Airline Pricing and Capacity Behavior

7/29/2019 Airline Pricing and Capacity Behavior

http://slidepdf.com/reader/full/airline-pricing-and-capacity-behavior 1/376

Copyright

by

William Joseph Brennan

2005



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




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



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.



v

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.



vi


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



vii

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.



viii

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



ix

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



1

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.



2

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



3

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.



4

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



5

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.



6

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



7

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



8

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!



9

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



10

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.



11

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



12

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



13

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.



14

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).



15

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



16

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.



17

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



18

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.



19

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



20

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.



21

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



22

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.



23

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



24

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



25

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.



26

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



27

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.



28

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)





30

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



31

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.



32

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



33

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



34

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



35

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.



36

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



37

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.



38

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



39

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)



40

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



41

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).



42

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.



43

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



44

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



45

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



46

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



47

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.



48

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



49

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



50

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



51

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



52

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)



53

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



54

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 –



55

{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)



56

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)



57

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



58

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


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



59

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.



60

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 +

−++−−+−

−=


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)



61

= 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



62

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−⎥⎦

⎤⎢⎣

⎡ −−−−−

=⎥⎦

⎤⎢⎣

⎡

−+−

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 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



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


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.



64

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.



65

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)



66

))(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

)(



67

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):



68

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)



69

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



70

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.



71

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



72

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 ≥



73

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 <



74

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 −≥−+++



75

( ) ( ) ( ) ( )[ ]( ) ( )( )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.



76

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



77

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



78

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.



79

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



80

∂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



81

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.



82

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



83

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



84

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.



85

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.



86

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.



87

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 +

−++−−+−

−=


αθ

λ 1

111



88

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



89

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):



90

))(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



91

( ) ( )( ) ( )( )[ ]

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.



92

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



93

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):



94

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



95

=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:



96

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)



97

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

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

+−

−−−

−−−



(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)



98

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



99

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



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:



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

−−

−−−−−−−⎥⎦

⎤⎢⎣

⎡

−−−

−−

−

−



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



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⎥⎦

⎤⎢⎣

⎡

−


−− −

−

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.



104

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



105

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⎥⎦

⎤⎢⎣

⎡

−


−− −

−

)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:



106

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.



107

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



108

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):



109

)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) .



110

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



111

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



112

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−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(



113

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



114

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



115

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



116

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



117

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



118

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.



119

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(


α−β−θ−β−−+α−β−θ−α

=2])1n(n))(1n([

])1n(n)[cH(


β−−−α> 0 (32)

Figures 14 and 15 provide an example that demonstrates that ∂λ/∂θ > 0. When θ


101.818 in Figure 15.



120

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(


−θ−−θα−β−−+α−β−θ−θ

=2])1n(n))(1n([

)]1n()1n(n))(1n()[cH(


−θα+β−−+α−β−θ−θ

=2])1n(n))(1n([

])1n(n))(1n()[cH(


β−−+α+α−β−θ−θ

=2])1n(n))(1n([

])1n(n)1n()[cH(


β−−+β−θ−θ

=2])1n(n))(1n([

)]1)(1n(n)[cH(


θ−−β−−θ> 0 (35)

Figures 12 and 13 provide an example that demonstrates that ∂λ/∂α > 0. When α


73.1707 in Figure 13.



121

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



122

(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)



123

g = )(

)())(1(

)1(

)())(1(

)1)(1(

cL

cLn

cLn

H

H

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

+−

−−−

−−−



(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



124

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.



125

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



126

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




127

[ ][ ] [ ][ ]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.



128

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)



129

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


[ ][ ]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



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

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−

+−

−−−

−−−



(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



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.



132

Proof (9g):

g = )()(

))(1()1(

)())(1(

)1)(1(

cLcL

n

cLn

H

H

−⎥⎥⎥

⎥

⎦

⎤

⎢⎢⎢

⎢

⎣

⎡

−−−+−

−−−

−−−



(22)

=( )

( ))(

))()(1()1(

))()(1()1)(1(cL

cLcH

cLcH−⎥

⎦

⎤⎢⎣

⎡

−−−+−−−−−−−−−


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 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 β θ

[ ] ( )[ ]( )[ ]

)())()(1()1(

))()(1()1)(1())(1()1(2

cLcLcH

cLcH−⎥

⎦

⎤⎢⎣

⎡

−−−+−−

−−−−−−−−−−−−−+


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


β α θ θ β α

( ) ( )[ ]( )[ ] ( )[ ]

)(1/))()(1()1(

)1)(1)(()1()1(2

cLcLcH

cHcH−⎥

⎦

⎤⎢⎣

⎡

−−−−+−−

−−−−+−−−+


θα β θ β α θα β θα α

[ ]( )[ ] ( )[ ]

)(1/))()(1()1(

))()(1()())(()1(2

cLcLcH

cLcL−⎥

⎦

⎤⎢⎣

⎡

−−−−+−−

−−−−−−−−−+


β α θ θ β α β α θαθ α

=( )[ ]

( )[ ]( ) ( )[ ] ⎥

⎦

⎤⎢⎣

⎡−−−⎥

⎦

⎤⎢⎣

⎡

−−−+−−

−−−−+−−−

denr

numr

g

gcL

cLcH

LHLHθ θ


β β α α θ θ α β α α θ 1

))()(1()1(

))(1)((1))(1)((2

2

(43)



133

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



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



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



136

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



137

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



138

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



139

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 –



140

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.



