東京大学 公共政策大学院

ワーキング・ペーパーシリーズ
GraSPP Working Paper Series
The University of Tokyo

GraSPP-DP-E-07-004 and ITPU-DP-E-07-004

Sequential Peak-Load Pricing
In a Vertical Setting :The Case of Airports and Airlines

Leonardo J.Basso and Anming Zhang
April 2007

GraSPP Discussion Paper E-07-004

ITPU Discussion Paper E-07-004

GRADUATE SCHOOL OF PUBLIC POLICY
THE UNIVERSITY OF TOKYOHONGO, BUNKYO - KU, JAPANGraSPP-DP-E-07-004

Sequential Peak-Load Pricing in a Vertical Setting:
The Case of Airports and Airlines
Leonardo J. Basso*+ and Anming Zhang*
April 2007

* Sauder School of Business – University of British Columbia+ Civil Engineering Department – Universidad de Chile
2053 Main Mall, Vancouver, B.C., Canada, V6T 1Z2.
Phone: 1 (604) 822-8420
Fax: 1 (604) 822-9574.
basso@sauder.ubc.ca
lbasso@ing.uchile.cl
anming.zhang@sauder.ubc.ca

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Discussion Papers are a series of manuscripts in their draft form. They are not intended
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the author.

1. INTRODUCTION
During the last several years airlines and passengers have been suffering from congestion and delays at busy airports,
and airport delays have become a major public policy issue. Since the early work of Levine (1969), Carlin and Park
(1970) and Borins (1978), economists have approached airport runway congestion by calling for the use of pricemechanism, under which landing fees are based on a flight’s contribution to congestion. While congestion pricing is
economically desirable, it has not really been practiced. The existing landing fees depend on aircraft weight, and thefee rates are based on the accountancy principle of cost recovery required usually for a public enterprise.1 Airports
have traditionally been owned by governments, national or local. This is changing, however. Starting with theprivatization of seven airports in the UK to BAA plc. in 1987, many airports around the world have been, or are in theprocess of being, privatized.2 One of the leading arguments for airport privatization is that privatised airports mightwell shift toward peak-load congestion pricing of runway services they provide to airlines, thus reducing delays inpeak travel times (Poole, 1990; Gillen, 1994; Vasigh and Haririan, 1996). For example, Gillen (1994) argues thatprivatization does a better job of producing efficient runway pricing mechanisms compared to public ownership.

Taken together, today’s shortage of airport capacity has revived much of the recent discussions about peak-loadcongestion pricing and airport privatization. In this paper we carry out an analysis of peak-load congestion pricing fora private, unregulated airport, as well as for a public airport that maximizes social welfare. The comparison of the twocases then allows us to shed some light on their pricing policies and traffic allocations to the peak and-off peak
periods. We find that compared to a welfare-maximizing airport, a profit-maximizing airport would charge higher
peak and off-peak runway prices, as well as a higher peak/off-peak price differential. As a consequence, privatisation
would lead to both fewer total air passengers and fewer passengers using the premium peak hours of the day for their

1

Airport charges include landing fees, aircraft parking and hangar fees, passenger terminal fees and air traffic control
charges (if the service is provided by the airport authority), with landing fees being most dominant. The revenues
derived from these charges are referred to as aeronautical revenues. In addition, busy airports derive significant
revenues from non-aeronautical business, such as concessions and other commercial activities. As Daniel (2001)
pointed out, landing fees in the U.S. traditionally recovered the “residual” costs –those remaining after all otherrevenue sources are fully exploited– with the fee rate equalling the annual residual costs divided by the weight of allaircraft landing during the year.
2
A number of major airports in Europe, Australia, New Zealand and Asia were recently privatized, or are in theprocess of being privatized. In the U.S., on the other hand, the airports that are used by scheduled airlines are virtually
all publicly owned facilities run by the local (city) government or by an agency on behalf of the local government.
Canada may represent a middle-of-the-road case in which airports recently devolved from direct Federal control to
become autonomous entities and major airports, though still government-owned, are now managed by private not-forprofit (but subject to cost recovery) corporations.

2

travel, both of which reduce social welfare. Hence, the alleged benefits of privatization would seem to have beenachieved, because those passengers who still use the peak period indeed face less congestion delays; but overall it is
not economically efficient to have such a lower level of peak congestion. This suggests that airport privatization
cannot be judged based on its effect on congestion delays alone.

Our analysis also shows that whilst private, profit-maximizing airports will always use peak-load pricing, somewhat
surprisingly, a public airport may actually charge a peak price that is lower than the off-peak price. Here the public
airport, on the surface, is not practicing the peak-load pricing, but such pricing structure is nevertheless socially
optimal.

We further investigate a case where a private airport strategically collaborates with the airlines so that it maximizes
the joint airport-airline profits, since it has been often argued that greater airlines’ countervailing power or morestrategic collaboration between airports and airlines may improve efficiency of privatized airports by allowing a betteralignment of incentives. The analysis shows that while the airport’s pricing practices would induce a collusive
outcome in the airline market, they would also, owing to the elimination of “double marginalization,” induce greater
total traffic and greater peak traffic than a pure (no-collaboration) private airport. Nevertheless, assuming nodifferences in technical efficiency, both figures will still be smaller than those for a public airport.

As indicated above, the present paper investigates airport peak-load pricing (PLP) and analyzes both the price leveland price structure (peak vs. off-peak). This is in contrast to the majority of airport pricing studies which did not
address inter-temporal pricing across different travel periods. In these congestion pricing studies, there is only one
demand function (i.e., a single-period model) for the airport, which is obtained by aggregating the demands of many
agents –in this case, the airlines. Since airport runways are congestible, when an airline decides to schedule a newflight, it induces extra-delays on every other flight. The airline however would only internalize the delays it imposeson its own flights and not others. Congestion pricing then looks at the price the airport, or a regulatory authority,
should charge to the airline for the new flight, in order for the airline to internalize all the congestion it produces (e.g.,Morrison, 1987; Zhang and Zhang, 2003, 2006; Pels and Verhoef, 2004; Basso, 2005). Notice that under congestion
pricing, since time-varying congestion is absent, there is only one way for either the airport or the airlines tointernalize congestion: raising prices to suppress the demands. In a PLP framework, on the other hand, excess demand3

problems arise because of the variability of demands during the reference times of the day. If the same price wascharged throughout the day, there would be peak periods at which the demand would be much higher than at off-peakperiods. PLP looks at the optimal time-schedule of prices so as to flatten the demand curve and make better use ofexisting capacity. As discussed below, both airports and airlines may engage in such demand spreading by using PLP.
Note that in this PLP framework, the airport is still a congestible facility, which implies that in the resulting optimal
price-schedule, prices at peak periods would still have to correct for uninternalized congestion: peak-load prices will
have a congestion pricing component.3

Another major feature of our analysis lies in the basic model structure used, which has strong implications for peakload pricing. Here an airport, as an input provider, makes its price decisions prior to the airlines’ output decisions. Thisvertical structure gives rise to sequential PLP: The PLP schemes implemented by the downstream airlines induce a
different periodic demand for the upstream airport, with the shape of that demand depending on the number ofdownstream carriers and the type of competition they exert. The airport then would have an incentive to use PLP as
well, which in turn affects the way the downstream firms use PLP. Although several very useful models of airport
peak-load congestion pricing have been developed (e.g., Morrison, 1983; Morrison and Winston, 1989; Oum andZhang, 1990; Arnott, De Palma and Lindsey, 1993; Daniel, 1995, 2001), these studies considered PLP primarily at the
airport level. Brueckner (2002, 2005), on the other hand, investigated PLP primarily at the airline level. Most of thesestudies considered only a public airport that maximizes social welfare, making no assessments about the effects ofprivatization on airport price structures.

There is an extensive body of literature on peak-load pricing. The classical papers (Boiteaux 1949; Steiner 1957;
Hirschleifer 1958; Williamson 1966) focused on normative rules for pricing a public utility’s non-storable service
subject to periodic demands. Some of the usual assumptions were: (i) demand is constant within each pricing period;(ii) demand in one period is independent of demand in other periods; (iii) constant marginal costs; (iv) the length ofpricing periods is fixed and exogenous; (v) the number of pricing periods is exogenous; and (vi) peak time is known.
Many authors have since contributed to the generalization of PLP results by relaxing one or a group of these3

As demonstrated in the text, the PLP-congestion pricing distinction is also important in that a single-period
congestion toll is not optimal unless it is charged on top of the optimal charge in the off-peak period, which may not be
the marginal cost. In other words, restricting the analysis to the toll that should be charged during the peak hours
offers only a partial view of the problem.

