Airport Congestion and Endogenous Slot Allocation [ Preliminary ]

Transcription

1 Airport Congestion and Endogenous Slot Allocation [ Preliminary ] Alessandro Tampieri # and Xi Wan # # Université du Luxembourg alessandro.tampieri@uni.lu; xi.wan@uni.lu February 2015 Abstract This paper analyzes the optimal slot allocation in the presence of airport congestion. We model peak and o peak slots as vertically di erentiated products, and congestion limits the number of peak slots that the airport can allocate. Ine ciency emerges when the airport does not exploit all its slots. We show that for a private airport, ine ciency may arise if the airport is not too congested and the per-passenger fee is small enough, while with a public airport it does not emerge. Furthermore, a private airport tends to give di erent slots to ights with same destination, while a public airport more likely favors identical slots to ights with same destination. We conclude that a private airport may be less e cient in terms of use of slots, and its allocation choices satisfy a lower number of passegners than a public airport. JEL classi cation R41; H23; H21. Keywords Slot allocation; Airport congestion; Vertical di erentiation. 1

2 1 Introduction Airline tra c growth has outpaced capacities at many of the world s major airports in the past decades, leading them to operate in congestion conditions. For instance, over half of Europe s 50 largest airports have already reached or are close to their saturation points in terms of declared ground capacity (Madas and Zografos, 2008). Airport congestion has been managed through control and management of slots, including slot sales, slot trading and slot auctions. According to the European Council Regulation EEC95/93, a slot is the permission to use airport infrastructure for landing or take-o on a speci c time. Under a slot system, the airport authority determines the total number of slots to make available, and slots are distributed among the airlines according to some allocation rule. Congestion calls for management of slots. For example, FAA (Federal Aviation Administration) capped peak hour ight movements at New York La Guardia, J.F.Kennedy, and Neward airports. As for Chicago s O Hare airport, FAA persuaded two major airlines United and American Airlines to reduce peak ight activities while prohibiting smaller airlines from increasing ights to ll the gap. In the analysis of airport congestion, the economic literature focused mainly on congestion pricing (for which carriers pay a toll according to their contribution to congestion) as a regulatory tool to deal with congestion. 1 However, despite its theoretical feasibility, congestion pricing has not been practised in the real world. By contrast, slot allocation is the usual business in the management of airports. Thus, a theoretical analysis investigating the interaction between slot allocation and congestion seems higy policy relevant. In this paper we analyse slot allocation in the presence of airport congestion. We examine a setting where an airport sorts slots according to di erent departure ights, and airlines compete in the ights market. As in Brueckner (2002), we model peak and o -peak slots as products of di erent qualities in a model of vertical di erentiation. Peak slots are congested, mirroring the situations of capacity shortages at peak hours faced by many airports. We analyze 1 For an early contribution on congestion pricing see Levine (1969). Recent representative studies include Brueckner (2002, 2005). Under congestion pricing, carriers could place as many ights as they wish provided they pay the toll, thus the overall level of congestion is determined by airline decisions. 2

3 both a private and a public airport being restricted to levy a uniform perpassenger fee for ight activities, this being pre-determined by administrative bodies. The analysis of a private airport also seems relevant. Although airports have long been owned by governments, there has been a signi cant worldwide trend towards government facilities privatization beginning from the middle of the 1980s. 2 We consider separately the case where two ights towards a same destination (denoted as pairwise ights along the paper) are served by two airlines, and where they are owned by a monopoly. In this complete information setting, airlines know the total provision of slots and that each participating airline would a single slot. We de ne as allocation ine ciency the situation in which not all the slots available are exploited. Our approach di ers from Brueckner (2002) as follows. In Brueckner (2002) s framework, a monopoly airport chooses the critical points on the continuum that respectively de ne whether to y or not and whether to y in peak slots or o peak slots. Focusing on nding out the optimal congestion pricing, he implicity assumes that airport capacity is su cient to meet peak hour demands. Unlike Brueckner (2002), our interest stems from the scarcity of peak hour slots. Thus we focus on the allocation instead of using the pricing tool. The airport s ownership and the airline market s structure a ect the results. For a private airport and duopoly airline markets, the airport tends to assign di erent slots, unless the per-passenger fee is high enough, and the airport is not too busy, i.e., if the number of peak slots is higher than the number of destinations. In this case, allocation ine ciency arises when the airport is not too busy and the per-passenger fee is su ciently low. The results are driven by two e ects. First, an o -peak ight in one market would attract additional lower-valuation passengers, yielding a wider range of passengers to travel in total. Second, consumers bene t from harsher competition. Thus when airlines obtain peak/peak slots, they compete head-to-head and more passengers would be served. While on the contrary, in the presence of quality di erentiation, competition is less intense, and airlines provide passengers with di erent taste. When the per-passenger fee is small the rst e ect outweighs the second e ect. 2 Following UK many major airports in Europe, Australia and Asia have followed suit and have undergone privatization or are in the process of being privatized. In principle, privatization is characterized by the transfer of ownership structure from state-owned to private enterprises. 3

4 In the case of monopoly, the airport prefers to assign identical time slots to pairwise ights. This is due to the fact that, without di erences in terms of type of slots, it is possible to extract a higher mark-up. When the airport is public, allocation ine ciency does not emerge. Also, in a public airport it is more likely that pairwise ights are assigned to identical time slots. The intuition is simple. Airline markets with same slot quality are more favorable to passengers, given the higher intensity of competition. Since a public airport is interested in consumer surplus also, this type of slot con guration is usually preferred. The emergence of allocation ine ciency in our results corresponds to the common practice in airport management of declaring a number of slots being lower than an airport s full capacity (Mac Donald, 2007, and De Wit and Burghouwt, 2008). Indeed, as De Wit and Burghouwt (2008) point out, an ef- cient use of the slots at least requires a neutral and transparent determination of the declared capacity. So far, slot allocation has drawn relatively little interest in the economic literature, with few but noteworthy contributions. Barbot (2004) models slots for airline activities as products of either high or low quality, and carriers choose the number of ights they serve. She shows that slot allocation improves ef- ciency according to the criteria for assessment, and welfare in fact decreases after re-allocation. Unlike the present paper, in Barbot (2004) s model carriers could place as many ights as they want. Our paper limits the number of peak slots, in order to address congestion. Verhoef (2008) and Brueckner (2009) compare the pricing policy and slot policy regimes. They show that the rst best congestion pricing and slot trading/auctioning generate the same amount of passenger volume and total surplus. They investigate a single congested period. Their contributions do not distinguish between peak and o -peak hours, and allow the airport to allocate slots without charges. Although this seems a plausible description of some public airports, a non-pro t behavior does not seem likely in the presence of a private airport. Departing from Verhoef (2008) and Brueckner (2009), we assume that each airline operates a single ight. Our approach models certain time intervals that are most desired by all passengers as peak period. In particular, the total number of slots that an airport could place in peak period does not meet the demand of passengers. 4