141

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



142

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



143

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



144

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)



145

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



146

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



147

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



148

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.



149

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



150

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.



151

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.



152

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



153

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



154

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)



155

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:



156

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



157

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.



158

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.



159

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



160

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.



161

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



162

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



163

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



164

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



165

[ ] 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.



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



167

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



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



169

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)



170

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.



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.



172

[ ] 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.



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)



174

[ ] 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



175

( ) ( )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



176

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.



177

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



178

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.



179

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.



180

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



181

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.



182

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



183

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



184

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.



185

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



186

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



187

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.



188

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 −+=



189

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



190

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



191

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



192

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):



193

( ))(

)()(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):



194

( )[ ] )(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 .



195

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.



196

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



197


> 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



198

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



199

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



200

( )( )

( ) ( )[ ]

( )222

1

2

11221

1

112211

)()(

)()()()()()(

)()(

)()()()()()()(

cppq

pqcppqcppq

cppq

cppqpqpq

c

pF

ss

sssmm

s

ss

sssm

s

−+

+−+−−+−+

−+

−++++−−=

∂

∂

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



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



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 +

+−++−=

∂

∂

( )

( )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



203

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



204

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:



205

( )( )

( )

( )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



206

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.



207

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.



208

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


+++++−=

( ) ( )( )[ ]

( ) )(

)(22

1

2

11212

pR

pR

ss

ssss

α α α


+

++−++

( ) )(

)(222

1

2

2

121

2

12

2

1

2

1

2

1

pR

pR

ss

msssss

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)



209

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



210

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.



211

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 ∞



212

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


++

+−−−+++

=

[ ] [ ]( )

( )ss

s

sssss

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


+

−+++++=

The numerator of this fraction is positive and therefores

LF

α ∂

∂ )(1 is positive when:



213

( ) 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


−

−−−=

−

−−−−<

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



214

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



215

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



216

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 .



217

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

+++−−

++< and the

denominator is positive in the fraction. If the denominator is negative, then

01

1

1 <∂∂

−∂∂

s

FF

α α .



218

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


+−−

+++>

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


+−−

+++<

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 +

−−−−++−=

∂

∂−

[ ] [ ]( ) )(

)(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

++++++++−

=∂

∂−

∂

∂

(15)



219

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

+++−−

++> (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

++++−−

++>

( )

[ ]( ) [ ] )(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


++++−−

+++>

[ ] [ ]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


+++−−−−

+++++>

3

1

2

1

3

422

1

3

1

3

12

2

22

α α α α


+−−

+++>

ss

ssss (15c)

If the denominator is negative in (15c) then for 0)()( 1

1

1 >∂

∂−

∂

∂

s

pFpF

α α



220

3

1

2

1

3

422

1

3

1

3

12

2

22

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 .



221

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



222

*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



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


+++−

++−+>

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


+++−

++−+<

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

+

−< .



224

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


++++−

( ) )(

)(2)(2)(2

1

2

2

2212

2

1

2

121

2

11

pR

pRpRpF

ss

ssm

sss

s 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



++−+

>+++−



225

[ ] )(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


+++−

++−+> (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


+++−

++−+>

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.



226

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



227

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

++++−

+++−+

>

[ ]1

2

1

2

111

1

32

1

2

11

2

1

2

12

2)(2

2)(22



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



+++−−

++−+++>

3

1

2

1

22

1

3

1

3

12

2

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.



228

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



229

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



230

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



231

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



232

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



233

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:



234

( ) ( ) ( )

( )

( )

( )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



235

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

+

+−+⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −−=− ∩

)()(

)()()()(

)(

)()(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



236

( )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 .



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


( ) ( )

( )

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

+

+−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

+

++−+−++−

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

+

+++−=

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


( ) ( ) ( )( )

( )⎟⎟

⎠

⎞⎜⎜

⎝

⎛

+

+−

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

+

+++++++−=

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



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



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 +

+−+++−=

∂−∂ ∩

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

−−−++

−−<

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

−−−++

−−>

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



240

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 −−−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

−+

[ ])()()())(()()()(

)()( 2

121212

1

2

11 pRpRpRpR

pRpRs

mss

sss

m


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



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

−−−++

−−< (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 :



242

( )

( ) ( )( )

)(242)(3

)(22

1

121

211

21

21

21

2m

s

ssm

s

mss

pRpR

pR

α α


α α α α α

+−−−++

−−<

( ) ( )21

21

31

21

21

21

31

31

221

221

31

224233

2222

sssss

ssss



−−−++++

−−−−<

( )

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


−

+−=

+−

−−−=

+−

−−−<

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 :



243

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



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


−−−++++−−

+++−−−++>

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



−−−++++−−

+++−−−++<

32c) Generally( ) ( )

0)(1)(1 21

2

21 >∂

−∂−

∂

−∂ ∩∩

s

pFpF

α α at prices near pm with some exceptions

when α2 is small



245

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 +

+−+++−=

∂

−∂ ∩

( ) ( ) ( )

( )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


+

+−

+

+++++−=

( ) ( )

