dynamic pricing in the airline industry a dissertation ...kk032tn6364/... · abstract the...

DYNAMIC PRICING IN THE AIRLINE INDUSTRY

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL OF BUSINESS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

John Lazarev

June 2012

This dissertation is online at: http://purl.stanford.edu/kk032tn6364

© 2012 by John Lazarev. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

ii

http://purl.stanford.edu/kk032tn6364

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Peter Reiss, Primary Adviser


Charles Benkard, Co-Adviser


Andrzej Skrzypacz

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

The dissertation consists of two essays on different aspects of dynamic pricing with

applications to the U.S. airline industry.

The first essay studies how a firm’s ability to price discriminate over time affects

production, product quality, and product allocation among consumers. The theoret-

ical model has forward-looking heterogeneous consumers who face a monopoly firm.

The firm can affect the quality and quantity of the goods sold each period. I show

that the welfare effects of intertemporal price discrimination are ambiguous. I use this

model to study the time paths of prices for airline tickets offered on monopoly routes

in the U.S. Using estimates of the model’s demand and cost parameters, I compare

the welfare travelers receive under the current system to several alternative systems,

including one in which free resale of airline tickets is allowed. I find that free resale of

airline tickets would increase the average price of tickets bought by leisure travelers

by 54%

The second essay, motivated by pricing practices in the airline industry, studies

the incentives of players to publicly and independently limit the sets of actions they

play later in a game. I find that to benefit from self-restraint, players have to exclude

all actions that create deviations for them and keep some actions that can deter

deviations of others. I develop a set of conditions under which these strategies form

a subgame perfect equilibrium and show that in a Bertrand oligopoly, firms can

mutually gain from self-restraint, while in a Cournot oligopoly they cannot.

v

Acknowledgments

I am truly indebted and thankful to the members of my reading committee. I thank

Peter Reiss, Lanier Benkard, and Andy Skrzypacz for helping me to transition from

a student to an academic researcher. I was very lucky to write my dissertation under

their guidance. They have always had time to talk about research, to read every draft

of my papers and to give their honest feedback. They have motivated me to work

harder. They have showed me how much hard work can achieve. Without them, this

dissertation would not have been possible.

I thank Mike Ostrovsky for guiding me through every step of the PhD program.

His advice and support have been invaluable. Conversations with Bob Wilson helped

me a lot in understanding the microeconomic foundations of dynamic pricing.

I am grateful to Stanford professors for teaching amazing classes in microeconomic

theory, industrial organization, and econometrics: David Kreps, Ilya Segal, Jeremy

Bulow, Jon Levin, Liran Einav, Tim Bresnahan, Takeshi Amemiya, Han Hong, Joe

Romano.

Sergei Guriev and Konstantin Sonin from New Economic School introduced me

to modern economic research. Together with Andrei Bremzen and Omer Moav, they

made it possible for me to get into the best Ph.D. programs in the world. Jeremy

Bulow and Ilya Strebulaev helped me to make the right decision and choose the Ph.D.

program at Stanford GSB.

I have benefited from many conversations about research with fellow students Tim

Armstrong, Alex Frankel, Ben Golub, Alexander Gorbenko, and Przemek Jeziorski.

Many comments and suggestions have improved individual essays in this work;

these are gratefully acknowledged at the end of each chapter.

vi

I would like to thank Jaime Andrade for his support and encouragement during

the job market process. Without him, I would never achieve what I have done so far.

I thank my mother, Nataliya Lazareva, for her love and faith in me. None of this

would have been possible without her.

Finally, I thank my undergraduate adviser, Anna Vuros, for introducing me to the

field of Industrial Organization. Her encouragement, help, and guidance have been

essential throughout my academic life, and I dedicate this dissertation to her.

vii

Contents

Abstract v

Acknowledgments vi

1 Introduction 1

2 Intertemporal Price Discrimination 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Institutional Background . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 The Model of Optimal Fares . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Airline’s problem . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.2 Demand System and Consumer Welfare . . . . . . . . . . . . 17

2.3.3 Optimal Price Path . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Monopoly Markets . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.2 Data Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5.1 Econometric Specification . . . . . . . . . . . . . . . . . . . . 25

2.5.2 Moment Restrictions . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.2.1 Daily prices . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.2.2 Monthly traffic . . . . . . . . . . . . . . . . . . . . . 29

2.5.2.3 Quarterly sample of tickets . . . . . . . . . . . . . . 30

2.5.3 Estimation Method and Inference . . . . . . . . . . . . . . . . 31

2.5.4 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

viii

2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6.1 Demand and Cost Estimates . . . . . . . . . . . . . . . . . . . 34

2.6.2 Optimal Price Path and Price Elasticities . . . . . . . . . . . . 35

2.6.3 Welfare Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7 Counterfactual Simulations . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7.1 Costless resale . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.7.2 The role of cancellation fee . . . . . . . . . . . . . . . . . . . . 42

2.7.3 Direct price discrimination . . . . . . . . . . . . . . . . . . . . 44

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.9 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Getting More from Less 47

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Elements of a commitment equilibrium . . . . . . . . . . . . . . . . . 53

3.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3.2 Reward, temptation, and punishment . . . . . . . . . . . . . . 55

3.3.3 Relation to Nash equilibrium . . . . . . . . . . . . . . . . . . 56

3.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Subgame supermodular games . . . . . . . . . . . . . . . . . . . . . . 59

3.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4.2 Single-action deviation principle . . . . . . . . . . . . . . . . . 61

3.4.3 Subgame best response . . . . . . . . . . . . . . . . . . . . . . 62

3.4.4 Subgame equilibrium response . . . . . . . . . . . . . . . . . . 64

3.4.5 Credibility of punishment . . . . . . . . . . . . . . . . . . . . 66

3.5 Stackelberg set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Concluding comments . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

A Duopoly with differentiated products 77

ix

List of Figures

2.1 List of fares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Example Price Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Dynamics of active buyer . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Optimal price path . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.6 Distributions of travelers’ utilities . . . . . . . . . . . . . . . . . . . . 38

2.7 Resale (constant marginal costs) . . . . . . . . . . . . . . . . . . . . . 40

2.8 Resale (fixed capacity) . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.9 Zero cancellation fee . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.10 Third degree price discrimination . . . . . . . . . . . . . . . . . . . . 44

3.1 Subgame best responses I . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Subgame best responses II . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3 Stackelberg set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.1 Differentiated Bertrand Duopoly . . . . . . . . . . . . . . . . . . . . . 79

x

Chapter 1

Introduction

This dissertation theoretically and empirically studies several aspects of dynamic

pricing in the airline industry.

Dynamic pricing is the practice of charging different prices for the same product

in different periods of sale. The product usually has two key characteristics. First, its

value becomes zero at a point of time. Typical examples include airline tickets, concert

tickets, hotel rooms, cruises. Second, the total quantity of the product (”capacity”)

is fixed and the marginal costs of changing it are relatively high.

There are two reasons why a firm can benefit from changing the price of such a

product over time. First, by doing so, the firm can indirectly segment consumers based

on their sensitivity to the time of purchase. If the sensitivity to the time of purchase

is correlated with price sensitivity for different customers, the firm can extract more

surplus from customers with lower price sensitivity. Second, if the aggregate demand

for the product is uncertain, the firm has an incentive to adjust the price based on the

remaining capacity. If the actual sales are less than expected, the firm will decrease

the price in the next period of sale. If they are more than expected, the firm will

increase the price.

Chapter 2 of this dissertation studies the first reason for dynamic prices: intertem-

poral price discrimination. Although passengers often complain about not being able

to sell previously purchased tickets if they don’t need them, the industry explicitly

prohibits any form of ticket resale and has enough means to enforce it. The absence

1

2 CHAPTER 1. INTRODUCTION

of secondary markets leads to inefficiencies in the ex-post allocation of airline tickets.

However, it also allows airlines to price discriminate over time, which may increase

social welfare and be beneficial for some consumers.

This chapter shows that the welfare effects of intertemporal price discrimination

are theoretically ambiguous and require an empirical investigation. This essay de-

velops an econometric model of dynamic monopoly pricing that captures relevant

institutional details of the industry. In the context of this model, I show how to

identify multidimensional consumers’ preferences from observed price trajectories un-

der the assumption that the airline maximizes its profit. Based a manually collected

sample of airline fares offered by U.S. carriers in monopoly markets, I estimate the

model’s demand and cost parameters. Using these estimates, I find the equilibrium

of the model under different alternative assumptions: (1) free resale of airline tickets,

(2) zero cancellation fee, (3) third degree price discrimination.

Chapter 3 of the dissertation is a theoretical study that investigates the incentives

of players to voluntarily limit the set of actions that they will later play in a game.

I consider a class of games that consist of two stages. In stage 0 (the commitment

stage), players can simultaneously and independently decide to constrain their actions.

These choices are then publicly observed. In stage 1 (the action stage), players

independently and simultaneously choose actions from their stage 0 constrained action

sets. Payoffs are then realized. The paper characterizes pure-strategy, subgame-

perfect Nash equilibria of such games and compares these equilibria to the pure-

strategy Nash equilibria of stage 1 games when players cannot restrict their actions.

Surprisingly, even though players in such a game cannot mutually commit to cer-

tain outcomes, or use repeated interactions, they may still benefit from independent

self-restraint. I develop a set of conditions under which self-restraint is a subgame

perfect equilibrium strategy. These results show, for example, why in a Bertrand

oligopoly firms can mutually gain from self-restraint while in a Cournot oligopoly

they cannot.

The tools developed in Chapter 3 allow us to analyze pricing competition in the

U.S. airline industry. Both the organization structure of airlines and their revenue-

management practices give airlines the ability to commit to a fixed set of fares. Almost

3

every major US airline has independent pricing and yield (revenue) management

departments. That operates as follows. The pricing department sets prices for each

seating class (e.g. up to 6 non-refundable economy class fares) starting many days

from the actual flight. These prices are subsequently updated rarely and this decision

never depends on a particular flight. The revenue management department treats

the prices as given but decides three times a day which of the fare classes to make

available for purchase and which to keep closed for each particular flight. According to

industry insiders, these departments do not actively interact with each other. Thus,

there exist two stages of decision making. Effectively, the pricing department commits

to a subset of prices, while the revenue management department chooses a price from

this subset. The Airline Tariff Publishing Company (ATPCO), jointly owned by

several airlines, collects the quoted fares from more than 500 airlines three times

a day and distributes this information to all airlines, travel agents, and reservation

systems. A Harvard case study known as American Airlines Value Pricing (1992)

confirms the prediction of the theoretical model.

The dissertation provides economists and policymakers with necessary tools to

analyze firms’ dynamic pricing decisions. It shows how to apply these tools to ad-

dress three questions about an important industry of the U.S. economy. Chapter 2

demonstrates that unlike in a static environment, firms’ dynamic pricing decisions de-

termine not only the total quantity sold but also the allocation of this quantity across

heterogeneous consumers. The benefits of secondary markets that lead to a better

allocation of products may be outweighed by their side effect as firms may decrease

the quantity produced or even exit the market. Policymakers should be aware of the

existence of this side effect, and take into account its magnitude, which can be quan-

tified using the estimation method proposed in the first essay. The tools developed in

Chapter 3 show that revenue-management practices currently employed by U.S. air-

lines together with the mechanism of distributing airline fares create anticompetitive

incentives that might facilitate collusive behavior in the industry.

Chapter 2

The Welfare Effects of

Intertemporal Price

Discrimination: An Empirical

Analysis of Airline Pricing in U.S.

Monopoly Market

2.1 Introduction

This essay estimates the welfare effects of intertemporal price discrimination using

new data on the time paths of prices from the U.S. airline industry. Who wins and

who loses as a result of this intertemporal price discrimination is an important policy

question because ticket resale among consumers is explicitly prohibited in the U.S.,

ostensibly for security reasons. Some airlines do allow consumers to ”sell” their tickets

back to them, but they also impose fees that can make the original ticket worthless.

Just what motivates these practices is a matter of public debate.1 Economic theory

1Consumer advocates speak out against these inflexible policies and question the legality of suchpractices. If you buy a ticket, they argue, it’s your property and you should be able to use it anyway you want, including giving it to a friend or selling it to a third party. For examples see Bly

4

2.1. INTRODUCTION 5

suggests that secondary markets are desirable because they facilitate more efficient

reallocations of goods. Yet the existence of resale markets also would frustrate airlines’

ability to price discriminate over time, which could potentially decrease overall social

welfare.

Theoretically, the welfare effects of price discrimination are ambiguous (Robinson,

1933). I focus on three channels through which price discrimination can affect social

welfare. First, price discrimination changes the quantity of output sold as some

buyers face higher prices and buy less, while other buyers face lower prices and buy

more.2 Second, price discrimination can affect the quality of the product (Mussa

and Rosen, 1978). For instance, a firm may deliberately degrade the quality of a

lower-priced product to keep people willing to pay a higher price from switching to

the lower-priced product (Deneckere and McAfee, 1996). Finally, price discrimination

can result in a misallocation of products among buyers. Since consumers potentially

face different prices, it is not necessarily true that customers willing to pay more for

the product will end up buying it.

Empirically, we know little about the costs and benefits of intertemporal price

discrimination.3 There are several reasons why there has been little work on this

problem. First, there is a lack of public data. In the airline industry, price and quan-

tity data that are necessary to estimate demand have been available to researchers

only at the quarterly level. Such data do not allow one to separate intertemporal

discrimination for a given seat on a given flight from variation for similar seats on

different days of departure. McAfee and te Velde (2007) is one of the few attempts

to use airline data to analyze intertemporal price discrimination. They had a sample

of price paths, but they did not have access to the corresponding quantities of seats

sold. I solve this problem by merging daily price data collected from the web with

quarterly quantity data using a structural model.

A second impediment to studying intertemporal price discrimination is that a

(2001), Curtis (2007), and Elliot(2011).2An increase in total output is a necessary condition for welfare improvement with third-degree

price discrimination by a monopolist. Schmalensee (1981), Varian (1985), Schwartz (1990), Aguirreet al (2010), and others have analyzed these welfare effects in varying degrees of generality.

3Exceptions include Hendel and Nevo (2011) and Nair (2007).

6 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION

structural model of dynamic oligopoly with intertemporal price discrimination would

necessarily be too complicated to estimate. Among other difficulties, one would have

to deal with the multiplicity of equilibrium predictions and account for multimarket

contact the presence of which is well documented in the industry (see e.g. Evans

and Kessides (1994)). I avoid these problems by focusing solely on monopoly routes.

Finally, I use institutional details of the way that prices are set in practice in the

industry to simplify the problem even further.

While I do observe the lowest available price on each day prior to departure, I

only observe the quantity of tickets purchased at each price on a quarterly basis. As a

result, it would be difficult to estimate demand and cost parameters directly. Instead,

I estimate the parameters of consumers’ preferences indirectly, based on a model of

optimal fares. In the model, a firm sells a product to several groups of forward-looking

consumers during a finite number of periods. Consumer groups differ in three ways:

what time they arrive in the market, how much they are willing to pay for a flight, and

how certain they are about their travel plans. The firm cannot identify and segregate

different consumer groups, but is able to charge different prices in different periods of

sale. There is no aggregate demand uncertainty.4 Under these assumptions, I show

that a set of fares with positive cancellation fees and advance purchase requirements

maximizes the firm’s profit. By contrast, the market-clearing fare without advance

purchase requirements or cancellation fees maximizes the social welfare defined as the

sum of the airline’s profit and consumers’ surplus.

For each value of the unknown parameters, my model predicts a unique profit-

maximizing path of fares as well as the corresponding quantities of tickets sold. I

match these predictions with data collected from 76 U.S. monopoly routes. For every

departure date in three quarters, I recorded all public fares published by airlines for

six weeks prior to departure. Since quantity data are not publicly available, I use the

model of optimal fares to predict quantities sold at each price level in each period. I

then aggregate these predictions to the quarterly level and match them to data from

4Aggregate demand uncertainty is another reason why an airline facing capacity constraints maybenefit from varying its prices over time (Gale and Holmes, 1993, Dana 1999). Puller et al (2009)found only modest support for the scarcity pricing theories in the ticket transaction data, while pricediscrimination explained much of the variation in ticket pricing.

