Competition in airline networks

Economics Letters 0165-1765/93/$06.00

42 (1993) 253-259 0 1993 Elsevier

Competition

253 Science

Publishers

B.V. All rights

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in airline networks

The case of constant elasticity demands Anming

Zhang *‘, Xin Wei

Department of Economics, Received Accepted

University of Victoria, P.O. Box 3050, Victoria, B.C. V8W 3P5, Canada

29 March 1993 1 June 1993

Abstract This paper is concerned with the effect of competition in airline hub-and-spoke networks. Using constant demand functions, we demonstrate that competition in a single market may generate positive or negative externalities, depending on price elasticities of demand.

elasticity network

1. Introduction Deregulation has had a profound impact on airlines’ route systems with most airlines having Brueckner and Spiller (1991) have pointed now adopted a ‘hub-and-spoke’ network. ’ Recently, out that the emergence of such networks can have important implications for antitrust policy towards airlines. In particular, they found that competition in a market served by a monopoly hub-and-spoke airline usually raises fares in all other markets in the network. This negative externality of competition may cause a reduction in total social surplus. Thus, in such a setting, a merger may be socially desirable. 2 It is noted that this negative externality result emerged from a model with linear demand functions. Essentially, with demand being linear, introducing competition into a market served by a monopoly hub airline reduces the monopolist’s output in the market. Given economies of density and the cost complementarities inherent to a hub-and-spoke network, this traffic leakage raises the marginal cost of a passenger on the affected spokes. While competitive pressure in the given market counteracts the higher marginal cost, thereby reducing fares, other markets that use the affected spokes do not benefit from added competition. As a result, the higher marginal costs lead to higher fares in these markets. * Corresponding

author

’ Financial support from the Social Science and Humanities Research Council of Canada is gratefully acknowledged. ’ A hub-and-spoke network concentrates most of an airline’s operations at one or a few ‘hub’ cities, serving virtually every other city in the network non-stop from the hub and providing predominantly one-stop or connecting service through the hub between cities on the ‘spokes’. Various researchers, including Kanafani and Ghobrial (1985), Morrison and Winston (1986), Levine (1987), Hendricks et al. (1992), and Oum Zhang, and Zhang (1992), have identified the phenomenon of hub-and-spoke networks and offered explanations for the dramatic emergence of such networks after deregulation. * The external effect of competition within hub-and-spoke networks is tested empirically in Brueckner et al. (1992). Other recent empirical studies on airline competition and mergers include Borenstein (1990) and Werden et al. (1991).

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Using constant elasticity demand functions, we demonstrate in this paper that competition in a single market may generate positive or negative network externalities, depending on price elasticities of demand. In particular, the positive externality arises because, with constant elasticity demands, competition in a market served by a monopoly network may result in increased output for the monopolist in that market. This traffic increase then lowers the monopolist’s marginal costs and leads to reduced fares in other markets in the network.

2. The monopoly

model

The paper uses the basic model developed by Brueckner and Spiller (1991). We consider a hub-and-spoke network that is likely the most simple structure in which the problem can be addressed. In this network a monopoly airline serves three cities: A, B and H, using H as the hub (Fig. 1). 3 The airline faces travel demand between any two of these cities, and demand is symmetric across city pairs. The (inverse) demand function for round-trip travel in each city-pair market is given by P,= D(Qi),with Q, representing the number of passengers in that market (i = AH, BH, AB). Since our principal objective is to examine the effect of competition when demand has constant elasticity, we consider P, = D(Q,)

= aQ;“’

(1)

.

In (1)) E represents the constant price elasticity of demand (a positive number), and cx represents the level of demand. The revenue function in market i is given by Z?(Qi) = Q,D(Q;). The cost specification is as follows. Although there are three city-pair markets, aircraft are flown only on two spoke routs, AH and BH, owing to the nature of hub-and-spoke systems. On a given spoke, say AH, aircraft carry both local (i.e. AH) passengers and connecting (i.e. AB) passengers. With cities A and B assumed to be equidistant from H, a common cost function, c(Q), applies to each of the spokes AH and BH. This function gives the round-trip cost of carrying Q passengers on the spoke. The cost function reflects increasing the returns to traffic density, with c(Q) satisfying c’(Q) > 0 and c”(Q) < 0. Furthermore, marginal cost takes the following linear form: c’(Q)=l-0Q,

(2)

A

H

I

<

B

Fig. 1. A simple

hub-and-spoke

system.

3 Brueckner and Spiller (1991) analyze the network of this paper extend to the four-city case.

Fig. 2. Interhub effect of competition

competition. using a four-city

model.

