transportation research part a · how to mix per-ﬂight and per-passenger based airport charges...

Transportation Research Part A 71 (2015) 77–95

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Transportation Research Part A

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How to mix per-flight and per-passenger based airport charges

http://dx.doi.org/10.1016/j.tra.2014.10.0140965-8564/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (A.I. Czerny), [email protected] (A. Zhang).

1 The International Civil Aviation Organisation (ICAO) proposes that landing charges as well as parking and hangar charges should be based on aircraformula (ICAO, 2012).

Achim I. Czerny a,⇑, Anming Zhang b,c

a Department of Spatial Economics, VU University Amsterdam, The Netherlandsb Sauder School of Business, University of British Columbia, Canadac China Academy of Financial Research, Shanghai Jiao Tong University, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 March 2014Received in revised form 9 September 2014Accepted 16 October 2014

Keywords:AirportPer-passenger chargePer-flight chargeSchedule delaysTime valuations

This paper investigates the questions of why carriers advocate for higher per-passengerairport charges and lower per-flight charges, and whether and when this proposal iswelfare-enhancing. Specifically, the paper compares the optimal mix of per-flight andper-passenger based airport charges from both a monopoly carriers’ and the social view-points conditional on airport cost recovery. It focuses on the trade-off between price andfrequency (i.e., schedule delays) when time valuations are uniform, or differ, between busi-ness and leisure passengers. We identify an easy test for the evaluation of the mix of per-passenger and per-flight based airport charges by policy makers, which is simply to checkwhether the carrier’s preferred per-flight charge is zero. Our analysis suggests that there isno need for immediate regulatory corrections of the current trend towards the strong useof per-passenger based airport charges.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

‘‘Many airport facilities are built and maintained for the benefit of airline passengers. It is in the interest of both the airport andthe airlines to recover these costs through passenger based charges instead of other aeronautical based charges.’’ (International AirTransport Association, IATA, 2010).

Traditionally, aeronautical charges are based on aircraft weight formula, which is especially true for landing, parking andhangar charges.1 But, as airport improvement fees, which are used to charge passengers for airport infrastructure developmentand/or debt repayment, have become a more important revenue source for airports (Zhang, 2012), airports worldwide derivetoday as much aeronautical revenues from per-passenger charges as from aircraft related (i.e., per-flight) charges (ACI,2008). Yet, the trade association for the world’s airlines (International Air Transport Association, IATA) seems to propose to fur-ther move away from per-flight related airport charges towards per-passenger related charges.

A recent empirical study of aeronautical charges at major US airports by Choo (2014) clearly shows that an increase in theper-passenger charges will be associated with a reduction of per-flight charges. But, why are carriers interested in raisingper-passenger airport charges relative to per-flight charges? Is the carriers’ proposal socially optimal? In this paper we inves-tigate these questions. The issues are addressed by comparing a monopoly carrier’s and the social viewpoints on the optimal

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78 A.I. Czerny, A. Zhang / Transportation Research Part A 71 (2015) 77–95

mix of airport-charges conditional on strict airport-cost recovery through revenues derived from airport per-flight and per-passenger charges.2 A crucial element of the model is that passengers experience schedule delays, which measure the absolutedifference between the passengers’ most preferred and their actual travel times. Following Douglas and Miller (1974) and morerecent work by, e.g., Bilotkach (2007), Brueckner (2004, 2010) and Brueckner and Flores-Fillol (2007), it is assumed that sche-dule delays are negatively related to the quantity of aircraft flights, i.e., frequency. This is because an increase in frequencyincreases the likelihood that passengers travel at their preferred times. Schedule delay costs then depend on frequency and timevaluations. In this scenario, passenger demands depend on ‘‘generalized prices,’’ which are composed of ticket price (fare) andschedule delay costs. The model therefore establishes a clear trade-off between low per-passenger charge, low fare, and lowfrequency versus low per-flight charge, high fare and high frequency.

Our analysis shows that an increase in the per-passenger airport charge and the associated reduction in the per-flightcharge can indeed increase carrier profit. Essentially, the reduction in schedule delay costs associated with a marginalincrease in frequency, which is fully internalized by the carrier when the passengers’ time valuations are uniform, exceedsthe marginal frequency cost when the per-flight charge is positive. The picture becomes more complex when passengerswith distinct time valuations exist. There is good evidence that business passengers exert high time valuations relative toleisure passengers (e.g., Morrison, 1987; Morrison and Winston, 1989; USDOT, 1997; Pels et al., 2003). Then, if the carriercharges a uniform price to business and leisure passengers, exact internalization of reductions in schedule delays is notensured. Specifically, if the average time valuations (defined as the arithmetic mean of all passengers’ time valuations)exceeds the marginal passengers’ time valuations (defined as the average time valuation of incremental passengers), the car-rier’s incentive to provide frequency is too low because it is concerned with the marginal time valuations. This translates intoa preference for low per-passenger charge and high per-flight charge relative to a carrier that would be concerned about theaverage time valuations. The distinction between the marginal and average time valuations becomes however less relevantwhen a carrier can price discriminate between business and leisure passengers in the form of third-degree price discrimina-tion by, for example, advanced purchase rebates.3

Turning to the social maximizer, she is concerned not only about producer surplus but also about consumer surplus. Addingthe consumers’ viewpoint into consideration means minimization of full fares conditional on airport cost recovery. Thus, thecarrier’s and social viewpoints can be identical only if the carrier’s preferred mix of airport charges minimizes full fares. Whilepolicy makers may have difficulties to identify the minimization of full fares, we further show that the carrier’s and the socialviewpoints are identical only if the carrier’s preferred per-flight charge is exactly equal to zero when time valuations are uni-form. Furthermore, the zero per-flight charge can still be optimal from the carrier’s and the social viewpoints if time valuationsdiffer between passenger groups, but only if ticket prices are discriminating. These results thus suggest an easy test for thepotential conflicts of interest between the carrier and the social maximizer, which is to simply check whether the carrier’s opti-mal mix of airport charges incorporates the zero per-flight charge. Numerical tests further indicate that the carrier’s preferredper-passenger charges are more likely to be excessive from the social viewpoint if time valuations are high.

Our paper contributes to several strands of the literature. The first is to the literature on airport pricing, which typicallyconcentrates on congestion pricing and the pricing of airport concession services based on passenger related charges.4 Forexample, Flores-Fillol (2010) analyzes airport congestion pricing when schedule delays are present but assumes that aggregatepassenger demand is fixed, normalizes the per-passenger airport charge to zero and therefore concentrates on frequency supplyand per-flight related congestion charges. Silva and Verhoef (2013) consider a congested airport and per-flight and per-passen-ger charges. They find that market power should be corrected by the per-passenger subsidy and that the per-flight chargeshould be used to control congestion. It is well known that subsidies may be required to reach the first-best welfare result(e.g., Pels and Verhoef, 2004). The problem here is that airports are often required to cover all or at least a large share of theircosts by own revenues (e.g., Zhang and Zhang, 2003), which means that the first-best result is often not achievable in practice.To our knowledge, our paper is the first that evaluates the mix of airport per-passenger and airport per-flight charges from thecarrier’s and the social viewpoints. Furthermore, it explicitly considers the empirically important airport cost-recovery con-straint, which further distinguishes the present study from previous ones.

Note that similar issues may also arise in other transportation infrastructures. In the port sector, for example, port chargesmay be levied on the ship and cargo. More specifically, prices charged for servicing a containership and its cargo at a portmay include: (i) (charged to the vessel) prices for pilotage, tuggage, dockage, line-handling, and vessel overtime; and (ii)(charged to container box) prices for wharfage, stevedoring, rental of terminal cranes, and number of containers movedon to and off the vessel. Furthermore, there have been discussions about the optimal pricing structures from the perspectivesof shipping lines, the port, shippers (i.e., cargo owners) and welfare (e.g., Talley, 2009). The insights derived in the presentpaper may therefore be relevant for port policies and, thus, adds to the seaport literature. Since schedule delays can be con-sidered as a quality dimension, this paper further adds to the literature on quality supplies, which was introduced by theseminal papers of Spence (1975) and Sheshinski (1976).

2 In a recent study, Lazarev (2013) identifies and analyzes 76 US origin–destination markets that are served by only one airline. We discuss the issue of airlinemarket structure further in the concluding remarks.

3 Airlines are a frequently used example for markets where price discrimination is prevalent (for example, Borenstein, 1985; Dana, 1999a,b and Cowan,2007). Stavins (2001) and Lazarev (2013) provide empirical evidence for airline third-degree price discrimination. To abstract away from self-selection, assumethat early booking is prohibitive for business passengers.

