Optimal Pricing of Flights and Passengers at Congested Airports: The Efficiency of Atomistic Charges

Transcription

1 TI /3 Tnbergen Insttute Dscusson Paper Optmal Prcng of Flghts and Passengers at Congested Arports: The Effcency of Atomstc Charges Hugo E. Slva Erk T. Verhoef* Faculty of Economcs and Busness Admnstraton, VU Unversty Amsterdam. * Tnbergen Insttute.

2 Tnbergen Insttute s the graduate school and research nsttute n economcs of Erasmus Unversty Rotterdam, the Unversty of Amsterdam and VU Unversty Amsterdam. More TI dscusson papers can be downloaded at Tnbergen Insttute has two locatons: Tnbergen Insttute Amsterdam Gustav Mahlerplen MS Amsterdam The Netherlands Tel.: +310) Tnbergen Insttute Rotterdam Burg. Oudlaan PA Rotterdam The Netherlands Tel.: +310) Fax: +310) Dusenberg school of fnance s a collaboraton of the Dutch fnancal sector and unverstes, wth the ambton to support nnovatve research and offer top qualty academc educaton n core areas of fnance. DSF research papers can be downloaded at: Dusenberg school of fnance Gustav Mahlerplen MS Amsterdam The Netherlands Tel.: +310)

3 Optmal prcng of flghts and passengers at congested arports: the effcency of atomstc charges Hugo E. Slva, Erk T. Verhoef 1 Department of Spatal Economcs, VU Unversty Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands Verson of 19 December, 2011 Abstract Ths paper nvestgates optmal arport prcng when arlnes provde mperfect substtutes products, and make decsons on capacty, schedulng and prcng. We show that the frst-best toll per flght may be hgher than the smple market-shares formulae that were recently derved for Cournot models, and approaches the atomstc toll whch gnores the arlnes nternalzaton of self-mposed congeston) as products become closer substtutes. Ths ncreases the relevance of congeston prcng and does not requre leadershp behavor. We also fnd that an arport requres two prcng nstruments to acheve the frst-best outcome: per-passenger subsdes to counteract arlnes market power, and per-flght tolls to correct congeston externaltes. We numercally analyze second-best polces of havng only one tax nstrument, as well as the performance of atomstc prcng, and fnd that the latter may offer a more attractve alternatve than what s suggested by smpler Cournot models. Keywords: Arport prcng, Congeston nternalzaton, Atomstc tolls JEL codes: H21, H23, L13, L93, R48 1. Introducton Delays at arports have been consstently ncreasng over the last years, becomng a major problem worldwde see, for example, Rupp 2009) and Santos and Robn 2010)). Besdes capacty enlargements, the prce mechansm has been wdely dscussed and proposed to manage congeston. Ths approach conssts of an arport authorty settng user charges, wth the possblty to charge passengers, arlnes or both. 2 Many papers have studed optmal arport prcng: for example, Brueckner 2002) frst showed that n olgopoly, arlnes competng n a Cournot fashon nternalze congeston mposed on themselves; therefore, the optmal charge should account only for the fracton of congeston that s mposed on compettors. We wll refer to ths as the Cournot toll, as opposed to the atomstc toll that consders margnal congeston costs mposed on all other flghts and passengers, regardless of the operator. One mportant mplcaton of a Cournot toll s that a domnant arlne should pay a lower charge per flght than small arlnes Brueckner, 2005). Pels and Verhoef 2004) extend the analyss by explctly consderng market power dstortons. They show that a welfare maxmzng arport has to correct two effects: arlnes market power by subsdzng them, and congeston externaltes wth the Cournot toll. Further extensons found that ths congeston prcng rule remans optmal. 3 However, optmal congeston toll has been subject of contnuous debate over the last decade, and the desrablty of the Cournot toll has been questoned, also from the emprcal sde. In contrast to the road case, where users behave atomstcally, the relevant queston n avaton markets s what share of congeston arlnes actually do nternalze when makng schedulng decsons of flghts. If they nternalze a hgh proporton of congeston costs, charges that optmally account for ths wll be relatvely low and thus should have a small mpact on flght patterns and socal welfare. Ths nternalzaton hypothess, based on Brueckner s analyss, s supported by emprcal evdence by Mayer and Sna 2003) wth U.S. data, and by Santos and Robn 2010) wth European data. Nevertheless, Danel 1995), who frst dentfed the potental for nternalzaton of congeston, argues wth a smulaton model that atomstc behavor may n fact be more pertnent from an emprcal pont of vew;.e., that arlnes do not take nto account self-mposed congeston when makng schedulng decsons. As We are very grateful to Vncent van den Berg and to Erc Pels for constructve and sharp comments whch have sgnfcantly mproved the paper. Fnancal support from ERC Advanced Grant OPTION #246969) s gratefully acknowledged. Correspondng author Emal addresses: h.slvamontalva@vu.nl Hugo E. Slva), e.t.verhoef@vu.nl Erk T. Verhoef) 1 Afflated to the Tnbergen Insttute, Roetersstraat 31, 1018 WB Amsterdam, The Netherlands. 2 Another approach to manage arport congeston s to use slot sales and slot tradng; see Brueckner 2009), Basso and Zhang 2010) and Verhoef 2010). 3 Zhang and Zhang 2006) explctly ncorporate arport s costs and capacty decsons, and Basso 2008) ncludes arlne dfferentaton, orgn and destnaton arports and schedule delay costs. Basso and Zhang 2007) analyze rvalry between two congestble facltes, whch agan could be arports.