We note that the results

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Letters 42 (1993) 253-259

2.55

where the intercept of marginal cost is normalized to one, and the extent of increasing returns is measured by 0( >O). The linear form (2) is used in Brueckner and Spiller (1991). To focus on the effect of demand specification on the nature of network externalities, we use the same linear form. We note, however, that although our analysis uses (2), in which c”‘=O, the results of the paper can be shown to hold so long as c”’ I 0. Given these specifications, the monopolist’s profit maximization can be represented by the problem of choosing QAH, Q,, and Q,, to maximize

II” = R(Q&

+ W&m

+ W&m) - 4Q.m + Q/us) - c(Qm + Q.d

(the superscript conditions

m stands

for monopoly).

The optimal

solution

4l-

WNQ,:’

= 1 - e(Q,,

+ Q,,)

,

cdl- W>)Q,:’

= 1 - e(Q,,

+ Q,,)

,

cdl- WNQ,:’

= 2 - e(Q,,

+ Q,,) - e(Q,,

is characterized

by the first-order

(4)

+ Q,m)

9

(5)

and the second-order conditions that the Hessian matrix (??17”/aQi aQ,) is negative definite when evaluated at the solution (i, j = AH, BH, AB). Marginal revenue in each market is which is set to equal the marginal cost of represented by the left-hand side of Eqs. (3)-(5), serving a passenger in that market. The cost complementarities inherent to the hub-and-spoke network are evident in these conditions. Referring to (3), for example, it is clear that the marginal cost of serving a passenger in the AH market falls when QAB increases. can be obtained by solving the system The monopoly solution, denoted (Q’&, Qi,, Qz,), (3)-(5). We consider the solution that has positive quantities and marginal revenues (costs). Since the elasticity parameter, E, must be greater than one if marginal revenues are to be positive, we restrict E to be greater than one.

3. The effects of interhub

competition

We now introduce competition into the monopoly hub-and-spoke network. The type of competition considered here is one in which a competing airline serves cities A and B through a different hub, K providing ‘interhub competition’ in the AB market (Fig. 2). As in Brueckner and Spiller (1991), we assume that the other airline has the same cost function and faces the same city-pair demand function as the original airline, and that there is no demand for travel in city-pair HK. The first assumption ensures that the two firms are symmetric, whereas the second assumption allows a straightforward comparison of equilibria under monopoly and interhub competition. 4 In this setting, the two firms play a Cournot game in the AB market while setting monopoly ’ Another

way to introduce competition in the AB market is for a small airline to provide direct service in AB. As in the case of interhub competition, we can examine the effects of competition by comparing the monopoly solution to the solution under this ‘direct competition’. However, direct competition lacks the symmetry of interhub competition; as a result, the solution is complex and comparison with the monopoly solution must be carried out by computer simulation. Our computer simulation suggests that consideration of this form of competition would lead to results qualitatively similar to those derived under interhub competition.

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Letters 42 (1993) 253-259

prices on respective monopoly routes. ’ Letting firm 1 be the original additional hub airline, firm l’s profit can be written as nd =

W&m)

+ W&m)

+ Rd(Qkm Q’,,> - 4Q,,

airline

and firm 2 be the

+ Qkd - c(Qm, + Q:,> 7

Qi,) = Q i,D( QL, + Q’,,) (the superscript d stands for duopoly). with R”(Qin, conditions for firm 1 are the same as before except that (5) is now changed to

41- (W
+ Q”,,)>)(Q;,

+ Qi,> - ‘4Q,,

The first-order

+ Q’,,>-“’

+ Qhd.

(6)

Given the symmetry of the model, we concentrate on the symmetric Cournot equilibrium. This can be found by solving the system (3), (4), and (6) under the requirement that Qi, = Qk,. Denoting the solution by Q”,,, Qt, and Qh”, = Qyn = Qi, (there is a symmetric solution for firm 2), the second-order conditions hold if the Hessian matrix (a211dlaQ, aQ,) is negative definite at (Qi,, Q&, Qi,>. Below we examine the effects of interhub competition by comparing the traffic levels between the monopoly and Cournot solutions. 6 From the structures of the first-order conditions in both cases, it is clear that Qsu = QAu so we need only examine changes in (say) Q,,. Furthermore, Eq. (3) implicitly determines QAu as a function of QAB. This function, denoted QArr (Q,,), satisfies the following property: Lemma

1. Q,,(

QAB) is monotonically increasing between QT,

and Q”,,.

