# Price Discrimination in International Airline Markets††thanks: We are very
grateful to the U.S. Department of Commerce for providing the data for our
analysis. We are thankful to research computing facilities at Boston College,
UNC and UVA, with special thanks to Sandeep Sarangi at UNC research computing.
Seminar (Arizona, BC, MIT, Penn State, Rice, Richmond Fed, Rochester, Texas
A&M Toulouse, UC-Irvine, UNC-Chapel Hill, UWO, UWSTL, Vanderbilt, Virginia)
and conference (SEA 2014, EARIE 2015, IIOC 2016, ASSA 2016, DSE 2019, SITE
2019) participants provided helpful comments. We thank Jan Brueckner, Ben
Eden, Gautam Gowrisankaran, Jacob Gramlich, Qihong Liu, Brian McManus, Nancy
Rose, Nick Rupp for insightful suggestions. All remaining errors are our own.

Gaurab Aryal
Department of Economics, University of Virginia<EMAIL_ADDRESS>Charles
Murry
Department of Economics, Boston College<EMAIL_ADDRESS>Jonathan W.
Williams
Department of Economics, University of North Carolina - Chapel Hill,
<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

We develop a model of inter-temporal and intra-temporal price discrimination
by monopoly airlines to study the ability of different discriminatory pricing
mechanisms to increase efficiency and the associated distributional
implications. To estimate the model, we use unique data from international
airline markets with flight-level variation in prices across time, cabins, and
markets, as well as information on passengers’ reasons for travel and time of
purchase. We find that the ability to screen passengers across cabins every
period increases total surplus by 35% relative to choosing only one price per
period, with both the airline and passengers benefiting. However, further
discrimination based on passenger’s reason to traveling improve airline
surplus at the expense of total efficiency. We also find that the current
pricing practice yields approximately 89% of the first-best welfare. The
source of this inefficiency arises mostly from dynamic uncertainty about
demand, not private information about passenger valuations.

##  1 Introduction

Firms with market power often use discriminatory prices to increase their
profits. However, such price discrimination can have ambiguous implications on
total welfare. Enhanced price discrimination may increase welfare by reducing
allocative inefficiencies but may also reduce consumer welfare. So, an
essential aspect of economic- and public-policy towards price discrimination
is to understand how well various discriminatory prices perform in terms of
the total welfare and its distribution, relative to each other and the first-
best, (e.g., Pigou, 1920; Varian, 1985; Council of Economic Advisors, 2015).

We evaluate the welfare consequences of price discrimination and quantify
sources of inefficiencies in a large and economically important setting,
international air travel markets. To that end, we develop and estimate a model
of inter-temporal and intra-temporal price discrimination by a monopoly
airline and study the ability of different discriminatory mechanisms to
increase welfare and the associated distributional implications. The model
incorporates a rich specification of passenger valuations for two vertically
differentiated seat classes on international flights, and a capacity-
constrained airline that faces stochastic and time-varying demand. The airline
screens passengers between two cabins while updating prices and seat offerings
over time. Using the model estimates, we implement various counterfactuals in
the spirit of Bergemann, Brooks, and Morris (2015), where we change the
information the airline has about preferences and the timing of arrivals and
measure the welfare under various discriminatory pricing strategies. Our
counterfactual pricing strategies are motivated by recent airline practices
intended to raise profits by reducing allocative inefficiencies, including
attempts to solicit passengers’ reason to travel and use of auctions (e.g.,
Nicas, 2013; Vora, 2014; Tully, 2015; McCartney, 2016).

We find that the ability to screen passengers across cabins increases the
total surplus by 35% relative to choosing a single price each period (i.e.,
“shutting down” second-degree price discrimination across cabins), with both
airline and passengers benefiting. However, further discriminatory practices
based on passengers’ reason to travel improve the airline’s surplus but lower
the total surplus. We also find that the current pricing practice yields
approximately 89% of the first-best welfare, and that the source of this
remaining inefficiency is mostly due to the dynamic uncertainty about demand,
not the private information about passengers’ valuations.111 Here, the “first-
best” welfare is defined in the context of our model. There may be other
effects worth exploring, for example ticket resale in Lazarev (2013), that our
model does not capture. This suggests that airlines’ attempts to improve
dynamic allocations could provide efficiency improvements.

Our empirical strategy centers around a novel dataset of international air
travel from the U.S. Department of Commerce’s Survey of International Air
Travelers. Compared to the extant literature, the novelty of these data is
that we observe both the date of transactions and passenger characteristics
for dozens of airlines in hundreds of markets. We document the late arrival of
passengers traveling for business, who tend to have inelastic demand, and the
associated changes in prices. Although business travelers’ late arrival puts
upward pressure on fares, fares do not increase monotonically for every
flight. This pattern suggests that the underlying demand for air travel is
stochastic and non-stationary.

To capture these salient data features, we propose a flexible but tractable
demand system. Each period before a flight departs, a random number of
potential passengers arrive and purchase a first-class ticket, an economy
class ticket, or decide not to fly at all. Passengers’ willingness-to-pay
depends on the seat class and passenger’s reason to travel. We allow
passengers to have different willingness-to-pay for first-class, so, for some
passengers, the two cabins are close substitutes but not for others.
Furthermore, we allow the mix of the two types of passengers–business and
leisure–to vary over time.

On the supply side, we model a monopoly airline’s problem of selling a fixed
number of economy and first-class seats. The airline knows the distribution of
passengers’ valuations and the expected number of arrivals each period but
chooses prices and seats to release before it realizes actual demand.222 We
model the airline committing to a seat release policy to mimic the “fare
bucket” strategy used by airlines in practice. See, for example, Alderighi,
Nicolini, and Piga (2015) for more on these buckets.  At any time before the
flight, the airline balances expected profit from selling a seat today against
the forgone future expected profit. This inter-temporal trade-off results in a
time-specific endogenous opportunity cost for each seat that varies with the
expected future demand and number of unsold seats. Besides this temporal
consideration, each period, the airline screens passengers between the two
cabins. Thus, our model captures both the inter-temporal and intra-temporal
aspects of price discrimination by airlines.

Estimation of our model presents numerous challenges. The richness of the
demand, and our supply specification, result in a non-stationary dynamic
programming problem that involves solving a mixed-integer nonlinear program
for each state. We solve this problem to determine optimal prices and seat-
release policies for each combination of unsold seats and days until
departure. Moreover, our data include hundreds of flights across hundreds of
routes, so not only do we allow for heterogeneity in preferences across
passengers _within a flight_ , we also allow different flights to have
different distributions of passenger preferences. To estimate the model and
recover the distribution of preferences across flights, we use a simulated
method of moments approach based on Ackerberg (2009). Similar approaches to
estimate a random coefficient specification has recently been used by Fox,
Kim, and Yang (2016), Nevo, Turner, and Williams (2016), and Blundell,
Gowrisankaran, and Langer (2020). Like these papers, we match empirical
moments describing within-flight and across-flight variation in fares and
purchases to a mixture of moments implied by our model.

Our estimates suggest that there is substantial heterogeneity across
passengers within a flight and substantial heterogeneity across flights. The
estimated marginal distributions of willingness-to-pay for business and
leisure travelers are consistent with the observed distribution of fares. We
estimate that the average willingness-to-pay for an economy seat in our data
by leisure and business passengers is $413 and $506, respectively.
Furthermore, on average, passengers value a first-class seat 23% more than an
economy seat, which implies meaningful cross-cabin substitution. We also find
declining arrivals of passengers overall, but with an increasing fraction of
business travelers. Using the model estimates, we calculate the unobserved
time-varying opportunity cost of selling a seat, which provides novel insight
into airlines’ dynamic incentives.

Using the estimates and the model, we characterize the level of efficiency and
the associated distribution of surplus for alternative pricing mechanisms that
provide new insights on the welfare consequences of price discrimination. In
terms of efficiency, we find that airlines’ current pricing practices increase
total welfare by 35%, compared to a scenario where we prohibit airlines from
charging multiple prices across cabins. We also find that the current pricing
achieves 89% of the first-best welfare, and almost all of this inefficiency is
due to the uncertainty about passengers’ arrivals. However, greater
discrimination based on the reason to travel –business versus leisure–
improves airlines’ surplus but slightly lowers total surplus.

In terms of the surplus distribution between airlines and passengers, we find
that price discrimination skews the distribution of surplus in favor of the
airlines. In particular, the gap between producer surplus and consumer surplus
increases by approximately 37% when airlines price differently across cabins,
compared to setting only one price per period. Additional price discrimination
based on passengers’ reasons to travel or willingness-to-pay leads to a higher
airline surplus but a lower total surplus. We illustrate this “monopoly
externality” by determining surplus division when the airline uses a Vickery-
Clarke-Grove (VCG) auction, period-by-period. We find that using VCG more than
doubles the consumer surplus, compared to the current pricing practices, while
still achieving the same efficiency as eliminating all static informational
frictions.

##### Contribution and Related Literature.

Our paper relates to a vast research on the economics of price discrimination
and research in the empirical industrial organization on estimating the
efficiency and division of welfare under asymmetric information. Most of these
empirical papers focus on either cross-sectional price discrimination (e.g.,
Ivaldi and Martimort, 1994; Leslie, 2004; Busse and Rysman, 2005; Crawford and
Shum, 2006; McManus, 2007; Aryal and Gabrielli, 2019) or inter-temporal price
discrimination dynamics (e.g., Nevo and Wolfram, 2002; Nair, 2007; Escobari,
2012; Jian, 2012; Hendel and Nevo, 2013; Lazarev, 2013; Cho et al., 2018;
Kehoe, Larsen, and Pastorino, 2020).

There is also a literature that focuses on dynamic pricing (e.g., Graddy and
Hall, 2011; Sweeting, 2010; Cho et al., 2018; Williams, 2020). However, none
study intra-temporal price discrimination, inter-temporal price discrimination
and dynamic pricing together, even though many industries involve all three.
We contribute to this research by developing an empirical framework where both
static discriminative pricing and dynamic pricing incentives are present and
obtain results that characterize the welfare implications.333 There is a long
and active theoretical literature on static and inter-temporal price
discrimination, see Stokey (1979); Gale and Holmes (1993); Wilson (1993); Dana
(1999); Courty and Li (2000); Armstrong (2006) and references therein. Airline
pricing has also been studied extensively from the perspective of _revenue
management_ ; see van Ryzin and Talluri (2005).

Additionally, we complement recent research related specifically to airline
pricing, particularly Lazarev (2013), Li, Granados, and Netessine (2014), and
Williams (2020).444 Relatedly, Borenstein and Rose (1994), Puller, Sengupta,
and Wiggins (2012); Chandra and Lederman (2018) use a regression approach to
study effects of oligopoly on price dispersion and price discrimination in
airline markets. Lazarev (2013) considers a model of inter-temporal price
discrimination with one service cabin and finds large potential gains from
allowing reallocation among passengers arriving at different times before the
flight (through ticket resale). Williams (2020) further allows for dynamic
adjustment of prices in response to stochastic demand and finds that dynamic
pricing (relative to a single price) increases total welfare at the expense of
business passengers. Li, Granados, and Netessine (2014) use an instrumental
variables strategy to study strategic behavior of passengers and infer that
between 5% and 20% of passengers wait to purchase in a sample of domestic
markets, with the share decreasing in market distance.555 We expect the share
to be small in our context because we study long-haul international markets.
Incorporating this strategic consumer behavior into a structural model of
pricing like Lazarev (2013), Williams (2020), and ours remains unexplored due
to theoretical and computational difficulties, although Lazarev (2013) allows
passengers to cancel their ticket.

Building on Lazarev (2013) and Williams (2020), we allow the airline to manage
revenue by optimally choosing the number of seats to release and by screening
passengers between two cabins. Through counterfactuals, our approach allows us
to measure the importance of different channels through which inefficiencies
arise in airline markets. In terms of estimation, we also allow a rich
specification for unobserved heterogeneity across markets via a random
coefficients approach to estimation. The richness of random coefficients
allows us capture variation in demand parameters due to permanent and
unobserved differences across markets.

##  2 Data

The Department of Commerce’s Survey of International Air Travelers (SIAT)
gathers information on international air passengers traveling to and from the
U.S. Passengers are asked detailed questions about their flight itinerary,
either during the flight or at the gate area before the flight. The SIAT
targets both U.S. residents traveling abroad and non-residents visiting the
U.S. Passengers in our sample are from randomly chosen flights from among more
than 70 participating U.S. and international airlines, including some charter
carriers. The survey contains ticket information, which includes the cabin
class (first, business, or economy), date of purchase, total fare, and the
trip’s purpose (business or leisure). We combine fares that are reported as
business class and first-class into a single cabin class that we label “first-
class.” This richness distinguishes the SIAT data from other data like the
Origin and Destination Survey (DB1B) conducted by the Department of
Transportation. In particular, the additional detail about passengers (e.g.,
time of purchase, individual ticket fares, and reason for travel) make the
SIAT dataset ideal for studying price discrimination.

We create a dataset from the survey where a unit of observation is a single
ticket purchased by a passenger flying a nonstop (or direct) route. We then
use fares and purchase (calendar) dates associated with these tickets to
estimate “price paths” for each flight in our data, where a flight is a single
instance of a plane serving a particular route. For example, in our sample, we
observe some nonstop passengers flying United Airlines from SEA to TPE on
August 12, 2010, departing at 5:10 pm, then we say that this is one flight.
From the data on fares and dates for this flight, we use kernel regression to
estimate price paths for economy seats and first-class seats leading up to
August 12, 2010. In this section, we detail how we selected the sample and
display descriptive statistics that motivate our model and analysis.