4

assumptions, including Pressman (1970), Panzar (1976), Dansby (1978), Craven (1971, 1985), Crew and Kleindorfer(1986, 1991), Gersten (1986), De Palma and Lindsey (1998), Dana (1999), Laffont and Tirole (2000), Shy (2001) andCalzada (2003).4 However, the case of sequential peak-load pricing, be it for public or private utilities, has yet been
analyzed. In the telecommunications research, for instance, Laffont and Tirole looked at PLP only at the upstream
level (the network access charge) whilst Calzada considered PLP only at the downstream level. Because of this, wethink our paper could be a contribution to the general peak-load pricing literature as well.

The paper is organized as follows. In the next section we set out the basic model. In Section 3 we analyze andcharacterize the output-market equilibrium, paying particular attention to the peak and off-peak derived demands for
airport services. Section 4 examines the airport’s decisions and discusses how the airport ownership influences thepeak and off-peak runway prices, traffic volumes, delays and welfare. Section 5 examines the case where a private
airport maximizes the joint airport-airline profits. Section 6 contains concluding remarks.

2. THE MODEL
We consider a two-stage model of airport and airline behavior, in which N air carriers service a congestible airport. In
the first stage the airport decides on its runway charges on airlines, and in the second stage each carrier chooses its
output in terms of the number of flights.

We shall consider a discrete choice model in which the consumer chooses between three mutually exclusivealternatives, namely: h=p, travel during peak hours of a day; h=o, off-peak period travel; and h=n, not traveling. Thereis a continuum of consumers labelled by θ. Denoting B h (θ ) the gross benefit for consumer θ from traveling inperiod h, consumers’ utility function may be written as:

U ( x, B h (θ ),− D h )(1)

In (1), x is consumption expenditure and Dh denotes the flight delay associated with travel in period h. We assume thatfor any given θ,4

See Crew, Fernando and Kleindorfer (1995) for an excellent survey of the peak-load pricing literature.

5

B p (θ ) > Bo (θ ) > Bn (θ ) = 0(2)

where Bn (θ ) is, for convenience, normalized to be zero. The inequalities say that if travel were free and without
congestion, the consumer would always prefer traveling to non-traveling. Furthermore, with identical airfares anddelays, consumers would always prefer traveling in the peak period to off-peak traveling. Thus, peak travel and offpeak travel are vertically differentiated: Controlling for fares and delay costs, passengers regard a peak flight as abetter product than an off-peak flight. This vertical-differentiation feature of air travel can arise if the peak period
represents the day’s more desirable travel times. Since people want to travel in those “popular” hours, the (unfettered)demand approaches or exceeds the capacity of the existing infrastructure, thereby resulting in (potential) congestion
during the peak hours. Note that our formulation (2) is different from the one used in Brueckner (2002, 2005), in
which he assumed a “single crossing property” in that the benefit functions intersect at an intermediate value of θ ,thus indicating that B p (θ ) > Bo (θ ) for large values of θ but B p (θ ) < Bo (θ ) for small values of θ . The singlecrossing condition was imposed to avoid a degenerate (corner) equilibrium in his analysis.

For simplicity we assume that B h (θ ) takes the simple linear form of Bh ⋅ θ , with Bh being constant. Condition (2)then implies B p > Bo > 0 , indicating that consumers differ in their travel benefits, with small θ indexing consumers[ ]with low benefit values. For simplicity, we assume θ is distributed uniformly on Θ = θ ; θ and normalize the numberof total consumers to θ − θ , so the number of passengers with type belonging to [θ 1 , θ 2 ] is directly given byθ 2 − θ 1 . We further assume that the utility function U is linear, and that consumers maximize it by choosing x andh ∈ { p, o, n} subject to the budget constraint x + t h ≤ I (θ ) , where th is the ticket price (airfare) of traveling inperiod h, and I(θ) is consumer θ’s income. We can then focus on the part of the utility function that determines thediscrete choice. This conditional indirect utility function is given by:

Vh (θ ) = Bhθ − αDh − th6(3)In (3), while θ indexes consumers according to their gross travel benefits, the positive parameter α represents the
subjective value of time savings and so αDh represents monetary costs of delays to passengers. Note that our demand
problem is identical to the one that will result if we fix θ but allow the value of time α to have a distribution among
consumers (in simple models with endogenous hours of work, the consumers’ “opportunity cost” of time lost in delays
is proportional to their wages). One could also argue that θ and α are related (Yuen and Zhang, 2005), but we do not
do this here.

As for the delay itself, the flight delay at period h, for h=p,o, may be given by Dh = D(Qh ; Lh , K ) , where Qh is thetotal number of flights in the period, Lh is the length (duration) of the pricing period, and K is the airport’s runway
capacity (measured in terms of the maximum number of flights that the airport’s runways can handle per hour). In thispaper we consider that K and Lh are exogenously given. 5 We further assume Lo is sufficiently long so that

D(Qo ; Lo , K ) = 0 throughout the relevant range of our analysis. In other words, whilst the narrow peak period iscongestible, congestion never arises in the broader off-peak period. 6 For the peak delay function, we make thestandard assumption that D p = D(Q p ) is differentiable in Q p andD 'p =

dD
> 0,
dQ p

D 'p' =

d 2D
≥0
dQ p2

(4)

This assumption is quite general, requiring only that for given airport capacity, increasing peak traffic will increase
congestion of the peak period and the effect is more pronounced when there is more congestion; that is, for given
capacity, the peak delay is convex in traffic volume. The assumption is certainly satisfied under a linear delay
function, D(Q p ; L p , K ) = δ ⋅ Q p /( L p K ) –which has been used by, e.g., Pels and Verhoef (2004)– or under the[functional form suggested by Lave and de Salvo (1968), that is, D (Q p ; L p , K ) = Q p L p K ( K − (Q p / L p ))

5

] .

−1 7

The case of variable and endogenous capacity is examined in Basso (2005) and Zhang and Zhang (2006) in acongestion-pricing framework.
6
This is similar to the two-period (peak/off-peak) formulation developed in Brueckner (2002).
7
This functional form was previously estimated from steady-state queuing theory and is further discussed in U.S.
Federal Aviation Administration (1969) and Horonjeff and McKelvey (1983). It has been used by, e.g., Morrison(1987), Zhang and Zhang (2003), and Basso (2005).

7

To obtain the consumer demands for peak and off-peak travel, we first note the following characteristics about theallocation of consumers: (i) if consumer θ1 flies, then consumers θ ≥ θ1 fly; (ii) if consumer θ1 does not fly, thenconsumers θ < θ1 do not fly; and (iii) if θ* denotes the consumer who is indifferent between traveling in the peak andoff-peak periods, then passengers θ ≥ θ* choose peak travel whereas passengers θ < θ* choose off-peak travel or nontravel.8 Hence, if we denote θ f the consumer who is indifferent between flying and not flying, (i), (ii) and (iii) above
imply, in the case of an interior solution, that θ < θf< θ * < θ . We assume for now the allocation is interior, butlater shall find conditions on the parameters for this to hold. Using qh to denote the total number of passengers inperiod h (h=p,o), then q p = θ − θ and qo = θ**−θ f .

Since runway charges are imposed on aircraft (flights), we need to transform the passenger-based demands q p andqo into per-flight demand functions. As in Brueckner (2002), Pels and Verhoef (2004), Basso (2005) and Zhang andZhang (2006), we make a “fixed proportions” assumption, i.e., S = Aircraft Size × Load Factor, is constant and thesame across carriers. 9 It then follows immediately that q p = Q p S = θ − θ*

and q o = Qo S = θ*− θ f , orequivalently,θ * = θ − Qp S ,θ f = θ * − Qo S

(5)From (3), the indifferent flyer θ is determined by θ ( B p − Bo ) − αD p = t p − t o (recall that Do = 0 ). This**says that a passenger’s gain θ ( B p − Bo ) , net of the delay cost αD p when shifting from the off-peak to peak*

8

Proof: (i) If θ1 flies, then θ 1 B h − αD h − t h ≥ 0 for h=p,o. If θ ≥ θ1, θB h − αD h − t h ≥ θ 1 B h − αD h − t h ≥ 0 and soθ flies. (ii) is analogous. (iii) Let ∆V (θ ) ≡ V p (θ ) − Vo (θ ) = ( B p − B o )θ − α ( D p − D o ) − (t p − t o ) and suppose θflies. Then if ∆V (θ ) ≥ 0 , θ chooses to fly in the peak period. If ∆V (θ ) < 0 , θ chooses to fly in the off-peak period.
Now, suppose that there exists θ * such that ∆V (θ *) = 0 (interior solution). Then it follows, since ∆V ' (θ ) > 0 , thatif θ ≥ θ * , θ chooses peak travel and if θ < θ * , θ chooses off-peak travel or non-travel. ■9That is, the number of passengers in each flight is constant. This assumption also allows us to abstract away from theissue of weight-based pricing as aircraft here have the same weight, and thereby focus on the main issue of peak-loadcongestion pricing.