( )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


+

−−−

+

+

+++−−=

( )=

∂−∂

− ∩

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 −−−⎟

⎟

⎠

⎞⎜⎜

⎝

⎛

+

−−

[ ])()()())(()()()(

)()( 2

121212

1

2

11 pRpRpRpR

pRpRs

mss

sss

m


α α +−++⎟

⎟ ⎠

⎞⎜⎜⎝

⎛

+

−−



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 +

−−−−=

∂

−∂− ∩

( )221

3

23

12

2122

122

13

1

)()()()(32272

pRpRpR

ss

mssss


++++++

( )22

13

22212

212

31

)()(

)(242

pR

pR

ss

ss

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

+

−−−+

+

+++−−=

∂−∂ ∩

( ) ( )=

∂−∂

−∂

−∂ ∩∩

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


+

−−−−

( )22

13

23

13

13

12

2122

122

1

)()(

)()(34257

pR

pRpR

ss

msssss

α α α


+

++++++

( )22

13

231

221

31

2212

212

31

)()(

)(2242

pR

pR

ss

sssss

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


( ) ( )( ) 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


+++

−−−++>



247

( ) ( ) ( )( ) ( ) ( ) 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

−−−++++−−

+++−−−++

>

( ) ( ) ( )( ) ( ) ( ) 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



−−−++++−−

+++−−−++> (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



−−−++++−−

+++−−−++> (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.



248

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



249

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

−−−++++++−−

++++−−−+++

>

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


( ) 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

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++−++++++

−−−−−−



( ) 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


−

+=

−

++=

−

++>

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 .



250

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



251

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


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

+++++−−−

−−−+++<



where ( ) 2321

21

312 )(452 pRK sss α α α α α α −−−−=


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


−−

+++> 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



252

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


( ) ( )( )

( ) ( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

+

++−++

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

+++−−−−=

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


( ) ( )22

13

22

31

2212

21

3121

)()(

)(232)(1

pR

pRpF

ss

msss

s α α α


α +

−−−−=

∂

−∂− ∩

( )22

13

23

12

2122

122

13

1

)()(

)()(32272

pR

pRpR

ss

mssss

α α α


+

+++++

( )22

13

22212

212

31

)()(

)(242

pR

pR

ss

ss

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

+

−−−−−−

=∂

−∂−

∂

−∂ ∩∩

( )22

13

31

322

31

221

2212

21

31

)()(

)()(236392

pR

pRpR

ss

mssssss

α α α


+

+++++++

( )22

13

232

2212

212

31

)()(

)(452

pR

pR

ss

sss

α α α


+

−−−−+ (19)



253

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


( ) ( ) )()(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 (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.



254

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



255

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



−−−−−−+

++++++++=

( ) 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



−−−−+

++++++−−−

( )( )( ) 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




−−−−+

++++++++

−−−−−−−−−=

( ) 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


−−

+++>

321

31

431

221

31

22

2552

ss

ssss

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


−−

+++< (19d)



256

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 ∞



257

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



258

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 +=



259

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)



260

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


1

2211211)(



261

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

+

−−=−

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)



262

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:


( ))(

)()(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



263

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:


( ) ( )( ) )()(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

( )( )

)(

)(

)(

)()()(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 −

+

+=⎥

⎦

⎤⎢⎣

⎡ −−

−

−



264

( ) ( )

( )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



265

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.



266

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



267


> 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


> 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 .



268

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



269

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



270

)( )

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−

−

+−+−−+−+


( ) ( ) [ ]( )( )2

2221

1)()()()(

D

cppqcpcppqpqcppqN

s

m

s

m

ss

⎥⎥⎦

⎤

⎢⎢⎣

⎡

− +−−−++

α α α α α α



271

( ) ( ) ( )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.



272

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



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 −

−

+−+−=



274

( ) ( )[ ]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


+−

−+++−++−−+−=

11

11212121

1

2)(2)()(1

( )( ) ( ) ( )( ) ( )

( ) ( )( ) ( )s

ss

s

msss

mss

Dn

npR

Dn

npRpRn

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


+−

−++

+

+−

−−−+−−−=

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



275

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


>∂∂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



276

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



277

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



278

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


=∂∂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



279

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

+−++−++−+−−

−−<

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

+−++−++−+−−

−−>

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

−+−+−

++<



280

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−

−

++−++


( ) ( )[ ]( )

2

1

2

112211

2)()()(

D

K n

npRpRpR n

n

ssm

s−

−

−−

++−+−


( ) ( )( )[ ]( )ss

mssss

D

pR

α α α


+

++++−

=1

11111 )(2

( ) ( )( )[ ] ( )

( )ss

sssss

Dn

nNpR

α α α


+−−

+−++−++

1

1112121

2)(2



281

( )ss

msssss

D

pR

α α α


+

++++−−=

1

2

121

2

12

2

1

2

1

2

1 )(22

[ ] ( )( )ss

ssssss

D

n

n

NpR

α α α


+−−

+−−−−−++1

1

2

221212

2

1

2

2211

2

)(22

[ ] [ ] ( )

( )ss

sssm

sss

D

n

nNpRpR

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


+−−

−−−−+++=

1

2

121

2

12

2

1

2

121

2

11

2)()(2

[ ] [ ]