###  2.1 Sample Selection

Our sample from the DOC includes 413,309 passenger responses for 2009-2011. We
clean the data in order to remove contaminated and missing observations and to
construct a sample of flights that will inform our model of airline pricing
which we specify in the following section. We detail our sample selection
procedure in Appendix A.1, but, for example, we exclude responses that do not
report a fare, are part of a group travel package, or are non-revenue tickets.
We supplement our data with schedule data from the Official Aviation Guide of
the Airways (OAG) company, which reports cabin-specific capacities, by flight
number. Using the flight date and flight number in SIAT we can merge the two
data sets. We include flights for which we observe at least ten nonstop
tickets after applying the sample selection criteria.

##### Nonstop Markets and Capacity.

Like other studies that model discriminatory pricing by airlines (e.g.,
Lazarev (2013); Puller, Sengupta, and Wiggins (2012); Williams (2020)), we
focus on nonstop travel in monopoly markets, where we define monopoly market
criteria below. Although a physical flight will have both connecting and
nonstop passengers, we assume that an airline devotes a specific portion of a
plane to nonstop passengers before it starts selling tickets, and the airline
does not change the plane’s apportionment. We make this assumption to keep our
model tractable because modeling airlines’ pricing strategies for both nonstop
and connecting passengers is a high-dimensional optimization problem that has
to balance the cross-elasticities of passengers on all potential itineraries
in the airline’s network that could use one of the flights in our sample as a
leg.

Specifically, we determine the ratio of nonstop travel on a flight in our
survey by first determining the ratio of nonstop travel for the route in a
quarter from the Department of Transportation’s DB1B data. We then apply this
ratio to the total capacity of the equipment used (which we observe and we
describe below) in our sample to arrive at a nonstop capacity for each flight.
As an illustration, consider the SEA to TPE flight on August 12, 2010. Suppose
in this flight, United Airlines used a Boeing 757 with 235 economy seats, and
according to the DB1B data in Q3:2010, only 45 percent of tickets for this
market are nonstop tickets. Then, for us, this means that United Airlines
reserved only 110 economy seats (out of 235) for nonstop travel, and so the
initial economy class capacity for this flight market is 110. We repeat this
exercise for the first-class seats.666 Few flights are not in the DB1B
dataset, and for them, we use the nonstop ratio from the SIAT survey.

We observe the entire itinerary of a passenger, so to select ticket purchases
for our final sample, we discard all passengers using a flight as part of a
connecting itinerary. For example, if we see a passenger report flying from
Cleveland to Taipei via Seattle, we drop them from our sample. The same is
true if we observe a passenger flying from Toulouse to Charlotte via Paris.

##### “Monopoly” Markets.

As mentioned earlier, we focus on monopoly markets. In international air
travel, nonstop markets tend to be concentrated for all but the few busiest
airport-pairs. We classify a market as a monopoly-market if it satisfies one
of following two criteria: (i) one airline flies at least 95% of the total
capacity on the route (where the capacity is measured using the OAG data); or
(ii) there is a US carrier and foreign carrier that operate on the market with
antitrust immunity from the U.S. Department Of Justice.

These antitrust exemptions come from market access treaties signed between the
U.S. and the foreign country that specify a local foreign carrier (usually an
alliance partner of the U.S. airline) that will share the route. For example,
on July 20, 2010 antitrust exemption was granted to OneWorld alliance, which
includes American Airlines, British Airways, Iberia, Finnair and Royal
Jordanian, for 10 years subject to a slot remedy.777 To determine such
markets, we use information from DOT and carriers’ 10K reports filed with the
SEC.

In a few cases, we define markets at the city-pair level because we are
concerned that within-city airports are substitutable. The airports that we
aggregate up to a city-pairs market definition include airports in the New
York, London, and Tokyo metropolitan. Thus, we treat a flight from New York
JFK to London Heathrow to be in the same market as a flight from Newark EWR to
London Gatwick.

Table 1: Top 20 Markets, ordered by sample representation

| Unique |  | Distance |  |  | Unique |  | Distance
---|---|---|---|---|---|---|---|---
Market | Flights | Obs. | (miles) |  | Market | Flights | Obs. | (miles)
LAX-PVG | 53 | 1,060 | 6,485 |  | JFK-SCL | 16 | 239 | 5,096
JFK-HEL | 35 | 581 | 4,117 |  | SFB-BHX | 10 | 234 | 4,249
SFO-AKL | 28 | 497 | 6,516 |  | JFK-NCE | 9 | 217 | 3,991
JFK-VIE | 29 | 469 | 4,239 |  | PHX-SJD | 15 | 211 | 721
JFK-EZE | 24 | 444 | 5,281 |  | JFK-POS | 13 | 208 | 2,204
JFK-WAW | 34 | 434 | 4,267 |  | JFK-PLS | 8 | 205 | 1,303
JFK-JNB | 26 | 368 | 7,967 |  | SFO-HND | 4 | 198 | 5,161
JFK-CCS | 20 | 360 | 2,109 |  | BOS-KEF | 12 | 178 | 2,413
JFK-CMN | 21 | 330 | 3,609 |  | SFO-CDG | 10 | 162 | 5,583
SEA-TPE | 21 | 328 | 6,074 |  | JFK-MTY | 9 | 148 | 1,826

Note: The table displays the Top 25 markets, their representation in our
sample, their distance (air miles) from U.S. Data from the Survey of
International Air Travelers and sample described in the text.

##### Description of Markets and Carriers.

After our initial selection and restriction on nonstop monopoly markets, we
have 14,930 observations representing 224 markets and 64 carriers. We list the
top 20 markets based on sample representation in Table 1, along with the
number of unique flights and the total observations. The most common U.S.
airports in our final sample are San Francisco (SFO), New York JFK (JFK), and
Phoenix (PHX), and the three most common routes are SFO to Auckland, New
Zealand (AKL), JFK to Helsinki, Finland (HEL), and JFK to Johannesburg, South
Africa (JNB). The median flight has a distance of 4,229 miles, and the inter-
quartile range of $2,287$ to $5,583$ miles. In Table 2 we display the top ten
carriers from our final sample, which represent approximately 60% of our final
observations.

Table 2: Top Ten Carriers, ordered by sample representation

Carrier | Unique Flights | Obs.
---|---|---
American Airlines | 88 | 1526
Continental | 98 | 1483
US Airways | 72 | 1200
Eastern China | 53 | 1060
Finnair | 35 | 581
Austrian | 35 | 573
Lufthansa | 40 | 558
Delta | 35 | 550
Air New Zealand | 30 | 520
ThomsonFly | 20 | 470

Note: The table displays the top ten carriers, ordered by their frequency in
the final sample. Data from the Survey of International Air Travelers and
sample described in the text.

###  2.2 Passenger Arrivals and Ticket Sale Process

Passengers differ in terms of time of purchases and reasons for travel and
prices vary over time and across cabins. In this subsection we present key
features in our data pertaining the passengers and prices.

##### Timing of Purchase.

Airlines typically start selling tickets a year before the flight date.
Although passengers can buy their tickets throughout the year, in our sample
most passengers buy in the last 120 days. To keep the model and estimation
tractable, we classify the purchase day into a fixed number of bins. There are
at least two factors that motivate our choice of bin sizes. First, it appears
that airlines typically adjust fares frequently in the last few weeks before
the flight date, but less often farther away from the flight date. Second,
there is usually a spike in passengers buying tickets at focal points, like 30
days, 60 days etc. In Table 3 we present eight fixed bins, and the number of
observations in each bin, and as we can see we use narrower bins when we are
closer to the flight date. Each of these eight bins correspond to one period,
giving us a total of eight periods.

Table 3: Distribution of Advance Purchase

Days Until Flight | Obs.
---|---
$0-3$ days | 597
$4-7$ days | 753
$8-14$ days | 1,038
$15-29$ days | 1,555
$30-44$ days | 2,485
$45-60$ days | 2,561
$61-100$ days | 2,438
$101+$ days | 3,400

Note: The table displays the distribution of advance purchases. First column
is the number of days before flight and the second column shows how many
passengers bought their tickets in those days.

##### Passenger Characteristics.

We classify each passenger as either a business traveler or a leisure traveler
based on the reason to travel. Business includes business, conference, and
government/military, while leisure includes visiting family, vacation,
religious purposes, study/teaching, health, and other, see Table A.1.1 for
more.888 We also observe the channel through which the tickets were purchased:
travel agents (36.47%), personal computer (39.76%), airlines (13.36%), company
travel department (3.99%), and others (6.42%). We classify service cabins into
economy class and first-class, where we classify every premium service cabin
as latter.

In Table 4 we display some key statistics for relevant ticket characteristics
in our sample. As is common in the literature, to make one-way and round-trip
fares comparable we divide round-trip fares by two. Approximately 4.5%
passengers report to have bought a one-way ticket, see Table A.1.1. From the
top panel, in our sample, 92.5% of the passengers purchased economy class
tickets, and the average fare was $447 whereas 7.5% of passengers purchased
first-class tickets and paid an average of $897. The standard deviations in
fares, which quantify both the across-market and within-flight variation in
fares, are large, with the coefficient of variation 0.85 for economy and 1.06
for first-class.

Table 4: Summary Statistics from SIAT, Ticket Characteristics

| Proportion | Fare
---|---|---
Ticket Class | of Sample | Mean | S.D.
Economy | 92.50 | 447 | 382
First | 7.50 | 897 | 956
Advance Purchase |  |  |
0-3 Days | 4.03 | 617 | 636
4-7 Days | 5.08 | 632 | 679
8-14 Days | 7.00 | 571 | 599
15-29 Days | 10.49 | 553 | 567
30-44 Days | 16.76 | 478 | 429
45-60 Days | 17.27 | 467 | 432
61-100 Days | 16.44 | 414 | 315
101+ Days | 22.93 | 419 | 387
Travel Purpose |  |  |
Leisure | 85.57 | 446 | 400
Business | 14.43 | 684 | 716

Note: Data from the Survey of International Air Travelers. Sample described in
the text.

In the second panel of Table 4, we display the same statistics by the number
of days in advance of a flight’s departure that the ticket was purchased
(aggregated to eight “periods”). We see that 4% of the passengers bought their
ticket in last three days before the flight; 5.08% bought 4-7 days; 7% bought
8-14 days; 10.4% bought 15-29 days; 16.7% bought 30-44 days; 17.2% bought
45-60 days, 16.4% bought 61-100 days and the rest 23% bought at least 100 days
before the flight. While the average fare increases for tickets purchased
closer to the departure date, so does the standard deviation.

Similarly, at the bottom panel of Table 4, we report price statistics by the
passenger’s trip purpose. About 14% of the passengers in our sample flew for
business purposes, and these passengers paid an average price of $684 for one
direction of their itinerary. Leisure passengers paid an average of $446. This
price difference arises for at least three reasons: business travelers tend to
buy their tickets much closer to the flight date, they prefer first-class
seats, and they fly different types of markets.

Figure 1: Business versus Leisure Passengers before the Flight Date

(a) Average Economy Fares Prior to Flight

(b) Business and Leisure Travelers

In Figure 1(a) we plot the average price for economy fares as a function when
the ticket was purchased. Both business and leisure travelers pay more if they
buy the ticket closer to the flight date, but the increase is more substantial
for the business travelers. The solid line in Figure 1(a) reflects the average
price across both reasons for travel. At earlier dates, the total average
price is closer to the average price paid by leisure travelers, while it gets
closer to the average price paid by the business travelers as the date of the
flight nears. In Figure 1(b), we display the proportion of business to leisure
travelers across all flights, by the advance purchase categories. In the last
two months before flight, the share of passengers traveling for leisure is
approximately 90%, which decreases to 65% a week before flight. Taken
together, business travelers purchase closer to the flight date than leisure
travelers, and markets with a greater proportion of business travelers have a
steeper price gradient.

Figure 2: Histogram of Percent of Business Passengers by Flight

Note: Histogram of business-travel index (BTI). The business-traveler index is
the flight-specific ratio of self-reported business travelers to leisure
travelers. The mean is 0.154 and the standard deviation is 0.210.

Observing the purpose of travel plays an important role in our empirical
analysis, reflecting substantial differences in the behavior and preferences
of business and leisure passengers. This passenger heterogeneity across
markets drives variation in pricing, and this covariation permits us to
estimate a model with richer consumer heterogeneity than the existing
literature like Berry, Carnall, and Spiller (2006) and Ciliberto and Williams
(2014). Further, a clean taxonomy of passenger types allows a straightforward
exploration of the role of asymmetric information in determining
inefficiencies and the distribution of surplus that arises from discriminatory
pricing of different forms.999 In the raw data, 5.46% of passengers report
that the main reason why they choose their airline is frequent flyer miles. In
our final (estimation) sample, 1.7% passengers get upgraded, i.e., they buy
economy but fly first-class. Our working hypothesis is that those upgrades
materialized only on the day of travel.

To further explore the influence that this source of observable passenger
heterogeneity has on fares, we present statistics on across-market variation
in the dynamics of fares. Specifically, we first calculate the proportion of
business travelers in each market, i.e., across all flights with the same
origin and destination. Like Borenstein (2010), we call this market-specific
ratio the business-traveler index (BTI). In Figure 2, we present the histogram
of the BTI across markets in our data. If airlines know of this across-market
heterogeneity and use it as a basis to discriminate both intra-temporally
(across cabins) and inter-temporally (across time before a flight departs),
different within-flight temporal patterns in fares should arise for different
values of the BTI.

