PRICING VS SLOT POLICIES WHEN AIRPORT PROFITS MATTER

Transcription

1 Workng Paper 009- PRICING VS SLOT POLICIES WHEN AIRPORT PROFITS MATTER Leonardo J. Basso Department o Cvl Engneerng, Unversdad de Chle Caslla 8-3, Santago, Chle Ph , Fax lbasso@ng.uchle.cl Anmng Zhang Sauder School o Busness, The Unversty o Brtsh Columba 053 Man Mall, Vancouver, BC, Canada, V6T Z Ph , Fax anmng.zhang@sauder.ubc.ca Copyrght 009 by Centre or Transportaton Studes

2 March 009 Revsed: September 009 Abstract: Ths paper analyzes prcng and slot-allocaton mechansms to manage arport capacty when prots are mportant to an arport, owng to budget constrants or prot maxmzaton. We nd that congeston prcng and slot tradng/slot auctonng do not lead to the same results. Total trac s hgher under slot auctons than under congeston prcng. Furthermore, arport prots matter ust margnally, then slot auctons wll outperorm congeston prcng n terms o achevng a hgher obectve-uncton value. On the other hand, arport prots matter sucently hghly, whch mechansm s better s then very much dependent on parameter values. In partcular, congeston prcng may be strongly preerred over slot auctons or certan parameter values. The mpact o congeston remedy mechansms on ndvdual carrers s also examned. Keywords: congeston prcng, slot aucton, slot tradng, arport prcng

3 . Introducton Trac growth has outpaced capacty ncreases at maor arports around the world. For the last several years arlnes and passengers have ncreasngly been suerng rom lght delays: t s then no surprse that congeston delays have become a maor publc polcy ssue. Other than capacty expansons, perhaps the most suggested and dscussed congeston remedy has been the prcng mechansm, where the socal planner works to correct congeston externaltes by settng congeston tolls. In the context o arports, recent lterature has recognzed the mportance o market power that ar carrers may have at a gven arport: wth non-atomstc arlnes optmal Pgouvan tolls would only charge a carrer or the congeston t mposed on other carrers, owng to the nternalzaton o lght congeston. In partcular, a hgher toll should be mposed on a smaller carrer, snce t nternalzes a smaller amount o congeston than a larger carrer, somethng that s potentally controversal n mplementaton as t would be poltcally senstve to establsh derentated tolls (Brueckner, 00; Morrson and Wnston, 007. In addton to prcng, other wdely dscussed congeston remedes are related to control and management o arport slots, ncludng slot sales, slot tradng and slot auctons. 3 Proponents o these solutons have argued that by settng a number o slots, congeston problems would be obvously solved whle, by allowng tradng o slots or by auctonng them, t would be ensured that each slot goes to the arlne whch values t most. Thereore, these mechansms would crcumvent the problem o perceved unarness that congeston prcng has, whle usng secondary markets to acheve ecency. An obvous queston that arses s whether these two mechansms, prcng vs. slot polces, are equvalent. Recently, Verhoe (008 and Brueckner (009 compared analytcally the prcng and slot-allocaton remedes. They ound that the st-best congeston prcng and slot tradng/slot auctonng are equvalent n terms o both the amount o trac generated and total socal welare, as long as the xed number o slots s optmally chosen. On the other hand, slot sales whch s actually treated as a unorm congeston toll s neror to congeston prcng and slot tradng/slot auctons except when the arlnes are symmetrc n sze. Czerny (008, on the other hand, provded a graphcal analyss o the prcng and slot-based approaches to arport congeston when there s uncertanty on congeston costs. He ound that wth uncertanty, congeston prcng may be more preerable than slot constrants. 4 Ths paper generalzes the analyss o Verhoe (008 and Brueckner (009 n whch the uncertanty ssue s abstracted away by consderng a stuaton where arport revenues are mportant, that s, they are not smply transers rom one type o agent to another. In other words, we explctly consder the role o arport prots n the comparson Fgures and acts regardng arport congeston can be ound, or example, n Krby (008. See, e.g., Brueckner (00, 005, Pels and Verhoe (004, Zhang and Zhang (006, Basso (008, Basso and Zhang (008a, Johnson and Savage (006, Morrson and Wnston (007, Brueckner and Van Dender (008. See also Danel (995 who rased the possblty o sel nternalzaton by arlnes at an arport. 3 See, e.g., Jones, Veho and Marks (993 and Forsyth and Nemeer (008 or general dscussons o the economcs o arport slots. 4 In a more recent study, Czerny (009 shows analytcally that arport networks (as opposed to the case o sngle arports ncrease the relatve welare benets.

