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Airline network structure and the gravity model
Transportation Research Part E 37 (2001) 267±279
www.elsevier.com/locate/tre
Airline network structure and the gravity model Oliver W. Wojahn a,b,* a
Department of Economics, University of Hamburg, von-Melle-Park 5, 20146 Hamburg, Germany b Lufthansa Technik AG, Business Development HAM WR 1, 22313 Hamburg, Germany Received 25 September 2000; accepted 8 November 2000
Abstract In this paper, we determine characteristics of the cost-minimizing airline network under economies of density. Airline demand is asymmetric and governed by the gravity model. Airline networks are restricted to those where each spoke city is assigned to a single hub and where hub cities are fully interconnected. The cost-minimizing network is a mixture of a point-to-point and a single hub network. Multi-hub networks where passengers change planes at more than one airport are found to be suboptimal. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Airline networks; Hub-and-spoke networks; Economies of density; Gravity model
1. Introduction Demand and cost conditions have been identi®ed as the main determinants of airline network topology in the literature. The purpose of this paper is to determine the characteristics of the costminimizing airline network if technology exhibits economies of density and asymmetric demand follows the gravity model. While there are many theoretical papers on the structure of airline networks, most contain an a priori con®nement to the polar cases of a simple hub-and-spoke network and a point-to-point network. 1 Work considering networks without restrictions on topology is scarce: Hendricks et al. (1995) show that under symmetric elastic demand, uniform ®xed costs of setting up a connection, and costs in trac density that are symmetric and concave, the airline will operate a hub-and-
*
Corresponding author. Tel.: +49-40-42838-2924; fax: +49-40-42838-6272. E-mail address: [email protected] (O.W. Wojahn). 1 In the simple hub-and-spoke network, there is a single hub city that has a direct connection to every spoke city, and the spoke cities are only connected to the hub city. In the point-to-point network, all cities are connected directly. 1366-5545/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 6 - 5 5 4 5 ( 0 0 ) 0 0 0 2 6 - 0
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spoke network if ®xed costs are high (i.e., strong economies of density exist). Starr and Stinchcombe (1992) consider the case of asymmetric inelastic demand, uniform ®xed costs of setting up a connection, and constant marginal costs per passenger kilometer travelled. If ®xed costs are suciently small, then the cost-minimizing network is point-to-point. If marginal costs are suf®ciently small, then it is either a cycle, a tree (if only symmetric solutions are considered), or a hub-and-spoke network (if passengers make at most one transfer). Empirical studies have found that airline costs are characterized by signi®cant economies of trac density, so that average costs on a connection decline as the number of passengers grows (Caves et al., 1984; Gillen et al., 1990; Brueckner and Spiller, 1994). Given the results of Hendricks et al. (1995) and Starr and Stinchcombe (1992), this provides a reason why many airlines under deregulation have reorganized their networks in a hub-and-spoke manner. The case for hubbing gets even stronger if the size of the network grows, which can be attributed to the network externalities or spillover aects of additional spokes (Nero, 1999). While the above models do build a strong case for single hub networks, they do not leave room for the kind of multi-hub networks that many major air carriers operate. Even the tree structure that might evolve in the model of Starr and Stinchcombe does not provide a satisfactory explanation: ®rst, it does not closely resemble actual multi-hub networks since it does not allow for hubs to be connected directly, and second, it results from strong assumptions on the cost structure. If airline network structure is indeed determined by cost and demand conditions, something seems to be missing in the above models. The evident question is whether the multi-hub networks could be the result of departures from the symmetric demand assumption of Hendricks et al., together with a more general cost structure than that considered by Starr and Stinchcombe. We try to answer this question in a model where cities have arbitrary sizes. Inelastic airline demand between cities follows from the gravity model and thus is proportional to the product of city sizes, presumably giving a predilection to some bigger cities to become hubs. 2 Space is symmetric in the sense that distance does not aect demand and costs. Costs on a connection are strictly increasing and concave in trac volume, re¯ecting economies of density. In Section 2, we present the formal framework of the analysis. We restrict attention to a certain class of multi-hub networks, where spoke cities are assigned to a single hub only, and the possible hubs are fully interconnected. In Section 3, we consider the case where possible hub cities are given exogenously. The costminimizing network then is a mixture of point-to-point and single hub network. Multi-hub networks where passengers change planes at more than one airport are found to be suboptimal. In Section 4, we discuss the case of endogenous determination of the identity of the hub city. It is shown that the single hub will be located in the largest city. Conclusions are found in Section 5.
2. The model Consider the set N f1; 2; . . . ; ng of n P 3 cities served by an airline. Within the airline network, cities are either connected or not. If cities i and j are connected, then air travel is possible on 2
For a detailed discussion of the gravity model, see Sen and Smith (1995) and Gauthier and Taae (1973).
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both directed connections (sectors) between them: i; j and j; i. If they are not connected, then direct travel between them is impossible. Denote the population share of city i by pi > 0, so that P p i2N i 1. Assumption 1. Demand is governed by the gravity model. Given Assumption 1, demand from city i to city j, qij , is proportional to the product of city sizes. Without loss of generality, we omit the scale factor and get qij pi pj :
1
In a world where space does not matter, distance does not determine demand. Rather, interaction is solely determined by population, which is reasonable if the travel time between any two cities does not vary strongly. This will hold if ®xed travel time components are large. Assumption 1 establishes inelastic, asymmetric demand, whereas demand in the model of Hendricks et al. (1995) is elastic and symmetric. Cities are either potential hub cities where passengers may change planes, or spoke cities where passengers only start or terminate travel. Assumption 2. Each spoke city is assigned to a single potential hub city. Assumption 3. Potential hub cities are fully interconnected. A dummy hub is a potential hub city without any spokes assigned, otherwise the hub city is a real hub. We will call a network con®guration a multi-hub network if and only if it contains at least two real hubs. At real hubs, passengers make transfers; at dummy hubs, they do not. Fig. 1 depicts a multi-hub network with 12 cities. There are four potential hubs: cities 1, 4 and 12 are real hubs, city 10 is a dummy hub. As the potential hubs are fully interconnected, non-stop travel is possible between any two of them (Assumption 3). Each spoke city is assigned to a single hub, for example, spoke city 5 is assigned only to hub city 4 (Assumption 2). This implies that all trac originating or terminating in city 5 connects, originates, or terminates at hub city 4. Assumption 4. Passengers travel on the routing with the least number of sectors.
Fig. 1. Multi-hub network.