Figure 3: Proportion of Business Travelers by Ticket Class

(a) Economy Class

(b) First-Class

In Figure 3 we present the results of a bivariate kernel regression where we
regress an indicator for whether a passenger is traveling for business on the
BTI in that market and number of days the ticket was purchased in advance of
the flight’s departure. Figures 3(a) and 3(b) present the results for economy
and first-class passengers, respectively. There are two important
observations. First, across all values of the BTI, business passengers arrive
later than leisure passengers. Second, business passengers disproportionately
choose first-class seats. To capture this feature, in Section 3, we model the
difference between business and leisure passengers in terms of the timing of
purchases and the preference for quality by allowing the passenger mix to
change as the flight date approaches, resulting in a non-stationary demand
process.

Figure 4: Across-Market Variation in Fares

(a) Economy

(b) First-Class

The influence of business passengers is evident on prices. Like Figure 3,
Figure 4(a) and Figure 4(b) present the results of a kernel regression with
fare paid as the dependent variable for economy and first-class cabins,
respectively. In both, we present cross-sections of these estimated surfaces
for the $25^{th}$, $50^{th}$, and $75^{th}$ percentile values of the BTI. For
both cabins, greater values of the BTI are associated with substantially
higher fares. Further, there is a positive relationship between the rate of
increase as the flight date approaches and the BTI, and this rate is positive
as the flight date approaches only in markets with non-zero BTI. This pattern
is most evident in first-class fares. Thus, the presence of business travelers
is associated with both greater average fares and steeper increases in fares
as the flight date approaches for both cabins. The larger increase in first-
class fares as the flight date approaches, relative to economy fares, is
consistent with the strong selection of business travelers into the first-
class cabin.

While there are clear patterns in how the dynamics of average fares vary with
the BTI, there is also substantial heterogeneity across flights in how fares
change as the flight date approaches. To see the heterogeneity in temporal
patterns for individual flights that Figure 3 masks, Figure 5 presents the
time-paths of economy fares for all flights in our data. Specifically, for
each flight, we estimate a smooth relationship between economy fares and time
before departure using a kernel regression, and then normalize the path
relative to the initial fare for that flight. Each line is a single flight
from our data, and begins when we first observe a fare for that flight, and
ends at $1$, the day of the flight.

Figure 5: Flight-Level Dispersion in Fares

Note: The figure presents the estimated fare paths from every flight we use in
our estimation. Each line represents a flight. We normalize the fares to unity
by dividing fares by the first fare we observe for the flight. The horizontal
axis represents the number of days until the flight, where zero is the flight
date.

For most flights we observe little movement in fares until approximately 100
days before departure. Yet, for a small proportion of flights, there are
substantial decreases and increases in fares as much as 5 months before
departure. Further, by the date of departure, the interquartile range of the
ratio of current fare to initial fare is 0.75 to 1.85. Thus, 25% of flights
experience a decrease of more than 25%, while 25% of flights experience an
increase of greater than 85%. The variation in the temporal patterns in fares
across flights is attributable to both the across-market heterogeneity in the
mix of passengers, and how airlines respond to demand uncertainty.

###  2.3 Aircraft Characteristics

Airlines’ fares, and their responsiveness to realized demand, depend on the
number of unsold seats. In Figure 6(a) we display the joint density of initial
capacity of first and economy class in our sample. The median capacity of an
aircraft in our sample is 116 economy seats and 15 first-class seats, and the
mode is 138 economy and 16 first-class seats.

Figure 6: Initial Capacity and Load Factor

(a) Density of Initial Capacities

(b) Histogram of Load Factor

Note: In part (a) this figure presents the Parzen-Rosenblatt Kernel density
estimate of the joint-density of initial capacities available for nonstop
travel. In part (b) this figure presents the histogram of the passenger load
factor across our sample.

The three most common aircraft types in our sample are a Boeing 777, 747, and
737 (36% of flights in our sample). The 777 and 747 are wide-body jets. The
777 has a typical seating of around 350 seats and the 737 has a typical
seating of around 160 seats (before adjusting for non-stop versus connecting
passengers). The most common Airbus equipment is the A330, which makes up
about 4% of the flights in our sample.

Across all flights, on average 88% of all seats are economy class. We merge
the SIAT data with the Department of Transportation’s T-100 segment data to
get a measure of the load factor for our SIAT flights. From the T100, we know
the average load factor across a _month_ for a particular route flown by a
particular type of equipment. In Figure 6(b) we display the density of load
factor across flights in our sample. The median load factor is 82%, but there
is substantial heterogeneity across flights.

Overall, our descriptive analysis reveals a number of salient features that we
capture in our model. We find that a business-leisure taxonomy of passenger
types is useful to capture differences in the timing of purchase, willingness-
to-pay, and preference for quality. Further, we find substantial heterogeneity
in the business or leisure mix of passengers across markets, which airlines
are aware of and responsive to, creating variation in both the level and
temporal patterns of fares across markets. Finally, across flights we observe
considerable heterogeneity in fare paths as the flight date approaches.
Together, these features motivate our model of non-stationary and stochastic
demand and dynamic pricing by airlines that we present in Section 3, as well
as the estimation approach in Section 4.

##  3 Model

In this section, we present a model of dynamic pricing by a profit-maximizing
multi-product monopoly airline that sells a fixed number of economy ($0\leq
K^{e}<\infty$) and first-class ($0\leq K^{f}<\infty$) seats. We assume
passengers with heterogeneous and privately known preferences arrive before
the date of departure ($t\in\\{0,\ldots,T\\}$) for a nonstop flight. Every
period the airline has to choose the ticket prices and the maximum number of
unsold seats to sell at those prices before it realizes the demand (for that
period).

Our data indicate essential sources of heterogeneity in preferences that
differ by reason for travel: willingness-to-pay, valuation of quality, and
purchase timing. Further, variability and non-monotonicity in fares suggest a
role for uncertain demand. Our model’s demand-side seeks to flexibly capture
this multi-dimensional heterogeneity and uncertainty that serves as an input
into the airline’s dynamic-pricing problem. Furthermore, our model’s supply-
side seeks to capture the inter-temporal and intra-temporal tradeoffs faced by
an airline in choosing its optimal policy.

###  3.1 Demand

Let $N_{t}$ denote the number of individuals that arrive in period
$t\in\\{1,\ldots,T\\}$ to consider buying a ticket. We model $N_{t}$ as a
Poisson random variable with parameter $\lambda_{t}\in\mathbb{N}$, i.e.,
$\mathbb{E}(N_{t})=\lambda_{t}$. The airline knows $\lambda_{t}$ for
$t\in\\{1,\ldots,T\\}$, but must make pricing and seat-release decisions
before the uncertainty over the number of arrivals is resolved each period.
The arrivals are one of two types, for-business or for-leisure. The
probability that a given individual is for-business varies across time before
departure and denoted by $\theta_{t}\in[0,1]$.

For a given individual, let $v\subset\mathbb{R}_{+}$ denote the value this
person assigns to flying in economy cabin, and let the indirect utility of
this individual from flying economy and first-class at price $p$,
respectively, be

$\displaystyle u^{e}(v,p,\xi)=v-p;\quad
u^{f}(v,p,\xi)=v\times\xi-p,\qquad\xi\in[1,\infty).$

Thus, $\xi$ is the (utility) premium associated with flying in a first-class
seat that captures the vertical quality differences between the two cabins.
Arrivals are heterogeneous in terms of their $v$ and $\xi$ that are mutually
independent and privately known to the individual.

We assume that the distribution of these preferences across arrivals are
realizations from type-specific distributions. Specifically, the $v$ of for-
business and for-leisure arrivals are drawn from $F^{b}_{v}(\cdot)$ and
$F^{l}_{v}(\cdot)$, respectively, and $\xi$ is drawn from $F_{\xi}(\cdot)$.
Together with the arrival process, the type-specific distribution of
valuations creates a stochastic and non-stationary demand process that we
assume is known to the airline.

Figure 7: Realization of Demand

Airline Chooses:
$(p_{t}^{e},\overline{q}_{t}^{e}),(p_{t}^{f},\overline{q}_{t}^{f})$Individuals
Arrive : $N_{t}\sim{\mathcal{P}}(\lambda_{t})$Business Arrivals:
$N_{t}^{b}$Leisure Arrivals: $N_{t}^{l}$ Business: $\theta_{t}$ Leisure:
$1-\theta_{t}$ $(v_{i},\xi_{i})\sim F^{b}_{v}\times
F_{\xi},i=1,\ldots,N_{t}^{b}$$(v_{i},\xi_{i})\sim F^{l}_{v}\times
F_{\xi},i=1,\ldots,N_{t}^{l}$Buy: $q_{t}^{e},q_{t}^{f}$; Not Buy:
$q_{t}^{o}$$\begin{array}[]{c}\overline{q}_{t}^{e}\geq q_{t}^{e}\\\
\overline{q}_{t}^{f}\geq
q_{t}^{f}\end{array}$$\begin{array}[]{c}\overline{q}_{t}^{e}<q_{t}^{e}\\\
\overline{q}_{t}^{f}\geq
q_{t}^{f}\end{array}$$\begin{array}[]{c}\overline{q}_{t}^{e}\geq q_{t}^{e}\\\
\overline{q}_{t}^{f}<q_{t}^{f}\end{array}$$\begin{array}[]{c}\overline{q}_{t}^{e}<q_{t}^{e}\\\
\overline{q}_{t}^{f}<q_{t}^{f}\end{array}$ (Case A) Not binding (Case B)
Economy-class binding (Case C) First-class binding (Case D) Both-classes
binding

Note: A schematic representation of the timing of demand.

At given prices and a given number of seats available at those prices, Figure
7 summarizes a realization of the demand process for period $t$. Specifically,
the realization of demand and timing of information known by the airline
leading up to a flight’s departure is as follows:

  1. (i)

Airline chooses a price and seat-release policy for economy cabin,
$(p_{t}^{e},\overline{q}_{t}^{e})$, and the first-class cabin,
$(p_{t}^{f},\overline{q}_{t}^{f})$, that determine the prices at which a
maximum number of seats in the two cabins may be sold.

  2. (ii)

$N_{t}$ many individuals arrive, the number being drawn from a Poisson
distribution with parameter $\lambda_{t}$. Each arrival realizes their reason
to fly from a Bernoulli distribution with parameter $\theta_{t}$ (i.e., for-
business equals one).

  3. (iii)

Each arrival observes their own $(v,\xi)$, drawn from the respective
distributions, $F^{b}_{v}(\cdot)$, $F^{l}_{v}(\cdot)$, and $F_{\xi}(\cdot)$.

  4. (iv)

If neither seat-release policy is binding (realized demand does not exceed the
number of seats released in either cabin), arrivals select their most
preferred cabin: first-class if $v\times\xi-
p_{t}^{f}\geq\max\\{0,v-p_{t}^{e}\\}$, economy if
$v-p_{t}^{e}\geq\max\\{0,v\times\xi-p_{t}^{f}\\}$, and no purchase if
$0\geq\max\\{v\times\xi-p_{t}^{f},v-p_{t}^{e}\\}$. Those arrivals choosing the
no-purchase option leave the market. If the seat-release policy is binding in
either one or both cabins, we assume that arrivals make sequential decisions
in a randomized order until either none remaining wishes to travel in the
cabin with capacity remaining or all available seats are allocated.

  5. (v)

Steps (i)-(iv) repeat until the date of departure, $t=T$, or all of the seats
are allocated.

In any given period ($t$), there are four possible outcomes given a demand
realization: neither seat-release policy is binding, either one of the two
seat-release policies is binding, or both are binding. If the seat-release
policy is not binding for either of the two cabins, then the expected demand
for the respective cabins in period $t$ when the airline chooses policy
$\chi_{t}:=(p_{t}^{e},\overline{q}_{t}^{e},p_{t}^{f},\overline{q}_{t}^{f})$ is

$\displaystyle\mathbb{E}_{t}(q^{e};\chi_{t}):=\sum_{n=0}^{\infty}\left\\{n\times\Pr(N_{t}=n)\underbrace{\Pr(v-p_{t}^{e}\geq\max\\{0,v\times\xi-
p_{t}^{f}\\})}_{:=P_{t}^{e}(\chi_{t})}\right\\}=\lambda_{t}\times
P_{t}^{e}(\chi_{t});$
$\displaystyle\mathbb{E}_{t}(q^{f};\chi_{t}):=\sum_{n=0}^{\infty}\left\\{n\times\Pr(N_{t}=n)\underbrace{\Pr(v\times\xi-
p_{t}^{f}\geq\max\\{0,v-p_{t}^{e}\\})}_{:=P_{t}^{f}(\chi_{t})}\right\\}=\lambda_{t}\times
P_{t}^{f}(\chi_{t}).$

If one or both of the seat-release policies are binding, the rationing process
creates the possibility for inefficiencies to arise both in terms of exclusion
of passengers with a greater willingness-to-pay than those that are allocated
a seat, as well as misallocations of passengers across cabins.

In Figure 8, we present a simple example to illustrate inefficiency arising
from asymmetric information in this environment under random allocation.
Assume the airline has one first-class and two economy seats remaining and
chooses to release one seat in each cabin at $p^{f}=2000$ and $p^{e}=500$.
Suppose three passengers arrive with values $v_{1}=2500$, $v_{2}=1600$, and
$v_{3}=5000$, with $\xi_{1}=\xi_{2}=2$ and $\xi_{3}=1$. Arrivals 1 and 2 are
willing to pay twice as much for a first-class seat as an economy seat,
whereas arrival 3 values the two cabins equally. Suppose that under the random
allocation rule, arrival 2 gets to choose first and arrival 3 is the last. As
shown in Figure 8, the final allocation is inefficient because: a) arrival 2
gets first-class even though 1 values it more; and b) arrival 1 gets economy
even though 3 values it more. This difference in arrival timing creates the
possibility for multiple welfare-enhancing trades. Given the limited
opportunity for coordination amongst arrivals to make such trades, and the
legal/administrative barriers to doing so, we believe random rationing is a
reasonable way to allocate seats within a period.

