論文 / 著書情報
Article / Book Information
題目(和文)

民営化空港の市場支配力濫用と容量不足に対する施策の評価

Title(English)

Assessment of countermeasures against the abuse of market powerand capacity deficiency of privatized airports

著者(和文)

許方舟

Author(English)

Fangzhou Xu

出典(和文)

学位:博士(学術),学位授与機関:東京工業大学,
報告番号:甲第12018号,
授与年月日:2021年3月26日,
学位の種別:課程博士,
審査員:花岡 伸也,朝倉 康夫,屋井 鉄雄,室町 泰徳,阿部 直也,大西 正光

Citation(English)

Degree:Doctor (Academic),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第12018号,
Conferred date:2021/3/26,
Degree Type:Course doctor,
Examiner:,,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Science Tokyo Research Repository)

Assessment of countermeasures against the abuse of
market power and capacity deficiency of privatizedairports

A Dissertation
Submitted to the Department of Transdisciplinary Science and Engineeringin Partial Fulfilment of the Requirements of the Degree ofDoctor of Philosophy

Fangzhou XuGlobal Engineering for Development, Environment and Society
Department of Transdisciplinary Science and Engineering
School of Environment and Society
Tokyo Institute of Technology

2021

Acknowledgements
I am deeply indebted to my supervisor Professor Shinya Hanaoka for his invaluableguidance and patient support throughout my master and doctor course. His influences onme are long-lasting and multi-dimensional, not only academically, but also in humanity.
My gratitude to him is far beyond words.
Besides my supervisor, I would like to thank the rest of my thesis committee: Professor
Yasuo Asakura, Professor Tetsuo Yai, Associate Professor Naoya Abe, Associate
Professor Yasunori Muromachi and Associate Professor Masamitsu Onishi, for theirconstructive comments and suggestions, which not only help me improve this thesis, butalso provide great motivation for my future study.
I thank Dr. Kawasaki, Dr. Nakamichi, Dr. Choi, Ms. Hattori, and all other lab mates fortheir kind support. I also thank my friends for sharing the joy and pain. I am grateful tothe staffs of the policy management center in Japan Airlines who provided incredibly
patient help to me in a very difficult situation.
Above all, I would like to express my deepest gratitude to my parents for theirunconditional love which I can never repay. Without their endless support, I definitely
cannot be who I am.

iAbstract
Airports are seeking structural changes, such as privatization, to meet and satisfy theincreasing demand. On the one hand, there are open issues on whether privatization canstimulate the capacity investment which is the ultimate countermeasure against capacity
shortage. On the other hand, privatization raises the concern regarding the abuse of market
power. Based on the background, the objectives of the thesis are, first, assessing theplausibility of the introduction of terminal competition as an alternative countermeasureagainst the abuse of market power; second, figuring out the relationship between
privatization and the investment decisions from some new perspectives.
We first investigate the effect of terminal competition on the pricing and social surplus.
We define several business models to characterize the organizational structures of the
airport before and after the introduction of terminal competition. Comparing theequilibrium outcomes in each business model, we find that, in most cases, having
competing terminals can neither lower the prices nor enhance the social surplus if theoperation of the terminal and airfield facilities are not completely separated in the existing
business model, whether or not airlines have the freedom to change base terminal in
response to the prices. The complementarity between the airfield and terminal service,
which is originally internalized by the joint operation, will become a negative effect thatcannot be offset even by a strong degree of substitution between terminals, if the twoservices are provided respectively by independent operators. In contrast, if the operationof the two sections has been completely separated in the existing business model, or the
airfield operator can be strictly regulated, having competing terminals can result in ahigher social surplus, as it will not increase the negative complementary effect. Instead,it creates a duopoly of substitute goods that can offset the complementary effect.
Second, we analyze the bi-projects investment problem at an airport. Adopting a realoption model, we obtain the optimal rule of the timings for the investment in capacity
expansion project and the investment in cost reduction project. We also estimate the loss
due to the suboptimal investment. Numerical results suggest that the decision-maker who
places a higher premium on social surplus always tends to invest earlier in both projects.
Other factors that stimulate an earlier investment include a lower cross-price effect, aii

greater total number of airlines, a higher drift, and a lower volatility in demand. The
optimal scales of projects are more sensitive to the change in drift and volatility rather
than the change in other factors. When the composition of the objective function changes,
the pattern of the change in optimal timing differs by project. The loss due to NPV-based
investment is much greater than that caused by the adoption of the deterministic model,
and its change with the change in the composition of the objective function can have amountain-shaped pattern, where the minimum can be reached when specific conditions
are met.
Third, we investigate the relationship between the privatization decisions and theinvestment decisions of an airport, and the effect of various factors on the relationship.
We consider two scenarios differing in the availability of governmental subsidiary to theinvestment. We find that the optimal capacity volume does not necessarily decline as theprivate share increases, if the governmental subsidiary is available. If the governmental
subsidiary is unavailable, three clear-cut regions of the investment decisions with regard
to the private share can be observed: If the private share is low, no expansion will be
carried out; if the private share is moderate, a gap-filling expansion will be carried out; if
the private share is high, the expansion will not only fill the current capacity gap, but also
leave some vacant capacity preparing for the future demand growth. Maximum
aggregated social surplus can be achieved by privatizing a certain share which is justconsistent with the value of one of the two breakpoints that connect two adjacent regions,while which breakpoint can maximize the aggregated social surplus depends on thetiming of the privatization.
Some findings of this thesis might have policy implications. The results regarding theeffect of terminal competition indicate that it cannot work as an alternative to the regular
regulation to restrain the abuse of market power, and a respective privatization of airport
facilities should be avoided. The results of the bi-project investment problem suggest thatthe regulatory adjustment of the timings, which aims to enhance social surplus, should beenacted in different rule for different type of project. The results regarding the effect of
airport privatization on the capacity expansion imply that the optimal share to privatize
in terms of aggregated social surplus can change as the timing for privatization change,iii

and there is a trade-off between the social surplus within the airport and the spill-oversocial benefit (capacity) in the case of an early privatization.

iv

Contents
Acknowledgements ........................................................................................................... i
Abstract ............................................................................................................................. ii
Chapter 1 Introduction ...................................................................................................... 1
1.1 Background ............................................................................................................. 1
1.2 Research Objectives ............................................................................................... 3
1.3 Scope and Limitations ............................................................................................ 4
1.4 Outline .................................................................................................................... 5
1.5 Contributions .......................................................................................................... 6
Chapter 2 Literature review .............................................................................................. 8
2.1 The impacts of airport privatization ....................................................................... 8
2.2 Intra-airport terminal competition .......................................................................... 9
2.2.1 The abuse of market power by the airport ....................................................... 9
2.2.2 Countermeasures ........................................................................................... 10
2.2.3 Terminal competition: History and current status ......................................... 13
2.3 Relevant studies regarding terminal competition ................................................. 15
2.4 Capacity shortage at airports ................................................................................ 17
2.5 Studies on airport capacity expansion .................................................................. 19
Chapter 3 Effects of airport terminal competition: A vertical structure approach ......... 22
3.1 Introduction .......................................................................................................... 22
3.2 The basic model .................................................................................................... 23
3.2.1 Description of the problem ............................................................................ 23
3.2.2 Objective functions ........................................................................................ 28
3.2.3 Case 1: airlines cannot change terminal ........................................................ 31
3.2.4 Case 2: airlines can change terminal ............................................................. 38
3.3 Extensions............................................................................................................. 41
3.3.1 Asymmetric terminals.................................................................................... 41
3.3.2 Airline in the upstream market ...................................................................... 43
3.3.3 Terminal competition on both price and service level .................................. 49
3.3.4 Regulated terminals ....................................................................................... 50
3.4. Conclusion ........................................................................................................... 54
Chapter 4 Expand Capacity or Reduce Cost? An Analysis of the Optimal Timings for1

Two Types of Projects at an Airport .............................................................................. 57
4.1 Introduction .......................................................................................................... 57
4.2. Uncertainty test .................................................................................................... 59
4.3 The model ............................................................................................................. 62
4.3.1 Description of the problem ............................................................................ 62
4.3.2 Formulation ................................................................................................... 64
4.4 Numerical examples ............................................................................................. 73
4.4.1 Settings .......................................................................................................... 73
4.4.2 Comparation of the optimal rules .................................................................. 75
4.4.3 Results of the optimal timings ....................................................................... 76
4.4.4 Loss due to investments following suboptimal rules..................................... 81
4.5 Conclusion ............................................................................................................ 85
Chapter 5 Effects of airport’s privatization decisions on capacity expansion: A real option
approach.......................................................................................................................... 87
5.1 Introduction .......................................................................................................... 87
5.2 The model ............................................................................................................. 89
5.2.1 Description of the problem ............................................................................ 89
5.2.2 The optimal charging problem ...................................................................... 92
5.2.3 The capacity expansion problem ................................................................... 96
5.2.4 Calculation of aggregated social surplus in Scenario II ................................ 98
5.3 Numerical examples ............................................................................................. 99
5.3.1 Scenario I ..................................................................................................... 100
5.3.2 Scenario II ................................................................................................... 102
5.4 Conclusion .......................................................................................................... 108
Chapter 6 Conclusion ................................................................................................... 110
6.1 Summary and conclusions .................................................................................. 110
6.2 Future scope........................................................................................................ 113
Appendices ................................................................................................................... 116
Appendix A A discussion on Business Model II...................................................... 116
Appendix B Solving process and proof of Section 3.2 ............................................ 117
Appendix C Results and proofs of Section 3.2.3 ...................................................... 119
Appendix D Effect of the relaxation of the perfect substitute assumption ............... 124
Appendix E Results and Proofs of Section 3.2.4 ...................................................... 127
2

Appendix F Results of Section 3.3.1 ........................................................................ 129
References .................................................................................................................... 131

3

List of figures
Fig 1.1 Linkage between the chapters .............................................................................. 4
Fig 2.1 Catchment areas of airports in Ireland ............................................................... 12
Fig 3.1 Illustration of the air transport market in Case 1 ................................................ 24
Fig 3.2 Illustration of business models ........................................................................... 26
Fig 3.3 A detailed illustration of the model’s framework .............................................. 27
Fig 3.4 Numerical results of 𝑆𝐼𝐼 − 𝑆𝐼𝐼𝐼 ........................................................................... 37
Fig 3.5 Illustration of the air transport market in Case 2 ................................................ 38
Fig 3.6 Visualization of price and social surplus comparison ........................................ 42
Fig 3.7 Illustration of Business Models with airlines in upstream ................................. 45
Fig 3.8 Comparison of equilibrium total prices in each Model ...................................... 46
Fig 3.9 Comparison of the social surplus of each Model ............................................... 47
Fig 3.10 Comparison of total prices and the social surplus of each Model .................... 53
Fig 4.1 Evolutions of traffic at the selected airports....................................................... 61
Fig 4.2 Diagram of the multi-stage game (Numbers in the circles indicate the sequenceof decision-making) ........................................................................................................ 62
Fig 4.3 Value function 𝒱 in each case ............................................................................ 73
Fig 4.4 Loss due to the optimal rule proposed by Truong et al. (2018) ......................... 75
Fig 4.5 Optimal timings for the investments with different b and different θ ................ 76
Fig 4.6 Changing rate of optimal timings ....................................................................... 76
Fig 4.7 Optimal timings for the investments with different m, n and different θ........... 78
Fig 4.8 Effect of α and σ on the optimal scales and timings (The size of the marker
indicates the scale of project) ......................................................................................... 79
Fig 4.9 Results of the loss due to suboptimal investment (Absolute) ............................ 83
Fig 4.10 Results of the loss due to suboptimal investment (Relative) ........................... 83
Fig 5.1 The sequence of the events in Scenario II .......................................................... 89
Fig 5.2 Effect of private share θ on the optimal investment decisions ......................... 101
Fig 5.3 The effect of the private share and the time of privatization on the optimal
decisions for the capacity expansion ............................................................................ 103
Fig 5.4 The effect of the private share and the time of privatization on the aggregatedsocial surplus ................................................................................................................ 103
Fig 5.5 The effect of the private share and volatility 𝜎 on the optimal decisions for the4capacity expansion ........................................................................................................ 104
Fig 5.6 The effect of the private share and volatility 𝜎 on the aggregated social surplus...................................................................................................................................... 105
Fig 5.7 The effect of the private share and drift 𝛼 on the optimal decisions for thecapacity expansion ........................................................................................................ 106
Fig 5.8 The effect of the private share and discount rate 𝑟 on the optimal decisions forthe capacity expansion .................................................................................................. 106
Fig 5.9 The effect of the private share and airline number 𝑛 on the optimal decisions forthe capacity expansion .................................................................................................. 106
Fig 5.10 The effect of the private share and the cross-price effect 𝑙 on the optimal
decisions for the capacity expansion ............................................................................ 107
Fig A.1 Numerical results of 𝑆𝐼𝐼𝐼 − 𝑆𝐼𝑉 ........................................................................ 123
Fig A.2 Numerical results of 𝑆𝐼𝑉 − 𝑆𝑉 ......................................................................... 123
Fig A.3 Numerical result of the comparison of social surplus between Models .......... 126
Fig A.4 Curves of airlines’ maximum profit given optimal charges in Case 2 ............ 127
Fig A.5 Contour graph of 𝑆 .......................................................................................... 128
Fig A.6 Graph of 𝑆𝐼𝐼 − 𝑆𝐼𝐼𝐼 in Case 2 ........................................................................... 129
Fig A.7 Numerical example of social surplus comparison in Extension 1................... 130

5

List of tables
Table 2.1 Examples of airport terminal competition ...................................................... 14
Table 2.2 Examples of investments in capacity expansion at airports ........................... 19
Table 3.1 Notations for Chapter 3 .................................................................................. 28
Table 3.2 Equilibrium total prices of each business model in Case 1 ............................ 32
Table 3.3 Values of parameters for numerical examples ............................................... 36
Table 3.4 Optimal total prices of each Business Model in Case 2 ................................. 38
Table 4.1 Basic information of airports selected for the test .......................................... 61
Table 4.2 Results of the Dickey-Fuller test .................................................................... 61
Table 4.3 Notations for Chapter 4 .................................................................................. 63
Table 4.4 Values of parameters for the numerical examples ......................................... 74
Table 4.5 The scale of each project ................................................................................ 74
Table 5.1 Notations for Chapter 5 .................................................................................. 92
Table 5.2 Values of the parameters for the numerical computation ............................... 99
Table A.1 Objective functions and first order conditions of each business model ...... 118

6

Chapter 1 Introduction
1.1 BackgroundIn the past decades, air transport market has seen significant development, which ismainly driven by the rapid growth in the demand for air transport. To maintain suchprosperity, two salient issues, namely how to meet the growing demand in terms of
capacity, and how to satisfy the demand in terms of service, are therefore necessary to beaddressed. Various development, which partially help address these issues, has takenplace in the airline industry in a fairly early stage, thanks to the trend of airline
deregulation. Meanwhile, as a participant in the upstream of the supply chain, airport alsoneed to take corresponding actions. Otherwise, the airport industry would become thebottleneck of the expected improvement in supply, since airport’s capacity forms theupper bound of supply for air transport, and the quality and price of the service provided
by an airport have a direct and immense implication on the service of downstream
suppliers.
Facing the increasingly complexity of the market brought by the growing demand and theradical change in the airline industry, public authorities or governments who has operated
airports for a longtime gradually find it difficult to address these issues in the traditional
way. Many of them thus began to operate their airports in a more commercialized
approach, and some even sought a partial or full privatization of airports. Rikhy et al.
(2014) summarized that governments privatize airports aiming for developing traffic
demand or meeting such demand; financing large-scale airport infrastructure; andbringing efficiency to the design and operations. Indeed, some of these aims can beachieved through commercialization and privatization in many cases. For instance,
empirical evidences show that airports with majority private share outperform those withdominant public share in terms of efficiency (Oum et al., 2006; Oum et al., 2008).
However, regarding some aspects of the government’s motivation for these structuralchanges, it is still ambiguous whether the expected effects can be achieved. For instance,
there are opposing views on the effect of privatization on the infrastructure investment(Graham, 2020). Numerous studies have tried to figure out the relationship between
1

privatization and infrastructure investment from various perspective, while there are stillsome aspects needing further investigation. For instance, most studies only focus on thevolume choice of capacity expansion, and did not consider the timing for the investment.
Investment in airport infrastructure can be a huge project, and has a long-term impact, so
it is essential to address the timing issue to avoid the loss caused by the untimely
investment. Further, for some airports, non-aeronautical business has already become themost significant revenue source; many airports also welcome the adoption of newtechnology to enhance efficiency and reduce cost. A holistic view on the infrastructure
investments, not only the investment in capacity expansion, but also the investment in the
non-aeronautical facilities and the efficiency-enhancing project, is also necessary.
On the other hand, the structural changes more or less make the airport’s operation more
profit-oriented, thus raising a concern that airport might abuse its substantial market
power to provide service with higher price (Graham, 2011). Airports are traditionallydefined as natural monopolies, whereas the concern of the abuse of market power seldom
appear at that time, as they are treated as public utilities. In recent years, although thenatural monopoly aspect of airport has been eroded for many reasons, such as thediseconomies of scale, and a fiercer inter-airport competition (e.g., Gillen et al., 2001;
Tretheway, 2001; Kamp et al., 2005), many airports are still monopolies or quasimonopolies for various reasons. For instance, an airport with a catchment area which
never overlap with that of other airports would seldom face competition on departing andarriving passengers. Moreover, if not competent to be an international hub, it would notneed to consider how to attract transiting passengers. Clearly, such an airport wields
substantial market power. If it is operated in a profit-oriented way as a result ofcommercialization or privatization, the concern might come true. Traditionalcountermeasure against this problem is to impose regulation, which has been substantiallydiscussed in the previous studies. However, regulation can be costly and controversial insome cases, so it is worthy to figure out whether there any effective alternative toregulation. An introduction of intra-airport terminal competition, namely a separate
privatization of airport, attracts some attention, while its effect has not been fully
investigated.

2

1.2 Research Objectives
This thesis thus aims to take a closer look at some aspects of the effects of the structuralchange (Fig 1.1). We focus on privatization, which is the most radical structural change.
As mentioned, privatized airports can have some new concerns (high charge) calling forcountermeasures, and privatization can affect the implementation of the countermeasuresagainst the old concerns (capacity deficiency). Therefore, we first assess the plausibilityof airport terminal competition as a countermeasure against the abuse of market power ofprivate airport. We then aim to investigate how the “privatized extent” of an airport affectits decision-makings regarding infrastructure investment, the ultimate countermeasureagainst capacity shortage. To break down, the objectives to be achieved are as follows:(1) To figure out the effect of intra-airport terminal competition on price and social
surplus. Multiple business models characterizing potential organizational
structures of the airport before and after the introduction of terminal competition
are considered and compared, and several special cases are investigated (Chapter
3).
(2) To figure out the optimal investment rule for a sequential investment in two typesof projects, namely capacity expansion and cost reduction. To analyze the effectof airport’s ownership, the structure of downstream market, and demand
uncertainty on the timings and scales of the bi-project investment. We also
investigate how these factors affect the loss due to a suboptimal investment(Chapter 4).
(3) To investigate how the ownership structure of the airport affect its capacity
expansion behavior, considering two cases differing in the availability of thegovernmental subsidiary for the expansion (Chapter 5).

3ComplicatedenvironmentMore
CapacityHighchargeStructuralchange(Privatizedairport)Betterservice
Increasing
demandHigherefficiency

Fig 1.1 Linkage between the chapters

1.3 Scope and Limitations
This thesis focuses on one single airport. We do not consider the interaction with regard
to the countermeasures between multiple airports within a metropolitan area or a networkin our discussions, otherwise the complexity of modelling might escalate. Nevertheless,further consideration of multiple airports can be an interesting topic for future research.
This study mainly focuses on “private airport” instead of “privatization” itself as a process.
In Chapter 3 and Chapter 4, we treat “privatization” as a status-quo, and investigate thecountermeasures against high charge and capacity shortage. Therefore, we do not discuss
the reasonableness and feasibility of the privatization.
This thesis mainly employs analytical modeling approach to the analyze of the researchquestions. Most of the results are derived analytically, while in some cases numericalcomputations are resorted to for the comparison among these results. The advantage ofanalytical modeling is that solutions can be derived in closed forms without predefining
values for parameters in most cases. Hence, the results can be general without being
subject to a specific context. Although sometimes it might be difficult to havecomparative statics for the results or even closed-form final solutions, the friendly
properties (e.g., concavity) of the objective functions enable a quicker and more accurate
calculation rather than the optimization of complicated objective function throughheuristic algorithm. On the other hand, functions adopted in analytical model usually have
simple forms (i.e., demand functions are always linear). Reality is somewhat sacrificed
for the sake of tractability. An analytical modelling without explicitly defining the formof functions, which can further enhance the generality of results, proves to be a4

meaningful attempt in future work.
The study in Chapter 3 can be regarded as a normative study where we discuss theplausibility of the introduction of terminal competition. On the other hand, the studies in
Chapter 4 and 5 are generally descriptive. Our main focus is “if A changes, how will B
change?” Its result can provide some references to the trade-off for the decision-making,yet it might be difficult to claim that “the policy maker should …”, as we did not take
into account all the aspects of the motivation which needs to be considered
comprehensively in policy making.
Other notable limitations include:a. In the study of terminal competition (Chap. 3), we do not consider its effect on
investment.
b. In the study of bi-project investment (Chap. 4), we assume that the decision-makercan choose the scales of the projects discretely, whereas the volume of each scale
is exogenously determined, instead of being optimized by investor.
c. In the studies of airport investment (Chap. 4 and 5), lead times of projects are notconsidered.
d. In the studies of airport investment (Chap. 4 and 5), hypothetical values are set
for parameters in the numerical calculation sections, due to the lack of real-worlddata.
1.4 Outline
Chapter 1 briefly explains the motivations behind the study. Objectives, contributions,and limitations of the study are then stated.
Chapter 2 presents the impact of the structural change, showing the problems, introducing
the countermeasures, and reviewing the previous studies.
Chapter 3 investigates the effect of the introduction of terminal competition on price andsocial surplus. In the basic model, we consider two cases varying the freedom of airlines
to change their base terminal. Under each case, we compare six scenarios, each of whichcorresponds to one potential organizational structure of the airport before or after the
introduction of competing terminals, to figure out whether terminal competition can5reduce price and improve social surplus, and how should terminal competition be induced.
We then extend the study to see the effect of terminal competition in four special cases,
where terminals have different allocative efficiency, airlines participate in upstream
market, terminal compete on both price and service level, and terminals are regulated.
Chapter 4 studies a case that airport has the option to carry out two types of investment,
namely capacity expansion and cost reduction. We adopt a real option approach to themodelling of investment decision; the investor can choose the timing and scale forinvestments respectively to maximize its expected payoff. Considering the interrelationship between the two projects, we derive the optimal rule for the sequential
investment. We investigate the effect of airport’s ownership, the structure of downstream
market, and demand uncertainty on the timings and scales of the investments. We also
estimate the loss due to suboptimal investment and how it is affected by various factors.
Chapter 5 investigates the relationship between the privatization decisions and capacity
expansion decisions of an airport. In a real option approach, we analyze the effect of time
and the share to sell of privatization on the optimal timing and volume for the capacity
expansion, and the aggregated social surplus.
Chapter 6 summarizes the results and findings of this study, and propose the future scope.
1.5 Contributions
We would like to illustrate the contributions of the study chapter by chapter as follows:
Chapter 3: This study sheds light on the important yet unanswered question of whether
the introduction of terminal competition can restrain the abuse of market power by amonopoly airport and lead to lower prices and higher social surplus, in various potential
business models, considering the participations of airlines in the upstream and theproperties of operators. This study can also help support relevant decision-making, as
current empirical evidence is insufficient, and related discussions are always mixed with
political issues and conflicts of interest.
Chapter 4: We investigate the bi-project investment problem at an airport in a real option
approach, which has not been addressed in the previous studies, to the best of ourknowledge. The downstream market is explicitly modeled, which enables us to examine
6the effects of its features on the investment decisions. We formally prove the interrelationship between the two types of projects. Based on the inter-relationship, we correct
the optimal rule for the sequential investment proposed by the previous studies.
Chapter 5: By investigating the relationship between the privatization decisions and theinvestment decisions, and analyzing how these decisions affect the long-term social
surplus, we provide some references for the decision-makers who need to balance various
objectives when planning the privatization. We find that the effect of ownership on theinvestment decision might not be as simple as suggested by the previous studies.

7

Chapter 2 Literature review
2.1 The impacts of airport privatization
In the airport industry, the private sector involvement has become an important trend forseveral years. As per ACI (2018), 51% of the top 100 busiest airports more or less haveprivate participation, representing 41% of all passengers around the world. Privatization
has brought various impacts, and the most discussed is the impact on efficiency. Graham(2020) has extensively reviewed the relevant studies on this issue. In summary, moststudies find that privatization at least would not cause an efficiency decline. Moreover, interms of the effect of mixed ownership, Oum et al. (2006), Oum et al. (2008) and Adler
Adler and Liebert (2014) reach a common conclusion that airport with a minority private
share is most inefficient. The impacts on service quality and non-aeronautical businessare also important points to be investigated, but there lack convincing studies. Regarding
the impact on price, it is obvious that the privatized airport is very likely to levy a higher
charge than the public airport, constituting the reason why regulation on privatized airport
with great market power is essential. However, is there any effective alternative? We will
introduce one potential alternative in Section 2.2 and 2.3, and investigated its
effectiveness in Chapter 3. Regarding infrastructure investment, the impact might still beambiguous, as presented by the opposing views. ACI (2018) claims that ‘… privatization
has been shown to be a successful means by which to fund infrastructure development.
Indeed, ACI World estimates that airports with private sector participation indeed
invested 14% more in CAPEX measured as compared to their public counterparts and 12%
more than the global average in the last five years’. By contrast, IATA (2018) contends
that ‘… we could not see any gains in efficiency or levels of investment’. The
contradictory opinions and the lack of clarity of the effect necessitate further investigationof the issue. Chapter 4 and 5 investigate the relationship between privatization andinvestment from new perspectives to supplement those put forward by previous studies.
In prior to the study, the current situation of capacity deficiency is introduced in Section
2.4, and relevant studies in Section 2.5.

8

2.2 Intra-airport terminal competition
2.2.1 The abuse of market power by the airport
Market power, which generally refers to the ability of a firm to charge a price above
marginal cost and earn a positive profit, is wielded by a monopoly hardly facing
competition (Perloff, 2014). In terms of an airport, the monopolistic status, namely thelack of competition, may come about for two types of reasons: locational reasons andnatural monopoly reasons (Forsyth et al., 2010). The locational reasons say that there arevarious geographical and political barriers for multiple airports to coexist adjacently in
one region to compete with each other. Besides, some airports are not situated in afavorable location for the establishment of a hub, making the emergence of hub
competition unrealistic. The natural monopoly reasons argue that the entry of newcompetitor will be precluded, as the colossal sunk cost and the economies of scale forma huge barrier. Many markets are indeed showing these characteristics, so the airports
thereof are regarded as monopolies with market powers. Nevertheless, monopoly does
not necessarily result in the abuse of market power, if the airport is not profit-oriented1.
In earlier decades, given a lower aviation demand and a more conservative regulatory
regime, the monopolistic characteristic of the airport ought to be much stronger than how
it is now. Yet the concern regarding the abuse of market power were seldom raised at thattime, for airports were under public sectors which would not place a high premium on
profit. However, the trend of airport commercialization and privatization makes airports
more profit-oriented, and thus more inclined to abuse market power. Although themonopolistic characteristics of the airports are becoming weaker thanks to factors, suchas the growing demand and the emergence of LCC, it cannot outweigh the increasing
extent of the abuse of market power in many circumstances. The abuse of market power
often results in a high charge that harms the social benefit. Several empirical studies
investigate the abuse of market power occurred in recent years, for example, Bel andFageda (2010) find that private airports not regulated charge higher prices than public or
regulated airports.

1 Non-profit-oriented airport may also levy high charges, but this is mainly due to the inaction on reducing cost rather

than the abuse of market power.

9

2.2.2 Countermeasures
It is literally plain that there are two ways to tackle the problem regarding the abuse of
market power: to reduce the extent of the abuse, or to eliminate market power as much as
possible. The former refers to the restriction on the pricing right of the airport, namely aregulation, while the latter is to weaken the extent of monopoly of the airport, namely an
introduction of competition.
Various regulations are widely employed to counter the high charges levied by airport;
they can be generally categorized into three types (Graham and Morrell, 2017):
Rate of return (ROR) or cost-based regulation: Under this type of regulation, airport ispermitted to cover its cost and gain some profit by levying the charge. In other words, it
defines an upper bound for the rate of return on the asset base. Besides preventing theabuse of market power, ROR regulation can also work to encourage the capital investment,
for the larger the whole pie, the larger the airport can get, given a fixed share. However,a big difficulty regarding its implementation is the determination of the assets that should
be included in the asset base. In addition, a costly scrutiny on financial date is necessary
to prevent cost inefficiency and overinvestment. Airports where this type of regulation isadopted include Amsterdam, Athens, etc.
Incentive or price-cap regulation: Under this type of regulation, a price-cap is determinedbased on the formula CPI – X + Y where CPI stands for the consumer price index, X theefficiency gain target and Y the external cost. Since any efficiency gains that the airport
can make in excess of the required X will directly benefit the airport, this regulation can
encourage the operator to enhance efficiency. The cost of scrutiny for it is alsosignificantly lower than that of ROR regulation. However, this regulation mightundermine the operator’s incentive to carry out long-term investment, since it focused on
short-term operational efficiency gain within each price control period. Airports where
this type of regulation is adopted include Vienna, Heathrow, etc.
Price monitoring or light-handed regulation: This type of regulation will be imposed on
the airport only when the abuse of market power has been affirmed. The market power of
the airport is constrained by the threat of regulation, rather than actual regulation. It isconsidered as a more proportionate approach for dealing with the modern day, more
10

competitive, airport industry, due to its compatibility and flexibility (ACI Europe, 2014).
The main challenge for its implementation would be the establishment of the appropriate
trigger criteria. Airports where this type of regulation is adopted include Sydney,
Auckland, etc.
There are three common concerns whatever the type of regulation is. First, a dearth of acommon method for the assessment of market power makes it difficult to judge whether
a regulation is necessary for an airport (Bilotkach and Mueller, 2012). Second, while mostregulations focus on pricing and efficiency, the establishment of service standards orappropriate quality-monitoring systems are often overlooked. Third, heavy regulation that
implicates government intervention might lead to strong protests from stakeholders,which can evolve into time-consuming litigation (e.g., Aer Rianta v. Commission for
Aviation Regulation [CAR] in Ireland, 2002).
As a market-based anti-monopoly alternative to regulation, the introduction ofcompetition could be politically less controversial. The so-called ‘new view on airport
regulation’ (Gillen et al., 2001; Tretheway, 2001) argues that airports are no longer
natural monopolies and that more competition would be preferable to traditional
regulation. ACI Europe (1999) identified six different forms of competition or‘perceptions of competition’ between airports:
1. Competition to attract new services.
2. Competition between airports with overlapping hinterlands.
3. Competition for a role as a hub airport and for transfer traffic between hubs.
4. Competition between airports within urban areas.
5. Competition for the provision of services at airports.
6. Competition between airport terminals.
The first form refers to the situation that airports with remaining capacity compete witheach other to attract airlines to involve them into their airway networks. However, in thiscase, rather than airports themselves, the demand levels behind the airports play a more
important role in the airlines’ decision-makings. The second form and the fourth form canbe actually combined together under the current context that new means of transport suchas HSR has greatly improved passengers’ accessibility to the airport. This kind of11competition can take place between a primary airport and a secondary airport which has
seen strong development contributed by LCCs (e.g., Brussels and Charleroi, see Barbot(2006)), or between two main airports with different functions (e.g., Narita and Haneda).
The competition between Incheon and Narita for China-U.S. transit traffic exactly
exemplifies the third form. The last two are not competition between airports, but between
service providers within one airport.

Fig 2.1 Catchment areas of airports in Ireland

However, some of the above-mentioned forms can be unrealistic for some airports. In
terms of hub competition, a good geographical location proves to be an important
advantage, yet many airports do not have such merit. In some overcentralized countries,
there lacks the conditions for the competition between airports with overlapping
hinterland. A typical example is Dublin airport in Ireland. From Fig 2.1, we can see thatthe catchment area of Dublin Airport does not overlap with that of other Irish airports.
12This, together with the huge edge in the frequency of flight and the number of routes overother Irish airports, making Dublin Airport hardly face any domestic challenge. In
addition, a deliberate introduction of a competing airport (e.g., construction of a new
airport) could be fraught with many obstacles such as proper funding and land-use
restrictions. Hence, for airports located in remote areas with an independent hinterland, acompromised approach, introduction of intra-airport competition (e.g., competition
between terminals) is conceived, as a terminal is somewhat easier than an entire airport
to duplicate (Kuchinke and Sickmann, 2005).
2.2.3 Terminal competition: History and current status
Terminals within an airport can compete on price and service quality (Graham, 2018).
Price generally refers to the passenger service facility fee. Terminals compete forpassengers via airlines, as “terminal service” is one of the inputs for the airline’s
production of “air travel service”. Given a cheap and high-quality input, the airlines areable to produce cheap and high-quality output to attract consumers.
There are few industry examples of terminal competition (See Table 2.1). In Canada,
Terminal 3 at Toronto Airport was once taken over by a private consortium in 1987 toprovide new investment, and was permitted to levy higher fees for cost recovery (Juan,
1996). However, the practice did not last long as the terminal was brought back to nonprofit authority along with new policy development in 1996. Similar cases can beobserved at Birmingham Airport, Brussels Airport, and Ninoy Aquino International
Airport (Manila), where terminals were at one point operated separately through buildoperate-transfer (BOT) contracts but eventually all returned to centralized operation(Graham, 2018). At some airports (e.g., New York JFK, Munich, Riga), leader airlines
run terminals, ostensibly creating terminal competition, but in fact merely creatingcompetition among the leader airlines.

13

Table 2.1 Examples of airport terminal competition

Country

Airport/Terminal

Form

PeriodCanadaTerminal 3,

BOT1991 - 1996

BOT

1989 ~ ?

TorontoU.K.

Eurohub,
Birmingham

Philippines(terminated)

Intl. terminal,

BOT

1999 - 2005

ManilaU.S.