2.1. INTRODUCTION 7

the well-known quarterly sample of airline tickets. To estimate demand and cost

parameters, I use a two-step generalized method of moments based on restrictions for

daily prices, monthly quantities and the quarterly distribution of tickets derived from

the model of optimal fares.

For markets in my data sample, the estimates suggest that, on average, 76%

of passengers travel for leisure purposes. More than 90% of leisure travelers start

searching for a ticket at least six weeks prior to departure. By contrast, 83% of

business travelers begin their search in the last week. Business travelers are willing

to pay up to six times more for a seat and they are significantly less price-elastic.

Business travelers tend to avoid tickets with a cancellation fee as the probability that

they have to cancel a ticket is higher.

These estimates allow me to assess the welfare effects of intertemporal price dis-

crimination. Compared to an ideal allocation that maximizes social welfare, the

profit-maximizing allocation results in a 21% loss of the total gains from trade. To

understand to what extent intertemporal price discrimination contributes to this loss,

I use the estimates to calculate the equilibrium sets of fares for three alternative de-

signs of the market.

The first scenario assesses the potential benefits and costs of allowing unrestricted

airline ticket resale.5 I model resale by assuming that there are an unlimited number

of price-taking arbitrageurs who can buy tickets in any period in order to resell

them later. Under this assumption, the profit-maximizing price path is flat. The

welfare effects of a secondary market, however, are ambiguous. On the one hand,

the secondary market increases the quality of tickets and eliminates misallocations

among consumers. On the other hand, the secondary market can – and, for the

markets I consider, does – reduce the total quantity of tickets sold in the primary

market. I find that the average price of tickets bought by leisure travelers would

increase from $77 to $118, and the number of tickets they buy would decrease by

10%. However, business travelers would face an average price decrease from $382

to $118, with quantity increasing by 49%. The consumer surplus of leisure travelers

5Recent empirical literature on resale and the welfare effects of actual secondary markets includesLeslie and Sorensen (2009), Sweeting (2010), Chen et al (2011), Esteban and Shum (2007), Gavazzaet al (2011). Ticket resale is explicitly prohibited in the U.S. airline industry.


would decline by 16%, the consumer surplus of business travelers would increase by

almost 100%, and the airline’s profit would decrease by 28%. Overall, social welfare

on the average route would increase by 12%, even though the total quantity of tickets

sold would go down.

In a second scenario, I return to a market without resale and assume that the

monopolist is not allowed to alter the quality of tickets by imposing a cancellation

fee but can still charge different prices in different periods. I find that the monopolist

would still discriminate over time but the equilibrium price path would become flatter,

which would reduce misallocations of tickets among consumers. The average ticket

price would go up from $137 to $157. Leisure travelers would benefit due to the

increase in the quality of tickets but would lose from the increase in prices. The net

effect on their consumer surplus would be still positive. Overall, social welfare would

slightly increase.

Finally, the third scenario compares the welfare properties of intertemporal and

third-degree price discrimination. Third degree price discrimination implies that the

airline can identify the customers’ types and is able to set different prices to different

types. By varying the price over time, the airline captures more than 90% of the

profit that it would receive if third degree price discrimination was possible. Sur-

prisingly, the estimates show that some customer groups would prefer third-degree

price discrimination to intertemporal price discrimination. Total social welfare is also

higher under third degree price discrimination.

The essay informs three important empirical literatures. First, it contributes to

the empirical price discrimination literature. Shepard (1991) considered prices of

full and self service options at gas stations. Verboven (1996) studied differences in

automobile prices across European countries. Leslie (2004) quantified the welfare

effects of price discrimination in the Broadway theater industry. Villas-Boas (2009)

analyzed wholesale price discrimination in the German coffee market. Second, it con-

nects to empirical studies of durable goods monopoly. Nair (2007) estimated a model

of intertemporal price discrimination for the market of console video games. Hendel

and Nevo (2011) estimated that intertemporal price discrimination in storable goods

markets increases total welfare. This essay arrives at a different conclusion for airline

2.2. INSTITUTIONAL BACKGROUND 9

tickets. Finally, there are several related papers that analyze price dispersion in the

U.S. airline industry (Borenstein and Rose, 1994, Stavins, 2001, Gerardi and Shapiro,

2009). To the best of my knowledge, this is the first paper to emirically estimate the

welfare effects of intertemporal price discrimination in the airline industry.

The rest of the essay proceeds as follows. Section 2.2 gives background information

on airline pricing. Section 2.3 presents a model of optimal fares. Section 2.4 describes

the data used in the analysis. In Section 2.5, I show how to use the model of optimal

fares to infer demand and supply parameters from the collected data. Section 2.6

presents the results of estimation. In Section 2.7, I formally describe the alternative

market designs and present the results of counterfactual simulations. Section 2.8

concludes.

2.2 Institutional Background

An airline can start selling tickets on a scheduled flight as early as 330 days before

departure. At any given moment, the price of a ticket is determined by the decisions

of two airline departments, the pricing department and the revenue management de-

partment. The pricing department moves first and develops a discrete set of fares that

can be used between any two airports served by the airline. The revenue management

department moves second and chooses which of the fares from this set to offer on a

given day.

The pricing department offers fares with different ”qualities” to discriminate be-

tween leisure and business travelers. High-quality fares are unrestricted. Low-quality

fares come with a set restrictions such as advance purchase requirements and can-

cellation fees. To secure cheaper fares, a traveler typically has to buy a ticket early,

usually a few weeks before her departure date. If her travel plans later change, she

may have to pay a substantial cancellation fee, which often could make the purchased

ticket worthless. These restrictions exploit the fact that business travelers are usually

more uncertain about their travel plans than leisure travelers.

Figure 2.1 gives a snapshot of all coach-class fares that were published by American

Airlines’ pricing department for Dallas – Roswell flights departing on March 1st, 2011,


Figure 2.1: List of available fares from Dallas, TX to Roswell, NM for 03/11/2011,six weeks before departure

six weeks prior the departure. Fares with advance purchase requirements include

a cancellation fee of $150. Fares without advance purchase requirements are fully

refundable.

The fact that the pricing department has published a fare does not imply that a

traveler will be able to get that fare on the specific flight. The flight needs to have

available seats in the booking class that corresponds to that fare. How many seats

to assign to each booking class in each flight is the primary decision of the revenue

management department.

Figure 2.2 shows the paths of coach-class prices for flights from Dallas, TX to

2.2. INSTITUTIONAL BACKGROUND 11

Figure 2.2: Example Price Path. Route: Dallas, TX - Roswell, NM. Departure Date:03/01/11

Roswell, NM on Tuesday, March 1st, 2011. American Airlines is the only carrier that

serves this route; there are three flights available during that day.

The behavior of ticket prices depicted is representative of monopoly markets in

my data. There are three main stylized facts in the data. First, prices increase in

discrete jumps. Second, there are several distinct times when the lowest price for

all flights jumps up simultaneously. As in the figure, these times typically occur 6,

13 and 20 days before departure. Third, between these jumps, prices are relatively

stable.

This behavior results largely because of the institutional details surrounding the


way airlines set ticket prices. The lowest price of a ticket for a given flight is de-

termined by the lowest fare with available seats in the corresponding booking class.

There are three reasons that the lowest price of an airline ticket for a given flight may

change over time. First, if the number of days before departure is less than the APR,

travelers cannot use that fare to buy a ticket. Less restrictive fares are usually more

expensive, which results in a price increase. If we look at Figure 2.1 again, we can see

that the first major price increase occurred 20 days before departure: the price went

up from $138 to $154. This was the day when the advance purchase requirement for

the two lowest fares became binding.

Second, the decision of the revenue management department to open or close

availability in a certain booking class may change the lowest price. Eighteen days

before departure, the revenue management department of American Airlines closed

booking class S for flight AA 2705 but kept booking class G open. As a result, the

lowest price for this flight went up from $154 to $211.

Finally, the pricing department can add a new fare, as well as update or remove

an existing one. On very competitive routes, airline pricing analysts monitor their

competitors very closely: pricing departments respond to competitor’s price moves

very quickly, often responding on the same day (Talluri and van Ryzin, 2005). On

routes with few operating carriers, the set of fares is usually stable. For example,

during the time period depicted on Figure 2.2, the pricing department of American

Airlines did not update fares for flights from Dallas to Roswell departing on March 1st,

2011. Changes in prices were caused primarily by APR restrictions or the decisions

of the revenue-management department.

2.3 The Model of Optimal Fares

To calculate the effect of intertemporal price discrimination on consumer welfare, we

need to estimate consumers’ demand functions. The demand system is estimated

using assumptions about pricing and the supply side. To recover consumers’ prefer-

ences (or, to be precise, the airline’s expectations about consumers’ preferences), I

develop a model that shows how a set of parameters reflecting travelers’ preferences

2.3. THE MODEL OF OPTIMAL FARES 13

transforms into a path of profit-maximizing fares.6

A theoretical model that is able to generate the stylized facts listed in Section 2.2

has to include the decision problems of both the pricing and revenue-management

departments. The solution of the pricing department’s problem is a finite set of fares

that include advance purchase requirements. To construct an optimal set of fares,

the pricing department has to calculate the value of the airline’s expected profit for

each possible set of fares. This value, in turn, depends on the strategy of the revenue

management department that takes the set of fares as given and updates availability

of each booking class in real time. Another complication comes from the fact that the

airline has to take into account not only direct passengers that travel on a particular

route but also passengers for whom this route is only a part of their trip. I will call

them ”direct passengers” and ”connecting passengers”, respectively. The model is

initially formulated for a representative origin and destination and a representative

departure date.

2.3.1 Airline’s problem

Consider a representative market that is defined by three elements: origin, destination

and travel date. The airline is the only producer in the market. It can offer up to

C seats on its flights from the origin to the destination. It flies both direct and

connecting passengers. For direct passengers, the origin is the initial point of their

trip and the destination is the final point of their trip. For connecting passengers,

this flight is only a part of their trip.

The airline is selling tickets during a fixed period of time. Advance purchase re-

quirements divide this period into T periods of sale. At the beginning of the first

period of sale, the airline’s pricing department sets a menu of fares for this market

p = (p1, ..., pT ) and for all markets that connecting passengers fly pj = (pj1, ..., pjT ).

The price pt is the price of the cheapest fare that satisfies the advance purchase

6I do not consider a more general problem of finding a profit-maximizing mechanism since themechanism observed in the data is implemented through publicly posted prices. This problem hasbeen studied by Gershkov and Moldovanu (2009), Board and Skrzypacz (2011), and Hoerner andSamuelson (2011), among others.


requirement for period of sale t. In the empirical application, advance period require-

ments observed define five periods of sale: 21 days and more, from 14 to 20 days,

from 7 to 13 days, from 3 to 6 days, and less than 3 days before departure.

The revenue management department at each moment of time decides which of

the fares that satisfy the advance purchase requirements to offer for purchase based

on the information ξt. Denote by Dt (p, ξt) the number of tickets that the airline

sells at price pt. Not all passengers that bought tickets will end up flying. Denote by

Qt (p, ξT ) the number of seats that that will be occupied by passengers who bought

tickets at price pt. Both Dt and Qt are the solutions of the revenue management

department’s problem. I will not solve this problem explicitly. Instead, I rely on the

fact that the pricing department is able to predict how p affects the number of sold

tickets Dt and the number of occupied seats Qt.

The airline’s revenue comes from selling tickets and collecting cancellation fees. If

a traveler needs to cancel a ticket, she has to pay a cancellation fee f . The fee f ≥ 0 is

taken to be exogenous because in practice U.S. airlines have only one cancellation fee

that applies to all domestic routes. The airline’s operational cost, ϕ (·), depends on

the total number of enplaned passengers. Thus, the airline’s profit takes the following

form:

π = R +∑j

Rj − ϕ

(Q+

∑j

Qj

),

where

R =T∑t=1

(ptQt + min (f , pt)

(Dt − Qt

))revenue from direct passengers,

Rj =T∑t=1

(pjtQjt + min (f, pjt)

(Djt − Qjt

))revenue from connecting passengers,

Q =T∑t=1

Qt the number of seats occupied by direct passengers,

Qj =T∑t=1

Qjt the number of seats occupied connecting passengers from market j.


The pricing department chooses menus of direct fares p and connecting fares pj to

maximize the expected value of the profit function subject to the capacity constraint.

Formally, the profit maximization problem takes the following form:

maxp,pj

E0π s.t. Q+∑j

Qj ≤ C.

The expectation is taken with respect to all information available at the beginning of

the first period of sale.

I will simplify the problem in three steps. First, the constrained optimization

problem can be written as unconstrained using the method of Lagrange multipliers.

Let φ (C) denote the value of the Lagrange multiplier that corresponds to the capacity

constraint. Then the unconstrained profit function takes the following form:

π = R +∑j

Rj − ϕ

(Q+

∑j

Qj

)− φ (C)

[Q+

∑j

Qj − C

].

The last two components of the profit function represent the economic cost of

the airline. The ϕ (·) term is the operational cost, the φ (·) term is the shadow cost

of capacity. Denote by c the value of the marginal economic cost evaluated at the

profit-maximizing level. Then, the solution of the original profit maximizing problem

coincides with the solution of the following problem:

maxp,pj

E0

[R +

∑j

Rj − c ·

(Q+

∑j

Qj

)].

The last problem is separable with respect to p and pj, i.e.

E0

[R +

∑Rj − c ·

(Q+

∑j

Qj

)]= E0

[R− cQ

]+∑j

E0

[Rj − cQj

].

Thus, if the value of the expected marginal cost c is given, then it is sufficient to solve

the profit-maximization problem for direct passengers without looking at the fares

set for connecting passengers or knowing the value of the capacity constraint. The


value of c can be interpreted in two ways. First, it reflects the expected marginal

revenue of adding an additional unit of capacity to the market. Second, it is equal to

the marginal revenue of flying connecting passengers.

Finally, consider the profit-maximization problem for direct passengers:

maxp

E0

[R− cQ

]= max

pE0

[T∑t=1

ptQt + min (f, pt)(Dt − Qt

)− cQt

].

By the law of iterated expectations, we can rewrite this problem as:

maxp

T∑t=1

[(pt − c)Qt + min (f, pt) (Dt −Qt)] , .

where Qt = E0Qt and Dt = E0Dt. The function Dt is the expected number of tickets

that will be sold at price pt if the pricing department offers the menu of fares p and

then the revenue management department behaves optimally given this menu. The

function Qt is the corresponding expected number of occupied seats.

To calculate the welfare effects of intertemporal price discrimination, we need to

know how the quantity of sold tickets and the number of occupied seats respond to

changes in the menu of fares and the cancellation fee. In other words, we need to

know the elasticities of demand with respect to the prices of all available fares and

the cancellation fee. Three limitations of the data do not allow us to estimate these

elasticities directly. The number of occupied seats for each fare pt is not available

for each individual flight or departure date. The data include only a 10% random

sample of the quantity data aggregated to the quarterly level. Second, the data do

not record tickets that were sold but later cancelled. Third, it would be hard to find

a source of exogenous variation that comes from the supply-side and would affect the

components of the fare menu differently. The form of the profit function suggests that

any variation in the cost function affects the entire menu of fares in a very specific

way. From the pricing department’s point of view, the value of the expected marginal

cost of flying an additional passenger is the same in all periods of sale. Finally, there

is almost no variation in the cancellation fee in the data. Almost all airlines charged


$150 in all domestic markets.

Given these limitations, I follow a different approach. I assume that the market

demand defined by Qt and Dt reflects the optimal decision of strategic consumers

whose preferences with respect to the price and time of purchase depend on a vector

of demand parameters θ. The vector of demand parameters θ determines the level of

consumer heterogeneity, their willingness to pay for an airline ticket, their aversion

of the imposed cancellation fee. The airline’s pricing department knows the value of

θ and chooses a menu of fares p to maximize the airline’s profit defined by functions

Qt and Dt that in turn depend on θ and c. Using daily price data and quarterly

aggregated quantity data, I will recover these parameters assuming that the observed

prices maximize the airline’s profit for these parameters.