4 o prcng vs. slot approaches to arport management. Ths emphass on arport behavor s derent rom the approach o Verhoe (008 and Brueckner (009 n whch the arport, whle t may derve prots rom ts operaton, s not consdered n the welare analyss. Ther approach s approprate, or example, the arport as a decson maker s publc owned and has no budget restrctons, and so t maxmzes arlnes and passengers surplus. In that case, arlne payments or the use o arport servces are n act ust transers between the arlnes and arport. Arport prots can be mportant or a number o reasons. Frst, t would be o central nterest or any prvate arport and note that, ollowng the prvatzaton o the Brtsh Arports Authorty (BAA n Brtan n 987, more and more countres have decded to partally or completely prvatze ther arports. Whle most o these prvatzed arports reman prce regulated, some have been deregulated and others have been subected to prce montorng only rather than ormal (ex ante regulatons (e.g., Forsyth, 997; Starke, 00; Productvty Commsson, 00. Second, wth the arport corporatzaton (or commercalzaton movement n recent years, even publc arports have been under growng pressure rom governments to be more nancally sel-sucent and less relant on government support. 5 In other words, prots can also matter to a publc arport that s subect to a certan degree o cost recovery (e.g., Zhang and Zhang, 003. We wll rst show that, whle achevng the socal optmum, congeston prcng, slot tradng and slot auctons do generate derent amounts o revenue to an arport, wth slot auctons generatng the hghest revenues. We then nvestgate two questons: rst, the obectve uncton to be maxmzed s a weghted uncton o arport prot and arlnes and passengers surplus, s congeston prcng stll equvalent to slot tradng or slot auctons n terms o trac and overall obectve-uncton value? Second, whch mechansm wll yeld hgher trac volumes and arport revenues as well as hgher overall obectveuncton values? We nd that when prots matter to an arport, prcng and slot tradng/slot auctonng do not lead to the same results. Frst, total trac s hgher under slot auctons than under congeston prcng. Second, the mportance o arport prots s not too large, then slot auctons wll always outperorm congeston prcng n terms o achevng a hgher obectve-uncton value. Thrd, the mportance o arport prots matters enough, whch mechansm s better s then very much dependent on parameter values. In partcular, prcng mght be strongly preerred over slot auctons or some parameter combnatons. The paper s organzed as ollows. Secton sets up the basc model and examnes the eect o congeston remedes (prcng vs. slot allocatons on arport prots. Secton 3 nvestgates the equvalence result when arport prots matter, and Secton 4 urther compares the alternatve mechansms n terms o trac, arport revenue and overall obectve-uncton value. Secton 5 contans the concludng remarks. 5 See, e.g., Saulny (008. 3

5 . The Model and Eect o Congeston Remedes on Arport Revenue One o the man results on arport congeston prcng s that the optmal toll to be charged to an arlne has two components (see Pels and Verhoe, 004, Zhang and Zhang, 006 and Basso, 008. The congeston component charges or un-nternalzed congeston and s thereore amed at reducng an arlne s producton, as s the usual case o Pgouvan taxes. Yet, snce arlnes are non-atomstc (.e., each produces more than one lght, ths charge consders only the congeston mposed on other arlnes lghts but not on own lghts. The second component o the toll s actually a subsdy, amed at ncreasng an arlne s producton. Ths s optmal because ceters parbus, arlnes reduce producton levels below what s optmal gven that they usually have market power. Thus, whether the nal toll wll be postve or negatve depends on whch o the components domnates. Importantly, the market power eect domnates, ths mples that n the absence o tolls there s no problem o congeston but a problem o not enough lghts. As ndcated above, the two theoretcal studes ndng equvalence between prcng and slot tradng/slot auctons are Verhoe (008 and Brueckner (009. In the ormer study, arlnes oer a homogenous product but der n costs, and t s assumed that the congeston eect domnates the market power eect. In the optmum then, only one arlne operates. Brueckner (009 nstead ocused on the case o perectly elastc demands so that the market-power eect s suppressed, whch allows a pure comparson to be made between congeston prcng and slot allocaton polces. Arlnes oer ndependent products yet congeston lnks arlnes cost unctons whch mples that at the optmum the two arlnes usually operate. We wll ollow the setup o Brueckner (009; the model s as ollows. There are two arlnes servng a congested arport. Wth perectly elastc demands, passengers o arlnes and are wllng to pay ull prces.e. sum o tcket plus congeston costs P and P, respectvely, or travel n and out o the arport. The two arlnes may be asymmetrc n the sense that, wthout loss o generalty, P P and thereore arlne serves the hgher-prce market and s a larger carrer P > P, whle the two carrers are symmetrc P = P. Let denote the lght volume o carrer, T ( the carrer s margnal cost uncton, and Φ the arport (runway charges pad by carrer. Then each arlne s prot uncton s gven by: [ P T ( ] c( Φ π (, =, =, ( where c ( s the congeston cost uncton, whch represents the sum o passengers and arlnes costs caused by arport congeston (here, and below, the ndces and appear n the same expresson, then t s to be understood that. Followng Brueckner (009, t s assumed that T( 0 > 0, T '( > 0 and T ''( 0, and that c( 0 = 0, c '( 0 6 and c' '( 0. Furthermore, n speccaton ( the number o passengers per lght has, 6 Note that the assumpton on T mples that an arlne operates under dseconomes o scale. As ndcated n Brueckner (009, p. 683, ths assumpton s needed to generate sensble results n the presence o perectly 4