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As passengers value time and dislike changing planes, this seems to be a reasonable assumption as long as passengers are free to choose their routing. 3 But if the airline has the power to choose the routing that passengers travel, then under certain cost conditions it turns out that it is favorable not to route passengers on the shortest path in order to minimize costs. For example, in the cost-minimizing cyclical network mentioned in the introduction, passengers travel in the same direction around the cycle (Starr and Stinchcombe, 1992), thereby violating Assumption 4. In conjunction with Assumptions 2 and 3, Assumption 4 ensures that trac ¯ows are symmetric and unambiguous. It is assumed that the airline serves total demand and minimizes the costs in doing so. Costs on the sector from city i to city j, cij , given Qij passengers who travel on sector i; j, are cij w Qij :
2
Assumption 5. The sector cost function w : R ! R is strictly increasing and concave, thus continuous in trac density and does not depend on distance. Concavity of the sector cost function establishes economies of trac density. The fact that the sector cost function does not depend on distance and thus is identical across city pairs again re¯ects a spatially symmetric (or spaceless) world. In contrast, costs in the model of Starr and Stinchcombe (1992) depend on both distance and trac density, but they are restricted to linear relations. Note that sector costs may embody ®xed costs of setting up a connection, so that w 0 > 0. Now consider a network with m potential hubs, 1 6 m 6 n. Let H be the index set of all potential S i Si [ fig. b Si hub cities and Si the index set of those spoke cities assigned to hub i. Denote b establishes a cluster of cities associated with hub i. In Fig. 1, H f1; 4; 10; 12g establishes four clusters, one of them is b S 4 f4; 5; 6; 9; 11g. Let xi denote the total population of cluster i: X pj : 3 xi j2b Si By Assumptions 2±4, all passengers from cluster r to cluster s travel on the inter hub sector r; s. Hence trac density on the inter hub sector r; s is XX pi pj xr xs : 4 Qrs i2b S r j2b Ss By Assumption 2, trac density on a spoke sector is independent of other network characteristics. All trac originating or terminating at the spoke airport travels on the two according spoke sectors. The trac density on the spoke sectors associated with city i is pi 1 pi in each direction. 3
In a model with pricing, Assumption 4 need not be consistent with utility-maximization of passengers. For example, if the cost-savings on indirect routings lead to pro®t-maximizing prices that are well below those of direct routings, then the lower price may overcompensate the disutility of longer travel times and the inconvenience of changing planes.
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Total network costs c depend on how many and which cities become hubs, and to which hub each spoke city is assigned: X X XX w xi xj 2w pj 1 pj : 5 c H ; S1 ; . . . ; Sm i2H j2H nfig
i2H
j2Si
The ®rst double sum represents the costs on the inter hub sectors; the second represents the costs on the spoke sectors. 3. Results with exogenous hubs First consider the case where the set of potential hub airports is given. Then total costs on the spoke sectors are ®xed; denote them by Csp . With x the m-vector of cluster sizes, network costs can be written as X X w xi xj Csp : 6 c x i2H j2H nfig
The problem of the airline is to ®nd an assignment of spoke cities to hub cities that minimizes costs. Neglecting indivisibility of cities, the feasible set of cluster sizes is given by ( ) X x 1; xi P pi for all i 2 H : 7 Tm x 2 Rm i2H i The ®rst restriction ensures that total population is served, the second that each cluster at least contains the according hub city. Tm is a closed and bounded, thus a compact, polyhedral set. The problem of the airline is min c x:
x2Tm
8
The following theorem characterizes a cost-minimizing network: Theorem 1. Suppose that Assumptions 1±5 hold and c is quasi-concave. Then there exists a costminimizing network that has 0 or 1 real hubs (i.e., the network is not a multi-hub network). Proof. As a sum of continuous functions, c is continuous. The problem to minimize a continuous and quasi-concave function c on the compact polyhedral set Tm has an optimal solution x , where x is an extreme point of Tm (Bazaraa and Shetty, 1979). P The set of extreme points of Tm is characterized by xi 1 j2H nfig pj and xj pj for all j 2 H n fig. If m < n, then city i is a real hub, cities j are dummy hubs and the network has a single real hub. If m n, then every city is a dummy hub. Note that we have obtained a feasible solution despite relaxing the indivisibility constraint of cities. The essence of Theorem 1 is: if the number of potential hubs and the number of cities coincide, then by Assumption 3 the network is fully connected (point-to-point) and every hub is a dummy hub. If the number of potential hubs is smaller than the number of cities,
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Fig. 2. Networks with four cities and two potential hubs.
then there is only a single real hub, i.e., a single city that serves as a transfer point. All other hubs have no spokes attached and thus are dummy hubs. The cost-minimizing network, given m, thus is a combination of a single hub serving all spoke airports and a point-to-point network. Each of the Assumptions 1±5 is crucial to obtain Theorem 1. To see the importance of the gravity model in establishing Theorem 1, consider the simplest case: a network of size n 4 with H f1; 2g. All possible network con®gurations under Assumptions 2 and 3 are displayed in Fig. 2. The costs on the inter hub sector 1; 2 are included in Fig. 2, and they coincide with the costs on the inter hub sector 2; 1. The costs on the spoke sectors are independent of the network con®guration. Hence the con®guration minimizing total costs is the one minimizing costs on 1; 2. Networks I and II have two real hubs each; III and IV have one real hub. Now assume that network I minimizes costs. First compare with III: because w is strictly increasing, it has to hold that p1 p4 p3 p4 6 p4 p2 . This implies p1 < p2 . Comparing I with IV yields the condition p3 p2 p3 p4 6 p1 p3 which implies p2 < p1 ; a contradiction so network I cannot minimize costs. A similar argument can be made for network II, so that none of the multi-hub networks can be cost-minimizing. More generally, connecting all spokes with a single hub is favorable because connecting trac between spoke cities does not have to travel on any inter hub routes. At the same time, the gravitational law implies that redistribution of trac between spoke and hub cities on inter hub routes does not overcompensate this eect. Consider the following counter example, where we relax Assumption 1 and as a consequence Theorem 1 does not hold. 4 Assume that demand is given by q12 q21 q13 q31 q24 q42 1 and q14 q41 q23 q32 q34 q43 e, where 1 > e > 0. Then it is readily checked that network I (with 2 real hubs) minimizes network costs for e suciently small. The reason is that demand does not obey the gravitational law. Demand between cities 1 and 3 is high and between cities 1 and 4 is low, which under the gravity model implies that city 3 is larger than city 4. Furthermore, demand between cities 2 and 3 is low and between cities 2 and 4 is high, which implies that city 4 is larger than city 3, a contradiction.
4
Unlike Newton's `law of universal gravitation' which can be safely accepted on the basis of its empirical accuracy, the same has never been true of `demographic gravitation (Sen and Smith, 1995, p. 3).
