Revenue sharing with multiple airlines and airports

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1 See dscussons, stats, and author profles for ths publcaton at: Revenue sharng wth multple arlnes and arports ARTICLE n TRANSPORTATION RESEARCH PART B METHODOLOGICAL SEPTEMBER 2010 Impact Factor: 2.95 DOI: /j.trb CITATIONS 28 READS 93 3 AUTHORS, INCLUDING: Anmng Zhang Unversty of Brtsh Columba - Vanco 132 PUBLICATIONS 2,670 CITATIONS Hangjun (Gavn) Yang Unversty of Internatonal Busness an 17 PUBLICATIONS 112 CITATIONS SEE PROFILE SEE PROFILE All n-text references underlned n blue are lnked to publcatons on ResearchGate, lettng you access and read them mmedately. Avalable from: Hangjun (Gavn) Yang Retreved on: 29 January 2016

2 Transportaton Research Part B 44 (2010) Contents lsts avalable at ScenceDrect Transportaton Research Part B journal homepage: Revenue sharng wth multple arlnes and arports Anmng Zhang a, *, Xaowen Fu b, Hangjun (Gavn) Yang a a Sauder School of Busness, Unversty of Brtsh Columba, 2053 Man Mall, Vancouver, BC, Canada V6T 1Z2 b Faculty of Busness, Hong Kong Polytechnc Unversty, Hong Kong artcle nfo abstract Artcle hstory: Receved 10 September 2009 Receved n revsed form 1 February 2010 Accepted 1 February 2010 Keywords: Concesson revenues Revenue sharng Arport competton Arport arlne vertcal cooperaton Non-atomstc carrers Ths paper nvestgates the effects of concesson revenue sharng between an arport and ts arlnes. It s found that the degree of revenue sharng wll be affected by how arlnes servces are related to each other (complements, ndependent, or substtutes). In partcular, when carrers provde strongly substtutable servces to each other, the arport has ncentve to charge arlnes, rather than to pay arlnes, a share of concesson revenue. In these stuatons, whle revenue sharng mproves proft, t reduces socal welfare. It s further found that arport competton results n a hgher degree of revenue sharng than would be had n the case of sngle arports. The arport arlne chans may nevertheless derve lower profts through the revenue-sharng rvalry, and the stuaton s smlar to a Prsoners Dlemma. As the chans move further away from ther jont proft maxmum, welfare rses beyond the level achevable by sngle arports. The (equlbrum) revenue-sharng proporton at an arport s also shown to decrease n the number of ts carrers, and to ncrease n the number of carrers at competng arports. Fnally, the effects of a pure sharng contract are compared to those of the two-part sharng contract. It s found that whether an arport s subject to competton s crtcal to the welfare consequences of alternatve revenue sharng arrangements. Ó 2010 Elsever Ltd. All rghts reserved. 1. Introducton An arport derves revenue from two facets of ts busness: the tradtonal aeronautcal operatons and the commercal (concesson) operatons. The former refer to avaton actvtes assocated wth runways, arcraft parkng and termnals, whereas the latter refer to non-aeronautcal actvtes occurrng wthn termnals and on arport land, ncludng termnal concessons (duty-free shops, restaurants, etc.), car rental and car parkng. For the last two decades, commercal revenues have grown faster than aeronautcal revenues and, as a result, have become the man ncome source of many arports. At medum to large US arports, for nstance, commercal busness represents 75 80% of the total arport revenue (Dogans, 1992). ATRS (2008) studed 142 arports worldwde and found a majorty of these arports derved 40 75% of ther revenues from non-avaton servces, a major part of whch s revenue from concesson servces (wth large hub arports relyng, on average, even more on concesson ncome). Further, commercal operatons tend to be more proftable than aeronautcal operatons (e.g. Jones et al., 1993; Starke, 2001; Francs et al., 2004), owng n part to prevalng regulatons and chargng mechansms (e.g. Starke, 2001). Parallelng the growth of concesson revenues, revenue sharng between arports and arlnes s gettng popular n practce. As documented n Fu and Zhang (forthcomng), there are cases, such as Tampa Internatonal Arport n the US * Correspondng author. Tel.: ; fax: E-mal addresses: anmng.zhang@sauder.ubc.ca (A. Zhang), lgtxfu@polyu.edu.hk (X. Fu), gavn.yang@sauder.ubc.ca (Hangjun (Gavn) Yang) /$ - see front matter Ó 2010 Elsever Ltd. All rghts reserved. do: /j.trb

3 A. Zhang et al. / Transportaton Research Part B 44 (2010) and Ryanar n Europe, where arports and arlnes share concesson revenues. 1 In many other cases, revenue sharng s n effect when arports allow arlnes to hold shares or control arport facltes. For example, Termnal 2 of Munch arport was jontly nvested by the arport operatng company FMG (60%) and the arport s domnant carrer, Lufthansa (40%) (Kuchnke and Sckmann, 2005). Commercal profts generated from ths termnal are thus shared between FMG and Lufthansa. Fu and Zhang (forthcomng) found that concesson revenue sharng has mportant compettve and welfare mplcatons: t allows the arport and arlnes to nternalze a mult-output complementarty between the passenger flghts and the concesson consumpton brought about by the flghts, whch may mprove socal welfare. Essentally, passengers travelng through the arport also create a demand for concesson consumpton. As an arport depends on arlnes to brng n passengers, concesson revenue sharng wll encourage the carrers to expand output, whch may n turn mprove proft for the whole arport arlnes chan (as well as mprove welfare). However, revenue sharng can cause a negatve effect on arlne competton as an arport may strategcally share revenue wth ts domnant carrers, further strengthenng these arlnes market power. The US Federal Avaton Admnstraton (FAA) has expressed concerns over arports practce of offerng partcular arlnes favorable terms, on the ground that such a specal treatment may harm competton n the arlne market downstream (FAA, 1999). 2 Snce 1995, the EU s competton authortes have ruled aganst several major arports n Belgum, Fnland, and Portugal concernng ther practces of chargng lower prces to home carrers (Barbot, 2006, 2009a). For the last several years, the effects of vertcal relatonshps between arports and arlnes have receved growng attenton from researchers. In addton to Fu and Zhang (forthcomng), Auerbach and Koch (2007) and Barbot (2009a,b) found that cooperaton between an arport and ts arlnes can brng benefts to the allance members n terms of ncreased traffc volume and operaton effcency. In ths paper we extend ths lterature on arport arlne vertcal cooperaton, focusng on the effects of concesson revenue sharng. More specfcally, we consder that carrers may provde complementary, ndependent or substtutable servces to each other, and that the proportons of revenue sharng may be outsde of the [0, 1] range. The latter allows us to compare alternatve sharng arrangements. Further, unlke the prevous studes, our analyss s mostly conducted under general demand and cost functonal forms. Moreover, as elaborated below, our work also extends the exstng lterature to the general case of multple competng arports wth each havng an arbtrary number of carrers. We fnd that the degree of revenue sharng wll be affected by how arlnes servces are related to each other. In partcular, when carrers provde substtutable servces to each other, they mght need to pay to the arport a share of concesson revenue (so-called the negatve sharng ) f substtutablty s suffcently strong and the fxed (transfer) payments between the arport and carrers are feasble (referred to as the two-part revenue sharng ). The negatve sharng allows the arport to penalze the over-competng carrers so as to support arfares downstream and mprove proft. In these stuatons, whle revenue sharng mproves total arport arlnes channel proft, t reduces socal welfare. If the fxed payments are not feasble, under the resultng pure revenue sharng the arport wll, for the cases of ndependent or complementary servces, share less concesson revenue wth ts carrers than would be under the two-part revenue sharng. For the substtutes case, however, the sharng-proportons comparson between the two types s n general ambguous. In the specal case of negatve sharng, the pure revenue sharng wll, for