( )ss

ssss

D

n

npRpR

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


+

⎥⎦

⎤⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −−

−⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−

=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


+

⎥⎦

⎤⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −−

−−+⎟ ⎠

⎞⎜⎝

⎛ −−

−−+⎟ ⎠

⎞⎜⎝

⎛ −−

−−

+ 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



282

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

+−++−++−+−−

−−< (26a)

At p = pm =∂

∂ −

s

mpF

α

)(1

( )ss

msss

D

pR

n

n

n

n

n

n

n

n

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



283

)(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



284

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



285

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


+

⎥⎦

⎤⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −−

−⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−

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


++

⎥⎦

⎤⎢⎣

⎡⎟

⎠

⎞⎜

⎝

⎛

−

−−−+⎟

⎠

⎞⎜

⎝

⎛

−

−−−+⎟

⎠

⎞⎜

⎝

⎛

−

−−−

+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


+

⎥⎦

⎤⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −−

−−+⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−

=

( )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


+

⎥⎦

⎤⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−−

+

( )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



286

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 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.



287

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



288

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:


( )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:




289

⎟⎟ ⎠

⎞⎜⎜⎝

⎛

+

−=− −

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


(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



290

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



291

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



−+−+−+−+−

−−−++>

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



−+−+−+−+−

−−−++<

58c) At high prices near p= pm, ( )s

pFn

pF

α α ∂

∂−−

∂

∂ )(1

)( 1

2

1 >0



292

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 +

+++−=

++−=

∂

∂

1

211111

2

1 )()()()()(

( ) ( )( )ss

sssm

ss

D

pRpRpF

α α α


α +

+++−−=

∂

∂

1

321

21

21

21

2

1 )(2)()(

( )[ ] ( ) ( ) ( )[ ]{ }( ) ( )ss

ssm

s

s

Dn

K pRnnnpRnn

pF

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

+

−+−++−++−+−−−

=∂

∂−−

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

+

−+−++−++−+−−−+

+

+++−−

=∂

∂−−

∂

∂

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


+

−+−++−++−+−−−=

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 −−−++=



293

( )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



−+−+−+−+−

−−−++> (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

−+−+−++−+−

−−−+++

>

( ) ( ) ( ) 21

21

31

31

221

21

238453

2

ss

sss

nnn α α α α α


−+−+−

++>



294

( ) ( ) ( ) 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 ∞



295

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

+−+−+−+−−−−

−+−+>



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

+−+−+−+−−−−

−+−+<



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



296

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

+−

−+++

+−

−−−+−−−=

∂

∂

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

+−

−+++

+−

−−−+−−−=

∂

∂

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

+−

−+−++−++−+−−−

=∂

∂−

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

+−

−−+

+−

−+−+−+−+−

=∂

∂−

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

+−

−−−−+

+−

−+++−−−−=

∂

∂

11

221

31

11

321

21

21

212

1

1

1

)(232

1

)(22)(232)(



297

( ) ( ) ( )[ ]{ }

( ) ( )( ) ( ) ( ) ( )[ ]{ }

( ) ( )

( ) ( )[ ]( ) ( )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

+−

−−−−−+

+−

−+−+−+−+

+−

−+−−−−

=∂

∂−

∂

∂

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

( ) ( )

( ) ( ) ( )[ ] ( ) ( )[ ] )(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

+−+−+−+−−−−

−+−+>



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


−+−+−+−

−+−+> (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,



298

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



299

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

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



( ) ( )

( ) ( ) ( )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



300

( )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



301

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



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



303

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



304

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 +

−+=

+

++−=

∂−∂

−∂

−∂ −−



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



306

( ) ( )

( )

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

⎟⎟⎟

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎜⎜⎜⎜

⎝

⎛

⎥⎦⎤⎢

⎣⎡ −+

+−+−=−

−

−∩ −

)(

)()(

)()()()(

)()(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



307

( ) ( )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



308

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


( ) ( )( ) ( ) ( )( ) ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −⎟⎟ ⎠

⎞⎜⎜⎝

⎛

+−

+−−++−−

)(

)()(

1

)(1)(1

11

2

12112

2

1

pR

pRpR

Dn

pRnpRn

s

m

s

sm

ss

α α α α




309

( )( ) ( )( )( ) ( )

( )( ) ( )( )[ ]

( ) ( )( )( )

( ) ( )

( )( )( ) ( )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


+−

−++−

+−

−+++

+−

−++−++

+−

−−−+−−−=

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


+−

−++−

+−

−−−−−−

+−

−−−−−

+−

−+++−

( )( ) ( )( )( ) ( ) )(1

)(3243

11

2312

31

2212

21

pRDn

pRnn

ss

msss

α α α α


+−

−−−+−−−=

( )( ) ( )( ) ( )( ) ( )( )( ) ( ) )(1

)()(326443107

11

221

312

31

2212

21

pRDnpRpRnnnn

ss

mssss


+−−++−+−+−+

( )( )( ) ( ) )(1

)(322

11

22212

212

31

pRDn

pRn

ss

ss

α α α α


+−

−++− (32a)

QED

Suppose that firm two’s loyal customers α2 are changed. The effect on )(1 121pF n−∩

− is:



310

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

+−++−=

∂

−∂ −∩ (33a)



