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-07-003 and ITPU-DP-E-07-003

Congestible Facility Rivalry
In Vertical Structures

Leonardo J.Basso and Anming Zhang
April 2007

GraSPP Discussion Paper E-07-003

ITPU Discussion Paper E-07-003

GRADUATE SCHOOL OF PUBLIC POLICY
THE UNIVERSITY OF TOKYOHONGO, BUNKYO - KU, JAPANGraSPP-DP-E-07-003

Congestible Facility Rivalry in Vertical Structures
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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1. INTRODUCTION
This paper investigates the rivalry between two congestible facilities –such as airports andseaports– and its effect on facility charges, capacities and congestion delays. A number ofauthors have studied duopolistic interactions between congestible facilities: Braid (1986) andVan Dender (2005) examined competition between fixed-capacity facilities, whereas De Palmaand Leruth (1989), Baake and Mitusch (2004) and De Borger and Van Dender (2006) examined
the rivalry between facilities that are able to adjust capacities. All of these studies haveconsidered the facilities as service providers to final consumers. In particular, De Borger and
Van Dender (2006), hereafter DBVD, studied duopolistic interaction between congestible
facilities that first decide on capacities and then on prices. They found, among other results, that(i) the duopolists offer lower prices but longer congestion delays –i.e., lower service quality–
than the monopolist; (ii) conditional on facility charges, the monopolist has the same rules forcapacity investment as a central planner who maximizes social welfare; and (iii) the monopolist
offers the same service quality as in the social optimum.
DBVD indicated that their analysis may apply to seaports, airports, internet access providers androads. Whilst roads and internet access providers may provide services directly to final
consumers, seaports and airports are input providers that reach final consumers only throughcarriers: these facilities are in an intermediate market and not in the final market. In this paper
we extend the existing literature, especially the analysis of DBVD, by considering a ‘verticalstructure’ setting: Each facility is an upstream firm that provides input service to downstream
firms (‘carriers’ hereafter), which in turn produce output for final consumers. 1 We shall allowthat these carriers may possess market power in the output market: as argued by Brueckner(2002), Pels and Verhoef (2004) and others, airlines at congested airports usually are notatomistic and hence they are not price-takers.
We find that (i) the duopoly facilities have lower prices than the monopolist (as in DBVD), butthey offer lower service quality only if the facilities first decide on capacities and then on prices.
When the capacity and price decisions are made simultaneously, the duopolists will provide thesame service quality as the monopolist. (ii) Conditional on facility charges, the monopolist will
have the same capacity investment rules as the central planner if and only if the downstream
carrier markets are perfectly competitive at both facilities. If there is (at least) one downstream
market that is imperfectly competitive, the monopoly capacity rules will be different from thesocially optimal capacity rules. (iii) At a facility, the monopolist will offer the same service
quality as the central planner if the carrier market (at that facility) is perfectly competitive.
Otherwise, the monopolist provides a higher service level than the central planner.
Importantly, since we have explicitly considered the carriers’ market, this allows us to see how
the equilibrium prices change with characteristics of this intermediate market. We find that forgiven capacities, (a) the duopolists’ equilibrium prices increase with both the consumers’ valueof time and the carriers’ cost sensitivity to delays; (b) entrance of a new carrier to any of the
1

As to be elaborated in the text, two other major departures from DBVD are: First, while DBVD considered thatfacilities supply perfect substitutes in the eyes of final consumers, we shall consider that competing facilities provide
differentiated services, with homogeneous facilities being a special case. Second, while DBVD looked at a closedloop duopoly game where capacities are decided prior to prices, we investigate both the closed-loop game and theopen-loop game (in which capacities and input prices are decided simultaneously) and compare their results.

facilities depresses the prices charged by both facilities; and (c) lower marginal cost of thecarriers at one facility induces a lower price at that facility but a higher price at the other facility.
Finally, our analysis shows that when the monopolist vertically integrates with the carriers at its
facilities, it would provide the same service level as the central planner. Nevertheless, thisservice level is not socially optimal in a second-best sense; in effect, in the fully ex-ante
symmetric case, it is too low with respect to the second best.
Incorporating the ‘vertical structure’ into the analysis of airport congestion, congestion pricing
and capacity investment has been done by several recent papers. These papers examine either thecase of a single airport (e.g., Brueckner, 2002; Zhang and Zhang, 2006) or the case of noncompeting airports. For the latter, Pels and Verhoef (2004), Brueckner (2005) and Basso (2005)
considered multiple airports but these airports are complementary to each other: passengers
travel from one airport to another (and back) so the airports produce complements, notsubstitutes; moreover, only Basso (2005) looked at the case of private airports. In general, very
few papers in the airport literature have examined the case of competing airports analytically. 2
This is understandable given the local monopoly nature of an airport. The situation is changing,
however. The world has experienced a rapid growth in air transport demand since the 1970s, andmany airports have been built or expanded as a result. This has led to a number of multi-airport
regions such as greater London and the San Francisco Bay Area, within which airports maycompete for air travelers. 3 At the same time, the dramatic growth of low cost carriers (e.g.,
Southwest Airlines and Jet Blue in the United States) has enabled some smaller and peripheral
airports to cut into the catchment areas of large airports. Taken together, these two developments
have significantly increased the degree of competition between certain airports. Furthermore,
airports susceptible to competition are usually prime candidates for congestion. In the U.S., forexample, the three multi-airport markets –Chicago, New York, and Washington metropolitan
areas– contain the four airports that are officially designated by the Federal Aviation
Administration (FAA) as ‘slot controlled.’ The description also applies to several of the 23
airports identified by the FAA as ‘delay-problem airports’ –these airports are in the metropolitan
areas containing one or more other airports with airline service (e.g., Dallas, Detroit, Huston, Los
Angeles, and San Francisco). In this context, our paper intends to extend the recent airport
congestion pricing literature that incorporates the vertical airport-airline structure to an
environment of competing airports, and compare the results with those of the single-airport case.
The competing-facilities case is also highly relevant in the study of ports, as there are many
multi-seaport regions around the world.
The paper is organized as follows. Section 2 sets up the model. Section 3 examines rivalry
between the two facilities, each of which chooses its capacity and price to maximize profit.
Section 4 investigates the monopoly case and the social optimum, and compares them with the
duopoly case. Section 5 examines the case in which the monopolist vertically integrates with thecarriers at its facilities, and Section 6 contains concluding remarks.

2

One exception is Gillen and Morrison (2003), who examined two competing airports in the context of a full-service
carrier and a low cost carrier. But they did not address the issue of congestion and capacity decisions, nor airline
competition within each airport.
3
De Neufville (1995) identified 26 multi-airport regions in different parts of the world as of the early 1990s. These
multi-airport regions cover large territorial size, with some spanning over 100 kilometres, and have high passenger
generating capacity (10 million or more annual originating air passengers).