Figure 8: Illustration of Random Rationing Rule.

Capacity: $\quad
K^{f}=1,K^{e}=2$$p^{f}=2000;\overline{q}^{f}=1$$p^{e}=500,\overline{q}^{e}=1$$\begin{array}[]{ccc}v_{1}=1800&v_{2}=1600,&v_{3}=1900\\\
\xi_{1}=2,&\xi_{2}=2,&\xi_{3}=1\end{array}$$\begin{array}[]{cc}\texttt{passenger-
id}&\texttt{preference}\\\ 1&f\succ e\succ o\\\ 2&f\succ e\succ o\\\ 3&e\succ
o\succ f\end{array}$$\begin{array}[]{cc}\texttt{passenger-
order}&\texttt{allocation}\\\ 2&f\\\ 1&e\\\
3&o\end{array}$pricesseatsdemandrealizationpreferenceorderingrandomallocation

Note: Example to demonstrate how random-rationing rule can generate
inefficiency in the model.

###  3.2 Supply

The airline has $T$ periods, before the departure, to sell $K^{e}$ and $K^{f}$
economy and first-class seats, respectively. Each period, the airline chooses
prices $\\{p_{t}^{e},p_{t}^{f}\\}$ and commits to selling no more than
$\\{\overline{q}_{t}^{e},\overline{q}_{t}^{f}\\}\leq\omega_{t}$ seats at those
prices, where $\omega_{t}:=(K_{t}^{e},K_{t}^{f})$ is the number of unsold
seats in each cabin. We model that airlines must commit to a seat release
policy to mimic the “fare bucket” strategy that airlines use in practice
(e.g., Alderighi, Nicolini, and Piga, 2015), which helps insure the airline
against a “good” demand shock where too many seats are sold today at the
expense of future higher willingness to pay passengers.101010 Our modeling
choices, in particular seat release policies and random assignments, imply
that there may be instances when a passenger with a high willingness-to-pay
shows up to the market and cannot be served. If, instead, we had used an
optimal rationing, where seats are assigned in the order willingness-to-pay,
it would lead to higher baseline efficiency. There still may be other modeling
choices, but overall, we believe that our model is a reasonable given our
sample and our final goal. One of this market’s defining characteristics is
that the airline must commit to policies this period before realizing the
current and future demand. The airline does not observe a passenger’s reason
to fly or valuations $(v,\xi)$; however, the airline knows the underlying
stochastic process that governs demand and uses the information to price
discriminate, both within a period and across periods.111111 See Barnhart,
Belobaba, and Odoni (2003) for an overview of forecasting airline demand.

Let $c^{e}$ and $c^{f}$ denote the constant cost of servicing a passenger in
the respective cabins. These marginal costs, or so-called “peanut costs,”
capture variable costs like food and beverage service that do not vary with
the timing of the purchase but may vary with the different levels of service
in the two cabins. Let
$\Psi:=(\\{F^{b}_{v},F^{l}_{v},F_{\xi},c^{f},c^{e}\\},\\{\lambda_{t},\theta_{t}\\}_{t=1}^{T})$
denote the vector of demand and cost primitives.

The airline maximizes the sum of discounted expected profits by choosing price
and seat-release policies for each cabin,
$\chi_{t}=\left(p_{t}^{e},p_{t}^{f},\overline{q}_{t}^{e},\overline{q}_{t}^{f}\right)$,
in each period $t=1,\ldots,T$ given $\omega_{t}$. The optimal policy is a
vector $\\{\chi_{t}:t=1,\ldots,T\\}$ that maximizes expected profit

$\displaystyle\sum_{t=1}^{T}\mathbb{E}_{t}\left\\{\pi(\chi_{t},\omega_{t};\Psi_{t})\right\\},$

where
$\pi(\chi_{t},\omega_{t};\Psi_{t})=(p_{t}^{f}-c^{f})q_{t}^{f}+(p_{t}^{e}-c^{e})q_{t}^{e}$
is the per-period profit after the demand for each cabin is realized
($q_{t}^{e}$ and $q_{t}^{f}$) and
$\Psi_{t}=(\\{F^{b}_{v},F^{l}_{v},F_{\xi},c^{f},c^{e}\\},\\{\lambda_{t},\theta_{t}\\})$.
The airline observes the unsold capacity ($\omega_{t}$) at the time of
choosing its policy, but not the particular realization of passenger
valuations that determine the realized demand. The optimal seat-release policy
must satisfy $\overline{q}_{t}^{e}\leq K_{t}^{e}$ and
$\overline{q}_{t}^{f}\leq K_{t}^{f}$ and take on integer values.

The stochastic process for demand, capacity-rationing algorithm, and optimally
chosen seat-release and pricing policies induce a non-stationary transition
process between states, $Q_{t}(\omega_{t+1}|\chi_{t},\omega_{t},\Psi_{t})$.
The optimal policy in periods $t\in\\{1,\ldots,T-1\\}$ is characterized by the
solution to the Bellman equation,

$\displaystyle
V_{t}(\omega_{t},\Psi)=\max_{\chi_{t}}\mathbb{E}_{t}\left\\{\pi(\chi_{t},\omega_{t};\Psi_{t})+\sum_{\omega\in\Omega_{t+1}}V_{t+1}(\omega_{t+1},\Psi)\times
Q_{t}(\omega_{t+1}|\chi_{t},\omega_{t},\Psi_{t})\right\\},$ (1)

where $\Omega_{t+1}$ represents the set of reachable states in period $t+1$
given $\omega_{t}$ and $\chi_{t}$. The expectation, $\mathbb{E}_{t}$, is over
realizations from the demand process ($\Psi_{t}$) from period $t$ to the date
of departure $T$. In period $T$, optimal prices maximize

$\displaystyle
V_{T}(\omega_{T},\Psi_{T})=\max_{\chi_{T}}\mathbb{E}_{T}\pi(\chi_{T},\omega_{T};\Psi_{T}),$

because the firm no longer faces any inter-temporal tradeoffs.121212 For model
tractability, we assume that passengers cannot strategically time their
purchases. So, their arrival times and their purchase times are the same and
do not depend on the price path. This assumption is also used by Williams
(2020) to model dynamic pricing. Board and Skrzypacz (2016) allow consumers to
be strategic under an additional assumption that the seller has full
commitment and chooses its supply function only once, in the first period. In
the airline industry, however, the assumption that airlines choose their fares
only once at the beginning is too strong. The dynamic programming that
characterizes an airline’s problem is useful for identifying the airline’s
tradeoffs and identifying useful sources of variation in our data.131313
Although we focus only on one flight, an airline may also consider future
flights. In the latter, fares across different flights can be interlinked. We
conjecture that we can then approximate an airline’s pricing problem with a
non-zero continuation value in the last period. However, to estimate such a
model, the SIAT survey data is insufficient because we would need to sample
every flight sufficiently many times.

The optimal pricing strategy includes both inter-temporal and intra-temporal
price discrimination. First, given the limited capacity, the airline must
weigh allocating a seat to a passenger today versus a passenger tomorrow, who
may have a higher mean willingness-to-pay because the fraction of for-business
passengers increases as it gets closer to the flight date. This decision is
difficult because both the volume ($\lambda_{t}$) and composition
($\theta_{t}$) of demand changes as the date of departure nears. Thus, the
good’s perishable nature does not necessarily generate declining price paths
like Sweeting (2010). Simultaneously, every period, the airline must allocate
passengers across the two cabins by choosing $\chi_{t}$ such that the price
and supply restriction-induced selection into cabins is optimal.

To illustrate the problem further, consider the trade-off faced by an airline
from increasing the price for economy seats today: (i) decreases the expected
number of economy seat purchases but increases the revenue associated with
each purchase; (ii) increases the expected number of first-class seat
purchases but no change to revenue associated with each purchase; (iii)
increases the expected number of economy seats and decreases the expected
number of first-class seats available to sell in future periods. Effects (i)
and (ii) capture the multi-product tradeoff faced by the firm, while (iii)
captures the inter-temporal tradeoff. More generally, differentiating Equation
1 with respect to the two prices gives two first-order conditions that
characterize optimal prices given a particular seat-release policy:

$\displaystyle\left(\begin{array}[]{c}\mathbb{E}_{t}(q^{e};\chi_{t})\\\
\mathbb{E}_{t}(q^{f};\chi_{t})\end{array}\right)+\left[\begin{array}[]{cc}\frac{\partial\mathbb{E}_{t}(q^{e};\chi_{t})}{\partial
p_{t}^{e}}&-\frac{\partial\mathbb{E}_{t}(q^{f};\chi_{t})}{\partial
p_{t}^{e}}\\\ -\frac{\partial\mathbb{E}_{t}(q^{e};\chi_{t})}{\partial
p_{t}^{f}}&\frac{\partial\mathbb{E}_{t}(q^{f};\chi_{t})}{\partial
p_{t}^{f}}\end{array}\right]\left(\begin{array}[]{c}p_{t}^{e}-c^{e}\\\
p_{t}^{f}-c^{f}\end{array}\right)=\left(\begin{array}[]{c}\frac{\partial{\mathbb{E}}_{t}V_{t+1}}{\partial
p_{t}^{e}}\\\ \frac{\partial{\mathbb{E}}_{t}V_{t+1}}{\partial
p_{t}^{f}}\end{array}\right).$ (10)

The left side is the contemporaneous marginal benefit net of static costs,
while the right side is the discounted future benefit.

Equation 10 makes clear the two components of marginal cost: (i) the constant
variable cost, or “peanut” cost, associated with servicing seats occupied by
passengers; (ii) the opportunity cost of selling additional seats in the
current period rather than in future periods. We refer to (iii), the vector on
the ride side of the Equation 10, as the shadow cost of a seat in the
respective cabins. These shadow costs depend on the firm’s expectation
regarding future demand (i.e., variation in volume of passengers and
business/leisure mix as flight date nears), and the number of seats remaining
in each cabin (i.e., $K_{t}^{f}$ and $K_{t}^{e}$). The stochastic nature of
demand drives variation in the shadow costs, which can lead to equilibrium
price paths that are non-monotonic in time. This flexibility is crucial given
the variation observed in our data (see Figure 5).141414 The model implies a
mapping between prices and the unobserved state (i.e., the number of remaining
seats in each cabin). This rules out the inclusion of serially correlated
unobservables (to the researcher) that shifts demand or costs that could
otherwise explain variation in prices.

The airline can use its seat-release policy to dampen both intra-temporal and
inter-temporal tradeoffs associated with altering prices. For example, the
airline can force everyone to buy economy by not releasing first-class seats
in a period and then appropriately adjust prices to capture rents from
consumers. Consider the problem of choosing the number of seats to release at
each period
$\overline{q}_{t};=(\overline{q}_{t}^{e},\overline{q}_{t}^{f})\leq\omega_{t}$.
For a choice of $\overline{q}_{t}$ in period $t$, let
$p_{t}(\overline{q}_{t}):=\\{p_{t}^{e}(\overline{q}_{t}),p_{t}^{f}(\overline{q}_{t})\\}$
denote the optimal pricing functions as a function of the number of seats
released. Then, the value function can be expressed recursively as

$V_{t}(\omega_{t},\Psi)=\max_{\overline{q}_{t}\leq\omega_{t}}\left\\{\pi_{t}((p_{t}(\overline{q}_{t}),\overline{q}_{t}),\omega_{t};\Psi_{t})+\sum_{\omega_{t+1}\in\Omega}V_{t+1}(\omega_{t+1},\Psi)\times
Q_{t}(\omega_{t+1}|(p_{t}(\overline{q}_{t}),\overline{q}_{t}),\omega_{t},\Psi_{t})\right\\}.$

The profit function is bounded, so this recursive formula is well defined, and
under some regularity conditions, we can show that it is has a unique optimal
policy. We present these regularity conditions and the proof of uniqueness in
Appendix A.4.

##  4 Estimation and Identification

In this section, we discuss the parametrization of the model, (method of
moments) estimation methodology, and the sources of identifying variation. The
model’s parametrization balances the dimensionality of the parameters and the
desired richness of the demand structure, and the estimation algorithm seeks
to limit the number of times we have to solve our model due to its
computational burden. At the same time, we seek to avoid strong assumptions on
the relationship between model primitives and both observable (e.g., business
travelers) and unobservable market-specific factors. Our identification
discussion provides details of the moments we use in the estimation and how
they identify each parameter.

###  4.1 Model Parametrization and Solution

Recall our model primitives,
${\Psi}=(\\{F_{b},F_{l},F_{\xi},c^{f},c^{e}\\},\\{\lambda_{t},\theta_{t}\\}_{t=1}^{T})$,
include distributions of valuations for business and leisure passengers,
$(F_{b},F_{l})$, distribution of valuations for 1st-class premium, $F_{\xi}$,
marginal costs for economy and 1st-class, $(c^{f},c^{e})$, and the time-
varying Poisson arrival rate of passengers, $\lambda_{t}$, and the fraction of
business passengers, $\theta_{t}$.

Motivated by our data, we choose $T=8$ to capture temporal trends in fares and
passenger’s reason for travel, where each period is defined as in Table 4.
There are two demand primitives, $\lambda_{t}$ and $\theta_{t}$, that vary as
the flight date approaches. To permit flexibility in the relationship between
time before departure and these parameters, we use a linear parameterization,

$\theta_{t}:=\min\left\\{\Delta^{\theta}\times(t-1),1\right\\};\qquad\lambda_{t}:=\lambda+\Delta^{\lambda}\times(t-1)$

where $\Delta^{\theta},\lambda$, and $\Delta^{\lambda}$ are scalar constants.
This parametrization of the arrival process permits the volume ($\lambda$ and
$\Delta^{\lambda}$) and composition ($\Delta^{\theta}$) of demand to change as
the flight date approaches, while also limiting the number of parameters to
estimate.