8periods, is balanced by the fare differential t p − t o . The final flyer θ
and θ

f

f

is determined by θ Bo = t o . Replacing θf*in (5) we obtain:t o (Qo , Q p ) = Boθ − Bo SQo − Bo SQ p

(6)t p (Qo , Q p ) = B pθ − Bo SQo − B p SQ p − αD (Q p )(7)Equation (6) is the (inverse) consumer demand function faced by the airlines for the off-peak period, whereas (7) is
the consumer demand function for the peak period. Note that this demand system is not linear if D is not. Further, thepeak and off-peak flights are substitutes for the final passengers, which gives the room for airlines to “spread thedemand” across the peak and off-peak travel periods by using peak-load airfares. The analytical expressions of thecross elasticity of demand between peak travel and off-peak travel can be obtained from (6) and (7).

We now turn to the airlines. They have identical cost functions, given by:

c iA (Qhi , Q −h i , Ph ) = ∑ [c + Ph + β D(Qh )]Qhi(8)h∈ p , o−iiwhere Qh is the number of airline i’s flights in period h, Q h denotes the vector of flights of airlines other than i, c isthe airline’s operating cost per flight, and Ph is the airport landing fee in period h. 10 Further, parameter β (>0)
measures the delay costs to an airline per flight, which may include wasted fuel burned while taxiing in line or
holding/circling in the air, extra wear and tear on the aircraft, and salaries of flight crews. Airlines’ profit functions
can then be written as:

φ i (Qhi , Q h−i , Ph ) = ∑ th (Qo , Q p )Qhi S − c iA (Qhi , Q h−i , Ph )(9)h∈ p , oWith these functions at hand, we shall investigate the subgame perfect equilibrium of our two-stage airport-airlines
game.

10

As indicated earlier, airport charges usually include landing and terminal charges (charges for aircraft parking areminor). While landing fees are based on aircraft movements, terminal charges are typically per-passenger based. Since
the present paper is concerned with runway congestion, we shall focus on landing fees.

93. ANALYSIS OF OUTPUT-MARKET EQUILIBRIUM

To solve for the subgame perfect equilibrium we start with the analysis of the second-stage airline competition. Given
the airport’s runway charges Pp and Po , the N carriers choose their quantities to maximize profits, and the Cournot
equilibrium is characterized by the first-order conditions,∂φ i / ∂Qhi = 0 ,h=p,o(note the second-order conditions are satisfied).11 Imposing symmetry Qh = Qh / N and re-arranging, the first-orderi

conditions can be expressed as:Ωo (Qo , Q p , Po , N ) ≡ ( Boθ S − c − Po ) − Qo

Bo S 2 ( N + 1)
B S 2 ( N + 1)
− Qp o=0NN

B p S ( N + 1)
Bo S 2 ( N + 1)
− Qp
N
N
Q


− (αS + β ) D(Q p ) + p D' (Q p )  = 0
N



(10)

2

Ω p (Qo , Q p , Pp , N ) ≡ ( B pθ S − c − Pp ) − Qo

(11)

As demonstrated in the Appendix (Proposition A.1), there exist conditions on the parameters that guarantee interior
solutions, that is, θ < θf< θ * < θ or equivalently, Q p , Qo , Qn > 0 . For example, the peak period is used if the

per-passenger airport peak/off-peak price differential is smaller than the incremental gross benefit, for the highest
consumer type θ , of shifting from off-peak travel to peak travel. In particular, when the airport does not practicepeak-load pricing (so Pp = Po ), the peak period is always used. The proof also reveals that a smaller airport peak/offpeak price differential increases the likelihood of both the peak and off-peak periods being used, and that the off-peakperiod is always used if θ is large enough.12 In the remainder of the paper we shall restrict our attention to interior
allocations.

11

We have assumed a Cournot game in the output-market competition. Brander and Zhang (1990, 1993), for example,
find some empirical evidence that rivalry between duopoly airlines is consistent with Cournot behaviour.
12
These results suggest that Brueckner (2002, 2005)’s single crossing property, which was introduced to guarantee theexistence of a non-empty peak/off-peak interior solution, may not be needed. This may be desirable because the10A useful equation obtained from (10) and (11) is:o( B p − Bo ) S 2 ( N + 1)

Qp+ (αS + β ) D(Q p ) +D' (Q p ) NN+ ( Pp − Po ) − θ S ( B p − Bo ) = 0− Ω + Ω = Qpp

(12)

Since equation (12) depends on Qp but not on Qo, it implicitly defines Q p as a function of Po , Pp and N .
Substituting this function into (10), equation (10) then implicitly defines Qo as a function of Po , Pp and N , leadingto:Q p = Q p ( Po , Pp ; N ),Qo = Qo ( Po , Pp ; N )(13)

Equations (13) are the airport’s demands for the use of its peak and off-peak periods, respectively. Here it is worth
stressing that, while to (Qo , Q p ) and t p (Qo , Q p ) –defined by equations (6) and (7)– capture the final consumer(inverse) demands for air travel, Qo ( Po , Pp ; N ) and Q p ( Po , Pp ; N ) are the derived demands faced by the airport.

We now characterize the airport’s demands Q p ( Po , Pp ; N ) and Qo ( Po , Pp ; N ) . Totally differentiating (10) and (12)with respect to Pp yields:

∂Q p
∂Pp

=−

∂ (−Ω p + Ωo ) / ∂Pp∂ (−Ω p + Ωo ) / ∂Q p

N
=−
<0
2
( B p − Bo ) S ( N + 1) + (αS + β )(( N + 1) D' (Q p ) + Q p D' ' (Q p ) )(14)

where the inequality follows from conditions (2) and (4). So the airport’s demand for the peak period is, as expected,
downward-sloping in the peak charge. Similarly, we can obtain:
property implies, if using Brueckner’s interpretation of θ as an index of a passenger’s tendency to travel on business(as opposed to leisure travel), that peak benefits are higher than off-peak benefits for business travelers, but are lower
than off-peak benefits for leisure travelers. This appears contradictory with the idea that the peak and off-peak periods
are vertically differentiated.

11

∂Q p
∂Pp

∂Q p

< 0,

∂Po

=−

∂Q p
∂Pp

> 0,

∂Q
∂Q p
∂Qo
=− p =
> 0,
∂Pp
∂Pp
∂Po

∂Q
∂Qo
N
=− p −< 0,
2∂Po∂Po Bo S ( N + 1)

∂ (Qo + Q p )∂ (Qo + Q p )

∂Pp

= 0,

∂Po

=−

(15)

N
<0
Bo S ( N + 1)
2

We can see that, ceteris paribus, the airport peak charge does not influence total traffic but only the allocation of
traffic to the peak and off-peak periods. Furthermore, from (12) and (14) we get:

∂Q p∂∆Pp − o

=

∂Q p
∂Pp

<0

(16)

where ∆Pp − o ≡ Pp − Po . The above results (15) and (16) lead to:

Proposition 1: The airport’s demands Q p ( Po , Pp ; N ) and Qo ( Po , Pp ; N ) have the following properties:(i) They are downward-sloping in own prices;(ii) The peak and off-peak periods are gross substitutes;(iii) The off-peak runway charge (Po) determines the amount of total traffic, while the difference between the peakand off-peak charges ( ∆Pp − o ) determines the partition of that traffic into the two periods, with peak traffic
declining with the charge differential.

Notice that Part (ii) of Proposition 1 shows that the airport has the room to “spread the flights” across the peak andoff-peak periods by using peak-load landing fees. Together with the discussion following equations (6) and (7),therefore, our vertical airport-airline structure gives rise to a possible sequential PLP: the PLP schemes implemented
by the downstream airlines (higher peak airfare) induce a different periodic demand for the upstream airport. As
shown in Proposition 2 below, the shape of that demand depends on the number of downstream carriers and the type12

of competition they exert. Our analysis conducted later in Section 4 shows that indeed, the airport then would have an
incentive to use PLP as well, which in turn affects the way the downstream firms use PLP.