5 The remainder of the paper is organized as follows. Section 2 introduces the model. Section 3 presents the baseline results, in which the airport is assumed to be private and an airline duopoly serves each destination. Section 2 and 5 show the changes in the results when either a monopoly airline is operating for each destination or the airport is publicly owned, respectively. Section 6 extends the analysis to the case where per-passenger fee is endogenously determined by the airport. Section 7 discusses some extensions. In particular, we show the changes in the results when collusion emerges between the airport and one of the airlines in the duopoly case and density is heterogenous among di erent ights. Section 8 concludes. 2 A model of slot quality We consider an economy with one private airport, where trips to N destinations d 2 D = f1; 2; ; Ng are served by 2N airlines. 3 Each destination, also called a market, is served by two ights being run by separate airlines. 4 Denote a ight as f f 2 F = f1; 2; ; 2Ng. Formally this can be depicted as a mapping I from ights f to destinations d I(f) = d. In turn, we write the inverse mapping function from ights to destinations as A(d) = ff I(f) = dg. We assume that airports at destinations are uncongested, so that the allocation decisions do not a ect the ight scheduling of destination airports. Furthermore, we focus on the case of solely single trip departing ights. Return trip ights can be dealt with by either an identical analysis with two runways, or simply by discounting a scale factor if there is a single runway. Destinations are independent in the sense that they are neither substitutes nor complements, therefore the demand for one destination is irrelevant to demands for other destinations. We assume that the quality di erential is characterized only by the departing time. Although the quality of an airline depends on many factors, this approach would allow us to concentrate on the congestion issue. There are two travel periods, denoted as peak and o -peak. A peak period represents the time window that consists of the most desirable travel times in a day, for instance early morning and late afternoon. The peak period may contain a collection 3 Section 5 analyses the case with a public airport. 4 Section 4 extends the analysis to the case where a private airport interacts with airline monopolies. 5

6 of disjoint time intervals like and An o -peak period, by contrast, contains all other time intervals that do not belong to the collection of peak period. In order to address the problem of peak slots congestion, the o -peak period is assumed to be uncongested, i.e., airport capacity can serve all ights within o -peak period time intervals. Conversely, the peak period is congested in the sense that airport capacity cannot serve all ights within peak period time intervals. This assumption captures tra c patterns at many airports such as Paris Charles de Gaulle, Amsterdam Schiphol, Munich and Brussels airport (NERA, 2004). All potential passengers perceive and agree with the slot associated with time slot, and agree over peak load hours (denoted as subscripts h) are more preferable than the o -peak load hours (denoted as subscripts l) at equal price. At peak hours the demand to use airport runways is much higher than o -peak hours. Thus slot qualities s l and s h are exogenously perceived s h > s l > 0. An airline o ering mid-day schedules are more favorable to all passengers than another airline that o ers late evening airlines. Finally, a slot allocation is de ned as the mapping m from airline f to a slot type i, m F! fl; hg, so that m(f) = i. For instance, m(f) = h reads as airline f is allocated peak time slot. We assume in each market the airlines engage in quantity competition. 5 We denote p f ii 0 as the price charged by an airline f ying to destination d = I(f) that takes o at slot i while its competitor on the same destination takes o at slot i 0, i; i 0 = fl; hg. Similarly, q f ii 0 denotes the number of passenger served by this ight. Following the general framework of vertical di erentiation (Gabszewicz and Thisse, 1979), in each market the demand is generated from a unit mass of passengers indexed by a type parameter v. Passengers di er in tastes, the taste parameter is described by v 2 [0; 1], v being uniformly distributed with unit density. Demand addressed to ight f is de ned by the set of passengers who maximize their utility when ying with airline f, rather than ight f 0 to the same destination or refraining from ying. Accordingly, each passenger ying to destination d purchases a ight ticket of airline f if her utility vs i p f ii is higher 0 than the utility vs i 0 p f 0 i 0 i of ying with the competitor and higher than the utility of not traveling. We assume each passenger ies at most once, and, if a passenger 5 The assumption of quantity competition is common in the airline economics literature. See Brueckner (2002), Pels and Verhoef (2004). Brander and Zhang (1990) nd empirical evidence that the rivalry between duopoly airlines is consistent with Cournot behavior. 6

7 does not y, her reservation utility is zero. It follows that the more convenient the slot of departing time slot, the higher the utility reached by passengers for a given price. Formally, a potential passenger in the destination market d with the airlines f and f 0, d = I(f) = I(f 0 ), has the following preferences 8 vs >< i p f ii if she ies with airline f in slot s 0 i at price p f ii 0 U d = vs i 0 p f 0 i 0 i if she ies with airline f 0 in slot s i 0 at price p f 0 i 0 i > 0 if she does not y. The airlines choose the number of seats in order to maximize pro ts. Airline costs include airport per-passenger charge, while marginal operating costs are normalized to zero for the moment. 6 We do not consider the entry of airlines in the airport and assume that xed costs are sunk. Thus the pro t of an airline f competing in the destination market d with another airline f 0, d = I(f) = I(f 0 ), is given by 8 < f = p f (qf ; 0 qf ) q f if m(f) = l and m(f 0 ) = h ; (1) f 0 = p f 0 (qf ; 0 qf ) q f 0 if m(f) = h and m(f 0 ) = l for an o -peak/on peak slot con guration and 8 < f ii = p f ii (qf ii ; 0 qf ii ) q f ii if m(f) = m(f 0 ) = i 2 fl; hg f 0 p f 0 ii (qf ii ; 0 qf ii ) q f 0 ii if m(f) = m(f 0 ) = i 2 fl; hg ii = (2) for the on peak/on peak and o -peak/o -peak slot con guration. The airport earns the charge for each passenger. It chooses the slot allocation mapping m() that maximizes it pro ts. We get the following program max = m() N d=1 q f m(f);m(f 0 ) + 0 qf m(f 0 );m(f) where I(f) = I(f 0 ) = d (3) 6 In Section 3.3 we also consider positive operating costs. These have been assumed in some contributions, such as Pels and Verhoef (2004), Brueckner and Van Dender (2008), and Basso (2008). 7