JFK (New York)

Vertical integration

Ongoing

Australia

Domestic

Vertical integration

Ongoing

terminals, several
airports

In the United Kingdom and Ireland, issues regarding terminal competition had been
seriously considered, and were brought into discussions several times. In a decision
document issued by the Department for Transport (2009), the UK government proposedthat for airports with substantial market power, the regulatory framework should notpreclude the possibility of separating the operation and development of terminals. Whilst
the proposal was advocated by airlines such as British Airways, it was opposed mainly
by airports for reasons such as difficulties in coordination and lack of evidence on interterminal competition. Finally, the government simply reiterated that the new regulatory
regime should not prohibit the development and operation of competing terminals. There
was no further progress as it was not convinced that the risks and shortcomings would
outweigh the potential benefits, such as reducing regulatory costs and improving thepassenger experience. In Ireland, the introduction of terminal competition at Dublin
Airport has been put on the agenda previously. Strong conflicts of interest among
stakeholders and ministerial turnover eventually led to the shelving of the plan, although
the expert panel “was favourably [sic] disposed to the idea of an independent terminal on
operational and technical feasibility grounds” (Irish Department of Transport, 2003;
Reynolds-Feighan, 2010). To summarize, no consensus has been reached so far, since no
stakeholder can give opinions persuasive enough to convince their opponents. The door
14

is not completely closed, though, as the Department for Transport still insists that “…thenew regulatory framework will not preclude the development of inter-terminal
competition…” even after the unbundling of airports operator BAA (Department forTransport, p.69, 2013).
2.3 Relevant studies regarding terminal competition
Few studies examine intra-airport terminal competition. McLay and Reynolds-Feighan(2006) elaborate on the incentives, processes, and potential implications of terminal
competition, taking the Irish case as an example. Dublin Airport, the gateway of Ireland,
wields significant market power due to its indispensability and independent catchment
area. The entry of a new competitor seems unrealistic because potential developers of
new airports would face substantial regulatory obstacles. Regulatory efforts to restrain
the incumbent monopolist resulted in strong protest; thus, the introduction of terminal
competition was proposed as an alternative. The authors raise several issues that needfurther consideration in terms of terminal competition. They suggest that the net welfare
effect of introducing terminal competition will depend upon the aggregate of individual
effects on each type of efficiency. They also note that airport service as a whole is abundle of terminal service and airfield service, and attention should be paid to thischaracteristic when deciding on a business model. Finally, they touch upon the investment
implications of terminal competition. However, they do not further analyze the impactsof various business models on competing terminals to see whether they can work to
improve social welfare; they only suggest that “it may be unreasonable to expect
competition between airport terminals to deliver unambiguous welfare improvements if
it is not accompanied by changes to the structure of the incumbents business” (p. 199). In
a subsequent study, Reynolds-Feighan (2010) reviews the development of terminal
competition in Dublin Airport, which went nowhere. Although he concludes that the Irish
case is now added to the shortlist of examples cited by monopoly airport operators
claiming that terminal competition is unworkable, the key questions of whether
competing terminals can really restrict the abuse of market power of airports and improvesocial welfare remain unclear.
Issues regarding competition among rival facilities are widely investigated due to their15

significance in transportation studies (i.e., De Borger and Van Dender, 2006; Basso andZhang, 2007; Barbot, 2009; Zhang et al., 2010; Saeed and Larsen, 2010; Matsumura andMatsushima, 2012; D’Alfonso and Nastasi, 2012; Benoot et al., 2013; Xiao et al., 2013;
Teraji and Morimoto, 2014; Noruzoliaee et al., 2015). We categorize these studies into
three general types. The first type investigates the impact of duopolistic competition
between rival facilities. The competition is purely “substitutive” and the downstream
market is assumed to be under perfect competition, but there are multiple decisions tomake, so the impact is not self-evident. For example, De Borger and Van Dender (2006)
study the duopolistic interaction between congestible facilities that supply perfect
substitutes and make sequential decisions on capacities and prices, and compare theresults to monopoly and first-best outcomes. They find that the service level provided
under a monopoly is the same as that in the social optimum while surpassing that provided
under a duopoly. Other similar studies include Noruzoliaee et al. (2015), which considers
competitive airports with different objectives, and Randrianarisoa and Zhang (2019),
which considers competing ports under uncertainty. The second type of study relaxes theperfect competition assumption of the first type by modelling downstream market, while
the upstream market is still oligopoly of substitute goods. For example, Basso and Zhang(2007) extends De Borger and Van Dender (2006) by further considering the verticalstructure of the market and suggest that the conclusion of De Borger and Van Dender(2006) will hold only if the downstream market is perfectly competitive. Benoot et al.
(2013) extends Basso and Zhang (2007) by specifically characterizing intercontinental
airport regulators. Barbot (2009) analyses the incentives for vertical collusion between
one airport and one airline that compete with another airport and another airline and findsthat the incentives exist when airports and airlines have different market sizes. Zhang etal. (2010) study the effect of the duopolistic competition of airports on their revenue
sharing with airlines and finds that competition results in a higher degree of verticalcooperation between an airport and its home carriers and improves social welfare. Other
similar studies include D’Alfonso and Nastasi (2012), which extends Barbot (2009) byinvolving other types of vertical agreement besides vertical collusion, and Saraswati and
Hanaoka (2014), which extends Zhang et al. (2010) by considering a network with hub
and spokes. The third type considers competing facilities with complementarity. For
example, Wan et al. (2016) investigate the strategic investment decisions of local
16

governments on regional landside accessibility in the context of seaport competition
where complementarity exists between the inland and port regions, and discusses theconditions under which the social-optimal grand coalition can be achieved; However, thedecision makers regarding the investment in accessibility are all social-surplus maximizer,
which is different from our problem. Teraji and Morimoto (2014) address the question of
how airport competition affects the carriers’ network choice where complementarity
originates from connecting routes.
In summary, we can find that, (a), in many studies, facilities that compete with each others
are pure substitutes and downstream carriers do not have market power (first type); (b),some studies model vertical structure and the complementarity between upstream anddownstream can be strengthened through considering vertical arrangement, while theupstream market is still an oligopoly of pure substitute goods (second type); (c), in studies
where complementarity and substitute coexist in upstream markets, downstream markets
are not (or are only roughly) modeled (third type). To answer the question of our study,
A model that involves the coexistence of complementarity and substitutes in upstream
markets and a relatively comprehensive modeling of downstream markets is needed. We
can hardly refer to the conclusion of previous studies directly. For instance, despite thesimilarity to terminal competition, airport competition is basically between substitute
goods. Passengers only need to choose one airport among the competitors and buy theairline service therein. However, the service at an airport consists of the service atterminals and the service at airfield facilities. One can choose a terminal service(equivalently an airline service thereof) if multiple terminals exist, while the purchase of
the single (bundle of) airfield service is compulsory, regardless of the terminal choice.
This feature makes the intra-airport competition different from that between airports andtherefore a new model is necessary.
2.4 Capacity shortage at airports
The air transport market has seen unprecedented demand growth in recent years. The
International Air Transport Association suggests that air passenger numbers could double
to 8 billion per year in 2039. By region, Asia Pacific is predicted to see the strongest
annual growth rate of 5%, while the lowest number is 2.2% for North America and Europe.
17

Although COVID-19 is having a profound impact on near-term demand, strong recovery
can be expected in the next few years.
It is questionable whether the existing capacity and the capacity planned to be increased
can meet the rapid growth of air traffic. In Europe, although a 16% increase in airport
capacity has been planned (As of 2018), it is reckoned not adequate to meet the demand.
In the optimistic scenario, it is estimated that 360 million passengers, that is, 16% of thedemand, will not be accommodated in 2040, given that no additional expansion plan will
be carried out. Even in a fairly conservative estimation, the capacity gap in 2040 willreach to 8%. In the optimistic scenario, almost all countries in Europe will more or lessencounter capacity deficiency. Even in the conservative scenario, countries such as
Turkey and Britain are unlikely to be free of capacity concerns. The capacity gap, as aresult, might cause the total delay to increase from 12.3 minutes to 20.1 minutes on
average, per flight. There were 6 airports at a level of congestion that 80% or more of thecapacity are occupied for at least 6 consecutive hours in 2016, while such congested,'Heathrow-like’ airports will reach a number of 16 by 2040 in the conservative scenario,or even 28 in the optimistic scenario (Eurocontrol, 2018).
Gelhausen et al. (2019) adopt the Capacity Utilization Index (CUI), namely the ratio of
the average hour traffic volume to the 5% peak hour traffic volume, to calculate thecapacity utilization at airports worldwide. As per their yardstick that airports with CUI
values higher than 0.65 are those suffering significant congestion problem2, significantly
congested airports already cover about 35% of the total global flights. Furthermore, theirglobal analysis shows that 35 airports are more or less constrained in terms of capacity in2016. These “constrained” airports are the primary airports of the global network; theyhandled a traffic volume of 13.3 million aircraft movements, corresponding to a share ofnearly 19% of the global traffic in 2016. Finally, they estimate that there will be a
substantial shortage of capacity in 2040, with almost 256 million passengers cannot be
served due to capacity constraints. The problem might be especially severe in Asia, with
Delhi Indira Gandhi, Mumbai Chhatrapati Shivaji and Jakarta Soekarno-Hatta ranked as
the top three constrained airports, due to the strong growth in demand.

2

If the corresponding 5% peak hour volumes reach the near-capacity values.

18Consequently, as Alexandre de Juniac, Director General and CEO of IATA said in 2018,“we are in a capacity crisis. And we don't see the required airport infrastructure
investment to solve it”3. There are indeed various ways to mitigate the capacity crisis,such as the development of a more efficient air traffic control system, or the adoption ofa more reasonable slot allocation. However, capacity expansion proves to be the mostdirect and effective countermeasure from a long-term perspective. It can be a constructionof a new runway, a new terminal, or even a new airport. As per Gelhausen et al. (2019),
the number of runways at airports of the global network has increased by 7.9% from 2008
to 2016. In Asia, the increasing rate is outstandingly high reaching 23.1%. More capacity
expansion projects has been planned and will be started henceforward. Table 2.2 lists
some representative examples across the globe.
Table 2.2 Examples of investments in capacity expansion at airports

AirportCountry

Major

Expected to be

infrastructurescompleted inCape Town

South Africa

Runway2023

Chongqing

China

Runway & terminal

2024

Hong Kong

China

Runway2024

Melbourne

Australia

Runway2025

DenverU.S.

Terminal

2025

NaritaJapan

Runway2029

ViennaAustria

Runway?

Hanoi

Vietnam

Runway & terminal?

2.5 Studies on airport capacity expansion
In general, quantitative studies on airport capacity construction/expansion can be
categorized into two major types. One major type of studies aims to develop new
methodologies for the decision-makings on the capacity expansion. Equivalently, they try
to devise new decision-making techniques. For instance, Smit (2003) developed a3

https://www.france24.com/en/20180604-global-airport-capacity-crisis-amid-passenger-boom-iata19

discrete-time binomial analysis under a combined real option and game theory framework,of which the modularity enables the consideration of many strategic features. Jorge andde Rus (2004) developed a cost–benefit analysis approach which does not entail muchdata and time for the investment in airport infrastructure. Their emphasis is placed in theconsistency across projects in deciding whether a given project is a good or bad
investment, rather than on the accuracy of project return estimates. Sun and Schonfeld(2015) proposed an “Out-Approximation” method which can enhance the efficiency ofthe optimization for the airport capacity expansion model.
Another major type focuses on the effects of various factors on the decision-makings
regarding the capacity expansion using existing methods, to which most studies belong.
We can further subdivide this major type into two types: the orthodox approach and thereal option approach, based on if the opportunity cost, or the option is taken into account.
The orthodox approach: Studies belonging to this type generally ignore the optimal timingof the expansion. Instead, they mainly focus on the optimal volume of the capacity.
Zhang and Zhang (2011) investigated the effects of the airport’s objective and the formof regulation on the capacity investment decisions. They found that profit-maximizing
airports and regulated social surplus-maximizing airports tend to over-invest inaeronautical capacity. One limitation of their study is that they did not consider the casethat the airport has a mixed objective, neither pure profit nor pure social surplus. Xiao etal. (2012) considered the mixed objective in token of the public-private ownership.
Although the subject is seaport, their model can also be compatible with airports thanks
to the similarity between them. This kind of single-airport investment problem has been
extended in various directions: Zhang and Zhang (2006) examined how can airlines’
behavior of self-internalization affect the investment decision; Lin and Zhang (2017)considered a hub-spoke network and the scheduling of airlines to investigate the effect ofdifferent charging form on the capacity expansion decision; Kidokoro et al. (2016)
included the investment in non-aeronautical infrastructure. However, these abovementioned studies did not address the demand uncertainty which can be non-negligible
in many cases in the air transport industry. Xiao et al. (2013) studied on the airport’s
capacity choice addressing the demand uncertainty by a uniform distribution. Xiao et al.
(2016) further enlarged their previous scope by investigating the effect of airport-airline
20

vertical structure on the capacity choice. Several studies also investigated the capacity
expansion problem of two competing airports. For instance, Basso (2008) examined theeffect of airport deregulation on the capacity investment of two congestible airports;
Noruzoliaee et al. (2015) studied the capacity and pricing choice of two congestible
airports in a multi-airport metropolitan region, under transition from a pure public,
centralized airport system to partial or full privatization. However, most of these studies
did not address the demand uncertainty as well.
The real option approach: Studies belonging to this type always explicitly address theissue of timing, continuously or discretely. For example, Xiao et al. (2017) modelled thereal option by supposing that the airport has another opportunity to expand its capacity atsome time in the future, on the reserved land purchased before. Balliauw and Onghena(2020) investigated the effect of various factors on the optimal timing and volume ofcapacity expansion at a profit-oriented airport. Balliauw et al. (2019) can be regarded as
an enhanced version of the former, with facility rivalry addressed through a game theorymodelling, although the subject was seaport. Zheng et al. (2020) studied the timing
decision of airlines’ investments in exclusive airport facilities in the presence of demandambiguity and competition.
However, despite the great number of the relevant literatures, there still lacks studies that
focus on the relationship between privatization and investment considering timing by areal option approach. Real option approach greatly underlines the opportunity cost ofinvestment, which is not addressed by the traditional cost-benefit analysis. If an
irreversible investment is carried out, the cost incurred can hardly be redeemed. Whenthe uncertainty of the market is high, its natural that the investor tends to wait more to see
how will the price or the demand changes, in order to avoid the loss caused by a potential
negative shock. Therefore, it is more reasonable for the investor to invest only when the
NPV exceeds the cost by some degree, as suggested by the real option approach. The
project at an airport, such as capacity expansion, is always costly and lumpy, and the air
travel demand can be very vulnerable to catastrophes, making the opportunity cost nonnegligible. Thus, adopting real option approach to model the decision-making regarding
the investment in airport projects can be more reasonable. In Chapter 4 and Chapter 5, weinvestigate the investment behavior of private airport employing real option approach.
21

Chapter 3 Effects of airport terminal competition: A verticalstructure approach
3.1 Introduction
As mentioned in Section 2.1, the introduction of intra-airport terminal competition isconceived as a potential alternative to the regular countermeasures, such as regulation andthe introduction of airport competition, against the abuse of market power by monopolist
airport. In the United Kingdom and Ireland, regarding issues had been intensively
discussed for several times, yet no consensus has been reached mainly due to conflict ofinterest and other political factors. Nevertheless, the belief that terminal competition canwork to restrain the abuse of market power was seldom doubted; the British document
highlights that the target airports for the introduction of terminal competition are thosewith such power.
However, it is far from certain that an airport with competing terminals will levy lower
charges. Since the services of different terminals are substitutes, whereas terminal
services and runway services are complementary, splitting the operation of an airport byintroducing terminal competition will not only create a substitutive duopoly between
terminals but also create at least one complementary duopoly between terminal andairfield facilities. Economic theory suggests that compared to monopoly, a substitutive
duopoly can reduce prices, while a complementary duopoly will raise prices, giving riseto an interesting question as to which effect will dominate the other when they coexist.
This is one of the most crucial issues with respect to terminal competition, and no
academic study has hitherto investigated it.
Thus, to bridge this research gap, this study aims to investigate the impact of terminal
competition on pricing and social surplus under different business models. We adopt ananalytical model with a vertical structure and three market participants including anairfield and two terminals in the oligopolistic upstream market. Competition among
airlines, interactions between downstream and upstream markets, and competition
between terminals are formulated, and the airport’s equilibrium prices are solved. We
compare and analyze the results of six different business models in two cases differing in
22

airlines’ freedom to change terminals. We further investigate four extended cases
addressing terminals with different allocative efficiency level, airlines’ participation in
the upstream market, multi-dimensional competition, and regulated terminals.
Contrary to the common belief, we find that the introduction of terminal competition
cannot restrict the abuse of market power by a profit-oriented airport. In other words, theprice of the service provided by the airport with competing terminals can be higher,
causing a loss in the social surplus. This is because the existence of the airfield facilities,
which is complementary to the terminals. If we introduce terminal competition, not only
the substitutive duopoly, but also at least one complementary duopoly will be created.
When these two types of duopoly coexist, the price reducing effect of the former will bedominated by the price increasing effect of the latter, resulting in the negative outcomes.
The contribution of the present paper is twofold. From the perspective of modelling, westudy a differentiated Bertrand model of substitute and complementary goods with avertical structure. We investigate the effect of firm number and the product combinationof each firm on prices and social surplus. This issue is seldom addressed in previous
studies. Practically, this study sheds light on the important yet unanswered question ofwhether the introduction of terminal competition can restrain the abuse of market powerby a monopoly airport, or is it better to privatize an airport’s facilities separately comparedwith the all-in-one privatization. This study can also help support relevant decisionmaking, as current empirical evidence is insufficient, and related discussions are always
mixed with political issues and conflicts of interest.
3.2 The basic model
3.2.1 Description of the problem
Stakeholders: We focus on one section of the entire supply chain of the air transportindustry. In the upstream, we focus on one single airport. Inside the airport, there are three“facility group” denoted as airfield, terminal A, and terminal B. Airfield refers to thefacilities which are prepared for the movement of aircrafts (e.g., runway, taxiway).
Terminal A or B does not necessarily refer to one terminal; instead, it can be a group of
several terminals. We assume that each “facility group” cannot be divided and operated

23

by multiple operators. In the downstream, there are 𝑁 = 𝑛𝐴 + 𝑛𝐵 airlines, with 𝑛𝐴 atterminal A, and 𝑛𝐵 at terminal B. We ignore the entry and exit of airlines.
Decision-making process: In the upstream, each “facility group” has an action which isthe price to levy. For airfield, the action is landing charge 𝑇 per flight; for terminal, theaction is terminal fee 𝑤 per passenger. The three actions (prices) are determined in a
Bertrand fashion4. Landing charge is levied to all airlines, while terminal fee is levied tothe passengers of the corresponding terminal. In the downstream, 𝑁 airlines choose theirquantities (passenger) in a Cournot fashion to maximize their respective profits
respectively, given the decisions of the upstream5. The relation between the upstream anddownstream will be modelled as a Stackelberg game, where the upstream acts at first(stage I), and then the downstream follows (stage II) (Fig 3.1).
Passenger fee 𝑤𝐴

Terminal A

Airfield

Terminal B

𝑛𝐴 Airlines

Quantity(airfare)𝑛𝐵 Airlines

Quantity(airfare)Charge 𝑇

Passenger

Passenger fee 𝑤𝐵

Fig 3.1 Decision making process

Cases: In the basic analysis, we consider two cases. In case 1, we assume that theallocation of airlines at the terminals is fixed. In case 2, we assume that the allocation ofairlines can be influenced by the price. Airlines have the freedom to choose their baseterminals according to the terminal fee. In each case, we consider six Business Models toreflect various situations before and after the introduction of terminal competition (Fig4

If we assume a Cournot model in the upstream market, (i.e., terminals compete on outputs [quantities ofpassengers]), one terminal will be driven out of market in some cases (i.e., Model II introduced in followingpart). Thus, to ensure that all Models investigated are at least feasible, we adopt a Bertrand model in
upstream market. In fact, there are hardly any study regarding airport competition assuming a Cournot
model to the best of our knowledge.
5
Brander and Zhang (1990) and Oum et al. (1993) find that Cournot model seems much more consistent
with the empirical data of U.S. airline market. Many recent studies also assume Cournot behaviour among
airlines under imperfect competition (e.g., Czerny and Zhang, 2011; Gillen and Mantin, 2014; Xiao et al.,2017; Zheng et al., 2020).

243.2). The description of each Model is as follows:Business Model I (Centralized operation): A single profit-maximizing operator
operates all facilities including the airfield and both Terminals A and B in the airport. It
may correspond to an airport which is privatized in an all-in-one approach. Suchprivatization can have various motivations, such as enhancing efficiency, or simply
producing financial gain (Graham, 2011). This Model serves as a yardstick which enables
us to investigate whether a “separate privatization” has the function of a regulation,
leading to a better outcome.
Business Model II (One independent terminal): Operator 1 operates the airfield and
one terminal (e.g., Terminal A) while the remaining terminal (e.g., Terminal B) is
operated by operator 2. All operators aim to maximize profit6. Historically, the cases ofToronto airport and Manila airport might be similar to Model II, while none of them lasted
long. Currently, there are no such evidence, to the best of our knowledge (Graham, 2018).
Nevertheless, this Model represents a highly likely form of terminal competition and it
addresses the issue of runway service allocation where the incumbent who operates one
terminal and airfield facilities jointly can “bully” the independent terminal operator by
over-biding runway access.
Business Model III (All independent): There are three operators. Each operates one
facility independently. All operators aim to maximize profit. We investigate this Modelto see whether it can serve as a possible method to tackle the runway service allocation
problem as the integrated terminal-airfield operation of the incumbent is partitioned aswell.
Business Model IV (Independent airfield): Operator 1 operates two terminals while
operator 2 operates the airfield. All operators aim to maximize profit. This Model canshow the effect of a pure complementary duopoly.
Business Model V (Regulated airfield): Operator 1 operates two terminals, while
operator 2 operates the airfield. Operator 2 follows the Ramsey pricing, which maximizessocial surplus subject to a non-negative profit constraint, while operator 1 aims to
6

The question whether Terminal B will be driven out of the market in this Business Model is discussed inAppendix A.

25

maximize profit. This Model is a variant of Model IV with complementarity neutralized.
Business Model VI (Regulated airfield, competing terminals): Operators 1 and 2
operate Terminals A and B, respectively, while operator 3 operates the airfield. Operator
3 follows the Ramsey pricing, which maximizes social surplus subject to a non-negative
profit constraint, whereas operators 1 and 2 aim to maximize profit. This Model can be
regarded as another method to resolve the runway access allocation problem by imposing
a price-cap on the price of runway access.
Case 2

Case 1Business model I
Terminal A

Terminal B

Airfield

Terminal A

Terminal B

Terminal A

Terminal B

Airfield

Operator 2

Operator 1

Operator 2

Operator 1

Business model IV
Terminal A

Terminal B

Operator 1

Airfield
Operator 3

Operator 1

Business model V
Terminal A

Terminal B

Operator 1

Airfield
Operator 2

Business model III

Business model II

Business model VI

Terminal A

Terminal B

Operator 1

Operator 2

Airfield
Operator 2

Airfield
Operator 3

Red: profit-maximizing operator Green: social surplus maximizing operator

Fig 3.2 Illustration of business models

The action set of each player (operator) in each Model depends on the facility groups it
operates. Fig 3.3 shows a more concrete demonstration of the decision-making process
taking Business Model II as an example. The effect of terminal competition thus can be
figured out by comparing the equilibrium outcome of each Business Model7.

7

We only focus the payoff, profit or social surplus, in one period, not in a dynamic approach. We can seethat some of the ranking of price and social surplus in each Business Model will not change, given any
parameters including that reflects the demand dynamics, so the ranking of one-period social surplus andthe ranking of aggregated net social surplus will be same. Therefore, a one-period consideration might not
be implausible.

26

Example: Business Model II
𝑇: Airfield charge; 𝑤 : Terminal fee; : Passenger number

Player 2

Player 1
Airfield

𝑇 𝑤𝐴

Payoff:𝑤𝐵

𝑇 𝑤𝐴 𝑤𝐵

BertrandActions: 𝑇 𝑤𝐴

𝑇 𝑤𝐴 𝑤𝐵

Payoff:Actions: 𝑤𝐵

𝑇
𝐴 𝑇 𝑤

𝑤

𝑤𝐵

𝑇 𝑤𝐴

Terminal A

Airline B2
𝐵

Airline A𝑛𝐴
For each𝐴

Action:

𝐴

𝐴

𝐴

𝑤

Airline B1

𝐴

Airline A2

Payoff:

𝐵 𝑇 𝑤

Terminal B

Airline A1
𝐴

Upstream

Terminal B

Terminal A

𝑇 𝑤𝐴

𝐵

Downstream

Airline B𝑛𝐵

Cournot

For each
Payoff:𝐵

Actions:

𝐵

𝐵

𝐵

𝑇 𝑤𝐵

Fig 3.3 A detailed illustration of the model’s framework

We also consider some special cases. They are either uncommon in the real world, ordifficult to be investigated for all six business models in the basic analysis section.
Therefore, studies on them are relegated to the extension section. Extension 1 considersa case that the unit cost or unit non-aeronautical profit of each terminal is different.
Although such case might well exist in the reality, it is difficult to deal with all six Models
here, so we only investigate Model I and II in this section. Extension 2 considers a casethat airline(s) participate in the upstream by operating or holding share of the terminal(s).
Although such situation can be observed in some countries such as U.S. and Australia, asa practice it is far from common. Extension 3 considers a case that the terminals cancompete not only on price, but also on service level. As is with the case of Extension 1,
it is difficult to treat all six Models, so we only compare Model I, II and III. Extension 4
focus on a case that the airfield is public, while the terminals are not purely social surplus
oriented, which corresponds to the potential situation that terminals are to some degree
allowed to pursue profit, in terms of a public airport.

27

The notations of the variables and parameters are shown in Table 3.1
Table 3.1 Notations for Chapter 3

Decision variables
Number of passengers of airline at terminal ℎ𝑇Landing charge𝑤

Usage fee of terminal ℎ𝑣 (extension 3)

Service level of terminal ℎParameters𝑎Own-price effect𝑙 0≤𝑙≤1

Cross-price effect coefficient𝑠

Aircraft capacity * load factor𝑐Unit cost per flight for airline𝛾

Unit cost per flight for airfield facility𝑛Number of airlines at terminal ℎ𝑟Difference between unit non-aeronautical profit and unit
cost per passenger at terminal ℎ𝛼Delay cost for passenger𝛽Delay cost for airlinek

Airport capacity

σ (extension 3)

Unit cost of service provision per passenger

η (extension 3)

Own-price effect in terms of service levelm 0≤𝑚≤1(extension 3)

Cross-price effect coefficient in terms of service level

3.2.2 Objective functions
We use a quadratic utility function to represent consumer’s utility with differentiatedgoods as (e.g., Vives, 1999)8:8

All “utility” presented in this thesis refers to the cardinal utility.

28

where𝑛= ∑ =ℎ

𝑈=

𝐴+

,ℎ∈{

}.

𝐵−

𝑎 𝐴 + 2𝑎𝑙 𝐴 𝐵 + 𝑎 𝐵

3.1

denotes the number of passengers of airline at

Terminal ℎ. 𝑎 denotes own price effect, and 𝑙 denotes cross-price effect or degree ofsubstitution between airlines at different terminals and the product differentiation ofterminal services. 𝑎 > 0, and 𝑙 can be any value within the interval [0 1]. 𝑙 = 1 meansthat the two terminals and their airlines are perfect substitutes, while 𝑙 = 0 indicates that
services provided by airlines at different terminals are independent (e.g., a domestic
terminal and an international terminal). If 0 < 𝑙 < 1, airlines and services at different
terminals are somewhat substitutable. We assume that airlines in the same terminal are
perfect substitutes. Thus, the cross-price effect among them is also denoted by 𝑎 in thebasic cases. Maximizing utility, we have the following demand function:
𝜕𝑈𝜌 = 𝑝 + 𝑤 + 𝛼𝛿 = 𝜕𝑞 = 1 − 𝑎ℎ𝑖𝑛where 𝛿=𝐹/𝑘, 𝐹 = 𝐹𝐴 + 𝐹𝐵 , 𝐹 = ∑ =ℎ 𝑓 , 𝑓= /𝑠, ℎ =

+𝑙

3.2

, and −ℎ labels the terminal

besides ℎ. 𝜌 denotes the total cost that a passenger faces including airfare 𝑝 from theairline, passenger fee 𝑤 from the terminal, and time cost 𝛼𝛿, where 𝛼 is the time-value
parameter, and 𝛿 denotes the congestion delay at the runway. 𝑘 denotes the capacity of
airfield facilities. 𝑓 denotes the number of flights of airline at Terminal ℎ. 𝑠 denotes
the product of average aircraft capacity and load factor. For simplicity, we assume thatthe 𝑠 is identical for all airlines9. Besides, we do not consider the congestion cost at theterminals
Some may argue that it is unnecessary to include the congestion cost in the demand
function, since the passengers have to use the airport in many cases; they have no otherchoices even if the congestion therein is very severe. This setting can be justified asfollows. Due to the report of IATA, the air travel demand elasticity is estimated to be 0.6 at supra-national level in 200710. Although the value implies that the air travel demandis relatively inelastic, we cannot assert that the price has no effect on the decision-makings

9

For airlines with similar business model, this assumption is not far away from some real cases. For
example, the s values of U.S.’s legacy airlines in 2019 is 160 for American, 165 for Delta, and 167 forUnited (Source: http://web.mit.edu/airlinedata/www/default.html).
10
https://www.travelready.org/PDF%20Files/Travel%20-%20IATA%20-%20Air%20Travel%20Demand.
pdf

29

of the passengers. As an economic man, which is implicitly assumed in the modelling,
the passenger will consider the total price of the travel including the monetary cost andtime cost, so the congestion delay cost can have an effect on her decision-making
whatever the extent is. If the effect is weak, we can express it by a small time value
parameter α, and doing so would not affect our analytical results, but it might not beappropriate to completely exclude it.
The profits of airlines in each terminal are formulated as:=𝑝−𝑓

𝑐 + 𝑇 + 𝛽𝛿

∀ℎ

3.3

where 𝑐 denotes the unit cost per flight for the airline, 𝑇 denotes the charge for runway
service from the airfield facilities, and 𝛽 denotes the time cost of the congestion delay forairlines11.
The profit functions of airfield and terminal operators are:
𝑅 =𝐹 𝑇−𝛾

=

𝑤 +𝑟

3.4
∀ℎ

3.5

Where 𝑅 labels airfield, 𝛾 denotes the marginal cost per flight for the airfield facilities,𝑟 denotes the difference between unit commercial profit and marginal cost per passenger
for Terminal ℎ. For simplicity, we assume 𝑟𝐴 = 𝑟𝐵 = 𝑟 in the basic case and leave theanalysis for asymmetric 𝑟 to Section 4.1. The objective functions of operators areformulated as:
𝑗 =𝑥 𝑅+𝑦 𝐴+𝑧 𝐵

3.6{𝑥 𝑦 𝑧} have different values depending on the Business Model and operator 𝑗. For
example, in Model II, {𝑥 𝑦 𝑧} = {1 1 0} when 𝑗 = 1, and {𝑥 𝑦 𝑧} = {0 0 1} when 𝑗 = 2.
Readers can learn the values of {𝑥 𝑦 𝑧} in different cases from Fig. 2. Social surplus is

11

If we ignore the complicated problem of fleet rotation and allocation in the whole network of an airline,
and assume each route to be independent, then the maximization of the airline’s gross profit is equivalent
to the profit maximization for each route. In this case, our setting can be partially justified by assuming thecharge of airport in the other end of each route is exogenous and fixed (e.g., cost-covering charge). Under
such assumption, the charge in other airports can be regarded just as a part of the airline’s cost which will
not change as the airline’s decision changes.

30defined as the difference between utility and total cost:
1
𝑆 = 𝑠 − 𝛼𝛿𝑠 − 𝑐 − 𝛽𝛿 − 𝛾 𝐹 − 𝑎𝑠 ( 𝐹𝐴
2

Note that in Model V and VI,+ 𝐹𝐵 ) − 𝑎𝑙𝑠 𝐹𝐴 𝐹𝐵 + ∑ 𝑟 𝑠𝐹

3.7

=𝐴 𝐵

𝑅 = 𝑆 in Eqn. (3.6) with non-negative profit constraint.
𝑐+𝛾 𝑐 𝛾Moreover, to simplify the following analysis, we assume that 𝑠 > max{ +𝑟

𝑟

} and

𝛼𝑠 + 𝛽 < 𝑎𝑘𝑠 . Backward induction is employed to solve the problem, and details areshown in Appendix B1.
Flight of airline at terminal ℎ and total output under subgame perfect equilibrium canbe then derived as:𝑓

𝑇 𝑇

𝑇𝐴 𝑇𝐵 ==𝑘 (𝐶 𝑛+1 𝑇 +𝑐−𝑠 −𝐶 𝑛
𝐶 𝑛 𝑛 −𝐶 𝑛 +1 𝑛

𝐶1 𝐶2 𝑘 2 𝑠𝑛𝐴 𝑛𝐵 (𝑇𝐴 +𝑇𝐵 +𝑇 +𝑐−𝑠 )+1𝑐 𝑠 )+𝐶1 𝑘 2 s(𝑛𝐴 𝑇𝐴 +𝑐 𝑠 +𝑛𝐵 𝑇𝐵 +𝑐 𝑠 )𝐶22 𝑛𝐴 𝑛𝐵

𝐶12 𝑛𝐴+

𝑛𝐵 +

3.8

3.9

Where 𝑇 = 𝑇 + 𝑠𝑤 denotes the total price that a flight of airline at terminal ℎ bears.
This integrated variable will be frequently used in the subsequent analysis forconvenience. 𝐶 = 𝑘 𝑠 𝑎𝑘𝑠 + 𝛼 + 𝛽 and 𝐶 = 𝑘 𝑠 𝑎𝑙𝑘𝑠 + 𝛼 + 𝛽 ; thus 𝐶 ≥ 𝐶 .
The effects of prices on outputs can be summarized as follows:
Lemma 3.1 If airlines cannot change the base terminal, and the number of airlines in eachterminal is fixed, then
𝜕𝑓
𝜕𝑓
<0
𝜕𝑇
𝜕𝑇

>0

𝜕
𝜕
<0
<0
𝜕𝑇𝐴
𝜕𝑇𝐵

The proof is presented in Appendix B2. The effects are quite intuitive in that the output(the number of flights) decreases as the price of the airline’s own terminal increases, while
the output increases with the price of the rival terminal due to the relation of substitutes,
and the total output always decreases as the prices increase.
3.2.3 Case 1: airlines cannot change terminal
FOCs of the airfield and facilities in each business model are shown in Appendix B1. Bysolving these FOCs, we have the equilibrium airfield charge and terminal fees in each31

Business Model, as shown in Appendix C1. Combining them by 𝑇 = 𝑇 + 𝑠𝑤 , wederive the equilibrium total prices in each Business Model as summarized in Table 3.2.
Comparing total prices in each business model and denoting the equilibrium value in eachBusiness Model by a superscript of its label (i.e., 𝑇𝐴𝐼 denotes equilibrium total price ofTerminal A in Model I), we have:
Table 3.2 Equilibrium total prices of each business model in Case 1

Business

𝑇𝐴

𝑇𝐵

Model
First-best[𝑛𝐴 2𝑘 𝑠 − 𝑐 𝛽 + 𝛼𝑠 − 𝐶 +[𝑛𝐵 2𝑘 𝑠 − 𝑐 𝛽 + 𝛼𝑠 − 𝐶 +𝐶𝐶𝑟𝑠 − 𝛾−𝐶𝑟 + 1 𝑠 − 𝑐 − 𝛾 ]/𝑛𝐴 𝐶 + 𝐶 + 2𝑘 𝛽 + 𝛼𝑠
1
2
1
2

I
II

1
2

1−𝑟 𝑠−𝑐+𝛾
1−𝑟 𝑠−𝑐+𝛾VVI

𝑟 + 1 𝑠 − 𝑐 − 𝛾 ]/1−𝑟 𝑠−𝑐+𝛾[𝑐 𝐶 𝑛𝐴 − 4𝐶 𝑛𝐴 + 1+

𝐶 𝑛𝐴 − −𝛾 + 𝑟𝑠 + 𝑠− 2𝐶 𝑛𝐴 +𝑟 − 2 𝑠 − 𝛾 ]/6𝐶 𝑛𝐴 + 1[−2𝐶 3 𝑛𝐴 + 1 𝑛𝐵 + 1 𝑛𝐴 2𝑛𝐵 + 1 +[−2𝐶 3 𝑛𝐴 + 1 𝑛𝐵 + 1 𝑛𝐴 2𝑛𝐵 + 1 +𝑛𝐵 2𝑐 − 𝛾 + 𝑟 − 2 𝑠 + 𝐶 𝐶 𝑛𝐴 +𝑛𝐵 2𝑐 − 𝛾 + 𝑟 − 2 𝑠 + 𝐶 𝐶 𝑛𝐴 𝑛𝐵 +1 𝑛𝐵 𝑛𝐴 2𝑛𝐵 4𝑐 − 𝛾 + 𝑟 − 4 𝑠 +1 𝑛𝐴 2𝑛𝐵 4𝑐 − 𝛾 + 𝑟 − 4 𝑠 + 𝑐 +7𝑐 − 3𝛾 + 3𝑟𝑠 − 7𝑠 + 𝑛𝐵 𝑐 + 𝛾 − 𝑟 +𝛾 − 𝑟 + 1 𝑠 + 𝑛𝐵 7𝑐 − 3𝛾 + 3𝑟𝑠 −

1 𝑠

7𝑠

+ 2𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 𝑛𝐵 𝑐 − 𝛾 +𝑟 − 1 𝑠 + 𝑐 − 𝑠 + 𝑛𝐵 𝑟𝑠 − 𝛾IV−𝐶𝑛𝐵 𝐶 + 𝐶 + 2𝑘 𝛽 + 𝛼𝑠1III𝑟𝑠 − 𝛾++ 2𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 𝑛𝐵 𝑐 − 𝛾 + 𝑟 −1 𝑠 − 𝛾 + 𝑟𝑠 + 𝑛𝐵 𝑐 − 𝑠+

2𝐶 3 𝑛𝐴 𝑛𝐵 𝑠 − 𝑐 ]/[2 3𝐶 3 𝑛𝐴 + 1 𝑛𝐵 +2𝐶 3 𝑛𝐴 𝑛𝐵 𝑠 − 𝑐 ]/[2 3𝐶 3 𝑛𝐴 + 1 𝑛𝐵 +1 𝑛𝐴 2𝑛𝐵 + 1 + 𝑛𝐵 − 5𝐶 𝐶 𝑛𝐴 𝑛𝐴 +1 𝑛𝐴 2𝑛𝐵 + 1 + 𝑛𝐵 − 5𝐶 𝐶 𝑛𝐴 𝑛𝐴 +1 𝑛𝐵 𝑛𝐵 + 1 − 𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 2𝑛𝐵 +1 𝑛𝐵 𝑛𝐵 + 1 − 𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 2𝑛𝐵 +1 + 𝑛𝐵 + 𝐶 3 𝑛𝐴 𝑛𝐵 ]1 + 𝑛𝐵 + 𝐶 3 𝑛𝐴 𝑛𝐵 ]1
2 − 𝑟 𝑠 − 2𝑐 + 𝛾
3
1
1−𝑟 𝑠−𝑐+𝛾2[2𝐶 𝑛𝐴 + 1 𝑛𝐵 + 1

1
2 − 𝑟 𝑠 − 2𝑐 + 𝛾
3
1
1−𝑟 𝑠−𝑐+𝛾2

𝑟−1 𝑠−𝑐−[2𝐶 𝑛𝐴 + 1 𝑛𝐵 + 1

1−𝑟 𝑠−𝑐−𝛾 + 𝐶 𝐶 𝑛𝐴 + 1 𝑛𝐵 𝑐 + 𝛾 − 𝑟 +𝛾 + 𝐶 𝐶 𝑛𝐴 𝑛𝐵 + 1 𝑐 + 𝛾 − 𝑟 +1 𝑠 + 𝐶 𝑛𝐴 𝑛𝐵 𝑐 + 𝛾 − 𝑠 ]/[4𝐶 𝑛𝐴 +1 𝑠 + 𝐶 𝑛𝐴 𝑛𝐵 𝑐 + 𝛾 − 𝑠 ]/[4𝐶 𝑛𝐴 +1 𝑛𝐵 + 1 − 𝐶 𝑛𝐴 𝑛𝐵 ] + 𝛾

