Hub congestion pricing: Discriminatory passenger charges

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Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Economics of Transportation journal homepage: www.elsevier.com/locate/ecotra

Hub congestion pricing: Discriminatory passenger charges Ming Hsin Lin a,n, Anming Zhang b a b

Faculty of Economics, Osaka University of Economics, Japan Sauder School of Business, University of British Columbia, Canada

art ic l e i nf o

a b s t r a c t

Article history: Received 17 June 2015 Received in revised form 14 February 2016 Accepted 24 February 2016

This paper investigates airport determination of per-ﬂight and per-passenger charges in a hub-spoke network. The hub airport is congestible and it levies a per-ﬂight charge on its carriers and discriminatory per-passenger charges on the local and connecting passengers. Our main results are: (i) the socially optimal per-passenger charges should take the higher congestion contribution by connecting passengers into account, leading to a higher charge on a connecting passenger than on a local passenger; (ii) generally, the social optimum cannot be achieved when the hub only levies a per-ﬂight charge on carriers; (iii) the optimal per-connecting passenger charge should be lower (higher, respectively) than the perlocal passenger charge when the per-ﬂight charge is large (small, respectively); and (iv) a proﬁtmaximizing hub can impose lower per-connecting passenger charges as compared to per-local passenger charges, owing to its market power, and this possibility is further strengthened by economies of trafﬁc density. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Hub-spoke network Hub congestion Airport pricing Per-ﬂight and per-passenger charges Airport privatization Economies of trafﬁc density

1. Introduction The worldwide airline deregulation/liberalization has led to a number of strategic actions being taken by airlines, including the formation of hub-spoke networks. As a result, a large number of passengers need to transit/transfer at a hub airport in order to reach their ﬁnal destinations.1 For example, the proportion of transfer passengers is more than 50% at a number of hub airports in the United States (Table 1A). The presence of transfer passengers at the major (hub) airports of other countries is also signiﬁcant (Table 1B). As carriers move from the point-to-point to hub-spoke networks, both the connecting trafﬁc and total trafﬁc have risen to the extent that the high trafﬁc volume (relative to runway capacity) has caused congestion and delays at many hub airports.2 According to the On-time Performance Report by FlightStats, the average on-time departure performance among the topn

Corresponding author. Fax: þ 81 6 6328 1825. E-mail address: [email protected] (M.H. Lin). 1 Usually, “transit” passengers refer to the passengers who arrive at, and depart from, the hub on the same ﬂight, whilst “transfer” passengers refer to those who need to change to another ﬂight at the hub. Both the transit and transfer passengers use the hub runways twice, one for landing and the other for take-off, and may be referred to as “connecting trafﬁc.” We shall use the three words (transit, transfer, connecting) interchangeably in this paper. 2 As demonstrated by Zhang (2010), airport capacity required under a hubspoke network would be more than twice as large as the capacity required under a point-to-point network.

35 international airports was 69.3% in July 2013.3 For the same year, the average on-time departure performance of the 29 major U.S. airports was 76.6% (U.S. Department of Transportation): The best performer, Salt Lake City airport, was 86.7%, while the worst performer, Chicago Midway, was 66.6%. In particular, Chicago O’Hare, a hub for both American Airlines and United Airlines, was 70.4%; in other words, 29.6% of the ﬂights were delayed. What can be done about runway congestion and delays? An “obvious” solution is to add more runway capacity, which is lumpy and time-consuming and involves large expenditures. Economists have, on the other hand, advocated the use of price mechanisms to balance the demand with the limited capacity, with early analyses by, e.g., Levine (1969), Carlin and Park (1970) and Borins (1978). These early pricing models were, understandably, developed along a line similarly to dealing with road congestion. As such, ﬂights were treated as “atomistic” (like individual drivers in the road case). The recent literature (e.g., Daniel, 1995; Brueckner, 2002) has incorporated the fact that a congested airport is usually dominated by only a few carriers, each of which runs a large number of ﬂights at the airport and has market power. The main insight is that congestion pricing has a partial place at an airport when carriers have market power, since carriers themselves will internalize congestion. Following this recent literature, the present 3 Typically, a ﬂight is considered as on-time when the actual departure time is within 15 min of the scheduled departure time. The on-time arrival data are similar to the on-time departure data.

http://dx.doi.org/10.1016/j.ecotra.2016.02.001 2212-0122/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Lin, M.H., Zhang, A., Hub congestion pricing: Discriminatory passenger charges. Economics of Transportation (2016), http://dx.doi.org/10.1016/j.ecotra.2016.02.001i

M.H. Lin, A. Zhang / Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

Table 1A The proportion of transfer passengers at U.S. airports (at least 50%, 2008). Source: U.S. DOT, Databank 1B, 2008. Airport code

Airport name

Percentage

CVG CLT MEM ATL DFW IAH MSP SLC ORD DTW

Cincinnati/Northern Kentucky Int'l Airport Charlotte Douglas Int'l Airport Memphis Int'l Airport Hartsﬁeld–Jackson Atlanta Int'l Airport Dallas/Fort Worth Int'l Airport George Bush Intercontinental Airport Minneapolis–Saint Paul Int'l Airport Salt Lake City Int'l Airport Chicago O'Hare Int'l Airport Detroit Metro Airport

0.732 0.724 0.659 0.636 0.559 0.544 0.517 0.514 0.503 0.500

Table 1B The proportion of transfer passengers at major non-U.S. airports (in descending order). Airport code

Airport name

Percentage (Year)

Data sources

FRA CDG DXB AMS MUC LHR SIN HKG CPH ICN NRT

Frankfurt Paris Charles de Gaulle Dubai Int'l Amsterdam Schiphol Munich, Franz Josef Strauss Int'l London heathrow Singapore changi Int'l Hong Kong Int'l Copenhagen Kastrup Incheon Int'l Narita Int'l

0.54 (2007) 0.52 (2011) Approximately 0.52 (current) 0.419 (2013) 0.39 (2012, 2013) 0.37 (2012, 2013) 0.30 (2013) 0.26 (2013) 0.208 (2014) 0.19 (2013) 0.185 (2013)

Civil aviation authority, Nov. 2008 Global business with reuters, March 29, 2012 Dubai airport, ofﬁcial report Schiphol group annual report 2013 Munich airport, annual trafﬁc report 2013 CAA 2013 air passenger survey Changi airport, ofﬁcial website ICF report 2013 Airport region mediation competence center http://www.ﬂightglobal.com/ Narita Int'l airport, ofﬁcial website

Copenhagen Kastrup (majority private), effective from October 1, 2009 to March 31, 2015 Passenger Service Passenger Security SerCharge (PSC) vice Charge (PSSC) Domestic departing DKK 28.81 DKK 32.43 passengers Transfer to domestic airport DKK 23.81 DKK 21.41 Int'l Departing passengers DKK 103.75 DKK 32.43 Transfer to int'l airport DKK 41.65 DKK 21.41 Source: CHARGES REGULATIONS applying to Copenhagen, Approved by SLV.

examine airport pricing in a hub-spoke network, which allows us to treat the connecting passengers differently from the “local passengers” who ﬂy between the hub and local airports.4 Our analysis is based on the observation that in the airport pricing practice, major (hub) airports impose both a ﬂight-based charge (e.g., a take-off and landing fee and parking charges) and a per-passenger charge. For the passenger charges, an interesting fact is that while U.S. major airports charge a uniform PFC (passenger facility charge) per passenger (Zhang, 2012, Ch. 13, Table 13.4), a number of hub airports in Canada (Toronto, Vancouver), Europe and Asia impose discriminatory PFCs on local passengers and connecting passengers (Tables 2A and 2B). In particular, they charge a lower PFC for connecting (transit, transfer) passengers, with some airports even waiving such fees entirely (e.g., Dubai and Hong Kong). In addition, Copenhagen and Singapore's Changi airports impose a lower fee on connecting passengers not only for PFCs but also for security charges, while Dubai waives the security charges on transit/transfer passengers. This discriminatory-charging strategy seems feasible and reasonable because: (i) a hub airport generally has a large number of connecting passengers as shown above; (ii) it is easy for a hub airport to distinguish the local and connecting passengers; (iii) a lower charge on connecting passengers attracts more such passengers to ﬂy through the hub, so as to gain from the economies of agglomeration and concession revenues, the so-called “concessions effect” in the literature (e.g., Zhang and Czerny, 2012); and (iv) the hub airport’s marginal cost for serving a connecting passenger may be smaller than that for a local passenger. For instance,

paper examines airport pricing for a congestible hub, taking carrier market power into account. As to be seen below, a major difference between our paper and existing studies is that we will

4 Airport congestion has also been examined in, among others, Morrison (1983, 1987), Morrison and Winston (1989), Oum and Zhang (1990), Fan (2003), Daniel (2011), Vaze and Barnhart (2012a, 2012b), Yan and Winston (2014), and Jacquillat and Odoni (2014). For recent surveys of the literature see, e.g., Basso and Zhang (2007), Barnhart et al. (2012), and Zhang and Czerny (2012); and for studies that are more closely related the present paper, see the discussion below.