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We have pointed out that if the assumption that passengers travel on the routing with the smallest number of sectors is relaxed, then cyclical networks may be cost-minimizing, resulting in passengers travelling in the same direction around the circle. In this case passengers change planes in more than one city. Likewise, it is possible to conceive of examples where a departure from any of the Assumptions 2, 3, or 5 results in optimality of multi-hub networks. For exposition, take the case of several geographical clusters of cities, where all intra-cluster distances and costs are small and all inter-cluster distances and costs are large, which represent a departure from Assumption 5. Then the cost-minimizing network may include a single hub in each of the clusters. Intra-cluster trac is served via the according hub, and inter-cluster trac is served via the inter hub sectors. Even if Assumptions 1±5 were to hold, rationalization of multi-hub networks may be found outside our simple cost-minimizing framework. Berechman and Shy (1996) and Brueckner and Zhang (1999) consider the value of travel time and ¯ight frequencies to passengers and their impact on the network decision. Both ®nd that hub-and-spoke networks are more likely to be pro®t-maximizing if the value of ¯ight frequency is high and the disutility of longer travel times and connecting is low. Barla and Constantatos (2000) examine the role of stochastic demand on the network decision. They ®nd that demand uncertainty bene®ts a hub-and-spoke network, because it allows a reallocation of capacities after demand has been revealed. Despite the fact that the above studies are con®ned to three cities, intuitively it seems plausible that the additional bene®ts of allocational ¯exibility or higher ¯ight frequencies taper o at a certain hub size, thereby providing some rationale for operating multiple hubs. Our model does not consider competition. Hendricks et al. (1999) demonstrate that competition of two large carriers may result in duopoly equilibria that resemble multi-hub networks, providing circumstantial evidence that strategic behavior might result in multi-hub networks. 5 Finally, airport congestion may rationalize multi-hub networks, where the unrestricted airline would operate a single hub. To prove Theorem 1, we had to make the additional assumption of quasi-concavity of c. Because demand Q xi ; xj xi xj is quasi-concave in xi and xj , and because w is a strictly increasing function, w Q xi ; xj is quasi-concave in xi and xj . But it is well known that the sum of quasiconcave functions is not necessarily quasi-concave; hence c is not necessarily quasi-concave. Therefore the classes of sector cost functions w that establish quasi-concave network costs c are of interest. It can be shown that linear and logarithmic sector cost functions both establish quasiconcave network costs; for a proof and further results, see Appendix A.
4. Results with endogenous hubs In the preceding section, we have assumed that the potential hub cities are exogenously given. Now we turn to the problem of determining the optimal number and identity of hubs. From Theorem 1 we know that at most a single hub will be a real hub. The following results characterize the real hub (proofs of this section are provided in Appendix B): 5
For further results on the eects of competition on airline network choice, see Oum et al. (1995) and Hendricks et al. (1997).
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Lemma 1. Suppose that Assumptions 1, 2 and 5 hold. Then in a cost-minimizing network, a spoke city is not larger than the hub city to which it is assigned. If in Fig. 1 city 5 is larger than city 4, then network costs decrease if we make city 5 the hub city of the cluster: H f1; 5; 10; 12g and S5 f4; 6; 9; 11g. The cost eects of interchanging cities 4 and 5 are con®ned to spoke sectors 4; 5 and 5; 4. The smaller city should be the spoke city in order to achieve a lower trac density on the spoke. Lemma 2. Suppose Assumptions 1±5 hold. Then in a cost-minimizing network, the spokes are assigned to the largest potential hub. Consider the network without spoke cities. The highest trac densities can be found on the inter hub sectors originating or terminating at the largest city. Due to the concavity of sector costs, the additional costs of accommodating the incremental ¯ows generated by assigning the spoke cities to a potential hub are lower if trac densities are high. So the incremental costs of the trac generated by the spoke cities are minimal if they are assigned to the largest potential hub. Lemma 1 implies that in a cost-minimizing network the real hub city is at least as large as any spoke city, and Lemma 2 implies that the real hub city is at least as large as any other hub city. Combining them yields: Theorem 2. Suppose Assumptions 1±5 hold. Then in a cost-minimizing network, the real hub city is the largest city. If the airline minimizes costs, Theorem 2 implies that the largest city is the only city where passengers connect. Factual evidence indeed indicates that airlines often choose to locate hubs at large cities with strong local demand. But in some instances, airlines have built hubs at relatively small cities, for example Delta Air Lines at Salt Lake City and Northwest Airlines at Memphis (Borenstein, 1992, p. 55, Table 3). In these cases, the location towards the center of the US and the resulting geographically ecient routings for connecting trac seem to outweigh the disadvantage of weak local demand. So far we have not addressed the following two questions: (i) What is the cost-minimizing number of potential hubs m? and (ii) Given the number of hubs, which cities should be spoke cities and which cities should be dummy hub cities? To both questions, there is no easy answer. To see this, consider the case of n 4, where p1 0; 5, p2 p3 0; 2, p4 0; 1, and w x xa . There are four possible networks conforming to Theorems 1 and 2; they are displayed in Fig. 3, together with the costs associated with dierent levels of a. Now the answer to question (i) depends on a: for a 1, network I with four potential hubs minimizes costs; for a 0; 72, network II with three potential hubs minimizes costs; and for a 0; 5, network IV with a single potential hub or two potential hubs (both yielding the same network structure) minimizes costs. So the cost-minimizing number of potential hubs depends on the parameter values of the sector cost function. It is obvious that the bang-bang solution valid under symmetric demand (either point-to-point or single hub, Hendricks et al., 1995) does not hold in the context of asymmetric demand.
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Fig. 3. Network costs.
To deal with question (ii), consider only the networks with three potential hubs: II and III. The decision which city should become a spoke city depends on a: for a 1 or a 0; 72, the small city should become a spoke city; for a 0; 5, one of the two large cities should become a spoke city. In summary, conclusions about network structure beyond those of Theorems 1 and 2 depend on the speci®cs of the sector cost function and on the distribution of city sizes. The sector cost function or economies of density have been estimated repeatedly (Caves et al., 1984; Gillen et al., 1990; Brueckner and Spiller, 1994). The distribution of city sizes can be described by one of the most accurate laws in economics, which `appears to hold in virtually all countries and dates for which there are data, even the United States in 1790 and India in 1911' (Gabaix, 1999, p. 129). 6 Exploring the implications of these regularities on airline networks should warrant further research. 5. Conclusion We have addressed the question: Are observed multi-hub airline networks the consequence of cost-minimizing behavior under economies of density? Within the framework of our spaceless model, the answer is no: by Theorem 1, we can eliminate multi-hub networks from the class of cost-minimizing networks. So the broader question remains: Why do airlines or airline alliances operate multi-hub networks? We have discussed some reasons for deviation from Theorem 1, like asymmetric costs (geography), non-validity of the gravity model, airport congestion, or competition. Moreover, in international air transport, regulatory constraints prevent airlines or airline alliances from freely building their networks. If economic and operating forces determine network patterns, then we expect that at least some spatially concentrated multi-hub systems will be abandoned in favor of single hub operations. The recent deregulation of the EU market will provide some indication if and how these predicted patterns indeed emerge. Acknowledgements I would like to thank the editor-in-chief and two anonymous referees for their helpful comments. 6
Zipf's Law states that if the largest city has size 1, then the second largest has size 1/2, the third largest has size 1/3, and so forth.