1 𝑛𝐵 + 1 − 𝐶 𝑛𝐴 𝑛𝐵 ] + 𝛾

32Proposition 3.1 If airlines cannot change the base terminal, and the number of airlines ineach terminal is fixed, then(i) 𝑇𝐴𝐼𝑉 > 𝑇𝐴𝐼𝐼𝐼 > 𝑇𝐴𝐼𝐼 = 𝑇𝐴𝐼 = 𝑇𝐴𝑉 > 𝑇𝐴𝑉𝐼 ; 𝑇𝐵𝐼𝑉 > 𝑇𝐵𝐼𝐼𝐼 > 𝑇𝐵𝐼𝐼 > 𝑇𝐵𝐼 = 𝑇𝐵𝑉 > 𝑇𝐵𝑉𝐼
𝜕𝑤 𝐼𝐼

𝜕𝑤 𝐼𝐼

𝜕𝑇 𝐼𝐼

𝜕𝑇 𝐼𝐼

𝜕𝑇 𝐼𝐼

𝜕𝑇 𝐼𝐼

𝜕𝑇 𝐼𝐼

𝐴

𝐴

𝐴

𝐴

𝐵

𝐵

𝐴

𝜕𝑇 𝐼𝐼

𝜕𝑇 𝐼𝐼

𝜕𝑇𝐴𝐼𝐼𝐼(ii) 𝜕𝑛𝐴 < 0, 𝜕𝑛𝐵 < 0, 𝜕𝑛 > 0; 𝜕𝑛𝐵 < 0, 𝜕𝑛𝐵 = 𝜕𝑛𝐴 = 𝜕𝑛𝐴 = 0;(iii) 𝜕𝑙𝐴 = 0; 𝜕𝑙𝐵 < 0;
(iv)

𝑉𝐼

>

𝐼

=

𝑉

>

𝜕𝑙
𝐼𝐼

< 0;

>

𝜕𝑇𝐵𝐼𝐼𝐼

𝐼𝐼𝐼

𝜕𝑙

>

<0
𝐼𝑉

Proof: See Appendix C2.
From Proposition 3.1, we find that inducing inter-terminal competition or splitting
terminals from the centralized operation leads to higher prices. When one of the twoterminals is operated independently (Business Model II), the integrated operator, which
operates one terminal and the airfield facilities, can intentionally set a fairly high charge
𝑇 𝐼𝐼 for runway service, making the reaction functions of both operators move inward
toward the origin. Due to the effect of uninternalized complementarity with airfield, theindependent operator (Terminal B) cannot have its reaction function moving as much asthat of the integrated operator. The integrated operator thus gains a price advantage overits opponent: the equilibrium terminal fee of Terminal A (𝑤𝐴𝐼𝐼 ) becomes much lower thanthat of Terminal B (𝑤𝐵𝐼𝐼 ) (see Appendix C1). This supports the concern raised by McLay
and Reynolds-Feighan (2006) that “it may be attractive for the runway/incumbent
terminal business to ‘over-bid’ for runway access, and to effectively subsidize its terminal
operations from the inflated proceeds of the sale of runway access.” The prerogative tocharge the runway service works as a strong tool for the integrated operator to place its
competitor at a disadvantage, and also helps the integrated operator maintain profitability
despite its low terminal fee. As a result, the total price using Terminal A, 𝑇𝐴𝐼𝐼 , remains at
the same level compared with the centralized case (Business Model I), whereas the total
price using Terminal B, 𝑇𝐵𝐼𝐼 , increases. Further, as the number of airlines 𝑛𝐴 goes greater,
Terminal A would gain greater superiority over Terminal B in the competition. In such asituation, Terminal B has to lower its terminal fee 𝑤𝐵𝐼𝐼 to maintain market share. To
counter, operator 1 raises the airfield charge, on the one hand, to offset Terminal B’s low
33fee. It then lowers its own terminal fee 𝑤𝐴𝐼𝐼 , on the other hand, to keep the total price ofTerminal A unchanged. Nevertheless, a greater airline number at Terminal A will reduce
𝜕𝑇 𝐼𝐼

𝜕𝑇 𝐼𝐼

𝐴

𝐴

the difference between the total price of Terminal B and A, as 𝜕𝑛𝐵 < 0 and 𝜕𝑛𝐴 = 0.
Interestingly, the number of airlines 𝑛𝐵 at the independent terminal will not affect theprices of both terminals. Consequently, although an increase in the number of airlines
might lead to a rise in substitutability between the two terminals, which can lead to lower
prices, the impact of substitutes can never outweigh that of complementarity between theterminal and airfield to lower prices. Moreover, a higher cross-effect parameter 𝑙 willcause a universal shrinkage of downstream demand, driving down prices.
When both terminals are split and operated by independent operators (Business Model
III), both 𝑇𝐴 and 𝑇𝐵 go higher as additional uninternalized complementarity emerges,
Higher 𝑙 can lower the prices; however, the prices will always be higher than that in thecentralized and partially separated operation (Business Models I and II). When the airfield
is operated independently while the two terminals are operated jointly (Business Model
IV), the prices become highest, since all that remains is the uninternalized
complementarity, and the competition of substitute goods is completely neutralized in the
upstream market.
When the ability for the airfield to use the “tool of complementarity” is restricted, or a
price-cap is equivalently imposed on airfield service, as suggested by McLay andReynolds-Feighan (2006) (Business Models V and VI), even without terminal
competition, the prices will not be high. Thus, 𝑇𝐴𝑉 and 𝑇𝐵𝑉 are at the same level with 𝑇𝐴𝐼
and 𝑇𝐵𝐼 , and the introduction of terminal competition in this situation would lead to lowest
prices (𝑇𝐴𝑉𝐼 and 𝑇𝐵𝑉𝐼 ) among all business models. Finally, we find that the introduction ofinter-terminal competition generally lowers the total output, which declines as theprices increase.
To summarize, if the operation of the terminal service section is completely separated
from that of the runway service section in the existing business model, or the pricing of
airfield operator is strictly regulated, having competing terminals can lower the prices andincrease the total output (e.g., from Business Model IV to III, from Business Model V toVI), since the complementarity has already been fully “externalized” or neutralized in34these cases, so the separation of terminal operation will not further aggravate the negative
effect of complementarity. Otherwise, the introduction of terminal competition would
only increase the prices and lower the total output (e.g., from Business Model I to II or
III). These results are generally in line with that of De Borger and De Bruyne (2011)
regarding the integration of firms that produce complementary good; they study the effectof the vertical integration of port terminal operators and hinterland transport firms on
government’s policy. They show that vertical integrations make government levy higher
port access charges, since lower price can be achieved through the elimination of double
marginalization so correction by government is no longer necessary. This is in line with
our result in terms of the integration of firms producing complementary goods. While
their modeling assumes balanced and symmetric vertical interactions, our result show thatbiased and asymmetric internalization of complementarity can also lead to lower priceunder some conditions (from Model III to II). Regarding the social surplus with optimal
prices in each Business Model, we have:
Proposition 3.2 If airlines cannot change the base terminal and the number of airlines ineach terminal is fixed, then𝑆 𝐼 = 𝑆 𝑉 > 𝑆 𝐼𝐼 𝑆 𝐼 > 𝑆 𝐼𝑉
Proof: See Appendix C3.
For the social surpluses of other business models, as it is arduous to compare them
analytically, we resort to numerical examples. We change airline number 𝑛𝐴 and 𝑛𝐵 in
each terminal and the cross-effect 𝑙, and fix other parameters. Parameters are set basedon the realistic data as much as possible, and the sources are as follows. In The UnitedState, the average CASM (Cost per Available Seat Mile) of airlines is USD 0.126 in 2019.
Dividing this by the average load factor (0.84), we can get the cost per passenger mile as
USD 0.15. Multiplying it by the average stage length (1166 miles), we have the marginal
cost per passenger as USD 175. Multiplying the average seat number 179 by the loadfactor, we have the average passenger number per flight as 150. Finally, multiplying the
average passenger number by marginal cost per passenger, we obtain the averagemarginal cost per flight as USD 26319. All data above are collected from the airline data35

project,12 and the unit of monetary values is the US dollar in 2019. For passengers, thetime value of delay is USD 37.6/hour in 2007 US dollars (NEXTOR, 2010). The average
delay cost of US airlines is estimated to be USD 391 per flight per hour in 2007 US dollars(Ferguson et al., 2013). Regarding the airport’s cost and revenue, Martín and VoltesDorta (2011) make a careful estimation based on the data of 161 airports worldwide
between 1991 and 2008. Their results show that the average marginal cost is USD 5.33
per domestic passenger and USD 6.19 per international passenger, respectively, in 2008
US dollars. The average marginal cost per flight in the Asia-Pacific region is about USD
450. The average unit commercial profit is USD 3.49 per passenger. The unit of all
monetary values mentioned above is converted to US dollars in 2019 and scaled by
dividing the actual value of maximum willingness-to-pay which is normalized to 1 in the
demand function. We choose 2000 for the maximum willingness-to-pay, and 0.15 for theown-price effect (Basso and Zhang, 2008). Finally, the parameters for numericalexamples are determined, as shown in Table 3.3.
Table 3.3 Values of parameters for numerical examples𝑎
7.5 × 10 5

𝑐
12.5

𝛼
2.25 × 10

𝛽𝛾𝑟0.24 0.275 −1.25 × 10 3

𝑘
120

𝑠
150

Fig. 3.4 shows the results of 𝑆 𝐼𝐼 − 𝑆 𝐼𝐼𝐼 . Under the pre-set parameters, 𝑆 𝐼𝐼 is always higher
than 𝑆 𝐼𝐼𝐼 . The reason might be that, compared with Business model II, an additional
complementary duopoly emerges between terminal and airfield while the substitutive
duopoly between terminals is kept unchanged in Business model III. Loss in social
surplus occurs as the complementarity can no longer be internalized by a single operator.
Further, we can find that 𝑆 𝐼𝐼 − 𝑆 𝐼𝐼𝐼 increases with 𝑛𝐴 , yet decreases as 𝑙 increases; thegreater the 𝑛𝐴 , the stronger the complementary effect, thus the greater the surplus loss
when complementarity are not internalized. The greater the cross-effect 𝑙, the smaller theuniversal downstream demand, thus smaller the output difference between II and III,
leading to a smaller difference between 𝑆 𝐼𝐼 and 𝑆 𝐼𝐼𝐼 . Other comparison results are shownin Appendix C4. Consequently, 𝑆 𝑉𝐼 > 𝑆 𝑉 = 𝑆 𝐼 > 𝑆 𝐼𝐼 > 𝑆 𝐼𝐼𝐼 > 𝑆 𝐼𝑉 , which follows thepattern that the lower the prices, the higher the social surplus. Having competing terminals
12<http://web.mit.edu/airlinedata/www/default.html>

36

can enhance social surplus if the terminal service and runway service sectors are operated
separately in the existing business model (e.g., from IV to III, from V to VI). Otherwise,
the introduction of terminal competition would have a negative effect on social surplus(e.g., from I to II and from II to III).

Fig 3.4 Numerical results of 𝑺𝑰𝑰 − 𝑺𝑰𝑰𝑰

In addition, we try an alternative modelling by relaxing the assumption that airlines insame terminal are perfect substitutes. Results are relegated to Appendix D. The rankingof Business Models in terms of social surplus will not change by relaxing this assumption.
What changes is the extent of the difference in pricing and social surplus between
different Business Models.

37

3.2.4 Case 2: airlines can change terminal
Passenger fee 𝑤𝐴

Terminal A

Charge 𝑇

Airfield

N
Airlines𝑛𝐴
𝑛𝐵

Airlines at
T_A

Quantity(airfare)

Airlines at
T_B

Quantity(airfare)Passenger

Passenger fee 𝑤𝐵

Terminal B

Fig 3.5 Illustration of the air transport market in Case 2

Table 3.4 Optimal total prices of each Business Model in Case 2

Business
ModelFirstbest
I
II
III
IV
V
VI

𝑻𝑨
𝐶

𝑻𝑩

𝑁 + 2 𝛾 − 𝑠 𝑁 + 2 𝑟 + 2 + 2𝑐 − 2𝑁𝑘 𝑐 − 𝑠 𝛽 + 𝛼𝑠 + 𝑁𝐶 𝛾 − 𝑟𝑠
𝑁 𝐶 + 𝐶 + 2𝑘 𝛽 + 𝛼𝑠
1𝑠 − 𝑟𝑠 − 𝑐 + 𝛾
2

𝐶1 𝑠 4𝑁 𝑁𝑟 3𝑟+3 + 𝑁+3 𝛾 4𝑁+3 𝑐 𝑁𝐶2 𝑟+ 𝑠 𝑐 𝛾
1
𝑠 − 𝑟𝑠 − 𝑐 + 𝛾
6 𝑁+ 𝐶1
2
𝐶 1 − 𝑁 𝑟 − 2 − 𝑟 𝑠 − 2𝑁 + 1 𝑐 + 𝑁 + 1 𝛾 − 𝑁𝐶 𝑠 − 𝑐
3𝑁 + 2 𝐶 − 𝑁𝐶
1
2 − 𝑟 𝑠 − 2𝑐 + 𝛾
31
𝑠 − 𝑟𝑠 − 𝑐 + 𝛾
2
𝐶 𝑁 1 − 𝑟 𝑠 − 𝑐 + 𝛾 + 𝛾 − 𝑟𝑠 − 𝑁𝐶 𝑠 − 𝑐
2𝑁 + 1 𝐶 − 𝑁𝐶

In Case 2, an airline can move to the terminal where it can make the highest profit; thus,
the number of airlines at each terminal can change according to the prices. The freedomof changing terminal further increases the degree of substitution between the twoterminals, allowing us to examine whether the effect of complementarity can be
dominated, and lower prices can be realized through the introduction of terminal
competition. The interactions among market participants can be modeled as a three-stage
game: The airfield and terminals set prices first, then airlines choose their base terminals;
finally, airlines compete in Cournot fashion (Fig 3.5). To induce backward, we start fromthe second stage. The optimal profits of each airline are as follows:
38

𝑇𝑤 𝑤

𝐶 𝑘 2 (𝐶1 𝑛−ℎ +

= 1

𝑠 𝑤ℎ

+𝑇 +𝑐 𝐶1 𝑛−ℎ 𝐶2 𝑛−ℎ +𝐶1(𝐶22 𝑛ℎ 𝑛−ℎ 𝐶12 𝑛ℎ +𝐶2 𝑛−ℎ 𝑠 𝑤−ℎ

2

+𝑇 )

2

3.10

𝑛−ℎ + )

By simply observing the form of the airlines’ optimal profit function, we can see thatobtaining the precise number of airlines at each terminal in equilibrium appears to bedifficult. Thus, we make several assumptions to simplify the problem:
Assumptions for Case 2: (1) Airlines believe that, in each terminal, the profit of eachairline decreases as the number of airlines increases; (2) the cost for changing terminals
is ignored (e.g., airlines have long-term considerations); (3) the number of airlines istreated as continuous.
Then, the equilibrium allocation of airlines can be achieved when=

. By solving

this equation, we can derive the equilibrium numbers as:𝑛𝐵 =𝑁(𝐶 𝑠 𝑤𝐵 − 1 + 𝑇 − 𝐶 𝑠 𝑤𝐴 − 1 + 𝑇 + 𝑐 𝐶 − 𝐶 ) + 𝐶 𝑠 𝑤𝐵 − 𝑤𝐴
𝐶 − 𝐶 𝑠 𝑤𝐴 + 𝑤𝐵 − 2 + 2𝑐 + 2𝑇

3.11

𝑛𝐴 = 𝑁 − 𝑛𝐵

3.12

Substituting the equilibrium numbers into outputs, we derive the equilibrium flights andtotal output as functions of the total prices:
𝑘 2𝑠 − 2𝑐 − 𝑇 − 𝑇
𝑁 + 2 𝐶 + 𝑁𝐶

3.13

𝑘 s 𝑛𝐴 + 𝑛𝐵 2𝑠 − 2𝑐 − 𝑇𝐴 − 𝑇𝐵
𝑁 + 2 𝐶 + 𝑁𝐶

3.14

𝑓 =

=

We can then summarize the effects of prices on the flight of each airline and total output
as:Lemma 3.2 If airlines can change the base terminal and the number of airlines in eachterminal is obtained according to Assumptions for Case 2, we have
∂𝑓ℎ𝑖
∂𝑇ℎ

∂𝑓

∂𝑄

∂𝑄

𝐴

𝐵

= ∂𝑇 ℎ𝑖 < 0; ∂𝑇 = ∂𝑇 < 0.
−ℎ

An increase (decrease) in the price of any terminal will result in a flight declining(increasing) for all airlines at both terminals. A lower price in Terminal ℎ not only
increases the flights of airlines therein, but also attracts newcomers from Terminal −ℎ.
The share increasing effect of a declining number of airlines at Terminal −ℎ outweighs
39the flight decreasing effect of a relatively high price compared with Terminal ℎ .
Substituting the subgame equilibrium output and airline numbers into the objectivefunctions and solving the FOCs in Table A1, we can derive the optimal prices as shownin Table 3.4. Under these optimal prices, the curves of airlines’ profits generally conformto (1) of Assumptions for Case 2 (see Appendix E1). Comparing these prices, we cansummarize:
Proposition 3.3 If airlines can change the base terminal, and the number of airlines ineach terminal is obtained according to Assumptions for Case 2, we have(i) 𝑇𝐴𝐼𝑉 > 𝑇𝐴𝐼𝐼𝐼 > 𝑇𝐴𝐼𝐼 = 𝑇𝐴𝐼 = 𝑇𝐴𝑉 > 𝑇𝐴𝑉𝐼 ; 𝑇𝐵𝐼𝑉 > 𝑇𝐵𝐼𝐼𝐼 > 𝑇𝐵𝐼𝐼 > 𝑇𝐵𝐼 = 𝑇𝐵𝑉 > 𝑇𝐵𝑉𝐼
(ii)

𝐼𝑉

<

𝐼𝐼𝐼

<

𝐼𝐼

<

𝐼

=

𝑉

<

𝑉𝐼

Proposition 3.3 can be proved in the same approach as that of Proposition 3.1, so we omit
the discussion here. The results are the same as in Case 1. A greater degree of substitution
between terminals cannot offset the effect of uninternalized complementarity, so having
competing terminals can lower the prices and increase total output only when“complementary duopoly” already exists in the existing business model (e.g., from IV toIII, from V to VI). Comparison of the social surplus in each business model leads to thefollowing finding:
Proposition 3.4 If airlines can change the base terminal and the number of airlines ineach terminal is obtained according to Assumptions for Case 2, we have
𝑆 𝐼𝐼 < 𝑆 𝐼 < 𝑆 𝑉𝐼 ; 𝑆 𝐼𝑉 < 𝑆 𝐼𝐼𝐼 < 𝑆 𝐼 = 𝑆 𝑉
Proof: See Appendix E2.
The comparison between 𝑆 𝐼𝐼 and 𝑆 𝐼𝐼𝐼 is investigated by numerical example (seeAppendix E3). As a result, the social surplus in Case 2 ranks as 𝑆 𝑉𝐼 > 𝑆 𝑉 = 𝑆 𝐼 > 𝑆 𝐼𝐼 >𝑆 𝐼𝐼𝐼 > 𝑆 𝐼𝑉 , which is the same as that in Case 1. The increasing degree of substitutecontributed by airlines’ freedom to change the base terminal cannot make the Business
Model with competing terminals better-off.

40

3.3 Extensions
3.3.1 Asymmetric terminals
In this case, we consider terminals with different marginal costs and non-aeronauticalprofitability. We only investigate Business Models I and II as Business Model II is most
likely to occur in reality, and the tractability of the model under other business models in
extended cases is questionable. The two Business Models are re-labeled as Model HI and
HII in this case. By solving the FOCs shown in Table 3.1, we have the optimal prices asfollows:
𝑤𝐴𝐻𝐼𝐼 =𝐶 𝑛𝐴 + 1 𝑠 1 − 3𝑟𝐴 − 2𝑟𝐵 − 𝛾 − 𝐶 𝑛𝐴 𝑠𝑟𝐴 − 𝛾 + 𝑠 − 𝑐 𝐶 𝑛𝐴 − 𝐶 𝑛𝐴 + 𝐶
6𝐶 𝑠 𝑛𝐴 + 1

3.15

𝐶 𝑛𝐴 + 1 𝑠 − 2𝑠𝑟𝐵 − 𝛾 − 𝐶 𝑛𝐴 𝑠𝑟𝐴 − 𝛾 + 𝑠 − 𝑐 𝐶 𝑛𝐴 − 𝐶 𝑛𝐴 + 𝐶
3𝐶 𝑠 𝑛𝐴 + 1

3.16

𝐶 𝑛𝐴 𝑠 − 𝛾 + 𝑠𝑟𝐴 + 2𝐶 1 + 𝑛𝐴 𝑠 + 2𝛾 + 𝑠𝑟𝐵 − 𝑐(𝐶 𝑛𝐴 + 2𝐶 1 + 𝑛𝐴 )6𝐶 1 + 𝑛𝐴

3.17

𝑤𝐵𝐻𝐼𝐼 =𝑇 𝐻𝐼𝐼 =𝑇𝐴𝐻𝐼 = 𝑇𝐴𝐻𝐼𝐼 =𝑇𝐵𝐻𝐼 =𝑇𝐵𝐻𝐼𝐼 =

1𝑠 − 𝑐 + 𝛾 − 𝑠𝑟𝐴
2

3.18

1
𝑠 − 𝑐 + 𝛾 − 𝑠𝑟𝐵2

3.19

2𝐶 𝑛𝐴 + 1 𝑠 2 − 𝑟𝐵 + 𝛾 − 𝐶 𝑛𝐴 𝑠𝑟𝐴 − 𝛾 + 𝑠 + 𝑐(𝐶 𝑛𝐴 − 4𝐶 𝑛𝐴 + 1 )6𝐶 𝑛𝐴 + 1

3.20

Comparing these prices, we can derive the condition that 𝑇𝐵𝐻𝐼𝐼 < 𝑇𝐵𝐻𝐼 (i.e., the condition
that Business Model HII can result in lower prices compared with Model HI, as 𝑇𝐴𝐻𝐼𝐼
always equals 𝑇𝐴𝐻𝐼 ). The condition is 𝑟𝐵 <=−𝐶1 𝑛𝐴 +𝐶1 𝐶2 𝑛𝐴 𝑠 𝑐 𝛾
𝐶1 𝑠 𝑛𝐴 +The equation 𝑟𝐵 =< 0;

+ 𝑟𝐴 , where

𝐶 𝑛= 𝐶 𝑛2 𝐴+ < 1.
1

𝐴

+ 𝑟𝐴 intersects with the line 𝑟𝐵 = 𝑟𝐴 at (that 𝑟𝐴 or 𝑟𝐵 is lower than

𝑐+𝛾 𝑠
𝑠

𝑐+𝛾 𝑠 𝑐+𝛾 𝑠𝑠,𝑠). The case

, where the profitability of terminal cannot be ensured, is

excluded form the discussion. We find that 𝑇𝐵𝐻𝐼𝐼 can be lower than 𝑇𝐵𝐻𝐼 if 𝑟𝐴 is greaterthan 𝑟𝐵 , and the difference is above a certain degree. The intuition behind the findings is41

that, when 𝑟𝐴 is much greater than 𝑟𝐵 , the integrated operator, who operates the airfield
and Terminal A, can lower its price much further than the independent operator who
operates Terminal B thanks to the “cost advantage”, so it does not need to set a runway
charge as high as that in Model II to obtain the price advantage additionally (note that
𝜕𝑇𝐻𝐼𝐼
𝜕𝑟𝐴

𝜕𝑇𝐻𝐼𝐼

< 𝜕𝑟 ). As a consequence, the total price using Terminal B in Model HII will be𝐵lower than in Model HI, though still higher than 𝑇𝐴𝐻𝐼𝐼 .

𝑟𝐴 = 𝑟𝐵

𝑇𝐵𝐻𝐼 = 𝑇̂𝐵𝐻𝐼𝐼𝑇𝐵𝐻𝐼𝐼 − 𝑇̂𝐵𝐻𝐼𝐼
= 𝑇̂𝐵𝐻𝐼𝐼 − 𝑇𝐵𝐻𝐼𝑇𝐵𝐻𝐼𝐼 = 𝑇𝐵𝐻𝐼𝑇𝐵𝐻𝐼 < 𝑇̂𝐵𝐻𝐼𝐼𝑇𝐵𝐻𝐼𝐼 < 𝑇𝐵𝐻𝐼

𝑐+𝛾 𝑠 𝑐+𝛾 𝑠,𝑠𝑠Fig 3.6 Visualization of price and social surplus comparison

However, lower prices do not necessarily enhance social surplus. Since 𝑇𝐴𝐻𝐼 = 𝑇𝐴𝐻𝐼𝐼 , tohave 𝑆 𝐻𝐼𝐼 > 𝑆 𝐻𝐼 , |𝑇𝐵𝐻𝐼𝐼 − 𝑇̂𝐵𝐻𝐼𝐼 | < |𝑇𝐵𝐻𝐼 − 𝑇̂𝐵𝐻𝐼𝐼 | should be satisfied, where 𝑇̂𝐵𝐻𝐼𝐼 is thestationary point of function 𝑆 𝑇𝐴 = 𝑇𝐴𝐻𝐼 = 𝑇𝐴𝐻𝐼𝐼 . To identify the area of 𝒓 that satisfies
the condition, we further derive lines 𝑇𝐵𝐻𝐼𝐼 − 𝑇̂𝐵𝐻𝐼𝐼 = 𝑇̂𝐵𝐻𝐼𝐼 − 𝑇𝐵𝐻𝐼 and 𝑇𝐵𝐻𝐼 = 𝑇̂𝐵𝐻𝐼𝐼 .
Interestingly, these two lines and 𝑇𝐵𝐻𝐼𝐼 = 𝑇𝐵𝐻𝐼 and 𝑟𝐵 = 𝑟𝐴 all intersect at (𝑐+𝛾 𝑠 𝑐+𝛾 𝑠
𝑠

,

𝑠

),

and by comparing their intercept (see Appendix F1) at 𝑟𝐵 axis, we can draw a graph asshown in Fig 3.6. From the graph, it is not difficult to find that 𝑆 𝐻𝐼𝐼 > 𝑆 𝐻𝐼 can occur only
when both 𝒓 are in the hatched area. The lower price 𝑇𝐵𝐻𝐼𝐼 only proves to be a result of
underpricing. Since the social optimal prices are low prices which are balanced between
42the two terminals regarding airline numbers, to get closer to such prices rather than 𝑇𝐴𝐻𝐼
and 𝑇𝐵𝐻𝐼 , an appropriate difference between 𝑟𝐴 and 𝑟𝐵 should be ensured in Model HII;
otherwise, the prices will be either too high or biased. The above analysis can be verified
by a numerical example shown in Appendix F2. The numerical example demonstrates
that even in the case where 𝑆 𝐻𝐼𝐼 > 𝑆 𝐻𝐼 , the difference is quite limited. To summarize, theconditions under which the introduction of terminal competition can enhance social
surplus are quite stringent, and the improvement is modest even if the conditions can besatisfied. Finally, we summarize these finding as follows:
Proposition 3.5 If terminals have different non-aeronautical profitability and unit costs,
to have 𝑇𝐵𝐻𝐼𝐼 < 𝑇𝐵𝐻𝐼 , 𝑟𝐵 <+ 𝑟𝐴 , where< 0 and< 1, should be satisfied. 𝑆 𝐻𝐼𝐼 >𝑆 𝐻𝐼 can be true if ε 𝑛𝐴 𝑛𝐵 < 𝑟𝐴 − 𝑟𝐵 < ε+ 𝑛𝐴 𝑛𝐵 where ε 𝑛𝐴 𝑛𝐵 and ε+ 𝑛𝐴 𝑛𝐵
are positive.
3.3.2 Airline in the upstream market
In recent years, airports and airlines have begun to have cooperative interactions witheach other frequently to gain competitive advantage, given an increasingly
commercialized air transport market (D’Alfonso and Nastasi, 2014). Various forms ofvertical arrangement can be observed around the world (Fu et al., 2011; Fu and Yang,2017). In the context of this study, we roughly categorize vertical arrangement into two
types based on whether the ownership or operation of key airport facilities which provide
indispensable services to multiple consumers (airlines) are shared with airlines. An
arrangement belongs to Type I if yes and Type II if no. Type I vertical arrangements
mainly include airlines’ control of and investment in airport facilities (e.g., Terminal 2 ofMunich Airport), long-term usage contract (e.g., Melbourne Airport and Virgin Blue),
airport-airline consortium (e.g., New York JFK Airport), and common ownership (e.g.,
Changi Airport). Type II vertical arrangements mainly include signatory airline
agreement (e.g., Seattle-Tacoma International Airport), revenue sharing (e.g., Tampa
International Airport), information sharing and joint marketing, and incentive programs.
Since Type I might create intra-airport competition via the participation of dominant
airline in upstream, which is highly relevant to this study, we further investigate this issue
by introducing and analyzing several new potential Business Models with the vertical43

arrangement.
We first characterize the cases that airlines directly and fully operate terminals, which cancorrespond to the long-term lease in Australia or some BOT contracts (Forsyth, 2008;
Starkie, 2008). In Business Model DI, Terminal B is operated independently by an airline
while Terminal A and airfield facility is operated jointly by a profit-maximizing operator.
In Model DII, Terminal B is operated independently by an airline while Terminal A andairfield facilities are operated by two operators separately. In Model DIII, two terminals
are operated by two dominant airlines separately, while a profit-maximizing operator
operates airfield facilities. In Model DIV and DV, public operators operate airfield
facilities. The difference is that terminals are operated by an airline and a profitmaximizing operator, respectively, in the former Model; in the latter Model, terminals areoperated by two airlines, respectively. The modelling approach is similar to that of Barbot(2011): in the upstream market, airfield operator, terminal operator and/or terminal
operating airlines first decide airfield charge 𝑇, terminal fee 𝒘, and the quantities ofterminal operating airlines 𝑓𝐷 . Given these prices and judging the remaining demand,
airlines in the downstream market compete in a Cournot fashion to clear the market. Next,
we characterize the case that terminal is operated by a join venture of an airline and aterminal operator as Model DVI, which bear some similarity with the vertical merger caseillustrated in Barbot (2011). In this model, the airline has a share 𝜃 of Terminal B and canacquire a corresponding part of its profit, However, the terminal is operated by theterminal operator. Airfield and terminal operators decide prices, then the shareholder
airline chooses the quantity, and, finally, other airlines compete on quantities. Fig 3.7
provides an intuitive illustration of these Business Models.

44

Business Model DI

Business Model DII

Business Model DIII

Terminal A

Terminal B

Terminal A

Terminal B

Terminal A

Terminal B

Airfield

Airline 1

Operator 1

Airline 1

Airline 1

Airline 2

Airfield
Operator 2

Operator 1

Business Model DV

Business Model DVI

Operator 1
Business Model DIV

Airfield

Terminal A

Terminal B

Terminal A

Terminal B

Terminal A

Terminal B

Operator 1

Airline 1

Airline 1

Airline 2

Airfield

Operator 2

Airfield
Operator 2

Airfield
Operator 1

Operator 1

Joint ventureof airline andterminal
operator

Red: profit-maximizing operator Blue: with participation of airline Green: social surplus maximizing operator

Fig 3.7 Illustration of Business Models with airlines in upstream

The demonstration of the tedious solving process, merely a similar variant of that in thebasic modeling, and the lengthy equilibrium solutions are omitted. Note that in themodeling of Barbot (2011), if the leader airline has the pricing right of the terminal, allother airlines will be driven out of the market. In our modeling, this can be avoided byremoving the terminal fee from the total price that the passengers of the terminal operating
airline are facing or, equivalently, by allowing airfare differentiation between upstream
airline and downstream airlines in the same terminal. By doing so, the total quantity ofthe downstream airlines in that terminal (Terminal B) becomes𝑘 2 𝑠2 𝑟 𝑛𝐵
𝐶1

in Business

Model DI, DII, DIII, DIV, and DV. Thus, as long as 𝑟 > 0, they will get a market share.
𝐶2 𝑛 +In Model DVI, the share parameter should satisfy 0 ≤ 𝜃 ≤ 𝐶12 𝑛𝐴+
1

to ensure a market share of downstream airlines.

45

𝐴

𝑛𝐵

𝐶22 𝑛𝐴 𝑛𝐵𝑛𝐵 +𝐶22 𝑛𝐴 𝑛𝐵

Fig 3.8 Comparison of equilibrium total prices in each Model

46

Fig 3.9 Comparison of the social surplus of each Model

Numerical examples of equilibrium prices and social surplus are shown in Fig 3.8 and
3.9, respectively. Same values are defined for parameters except that 𝑟 is changed to
1/4000. From the plots of prices, we can observe that total prices of Business Models withairline(s) in the upstream market (DI, DII, DIII, DVI) are generally higher than that of theordinary Models (I, II, III), when a private operator operates the airfield facilities, andvice versa (V and VI versus DIV and DV). The latter indicates that airlines’ participation47

in terminal operations originally has a price-reducing effect due to a smaller share offollower airlines whose passengers are the only payers of terminal fees and the lessintense competition in the downstream market (number of airlines are 𝑛 − 1). However,when the airfield operator is private, the effect of uninternalized complementary will
greatly raise the prices, especially the charge for airfield facilities. Since the potential
demand can be greater due to the partial elimination of double marginalization when an
airline operates terminal, the effect of complementarity works more prominently than inordinary Models, thereby causing the reverse results.
However, the high prices do not necessarily lead to a decline in social surplus when theairfield operator is private. Comparing the Models with similar configuration (II and DI;
III and DII, DIII), we can find that Models with upstream airlines outperform theircounterparts in social surplus (𝑆 𝐷𝐼 > 𝑆 𝐼𝐼 𝑆 𝐷𝐼𝐼 > 𝑆 𝐼𝐼𝐼 𝑆 𝐷𝐼𝐼𝐼 > 𝑆 𝐼𝐼𝐼 ). The intuition behind
the result is that the high prices in Models with upstream airlines are merely imposed onfollower airlines and their passengers fully, while the terminal operating airline(s) only
bears the airfield charge. The decline in the quantities of follower airlines is offset by theincrease in the quantities in the terminal operating airline(s), as compared to the ordinary
Models. Therefore, the social surplus does not go down with high prices. Together, theincrease in total quantities and airfield charge generates a higher profit of airfield operator.
It is contrary to the findings of Kaselimi et al. (2011), where vertical integration onlymeans a “virtual exit” of the dedicated terminal, which cannot exploit the price advantage
to expand market share as done by the terminal operating airline in their study. In the casethat a public operator operates airfield facilities, it is straightforward that lower prices
lead to higher social surplus. Thus, we have 𝑆 𝐷𝐼𝑉 > 𝑆 𝑉𝐼 and 𝑆 𝐷𝑉 > 𝑆 𝑉𝐼 .
Consequently, we can summarize the results as follows. If a private operator operates theairfield facilities (i.e., the effect of uninternalized complementarity exists), almost no
Business Models with the upstream airline can work as an alternative introduction ofterminal competition to improve social surplus. Only in extremely limited case (e.g.,when the cross-effect or the degree of substitute 𝑙 is very large) can Model DVI yieldhigher social surplus than the integrated operation, Model I, thanks to the joint effect of
the shift from pure Cournot to Cournot-Stackelberg model (Daughety, 1990), less doublemarginalization, a stronger increase in quantity from monopoly due to higher degree of48

substitution, and a weaker effect of complementarity. Nonetheless, in Models II and III,
if the independent terminal(s) are operated by airline(s), the social surplus can be
improved despite a rise in price. On the other hand, if the airfield facilities are public
while terminals are privatized, having airlines to operate the terminals can always lower
the prices and enhance social surplus. The more the terminals operated by airlines, thestronger the positive effect, which is consistent with the finding of D’Alfonso and Nastasi(2012). Finally, we must restate that these Business Models with the upstream airline(s)are meaningful only when 𝑟 (the difference between unit non-aeronautical profit and themarginal cost of terminal) is positive.
3.3.3 Terminal competition on both price and service level
In this section, we try an alternative modeling where terminal operators can determine
their service levels. In this case, product differentiation between terminals is characterized
by service level instead of the cross-effect parameter 𝑙, which is now used to address thedegree of substitution among all airlines regardless of which terminal they are situated.
The inverse demand function is defined as follows:
𝑝

=1−𝑎

− 𝑎𝑙

−

− 𝑤 − 𝛼 ( ) + 𝜂𝑣 − 𝜂𝑚𝑣
𝑘

3.21

Where 𝑣 denotes the service level of terminal ℎ, 𝜂 denotes the passengers’ valuing of
the service, or own service level effect, and 𝑚 denotes the degree of substitute between
the service of different terminal (0 ≤ 𝑚 ≤ 1 . This type of demand function is widelyadopted in the literature (e.g., Boyaci and Ray, 2003; Wang et al., 2017).
Accordingly, the profit function of Terminal ℎ becomes𝑤 𝑣= 𝑤 +𝑟−𝑣 𝜎

𝒘𝑇𝒗

3.22

where 𝜎 denotes the unit cost of service provision. Adopting backward induction, we canobtain the equilibrium prices and service levels of terminals in Business Model I, II and
III as:
𝑇𝐴𝐼 = 𝑇𝐴𝐼𝐼 = 𝑇𝐴𝐼𝐼𝐼 =𝜎 𝑠 − 𝑐 + 𝜂 1 − 𝑚 𝑟𝑠 − 𝛾
𝜂 𝑚−1 +𝜎49

3.23

𝜎 𝑠 − 𝑐 + 𝜂 1 − 𝑚 𝑟𝑠 − 𝛾
𝜂 𝑚−1 +𝜎3.24𝑣𝐴𝐼 = 𝑣𝐴𝐼𝐼 = 𝑣𝐴𝐼𝐼𝐼 =𝑠 𝑟+1 −𝑐−𝛾𝑠 𝜂 𝑚−1 +𝜎

3.25

𝑣𝐵𝐼 = 𝑣𝐵𝐼𝐼 = 𝑣𝐵𝐼𝐼𝐼 =𝑠 𝑟+1 −𝑐−𝛾𝑠 𝜂 𝑚−1 +𝜎

3.26

𝑇𝐵𝐼 = 𝑇𝐵𝐼𝐼 = 𝑇𝐵𝐼𝐼𝐼 =Interestingly, when competing both on price and service level, the two terminals will
make identical decisions regardless of being bundled with airfield facilities or operated
independently. With the same equilibrium decisions, the social surplus of these three
Business Models will also be identical. These results are partially consistent with thefindings of Basso and Zhang (2007) that duopolists will offer same service quality as themonopolist if price and service level are determined simultaneously. The difference isthat Basso and Zhang (2007) find that price set by duopolists will be lower, while theprice remains the same in our results. A possible explanation is that the price-increasing
effect of uninternalized complementary can only be partially offset by multi-dimensional
competition between substitute products. Thus, the price will not be as high as those in
the pure price competition; however, it cannot go further lower to the level of substituteduopoly. These findings indicate that the introduction of terminal competition might not
be worse-off in terms of social surplus as long as terminals can decide their service levels
in addition to prices, while no improvement can be achieved both in social surplus andservice level.
3.3.4 Regulated terminals
This section considers Business Models where the airfield operator aims to maximizesocial surplus by marginal-cost pricing, while the objective functions of terminals are a
mixture of profit and social surplus. That is, the airfield operation is strictly regulated,
whereas terminals can retain profitability to some extent. Terminals, in this case, maycorrespond to those under some kind of price-cap regulation, or those with mixed publicprivate ownership. Two new business models are labeled respectively as Model 𝑉̅ , where
two terminals are jointly operated, and Model ̅̅̅
𝑉𝐼 , where two terminals are separately
operated by two independent operators, with a mixed objective of profit and social surplus.