Table 2A Airport passenger charges (per passenger) at selected European airports. London heathrow (majority private) Departing passengers Final proposed Proposed 2012/13 2013/14 d GBP d GBP Europe – destination 24.55 28.30 Other – destination 34.49 39.75 Europe – transfer/transit 18.41 21.23 Other – transfer 25.87 29.82 Source: Consultation Document Prepared by Heathrow Airport Limited, Chapter 7 – Proposed Airport Charges Tariffs for 2013/14. Date: October 26, 2012 Munich (multi-level government owned), effective Domestic ﬂight For local boarding For transfer and transit European ﬂight [EU] incl. For local boarding Iceland, Liechtenstein, NorFor transfer and way, Switzerland transit Int’l ﬂights[Non-EU] For local boarding For transfer and transit Source: Munich Airport, Tariff regulation, Part 1

from January 1, 2015 17.99 EUR 15.11 EUR 17.99 EUR 15.11 EUR 18.89 EUR 15.57 EUR

Please cite this article as: Lin, M.H., Zhang, A., Hub congestion pricing: Discriminatory passenger charges. Economics of Transportation (2016), http://dx.doi.org/10.1016/j.ecotra.2016.02.001i

M.H. Lin, A. Zhang / Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 2B Airport passenger charge (per passenger) at selected Asian airports. Narita Int'l Airport (authority/public corporation), effective from December 10, 2014 Passengers/terminals Passenger Service Passenger Security Charge (PSC) Service Charge (PSSC) Departing passengers at ¥2090 (Adult) ¥520 Terminal 1, 2 ¥1050 (Children) Departing Transfer and ¥1050 (Adult) ¥520 Transit Passengers at ¥520 (Children) Terminal 1, 2 Departing passengers at ¥1020 (Adult) ¥520 Terminal 3 ¥510 (Children) ¥520 Departing transfer and tran- ¥510 (Adult) sit passengers at Terminal ¥250 (Children) 3 Incheon Int'l Airport (public corporation), Current Int’l departing passengers KRW 28000 Int’l transfer passengers KRW 10,000 Hong Kong Int'l Airport (authority) Air Passenger Departure Tax (APDT), current Departing passengers HK$120.00 Departing Transit Passengers Exemption

Passenger security charge, from June 1, 2014 HK$45.00 HK$45.00

Dubai Int'l Airport (government owned), effective from October 26, 2014 Passenger Service Passenger Security & Charges (PSC) Safety Fee (PSSF) Departing passengers AED 75.00 AED 5.00 Transit/Transfer Passengers Exemption Exemption Source: Ofﬁcial Website of Narita, Incheon, Hong Kong and Dubai Int’l airports Changi Int'l Airport (government owned), effective from April 1, 2013 Passengers/Terminals Passenger Service Passenger Security Charge (PSC) Service Charge (PSSC) Departing passengers at S$19.90 S$8.00 Terminal 1, 2, and 3 Departing Transfer and S$9.00 S$3.00 Transit Passengers Source: List of Fees and Charges Applicable at Changi Airport (updated as at October 14, 2014) by Changi Airport Group.

Graham (2014, Ch. 4), argued that the airport cost of serving a connecting passenger may be lower than that of serving a local passenger, since connecting passengers have no surface access/ egress requirements and associated meeters/greeters, and very often do not need check-in, security and immigration facilities either. On the other hand, serving connecting passengers might be more costly in some aspects as they require baggage handling and special facilities for a rapid transfer at the hub (Graham, 2014, Ch. 4). In this paper, we abstract away the concessions effect and further assume that the two types of passenger impose the same airport costs.5 An interesting question then is: Is it socially desirable to charge connecting passengers lower than local passengers? The question is policy relevant, and seems natural since we observe that connecting passengers must use the hub runway twice – one landing and one take-off – for their travel whilst local passengers use the runway just once. As a consequence, efﬁcient infrastructure pricing should reﬂect this differential in the users' contributions to congestion. To investigate this question more formally, we consider a simple hub-spoke network, and model the behavior of the hub airport and carriers as a two-stage game: First, the hub airport sets its charges to maximize social welfare (airport proﬁt, respectively) if it is a public airport (a private airport, 5 We shall discuss the issue of positive effects of hubbing, including the concessions effect and the schedule-delay reduction, in the concluding remarks.

3

respectively). In the second stage, each hub carrier chooses output to maximize its proﬁt under a “ﬁxed proportions” assumption. We ﬁrst consider hub pricing with a pure per-passenger charge, and ﬁnd that the socially optimal charges should levy double delay costs on the connecting passengers, leading in general to a higher per-connecting passenger charge than a per-local passenger charge. The ﬁnding is intuitive because a connecting passenger contributes higher runway congestion at the hub than a local passenger. Second, we ﬁnd that levying discriminatory perpassenger charges can lead to the social optimum, but in general, levying per-ﬂight charge only cannot achieve the ﬁrst-best outcome. The two ﬁndings are particularly interesting given that we make a “ﬁxed proportions” assumption so that normally per-ﬂight and per-passenger charges are equivalent in a single-airport case. But once the hub-spoke network aspect is introduced, they are not equivalent anymore. Speciﬁcally, the discriminatory per-local and per-connecting passenger charges (two variables) can correct the hub congestion and correct the distortions caused by carriers' market power, which emerge in the local and connecting markets, and hence, this per-passenger charging scheme can lead to the social optimum. However, although the per-ﬂight charge (one variable) can correct the hub congestion, too, it cannot appropriately correct the local and connecting market inefﬁciencies since it is aircraft-based and is levied on the carriers' ﬂight nondiscriminately with respect to passenger types, and hence, the ﬁrst-best outcome cannot achieve under the per-ﬂight charging scheme. Third, we consider combined per-ﬂight and per-passenger charges and ﬁnd that at the social optimum, the per-connecting passenger charge should be lower (higher, respectively) than the per-local passenger charge when the per-ﬂight charge is large (small, respectively). This ﬁnding may explain why major hub airports charge connecting passengers lower than local passengers, such as those shown in Table 2. This may occur because these airports currently levy a relatively high proportion of their total charges on per-ﬂight charges, and thus the connecting passengers who use two ﬂights have to pay (via the per-ﬂight charges) more than the local passengers who use one ﬂight only. However, as a current trend indicated by IATA (2010), if a hub airport shifts away from per-ﬂight related charges towards per-passenger related charges which leads to a low (or even zero) aircraft-based charge, then it should charge its connecting passengers higher than its local passengers. Fourth, by including the analysis for a proﬁt-maximizing hub airport and assuming a pure per-passenger charging scheme, we ﬁnd that a welfare-maximizing hub charges a connecting passenger higher than a local passenger, but a proﬁt-maximizing hub can impose lower per-connecting passenger charges (as compared to per-local passenger charges) and, very interestingly, this possibility is further strengthened by economies of trafﬁc density which is an important aspect of a hub-spoke network. This result provides another potential explanation for why some hub airports charge a connecting passenger lower than a local passenger. As discussed, a look at the ownership (or the proﬁt orientation) of the airports in Tables 2A and 2B and their charging practice suggests some consistency with the result. Previous papers by, e.g., Oum et al. (1996), Brueckner (2005), Czerny (2010), Flores-Fillol (2010), and Lin (2013), have studied airport pricing under a hub-spoke network. Highlighting the feature of per-ﬂight and per-passenger charges, the recent studies by Silva and Verhoef (2013) and Czerny and Zhang (2015) examined the issue in a single airport setting, rather than in a hub-spoke network. In particular, Silva and Verhoef showed that an airport requires both per-passenger and per-ﬂight charges to achieve the ﬁrst-best outcome: Congestion externalities need to be addressed

Please cite this article as: Lin, M.H., Zhang, A., Hub congestion pricing: Discriminatory passenger charges. Economics of Transportation (2016), http://dx.doi.org/10.1016/j.ecotra.2016.02.001i

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4

Fig. 1. Airports and airlines in a simple hub-spoke network.

through per-ﬂight tolls whereas the inefﬁciency caused by airlines' market power must be corrected with per-passenger subsidies. None of existing papers have, to our knowledge, paid attention to the discriminatory passenger charges of hub airports on congestion pricing and derived results in a (predominantly) analytical fashion.6 The remainder of the paper is organized as follows: Section 2 presents the model and basic analysis. Section 3 examines socially optimal airport charges for the sub-cases of: (i) pure per-passenger charges, (ii) pure per-ﬂight charge, and (iii) combined perpassenger and per-ﬂight charges. Section 4 studies the pure perpassenger charges of a proﬁt-maximizing hub and compares them with those of a welfare-maximizing hub. Section 5 solves the closed-form solutions of our model for the linear demands, quadratic costs with economies of trafﬁc density and linear/ quadratic delay functions. Finally, Section 6 contains the concluding remark.