Next we examine the airport demands change with the number of airlines, N. We obtain the following comparative
static results:

Proposition 2: At the sub-game Cournot equilibrium,(i) 0 < ∂Q p / ∂N < Q p / ( N ( N + 1) ) , so the number of passengers traveling in the peak period increases with N;(ii) ∂ (Qo + Q p ) / ∂N = (Qo + Q p ) / ( N ( N + 1) ) > 0 , so the total number of traveling passengers increases withN;(iii) ∂Qo / ∂N > Qo / ( N ( N + 1) ) , so if the off-peak period is used ( Qo > 0 ) then the number of passengers
traveling in the off-peak period increase with N.

The proof of Proposition 2 is provided in the Appendix. Given that we consider interior solutions, conditional on
runway fees Pp and Po , both the peak and off-peak traffic volumes increase with the number of firms in the output
market. The proposition also shows that the (positive) elasticity of total traffic with respect to N, is equal to

1 /( N + 1) , whereas the elasticities of the peak traffic and off-peak traffic (with respect to N) are, respectively,
smaller and larger than 1 /( N + 1) . All three elasticities become smaller as N increases.

The final ingredient to characterize the Cournot equilibrium in the output market is related to the important issue of
airfares: For given airport charges, how do the peak and off-peak airfares compare with each other? From (6) and (7) it
follows that∆t p − o ≡ t p − to = θ ( B p − Bo ) − Q p S ( B p − Bo ) − αD(Q p )(17)

From the equilibrium condition (12) we obtain an expression for θ ( B p − Bo ) . Replacing that expression in (17)gives rise to the following airfare-differential formula, evaluated at the Cournot equilibrium:13∆t p − oCournot eq=

Pp − Po
S
+

+

β Qp
S N

β
S

D(Q p )

D' (Q p ) + α

QpN

D' (Q p ) + Q p( B p − Bo ) S

(18)

N

It is clear from (18) that if Pp ≥ Po , then ∆t p − o > 0 , that is, if the airport uses peak-load pricing (in the sense that it
charges a higher landing fee in the peak than in the off-peak), airlines will also use PLP (i.e., higher peak airfares) inequilibrium. More interesting perhaps is the fact that, even if the airport price the periods backwards, i.e., Pp < Po ,airlines may still use peak-load pricing in equilibrium, because the remaining four terms in (18) are all positive.

To further interpret (18), first note that holding Pp and Po constant, ∂∆t p − o / ∂N is negative, which can be seen bydifferentiating (17) and recalling, from Proposition 2, that sub-game equilibrium Qp and Qo increase in N. This implies
that a monopoly airline would have the largest airfare differential. Since, from (6), ∂t o / ∂N is also negative, thelower the N, the larger the off-peak fare. These two observations are consistent with what we have already shown in
Proposition 2 with respect to total peak and off-peak traffic. Next, it can be seen that for very large N, the airfare
differential approaches to the difference between an airline’s peak and off-peak per-passenger average costs, i.e., thefirst and second terms on the right-hand side (RHS) of (18). When there is an oligopoly, however, three extra terms
are added. Specifically, the third term on the RHS of (18) is the cost of extra congestion on an airline’s own flights
and caused by an additional passenger flying in the peak period. Thus, the first three terms on the RHS of (18)represent the difference between an airline’s peak and off-peak marginal costs. The fourth term represents the money
value of extra congestion to an airline’s passengers when a new passenger chooses to fly in the peak period, whereas
the fifth term is the mark-up term that arises from the oligopoly airlines’ exploitation of market power. Hence, as it is
now known, the airlines in oligopoly only internalize (charge for) the congestion they impose on their own flights,
which has two cost components: extra operating costs for the airline, and extra delay costs for its passengers(Brueckner, 2002). When there is a monopoly airline, congestion is perfectly internalized but exploitation of market
power is at its highest degree. When N is large, exploitation of the market power is small but congestion is imperfectlyinternalized.

14

These points can be made more clearly if the Cournot case is compared to the case in which a social plannermaximizes total surplus in the second-stage game. To do this we first need a measure of consumer surplus (CS). Given
the linearity of our conditional indirect utility function in (3), CS is given by:
θ

[

θ*

]

[

]

CS = ∫ θB p − αD(Q p ) − t p (Q p , Qo ) f (θ )dθ + ∫ θBo − to (Q p , Qo ) f (θ )dθ
θ

*

θ

(19)

f

where f (θ ) is the density function. Using (6) and (7) for to and tp, solving the integrals and replacing θ and θ*fwith (5), we finally obtain:

CS =

(

S2
Bo Qo2 + 2 Bo Qo Q p + B p Q p2
2

)

(20)

As said, we now examine the case in which the planner maximizes, for given airport charges, the sum of consumer
surplus and airline profits:

CS + Φ ≡ CS + ∑i =1φ iN(21)where Φ denotes the aggregate airline (equilibrium) profits. The first-order conditions of (21) with respect to airline
quantities, together with the imposition of symmetry, then lead to two equations, analogous to (10) and (11), whichcharacterize the optimum. Subtracting the two equations from each other yields:

Q p ( B p − Bo ) S 2 + (αS + β )(D(Q p ) + Q p D' (Q p ) ) + ( Pp − Po ) − θ S ( B p − Bo ) = 0(22)

Using (22) to obtain a new expression for θ ( B p − Bo ) and replacing the term in (17), we get:∆t p − o efficient =outputPp − PoS+βS

D(Q p ) +αS + βS

Q p D' (Q p )(23)

Conditional on the airport charges and the airline market structure, (23) gives the socially efficient difference between
the peak and off-peak airfares. This fare differential is equal to the difference between an airline’s peak and off-peakaverage costs (the first and second terms on the RHS of (23)), plus all the external costs associated with a new flyer in15the peak period, with the latter being the extra congestion cost of all the airlines and passengers, not just that of theairline that carries the new peak passenger. Obviously, the last two terms represent the portion of the optimal airfare
differential that is not directly affected by the airport’s pricing practices.

It is also insightful –and useful later in Section 5– to compare the Cournot case with the “cartel” case in which theairlines choose outputs to maximize their joint profit Φ =∑ φ . The first-order conditions in the cartel case,
N

i

i =1

together with the imposition of symmetry, lead to two equations, analogous to (10) and (11), which characterize thecartel optimal solution. Subtracting the two equations then yields:

2Q p ( B p − Bo ) S 2 + (αS + β )(D(Q p ) + Q p D' (Q p ) ) + ( Pp − Po ) − θ S ( B p − Bo ) = 0(24)

Using (24) to obtain a new expression for θ ( B p − Bo ) to replace the term in (17), we find the difference between
the peak and off-peak airfares in the cartel case:∆t p − ocartel output=

Pp − PoS+βS

D(Q p ) +αS + βS

Q p D' (Q p ) + Q p ( B p − Bo ) S(25)Here, the fare differential is equal to the difference between an airline’s peak and off-peak average costs (the first andsecond terms on the RHS of (26)), plus all the external costs associated with a new flyer in the peak period.
Comparing (25) with (23) reveals that the cartel, as in the social-planner case, internalizes the congestion costs of all
the carriers and passengers. Here however, there is a fourth term which will increase the fare differential. This term isrelated to the “business stealing” externality: Since oligopoly carriers behave in non-cooperative fashion, they produce
too much with respect to the optimum for the airlines as a whole. This is so because they fail to consider the profits
lost by the other airlines when they increase own output, depressing airfares: the fare differential of an oligopoly is
insufficiently large from the cartel’s point of view, and this problem worsens the “looser” the oligopoly is (i.e., thelarger N is). Consequently, the cartel, as a monopoly, is interested in having a less used peak period. In effect, thecartel airfare differential is identical to the monopoly’s; see (18) by imposing N=1. The cartel and monopoly traffic
volumes will differ however, since cost functions are convex and not flat.

16

4. AIRPORT PRICING, TRAFFIC, DELAY AND WELFARE COMPARISONS

We have shown that the airport decisions, namely, Pp and Po , can influence the subsequent output-market
competition among airlines. When deciding its runway charges in the first stage, therefore, the airport will take thesecond-stage equilibrium output into account. These decisions may in reality be set by a public airport or a privatizedairport. Consequently, the objective of an airport may be to maximize welfare or to maximize profit. In this section,
we compare airport charges and consequent airfares for these two airport types.