50

̅̅̅
Since the public airfield operator will always levy a cost recovery charge 𝑇 𝑉̅ = 𝑇 𝑉𝐼
= 𝛾,

we only need to focus on the equilibrium pricing of terminals. In Model 𝑉̅ , the joint
objective function of the terminal operator is formulated as follows:
̅
𝑉

= 1−𝜇

𝐴+ 𝐵

+𝜇 𝑆

3.27

where 𝜇 denotes the share of social surplus in the objective with 0 ≤ 𝜇 ≤ 1 . The
Lagrange function is
𝐿𝑉̅ =

̅
𝑉

+𝜆

𝐴+ 𝐵

3.28

The KKT conditions are
𝜕𝐿𝑉̅
𝜕𝐿𝑉̅
𝜕𝐿𝑉̅
𝜕𝐿𝑉̅
=0
=0
≥0 𝜆≥0 𝜆
=0
𝜕𝑤𝐴
𝜕𝑤𝐵
𝜕𝜆
𝜕𝜆

3.29

Solving them, we have two cases as follows: if 𝜆 > 0 𝒘𝑉̅ = 𝒘
̅ 𝑉̅ , which is the solution
𝜕𝐿𝑉

̅

𝜕𝐿𝑉

̅

𝜕Γ𝑉

̅

𝜕Γ𝑉

̅

𝐴

𝐵

𝐴

𝐵

of 𝜕𝑤 = 0, 𝜕𝑤 = 0; otherwise, 𝒘𝑉̅ = 𝒘
̂ 𝑉̅ , which is the solution of 𝜕𝑤 = 0, 𝜕𝑤 = 0.
Numerical results are obtained, as the problem is intractable analytically.
̅̅̅ , the objective function of terminal h is formulated as
In Model 𝑉𝐼
̅̅̅
𝑉𝐼

= 1−𝜇

+𝜇 𝑆

3.30

The Lagrange function for terminal h is
̅̅̅
𝐿𝑉𝐼
=

̅̅̅
𝑉𝐼

+𝜆

3.31

The KKT conditions are
̅̅̅
̅̅̅
̅̅̅
𝜕𝐿𝑉𝐼
𝜕𝐿𝑉𝐼
𝜕𝐿𝑉𝐼
=0
≥0 𝜆 ≥0 𝜆
=0 ℎ=
𝜕𝑤
𝜕𝜆
𝜕𝜆

Solving them, we have four cases as follows:(i). If 𝜆𝐴 > 0 and 𝜆𝐵 > 0
𝑉𝐼
𝑉𝐼
𝑤𝐴̅̅̅
= 𝑤𝐵̅̅̅
= −𝑟,
̅̅̅̅

̅̅̅̅

𝜕𝐿𝑉𝐼

𝜕𝐿𝑉𝐼

ℎ

ℎ

where 𝜆𝐴 and 𝜆𝐵 are the solutions of 𝜕𝑤ℎ = 0, 𝜕𝜆ℎ = 0.

51

3.32

(ii). If 𝜆𝐴 ≤ 0 or 𝜆𝐵 ≤ 0, 𝜆𝐴 ≤ 0 and 𝜆𝐵 ≤ 0,
𝑉𝐼
𝑉𝐼
𝑉𝐼
𝑉𝐼
𝑤𝐴̅̅̅
= 𝑤𝐴̅̅̅
; 𝑤𝐵̅̅̅
= 𝑤𝐵̅̅̅
.
𝜕𝐿𝑉𝐼

̅̅̅̅

𝜕𝐿𝑉𝐼

̅̅̅̅

𝜕Γ𝑉𝐼

̅̅̅̅

ℎ

ℎ

−ℎ

̅̅̅

where 𝜆 is the solution of 𝜕𝜆ℎ = 0 , 𝜕𝑤ℎ = 0 , 𝜕𝑤−ℎ = 0 . 𝑤 𝑉𝐼

is the solution of

̅̅̅̅

𝜕Γ𝑉𝐼
ℎ
𝜕𝑤ℎ

= 0.

̅̅̅
(iii). If 𝜆𝐴 ≤ 0 or 𝜆𝐵 ≤ 0, 𝜆𝐴 > 0 and 𝑤𝐵𝑉𝐼
> −𝑟,
𝑉𝐼
𝑉𝐼
𝑉𝐼
𝑤𝐴̅̅̅
= −𝑟; 𝑤𝐵̅̅̅
= 𝑤𝐵̅̅̅
.
𝑉𝐼(iv). If 𝜆𝐴 ≤ 0 or 𝜆𝐵 ≤ 0, 𝜆𝐵 > 0 and 𝑤𝐴̅̅̅
> −𝑟,
̅̅̅
̅̅̅
̅̅̅
𝑤𝐴𝑉𝐼
= 𝑤𝐴𝑉𝐼; 𝑤𝐵𝑉𝐼
= −𝑟,
̅̅̅

𝜕Γ𝑉𝐼

̅̅̅̅

𝜕𝐿𝑉𝐼

̅̅̅̅

𝜕𝐿𝑉𝐼

̅̅̅̅

ℎ

−ℎ

−ℎ

where 𝑤 𝑉𝐼 is the solution of 𝜕𝑤ℎ = 0, 𝜕𝜆−ℎ = 0, 𝜕𝑤−ℎ = 0.
The equilibrium solutions are quite lengthy so we omit the demonstration. Hence, using
the same parameter values defined in previous section, numerical examples with different
𝜇 (the share of social surplus), 𝑙 (the cross effect between terminals), and 𝑛 (the numberof airlines in each terminal) are shown in Fig 3.10. We can find that if the numbers ofairlines in both terminals are the same, the social surplus of Model 𝑉̅ will never exceed
̅̅̅ regardless of the social surplus share 𝜇. The lower the 𝜇, the greater thethat of Model 𝑉𝐼
difference in social surplus. If terminals have different numbers of airlines, when 𝜇 is not
so high, Model ̅̅̅
𝑉𝐼 outperforms 𝑉̅ in the social surplus and vice versa.

52

Fig 3.10 Comparison of total prices and the social surplus of each Model

The reason can be found from the comparison of terminal fees. In Model 𝑉̅ withasymmetric terminals regarding the airline number, when 𝜇 is so high that the non53

negative profit constraint is bound, the operator can set out a flexible “high-low” or“positive-negative” pricing strategy to balance the quantities of two terminals in pursuitof higher social surplus since the centralized operation allows for cross-subsidization
between terminals to ensure non-negative profits of terminals. This cannot be realized in
the separated Model ̅̅̅
𝑉𝐼 where both operators have to ensure the profit of terminals
respectively and a negative terminal fee (equivalently an incentive) therefore seems
infeasible. On the other hand, as the comparison between Model V and VI shows, profitmaximizing duopoly of substitutes can always lead to lower prices and higher social
surplus than monopoly. Therefore, when 𝜇 is low, the positive effect of competing
substitutes outweighs the negative effect of separated social surplus maximization, thus̅̅̅ . These results indicate that when a heavy
resulting in the advantage of Model 𝑉𝐼
regulation or a high public share cannot be achieved, the introduction of terminal
competition can serve as a means to restrain market power, when the airfield is public.
This is also in line with McLay and Reynolds-Feighan (2006) that terminal competitionis only a fallback option when regulation is difficult to impose.
3.4. Conclusion
The introduction of inter-terminal competition is viewed as an alternative to regulation to
restrain the abuse of market power by airports with specific characteristics. Although theidea has been given much consideration in some countries, it is still far from realized.
Discussions about relevant issues are always dominated by conflicts of interest among
stakeholders, while in-depth investigations of some critical questions, such as whether
terminal competition can truly restrain substantial market power, is lacking. Thus, tobridge this gap and help support decision-making, this study investigates the impact of
introducing terminal competition on pricing and social surplus. We solve an analyticalmodel and compare results across several cases.
Our results suggest that, in most cases, having competing terminals can neither lower the
prices nor enhance the social surplus if the operation of the terminal and airfield facilities
are not completely separated in the existing business model (e.g., the incumbent operator
operates both the airfield facilities and terminals), whether or not airlines have thefreedom to change base terminal in response to the prices. The complementarity between
54the airfield and terminal service, which is originally internalized by the joint operation,
will become a negative effect that cannot be offset even by a strong degree of substitution
between terminals, if the two services are provided respectively by independent operators.
When the complementary duopoly and the substitutive duopoly coexist, the pricereducing effect of the latter is dominated by the price increasing effect of the former. In
contrast, if the operation of the two sections has been completely separated in the existing
business model (e.g., one operator operates the terminals while another operates theairfield), or the airfield operator can be strictly regulated, having competing terminals canresult in a higher social surplus, as it will not increase the negative complementary effect.
Instead, it creates a duopoly of substitute goods that can offset the complementary effect.
The runway service allocation problem regarding terminal competition, raised by McLay
and Reynolds-Feighan (2006), that the incumbent operating one terminal and airfield
facilities jointly can “bully” the independent terminal operator by over-biding runway
access, can only be tackled by the imposition of price-cap on the airfield service. Split of
the joint operation will cause a further adverse outcome.
If the terminals have different marginal costs and non-aeronautical profitability, stringent
conditions must be satisfied to make the business model with competing terminals betteroff. However, the improvement in social surplus might be modest. Regarding terminal
competition with airlines’ participation in the upstream market, social surplus can be
improved only in the case that an airline owns a share of a terminal instead of directly
operating it, if the airfield operator is not regulated. If terminals can compete on bothprice and service level, the introduction of terminal competition will not have any effecton the price, service level, and social surplus. Finally, if airfield operation is strictly
regulated while terminals can retain profitability to some extent, competing terminals canbe better-off as long as the regulation on terminals is not too heavy.
Overall, our findings indicate that the introduction of terminal competition fails to restrain
the abuse of market power by monopoly airport, and improve social surplus in most cases,
contrary to common belief. If we further take into account other potential problems ofcompeting terminals, such as the safety concern and loss of economies of scale, we canonly say that having competing terminals without any restrictions may not be a reasonable
solution. If an airport is to be privatized, all-in-one privatization can lead to better results,
55

and if a new terminal is to be constructed, it should be operated by the incumbent operator.
If intra-airport competition between terminals becomes inevitable, a well-designed
regulation imposed on operators is necessary to avoid welfare loss, but at the same time,
this creates regulatory costs for the operators and the regulator, which is just one of thereasons governments want to find a new approach to replace regulations.
The findings of this study can also be extended to other transport facilities with similar
structures. For instance, in terms of the seaport, intra-port terminal competition is not rarein the reality (e.g., Rotterdam, ECT vs APMT), and, apart from the substitutive container
handing services provided by terminals, there also has navigation service provided by the
port authority which can be complementary with the handling services. Since in mostcases the supplier of the complementary good, namely the port authority, is public, partof our findings can be extended to the seaport case that the intra-terminal competition canbe better-off as long as the pricing of the complementary good is regulated under certain
conditions, which is partially consistent with the conclusion of De Langen and Pallis(2006).
The study has some limitations. First, the impact of introducing terminal competition on
long-term infrastructure investment, which is also a very important issue in this context,
is not considered. McLay and Reynolds-Feighan (2006) note that the fragmentation ofcommercial information-gathering across multiple terminals may obscure the evidence ofpotential demand growth and existing terminal capacity constraints, making decisionmaking for investment more complex. Further, related to the terminal capacity problem,
terminal congestion is also ignored. This issue should be further addressed together withpricing in a future study. Second, we employ Bertrand model to formulate the competitionin the upstream market, but its plausibility needs further verification. Third, the diversityof airlines is not fully addressed; in reality, airlines with different market positioning canhave different operating objectives and efficiency level (e.g., marginal cost). Theselimitations offer directions for additional study.

56

Chapter 4 Expand Capacity or Reduce Cost? An Analysis of
the Optimal Timings for Two Types of Projects at an Airport
4.1 Introduction
As mentioned in Section 2.3, many airports in the world are encountering or going toencounter capacity crisis. Whilst there are plenty of measures that can alleviate thedisutility caused by capacity shortage temporarily, a capital investment in the airport
infrastructure proves to be a “surgical” solution that can tackle the problem for a longterm.
On the other hand, airports are embracing innovation, technology adoption, and newservices aiming to improve security, operational efficiency, and passenger experience.
Among them, improvements in operational efficiency have been paid much attention byboth public and private airports. For public airports, higher operational efficiency could
mean lower cost, higher financial sustainability, and better social surplus. For private
airports, since regulations are always imposed based on efficiency level (for example,
incentive regulation or CPI +/- X), higher operational efficiency means higher “freeprofit.” Many examples of airports investing in innovation for efficiency improvement
can be observed: a digital twin, which can create a virtual replica of airport assets throughreal-time data, was built in Amsterdam Airport Schiphol in 201713. The International Data
Corporation projects that cost saving up to 25 per cent can be realized through the
adoption of a digital twin14. Heathrow Airport invested £50 million in facial recognition
to greatly improve the efficiency of various checks15.
In addition to improving operational efficiency, there is another approach that could
indirectly “reduce” operational costs: increasing non-aeronautical revenue. Nonaeronautical revenue has become a major component of airports’ revenue streams. In
some airports, non-aeronautical revenues even exceed aeronautical revenues (for example,
Atlanta and Singapore) (Graham, 2018). As the significance of non-aeronautical activities
13

https://www.esri.com/about/newsroom/arcuser/digital-twin-helps-airport-optimize-operations/14

https://www.altran.com/as-content/uploads/sites/7/2019/09/digital-twin-pov-whitepaper_v7.pdf

15

https://mediacentre.heathrow.com/pressrelease/details/81/Brand-News-22/10209

57

is unlikely to decline in the future (Flores-Fillol et al., 2018), many airports haveproactively conducted projects to renovate existing commercial facilities, or construct
new ones, in recent years. For instance, Helsinki Airport opened a new commercial area
called “Aukio” in 201916; retail and hospitality expansion of Terminal 2 in Melbourne
Airport is expected to be completed in 202217. Although it will be up to the concessionaire
to provide the investment for fitting out the facility, it was found that an increase of 1 percent in the commercial area space led to a 0.2 per cent growth in commercial revenue18.
Facing two types of investments, capacity expansion and cost reduction (including the
expansion of non-aeronautical facilities in the context of this paper), how should airports
make decisions that could bring the greatest benefit? Three points need to be addressedcarefully. First, decision-makers should have a holistic consideration. Since aeronauticaland non-aeronautical services are complementary, and the advantages of lower costcannot be fully exploited if passenger growth is hindered by capacity shortage, a partial
consideration might result in biased investment that fails to bring expected benefits.
Second, the timing of each type of investment should be considered. Investment ininfrastructure is lumpy, costly, and irreversible. An untimely investment decision mightcause substantial losses that cannot be redeemed. Third, demand uncertainty should not
be ignored. Aviation demand is generally growing year by year in most markets, but it
can be highly vulnerable to some kind of catastrophe (for example, 9/11, the GreatRecession, or COVID-19). Decision-makers should carefully estimate the risk caused bydemand shock when planning projects in airports. Although numerous studies regarding
airport investment can be found, most of them only focus on capacity expansion, andmany of them do not consider timing and demand uncertainty. Only few studies consider
multiple types of investments in airports. Kidokoro et al. (2016) study the airport’s
decision on the size choices for both aeronautical and non-aeronautical services
employing a general-equilibrium model. However, their analysis is based on deterministic
modeling. Sun and Schonfeld (2015) developed an airport capacity expansion model for16

https://www.finavia.fi/en/newsroom/2019/helsinki-airports-new-aukio-central-plaza-showcases-

finnish-brands-long-haul17

https://my.melbourneairport.com/t2-retail-expansion

18

https://www.moodiedavittreport.com/aci-underlines-role-of-service-quality-on-commercial-revenues/

58three types of facilities—airfield, terminal, and cargo—considering demand uncertainty.
While they address the timing issue in the simulation, the investment period is partitioned
discretely, and the value of the option is not considered. For studies regarding multi-type
investments in other areas, Décamps et al. (2006) provide an excellent discussion on the
problem of investment in alternative projects with real options, which supersedes theclassical theory of Dixit (1993). However, in their settings, two projects are each
exclusive in that they are not allowed to coexist, and the investment sequence ispredetermined. Truong et al. (2018) study the investment decisions of multiple climatechange adaptation projects in the real option approach. However, they do not address thecase in which several projects are invested in simultaneously.
To bridge the research gaps, this study aims to investigate the effect of various external
factors on the timings and scales of airport investment considering two types ofinvestments, namely capacity expansion and cost reduction. First, we examine theplausibility of modeling airport traffic evolution as a geometric Brownian process. Next,
we build an analytical model to present the vertical structure of the market and derive theequilibrium output. Employing a real option approach to the decision-making forinvestments, we derive an optimal rule for the sequential investment taking into accountof the inter-relationship between the projects. We then investigate how the profit-oriented
degree, together with the downstream market structure and the dynamics of demand,
affect the timing and scale of investments through numerical examples. Finally, weinvestigate the effects of those factors on the loss incurred when investments are carriedout in sub-optimal timings following net present value (NPV) and deterministic rules.
4.2. Uncertainty test
Before modeling, we need to determine the type of stochastic process that is used toaddress the growing trend and uncertainty of airport traffic. Owing to its friendly
mathematical properties, geometric Brownian motion (GBM) is widely used in various
areas, including some recent studies regarding investments in transportation
infrastructures (for example, Balliauw et al., 2019; Balliauw and Onghena, 2020; Zheng
et al., 2020). A standard GBM is formulated as 𝑑𝑥 = 𝛼𝑥𝑑𝑡 + 𝜎𝑥𝑑𝑧, where 𝑥 denotes thestochastic variable, 𝛼 and 𝜎 denote drift and variance, respectively, and 𝑡 refers to time.
59

𝑧 is a Wiener process with 𝑑𝑧 = 𝜖𝑡 √𝑑𝑡, where 𝜖𝑡 is a standard normal deviate. Whiledemand is assumed to follow GBM in the studies mentioned above, plausibility of suchan assumption is seldom examined using realistic data. In this section, we follow theapproach of Hamilton (1994) and also Pindyck and Rubinfeld (1998) to perform a
Dickey-Fuller test to verify whether it is reasonable to formulate airport traffic as GBM.
The process is as follows:
1. Take logarithm of the data. If the data follows GBM, its logarithm will follow
Brownian motion.
2. Define the null hypothesis, 𝐻0 , as 𝑥𝑡 = 𝛼 + 𝑥𝑡

+ 𝑢𝑡 with 𝑢𝑡 i.i.d. with mean zero. It

corresponds to a random walk whose limit is Brownian motion. Define the alternative
hypothesis, 𝐻𝐴 , as 𝑥𝑡 = 𝛼 + 𝛽𝑡 + 𝜌𝑥𝑡+ 𝑢𝑡 , which corresponds to a first-order

autoregressive process.
3. Run regressions. Regression for 𝐻0 and 𝐻𝐴 is called restricted and unrestricted
regression, respectively. Then, calculate 𝐹 and 𝑡 statistics, and refer to the values in the
Dickey-Fuller table. 𝐹 is calculated by 𝐹 = 𝑁 − 𝑘 𝐸𝑆𝑆𝑅 − 𝐸𝑆𝑆𝑈𝑅 / 𝐸𝑆𝑆𝑈𝑅 , where 𝑁
is the sample size, 𝑘 denotes the number of estimated parameters in the unrestricted
regression (𝐻𝐴 ), anddenotes the number of parameter restrictions. 𝐸𝑆𝑆𝑅 and 𝐸𝑆𝑆𝑈𝑅

denote the sums of squared residuals in the restricted and unrestricted regressions,
respectively. The 𝑡 statistic is calculated by 𝑡 = 𝜌̂ − 1 /𝜎̂𝜌̂ , where 𝜌̂ is the estimated 𝜌and 𝜎̂𝜌̂ denotes the standard errors.
We use the historical traffic data of four airports, as shown in Table 4.1 and Fig. 4.1. Eachof the four represents one type of airport. As one of the busiest airports in the world,
Heathrow is a typical international hub. While Sydney is also a hub airport with hightraffic, its “hubness” is significantly lower than that of Heathrow (Cheung et al., 2020).
Bergen, with much lower traffic than the previously mentioned two large airports, is aninternational airport of medium scale, while Ljubljana is an emerging regional hub with
low traffic in the current stage. The test results are shown in Table 4.2. In terms of the Ftest, the null hypothesis that the traffic follows a random walk cannot be rejected at the0.01 level for Heathrow and at the 0.1 level for the other three airports. In terms of the ttest, for all airports, the null hypothesis cannot be rejected at the 0.1 level. These results
60suggest that it is not unacceptable to describe airport traffic as GBM.
Table 4.1 Basic information of airports selected for the test

AirportCountry

Traffic in 2019

HeathrowUK

80,884,310

SydneyAustralia

44,428,845

BergenNorway5,964,341

Ljubljana

Slovenia

1,721,355

Fig 4.1 Evolutions of traffic at the selected airports
Table 4.2 Results of the Dickey-Fuller test

Cannot be

SampleDickey-size (N)

Fuller F

Heathrow

32

8.712531%-2.9046

0.175582

Sydney33

3.3385110%-2.04189

0.555857

Bergen28

2.1019710%-1.8791

0.637216

Ljubljana33

2.4465310%-2.17775

0.484232

Airportrejected at(F)

61

DickeyFuller t

P-value (t)4.3 The model
4.3.1 Description of the problem
Stakeholders: We focus on one section of the air transport supply chain for our discussion.
It comprises the upstream market where the one single airport sells airport service toairlines and the downstream market where the airlines sell air service, which is a bundlingof airport service and airline service, to passengers. We consider two types of airlines inthe downstream: Leader airlines and follower airlines. We ignore the entry and exit ofairlines.
Decision-making process: The airport has two types of decisions, investing decision andpricing decision, to make. The investing decision embodies the time and the scale for theinvestment in each project, and the pricing decision refers to the charge airport levies tothe airlines. The investment decision is made in prior to the pricing decision. In thedownstream, given the decisions from the upstream, the leader airlines choose their
outputs (passenger) at first to maximize their profits respectively. Next, observing theremaining market, the follower airlines decide their outputs. Under such setting, theoutput of each leader airline will be higher than that of each follower airline, which canreflect the situation that hub airlines always have higher market shares than other airlines
at an airport. The multi-stage game is illustrated by Fig 4.2.
1

AirportScales andtimings forinvestments

Airport2

ChargeLeader
Airlines

Unsatisfied
demandSupply 3

Charge2

Follower
Airlines

Supply 4

Passenger

Fig 4.2 Diagram of the multi-stage game (Numbers in the circles indicate the sequence ofdecision-making)62

Table 4.3 Notations for Chapter 4

Decision variables
Number of passengers of airline
𝑤
𝑌𝐶
𝑐
𝑌𝐾
𝑘

Airport charge per passenger
Optimal timing (demand level) for the investment in cost
reduction
New cost level after the investment in cost reduction
Optimal timing (demand level) for the investment in
expansion
New capacity level after the investment in expansion

Parameters𝑎Delay cost parameter𝑏 0≤𝑏≤1

Cross-price effect coefficient𝑋

Maximum willingness to pay of the passenger

𝑌

Reverse of the own price effect𝑚

Number of leader airlines𝑛Number of follower airlines

𝐼𝐾

Fixed cost for the investment in capacity expansion

𝑐𝐾

Unit cost for the investment in capacity expansion

𝑐𝐶

Unit cost for the investment in cost reduction

𝑐0

Cost level before the investment

𝑘0

Capacity level before the investment

𝜃Proportion of profit in the airport’s objective function

𝛼

Drift

𝜎

Volatility𝑟Discount rate

Several important assumptions are made as follows for the sake of tractability or
simplicity:a. The decision-maker cannot optimize the volumes of the projects. Instead, he or she can63select the scales of the projects from several pre-determined discrete levels to maximize
the payoff.
b. Lead times of the investments are ignored. Alternatively, this can be explained by thefact that timing refers to the time when the project is expected to be finished, and the costrefers to the aggregated retrospective cost (for example, if the span is three years, the cost
stands at 𝐼 𝑒 3𝑟 instead of 𝐼) incurred in the entire investment span.
c. Incremental investments are not considered.
Table 4.3 shows the notations of variables and parameters.
4.3.2 Formulation
We adopt backward induction to obtain the optimal decisions for each market participant.
A quadratic function is employed to formulate the passengers’ utility (for example, Vives,
1999). It is formulated as
𝑈=𝑋 −

1
2𝑌

𝐿 + 2𝑏 𝐿

𝐹+

4.1

𝐹

where 𝑈 denotes utility, 𝑋 denotes the maximum level of willingness-to-pay ofpassengers, and 1/𝑌 denotes the own-price effect.

𝑚
𝐿 =∑=

𝐿

and

𝑛
𝐹 = ∑𝑗=𝐹𝑗denote the total passenger numbers (quantities) of the leader airlines and follower airlines,
respectively. 𝑏 denotes the cross-price effect with 0 ≤ 𝑏 ≤ 1.

=

𝐿+

𝐹 . Maximizing

the utility function, we can derive the inverse demand functions for passengers as𝑎
𝜕𝑈1
𝜌 = 𝑝 + ( )𝛿 ==𝑋−( )
𝑌
𝜕
𝑌

𝑏
−( )
𝑌

4.2

where ℎ = 𝐿 𝐹; 𝑝 denotes the airfare of airline that belongs to type ℎ; 𝑤 denotes the
charge from airport; 𝑎 denotes the unit delay cost; and 𝛿 = /𝑘 denotes the delay time,
where 𝑘 is the capacity level. It is clear that both 𝑋 and 𝑌 are external factors that candetermine the demand level, and most similar studies let 𝑋 follow a stochastic process todescribe demand uncertainty. In contrast, we set 𝑌 as the stochastic variable in this study.
Doing so ensures a linear payoff function of the stochastic variable, which greatly
simplifies the calculation, as shown in the following section. As examined in Section 3,
GBM is not inappropriate to describe airport traffic, so we formulate the evolution of 𝑌64as𝑑𝑌 = 𝛼𝑌𝑑𝑡 + 𝜎𝑌𝑑𝑧

4.3

where the meanings of the parameters have been explained in Section 3. Some may
question that while GBM has been verified to be reasonable for describing airport traffic,
it is employed here to describe airport demand. This doubt can be resolved by obtaining
the final output of traffic.
Next, we solve the subgame perfect equilibrium of the downstream market. The profit
function of the follower airline is
𝜋𝐹

𝐹

𝒒𝐹

𝑤 𝑘

𝐿 𝑌

= [𝑝𝐹

𝐹

𝒒𝐹

𝑘

𝐿 𝑌

− 𝑤] 𝐹

4.4

where the operational cost was normalized to 0. For simplicity, we assume that airlines
do not incur congestion costs following Wan et al. (2015). This can be explained by thefact that the cost is transferred to the passengers. Solving the first-order condition (FOC)of Eq. (4) and imposing symmetry, we have

𝐹

𝑤 𝑘

𝐿 𝑌

=

𝑘𝑌 𝑋 − 𝑤 − 𝑎 + 𝑏𝑘
𝑛+1 𝑎+𝑘𝐿

4.5

Substituting Eq. (5) into the profit function of the leader airline and solving the FOC of
𝜋𝐿

𝐿

𝒒𝐿

𝑤 𝑘

𝐹 𝑌

= [𝑝𝐿

𝐿

𝒒𝐿

𝑘

𝐹 𝑌

− 𝑤] 𝐿 , we have the subgame

perfect equilibria of outputs as
𝐿

𝑤 𝑘 𝑌 = 𝐶𝐿 𝑌 𝑋 − 𝑤

4.6

𝐹𝑤 𝑘 𝑌 = 𝐶𝐹 𝑌 𝑋 − 𝑤

4.7

where:𝐶𝐿 =𝐶𝐹 =𝑘𝑚[𝑎 + 𝑘 1 + 1 − 𝑏 𝑛 ]𝑚 + 1 [𝑘 ( 1 − 𝑏 𝑛 + 1) + 𝑎 + 2𝑎𝑘( 1 − 𝑏 𝑛 + 1)]

4.8𝑘𝑛
1 − 𝑏 𝑘𝑚
1
[
+
] 4.9
𝑚 + 1 𝑘 ( 1 − 𝑏 𝑛 + 1) + 𝑎 + 2𝑎𝑘( 1 − 𝑏 𝑛 + 1)
𝑛+1 𝑎+𝑘𝑚 and 𝑛 denote the number of leader and follower airlines respectively.
The airport’s objective function is formulated as
𝛤 𝑤 𝑘 𝑐 𝑌 =𝜃𝑤 𝑘 𝑐 𝑌 + 1−𝜃 𝑆 𝑤 𝑘 𝑐 𝑌

65

4.10

where:𝑤 𝑘𝑐 𝑌 =𝑎
𝑆 𝑤 𝑘 𝑐 𝑌 = (𝑋 − 𝑐 − ( ) 𝛿
𝑌

𝑘 )

𝑤−𝑐
−

1
2𝑌

𝐿

4.11
+ 2𝑏 𝐿

𝐹

+

𝐹

4.12

𝜃 is a coefficient that determines the premium the airport places on profit with 0 ≤ 𝜃 ≤1. It can also be interpreted as the amount of private share in the airport’s ownership. The
higher the 𝜃, the more profit-oriented the airport is. If 𝜃 = 0, the airport will never takeadvantage of its market power; it will maximize the social surplus. 𝑐 denotes thedifference between unit cost and unit non-aeronautical profit per passenger.
We assume that the airport must ensure financial break-even when setting the optimal
charge, so a non-negative profit constraint should be attached to Eq. (4.10). From Eq.
(4.11), we know that the non-negative profit constraint is equivalent to 𝑤 ≥ 𝑐, so the
Lagrange function becomes𝐿 𝑤 𝜆 𝑘 𝑐 𝑌 =𝛤 𝑤 𝑘 𝑐 𝑌 +𝜆 𝑤−𝑐

4.13

𝜕𝐿

The Karush-Kuhn-Tucker condition for the problem is 𝜕𝑤 = 0 𝑤 ≥ 𝑐 𝜆 ≥ 0 𝜆 𝑤 −𝑐 = 0. Solving these conditions, we obtain the optimal charge as
𝑤 𝑘𝑐 =𝑐0≤𝜃<𝜃{𝑘 𝑋 𝑐 𝐶𝐹 +𝐶𝐿
𝐶𝐹 ( 𝑏𝑘+ 𝑎 𝜃

𝐶𝐹2

𝐶𝐿 𝑘𝜃)+ 𝜃

𝑎+𝑘 +𝐶𝐿 ( 𝜃

𝐶𝐿

𝑎+𝑘

𝑘𝜃)

+𝑋 𝜃 ≤𝜃 ≤1

4.14

Maximum payoff is accordingly derived as𝛤 𝑘𝑐 𝑌 =𝜃

𝑌 𝑋 𝑐 2 ( 𝑎 𝐶𝐹 +𝐶𝐿 2 +𝑘( 𝐶𝐹 𝑏𝐶𝐿+𝐶𝐹2 + 𝐶𝐿

𝐶𝐿 ))𝑘
𝑘𝑌 𝑋 𝑐 2 𝐶𝐹 +𝐶𝐿 2{
4𝑎𝜃 𝐶𝐹 +𝐶𝐿 2

where 𝜃 = 𝑘( 𝐶𝑘𝜃( 𝐶𝐹 𝑏𝐶𝐿

𝐹 𝑏𝐶𝐿+𝐶𝐹2 + 𝐶𝐿𝐶𝐿 )+ 𝑘( 𝑏𝐶𝐹 𝐶𝐿 +𝐶𝐹2 +𝐶𝐿2 )𝑘 𝐶𝐹 +𝐶𝐿+𝐶𝐹2 + 𝐶𝐿𝐶𝐿 )+ 𝑎 𝐶𝐹 +𝐶𝐿 2

0≤𝜃<𝜃

4.15

𝜃 ≤𝜃≤1+ 1. From Eq. (4.14), it is clear that the

optimal charge is not a function of 𝑌 . The equilibrium output (traffic), derived by66

substituting the optimal charge into Eq. (4.6) and (4.7), is merely a linear function of 𝑌
without an intercept. By Eq. (4.3), it is known that if 𝑌 follows GBM, the product of 𝑌
and a constant will follow GBM as well. Hence, the equilibrium output follows GBM.
Let the initial levels of capacity and cost be 𝑘0 and 𝑐0 , respectively. Improvement in the
maximum payoff by expanding capacity from 𝑘0 to 𝑘 at 𝑌 is denoted as
𝑅𝐾 𝑌 = 𝛤 𝑘 𝑐0 𝑌 − 𝛤 𝑘0 𝑐0 𝑌

4.16

Improvement in the maximum payoff by reducing cost from 𝑐0 to 𝑐 at 𝑌 is denoted as
𝑅𝐶 𝑌 = 𝛤 𝑘0 𝑐 𝑌 − 𝛤 𝑘0 𝑐0 𝑌

4.17

Improvement in the maximum payoff by carrying out two projects simultaneously at 𝑌 isdenoted as
𝑅𝐴 𝑌 = 𝛤 𝑘 𝑐 𝑌 − 𝛤 𝑘0 𝑐0 𝑌

4.18

We then have:
Lemma 4.1
For any 𝑘 > 𝑘0 > 0 and 𝑐0 > 𝑐 > 0, the improvement in payoff by carrying out two
types of projects simultaneously will always be greater than the summation of the two
respective improvements in payoffs, namely, 𝑅𝐴 > 𝑅𝐾 + 𝑅𝐶 .
Proof: The objective function 𝛤 𝑘 𝑐 𝑌 can be written as a multiplicative function of
three one-variable functions, namely, 𝛤 𝑘 𝑐 𝑌 = 𝛤𝐾 𝑘
𝑅𝐶 = [𝛤𝐾 𝑘− 𝛤𝐾 𝑘0 ] 𝛤𝐶 𝑐0 + [𝛤𝐶 𝑐

𝑅𝐴 = 𝛤𝐾 𝑘

𝛤𝐶 𝑐− 𝛤𝐾 𝑘0

𝑅𝐴 − 𝑅𝐾 + 𝑅𝐶 = [𝛤𝐾 𝑘𝑑𝛤𝐶 𝑐

To prove
we have

𝑑𝛤𝐾 𝑘
𝑑𝑘

𝑑𝑐

𝛤𝐶 𝑐0 , and− 𝛤𝐾 𝑘0 ] [𝛤𝐶 𝑐𝜕𝑘=− 𝛤𝐶 𝑐0 ].
𝑑𝛤𝐶 𝑐𝜕𝐿 𝑤 𝜆 𝑘 𝑐 𝑌𝜕𝑘|𝜕𝛤 𝑘 𝑐 𝑌

𝑤=𝑤 𝜆=𝜆

𝜕𝑘

=

𝜕𝑘

𝑑𝛤𝐾 𝑘
𝑑𝑘

𝑑𝛤𝐾 𝑘
𝑑𝑘

have different signs. It is> 0 as follows:> 0. Following the envelope theorem,𝜕𝛤 𝑤 𝑘

67

and

𝑑𝑐

< 0. We can then prove that

> 0, it suffices to show𝜕𝛤 𝑘 𝑐 𝑌𝑌, so we have 𝑅𝐾 +− 𝛤𝐶 𝑐0 ] 𝛤𝐾 𝑘0

To complete the proof, it suffices to prove that
self-evident that

𝛤𝐶 𝑐

|

𝑤=𝑤

. Since𝜕𝑄𝐹 𝑤 𝑘

𝜕𝐶

𝑤=𝑤

𝑎
𝑎+𝑘 2
𝜕𝑄𝐿 𝑤 𝑘
𝜕𝑘

(

+𝑛

)[

𝑎

𝑏 + )

𝑏 𝑘𝑚(𝑎+𝑘 𝑛( 𝑎+𝑘 2 +𝑎+𝑏𝑘+𝑘 )

𝑏 𝑘𝑛

2

+

𝜕𝐶

= 𝑋 − 𝑤 𝑌 𝜕𝑘𝐿 =𝑤=𝑤

𝑎𝑚 𝑋 𝑤 𝑌
𝑚+

Since

𝑚+

] > 0 and|

we have𝑛 𝑋 𝑤 𝑌= 𝑋 − 𝑤 𝑌 𝜕𝑘𝐹 = (

|

𝜕𝑘

)[

𝑎+𝑘 2 +

𝑎+𝑘 2 +

𝜕𝛱 𝑤 𝑘

|

𝜕𝑘

𝜕𝑆 𝑘
𝜕𝑄𝐹

𝑏 𝑘

𝑤=𝑤

|

𝑏 𝑘

= 𝑤 −𝑐 [= 𝑝𝐹 𝑤𝑏 2 𝑘 2 𝑛2

𝑎+ 3 𝑏 𝑘 𝑛+

𝑎+𝑘+𝑏𝑘 𝑛 2
𝜕𝑄𝐹 𝑤 𝑘𝜕𝑘|𝑤=𝑤

] > 0,
𝜕𝑄𝐿 𝑤 𝑘+𝜕𝑘𝑎(𝑄𝐹 𝑤 +𝑄𝐿 𝑤 )

−𝑐−

𝑤=𝑤
𝑎(𝑄𝐹 𝑤 +𝑄𝐿 𝑤 ), we have𝑘𝑌] > 0.