4 See Zhang and Czerny (2012) for a literature survey on airport congestion pricing and airport concession revenues.

A.I. Czerny, A. Zhang / Transportation Research Part A 71 (2015) 77–95 79

The paper is organized as follows. Section 2 develops the basic model, which is used to derive the price and frequencyeffects of a marginal increase in per-passenger charges in Section 3. Price and frequency effects are crucial for the evaluationof the mix of airport charges from the carrier’s and the social viewpoints. This section also provides an example, which makesuse of specific functional forms and numerical parameter instances to illustrate the distinct viewpoints on the mix of airportcharges. Passenger types with distinct time valuations are considered in Section 4, which also includes an example to illus-trate the effect of uniform versus discriminating fares on the optimal mix of airport charges. Section 5 concludes and dis-cusses avenues for future research.

2. Basic model

The supply side is described first. There is an upstream airport that provides an essential input (e.g., runway and terminalcapacity) to a downstream monopoly carrier. Airport charges are of two types: a per-passenger charge, denoted sq, and a per-flight charge (e.g., arrival and departure of an aircraft) denoted sf ; both of which are charged to the carrier. Notice, since thepayment of airport charges is separable from other social costs, the assumption that the per-passenger charge is levied to thecarrier (not passengers) is without loss of generality.5 Denote the carrier’s passenger quantity by q P 0ð Þ and the quantity ofvehicle flights by f > 0ð Þ. Aircraft are all of the same size with a non-binding maximum capacity.6 This leads to an infrastructure(airport) revenue of sqqþ sf f . Normalizing the airport’s per-passenger and per-flight costs to zero, the airport profit can be writ-ten as

5 ICAincreasi

6 Lett7 We

achieve8 As p9 Acc

differenone’s prdelay a

10 See

P ¼ sqqþ sf f � F; ð1Þ

where F > 0ð Þ denotes the fixed infrastructure cost.7 To ensure that airport cost recovery can be achieved, assume that F is suf-ficiently small. The carrier’s per-passenger cost is normalized to zero as well, whilst the carrier’s per-flight cost is denoted asc > 0ð Þ. 8 Thus, the carrier’s costs include both airport payments and its per-flight operating cost, and can be written assqqþ sf þ c

� �f . The carrier charges fare p to passengers, which yields carrier profit

p ¼ p� sq� �

q� sf þ c� �

f : ð2Þ

Turning to the demand side, let B denote the (gross) travel benefit to passengers with B ¼ BðqÞ. The benefit function has itsusual property of B0ðqÞ > 0 and B00ðqÞ < 0, indicating that the benefit is strictly concave in the passenger quantity. The gen-eralized passenger cost (‘‘generalized price’’) is composed of the fare, p, and the per-passenger ‘‘schedule delay cost,’’ withthe latter being the cost that a passenger incurs for the time between the passenger’s desired departure and the actual depar-ture time. (We assume, as is common in the literature, that consumers are able to place a monetary value on non-price ser-vice attributes.) The schedule delay costs of all passengers are obtained by the multiplication of schedule delays, denoted C,and time valuation, denoted v. Letting g denote the generalized price, it holds that g ¼ pþ vC. Schedule delays are decreas-ing in flight frequency, that is C ¼ Cðf Þ with C0ðf Þ < 0. 9 To characterize the schedule delay function, let nðf Þ denote the elas-ticity of the slope of schedule delay Cwith respect to frequency f, i.e., nðf Þ ¼ �f � C00=C0. Specifically, nðf Þmeasures the curvature(convexity) of the schedule delay function, which will play a major role in the analysis presented below. If nðf Þ > 0, this meansthat schedule delays are strictly convex.

Consumer surplus, denoted CS, and social welfare, denoted W, can now be written as

CS ¼ B� gq and W ¼ B� qvC� cf � F; ð3Þ

respectively. Furthermore, passenger demand is determined by the equilibrium condition @CS=@q ¼ 0, which impliesB0ðqÞ ¼ g, where the left-hand side is the marginal benefit of flying, while the right-hand side is the generalized price. Totallydifferentiating this equilibrium condition with respect to fare and frequency yields, respectively,

@q@p¼ 1

B00ðqÞ< 0; and

@q@f¼ vC0ðf Þ

B00ðqÞ> 0: ð4Þ

Clearly, the demand is decreasing in fare and increasing in frequency because an increase in fare raises the generalized price,while an increase in frequency reduces the generalized price. Let D denote the demand with q ¼ DðgÞ and xðDÞ denote theelasticity of the slope of the D function with respect to fare p. That is, xðDÞ ¼ �p � D00=D0, where xðDÞmeasures the convexityof the demand function.10 These definitions lead to:

O (2012) proposes that airport passenger-service charges should be levied through aircraft operators rather than passengers for the purpose ofng the efficiency of collecting airport charges (noting that there are much less carriers than passengers).ing q denote the maximum passenger quantity per flight, load factors can be derived as q=fð Þ=q, where q=f is the quantity of passengers per flight.employ the interpretation of fixed costs for F, while the parameter F could have the more general interpretation of a fixed revenue the airport has to

, which could potentially exceed fixed airport costs and ensure a positive but fixed airport profit.ointed in, e.g., Brander and Zhang (1990), once a flight is committed to service, costs per passenger are usually rather small.

ording to Douglas and Miller (1974), the schedule delay may be decomposed into ‘‘frequency delay’’ and ‘‘stochastic delay.’’ The former refers to thece between one’s desired departure time and the closest scheduled departure by the airline, whereas the latter is the delay caused by excess demand foreferred flight(s). Both delays are dependent on flight frequency. Appendix C shows that C can also be considered as a combined measure for schedule

nd congestion delay. The basic model is therefore quite general and the results can be applied to congested airports as well.Aguirre et al. (2010) for a discussion of convexity measures for demands.


Assumption 1. For D; f > 0, (i) xðDÞ < min 1; p= p� sq� ��

and (ii) nðf Þ > �f � vC0= p� sq� �

at the carrier optimum.

Recall that the demand q ¼ DðgÞ is implicitly determined by B0ðqÞ ¼ g; consequently, D0ðgÞ ¼ 1=B00ðqÞ < 0. Assumption 1limits the convexity of the demand function from above and the convexity of the schedule delay function from below, whichare useful in ensuring the existence of solutions for the carrier’s behavior in terms of fare and frequency.11

The decision-making structure consists of two stages. In the first stage, the per-passenger and per-flight charges aredetermined by the airport in order to ensure airport cost recovery (i.e., P P 0). Thus, carriers cannot directly control airportcharges, while in reality they seem to influence airport charges indirectly. To analyze the distinct views on per-passengercharges from the carrier’s and the social viewpoints, we consider the airport’s unifying objective

11 It isimpose

12 Witprice.

V ¼ pþ / � Pþ CSð Þ; ð5Þ

where / is called the viewpoint parameter with / 2 0;1½ �. Analysis of the scenario with zero viewpoint parameter (i.e., / ¼ 0)helps to shed some light on the carrier’s interests, while the social interests are considered when / ¼ 1. In the second stage,the carrier chooses fare and frequency simultaneously.12

3. Optimal mix of airport charges

In order to gradually increase complexity of the analysis, we concentrate on the carrier’s viewpoint by assuming / ¼ 0 ina first step. In a second step, the effect of an increase in the viewpoint parameter on the optimal mix of per-flight and per-passenger charges is considered in order to identify potential conflicts of interest between the carrier and the social maxi-mizer. The results may then be used to evaluate whether the carriers’ desire for per-passenger based airport charges shouldbe of concern to policy makers.

3.1. Carrier’s incentives

Consider the second-stage first. The carrier’s interests are determined by the first-order conditions for the fare, @p=@p ¼ 0,and the frequency, @p=@f ¼ 0. From profit expression (2), the first-order condition for fare can be written as (superscript Mfor the monopoly solution),

Dþ pM � sq� �

D0 ¼ 0; ð6Þ

where Assumption 1 part (i) ensures that @p2=@p2 < 0 is satisfied at optimum. The first-order condition for frequency can bewritten as

pM � sq� �

D0vC0 � sf þ c� �

¼ 0; ð7Þ

where Assumption 1 parts (i) and (ii) jointly ensure that @p2=@f 2< 0 is satisfied at optimum. Using (6), Eq. (7) can be rewrit-

ten as

�DvC0 � sf þ c� �

¼ 0; ð8Þ

which shows that at the carrier optimum, the marginal reduction in schedule delay costs is just offset by the carrier’s mar-ginal frequency costs (c þ sf ). This is intuitive, since the carrier can fully internalize any reduction in schedule delay costs byan increase in fare.