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

+−−−−−−++−

++−−<



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

+−−−−−−++−

++−−>



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



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

+

⎥⎦

⎤⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −−

−⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−+⎟ ⎠

⎞⎜⎝

⎛ −−

−⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

=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

+

⎥⎦

⎤⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −−

−−+⎟ ⎠

⎞⎜⎝

⎛ −−

−−+⎟ ⎠

⎞⎜⎝

⎛ −−

−−⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

+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

+

⎥⎦

⎤⎢⎣

⎡⎟

⎠

⎞⎜

⎝

⎛

−

−−+⎟

⎠

⎞⎜

⎝

⎛

−

−−+⎟

⎠

⎞⎜

⎝

⎛

−

−−+⎟

⎠

⎞⎜

⎝

⎛

−

−−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −

=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

+

⎥⎦

⎤⎢⎣

⎡⎟ ⎠

⎞⎜⎝

⎛ −−

−−+⎟ ⎠

⎞⎜⎝

⎛ −−

−−+⎟ ⎠

⎞⎜⎝

⎛ −−

−−⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

+1

2

2212

2

111 )(

1

22

1

224

1

22

)(

)()(



313

( )[ ]

( ) ( )ss

msss

s

m

Dn

pRnnnpR

pRpR

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

+−

+−++−++−⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −

+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


+−

++−+=

( )

( ) ( )

( ) ( ) ( )[ ]( ) ( ) )(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


+−+−++−++−+

+−

−−−−−+

( ) ( ) ( )

( ) ( ) )(1

)(438643

1

2

22

212

2

12

3

1

pRDn

pRnnn

ss

ss

α α α


+−

−+−+−+

( )

( ) ( ) )(1

)(12

1

2

22

3

1

22

12

2

1

3

1

pRDn

pRnnn

ss

msss

α α α


+−

++−+=

( ) ( ) ( )

( ) ( ) )(1

)()(433398

12

2

212

3

1

22

12

2

1

3

1

pRDn

pRpRnnnnn

ss

mssss

α α α


+−

−−−−−−−−+

( ) ( ) ( )

( ) ( ) )(1

)(438643

1

2

22

212

2

12

3

1

pRDn

pRnnn

ss

ss

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


)()()(22

1

3

1222

1

3

1 pRpRnnpRnn mss

mss α α α α α α α α ++−−



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

+−−−−−−++−

++−−<



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 (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



−+−+−+−−−−−−

++−−<

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



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.


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



316

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

++−−−−−−+++−

++++−−<



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

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)



317

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



318

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:





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

+−++−=

∂

−∂ −∩

)

( ) ( )( ) ( )( )

( ) )(

)()()(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

+

++−++++−

=∂

−∂ −∩



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


+

−−−−+

+

++++−−=

( )

[ ]( ) )(

)(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


+

−−−+

+

+++−−=

( )( ) ( )

( ) )(

)(12)(11

1

2

22

3

1

22

12

2

1

3

121 1

pRD

pRnnnpFn

ss

msss

s

n

α α α


α +

++−+−=

∂

−∂−−

−∩

( ) ( ) ( )

( ) )(

)()(433398

1

2

2

212

3

1

22

12

2

1

3

1

pRD

pRpRnnnnn

ss

mssss

α α α


+

−−−−−−−−−+

( ) ( ) ( )

( ) )(

)(438643

1

2

22

212

2

12

3

1

pRD

pRnnn

ss

ss

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

+

−−−+−+−

=∂

−∂−−

∂

−∂ −− ∩∩

(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


+

−−−−−−−−−+

+

−+−+++−++++

( )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


+++

+−+−−++++

>⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−−−−+

−+−+−+−−−



322

( )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


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

+−+−+−+−−−

++−+−−++++>



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


−−−−−−+−+−+−

+++−−−>

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).



323


( )( )

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 ∞



324

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

++−+−+−++−−−

+++−+−−+++++>





325

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

+−−−−−−−−−

++++++++++++>



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


where

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

−−−−−−

−+−+−+−+−+−=

221

31

41

31

31

221

221

41

31

13

438643

433398433398

ss

sssss

nnn

nnnnnnK

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


−+−+−

++>

( )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).



326

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



327

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




328

(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

+−+−+−−−−−

−−−−−+−+−+<



(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


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



329

Proof:

1

21)(1 1

α ∂

−∂ −∩pF n ( )( ) ( )( )

( ) ( ) )(1

)(3243

11

2312

31

2212

21

pRDn

pRnn

ss

msss

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


+−

−++−+−+−+

( )( )( ) ( ) )(1

)(322

11

22212

212

31

pRDn

pRn

ss

ss

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


+−

−−−+−−−=

( )( ) ( )( ) ( )( )

( ) ( ) )(1

)()(6443107

12

1

23

132

12

22

1

pRDn

pRpRnnn

ss

msss

α α α α


+−

−+−+−+

( )( ) ( )( )

( ) ( ) )(1

)(322)()(32

12

1

2321

22

212

31

321

231

pRDn

pRnpRpRn

ss

sssm

ss

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


+−

++−+=

( ) ( ) ( )

( ) ( ) )(1

)()(433398

1

2

2

212

3

1

22

12

2

1

3

1

pRDn

pRpRnnnnn

ss

mssss

α α α


+−

−−−−−−−−+

( ) ( ) ( )