𝑤=𝑤

𝜕𝑆 𝑤 𝑘

,

𝑘𝑌

|

𝜕𝑄𝐿

|
𝑤=𝑤

= 𝑝𝐿 𝑤−𝑐−

𝜕𝑆 𝑤 𝑘𝜕𝑆 𝑤 𝐿 𝐹 𝑘 𝜕 𝐹 𝑤 𝑘
𝜕𝑆 𝑤 𝐿 𝐹 𝑘 𝜕 𝐿 𝑤 𝑘
|
=[
+
𝜕𝑘
𝜕 𝐹
𝜕𝑘
𝜕 𝐿
𝜕𝑘
𝑤=𝑤
𝜕𝑆 𝑤 𝐿 𝐹 𝑘
+
]|
𝜕𝑘
𝑤=𝑤
𝜕 𝐹 𝑤𝑘𝜕 𝐿 𝑤𝑘= 𝑝𝐹 𝑤 − 𝑐+ 𝑝𝐿 𝑤 − 𝑐
|
|
𝜕𝑘
𝜕𝑘
𝑤=𝑤
𝑤=𝑤
+

𝑎( 𝐹 𝑤+ 𝐿 𝑤 )
𝜕
−(
𝑘 𝑌

𝐹 𝑤

+
𝜕𝑘

The first and the second terms are positive ( 𝑝𝑗 𝑤

𝐿 𝑤

)(

𝑎( 𝐹 𝑤+
𝑘𝑌

𝐶 +𝐶

can be shown that𝐶 = 𝑛+

𝑛
𝑎+𝑘 2

𝑘

−

𝜕 𝐶𝐹 +𝐶𝐿𝑑𝑘

3+ 1 + 𝑘 2 1 − 𝑏 𝑛 + 1 ( 1 − 𝑏 𝑛 + 1)] > 0,

𝐶 +𝐶

𝜕 𝐶𝐹 +𝐶𝐿

𝑑𝛤𝐾 𝑘

. It

= +𝑚 𝐶 + 𝐶 , where+ 1 − 𝑏 𝑘𝑛 2𝑎 + 𝑏𝑘 + 𝑘𝜕𝑘> 0, andConsequently, we can obtain
so

𝜕𝑘

𝐶2

𝑎+𝑘

Hence, 𝐹 𝑘 𝐿 −𝜕 𝐶𝐹 +𝐶𝐿> 0,

𝐶 = 𝑚[𝑎 + 2𝑎𝑘 𝑛 1 − 𝑏
𝐶3 =

𝜕𝑘

𝑘

)
)

> 𝑤 ≥ 𝑐 . We then need toinvestigate the sign of the remnant, which is equivalent to the sign of 𝐹 𝑘 𝐿 −𝐶𝐹 +𝐶𝐿

𝐿 𝑤

𝜕𝑆 𝑤 𝑘|𝜕𝑘𝑤=𝑤

𝜕𝛤 𝑤 𝑘
𝜕𝑘

|

𝑤=𝑤

> 0.
> 0.

= 1−𝜃

𝜕𝑆 𝑤 𝑘
𝜕𝑘

|

𝑤=𝑤

+𝜃

𝜕𝛱 𝑤 𝑘
𝜕𝑘

|

𝑤=𝑤

> 0,

> 0.

Therefore, 𝑅𝐴 > 𝑅𝐶 + 𝑅𝐾 . ∎We then solve the optimal timing problem in a real option approach. As mentioned in
68Chapter 2, real option approach greatly underlines the opportunity cost of investment,
which can be colossal in the case of airport. Hence, it might be more appropriate to modelthe decision-making regarding the investment in airport projects in a real option approach.
The optimal timing problem can be formulated as
𝐹𝐶𝐾 𝑌 = max [𝑉𝐶 𝑌 + 𝔼[𝑒 𝑟𝑡𝐶𝐾 (𝑉𝐴 𝑌𝐶𝐾 − 𝑉𝐶 𝑌𝐶𝐾 )]]4.19

𝐹𝐾𝐶 𝑌 = max [𝑉𝐾 𝑌 + 𝔼[𝑒 𝑟𝑡𝐾𝐶 (𝑉𝐴 𝑌𝐾𝐶 − 𝑉𝐾 𝑌𝐾𝐶 )]]4.20

𝐹 𝑠 𝑌 = max{𝐹𝐶𝐾 𝑌 𝐹𝐾𝐶 𝑌 }4.21

𝐹 𝑌 = max 𝔼[𝑒 𝑟𝑡𝑠 𝐹 𝑠 𝑌𝑠 ]

4.22

𝑡𝐶𝐾

𝑡𝐾𝐶

𝑡𝑠

where 𝐹𝑥𝑦 𝑌 denotes the maximum expected value of the entire investment in asequence that project 𝑥 is started first at 𝑌 and project 𝑦 will follow subsequently.
Similarly, 𝑡𝑥𝑦 denotes the time to invest in project 𝑦 , given that project 𝑥 has been
finished and 𝑌𝑥𝑦 = 𝑌 𝑡𝑥𝑦 . Subscripts 𝐾 and 𝐶 denote capacity expansion and cost
reduction, respectively. 𝑉𝑧 𝑌 denotes the expected value of project 𝑧 at 𝑌, and subscript
denotes the case in which two types of projects are started simultaneously. 𝐹 𝑌
denotes the value of the option for the entire investment, which can be carried out in any
sequence (KC, CK, or A), given that the investment has not started yet. 𝑡𝑠 denotes thetime to start the entire project. 𝑟 denotes discount rate.
The problem can be solved using a backward induction approach (from 𝑡𝑥𝑦 to 𝑡𝑠 ). We
take the sequence CK as an example to illustrate the solving process. First, we calculate
the expected value 𝑉𝐴 at 𝑌 as∞

𝑉𝐴 𝑌 = 𝔼 [∫ 𝑅𝐴 𝑌𝑡 𝑒 𝑟𝑡 𝑑𝑡] − 𝐼𝐾 − 𝐼𝐶 =0𝑅𝐴 𝑌
− 𝐼𝐾 − 𝐼𝐶
𝑟−𝛼

4.23

where 𝐼𝐾 and 𝐼𝐶 denote the investment cost of project 𝐾 and 𝐶, respectively. Then, we
obtain ℱ𝐶𝐾 𝑌 = 𝐹𝐶𝐾 𝑌 + 𝐼𝐶 by the Bellman function as follows:ℱ𝐶𝐾 𝑌 = 𝑅𝐶 𝑌𝑑𝑡 + 𝔼[ℱ𝐶𝐾 𝑌 + 𝑑𝑌 𝑒 𝑟𝑑𝑡 ]4.24

Expanding Eq. (4.24) by Ito’s lemma, we have the following differential equation:
1
𝜎 𝑌 ℱ𝐶𝐾 ′′ 𝑌 + 𝛼𝑌ℱ𝐶𝐾 ′ 𝑌 − 𝑟ℱ𝐶𝐾 𝑌 + 𝑅𝐶 𝑌 = 0
2
69

4.25

Solving Eq. (4.25) and ruling out worthless terms, we have
𝐹𝐶𝐾 𝑌 = ℱ𝐶𝐾 𝑌 − 𝐼𝐶 =

where

𝑅𝐶 𝑌
− 𝐼𝐶 +𝑟−𝛼

𝐶𝐾 is a constant that is yet to be determined.

The optimal timing (demand level) 𝑌𝐶𝐾 and

𝛽
𝐶𝐾 𝑌

𝛼

4.26
𝑟

𝛼

𝛽 = √( − 𝜎2 ) + 𝜎2 − 𝜎2 + .

𝐶𝐾 canbe obtained by the following

boundary conditions (Dixit and Pindyck, 1994):
𝐹𝐶𝐾 𝑌𝐶𝐾 = 𝑉𝐴 𝑌𝐶𝐾

4.27

′
𝐹𝐶𝐾
𝑌𝐶𝐾 = 𝑉𝐴′ 𝑌𝐶𝐾

4.28

where Eq. (4.27) is the value-matching condition and Eq. (4.28) is the smooth-pasting
condition. Solving them, we have𝛽𝐼𝐾 𝑟 − 𝛼𝛽 − 1 𝑅𝐴 − 𝑅𝐶

𝑌𝐶𝐾 =

𝛽−1 𝛽
(
)
=
𝐶𝐾
𝐼𝐾

𝑅𝐴 − 𝑅𝐶 𝛽
(
)
𝛽 𝑟−𝛼

4.29
4.30

If 𝐹 𝑠 𝑌 = 𝐹𝐶𝐾 𝑌 , by a similar approach we can derive the option value 𝐹 𝑌 as
𝐹 𝑌 =
where

𝛽
𝐶 =( 𝐼

𝛽

𝐾

)

𝑅

𝑅

𝛽

𝛽

𝛽

(𝛽 𝐴𝑟 𝛼𝐶 ) + ( 𝐼 )
𝐶

𝛽
𝐶𝑌

4.31

𝑅

(𝛽 𝑟 𝐶 𝛼 ) . The optimal timing (demandlevel) to start the entire project can thus be obtained as
𝑌𝐶 =𝛽𝐼𝐶 𝑟 − 𝛼𝛽 − 1 𝑅𝐶

4.32

As a result, we can summarize the value of the investment in sequence CK as
𝑅𝐴 𝑌− 𝐼𝐾 − 𝐼𝐶𝑌 ≥ 𝑌𝐶𝐾𝑟−𝛼
𝑅𝐶 𝑌
𝒱𝐶𝐾 𝑌 =𝐹𝐶𝐾 𝑌 =− 𝐼𝐶 + 𝐶𝐾 𝑌𝛽 𝑌𝐶 ≤ 𝑌 < 𝑌𝐶𝐾𝑟−𝛼
𝑌 < 𝑌𝐶{ 𝐹 𝑌 = 𝐶 𝑌𝛽
𝑉𝐴 𝑌 =

4.33

Similarly, we derive the optimal timing (demand level) of investment in other sequences(KC, A) as

70

𝛽𝐼𝐾 𝑟 − 𝛼
𝛽 − 1 𝑅𝐾

4.34

𝑌𝐾𝐶 =

𝛽𝐼𝐶 𝑟 − 𝛼𝛽 − 1 𝑅𝐴 − 𝑅𝐾

4.35

𝑌𝐴 =

𝛽 𝐼𝐾 + 𝐼𝐶 𝑟 − 𝛼
𝛽 − 1 𝑅𝐴

4.36

𝑌𝐾 =

We then obtain the optimal rule of the investment as:
Proposition 4.1
𝑅

𝑅

𝐶

𝐶

𝑅

𝐴
If and only if 𝐼 𝐶 > 𝐼 +𝐼
> 𝐼 𝐾, it is optimal to invest in cost reduction (C) once Y reaches
𝐾

𝐾

𝑌𝐶 , and then invest in capacity expansion (K) once Y reaches 𝑌𝐶𝐾 . If and only if
𝑅𝐴
𝐼𝐶 +𝐼𝐾

𝑅𝐾
𝐼𝐾

>

𝑅

> 𝐼 𝐶, it is optimal to invest in capacity expansion (K) once Y reaches 𝑌𝐾 , and then𝐶

invest in cost reduction (C) once Y reaches 𝑌𝐾𝐶 . Otherwise, it is optimal to invest in thetwo projects simultaneously (A) when Y reaches 𝑌𝐴 .
Proof: Comparing the optimal timings derived, we have following four cases:
Case 1. 𝑌𝐾𝐶 > 𝑌𝐾 and 𝑌𝐶𝐾 > 𝑌𝐶
In this case, rearranging the inequalities, we have 𝑅𝐾 + 𝑅𝐶 > 𝑅𝐴 . Lemma 4.1 has proventhat 𝑅𝐾 + 𝑅𝐶 > 𝑅𝐴 is false. By modus tollens, 𝑌𝐾𝐶 > 𝑌𝐾 and 𝑌𝐶𝐾 > 𝑌𝐶 are thus false. It is
impossible for Case 1 to happen.
Case 2. 𝑌𝐾𝐶 > 𝑌𝐾 and 𝑌𝐶𝐾 ≤ 𝑌𝐶 (Fig. 4.3a)
In this case, the decision-maker has two alternatives: investing in sequence KC or A.
Since it can be proved that 𝑌𝐴 < 𝑌𝐾𝐶 in this case, KC is greater than A in terms of thevalue of the investment at 𝑌𝐴 , namely, 𝒱𝐾𝐶 𝑌𝐴 > 𝒱𝐴 𝑌𝐴 . It is self-evident that
𝒱𝐾𝐶 𝑌 ≥ 𝒱𝐴 𝑌 for any 𝑌 satisfying 𝑌 > 𝑌𝐴 . To prove that 𝒱𝐾𝐶 𝑌 > 𝒱𝐴 𝑌 for any 𝑌
satisfying 𝑌 < 𝑌𝐴 , it suffices to show that𝐶 >𝐴 . If𝐶 ≤𝐴 , as the slope of 𝒱𝐴 in

region 𝑌 < 𝑌𝐾 will not grow slower than that of 𝒱𝐶 , and their starting values and slopes
are all zero, 𝒱𝐾𝐶 𝑌𝐾 will not be greater than 𝒱𝐴 𝑌𝐾 . Then, as the region changes from𝑌 < 𝑌𝐾 to 𝑌𝐾 ≤ 𝑌 < 𝑌𝐾𝐶 , the growing speed of the slope of 𝒱𝐾𝐶 𝑌 will decline, so
71

𝒱𝐴 𝑌 − 𝒱𝐾𝐶 𝑌 will be larger and 𝒱𝐴 𝑌𝐴 should definitely be greater than 𝒱𝐾𝐶 𝑌𝐴 .
This is false, and so

𝐶 >𝐴 is true. As a result, 𝒱𝐾𝐶 𝑌≥ 𝒱𝐴 𝑌 will hold given any

positive 𝑌 , so the decision-maker will never invest in sequence A. Rearranging theinequalities, we can easily find that 𝑌𝐶𝐾 ≤ 𝑌𝐶 𝑌𝐾𝐶 > 𝑌𝐾 is necessary and sufficient for
𝑅𝐾
𝐼𝐾

𝑅

𝑅

> 𝐼 +𝐼𝐴 > 𝐼 𝐶.
𝐶𝐾𝐶Case 3. 𝑌𝐾𝐶 ≤ 𝑌𝐾 and 𝑌𝐶𝐾 > 𝑌𝐶 (Fig. 4.3b)This is symmetrical to Case 2. Using the same method, we find that 𝑌𝐾𝐶 ≤ 𝑌𝐾 𝑌𝐶𝐾 > 𝑌𝐶
𝑅

𝑅

𝐶

𝐶

𝑅

𝐴
is necessary and sufficient for 𝐼 𝐶 > 𝐼 +𝐼
> 𝐼 𝐾.
𝐾

𝐾

Case 4. 𝑌𝐾𝐶 ≤ 𝑌𝐾 and 𝑌𝐶𝐾 ≤ 𝑌𝐶 (Fig. 4.3c)
The only alternative for the decision-maker is to invest in two projects simultaneously. It𝑅

is straightforward that 𝑌𝐾𝐶 ≤ 𝑌𝐾 𝑌𝐶𝐾 ≤ 𝑌𝐶 is necessary and sufficient for 𝐼 +𝐼𝐴 >
𝐶

𝑅𝐾

𝑅𝐴

𝐼𝐾 𝐼𝐶 +𝐼𝐾𝐾𝑅

> 𝐼 𝐶. ∎𝐶

As a result, the optimal rule proves to be very simple: investing in the project with thegreatest gain-cost ratio first. The sequence of investments depends on the ranking of thegain-cost ratio. The optimal timing for the investment in each project can be expressed
as:
𝑌𝐶𝐾
𝑌𝐾 =

𝑌𝐾
𝑌
{𝐴
𝑌𝐶

𝑌𝐶 =

𝑌𝐾𝐶
𝑌
{𝐴

𝑅𝐶
𝑅𝐴
𝑅𝐾
>
>
𝐼𝐶
𝐼𝐶 + 𝐼𝐾
𝐼𝐾
𝑅𝐾
𝑅𝐴
𝑅𝐶
>
>
𝐼𝐾
𝐼𝐶 + 𝐼𝐾
𝐼𝐶
𝑅𝐴
𝑅𝐾
𝑅𝐴
𝑅𝐶
>
>
𝐼𝐶 + 𝐼𝐾𝐼𝐾 𝐼𝐶 + 𝐼𝐾
𝐼𝐶

4.37

𝑅𝐶
𝑅𝐴
𝑅𝐾
>
>
𝐼𝐶
𝐼𝐶 + 𝐼𝐾
𝐼𝐾
𝑅𝐾
𝑅𝐴
𝑅𝐶
>
>
𝐼𝐾
𝐼𝐶 + 𝐼𝐾
𝐼𝐶
𝑅𝐴
𝑅𝐾
𝑅𝐴
𝑅𝐶
>
>
𝐼𝐶 + 𝐼𝐾𝐼𝐾 𝐼𝐶 + 𝐼𝐾
𝐼𝐶

4.38

72

As discussed, under our setting, the output will be a linear function of Y without anintercept. Thus, in practice, we can determine the time to invest just by observing thenumber of passengers (output) using the airport. For example, let the current daily numberof passengers be 10000, and obtain the current value of Y as 10. If the optimal demandlevel (timing) is derived as 20, the decision-maker should invest as long as the number ofpassengers reaches 20000. Following Proposition 1, we can write the value of the wholeinvestment as:
𝑅𝐶

𝒱𝐶𝐾 𝑌

𝐼𝐶
𝑅𝐾

𝒱 𝑌 = 𝒱𝐾𝐶 𝑌>>𝑅𝐴𝐼𝐶 +𝐼𝐾
𝑅𝐴

>
>

𝑅𝐾
𝐼𝐾
𝑅𝐶

4.39

𝐼𝐾
𝐼𝐶 +𝐼𝐾
𝐼𝐶
𝑅𝐴
𝑅𝐾
𝑅𝐴
𝑅𝐶
> 𝐼 𝐼 +𝐼 > 𝐼
𝐼𝐶 +𝐼𝐾
𝐾
𝐶 𝐾
𝐶

{𝒱𝐴 𝑌

The optimal rule will then be applied to the numerical simulations in following sections.

(a)

(b)

(c)

Fig 4.3 Value function 𝓥 in each case: (a) 𝓥𝑲𝑪 , (b) 𝓥𝑪𝑲 , and (c) 𝓥𝑨

4.4 Numerical examples
4.4.1 Settings
In this section, we present several numerical examples to see the effect of 𝜃 (the private
share), 𝑏 (the cross-price effect), 𝑚 and 𝑛 (the number of airlines), 𝛼 (drift), and 𝜎(volatility) on the optimal timings and scales for the investments. For each combinationof project scale (for example, small-scale project of capacity expansion plus large-scale
project of cost reduction), we obtain its optimal timing of each type of investment basedon the abovementioned rules, and then calculate the value of the investment according to
Eq. (4.33). Combination with the highest value is chosen, and the optimal timings thereof73are presented as the final results.
Values of parameters that will not change throughout this section, if not specificallymentioned, are set as shown in Table 4.4. The value of 𝑋 is set based on Basso and Zhang(2008), the marginal operating cost of airport is set based on Martín and Voltes-Dorta(2011), and the cost of the investment in capacity expansion is set based on Sun andSchonfeld (2015). Following Petrakis and Roy (1999), we define the cost of theinvestment in cost reduction as a convex function with regard to the volume of reduction.
Other values are set hypothetically due to the lack of data. The scale of each project has
five levels as shown in Table 4.5.
Table 4.4 Values of parameters for the numerical examples

Parameter

Value

Parameter

Value

𝑋

2000

𝐼𝐶

𝑐𝐶

𝑎

300

𝑐𝐾

2 105

𝑏

0.5

𝑓𝐾

6.5 107

𝑚

2

𝑐𝐶

8 107

𝑛

2

𝜎𝑦𝑒𝑎𝑟 (annual)

0.1

𝑘0

10000

𝛼𝑦𝑒𝑎𝑟 (annual)

0.04

𝑐0

8

𝑟𝑦𝑒𝑎𝑟 (annual)

0.1

𝐼𝐾

𝑓𝐾 + 𝑐𝐾

𝑌0 (Starting time)2𝑘 − 𝑘0𝑐0 − 𝑐Table 4.5 The scale of each project

Very small

Small

MediumLarge

Very large

Capacity
expansion
(𝒌𝟏 )

12000

14000

16000

18000

20000

(+20%)(+40%)(+60%)(+80%)(+100%)

Cost
reduction
(𝒄𝟏 )

7.6

7.2

6.8

6.4

6

(-5%)

(-10%)(-15%)(-20%)(-25%)

74

Note that the payoff will be calculated on a monthly basis, so the monthly drift, volatility
𝜎𝑦𝑒𝑎𝑟 2and discount rate are obtained by 𝛼 = 𝛼𝑦𝑒𝑎𝑟 /12, 𝜎 = √and 𝑟 = 12√1 + 𝑟𝑦𝑒𝑎𝑟 − 1,

respectively.
4.4.2 Comparation of the optimal rules
Before conducting the formal analysis, we would like to compare the optimal rules
through a numerical experiment. Truong et al. (2018) gives an optimal rule for thesequential investment in inter-related projects. Their rule prescribes that the decisionmaker should select the sequence which yields the maximum value, but they did notconsider a case that the optimal timings therein can be just opposed to the selectedsequence (in our context, for example, the optimal sequence is CK, while 𝑌𝐶𝐾 < 𝑌𝐶 ,namely the optimal timing to invest in K is prior to C). Setting the scale of capacity
expansion as medium, and the scale of cost reduction very large, numerical experimentshows that the investment following their “optimal” rule will cause some loss in the value
compared with ours, although the loss is not significant (Fig 4.4).

Fig 4.4 Loss due to the optimal rule proposed by Truong et al. (2018)75

Fig 4.5 Optimal timings (demand level Y) for the investments with different b anddifferent θ

Fig 4.6 Changing rate of optimal timings

4.4.3 Results of the optimal timings
Fig. 4.5 shows how the optimal timings (demand level 𝑌) for the investments change as76

𝑏 and 𝜃 change. First, it is obvious that the optimal timings for the investments in bothprojects become earlier as the private share goes lower. A social surplus-oriented airport
always has stronger inclination to invest earlier than a profit-oriented airport in spite of
the type of project. Next, we can see that the higher the 𝑏 (cross-price effect between
leader and follower airlines), the later the optimal timings. A higher 𝑏 will lead to a lower
output of the follower airline, while the leader airline’s output is a convex function withregard to 𝑏: It increases with 𝑏 when 𝑏 is high, and vice versa. Anyway, the total output
will always decrease as 𝑏 goes higher. A lower output leads to a lower payoff, making
the demand level (timing) which can justify the investment higher. However, the scale ofeach project does not change as 𝜃 and 𝑏 change. In current setting, the optimal scale ofcapacity expansion is very large, while the optimal scale of cost reduction is very small(see Table 4.5). Given higher 𝜃 and 𝑏, the decision maker would rather postpone theinvestment than reduce the scale, and vice-versa.
The priority between the two investments can change as the gain-cost ratio differs. Thechanging rate of the optimal timings with regard to the change in 𝜃 (private share) is aninteresting issue to investigate. To make things clearer, we further present the numerical𝑑𝑌

examples that describe 𝑑𝜃 (Fig 4.6). We can observe that in the non-binding region (recall
that the charge will be bound to unit cost if private share is too low, see Eq. (4.14)), thetiming for cost reduction changes uniformly along with 𝜃, while the timing for capacity
expansion seems to be convex with regard to 𝜃. In other words, the changing rate of the
optimal timing for capacity expansion will grow faster as 𝜃 goes higher. Regarding the
optimal timing for cost reduction, the following result can be deduced analytically.
Proposition 4.2
In the region where the non-negative profit constraint is not binding (that is, 𝜆 = 0) andtwo types of investments are not carried out simultaneously, given a fixed scale of project,
the optimal timing for the investment in cost reduction changes uniformly as 𝜃 changes,
𝑑2 𝑌

namely, 𝑑𝜃2 = 0.
Proof: In the case where the investment in cost reduction is carried out first, 𝑅𝐶𝑈 can be
rewritten as 𝑅𝐶𝑈 = 𝑍 𝜃 𝑘0 [ 𝑋 − 𝑐− 𝑋 − 𝑐0 ] referring to Eq. (17), where77𝑍 𝜃 𝑘 = 4𝑎𝑘 𝐶𝐹 +𝐶𝐿 2
𝜃 𝐶𝐹 +𝐶𝐿2

𝜃𝑘( 𝐶𝐹 𝑏𝐶𝐿+𝐶𝐹2 + 𝐶𝐿𝐶𝐿 )+ 𝑘( 𝑏𝐶𝐹 𝐶𝐿 +𝐶𝐹2 +𝐶𝐿2 )subscript 𝑈 denotes the unbounded region. Hence,𝑑2⁄𝑍 𝜃 𝑘0
𝑑𝜃2

𝑑2 𝑌𝐶𝑈
𝑑𝜃2

= 𝛽

𝑑2

⁄𝑍 𝜃 𝑘0. Because 1⁄𝑍 𝜃 𝑘0 is a linear function of 𝜃,

𝑑𝜃2

and

𝛽𝐼𝐶 𝑟 𝛼[ 𝑋 𝑐1 2 𝑋 𝑐0 2 ]𝑑2𝛽[ 𝑋 𝑐1 2

𝑋 𝑐0 2 ]⁄𝑍 𝜃 𝑘1𝑑𝜃2, so

𝑑2 𝑌𝐾𝐶𝑈𝑑𝜃2𝑑2 𝑌= 0 and 𝑑𝜃𝐶𝑈=
20 . For the case in which the investment in capacity is carried out first,
𝛽𝐼𝑐 𝑟 𝛼the𝑑2 𝑌𝐾𝐶𝑈𝑑𝜃2== 0 also holds. ∎Fig 4.7 Optimal timings (demand level Y) for the investments with different m, n anddifferent θ

Fig 4.7 shows how the optimal timings (demand levels) for the investments change as 𝑚(number leader airlines) and 𝑛 (number follower airlines) change. Results indicate that78

the larger the total number of airlines, the earlier the optimal timings, while the respective
number of leader and follower airlines does not have a significant impact, as long as thetotal number keeps unchanged. This is not difficult to understand, as the more the numberof airlines, the fiercer the competition in the downstream market, and the greater thedownstream demand. However, the total downstream demand would not change greatly
given the total number of airlines 𝑚 + 𝑛 remains unchanged, although the allocation ofdemand might differ as 𝑚 and 𝑛 vary. As with the case changing 𝑏, the optimal scale ofproject is not sensitive to the structure of the downstream.

Fig 4.8 Effect of α and σ on the optimal scales and timings (demand level Y) (The size ofthe marker indicates the scale of project)

Fig 4.8 shows how do 𝛼 (drift) and 𝜎 (volatility) affect the optimal timings and scales.
Results indicate that the optimal scale of project increases as 𝛼 and 𝜎 go higher, and an79increase in the scale also postpones its investment. The scale of cost reduction jumps fromthe smallest to the largest scale when 𝛼 or 𝜎 reaches a certain level, while the scale ofcapacity expansion increases gradually. It is difficult to figure out the actual reason of thedifference. However, it might originate from the following two aspects. First, cost
reduction and capacity expansion affect the payoff function in different ways. The
capacity level 𝑘 is involved in the subgame equilibrium, while the cost level 𝑐 only
appears in the airport’s objective function. Second, the investment cost function is linear
in terms of capacity expansion project, while convex in terms of cost reduction project.
Anyway, given same project scale, the higher the 𝛼, the earlier the investments. This isconsistent with the common sense that a sound prospect of the industry could stimulate
investment. Similarly, a higher 𝜎 results in a later investment. This is because the investor
always tends to wait when being uncertain about the prospect so that she can avoid loss
if demand goes down and retain the opportunity to invest when demand goes up (Dixit
and Pindyck, 1994).
In summary, these results imply that, to restrain the abuse of market power by airports,that is, to improve social surplus, adjustment to the timing for investment might be asimportant as price regulation. The government can use instruments such as subsidization
to align the timings for investments with the social optimum (e.g., Zheng et al., 2020).
Depending on the goal of improvement in social surplus to be achieved, the optimal
adjustment to the timing for different projects can vary by project and external factors.
Generally, projects with lower gain-cost ratios should be brought forward by a longer
span than the other projects. While the span to bring forward can be consistent for cost
reduction in the non-binding region if the goal ∆𝜃 is the same, the span for capacity
expansion is always decreasing as 𝜃 decreases. For example, in terms of cost reduction,
the spans to bring forward can be the same for reducing 𝜃 from 0.5 to 0.4, and reducing
𝜃 from 0.4 to 0.3. However, to achieve the former, the span to bring forward should be
longer than achieving the latter in terms of capacity expansion (see Fig. 4.5). External
factors including cross-price effect, number of airlines, drift and volatility also have nonnegligible effect on the optimal times and the changing rate thereof as 𝜃 changes. The
optimal scale is more sensitive to drift and volatility rather than other factors.

80

4.4.4 Loss due to investments following suboptimal rules
In reality, as there are several obstacles to adopting the real option rule (Zhang andBabovic, 2011), many investments are still planned based on traditional rules such as thenet present value (NPV) rule. In this section, we follow the approach of Truong et al.
(2018) to estimate the loss incurred due to investment following two suboptimal rules,
namely, the NPV rule and the deterministic model. The incentive to perform thisestimation is to see the case in which the decision-maker should avoid suboptimal
investment as much as possible.
By the NPV rule, the decision-maker ignores the value of the option to invest and theopportunity cost incurred when investment is carried out. He or she will invest
immediately as soon as the NPV becomes positive, that is, the expected gain exceeds thetotal cost. The NPV of a project can be formulated as∞

𝑁𝑃𝑉 𝑌 = 𝔼 [∫ 𝑅 𝑌𝑒 𝑟𝑡 𝑑𝑡] − 𝐼

=𝐾 𝐶

4.40

0

Solving 𝑁𝑃𝑉 𝑌 = 0 for each sequence, we can derive the optimal timings (for example,
𝐼

𝑌𝑁𝐶 = 𝐶𝑅𝐶 𝑟 𝛼, 𝑌𝑁𝐶𝐾 =𝐼𝐾
𝑅𝐴 𝑅𝐶 𝑟 𝛼).

The sequencing rule for the investment is the same as in Proposition 4.1. We then obtainthe loss due to the NPV rule as
𝒱 𝑌𝑁𝐶𝑌⁄𝑌𝑁𝐶 𝛽𝑌 < 𝑌𝑁𝐶
𝐿𝑜𝑠𝑠𝑁𝐶𝐾 𝑌 = {𝒱 𝑌 − 𝑁𝑃𝑉𝐶 𝑌
𝑌𝑁𝐶 ≤ 𝑌 < 𝑌𝑁𝐶𝐾𝒱 𝑌 − 𝑁𝑃𝑉𝐴 𝑌
𝑌 ≥ 𝑌𝑁𝐶𝐾

4.41

𝒱 𝑌𝑁𝐾
𝑌⁄𝑌𝑁𝐾 𝛽𝑌 < 𝑌𝑁𝐾
𝐿𝑜𝑠𝑠𝑁𝐶𝐾 𝑌 = {𝒱 𝑌 − 𝑁𝑃𝑉𝐾 𝑌𝑌𝑁𝐾 ≤ 𝑌 < 𝑌𝑁𝐾𝐶
𝒱 𝑌 − 𝑁𝑃𝑉𝐴 𝑌
𝑌 ≥ 𝑌𝑁𝐾𝐶

4.42

𝒱 𝑌𝑁𝐴
𝑌⁄𝑌𝑁𝐴 𝛽𝒱 𝑌 − 𝑁𝑃𝑉𝐴 𝑌

4.43

𝐿𝑜𝑠𝑠𝑁𝐴 𝑌 = {𝑌 < 𝑌𝑁𝐴𝑌 ≥ 𝑌𝑁𝐴𝑅𝐶

𝐿𝑜𝑠𝑠𝑁𝐶𝐾 𝑌𝐼𝐶𝑅𝐾𝐿𝑜𝑠𝑠𝑁 𝑌 = 𝐿𝑜𝑠𝑠𝑁𝐾𝐶 𝑌

𝑅𝑅> 𝐼 +𝐼𝐴 > 𝐼 𝐾
𝐶

>

𝑅𝐴

𝐾

𝐾

>

𝑅𝐶

𝐼𝐾
𝐼𝐶 +𝐼𝐾
𝐼𝐶
𝑅𝐴
𝑅𝐾
𝑅𝐴
𝑅𝐶
> 𝐼 𝐼 +𝐼 > 𝐼
𝐼𝐶 +𝐼𝐾𝐾
𝐶 𝐾
𝐶{𝐿𝑜𝑠𝑠𝑁𝐴 𝑌
81

4.44

Using a deterministic model, the decision-maker will not consider demand uncertainty,
that is, 𝜎 = 0. Investment will be carried out at the time when the NPV of the wholeproject reaches its maximum. The problem is formulated as∞

𝐷𝑉𝑗 𝑌 = max ∫𝑡 𝑅 𝑌𝑒
𝑡𝑖 𝑡𝑗

𝑖

𝑟 𝛼 𝑡

∞

𝑑𝑡 − 𝑒 𝑟𝑡𝑖 𝐼 + ∫𝑡 𝑅𝐴 − 𝑅 𝑌𝑒

𝑟 𝛼 𝑡

𝑗

𝑑𝑡 − 𝑒 𝑟𝑡𝑗 𝐼𝑗

𝑟𝐼

𝑟𝐼𝑗

𝑖

𝐴

The optimal times can then be derived as 𝑡 = 𝛼 ln 𝑅 𝑖 and 𝑡𝑗 = 𝛼 ln 𝑅
optimal demand levels are 𝑌𝐷 = 𝑌 𝑡

𝑟𝐼

𝑟𝐼𝑗

𝑖

𝐴

= 𝑅 𝑖 and 𝑌𝐷 𝑗 = 𝑌(𝑡𝑗 ) = 𝑅

𝑅𝑖

𝑅𝑖

4.45

, and the. The loss due toadopting a deterministic model is
𝒱 𝑌 − 𝐷𝑉𝐶 𝑌𝐷𝐶 𝑌⁄𝑌𝐷𝐶 𝛽
𝑌 < 𝑌𝐷𝐶𝛽
𝐿𝑜𝑠𝑠𝐷𝐶𝐾 𝑌 = {𝒱 𝑌 − 𝐷𝑉𝐶 𝑌 − 𝐷𝑉𝐶𝐾 𝑌𝐷𝐶𝐾 𝑌⁄𝑌𝐷𝐶𝐾 𝑌𝐷𝐶 ≤ 𝑌 < 𝑌𝐷𝐶𝐾𝒱 𝑌 − 𝐷𝑉𝐶 𝑌 − 𝐷𝑉𝐶𝐾 𝑌
𝑌 ≥ 𝑌𝐷𝐶𝐾

4.46

𝒱 𝑌 − 𝐷𝑉𝐾 𝑌𝐷𝐾 𝑌⁄𝑌𝐷𝐾 𝛽
𝑌 < 𝑌𝐷𝐾𝛽𝐿𝑜𝑠𝑠𝐷𝐾𝐶 𝑌 = {𝒱 𝑌 − 𝐷𝑉𝐾 𝑌 − 𝐷𝑉𝐾𝐶 𝑌𝐷𝐾𝐶 𝑌⁄𝑌𝐷𝐾𝐶 𝑌𝐷𝐾 ≤ 𝑌 < 𝑌𝐷𝐾𝐶
𝒱 𝑌 − 𝐷𝑉𝐾 𝑌 − 𝐷𝑉𝐾𝐶 𝑌
𝑌 ≥ 𝑌𝐷𝐾𝐶

4.47

𝐿𝑜𝑠𝑠𝐷𝐴 𝑌 = {𝒱 𝑌 − 𝐷𝑉𝐴 𝑌𝐷𝐴 𝑌⁄𝑌𝐷𝐴 𝛽 𝑌 < 𝑌𝐷𝐴
𝒱 𝑌 − 𝐷𝑉𝐴 𝑌
𝑌 ≥ 𝑌𝑁𝐴
𝑅𝐶

𝐿𝑜𝑠𝑠𝐷𝐶𝐾 𝑌𝐼𝐶𝑅𝐾𝐿𝑜𝑠𝑠𝐷 𝑌 = 𝐿𝑜𝑠𝑠𝐷𝐾𝐶 𝑌
𝑅𝐴{𝐿𝑜𝑠𝑠𝐷𝐴 𝑌

𝐼𝐶 +𝐼𝐾

82

𝑅

𝑅> 𝐼 +𝐼𝐴 > 𝐼 𝐾
𝐶

>

𝐾

𝑅𝐴

𝐼𝐾
𝐼𝐶 +𝐼𝐾
𝑅𝐾
𝑅𝐴

>𝐼

4.48

𝐾

𝐼𝐶 +𝐼𝐾

𝐾

𝑅𝐶

>𝐼

𝐶

𝑅

> 𝐼𝐶
𝐶

4.49

(a)

(b)

(c)

(d)

Fig 4.9 Results of the loss due to suboptimal investment (Absolute)

Fig 4.10 Results of the loss due to suboptimal investment (Relative)

83

Fig 4.9 shows the effects of various factors on the absolute loss due to suboptimal
investment. We can find that the loss due to the NPV rule changes significantly as theprivate share 𝜃 varies. Particularly, in some cases, a minimum (a reverse peak) can appear
with specific value of 𝜃19. Defining the 𝜃 that leads to the minimum as 𝜃̂, this can beinterpreted as follows. When 𝜃 is larger than 𝜃̂ , the optimal timing (demand level) to
carry out the investment based on the NPV rule would not be reached, and the loss would
merely be the discounted expected value of the option in terms of the optimal rule (seeEq. (4.39–4.41) and Truong et al. (2018)). This value increases as 𝜃 declines, as the moresocial surplus-oriented the airport, the earlier it is likely to invest, thus the lesser the loss
discounted. In contrast, when 𝜃 is smaller than 𝜃̂, the investment based on the NPV ruleshould have been finished. Its value increases as 𝜃 declines, and the growing speed
exceeds that of the value in terms of the optimal rule because the latter is merely an option
value with the investment not started yet (investment based on the NPV rule always startearlier, see Dixit and Pindyck, 1994). Therefore, when smaller than 𝜃̂, as 𝜃 declines, thedifference between the two values will decrease. 𝜃̂ can change as the cross-price effect 𝑏
and the number of airlines 𝑚 and 𝑛 change, while the depths of the minima will notchange (Fig 4.9a, 4.9b). However, when the drift 𝛼 changes, both the height and positionof the minimum change (Fig 4.9c). The depth of the minimum changes when volatility 𝜎changes, but the position remains unchanged (Fig 4.9d). In contrast, the loss due to theinvestment adopting the deterministic model is quite modest compared with that causedby the NPV rule, as long as the volatility is not very high. Fig 4.10 shows the relative loss𝑉( 𝑜𝑝𝑡𝑖𝑚𝑎𝑙𝑉

𝑉𝑠𝑢𝑏𝑜𝑝𝑡𝑖𝑚𝑎𝑙

𝑜𝑝𝑡𝑖𝑚𝑎𝑙) due to suboptimal investment. It can be found that the loss is 100%

when 𝜃 is high. This is because the optimal timing of NPV-based investment will go lateras 𝜃 increases, and it might exceed the starting time when 𝜃 is high enough. In this case,following NPV rule, the decision-maker will invest once the demand level reaches a levelthat makes the NPV equals to 0, so the value of the investment in this case also 0. By
contrast, the optimal timing of the optimal rule will always be later than that of the NPV

19

To demonstrate such cases, we reset the values of some parameters (the scales of projects are fixed):

𝑘0 = 2000 𝑘 = 3000 𝑐 = 6 𝑓𝐾 = 2 107 𝑐𝐾 = 1 105 𝑐𝐶 = 2 107 𝑎 = 60.

