+−
−++−
+−
−−−−−−
+−
−−−−−
+−
−+++−
( )( ) ( )( )( ) ( ) )(1
)(3243
11
2312
31
2212
21
pRDn
pRnn
ss
msss
α α α α
α α α α α α α α α
+−
−−−+−−−=
( )( ) ( )( ) ( )( ) ( )( )( ) ( ) )(1
)()(326443107
11
221
312
31
2212
21
pRDnpRpRnnnn
ss
mssss
α α α α α α α α α α α α α α α α
+−−++−+−+−+
( )( )( ) ( ) )(1
)(322
11
22212
212
31
pRDn
pRn
ss
ss
α α α α
α α α α α α α α
+−
−++− (32a)
QED
Suppose that firm two’s loyal customers α2 are changed. The effect on )(1 121pF n−∩
− is:
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Lemma 70:
70a) 2
21)(1 1
α ∂
−∂ −∩pF n
> 0 for p in [L, pm
)
70b) 2
21)(1 1
α ∂
−∂ −∩pF n
= 0 at p = pm
Proof:
( )⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∂
∂=
∂
−∂ −∩
)(
)()()()(111
2
1
2
21 1
pR
pRpRpFpF
s
mn
α
α α
α α
( ) ( )
( )
( )0
)(
)()()(
)()(
1
2
111
1221
⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡ −+
+−+−
−
−
n
n
s
m
ss
sm
s
pR
pRpRpR
pRpR
α
α α α α α
α α α α α α (33)
By lemma 46 0)(
2
1 ≥∂
∂
α
pF.
0)(1
2
211
>∂
−∂ −
∩α
pF n
for all p except at p = pm, where 0)(1
2
211
=∂
−∂ −
∩α
pF n
.
The smaller firms’ cumulative probability distribution functions do not change when α2
changes. Thus firm one’s cumulative probability distribution function relationship with
α2 provides the reason why this derivative function is positive.
Expanding and simplifying equation (33):
( ) ( )⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ++−=∂
−∂ −∩
)(
)()()()()(1 1111
2
21 1
pR
pRpR
D
pRpRpF
s
ms
mn
α
α α α α α
α
( ) ( ) ( ))(
)()()(2)()(1 21
211
21
221
2
21 1
pRD
pRpRpRpRpF
s
sm
sm
n
α
α α α α α α α
α
+−++−=
∂
−∂ −∩ (33a)
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311
QED
Suppose now the shoppers αs are changed. This affects the probability that at
least one price is lower )(1 121pF n−∩
− by:
Lemma 71:
71a)s
pF n
α ∂
−∂ −∩)(1 121 > 0 iff
( )[ ] ( ) ( ) ( )[ ] 7
22
1122
11
2
1
2
122
1
2
12
)()(433398)(12
)()()(
K pRpRnnnpRn
pRpRnnpRnnm
ssm
s
mss
mss
+−−−−−−++−
++−−<
α α α α α α α
α α α α α α α α α
71b) s
pF n
α ∂
−∂ −∩)(1 121 < 0 iff
( )[ ] ( ) ( ) ( )[ ] 7
22
1122
11
2
1
2
122
1
2
12
)()(433398)(12
)()()(
K pRpRnnnpRn
pRpRnnpRnnm
ssm
s
mss
mss
+−−−−−−++−
++−−>
α α α α α α α
α α α α α α α α α
where ( ) ( ) ( ) 221
217 )(438643 pRnnnK ss α α α α −+−+−=
71c) Generallys
pF n
α ∂
−∂ −∩)(1 121 > 0 except when α1 is very large, and αs is very small
relative to α2
71d) There are more parameter values supportings
pF n
α ∂
−∂ −
∩
)(1 1
21 > 0 when the number
of firms is higher
71e) At n = 3 and n = 11, there are more parameter values that support
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312
s
pF n
α ∂
−∂ −∩)(1 121 > 0 at prices near L than at p
m
Proof:
)⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∂
∂=
∂
−∂ −∩
)(
)()()()(111121 1
pR
pRpRpFpF
s
m
ss
n
α
α α
α α
( ) ( )
( )
⎟⎟
⎠
⎞⎜⎜
⎝
⎛ +−
⎟⎟⎟
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎜⎜⎜⎜
⎝
⎛
⎥⎦⎤⎢
⎣⎡ −+
+−+−
−
−)(
)()(
)()()()(
)()(2
11
1
2
111
1221
pR
pRpR
pRpRpRpR
pRpR
s
m
n
n
s
m
ss
sm
s
α
α α
α
α α α α α
α α α α α α
( )ss
msss
s
m
D
pRn
n
n
n
n
n
n
n
pR
pRpR
α α α
α α α α α α α α α α
α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−+⎟ ⎠
⎞⎜⎝
⎛ −−
−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
=1
2
2
1
2
121
2
111 )(
1
2
1
21
1
22
1
21
)(
)()(
( )ss
ss
s
m
D
pRn
n
n
n
n
n
pR
pRpR
α α α
α α α α α α α α
α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
+1
2
2212
2
111 )(
1
21
1
222
1
21
)(
)()(
( )
( )ss
s
s
m
D
NpR
pRpR
α α α
α α α
α α
+
+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
+1
111
)(
)()(
( )ss
msss
s
m
D
pR
n
n
n
n
n
n
n
n
pR
pRpR
α α α
α α α α α α α α α
α
α α
+
⎥⎦
⎤⎢⎣
⎡⎟
⎠
⎞⎜
⎝
⎛
−
−−+⎟
⎠
⎞⎜
⎝
⎛
−
−−+⎟
⎠
⎞⎜
⎝
⎛
−
−−+⎟
⎠
⎞⎜
⎝
⎛
−
−−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
=1
2
2
1
2
121
2
111 )(
1
21
1
22
1
23
1
22
)(
)()(
( )ss
sss
m
D
pRn
n
n
n
n
n
pR
pRpR
α α α
α α α α α α α α
α α
+
⎥⎦
⎤⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−+⎟ ⎠
⎞⎜⎝
⎛ −−
−−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
+1
2
2212
2
111 )(
1
22
1
224
1
22
)(
)()(
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313
( )[ ]
( ) ( )ss
msss
s
m
Dn
pRnnnpR
pRpR
α α α
α α α α α α α α α α
α α
+−
++−+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
=1
2
2
1
2
121
2
111
1
)(12)(
)()(
( ) ( ) ( )[ ]
( ) ( )ss
ss
s
m
Dn
pRnnnpR
pRpR
α α α
α α α α α α α α
α α
+−
+−++−++−⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
+1
2
2212
2
111
1
)(438643)(
)()(
( )
( ) ( ) )(1
)(12
1
2
22
3
1
22
12
2
1
3
1
pRDn
pRnnn
ss
msss
α α α
α α α α α α α α α
+−
++−+=
( )
( ) ( )
( ) ( ) ( )[ ]( ) ( ) )(1)()(438643
)(1
)()(12
1
2
2
212
2
12
3
1
1
2
2
3
1
22
12
2
1
3
1
pRDnpRpRnnn
pRDn
pRpRnnn
ss
mss
ss
msss
α α α
α α α α α α α α
α α α
α α α α α α α α α
+−+−++−++−+
+−
−−−−−+
( ) ( ) ( )
( ) ( ) )(1
)(438643
1
2
22
212
2
12
3
1
pRDn
pRnnn
ss
ss
α α α
α α α α α α α α
+−
−+−+−+
( )
( ) ( ) )(1
)(12
1
2
22
3
1
22
12
2
1
3
1
pRDn
pRnnn
ss
msss
α α α
α α α α α α α α α
+−
++−+=
( ) ( ) ( )
( ) ( ) )(1
)()(433398
12
2
212
3
1
22
12
2
1
3
1
pRDn
pRpRnnnnn
ss
mssss
α α α
α α α α α α α α α α α α
+−
−−−−−−−−+
( ) ( ) ( )
( ) ( ) )(1
)(438643
1
2
22
212
2
12
3
1
pRDn
pRnnn
ss
ss
α α α
α α α α α α α α
+−
−+−+−+ (34)
s
pF n
α ∂
−∂ −∩)(1 121 is positive if the numerator of equation (34) is positive.