( ) ( ) )(1

)(438643

1

2

22

212

2

12

3

1

pRDn

pRnnn

ss

ss

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


+−

++−+=

( ) ( ) ( )

( ) ( ) )(1

)()(433398

12

1

2

2

2

12

4

1

23

12

3

1

4

1

pRDn

pRpRnnnnn

ss

mssss

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


+−

−+−+−+



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


+− −−−+−−−=

( )

( ) ( ) )(1

)(12

1

2

1

22

4

1

23

12

3

1

4

1

pRDn

pRnnn

ss

msss

α α α α


+−

++−+−

( )( ) ( )( ) ( )( )

( ) ( ) )(1

)()(6443107

12

1

23

132

12

22

1

pRDn

pRpRnnn

ss

msss

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


+−

−−−−−−−−−

( )( ) ( )( )

( ) ( ) )(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


+−

−++−−++

( ) ( ) ( )

( ) ( ) )(1

)(438643

1

2

1

22

2

2

12

3

12

4

1

pRDn

pRnnn

ss

ss

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


+−

−−−−−−−−+−=

( ) ( ) ( ) ( )

( ) ( ) )(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


+−

−+−+−+−++

( ) ( )

( ) ( ) )(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


+−

−−−−−−−−+

s

pFpF nn

α α ∂−∂−

∂−∂ −− ∩∩ )(1)(1 11 21

1

21 is positive iff the numerator of equation (36) is positive.



331

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

−−−−−+−+−+

>

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−−−−−−−−+

−+−+−+−+

−−−−−

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

where( ) ( )[ ]

( ) ( ) ( ) ( )[ ] ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−−−−−−+

−+−=

23

1

22

1

3

1

4

1

3

1

22

1

14)(3210711843

)()(321410

pRnnnn

pRpRnnK

sss

mss


α α α α

( ) ( ) ( ) ( )

( ) ( )[ ] ( ) ( )[ ] 1431212312121

31

221

31

2221

31

31

2 )()(331512)(4443

)()(4333)(3343

K pRpRnnpRnn

pRpRnnnpRnnn

msmss

msss

msss

+−+−+−−−−−

−−−−−+−+−+>



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



332

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.



333


( )( )

s

mm pFn

pF nn

α α ∂

−∂−−

∂

−∂ −− ∩∩)(1

1)(1 11 21

1

21

α1 αs n R(p

m


(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


(36a)

( )

( )( )

s

m

m

pF

n

pF

n

n

α

α

∂

−∂

−−

∂

−∂

−

−

∩

∩

)(1

1

)(1

1

1

21

1

21



334

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

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 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

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


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

( ) ( ) ( )( ) ( ) ( ) 42

133

124

15

1

521

431

341

251

2128453

437634

ssss

ssss

nnn

nnnn



−−−−+−

−+−+−+>



335

( ) ( ) ( )

( ) ( ) ( ) 32

1

2

1

3

1

43

1

22

1

3

12

128453

437634

sss

ssss

nnn

nnnn

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


−−−−+−

−+−+−+< (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.



336

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



337

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.



338

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



339

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



340

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



341

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.



342

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



343

200 400 600 800 1000 Pri ce

0. 002

0. 004

0. 006

0. 008


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



344

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



345

500 1000 1500 2000 Pri ce

0. 002

0. 004

0. 006

0. 008


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





347

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



348

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


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



349

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


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



350

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





352

Figures 31e and 31f

0

. 0 0 5

. 0 1

. 0 1 5

D e n s i t y

0 100 200 300 400 500nycmcof

NYC – MCO Kiwi YD

0

. 0 0 5

. 0 1

. 0 1 5

D e n s i t y

0 100 200 300 400nycmcof

NYC – MCO Trans World YD



353

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

0

. 0 0 5

. 0 1

D e n s i t y

0 200 400 600 800nycmcof

NYC – MCO US Airways YD



354

Figures 31i and 31j

0

. 0 0 5

. 0 1

. 0 1 5

D e n s i t y

0 100 200 300 400nycmcof

NYC – MCO Continental Y

0

. 0 0 5

. 0 1

. 0 1 5

. 0 2

D e n s i t y

0 200 400 600nycmcof

NYC – MCO Delta Y



355

Figure 31k

0

. 0 0 5

. 0 1

. 0 1 5

. 0 2

. 0 2 5

D e n s i t y

0 100 200 300 400 500nycmcof

NYC – MCO US Airways Y



356

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



357

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

0

. 0 0 1

. 0 0 2

. 0 0 3

. 0 0 4

. 0 0 5

D e n s i t y

0 200 400 600 800 1000mspatlf

YD both major carriers



358

Figures 32c and 32d

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

YD Northwest Airlines

0

. 0 0 1

. 0 0 2

. 0 0 3

. 0 0 4

. 0 0 5

D e n s i t y

0 200 400 600 800mspatlf

YD Delta Airlines



359

Figures 32e and 32f

0

. 0 2

. 0 4

. 0 6

D e n s i t y

0 200 400 600 800 1000mspatlf

Y Delta Airlines

0

. 0 0 1

. 0 0 2

. 0 0 3

. 0 0 4

. 0 0 5

D e n s i t y

0 200 400 600mspatlf

Y Northwest Airlines



360

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


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



361

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



362

Figure 35

0

. 0 0 0 5

. 0 0 1

. 0 0 1 5

. 0 0 2

. 0 0 2 5

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



363

Bibliography:

Aguilar, Louis. “In Frontier Sights; Airline Targets United’s Elite Customers,” TheDenver Post: February 27, 2004, Business, p. C01.