84

rule, so the value must be positive, causing a 100% loss. In summary, these results
indicate that investment based on the NPV rule should be avoided, especially when theexternal factors satisfy certain conditions.
4.5 Conclusion
We study the optimal timings for investment in two types of projects, namely, capacity
expansion and cost reduction, at an airport. A real option model considering demand
uncertainty is developed, and the values of projects are obtained endogenously through amulti-stage Stackelberg model. We formally prove the inter-relationship of the two
projects, and obtain the optimal rule for the investment. The effects of various external
factors on the optimal timings for each project are investigated, and the loss due to theinvestment following suboptimal rules is estimated. We find that the decision-maker who
places a higher premium on social surplus always tends to invest earlier in both projects.
Other factors that stimulate an earlier investment include a lower cross-price effect, agreater total number of airlines, a higher drift, and a lower volatility in demand. The
optimal scales of projects are more sensitive to the change in drift and volatility rather
than the change in other factors. When the composition of the objective function changes,
the pattern of the change in optimal timing differs by project: for cost reduction, thechanging rate does not change with the share of profit in most cases, while the optimal
timing is convex with regard to the share for capacity expansion. The loss due to NPVbased investment is much greater than that caused by the adoption of the deterministic
model, and its change with the change in the composition of the objective function has amountain-shaped pattern, where the minimum can be reached when specific conditions
are met.
The contribution of this study is two-fold. Theoretically, we improve the optimal rule fora sequential investment in inter-related projects proposed by the previous study (Truong
et al., 2018). Practically, the results might provide following implications. First, toimprove social surplus, adjustment to the timing for investment might be as important as
price regulation. To improve social surplus, the timings of investments in both projects
should be brought forward. Second, the adjustment for different types of projects should
be distinguished, as their pattern of change in timings is not the same. Third, even when85the real option rule is not applicable, NPV-based investment should be avoided in mostcases.
This study has several limitations. First, we focus only on the optimal timings and scales
for the investment. The volume of each scale is pre-determined instead of being optimized
endogenously. Second, we only consider one-time investment in the modeling. Piecewise
investment is beyond the scope of this study. Third, the numerical calculations are
performed using hypothetical values of parameters because we are unable to collectrealistic data from the industry. Real data from industrial practice can make the results
more persuasive. Fourth, we only show that the optimal rule derived from a real option
approach benefit the airport. Whether other stakeholders can be benefited by the realoption based optimal rule is questionable, which is worthy to be investigated further. Fifth,
the demand trend of an airport might not simply follow a single stochastic process. It canbe more complicated, especially when some catastrophes occur (e.g., Covid-19). These
shortcomings provide directions for future work.

86Chapter 5 Effects of airport’s privatization decisions on
capacity expansion: A real option approach
5.1 Introduction
In the recent decades, airport privatization has become a worldwide trend. Despite thevarious form, it can be generally categorized into two types based on whether the airport
is privatized with a transfer of ownership. The type without ownership transfer includes
concession, project finance/BOT, and management contract. The former two are also
called public–private partnerships (PPP). With this type of privatization, the government
maintains the ownership of the airport, while the operating right is leased to the private
investor for a certain period. Another type includes share floatation and trade sale, which
results in a permanent transfer of ownership from to the public sector to the shareholders
or the consortium that comprises various investors (Graham and Morrell, 2017)20.
Privatization can have immense impacts on the future of the airport and the public sector.
On the one hand, the potential positive impacts are inferred by the objectives for airport
privatization. As summarized by Graham (2011), six most significant objectives forprivatization are identified as: improving efficiency and performance; providing newinvestment funds; improving the quality of management and encouraging diversification;
improving service quality; producing financial gains for the public sector; lessening the
public sector influence. On the other hand, airport privatization might lead to several
potential negative impacts such as the loss of social benefit due to the incompatibility ofthe public and private interests, and the loss of control over the vital national assets.
Besides, the impacts on some aspects are not clear yet. For instance, according to theprevious practices, privatization did not necessarily lead to the improvement of efficiency(Oum et al., 2006). Therefore, the decision-making regarding privatization is highly
associated with the trade-off among these impacts. In order to make good decisions on
privatization - in other words, to make reasonable trade-offs, a comprehensive
understanding of these impacts is necessary.

20

In some cases, the transfer of ownership has a period, but the period is always very long compared withthat of concession or BOT (e.g., the period for the privatization of Sydney airport is 50+49 year).

87Different type of privatization has different emphasis on the above-mentioned objectives,or the positive impacts. For instance, the main incentive for the privatization with anownership transfer can be a financial gain which can either be directed to other sectors orserve as the funds for new investment, or improving the efficiency. Much academic
attention has been paid to these impacts: Oum et al. (2006) investigated the effect of
airport ownership form on the productive efficiency, finding that airports owned andmanaged by a mixed enterprise with a government-owned majority is significantly lessefficient than 100% publicly owned and operated airports; this finding was further
reinforced by their subsequent study (Oum et al., 2008); Adler and Liebert (2014)
investigated similar issues taking into account airport competition, while the previous
finding in the ownership’s effect on efficiency was hardly invalidated. However, thesecondary impacts, which might not be the main incentive for the privatization, are always
overlooked. For example, a trade sale privatization might not be designed for the sake ofthe facilitation of capacity expansion if the proceeds are not directed to the airport, but
capacity expansion is not impossible to take place in the future as long as it is in line withthe operator’s interest. Some examples can be observed in the real world: Liverpool
Airport, with 90% of its interest held by private investors, plans to expand runway andterminal in the future decades21; Brussels Airport, with 75% of its share held by a private
consortium, aims to complete the construction of two new piers by 2040 22 . To
comprehend the long-term effect of such kind of privatization, the secondary impacts
such as those regarding the capacity expansion needs further investigation. There aresome studies on this issue (e.g., Zhang and Zhang (2003)), but the timing of investment
and the long-term effect on social surplus were not explicitly discussed.
To bridge the research gaps, this study aims to investigate the effect of airport
privatization on the decisions of its subsequent capacity expansion, namely the timing
and the volume choice. We consider two scenarios. In Scenario I, while the budget
constraint (i.e., profit should not be negative) is addressed in terms of charging, it is nottaken into account for the decision-makings regarding capacity expansion. In other words,
the governmental subsidiary to the investment is available. In Scenario II, the budget

21
22

https://www.liverpoolairport.com/about-ljla/liverpool-john-lennon-airport-master-plan-to-2050
https://www.brusselsairport2040.be/en/vision-2040/48/terminal-and-gate-capacity

88constraint is considered in both decision-making phases. In other words, thegovernmental subsidiary to the investment is unavailable. To achieve the objectives, we
adopt a real-option model with demand growth and demand uncertainty considered to
optimize the decisions on capacity expansion; a Stackelberg model is employed to address
the charging behavior of the airport and the competition among downstream airlines.
5.2 The model
5.2.1 Description of the problem
𝑡0

𝑡𝑘

𝑡

Sells 𝜃 of the
airport to an
investor

Owner

Expands capacity
by 𝑘 to maximize
its profit

Investor

Operator

Airlines

Time

Sets charge 𝑤 to
maximize social
surplus given 𝑘0

Maximizes the mixtureof profit and social
surplus given 𝑘0

Maximizes the mixtureof profit and social
surplus given 𝑘0 + 𝑘

Compete on quantity in Cournot fashion

Fig 5.1 The sequence of the events in Scenario II

Stakeholders: We focus on one section of the air transport supply chain for our discussion.
In the upstream, there is one single airport owned by the government, or jointly ownedby the government and the private investor. The operator is in charge of setting the charge,
while the investor is in charge of making capacity expansion decisions for the airport.
Depending on the scenario introduced later, the operator and the investor can have sameor different objective. In the downstream, there are 𝑛 symmetric airlines selling air travel
service to the passengers. We ignore the entry and exit of airlines.
Scenarios:▪

Scenario I: We investigate how the ownership structure of an airport affects thetiming and volume choice for the capacity expansion. We assume that the
governmental subsidiary is available for the investment. In other word, the89

constraint that the budget used for the investment should only originate from theairport’s profit is removed. In this case, since the investment does not necessarily
take place after the privatization, we do not consider the effect of the timing ofprivatization for simplicity. We focus on a privatized airport. The operator and theinvestor are same.
▪

Scenario II: We assume that the governmental subsidiary is unavailable for theinvestment, and the proceeds from the privatization will be directed towards otherpublic projects. We further assume that the social surplus-oriented government
would not obtain any profit form the operation of the airport, so when thegovernment is the operator of the airport, namely before the privatization, theinvestment cannot take place. After privatization, with the financing of the private
investor, the expansion can be carried out at some time in the future. In this case,the operator and the investor are not same after the privatization. The operator set
charge based on the mixed benefit of all shareholders, while the investor makes
decisions based on its own profit, as it bears all the cost of the expansion. Under
such settings, we investigate how the privatization decision affect the investment
decision of the airport and the aggregated social surplus. The timing of investment
decision is considered as well as the share to sell.

Decision-making process:▪

Scenario I: A multi-stage Stackelberg game is modelled to reflect the hierarchicalinteractions among the market participant: In the first stage, the investor makes
decisions regarding capacity expansion by choosing the time and the volume. In
the second stage, the operator sets the charge, given the current level of capacity.
The objectives of the investor and the operator are same, that is, to maximize themixture of profit and social surplus depending on the ownership structure. In thethird stage, given the charge and the capacity level, airlines in the downstream
compete in a Cournot fashion.

▪

Scenario II: The airport is assumed to be fully controlled by the government
originally, and the operator sets a charge that maximizes the social surplus subject
to the non-negative profit constraint. At time 𝑡 , the public owner sells 𝜃 (0 <90𝜃 ≤ 1) of the airport’s ownership to a private investor23. The new operator thensets a new charge that maximize the mixture of profit and social surplus, which
reconciles the different objectives of the private and public shareholders. At time
𝑡𝑘 , the investor expands the capacity of the airport by 𝑘 , making the airport’s
capacity 𝑘 + 𝑘0 . The operator then adjusts its charge based on the new condition.
In the downstream, 𝑛 airlines compete with each other in a Cournot fashion to
maximize their respective profit, given the capacity level and charge. Fig 5.1
presents an intuitive illustration of the whole sequence of the events.
Several assumptions are made as follows for the sake of tractability or simplicity:a. The proceeds from the privatization will be directed towards other public projects,
instead of being directed to airport. In the ICAO (International Civil Aviation
Organization)’s document regarding airport privatization, it writes: “In several
States, such as in European States, Australia and New Zealand, funds generated
through private participation and privatization in the provision of airports are
credited to the treasury without any commitment to use them for the developmentof the aviation industry.”24 Therefore, this assumption can be plausible in somecases.
b. Lead times of the investments are ignored. Alternatively, this can be explainedthat the timing refers to the time when the project is expected to be finished, andthe cost refers to the aggregated retrospective cost (e.g., if the span is 3 year, thecost stands for 𝐼 𝑒 3𝑟 instead of 𝐼) incurred in the whole investment span.
c. Incremental investments are not considered. Airport capacity investment is a
lumpy project which is unlikely to be carried out twice in a relatively short timespan (e.g., Amsterdam Airport’s first large-scale expansion took place in 1960s,
the second happened in 2000s). The value in the far future when the expansion
23

It is not implausible that an airport sells part of its ownership to the private investor(s). Generally, it cantake place in the case that the government find it difficult to finance the airport without private capital, but
it still wants to retain some control of the airport since it can be a strategic infrastructure. In this case, there
are two approach. The first is share floatation, which is to issue and trade the share on the stock market. For
example, the New Zealand government sold Auckland airport’s 52 percent shareholders through an IPO(initial public offering) in 1998. The second is trade sale, which is to sell part of the ownership to a
consortium. For example, 60% of Lyon airport’s ownership was sold to a consortium in 2016.
24
http://www.aviationchief.com/uploads/9/2/0/9/92098238/icao_doc_9980__manual_on_privatization_of_airports_and_ans_1.pdf

91

can take place again will be greatly discounted, so we ignore it.
d. The composition of ownership will not change after the capacity expansion25.
Notations of the decision variables and parameters are shown in Table 5.1
Table 5.1 Notations for Chapter 5

Decision variables
Number of passengers of airline𝑤

Airport charge per passenger

𝑎𝑘

Timing (demand level) for the capacity expansion𝑘Capacity level to expand

Parameters𝑎Maximum willingness to pay𝑏 0≤𝑏≤1

Own-price effect𝑙 0≤𝑙≤1

Cross-price effect𝑛Number of airlines𝛾

Unit cost per passenger of airline𝑐

Unit cost per passenger of airport

𝑘0

Original capacity level

𝑐𝐾

Unit cost for the capacity expansion

𝑓𝐾

Fixed cost for the capacity expansion𝑎Timing (demand level) for the privatization

𝜃Private share

𝛼

Drift

𝜎

Volatility𝑟Discount rate

5.2.2 The optimal charging problem
We use the backward induction to solve the optimal charging problem. We first derive25

We cannot find counter examples.

92

the demand function (Eq. (5.2)) from a quadratic utility function (Eq. (5.1)).
𝑛

𝑈

𝑗

1
= 𝑎 − [𝑏 (∑2𝑝 =𝑎−𝑏Where) + 2𝑏𝑙 (∑

𝑗 )]

5.1

>𝑗

− 𝑏𝑙

−

5.2
= ∑𝑛=

denotes the number of passengers of airline and; 𝑝 denotes the

airfare of airline ; 𝑎 denotes the maximum willingness-to-pay. 𝑏 and 𝑙 denote the own
price effect and the cross-price effect respectively with 0 ≤ 𝑙 ≤ 1.
The demand uncertainty is described by the Geometric Brownian motion (GBM):
𝑑𝑎 = 𝛼𝑎𝑑𝑡 + 𝜎𝑎𝑑𝑧

5.3

Where 𝛼 denotes the drift; 𝜎 denotes the volatility.
The airlines’ Cournot competition problem is formulated as:
max 𝜋 = 𝑝

−𝛾−𝑤

𝑞𝑖

s. t.

∀

5.4

≤𝑘

Where 𝛾 denotes the marginal cost of airline; 𝑤 denotes the charge. The solution is:0𝑎−𝛾−𝑤𝑤 𝑘 = 𝑏 𝑙 𝑛−1 +2𝑘{𝑛Where 𝑤𝑏 = 𝑎 − 𝛾 −

𝑏𝑘

+𝑙𝑛 𝑙
𝑛

𝑤 ∈ [𝑎 − 𝛾 ∞𝑤 ∈ 𝑤𝑏 𝑎 − 𝛾

5.5

𝑤 ∈ 0 𝑤𝑏 ]is the charge that makes the optimal outputs hit thecapacity bound. If the charge is very high, airlines will stop operation to avoid loss (thefirst row of Eq. (5.5)). If the charge is lower than 𝑤𝑏 , airlines’ real optimal outputs will
exceed the capacity constraint, so they have to choose the suboptimal outputs that fully
occupy the capacity loss (the third row of Eq. (5.5)). The real optimum can only beachieved when the charge is neither too high nor to low (the second row of Eq. (5.5)).
The airport operator’s optimal charging problem is formulated as:

93

max 𝛬 𝑤 𝑘 = 𝜃𝛤 𝑤 𝑘 + 1 − 𝜃 𝑆 𝑤 𝑘

5.6

𝑤

s. t. 𝛤 𝑤 𝑘 ≥ 0
Where 𝜃 denotes the private share26. 𝛤 𝑤 𝑘 , the profit of airport, is formulated as:
𝛤 𝑤 𝑘 = 𝑤−𝑐

𝑤 𝑘

5.7

Where 𝑐 denotes airport’s marginal cost. 𝑆 𝑤 𝑘 , the social surplus, is formulated as:
𝑆 𝑤 𝑘 = 𝑈(𝑤𝑘𝑗𝑤 𝑘 )− 𝑐+𝛾

𝑤𝑘

5.8

To solve the problem, we first see the case that the capacity constraint is not binding. In
this case, substituting the second row of Eq. (5.5) into Eq. (5.6), and solving the KKT
conditions 𝛤 𝑤 𝑘 ≥ 0 𝜆 ≥ 0 𝜆 𝛤 𝑤 𝑘 = 0𝜕 𝛬 𝑤 𝑘 +𝜆 𝛤 𝑤 𝑘𝜕𝑤𝑐
𝑤𝑢 = {𝜃 𝑎 − 𝛾 𝑙 𝑛 − 1 + 3 − 𝑎 + 𝑐 𝑙 𝑛 − 1 + 2 + 𝛾
3𝜃 + 𝜃 + 1 𝑙 𝑛 − 1 + 1
Where 𝜃̅ = 𝑙 𝑛+3= 0, we have:𝜃 ∈ 0 𝜃̅𝜃 ∈ [𝜃̅ 1]

5.9, the subscript 𝑢 refers to being unbound in terms of capacity(abbreviated as “capacity-unbound” hereinafter).
Next, we discuss the two cases with regard to different region of 𝜃 respectively. In the“profit-binding” (𝜃 ∈ 0 𝜃̅ ) case, comparing the maximized payoff and the optimal
charge, we have the following four cases: (i), 𝑤𝑢 > 𝑤𝑏 and 𝛬 𝑤𝑢 𝑘 > 𝛬 𝑤𝑏 𝑘 ; (ii),𝑤𝑢 > 𝑤𝑏 and 𝛬 𝑤𝑢 𝑘 ≤ 𝛬 𝑤𝑏 𝑘 ; (iii), 𝑤𝑢 ≤ 𝑤𝑏 and 𝛬 𝑤𝑢 𝑘 > 𝛬 𝑤𝑏 𝑘 ; (iv), 𝑤𝑢 ≤𝑤𝑏 and 𝛬 𝑤𝑢 𝑘 ≤ 𝛬 𝑤𝑏 𝑘 . For (i) and (iii), it seems that the optimal choice is to set
the charge as 𝑤𝑢 , since 𝛬 𝑤𝑢 𝑘 is greater. However, this is only available for (i); for (iii),
the final outputs will not change as long as the charge is not higher than 𝑤𝑏 because of26

To justify such setting, we can assume that the board comprises outside directors who focuses on thereconciled total benefit of the firm instead of favoring certain group of shareholders, and the fair public
sector will not abuse its voting right (i.e., not veto the fair resolutions proposed by the board). In this case,the board is assumed to make decisions that maximizes the mixed objective coordinated by θ. If the public
sector has majority share, although the decision is not expected to maximize its own objective (social
surplus), it would not veto the decision otherwise the private investors cannot get any dividend. If theprivate investors have majority share, the decision will be fairly profit-oriented. The fair public sector
certainly would not veto it. On the other hand, although the decision is not purely profit-maximizing, theprivate shareholders can get expected dividend, since all profit goes to them. Hence, we assume they would
not veto as well.

94the capacity bound. Similarly, for (ii) and (iv), choosing 𝑤𝑏 is only available for (iv); for(ii), the non-negative profit constraint will be violated by doing so. Consequently, in the“profit-binding” region, the optimal charge is the highest one among 𝑤𝑢 and 𝑤𝑏 . For the“profit-unbounded” region, the optimal charge can be derived in a similar way. To
organize, we have:
Proposition 5.1
Let 𝑔 = 𝑐 + 𝛾,(i)

If 0 < 𝜃 < 𝜃̅ , the non-negative profit constraint is binding. In this case, if 𝑎 ≤ 𝑔,
the operation will terminate, the payoff of the expansion decision-maker𝑅

𝑎 𝑘 is 0 for both Scenario I and II; if 𝑔 < 𝑎 <𝑏𝑘 𝑙 𝑛+𝑛+ 𝑔, the optimal

charge is 𝑤 = 𝑐 ; the payoff of the expansion decision-maker 𝑅+3 𝑎 𝑔 2

𝑛 𝑙 𝑛
𝑏 𝑙 𝑛

+ 2

for Scenario I, 0 for Scenario II. If 𝑎 ≥optimal charge is 𝑤 = 𝑎 − 𝛾 −maker 𝑅 3 𝑎 𝑘 is
𝑙 𝑛

+
𝑛

(ii)

𝑘

𝑛 𝑎 𝑔

𝑏𝑘

+𝑙𝑛 𝑙
𝑛

𝑏𝑘 𝑙 𝑛+𝑛𝑎 𝑘 is+ 𝑔, the; the payoff of the expansion decision-

𝑏𝑘 3𝜃+ 𝜃+ 𝑙 𝑛+𝑛

for Scenario I, 𝑘 𝑎 − 𝑔 −for Scenario II.

If 𝜃̅ ≤ 𝜃 ≤ 1, the constraint is not binding. In this case, if 𝑎 ≤ 𝑔, the operation
will terminate, the payoff of the expansion decision-maker 𝑅
Scenario I and II; if 𝑔 < 𝑎 < 𝑔 +𝑤 =𝜃 𝑎 𝛾 𝑙 𝑛

maker 𝑅+3

3𝜃+ 𝜃+

𝑎+𝑐 𝑙 𝑛
𝑙 𝑛

+𝑙𝑛 𝑙
𝑛

𝑘

𝑛 𝑎 𝑔

𝑛

+ +𝛾

+
𝑛 𝑎 𝑔 2𝑎 𝑘 is 𝑏 3𝜃+ 𝜃+

Scenario II; if 𝑎 ≥ 𝑔 +𝑏𝑘

𝑏𝑘 3𝜃+ 𝜃+ 𝑙 𝑛+

𝑎 𝑘 is 0 for both, the optimal charge is, the payoff of the expansion decision𝑛 𝑎 𝑔 2 𝜃 𝑙 𝑛

𝑙 𝑛+

𝑏𝑘 3𝜃+ 𝜃+ 𝑙 𝑛𝑛

for Scenario I, 𝑏 3𝜃+ 𝜃+
+

+3

𝑙 𝑛

+ 2

, the optimal charge is 𝑤 = 𝑎 − 𝛾 −; the payoff of the expansion decision-maker 𝑅 3 𝑎 𝑘
𝑏𝑘 3𝜃+ 𝜃+ 𝑙 𝑛𝑛+

for Scenario I, 𝑘 𝑎 − 𝑔 −

II.

95

for

𝑙 𝑛

+
𝑛

is

for Scenario

5.2.3 The capacity expansion problem
Following Balliauw and Onghena (2020), we first derive the net value of the airport fromthe perspective of the private investor in each case. Solving the Bellman function
𝑉𝑥𝑦 𝑎 𝑘 = 𝑅𝑥𝑦 𝑎 𝑘 𝑑𝑡 + 𝐄[𝑉𝑥𝑦 𝑎 + 𝑑𝑎 𝑘 𝑒 𝑟𝑑𝑡 ] for each 𝑥 ∈ 1 2 and 𝑦 ∈ 1 2,3, andremoving the meaningless terms, we have:𝑉 𝑎 𝑘 =
𝑘 𝑎 𝛽1
𝑉 𝑎 𝑘 =
𝑘 𝑎 𝛽1𝑎𝑘𝑇
𝑇 𝑣 𝑘 − 𝑔𝑘
𝑘 𝑎𝛽2 ++𝑟−𝛼𝑟

𝑉 𝑎 𝑘 ={

𝑉3 𝑎 𝑘 =𝐶𝑉
𝑉 𝑎 𝑘 = 𝑉𝑘 𝑎 𝛽1 +𝑎 𝑘 =𝑉3 𝑎 𝑘 =𝐶{𝛼Where 𝛽 = − 𝜎2 + √−𝑘 𝑏𝑘 𝑙 𝑛𝑛

𝑘 𝑎𝛽2 + 𝑇𝑣

+

𝛼

− 𝜎2
𝑛 𝜃 𝑙 𝑛, 𝑣 = 𝑏 3𝜃+ 𝜃+𝑎 ∈ [0 𝑔]
𝑎∈ 𝑔 𝑎̅̅̅ 𝑘𝑘 𝑎𝛽1 𝑎 ∈ [0 𝑔]

𝑎 𝑘 =
𝑎2

+

𝑎𝑔

+

𝛼+𝑟 𝜎 2
𝛼 𝑟
𝑎𝑘𝑇
𝑇 𝑣1 𝑘
𝛽2
𝑘 𝑎 +
+
𝑟 𝛼
𝑟

𝑟

𝛼

+ 𝜎2 , 𝛽 = − 𝜎2 − √
+3
𝑙 𝑛

𝑔2

𝑎∈ 𝑔 𝑎
̅̅̅ 𝑘

𝑟
𝑔𝑘

5.11

𝑎 ∈ [𝑎̅̅̅ 𝑘 ∞𝛼− 𝜎2

𝑟+ 𝜎2 , 𝑣 𝑘 =. 𝑟 denotes the discount rate, and 𝑇 denotes+ 2

the number of the periods when 𝑎 remains the same27. 𝑎
̅̅̅ 𝑘 =𝑔+

5.10

𝑎 ∈ [𝑎̅̅̅ 𝑘 ∞

𝑏𝑘 𝑙 𝑛−1 +2𝑛+ 𝑔, 𝑎̅̅̅ 𝑘 =

𝑏𝑘 3𝜃+ 𝜃+1 𝑙 𝑛−1 +1.
𝑛Unknown parameters

𝑘

𝑘 𝐶

𝑘

𝑘

𝑘

𝑘 𝐶

𝑘

can bederived via value-matching and smooth-pasting conditions (Dixit and Pindyck, 1994),
their lengthy expressions are omitted here.
The gain through carrying out the capacity expansion can thus be expressed as:

𝐺 𝑎 𝑘

𝐺𝑎 𝑎 𝑘
𝐺𝑏 𝑎 𝑘=𝐺𝑐 𝑎 𝑘
{𝐺 𝑑 𝑎 𝑘= 𝑉 𝑎 𝑘 + 𝑘0 − 𝑉 𝑎 𝑘0
= 𝑉 𝑎 𝑘 + 𝑘0 − 𝑉 𝑎 𝑘0
= 𝑉 𝑎 𝑘 + 𝑘0 − 𝑉 3 𝑎 𝑘0
= 𝑉 3 𝑎 𝑘 + 𝑘0 − 𝑉 3 𝑎 𝑘0

𝑎 ∈ [0 𝑔]𝑎∈ 𝑔𝑎
̅̅̅ 𝑘0𝑎 ∈ [𝑎̅̅̅ 𝑘0 𝑎
̅̅̅ 𝑘0 + 𝑘𝑎 ∈ [𝑎̅̅̅ 𝑘0 + 𝑘 ∞

5.12

𝐺 𝑎 𝑘

𝐺𝑎 𝑎 𝑘
𝐺𝑏 𝑎 𝑘=𝐺𝑐 𝑎 𝑘
{𝐺 𝑑 𝑎 𝑘= 𝑉 𝑎 𝑘 + 𝑘0 − 𝑉 𝑎 𝑘0
= 𝑉 𝑎 𝑘 + 𝑘0 − 𝑉 𝑎 𝑘0
= 𝑉 𝑎 𝑘 + 𝑘0 − 𝑉 3 𝑎 𝑘0
= 𝑉 3 𝑎 𝑘 + 𝑘0 − 𝑉 3 𝑎 𝑘0

𝑎 ∈ [0 𝑔]
𝑎 ∈ 𝑔 ̅̅̅𝑎 𝑘0
𝑎 ∈ [𝑎̅̅̅ 𝑘0 ̅̅̅
𝑎 𝑘0 + 𝑘𝑎 ∈ [𝑎̅̅̅ 𝑘0 + 𝑘 ∞

5.13

27

For example, if 𝛼 𝜎 𝑟 denote monthly drift, volatility and discount rate respectively, and the payoff (Eq.
(5.6)) is calculated on a daily basis, 𝑇 = 30.

96

𝐺 𝑎 𝑘={𝜃 ∈ 0 𝜃̅𝜃 ∈ [𝜃̅ 1]

𝐺 𝑎 𝑘
𝐺 𝑎 𝑘

5.14

Next, we need to maximize the gain by choosing the optimal timing 𝑎𝑘 and the optimal
volume 𝑘 for the expansion. According to Dangl (1999), the problem can be formulatedas:𝐹 𝑎 = max {e 𝑟𝑑𝑡 𝐄[𝐹 𝑎 + 𝑑𝐹 𝑎 ] max[𝐺 𝑎 𝑘
𝑘1

Where 𝐼 𝑘

− 𝐼 𝑘 ]}

5.15

denotes the total cost of the expansion. The inner optimization can be solvedby the first order condition, and the outer optimization can be solved by the boundary
conditions. Consequently, 𝑎𝑘 and 𝑘 can be obtained by solving the following system of
equations:
𝐺 𝑎𝑘 𝑘

=(

−𝐼 𝑘

𝑎𝑘
𝜕[𝐺 𝑎 𝑘 − 𝐼 𝑘 ]
) (
)
|
𝛽
𝜕𝑎
𝑎=𝑎 𝑘 =𝑘𝑘

1

1

𝜕[𝐺 𝑎 𝑘 − 𝐼 𝑘 ]|=0𝜕𝑘
𝑎=𝑎 𝑘 =𝑘
𝑘

1

5.16

1

As per Hagspiel et al. (2016), under this kind of setting, the investor will never carry out
the investment when 𝑎 is in the regions where it cannot improve its profit, so we only
need to focus on 𝐺 𝑐 𝑎 𝑘 , 𝐺 𝑑 𝑎 𝑘 , 𝐺 𝑐 𝑎 𝑘
𝐺 𝑑 𝑎 𝑘 ,𝐺 𝑐 𝑎 𝑘

and 𝐺 𝑑 𝑎 𝑘

and 𝐺 𝑑 𝑎 𝑘

for Scenario I, andfor Scenario II. It is difficult to derive the precise

solutions through solving Eq. (5.16). An approximate algorithm is proposed utilizing theintersection point-finding function of the software Mathematica. The brief procedure is
as follows:
Step 1: Set the starting value of 𝑎 as 𝑎 = 𝑎̅ 𝑘0 . Define the values of 𝑎̂ and ∆𝑎.
Step 2: Solve𝜕[𝐺 𝑎 𝑘1

𝐼 𝑘1 ]

𝜕𝑘1= 0 for 𝑘 , and exclude the negative solutions.

Step 3: Check whether the remaining solution 𝑘
𝜕 2 𝐺 𝑎 𝑘1

𝐼 𝑘1

𝜕𝑘12

maximizes the objective function. If< 0, go to Step 4; otherwise, go to Step 5.

Step 4: Check whether 𝑘

is within the feasible domain; if yes, go to Step 6; otherwise,

go to Step 5.

97

Step 5: Compare the boundary values. Set 𝑘

to the value which maximizes the objectivefunction.
Step 6: Set 𝑘

𝑎 = 𝑘 . Calculate 𝐺 𝑎 𝑘

𝑎

−𝐼 𝑘

𝑎 .

Step 7: If 𝑎 < 𝑎̂, set 𝑎 = 𝑎 + ∆𝑎 and return to Step 1; otherwise, plot 𝐺 𝑎 𝑘
𝐼 𝑘𝑎

𝐼 𝑘1 ]𝑎𝜕[𝐺 𝑎 𝑘11

𝜕𝑎and (𝛽 ) (|𝑘1 =𝑘1 𝑎𝑎−) with 𝑎 as the horizontal axis, and findthe intersection point which is regarded as 𝑎𝑘 .
Step 8: Set 𝑎 = 𝑎𝑘 and return to Step 1. Stop at Step 6 and set 𝑘 = 𝑘

𝑎𝑘 . Output 𝑘

and 𝑎𝑘 as the final results.
Following this procedure, we can derive the optimal solutions for 𝐺 𝑐 𝑎 𝑘
𝐺𝑐 𝑎𝑘

𝐺𝑑 𝑎𝑘 ,and 𝐺 𝑑 𝑎 𝑘 , respectively. Note that we need to compare the optimal valueof 𝐺𝑐 𝑎 𝑘

to obtain the final solution for the “profit-unbound” region.

and 𝐺𝑑 𝑎 𝑘

5.2.4 Calculation of aggregated social surplus in Scenario II
We first divide the whole time span into three periods:▪

Period 1 (𝑡0 ~𝑡 ): Pre-privatization.

▪

Period 2 (𝑡 ~𝑡𝑘 ): Post-privatization, pre-expansion.

▪

Period 3 (𝑡𝑘 ~): Post-expansion.

For Period 1, there are three possible cases:▪

Case 1.1: During the whole period, the capacity constraint is binding.

▪

Case 1.2: During the whole period, the capacity constraint is non-binding.

▪

Case 1.3: The capacity is originally non-binding, while it becomes binding at 𝑡̃ ∈𝑡0 𝑡

as a result of the demand growth.

The expected social surplus of Period 1 in each case can be formulated as 𝑆𝑡

𝔼 [∫0 𝑝 𝑆𝑏 𝑎𝑡 𝑒 𝑟𝑡 𝑑𝑡]
𝑡̃

𝑆

,

𝑡

= 𝔼 [∫0 𝑝 𝑆𝑢 𝑎𝑡 𝑒 𝑟𝑡 𝑑𝑡]

,

and

=

𝑆3=

𝑡

𝔼 [∫0 1 𝑆𝑏 𝑎𝑡 𝑒 𝑟𝑡 𝑑𝑡] + 𝔼 [∫𝑡̃𝑝 𝑆𝑢 𝑎𝑡 𝑒 𝑟𝑡 𝑑𝑡] , where the subscripts 𝑢 and 𝑏 denote1

non-binding and binding respectively. 𝑆 𝑎
98

can then be confirmed based on the case.

For Period 2 and 3, there are two possible cases, as the investor will never invest whencapacity is not binding:▪

Case 2(3).1: During the whole period, the capacity constraint is binding.

▪

Case 2(3).2: The capacity is originally non-binding, while it becomes binding at
some time.