6 as s common n the lterature, been assumed to be constant, whch s urther normalzed to unty. Consequently, represents both the number o lghts operated by carrer and the number o passengers carred by. Consder rst that a publc arport chooses arport congeston management methods (prcng or slot polces to maxmze socal welare and let us denote the arport s obectve Pub uncton as OF. Snce consumer surplus s zero under perectly elastc demands, total surplus s ust the sum o arlne and arport prots. For smplcty, we assume that the arport s margnal costs are constant and normalzed to zero, and that there are no xed costs. As a consequence, the arport s prot equals ts total revenue Φ = Φ Φ, and Pub OF s gven by: = = Φ ( π [ ] [ ] OF Pub W π = P T ( P T ( ( c( ( where W denotes total surplus. As can be seen rom (, any arport charges, whch make up arport revenues, wll come rom the arlnes, and thus they cancel up n total surplus. As a result, W s ust equal to the arlnes pre-arport-payment prots, and the arport s nterests can be gnored n the welare analyss as done n Brueckner (009 and Verhoe Pub (008 or or that matter, n the analyss o behavor o a publc arport wth OF. Pub Maxmzng OF wth respect to and, the ollowng rst-order condtons are obtaned: P T( T '( c( ( c'( = 0, =, (3 Assumng the second-order condtons to hold, equatons (3 mply the rst best (.e., welare-maxmzng trac. As shown n Brueckner (009, the rst-best trac s mplementable through Pgouvan taxes, whch wll take derent values or asymmetrc arlnes. In ths setup, slot tradng (herenater, ST and slot auctons (herenater, SA acheve the same equlbrum as the Pgouvan taxes. In the case o ST, arlnes negotate untl they agree on the number o slots to be traded and the prce per slot, startng rom some ntal number o slots gven or ree by the arport. In the second case, Brueckner (009 assumes that the arlnes bd or each ncremental slot ther margnal benet, yet the process o auctonng s accordng to a unorm-prce,.e. at the end, there s only one prce y to be pad or each o the slots. Ths prce s chosen such that carrers bddng at least y or ncremental slots receve them and a total o n slots s allocated. 7 Ths last pont s o vtal mportance: n both cases, ST and SA, the total number o slots n whch s the same elastc demands when ull prces der across carrers. Brueckner urther argues that decreasng returns may be a plausble assumpton or a carrer operatng at a congested arport. 7 O course, there are a number o derent bddng mechansms one can thnk o. The mechansm used by Brueckner (009 and here ndeed s ust one possblty, whch we deem to be a good startng pont and that reles on the assumpton o non-manpulatve behavor rom the part o the arlnes; see Brueckner (009 or urther dscusson. 5

7 as the total lght volume s taken as parametrc by the arlnes when they make ther decsons. 8 As a result, the equlbrum condtons or both SA and ST are: P T ( T '( c( n = y, =, (4 = n (5 where y s the prce o slot, whch goes to the arport n the case o SA, or s exchanged between the arlnes n the case o ST. What Brueckner has ponted out s that n both the rst-best and ST/SA cases, the let-hand sde o equaton (4 s set to equal a common value, ndependent o carrer. Ths pont can, here, be seen by comparng (4-(5 wth (3. Thereore, n addton n s set to equal, where superscrpt denotes the rstbest equlbrum, then the congeston prcng, ST and SA all lead to the same trac levels and, hence, also to the same level o socal welare: the three congeston remedes are equvalent. Here, we examne rst a ollow-up queston: are these methods equvalent n the amount o arport revenues generated? and not, whch mechansm delvers a larger arport prot? Agan usng Brueckner s setup and lettng z be the prce that arlne must pay wth congeston prcng, then z = c'(,.e. un-nternalzed congeston. Thus, arport revenues Φ under the rst-best congeston prcng soluton are equal to: FB Φ = z z = c'( (6 where FB stands or the rst best congeston prcng. Under Brueckner s assumpton, ST does not provde the arport wth revenues. SA does, however, and s equal to SA Φ = y (, where y s gven by ether o the two expressons n (4 and must be evaluated at the rst-best requences, and n s gven by. Thus, rom (4 and (5 we obtan y = P T ( T '( c( It ollows rom (3 that are gven by: Φ SA = = y = ( c'( [( ( ] c'( FB FB [( ( ] c'( Φ > Φ. Thereore, arport revenues under SA c' ( where the second equalty ollows rom the use o (6. The above analyss leads to: (7 8 As argued n Brueckner (009, the arport announces the slot total whle allocatng ths xed number o total slots to ndvdual ar carrers, wth the allocaton acheved ether by ST or through SA. As a consequence, t s proper or the arlnes to vew the total as xed when each decdes on ts lght level. 6