2. Basic analysis 2.1. Model setup We consider a hub-spoke network that is likely the simplest structure in which our questions can be addressed. As depicted in Fig. 1, there are three cities, with a hub airport H linking two local airports A and B. Individuals living in each city wish to travel to the other cities.7 Passengers traveling on spoke routes AH and BH (“AH and BH passengers”) use a carrier’s nonstop service, whereas passengers on the rim route, AB, (“AB passengers”) use the onestop service connected at H. There are n symmetric carriers at H providing service in this hub-spoke network. A representative carrier, airline i (i ¼ 1; 2; ::; n), operates two direct ﬂights on routes AH and BH at frequencies f iA and f iB , respectively, to serve the three city-pair markets. Following the usual practice in the airport-pricing literature we make a “ﬁxed proportions” assumption: the number of passengers per 6 Similar insights arise in a recent discussion paper by Van der Weijde (2014) in the context of short- vs. long-haul trafﬁc in a rail network. Since long-haul trafﬁc use local links twice as compared to local short-haul trafﬁc, discriminatory charges should be imposed. In addition, an anonymous referee brought a very relevant paper by Silva et al. (2014) to our attention. Silva et al. study welfare-maximizing airport pricing in the context of networks (including hub-spoke). They have discriminatory charges for connecting and non-connecting passengers, and their optimal congestion tolls also include a double delay term for connecting passengers, although through per-ﬂight tolls. As they deal with a very general and complicated setting, they have to rely on numerical simulations for much of the analysis. In comparison, we study the behavior of both a welfare-maximizing airport and a proﬁt-maximizing airport, and we characterize the optimal airport charges under quite general forms of the demand, airline cost and delay functions. We will discuss Silva et al. (2014) in several parts of the text. 7 Passengers will also travel back to their original cities. For simplicity (but without loss of generality) our modeling considers just one-way trips. This simple network structure has been used in other papers; for example, Oum et al. (1995) and Zhang (1996) use it to study the strategic effects of hub-spoke network.

ﬂight, namely, s ¼aircraft size load factor, is constant and the same across carriers, with load factor being further taken as 100% 8 (for simplicity). As a consequence, the relations qiA þ qio ¼ sf iA and qiB þ qio ¼ sf iB hold, with subscripts A, B and o denoting markets AH, BH and AB respectively.9 The aggregate demand in P P P each market is deﬁned by Q A ≔ ni qiA , Q B ≔ ni qiB and Q o ≔ ni qio , and the aggregate Pn frequency of a spoke route is deﬁned by F A ≔ P n i f iA and F B ≔ i f iB . Our model of airline and airport behavior is based on a two-stage game: ﬁrst, the hub sets airport charges to maximize social welfare (airport proﬁt, respectively) when it is a public airport (a private airport, respectively). In the second stage carriers are assumed, as common in the recent airport-pricing literature, to engage in output competition in the three markets (AH, BH, and AB). Two types of airport charge are considered: a per-passenger charge and a per-ﬂight charge. The former is passenger-speciﬁc: It is denoted as u if the passenger in concern is a “local” (AH and BH) passenger, while it is denoted as v for a “connecting” (AB) passenger. The per-ﬂight charge, denoted by t, is aircraft-based and is levied on the carriers’ ﬂights nondiscriminately with respect to passenger types. We examine the subgame perfect Nash equilibrium of this two-stage game. Given our main concern is the hub's congestion pricing, we assume that the two spoke airports suffer no congestion and their behavior is exogenous to the present model.10 2.2. Passenger demand functions Oum et al. (1996) we consider an aggregate demand Following Q j ρj in each market, where ρj represents the “full price” faced by passengers ðj ¼ A; B; oÞ. The full prices are given by

ρA ¼ P A þ u þ DðF A þ F B Þ;

ð1Þ

ρB ¼ P B þ u þDðF A þ F B Þ;

ð2Þ

ρo ¼ P o þ v þ 2DðF A þ F B Þ;

ð3Þ

where P j is the airline ticket price in market j (airline services are assumed homogenous). In Eq. (1), DðF A þ F B Þ is the cost of congestion delay occurring at the hub, and this cost depends on the hub’s total trafﬁc ðF A þF B Þ.11 A similar interpretation applies to Eq. (2) for market BH. As for connecting market AB, connecting passengers suffer a congestion-delay cost that is twice the cost incurred by local passengers, since they need two ﬂight movements at the hub (one landing and one subsequent take-off). The speciﬁcation of “2D” in Eq. (3) makes sense because the data for on-time departure performance at congested hub airports are quite similar to those for on-time arrival performance. For congestion-delay cost DðF A þF B Þ we assume D0 4 0 and D00 Z0, i.e., higher frequency (and with the ﬁxed-proportions assumption, higher trafﬁc volume) at the hub will, as expected, increase its congestion, and the effect is (at least) more pronounced when there is more congestion.12 8 See, e.g., Brueckner (2002), Pels and Verhoef (2004), Zhang and Zhang (2006), Basso (2008), Basso and Zhang (2008), and Lin (2013). The ﬁxed-proportions assumption allows us to focus on the main issues to be examined analytically. This assumption also implies that our analysis is perhaps more suited for a short- to medium-run situation where the aircraft ﬂeet is more or less given to a carrier. We discuss the issue further in the concluding remarks. 9 This holds because AB passengers are jointly served with AH and BH passengers using the same aircraft. Given this relation, once airline i's market output is decided, its frequencies on the spoke routes are determined accordingly. 10 For a study that looks at the strategic interaction on airport pricing among a congested hub and two local airports, see Lin (2013). 11 Note that the hub capacity is taken as given in this paper. 12 For simplicity we have ignored the transfer costs for connecting passengers. Due to additional descent and ascent at the hub and to extra cruise time required for the circuitous routing, connecting passengers incur an additional cost as

Please cite this article as: Lin, M.H., Zhang, A., Hub congestion pricing: Discriminatory passenger charges. Economics of Transportation (2016), http://dx.doi.org/10.1016/j.ecotra.2016.02.001i

M.H. Lin, A. Zhang / Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2.3. Airline costs and airport charges The effect of hubbing on airline costs has been well researched in the literature; in particular, a hub-spoke network allows the adopting carrier to better exploit “economies of trafﬁc density” (e.g., Brueckner and Spiller, 1991, Hendricks et al., 1995). The latter refers to that the carrier's average cost on a nonstop route falls as the number of passengers traveling on the route rises.13 To include this hub-spoke network effect on the cost side, we use ciA qiA þ qio to represent airline i's total operating cost of carrying qiA þ qio passengers on spoke route AH, with the cost function satisfying c0iA 4 0 (positive marginal cost) and c00iA o 0. Inequality c00iA o 0 implies a declining marginal cost at the route level, and is employed to capture economies of trafﬁc density in this paper.14 A similar assumption applies to spoke route BH. Following Lin (2013), the aeronautical charges paid to the hub airport by airline i are given by C i ¼ wtf iA þ wtf iB , where t is the hub's per-ﬂight charge, and parameter w is introduced to distinguish two types of the per-ﬂight charge: w ¼ 1 indicates that the airport charges are movement-related and independent of aircraft size, as the setting in Brueckner (2004) and Flores-Fillol (2010); whereas w ¼ s indicates that these charges are weight-related and proportional to s, as the setting in, e.g., Lin (2013). We focus on the 15 case of w ¼ s throughout the paper. Using qiA þ qio ¼ sf iA and qiB þ qio ¼ sf iB , the airline costs can be rewritten as, TC i ¼ t qiA þ qio þ t qiB þ qio þ ciA qiA þ qio þ ciB qiB þ qio : ð4Þ Note that although there are three markets, airline operations, and hence costs, occur only on the two routes, AH and BH.

2.4. Airline competition

practice in models of quantity competition (e.g., Tirole, 1988) we assume that a carrier’s marginal proﬁt declines in another carrier's output, implying that the carriers' outputs are “strategic substitutes” (Bulow et al., 1985). We further assume that the equilibrium is locally, strictly stable (e.g., Zhang and Zhang, 1996).

3. Socially optimal airport charges Social welfare in our three-market network is deﬁned as: "Z # "Z # Q Qo W ¼2 ρ ξ dξ ρQ þ ρo ξ dξ ρo Q o 0

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ðiÞ

The ﬁrst-order conditions of airline i’s proﬁt maximization are ∂π i ¼ 2 ρ0 qi þ ρ D D0 qi þ qio ðt þ uÞ c0i ¼ 0; 8 i; ∂qi

ð6Þ

∂π i ¼ ρ0o qio þ ρo 2D 2D0 qi þqio ð2t þvÞ 2c0i ¼ 0; 8 i: ∂qio

ð7Þ

The marginal airport charge to a local passenger is ðt þ uÞ in total, and that to a connecting passenger, who uses both the ﬂights on AH and BH routes, is ð2t þ vÞ. The second-order conditions are assumed to hold for the entire range relevant to our analysis, and are given in Appendix A. Furthermore, following the standard (footnote continued) compared to local passengers. If such transfer costs were added and modeled as a ﬁxed term (as in Oum et al. (1995) and Silva et al. (2014)), our main results would still hold. 13 Caves et al. (1984) and Brueckner and Spiller (1994) have empirically shown the presence of economies of trafﬁc density in the US market. The density economies are one of main drivers for the emergence of hub-spoke networks. 14 Using a declining marginal cost to capture economies of density has been considered in a number of studies including Brueckner and Spiller (1991) and Zhang (1996). Other studies, e.g. Hendricks et al. (1995) and Silva et al. (2014), capture the concept by using a declining average cost. 15 We note that our results also hold in the case of w ¼ 1 and so are more general than the case of w ¼ s analyzed in the paper.