4 a consequence, the optmal toll should be the so-called atomstc toll that gnores any nternalzaton, and that s equal to total margnal congeston costs. The atomstc behavor of arlnes s further supported by emprcal evdence by Danel and Harback 2008) and by Rupp 2009), who contrbutes to the nternalzaton debate by showng that arlnes are not nternalzng passenger delay cost, and are hence behavng more lke atomstc compettors p. 26). Moreover, from a dfferent perspectve, Morrson and Wnston 2007) argue n favor of levyng atomstc tolls at congested arports, because they fnd a small net beneft loss when an arport charges the atomstc toll nstead of the Cournot toll. Ths dvergence between vews on the extent to whch atomstc versus Cournot tolls are more desrable has been addressed before. For example, Brueckner and Van Dender 2008) present a theoretcal analyss showng that when one arlne acts as a Stackelberg leader and nteracts wth a large number of frnge carrers, the leader behaves atomstcally as long as the products are perfect substtutes. In ths case, the optmal congeston toll would be the atomstc charge, because congeston s not beng nternalzed by any carrer. Although theoretcally vald, the Stackelberg settng wth carrers behavng n a perfectly compettve fashon seems to be restrctve. Czerny and Zhang 2011) present a dfferent argument and state that, when travelers wth dfferent values of tme are consdered, and n absence of prce dscrmnaton by carrers, t mght be useful to ncrease the arport charge towards the atomstc toll. Ths s to protect passengers wth a relatvely hgh value of tme from congeston caused by passengers wth a relatvely low value of tme. Obvously, ther analyss cannot explan dfferences n fndngs between models that consder homogeneous values of tme. The purpose of ths paper s to show that the optmal congeston charge per flght can be hgher than the Cournot toll and can be sgnfcantly close to the atomstc toll, under rather plausble assumptons. We fnd that arlnes nternalzaton of congeston can be close to atomstc behavor, even when arlnes fully recognze the mpact of ther flght schedulng on arport congeston, and n absence of leadershp behavor n contrast to Brueckner and Van Dender 2008)), and wthout heterogenety n values of tme n contrast to Czerny and Zhang 2011)). We show, n a duopoly settng where outputs are mperfect substtutes, that arlnes can nternalze less than the self-mposed congeston, because they take nto account the fact that an extra flght mposes congeston on ts compettor s passengers, affectng postvely ts own demand and proft. Ths new schedulng ncentve for arlnes, yelds an optmal congeston toll that les between the margnal congeston cost mposed on the compettors passengers the Cournot toll) and the atomstc toll. We also fnd that the sze of the devaton from the self-mposed nternalzaton result of Cournot competton depends only on the degree of product dfferentaton. The entre range of tolls can be optmal: when the dfferentaton s strong, the optmal congeston charge s close to the Cournot toll. Conversely, f the dfferentaton s weak, arlnes should be charged a toll very close to the atomstc one, even when one of them s a domnant arlne wth a hgh market share. Ths result may help explanng why the emprcal and smulaton studes provde such a wde range of estmatons regardng nternalzaton, and ths fndng provdes new arguments n favor of applyng congeston prcng at arports. The result holds n a settng where arlnes compete smultaneously takng rval s decsons arcraft sze, frequency and fares) as gven; but also when arlnes are engaged n a competton where fares are chosen n the last stage, whle frequency and arcraft sze competton occurs n prevous stages, as long as they are unable to observe rval s play. On the other hand, when they are engaged n ths sequental competton, but they observe the compettor s play, the outcome s the same as f they take rval s traffc quantty) as gven, therefore the optmal toll s the Cournot toll. We also reproduce the result of Brueckner and Van Dender 2008) for a Stackelberg leader wth a Cournot follower, and extend t to the case of a Stackelberg leader wth a Bertrand follower, fndng optmal tolls that le between the Cournot toll and the atomstc toll for both players. In ths last settng, the optmal toll agan approaches the atomstc toll as the dfferentaton s weaker. Varous polcy mplcatons follow from the analyss. If the frms behave takng the rval s fare as gven nstead of traffc), the optmal toll can be anywhere n between the margnal congeston cost mposed on the compettors passengers, and the atomstc toll. Perloff et al. 2007), n an emprcal study on arlnes conduct n a duopoly market, estmate that ths behavor, n fact, better descrbes the ndustry. 4 If ths s the case, congeston prcng may brng more sgnfcant welfare gans and congeston reductons. We present some numercal examples to assess the relatve effcency of second-best polces, and, for example, fnd that only usng a congeston per-flght charge and levyng atomstc tolls yeld substantal benefts when arlnes do not behave n a Cournot fashon, and when the degree of product dfferentaton s not too strong. The paper s organzed as follows. Frst, n Secton 2, we ntroduce the model that ncludes arcraft sze, fare and frequency decsons n an olgopolstc arlne market, and that formally takes nto account market power exerton and potental) congeston nternalzaton. In Secton 3 we derve analytcal solutons for the arports problem, specfcally frst-best tolls and optmal capacty nvestment. We show that a welfare maxmzng arport can only reach the frst-best outcome by usng two tax nstruments, namely per-flght and per-passenger tolls. Moreover, congeston and market power effects are separate: the market power exerton can only be corrected by means of a per-passenger subsdy, whle the optmal congeston charge should only be charged wth the per-flght toll. Secton 4 presents numercal exercses to quantfy the analytcal results, to assess the effcency of second-best polces, and to study the performance of levyng atomstc tolls. Fnally, Secton 5 concludes. 4 Ths s found usng the dataset of Brander and Zhang 1990) and Oum et al. 1993), that has been used to support Cournot behavor. 2

