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WORKING 

PAPERS

DYNAMICS AND EQUILIBRIUM IN A STRUCTURAL MODEL

 OF COMMERCIAL AIRCRAFT OWNERSHIP

Loren Smith

WORKING PAPER NO. 280

December 2005

FTC Bureau of Economics working papers are preliminary materials circulated to stimulate discussion 

and critical comment. The analyses and conclusions set forth are those of the authors and do not 

necessarily reflect the views of other members of the Bureau of Economics, other Commission staff, or 

the Commission itself. Upon request, single copies of the paper will be provided. References in 

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author to protect the tentative character of these papers. 

BUREAU OF ECONOMICS 

FEDERAL TRADE COMMISSION 

WASHINGTON, DC 20580 
"Dynamics and Equilibrium in a Structural Model

of Commercial Aircraft Ownership"

Loren Smith∗

December, 2005

Abstract

This paper develops and estimates a dynamic equilibrium model of the market for commercial aircraft. Airline choices are modeled as the solution to a

discrete time dynamic programming problem, where in each period, each airline

chooses one or more of various models of new and used aircraft. In equilibrium aircraft prices are such that no airline would benefit from buying, selling,

trading or scrapping aircraft. The parameters of the model are estimated by

maximum simulated likelihood using a new dataset that contains all aircraft

transactions made in the twenty-year period 1978-1997. The transaction data

is merged with a dataset containing aircraft prices. The estimated model is

used to show that a 10 percent investment tax credit on the purchase of new

aircraft has only a small effect on airline behavior and that the demand for new

durable goods is more elastic than previous studies have shown. In addition,

forcing the modernization of older aircraft causes U.S. airlines to reduce the

number of older aircraft they operate by approximately 4 percent, and it is

shown that the new aircraft are the poorest substitute for older aircraft.

JEL Classification: L20, C51

Key Words: Dynamic structural model, secondary market equilibrium,

durable goods, airline economics, simulation estimation

∗I would like to thank my advisors at the University of Virginia Steven Stern, Simon Anderson,

and John Pepper for their helpful comments, suggestions, and support. Any remaining errors are

mine. I thank Ronald Wilcox, Paul Farris, and the Darden Business School for their assistance

acquiring and purchasing the data used in this research. I also thank Todd Pulvino at the Kellogg

School of Management for giving me much of the price data. Funding from the Bankard Fund for

Political Economy at the University of Virginia and the University of Virginia College of Arts &

Sciences is gratefully acknowledged. The opinions expressed in this paper are those of the author

and do not necessarily represent those of the FTC or its individual Commissioners. Correspondence:

Federal Trade Commission, 601 New Jersey Ave., Washington, DC 20001. Email: lsmith2@ftc.gov.

1
1 Introduction

This article examines the effects of durability, secondary markets, and multi-unit ownership on

demand for new and used wide-body commercial aircraft. Theoretical models of durable goods

markets have shown that a good’s durability and resellability can have meaningful effects on its

value. Durable goods are often useful for several periods, and therefore, a good’s expected viability

in the future impacts its value to consumers in the current period. Furthermore, buyers of new

and used durable goods are often sellers of used goods as well, and therefore the expected future

market prices of durable goods impact the opportunity cost of keeping a used durable in the current

period.

Several recent empirical models of demand for durable goods have employed variations of a static

discrete-choice logit framework, and have focused on developing methods that allow for very general

patterns of substitution across differentiated products and heterogeneity in consumer preferences.1

However, the static frameworks typically ignore features that are central to many theoretical models

if durable goods markets, like intertemporal dependence across consumers’ ownership decisions and

secondary markets for used goods.2 The dynamic model developed in this paper allows consumers

to consider the effects decisions made in the current period have on the (expected) payoffs they

will receive in the future, and assumes that consumers participate as both buyers and sellers in the

market for used goods.

Due to computational complexity and/or the lack of good data, estimation of structural dynamic

models is rare. Rust (1987) developed a framework for estimating a dynamic model of a single agent’s

decision to purchase or replace a single durable good (a commercial bus engine).3 The estimated

parameters of the discrete time, discrete choice, dynamic model showed that the agent’s behavior

was consistent with the solution to a regenerative optimal stopping problem that formalized the

trade-off between minimizing expenditures on maintenance and replacements and minimizing the

occurrence of break-downs in the service the good provided.

In spite of the large volume of trade in most markets for used durable goods,4 there have been

few empirical studies that account for the effects of used markets on demand for durable goods.

In a static frameworks, Manski (1983) and Berkovec (1985) estimated models of the market for

1 See, for example, Berry, Levinsohn and Pakes (1995), Goldberg (1995), and Petrin (2002). 2There is a large body of theoretical work on the impact of good durability and secondary markets for durable

goods on demand that is not covered in this paper. See Smith (2005) for a review of this literature. 3Kennet (1994) and Cho (2004) used Rust’s framework to study the optimal timing of the replacement of aircraft

engines and the optimal timing of repair or replacement of mainframe computers respectively. 4For the period 1978-1997 approximately 61% of all aircraft sold were used. And active secondary markets are

not unique to commercial aircraft. For example, approximately 67% of all automobile sales take place in secondary

markets.

1
new automobiles while controlling for the stock of used automobiles. Both authors developed

computable general equilibrium models that controlled for the effects markets for used automobiles

have on equilibrium in the market for new automobiles (and visa versa) in each period. Although

the modelling approach has several nice properties and is easy to compute, it does not explicitly

account for the intertemporal linkages in individual consumer purchase decisions the way the dynamic

structural model developed in this work does. In more recent work, Engers, Hartmann and Stern

(2005) use a dynamic structural model to quantify the equilibrium effects of lemons in the market

for used cars.5 Similarly to Rust (1987), Engers et. al. use a single agent, single asset framework

where car owners are permitted to trade-in the used car they currently own for another new or used

car in every period.

Similarly to previous work, the dynamic model developed below allows for intertemporal substitution by consumers and, in addition, expands the set of choices consumers can make in each period.

In contrast to owners of cars or boats, owners of goods like buses, airplanes, or computers maximize

their expected discounted present value by choosing or maintaining the entire portfolio of several

different models and ages of a good in each time period.6 Specifically, in each period each airline

chooses a fleet of aircraft to maximize the expected discounted present value of its profit flows by

deciding whether to engage in one or more of the following transactions: selling one or more used

aircraft, buying one or more used aircraft, and buying one or more new aircraft. Airline choices are

made simultaneously, and equilibrium is attained when, given the allocation of aircraft and aircraft

prices, no airline wants to buy, sell or trade any aircraft. It is shown below that, given the specification of the model, as long as transactions costs are low enough and/or there are sufficient taste

differences across airlines, a market for both new and used commercial aircraft exists.

The estimates of the structural model show that transaction costs and adjustment costs are

substantial. In fact, it is not uncommon for the purchase of a single aircraft to have costs associated

with the transaction and fleet adjustment that exceed the aircraft’s market value. Additionally,

differences in airlines’ strategic choices of route structures are reflected in their varying preferences

for scale of operation, rate of growth, and the types of aircraft they operate. Specifically, Large U.S.

airlines have exhibited the highest rate of growth in the overall use of wide-body aircraft; airlines

outside of the U.S. have the greatest relative preference for newer aircraft; and Boeing aircraft are

appealing to airlines throughout the world, while Douglas and Lockheed aircraft appeal primarily

to U.S. airlines, and Airbus aircraft are most popular outside of the U.S.

5Esteban and Shum (2004), and Benkard (2004) develop dynamic models that explicitly consider the impact of

used product markets on the strategies employed by manufacturers of durable goods. In contrast to this research, all

of the dynamics in Esteban and Shum (2004), and Benkard (2004) are on the supply side of the model. 6Hendel (1999) shows that considering the decisions made by multi-unit owners in each period to be independent

can systematically bias parameter estimates.

2
One of the benefits of estimating a structural model is that one can evaluate the effects of

changes in market characteristics and policies on equilibrium outcomes. In addition, the evaluation

of economic policies that effect the airline industry has intrinsic value. The sale of commercial

aircraft generates billions of dollars every year and therefore changes in economic policy that affect

the market for commercial aircraft can have profound effects on national and even global economies.7

The estimates of the structural model developed below are used to analyze the effects of two economic

policies on airline investment in new and used aircraft. First, the effects of instituting a 10 percent

tax credit for investment in new capital goods are analyzed. The 10 percent investment tax credit

is identical to the one removed by the Tax Reform Act of 1986. Second, the effects of implementing

stricter safety and/or noise abatement legislation are evaluated.

The results of the first counterfactual experiment show that the implementation of an investment

tax credit generates only small increases in total aircraft ownership by U.S. airlines, and slightly

decreases aircraft ownership in other parts of the world. Additionally, the policy induces a small

amount of substitution toward new aircraft from used aircraft. The experiment indicates that

demand for new commercial aircraft is inelastic, although elasticity estimates are quantitatively

larger than those from previous studies of durable goods industries. The increase in price elasticity

is likely due to the model accounting for both used goods markets and intertemporal substitution by

consumers, and both features of the market increase consumers’ sensitivity to changes in the price

of new goods.

Stricter noise safety and/or noise abatement policy is simulated by imposing a one-time additional

cost to operating older aircraft. Newer generations of aircraft are designed to be safer and quieter

than previous generations. The design of the experiment mimics the hushkits and improvements

airlines purchase to modernize older aircraft. The hushkit mandate reduces the number of older

aircraft operated by U.S. airlines. Interestingly, the number of older aircraft operated by airlines

outside of the U.S. increases slightly. However, the total number of older aircraft scrapped does

increase. New aircraft are shown to be the worst substitute for the oldest vintages of aircraft. Due

to the dynamics in the model, U.S. airlines reduce the number of older aircraft they operate well in

advance of the mandated policy.

The remainder of the paper is organized as follows. The next section provides a brief history

of the market for wide-body commercial aircraft. Section 3 describes the data used to estimate

the structural model. Section 4 details the structural model and discusses some of the implications

of its specification, including the existence of equilibrium. The model solution and estimation

techniques employed in this research are discussed in Sections 5 and 6 respectively. Section 7

7The price of a new wide-body commercial aircraft range from $10 million to over $100 million in my sample

period, and the sale of wide-body commercial aircraft produced $36 billion in revenue in 1997.

3
discusses the estimation results, and Section 8 performs two counterfactual simulations using the

estimated structural model. Finally, Section 9 concludes.

2 The Market for Wide-body Commercial Aircraft

In 1970, Pan American Airlines introduced the 500-passenger Boeing 747 aircraft as the “wave of

the future” for long-distance passenger flights. Although initially other major carriers were more

reserved in their optimism for the bigger-is-better notion that was being promoted by Pan Am

and Boeing, eventually all major carriers were using wide-body jets to service their longer routes.

Unfortunately for wide-body commercial jet producers, the introduction of wide-body jets occurred

at a very tenuous time for many airlines. As airlines were introducing additional capacity to carry

passengers through wide-body jets, growth in air-travel was stagnating.

After a number of airlines appealed to the Civil Aeronautics Board (CAB), the authority that

regulated the setting of fares prior to deregulation, the CAB granted a general increase in airfares

(about 6 percent) in the hope that airline revenues would increase. However, after the increase in

fares, growth in passenger traffic decreased even more.

Internationally, Pan Am and TWA were feeling the pressure of competition, as low-fare charter

services were increasing their market share, especially on the transatlantic routes. After a series of

discussions that lasted through 1971, all the major transatlantic carriers agreed to fix fares for the

transatlantic route, making the average flight across the Atlantic Ocean about $204. The lower fares

introduced intercontinental travel to a whole new social class making transatlantic flights affordable

for the middle class.

In spite of the facts that the oil crisis of the early 1970s raised airline fares significantly and

decreased the rate of growth in passenger flights, most of the major airlines were able to survive.

Air traffic growth slowed throughout the 1970s, but there was never a decline in the number of

flights. In fact, the number of passengers continued to grow despite the fare increases, evidence of

the public’s increasing reliance on flying, now synonymous with jet aircraft, as a routine activity.

Fascination with the Boeing 747 generated enough momentum to produce two additional widebody jets. The Douglas DC-10 was introduced by American Airlines in 1971, and the Lockheed

L-1011 was introduced by Eastern Airlines in 1972. Both aircraft were capable of carrying about

300 passengers and had a range of 2,500 to 3,000 miles. During the period 1978 through 1997, there

were a total of ten different wide-body aircraft introduced by four manufacturers: Boeing, Airbus,

Lockheed-Martin, and McDonnell-Douglas. Even though there have never been more than four

manufacturers of wide-body jets, competition among the manufacturers and from used aircraft sold

in secondary markets have made it tough for manufacturers to be profitable. Currently, there are

4
only two manufacturers producing wide-body jets. Lockheed-Martin recorded negative profits in

every year its L-1011 was manufactured and finally exited the market in 1985. McDonnell-Douglas,

the manufacturer of the DC-10 and MD-11 aircraft, was acquired by Boeing in 1997.

Since their introduction, sales of wide-body commercial aircraft have increased steadily. In 1997,

the sale of new wide-body commercial aircraft generated $36 billion in revenue, which accounted for

approximately 60% of the revenue and 30% of the number of units sold in the industry. Currently,

commercial aircraft are one of the Unites State’s largest net exports, accounting for an average of

$25-30 billion in trade surpluses in the 1990s. U.S. airline consumption of wide-body jets has been

consistent, and there has been significant growth in sales to foreign airlines. In total, over 500

passenger, charter, and freight airlines have, at one time or another, owned a wide-body jet.

In addition to the sale of new wide-body jets, an active secondary market for used wide-body

commercial aircraft has developed and grown in the past 30 years. Used wide-bodies made up about

50% of the total number of units sold in 1978 and grew steadily until they made up approximately

85% of units sold in 1997. Growth in the volume of used aircraft sales is due primarily to the market

for wide body jets being relatively young and aircraft being extremely durable goods.8 Therefore,

new wide bodies have continued to enter the market and very few used wide-body jets have been

scrapped.

The market for used wide-bodies has similarities to and differences from the market for used

cars. Like automobile owners, when an airline wishes to sell one or more aircraft, it publishes

an advertisement in one of many trade journals and sells the aircraft to the highest bidder. One

significant difference between the market for wide-body jets and automobiles is that there is likely

to be little or no asymmetric information about the quality of used commercial aircraft. There

are a couple of reasons why this conjecture is likely to be true. First, wide-body aircraft are very

expensive and obtaining information about their quality is relatively inexpensive. In fact, the data

that I use for this project contains information on almost everything one could possibly want to know

about an aircraft, from when it was damaged on a landing to when a row of seats was changed.

Second, sellers in this market cannot afford to get a reputation for selling poor quality aircraft.

Everyone knows who the sellers are, and, in general, commercial aircraft owners sell in secondary

markets with greater frequency than do automobile owners. So selling a poor quality aircraft and

withholding information has the potential to be particularly costly to airlines.

8The average wide-body jet has a life-cycle of 25-35 years.

5
3 Data

In this section I present an overview of the data. I present some descriptive statistics and information

about how the data were compiled and aggregated to estimate the structural model. The aircraft

transaction data were provided by Back Aviation Solutions (BAS) and transaction prices and aircraft

appraisals were provided by Avmark Incorporated (AI).9

3.1 Transaction data

The transaction data include detailed information about every aircraft that was registered to fly

in the past 60 years at several points in time. The data provide a rich set of information about

each change in aircraft operator or owner for each aircraft in the sample, including the day of the

transaction, the type of transaction (e.g. sale, lease, sublease, or retirement), the identities of the

buyer and seller of the aircraft, and the condition of the aircraft when it was sold.