AirTran Press Releases August 17, 2004:AirTran Airways Launches 48-Hour Sale for Travel To and From Philadelphia AirTran Airways Launches 48-Hour Sale for Travel To and From Baltimore

Air Transport Association 2004 Economic Report.http://www.airlines.org/home/default.aspx http://www.airlines.org/attachments/bin/vday/2004AnnualReport.pdf

American Airlines Press Release: April 27 ,2005:http://www.aa.com/content/amrcorp/pressReleases/2005_04/27_mothers_day_sal

e.jhtml

American Express Business Travel Monitor Press Release: November 22, 2004.http://home3.americanexpress.com/corp/pc/2004/BTM_print.asp

Anderson, John H. Jr. “Changes in Airfares, Service, and Safety Since Deregulation”United States General Accounting Office testimony before the US SenateCommittee on Commerce, Science, and Transportation: Thursday, April 25, 1996.http://www.gao.gov/archive/1996/rc96126t.pdf

Bailey, Elizabeth E., Graham, David R., and Kaplan, Daniel P. Deregulating the

Airlines. Cambridge, Massachusetts: The MIT Press, 1991.

Baye, Michael R. and John Morgan. “Necessary and Sufficient Conditions for Existenceand Uniqueness of Bertrand Paradox Outcomes,” Woodrow Wilson School of Public and International Affairs Economics Working Papers #186, PrincetonUniversity, 1997.

Bjorhus, Jennifer. “Business Travelers Hold the Wild Card,” Pioneer Press: SundayFebruary 23, 2003.

Borenstein, Severin. “Hubs and High Fares: Dominance and Market Power in the

Airline Industry, RAND J ournal of Economics: Vol. 20, No. 3, Autumn 1989, p.344 – 365.

Brannigan, Martha, Carey, Susan, and McCartney, Scott. “Fed Up with Airlines,Business Travelers Start to Fight Back,” Wall Street Journal:Tuesday August 28,2001, p. A1 column 1.



364

Burdett and Judd. “Equilibrium Price Dispersion,” Econometrica: Volume 51, Number 4, July 1983, p. 955 – 969.

Butters, Gerald. “Equilibrium Distribution of Sales and Advertising Prices,”

Review of Economic Studies: Volume 44, October 1977, p. 465 – 91.

Carlson, John A. and R. Preston McAfee. "Discrete Equilibrium Price Dispersion," J ournal of Political Economy: 1983, Volume 91, Number 3, p. 480 - 493.

Delta Airlines Press Release January 4, 2005:Delta Slashes Everyday Fares Up To 50 Percent As Airline IntroducesSimpliFares™ Nationwide (04 Jan 2005)

Deneckere, Ray and Preston R. McAfee. "Damaged Goods," J ournal of Economics andManagement Strategy, Volume 5, No. 2, Summer 1996, p. 149 - 74.

Diamond, Peter A. "A Model of Price Adjustment," J ournal of Economic Theory: June1971, Volume 3, p. 156 - 68.

Fedor, Liz. “Airlines Fight for Business Travelers,” Minneapolis-St. Paul Star Tribune:reprinted in The Sun: October 9, 2003.

Grantham, Russell. “Plane Fares Creep Higher; Switcheroo: Low-cost Carriers Able toRaise Prices, Unlike Big Counterparts, The Atlanta J ournal-Constitution:Thursday March 11, 2004, Business, p. 1F.

Hamburger, Tom. “Airline Mergers May Face Heavy Flack; Airline Deregulation in the1980’s Prompted a Wave of Mergers, But the Regulatory Climate Has ChangedSince Then,” Star Tribune: Monday, June 12, 2000, News, p. 1A.

Hamilton, Martha M. “Airline Pricing: Highly Complex, Hotly Competitive,” TheWashington Post: November 20, 1988, Financial, p. H1.

Heller, Jean. “Airlines Return Perks At a Price,” St. Petersburg Times: March 12, 2004,Business, On-Line.

http://www.aa.com

http://www.delta.com http://news.delta.com/news.index.cfm

http://www.flyi.com http://www.flyi.com/company/pressarchive/default.aspx 08/02/05: Independence Air Fare Sale Offers Cool Fall Travel Deal



365

http://www.webflyer.com http://www.webflyer.com/company/press_room/facts_and_stats

Jevons, William S. The Theory of Political Economy. First Edition, London, England:Macmillian and Co., 1871.

Koenig, David. “Southwest Lowers Last Minute Fares to Lure Business Travelers,” TheAssociated Press: reprinted in the Seattle Post-Intellinger: Friday, August 23,2002.

Kreps, David M. and José A. Scheinkman. “Quantity Precommitment and BertrandCompetition Yield Cournot Outcomes,” Bell J ournal of Economics, Autumn1983, p. 326 – 337.

Maynard, Micheline. “Budget Airlines Set Off On a Cross-Country Joy Ride,” The New York Times: Sunday April 3, 2005, Section 3, Sunday Business, p. 7.