Consider first a public airport that chooses Pp and Po to maximize welfare. With three agents –namely, airport,
airlines, and passengers– social welfare (SW) is the sum of their payoffs:

SW ( Po , Pp ; N ) = π ( Po , Pp ; N ) + CS + Φ(26)

π ( Po , Pp ; N ) = PoQo + PpQ p − C ⋅ (Qo + Q p ) − rK

(27)

where the airport’s profit, π , is given by

In (27), Qo = Qo ( Po , Pp ; N ) and Q p = Q p ( Po , Pp ; N ) are the airport’s demands for the peak and off-peakperiods, respectively, and are given by (13). In addition, C is the unit runway operating cost of the airport and r is its
unit cost of capital. Note that we have assumed, as is common in the literature, that the operational and capital costs
are separable and that the marginal operating cost is constant. For the latter, the estimation of cost function showedthat airport runways have relatively constant return to scale (e.g., Morrison, 1983; Pels, Nijkamp and Rietveld, 2003).
Consumer surplus, CS, is given by (19), whereas the aggregate airlines’ profit, Φ =∑ φ , is introduced in (21).
N

i

i =1

(

)

Since the downstream equilibrium is symmetric, φ (Qh , Q h , Ph ) = φ Qo ( Po , Pp ; N ), Q p ( Po , Pp ; N ), Ph is each
i

−i

i

1

airline’s equilibrium profit. We can then easily calculate Φ as Φ ( Po , Pp ; N ) = N ⋅ φ ( Po , Pp ; N ) , that is,1()

Φ ( Po , Pp ; N ) = θ S ( B pQ p + BoQo ) − S 2 BoQo2 + 2 BoQoQ p + B pQ p2 − (αS + β )Q p D(Q p )− (c + Po )Qo − (c + Pp )Q p

17

(28)

We do not include a budget constraint in the public airport problem, noting that fixed fees may solve the problem ofbudget adequacy. If lump-sum transfers are not feasible, then Ramsey-Boiteaux prices should be considered (seeBasso, 2005, for more discussion on this).

Substituting Φ, π , and CS into (26) we obtain:

SW = θ S ( B pQ p + BoQo ) − c ⋅ (Q p + Qo ) − C ⋅ (Q p + Qo ) − rK(29)

− S 2 ( BoQo2 + 2 BoQoQ p + B pQ p2 ) / 2 − (αS + β )Q p D(Q p )Derivation of the pricing formulas then follows from the first-order conditions. More specifically, we obtain, using
WW

equations (10), (12) and (15) ( Po , Pp denoting the welfare-maximizing runway charges):2

PoW = C −Qo S 2 Bo Q p S Bo
−
N
N

(30)

Q p S ( B p − Bo )N −1=
(αS + β )Q p D' (Q p ) −
N
N
2

∆P

W
p −o

(31)

The above welfare-maximizing airport pricing may be seen as if the fees were determined in two phases. First, choiceWWof an off-peak price Po induces the (socially) right amount of total traffic; as can be seen from (30), Po is below
the airport’s marginal cost. This is needed because exploitation of market power in the airline market would induce
allocative inefficiencies by producing too little output. A welfare-maximizing airport fixes this inefficiency byproviding a “subsidy” to the airlines and hence lowering their marginal costs in the off-peak period. The exact amountof the subsidy depends in part on the extent of market power, which here is captured by N. Once the total traffic is set
to its optimal level, the next phase is concerned with the optimal allocation of this traffic to the peak and off-peakperiods, which is, as indicated earlier, determined by ∆Pp − o . In particular, the public airport sets the peak/off-peakprice differential to ∆Pp − o that will induce the optimal airfare differential downstream. This is apparent fromW

substituting (31) into (18), which yields∆t Wp − oCournot eq=

βS

D(Q p ) +18

αS + βS

Q p D' (Q p ) > 0(32)

The RHS of (32) is equal to the optimal airfare differential that is not directly affected by the airport’s pricing
practices, as discussed in (23). Hence, the outcome is the same as if the airport were to set Po = Pp , which is optimal
because there are no differences in costs, and then social welfare is maximized in the airline market.

Brueckner (2002) identified the first term in (31) as the per-flight toll that should be charged by the airport authorities
to address the problem of uninternalized congestion (note that when N=1, this toll is equal to zero). Pels and Verhoef(2004), Basso (2005) and Zhang and Zhang (2006) pointed out that the optimal toll should also include the second
term, the market-power effect; 13 they did this, however, using models of congestion pricing (one period), while
Brueckner (2002) and the present paper use a model of peak-load pricing. This distinction is important because a tollequal to the two terms, thereby capturing both the congestion and market power effects, will not be optimal unless it ischarged on top of the optimal charge in the off-peak period, which is not the marginal cost. In other words, restricting
the analysis to the toll that should be charged during the peak hours offers only a partial view of the problem.

But note further that the charge differential, ∆Pp − o , given in (31), is not signed a priori. Hence, it may happen thatW

the airport charge is smaller in the peak period than in the off-peak period. More specifically, the airport charge
differential will be negative for small N. This is so because a “tight” airline oligopoly has an airfare differential that is
too large due to strong market power, while congestion is reasonably internalized. As a consequence, the airport pricedifferential is driven predominantly by the market-power effect. When N is large, on the other hand, the airport pricedifferential will be positive. This is so because a “loose” oligopoly would have an airfare differential that is too small
due to uninternalized congestion, whereas market power is relatively weak. The airport charge differential is thenWdriven by the congestion effect. Note from (32) that although Pp (the welfare-maximizing peak charge) may be lessWthan Po , final passengers will, nevertheless, always pay higher peak airfare than off-peak airfare.

The above discussion may be summarised in the following proposition:

13

To be fair, although Brueckner did not formally consider the second term in the toll to be charged, he did point out
that, depending on the size of the market-power term, a pure congestion toll could be detrimental for social welfare.

19

Proposition 3: For a public, welfare-maximizing airport, (i) the off-peak runway charge is below its marginal cost; (ii)for small N, the off-peak runway charge may be greater than its peak runway charge; in this sense, it appears that the
airport does not use peak-load pricing; (iii) although the airport’s peak charge may be less than its off-peak charge,
final passengers will nevertheless always pay higher peak airfare than off-peak airfare.

Note that if lump-sum transfers (two-part tariffs) are unfeasible, the pricing rules previously discussed may lead to
airport’s budget inadequacy. If budget adequacy has to be ensured but lump sums are not feasible, then the first best
may not be attainable: Marginal prices would have to do both –namely, aligning incentives and transferring surplus–
making the airport fall short of “control instruments” (Mathewson and Winter, 1984).

Next, consider a private, unregulated airport. The airport’s profit is given by (27). The airport will choose Pp and Po toππmaximize its profit, and the first-order conditions lead to ( Po , Pp denoting the profit-maximizing airport charges):

Poπ − C =Ppπ − C =where

ε oo ≡ −(∂Qo / ∂Po )( Po / Qo )

is

the

Poπ

ε oo

Ppπ

ε pp

(positive)++( Ppπ − C )Q pε po(33)

Qoε oo( Poπ − C )Qo ε op

price

(34)

Q p ε ppelasticity

ofoff-peakairportdemand,ε po ≡ (∂Q p / ∂Po )( Po / Q p ) is a cross-price elasticity, and ε pp and ε op are defined analogously. Since∂Q p / ∂Po > 0 and ∂Qo / ∂Pp > 0 –see (15) or Proposition 1– both ε op and ε po are positive, implying that the
airport charges are higher than would be if the peak and off-peak charges were chosen independently (in which casethe mark-ups would be proportional to the inverse of own-price demand elasticities only). This is a well-known result
for multi-product monopolies that produce substitutes.

20

π

π

We can simplify the pricing equations and show that Pp > Po . To do this, replace the elasticities’ definitions andsimplify, using the fact that ∂Qo / ∂Pp = −∂Q p / ∂Pp in (15) and then using equation (14). We obtain the following
charging formulas:

Qo S 2 Bo ( N + 1) Q p S Bo ( N + 1)+
NN2

Poπ = C +π

∆Pp − o =αS + βN[]

Q p ( N + 1) D' (Q p ) + Q p D" (Q p ) +π

Q p ( B p − Bo ) S 2 ( N + 1)
N

(35)

(36)

π

The RHS of (36) is, by (2) and (4), positive and hence Pp > Po : The private airport charges higher runway fees in
the peak period than in the off-peak period, and this is true for any N. Thus, a profit-maximizing airport has anincentive to use peak-load pricing. Further, note that, since (αS + β )Q p ( N − 1) / N is the extra cost each airline
induces by not fully internalizing congestion, the first term on the RHS of (36) shows that the private airport will
overcharge for congestion. Moreover, notice from (35) that the off-peak charge, which determines the amount of total
traffic, is above marginal cost. This is a result of monopoly power on the part of the airport. There is, therefore, a“double marginalization” problem, which is typical of an uncoordinated vertical structure. The discussion leads to thefollowing proposition:

Proposition 4: A private, profit-maximizing airport would use peak-load pricing but would charge more than the costof uninternalized congestion. Further, it would charge an off-peak runway fee that is above its marginal cost.