The BAS data contains information for all types of aircraft from small private jets to large

commercial aircraft. This research is limited to the just over 3500 wide-body commercial aircraft

used during the twenty year period 1978-1997. I focus on wide-body jets because (1) a model

of the entire commercial aircraft market would be computationally infeasible, and (2) focusing on

wide-body commercial jets is consistent with other work in the literature (e.g. Benkard (2004)).

Moreover, the market for wide-body commercial jets is appealing because it generates more revenue

in both primary and secondary markets than all other types of commercial aircraft combined.

In the theoretical model of airline behavior, airlines make ownership decisions in several discrete

time periods. I choose a length of time for each decision period that is both consistent with the

theoretical model, and that is tractable from a model solution and estimation standpoint. The

BAS data is extremely detailed and provides the day of each transaction. However, the time cost

associated with solving and estimating the model increases dramatically as the number of time

periods increases. Therefore, I consider only changes in ownership that last a year or longer sales.

One year decision periods allow the model to be solved and estimated, and are consistent with other

models in the literature on dynamic decision problems. Obviously aggregating time periods in this

way disregards any information provided by ownership spells lasting less than a year, and the model’s

estimates may therefore underpredict changes in aircraft ownership. However, some of the extremely

short ownership spells observed in the data do not indicate actual changes in aircraft ownership.10

Some sell and buy-back transactions may simply be a form of refinancing or outsourcing of resources

by the original owner of the aircraft. For example, some of the sell and buy-back decisions occur

9 I thank Todd Pulvino at the Kellog Graduate School of Management for providing me with the Avmark data. 1 0 Some ownership spells are as short as one day.

6
seasonally, and it is hard to differentiate the sale and buy-back from the original owner hiring the

buyer to fly the planes on a different set of routes for a short period of time.

The data provided by BAS also include measures of several observed attributes of each aircraft in

the sample. The attributes included are as general as the aircraft’s make and model and as specific

as the height of the aircraft’s cargo door in centimeters. Airlines consider carefully the technical

specifications of an aircraft prior to making a purchase, and monitor each aircraft’s condition as it

ages. Therefore, it is reasonable to assume that the observed attributes of aircraft influence the

value of aircraft to airlines. However, I sacrifice some of the information provided by the detailed

data on observed differences across products in order to facilitate model solution and estimation.

Specifically, I aggregate the over 3500 aircraft into 20 distinct types, where a aircraft’s type is

determined by its model and vintage.

3.2 Price data

The transaction data are merged with observed prices and appraisals provided by Avmark Incorporated (AI). The data include actual transaction prices for 1165 deliveries and sales for the period

1978-1993. Prior to 1993, airlines in the United States were required by law to report the price of

any new or used aircraft they purchased or sold. The data was originally reported to the Department of Transportation and the Federal Aviation Association and was then compiled by Avmark. In

addition to observed transaction prices, the Avmark data also contains appraisal values for aircraft

for the period 1978-1997. The appraisals are computed using observed aircraft prices. In fact,

Avmark’s appraisals for the period 1978-1993 can be duplicated using cell means from the observed

price data. Pricing information on particular aircraft is sacrificed for the period 1994-1997 because

airlines were not required to divulge the price of aircraft that they purchased or sold during that

time.

Observed aircraft prices are used to create a value for every type of aircraft in every period. I

refer to this single price for each type of aircraft in each period as a price index. There are two

reasons for creating a price index. First, there are several transactions for which I do not observe

a price. For example, leases are considered a change in ownership in the theoretical model, but I

do not observe a transaction price for ownership changes that involve a lease. Second, aircraft are

aggregated into types and all aircraft of the same type are assumed to be identical. Observed prices

and appraisal values are used to construct the Specifically, I run a regression of observed log prices

on aircraft make, model, age, aircraft characteristics, and a full set of age and time dummies. The

fitted values of the regression are then used to create a price for each type of aircraft in each time

period. The robustness of the imputation is checked by comparing the prices to aircraft appraisals

7
computed by Avmark Incorporated.

3.3 Descriptive Statistics

Table 1 gives the frequency of each model of aircraft observed in the sample (all tables and figures

are in Appendix E. The most frequently observed aircraft in the sample is the Boeing 747 with

1078 aircraft. As was mentioned above, the 747 was also the first wide-body aircraft to enter the

market, making its first passenger flight in early 1970 for Pan Am Airlines. The model with the

fewest aircraft in the sample, the Boeing 777, was also the most recent introduction, making its first

passenger flight in 1995.

Table 2 gives the means of selected aircraft characteristics for aircraft observed in the sample

period 1978-1997. The mean length of a wide-body commercial jet is approximately 59 meters.

The Boeing 747 is the longest model of aircraft at over 70 meters, whereas some versions of the

Airbus A310 were just under 50 meters long. The average aircraft in the sample carries about 410

passengers. The Boeing 747 can be outfitted to carry 600 passengers, whereas the Boeing 767 has a

maximum seating capacity of 280 passengers. The mean fuel capacity for aircraft in the sample is

128,000 gallons, and the average range of flight when at full capacity is approximately 5,900 miles.

This implies that the average aircraft in the sample requires 100 gallons of fuel to fly just over 4

miles. Being the largest aircraft, the Boeing 747 also has the greatest fuel capacity and range of

flight at approximately 400,000 gallons and 9,000 miles respectively.

Figures 1 and 2 describe patterns of aircraft movement from owner to owner observed in the data.

Each line in Figure 1 measures the proportion of each make aircraft of a particular age that were sold

in secondary markets in each period. Perhaps intuitively, aircraft tend to be sold more frequently

as they age. For example, only 1.5% of Boeing aircraft in their fifth year are sold in secondary

markets, while 6.4% of Boeing aircraft in their fifteenth year are sold in secondary markets; trends

are similar for Airbus, McDonnell-Douglas, and Lockheed-Martin aircraft. Airbus aircraft were sold

the most frequently for 13 of the 19 age categories, while Boeing aircraft were sold with the least

frequency for ten of the first 11 age categories. Assuming there is no asymmetric information across

airlines about the quality of aircraft,11 there are a few possible explanations for one type of aircraft

being sold more frequently than others. Aircraft may be sold more frequently in secondary markets

because they physically degrade and become relatively ineffective more quickly than other aircraft.

If an aircraft degrades rapidly, it rapidly moves out of the range of desired usefulness to its current

owner, and therefore the aircraft will be sold to another airline relatively quickly. Another possible

1 1This assumption is reasonable because airlines can obtain almost costlessly a complete record for any aircraft.

8
reason for one aircraft being traded relatively often in secondary markets is that the aircraft has

low transaction costs (e.g. costs associated with learning to operate the aircraft, cost of adapting

the aircraft for different tasks, or a lack of warranties or guarantees). As is discussed in Anderson

and Ginsburgh (1994), and Konishi and Sandfort (2002), an aircraft with relatively low transaction

costs is likely to be sold with greater frequency in secondary markets.

Figure 2 shows the proportion of aircraft sold in secondary markets that are sold to passenger

airlines. As might be expected, in general older aircraft are less likely to be purchased by a

passenger airline and are more likely to be purchased by a freight or charter airline. Of Boeing

aircraft sold in secondary markets, approximately 84% of the newest age range are sold to passenger

airlines, while 70% of the oldest age range are sold to passenger airlines. The drop in percentage of

secondary market purchases made by passenger airlines is much more dramatic for Airbus aircraft.

Approximately 87% of the newest category of Airbus aircraft sold in secondary markets were sold to

passenger airlines, while only 41% of the oldest category of Airbus aircraft sold in secondary markets

were sold to passenger airlines. The percentage of McDonnell-Douglas aircraft sold to passenger

airlines is very consistent, staying at about 80% for all age categories, while the percentage of

Lockheed-Martin aircraft sold to passenger airlines is, in general, lower averaging between 50% and

60%. An aircraft is sold to freight or non-passenger airlines earlier in its life-cycle because (1) freight

and non-passenger airlines care less about the aircraft’s level of degradation than passenger airlines,

and the aircraft physically depreciates more quickly than other aircraft, or (2) the aircraft is more

easily converted to serve the purposes of freight or non-passenger airlines (i.e. lower transaction

costs).

Figure 3 shows trends in the volume of trade in markets for new and used wide-body commercial

aircraft. The growth in the number of new aircraft delivered into the market over time is due

primarily to an increase in the number of new aircraft delivered to non-U.S. airlines. The figure

shows very clearly that the trend in the number of deliveries of wide-body aircraft taken by US

passenger airlines is relatively flat, while the trend in total deliveries of wide-bodies is slightly

upward sloping. The trend in secondary market trades per period is steeply upward sloping. The

increase in secondary market activity is not surprising because the secondary market for wide-body

commercial aircraft is still relatively young (the oldest aircraft in the sample are about 25 years old),

which implies that aircraft have continuously entered the market but very few aircraft have been

scrapped.

4 Structural Model

9
At each time t, each airline maximizes the expected discounted present value of its current and

future profit flows by choosing a fleet of aircraft to own and operate. An airline may change its fleet

by selling one or more used aircraft, buying one or more used aircraft, and/or buying one or more

new aircraft. Airline fleets are chosen from an exogenously determined quantity of new aircraft

that enter the market in each period, and a stock of used aircraft is carried over from the previous

period. Airline preferences vary over the different makes, models, and ages of aircraft; and, in

addition, airlines may prefer different scales of operation. All airlines make ownership decisions

simultaneously which determines the equilibrium quantity and price for all new and used aircraft.

Specifically, airline decisions are consistent with an equilibrium where no airline wants to buy, sell,

or trade any of its aircraft given the current prices and allocation of aircraft.

Typically, discrete choice models of demand limit the choices of buyers to purchasing a single

unit of one of the products available or choosing an outside option (generally, the outside option

is choosing to not purchase any of the available products). An agent that purchases more than

one unit of the available products in a single period is assumed to have made each purchase decision independently. The independent discrete choice assumption is made because (1) the modeler

believes the assumption is reasonably close to reality, and (2) the assumption greatly reduces the

computational burden of estimating the model. However, in the case of commercial aircraft, airlines

are known to make multiple ownership decisions simultaneously in each decision period. Therefore, ignoring the multiple-discrete choices made by airlines would discard valuable information and

would likely distort estimation results. The model specified here allows agents to choose multiple aircraft in each period. Hendel (1999) refers to a model that allows agents to simultaneously

choose multiple products as a multiple-discrete choice model.12 Allowing airlines to choose combinations of multiple units greatly increases the total number of possible outcomes in each period,

and therefore greatly increases the computational complexity of solving and estimating the model.

I mitigate much of computational burden created by the increase in the number of choices afforded

to airlines, by imposing enough structure on airline value functions to derive an expression for the

marginal contribution of each aircraft to the airline’s fleet. These marginal values are then used to

derive a tractable number of equilibrium conditions which are used to estimate the parameters of

the model.13

The market for commercial aircraft consists of manufacturers, airlines, and scrappers (who are

responsible for disposing of unwanted aircraft). Admittedly, the models assumptions greatly simplify

1 2 Similar modelling approaches have been used in other fields. For example, Howell (2004) models the application

stage of a students college enrollment decision as a multiple discrete choice over all postsecondary institutions. That

is, students are allowed to apply to multiple institutions contemporaneously. 1 3The equilibrium concept used in this paper is similar to the equilibrium assumed by Howell (2004), in her structural

model of students’ college application and admission decisions.

10
the behavior of manufacturers and scrappers.14 However, the focus of this work is on the dynamic

equilibrium behavior of airlines, and the simplifications are necessary to make the model tractable.

4.1 Scrappers

Following Rust (1985), I assume that scrappers have an infinitely elastic demand for each type of

aircraft at a fixed price Pj . That is, scrappers are always willing to acquire and scrap any number

of aircraft at a price of Pj . For the empirical work, I assume Pj = 0 for all j, regardless of the

aircraft’s type or level of degradation.

15

4.2 Manufacturers

Let there be an arbitrarily large but finite number of discrete time periods indexed by t = 1, 2, ...T.

Furthermore, let m = 1, .., M index the different makes of aircraft available in the market, and at

each time t, assume there are Jt types of aircraft available, where types are indexed j = 1, ..., Jt. I

assume all aircraft of the same type (i.e. model and vintage) are identical. The total number of

distinct types Jt vary across time periods because new vintages of aircraft enter the market in each

period and old vintages of aircraft stay in the market for several periods. I assume that the number

of new type j aircraft supplied to the market at time t is exogenously specified. Once new aircraft

enter the market, they are allocated to airlines or the scrapper by an efficient auction mechanism.

4.3 Airlines

Used aircraft carried over from the previous period and the exogenously determined quantity of new

aircraft supplied to the market comprise the stock of aircraft available to airlines in each period.

Each new and used aircraft available in each period is acquired by either an airline or the scrapper.

If an aircraft is acquired by an airline, it will again be available to airlines for purchase in the

following period. If an aircraft is acquired by the scrapper, it is destroyed and leaves the market

forever.

I airlines, indexed i = 1, ..., I, are in the market at each time t. The only non-airline consumer

of aircraft, the scrapper, is indexed by i = 0. In addition, I assume that there are K<I different

groups of airlines indexed k = 1, ..., K that have similar preferences for aircraft. Airlines are

separated into groups by size and the service they provide (e.g. passenger airline or freight airline).

1 4 Benkard (2004) focuses on the strategic interactions of aircraft manufacturers. He uses the estimates of his model

to analyze industry pricing, industry performance, and optimal industry policy. Benkard estimates a static discrete

choice demand system and treats used aircraft as additional differentiated products. 1 5Assuming a price of zero for scrap aircraft is not a significant deviation from reality. New and used aircraft

that are still in use are generally exchanged for millions of dollars, whereas scrappers can typically acquire unwanted

aircraft for less than $100,000.

11
Airline choices at time t are summarized by the vector qit = (qi1t, ..., qiJtt), where qijt is the

quantity of type j aircraft that airline i owns at time t. Airline i may alter qit by buying aircraft,

selling aircraft, and/or scrapping aircraft. There is a finite quantity of type j available at time t, qjt.

Therefore, the set of feasible airline choices are all possible permutations of the vector (qi1t, ..., qiJtt)

such that 0 ≤ qijt ≤ qjt. Airlines choose a fleet of aircraft to maximize the expected discounted

present value of their profit flows.

4.3.1 Profit flow equations

The profit flow generated by airline i at time t is a function of the state vectors sit and ξit. sit

is a vector of observed state variables which include airline i’s fleet choice in the previous period

qit−1 = ¡

qi1t−1, ..., qiJt−1t−1

¢

, aircraft prices Pt = (P1t, ..., PJtt) in the current period, and the time t.

Each qit−1 was determined the previous period, and Pt is determined by equilibrium (the equilibrium

of the model is described in detail below). ξit = (ηt, εi1t, ..., εijt, ..., εiJtt) is a vector of random state

variables, where ηkt is a unobserved airline group-specific random variable which captures the effect

of time trends and shocks on the demand for air-travel,16 and εijt is a unobserved shock to the

match i, j at time t. ξit captures all influences on airline profit not observed by the econometrician.

Agents in the model know the current realization of ξit, but know only the distribution of future

realizations ξit+1, ξit+2, .... A detailed description of the joint distribution of the components of ξit

is given in the estimation section.

Airline i’s profit flow at time t is determined by the attributes of the individual aircraft they

own, scale effects, transaction costs, and adjustment costs. Xjt is a N-dimensional vector of aircraft j’s observed attributes in period t, where the elements of Xjt are limited to variables that are

deterministic and have known paths (e.g. seating capacity and age). The effect of Xjt on airline

i’s profit is a function of a parameter vector representing airline i’s group-specific preferences. Airline profit flow is a quadratic function of the total number of aircraft airline i operates at time t.