Denote ðpM ; f MÞ to be the solution of the second stage, which is a function of first-stage airport charges ðsq; sf Þ. Totallydifferentiating (6) and (7) with respect to per-passenger charge sq and applying Cramer’s rule yields the comparative-staticrelationships (the proofs of the paper’s lemmas and propositions are relegated to Appendix A):

Lemma 1. An increase in the per-passenger charge reduces the flight frequency, while increasing the fare. On the other hand, anincrease in the per-flight charge reduces both the fare and frequency.

Since an increase in the per-passenger charge increases the carrier’s marginal passenger costs, it seems natural that thisalso increases the fare. However, an increase in the per-passenger charge reduces frequency, which exerts a downward pres-sure on the fare. The overall effect of an increase in the per-passenger charge is therefore ambiguous in sign. Yet, if the sche-dule delay function is sufficiently convex (as explicitly specified by Assumption 1), an increase in the per-passenger chargehas a clear-cut and positive effect on the fare. That an increase in the per-flight charge increases per-flight costs and thusreduces frequency, while that a reduced frequency increases schedule delays and thus reduces demand and fare is intuitive.

We now examine the carrier’s preferred per-passenger charges. The airport cost-recovery constraint clearly is bindingfrom the carrier’s perspective, since any reduction of payments to the airport would increase the carrier profit. This impliesthat airport profit should be zero from the carrier’s perspective and, from (1), that sqqþ sf f � F ¼ 0. Fig. 1 displays a stylized

standard to assume that the convexity of demand is less than one, while the potential use of per-passenger subsidies in the present model frameworks stronger limits on the convexity of the demand.h a monopoly carrier, the simultaneous choice of price and frequency is without loss of generality relative to the sequential choice of frequency and

Fig. 1. Stylized zero airport-profit contour.


zero airport-profit contour in the sq � sf space and thus all the combinations of per-passenger and per-flight charges thatimply zero airport-profit. From the carrier’s perspective only the downward sloping part of the contour is relevant, andwe will show below that the same is true from the social perspective. We can therefore concentrate on those combinationsof per-passenger and per-flight charges where an increase in the per-passenger charge is associated with a reduction in theper-flight charge, i.e., dsf =dsq < 0. 13

Substituting sqqþ sf f ¼ F into (2) (and replacing q with D), the carrier profit can be rewritten as p ¼ pMD� cf M � F. Thisis done by considering the per-flight charge as a function of the per-passenger charge (i.e., sf ¼ sf ðsqÞ) and looking at thefirst-order condition @p=@sq ¼ 0. This leads to

13 The

DpDþ pMD0 � Dp þ Df vC0� �|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

Dg¼

�Df c ¼ 0 ð9Þ

with

Dp ¼@p@sqþ @p@sf

dsf

dsq; and Df ¼

@f@sqþ @f@sf

dsf

dsqð10Þ

and @2p=@s2q < 0 by the second-order condition.

An increase in the per-passenger charge leads to a ‘‘fare effect,’’ denoted as Dp, and a ‘‘frequency effect,’’ denoted as Df ,which measure how an increase in sq and the corresponding change in sf change fare and frequency, respectively. The fareeffect is clear-cut and positive in sign. This is because an increase in sq increases per-passenger costs and thus directlyincreases the fare and because the fare is further increased by the reduction in sf , which stimulates frequency supply. Onthe other hand, the frequency effect can be positive or negative in sign because an increase in sq directly reduces frequency,while it indirectly increases frequency as it leads to a reduction in the per-flight charge.

Using these definitions of the fare and frequency effects, there is a clear interpretation for the left-hand side of (9). Thefirst two terms on the left-hand side show the change in revenues associated with a marginal increase in the per-passengercharge, which is composed of (i) the change in revenues associated with the change in the price, and (ii) the change in rev-enues associated with the change in the generalized price, which is denoted as Dg with Dg ¼ Dp þ Df vC0, multiplied with thecorresponding change in the passenger quantity. The third term shows the change in frequency costs associated with a mar-ginal increase in the per-passenger charge. Then, the per-passenger charge is optimal when the revenue and cost effects areequalized in absolute values.

To see when the carrier can benefit from a strictly positive per-passenger charge, expand the left-hand side of (9), andnote that the new first two terms, Dp � Dþ pMD0

� �, are zero when the per-passenger charge is zero (and the per-flight charge

is strictly positive), which holds true by the first-order condition in (6). Then, whether the carrier would be better off with anincrease in sq depends on the sign of the frequency effect alone. Specifically, we have the following results:

Proposition 1. The optimal per-passenger charge from the carrier’s viewpoint is strictly positive if the frequency effect, Df , isstrictly positive.

Thus, the carrier benefits from a strictly positive per-passenger charge if the associated reduction in the per-flight chargeincreases frequency. This is intuitive, since schedule delay costs exceed the additional frequency cost when the per-flightcharge is strictly positive (see first-order condition (6)).

negative relationship between per-passenger and per-flight airport charges has been found empirically by Choo (2014).


Proposition 1 is consistent with current carrier policies and thus provides an explanation for why carriers may be inter-ested in a strictly positive per-passenger charge in reality. The case where Df > 0, therefore seems to be of special practicalrelevance, since Df < 0 would be more consistent with a situation where carriers lobby for an increase in the per-flightcharge. The following assumption is thus imposed in our discussion of the carrier’s and social incentives, so as to focuson the more relevant case where the frequency effect is positive:

Assumption 2. An increase in the per-passenger charge increases optimal frequency in the sense that Df > 0.

3.2. Carrier’s versus social incentives

We first demonstrate that the airport cost-recovery constraint is binding from the social viewpoint. To do this, it is usefulto analyze the carrier behavior from the social viewpoint because the social optimizer can indirectly influence fare and fre-quency by the choice of airport charges. The welfare-optimal fare is determined by the first-order conditions@W=@p ¼ p � ð@q=@pÞ ¼ 0. Since the demand is, by (4), strictly decreasing in price for q > 0, the welfare-optimal fare and asso-ciated carrier revenue are zero (which is natural, since the marginal passenger cost is normalized to zero). Furthermore, toimplement the welfare-optimal fare of zero, passenger traffic must be subsidized by Lemma 1. The welfare-optimal fre-quency is determined by the first-order condition @W=@f ¼ 0. Substituting p by zero, this condition @W=@f ¼ 0 can be writ-ten as �DvC0 � c ¼ 0. 14 Together with condition (8), this reveals that the carrier’s frequency choice is optimal from the socialviewpoint if and only if the per-flight charge is zero. Since the carrier must be subsidized to achieve the first-best price, whilethe first-best per-flight charge is zero, the airport cost-recovery constraint is strictly binding from the carrier’s and the socialviewpoints.

It is useful to understand that the frequency supply can be considered as a quality dimension (high frequency reducespassenger schedule delays) and thus the finding that monopoly frequency supply is socially optimal is anticipated by theanalyses of Spence (1975) and Sheshinski (1976), who showed that the monopoly quality supply is optimal from the socialviewpoint if the quality valuation of the marginal customer is representative for all customers. Since time valuations areassumed to be the same for all passengers in our basic set-up, monopoly frequency supply is socially optimal (for givenquantity) when the per-flight charge is zero.

Recall the unifying objective V, which reduces to V ¼ pþ /CS, since the airport cost-recovery constraint is binding fromthe carrier’s and the social viewpoints. Then, to identify the differences between the carrier’s and the social viewpoints, con-sider the sign of the cross-derivative of V with respect to the viewpoint parameter and the per-passenger charge, which canbe written as

14 Theensures

@2V@/@sq

¼ @CS@sq¼ �DgD: ð11Þ

Notice that Dg can be positive or negative in sign, while consumer surplus would be maximized when the generalized priceis minimized in the sense that the marginal effect of a change in the per-passenger charge on the generalized price condi-tional on airport cost recovery is zero.

This leads to:

Proposition 2. The monopoly carrier’s preference for a zero per-flight charge clearly shows that the carrier’s and the socialviewpoints on the optimal mix of per-passenger and per-flight charges are identical. On the other hand, the carrier’s preferred per-passenger charge is excessive (too low, respectively) from the social viewpoint when the carrier’s preferred per-flight charge ispositive (negative, respectively).