8 Figure 1 Timing subject to the peak slot capacity constraint #ff m(f) = hg M; (4) where M is the total number of peak slots. To avoid a cumbersome discussion of ties, we assume that M is even. Constraint (4) implies that the overall allocated peak slots cannot exceed the total number of available peak slots. Peak capacity could not accommodate all ights M < 2N. By constrast, the o -peak capacity could accommodate all ights (there is no constraint for the o -peak slots). We then de ne allocation ine ciency as follows. De nition 1 Allocation ine ciency emerges when at least one peak slot is not used. This assumption seems natural. In the presence of airport congestion, leaving some peak slots unused represents some degree of ine ciency. Figure 1 shows the timing of the game. In the rst stage the airport allocates peak and o -peak slots for a given fee. In the second stage airlines choose number of seats to supply q f ii based on slot allocation. In the third stage 0 passengers in each destination decide whether to y with a peak period airline, an o -peak period airline, or not y at all. The equilibrium concept is the subgame perfect equilibrium by backward induction. 8

9 3 Baseline results In this section we show the baseline results of the model. In particular, we consider the scenario where per-passenger fee is uniform. For the sake of simplicity we omit superscript d in the ensuing analysis, as the demand in one market is not in uenced by the demand in other markets. 3.1 Passengers In the third stage, a passenger decides whether to y and, if so, the time of ying. Let us consider the destination market d with two airline competitors f and f 0, i.e. d = I(f) = I(f 0 ). If the two ights in this market obtain a peak and an o -peak slot, the passenger with a taste parameter v such that vs l p f = vs h p f is indi erent between ying at peak load time and at o -peak load time. Likewise, the passenger with a taste parameter v such that vs l p f = 0 is indi erent between ying at o -peak time and not ying at all. Thus airline markets are partially covered. 7 In this case, if two ights in a single market obtain slots of same quality, the passenger with a taste parameter v 0 such that v 0 s l p f ll = 0 with p f ll = p f 0 ll, or v 0s h p f hh = 0 with pf hh = 0 pf hh is indi erent between ying and not ying. Three market con gurations may arise at equilibrium (i) the two airlines obtain peak/o -peak slots; and both obtain either (ii) o -peak slots, or (iii) peak slots. They are characterized by the following demand function, respectively 8 0 >< q f (pf ; 0 pf ) = pf p f p f (i) s h s l s l > q f 0 (pf ; 0 pf ) = 1 p f 0 p f ; (5) s h s l (ii) q f ll (p ll) + q f 0 ll (p ll) = 1 (iii) q f hh (p hh) + q f 0 hh (p hh) = 1 p ll ; s l (6) p hh s h (7) 7 Motta (1993) shows that Cournot competition can be studied only with partial market coverage, since the demand function can not be inverted with full market coverage. 9

10 In turn, the inverse demand functions corresponding to the three possible market structures are given respectively as follows 3.2 Airlines (i) 8 < p f p f 0 = s l 1 q f q f 0 = s h 1 ; (8) s l s h q f q f 0 (ii) p ll = s l 1 q f ll q f 0 ; (9) ll (iii) p hh = s h 1 q f hh q f 0 hh (10) In the second stage, airlines set their optimal supply of seats. We analyze each possible con guration (peak/o -peak; peak/peak; o -peak/o -peak) separately. Consider rst a market where pairwise ights obtain di erent slots, then according to (1), airlines pro ts are expressed by (i) 8 < f = f 0 h s l (1 q f q f 0 ) i q f = (s h s l q f s h q f 0 )q f 0 Airlines choose the number of seats to maximise pro ts, for any given. The rst-order conditions f = + (1 q f q f 0 )s l q f s l = 0; f f 0 = + s h 2q f 0 s h q f s l = 0 (12) Solving (11) and (12) simultaneously with respect to q f and qf 0 yields q f = s hs l (2s h s l ) (4s h s l )s l ; (13) q f 0 = 2s h s l 4s h s l (14) 10

11 To ensure interior solutions, we assume the condition 0 < < 1 s hs l. 2s h s l Note that q f < 0 qf for all 0 < < 1. If a market obtains di erent slots, then the airline with peak slot serve more passengers in equilibrium than its o -peak competitor. Plugging (13) and (14) into (8) yields p f = s h(2 + s l ) 4s h s l ; (15) being always positive, and where p f 0 = 2s2 h + s h(3 s l ) s l 4s h s l ; (16) p f 0 p f = (s h s l ) (s h + 2) 4s h s l > 0 Since q f < 0 qf ; pf < 0 pf ; then the airline ying during the peak slot yields higher pro t than its o -peak slot counterpart. Consider next the optimal number of seats provided in markets (ii) and (iii) where both airlines obtain the same slots i = m(f) = m(f 0 ). The airlines face the demand q f ii + qf 0 ii = 1 Plugging (17) and (19) into airline pro ts (2) yields The rst order conditions of f ii 8 h < f ii = (1 q f ii q f 0 ii )s i h f 0 ii = (1 q f ii q f 0 ii )s i p i s i (17) i i q f ii q f 0 ii (18) 0 and f ii with respect to q f 0 ii and qf ii ; respectively, yield 8 < q f ii s i + s i (1 q f ii q f 0 ii ) = 0 q f 0 ii s i + s i (1 q f ii q f 0 ii ) = 0 By solving the above two equations for q f 0 ii and qf ii we obtain the optimal number of seats served by two airlines, which are identical due to symmetry q f ii = qf 0 ii = s i 3s i (19) 11