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Appendix A Denote partial derivatives by a subscript. If not indicated otherwise, counters in sums run over the set of hubs HP. Without loss of generality in the proofs, we may ignore the constant term in c, P yielding c x i j6i w xi xj . Lemma 3. c x is quasi-concave on Tm if (i) w x a b ln x, or (ii) w x a bxa , where b P 0, 0 6 a 6 12. Proof. Let w x a b ln x. Then wii xi xj b=x2i , wjj xi xj b=x2j , and wij xi xj 0. Hence wii 6 0, wjj 6 0, and wii wjj w2ij P 0, so that w xi xj is concave in xi and xj . A sum of concave functions is concave, so c is concave and thus quasi-concave. Case (ii) can be shown in a similar manner. Lemma 4. Assume that w is such that for all x; y 2 Tm , XX w xi yj P minfc x; c yg: i
A:1
j6i
Then c x is quasi-concave on Tm . Proof. By de®nition, c x is quasi-concave on Tm if and only if for all x; y 2 Tm and all k 2 0; 1, c kx 1
ky P minfc x; c yg:
Now we can expand c kx 1
ky
XX i
j6i
XX i
kyi kxj 1
w k2 xi xj 2k 1
kxi yj 1
k2 1, we can apply Jensen's inequality: XX k2 c y 2k 1 k w xi yj :
ky P k2 c x 1
Case 1. c x P c y. Then by assumption, ky P k2 c y 1
ky P k2 c x 1
P P i
j6i
A:3
P P i
j6i
kc y c y:
A:5
w xi yj P c x, and we get
k2 c x 2k 1
Combining cases 1 and 2 yields c kx 1
A:4
j6i
w xi yj P c y, and we get
k2 c y 2k 1
Case 2. c y P c x. Then by assumption, c kx 1
k2 yi yj :
k 1
i
c kx 1
kyj
j6i
As w is concave and k2 2k 1 c kx 1
w kxi 1
A:2
kc x c x:
ky P minfc x; c yg.
A:6
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Lemma 5. c x is quasi-concave on Tm if w x ax b, a > 0. Proof. For x; y 2 Tm , it holds that XX X xi yj 1 xi yi : i
P i
xi
P
i yi
1. This yields
P P i
j
xi yj 1, hence A:7
i
j6i
By Cauchy±Schwarz's inequality, v" #" # u X X u X x y P1 t x2 y2 : 1 i i
i
i
i
A:8
i
i
P P Case P 1. c x 6Pc y. With c x a i j6i xi xj n n 1b a 1 plies i x2i P i yi2 , thus v #" # u" X XX X u X xi yj P 1 t x2 x2 1 x2 ; i
and
XX i
i
j6i
i
i
i
i
i
w xi yj P c x:
and
x2i n n
1b, this im-
A:9
A:10
i
j6i
XX i
i
j6i
P P Case 2. c x P c y. This implies i yi2 P i x2i , thus v #" # u" u X X X XX xi yj P 1 t yi2 yi2 1 yi2 ; i
P
i
w xi yj P c y:
j6i
Combining cases 1 and 2 yields quasi-concave.
A:11
i
A:12 P P i
j6i
w xi yj P minfc x; c yg; hence by Lemma 4, c x is
We conjecture that c x is quasi-concave on Tm for any w that is strictly increasing and concave. Appendix B Proof of Lemma 1. Consider a spoke s and a hub h. Assume that ps > ph . We want to show that the cost savings of interchanging the two cities, d, are strictly positive. By Assumptions 1 and 2, the trac density on a spoke sector where the spoke city has population p is r p p 1 p. The cost eects of interchanging the two cities are local on the spoke sector: d 2w r ps
w r ph :
B:1
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By Assumption 5, w is strictly increasing, so that cost savings are strictly positive if and only if r ps > r ph . Case 1. ps < 0; 5. Because r is strictly increasing for p < 0; 5, and because ps > ph it follows that r ps > r ph . Case 2. ps P 0; 5. Because total population is 1 and n P 3, it follows that ph 1 ps e, where e > 0. Due to symmetry of r about p 0; 5, it follows that r ps > r 1 ps e r ph . Proof of Lemma 2. Consider two potential hub cities, without loss of generality cities 1 and 2. Assume that city 2 is larger than city 1, p2 > p1 , and that the total population of all spoke cities is ps . Denote the costs of the network with the spokes assigned to hub i by Ci . We want to show that C2 < C1 , that is the costs are lower if the spokes are assigned to the larger hub. Applying Eq. (6), we can write C1
C2 2w p1 ps p2 w p2 ps p1 X 2 w p1 ps pj w p1 pj
B:2a fw p2 ps pj
w p2 pj g:
B:2b
j2H nf1;2g
Term (B.2a) is strictly positive because by assumption city 2 is larger, thus p1 ps p2 > p2 ps p1 . Let D x; h w x h w x=h, h > 0, be the dierence quotient giving the slope of the secant line passing through w x and w x h. From concavity of w it follows that Dx x; h w0 x h w0 x=h < 0. Because p2 > p1 , this implies D p1 pj ; ps pj D p2 pj ; ps pj > 0, or rewriting w p1 pj ps pj ps pj
w p1 pj
w p2 pj ps pj ps pj
w p2 pj
> 0:
B:3
Hence each term in Eq. (B.2b) is positive, and so is Eq. (B.2b). From Eq. (B.2a) > 0 and Eq. (B.2b) > 0, we conclude C1 C2 > 0. References Barla, P., Constantatos, C., 2000. Airline network structure under demand uncertainty. Transportation Research Part E 36 (3), 173±180. Bazaraa, M.S., Shetty, C.M., 1979. Nonlinear Programming. Wiley, New York. Berechman, J., Shy, O., 1996. The structure of airline equilibrium networks. In: van den Bergh, J.C.J.M., Nijkamp, P., Rietveld, P. (Eds.), Recent advances in spatial equilibrium modelling. Springer, Berlin, pp. 138±155. Borenstein, S., 1992. The evolution of US airline competition. Journal of Economic Perspectives 6 (2), 45±73. Brueckner, J.K., Spiller, P.T., 1994. Economies of trac density in deregulated airline markets. Journal of Law and Economics 37 (2), 379±415. Brueckner, J.K., Zhang, Y., 1999. Scheduling Decisions in an Airline Network: A Hub-and-Spoke System's Eect on Flight Frequency, Fares and Welfare. Oce of Research Working Paper 99/0110, University of Illinois at UrbanaChampaign.. Caves, D., Christensen, L., Tretheway, M., 1984. Economies of density versus economies of scale: Why trunk and local service airline costs dier. Rand Journal of Economics 15 (4), 471±489.