For an intuitive explanation for why the carrier’s and the social viewpoints can be conflicting, note that the social opti-mizer attaches a higher weight (in absolute values) to changes in the generalized price than the carrier because the socialoptimizer also cares about consumer surplus. Specifically, if an increase in the per-passenger charge and the associatedreduction in schedule delays reduces the generalized price, the carrier accounts for the revenue effect of the increase inthe passenger quantity but not the increase in consumer surplus. This is why the social optimizer’s optimal per-passengercharge is high relative to the carrier’s optimal passenger charge in this situation. On the other hand, if an increase in the per-passenger charge and the associated reduction in schedule delays increases the generalized price, the carrier accounts, again,for the revenue effect of the reduction in the passenger quantity but not the reduction in consumer surplus. This is why thesocial maximizer’s optimal per-passenger charge is low relative to the carrier’s optimal passenger charge in this situation.But, the carrier’s and the social viewpoints are not always conflicting, and the carrier’s preference for the zero per-flightcharge indicates whether the carrier’s and the social viewpoints may be identical or conflicting by Proposition 2. Specifically,if a zero per-flight charge is optimal from the carrier’s viewpoint, then the carrier’s and the social viewpoints must be iden-

Hessian of welfare with respect to fare and frequency is negative definite at optimum with strictly positive determinant vC0� �2

=B00 � qvC00� �

=B00 , whichthat the second-order conditions for a welfare maximum are satisfied.

Fig. 2. Iso-carrier profit (dashed line; carrier profit is 10.64), iso-airport profit (thin solid line; airport profit is zero), and iso-welfare (thick solid line; welfareis 17.49) curves: The carrier’s preferred per-passenger charge is too low from the social viewpoint.


tical because the marginal effect of a change in the per-passenger charge on the generalized price is zero and thus the mar-ginal effect on the passenger quantity and consumer surplus is also zero; there is no source for a conflict of interest in thissituation, which means that the incentives are the same.

Example 1. This example is used to numerically analyze the effect of time valuations on the optimal mix of airport chargesfrom the carrier’s and the social viewpoints. To do this, assume that schedule delays are given by C ¼ 1� f þ f 2

=2. Scheduledelays are a strictly convex function of frequency for f < 1, since n ¼ f � 1� fð Þ. 15 Furthermore, assume that the benefitfunction is BðqÞ ¼ 10q� q2, the time valuations are given by v 2 1=8;1=4;1=2f g, the marginal frequency cost by c ¼ 1=4, and thefixed airport cost by F ¼ 3=2. 16

Figs. 2–4 display iso-carrier profit (dashed line), iso-airport profit (thin solid line) and iso-welfare (thick solid line) curvesin the sq-sf space. The iso-airport curve displays the combinations of per-passenger and per-flight charges that yield exactairport cost recovery (P ¼ 0). Since a reduction in airport charges increases carrier profit and welfare, the optimal mix ofairport charges can be derived by identification of the iso-carrier and iso-welfare curves, which are tangent to the iso-airportprofit curve. All iso-curves are downward sloping in the relevant regions, which illustrates the trade-off between increases inper-passenger or per-flight charges. In these instances, it holds that the carrier’s optimal per-passenger charge is too lowfrom the social viewpoint when v ¼ 1=8, they are exactly the same when v ¼ 1=4 (i.e., optimal mix of airport charges min-imizes the generalized price conditional on airport cost recovery in this parameter instance) and the carrier’s optimal per-passenger charge is excessive from the social viewpoint when v ¼ 1=2. In this sense, the carrier’s optimal per-passengercharge is more likely to be excessive from the social viewpoint if time valuations are high.

For an intuition, it is useful to understand that the socially optimal mix of airport charges maximizes total surplus con-ditional on airport-cost recovery and thus can have the interpretation of Ramsey charges. It is well known that, with Ramseycharges, the markups on marginal costs are determined by demand elasticities; specifically, markups are high for marketswith relatively low demand elasticities. In the present paper, the relevant elasticities are the elasticity of frequency supplywith respect to the per-flight charge (i.e., @f=@sf � sf =f ) and the elasticity of passenger supply with respect to the per-passenger charge (i.e., @q=@sq � sq=q). It can be shown that frequency supply is increasing in time valuations.17 The numericalsimulations in this example further indicate that an increase in time valuations reduces elasticity @f=@sf � sf =f relative to elas-ticity @q=@sq � sq=q in absolute values, which provides an intuition for why the optimal per-flight charges are negative from thecarrier’s and the social viewpoints when time valuations are relatively low and positive when they are relatively high. If timevaluations are sufficiently high, the socially optimal per-passenger charge can even become negative.18

15 Since schedule delays are highly non-linear in frequency, the analytical solutions are difficult to interpret. The following therefore relies on specificparameter instances (the same is true for Example 2 below).

16 The qualitative results that are discussed in Example 1 do not hinge upon the specific choice of parameter instances.17 The proof is analogue to the proof of Lemma 1 in the appendix.18 Appendix B provides an example where the optimal per-passenger charge is negative from the social viewpoint.

Fig. 3. Iso-carrier profit (dashed line; carrier profit is 10.45), iso-airport profit (thin solid line; airport profit is zero), and iso-welfare (thick solid line; welfareis 17.23) curves: The carrier’s and the social viewpoints are in line.

Fig. 4. Iso-carrier profit (dashed line; carrier profit is 10.11), iso-airport profit (thin solid line; airport profit is zero), and iso-welfare (thick solid line; welfareis 16.78) curves: The carrier’s preferred per-passenger charge is excessive from the social viewpoint.


4. Passenger types

This part considers passenger types with distinct time valuations and shows that this changes the carrier’s incentives toprovide flight frequency and, hence, the optimal mix of airport charges. Specifically, there are two types of passengers calledbusiness and leisure passengers. Letting qB and qL denote the business and leisure passenger quantities (simply called thebusiness and leisure quantities), respectively, the travel benefits now become B ¼ BðqL; qBÞ. The Hessian of the benefits withrespect to the business and leisure quantities is assumed to be negative definite, which means that the benefits are a strictlyconcave function of the passenger quantities. Furthermore, to ensure that group-specific benefits are separable,@2BB=@qB@qL ¼ 0 is assumed to hold.19 Since business passengers typically have a greater value of time, assume that the

19 Separability is important to ensure that changes in the business passenger quantity can be independent of changes in the leisure passenger quantity. Forexample, this rules out that time valuations depend on whether passengers buy a business or a leisure ticket.


business generalized price is gB ¼ pþ vBC and leisure generalized price is gL ¼ pþ vLC, where vB is the business passengers’time valuation, vL is the leisure passengers’ time valuation with vB P vLðP 0Þ, and p is the uniform fare charged to both thebusiness and leisure passengers.20 Consumer surplus can now be written as CS ¼ B� gBqB � gLqL.

The demand equilibrium is determined by the conditions @CS=qB ¼ 0 and @CS=@qL ¼ 0, which implies @B=@qB ¼ gB and@B=@qL ¼ gL. Totally differentiating these equilibrium conditions with respect to the price and, respectively, frequency, yieldsthe intuitive result:

Lemma 2. Aggregate demand is decreasing in the fare and increasing in frequency.

4.1. Uniform fare

4.1.1. Carrier’s incentivesDenote the business and leisure demands as DB ¼ DBðgBÞ ¼ qBðgBÞ and DL ¼ DLðgLÞ ¼ qLðgLÞ, respectively. This yields

aggregate demand D ¼ DB þ DL. To understand the effect of changes in airport charges on the carrier’s behavior in termsof price and frequency, it is useful to understand that an increase in the fare has the same effect on the generalized pricesfor business and leisure passengers, while the effect of a frequency change differs between passenger types when their timevaluations are distinct. To capture the effect of time valuations on demands, it is helpful to apply a new measure for the con-vexity of aggregate demand D (besides xðDÞ ¼ �p � D00=D0) denoted as xv ðDÞ with xvðDÞ ¼ vBD00B þ vLD00L

� �= vBD0B þ vLD0L� �

,which accounts for the differences in time valuations. These convexity measures are interdependent because:xv ðDÞ �xðDÞ > 0) xðDBÞ < xðDLÞ for vB > vL. Specifically, this means that xvðDÞ �xðDÞ > 0 if business demand is moreconcave relative to leisure demand. Furthermore, denote bv ¼ vBD0B þ vLD0L

� �=D0, where the right-hand side defines average

time valuation of incremental passengers called the ‘‘marginal time valuations’’ (see, Czerny and Zhang (2014a,b), who con-centrate on congestion effects). This leads to:

Assumption 3. For DB;DL; f > 0, (i) xðDBÞ;xðDLÞ < min 1; p= p� sq� ��

and (ii) nðf Þ > �f � bvC0= p� sq� �

in optimum.Compared to Assumption 1, Assumption 3 applies the first part to both business and leisure demands. Part (i) implies that

the same condition is satisfied for aggregate demands and the new convexity measure (i.e.,xðDÞ;xv ðDÞ < min 1; p= p� sq

� �� ). Furthermore, Assumption 3 part (ii) uses marginal time valuations.