12 To ensure interior solutions, we make the assumption 0 < < s i with s i > 1 ; i 2 fh; lg. Hence 0 < < 1 is su cient condition for q f ; qf ; qf ii 0 to be positive. For all 0 < < 1 ; we have q f > qf hh > qf ll > qf (10) yields 3.3 Airport p ll = s l Plugging (19) into (18) and ; p hh = s h + 2 (20) 3 In the rst stage, the airport maximizes its pro t by allocating peak slots subject to congestion. In this setting, the presence of congestion implies that the number of peak slots is lower than the total number of ights. Hence, allocating one peak slot to a market implies to take that peak slot away from another market. This in turn implies that, since the type of market allocation in uences another market, then the airport needs to consider the di erences in the allocation of two peak slots, in the con guration of two markets together. For example, putting two peak slots to the same market might leave another market with two o -peak slots. If con guration peak/peak is preferred to peak/o -peak, but the combination peak/peak + o -peak/o -peak gets less passengers than two con gurations peak/o -peak, then the best allocation is one peak slot in each market. This despite the fact that, in a single market, the con guration peak/peak reaches the best outcome. Begin the analysis by comparing the number of seats in each con guration for a single market. When two ights in a market obtain peak/o -peak slots, according to (13) and (14), the number of passengers in a market is given by q f + 0 qf = s h(3s l 2) s 2 l (21) s l (4s h s l ) whereas with con guration peak/peak, the number of passenger in a market is q f ii + qf 0 ii = 2(s i ) 3s i ; i 2 fh; lg (22) Note that for all s h > s l > 0; q f ll + 0 qf ll < min n q f + 0 qf ; qf hh + 0 qf hh o. In words, o -peak/o -peak is strictly dominated by peak/peak and peak/o -peak in terms of number of passengers, which implies that in a market having either 12

13 one or two peak slots is better than having only o -peak slots. Conversely, by comparing the values of q f + 0 qf in (21) with q f hh + 0 qf hh given by (22) s h (3s l 2) s 2 l s l (4s h s l ) 2(s h ) 3s h if and only if 2 s ls h 6s h 2s l, where 1 2 = s l (2s h s l ) 2 (3s h s l ) > 0 Consider next the di erences in the allocation of two peak slots. Since two peak slots can be allocated either in one market (thus leaving another market with two o -peak allocation) or in two market, we must also compare Q 1 = q f + qf 0 to Q 2 + Q 3 = q f hh + qf 0 ll, yielding q f + qf 0 q f hh q f ll = 1 3 (s h s l ) (2s h s l ) s h s l s h s l (4s h s l ) > 0 for < 1 Figure 2 compares the con gurations in a single market according to airport fees, by illustrating how Q 1 = q f + 0 qf ; 2Q 2 = q f hh + 0 qf hh and 2Q 3 = q f ll + 0 qf ll 13

14 are linear decreasing functions in with 2 (0; 1 ). Two e ects emerge from Figure 2. The rst e ect entails that having an o -peak ight in one market, depicted by Q 1, would attract additional passengers with low v, yielding a wider range of passengers to travel. The second e ect entails that consumers bene t from harsher competition. This is to say, when airlines obtain peak/peak slots, depicted by 2Q 2, they compete head-to-head and thus the outcome is more bene cial to passengers, since more passengers would be served. On the contrary, in the presence of slot di erentiation, competition is less intense, and airlines provide passengers with di erent taste. When is small the rst e ect outweighs the second e ect, so that Q 1 is greater than 2Q 2. On the other hand, con guration Q 2 + Q 3 is always dominated by the alternatives 2Q 2 and Q 1. In this set-up, destination markets can have only three types of slot allocations. There are n 1 destination markets with peak/o -peak allocations such that n 1 = #fd m(f) 6= n(f 0 ) and d = I(f) = I(f 0 )g. The peak/peak allocations includes n 2 destinations markets where n 2 = #fd m(f) = m(f 0 ) = h and d = I(f) = I(f 0 )g. Finally the number of destination markets that receive o -peak/o -peak allocations is given by n 3 = #fd m(f) = m(f 0 ) = l and d = I(f) = I(f 0 )g. As a consequence the airport allocation problem (3) simpli es to the following linear programming problem subject to max n 1 q f n 1 ;n 2 ;n + 0 qf + n 2 q f hh + 0 qf hh + n 3 q f ll + 0 qf ll 3 (23) n 1 + n 2 + n 3 = N (24) n 1 + 2n 2 M (25) 0 n 1 ; n 2 ; n 3 N (26) The solution of this problem is found in Appendix A. This is (i) n 1 = minfm; Ng; n 2 = 0; n 3 = N n 1 if < 2 - case 1; (ii) n 1 = M, n 2 = 0 and n 3 = N M if 2 < < 1 and N M - case 3; (iii) n 1 = 2N M; n 2 = M N; n 3 = 0 if 2 < < 1 and M > N - case 4. 14