suffcently symmetrc carrers, result n not only a hgher sharng proporton, but also a hgher welfare level, than the two-part revenue sharng. Our second objectve n ths paper s to extend the arport lterature by nvestgatng revenue sharng for multple, competng arports. Very few papers have examned competng arports analytcally. For example, Fu and Zhang (forthcomng) examned revenue sharng only for a monopoly arport. The few exceptons nclude Gllen and Morrson (2003), who examned two competng arports n the context of a full-servce carrer and a low-cost carrer. More recently, Basso and Zhang (2007) provded a more general examnaton of arport competton wth congeston and non-atomstc arlnes at each arport, and Barbot (2009a) examned arport arlne nteractons (colluson, n partcular) usng a spatal model smlar to that of Basso and Zhang. The ssues of concesson revenues and revenue sharng were not consdered n these papers, however. Ths lack of analytcal studes on arport competton s understandable gven the local monopoly nature of an arport. The stuaton s changng, however. The world has experenced a rapd growth n ar transport demand snce the 1970s, and many arports have been bult or expanded as a result. Ths has led to a number of mult-arport regons such as greater London n the UK and several metropoltan areas n the US (e.g. San Francsco, Chcago, New York, Washngton, Dallas, Detrot, Huston, and Los Angeles) wthn whch arports may compete wth each other. At the same tme, the dramatc growth of low-cost carrers (e.g. Southwest n the US and Ryanar n Europe) has enabled some smaller and perpheral arports to cut nto the catchment areas of large arports. Starke (2008) conducted an overvew of UK arports from the perspectve of a busness enterprse. He concluded that effectve competton across arports s possble and a compettve arport ndustry can be fnancally vable. Taken together, these observatons suggest that t s mportant to nvestgate the effects of revenue sharng n the context of multple, competng arports. 1 Tampa has been sharng revenue wth ts carrers for several years. In 2004, t shared $7 mllon out of a total budget of $30 mllon (see the 2004 Annual Report of Tampa Internatonal Arport). On the other hand, Ryanar has dentfed arport car parkng as one of ts busness opportuntes and cooperated wth the leadng arport parkng company BCP. In ts negotatons wth some arports, Ryanar asked for parkng revenue sharng as a condton to serve the arports. 2 Prevous studes (e.g. FAA, 1999; GAO, 1997; Dresner et al., 2002; see also Hartmann, 2006, for a useful revew on the topc) suggest that arlne entry may be deterred f the domnant arlne controls key arport facltes. Apparently, such a strategy by the domnant carrer would requre at least mplct consent/ cooperaton from the arport. In the US, large and medum arports that meet a certan threshold of arlne concentraton are now requred to submt competton plans as mandated by the Wendell H. Ford Avaton Investment and Reform Act for the 21 st Century legslated n 2000.

4 946 A. Zhang et al. / Transportaton Research Part B 44 (2010) We fnd that arport competton wll result n a hgher degree of revenue sharng than would be had n the case of sngle arports. Nevertheless, the arport arlne chans may derve lower profts through ths revenue-sharng rvalry. As the arport arlne chans move further away from ther jont proft maxmum, socal welfare rses beyond the level achevable by sngle arports. Our analyss also shows that the arlne market structure can have a bearng on revenue sharng arrangements not only at the arport n queston, but also at ts competng arports. Specfcally, the (equlbrum) revenue-sharng proporton at an arport decreases n the number of ts carrers, but ncreases n the number of carrers at the competng arport. In terms of the welfare consequences of alternatve revenue sharng arrangements, whether an arport s subject to competton s crtcal: for competng arports, no sharng s worse than pure sharng whch s n turn worse than the two-part sharng. For sngle arports however, both no-sharng and pure-sharng mght be better than the two-part sharng when arlnes provde substtutable servces to each other. The paper s organzed as follows. Secton 2 sets out the basc model and examnes the revenue-sharng equlbrum for a sngle arport wth multple arlnes. Secton 3 examnes revenue sharng for the general case of competng arports wth each havng an arbtrary number of carrers. Secton 4 nvestgates the pure revenue sharng and compares ts effects wth those of the two-part revenue sharng. Secton 5 contans concludng remarks. 2. Sngle arport wth multple arlnes 2.1. Basc model Consder, n ths secton, that a sngle arport provdes aeronautcal servce to arlnes, for whch t mposes a charge. In our modelng ths charge s represented by a per-passenger fee w (>0), and s regulated and cannot be changed unlaterally by ether the arport or arlnes. 3 We have two carrers, labeled as = 1,2, operatng from the arport, although the analyss and results extend mmedately to the n-carrer case (see, e.g. Secton 3.2). They face nverse demands p (q 1,q 2 ), whch satsfy the usual propertes of p < 0 and p1 1 p2 2 p1 2 p2 1 > 0 wth subscrpts denotng partal dervatves.4 The arlnes revenue from provdng avaton servce s then gven by R (q 1,q 2 )=p (q 1,q 2 )q. The revenue functons can be used to defne how one arlne s output s related to the other s. There are three possble cases: () Complements: two carrers offer complementary servces n the sense that R j ðq 1; q 2 Þ¼p j ðq 1; q 2 Þq > 0; R j ðq 1; q 2 Þ > 0: ð1þ That s, ncreasng carrer j s output ncreases both the total and margnal revenues of carrer (here, and below, f the ndces and j appear n the same expresson, then t s to be understood that j). 5 In the present context, servces provded by a trunk arlne and a feeder arlne wth ther passengers connectng at the arport may be consdered as complements. Another example would be that two arlnes engage n some form of strategc allances or code-sharng arrangements (e.g. Brueckner, 2001; Brueckner and Whalen, 2000). () Independent servces: two carrers servces are unrelated n demand as R j ðq 1; q 2 Þ¼p j ðq 1; q 2 Þq ¼ 0: ð2þ Note, n ths case, that R j ¼ 0 mples R j ¼ 0. () Substtutes: rasng a carrer s output reduces the other s total and margnal revenues, R j ðq 1; q 2 Þ¼p j ðq 1; q 2 Þq < 0; R j ðq 1; q 2 Þ < 0; ð3þ For nstance, two competng trunk carrers lkely provde substtutes at an arport, and so do two competng feeder carrers. We consder that for each passenger gong through the arport, a concesson revenue h (>0) s derved. Assumng further (for smplcty) zero costs for provdng concesson servces by the arport, then h represents a net surplus per passenger. 6 3 Snce prce dscrmnaton (on aeronautcal charges) by an arport s prohbted by the Internatonal Ar Transport Assocaton (IATA) rules, all arlnes servng the arport face the same w. 4 Whle p < 0 ndcates the usual property of downward-slopng demands, p1 1 p2 2 p1 2 p2 1 > 0 refers to the property of own effects domnatng cross effects n demand functons. As noted by Dxt (1986, p. 108), the domnance of own-effects over cross-effects s a standard assumpton n models of olgopoly. 5 The frst nequalty n (1) shows (gross) complements between the arlne servces, whereas the second nequalty mples strategc complements (Bulow et al., 1985). That the former mples the latter holds for most (but not all) plausble demand structures; t certanly holds when demand functons are lnear. In other words, the fact that servces are complements s conducve to ther strategc complementarty. Restrctng attenton to strategc complementarty s a standard practce n olgopoly models (Dxt, 1986; Trole, 1988). Smlar observatons on substtutes and strategc substtutes hold for the substtutes case dscussed next. We shall, as s common n the lterature, refer to these two cases smply as complements and substtutes. 6 Ths formulaton of concesson surplus has been used n, e.g. Zhang and Zhang (1997), Zhang and Zhang (2003), Oum et al. (2004) and Fu and Zhang (forthcomng). It s, nevertheless, a smple representaton where concesson surplus s strctly complementary to passenger volume. For an alternatve and perhaps more realstc formulaton, see Czerny (2006).