The social surplus in each case and each period can be calculated in a similar way as done
for period 1. Consequently, summing up the results of all periods, we can get theaggregated social surplus as:
𝑆(𝑎 𝑎𝑘 𝑘 ) = 𝑆 (𝑎 ) + 𝑆 (𝑎 𝑎𝑘 ) + 𝑆3 𝑎𝑘 𝑘

− 𝔼[𝑒 𝑟𝑡𝑘 ]𝐼 𝑘

5.17

5.3 Numerical examples
The values of the parameters for the numerical computation are set as shown in Table 5.2.
These values will not change throughout this section if not specifically mentioned. The
marginal operating cost of airline 𝛾 is set based on the airline data project 28 , and themarginal operating cost of airport 𝑐 is set based on Martín and Voltes-Dorta (2011). The
cost of the investment in capacity expansion is set based on Sun and Schonfeld (2015).
Other values are set hypothetically due to the lack of data.
Table 5.2 Values of the parameters for the numerical computation

28

Parameter

Value

Parameter

Value

𝑎0

1000

𝐼

𝑓𝐾 + 𝑐𝐾 𝑘

𝑏

0.25

𝑐𝐾

1 105

𝑙

0.5

𝑓𝐾

6.5 107𝑛2

𝛼𝑦𝑒𝑎𝑟 (annual)

0.01

𝑐

8.5

𝜎𝑦𝑒𝑎𝑟 (annual)

0.05

𝛾

175

𝑟𝑦𝑒𝑎𝑟 (annual)

0.1

𝑘0

3000

𝑇

30

http://web.mit.edu/airlinedata/www/default.html

99

5.3.1 Scenario I
Fig 5.2 shows how the private share 𝜃 affects the optimal decisions for the capacity
expansion. As opposed to the finding of Xiao et al., (2012), our results show that the
optimal volume does not monotonically decrease as 𝜃 increases. Instead, the change ofthe optimal volume with regard to 𝜃 can have an irregular pattern. The intuition behind
the result can be explained as follows. There are two countervailing effects driving thechange of the optimal volume with regard to 𝜃. On the one hand, as 𝜃 goes higher, themaximum number of passengers will decline (𝜕𝑄 𝑎 𝜃
𝜕𝜃

𝑎 𝜃 =

𝑛 𝑎 𝑔
𝑏 3𝜃+ 𝜃+ 𝑙 𝑛+

and< 0 for any 𝜃 ∈ [0 1]), so the airport can expect to accommodate all potential

passengers with a lower volume. On the other hand, when 𝜃 is low, even if the capacity
is no longer enough at one time, the loss is not significant; however, the loss due to atransfer from the capacity non-binding region to the binding region will grow as 𝜃
increases, namely

𝜕𝑅12 𝑎 𝜃

𝜕𝑅13 𝑎 𝜃𝜕𝜃> 0 and𝜕𝑅22 𝑎 𝜃

𝜕𝑅23 𝑎 𝜃𝜕𝜃> 0, thus, in order toavoid the loss, or to make the loss insignificant in the current view, the airport tends toincrease the volume to postpone the loss as much as possible. The intensity of each effectvaries by the value of parameters, making the pattern of the optimal volume different ineach case. Moreover, the pattern can change greatly due to the change of region, since thepayoff function also changes. The change of the optimal timing 𝑎𝑘 29 basically follows
one pattern; as 𝜃 increases, it becomes earlier in the charge-binding region, while goeslater in the non-binding region.
Regarding the effect of parameters, we can find that a lower cross-price effect 𝑙 and ahigher airline number 𝑛 do not necessarily lead to a greater capacity volume, but mightpush forward the expansion greatly. A higher drift 𝛼, a higher volatility 𝜎 and a lower
discount rate 𝑟 can lead to a greater capacity volume and a later expansion. Generally,factors regarding the airline market (𝑙 𝑚) affect the timing more than the volume, while
factors regarding the dynamics of demand (𝛼 𝜎) and discount factor 𝑟 have stronger

In fact, 𝑎𝑘 should be written as 𝑎 𝑡𝑘 . 𝑎 denotes the demand level (maximum willingness-to-pay) which
is a function of time 𝑡. 𝑡𝑘 is the time when the investment should be carried out, yet with uncertainty it does
not have a fixed value, so we use the demand level to denote the “timing” instead of a precise “time”.
29100

effect on the volume rather than the timing.

Fig 5.2 Effect of private share θ on the optimal investment decisions

1015.3.2 Scenario II
Fig 5.3 shows the effect of the private share 𝜃 and the timing (demand level) ofprivatization 𝑎 on the optimal decisions for the capacity expansion. We can observe
three clear-cut regions with regard to the private share. If 𝜃 is low, capacity expansion
will never be carried out (no-expansion region). If 𝜃 is moderate, capacity expansion will
be carried out at some time in the future, while the volume, which just fill a part of thecapacity gap, will not be very great (gap-filling region). As 𝜃 goes high, a jump of the
optimal volume can be observed. When 𝜃 has a high value, the capacity expansion will
be carried out in a more extensive way; the volume will be much greater than that of themoderate-𝜃 case, and the expansion will take place a little earlier. The high-volume
capacity is expanded not only for the gap-filling, but as a preparation for the future
demand growth, as it will not be fully occupied after the expansion (future-preparing
region). In each region, the optimal volume and timing only change slight as 𝜃 changes.
On the other hand, the timing of the privatization has no effect on the optimal decisions
for the expansion as long as it precedes the expansion time.
Fig 5.4 shows the effect of the private share and the time of privatization on theaggregated social surplus. It can be found that if the privatization largely precedes the
expansion (i.e., 𝑎 is low), the maximum of the aggregated social surplus can be achievedby a fairly low 𝜃 which is just the breakpoint between no-expansion region and gapfilling region. However, if the expansion does not happen much later than theprivatization (i.e., 𝑎 is high), the peak-point of the aggregated social surplus will transfer;
it will be the breakpoint between gap-filling region and future-preparing region,
characterized by a fairly high 𝜃. This is because of the trade-off between the socialsurplus improving effect of the expansion, and the social-surplus reducing effect of theprivatization. When the privatization takes place at a late time, the loss of social surplus
due to the privatization is greatly discounted, making its negative effect overweighed by
the positive effect of the capacity expansion, so a high-share privatization which can leadto an extensive expansion is favored in this case, and vice-versa.

102

Fig 5.3 The effect of the private share and the time of privatization on the optimal
decisions for the capacity expansion

Fig 5.4 The effect of the private share and the time of privatization on the aggregatedsocial surplus

103Fig 5.5 shows how volatility 𝜎 affect the optimal decisions for the capacity expansion.
As expected, the higher the uncertainty, the larger the optimal volume, and the later the
expansion. Moreover, it can be observed that the breakpoint between gap-filling region
and future-preparing region moves leftward, that is, moves to a lower 𝜃 as 𝜎 goes higher,
while the breakpoint between no-expansion region and gap-filling region hardly changes.
Fig 5.6 shows the effect of the private share and volatility on the aggregated social surplus
given a privatization time 𝑎 , indicating that some other factors can also greatly affectthe relation between the private share and the aggregated social surplus. When 𝜎 is low,
although the period between the privatization and the expansion is shortened comparedwith other case, the peak point is not the breakpoint between gap-filling region and futurepreparing region. This is because the difference in the volume of expansion between thetwo regions in this case is fairly low, so the increase of volume realized by the high-shareprivatization is not enough to offset the social surplus loss it causes.

Fig 5.5 The effect of the private share and volatility 𝝈 on the optimal decisions for thecapacity expansion (𝒂𝒑 = 𝟏𝟔𝟎𝟎

104

Fig 5.6 The effect of the private share and volatility 𝝈 on the aggregated social surplus(𝒂𝒑 = 𝟏𝟖𝟎𝟎

Fig 5.7 shows the joint effect of the private share and drift 𝛼, which is similar as that ofvolatility: The stronger the growth, the larger the optimal volume, and the later the
expansion. Also, the breakpoint between gap-filling region and future-preparing region
moves leftward as 𝛼 goes higher, while the breakpoint between no-expansion region andgap-filling region hardly changes. Fig 5.8 shows the joint effect of the private share anddiscount rate 𝑟. A lower 𝑟 has very similar effects as what a higher 𝛼 has. Fig 5.9 presents
the joint effect of private share and airline number 𝑛. Interestingly, a fiercer downstream
competition will not necessarily result in an expansion of high volume; instead, theinvestor would rather choose to push the expansion forward. The change of the breakpoint
between gap-filling region and future-preparing region is not so conspicuous, while an
obvious leftward movement of the breakpoint between no-expansion region and gapfilling region can be observed as 𝑛 becomes greater. Similar phenomenon can beobserved regarding the effect of the cross-price effect 𝑙 (Fig. 5.10). A low 𝑙 makes thevolume slight smaller, yet the expansion will be greatly pushed forward.

105

Fig 5.7 The effect of the private share and drift 𝜶 on the optimal decisions for the capacity
expansion (𝒂𝒑 = 𝟏𝟔𝟎𝟎

Fig 5.8 The effect of the private share and discount rate 𝒓 on the optimal decisions for thecapacity expansion (𝒂𝒑 = 𝟏𝟔𝟎𝟎

Fig 5.9 The effect of the private share and airline number 𝒏 on the optimal decisions forthe capacity expansion (𝒂𝒑 = 𝟏𝟐𝟎𝟎

106

Fig 5.10 The effect of the private share and the cross-price effect 𝒍 on the optimal decisions
for the capacity expansion (𝒂𝒑 = 𝟏𝟔𝟎𝟎

In summary, there are three regions of the private share in terms of the investor’s capacityexpanding behaviour. The volume and the timing vary greatly by region, while the change
is modest within each region. Therefore, we can have an optimal value of the private sharein terms of the aggregated social surplus, and that value, in many cases, is consistent with
one of the breakpoints that link two regions. Which breakpoint maximize the aggregatedsocial surplus depends on a joint effect of the privatization time and some other factors.
Moreover, the breakpoints can change given different external factors. The breakpoint
between no-expansion region and gap-filling region is more sensitive to the change ofthe situation of the downstream airlines, while the breakpoint between gap-filling region
and future-preparing region is more sensitive to the change of the dynamics of demand
and the discount rate. These findings can provide some references for the government todetermine the privatization time and the private share. For instance, if the degree of
product differentiation between the downstream airlines is fairly low (i.e., 𝑙 is low), andthe social-surplus oriented government want to privatization the airport earlier, althougha fairly low share might be appropriate, its should not be too low because the breakpoint
has been drawn rightward by the low 𝑙. However, if the government place higher premiumon the capacity rather than the aggregated social surplus (this might happen in the casethat the spill-over benefit brought by the increasing passengers is highly valued), a highprivate share might be better-off, but a very high share (e.g., an utter transfer of theownership) is unnecessary.

1075.4 Conclusion
Privatization of an airport would have immense and long-term impacts on various aspects.
A comprehensive understanding of these impacts thus proves to be important for areasonable decision-making regarding airport privatization. This study investigates theeffect of the airport privatization with ownership transfer on the investor’s decisions of
the capacity expansion and the long-term aggregated social surplus. We consider two
scenarios. In Scenario I, while the budget constraint (i.e., profit should not be negative)
is addressed in terms of charging, it is not taken into account for the decision-makings
regarding capacity expansion. In other words, the governmental subsidiary to theinvestment is available. In Scenario II, the budget constraint is considered in bothdecision-making phases. A Stackelberg model is employed to present the optimal pricing
behavior of the airport, and the decision-makings on the capacity expansion is modelled
in a real option approach. Numerical computation is performed to show the relationshipbetween the privatization decisions and the investment decisions, and the effect of various
external factors on that relationship.
The main findings can be concluded as follows. In Scenario I, the optimal volume andtiming for the investment do not monotonically increase or decrease with regard to theprivate share. Their changing patterns are not simple due to two countervailing effects.
Generally, the earliest and largest investment can be achieved by a fairly low private share.
Factors regarding the airline market affect the timing more than the volume, while factors
regarding the dynamics of demand and the discount factor have stronger effect on thevolume rather than the timing. In Scenario II, there are three clear-cut regions of theinvestment decisions with regard to the private share: If the private share is low, no
expansion will be carried out; if the private share is moderate, a gap-filling expansion will
be carried out; if the private share is high, the expansion will not only fill the current
capacity gap, but also leave some vacant capacity preparing for the future demand growth.
The maximization of the long-term aggregated social surplus can be achieved byprivatizing a certain share which is just consistent with the value of one of the two
breakpoints that link two adjacent regions, while which breakpoint can maximize theaggregated social surplus depends on the timing of the privatization. While factors
regarding the downstream airlines might have greater effects on the timings rather than108

the volume of the investment, factors regarding the demand dynamics and the discountrate can have strong effect on both timing and volume. Moreover, the effects on the
breakpoint also vary by the type of the factors.
These findings can provide some references for the government to determine theprivatization time and the private share. For instance, if governmental subsidiary isavailable, the earliest and largest expansion can be achieved by a fairly low private share.
By contrast, a low private share should be avoided, if governmental subsidiary isunavailable. In this case, there is a trade-off between the social surplus within the airport
and the spill-over social benefit (capacity) in terms of an early privatization. A fairly low
private share can maximize the aggregated social surplus, while the capacity to be
expanded will not be large.
This study has several limitations. First, we merely regard capacity as an “inelastic” upperbound for the output; no congestion cost or disutility will occur as long as the capacity is
not fully occupied. This might not be the case in reality. Second, we do not consider the
possibility of incremental investment in capacity. However, as a matter of fact, an airport
can be expanded for several times. Third, the assumption that the social-surplus oriented
government will obtain no profit is fairly strong. When capacity is binding, thegovernment does not necessarily just levy a cost-recovering charge. Fourth, theprivatization decisions are exogenous to the private investor. In fact, the private investor
is willing to purchase some share only when the privatization proves to be beneficial. A
bargaining between the government and the investor for the decisions on the privatization
should be modelled. Fifth, regarding the public sector’s objective, we only focus on thesocial surplus within the airport. As a matter of fact, airport capacity investment can havevarious externalities, such as the noise problem. Such aspects should be explored further.
These limitations provide the directions for the future improvement.

109

Chapter 6 Conclusion
6.1 Summary and conclusions
In the recent decades, airports are seeking structural changes, such as privatization, tomeet and satisfy the increasing demand. On the one hand, there are open issues on whether
privatization can stimulate the capacity investment which is the ultimate countermeasureagainst capacity shortage. On the other hand, privatization raises the concern regarding
the abuse of market power. Based on the background, the objectives of the thesis are, first,
assessing the plausibility of the introduction of terminal competition as an alternative
countermeasure against the abuse of market power; second, figuring out the relationshipbetween privatization and the capacity expansion from some new perspectives. Mainfindings of this thesis are summarized by chapter as follows:
In Chapter 2, we explain why the introduction of intra-airport terminal competition is
proposed, presenting the limited practical examples, and showing the remarks of official
discussions on it. We then review the studies on competing facilities, clarifying thenecessity of a new modelling. Next, we introduce the current status of airport capacity
shortage and expansion in the real world. We review the relevant studies, confirming that
our study can supplement the existing findings.
Chapter 3 studies the effect of intra-airport terminal competition on the pricing and social
surplus. We consider several Business Models characterizing the potential situations
before and after the introduction of terminal competition, and address the freedom ofairlines to change their base terminals in response to the charges. An analytical model isdeveloped considering the competition of downstream airlines. Comparing the charges
and the social surplus in equilibrium in each Business Model, we discuss the effect ofterminal competition. We further extend the study by considering four special cases: (i),terminals have different levels of marginal cost and unit non-aeronautical profit; (ii),
leader airlines participate in the upstream via directly operating or owing terminals; (iii),
terminals not only compete on price, but also compete on service level (quality); (iv),terminals are regulated. We investigate whether an airport with terminal competition canbe better-off in these special cases.

110

We find that, in most cases, having competing terminals can neither lower the prices norenhance the social surplus if the operation of the terminal and airfield facilities are notcompletely separated in the existing business model (e.g., the incumbent operator
operates both the airfield facilities and terminals), whether or not airlines have thefreedom to change base terminal in response to the prices. The complementarity between
the airfield and terminal service, which is originally internalized by the joint operation,
will become a negative effect that cannot be offset even by a strong degree of substitution
between terminals, if the two services are provided respectively by independent operators.
In contrast, if the operation of the two sections has been completely separated in theexisting business model (e.g., one operator operates the terminals while another operates
the airfield), or the airfield operator can be strictly regulated, having competing terminals
can result in a higher social surplus, as it will not increase the negative complementary
effect. Instead, it creates a duopoly of substitute goods that can offset the complementary
effect. Moreover, only few special cases can make the introduction of terminal
competition better-off in terms of social surplus.
Our findings indicate that the introduction of terminal competition fails to restrain theabuse of market power by monopoly airport, and improve social surplus in most cases,
contrary to common belief. If we further take into account other potential problems ofcompeting terminals, such as the difficulty of coordination and loss of economies of scale,
we can only contend that having competing terminals without any restrictions may not bea reasonable solution.
Chapter 4 analyzes the optimal timing problem of investment when the airport faces two
types of project, namely capacity expansion and cost reduction. We adopt the Stackelberg
model to solve the optimal pricing problem for the airport, and formally prove the interrelationship between the projects. Then, employing the real option approach, we derive
the optimal rule of the timings for the bi-projects investments which improves the rule
proposed by the previous study. We also investigate the loss of the investments due tosuboptimal rules.
The results of the numerical examples suggest that the decision-maker who places ahigher premium on social surplus always tends to invest earlier in both projects. Other111

factors that stimulate an earlier investment include a lower cross-price effect, a greater
total number of airlines, a higher drift, and a lower volatility in demand. The optimal
scales of projects are more sensitive to the change in drift and volatility rather than thechange in other factors. When the composition of the objective function changes, thepattern of the change in optimal timing differs by project: for cost reduction, the changing
rate does not change with the share of profit in most cases, while the optimal timing isconvex with regard to the share for capacity expansion. The loss due to NPV-based
investment is much greater than that caused by the adoption of the deterministic model,
and its change with the change in the composition of the objective function has amountain-shaped pattern, where the minimum can be reached when specific conditions
are met.
Our results imply that, to improve social surplus, the timings of investments in bothprojects should be brought forward. However, the adjustment for different types ofprojects should be distinguished, as their pattern of change in timings is not the same.
Besides, NPV-based investment should be avoided in most cases even when the real
option rule is not applicable.
Chapter 5 investigates the relationship between the privatization decision and theinvestment decision. We consider two scenarios differing in the availability of
governmental subsidiary to the investment. Similar with Chapter 4, we use theStackelberg model to derive the optimal charge of the airport, and then obtain the optimal
timing and volume for the capacity expansion by a real option approach. Numericalcomputations are performed with different privatization timings and the private shares.
The results suggest that the optimal volume and timing for the investment do notmonotonically increase or decrease with regard to the private share. Their changing
patterns are not simple due to two countervailing effects, if the governmental subsidiary
to the investment is available. If the governmental subsidiary to the investment isunavailable, three clear-cut regions of the investment decisions with regard to the private
share can be found: If the private share is low, no expansion will be carried out; if theprivate share is moderate, a gap-filling expansion will be carried out; if the private shareis high, the expansion will not only fill the current capacity gap, but also leave some112

vacant capacity preparing for the future demand growth. Maximum aggregated social
surplus can be achieved by privatizing a certain share which is just consistent with thevalue of one of the two breakpoints that connect two adjacent regions, while which
breakpoint can maximize the aggregated social surplus depends on the timing of theprivatization. Whilst factors regarding the downstream airlines might have greater effects
on the timings rather than the volume of the investment, factors regarding the demand
dynamics can have strong effect on both timing and volume. Moreover, the effects on the
breakpoint also vary by the type of the factors.
These findings can provide some references for the government to determine theprivatization time and the private share. For instance, if governmental subsidiary isavailable, the earliest and largest expansion can be achieved by a fairly low private share.
By contrast, a low private share should be avoided, if governmental subsidiary isunavailable.
6.2 Future scope
A multitude of aspects of the thesis can be improved. We first point out the common
limitations of all studies involved, and then summarize the flaws of each chapter
respectively. Those are followed by the recommendations for the future works.
Common limitations
1. All studies in the thesis only focus on one single airport. However, many changes
in the recent years, such as the deregulation of airlines and the emergence of LCC,
have made inter-airport competition no longer an ignorable factor. Future study
can extend the works in this thesis by taking into account multiple airports which
can be competing hubs or airport-pair in a metropolitan area.
2. This thesis mainly employs analytical model which, to some degree, ensures thegenerality of result, due to the lack of realistic data. However, several factors are
excluded from the modelling for the sake of tractability. A more comprehensive
and explicit modelling using realistic data is expected to examine the robustnessof the findings of this thesis.
Respective flaws
113

Chapter 3
1. We did not consider LCC terminal which should have fairly different featurescompared with the normal terminal. Future study can address this issue by
investigating whether an independently operated LCC terminal should exist interms of various aspects.
2. We did not address the issues regarding investment. In reality, a new terminal atan airport was constructed through a BOT project in many cases, so the terminal
competition might not exist forever. Future study can investigate issues such as
the effect of the details of BOT contract on the aggregated social surplus.
Chapter 4
1. Capacity is not treated as a “hard” bound in the setting, meaning that there is no
upper bound for the output. A shortcoming of such setting is that a congestion cost
will still occur even if the output is well under the capacity, which is not realistic.
Analysis using other capacity setting might should be tried.
2. Incremental investment is not considered. In reality, especially the project for cost
reduction, can be invested in many times. Future improvement should address thisissue.
3. The airport can decide the scale of each project discretely, while the volume ofeach scale is pre-determined. A “pure” optimization of both timing and volume of
the investment is desired.
4. Lack of real data, particularly the data regarding the cost for the investment,
renders the results less persuasive. Case studies with real-world data are desired.
Chapter 5
1. The modelling of the privatization decisions is fairly rough. In reality, theprivatization contract and the process to reach the agreement can be quite
sophisticated. Future study should consider more details of the contract, andaddress the complex process of bargaining.
2. Several assumptions in the settings are fairly strong, such as the non-profiting
assumption for the social surplus-oriented government, as mentioned. Theseassumptions should be relaxed as much as possible in the future improvement.
114

3. Comparison of the effects on investment of different forms of privatization can bea meaningful extension for the future study.
4. Regarding the public sector’s objective, we only focus on the social surplus within
the airport. As a matter of fact, airport capacity investment can have various
externalities, such as the noise problem. Such aspects should be explored further.
5. Private share does not necessarily affect the composition of the objective functionin this continuous way. For instance, 51% and 49% can be very different in termsof the decision-makings. Future study should seriously address such difference.

115Appendices
Appendix A A discussion on Business Model II
The concern that Terminal B will be driven out of the market in Business Model II might
be true if the profit function of operators can be written as follows.
==
And the condition

𝜏 −𝒸

𝐴
𝐵

+

𝐵

𝜏

𝜏 −𝒸 −𝜏

1/𝑎 − 𝒸 / 1/𝑎 − 𝒸≥ 𝑙 should be satisfied (Arya et al., 2008).

Where 𝜏 , 𝜏 are the prices of whole airport service (terminal plus airfield) per passenger
set by operator 1 and 2 respectively, 𝜏

is the price of runway access that operator 2 has

to purchase in order to produce the final good (airport service). 𝑙 is the cross-price effectbetween terminals.
However, a crucial difference should be noticed comparing this with our modeling. In our
setting, there are no direct trades between the two upstream operators, all goods are soldto the downstream airlines, albeit an indispensable input (airfield service) for all airlines
is only produced by operator 1. In contrast, in the model shown above, the runway accesswill be sold by operator 1 to 2, indicating a price-discrimination in the homogenous
airfield goods due to double marginalization. Moreover, if we assume identical cost, thecondition

1/𝑎 − 𝒸 / 1/𝑎 − 𝒸≥ 𝑙 can only be satisfied when two terminals are

perfect substitutes, which merely corresponds to an extreme case in our setting.
Consequently, in this study, different from the results of Barbot (2011) and D’Alfonsoand Nastasi (2012), operator 2 will not be driven out as long as the competition in the
upstream market is in a differentiated Bertrand form (if two operators compete on quantity,
operator 2 will indeed be driven out). Findings in Chen (2001) are also in line with thisresult.
Besides, observing the equilibrium output of Terminal B in Model II, we can find𝑘 2 𝑠𝑛𝐵 𝐶1 𝑛𝐴 𝐶2 𝑛𝐴 +𝐶1
3𝐶12 𝑛𝐴 +𝑛𝐵 +𝑟+ 𝑠 𝛾 𝑐
3𝐶22 𝑛𝐴 𝑛𝐵𝑐+𝛾 𝑐 𝛾

> 0 as 𝐶 ≥ 𝐶 and 𝑠 > max{ +𝑟

𝑟

𝐼𝐼
𝐵 =

} defined as basic

assumption. Operator 2 can still ensure a share and thus will not exit. If operator 1 forces
operator

2

to

exit

by

setting

a
116

much

higherairfieldcharge𝑇=

𝐶1 𝑛𝐴 +𝐶2 𝑛𝐴

𝑟+ 𝑠 𝑐 +𝛾𝐶2 𝑛𝐴

and lowering terminal fee accordingly, its own profit

𝐶1 𝑛𝐴 +

will

also

𝑘 2 𝑛𝐵 𝐶1 𝑛𝐴 𝐶2 𝑛𝐴 +𝐶1

sustain𝑟+ 𝑠 𝑐 𝛾
9𝐶1 𝑛𝐴 +𝐶1 𝑛𝐴
2
𝐶1 𝑛𝐴 +a
𝐶2 𝑛𝐴 +𝐶1

loss

of

𝑟+ 𝑠 𝑐 𝛾 +3𝐶1 𝜆𝑠 𝑛𝐴 +𝐶22 𝑛𝐴 𝑛𝐵

𝑛𝐵 +

∆=

> 0 , causing alose-lose outcome.
Appendix B Solving process and proof of Section 3.2
B1. Solving process of the multi-stage games
First order conditions (FOCs) of the subgame equilibrium are:𝜕𝛱ℎ𝑖=𝑠 1−𝑤
𝜕𝑓ℎ𝑖− 𝑠 𝑎𝐹 + 𝑎𝑙𝐹

𝛼𝛽− 𝛼𝛿 − 𝑐 − 𝑇 − [𝑠 (𝑎𝑠 + ) + ] 𝑓 − 𝛽𝛿 = 0𝑘𝑘.1

Adopting symmetry and solving the system, we have the equilibrium outputs as Eq. (3.8)
and (3.9) as functions of total prices, or as functions of airfield charge and terminal fees
as:
𝑓

𝑇𝑤 𝑤

𝑇 𝑤𝐴 𝑤𝐵 ==𝑘 2 (𝐶1 𝑛−ℎ +𝑇+𝑤ℎ 𝑠+𝑐 𝑠

𝐶2 𝑛−ℎ 𝑇+𝑤−ℎ 𝑠+𝑐 𝑠 )𝐶22 𝑛ℎ 𝑛−ℎ 𝐶12 𝑛ℎ +𝐶1 𝐶2 𝑘 2 𝑛𝐴 𝑛𝐵 ( 𝑤𝐴 +𝑤𝐵 𝑠+

𝑛−ℎ +𝑇+𝑐 𝑠 )+𝐶1 (∑ℎ=𝐴 𝐵 𝑛ℎ 𝑇+𝑤ℎ 𝑠+𝑐 𝑠 )𝐶22 𝑛𝐴 𝑛𝐵 𝐶12 𝑛𝐴 +

𝑛𝐵 +

.2
.3

Substituting Eq. (3.8) and (3.9) into the Eq. (3.7), we first check the concavity of social
surplus function with respect to the total prices. Second order conditions are derived asfollows.
|𝐻 | =

2𝑘 2 𝑛𝐴
𝐶13 𝑛𝐵 + 2 𝑘 𝛼𝑠+𝛽 𝐶1 𝑛𝐵 𝐶2 𝑛𝐵 +𝐶1 2 )(𝐶22 𝐶1 𝑛𝐵 𝑛𝐵 +
𝜕2 𝑆
=
<0
2
2
𝜕𝑇𝐴
(𝐶22 𝑛𝐴 𝑛𝐵 𝐶12 𝑛𝐴 + 𝑛𝐵 + )|𝐻 | = |

𝜕2 𝑆
𝜕𝑇𝐴2

𝜕2 𝑆
𝜕𝑇𝐴 𝜕𝑇𝐵

𝜕2 𝑆

𝜕2 𝑆

𝜕𝑇𝐵 𝜕𝑇𝐴

𝜕𝑇𝐵2

|=

2 2
𝐶1 𝐶2 𝑘 4 𝑛𝐴𝑛𝐵 (𝐶1 +𝐶2 + 𝑘 𝛽+𝛼𝑠 )(𝐶22 𝑛𝐴 𝑛𝐵 𝐶12 𝑛𝐴 +𝑛𝐵 + )

2

>0

.4

.5

Next, substituting the equilibrium outputs of the downstream subgame into the objectivefunctions of each Business Model respectively, we can obtain the equilibrium prices. Notethat for the first-best benchmark and Business Model I, the objective function must be
converted into a form with two decision variables, which are the total price 𝑇𝐴 and 𝑇𝐵 , toensure the non-singularity of the coefficient matrix of FOCs. While for other Models,117objective functions with three decision variables, namely airfield charge 𝑇, terminal fees
𝑤𝐴 and 𝑤𝐵 , can be retained. FOCs of each Models are summarized in Table A.1.
Table A.1 Objective functions and first order conditions of each business model

ModelsObjective functionsFirst order conditions
𝜕𝑆
𝜕𝑆= 0; 𝜕𝑇 = 0
𝜕𝑇𝐴𝐵First-bestI

𝑗 = 1 {𝑥 𝑦 𝑧} = {1 1 1}

II

𝑗 = 1 {𝑥 𝑦 𝑧} = {1 1 0}
𝑗 = 2 {𝑥 𝑦 𝑧} = {0 0 1}

𝜕𝛤1𝜕𝛤= 0; 1 = 0
𝜕𝑇𝐴
𝜕𝑇𝐵

𝜕𝛤1
𝜕𝛤
𝜕𝛤
= 0; 1 = 0; 2 = 0
𝜕𝑇
𝜕𝑤𝐴
𝜕𝑤𝐵

𝑗 = 1 {𝑥 𝑦 𝑧} = {0 1 0}

III

𝑗 = 2 {𝑥 𝑦 𝑧} = {0 0 1}

𝜕𝛤1
𝜕𝛤
𝜕𝛤
= 0; 𝜕𝑤2 = 0; 𝜕𝑇3 = 0
𝜕𝑤𝐴𝐵

𝑗 = 3 {𝑥 𝑦 𝑧} = {1 0 0}
𝑗 = 1 {𝑥 𝑦 𝑧} = {0 1 1}

IV

𝑗 = 2 {𝑥 𝑦 𝑧} = {1 0 0}

V

𝑗 = 1 {𝑥 𝑦 𝑧} = {0 1 1}
𝑗 = 2 {𝑥 𝑦 𝑧} = {1 0 0}

𝑅 =𝑆+𝜆 𝑅

𝑗 = 1 {𝑥 𝑦 𝑧} = {0 1 0}

VI

𝑗 = 2 {𝑥 𝑦 𝑧} = {0 0 1}
𝑗 = 3 {𝑥 𝑦 𝑧} = {1 0 0}

𝑅 =𝑆+𝜆 𝑅𝜕𝛤1𝜕𝛤𝜕𝛤= 0; 1 = 0; 2 = 0
𝜕𝑤𝐴
𝜕𝑤𝐵
𝜕𝑇

𝜕𝛤1
𝜕𝛤
𝜕𝛤
= 0; 𝜕𝑤1 = 0; 𝜕𝑇2 = 0;
𝜕𝑤𝐴
𝐵
𝜕𝛤2
𝜕𝛤
≥ 0; 𝜆 ≥ 0; 𝜕𝜆2 𝜆 = 0
𝜕𝜆

𝜕𝛤1
𝜕𝛤
𝜕𝛤
= 0; 𝜕𝑤2 = 0; 𝜕𝑇3 = 0;
𝜕𝑤𝐴
𝐵
𝜕𝛤3
𝜕𝛤
≥ 0; 𝜆 ≥ 0; 3 𝜆 = 0𝜕𝜆𝜕𝜆

A2. Proof of Lemma 1
𝜕𝑓
𝐶 𝑘 𝑛 +1=−
𝜕𝑇
𝐶 𝑛 +1 𝑛 +1 −𝐶 𝑛 𝑛

<0

.6

𝜕𝑓
𝜕𝑇

>0

.7

=

𝐶 𝑘 𝑛
𝐶 𝑛 +1 𝑛 +1 −𝐶 𝑛 𝑛118𝜕
𝑘 𝑠𝑛𝐴 ( 𝐶 − 𝐶 𝑛𝐵 + 𝐶 )
=−
<0
𝜕𝑇𝐴
𝐶 𝑛𝐴 + 1 𝑛𝐵 + 1 − 𝐶 𝑛𝐴 𝑛𝐵.8𝜕𝑘 𝑠𝑛𝐵 ( 𝐶 − 𝐶 𝑛𝐴 + 𝐶 )
=−
<0
𝜕𝑇𝐵
𝐶 𝑛𝐴 + 1 𝑛𝐵 + 1 − 𝐶 𝑛𝐴 𝑛𝐵.9Since 𝐶 ≥ 𝐶 > 0, these signs can be judged easily.
Appendix C Results and proofs of Section 3.2.3
C1. Equilibrium airfield charge and terminal fees in each Business Model𝑤𝐴𝐼𝐼 =𝐶 𝑛𝐴 + 1 𝑠 − 5𝑟𝑠 − 𝛾 − 𝐶 𝑛𝐴 𝑠 + 𝑟𝑠 − 𝛾 − 𝑐 𝐶 𝑛𝐴 − 𝐶 𝑛𝐴 + 𝐶
6𝐶 𝑠 𝑛𝐴 + 1

𝐶. 1

𝑤𝐵𝐼𝐼 =𝐶 𝑛𝐴 + 1 𝑠 − 2𝑟𝑠 − 𝛾 − 𝐶 𝑛𝐴 𝑠 + 𝑟𝑠 − 𝛾 − 𝑐 𝐶 𝑛𝐴 − 𝐶 𝑛𝐴 + 𝐶
3𝐶 𝑠 𝑛𝐴 + 1

𝐶. 2

2𝐶 𝑛𝐴 + 1 + 𝐶 𝑛𝐴 ( 𝑟 + 1 𝑠 − 𝑐) + 𝛾 4𝐶 𝑛𝐴 + 1 − 𝐶 𝑛𝐴
6𝐶 𝑛𝐴 + 1

𝐶. 3

𝑇 𝐼𝐼 =𝑤𝐴𝐼𝐼𝐼 = [2𝐶 3 𝑛𝐴 + 1 𝑛𝐵 + 1 𝑛𝐴 2𝑛𝐵 + 1 + 𝑛𝐵 𝑠 − 2𝑟𝑠 − 𝑐 − 𝛾 + 𝐶 𝐶 𝑛𝐴 +1 𝑛𝐵 𝑛𝐵 𝑛𝐴 6𝑐 + 6𝛾 + 4𝑟𝑠 − 6𝑠 + 𝑐 + 𝛾 − 𝑟 + 1 𝑠 + 5𝑛𝐴 𝑐 + 𝛾 + 𝑟 − 1 𝑠
𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 4𝑟𝑠𝑛𝐵 + 𝑐 + 𝛾 + 𝑟 − 1 𝑠 − 𝑛𝐵 𝑐 + 𝛾 − 3𝑟𝑠 − 𝑠+3− 2𝐶 𝑛𝐴 𝑛𝐵 𝑐 + 𝛾 − 𝑠 ]/

𝑠𝐷0

𝐶. 4

𝑤𝐵𝐼𝐼𝐼 = [ 2𝐶 3 𝑛𝐴 + 1 𝑛𝐵 + 1 𝑛𝐴 2𝑛𝐵 + 1 + 𝑛𝐵 𝑠 − 2𝑟𝑠 − 𝑐 − 𝛾 + 𝐶 𝐶 𝑛𝐴 𝑛𝐵 +1 𝑛𝐴 𝑛𝐵 6𝑐 + 6𝛾 + 4𝑟𝑠 − 6𝑠 + 𝑐 + 𝛾 − 𝑟 + 1 𝑠 + 5𝑛𝐵 𝑐 + 𝛾 + 𝑟 − 1 𝑠
𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 4𝑟𝑠𝑛𝐵 − 𝑐 − 𝛾 + 3𝑟𝑠 + 𝑠 + 𝑛𝐵 𝑐 + 𝛾 + 𝑟 − 1 𝑠
𝑠𝐷0+− 2𝐶 3 𝑛𝐴 𝑛𝐵 𝑐 + 𝛾 − 𝑠 ]/𝐶. 5

𝑇 𝐼𝐼𝐼 = [ 2𝐶 3 𝑛𝐴 + 1 𝑛𝐵 + 1 𝑛𝐴 2𝑛𝐵 + 1 + 𝑛𝐵

𝑟 + 1 𝑠 − 𝑐 + 2𝛾 + 2𝐶 𝐶 𝑛𝐴 𝑛𝐴 +1 𝑛𝐵 𝑛𝐵 + 1 𝑐 − 4𝛾 − 𝑟 + 1 𝑠 + 𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 2𝑛𝐵 + 1 + 𝑛𝐵 𝑐 − 𝛾 − 𝑟 + 1 𝑠 +2𝛾𝐶 3 𝑛𝐴 𝑛𝐵 ]/𝐷0