The behavior of airlines observed in the data is only individually rational and consistent with the

assumptions of the model if the discounted value of airline profit exhibits decreasing returns to scale

at the relevant margin. Buyers of aircraft incur a transaction cost for each aircraft they purchase.

The transaction cost varies across aircraft make and captures costs associated with reconfiguring

the aircraft, learning to operate the aircraft, and/or forgone warranties or guarantees. Additionally,

airlines incur adjustment costs that are quadratic in the difference between qijt and qijt−1 for all

j. Adjustment costs capture the cost of changing an airlines route structure. Transaction and

1 6 I assume that different groups of airlines have different preferences for different types of aircraft because they

construct their routes in significantly different ways to maximize their ob jective function. Similarly, time-specific

shocks to the demand for air-travel differ across groups of airlines because, for example, freight airlines may be

impacted by the introduction of alternative shipping mechanisms that do not affect passenger airlines.

12
adjustment costs explain the patterns of aircraft purchases and sales observed in the data. Rust

(1985) shows that, in the absence of market frictions, agents replace their entire stock of assets in

every period. As a result, a model of demand for commercial aircraft that ignores market frictions is

likely to over-predict the number of changes airlines make to their fleets in every period. Finally, a

airline group-specific time trend is included to capture the growth in the demand for travel on routes

serviced by wide-body aircraft (and therefore the demand for wide-bodies).17 Differences in rates

of growth across the different groups of airlines reflect differences in strategies employed by airlines

when developing their route structure. In particular, it is possible that one group of airlines has

grown by acquiring more international routes, while another group of airlines has expanded mostly

in the number of domestic routes they operate.

Using the variables described above, airline profit flow is specified as

X

Jt

π (qit, sit, ξit) =

j=1

{qijt (Xjtγk + ηkt + εijt) (1)

+1 (qijt > qijt−1) λm (qijt − qijt−1)}

⎛ ⎞2

−c (qijt − qijt−1)

2 − (qijt − qijt−1) Pjt − δk ⎝X

Jt

qijt⎠ (2)

j=1

Airline i’s preferred scale of operation is shifted by the airline group-specific shock ηkt, and the

idiosyncratic error εijt. γk is a N-dimensional vector measuring the intensity of a group-k airline’s

preferences for the observed attributes of aircraft. λm is a transaction cost incurred by consumers

when purchasing an aircraft. The m subscript indicates that transaction costs may differ across

different aircraft makes. The indicator function 1 (.) equals one if its argument is satisfied and zero

otherwise, so that transaction costs are incurred only if the airline increases the number of type

j aircraft it owns in period t. c measures the penalty an airline pays when making adjustments

its fleet in each period. The penalty is increasing in the magnitude of the changes the airline

makes. Finally δk measures the rate at which marginal returns to a airline from group k’s profit

is decreasing in the number of aircraft the airline operates. γk, λm, c, δk and the parameters of the

joint distributions of ηkt and εijt are estimated.

1 7The average size of airline fleets has grown in each year of the sample. One possible explanation for this growth

is that the market for wide-body commercial aircraft has not yet matured. In particular, the first wide-body aircraft

was delivered in 1969 and new wide-body aircraft have been entering the market ever since, but since aircraft have

useful lives of 25-35 years very few aircraft have been scrapped during the same time period. Additionally, during

the sample period population growth, a growing familiarity with flying, the globalization of the economy, etc. have

caused a steady increase in the demand for air travel.

13
4.3.2 Value Function

Airline i’s value function at time t is the expected discounted present value of the profit flow it

receives in the current and future periods. That is,

( ) X

T

V (qit, sit, ξit) = Et max βs−t

π (qis, sis, ξis) , (3) qit+1,... s=t

where 0 <β< 1 is a common discount factor. The expectation is taken over the joint distribution

of ξit for all future time periods. The value function given by equation (3) can be expressed in

Bellman equation form as

1⁄2 3⁄4

V (qit, sit, ξit) = π (qit, sit, ξit) + βEt max V ¡

qit+1, sit+1, ξit+1¢ (4) qit+1∈C(sit+1)

where the last term on the right-hand side is the expected value of airline i’s best choice in the next

period and C (sit) is airline i’s set of feasible fleet choices given the observed state. The choice set

C (sit) contains only a finite number of choices since at equilibrium prices each airline wants only a

finite number of aircraft. That is, if there is excess demand for a particular type of aircraft at some

price Pjt, Pjt is too low to be an equilibrium price.

4.4 Equilibrium

Conditional on the value functions described above, equilibrium is defined by a aircraft allocation

and prices where no airline wants buy, sell, or trade an aircraft. Each airline maximizes its value

function by acquiring the airline fleet that maximizes its value function given aircraft prices, and

prices are determined by an equilibrium in the actions of all firms. Specifically, airlines’ optimization

problems are solved simultaneously, which implies a mapping between airline quantity choices and

aircraft prices, and the equilibrium quantities and prices of the model are a fixed point to this

mapping. More formally, equilibrium is defined as follows.

Definition 1 At each time t, a vector of aircraft prices P∗ , ..., P∗ .., P∗ and a vector of airline ∗ ¡ ∗ ∗ ∗ ¢

1t jt, . Jtt

quantity choices qt = q11t, ..., qijt, ..., qIJtt are an equilibrium if

∗qit = arg max V (qit, sit, ξit) for all i, (5) qit

and the market clearing conditions

X if P∗ > 0 then q∗ or (6) jt ijt = qjt

X

i

if P∗ = 0 then q∗ jt ijt ≤ qjt

i

are satisfied for all aircraft types j.

14
Equation (5) says that, in equilibrium, all airlines must be maximizing their value function. The

market clearing conditions given by (6) say that if prices are positive the supply of type j aircraft

must exactly equal the demand for type j aircraft, and establishes a lower bound of zero for all

aircraft prices. If the final equation is satisfied with strict inequality, then the excess aircraft are

acquired and destroyed by the scrapper.

4.5 Model Implications

This section presents propositions that assert some properties of the value functions that ultimately

ensure the existence of the equilibrium defined above. The following assumptions are necessary for

the model to have the properties described below.

A1 : The joint distribution of the errors of the model is continuous.

A2 : The common discount factor β is positive, constant over time, and less than one.

Assumption A1 ensures for each vector of aircraft prices, there is a combination of errors ξit =

(ηkt, εi1t, ..., εijt, ..., εiJt) that corresponds to every feasible combination of aircraft the airline could

own, and that the boundaries that separate adjacent fleets are well-defined in terms of the possible

ξit (this point is discussed in greater detail below). Assumption A2 ensures that the value functions

are finite.

Additionally, to prove the propositions of this section, it is useful to define the marginal value

of a type j aircraft to airline i’s fleet as the difference in airline i’s value function with qijt type j

aircraft and airline i’s value function with qijt − 1 type j aircraft, while leaving the quantity of all

other types of aircraft unchanged. Define ej to be a vector with a one in the jth position and zeros

everywhere else. Then the marginal value of a type j aircraft to airline i at time t is expressed

Mijt (qit) = V (qit, sit, ξit) − V (qit − ej , sit, ξit) (7)

= μk + ηkt + Xjtγk + εijt − 1 (qijt > qijt−1) λm − ⎡ ⎛ ⎞ ⎤ X

Jt

(2qijt − 2qijt−1 − 1) c − Pjt − ⎣2 ⎝ qijt⎠ − 1⎦δk

j=1

( £ ¤

)

0 0 +βEξ max V [qit+1, sit+1, ξ] − max V qit+1, sit+1, ξ , qit+1∈Ci(sit+1) q0 ∈Ci(s0 it+1 it+1)

0 0 where sit+1 and qit+1 are respectively the observed state and expected best choice in period t + 1

conditional on airline i choosing qit − ej (instead of qit) at time t. Note that the marginal value of

each type of aircraft to airline i at time t depends on the entire vector of aircraft the airline owns,

qit.

15
It is easy to see that the marginal value of an aircraft is increasing in the group-specific return

shifter μk, the aggregate shock ηkt, and the idiosyncratic airline-aircraft match εijt; and it is also

straight-forward to see that the marginal value of an aircraft is decreasing in its price Pjt and the

non-linear scale effect δk. However, since both current and future profit flows depend on qijt, and qijt

indirectly influences future choices qi1t+1, ..., qijt+1, ..., qiJt+1, it is not as obvious that the marginal

value of an aircraft is decreasing in qijt. The first proposition shows that, given the specification of

the model, the value function is finite at mean values of ξit, and that the marginal value of aircraft

is decreasing in the number of each type of aircraft the airline owns.

Proposition 1 V (qit, sit, 0) is finite, and Mijt (qit, sit, ξit) > Mijt (qit + ej , sit, ξit) for all j, qit, sit,

and ξit.

Proof. All proofs are in Appendix A

Proposition 1 states that the value function is finite for finite values of qit, sit and ξit. Additionally, the fact that the marginal value of aircraft is decreasing implies that, given the observed state vector, sit, and the vector containing all other random components of the model,

ξi−jt = (ηkt, εi1t, ...εij−1t, εij+1t, ..., εiJt), there exists a reservation value εijt

∗ that separates the

higher values of εijt where airline i chooses qijt aircraft j in period t from the lower values of εijt

where i chooses qijt − 1 aircraft j. That is,

for each observed state, sit, and vector of all other random components, ξi−jt = (ηkt, εi1t, ...εij−1t, εij+1t, ..., εiJt),

there exist reservation values εijt, where if εijt > εijt then V qit, sit, ξi−jt, εijt > V qit − ej , sit, ξi−jt, εijt ,

∗ ¡

∗

¢ ¡

∗ ¡

¢

¢ ¡ ¢

and if εijt < εijt then V qit, sit, ξi−jt, εijt < V qit − ej , sit, ξi−jt, εijt for all i, j, t and qit.

This result facilitates the derivation of the likelihood function used to estimate the parameters

of the model. Specifically, the equilibrium conditions of the model can be expressed as inequalities

in terms of the εijt, and the probability of observing an equilibrium outcome are computed by

integrating over the joint distribution of ξit.

Given the number of choices available to airlines in each period, searching over all possible

quantity vectors to find the quantity vector that satisfies equation (5) for each airline requires the

evaluation of V (qit, sit, ξit) at a extremely large number of points. Furthermore, the model must be

solved several times to estimate the parameters. Therefore, searching for equilibrium in this manner

is computationally infeasible. To mitigate some of the computational burden associated with solving

the model, I use the model’s theoretical properties to reduce the number of quantity vectors that

16
must be considered in each period to find equilibrium. The following proposition states that, given

the specification of the model, if an airline’s value function does not increase by choosing any of the

vectors of aircraft that are adjacent to the vector of aircraft it currently holds, then there are no

alternative fleets of aircraft that are an improvement over its current fleet. That is, if an airline

∗ choosing quantity vector qit does not want to add an aircraft to its fleet, subtract an aircraft from

its fleet, or swap one aircraft out of its fleet for another aircraft, there are no alternative quantity

vectors that airline i prefers to q∗

it.

Proposition 2 Given equilibrium prices, Pt

∗,

∗qit = arg max V (qit, sit, ξit) qit

if and only if

M ∗ ijt (qit) > 0 ∀j, (8)

Mijt (q∗ ∀j, (9) it + ej ) < 0

and

M ∗ ∗ ijt (qit) − Mij0t (qit + ej0 − ej ) > (10)

0 ∀j, j0 and 6 j0 j = . (11)

The conditions given by equations (8) and (9) can be thought of as a discrete-choice analog to a

set of first order conditions for airline i’s optimization problem at time t. The last condition states

formally the revealed preference argument that the last aircraft that an airline acquires must be more

beneficial than any alternative aircraft. This condition together with the conditions of Proposition

1 are a discrete-choice analog to the an airline satisfying its second order conditions. An important

implication of Proposition 2 is that, given a vector of equilibrium prices Pt

∗, satisfaction of the

conditions on q∗ given by equations (8), (9), and (10) for all i is both necessary and sufficient for it

∗ ∗ ∗ qt = (q1t, ..., qIt) to be an equilibrium as defined above.

4.5.1 Existence of Equilibrium

Given the assumptions of the model, a equilibrium in aircraft prices and airline quantity vectors

exists. Equilibrium in each period relies on the ability of airlines to compute the discounted expected

value of any fleet of aircraft in the current period. This implies that airlines can compute expected

equilibrium outcomes in future periods given each possible choice it makes in the current period.

17
Equilibrium in the quantities of aircraft airlines choose in each period depend on the appropriate

vector of aircraft prices. That is, given prices, if each airline chooses its best fleet of aircraft, and

the number of aircraft demanded does not equal to the number of aircraft supplied for each type of

aircraft, then the market has not yet reached equilibrium.18 A vector of aircraft prices exists that

are consistent with a equilibrium in aircraft quantities.

The following proposition states formally the assertions of the existence of equilibrium. Proof

of the proposition is by construction. Specifically, I will show that there is a auction mechanism

that generates an allocation of aircraft and a vector of aircraft prices that is an equilibrium with

the properties given in Proposition 2. In each period, aircraft quantities and prices are determined

using the following simple auction mechanism.

To initialize, in each period (t subscripts are suppressed) start with a finite number of aircraft ,qj

, of each aircraft type j available to airlines, set all prices Pj (0) = 0, and order aircraft arbitrarily

from j = 1, ..., J.

Starting with iteration k = 1.

1. Given the current vector of prices P (k − 1), allow airlines to choose their highest valued

airline fleets. This can be done by allowing each airline to choose aircraft one at a time in order of

their highest marginal contribution to the airline’s fleet.

PI PI 2. If i=1 qij (k) = qj , or Pj (k)=0 and i=1 qij (k) < qj , for all j then go to 5. Otherwise,

PI PI for all j, starting with aircraft j = 1, if i=1 qij (k) > qj , increase Pj (k) until i=1 qij (k) = qj (the

fact that the quantity demanded will be reduced each time price is increased follows from Proposition

PI 1). Alternatively, if i=1 qij (k) < qj leave Pj (k) = Pj (k − 1) = 0.

3. P (k) is the new vector of prices.

4. Set k = k + 1 and go to 1.

5. P (k) is the vector of equilibrium prices and q (k) is the vector of equilibrium quantities.

Proposition 3 - Existence Given assumptions A1 and A2 and Proposition 1, a equilibrium in

airline quantity choices and aircraft prices that satisfies the conditions of Proposition 2 exists.

Proof of the proposition follows from the properties of the linear operator

∗

∙ 1⁄2 ¡ ¢3⁄4 ̧ qit = arg max π (qit, sit, ξit) + βEt max V qit+1, sit+1, ξit+1 qit qit+1∈C(sit+1)

1 8The notable exception to this statement is the case of aircraft with a price of zero. When the price of an aircraft

is zero, the demand for that type of aircraft may exceed the supply of that type of aircraft, in which case the excess

aircraft are scrapped.

18
and the pricing mechanism described above, which lead to the four lemmas written below. Proofs

of the lemmas are in the appendix.

Lemma P3-1 The bid prices implied by the algorithm are non-negative and finite.

Prices are non-negative by the assumption that aircraft can always be scrapped for free. The

fact that prices are finite follows from the assumptions that (1) each airline chooses the aircraft that

maximize its value function, (2) there is free disposal of aircraft, and (3) airlines’ value functions are

finite.

Lemma P3-2 If P

i

I

=1 qij (k) = qj , given the vector of prices {P1 (k), ..., Pj (k),

PI Pj+1 (k − 1), ..., PJ (k − 1)}, then i=1 qij (k) ≥ qj given prices {P1 (k), ..., Pj (k),

Pj+1 (k), ..., PJ (k)} , where Pj (k) ≥ Pj (k − 1) ∀j.