5 A. Zhang et al. / Transportaton Research Part B 44 (2010) How total concesson revenue hq s shared between the arport and arlnes s modeled as a two-stage game. In the frst stage, the arport offers carrer to share proporton r of revenue hq n exchange for a fxed fee f, subject to the carrer s partcpaton constrant. No restrcton s mposed on r, and so r can be less than zero or greater than one. In the second stage, arlnes choose quanttes to maxmze ndvdual profts. 7 The subgame perfect equlbrum of ths two-stage game s referred to as the revenue sharng equlbrum. Pror to examnaton of the revenue-sharng equlbrum, two thngs about the sharng contract (r,f ), = 1,2, are worth notng. Frst, assumng that arlne gets a share of revenue hq means that the arport s able to know who s flyng n whch arlne. Ths can happen after boardng, as passengers may present ther boardng card, but not necessarly before boardng. Second, the contract beng a par (r,f ) suggests a two part revenue-sharng scheme under whch fxed payments are possble. Such a model can be used to examne the ncentve for vertcal arport arlne cooperaton.e. takng account of the proft for the arport arlne channel as a whole and may also be consstent wth stuatons n whch arports and arlnes can commt to medum-/long-term cooperaton. Nonetheless, such fxed payments between arports and arlnes mght not be feasble n certan stuatons, owng to the dffculty n ther agreeng to the rght amount of payments, or to the preference for smpler revenue-sharng arrangements that do not nvolve any medum-/long-term commtment. In Secton 4, we wll examne a pure sharng contract that restrcts fxed payments f to zero Revenue-sharng equlbrum The revenue-sharng equlbrum s solved n the standard backward nducton. Stage two: gven sharng contract (r,f ), each carrer s proft s: p ðq 1 ; q 2 Þ¼R ðq 1 ; q 2 ÞC ðq Þwq þ r hq f ; where C (q ) denotes carrer s cost. Thus for carrer, the total operatng cost net of fxed payment f equals C (q )+wq. The Cournot Nash equlbrum s characterzed by the frst-order condtons, p ðq 1; q 2 Þ¼R ðq 1; q 2 ÞC 0 ðq Þwþr h ¼ 0; ð5þ and the second-order condtons p ðq 1; q 2 Þ¼R ðq 1; q 2 ÞC 00 ðq Þ < 0. Both the second-order condtons and the stablty condton, J p 1 11p 2 22 p1 12p 2 21 > 0, are assumed to hold over the entre regon of nterest.8 The soluton to (5) yelds the second-stage equlbrum quanttes, whch are functons of the frst-stage varables (r 1,r 2 ). Snce fxed payments f 1 and f 2 enter the arlnes proft functons (4) as constants, they wll not affect the equlbrum quanttes. Denotng the equlbrum quanttes as q ðr 1; r 2 Þ, substtutng them nto (5) and totally dfferentatng the resultng dentty wth respect to r, we =@r ¼hp j =J; j =@r ¼ hp j j =J: ð6þ It follows mmedately =@r > 0, j =@r havng the same sgn as p j ¼ j Rj j, whch by (1) (3) leads to: Lemma 1. =@r > 0 and j =@r > 0; ¼ 0, and <0 for carrers producng complements, ndependent servces, and substtutes, respectvely. Thus an ncrease n the share of concesson revenue to carrer ncreases s output. The reason s that an ncrease n r wll mprove carrer s margnal proftablty, owng to the mult-output complementarty between passenger flghts and concesson consumpton. Furthermore, an ncrease n r ncreases, not affects, and decreases carrer j s output f the carrers offer complementary, ndependent, and substtutable servces, respectvely. For the case of substtutes, snce that p j ¼ j Rj < 0 ensures a downward-slopng best reply functon for each carrer (defned by (5) n the output space), an ncrease n r wll, by j ncreasng carrer s margnal proft, shft ts best-reply functon outward. Ths wll move the equlbrum quanttes downward along j s best-reply functon, thereby ncreasng q and decreasng q j. For complements, on the other hand, that p j ¼ j Rj > 0 ensures an upward-slopng best-reply functon for each carrer. An ncrease n r j wll agan shft s best-reply functon outward, movng the equlbrum quanttes upward along j s best-reply functon, thereby ncreasng both q and q j. Fnally, f the servces are ndependent, then an ncrease n r does not affect q j, as expected. Stage one: revenue-sharng structures therefore nfluence subsequent arlne quanttes, whch n turn wll affect the arport s proft. Assume, for smplcty, that the arport s fxed cost s zero and ts margnal cost s constant and normalzed to zero. The arport s proft s then gven by: P ¼ w q 1 þ q 2 þ ð1 r1 Þhq 1 þð1r 2Þhq 2 þ f1 þ f 2 ; ð7þ ð4þ 7 Ths mples carrers nteract wth each other n Cournot fashon. Recent studes on arport prcng and capacty nvestment that have ncorporated mperfect competton of ar carrers at an arport (e.g. Brueckner, 2002; Pels and Verhoef, 2004; Zhang and Zhang, 2006; Basso, 2008) have assumed Cournot behavor. Brander and Zhang (1990, 1993), for example, fnd some emprcal evdence that rvalry between duopoly arlnes s consstent wth Cournot behavor. 8 Ths assumpton mples that the Cournot equlbrum exsts and s unque (e.g. Fredman, 1977). Note that f carrers face lnear demands, then all these condtons wll be satsfed.