15 The description of single cases is in Appendix A. In solution (i) and (ii), the airport never allocates two ights to the same destination in the peak slots. It rather separates the airlines in di erent slots. Note that solution (i) and (ii) occur always when the airport is busy (destinations are more than peak slots, N > M). In this case, the relative scarcity of peak slots implies that allocating one slot in one market always subtracts it to another market. Then the best strategy for the airport is to compare the con gurations of two markets peak/peak+o -peak/o -peak against peak/o -peak. Since Q 1 > Q 2 + Q 3, the airport favors the peak/o -peak con guration. When the airport is not busy (M > N) and con guration peak/o -peak yields a higher pro t than con- guration peak/peak ( < 2 ), then solution (i) occurs. In this case, the airport under-uses the peak slot capacity, leading to allocation ine ciency. Indeed the number of peak slots exploited is N; whereas M N slots would be left unused. In solution (iii), the airport is not busy (M > N) and con guration peak/peak yields a higher pro t than con guration peak/o -peak ( > 2 ). Given the relative abundance of peak slots, allocating one slot in one market not always subtracts it to another market, thus for some markets the best strategy is to compare single market con gurations. The trade-o between the constraint due to the number of peak slots (for which the condition Q 1 > Q 2 + Q 3 matters) and the fact that 2Q 2 > Q 1 in the single market results in a combination of peak/peak (M N) and peak/o -peak (2N M) con gurations in equilibrium. The foregoing discussion can be summarized in the following proposition. Proposition 1 Suppose all markets are served by duopoly airlines, and the airport is private. For M N, the airport uses all available peak slots, and favors peak/o -peak con guration. For M > N and 2 (0; 2 ], the airport does not use all available peak slots (ine ciency), and the con guration is peak/o -peak in each market; 2 [ 2 ; 1 ), the airport uses all available peak slots, and implements (M N) peak/peak over (2N M) peak/o -peak con guration. Figure 3.a describes the equilibria in the cases of private airport with duopoly markets. The numbers in brackets refer to the notation adopted in Appendix A. Proposition 1 higights an important issue. When peak slots are not scarce 15

16 relative to the number of markets (M > N), a private airport would leave a number of peak slots unused when the pre-determined fee is small, thereby resulting in allocative ine ciencies. In reality, such behavior may be expressed by misreporting true airport handling capacity. This is in line with De Wit and Burghouwt (2008), who nd that e cient slot use can be a ected by capping available slots through capacity declaration. To obtain some intuition as to why airport has a propensity to leave some peak slots unused rather than using all, recall in Figure 2 the ranking of Q 1 and Q 2 is driven by two e ects in a market. More speci cally, e ect 1 outweighs e ect 2 when 2 (0; 2 ] In this case the airport favors peak/o -peak and thus determining allocation ine ciency. When 2 [ 2 ; 1 ) ; the airport favors peak/peak and as a consequence no peak slots would be optimally left unused. On the contrary, in a very busy airport where available peak slots are scarce relative to total demand (M N), it is optimal to allocate all available peak slots. Positive airline operating costs For the sake of completeness, we end the section by discussing the case where airlines have identical operating costs 16

17 c > 0. The analysis can be developed in a similar vein as before, where the airline marginal cost is now c + rather than only. Naturally, in both con gurations the volume of passengers is larger without operating cost. The conclusion drawn from the comparison under peak-o peak con guration also applies here. It follows that, with positive operating costs airport pro t is also smaller in each con guration. The condition required to guarantee positive passenger volumes in equilibrium is 0 < < 0 1 s hs l 2s h s l c; while the threshold determining the preference between peak/peak and peak/o peak is 0 2 s hs l 6s h 2s l c Therefore, the above proposition now reads with 0 1 and 0 2 substituting for 1 and 2. If the cost c is small enough so that 0 2 > 0, the proposition presents the same con gurations and the same issue of allocation ine ciency. The con guration peak-o /peak induces more passenger volume than con guration peak/peak so that the airport does not distribute all available peak slots and ine ciency arises. However, if c is large enough so that 0 2 becomes negative, all available peak slots are distributed and allocation ine ciency never arises. 4 Airline monopolies Having examined competition between duopolists in each market, we shall now investigate the case in which each market is served by a single airline that acts as a monopolist operating two ights in the same destination markets. As with the baseline model, we analyze the second stage in each possible con guration separately, while the analysis of the third stage remains unchanged. Beginning with the peak/o -peak con guration, the airline pro t is M = s h s l q M s h q M q M + s l (1 q M q M ) q M ; 17

18 where the superscript M stands for monopoly. The rst-order conditions M = (1 2q M 2q M )s l q M s h = 0; M = s h q M (s h + 2s l ) = 0 (28) Solving the rst-order conditions for passenger volumes, one obtains q M = s h ; (29) s h + 2s l q M = s l (2s l s h ) s h (2s l + s h ) 2 (30) The condition 0 < < M s l(2s l s h ) s h is su cient to ensure interior solutions, implying also 2s l > s h In contrast to the duopoly case, now q M > qm for all 0 < < M. Next, compare the monopoly outcome (29) and (30) and the duopoly outcome (13) and (14). q M q f < 0 and qm q f > 0; for 0 < < M (see computation in Appendix B). Comparing total passenger volume in two cases shows that q M + q M q f + qf < 0; for 0 < < M As expected, the total number of seats provided by the monopolist is smaller than the duopolists. Replacing q f 0 and qf in (8) with (29) and (30) yields p M = s l [2 (s h + s l ) + s l (3s h + 2s l )] (s h + 2s l ) 2 ; (31) p M = s2 h ( + s h) + s h s l (4s h + ) + 2s 2 l (s h + 2s l ) 2 (32) Comparing prices with the duopoly case, one gets 18

19 p M p f = s hs 2 l (s l 2) 2s 3 l ( + s l) + 8s 2 h s2 l s 3 h (2 + s l) (4s h s l ) (s h + 2s l ) 2 > 0; p M p f 0 = s3 h 8s 2 h s l s h s 2 l + 2s3 l + 2s4 h + 8s3 h s l 8s 2 h s2 l + 4s hs 3 l (4s h s l )(s h + 2s l ) 2 > 0 Of course, prices are higher in monopoly than in duopoly. These preliminary results can be summarized in the following lemma. Lemma 1 In a market of peak/o -peak slots, the airline monopolist (i) serves more passengers in the o -peak period than in the peak period, and (ii) provides less peak period seats, more o -peak period seats, and less total seats than in the duopoly case. The price of travelling is higher under monopoly at any time. Consider next the numbers of seat provided in markets (ii) and (iii) where a monopoly airline obtains identical slots. In this case, we denote the slot type (h or l) the monopolist obtains by i, it follows that the monopolist s pro t is given by M i = (1 q M i )s i q M i ; with i 2 fh; lg Taking the rst-order condition for number of seats we get the equilibrium number of seats the monopoly would provide q M l = s l 2s l ; q M h = s h 2s h (33) To ensure interior solutions, suppose condition 0 < < s l is satis ed. It can be veri ed that M < s l ; hence < M is su cient condition for interior solutions in all con gurations. Plugging (33 ) into (18) and (10) yields p M l = s l + 2 ; p M h = s h + (34) 2 We can compare the outcomes of di erent con gurations of a single market. From (29), (30) and (33), one obtains q M + q M q M h < 0; q M + q M q M l < 0; and q M h q M l > 0 19