Having derived and characterized the pricing structures for both the public and private airports, we now want tocompare them. To have a clearer picture about their performance differences, we shall compare not only the off-peakrunway fees and the peak/off-peak fee differentials, but also the induced traffic levels, delays and total surplus levels.
Moreover, we want to assess how these differences (if any) change with the number of airlines, N, which isexogenously given and may be considered as a proxy for airline market structure. We summarize our findings in thefollowing proposition (the proof is provided in the Appendix):21

Proposition 5: Comparisons of airport pricing, traffic, delay and welfare between the private and public airports are as

follows:

dPoW dPoπ
>
=0;
dN
dN

(i)

PoW < Poπ and

(ii)

∆P

(iii)

Q

W
p

(iv)

dQtπ dQtW
Q > Qt and>= 0 , where Qt ≡ Q p + Qo is total traffic volume;
dN
dN

(v)

D Wp > D πp and

(vi)

W
p −o

< ∆Pp − o andπ> Q p and

dN

dQ πp
dN

>

dQ Wp
dN

> 0 . If the delay function is linear, thend∆Ppπ− o
dN

= 0;

= 0;

π

W
t

SW

d∆PpW− o

π

W

π

dD πp
dN

> SW and

>

dD Wp
dN

dSW π
dN

>

= 0;
dSW W
dN

= 0.

From Proposition 5 we see that a private, profit-maximizing airport would induce too small total traffic as compared to
the first-best outcome, thereby resulting in allocative inefficiencies. Additionally, a private airport has a greaterpeak/off-peak runway charge differential than a public airport. Hence, with a private airport, the peak period would beunderused not only because the airport has smaller total traffic, but also because its charge differential is too large.
This reduction in peak traffic volume is welfare-reducing since passengers view, other things being equal, traveling in
the peak times as a higher quality product than traveling in the off-peak times. Less peak traffic then means fewer
consumers will enjoy a premium product. And although those passengers who still use the peak period benefit fromless delays as part (v) enounces, overall it is not economically efficient to have such a lower level of peak congestion
because total welfare is in fact reduced as shown in part (vi) of Proposition 5. To help better understand this
proposition, we also offer a schematic representation of the findings in Figure 1.

22

A numerical simulation helps to better see these deadweight losses.14 In Table 1, we show total welfare for a private
airport as a percentage of the total welfare obtained by a public airport (as can be seen from part (vi), SWW

does notdepend on N), and for different values of N. Because lump-sum transfers may not be feasible, we also provide thesocial welfare levels attained by a public airport that must achieve budget adequacy and therefore uses RamseyBoiteaux prices.
Table 1: Percentage of maximum social welfare attainable

N =1

N=3

N=5

N = 10Private airport

49

65

69

74

Budget-constrained public airport

81

96

98

99

This discussion and the results in Table 1 highlight an important issue: one of the main ideas behind airport
privatization has been that it would allow airports to use peak-load pricing and thus help solve the congestion
problems. But if privatization is measured solely by its effect on congestion delays, it may be seen as a better idea thanit actually is, and important deadweight losses may be overlooked. This result, which holds here for a fixed capacity–peak-load pricing model, was also found by Basso (2005) in a congestion pricing model with variable (endogenous)
capacity.

Qp,t , SW , D

Po , ∆Pp −o

Poπ , ∆Ppπ− o

Q Wp ,t , SW W , D W

PoW , ∆PpW− oQ πp ,t , SW π , D π

NNFigure 1: Schematic representation of the results in Proposition 5

14

Details of the simulation are available upon request.

23

We have seen that the public airport is indifferent between values of N –although Basso (2005) showed that this may
not be the case if airlines are not homogenous or if passengers are affected by schedule delay cost. Given that the(welfare) performance of a private airport improves as the number of airlines rises (see Proposition 5 and Figure 1), it
seems important to know what would be the preferred N of a private airport itself. From (34) we have:

∂Q p
∂Qo
dπ
∂π
=
= Poπ − C
+ Ppπ − C>0
dN Poπ , Ppπ∂N Poπ , Ppπ
∂N
∂N

(

)

(

)

where the first equality follows from the envelope theorem and the inequality follows from Proposition 2 and the firstorder conditions (33) and (34), which indicate that prices are above marginal costs. Thus, the private airport prefers alarge N, which is a desirable property, given the findings of Proposition 5.

5. Airport-Airline Joint Profit Maximization

In this section, we shall consider an airport that has some sort of strategic agreement with the airlines using it. The
reasons why it is interesting to look at this case are two-fold: on one hand, a simple pricing mechanism, two-part tariff,
may be enough for the outcome of joint profit maximization to arise. On the other hand, it has been often argued that
greater airlines’ countervailing power or more strategic collaboration between airlines and airports may improve
efficiency of privatized airports by allowing a better alignment of incentives, and even may make price regulation
unnecessary (see, e.g., Beesley, 1999; Condie 2000; Forsyth, 1997; Starkie, 2001; Productivity Commission, 2002;
Civil Aviation Authority UK, 2004). The analysis of joint profit maximization may then serve as a benchmark case.

The objective faced by this airport is to maximize the sum of the airport’s profit and airlines’ profits. Using π in (27)and Φ in (28), the problem can be re-written as:

max π + Φ = θ S ( B pQ p + BoQo ) − c ⋅ (Q p + Qo ) − C ⋅ (Q p + Qo ) − rK
Po , Pp− S 2 ( BoQo2 + 2 BoQoQ p + B pQ p2 ) − (αS + β )Q p D(Q p )Derivation of the pricing formulas follows from the first-order conditions of the above problem, using equations (10),JPJP(12) and (15) and rearranging ( Po , Pp

denoting the joint profit-maximizing airport charges),24

Qo S 2 Bo ( N − 1) Q p S Bo ( N − 1)
+
N
N

(37)

Q p S 2 ( B p − Bo )( N − 1)
N −1=
(αS + β )Q p D' (Q p ) +
N
N

(38)

2

PoJP = C +

∆P

JP
p −o

The interpretation of the JP airport’s pricing rules (37) and (38) is as follows. As before, this airport may be seen as
deciding its runway fees in two phases; first, it induces a contraction of total traffic by choosing an off-peak price

PoJP above its marginal cost. It does so because the failure of coordination among the airlines results in themproducing too much with respect to what would be best for them as a whole. The amount of excess production
depends on how tight the oligopoly is, which is why the off-peak mark-up decreases with N. In particular, when theJPairline market is monopolized, (37) shows that Po= C so the airport does not need the mark-up at all. Comparisonof (37) and (35), however, shows that the total traffic contraction in the JP case is smaller than that in the pure private
case; indeed, coordination of the vertical chain resolves the double-marginalization problem.

In the second phase, the airport chooses the (non-negative) price differential ∆Pp − o that will induce the airlines’JPcartel outcome, destroying airline competition downstream. This is apparent from substituting (38) into (18), andnoting that the result is equal to the cartel’s airfare differential not directly affected by the airport pricing practice.
Hence, the outcome is the same as if the airport were to set Po = Pp and a cartel were running the airline market.
This result, which was obtained by Basso (2005) in a congestion pricing setting, has different intuitions depending on
why the maximization of joint profits is the relevant case. With two-part tariffs, the private airport use the variable
prices, peak and off-peak, to destroy competition downstream in order to maximize the profits of airlines, which arelater captured by the airport through the fixed fee. When the joint profit maximization arises because of collaboration
between airlines and the airport, what happens is that the airlines would like to collude in order to increase profits, but
are unable to do so themselves because of the incentives to defect on any possible agreement. What they manage todo, however, is to “capture” an input provider to run the cartel for them. By altering the prices of the inputs (runway
services) and hence the downstream marginal costs in both the peak and the off-peak periods, the input provider(airport) induces both the collusive total output and the “right” (to the airlines) allocation of passengers to the peak and25

off-peak periods. The upstream firm is then rewarded with part of the collusive profits, which is where bargaining
power enters the picture.

Note also that the airport pricing rules (37) and (38) take into account both the congestion externality and the businessstealing externality, at both pricing phases: the airport’s price differential has two parts.15 When N=1, there is nobusiness-stealing effect and congestion is perfectly internalized by the monopolist. Consequently, both terms vanish:with a monopoly airline, the airport will not use peak-load pricing.