2.3.2 Demand System and Consumer Welfare

This subsection describes how the vector of demand parameters θ determines the

relationship between the expected quantities of sold tickets Dt, the occupied seats

Qt, and the menu of offered fares p. It can be viewed as a micro model of the market

demand functions Qt

(p; θ)

and Dt

(p; θ)

. Since these functions by construction

represent expected quantities, the model does not allow any demand uncertainty at

the market level.

Types, Arrival and Exit The population of potential direct passengers of size M

consists of I discrete types; types are indexed by i = 1, ..., I. (In the estimation, I

assume that I = 2: leisure and business travelers.) The sizes of different types of

potential buyers change over time for three reasons. First, each period new travelers

arrive to the market.7 The mass of new buyers of type i who arrive at time t is equal

to Mit = λit · γi · M , where γi is the weight of each type in the population and λit is

the type-specific arrival rate. Second, those travelers who bought tickets in previous

periods are not interested in purchasing additional ones. Third, each period a fraction

7Without this assumption, the profit-maximizing monopolist would forgo the opportunity todiscriminate over time (Stokey, 1979). Board (2008) analyzes the profit-maximizing behavior of adurable goods monopolist when incoming demand varies over time.


of travelers who arrived in the previous periods learn that they will not be able to fly

due to some contingency, so they cancel the ticket (if purchased) and exit the market.

The probability that a traveler of type i learns that she will not be able to fly is equal

to (1− δi) in every period.

Preferences Travelers know their utilities conditional on flying but are uncertain

if they are able to fly. If a traveler ι of type i buys a ticket in period t, she pays the

price pt and, conditional on flying, receives:

uιit ≡ µi + σi (ειit − ειi0) , (2.1)

where µi is type-i’s mean utility from flying on this route measured in dollar terms,

ειit are i.i.d. Type-1 extreme value terms that shift traveler ι’s utility in each period,

and σi is a normalizing coefficient that controls the variance of ειit. The error term

ειit reflects idiosyncratic customers’ preferences with respect to the time of purchase.

They may reflect customers’ tastes with regard to other characteristics of restricted

fares or their idiosyncratic level of uncertainty about their travel plans. The errors

represent the consumer tastes that the airline and researcher do not observe. This

coefficient σi captures the slope of the demand curve and hence the price sensitivity

across the population of type-i travelers: the lower the coefficient, the less sensitive

are type-i travelers. The traveler learns all components of their utilities defined in

equation (2.1) at the beginning of the period she arrived in the market.8

After purchase, the traveler can cancel a ticket. If she cancels a ticket in period t′,

she loses the price she paid, pt, but may receive a monetary refund if the cancellation

fee does not exceed the price. The refund is equal to max (pt − f , 0). Since the refund

does not exceed the price of the ticket, the traveler will cancel her ticket only if she

learns that she is not able to fly. If the traveler doesn’t fly, her utility is normalized

to zero.

8An alternative assumption would be for travelers to learn a component of ειi before each periodof sale.Under this assumption each customer would compare the current value of the term with itsexpected future values. Under the original assumpton each customer would compare this value withits actual future values. Qualitatively we would receive the same results. However, the demandfunction will not have a closed form solution.


Travelers are forward-looking and make purchase decisions to maximize their ex-

pected utility. They face the following tradeoff: if they wait, they will receive more

information about their travel plans but may have to pay a higher prices if the airline

increases prices over time.

Individual demand Consider the utility-maximization problem of a type-i traveler

who is in the market at time τ . She has T − τ periods to buy a ticket. She buys

a ticket at time τ only if it gives a higher utility than buying a ticket in subsequent

periods or not buying a ticket at all. If she buys a ticket in period τ , then her net

expected utility is given by:

[δT−τi uiτ +Riτ

]− pτ ,

where ρiτ denotes the expected value of the refund:

ρiτ =(1− δT−τi

)max (pτ − f , 0) .

Suppose the traveler decides to wait until period τ ′. Then with probability(1− δτ ′−τi

)she learns about a travel emergency and exits the market. With the

remaining probability δτ′−τi she stays in the market. If she buys a ticket, she receives

δT−τ′

i [µi + σi (ειiτ ′ − ειi0)] + ρiτ ′ − pτ ′ . In this case, her expected utility is equal to

δT−τi [µi + σi (ειiτ ′ − ειi0)] + δτ′−τi (ρiτ ′ − pτ ′) .

Thus, the traveler buys a ticket in period τ if the following set of inequalities

holds:

δT−τi [µi + σi (ειiτ − ειi0)] + ρiτ − pτ > δT−τi [µi + σi (ειiτ ′ − ειi0)] + δτ′−τi (ρiτ ′ − pτ ′)

for all τ < τ ′ ≤ T and

δT−τi [µi + σi (ειiτ − ειi0)] + ρiτ − pτ > 0.


These inequalities can be rewritten in a more convenient way:

δT−τi µi + ρiτ − pτσiδ

T−τi

+ ειiτ >δT−τ

′

i µi + ρiτ ′ − pτ ′σiδ

T−τ ′i

+ ειiτ ′ for all τ < τ ′ ≤ T and(2.2)

δT−τi µi + ρiτ − pτσiδ

T−τi

+ ειiτ > ειi0.

Market demand for airline tickets To calculate the firm’s expected demand for

tickets, we need to know the demand of each traveler type as well as the size of each

type in a given period. Denote by sitτ the share of type-i buyers who arrived in period

τ and purchase a ticket in period t conditional on not exiting the market. This share

corresponds to the probability that traveler ι has a realization of ειit, t = τ, ..., T that

satisfies inequalities defined in (2.2). Under the assumption that ειiτ is extreme value,

this share is equal to

sitτ =exp

(δT−ti µi+ρit−pt

σiδT−ti

)1 +

∑Tk=τ exp

(δT−ki µi+ρik−pk

δT−ki σi

) .

Consider the size of type-i buyers who arrived in period τ . By time t, only δt−τi

of the initial size has not exited the market due to a realized emergency. Thus, the

total demand of type-i travelers is equal to:

Dit =t∑

τ=1

sitτδt−τi Miτ ;

the market demand for tickets in period t is given by:

Dt =I∑i=1

Dit.

Thus, the vector of demand parameters θ includes the following parameters: shares

of each customer type γi, the mean utilities µi, the price sensitivity σi, the probability

of cancellation δi, the arrival parameters λit.


Number of occupied seats The probability of not cancelling a trip for traveller

of type i who bought a ticket in period t by the time of departure is given by δT−ti .

Thus the number of occupied seats is equal to

Q =T∑t=1

Qt, where Qt =I∑i=1

δT−ti Dit.

Welfare9 For each price path p, we can calculate the sum of utilities for each type

of travelers. Consider the group of type-i travelers who arrived at time τ and define

the average aggregate utility of this group by viτ (p). Then,

viτ (p) =

∫ι

maxτ≤τ ′≤T

{δT−τi [µi + σi (ειiτ ′ − ειi0)] + δτ

′−τi (ρiτ ′ − pτ ′) , 0

}dι.

Integrating with respect to the extreme value distribution, we get:

viτ (p) = δT−τi σi log

(1 +

T∑t=τ

exp

(δT−ti µi + ρiτ − pτ

δT−ti σi

)).

Then, the total sum of traveler’s utilities equals:

V (p) =I∑i=1

T∑τ=1

viτ (p) Miτ .

Define social welfare as the sum of travelers’ ex-post utilities and the airline’s

profit. The supply and allocation of seats among travelers are efficient if they maxi-

mize social welfare. A price path p is called efficient if it induces an efficient supply

and allocation of seats. By the First Welfare Theorem, the allocation of seats will

be efficient only if all consumers take the same prices into account. If it is not the

case, then there could be two customers who would be willing to trade with each

other right before departure. The reason why the customer who wants to buy the

ticket now didn’t buy it before was his higher probability of cancellation. Therefore,

9Given the data limitations, I can only estimate the welfare effects of intertemporal price dis-crimination on direct passengers.


there is always some positive probability that the ex-post allocation is not efficient,

therefore any price path with a positive cancellation fee is not efficient.

Thus, there are three conditions for efficient supply and allocation of seats. First,

the price path has to be flat. Second, it has to equal to the value of the marginal costs

c. Third, the cancellation fee has to be zero. If the cancellation fee is positive, then

the expected value of the refund is different for customers of different types. This fact

implies that even thought the airline offers the same menu of fares to all customers,

the effective ex-post price is different for different customer types.

These conditions illustrate two impediments to the efficient supply and allocation

of seats: market power and dynamic pricing. First, if price exceeds marginal cost,

then the number of seats sold by the airline is lower than the socially efficient level.

As a result, social welfare is lower than its maximum level due to inefficiency in the

quantity of production. Second, if the price path is not flat, then the airline charges

different prices in different time periods, which results in a misallocation of seats

among travelers. In this case, social welfare does not achieve its maximum level due

to inefficiency in allocation. A positive cancellation fee makes a ticket less attractive

to travelers. For this reason, I refer to it as a measure of ticket quality. A positive

cancellation fee thus implies inefficiency in the quality of production. Inefficiency

in quality of production, inefficiency in quantity of production, and inefficiency in

allocation are the three reasons why a price path may not induce an efficient outcome.

2.3.3 Optimal Price Path

A price path p is called optimal if it maximizes the airline’s profit π (p):

π (p) =T∑t=1

[(pt − c)Qt + min (f, pt) (Dt −Qt)]

Denote by p∗(θ, c)

the optimal price path as a function of the demand parameter θ

and the cost parameter c.

Except for a knife-edge realization of the demand and cost parameters, the optimal

price path implies intertemporal price discrimination, i.e. prices differ in different

2.4. DATA 23

periods. Furthermore, in practice, airlines often impose a positive cancellation fee

for lower fares. Even though a positive cancellation fee diminishes the quality for all

traveler groups, travelers with a higher probability of cancellation suffer from it more.

If the probability of cancellation is positively correlated with the utility from flying,

the fee screens travelers by their type.

Thus, our theoretical analysis suggests that price paths observed in practice lead

to all three types of inefficiency identified in the previous subsection: inefficiency

in quality of production, inefficiency in quantity of production, and inefficiency in

allocation. To evaluate the welfare losses associated with each type of inefficiency, we

need to know the estimates of the demand parameter θ and cost parameter c. I will

estimate these parameters using a sample of optimal price paths and corresponding

quantities.

2.4 Data

2.4.1 Monopoly Markets

A market is defined by three elements: origin airport, destination airport and depar-

ture date. A product is an airline ticket that gives a passenger the right to occupy a

seat on a flight from the origin to the destination departing on a particular date.

To be included in my dataset, a domestic route has to satisfy five criteria. First,

the operating carrier on the route was the only scheduled carrier in the time period I

consider. Second, the carrier had to have been the dominant firm for at least a year

before the period I consider. Specifically, its share in total market traffic had to be

at least 95% in each month prior to the period of study. Third, at least 90% of the

passengers flying from the origin to the destination must fly nonstop. Fourth, total

market traffic on the route must be at least 1000 passengers per quarter. Fifth, there

should be no alternative airports that a traveler willing to fly this route can choose.

I do not include routes to/from Alaska or Hawaii. These criteria were chosen to limit

ambiguities in markets and to ensure the markets were nontrivial.

In all, I have 76 directional routes that satisfy these criteria. A typical route has a


Table 2.1: Monopoly routes: summary statistics

mean st.d.distance 401 213median family income $71,942 $8,432average ticket price $205 $236quarterly traffic, passengers 16,663 11,854share of major airline, traffic 0.9953 0.0188share of nonstop passengers 0.9772 0.0255share of connecting passengers 0.6511 0.2616load factor 0.7104 0.0896

major airline hub as either its origin or destination. There are six monopoly airlines in

the dataset: American Airlines (26 routes to or from Dallas/Fort Worth, TX), Alaska

Airlines (26 routes mainly to or from Seattle, WA), United/Continental Airlines (8

routes to or from Houston, TX), AirTran Airways (4 routes to or from Atlanta, GA),

Spirit Airlines (6 routes to or from Fort Lauderdale, FL), and US Airways (6 routes

to or from Phoenix, AZ). Table 2.1 gives summary statistics of route characteristics.

2.4.2 Data Sources

Fares are distributed by the Airline Tariff Publishing Company10 (ATPCO), an orga-

nization that receives fares from all airlines’ pricing departments. It publishes North

American fares three times a day on weekdays, and once a day on weekends and hol-

idays11. Until recently, the general public did not have access to information stored

in global distribution systems. Yet a few websites have provided travelers with rec-

ommendations on when is the best time to book a ticket based on this information.

In 2004, travelers received direct access to public fares and booking class availabili-

ties through several new websites and applications. I recorded fares manually from

a website that has access to global distribution systems subscribed to ATPCO data.

10Until recently, ATPCO was the only agency distributing fares in North America. In March 2011,SITA, the only international competitor of ATPCO, received an approval from the US Departmentof Transport and the Canadian Transportation Agency to distribute data for airlines operating inthe region.

11On weekdays, the fares are published at 10 am, 1 pm and 8 pm ET. On weekends, the fares arepublished at 5 pm. In October 2011, ATPCO added a fourth filing feed on weekdays – at 4 pm ET.

2.5. ESTIMATION 25

This website is widely known among industry experts and regarded as a reliable and

accurate source of public fares12. I recorded fares that were published six weeks before

departure. The period of six weeks is motivated by three facts. First, few tickets are

sold earlier than that period. Second, most travel websites recommend searching for

cheap tickets six to eight weeks before departure. Third, when a pricing department

updates fares it takes into account flights that depart in the next several weeks rather

than flights that depart in the next several days. Thus, I believe that it is reasonable

to assume that fares posted six weeks before departure reflect the optimal decision of

pricing departments.

I consider three quarters of departure dates between October 1, 2010 and June 30,

2011. Besides the data on daily fares described above, I use monthly traffic data from

the T-100 Domestic Market database and the Airline Origin and Destination Survey

Databank 1B that contains a 10% random sample of airline tickets issued in the U.S.

within a given quarter. Both datasets are reported to the U.S. Department of Trans-

portation by air carriers and are freely available to the public. In the estimation, I

control for several route characteristics, which allows me to compare different mar-

kets with each other. These characteristics include route distance, median household

income in the Metropolitan Statistical Areas to which origin and destination airports

belong, and population in the areas.

2.5 Estimation

2.5.1 Econometric Specification

My empirical model allows for two types of travelers. I refer to the first type as leisure

travelers (L), and to the second type as business travelers (B). Leisure travelers are

highly price sensitive customers who are willing to book earlier and are more willing

to accept ticket restrictions. Business travelers, on the other hand, are less price

12In addition to public fares that are available to any traveler, airlines can offer private fares.Private fares are discounts or special rates given to important travel agencies, wholesalers, or cor-porations. Private fares can be sold via a GDS that requires a special code to access them or as anoffline paper agreement. In the United States, the majority of sold fares are public.


sensitive, book their trips later and less likely to accept restrictions.13 The demand

parameters of the model of optimal fares are able to capture these distinctions.

For a given departure date d = 1, ..., D and a given route n = 1, ..., N , the de-

mand parameters θnd and the cost parameter cnd determine the optimal price path

p∗(θnd, cnd

). These parameters are known to the airline but unknown to the re-

searcher. The goal of the estimation routine is to recover θnd and cnd for each date

and route from the observed price and quantity data. Given the limitations of the

dataset, I need to reduce the dimension of the unknown parameters. To do this, I

restrict both observed and unobserved variation in the parameters within and across

markets.

The shares of each type, γi, are assumed to be the same in all routes and all

departure dates. Type-specific mean utilities from flying, µi, are proportional to the

route distance. The proportionality coefficient in turn linearly depends on the route

median income. These coefficients do not vary with the departure date. Thus,

µind = µ1i + (µ2i + µ3i · incomen) · distn.

The variance of the type-I error (σi) that controls intertemporal utility variation

within a type is the same in all markets and all departure dates. The probability

of having to cancel the trip, 1 − δi, is also the same in all routes but varies with

the departure date. It can take two type-specific values: one for regular season and

one for holiday seasons. Holiday season departure dates correspond to Thanksgiving,

Christmas, New Year’s and Spring Break. The probability of canceling a trip is

different during these periods as travelers may be more certain about their holiday

trips than about their regular trips. If we denote by hd the holiday season dummy

variable, then

δind = δholidayi · hd + δregulari · (1− hd) .