12. THE MODEL
We consider an infinite linear city, where potential consumers are distributed uniformly with adensity of one consumer per unit of length. There are two congestible facilities located at 0 and 1
with, respectively, N0 and N1 carriers offering services (Figure 1). The locations of the facilities,
the number of carriers and the facility from which they produce are exogenous. At each facility,
carriers are ex-ante symmetric and offer a homogenous good/service, which is to be referred to asa ‘product’ hereafter. We will use the term ‘fare’ to indicate the price of the final product,
reserving the terms ‘price’ and ‘charge’ for the facilities’ price. Given the homogeneity andsymmetry, the fare at a given facility will, in equilibrium, be the same for every carrier.
The vertical structure of facility-carrier behavior is represented by a multistage game: (i) the
facilities choose their capacities and prices for the input to be used by carriers; (ii) given thefacilities’ decisions, carriers compete with one another in the output market; and (iii) final
consumers decide whether to consume the product and if so, which facility to go.
We investigate the subgame perfect Nash equilibrium of this facility-rivalry game. For thispurpose, we first specify and solve the consumers’ problem. Potential consumers have unit
demands for the product, and they care for its ‘full fare.’ The full fare faced by a consumer
located at 0 ≤ z ≤ 1 , and who goes to facility 0, is given by:
f 0 + αD(Q0 , K 0 ) + (t / 4) z ,where f 0 is the (equilibrium) fare at facility 0, D is its congestion delay time –which depends on
total carriers’ production at the facility, Q0, and its capacity K0–, α is the consumers’ value oftime, and t/4 is a parameter capturing consumers’ transportation cost. 4 Thus the full fare is thesum of fare, facility congestion cost, and travel cost to the facility. If the product is consumed, theconsumer derives a net benefit (utility):
U 0 = V − f 0 − αD(Q0 , K 0 ) − (t / 4) z ,with V denoting a gross benefit. Similarly, if the consumer goes to facility 1, then she derives a
net benefit:

U 1 = V − f1 − αD(Q1 , K1 ) − (t / 4)(1 − z ) .
This is an ‘address model’ with linear transportation costs, and the differentiation of the two
facilities is captured by consumer transportation cost (i.e., positive t). Within a multi-airport
region, for example, passengers may not necessarily choose an airport with cheaper fare, but may
go to an airport that is nearer and has shorter total travel time. Indeed, the access time has beenshown empirically to be one of the main determinants of airport choice (e.g., Pels, et al., 2001;
Ishii, et al., 2005; Fournier, et al., 2006). Our ‘differentiated facilities’ formulation extends the4

The parameter t/4 is chosen for it will simplify most of the equations in the paper (see, e.g., equations (4)).

2‘homogenous facilities’ formulation in DBVD, noting that by setting t to zero we get thehomogenous case. 5
Assuming that everyone in the [0,1] interval consumes and both facilities receive consumers
from [0,1], 6 then the indifferent consumer ~z ∈ (0,1) is determined by U 0 = U 1 , or

1 f + αD(Q1 , K1 ) − f 0 − αD(Q0 , K 0 )~z= + 1
.
2
t/2

(1)

Thus, the number of [0,1] consumers going to facility 0 (rather than facility 1) increases in
f 1 + αD(Q1 , K 1 ) and decreases in f 0 + αD(Q0 , K 0 ) . Since facility 0 also captures the consumers
at its immediate left side, define z l as the last consumer on the left side of the city, who
consumes and goes to facility 0. Similarly, define z r as the last consumer on the right side of thecity, who consumes and goes to facility 1. With the uniformity and unit density of consumers, z land z r can be obtained as:zl = −

V − f 0 − αD(Q0 , K 0 ),t/4

zr = 1+

V − f1 − αD(Q1 , K 1 )
.
t/4

(2)

These points, along with ~z , also define the catchment areas of each facility as shown in Figure 1.
Hence, the consumer demands are given by Q0 = ~z + z l and Q1 = (1 − ~z ) + ( z r − 1) . Replacing~z from (1) and z l, z r from (2) then yields:(t / 4) + 2V f 1 + αD1 − 3( f 0 + αD0 )
+
t/2
t/4
(t / 4) + 2V f 0 + αD0 − 3( f 1 + αD1 )
Q1 =
+
t/2
t/2
Q0 =

(3)

where D h ≡ D(Q h , K h ) . It is clear that the consumer demands depend not only on the fares, butalso on the delays at the two facilities. Notice that in order to have both facilities receiving
consumers from [0,1], we need f1 + αD1 − f 0 − αD0 < t / 4 , whereas in order to have everyone in
the [0,1] interval consuming, we need 2V ≥ f 1 + αD1 + f 0 + αD0 + (t / 4) , both of which are ourmaintained assumptions.

5

In addition to distance, other aspects of facility differentiation may be captured by extending the present
formulation. For instance, Pels, et al. (2000, 2001, 2003) have shown, using a hypothetical example and later the San
Francisco Bay Area case study, that ground accessibility of an airport is the most important factor in affecting airport
choices in a multi-airport market. We could further address the differential ground access costs by introducing aparameter to the net-benefit function such that U 1 = V − f 1 − αD (Q1 , K 1 ) − tλ1 (1 − z ) / 4 , where λ1 > 1 ( 0 < λ1 < 1 ,respectively) if facility 1 has a higher (lower, respectively) access cost for consumers than facility 0.
6
For the conditions for both assumptions to hold, see the analysis below.

3In the output market we assume Cournot behavior in modeling carrier competition.7 Inverting thedemand system (3) in ( f 0 , f 1 ) , we obtain the inverse demand functions that carriers at eachfacility face:

f 0 (Q0 , Q1 , K 0 ) = 2t + V − 3tQ0 − tQ1 − αD0 (Q0 , K 0 )f1 (Q0 , Q1 , K 1 ) = 2t + V − 3tQ1 − tQ0 − αD1 (Q1 , K1 )(4)

Thus, in the output market, although any given carrier faces direct competition from the othercarries at the same facility, it would also take into account what happens at the other facility: thedemands depend on both Q0 and Q1. From (4) it may also seem that carries would care about thecongestion only at their own facility, but this is not the case. Recall that in the direct demandsystem (3), the demands depend on the delays at both facilities.
Since we consider ex-ante symmetric carriers at each facility, the cost function of carrier i at
facility h is given by:
C ih (Q hi , Q −h i ) = (c h + Ph + β h D (Q h , K h ) )Q hi , h = 0,1(5)

where ch is the (constant) marginal operating cost, Ph is the facility charge (an input price), and βhis the (positive) delay cost parameter for carriers at facility h. Thus, congestion at a facility
affects not only its final consumers as discussed above, but its carriers as well. Further, the costfunction C ih depends not only on its own output level Qhi , but also on the output of other carriers
at the facility, Q −h i , through the congestion term and Qh = ∑i Qhi . It does not depend on the
output of carriers at the other facility, however.
Having specified demand and cost functions, we now turn to the delay function. We shall use thesame delay function as the one in De Borger and Van Dender (2006):
D (Q, K ) = a (Q / K ) ,(6)

where a is a positive parameter. Use of this linear delay function may nevertheless lead to the
problem of the first-order condition approach prescribing a solution in which capacity is
exceeded, something that does not happen when delay functions are convex enough (e.g., when−1
D (Q, K ) = Q[K ( K − Q )] , delays approach infinity when output approaches capacity). There are
two ways around this problem: (i) we can assume an interior solution and later find conditions forthis to be true; or (ii) we can impose a priori a capacity-rationing rule for the case in which
capacity is reached. In this paper we shall take the first approach.
With these specifications, the profit for carrier i at facility 0 is:

7

Earlier studies that have incorporated imperfect competition of carriers at a congestible airport (e.g., Brueckner,
2002, 2005; Pels and Verhoef, 2004; Basso, 2005; Zhang and Zhang, 2006; Basso and Zhang, 2006) have assumed
Cournot behavior. Brander and Zhang (1990, 1993), for example, find some empirical evidence that rivalry between
duopoly airlines is consistent with Cournot behaviour.