5 2. Arlnes duopoly model For the analyss, we consder a vertcal settng on a sngle market,.e., a sngle orgn-destnaton par. In the frst stage an arport chooses capacty, toll per flght and toll per passenger charged to the carrers that use the faclty. In the second stage, a duopoly of arlnes compete wth arcraft sze, frequency and fare as decson varables. Followng Zhang and Zhang 2006), we model only one arport for analytcal smplcty, but the conclusons reman the same f the other arport s ncluded, as long as the arports share the objectve functon e.g. they perform jont welfare maxmzaton). 5 For the arlnes market, we consder the dfferentated duopoly proposed by Dxt 1979), assumng that demands arse from the followng quadratc utlty functon: Uq, q j ) = A q + q j ) B q E q q j + B q 2 j )/2, 1) where q s the amount of good hereafter, when subscrpt j appears n the same expresson wth, t refers to the rval arlne), A, B and E are postve parameters wth B > E > 0 so that goods are mperfect substtutes. 6 Ths utlty functon gves rse to a lnear demand structure, wth nverse and drect demands: θ = A B q E q j, 2) q = a b θ + e θ j, 3) where θ s the full prce of good and parameters a, b and e satsfy a = A/B + E), b = B/B 2 E 2 ) and e = E/B 2 E 2 ). The full prce of travelng wth arlne s assumed to be: θ = p + D + g, 4) where p s the fare and the passengers cost of congeston s represented by D: the congeston delay experenced at arports. D depends on arport capacty K) and on the total number of take offs and landngs at the congested arport F = f + f j ). Fnally, g s the schedule delay cost faced by a passenger that travels wth arlne, whch depends only on the flght frequency of the arlne f ). The fact that schedule delay does not depend on rval s frequency, as congeston does, reflects our assumpton that n the dfferentated duopoly, frequency s perceved as an arlne-specfc attrbute. We make the plausble assumptons that D s dfferentable n F, that g s dfferentable n f and that: D f > 0, 2 D f 2 2 D K f < 0, = 2 D f f j 0, g f < 0, 2 g f 2 D K > 0,. < 0, 5) Congeston thus ncreases wth the number of flghts and the effect s stronger when congeston s more severe; congeston decreases wth arport s capacty; schedule delay cost decreases wth arlne-specfc frequency, and that effect s smaller when frequency s hgher. 7 Followng Brueckner 2004), we model arlnes cost C ) as a functon of arcraft sze and frequency n the followng way: C = f γ f + γ s s ), 6) where γ f and γ s are postve cost parameters and s s the number of seats per flght. The underlyng assumpton s that cost per flght s a lnear functon of the number of seats, a relaton that has been also found n a cost-engneerng study for arlnes by Swan and Adler 2006). 8 Congeston costs for arlnes are not consdered n the analyss because we focus on 5 If arports are not regulated by the same authorty or f arports ndependently maxmze profts both arports have to be formally modelled. For a dscusson on the mplcatons of two local welfare maxmzng arports see Pels and Verhoef 2004) and for a dscusson on ndependent proft maxmzaton see Basso 2008). 6 The mperfect substtuton allows to have arlnes n equlbrum wth dfferent generalzed prce and traffc. The fact that an arlne can have demand despte havng a hgher generalzed prce than the compettor has been also modelled wth a brand-loyalty varable that gves the addtonal gan from travellng wth a specfc arlne relatve to travel wth the other arlne Brueckner and Flores-Fllol, 2007; Flores-Fllol, 2010). In our model t s also possble to model a preference for a specfc carrer by lettng the demand parameters B and E) vary across carrers. 7 Ths set of assumptons s common n the lterature: the lnear delay functon used by Pels and Verhoef 2004) and the convex functon used by Zhang and Zhang 2006) satsfy the assumptons regardng D. The schedule delay functon that s nversely proportonal to the arlne frequency satsfes the condtons for g see Brueckner 2004) and Basso 2008) for a dscusson). 8 In ths work, a cost functon per trp s calbrated usng dstance and arcraft sze as explanatory varables, then holdng dstance fxed the functon s lnear n number of seats. 3

6 passengers congeston, but ncludng them would not change the results n any essental way because t only changes C and does not modfy the demand structure. 9 Wth the cost functon defned, we can now wrte the proft of arlne as: π = q p f γ f + γ s s ) f τ f q τ q, 7) where τ f s the per-flght toll charged by the arport and τ q the toll charged per passenger. One of the goals of ths paper s to assess the mpact of dfferent knds of strategc nteracton. In modellng the arlnes competton, we study a closed-loop game where arlnes, frst, smultaneously decde frequency and arcraft sze, and fares n a second stage. We thereafter look at an open-loop game, where arlnes do not observe the play of the others or, equvalently, arcraft sze, frequency and fares are decded smultaneously. Fnally, we study two Stackelberg settngs, where the leader chooses all the relevant varables pror to a follower who takes the rval s traffc volume as gven, or the rval s fare as gven Closed-Loop In ths game settng, there are two stages: arlnes frst smultaneously choose arcraft sze and frequency and, n the second stage, they compete on fares note that an arlne can transport at most f s passengers, hence n the frst stage they also make a capacty decson). We solve ths game usng backward nducton, fndng that arlnes unque equlbrum s to choose frequency, fares and arcraft sze accordng to the followng condtons the complete dervatons are presented n Appendx A). p = γ s + τ q + q B, 8) γ f + τ f D = q + g ), 9) f f q = s f. 10) Equaton 8) states that the fare charged by an arlne has three terms: ) the margnal cost per capacty unt γ s ); ) the arport charge per passenger τ q ) and ) a conventonal monopolstc markup reflectng carrer s market power, whch s related to the senstvty of demand and own demand q B). Equaton 9) states that arlne s margnal cost per flght equals margnal benefts for own passengers margnal congeston savngs plus margnal schedule delay benefts); therefore, arlnes nternalze own-passenger congeston. The last equaton show that arlnes do not have dle capacty. These rules bascally descrbe that arlnes nternalze congeston on ther own passengers and charge a markup whch equals q θ / q, a result analogous to the rules obtaned prevously n Cournot competton e.g. Pels and Verhoef, 2004; Zhang and Zhang, 2006; Basso, 2008). 10 Moreover, solvng a game wth arcraft sze s ), frequency f ) and traffc q ) as decson varables, we also obtan equatons 8) 10) as frst-order condtons. Therefore, ths closed-loop game s equvalent to assumng that arlnes act smultaneously takng compettor s quantty as gven. 11 From now on, we refer to Cournot nternalzaton to the result obtaned n ths game settng,.e. nternalzaton of congeston mposed on own-passengers. Ths closed-loop settng s also equvalent wth a three stage game where arlnes make the decson of arcraft sze n the frst stage, because, once the fare and frequency are chosen, demands are already set. Hence, dle seat capacty s not optmal. It s worth notng some aspects about arlnes behavor that arses from ths model. Arlnes do not charge passengers for congeston because they set frequency and fare separately; for any gven demand, they nternalze own-passengers congeston by settng frequency accordng to 9) and adjustng arcraft sze to accommodate the passengers. In a fxedproportons model, an addtonal passenger necessarly ncreases delays and the only way to nternalze ths s by chargng self-mposed margnal congeston costs to passengers. But, when arcraft sze s a strategc varable, ths s no longer desrable, because they can accommodate a new passenger by ncreasng arcraft sze by 1/ f at a cost of γ s whch they do charge to passengers, see 8)) wthout rasng delays. Ths also explans why the per-flght toll τ f ) s absent n the arlnes fare: t affects frequency and arcraft sze settng, whle keepng the cost per passenger constant at γ s + τ q. 9 Congeston mposed on arlnes works out n a way smlar to congeston mposed on passengers. If congeston costs are ncluded, arlnes would not nternalze congeston costs mposed on the compettors flghts and ths should be corrected n frst-best tolls, as found by Basso 2008), Brueckner 2009) and Verhoef 2010). 10 The second term n the rght-hand sde of equaton 9) s present n prevous studes ncludng schedule delay cost. It s n Brueckner 2004) for a monopoly and t s n Basso 2008), but does not appear n the prcng rule of arports because t s set optmally by a prvate arlne from a socal welfare perspectve. 11 The ntuton of ths result comes from Kreps and Schenkman 1983) who show that under mld demand assumptons, Cournot outcomes are the unque equlbrum n a two-stage game where frms frst choose capacty and then prces. In our game, the key dfferences are the mperfect substtutablty and the fact that frst stage s more than capacty nvestment snce t affects generalzed prce through frequency. Proof s n Appendx A. 4