The share of new passengers who arrive in period τ , has the following parametric

13See, Phillips (2005).

2.5. ESTIMATION 27

representation:

λiτnd = λ (τ, T, αi) + ελτnd =( τT

)αi

−(τ − 1

T

)αi

+ ελτnd,

where ελ1nd is normalized to 0 and ελ2nd, ..., ελTnd are unobserved i.i.d. mean-zero

errors. The parameter αi determines the time when the majority of type-i consumers

start searching for a ticket: types with low values of αi begin their search early, types

with high values of αi arrive to the market only a few days before departure. These

parameters are the same for all routes and departure dates. The unobserved error

ελτnd randomly shifts the arrival probabilities. Since the airline observes these errors

before it determines its price path, these errors explain a part of the daily variation

in observed fares. The sum of the errors does not affect the optimal price path and

thus is not identified from the observed fares. For this reason, I normalize the value

of the first error to zero.

The value of the expected marginal costs cnd, by construction, is equal to the

derivative of the total economic costs evaluated at the profit-maximizing level of the

total quantity of occupied seats. The economic costs include both the operational

costs and the shadow costs of capacity. If the total quantity of occupied seats were

available, then the most natural way to estimate c would be as nonparametric function

of the total quantity. I do not observe this quantity, so I estimate the average value of

the marginal costs by assuming that c = c+ εcnd where εcnd is a mean-zero deviation

of the actual value from its mean. The unobserved error εcnd randomly shifts the

opportunity cost of flying a passenger each day and in each route and also explains

a part of the daily variation in observed fares. It captures factors that affect both

the operational costs (such us distance, capacity, etc.), and the shadow cost of the

capacity constraint (the demand of connecting passengers etc.). This error shifts the

entire time path of prices, while ελτnd affects relative levels of the prices in the path.

The total number of potential travelers is different for each route and each depar-

ture date. I denote by Mn the mean number of travelers on route n and assume that

the deviations from these means, the arrival errors ελτnd, and the cost errors εcnd are

jointly independent.


Together, we can divide all demand and cost parameters known to the airline into

three groups: estimated coefficients θ = (γ, µ, σ, δ, α) , c, and Mn, errors unobserved

to the researcher εnd = (ελnd, εnd), and market specific covariates (hd, xn), where xn

denotes route characteristics such as (distn, incomen). These restrictions allow me to

estimate the coefficients jointly for all markets in my sample.

2.5.2 Moment Restrictions

To estimate the demand parameter θ and cost parameters c, I follow the standard

practice of using both price and quantity data. However, I face the nonstandard com-

plication that these data are observed with different frequencies: prices are observed

daily, quantities are observed quarterly. Only having quarterly quantity data means

that they contain two sources of variation: variation due to different departure dates

and variation due to different purchase dates. I use the model of optimal fares to

distinguish between these two sources of variation.

2.5.2.1 Daily prices

Define by ptnd the lowest fare satisfying the advance purchase requirement for period

of sale t for route n and departure date d. Since the posted fares should be equal to

the optimal fares predicted by the model, the posted fares should satisfy the system

of first order conditions:

G(p, θ)

=

∂π(p; θ)

∂p1, ...,

∂π(p; θ)

∂pT

′

.

To construct moment restrictions that correspond to the posted prices, we need to

invert the system of equations to derive an expression for the unobserved error term

εnd.It turns out that there exists a unique mapping gP : RT × Rdim(θ) × Rdim(hd) ×Rdim(xn) → RT , such that for any θ, it holds that G (pnd, θ, hd, xn , gP (pnd, θ, hd, xn)) =

0. The proof of this statement follows from the fact that the system of first order

conditions is triangular and linear with respect to the errors. The first equation

2.5. ESTIMATION 29

includes only εcnd, the second equation includes εcnd and ελ2nd., etc. Thus, we can

invert the system by the substitution method: derive the value of εcnd from the first

equation and plug it into the second one, etc.

Since we assumed that εnd has zero mean, the moment restrictions that correspond

to the observed prices take the following form:

Eεnd = Egp (pnd, θ, hd, xn) = 0.

I use these restrictions as the basis for the first set of sample moment conditions.

2.5.2.2 Monthly traffic

The model predicts the expected total number of direct passengers for departure

date d and route n is equal to∑T

t=1Qndt

(pnd, θ

). In the data, we observe the actual

number of flying passengers. Denote by Qtrafficnm the total number of enplaned direct

passengers observed in the data for route n and month m. Thus, the predicted number

of enplaned passengers is equal to

∑d∈month(m)

I∑i=1

T∑t=1

Qndit

(pnd, θ

).

Denote by gM

(pnd, θ,Mnm

)=∑

d∈month(m)

∑Ii=1

∑Tt=1 δ

T−tid Dit

(pnd, θ

)− Qtraffic

nm .

This error comes from the fact that the revenue-management department due to

the stochastic nature of the demand cannot perfectly implement the plan designed by

the pricing department. Sometimes it allocates more seats to a certain class, some-

times less. The goal of the revenue management department, however, is to get as

close to the target level as possible. Therefore, it is not unreasonable to assume that

the variance of the error is bounded and its expected value is equal to zero. Then, a

moment restriction that corresponds to the observed number of enplaned passengers

is given by:

EgM(pnd, θ, Q

trafficnm

)= 0.

I use this restriction as to define the second set of sample moment conditions.


2.5.2.3 Quarterly sample of tickets

Denote by rlnq a ticket issued for market n in quarter q and let p (rlnq) and f (rlnq)

denote the corresponding one-way fare and number of traveling passengers.14 The

quarterly ticket data have several potential sources of measurement error. These

data include special fares, frequent flier fares, military and government fares, etc. To

reduce the impact of these special fares, I do the following. First, I divide the range

of possible prices into B + 1 non-overlapping intervals:15 [pb, pb+1], b = 0, ...., B. For

each interval, the model predicts the total number of tickets sold during the quarter.

Hence, we can calculate the model-predicted probability of drawing a ticket from each

interval. Denote by wbnq the probability of drawing a ticket with a price that belongs

to interval [pb, pb+1] for market n in quarter q. This probability equals:

wbnq

(pnd, θ

)=

∑d∈quarter(q)

∑Ii=1

∑Tt=1Qit

(pnd, θ

)· 1 {ptnd ∈ [pb, pb+1]}∑

d∈quarter(q)∑I

i=1

∑Tt=1Qit

(pnd, θ

) ,

Similarly, we can calculate the relative frequency of observing a ticket within

a given price range using the 10% sample of airline tickets. I treat a ticket with

multiple passengers as multiple tickets with one passenger each. If a ticket has a

round-trip trip fare, I assume that I observe two tickets with two equal one-way fares.

Finally, I only take into account those intervals for which the model predicts non-

zero probabilities. Denote these frequencies as wbnq and define gW

(pnd, θ, rnd

)=

[w1nq − w1nq, ..., wBnq − wBnq]′.

Assuming that the 10% sample is drawn at random, we can derive the third part

of the moment restriction set from the population moment conditions for each price

interval:

EgW(pnd, θ, rnm

)= 0.

To avoid linear dependence of the moment restrictions, I exclude the last interval.

14I manually removed the taxes to get the published fares. The details are in Appendix B.15I estimate the model using the following 17 price thresholds: 20, 50, 80, 100, 120, 135, 150, 170,

190, 210, 220, 240, 270, 300, 330, 360, 410.

2.5. ESTIMATION 31

Figure 2.3: Identification

2.5.3 Estimation Method and Inference

I use a two-step generalized method of moments. The optimal weighting matrix is

estimated using unweighted moments. For computational purposes, I optimize the

objective function for a monotone transformation of the parameters. This transfor-

mation guarantees that the estimates will be positive and, where necessary, less than

one. The standard errors are calculated using the asymptotic variance matrix for a

two-step optimal GMM estimator.

2.5.4 Identification

Section 2.5.2 established T moment restrictions based on the daily fare data, one

restriction based on the monthly traffic data and B restrictions based on the quarterly

ticket data. I use these T +B + 1 = 5 + 17 + 1 = 23 moment conditions to estimate

the 15 parameters that define θ and c. These parameters are identified from the joint

distribution of daily optimal prices and quantities aggregated to the quarterly level.

To show identification formally, I would need to prove that the T moment restrictions

can be satisfied only under the true parameter θ0. This fact is rarely possible to prove

without knowing the true distribution of the data.


To gain intuition on what properties of the joint distribution identify each com-

ponent of the parameter θ, I performed two simulation exercises using the model of

optimal fares. The first exercise shows how a change in each component of the demand

and cost parameter θ affects the profit maximizing vectors of prices and quantities.

The second exercise does the opposite. After changing a component of the price-

quantity vector, I find a vector of parameters θ under which the new price-quantity

vector would maximize the airline’s profit. Based on these results, I can provide an

intuitive explanation for how the joint distribution of the data may identify the pa-

rameters of the model. The explanation is, by all means, heuristic as we should keep

in mind that whenever we change one parameter of the model, all components of the

profit-maximizing prices and quantities will necessarily change.

Consider a representative market. The solid line in Figure 2.3 shows a typical price

path that we observe in the data. For the sake of argument, suppose we also observe

the corresponding quantities of sold tickets for this departure day. These quantities

are depicted by the bar graph on Figure 2.3. Thus, we know two profit maximizing

vectors p = (p1, p2, p3, p4, p5) and q = (q1, q2, q3, q4, q5). From these vectors, we need

to infer the following demand and cost parameters: a share of each type γ, the mean

utilities µi, the within-type heterogeneity parameter σi, the probability of cancellation

δi, the arrival parameters αi, and the cost parameter c.

The behavior of the typical price path can be described as follows. In the first

two periods, the price rises but at a relatively slow level. Then in period 3 or 4,

the price jumps up and continues to increase but, again, with a slower speed. To

understand this behavior, consider the tradeoff that the airline has. Recall that it

faces two heterogeneous groups of customers with different marginal willingness to

pay: business travelers are willing to pay more than leisure travelers. Therefore, the

airline can charge a high price and receive a low quantity as most leisure travelers

cannot afford to fly. Alternatively, it can charge a low price but receive a high quantity.

The price path suggests that it should be profit maximizing for the airline to charge

a low price in the first periods and then switch to a high price.

Having this intuition in mind, we can infer that most customers buying early are

leisure (type 1) travelers, while customers who are buying later, at a higher price, are

2.5. ESTIMATION 33

business (type 2) travelers. The exact level of the prices in early periods is determined

by the elasticity of leisure travelers, while the price level in later periods is determined

by the elasticity of business travelers. The elasticity of each group in turn depends on

the price-sensitivity parameter σi. Similarly, the quantities sold in early periods reveal

information about the mean utility of leisure travelers (µL), while the quantities sold

in later periods depend on the mean utility of business travelers (µB). By comparing

the sum of quantities sold in early periods with the total sum of quantities, and taking

into account the profit maximizing conditions, we can infer the share of leisure type

(γ).

The increase in prices in period 2 compared to period 1 is determined by the

probability of cancellation. After the first period, customers became more certain

about their travel plans since there are fewer periods during which they can learn

that they won’t be able to fly. As a result, they are willing to pay more. The airline

realizes this change and increases price. Since most customers who are buying tickets

in the first two periods are leisure travelers, the change in these two prices identifies

the probability of cancellation for leisure travelers (δL). Similarly, the probability

of cancellation for business travelers (δB) is identified from the change in the last

two prices. Further, if no new customers arrived in period 2, the profit-maximizing

quantities in period 1 and 2 would be the same. Customers with a high first-period

shock ειi1 would buy in period 2, customers with a high second-period shock ειi2

would buy in the second period. The picture suggests that this is not the case. The

reason why the quantity in period 2 is higher is the arrival of new customers. For the

same reason, quantities in period 4 and 5 are also different. Thus, the exact difference

between the two quantities reveals the value of the arrival parameter αi.

Finally, the period in which the price jump occurs identifies the value of the cost

parameter c. Intuitively, in the equilibrium, the marginal revenue that the airline

receives from business travelers should be equal to the marginal revenue it receives

from leisure travelers and both should be equal to the value of marginal cost. If the

costs are high, then the marginal revenue the airline receives from leisure travelers

has to be higher. Therefore, fewer leisure travelers will be served in the equilibrium,

so the airline has to switch to business travelers sooner. If the costs are low, then


the marginal revenue from leisure travelers has to be low, so the airline will offer the

lower price longer.

If the menus of fares are the same for all travel dates within a quarter, we can just

divide the quarterly aggregated quantities by the number of travel dates and apply

this intuition directly. Suppose that the menus of fares are the same except for one

travel date, say, Thanksgiving. Then, this travel date has its own menu of fares, at

least one price of which is different from the rest. We can look at the quantity that

is associated with this price, and based on it and the model of optimal fares, deduce

the quantities for other fares from these menus. After subtracting these quantities

from the aggregated data, we are back in the original setting when the fares are the

same for the remaining travel dates. This intuitive explanation suggests that the

aggregated quantity data provide us with informative moment conditions.

2.6 Results

2.6.1 Demand and Cost Estimates

Table 2.2 presents the optimal GMM estimates of the demand and cost parameters.

Based on these estimates and the model of optimal fares, I calculate that 76% of

passengers travel for leisure purposes. Business travelers are willing to pay up to six

times more for a seat on the average route in my data sample and they are less price

sensitive. If fares in all periods go up by 1%, the total demand of leisure travelers

goes down by 1.3%, while the total demand of business travelers goes down by 0.8%.

Business travelers tend to avoid tickets with a cancellation fee as the probability that

they have to cancel a ticket is high.

The dynamics of arrival of each traveler type for the estimate of the arrival process

αi is depicted by dotted lines in Figure 2.4. A significant share of leisure travelers

start searching for a ticket at least six weeks prior to departure. By contrast, 83%

of business travelers begin their search in the last week. The bar graph in Figure 2.4

demonstrates how the number of active buyers changes over time. In the first few

periods, the number of active buyers goes down as travelers buy tickets or learn that

2.6. RESULTS 35

Table 2.2: Estimates of demand and cost parameters

Leisure Travelers Business TravelersShare of Traveler Type γi 79.71%

(0.20%)20.29%(0.20%)

Mean Utility µi $43.63(1.05)

+

[$7.11(0.01)

+ 0.89(0.05)

incomen

]distn $320.23

(19.35)+

[$27.89(4.95)

+ 2.54(1.54)

incomen

]distn

Price sensitivity σi 0.34(0.007)

2.46(0.06)

Probability of cancellationregular season / holiday season

1− δi 9.95%(0.11%)

/ 0.79%(0.01%)

12.33%(0.13%)

Arrival process parameter αi 0.02(0.09)

7.85(1.82)

Marginal cost c $4.00($12.36)

Note: incomen is in $ 100,000, distn is in 100 miles.

they will not be able to fly. The arrival of new travelers does not counteract this

decrease. A week before departure, most business travelers start searching for tickets,

and the number of active ticket buyers goes up.

2.6.2 Optimal Price Path and Price Elasticities

To put these estimates into perspective, I use the model of optimal fares to calculate

the price path for flights on a route with median characteristics on a non-holiday

departure date. Figure 2.4 shows this path together with the quantities of tickets

purchased in each period by leisure and business travelers. The figure shows that

leisure travelers usually purchase tickets up until seven days before departure, prior

to the moment when most business travelers arrive in the market. When business

travelers arrive, the airline significantly increases the price, trying to extract more

surplus from travelers who are willing to pay more.

Table 3 presents the estimates of price elasticities evaluated at the optimal price

path. The estimates show that in periods 1 and 5 the airline extracts almost the

maximum amount of revenue from travelers as the elasticities are close to one. In

both periods, the buyers are almost homogenous. In period 1, the majority of active

buyers are leisure travelers. In period 5, the price is so high that only business

travelers can afford it. By contrast, in periods 3 and 4, the estimates of elasticities

indicate that the maximum revenue is not achieved. As we can see from the quantity

estimates in Figure 2.5, both groups are buying tickets at the optimal prices in these

periods.