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Gabaix, X., 1999. Zipf's law and the growth of cities. AEA Papers and Proceedings 89 (2), 129±132. Gillen, D.W., Oum, T.H., Tretheway, M., 1990. Airline cost structure and policy implications: A multi-product approach for Canadian airlines. Journal of Transport Economics and Policy 24 (1), 9±34. Gauthier, H.L., Taae, E.J., 1973. Geography of Transportation. Prentice-Hall, Englewood Clis, NJ. Hendricks, K., Piccione, M., Tan, G., 1995. The economics of hubs: The case of monopoly. Review of Economic Studies 62 (1), 83±99. Hendricks, K., Piccione, M., Tan, G., 1997. Entry and exit in hub-spoke networks. Rand Journal of Economics 28 (2), 291±303. Hendricks, K., Piccione, M., Tan, G., 1999. Equilibria in networks. Econometrica 67 (6), 1407±1434. Nero, G., 1999. A note on the competitive advantage of large hub-and-spoke networks. Transportation Research Part E 35 (4), 225±239. Oum, T.H., Zhang, A., Zhang, Y., 1995. Airline network rivalry. Canadian Journal of Economics 28 (4), 836±857. Sen, A., Smith, T.E., 1995. Gravity Models of Spatial Interaction. Springer, Berlin. Starr, R.M., Stinchcombe, M.B., 1992. Ecient Transportation and Natural Monopoly in the Airline Industry: An Economic Analysis of Hub-Spoke Related Systems. Department of Economics Working Paper 92±25, University of California at San Diego.
www.elsevier.com/locate/tre
Airline network structure and the gravity model Oliver W. Wojahn a,b,* a
Department of Economics, University of Hamburg, von-Melle-Park 5, 20146 Hamburg, Germany b Lufthansa Technik AG, Business Development HAM WR 1, 22313 Hamburg, Germany Received 25 September 2000; accepted 8 November 2000
Abstract In this paper, we determine characteristics of the cost-minimizing airline network under economies of density. Airline demand is asymmetric and governed by the gravity model. Airline networks are restricted to those where each spoke city is assigned to a single hub and where hub cities are fully interconnected. The cost-minimizing network is a mixture of a point-to-point and a single hub network. Multi-hub networks where passengers change planes at more than one airport are found to be suboptimal. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Airline networks; Hub-and-spoke networks; Economies of density; Gravity model
1. Introduction Demand and cost conditions have been identi®ed as the main determinants of airline network topology in the literature. The purpose of this paper is to determine the characteristics of the costminimizing airline network if technology exhibits economies of density and asymmetric demand follows the gravity model. While there are many theoretical papers on the structure of airline networks, most contain an a priori con®nement to the polar cases of a simple hub-and-spoke network and a point-to-point network. 1 Work considering networks without restrictions on topology is scarce: Hendricks et al. (1995) show that under symmetric elastic demand, uniform ®xed costs of setting up a connection, and costs in trac density that are symmetric and concave, the airline will operate a hub-and-
*
Corresponding author. Tel.: +49-40-42838-2924; fax: +49-40-42838-6272. E-mail address: [email protected] (O.W. Wojahn). 1 In the simple hub-and-spoke network, there is a single hub city that has a direct connection to every spoke city, and the spoke cities are only connected to the hub city. In the point-to-point network, all cities are connected directly. 1366-5545/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 6 - 5 5 4 5 ( 0 0 ) 0 0 0 2 6 - 0
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spoke network if ®xed costs are high (i.e., strong economies of density exist). Starr and Stinchcombe (1992) consider the case of asymmetric inelastic demand, uniform ®xed costs of setting up a connection, and constant marginal costs per passenger kilometer travelled. If ®xed costs are suciently small, then the cost-minimizing network is point-to-point. If marginal costs are suf®ciently small, then it is either a cycle, a tree (if only symmetric solutions are considered), or a hub-and-spoke network (if passengers make at most one transfer). Empirical studies have found that airline costs are characterized by signi®cant economies of trac density, so that average costs on a connection decline as the number of passengers grows (Caves et al., 1984; Gillen et al., 1990; Brueckner and Spiller, 1994). Given the results of Hendricks et al. (1995) and Starr and Stinchcombe (1992), this provides a reason why many airlines under deregulation have reorganized their networks in a hub-and-spoke manner. The case for hubbing gets even stronger if the size of the network grows, which can be attributed to the network externalities or spillover aects of additional spokes (Nero, 1999). While the above models do build a strong case for single hub networks, they do not leave room for the kind of multi-hub networks that many major air carriers operate. Even the tree structure that might evolve in the model of Starr and Stinchcombe does not provide a satisfactory explanation: ®rst, it does not closely resemble actual multi-hub networks since it does not allow for hubs to be connected directly, and second, it results from strong assumptions on the cost structure. If airline network structure is indeed determined by cost and demand conditions, something seems to be missing in the above models. The evident question is whether the multi-hub networks could be the result of departures from the symmetric demand assumption of Hendricks et al., together with a more general cost structure than that considered by Starr and Stinchcombe. We try to answer this question in a model where cities have arbitrary sizes. Inelastic airline demand between cities follows from the gravity model and thus is proportional to the product of city sizes, presumably giving a predilection to some bigger cities to become hubs. 2 Space is symmetric in the sense that distance does not aect demand and costs. Costs on a connection are strictly increasing and concave in trac volume, re¯ecting economies of density. In Section 2, we present the formal framework of the analysis. We restrict attention to a certain class of multi-hub networks, where spoke cities are assigned to a single hub only, and the possible hubs are fully interconnected. In Section 3, we consider the case where possible hub cities are given exogenously. The costminimizing network then is a mixture of point-to-point and single hub network. Multi-hub networks where passengers change planes at more than one airport are found to be suboptimal. In Section 4, we discuss the case of endogenous determination of the identity of the hub city. It is shown that the single hub will be located in the largest city. Conclusions are found in Section 5.
2. The model Consider the set N f1; 2; . . . ; ng of n P 3 cities served by an airline. Within the airline network, cities are either connected or not. If cities i and j are connected, then air travel is possible on 2
For a detailed discussion of the gravity model, see Sen and Smith (1995) and Gauthier and Taae (1973).
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both directed connections (sectors) between them: i; j and j; i. If they are not connected, then direct travel between them is impossible. Denote the population share of city i by pi > 0, so that P p i2N i 1. Assumption 1. Demand is governed by the gravity model. Given Assumption 1, demand from city i to city j, qij , is proportional to the product of city sizes. Without loss of generality, we omit the scale factor and get qij pi pj :
1
In a world where space does not matter, distance does not determine demand. Rather, interaction is solely determined by population, which is reasonable if the travel time between any two cities does not vary strongly. This will hold if ®xed travel time components are large. Assumption 1 establishes inelastic, asymmetric demand, whereas demand in the model of Hendricks et al. (1995) is elastic and symmetric. Cities are either potential hub cities where passengers may change planes, or spoke cities where passengers only start or terminate travel. Assumption 2. Each spoke city is assigned to a single potential hub city. Assumption 3. Potential hub cities are fully interconnected. A dummy hub is a potential hub city without any spokes assigned, otherwise the hub city is a real hub. We will call a network con®guration a multi-hub network if and only if it contains at least two real hubs. At real hubs, passengers make transfers; at dummy hubs, they do not. Fig. 1 depicts a multi-hub network with 12 cities. There are four potential hubs: cities 1, 4 and 12 are real hubs, city 10 is a dummy hub. As the potential hubs are fully interconnected, non-stop travel is possible between any two of them (Assumption 3). Each spoke city is assigned to a single hub, for example, spoke city 5 is assigned only to hub city 4 (Assumption 2). This implies that all trac originating or terminating in city 5 connects, originates, or terminates at hub city 4. Assumption 4. Passengers travel on the routing with the least number of sectors.