7 2.2. Open-Loop The open-loop game can take the same two-stage dynamc game structure as the closed-loop, but arlnes cannot observe rval s play. Ths settng s analytcally equvalent to smultaneous decsons. Therefore, the problem faced by an arlne s to maxmze proft equaton 7) takng rval s frequency, arcraft sze and fare as gven. Because havng dle seat capacty only decreases proft, t s straghtforward that an arlne wll set the number of seat such that arcraft s flled s = q / f ); ths allows us to express proft n terms of fare and frequency. Usng the same argument, ths open-loop game s equvalent wth a two-stage game where arlnes frst compete on arcraft sze and then on frequency and fares. Rewrtng equaton 7) and explctly ncludng the functons arguments wthout ncludng rval s varables snce they are taken as gven) we get: Then, frst-order condtons wth respect to fare and frequency yeld: π p, f ) = q p, f ) p γ s τ q ) f γ f + τ f ). 11) π p = 0 p = γ s + τ q + q b, 12) π = 0 γ f + τ f = p γ s τ q f ) q. 13) f Agan, the fare charged by an arlne ncludes the margnal cost per capacty unt γ s ), the arport charge per passenger, and a market-power markup. The dfference wth the closed-loop game s that the market power effect s now weaker 1/b < b/b 2 e 2 ) = B). The equvalence between the closed-loop game and Cournot competton mentoned earler helps explanng why the market power effect s stronger n the closed-loop game: when arlnes take rval s prce as gven, the outcome s more compettve than when they take rval s quantty as gven see Sngh and Vves 1984) for a dscusson n a general context). From the frequency frst-order condton 13), we get that frequency s set optmally by equatng margnal cost per flght to the revenue gan from an extra flght, as found by Brueckner 2004). For further nterpretaton, note that usng equaton 3) we can rewrte q / f as: q = b θ + e θ j D = b + g ) + e D. 14) f f f f f f Ths expresson shows that the effect of an ncrease n the arlne s number of flghts, has two effects on ts demand: t changes both schedule delay cost and congeston for own passengers, but t also ncreases the congeston experenced by compettor s passengers when ts frequency s fxed or taken as gven). The second effect has a postve mpact for the arlne, snce ncreasng compettor s congeston rases own demand due to the fact that arlnes offer mperfect) substtute outputs. Usng p γ s τ q = q /b from equaton 12) and equaton 14), we can rewrte 13) as: γ f + τ f = q D 1 e ) + g f b f ). 15) Ths equaton defnes how an arlne sets frequency. It dffers from equaton 9) for the closed-loop game n the term multplyng margnal congeston costs. In ths game, an arlne nternalzes congeston mposed on ts own passengers but also takes nto account the congeston mposed on ts compettor, as explaned above. Ths s represented by the degree of dfferentaton e/b that appears n equaton 15). Ths term causes a dfference wth the common nternalzaton fndng, and wth the closed-loop game result, because now arlnes are not takng the compettor s traffc as gven. In Cournot competton, arlnes beleve that they are not able to nfluence the compettor s traffc by rasng congeston, smply because they do not see the effect by assumpton. On the other hand, when takng the compettor s fare together wth frequency as gven, traffc s the result of settng the generalzed prce through the two varables. Therefore, arlnes realze that they can nfluence compettor s traffc, or ncrease own demand, by rasng the rval s congeston. The sze of the devaton from the tradtonal result of nternalzaton depends drectly on the degree of dfferentaton e/b. Ths rato ranges from 0 when products are completely ndependent to 1 when products are perfect substtutes. The fracton of runway congeston consdered by an arlne, when settng frequency, s gven by the rato of congeston terms from equaton 15) and total margnal congeston costs: q D 1 e/b) q = 1 e ). 16) q + q j ) D q + q j b 5