0

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ðiiÞ

þ2 ρ u D Q þ ρo v 2D Q o |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ðiiiÞ

ðivÞ

Xn

2 c 2t ðQ þQ o Þ þ ½2ðt þ uÞQ þ ð2t þ vÞQ o : i¼1 i |ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ ﬄ} |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ðviÞ

ðvÞ

ð8Þ

ðviiÞ

In Eq. (8), the noted terms (i) and (ii) denote consumer surplus of the two local markets and the connecting market, respectively. Terms (iii) and (iv) are the total airline revenues from the two local markets and the connecting market, respectively, and term (v) is the total airline costs. Term (vi) is the total aircraft-based charge paid by carriers to the hub airport, and term (vii) is the hub’s total revenue (proﬁt) from the airlines and passengers. Note that zero costs are assumed for the airport.16 The function, W, can be simpliﬁed to Z Qo Z Q ρ ξ dξ þ ρo ξ dξ 2DðQ þ Q o Þ 2nc: W ¼2 0

We start the analysis with airline competition in the second stage of the game. Here each carrier chooses its quantities qiA , qiB and qio to maximize proﬁt, given the quantities of the other carriers. For simplicity, we assume that the two spoke markets are symmetric. Let qiA ¼ qiB ≔qi , ciA ¼ ciB ≔ci Q A ¼ Q B ≔Q , F A ¼ F B ≔F, P A ¼ P B ≔P, and ρA ¼ ρB ≔ρ. By these symmetries, the proﬁt function of airline i can be simpliﬁed into π i ¼ 2 ρ D u qi þ ρo 2D v qio 2t qi þ qio 2ci : ð5Þ

5

0

That is, welfare is the sum of the passengers' gross beneﬁts, net of the congestion delay cost and total airline costs (with ci ¼ c). 3.1. Per-passenger charges only (t ¼ 0; u; v Z 0): ﬁrst-best outcome Given that the International Air Transport Association (IATA) appears to propose to move away from the current predominantly per-ﬂight charges towards the per-passenger chargers,17 this subsection focuses on the case of discriminatory per-passenger charges by assuming zero per-ﬂight charges (i.e., t ¼ 0). The pure per-ﬂight charge will be investigated in Section 3.2, whereas the combined per-ﬂight and per-passenger charges will be discussed in Section 3.3. The ﬁrst-order conditions for the optimal per-passenger charges are

∂W ∂W ∂Q ∂W ∂Q o ¼ 2c0 þ ¼ 0; x ¼ u; v: ð9Þ 2c0 ∂x ∂Q ∂x ∂Q o ∂x From the airlines' ﬁrst-order conditions Eqs. (6) and (7), with t ¼ 0, we have 1 P ¼ ρ0 q þ D0 ðQ þQ o Þ þ u þc0 ; n

ð10Þ

2 P o ¼ ρ0o qo þ D0 ðQ þQ o Þ þ v þ 2c0 : n

ð11Þ

16 This assumption is for simplicity only: for non-zero airport costs, our results continue to hold as long as the marginal costs are constant. 17 For example, this may be reﬂected from its statement: ‘‘many airport facilities are built and maintained for the beneﬁt of airline passengers. It is in the interest of both the airport and the airlines to recover these costs through passenger based charges instead of other aeronautical based charges.’’ (IATA, 2010). See Czerny and Zhang (2015) for further discussion.

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6

∂Q o Using the comparative-static effects of ∂Q ∂x and ∂x (shown in Appendix B), and Eqs. (10) and (11) to solve the equation system of Eq. (9), we obtain the optimal pricing rule of the per-local and per-connecting passenger charges as follows:

n1 0 D ðQ þ Q o Þ þ ρ0 q; u ¼ ð12Þ n

n1 0 D ðQ þ Q o Þ þ ρ0o qo : v ¼ 2 n

ð13Þ

dρ dQ o dρo Let ε≔ dQ Q = ρ and εo ≔ Q o = ρo denote the (positive) full-price elasticities of demands for the local and connecting markets, ρ respectively. We have ρ0 q ¼ 1n ρε and ρ0o qo ¼ 1n εoo . Then, Eqs. (12) and (13) can be rewritten, respectively, as

1 0 1ρ u ¼ 1 D ðQ þ Q o Þ ; ð14Þ n nε |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄ{zﬄﬄﬄ} congestion delay cost

airline markup

1 0 1 ρo : v ¼ 2 1 D ðQ þ Q o Þ n n εo |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} congestion dealy cost

ð15Þ

airline markup

Each of Eqs. (14) and (15) indicates that the optimal perpassenger charge is equal to the sum of the share of the congestion delay (net of the internalized portion 1n by carriers), minus a rebate term reﬂecting carriers' market power at the hub. As noted in the equations, the ﬁrst terms on the right-hand side (RHS) of Eqs. (14) and (15) reﬂect the distortion from hub congestion externality (which disappears when n ¼ 1), whereas the second terms reﬂect the distortion from carriers’ market power (which disappears when n-1). These two components in optimal charges are consistent with the ﬁndings of Brueckner (2002), Pels and Verhoef (2004), and Zhang and Zhang (2006), each of which focused on the case of a single airport. The new insights here are that, ﬁrst, the existing result remains valid in a hub-spoke setting; and second, absent of carriers’ market power, a welfaremaximizing hub airport charges a connecting passenger twice higher as it charges a local passenger. We have: Proposition 1. Assuming per-passenger charges only,

as P ¼ c0 þ D0 ðQ þ Q o Þ;

ð16Þ

P o ¼ 2c0 þ 2D0 ðQ þ Q o Þ:

ð17Þ

(Eqs. (16) and 17) show that the socially optimal fares should equal the social marginal costs, as expected. The latter are the sum of airline marginal cost and marginal delay cost which depends on the aggregated trafﬁc of both local and connecting passengers. More interestingly, the relation P o ¼ 2P holds: essentially, the connecting passengers use both spoke ﬂights and cause congestion costs twice of those by the local passengers. It is also worth noting that airline marginal cost c0 is included in the ticket prices, but not in the optimal pricing rule shown in Eqs. (14) and (15), implying that the hub-spoke network effect on the cost side (economies of trafﬁc density) has no impact on the optimal congestion pricing rule of a welfare-maximizing hub airport. However, we will show that it plays an important role in the proﬁtmaximizing hub case in Sections 4 and 5. 3.2. Per-ﬂight charges only (t Z 0; u; v ¼ 0): second-best outcome This sub-section investigates the scheme of a per-ﬂight charge imposed on carriers (i.e., let u ¼ v ¼ 0). This is the pricing policy considered by Lin (2013) where per-passenger charges are excluded (and only speciﬁc functions for both the demands and delay costs are employed in his analysis). The ﬁrst-order condition for this optimal charge is

∂W ∂W ∂Q ∂W ∂Q o ¼ 2c0 þ ¼ 0: ð18Þ 2c0 ∂t ∂Q ∂t ∂Q o ∂t The corresponding comparative-static effects are ∂Q n ¼ ð1 þ nÞρ0o þ ρ00o Q o o 0; ∂t Λ |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ≔X o

∂Q o 2n ¼ ð1 þ nÞρ0 þ ρ00 Q o 0: ∂t Λ |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ≔X

i) Absent of carrier market power (n-1), the socially optimal congestion toll on a connecting passenger is twice that on a local passenger, i.e. v ¼ 2u ; and ii) The presence of carriers’ market power (n being ﬁnite) can still lead to v 4 u ; in effect, it is likely that v 4 2u . For part ii) of the proposition, note that with ﬁnite n, the second terms onthe RHS of Eqs. (14) and (15) are both negative, but it ρ is likely that εoo o ρε : it seems natural to consider εo 4 ε, since the connecting passengers are likely to have more alternative options for their travel as compared to the local passengers. As a consequence, with carrier market power (n being ﬁnite) it is likely, by Eqs. (14) and (15), that v 4 2u , provided ρ and ρo are about the same value. Proposition 1 suggests that the current discriminatory per-passenger charges indicated in Table 2 (charging connecting passengers a lower fee than local passengers) may not be socially desirable if the hub is congestible, ignoring other potentially positive effects brought by connecting passengers (such as the schedule-delay reduction and the concessions effect; see the discussion in the concluding remarks). In other words, in order to achieve the ﬁrst-best outcome, the hub airport should charge connecting passengers more (as compared to local passengers) as they contribute more runway congestion to the hub. Furthermore, substituting Eqs. (12) and (13) into (10) and (11) and rearranging the terms, we derive the socially optimal airfares

Using these values and Eqs. (10) and (11) and solving the ﬁrstorder condition Eq. (18) yield the optimal per-ﬂight charge written as:

1 1 X ρo Xo ρ t 2nd ¼ 1 D0 ðQ þ Q o Þ þ ; ð19Þ n n 2X þ X o εo 2X þ X o ε |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} congestion delay cost

weighted average of airline markup

where superscript “2nd*” stands for the sub-optimal outcome. Eq. (19) indicates that the optimal per-ﬂight charge is the marginal external cost of congestion and a weighted average of the airline markups from the local and connecting markets. Comparing Eq. (19) with Eqs. (14) and (15), we see that the optimal perﬂight charge also includes two correction terms: one for correcting the congestion externality and the other for correcting the market inefﬁciency.18 However, the important difference here is that the optimal per-ﬂight charge can only correct the two markets’ inefﬁciencies in a weighted average level since it is aircraft-based and is levied on the carriers’ ﬂight non-discriminately with respect to passenger types. We have the following proposition: 18 By noting that a connecting passenger uses two ﬂights and pays 2t, the magnitude of congestion externality should be corrected by the ﬁrst term in Eq. (19), which is identical with that by Eqs. (14) and (15).