Lemma P3-2 asserts that aircraft are gross substitutes for one another. That is, when the

price of one aircraft type goes up the demand for another type will not go down. Type j0 aircraft

contribute to the marginal value of type j aircraft only through its effect on the aggregate number

of aircraft airline i owns. When the number of type j0 aircraft airline i owns is decreased, the total

number of aircraft airline i owns is decreased as well, which cannot decrease the marginal value of

type j aircraft to airline i. This proposition ensures that prices are non-decreasing for all types of

aircraft in successive bidding rounds.

Lemma P3-3 Prices will eventually stop increasing at allocation where PI

i=1 qij (k) = qj and

Pj (k) ≥ 0 or PI

i=1 qij (k) < qj and Pj (k)=0 for all aircraft types j.

Lemmas P3-1 and P3-2 imply that prices are non-decreasing and finite, which leads to the result

given in Lemma P3-3. The following lemma implies that a equilibrium aircraft allocation and prices

will occur in a finite number of bidding rounds, and completes the proof of the proposition. The final

lemma stated below, ties the existence result to the equilibrium conditions described in Proposition

2.

19
Lemma P3-4 The equilibrium allocation and prices found using the above auction algorithm has

the properties described in Proposition 2.

The proof of the final lemma completes the proof of the proposition.

4.5.2 A Note on Uniqueness

Given a starting point, the algorithm described above will always locate the same equilibrium.

However, I have been unable to show that there are not other allocations of aircraft and aircraft

prices that are an equilibrium as defined above that would result if aircraft were ordered differently

to start the algorithm.

5 Model Solution

5.1 Computing the value functions

This section discusses in detail the stochastic algorithm used to solve the dynamic model described

above. A common way of solving dynamic decision problems is to designate a final decision period

and solve the model backwards recursively using the value function equations

V (qit, sit, ξit) = π (qit, sit, ξit) + βEt

∙

max {V (qit+1, sit+1, ξ)}

 ̧

. (12) qit+1∈C(sit+1)

Equation (12) implies that to find its best choice in period t airline i must compute the expected

value of its best choice in period t + 1 for all feasible qit. To facilitate exposition, I define the

following notation

EMAX (qit) = Et

∙

max {V (qit+1, sit+1, ξ)|qit}

 ̧

. qit+1∈C(sit+1)

Computing EMAX (qit) at each possible qit is computationally expensive for two reasons. First,

EMAX (qit) is, in general, a multivariate integral. Therefore, depending on the distribution of the

errors, evaluating EMAX (qit) numerically for a given qit can be difficult or impossible. Second,

EMAX (qit) must be computed for every feasible qit, and for any reasonable specification of the

model, the number of possible qit will be quite large. In addition, the computational burden of solving the dynamic model backwards recursively grows exponentially as the number of decision periods

grows since V (qit+1, sit+1, ξ) relies on the computation of EMAX (qit+1), and V (qit+2, sit+2, ξ) depends on EMAX (qit+2) and so on. In the literature on discrete choice dynamic decision problems,

the increase in the computational burden of solving a dynamic model that is due to an increase

in the number of possible states is known as the ‘curse’ of dimensionality. It is well documented

that the curse of dimensionality can make solving dynamic decision problems infeasible. Given the

20
number of choices that airlines are allowed in each period, solving model presented in this paper

using standard backward solution methods is not possible.

Recent works by Rust (1997) and Pakes and McGuire (2001) develop methods that use simulation

to break the curse of dimensionality. The basic idea of a stochastic algorithm is to compute future

outcomes several times for randomly simulated draws of the errors of the model, and treat the

resulting outcomes as possible realizations of the future. Then the possible realizations of the

future at each state are averaged to approximate the expected value of optimal decisions in the

future. The Pakes and McGuire algorithm has two key features that reduce the computational

burden of solving the dynamic equilibrium model presented in this paper. First, the expected future

value terms given by EMAX(qit) are never actually computed. Instead, the EMAX(qit) terms are

approximated using the average of equilibrium outcomes of the model that occur at simulation draws

of the random components of the model. Second, given an airline’s current guess of the EMAX

term for each state and a draw of the random components of the model, the algorithm solves for

a single sequence of states in periods t = 1, 2, .... Solving the model at only a single sequence of

states has two advantages: (1) each iteration of the algorithm can be performed very quickly, and (2)

the choices that are more likely to occur given the parameters of the model will occur with greater

frequency than other choices, and therefore the approximations of the EMAX (qit) will be more

accurate at the “important” qit. The computational benefits of the stochastic algorithm are not

without costs. The stochastic algorithm is not as precise as backward solution methods. However,

the precision of the approximation of the EMAX terms increases with number of times the model is

solved, and since the number of times the model must be solved to obtain a accurate approximation

of the EMAX terms is not necessarily related to the size of the state space, the stochastic algorithm

may mitigate the curse of dimensionality entirely.

In this application, a vector of approximations of the EMAX(qit) terms are kept in storage and

updated at each iteration of the algorithm. At each iteration of the algorithm, the most recent

approximation of the EMAX(qit) terms and a simulation draw of the random vectors ξ1, ..., ξT are

used to solve for equilibrium in periods t = 1, ..., T. Then the approximations of the EMAX (qit)

at the qit that occurred in equilibrium at times t = 1, ..., T − 1 are updated by averaging the value

functions computed at times t = 2, ..., T with the most recent approximations of the EMAX(qit)

terms. The algorithm can be thought of as “what would happen if airlines, given their best guess

about the future and a draw of the random vectors ξ1, ..., ξT , simultaneously choose an optimal

sequence of airline fleets qi1, ..., qiT .” Then each airline takes the outcome that occurs and averages

the information with its previous knowledge to improve its guess about the future.

As was mentioned above, the equilibrium of the model must be computed several times at each

21
time t to get accurate approximations of the EMAX terms, and in this research, the model’s

equilibrium also found by an iterative procedure. That is, the model solution algorithm is a nested

algorithm, where the “inner” loop of the algorithm takes airlines’ most recent approximations of

the EMAX terms at time t and a draw of the random components of the model and solves for

equilibrium, and the “outer” loop of the algorithm takes the value functions from the solution to

the inner loop and averages them with previous outcomes to update airlines’ guesses of the EMAX

terms. The nested algorithm converges when airlines’ approximations of the EMAX terms are

changing within some prespecified tolerance level. The next section discusses the algorithm used

to solve for equilibrium at each time t. Appendix B provides additional details of both the “inner”

loop and the “outer” loop of the nested algorithm used to solve the model.

5.2 Solving for Equilibrium

Given the most recent approximation of the EMAX terms and a draw of the random components

ξ1, ..., ξT , the equilibrium of the model at each time t is found using a algorithm very similar to

the simple auction mechanism described just before Proposition 3 above. Bertsekas, Castanon, and

Tsaknakis (1993) prove that there exists a forward and reverse auction algorithm that is equivalent

to the above algorithm, and arrives at equilibrium much more quickly. The state at time t is a

function of the equilibrium outcome at time t − 1, and the state at time t + 1 is a function of

the equilibrium outcome at time t, etc. In the algorithm, airlines increase their bids to obtain the

aircraft they desire the most (this is called a forward step), and then, when necessary, the price of

an aircraft that is dropped by its owner may be reduced to attract a new owner (this is called a

reverse step). The details of how the bids for aircraft are determined and equilibrium is found using

the forward and reverse auction algorithm are relegated to Appendix B.

6 Estimation

The parameters of the structural model are estimated by maximum simulated likelihood estimation

using data on observed airline ownership choices from Back Aviation Solutions and prices from

Avmark Incorporated. Construction of the likelihood function follows directly from the solution of

the model and the distributions assumed for the structural errors. Simulation methods are then

used to evaluate the high dimensional integrals of the likelihood function.

The first part of this section defines the error distributions assumed for ηkt and εijt. Next, I

provide a informal discussion of how the data identifies the parameters of the structural model.

Then, the outcome probabilities that compose the likelihood function are derived. Finally, I discuss

22
the Maximum Simulated Likelihood estimation algorithm used to estimate the parameters of the

model.

6.1 Error distributions

ηkt a common shock that influences the average size of group-k airlines’ fleets at time t. ηkt can

be attributed to growth in the economy, war, the threat of terror, etc. As was mentioned above,

different groups of airlines may have different growth rates in the segment of their fleet operated

by wide-bodied aircraft because of the way they’ve chosen to structure their routes. In this initial

specification of the model, I assume

ηkt = αkt + φD1986,t,

where the αk are estimated as fixed parameters, and D1986,t is a dummy variable that equals one if

the purchase took place in or after 1986 so that φ captures any additional effects the Tax Reform Act

of 1986 may have had on airlines’ investment behavior. Alternative distributional assumptions, such

as the addition of a aggregate random effect, could very easily be employed in future specifications.

εijt is an idiosyncratic shock to the productivity of a particular airline-aircraft match in time t. I

assume that

εijt ∼ iidN (0, σε)

across airlines, aircraft and time.

6.2 Identification

I now provide an informal discussion of how variation in transaction and price data identify the

parameters of the structural model. The set of parameters to be estimated is

θ = ¡

μ1, ..., μK, γ1, ..., γK, α, δ1, ..., δK, λ1, ..., λ4, c, σ2

ε, α1, ..., αK, φ¢

and the data used to estimate the parameters is given by

{q111, ..., qijt, ..., qIJT ; X11, ..., Xjt, ..., XJT ;

P11, ..., Pjt, ..., PJT ; 1, ..., t, ..., T } .

Referring to the profit flow equation (1), the αk are identified by within-group variation in the

average size of airline fleets over time, while the airline specific parameter μk is identified by variation

in fleet-size across airlines within each time period, φ is identified by variation in the composition of

fleets before and after the Tax Reform Act of 1986. The parameters δk measure the rate at which

marginal returns to aircraft is decreasing and is identified by variation in fleet size. The airline-type

23
preference parameters γk are identified by covariation in the observed characteristics of aircraft,

ownership choices of different types of airlines, and aircraft prices. The parameters λm measure

transaction costs, and are identified by variation across different aircraft makes in the length of time

an aircraft is held between sales. c, which measures adjustment costs is identified by covariation

in the number of adjustments an airlines make to a particular model of aircraft in its fleet and the

overall number of changes airlines make to their fleet in each period.

The discount factor β is weakly identified by changes in investment patterns over time since it

is assumed that airline preferences do not change across time periods. However, I do not attempt

to identify β, but instead assume it is constant and equal to 0.9. Transaction costs and adjustment

costs are separately identified non-parametrically, the assumptions that transaction costs enter the

profit flow equation linearly and adjustment costs enter quadratically are made to capture the facts

that purchasing new aircraft is costly, but making large adjustments to the composition of an airline

fleet requires the airline to acquire or sell gates and routes to accommodate its new fleet structure.

6.3 The Likelihood Function

The estimation method assumes that observed prices and quantities are consistent with the definition

of the market equilibrium given by equations (5) and (6). Given aircraft prices and quantities, the

likelihood of the observed outcome in each period can be expressed in terms of the necessary and

sufficient conditions for marginal aircraft given in Proposition 2. As can be seen in the profit

flow equation (1), airline profit flow is additively separable in the errors εijt, i.e. π (qit, sit, εit) =

P π (qit, sit) + j qijtεijt. Therefore, the marginal value of aircraft j to airline i’s fleet, given by

equation (7) is also additively separable in εijt and can be expressed Mijt (qit) = Mijt (qit) + εijt.

The following terms are defined to facilitate the construction of the outcome probabilities that are

used to derive the likelihood function:

∆+

ijt ≡ −Mijt (qit + ej ), (13)

∆−

ijt ≡ −Mijt (qit) and (14)

∆ij0jt ≡ Mijt (qit) − Mij0t (qit + ej0 − ej ). (15)

Using the definitions (13), (14) and (15), the conditions given by equations (8), (9) and (10) in

24
Proposition 2 can be rewritten in terms of the errors εijt as

∆− εijt > ijt, (16)

∆+ εijt < ijt, (17)

εijt < ∆ijj0t + εij0t for all i, j, j0 and t. (18)

Ordering aircraft from (1) to (J) (where the parentheses indicate that an aircraft’s type index j is

not necessarily the same as its position in the ordering of aircraft)19 Combining the conditions given

by equations (16), (17), and (18), the upper and lower bounds of each εijt are given by

∆+ bijt = min ¡

ijt, εi1t + ∆ij1t, ..., εij−1t + ∆ijj−1t, (19)

εij+1t + ∆ijj+1t, ..., εiJt + ∆ijJt) and

= max ¡

∆− b , εi1t − ∆i1jt, ..., εij−1t − ∆ij−1jt, ijt ijt

εij+1t − ∆ij+1jjt, ..., εiJt − ∆iJjt).

The ∗ ∗ ∗ probability that a vector of airline quantity choices qit = (qi1t, ..., qiJt) are optimal for airline i

at time t can be expressed in terms of the errors εi(j)t, and the bounds bi(j)t and bi(j)t as

£ ¤ ∗ ∗ Qit = Pr bijt (qit) < εijt < bijt (qit) ∀j ,

which, given that the εijt are iidN ¡

0, σ2¢

ε , can be written

Z Z Y

J μ ¶ ¡ ¢ ∗ ∗ 1 εijt Qit = ··· 1 bijt (qit) < εijt < bijt (qit) σ2

ε

φ σε

dεijt, (20)

j=1

where 1 (.) is an indicator variable that equals one if its argument is satisfied and zero otherwise,

and φ is the standard normal pdf.

Since the errors εijt are independent across airlines the log-likelihood of observing a sequence

of equilibrium outcomes in periods t = 1, ..., T, can be calculated as the sum over i of the natural

logarithm of the outcome probabilities given by equation (20), or

XX

T I

L(θ) = ln [Qit] , (21)

t=1 i=1

where θ = (μk, γk, σε, λm, δk, c, αk, φ) is the vector of parameters to be estimated.

6.4 Maximum Simulated Likelihood

A frequency simulator of Qit as it is expressed in (20) can be written

R J 1 X Y ¡ ¢ b ∗ ∗ Qit (θ) = 1 bijt (q it) < εr it, εr , it, εr ijt < bijt (q it) R r=1 j=1

1 9The method used to order aircraft is discussed in detail in the appendix that describes the importance sampling

algorithm used to compute the integrals of the likelihood function.

25
where εr = (εr , ..., εr ) is a vector of simulation draws from the iidN (0, σε) distribution. The it i1t iJt

model can be estimated using a frequency simulator, but there are a few significant costs to doing

so. First, given the way the model is constructed it is likely (especially for early guesses of the

parameters) that many vectors of random simulation draws of the errors will be inconsistent with

equilibrium, and therefore many draws of the errors may be necessary to produce an accurate

simulator of Qit. A second problem with a frequency simulator is that it is not continuous in

the parameters of the structural model. That is, for changes in the parameters of the model the

frequency simulator of outcome probabilities may make discrete jumps.

To avoid the problems associated with using a frequency simulator, I construct a importance

sampling simulator that focuses its attention on the regions of the error distribution that are consistent with equilibrium, and is continuous in the parameters of the model. The simulator developed

in this research is similar to the well known GHK simulator (Geweke (1991), Hajivassiliou and McFadden (1990), and Keane (1990)). Details of how Qit is simulated, and properties of the simulator

are given in Appendix C.

The maximum simulated likelihood estimator replaces Qit with the simulated probability Qbit, to

get

XX

T I

3  ́

Lb (θ) = ln Qbit . (22)

t=1 i=1

maximum simulated likelihood estimators are consistent but biased for a finite number of simulation

3  ́ 3  ́ draws because although E Qbit = Qit, E ln Qbit = ln 6 Qit. Despite the biased evaluation of

ln (Qit), Borsch-Supan and Hajivassilou (1993) use Monte Carlo experiments to show that maximum

simulated likelihood estimators that use importance simulation techniques, such as the well known

GHK simulator or the Stern (1994) simulator, perform well relative to alternative estimators.