8 As e < b, carrers act as f they nternalze less congeston than what s mposed on ther own passengers, as n the closedloop game and n Brueckner s 2002) Cournot model. Only when products are close to be ndependent, the effectve nternalzaton approaches the market share. When they are close to be homogeneous, arlne behavor approaches atomstc behavor. For example, f the output s symmetrc and the rato of dfferentaton s 0.5, arlnes nternalze only 25 percent of congeston costs, nstead of one half. Whch one of the settngs presented so far s more approprate depends on market-specfc condtons. One argument n favor of the closed-loop settng s the contrast between the hgh number of fare adjustments performed by arlnes and the comparatvely slow process of changng frequency and arcraft sze. On the other hand, the prvate nature of arlnes and the fact that carrers serve several routes can be used to support the dffculty of observng rval s capacty decsons on a specfc market, whch leads to the open-loop game outcome. There are few emprcal studes that estmate arlnes strategy. Brander and Zhang 1990) and Oum et al. 1993) estmatons whch have been used to support Cournot behavor are well summarzed by the latter s concluson that the overall results ndcate that the duopolsts conduct may be descrbed as somewhere between Bertrand and Cournot behavor, but much closer to Cournot, n the majorty of the sample observatons p. 189). Ths means that fares generally exceed margnal costs Bertrand outcome) and by less than the Cournot markup n our notaton q θ / q = q B). The results of the closed-loop game are consstent wth Cournot behavor whle the open-loop fare equaton 12) s somewhere between margnal cost and Cournot fare, because arlnes take rvals fare as gven n a context of mperfect substtuton. In a recent work, Perloff et al. 2007) estmate arlnes prcng strateges usng the same database as the prevous studes, but wth several methods and allowng for the possblty that arlnes provde dfferentated servces on a route. They show that the average Bertrand Lerner ndex s vrtually the same as the average Lerner ndex estmated from the data wth all the methods, whle the Cournot and collusve ndexes are much hgher. Ths provdes emprcal evdence that supports the open-loop settng as representatve Stackelberg leader and Cournot follower In the Cournot-Stackelberg settng we suppose that arlne s a leader and arlne j the follower that chooses traffc q j ), frequency f j ) and arcraft sze s j ) takng the leader s strategc varables as gven. As we explaned n Secton 2.1, ths s equvalent to the closed-loop settng and therefore the follower s behavor s characterzed by frst-order condtons 8)-10). The leader maxmzes proft knowng the response of the follower to ts own decsons. As n prevous settngs, t s not optmal to have dle capacty, so the proft functon of the leader s: π q, f ) = q [p q, f, q j, f j ) γ s τ q ] f γ f + τ f ), 17) where both q j and f j depend on the leader s choce varables. Frst-order condtons wth respect to traffc and frequency yeld: π q = 0 p = γ s + τ q + q π f = 0 γ f + τ f = q B + E q j [ D 1 + f j f f + D f ) j, q f q 18) ) + g + E q ] j. f f 19) Snce the follower s responses are downward-slopng the proof s n Appendx B), the leader sets a hgher quantty than an arlne that takes rval s traffc as gven as the follower does). As a consequence, the leader s market power effect s weaker for a gven frequency, hgher traffc mples a lower fare). Regardng frequency settng frst-order condton 19), as the leader antcpates the way the follower reacts, the ncentves to reduce frequency are dfferent from those n the Cournot case, because of two effects. Frst, the leader predcts that any frequency reducton s partally offset by an ncrease on number of flghts by the follower f j / f < 0). As can be seen n 19) the term nvolvng margnal congeston s cut by ths expresson whch stuates leader s nternalzaton n between the Cournot case of self-mposed congeston and the atomstc behavor, just as ponted out by Brueckner and Van Dender 2008). The second effect not drectly related to margnal congeston s the last term multplyng q on the rght handsde of equaton 19), whch further reduces the nternalzaton. The leader realzes that any frequency ncrease nduces a reducton on follower s traffc, therefore the frequency reducton ncentve s dmnshed. The overall effectve nternalzaton s n between congeston mposed on own passengers and atomstc behavor, but the magntude of the devaton depends, among other thngs, on the degree of dfferentaton. In Secton 4, we quantfy the devaton and expand more on ths matter. 6

9 2.4. Stackelberg leader and Bertrand follower Fnally we solve a Stackelberg game wth arlne as a leader and arlne j as a follower that chooses fare p j ), frequency f j ) and arcraft sze s j ). The only dfference wth the prevous settng s that the follower takes the leader s fare as gven nstead of traffc. We refer to ths settng as the Bertrand-Stackelberg game. The problem for the follower s the same as n the open-loop game,.e. maxmze proft equaton 11) wth respect to fare and frequency, yeldng the same frst-order condtons equatons 12 and 13). For the leader, the proft functon now s: Frst-order condtons wth respect to fare and frequency yeld: π p, f ) = q p, f, p j, f j ) p γ s τ q ) f γ f + τ f ). 20) [ π = 0 γ f + τ f D = q 1 e ) 1 + f j f f b f π = 0 p = γ s + τ q + q p b, 21) ) + g f e b p j f + g j f j f j f )] b b, 22) where b b e p j / p q / f j f j / p. In ths game, follower s reacton functons wth respect to frequency are downward-slopng whle the reacton functons wth respect to fare are upward-slopng see Appendx B). Ths feature makes that n both frst-order condtons for the leader equatons 21 and 22), there are opposte effects that prevent from quantfyng analytcally the effects. In fare settng t s not possble to state whether the market power effect s hgher or lower n comparson wth the open-loop game. Concernng frequency choce, we fnd the same effect that reduces nternalzaton due to the follower at least partly offsettng reductons n the number of flghts f j / f < 0). But, n ths case, the sgn of the second effect last term n brackets on the rght-hand sde of equaton 22) cannot be determned a pror, so the nternalzaton reducton wth respect to the open-loop mght well be counteracted. In Secton 4 we study the magntude of these effects wth numercal examples, suggestng that the leader nternalzaton can be less than n the open-loop game, but stll n between Cournot and atomstc behavor. As we dscuss n Secton 1, there are emprcal studes that support the nternalzaton hypothess as well as studes that reject t. We beleve that our model helps n explanng such a wde range of fndngs. When arlnes take rval s fare, frequency and arcraft sze as gven open-loop and Bertrand-Stackelberg settng) the degree of nternalzaton s n between non-nternalzaton and the self-mposed congeston. Moreover, t depends on demand-structure parameters the rato of dfferentaton e/b) and there s emprcal evdence supportng ths behavor. Therefore, t s possble that n some market an arlne behaves almost atomstcally, even f t s domnant, whle n others t nternalzes a bg share of congeston. Wth the arlnes market characterzed, we now analyze arport prcng and capacty nvestments for the arlnes model we propose. 3. Arport prcng and capacty nvestment We consder the frst-best case of a unweghted) welfare maxmzng arport wth capacty and both per-flght and perpassenger toll as nstruments. We solve the arport maxmzaton problem analytcally n ths secton, and numercally n Secton 4. The dervaton of frst-order condtons presented n ths secton s n Appendx C. Socal welfare s defned as the sum of benefts for each agent: consumer surplus, arlnes profts and arport s proft. The frst of ths, wth quanttes and full-prces beng q, q j, θ and θ j, s just Uq, q j ) θ q θ j q j. Usng 1) and 2), straghtforward calculatons yeld the followng expresson: CS = B 2 q2 1 + q2 2 ) + E q 1 q 2. 23) We assume that arport s costs are separable and proportonal to traffc, frequency and capacty, shapng arport profts n the followng way: Π = q τ q c q ) + f τ f c f ) ) K r, 24) where c q s the constant) operatng cost per-passenger, c f the constant) operatng cost per-flght, r the cost of captal and K the capacty of the arport. Addng arlnes proft equaton 7) we can express socal welfare as a functon of traffc, frequences and fares: 7