Now, despite the fact that the result is as if airlines were colluding, this case is not worse, in terms of social welfare,than a private airport charging linear prices as before. This is because, here, the two other harmful externalities,
namely, the vertical double marginalization and the congestion externality, have been dealt with. In effect, we canshow that the JP case (where the airport has strategic agreements with airlines) represents a middle-of-the-road case:in Proposition 5, the runway fees, traffic volumes, delays and social welfare will be in between those of the private
and public airport cases. And in Figure 1, the curves pertaining to the JP case would be parallel displacements of the
public airport curves, lying in between the two existing public and private curves. Strategic collaboration between the
airport and the airlines smoothes the airport-charge problem. But recall that the downstream airfares would be as if theairlines were engaging in collusion, so we cannot expect that the JP ends up being very close to the first best. In effect,a numerical simulation has shown that the JP would correspond to 78% of the maximum social welfare attainable.16
The performance of the JP airport gets closer the performance of a budget-constrained public airport (see Table 1), yet
it is still worse. The difference is smallest when there is a monopoly airline because, in that case, the subsidies
required for the first best are higher, and hence the social welfare attained by the second-best public airport is thelowest. But a slight increase in the number of airlines is enough to increase the gap in an important way.

15

This idea of an upstream firm running the cartel for the downstream firms has been discussed in the vertical control
literature and, particularly, in the input joint-venture case. For example, Chen and Ross (2003) formalized theconjecture that input joint-ventures can facilitate collusion and push a market toward the monopoly outcome. If airport
provision was seen as an input joint-venture by the airlines, our results show three things in addition to what Chen andRoss have found. First, the results hold even in a peak-load pricing setting, i.e., when demand is periodic. Second, if
there are externalities, the input prices are, additionally, used to force their internalization by downstream competitors.
Third, when marginal costs downstream are not constant, the outcome is not as in a monopoly or a downstream
merger, but as in a cartel.
16
The proofs of the results discussed in this paragraph, and details of the simulation are available upon request.

26

6. CONCLUDING REMARKSIn this paper, we have analyzed the sequential peak-load pricing (PLP) problem that arises when airports arerecognized as input providers for final consumer markets facing periodic demands. We have analyzed this PLP
problem for a private unregulated airport, for a public airport that maximizes social welfare, and for an airport thatstrategically collaborates with the airlines and hence maximizes their joint profits. We found that privatization would
not induce efficient peak-load pricing schemes as it has been argued in some studies. While a private airport always
has an incentive to use PLP –higher runway fees in the peak than off-peak periods, even when the airlines have used
PLP themselves and irrespective of the number of airlines servicing the airport– its pricing structure would induce
insufficient total traffic and insufficient peak traffic. Private airports will overcharge for congestion. Somewhat
surprisingly, depending on the degree of market power (captured here by the number of carriers at the airport), a
public airport may choose a peak runway charge that is lower than the off-peak charge, so as to offset the market
power downstream at the airline level. Here, the public airport, on the surface, is not practicing the peak-load pricing,but such pricing structure is nevertheless socially optimal. Finally, a private airport that strategically collaborates withthe airlines would induce greater total traffic and greater peak traffic than a pure private airport, but both figures will
still be smaller than those for a public airport. If the airport collaborates with a monopoly airline, it would not use
peak-load pricing.

Although the airline industry is chosen for analysis, our basic model structure, in which airports, as input providers,make their pricing decisions prior to airlines’ strategic interactions in the final output market, is highly relevant toseveral other industries including electricity, telecommunications, and transport terminals (e.g., the vertical chain ofports-sea carriers-shippers). In telecommunications, for example, at the upstream level there are the network owners,
while downstream there are carriers who must use the network in order to produce the final good (telephone calls).
Like airports, these industries are undergoing privatization in a number of countries. We note that the sequential PLP
method used in the present paper may be useful in examining similar issues in those sectors as well.

27

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APPENDIX•

Proposition A.1: Conditions for an interior allocation of consumers(i) If ( Pp − Po ) / S < θ ( B p − Bo ) , then the peak period is used, that is θ < θ .
*(ii) If θ Bo < (c + Po ) / S , then some consumers will not fly, that is θ

f

>θ .

(iii) If θ is large enough, then the off-peak period is used, that is θ > θ .
*f

Proof:First, equivalent conditions for interior allocations, but in terms of Qp and Qo are:
The peak is used: θ < θ ⇔ (θ − θ ) / S > 0 ⇔ Qo > 0**Some consumers do not fly: θThe off-peak is used: θ > θ*ff> θ ⇔ (θ − θ f ) / S < (θ − θ ) / S ⇔ Qo + Q p < (θ − θ ) / S⇔ (θ * − θ f ) / S > 0 ⇔ Qo > 0With this, the proofs of each part are:(i) Note that ( −Ω + Ω ) in (12) is strictly increasing in Qp , andp(− Ω + Ω )(− Ω + Ω )
p

o

p

o

o

Q p =o= ( Pp − Po ) − θ S ( B p − Bo )Q p =o< 0 and Q p > 0 .

.

Hence,

if(− Ω + Ω )po

Q p →∞

Pp − Po < θ S ( B p − Bo ) ,■30> 0 . Also,then(ii) From Ωo=0 in (10) we get that (Qo + Q p ) Bo S ( N + 1) / N = Boθ S − c − Po , This imply that2

Qo + Q p < ( Boθ S − c − Po ) /( Bo S 2 ) . Hence, a sufficient condition for Qo + Q p <( Boθ S − c − Po ) /( Bo S 2 ) < θ − θ / S , which leads to θ Bo < (c + Po ) / S .
(iii) From Ωo=0 we know thatQp <Qo + Q p =

θ −θ
S

is:

■

( Boθ S − c − Po ) N. Hence, Qo > 0 is equivalent to
Bo S 2 ( N + 1)

( Boθ S − c − Po ) N ~~≡ Q p . In order to ensure Q p < Q p , we need that − Ω p + Ω o Q p =Q~ p > 0 (see2
Bo S ( N + 1)

proof

of

part

i).

Straightforward()

algebra

gives

us

~
 ~
( Pp − Po ) ( B p − Bo )(c + Po ) Qp~~−(−Ω + Ω )(Q p ) = (αS + β )  D(Q p ) +D' (Q p ) + , so that a(αS + β )NBo (αS + β ) 

p

o

(

sufficient condition for − Ωp)+ Ω o Q~ p > 0 is

( B p − Bo )(c + Po ) ( Pp − Po )~~D(Q p ) >−. And since ∂Q p / ∂θ > 0 , the condition is always fulfilled for(αS + β )Bo (αS + β )θ large enough.

■Part (i) says that the peak period is used if the airport price differential between peak and off-peak is not too large.
Specifically, the per-passenger airport price differential has to be smaller than the incremental benefit, for the highest
consumer type, of changing from the off-peak to the peak. Clearly, when the airport does not practice PLP, the peak isalways used. Part (ii) says that if θ is low enough, then some consumers will not fly. In particular, the lowest
consumer type must have a willingness to pay for off-peak travel that is smaller than the airlines’ per-passenger
marginal cost for an off-peak flight. Finally, part (iii) implies that Brueckner (2002, 2005)’s single crossing property,
which imposes that B p (θ ) < Bo (θ ) for small θ values, may not be needed to have a non-empty off-peak, and that asmaller airport price differential between peak and off-peak increases the likelihood of the off-peak been used. The
lower bound for θ cannot be made explicit because of the non-linearity of the delay function. For a linear delay
functionD (Q p , K ) = δQ p / K

,

the

lower

bound

on

θ

is

given

by

SK ( N + 1)(( B p − Bo )(c + Po ) − Bo ( Pp − Po )) + c + Po , while a lower bound not depending on N,
δBo (αS + β ) N~
would be 2θ N /( N + 1) .
~

θ =

31

•

Proof of Proposition 2(i) Differentiating (12) with respect to N we get:

∂Q p
∂N

Q p S 2 ( B p − Bo )
=

+

∂Q p
∂N

=−

∂ (−Ω p + Ω o ) / ∂N
. This leads to∂ (−Ω p + Ω o ) / ∂Q p

(αS + β )Q p DQ (Q p )N2
N2, which can be written as
Qp
S 2 ( N + 1)( B p − Bo )
 N +1+ (αS + β )
D' ' (Q p ) D' (Q p ) +
N
N
N







S 2 ( B p − Bo ) + (αS + β ) DQ (Q p )=(αS + β )∂N
N ( N + 1)  2
Q p D ' ' (Q p )  S ( B p − Bo ) + (αS + β ) D' (Q p ) +
N +1



∂Q p

Qp

,

0 < ∂Q p / ∂N < Q p / ( N ( N + 1) ) follows.
(ii) From Ωo=0 in (10) we know that