Fig. 1. Multi-hub network.
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As passengers value time and dislike changing planes, this seems to be a reasonable assumption as long as passengers are free to choose their routing. 3 But if the airline has the power to choose the routing that passengers travel, then under certain cost conditions it turns out that it is favorable not to route passengers on the shortest path in order to minimize costs. For example, in the cost-minimizing cyclical network mentioned in the introduction, passengers travel in the same direction around the cycle (Starr and Stinchcombe, 1992), thereby violating Assumption 4. In conjunction with Assumptions 2 and 3, Assumption 4 ensures that trac ¯ows are symmetric and unambiguous. It is assumed that the airline serves total demand and minimizes the costs in doing so. Costs on the sector from city i to city j, cij , given Qij passengers who travel on sector i; j, are cij w Qij :
2
Assumption 5. The sector cost function w : R ! R is strictly increasing and concave, thus continuous in trac density and does not depend on distance. Concavity of the sector cost function establishes economies of trac density. The fact that the sector cost function does not depend on distance and thus is identical across city pairs again re¯ects a spatially symmetric (or spaceless) world. In contrast, costs in the model of Starr and Stinchcombe (1992) depend on both distance and trac density, but they are restricted to linear relations. Note that sector costs may embody ®xed costs of setting up a connection, so that w 0 > 0. Now consider a network with m potential hubs, 1 6 m 6 n. Let H be the index set of all potential S i Si [ fig. b Si hub cities and Si the index set of those spoke cities assigned to hub i. Denote b establishes a cluster of cities associated with hub i. In Fig. 1, H f1; 4; 10; 12g establishes four clusters, one of them is b S 4 f4; 5; 6; 9; 11g. Let xi denote the total population of cluster i: X pj : 3 xi j2b Si By Assumptions 2±4, all passengers from cluster r to cluster s travel on the inter hub sector r; s. Hence trac density on the inter hub sector r; s is XX pi pj xr xs : 4 Qrs i2b S r j2b Ss By Assumption 2, trac density on a spoke sector is independent of other network characteristics. All trac originating or terminating at the spoke airport travels on the two according spoke sectors. The trac density on the spoke sectors associated with city i is pi 1 pi in each direction. 3
In a model with pricing, Assumption 4 need not be consistent with utility-maximization of passengers. For example, if the cost-savings on indirect routings lead to pro®t-maximizing prices that are well below those of direct routings, then the lower price may overcompensate the disutility of longer travel times and the inconvenience of changing planes.
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Total network costs c depend on how many and which cities become hubs, and to which hub each spoke city is assigned: X X XX w xi xj 2w pj 1 pj : 5 c H ; S1 ; . . . ; Sm i2H j2H nfig
i2H
j2Si
The ®rst double sum represents the costs on the inter hub sectors; the second represents the costs on the spoke sectors. 3. Results with exogenous hubs First consider the case where the set of potential hub airports is given. Then total costs on the spoke sectors are ®xed; denote them by Csp . With x the m-vector of cluster sizes, network costs can be written as X X w xi xj Csp : 6 c x i2H j2H nfig
The problem of the airline is to ®nd an assignment of spoke cities to hub cities that minimizes costs. Neglecting indivisibility of cities, the feasible set of cluster sizes is given by ( ) X x 1; xi P pi for all i 2 H : 7 Tm x 2 Rm i2H i The ®rst restriction ensures that total population is served, the second that each cluster at least contains the according hub city. Tm is a closed and bounded, thus a compact, polyhedral set. The problem of the airline is min c x:
x2Tm
8
The following theorem characterizes a cost-minimizing network: Theorem 1. Suppose that Assumptions 1±5 hold and c is quasi-concave. Then there exists a costminimizing network that has 0 or 1 real hubs (i.e., the network is not a multi-hub network). Proof. As a sum of continuous functions, c is continuous. The problem to minimize a continuous and quasi-concave function c on the compact polyhedral set Tm has an optimal solution x , where x is an extreme point of Tm (Bazaraa and Shetty, 1979). P The set of extreme points of Tm is characterized by xi 1 j2H nfig pj and xj pj for all j 2 H n fig. If m < n, then city i is a real hub, cities j are dummy hubs and the network has a single real hub. If m n, then every city is a dummy hub. Note that we have obtained a feasible solution despite relaxing the indivisibility constraint of cities. The essence of Theorem 1 is: if the number of potential hubs and the number of cities coincide, then by Assumption 3 the network is fully connected (point-to-point) and every hub is a dummy hub. If the number of potential hubs is smaller than the number of cities,
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Fig. 2. Networks with four cities and two potential hubs.
then there is only a single real hub, i.e., a single city that serves as a transfer point. All other hubs have no spokes attached and thus are dummy hubs. The cost-minimizing network, given m, thus is a combination of a single hub serving all spoke airports and a point-to-point network. Each of the Assumptions 1±5 is crucial to obtain Theorem 1. To see the importance of the gravity model in establishing Theorem 1, consider the simplest case: a network of size n 4 with H f1; 2g. All possible network con®gurations under Assumptions 2 and 3 are displayed in Fig. 2. The costs on the inter hub sector 1; 2 are included in Fig. 2, and they coincide with the costs on the inter hub sector 2; 1. The costs on the spoke sectors are independent of the network con®guration. Hence the con®guration minimizing total costs is the one minimizing costs on 1; 2. Networks I and II have two real hubs each; III and IV have one real hub. Now assume that network I minimizes costs. First compare with III: because w is strictly increasing, it has to hold that p1 p4 p3 p4 6 p4 p2 . This implies p1 < p2 . Comparing I with IV yields the condition p3 p2 p3 p4 6 p1 p3 which implies p2 < p1 ; a contradiction so network I cannot minimize costs. A similar argument can be made for network II, so that none of the multi-hub networks can be cost-minimizing. More generally, connecting all spokes with a single hub is favorable because connecting trac between spoke cities does not have to travel on any inter hub routes. At the same time, the gravitational law implies that redistribution of trac between spoke and hub cities on inter hub routes does not overcompensate this eect. Consider the following counter example, where we relax Assumption 1 and as a consequence Theorem 1 does not hold. 4 Assume that demand is given by q12 q21 q13 q31 q24 q42 1 and q14 q41 q23 q32 q34 q43 e, where 1 > e > 0. Then it is readily checked that network I (with 2 real hubs) minimizes network costs for e suciently small. The reason is that demand does not obey the gravitational law. Demand between cities 1 and 3 is high and between cities 1 and 4 is low, which under the gravity model implies that city 3 is larger than city 4. Furthermore, demand between cities 2 and 3 is low and between cities 2 and 4 is high, which implies that city 4 is larger than city 3, a contradiction.
4
Unlike Newton's `law of universal gravitation' which can be safely accepted on the basis of its empirical accuracy, the same has never been true of `demographic gravitation (Sen and Smith, 1995, p. 3).