Rearranging the numerator of equation (34):
( )[ ] ( ) ( ) ( )[ ]( ) ( ) ( )[ ]
>⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−+−+−+
−−−−−−++−22
1
2
1
3
1
21
31
21
231
21
2
)(438643
)()(433398)(12
pRnnn
pRpRnnnpRn
ss
mss
ms
α α α α α
α α α α α α α α α
)()()(22
1
3
1222
1
3
1 pRpRnnpRnn mss
mss α α α α α α α α ++−−
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314
( )[ ] ( ) ( ) ( )[ ] 7
2
1
3
1
2
123
1
2
1
22
1
3
1222
1
3
12
)()(433398)(12
)()()(
K pRpRnnnpRn
pRpRnnpRnnm
ssm
s
mss
mss
+−−−−−−++−
++−−<
α α α α α α α α
α α α α α α α α α
where( ) ( ) ( )
22
1
2
1
3
17)(438643 pRnnnK
ssα α α α α −+−+−=
The inequality sign switches because the denominator is a negative number.
s
pF n
α ∂
−∂ −∩)(1 121 > 0 iff
( )[ ] ( ) ( ) ( )[ ] 7
22
1122
11
2
1
2
122
1
2
12
)()(433398)(12
)()()(
K pRpRnnnpRn
pRpRnnpRnnm
ssm
s
mss
mss
+−−−−−−++−
++−−<
α α α α α α α
α α α α α α α α α (34a)
where ( ) ( ) ( )22
1
2
17 )(438643 pRnnnK ss α α α α −+−+−=
At p= pm
for ( )
0)(1 121 >
∂
−∂ −∩
s
mpF n
α
( ) ( ) ( )[ ] ( ) ( ) ( )[ ] 22
1
2
1222
11
22
1
2
122
1
2
12
)(438643)(434386
)()(
mss
mss
mss
mss
pRnnnpRnnn
pRnnpRnn
α α α α α α α α
α α α α α α α α α
−+−+−+−−−−−−
++−−<
Unfortunately, the right-hand side of the above equation reduces tozero
zero. This makes
interpretation by normal inspection impossible. However, running calculations by
equation (34a) and letting R(p) get close to R(pm
) is a second – best approach to finding
the sign of s
pF n
α ∂
−∂ −∩)(1 121 at high prices near p = pm. Table 26 also reveals that
equation (34a) does not hold or s
pF n
α ∂
−∂ −∩)(1 121 < 0 at high prices if α1 is extremely high
and αs is extremely low and the number of firms is small. The 11 and 101 firm cases
reveal that only very extreme values of α1 and αs causes
pF n
α ∂
−∂ −∩)(1 121 < 0 at high
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315
prices. There is a pattern that there are more positive values of ( )
s
mpF n
α ∂
−∂ −∩)(1 121 for
higher values of n. For instance, at n = 101 firms, only positive values of
( )s
mpF n
α ∂
−∂ −∩)(1 121 exist when α2 equals 0.001. At n= 101 firms, positive values exist for
( )s
mpF n
α ∂
−∂ −∩)(1 121 at all values α1 and αs when α2 equals a much larger 0.004 except for
when α1 = 0.59 and αs = 0.01. The 101 firm case contrasts a similar size of nα2 at n = 3
firms. At n = 3 firms, all values ( )s
mpF n
α ∂−∂ −∩ )(1 121 at α2= 0.2 are negative.
Table 26: Signing( )
s
mpF n
α ∂
−∂ −∩)(1 121
α1 αs R(pm) R(p) # of
firms
α2 RHS of eq (34a) ( )s
mpF n
α ∂
−∂ −∩)(1 121
0.8 0.1 20 19.999 3 0.05 0.0717656 Positive Æ
0.85 0.05 20 19.999 3 0.05 0.0368403 Negative ∞
0.59 0.01 20 19.999 3 0.20 0.00746673 Negative Æ
0.85 0.05 20 19.999 11 0.01 0.0192353 Positive Æ
0.89 0.01 20 19.999 11 0.01 0.00391429 Negative
0.1 0.5 20 19.999 11 0.04 0.0351832 Negative
0.2 0.4 20 19.999 11 0.04 0.0605541 Positive ∞
0.4 0.2 20 19.999 11 0.04 0.0613991 Positive Æ
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0.5 0.1 20 19.999 11 0.04 0.0361071 Negative ∞
0.89 0.01 20 19.999 101 0.001 0.00337655 Positive Æ
0.55 0.05 20 19.999 101 0.004 0.0163696 Positive Æ
0.59 0.01 20 19.999 101 0.004 0.00336982 Negative
α1 αs R(pm) R(p) # of
firms
α2 RHS of eq (34a) ( )s
mpF n
α ∂
−∂ −∩)(1 121
At p = L,
)s
pF n
α ∂
−∂ −∩)(1 121
> 0 iff a modified version of equation (34a) holds.
Substituting( )s
mpRLR
α α
α
+=
1
1 )()( into equation (34a):
( ) ( )
( )[ ]( ) ( ) ( ) ( )[ ]( ) 82
1
2
1
3
1
2
122
1
2
11
21
22
1
3
122
1
2
1
2
12
)(433398)(12
)()(
K pRnnnpRn
pRnnpRnnm
sssm
ss
msss
msss
++−−−−−−+++−
++++−−<
α α α α α α α α α α α α
α α α α α α α α α α α α α
where ( ) ( ) ( ) 2221
31
418 )(438643 m
ss pRnnnK α α α α α −+−+−=
( ) ( ) ( )[ ] 89
22
1
3
1
3
1
22
1
4
1
3
1
321
231
231
41
41
321
321
231
231
41
21222412
22
K K nnn
nnnnnnnnnn
sssss
ssssssssss
+++−++−++−
++++−−−−−−<
α α α α α α α α α α α
α α α α α α α α α α α α α α α α α α α α α
where ( ) ( ) ( ) 221
31
418 438643 ss nnnK α α α α α −+−+−=
where
( ) ( ) ( ) ( ) ( ) ( ) 3
1
3
1
22
1
22
1
4
1
3
19 433398433398 sssss nnnnnnK α α α α α α α α α α α −−−−−−−−−−−−=
( ) ( ) ( ) 21
21
32
1
2
1
2 38453
2
ss
sss
nnn
nnn
α α α α
α α α α α
α −+−+−
++
<(34b)
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Table 27: Signings
LF n
α ∂
−∂ −∩)(1 121
α1 αs # of firms
α2
RHS of eq (34b)
s
LF n
α ∂
−∂ −
∩
)(1 1
21
0.8 0.1 3 0.05 0.084375 Positive Æ
0.85 0.05 3 0.05 0.0397059 Negative ∞
0.4 0.2 3 0.20 0.225 Positive Æ
0.5 0.1 3 0.20 0.09 Negative ∞
0.85 0.05 11 0.01 0.0204545 Positive Æ
0.89 0.01 11 0.01 0.00396 Negative
0.5 0.1 11 0.04 0.0445946 Positive Æ
0.55 0.05 11 0.04 0.0208861 Negative ∞
0.89 0.01 101 0.001 0.00341473 Positive Æ
0.55 0.05 101 0.004 0.0179502 Positive Æ
0.59 0.01 101 0.004 0.0034276 Negative
α1 αs # of firms α2 RHS of eq (34b) )s
LF n
α ∂
−∂ −∩)(1 121
From the above table and equation (34b),s
LF n
α ∂−∂ −∩ )1 121 is only going to be
negative when α1 is very high and αs is very low. There are also more outcomes of
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s
LF n
α ∂
−∂ −∩)(1 121 > 0 if the number of firms is larger. There are more parameter values
that support )s
LF n
α ∂−∂ −∩ )(1 121 being positive than ( )s
m
pF n
α ∂−∂ −∩ )(1 121 , especially at n = 3 and
11 firms and when na2 is high.