6 948 A. Zhang et al. / Transportaton Research Part B 44 (2010) where the second-stage equlbrum outputs are taken nto account. (Throughout the paper, we use captal letter P to denote arport proft, whle lower case p denotng arlne proft.) There are three components n P: () the aeronautcal revenue (proft) gven by w (q 1 + q 2 ); () the resdual concesson revenue gven by the bracketed term n (7) and () the fxed payment collected from carrers, f 1 + f 2. The arport chooses (r,f ), = 1,2, to maxmze P. Whle f wll not, as ndcated above, affect the second-stage equlbrum outputs, op/@f =1by(7). Consequently, the arport should, gven ts Stackelberg leader s role, charge the arlnes a fee as hgh as possble subject to ther partcpaton constrants p P p 0, wth p 0 beng carrer s reservaton proft. Assume, wthout loss of generalty, that each carrer receves ts reservaton proft. 9 Ths partcpaton constrant mples, usng (4), that f ¼ R q 1 ; q 2 C q wq þ r hq p 0 ; ¼ 1; 2; ð8þ where equlbrum outputs q are functons of r 1 and r 2. Wth (8), arport proft (7) becomes: Pðr 1 ; r 2 Þ¼ X h R q 1 ; q 2 C q þ hq p 0 v q 1 ðr 1; r 2 Þ; q 2 ðr 1; r 2 Þ : ð9þ Thus, the revenue-sharng equlbrum s characterzed by the frst-order ¼ =@r þ v j =@r ¼ 0; ¼ 1; 2; ð10þ where v Þ¼R q 1 ; q 2 C 0 q þ h þ R j q 1 ; q 2.By(5), v can be rewrtten as: v ¼ w þð1r Þh þ R j q 1 ; q 2 : ð11þ Consder frst the case where carrers servces are ndependent. It can be easly seen from (10), (11), (2) and Lemma 1 that the equlbrum sharng proportons are gven by (superscrpt I for ndependent servces ): r I ¼ 1 þðw=hþ; ¼ 1; 2; ð12þ whch are strctly postve. Revenue sharng therefore mproves the arport s proft here, the proft gan s due to the nternalzaton of a demand complementarty between the flghts and concesson consumpton. Further, even when r = 1, the proft wll rse wth r gong beyond the full share. Bascally, the two-part revenue sharng resolves the well-known double margnalzaton problem n a vertcal structure (e.g. Trole, 1988). The ndependent-servces case turns out to be a useful benchmark for the cases of substtutes and complements. By frstorder condtons (10) t follows: v 1 =@r 1 þ v 2 =@r 1 ¼ 0; ð13:1þ v 1 =@r 2 þ v 2 =@r 2 ¼ 0; ð13:2þ whch gve rse to v 1 =@r 2 =@r 1 =@r 2 =@r 2 ¼ 0. Ths equaton, by (6), reduces further to v 2 h 2 p 1 12p 2 21 p1 11p 2 22 =J ¼ 0 )v2 h 2 ¼ 0 ) v 2 ¼ 0. Pluggng v 2 = 0 nto (13.1) we mmedately have v 1 = 0. It follows from (11) that r ¼½1þðw=hÞŠ þ R j =h ; ¼ 1; 2: ð14þ If arlne servces are complements, then R j > 0; consequently (superscrpt C for complements ), r C > 1 þðw=hþ ¼r I ; ¼ 1; 2: ð15þ If arlne servces are substtutes, then R j < 0 and so Eq. (14) yelds (superscrpt S for substtutes ) r S < 1 þðw=hþ ¼r I ; ¼ 1; 2; ð16þ leadng, therefore, to: Proposton 1. At the revenue-sharng equlbrum wth a sngle arport, the sharng proportons are r I ¼ 1 þðw=hþ when arlnes servces are ndependent, = 1,2. The sharng proportons are greater (smaller, respectvely) than r I when arlnes provde complementary (substtutable, respectvely) servces to each other. 9 The assumpton that the arport chooses the fees as hgh as possble subject to carrers partcpaton constrants mples that all the benefts from mprovements n performance go to the arport. Ths s due to our arport arlnes relatonshp wth the arport beng a Stackelberg leader. Such a vertcal structure has been a standard set-up n the recent lterature on arport prcng and capacty nvestment that ncorporates mperfect competton of arlnes at an arport (see, e.g. Basso and Zhang, 2008, for a survey). As ponted out by an anonymous referee, Bowley (1928) has ths dea (not the expresson) for a two frms game n a vertcal context (wth one buyer and one seller, when the seller has more market power). In the present context, maybe a sngle arport ( monopolst ) fts the set-up better than multple arports: snce the arport has more market power than arlnes, ts frst mover advantage n choosng the fees may be due to ts monopolst poston. We dscuss the ssue further n the concludng remarks.

7 A. Zhang et al. / Transportaton Research Part B 44 (2010) The explanatons for the devatons from the ndependent-servces benchmark are as follows. When servces are complementary to each other, both carrers are nterested n ncreasng passengers numbers but are unable to nternalze such complementarty by themselves. The arport, as a frst mover, can acheve ths by manpulatng revenue-sharng proportons here, by ncreasng the sharng proportons beyond r I and ths n turn wll ncrease the arport s proft. Conversely, substtutablty between arlnes servces wll lead to a falure of coordnaton between competng arlnes, resultng n ther provdng more servce than what would be optmal for them as a whole. Antcpatng ths, the arport uses revenue sharng as a devce to coordnate arlne competton downstream. In partcular, a smaller sharng proporton than the ndependentservces benchmark wll, by Lemma 1, reduce ndustry output, thus lessenng excessve servces by carrers. 10 It s mportant to pont out that for the substtutes case, the sharng proportons mght become negatve (.e. r S < 0). Ths s because arlne-servce substtutablty works n an opposte drecton of the flghts-concessons demand complementarty dscussed above, n terms of the amount of arlne servces the arport would lke to nduce. The optmal level of revenue sharng, r S, s set to balance these two effects. Thus, rs depends on the degree of substtutablty between carrers servces. Numercal examples are constructed at the end of ths secton, n whch arlne-servce substtutablty s so strong that r S becomes negatve. Such a negatve revenue sharng allows the arport to penalze the over-competng arlnes so as to support prces n the output market and mprove the proft of ( coordnate ) the whole arport arlnes chan Comparson wth the no-sharng regme Our concern now s to compare the revenue-sharng equlbrum wth the stuaton where arport arlne revenue sharng s not allowed, characterzed by r = f = 0. Frst, for the cases of complements and ndependent servces, t s clear from Lemma 1, Proposton 1, (10) and (11) that revenue sharng wll ncrease arlne output and mprove arport proft. Defne socal welfare as the sum of the arport arlne proft and consumer (passenger) surplus: Wðr 1 ; r 2 Þ¼Uq 1 ; q 2 C1 q 1 C2 q 2 þ hq 1 þ hq 2 u q 1 ; q 2 ; ð17þ where U(q 1,q 2 ) s the consumer utlty functon n the usual ndustry (partal equlbrum) analyss, = p. Although passengers may derve surplus also from ther concesson consumpton, such surplus per passenger s assumed constant and further normalzed to zero, thus gvng rse to formulaton (17). Dfferentatng W wth respect to r ¼ p C 0 þ =@r þ p j C 0 j þ j =@r : ð18þ Snce p C 0 > 0 (postve markups n olgopoly), the output expanson dentfed above leads mmedately W/@r > 0 and thus, revenue sharng mproves welfare. As for prces, t can be easly seen (from below) that they wll fall f carrers servces are ndependent. For the complements case, the effect s not as straghtforward. Dfferentatng p q 1 ; q 2 wth respect to r and r j yelds þ =@r ¼ =@r j ¼ =@r j þ p j j =@r ; j =@r j ; ð19:2þ respectvely. Wth carrers servces beng complementary, the frst term on the rght-hand sde (RHS) of (19.1) s negatve (recall p < 0 and Lemma 1) whlst the second term s postve. Smlarly, the frst term on the RHS of (19.2) s negatve whlst the second term s postve. Under symmetry however, the overall effects wll be negatve for both (19.1) and (19.2), ass shown below (Proposton 2). By symmetry we mean () carrers have dentcal cost functons and face symmetrc demands, and () at the equlbrum, carrers have the same sharng contract (.e. r 1 = r 2,f 1 = f 2 ). The symmetry condton wll also be used n the comparson for the substtutes case (see Proposton 2). Proposton 2. At the revenue-sharng equlbrum wth a sngle arport, 1. when arlnes provde ndependent or complementary servces to each other, () outputs and welfare are greater and () under symmetry, prces are lower, than n the absence of revenue sharng and 2. when arlnes provde substtutable servces to each other and are symmetrc, () outputs and welfare are greater (smaller, respectvely) and () prces are lower (hgher, respectvely), than n the absence of revenue sharng f r S > 0 (r S < 0, respectvely). Proof 1. We only need to show the prce effect for the complements case (the other parts have been shown n the text). Use D to denote any dfference of varables between the revenue-sharng regme and no-sharng regme. Applyng the mean value theorem (MVT) to the functon p q 1 ; q 2 yelds: Dp ¼ p Dq þ p j Dq j ; 10 An alternatve explanaton for the substtutes case s that substtutablty ncreases one carrer s passengers at the expense of another carrer s passengers. The arport must balance ths trade-off and wll not allow for a very hgh r.