20 for all 2s l > s h > s l > 0 and 0 < < M (see computation in Appendix B). Taken together, a ranking of con guration is straightforward in terms of passenger volume peak/peak is superior to o -peak/o -peak, which is better than peak/o -peak. The airport has the same problem as (23) with the new values for q M ii 0. The solution is given by (see Appendix A) n 1 = 0, n 2 = minfm=2; Ng, n 3 = N n 2 - case 6. Therefore, the airport concentrates the competitors in the same slots and extensively uses the peak slot capacity. Proposition 2 Suppose all markets are served by monoply airlines. Then a private airport uses all available peak slots for peak/peak market con gurations. Figure 3.b describes the equilibria in the case of private airport with monopoly markets. The equilibrium type is the same in the entire parameter range. Here it is worth stressing that, unlike the duopoly airline case, the allocation of slots is not in uenced by whether the number of slots is relatively abundant or not compared to the number of markets. Speci cally, since o -peak/o -peak dominates peak/o -peak, the airport would either allocate two peak slots to a market, or no peak slots to a market at all. 5 Public airport We now investigate the case of a pubic, welfare-maximizing airport. This allows us to obtain some insights on how the airport s ownership in uences slot allocation. In this regard, social welfare W is represented by the sum of airport pro ts, passenger surplus (denoted as CS) and airlines pro ts W = + CS + n 1 ( + ) + 2 (n 2 hh + n 3 ll ) Consider an airline duopoly in each market. Since airport and airlines operating costs are normalized to zero, then airport pro ts come from total per-passenger fees, whereas airline pro ts are the ticket income less total per-passenger fees paid to the airport. In turn, passenger surplus is represented by the total gross utility generated from ying minus all ticket payments. Since monetary transfers between airlines and airport cancel out, and so do transfers between 20

21 passengers and airlines, then social welfare equals the sum of passengers gross utility in all 2N markets. Thus W can be rewritten as where Z v W = n 1 vs l dv + v Z 1 v Z 1 vs h dv + n 2 vs h dv + n 3 Z 1 vs l dv; (35) v 0 v 0 0 v = pf ; v = pf p f p f ; v 0 = pf hh ; v 0 = pf ll (36) s l s h s l s l s h s l The analysis of the second and third stage remains the same as in the baseline model. For notational simplicity we de ne B + R v vs v ldv + R 1 vs v hdv, B hh = R 1 vs v 0 h dv and B ll = R 1 vs v 0 l dv; so that (35) becomes W = n 1 B + + n 2 B hh + n 3 B ll ; We will consider rst the case with duopoly airlines and then the case with monopoly airlines. 5.1 Airline duopolies Putting (15), (16) and (20) together with (36) yields v = s h (s l + 2) s l (4s h s l ) ; v = 2s h + ; v 0 = s h 2 ; v 0 = s l 2 (37) 4s h s l 3s h 3s l Assuming < P s l ensures that the market is partially covered. Substituting (37) into (35) and solving the integrals 2 yields and B + = 4s hs l (s l 2s h ) + s h s l (12s 2 h 5s h s l + s 2 l ) 2 (4s 2 h + s hs l s 2 l ) 2s l (4s h s l ) 2 ; B ii = 2( + s i)(2s i ) ; i 2 fh; lg 9s i By comparing the number of seats obtained in each con guration, we get (see Appendix B) B hh > B + > B ll > 0 and (B hh + B ll ) =2 < B +. Therefore we now arrive at a situation similiar to the private airport case with 2 [ 2 ; 1 ) The following proposition summarizes the features of the equilibrium. 21

22 Proposition 3 Suppose all markets are served by duopoly airlines, and the airport is public. For M N, the airport uses all available peak slots, and favors peak/o -peak con guration. For M > N the airport implements (M N) peak/peak and (2N M) peak/o -peak con gurations. The results are qualitatively similar to the case with private airport for 2 < < 1. However, now the public airport would use all available peak slots in any case, hence, ine ciency does not emerge when the airport is public. 5.2 Airline monopolies We now turn to the interplay between a public airport and airline monopolies. Putting (31), (32) and (34) together with (36) yields v = s h(5s l + ) + 2s 2 l + s2 h (s h + 2s l ) 2 ; v = s2 h + s h(5s l + ) + 2s 2 l ; v 0 = s h + (s h + 2s l ) 2 2s h Substituting (38) into (35) and solving the integrals yields ; v 0 = s l + 2s l (38) B + = s hs l (20s 3 l + 5s2 h s l + 32s h s 2 l s 3 h ) 2 (s4 h + 4s3 h s l + 3s 2 h s2 l + 8s hs 3 l + 4s4 l ) 2(s h + 2s l ) 4 and 2 (s 3 h + 3s2 h s l + 8s h s 2 l + 4s3 l ) 2(s h + 2s l ) 4 ; B ii = (s i )(3s i + ) ; i 2 fh; lg 8s i Comparing the number of seats obtained in each con guration, we get (see Appendix B) B hh > max fb ll ; B + g Now we compare B + and B ll We show that B + B ll > 0 for s l < s h < 15s l and < M p ; or 15s l < s h < 2s l and 0 < < M p, where M p s l (7s 3 h + 18s2 h s l 20s h s 2 l 24s 3 l ) s h (s 2 h + 4s hs l + 12s 2 l ) As before, we then evaluate the di erences in the allocation of two peak 22