4

φ i 0 (Q0i , Q 0− i , Q1 , K 0 ) = f 0 (Q0 , Q1 , K 0 )Q0i − (c 0 + P0 + β 0 D (Q0 , K 0 ) )Q0i ,i=1,…,N0(7)

and the carrier profits at facility 1 can be similarly written. As can be seen from (7), these profit
functions depend on the outputs of carriers at both facilities. 8 The Cournot equilibrium ischaracterized by first-order conditions,∂φ ih / ∂Qhi = 0 ,i=1,…,Nh,h=0,1.

(8)

Solving (8) we can obtain the derived demands for the two facilities. Specifically, the derived
demand for facility 0 is:

Q0 (P, K ) =t (c1 + P1 − 2t − V ) − g1 (c0 + P0 − 2t − V )g 0 g1 − t 2(9)where P ≡ ( P0 , P1 ) , K ≡ ( K 0 , K 1 ) , andg0 =N 0 +1 ⎛α + β0 ⎞⎜⎜ 3t + a⎟.
N0 ⎝K 0 ⎟⎠(10)

The expressions for facility 1 are analogous. 9 Thus, the demands faced by the facilities depend
directly on facility prices and capacities: they are linear in P0, P1, but non-linear in K0, K1. Noticethat g0 consists of two parts: the first part is related to transportation cost t (which leads to the two
facilities being differentiated), and the second part is related to K0. It depends on the carrier
market structure: g0 is largest with a monopoly carrier ( N 0 = 1 ) and smallest with atomistic
carriers ( N 0 → ∞ ). It can be easily shown that g h > t and so the denominator of (9) is positive.
Furthermore, g h increases in α and β h , and decreases in K h and N h .
Taking the perspective of facility 0, we can characterize the facility demands through thefollowing comparative statics:∂Q0 ∂Q0
g1
=
=−
<0
∂P0
∂c0
g 0 g1 − t 2(11)∂Q0 ∂Q0
t
=
=
>0
∂P1
∂c1
g 0 g1 − t 2

(12)

8

Since each consumer in the model consumes one unit of the carriers’ products, this would imply, in the airportairlines case, one flight per person. The simplest way to obtain the real case of many passengers per flight would bethrough a ‘fixed proportions’ assumption: let S be the number of consumers in a flight and then assume S is constantand the same across the airlines. The only change in our results would be that a parameter S would be included. The
fixed-proportions condition has been assumed in Brueckner (2002, 2005), Pels and Verhoef (2004), Basso (2005),
Zhang and Zhang (2006), and Basso and Zhang (2006).
9
To save notations, in what follows various expressions may be written for facility 0 only, rather than for bothfacilities 0 and 1. If this is the case, the corresponding expressions for facility 1 will be analogous.

5∂Q0
Q g (∂g 0 / ∂η )=− 0 1>0,∂η
g 0 g1 − t 2for η ∈ {K 0 , N 0 − β 0 }(13)∂Q0 Q1 t (∂g 1 / ∂η )
=
< 0,
∂η
g 0 g1 − t 2for η ∈ {K1 , N1 ,− β1 }(14)

∂Q0 Q1 t (∂g1 / ∂α ) − Q0 g1 (∂g 0 / ∂α )=.
∂αg 0 g1 − t 2(15)

All the signs in (11)-(14) are as expected: e.g., inequality (11) is equivalent to the demand
functions being downward sloping, whereas (12) shows that the facilities are ‘gross’ substitutes.
The effects of carriers’ marginal costs on the facility demands are the same as those of prices;
after all, for the carriers, the facility charge is part of its marginal cost. Further, (13) and (14)indicate that the demand for a facility increases in own capacity, but decreases in the rival’s
capacity. Moreover, a facility’s demand rises the greater the number of carriers it has, and theless its carriers care about congestion (i.e., the lower the value β h is). Its demand also rises thefewer carriers there are at the other facility, and the more they care about congestion. These
comparative-static results are sensible, and will be used in our subsequent analysis.

3. EQUILIBRIA OF DUOPOLY FACILITIESHaving characterized the output-market equilibrium and the facilities’ demands, we now analyze
the facility market. This section investigates rivalry between the two facilities, each of whichchooses its capacity and price to maximize own profit. Following DBVD we shall, initially,
investigate a closed-loop game in which capacities are chosen prior to prices. We later compare
this case with an open-loop game in which capacities and prices are decided simultaneously.
Assume that the facilities’ operational and capacity costs are separable and their marginal costs
are constant. Without loss of generality we further set the operational marginal costs to zero, so
the profit of facility h can be written as:

π h ( P, K) = Q h ( P, K ) Ph − m h K h ,h = 0,1(16)

where m h denotes the marginal cost of capacity.
3.1 Closed-loop Duopoly
Pricing Stage
The closed-loop game is solved by backward induction, that is, price rivalry is analyzed first.
Specifically, given capacities K ≡ ( K 0 , K 1 ) , the facilities simultaneously choose their prices Ph

6

to maximize profit (16). After taking the carriers’ competition into account (e.g., equation (11)),the first-order conditions lead to the following pricing rules:
P0 = Q0 g 0 − (Q0 t 2 / g 1 ) ,P1 = Q1 g 1 − (Q1t 2 / g 0 ) .

(17)

Using (6) and (10) these pricing rules can be further written as, for facility 0:
P0 = (α + β 0 ) D0 (1 + s 0 ) + 3tQ0 (1 + s 0 ) − (Q0 t 2 / g 1 )(18)

where s 0 = 1 / N 0 is a carrier’s market share at facility 0. The first term on the RHS of equation(18) is a congestion toll. But here, the duopoly facilities charge more than just the pure uninternalized congestion of each carrier (which would have been factored by 1 – sh, rather than 1 +sh): this is caused by the failure of coordination in the vertical structures. 10 The second term in(18) is a mark-up from the exploitation of market power that arises from the locational
preferences of consumers and travelling cost (positive t). The third term is a mark-down, owing
to facility competition: As the other facility becomes more attractive –i.e., as K1 rises and hence
g 1 falls– the mark-down for facility 0 increases, reducing its price. 11
Each pricing rule in (18) implicitly defines a ‘best reply’ function and the pricing equilibrium,
denoted P * = ( P0* , P1* ) , is at the intersection of the two best-reply functions. To see how P * areaffected by both the capacities and features of the consumer demand and the carrier market, wefirst examine whether P * are unique and stable. From (11)-(12) and g h > t , we have∂Q0 ∂Q0
g1 − t
+
=−
< 0,
∂P0
∂P1
g 0 g1 − t 2that is, own-price effects on demand dominate cross-price effects. This condition is equivalent tothe stability condition for P * and, together with downward-sloping demands (11), further implies
the uniqueness of P * (see, e.g., Dixit, 1986). The uniqueness and stability of P * then allows us
to conduct the comparative statics; the results are reported in Proposition 1.
Proposition 1: In a closed-loop duopoly,(i) ∂P0* / ∂K 0 < 0 , ∂P0* / ∂K 1 < 0 , i.e., higher capacities imply smaller equilibrium prices;(ii) For given capacities, (a) the (equilibrium) prices increase with the time costs –either
consumers’ cost (α) or carriers’ cost ( β 0 , β1 ); (b) entrance of a new carrier to any of the
facilities depresses the prices charged by both facilities; and (c) lower marginal cost of thecarriers at one facility induces a lower price at that facility but a higher price at the otherfacility.
10

As is to be seen in Sections 4 and 5, this coordination failure is not resolved even if both facilities are owned by asingle firm (monopolist), but would be resolved by a vertical integration of facilities and carriers.
11This expression has a different flavor than the one obtained in a non-vertical setting –e.g., De Borger and Van
Dender (2006, equation (9))– even if we assume away facility differentiation. In the non-vertical case, the third part
would be positive, although a less attractive facility 1 would reduce the mark-up.