8 950 A. Zhang et al. / Transportaton Research Part B 44 (2010) where p and p j are evaluated at some pont between qo 1 ; qo 2 and q C 1 ; q C 2, wth superscrpt O denotng varables assocated wth the no-sharng regme. Under symmetry, Dq ¼ Dq j > 0 from part (). Consequently, Dp ¼ p þ p j Dq < 0 wth the nequalty followng from the condton p 1 1 p2 2 p1 2 p2 1 > 0 and symmetry: notng that symmetry mples p þ p j p p j > 0. Snce p p j < 0 for complements and ndependent servces, t follows that p þ p j < () Applyng MVT to q ðr 1; r 2 Þ yelds Dq =@r Dr =@r j Drj, =@r =@r j evaluated at some pont between r O 1 ; ro 2 and r S 1 ; r S 2. Under symmetry, D r = Dr j =@r j j =@r ; consequently, h h Dq q þ q j =@r Dr ¼ h p j j pj jj =J Dr Fg. 1. Revenue sharng vs. no-sharng: sngle arport wth two carrers (parameter values: b = , c 1 = c 2 = 0.45, w = 0.05, h = 0.05).

9 A. Zhang et al. / Transportaton Research Part B 44 (2010) Snce p j j pj jj > 0 and J > 0 under the second-order and substtutes condtons and the stablty condton, respectvely, Dq must have the same sgn as Dr r S r O ¼ r S (recall r O ¼ 0). For the welfare comparson, applyng MVT to u q 1 ; q 2 n (17) yelds Du ¼ u Dq þ u j Dq j, where u and u j are evaluated at some pont between q O 1 ; qo 2 and q S 1 ; qs 2. Under symmetry, Dq ¼ Dq j. Consequently, Du ¼ðu þ u j ÞDq has the same sgn as Dq, because u ¼ p C 0 þ h > 0. The welfare result then follows from the above quantty comparson. () Applyng MVT to p q 1 ; q 2 yelds Dp ¼ p Dq þ p j Dq j, where p and p j are evaluated at some pont between qo 1 ; qo 2 and q S 1 ; qs 2. Wth Dq ¼ Dq j under symmetry, Dp ¼ p þ p j Dq has the opposte sgn as Dq, because p < 0 and, by (3), p j < 0. The result then follows from the above quantty comparson. h Three comments about Proposton 2 are worth makng. Frst, although the proposton does not say anythng about arport arlnes profts, the jont profts of the arport and arlnes are always hgher at the revenue sharng equlbrum when carrers servces are ndependent, as expected. When carrers provde substtutable and complementary servces, the jont profts are hgher at the revenue sharng equlbrum for lnear demand and cost functons, but we are unable to prove the result for general demand and cost functons. 11 Second, although some of the comparsons n Proposton 2 are carred out under perfect symmetry between arlnes, a closer look at the above proof ndcates that small asymmetres wll not undermne the results. Thrd, Proposton 2 shows that when carrers offer complementary and unrelated servces to each other, revenue sharng between an arport and ts arlnes mproves welfare. The welfare mprovement arses because prces exceed margnal costs n the olgopolstc arlne market and revenue sharng reduces prces (or equvalently, expands outputs). When carrers provde substtutable servces to each other, revenue sharng may or may not mprove welfare, dependng on the sgn of equlbrum sharng proportons r S. As ndcated above, the sgn of rs wll n turn depend on the degree of substtutablty between carrers servces. To capture such an mpact, we need to mpose more structure on the model. 12 Specfcally, a lnear (nverse) demand s specfed: p ¼ 1 bq kq j ; wth b > 0 and k 2 (-b,b), whch ensure downward-slopng demands and the property of own-prce effects domnatng cross-prce effects. It s clear that carrers servces are complements, ndependent and substtutes when k < 0, = 0 and >0, respectvely. Carrers margnal costs c 1 and c 2 are constant and c 1 = c 2. In the smulaton, parameters are chosen to ensure postve outputs and margnal revenues. Fg. 1 reports the effects of arlne-servce substtutablty, where we defne k = m b wth m 2 (-0.1, 1). Thus, negatve m ndcates complementarty between arlnes servces, whlst for postve m, larger m s mean ncreasngly substtutable servces. As expected, for complementary servces (m < 0), the arport shares a hgh percentage of concesson revenue wth arlnes (r > 1) so as to nternalze arlne-servce complementarty. On the other hand, the (equlbrum) sharng proportons r S fall when arlne servces become ncreasngly substtutable. When arlne-servce substtutablty becomes suffcently strong, r S turns nto a negatve value, mplyng carrers pay a hgher prce (than arport charge w) per unt of output. The fgure shows that the fxed fees become negatve n ths case, ndcatng carrers are compensated for wth fxed payments from the arport. 13 In such a case, the output and welfare (not shown n Fg. 1) wth revenue sharng (the sold lne n the fgure) are less than those n the no-sharng case (the dotted lne), as predcted by Proposton 2. Here, whle revenue sharng mproves the total channel proft (see the fgure), t mght reduce socal welfare. 11 A smlar comment about the proft comparson apples to Proposton 6 below. 12 Examnng how equlbrum results change wth substtutablty (.e. when arlne servces become more substtutable to each other) s also mportant, snce there are stuatons n whch arports or polcy makers can moderate such substtutablty. For example, only a few Asan ctes are served by multple arports and as a result, low-cost carrers (LCCs) are often forced to use the same arport as competng full-servce arlnes (FSAs). Recently, arports n, e.g. Kuala Lumpur and Sngapore, chose to buld separate LCC termnals whch offer lower qualty of arport servce wth less charge (Zhang et al., 2009). Such a measure would make LCCs servces less substtutable to the servces provded by FSAs. 13 The combnaton of negatve sharng and arports transfer payments to arlnes may also be observed n practce. There are cases, for nstance, where arports may make one-shot nvestments (for carrers) to offset hgh arport charges. For example, Federal Express (FedEx) had been plannng to move ts Asa Pacfc operatng center from Subc Bay n the Phlppnes to Guangzhou n Chna snce However, FedEx was concerned about the hgh operatng costs n Guangzhou arport due to ts hgh charges for fuel, arport and ar traffc control (ATC) servces whch are regulated by the central government. To offset these hgh servce charges and attract FedEx, the arport agreed to nvest US $300 mllon on nfrastructures ncludng exclusve arcraft parkng space and tax runways for the usage of FedEx. FedEx opened ts Asa Pacfc operatng center n Guangzhou n February 2009, and wthn half a year t operates 136 flghts per week at the arport. Notce, here, a negatve relatonshp between arport charge w and revenue-sharng proporton r. For nstance, the value of r, for ndependent arlne servces, s equal to 1 + (h/w); consequently, dw/dr < 0:.e. the hgher w s, the smaller wll r be. A smaller or negatve sharng would then be compensated wth hgher fxed payments whch, n the present example, are represented by the nvestments that the arport made to attract carrers (.e. arport qualty s n a sense nterpreted as negatve fees offered to arlnes). Our results also suggest that when the downstream arlne market s extremely compettve, the arport may prefer smlar negatve sharng strategy to the choce of encouragng addtonal outputs va postve revenue sharng, n order to coordnate the arport arlnes chan. We dscuss the ssue further n the concludng remarks.