7 Results and Model Fit

This section presents parameter estimates for the structural model developed in Section 6. After

the parameter estimates are discussed, the structural model’s fit to the data is analyzed. In addition

to performing chi-square goodness-of-fit tests, several figures that illustrate the model’s ability fit to

the data are provided.

7.1 Parameter estimates

Estimation results are given on Tables 3 and 4. Recall that aircraft prices are measured in the

natural log of real U.S. dollars. Therefore, the impact of all parameters on airlines’ value functions

are either discussed in terms of the percentage impact they have on the value of airline fleets or

26
are translated into real dollars. The first set of parameter estimates, presented on Table 3, are

parameters that are common across the different types of airlines, including parameters measuring

industry-wide time trends and market frictions. Table 4 presents parameters that determine an

airlines’ preferred scale of operation and fleet composition. The standard errors of the parameter

estimates are in parentheses. Almost all of the parameter estimates differ from zero at a high level

of statistical significance.20

Referring to Table 3, φ is a dummy variable that equals one in periods after 1986.21 The estimate

of φ does not significantly differ from zero, which indicates that there is not a significant difference

in the rate of airline growth before and after 1986. This does not necessarily imply that the Tax

Reform Act of 1986 had no effect on investment, but if the tax change did discourage investment in

new aircraft, as is commonly believed, the dampening effect was offset by other unobserved factors.

The effects of the Tax Reform Act of 1986 are analyzed in greater detail in the next section.

The standard deviation of the airline-aircraft-time specific error, σε, has a estimate of 3.0268,

which translates into roughly $21 million. This amount is a little less than the average transaction

price of aircraft sold during the sample period.

Even though the composition of airline fleets changes over time, there is a significant amount of

persistence in aircraft ownership. In fact, it is not unusual for an aircraft to stay with the same owner

for more than 15 years. The λ parameters, which measure transaction costs incurred by buyers

when a sale is made, and c, which measures the cost to an airline of adjusting the composition

of its fleet, capture the persistence in aircraft ownership observed in the data. The magnitude

of the estimates of the cost parameters imply that there are significant frictions in the market for

commercial aircraft. The estimates of the λ are similar across the different makes of aircraft, and the

estimates indicate that when a airline purchases a new $100 million aircraft, the cost of reconfiguring

the aircraft, learning to operate the aircraft, etc. are about $62 million. Adjustment costs, c, on

the other hand, multiply the sum of the squares of the differences from one period to the next in

the number of aircraft of each type an airline owns. Therefore, if an airline increases or decreases

the number of a particular type of aircraft it operates by only one unit from one period to the next,

the marginal value of each aircraft of that type decreases by approximately 13 percent. However, if

the increase or decrease in the quantity of aircraft the airline owns is larger, for example four units,

the marginal value of each aircraft of that type is decreased by more than 50 percent.

Heterogeneity in airlines’ choices of route structures is reflected in their preferred scales of op2 0Typically, t-statistics in non-linear structural models are quite large. See, for example, Rust (1987) or Brien,

Lillard and Stern (2005). 2 1There was a major change in the U.S. tax code in 1986, which is believed to have affected investment in captital

goods like aircraft. This tax reform, known as the Tax Reform Act of 1986, is discussed in greater detail in the next

section.

27
eration and tastes for the different types of aircraft. For example, some airlines choose to offer

a large number of routes to Europe, while others focus their attention on routes that service Asia

or the Middle East, and the different route structures require different quantities of aircraft with

different attributes. There are three types of airlines considered in this work, large U.S. passenger

airlines,22 small U.S. passenger airlines,23 and all other airlines. The three types are referred to

as Large U.S. Airlines, Small U.S. Airlines, and “Other” Airlines hereafter. Referring to Table

4, the parameters αk multiply a linear time trend, where 1978 is period one, 1979 is period two,

etc. In the profit flow equation (1), the interaction αkt multiplies the total number of aircraft an

airline operates in the current period, so that the marginal value of each aircraft a airline operates is

increased by αkt at time t. For Large U.S. Airlines, the coefficient estimate of 0.2663 implies that,

ceterus paribus, the marginal value of aircraft to airlines increased approximately 26.6% percent in

each period. For Small U.S. Airlines and “Other" Airlines the increase in value aircraft value is

much smaller, approximately 7.3% and 9.5% respectively. Obviously, this increase in aircraft value

over time does not translate directly into a increase in the market value of aircraft because the effect

is mitigated to some degree by a contemporaneous increase in the average size of airline fleets.

Table 4 also shows that the different types of airlines also have significantly different preferences

for scale of operation and the types of aircraft they prefer to operate. The parameter δk, which

captures the rate of decreasing returns for each of the different types of airlines, enters the airlines

profit flow equation negatively and multiplies the square of the total number of aircraft the airline

owns. Therefore, if, for example, a Large U.S. Airline owns 100 wide-bodied aircraft, the marginal

value of each of the aircraft the airline owns is decreased by 1.95. Translated into millions of U.S.

dollars, if the marginal value of an aircraft in the airline’s fleet is $100 million without scale effects,

its marginal value is about $14 million after imposing the effect δk. The scale effects are much more

severe for Small U.S. Airlines. For example, a Small U.S. Airline would only have to own 18 and 19

aircraft for the marginal value of each of the aircraft it owns decreased by the same amount, 2.434.

Naturally, “Other” Airlines, which encompass all other airlines in the world, have scale effects that

are much less severe. “Other” Airlines could own about 10,200 aircraft before the marginal value

of each of the aircraft is decreased by 1.95.

The parameters γ1 −γ3 capture each airline’s relative preference for different vintages of aircraft.

Aircraft manufactured prior to 1978 are the comparison group. Small U.S. airlines place about

the same value on a newer aircraft as they do on a comparably equipped older model. Large U.S.

Airlines, on the other hand, value newer aircraft more than comparably equipped older aircraft. For

example, if the marginal value of a aircraft in the oldest category is $25 million to a Large U.S. Airline,

2 2The large U.S. scheduled airlines are Delta, United, and American. 2 3The small U.S. passenger airlines are TWA, Eastern, Northwest, Continental, Piedmont, and U.S. Air.

28
then, ceterus paribus, the marginal values of a comparably equipped aircraft manufactured from

1978-1984, 1985-1991, and 1992-1997 are $29.6 million, $30.8 million, and $32.4 million respectively.

Even more striking is the premium airlines in other parts of the world place on newer aircraft. If

the marginal value of a aircraft to “Other” Airlines in the oldest category is $25 million, then ceterus

paribus the marginal values of a comparably equipped aircraft manufactured from 1978-1984, 1985-

1991, and 1992-1997 are $55 million, $64.5 million, and $222.6 million respectively. The estimates

of the vintage parameters reported above are consistent with the delivery trends displayed in Figure

3. Specifically, as can be seen in the figure, the trend in the number of new aircraft delivered to

airlines in the U.S. is relatively flat, while the trend in new aircraft deliveries to airlines in other

parts of the world is upward sloping. Therefore, a greater proportion of new aircraft are delivered

to “Other” Airlines as time goes on, which is consistent with their relatively high preference for the

newer generations of aircraft.

The parameters γ4 − γ9 measure airlines’ relative preferences for different models of aircraft. It

is obvious by looking at the estimates that the different types of airlines prefer different models of

aircraft. Large U.S. Airlines have a relatively high preference for the Boeing 767 and the McDonnellDouglas DC-10; while Small U.S. Airlines prefer the Boeing 747, the DC-10, and the Lockheed L1011; and “Other” Airlines have a strong preference for Airbus aircraft. Comparing preferences for

the same model of aircraft across the different types of airlines is complicated by airlines’ varying

trends in growth over time, scale effects, and preferences for different vintages of aircraft. For

example, neglecting the airline’s scale, transaction price, transaction costs, and adjustment costs,

the value of adding a Boeing 767 produced in the period 1978-1984 to its fleet in 1997 is 5.16 to a

Large U.S. Airline’s value function, while the same model and vintage adds 4.39 to the value function

of “Other” Airlines . In contrast, the same model produced in the period 1992-1997 adds 5.25 to

a Large U.S. Airline’s value function, and 5.79 to the value function of “Other” Airlines in 1997.

The parameter estimates of airline preferences are broadly consistent with the relative popularity of

each model of aircraft (see Table 1). For example, the fact that the Boeing 747 and 767 aircraft

have been owned and operated with great frequency world-wide is consistent with airlines’ relatively

high preference for these models, while the fact that Douglas DC-10 is very popular among U.S.

airlines is also supported by the parameter estimates. The Douglas DC-10 and the Lockheed L-1011

have similar technical specifications, but according to the parameter estimates, the DC-10 was much

more popular among the big three U.S. carriers, Delta, United, and American, while the L-1011 was

popular mainly among the Small U.S. carriers. The fact that Large U.S. and international carriers

have grown at a faster rate than have Small U.S. carriers provides one possible factor in the eventual

demise of the Lockheed L-1011.

29
7.2 Model fit

In this subsection, I test how well the equilibrium quantities of aircraft generated by the model

match those observed in the data. After discussing the statistical significance of the model’s fit, I

provide a brief discussion of some figures that show that even though the predictions of the model

are often rejected by the data statistically, the model does a reasonable job of predicting airline

decisions.

The first test statistic is computed using the total number of aircraft owned by each type of

airline in each period. There are K types of airlines that can possibly acquire aircraft in each

o period. Let qkt be the observed quantity of aircraft owned by type k airlines at time t, and let qkt

be the quantity of aircraft owned by type k airlines at time t predicted by the model. The test

statistic

X

K o (qkt − qkt)2

Wt = (23) qkt k=1

has a chi-square distribution with K − 1 degrees of freedom.

Table 5 presents one chi-square statistic, Wt, for each of the twenty time periods 1978-1997. The

null hypothesis is that the quantities predicted by the model are equal to the quantities observed

in the data. Since there are only three types of airlines, each statistic Wt given in equation (23) is

distributed χ2 (2). Therefore, if Wt is above the critical value of 5.99 the data rejects the model at

the 5 percent level of significance. As can be seen in the table, the null hypothesis is rejected in

sixteen of the twenty time periods.

Next, I compute the same test statistic disaggregated by model. There are M models of aircraft

available at time t. Therefore, I compute M ∗ T test statistics

K o − qkmt) X

2 (q Wmt = kmt

qkmt k=1

each of which has a chi-square distribution with K − 1 degrees of freedom.

In the interest of brevity, I report the results for only one model of aircraft, the Boeing 747.

Similarly to Table 5, Table 6 presents twenty chi-square statistics. And again, since there are only

three types of airlines, each statistic Wmt given in equation (23) is distributed χ2 (2). Similarly to

the results discussed above, the null hypothesis that the simulated and observed quantities are the

same is rejected in thirteen of the twenty time periods.

Figures 4-6 show that even though the model is often rejected by the data in statistical tests, the

quantities predicted by the model are qualitatively similar to the quantities observed in the data.

Figure 4 shows the predicted and observed values of the total quantity of aircraft operated by each

type of airline in each period. As can be seen in the figure, because the trend in growth in airline

30
fleets is assumed to be linear, the predicts a very flat trend of growth fro Small U.S. airlines, and

therefore under-predicts the number of aircraft owned in the first several periods and over-predicts

the number of aircraft owned in the last few periods. Finally, the model tends to fit the quantity

choices of Large U.S. and “Other” Airlines quite well.

Figures 5 and 6 show the predicted and observed quantities owned in each time period of the

Boeing 747 and the Douglas DC-10 respectively. Unlike Figure 4, the model’s mistakes in predicting

the ownership behavior of Small U.S. airlines observed in Figures 5 and 6 is a little more sporadic.

For both models of aircraft, the models predictions follow a over-under-over-under pattern over the

sample period. Again, the model tends to fit the quantity choices of Large U.S. and “Other” Airlines

quite well.

8 Counterfactual Simulations

Structural modelling and estimation provides the researcher with the necessary tools for performing

experiments to answer hypothetical questions about agent behavior and potential market outcomes.

The estimated parameters are primitives to a economic model of agent behavior, which allows the

researcher to evaluate the total effect of a counterfactual change in policy or market structure as

opposed to relying on analyses that are based on nonstructural estimation. For example, one might

take a positive parameter estimate on the dummy variable “1986” interacted with age in a log-price

regression to mean that the relative (to a new aircraft) value of a ten year-old aircraft increased ten

percent after the tax reform. However, this conclusion is unlikely to be correct because the tax

change impacted the market equilibrium, and the price regression captures only the partial effect of

the tax change on prices (holding quantities constant). In contrast, the counterfactual experiments

performed below analyze the equilibrium changes in quantity and price that occur given a change

in tax policy. In the remainder of the section, the parameter estimates of the model are used to

investigate the effects on the market equilibrium of (1) instituting a 10 percent investment tax credit

like the one repealed by the Tax Reform Act of 1986, and (2) imposing stricter safety and/or noise

abatement policy.

8.1 Tax reform

Prior to 1986 airlines received a tax credit on purchases of new aircraft equal to ten percent of the

aircraft’s purchase price.24 The investment tax credit was designed to increase investment in new

2 4There were several other reforms made to the corporate tax code in 1986. Most noteably, the corporate tax rate

was reduced from 46% to 35%.

31
durable assets. This would suggest that the elimination of the tax credit would cause investment

in new aircraft to slow. In addition, since new and used aircraft are substitutes, one would expect

holdings of used aircraft to increase more than they otherwise would have when the investment tax

credit was removed. To assess the effects of reinstituting the investment tax credit repealed by the

Tax Reform Act of 1986, I use the parameter estimates of the model to compare airline quantity

choices five periods into the future under two regimes: (1) no change in tax policy, and (2) the

implementation of a investment tax credit of 10 percent paid to U.S. airlines on the purchase of new

aircraft.

Table 7 summarizes the results of the counterfactual experiment. The tax credit induces only

small increases in total aircraft ownership among U.S. airlines and induces small decreases in aircraft ownership among “Other” Airlines. Looking at airlines’ aircraft ownership choices across the

different vintages of aircraft, one change stands out. U.S. airlines increase the number of 1978-1991

vintage aircraft that they own, and the “Other” Airlines reduce the number of aircraft they own

of the same vintage. This result, together with the slight increase in overall ownership by U.S.

airlines, implies substitution effects between new and used aircraft are small, and that the tax credit

produces a small increase in the relative purchasing power of U.S. airlines.

Large U.S. airlines have an average price elasticity of demand of approximately -0.75 for 1998-

2002 vintage aircraft. The inelastic demand implied by the structural model is consistent with

previous literature on capital good investment (see for example Auerbach and Hassett (1990, 1992)).

However, perhaps due to the specification of a dynamic model and the explicit inclusion of markets

for used aircraft, the elasticities implied by the experiment are quantitatively larger than previous

studies.

8.2 Safety restrictions and noise abatement

Some industry experts claim that increasingly strict safety, noise abatement and environmental

policies have greatly influenced the development and implementation of technological advances in

aircraft. From collision avoidance equipment and stronger fuselage materials to quieter engines,

research and development that improves an aircraft’s safety and reduces its noise output impacts

almost every component of an aircraft, and older aircraft are more likely to not meet increasingly

strict safety or noise abatement policies than are new aircraft. And since policies have continued

to get more strict,25 airlines have been forced to overhaul or scrap their older aircraft.

The counterfactual experiment presented here evaluates the hypothesis that a policy that forces

airlines to modernize older aircraft alters the equilibrium quantities of all new and used aircraft

2 5There have been four increasingly strict noise abatement policies instituted in the U.S. in the past 20 years.