10 [ B ] S W = 2 q2 1 + q2 2 ) + E q 1 q 2 + q p c q ) f γ f + s γ s + c f ) [K r], 25) where the frst bracketed term s consumer surplus, the second bracketed term s arlnes and arport s per-passenger revenues mnus costs arport s revenues from tolls cancel out aganst arlnes costs from tolls), the thrd s per-flght revenues mnus costs agan arport s tolls cancel out) and the last bracketed term s capacty costs. Both traffc volumes are a functon of fares and frequences, but we omt the arguments. Straghtforward calculatons lead to the followng condtons for optmal fares. p γ s c q = 0,. 26) Ths result states that optmal fare must equal arlnes margnal cost per capacty unt plus arport margnal operatng cost per passenger. The welfare maxmzng fare that should be charged to passengers does not nclude any congeston term because as explaned n the prevous secton arlnes take congeston nto account only n ther frequency settng. From frst-order condtons for frequency we obtan the followng: q + q j ) D f q g f = c f + γ f,. 27) Ths means that the optmal frequency must be such that the margnal cost per flght rght-hand sde) equals margnal net benefts of all passengers from congeston plus schedule delay savngs left-hand sde). We defne total margnal congeston costs as the congeston cost that an extra flght mposes on all passengers D/ f q +q j )). If arlnes do not nternalze any congeston at all, they should be charged ths amount plus the arport s margnal operatng cost per flght c f ), the so-called atomstc toll. Fnally, the optmal nvestment rule for the arport s: q + q j ) D K = r. 28) Ths shows that arport capacty s ncreased untl margnal cost equals margnal benefts from congeston reductons. Havng establshed the frst-order condtons for socal optmal fares and frequences, we can now derve the optmal toll per passenger τ q, τq f j ) and per flght τ, τ f j ) by usng arlnes fare and frequency frst-order condtons of each game e.g., for the open-loop game, we use equatons 12), 15), 26) and 27)). Results now follow treated by game type. Closed-loop game For the closed-loop game, equvalent to Cournot behavor, t can be expected that optmal tolls are consstent wth the earler arport prcng lterature. Indeed, the per-passenger toll n 29) s margnal operatng cost plus a subsdy equal to market power markup, and the per-flght toll n 30) equals margnal operatng cost plus congeston mposed on the rval arlne passengers. τ q = c q q B, 29) τ f = c f + q j D. 30) However, the subsdy s separate from the congeston toll, and the per-passenger frst-best toll s negatve when the market power effect s bgger than arport per-passenger margnal operatng cost. Hereafter, we use the term Cournot toll for the charge that accounts only for the congeston costs mposed on the compettor s passengers equaton 30)). Open-loop game For the open-loop game, we fnd: τ q = c q q 1 b, 31) τ f = c f + q j + e ) b q D. 32) The per-passenger toll n the open-loop game s margnal operatng cost plus a subsdy equal to the market power markup. For the per-flght toll, we obtan that t s margnal operatng cost per flght plus a congeston toll, whch s however dfferent 8