There are three distributions $(F_{b},F_{l},F_{\xi})$ that determine passenger
preferences. We assume that business and leisure passenger valuations are
truncated Normal random variables, $F_{b}$ and $F_{l}$, respectively, left-
truncated at zero. Given the disparity in average fares paid by business and
leisure passengers, we assume $\mu^{b}\geq\mu^{l}$, which we model by letting
$\mu^{b}=\mu^{l}\times(1+\delta^{b})$ with $\delta^{b}\geq 0$. The two cabins
are vertically differentiated, and passengers weakly prefer first-class to the
economy. To capture this product-differentiation, we assume that the quality
premium, $\xi$, equals one plus an Exponential random variable with mean
$\mu^{\xi}$

Finally, we fix the marginal cost of supplying a first-class and economy seat,
$c^{f}$ and $c^{e}$, respectively, to equal industry estimates of marginal
costs for servicing passengers. Specifically, we set $c^{f}=40$ and $c^{e}=14$
based on information from the International Civil Aviation Organization,
Association of Asia Pacific Airlines, and Doganis (2002).151515 Doganis (2002)
finds “peanut costs” in first-class are 2.9 times that of economy, and the
average overall cost is $17.8 (after adjusting for inflation). Given the
relative sizes of the economy and first-class cabins for aircraft in our data,
this implies costs for servicing one first-class and economy of $40 and $14,
respectively. Our estimates and counterfactuals are robust to other values for
these costs because the price variation is primarily due to inter-temporal and
intra-temporal changes in the endogenous _shadow costs_ of seats. In
international travel, where the average fare is substantially greater than in
domestic travel, these _shadow costs_ are more important than passenger-
related services’ direct costs.

Given this parametrization of the model, the demand process can be described
by a vector of parameters,
${\Psi}=\left(\mu^{l},\sigma^{l},\delta^{b},\sigma^{b},\mu^{\xi},\lambda,\Delta^{\lambda},\Delta^{\theta}\right)\in[\underline{\Psi},\overline{\Psi}]\subset\mathbb{R}^{8}$.
The model is a finite period non-stationary dynamic program. We solve the
model for state-dependent pricing and seat-release policies by working
backward; computing expected values for every state in the state space, where
the state is the number of seats remaining in each cabin. At each state, the
optimal policy is the solution to a mixed-integer non-linear program (MINLP)
because seats are discrete and prices are continuous controls.161616 We
perform calculations in MATLAB R2020a, using MIDACO’s MINLP solver (Schlüter,
Gerdts, and Rückmann, 2012). For speed and accuracy, we use warm start points
by (a) solving the no-price-discrimination problem first and (b) starting the
MINLP problem for each state at the solution to an adjacent state.

###  4.2 Estimation

Although the parameterization above in Section 4.1 is for a single flight, or
a specific flight at a specific time between two airports, our data represent
many diverse fights, and there may be many observed and unobserved factors
that impact model primitives in an unknown way. For example the distance or
commerce between cities may affect willingness-to-pay for a first-class seat.
Instead of further parameterizing the model as a function of observables, we
propose a flexible approach to estimate the distribution of flight-level
heterogeneity. The approach has the added benefit of limiting the number of
times the model is solved.

To illustrate our approach, consider the following example. We have many
instances of the SEA-TPE route in our data, and we treat the prices and
quantities sold on each instance of this route as a separate flight. Demand in
such routes may vary across seasons; for example, there may be a higher
willingness to pay for flights in the summer during the tourist season than in
the winter. One approach would be to incorporate the observable
characteristics of different flights (e.g., season, sporting events, college
attendance) and allow them to affect the willingness-to-pay through some
functional form.

Instead of relying on any such functional form assumption, we take a different
approach and instead estimate a random coefficients model to estimate a
distribution of demand primitives across flights. So, two different instances
of the SEA-TPE route (two different flights) are allowed to differ in their
demand primitives without us imposing any restriction, and the differences due
to seasonality in demand will be captured by (parameters of) the distribution.
We take this approach because (1) including enough observables to capture
differences across flights would result in too many parameters to feasibly
estimate the model, and (2) for our counterfactuals, our primary goal is to
learn the distribution of demand, and not so much about the relationship
between prices and flight-market observable characteristics.

Our approach combines the methodologies of Ackerberg (2009), Fox, Kim, and
Yang (2016), Nevo, Turner, and Williams (2016), and Blundell, Gowrisankaran,
and Langer (2020). We posit that empirical moments are a mixture of
theoretical moments, with a mixing distribution known up to a finite-
dimensional vector of parameters. To limit the computational burden of
estimating these parameters that describe the mixing distribution, we rely on
the importance sampling procedure of Ackerberg (2009). Our estimation proceeds
in three steps. First, we calculate moments from the data to summarize the
heterogeneity in equilibrium outcomes within and across flights. Second, we
solve the model once, at $S$ different parameter values that cover the
parameter space $[\underline{\Psi},\overline{\Psi}]$. Third, we optimize an
objective function that matches the empirical moments to the analogous moments
for a mixture of candidate data-generating processes. The mixing density that
describes across-market heterogeneity in our data is the object of inference.

Specifically, for a given level of observed initial capacity,
$\omega_{1}:=(K^{f}_{1},K^{e}_{1})$, our model produces a data-generating
process characterized by parameters that describe demand and costs,
$\Psi=\left(\mu^{l},\sigma^{l},\delta^{b},\sigma^{b},\mu^{\xi},\lambda,\Delta^{\lambda},\Delta^{\theta}\right)$.
This data-generating process can be described by a set of $N_{\rho}$–many
moment conditions that we denote by $\boldsymbol{\rho}(\omega_{1};\Psi)$. We
assume that the analogous empirical moment conditions, $\rho(\omega_{1})$, can
be written as a mixture of candidate moment conditions, i.e.,

$\displaystyle\rho(\omega_{1})=\int_{\underline{\Psi}}^{\overline{\Psi}}\boldsymbol{\rho}(\omega_{1};{\Psi})h(\Psi|\omega_{1})d\Psi,$
(11)

where $h(\Psi|\omega_{1})$ is the conditional (on initial capacity
$\omega_{1}$) density of the parameters $\Psi$.171717 Thus, we estimate a
separate density for each initial capacity $\omega_{1}$. This captures the
possibility that the observed initial capacities may be correlated with the
unobserved demand.

The goal is to estimate the mixing density, $h(\Psi|\omega_{1})$, that best
matches the empirical moments (left side of Equation 11) to the expectation of
the theoretical moments (right side of Equation 11). To identify the mixing
density, we assume a particular parametric form for $h(\Psi|\omega_{1})$ that
reduces the matching of empirical and theoretical moments to a finite-
dimensional nonlinear search. Specifically, we let the distribution of $\Psi$
conditional on $\omega_{1}$ be a truncated multivariate normal distribution,
i.e.,

$\Psi|\omega_{1}\sim h(\Psi|\omega_{1};{\mu_{\Psi}},{\Sigma_{\Psi}}),$

where $\mu_{\Psi}$ and $\Sigma_{\Psi}$ are the vector of means and covariance
matrix, respectively, of the non-truncated distribution. We choose our
estimates based on a least-squares criterion

$\displaystyle\left(\hat{\mu}_{\Psi}(\omega_{1}),\hat{\Sigma}_{\Psi}(\omega_{1})\right)=\arg\min_{\left(\mu_{\Psi},\Sigma_{\Psi}\right)}\left(\hat{\rho}(\omega_{1})-\widetilde{\mathbb{E}}({\boldsymbol{\rho}}(\omega_{1};\mu_{\Psi},\Sigma_{\Psi}))\right)^{\top}\left(\hat{\rho}(\omega_{1})-\widetilde{\mathbb{E}}({\boldsymbol{\rho}}(\omega_{1};\mu_{\Psi},\Sigma_{\Psi}))\right)$
(12)

where $\hat{\rho}(\omega_{1})$ is an estimate of the ($M\times 1$) vector of
empirical moments and
$\widetilde{\mathbb{E}}({\boldsymbol{\rho}}(\omega_{1};\mu_{\Psi},\Sigma_{\Psi}))$
is a Monte Carlo simulation estimate of
$\int_{\underline{\Psi}}^{\overline{\Psi}}\boldsymbol{\rho}(\omega_{1};{\Psi})h(\Psi|\omega_{1};\mu_{\Psi},\Sigma_{\Psi})d\Psi$
equal to $\frac{1}{S}\sum_{j=1}^{S}\boldsymbol{\rho}(\omega_{1};{\Psi}_{j})$
with the $S$ draws of $\Psi$ taken from
$h(\Psi|\omega_{1};\mu_{\Psi},\Sigma_{\Psi})$.181818 For the consistency of
our estimator we assume that, for each initial capacity $\omega_{1}$, the
number of flights and the number of passengers in those flights are
sufficiently large so that $\hat{\rho}(\omega_{1})$ is a consistent estimator
of the true moment ${\rho}(\omega_{1})$, and our importance sampling procedure
to determine
$\widetilde{\mathbb{E}}({\boldsymbol{\rho}}(\omega_{1};\mu_{\Psi},\Sigma_{\Psi}))$
is also consistent. For a formal analysis of the subject see Gourieroux,
Monfort, and Renault (1993) Proposition 1.

The dimensionality of the integral we approximate through simulation requires
a large number of draws. After some experimentation to ensure simulation error
is limited for a wide range of parameter values, we let $S=10,000$. Thus, the
most straightforward approach to optimization of Equation 12 would require
solving the model $S=10,000$ times for each value of
$(\mu_{\Psi},\Sigma_{\Psi})$ until a minimum is found. Our model is complex,
and the dimensionality of the parameter space to search over makes such an
option prohibitive. For this reason, we appeal to the importance sampling
methodology of Ackerberg (2009).

The integral in Equation 11 can be rewritten as

$\displaystyle\int_{\underline{\Psi}}^{\overline{\Psi}}\boldsymbol{\rho}(\omega_{1};{\Psi})\frac{h(\Psi|\omega_{1};\mu_{\Psi},\Sigma_{\Psi})}{g(\Psi)}g(\Psi)d\Psi,$

where $g(\Psi)$ is a known well-defined probability density with strictly
positive support for $\Psi\in\left[\underline{\Psi},\overline{\Psi}\right]$
and zero elsewhere like $h(\Psi|\omega_{1};\mu_{\Psi},\Sigma_{\Psi})$.
Recognizing this, one can use importance sampling to approximate this integral
with

$\displaystyle\frac{1}{S}\sum_{j=1}^{S}\boldsymbol{\rho}(\omega_{1};{\Psi}_{j})\frac{h(\Psi_{j}|\omega_{1};\mu_{\Psi},\Sigma_{\Psi})}{g(\Psi_{j})}$

where the $S$ draws of $\Psi$ are taken from $g(\Psi)$. Thus, the importance
sampling serves to correct the sampling frequencies so that it is as though
the sampling was done from $h(\Psi|\omega_{1};\mu,\Sigma)$.

The crucial insight of Ackerberg (2009) is that this importance-sampling
procedure serves to separate the problem of solving the model from the
optimization of the econometric objective function. That is, we solve the
model for a fixed number of $S$ draws of $\Psi$ from $g(\Psi)$, and then
$\boldsymbol{\rho}(\omega_{1};{\Psi}_{j})$ is calculated once for each draw.
After these calculations, optimization of the objective function to determine
$(\hat{\mu}_{\Psi}(\omega_{1}),\hat{\Sigma}_{\Psi}(\omega_{1}))$ simply
requires repeatedly calculating the ratio of two densities,
$\frac{h(\Psi_{j}|\omega_{1};\mu_{\Psi},\Sigma_{\Psi})}{g(\Psi)}$. To simplify
the importance sampling process, we fix the support of $g(\cdot)$ and
$h(\cdot)$ to be the same, and let $g(\cdot)$ be a multivariate uniform
distribution with the support [$\underline{\Psi}$,$\overline{\Psi}$] chosen
after substantial experimentation to ensure it encompasses those patterns
observed in our data.

To solve Equation 12, we use a combination of global search algorithms and
multiple starting values. We repeat this optimization for each $\omega_{1}$
which provides an estimate of the parameters of the distribution of market
heterogeneity,
$(\hat{\mu}_{\Psi}(\omega_{1}),\hat{\Sigma}_{\Psi}(\omega_{1}))$. To calculate
the distribution of demand parameters across all flights, we then
appropriately weight each estimate by the probability mass associated with
that value of $\omega_{1}$ (Figure 6). We calculate standard errors for the
estimates and the counterfactuals by re-sampling the individual passenger
observations in the SIAT data. This procedure accounts for error in survey
responses as well as variation in our moments across flights. However, this
procedure does not account for numerical error coming from the importance-
sampling draws, but we argue that $S=10,000$ is large enough for that to
matter.

###  4.3 Identification

In this section, we introduce the moments we use in Equation 11 to estimate
the market heterogeneity, $\Psi\sim
h(\cdot|\omega_{1};\mu_{\Psi},\Sigma_{\Psi})$, and present the identification
argument that guides our choice. To that end, we present arguments that our
moments vary uniquely with each element of $\Psi$, under the assumption that
our data is generated from our model described in Section 3. In showing
identification, we use several modeling and parametric assumptions, some of
which are necessary, and some are to ease the computational burden and could
be relaxed in principle.