The first-order conditions @p=@p ¼ 0 and @p=@f ¼ 0 can be written as

20 Moconcenthigher t

21 Subconditiopasseng

Dþ pT � sq� �

D0 ¼ 0 ð12Þ

and

pT � sq� �

D0bvC0 � c þ sf� �

¼ 0; ð13Þ

where pT is the monopoly fare (superscript T indicates the scenario where time valuations depend on the passenger’s type,while fares are uniform). Using (12), condition (13) can be rewritten as �DbvC0 � c þ sf

� �¼ 0. This shows that the monopoly

carrier evaluates changes in schedule delays with the marginal time valuations. Since schedule delays can be considered as aquality dimension, the fact that the monopoly carrier is concerned about the marginal time valuations is consistent with thefindings by Spence (1975) and Sheshinski (1976), who showed that a monopoly supplier is concerned about the marginalcustomers’ quality valuations.

Letting v denote the average time valuations with v ¼ vBDB þ vLDLð Þ=DT , the remainder assumes that the elasticitycondition

D0L=DL < D0B=DB ðElasticity conditionÞ

is satisfied, which implies that marginal time valuations are less than average time valuations, i.e., bv < v . 21

To derive the optimal mix of airport charges from the carrier’s viewpoint, one needs to understand the comparative-staticrelationships between airport charges and the uniform fare. These can be derived as:

Lemma 3. If passengers with distinct time valuations exist (i.e., vL < vB) and fares are uniform, it holds that: (i) An increase in theper-passenger charge increases the equilibrium fare, while frequency is also increased if the business passengers’ demand issufficiently concave in the sense that xvðDÞ �xðDÞ > p= p� sq

� �(evaluated at the optimum). (ii) An increase in the per-flight

charge reduces both the fare and frequency.

rrison (1987), Morrison and Winston (1989), USDOT (1997), Pels et al. (2003) found that time valuations related to congestion are distinct, while werate on schedule delays. Based on this empirical evidence, it seems however sensible to assume that business passengers’ time valuation may also behan leisure passengers’ time valuation when schedule delays are considered.stituting 1� D0L=D0

� �for D0B=D0 and 1� DL=Dð Þ for DB=D, one can show that for DB;DL > 0; bv < v when D0L=D0 > DL=D, which is equivalent to the elasticity

n. In words, the elasticity condition implies that incremental passengers exert a high proportion of leisure passengers relative to inframarginalers.


The main difference relative to the case with uniform time valuation is that an increase in the per-passenger charge maylead to an increase in flight frequency when passengers with distinct time valuations exist. The intuition is related to the factthat an increase in the per-passenger charge may be associated with a reduction in the aggregate passenger quantity and anincrease in marginal time valuation, which increases the carrier’s incentive to provide frequency. Notice that the reduction inaggregate quantity may be particularly large when leisure demands are convex, while the reduction in business quantitymay be relatively low when business demands are concave evaluated at the optimum, which implies xvðDÞ �xðDÞ � 0and can altogether lead to a sharp increase in marginal time valuations and, hence, in the incentives to provide frequency.

The first-order condition @p=@sq ¼ 0 leads to

22 Sinscenarito optim

23 The

DpDþ pT D0 � Dp þ Df bvC0� �

� Df c ¼ 0 ð14Þ

and @2p=@s2q < 0 by the second-order condition. As before (with uniform time valuations), an increase in the per-passenger

charge leads to a fare effect, denoted as Dp, and a frequency effect, denoted as Df . For similar reasons as the ones described forthe case of uniform time valuations in Section 3, assume that the fare and the frequency effects are both strictly positive insign.22

Note that the first-order condition (14) weights the frequency effect at marginal time valuations. Thus, the carrierattaches a relatively low weight to the frequency effect, since marginal time valuations are low relative to average time val-uations. Still, the carrier’s optimal per-passenger charge is strictly positive when the frequency effect is positive; thus, Prop-osition 1 carriers over to the case of uniform fares in an environment where time valuations may depend on whetherpassengers are of the business or leisure type.23

4.1.2. Carrier’s versus social incentivesTo show that the airport cost-recovery constraint is binding from the social perspective, we write welfare as

W ¼ B� DvC� cf � F. Assume that the welfare optimal price and the welfare optimal frequency are determined by thefirst-order conditions @W=@p ¼ 0 and @W=@f ¼ 0, respectively. These imply D0 � p ¼ 0 and �DvC0 � c ¼ 0, which shows thatstrictly negative per-passenger and per-flight charges are required to implement the first-best solution. Recall, with uniformtime valuations, a zero per-flight charge was required to implement the first-best result. In this sense, the airport cost-recov-ery constraint is ‘‘more’’ binding when passengers with distinct time valuations exist relative to a scenario with uniform timevaluations because the carrier is concerned with marginal time valuations.

Anticipating that the airport cost-recovery constraint is binding from the carrier’s and the social viewpoints, we againconcentrate on the objective V ¼ pþ /CS, where the second term on the right-hand side involves consumer surplusCS ¼ B� gBDB � gLDL. The sign of the cross-derivative @2V=@/@sq is determined by the sign of

@CS@sq¼ � Dp þ Df vC0

� �� D: ð15Þ

Incorporating the consumers’ perspectives thus means that generalized prices evaluated at average time valuations shouldbe minimized. The optimal per-passenger charges are determined by the first-order condition @V=@sq ¼ 0, and (15) can beused to derive:

Proposition 3. If passengers with distinct time valuations exist and fares are uniform, the optimal mix of airport charges from thecarrier’s and the social viewpoints can be identical only if they incorporate a positive per-flight charge and a positive per-passengercharge.

The intuition is based on the fact that the carrier’s incentives to reduce the per-flight charge are lowered by the existenceof passenger types with distinct time valuations. This is because the associated profit gains from increased frequency areevaluated by marginal time valuations, which are low relative to average time valuations. Therefore, the carriers’ and thesocial incentives can be identical only if the optimal mix of airport charges incorporates a positive per-flight charge. Yet,the fact that the carrier’s incentives to increase frequency are too low from the social viewpoint means that the social max-imizer may want to charge a relatively high per-passenger charge, which can be used to reduce the per-flight charge andstimulate frequency supply. Altogether, this implies that the carrier’s preferred per-passenger charge is, in a sense, less likelyto be excessive from the social viewpoint, but may rather be too low.

From this discussion it is also clear that, if the optimal per-flight charge and the optimal per-passenger charge are bothpositive from the carrier’s viewpoint, this is not a sufficient condition for the carrier’s and the social viewpoints to be iden-tical because the social optimizer may well prefer a negative per-flight charge under these conditions. Example 2 below isused to illustrate this.

ce an increase the per-passenger charge may increase frequency supply by Lemma 3, this seems a more restrictive assumption here relative to theo with uniform time valuations. To demonstrate how the existence of passenger types with distinct time valuations can change the picture with respect

al airport-charges structures, it is however sufficient to concentrate on the cases where Dp > 0 and Df > 0.proof is analogous to the proof of Proposition 1 and omitted here.


4.2. Discriminating fares

4.2.1. Carrier’s incentivesTo consider discriminating fares in the sense of third-degree price discrimination when all markets are covered, assume

that the carrier can charge fares pB to business passengers and pL to leisure passengers (e.g., Czerny and Zhang 2014a). Thisleads to generalized prices gB ¼ pB þ vBC and gL ¼ pL þ vLC. The carrier profit may be written as

24 Cze25 Spe

in optim

p ¼ pB � sq� �

DB þ pL � sq� �

DL � c þ sf

� �f : ð16Þ

First-order conditions for fares are @p=@pB ¼ 0 and @p=@pL ¼ 0, which can be written as

DB þ pDB � sq

� �D0B ¼ 0 and DL þ pD

L � sq� �

D0L ¼ 0; ð17Þ

where the elasticity condition implies that business fares are high relative to the leisure fares (superscript D indicates dis-criminating fares).