McAfee, R. Preston. “Endogenous Availability, Cartels, and Merger in an EquilibriumPrice Dispersion,” J ournal of Economic Theory: Volume 62, Number 1, February1994, p. 24 – 47.

McAfee, R. Preston. "Multiproduct Equilibrium Price Dispersion," J ournal of Economic Theory: October 1995, Volume 67, Number 1, p. 83 - 105.

Nahata, Babu, Krzysztof Ostaszewski and P.X. Sahoo. “Direction of Price Changes in

Third Degree Price Discrimination,” American Economic Review: Volume 80, Number 5, December 1990: p. 1254 - 1258.

Nazareno, Analisa. “A Special Offer; Retailers Still Offer Coupons Even Though Their Use by Consumers Has Been Declining,” San Antonio Express-News: November 1, 2003, Metro: Business Express, p. 8H.

Reed, Dan. “Cost of Flying for Business Gets Cheaper,” USA Today: Friday July 18,2003, News, p. 1A.

Rob, Rafael. "Equilibrium Price Distributions," Review of Economic Studies: July 1984,

Volume 52, Issue 3: p. 487 - 504.

Robert, Jacques and Dale O. Stahl II. “Informative Price Advertising in a SequentialSearch Model,” Econometrica: Volume 61, Issue 3, May 1993, p. 657 – 686.



366

Rothschild, Michael. “Models of Market Organization with Imperfect Information: ASurvey,” J ournal of Political Economy: Volume 81, Number 6, November/December 1973: p,1283 – 1308.

Salop, Steven. "The Noisy Monopolist: Imperfect Information, Price Dispersion, andPrice Discrimiation," Review of Economic Studies: 1977, Volume 44.

Salop, Steven and Joseph Stiglitz. "Bargains and Ripoffs: A Model of MonopolisticallyCompetitive Price Dispersion," Review of Economic Studies: October 1977,Volume 44: p. 493 - 510.

Salpukas, Agis. “American: Disciplinarian of the Skies,” The New York Times: FridayMay 29, 1992, p. D1.

Schlesinger, Jacob M. "If E-Commerce Helps Kill Inflation, Why Did Prices Just

Spike?", Wall Street Journal: October 18, 1999: p. A1 col 6.

Schmalensee, Richard. “Output and Welfare Implications of Monopolistic Third DegreePrice Discrimination,” American Economic Review: Volume 71, March 1981, p.242 – 7.

Schmitt, Eric. “The Art of Devising Air Fares,” The New York Times: Wednesday,March 4, 1987, Section D, page 1, Column 3.

Sloane, Leonard. “Why an $873 Flight Can Also Cost $124,” The New York Times:Saturday, April 2, 1988, Section 1, Page 28, Column 3.

Southwest Airlines Press Release: February 28, 2005:http://www.southwest.com/about_swa/press/prindex.html Southwest Airlines Sends Hottest Fares Directly to Customers' Computer Desktops

Southwest Airlines Press Release: July 5, 2005:http://www.southwest.com/about_swa/press/prindex.html

Southwest Airlines Offers $39 One-Way Fares From Phoenix and SouthernCalifornia Cities to Las Vegas Southwest Airlines Offers $39 - $79 One-Way Fares in Select Florida Markets

Southwest Airlines Offers $39 One-Way Fares in Select Midwest Markets Southwest Airlines Offers $39 - $69 One-Way Fares in Select Texas andHeartland MarketsSouthwest Airlines Offers $39 - $89 One-Way Fares in Select Northwest Markets Southwest Airlines Offers $49 One-Way Fares in California Southwest Airlines Offers Cheap Airfares as Low as $39 One-Way



Stahl, Dale O. II. "Oligopolistic Pricing with Sequential Consumer Search," AmericanEconomic Review: September 1989, Volume 79, Number 4, p. 700 - 712.

Stigler, George J. "The Economics of Information," The Journal of Political Economy:

Volume 69, Number 3, p. 213 - 225.

Stoller, Gary. “Airlines’ ‘Million Milers’ Get Quiet Perks,” USA Today: February 22,2005, Money, p. 8B.

United Airlines Press Release: October 20, 1999.Current internet site documenting the original press release:http://www.findarticles.com/p/articles/mi_m0CWU/is_1999_Oct_25/ai_56752449

Varian, Hal. “A Model of Sales,” American Economic Review: Volume 70, Number 4,

September 1980, p. 651 – 9.

Varian, Hal. “Price Discrimination and Social Welfare,” American Economic Review:Volume 75, September 1985, p. 870 – 75.

Walsh, Chris. “Airline Jobs Descend; But Low-Cost Carriers Lead Hiring Rebound – AtWages’ Expense,” Rocky Mountain News: Tuesday March 15, 2005, Business, p.1B.

Wiesmeth, Hans. "Price Discrimination Based on Imperfect Information: Necessary andSufficient Conditions," Review of Economic Studies: Volume 49, 1982, p. 391 –

402.

Wilensky, Dawn. “Targeting: Key Strategy for Coupons,” BrandMarketing Supplementto Supermarket News: Vol 5, No. 9, September 1998, p. 20.

Williams, Ian. “Innovation: ‘Dark Science’ Brings a Boost to Airline Profits,” The Times (London): Sunday, November 22, 1987, Issue 8520.

Documents

Airline Pricing and Capacity Behavior