8 Proposton. To acheve the socal optmum, the three congeston remedes namely, congeston prcng, slot tradng and slot auctonng generate derent amounts o arport revenue. Slot auctonng generates greater revenue to the arport than congeston prcng, whch n turn generates greater arport revenue than slot tradng. A varant o Proposton s contaned n Brueckner (009, Proposton 6, p. 688 when he examnes whether the arport revenues rased under the derent prce and slot polces cover the arport cost. The ntuton or ths result s as ollows: n the slot case an arlne knows that t gets an extra slot, congeston wll not worsen snce an extra lght or one arlne necessarly means a lght less or the other. Hence, ts margnal wllngness to pay or an extra lght (slot s larger than n the prcng regme, where wth each extra lght, congeston ncreases margnally. And snce n the slot case, arlnes pay accordng to ther wllngness to pay, revenues wll be larger. The derental arport revenues/prots are o no materal consequence the arport s a publc/welare-maxmzng arport, as ndcated earler n (. However, as ponted out n the ntroducton, prots are mportant, or example, or a prvate arport or or a publc arport subect to cost recovery. In ths context then, one would lke to know whch arport management mechansm perorms better n terms o the obectve o the arport, and what the consequences o choosng ether mechansm are n terms o arport revenues, trac and so on. Consequently, we consder the ollowng general arport obectve uncton: OF = Φ ( π π, (8 ( Here, the parameter captures how mportant arport prots (revenues are to the Pub decson maker. More speccally, when =, OF ( reduces to OF n (, or Brueckner s welare uncton. In ths sense, ormulaton (8 nests Brueckner s settng (recall consumer surplus contnues to be zero. Because the most mportant nsghts n ths paper occur or >, we usty ths modelng n three derent ways. Frst, would be greater than one the arport needs to cover costs, or to create revenues or the local cty. In any o those cases, a nancal constrant n the optmzaton problem would generate a Lagrange uncton where arport revenues are multpled by plus the (postve Lagrange multpler; ths sum would be. A second ustcaton s that may capture the degree o commercalzaton/prvatzaton, wth a larger value ndcatng a hgher degree o commercalzaton/prvatzaton. Obvously, the lmtng case corresponds to a pure unregulated prvate arport. For example, n a case lke ths, may capture the result o barganng over the obectve uncton between a prvate arport and publc partes (whch may be modeled as the result o a Nash-barganng process, where captures the degree o barganng power o the arport wth respect to the publc partes nvolved. Fnally, one can resort to a poltcal economy explanaton: most arports are stll wholly or maorty publc owned and are managed by government departments, or ther delegated agents who behave lke government bureaucrats. One could then appeal to the lterature on budget-maxmzng 7

9 bureaucrats, whch oten portrays these ndvduals as maxmzng a weghted sum o ther budget and consumer utlty (here, consumers o an arport are arlnes as well as passengers, n the same sprt as our treatment o beng between and nnty n the obectve uncton (8. 9 Wth the obectve uncton n (8 at hand we can then attempt to answer our two prncpal research questons: Queston : Consderng that the arport, or a regulator, chooses a lght-allocaton mechansm to maxmze OF ( gven n (8, s prcng stll equvalent to slot tradng or slot auctons n terms o trac volumes s and the OF value gven by (8? Queston : Whch mechansm wll yeld hgher trac volumes and arport revenues as well as hgher overall OF values? Beore nvestgatng these two questons n Sectons 3 and 4 respectvely, we note that or > t may not be approprate to call the prcng remedy the rst best congeston prcng, as there s now market power rom the part o the aport that starts to kck n and, thereore, prces may end up above socal margnal cost. 0 Morover, snce even or = assymetrc arlnes would lead to derent rst-best congeston prcng tolls, as ncreases the prcng remedy may be consdered a prce-dscrmnaton mechansm. Hence, we wll use PR or the prce mechansm whle, or smplcty, we keep to denote ts resultng values. Furthermore, note that snce the slot-sales mechansm actually corresponds to unorm prcng, t smply precludes the possblty o prce dscrmnaton (.e. the arport s chargng asymmetrc arlnes derentated tolls and hence represents a more restrcted optmzaton. As a consequence, t wll always be domnated by PR. For ths reason, we wll ocus only on the comparson between PR prcng and ST/SA n the remander o the paper. 3. Equvalence o Prcng and Slot Polces When Arport Prots Matter Ths secton examnes whether congeston prcng and slot tradng/slot auctons reman equvalent when arport prots matter. Consder rst the equlbrum under SA/ST. It s qute obvous that or any gven number o slots beng traded or auctoned, the (condtonal on n trac levels under SA or ST are stll gven by equatng the two equatons gven n (4. That s, the rato o trac levels under SA or ST s, or all, gven by P T T '( = P T ( T '( (9 ( 9 See, e.g. Moene (986; equaton 9, n partcular. We thank Jan Brueckner or pontng out ths motvaton to us. 0 A prvate road would charge, as part o ts prce, un-nternalzed congeston but wll add to ths a market power mark-up (see, e.g., Small and Verhoe, 007. In a vertcal structure though, a prvate arport would charge more than the un-nternalzed congeston o each carrer (see Basso,