7

Proof. See the Appendix .
One implication of Proposition 1 (i) is that the more congestible the system is, the higher theequilibrium prices are. This result was also found in the non-vertical setting of DBVD, which is
not surprising because our derived demands for the facilities react to changes in prices andcapacities in the same fashion as the demands they assumed in their final market. Anotherinteresting result is that ∂P0* / ∂α > 0 –part (a) of (ii). From (15), higher time value α will, in a
fully symmetric case with respect to both facilities and carriers, reduce the demands for bothfacilities. However, the demand for a facility may increase with α in asymmetric cases. 12 Despitethat the demand for a facility may increase or decrease in α, the equilibrium prices will always
increase in α. Just as a more congestible system leads to higher facility prices, a higher consumer
time cost also induces higher facility prices.
But perhaps more interesting results from Proposition 1 pertain to the price effects of changes incharacteristics of the intermediate market such as variables βh, Nh or ch, as this is our main
departure from the literature. In particular, Proposition 1 suggests that for given capacities, alower marginal cost of the carriers at a facility would induce a lower price at that facility, but ahigher price at the other facility. For example, if we start from a situation in which airlines havethe same marginal costs c0 = c1 and we replace the airlines of one facility by lower marginal-costcarriers, then the airport charge would fall at the airport with low-cost carriers, while the charge
at the other airport would rise. This might serve as a testable implication for empirical studies.
Note, here, that the number of airlines at each airport does not need to be the same.
Capacity StageIn the closed-loop game, each facility chooses its capacity taking price equilibrium P * (K ) into
account. The reduced-form profit of each facility is:()Π h (K ) = π h P * (K ), K ,h = 0,1(19)

where π h (P, K ) is given by (16). The capacity equilibrium is characterized by first-orderconditions,Π 00 ≡∂Π 0 (K ) ∂π 0 ∂P0* ∂π 0 ∂P1* ∂π 0 ∂π 0 ∂P1* ∂π 0
=
+
+
=
+
=0
∂K 0
∂P0 ∂K 0 ∂P1 ∂K 0 ∂K 0
∂P1 ∂K 0 ∂K 0

(20)

(where the second equality follows from the envelope theorem) and Π 11 = 0 . We assume thesecond-order conditions Π hhh < 0 hold for the entire range of interest.

12

For example, if everything is symmetric except for carriers’ marginal costs, and (c0 – c1) is large enough, thenhigher c0 implies that carriers at facility 0 would have, ceteris paribus, higher fares and hence the facility will have asmaller demand but also less congestion. A marginal increase in α would induce a shift by consumers towards theless congested facility, thereby increasing its demand.

8

Notice that equation (20) can be rewritten as:∂Q0 ∂P1*∂Q0
P+ [ P0*− m0 ] = 0 .
{ ∂P ∂K
∂K 0
1 {
0
+ {
*
0

+

(21)

−

The bracketed term in (21) contains the direct effect of capacity: At cost m0 , a marginal increasein capacity will enhance own demand –recall ∂Q0 / ∂K 0 > 0 by (13)– and hence own revenue.
The indirect effect –the first term in (21)– indicates that a marginal increase in own capacity willlead to a reduction in the rival facility’s price (recall Proposition 1) which in turn will, by (12),reduce own demand. As indicated, this ‘strategic’ effect is negative to the facility’s profit.
3.2 Open-loop Duopoly and Comparison with Closed-loop Duopoly

In an open-loop game, the problem faced by facility h is:max π h (P, K)
Ph , K hwhere π h is given by (16). The corresponding first-order conditions will give rise to the pricing
and capacity rules. The pricing rules remain the same as those given in (17) or (18), whereas thecapacity rules can be derived, using (13), as:⎛ a (α + β h )( N h + 1) ⎞⎟⎟K = ⎜⎜
mh N h
⎠
⎝
o
h

1/ 2

Qho ,

h = 0,1(22)

where superscript o stands for the open-loop game. (22) implies that in equilibrium, the delay
time at facility h equals:⎞
Qho ⎛a mh N h⎟
D (Q , K ) = a o = ⎜⎜K h ⎝ (α + β h )( N h + 1) ⎟⎠
o
h

o
h

1/ 2

,

h = 0,1.

(23)

At the open-loop equilibrium, therefore, congestion delay at facility h increases in N h and m h ,but decreases in α (consumers’ time cost) and β h (facility h carriers’ time cost). Equations (23)also define a sufficient condition for an ‘interior solution:’ that Q h ≤ K h if and only if
m h ≤ a(α + β h ) ,h = 0,1.

(24)

Hence, if the capacity costs are low enough, or if the time costs are high enough, the open-loop
game will have an interior solution.
Next, compare the results between the closed- and open-loop games. We first show that thefacilities invest less in capacity in the closed-loop game than in the open-loop game. Recall that9the closed-loop capacity rule is given by (20), that is, Π 00

K c0

= 0 , where superscript c stands forthe closed-loop game. Evaluating Π 00 at the open-loop capacity yields:
Π 00

=(
o

K0

∂Q0 ∂P1*∂π 0 ∂P1* ∂π 0
∂π 0 ∂P1*+) =() = P0*< 0.
∂P1 ∂K 0 ∂K 0 K o∂P1 ∂K 0 K o∂P1 ∂K 0
0

(25)

0

In (25), the second equality follows from the capacity first-order condition in the open-loop game,whereas the inequality has already been indicated in (21). Then K 0c < K 0o and, similarly,
K1c < K1o follow by the concavity of the profit functions.
The intuition behind the above result is clear. In the closed-loop game, according to thenomenclature of Fudenberg and Tirole (1984), the facilities invest less in capacity following‘puppy dog’ strategies: Investment in capacity would make a facility tough, in that it decreases
the facility’s price hurting the rival (recall that prices are higher in more congestible systems).
But that would trigger a harsh pricing reaction from the rival facility, since the prices are strategic
complements. Hence, the facilities will try to soften the price competition by committing tosmaller capacities in the first stage: they want to look small and inoffensive. This also directly
leads to higher prices.
However, the fact that capacities are smaller when they are chosen prior to prices does notdirectly imply that delays will also be longer. This is because, on one hand, capacity levels
directly affect demands and, on the other hand, we now have higher prices. Yet, it can be shownthat the delays do increase. From (25) and (21) we have, at the closed-loop equilibrium,
( P0*

∂Q0
− m0 ) > 0 .
∂K 0
c

(26)

Since equation (17) must hold at this equilibrium, we can replace P0* in (26) with
Q0 g 0 − (Q0 t 2 / g 1 ) . Further replacing ∂Q0 / ∂K 0 with (13), calculating ∂g 0 / ∂K 0 with (10), andrearranging, (26) becomes:⎞
Qc ⎛a m0 N 0⎟
a 0c > ⎜⎜K 0 ⎝ (α + β 0 )( N 0 + 1) ⎟⎠1/ 2⇒ D (Q0c , K 0c ) > D (Q0o , K 0o )(27)

where (23) is used in the second part of (27). The above comparisons thus lead to:
Proposition 2: A closed-loop duopoly invests less in capacity, charges higher facility prices, andhas longer congestion delays than an open-loop duopoly.