Proof. From (3), we can explicitly express QAn in terms of Q,n: QAB = --[a(1 - (l/~))Qi;' f’(QAn) = -[-a(lle)(l -- (ll,))Qi;--f”t) + fJ]/0 = -g(Q,,/R 1 + 8Q,,]/8 -f(Q,,). Hence, Since g’(QAH) = (~(l/e)(l - (~/E~))Q~~-(~“) > 0, and g( Qz,) < 0 and g(Qd,,) < 0 by the second< 0 at their respective solutions, it follows order conditions that ~211”/~Q~, < 0 and ~211d/aQ~, that g(Q,,) < 0 between Qz, and Qi,. Consequently, f(Q,n is monotonically increasing between Q’J, and Qi,. The inverse function, f -‘, therefore exists, f-‘(QA,) = Q,,(Q,,), and it is monotonically increasing between Qz, and Qi,. Q.E.D. Given Q,, be written as

= QAn(QAB),

mc(Q,,> Using

(7), then

firm l’s marginal

= 2(1- ~(Q,dQ.d

- Q%(mc(Qh)>’

with p = (E - (1/2))
a passenger

in the AB market

may

(7)

+ Q,,>) .

(5) and (6) can be manipulated

Q”,B(m4Qh>>e

cost of serving

to yield

= (P - l>Qh’

the mean-value

theorem

”

to Q,,(mc(Q,,))‘,

(8) the left-

5 For concreteness we assume a Cournot game in the duopoly market. Brander and Zhang (1990) find some evidence. that rivalry between duopoly airlines is consistent with Cournot behaviour. The Cournot assumption is not critical, however, and a solution using a non-zero ‘conjectural variation’ would yield qualitatively similar results. ’ It is noted that under both solutions, an arbitrage condition needs to be imposed under which the fare in the AB market travellers would have an incentive to cannot exceed the sum of the separate fares for the two spokes. Otherwise, purchase the spoke tickets separately. It can be easily verified that this arbitrage condition holds in both cases.

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A. Zhang and X. Wei I Economics Letters 42 (1993) 253-259


is some point

Q,,

between

+

d?,,mc’(&,>>= (P - l>QXmc
and Q”,, . Using

Qz,

Eq. (9) and Lemma

9

(9)

1 we can show:

Proposition 1. Under interhub competition in market A B, the original airline’s trafjic in AB is higher (lower) than under monopoly as E < ( >)c*, where E* is determined by the equation E = 2l”(E - 1) + (112) and can be computed as (approximately) equal to 2.73. Proof. We show that both the second term, mc( $,,), and the third term on the left-hand side of (9) are positive. From (7) and Lemma 1, it follows that mc’(Q,,) = -2f3(Qi,(Q,,) + 1) < 0 for QAB between Qz, and Qi,. Hence, mc($AB)z-mc(QT\B)>O, with QJ&=max{Q~,, Qi,>. Next, using (7) and (3), the third term can be expressed as

(10) By the second-order conditions, the determinants negative at the monopoly and Cournot solutions,

(I-

Since dition

e(Q;,

+ Q”,,) - d'Q:,,($+)

1 - e
1 - e( Q,, by (11) and (13).

between + Q,,)

Using

Q& - E0$,”

condition

(10) > 2( 1 - 0( Q,,

lBQiH

> 0.

and Qi,, - E8&,,

and,

(d211dlaQj aQj) are

(12) > 0 by the

second-order

con-

(13)

+ QAB) - leQ,,tQ,,l

= I- ~ + l)
- E~Q;,

-

and to

‘O.

- E~Q:B

- 2) < 1 and 1 - 0(Qi, + Qi,) < 0, it follows from (12) that

0 < (36 - 1)/(4~ that ~211dl~Q~,

Letting e)(Q;,(Q,,)

of (a*fl”/aQ, aQ,) respectively, leading

- loQAB, then h’(QAB) = -0(1-t

hence,

> 0

(14)

(14) then yields

+ OAB) - E8$,,

- E&?,,)

> 0 .