10 In ths sense, can only play a role once we look or the optmal n. In other words, the change n the obectve uncton does not aect arlnes bddng or tradng behavor, a qute reasonable outcome. Next consder the equlbrum under PR. In ths case, we consder that the arport wll charge, per lght, and to arlnes and, respectvely, and the arlnes take these charges as gven and decde on the number o lghts n order to maxmze ther prots. z z The result o the Nash equlbrum are arport trac levels condtonal on arport charges; these are the derved demands that the arport aces (see, e.g., Basso and Zhang, 008b or more dscusson on derved demands wthn the vertcal structure approach to arport prcng. The downstream arlne equlbrum, obtaned ater computng the arlnes rst-order condtons, gve us: '( ( '( ( c c T T P z =, (0 =, The two equatons n (0 dene, mplctly, the nverse derved demands or the arport and. Solvng the system o equatons, one can obtan the correspondng drect demands., ( z, ( z The problem aced by the arport s, usng (8, then: ( [ ] = =, ( ( ( c z T P z z OF Max ( where and are gven by (0. Note that we set the problem n terms o nverse demands or smplcty but obvously the results do not change we use the drect demands. From (, the rst-order condtons can be derved n a straghtorward manner: takng dervatve wth respect to one obtans: z z 0 ''( ''( ''( ''( ''( ''( '( 3 '( '( '( 3 '( ( ( = T T c c c c T T c c c T c P Then, reorganzng terms we get Note that ths per-lght arport charge scheme has ruled out a two-part tar scheme whch, by havng more nstruments, obvously leads to hgher obectve uncton values. We wll dscuss the ssue urther n the concludng remarks. 9

11 [ P T ( T '( c( ] ( c'( ( [ T '( T ''( ] ( ( [(3 c'( ( c''( ] = 0 and the rst-order condton or s gven by the analogous expresson wth the and subscrpts nterchanged. From these two equatons, the rato o trac levels under PR can then be obtaned rom: [ P T ( T '( ] ( [( T '( T ''( (3 c'( ] [ P T ( T '( ] ( [( T '( T ''( (3 c'( ] = A comparson between (3 and (9 gves us a hnt that trac levels wll be derent a prcng regme s used by the arport, than a slot aucton/slot tradng regme s used, whch mples that the lght allocaton mechansms would be no longer equvalent. A proposton summarzng ths non-equvalence result and ts proo ollows. Proposton. ( I arport prots matter so that a lght-allocaton mechansm s chosen to maxmze OF (, then sucent condtons or the prcng and slot tradng/slot auctonng mechansms to lead to derent results are >, P P and that T ( s lnear or not too concave. ( Prcng and slot tradng/slot auctonng are equvalent when = and then n s set to (.e. the rst-best total trac level. In that case, snce arlne payments are ust transers between the arlnes and arport, t would be also true CP ST / SA that OF ( = = OF ( =. Proo: ( Trac levels under PR ulll equaton (3. Trac levels under SA or ST, on the other hand, ulll equaton (9 leadng to SA SA SA ( T ( T '( SA SA SA P P = T ( T '( And snce T '( > 0 and T ''( 0, then T ( x xt '( x s ncreasng n x and t ollows that SA SA P > P > P (3. We now assume wthout loss o generalty that P > and SA SA SA SA thereore >, and show that assumng =, = leads to a contradcton. Frst, the SA and PR trac levels are equal, then rom (9 and (3 t would be true that: SA SA SA ( T '( T ''( (3 c' ( SA SA SA SA SA SA SA SA ( T '( T ''( (3 c'( SA = SA SA SA SA 0

12 whch s equvalent to SA SA SA SA SA SA SA SA ( T '( T ''( ( T '( T ''( ( SA SA SA SA c'( = Next, T ' s lnear (and so T ''( x = 0 or s not too concave such that T ''( x s not too negatve then ( T '( x xt ''( x x s ncreasng n x, and snce the let-hand sde o the last equaton s postve by P >, the only way that the rght-hand sde can be postve s P SA SA, whch s a contradcton. ( The proo o the second part o the proposton s n Brueckner (009 and uses the denton o OF(. > Note that P = P does not mply that the mechansms are equvalent; but the nonequvalence n that case has to reer to the actual number o slots n chosen, somethng that we cannot analyze n general (.e. or the case o T' '( x 0. In the next secton however, we show that T s lnear (and so T ''( x = 0, then the non-equvalence holds even P = P. 4. Comparson o Trac, Arport Revenue and Obectve-Functon Value Gven that prcng and slot polces are not equvalent when arport revenues (prots matter, we now nvestgate the second research queston: Whch congestonremedy mechansm wll yeld hgher trac volumes and arport revenues as well as hgher overall OF values? Note, rst, that whle trac volumes are not an obectve n tsel, t s stll a useul comparson. Ths s because derences n ndvdual carrers trac volumes ndcate the extent o carrer asymmetry, whch s an mportant startng pont o the nvestgaton, whereas derences n total trac reveal the extent o arport congeston and delays, whch s a man perormance ndcator or alternatve congeston management mechansms. Second, because slot tradng does not generate revenue or the arport under the modelng assumptons, we wll ocus on the comparson between prcng and slot auctons. Thrd, we assume rom now on that both the congeston cost uncton and the arlne margnal cost uncton are lnear: c ( n = B n, T ( x = t a x (4 wth B 0, t > 0 and a > 0. The smplcaton s necessary to push the analyss urther but, as t wll become clearer, t s more than sucent to show some nterestng results whch clearly do not hnge on these lnearty assumptons. Prcng soluton