10 952 A. Zhang et al. / Transportaton Research Part B 44 (2010) Competng arports 3.1. Strategc revenue sharng We now consder two arports, represented by = 1,2, begnnng wth a stuaton of one carrer at each arport. (The case of multple arlnes wll be consdered n Secton 3.2.) To save notaton we contnue to use p (q 1,q 2 ) for the nverse demands faced by carrers, wth denotng the th arport s carrer (and q ts output). The two arports compete wth each other n the sense that ther arlnes servces are substtutes n the eyes of passenger: thus arlnes compete wth each other even f they operate at dfferent arports. More specfcally, arlne revenue functons R (q 1,q 2 )(=p (q 1,q 2 )q ) satsfy the substtutes condton (3). Arport arlne behavor s modeled agan as a two-stage game: In the frst stage, each arport offers ts carrer to share proporton r of concesson revenue hq n exchange for fxed fee f, subject to the carrer s partcpaton constrant. In the second stage, arlnes compete n Cournot fashon wth ther profts gven by (4). Ths arport arlne vertcal structure has also been assumed n other analytcal studes on competng arports mentoned n the ntroducton. Gven ths set-up, the second-stage equlbrum s characterzed by (5), the same condton as n the sngle-arport case. Further, the equlbrum quanttes denoted agan as q ðr 1; r 2 Þ have the comparatve-statc propertes of Lemma 1:.e. an ncrease n the sharng proporton by arport wll ncrease ts carrer s output whle reducng output of the competng arport s carrer. Takng the second-stage equlbrum outputs nto account, each arport s proft n stage one are expressed, P ¼ wq þð1 r Þhq þ f ; ¼ 1; 2: ð20þ The subgame perfect equlbrum then arses when each arport chooses ts sharng contract (r,f ) to maxmze P, takng ts rval s sharng contract at the equlbrum values. Ths revenue-sharng equlbrum wth arport competton wll be referred to as the rvalry (revenue sharng) equlbrum, where rvalry refers to arport rvalry. Wthout loss of generalty the carrers are agan assumed to receve ther reservaton profts p 0 ; ¼ 1; 2; consequently, each arport s proft can be rewrtten as: P ðr 1 ; r 2 Þ¼R q 1 ; q 2 C q þ hq p 0 v q 1 ðr 1; r 2 Þ; q 2 ðr 1; r 2 Þ : ð21þ The rvalry equlbrum s characterzed by the frst-order condtons, P ¼ =@r þ j =@r ¼ 0; ¼ 1; 2; ð22þ j where subscrpts agan denote partal dervatves (e.g. =@r ; =@q and v =@q j ). From (21), v ¼ R q 1 ; q 2 C 0 q þ h whch can by (5) be rewrtten as: v ¼ w þð1r Þh: For the rvalry equlbrum, snce v j ¼ R j < 0;@q =@r > 0 j =@r < 0, t follows by (22) that v equlbrum sharng proportons satsfy (superscrpt R for rvalry equlbrum ), ð23þ < 0. Thus by (23), the r R > 1 þðw=hþ; ¼ 1; 2: ð24þ It s nterestng to compare ths rvalry equlbrum wth the non-rvalry (revenue sharng) soluton, whch s obtaned when the two arports were perceved as ndependent n the sense that p j ðq 1; q 2 Þ¼0. It can be easly seen from (22) and (23) that the non-rvalry sharng proportons are gven by (superscrpt N for non-rvalry soluton ): r N ¼ 1 þðw=hþ; ¼ 1; 2: ð25þ Comparng (25) wth (24) leads to: Proposton 3. The revenue-sharng proportons are greater at the rvalry revenue-sharng equlbrum than under the non-rvalry revenue-sharng soluton,.e. r R > r N for = 1,2. The non-rvalry regme s, from (25) and (12), smlar to the case of a sngle (monopoly) arport examned n Secton 2, as expected: Lke a monopoly arport, each arport n the non-rvalry regme shares postve proporton r N ¼ 1 þðw=hþ of concesson revenue wth ts carrer. Whle r N nternalzes the flghts-concessons demand complementarty, the rvalry revenue sharng nvolves an addtonal term d.e. r R ¼ r N þ d whch s unque to the case of competng arports. Snce ths addtonal effect works by ndrectly nfluencng the behavor of the rval arport arlne par whch n turn wll mprove proft of the arport arlne par n queston the rvalry revenue sharng may be referred to as the strategc revenue sharng. Proposton 3 therefore shows that arport competton wll, owng to ths strategc effect, result n a hgher degree of revenue sharng than would be had n the case of sngle arports. Next, the rvalry equlbrum s compared to the non-rvalry soluton n terms of output, prce, proft and socal welfare. Here, welfare s the sum of passenger surplus and profts of the two arport arlne pars; hence, t takes the same form as (17). The comparson results are stated as follows:

11 A. Zhang et al. / Transportaton Research Part B 44 (2010) Proposton 4. Under symmetry, at the rvalry revenue-sharng equlbrum, () outputs are greater, () prces are lower, () arport profts are lower, and (v) socal welfare s hgher, than at the non-rvalry revenue-sharng soluton. Proof. Use D to denote any dfference of varables between the rvalry equlbrum and the non-rvalry soluton. Here, we just show parts () and (); the proofs for parts () and (v) are smlar to those of Proposton 2. () Applyng the mean value theorem (MVT) to q ðr 1; r 2 Þ yelds Dq =@r Dr =@r j Drj, =@r =@r j evaluated at some pont between r N 1 ; rn 2 and r R 1 ; r R 2. Under symmetry, Dr = Dr j =@r j j =@r ; consequently, h h Dq q þ q j =@r Dr ¼ h p j j pj jj =J Dr Snce p j j pj jj > 0 and J > 0 under the second-order and substtutes condtons and the stablty condton, respectvely, Dq q R q N must have the same sgn as Dr r R r N.ByProposton 3, rr > r N and hence q R > q N. () Applyng MVT to P (r 1,r 2 ), gven by (21), yelds DP ¼ P Dr þ P j Dr j, where p and P j