(15)

that is, the third term on the left-hand side of (9) is positive. Since the second term also is positive, we conclude from (9) that Q i, >, = , and , =, and ~1. WithP = (e - (l/2))‘/ 2(~-l)~,thelatterinturnisequivalentto~>,=,and, =, and E*, respectively, with E* being determined by E = y(e). First notice that, as 2.5 > 2.48 = ~(2.5) and 3 < 3.02 = y(3), there exists a solution to E = y(e) between 2.5 and 3. Denoting this solution as E*, we next show that E* is also the unique solution. Since y’(e) = 2l”(l - (ln2)(e - 1)/e2) > 0 and y”(e) = -2l”(ln 2)((2 In 2)~ + In 2)/e4 < 0, y( E) is increasing and strictly concave so there cannot be more than two solutions to E = y(e). Suppose now that there is a second solution, l0 # E*. Without loss of generality, assuming l0 > E* (the analysis when l,, < E* is similar), then there exists an Ed, E* < l1 < Ed, such that y’(el) = 1 and y’(e,,) < y’(eI) = 1. Moreover, y’(e) % y’(eO) < 1 for all E 2 l0;

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consequently, lim,,, y’(e) < 1. But this contradicts the fact that (In 2)(6 - 1)/e2) = 1, and the contradiction establishes uniqueness.

lim,,, y’(e) = lim,,, Q.E.D.

2l”(l

-

Proposition 1 shows that with constant elasticity demands, competition in a market served by a monopoly network may increase or decrease the monopolist’s traffic in that market, depending on the demand elasticity, E. In particular, for 1 < E < 2.73, competition will result in increased output for the monopolist. 7 This is similar to the general result that entry into an industry may result in increased output per firm, which is obtained by Seade (1980) in the context of the one product per firm case. In our case, however, each carrier is a multiproduct firm with its products corresponding to travel in particular city-pair markets. With economies of density and the cost complementarities inherent to the hub-and-spoke network, higher (lower) AB traffic leads to lower (higher) marginal costs for the monopolist on the AH and BH spokes, thereby raising (reducing) traffic levels in these markets. Formally, applying Lemma 1, the following result can be derived directly from Proposition 1: Proposition 2. In markets AH and BH traffic levels are higher (lower) and fares are lower (higher) under interhub competition than under monopoly as E < (>) 2.73. Proposition 2 is in contrast to the negative externality result of Brueckner and Spiller (1991). Using the linear demand functions it can be easily shown that interhub competition will reduce traffic (and raise fares) in markets AH and BH. However, with constant elasticity demands, competition may generate positive or negative externalities outside the market where it occurs. In effect, for 1 < E < 2.73, competition generates positive network externalities, leading to an increase in traffic throughout the network. Thus, whether competition in a single market creates positive or negative network externalities can critically depend on the demand specification. Our analysis suggests that a careful examination of demand specification may be warranted when one applies Brueckner and Spiller’s (1991) analysis to the evaluation of mergers in the airline industry and other industries.

References Borenstein, S., 1990, Airline mergers, airport dominance, and market power, American Economic Review 80, 400-404. Brander, J.A. and A. Zhang, 1990, Market conduct in the airline industry: An empirical investigation, Rand Journal of Economics 21, 567-583. Brueckner, J.K. and P.T. Spiller, 1991, Competition and mergers in airline networks, International Journal of Industrial Organization 9, 323-342. Brueckner, J.K., N.J. Dyer and P.T. Spiller, 1992, Fare determination in airline hub-and-spoke networks, Rand Journal of Economics 23, 309-333. Hendricks, K., M. Piccione and G. Tan, 1992, The economics of hubs: A case of monopoly, Discussion paper 9209, Department of Economics, University of British Columbia, Vancouver, BC. Kanafani, A. and A.A. Ghobrial, 1985, Airline hubbingsome implications for airport economics, Transportation Research A (General) 19, 15-27. Levine, M.E., 1987, Airline competition in deregulated markets: Theory, firm strategy, and public policy, Yale Journal on Regulation 4, 393-494. Morrison, S.A. and C. Winston, 1986, The economic effects of airline deregulation (Brooking% Washington, DC).

’ The elasticity range of 1< E < 2.73 appears to be consistent with the majority of empirical estimates of air passenger travel [see, for example, the survey of airline demand studies by Oum, Waters,

of demand elasticities and Yong (1992)].

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and X. Wei i Economics

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Oum, T.H., W.G. Waters II and J. Yang, 1992, Concepts of price elasticities of transport demand and recent empirial estimates, Journal of Transportation Economics and Policy 30, 139-154. Oum, T.H., A. Zhang and Y. Zhang, 1992, Airline network rivalry, mimeo, Department of Economics, University of Victoria, Victoria, BC. Seade, J., 1980, On the effects of entry, Econometrica 48, 479-489. Werden, G.J., A.S. Joskow and R.L. Johnson, 1991, The effects of mergers on price and output: Two case studies from the airline industry, Managerial and Decision Economics 12, 341-352.