Thus, in terms of facility price and service quality, the open-loop duopoly dominates the closedloop duopoly as it has both lower facility prices and shorter delays.

104. MONOPOLY AND THE SOCIAL OPTIMUM
Having examined the duopoly case, we shall in this section investigate the monopoly case –in
which a monopolist owns both facilities– and the social optimum, emphasizing comparisonsamong the three cases. Note that in both the monopoly case and the social optimum, the results
remain the same whether the capacity and price decisions are made simultaneously orsequentially.
4.1 Monopoly

The monopolist’s problem is:
max π 0 (P, K ) + π 1 (P, K ) = max ∑ (Qh (P, K ) Ph − m h K h ) .
P ,K

P ,K

h

Taking the first-order conditions for prices, and using (6), (10) and (11), we obtain (superscript M
stands for monopoly):
P0M = (α + β 0 ) D0 (1 + s 0 ) + 3tQ0 (1 + s 0 ) + Q1t .

(28)

and the expression for facility 1 is analogous. These monopoly pricing rules can be compared tothe duopoly pricing rules (18). The first two terms on the RHS of (28) are the same as those in(18), although they are evaluated at different prices (and capacities). The first term is related tothe congestion toll, but the monopoly facilities, like the duopolists, charge more than just the pure
un-internalized congestion of each carrier (which would have been 1 – s0, rather than 1 + s0). The
second term is the mark-up from the exploitation of market power, which arises from theconsumers’ locational preferences and their traveling cost. As for the third term in (28), contrary
to what happens with the duopoly, the monopoly has a mark-up –rather than a mark-down as in(18)– owing to the absence of facility competition. Here, when raising the price for one facility,
the monopolist takes into consideration that it is actually increasing the demand for the otherfacility, with the resulting profit accruing to itself. In other words, the monopolist has internalized
the interrelation of demands, and hence the facility competition.
From (28) and (18) it is not immediate, however, to conclude that monopoly facility charges arehigher than duopoly charges. The reasons are two-folds. First, both (28) and (18) are actually asystem of fixed points, since Q0 and Q1 depend on both P0 and P1. Second, perhaps more
fundamentally, capacities will likely differ in the two cases; prices and capacities are decided
simultaneously. Prices can therefore be compared in two ways (see Spence, 1975; Basso, 2005):(i) compare prices as if capacities were fixed; and (ii) compare actual prices, taking the capacity
difference (if any) into consideration. The first one is useful because it may represent a shortterm case, but it is also useful in performing the second comparison. In what follows we will,when feasible, perform both comparisons.
Prior to the price comparisons, we first look at the capacity rules under monopoly. Taking thefirst-order conditions for capacities, and using (10), (13) and (28), we get:

11

K

M
h

⎛ a (α + β h )( N h + 1) ⎞
⎟⎟
= ⎜⎜
m
N
h
h
⎝
⎠

1/ 2

QhM ,

h = 0,1.

(29)

The monopoly capacity rule (29) is identical to the open-loop duopoly capacity rule (22).
Obviously, since their pricing rules are different, the consumption levels, and hence actual
capacities, will be different in the two cases. Delay times will be equal however, given thelinearity assumption of the delay function. From (29) we have:⎞QM ⎛a m0 N 0⎟⎟
D (Q , K ) = a 0M = ⎜⎜K0
⎝ (α + β 0 )( N 0 + 1) ⎠
M
0

M
0

1/ 2

= D (Q0o , K 0o ) ,(30)

where the last equality follows from (23). Note that the sufficient condition for an interior
solution in the monopoly case remains the same as (24). De Borger and Van Dender (2006)
found that the duopolists offer lower service quality, in terms of longer delays, than themonopolist. Here we find that this is the case only if capacity decisions are made prior to pricedecisions (which is the situation analyzed in DBVD). When the capacity and price decisions are
made simultaneously, or when capacity investments are not observable prior to price decisions,the duopolists will provide the same service quality as the monopolist.
Next, compare the monopoly and duopoly prices for given capacities. Obtaining monopoly prices
for given capacities involves solving system (28), which leads to:
P0M (K ) = (2t + V − c 0 ) / 2 ,P1M (K ) = (2t + V − c1 ) / 2 .

(31)

Thus, given the capacities, the monopoly prices are, somewhat surprisingly, actually independentof the capacities! This means that the monopolist would charge prices (31) independently ofwhether it can choose capacities or not (provided, of course, that it leads to an interior solution).
A facility’s price decreases with the marginal cost of carriers at that facility, but is independent of
the marginal cost of carriers at the other facility. This is in contrast to the duopoly case, where afall in carriers’ marginal cost at one facility induces a price increase at the other facility. It isnoted that this distinction between the duopoly and monopoly pricing might serve as anempirically testable hypothesis.
The result that the monopoly pricing rules do not depend on capacities also allows us to showthat the monopoly prices are indeed higher than the actual duopoly prices. The monopolyduopoly comparisons are reported in Proposition 3:
Proposition 3: For facility h (h = 0,1),(i) Pho < Phc < PhM , i.e., an open-loop duopoly has lower facility prices than a closed-loop
duopoly, which in turn has lower facility prices than a monopoly; and(ii) Dho = DhM < Dhc , i.e., duopoly facilities offer lower service quality, in terms of longer
delays, than the monopolist only if capacity decisions are made prior to price decisions. If
the capacity and price decisions are made simultaneously, the duopolists offer the same
service level as the monopolist.