are evaluated at ðr 1 ; r 2 Þ wth r N < r < r R (usng Proposton 3). Snce D r = Dr j under symmetry and Dr r R r N > 0, t follows that DP ¼ P þ P j r R r N and hence DP < 0 f (and only f) P þ P j < 0. By (21) and symmetry, t follows that P þ P j ¼ v þ v q 1 þ q 2 =@r ; where v and v j are evaluated at q ¼ q ðr 1; r 2 Þ.By(23), v ¼ w þð1r Þh whch s negatve gven that r > r N Furthermore, snce v j ¼ R j < 0 (substtutable arports) q þ q j =@r ¼ h Therefore, DP P R -P N <0. h ¼ 1 þðw=hþ. p j j pj jj =J > 0, t follows that P þ P j < 0. Perhaps the most surprsng result from Proposton 4 (especally as compared to the sngle-arport case) s related to proft comparson: both arport arlne pars wll derve lower profts through ths revenue-sharng rvalry. In effect, the arport arlne pars are trapped by the ncentve structure of the envronment. If one arport arlne par gnores the possblty of strategc use of revenue-sharng contracts whle the other par shares revenue strategcally, the frst par loses whle the second par gans relatve to the non-strategc sharng arrangement. Here the stuaton s smlar to a classc Prsoners Dlemma. As the pars move further away from ther jont proft maxmum through such a revenue-sharng rvalry, socal welfare nevertheless rses beyond the level achevable by sngle arports Multple arlnes Secton 3.1 studes the case of one carrer per arport. We now extend the analyss to a stuaton where there may be multple competng arlnes at each arport. Our second objectve n ths secton s to show that the general demand structure used n Secton 3.1 can be generated through explct consderatons of passenger behavor. More specfcally, our demand dervaton follows Basso and Zhang (2007) by consderng an nfnte lnear cty, where potental consumers are dstrbuted unformly wth a densty of one consumer per unt of length. Two competng arports are located at 0 (arport 1) and 1 (arport 2) and there are n carrers at arport, = 1,2 (see Fg. 2). At each arport, carrers provde homogeneous output, wth total output Q ¼ P n q k¼1 k and market prce p. The full prce faced by a consumer located at 0 6 z 6 1 and who goes to arport 1 s gven by p 1 +(4t) z, where 4t (>0) represents the consumer s transportaton cost from z to locaton 0. By choosng arport 1 or arport 2 (but not both) the consumer derves the followng respectve net utltes: U 1 ¼ V p 1 ð4tþz; U 2 ¼ V p 2 ð4tþð1 zþ; ð26þ $ $ V V 1 p 2 p l z 0 z ~ 1 r z z Fg. 2. Consumer dstrbuton and arports catchment areas (adapted from Basso and Zhang, 2007).

12 954 A. Zhang et al. / Transportaton Research Part B 44 (2010) where V denotes (gross) beneft from ar travel. 14 Assumng everyone n the [0,1] nterval consumes, then the ndfferent passenger ~z 2ð0; 1Þ s determned by settng U 1 = U 2,or ~z ¼ð1=2Þþðp 2 p 1 Þ=8t: ð27þ Gven that arport 1 also captures consumers at ts mmedate left sde, defne z l as the last passenger on the left sde of the cty who goes to arport 1. Smlarly, defne z r as the last passenger on the rght sde of the cty who goes to arport 2. Wth the unformty and unt densty of consumers, z l and z r are computed as: z l ¼ðV p 1 Þ=4t z r ¼ 1 þðv p 2 Þ=4t ð28þ The arports catchment areas are shown n Fg. 2, and ther demands are computed as: Q 1 ¼ ~z þjz l j¼ 1 2 þ p2 p 1 þ 8t Q 2 ¼ð1 ~zþþðz r 1Þ ¼ 1 2 p2 p 1 þ 8t From (29) the nverse demands are gven by V p1 ; ð29:1þ 4t V p2 : ð29:2þ 4t p ðq 1 ; Q 2 Þ¼ð2t þ VÞ3tQ tq j ; ; j ¼ 1; 2: ð30þ whch take the lnear functonal forms. Ths demand system has the propertes of p ¼3t < 0; p1 1 p2 2 p1 2 p2 1 ¼ 8t2 > 0, and substtutes condton (3). To solve the two-stage arport competton game, we begn wth an analyss of the second stage when arlnes engage n ntra- and nter-arport competton. Suppose for smplcty that carrers have lnear costs C(q)=F + cq. Consder frst that the two arports have the same number of carrers,.e. n 1 = n 2 n. Then arlne profts can be wrtten as: p k ðq 1 ; Q 2 ; q k Þ¼p ðq 1 ; Q 2 Þq k F cq k wq k þ r hq k f : The Cournot Nash equlbrum s characterzed by frst-order k ðq 1 ; Q 2 ; q k Þ=@q k ¼ p 3tq k c w þ r h ¼ 0; k ¼ 1;...; n; ¼ 1; 2; ð32þ (and the correspondng second-order condtons, whch hold 2 p k =@q 2 k ¼6t < 0). Gven the underlyng symmetry of ths set-up, the equlbrum quanttes are easly obtaned: q k ðr 1; r 2 Þ¼ ½3ðn þ 1Þr nr j Šh ð2n þ 3Þð4n þ 3Þt Back to the frst stage of the game, each arport s proft s: 2t þ V c w þ ; k ¼ 1;...; n; ¼ 1; 2: ð33þ ð4n þ 3Þt P ¼ wq þð1 r ÞhQ þ nf ; ¼ 1; 2: ð34þ Wth the arlne partcpaton constrants, these profts can be rewrtten as, P ðr 1 ; r 2 Þ¼ p Q 1 ; Q 2 c þ h Q n F þ p 0 : ð35þ Hence the rvalry equlbrum s characterzed by frst-order condtons, P ¼ w þð1r Þh 3t Q q =@r j =@r ¼ 0; ¼ 1; 2: ð36þ From (36) the equlbrum sharng proportons are obtaned as, r R ðnþ ¼1 þðw=hþð8n2 9Þð2t þ V þ h cþ=nð20n þ 21Þh; ¼ 1; 2: ð37þ Notce from (37) that f n = 1 (each arport has one carrer) then r R > 1 þðw=hþ ¼r N, a result obtaned n Secton 3.1 (see Proposton 3). 15 Further, t follows from (37) that dr R =dn < 0; ¼ 1; 2:.e. the sharng proportons decrease n the number of carrers servng the arports. For the general case where arports 1 and 2 have n 1 and n 2 carrers, respectvely, the nverse demands are gven by (30), where Q ¼ P n q k¼1 k s the aggregate demand at arport. Solvng the two-stage game yelds: ð31þ 14 Ths s an address model wth postve lnear transportaton costs, and the dfferentaton of the two arports s captured by consumer transportaton cost. Wthn a mult-arport regon, passengers may not necessarly choose an arport wth cheaper arfare, but may go to a nearer arport see the emprcal studes by, e.g. Pels et al. (2001), Fourner et al. (2007) and Ish et al. (2009). In addton to dstance, other aspects of arport dfferentaton may be captured by extendng the present formulaton. For nstance, Pels et al. (2000, 2001, 2003) have shown, usng a hypothetcal example and later the San Francsco Bay Area case study, that ground accessblty of an arport s the most mportant factor n affectng arport choces n a mult-arport market. Such dfferental ground access costs could be addressed by ntroducng a new parameter to the net-beneft functons (26). 15 Ths result can also be shown usng demand functons (30) and the property of ther assocated revenue functons R j ¼ p j q ¼tq < 0.