Figure 2.4: Dynamics of active buyers on a route with median income and distance

Table 2.3: Estimates of price elasticities

Market Demand in Period:Price in Period: t = 1 t = 2 t = 3 t = 4 t = 5t = 1 −2.634 0.598 0.647 0.562 0.013t = 2 0.549 −6.178 1.596 1.388 0.033t = 3 0.546 1.467 −10.923 2.707 0.072t = 4 0.448 1.207 2.560 −16.538 0.193t = 5 0.034 0.099 0.241 0.695 −2.654

2.7. COUNTERFACTUAL SIMULATIONS 37

Figure 2.5: Optimal price path for a route with median distance and income

2.6.3 Welfare Estimates

Compared to the efficient supply and allocation of seats, the model’s profit-maximizing

ticket allocation predicts that travelers and the firm attain 79% of the maximum gains

from trade. That the gains are below 100% is due market power distortions and mis-

allocations due to price discrimination. Figure 2.6 shows the distribution of utilities

for two groups of travelers who are able to fly on the day of departure. The first group

includes travelers who bought tickets, the second group are travelers who didn’t buy

tickets because of high prices. If the allocation was efficient, only travelers who value

a ticket more would end up buying it. As we can see from the figure, there is an over-

lap in the supports of these two distributions. This fact indicates that the optimal

price path leads to misallocations of seats.

2.7 Counterfactual Simulations

In the counterfactual simulations, I consider three alternative market designs that

can eliminate some types of inefficiency caused by intertemporal price discrimination.

The first scenario allows costless resale in the presence of market arbitrageurs. Under

this assumption, two types of inefficiencies would disappear: quality distortions and


Figure 2.6: Distributions of travelers’ utilities under the optimal allocation of seats


misallocations among the consumers. On the other hand, the third type of inefficiency,

inefficiency in the quantity of production, could increase. In the second scenario, the

airline is allowed to sell only fully refundable tickets. This restriction eliminates one

type of inefficiency, quality distortions. By doing so, it reduces the firm’s ability

to price discriminate, and therefore, decreases allocative inefficiency. However, the

restriction can increase inefficiency in the quantity of production. The last scenario

considers the case of direct price-discrimination when the airline can perfectly identify

customers’ types and set prices contingent on them.

2.7.1 Costless resale

To study the effects of a potential secondary market, I modify the fare model in

the following way. In addition to travelers and the airline, I assume there exists an

unlimited number of arbitrageurs. In any period, an arbitrageur can buy a ticket

from the airline and then sell it to travelers later. The arbitrageurs are price-takers.

Their goal is to maximize the difference between the price at which they buy a ticket

and the price they sell a ticket later. Under these assumptions, the optimal price

path has to be flat. To see that, first, note that for any optimal sequence of prices,

the maximum profit of each arbitrageur is zero. Indeed, if an arbitrageur is able to

extract some profit then the airline can repeat her actions and increase its profit,

which would violate the condition of profit-maximization. Since the maximum profit

of each arbitrageur is zero, the optimal price path cannot be increasing. But could it

be profitable for the airline to decrease the prices? Only if it did so without resale.

Thus, if the price path without resale is increasing, then the optimal price path in a

market with costless resale is flat.

To calculate the optimal fare in this counterfactual scenario, it is sufficient to

consider the profit maximization problem assuming that the price path is flat. The

share of type-i buyers who arrive in period τ and purchase a ticket in period t becomes:

sitτ =exp

(µi−pσi

)1 +

∑Tk=τ exp

(µi−pσi

) =exp

(µi−pσi

)1 + (T − τ + 1) exp

(µi−pσi

) .


Figure 2.7: Resale (constant marginal costs)

This share is the same for all purchase periods t since travelers pay the same price in

all periods and can get a full refund if they have to cancel their tickets. The airline’s

profit is equal to:

π(p; θ)

= (p− c)I∑i=1

T∑t=1

δT−ti Dit.

Since the value of the expected marginal costs is identified only at the profit-

maximizing level, we need to make an assumption about its value in the counterfactual

scenario. I will make two alternative assumptions. In the first case, I assume that

the expected value of the marginal costs is flat. This assumption corresponds to an

ideal situation in which the airline is able to adjust its capacity continuously. The

value of c will represent the minimum expected value of the average costs, which is

the value of the expected marginal costs evaluated at the minimum efficient scale. In

the second case, I assume that the graph of the marginal costs is a vertical line, i.e.

the airline cannot adjust their capacity.

In both cases, the welfare effects of ticket resale are unclear because the ability to

resell tickets eliminates the inefficiency in quality of production and the flat optimal

price eliminates inefficiency in allocation. However, inefficiency in the quantity of

production may go up since the airline is not able to price discriminate. To quantify

the net effect on social welfare, I again use the value of demand parameters that


Figure 2.8: Resale (fixed capacity)

correspond to a route with median characteristics and a non-holiday travel date.

Figure 2.7 shows the optimal price path for the first case in which the expected

marginal costs are fixed. If resale were possible, the average price of a ticket bought

by leisure travelers would increase from $77 to $118, while the average price of a ticket

purchased by business travelers would decrease from $318 to $118. The effect on the

business traveler is unambiguous: they pay a lower price and buy a higher quality

product. The effect on the leisure travelers is theoretically ambiguous. The price for

them increases for two reasons. First, they compete against customers who are willing

to pay more. Second, they are willing to pay more for a higher quality product. The

estimates suggest that the first effect dominates: their consumer welfare goes down

by 20%. The number of seats occupied by them would correspondingly decrease by

10%. The number of seats occupied by business travelers would go up by 50% and the

consumer surplus of business travelers increases by almost 100%. The airline’s profit

decreases by 28%. Overall, social welfare on the average route increases by 12%. The

decrease in the airline’s profit may force the airline to exit from the market, which

will decrease the social welfare to zero. Since the fixed costs of the airline are not

identified without observing any variation in entry-exit behavior, I cannot evaluate

how plausible such an outcome may be.

In the first case, the total number of occupied seats goes up. Therefore, to consider


the case in which the airline cannot adjust their capacity, I increased the value of the

marginal costs until the number of occupied seats in the counterfactual scenario is

equal to its initial level. Figure 2.8 shows that qualitatively the welfare effects of

intertemporal price discrimination remain the same. The average price goes up even

more, the median price goes down. The airline’s profit decreases even further. The

gains for the business travelers outweighs the losses of leisure travelers and the airline.

In this counterfactual, the inefficiency in production is fixed since the total quantity

remains the same. The increase in the social welfare (+6%) comes from elimination

inefficiency in allocation of seats caused by intertemporal price discrimination.

2.7.2 The role of cancellation fee

The cancellation fee has two effects on social welfare. Directly, it affects the quality of

production. Indirectly, it also affects the allocation and supply of tickets as it changes

the airline’s ability to price discriminate over time. A zero cancellation fee achieves

the socially optimal level of ticket quality. On the other hand, the airline loses one of

its screening tools, which makes price discrimination more difficult.

With a zero cancellation fee, the expected value of a refund is equal to Riτ =(1− δT−τi

)pτ , changing both individual demand functions and the airline’s profit.

The share of type-i buyers who arrived in period τ and purchase a ticket in period t

now becomes:

sitτ =exp

(µi−ptσi

)1 +

∑Tk=τ exp

(µi−pkσi

) ,

while the airline’s profit is equal to:

π(p; θ)

=I∑i=1

T∑t=1

δT−ti (pt − c)Dit.

With a zero cancellation fee, the optimal price path becomes flatter. As a result

the inefficiency in allocation goes down but inefficiency in the quantity of production

may go up. The net effect on social welfare is theoretically ambiguous and depends

on the value of demand and cost parameters.


Figure 2.9: Zero cancellation fee

Figure 2.9 shows the optimal price path on a route with median distance and

income departing on a non-holiday date. With zero cancellation fee, the difference

between average prices paid by business and leisure travelers would go down from

$305 to $273. This decrease is mainly caused by the fact that the average price that

leisure travelers pay goes up. The reason why leisure travelers would be willing to

accept higher prices is the better quality of airline tickets. The consumer surplus of

both groups would go up slightly while the airline’s profit would go down. Overall,

social welfare would increase, but by a relatively small amount (less than 1%). This

result is not too surprising as the airline does not really need to separate business and

leisure travelers, as most business travelers are estimated to arrive later than leisure

travelers.

This counterfactual assumes that the time when travelers start searching for the

ticket is exogenous and therefore does not depend on the value of the cancellation

fee. The exogeneity of customers’ arrival to the market is the reason why the airline

is able to price discriminate. This assumption, however, may not hold in reality. If

there is no cost associated with booking tickets early, business travelers might start

arriving to the market early and book preemptively. This assumption quickly brings

us to the case of costless resale.


Figure 2.10: Third degree price discrimination

2.7.3 Direct price discrimination

The last counterfactual evaluates the effectiveness of the intertemporal price dis-

crimination strategy. Suppose the airline can recognize a customer type and charge

different prices to different customer types. Then there will be two price paths: one

for business travelers, another for leisure travelers. The airline will not impose a

cancellation fee to separate customers within its type, since there is no within type

variation in the value of the cancellation probability. Therefore, in this counterfactual

I set the cancellation fee to zero. Figure 2.10 presents the optimal price paths and

the corresponding quantities of sold tickets.

By using intertemporal price discrimination, the airline captures more than 90%

of the profit that it could achieve if type-specific prices were possible. Surprisingly,

leisure travelers would prefer to see type-specific prices. There are two reasons for

that. First, the airline does not have to damage the product by imposing a cancella-

tion fee. Second, leisure travelers do not compete directly or indirectly with business

travelers. As the result, the airline can offer a lower price to leisure travelers, not

fearing to lose the price margin on business travelers. Business travelers lose from

third-degree price discrimination but their loss is smaller than the total gain of leisure

travelers and the airline.

2.8. CONCLUSION 45

2.8 Conclusion

In this essay, I developed an empirical model of optimal fares and estimated it using

new data on daily ticket prices from domestic monopoly markets. The estimates

of demand and cost parameters for monopoly routes allowed me to quantify the

costs and benefits of intertemporal price discrimination. I found that intertemporal

price discrimination results in a lower ticket quality for leisure travelers, higher prices

for business travelers, lower supply of tickets for business travelers, lower overall

supply and misallocations of tickets among travelers. On the other hand, the benefits

of intertemporal price discrimination are lower prices and higher supply for leisure

travelers.

I also found that free resale of airline tickets would reduce airlines’ ability to price

discriminate over time. As a result, business travelers would win from resale and

leisure travelers would lose, even though the quality of tickets would improve. Overall,

the short-run effect of ticket resale on social welfare is positive. However, since the

airline’s profit goes down, it may choose to exit from the market in the long run.

The effect of the cancellation fee on social welfare is small. The estimated increase in

prices is mainly caused by an increase in ticket quality, which does not affect social

welfare. Finally, I found that intertemporal price discrimination allows the airlines

to achieve more than 90% of the profit that third degree price discrimination would

generate.

The study focuses on the set of monopoly markets. There are two potential

difficulties with generalizing its results to more competitive markets. First, one may

worry about special characteristics of isolated monopoly markets. As the result, the

estimated demand parameters may not be representative of the entire industry. Unless

the difference between monopoly markets and the rest of the industry is solely caused

by the number of potential travelers, this is a valid concern. The second problem is

the impact of competition. Dynamic oligopoly models do not generally have a unique

equilibrium prediction. As a result, in may be very difficult to compare equilibria with

and without price discrimination. In particular, if resale were allowed, we will have to

consider an equilibrium in which a travel agency buys all tickets from the competing


airlines at the beginning of sale and then acts as a monopoly in the secondary market.

Whether this outcome is plausible is a question for future research.

2.9 Acknowledgments

I thank Lanier Benkard and Peter Reiss for their invaluable guidance and advice.

I am grateful to Tim Armstrong, Jeremy Bulow, Liran Einav, Alex Frankel, Ben

Golub, Michael Harrison, Jakub Kastl, Jon Levin, Trevor Martin, Michael Ostrovsky,

Mar Reguant, Andrzej Skrzypacz, Alan Sorensen, Bob Wilson, Ali Yurukoglu and

participants of the Stanford Structural IO lunch seminar for helpful comments and

discussions.

Chapter 3

How Firms Can Get More from

Less: An Airline Pricing Puzzle

3.1 Introduction

The idea that the ability to commit to a specific action can be beneficial for an agent

is a classic observation of game theory (Schelling, 1960). The contribution of this

chapter is to show that in a broader set of situations, agents can benefit if they can

commit to a menu of actions.

The result of the chapter is more than a theoretical curiosity. It offers a solution

to an airline pricing puzzle. It is well known that pricing in the airline industry is

complex. What is less known is that at any given moment, the price of a flight ticket

is determined by the decisions of not one but two airline departments, the pricing

department and the revenue management department. The pricing department sets

a menu of fares starting many days from the actual flight. This menu is subsequently

updated very rarely. The revenue management department treats the menu as given

but decides in each moment of time which part of the menu to make available for

purchase and which to keep closed. The revenue management department cannot

add new fares or change the prices of the existing ones. As the result, the price of

a ticket for a given flight can take only a limited number of values, not more than

the number of fares in the menu. Why would airlines introduce such a complex

47

48 CHAPTER 3. GETTING MORE FROM LESS

structure, in which they effectively restrict their flexibility in changing the prices

over time? At first thought it might seem they can do better by just changing the

price over time without precommitting to a set of fares. One answer may be that,

as long as the fare grid is fine enough, airlines do not lose much from significantly

restricting the set of possible prices. This chapter offers an alternative explanation.

It demonstrates that, in a strategic environment, firms can benefit from restricting

the set of available actions as long as they can credibly commit to doing so. Creating

a separate department that restricts the set of prices by choosing a menu of fares

might be a way to establish and maintain the credibility of such a commitment.

As a simple theoretical example of why menus may be desirable, consider a

Bertand game between two firms that sell identical products. The costs of produc-

tion are zero. The firm that charges the lower price gets the market demand. If the

prices are the same, the market demand is evenly divided between the firms. In this

example, the equilibrium payoffs are zero. The ability to commit cannot increase the

profit of an individual firm. If it commits to a price other than zero, the competitor

can steal the market by charging a slightly lower price and getting the entire market.

Consider now the following modification. The game consists of two stages: the

commitment stage and the action stage. At the commitment stage, firms can simulta-

neously and independently decide to constrain themselves to a menu of prices. Their

choices are then publicly observed. Then, at the action stage, firms independently and

simultaneously choose prices from their menus. A price outside the restricted menu

cannot be chosen by firms. The profits are determined according to the Bertrand

game described above.

Unlike in the original game, the equilibrium payoffs in this modification can be

positive. Moreover, there exists a subgame perfect equilibrium in which each firm

gets a half of the monopoly profit. The following pair of symmetric strategies form

this equilibrium. At the commitment stage, each firm chooses a menu that consists

of two prices, the monopoly price and zero. If both firms chose these menus, then

charging the monopoly price is an equilibrium of this subgame. Indeed, the only

deviation at the action stage is charging zero, which reduces the profit from the half

of the monopoly profit to zero. There are many deviations at the commitment stage

3.1. INTRODUCTION 49

but to be profitable they have to include a price that is higher than zero and less

than the monopoly price. If a firm attempts such deviation, then the competitor will

be indifferent between charging zero or the monopoly price at the action stage. If it

charges zero, then the deviator will end up receiving zero. Thus, deviations at the

commitment stage cannot be profitable either.

Thus, as we can see, Bertrand competitors can still achieve the monopoly outcome

without signing an enforceable contract or using repeated interactions to enforce

monopoly pricing. What they need to do is to exclude a set of profitable deviations

from the monopoly outcome (”temptations”) and keep ”punishment” actions in case

their competitors misbehave at the commitment stage.

This example is in a certain sense limited. Even when a firm undercuts by a very

small amount, the profit of its competitor decreases to zero. This discontinuity of

the payoffs guarantees that the punishment by zero price is a credible threat. If the

competitors deviate even by a small amount, the profit of the opponent drops to zero.

For continuous payoffs, these pair of strategies would not be equilibrium anymore.

This chapter focuses on a class of games with continuous payoffs and characterizes

pure-strategy subgame-perfect Nash equilibria of a game that has two stages described

above.

The two-stage construction of the chapter has four crucial components. First, the

agents are free to choose any subsets of the unrestricted action spaces. In particular,

these subsets can be non-convex or include isolated actions. Non-convex subsets, as

we will see later, allow players to achieve payoffs that dominate Nash equilibrium ones.