12

Proof. See the Appendix.
Thus, in terms of facility price and service quality, the open-loop duopoly dominates monopoly
as it has both lower prices and shorter delays. It is not clear, however, whether the closed-loop
duopoly is superior to monopoly: while having lower prices, it has longer delays than monopoly.
4.2 The Social Optimum

The social optimum arises when a central planner chooses facility prices P and capacities K to
maximize social welfare. Since our setting extends DBVD’s by introducing the downstream
carriers, we now have the surplus of three types of agents –namely, facilities, carriers, and final
consumers– to consider, rather than just two types of agents (facilities and final consumers as inDBVD). 13 This gives rise to the following social-welfare function:
SW (P, K ) = CS + (Φ 0 + Φ 1 ) + (π 0 + π 1 )(32)

where CS is consumer surplus, Φ h = ∑i φ ih is the aggregate (equilibrium) profit for carriers at
facility h, and π h is the (equilibrium) profit of facility h.
The first-order conditions with respect to P give rise to the social pricing rules. The derivation islong but straightforward and hence is given in the Appendix. We obtain:
P0W = (α + β 0 ) D0 (1 − s 0 ) − 3tQ0 s 0 ,P1W = (α + β 1 ) D1 (1 − s1 ) − 3tQ1 s1(33)

where superscript W stands for welfare maximization. This pricing rule for each facility isconceptually similar to the ones obtained by Brueckner (2002), Pels and Verhoef (2004), Basso(2005) and Zhang and Zhang (2006). The socially optimal price at a facility consists of acongestion term –by which the facility charges each carrier for the un-internalized congestion it
produces– and a mark-down, the market-power term, by which the facilities ‘subsidize’ carriers
so as to countervail the exploitation of market power by monopoly or oligopoly carriers andinduce the allocatively efficient output. One consequence of such subsidy is that the facilities
may not recover their costs if the carriers’ market power, and hence the market-power term, islarge, even though we have constant returns to scale in the provision of capacity and linear delay
functions. This is in contrast to DBVD, and earlier studies (e.g., Morrison, 1983; Zhang andZhang, 1997), that have shown that under the constant returns to scale and linear delay functions,the optimal pricing and optimal provision of capacity lead to exact cost recovery for a congestible
facility (e.g., airport). These studies did not consider imperfect competition in the carriers’
market. 14
13

As demonstrated by Basso (2005, 2006), this distinction is relevant and important in the derivation of welfaremaximization results. This is also seen from the analysis below.
14
The non-cost recovery result has also been obtained by Brueckner (2002), Basso (2005) and Zhang and Zhang(2006) under different model settings. All of these studies have explicitly considered imperfect competition in the
carriers’ market. The issue of budget adequacy is further discussed by Zhang and Zhang (2006) for the single-airportcase and by Basso (2005) in the context two distant airports, but the conclusions there apply to this competingfacilities case as well: two-part tariffs, or cost-recovery two-part tariffs if the carriers do not make enough profits,13

Equations (33) also show that when there is a single carrier at a facility ( s h = 1 ), the congestion
term at that facility becomes zero. This is because the monopoly carrier perfectly internalizes
congestion and consequently there is no need to correct for congestion. On the other hand, withatomistic carriers ( s h → 0 ) the market-power term at facility h will vanish. These results havealready been obtained by Brueckner (2002) and others in the context of non-competing airports.
With competing facilities, (33) further shows that absent of any facility differentiation (i.e., t=0)
the market-power terms at both facilities will vanish, irrespective of the carrier market structure
at either facility. This result is related to our Bertrand assumption of facility competition; it also
extends the analysis of DBVD who considered Bertrand rivalry between perfectly substitutablefacilities: With t=0, the market-power term disappears and so the socially-optimal charges will
equal PhW = (α + β h ) D h (1 − s h ) , i.e., the (un-internalized) congestion tolls. Finally, (33) is easily
comparable to both the monopoly pricing equation (28) and the duopoly pricing equation (18).
The first-order conditions with respect to K give rise to the following social capacity rules (thederivation is given in the Appendix):KWh⎛ a (α + β h ) ⎞
⎟⎟
= ⎜⎜
m
h
⎝
⎠

1/ 2

QhW ,

h = 0,1.

(34)

These capacity rules can be compared to the monopoly capacity rules (29), giving rise to:
Proposition 4: Conditional on facility charges, the monopoly capacity rules are the same as thesocially-optimal capacity rules if and only if N 0 , N1 → ∞ , i.e., the downstream carrier markets
are perfectly competitive at both facilities.

Thus, if there is (at least) one downstream market that is imperfectly competitive, the monopoly
capacity rules will be different from the social capacity rules. This is in contrast to what was
found by DBVD in their analysis without an intermediate carrier market; they found that themonopoly and socially optimal capacity rules are identical. Their result had a precedent in Oum,et al. (2004) who analyzed the price and capacity decisions by a single congestible airport. Since
they did not formally derive the airport’s demand from the equilibrium of the airline market, their
setting is actually quite close to DBVD’s, with the exception that DBVD had two facilities withinterdependent demands. Proposition 4 shows that, when one takes into consideration that thecongestible facilities may be upstream providers of an input, which is the case for airports,
seaports, and perhaps railroad tracks or telecommunication networks, the monopoly capacity
rules will coincide with the socially optimal rules only when carriers are atomistic. 15
Both Oum et al. and DBVD correctly pointed out that, since the pricing rules are different, theconsumption levels and, hence, actual capacities will be different. However, taking advantage ofthe assumption of a linear delay function, DBVD showed that, in their case where the facilities
may resolve the problem. If fixed fees are not feasible for some reason, the less efficient alternative of RamseyBoiteaux prices is called for.
15
See also Basso (2005) and Zhang and Zhang (2006) for a similar result in the context of non-competing airports.

14

interact directly with final consumers, the monopolist offers exactly the same service quality –i.e.,the same level of delays– as welfare-maximizing facilities. In our case, from (34) and (30) it
follows that 16⎛ a mh ⎞⎟⎟D(Q , K ) = a W = ⎜⎜Kh⎝α + βh ⎠
W
h

W
h

Q hW

1/ 2

⎛
⎞
a mh N h⎟⎟> ⎜⎜⎝ (α + β h )( N h + 1) ⎠1/ 2

= D(Q hM , K hM )(35)

and D (QhW , K hW ) = D (QhM , K hM ) only if N h → ∞ . Thus, DBVD’s result emerges as a special
case of our comparison (i.e., when carriers are atomistic). Proposition 5 summarizes the result:
Proposition 5: At a facility, the monopolist will offer the same service quality –in terms ofcongestion delays– as the central planner if the carrier market at the facility is perfectlycompetitive. Otherwise, the monopolist offers a higher service level than the central planner.

5. VERTICAL INTEGRATION
Proposition 5 shows that when the facilities are input providers to an imperfectly competitive
output market, the monopolist would no longer provide exactly the same service quality as thecentral planner. However, an important issue is: if the congestion levels were equal, does thatmean that the monopolist is providing the socially optimal level of service quality? To address
the issue, we analyze the case in which the monopolist vertically integrates with the carriers at
the facilities: As is to be seen below, in this case the monopolist will have exactly the same
service level as the central planner, just as in the case of DBVD. The vertical-integration case isalso relevant in the real world. For example, in the case of airports, it has often been argued thatstrategic collaboration between airports and airlines would solve the incentive and coordination
problems regarding capacity and pricing in the vertical structure (see, e.g., Beesley, 1999;
Forsyth, 1997; Starkie, 2001; for a more complete list see Basso, 2005).
Under vertical integration, our hyper-monopolist’s problem is:
max ∑ (Φ h + π h ) .
P ,KhThe pricing rules are given by:
P0VI = (α + β 0 ) D0 (1 − s 0 ) + [3tQ0 (1 − s 0 ) + Q1t ](36)

where superscript VI stands for vertical integration. This pricing equation is conceptually similar
to the ones obtained by Basso (2005) for the case of two distant airports that vertically integratewith airlines, and by Basso and Zhang (2006) for the case of peak-period pricing by a verticallyintegrated airport. It shows that the price consists of a congestion toll –by which the facility
16

Note that the sufficient condition for an interior solution at the social optimum remains the same as (24), that is,m h ≤ a (α + β h ) .