∂ (Qo + Q p )
∂N

=

from

where

■

Qo + Q p =( Boθ S − c − Po ) N, from where it is direct that
Bo S 2 ( N + 1)

( Boθ S − c − Po ) Qo + Q p=>0
N ( N + 1)
Bo S 2 ( N + 1) 2■(iii) From parts (i) and (ii) we know that ∂Qo / ∂N > Qo / ( N ( N + 1) ) . If the off-peak is used for all N, thenQo > 0 and therefore ∂Qo / ∂N > 0 .
•■Proof of Proposition 5

To prove parts (i) and (ii), it is useful to first state the following Lemma:
Lemma A.2: If two prices P1 and P2 are given by the fixed points P1 = f (Q ( P1 )) and P2 = g (Q ( P2 ))respectively, where f is continuously differentiable in(Q ( P1 ); Q ( P2 ) ) , Q is continuously differentiable in( P1 ; P2 ) , f (Q ( P2 )) > g (Q ( P2 )) , and either Q (⋅) is non-increasing and f (⋅) is non-decreasing, or Q(⋅) isnon-decreasing andf (⋅) is non-increasing, then P1 > P2 .
~Proof: We prove this by contradiction. Suppose that P1 ≤ P2 . Denote P2 = f (Q ( P2 )) . Applying the mean-value~

theorem to P1 = f (Q ( P1 )) and P2 = f (Q ( P2 )) yields:~
P1 − P2 = f ' (Q )(Q( P1 ) − Q( P2 ))32

where Q is some point between Q ( P1 ) and Q ( P2 ) . Further applying the mean-value theorem to Q ( P1 ) andQ ( P2 ) , the above equation becomes:~
P1 − P2 = f ' (Q )(Q( P1 ) − Q( P2 )) = f ' (Q )Q' ( P )( P1 − P2 ) ≥ 0~

where the inequality arises because f ' (Q )Q ' ( P ) ≤ 0 and the assumption that P1 ≤ P2 . Thus, P1 ≥ P2 . But~

since by assumption f (Q ( P2 )) > g (Q ( P2 )) or, equivalently, P2 > P2 , we obtain P1 > P2 , thus resulting in■a contradiction.
Now, we can prove Proposition 5.
π(i) That Po < PoW

πfollows from writing the pricing rules (30) and (35) as Po = f o (Qo + Q p ) and

PoW = g o (Qo + Q p ) . Since total traffic Q0 + Q p is, by (15), downward sloping in P0, f o (⋅) is increasingπ

and f o (⋅) > g o (⋅) , then Po < Po by Lemma A.2.
W

Next, taking derivative of (34) and (29) with respect to N gives us:

 Qo + Q p ∂ (Qo + Q p ) 1 dPoW= − S 2 Bo  −
+
 and
∂N
N
dN
N2

 Qo + Q p ∂ (Qo + Q p )  N + 1 dPoπ
= S 2 Bo −
+

 .
∂N
dN
N2
 N 

But, from Proposition 2.2 we know that ∂ (Qo + Q p ) / ∂N = (Qo + Q p ) / ( N ( N + 1) ) . Therefore

dPoW S Bo (Qo + Q p )> 0 and dPoπ / dN = 0 .
=dN
N ( N + 1)
2

π

■
π

(ii) That ∆Pp − o < ∆Pp − o follows from writing the pricing rules (31) and (36) as ∆Pp − o = f p − o (Q p ) andW∆PpW− o = g p − o (Q p ) . Since peak traffic Q p is, by (16), downward sloping in ∆Pp −o , f p −o (⋅) is nondecreasing as long as the (unsigned) term D ' ' ' (⋅) is non-negative (or if is negative, its magnitude is not too
πlarge) and f p − o (⋅) > g g − o (⋅) , then ∆Pp − o < ∆Pp − o by Lemma A.2.
W

d∆PpW− odN> 0 follows from differentiation of (31) with respect to N. We get:33

d∆PpW− odN Q p D' (Q p ) ∂Q p ( N − 1)
( N − 1) D' (Q p ) + Q p D' ' (Q p )= (αS + β ) 
+

2
N
N 
∂N
N

 Q p ∂Q p 1 + S 2 ( B p − Bo )  2 −
∂N N N∂Q p / ∂N > 0 by Proposition 2.1, the first term in the RHS is positive. Since

Since∂Q p / ∂N < Q p / ( N ( N + 1) ) by Proposition 2.1,

Qp

∂Q p 1

N

∂N N

>
2

making the second term in the RHS

positive as well. Therefore, d∆Pp − o / dN > 0 .
Wd∆Ppπ− odN= 0 if the delay function is linear, follows from differentiating (36) and then imposing

D' ' (Q p ) = 0 , D' ' ' (Q p ) = 0 . We get:d∆Ppπ− odN Q p ∂Q p ( N + 1) = (αS + β ) D' (Q p ) + ( B p − Bo ) S 2 − 2 + , where D' (Q p ) is a constant.
N 
∂N
 N

(

)

With a linear delay function Proposition (2.1) changes to ∂Q p / ∂N = Q p / ( N ( N + 1) ) , makingd∆Ppπ−o / dN = 0 . If we consider D' ' (Q p ) > 0 , the sign is then undetermined and depends on the values of,for example, Bp, Bo and D ' ' ' (Q p ) .
(iii) QWp■π

> Q p flows from part (ii), and the comparative statics in (16) or Proposition (1.3).

dQ πp
dN

>

dQ Wp
dN

=0

π

follows from replacing ∆Pp − o in the sub-game equilibrium equation (12); we get:(− Ω + Ω )(∆P ) =poπ

2Q πp ( B p − Bo ) S 2 ( N + 1)

p −o

N

− θ S ( B p − Bo )2
N +2 π+ (αS + β ) D(Q πp ) +Q p D' (Q πp ) + Q πp D' ' (Q πp )  = 0
N



( )

As in the proof of Proposition 2.1, use this to calculate

dQ πp
dN

=−

∂ (−Ω p + Ω o )(∆Ppπ−o ) / ∂Nπ∂ (−Ω + Ω )(∆Pp −o ) / ∂Q ppoand to prove that 0 < dQ p / dN < Q p / ( N ( N + 1) ) , whichπshows that peak traffic increases with N.

34πSimilarly,

replacing∆PpW−ointhesub-gameequilibrium

equation

(12),

we

get:

(− Ω + Ω )(∆P ) = Q ( B − B )S − θ S ( B − B )+ (αS + β )(D(Q ) − Q D' (Q ) ) = 0
p

o

W
p −o

W
p

2

p

o

p

W
p

o

W
p

W
p

Which does not depend on N, hence dQ p / dN = 0 .
Wπ(iv) Qt > Qt
W■flows from part (i) and the comparative statics in (15) or Proposition (1.1). To prove

dQtπ dQtW
dQoπ dQoW>= 0 , use the sub-game equilibrium equation (10) to prove>= 0 in an analogous
dN
dN
dN
dN
way as in part (iii). The result then follows directly from this and part (iii).
(v) Direct from part (iii) and D ' (Q p ) > 0 (equation 4).

■(vi) Consider the SW function in (29). We can rewrite it in terms of total traffic, Qt, and peak-traffic, by replacing

Qo = Qt − Q p . This gives us:
SW (Qt , Q p ) = θ S ( B p Q p + Bo (Qt − Q p )) − c(Qt ) − C (Qt ) − Kr
S2−
( Bo (Qt − Q p ) 2 + 2 Bo (Qt − Q p )Q p + B p Q p2 ) − (αS + β )Q p D(Q p )2
Now, SW is globally concave in (Qt , Q p ) because∂ 2 SW∂Qt

2= − Bo S < 0 ,2∂ 2 SW∂Q p

2= −( B p − Bo ) S 2 − (αS + β )(2 D' (Q p ) + Q p D' ' (Q p ) ) < 0 and2∂ 2 SW ∂ 2 SW  ∂ 2 SW ⋅−= Bo S 2 ( B p − Bo ) S 2 + (αS + β )(2 D' (Q p ) + Q p D' ' (Q p ) ) > 0 .
2
2


∂
Q
∂
Q
∂Qt
∂Q p
 t p

[]ππ

Since (Qt , Q p ) maximizes SW, and from parts (iii) and (iv), (Qt , Q p ) > (Qt , Q p ) , then SW
W

Finally,

W

W

since

(Qtπ , Q πp )increases

with

N,

SW W does not change with N because (QtW , Q Wp ) does not.

35

W

SW π

increases

with
■

W

> SW π .

N,

while