13 r R A. Zhang et al. / Transportaton Research Part B 44 (2010) ¼ 1 þ w h ½ð8n j þ 9Þn 9n j 9Šð14n þ 15Þð2t þ V þ h cþ ; ¼ 1; 2: ð38þ n ð280n n j þ 297n þ 297n j þ 315Þh Note that when n 1 = n 2 = n, the above expresson reduces to expresson (37). From (38) t s straghtforward to show that dr R =dn < 0 and dr R =dn j > 0, leadng to: Proposton 5. At the rvalry equlbrum wth n 1 and n 2 carrers at arports 1 and 2, respectvely, dr R =dn < 0 and dr R =dn j > 0:.e. the revenue-sharng proporton of an arport arlnes chan decreases n the number of carrers at ts arport, and ncreases n the number of carrers at the competng arport. If n 1 =n 2 = n, then dr R =dn < 0. The ntuton behnd dr R =dn < 0 s smlar to that of Proposton 1 (the substtutes case): as n rses (whle holdng n j constant) and arlne competton ntensfes, total output becomes ncreasngly excessve (relatve to proft maxmzaton) for the th arport arlnes chan. Antcpatng ths, arport wll have a greater ncentve to dscourage such competton, whch can be acheved by a smaller sharng proporton. 16 Whle ths result s largely expected, the other result, dr R =dn j > 0, s not obvous. Here, the explanaton s related to the number of compettors effect: An ncrease n the number of arlnes servng at arport j, whle holdng n unchanged, would ncrease arport j s output share n the two-arport market. 17 To counter the effect, arport strategcally rases the sharng proporton so as to nduce ts carrers to commt to greater output. Ths would credbly deter arport j s carrers from provdng more servce, whch n turn mproves proft of the th arport arlne chan. 18 Fnally, dr R =dn < 0 for n 1 = n 2 n, ndcatng that as n rses, the (negatve) excessve-output effect domnates the numberof-compettors effect. Lke Secton 3.1 (whch consders one carrer at each arport) we can compare the rvalry equlbrum wth the non-rvalry soluton n the present case however, each arport has multple carrers. It can be easly calculated that the non-rvalry sharng proportons are equal to: r N ¼ 1 þðw=hþðn 1Þð2t þ V þ h cþ=2n h; ¼ 1; 2: ð39þ Note, frst, that f n =1,(39) reduces to (25) and so t extends formula (25) to the case of multple arlnes. Second, usng (39) we obtan: dr N =dn < 0; ¼ 1; 2: ð40þ that s, as the number of arlnes at a sngle arport ncreases and hence (uncoordnated) output gets ncreasngly excessve for the carrers jont-proft maxmzaton, the arport then has a greater ncentve to curb output by usng a smaller sharng proporton. 19 Ths result s a clear extenson of Proposton 1 whch consders the effect of movng from one carrer to two carrers. Fnally, comparng (39) wth (38) yelds that r R > r N for any n and n j (n and n j can take dfferent values,,j = 1,2):.e. the revenue-sharng proportons are greater at the rvalry revenue-sharng equlbrum than under the non-rvalry revenuesharng soluton. Ths extends Proposton 3 of Secton 3.1 to the general case of multple arports wth each havng an arbtrary number of carrers Pure revenue-sharng contract So far our approach to revenue sharng has focused on a two part scheme under whch an arport chooses both a sharng proporton and a lump-sum fee on ts carrers for the rght to share concesson revenue. In ths secton we consder a pure sharng contract under whch the fxed fee s constraned to zero, whle keepng the rest of the model unchanged. Usng hat to denote the pure revenue-sharng equlbrum.e. ð^r 1 ; ^r 2 Þ these sharng proportons are constraned by the carrers partcpaton constrants. Unlke the two-part sharng scheme, therefore, negatve sharng s not possble snce these carrers cannot be compensated for wth any fxed payments by the arport, ndcatng ^r P 0. Gven these observatons, the effects of the pure revenue-sharng contract wll be compared to those of the two-part scheme as well as the no-sharng regme Sngle arport Consder frst a sngle arport served by two carrers, whch provde complementary, ndependent or substtutable servces to each other. The arport offers carrer a pure-sharng contract wth sharng proporton r, and the carrers compete by choosng quanttes q. We can show (see Appendx A): 16 Whle the two results have smlar ntutons, the present result s nevertheless obtaned n an envronment of competng arports. 17 Ths number of compettors effect s related to a well-known result found by Salant et al. (1983): n a Cournot market, a merger of two frms nto one entty reduces the merger partners proft (unless the merger leads to a monopoly). By nternalzng part of the effect that a frm s quantty decson has on the rvals proft, the merged entty sets ts quantty too low, thereby yeldng market share to the non-partcpatng frms. 18 As noted by an anonymous referee, Proposton 5 s rather nterestng n that t recalls the case of some LCCs (for nstance, Ryanar) at some secondary arports an LCC domnates. Whle competng wth man arports wth many arlnes, the LCC wll ask for a hgher r and the arport wll be nterested n ths. 19 An alternatve explanaton for the ntuton behnd ths result, suggested by an anonymous referee, s the followng: as the number of arlnes n an arport ncreases, the arport has more market power and may rase ts share 1-r. Moreover, the competton effect (or the ncrease of the output, recallng that w s fxed, so that wq ncreases wth q) s done n the downstream market by a more ntense competton amongst arlnes. Then the arport may ncrease 1-r wthout decreasng much the output. 20 Smlarly, Proposton 4 (ncludng the Prsoners Dlemma result) can be extended to the n-carrer case. The dervaton s avalable upon request.