15charges each carrier for the un-internalized congestion it produces– and a mark-up. The
congestion toll –represented by the first term on the RHS of (36)– depends on the number ofcarriers at the facility in the same fashion as in the central-planner case of (33). By the mark-up –which is represented by the bracketed term in (36)– the facility induces the cartel level of output,
and hence maximizes the carriers’ joint profit, by raising their marginal costs through a higher
facility charge. In particular, the mark-up takes into account both the intra-facility carrier
competition –represented by 3tQ0 (1 − s 0 ) , which vanishes when there is a single carrier– and theinter-facility carrier competition ( Q1t , which vanishes when t=0).
It turns out that the hyper-monopolist’s capacity rule is the same as the central-planner rule (34).
As a result, the level of service quality (delays) is also the same:⎛ a m0 ⎞⎟⎟D (Q , K ) = a VI = ⎜⎜K0⎝α + β0 ⎠
VI
0

VI
0

Q0VI

1/ 2

= D(Q0W , K 0W ) ,(37)

where the last equality is based on (35). While the level of delays in (37) is optimal given thesocially optimal pricing, if a constrained central planner is forced to price as the hypermonopolist does, it would choose a different capacity rule and thereby induce a different –in fact,
probably superior– service level. We formally state the result in Proposition 6:
Proposition 6: When the monopolist vertically integrates with the carriers at its facilities, itwould provide the same service level as the central planner. Nevertheless, the hyper-monopoly
service level is not socially optimal in a second-best sense; in effect, in the fully ex-ante
symmetric case, it is too low with respect to the second best.

Proof. See the Appendix.
Hence, the service level provided by the hyper-monopolist is not optimal: at the very least, in the
fully symmetric case, a central planner who is forced to use the same pricing rule as the hypermonopolist would choose a higher service level. The congestion delays of the hyper-monopolist
are not second best. The intuition is simple: given that the prices will be higher, the constrained
central planner will compensate consumers and carriers by providing a higher quality service.
More generally, as has been pointed out, capacities and prices are decided jointly so they cannot
be analyzed separately. And since service quality is a result of both the level of demands induced
and the capacities chosen, it cannot be looked at on its own; just as with capacity, delays must be
analyzed together with pricing.

6. CONCLUDING REMARKSOur main objective in writing this paper is to contribute to the understanding of rivalry between
congestible facilities, but for the case in which the facilities provide an input for downstream
firms, who sell the final product to consumers. This is the case for airports, seaports, and perhaps
railroad tracks or telecommunication networks. By explicitly incorporating the behaviour ofoligopolistic downstream carriers and final consumers into the analysis of a duopolistic facility

16rivalry, we found that for given capacities, (a) the duopolists’ equilibrium prices increase withboth the consumers’ value of time and the carriers’ cost sensitivity to congestion delays; (b)entrance of a new carrier to any of the facilities depresses the prices charged by both facilities;and (c) lower marginal cost of the carriers at one facility induces a lower price at that facility but
a higher price at the other facility. As discussed in the text, some of these results might serve as
testable implications for empirical studies.
We further found that: (i) although the duopoly facilities have lower prices than the monopolist,
they offer lower service quality only if capacity decisions are made prior to the facility pricing
decisions (i.e., if the duopoly rivalry is a closed-loop game, as in DBVD). When the capacity andpricing decisions are made simultaneously (i.e., an open-loop game), the duopolists will provide
the same service quality as the monopolist. (ii) A closed-loop duopoly invests less in capacity
and charges higher facility prices than an open-loop duopoly. Here, the closed-loop duopolists
follow a ‘puppy dog’ strategy: they try to soften the price competition by committing to smaller
capacities in the first stage. This directly leads to higher prices. (iii) Conditional on facility
charges, the monopolist will have the same capacity rules as the central planner if and only if thedownstream carrier markets are perfectly competitive at both facilities. If there is (at least) one
downstream market that is imperfectly competitive, the monopoly capacity rules will be different
from the socially optimal capacity rules. (iv) At a facility, the monopolist will provide the same
service quality as the central planner if the carrier market is perfectly competitive; otherwise, themonopolist provides a higher service level than the central planner. Finally, our analysis showedthat when the monopolist vertically integrates with the carriers at its facilities, it would provide
the same congestion level as the central planner. Nevertheless, this integrated monopoly service
level is not socially optimal in a second-best sense. In effect, in the fully symmetric case, it is too
low with respect to the second best.

17

$

$

V

V

f1 + αD1

f 0 + αD 0
z

l

0

~z

1

z

r

z

Figure 1. Consumer distribution and facilities’ catchment areas

18

Appendix
Proof of Proposition 1: (i) Solving (17) for P * and using of (9) and (10), we find:g 0 t (c1 − 2t − V ) − (c 0 − 2t − V )(2 g 1 g 0 − t 2 )
*
,
(A.1)
P0 (K ; x) =4g 0 g1 − t 2
where x ≡ (α , β 0 , β1 , N 0 , N1 , c0 , c1 ) is the vector of exogenous parameters. Differentiating P0*
with respect to K 0 , K 1 yields:∂P0* t 2 (Q0 g 0 + Q1t ) ∂g 0=
< 0,
∂K 0 g 0 (4 g 0 g 1 − t 2 ) ∂K 0

∂P0* 2 g 0 t (Q1 g 1 + Q0 t ) ∂g 1
=
<0
∂K 1
g 1 (4 g 0 g 1 − t 2 ) ∂K 1

(A.2)

where the inequalities arise from g h > t and the use of (10).
(ii) From (A.1) we can show, for h=0,1,
∂P0*
∂P0*
∂P0*
> 0,
> 0,
<0
∂α
∂β h
∂N h
which give rise to parts a) and b). Part c) follows from the following inequalities:
dP0*
g 0t
dP0*
2g g − t 2=− 0 1 2 <0,=> 0.
Q.E.D.
dc0
4 g 0 g1 − t
dc1 4 g 0 g1 − t 2
Proof of Proposition 3: Part (ii) has been proved in the text and by using Proposition 2. For the
proof of part (i), it is sufficient, using Proposition 2, to prove P0c < P0M . From (A.1) and (31) we
can easily show that for given capacities, the duopoly prices are, as expected, smaller than themonopoly prices:t [t (2t + V − c 0 ) + 2 g 0 (2t + V − c1 )]P0M (K ) − P0* (K ) =
> 0.
(A.3)
2 4 g 0 g1 − t 2
Hence,(A.4)
P0c = P0* (K c ) < P0M (K c ) = P0M (K M ) = P0M ⇒ P0c < P0M()

where the first inequality follows from (A.3), and the equality P0M (K c ) = P0M (K M ) follows from(31), i.e., the monopoly pricing rules do not depend on capacities.
Q.E.D.
Derivations of Social Pricing and Capacity Rules: We first specify the welfare function. With
consumers being uniformly distributed with density one per unit of length, the consumers’
surplus is given by (see Figure 1):
zl

~
z

0

0

CS = ∫ [V − p 0 (Q0 , Q1 ) − αD0 − tz )]dz + ∫ [V − p 0 (Q0 , Q1 ) − αD0 − tz )]dz
1− ~z

z r −100+ ∫ [V − p1 (Q0 , Q1 ) − αD1 − tz )]dz + ∫ [V − p1 (Q0 , Q1 ) − αD1 − tz )]dz
Note that Q0 and Q1 –which are given by (3)– do not depend on z, whereas z l, z r and ~z dependon Q0 and Q1. Hence, we will obtain an expression dependent on Q0 and Q1. Using (4) to replacep0 and p1 both in the integrands and in z l, z r and ~z , and solving the integrals we get:

19

(

)

CS = t 3Q02 + 2Q0 Q1 + 3Q12 − 4 / 2 .