Key to our identification is the shadow cost associated with each seat in the
current period, which equals the expected revenue loss from selling the seat
today instead of a future period. These shadow-costs depend on the demand and
airline’s capacity and can vary substantially across time for a flight due to
the stochastic nature of demand.191919 Heuristically, our identification
strategy is similar to that of Nevo, Turner, and Williams (2016), who study
households that optimize their usage of telecommunications services when
facing nonlinear pricing (fixed fee, allowance, and overage price) and
uncertainty about their future usage. This uncertainty introduces a shadow-
price for current usage that is a function of the overage price and
probability of exceeding the usage allowance by the end of a billing cycle. If
uncertainty is substantial and varies from month to month, it creates
variation in shadow costs that provide useful variation to identify the
household’s preferences. Our model maps these shadow-costs to observables like
prices of economy and first-class seats, price-paths, the timing of the
purchase, passenger volumes, and business passengers’ share. We use this
mapping to construct flight-specific moments for each of these outcomes, which
we then pool across flights with similar levels of capacity to construct
aggregate moments.202020 To construct aggregate moments from flight-specific
moments, we use Kernel density weights. These weights are higher
(respectively, lower) for flights with similar (respectively, dissimilar)
capacities. This results in a set of empirical moments for each capacity,
$\hat{\rho}(\omega_{1})$, that we seek to match.

For a given initial capacity $\omega_{1}$ and each period prior to the
departure, we use the following moments conditions: (i) the fares for economy
and first-class tickets, for various levels of BTI, which is shown in Figure
4; and (ii) the distribution of the maximum and minimum differences in first-
class and economy fares over time, i.e.,
$\max_{t=1,\ldots,T}\\{p^{f}_{t}-p^{e}_{t}\\}$ and
$\min_{t=1,\ldots,T}\\{p^{f}_{t}-p^{e}_{t}\\}$, respectively; (iii) the
proportion of business traveler in each period and the economy/first-class
fares, as shown in Figure 3; (iv) the joint distribution of flight-BTI and
proportion of total arrivals for different periods; (v) the quantiles of
passenger load factor which is shown in Figure 6(b); (vi) number of tickets,
for each class, sold at various levels of BTI, which is similar to Figure 3
with the number of seats on the z-axis; and (vii) overall proportion of
business travelers, see Figure 2.

Next, we explain why we chose these moments and determine conditions under
which a unique set of model parameters rationalizes the data. In particular,
we explain how the moments (i) and (ii) identify the willingness-to-pay
parameters (i.e., $\mu^{l}$, $\sigma^{l}$, $\delta^{b}$, $\sigma^{b}$,
$\mu^{\xi}$) and how the remaining moments from (iii)-(vii) identify the
arrival process and passenger mix parameters (i.e.,
$\lambda,\Delta^{\lambda},\Delta^{\theta}$). For notational ease, we suppress
the dependence on $\omega_{1}$.

####  4.3.1 Willingness-to-Pay

Moments (i) describe the variation in prices, both within and across flights,
and provide information that identifies the parameters that determine the
distribution of willingness-to-pay. To see this, consider the decision of an
individual of type $(v,\xi)$ who arrives in period $t$ and faces prices
$(p_{t}^{f},p_{t}^{e})$. If we assume that the seat-release policy is not
binding, the passenger’s optimal choice is given by

first-class, if $\displaystyle
v_{i}\times\xi-p^{f}\geq\max\\{0,v_{i}-p_{t}^{e}\\}$ economy, if
$\displaystyle v_{i}-p^{e}\geq\max\\{0,v_{i}\times\xi-p_{t}^{f}\\}$ do not
buy, if $\displaystyle\max\\{v_{i}\times\xi-p_{t}^{f},v_{i}-p_{t}^{e}\\}\leq
0.$

Therefore, the probability of purchase is decreasing in prices, and the rate
of decrease depends on the distribution of $v$. Conditional on purchase, the
fraction of passengers buying first-class in a flight at time $t$ is the
probability that $v\geq(p_{t}^{f}-p_{t}^{e})/(\xi-1)$, and the fraction buying
economy is the probability that $v\leq(p_{t}^{f}-p_{t}^{e})/(\xi-1)$. Because
$F_{b}$ and $F_{l}$ are time invariant, conditional on knowing the
distribution of $\xi$, the variation in fares and the resulting differences in
these probabilities by reason for travel, which in turn vary with flight date,
trace the distributions $F_{b}$ and $F_{l}$ and reveal ($\mu^{l}$,
$\sigma^{l}$, $\delta^{b}$, $\sigma^{b}$). Note this implies that we treat
both anticipated and unanticipated “demand shocks” as the same, and that any
seasonality in our data will affect the variance of the estimate of parameter
density.

Next we consider the identification of preference parameters ($\mu^{l}$,
$\sigma^{l}$, $\delta^{b}$, $\sigma^{b}$) when the seat-release policies are
binding. When seats bind in period $t$, we possibly only observe a subset of
passengers. However, the fact that $F_{b}$ and $F_{l}$ are time-invariant
means that variation in the fares over time is sufficient for the
identification of the preference parameters, and thus rationing only affects
the identification of the parameters that govern the arrival process
$(\lambda_{t},\theta_{t})$. Moreover, conditional on identifying
$(\lambda_{t},\theta_{t},F_{\xi})$, we also have variation in prices across-
markets with similar $\omega_{1}$, which are informative about these
parameters. For instance, if there is an increase in the demand for economy
tickets relative to business (e.g., Christmas seasonal effect), making the
change in fares greater for the economy class than for the first-class
tickets, then as long as there is sufficient variation in fares this surge
will affect the size of the market, not the willingness-to-pay.

Next, we consider the identification of the distribution
$F_{\xi}(\cdot;\mu^{\xi})$, under the assumption that the distribution is
known up to the mean parameter, $\mu^{\xi}$. The moments (ii) use the
variation in the extreme differences of fares across cabins and help identify
the mean of the quality premium $(\xi-1)$. Note that for a passenger with
$(v,\xi)$ who buys first-class, $\xi$ must be at least
$(p_{t}^{f}-p_{t}^{e})/v$, and for a passenger with $(v,\xi)$ who buys economy
$\xi$ must be at most $(p_{t}^{f}-p_{t}^{e})/v$. Comparing across all
passengers and all times gives

$\displaystyle\frac{\max_{t}(p_{t}^{f}-p_{t}^{e})}{\min\\{v:\texttt{bought
first-
class}\\}}\leq(\xi-1)\leq\frac{\min_{t}(p_{t}^{f}-p_{t}^{e})}{\max\\{v:\texttt{bought
economy}\\}},$

where, for example, $\min\\{v:\texttt{bought first-class}\\}$ and
$\max\\{v:\texttt{bought economy}\\}$ are the minimum and maximum value among
those who buy first-class and economy, respectively. Thus, moments capturing
the covariation between cabin-specific quantities and the price differential
across cabins identify $\mu^{\xi}$. Implicitly, we are assuming that for some
$t$, the expected maximum and the expected minimum fare differences are the
same, in which case the above inequalities become equal. Although we assume
that $\xi$ is an Exponential, this assumption is not necessary, we could have
used other distribution, but we can only identify the mean $\mu^{\xi}$.

####  4.3.2 Arrival Process and Passenger Mix

Moments (iii)-(v) capture within-flight price dispersion as the flight date
approaches and identify the arrival rate. Recall our assumption that
$\lambda_{t}=\lambda+\Delta^{\lambda}\times(t-1)$ so at the initial period
$\lambda_{1}=\lambda$, and given our Poisson assumption $\lambda$ is an
average number of passengers that arrive in the first period, where seat-
release policy is less likely to be binding. Although we do not observe the
number of passengers that arrive in $t=1$, we can exploit the observed price-
path, in particular, its value at $t=1$ and its slope because, all else equal,
the shadow cost of a seat is proportional to $\lambda$ and $\Delta^{\lambda}$.
We also use the fact that while the arrival process is linear in $t$, prices
are not.

Heuristically, higher demand (relative to the capacity) manifests itself in
the form of a more significant increase in prices, over time, because it
implies more significant variation in the opportunity cost of a seat due to
more substantial “surprises” in the number of sales at given prices. Thus, the
monotonic relationship between the size of the demand and variability in price
paths for a given flight suggests that we can use the dispersion of price
paths from their initial levels to identify $(\lambda,\Delta^{\lambda})$. For
instance, suppose $\lambda$ is small but $\Delta^{\lambda}$ is high.

Thus for the identification, we have relied on the assumption that the arrival
process is Poisson, and passengers are myopic because the latter assumption
implies that higher dispersion in paid fares as the departure gets closer is
only due to the increasing share of business passengers and their preferences.
This data feature suggests that while we can relax the Poisson assumption and
use the Negative Binomial, say, to model the arrival process, it is essential
that the rate of change in arrival, $\Delta^{\lambda}$, is constant and
passengers are myopic.

To identify the passenger mix, $\theta_{t}$, we use both the reason to travel
and the covariation between deviations of fares from their initial level,
which is captured by the moment conditions (vi) and (vii). To see why we need
the former, note that we have

$\Pr(\texttt{business})=\Pr(\texttt{business|buy})\times\Pr(\texttt{buy})+\Pr(\texttt{business|not-
buy)}\times\Pr(\texttt{not-buy}),$

where $\Pr(\texttt{business|buy})$ is estimable, and given $\lambda$ we know
$\Pr(\texttt{buy})$ and $\Pr(\texttt{not-buy})$, but
$\Pr(\texttt{business|not-buy})$ is unknown. Business travelers have higher
mean willingness-to-pay than leisure travelers, so $\Pr(\texttt{leisure|not-
buy})>\Pr(\texttt{business|not-buy})$. This selection is, however, the
smallest at $t=1$ (Figure 1(b)) and we assume that $\Pr(\texttt{business|not-
buy})=0$ at $t=1$. Then, using the assumption that
$\theta_{t}=\min\\{\Delta^{\theta}(t-1),1\\}$, which gives us $\theta_{1}=0$,
and the number of sales in $t=1$ we can determine $\Pr(\texttt{buy})$. In
other words, we rely on the assumption that at the start $t=1$ there are no
business travelers – see Figure 1. This probability varies over time, and even
though it is a function of both preferences and $\Delta^{\theta}$, the only
reason for its time variation is $\Delta^{\theta}$, which in turn is captured
in variation in price paths (Figures 3 and 5). So the distribution of changes
in fares relative to the initial fare is informative about $\Delta^{\theta}$.

##  5 Results

In this section, we present our estimation results. First, we discuss how our
estimates capture sources of across-market heterogeneity. Second, we calculate
the distribution of opportunity costs for a seat and show how they vary across
cabins and time until departure. We discuss model fit in Appendix A.3.

###  5.1 Market Heterogeneity

Recall that we estimate means and covariances of the parameters’ distribution
across all markets in our sample. In Table 5 we present the means (i.e., the
first moments) of these distributions averaged across all of our markets, and
means for the market with the modal initial-capacity, of 117 economy seats and
15 first-class seats. The mean willingness-to-pay for a (one-way) economy
class seat across our entire sample is $413, and the modal market is $508. The
mean coefficient of variation of willingness-to-pay is 0.608, and the mean
business traveler values an economy class ticket 22.6% more than the mean
leisure passenger. We estimate that, on average, about 23 passengers show up
in the first period (110+ days before the flight) and, on average, arrivals
decrease by 7% each period while the fraction of business passengers increases
by 7% each period (across eight periods).

Table 5: Demand Estimates

Demand | Mean | Mean
---|---|---
Parameters | All Markets | Markets with Modal Capacities
| (1) | (2)
$\mu^{l}$ | 413.234 | 508.054
| (19.539) | (30.942)
$\frac{\sigma^{l}}{\mu^{l}}$ | 0.608 | 0.247
| (0.04) | (0.165)
$\delta^{b}$ | 0.226 | 0.116
| (0.011) | (0.04)
$\frac{\sigma^{b}}{\mu^{b}}$ | 0.353 | 0.371
| (0.044) | (0.145)
$\mu^{\xi}$ | 0.230 | 0.267
| (0.042) | (0.004)
$\lambda$ | 23.318 | 18.119
| ( 1.019) | ( 3.385)
$\Delta^{\lambda}$ | -0.071 | -0.052
| ( 0.003) | (0.011)
$\Delta^{\theta}$ | 0.077 | 0.071
| ( 0.003) | (0.019)

Note: In this table we present the mean demand parameters from the marginal
distribution $h(\Psi;\widehat{\mu}_{\Psi},\widehat{\Sigma}_{\Psi})$ (in column
(1)) and the conditional density
$h(\Psi|\omega_{1}^{*};\widehat{\mu_{\Psi}},\widehat{\Sigma_{\Psi}})$, where
$\omega_{1}^{*}$ is the modal capacities (in column (2)). In our sample, a
market with modal capacities has 115 economy class seats and 14 1st-class
seats. Bootstrapped standard errors are in the parentheses.

Figure 9: Market Heterogeneity: Marginal Densities of Demand Parameters

(a) PDF of $\mu^{l}$

(b) PDF of $\delta^{b}$

(c) PDF of $\Delta^{\theta}$

(d) PDF of $\mu^{\xi}$

Note: This figure displays the marginal densities for (a) leisure travelers’
average willingness-to-pay for an economy seat ($\mu^{l}$); (b) difference in
business travelers’ average willingness-to-pay for an economy seat $\mu^{b}$
and $\mu^{l}$ expressed as a ratio of $\mu^{l}$ ($\delta^{b}$), (d) change in
the share of business travelers each period ($\Delta^{\theta}$), and (d) the
average additional valuation for a first-class seat relative to an economy
seat ($\mu^{\xi}$).