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Proposition 2. The socially optimal outcome cannot be achieved when the hub airport only levies a per-ﬂight charge on carriers, except for the special cases where 2ρε ¼ ρεoo or n-1. Proposition 2 is important because it clearly indicates that the socially optimal charge based on per-ﬂight only (as derived in Lin (2013), Proposition 2) is, in effect, not the socially optimal (ﬁrstbest) outcomes. Essentially, the per-ﬂight charging scheme cannot treat the local and connecting passengers distinctly which is required at the social optimum. In other words, together with Proposition 1, the new insight here is that introducing discriminatory user charges between the local and connecting passengers can achieve the socially optimal outcomes in hub-spoke network markets. To the best of our knowledge, the existing literature does not clarify this property. This is particularly important in a hub-spoke network because although the hub cannot discriminate passengers if it levies an aircraft-based charge on each ﬂight or runway movement (in which both the local and connecting passengers sit in the same aircraft), it can easily discriminate them with a per-passenger charging scheme. 3.3. Combined per-ﬂight and per-passenger charges (t; u; v Z0): ﬁrst-best outcome Suppose that the hub airport can choose a set of airport charges, including a per-ﬂight charge and discriminatory perpassenger charges to maximize welfare. Namely, the hub chooses a set of charge ðt þ uÞ≔t u on local passengers and another set of charge ð2t þ vÞ≔t v on connecting passengers who use both AH and BH ﬂights of an airline.19 Here, although the ﬁxed-proportions assumption indicated earlier may not be the most natural assumption for the topic, this assumption in our model implies substitutability between per-passenger and per-ﬂight charges which enables us to highlight the different roles played by perﬂight and per-passenger charges under a hub-spoke network. The socially optimal set of airport charges can be derived as:

1 1ρ ; ð20Þ ðt þ uÞ ¼ 1 D0 ðQ þ Q o Þ n nε

1 1 ρo : ð2t þvÞ ¼ 2 1 D0 ðQ þ Q o Þ n n εo

ð21Þ

Note that the set of socially optimal charges Eqs. (20) and (21) are the same with Eqs. (14) and (15), implying that the ﬁrst-best outcome can be achieved under the combined per-ﬂight and discriminatory per-passenger charges. Quite interestingly, we see the following relationship holds:

1 1ρ 1 ρo ≔t^ 2v⋛u: t⋚ 1 D0 ðQ þ Q o Þ þ n nε n εo This relationship holds for any t^ ⋛0. We therefore have the following results: Proposition 3. Under the socially optimal pricing rules: If the chosen per-ﬂight charge is small (in the sense that t o t^ ), then the perconnecting passenger charge is higher than the per-local passenger 19 This approach is similar, in spirit, to Lin (2013) who considered an optimal set of airport charges as a combination of charges by the hub and local airports. In our model, t and u (and v) cannot be decided separately because the marginal airport charge of a local passenger is ðt þ uÞ, and hence the effect of an increase in t is proportional to the increase in u. A similar reason applies to the connecting market. We expect that, as a future study, if we extend our analysis to a long-run situation where the hub carriers can simultaneously choose outputs and frequencies, then the per-ﬂight and per-passenger charges could be decided separately, and different results might emerge.

7

charge, i.e. v 4 u. However, if the chosen per-ﬂight charge is large (t 4 t^ ), then v o u. Proposition 3 holds because a connecting passenger uses two ﬂights and pays 2t, and hence if t is sufﬁciently large, v can be small. The result is not that surprising given there are two sources of inefﬁciency (hub congestion, carrier market power) and only two charges are needed to correct the distortions. By using three charges there is an extra degree of freedom.20 The result has meaningful policy implications. Currently, between the per-ﬂight and per-passenger charges most major airports levy a relatively high proportion of their total charges with per-ﬂight charges, and thus their per-passenger pricing strategy (i.e., charging the connecting passengers lower than the local passengers, such as the examples shown in Table 2) seems sensible. However, as mentioned earlier, the current trend appears for a hub airport to shift away from per-ﬂight related charges to per-passenger related charges. Then with a low aircraft-based charge, the hub should charge its connecting passengers higher than its local passengers. Our analysis further suggests that there is no need for immediate regulatory corrections of the current trend towards the strong use of per-passenger based airport charges, a result that is consistent with the one found by Czerny and Zhang (2015).

4. Per-passenger charges only: proﬁt-maximizing hub Over the last two decades, airport privatization has become a worldwide trend (e.g., Zhang and Czerny, 2012; Lin, 2013). Indeed, some of the airport examples in Table 2, such as London Heathrow and Copenhagen Kastrup, are majority privately owned. This section extends our analysis to the case of a proﬁt-maximizing hub airport. This consideration can also be important even for majority public airports (Dubai, Hong Kong, Singapore, etc.): While publicly owned, these airports also emphasize on proﬁt. From Eq. (8), the hub's total revenue (proﬁt) derived from the airlines and passengers is 2ðt þ uÞQ þ ð2t þ vÞQ o . To facilitate the comparison between the public (welfare-maximizing) and private (proﬁt-maximizing) airports we will, for the remainder of this paper, focus on the pure per-passenger charges by letting t ¼ 0. Solving the proﬁt-maximization problem of a private hub yields the “private” optimal charges, which can be written as

1 0 1 ρ upri ¼ c00 q þ qo þ 1 þ D ðQ þ Q o Þ þ 1 þ n |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} n ε |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄ} 40 o0 40

1 Q2 þ D00 ðQ þ Q o Þ2 ρ00 ; n|ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ} n

ð22Þ

40

1 0 1 ρo vpri ¼ 2c00 q þ qo þ 2 1 þ D ðQ þ Q o Þ þ 1 þ n n εo |ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ} o0

2 Q2 þ D00 ðQ þ Q o Þ2 ρ00o o ; n n

ð23Þ

Note that the negativity for c00 in Eqs. (22) and (23) is due to the trafﬁc-density effect associated with a hub-spoke network. The optimal per-local passenger (per-connecting passenger, respectively) charge is positive – i.e., is strictly above the airport’s zero marginal cost – if c00 is small in magnitude (small density effect) and ρ00 (ρ00o , respectively) is non-positive or is small if it is positive. 20 An area of future research is to introduce a non-negative proﬁt constraint into the hub airport, in the spirit of Czerny and Zhang (2015). This would add an important practical aspect to the analysis and also exhaust this extra degree of freedom.

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8

Table 3 Equilibrium outcomes of welfare- and proﬁt-maximizing hubs with duopoly airlines: linear demand and linear congestion delay functions. Carrier i's input demand functions, a Z ao 4 2 and g≔γs. 2θÞ ðao vÞð6g θÞ 3 θÞ 2ða uÞð6g θÞ 6 , qio ðu; vÞ ¼ ðao vÞð3 þ 6g9ð1 . qi ðu; vÞ ¼ ða uÞð3 þ 12g9ð1 þ 6cg θÞ þ 6g θÞ

Welfare-maximizing hub airport u ¼

Per-passenger charges

v ¼

Market output

Proﬁt-maximizing hub airport

2ð 1 þ a þ 6ag 6ð 1 þ ao ÞgÞ þ ð2a ao Þθ , 4 þ 48g 6θ 4 2ao þ 24ð 1 þ aÞg ð2a ao Þθ . 4 þ 48g 6θ

upri ¼ ða 2 1Þ;, vpri ¼ ðao 2 2Þ.

The second-order condition requires ð2 þ 24g Þ4 3θ.

The second-order condition requires ð1 þ 6gÞ 4θ.

4ao gÞ ð2a ao Þθ Q ¼ 2ð 1 þ a þ 28ag , þ 24g 3θ

4ag 2ao gÞ ð2a ao Þθ , Q pri ¼ 3ð 1 þ a þ 9ð1 þ 6g θÞ

Q o

4ag þ 2ao gÞ þ ð2a ao Þθ ¼ 3ð 2 þ ao 9ð1 . Q pri o þ 6g θÞ

þ 4ao gÞ þ ð2a ao Þθ ¼ 2ð 2 þ ao 28ag . þ 24g 3θ 2ð1 þ a þ 10ag 2ð 9 þ ao ÞgÞ ð4a þ ao Þθ P ¼ , 4 þ 48g 6θ þ 8ao gÞ ð2a þ 5ao Þθ . P o ¼ 2ð2 þ ao 4ð 94þþaÞg 48g 6θ

Ticket price

þ 6gÞ ð9 þ 5a þ 2ao Þθ , P pri ¼ 3ð5 þ aÞð118ð1 þ 6g θÞ þ 6gÞ ð18 þ 4a þ 7ao Þθ ¼ 3ð10 þ ao Þð118ð1 . P pri o þ 6g θÞ

Note: all equilibrium outputs and ticket prices are positive for 9 þ aÞg þ 8ao gÞ o θ o 2ð2 þ ao 4ð and g o 6a ao3ao . 2a þ 5ao

When hub carriers are atomistic (i.e., n-1), Eqs. (22) and (23) become

ρ

upri n-1 ¼ c00 q þqo þ D0 ðQ þ Q o Þ þ ; ð24Þ

ε

ρ vpri

n-1 ¼ 2c00 q þ qo þ 2D0 ðQ þ Q o Þ þ o :

ð25Þ

εo

This private optimal pricing rule is consistent with the earlier results by, e.g., Basso (2008) and Basso and Zhang (2008). In particular, the proﬁt-maximizing hub airport captures the full-price ρ markups ρε from the local passengers and εoo from the connecting passengers, owing to its monopoly power. It is also interesting to note that the airline cost term is in general included in the proﬁtmaximizing optimal pricing rule of Eqs. (24) and (25), except the case when c00 ¼ 0 (i.e., no trafﬁc-density effects). This is in contrast to the socially optimal pricing rule of Eqs. (14) and (15), in which the airline cost term does not appear no matter c00 ¼ 0 or not. These results extend those in the single-airport congestion pricing literature (e.g., Zhang and Zhang, 2006) that assumes away trafﬁcdensity effects. Our analysis shows that the hub-spoke network effects (here via economies of trafﬁc density) can play an important role in private-airport pricing. Another important novelty here is the difference between the two per-passenger charges: Comparing Eq. (22) from (23) and Eq. (24) from (25), we have

1 0 vpri upri ¼ c00 q þ qo þ 1 þ D ðQ þQ o Þ n |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ}

1 þ 1þ n

o0

40

!