Both the past two tables reveal that generallys
pF n
α ∂
−∂ −∩)(1 121 > 0 , especially
when the number of firms is high. Only extreme parameter values make this negative.
As there are more firms, competition among firms becomes more intense over a greater
number of shoppers thus increasing the probability of the minimum price. When there
are more shoppers, the prize of winning the shoppers is larger, so smaller firms place less
emphasis on their loyal customers and more towards randomizing toward the larger group
of shoppers. With fewer shoppers, this is the reverse case. Fewer shoppers mean less
competition between the many smaller firms and the probability of the minimum price
decreases as firms have to discount less to win the shoppers. When there are fewer firms,
they are more interested in the prize of shoppers as much because competition is less and
thus randomize more toward them and less toward their loyal customers.
QED
Suppose that the number of firms is changed. That affects the probability of at
least one price being smaller )(1 121pF n−∩
− by:
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320
where 232
1
2
110 )(2 pRK sss α α α α α ++=
where ( ) ( ) ( ) 22
1
2
111 )(438643 pRnnnK ss α α α α −−−−−−=
73c) ( )s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 > 0 at high prices if n is small and α1 is very
large and αs is very small or if n is small and α1 is very small and αs is very large
73d) ( )s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 > 0 at low prices if n is small and α1 is very
large and αs is very small
73e) ( )s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 < 0 if n is large, except in very extreme cases
73f) The instances of )
( ))
s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 < 0 increases when there are
more firms
73g) There are more instances of ( )s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 < 0 at low prices
than at high prices
Proof:
( ) ( ) ( ))(
)()()(2)()(1 21
211
21
221
2
21 1
pRD
pRpRpRpRpF
s
sm
sm
n
α
α α α α α α α
α
+−++−=
∂
−∂ −∩
)
( ) ( )( ) ( )( )
( ) )(
)()()(2)(
)(1
12
21
21
211
21
21
21
21
2
21 1
pRD
pRpRpRpR
pF
ss
sssm
sssm
ss
n
α α α
α α α α α α α α α α α α α α α α
α
+
++−++++−
=∂
−∂ −∩
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321
( )
[ ]( ) )(
)(
)(
)()(22)(
12
231
221
221
31
1
2
31
221
221
31
2221
31
pRD
pR
pRD
pRpRpR
ss
ssss
ss
mssss
mss
α α α
α α α α α α α α
α α α
α α α α α α α α α α α α
+
−−−−+
+
++++−−=
( )
[ ]( ) )(
)(2
)(
)()(32)(
1
2
23
1
22
1
3
1
12
31
221
31
2221
31
pRD
pR
pRD
pRpRpR
ss
sss
ss
msss
mss
α α α
α α α α α α
α α α
α α α α α α α α α α
+
−−−+
+
+++−−=
( )( ) ( )
( ) )(
)(12)(11
1
2
22
3
1
22
12
2
1
3
121 1
pRD
pRnnnpFn
ss
msss
s
n
α α α
α α α α α α α α α
α +
++−+−=
∂
−∂−−
−∩
( ) ( ) ( )
( ) )(
)()(433398
1
2
2
212
3
1
22
12
2
1
3
1
pRD
pRpRnnnnn
ss
mssss
α α α
α α α α α α α α α α α α
+
−−−−−−−−−+
( ) ( ) ( )
( ) )(
)(438643
1
2
22
212
2
12
3
1
pRD
pRnnn
ss
ss
α α α
α α α α α α α α
+
−+−+−−+
( )
( ) ( ) ( )[ ]( ) )(
)(1211
)(11
)(1
12
22
312
21
221
31
21
2
21 11
pRD
pRnnn
pFn
pF
ss
msss
s
nn
α α α
α α α α α α α α α
α α
+
−−−+−+−
=∂
−∂−−
∂
−∂ −− ∩∩
(35)
( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( )[ ]( ) )(
)(4386432
)(
)()(43333982
1
2
22
212
2
12
3
1
3
1
22
1
3
1
1
2
2
212
3
1
22
12
2
1
3
1
3
1
pRD
pRnnn
pRD
pRpRnnnnn
ss
sssss
ss
msssss
α α α
α α α α α α α α α α α α α α
α α α
α α α α α α α α α α α α α α
+
−−−−−−−−−+
+
−+−+++−++++
( )s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 > 0 if the numerator of equation (35) is positive.
Rearranging the numerator of equation (35):
( )[ ] ( ) ( ) ( )[ ]( ) ( ) ( )[ ]
( ) ( )[ ] ( ) ( )[ ][ ] 23
122
13
1
221
31
31
2221
31
221
21
31
21
31
21
231
21
2
)(2
)()(32)(11
)(438643
)()(433398)(12
pR
pRpRnnpRnn
pRnnn
pRpRnnnpRn
sss
msss
mss
ss
mss
ms
α α α α α α
α α α α α α α α α α
α α α α α
α α α α α α α α α
+++
+−+−−++++
>⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−−−−−+
−+−+−+−−−
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( )s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 > 0 iff
( ) ( ) ( ) ( )
( )[ ] ( ) ( ) ( )[ ] 11
2
1
3
1
2
123
1
2
1
10
22
1
3
1
3
1
222
1
3
12)()(433398)(12
)()(32)(11
K pRpRnnnpRn
K pRpRnnpRnnm
ssm
s
m
sss
m
ss
+−+−+−+−−−
++−+−−++++> α α α α α α α α
α α α α α α α α α α α
where 231
221
3110 )(2 pRK sss α α α α α α ++=
where ( ) ( ) ( ) 22
1
2
1
3
111 )(438643 pRnnnK ss α α α α α −−−−−−=
The denominator is the same as in equation (34) except it has the opposite sign. Thus the
denominator is positive and the inequality sign is signed positive here instead of negative
in equation (34).
( ) ( ) ( ) ( )
( )[ ] ( ) ( ) ( )[ ] 1122
1122
11
102
12
1322
12
12
)()(433398)(12
)()(32)(11
K pRpRnnnpRn
K pRpRnnpRnnm
ssm
s
msss
mss
+−+−+−+−−−
++−+−−++++>
α α α α α α α
α α α α α α α α α α
where 2321
2110 )(2 pRK sss α α α α α ++= (35a)
where ( ) ( ) ( ) 22
1
2
111 )(438643 pRnnnK ss α α α α −−−−−−=
At p = pm,( )
( )( )
s
mm pFn
pF nn
α α ∂
−∂−−∂
−∂ −−
∩∩
)(11
)(1 11
21
2
21 > 0 iff equation (35a) is modified to:
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]21
21
31
21
31
21
31
221
31
31
221
31
2438643434386
22
ssss
ssssss
nnnnnn α α α α α α α α α α
α α α α α α α α α α α α α
−−−−−−+−+−+−
+++−−−>
Like the monopoly case of lemma 30, the above equation reduces tozero
zero. Thus
inspection as a first option does not work. As a second best option, numerical
calculations using equation (35a) with values of R(p) approaching R(pm) can be used to
determine trends in prices near p = pm. Below is Table 28 of the calculations of different
parameters of α1, αs, n, α2, R(pm), and R(p).