23 slots. In this case (see Appendix B) Bhh M + BM ll > 2B+ M. We get the following solution (Appendix A) n 1 = 0, n 2 = minfm=2; Ng,n 3 = N n 2. - case 6 ( P M > > P 1 ) and case 2 (for ( < P M ), yielding Proposition 4 Suppose all markets are served by monopoly airlines. Then a public airport uses all available peak slots for peak/peak market con gurations. Figure 4.b describes the equilibria in the case of public airport with monopoly markets. The results are qualitatively similar to the case with private airport and airline monopolies. The allocation of slots is not a ected by the relationship between the number of destinations and the number of peak slots, the airport would either allocate two peak slots to a market, or no peak slots to a market at all. 6 Endogenous Per-Passenger Fee In the above analysis, per-passenger fee was taken as given by the airport, re ecting the fact that an airport is generally subject to government regulation. We now relax this assumption by allowing the airport to choose. In particular, 23

24 in the rst stage of the game, the airport both sets and allocates slots so as to maximize its pro ts, respectively. The last two stages of the game do not change compared to the previous analysis in the second stage, airlines compete in quantities, in the third stage, passengers buy (or not) one ticket for their destinations. In what follows we focus on introducing endogenous per-passenger fees in the baseline case. For the sake of completeness, the other cases (private airport and airline monopolies, public airport and airline duopolies/monopolies) can be found in the appendix. Begin by noting that the airport s allocation choice does not change compared to the baseline case, so that Proposition 1 still holds. Let us consider the three cases of Proposition 1 8 >< a () for (; N) 2 [0; 2 ] R + () = b () for (; N) 2 [ 2 ; 1 ] [M; 1] > c () for (; N) 2 [ 2 ; 1 ] [0; M) where h a () = minfm; Ng q f + 0 qf + (N minfm; Ng) q f ll + 0 qf ll b () = M q f + 0 qf + (M N) q f ll + 0 qf ll ; c () = (2N M) q f + 0 qf + (M N) q f hh + 0 qf hh ; i ; For each possible allocation choice in Proposition 1, the airport (i) maximizes its objective function with respect to per-passenger fees ; (ii) veri es if the optimal level of per passenger fees is consistent with the allocation choice, (that is, if lies in the range determined by the equilibrium allocation) and (iii) evaluates whether the optimal ensures an interior solution in the market stage ( < 1 ). These steps are developed in the appendix. The following proposition summarizes the equilibrium per-passenger fee. For convenience, denote g = M=N. The optimal per-passenger fee exhibits these features Proposition 5 Suppose all markets are served by duopoly airlines, and the airport is private. Then the airport never sets the per-passenger fees below 24

25 2 s ls h 6s h 2s l. Moreover, the optimal per-passenger fee is 8 > 1 for g c < g < 2, and s l 2 s b ; 2s 3 h ; or g b < g < 2 and s l 2 (0; s b ) ; >< for 1 < g < g c and s l 2 (g) = c or 1 < g < 2 and s l > 2s 3 h; (5 p 17) 2 s h ; 2 3 s h b for 0 < g g b and s l < s b ; or 0 < g < 1 and s l 2 (s b ; s h ) ; ; where g b 2s l (4s h s l ) ; g 6s 2 h 7s h s l + s 2 c l 2s h (2s h + s l ) (10s h 3s l ) (s h s l ) ; s b 5 p 17 2 s h Figure 5 depicts the Equilibria. The study of function b and c can be found in the appendix. Proposition 5 shows that, when the airport has the power of setting per-passenger fees, it never chooses a too low per-passenger fee (case a). According to Proposition 1 case a would cause an under-use of available slots. In other words, allowing the airport to determine per-passenger fee prevents the emergence of allocation ine ciency. 7 Extensions 7.1 Airport-Airline Collusion This section investigates the consequences of collusion between the airport and one airline. Since airports and airlines are vertically connected, the targets of both on passenger volumes are coincident, despite the divergent interests in perpassenger fees. Hence it is not surprising that an airport establishes vertical agreements similiar to vertical mergers with one or multiple airlines that are not in a same city pair. This implies that the airport and the airline in the agreement maximize joint pro ts. The third stage does not change compared to the baseline case. In the second stage, airlines set their optimal supply of seats. Note that, when a peak/o peak con guration emerges, the airport might collude with the airline obtaining 25

26 either the peak or the o -peak slot. We analyze each possible con guration separately. Peak/o -peak market with peak slot airline colluding Consider a market where pairwise ights obtain di erent slots and the peak slot airline colludes with the airport. Denote the joint pro ts of airport-airline as p p + f 0 Then according to (1), pro ts are expressed by 8 h i < f (i) = s l (1 q f q f 0 ) q f p = (s h s l q f s h q f 0 )q f 0 + q f + 0 qf The former expression shows the pro t of the non colluding airline, while the later shows the joint pro t of the peak slot airline and the airport. We now characterize the airlines pro t-maximizing numbers of seats for any given. The procedure is the same as in the baseline case. The airlines choose q f 0 and qf to maximize their pro ts, and the rst-order conditions f = + (1 q f q f 0 )s l q f s l = 0; (39) 26

27 @ f 0 = s h 2q f 0 s h q f s l = 0 (40) Solving (39) and (40) simultaneously with respect to q f q f = s h (s l 2) s l (4s h s l ) ; = 2s h + s l 4s h s l q f 0 and qf 0 yields Condition 0 < < s l ensures interior solutions. Note that 2 qf < 0 qf for all > 0. In this case the peak slot airline serves more passengers in equilibrium than its o -peak counterpart. We compare next q f 0 and qf in the case of collusion with those in the basic model (13) and (14). The colluding airline while its counterpart 2s h + s l 2s h + s l = 2 > 0; 4s h s l s l 4s h 4s h s l s h (s l 2) s l (4s h s l ) s h s l (2s h s l ) (4s h s l )s l = s l 4s h < 0 It s hardly surprising to see that the colluding airline serves more passengers than it would do if there were no collusion, and its non-colluding counterpart serves fewer passengers than it would do if there were no collusion. Comparing pro ts yields p f = s h [(s h s l ) s l + 3(s l )] (4s h s l )s l > 0; for 0 < < s l ; implying that the joint pro t of colluding airport-airline is larger 2 than that of the non-colluding airline. Peak/o -peak market with o -peak slot airline colluding We turn now to the case where pairwise ights still obtain di erent slots, with the o -peak slot airline now colludes with the airport. Denote the joint pro t of airport-airline as o o + f Using this notation, from (1) it 27