ρo ρ ðQ þQ o Þ2 Q2 Q 2 þ D00 þ ρ00 ρo 00 o : εo ε n n n |ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

ð26Þ

40

0 ρ ρ vpri j n-1 upri j n-1 ¼ c00 q þ qo þ D ðQ þ Q o Þ þ o : εo ε |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} o0

ð27Þ

1 0 1 ρ ρo v u ¼ 1 D ðQ þ Q o Þ þ ; n |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} n ε εo

ð28Þ

40 0

40

Note: all equilibrium outputs and ticket prices are positive for 3ð2 ao þ 4ag 2ao gÞ þ ao Þð1 þ 6gÞ o θ o 3ð10 2a ao 18 þ 4a þ 7ao

þ ao Þ and g o 3ð2 8a 4ao .

hub case, vpri n-1 o upri jn-1 may hold if ρεoo o ρε and the dif ρ ference is large. As discussed earlier in Section 3.1, εoo o ρε is likely to hold due to εo 4 ε. Now introduce the density effect: As can be seen from (27), the negative c00 q þ qo will make

pri pri jn-1 more likely to hold. The discussion leads to: v n-1 o u

Proposition 4. Consider the per-passenger charges only and no carrier market power (n-1). A welfare-maximizing hub airport imposes higher per-connecting passenger charges as compared to per-local passenger charges. On the other hand, a proﬁt-maximizing hub airport can impose lower per-connecting passenger charges (as compared to per-local passenger charges) and this possibility is strengthened by economies of trafﬁc density. Proposition 4 offers interesting insights. In the public hub case, the correction terms for carrier market power (to the local and connecting passengers) disappear when carriers are atomistic; see Eqs. (28) and (29). In the private hub case, however, these price markup terms remain even when carriers are atomistic (see Eqs. (26) and (27)), because the private hub has a full monopoly power. As a consequence, the difference between ρε and ρεoo may lead to vpri o upri . In addition, a strong density effect further contributes to the rise of inequality vpri o upri . Basically, in a hub-spoke network, a connection passenger has a positive effect on airline costs (i.e., cost reduction), which would put a downward pressure on the proﬁt-maximizing airport’s charge on them (but not on the welfare-maximizing airport as the airline cost term does not enter the hub pricing rules). Proposition 4 thus provides yet another explanation for why some hub airports may levy a connecting passenger lower than it would levy a local passenger. In other words, the hub-spoke network effect on the cost side and the monopoly power of a privatized hub may lead to such a pricing strategy.

40

In comparison, in the public hub-airport case, from Eqs. (14) and (15) we have

v j n-1 u j n-1 ¼ D ðQ þ Q o Þ : |ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ}

2ð2 ao þ 8ag 4ao gÞ 2a ao

ð29Þ

Consider ﬁrst the absence of trafﬁc-density effects (c00 ¼ 0). Comparing Eq. (29) with (27) we see that in the case of a public hub airport, v jn-1 4u jn-1 always hold, whilst in the private

5. Per-passenger charges under speciﬁc functions This section will examine hub pricing (per-passenger charges only, i.e., t ¼ 0) of a public and a private airport under linear demand functions combined with either a linear or quadratic function of congestion-delay costs, and an airline cost function that reﬂects economies of trafﬁc density. Since the previous comparison result (Proposition 4) is only concerned with the case of (n-1), we now considers a duopoly airline market (n ¼ 2). Furthermore, the use of speciﬁc functions allows us to obtain closed-form solutions and further clarify some interesting

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M.H. Lin, A. Zhang / Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

9

Table 4A Equilibrium outcomes of a welfare-maximizing hub with duopoly airlines: linear demand and quadratic congestion delay functions. Carrier i's input demand functions, a Z ao 4 2, and g≔sγ2 . pﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 128g ð 3 þ a þ ao u vÞ þ 3ð 1 þ θÞ2 3 128g ð 3 þ a þ ao u vÞ þ 3ð 1 þ θÞ2 , qio ðu; vÞ ¼ ð 3 64ða uÞg þ 32ðao vÞgÞ þ 3θ þ 288g . qi ðu; vÞ ¼ ð 3 þ 128ða uÞg 64ðao vÞgÞ þ 3θ þ 576g

Per-passenger charges

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 144ð6 þ 4a 5ao Þg ð5 3θÞ 576ð 3 þ a þ ao Þg þ ð2 3θÞ2 , u~ ¼ ð2 3θÞð5 3θÞ2592g 2592g pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 72ð 12 þ 10a þ ao Þg ð5 3θÞ 576ð 3 þ a þ ao Þg þ ð2 3θÞ2 ~v ¼ ð2 3θÞð5 3θÞ þ1296g . 1296g The second-order condition for the welfare-maximization problem: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 16 1152ð 3 þ a þ ao Þg þ 2ð2 3θÞ2 þ ð5 3θÞ 576ð 3 þ a þ ao Þg þ ð2 3θÞ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ3=2 40 for ða þ ao Þ 43 and θ o 5=3. 41 78θ þ 45θ2 þ 2304ð 3 þ a þ ao Þg þ 4ð5 3θÞ

576ð 3 þ a þ ao Þg þ ð2 3θÞ

The second-order condition for carrier i’s proﬁt-maximization problem: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 3 27θ þ 5 41 78θ þ 45θ2 þ 2304ð 3 þ aþ ao Þg þ 4ð5 3θÞ 576ð 3 þ a þ ao Þg þ ð2 3θÞ2 40 2

Market output

3θÞ for 0 o θ r 23; and f or 23 o θ o 53 and g 4 600ðð23 þ a þ ao Þ. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 288ð2a ao Þg 9ð1 θÞ þ 41 78θ þ 45θ2 þ 2304ð 3 þ a þ ao Þg þ 4ð5 3θÞ 576ð 3 þ a þ ao Þg þ ð2 3θÞ2 Q ¼ 4 0, 864g

Note that the numerator of Q is increasing in g, and it is zero when g ¼ 0, thus Q 40. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

144ðao 2aÞg 9ð1 θÞ þ 41 78θ þ 45θ2 þ 2304ð 3 þ a þ ao Þg þ 4ð5 3θÞ 576ð 3 þ a þ ao Þg þ ð2 3θÞ2 θÞ þ 2aθ 4 Z 0 for g r ao ð212ð2a ≔g Q ¼ 0 . Q o ¼ 432g ao Þ2 o pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ 2 2 576ð 3 þ a þ ao Þg þ ð2 3θÞ ð5 3θÞ 3ð1 þ 3θÞ 41 þ 2304ð 3 þ a þ ao Þg 78θ þ 45θ2 þ 4 576ð 3 þ a þ ao Þg þ ð2 3θÞ ð5 3θÞ ao Þg þ 96θ 99θ2 P ¼ 7 þ 576ð12 þ 8a þ , 25184g 10368g 33456g

Ticket price

P 40 for 0 r θ r 24að1þþaaoÞ; and P Z 0 for 24að1þþaaoÞ o θ o 5=3 and g Z ð 1 þ 6θÞð 2 þ ao θ þ að 2 2 þ 4θÞÞ. 8ð 12 8a þ ao Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 576ð 3 þ a þ ao Þg þ ð2 3θÞ2 ð5 3θÞ 3ð1 þ 3θÞ 41 þ 2304ð 3 þ a þ ao Þg 78θ þ 45θ2 þ 4 576ð 3 þ a þ ao Þg þ ð2 3θÞ ð5 3θÞ 7ao Þg þ 96θ 99θ2 P o ¼ 7 288ð 24 þ 2a þ , 22592g 5184g

31728g

ao Þ 2ð2 þ ao Þ ð 1 þ 6θÞð 4 þ 2aθ þ ao ð 2 þ 5θÞÞ P o 40 for 0r θ r 22að2þþ5a . ; and P Z 0 for o θ o 5=3 and g Z ≔g

2 o 2a þ 5ao o P ¼0 4ð24 2a þ 7ao Þ

o

Table 4B Equilibrium outcomes of a proﬁt-maximizing hub with duopoly airlines: linear demand and quadratic congestion delay functions. Per-passenger charges