Source - U.S. Department of Transportation.

32
airlines own. To imitate the effect of stricter safety, noise, and environmental policies, I enforce a

hypothetical policy that in order for airlines to continue operating aircraft over 19 years of age they

must pay a one time $2 million fee. The fee mimics hushkitting and other improvements that can

be made to modernize older aircraft.26

Table 8 gives the results of the experiment, in which all aircraft over the age of 19 must be

modernized. It is evident from the table that the policy decreases the number of older aircraft

operated by U.S. airlines, but slightly increases the number of older aircraft operated by “Other"

airlines. By the end of five years of the policy, the number of the oldest vintage of aircraft operated by

U.S. Airlines has been reduced by approximately 4 percent, and the “Other” Airlines have increased

the quantity of aircraft they operate of the same vintage by under 2 percent. This result indicates

that new and used aircraft are closer substitutes for U.S. airlines than they are for airlines in other

parts of the world.

Ownership of aircraft manufactured in the period 1978-1984 declines not only because of the

actual implementation of the policy but also because of airlines’ anticipate the policy and know the

aircraft will soon be worth less in the secondary market. Therefore, airlines that would, in the

absence of the policy, sell older aircraft in the secondary market at a higher price, are now induced

to sell or scrap the aircraft a few periods in advance of when they otherwise would. Large U.S.

Airlines even start to sell or scrap 1985-1991 vintage aircraft in 2002 in anticipation of their future

mandated hushkitting. The “Other” Airlines take advantage of the U.S. Airlines’ fire sale of older

aircraft and increase their ownership of 1985-1991 vintage aircraft.

Small U.S. Airlines partially replace the older aircraft they scrap by increasing the number of

newer aircraft, those manufactured after 1985-1997, they own. Large U.S. Airlines, on the other

hand, replace most of the older aircraft with aircraft produced 1992-1997, and finally, the “Other”

Airlines, with a few exceptions, actually decrease the number of newer versions of aircraft they own.

It is interesting to note that as a result of the policy, all airlines reduce the number of the newest

vintage of aircraft they own. This indicates that new aircraft are the worst substitutes for aircraft

produced before 1985.

9 Conclusions

This article studied purchasing and selling behavior in primary and secondary markets for commercial

aircraft. The focus of this work was to develop a framework for analyzing demand behavior that

allowed for both intertemporal dependence and intratemporal dependence in markets for durable

2 6Hushkits for widebody aircraft typically run $500,000 to $4 million - source Metropolitan Airports Commission

in cooperatioin with ACI - NA.

33
goods. The data used to estimate the structural model came from two private sources. The

data on aircraft ownership and aircraft characteristics came from Back Aviation Solutions, and the

transaction price data came from Avmark Incorporated.

Data on the locations (owners) of aircraft in the current period, the locations of aircraft in the

previous period, and the market price of aircraft in the current period were used to identify the

parameters of the model. Airline value functions and the equilibrium conditions of the model were

used to derive the likelihood of events observed in the data as a function of the parameters of the

model. The model’s specification allowed airlines to have different preferences for scale of operation

and the age and model of aircraft owned.

The parameter estimates of the structural model indicated that there are significant transaction and adjustment costs that affect transaction patterns in primary and secondary markets for

commercial aircraft. These costs explain the observed persistence in aircraft ownership over time.

Differences in the length, frequency, and passenger volume of the routes the airlines choose to service are reflected in the heterogeneity in airline preferences for the size and composition of their

airline fleets. The estimated model showed that Boeing aircraft are relatively popular to all types

of airlines, while Douglas and Lockheed models mainly appeal to U.S. airlines, and Airbus aircraft

are most popular among European and Asian airlines.

The parameters of the structural model were used to perform two counterfactual experiments.

First, the impact of reinstituting a 10 percent income tax credit on the purchase of new aircraft by

U.S. airlines was analyzed. The hypothetical tax credit used in the experiment was identical to

the one that was removed by the Tax Reform Act of 1986. The results of the experiment showed

that the implementation of the tax credit would have a small impact on equilibrium quantities.

Specifically, there would be only a small amount of growth in the size airline fleets and a small

amount of substitution toward new aircraft by U.S. airlines as a result of the policy. In addition,

the estimated demand elasticities averaged about -.75 for Large U.S. airlines, which is slightly higher

than elasticities computed in previous studies of markets for durable capital goods.

Next, I performed an experiment illustrating the effects of stricter noise abatement and/or safety

policies. The results of the experiments showed that mandating the modernization of older aircraft

reduces the size of the number of older aircraft operated by U.S. airlines. Interestingly, airlines in

other parts of the world actually increase the number of older aircraft they operate as a result of

the policy indicating that new and used aircraft are closer substitutes to U.S. airlines.

In future work in markets where competing firms purchase durable assets, I plan to develop a

model that allows for more strategic interaction the buyers. Firms that buy capital and labor often

appear to compete with other firms by buying or maintaining ownership of assets to keep them from

34
the competition. The results of such behavior, like buying a store-front and leaving it closed to

keep a competitor from reaping the benefits of owning it, are pure rent-seeking. Firms that own

durable goods may engage in similar behavior by buying or maintaining ownership of a good and

storing it so that their competitors cannot use it. By developing a model of behavior that allows

for this type of strategic behavior by firms, I hope to identify the social costs of such behavior.

In addition, this research develops methodologies for estimating a model of dynamic - equilibrium

purchasing and selling behavior in the market for commercial aircraft. However, the modelling

and estimation techniques can be applied much more broadly. Commercial and residential real

estate markets are comprised of a limited number of durable assets that are often sold several

times and appreciate in value over time. Rare consumer durables (e.g. paintings) are also likely

candidates for the methodologies developed in this research. In addition, labor markets that contain

a relatively small supply of workers with unique talents, like athletes in professional sports leagues

or college professors, would be an intriguing application of the above methodologies. Each of the

aforementioned markets are characterized by goods or labor that are useful for several periods, and

purchase and sales decisions of each potential owner that depend on the purchase and sales decisions

of other owners in the market. The methodologies developed in this work can be used to estimate

models of behavior in these and similar markets, and the estimated models can be used to analyze

potential changes in economic policy or market structure.

35
A Proofs

Proof of Proposition 1.

X

Jt

V (qit, sit, 0) =

j=1

{qijtXjtγk + ηkt (24)

−1 (qijt > qijt−1) λm (qijt − qijt−1)}

⎛ ⎞2

X

Jt

−c (qijt − qijt−1)

2 − (qijt − qijt−1) Pjt − δk ⎝ qijt⎠

j=1

∙

i, sit+1 (qit), ξ)} 0

 ̧

+βEt max qi

0∈C(sit+1(qit)) {V (q

Each of the terms in the first three lines of equation (24) are finite. Next, consider the final term

∙  ̧

(25) 0

i, sit+1 (qit), ξ)}

, s (q ), ξ)} f (ξ) dξ, i it+1 it 0

Et max {V (q q0∈C(sit+1(qit))

Z iZ

= ··· max {V (q q0∈C(sit+1(qit)) i

where f (.) is the joint density of ξit+1, which is the product of independent normal densities. Since

i, sit+1 (qit), ξ)} grows linearly in each element of ξit+1

faster than exponentially in each element of ξit+1, equation (25) is finite. Thus, V (qit, sit, 0) is also

finite.

0

Now, assume there exists a time t0 after which airlines make the same choice in every period,

implying

V (qit0 , sit0 , ξit0 ) = π (qit0 , sit0 , ξit0 ) + B

⎛ ⎞ ⎛ ⎞2

X

Jt X

Jt

= (μk + ηkt0 ) ⎝ qijt0⎠ − δk ⎝ qijt0⎠ +

j=1 j=1

X

Jt h

qijt0 (Xjt0γk + εijt0 ) − c (qijt0 − qijt0−1)

2

j=1

− (qijt0 − qijt0−1) Pjt0 ] + B,

X

Jt

V (qit0 , sit0 , ξit0 ) = {qijt0 (Xjt0γk + ηkt0 + εijt0 ) (26)

j=1

−1 (qijt0 > qijt0−1) λm (qijt0 − qijt0−1)}

⎛ ⎞2

X

Jt

−c (qijt0 − qijt0−1)

2 − (qijt0 − qijt0−1) Pjt − δk ⎝ qijt0⎠ + B.

j=1

max 0 q ∈C(sit+1(qit)) {V (q i and f (.) declines

36
Note that

M (qit0 + ej , sit0 , ξit0 ) − M (qit0 , sit0 , ξit0 ) = [1 (qijt0 + 1 > qijt0−1) − 1 (qijt0 > qijt0−1)] λm − 2 (δk + c)

≤ −1 (qijt0 > qijt0−1) λm − 2 (δk + c) < 0,

and π (qit0 , sit0 , ξit0 ) is concave in qijt0−1 since

∂2π (qit0 , sit0 , ξit0 ) = −2c. ∂2qijt0−1

Moving backward to period t

0 − 1,

¡ ¢ X

Jt © ¡ ¢ V qit0−1, sit0−1, ξit0−1 = qijt0−1 Xjt0γk + μk + ηkt0−1 + εijt0−1 (27)

j=1

−1 (qijt0−1 > qijt0−2) λm (qijt0−1 − qijt0−2)}

−c (qijt0−1 − qijt0−2)

2 − (qijt0−1 − qijt0−2) Pjt − ⎛ ⎞2

" # X

Jt

δk ⎝ ⎠ + βEt0−1 max π (qit0 , sit0 , ξit qijt0−1 0 )

j=1 qit0∈C(qit0−1)

and

⎡

M ¡qit0−1 + ej , sit0−1, ξit0−1

¢

− M ¡qit0−1, sit0−1, ξit0−1

¢

≤ −1 (qijt0−1 > qijt0−2) λm − 2 (δk + c) +

⎤

⎥

⎥

⎥

⎦

it0∈C(qit0−1+ej ) π (q max 0 0 q0 it0 , sit0 (qit0−1 + ej ), ξit0 ) +

00 00 it0 , sit0 (qit βEt0−1 0−1 −

⎢

⎢

⎢

⎣

maxq

3it0∈C(qit0−1−ej ) 00 π (q ej ), ξit

 ́

0 ) −

2 maxqit0∈C(qit0−1) π (qit0 , sit0 (qit0−1), ξit0 )

< 0

⎡ ⎤

⎥

⎥

⎥

⎦

max 0 it0 , sit0 (qit0−1 + ej ), ξit q 0 ) + it0∈C(qit0−1+ej ) π (q0 0

00 00 it0 , sit0 (qit0−1 −

⎢

⎢

⎢

⎣

because E < 0, since π (qit0 , sit0 , ξit e 0 ) is j ), ξit

 ́ −e 0 ) − j ) t0−1 π (q

2 maxqit0∈C(qit0−1) π (qit0 , sit0 (qit0−1), ξit0 )

concave in qit0−1. Furthermore,

∂2

V

V

¡

¡

qit0−1, sit0−1, ξit0−1

¢

¢

is concave in qijt0−2 since

qit0−1, sit0−1, ξit0−1 = −2c < 0. ∂q2

ijt0−2

max 00

3it0∈C(qit q 0−1

37
Finally, at some arbitrary time t<t0

X

Jt

V (qit, sit, ξit) = {qijt (Xjtγk + μk + ηkt + εijt) (28)

j=1

−1 (qijt > qijt−1) λm (qijt − qijt−1)}

⎛ ⎞

⎠

2

X

Jt

−c (qijt − qijt−1)

2 − (qijt − qijt−1) Pjt − δk ⎝ qijt +

j=1

∙

βEt max {V (qi

0

, sit+1 (qit), ξ)}

 ̧

0 qi∈C(sit+1(qit))

and

M (qit + ej , sit, ξit) − M (qit, sit, ξit)

≤ −1 (qijt > qijt−1) λ − ⎡ m 2 (δk + c) +

⎤ ¡

¡

¢

¢

0 0 max 0 qit+1∈C(qit+ej ) V qit+1, sit+1 (qit + ej ), ξit+1 +

00 00 max 00 q ∈C(qit−ej ) V qit+1, sit+1 (qit − ej ), ξ it+1 it+1

maxqit+1∈C(qit) V

⎢

⎢

⎢

⎣

⎥

⎥

⎥

⎦

βEt − ¡ ¡ ¢¢ 2 qit+1, sit+1 (qit), ξit+1

< 0

⎡ ⎤ ¡ ¢ max 0 0 q0 ∈C(qit+ej ) V qit+1, sit+1 (qit + ej ), ξit+1 + it+1

¡ ¢ 00 00 max 00 q ∈C(qit−ej ) V qit+1, sit+1 (qit − ej ), ξ it+1 it+1

maxqit+1∈C(qit) V qit+1, sit+1 (qit), ξit+1

⎢

⎢

⎢

⎣

⎥

⎥

⎥

⎦

because Et − < 0, since

¡ ¢¢

qit+1, sit+1 (qit), ξit+1

¡

is concave in qijt.

2

¡ ¢

f o

V

∗

it Proo f Proposition 2. Conditions (8),(9) and (10) are by definition necessary for q to be

optimal. Condition (8) and Proposition 1 imply that i would not benefit from subtracting any

quantity of aircraft. Condition (9) and Proposition 1 imply that i would not benefit from adding

∗ any quantity of aircraft. Finally, assume qit satisfies the conditions given by (8),(9) and (10). Now,

∗ PJt suppose there is an alternative fleet of aircraft qit + j=1 aj ej such that

X

Jt

∗qit + aj ej = arg max V (qit), qit j=1

such that aj0 is a negative integer and aj00 is a positive integer for some j0 and j00. First note that

given (8),(9) and the conditions of Proposition 1, if

X

Jt

aj ej > 0

j=1

then

⎛ ⎞

X

Jt

M ∗ j00 ⎝qit + aj ej⎠ < 0

j=1

38
and if

X

Jt

aj ej < 0

j=1

then

⎛ ⎞

X

Jt

Mj0 ⎝q∗ aj ej⎠ > 0. it +

j=1

So consider any vector q∗ PJt

j where it + j=1 aj e

X

Jt

aj ej = 0.

j=1

3  ́

If q∗ PJt

j is optimal, it must be true that Mj00 q∗ PJt

it + >

3

j=1 aj e

 ́

it + j=1 aj ej

M ∗ PJt ∗ ∗ j0 qit + j ej + ej0 − ej00 . However, since Mj0 (qit) > Mj00 (qit − ej0 + ej00 ), and under j=1 a

3  ́ ∗ PJt ∗ ∗ the conditions of Proposition 1 Mj0 qit + j=1 aj ej + ej0 − ej00 > Mj0 (qit) and Mj00 (qit − ej0 + ej00 ) >

3  ́

M ∗ PJt ∗ PJt ∗ j00 qit + j=1 aj ej ; qit + j=1 aj ej cannot be optimal. Since this is true for all possible a, qit

must be optimal given satisfaction of (8),(9) and (10).

Proof of Proposition 3 Proof of Lemma P3-1. This is an obvious result of Proposition 1

and the assumption that airlines are profit maximizing.

Proof of Lemma P3-2. This follows from the properties of the value function. When the price of

a type j aircraft goes up the marginal value of type j aircraft goes down. This implies that airlines

will want to reduce the quantity of type j aircraft they own if they change the quantity at all. Since

the marginal value of each type of aircraft is decreasing in the total number of aircraft the airline

owns, this implies that the marginal value of all other types of aircraft will increase. Therefore, no

airline will reduce the number of any other type j0 aircraft they demand when the price of type j

aircraft is increased.

One important result of this lemma is that prices will never be reduced once they are increased.