However, we have many different markets in our sample. To get a sense of the
heterogeneity across these markets, in Figure 9 we show the marginal densities
of four parameters (out of the eight parameters in $\Psi$). The main takeaway
is that there is, indeed, substantial heterogeneity in demand across markets.
For example, from Figure 9-(a) we can see that the mode value of $\mu^{\ell}$
is approximately $300, which is close to the market average of $413 as shown
in Table 5(1). However, there is a long upper tail, with a positive mass at
the upper bound value of $1,000. In Figure 9(b), we present the density of
$\delta^{b}$ that determines the difference in willingness-to-pay between
leisure and business passengers as $\mu^{b}=\mu^{l}\times(1+\delta^{b})$. We
present the rate of change in the fraction of business passenger arrivals in
panel (c). This distribution implies substantial heterogeneity in the fraction
of business arrivals.

We present the density of the taste for first-class service ($\xi$) in panel
(d). On average, passengers’ willingness-to-pay for a first-class seat is 23%
more than for an economy seat, with substantial heterogeneity across
passengers. There is a mass close to zero, which implies thre is meaningful
cross-cabin substitution, but also a long tail, which implies there is
substantial potential surplus for airlines to capture.

In Appendix A.2, Table A.2.1 we display the full variance-covariance estimates
for the modal market. And in Tables A.2.3 and A.2.2 we display mean of the
demand parameters across all twenty initial capacities for which we estimated
the model.

###  5.2 Implications of Demand Estimates

Using our estimates, we can determine the implied densities of the
willingness-to-pay for an economy seat and a first-class seat and how these
densities change over time. Recall that the density of the willingness-to-pay
for an economy seat in period $t$ is the mixture $\theta_{t}\times
f_{v}^{b}(\cdot)+(1-\theta_{t})\times f_{v}^{l}(\cdot)$, and the density of
the willingness-to-pay for an first-class seat in period $t$ is similar to the
economy seat augmented by $f_{\xi}(\cdot)$. These densities, for periods
$t=1,3,5,8$ are displayed shown in Figure 10. As expected, densities of
willingness-to-pay for a first-class seat is “shifted” to the right of the
densities of willingness-to-pay for an economy seat by $\xi$. Using the
estimates for $\lambda_{t}$ and $\theta_{t}$ from column 2 of Table 5, we can
also determine the average number of arrivals by their reason to travel. From
the definition of $\lambda_{t}=\lambda+\Delta^{\lambda}\times(t-1)$ we find
the average number of arrival is 18 in period $t=1$, and every period
decreases at the rate of $\Delta^{\lambda}=-0.052$. The share of business
travelers increases at a rate of $\Delta^{\theta}=0.071$.

Figure 10: Willingness-to-Pay for a Seat by Cabin and the Arrivals, for Modal
Capacity

(a) PDF of WTP for an Economy Seat

(b) PDF of WTP for a First-Class Seat

(c) Number of Arrivals, over time

Note: This figure displays, for the modal capacity and mean market, the
implied densities of WTP for an economy seat and WTP for a first-class seat,
and the number of arrivals by their purpose of travel

It is also illustrative to consider what these parameters imply about the
total (opportunity or shadow) marginal cost of a seat. The total marginal cost
of a seat comprises its “peanut” cost, which is constant, and the opportunity
cost varies over time depending on the state’s evolution, i.e., the number of
unsold economy seats and first-class seats. The shadow cost is the right-hand
side of Equation 10, the change in expected value for a change in today’s
price. In other words, the shadow cost is the cost of future revenues to the
airline of selling an additional seat today.

In Figure 11 we present the state’s evolution, in terms of the contours
corresponding to the state’s joint density, as implied by our model estimates.
Consider $\omega_{1}$, which is the initial capacity for this modal capacity
market. So, when we move to the next few periods, we see that the uncertainty
increases. However, as we get closer to the departure time, the contours move
towards the origin, which denotes that fewer seats might remain unsold with
time. The contour of the state at the time of departure
($\omega_{\texttt{dept}}$) denotes the distribution of the state at the time
of departure. The fact that even at the time of departure, the contours are
not degenerate is consistent with the load factor observed in our sample, see
Figure 6(b). Thus we can conclude that there is substantial uncertainty (or
volatility) about demand.

Figure 11: Evolution of Seats Remaining

Note: The figure displays the contours corresponding to the joint density of
unsold seats for every period, over all capacities and markets.

One of the implications of this demand volatility is the implied volatility in
the value of a seat to the airline, i.e., the seat’s opportunity cost. In
Figure 12, we present the distributions of the marginal cost for an economy
and first-class seat that are realized in equilibrium, averaged across all
markets and all capacities. This feature graphically relates the state
transitions to the shadow cost of a seat. In particular, in Figure 12, we take
the distribution of states realized in a given period (Figure 11) and sample
the total marginal costs (sum of the derivative of value functions with
respect to price and the “peanut” cost) based on those frequencies and then
plot the distributions. In panel (a), we present the distributions for an
economy seat, and in panel (b), we present the distributions of a first-class
seat. As can be seen, there is a significant variation in the costs. These
variations are crucial for our identification as they are the underlying
reason for dispersion in the observed fares.

Figure 12: Distributions of marginal (opportunity) cost of a seat

(a) Economy

(b) First-Class

Note: Panels (a) and (b) display the distributions of opportunity cost
distributions at each time, for an economy and a first-class seat,
respectively.

In the first period, $t=1$, there is no uncertainty about the state, which in
turn means the marginal cost is degenerate at $57.64 for an economy seat and
$217.74 for a first-class seat. With $t>1$, the distributions become more
dispersed but with little change in the mean. For instance, the means of the
marginal costs for an economy seat in periods $t=3,4,5$ and $7$ are $84.54,
$84.84, $84.98, and $84.87, respectively. In contrast, the variances increase
substantially from $68 to $100 to $132 to $298, during the same periods. We
observe similar pattern for a first-class seat; the mean marginal costs are,
approximately, $245, $239, $243 and $238 in periods $t\in\\{3,4,5,7\\}$,
respectively. And in the same period the variance increases from $570 in $t=3$
to $851 in $t=7$. Finally, in the last period, $t=8$, the opportunity cost of
a seat is zero as the marginal cost is only the peanut cost.

##  6 Inefficiency and Welfare

There are two sources of informational frictions in this market that
contribute to inefficiency: asymmetric information about passengers’
valuations and uncertainty about future demand. A passenger’s valuation may be
due to idiosyncratic preferences and may also be associated with their reason
for travel. Airlines’ inability to price based on a passenger’s reason for
travel or even the idiosyncratic valuation can distort the seats’ final
allocation.212121 This source of inefficiency is present in a static monopoly
setting when the monopolist chooses one price. However, in our setting, the
monopolist must consider the unknown valuations of future arriving passengers.
The second source leads to inefficient allocations of limited capacity because
the airline chooses its prices and seat-release policies before the demand is
realized. Intra-temporal and inter-temporal misallocation introduced by these
frictions represent opportunities for welfare-improving trade. Using
counterfactual pricing and allocation mechanisms, we quantify the
inefficiencies attributable to these sources. We first show how to visualize
these sources of inefficiencies using a schematic representation of a welfare
triangle. We then present and discuss our results.

###  6.1 Welfare Triangle

Consider the first-best allocation: seats are allocated to the highest
valuation passengers $(v,\xi)$ regardless of the timing of their arrival to
the market. Under this allocation, the division of surplus would depend on the
prices. Figure 13 shows the line between A and B associated with this
efficient benchmark and forms the welfare triangle (OAB). Point A represents
the full extraction of consumer surplus (i.e., price equals valuation), and
point B represents maximum consumer surplus (i.e., prices equal the peanut
costs).

Figure 13: Welfare Triangle

Producer SurplusConsumer SurplusA: ($1^{st}$-best; full extraction)B:
($1^{st}$-best; zero price)C:(Data)F($1^{st}$-degree;zero price)E:
($1^{st}$-degree)D:($3^{rd}$-degree)G:(VCG)OH:(Only Dynamic)

Note: A schematic representation of the welfare triangle under a different
pricing regime.

Point C (“Data”) in Figure 13, which is in the interior of the triangle OAB,
denotes the division of surplus resulting from current pricing practices by
airlines that we observe in our data and that we use to estimate our model.
This outcome is preferable to no trade for both the airline and consumers but
is strictly inside the welfare frontier due to the two inefficiency sources
discussed above. The distance of C from A–B illustrates the magnitude of
welfare-improving opportunities relative to current practice.

Suppose the airline did not employ second-degree price discrimination each
period before the flight, and only changed prices across time. Or, in other
words, the airline did not exploit the difference in quality between an
economy seat and a first-class seat when choosing its prices, and each period
chose one price for both cabins.222222 Formally, airline solves
$\max_{\\{p_{t},q_{t}\\}_{t=1}^{T}}\sum_{t=1}^{T}\mathbb{E}_{t}\left\\{\pi(\chi_{t},\omega_{t};\Psi_{t})\right\\}$,
where the flow profit is
$\pi(\chi_{t},\omega_{t};\Psi_{t})=(p_{t}-c^{f})\times\mathbb{E}(q_{t}^{f}|(p_{t},q_{t}))+(p_{t}-c^{e})\times\mathbb{E}(q_{t}^{e}|(p_{t},q_{t}))$,
and $\mathbb{E}(q_{t}^{f}|(p_{t},q_{t}))$ and
$\mathbb{E}(q_{t}^{e}|(p_{t},q_{t}))$ are expected demands. The airline would
still adjust the price each period depending on the opportunity cost of a
seat. This counterfactual of choosing one price across cabins corresponds to
point H in Figure 13 (“Only Dynamic”). While the producer surplus under H will
be lower than the producer surplus under C, the effect on consumer surplus is
theoretically ambiguous. Choosing one price across two cabins should improve
welfare for those who buy first-class under the current prices, but it should
lower welfare because economy class seats become expensive and total sales
will adjust.

Airlines do not observe arrivals’ reasons for travel, limiting their ability
to price based on the difference between business and leisure arrivals’
willingness-to-pay, possibly resulting in the exclusion of leisure arrivals on
account of expectations of greater demand from business arrivals. Permitting
the airline to price based on the reason for travel, i.e., third-degree price
discrimination, can increase profits for the airline, but the implication for
passengers is ambiguous. Leisure passengers may benefit, but it may come at
the cost of business passengers. Since leisure and business travelers arrive
at different times, and the airline faces capacity concerns, the change in
consumer surplus depends on the entire demand process. Furthermore, the number
of total seats sold may increase or decrease, affecting the total welfare.
Point D (“$3^{rd}$-degree”) in Figure 13 represents a division of welfare when
the airline can charge different prices based on passenger’s reason for travel
and one seat-release policy for each cabin.

Even with the airline’s ability to price based on the reason for travel,
asymmetric information about idiosyncratic valuations can create
inefficiencies. For example, some leisure passengers may have unusually high
valuations, and some business travelers may have unusually low valuations. To
ascertain the importance of this information asymmetry, we consider a setting
where the airline practices first-degree price discrimination. The airline
observes valuations each period and decides which arrivals to accommodate,
charging each arrival its valuation. However, the airline is still uncertain
about future demand realizations. This outcome corresponds to point E in
Figure 13. Likewise, point F in Figure 13 corresponds to the first-degree
allocation of seats but with the price equal to the peanut cost.

The (dotted) line that joins E and F is informative about the extent of
dynamic inefficiency in the market. In particular, the line E–F represents the
frontier of the welfare triangle (OEF) when the airline knows $(v,\xi)$ for
passengers in a given period but cannot foresee future realizations of the
demand process. One way to divide the surplus along the E-F frontier is by
implementing Vickery-Clarke-Groves auctions every period. Such a division of
surplus is denoted by point G (“VCG”) in Figure 13. Thus, the set of potential
outcomes in OAB but not in OEF represents lost surplus due to inter-temporal
demand uncertainty. One could envision a secondary-market run by the airline
that could resolve these dynamic inefficiencies, and our estimates provide the
value that could be created by such an exchange.

###  6.2 Counterfactual Results

Table 6 presents the welfare estimates under all of these alternative pricing
strategies, averaged across all markets and capacities observed in our data.
Columns of Table 6 are indexed by a letter (e.g., C) that corresponds to the
point in the welfare triangle of Figure 13.

##### Current Pricing.

Recall that point C in Figure 13 denotes the division of surplus resulting
from the airlines’ current pricing practices. The surplus associated with this
pricing strategy is in the first column of Table 6. Total surplus is $56,050
for the average flight, with 69% of the surplus going to the airline. This
outcome is preferable to no trade for both the airline and consumers but is
strictly inside the welfare frontier due to the two inefficiency sources
discussed above. Comparing the total surpluses under columns C and A of Table
6 we find that the surplus associated with current pricing represents 89% of
the average market’s potential attainable surplus.

Table 6: Price Discrimination Counterfactuals

Data Only Dynamic $3^{rd}$-Degree $1^{st}$-Degree VCG $1^{st}$-Best C H D E G
$\overline{\text{A}\text{B}}$ Producer Surplus 38,654 28,475 38,903 56,109
14,987 - ( 2,297) ( 1,571) ( 2,315) ( 4,031) ( 1,564) Consumer Surplus 17,396
12,977 17,042 - 41,122 - ( 1,714) ( 1,127) ( 1,697) ( 2,886) – Business 5,416
4,621 5,192 - 10,516 - ( 474) ( 417) ( 438) ( 587) – Leisure 11,980 8,355
11,850 - 30,605 - ( 1,510) ( 1,021) ( 1,388) ( 2,724) Total Surplus 56,050
41,452 55,946 56,109 56,109 63,226 ( 3,917) ( 2,619) ( 3,904) ( 4,031) (
4,031) ( 4,728) • Note: In this table we present measures of welfare for six
different outcomes, corresponding to points A-H in Figure 13, where Column C
is the data. These calculations are performed for all market types receiving
positive weight for a given capacity, then averaged across types and
capacities. Bootstrapped standard errors are in parentheses.