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ð32ð 3 þ a þ ao Þg þ ð 1 þ θÞ2 Þ , pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pri ð1 θÞ2 þ 8ð 24 þ 2a þ 11ao Þg ð1 θÞ ð32ð 3 þ a þ ao Þg þ ð 1 þ θÞ2 Þ ¼ . v~ 144g 2

u~ pri ¼ ð1 θÞ

þ 16ð 12 þ 10a þ ao Þg ð1 θÞ 288g

The second-order condition for the hub’s proﬁt-maximization problem: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 32 64ð 3 þ a þ ao Þg þ ð1 θÞ 2ð1 θÞ þ 32ð 3 þ a þ ao Þg þ ð 1 þ θÞ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ3=2 40 for ða þ ao Þ 43 and θ o 1. 128ð 3 þ a þ ao Þg þ ð1 θÞ 5ð1 θÞ þ 4

32ð 3 þ a þ ao Þg þ ð 1 þ θÞ

The second-order condition for carrier i’s proﬁt-maximization problem: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ! 40 1 9θ þ 5 128ð 3 þ aþ ao Þg þ ð1 θÞ 5ð1 θÞ þ 4 32ð 3þ a þ ao Þg þ ð 1þ θÞ2

Market output

pﬃﬃﬃ pﬃﬃﬃ ð1 þ θÞð 2 þ 3θÞ 1 1 29 10 5 ; and for 11 29 10 5 o θ o 1 and g 4 100ð for 0 o θ o 11 3 þ a þ ao Þ . qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ Q pri ¼

32ð2a ao Þg 3ð1 θÞ þ

128ð 3 þ a þ ao Þg 5 þ 4

32ð 3 þ a þ ao Þg þ ð 1 þ θÞ 5θ ð 1 þ θÞ

.

288g

Note that the numerator of Q pri is increasing in g, and it is zero when g ¼ 0, thus Q pri 4 0. qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Q pri ¼ o

16ðao 2aÞg 3ð1 θÞ þ

128ð 3 þ a þ ao Þg 5 þ 4

Ticket price 2

aÞg þ 10θ 11θ P pri ¼ 1 þ 192ð5 þ1152g þ2

θÞ þ 2aθ 6 Z 0 for g r ao ð34ð2a ≔g Q pri ¼ 0 . ao Þ2 o qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

32ð 3 þ a þ ao Þg þ ð 1 þ θÞ2 5θ ð 1 þ θÞ

288g

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2ﬃ

32ð 3 þ a þ ao Þg þ ð 1 þ θÞ ð1 θÞ 2576g

128ð 3 þ a þ ao Þg 5 þ 4

32ð 3 þ a þ ao Þg þ ð 1 þ θÞ2 5θ ð 1 þ θÞð1 þ 3θÞ

1152g

pri 5 þ aÞ 3ð5 þ aÞ θð 15 þ ð9 þ 2ao Þθ þ að 3 þ 5θÞÞ P pri 40 for 0 r θ r 9 þ3ð5a ; and P Z 0 for o θ o 1 and g r ≔g P pri ¼ 0 . 2 þ 2ao 9 þ 5a þ 2ao 36ð5 þ aÞ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2

o Þg þ 10θ 11θ ¼ 1 þ 96ð10 þ a576g þ2 P pri o

P pri 40 o

for

ð10 þ ao Þ 0 r θ r 183þ 4a þ 7ao ;

32ð 3 þ a þ ao Þg þ ð 1 þ θÞ2 ð1 θÞ 2288g

and

P pri Z0 o

for

relationships between per-local and per-connecting passenger charges. Similar to Pels and Verhoef (2004) and Lin (2013) we use the linear demands ρj ¼ aj Q j , with aj representing the maximum reservation price (market size). The inverse demand functions Eqs. (1)–(3) can then be expressed as: P A ¼ aA Q A u γ ðF A þ F B Þβ ; P B ¼ aB Q B u γ ðF A þF B Þβ ; P o ¼ ao Q o v 2γ ðF A þ F B Þβ ; where γ ð Z 0Þ is the passengers’ monetary valuation parameter,

128ð 3 þ a þ ao Þg 5 þ 4

3ð10 þ ao Þ 18 þ 4a þ 7ao o θ o 1

32ð 3 þ a þ ao Þg þ ð 1 þ θÞ 5θ ð 1 þ θÞð1 þ 3θÞ 576g

and

,

,

g Z θð 30 þ 2ð9 þ 2aÞθ þ a2o ð 3 þ 7θÞÞ≔g P pri ¼ 0 . 18ð10 þ ao Þ o

and β ¼ 1; 2 is the power of delay function. Under the assumption of symmetric local markets, we let aA ¼ aB ≔a: Following Brueckner and Spiller (1991) we assume that on each spoke route, the mar ginal cost of airline i is linear and given by c0i ¼ 1 θ qi þ qio with θ Z 0. Hence, θ ¼ 0 indicates the absence of trafﬁc density effects, while higher values of θ represent more signiﬁcant density effects on each spoke route. We assume a general situation where the local market size is larger than, or equal to, the connecting market (i.e., a Z ao ). By ﬁxing the intercept of the marginal cost of serving a connecting passenger at 2, we consider a natural case where ao 4 2. The equilibrium outcomes of a welfare-maximizing and a proﬁtmaximizing hub are shown in Table 3 for the case of linear delay

Please cite this article as: Lin, M.H., Zhang, A., Hub congestion pricing: Discriminatory passenger charges. Economics of Transportation (2016), http://dx.doi.org/10.1016/j.ecotra.2016.02.001i

M.H. Lin, A. Zhang / Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

function (β ¼ 1Þ, and in Tables 4A and 4B for the case of quadratic delay function (β ¼ 2). In all the tables, we consider the equilibria that have both positive market outputs and ticket prices. 5.1. Linear delay function (β ¼ 1Þ According to Table 3, the relationship between the optimal perpassenger charges in the public hub case is 1 þ a 6g þ 18ag ao ð1 þ 6g Þ ð2a ao Þθ ⋛02 2 þ 24g 3θ 1 þ a 6g þ 18ag ao ð1 þ6g Þ θ⋚ ≔θjv ¼ u : 2a ao

v u ¼

ð30Þ

Note that the positivity of the connecting-passenger fare 9 þ aÞg þ 8ao gÞ , and this critical value is (P o 4 0) requires θ o 2ð2 þ ao 4ð 2a þ 5a0 smaller than θjv ¼ u . We therefore have v u 4 0. In the case of a private hub airport, the relationship between the optimal per-passenger charges is: vpri upri ¼

ð1 þ a ao Þ o 0: 2

ð31Þ

Note that the RHS of Eq. (31) is strictly negative by a Zao , and the relationship is independent of g ≔γs and θ. The following proposition summarizes these results: Proposition 5a. Under a linear delay function, linear demand functions in which the local market size is larger than the connecting market size, and duopoly airlines,

Fig. 2. Relationship between socially optimal per-local and per-connecting passenger charges.

i) A welfare-maximizing hub imposes higher per-connecting passenger charges as compared to per-local passenger charges, whilst ii) A proﬁt-maximizing hub imposes lower per-connecting passenger charges as compared to per-local passenger charges. Part (i) of the proposition demonstrates one speciﬁc case for Proposition 1 (part ii), with n ¼ 2. Part (ii) is consistent with Eq. (27). Although the relationship shown in (31) is independent of economies of trafﬁc density and the congestion delay cost (due to the speciﬁc functions), it depends on the demand elasticities ρo ρ εo ε , the last term in Eq. (27). The familiar economic intuition here is: when the connecting market is smaller (than the local market) and has more elastic demand, it must have the lower charge. 5.2. Quadratic delay function (β ¼ 2Þ We further examine how Proposition 5a would change if the congestion delay is more signiﬁcant, represented by the case of β ¼ 2 (i.e., quadratic function). Tables 4A and 4B show the equilibrium outcomes of a welfare-maximizing hub and those of a proﬁt-maximizing hub, respectively. Note that passengers’ congestion valuation is g≔sγ2 in this case. Given the socially optimal charges, we have (“ ” denotes the case of quadratic delay function): 0 1 @ 288ð 3 þ 7a 2ao Þg v~ u~ ¼ 2592g |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} 40

1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 576ð 3 þ a þ ao Þg þ 2 3θ 2 þ3θ A: 5 3θ |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} o0

ð32Þ The ﬁrst noted positivity and second noted negativity in Eq. (32) hold when the (sufﬁcient) second-order conditions, ða þ ao Þ 4

Fig. 3. Relationship between private optimal per-local and per-connecting passenger charges.