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Table 28: Signing( )
( )( )
s
mm pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21
α1 αs n R(p
m
) R(p) α2 RHS of eq
(35a)( )
( )( )
s
m
m
pFn
pF
n
n
α
α
∂
−∂−−
∂−∂
−
−
∩
∩
)(11
)(1
1
1
21
2
21
0.2 0.7 3 20 19.999 0.05 -0.0190258 Positive Æ
0.3 0.6 3 20 19.999 0.05 0.0529526 Negative ∞
0.8 0.1 3 20 19.999 0.05 0.0687767 Negative Æ
0.85 0.05 3 20 19.999 0.05 0.0361186 Positive ∞
0.59 0.01 3 20 19.999 0.20 0.00742467 Positive Æ
0.1 0.8 11 20 19.999 0.01 0.00991028 Positive
0.2 0.7 11 20 19.999 0.01 0.0460791 Negative ∞
0.85 0.05 11 20 19.999 0.01 0.0191325 Negative Æ
0.89 0.01 11 20 19.999 0.01 0.00391031 Positive
0.1 0.5 11 20 19.999 0.04 0.0191918 Positive
0.2 0.4 11 20 19.999 0.04 0.0495451 Negative ∞
0.4 0.2 11 20 19.999 0.04 0.0586086 Negative Æ
0.5 0.1 11 20 19.999 0.04 0.0354508 Positive ∞
0.01 0.89 101 20 19.999 0.001 0.000400002 Positive
0.1 0.8 101 20 19.999 0.001 0.0296169 Negative ∞
0.1 0.5 101 20 19.999 0.004 0.029516 Negative ∞
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0.55 0.05 101 20 19.999 0.004 0.0163549 Negative Æ
0.59 0.01 101 20 19.999 0.004 0.00336926 Positive
α1 αs n R(pm) R(p) α2 RHS of eq
(35a)
( )
( )( )
s
m
m
pFn
pF
n
n
α
α
∂
−∂−−
∂−∂
−
−
∩
∩
)(11
)(1
1
1
21
2
21
The above table reveals that( )
( )( )
s
mm pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 is negative
in more instances as n increases. For instance, only in the most extreme case in the 101
firm case is( )
( )( )
s
mm pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 positive. Whereas, in the three
firm case there are many cases, especially when α2 = 0.2 (ie α2 is large), when
( )( )
( )s
mm pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 is positive. Generally a extremely large α1 and
small αs or extremely large αs and extremely small α1 at high prices causes
( )( )
( )s
mm pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 to be positive. The ranges of α1 and αs for
( )( )
( )s
mm pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 > 0 expand as the number of firms increases
and as α2 decreases.
At p = L equation (35a) can be simplified to:
( ) ( ) ( ) ( ) ( ) ( )
( )[ ]( ) ( ) ( ) ( )[ ]( ) 131
2
1
3
1
2
1
2
1
2
11
121
22
1
3
1
3
1
2
1
2
1
2
12
43339812
3211
K nnnn
K nnnn
sssss
sssssss
++−+−+−++−−−
+++−+−−+++++>
α α α α α α α α α α α α
α α α α α α α α α α α α α α α
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where 32
1
23
1
4
112 2 sssK α α α α α α ++=
where ( ) ( ) ( ) 22
1
3
1
4
113 438643 ss nnnK α α α α α −−−−−−=
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )[ ] 13
22
1
3
1
3
1
22
1
4
1
3
1
12
4
1
32
1
32
1
23
1
23
1
4
12
1222412
11222211
K nnn
K nnnnnn
sssss
ssssss
+−−−−−−−−−
++++++++++++>
α α α α α α α α α α α
α α α α α α α α α α α α α
where( ) ( ) ( ) ( )
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+++
+−+−−+−+−−=
32
1
23
1
4
1
32
1
23
1
4
1
23
1
4
1
32
1
122
3232
sss
ssssss nnnnK
α α α α α α
α α α α α α α α α α α α
where
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−−−−−
−+−+−+−+−+−=
221
31
41
31
31
221
221
41
31
13
438643
433398433398
ss
sssss
nnn
nnnnnnK
α α α α α
α α α α α α α α α α α
( ) ( ) ( ) 3
1
22
1
3
1
4
1
32
1
23
12
38453
2
sss
sss
nnn
nnn
α α α α α α
α α α α α α α
−+−+−
++>
( )s
LFn
LF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 > 0 iff
( ) ( ) ( )
2
1
2
1
32
1
2
12
38453
2
ss
sss
nnn
nnn
α α α α
α α α α α α
−+−+−
++> (35b)
Table 29 below reveals some different values for the parameters α1, αs, α2, and a
calculated equation (35b).
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Table 29: Signing ( )s
LFn
LF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21
α1 αs n
α2 RHS of (35b)
( )( )
s
LFn
LF
n
n
α
α
∂
−∂−−
∂
−∂
−
−
∩
∩
)(11
)(1
1
1
21
2
21
0.8 0.1 3 0.05 0.084375 Negative Æ
0.85 0.05 3 0.05 0.0397059 Positive ∞
0.25 0.35 3 0.20 0.63 Negative ∞
0.4 0.2 3 0.20 0.225 Negative Æ
0.5 0.1 3 0.20 0.09 Positive ∞
0.05 0.85 11 0.01 1.02622 Negative ∞
0.85 0.05 11 0.01 0.0204545 Negative Æ
0.89 0.01 11 0.01 0.00396 Positive
0.1 0.5 11 0.04 0.485294 Negative ∞
0.5 0.1 11 0.04 0.0445946 Negative Æ
0.55 0.05 11 0.04 0.0208861 Positive ∞
0.01 0.89 101 0.001 0.896907 Negative ∞
0.89 0.01 101 0.001 0.00341473 Negative Æ
0.55 0.05 101 0.004 0.0179502 Negative Æ
0.59 0.01 101 0.004 0.0034276 Positive
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Again these values reveal that ( )s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 is generally
negative at lower prices as the number of firms increases. Only extreme high value of α1
and extremely low value of αs cause ( )s
LFn
LF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 to be positive
at low prices. As the number of firms decreases and α2 increases, the threshold at the
upper range for α1 lowers and the threshold at the lower range for αs increases for
( )s
LFn
LF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 to be positive. Because
2
21)(1 1
α ∂
−∂ −∩pF n
term
carries much less influence than the adjusted ( )s
pFn
n
α ∂
−∂−
−∩)(1
1121 term, the results found
in lemma 73 involving ( )s
pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
2
21 are nearly the opposite
sign of those found in the case of lemma 71 wheres
pF n
α ∂
−∂ −∩)(1 121 is positive in sign.