Frequency is determined by the first-order condition @p=@f ¼ 0. Using the first-order conditions (17), the first-order con-dition for frequency can be written as �DvC0 � c þ sf

� �¼ 0, which shows that the carrier is concerned about average time

valuations when fares are discriminating. This is because with price discrimination the carrier can internalize any reductionsin schedule delay costs by changes in passenger type-specific fares. Interestingly, the overall effect of fare discrimination onthe surplus of business passengers can therefore be positive or negative: The business fares are clearly increased by fare dis-crimination relative to uniform fares when frequency is given, while the potential increase in frequency supply reduces thebusiness generalized price.

Totally differentiating the first-order conditions for business and leisure fares as well as frequency with respect to theper-flight charge yields:

Lemma 4. An increase in the per-flight charge reduces the discriminating business and leisure fares as well as frequency.

Since an increase in the per-flight charge increases the cost of frequency supply and frequency is a quality dimension forpassengers, this is intuitive. On the other hand, the effect of an increase in the per-passenger charge on business and leisurefares as well as frequency is ambiguous in sign for all relationships. This is because an increase in the per-passenger chargechanges passenger quantities and average time valuations, which altogether can lead to a strong reduction in frequency sup-ply (relative to the case with uniform time valuations, where an increase in the per-passenger charge always increases thefare) and a reduction in fares.

Turning to the airport charges structure, assume that the optimal per-passenger charges from the carrier’s perspective aredetermined by the first-order condition @p=@sq ¼ 0, which leads to
X
x¼B;L

Dxp � Dx þ Dgx � pD

x D0x� �

� Df � c ¼ 0 ð18Þ

for x ¼ B; L with

DBp ¼

@pDB

@sqþ @pD

B

@sf

@sq

@sf; DL

p ¼@pD

L

@sqþ @pD

L

@sf

@sq

@sf; ð19Þ

DgB ¼ DBp þ Df vBC

0; DgL ¼ DLp þ Df vLC

0: ð20Þ

In this scenario, an increase in the per-passenger charges leads to two fare effects, denoted as DBp and DL

p, for business and lei-sure passengers, respectively. Similarly to the scenarios with uniform time valuations or distinct time valuations but uniformfares, the optimal per-passenger charge from the carrier’s viewpoint is strictly positive when the frequency effect is positive.

4.2.2. Carrier’s versus social incentivesThe possibility to discriminate between business and leisure passengers leaves the pricing behavior of a social maximizer

unchanged because the welfare-optimal fares are uniform.24 Furthermore, since the carrier is concerned about average timevaluations when fares are discriminating between business and leisure passengers, there is no need for the social maximizer tosubsidize or penalize frequency supply. The airport cost-recovery constraint is therefore binding from the social perspectivewhether fares are uniform or discriminating.

To analyze the distinct views of the carrier and the social maximizer on the optimal mix of per-passenger and per-flightcharges, denote the average change in the generalized prices of business and leisure passengers associated with a marginalincrease in the per-passenger charge as Dg, (i.e., Dg ¼ DBDgB þ DLDgLð Þ=D). With discriminating fares, the cross-derivative@2V=@/@sq can then be written as @CS=@sq ¼ �Dg � D. Thus, the carrier’s and the social viewpoints are identical if the optimalmix of per-passenger and per-flight charges minimizes the average generalized price. Furthermore, the optimal mix of air-port charges may minimize the average generalized price and incorporate a zero per-flight charge.25 In this sense, this partly

rny and Zhang (2011, 2014a,b) showed a similar result for a congested airport.cifically, the optimal mix of airport charges minimizes the average generalized price and incorporates a zero per-flight charge if DgBDBeB þ DgLDLeL ¼ 0um where ex is the elasticity of the business and passenger demands with respect to business and leisure fares, respectively, (i.e., ex ¼ pxD0x=Dx).

Fig. 5. Iso-carrier profit (dashed line; carrier profit is approximately 0.64), iso-airport profit (thin solid line; airport profit is zero), and iso-welfare (thicksolid line; welfare is approximately 2.22) curves with uniform fares: The carrier’s optimal per-passenger charge is too low from the social viewpoint.

Fig. 6. Iso-carrier profit (dashed line; carrier profit is approximately 1.33), iso-airport profit (thin solid line; airport profit is zero), and iso-welfare (thicksolid line; welfare is approximately 2.18) curves with discriminating fares: The carrier’s optimal perpassenger charge is excessive from the social viewpoint.


restores the results derived by the initial analysis, which concentrates on uniform time valuations, where the carrier’s and thesocial view on frequency were fully identical only if the optimal per-passenger charge would be zero.

Example 2. This example illustrates that the optimal per-passenger charge may be too low from the social viewpoint whenthe carrier charges a uniform fare to business and leisure passengers, while the reverse would be true under farediscrimination. To show this, assume that benefits are

B ¼Xx¼B;L

axqx � bxq2

x

2

ð21Þ

with ax; bx > 0 and aB sufficiently high in order to ensure that the carrier would like to charge a higher fare to business pas-sengers relative to leisure passengers. For schedule delays, it holds C ¼ 1=f .

Figs. 5 and 6 display iso-carrier profit (dashed lines), iso-airport profit (thin solid line) and iso-welfare (thick solid line)curves in the sq-sf -space for parameters aB ¼ 4; bB ¼ 2; aL ¼ 1; bL ¼ 1=2; vB ¼ 1; vL ¼ 0; c ¼ 1=10 and F ¼ 1=2. Fig. 5


displays the optimal mix of airport charges when fares are uniform. In this situation, the carrier’s incentives to provide fre-quency are too low from the social viewpoint; thus, to increase frequency supply, the socially optimal per-passenger chargeis high relative to the carrier’s preferred per-passenger charge. The opposite is, however, true when the carrier charges dis-criminating fares to business and leisure passengers. This is because discriminating fares increase the carrier’s incentives toprovide frequency as can be seen by inspection of Fig. 6.

5. Conclusions

Airport charges are largely determined by the quantity of aircraft flights, while carriers recently raised the point that pas-senger related charges should be used to ensure airport cost recovery. This paper has addressed this issue by comparing thecarriers’ and the social viewpoints on the optimal mix of airport charges conditional on airport cost recovery through rev-enues derived from airport per-passenger and per-flight charges. A crucial element of the model is that passengers experi-ence schedule delays, which are negatively related to the quantity of aircraft flights, i.e., frequency. In this case, passengerdemands depend on full fares, which are composed of fares and schedule delay costs. The model therefore established a cleartrade-off between low per-passenger charge, low fare, high load factor and low frequency versus low per-flight charge, highfare, low load factor and high frequency.

To analyze this trade-off, a basic scenario with a monopoly carrier and a single passenger type, and later an extensionincluding passenger types (i.e., business and leisure) with differences in time valuations, have been considered. Our analysisshowed that carriers can indeed benefit from a positive per-passenger charge relative to zero per-passenger charge becausethe associated reduction in the per-flight charge can increase frequency and profit. On the other hand, if passengers havedistinct time valuations, the carrier’s incentive to provide frequency and the profit gain from reductions in the per-flightcharge may be reduced when fares are uniform. This is because carriers are concerned with marginal time valuations underthese conditions. This effect however disappears when the business and leisure prices are discriminating. Altogether, thepresent analysis helped one to understand the carriers’ effort to strengthen the role of per-passenger based airport charges.

Turning to the social maximizer, she cares about carrier profits and consumers, and incorporating the consumers’ view-point means minimization of full fares conditional on airport cost recovery. In effect, the carrier’s and the social viewpointscan be identical only if the carrier’s preferred mix of airport charges minimizes full fares conditional on airport cost recovery.We further showed that the carrier’s and the social preferences may be identical only if the carrier’s preferred per-flightcharge is exactly equal to zero. This result can facilitate the policy makers’ effort to identify potential conflicts of interestwith carriers. Specifically, the carriers’ interest in positive per-passenger charges are consistent with the zero per-flightcharge and thus with a situation where the carriers’ and the social viewpoints are indeed identical. Thus, we did not findstrong evidence for a conflict of interest between the carriers’ and the social maximizer’s interests. Our analysis thereforesuggests that the current trend towards the increased use of per-passenger based airport charges requires no immediate reg-ulatory corrections.

Our analysis assumed that frequency supply reduces schedule delay costs, but abstracted away from the possibility that itincurs congestion and raises passenger and carrier costs under conditions of scarce runway capacity. But, our analysis is suf-ficiently general so that schedule delays can have the interpretation of a combined measure for schedule and runway con-gestion delays, which we show in Appendix C. Our results are therefore robust in the sense that they hold for uncongestedand congested airports.