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We have pointed out that if the assumption that passengers travel on the routing with the smallest number of sectors is relaxed, then cyclical networks may be cost-minimizing, resulting in passengers travelling in the same direction around the circle. In this case passengers change planes in more than one city. Likewise, it is possible to conceive of examples where a departure from any of the Assumptions 2, 3, or 5 results in optimality of multi-hub networks. For exposition, take the case of several geographical clusters of cities, where all intra-cluster distances and costs are small and all inter-cluster distances and costs are large, which represent a departure from Assumption 5. Then the cost-minimizing network may include a single hub in each of the clusters. Intra-cluster trac is served via the according hub, and inter-cluster trac is served via the inter hub sectors. Even if Assumptions 1±5 were to hold, rationalization of multi-hub networks may be found outside our simple cost-minimizing framework. Berechman and Shy (1996) and Brueckner and Zhang (1999) consider the value of travel time and ¯ight frequencies to passengers and their impact on the network decision. Both ®nd that hub-and-spoke networks are more likely to be pro®t-maximizing if the value of ¯ight frequency is high and the disutility of longer travel times and connecting is low. Barla and Constantatos (2000) examine the role of stochastic demand on the network decision. They ®nd that demand uncertainty bene®ts a hub-and-spoke network, because it allows a reallocation of capacities after demand has been revealed. Despite the fact that the above studies are con®ned to three cities, intuitively it seems plausible that the additional bene®ts of allocational ¯exibility or higher ¯ight frequencies taper o at a certain hub size, thereby providing some rationale for operating multiple hubs. Our model does not consider competition. Hendricks et al. (1999) demonstrate that competition of two large carriers may result in duopoly equilibria that resemble multi-hub networks, providing circumstantial evidence that strategic behavior might result in multi-hub networks. 5 Finally, airport congestion may rationalize multi-hub networks, where the unrestricted airline would operate a single hub. To prove Theorem 1, we had to make the additional assumption of quasi-concavity of c. Because demand Q xi ; xj xi xj is quasi-concave in xi and xj , and because w is a strictly increasing function, w Q xi ; xj is quasi-concave in xi and xj . But it is well known that the sum of quasiconcave functions is not necessarily quasi-concave; hence c is not necessarily quasi-concave. Therefore the classes of sector cost functions w that establish quasi-concave network costs c are of interest. It can be shown that linear and logarithmic sector cost functions both establish quasiconcave network costs; for a proof and further results, see Appendix A.
4. Results with endogenous hubs In the preceding section, we have assumed that the potential hub cities are exogenously given. Now we turn to the problem of determining the optimal number and identity of hubs. From Theorem 1 we know that at most a single hub will be a real hub. The following results characterize the real hub (proofs of this section are provided in Appendix B): 5
For further results on the eects of competition on airline network choice, see Oum et al. (1995) and Hendricks et al. (1997).
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Lemma 1. Suppose that Assumptions 1, 2 and 5 hold. Then in a cost-minimizing network, a spoke city is not larger than the hub city to which it is assigned. If in Fig. 1 city 5 is larger than city 4, then network costs decrease if we make city 5 the hub city of the cluster: H f1; 5; 10; 12g and S5 f4; 6; 9; 11g. The cost eects of interchanging cities 4 and 5 are con®ned to spoke sectors 4; 5 and 5; 4. The smaller city should be the spoke city in order to achieve a lower trac density on the spoke. Lemma 2. Suppose Assumptions 1±5 hold. Then in a cost-minimizing network, the spokes are assigned to the largest potential hub. Consider the network without spoke cities. The highest trac densities can be found on the inter hub sectors originating or terminating at the largest city. Due to the concavity of sector costs, the additional costs of accommodating the incremental ¯ows generated by assigning the spoke cities to a potential hub are lower if trac densities are high. So the incremental costs of the trac generated by the spoke cities are minimal if they are assigned to the largest potential hub. Lemma 1 implies that in a cost-minimizing network the real hub city is at least as large as any spoke city, and Lemma 2 implies that the real hub city is at least as large as any other hub city. Combining them yields: Theorem 2. Suppose Assumptions 1±5 hold. Then in a cost-minimizing network, the real hub city is the largest city. If the airline minimizes costs, Theorem 2 implies that the largest city is the only city where passengers connect. Factual evidence indeed indicates that airlines often choose to locate hubs at large cities with strong local demand. But in some instances, airlines have built hubs at relatively small cities, for example Delta Air Lines at Salt Lake City and Northwest Airlines at Memphis (Borenstein, 1992, p. 55, Table 3). In these cases, the location towards the center of the US and the resulting geographically ecient routings for connecting trac seem to outweigh the disadvantage of weak local demand. So far we have not addressed the following two questions: (i) What is the cost-minimizing number of potential hubs m? and (ii) Given the number of hubs, which cities should be spoke cities and which cities should be dummy hub cities? To both questions, there is no easy answer. To see this, consider the case of n 4, where p1 0; 5, p2 p3 0; 2, p4 0; 1, and w x xa . There are four possible networks conforming to Theorems 1 and 2; they are displayed in Fig. 3, together with the costs associated with dierent levels of a. Now the answer to question (i) depends on a: for a 1, network I with four potential hubs minimizes costs; for a 0; 72, network II with three potential hubs minimizes costs; and for a 0; 5, network IV with a single potential hub or two potential hubs (both yielding the same network structure) minimizes costs. So the cost-minimizing number of potential hubs depends on the parameter values of the sector cost function. It is obvious that the bang-bang solution valid under symmetric demand (either point-to-point or single hub, Hendricks et al., 1995) does not hold in the context of asymmetric demand.
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Fig. 3. Network costs.
To deal with question (ii), consider only the networks with three potential hubs: II and III. The decision which city should become a spoke city depends on a: for a 1 or a 0; 72, the small city should become a spoke city; for a 0; 5, one of the two large cities should become a spoke city. In summary, conclusions about network structure beyond those of Theorems 1 and 2 depend on the speci®cs of the sector cost function and on the distribution of city sizes. The sector cost function or economies of density have been estimated repeatedly (Caves et al., 1984; Gillen et al., 1990; Brueckner and Spiller, 1994). The distribution of city sizes can be described by one of the most accurate laws in economics, which `appears to hold in virtually all countries and dates for which there are data, even the United States in 1790 and India in 1911' (Gabaix, 1999, p. 129). 6 Exploring the implications of these regularities on airline networks should warrant further research. 5. Conclusion We have addressed the question: Are observed multi-hub airline networks the consequence of cost-minimizing behavior under economies of density? Within the framework of our spaceless model, the answer is no: by Theorem 1, we can eliminate multi-hub networks from the class of cost-minimizing networks. So the broader question remains: Why do airlines or airline alliances operate multi-hub networks? We have discussed some reasons for deviation from Theorem 1, like asymmetric costs (geography), non-validity of the gravity model, airport congestion, or competition. Moreover, in international air transport, regulatory constraints prevent airlines or airline alliances from freely building their networks. If economic and operating forces determine network patterns, then we expect that at least some spatially concentrated multi-hub systems will be abandoned in favor of single hub operations. The recent deregulation of the EU market will provide some indication if and how these predicted patterns indeed emerge. Acknowledgements I would like to thank the editor-in-chief and two anonymous referees for their helpful comments. 6
Zipf's Law states that if the largest city has size 1, then the second largest has size 1/2, the third largest has size 1/3, and so forth.