QED
Lemma 74:
74a) s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 > 0 iff
( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( )[ ] 143
12
123
12
12
1
3
1
22
1
3
1
222
1
3
1
3
12)()(331512)(4443
)()(4333)(3343
K pRpRnnpRnn
pRpRnnnpRnnnm
sm
ss
m
sss
m
sss+−+−+−−−−−
−−−−−+−+−+> α α α α α α α α
α α α α α α α α α α α α α
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(the denominator is positive in more cases than not; the > sign reverses to < in the cases
where the denominator is negative which is found near the extreme ends of the
distribution)
74b) s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 < 0 iff
( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( )[ ] 143
12
123
12
12
1
31
221
31
2221
31
31
2)()(331512)(4443
)()(4333)(3343
K pRpRnnpRnn
pRpRnnnpRnnnm
sm
ss
msss
msss
+−+−+−−−−−
−−−−−+−+−+<
α α α α α α α α
α α α α α α α α α α α α α
(the denominator is positive in more cases than not; the < sign reverses to > in the cases
where the denominator is negative which is found near the extreme ends of the
distribution)
where ( ) ( )
( ) ( ) ( ) ( )[ ] ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−−−−−−−+
−+−=
2321
21
31
321
14)(3210711843
)()(321410
pRnnnn
pRpRnnK
sss
mss
α α α α α α
α α α
74c) Generally( ) ( )
s
mm pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 < 0 except when as is low compared
with a2
74d) s
LFLF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 < 0 except when α1 is extremely high and αs is
extremely low
74e) Generallys
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 < 0 except for the extreme cases
described above
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Proof:
1
21)(1 1
α ∂
−∂ −∩pF n ( )( ) ( )( )
( ) ( ) )(1
)(3243
11
2312
31
2212
21
pRDn
pRnn
ss
msss
α α α α
α α α α α α α α α
+−
−−−+−−−=
( )( ) ( )( ) ( )( ) ( )( )( ) ( ) )(1
)()(326443107
11
2
21
3
12
3
1
22
12
2
1
pRDn
pRpRnnnn
ss
mssss
α α α α
α α α α α α α α α α α α
+−
−++−+−+−+
( )( )( ) ( ) )(1
)(322
11
22212
212
31
pRDn
pRn
ss
ss
α α α α
α α α α α α α α
+−
−++−
1
21)(1 1
α ∂
−∂ −∩pF n ( )( ) ( )( )
( ) ( ) )(1
)(3243
12
1
22312
31
321
22
21
pRDn
pRnn
ss
mssss
α α α α
α α α α α α α α α α
+−
−−−+−−−=
( )( ) ( )( ) ( )( )
( ) ( ) )(1
)()(6443107
12
1
23
132
12
22
1
pRDn
pRpRnnn
ss
msss
α α α α
α α α α α α α α
+−
−+−+−+
( )( ) ( )( )
( ) ( ) )(1
)(322)()(32
12
1
2321
22
212
31
321
231
pRDn
pRnpRpRn
ss
sssm
ss
α α α α
α α α α α α α α α α α α α α
+−
−++−−++
s
pF n
α ∂
−∂ −∩)(1 121 ( )
( ) ( ) )(1
)(12
1
2
22
3
1
22
12
2
1
3
1
pRDn
pRnnn
ss
msss
α α α
α α α α α α α α α
+−
++−+=
( ) ( ) ( )
( ) ( ) )(1
)()(433398
1
2
2
212
3
1
22
12
2
1
3
1
pRDn
pRpRnnnnn
ss
mssss
α α α
α α α α α α α α α α α α
+−
−−−−−−−−+
( ) ( ) ( )
( ) ( ) )(1
)(438643
1
2
22
212
2
12
3
1
pRDn
pRnnn
ss
ss
α α α
α α α α α α α α
+−
−+−+−+
s
pF n
α ∂
−∂ −∩)(1 121 ( )
( ) ( ) )(1
)(12
1
2
1
22
4
1
23
12
3
1
4
1
pRDn
pRnnn
ss
msss
α α α α
α α α α α α α α α
+−
++−+=
( ) ( ) ( )
( ) ( ) )(1
)()(433398
12
1
2
2
2
12
4
1
23
12
3
1
4
1
pRDn
pRpRnnnnn
ss
mssss
α α α α
α α α α α α α α α α α α
+−
−−−−−−−−
+
( ) ( ) ( )
( ) ( ) )(1
)(438643
1
2
1
22
2
2
12
3
12
4
1
pRDn
pRnnn
ss
ss
α α α α
α α α α α α α α
+−
−+−+−+
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330
s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21
( )( ) ( )( )( ) ( ) )(1)(3243
12
1
223
12
3
1
32
1
2
2
2
1
pRDnpRnn
ss
m
ssss
α α α α α α α α α α α α α α
+− −−−+−−−=
( )
( ) ( ) )(1
)(12
1
2
1
22
4
1
23
12
3
1
4
1
pRDn
pRnnn
ss
msss
α α α α
α α α α α α α α α
+−
++−+−
( )( ) ( )( ) ( )( )
( ) ( ) )(1
)()(6443107
12
1
23
132
12
22
1
pRDn
pRpRnnn
ss
msss
α α α α
α α α α α α α α
+−
−+−+−+
( ) ( ) ( )
( ) ( ) )(1
)()(433398
12
1
2
2
2
12
4
1
23
12
3
1
4
1
pRDn
pRpRnnnnn
ss
mssss
α α α α
α α α α α α α α α α α α
+−
−−−−−−−−−
( )( ) ( )( )
( ) ( ) )(1
)(322)()(32
1
2
1
23
21
2
2
2
12
3
1
3
21
23
1
pRDn
pRnpRpRn
ss
sssm
ss
α α α α
α α α α α α α α α α α α α α
+−
−++−−++
( ) ( ) ( )
( ) ( ) )(1
)(438643
1
2
1
22
2
2
12
3
12
4
1
pRDn
pRnnn
ss
ss
α α α α
α α α α α α α α
+−
−+−+−−
( )( ) ( ) ( )
( ) ( ) )(1
)(334443
1
2
1
22
4
1
23
12
3
1
32
1
2
2
2
1
4
1
pRDn
pRnnnn
ss
msssss
α α α α
α α α α α α α α α α α α α α
+−
−−−−−−−−+−=
( ) ( ) ( ) ( )
( ) ( ) )(1
)()(141033331512
1
2
1
2
2
2
12
4
1
23
12
3
1
4
1
pRDn
pRpRnnnnn
ss
mssss
α α α α
α α α α α α α α α α α α
+−
−+−+−+−++
( ) ( )
( ) ( ) )(1
)()(3243
1
2
1
3
21
32
1
pRDn
pRpRnn
ss
mss
α α α α
α α α α α
+−
−+−+ (36)
( ) ( ) ( ) ( )
( ) ( ) )(1
)(3210711843
1
2
1
23
21
2
2
2
12
3
12
4
1
pRDn
pRnnnn
ss
sss
α α α α
α α α α α α α α α α α
+−
−−−−−−−−+
s
pFpF nn
α α ∂−∂−
∂−∂ −− ∩∩ )(1)(1 11 21
1
21 is positive iff the numerator of equation (36) is positive.
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Rearranging:
( ) ( )
( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ]
( ) ( )[ ] ( ) ( )[ ] )()(4333)(3343
)(3210711843
)()(321410331512
)(4443
32
1
23
1
4
1223
1
32
1
4
1
23
1
22
1
3
1
4
1
3
1
22
1
4
1
3
1
24
1
3
1
22
1
2
pRpRnnnpRnnn
pRnnnn
pRpRnnnn
pRnn
msss
msss
sss
msss
mss
α α α α α α α α α α α α
α α α α α α α
α α α α α α α
α α α α α
α
−−−−−+−+−+
>
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
−−−−−−−−+
−+−+−+−+
−−−−−
Assuming the denominator is positive, which in many cases is the result. The extreme
ends of the distribution give some negative values of the denominator. Thus the > sign
switches to < in the cases where the denominator is negative for
s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 to be positive.