𝐶. 6
𝑤𝐴𝐼𝑉 = 𝑤𝐵𝐼𝑉 =𝑠 − 2𝑟𝑠 − 𝑐 − 𝛾
3𝑠

119

𝐶. 7

𝑇 𝐼𝑉 =

1𝑠 + 𝑟𝑠 − 𝑐 + 2𝛾
3.8𝑠 − 𝑟𝑠 − 𝑐 − 𝛾
2𝑠

𝐶. 9

𝑤𝐴𝑉 = 𝑤𝐵𝑉 =𝑇𝑉 = 𝛾𝑤𝐴𝑉𝐼 =𝑤𝐵𝑉𝐼 =

𝐶12 𝑛𝐴+

𝐶12 𝑛𝐴+𝑛𝐵 +𝑠 𝑟𝑠 𝑐 𝛾

𝐶2 𝐶1 𝑛𝐴 +2
𝑠(4𝐶1 𝑛𝐴 + 𝑛𝐵 +

𝑛𝐵 +

𝐶. 10

𝑛𝐵 ( 𝑟+ 𝑠 𝑐+𝛾) 𝐶22 𝑛𝐴 𝑛𝐵 𝑠 𝑐 𝛾

𝑠 𝑟𝑠 𝑐 𝛾

𝐶2 𝐶1 𝑛𝐴 𝑛𝐵 +𝑠(4𝐶12 𝑛𝐴 + 𝑛𝐵 +𝐶22 𝑛𝐴 𝑛𝐵 )( 𝑟+ 𝑠 𝑐+𝛾) 𝐶22 𝑛𝐴 𝑛𝐵 𝑠 𝑐 𝛾
𝐶22 𝑛𝐴 𝑛𝐵 )𝑇 𝑉𝐼 = 𝛾

𝐶. 11
𝐶. 12
𝐶. 13

𝐷0 = 2 3𝐶 3 𝑛𝐴 + 1 𝑛𝐵 + 1 𝑛𝐴 2𝑛𝐵 + 1 + 𝑛𝐵 − 5𝐶 𝐶 𝑛𝐴 𝑛𝐴 +Where

1 𝑛𝐵 𝑛𝐵 + 1 − 𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 2𝑛𝐵 + 1 + 𝑛𝐵 + 𝐶 3 𝑛𝐴 𝑛𝐵
C2. Proof of Proposition 1
For (i), we can confirm that𝑇𝐴𝐼𝐼𝐼 − 𝑇𝐴𝐼𝐼 = 𝐶 𝑛𝐵 − 𝐶 𝑛𝐵 + 𝐶
𝐶+ 𝐶 𝑛𝐵𝑛𝐴 3𝐶 − 𝐶 𝐶 𝑛𝐵 + 𝐶+ 𝑛𝐴 2𝐶 − 𝐶 𝐶 − 𝐶 𝑛𝐵 +𝑟 + 1 𝑠 − 𝑐 − 𝛾 /2𝐷 > 0

𝐶. 14

𝑇𝐴𝐼𝑉 − 𝑇𝐴𝐼𝐼𝐼 = [𝐶 𝑛𝐵 𝑟 + 1 𝑠 − 𝑐 − 𝛾 𝑛𝐴 7𝐶 − 4𝐶 𝐶 𝑛𝐵 + 𝐶
2𝐶 𝑛𝐵 + 𝐶 + 2𝐶 𝐶

+ 3𝐶 𝑛𝐵 ]/6𝐷 > 0

𝑇𝐴𝑉 − 𝑇𝐴𝑉𝐼 =𝐶. 15

𝐶 𝑛𝐵 (𝐶 𝑛𝐴 + 2𝐶 𝑛𝐴 + 1 )( 𝑟 + 1 𝑠 − 𝑐 − 𝛾)>0
8𝐶 𝑛𝐴 + 1 𝑛𝐵 + 1 − 2𝐶 𝑛𝐴 𝑛𝐵

𝐶. 16( 𝐶 − 𝐶 𝑛𝐴 + 𝐶 )( 𝑟 + 1 𝑠 − 𝑐 − 𝛾)>0
6𝐶 𝑛𝐴 + 1

𝐶. 17

𝑇𝐵𝐼𝐼 − 𝑇𝐵𝐼 =𝑇𝐵𝐼𝐼𝐼 − 𝑇𝐵𝐼𝐼 =+ 𝑛𝐴 4𝐶 − 2𝐶 𝐶 −∏ℎ=𝐴 𝐵[𝑛ℎ 𝐶1 𝑛ℎ +𝐶2 𝑛ℎ ](𝐶2 𝑛𝐴 + 𝐶1 𝑛𝐴 + )( 𝑟+𝑠 𝑐 𝛾)𝐶2

6𝐶1 𝐷1 𝑛𝐴 +

>0

𝐶. 18

𝑇𝐵𝐼𝑉 − 𝑇𝐵𝐼𝐼𝐼 = [𝐶 𝑛𝐴 𝑟 + 1 𝑠 − 𝑐 − 𝛾 𝑛𝐴 4𝐶 − 2𝐶 𝐶 − 2𝐶 𝑛𝐵 + 7𝐶 − 4𝐶 𝐶 𝑛𝐵 +3𝐶

+ 𝐶 + 2𝐶 𝐶 𝑛𝐵 + 𝐶 𝑛𝐵 ]/6𝐷 > 0
𝑇𝐵𝑉 − 𝑇𝐵𝑉𝐼 =

Where

𝐶. 19

𝐶 𝑛𝐴 (𝐶 𝑛𝐵 + 2𝐶 𝑛𝐵 + 1 )( 𝑟 + 1 𝑠 − 𝑐 − 𝛾)>0
8𝐶 𝑛𝐴 + 1 𝑛𝐵 + 1 − 2𝐶 𝑛𝐴 𝑛𝐵

𝐷 = 3𝐶 3 𝑛𝐴 + 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵 + 9𝐶 3 − 5𝐶 𝐶 − 𝐶 𝐶
120

𝐶. 20

𝑛𝐴 𝑛𝐵 + 𝑛𝐴 𝑛𝐵 +𝐶 −𝐶

3𝐶 − 𝐶

2𝐶 + 𝐶 𝑛𝐴 𝑛𝐵 + 12𝐶 3 − 5𝐶 𝐶 𝑛𝐴 𝑛𝐵 > 0.

Remaining equations/inequations are self-evident.
For (ii), we have𝜕𝑤𝐴𝐼𝐼
𝐶 ( 𝑟 + 1 𝑠 − 𝑐 − 𝛾)
=−
<0
𝜕𝑛𝐴
6𝐶 𝑠 𝑛𝐴 + 1

𝐶. 21

𝜕𝑤𝐵𝐼𝐼
𝐶 ( 𝑟 + 1 𝑠 − 𝑐 − 𝛾)
=−
<0
𝜕𝑛𝐴
3𝐶 𝑠 𝑛𝐴 + 1

𝐶. 22

𝜕𝑇 𝐼𝐼 𝐶 ( 𝑟 + 1 𝑠 − 𝑐 − 𝛾)=>0𝜕𝑛𝐴
6𝐶 𝑛𝐴 + 1

𝐶. 23

𝜕𝑇𝐵𝐼𝐼
𝐶 ( 𝑟 + 1 𝑠 − 𝑐 − 𝛾)
=−
<0
𝜕𝑛𝐴
6𝐶 𝑛𝐴 + 1

𝐶. 24

Remaining equations/inequations are self-evident.
𝜕𝑇 𝑀

𝜕𝑇 𝑀 𝜕𝐶2

For (iii), since 𝜕𝑙ℎ = 𝜕𝐶ℎ

2

of

𝜕𝑇ℎ𝑀
𝜕𝐶2

𝜕𝑙

𝜕𝐶

and 𝜕𝑙2 = 𝑎𝑘 𝑠 > 0, we only need to investigate the signfor each case.
𝜕𝑇𝐵𝐼𝐼𝑛𝐴 ( 𝑟 + 1 𝑠 − 𝑐 − 𝛾)=−<0𝜕𝐶
6𝐶 𝑛𝐴 + 1

𝜕𝑇𝐴𝐼𝐼𝐼
= −[𝐶
𝜕𝐶2

𝐶

𝑛𝐴 + 1 𝑛𝐵 𝑛𝐴3 2 𝐶 − 𝐶

𝐶. 25

7𝐶 + 5𝐶 𝐶 − 3𝐶 𝐶 𝑛𝐵 + 2 −2𝐶 + 𝐶 𝐶 +𝑛𝐵3 + 7𝐶 4 + 8𝐶 𝐶 3 − 9𝐶 𝐶 𝑛𝐵 + 𝐶 4 + 4𝐶 𝐶 3 + 𝑛𝐴 2 𝐶 − 𝐶

9𝐶 − 𝐶 𝐶 +𝐶 𝐶 𝑛𝐵3 + 28𝐶 4 − 24𝐶 𝐶 3 + 2𝐶 𝐶 𝑛𝐵 + 11𝐶 4 − 4𝐶 3 𝐶 𝑛𝐵 + 𝐶 4 + 𝑛𝐴 4𝐶 4 𝑛𝐵 +13𝐶 4 − 8𝐶 𝐶 3 + 𝐶 𝐶 𝑛𝐵3 + 17𝐶 4 − 8𝐶 3 𝐶 𝑛𝐵 + 3𝐶 4 𝑛𝐵3 + 3𝐶 4 𝑛𝐵

𝑟 + 1 𝑠 − 𝑐 − 𝛾 ]/2𝐷 < 0𝜕𝑇𝐵𝐼𝐼𝐼= −[𝐶 𝑛𝐴 𝑛𝐵 + 1
𝜕𝐶2

𝐶

𝐶. 26
𝑛𝐴3 2 𝐶 − 𝐶

9𝐶 − 𝐶 𝐶 + 𝐶 𝐶 𝑛𝐵 + 2 −2𝐶 + 𝐶 𝐶 +𝑛𝐵3 + 13𝐶 4 − 8𝐶 𝐶 3 + 𝐶 𝐶 𝑛𝐵 + 3𝐶 4 + 𝑛𝐴 2 𝐶 − 𝐶

7𝐶 + 5𝐶 𝐶 − 3𝐶 𝐶 𝑛𝐵3 +28𝐶 4 − 24𝐶 𝐶 3 + 2𝐶 𝐶 𝑛𝐵 + 17𝐶 4 − 8𝐶 3 𝐶 𝑛𝐵 + 3𝐶 4 + 𝑛𝐴 4𝐶 4 𝑛𝐵 + 7𝐶 4 +8𝐶 𝐶 3 − 9𝐶 𝐶 𝑛𝐵3 + 11𝐶 4 − 4𝐶 3 𝐶 𝑛𝐵 + 𝐶 4 𝑛𝐵 + 𝐶 4 + 4𝐶 𝐶 3 𝑛𝐵3
2𝐷 < 0𝑟 + 1 𝑠 − 𝑐 − 𝛾 ]/
𝐶. 27

121

Remaining equation is self-evident.
𝜕𝑄

𝜕𝑄

𝐴

𝐵

For (iv), recalling Lemma 1 that 𝜕𝑇 < 0 and 𝜕𝑇 < 0, and (i) of Proposition 1, (iv)
𝜕𝑄

𝜕𝑄

𝐴

𝐵

becomes self-evident, since 𝑑 = 𝜕𝑇 𝑑𝑇𝐴 + 𝜕𝑇 𝑑𝑇𝐵 .
C3. Proof of Proposition 3.2
𝜕𝑆

To prove 𝑆 𝐼 > 𝑆 𝐼𝐼 , we first apply the mean value theorem (MVT) to 𝜕𝑇 , we then have
𝐵

∆
𝜕2𝑆

𝜕𝑆
𝜕 𝑆
𝜕 𝑆
= ∆𝑇𝐴
+ ∆𝑇𝐵
𝜕𝑇𝐵
𝜕𝑇𝐵 𝜕𝑇𝐴
𝜕𝑇𝐵

𝜕2𝑆

𝜕𝑆

𝐵

𝐵

𝐶. 28
𝜕𝑆

𝜕𝑆

Where 𝜕𝑇 𝜕𝑇 and 𝜕𝑇 2 are evaluated at 𝑇𝐴𝐼 𝑇̃𝐵 , ∆ 𝜕𝑇 = 𝜕𝑇 |𝐵

𝐴

𝐵 𝑻= 𝑇 𝐼 𝑇̅𝐴 𝐵− 𝜕𝑇 |

𝐵 𝑻= 𝑇 𝐼 𝑇 𝐼
𝐴 𝐵,∆𝑇𝐵 = 𝑇̅𝐵 − 𝑇𝐵𝐼 ∆𝑇𝐴 = 0, 𝑇̃𝐵 is some value between 𝑇𝐵𝐼 and 𝑇̅𝐵 , while 𝑇̅𝐵 can have any
𝜕2𝑆value between 𝑇𝐵𝐼 and 𝑇𝐵𝐼𝐼 . Since 𝜕𝑇 2 < 0 (second-order condition), for any 𝑇̅𝐵 , we have𝐵𝜕𝑆

∆ 𝜕𝑇 < 0 . Adopting the assumption 𝛼𝑠 + 𝛽 < 𝑎𝑘𝑠 mentioned before, we can𝐵confirm that𝜕𝑆|

𝜕𝑇𝐵 𝑻= 𝑇 𝐼 𝑇 𝐼< 0,thus

𝐴 𝐵

𝜕𝑆

|

𝜕𝑇𝐵 𝑻= 𝑇 𝐼 𝑇̅

< 0.

𝐵

𝐴

Next, applying the MVT to 𝑆, we have
∆𝑆 = ∆𝑇𝐴

𝜕𝑆
𝜕𝑆
+ ∆𝑇𝐵
𝜕𝑇𝐴
𝜕𝑇𝐵

𝐶. 29
𝜕𝑆

Where ∆𝑇𝐴 = 𝑇𝐴𝐼𝐼 − 𝑇𝐴𝐼 = 0, ∆𝑇𝐵 = 𝑇𝐵𝐼𝐼 − 𝑇𝐵𝐼 > 0. Since 𝜕𝑇 |𝐵 𝑻= 𝑇 𝐼 𝑇̅𝐴 𝐵< 0, thus ∆𝑆 =𝑆 𝑇𝐵𝐼𝐼 𝑇𝐵𝐼𝐼 − 𝑆 𝑇𝐵𝐼 𝑇𝐴𝐼 < 0, 𝑆 𝐼 > 𝑆 𝐼𝐼 .
To prove 𝑆 𝐼 > 𝑆 𝐼𝑉 , we first convert 𝑆 𝑇𝐴 𝑇𝐵 into a function with single variable 𝑆 𝑇𝑈
by setting 𝑇𝑈 = 𝑇𝐴 = 𝑇𝐵 . Then we can obtain the stationary point of 𝑆 𝑇𝑈 as
𝑇𝑈 = [𝑐 −𝐶 2𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 + 𝑛𝐵 + 2 + 𝑘 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵
𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝐶 𝑛𝐴 + 𝑛𝐵 + 4𝑘 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵 𝛽 + 𝛼𝑠
𝐶 3 𝑛𝐴 𝑛𝐵 𝑛𝐴 + 𝑛𝐵 + 4 + 𝑛𝐴 + 𝑛𝐵
𝛾 𝑛𝐴 + 1 𝑛𝐵 + 1

𝛽 + 𝛼𝑠+− 4𝐶 𝑘𝑛𝐴 𝑛𝐵 𝛽 + 𝛼𝑠 ++ 𝐶 2𝐶 𝑛𝐴 𝑛𝐵 𝑠 𝑟 𝑛𝐴 + 1 𝑛𝐵 + 1 + 𝑛𝐴 + 𝑛𝐵 + 2 −+ 𝑘𝑠 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵

𝛽 + 𝛼𝑠+ 𝐶 𝐶 𝑛𝐴 𝑛𝐵 −𝛾𝐶 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 +𝑛𝐵 + 𝐶 𝑠 𝑛𝐴 2𝑟𝑛𝐵 + 𝑟 − 1 + 𝑟 − 1 𝑛𝐵 − 4𝑘𝑠 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵 𝛽 + 𝛼𝑠

122+

2𝐶 𝑛𝐴 𝑛𝐵 𝐶 𝛾 − 𝑟𝑠 + 2𝑘𝑠 𝛽 + 𝛼𝑠

+ 𝐶 3 𝛾 𝑛𝐴 + 1 𝑛𝐵 + 1 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵 +𝑠 −𝑟 𝑛𝐴 + 1 𝑛𝐵 + 1 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵 + 𝑛𝐴 −𝑛𝐵 𝑛𝐴 + 𝑛𝐵 + 4 − 𝑛𝐴 − 𝑛𝐵 ]/𝐶 𝑘 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵

𝛽 + 𝛼𝑠 − 2𝐶 𝑛𝐴 𝑛𝐵 𝑛𝐴 𝑛𝐵 − 1

𝑛𝐵 + 2𝑘 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵 𝛽 + 𝛼𝑠
1 + 1 + 2𝑛𝐴 𝑛𝐵 + 𝑛𝐵

− 2𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝐶 𝑛𝐴 𝑛𝐵 + 𝑛𝐴 ++ 2𝐶 𝑛𝐴 𝑛𝐵 𝐶 + 2𝑘 𝛽 + 𝛼𝑠

+ 𝐶 3 𝑛𝐴 2𝑛𝐵 𝑛𝐵 +𝐶. 30

we can then confirm that 𝑇𝑈 < 𝑇𝑈𝐼 < 𝑇𝑈𝐼𝑉 . We can also confirm that 𝑆 𝑇𝑈 is concave.
Thus, 𝑆 𝐼 > 𝑆 𝐼𝑉 . We omit the tedious process.
C4. Numerical results of the social surplus comparison in Case 1

Fig A.1 Numerical results of 𝑺𝑰𝑰𝑰 − 𝑺𝑰𝑽

Fig A.2 Numerical results of 𝑺𝑽𝑰 − 𝑺𝑽123

Appendix DEffect of the relaxation of the perfect substitute assumption
Let 𝑙 denotes the cross effect between every two airlines. For airlines in different
terminals, their cross effect is 𝑙 𝑎, and this is also true for airlines in same terminal,
unlike the basic modelling. From the perspective of passengers, airlines provide
heterogeneous services, and no airline can be perfect substitute to others, as long as 𝑙 ≠1.
In such a setting, the quadratic utility function becomes:
𝑈=
Inverse𝐴+

demand𝜌𝐵−function𝑛[𝑎(∑ 𝐴of𝑛𝐴+ ∑𝑗 𝐵

airline𝐵𝑗 ) + 2𝑎𝑙(∑𝒾>𝒿 𝒾 𝒿 )]atterminal

𝜕𝑈= 𝑤 +𝑝 +𝛼( ) =
=1−𝑎
𝑘
𝜕

− 𝑎𝑙

ℎ

is

𝐷. 1

denoted

−

as:
𝐷. 2

Adopting the same processes of backward induction, we can obtain the equilibrium
pricings of each business model:
𝑇𝐴𝐼 = 𝑇𝐴𝐼𝐼 = 𝑇𝐵𝐼 =𝑇𝐵𝐼𝐼 =

1𝑠 + 𝛾 − 𝑐 − 𝑟𝑠2

𝑐𝐶 4 − 3𝑛𝐴 + 𝛾𝐶 3𝑛𝐴 − 2 + 𝐶 𝑠 −3𝑟𝑛𝐴 + 3𝑛𝐴 + 2𝑟 − 4 − 8𝑐𝐶 + 4𝛾𝐶 − 4𝐶 𝑟 − 2 𝑠
6 𝐶 𝑛𝐴 − 1 + 2𝐶

𝑇𝐴𝐼𝐼𝐼 = 𝑐 −16𝐶 𝑛𝐴 + 𝑛𝐵 − 2𝐶 𝐶

9𝑛𝐴 − 8 𝑛𝐵 + 4 𝑛𝐴 − 2 𝑛𝐴 + 3𝑛𝐵 + 𝐶 −3 𝑛𝐴 − 1 𝑛𝐵 +−3 𝑛𝐴 − 3 𝑛𝐴 − 4 𝑛𝐵 + 4 𝑛𝐴 − 1 𝑛𝐴

− 8𝐶 𝑛𝐴 + 𝑛𝐵

𝑟 − 2 𝑠 − 𝛾 + 2𝐶 𝐶 𝛾 𝑛𝐴 + 𝑛𝐵 2𝑛𝐴 +3𝑛𝐵 − 4 + 𝑠 −𝑟 𝑛𝐴 + 𝑛𝐵 2𝑛𝐴 + 3𝑛𝐵 − 4 + 9𝑛𝐴 𝑛𝐵 + 4𝑛𝐴 − 8𝑛𝐴 + 3𝑛𝐵 − 8𝑛𝐵+ 𝐶 𝛾 𝑛𝐴 −1 3𝑛𝐵 − 2 𝑛𝐴 + 𝑛𝐵 + 𝑠 −𝑟 𝑛𝐴 − 1 3𝑛𝐵 − 2 𝑛𝐴 + 𝑛𝐵 + 3𝑛𝐴 𝑛𝐵 + 3𝑛𝐴 𝑛𝐵 − 9𝑛𝐴 𝑛𝐵 − 4𝑛𝐴 +4𝑛𝐴 − 3𝑛𝐵 + 4𝑛𝐵/ 24𝐶 𝑛𝐴 + 𝑛𝐵 + 4𝐶 𝐶 𝑛𝐴 7𝑛𝐵 − 6 + 3𝑛𝐴 + 3 𝑛𝐵 − 2 𝑛𝐵 + 2𝐶 3𝑛𝐴 𝑛𝐵 −1 + 𝑛𝐴 𝑛𝐵 3𝑛𝐵 − 7 + 3 − 3 𝑛𝐵 − 1 𝑛𝐵
𝑇𝐵𝐼𝐼𝐼 = 𝑐 −16𝐶 𝑛𝐴 + 𝑛𝐵 − 2𝐶 𝐶 𝑛𝐴 9𝑛𝐵 − 8 + 3𝑛𝐴 + 4 𝑛𝐵 − 2 𝑛𝐵 + 𝐶 −3𝑛𝐴 𝑛𝐵 − 1 +𝑛𝐴 −3 𝑛𝐵 − 3 𝑛𝐵 − 4 + 4 𝑛𝐵 − 1 𝑛𝐵

− 8𝐶 𝑛𝐴 + 𝑛𝐵

𝑟 − 2 𝑠 − 𝛾 + 2𝐶 𝐶 𝛾 𝑛𝐴 + 𝑛𝐵 3𝑛𝐴 +2𝑛𝐵 − 4 + 𝑠 𝑛𝐴 −5𝑟𝑛𝐵 + 9𝑛𝐵 + 4𝑟 − 8 − 3 𝑟 − 1 𝑛𝐴 − 2 𝑟 − 2 𝑛𝐵 − 2 𝑛𝐵

+ 𝐶 𝛾 3𝑛𝐴 −2 𝑛𝐵 − 1 𝑛𝐴 + 𝑛𝐵 + 𝑠 3𝑛𝐴 𝑟 −𝑛𝐵 + 𝑛𝐵 + 𝑟 − 1 + 𝑛𝐴 𝑛𝐵 −3𝑟𝑛𝐵 + 3𝑛𝐵 + 5𝑟 − 9 − 2𝑟 + 4 +2 𝑟 − 2 𝑛𝐵 − 1 𝑛𝐵

/ 24𝐶 𝑛𝐴 + 𝑛𝐵 + 4𝐶 𝐶 𝑛𝐴 7𝑛𝐵 − 6 + 3𝑛𝐴 + 3 𝑛𝐵 − 2 𝑛𝐵 +2𝐶 3𝑛𝐴 𝑛𝐵 − 1 + 𝑛𝐴 𝑛𝐵 3𝑛𝐵 − 7 + 3 − 3 𝑛𝐵 − 1 𝑛𝐵

124

𝑇𝐴𝐼𝑉 =

1
𝛾 + 2 − 𝑟 𝑠 − 2𝑐
3

𝑇𝐵𝐼𝑉 =

1
𝛾 + 2 − 𝑟 𝑠 − 2𝑐
3

𝑇𝐴𝑉 =

1
𝑠 + 𝛾 − 𝑐 − 𝑟𝑠
2

𝑇𝐵𝑉 =

1
𝑠 + 𝛾 − 𝑐 − 𝑟𝑠
2

𝑇𝐴𝑉𝐼 = −𝑐 2𝐶 − 𝐶

𝐶 2𝑛𝐴 + 𝑛𝐵 − 2 + 4𝐶

𝑟𝑠 𝐶 𝑛𝐴 − 1 + 2𝐶

𝐶 3𝑛𝐵 − 2 + 4𝐶

+ 𝛾 𝐶 𝑛𝐴 − 1 + 2𝐶

𝐶 3𝑛𝐵 − 2 + 4𝐶−

+ 2𝐶 − 𝐶 𝑠 𝐶 2𝑛𝐴 + 𝑛𝐵 − 2 + 4𝐶 /8𝐶 𝐶 𝑛𝐴 +𝑛𝐵 − 2 + 𝐶 𝑛𝐴 3𝑛𝐵 − 4 − 4𝑛𝐵 + 4 + 16𝐶
𝑇𝐵𝑉𝐼 = −𝑐 2𝐶 − 𝐶

𝐶 𝑛𝐴 + 2𝑛𝐵 − 2 + 4𝐶

𝑟𝑠 𝐶 3𝑛𝐴 − 2 + 4𝐶

𝐶 𝑛𝐵 − 1 + 2𝐶

+ 𝛾 𝐶 3𝑛𝐴 − 2 + 4𝐶

𝐶 𝑛𝐵 − 1 + 2𝐶−

+ 2𝐶 − 𝐶 𝑠 𝐶 𝑛𝐴 + 2𝑛𝐵 − 2 + 4𝐶 /8𝐶 𝐶 𝑛𝐴 +𝑛𝐵 − 2 + 𝐶 𝑛𝐴 3𝑛𝐵 − 4 − 4𝑛𝐵 + 4 + 16𝐶

Adopting similar approach in Appendix C, we can easily prove that 𝑇𝐴𝐼𝑉 > 𝑇𝐴𝐼𝐼𝐼 > 𝑇𝐴𝐼𝐼 =𝑇𝐴𝐼 = 𝑇𝐴𝑉 > 𝑇𝐴𝑉𝐼 ; 𝑇𝐵𝐼𝑉 > 𝑇𝐵𝐼𝐼𝐼 > 𝑇𝐵𝐼𝐼 > 𝑇𝐵𝐼 = 𝑇𝐵𝑉 > 𝑇𝐵𝑉𝐼 , which has the same order with thatderived from the original modelling.
Ranking of social surplus can also be proved in similar approach used in originalmodelling as 𝑆 𝐼 = 𝑆 𝑉 > 𝑆 𝐼𝐼 𝑆 𝐼 > 𝑆 𝐼𝑉 . Comparison of social surplus between otherbusiness models are conducted through numerical example (Fig A.3).

125

Fig A.3 Numerical result of the comparison of social surplus between Models

As a result, the ranking of business models in terms of social surplus also coincides withthat in the original modelling. Distinction is in the extent of the difference of pricing andsocial surplus between different business models. For example, in the result of 𝑆 𝐼𝐼 − 𝑆 𝐼𝐼𝐼 ,the surplus difference of the alternative modelling is much greater than that of the originalmodelling when 𝑙 is small. The reason is that a lower cross-price effect will lead to greater
quantities. When 𝑙 = 0, for instance, the effect between airlines in same terminal remains
full-scale in the original modelling, while the cross-price effect between all pairs vanishes
in the all-different modelling, so the quantity level thereof is much higher. A higher
quantity level makes the quantity loss, consequentially the social surplus loss switchingfrom model II to III greater. In conclusion, we can say that the assumption of perfect126

substitutes would not affect the primary results of the study
Appendix E Results and Proofs of Section 3.2.4
E1. Curves of maximum profit given optimal charges in Case 2

Fig A.4 Curves of airlines’ maximum profit given optimal charges in Case 2

E2. Proof of Proposition 3.4
Concavity of 𝑆 𝑇𝐴 𝑇𝐵 can be confirmed from following SOCs:|𝐻 | =𝑘 2 (𝑁𝐶1 ( 𝑁+
𝜕2 𝑆
=−
2
𝜕𝑇𝐴

|𝐻 | = |+ 𝐶12 )

𝐶2 +𝑁𝑘 𝛽+𝛼𝑠 ) 𝑁2𝐶2 k 𝛽+𝛼𝑠 + 𝑁 𝑁+
𝐶1 𝐶2 ( 𝑁+

𝜕2 𝑆
𝜕𝑇𝐴2

𝜕2 𝑆
𝜕𝑇𝐴 𝜕𝑇𝐵

𝜕2 𝑆

𝜕2 𝑆

𝜕𝑇𝐵 𝜕𝑇𝐴

𝜕𝑇𝐵2

|=

2

𝐶1 +𝑁𝐶2 )𝑁 2 𝑘 4 (𝐶1 +𝐶2 + 𝑘 𝛽+𝛼𝑠 )𝐶1 𝐶2 ( 𝑁+2

𝐶1 +𝑁𝐶2 )

>0

<0

𝐸. 1

𝐸. 2

As a quadratic function, 𝑆 𝑇𝐴 𝑇𝐵 can be written as:
𝑆 = 𝒙𝑻 𝓐𝒙 + 𝓫𝑻 𝒙 + 𝒸
Where 𝒙𝑻 = [𝑇𝐴 𝑇𝐵 ]Since 𝑆 𝑇𝐴 𝑇𝐵 is concave, its contours are ellipses. Directions of ellipses’ axes are same127

with directions of eigenvectors of matrix 𝓐, obtained as:
𝒗𝟏 = [1 1] 𝒗𝟐 = [−1 1]
We then rebuild the coordinate system of contour graph of 𝑆 by setting maximum pointof 𝑆 as new origin and the eigenvectors as new axes (Fig A.5). In this new system, if avector can be written as the linear combination of unit vectors with coefficients that havesame signs with the coordinates of its quadrant, it is a descending direction. Therefore,
we can intuitively find that 𝑆 𝐼𝐼 < 𝑆 𝐼 < 𝑆 𝑉𝐼 and 𝑆 𝐼𝑉 < 𝑆 𝐼𝐼𝐼 < 𝑆 𝐼 = 𝑆 𝑉 from Fig A.5.

𝑇𝐵

𝑻𝑰𝑽
𝑻𝑰𝑰

𝑻𝑰𝑰𝑰

𝑻𝑰 𝑻𝑽
𝑽𝑰

𝑻

𝑇𝐴

Fig A.5 Contour graph of 𝑺

E3. Numerical example of the comparison between 𝑆 𝐼𝐼 and 𝑆 𝐼𝐼𝐼 in Case 2

128

Fig A.6 Graph of 𝑺𝑰𝑰 − 𝑺𝑰𝑰𝑰 in Case 2

Appendix F Results of Section 3.3.1
F1. Important values of Fig. 5
𝑇̂𝐵𝐻𝐼𝐼 = 𝑐 −𝐶 𝐶 𝑛𝐴 𝑛𝐴 𝑛𝐵 + 2 + 2𝑛𝐵 + 3 + 𝑘 𝑛𝐴 + 1 3𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 2𝑛𝐵 𝛽 + 𝛼𝑠
𝐶 𝐶 𝑛𝐴 𝑘 6𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 5𝑛𝐵 𝛽 + 𝛼𝑠 + 2𝐶 𝑛𝐵 + 𝐶 𝑛𝐴 𝑛𝐵 𝐶 − 3𝑘 𝛽 + 𝛼𝑠
1

+ 2𝐶 3 𝑛𝐴 ++ 𝐶 𝐶 𝑛𝐴 𝑛𝐴 𝑛𝐵 + 2 + 2𝑛𝐵 + 3 𝑠𝑟𝐴 − 𝛾 + 𝑠 + 𝑘 𝑛𝐴 + 1 𝛽 + 𝛼𝑠 −𝛾𝑛𝐴 𝑛𝐵 +1 + 𝑠𝑛𝐴 𝑟𝐴 + 3 𝑛𝐵 + 𝑟𝐴 + 1 + 2𝑠𝑛𝐵
1+

+ 𝐶 𝐶 𝑛𝐴 2𝐶 𝑛𝐵 𝑠 𝑛𝐴 𝑟𝐵 + 𝑟𝐵 − 1 − 𝛾 𝑛𝐴 ++ 𝑘 𝛽 + 𝛼𝑠 𝛾 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 𝑛𝐵 + 𝑠 𝑟𝐴 + 5 −𝑛𝐵 − 𝑛𝐴 2 𝑟𝐴 + 3 𝑛𝐵 + 𝑟𝐴 + 1

𝐶 𝑛𝐴 𝑛𝐵 𝐶 𝑠 −𝑟𝐴 − 1 + 𝛾 + 𝑘 𝛽 + 𝛼𝑠 𝑠 𝑟𝐴 + 3 − 𝛾
1 − 𝛾 𝑛𝐵 + 1
𝛼𝑠/ 2𝑛𝐵 𝐶 𝑘 𝑛𝐴 + 1− 2𝐶 3 𝑛𝐴 + 1+

𝑠 𝑛𝐵 𝑟𝐵 + 𝑟𝐵 +𝛽 + 𝛼𝑠 − 𝐶 𝐶 𝑛𝐴 𝐶 𝑛𝐴 + 2 + 2𝑘 𝑛𝐴 + 1 𝛽 ++ 𝐶 𝑘𝑛𝐴 𝛽 + 𝛼𝑠 + 𝐶 3 𝑛𝐴 + 1

𝐹. 1

The intercept of 𝑇𝐵𝐻𝐼𝐼 = 𝑇𝐵𝐻𝐼 axis 𝑟𝐵 is
𝐼 =−𝐶 𝑛𝐴 + 𝐶 − 𝐶 𝑛𝐴 𝑠 − 𝑐 − 𝛾
𝐶 𝑠 𝑛𝐴 + 1

𝐹. 2

The intercept of 𝑇𝐵𝐻𝐼 − 𝑇̂𝐵𝐻𝐼 = 𝑇̂𝐵𝐻𝐼 − 𝑇𝐵𝐻𝐼 at axis 𝑟𝐵 is:𝐼 =𝑐 + 𝛾 − 𝑠 𝐶 3 𝑛𝐴 + 1 −𝐶 𝑛𝐴 7𝑛𝐴 𝑛𝐵 + 12𝑛𝐴 + 13𝑛𝐵 + 18 − 𝑘 𝑛𝐴 + 1 11𝑛𝐴 𝑛𝐵 +6𝑛𝐴 + 5𝑛𝐵 𝛽 + 𝛼𝑠
𝛼𝑠

− 𝐶 𝐶 𝑛𝐴 𝑛𝐴 + 1 𝐶 7𝑛𝐴 + 2 𝑛𝐵 − 3𝑘 7𝑛𝐴 𝑛𝐵 + 2𝑛𝐴 + 5𝑛𝐵 𝛽 ++ 𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝐶 7𝑛𝐴 + 8 − 9𝑘 𝑛𝐴 + 1 𝛽 + 𝛼𝑠

1 3 7𝑛𝐵 + 12

− 𝐶 3 𝑘𝑛𝐴3 𝑛𝐵 𝛽 + 𝛼𝑠 + 𝐶 4 𝑛𝐴 +/ 𝐶 𝑠 𝑛𝐴 + 1 −5𝐶 𝑘 𝑛𝐴 + 1 𝑛𝐵 𝛽 + 𝛼𝑠 + 𝐶 𝐶 𝑛𝐴 𝑛𝐵 10𝑘 𝑛𝐴 + 1 𝛽 +𝛼𝑠 − 𝐶 7𝑛𝐴 + 2− 5𝐶 𝑘𝑛𝐴 𝑛𝐵 𝛽 + 𝛼𝑠 + 𝐶 3 𝑛𝐴 + 1

129

7𝑛𝐵 + 12

𝐹. 3

The intercept of 𝑇𝐵𝐻𝐼 = 𝑇̂𝐵𝐻𝐼𝐼 at axis 𝑟𝐵 is:𝐼3 = −

𝑐 + 𝛾 − 𝑠 𝐶 −𝐶 𝑛𝐴 𝑛𝐴 + 2 𝑛𝐵 + 2𝑛𝐴 + 3 − 𝑘 𝑛𝐴 + 1 2𝑛𝐴 𝑛𝐵 + 𝑛𝐴 +𝑛𝐵 𝛽 + 𝛼𝑠
𝛼𝑠+ 𝐶 𝐶 𝑛𝐴 𝑘 4𝑛𝐴 𝑛𝐵 + 𝑛𝐴 + 3𝑛𝐵 𝛽 + 𝛼𝑠 − 𝐶 𝑛𝐴 𝑛𝐵 + 𝐶 𝑛𝐴 𝑛𝐵 𝐶 − 2𝑘 𝛽 ++ 𝐶 3 𝑛𝐴 + 1

1 𝛽 + 𝛼𝑠𝑛𝐵 + 2/ 𝑠 𝐶 𝑘 𝑛𝐴 + 1 𝑛𝐵 𝛽 + 𝛼𝑠 + 𝐶 𝐶 𝑛𝐴 𝑛𝐵 𝐶 𝑛𝐴 − 2𝑘 𝑛𝐴 ++ 𝐶 𝑘𝑛𝐴 𝑛𝐵 𝛽 + 𝛼𝑠 + 𝐶 3 − 𝑛𝐴 + 1

𝑛𝐵 + 2

𝐹. 4

By judging their differences, we can confirm that 𝐼 < 𝐼 < 𝐼3 . We omit the detailed
process as it’s arduous.
F2. Numerical example of social surplus comparison
Parameters: in addition to these used previously, we set 𝑙 = 0.8, 𝑛𝐴 = 4 𝑛𝐵 = 8.

𝑆 𝐻𝐼𝐼 − 𝑆 𝐻𝐼

Fig A.7 Numerical example of social surplus comparison in Extension 1

130

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