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Appendix A Denote partial derivatives by a subscript. If not indicated otherwise, counters in sums run over the set of hubs HP. Without loss of generality in the proofs, we may ignore the constant term in c, P yielding c x i j6i w xi xj . Lemma 3. c x is quasi-concave on Tm if (i) w x a b ln x, or (ii) w x a bxa , where b P 0, 0 6 a 6 12. Proof. Let w x a b ln x. Then wii xi xj b=x2i , wjj xi xj b=x2j , and wij xi xj 0. Hence wii 6 0, wjj 6 0, and wii wjj w2ij P 0, so that w xi xj is concave in xi and xj . A sum of concave functions is concave, so c is concave and thus quasi-concave. Case (ii) can be shown in a similar manner. Lemma 4. Assume that w is such that for all x; y 2 Tm , XX w xi yj P minfc x; c yg: i
A:1
j6i
Then c x is quasi-concave on Tm . Proof. By de®nition, c x is quasi-concave on Tm if and only if for all x; y 2 Tm and all k 2 0; 1, c kx 1
ky P minfc x; c yg:
Now we can expand c kx 1
ky
XX i
j6i
XX i
kyi kxj 1
w k2 xi xj 2k 1
kxi yj 1
k2 1, we can apply Jensen's inequality: XX k2 c y 2k 1 k w xi yj :
ky P k2 c x 1
Case 1. c x P c y. Then by assumption, ky P k2 c y 1
ky P k2 c x 1
P P i
j6i
A:3
P P i
j6i
kc y c y:
A:5
w xi yj P c x, and we get
k2 c x 2k 1
Combining cases 1 and 2 yields c kx 1
A:4
j6i
w xi yj P c y, and we get
k2 c y 2k 1
Case 2. c y P c x. Then by assumption, c kx 1
k2 yi yj :
k 1
i
c kx 1
kyj
j6i
As w is concave and k2 2k 1 c kx 1
w kxi 1
A:2
kc x c x:
ky P minfc x; c yg.
A:6
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Lemma 5. c x is quasi-concave on Tm if w x ax b, a > 0. Proof. For x; y 2 Tm , it holds that XX X xi yj 1 xi yi : i
P i
xi
P
i yi
1. This yields
P P i
j
xi yj 1, hence A:7
i
j6i
By Cauchy±Schwarz's inequality, v" #" # u X X u X x y P1 t x2 y2 : 1 i i
i
i
i
A:8
i
i
P P Case P 1. c x 6Pc y. With c x a i j6i xi xj n n 1b a 1 plies i x2i P i yi2 , thus v #" # u" X XX X u X xi yj P 1 t x2 x2 1 x2 ; i
and
XX i
i
j6i
i
i
i
i
i
w xi yj P c x:
and
x2i n n
1b, this im-
A:9
A:10
i
j6i
XX i
i
j6i
P P Case 2. c x P c y. This implies i yi2 P i x2i , thus v #" # u" u X X X XX xi yj P 1 t yi2 yi2 1 yi2 ; i
P
i
w xi yj P c y:
j6i
Combining cases 1 and 2 yields quasi-concave.
A:11
i
A:12 P P i
j6i
w xi yj P minfc x; c yg; hence by Lemma 4, c x is
We conjecture that c x is quasi-concave on Tm for any w that is strictly increasing and concave. Appendix B Proof of Lemma 1. Consider a spoke s and a hub h. Assume that ps > ph . We want to show that the cost savings of interchanging the two cities, d, are strictly positive. By Assumptions 1 and 2, the trac density on a spoke sector where the spoke city has population p is r p p 1 p. The cost eects of interchanging the two cities are local on the spoke sector: d 2w r ps
w r ph :
B:1
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By Assumption 5, w is strictly increasing, so that cost savings are strictly positive if and only if r ps > r ph . Case 1. ps < 0; 5. Because r is strictly increasing for p < 0; 5, and because ps > ph it follows that r ps > r ph . Case 2. ps P 0; 5. Because total population is 1 and n P 3, it follows that ph 1 ps e, where e > 0. Due to symmetry of r about p 0; 5, it follows that r ps > r 1 ps e r ph . Proof of Lemma 2. Consider two potential hub cities, without loss of generality cities 1 and 2. Assume that city 2 is larger than city 1, p2 > p1 , and that the total population of all spoke cities is ps . Denote the costs of the network with the spokes assigned to hub i by Ci . We want to show that C2 < C1 , that is the costs are lower if the spokes are assigned to the larger hub. Applying Eq. (6), we can write C1
C2 2w p1 ps p2 w p2 ps p1 X 2 w p1 ps pj w p1 pj
B:2a fw p2 ps pj
w p2 pj g:
B:2b
j2H nf1;2g
Term (B.2a) is strictly positive because by assumption city 2 is larger, thus p1 ps p2 > p2 ps p1 . Let D x; h w x h w x=h, h > 0, be the dierence quotient giving the slope of the secant line passing through w x and w x h. From concavity of w it follows that Dx x; h w0 x h w0 x=h < 0. Because p2 > p1 , this implies D p1 pj ; ps pj D p2 pj ; ps pj > 0, or rewriting w p1 pj ps pj ps pj
w p1 pj
w p2 pj ps pj ps pj
w p2 pj
> 0:
B:3
Hence each term in Eq. (B.2b) is positive, and so is Eq. (B.2b). From Eq. (B.2a) > 0 and Eq. (B.2b) > 0, we conclude C1 C2 > 0. References Barla, P., Constantatos, C., 2000. Airline network structure under demand uncertainty. Transportation Research Part E 36 (3), 173±180. Bazaraa, M.S., Shetty, C.M., 1979. Nonlinear Programming. Wiley, New York. Berechman, J., Shy, O., 1996. The structure of airline equilibrium networks. In: van den Bergh, J.C.J.M., Nijkamp, P., Rietveld, P. (Eds.), Recent advances in spatial equilibrium modelling. Springer, Berlin, pp. 138±155. Borenstein, S., 1992. The evolution of US airline competition. Journal of Economic Perspectives 6 (2), 45±73. Brueckner, J.K., Spiller, P.T., 1994. Economies of trac density in deregulated airline markets. Journal of Law and Economics 37 (2), 379±415. Brueckner, J.K., Zhang, Y., 1999. Scheduling Decisions in an Airline Network: A Hub-and-Spoke System's Eect on Flight Frequency, Fares and Welfare. Oce of Research Working Paper 99/0110, University of Illinois at UrbanaChampaign.. Caves, D., Christensen, L., Tretheway, M., 1984. Economies of density versus economies of scale: Why trunk and local service airline costs dier. Rand Journal of Economics 15 (4), 471±489.
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