3 and 5 3θ 4 0, are satisﬁed. From Eq. (32), we have

5 þ 3θ 1 þ a ao ð2a ao Þθ sign v~ u~ ⋛02g⋛ ≔g v~ ¼ u~ : 16ð3 7a þ 2ao Þ2 Fig. 2 shows two cases of this relationship by taking the nonnegativity conditions into account. On the other hand, for the proﬁt-maximizing hub case, we have 0 1 @ pri pri v~ 32ð6 þ 4a 5ao Þg u~ ¼ 288g |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} basically o 0

1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 32ð 3 þ a þ ao Þg þ 1 θ 1 θ A: 1θ |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} o0

ð33Þ The second noted negativity hold when the second-order conditions, ða þ ao Þ 4 3 and 1 θ 4 0, are satisﬁed. The ﬁrst noted negativity also holds in a general situation where the local market size is relatively larger than that of the connecting market

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M.H. Lin, A. Zhang / Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

(speciﬁcally, 6 þ 4a 5ao 40). However, if the market sizes are such that 6 þ 4a 5ao o 0, we then have 9ð1 þa ao Þð 1 þ θÞ2

≔g v~ pri ¼ u~ pri : sign v~ pri u~ pri ⋛02g⋛ 32ð6 þ4a 5ao Þ2 Fig. 3 shows the relationships for this interesting case. Proposition 5b. Under a quadratic delay function, linear demand functions in which the local market size is larger than the connecting market size, and duopoly airlines, i) A welfare-maximizing hub charges a connecting passenger higher than a local passenger if passengers’ congestion valuation is sufﬁciently large; ii) A proﬁt-maximizing hub airport charges a connecting passenger lower than a local passenger. However, if the two market sizes are similar, then this relationship holds (reverses, respectively) when both the passengers’ congestion valuation and economies of trafﬁc density are small (large, respectively). The ﬁrst part demonstrates another speciﬁc case for Proposition 1 (part ii), with n ¼ 2. The second part is again, consistent with Eq. (27), and the relationship shown in Eq. (33) depends on economies of trafﬁc density, the congestion delay cost and the demand elasticities. Therefore, in accordance with the explanation of Proposition 4, a proﬁt-maximizing hub airport basically charges a connecting passenger lower than a local passenger, except the two market sizes are similar and the congestion delay cost is dominant. In this exception, the hub airport charges a connecting passenger higher because the hub congestion caused by the connecting passenger is more signiﬁcant. Proposition 5 offers another explanation for why some hub airports, such as London Heathrow and Copenhagen Kastrup with majority private ownership or Dubai, Hong Kong, Singapore that, while being publicly owned, emphasize on deriving proﬁt from their high percentage of international connecting passenger, may levy per-connecting passenger charges lower than per-local passenger charges as shown in Tables 2A and 2B. Basically, the monopoly power of a privatized, or proﬁt-maximizing/emphasizing, hub airport may lead to this pricing strategy.

11

highlight the different roles played by per-ﬂight and per-passenger charges under a hub-spoke network. The assumption, on the other hand, implies that our analysis perhaps is more suited for a shortto medium-run situation where the aircraft ﬂeet is given to a carrier. In the longer run, carriers may respond to airport charges by changing their aircraft size and frequency (with load factor being a ﬁxed or residual variable). A number of studies have discussed the implications of relaxing the ﬁxed-proportions assumption in the literature (e.g., Brueckner, 2004; Wei and Hansen, 2007; Flores-Fillol, 2010; Daniel, 2011; Vaze and Barnhart, 2012a; Yan and Winston, 2014). Silva and Verhoef (2013) addressed the congestion pricing issue, but they only considered one single airport and hence just one type of passengers. Silva et al. (2014) did not have the ﬁxed-proportions assumption in their examination of socially optimal per-passenger and per-ﬂight charges. Extending our analysis to a variable-proportions setting along the line of Silva et al. (2014) (but aiming for more analytical and general results) is a promising future area of research. Third, the hub-spoke network was considered as given in the present model. However, if congestion at the hub airport is signiﬁcant and the hub’s charge on connecting passengers is high, a hub carrier may consider to switch its network to a point-to-point type (that is, de-hubbing). Note that this network choice can, as shown in Silva et al. (2014), introduce another source of inefﬁciency. Fourth, we may add a positive effect of hubbing to the analysis: a large frequency may reduce passengers' schedule-delay cost.21 Incorporating the frequency effect won't change our results qualitatively, however. Another positive effect is that a large frequency may derive more concession revenues and the economies of agglomeration for the hub airport; incorporating this effect is, we believe, a more substantial undertaking in terms of analytical work. Finally, according to The Independent Travel News on 3 January 2015, British passengers who live outside southeast England and beyond easy reach of Heathrow, Gatwick and Stansted are now more likely to transfer at Amsterdam or Dubai, so as to avoid the UK's busiest airports. This implies that airport charges with respect to local and connecting passengers can depend on the degree of hub airport competition. We consider that extending the analysis proposed here along these lines would be interesting projects, although beyond the scope of the present paper.

6. Concluding remarks Acknowledgment We have investigated airport pricing in a hub-spoke network with hub congestion and carrier market power. This investigation is important given that most congested airports are hubs that serve as trafﬁc collection and distribution centers. We found that the social optimum may require that the hub impose higher perconnecting passenger charges (as compared to per-local passenger charges) when the per-ﬂight charge is small. On the other hand, a proﬁt-maximizing airport can charge lower per-connecting passenger charges (as compared to per-local passenger charges) and more interestingly, this possibility is augmented by economies of trafﬁc density, an important aspect of a hub-spoke network. We also found that while charging per-local and per-connecting passenger differently at hub airports leads to the social optimum, in general levying a pure per-ﬂight charge cannot achieve the ﬁrstbest outcome. In addition to meaningful policy implications, our analysis sheds new insights about potential reasons for why, in practice, some hub airports charge a connecting passenger lower than a local passenger. The paper has also raised a number of other issues and avenues for future research. First, the hub's capacity was treated as ﬁxed. It would be important to examine the variable capacity case. Second, we used the ﬁxed-proportions assumption which allows us to

We would like to thank two anonymous referees for constructive comments and suggestions that led to a signiﬁcant improvement of the paper. We also thank Jan Brueckner, Achim Czerny, Bruno de Borger, Amihai Glazer, Anne Graham, Kenneth Small, Harry van der Weijde, Yimin Zhang, and the seminar participants at University of British Columbia (the Center for Transport Study), UC Irvine, OPTION conference (organized by Tinbergen Institute and VU University Amsterdam), Canadian Economic Association (49th) annual meetings, Western Economic Association International (90th) annual conference, and Hong Kong Polytechnic University (Department of Logistics and Maritime Study) for very helpful comments and discussions. This work was supported by the MEXT/JSPS KAKENHI Grant (no. 24530281) and (partially) by the Social Science and Humanities Research Council of Canada (SSHRC, no. 410-2011-0569). 21 In the airline literature, it is well known that higher frequency on a route gives passengers a wider choice of departure/arrival times and thus reduces the cost of “schedule delay,” the difference between the desired and actual departure/ arrival times (e.g., Brueckner, 2004, Heimer and Shy, 2006, Kawasaki, 2008). See Silva et al. (2014) for a related analysis that incorporates this frequency effect.

Please cite this article as: Lin, M.H., Zhang, A., Hub congestion pricing: Discriminatory passenger charges. Economics of Transportation (2016), http://dx.doi.org/10.1016/j.ecotra.2016.02.001i

M.H. Lin, A. Zhang / Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

12

Appendix A. The second-order conditions of airline competition ∂2 π ¼ 2 ρ00 qi þ 2ρ0 4D0 2D00 qi þ qio 2c00i o 0; ∂qi ∂qi ∂2 π ¼ ρ00o qio þ 2ρ0o 4D0 2D00 qi þ qio 2c00i o 0; ∂qio ∂qio ∂2 π ∂2 π ¼ ¼ 4D0 2D00 qi þ qio 2c00i o 0; ∂qi ∂qio ∂qio ∂qi ∂2 π ∂2 π ∂2 π ∂2 π 4 0; 8 i: ∂qi ∂qi ∂qio ∂qio ∂qi ∂qio ∂qio ∂qi B. Comparative-static effects of airport charges We differentiate both sides of Eqs. (6) and (7) with respect to u and v respectively, and solve the resulting equations to yield the following comparative-static effects: 0

1

0 ∂Q n B 0 C ¼ @ð1 þ nÞρo þ ρ00o Q o 2c 00 2ðn þ1ÞD 2D00 ðQ þ Q o Þ A o 0; |ﬄ{zﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ∂u Λ |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

o0

≔X o o 0

o0

0

1

0 ∂Q o n B 0 0 C ¼ @ð1þ nÞρ þ ρ00 Q c ðnþ 1ÞD D00 ðQ þ Q o Þ Ao 0; ∂v Λ |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |{z} |ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

o0

≔X o o 0

∂Q o ∂u

∂Q ¼2 ∂v

ð34Þ

o0

o0

ð35Þ

o0

0

1

0 2nB C ¼ @ c00 þ ðnþ 1ÞD þ D00 ðQ þ Q o Þ A 4 0; Λ |{z} |ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

40

40

ð36Þ

40

where Λ ¼ XX o ð2X þX o Þ c00 þ ð1 þ nÞD0 þ D00 ðQ þ Q o Þ 4 0 by the stability condition. In addition, the ﬁrst noted terms in Eqs. (34) and (35), deﬁned as X o and X respectively, are the slopes of carriers’ marginal revenues from markets AB and AH (BH), which are generally negative. Thus, the inequalities in Eqs. (34)–(36) hold. The result shown in Eq. (34) is sensible: the total equilibrium output in local market decreases in local-passenger charges. A similar interpretation applies to Eq. (35) for the connecting market. Eq. (36) is the effect through the congested hub: a higher airport charge on local (connecting) passengers reduces the market output, which relieves congestion at the hub. This in turn will beneﬁt connecting (local) passengers, thus raising the connecting (local) output.

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Please cite this article as: Lin, M.H., Zhang, A., Hub congestion pricing: Discriminatory passenger charges. Economics of Transportation (2016), http://dx.doi.org/10.1016/j.ecotra.2016.02.001i

Hub congestion pricing: Discriminatory passenger charges

Hub congestion pricing: Discriminatory passenger charges

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