Proof of Lemma P3-3. This lemma is a straightforward application of Tarsky’s fixed point

theorem. That is, the set of possible prices for every type of aircraft is bounded on the interval

£ ¤ 0, P , and the auction mechanism described in the text is a function that maps bid prices Pj (r)

into Pj (r + 1) such that Pj (r + 1) ≥ Pj (r).

Proof of Lemma P3-4. It is obvious by the way airlines’ highest valued fleets are constructed

that the selected fleets satisfy the conditions of Proposition 2.

39
B Model Solution Algorithms

B.1 EMAX Updates - “Outer Loop”

At each iteration r of the outer loop of the model solution algorithm, the approximations EMAX \ (qit, r)

r of EMAX (qit) is updated for all qit. The outcomes qit for periods t = 2, ..., T that occur in the

inner loop of the algorithm are the outcomes of the model given the most recent approximations

EMAX \ (qit, r) for all qit, and a draw of the random components ξr

1, ..., ξrT . The value function of

the equilibrium outcome at time t + 1 given qr

it is averaged with previous outcomes of the algorithm

to obtain the updated value of EMAX (qit). That is,

¡ ¢ q

where the state sr depends on the vector qr , and n(qr ) is the number of times a outcome qr has it+1 it it it

occurred (up to and including iteration r). If, on the other hand, a vector qit does not occur in the

inner loop of the algorithm, the approximations of EMAX (qit) do not change so that

EMAX \ (qit, r + 1) = EMAX \ (qit, r).

EMAX \ (q , ξr

it|q r r

it, r) + V r rit+1, s r

\ (n(q ) − 1) ∗ EMAX (qr it it+1 it

it, r + 1) = r n(qit)

°

°

°

The algorithm described above has converged when for the vector q = (q11, ..., qit, ..., qIT ) the distance

EMAX \ (qt, r + 1) − EMAX \ (qt, r)

°

° measure

°

is less than a predetermined threshold  > 0.

B.2 Equilibrium - “Inner Loop”

t subscripts are suppressed because the algorithm that determines equilibrium aircraft prices and

quantities is the same for all t. To compute the price in bidding round b of a type j aircraft, the

algorithm calculates the “buy threshold” (i.e. maximum willingness-to-pay) for an additional type j

aircraft for each airline, and also the “keep threshold” (i.e. marginal value to airlines of keeping) for

currently held type j aircraft, given the current prices and quantities. If the maximum (over airlines)

buy threshold exceeds the minimum keep threshold for type j aircraft, a type j aircraft is sold from

the airline with the minimum keep threshold to the airline with the maximum buy threshold, and

the price of type j aircraft is set equal to the minimum keep threshold plus a arbitrarily small bid

increment. If, on the other hand, no type j aircraft are sold, the price of a type j aircraft does not

change.

Airline i’s buy threshold for a type j aircraft is denoted Bij , and is equal to the maximum of

the current price of a type j aircraft plus the larger of (1) the marginal value of adding a type j

aircraft to its current fleet or (2) the marginal value of swapping another type of aircraft for a type

40
j aircraft. That is,

Bij (b) = max {Mij (qi (b) + ej ), Mij (qi (b) + ej − ej0 )} + Pj (b − 1).

The only caveat to this computation is that the buy threshold of the scrapper is always zero.

The amount an airline is willing to pay to keep a type j aircraft, denoted Kij , is equal to the

amount the airline loses if you take away a type j aircraft plus the current price of type j aircraft.

That is,

Kij (b) = Mij (qi (b)) + Pj (b − 1).

Keep thresholds are only calculated for those airlines that own at least one type j aircraft.

A sale of a type j aircraft occurs if and only if at least one airline has a buy threshold that is

greater than another airline’s keep threshold. That is, if

max {Bij (b)} > min {Kij (b)} i:qij<qj i:qij>0

then a type j aircraft is sold from the airline with the minimum keep value to that airline with the

maximum buy threshold. The price of a aircraft j in bidding round b is given by

Pj (b) = Pj (b − 1) if max {Bij (b)} < min {Kij (b)} and

i:qij<qj i:qij>0

Pj (b) = min {Kij (b)} + τ if max {Bij (b)} > min {Kij (b)} , i:qij>0 i:qij<qj i:qij>0

where τ > 0 is an arbitrarily small bid increment.

B.2.1 Reverse Step:

An airline determines its buy threshold for a type j aircraft by determining how much it would be

willing to pay to add a type j aircraft to its current fleet, and how much it would be willing to pay

to get rid of a type j0 aircraft from its current fleet and acquire a type j aircraft, for some j0

. In the

case where the airline with the highest buy threshold for type j aircraft is determined by a airline

that is best off swapping out a type j0 aircraft for a type j aircraft, the type j0 aircraft becomes

unowned. In this case the algorithm immediately reduces the price of type j0 aircraft until the

aircraft is acquired by an airline or is scrapped. Specifically, if

max {Bij (b)} = Mij (qi (b) + ej − ej0 ) + Pj (b − 1) for some i, i:qij<qj

then the price of j0 is lowered to the maximum marginal value across airlines of adding a type j0

aircraft less an arbitrarily small amount. That is,

h i

Pj0 (b) = max max {Mij (qi (b) + ej0 ) + Pj0 (b − 1)} − , 0 . i

41
At each time t, the algorithm for finding equilibrium prices and quantities can be written concisely

as follows.

Initialization: Set b = 1, j = 1, qt (1) = qt−1, and Pjt (1) = 0 for all j.

(1) For j, b compute Cijt (b) and Kijt (b).

(2) If

max {Bijt (b)} > min {Kijt (b)} and

i:qijt<qjt i:qijt>0

Mijt (qit (b) + ej − ej0 ) + Pjt (b − 1) = max {Bijt (b), 0} for some i

i

remove one type j aircraft from the airline with the minimum keep threshold and give it to the airline

with the highest buy threshold, set

Pjt (b) = mini:qijt>0 {Kijt (b)} + τ, and go to reverse auction step (3). If, on the other hand

max {Bijt (b)} > min {Kijt (b)} and

i:qijt<qjt i:qijt>0

Mijt (qit (b) + ej ) + Pjt (b − 1) = max {Bijt (b), 0} for some i

i:qijt<qjt

remove one type j aircraft from the airline with the minimum keep threshold and give it to the airline

with the highest buy threshold, set

Pjt (b) = mini:qijt>0 {Kijt (b)} + τ, and go to step (4). Finally, if

max {Bijt (b), 0} ≤ min {Kijt (b)} i i:qijt>0

set Pjt (b) = Pjt (b − 1) and go to (4).

(3) Set

h i

Pj0t (b) = max max {Mij0t (qit (b) + ej0 )} − τ, 0 + Pj0t (b − 1) i

and allocate the free type j0 aircraft to the airline i with

Mij0t (qit (b) + ej0 ) = maxi {Mij0t (qit (b) + ej0 )} or to the scrapper if

maxi {M (qit (b) + ej0 )} < 0. Go to step (4)

(4) If j<Jt, set j = j + 1, b = b + 1, and return to (1). If j = Jt and

Pt (b) 6= Pt (b − Jt), set j = 1, b = b + 1, and return to (1). If j = Jt and

Pt (b) = Pt (b − Jt) the algorithm has converged and qt (b) and Pt (b) are an equilibrium.

42
C Simulation

C.1 Simulation Algorithm

This subsection details the algorithm used to simulate Qit. The simulator uses the law of total

probability and strategically orders the errors of the model to maximize the amount of time the

algorithm spends in the important regions of the errors given the parameters of the model, and

therefore reduces the number of simulation draws that are necessary to simulate Qit accurately.

Ordering aircraft in some way ((1), ...,(j), ...,(J)) the probability

£ ¤ ∗ ∗ Qit (ηt) = Pr bijt (qit, ηt) < εijt < bijt (qit, ηt) ∀j

can be rewritten

Y h ∗ i

Qit (ηt) = Pr b

∗

i(j)t < εi(j)t < bi(j)t|εi(1)t, ..., εi(j−1)t , (29)

j=1

where

3 ∗ ∆+ bi(j)t = min i(j)t, εi(1)t + ∆i(j)(1)t, ..., εi(j−1)t + ∆i(j)(j−1)t, (30)

 ́

∆+

i(j+1)t + ∆i(j)(j+1)t, ..., ∆+i(J)t + ∆i(j)(J)t and

3

b

∗

i(j)t = min ∆−

i(j)t, εi(1)t − ∆i(1)(j)t, ..., εi(j−1)t − ∆i(j−1)(j)t,

 ́

∆− − ∆ij+1jt, ..., ∆− − ∆iJjt . i(j+1)t i(J)t

The simulation algorithm described below constructs the probability Qit (ηt) as suggested by equation (29) by sequentially drawing errors εi(j)t, where the draw εr

i(j)t is conditional on the draws

εr

i(1)t, ..., εri(j−1)t. © a (1) Order aircraft j : qjt > 0 according to some criterion (the criterion used in practice is

© a described below). Let (j) be the jth element of the ordered set. Let J∗ = # j : qjt > 0 .

(2) Initialize Pi

r = 1.

(3) For each (j) ≤ J∗,

(a) Let

⎛

⎜⎜⎜⎝

⎞

∆+ , εr

3

+ ∆i(j)(1)t, ..., ε

 ́

r + ∆i(j)(j−1)t, i(j)t i(1)t i(j−1)t

⎟⎟⎟⎠

∗ ε , (31) r , ..., εr i(1)t i(j−1)t b = min εi(j+1)t + ∆i(j)(j+1)t, ...,  ́

i(j)t

3

εi(J)t εr

i(1)t, ..., εri(j−1)t + ∆i(j)(J)t

be a upper bound for εi(j)t, where

3  ́ 3  ́

εr ∆+ εi(j+1)t i(1)t, ..., εr

i(j−1)t = min i(j+1)t, εri(1)t + ∆i(j+1)(1)t, ..., εri(j−1)t + ∆i(j+1)(j−1)t .

43
Also let

⎛ ⎞

⎜⎜⎜⎝

∆−

i(j)t, εi

r

(1)t − ∆i(1)(j)t, ..., εi

r

(j−1)t − ∆i(j−1)(j)t, 3  ́

εi(j+1)t ε

3

i

r

(1)t, ..., εi

r

(j−1)t

 ́

− ∆ij+1jt, ...,

⎟⎟⎟⎠

b

∗

i(j)t = max

εi(J)t εr

i(1)t, ..., εri(j−1)t − ∆iJjt

be a lower bound, where

3  ́ 3  ́

εr , ..., εr = max ∆− , εr − ∆i(1)(j+1)t, ..., εr − ∆i(j−1)(j+1)t ε . i(j+1)t i(1)t i(j−1)t i(j+1)t i(1)t i(j−1)t

(b) Update

" Ã ! μ ¶# bi(j)t bi(j)t Pi

r = Pir Φ − Φ σε σε

(c) Draw εr

i(j)t conditional on bi(j)t ≤ εi

r

(j)t ≤ bi(j)t as

(" Ã ! μ ¶# μ ¶)

εr = σεΦ−1 Φ

bi(j)t − Φ

bi(j)t

ur + Φ

bi(j)t , i(j)t σε σε σε

where ur ∼ U (0, 1).

27

(4) if (j) = J∗ Pi

r is the simulator, otherwise set (j)=(j)+1 and return to (3).

C.2 Properties of the Simulator

Next I discuss some properties of the above simulator of Qit, namely that it satisfies the criterion

of a importance sampling simulator. Let

Y 3  ́

h (εit) = 1 bi(j)t < εi(j)t < bi(j)t

(j)

be a function that is equal to one if the vector εit = ¡

εi(1)t, ..., εi(J)t

¢

is consistent with equilibrium

and zero otherwise. The likelihood contribution of each airline is given by

Z Z Qit = ··· h (εit) f (εit) dεit, (32)

where

Y μ ¶ 1 εi(j)t f (εit) = σ2 φ σε (j) ε

is the joint density of the εi(j)t. Note that since the integrand is zero when the errors are outside

of the bounds defined by equilibrium. Therefore, the value of the right hand side of equation (32)

2 7For the first couple of guesses of the parameters of the model, both bi(j)t and bi(j)t may be in the ex- μ ¶ 3  ́

treme tail of the distribution. In cases where both Φ bi(j)t and Φ bi(j)t are very close to zero or one σε σε

the estimation

1⁄2

algorithm may crash. In such cases, I use L’Hopital’s rule to derive the alternative simulator

∙ μ ¶ 3  ́ ̧ 3  ́3⁄4

εr = φ−1 φ bi(j)t − φ bi(j)t ur + φ bi(j)t . i(j)t σε σε σε

44
does not change if we integrate only over the portion of the support of f (.) that is consistent with

equilibrium. That is,

Z Z Z Z ··· h (εit) f (εit) dεit = ··· h (εit) f (εit) dεit. bi(j)t<εi(j)t<bi(j)t

Now, rewrite equation (32) as

Z Z h (εit) f (εit) ··· g (εit) dεit, bi(j)t<εi(j)t<bi(j)t g (εit)

where

3  ́ Y σ

1

2 φ εi

σ

(j

ε

)t

g (εit) = 3 ε  ́ 3  ́, (33)

(j) Φ bi

σ

(j

ε

)t − Φ bi

σ

(j

ε

)t

3  ́ is the joint pdf of independent truncated normal random variables with truncation points bi(j)t, bi(j)t .

Note that g (εit) can be written as

3  ́ 3  ́ Y

1 bi(j)t < εi(j)t < bi(j)t σ

1

2 φ εi(j)t

σε

g (εit) = 3  ́ 3 ε  ́ .

(j) Φ bi

σ

(j

ε

)t − Φ bi

σ

(j

ε

)t

g (εit) satisfies the four features of a good importance sampling distribution: (1) g (.) is easy to

draw from, (2) has the same support as f (εit), (3) h(ε

g

it

(

)

ε

f

it

(

)

εit)

h (εit) f (εit) Y" Ã

bi(j)t

! μbi(j)t

¶#

= Φ − Φ , g (εit) σε σε (j)

is easy to evaluate, and (4) h(ε

g

it

(

)

ε

f

it

(

)

εit) is bounded between 0 and 1, and is smooth in the parameters

of the model. Thus, the algorithm described above is an importance simulator that simulates

E h(ε

g

it

(

)

ε

f

it

(

)

εit)

, where εit ∼ g. The fact that I can write the simulator as an importance sampling

simulator means that it is unbiased.

Finally, I discuss the ordering criterion used in practice in step (1) of the simulation algorithm.

Since I am using MSL it is very important that I reduce the variance of my simulator as much as

possible. One way to improve the variance of the simulator is to strategically order {j : qjt > 0}

in Step 1 of the above algorithm to reduce the variance of the simulator. Consider the case where

J∗ = 2. In Figure 4, the area within the six-sided figure represents the values of ε1 and ε2 that are

consistent with equilibrium, and the dotted circles represent the level sets of the joint distribution

of ε1 and ε2. The GHK algorithm first computes the probability that b(1) ≤ ε(1) ≤ b(1). Then it

simulates a value of ε(1) conditional on b(1) ≤ ε(1) ≤ b(1). Finally, it computes the probability that

ε(2) is such that the simulated values of ε(1) and ε(2) are within the area consistent with equilibrium.