##### Only Dynamic Pricing.

When we restrict the airline to choose only one price each period for both
cabins, as expected, it lowers the total welfare. Note that the airline can
still change prices over time. Producer surplus is 74% of the baseline (Column
C), and the total surplus is 74% of the baseline. Although the airline’s
ability to use second-degree price discrimination to screen passengers between
cabins (from H to C) increases the total surplus, airlines capture roughly
two-thirds of the additional surplus.

##### Group Pricing: Business versus Leisure.

Column D of Table 6 provides surplus estimates when the airlines are permitted
to price based on the reason for travel, i.e., third-degree price
discrimination. Relative to current pricing practice (i.e., Column C), we find
airline surplus increases by about one percent and consumer surplus falls by
about the same, leaving total surplus nearly unchanged. Thus, group pricing
based on reason to travel slightly increases revenue but lowers the total
surplus.

##### Static versus Dynamic Inefficiencies.

We begin with the surplus under the first-degree price discrimination in
column E of Table 6, where the airline can price equal to the arrivals’
willingness-to-pay but still faces uncertain future demand. By construction,
the airline can capture the entire surplus. However, the total surplus is only
slightly higher than the second and third-degree price discrimination (C and D
of Table 6, respectively). We find that a VCG auction would result in
relatively low prices, or that consumers would capture 73% of the surplus in
the presence of a period-by-period VCG auction. Thus, these results suggest
that airlines’ increased effort to learn passenger information to group price
discrimination does not increase total surplus in the market and instead only
transfers surplus from passengers to the airline. Comparing total surpluses
under C and A, we find that stochastic demand and asymmetric information lead
to approximately 11% loss of welfare. Comparing E and A, we find that almost
all of this inefficiency is due to stochastic demand.

##  7 Conclusion

We develop a model of intra-temporal and inter-temporal price discrimination
by airlines that sell a fixed number of seats of different quality to
heterogenous consumers arriving before a flight. We specify demand as non-
stationary and stochastic, which accommodates the salient features of airline
pricing. Using unique data from international airline markets, we flexibly
estimate the distribution of preferences for flights. The estimation exploits
the relationship between a passenger’s seat chosen, timing of purchases,
reasons for travel, and the fare paid to identify how effectively airlines
discriminate using sources of passenger heterogeneity. We find that the
flexibility of the model and estimation algorithm are successful in capturing
key features of our data.

Next, through several counterfactual exercises, we use the estimates to
explore the role that stochastic demand and asymmetric information have on
efficiency and the distribution of surplus. We find that current pricing
practices result in substantial inefficiencies relative to the first-best
outcome. In particular, total welfare is only 89% of the welfare without
demand uncertainty and asymmetric information. To isolate the role of
different sources of asymmetric information in determining welfare, we solve
for optimal seat-release and prices when the airline can discriminate based on
passengers’ reason to travel, and also when the airline can observe their
preferences. The first case (i.e., third-degree price discrimination) achieves
88% of the first-best welfare, representing a 1% decrease from current
practices. Business passengers’ and leisure passengers’ surpluses decrease due
to the loss of informational rent and reduction in seats sold. The second case
(i.e., first-degree price discrimination) where the only remaining source of
inefficiency is inter-temporal demand uncertainty, has an insignificant effect
on welfare compared to the first case. Thus, demand uncertainty accounts for
almost all of the total welfare loss, while asymmetric information accounts
for none.

There are many avenues for future research on related topics. First, like
other studies of dynamic pricing, we model a monopolistic market structure
that accurately reflects our data. This limits our ability to examine the
impact of competition on discriminatory-pricing practices. Another interesting
path for future research is to consider the possibility that consumers are
strategic in their purchasing decisions. While this is difficult to conclude
with our data, purchases of numerous goods are increasingly made online, which
allows firms to track search behavior and adapt pricing accordingly. Given the
growing theoretical literature on this topic (e.g., Board and Skrzypacz (2016)
and Dilme and Li (Forthcoming)) that yield testable implications from
strategic behavior by consumers, empirical studies like ours and Sweeting
(2010) represent an opportunity to offer insight to future modeling efforts.

Relatedly, as firms gather more information about the preferences and
purchasing habits of consumers, exploitation of this information becomes an
important concern. For more on the role of privacy and efficiency, see
Hirshleifer (1971) and Posner (1981). Although there are few papers that study
the role of privacy (Taylor, 2004; Calzolari and Pavan, 2006) in price
discrimination, more empirical research in this topic is needed. Future
research should seek to understand the trade-off between efficiency and
privacy, especially in industries where firms have greater access to such
information.

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Appendix

##  A.1 Survey of International Air Travel

We present additional details about the SIAT data. As we mentioned in Section
2, the data are collected by the Department of Commerce. The DOC contracts
with a private survey firm, CIC Research Inc. We use data from the surveys
conducted in 2009, 2010, and 2011. There are two data collection methods: (1)
Direct participation of the airlines, which arrange for their flight crews to
distribute and collect surveys on-board; (2) Use of sub-contractors to
distribute and collect the questionnaires in the airport departure gate area.
According to the SIAT, in 2009, these two methods accounted for 60% and 40% of
all collections, respectively. The dataset can be purchased at
https://rb.gy/fop8cc. A copy of the survey questionnaire is available at
http://charliemurry.github.io/files/SIAT_Data_Doc_2009.pdf.232323 DOC
processes the surveys and implements a quality control process to ensures data
integrity. Any foreign language responses are translated into English.

There are 413,309 survey responses in the data we receive from the Department
of Commerce. We impose many restrictions to arrive at our final sample. In
Table A.1.1, we display summary statistics at four stages of the sample
selection process: (1) the original data, (2) after we drop responses that do
not report a price, (3) after we make additional selection criteria, like
dropping flights with less than 10 responses, responses with other partial
information, non-revenue and other exotic tickets, and connecting tickets and
(4) our final sample after we select monopoly markets and merge with auxiliary
data on capacities. Approximately 38% of the original survey responses do not
have information about fares, and we drop those. Out of the remaining 62% who
report fares, approximately 53% report traveling with at least one companion.
If there are multiple people traveling together, e.g., a family, the survey is
intended to be administered to one person in the group. When a respondent
reports flying with other passengers, we duplicate the ticket data for each
passenger they report flying with. We exclude respondents who report buying
their tickets as a part of a tour package, or using airlines miles, or through
any other discounted fare. We also restrict our sample to responses that
report traveling with at most 10 people in their group (which is 98.23% of the
original sample) to minimize the chances that the tickets were bought as part
of some tour package.

Table A.1.1: Summary Statistics

Variable | Mean | Min | Median | Max | Non-missing |  | Mean | Min | Median | Max | Non-missing
---|---|---|---|---|---|---|---|---|---|---|---
| (1) Raw Data | (2) Non-missing Fares |
Business Passenger | 0.17 | – | – | – | 413,309 |  | 0.19 | – | – | – | 256,843
– busines | 0.04 | – | – | – | 413,309 |  | 0.04 | – | – | – | 256,843
– convention | 0.02 | – | – | – | 413,309 |  | 0.02 | – | – | – | 256,843
– government | 0.00 | – | – | – | 413,309 |  | 0.00 | – | – | – | 256,843
– teaching | 0.02 | – | – | – | 413,309 |  | 0.02 | – | – | – | 256,843
– vacation | 0.17 | – | – | – | 413,309 |  | 0.20 | – | – | – | 256,843
– visit friends/family | 0.12 | – | – | – | 413,309 |  | 0.15 | – | – | – | 256,843
– religion | 0.01 | – | – | – | 413,309 |  | 0.01 | – | – | – | 256,843
– health | 0.01 | – | – | – | 413,309 |  | 0.01 | – | – | – | 256,843
Nonstop | 0.49 | – | – | – | 413,309 |  | 0.46 | – | – | – | 256,843
Economy Class | 0.90 | – | – | – | 413,309 |  | 0.90 | – | – | – | 256,843
Advanced Purchase | 72.37 | 0.00 | 45.00 | 365.00 | 413,309 |  | 67.39 | 0.00 | 45.00 | 365.00 | 256,843
Fare (on-way adjusted) | 567.97 | 0.00 | 400.00 | 24,258.00 | 256,843 |  | 567.97 | 0.00 | 400.00 | 24,258.00 | 256,843
Total travelers | 2.98 | 1.00 | 2.00 | 10.00 | 413,309 |  | 2.80 | 1.00 | 2.00 | 10.00 | 256,843
Trav. with Family* | 0.36 | – | – | – | 413,309 |  | 0.34 | – | – | – | 256,843
US Resident* | 0.48 | – | – | – | 413,309 |  | 0.51 | – | – | – | 256,843
Male* | 1.54 | 1.00 | 2.00 | 2.00 | 363,586 |  | 1.56 | 1.00 | 2.00 | 2.00 | 236,658
Age* | 43.01 | 18.00 | 42.00 | 99.00 | 234,546 |  | 43.10 | 18.00 | 42.00 | 92.00 | 153,092
Income Bin* | 5.68 | 1.00 | 5.00 | 11.00 | 305,647 |  | 5.77 | 1.00 | 5.00 | 11.00 | 213,284
| (3) Selection On Additional Variables | (4) Final Sample |
Business Passenger | 0.16 | – | – | – | 25,945 |  | 0.14 | – | – | – | 14,930
– busines | 0.04 | – | – | – | 25,945 |  | 0.03 | – | – | – | 14,930
– convention | 0.02 | – | – | – | 25,945 |  | 0.02 | – | – | – | 14,930
– government | 0.00 | – | – | – | 25,945 |  | 0.00 | – | – | – | 14,930
– teaching | 0.02 | – | – | – | 25,945 |  | 0.01 | – | – | – | 14,930
– vacation | 0.19 | – | – | – | 25,945 |  | 0.17 | – | – | – | 14,930
– visit friends/family | 0.14 | – | – | – | 25,945 |  | 0.13 | – | – | – | 14,930
– religion | 0.01 | – | – | – | 25,945 |  | 0.00 | – | – | – | 14,930
– health | 0.01 | – | – | – | 25,945 |  | 0.01 | – | – | – | 14,930
Nonstop | 0.58 | – | – | – | 25,945 |  | 1.00 | – | – | – | 14,930
Economy Class | 0.93 | – | – | – | 25,945 |  | 0.92 | – | – | – | 14,930
Advanced Purchase | 73.65 | 0.00 | 60.00 | 365.00 | 25,945 |  | 75.25 | 0.00 | 60.00 | 365.00 | 14,930
Fare (on-way adjusted) | 528.43 | 70.00 | 400.00 | 5000.00 | 25,945 |  | 480.37 | 70.00 | 375.00 | 5000.00 | 14,930
Total travelers | 2.90 | 1.00 | 2.00 | 10.00 | 25,945 |  | 3.06 | 1.00 | 2.00 | 10.00 | 14,930
Travel with Family* | 0.38 | – | – | – | 25,945 |  | 0.42 | – | – | – | 14,930
US Resident* | 0.51 | – | – | – | 25,945 |  | 0.47 | – | – | – | 14,930
Male* | 1.54 | 1.00 | 2.00 | 2.00 | 23,698 |  | 1.54 | 1.00 | 2.00 | 2.00 | 13,604
Age* | 43.67 | 18.00 | 43.00 | 88.00 | 16,209 |  | 43.34 | 18.00 | 42.00 | 88.00 | 9,434
Income Bin* | 5.69 | 1.00 | 5.00 | 11.00 | 21,247 |  | 5.70 | 1.00 | 5.00 | 11.00 | 12,011

  * •

Note: In this table we present the summary statistics of variables that we
observe from the SIAT survey. Variables with an asterisk (*) denote the
variables that are not used in our empirical analysis. The table is separated
into four sub-tables, where each sub-table displays the summary statistics
after each round of sample selection. Sub-table (1) denotes the original data;
Sub-table (2) denotes the sample after we have dropped any observation that
does not report fares; Sub-table (3) denotes the sample after we drop flights
with less than 10 responses, responses with partial information for our
purposes, and non-revenue tickets, and connecting tickets; and sub-table (4)
denotes the sample after we restrict to monopoly markets.

##  A.2 Additional Estimation Results

Table A.2.1: Mean and Variance-Covariance of joint truncated Normal
distribution for Markets with Modal Capacity

Parameter | Mean ($\mu_{\Psi}$) | Variance-Covariance Matrix ($\Sigma_{\Psi}$)
---|---|---
$\mu^{l}$ | $\frac{\sigma^{l}}{\mu^{l}}$ | $\delta^{b}$ | $\frac{\sigma^{b}}{\mu^{b}}$ | $\mu^{\xi}$ | $\lambda$ | $\Delta^{\lambda}$ | $\Delta^{\theta}$
$\mu^{l}$ | 427.8 | 61,315 | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$
| (70.344) | (107.083) |  |  |  |  |  |  |
$\frac{\sigma^{l}}{\mu^{l}}$ | $0.523$ |  | $0.428$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$ | $\mathchoice{\mathbin{\vbox{\hbox{{$\displaystyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\textstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptstyle\bullet$}}}}}{\mathbin{\vbox{\hbox{{$\scriptscriptstyle\bullet$}}}}}$
| (0.244) |  | (0.199) |  |  |  |  |  |