Second, it is important that the players are able to commit not to play actions outside

of the chosen subsets. Without such commitment, the initial stage will not change

the incentives of the players. Third, the chosen restricted set must become public

knowledge. Finally, the players choose these subsets simultaneously. If players move

sequentially, then some outcomes may not be an equilibrium to the sequential-move

game.

Pricing in the airline industry has all these four components. Airline fares are

discrete. The separation of the two departments creates some commitment power.

Indeed, according to industry insiders, these departments do not actively interact


with each other. Next, it turns out that sixteen major airlines own a company, the

Airline Tariff Publishing Company (ATPCO), whose role is to collect fares from more

than 500 airlines and distributes them four times a day to all airlines, travel agents,

and reservation systems. Thus, the information about the chosen subsets of fares

becomes public and the decisions are made simultaneously. Therefore, all necessary

features implied by the commitment concept are present in the pricing competition

among U.S. airlines.

The chapter has three main theoretical results. The first result shows that to

support other than the Nash equilibrium outcomes of a one stage game, players have

to constrain their action space. Moreover, at least one player has to choose a subset

with several actions at the commitment stage. The second result provides a necessary

and sufficient condition for an outcome to be supported by a commitment equilibrium.

An important corollary from this result shows that in a Bertrand oligopoly firms may

mutually gain from self-restraint while in Cournot they cannot. For a subset of

games, the third result demonstrates that there is a whole set of outcomes that can

be supported by an equilibrium of the two-stage model. This set, among others,

includes Pareto-efficient outcomes. The proof of the third result is constructive.

Surprisingly, a Harvard case study known as American Airlines Value Pricing

(1992) confirms the theoretical prediction of the model, namely the first result of the

chapter. In April 1992, American Airlines tried to abandon revenue management

system. They thought that fares were too complex, so the idea was to have one fare

that reflects the ”true value”. In terms of the model, they tried to commit to a set

that included only one action. Within a week, most major carriers (United, Delta,

Continental, Northwest) adopted the same pricing structure. However, in a week, the

airlines started a fare war. That is what exactly the model predicts: if players do not

include any punishment actions, a Bertrand-type price war (the unique static Nash

equilibrium) is the only subgame equilibrium outcome. By November 1992, American

Airlines acknowledged that the plan had ”clearly failed” and decided to come ”back

to setting and manipulating thousands of fares throughout the system”.

The rest of the chapter is organized as follows. Section 3.2 reviews related liter-

atures. Section 3.3 presents notation and definitions, and shows that the two-stage

3.2. RELATED LITERATURE 51

modification is a coarsening concept and can have as an equilibrium both Pareto

better and Pareto worse outcomes compared to the Nash equilibrium outcome. In

Section 3.4, I show that supermodular games is a natural class in which we can study

the two-stage modification as it is the class in which a (pure-strategy) subgame per-

fect equilibrium is guaranteed to exist. Section 3.5 shows that there is a nontrivial set

of outcomes that can be supported as an equilibrium in the two-stage game. Section

3.6 concludes.

3.2 Related Literature

This paper contributes to several literatures. First, the paper extends our under-

standing of the role of commitment in strategic interactions originally developed by

Schelling (1960). In his work, Schelling gives examples of how a player may benefit

from reducing her flexibility. In these examples, a player receives a strategic advan-

tage by moving first and committing to a particular action. No matter what the other

players do, she will not play a different action. In contrast, in this paper, the players

move simultaneously. As a result, they must recognize that they need some flexibility

in their actions in case other players do not restrict their actions or otherwise deviate.

For example, if a player commits to a single action, then she will have no punishment

should other players deviate. When a player commits to a subset of actions, then

her punishment action may differ from her reward action. Moreover, the punishment

may depend on the exact deviation chosen by an opponent.

At the same time, punishments have to be credible. Therefore, players do not

want to include too many actions as their potential punishments. A punishment

is effective only in the case when those who are supposed to punish will have an

incentive to execute this punishment. The smaller is the set of available actions,

the more likely the punishment is to be executed because less alternative actions are

available to the player. Thus, this paper studies the trade-off between the flexibility

and the credibility of available punishments. Committing to a single action is one

of the extremes in this trade-off, when punishment is fully credible but completely

inflexible.


Ideally, players would like to sign an enforceable contract that specifies what action

each agent will play. Of course, it may be hard to specify exactly what action each

agent should play. Hart and Moore (2004) recognize this fact and view a contract

as a mutual commitment not to play outcomes that are ruled out by the signed

contract. Bernheim and Whinston (1998) show that optimal contracts, in fact, have

to be incomplete and include more than one observable outcome if some aspects of

performance cannot be verified. Both papers view contracts as a means to constrain

mutually the set of available actions. This paper assumes that players cannot sign

such contracts. If they choose to restrict their set of available actions, they can only

do it independently of each other.

By maintaining independence, this paper is more similar in the spirit to the liter-

ature that models tacit collusion. It is well known that if players interact with each

other during several periods, they can achieve a certain level of cooperation even

if they act independently of each other. In finite-horizon games, deviations can be

deterred by a threat to switch to a worse equilibrium in later periods (Benoit and Kr-

ishna, 1985). Similarly, in infinite-horizon games, players can use a significantly lower

continuation value as a punishment that supports an equilibrium in the subgame in-

duced by a deviation. This result, known as the Folk theorem, states that if players

are patient enough any individually rational outcome can be supported by a subgame

perfect Nash equilibrium (see Abreu et al. (1990) and Fudenberg and Maskin (1986),

among others). In contrast, this paper assumes that the game is played only once.

As a result, players cannot use future outcomes to punish deviations. Instead, they

strategically choose a subset of actions that must include credible punishments suffi-

cient to deter deviations from the proposed equilibrium. These punishments must be

executed immediately to affect any cooperation.

Fershtman and Judd (1987) and Fershtman et al. (1991) studied delegation games

in which players can modify their payoff functions by signing contracts with agents

who act on their behalf. By doing so, the players can change their best responses and

therefore play other than a Nash equilibrium strategy, which may result in achieving a

Pareto efficient outcome. This study takes a different approach. Instead of modifying

their payoff functions, players can modify their action sets in a very specific way. They

3.3. ELEMENTS OF A COMMITMENT EQUILIBRIUM 53

can exclude a subset of actions but cannot include anything else. Thus, the model

of this paper may be viewed as a specific case of payoff modification: the players can

assign a large negative value to a subset of actions but cannot modify the payoffs in

any other way.

The closest papers to this paper are Bade et al. (2009) and Renou (2009). They

develop a similar two-stage construction in which commitment stage is followed by

a one-shot game. The constructions of their paper, however, are different in several

important aspects. In the first paper, players can only commit to a convex subset

of actions. This paper, however, allows players to choose any subset of actions. If

players can choose only a convex subset of actions, then they often cannot get rid

of temptations and keep actions that can be used as punishments, which are key

to the results of this paper. The second paper considers finite games, while this

paper studies supermodular games without making any restrictions on the number of

available actions.

There are a number of papers that endogenize players’ commitment opportuni-

ties.1 This paper does not address this question. The ability of firms to voluntarily

restrict their action space is assumed. This assumption, however, is motivated by real

world practices used by firms in an important industry: airlines.

3.3 Elements of a commitment equilibrium

3.3.1 Definitions

The game G Economic agents play a one-shot normal-form game, G. The game has

three elements: a set of players I = {1, 2, ..., n}, a collection of action spaces {Ai}i∈I ,and a collection of payoff functions {πi : A1 ×A2 × ...×An −→ R}i∈I . Thus, G =

(I,A, π) , where A = A1 × A2 × ... × An and π = π1 × π2 × ... × πn. (In general,

the omission of a subscript indicates the cross product over all players. Subscript −idenotes the cross product over all players excluding i). The agents make decisions

1See Rosenthal (1991), Van Damme and Hurkens (1996), and Caruana and Einav (2008), amongothers.


simultaneously and independently, which determines their payoffs. I refer to a ∈ Aas an outcome of G and to π (a) ∈ Rn as a payoff of G.

I assume that for every player i: a) Ai is a compact subset of R, and b) πi is upper

semi-continuous in ai, a−i.

An outcome a∗ ∈ A is called a Nash equilibrium (NE) outcome if there exists a

pure-strategy Nash equilibrium that supports a∗ ∈ A. In other words, a∗ ∈ A is a

NE outcome if and only if for all i ∈ I the following inequality holds:

πi(a∗i , a

∗−i)≥ πi

(ai, a

∗−i)

for any ai ∈ Ai.

Denote the set of all NE outcomes by EG and the set of corresponding NE payoffs by

ΠG.

The two-stage game C (G) Consider the following modification of game G. Define

C (G) as the following two-stage game:

Stage 0 (commitment stage). Each agent i ∈ I simultaneously and independently

chooses a non-empty compact subset Ai of her action space Ai. Once chosen, the

subsets are publicly observed.

Stage 1 (action stage). The agents play game GC = (I, A, π) where A = A1 ×A2 × ... × An. In other words, the players can choose actions that only belong to

the subsets chosen at stage 0. The actions outside the chosen subsets Ai can not be

played.

In this paper, I study pure-strategy subgame perfect Nash equilibria of C (G).

A strategy for player i in the game C (G) is a set Ai and a function σi which

selects, for any subsets of actions chosen by players other than i, an element of Ai, i.e.

σi : A−i −→ Ai. A pure-strategy subgame perfect Nash equilibrium of C (G) is called

an independent simultaneous self-restraint (commitment) equilibrium of the game G.

Formally, a commitment equilibrium is an n-tuple of strategies, (A∗, σ∗), such that

(i) for all i and any set of actions Ai ∈ Ai,

πi (a∗) ≥ πi

(σ∗1, ..., σ

∗i−1, ai, σ

∗i+1, ..., σ

∗n

)for any ai ∈ Ai,


where σ∗j = σ∗j

(A∗−j

)and A∗−j = A∗1 × ...A∗j−1 × A∗j+1 × ...A∗i−1 × Ai ×

A∗i+1...× A∗n;

(ii) for all i and any Ai, there exists ai ∈ Ai, such that(ai, σ

∗−i)

is a pure

strategy Nash equilibrium of game G =(I, A∗1 × ...A∗i−1 × Ai × A∗i+1...× A∗n, π

).

An outcome a∗ ∈ A is called a commitment equilibrium outcome if there exists a

commitment equilibrium that supports a∗ ∈ A, i.e. a∗ =(σ∗1(A∗−1

), ..., σ∗n

(A∗−n

)).

Denote the set of all commitment equilibrium outcomes by EC and the set of corre-

sponding commitment equilibrium payoffs by ΠC .

3.3.2 Reward, temptation, and punishment

To better describe the structure of a commitment equilibrium, I introduce the con-

cepts of reward, temptation and punishment actions. Take any commitment equilib-

rium, let a∗i = σ∗i(A∗−i)

denote the reward action of player i.

For an outcome a∗ ∈ Ai, an action aTi ∈ Ai is called a temptation in Ai of player

i if πi(aTi , a

∗−i)> πi

(a∗i , a

∗−i). Define the set of temptations Ti (a

∗|Ai) as:

Ti (a∗|Ai) =

{aTi ∈ Ai : πi

(aTi , a

∗−i)> πi

(a∗i , a

∗−i)}.

It follows from the definition of commitment equilibrium that if a∗ is an outcome

supported by a commitment equilibrium (A∗, σ∗), then none of the players has a

temptation in A∗i ⊆ Ai. In other words, Ti (a∗|A∗i ) = ∅ for all i. Obviously, a∗ is a

Nash equilibrium outcome if and only if none of the players has a temptation in Ai.A commitment equilibrium can include not only actions that are played along the

equilibrium path but also punishment actions that deter profitable deviations from

the intended outcome. Let APi = A∗i \ {a∗i } and aPi ∈ APi denote a punishment set and

a punishment action, respectively. Depending on the payoff function and the outcome

that the equilibrium intends to support, players may need to use different punishments

against different deviations. However, even one punishment may give the players a

powerful device that allows them to coordinate on better outcomes. Of course, APi


may be empty for one or all players. In the latter case, a commitment equilibrium

outcome will coincide with a NE outcome, as Proposition 3.1 will demonstrate.

Therefore, the intuition behind commitment equilibria is the following. On the

one hand, players want to exclude all temptations from their action space when they

choose to restrain their sets of available actions. On the other hand, since they

cannot promise to play the reward actions at the action stage, the players need to

include some actions that can be used as punishments should other players deviate

and fail to remove their temptations. To determine under what conditions the chosen

punishments can deter other players from deviating, we need to define and analyze

the subgames induced by players’ choices at the commitment stage.

3.3.3 Relation to Nash equilibrium

We now show that any NE outcome can be supported by a commitment equilibrium.

Take any Nash equilibrium. Suppose that at the commitment stage all players choose

their NE action, excluding all other actions from their sets. At the action stage, there

are no profitable deviations since there is only one action available to play. For this to

be a commitment equilibrium, we need to show that there are no deviations available

at the commitment stage. But this result follows almost immediately. Indeed, if all

but one player chose their NE action, then the remaining player cannot choose any

other subset and find it profitable to deviate by doing so. To be complete, I must

show that the deviator’s payoff attains its maximum on any subset. This requires

assuming that the payoff functions are upper semi-continuous and the action subsets

are compact, which was a part of the definition of the game.

Commitment equilibrium is a coarsening concept. In order to support a Nash

equilibrium, players have to restrict their subsets of actions at the commitment stage

to a singleton. If one player fails to do so and leaves her action set unconstrained,

the other player might find it beneficial to constrain himself to playing his Stackel-

berg action, i.e. the action that maximizes his payoff subject to his opponent’s best

response.

The main question of this paper, however, is whether commitment equilibria can


achieve better (or worse) payoffs compared to NE outcomes. It turns out that the

number of actions that the players keep at the commitment stage is a key factor

that determines what outcomes can be supported in a commitment equilibrium. It is

not sufficient to exclude deviations that will be profitable. Players have to have the

ability to carry out punishments in order to motivate other players not to deviate at

the commitment stage. The next proposition formalizes this idea.

Proposition 3.1. (To get a carrot, one needs to publicly carry a stick) Sup-

pose at stage 0 players can only choose subsets Ai that include only one element, i.e.

Ai = {Ai : |Ai| = 1}. Then commitment at stage 0 does not produce new equilibrium

outcomes, i.e. EG = EC .

Proof. The proof shows that if a0 is not a NE outcome, then it cannot be a com-

mitment equilibrium outcome in the case when |Ai| = 1. If a0 is not a NE, then

there exists a player j and an action a′j such that πj(a0j , a

0−j)< πj

(a′j, a

0−j). Sup-

pose a0 is in fact a commitment equilibrium. Since for all i, |Ai| = 1, it holds that

σ∗i({a0j},{a0−j})

= σ∗i({a′j},{a0−j})

= a0i . But then

πj (a∗) = πj(a0j , a

0−j)< πj

(a′j, a

0−j)

= πj(σ∗1, ..., σ

∗j−1, a

′j, σ∗j+1, ..., σ

∗n

),

which violates the first condition of commitment equilibrium. Thus, EG ⊆ EC .

In other words, to support an outcome outside EC , players have to choose more

than one action at the commitment stage. If players want to achieve payoffs outside

the NE set, they have to constrain themselves, but not too much.

3.3.4 Examples

The following example shows that it is in fact possible to support an outcome that

Pareto dominates the NE one.

Example 3.1 (Commitment equilibrium that Pareto dominates NE). Consider the


following two-by-two game:

C1 C2

R1 1, 1 −1,−1

R2 2,−1 0, 0

Row player has a strictly dominant strategy R2. Column player will respond by playing

C2. Thus, (R2,C2) is the only NE outcome. However, outcome (R1,C1) can be

supported by a commitment equilibrium. Indeed, let A1 = (R1), A2 = (C1,C2). At

stage 1, Row player plays R1 and Column player plays C1 if Row player plays C1,

otherwise she plays C2. It is easy to verify that this set of strategies is a commitment

equilibrium.

Two things are important in Example 1. First, Row player commits not to use

his dominant action R2. Second, Column player has an ability to punish Row player

by playing C2 if she decides to deviate. This punishment must be included at stage

0 and this fact must be publicly known.