( ) ( )[ ] ( ) ( )[ ]( ) ( )[ ] ( ) ( )[ ] 14
4
1
3
124
1
3
1
22
1
32
1
23
1
4
1223
1
32
1
4
1
2
)()(331512)(4443
)()(4333)(3343
K pRpRnnpRnn
pRpRnnnpRnnnm
sm
ss
msss
msss
+−+−+−−−−−
−−−−−+−+−+
>
α α α α α α α α
α α α α α α α α α α α α
α
where( ) ( )[ ]
( ) ( ) ( ) ( )[ ] ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−−−−−−−+
−+−=
23
1
22
1
3
1
4
1
3
1
22
1
14)(3210711843
)()(321410
pRnnnn
pRpRnnK
sss
mss
α α α α α α α
α α α α
( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( )[ ] 1431212312121
31
221
31
2221
31
31
2 )()(331512)(4443
)()(4333)(3343
K pRpRnnpRnn
pRpRnnnpRnnn
msmss
msss
msss
+−+−+−−−−−
−−−−−+−+−+>
α α α α α α α α
α α α α α α α α α α α α α
where ( ) ( )[ ]( ) ( ) ( ) ( )[ ] ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−−−−−−−+
−+−=
2321
21
31
321
14)(3210711843
)()(321410
pRnnnn
pRpRnnK
sss
mss
α α α α α α
α α α (36a)
At p = pm
for s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 to be positive at high prices, equation
(36a) reduces to undefinedzero
zero=>2α . (36b)
Since equation (36b) is undefined by traditional means, a second best method of using
various numerical values of α1, α2, αs, n, R(p), and R(pm) to see if equation (36a) holds
for values near p = pm. Table 30 below summarizes these various parameter values plus
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the right –hand side of equation (36a) and whether s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 is
positive or negative at high prices. Generally,s
pFpF nn
α α ∂
−∂−∂
−∂ −−
∩∩
)(1)(1 11
21
1
21 is
negative at high prices near p = pm except when αs is small in size to α2. There are more
instances of s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 < 0 as the number of firms increases.
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Table 30: Signing( )
( )( )
s
mm pFn
pF nn
α α ∂
−∂−−
∂
−∂ −− ∩∩)(1
1)(1 11 21
1
21
α1 αs n R(p
m
) R(p) α2 RHS of eq
(36a)
( )
( )( )
s
m
m
pFn
pF
n
n
α
α
∂
−∂−−
∂−∂
−
−
∩
∩
)(11
)(1
1
1
21
1
21
0.8 0.1 3 20 19.999 0.05 0.0810311 Negative Æ
0.85 0.05 3 20 19.999 0.05 0.0390508 Positive ∞
0.3 0.3 3 20 19.999 0.20 0.233354 Negative Æ
0.4 0.2 3 20 19.999 0.20 0.173149 Positive ∞
0.85 0.05 11 20 19.999 0.01 0.0212224 Negative Æ
0.89 0.01 11 20 19.999 0.01 0.00399054 Positive
0.5 0.1 11 20 19.999 0.04 0.0489889 Negative Æ
0.55 0.05 11 20 19.999 0.04 0.0220367 Positive ∞
0.01 0.89 101 20 19.999 0.001 0.0148583 Negative ∞
0.89 0.01 101 20 19.999 0.001 0.00345157 Negative Æ
0.55 0.05 101 20 19.999 0.004 0.0193688 Negative Æ
0.59 0.01 11 20 19.999 0.004 0.00348296 Positive
α1 αs n R(pm
) R(p) α2 RHS of eq
(36a)
( )
( )( )
s
m
m
pF
n
pF
n
n
α
α
∂
−∂
−−
∂
−∂
−
−
∩
∩
)(1
1
)(1
1
1
21
1
21
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Generally, adding more loyal shoppers at high prices and taking away shoppers
reduces the weight of the probability of the minimum price at high prices. The largest
firm is concerned about its monopoly profits at its atom. The largest firm randomizes
less towards the shoppers. The rest of the smaller firms compete less intensely for the
shoppers and instead concentrate on its uninformed shoppers.
At p = L for ) )
s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 to be positive equation (36a) reduces
to:
( ) ( )[ ]( ) ( ) ( )[ ]( )
( ) ( )[ ]( ) ( ) ( )[ ]( ) 1615
14
12
14
13
122
1
133
124
15
12
123
132
14
1
2
3315124443
43333343
K nnnn
nnnnnn
sssss
ssssssss
++−+−++−−−−−
+−−−−−++−+−+
>
α α α α α α α α α α α α
α α α α α α α α α α α α α α α α
α
where( ) ( )[ ]( )
( ) ( ) ( ) ( )[ ]⎪⎭⎪⎬⎫
⎪⎩
⎪⎨⎧
−−−−−−−−+
+−+−=
33
1
24
1
5
1
6
1
1
32
1
23
1
163210711843
321410
sss
sss
nnnn
nnK
α α α α α α α
α α α α α α and again
assuming the denominator in (36a) is positive.
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 16
5
1
24
1
33
1
6
1
5
1
24
1
1534
143
125
125
134
16
1
2
288864443
668623343
K nnnn
K nnnnnn
sssss
ssssss
+−−−−−−−−−−
+−+−++−+−+
>
α α α α α α α α α α α
α α α α α α α α α α α α
α
where( ) ( ) ( ) ( )
( ) ( ) ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−−−−
−−−−−−+−+=
43
1
34
1
25
1
34
1
25
1
6
1
43
1
52
1
34
1
15
4333
43333343
sss
ssssss
nnn
nnnnnnK
α α α α α α
α α α α α α α α α α α α
where
( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ⎪
⎪
⎭
⎪⎪
⎬
⎫
⎪
⎪
⎩
⎪⎪
⎨
⎧
−−−−−−−−
−+−+−+−+
−+−+−+−+
−−−−−
=
3312415161
42
1
33
1
5
1
24
1
33
1
24
1
6
1
5
1
24
1
33
1
42
1
16
3210711843
321410331512
321410331512
4443
sss
ssss
sss
sss
nnnn
nnnn
nnnn
nn
K
α α α α α α α
α α α α α α α α
α α α α α α α
α α α α α α
( ) ( ) ( )( ) ( ) ( ) 42
133
124
15
1
521
431
341
251
2128453
437634
ssss
ssss
nnn
nnnn
α α α α α α α α
α α α α α α α α α
−−−−+−
−+−+−+>
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( ) ( ) ( )
( ) ( ) ( ) 32
1
2
1
3
1
43
1
22
1
3
12
128453
437634
sss
ssss
nnn
nnnn
α α α α α α
α α α α α α α α
−−−−+−
−+−+−+> (36c)
Since at p = L the denominator can be negative, the condition is also rewritten when α1 is
low and αs is high:
( ) ( ) ( )
( ) ( ) ( ) 32
1
2
1
3
1
43
1
22
1
3
12
128453
437634
sss
ssss
nnn
nnnn
α α α α α α
α α α α α α α α
−−−−+−
−+−+−+< (36d)
Table 31 below lists different values for α1, αs, n, α2, and the right-hand side of equation
(36c). Generallys
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 is negative at low prices with the
exception being if α1 being extremely high and αs being extremely low. At the midpoint
between L and pm
for the various parameter combinations of α1, αs, and α2,
s
pFpF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21 is negative at almost all possibilities for three firms,
and negative except for the same exceptions found at p = L for n = 11 and n = 101 firms.
Thuss
pFpF nn
α α ∂−∂−
∂−∂ −− ∩∩ )(1)(1 11 21
1
21 is generally negative with these few exceptions.