The present paper has intentionally abstracted away from oligopolistic airline markets, which are considered in a com-panion paper (Czerny and Zhang, 2014c). An important feature of oligopolistic markets is that the time structure of decisionsbecomes important. Specifically, airline ticket prices often change on a daily basis, while flight schedules typically hold for aminimum of half a year. The companion paper therefore employs the assumption that carriers choose frequencies prior tofares. Importantly, the sequential time structure allows us to distinguish between the short- and long-run effects of airportcharges structures, where the short-run effects are derived for fixed frequency supplies, while fares and frequencies are con-sidered as endogenous in the long run. Interestingly, also the analysis of the oligopolistic carrier markets provides no strongevidence for a conflict of interest between the carrier’s and the social maximizer’s positions.

There two important avenues for future research. First, we assumed perfect information, while passenger demand ishighly uncertain. A useful extension of the present analysis would therefore be to derive a better understanding of howdemand uncertainty can affect the efficiency of per-passenger and per-flight based airport charges from the carrier’s andthe social viewpoints. Second, it would clearly be useful to derive empirical evidence for how the increased importance ofper-passenger charges has changed fares and frequency supply in reality.

Acknowledgements

We thank Leo Basso, Mogens Fosgerau, Robin Lindsey, Benny Mantin, Tae Oum, Sarah Wan, seminar participants at theCentre for Transportation Studies Seminar (University of British Columbia), the Department of Logistics and Maritime Stud-ies of the Hong Kong Polytechnic University and conference participants at the ITEA Conference on Transportation Econom-ics (Kuhmo Nectar) 2014 for helpful comments and suggestions. We are especially grateful to the organizers of the ITEA(International Transportation Economics Association) Conference 2014, who kindly awarded an earlier version of this paper


with the Best Overall Paper Prize, and the editor John Rose and two anonymous referees for helpful comments and sugges-tions. Partial financial support from the Social Science and Humanities Research Council of Canada (SSHRC) and the EuropeanResearch Council (ERC, AdG Grant #246969 OPTION) is gratefully acknowledged.

Appendix A. Proofs

Lemma 1. Cramer’s rule is applied in the following in order to identify the effect of airport charges on price and frequency.Let X denote the determinant of the Hessian of profit with respect to price and frequency:

X ¼ det@2p=@p2 @2p=@p@f

@2p=@p@f @2p=@f 2

!ð22aÞ

¼ det2� p�sq

p �xðDÞ� �

� D0 1� p�sq

p �xðDÞ� �

� D0vC0

1� p�sq

p �xðDÞ� �

� D0vC0 � xðDÞ þ pf vC0 � nðf Þ

� �� p�sq

p D0 vC0� �2

0B@1CA ð22bÞ

¼ 2�p� sq� �

p�xðDÞ

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

>1

p� sq� �

f � �vC0� � � nðf Þ � 1

0BBB@1CCCA D0 � vC0� �2

: ð22cÞ

Note that 2� p� sq� �

=p �x > 1 and n= f � �vC0� ��

> 1 by Assumption 1, which together imply that the right-hand side of(22c) is strictly positive at optimum, which is a necessary condition for a maximum.

Cramer’s rule can now be applied to derive the effect of an increase in the per-passenger charge on price and frequency,which yields:

dpM

dsq¼ 1

Xp� sq� �

f � �vC0� � nðf Þ � 1

!D0vC0� �2 ð23Þ

and

df M

dsq¼ 1

XD0� �2vC0: ð24Þ

The right-hand side of (23) is negative by Assumption 1. The right-hand side of (24) clearly is also negative.Analogously, the comparative-static relationships between price, frequency and the per-flight charge can be derived as:

dpM

dsf¼ � 1

X@2p@p@f

anddf M

dsf¼ 1

X@2p@p2 : ð25Þ

The first relationship is negative by Assumption 1, while the second relationship is negative by the second-order conditions(i.e., the negative definiteness of the Hessian of profit with respect to price and frequency).

Proposition 1. Suppose that the per-passenger charge is zero at the carrier optimum. In this situation, the first-ordercondition in (9) can be rewritten as

�DvC0 � c� �

Df ¼ 0: ð26Þ

Airport cost recovery then requires that the per-flight charge is strictly positive, which means that the bracketed term isstrictly positive by the first-order condition (7) or (8). If the frequency effect, Df , is strictly positive, this is a contradiction.Furthermore, since the left-hand side of Eq. (26) is strictly positive when the per-passenger charge is zero and the frequencyeffect is positive, the optimal per-passenger charge is strictly positive by the second-order condition @2p=@s2

q < 0.

Proposition 2. Consider the first-order condition @V=@sq ¼ 0, which can be written as

DpD� Df c þ Dg pMD0 � /D� �

¼ 0: ð27Þ

Since pMD0 � /D� �

is strictly negative in sign, the carrier’s and the social viewpoints can be identical only if generalized pricesare minimized conditional on airport cost recovery, i.e., Dg ¼ 0. Furthermore, suppose the generalized prices are minimizedat the carrier optimum, which implies Dp ¼ �Df vC0. Substituting �Df vC0 for Dp in the first-order condition (27) yields

Df � �vC0D� c� �

¼ 0; ð28Þ

which shows that the zero per-flight charge is a necessary condition for the carrier’s and the social viewpoints to be identicalby the first-order condition (8).


To show that the zero per-flight charge is not only a necessary but also a sufficient condition for the carrier’s and thesocial viewpoints to be identical asks the question whether a zero per-flight charge may be optimal from the carrier’s view-point when generalized prices are not minimized. Consider the first-order condition (8), again. Substituting c by�D � vC0 � sf , the expression DpD� Df c can be rewritten as Dg � Dþ sf D. The first-order condition (27) can then be rewrittenas

Dg � Dþ pMD0� �

þ sf D ¼ 0 ð29Þ

for / ¼ 0 (the latter is used because only the carrier’s viewpoint is relevant here). Since Dþ pMD0� �

¼ sf ¼ 0 is a contradic-tion because it implicitly assumes a zero per-passenger charge by the first-order condition in (6), the left-hand side showsthat a zero per-flight charge can be optimal from the carrier’s viewpoint only when generalized prices are minimized.

The first-order condition (29) can be used to infer whether the carrier’s optimal per-passenger charge may be excessive ortoo low from the social viewpoint. Recall that the carrier’s preferred per-passenger charge is always strictly positive. Then, ifthe optimal per-passenger and per-flight charges are positive from the carrier’s viewpoint, Dg > 0 and Dþ pMD0

� �< 0 in

the carrier optimum by the first-order condition (6), which means that the carrier’s optimal per-passenger charges are exces-sive from the social viewpoint by the first-order condition (27). This is because an increase in the per-passenger chargeincreases full fares in the carrier optimum, which is more important for the social optimizer than for the carrier. On the otherhand, if the carrier’s optimal airport-charges structures incorporates a negative per-flight charge and a positive per-passen-ger charge, (29) implies that Dg < 0, which means that the carrier’s optimal per-passenger charge is too low from the socialviewpoint.

Lemma 2. Totally differentiating the equilibrium conditions @CS=@qB ¼ gB and @CS=@qL ¼ gL with respect to generalizedprices leads to

dqB

dgB¼ 1

U� @

2B@q2

L

anddqL

dgL¼ 1

U� @

2B@q2

B

ð30Þ

with

U ¼ det@2B=@q2

B 0

0 @2B=@q2L

!¼ @

2B@q2

B

� @2B@q2

L

; ð31Þ

where the right-hand side of (31) is positive by the negative definiteness of the Hessian of B. The relationships in (30) can beused to derive how passenger quantities change in the uniform fare and frequency:

@q@p¼ dqB

dgBþ dqL

dgL¼ @

2B@q2

B

þ @2B@q2

L

ð32Þ

and

@q@f¼ vB

dqB

dgBþ vL

dqL

dgL

� C0 ¼ vB

@2B@q2

B

þ vL@2B@q2

L

!� C0: ð33Þ

Lemma 3. The determinant of the Hessian of profit with respect to the uniform price and frequency, X, can be written as

X¼det2� p�sq

p xðDÞ� �

D0 1� p�sq

p �xvðDÞ� �

D0bvC0

1� p�sq

p xvðDÞ� �

D0bvC0 � xv ðDTÞþ pfC0 nðf Þ

� �p�sq

p D0bv C0� �2

0B@1CA ð34aÞ

¼ 2vLxðDLÞ�vBxðDBÞ

vB�vLþp�sq

pxðDLÞxðDBÞ

� �D0BD0L

pvB�vLð Þ2þ 2�p�sq

pxðDÞ

p�sq

f � �bvC0� �nðf Þ�1

" #D0bvC0� �2

; ð34bÞ

where the right-hand side is strictly positive if schedule delay is sufficiently convex. Note that the first term on the right-hand side is zero when time valuations are uniform. Otherwise, the sign of the first term depends on the convexity of busi-ness and leisure demands weighted by time valuations. The second term on the right-hand side is positive by Assumption 3.