13 Substtutng (4 nto the rst-order condtons (, the PR trac levels are obtaned by solvng the resultng system o equatons. Straghtorward algebra leads to: ( a B( ( P t B ( P t = (5 ( a a B ( B (3 ( ( a B( ( P t B ( P t = (6 ( a a B ( B (3 ( Snce ( (3 0, the denomnators n (5 and (6 are postve, ndcatng that the second-order condtons or the arport problem ( hold. Further, gven that, P and a B > B, t s clear that s postve and P are greater than t. Snce P P > t are requred or the cost uncton to start below the ull prce, t ollows that > 0. Whlst > 0 n the model, s postve only parameter a s sucently large. Speccally, > 0 P [( t P ( P ] B a > a0 ( (7 ( P t ( From (7 we have a = B( P P /( P t 0, and a ( = Bt /( P t( 0,.e. a0 P = P 0 ( 0 ' > rses as ncreases. Also notce that when the two arlnes are dentcal (.e. then, as expected, (= s always postve. Total trac n the PR case s easly obtaned by addng (5 and (6: ( P P t T = (8 a ( B(3 As can be seen, T s always postve. Further, takng the dervatve o (8 wth respect to, t s easy to see that total trac alls as ncreases: As expected, as arport revenues matter more and more, the arport wll exercse, n an ncreasng ashon, ts market power, nducng output contracton. Ths outcome s a manestaton o the classc double margnalzaton problem, whch s typcal o an uncoordnated vertcal arport-arlnes structure (e.g. Basso and Zhang, 008a. Takng the lmt wth approachng to, one can see that a prvate arport would charge n such a way that t wll nduce total trac gven by P P t /(4a 6. ( B Slot-auctons soluton Frst, we solve or trac levels o the two arlnes gven an arbtrary number n o slots beng auctoned. These trac levels are obtaned by solvng the system gven by equatons (5 and (9, leadng to:

14 As expected, both trac levels ncrease wth n. P P a n SA ( n = (9 4 a P P a n SA ( n = (0 4 a Next, we can look at how trac levels would compare between mechansms, the number o slots s set to what s best under prcng (PR, somethng that s only optmal when =. Replacng n n (9 and (0 by n (8, one can easly show that >, SA SA then ( T > and ( T <. Thereore, or >, the arport sets wrongly the total number o slots to be auctoned to what s best under PR prcng, then slot auctonng would lead to a larger trac volume or, but a smaller, than PR prcng. I =, then the s are equal as expected. We now calculate the optmal number o slots to be auctoned. Frst, the prce that each slot ends up yeldng, as a uncton o n, s gven by (4,.e. T y( n = P = P T ( T ( ( n ( n ( n T '( ( n T '( ( n c ( n ( n c ( n ( Then the obectve uncton o the arport s: P T ( ( n y( n c( n ( n = [ ( n ( n ] [ Max y( n ] ( n where ( n and ( n are gven by (9 and (0. The rst-order condton rom ( leads to: n ( P P t SA = a ( 4 (3 B From (3, total trac under SA decreases wth, smlar to Comparsons between mechansms T n the PR case. As can be seen rom (3, or a prot-maxmzng arport (.e. total trac SA under SA, n, converges to ( P P t / 4( a B. Ths trac level s greater than P P t /(4a 6, the lmtng trac under prcng. Drect comparson o (3 and ( B SA SA (8 urther reveals that n = T when =, but n > T or all >. Thus, there wll always be more arport delays under slot auctons than under prcng (derentated tolls, 3

15 except when =, the case n whch they are equal. Furthermore, the derence between SA n and T ncreases as ncreases, reachng ts maxmum or a prvate, protmaxmzng arport. We now compare the actual trac levels o ndvdual arlnes under the two lghtallocaton mechansms. That s, we compare, or >, SA ( SA n wth. Frst, snce both ( n and ( n ncrease wth n, t s always true that under slot auctons, both ndvdual trac volumes are greater when the optmal number o slots s used, than when SA SA SA SA SA SA the optmal PR number o slots s used,.e. ( n > ( and ( n > (. T T SA SA SA Second, by ( > as shown above, t ollows that ( n > or all >. T SA T However, snce ( <, there are two opposng eects or the case o arlne. In eect, whether nally s larger under one mechansm or the other wll depend on the model s parameter values. Three numercal examples are gven n Fgure, where the SA SA horzontal lne s set at ( n =. One example (the P = 9. 5 curve shows SA SA SA SA ( n > or all >, whlst the other two examples show ( n < or all >. Fgure about here Summarzng the above comparsons yelds Proposton 3. I arport prots matter so that a lght-allocaton mechansm s chosen to SA SA maxmze OF (, then n = T or = ; but or >, n > T,.e. total trac and arport delays are larger under slot auctonng than under prcng; urthermore, the derence between the two ncreases as ncreases. Also, when >, arlne s trac s greater under slot auctons than under prcng, whle arlne s trac may or may not be greater under slot auctons than under prcng. Now that we have shown that the mechansms are equvalent only when =, and that we have establshed that n general SA leads to hgher trac levels, we would lke to see whch mechansm s better n terms o leadng to a hgher obectve-uncton value. What we have known rom Proposton s that or =, arport revenues are larger wth SA than wth PR. It s clear, thereore, that or values o slghtly above, SA wll outperorm PR. The nterestng queston then s whether ths result wll hold or larger values o. To study ths, t s sucent to replace the expressons we have obtaned above and calculate the resultng values o OF ( n ( and ( respectvely. We calculated the rato o the evaluated obectve unctons but the resultng expresson was too complex to obtan an analytcal result. We turned to numercal methods and ound, somewhat surprsngly, that even wth our lnearty assumptons about the unctonal orms, whch mechansm perorms better s very much dependent on parameter values. To show ths 4