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Table 31: Signings
LFLF nn
α α ∂
−∂−
∂
−∂ −− ∩∩)(1)(1 11 21
1
21
α1 αs n
α2 RHS of
(36c)/(36d) ( )s
LF
LF
n
n
α
α
∂
−∂−
∂
−∂
−
−
∩
∩
)(1
)(1
1
1
21
1
21
0.05 0.85 3 0.05 -2.2953 Negative ∞
0.8 0.1 3 0.05 0.0964567 Negative Æ
0.85 0.05 3 0.05 0.042201 Positive ∞
0.25 0.35 3 0.20 92.75 Negative ∞
0.4 0.2 3 0.20 0.414286 Negative Æ
0.5 0.1 3 0.20 0.112245 Positive ∞
0.05 0.85 11 0.01 -2.77575 Negative ∞
0.85 0.05 11 0.01 0.0226211 Negative Æ
0.89 0.01 11 0.01 0.00403748 Positive
0.5 0.1 11 0.04 0.0621918 Negative Æ
0.55 0.05 11 0.04 0.0243797 Positive ∞
0.01 0.89 101 0.001 -2.72221 Negative ∞
0.89 0.01 101 0.001 0.00349089 Negative Æ
0.55 0.05 101 0.004 0.0213576 Negative Æ
0.59 0.01 101 0.004 0.00354336 Positive
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QED
Lemma 75: ( ) ) )2
21
1
21 )(1)(1111
α α ∂−∂−
∂−∂−
−− ∩∩ pFpFnnn
< 0.
Proof:
By lemma 691
21)(1 1
α ∂
−∂ −∩pF n
< 0.
By lemma 702
21)(1 1
α ∂
−∂ −∩pF n
> 0.
Thus ( )2
21
1
21)(1)(1
111
α α ∂
−∂−
∂
−∂−
−− ∩∩pFpF
nnn
< 0.
QED
3-6. Conclusion
Varian (1980) demonstrated that firms equal in size randomize in prices between
two groups of customers: those loyal to a particular firm and the shoppers, just loyal to
the lowest possible price. The model developed here builds upon Varian in a two firm
and a modified n – firm case where there is one large firm and either one smaller or n – 1
equally sized smaller firms competing for the same group of shoppers. Each firm, like
Varian (1980), has a loyal group of customers willing to pay whatever price at hand.
From this setup of this model, there can be several conclusions drawn.
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Just like Varian (1980), as there is no price that has probability one and all prices
have some probability greater than zero. Randomization occurs as firms are pricing to
obtain the highest possible price possible from the uninformed or loyal customers while
on the other hand discounting as much as possible to increase the chances of obtaining
the group of shoppers that see all of the prices of the firms. As a second conclusion, the
largest of these firms is able to place an atom of probability at its highest price in the
distribution and collect more profits from its loyal customers. The smaller firm or firms
do not have this ability and instead compete for the shoppers at all prices in their
distribution. The size of the atom is the difference of the loyal customer base between
the larger and individual smaller firm divided by the total number of customers the larger
firm can capture: loyal customers of the larger firm plus the group of shoppers.
Firms randomize less towards obtaining the shoppers or the cumulative function
decreases when the proportion of loyal or uninformed firm one customers increases.
When there are more shoppers than loyal customers of the smaller firms, the largest firm
places more weight on its atom instead as it is more profitable to earn more monopoly
profits. Expected profits for the largest firm increase. The smaller firms reacting to the
largest firm do not need to compete as intensely for the shoppers to win them.
The largest firm randomizes more toward obtaining the shoppers or the
cumulative function of firm one increases when the proportion of loyal or uninformed
firm two customers increases. This is because the size of the atom decreases for the
largest firm when these smaller firm(s) loyal customers increase. The size of the smaller
firm(s) loyal customer base has no effect on the cumulative distribution function for the
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smaller firm(s) as the smaller firm(s) are competing for shoppers at every price in their
distribution. The profits for the smaller firms, however, may increase. This occurs when
their share of loyal customers increases at the expense of the shoppers.
Generally in the two firm case, the largest firm tries to compete for the shoppers
more or the cumulative distribution function for firm one increases when the shoppers
increase. As the number of firms increases, it becomes less likely that increasing the
number of shoppers will result in the largest firm randomizing more to attract the
shoppers at high prices. There are more instances of the firm one’s cumulative
distribution function increasing when there more shoppers at low prices.
Unlike the complexity of the larger firm when shoppers increase, the smaller
firms have a clear result. For a smaller firm, increasing the size of shoppers’ base results
in more discounting to attract the shoppers. All firms have a clear result when more
firms enter the marketplace: there is less discounting to attract the shoppers. Since it
becomes more difficult to win the shoppers with more firms in the marketplace, firms
rely upon their loyal customers and expected profits decrease for the smaller firms,
assuming that the total market size of the smaller firms is constant or that the increase in
market size of the smaller firms comes at the expense of shoppers.
Increasing one group of customers at the expense of another group of customers
also provides some interesting results. The clearest of the results is the case of increasing
the largest firm’s loyal customers at the same rate that the smaller firm(s) loyal customers
are decreased. This causes the largest firm to discount less. The largest firm has an
increase in monopoly profits at its atom and does not need to discount as much to win
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over the shoppers thus the discounting is less than before. Increasing the larger firm’s
loyal customer base at the expense of the smaller firms’ loyal customer base also causes
the smaller firm’s cumulative distribution function to decrease.
Increasing firm one’s loyal customer base at the expense of the group of shoppers
generally causes firm one’s cumulative distribution function to decrease. As the number
of firms increases, there are more instances where this holds true. As the number of firms
increases, the atom for firm one in this case grows larger. The smaller firms’ cumulative
distribution function decreases as firm one is not aggressively competing with the smaller
firms for the shoppers.
Increasing the smaller firms’ loyal customers at the expense of the shoppers
causes two changes in firm one’s cumulative distribution function. At higher prices, the
cumulative distribution function increases. The atom decreases, shifting some of that
weight to the higher prices. The largest firm’s cumulative distribution function decreases
at low prices. The smaller firms’ cumulative distribution function decreases throughout
the price range. The lowest price between the two firms increases.
The probability of the minimum price is a statistic that can be created to combine
the largest and smaller firm(s) cumulative distribution functions. From this exercise, it
was found that this probability function behaved very similar to the largest firm’s
distribution function: decreasing when more loyal customers of the largest firm are
increased and when more firms are increased, being positive when more loyal customers
of the smaller firm(s) are increased, and generally positive when the proportion of
shoppers are increased. The smaller firm(s) cumulative distribution function in this
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minimum cumulative price distribution help ease the conditions for when an increase in
the proportion of shoppers causes the probability of the minimum price to increase.
Increasing the number of firms causes the minimum price cumulative distribution to
decrease. In the case of moving one group of customers over another, the same signs
hold as firm one except for the case when the smaller firms’ loyalty share is increased at
the expense of the shoppers. In that case, there are more cases that the cumulative
distribution function decreases.
The model created does extend the Varian (1980) model to the basic asymmetric
case where there is one larger firm and one smaller firm or n – 1 equally sized smaller
sized firms. Like Varian, there is still randomization. Unlike Varian, the largest firm
now has an atom of probability at its monopoly price. Comparative statics generally
reveal clear results except for the case of the change of shoppers on the largest firm’s
cumulative distribution function. Future questions could include changing costs of firms
and extending the model beyond one period.
