dynamic pricing in the airline industry a dissertation ...kk032tn6364/... · abstract the...
TRANSCRIPT
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
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
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.
Charles Benkard, Co-Adviser
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.
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
iv
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.
8 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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,
10 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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
12 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
14 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
2.3. THE MODEL OF OPTIMAL FARES 15
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
16 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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
2.3. THE MODEL OF OPTIMAL FARES 17
$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.
18 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
2.3. THE MODEL OF OPTIMAL FARES 19
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.
20 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
2.3. THE MODEL OF OPTIMAL FARES 21
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.
22 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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
24 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
26 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
28 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
30 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
32 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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
34 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
36 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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
38 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
Figure 2.6: Distributions of travelers’ utilities under the optimal allocation of seats
2.7. COUNTERFACTUAL SIMULATIONS 39
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
) .
40 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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
2.7. COUNTERFACTUAL SIMULATIONS 41
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
42 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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.
2.7. COUNTERFACTUAL SIMULATIONS 43
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.
44 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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
46 CHAPTER 2. INTERTEMPORAL PRICE DISCRIMINATION
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
50 CHAPTER 3. GETTING MORE FROM LESS
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.
52 CHAPTER 3. GETTING MORE FROM LESS
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.
54 CHAPTER 3. GETTING MORE FROM LESS
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,
3.3. ELEMENTS OF A COMMITMENT EQUILIBRIUM 55
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
56 CHAPTER 3. GETTING MORE FROM LESS
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
3.3. ELEMENTS OF A COMMITMENT EQUILIBRIUM 57
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
58 CHAPTER 3. GETTING MORE FROM LESS
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
60 CHAPTER 3. GETTING MORE FROM LESS
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′.
3.4. SUBGAME SUPERMODULAR GAMES 61
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
62 CHAPTER 3. GETTING MORE FROM LESS
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
3.4. SUBGAME SUPERMODULAR GAMES 63
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.
64 CHAPTER 3. GETTING MORE FROM LESS
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.
3.4. SUBGAME SUPERMODULAR GAMES 65
p1
p2
p1H
p2i
p1L
BR1U
BR1R
Isoprofit 1
Figure 3.2: Firm 1’s restricted and unrestricted subgame best responses II
66 CHAPTER 3. GETTING MORE FROM LESS
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.
68 CHAPTER 3. GETTING MORE FROM LESS
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.
70 CHAPTER 3. GETTING MORE FROM LESS
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.