Cramer’s rule can be applied to derive the relationships between the uniform fare and the per-passenger charge as well asfrequency and the per-passenger charge as

dpT

dsq¼ 1

Xdet

�@2p=@p@sq @2p=@p@f

�@2p=@f@sq @2p=@f 2

!ð35Þ


with

@2p@p@sq

¼ �D0 > 0;@2p@f@sq

¼ @2p@p@sq

� bvC0 < 0: ð36Þ

The right-hand side of (35) is clear-cut and positive in sign by Assumption 3 and the second-order conditions.The effect of a marginal increase in the per-passenger charge can be derived as

df T

dsq¼ 1

Xdet

@2p=@p2 D0

@2p=@p@f D0bvC0

!ð37aÞ

¼ 1þ p� sq

p� xvðDÞ �xðDÞð Þ

� D0� �2bvC0: ð37bÞ

Since xv ðDÞ > xðDÞ means that xv ðDBÞ < xðDLÞ for vB > vL, the right-hand side of (37b) implies that an increase in the per-passenger charge increases frequency when business demands are sufficiently concave. Clearly, if time valuations are uni-form, an increase in the per-passenger charges always reduces frequency because xvðDÞ ¼ xðDÞ in this situation.

The effect of an increase in the per-flight charge is clear-cut and negative in sign for both uniform fare and frequency:

dpT

dsf¼ 1

Xdet

�@2p=@p@sf @2p=@p@f

�@2p=@f@sf @2p=@f 2

!ð38aÞ

¼ 1X

det0 @2p=@p@f

1 @2p=@f 2

!< 0 ð38bÞ

and

df T

dsf¼ 1

Xdet

@2p=@p2 0@2p=@p@f 1

!< 0: ð39Þ

Proposition 3. To establish part (i), consider the first-order condition @V=@sq ¼ 0, which can be written as

DpD� Df c þ pT D0 � Dp þ Df � bvC0� �

� /D � Dp þ Df � vC0� ��

¼ 0: ð40Þ

Since D is strictly positive in sign, the carrier’s and the social viewpoints can be identical only if the generalized prices eval-uated at average time valuations are minimized conditional on airport cost recovery.

To establish part (ii), suppose that the generalized prices are minimized at the carrier optimum in the sense thatDp ¼ �Df � vC0. Substituting �Df � vC0 for Dp in the first-order condition (40) and expanding by Df � DbvC0, yields

�DbvC0 � c� �

� pT � D0 þ D� �

� v � bv� �� C0 ¼ 0: ð41Þ

Suppose that the optimal per-flight charge is negative in sign, while the optimal per-passenger charge is positive in sign. Thefirst term on the left-hand side is then negative in sign, while the second term is also negative in sign; a contradiction. Sup-pose instead that the optimal per-flight charge is positive in sign, while the optimal per-passenger charge is negative in sign.In this situation, the first term on the left-hand side is positive in sign, while the second term is also positive in sign; anothercontradiction. Finally, suppose that both the optimal per-flight charge and the optimal per-passenger charge are positive insign. Then, the first term on the left-hand side is positive in sign, while the second term is negative in sign, which thus estab-lishes the necessary condition for the carrier’s and the socially optimal behavior to be identical.

Lemma 4. Using @2p=@px@py ¼ 0 for x ¼ B; L and y – x, this leads to, the determinant of the Hessian with respect todiscriminating business and leisure fares as well as frequency can be written as

! ¼ det

@2p=@p2B 0 @2p=@pB@f

0 @2p=@p2L @2p=@pL@f

@2p=@pB@f @2p=@pL@f @2p=@f 2

0B@1CA ð42aÞ

¼ 2 � @2p@p2

B

@2p@p2

L

@2p@f 2 �

@2p@p2

B

� @2p@pL@f

!2

� @2p@p2

L

� @2p@pB@f

!2

ð42bÞ

with second- and cross-derivatives

26 Sim


@2p@p2

x¼ 2� px � sq

px�xðDxÞ

� D0x ð43Þ

@2p@f 2 ¼ � C0

� �2 �X

x

xðDxÞ þpx

f � vxC0 � nðf Þ

� px � sq

pxv2

x D0x ð44Þ

@2p@px@f

¼ 1� px � sq

px�xðDxÞ

� D0xvxC

0: ð45Þ

The second derivatives are negative, while the cross-derivatives are positive in sign by Assumption 3. Furthermore, the right-hand side (42b) is negative in optimum by the second-order condition. Cramer’s rule can now be applied to yield the com-parative-static relationships between business fares, leisure fares and frequency:

@pDB

@sf¼ 1

!det

�@2p=@pB@sf 0 @2p=@pB@f

�@2p=@pL@sf @2p=@p2L @2p=@pL@f

�@2p=@f@sf @2p=@pL@f @2p=@f 2

0BB@1CCA ð46aÞ

¼ 1!

det0 0 @2p=@pB@f

0 @2p=@p2L @2p=@pL@f

1 @2p=@pL@f @2p=@f 2

0B@1CA ð46bÞ

¼ 1!

det0 @2p=@pB@f

@2p=@p2L @2p=@pL@f

!< 0; ð46cÞ

@pDL

@sf¼ � 1

!det

@2p=@p2B @2p=@pB@f

0 @2p=@pL@f

!< 0 ð47aÞ

and

@f D

@sf¼ 1

!det

@2p=@p2B 0

0 @2p=@p2L

! !< 0: ð48aÞ

Appendix B. Negative per-passenger charge

Fig. 7 displays iso-carrier profit (dashed lines), iso-airport profit (thin solid line) and iso-welfare (thick solid line) curves inthe sq-sf -space for parameters aB ¼ 4; bB ¼ 2; aL ¼ 1; bL ¼ 1=2;vB ¼ 3;vL ¼ 12=5; c ¼ 1=10 and F ¼ 1=2. Furthermore, faresare discriminating. In this parameter instance, the business and leisure passengers’ time valuations are high relative tothe instance considered in Example 2. The carrier’s frequency supply is therefore sufficiently inelastic relative to passengersupply so that the optimal per-passenger charge is negative from the social viewpoint.

Appendix C. Congestion

This part shows that our basic model is sufficiently general to capture not only schedule delays but also runway conges-tion.26 For this, denote the carrier’s average cost due to runway congestion as Kp with Kp ¼ Kpðf Þ and K0p > 0. The carrier profitcan be written as p ¼ p� sq

� �q� sf þ c þKp

� �f . Furthermore, denote the passengers’ average cost due to runway congestion as

Kq with Kq ¼ Kqðf Þ and K0q > 0. Letting Wdenote the per-passenger schedule delay cost with W ¼ Wðf Þ and W0 < 0, the general-ized price can be written as g ¼ pþWþKq. Using the equilibrium condition B0 ¼ g, the inverse demand in the fare can be writ-ten as B0 �W�Kq. Thus, profit can be rewritten as

p ¼ B0 �W�Kq � sq� �

q� sf þ c þKp� �

f : ð49Þ

For vCðf Þ ¼ Wðf Þ þKqðf Þ and Kp ¼ 0. Note that the right-hand side may become positive if congestion cost to passengers aresufficiently high, while C0 was assumed to be negative. The right-hand side in (49) therefore almost reduces to the profit inthe basic model version. However, our results do not hinge upon the non-negativity of C0. Thus, vCcan have the interpreta-tion of a combined measure for per-passenger schedule delay and runway congestion cost. If the carrier’s runway congestioncosts are positive, i.e., Kp > 0 , the only change relative to the basic model is that the carrier’s marginal frequency cost are anincreasing function in frequency and thus not constant anymore. While this would complicate the analysis, it would notchange our main results.

One may wonder whether the existence of congestion effects has some effect on whether the airport cost-recovery con-straint is binding from the social viewpoint. This could be true if an increase in airport charges would be useful in order to

ilar results also hold for the extended model version with business and leisure passengers.

Fig. 7. Optimal per-passenger charge is negative from the social viewpoint.


internalize congestion externalities. However, congestion externalities are not present in our model with a monopoly carrier.Thus, there is no need to internalize congestion externalities (which could potentially increase airport revenues) under theseconditions.27

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transportation research part a · how to mix per-ﬂight and per-passenger based airport charges...

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