45
The simulated probability is a product of two probabilities, and the variance of the simulator is

proportional to the variance of the second probability, which is a function of the simulated value

of ε(1). In order to reduce the variance of the simulator as much as possible, I would arrange

{j : qjt > 0} in descending order of the variance of the contribution of the simulator with respect to

those elements of ε that precede it. However, such a rule is too costly to construct. Alternatively, I

develop a simpler rule that is similar to the above rule that is less costly to implement. Specifically,

I order {j : qjt > 0} in increasing order of

φ

Φ

μ

μ

¶

¶

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

 ̄

,

∗ 3  ́ b(j)

σε

∗

b∗

(j) − φ σε

3  ́ b∗

(j) b(j)

σ − Φ ε σε

where

⎛ ⎞

∆+ , ∆+

b(

∗

j) = min ⎝

(j) (1) + ∆(j)(1), ..., ∆+

(j−1) + ∆(j)(j−1), ⎠

+ ∆(j)(j+1), ..., ∆+ ∆ + +

(j+1) (J) ∆(j)(J)

⎛ ⎞

∆− , ∆− − ∆(1)(j), ..., ∆− − ∆(j−1)(j), b

∗

(j) = max ⎝

(j) (1) (j−1)

⎠ .

∆− − ∆(j+1)(j), ..., ∆− − ∆(J)(j) (j+1) (J)

46
D References

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50
Table 1 

Aircraft

Model N 

Boeing 747 1078 

Boeing 767 690 

Boeing 777 104 

Airbus 300/310 682 

Airbus 330/340 191 

Douglas DC-10 371 

Douglas MD-11 174 

Lockheed L-1011 236 

Total 3526 

Table 2 

Observed Aircraft Attributes

Attribute Mean Standard Minimum Maximum 

Deviation 

Length 

(meters) 

Seating 

Capacity 

Fuel Capacity 

(1000 gallons) 

Payload Range 

(1000 miles) 

59.44 8.33 46.7 70.7 

410.93 98.98 253 550 

128.05 57.65 44.01 219.62 

5.87 2.01 1.04 9.21 
Table 3 

Structural Estimates: 

Industry and Cost

φ : Tax Reform -0.0013 

(0.0159) 

c : Adjustment cost 0.06495 

(0.0299) 

σ ε : Error standard deviation 3.0368 

(0.0677) 

λ1 : Boeing transaction cost 0.9650 

(0.0628) 

λ2 : Airbus transaction cost 0.9566 

(0.0284) 

λ3 : McDonnell-Douglas transaction cost 0.9747 

(0.1946) 

λ4 : Lockheed-Martin transaction cost 0.9445 

(0.0506) 

Standard Errors are in parentheses. 
Table 4 

Structural Estimates: 

Airline Preferences 

Large U.S. passenger airline: Small U.S. passenger Other airline: 

airline: 

Trend ( ) α : 0.2663 Trend (α ) : 0.0728 Trend ( ) α : 0.0951 

(3.37e-04) (7.29e-04) (2.80e-04) 

Rate of 0.00976 Rate of 0.05223 Rate of 9.58e-05 

decreasing (3.92e-05) decreasing (0.0013) decreasing (9.29e-05) 

returns ( ) δ : returns (δ ): returns (δ ): 

Aircraft 0.1702 Aircraft 0.0047 Aircraft 0.7898 

manufactured (0.0676) manufactured (0.0029) manufactured (0.0096) 

1978-84 ( ) 1 γ : 1978-84 ( 1 γ ): 1978-84 ( ) 1 γ : 

Aircraft 0.2101 Aircraft 0.0099 Aircraft 0.9474 

manufactured (0.0708) manufactured (0.1797) manufactured (0.0232) 

1985-91 ( ) 2 γ : 1985-91 ( 2 γ ): 1985-91 ( ) 2 γ : 

Aircraft 0.2593 Aircraft 0.0196 Aircraft 2.1857 

manufactured (0.0191) manufactured (0.0721) manufactured (0.2239) 

1992-97 ( ) 3 γ : 1992-97 ( 3 γ ): 1992-97 ( ) 3 γ : 

Boeing 747 -0.9780 Boeing 747 4.0334 Boeing 747 2.5071 

( ) 4 γ : (0.0912) ( 4 γ ): (0.085) ( 4 γ ): (0.0215) 

Boeing 767 -0.3374 Boeing 767 1.4712 Boeing 767 1.7003 

( ) 5 γ : (0.2002) ( 5 γ ): (0.2018) ( 5 γ ): (0.0278) 

Airbus 300/310 -1.7071 Airbus 300/310 2.9528 Airbus 300/310 4.2430 

( ) 6 γ : (0.0789) ( 6 γ ): (0.0604) ( 6 γ ): (0.2986) 

Douglas DC-10 0.7132 Douglas DC-10 4.1726 Douglas DC-10 0.9427 

( ) 7 γ : (0.0515) ( 7 γ ): (0.2326) ( 7 γ ): (0.0857) 

Douglas MD-11 -2.2373 Douglas MD- 1.9768 Douglas MD- 1.4847 

( ) 8 γ : (0.1601) 11 ( 8 γ ): (0.1732) 11 ( 8 γ ): (0.2267) 

Lockheed -3.2784 Lockheed 4.1864 Lockheed 1.0731 

L-1011 ( ) 9 γ : (0.0111) L-1011 ( 9 γ ): (0.5171) L-1011 ( ) 9 γ : (0.0299) 

Standard errors are in parentheses. 
Table 5 

Chi-Square Tests for the Total Quantity of Aircraft Owned 

Small U.S. Passenger 

Airlines 

Large U.S. Passenger 

Airlines 

“Other” Airlines 

Period Predicted Observed Predicted Observed Predicted Observed χ2 -Statistic 

1978 

1979 

1980 

1981 

1982 

1983 

1984 

1985 

1986 

1987 

1988 

1989 

1990 

1991 

1992 

1993 

1994 

1995 

1996 

1997 

225.2 189 

292 218 

283.8 268 

272.8 292 

265.8 292 

259.2 298 

267.2 296 

255 325 

257.2 301 

239.2 309 

267.4 311 

256.4 295 

269 304 

298.2 220 

283.4 211 

280.6 220 

267.2 229 

256.2 221 

243.6 247 

240 242 

102.4 119 

131.8 130 

151 140 

185.4 150 

210.2 173 

216 202 

211.8 201 

223.2 205 

243 243 

242.2 270 

281.2 302 

286 312 

330.6 327 

392.2 395 

419.4 446 

459.4 477 

444.2 465 

440.4 449 

472 447 

476.6 437 

489 486 

591.4 594 

681.6 710 

777.4 819 

838.6 885 

968.4 968 

1000.2 1058 

1181 1126 

1228.6 1216 

1341.8 1283 

1383.2 1372 

1523.8 1512 

1606.8 1648 

1740 1800 

1947.2 1950 

2083.8 2071 

2191.4 2172 

2297.8 2273 

2364.2 2340 

2446.8 2427 

8.53 

18.79 

2.86* 

9.13 

11.73 

6.72 

7.00 

23.26 

7.59 

26.14 

8.74 

8.27 

5.65* 

22.60 

20.19 

13.84 

6.61 

5.27* 

1.62* 

3.47* 

The test statistics are distributed χ2 (2). 

The * indicates that the model fits the data at the 5% level of significance. 
Table 6 

Chi-Square Tests for the Quantity of Boeing 747 Aircraft Owned 

Small U.S. Passenger 

Airlines 

Large U.S. Passenger 

Airlines 

“Other” Airlines 

Period Predicted Observed Predicted Observed Predicted Observed χ2 -Statistic 

1978 

1979 

1980 

1981 

1982 

1983 

1984 

1985 

1986 

1987 

1988 

1989 

1990 

1991 

1992 

1993 

1994 

1995 

1996 

1997 

93.4 85 

139.6 102 

129.6 117 

115.6 113 

104.2 113 

94.6 116 

102.4 132 

92.4 144 

88 125 

101 132 

108.2 130 

90.4 117 

135.6 119 

133 86 

135 84 

118.8 90 

108 101 

100.8 100 

98.4 112 

97.4 112 

20.2 40 

40.8 40 

33.2 41 

38.4 44 

44.4 44 

35.8 43 

35 31 

25.2 25 

36.6 38 

18.8 44 

38.4 45 

51.8 48 

58 55 

70.2 64 

68.8 69 

92.8 70 

89.8 73 

79.6 69 

78.8 71 

74.4 73 

288.8 261 

316.8 315 

346.6 367 

388.4 412 

410.4 438 

448.6 452 

446.6 461 

508.8 475 

527.4 508 

553.2 521 

549 542 

611 599 

634 657 

686.2 712 

753 772 

799.6 810 

837.4 831 

878.2 850 

906 865 

934.8 897 

22.84 

10.15 

4.26* 

2.31* 

2.60* 

6.31 

9.48 

30.62 

15.61 

43.37 

5.75* 

12.66 

5.65* 

26.75 

41.24 

20.87 

6.77 

2.33* 

4.51* 

3.74* 

The test statistics are distributed χ2 (2). 

The * indicates that the model fits the data at the 5% level of significance. 
Table 7 

Predicted Quantities of Aircraft Owned by Airlines 1998-2002 with and without a 10% 

Investment Tax Credit 

Small U.S. 

Airlines: 

No Policy 

Year 

1998 

1999 

2000 

2001 

2002 

1969-1977 

77.7 

77.3 

46.3 

45.3 

58 

Year of Manufacture 

1978-1984 1985-1991 1992-1997 

62.4 31.4 62 

50 17.6 39.1 

63.7 26.3 48 

90.6 25.3 35.4 

82 18.9 31.9 

1998-2002 

19.4 

21.3 

16.4 

14.4 

12.3 

Total 

252.9 

205.3 

200.7 

211 

201.3 

10% Tax 

Credit 

1998 

1999 

2000 

2001 

2002 

73.6 

74.7 

43 

41 

51.9 

64.7 

55.3 

67.3 

85.1 

81.3 

32.3 

16.4 

27.6 

29.1 

23.3 

64 

38.7 

43.6 

36.6 

29.4 

19.3 

22 

12.7 

14.7 

13 

253.9 

207.1 

194.2 

206.5 

198.9 

Large U.S. 

Airlines: 

No Policy 

1998 

1999 

2000 

2001 

2002 

98.3 

95.9 

105.9 

127 

111.7 

98.3 

94.4 

93.7 

85.7 

87.3 

111.3 

144.9 

146.3 

129.7 

132.7 

241 

241.7 

208 

230 

223.6 

41.7 

18.1 

20.6 

24.7 

35.4 

590.6 

595 

574.5 

597.1 

590.7 

10% Tax 

Credit 

1998 

1999 

2000 

2001 

2002 

92 

90 

97.6 

125 

106.1 

96 

88.3 

87.7 

81.9 

80.3 

110.3 

148 

150.7 

129.7 

138.6 

241.3 

245.7 

212.6 

228.3 

221.1 

43.4 

23 

16.4 

29 

39.1 

583 

595 

565 

593.9 

586.2 

“Other” 

Airlines: 

No Policy 

1998 

1999 

2000 

2001 

2002 

507.1 

503.1 

523.2 

491.7 

492 

493.6 

507.7 

489.8 

470 

475.6 

445.9 

424.4 

411.1 

424.4 

421 

498.1 

517.3 

540.6 

529 

537.4 

393.9 

414 

423 

427.7 

440.7 

2338.6 

2367 

2387.9 

2342.8 

2367.7 

10% Tax 

Credit 

1998 

1999 

2000 

2001 

2002 

518 

512 

535.3 

498 

505.9 

493.6 

508.7 

494.1 

479.7 

484 

446 

422.4 

404.7 

420.3 

411.7 

498 

517.6 

544.7 

535.3 

547.3 

393.4 

409.7 

412.7 

418.3 

432.4 

2349 

2370.4 

2391.5 

2351.6 

2381.3 
Table 8 

Predicted Quantities of Aircraft Owned by Airlines 1998-2002 – Mandated Modernization 

for Aircraft Over 20 Years Old 

Small U.S. 

Airlines: 

No Policy 

Year 

1998 

1999 

2000 

2001 

2002 

1969-

1977 

77.7 

77.3 

46.3 

45.3 

58 

Year of Manufacture 

1978-

1984 

1985-

1991 

1992-

1997 

62.4 31.4 62 

50 17.6 39.1 

63.7 26.3 48 

90.6 25.3 35.4 

82 18.9 31.9 

1998-

2002 

19.4 

21.3 

16.4 

14.4 

12.3 

Total 

252.9 

205.3 

200.7 

211 

201.3 

Modernization 

Mandate 

1998 

1999 

2000 

2001 

2002 

73 

70.6 

43 

40.7 

55.6 

64.4 

53.3 

64 

90.4 

87 

33 

17.9 

26.3 

27.7 

19.4 

63.3 

38.6 

48 

35.4 

36.6 

19.3 

23.1 

17 

13.7 

8 

253 

203.5 

198.3 

207.9 

206.6 

Large U.S. 

Airlines: 

No Policy 

1998 

1999 

2000 

2001 

2002 

98.3 

95.9 

105.9 

127 

111.7 

98.3 

94.4 

93.7 

85.7 

87.3 

111.3 

144.9 

146.3 

129.7 

132.7 

241 

241.7 

208 

230 

223.6 

41.7 

18.1 

20.6 

24.7 

35.4 

590.6 

595 

574.5 

597.1 

590.7 

Modernization 

Mandate 

1998 

1999 

2000 

2001 

2002 

90.7 

91.9 

100.4 

126.3 

107.6 

96.3 

92.4 

91.3 

82 

83.4 

106 

141 

143.1 

123.2 

135.4 

241.4 

246.7 

213 

237 

224.7 

42.6 

18.3 

20.9 

25.7 

32.3 

577 

590.3 

568.7 

594.2 

583.4 

“Other” 

Airlines: 

No Policy 

1998 

1999 

2000 

2001 

2002 

507.1 

503.1 

523.2 

491.7 

492 

493.6 

507.7 

489.8 

470 

475.6 

445.9 

424.4 

411.1 

424.4 

421 

498.1 

517.3 

540.6 

529 

537.4 

393.9 

414 

423 

427.7 

440.7 

2338.6 

2367 

2387.9 

2342.8 

2367.7 

Modernization 

Mandate 

1998 

1999 

2000 

2001 

2002 

519.3 

514 

533 

497.7 

501 

493.6 

507 

492.8 

474.3 

475.3 

449.3 

428.4 

414.4 

428.4 

421.1 

498 

514.3 

536.7 

523.1 

532.4 

392.7 

408.6 

411 

415.7 

427.7 

2352.9 

2372.3 

2387.9 

2339.2 

2357.5 
Figure 1 

Proportion of Aircraft Sold in Secondary Markets

Proportion Sold

0.3 

0.25 

0.2 

0.15 

0.1 

0.05 

0 

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 

Age 
Figure 2 

Proportion of Aircraft Sold to Passenger Airlines

Boeing Airbus Douglas Lockheed 

0 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 

1 
19781979198019811982198319841985198619871988198919901991199219931994199519961997 

Figure 3 

Deliveries and Sales per Period 

0 

50 

100 150 200 250 300 350 

No. of Aircraft Seco

TotaDeliv

Year 
Figure 4 

Total Quantity of Aircraft Owned by Small U.S. Airlines

400300200100

0

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 

Total Quantity of Aircraft Owned by Large U.S. Airlines 

600500400300200

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 

Total Quantity of Aircraft Owned by "Other" Airlines 

100 

0

30002500

200015001000

500

0

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 
Figure 5 

Boeing 747 Aircraft Owned by Small U.S. Airlines

100

80604020

0

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 

160140120

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 

Boeing 747 Aircraft Owned by "Other" Airlines 

100

Boeing 747 Aircraft Owned by Large U.S. Airlines 

80604020

0

1000

800

600400200

0

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 
Figure 6 

DC-10 Aircraft Owned by Small U.S. Airlines

100

80604020

0

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 

DC-10 Aircraft Owned by Large U.S. Airlines 

140120100

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 

DC-10 Aircraft Owned by "Other" Airlines 

80 604020

0

250200150100

50

0

1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995