28 follows that airlines pro ts are (i) 8 < o = f 0 h s l (1 q f q f 0 ) = (s h s l q f s h q f 0 )q f 0 i q f + q f + 0 qf Given ; airlines choose the number of seats q f The rst-order conditions are and qf 0 to maximize pro f = (1 q f q f 0 )s l q f s l = f f 0 = + s h 2q f 0 s h q f s l = 0 The numbers of passengers in equilibrium are q f = s h + 4s h s l ; q f 0 = 2 (s h ) s l 4s h s l As before, we compare q f 0 and qf with the numbers of passengers in the case without collusion. For the colluding airline and for its counterpart s h + s h s l (2s h s l ) = 4s h s l (4s h s l )s l 2 (s h ) s l 4s h s l 2s h + s l s l 4s h = 2s h (4s h s l )s l > 0; s l 4s h < 0 Hence, the colluding airline now serves more passengers than the previous situation where collusion is absent, and its non-colluding counterpart serves fewer passengers than before. One can show that o > f 0 for either s h > 25s l ; and 0 < < s l or 2 s h 25s l and < s h s l Conversely, 3 o < f 0 for s h < 25s l and s h s l < < s l 3 2 Therefore, the colluding o -peak airline does not necessarily yield higher pro ts than its non-colluding counterpart. 28

29 Identical slots market with one airline colluding Consider next a market where both airlines obtain slots type (i; i 0 ) 2 f(h; h) ; (l; l)g De ne the joint pro t of airport-airline in a market with slot type i as i i + f i 0 i. From (2), it follows that airlines pro ts are 8 h < i = 1 q f ii q f 0 0 i 0 i h f 0 i 0 i = 1 q f ii q f 0 0 i 0 i s i s i 0 i i q f ii + i 0 q f 0 i 0 i ; q f ii 0 q f 0 i 0 i with (i; i 0 ) 2 f(h; h) ; (l; l)g. The rst-order conditions of i and f 0 i 0 i with respect to q f ii and q f 0 0 i 0 i respectively, yields 8 < q f ii s 0 i + s i 1 q f ii q f 0 0 i 0 i = 0 q f 0 i 0 i s i + s i 1 q f ii q f 0 0 i 0 i = 0 Solving the two equations for q f ii and q f 0, we obtain 0 i 0 i q f ii = s i + ; 0 3s i q f 0 i 0 i = s i 2 3s i Note that i f 0 i 0 i = (s i ) s i > 0; that is, the colluding airline has a higher pro t than its non-colluding counterpart. We are now in a position to compare joint colluding pro ts in these four con gurations p ; o ; h and l. One can check that (i) p o > 0, (ii) p l > 0, (iii) p h > 0, implying that 1 dominates all other three joint pro ts. The discussion is summarized in the next proposition. Proposition 6 Suppose all markets are served by duopoly airlines, and the airport is private. Also, suppose that the airport collude with one airline in a market. Then the optimal airport allocation is peak/o -peak, with the collud ing airline being assigned to a peak slot. 29

30 7.2 Heterogeneous Density In this section, we assume heterogeneus density across markets. For simplicity we study an economy with only two markets that exhibit di erent densities the small market 1 has a lower density 1, the large market 2 has a higher density 2 > 1. We discuss three examples the airport allocates either, one, two or three peak slots in the two markets. To begin with, we rst look at the case where peak slots are higy scarce, where one peak slot is available to four slot demands, M = 1; N = 2. Since, for every market, o -peak/o -peak is strictly dominated by peak/peak and peak/o -peak (see section 3.3), there are two ways to allocate the peak slot (1) one of the two airlines from the small market, and (2) one of the two airlines from the large market. The demand functions for each airline in the two possible con gurations are 1. Con guration 1. Market 1 q 1f = 1 p 1f 0 p 1f p 1f s h s l s l! ; q 1f 0 = 1 1 p f 0 s h p f s l! ; market 2 q 2f ll + q 2f 0 ll = 2 1 p 2 ll s l 2. Con guration 2. Market 1 q 1f ll + q 1f 0 ll = 1 1 p 1 ll ; s l market 2 q 2f = 2! p 2f p 2f s h s l p 2f 0 s l ; q 2f 0 = 2 1 p 2f 0 s h p 2f s l!. The number in the superscript denotes the market considered. In the similar manner with (13), (14) and (19), we could derive the optimal passenger volumes served by each airline and consequently each market in equilibrium. A comparison of the equilibrium passenger volumes in the two con gurations 30

31 yields q 1f + 0 q1f + q 2f ll + q 2f 0 ll q 1f ll + q 1f 0 ll + q 2f + 0 q2f = ( 1 2 ) [s l (9 + s h + 2s l ) + 2 (s h s l )] 3s l (4s h s l ) < 0 implying con guration 2 yields a higher number of passenger than con guration 1. The ranking in this simple framework suggests Proposition 7 Suppose an economy with two duopolies with di erent density levels, a private airport and a single peak slot. Then the airport allocates the peak slot to one of the airlines in the large market, and ine ciency would not arise. We investigate next the case where peak slots have a moderate scarcity at the airport. In this setting there are two peak slots available for two markets, M = N = 2. It is straightforward to see that allocation two peak slots to market 2 dominates allocation two peak slots to market 1. Indeed, the large market has a bigger multiplier for density 2 > 1. This, together to the fact that o -peak/o -peak is strictly dominated by peak/peak and peak/o -peak, implies that there are two possible allocations (1) two peak slots to the large market, and (2) one peak slot to each market. The demand functions for each airline in these two con gurations are 1. Con guration 1. Market 1 q 1f ll + q 1f 0 ll = 1 1 p 1 ll ; s l market 2 q 2f hh + 0 q2f hh = 2 1 p 2 hh s h 2. Con guration 2. Market 1 q 1f = 1 p 1f 0 p 1f p 1f s h s l s l! ; q 1f 0 = 1 1 p 1f 0 s h p 1f s l! ; 31