(A.5)

It might seem that CS increases in t, and it is negative if there is no consumption. However, it isimportant to recall that both Q0 and Q1 are equilibrium values, so they depend on the level ofcongestion and on the value of t. Indeed, an examination of (3) reveals that Q0 and Q1 will rise ast falls, so the overall result is that CS actually falls as t increases, as expected. Also, recall that themaintained assumption has been that V is sufficiently large so that everyone in the [0,1] interval
consumes. This implies that the minimum values of Q0 and Q1 for which the above CS expression
is valid are when both are equal to 1 (each facility gets ½ consumer from each side, left and right).
Therefore, CS is never less than 2t.
Regarding the carriers’ profit, it is straightforward, from (7), (4) and symmetry, to obtain:
Φ 0 ( P, K ) = ( 2t + V − c 0 )Q0 − Q0 P0 − 3tQ02 − tQ0 Q1 − (α + β 0 )Q0 D0 .

(A.6)

With CS, Φ h and the facilities’ profits by (16), the welfare function (32) can be written as:
SW (P, K ) = (2t + V − c0 )Q0 + (2t + V − c1 )Q1 − (t / 2)(3Q02 + 2Q0 Q1 + 3Q12 ) − 2t

(A.7)
− (α + β 0 )Q0 D0 − (α + β1 )Q1 D1 − m0 K 0 − m1 K1
Notice that SW above is not directly a function of prices; instead, it is a function of Q0, Q1 and,through them, a function of P0, P1.

The first-order condition with respect to P0 is,
dSW ∂SW ∂Q0 ∂SW ∂Q1 ∂SW
=
+
+
= 0.
dP0
∂Q0 ∂P0∂Q1 ∂P0∂P0
Calculating this –noticing ∂SW / ∂P0 = 0 and using (A.7), (A.1), (11) and (12) – we get:⎡Q
Q ⎛ N − 1 ⎞⎤Q1
Q ⎛ N − 1 ⎞⎤ ⎡⎟⎥ g 0 .
− a (α + β 1 ) 1 ⎜⎜ 1 ⎟⎟⎥t = ⎢ P0 + 3t 0 − a (α + β 0 ) 0 ⎜⎜ 0
⎢ P1 + 3t
N1
K 1 ⎝ N 1 ⎠⎦ ⎣⎢N0
K 0 ⎝ N 0 ⎟⎠⎦⎥⎣
Similarly, the first-order condition with respect to P1 leads to:⎡⎡Q
Q ⎛ N − 1 ⎞⎤Q1
Q ⎛ N − 1 ⎞⎤⎟⎟⎥t .
− a (α + β 1 ) 1 ⎜⎜ 1 ⎟⎟⎥ g 1 = ⎢ P0 + 3t 0 − a(α + β 0 ) 0 ⎜⎜ 0
⎢ P1 + 3t
N
K
N
N
K
N
⎢
1
1 ⎝
1
0
0 ⎝
0
⎠⎦
⎠⎥⎦
⎣
⎣

(A.8)

(A.9)

Since the bracketed terms on the LHS of (A.8) and (A.9) are the same, the bracketed terms on the
RHS of (A.8) and (A.9) are the same, g 0 , g1 > t and g 0 ≠ g1 , equations (A.8) and (A.9) hold
only if each of the bracketed terms is zero. Using (6), this gives rise to the social pricing rules(33).
To derive the social capacity rules, it is useful to point out that the pricing rules (33) are obtained
as if we were maximizing directly in terms of (Q0, Q1) rather than (P0, P1), because the pricing
rules are in fact derived from ∂SW / ∂Qh = 0 . Hence:dSW ∂SW ∂Q0 ∂SW ∂Q1 ∂SW ∂SW
=
+
+
=
= 0.
(A.10)dK h∂Q0 ∂K h ∂Q1 ∂K h ∂K h
∂K h
From (A.10), (A.7), it follows immediately that the social capacity rules are given by (34).

20

Proof of Proposition 6: The first part of the proposition has been indicated by (37). To prove theremaining parts, consider the fully ex-ante symmetric case, namely, N 0 = N 1 = N , β 0 = β1 = β~
and m0 = m1 = m . Let S W (K ) ≡ SW ( P VI (K )) be a second-best social welfare function, where
P VI (K ) represents the hyper-monopolist pricing rule. Hence we have:
~
dS W
dSW
=0⇔
= 0.
dK 0
dK 0 PVI ( K )CalculatingdSW ∂SW ∂Q0 ∂SW ∂Q1 ∂SW=++from (A.7), we get:dK 0
∂Q0 ∂K 0 ∂Q1 ∂K 0 ∂K 0⎡
Q0
Q ⎛ N − 1 ⎞⎤ ∂Q0 ⎡Q
Q ⎛ N − 1 ⎞⎤ ∂Q1⎟⎟⎥⎟⎟⎥− a (α + β 0 ) 0 ⎜⎜ 0
+ ⎢ P1 + 3t 1 − a (α + β1 ) 1 ⎜⎜ 1
⎢ P0 + 3t
∂
N
K
N
K
N
K
N
0
0
0
0
1
1
1
⎝
⎠⎦ ∂K 0
⎝
⎠
⎣
⎣
⎦
2

⎛Q ⎞
+ a (α + β 0 )⎜⎜ 0 ⎟⎟ − m0 = 0.
⎝ K0 ⎠
Evaluating this at P VI (K ) , which is given by (36) an its counterpart for facility 1, yields:
2

⎛Q ⎞
∂Q
∂Q
dSW
= [3tQ0 + tQ1 ] 0 + [3tQ1 + tQ0 ] 1 + a(α + β 0 )⎜⎜ 0 ⎟⎟ − m0 = 0 .
dK 0 PVI ( K )
∂K 0
∂K 0
⎝ K0 ⎠(A.11)

From (13) and (14) it follows that:∂Q0 g1 ( N 0 + 1) a(α + β 0 ) Q0
∂Q0t ( N 0 + 1) a(α + β 0 ) Q0
=
=−
,
.
2
2
2
∂K 0
∂K 0
N 0 ( g 0 g1 − t ) K 0
N 0 ( g 0 g1 − t 2 ) K 0
Replacing these in (A.11) and then looking into the symmetric-capacities solution for the fully
symmetric case, we obtain:⎡⎤
Q 2 ⎢ ( N + 1)4t ⎥+ 1⎥ − m = 0 .
a(α + β ) 2 ⎢N (g + t) ⎥
K ⎢1
424
3
Ψ
⎣
⎦
Since Ψ>0, we can conclude that:
1/ 2
~
⎞
QW ⎛
am
~W ~ W⎟⎟ < D(Q W , K W ) = D(Q VI , K VI ) .
D(Q , K ) = a ~ W = ⎜⎜(A.12)
(
+
)(
1
+
Ψ
)
α
β
K
⎝
⎠
Q.E.D.

21

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