16 more clearly we provde two sets o examples, presented n Fgures and 3 respectvely. In each o these cases we graph the rato o OF s aganst, showng derent curves or derent values o P and t. As can be seen, or some parameter values, SA s preerred over PR and the derence n the obectve-uncton values between SA and PR s up to 0%. For other parameter values, however, PR s avored by up to 0% ater reaches some threshold value. Fgure about here Fgure 3 about here We summarze these ndngs n the ollowng proposton Proposton 4. I arport prots matter so that a lght-allocaton mechansm s chosen to maxmze OF (, then or values o slghtly above, slot auctons wll outperorm congeston prcng n terms o leadng to a hgher obectve-uncton ( OF value. For larger values o, however, whch mechansm s better s very much dependent on parameter values. In partcular, or certan parameter values, prcng outperorms slot auctons by up to 0% ater reaches some threshold value. Fgure shows a pcture where SA perorms better n terms o obectve uncton value the more alke the two arlnes are,.e. when P and P are close. Ths s an mportant pont that allows one to provde ntuton or a number o the results we have so ar. What happens when arlnes are very asymmetrc s that the arport would lke to choose derent trac levels or the two arlnes. The arport can actually acheve these trac levels through the prcng mechansm snce t allows dscrmnaton (.e. derent tolls or each arlne. The slot aucton mechansm however cannot acheve the same asymmetrc outcome because there, the levels o trac are acheved through the arlnes bddng behavour, whch depends on the wllngness to pay, somethng leadng to derent ratos o trac (Proposton. In act, as we have shown, SA wll, n general, allow arlne to acheve hgher trac levels than through prcng. It s because o ths, then, that under auctonng the arport needs to choose a number o slots that s larger than the total trac under prcng (Proposton 3: t s ts way to take nto account the sub-optmal trac ratos between the arlnes. On the other hand, and as dscussed beore, wth auctonng the arlnes wllngness to pay or an extra slot s larger than under prcng because wth a xed number o slots to be auctoned, an extra lght or an arlne causes no margnal congeston cost. Thereore, there are two opposng eects: prcng allows better control o the trac ratos, but auctonng leads to an ncrease o the arlnes wllngness to pay, whch are then captured In eect, when P = 0 and P = 9, SA s always better than PR. On the other hand, when P = 0 and P = 4, PR s better than SA or values o that are above a certan threshold. Values o P between 4 and 9 lead to curves that le n between the two curves shown. 5

17 by the arport. Thus, when arlnes are very symmetrc, the eect o hgher wllngness to pay domnates. I arlnes are very asymmetrc, then prcng wth ts good control over trac ratos wll domnate. Gven these results, t s natural to ask what happens to arport revenues by themselves; but gven the ntuton we have bult, t s also evdent that whch mechansm leads to larger arport revenues depends also on parameter values, as shown n Fgures 4 and 5, and arport revenues are hgher when arlnes are more symmetrc. What s new, s that the threshold values o n the case o OF do not concde wth the threshold values o or arport revenues. Fgure 4 about here Fgure 5 about here Overall, t s qute complex to know whch mechansm wll perorm better as t would requre a lot o normaton on parameter values. For some parameter values, one mechansm may lead to larger revenues or the arport, whle at the same tme reducng arlnes prots and the value o OF(. Ths may mply that a regulator asks arport managers to maxmze OF( but leaves the choce o a lght-allocaton mechansm to the arport, there may be socal welare losses. It s nterestng to note that all these results occur n a context whch, or all >, SA leads to larger total trac volumes than PR. 5. Concludng Remarks Ths paper analyzes prcng and slot-allocaton mechansms when prots are mportant to an arport, owng to budget constrants or prot maxmzaton. We nd that prcng and slot tradng/slot auctonng do not lead to the same results. Total trac s hgher under slot auctons than under congeston prcng. Furthermore, arport prots matter ust margnally, then slot auctons wll outperorm prcng n terms o achevng a hgher obectve-uncton value. On the other hand, arport prots matter sucently hghly, whch mechansm s better s then very much dependent on parameter values. In partcular, prcng may be strongly preerred over slot auctons or certan parameter values (specally when arlnes are very asymmetrc. Our analyss suggests that strategc behavor on the part o the arport that cares about ts prot can have a sgncant bearng on the comparson o prce vs. slot-based approaches to congeston management, dependng on what s asked rom the arport, and what matters to the arport. Ths s because, or some parameter values, one mechansm may lead to larger arport prots whle at the same tme reducng arlnes prots and the value o the overall obectve uncton. Hence, a regulator asks arport managers to maxmze a certan obectve uncton, but the choce o a lght-allocaton mechansm s let to the arport, there may be socal welare losses, wth the extent o welare losses dependng n general on cost and demand parameters. Our results thus mply that, arport prots matter, then there s no smple soluton as to whch mechansm should be employed or mplemented. 6