11 Brueckner s 2002) Cournot toll. A frst-best arport charges the congeston costs mposed on rval s passengers q j D ) plus an addtonal term. Ths new term s the own-passengers margnal congeston cost q D ) tmes the degree of dfferentaton e/b). When ths rato s zero, the goods are ndependent and the optmal per-flght toll s the congeston mposed on the rval s passengers. When t s one, goods are perfect substtutes, and the frst-best toll s the so-called atomstc toll. Any other feasble value of e/b between zero and one) yelds a charge that s somewhere n between the atomstc toll and the Cournot toll. The toll reflects that arlnes are nternalzng less than self-mposed congeston. Hence, frst-best tolls are closer to the atomstc toll than n Cournot competton; and, the weaker the dfferentaton s, the hgher the frst-best toll should be. The ntuton of ths result, whch to the best of our knowledge s new n the arport prcng lterature, s that when goods are ndependent, there are two monopoles usng the same faclty n order to serve two ndependent markets. Therefore, a global welfare maxmzng arport charges to each carrer the congeston mposed on the compettor, whch was entrely gnored by the operators because of the ndependence. In the other extreme, where the rato equals one, goods are perfect substtutes and snce arlnes take the rval s fare as gven the Nash equlbrum s the perfectly compettve outcome as n Bertrand competton wth homogeneous goods). In ths extreme case, an arlne wll expect that any reducton n ts own flght volume wll be offset by an equally bg ncrease n the compettor s flght volume. As a consequence, the total number of flghts, thus arport congeston, wll reman unchanged, and nternalzaton of own congeston makes no sense. Ths new result on optmal congeston prcng suggests that welfare maxmzng per-flght tolls mght be hgher than what s suggested by the Cournot model, and hence have a more substantal mpact n arlnes schedulng decsons, and therefore mght yeld more szable welfare gans. Even f the market s hghly concentrated, the domnant arlne has to be charged for a hgh proporton of total margnal congeston costs f the dfferentaton between the arlnes s not too strong. Stackelberg leader and Cournot follower game Snce the frst-order condtons for the follower are the same as n the closed-loop settng, the frst-best tollng rules also concde. The per-passenger toll s margnal operatng cost plus the market power subsdy see equaton 29) and the per-flght toll s margnal operatng cost plus congeston mposed on the rval equaton 30). For the leader ), the frst-best tolls are: τ q = c q q B, 33) τ f = c f + q j + q f ) j D + q f E q j f, 34) where B = B + E q j / q + D f j / q and dervatves n absolute value are negatve see Appendx B). The nterpretaton of these tolls s the same as n prevous settngs. The frst-best per-passenger toll s the margnal operatng cost plus a subsdy equal to the market power markup, and the per-flght toll corrects for unnternalzed congeston. Because the leader when consderng the effect of ts own decsons on the followers alters nternalzaton, the optmal congeston toll charged s n between the Cournot and the atomstc toll. Ths optmal congeston toll reproduces the result of Brueckner and Van Dender 2008) n ther Stackelberg behavor wth a Cournot follower game. Stackelberg leader and Bertrand follower game The optmal tollng rules for the follower are dentcal to the open-loop tolls equatons 31 and 32), although the actual tolls wll dffer because the equlbrum s dfferent. On the other hand, the optmal tolls charged to the leader ) are: τ q = c q q 1 b, 35) ) τ f e b b ) = c f + q j + q D e b + q D + g + D f j ẽ p j f + g j j ), 36) b b b f b f f where b b e p j / p q / f j f j / p. The optmal toll per passenger s arport s margnal operatng cost per passenger plus the subsdy that corrects market power. Snce b s the dervatve of own traffc wth respect to own fare takng nto account the effect on the follower) the nterpretaton s the same. The per-flght toll corrects the frequency settng so that the leader sets the welfare maxmzng frequency. As the sgn of b b cannot be determned a pror, the per-flght toll for the leader has to be studed numercally. In Secton 4 we do ths, fndng that n the numercal examples the optmal toll s n between the Cournot and the atomstc toll, and can be above or below the open-loop toll dependng on the degree of dfferentaton. We have shown n ths secton that a welfare maxmzng arport needs to use two taxes, namely per-passenger and perflght tolls, to reach the frst-best outcome. It corrects the market power effect wth a per-passenger toll and the frequency 9

12 neffcency wth a per-flght charge. The former τ q ) s below arport margnal operatng cost per passenger because t counteracts the arlne market power exerton by means of a subsdy. As a consequence, ths toll s negatve when arlnes markups exceed arport s margnal operatng costs. Conversely, the frst-best per-flght toll s always above arport margnal operatng cost, because arlnes partally nternalze congeston. If only one tax can be appled, the arport s facng a second-best problem, and whch nstrument s better to apply depends on market specfc condtons. If the arlnes market power effect s stronger than the congeston effect, then usng only per-passenger tolls s more effcent than chargng only per-flght, and vce versa. In the extreme case of monopoly operaton, the per-flght toll s unnecessary, and the frst-best s attaned wth per-passenger subsdes only. The results also mply that the frst-best outcome cannot be reached by only chargng passengers, because also charges per movement are necessary. If the authorty or the faclty wants to charge arlnes per flght and passengers per trp, a postve tax for the passengers s not consstent wth welfare maxmzaton f arlnes have market power. We further expand on ths n the numercal analyss below. Two recent analyses have put a queston mark on the desrablty of the tradtonal Cournot congeston toll,.e. total margnal congeston costs tmes the market share of each arlne n our model the closed-loop toll n equaton 30). Morrson and Wnston 2007) fnd a small dfference between the net benefts of chargng the Cournot toll versus the atomstc toll that gnores any nternalzaton. Then, Brueckner and Van Dender 2008) shows that Stackelberg behavor wth a Cournot follower yelds optmal arport tolls that le n between of both polces. The results of ths model gve new nsghts nto the debate concernng the desrablty of the tradtonal Cournot congeston toll: frst, the optmal toll mght well be close to the atomstc toll wthout assumng leadershp behavor and wthout abstractng from arlnes market power exerton. Ths s the open-loop case, smultaneous competton wth arcraft sze, fare and frequency as strategc varables, and the closed-loop settng f moves cannot be observed by the compettor. From equaton 32), t s straghtforward that the closeness of the optmal toll to the atomstc toll depends on market-specfc characterstcs rato of dfferentaton e/b). Therefore, wthn the same settng, the entre range of tolls can be optmal. We also confrm the nternalzaton result of Brueckner and Van Dender 2008) for a Stackelberg leader wth a Cournot follower and extend t to the case of a Stackelberg leader wth a Bertrand follower, fndng even hgher optmal tolls. These fndngs may lead to optmsm on the relatve effcency of atomstc congeston prcng n avaton markets. As we cannot compare welfare and equlbrum values analytcally, n Secton 4 we solve numercally the equlbrum for an arport chargng the atomstc toll and make the comparsons wth the frst-best to assess the relatve effcency. 4. Numercal analyss In ths secton we present a numercal analyss that allows makng comparsons that are not possble analytcally. We also analyze the performance of second-best polces. Despte the smplfed structure of the model, we use parameters that are as much as possble calbrated so as to reflect realstc values. We use the followng functonal forms for the schedule delay cost g ) and passengers congeston cost D): g = γ 1, f 37) ) f + f β j D = α, K 38) where the schedule delay cost n 37) s nversely proportonal to the arlne frequency and γ s a constant representng the monetary value of a unt of schedule delay tme as t would be wth unformly dstrbuted desred departure tmes, and equally spaced flghts), as s usual n the lterature e.g. Brueckner, 2004; Basso, 2008). The congeston delay at the arport n 38) s a functon of the volume capacty rato, wth α beng proportonal to the passengers value of travel tme, K the capacty, and β the power of the functon. Our reference scenaro for calbraton has symmetrc arlnes and assumes a margnal-operatng-cost prcng arport,.e. a toll per passenger of c q and a toll per flght of c f. We use the closed-loop and open-loop settngs for calbraton purpose, and leave the Stackelberg settngs for analyses. The parameters and equlbrum outputs are: Table 1: Parameter values. Demand Cost parameters parameters Arlnes Arport A 1250 a 750 γ 30 c c q 10 B 1.04 b 1.5 α 26 c c f 1000 E 0.63 e 0.9 β 3 r