In Example 1, the commitment equilibrium outcome is better for both players

than the unique Nash equilibrium of the original game. The next example shows that

the opposite can be also true: an outcome that is strictly worse for both players than

the unique Nash equilibrium can be supported by a commitment equilibrium.

Example 3.2 (Commitment equilibrium that is Pareto dominated by NE). Consider

the following three-by-three game:

C1 C2 C3

R1 2, 2 2,−1 0, 0

R2 −1, 2 1, 1 −1,−1

R3 0, 0 −1,−1 −1,−1

Row player has a strictly dominant strategy R1. Column player has a strictly dom-

inant strategy C1. Thus, (R1,C1) is the unique NE outcome. However, outcome

(R2,C2) can be supported by a commitment equilibrium. Let A1 = (R2,R3), A2 =

3.4. SUBGAME SUPERMODULAR GAMES 59

(C2,C3). At stage 1, Row player plays R2 and Column player plays C2 if the op-

ponents chose the equilibrium subsets. They play R3 or C3 otherwise. This set of

strategies is a commitment equilibrium that leads to an outcome that is strictly worse

than the unique Nash equilibrium outcome for both players.

The next example shows that in some cases a player has to have several punish-

ments to achieve a better equilibrium outcome.

Example 3.3 (To get the carrot, one may need several sticks). Consider the following

three-by-three game:

C1 C2 C3

R1 0, 0 0,−1 −1, 0

R2 −1, 3 1, 1 0, 2

R3 −1, 3 0,−2 2,−1

Column player has a dominant strategy C1. For Row player, R1 is a unique best

response. Thus, (R1,C1) is the unique NE outcome. However, outcome (R2,C2) can

be supported by a commitment equilibrium. Let A1 = (R1,R2,R3), A2 = (C2). At

stage 1, Row player plays R2 if Column player chose A2, R3 if Column player chose

C3 or {C2,C3} and R1 otherwise. Column player plays C2. This set of strategies

is a commitment equilibrium. Note that (R2,C2) can be supported by an equilibrium

only when Row player is able to include all three actions at stage 0.

Thus, to support an outcome outside the NE set, at least one player must include

at least two actions in her subset at the commitment stage. In some cases, players

have to include multiple punishments. One punishment may be effective and credible

against one deviation, another may be effective and credible against other deviations.

3.4 Subgame supermodular games

3.4.1 Definitions

In the previous section, we showed that any pure-strategy Nash equilibrium outcome

can be supported by a commitment equilibrium. Therefore, if a pure strategy Nash


equilibrium exists in the original game G, then a commitment equilibrium also exists

in the modified game C (G). However, Proposition 3.1 also showed that to support

an outcome outside of the set of NE outcomes, at least one of the players has to

announce more than one action at the commitment stage. Since some other player

can deviate to a subset that includes more than one action, a subgame induced by

such deviation, in principle, may not possess a pure-strategy Nash equilibrium, which

will violate the second requirement of a commitment equilibrium. Thus, to guarantee

the existence of a commitment equilibrium that can support other than NE outcomes,

we need to make sure that a pure-strategy Nash equilibrium exists in any subgame

following the commitment stage.

There are two approaches in the literature that establish the existence of pure-

strategy equilibria. The first approach derives from the theorem of Nash (1950). The

conditions of the theorem require the action sets be nonempty, convex and compact,

and the payoff functions to be continuous in actions of all players and quasiconcave

in its own actions. Reny (1999) relaxed the assumption of continuity by introducing

an additional condition on the payoff functions known as better-reply security. The

second approach was introduced by Topkis (1979) and further developed by Vives

(1990) and Milgrom and Roberts (1990),2 among others. They proved that any

supermodular game has at least one pure-strategy Nash equilibrium.

The first approach requires the action space be convex, while the second approach

places no such restrictions. Even if the unconstrained action space was convex, players

could choose a non-convex subset at the commitment stage. As the result, there may

exist a subgame induced by a unilateral deviation that has no pure-strategy Nash

equilibria. Therefore, to guarantee the existence of an equilibrium in any subgame,

we will follow the second approach and focus our attention on supermodular games.

Game G is called supermodular if for every player i, πi has increasing differences

in (ai, a−i) .3 Game C(G) is called subgame supermodular if any subgame induced by

a unilateral deviation at the commitment stage is supermodular. It is easy to show

2Migrom and Roberts (1990) lists a number of economic models that are based on supermodulargames.

3A function f : S × T → R has increasing differences in its arguments (s, t) if f (s, t) − f (s, t′)is increasing in s for all t ≥ t′.


that C(G) is subgame supermodular if and only if G is supermodular. First, suppose

that C(G) is subgame supermodular. Consider a subgame in which players did not

restrict their actions in the first stage. By definition, this subgame is supermodular,

therefore G is supermodular. Now, suppose that G is supermodular, consider any

subgame of C(G). For any non-empty compact subsets Ai the payoff functions πi

still have increasing differences (See Topkis, 1979). Therefore, C(G) is subgame

supermodular. The existence of a commitment equilibrium follows from the fact that

game G has a pure-strategy Nash equilibrium since it is a supermodular game.

3.4.2 Single-action deviation principle

To prove that a collection of restricted action sets together with a profile of proposed

actions form a subgame perfect equilibrium, we need to show that no profitable de-

viation exists. The absence of profitable deviations at the action stage is easy to

establish. The only condition we need to verify is to check if the players excluded all

temptations from their action sets. The commitment stage is more involved since a

player can possibly deviate to any subset of the unrestricted action set. Luckily, the

following result demonstrates that it is sufficient to check if there exists a profitable

deviation to subsets that includes only one action.

Lemma 3.1. In a subgame supermodular game, an outcome a∗ from a collection of

restricted action sets A∗ can be supported by a commitment equilibrium if and only

if no player can profitably deviate to a singleton. Formally, for each player i and each

her action ai, there exists an equilibrium outcome a in a subgame induced by sets

Ai = {ai} and A∗−i such that πi (a) ≤ πi (a∗).

Proof. If a player can profitably deviate to a singleton at the commitment stage, then

the original construction is not a commitment equilibrium. Suppose now that a player

can profitably deviate to a set that includes more than one action. Then there exists

an equilibrium in the subgame that is induced by the deviation that gives the player

a higher payoff. Suppose instead of deviating to the set the player chooses only the

action that is played in that equilibrium. It is easy to see that this outcome will still be

equilibrium. This player does not have any deviations and the other players can still


play the same equilibrium actions: elimination of irrelevant actions cannot generate

profitable deviations in this subgame. Thus, if there exists a profitable deviation to

a set, then a profitable deviation to a singleton has to exist.

Thus, to find commitment equilibrium outcomes, it is enough to consider devia-

tions to singletons. To do that, it is convenient to work with best response functions

defined for subgames induced by deviations at the commitment stage.

3.4.3 Subgame best response

Suppose at the commitment stage player i chose a subset of actions Ai. If player

i follows a commitment equilibrium strategy, then she must choose an action from

her subset that maximizes her payoff in the subgame. Formally, if the other players

play a−i ∈ A−i, then her equilibrium strategy has to choose an action σi (A−i) ∈BRi (a−i|Ai), where:

BRi (a−i|Ai) = Arg maxai∈Ai

πi (ai, a−i) .

I will call BRi (a−i|Ai) player ı’s subgame best response. Note that BRi (a−i|Ai) is a

standard best-response correspondence of player i in game G.

Several properties of subgame best responses follow from continuity of the pay-

off functions. By the Maximum Theorem of Berge (1979), the subgame best re-

sponse BRi (a−i|Ai) is non-empty, compact-valued and upper semi-continuous for any

nonempty set Ai. If game G is supermodular, then in addition to these properties,

the subgame best response is nondecreasing (Topkis, 1979).

Based on the continuity of the subgame best-response, we can analyze the relation

between subgame best responses for unrestricted and restricted action sets. Denote

by BRUi (a−i) the subgame best response for an unrestricted set. Consider a restricted

subgame best response for a set Ai.

First, suppose that Ai includes an interval [ai, ai] ∈ Ai that belongs to the image

of the unrestricted subgame best response BRUi (a−i). Then for any interior point of


p1

p2

p1Hp

1L

BR1U

BR1R

Figure 3.1: Firm 1’s restricted and unrestricted subgame best responses I

this interval ai ∈ (ai, ai), the inverse images of the restricted and unrestricted sub-

game best responses coincide: BR−1i (ai |Ai) = BR−1i (ai |Ai). Figure 3.1 illustrates

this property for the case of a differentiated Bertrand duopoly with linear demand

functions.4 The black line shows the unrestricted subgame best response for Firm 1,

i.e. it shows what price p1 Firm 1 should charge if Firm 2 charges price p2. If the

demand function is linear, then the unrestricted subgame best response function is

linear as well. Suppose Firm 1 keeps an interval of prices[pL1 , p

H1

]in its menu. Then

for any price p1 that in the interior of this interval, the restricted and unrestricted

subgame best responses coincide.

4See Appendix for the description of the model.


Second, suppose that two points from the image of the unconstrained subgame

best response BRi (A−i |Ai) belong to the restricted set Ai but the interval between

them (ai, ai) does not. Then there exists a vector of the competitors’ actions a−i

such that both points of the interval belong to the restricted subgame best response

for this vector: {ai; ai} ∈ BRi (a−i |Ai) and the agent is indifferent between playing

either of them πi (ai, a−i) = πi (ai, a−i). Figure 3.2 illustrates this result for the

differentiated Bertrand duopoly with linear demand functions. Suppose that Firm 1

kept two prices pL1 and pH1 but excluded the interval between them at the commitment

stage. Then, there exists a price pi2 such that if Firm 2 announces this price, Firm 1

will be indifferent between charging prices pL1 and pH2 . The isoprofit crossing points(pL1 , p

i2

)and

(pH1 , p

i2

)illustrates this fact.

3.4.4 Subgame equilibrium response

For games with two players it is enough to analyze restricted subgame best responses.

Indeed, it is enough to check that the equilibrium action of the player maximizes

her profit subject to the fact that the opponent plays his restricted subgame best

response. Formally, an outcome a∗ = (a∗1, a∗2) ∈ A1 × A2 can be supported by a

commitment equilibrium if and only if for each player i and her action ai, it holds

that πi (ai, a−i) ≤ πi (a∗) where a−i ∈ BR−i

(ai|A∗−i

).

To analyze games with three or more players, looking at best responses is not

enough. We need to find a new equilibrium in each subgame induced by a possible

deviation. Since more than one player will react to the deviation, the subgame best

responses have to take into account not only the action of the deviator but also the

reaction of the other players to this deviation. To take this into account, we define a

subgame equilibrium response as a function that for each possible deviation of player i

defines an equilibrium response of her opponents. Formally, function ER−i (ai|A−i) =

a−i, where (ai, a−i) is an equilibrium of the subgame for restricted sets Ai = {ai} and

A−i. If there are several equilibria in this subgame, ER−i chooses one for which

πi (ai, a−i) is smallest. Using the subgame equilibrium response function, we can

formulate a necessary and sufficient condition for a commitment equilibrium.


p1

p2

p1H

p2i

p1L

BR1U

BR1R

Isoprofit 1

Figure 3.2: Firm 1’s restricted and unrestricted subgame best responses II


Proposition 3.2. A collection of restricted sets A∗i , i = 1, ...n can support an out-

come a∗ ∈ A∗ by a commitment equilibrium if and only if for each agent i and action

ai, πi(a i, ER−i

(ai|A∗−i

))≤ πi

(a∗i , a

∗−i).

Proof. If the inequality does not hold for some ai, then choosing ai at the commitment

stage is a profitable deviation for player i. So, we need only to show that the reverse is

true. By Lemma 3.1, it is enough to show that no profitable deviation to a singleton

exists. Suppose it does and ai is such a deviation. Then in the subgame induced

by this deviation there is an equilibrium in which the opponents of the deviator play

a−i = ER−i(ai|A∗−i

). But since πi (ai, ER−i (ai|A∗i )) is less than the equilibrium

payoff for any ai, this deviation cannot be profitable.

Proposition 3.2 can be restated using the notion of a superior set. For each player

i define a superior set as Si (x) = {a ∈ A : πi (a) > πi (x)}. Then an outcome a∗ ∈ Acan be supported by a commitment equilibrium if and only if for all i the intersection

of Si (a∗) and the graph of ER−i (ai |A−i) is empty. The fact we just stated illustrates

the role the commitment stage plays. For a given outcome a ∈ A, the superior set

Si (a∗) is fixed but the best response equilibrium depends on the chosen restricted

sets. To support a commitment equilibrium, the players have to chose a collection of

restricted action sets such that the graph of the resulting equilibrium response does

not overlap with the superior set.

3.4.5 Credibility of punishment

For a subgame supermodular game, the subgame best response is a nondecreasing cor-

respondence. This fact implies that ER−i (ai|A−i) is a nondecreasing function for any

A−i. Consider the incentives of player i to play an outcome a ∈ A as a commitment

equilibrium. Suppose he decides to deviate by playing a′i > ai. This deviation will

not be profitable if πi (a′i, ER−i (a

′i, A−i)) ≤ πi (a). The same equality should hold if

the deviator decides to increase its action. Thus, the payoff function cannot increase

or decrease in all its arguments around the commitment equilibrium point. Since the

equilibrium response function is nondecreasing, only those outcomes for which the

3.5. STACKELBERG SET 67

temptation set for each player lies below the reward action, i.e. Ti (ai|Ai) ≤ ai for all

i, can be supported by a commitment equilibrium.

For a Bertrand oligopoly with differentiated products this property holds if the

firms want to support higher than NE prices. Indeed, when the prices exceed the NE,

the profit of a potential deviator will increase if he reduces its price but decrease if

his competitors decide to do so.

Cournot duopolies,5 however, will violate this property. Suppose the firms want

to support a higher than monopoly price. To do so, they need to produce less. Thus,

the deviator’s profit will increase if he increases his output. To punish him, the firms

have to increase their output further, which will drive the price down. However, their

best response to an increase in total output is to decrease their own. Thus, in Cournot

oligopolies, punishments that could have supported a high price are not credible.

3.5 Stackelberg set

This subsection shows that there is, in fact, a set of payoff profiles that contains

Pareto efficient payoffs that can be supported by a commitment equilibrium. To

define this set, we need to use the concept of Stackelberg outcomes. An outcome

aLi is called a Stackelberg outcome for player i, if πi(aLii , ER−i

(aLii

))≥ πi (ai, a−i)

for any ai ∈ Ai. This outcome corresponds to a subgame perfect equilibrium in

the following modification of game G: player i moves first; the rest of the players

move simultaneously and independently of each other after observing the action of

player i. Similarly to a Nash equilibrium outcome, it is easy to establish that a

Stackelberg outcome for any player i can be supported by a commitment equilibrium.

This equilibrium would prescribe that players only choose their equilibrium action

at the commitment stage. If all players do that, then none will have an incentive to

deviate either at the commitment stage or at the action stage.

Define the payoff of player i that corresponds to a Stackelberg outcome by πLii .

5Generically, a Cournot oligopoly with three or more firms is not a supermodular game. However,it may become one under a set of reasonable assumptions placed on the market demand and costfunctions (see, Amir, 1996). The argument will still be true for games that satisfy these assumptions.


p1L

p2L

p1F

p2F

BR2

BR1

Isoprofit 2

Isoprofit 1

L

Figure 3.3: Stackelberg set

The Stackelberg set L is then defined as:

L ={a ∈ A: πi (a) ≥ πLi and Ti (a|Ai) ≤ ai for all i

}.

The previous literature has shown that the Stackelberg set L for a class of games is

not empty (Amir and Stepanova (2006) and Dowrick (1986)Dowrick (1986)). Figure

3.3 shows the Stackelberg set for the Bertrand duopoly with differentiated products

and linear demands.

The next proposition shows that any outcome from the Stackelberg set can be

supported by a commitment equilibrium if the players have an action that can serve

as a credible punishment. The proof of the proposition is constructive.