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Appendix. PDF, 3 Valuation Case, and Data Graphs
200 400 600 800 1000Pri ce
0. 002
0. 004
0. 006
0. 008
0. 01PDF FHL Fi gure 24a
Figures 24a&b: α = 0.4 θ = 0.3 H = 1000 c = 50 λ = 101.818
Pdf of Fig 15: β = 0.4 n = 3 L = 200 M = 550
999. 99 999. 992 999. 994 999. 996 999. 998Pri ce
0. 5
1
1. 5
2
2. 5
3
3. 5
4
PDF FHLFi gure 24b
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200 400 600 800 1000 Pri ce
0. 002
0. 004
0. 006
0. 008
0. 01PDF FHL Fi gure 25a
Figures 25a&b:
α = 0.8 θ = 0.3 H = 1000 c = 50 λ = 158.571
β = 0.3 n = 3 L = 200 M = 689.144
999. 99 999. 992 999. 994 999. 996 999. 998Pri ce
0. 5
1
1. 5
2
2. 5
3
3. 5
4
PDF FHLFi gure 25b
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500 1000 1500 2000Pri ce
0. 2
0. 4
0. 6
0. 8
1CDF FHL Fi gure 26
Figure 26: α = 0.4 β = 0.4 s = 0.5 θ1 = 0.2 θ2 = 0.4
θ3 = 0.4 H = 2000 L = 600 L1 = 280 c = 50
λ = 123.585 n = 3 M1 = 446.777 M2= 1700
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500 1000 1500 2000 Pri ce
0. 002
0. 004
0. 006
0. 008
0. 01PDF FHL Fi gure 27a
Figures 27a&b : α = 0.4 β = 0.4 s = 0.5 θ1 = 0.2 θ2 = 0.4
θ3 = 0.4 H = 2000 L = 600 L1 = 280 c = 50
λ = 123.585 n = 3 M1 = 446.777 M2= 1700
1999. 9 1999. 92 1999. 94 1999. 96 1999. 98Pri ce
0. 2
0. 4
0. 6
0. 8
1
PDF FHLFi gure 27b
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Figure 29
0
. 0 0
5
. 0 1
. 0 1 5
. 0 2
D e n s i t y
0 200 400 600 800mspordf
MSP - Chicago O’Hare
segcarr | Freq. Percent Cum.------------+-----------------------------------
AA | 1,343 18.16 18.16 NW | 3,350 45.29 63.64UA | 2,689 36.36 100.00
------------+-----------------------------------Total | 7,396 100.00
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Figure 30
0
. 0 0 5
. 0 1
. 0 1 5
. 0 2
D e n s i t y
0 100 200 300 400dcalgaf
Washington National – New York LaGuardia
segcarr | Freq. Percent Cum.------------+-----------------------------------
AA | 11 0.03 0.03
DL | 14,962 45.27 45.30TB | 18,069 54.67 99.97UA | 3 0.01 99.98US | 6 0.02 100.00
------------+-----------------------------------Total | 33,051 100.00
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Figure 31a
0
. 0 0 5
. 0 1
. 0 1 5
. 0 2
D e n s i t y
0 200 400 600 800nycmcof
NYC 3 Airports – MCO All Fares Both Ways
segcarr | Freq. Percent Cum.------------+-----------------------------------
AA | 62 0.19 0.19CO | 12,859 38.60 38.79DL | 9,471 28.43 67.22HP | 3 0.01 67.23JI | 47 0.14 67.37KP | 3,127 9.39 76.76KW | 1 0.00 76.76 NW | 7 0.02 76.78TW | 2,740 8.23 85.01TZ | 502 1.51 86.52UA | 49 0.15 86.66
US | 4,442 13.34 100.00------------+-----------------------------------
Total | 33,310 100.00
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Table 32: Segment Fare Basis Codes For Carriers between NYC and Orlando| segfareb
segcarr | F FR Y YD | Total-----------+--------------------------------------------+----------
AA | 0 0 0 62 | 62CO | 215 14 1,706 10,924 | 12,859DL | 197 184 644 8,446 | 9,471HP | 0 0 0 3 | 3JI | 0 0 0 47 | 47KP | 0 0 0 3,127 | 3,127KW | 0 0 0 1 | 1 NW | 0 0 0 7 | 7TW | 0 12 10 2,718 | 2,740TZ | 0 0 0 502 | 502UA | 0 0 0 49 | 49
US | 21 38 282 4,101 | 4,442-----------+--------------------------------------------+----------
Total | 433 248 2,642 29,987 | 33,310
Figure 31b
0
. 0 0 5
. 0 1
. 0 1 5
. 0 2
D e n s i t y
0 200 400 600 800nycmcof
NYC – MCO YD All Carriers
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Figures 31e and 31f
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D e n s i t y
0 100 200 300 400 500nycmcof
NYC – MCO Kiwi YD
0
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Figures 31g and 31h
0
. 0 5
. 1
. 1 5
D e n s i t y
65 70 75 80nycmcof
NYC – MCO American Trans Air YD
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0 200 400 600 800nycmcof
NYC – MCO US Airways YD
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Figures 31i and 31j
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D e n s i t y
0 100 200 300 400nycmcof
NYC – MCO Continental Y
0
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0 200 400 600nycmcof
NYC – MCO Delta Y
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Figure 31k
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0 100 200 300 400 500nycmcof
NYC – MCO US Airways Y
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Figure 32a
0
. 0 0 1
. 0 0 2
. 0 0 3
. 0 0 4
D e n s i t y
0 200 400 600 800 1000mspatlf
All fares in MSP – ATL Market Both Ways
Segcarr | Freq. Percent Cum.
------------+-----------------------------------BF | 1 0.02 0.02CO | 1 0.02 0.04DL | 1,939 40.27 40.31 NW | 2,871 59.63 99.94UA | 2 0.04 99.98UK | 1 0.02 100.00
------------+-----------------------------------Total | 4,815 100.00
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Table 33: Segment Fare Basis Codes For Carriers between MSP and Atlanta
segfareb | BF CO DL NW UA UK | Total-----------+-------------------------------------------------------+----------
CR | 0 0 0 1 0 0 | 1F | 0 0 46 1 0 0 | 47
FR | 0 0 65 14 1 0 | 80Y | 0 0 463 44 0 0 | 507
YD | 1 1 1,365 2,811 1 1 | 4,180-----------+-------------------------------------------------------+----------
Total | 1 1 1,939 2,871 2 1 | 4,815
Figure 32b
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. 0 0 2
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D e n s i t y
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Figures 32c and 32d
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0 200 400 600 800 1000mspatlf
YD Northwest Airlines
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Figures 32e and 32f
0
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0 200 400 600 800 1000mspatlf
Y Delta Airlines
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Figure 33
0
. 0 1
. 0 2
. 0 3
. 0 4
D e n s i t y
0 100 200 300 400laxlasf
LAX - LAS
segcarr | Freq. Percent Cum.------------+-----------------------------------
AA | 285 0.80 0.80BF | 92 0.26 1.06DL | 2,088 5.85 6.91HA | 914 2.56 9.47HP | 4,767 13.36 22.82UA | 4,962 13.90 36.73US | 158 0.44 37.18WN | 22,415 62.80 99.98
------------+-----------------------------------Total | 35,691 100.00
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Figure 34
0
. 0
0 5
. 0 1
. 0 1 5
D e n s i t y
0 200 400 600 800 1000nycchif
NYC’s three airports – Chicago’s two airports
segcarr | Freq. Percent Cum.
------------+-----------------------------------AA | 12,892 33.88 33.88BF | 1,297 3.41 37.29CO | 3,679 9.67 46.96DL | 276 0.73 47.68KP | 2,097 5.51 54.11 NW | 121 0.32 54.43TW | 1,498 3.94 58.37UA | 15,754 41.40 99.77
------------+-----------------------------------Total | 38,054 100.00
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Figure 35
0
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D e n s i t y
0 200 400 600 800auspvdf
AUS – Providence, RI
repcarr | Freq. Percent Cum.------------+-----------------------------------
AA | 102 47.00 47.00CO | 26 11.98 58.99DL | 24 11.06 70.05 NW | 14 6.45 76.50UA | 29 13.36 89.86US | 22 10.14 100.00
------------+-----------------------------------Total | 217 100.00
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