13 Table 2: Equlbrum outputs, margnal operatng cost prcng arport. Game Total Fare Total Arcraft K Arlnes settng traffc Frequency sze Proft Open-loop ,452 Closed-loop ,493 Table 3: Generalzed prce. Game Gen. Fare Congeston Schedule Demand settng prce share share delay share Elastcty Open-loop Closed-loop Arlnes operatng cost parameters are from Swan and Adler 2006), usng a reference dstance of km and an adjustment n the per-passenger cost as suggested by the authors. Values of tme are from Morrson and Wnston 1989), whle congeston, delay and demand parameters are set such that equlbrum elastcty wth respect to generalzed prce n both settngs s close to , the mean value of 204 observatons reported by Brons et al. 2002). Values are set to the same monetary unts U.S. dollars) throughout the calbraton process Internalzaton We frst solve the problem wth a welfare maxmzng arport and a duopoly of symmetrc arlnes, to assess the equlbrum share of congeston that s nternalzed by each player. In ths case, each carrer has an equlbrum share of 50 percent of passengers. Fgure 1 shows the percentage of congeston that s nternalzed by each agent for a feasble range of the rato of dfferentaton, e/b. Recall that a value of 0 means ndependent goods, whereas 1 represents pure substtutes. Accordng to the frst-order condtons derved n Secton 2, n a closed-loop settng each arlne wll nternalze half of the total margnal congeston cost see equaton 9). For the open-loop settng, nternalzaton s n between one half and zero, and decreases lnearly n the rato of dfferentaton; see 16). For the game wth a Stackelberg leader and a Cournot follower, the leader wll nternalze less than half of total margnal congeston and the follower wll act as n the closed-loop game, nternalzng exactly half of total margnal congeston costs. Fnally, Bertrand follower s nternalzaton s the same as n the open-loop game, whle the leader behavor cannot be quantfed a pror. The self-mposed nternalzaton of Cournot behavor and the lnear decrease of the nternalzaton for arlnes n an open-loop settng are shown n Fgure 1 n black lnes. To llustrate the possble mplcatons of our results, we may look at parameters estmated by Perloff et al. 2007) for a lnear demand structure as n equaton 3)) n a duopolstc competton. They fnd values for the rato e/b between 0.4 and 0.7, whch mples that arlnes would nternalze between 12 and 30 percent of congeston costs, and therefore should be charged for a remanng 70 to 88 percent of the total margnal congeston costs. Fgure 1: Internalzaton. For both Stackelberg games, we fnd that the leader nternalzaton s less than the self-mposed congeston, and decreases towards atomstc behavor as the rato e/b ncreases. In the case of Cournot behavor the two sold lnes), the leader 11

14 always nternalzes less than the follower, whose nternalzaton s always the congeston mposed on own passengers. For Bertrand behavor n the Stackelberg settng the two dashed lnes), we fnd that the leader nternalzes roughly the same congeston as the follower, both beng less than the self-mposed and approachng to zero as the rato of dfferentaton grows. Brueckner and Van Dender 2008) also analyze Stackelberg behavor wth a Cournot follower, but suppressng market-power by assumng perfectly elastc demand, and assumng that the outputs of the carrers are perfect substtutes. We extend ths to the case of prce-senstve demand and mperfect substtuton, fndng smlar results regardng nternalzaton of congeston: the leader nternalzes less than self-mposed congeston, but do not fully reach atomstc behavor. Ths s represented n Fgure 1 by the sold grey lne. From ths analyss t follows that, for Bertrand behavor n the open-loop game as well as n Stackelberg competton, optmal congeston charges are close to the atomstc charge when the dfferentaton s not too strong on the rght end of Fgure 1). For ths reason, optmal per-flght congeston tolls mght have a more sgnfcant mpact on arlnes schedulng decsons and, therefore, n allevatng congeston, than what was suggested by earler Cournot models Alternatve polces We next look at alternatve polces: ) the second-best case when an arport can use only one tax nstrument, and ) the relatve performance of atomstc prcng Usng one tax nstrument As shown n Secton 3, a welfare maxmzng arport needs two prcng nstruments to reach the frst-best outcome. It corrects the market power effect wth a per-passenger subsdy and the frequency neffcency wth a per-flght charge. We now look at what happens when t can only use one nstrument. For ths purpose, we defne the relatve effcency Ω p as the welfare gan due to a polcy p), relatve to the frst-best gan: Ω p = S W p S W mc, 39) S W f b S Wmc where superscrpt f b refers to the frst-best case, and mc to an arport chargng margnal operatng costs as n the reference scenaro for calbraton). Fgure 2a shows the relatve effcency of both second-best polces for the base calbraton. The results show that, n our calbraton, the market power effect domnates the congeston effect, yeldng a hgh relatve effcency for a per-passenger subsdy and a low one for the per-flght toll. The results also show a sgnfcant effect of game type on the performance of a polcy. As we dscuss n Secton 2, the market power exerton and the amount of nternalzed congeston are always hgher n a closed-loop settng than n an open-loop settng. Ths leads to a hgher relatve effcency of the per-passenger subsdy, and a lower effcency of the per-flght charge, n the closed-loop game than n the open-loop game. For the Stackelberg games, the relatve effcency les n between the closed-loop and open-loop, but the rankng s senstve to the parameters as s the degree of nternalzaton). The results furthermore show that, for the chosen parametrzaton, the socal gans for the two nstruments are nearly addtve. That s, the gans from ntroducng the one nstrument are almost nsenstve to the other nstrument beng n place already. Ths underlnes the lack of substtuton between the polces. a) Relatve effcency by game type. b) Relatve effcency for multple frms. Fgure 2: Second-best relatve effcency analyss. The second analyss we perform has the purpose of assessng the relatve effcency of both second-best polces when the market power and the congeston effect have dfferent comparatve mportance. We do ths by ncreasng the number 12