3.5. STACKELBERG SET 69

Proposition 3.3. Suppose for each player i, the payoff function πi is increasing in

all a−i. Then an outcome a∗ ∈ L can be supported by a commitment equilibrium if

for any i there exists an action aPi satisfying πi(aPi , a

∗−i)

= πi(a∗i , a

∗−i)

such that aPi

≤ BRi

(a∗−i |Ai

).

Proof. Consider the following set of strategies. At the commitment stage each player

includes in A∗i the following elements: a∗i , aPi and all actions below aPi . To prove

that a∗ can be supported by a commitment equilibrium with these restricted sets, it

is enough to show that for each player i the graph of the equilibrium best response

function and the superior set do not overlap.

First, consider the restricted subgame best responses. Based on the properties of

the restricted subgame best response established in Section 4.3, we can conclude that

for a−i < aP−i, the best response functions coincide. Then the restricted best response

is equal to aPi until a∗−i. At a∗−i the player is indifferent between playing aPi and a∗i .

Finally, for a−i > a∗−i, player i will be playing a∗i . Thus, the equilibrium response

function will be the following: for ai ≥ a∗i , the other players will play a∗−i, for ai < a∗i ,

they will play aP−i until ai ∈ BRi

(aP−i|Ai

)and ER−i (ai|A−i) otherwise.

Since a∗ ∈ L, the superior set of player i belongs to the superior set constructed

for the Stackelberg outcome for player i, which in turn does not overlap with the

unrestricted equilibrium response function. For ai < a∗i the graph of the restricted

equilibrium responses is bounded from above by the unrestricted equilibrium response

since aPi ≤ BRi

(a∗−i|Ai

). Therefore, we need to check that there are no overlaps for

ai > a∗i . However, since the temptation set lies below a∗i , such overlaps cannot

exist.

The proposition shows that the punishment action has to satisfy the following

property. Conditional on the fact that all the opponents play the reward action,

the player has to be indifferent between playing his reward action and playing the

punishment action. If a deviator follows his temptation and includes an action from

the temptation set, the other players will execute the punishment, which precludes

the deviator from receiving any benefits from not choosing the reward.


3.6 Concluding comments

Motivated by pricing practices in the airline industry, this paper shows that in a

competitive environment, agents may benefit from committing not to play certain

actions. For the commitment equilibrium construction to work, all players should

have commitment power and be able to play several actions after the commitment

stage. The main intuition of the paper is the following: to get the reward, the players

need to exclude all temptations but keep punishments to motivate their opponents.

Although the model is simple, its intuition can explain the otherwise puzzling menu

structure of airline fares.

There are several interesting extensions that could be further investigated. First,

the model assumed that the action stage was played only once. It is interesting to see

what the players can achieve when they can play the game a finite number of times.

The commitment stage in this case may provide punishments that players execute not

only after deviations at the commitment stage but also following deviations at the

preceding action stages. Second, the trade-off becomes more complicated when there

is some uncertainty that is resolved between the commitment and action stages. In

this case, at the commitment stage the players do not know the exact punishment that

they will need to include and may end up including a non-credible punishment or a

temptation. Therefore, there is always a probability that the award action will not be

an equilibrium at the action stage along the equilibrium path. The third extension is

more technical. Renou (2009) shows an example of a game in which a mixed strategy

Nash equilibrium does not survive in the two-stage commitment game. Thus, the

question is which mixed strategy equilibria can and which cannot be supported by a

commitment equilibrium.

3.7 Acknowledgments

I thank Peter Reiss and Andy Skrzypacz for their invaluable guidance and advice.

I am grateful to Lanier Benkard, Jeremy Bulow, Alex Frankel, Ben Golub, Michael

Harrison, Jon Levin, Trevor Martin, Michael Ostrovsky, Bob Wilson and participants

3.7. ACKNOWLEDGMENTS 71

of the Stanford Structural IO lunch seminar for helpful comments and discussions.

Bibliography

Abreu, D., D. Pearce, and E. Stacchetti (1990): “Toward a Theory of Dis-

counted Repeated Games with Imperfect Monitoring,” Econometrica, 58, 1041–

1063.

Aguirre, I., S. Cowan, and J. Vickers (2010): “Monopoly Price Discrimination

and Demand Curvature,” The American Economic Review, 100, 1601–1615.

Amir, R. (1996): “Cournot Oligopoly and the Theory of Supermodular Games,”

Games and Economic Behavior, 15, 132–148.

Amir, R. and A. Stepanova (2006): “Second-mover advantage and price leader-

ship in Bertrand duopoly,” Games and Economic Behavior, 55, 1–20.

Bade, S., G. Haeringer, and L. Renou (2009): “Bilateral commitment,” Jour-

nal of Economic Theory, 144, 1817–1831.

Benoit, J. and V. Krishna (1985): “Finitely Repeated Games,” Econometrica,

53, 905–922.

Bernheim, B. D. and M. D. Whinston (1998): “Incomplete Contracts and

Strategic Ambiguity,” The American Economic Review, 88, 902–932.

Bly, L. (2001): “Web Site for Resale of Air Tickets Shows Promise, but It’s Still a

Fledgling,” Los Angeles Times. June 3, 2001.

Board, S. (2008): “Durable Goods Monopoly with Varying Demand,” Review of

Economic Studies, 75, 391–413.

72

BIBLIOGRAPHY 73

Board, S. and A. Skrzypacz (2010): “Revenue Management with Forward-

Looking Buyers,” Tech. rep., Stanford University.

Borenstein, S. and N. L. Rose (1994): “Competition and Price Dispersion in

the U.S. Airline Industry,” Journal of Political Economy, 102, 653–683.

Caruana, G. and L. Einav (2008): “A Theory of Endogenous Commitment,”

Review of Economic Studies, 75, 99–116.

Chen, J., S. Esteban, and M. Shum (2010): “How much competition is a sec-

ondary market?” Working Paper.

Curtis, W. (2007): “World of Wonders,” The New York Times. December 22, 2007.

Deneckere, R. J. and R. Preston McAfee (1996): “Damaged Goods,” Journal

of Economics & Management Strategy, 5, 149–174.

Dowrick, S. (1986): “von Stackelberg and Cournot Duopoly: Choosing Roles,” The

RAND Journal of Economics, 17, 251–260.

Elliot, C. (2011): “Ridiculous or not? Your airline ticket isn’t transferrable,”

Consumer advocate Christopher Elliott’s site. March 16, 2011.

Esteban, S. and M. Shum (2007): “Durable-goods oligopoly with secondary mar-

kets: the case of automobiles,” The RAND Journal of Economics, 38, 332–354.

Evans, W. N. and I. N. Kessides (1994): “Living by the ”Golden Rule”: Multi-

market Contact in the U.S. Airline Industry,” The Quarterly Journal of Economics,

109, pp. 341–366.

Fershtman, C. and K. L. Judd (1987): “Equilibrium Incentives in Oligopoly,”

The American Economic Review, 77, 927–940.

Fershtman, C., K. L. Judd, and E. Kalai (1991): “Observable Contracts:

Strategic Delegation and Cooperation,” International Economic Review, 32, 551–

559.

74 BIBLIOGRAPHY

Fudenberg, D. and E. Maskin (1986): “The Folk Theorem in Repeated Games

with Discounting or with Incomplete Information,” Econometrica, 54, 533–554.

Gavazza, A., A. Lizzeri, and N. Roketskiy (2010): “A Quantitative Analysis

of the Used Car Market,” Tech. rep., mimeo, New York University.

Gerardi, K. S. and A. H. Shapiro (2009): “Does Competition Reduce Price Dis-

persion? New Evidence from the Airline Industry,” Journal of Political Economy,

117, 1–37.

Gershkov, A. and B. Moldovanu (2009): “Dynamic Revenue Maximization

with Heterogeneous Objects: A Mechanism Design Approach,” American Eco-

nomic Journal: Microeconomics, 1, 168–198.

Hart, O. and J. Moore (2004): “Agreeing Now to Agree Later: Contracts that

Rule Out but do not Rule In,” National Bureau of Economic Research Working

Paper Series, No. 10397.

Hendel, I. and A. Nevo (2011): “Intertemporal Price Discrimination in Storable

Goods Markets,” National Bureau of Economic Research Working Paper Series,

No. 16988.

Hoerner, J. and L. Samuelson (2011): “Managing Strategic Buyers,” Journal

of Political Economy, 119, 379–425.

Leslie, P. (2004): “Price Discrimination in Broadway Theater,” The RAND Journal

of Economics, 35, 520–541.

Leslie, P. and A. Sorensen (2009): “The Welfare Effects of Ticket Resale,”

National Bureau of Economic Research Working Paper Series, No. 15476.

McAfee, R. P. and V. te Velde (2007): “Dynamic Pricing in the Airline Indus-

try,” Handbook on Economics and Information Systems, 1.

Milgrom, P. and J. Roberts (1990): “Rationalizability, learning, and equilibrium

in games with strategic complementarities,” Econometrica, 58, 1255–1277.

BIBLIOGRAPHY 75

Mussa, M. and S. Rosen (1978): “Monopoly and product quality,” Journal of

Economic Theory, 18, 301–317.

Nair, H. (2007): “Intertemporal price discrimination with forward-looking con-

sumers: Application to the US market for console video-games,” Quantitative Mar-

keting and Economics, 5, 239–292.

Nash, J. F. (1950): “Equilibrium points in n-person games,” Proceedings of the

National Academy of Sciences of the United States of America, 48–49.

Phillips, R. L. (2005): Pricing and revenue optimization, Stanford University Press.

Renou, L. (2009): “Commitment games,” Games and Economic Behavior, 66, 488–

505.

Reny, P. J. (1999): “On the existence of pure and mixed strategy Nash equilibria

in discontinuous games,” Econometrica, 67, 1029–1056.

Robinson, J. (1933): “The Economics of Imperfect Competition,” .

Rosenthal, R. W. (1991): “A note on robustness of equilibria with respect to

commitment opportunities,” Games and Economic Behavior, 3, 237–243.

Schelling, T. C. (1960): The strategy of conflict, Harvard University Press.

Schmalensee, R. (1981): “Output and Welfare Implications of Monopolistic Third-

Degree Price Discrimination,” The American Economic Review, 71, 242–247.

Schwartz, M. (1990): “Third-Degree Price Discrimination and Output: Generaliz-

ing a Welfare Result,” The American Economic Review, 80, 1259–1262.

Shepard, A. (1991): “Price Discrimination and Retail Configuration,” Journal of

Political Economy, 99, 30–53.

Silk, A. and S. Michael (1993): “American Airlines Value Pricing (A),” Harvard

Business School Case Study 9-594-001. Cambridge, MA: Harvard Business School

Press.

76 BIBLIOGRAPHY

Stavins, J. (2001): “Price Discrimination in the Airline Market: The Effect of

Market Concentration,” The Review of Economics and Statistics, 83, 200–202.

Stokey, N. L. (1979): “Intertemporal Price Discrimination,” The Quarterly Journal

of Economics, 93, 355 –371.

Sweeting, A. (2010): “Dynamic Pricing Behavior in Perishable Goods Markets:

Evidence from Secondary Markets for Major League Baseball Tickets,” Tech. rep.,

mimeo.

Talluri, K. and G. Van Ryzin (2005): The theory and practice of revenue man-

agement, vol. 68, Springer Verlag.

Topkis, D. M. (1979): “Equilibrium points in nonzero-sum n-person submodular

games,” SIAM Journal on Control and Optimization, 17, 773.

van Damme, E. and S. Hurkens (1996): “Commitment Robust Equilibria and

Endogenous Timing,” Games and Economic Behavior, 15, 290–311.

Varian, H. R. (1985): “Price Discrimination and Social Welfare,” The American

Economic Review, 75, 870–875.

Verboven, F. (1996): “International Price Discrimination in the European Car

Market,” The RAND Journal of Economics, 27, 240–268.

Villas-Boas, S. B. (2009): “An empirical investigation of the welfare effects of

banning wholesale price discrimination,” The RAND Journal of Economics, 40,

20–46.

Vives, X. (1990): “Nash equilibrium with strategic complementarities,” Journal of

Mathematical Economics, 19, 305–321.

Appendix A

Commitment equilibrium in

Bertrand duopoly with

differentiated products

To illustrate the idea of a commitment game, consider the differentiated Bertrand

duopoly model with linear demand functions. Suppose two symmetric firms (i = 1,2)

produce differentiated products. The demand for each product depends both on its

price and the price of the competitor in a linear way:

q1 (p1, p2) = 1− p1 + αp2,

q2 (p1, p2) = 1− p2 + αp1,

where α ∈ (0, 1) is the parameter that characterizes the degree of products’ substi-

tutability: the higher is α, the more substitutable are the products.

If firms set their prices simultaneously and independently, then there exists a

unique Nash equilibrium in which both firms charge the following price:

pNE1 = pNE2 =1

2− α.

As a result, both firms receive profits equal to πNE1 = πNE2 = 1(2−α)2 .

77

78 APPENDIX A. DUOPOLY WITH DIFFERENTIATED PRODUCTS

It is easy to see that this pair of profits does not lie on the Pareto frontier. In

other words, if both firms decide to increase their price by a small amount (ε > 0),

then both profits will increase:

π1 = π2 =1

(2− α)2+

α

2− αε− (1− α) ε2 >

1

(2− α)2

for small ε > 0.

Thus, if the firms could agree to coordinate their actions, they could do better.

Yet, it is clear that neither firm unilaterally has an incentive to declare it will charge

the profit maximizing price pM1 = pM2 = 12(1−α) . To see why this pair of prices cannot

be a Nash equilibrium, consider the figure.

The figure depicts the isoprofit curves for each firm corresponding to the prices that

maximize the firms’ joint profit. These isoprofits touch each other at(

12(1−α) ,

12(1−α)

),

the point that maximizes the firms’ joint profit. Suppose firm 1 knows that firm 2

will charge pM2 . If firm 1 charges pM1 , then it gets πM1 = 14(1−α) . However, if firm 1

charges a different price, in particular, any price in the range(pP1 , p

M1

), then its profit

will be strictly higher than πM1 . In other words, firm 1 has a temptation to deviate

from the price it is supposed to charge and charge something lower. Unless firm 1 can

commit not to charge prices from the interval(pP1 , p

M1

), the pair of prices

(pM1 , p

M2

)cannot be an equilibrium.

Suppose now that firms have ability to restrain themselves independently of each

other. To be more precise, suppose that firms choose a subset of their prices first. We

saw that firm 1 can profitably deviate if it charges any price from(pP1 , p

M1

). Therefore,

in an equilibrium it needs to restrain itself and exclude all prices from that interval.

However, it has an incentive to cheat (a ”temptation”). Therefore firm 2 has to be

able to punish firm 1 if it includes prices from that interval. Hence, firm 2 has to

choose not only price pM2 that it is supposed to play in an equilibrium but also some

other prices that it will use as punishments should firm 1 not commit to exclude

prices in the interval(pP1 , p

M1

).

For reasons discussed in the main text, it is sufficient in this case for each firm to

choose just two prices: the pair of prices that will be played along the equilibrium

79

Figure A.1: Differentiated Bertrand Duopoly. How to Support a Joint-Profit Maxi-mizing Outcome

80 APPENDIX A. DUOPOLY WITH DIFFERENTIATED PRODUCTS

path(pM1 , p

M2

)and two prices that will be used as punishments, namely pP1 and pP2 .

To see why committing to just two prices in an initial stage will work, suppose firm

1 decides to deviate and restrains itself to some other price. It is easy to verify that

firm 2 will charge pM2 if firm 1 charges any price higher than pM1 . If firm 1 charges

any price lower than pM1 , then firm 2 should choose pP2 . Thus, firm 1 cannot benefit

from a deviation given firm 2 commits to{pM2 , p

P2

}since the isoprofit of firm 1 lies to

the north of firm 2’s optimal response. Perhaps, however, firm 1 could benefit from

choosing more than one price at the commitment stage? It turns out that no matter

what subset of prices the firm chooses, there will always exist a pure-strategy Nash

equilibrium in the corresponding subgame in which firm 2 will play either pM2 or pP2 .

In neither case, as we already have seen, can firm 1 benefit if it deviates.

Thus, in this widely studied game, if firms can restrain themselves independently

of each other, they can coordinate on the outcome that maximizes their joint profit.

dynamic pricing in the airline industry a dissertation ...kk032tn6364/... · abstract the...

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