18 The paper has also rased several other ssues and avenues or uture research. Frst, our prcng approach has constraned the xed entry ee to be zero and thus ruled out the possblty o a lump-sum tax leved by arports (n addton to a per-lght charge. As ponted out by an anonymous reeree, gven the obectve chosen (or > the best approach or an arport would be to charge rst-best congeston prces per lght, and then levy a lump-sum tax on the arlnes or the rght to be present at the arport (the entry ee, wth the tax beng set such that arlne prots are entrely skmmed o. Whle desrable or the arport, such a lump-sum tax s not commonly observed n actual arport charges, and the correspondng two-part tar scheme s rarely examned n the arport prcng lterature (an excepton s Basso, 008; see, Basso and Zhang, 007 or a revew o the lterature. Nonetheless, t would be nterestng to theoretcally nvestgate the propertes o the twopart tar scheme under the obectve uncton consdered n ths paper, and compare them wth the propertes o slot polces as well as the present second best prcng approach. Second, to acltate the analyss and to ocus on our man concern o demonstratng possble non-equvalence between prcng and slot polces when arport prots matter, we have ollowed Brueckner (009 by assumng that demand s perectly elastc. It s noted that the benchmark result n the present paper that the three polces o prcng, slot auctons and slot tradng produce the same lght levels crtcally hnged on that assumpton. We see comparson o prce and slot polces under more realstc demand speccatons as a natural extenson o the analyss presented here, although beyond the scope o the present artcle. 7

19 FIGURES SA SA Prcng. a=7 a = 7 B= B = P=0 P t t=0.4 =. P = a 0.9 P = P = 6.5 Fgure. Trac comparson or arlne under PR and SA 8

20 Fgure. Comparson o obectve-uncton value under PR and SA: varable P Fgure 3. Comparson o obectve-uncton value under PR and SA: varable t 9

21 Fgure 4. Comparson o arport revenues under PR and SA: varable P Fgure 5. Comparson o arport revenues under PR and SA: varable t 0

22 Acknowledgement: We are very grateul to anonymous reerees and Jan Brueckner or ther helpul comments. Fnancal support rom FONDECYT-Chle, Grant 09087, rom the Mllenum Insttute Complex Engneerng Systems and rom the Socal Scence and Humantes Research Councl o Canada (SSHRC s grateully acknowledged. Reerences Basso, Leonardo J., 008. Arport deregulaton: Eects on prcng and capacty. Internatonal Journal o Industral Organzaton 6, Basso, Leonardo J., Zhang, Anmng, 007. An nterpretatve survey o analytcal models o arport prcng. In Lee, D. (Ed., Advances n Arlne Economcs, Vol. : Elsever, Basso, Leonardo J., Zhang, Anmng, 008a. Sequental peak-load prcng n a vertcal settng: The case o arports and arlnes. Canadan Journal o Economcs 4, Basso, Leonardo J., Zhang, Anmng, 008b. On the relatonshp between arport prcng models. Transportaton Research Part B 4, Brueckner, Jan K., 00. Arport congeston when carrers have market power. Amercan Economc Revew 9, Brueckner, Jan K., 005. Internalzaton o arport congeston: A network analyss. Internatonal Journal o Industral Organzaton 3, Brueckner, Jan K., Van Dender, Kurt, 008. Atomstc congeston tolls at concentrated arports? Seekng a uned vew n the nternalzaton debate. Journal o Urban Economcs 64, Brueckner, Jan K., 009. Prce vs. quantty-based approaches to arport congeston management. Journal o Publc Economcs 93, Czerny, Achm I., 008. Managng congested arports under uncertanty. In: Czerny, A., Forsyth, P., Gllen, D., Nemeer, H.-M. (Eds., Arport Slots: Internatonal Experences and Optons or Reorm. Ashgate, Aldershot, UK. pp. -6. Czerny, Achm I., 009. Arport congeston management under uncertanty. Transportaton Research Part B, orthcomng. Danel, Joseph I., 995. Congeston prcng and capacty o large hub arports: A bottleneck model wth stochastc queues. Econometrca 63, Forsyth, Peter, 997. Prce regulaton o arports: Prncples wth Australan applcatons. Transportaton Research E 33,

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